Bay eu he G aduie enschule ü Ma hema ik und Na u wissenscha en
Fo ophysik syn he ische und biologische mul ich omopho e Sys eme
Cha ge and exci a ion-ene gy ans e
in ime-dependen densi y unc ional
heo y
Genehmig e Abhandlung
zu E langung des akademischen G ades eines Dok o s de
Na u wissenscha en (D . e . na .) im P omo ionsp og amm
Fo ophysik syn he ische und biologische mul ich omopho e Sys eme
de Bay eu he G aduie enschule ü Ma hema ik und
Na u wissenscha en
on
Di k Ho mann-Mees
geb. Ho mann
aus
Cobu g
E s gu ach e : P o . D . S ephan Kümmel
Zwei gu ach e : P o . D . V. Ma in Ax
Tag de Ein eichung: 06. Sep embe 2012
Tag des Kolloquiums: 19. Dezembe 2012
Abs ac
Lea ning abou and unde s anding he mechanisms and pa hways o cha ge and exci a ion-
ene gy ans e o na u al molecula complexes is a p omising app oach o he ailo ed
design o new a icial ene gy-con e ing ma e ials. The e o e, nex o ex ensi e expe i-
men al in es iga ions, a heo e ical me hod ha is able o eliably desc ibe and p edic
hese phenomena om  s p inciples is o p ac ical ele ance. In p inciple, densi y unc-
ional heo y (DFT) and ime-dependen densi y unc ional heo y (TDDFT) appea as
na u al choices o s udy he ele an sizable molecules on a  s -p inciples scale a bea -
able compu a ional cos . Howe e , he applica ion o s anda d local and semilocal densi y
unc ional app oxima ions sue s om well-known deciencies, in pa icula , as a as he
simula ion o cha ge- ans e phenomena is conce ned. The p esen hesis app oaches cha ge
and exci a ion-ene gy ans e wi h he objec i e o imp o ing he p edic i e powe and
ex ending he ange o applicabili y o (TD)DFT.
The deciencies o s anda d densi y unc ional app oxima ions ha e been ela ed o sel -
in e ac ion. Hence, one majo aspec o his wo k is he ex ension o he sel -in e ac ion
co ec ion in Kohn-Sham DFT ha is based on he gene alized op imized eec i e po en ial
o TDDFT using a eal- ime p opaga ion app oach. The mul iplica i e Kohn-Sham po en ial
allows o a anspa en analysis o he exchange-co ela ion po en ial du ing ime e olu ion.
I e eals equency-dependen eld-coun e ac ing beha io and s ep s uc u es ha appea
in dynamic cha ge- ans e si ua ions. The la e a e impo an o he p ope desc ip ion o
cha ge ans e . Sel -in e ac ion co ec ion allows o access many cases ha a e dicul o
s anda d TDDFT anging om chain-like sys ems o e exci onic exci a ions in semiconduc-
o nanoclus e s o sho - and long- ange cha ge- ans e exci a ions. A he same ime, i
does no spoil he easonable accu acy ha al eady (semi)local unc ionals exhibi o local
exci a ions. Mo eo e , he TDDFT pe spec i e on sel -in e ac ion co ec ion sheds new ligh
also on he g ound-s a e o malism. Complex deg ees o eedom in he ene gy-minimizing
ans o ma ion o he gene alized op imized eec i e po en ial app oach yield smoo he o -
bi al densi ies ha appea mo e easonable when inse ed in o app oxima e unc ionals in he
sel -in e ac ion co ec ion o malism. This wo k p o ides new insigh in o he use o die en
unc ional app oxima ions. Las bu no leas , he inuence o spin-symme y b eaking and
s ep s uc u es o he po en ial on he p e e ence o ans e in ege uni s o he elemen a y
elec ic cha ge be ween la gely sepa a ed dono and accep o moie ies is illus a ed when
s a ic ex e nal elec ic elds a e applied. This wo k has been epo ed in h ee publica ions
and one submi ed manusc ip .
In he eld o exci a ion-ene gy ans e , ecen disco e ies o quan um cohe ence eec s
shed new ligh on he mechanisms behind ene gy- ans e a es. The la e a e aec ed by
a numbe o die en p ope ies o he isola ed molecules, bu in ol e also eec s due o he
III
IV
Abs ac
en i onmen o he sys em. This hesis add esses exci a ion-ene gy ans e phenomena om
wo pe spec i es. Fi s , I use eal- ime p opaga ion TDDFT o in es iga e he in e molecula
coupling s eng h and he coupling mechanism be ween single agmen s o supe molecula
se ups. These in es iga ions base on s anda d closed quan um sys em TDDFT and exploi
he cohe en oscilla ion o exci a ion ene gy be ween sepa a ed molecules a e he ini ial
exci a ion p ocess. Second, I use open quan um sys em ideas in he amewo k o TDDFT
o s udy he inuence o he sys em's en i onmen on he ene gy- ans e ime scales and
pa hways in a ci cula a angemen o molecules using an eec i e ene gy-dissipa ion mecha-
nism. The  s pa o hese esul s is published. The second pa is p esen ed in his hesis
and includes wo k in p og ess.
Ku z assung
Ein iel e sp echende Ansa z, um küns liche Ma e ialien ü Zwecke de Ene gieumwand-
lung zu en wickeln, bes eh da in, neues Wissen übe die Mechanismen und Übe agungs-
wege on Ladung und An egungsene gie in Molekülkomplexen, die in de Na u o kommen,
zu e we ben. Da ü we den neben um ang eichen expe imen ellen Un e suchungen auch
Me hoden benö ig , mi denen man solche Phänomene zu e lässig heo e isch besch eiben
und o he sagen kann. P inzipiell bie en sich die Dich e unk ional heo ie (DFT) und die
zei abhängige Dich e unk ional heo ie an, um Sys eme de ele an en G öÿe mi agba em
nume ischem Au wand ausgehend on physikalischen G undp inzipien zu un e suchen. Alle -
dings wu den in Anwendungen de DFT mi lokalen und semilokalen Dich e unk ional-
Nähe ungen P obleme au gezeig , die insbesonde e in Simula ionen on Ladungs ans e -
phänomenen k i isch sind. Dahe un e such die o liegende Disse a ion Ladungs- und
Ene gie ans e phänomene mi dem Ziel, die Vo he sagen de (zei abhängigen) DFT zu
e besse n und de en Anwendungsbe eich zu e wei e n.
Die P obleme, die sich au zeigen, wenn man he kömmliche Dich e unk ional-Nähe ungen
e wende , wu den mi de sogenann en Selbs wechselwi kung in Ve bindung geb ach . Des-
halb is die E wei e ung de Me hode zu Selbs wechselwi kungsko ek u , die im Rahmen
de Kohn-Sham-DFT au dem Ve ah en de e allgemeine en op imie en eek i en Po-
en iale be uh , au den Be eich de zei abhängigen DFT eine de zen alen Aspek e diese
A bei . Die E wei e ung be uh au einem Ech zei p opaga ions e ah en und e wende
ein mul iplika i es Kohn-Sham-Po en ial, mi dem man au anspa en e A und Weise den
Zei e lau des Aus ausch-Ko ela ionspo en ials un e suchen kann. In de o liegenden A -
bei wi d au gezeig , dass dieses Po en ial equenzabhängiges Gegen eld e hal en au weis
und sich S u ens uk u en in Ladungs ans e simula ionen au bauen. Diese S uk u en sind
ü eine zu e lässige Besch eibung on Ladungs ans e phänomenen wich ig. Da übe hin-
aus e möglich das Selbs wechselwi kungsko ek u e ah en die Un e suchung iele Sys-
eme, die ü he kömmliche zei abhängige DFT als no o isch schwie ig gel en, ohne dabei
die Genauigkei zu e lie en, die be ei s (semi)lokale Funk ionale bei de Besch eibung lokale
An egungen au weisen. Un e diesen Sys emen benden sich ke en ö mige Moleküle, Halb-
lei e -Nanoclus e , de en An egungen als exzi onisch gel en, sowie ku z- und lang eichwei ige
Ladungs ans e an egungen. Wei e hin lie e n die E kenn nisse aus den Un e suchungen de
zei abhängigen DFT neue Einblicke in den G undzus ands o malismus. Die Ve wendung on
komplexwe igen F eihei sg aden zu Bes immung de ene gieminimie enden T ans o ma io-
nen im Ve ah en de e allgemeine en op imie en eek i en Po en iale üh zu gla e en
O bi aldich en. Diese O bi aldich en scheinen besse geeigne zu sein, um sie im Rahmen des
Selbs wechselwi kungsko ek u e ah ens in genähe e Funk ionale einzuse zen. In diese
A bei disku ie e ich neue Einsich en in die Ve wendung un e schiedliche Nähe ungen de
V

VI
Ku z assung
Funk ionale. Wei e hin e läu e e ich den Einuss on Spinsymme ieb echung und S u en-
s uk u en im Po en ial da au , ob beim Ladungs ans e du ch ex e ne elek ische Felde
zwischen zwei wei en e n en Dono - und Akzep o molekülen de Ladungs ans e ganz-
zahlige Viel ache de Elemen a ladung be o zug wi d ode nich . Die ge undenen Re-
sul a e sind in d ei Publika ionen e öen lich und ein wei e es Manusk ip wu de be ei s
einge eich .
Im Fo schungsgebie des An egungsene gie ans e s haben ak uelle E kenn nisse zu Rolle
sogenann e Quan enkohä enzen neue Einblicke in die Mechanismen des Ene gie ans e s
gelie e . Die ele an en Ene gie ans e a en we den on ielen e schiedenen Eigenscha en
de Moleküle und du ch Eek e de Umgebung des Sys ems beeinuss . In diese A bei un-
e suche ich den An egungsene gie ans e aus zwei Rich ungen. Eine sei s e wende ich
Ech zei p opaga ion im Rahmen de zei abhängigen DFT, um die Kopplungss ä ke und den
Kopplungsmechanismus zwischen zwei einzelnen Molekülen zu un e suchen. Diese S udie
lieg eine He angehensweise zug unde, die ein geschlossenes Quan ensys em benu z und in
de en Rahmen man kohä en e Oszilla ionen de An egungsene gie zwischen den Molekülen
beobach en kann. Ande e sei s nu ze ich einen Ansa z, oene Quan ensys eme in Kombi-
na ion mi dem Fo malismus de zei abhängigen DFT zu e wenden, um den Einuss de
Umgebung des Sys ems au die Zei skala und die Wege des An egungsene gie ans e s zu un-
e suchen. Ich habe dazu einen heu is ischen Dissipa ionsmechanismus en wickel und wende
diesen au eine ing ö mige Ano dnung on Molekülen an. De e s e Teil diese E gebnisse is
be ei s e öen lich , wäh end de zwei e Teil Un e suchungen en häl , die in diese A bei
e s mals p äsen ie we den.
Con en s
Abs ac III
Ku z assung V
Con en s VII
1 Mo i a ion 1
2 Densi y unc ional heo y and ime-dependen densi y unc ional heo y 3
2.1 Densi y unc ional heo y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.1 Basics o densi y unc ional heo y . . . . . . . . . . . . . . . . . . . . 4
2.1.2 Kohn-Sham densi y unc ional heo y . . . . . . . . . . . . . . . . . . . 4
2.2 Time-dependen densi y unc ional heo y . . . . . . . . . . . . . . . . . . . . 6
2.2.1 An in oduc ion o ime-dependen densi y unc ional heo y . . . . . 6
2.2.2 TDDFT linea esponse o malism . . . . . . . . . . . . . . . . . . . . 7
2.2.3 Real- ime p opaga ion TDDFT . . . . . . . . . . . . . . . . . . . . . . 8
2.3 The ansi ion densi y analysis ool . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Exac p ope ies and ea u es o he exchange-co ela ion po en ial . . . . . . 10
2.5 Exchange-co ela ion unc ionals . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.5.1 App oxima ions o he exchange-co ela ion ene gy unc ional . . . . 12
2.5.2 Exchange-co ela ion unc ionals in TDDFT . . . . . . . . . . . . . . . 14
2.6 Nume ical ealiza ion................................ 15
3 Sel -in e ac ion co ec ion 17
3.1 The sel -in e ac ion p oblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Sel -in e ac ion co ec ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3 The op imizied eec i e po en ial and i s gene aliza ion . . . . . . . . . . . . 19
3.3.1 The op imized eec i e po en ial me hod . . . . . . . . . . . . . . . . 19
3.3.2 Gene aliza ion o he OEP o uni a ily a ian unc ionals . . . . . . . 20
3.4 Uni a y op imiza ion wi h die en ypes o o bi als . . . . . . . . . . . . . . 21
3.5 Gene alized sel -in e ac ion co ec ion in DFT . . . . . . . . . . . . . . . . . . 22
3.6 Gene aliza ion o SIC in TDDFT . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.7 Pe o mance o gene alized SIC in TDDFT . . . . . . . . . . . . . . . . . . . 26
VII
VIII
CONTENTS
4 Cha ge ans e and cha ge- ans e exci a ion ene gies 29
4.1 The cha ge- ans e p oblem o DFT and TDDFT and some solu ion ideas . . 29
4.2 Sel -in e ac ion co ec ion and cha ge ans e . . . . . . . . . . . . . . . . . . 31
4.2.1 The in ege p e e ence o elec on jumps . . . . . . . . . . . . . . . . . 31
4.2.2 The pe o mance o GSIC on cha ge- ans e exci a ion ene gies . . . . 35
5 Exci a ion-ene gy ans e 37
5.1 In e molecula coupling and eal- ime TDDFT . . . . . . . . . . . . . . . . . 37
5.2 Open quan um sys ems in he densi y unc ional con ex . . . . . . . . . . . . 39
5.2.1 The S ochas ic Sch ödinge equa ion . . . . . . . . . . . . . . . . . . . 40
5.2.2 S ochas ic Sch ödinge equa ion and Kohn-Sham densi y unc ional
heo y.................................... 41
5.2.3 A single-pa icle app oach in p ac ice . . . . . . . . . . . . . . . . . . . 43
5.3 Pa hways and ime cons an s o exci a ion-ene gy ans e . . . . . . . . . . . 44
5.3.1 The exci a ion-ene gy ans e model sys em . . . . . . . . . . . . . . . 44
5.3.2 P ac ical simula ion app oach . . . . . . . . . . . . . . . . . . . . . . . 46
5.3.3 Resonan exci a ion sp ead and decay ime cons an s . . . . . . . . . . 48
5.3.4 Inuence o ene ge ic o- esonance . . . . . . . . . . . . . . . . . . . . 49
5.3.5 Inuence o he in asys em coupling . . . . . . . . . . . . . . . . . . . 50
5.4 Summa yandOu look............................... 51
Appendix 55
A Pola izabili y and s a ic cha ge- ans e p ope ies 55
A.1 Pola izabili y and eld-coun e ac ing po en ials in sel -in e ac ion ee densi y
unc ional heo y.................................. 55
A.2 Pola izabili y o polyace ylene chains . . . . . . . . . . . . . . . . . . . . . . . 59
A.3 S a ic cha ge ans e in a anspa en model sys em . . . . . . . . . . . . . . 60
B Mul ig id Poisson sol e 63
B.1 Mul ig id and de ec co ec ion . . . . . . . . . . . . . . . . . . . . . . . . . . 64
B.1.1 Themul ig ididea............................. 64
B.1.2 An in oduc ion o de ec co ec ion . . . . . . . . . . . . . . . . . . . 66
B.2 Fea u es o he PARSEC code in he mul ig id con ex . . . . . . . . . . . . . 67
B.3 The mul ig id implemen a ion in PARSEC . . . . . . . . . . . . . . . . . . . . 69
B.4 Assessmen o he mul ig id sol e . . . . . . . . . . . . . . . . . . . . . . . . 71
C Algo i hms o he uni a y op imiza ion 75
C.1 Algo i hmicp inciples ............................... 75
C.1.1 Fois loops based on he Pede son c i e ion . . . . . . . . . . . . . . . . 75
C.1.2 Ene gy g adien based algo i hm . . . . . . . . . . . . . . . . . . . . . 77
C.1.3 Ene gy g adien and line-sea ch op imiza ion . . . . . . . . . . . . . . 78
C.2 The uni a y op imiza ion algo i hms in PARSEC . . . . . . . . . . . . . . . . 80
C.2.1 Ini ializa ion o he op imiza ion s eps . . . . . . . . . . . . . . . . . . 80
C.2.2 The PARSEC implemen a ion . . . . . . . . . . . . . . . . . . . . . . . 82
CONTENTS
IX
D Fö s e - ype po en ials and s ochas ic ime-dependen densi y unc ional
heo y 85
D.1 Fö s e - ype po en ials and g id pa i ioning . . . . . . . . . . . . . . . . . . . 85
D.1.1 The Fö s e - ype po en ial expansion in Dono -Accep o sys ems . . . 85
D.1.2 Using Fö s e - ype po en ials in supe molecula sys ems . . . . . . . . 87
D.1.3 Pa i ion-selec i e exci a ion and obse a ion . . . . . . . . . . . . . . 88
D.2 Un a eling he coupling s eng h wi h TDDFT . . . . . . . . . . . . . . . . . 88
D.3 S ochas ic ime-dependen densi y unc ional heo y . . . . . . . . . . . . . . 90
D.3.1 An a emp owa ds a heo e ical jus ica ion o s ochas ic ime-dependen
densi y unc ional heo y wi h specic ba h ope a o s . . . . . . . . . 90
D.3.2 Ba h ope a o s in he single-pa icle KS amewo k and ela ed ea u es 95
D.3.3 PARSEC ea u es and inpu pa ame e s . . . . . . . . . . . . . . . . . 97
E PARSEC miscellaneous 99
E.1 Gene al commen s on he PARSEC inpu . . . . . . . . . . . . . . . . . . . . 99
E.2 De e mining he midpoin Hamil onian du ing p opaga ion ia ex apola ion 99
E.3 On he y Fou ie ans o ma ion o he ime-dependen densi y . . . . . . . . 101
E.4 P opaga ion miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
E.5 Adap a ion o he diagonaliza ion ole ance du ing he g ound-s a e p ocedu e 102
E.6 G ound-s a e miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Acknowledgmen 105
Lis o publica ions and manusc ip s 107
Lis o abb e ia ions, unc ionals, and me hods 109
Bibliog aphy 113
E klä ung 133
Publica ion 1 135
Publica ion 2 147
Publica ion 3 155
Publica ion 4 169
Manusc ip 1 189
6
CHAPTER 2. DENSITY FUNCTIONAL THEORY AND TIME-DEPENDENT
DENSITY FUNCTIONAL THEORY
2.2 Time-dependen densi y unc ional heo y
2.2.1 An in oduc ion o ime-dependen densi y unc ional heo y
Al hough he HK heo ems lay he ounda ion o a densi y ep esen a ion o he ully in e -
ac ing many-pa icle sys em, hey do no es ablish a di ec ela ion be ween he GS densi y
and uly dynamic o exci ed-s a e p ope ies. The in es iga ion o such p ope ies using
densi y unc ionals is based on ime-dependen densi y unc ional heo y. In TDDFT, he
Runge-G oss heo em
[RG84] is he complemen o he HK heo ems o GS DFT. Gi en
an ini ial s a e and pa icle-pa icle in e ac ion, i es ablishes a one- o-one co espondence
be ween he ime-dependen (TD) densi y
n(
, )
and he TD ex e nal po en ial
ex (
, )
up o a pu ely TD unc ion
c( )
. Wi h
ex (
, )
and he ini ial s a e
Ψ( 0)
, also he TD
wa e- unc ion is de e mined uniquely up o a TD phase ia solu ion o he TD Sch ödinge
equa ion. As expec a ion alues o any ope a o a e no sensi i e o he phase o he wa e
unc ion, in p inciple, each obse able is a unc ional o
n(
, )
and he ini ial s a e. The
Runge-G oss p oo has be ened by an Leeuwen [ L99] who co e s he
non-in e ac ing
-
ep esen abili y ques ion
o TD densi ies by a cons uc ion p ocedu e o he ex e nal po en ial
o he al e na i e e e ence sys em [ L99, MUN
+
06]. Mo eo e , he
ini ial-s a e dependence
o he densi y ep esen a ion has been discussed in Re s. [MB01] and [MBW02].
To de i e a calcula ion scheme o dynamic p ope ies, Runge and G oss [RG84] sugges ed
a a ia ional p inciple ha es s upon on an ac ion unc ional. Howe e , Re s. [ L98] and
[ L01] demons a e ha TDDFT based on he Runge-G oss ac ion leads o con adic ions
in he symme y and causali y equi emen s o one o he mos impo an ing edien s o
TDDFT
linea esponse heo y
, namely he xc ke nel
xc(
, ;
0, 0) = δ2Axc
δn(
, )δn(
0, 0),
(2.10)
whe e
Axc
is he xc pa o he Runge-G oss ac ion [RG84]. Van Leeuwen [ L98, L01]
sol ed his p oblem by in oducing a new ac ion unc ional ha is based on he ime con ou
me hod due o Keldysh ( o mo e de ails, see Re s. [ L98, L01, MUN
+
06, Mun07]). The
hus ob ained a ia ional p inciple yields a se o
ime-dependen Kohn-Sham equa ions
i∂
∂ ϕjσ(
, ) = hKS,σ(
, )ϕjσ(
, ),
(2.11)
whe e he TD KS Hamil onian eads
hKS,σ(
, ) = −∇2
2+ H(
, ) + xc,σ(
, ) + ex (
, ).
(2.12)
The TD xc po en ial
xc,σ(
, )
ollows om he unc ional de i a i e
xc,σ(
, ) = δAxc
δn(
, τ)n=nσ(
, )
(2.13)
o he xc pa o he new ac ion unc ional wi h espec o he densi y
n(
, τ)
whe e he ime
a iable
τ
is he Keldysh pseudo ime, bu he unc ional de i a i e is aken a he physical
TD densi y
n(
, )
[ L98, L01]. By he basic heo ems o TDDFT,
xc,σ(
, )
is a unc ional

2.2. TIME-DEPENDENT DENSITY FUNCTIONAL THEORY
7
o he TD densi y and i s en i e his o y, he ini ial in e ac ing wa e unc ion, and he ini ial
s a e o he KS sys em. Finally, based on xc app oxima ions, he TD densi y ollows om
he
Nσ
occupied o bi als pe spin channel o he TD KS sys em acco ding o
n(
, ) = X
σ=↑,↓
Nσ
X
j=1 |ϕjσ(
, )|2.
(2.14)
P ac ical calcula ions o TDDFT ely ei he on he
linea esponse o malism
o on
eal- ime
p opaga ion
o he KS sys em. Bo h app oaches a e in oduced in he ollowing.
2.2.2 TDDFT linea esponse o malism
Today, mos TDDFT exci a ion ene gy in es iga ions a e based on he linea esponse o -
malism. The linea esponse o he GS densi y o small pe u ba ions
δ ex (
0, 0)
o he
ex e nal po en ial eads
δn(
, ) = Z Z χ[nGS](
,
0, − 0)δ ex (
0, 0) d3 0d 0,
(2.15)
whe e
χ[nGS](
,
0, − 0)
is he linea esponse unc ion o he in e ac ing sys em. Fo he
sake o cla i y, I use a spin-independen no a ion he e and in he nex sec ion. Based on
he undamen al heo ems o TDDFT, he densi y esponse may be exp essed in e ms o
he linea esponse o he KS sys em due o changes o he KS po en ial [PGG96]. Hence, a
ela ion be ween
χ[nGS](
,
0, − 0)
and he linea esponse unc ion o he KS sys em
χKS
exis s [PGG96, MUN
+
06]. In equency space, he in e ac ing linea esponse unc ion is
χ(
,
0, ω) =χKS(
,
0, ω)
+Z Z χKS(
,
1, ω)1
| 1− 2|+ xc(
1,
2, ω)χ(
2,
0, ω) d3 1d3 2,
(2.16)
whe e
χKS(
,
0, ω)
is he equency-dependen linea esponse o he KS sys em eading
χKS(
,
0, ω) = 2 lim
η→0+X
i,a ξia(
)ξ∗
ia(
0)
ω−ωia + iη−ξ∗
ia(
)ξia(
0)
ω+ωia + iη.
(2.17)
I depends on he eigen alue die ences
ωia =εa−εi
be ween all possible combina ions o
occupied KS o bi als
i
and unoccupied KS o bi als
a
and on he o bi al p oduc s
ξia(
) =
ϕ∗
i(
)ϕa(
)
o he co esponding GS KS o bi als [MUN
+
06]. I has poles a equencies
ωia
.
The KS esponse con ibu ion o Eq. (2.16) includes only hose eec s ha a e encoded in
he single-pa icle GS KS sys em, whe eas he Ha ee-exchange-co ela ion (Hxc) ke nel
Hxc(
,
0, ω) = 1
| − 0|+ xc(
,
0, ω)
(2.18)
needs o co e all many-pa icle eec s beyond ha .
Finding he exci a ion ene gies o he in e ac ing sys ems in linea esponse amoun s
o nding he poles o
χ(
,
0, ω)
. Casida [Cas95, Cas96] de eloped a ma ix-equa ion o -
mula ion o p ac ical implemen a ions o his s a egy. In his app oach, solu ion o he
eigen alue p oblem [Cas96, MUN
+
06]
X
i0,a0
Ria,i0a0Fi0a0= Ω2Fia,
(2.19)
8
CHAPTER 2. DENSITY FUNCTIONAL THEORY AND TIME-DEPENDENT
DENSITY FUNCTIONAL THEORY
whe e
Ria,i0a0=ω2
iaδii0δaa0+ 4√ωiaωi0a0Kia,i0a0
(2.20)
and
Kia,i0a0(ω) = Z Z ξ∗
ia(
) Hxc(
,
0, ω)ξi0a0(
0) d3 d3 0,
(2.21)
yields he exci a ion ene gies
Ω
o he in e ac ing sys em. The co esponding oscilla o
s eng h is encoded in he eigen ec o s [Cas96]. This p ocedu e is well-known as he 
Casida
app oach
 o linea esponse TDDFT. I is mos equen ly used in oday's TDDFT based
applica ions and implemen ed in mos quan um chemis y codes.
An app oxima e app oach o nding
Ω
om TDDFT linea esponse p o ides means o
mo e insigh in o he inuence o he indi idual con ibu ions o Eq. (2.16), i.e., con ibu-
ions om he GS KS sys em and Ha ee-exchange-co ela ion eec s. I is based on he
single-pole app oxima ion [PGG96]: Assuming ha he ue exci a ion
˜
Ω
is domina ed by
one ansi ion om a single occupied o bi al
j
o a single unoccupied o bi al
b
, all o he
con ibu ions o he o al esponse can be neglec ed. Thus, one ob ains [PGG96, AGB03]
˜
Ω≈ωjb + 2 Re{Kjb,jb(ωjb)}.
(2.22)
Equa ion (2.22) is used in Sec. 4.1 o explain he CT p oblem o TDDFT.
2.2.3 Real- ime p opaga ion TDDFT
In his hesis, I mainly used he eal- ime (RT) p opaga ion app oach o TDDFT which does
no equi e explici linea esponse heo y, bu is di ec ly based on he TD KS equa ions o
Sec. 2.2.1. The cen al idea o his me hod is o compu e he ime e olu ion o he densi y
om RT p opaga ion o he TD KS sys em
ϕj(
, ) = U( , 0)ϕj(
, 0) = Texp −iZ
0
hKS(
, 0) d 0ϕj(
, )
(2.23)
ia applica ion o he p opaga o
U( , 0)
. All o he obse ables need o be ob ained om
n(
, )
. This is a c ucial poin because in some si ua ions i is dicul o ex ac in o ma ion
ha is easily a ailable in he Casida app oach om he ime e olu ion o he densi y, e.g.,
see Chap. 4. In such cases, new in es iga ion ideas need o be de eloped as o ins ance
he
ansi ion densi y analysis ool
o Sec. 2.3. Ye , some impo an obse ables a e explici
unc ionals o he densi y, in pa icula he TD dipole momen
d
( ) = Z
n(
, ) d3 .
(2.24)
Exci a ion ene gies eme ge as peaks in he spec um o he TD densi y a e some ini ial
exci a ion [ZS80, CRS97]. Mos impo an ly, op ical exci a ions ha a e sensi i e o he
dipole momen cause peaks a equencies
ω
in he Fou ie ans o ma ion
d
(ω)
o he TD
dipole momen ha co espond o op ical exci a ion ene gies. To compu e he la e , he
sys em is ypically exci ed by an ini ial momen boos and he esul ing TD dipole momen is
used as basic obse able [YB96, YB99a, YB99b, MCBR03, CAO
+
06, MK07, Mun07, Mun09].
He e, he boos is applied ini ially by mul iplying
exp i
·
p
boos
~
o he GS KS o bi als. This
in oduces an exci a ion ene gy
Eexci
ia he momen um
|
p
boos |=p2mEexci /N
. The
2.3. THE TRANSITION DENSITY ANALYSIS TOOL
9
Fou ie ans o ma ion o he dipole signal a e such a momen um boos can be ela ed o
he
dynamical pola izabili y
and he
dipole s eng h unc ion
[ZS80, YB96, CRS97, CAO
+
06],
whe e peak posi ions co espond o exci a ion ene gies and peak heigh s a e ela ed o he
oscilla o s eng h o he ansi ions. As long as one is in e es ed only in he posi ion o
exci a ion peaks, he dipole powe spec um [CRS97, Mun07, Mun09]
D(ω) =
3
X
j=1 |dj(ω)|2
(2.25)
gi es a eliable and ela i ely clean signal in he equency domain. O he applica ions
in ol e exci a ions ia TD ex e nal elds, e.g., lase elds. Such ex e nal pe u ba ions may
be included in e ms o explici ly TD ex e nal po en ials ha ac du ing p opaga ion. Mo e
insigh in o ex e nal lase elds, possible lase pulse shapes, and applica ions a e gi en in
Re s. [Mun07] and [Mun09].
The RT p opaga ion app oach comes along wi h some ad an ages in compa ison o he
explici linea esponse o mula ion. Fi s , he xc ke nel does no need o be compu ed, as RT
p opaga ion is based on he po en ial only. The de e mina ion o
xc
may be complica ed, in
pa icula , in case o xc app oxima ions whe e al eady he de e mina ion o
xc
is dicul .
Second, RT p opaga ion shows a mo e ad an ageous scaling [YB99b]. Howe e , o many
applica ions, his scaling a gumen holds only o eally la ge pa icle numbe s, because
mul iplie s ha a e modi ying he scaling beha io a e ypically no ably la ge in case o he
RT app oach. Thi d, RT p opaga ion is no limi ed o he linea esponse egime. Thus, i
can be applied o non-pe u ba i e and non-linea phenomena, as o ins ance exci a ion by
s ong lase elds. Finally, he RT p opaga ion me hod is applicable o gene al TD si ua ions
and, he e o e, illumina es he RT e olu ion o explici ly TD obse ables.
2.3 The ansi ion densi y analysis ool
A  s glance, RT p opaga ion does no seem o p o ide mo e in o ma ion abou he na u e
o exci a ions han hei ene ge ic posi ion and he co esponding oscilla o s eng h. F om
his alone, i appea s dicul o make clea s a emen s abou he pe o mance o die -
en unc ional app oxima ions by jus compa ing he compu ed abso p ion spec a. Fo a
us wo hy assessmen o exci a ion ene gy esul s, i is impo an o know he na u e o
he exci a ion peaks, i.e., o unde s and he cha ac e o he unde lying ansi ions. The
Casida app oach [Cas95, Cas96] allows his in a na u al way as i decomposes exci a ions
in o ansi ions om occupied o unoccupied o bi als wi h a ce ain weigh ing ac o . In
his way, one can, e.g., dis inguish be ween local and CT exci a ions. In p inciple his in o -
ma ion is also a ailable om RT p opaga ion, bu ex ac ing he in o ma ion om he TD
o bi als is edious. The e o e, I sugges ed an analysis ool o he RT signal ha is based
on he
ansi ion densi y
and allows o explo e he na u e o exci a ions. The ansi ion
densi y is di ec ly ela ed o he TD densi y, hus well dened in he amewo k o TDDFT
[KAR01, BCOR04, TK09].
The ansi ion densi y
ρω(
)
co esponding o an exci a ion a equency
ω
is p opo ional
o he nega i e o he imagina y pa o he Fou ie ans o ma ion
δn(
, ω)
o he TD densi y
uc ua ions
δn(
, ) = n(
, )−n(
,0)
acco ding o [BCOR04, TK09]
ρω(
)∝ −Im{δn(
, ω)}.
(2.26)
10
CHAPTER 2. DENSITY FUNCTIONAL THEORY AND TIME-DEPENDENT
DENSITY FUNCTIONAL THEORY
In PARSEC, I implemen ed a s ep by s ep Fou ie ans o ma ion du ing ime p opaga ion o
calcula e
ρω(
)
o p ese equencies (see Appendix E.3 o de ails). The ob ained ansi ion
densi ies a e a unique nge p in o he exci a ion and can be used o dis inguish be ween
exci a ions o die en cha ac e . Visual inspec ion o ansi ion densi ies helps o assign
exci a ions peaks o spec a compu ed by die en me hods o unc ional app oxima ions.
Examples o how ansi ion densi y nge p in s a e used o iden i y ansi ions o known
cha ac e in spec a om die en unc ionals a e gi en in Pub2 and Pub4.
Mo e insigh in o he na u e o ansi ions is a ailable i he con ibu ions o occupied o
unoccupied o bi al ansi ions a e known. In his case, he ansi ion densi y can be w i en
as a weigh ed sum o occupied and unoccupied o bi al p oduc s [BCOR04]
ρω(
)∝
occup.
X
i
unoccup.
X
j
aω
ijϕi(
)ϕ∗
j(
)
(2.27)
wi h weigh ing ac o s
aω
ij
. T ansi ion densi ies om RT p opaga ion a e compa ed o ansi-
ion densi ies om GS o bi al p oduc s in Pub2 and Pub4 o iden i y CT ansi ions and he
co esponding exci a ion ene gies in wo impo an model sys ems ha exhibi die en CT
cha ac e . Mo eo e , in an a emp o se he  isual inspec ion idea on a mo e objec i e
oo ing, I sugges ed wo quan i a i e compa ison c i e ia ha a e explained in Pub4.
2.4 Exac p ope ies and ea u es o he exchange-co ela ion
po en ial
The wo k p esen ed in his hesis is deeply oo ed in he KS amewo k o DFT. The KS
scheme p o ides some heo e ical and echnical ad an ages: i ullls exac condi ions as, e.g.,
Janak's ioniza ion-po en ial heo em [Jan78], exac KS eigen alues a e good app oxima ions
o elaxed e ical ioniza ion po en ials [CGB02, Kö 09], using local po en ials has nume ical
ad an ages, and he local mul iplica i e po en ial allows o a anspa en analysis o he
esponse beha io and in e p e abili y o o bi als and eigen alues [DKK
+
11]. Taking exac
p ope ies o he KS sys em and i s xc con ibu ion in o accoun , is an es ablished ou e o
imp o ing xc unc ional app oxima ions. The e o e, concluding he o e iew o DFT and
TDDFT basics, I commen on he
de i a i e discon inui y
and i s mani es a ion in he xc
po en ial and name some o he impo an ea u es and exac p ope ies o he KS app oach.
One p ope y o he DFT desc ip ion o xc eec s ha is impo an o his wo k is
ela ed o he beha io o he o al ene gy o a quan um sys em when i s o al elec on
numbe passes in ege alues. This si ua ion was s udied by Pe dew
e al.
[PPLB82] based
on a s a is ical mix u e o
N
- and
(N+1)
- elec on sys ems o ealized ac ionally occupied
sys ems wi h o al elec on numbe
N+ω
, whe e
0≤ω≤1
. The impo an nding o Re .
[PPLB82] is ha he GS ene gy
EN+ω
o ac ionally occupied sys ems a ies linea ly wi h
he ac ional occupa ion
ω
be ween he ene gies o he
N
- and
(N+ 1)
- elec on sys em as
EN+ω= (1 −ω)EN+ωEN+1.
(2.28)
This s aigh -line beha io o he o al ene gy be ween in ege elec on numbe s implies
discon inuous changes o i s slope wi h espec o he ac ional pa icle numbe
Z
when i
2.4. EXACT PROPERTIES AND FEATURES OF THE
EXCHANGE-CORRELATION POTENTIAL
11
passes in ege s
N
. Thus, also he
chemical po en ial
µ(Z) = ∂EZ/∂Z
jumps discon inuously
µ(Z) = (−IP (N) = EN−EN−1, N −1< Z < N
−EA(N) = EN+1 −EN, N < Z < N + 1
(2.29)
when
Z
passes in ege occupa ions
N
[PPLB82].
IP
and
EA
a e he ioniza ion po en ial
and he elec on ani y o he
N
-elec on sys em. The discon inui y a he in ege elec on
numbe
N
is gi en by
∆ = IP (N)−EA(N)
. This die ence is called he
undamen al gap
.
In KS DFT,
∆
may be sepa a ed in o wo con ibu ions
∆=∆KS + ∆xc,
(2.30)
he
KS gap
∆KS
and he
de i a i e discon inui y
o he xc po en ial
∆xc
[PL83]. The KS
gap is he die ence
εexac KS
LUMO −εexac KS
HOMO
o he HOMO (highes occupied molecula o bi al)
and he LUMO (lowes unoccupied molecula o bi al) eigen alue o he ye unknown exac
N
-elec on KS sys em.
∆xc
quan ies he in ege jump o he xc po en ial when in an open
sys em amewo k wi h non-in ege pa icle numbe s he elec on numbe passes in ege
alues [PPLB82, PL83, SS83, Pe 90, DG90]. Hence, i is dened as
∆xc = +
xc(
)− −
xc(
)=(IP (N)−EA(N)) −εexac KS
HOMO −εexac KS
LUMO ,
(2.31)
whe e he po en ials
+
xc(
)
and
−
xc(
)
co espond o he limi ing cases when he ac ional
elec on occupa ion app oaches
N
om abo e o om below acco ding o
+
xc(
) = lim
ω→0
δExc
δn(
)N+ω
,
(2.32)
−
xc(
) = lim
ω→0
δExc
δn(
)N−ω
.
(2.33)
The discon inuous beha io o he GS ene gy eec s he s ong endency o ue elec-
onic sys ems o ejec ac ional occupa ion [Pe 90]. Pe dew [Pe 90] ela ed he de i a i e
discon inui y o he
p inciple o in ege p e e ence
: In a sys em composed o sepa a e subsys-
ems, na u e always p e e s o loca e in ege cha ges on each objec . Al hough his concep
was in oduced in an ensemble o mula ion o quan um sys ems wi h ac ional occupa ions,
i also mani es s in he xc po en ial o sys ems wi h in ege elec on numbe s in e ms o
s ep-like s uc u es
[SP08, GGS09].
The dissocia ion p ocess o dia omic molecules is one such si ua ion whe e s ep-like s uc-
u es a e impo an o suppo in ege numbe o elec ons on agmen s o he sys em
[PPLB82, Pe 90, RPC
+
06, KAK09, TMM09, Kö 09, MKK11]: When wo die en a oms A
and B wi h die en elec onega i i y dissocia e, a s ep eme ges in he exac xc po en ial in
be ween he wo a oms as hey mo e apa . I is needed o align he eigen alues co espond-
ing o he HOMO o A and B in he inni ely sepa a ed case ia ela i e shi s o he A and
B po en ial wells in o de o a oid ac ional cha ge ans e du ing he dissocia ion p ocess
[KAK09]. A second kind o s ep-like s uc u e was obse ed in
xc
a he bounda ies o he
shells o he a omic shell s uc u e [KLI92, G LB94, LGB95]. S ep s uc u es appea also
du ing TD p ocesses [LK05, MK05], as o ins ance a he bounda y o an eme ging po en ial
pla eau du ing ioniza ion due o s ong ex e nal elds [LK05].

12
CHAPTER 2. DENSITY FUNCTIONAL THEORY AND TIME-DEPENDENT
DENSITY FUNCTIONAL THEORY
A die en mani es a ion o s ep-like s uc u es and he de i a i e discon inui y is he
eld-coun e ac ing beha io
o
xc
when ex e nal elds a e applied. A eld-coun e ac ing
end o he xc po en ial has been iden ied o be decisi e o he desc ip ion o s a ic
esponse p ope ies such as pola izabili ies [ GSG
+
99, G GSB00, KKP04, KMK08, KAK09].
Mo e insigh in o he s a ic eld-coun e ac ing beha io is gi en in Appendix A.1.
The de i a i e discon inui y, s ep-like s uc u es, and eld-coun e ac ing beha io a e
pa icula ly impo an also o Coulomb blockade eec s [CBKR07, KSK
+
10] and cha ge-
ans e in es iga ions [TFSB05, Mai05, HG12]. In Chap. 4, he la e opic is discussed in
g ea e de ail in s a ic and dynamic si ua ions.
Exac p ope ies o he GS unc ional [MMN
+
12] beside he de i a i e discon inui y in-
clude scaling ela ions and signs o he ene gy con ibu ions, he xc i ial heo em, eedom
om sel -in e ac ion in one-elec on sys ems [PZ81] (see Chap. 3 o mo e de ails), p ope -
ies o he xc hole, ze o- o ce and o que heo em o he xc po en ial, and he asymp o ic
beha io o he po en ial. In TDDFT, du ing ime p opaga ion undamen al conse a ion
laws should be espec ed. In pa icula , he o al ene gy should emain cons an when no
ex e nal pe u ba ion ac s. The o que and ze o- o ce heo ems [Vig95, GDP96] also apply
in TDDFT. The la e s a es ha he xc po en ial canno exe a ne o ce on he sys em as
Zn(
, )∇ xc(
, ) d3 = 0.
(2.34)
Mo e exac cons ain s on densi y unc ionals and hei ele ance in TDDFT a e discussed
in Re s. [HPB99, MUN
+
06, MMN
+
12].
2.5 Exchange-co ela ion unc ionals
2.5.1 App oxima ions o he exchange-co ela ion ene gy unc ional
The p ac ical usabili y and eliabili y o DFT and TDDFT s ongly depends on he quali y
o he used xc densi y unc ionals. This aspec has no been se led so a . I gi e an
in oduc ion in o he mos impo an and o his hesis mos ele an app oxima ions o
Exc
in he ollowing.
Local and semilocal unc ionals:
The ea lies and s ill one o he mos wide-sp ead
app oxima ions is he
local densi y app oxima ion
(LDA) [HK64] and i s ex ension o spin-
dependen cases, he
local spin-densi y app oxima ion
(LSDA) [ BH72]. The a ionale behind
LDA is o use he unc ionals o he exchange and co ela ion ene gies (
hom
x(nhom)
and
hom
c(nhom)
) o he homogeneous elec on gas and eplace he homogeneous elec on densi y
nhom
by he local densi y
n(
)
acco ding o
hom
xc [n( )] = hhom
x(n0) + hom
c(n0)in0=n( ).
(2.35)
The exchange pa o he homogeneous elec on gas xc ene gy has an analy ical exp ession.
The co ela ion con ibu ion is only known om highly accu a e quan um Mon e-Ca lo simu-
la ions [CA80] and needs o be pa ame ized o applica ion in DFT, e.g., he pa ame iza ion
o Pe dew and Wang [PW92]. The LDA xc ene gy eads
ELDA
xc [n] = Zhom
xc (n( ))n( ) d3 .
(2.36)
2.5. EXCHANGE-CORRELATION FUNCTIONALS
13
Semilocal unc ionals
a e he  s class o beyond-LDA unc ionals. His o ically, he  s
s ep beyond pu ely local unc ionals was o include also g adien s o he densi y in o he
xc unc ional. Howe e , consis en imp o emen s we e ob ained only when he so-called
gene alized g adien app oxima ions
(GGAs) [LM83, PY86, Pe 86] we e in oduced. One o
he mos popula GGAs, he non-empi ical GGA o Pe dew, Bu ke, and E nze ho (PBE)
[PBE96], is based on exac cons ain s, e.g., o he xc hole. O he unc ionals use ee
pa ame e s and  hose o da a se s om e e ence calcula ions o expe imen al ndings.
Fo ins ance, he semiempi ical BLYP unc ional combines Becke88 exchange [Bec88] wi h
he co ela ion unc ional o Lee, Yang, and Pa (LYP) [LYP88]. A second class o semilocal
unc ionals a e he so-called
me a-GGAs
ha may include also highe -o de de i a i es o
he densi y, he kine ic ene gy densi y, and g adien s o he la e . No e ha me a-GGAs
may al eady all in o he nex class o unc ionals, he so-called
o bi al unc ionals
, because
hey may include explici o bi al dependence al hough hey a e semilocal in na u e.
O bi al unc ionals:
O bi al-dependen unc ionals
comp ise explici dependence o
he o bi als o he KS sys em beyond semilocal con ibu ions [KK08]. They a e s ill implici
densi y unc ionals because he o bi als hemsel es a e implici unc ionals o he densi y.
P ominen ep esen a i es o his class o unc ionals a e he sel -in e ac ion co ec ion (SIC)
o Pe dew and Zunge [PZ81] and he exac exchange (EXX) unc ional
Ex[{ϕjτ }] = −1
2X
σ=↑,↓
Nσ
X
i,j=1 Z Z ϕ∗
iσ(
)ϕ∗
jσ(
0)ϕjσ(
)ϕiσ(
0)
|
−
0|d3 d3 0,
(2.37)
he Fock exchange in eg al known om Ha ee-Fock (HF) heo y compu ed wi h KS o -
bi als. The la e includes exac exchange only and nding a compa ible co ela ion unc-
ional is known o be dicul . In he SIC app oach, exchange and co ela ion a e based on
he unde lying xc unc ional app oxima ion on op o which he sel -in e ac ion co ec ion is
pe o med (see Pub3 and Chap. 3 o a discussion and mo e de ails).
The p ice one has o pay when using o bi al unc ionals a e dicul ies when compu ing
he xc po en ial ia he unc ional de i a i e o
Exc
wi h espec o he densi y, because one
does no know he explici densi y dependence o he o bi als. A solu ion o his p oblem,
he op imized eec i e po en ial (OEP) me hod [SH53, TS76, GKG97, KK08], yields a mul-
iplica i e xc po en ial in he KS sense. This me hod is in oduced in he con ex o SIC in
Chap. 3. As an al e na i e o he OEP, one may lea e he g ounds o KS heo y and ely on
he gene alized KS (GKS) app oach [SGV
+
96]. In he GKS scheme, he cons ain o s ic ly
non-in e ac ing e e ence sys ems is elaxed and in e ac ing e e ence sys ems ha use a sin-
gle Sla e de e minan a e allowed. The po en ial in he GKS app oach is no longe a local
bu an o bi al-specic one. Typically, unc ionals implemen ed ia he GKS me hod in ol e
a leas a ac ion o EXX, hus mos GKS po en ials include a ac ion o he nonlocal
Fock po en ial [KK10]. De ails abou he GKS app oach and die ences o he heo e ical
amewo k o he KS scheme a e discussed in Re s. [SGV
+
96, KK10, BLS10].
Hyb id unc ionals:
In he
hyb id unc ional
idea, basically a ac ion o he EXX
unc ional is mixed wi h some semilocal (sl) densi y unc ional. Fo ins ance, a one-pa ame e
hyb id can be w i en as
Ehyb
xc =aEx+ (1 −a)Esl
x+Esl
c
(2.38)
wi h he mixing pa ame e
a
. In his case, one mixes he semilocal exchange
Esl
x
wi h
exac exchange and akes he ull co ela ion
Esl
c
o he semilocal unc ional. Typically, he
14
CHAPTER 2. DENSITY FUNCTIONAL THEORY AND TIME-DEPENDENT
DENSITY FUNCTIONAL THEORY
mixing pa ame e o such hyb id unc ionals is chosen empi ically, o ins ance by  ing he
unc ional o a es se o a omic and molecula p ope ies. P obably he mos p ominen
hyb id unc ional is B3LYP [Bec93, SDCF94], a h ee-pa ame e hyb id unc ional based on
a weigh ed mix u e o LDA exchange and co ela ion, LYP co ela ion, Becke88 exchange,
and EXX. The h ee pa ame e s a e ob ained empi ically om  ing o a se o a omic
p ope ies. O he app oaches emphasize he densi y dependence o such mixing pa ame e s
and sugges app oaches o compu e mixing pa ame e s om he densi y alone [MVO
+
11].
Recen ly, he
ange-sepa a ed hyb id unc ional
idea became inc easingly popula . I es s
upon a ange-sepa a ion scheme [Sa 95] o he elec on-elec on in e ac ion in o a sho - ange
and a long- ange pa . In hose wo pa s, he elec on-elec on in e ac ion is ea ed wi h
die en unc ional app oxima ions. Each o hose app oxima ions is supposed o play a
specic ole: Typically semilocal unc ionals a e used in he sho - ange pa , whe eas EXX
is supposed o domina e he long- ange con ibu ion. The ansi ion be ween sho and long
ange is de e mined by a pa i ioning scheme and a ange-sepa a ion pa ame e
γ
[Sa 95,
VS06, LB07, BLS10, KSSB11]. The in e se o his pa ame e
1/γ
can be in e p e ed o be a
cha ac e is ic leng h scale ha dis inguishes be ween sho and long ang. The choice o
γ
is
he key elemen o he pe o mance o such ange-sepa a ion ideas. Fi s app oaches o his
kind we e based on empi ical ange-sepa a ion pa ame e s [YTH04, VS06, LB07, CHG07],
bu only ecen ly pa ame e uning o some addi ional heo e ical cons ain s [SKB09a,
SKB09b, BLS10, KSSB11] has been employed.
Tuned ange-sepa a ed hyb id unc ionals
, as
long as he unde lying xc unc ionals a e non-empi ical, do no ely on empi ical inpu da a.
Hyb id unc ionals in ol e an explici dependence on he o bi als. Thus, he dicul ies
wi h o bi al unc ionals al eady discussed in he p e ious sec ion apply again. Usually, hyb id
unc ionals a e implemen ed ia he GKS scheme [SGV
+
96, KSSB11]. An implemen a ion
wi hin he KS amewo k o DFT can be pe o med based on he OEP me hod.
2.5.2 Exchange-co ela ion unc ionals in TDDFT
The xc ac ion unc ional and he co esponding TD xc po en ial in TDDFT a e e y complex
quan i ies, p esumably e en mo e complex han hei s a ic coun e pa s. Ye , he alidi y
o esul s om TDDFT calcula ions s ongly depends on he quali y o he desc ip ion o xc
eec s. The e o e, al hough he ac ion unc ional o malism p o ides a solid s a ing poin
o de eloping unc ional app oxima ions, nding eliable TD xc po en ials o p ac ical
calcula ions can be edious, in pa icula , o dicul applica ions like cha ge ans e .
Today, mos applica ions o TDDFT ely on he so-called
adiaba ic app oxima ion
[GDP96, EBF07]. The a ionale behind his app oach is ha in cases whe e he ex e nal
po en ial a ies slowly enough in ime, he ime e olu ion o he sys em looses i s dependence
on he pas , and can be well desc ibed by he ins an aneous densi y. Thus, he adiaba ic
app oxima ion amoun s o using well-known unc ionals and he co esponding po en ials
om GS DFT as TD xc po en ials acco ding o
adia
xc,σ[n](
, ) = xc,σ[n ](
) = δExc[n ]
δn ,σ(
),
(2.39)
whe e he ime a iable
is conside ed as a pa ame e o he densi y
n
. Fo linea esponse
calcula ions he adiaba ic app oxima ions may also be applied o he xc ke nel. The adia-
ba ic xc ke nel is comple ely local in ime, hus ans o ming i in o Fou ie space yields a
equency-independen xc ke nel [MUN
+
06].
2.6. NUMERICAL REALIZATION
15
In p inciple, he adiaba ic app oxima ion in oduced so a may be applied o e e y
GS xc app oxima ion. Many applica ions o TDDFT ely on he adiaba ic local densi y
app oxima ion (ALDA, TDLDA), which is he simples ex ension o GS DFT o he TD
case. Su p isingly, ALDA wo ks qui e well a beyond i s ob ious ange o alidi y, namely
slowly a ying densi ies bo h in space and in ime [MUN
+
06]. O he applica ions use GGAs,
hyb id unc ionals, o o bi als unc ionals as in oduced in Sec. 2.5.1. In any case, one should
be ca e ul in choosing he igh unc ional o each applica ion depending on he sys em and
obse able one is in e es ed in.
Ca e should be aken when using he e m adiaba ic and conside ing i s implica ions on
he his o y dependence o such app oaches. Explici densi y unc ionals implemen ed ia Eq.
(2.39) neglec all memo y dependence o he sys em e olu ion o imes
0≤
. When i comes
o explici ly o bi al-dependen unc ionals, howe e , he TD KS o bi als in gene al depend
on he en i e his o y o he densi y
n(
, 0)
o
0≤
, hus eco e in a na u al way pa o
he memo y dependence ha is no in he TD densi y [GDP96, MBW02, MMN
+
12]. In his
case, he adiaba ic app oxima ion should be e be e med
o bi al-adiaba ic app oxima ion
in con as o
densi y-adiaba ic app oxima ion
.
2.6 Nume ical ealiza ion
Mos DFT and RT TDDFT in es iga ions p esen ed h oughou his hesis a e based on
he Bay eu h e sion [MK07, Mun07, Mun09] o he PARSEC p og am package [KMT
+
06].
PARSEC is a eal-space elec onic-s uc u e code ha uses no m-conse ing pseudopo en-
ials o T oullie -Ma ins [TM91] ype and a high-o de ni e die ence schemes o nume i-
cally ep esen ing he Laplacian ope a o [CTS94, KMT
+
06]. The GS e sion is designed o
sol ing he KS equa ions by nume ical diagonaliza ion o he KS Hamil onian. The Bay eu h
e sion includes solu ion o he OEP equa ion [KK08, Kö 09] ia he cons uc ion scheme o
Re s. [KP03a, KP03b] and app oxima ions o he OEP, as o ins ance he K iege -Li-Ia a e
(KLI) app oxima ion [KLI92] and he Sla e app oxima ion [Sla51].
The p ac ical ealiza ion o he RT p opaga ion idea is based on s epwise nume ical
p opaga ion wi h ime s eps
∆
using he p opaga o
U( +∆ , )
[CAO
+
06, Mun07, Mun09].
The PARSEC eal- ime TDDFT implemen a ion uses a Taylo se ies up o ou h o de
o nume ically expand
U( + ∆ , )
combined wi h he exponen ial midpoin ule [MK07,
Mun07, Mun09]. In his scheme, he po en ial needs o be de e mined wice pe ime s ep.
O he p opaga ion echniques a e explained in Re . [CMR04]. To a oid spu ious eec ion
o densi y ha mo es o he bounda y o he nume ical g id, RT PARSEC oe s abso bing
bounda ies [RSA
+
06, Mun07, Mun09]. Las bu no leas , he Bay eu h e sion o TD
PARSEC [MK07, Mun07] includes RT p opaga ion o o bi al unc ionals implemen ed ia
he ime-dependen KLI app oxima ion.
Fo he easibili y o mos RT calcula ions p esen ed in his hesis, nume ical op imiza-
ion o he o iginal PARSEC e sion and he implemen a ion o new algo i hms we e needed
o each accep able compu a ion imes. One o he mos ime-consuming s eps du ing ime
p opaga ion is he e alua ion o he Ha ee po en ial ia solu ion o Poisson's equa ion.
Nume ical eciency o he Poisson sol e is especially impo an when o bi al unc ionals
as, e.g., he EXX o he SIC a e used. The e o e, I implemen ed a mul ig id sol e as an
al e na i e o he exis ing conjuga e g adien sol e . De ails abou he nume ical ealiza-
22
CHAPTER 3. SELF-INTERACTION CORRECTION
(see Pub3 o mo e de ails). Howe e , s a ing om eal canonical KS o bi als, he e a e s ill
wo possible choices o he o bi als
{˜ϕiσ}
and he ans o ma ion
Uijσ
: They can be chosen
ei he bo h eal- alued o bo h complex- alued. Each choice co espond o one deni ion
o he uni a y ans o ma ion:
Uijσ
is dened acco ding o Eq. (3.12) wi h numbe s ei he
es ic ed o be eal o ee o be complex. These op ions inuence he shape o he esul ing
o bi al densi ies
˜niσ =|˜ϕiσ|2
and, he e o e, may exhibi die en pe o mances o he SIC.
This inuence is discussed in de ail in Pub3 and summa ized in Sec. 3.5. I deno e ene gy-
minimizing GSIC based on he GOEP, GKLI, and GSLA me hods by GOEP-SIC, GKLI-
SIC, and GSLA-SIC. I no s a ed explici ly, complex- alued ene gy minimiza ion is used
h oughou his hesis.
Spa ially localizing ans o ma ion:
A second deni ion o he uni a y ans o ma-
ion is based on he obse a ion ha ypically localized o
Fos e -Boys
(FOBO) o bi als
[Boy60, FB60, ER63, PM89] a e good app oxima ions o eal ene gy-minimizing o bi als
[KKM08]. This nding is a ionalized in Re . [Kö 11] and suppo ed by Pub3. Pe o ming
he FOBO uni a y op imiza ion is nume ically less expensi e, hus se es as a easonable
al e na i e o ene gy-minimizing ans o ma ions. The FOBO ans o ma ion can also be
pe o med wi h complex- alued o bi als. Ye , he FOBO c i e ion is no sensi i e o he
deg ees o eedom p o ided by complex numbe s. I deno e GSIC wi h FOBO localiza ion
based on he GKLI po en ial by FOBO-SIC.
The op imiza ion p ocedu e o he ene gy-minimiza ion and spa ial localiza ion c i e ia
is one o he mos ime-consuming s eps in GSIC calcula ions. The e o e, o gua an ee
easonable nume ical pe o mance, I implemen ed new algo i hms ha a e based on ene gy
g adien s (see Appendix C). Bes pe o mance can now be ob ained in PARSEC wi h an
algo i hm ha uses conjuga e g adien s, s ep-size op imiza ion, and ha akes he uni a y
cons ain o
Uijσ
in o accoun explici ly. This algo i hm was adap ed and implemen ed o
PARSEC in collabo a ion wi h Pe e and Simon Klüp el. Mo e de ails a e explained in Pub3
and Appendix C.1.3. The ex ension o exis ing algo i hms in PARSEC (see Re . [Kö 09]) o
complex- alued
{˜ϕiσ}
and
Uijσ
is p esen ed in Appendix C.1. In Appendix C.2, I compiled
an o e iew o PARSEC ea u es and use -inpu op ions ela ed o GSIC.
3.5 Gene alized sel -in e ac ion co ec ion in DFT
Many o he ea u es o GOEP and i s GKLI app oxima ion we e al eady in es iga ed in he
wo k o Thomas Kö zdö e [KKM08, Kö 09]. Bo h app oaches yield GS ene gies and spin
densi ies in close ag eemen [KKM08]. The ela ion be ween GKLI and GOEP is illus a ed
u he by he esul s in Appendix A. Ye , using complex deg ees o eedom in he uni a y
ans o ma ion yields a new pe spec i e on he GS GOEP-SIC app oach. The inuence o
complex- alued uni a y ans o ma ions and op imized o bi als is he con en o Pub3. In
he con ex o Pub4, I pe o med supplemen a y in es iga ions on he esponse beha io o
chain-like sys ems depending on die en app oxima ions o he GOEP and die en choices
o
Uijσ
. These ndings a e p esen ed in de ail in Appendix A.1. He e, I gi e a summa y o
he SIC and GSIC esul s o his hesis.
In es iga ions o o al ene gies and bond leng hs o a ep esen a i e se o dime s and
small molecules in Pub3 show ha he choice o he uni a y ans o ma ion has no able
inuence on he ou come o GKLI-based calcula ions, while s anda d KLI-SIC esul s may

3.5. GENERALIZED SELF-INTERACTION CORRECTION IN DFT
23
Figu e 3.1: O bi al densi ies o he ene gy-minimizing o bi als o H
2
O compu ed sel -
consis en ly wi h GKLI-SIC using eal- alued ene gy minimiza ion (uppe pa ) and
complex- alued ene gy minimiza ion (lowe pa ). The numbe ing o he op imized o bi al
densi ies is a bi a y as hei o de does no ca y physical meaning. One clea ly obse es
o he example o H
2
O ha he o bi al densi ies o eal- alued ene gy-minimizing o bi als
o m nodal planes, whe eas nodal planes a e a oided in he complex- alued case.
be comp omised by deciencies o he KLI po en ial (see Sec. 3.3.1). To al GSIC ene gies
dec ease wi h inc easing he deg ees o eedom o he uni a y ans o ma ion om eal-
alued o complex- alued numbe s. The inuence on he bond leng h die s o single-
bond and mul iple-bond sys ems. In case o single-bond sys ems, inc easing he deg ees
o eedom leads o inc easing bond-leng h unde es ima ion. Fo double- and iple-bond
sys ems, GKLI-SIC wi h a complex- alued ene gy-minimizing ans o ma ion imp o es upon
he known SIC bond-leng h unde es ima ion [GU97, VSP
+
06]. To al ene gies and bond
leng hs compu ed wi h FOBO-SIC a e close o eal- alued ene gy-minimizing GKLI-SIC.
The die ence be ween eal and complex ene gy minimizing o bi als is ela ed o he
o ma ion o nodal planes o he o bi al densi ies
˜niσ(
)
. The op imized o bi als
{˜ϕiσ}
need o be o hogonal like he canonical KS o bi als. Real- alued o bi als p ese e his
o hogonali y ia o ma ion o nodal planes ha a e passed on o he o bi al densi ies.
Howe e , using complex deg ees o eedom, nodal planes o he o bi al densi ies can be
a oided while p ese ing o hogonali y o he op imized o bi als. This nding is illus a ed
wi h he o bi al densi ies o H
2
O in Fig. 3.1 and wi h CO in Pub3. Thus,
˜niσ(
)
wi hou
nodal planes a e smoo he and, he e o e, close o GS densi ies. Such o bi al densi ies a e
impo an in he con ex o he ques ion whe he i is allowed o inse o bi al densi ies in o
GS xc ene gy unc ionals: Hope is high ha smoo h o bi al densi ies o he complex- alued
case a e close o he ealm whe e (semi)local unc ionals a e app op ia e. The educed
appea ance o nodal planes in complex- alued
˜niσ(
)
also se es as an explana ion why he
SIC o PBE is mo e sensi i e o changes om eal o complex numbe s han he SIC o
24
CHAPTER 3. SELF-INTERACTION CORRECTION
LSDA (see Pub3). The PBE unc ional is mo e sensi i e o la ge changes o he densi y due
o i s dependence on densi y g adien s. The e o e, allowing o a oiding nodal planes whe e
la ge densi y g adien s occu makes a mo e p onounced die ence o PBE han o LSDA.
Mo e insigh in o he pe o mance o die en (G)OEP app oxima ions is p o ided om
he esul s o Appendix A.1 and Appendix A.2, whe e ene gy minimiza ion in GKLI-SIC is
always pe o med wi h complex- alued o bi als. The esponse o hyd ogen chains o s a ic
elec ic elds shows ha SLA-SIC and KLI-SIC may exhibi a dis o ed esponse beha io .
This nding can be unde s ood om he uni a y in a iance o bo h he SIC ene gy exp ession
and he KLI po en ial app oxima ion. Ins ead, he GKLI-SIC po en ial is a easonably
good app oxima ion o ull GOEP-SIC and exhibi s he eld-coun e ac ing beha io o he
xc po en ial ha is impo an o a p ope desc ip ion o pola izabili ies (see Sec. 2.4).
GSLA-SIC gi es pola izabili ies close o GKLI-SIC o sho chains, bu he pola izabili y
de ia ions o GKLI-SIC inc ease wi h inc easing chain leng h. This nding is in line wi h
he obse a ion ha GSLA-SIC does no show a clea eld-coun e ac ing esponse beha io .
In case o hyd ogen chains, FOBO-SIC pola izabili ies a e close o GKLI-SIC, bu sligh ly
wo se. The si ua ion e e ses in polyace ylene chains, whe e FOBO-SIC pola izabili ies a e
sligh ly close o e e ence calcula ions han GKLI-SIC. The compa ison be ween hyd ogen
and polyace ylene chains indica es ha localiza ion eec s play die en oles in die en
kinds o sys ems. Only de ailed in es iga ions show i o bi al localiza ion o ull ene gy
minimiza ion gi es esul s close o e e ence calcula ions and o expe imen al ndings.
3.6 Gene aliza ion o SIC in TDDFT
One o he main aspec s o his wo k is he ex ension o GSIC om GS DFT o TDDFT
co e ed by Pub2 and Pub4. The mo i a ion o his ex ension was based on he p omising
ea u es o GKLI-SIC in DFT, see o ins ance Re s. [KKM08, KKMK09, Kö 09, DKK
+
11]
and Sec. 3.5. The GKLI-SIC pe o mance is good news o applica ion in TDDFT o
wo easons. Fi s , he ime-dependen OEP (TDOEP) is known o be e y demanding o
sol e [MK06, WU08]. The e o e, mos applica ions o o bi al unc ionals in TDDFT ely
on a TD ex ension o he KLI app oxima ion [UGG95, MK06]. Second, howe e , he
ime-
dependen KLI
(TDKLI) app oxima ion sue s om i s own p oblems beyond he deciencies
known o GS KLI: RT p opaga ion o he TDKLI po en ial shows s abili y issues and ze o-
o ce heo em iola ion [MK LR07, MDRS09b, MDRS11]. The e o e, s a ing wi h GSIC
in TDDFT, hope was high ha he addi ional deg ees o eedom p o ided by he uni a y
ans o ma ion oge he wi h he SIC unc ional could be exploi ed o suppo p opaga ion
s abili y. Then, he p omising ea u es o SIC could help in TDDFT o s udy ques ions ha
we e conside ed oo dicul o local and semilocal densi y unc ionals.
The ex ension o SIC o TDDFT by Tong and Chu [TC98b, TC98a] uses he PZ SIC
ene gy unc ional in an o bi al-adiaba ic sense (see Sec. 2.5.2) oge he wi h he TDKLI
app oxima ion. TD SIC wi h GKLI is based on a simila a ionale, i.e., he o bi als and
he uni a y ans o ma ion a e used a e e y ins an o ime o keep he sys em on a s able
ene gy pa h du ing p opaga ion. The GSIC ex ension o TDDFT esul s om h ee s eps.
Fi s , he gene alized o bi al-specic po en ials a e ans e ed o he TD case, hus eading
uG
xc,jσ(
, ) = 1
ϕ∗
jσ(
, )
Nσ
X
i
Uσ
ij( ) ˜ϕ∗
iσ(
, )˜ SIC
iσ (
, ),
(3.13)
3.6. GENERALIZATION OF SIC IN TDDFT
25
whe e
˜ SIC
iσ (
, ) = δESIC
xc [{˜nkτ }]
δ˜niσ(
, ).
(3.14)
Second, he s anda d o bi al-specic po en ials
uxc,jσ(
, )
need o be eplaced by he gen-
e alized ones in he TDKLI po en ial [TC98b, TC98a, MK06]
TDKLI
xc,σ (
, ) + σ(
, ) =
TDSLA
xc,σ (
, ) + 1
2nσ(
, )
Nσ
X
j=1
njσ(
, )¯ TDKLI
xc,jσ ( )−¯uxc,jσ( )+ c.c.,
(3.15)
whe e
σ(
, ) = −i
4nσ(
, )
Nσ
X
j=1 ∇2njσ(
, )Z
−∞ ¯uxc,jσ( 0)−¯u∗
xc,jσ( 0)d 0
(3.16)
is he so-called memo y e m and he TD Sla e (TDSLA) con ibu ion eads
TDSLA
xc,σ (
, ) = 1
2nσ(
, )
Nσ
X
j=1
njσ(
, )uxc,jσ(
, ) + u∗
xc,jσ(
, ).
(3.17)
O bi al-a e aged po en ials a e compu ed in analogy o Eqs. (3.7) and (3.8) o he g ound
s a e. Thi d, in he TD case, he uni a y ans o ma ion is a TD unc ion ha connec s he
TD KS o bi als and a second op imized o bi al se acco ding o
˜ϕiσ(
, ) =
Nσ
X
j=1
Uσ
ij( )ϕjσ(
, ).
(3.18)
Thus, a sui able choice o
Uσ
ij( )
concludes he
ime-dependen GKLI
(TDGKLI) app oxima-
ion o he SIC unc ional and es ablishes
ime-dependen GSIC
(TDGSIC). Replacing
uxc,jσ
by
uG
xc,jσ
in Eq. (3.17) yields he
ime-dependen GSLA
(TDGSLA) app oxima ion.
Al hough his ex ension appea s s aigh o wa d a  s glance, he e a e a lo o hidden
dicul ies and obs acles. The  s issue conce ns he choice o
Uσ
ij( )
ha is discussed
comp ehensi ely in Pub4. The cen al idea is o apply he ene gy-minimiza ion and he
spa ial localiza ion c i e ia known om GS DFT (see Sec. 3.4) a e e y ins an o ime.
This p oceeding is in line wi h he o bi al-adiaba ic use o he SIC ene gy unc ional. I
use he sho cu TDGKLI-SIC o complex- alued ene gy minimiza ion and TDFOBO-SIC
o spa ial localiza ion oge he wi h he TDGKLI app oxima ion. Ye , op imizing ei he
o he GS c i e ia o many en housands o ime s eps du ing ime p opaga ion ende s
applica ion o TDGSIC imp ac ical o sizable sys ems. When p opaga ing he GSIC g ound
s a e wi hou ex e nal pe u ba ion, I ealized ha pa o he TD ans o ma ion could also
be pe o med analy ically. This idea enabled he b eak h ough o TDGSIC. In such a se up,
Uσ
ij( )
e ol es om
Uσ
ij( −∆ )
o he p e ious ime s ep wi h he KS eigen alues
εj
, i.e.,
Uσ
ij( ) = eiεj∆ Uσ
ij( −∆ ).
(3.19)
This ime p opaga ion o he uni a y ans o ma ion can be used as an ini ial guess o
subsequen uni a y op imiza ion a e e y ins an o ime o as an app oxima ion o he
26
CHAPTER 3. SELF-INTERACTION CORRECTION
TD ans o ma ion. Mo e de ails a e explained in Pub4. All TDGSIC p opaga ions a e pe -
o med wi h complex TD o bi als and ans o ma ions. They a e s a ed om a g ound s a e
wi h complex- alued
Uσ
ij
op imiza ion because, o he wise, when s a ing om eal- alued op-
imized o bi als and ans o ma ions he addi ional deg ees o eedom o he complex- alued
TD ans o ma ion may ins an aneously change he g ound s a e e en wi hou ex e nal pe -
u ba ion.
Finally, he choice o he uni a y ans o ma ion is impo an o he explici memo y
dependence o he GKLI app oxima ion ia i s gene alized
σ(
, )
(
G
σ(
, )
) con ibu ion.
While
σ(
, )
anishes in TDKLI-SIC [UGG95, GDP96, TC98a, TC98b, MK06], i is no
clea om he s a ha
G
σ(
, )
anishes in TDGKLI-SIC. Pub4 demons a es ha he
con ibu ion o he TDGKLI memo y e m depends on he c i e ion o
Uσ
ij( )
. When ene gy
minimiza ion is used in he g ound s a e and du ing ime p opaga ion, he memo y e m
does no con ibu e o TDGKLI. Fo all o he cases, he memo y dependence ia
G
σ(
, )
emains unclea . Explici choices o
G
σ(
, )
a e explained in Pub4. No e ha no ma e
how
G
σ(
, )
con ibu es o he TDGKLI memo y, p opaga ion o SIC wi h TDGKLI includes
an implici dependence on he his o y o he densi y
n(
, 0)
o
0≤
ia he KS o bi als in
a na u al way [GDP96, MBW02, MMN
+
12].
3.7 Pe o mance o gene alized SIC in TDDFT
Fi s in es iga ions o TDGSIC [Ho 08] ocused on he p opaga ion s abili y ques ion o Na
5
,
a no o iously p oblema ic sys em o RT p opaga ion. He e, he p opaga ion ins abili y
mani es s in ze o- o ce heo em iola ion and no able d i s o he o al ene gy om i s
GS alue al hough no ex e nal eld ac s [MK LR07, Mun07]. Re e ence [Ho 08] indica es
ha p opaga ion s abili y can be inc eased by inc easing he nume ical accu acy o he
uni a y op imiza ion. Howe e , using he ini ial guess o Eq. (3.19) pu s his nding in o
a new pe spec i e. The appendix o Pub4 demons a es ha TDGSIC schemes show much
be e s abili y han p opaga ion wi h TDKLI-SIC e en o he dicul case o Na
5
. Fo
mos applica ions, easonably s able p opaga ion o TDGSIC can be pe o med in a ime
window ha is long enough o compu ing he dipole spec um. Howe e , as a wa ning, one
should be awa e ha ins abili y p oblems may occu and always check o s abili y issues.
Based on his eassu ing nding, TDGSIC is eady o applica ions o dynamic si ua ions
and in es iga ions o exci a ion ea u es. I summa ize he esul s on he pe o mance o
TDGSIC in he ollowing.
Hyd ogen chains a e a anspa en model sys em o s udy he esponse beha io o he
po en ial o die en app oxima ions o
xc(
, )
. Fo slowly a ying ex e nal pe u ba ions,
one expec s he TD xc po en ial o pe o m simila o GS xc po en ials, i.e., ha i coun-
e ac s he ex e nal eld (see Sec. 2.4). Pub4 demons a es ha he TD esponse o such
an ex e nal po en ial pa allels he esponse beha io known om GS xc po en ials:
xc(
, )
o TDLDA ollows he ex e nal eld, whe eas TDGKLI-SIC exhibi s a eld-coun e ac ing
beha io . A highe equencies o he ex e nal pe u ba ion, TDGKLI-SIC de elops compli-
ca ed ea u es ha can no be assigned o a simple equency-dependen esponse beha io .
Simila s udies o die en app oxima ions o TDGKLI-SIC a e also p esen ed in Pub4. In-
e es ingly, TDFOBO-SIC exhibi s a esponse beha io e y simila o TDGKLI-SIC. While
a clea assignmen o he ene gy-minimizing GSLA-SIC esponse was dicul in he GS case,
3.7. PERFORMANCE OF GENERALIZED SIC IN TDDFT
27
TDGSLA-SIC unambiguously ollows he ex e nal eld e en a ele a ed equencies o he
ex e nal pe u ba ion. Mo eo e , using only Eq. (3.19) o he uni a y ans o ma ion spoils
he eld-coun e ac ing beha io o he xc po en ial al eady a low equencies o he ex e nal
eld. Thus, he esul s o Pub4 show ha he esponse e m o he (G)KLI app oxima ion
and op imizing he uni a y ans o ma ion a e e y ins an o ime a e impo an o a p ope
eld-coun e ac ing esponse beha io .
The TD esponse beha io s udies a e complemen ed in Pub4 by in es iga ions o hy-
d ogen chain exci a ions ene gies. Whe eas TDKLI-SIC esul s sue om p opaga ion
ins abili y and do no imp o e upon TDLDA, TDGSIC no ably shi s he lowes exci a ion
ene gies o highe alues. TDGKLI-SIC exci a ion ene gies a e in close ag eemen o B3LYP
bu de ia e om app oxima e coupled-clus e singles-and-doubles model exci a ion ene gies
by a leas 0.9 eV. In conclusion, TDGKLI-SIC and i s TDFOBO-SIC app oxima ion show
p omising pe o mance on hyd ogen chains. Ye , al eady s a ic pola izabili ies o hyd ogen
chains exhibi la ge de ia ions be ween OEP and GOEP [KK11] ( o mo e backg ound, see
Appendix A.1) and a compa ison be ween TDOEP and TDGOEP is no a ailable. Because
o he peculia na u e o hyd ogen chains [ FdB L
+
02], he pe o mance o TDGSIC also
needs o be assessed o eal molecules.
Applica ion o me al clus e s in Pub4 shows ha TDGSIC does no spoil he good accu-
acy ha al eady TDLDA eaches. Howe e , TDGSIC imp o es in cases whe e (semi)local
unc ionals exhibi sys ema ic ailu es: In hyd ogena ed silicon clus e s, quan um conne-
men and exci onic eec s a e known o play an impo an ole [OCL97, RL98, RL00, ORR02,
VOC02, VOC06]. He e, low-lying op ical exci a ions a e unde es ima ed by s anda d unc-
ionals and TDKLI-SIC [MCR01], in pa icula o he e y small clus e s. Pub2 and Pub4
demons a e ha TDGSIC no ably imp o es and yields exci a ion ene gies in good ag ee-
men wi h esul s om he GW Be he-Salpe e equa ion app oach. Lowes exci a ion ene -
gies o oligo-ace ylenes in Pub4 also imp o e upon TDLDA and e eal die ences be ween
he die en possible choices o he uni a y ans o ma ion du ing ime p opaga ion. Las
bu no leas , TDGSIC also gi es p omising esul s o s a ic and dynamic CT si ua ions as
well as CT exci a ion ene gies. This is he opic o he nex chap e .

Chap e 4
Cha ge ans e and cha ge- ans e
exci a ion ene gies
The ob ious inabili y o p esen -day physics and chemis y o accoun
o such e en s is no eason a all o doub ing ha hey can be
accoun ed o by hose sciences.
E win Sch ödinge
Wha is li e? (1944)
A us wo hy desc ip ion and heo e ical p edic ion o cha ge ans e is one o he
well-known and longs anding p oblems o ( ime-dependen ) densi y unc ional heo y
[DWHG03, Toz03, Mai05, TFSB05, KBE06, TS07, KBY07, TS08, EVV09, LBBS12]. Typi-
cally, ene gies o elec onic exci a ions ha exhibi CT cha ac e a e unde es ima ed when
calcula ed wi h (semi)local xc unc ionals [DWHG03, Toz03, Mai05]. T anspo p ope -
ies as, e.g., conduc ance and
I
-
V
cha ac e is ics o molecula elec onic de ices may be
se e ely in e o [TFSB05, KBE06, TS07, KBY07, TS08, LBBS12] when calcula ed om he
Landaue -Bü ike app oach [Lan57, Bü 86] based on non-equilib ium G een's unc ion he-
o y in combina ion wi h DFT using s anda d xc unc ionals. Recen ly, also RT p opaga ion
has been used o s udy CT scena ios [KSA
+
05, CEVV06], bu he eliabili y o esul s is
limi ed by he quali y o xc unc ional app oxima ions. I p o ide mo e insigh in o he CT
p oblem o (TD)DFT in Sec. 4.1 and complemen his discussion by some solu ion ideas.
The CT ailu e is commonly ela ed o he sel -in e ac ion e o o (semi)local xc unc ionals
[TFSB05, TS07, KBY07, TS08] and he lack o a de i a i e discon inui y in such app oaches
[TFSB05, KBE06, TS07, TS08, LBBS12]. The e o e, sel -in e ac ion co ec ion is a p omis-
ing app oach o imp o ing he (TD)DFT desc ip ion o CT phenomena. I discuss in Sec. 4.2
how gene alized KS SIC in oduced in Chap. 3 pe o ms in s a ic and dynamic CT si ua ions
and how i imp o es he desc ip ion o CT exci a ion ene gies.
4.1 The cha ge- ans e p oblem o DFT and TDDFT and
some solu ion ideas
The discussion o he CT p oblem o TDDFT in he li e a u e [DWHG03, Toz03, DHG04,
GB04, Mai05, TFSB05, MT06, ZSK
+
09, Au 09, IHG10, FRM11, HG12] e eals he many
ace s o his in iguing issue. Among he many die en pe spec i es o app oach he CT
29
30
CHAPTER 4. CHARGE TRANSFER AND CHARGE-TRANSFER EXCITATION
ENERGIES
Figu e 4.1: Model CT si ua ion be ween a dono (D) and an accep o (A) sepa a ed by he
la ge dis ance
R
. See he main ex .
p oblem, in he ollowing, I p esen one ha is based on he long- ange CT be ween a dono
(D) and an accep o (A) subuni sepa a ed by he la ge dis ance
R
( o an illus a ion, see
Fig. 4.1). The exci a ion ene gy ha is needed o ans e one in ege uni o he elemen a y
elec ic cha ge
e
om D o A is gi en by Mulliken's ule [Mul50, SKB09a, SKB09b]
ΩM=IP D−EAA−1
R.
(4.1)
The ioniza ion po en ial
IP D
o D is he ene gy ha needs o be payed when emo ing
one elec on om D, and he elec on ani y
EAA
o A is he ene gy one gains when
adding one elec on o A. The hi d e m amoun s o he ene gy gain due o he Coulomb
in e ac ion be ween he addi ional nega i e cha ge on A and he posi i e cha ge o D. The
exci a ion ene gy needed o a e y long- ange cha ge ans e (
R→ ∞
) equals he die ence
IP D−EAA
be ween he ioniza ion po en ial o D and he elec on ani y o A.
The impo an ques ion now is how TDDFT beha es [DWHG03, Toz03, DHG04]. Fo a
anspa en illus a ion, I use he TDDFT linea esponse app oach in single-pole app ox-
ima ion (see Sec. 2.2.2). One assumes ha in he case o a well sepa a ed D and A, he
eigen alues and o bi als o he en i e sys em co espond o he eigen alues and o bi als o
he isola ed D and A, and he cha ge ans e is domina ed by he ansi ion om he HOMO
(H) loca ed on D o he LUMO (L) loca ed on A. Wi h hese assump ions, he exci a ion
ene gy eads
Ω = ωKS + 2 Z Z ϕD∗
H(
)ϕA
L(
) Hxc(
,
0, ωKS)ϕD
H(
0)ϕA∗
L(
0) d3 d3 0,
(4.2)
whe e
ωKS =εA
L−εD
H
abb e ia es he eigen alue die ence. The o e lap o KS o bi als
localized on D and o bi als localized on A dec eases exponen ially wi h he dis ance
R
, hus
he o bi al p oduc s o he las e m o Eq. (4.2) anish when
R→ ∞
. This e m con ibu es
o he CT exci a ion ene gy only when he ke nel di e ges o compensa e o he anishing
o e lap [GB04]. No e ha by deni ion he exac xc ke nel exhibi s all ea u es ha a e
equi ed o yield co ec CT exci a ion ene gies. Howe e , adiaba ic, (semi)local xc app ox-
ima ions do no show singula beha io o he xc ke nel. Resul ing CT exci a ion ene gies
a e domina ed by
ωKS
, which is ypically no a good app oxima ion o he undamen al gap
∆ = IP D−EAA
, and do no exhibi he
1/R
beha io .
This deciency o s anda d, explici ly densi y-dependen unc ionals has been analyzed
om die en pe spec i es. S ong equency dependence o he xc ke nel and a p ope
4.2. SELF-INTERACTION CORRECTION AND CHARGE TRANSFER
31
inclusion o he pa icle numbe discon inui y ha e been iden ied o be impo an o he
eliable desc ip ion o cha ge ans e [Toz03, Mai05, TFSB05, MT06, HG12]. Mo eo e ,
he CT p oblem has equen ly been ela ed o he sel -in e ac ion e o o many densi y
unc ional app oxima ions [TFSB05, TS07, KBY07, TS08].
Recen app oaches o imp o e upon he CT ailu e o (semi)local unc ionals include
o bi al and hyb id unc ionals. Exac exchange ha does no sue om he SIE yields he
co ec
1/R
dependence [DWHG03, IHG10] due o he non-local con ibu ion o he Fock
exchange unc ional, bu misses co ela ion eec s. S anda d hyb id unc ionals include only
a ac ion o EXX and, he e o e, only pa ially co ec o he CT p oblem. Ye , ange-
sepa a ed hyb id unc ionals whe e EXX domina es he long ange ha e been shown o
p edic CT exci a ions eliably [YTH04, CHG07, LB07, SKB09b, KSBK11].
Beside he jus p esen ed app oaches, I demons a e in he ollowing sec ion ha KS
(TD)GSIC imp o es he (TD)DFT desc ip ion o CT si ua ions. The ndings o Chap.
3 al eady e eal  s indica ions o his imp o emen : GSIC exhibi s a eld-coun e ac ing
esponse beha io o s a ic ex e nal elec ic elds and imp o es s a ic pola izabili ies o chain-
like sys ems (see Chap. 3.5 and Appendix A). The eld-coun e ac ing beha io is ela ed
o he de i a i e discon inui y and mani es s also in s ep-like s uc u es o he po en ial
(see Sec. 2.4 o an in oduc ion). Such kind o discon inuous s uc u es o he po en ial
lead o singula ea u es in he xc ke nel ha may compensa e o he anishing o bi al
o e lap. In ene gy-minimizing TDGKLI-SIC, he eld-coun e ac ing beha io ca ies o e
o he esponse a low- equency TD ex e nal pe u ba ions and exhibi s no able equency
dependence when ex e nal elds wi h highe equencies a e applied (see Sec. 3.7). Las
bu no leas , he SIC unc ional is non-local and i s xc po en ial shows he p ope
1/
asymp o ic decay [PZ81, GKKG00].
4.2 Sel -in e ac ion co ec ion and cha ge ans e
4.2.1 The in ege p e e ence o elec on jumps
In his sec ion, I demons a e how (TD)GSIC desc ibes he s a ic and dynamic cha ge ans-
e be ween wo well sepa a ed D and A moie ies. To his end, I chose a anspa en model
sys em [KBY07] o wo hyd ogen chains as depic ed in he le pa o Fig. 4.2. The chains
a e sepa a ed by a dis ance o 8 Å which is la ge enough o he o e lap o he elec onic
s uc u e o he wo chains o almos anish, bu which is a he same ime sho enough o
allow cha ge ans e d i en by ex e nal elds om one chain (D) o he o he (A). The e o e,
one expec s cha ge ans e o p oceed in jumps o in ege uni s o
e
om D o A when ex-
e nal elds wi h sucien eld s eng h a e applied. Man1 in es iga es his sys em in s a ic
and dynamic si ua ions a die en ex e nal eld s eng hs and e eals he peculia ea u es
o he (TD)GSIC xc po en ial ha suppo in ege p e e ence o he elec on jumps. He e, I
summa ize and complemen he ndings o Man1. Beyond ha , I p esen in Appendix A.3
one addi ional obse a ion o he o al ene gy o he CT sys em wi h espec o he ex e nal
elec ic eld s eng h. I s a wi h GS DFT in es iga ions be o e I u n o he dynamic
scena io.
The  s impo an esul o Man1 is ha GSIC exhibi s  in con as o (semi)local and
s anda d hyb id unc ionals 
in ege pa icle p e e ence
du ing he cha ge ans e om D
o A when s a ic ex e nal elds a e applied. This nding is illus a ed in he igh pa o
38
CHAPTER 5. EXCITATION-ENERGY TRANSFER
always check i s alidi y as, e.g., he applicabili y o Fö s e heo y depends on he unde lying
coupling mechanism be ween he dono and he accep o moie y. Fö s e heo y elies on
a dipole-dipole in e ac ion be ween he ansi ion dipole momen s on D and A. Based on a
o mula ion in he weak coupling limi , he Fö s e EET a e exhibi s a cha ac e is ic
1/R6
dependence. The e o e, when applying Fö s e heo y o ins ance as a spec oscopic ule
on he nanoscale [S 78], i should be cla ied i he assump ions on which Fö s e heo y is
based a e ullled.
The  s in es iga ions o he coupling s eng h and coupling mechanism [Ho 08] be ween
a dono and an accep o molecule ocused on he dis ance dependence o he coupling s eng h
in an a emp o dis inguish be ween Fö s e - and non-Fö s e - ype coupling in a esonan
si ua ion o wo equal molecules. This wo k is published in Pub1. I demons a es in a es
sys em o wo sodium dime s and a mo e ealis ic sys em o wo benzaldehyde molecules
a which dis ance he Fö s e - ype coupling app oxima ions b eaks down. To his end, a
mul ipole expansion is pe o med explici ly in he Ha ee con ibu ion o he KS po en ial
along wo ou es, namely s a ing om he Ha ee ene gy and s a ing di ec ly on he le el
o he Ha ee po en ial. This expansion is unca ed a e he dipole-dipole coupling e m.
Pub1 a gues ha he expansion ou e s a ing om he Ha ee po en ial yields a po en ial
wi h a mo e na u al beha io and, he e o e, uses his po en ial du ing ime p opaga ion. I
explain he po en ial expansion idea in g ea e de ail oge he wi h i s implemen a ion o
he PARSEC code in Appendix D.1. F om a compa ison be ween wo eal- ime e olu ions
o he coupled sys em o wo molecules, one wi h he ull Ha ee po en ial and one wi h
he Ha ee po en ial in mul ipole expansion and unca ed a e he dipole-dipole e m, one
can dis inguish be ween Fö s e - and non-Fö s e - ype coupling in bo h example sys ems.
Second, Pub1 sugges s a scheme o di ec ly compu e he in e molecula coupling s eng h
om he eal- ime p opaga ion o wo in e ac ing molecules. This in es iga ion is based on
he Da ydo spli ing and ex ac s he coupling-ma ix elemen be ween he ini ial and he
nal s a e o EET om he dipole momen ime e olu ion using a esonan wo-le el model.
I clea ly e eals in he sodium dime es case ha he coupling is o dipole-dipole ype o
dis ances abo e 25 boh , whe eas clea de ia ions om he dipole-dipole cha ac e can be
obse ed o dis ances below 20 boh .
Howe e , o also in es iga e he inuence o he ene ge ic alignmen o wo neighbo ing
molecules on he coupling s eng h one needs o go beyond he esonan coupling case. The
illus a ion in he ollowing is based on he well-app o ed Na
2
model sys em o Pub1.
1
Fo
exci a ions o ien ed along he bond axis, Na
2
is almos a single le el sys em as he e is
one p ominen exci a ion a 2.1 eV and a second exci a ion a 4.1 eV wi h no ably smalle
oscilla o s eng h. Na
2
exhibi s s ong dipola cha ac e and he elec onic s uc u e can
be modied easily by bond-leng h a ia ion s a ing om he expe imen al bond leng h o
5.78 boh . The ela ion be ween bond-leng h a ia ion and ene ge ic de uning is depic ed
in he le pa o Fig. 5.1. Due o a ia ions o he bond leng h on he o de o 1 boh ,
exci a ion-ene gy shi s o abou 0.25 eV can be induced.
In an o- esonan coupling si ua ion be ween wo isola ed exci ed molecules whe e wo
exci a ion ene gies a e close and all o he s a o, a wo-le el pic u e [Neu08] simila o Pub1
can be applied o de e mine he coupling-ma ix elemen . In analogy o he esonan case,
1
All calcula ions in his chap e a e based on he LDA unc ional. I used eal-space g ids wi h a g id
spacing o 0.7 boh , an LDA pseudopo en ial wi h a co e cu -o adius o
c
(Na) = 3.09 boh , and p opaga ion
ime s eps o 0.003 s o closed quan um sys em and 0.001 s o open quan um sys em in es iga ions.

5.2. OPEN QUANTUM SYSTEMS IN THE DENSITY FUNCTIONAL CONTEXT
39
Figu e 5.1: Le panel: Exci a ion-ene gy a ia ion o one sodium dime depending on a i-
a ions o he bond leng h. I compu ed his p ope y om a supe sys em calcula ion as ex-
plained in Appendix D.2 using 20 boh and 25 boh dime dis ances. Righ panel: Coupling-
ma ix elemen calcula ed in a sys em o wo Na
2
depending on he ene ge ic o- esonance
(shi o he exci a ion ene gy om i s alue a he GS geome y) o one o he dime s.
he coupling s eng h can be ex ac ed om a sui able obse a ion o he dipole momen s o
each subsys em ( o de ails, see Appendix D.2). The esul s o he coupling-ma ix elemen
as a unc ion o he ene ge ic o- esonance o he exci a ion ene gy o one o he dime s a e
shown in he igh pa o Fig. 5.1. One obse es a gene al end o he coupling-ma ix
elemen o dec ease wi h inc easing exci a ion ene gy wi hin he da a ange in es iga ed
he e. A clea esonance peak o he coupling s eng h appea s a he esonan coupling
si ua ion. I is by abou a ac o o 1.4 la ge han he coupling in o- esonan si ua ions. In
summa y, RT p opaga ion TDDFT p o ides a ool no only o dis inguish be ween Fö s e -
and non-Fö s e - ype coupling bu also o compu e he coupling-ma ix elemen in esonan
and o- esonan coupling si ua ions based on wo-le el models.
5.2 Open quan um sys ems in he densi y unc ional con ex
So- a , I ha e been discussing RT TDDFT me hods o app oach elec onic exci a ions and
he coupling be ween sepa a ed agmen s o supe molecula se ups. This closed quan um
sys em TDDFT o mula ion desc ibes he cohe en sys em e olu ion, bu does no include
decohe ence and dissipa ion eec s. The e o e, o s udy he ole o cohe en ene gy ans e
and he en i onmen o he sys em o EET pa hways and ime scales in he con ex o LH
sys ems, one needs o go beyond he closed quan um sys em o mula ion. Usually, such
s udies a e based on he densi y-ma ix o malism and mas e equa ions whe e he models
o Haken and S obl, he model o Redeld, pola on modica ions, o ela ed heo ies a e
employed [JJS02, YF02, JCRE08, IF09, RMAG09, KNOC11, Sil11]. The la e app oaches
ypically use inpu such as coupling pa ame e s and exci a ion ene gies om he expe imen
o om elec onic-s uc u e heo y. He e, I in oduce an al e na i e o mas e equa ions
ha add esses he EET p oblem di ec ly om he elec onic-s uc u e heo y pe spec i e.
I es s upon RT TDDFT and uses he s ochas ic TD KS equa ion [DVD07, DDV08, ADV09,
ADV11, MMN
+
12]. The a ionale behind his app oach is o use he s eng h o (TD)DFT
in desc ibing s a ic and dynamic p ope ies o he elec onic s uc u e, while he in e ac ion
wi h he sys em's en i onmen is ea ed eec i ely ia he open quan um sys em scheme.
The app oach opens a pa h o in es iga ing he inuence o elec onic-s uc u e p ope ies
on EET. I s a wi h an in oduc ion o he
s ochas ic Sch ödinge equa ion
(SSE) and ou line
ecen esea ch abou using SSEs in KS TDDFT.
40
CHAPTER 5. EXCITATION-ENERGY TRANSFER
5.2.1 The S ochas ic Sch ödinge equa ion
The s ochas ic Sch ödinge equa ion is an al e na i e o malism o quan um mas e equa ions
ha is able o deal wi h quan um sys ems in con ac wi h an ex e nal ba h [DCM92, GN99,
BP06, K07, Wei08, MMN
+
12]. I exhibi s some ad an ages in he TDDFT con ex as
I ou line a he end o his sec ion. In con as o quan um mas e equa ion app oaches,
he SSE does no wo k on he le el o densi y ma ices bu uses a s a is ical ensemble o
s a e ec o s o un a el he open quan um sys em dynamics di ec ly on he le el o wa e
unc ions. To his end, he so-called Feshbach p ojec ion-ope a o me hod can be employed
o sepa a e he sys em and ba h deg ees o eedom o a combined Hamil onian o sys em and
en i onmen [DCM92, GN99]. Such app oaches s a om a o al Hamil onian o sys em
and ba h whe e he sys em deg ees o eedom a e coupled o a bosonic en i onmen
H=HS⊗IB+IS⊗HB+λHSB.
(5.1)
IS
and
IB
deno e iden i ies in he sys em (S) and ba h (B) Hilbe spaces. He e, he sys em
includes all dynamics and obse ables o he co e sys em, e.g., o one molecula complex.
The ba h and sys em-ba h coupling desc ibe he en i onmen as o example su ounding
molecules and i s in e ac ions wi h he co e sys em pa . The sys em o in e es is desc ibed
by he many-pa icle Hamil onian
HS=
N
X
i=1 [pi+Aex (
i, )]2
2+ ex (
i, )+
N
X
i<j
W(
i−
j)
(5.2)
o he elec onic deg ees o eedom, whe e
p
is he momen um ope a o ,
Aex
an ex e nal
ec o po en ial,
ex
a scala ex e nal po en ial, and
W
desc ibes he pa icle-pa icle in e -
ac ion. Spin indices a e omi ed he e and in he ollowing. The en i onmen , gi en by
HB
,
induces a uc ua ing o ce ha d i es he sys em due o he in e ac ion be ween he sys em
and he ba h
HSB =X
α
Sα⊗Bα.
(5.3)
The la e is exp essed by many-pa icle ope a o s
Sα
and
Bα
in he mos gene al case,
whe e
Sα
ope a es on he sys em deg ees,
Bα
on he ba h deg ees o eedom, and
α
deno es
die en possible sys em-ba h coupling mechanisms. The ba h may ha e a complex s uc u e
and no all o i s mic oscopic de ails a e ele an o he sys em dynamics. The d i ing o ce
ha is induced by he ba h may, he e o e, be subsumed by a s ochas ic noise ha can
usually be cha ac e ized by mean alues and co ela ion unc ions [GN99].
λ
de e mines he
s eng h o he sys em-ba h coupling and se es as an expansion pa ame e .
The de i a ion o he SSE in he so-called Bo n-Ma ko limi in ol es some app oxima-
ions [DCM92, GN99, ADV11, MMN
+
12]: The Bo n app oxima ion assumes ha a pe u -
ba i e expansion up o second o de in he coupling pa ame e
λ
is sucien ly accu a e.
The desc ip ion o he en i onmen in e ms o he ba h elies on he ac ha he ba h
deg ees o eedom o m a dense ene gy spec um and emain in he mal equilib ium. The
Ma ko app oxima ion amoun s o assuming ha he ba h he maliza ion ime scales a e
much sho e han ele an sys em ime scales. Thus, he ba h does no e ain memo y o
he sys em-ba h in e ac ion a p e ious imes. Mo eo e , he de i a ion in okes ha he e
a e no ini ial co ela ions be ween sys em and ba h and he phases o he ba h deg ees o
5.2. OPEN QUANTUM SYSTEMS IN THE DENSITY FUNCTIONAL CONTEXT
41
eedom can be desc ibed by a andom-phase app oxima ion. Finally, one a i es a he SSE
in he Bo n-Ma ko limi [DCM92, GN99, ADV11, MMN
+
12]
i∂ ΨS( ) = HS( )ΨS( )−i
2X
α
S†
αSαΨS( ) + X
α
lα( )SαΨS( ),
(5.4)
whe e
lα( )
a e s ochas ic p ocesses wi h anishing ensemble a e age and
δ
- ime-co ela ion
lα( ) = 0, lα( )lβ( 0)=0, l∗
α( )lβ( 0) = δαβδ( − 0).
(5.5)
The ba deno es he s a is ical a e age o e an ensemble o s ochas ic p ocesses. Fo he sake
o con enience, he coupling s eng h pa ame e
λ
has been abso bed in he ba h ope a o
Sα
. The  s e m o he SSE (5.4) de e mines he usual uni a y sys em e olu ion unde
he ac ion o he Hamil onian
HS
. Al hough he SSE employs he sys em Hilbe space
only, he coupling o he ba h is s ill included by he second e m ha desc ibes dissipa ion
eec s due o he sys em-ba h in e ac ion. Finally, he hi d e m in oduces uc ua ions in
he ime e olu ion: Al hough he dissipa i e e m causes he p obabili y densi y o decay in
ime, he no m o he s a e ec o
Ψ( )
a e aged o e a s a is ical ensemble o ealiza ions
is conse ed up o ou h o de in he sys em-ba h coupling pa ame e
λ
. Fo he sake o
a clea no a ion, in he ollowing I use only a single ba h ope a o
S
and, he e o e, omi
indices a he ba h ope a o . A single ba h ope a o is sucien o he in es iga ions o
Sec. 5.3 as well.
As a esul o he s ochas ic na u e o Eq. (5.4) he sys em wa e unc ion can no be
simula ed by a single e olu ion o he SSE bu needs o be ep esen ed by a s a is ical
ensemble o wa e unc ions
{Ψs( )}
. S a ing om a pu e ini ial s a e, he ime e olu ion o
expec a ion alues
< OS>=<Ψ( )|OS|Ψ( )>
(5.6)
o physical obse ables
OS
can be calcula ed om he s a is ical a e age o e all ensem-
ble membe s
{Ψs( )}
. Ye , i is impo an o no e ha a eliable compu a ion o smoo h
obse ables equi es a la ge enough se o s ochas ic ealiza ions.
Finally, I add ess one o he no ewo hy die ences o he SSE and Redeld and simila
mas e -equa ion- ype app oaches. In he la e , posi i i y o he s a is ical ope a o may no
be gua an eed in cases o s ochas ic Hamil onians, TD Hamil onians, o TD ba h ope a o s
[FO05, DDV08, DDV09]. The SSE yields no malized ensemble-a e aged wa e unc ions
wi h a posi i e weigh o a bi a y TD ope a o s, hus posi i i y is gua an eed in any
case. This aspec is pa icula ly impo an i one in ends o use KS Hamil onians and DFT
app oxima ions ha depend on in e nal deg ees o eedom o he sys em o each ensemble
membe sepa a ely, as his ende s he Hamil onian s ochas ic [DVD07, DDV08, DDV09].
Hence, he SSE is a solid s a ing poin o a DFT heo y o open quan um sys ems.
5.2.2 S ochas ic Sch ödinge equa ion and Kohn-Sham densi y unc ional
heo y
In his sec ion, I begin wi h an in oduc ion o he open quan um sys em SSE app oach in he
amewo k o TD cu en densi y unc ional heo y (TDCDFT) o Re s. [DVD07, DDV08]. In
con as o s anda d TD(C)DFT, he open quan um sys em scheme uses ensemble-a e aged
quan i ies. The e o e, one in oduces he ensemble-a e aged pa icle densi y
n(
, ) = hn(
)i,
(5.7)
42
CHAPTER 5. EXCITATION-ENERGY TRANSFER
whe e he densi y ope a o is dened as
n(
) =
N
X
i=1
δ(
−
i),
(5.8)
and he ensemble-a e aged cu en densi y
j(
, ) = hj(
, )i,
(5.9)
whe e he cu en ope a o eads
j(
, ) = 1
2
N
X
i=1 {δ(
−
i),pi+Aex (
i, )}
(5.10)
and
{. , .}
deno es he an icommu a o b acke . The heo em o s ochas ic TDCDFT o
Re s. [DVD07, DDV08] s a es ha unde easonable physical condi ions o a gi en and
xed ba h ope a o
S
, many-pa icle in e ac ion
W
, and ini ial s a e
Ψ( 0= 0)
a one- o-one
co espondence be ween he ex e nal ec o po en ial
Aex (
, )
and he ensemble-a e aged
cu en densi y
j(
, )
exis s. The heo em pa es he way o a non-in e ac ing KS scheme o
open quan um sys em TDCDFT as i gua an ees he exis ence o a KS sys em ha yields
he same cu en densi y as he ue in e ac ing sys em. The KS Hamil onian eads
HKS ({
k}, ) =
N
X
i=1 [pi+Aex (
i, ) + Axc(
i, )]2
2+ ex (
i, ) + H(
i, )
(5.11)
wi h he xc ec o po en ial
Axc(
, )
and he Ha ee po en ial
H(
, )
. The KS Sla e
de e minan
ΦKS( )
e ol es acco ding o he open sys em KS equa ion [DVD07]
i∂ ΦKS( ) = HKSΦKS( )−i
2S†SΦKS( ) + l( )SΦKS( ).
(5.12)
The xc ec o po en ial in he open quan um sys em app oach may in gene al depend on
j(
, )
, he ini ial s a es
Ψ( 0)
and
ΦKS( 0)
, and he ba h ope a o
S
. Fo p ac ical calcula-
ions, howe e , one needs o ely on exis ing app oxima ions o he xc ec o po en ial as
he ue
Axc(
, )
, especially in open quan um sys ems, is no known [DVD07].
The p oo o he s ochas ic TDCDFT heo em is based on he easoning o well-known
p oo s o TDDFT [RG84, L99] and TDCDFT [Vig04] and discussed in de ail in Re s.
[DVD07, DDV08, ADV11]. The e o e, I do no ei e a e he en i e p oo , bu ocus on some
c ucial aspec s and commen s. Fi s , a no able die ence be ween he closed and he open
quan um sys em app oach is ha he usual con inui y equa ion be ween he densi y and he
cu en densi y does no hold in he open quan um sys em case. Due o he coupling o he
ex e nal ba h, addi ional e ms appea in he equa ion o mo ion o ensemble-a e aged ex-
pec a ion alues [F e90, DDV08]. The con inui y equa ion o he ensemble-a e aged densi y
and cu en densi y eads
∂ n(
, ) = −∇j(
, ) + FB(
, ),
(5.13)
whe e he densi y modula ion ha is induced by he ba h is desc ibed by
FB(
, ) = 1
2h2S†nS −S†Sn −nS†Si.
(5.14)
5.2. OPEN QUANTUM SYSTEMS IN THE DENSITY FUNCTIONAL CONTEXT
43
The p oo o he abo e s a ed one- o-one co espondence equi es implici ly ha
FB(
, )
is a
unc ional o
n(
, )
and
j(
, )
alone and ha Eq. (5.13) can be sol ed uniquely o de e mine
n(
, )
[DDV08, DDV09, ADV11]. Howe e , al hough he e is a unique ela ion be ween he
densi y and he cu en densi y, he p oo gua an ees only he one- o-one co espondence
be ween
j(
, )
and he ec o po en ial. The e o e, he ensemble-a e aged densi y o he
ue and he non-in e ac ing sys em a e no necessa ily equal.
Second, he eec o he scala po en ial
ex (
, )
is no discussed explici ly in Re s.
[DVD07, DDV08] as i can be elimina ed by a gauge ans o ma ion a all imes [Vig04].
In he con ex o a non-in e ac ing ep esen a ion o an in e ac ing many-pa icle sys em
he use o scala po en ials is accompanied by dicul ies: E en i a gi en cu en densi y is
- ep esen able in an in e ac ing many-pa icle sys em, he cu en densi y is no necessa ily
- ep esen able in he non-in e ac ing ep esen a ion o he same sys em [DV05].
One impo an applica ion o open quan um sys em schemes a e quan um anspo
p oblems whe e one is in e es ed in he spa ial dis ibu ion o he cu en densi y o in es-
iga e cha ge ow. Ye , he in es iga ion in he ollowing app oaches EET p ocesses whe e
no cha ge ans e occu s. Such in es iga ions can be based on he densi y and i s momen s
alone wi hou explici need o he cu en densi y. F om a p ac ical poin o iew, he num-
be o a ailable app oxima ions o
Axc(
, )
is limi ed, whe eas he e a e plen y o choices o
app oxima ions o he (TD)DFT xc po en ial. Also o nume ical easons a densi y-based
app oach is p e e able. Fo applica ions ha a e based on he ime e olu ion o he densi y
i is o g ea in e es o nd an open quan um sys em app oach in he amewo k o TDDFT.
Ex ensions o TDDFT o open quan um sys ems we e al eady pe o med by Bu ke
e
al.
[BCG05] and Yuen-Zhou
e al.
[YZTRRAG10] based on densi y ma ices and mas e
equa ions. Bo h s a egies aim a he ep esen a ion o he densi y o he in e ac ing open
quan um sys em by an auxilia y sys em wi h a die en pa icle-pa icle in e ac ion. To
p o e he alidi y o such app oaches, one needs o nd he co esponding ex e nal po en ial
o he auxilia y KS sys em ha p oduces he same densi y unde gi en condi ions. The p oo s
o Re s. [BCG05] and [YZTRRAG10] a e based on he easoning o Re s. [RG84, L99, Vig04]
and ely on assump ions abou he ac ion o he ba h ope a o . Ye , o he bes o my
unde s anding, de ails abou he consequences o hese assump ions, in pa icula wi h ega d
o p ac ical calcula ions and choices o he ba h ope a o , a e open ques ions in he eld,
and he use o app oxima e KS Hamil onians in mas e equa ions is deba ed. Mo eo e ,
a conclusi e ex ension o he p oo o s ochas ic TDCDFT ha abandons he ensemble-
a e aged cu en densi y and es ablishes an open quan um sys em TDDFT scheme based on
he ensemble-a e aged densi y alone is no a ailable ye .
5.2.3 A single-pa icle app oach in p ac ice
P ac ical applica ions o s ochas ic TDDFT so a ely on app oxima e ealiza ions o he
s ochas ic o malism in KS TDDFT. In he app oach o Re s. [ADV09, ADV11], a heu is-
ic ba h ope a o [PDV08] is used ha applies o single TD KS o bi als and allows o a
single-pa icle KS app oach o he open quan um sys em p oblem. In collabo a ion wi h
Massimiliano Di Ven a and Heiko Appel, I applied a s ochas ic TDDFT scheme acco ding
o his a ionale. The ollowing in es iga ions a e based on he s ochas ic single-pa icle KS
equa ions
i∂ ϕi(
, ) = hKS(
, )ϕi(
, )−i
2s†
isiϕi(
, ) + l( )siϕi(
, ),
(5.15)

44
CHAPTER 5. EXCITATION-ENERGY TRANSFER
he usual non-in e ac ing single-pa icle KS Hamil onian o Eq. (2.12), and specic single-
pa icle ba h ope a o s
si
[ADV09, ADV11]. In he ollowing, xc eec s a e app oxima ed
by he LDA. The heu is ic ba h ope a o o he in es iga ions p esen ed below is in oduced
and mo i a ed in Sec. 5.3.2. Beyond ha , I also in es iga ed he heo e ical jus ica ion o
his app oach. Mo e backg ound abou hese in es iga ions is p esen ed in Appendix D.3.1.
The open quan um sys em simula ions we e ca ied ou using he quan um-jump al-
go i hm [DCM92, GPZ92, BP95, BP06] ha has been in oduced in he con ex o open
quan um sys em KS equa ions in Re . [ADV11]. The algo i hm is implemen ed in a cus-
omized e sion o he PARSEC p og am package. De ails on he ba h ope a o implemen-
a ion a e ga he ed in Appendix D.3.2. The quan um-jump algo i hm elies on a piecewise
de e minis ic e olu ion o he no m-p ese ing single-pa icle equa ions
i∂ ϕi(
, ) = hKS(
, )ϕi(
, )−i
2s†
isiϕi(
, ) + i
2|siϕi(
, )|2ϕi(
, )
(5.16)
ha a e in e up ed by quan um jumps. These jumps occu in he en i e sys em and ep e-
sen he non-de e minis ic ac ion o he ba h. The poin s o ime whe e such a jump occu s
a e de e mined by a andom p ocess acco ding o a wai ing- ime dis ibu ion. Howe e ,
as in gene al his wai ing- ime dis ibu ion is no known be o ehand, i needs o be de e -
mined alongside he ac ual p opaga ion o Eq. (5.16) ( o de ails, see Re . [ADV11]): The
wai ing- ime dis ibu ion can be de e mined om he decay o he no m
η( ) = 1
N
N
X
i=1 Z|ϕaux
i(
, )|2d3
(5.17)
o an auxilia y sys em o
N
pa icles in con ac wi h he single-pa icle ba h ope a o s
si
s a ing om he same g ound s a e and boos exci a ion as he o iginal sys em and e ol ing
acco ding o
i∂ ϕaux
i(
, ) = hKS(
, )ϕaux
i(
, )−i
2s†
isiϕaux
i(
, ).
(5.18)
In his sys em, no m conse a ion is no build in explici ly. One ob ains single wai ing
imes ha ep esen quan um-jump imes by d awing andom numbe s in he in e al [0,1]
and choosing he ime
T
when
η(T)
d ops below his numbe . The wai ing- ime dis ibu ion
ollows om many samples o such single quan um-jump imes. Each o hese p ocesses yields
one KS o bi al se
{ϕi,s( )}
ha is one membe o he s a is ical ensemble o KS o bi al se s.
Physical obse ables a e calcula ed om he s a is ical a e age o e all ensemble membe s.
5.3 Pa hways and ime cons an s o exci a ion-ene gy ans e
5.3.1 The exci a ion-ene gy ans e model sys em
In he ollowing, I use he s ochas ic open quan um sys em o malism in oduced in he
p e ious sec ion wi h a heu is ic ba h ope a o o p ac ical applica ions in he a ea o EET
wi h he objec i e o s udying EET in a supe molecula a angemen o molecules: I s udy
a model sys em o ci cula ly a anged molecules ha is designed in analogy o ci cula LH
complexes o he an enna sys em o LH o ganisms (see Fig. 5.2). The aim is o in es iga e
he inuence o elec onic-s uc u e p ope ies on EET ime scales and pa hways in such
5.3. PATHWAYS AND TIME CONSTANTS OF EXCITATION-ENERGY
TRANSFER
45
Figu e 5.2: Ci cula a angemen o eigh
molecules m1, m2, ..., m8 wi h equal in-
e molecula dis ance. In p inciple, he
molecules may be chosen a bi a ily, bu o
he sake o cla i y, I use Na
2
as a model sys-
em. The exci a ion o he en i e ing se up is
pe o med ia boos applica ion a molecule
m5. I measu e he exci a ion sp ead in his
a angemen by applica ion o a dissipa i e
ba h ope a o ha se es as a measu emen
p ocess a molecule m1. Fo a anspa en in-
es iga ion o he inuence o he in e molec-
ula coupling and he ene ge ic alignmen o
he molecules on ene gy- ans e pa hways,
I in oduce de ec s in he molecules m3 and
m7. All o he molecules a e xed acco ding
o hei GS geome y.
supe molecules, especially he inuence o he coupling and he ene ge ic alignmen . Wi h
his in mind, he physical pic u e behind hese calcula ions is he ollowing: One molecule
o he complex ge s exci ed ia ligh abso p ion. Then, he exci a ion a els in he sys em
due o he elec onic in e ac ion be ween he molecules. As a esul he exci a ion sp eads
o e he en i e complex and I in end o measu e he ime scale o his exci a ion sp ead.
In a ci cula a angemen o molecules, he e a e a leas h ee ime scales ele an o
he signal obse ed on he ing: he ime scale ela ed o he ene gy o he exci ed s a es,
he ime scale due o in e molecula coupling (he e, mo e ime scales a e in ol ed i he
sys em is pa ly o in o al o- esonan ), and he ime scale due o dissipa i e ba h ac ion.
The la e p ocess is needed o b eak he cohe en e olu ion o he sys em (see Sec. 5.3.3
o an illus a ion). To allow o he in es iga ion o he inuence o elec onic-s uc u e
p ope ies on he exci a ion-ene gy sp ead, i s ime scale needs o be chosen such ha i
does no in e e e wi h he EET ime scale ha is de e mined by he coupling mechanism
and coupling s eng h. In his sense, he ba h mechanism ha I in oduce in he nex sec ion
plays he ole o a measu emen p ocess.
In he basic model se up consis ing o eigh molecules as indica ed in Fig. 5.2, all
molecules in he ci cula a angemen and all in e molecula dis ances a e equal. The ac ual
s udy uses he Na
2
model sys em, bu he conside a ions in he ollowing a e no es ic ed
o dime s and can be applied o mo e gene al sys ems. All sodium dime s a e aligned along
he
z
-axis and placed acco ding o he se up o Fig. 5.2. Thei cen e s o mass a e in he
x
-
y
-plane. Ini ially, s a ing om he g ound s a e o he se up I in oduce an exci a ion a
one o he molecules. To be explici in he assignmen , I chose molecule m5. I simula e he
exci a ion due o ligh abso p ion by a momen um boos o ien ed along he
z
-axis. In he
Na
2
case, his dominan ly amoun s o an exci a ion a 2.1 eV. In p ac ical calcula ions, he
boos is applied only in one sec ion o he eal-space g id, so ha only molecule m5 ge s
exci ed (see Appendix D.1.3 o de ails abou he implemen a ion). De ec s in e ms o a i-
a ions o he elec onic s uc u e can be in oduced easily by Na
2
bond-leng h a ia ion. To
46
CHAPTER 5. EXCITATION-ENERGY TRANSFER
gua an ee unambiguousness and anspa ency o he in es iga ions in he ollowing sec ions,
I modi y only molecules m3 and m7, and x all o he sys em componen s. The ele an
coupling pa ame e s o Na
2
a e discussed in de ail in Sec. 5.1.
5.3.2 P ac ical simula ion app oach
The ba h ope a o is supposed o ac as an eec i e measu emen p ocess ha allows o
measu e he ime an ini ial exci a ion a els wi hin he ing sys em. In he se up o Fig.
5.2, i ope a es on molecule m1 on he opposi e side o he ini ial exci a ion o measu e he
ime scale he exci a ion needs o a el hal way h ough he ing. The measu emen p ocess
is supposed o model deexci a ion, hus emo e he en i e exci a ion ene gy ou o he sys em
when he ini ial exci a ion has eached molecule m1. In con as o ue molecules whe e
he exci a ion mo es due o elaxa ion o lowe lying ene gy le els, he e he measu emen
p ocesses immedia ely b ings he supe molecule back o i s g ound s a e. The ba h ope a o
needs o model an incohe en mechanism wi hou back ans e om he ba h o he sys em.
I assume ha he ba h is sensi i e o dipola exci a ion and couples o he dipole momen
o molecule m1. Fo hese easons, he ba h ope a o has he ollowing s uc u e
si=√γ|d1( )−d1( 0)|
D|ϕi( 0)ihϕi( )|
(5.19)
wi h h ee specic con ibu ions. The  s ac o
√γ
is a ee pa ame e ha includes he
eec i e decay a e
γ
. The second ac o is sensi i e o local changes o he dipole momen
|d1( )−d1( 0)|
o molecule m1. He e, he index deno es molecule m1 and indica es ha
he dipole momen is calcula ed only in he sec ion o he g id ha co esponds o m1.
This ac o ensu es ha he ba h couples only o dipola exci a ions and i ende s he ba h
ope a o sensi i e o he dipola exci a ion ha has eached molecule m1.
D
deno es a
no maliza ion ac o o he dipole momen a ia ions and needs o be chosen easonably as
discussed below. The hi d ac o is a p ojec o ha akes he TD KS o bi als and p ojec s
he la e back on o hei co esponding GS o bi al. De ails abou he implemen a ion o
his ba h ope a o and examples o al e na i e heu is ic ba h ope a o s a e assembled in
Appendix D.3.2.
Ha ing chosen one specic deni ion o he ba h ope a o , I  s assess i s unc ionali y,
i.e., I adjus he no maliza ion ac o
D
and in es iga e i s pe o mance on a single Na
2
.
The aim o his in es iga ion is o nd
D
such ha he unc ionali y o he scaling ac o s
√γ
and
(|d1( )−d1( 0)|)/D
in on o he p ojec o is clea ly spli in o wo con ibu ions:
D
needs o be de e mined such ha he decay ime
τ
o he elaxa ion p ocess is de e mined
only by he decay a e
γ
, hus
τ= 1/γ
. The dipole dependen scaling ac o is in ended o
gua an ee o he coupling o he dipole momen o molecule m1 and should no in e e e
wi h he ole o
γ
. The e o e, he no maliza ion ac o needs o be adap ed o he dipole
oscilla ions o he isola ed model molecule m1, i.e., o one isola ed Na
2
.
In he p ocedu e o he de e mina ion o
D
, I calcula ed 100 s o he dipole momen
ime e olu ion o a single Na
2
in a closed quan um sys em a e an ini ial momen um boos
ha was applied along he bond axis o he dime . He e, only he dipole momen along his
axis ge s exci ed by he boos and he in es iga ion can be es ic ed o his dipole momen
componen . The e a e a ious die en op ions o de e mine he no maliza ion based on he
exis ing dipole momen da a:
D
may be chosen o be he  s maximum, he a e age o e
5.3. PATHWAYS AND TIME CONSTANTS OF EXCITATION-ENERGY
TRANSFER
47
Figu e 5.3: No m decay
η( )
(see Eq. (5.17))
o a single model molecule. The damping
is pe o med wi h die en decay- ime con-
s an s
τ
a e an ini ial boos exci a ion wi h
0.001 eV exci a ion ene gy. The no m de-
cay always ollows an exponen ial unc ion
exp(− /τ)
wi h he p ese ime
τ
. He e,
quan um jumps we e pe o med as he no m
d opped below 0.014 %. They mani es in
e ical lines whe e he no m jumps back o
one and s ays o he es o he ime e olu-
ion as applica ion o he ba h ope a o does
no change he g ound s a e.
all maxima, he absolu e a e age, and he absolu e squa e a e age o he dipole momen . I
ound by nume ical es s ha only no maliza ion acco ding o he a e age o he absolu e
squa e o he dipole momen gi es an exponen ial decay wi h ime cons an
τ
ha I aimed
a by se ing he decay a e. This obse a ion is obus wi h ega d o die en decay- ime
cons an s as Fig. 5.3 shows. All compu ed decay imes a e in acco dance wi h he p ese
decay a e.
Ano he inuencing ac o on he de e mina ion o
D
is he ene gy ha one in oduces
o he sys em by he boos exci a ion. I in e p e he exci a ion p ocedu e as a single
abso p ion p ocess o one ene gy po ion. The dissipa ion o such an ene gy po ion due o
he measu emen p ocess should be independen o he boos s eng h as long as he boos
s eng h co esponds o a single exci a ion p ocess. The e o e, in he single exci a ion case,
he no maliza ion ac o needs o be adap ed o he boos s eng h. I gua an ee o he boos
s eng h adap a ion by always de e mining
D
om a closed quan um sys em calcula ion wi h
he same ini ial boos exci a ion as in he open quan um sys em calcula ion ha I aim a .
In he ollowing, a boos wi h 0.001 eV exci a ion ene gy is used consis en ly. The hus
ob ained no maliza ion ac o yields a decay ime ha is independen o he boos s eng h.
The exci a ion p ocess could likewise be pe o med by applica ion o an ex e nal lase
eld. In his case, an ins an aneous exci a ion could be simula ed by a sho pulse and he
de e mina ion o he no maliza ion ac o should hen be pe o med using he dipole signal
a e he pulse exci a ion.
Figu e 5.4: Le , ensemble-a e aged dipole momen (
z
-componen ) and, igh , o al ene gy
o a single molecule in con ac wi h a dissipa i e ba h ha induces a decay ime o 10 s.
Bo h gu es a e calcula ed om an ensemble o 200 membe s.

Appendix A
Pola izabili y and s a ic
cha ge- ans e p ope ies
The  s appendix o his hesis includes a compila ion o ye unpublished esul s and ndings
supplemen ing he p e ious wo k. I s a in Sec. A.1 wi h esul s on model hyd ogen chains
and in es iga e die ences ha occu in longi udinal pola izabili ies when SIC is pe o med
ia die en me hods. Such obse a ions can be explained and unde s ood i one analyzes
he shape o he xc po en ial when ex e nal elds a e applied. Analysis o he con ibu ions
o he xc po en ials in app oxima ions o he (G)OEP e eals die ences in he way he KLI
and SLA app oxima ions on he one hand, and he gene alized me hods GKLI and GSLA
on he o he hand ea he so-called esponse pa o he xc po en ial. The e o e, his ans-
pa en sys em allows o add ess he ques ion abou he quali y o he (G)KLI and (G)SLA
app oxima ions compa ed o he ull (G)OEP. In Sec. A.2, I pe o m a simila pola izabil-
i y in es iga ion wi h a ealis ic es sys em, he polyace ylene chains, complemen ing he
esul s o Pub4. The compila ion o addi ional esul s and ndings closes in Sec. A.3 wi h
an obse a ion made in he CT model sys em o Man1, namely a discon inuous a ia ion o
he o al ene gy in GSIC whe e he LSDA o al ene gy a ies con inuously. Th oughou his
appendix, complex- alued ene gy minimizing ans o ma ions a e used. Fo he no a ion in
he gu es and ables, I use only he sho cu s o implemen a ion me hods because hey a e
all applied o he SIC unc ional. Fo ins ance, ins ead o KLI-SIC I w i e jus KLI.
A.1 Pola izabili y and eld-coun e ac ing po en ials in sel -
in e ac ion ee densi y unc ional heo y
Fo a anspa en and conclusi e in es iga ion o he inuence o xc po en ial app oxima-
ions on pola izabili ies and he shape o he xc po en ial [CMVA95, GSG
+
99, G GSB00,
MSWY03, KKP04, KK06, PSB08, RPC
+
08, KMK08, AKK08, CK09, KK11], I chose hyd o-
gen chains as a model sys em ha allows o a clea -cu analysis. He e, he hyd ogen chains
a e se up om H a oms wi h al e na ing H-dis ances o 2 boh and 3 boh .
1
I p o ide in Table A.1 longi udinal pola izabili ies o a se o hyd ogen chains along he
chain's backbone calcula ed om he change o he dipole momen when an ex e nal eld
1
I used a hyd ogen LDA pseudopo en ial wi h co e cu -o adius
c
(H) = 1.39 boh . The g id spacing
was 0.25 boh .
55
56
APPENDIX A. POLARIZABILITY AND STATIC CHARGE-TRANSFER
PROPERTIES
LDA KLI GKLI GOEP SLA GSLA FOBO CC MP4
H
4
38 20 34 33 20 36 34 29 29
H
6
73 62 64 62 65 68 65 51 52
H
8
116 99 99 93 103 107 100 74 76
H
10
165 152 136 132 174 150 138 99 101
H
12
217 196 175 170 243 196 178 124 127
Table A.1: Pola izabili y in a omic uni s o H chains wi h al e na ing H-dis ance o 2 boh
and 3 boh . In he  s pa o he able, I show LDA and (G)SIC esul s whe e he me hods
behind he la e a e o de ed acco ding o inc easing sophis ica ion. In he second pa o he
able, I compiled esul s om he Sla e app oxima ion and calcula ions based on Fos e -
Boys o bi als. The nal pa shows e e ence alues: CCSD(T) (CC) pola izabili ies a e
ob ained om Re . [CK09] and MP4 esul s om Re . [PSB08].
2
is applied [KKP04, KK06]. I compa e hese esul s o Coupled-Clus e (singles, doubles,
and pe u ba i e iples, CCSD(T)) and Mølle -Plesse pe u ba ion heo y o ou h o de
(MP4) e e ence alues. These benchma ks a e e y close in he case o hyd ogen chains,
hus p o ide a solid basis o compa ison. A  s sigh , one clea ly obse es he well-known
pola izabili y o e es ima ion o LDA. Howe e , in he hyd ogen chains p esen ed he e, ull
OEP-SIC is known o pe o m well and yield pola izabili ies close o he CCSD(T) bench-
ma k [KKM08, KMK08, KK11] (no shown he e). This ema kable pe o mance disappea s
i one uses he KLI o SLA app oxima ion and SLA-SIC is e en wo se han LDA in he
long-chain case. Pa o he OEP-SIC pe o mance howe e can be eco e ed by he GSIC
app oaches [KKM08, KK11]. Ye , in he hyd ogen chains, GOEP-SIC is no as close o
CCSD(T) esul s as ull OEP-SIC [KK11]. The calcula ions show beyond he esul s o Re .
[KK11] ha GKLI-SIC pola izabili ies a e compa ably close o GOEP-SIC, while GSLA-SIC
esul s al hough being no ably be e han ba e SLA-SIC a e u he o . FOBO localiza ion
gi es pola izabili ies close o complex ene gy-minimizing GKLI-SIC ha a e a he maxi-
mum 1.8 % la ge , as a as he da a p esen ed he e is conce ned. Mo eo e , i one compu es
he pola izabili ies pe H
2
epea uni , he de ia ion be ween GOEP-SIC, GKLI-SIC, and
FOBO-SIC emains almos he same, whe eas he dis ance o GSLA-SIC pola izabili ies
inc eases wi h inc easing chain leng h.
So- a , i is ob ious ha GSIC calcula ions in gene al imp o e upon LDA, bu GSLA-SIC
is no pe ec ly in line wi h he ndings o GOEP-SIC, GKLI-SIC, and FOBO-SIC. Analysis
o he shape o he xc po en ial and i s beha io when an ex e nal eld is applied gi es mo e
insigh in o he pe o mance o he die en SIC app oxima ions. He e, he ndings a e
exemplied wi h he H
8
chain. To his end, I in es iga ed he change o he xc po en ial when
an ex e nal eld
F
o eld s eng h 0.005 Ry/boh is applied o he g ound-s a e
xc
along
he chain's backbone in
x
-di ec ion. The GOEP, ha gi es esul s closes o he CCSD(T)
pola izabili ies, shows a p onounced eld-coun e ac ing beha io (see Fig. A.1 (a)), ha is
decisi e o a p ope desc ip ion o he esponse [ GSG
+
99, G GSB00, KKP04, KMK08].
LDA howe e misses he eld-coun e ac ing end comple ely. An in e es ing obse a ion
can be made in he KLI-SIC and SLA-SIC esponse o he ex e nal eld in Fig. A.1 (b).
Bo h po en ials do no ollow he esponse pa e n obse ed in GOEP-SIC bu show a a he
2
Benoî Champagne kindly p o ided a lis o hyd ogen chain CCSD(T) and MP4 pola izabili ies.
A.1. POLARIZABILITY AND FIELD-COUNTERACTING POTENTIALS IN
SELF-INTERACTION FREE DENSITY FUNCTIONAL THEORY
57
Figu e A.1: Each o he panels shows he change o he xc po en ial
xc(F)− xc(0)
when
an ex e nal eld wi h eld s eng h
F= 0.005
Ry/boh is applied compa ed o he g ound
s a e wi h
F= 0
Ry/boh . I depic LDA and GOEP-SIC in pa (a), KLI-SIC and SLA-SIC
in pa (b), GKLI-SIC and GOEP-SIC in pa (c), and GKLI-SIC, GSLA-SIC, and FOBO-
SIC in pa (d). The ex e nal po en ial is gi en h oughou . FOBO-SIC and GKLI-SIC a e
almos iden ical in pa (d). No e ha he calcula ions we e pe o med spin-dependen ly, bu
I show only one o he spin channels o he xc po en ial as bo h spin channels a e iden ical
in H
8
.
58
APPENDIX A. POLARIZABILITY AND STATIC CHARGE-TRANSFER
PROPERTIES
Figu e A.2: Le panel: Sla e con ibu ion o he xc po en ial o GS H
8
wi hou ex e nal
eld in sel -consis en KLI-SIC, SLA-SIC, GKLI-SIC, and GSLA-SIC calcula ions. The
Sla e con ibu ion is almos he same in he cases o KLI-SIC and SLA-SIC as well as in
he cases o GKLI-SIC and GSLA-SIC. Righ panel: Co esponding esponse con ibu ions
KLI
xc (x)− SLA
xc (x)
o he KLI-SIC and GKLI-SIC po en ial. The shape o he esponse
po en ial die s no ably be ween he s anda d and he gene alized KLI app oxima ion.
dis o ed esponse s uc u e. He e, a clea s a emen abou he eld-coun e ac ing beha io
is dicul . I unde s and his nding as a consequence o he SIC unc ional sue ing om
uni a y a iance p oblems and he KLI-SIC and SLA-SIC po en ials nei he being unc ional
de i a i es no being uni a y in a ian hemsel es ( o mo e de ails, see he discussion and
e e ences in Pub3 and Sec. 3.3.1).
The eld-coun e ac ing beha io howe e is eco e ed in he GKLI-SIC app oach. Fig.
A.1 (c) clea ly shows ha he GKLI-SIC po en ial models he GOEP-SIC beha io eason-
ably in he cen al egion bu is less a o able in he nea asymp o ics. Such die ences
ob iously cause he de ia ions o he pola izabili ies obse ed in Table A.1. As a as he xc
po en ial esponse is conce ned, FOBO localiza ion is a good app oxima ion o he ene gy-
minimizing GKLI-SIC and exhibi s only small de ia ions om he GKLI-SIC beha io (see
Fig. A.1 (d)). Howe e , he in e p e a ion o he GSLA-SIC esponse is no ob ious: I one
d aws a line h ough he peaks o he GSLA-SIC esponse coming om he nega i e side,
one nds o he  s h ee peaks a slope in acco dance wi h he ex e nal po en ial ollowed
by a eld-coun e ac ing beha io a he end o he chain. No e ha he eld-coun e ac ing
con ibu ion o he s anda d KLI po en ial is ypically a ibu ed o he so-called esponse
pa o
xc
, i.e., he die ence be ween ull KLI and i s SLA con ibu ion.
Howe e , his assignmen does no hold in he same way in he GSIC app oaches as
he o al GSLA-SIC esponse exhibi s a eld-coun e ac ing beha io in pa s. The e o e, I
isola ed he Sla e con ibu ion o he KLI-SIC and GKLI-SIC po en ials and in es iga ed
occu ing die ences as depic ed in he le panel o Fig. A.2. In bo h cases, he sel -consis en
Sla e only po en ial esembles he Sla e con ibu ion o he sel -consis en (G)KLI po en ial,
bu misses he (G)KLI esponse e m. Ye , one obse es ha he dis ibu ion be ween
Sla e and esponse con ibu ion o he (G)KLI po en ial no ably changes i one goes om
he s anda d KLI o an ene gy-minimizing GKLI po en ial. This eshuing o he Sla e
A.2. POLARIZABILITY OF POLYACETYLENE CHAINS
59
con ibu ion also aec s he esponse pa , as can be obse ed in he igh panel o Fig. A.2.
The esponse con ibu ion o he KLI po en ial, which is an app oxima ion o he s anda d
OEP, is known o be impo an o o bi al unc ionals in gene al. Howe e , in he case o
GSIC using GKLI i does no play an as p onounced ole.
I conclude ha app oxima ions o he s anda d OEP may exhibi a dis o ed esponse
i applied o he SIC unc ional. The e o e, I s ongly ecommend he use o gene alized
app oaches ha explici ly ake he possible uni a y eedom o o bi al unc ionals as o
ins ance o he SIC unc ional in o accoun , pa icula ly when i comes o app oxima ions
o he (G)OEP. In his con ex , he GKLI-SIC po en ial is a easonable app oxima ion o
he GOEP-SIC in g ound-s a e DFT and exhibi s a p ope eld-coun e ac ing beha io .
GSLA-SIC no ably imp o es upon SLA-SIC, bu does no show a clea eld-coun e ac ing
con ibu ion and, he e o e, yields pola izabili ies u he o.
Finally, no e ha al hough hyd ogen chains a e a anspa en es sys em ha allow
o a clea -cu analysis o he beha io o he xc po en ial, hey a e a icial sys ems and
o peculia na u e. Al eady in ea lie wo ks, i has been obse ed, e.g., ha cu en DFT
aking ul anonlocal xc eec s in o accoun leads o la ge pola izabili y imp o emen s o
eal conjuga ed polyme s, whe eas in hyd ogen chain models he imp o emen s a e small
[ FdB L
+
02]. Hence, ca e needs o be aken when d awing conclusions om absolu e num-
be s calcula ed in such es sys ems. The pe o mance in ealis ic molecules needs o be
in es iga ed independen ly. The e o e, in he ollowing sec ion I u n o eal polyace ylene
chains o u he in es iga e he dependence o he pola izabili y on unc ional app oxima-
ions and implemen a ion me hods.
A.2 Pola izabili y o polyace ylene chains
The polyace ylene chains a e an in e es ing eal es sys em because hey a e almos as
anspa en as hyd ogen chains, easy o modi y in e ms o he chain leng h, and e eal
die ences be ween die en (G)SIC me hods. Pub4 demons a es ha exci a ion ene gies
o polyace ylene chains die no ably be ween die en app oxima ions o he TDGOEP, in
pa icula TDFOBO-SIC calcula ions yield exci a ion ene gies ha a e close o he e e ence
alues han ene gy-minimizing TDGKLI-SIC.
He e, I complemen he da a on exci a ion ene gies wi h esul s o polyace ylene pola -
izabili ies and es i he obse a ions made on he pola izabili ies o hyd ogen chains also
hold in case o polyace ylene. To his end, I gi e he s a ic pola izabili y o a selec ion o
polyace ylene chains calcula ed wi h die en unc ionals in Table A.2. The da a con ms
he gene al nding ha LDA o e es ima es polyace ylene pola izabili ies. SIC calcula ions
based on s anda d KLI and SLA do no cu e his o e es ima ion: S anda d KLI-SIC e en
su passes LDA esul s and SLA-SIC is simila o LDA o sligh ly be e . Thus, i a all,
bo h me hods a e no able o imp o e much upon LDA a he end o he day. This nd-
ing pa allels he exci a ion-ene gy esul s. I is eec ed in an e oneous educ ion o he
lowes exci a ion ene gy in TDKLI-SIC. Howe e , all GSIC schemes educe he pola izabil-
i y o e es ima ion, hus cu e pa o he p oblem: Excep o he sho es chain, C
4
H
6
,
GKLI-SIC gi es pola izabili ies close o B3LYP. GSLA-SIC is close o GKLI-SIC esul s
bu ends o gi e sligh ly lowe , hus be e pola izabili ies. FOBO localiza ion gi es he
lowes pola izabili ies o all app oxima e SIC schemes add essed so a . He e, he Fos e -

60
APPENDIX A. POLARIZABILITY AND STATIC CHARGE-TRANSFER
PROPERTIES
LDA KLI GKLI GOEP SLA GSLA FOBO B3LYP HF MP2
C
4
H
6
76 77 68 62 77 68 67 84 75 64
C
6
H
8
173 175 160 140 171 158 154 164 142 112
C
8
H
10
294 297 273 222 283 277 262 274 229 187
C
10
H
12
457 466 426 322 445 435 406 413 332 267
Table A.2: Axial pola izabili y in a omic uni s o se e al polyace ylene chains calcula ed
wi h die en unc ionals. As a e e ence, I gi e MP2 esul s aken om Re . [TTdM
+
95],
HF aken om Re . [KTRH95], and B3LYP pola izabili ies ha we e calcula ed wi h TUR-
BOMOLE.
3
Boys c i e ion gua an ees o spa ial localiza ion ha is no necessa ily equal o he complex
ene gy-minimizing ans o ma ion.
Full GOEP-SIC calcula ions a e no ably be e han GKLI-SIC and FOBO-SIC pola iz-
abili ies: Fo he longe polyace ylene chains, GOEP-SIC seems o imp o e by abou hal he
dis ance be ween LDA and MP2 esul s. In e es ingly, he GOEP-SIC pola izabili ies p e-
sen ed he e a e close o, bu sligh ly be e han HF ones. Ye , he e is s ill a no iceable gap
o he MP2 e e ence. The e o e, I conclude ha al hough GSIC app oaches and especially
he ull GOEP-SIC imp o e upon LDA, he esul s do no coincide wi h pola izabili ies om
wa e- unc ion pe u ba ion heo y.
The obse a ions made on he basis o he pola izabili y da a a e in line wi h he end o
he exci a ion ene gies whe e TDFOBO-SIC p o ides he highes alues in bes ag eemen o
e e ence calcula ions and he expe imen . Hence, in case o polyace ylene chains he end
o he lowes exci a ion ene gies can be ela ed o he beha io o he s a ic pola izabili y.
The e o e, o sys ems ha a e simila o polyace ylene he pola izabili y could possibly be
used as an indica o o he pe o mance o exci a ion-ene gy e alua ions, whe e TDGKLI-
SIC and TDFOBO-SIC gi e imp o ed esul s, bu he de ia ion om e e ence alues is
s ill 0.3 eV a he maximum. Following he pola izabili y end one may expec e en be e
exci a ion ene gies om ull TDGOEP-SIC, as o GOEP-SIC a signican educ ion o he
pola izabili y can be obse ed. I nally conclude ha he (TD)GSIC scheme yields no able
imp o emen o he LDA desc ip ion o polyace ylene chains, whe e among he app oaches
a ailable he e FOBO localiza ion gi es he bes numbe s o exci a ion ene gies and GOEP-
SIC exhibi s he o e all mos p omising pola izabili y esul s.
A.3 S a ic cha ge ans e in a anspa en model sys em
Finally, I complemen he ndings on he CT model o Man1 and Sec. 4.2.1 by one addi ional
obse a ion on he o al ene gy o his sys em when s a ic ex e nal elds a e applied. I
one plo s as in Fig. A.3 he o al ene gy o he model sys em e sus he eld s eng h o
he ex e nal eld, one obse es almos s aigh line segmen s o he o al ene gy ha a e
in e up ed by discon inuous changes o he slope. Sligh de ia ions om hese s aigh lines
can be obse ed in he in e media e segmen . No e ha GOEP-SIC calcula ions o eld
s eng hs up o
3.5×109
V/m a e in almos pe ec ag eemen wi h he GKLI-SIC ndings.
The eld s eng hs whe e he slope changes discon inuously equal he eld s eng hs whe e
3
The B3LYP pola izabili ies we e kindly p o ided by And eas Ka olewski.
A.3. STATIC CHARGE TRANSFER IN A TRANSPARENT MODEL SYSTEM
61
Figu e A.3: Dependence o he o al ene gy
o he CT model sys em o he le pa o
Fig. 4.2 on he s eng h o he ex e nal eld.
I p esen he esul s compu ed wi h LSDA,
GKLI-SIC, and GOEP-SIC. Fo his illus-
a ion, I subsumed he degene a e ealiza-
ions o GKLI-SIC (see Sec. 4.2.1). As a
guide o he eye, I  ed s aigh lines wi h
a leas -squa es linea eg ession o he h ee
segmen s o he GKLI-SIC da a poin s om
0.0
V/m o
2.5×10−9
V/m, om
2.5×10−9
V/m o
5.5×10−9
V/m, and om
6.0×10−9
V/m o
8.0×10−9
V/m. Theses segmen s
co espond o he si ua ion whe e no elec on
is ans e ed, one elec on has jumped om
he dono o he accep o chain, and wo elec-
ons a e ans e ed.
he elec ons jump om he dono o he accep o chain acco ding o Man1 and he igh pa
o Fig. 4.2. Thus, he discon inuous changes o he slope o he o al ene gy can be a ibu ed
o hese elec on jumps. A he co esponding eld s eng hs, he densi y eshues no ably
so ha he eld ac s on e y die en densi y dis ibu ions be o e and a e he elec ons
ans e . In Fig. A.3, his mani es s in he die en slopes o he o al ene gy wi h espec
o he elec ic eld s eng h. De ails o he la e dependence emain o be in es iga ed.
A  s sigh , he s aigh line segmen s o he o al ene gy o Fig. A.3 appea eminiscen
o he s aigh line beha io o he o al ene gy wi h ac ional changes o he elec on numbe
ha Pe dew
e al.
epo ed in Re . [PPLB82] ( o an o e iew, see Sec. 2.4). Ye , he pic u e
ha eme ges he e is a die en one as he calcula ions do no in ol e ac ional cha ges o
he en i e sys em and Fig. A.3 shows he o al ene gy e sus he eld s eng h in con as
o ac ional cha ges ha a e depic ed in Re . [PPLB82]: Al hough some alues o he
ex e nal eld s eng h can be ela ed o specic in ege occupa ions o D and A in GKLI-
SIC calcula ions, none o he in e media e poin s co esponds o any ac ional occupa ion o
he wo hyd ogen chains. S ill, he ab up changes o he slope a he eld s eng hs whe e
elec ons jump a e a mani es a ion o he in ege elec on ans e beha io .
In p inciple, a ela ion be ween he ex e nal eld s eng h and he ( ac ional) cha ge o
he D o o he A chain may be se up in he case o LSDA using he ac ional occupa ions
o he single moie ies in he LSDA da a o he igh pa o Fig. 4.2. Ye , his pic u e
does no coincide wi h he ac ional occupa ion pic u e o he en i e sys em o Pe dew
e
al.
[PPLB82] despi e he s iking analogy, because he e he single moie ies a e no isola ed
and in e ac ions be ween D and A need o be aken in o accoun . In Fig. A.3, he LSDA
unc ional exhibi s a con inuous conca e shape o he o al ene gy wi h espec o he ex e nal
eld s eng h. This nding eec s he ac ional elec on ans e beha io o LSDA whe e
no ab up densi y eshuing occu s.
Appendix B
Mul ig id Poisson sol e
The solu ion o Poisson's p oblem is one o he key ea u es o he PARSEC code as i is
exploi ed o de e mine he Ha ee po en ial ins ead o compu ing he Ha ee po en ial in-
eg al di ec ly [KMT
+
06]. The s anda d me hod o sol e Poisson's equa ion in PARSEC is
he conjuga e g adien (CG) me hod [Saa03]. In a s anda d (semi)local densi y unc ional
PARSEC un, he solu ion o Poisson's equa ion is no o cen al limi ing cha ac e o he
o e all pe o mance o he code as i is equi ed only once pe GS sel -consis ency i e a ion
and once pe ime s ep i one assumes ha one needs o pe o m a single po en ial e alua ion
pe ime s ep. Ye , o o bi al-dependen unc ionals as o example SIC, EXX, and espe-
cially ene gy-minimizing GSIC, he eciency o he Poisson sol e is c ucial o he o e all
pe o mance. Fo ins ance, in SIC he Poisson sol e needs o be called
N+ 1
imes bo h
pe sel -consis ency i e a ion and pe ime s ep, whe e
N
is he numbe o o bi als in ol ed.
The si ua ion is e en mo e complica ed in ene gy-minimizing GSIC calcula ions. He e,
he Poisson sol e is needed a leas
N
imes in each s ep ha needs o be aken du ing
he i e a i e uni a y op imiza ion o he uni a y ans o ma ion in a GS calcula ion (see Ap-
pendix C). Mos Poisson calls, howe e , a e pe o med du ing TDGSIC ime p opaga ion.
Fo ins ance, p opaga ion o he 28 occupied o bi als o he DMABN molecule o Sec. 4.2.2
o 40 s equi es
5×104
ime s eps, whe e in he example pe ime s ep and o bi al on a e -
age
4.2
Poisson calls a e pe o med. In o al his makes abou
6×106
Poisson calls using he
mos ecien p opaga ion scheme and algo i hms a ailable in PARSEC a cu en e ms.
This example clea ly demons a es ha he solu ion o Poisson's equa ion is a pe o mance
limi ing ac o i one uses o bi al unc ionals in PARSEC. Fo he easibili y o many calcu-
la ions p esen ed in his hesis, I add essed his pe o mance bo leneck and implemen ed a
new me hod based on mul ig id (MG) and de ec co ec ion ideas o he solu ion o Poisson's
equa ion in he PARSEC p og am package.
In his appendix, I gi e an elemen a y in oduc ion o he basic MG and de ec co ec ion
concep s and explain he implemen a ion o he MG sol e in he PARSEC code. The new
algo i hm u ns ou o be much mo e ecien han he CG Poisson sol e . Mo eo e , he
pa alleliza ion o he MG Poisson sol e is adjus ed o he equi emen s o SIC and EXX
calcula ions, whe e o bi al densi y dependen Poisson p oblems can be pe o med in pa allel.
Fo a mo e de ailed in oduc ion o he MG me hod, I ecommend he book o T o enbe g,
Oos e lee, and Schülle [TOS01], he book o Hackbusch [Hac85], and he in oduc ion o
MG me hods in he Nume ical Recipes [PTVF92]. Pa s o he p esen a ion in he ollowing
a e based on hese books.
63
70
APPENDIX B. MULTIGRID POISSON SOLVER
pa ame e name ype op ions explana ion
poissonsol e in
0 (D) CG sol e
1 MG sol e (ob ained om he K onik g oup)
2 MG sol e
10 CG sol e wi h g id shi as in MG
MG_Con e gence dp
1.0 (D) scaling pa ame e o accu acy
MG_Relaxa ion_mode in
1 (D) s anda d ed-black Gauss-Seidel elaxa ion
2 ed-black Gauss-Seidel wi h o e elaxa ion
MG_Cycle_mode in
1 (D) V-cycle
2 F-cycle
3 V-cycle, de ec co ec ion on all g id le els
4 F-cycle, de ec co ec ion on all g id le els
5 wo F-cycles, hen V-cycles
MG_Laplace in
1-4 (D) Laplacian o de in de ec co ec ion
Table B.1: PARSEC inpu pa ame e s and inpu alues ela ed o he Poisson sol e . Each
pa ame e is desc ibed in b ie in he las column. The sho cu 
in
 deno es in ege and he
sho cu 
dp
 double p ecision da a ypes. De aul alues a e labeled by (D). All pa ame e s
wi h names ha begin wi h 
MG
 a e a ailable only in case o
poissonsol e
2 which is he
MG sol e in oduced in his appendix.
I employ his es ima e in PARSEC oge he wi h wo deni ions o he no m: (i) he squa e
oo o he in eg al o e he squa e o
˜τh
and (ii) he maximum alue o
|˜τh|
. This p oceeding
yields wo die en con e gence pa ame e s. To be on he sa e side, I scale bo h pa ame e s
by a ac o o one hal . Mo eo e , an addi ional scaling ac o o his es ima e can be applied
by he inpu pa ame e
MG_Con e gence
ha is mul iplied o he es ima e o
ε
. These
con e gence pa ame e s a e calcula ed on he y based on he nes g id and he  s g id
coa se han he nes g id du ing he  s call o he MG Poisson sol e . In all u he
calls, con e gence o he app oxima e solu ion
˜ h
is es ed e e y ime he nes g id le el is
eached.
Las bu no leas , he MG Poisson sol e is embedded in o he pa allel PARSEC en-
i onmen . The e o e, le me nally commen on pa alleliza ion o MG me hods in gene al
and he PARSEC implemen a ion in pa icula . Die en me hods and ideas o a MG pa -
alleliza ion a e a ailable in he li e a u e [B a81, MT96, Jun97, HJ97, Mi 97, FJ00, 
ST03,
DHL03, HKMR05, KSRS08]. Such pa alleliza ion s a egies add ess he pa alleliza ion o e
he g id o a single Poisson sol e call and in ol e g id pa i ioning and domain decomposi-
ioning echniques. In hese cases communica ion ope a ions need o be pe o med du ing
elaxa ion and in e pola ion on all g id le els o he coa se g id hie a chy when g id poin s
ha a e ea ed by die en p ocesso s a e in ol ed. Mo eo e , one has o nd an app op i-
a e dis ibu ion o g id poin s o e p ocesso s in he pa allel en i onmen o all g id le els.
This is an especially dicul ask on he coa se g id le els whe e he numbe o g id poin s
may be on he o de o he p ocesso s ha a e in ol ed. The e o e, o ob ain a easonable
pa allel speed-up, one needs o ca e ully design he managemen o g id le els and pa allel
dis ibu ion o nume ical load.
Ye , wi h espec o he special equi emen s o PARSEC whe e solu ions o Poisson's

B.4. ASSESSMENT OF THE MULTIGRID SOLVER
71
equa ion need o be pe o med o he o bi al densi ies o all
N
occupied o bi als, a die -
en pa alleliza ion s a egy wi h pa alleliza ion o e he o bi als lends i sel o be used. I
pa allelized he
N
Poisson sol e calls in PARSEC such ha he Ha ee po en ial o one
o bi al densi y is compu ed only by one p ocesso . The ad an age is ha such a single Pois-
son sol e call does no equi e any pa allel communica ion, hus he sequen ial MG sol e
can be used. Howe e , as he pa alleliza ion in all o he PARSEC ou ines is implemen ed
ia g id pa i ioning, he new s a egy equi es an ini ial edis ibu ion o o bi al densi ies
and a nal edis ibu ion o he co esponding o bi al-specic Ha ee po en ials om g id
pa i ioning o o bi al pa alleliza ion and back. This edis ibu ion is conduc ed by col-
lec i e message passing in e ace (MPI) ou ines. I is designed such ha i possible each
p ocesso pe o ms he same numbe o Poisson calls. Ye , i an equal dis ibu ion o he
nume ical load is no possible, some p ocesso s a e idle while he o he p ocesso s wo k on
hei o bi al-specic Poisson p oblems. Thus, in his pa alleliza ion s a egy he numbe o
p ocesso s should be adap ed o he numbe o occupied o bi als: Di iding
N
by he numbe
o p ocesso s should yield a numbe ha equals o is sligh ly smalle han an in ege . To
use he sequen ial Poisson sol e o compu ing he Ha ee po en ial o he o al densi y in
a pa allel en i onmen , he ollowing s eps a e pe o med:
(i) he densi y is dis ibu ed o all p ocesso s
(ii) each p ocesso pe o ms he same MG sol e call wi h he o al densi y as inpu and
he Ha ee po en ial on he en i e g id as ou pu
(iii) each p ocesso akes he app op ia e pa o he Ha ee po en ial acco ding o he g id
pa i ioning om i s own en i e-g id solu ion o sa e communica ion o e head
B.4 Assessmen o he mul ig id sol e
Ha ing implemen ed he MG sol e as desc ibed be o e, I nally assessed he pe o mance
o he sol e in e ms o accu acy o he solu ion and o compu a ion ime. Fo a easonable
compa ison o he CG me hod i was c ucial o use a PARSEC g id ha is iden ical in bo h
Poisson sol e cases. The e o e, I in oduced a new Poisson sol e me hod 
poissonsol e
=
10 ha is a CG sol e wi h he PARSEC g id shi ed as in he MG case (CG shi ed).
Using his new sol e se up, I  s compa ed he sequen ial sol e e sion in e ms o
accu acy o he eigen alues, Ha ee ene gy, and dipole momen as well as he compu a ion
ime o a es se o molecules ep esen a i e o he sys ems in es iga ed in he Kümmel
g oup: Eigen alue de ia ions we e ypically smalle han
1×10−3
%, Ha ee ene gy de ia-
ions smalle han
5×10−4
%, and dipole momen de ia ions smalle han
1×10−5
a omic
uni s. These de ia ions, i no able a all, a e smalle han he accu acy ha may be expec ed
om densi y unc ionals, hus con m he accu acy o he MG implemen a ion. Ye , a de-
ailed p esen a ion o he es se goes beyond he scope o his appendix. The e o e, I jus
ou line a single example ha gi es a a o o he pe o mance o he MG implemen a ion.
I chose he small silicon clus e Si
4
as a anspa en es sys em (see Table B.2 o de ails
o he sys em and he PARSEC se up). I is an ideal es case o pa alleliza ion as all eigh
occupied o bi als o Si
4
can be dis ibu ed o e he CPUs o an oc o-co e p ocesso . The
esul s o a KLI-SIC g ound-s a e un a e lis ed in Table B.3. Fi s , one obse es ha CG
shi ed and MG gi e almos iden ical o al ene gies and eigen alues. Ye , de ia ions on he
72
APPENDIX B. MULTIGRID POISSON SOLVER
es sys em PARSEC inpu MG g id se up
Si
4
Bounda y_Sphe e_Radius
: 18.0d0 numbe o g id
G id_Spacing
: 0.4d0 poin s pe
coo dina es o he nuclei [boh ]:
Expansion_O de
: 10 dimension:
0.000001, 2.246272, 0.000000
S a es_Num
: 10 - sphe e size: 93
3.739822, 0.000372, -0.000069
Spin_Pola iza ion
: . alse. - ne g id: 97
0.000001, -2.244924, 0.000001
Con e gence_C i e ion
: 1.0d-05 - coa se g id: 7
-3.739817, 0.000373, 0.000052
Diag_Tole ance
: 1.0d-07
Eigensol e
: diagla numbe o g id
occupied KS o bi als: 8
Mixing_Me hod
: Ande son le els: 5
Mixing_Pa am
: 0.3d0
g id poin s (CG se up): 382,336
Memo y_Pa am
: 3
g id poin s (MG se up): 38,915
Table B.2: Si
4
es sys em o assessing he pe o mance o he MG sol e . I assembled
impo an pa ame e s o Si
4
and he g id se up in he le column, ele an PARSEC inpu
pa ame e s in he cen al column, and an o e iew o e he numbe o g id poin s and
die en g id le els o he MG in he igh column.
o de o
1×10−3
% occu i one compa es CG calcula ions wi h g id shi s ela i e o each
o he . Such de ia ions a e solely ela ed o he nume ical ep esen a ion on he g id and can
be educed by dec easing he g id spacing. I pe o med simila es s on a coa se PARSEC
g id wi h a spacing o 0.6 boh ins ead o 0.4 boh and ound no ably la ge de ia ions o
ha kind. Howe e , de ia ions be ween CG shi ed and MG we e by almos h ee o de s o
magni ude smalle . Second, one nds in sequen ial uns ha Ha ee compu a ion imes a e
by abou a ac o o  e and KLI compu a ion imes a e by abou a ac o o six as e in MG
han in CG cases. This is a clea indica o o he ema kable speed-up o he MG sol e .
Al hough in he pa allel case he speed-up o single Ha ee calls is only abou a ac o o
1.5, KLI xc po en ial e alua ions a e almos a ac o o en as e when using he MG han
using he CG. This mani es also in a no able imp o emen o he o al compu a ion ime.
I assessed also he pe o mance o GSIC calcula ions wi h ene gy minimiza ion using he
sequen ial pa allel
E o
[Ry] HOMO [Ry]
TH
[s]
TKLI
[s]
T o
[s]
TH
[s]
TKLI
[s]
T o
[s]
CG
sh
-30.66069 -0.53198 7.91 47.88 2,275.35 2.14 13.19 503.15
CG -30.66176 -0.53236 7.71 47.66 2,061.81 2.44 14.82 518.44
MG -30.66068 -0.53198 1.51 7.61 1,745.35 1.47 1.32 374.64
Table B.3: Pe o mance assessmen o he sequen ial and pa allel MG sol e in a s anda d
SIC KS sel -consis en g ound-s a e calcula ion using Si
4
as a es sys em (see Table B.2).
The pa allel en i onmen uses eigh CPUs on he same co e. All PARSEC uns lis ed
he e equi ed 11 i e a ions. Mo eo e , I compiled he a e age CPU imes needed o he
Ha ee sol e (
TH
), he KLI po en ial p ocedu e (
TKLI
), and he o al CPU ime o he
sel -consis ency i e a ion (
T o
). The solu ion ob ained wi h he CG sol e whe e he g id is
shi ed as in he MG case (CG
sh
) se es as a e e ence.
B.4. ASSESSMENT OF THE MULTIGRID SOLVER
73
g adien line-sea ch op imiza ion algo i hm (see Sec. C.1.3). While eigen alues and o al
ene gies a e in easonable ag eemen , he MG sol e gains no able speed-up o he o al
sel -consis ency p ocedu e. In he sequen ial PARSEC es , he MG p ocedu e equi es 21
sel -consis ency i e a ions and 212 op imiza ion loops ins ead o 17 and 198 using he CG
sol e , bu ob ains con e gence by abou a ac o o ou as e (
TMG
o
= 5,743.6 s ins ead o
TCG shi ed
o
= 24,180.8 s). In he pa allel case, speed-up is e en be e as wi h a o al ime
o 1,126.5 s he MG e sion is by abou a ac o o 5.6 as e han he CG un (
TCG shi ed
o
= 6,259.8 s).
Finally, I s ess ha al hough al eady in GS calcula ions he gain o compu a ion ime
is ema kable, e en be e pe o mance enhancemen in e ms o o al compu a ion ime
may be expec ed du ing ime p opaga ion. The e o e, I p opaga ed Si
4
o 1.0 s wi h an
elec onic ime s ep o 0.001 s using he Taylo  p opaga ion algo i hm [CMR04, MKHM06,
Mun07, Mun09] and compa ed he o al CPU imes needed by he die en me hods. The
sequen ial p opaga ion based on he MG sol e equi ed 22,860 s CPU ime in compa ison
o 117,0442 s based on he CG shi ed sol e . Thus, in his case he MG e sion o he
sequen ial code is by a ac o o 5.1 as e han he CG e sion. E en be e pe o mance
enhancemen can be obse ed in he pa allel e sion: The p opaga ion akes 40,098 s using
he CG shi ed sol e , whe eas he MG e sion equi es only 7,300 s o he  s s. This
amoun s o a speed-up wi h a ac o o 5.5 o he en i e p opaga ion pa o he PARSEC
code due o he MG Poisson sol e implemen a ion.
Appendix C
Algo i hms o he uni a y
op imiza ion
The de e mina ion o he uni a y ans o ma ion is he mos ime c i ical s ep in GSIC
calcula ions. The uni a y op imiza ion in PARSEC is based on i e a i e algo i hms whe e
he ime need mani es s in wo ways:  s , he numbe o s eps ha a e needed o ob ain
con e gence o he i e a ion and, second, he nume ical load pe i e a ion s ep. In case o
ene gy-minimizing uni a y ans o ma ions, o bi al-specic Ha ee and xc po en ials need
o be compu ed pe o bi al in each s ep o he i e a ion. The e o e, a easonable pe o -
mance o he algo i hm equi es an ecien solu ion o Poisson's p oblem ha is behind he
compu a ion o Ha ee po en ials. This aspec is discussed in Appendix B. Fu he mo e,
op imiza ion can be ob ained by educing he numbe o i e a ion s eps equi ed o achie e
con e gence. To his end, I implemen ed die en algo i hms in PARSEC ha accomplish
uni a y op imiza ion along die en ideas. Mo eo e , o conduc uni a y op imiza ion wi h
complex- alued op imized o bi als, algo i hms ha we e a ailable o eal- alued o bi als
needed o be cus omized o he complex case. In he ollowing discussion, I ocus on he
mo e gene al complex- alued case. Based on he esul ing equa ions, one ob ains he algo-
i hms o eal- alued o bi al op imiza ion by neglec ing all he complex conjuga ion signs.
C.1 Algo i hmic p inciples
C.1.1 Fois loops based on he Pede son c i e ion
The  s algo i hm discussed in his sec ion is based on an idea o Fois, Penman, and Madden
[FPM93]. I uses he Pede son c i e ion (3.12) and s a s om an ini ial se o op imized
o bi als
{˜ϕ(0)
jσ }
. In each s ep
(k)
o he i e a i e p ocedu e a new uni a y ans o ma ion
U(k+1)
ijσ =
Nσ
X
n=1
S(k)
inσU(k)
njσ
(C.1)
and hus a new se o op imized o bi als
˜ϕ(k+1)
iσ (
) =
Nσ
X
j=1
S(k)
ijσ ˜ϕ(k)
jσ (
)
(C.2)
75

76
APPENDIX C. ALGORITHMS FOR THE UNITARY OPTIMIZATION
can be ob ained by uni a y o a ions wi h he ma ix
S(k)
σ
. This ans o ma ion is assumed
o pe o m only small changes. Thus, i a ies only sligh ly om he uni ma ix acco ding
o
Sno
ijσ =δij +τijσ,
(C.3)
whe e
τijσ
a e complex numbe s
|τijσ|  1
and
τijσ =−τ∗
jiσ
. As in his app oxima ion he
ans o ma ion
Sno
ijσ
is no s ic ly uni a y, Thomas Kö zdö e [Kö 09] sugges ed o explici ly
ake ca e o he loss o uni a i y by means o Löwdin's me hod o symme ic o hogonaliza ion
[Löw50, May02]: Using he Löwdin ma ix
Cijσ =h˜ϕno
iσ |˜ϕno
jσi−1/2
(C.4)
compu ed om he nono hogonal o bi als
˜ϕno
iσ =
Nσ
X
j
Sno
ijσ ˜ϕ(k)
jσ
(C.5)
one ob ains a new o hogonal o bi al se
˜ϕiσ =
Nσ
X
j,m
CimσSno
mjσ ˜ϕ(k)
jσ .
(C.6)
Inse ing his new o bi al se in o Pede son's symme y condi ion yields
Nσ
X
k,m=1
C∗
jmσSno ∗
mkσ
Nσ
X
l,n=1
CinσSno
nlσ h˜ϕ(k)
kσ |˜ SIC
jσ −˜ SIC
iσ |˜ϕ(k)
lσ i
| {z }
=:hk|j−i|li
= 0,
(C.7)
whe e I in oduced a sho cu o he b acke no a ion. The Löwdin ma ix (C.4) can be
app oxima ed [GU97, Kö 09] by
Cijσ ≈δij −1
2h˜ϕno
iσ |˜ϕno
jσi−δij=δij +1
2 Nσ
X
m
τ∗
imστ∗
mjσ!
(C.8)
and exp essed in e ms o
τijσ
ia Eqs. (C.5) and (C.3). Thus one eadily compu es
Nσ
X
n=1
CinσSno
nlσ =δil +τilσ +1
2
Nσ
X
p=1
τ∗
ipστ∗
pjσ +1
2
Nσ
X
p,j=1
τ∗
ipστ∗
pqστqlσ
| {z }
=:ωilσ
.
(C.9)
Following he de i a ion o Fois
e al.
[FPM93] and Kö zdö e [Kö 09] his esul needs o
be in oduced in o Eq. (C.7) and one ob ains he nal exp ession a e a li le algeb a
τijσ =1
2hj|j−i|ii+1
2
Nσ
X
l=1
[ωilσhj|j−i|li+δjlτilσ]
+1
2
Nσ
X
k=1 ω∗
jkσhk|j−i|ii−δikτ∗
jkσ+1
2
Nσ
X
k,l=1
ω∗
jkσωilσhk|j−i|li.
(C.10)
C.1. ALGORITHMIC PRINCIPLES
77
A nal o hogonaliza ion o he ans o ma ion o Eq. (C.3) using
τijσ
o Eq. (C.10) yields
he uni a y ans o ma ion
S(k)
ijσ
. Thus, one ob ains he new o bi al se and ans o ma ion
o he i e a ion p ocedu e om Eqs. (C.1) and (C.2). The algo i hmic implemen a ion o
his me hod is iden ical o he one desc ibed by Kö zdö e [Kö 09]. He e, con e gence is
de e mined by he Pede son c i e ion. I is eached when he absolu e alue o he la ges
ma ix elemen o he ma ix o Eq. (3.12) is smalle han a gi en h eshold which is yp-
ically less han
10−5
Ry. Nume ical es s o his me hod e ealed ha a no able numbe
o i e a ions is equi ed o ob ain con e gence o he Pede son c i e ion below a easonable
h eshold. Ye , an op imiza ion o his me hod in e ms o la ge s eps is dicul . The e-
o e, I conclude ha designing op imiza ion algo i hms based on he Pede son c i e ion o
Eq. (3.12) is legi ima e, bu algo i hms ha base di ec ly on he ene gy exp ession may be
mo e ecien . I discuss such kind o algo i hms in he ollowing sec ions.
C.1.2 Ene gy g adien based algo i hm
The second algo i hm p esen ed in his appendix uses an ene gy g adien sugges ed by Mes-
sud
e al.
[MDRS09a]. I is based on he ac ha he SIC ene gy a ies wi h espec o
uni a y ans o ma ions o he occupied KS o bi als
˜ϕiσ(
) =
Nσ
X
j=1
Uijσϕjσ(
).
(C.11)
Fo compu ing he g adien o Messud
e al.
[MDRS09a], one needs o ake he Lag agian
mul iplie ma ix in o accoun ha appea s when minimizing he SIC ene gy wi h espec
o he o bi als and gua an ees o hogonali y o he la e . This ma ix does no appea in
KS DFT. The e o e, he analogue KS ene gy g adien o he o al ene gy
ESIC
o
in he GSIC
o malism eads
Dijσ =∂U∗
ijσ ESIC
o =−hϕjσ|˜ SIC
iσ |˜ϕiσi.
(C.12)
I de e mines he di ec ion in which he uni a y ans o ma ion
Uijσ
needs o be changed
o nd he o al ene gy minimum s a ing om a xed se o occupied KS o bi als. The
g adien scaled by he g adien s ep size
η
can be used o nd a new ans o ma ion
Uno
ijσ =U(k)
ijσ −ηD(k)
ijσ
(C.13)
a each s ep
(k)
o a esul ing i e a i e p ocedu e s a ing om a p e ious ans o ma ion
and op imized o bi al se . As his ans o ma ion is nono hogonal, he new ans o ma ion
U(k+1)
ijσ
o he nex i e a ion needs o be de e mined by o hogonaliza ion o
Uno
ijσ
. In PAR-
SEC, his a pos e io i o hogonaliza ion can be pe o med ei he by G am-Schmid o by
symme ic Löwdin o hogonaliza ion.
In g adien algo i hms, he g adien s ep size is a na u al s a ing poin o pe o mance
enhancemen . Howe e , as applica ion o he g adien is always ollowed by an o hogonal-
iza ion o gua an ee o uni a i y o he ans o ma ion, he o iginal sea ch di ec ion gi en
by he g adien ge s los o some ex end. Mo eo e , he g adien s ep size no only scales he
applica ion o he g adien , i also has a signican inuence on he ou come o he a pos e-
io i o hogonaliza ion. This ende s he op imiza ion o he s ep size dicul . The e o e, in
he PARSEC implemen a ion o he ene gy-g adien algo i hm, I in oduced an adap a ion
78
APPENDIX C. ALGORITHMS FOR THE UNITARY OPTIMIZATION
o he g adien s ep size depending on how he g adien pe o ms du ing p e ious s eps o
he i e a ion: Du ing he i e a ion, I moni o i applica ion o he g adien imp o es he
con e gence c i e ion owa ds he p ese con e gence h eshold o no . I he con e gence
pa ame e imp o es du ing h ee succeeding cycles, he g adien s ep size is mul iplied by a
ac o o wo un il i eaches a p ese maximum alue. In his scheme, he ini ial g adien
s ep size is 100 and he maximum alue can be de e mined by he PARSEC inpu . On he
con a y, in each cycle whe e he con e gence pa ame e de e io a es, he g adien s ep size
is di ided by a ac o o wo and he code checks i applica ion o he old g adien scaled by
he new g adien s ep size imp o es o no . This down scaling o he g adien s ep size is
pe o med ei he un il he con e gence pa ame e imp o es again o un il a minimum o he
s ep size o
1×10−4
is eached. I applica ion o he g adien is no longe able o imp o e
he con e gence pa ame e , he algo i hm s ops wi h a wa ning. O he wise, he i e a ion
is pe o med un il con e gence. Fo a success ul applica ion o his algo i hm i is impo -
an ha changes o he uni a y ans o ma ion due o applica ion o he ene gy g adien
a e no de e io a ed by he a pos e io i o hogonaliza ion. The e o e, he o hogonaliza ion
p ocedu e needs o main ain he di ec ion o he uni a y o a ion o
Uno
σ
as close as possible.
To his end, I ecommend o use Löwdin's o hogonaliza ion me hod ha yields a uni a y
ans o ma ion
U(k+1)
σ
as close as possible o
Uno
σ
.
In case o ene gy-g adien algo i hms, a leas wo con e gence pa ame e s lend hem-
sel es o be used: (i) minimiza ion o he o al ene gy and (ii) educ ion o he Pede son
c i e ion below a gi en h eshold. As he minimum o he ene gy is no known, his c i e ion
needs o be exploi ed as a sel -consis ency c i e ion. Thus, ene gy con e gence is eached
i he ene gy changes o wo consecu i e i e a ion s eps a e lowe han he co esponding
con e gence pa ame e . Typical ene gy con e gence pa ame e s a e on he o de o
10−8
Ry. In he PARSEC implemen a ion, bo h con e gence pa ame e s can be combined o
used sepa a ely (see Table C.2). Mo eo e , he ea men o he con e gence pa ame e in
g ound-s a e calcula ions and du ing ime p opaga ion can be chosen independen ly.
Nume ical es s o his algo i hm showed ha i no ably imp o es in e ms o compu a ion
ime un il con e gence is eached in compa ison o he algo i hm o Sec. C.1.1. Ye , due o
he a pos e io i o hogonaliza ion la ge g adien s eps a e p ecluded. The e o e, in he nex
sec ion I explain ano he idea ha allows o la ge ene gy-g adien s eps due o s ep-size
op imiza ion.
C.1.3 Ene gy g adien and line-sea ch op imiza ion
The cen al idea o he hi d algo i hm o uni a y op imiza ion o he SIC ene gy imple-
men ed in PARSEC is o op imize he applica ion o he ene gy g adien so ha no explici
a pos e io i o hogonaliza ion is needed and consequen ly s ep size op imiza ion can be
pe o med ecien ly. I is based on a conjuga e g adien (CG) me hod whe e he uni a y
cons ain o he ene gy-minimizing ans o ma ion is aken in o accoun explici ly [AEK09].
He e, o he sake o comple eness, I gi e a sho in oduc ion and summa y o his algo i hm
ollowing he p esen a ion o Pub3 whe e i is explained in mo e de ail.
In con as o he g adien o Sec. C.1.2, he ene gy g adien o his sec ion is based on
a die en idea. He e, in each s ep
(k)
o he i e a i e p ocedu e one ies o nd a uni a y
C.1. ALGORITHMIC PRINCIPLES
79
ans o ma ion
S(k)
σ
ha de e mines a new o bi al se
{˜ϕ(k+1)
iσ }
acco ding o
˜ϕ(k+1)
iσ (
) =
Nσ
X
j=1
S(k)
ijσ ˜ϕ(k)
jσ (
)
(C.14)
wi h lowe SIC ene gy s a ing om a p e ious op imized o bi al se
{˜ϕ(k)
iσ }
. Thus,
S(k)
σ
also
needs o modi y he p e ious uni a y ans o ma ion
U(k)
σ
acco ding o
U(k+1)
ijσ =
Nσ
X
n=1
S(k)
inσU(k)
njσ
(C.15)
so ha
U(k+1)
σ
yields he new op imized o bi als om he occupied KS o bi als
˜ϕ(k+1)
iσ (
) =
Nσ
X
j=1
U(k+1)
ijσ ϕjσ(
).
(C.16)
In his case, one is in e es ed in SIC ene gy changes due o small a ia ions o he ans o -
ma ion
Sσ
om he uni ma ix s a ing om a xed o bi al se
{˜ϕ(k)
jσ }
[PFD
+
11]: Linea
o de changes o he SIC ene gy due o de ia ions o
Sσ
om he uni ma ix a e gi en by
he an i-He mi ian g adien ma ix
G(k)
ijσ =∂ESIC
xc [{˜ϕ(k)
nσ }, Smnσ]
∂S∗
ijσ S∗
ijσ=δij
=h˜ϕ(k)
jσ |˜ SIC (k)
iσ (
)−˜ SIC (k)
jσ (
)|˜ϕ(k)
iσ i.
(C.17)
The SIC po en ials
˜ SIC (k)
jσ (
)
a e compu ed om he ene gy-minimizing o bi als
˜ϕ(k)
jσ (
)
.
The g adien o Eq. (C.17) coincides wi h he Pede son condi ion.
The idea o Re . [AEK09] is o pe o m he g adien s ep in a educed space whe e he
uni a i y cons ain is gua an eed by choice o he pa ame e space [EAS98]. Thus, he
op imiza ion p oblem can be u ned in o an uncons ained one by sui able es ic ion o he
pa ame e space o he Lie g oup o
n×n
uni a y ma ices
U(n)
[AEK09]. Due o he
an i-He mi ici y o he g adien ma ix, he exponen ial map
S(k)
σ(l(k)
σ) = exp{−l(k)
σG(k)
σ}
(C.18)
wi h s ep size
l(k)
σ
cons i u es a uni a i y-conse ing ans o ma ion when i is applied o
he uni a y ans o ma ion
U(k)
σ
as in Eq. (C.15).
G(k)
σ
co esponds o he s eepes descen
sea ch di ec ion, bu due o he exponen ial map uni a i y o
S(k)
σ
is gua an eed. Thus, he
p oblem o a pos e io i o hogonaliza ion is a oided.
The nume ical pe o mance o his uni a i y-conse ing applica ion o he ene gy g adien
can be inc eased by means o CGs and op imiza ion o he g adien s ep size. The idea o he
CG is o use no only he in o ma ion o he sea ch di ec ion gi en by he ene gy g adien ,
bu o also ake in o ma ion o he p e ious sea ch di ec ion in o accoun . Thus, he new
sea ch di ec ion
H(k)
σ
is based on he cu en g adien
G(k)
σ
and he sea ch di ec ion
H(k−1)
σ
o he las s ep o he i e a ion. In he PARSEC CG implemen a ion, I use he new sea ch
di ec ion
H(k)
ijσ =G(k)
ijσ +γ(k)
σH(k−1)
ijσ ,
(C.19)
86
APPENDIX D. FÖRSTER-TYPE POTENTIALS AND STOCHASTIC
TIME-DEPENDENT DENSITY FUNCTIONAL THEORY
The Hea iside unc ion spli s he en i e eal space in o a D (
Θ(−x)
) and an A (
Θ(x)
) pa (see
Fig. 1 o Pub1). Hence, he supe sc ip s D and A indica e p ope ies ha a e compu ed only
on he D o A side, espec i ely. Based on his no a ion,
H[nD(A)]
is he Ha ee po en ial
o D(A) and
ddE,D(A)
H[nA(D)]
is he po en ial esul ing om he in e ac ion be ween D and
A in he D(A) hal space. Wi h he help o he mul ipole momen s o he D(A) densi y,
Ni=Zni(
i) d3 i,
d
i=Z
ini(
i) d3 i,
Qi
jk =Zni(
i)(3 i
j i
k− i2δjk) d3 i,
(D.2)
whe e
i= D,A
, he Fö s e po en ial ha he D densi y gene a es in he hal space o A is
ddE,A
H[nD](
A) =ND
|
R
|−(ND
A−
d
D)·
R
|
R
|3+
A·
d
D
|
R
|3+ND
2
3
X
j,k=1
(3 A
j A
k− A2δjk)RjRk
|
R
|5
+
3
X
j,k=1
QD
jkRjRk
2|
R
|5−3(
R
·
A)(
R
·
d
D)
|
R
|5.
(D.3)
The co esponding po en ial
ddE,D
H[nA](
D)
is ob ained om equa ion (D.3) by in e changing
D and A supe sc ip s and eplacing
R
by
−
R
.
Al e na i ely, he mul ipole expansion can be inse ed di ec ly in o he Ha ee po en ial
and one ob ains he po en ial
dd
H[nD, nA] = H[nD] + dd ,D
H[nA]Θ(−x)+ H[nA] + dd ,A
H[nD]Θ(x)
(D.4)
wi h he supe sc ip index ha deno es he po en ial ou e, whe e
dd ,A
H[nD] = ND
|
A+
R
|+
d
D·(
A+
R
)
|
A+
R
|3+
3
X
j,k=1
QD
jkRjRk
2|
A+
R
|5
(D.5)
and
dd ,D
H[nA] = NA
|
D−
R
|+
d
A·(
D−
R
)
|
D−
R
|3+
3
X
j,k=1
QA
jkRjRk
2|
D−
R
|5.
(D.6)
Pub1 demons a es ha hese wo app oaches esul in po en ials ha die quali a i ely
ou side he egions whe e he densi y is high. F om a compa ison o he wo po en ials
wi h he ue Ha ee po en ial, Pub1 concludes ha
dd
H[nD, nA]
is mo e app op ia e as i
exhibi s he asymp o ic decay o ze o expec ed na u ally om a physical po en ial.
Bo h ways o using Fö s e - ype po en ials a e implemen ed in he PARSEC code.
The Fö s e - ype po en ial expansion ea u e can be (de)ac i a ed using he boolean ag
Use_ oe s e
. The inpu pa ame e
oe s e _ ype
allows o he choice o he po en ial-
de e mina ion ou e along he p e ious lines: 
oe s e _ ype
= 1 is he de aul and uses Eq.
(D.4), whe eas 
oe s e _ ype
= 2 uses Eq. (D.1). The sepa a ion o he g id in o wo hal

D.1. FÖRSTER-TYPE POTENTIALS AND GRID PARTITIONING
87
spaces is always pe o med by he planes o he coo dina e sys em ha a e pe pendicula o
he coo dina e axes. To his end, one needs o indica e he desi ed coo dina e axis x, y, o
z ia he inpu pa ame e
oe s e _hal _space
(
s ing
) and he pa i ioning is pe o med
acco dingly. To gua an ee o consis en calcula ions, he selec ed scheme is used o he
g ound s a e and du ing eal- ime p opaga ion h oughou .
D.1.2 Using Fö s e - ype po en ials in supe molecula sys ems
Fo he in es iga ions on supe molecula sys ems o Chap. 5, I ex ended he Fö s e - ype
po en ial e alua ion o he p e ious sec ion o deal wi h mo e complica ed molecula a ange-
men s. This unc ionali y may be (de)ac i a ed by he boolean ag
Use_mul i_ oe s e
when

Use_ oe s e
= ue. The s a emen s o he p e ious sec ion conce ning
oe s e _ ype
ap-
ply acco dingly.
Two die en ypes o a angemen s, linea and ci cula ones, may be used o sepa a e
he en i e g id in o pa i ions, acco ding o which one may place single molecules. These
se ups can be add essed by he in ege inpu pa ame e
Foe s e _neighbo _ ype
: numbe s
1 o 3 co espond o linea a angemen s, whe eas numbe s 10 and 11 imply ci cula se ups.
The PARSEC inpu block
Foe s e _cen e s
was implemen ed o dene he posi ion o he
g id segmen s ia he coo dina es o hei cen e s. Fo he sake o clea ness, I use he e m
hub o hese poin s in he ollowing. No e ha he hubs always need o be lis ed in
consecu i e o de . In case o linea se ups one addi ionally needs o speci y he o ien a ion
o he alignmen in e ms o he coo dina e axis using he pa ame e
oe s e _hal _space
.
Then, he g id slicing is pe o med along each cen e line o neighbo ing hubs whe e cen e
lines a e pe pendicula o he specied coo dina e axis.
Ci cula se ups a e supposed o be a anged on a ci cle a ound he o igin in he
x
-
y
-
plane. The slicing is pe o med in e ms o angles. I hubs a e gi en ia he inpu block
Foe s e _cen e s
, PARSEC compu es he angles co esponding o he cen e lines ha un
h ough he o igin and h ough he middle o all neighbo ing hubs. He e, he numbe o
hubs should equal he numbe o wedges specied ia he in ege inpu pa ame e
Foe -
s e _wedge_numbe
. The g id pa i ioning is pe o med along planes gi en by hese cen e
lines and he
z
-axis. When no hubs a e dened, PARSEC pe o ms an equidis an ci cula
slicing while i assumes ha he  s molecule is posi ioned on he
x
-axis. In his case, one
needs o gi e he numbe o wedges using he pa ame e
Foe s e _wedge_numbe
.
Wi hin each o hese a angemen s, one may choose be ween die en ways o how he
in e ac ion be ween he subsys ems is implemen ed. I s a wi h he op ions ha a e a ail-
able in he cases o linea a angemen s. 
Foe s e _neighbo _ ype
= 1 co esponds o an
in e ac ion be ween nex neighbo s only o all momen s o he mul ipole se ies excep o he
monopole pa . The la e con ibu ion is by deni ion he mos long- ange one o he mul-
ipole se ies. I is impo an o easonable absolu e alues o he po en ial and, he e o e,
needs o be applied also beyond nex neighbo s. In case o 
Foe s e _neighbo _ ype
= 2,
PARSEC compu es he pai wise in e ac ion be ween all g id pa s. 
Foe s e _neighbo _ ype
= 3 models pe iodic bounda y condi ions: I compu es he mul ipole momen s o he las
( s ) molecule o linea a angemen s and places a c i ious molecule wi h he same p op-
e ies in on o he  s (behind he las ) molecule. The in e molecula in e ac ion wo ks
only be ween nex neighbo s and assumes equidis an molecula spacing.
As his p oceeding is jus a c ude app oxima ion o pe iodic sys ems, he idea wi h
88
APPENDIX D. FÖRSTER-TYPE POTENTIALS AND STOCHASTIC
TIME-DEPENDENT DENSITY FUNCTIONAL THEORY
ci cula a angemen s is o design closed sys ems whe e each molecule na u ally has nex
neighbo s on bo h sides and no explici pe iodic bounda y condi ions need o be applied. In
he ci cula se up o 
Foe s e _neighbo _ ype
= 10, he in e ac ion con ibu ions beyond
he monopole e m wo k only be ween nex neighbo s whe eas monopole in e ac ion applies
be ween all molecules. 
Foe s e _neighbo _ ype
= 11 uses pai wise in e ac ion be ween all
molecules o he se up.
D.1.3 Pa i ion-selec i e exci a ion and obse a ion
Fo TD in es iga ions, i may be ele an o exci e only specic subsys ems ins ead o he
en i e sys em. In cases whe e he p e iously explained pa i ioning is used, such subsys ems
can be ela ed o pa i ions o he g id. In D-A sys ems whe e he sepa a ion is pe o med
in o wo hal spaces, such a ea u e can be con olled by he boolean ag
Use_sel_boos
. I

Use_sel_boos
= ue, one may indica e ia
boos _hal _space
in which o he hal spaces
he exci a ion is supposed o wo k. This pa ame e eads a s ing wi h wo posi ions whe e
he  s posi ion disc imina es be ween he nega i e (n) and he posi i e (p) side o he
coo dina e axis and he second posi ion de e mines he axis (x, y, o z) along which he
sepa a ion occu s.
A simila ea u e is implemen ed in case o mo e complex supe molecua sys ems by
he in ege inpu pa ame e
Exci e_cen e
. I 
Exci e_cen e
= 0, he exci a ion is pe -
o med only in one o he hal spaces o he g id, hus one addi ionally needs o speci y
boos _hal _space
. Al e na i ely, one may add ess he g id pa i ion whe e exci a ion is pe -
o med ia he hubs gi en by
Foe s e _cen e s
. To his end, one needs o se
Exci e_cen e
o he numbe co esponding o he posi ion o he hub in he
Foe s e _cen e s
inpu lis .
Finally, he inpu op ion 
Exci e_cen e
= -1 deac i a es selec i e exci a ion and he exci a-
ion p ocess is applied in all pa i ions o he g id. No e ha bo h space-selec i e exci a ion
ea u es may also be used independen o he Fö s e - ype po en ial expansion.
In PARSEC calcula ions wi h mul iple cen e s, some impo an obse ables a e calcula ed
no only in he en i e sys em bu also o each sys em pa i ion sepa a ely. In cases o Fö s e -
ype po en ial app oxima ions o selec i e exci a ion, he dipole momen and he in eg al
o e he densi y co esponding o each g id pa i ion a e w i en o he le mul i_cen e .ou 
du ing ime p opaga ion. He e, he TD dipole momen is calcula ed ela i e o he cen e s
o mass a he ini ial ime
= 0
co esponding o each g id pa i ion.
D.2 Un a eling he coupling s eng h wi h TDDFT
In his sec ion, I b iey explain how he coupling s eng h and he ene ge ic o- esonance
in a sys em o one dono and one accep o molecule mani es in he TD dipole momen .
No e ha he esonan coupling case was al eady discussed in Re . [Ho 08] and Pub1. The
conside a ions employ a wo-le el model [Neu08] based on he assump ion ha he wa e
unc ion o he o al sys em can be sepa a ed in o D and A pa s due o negligible elec onic
coupling be ween D and A. To be able o employ he wo-le el pic u e, he exci a ion ene gies
o in e es o he ue sys em need o be well sepa a ed om he ene gies o o he exci ed
s a es. Ini ially, he accep o is in i s g ound s a e deno ed by
|Ai
and he dono is in an
exci ed s a e
|D∗i
. This amoun s o an ini ial p oduc s a e
|D∗Ai=|D∗i|Ai=|1i
. The
nal wa e unc ion co esponds o he in e se si ua ion, whe e
|DA∗i=|2i
. The s a es a e
D.2. UNRAVELING THE COUPLING STRENGTH WITH TDDFT
89
cha ac e ized by hei eigenene gies
E1
and
E2
. One can measu e he ene ge ic o- esonances
by he pa ame e
∆E=1
2(E1−E2).
(D.7)
The coupling be ween
|1i
and
|2i
is media ed by he Coulomb in e ac ion
VC
. I leads o
he coupling-ma ix elemen
V=hDA∗|VC|D∗Ai.
(D.8)
The ime e olu ion o he wo-s a e sys em wi h ini ial s a e
|Ψ(0)i=|1i
is gi en by
|Ψ( )i=a1( )|1i+a2( )|2i
(D.9)
wi h he coecien s
a1( )
and
a2( )
[CTDF99],
|a1( )|2=B+Acos2pV2+ ∆E2 ,
|a2( )|2=Asin2pV2+ ∆E2 ,
(D.10)
whe e
A=V2
V2+∆E2
and
B=∆E2
V2+∆E2
. This ime e olu ion o he coecien s co esponds
o an incomple e oscilla ion wi h bea equency
ωbea =√V2+ ∆E2
ha depends on he
coupling be ween he ini ial and he nal s a e as well as he ene ge ic o- esonance: The
occupa ion p obabili y o he ini ial s a e a ies a ound B wi h ampli ude A, while he
occupa ion p obabili y o he nal s a e oscilla es wi h ampli ude A a ound ze o.
The TD dipole momen
d
A( ) = hΨ( )|
A|Ψ( )i
o he accep o , whe e I ake he dipole
ope a o
A
in he space o he accep o only, can be calcula ed as
d
A( ) = |a1( )|2hA|
A|Ai+|a2( )|2hA∗|
A|A∗i.
(D.11)
He e, I exploi ed he o hogonali y o
|Di
and
|D∗i
. I he s a ic dipole momen
hA|
A|Ai
o A anishes, Eq. (D.11) simplies o
d
A( ) = |a2( )|2hA∗|
A|A∗i.
(D.12)
Acco dingly, wi h he assump ion ha he s a ic D dipole momen anishes, he TD dono
dipole momen eads
d
D( ) = |a1( )|2hD∗|
D|D∗i.
(D.13)
Thus, he esonance oscilla ion o he coecien s can be obse ed in he ime e olu ion o
he dipole momen s
d
A( )
and
d
D( )
. In p inciple bo h Eqs. (D.12) and (D.13) can be used
o de e mine
V
. O special impo ance is Eq. (D.13) oge he wi h Eq. (D.10): A  o
he absolu e o he ex ema o he
k
- h componen o he D dipole momen ime e olu ion
p o ides
Apk
,
Bpk
, and
ωbea
, whe e
pk=hD∗| D
k|D∗i
. I can be used o de e mine
V
,
∆E
,
and
pk
. One ob ains
V=sApk
Apk+Bpk
ωbea
(D.14)
and
∆E=sBpk
Apk+Bpk
ωbea .
(D.15)
90
APPENDIX D. FÖRSTER-TYPE POTENTIALS AND STOCHASTIC
TIME-DEPENDENT DENSITY FUNCTIONAL THEORY
Figu e D.1: Dono and accep o dipole mo-
men (
z
-componen ) in a se up o wo Na
2
,
whe e he bond leng h o he accep o Na
2
is
educed by 0.5 boh compa ed o he expe -
imen al bond leng h. I pe o med  s o he
en elope o he oscilla ion acco ding o he
model discussed in he ex .
A ypical ime e olu ion o he
z
-componen o he D and A dipole momen s o he model
sys em is depic ed in Fig. D.1 oge he wi h  s o he co esponding en elopes. No e ha he
wo-le el model quali a i ely  s o he dipole oscilla ion o he o- esonan , coupled sys em
o wo molecules. Howe e , he dipole momen en elopes do no pe ec ly ollow he
sin2
- and
cos2
-shape. I unde s and hese de ia ions as a consequence o he coupled sys em no being
pe ec ly sepa able in o D and A pa s as one assumes in he model. Fu he mo e, al hough
he second exci a ion o Na
2
wi h pola iza ion in z-di ec ion is ene ge ically a o and ca ies
no ably smalle oscilla o s eng h, Na
2
is no a pe ec single le el sys em. The e o e, he
sys em o wo sodium dime s does no pe ec ly  in o he wo-le el model. Ne e heless,
he app oach p o ides a ool o de e mine he coupling
V
and he ene ge ic o- esonance
∆E
. The alidi y o he model can be checked by compa ison o he hus ob ained
∆E
o TDDFT exci a ion ene gies. Two die en ways a e a ailable o calcula ing he la e :
ei he om s aigh o wa d TDDFT p opaga ion o one dime wi h shi ed bond leng h
o along he jus p esen ed ou e. I pe o med bo h me hods and ound good ag eemen .
The e o e, I assume easonable quali y o he coupling-ma ix elemen esul s o Fig. 5.1.
D.3 S ochas ic ime-dependen densi y unc ional heo y
D.3.1 An a emp owa ds a heo e ical jus ica ion o s ochas ic ime-
dependen densi y unc ional heo y wi h specic ba h ope a o s
Es ablishing an open quan um sys em scheme in he amewo k o TDDFT is highly ele an
o s udying la ge scale sys ems in con ac wi h some eec i e en i onmen . Howe e , i s
heo e ical ounda ion sue s om concep ual dicul ies as I explain in Sec. 5.2.2. In his
sec ion, I supplemen al eady exis ing app oaches o p o ing open quan um sys em schemes
in he TD(C)DFT amewo k. The unde lying idea is based on he SSE and anspa en ly
illus a es how he ange o possible ba h ope a o s is es ic ed o gua an ee o he exis ence
o an auxilia y sys em ha ep oduces he same densi y as he o iginal in e ac ing open
quan um sys em o in e es . The assump ions on he ange o ba h ope a o s a e guided
by he a emp o a oid cu en densi y dependen con ibu ions whe e
- ep esen abili y
is ques ionable [DV05]. The idea o p oo was de eloped o es ablish he exis ence o a
non-in e ac ing open quan um sys em ha ep oduces he densi y o an in e ac ing open
quan um sys em o he ba h ope a o ha I in oduced in Sec. 5.3.2. The applicabili y o
D.3. STOCHASTIC TIME-DEPENDENT DENSITY FUNCTIONAL THEORY
91
he idea o p oo o his ba h ope a o is discussed c i ically a he end o his sec ion.
The a emp owa ds a p oo o s ochas ic TDDFT s a s om he sys em Hamil onian
HS=
N
X
i=1 p2
i
2+ ex (
i, )+
N
X
i<j
W(
i−
j)
(D.16)
wi hou ec o po en ials. The ini ial s a e o he wa e unc ion
Ψ
a
= 0
is
Ψ0
. I
ollow he easoning o an Leeuwen [ L99] and conside he equa ions o mo ion o he
ensemble-a e aged densi y (see Eq. (5.13)) and o he ensemble-a e aged cu en densi y
∂ jk(
, ) = −n(
, )∂k ex (
, )−hψ( )|
3
X
i=1
∂iTik(
)|ψ( )i−hψ( )|Wk(
)|ψ( )i+GBk(
, ),
(D.17)
whe e he k- h componen o he cu en densi y modula ion ha is induced by he ba h is
desc ibed by
GBk(
, ) = 1
2h2S†jk(
, )S−S†Sjk(
, )−jk(
, )S†Si.
(D.18)
He e, I adop ed he no a ion o an Leeuwen [ L99] o he momen um-s ess enso
Tik(
)
and he g adien o he pa icle-pa icle in e ac ion
Wk(
)
. Mo eo e , I apply an Leeuwen's
physical assump ions on he shape o he ex e nal po en ial and i s analy ici y in ime [ L99].
By aking he ime de i a i e o Eq. (5.13) and he di e gence o Eq. (D.17) one ob ains
∂2
n(
, ) = ∇hn(
, )∇ ex (
, )i+q(
, )−∇GB(
, ) + ∂ FB(
, ),
(D.19)
whe e
q(
, ) = hψ( )|
3
X
i=1
3
X
k=1
∂i∂kTik(
) +
3
X
k=1
∂kWk(
)|ψ( )i.
(D.20)
To in es iga e he one- o-one co espondence be ween he ex e nal po en ial and he
ensemble-a e aged densi y, I pu some es ic ions on he unc ional dependence o
FB(
, )
and
GB(
, )
on
n(
, )
and
j(
, )
. I assume ha
FB(
, )
depends on
n(
, )
only and
GB(
, )
exhibi s he gene al o m
GB(
, ) = a[n(
, )]( )j(
, ) + b[n(
, )]( )j(
, 0) + c[n(
, )](
, ).
(D.21)
He e,
a
and
b
a e a bi a y scala unc ions ha may depend on
n(
, )
and
, whe eas
c
is
a gene al ec o ha may depend on
n(
, )
,
, and
. The ques ion whe he ba h ope a o s
ha ulll hese assump ions exis is discussed below. One ob ains
∂2
n(
, ) =∇hn(
, )∇ ex (
, )i+q(
, ) + ∂ FB(
, )
−a[n(
, )]( )∇j(
, )−b[n(
, )]( )∇j(
, 0)−∇c[n(
, )](
, )
(D.22)
by inse ing
GB(
, )
o Eq. (D.21) in o Eq. (D.19). Finally,
∇j(
, )
and
∇j(
, 0)
can be
subs i u ed using he con inui y equa ion (5.13) and one a i es a
∂2
n(
, ) =∇hn(
, )∇ ex (
, )i+q(
, ) + ∂ FB(
, )
+a[n(
, )]( )∂ n(
, )−a[n(
, )]( )FB(
, )
+b[n(
, )]( )∂ n(
, ) 0−b[n(
, )]( )FB(
, 0)−∇c[n(
, )](
, ).
(D.23)

92
APPENDIX D. FÖRSTER-TYPE POTENTIALS AND STOCHASTIC
TIME-DEPENDENT DENSITY FUNCTIONAL THEORY
Equa ion (D.23) di ec ly ela es he ex e nal po en ial o he ensemble-a e aged densi y
wi hou need o he cu en densi y.
Based on an equa ion simila o Eq. (D.23), an Leeuwen shows wi h his o iginal p oo
ha he densi y o a closed quan um sys em can be ep oduced by an auxilia y sys em wi h
Hamil onian
H0
S=
N
X
i=1 p2
i
2+ 0
ex (
i, )+
N
X
i<j
W0(
i−
j)
(D.24)
which con ains a die en pa icle-pa icle in e ac ion
W0
and ex e nal po en ial
0
ex
. The
auxilia y sys em s a s om he ini ial s a e
Φ0
and he po en ial
0
ex
anishes a inni y
as
ex
does. The an Leeuwen p oo elies on some ini ial and bounda y condi ions o he
densi y and cu en densi y: The densi y o bo h sys ems needs o be equal, i.e.,
hΦ0|n(
)|Φ0i=hΨ0|n(
)|Ψ0i.
(D.25)
Bo h sys ems need o s a om he same ime de i a i e o he densi y, i.e.,
∂ n0(
, ) =
∂ n(
, )
a
= 0
, and one ob ains ia he con inui y equa ion
hΦ0|∇j(
)|Φ0i=hΨ0|∇j(
)|Ψ0i.
(D.26)
To apply he an Leeuwen cons uc ion scheme o he ex e nal po en ial o he auxilia y
sys em in he open quan um sys em case wi h ba h ope a o
S0
, I adop equi alen ini ial
and bounda y condi ions o he co esponding ensemble-a e aged p ope ies. Howe e , due
o he p esence o
FB(
, )
and
GB(
, )
, some u he equi emen s on he ac ion o he ba h
ope a o need o be in oduced in he auxilia y sys em. I assume ha he modula ions o he
densi y equa ion o mo ion ha a e induced by he ba h need o be bo h densi y dependen
only and iden ical a
= 0
, i.e.,
FB(
, 0) = F0
B(
, 0)
. Mo eo e ,
G0
B(
, )
needs o ha e he
same s uc u e as
GB(
, )
, so ha
G0
B(
, ) = a0[n0(
, )]( )j0(
, ) + b0[n0(
, )]( )j0(
, 0) + c0[n0(
, )](
, ).
(D.27)
He e,
a0
,
b0
, and
c0
co espond o he quan i ies o he o iginal sys em in e ms o hei
dependence on
n0(
, )
,
, and
. These equi emen s imply ha one needs o nd a ba h
ope a o
S0
in he auxilia y sys em ha leads o a densi y-only dependen
F0
B(
, )
wi h
ini ial condi ion
FB(
, 0) = F0
B(
, 0)
, a linea cu en densi y dependence o
G0
B(
, )
, and
densi y dependen coecien s
a0
,
b0
, and
c0
. Thus, he auxilia y sys em ollows he equa ion
∂2
n0(
, ) =∇hn0(
, )∇ 0
ex (
, )i+q0(
, ) + ∂ F0
B(
, )
+a0[n0(
, )]( )∂ n0(
, )−a0[n0(
, )]( )F0
B(
, )
+b0[n0(
, )]( )∂ n0(
, ) 0−b0[n0(
, )]( )F0
B(
, 0)−∇c0[n0(
, )](
, )
(D.28)
which is simila o Eq. (D.23) wi h all sys em specic p ope ies being p imed.
I one assumes ha bo h sys ems ha e he same ensemble-a e aged densi y, i.e.,
n(
, ) =
n0(
, )
, one ob ains by sub ac ing Eqs. (D.23) and (D.28)
∇hn(
, )∇ω(
, )i=ζ(
, ),
(D.29)
D.3. STOCHASTIC TIME-DEPENDENT DENSITY FUNCTIONAL THEORY
93
whe e
ω(
, ) = ex (
, )− 0
ex (
, )
and
ζ(
, ) = q0(
, )−q(
, ) + ∂ F0
B(
, )−FB(
, )
+ha0[n(
, )]( )−a[n(
, )]( )i∂ n(
, )−a0[n(
, )]( )F0
B(
, )
+a[n(
, )]( )FB(
, ) + hb0[n(
, )]( )−b[n(
, )]( )i∂ n(
, ) 0
−b0[n(
, )]( )F0
B(
, 0) + b[n(
, )]( )FB(
, 0)
−∇c0[n(
, )](
, ) + ∇c[n(
, )](
, ).
(D.30)
The  s wo e ms o Eq. (D.30) equal he con ibu ions o he an Leeuwen cons uc ion,
whe eas all o he e ms come om he inuence o he ba h and depend only on he ensemble-
a e aged densi y. No e ha no explici ly cu en -densi y-dependen e ms occu as I aimed
a when se ing up he assump ions o he s uc u e o
FB
and
GB
. As Eq. (D.29) is o
S u m-Liou ille ype, a unique solu ion o
ω(
, )
exis s i
n(
, )
and
ζ(
, )
a e known
[ L99]. F om he e on, he idea is o ollow he a ionale o Re . [ L99]: Sol e Eq. (D.29) a
= 0
and consecu i ely compu e i s ime de i a i es o cons uc
0
ex (
, )
om i s Taylo
se ies o de by o de in ime wi hin he con e gence adius o he Taylo expansion. No e
ha he po en ial
0
ex (
, )+C( )
p oduces he same densi y as he sys em is no sensi i e o
a pu ely TD shi
C( )
o he po en ial. In he an Leeuwen cons uc ion scheme [ L99], he
choice o he bounda y condi ion o
ω(
, )
a inni y xes a pa icula gauge o he po en ial
0
ex (
, )
, hus de e mines he a bi a y cons an
C( )
.
In summa y, he jus p esen ed a ionale indica es: I one es ic s he ange o allowed
ba h ope a o s and uses easonable ini ial and bounda y condi ions he ensemble-a e aged
densi y
n(
, )
ob ained om an open quan um sys em wi h Hamil onian
HS
, ba h ope a o
S
, and ini ial s a e
Ψ0
can be ep oduced by an auxilia y open quan um sys em wi h di -
e en pa icle-pa icle in e ac ion
W0
and ini ial s a e
Φ0
. The ex e nal po en ial
0
ex (
, )
 de e mined up o a pu ely TD unc ion
C( )
 is uniquely dened as long as he ba h
ope a o
S0
is chosen adequa ely. Thus, wi hin he assump ions discussed abo e, a one- o-
one co espondence be ween he ex e nal po en ial and
n(
, )
exis s, and he app oach can
be used o open quan um KS sys ems wi h a TDDFT Hamil onian and a sui able ex e nal
po en ial.
The po en ial ob ained along his cons uc ion scheme may depend s ongly on he choice
o he ba h ope a o s
S
and
S0
. The e o e, I conside one special case o he p e ious
assump ions whe e he densi y dependence o he ac ion o he ba h ope a o on he equa ion
o mo ion o he ensemble-a e aged densi y and cu en densi y is he same in he o iginal
and he auxilia y sys em, i.e.,
FB(
, ) = F0
B(
, )
,
a=a0
,
b=b0
, and
c=c0
. In his case
Eq. (D.30) educes o
ζ(
, ) = q0(
, )−q(
, ),
(D.31)
hus equals he e m o an Leeuwen's closed quan um sys em p oo , ye wi h ensemble-
a e aged quan i ies. He e, he po en ial
0
ex (
, )
is de e mined by
ζ(
, )
only, as bo h ba h
ope a o s ha e by cons uc ion he same inuence on he equa ions o mo ion o
n(
, )
and
j(
, )
in he o iginal and he p imed sys em.
Finally, I discuss he p ac ical alue o he jus p esen ed s a egy. To his end, I conside
a ba h ope a o in he o iginal SSE ha is mo i a ed in analogy o he single-pa icle ba h
94
APPENDIX D. FÖRSTER-TYPE POTENTIALS AND STOCHASTIC
TIME-DEPENDENT DENSITY FUNCTIONAL THEORY
ope a o o Sec. 5.3.2: The ba h ope a o
S=√γ|Ψ( 0)ihΨ( )|
(D.32)
induces elaxa ion o he exci ed sys em back o i s g ound s a e wi h he decay a e
γ
.
Inse ing
S
in o he o mula o
FB(
, )
yields
FB(
, ) = γhhΨ( )|Ψ( )ihΨ( )|Ψ( )in(
, 0)−hΨ( )|Ψ( )ihΨ( )|n(
)|Ψ( )ii.
(D.33)
The wa e unc ions ha a e solu ions o he SSE a e no malized only in he ensemble a e age
up o ou h o de in he coupling pa ame e
λ
, i.e., no e e y ensemble membe i sel is
no malized. The e o e, ocusing on he second e m o Eq. (D.33), one canno simply spli
he s a is ical a e age o his e m in wo ac o s because in gene al
hΨ( )|Ψ( )ihΨ( )|n(
)|Ψ( )i 6=hΨ( )|Ψ( )i hΨ( )|n(
)|Ψ( )i= (1 + O(λ4))hΨ( )|n(
)|Ψ( )i.
(D.34)
The same p oblem occu s also when se ing up
F0
B(
, )
using he KS e e ence sys em o-
ge he wi h he ba h ope a o
S0=√γ|Φ( 0)ihΦ( )|
(D.35)
ha is supposed o model a ba h mechanism ha is equi alen o
S
in he s ochas ic TD
KS equa ion (5.12) using he s ochas ic TDDFT KS Hamil onian
H0
KS({
k}, ) =
N
X
i=1 −∇2
i
2+ 0
H(
i, ) + 0
xc(
i, ) + ex (
i, ),
(D.36)
whe e
0
xc
needs o be compu ed om
0
H
,
ex
, and he co esponding
0
ex
.
Fo he connec ion be ween he ba h ope a o s
S
and
S0
and he idea o p oo p esen ed
abo e, i emains o be a gued ha
FB(
, )
and
F0
B(
, )
a e pu e unc ionals o he ensemble-
a e aged densi y, i.e., one needs o es ablish a ela ion like
hΨ( )|Ψ( )ihΨ( )|n(
)|Ψ( )i= [n(
, )] hΨ( )|n(
)|Ψ( )i
(D.37)
and co esponding ela ions o he KS sys em, whe e
is some unique unc ional o
n(
, )
.
An analogous p oblem appea s also when one in es iga es he dependence o
GB(
, )
and
G0
B(
, )
on he ensemble-a e aged densi y and cu en densi y. Rela ions such as Eq. (D.37),
simila ela ions o s a is ical co ela ions ha eme ge in
GB(
, )
, and he co esponding
con ibu ions in he KS sys em emain o be in es iga ed.
F om a p ac ical poin o iew, one could consul he quan um-jump (qj) algo i hm
[DCM92, GPZ92, BP95, BP06] in o de o ge a g ip on he p oblem wi h he s a is ical co -
ela ions. The piecewise de e minis ic e olu ion pe o med in he quan um-jump algo i hm
p ese es he no m o each ensemble membe
Ψqj
. Thus, he p oblem wi h he s a is ical
co ela ions o he abo e discussion is a oided as
hΨqj( )|Ψqj( )i= 1
o each ensemble mem-
be .
FB(
, )
,
F0
B(
, )
,
GB(
, )
, and
G0
B(
, )
compu ed wi h
S
and
S0
would hen exhibi
he dependencies on
n(
, )
and
j(
, )
equi ed o he idea o p oo .
Fo sol ing he open sys em KS equa ion (5.12) based on he s ochas ic TDDFT Hamil-
onian o Eq. (D.36) wi h he quan um-jump algo i hm, one needs o p opaga e he no m-
p ese ing equa ion
i∂ Φ = H0
KSΦ−i
2S0†S0Φ + i
2||S0Φ||2Φ
(D.38)
D.3. STOCHASTIC TIME-DEPENDENT DENSITY FUNCTIONAL THEORY
95
and simul aneously also he auxilia y equa ion
i∂ Φaux =H0
KSΦaux −i
2S0†S0Φaux,
(D.39)
whe e he no m o he auxilia y sys em decays. The scheme o Re . [WT79] o ans e hese
equa ions o wo se s o
N
single-pa icle equa ions applies when one uses
S0
o Eq. (D.35).
One can demons a e ha he esul ing equa ions equal he equa ions o he single-pa icle
quan um-jump algo i hm o Sec. 5.2.2 i one akes he single-pa icle ba h ope a o
s0
i=√γ|ϕi( 0)ihϕi( )|.
(D.40)
In summa y, i is possible o nd a single-pa icle e sion o he quan um-jump algo i hm
acco ding o he scheme o Sec. 5.2.2 o simula ing he open quan um sys em KS equa ion
(5.12) when using he TDDFT Hamil onian o Eq. (D.36) and he ba h ope a o
S0
o Eq.
(D.35). No e ha he dipole-dependen ac o o he single-pa icle ba h ope a o o Eq.
(5.19) can be included s aigh o wa dly o his a ionale by in oducing a ime-dependen
damping ac o
˜γ( ) = γ|dk( )−dk( 0)|2/D2
o
s0
i
ins ead o
γ
ha akes he pu ely densi y-
dependen dipole momen in o accoun .
D.3.2 Ba h ope a o s in he single-pa icle KS amewo k and ela ed
ea u es
The choice o he ba h ope a o is a he hea o he applicabili y o he open quan um
sys em TDDFT scheme. On he one hand, he ba h ope a o models he unde lying physics
o he ba h mechanism, hus needs o be mo i a ed by physical p ocesses. On he o he hand,
i is impo an o he heo e ical ounda ion o he open quan um sys em TDDFT amewo k
and he exis ence o a single-pa icle scheme. He e, I in oduce ou heu is ically mo i a ed
ba h ope a o s. All ou ba h ope a o s a e designed o he single-pa icle amewo k o
KS TDDFT. They in ol e p ojec ion ope a o s and induce elaxa ion o he en i e exci ed
sys em back o i s g ound s a e ia p ojec ion o he occupied g ound-s a e KS o bi als. The
a ionale behind hese ope a o s and hei implemen a ion is explained in he ollowing.
The cen al idea behind he  s ba h ope a o is o in e p e he die ences be ween
eigen alues o occupied and unoccupied KS o bi als as exci a ion ene gies o he sys em. In
his sense, one ob ains, e.g., an exci ed s a e by eplacing one o he occupied o bi als o he
GS sys em by one o he highe lying unoccupied KS o bi als. Then, elaxa ion o he g ound
s a e may be pe o med i one de ec s he o e lap o he ime e olu ion o each o bi al o
his sys em wi h he space o all (o a leas he mos impo an ) unoccupied o bi als and
p ojec s back o he g ound s a e. To model his beha io wi h single-pa icle ope a o s,
he  s ba h ope a o
s(1)
i
uses a la ge basis o
M
occupied and unoccupied KS o bi als,
compu es he o e lap o all TD KS o bi als o hese basis unc ions, and p ojec s on o he
co esponding GS o bi al depending on he magni ude o he o e lap [PDV08, ADV11]. This
ba h ope a o eads
s(1)
i=√γ
M
X
j=i+1 |ϕi( 0)ihϕj( 0)|,
(D.41)
whe e I use all KS eigens a es wi h an eigen alue ha is ene ge ically abo e he eigen alue o
he e e ence o bi al
ϕi( 0)
and p ojec back o he la e o bi al. The ac o
√γ
includes he
102
APPENDIX E. PARSEC MISCELLANEOUS
he KS Hamil onian and ozen o bi als ha emain a hei GS alues need o be labeled by
0. Fi s nume ical es s o o bi al eezing on small molecules showed ha as soon as one o
he o bi als was ozen, he dipole signal die ed no ably om he ue dipole momen ime
e olu ion. Ye , his idea was ne e checked o sizable molecules whe e po en ially some o
he lowe lying o bi als con ibu e less o he TD dipole signal.
Finally, i one likes o use s a ic elec ic elds in dipole app oxima ion du ing ime p op-
aga ion, his may be achie ed by choosing 
Lase _shape
= ampnola in he PARSEC inpu
as implemen ed by Anne Klimach. In his case, he po en ial o he ex e nal eld may g ow
o unphysically la ge numbe s a he ou e pa o he eal-space g id. The e o e, in some
cases i may appea easonable o se he ex e nal po en ial cons an ou side a ce ain ol-
ume a ound he sys em. To his end, I implemen ed he physical PARSEC inpu pa ame e
Cons _eld_ adius
. I eads a dis ance o he o igin o he g id ha de e mines a bounda y
ou side which he po en ial is se cons an o he alue o he po en ial on his bounda y.
No e ha his ea u e wo ks only i he pola iza ion o he ex e nal eld is chosen o poin
exac ly in o he di ec ion o he coo dina e sys em. This op ion is u ned o (de aul ) when
Cons _eld_ adius
is se o be ze o.
E.5 Adap a ion o he diagonaliza ion ole ance du ing he
g ound-s a e p ocedu e
Mos o he compu a ion ime o he PARSEC g ound-s a e i e a ions goes in o he diago-
naliza ion o he Hamil onian ma ix. Compa ison o he pe o mances o PARSEC and he
oc opus code [MCBR03, CAO
+
06] in collabo a ion wi h Heiko Appel e ealed ha pa ic-
ula ly much ime ge s los in PARSEC when diagonaliza ion is pe o med du ing he  s
i e a ions o he sel -consis ency p ocedu e. The e is one s iking die ence be ween PAR-
SEC and oc opus: To my unde s anding, he oc opus code limi s he maximum numbe
o ma ix imes ec o mul iplica ions in he diagonaliza ion ou ine
2
. Thus, wi h a ea-
sonable choice o his limi , i accep s a educ ion o he accu acy o he diagonaliza ion
du ing he  s i e a ion s eps whe e ypically a la ge numbe o ma ix imes ec o mul-
iplica ions a e needed o high diagonaliza ion accu acy. PARSEC pe o ms high accu acy
diagonaliza ion om he  s i e a ion s ep on. Ye , du ing he  s s eps o he KS sel -
consis ency p ocedu e, he KS Hamil onian may no ably de ia e om he nal sel -consis en
KS Hamil onian. Hence, i appea s wo hless compu ing he o bi als al eady wi h he nal
diagonaliza ion accu acy as hey come om a Hamil onian ha is no con e ged.
The e o e, I implemen ed an adap a ion scheme ha adjus s he diagonaliza ion ole ance
o he sel -consis ency esidual e o measu ed ia he in eg al o e changes o he cha ge
weigh ed po en ial. F om he  s i e a ion un il he SRE d ops below
105
imes he p ese
con e gence h eshold
c
, a diagonaliza ion sol e ole ance
o
1×10−3
is used. Then, he
diagonaliza ion ole ance is educed owa ds he nal ole ance
in ou s eps
i
when he
SRE d ops below
105−ic
acco ding o
= 10−3+ i
4(3+log10( )).
(E.7)
2
The in o ma ion abou he limi a ion o ma ix imes ec o mul iplica ions in he oc opus code came
up in discussions wi h Heiko Appel.

E.6. GROUND-STATE MISCELLANEOUS
103
This choice o he SRE and he sol e - ole ance h esholds elies on p ac ical expe ience and
yields easonable pe o mance enhancemen , while o mos sys ems he nal esul s a e no
inuence by adap i e sol e se ings as long as sucien ly low con e gence pa ame e s a e
gua an eed in he end. The e o e, using his scheme he de ec ion o he con e gence o
he sel -consis ency p ocedu e is allowed only when he sol e ole ance has d opped o i s
nal alue
. This diagonaliza ion ole ance adap a ion can be ac i a ed by he boolean ag
Adap i e_diag_ ole ance
. Ye , i is deac i a ed au oma ically when he (G)OEP o GKLI
a e used, 
poissonsol e
= 1, o Fö s e - ype po en ial e alua ion is applied, because in
hese cases expe ience shows ha high accu acy o he diagonaliza ion is p e e able om he
s a .
E.6 G ound-s a e miscellaneous
The las sec ion o his appendix con ains a lis o ex ensions o he GS PARSEC code ha
a e no co e ed by he p e ious appendices.
In he o iginal e sion, PARSEC mo es all nuclei o he sys em o cen e he ou e mos
nuclei wi h espec o he planes o he coo dina e sys em. Ye , o some applica ions his
cen e ing p ocedu e o he nuclei posi ions is no desi ed. The e o e, I added he boolean ag
Cen e ing_a om_posi ion
o con ol he cen e ing op ion. The posi ion cen e ing is applied,
when he ag is se o i s de aul alue  ue and ice e sa.
To in es iga e all con ibu ions o he OEP po en ial and/o i s KLI app oxima ion
sepa a ely, I implemen ed he op ion
OEP_po en ial_ou pu
(
boolean
). When his ag is
 ue, he OEP o KLI po en ial is spli in o Sla e and all con ibu ions beyond Sla e .
These a e w i en o sepa a e ou pu les along all coo dina e axis, on he planes o he
coo dina e sys em, and as a cube le on he h ee dimensional g id.
Las bu no leas , a s a ic elec ic eld in dipole app oxima ion may be applied du ing
GS calcula ions using he boolean ag
Apply_elec ic_eld
. In he cu en implemen a ion,
his eld applies only in
x
-di ec ion o he coo dina e sys em. The eld s eng h may be
dened by he physical PARSEC inpu pa ame e
Elec ic_Field
.
Acknowledgmen
I am g a e ul o a lo o people who accompanied me on my way o his hesis. In g a i ude o
he encou agemen , suppo , and inspi a ion ha I ha e ecei e, I would like o acknowledge
in pa icula ...
... my supe iso S ephan Kümmel who guided my wo k wi h his sugges ions and ad ice,
by sc u inizing my ndings, and by sha ing his deep knowledge and unde s anding o physics
in gene al and especially densi y unc ional heo y wi h me. His en husiasm o science has
always been a g ea inspi a ion.
... he membe s and alumni o he Kümmel-g oup: Anne Klimach, Linn Leppe , Rod igo
Albuque que, Richa d A mien o, Ma hias Dau h, Sebas ian F ank, And eas Ka olewski,
Ma kus Knoll, Thomas Kö zdö e , Philipp Schahause , Tobias Schmid , Ma k Thiele, and
Sebas ian Wüs ne . I app ecia e g ea physics and p i a e discussions wi h my oce ma es,
c i ical ques ions and insigh du ing ou g oup semina s, and he iendly a mosphe e in
gene al. My hanks go o Linn, And eas, Ma hias, and Tobias o hei p oo eading and
help ul commen s on his manusc ip .
... all iends and ellows o he Eli e S udy P og am Mac omolecula Science and he
G aduie enkolleg 1640 Pho ophysics o Syn he ic and Biological Mul ich omopho ic Sys-
ems. Sha ing science wi h you in many ocial and p i a e occasions has always been a
pleasu e.
... my scien ic pa ne s and hos s: Massimiliano Di Ven a o insigh ul discussions
on open quan um sys ems and s ochas ic Sch ödinge equa ions in he con ex o TDDFT
as well as his hospi ali y in San Diego; Heiko Appel,  s o insigh and suppo on he
p ojec o exci a ion-ene gy ans e and s ochas ic TDDFT, second o many handy linux
hin s, and hi d o he g ea ime in San Diego; O Cohen and Leeo K onik o help ul
commen s and sha ing hei expe ience on mul ig id me hods; Simon and Pe e Klüp el
o sha ing hei knowledge abou op imiza ion me hods o de e mining ene gy-minimizing
uni a y ans o ma ions; Miguel A. L. Ma ques o commen s on TD p opaga ion schemes.
... Monika Bi kelbach, Claudia Masuch, Ma kus Hil , and Be nha d Winkle o helping
wi h adminis a i e ma e s and sol ing all compu e issues.
... all spo s ellows who sha e my en husiasm o Taekwondo and who ha e accompanied
me o many yea s. I ha e always enjoyed p ac icing spo s wi h you and elaxing my mind
in he gymn.
... my amily o p o iding all he oppo uni ies and challenges ha I enjoyed, o lo s o
suppo and ad ice, and o he us ha I expe ienced.
... Ka ina Mees o he e lo e, o he unde s anding o he many ups and downs du ing
his hesis, o he suppo and ad ice in all ma e s o li e, o he us , and o sha ing
he e li e wi h me.
105
Lis o publica ions and manusc ip s
1.
D. Ho mann
, T. Kö zdö e , and S. Kümmel,
Phys. Re . A
82
, 012509 (2010):
Ene gy ans e and Fö s e 's dipole coupling app oxima ion in es iga ed in a eal- ime
Kohn-Sham scheme
2.
D. Ho mann
, T. Kö zdö e , and S. Kümmel,
Phys. Re . Le .
108
, 146401 (2012):
Kohn-Sham Sel -In e ac ion Co ec ion in Real Time
3.
D. Ho mann
, S. Klüp el, P. Klüp el, and S. Kümmel,
Phys. Re . A
85
, 062514 (2012):
Using complex deg ees o eedom in he Kohn-Sham sel -in e ac ion co ec ion
4.
D. Ho mann
and S. Kümmel,
J. Chem. Phys.
137
, 064117 (2012):
Sel -in e ac ion co ec ion in a eal- ime Kohn-Sham scheme: access o dicul exci-
a ions in ime-dependen densi y unc ional heo y
5.
D. Ho mann
and S. Kümmel,
Phys. Re . Le ., submi ed (2012):
In ege pa icle p e e ence du ing cha ge ans e in Kohn-Sham heo y
107

Lis o abb e ia ions, unc ionals, and
me hods
A accep o
ALDA adiaba ic local densi y app oxima ion
a.u. a omic uni s
B ba h
B3LYP Becke h ee-pa ame e Lee-Yang-Pa hyb id unc ional
BLYP semiempi ical GGA unc ional ha combines Becke88 exchange
and Lee, Yang, and Pa co ela ion
CASPT2 comple e ac i e space pe u ba ion heo y o second o de
CC Coupled-Clus e
CCSD(T) Coupled-Clus e (singles, doubles and pe u ba i e iples)
CEDA common ene gy denomina o app oxima ion
CG conjuga e g adien
CT cha ge- ans e
D dono
DFT densi y unc ional heo y
e
elemen a y elec ic cha ge
EA elec on ani y
EET exci a ion-ene gy ans e
EXX exac exchange
FOBO Fos e -Boys
FOBO-SIC SIC unc ional implemen ed ia he GKLI app oxima ion using
FOBO ans o ma ions
s em osecond
GGA gene alized g adien app oxima ion
GKLI gene alized K iege , Li, and Ia a e app oxima ion
GKLI-SIC SIC unc ional implemen ed ia he GKLI app oxima ion using
ene gy-minimizing ans o ma ions
GKS gene alized Kohn-Sham app oach
GOEP gene alized op imized eec i e po en ial
GOEP-SIC SIC unc ional implemen ed ia he GOEP app oxima ion using
ene gy-minimizing ans o ma ions
GS g ound-s a e
GSIC gene alized sel -in e ac ion co ec ion
GSLA gene alized Sla e app oxima ion
109
110
LIST OF ABBREVIATIONS, FUNCTIONALS, AND METHODS
GSLA-SIC SIC unc ional implemen ed ia he GSLA app oxima ion using
ene gy-minimizing ans o ma ions
HF Ha ee-Fock
HK Hohenbe g-Kohn
HOMO, H highes occupied molecula o bi al
Hxc Ha ee-exchange-co ela ion
IP ioniza ion po en ial
KLI K iege , Li, and Ia a e app oxima ion
KLI-EXX EXX unc ional implemen ed ia he KLI app oxima ion
KLI-SIC SIC unc ional implemen ed ia he KLI app oxima ion
KS Kohn-Sham
LDA local densi y app oxima ion
LH ligh -ha es ing
LHF localized Ha ee-Fock
LHF-CEDA localized Ha ee-Fock - common ene gy denomina o app oxima ion
loc localized
LSDA local spin-densi y app oxima ion
LUMO, L lowes unoccupied molecula o bi al
LYP Lee, Yang, and Pa co ela ion unc ional
MG mul ig id
MPI message passing in e ace
MP2 Mølle -Plesse pe u ba ion heo y o second o de
MP4 Mølle -Plesse pe u ba ion heo y o ou h o de
OEP op imized eec i e po en ial
OEP-SIC SIC unc ional implemen ed ia he OEP app oxima ion
PARSEC pseudopo en ial algo i hm o eal-space elec onic s uc u e calcula ions
PBE Pe dew, Bu ke, and E nze ho unc ional app oxima ion
PZ Pe dew-Zunge
qj quan um-jump
RT eal- ime
TD ime-dependen
TDCDFT ime-dependen cu en densi y unc ional heo y
TDDFT ime-dependen densi y unc ional heo y
TDGKLI gene alized K iege , Li, and Ia a e app oxima ion in TDDFT
TDGKLI-SIC SIC unc ional implemen ed ia he TDGKLI app oxima ion using
complex- alued ene gy-minimizing ans o ma ions
TDGSLA gene alized Sla e app oxima ion in TDDFT
TDGSLA-SIC SIC unc ional implemen ed ia he TDGSLA app oxima ion using
complex- alued ene gy-minimizing ans o ma ions
TDGSIC gene alized sel -in e ac ion co ec ion in TDDFT
TDFOBO-SIC SIC unc ional implemen ed ia he TDGKLI app oxima ion using
spa ially localizing ans o ma ions
TDKLI K iege , Li, and Ia a e app oxima ion in TDDFT
TDL(S)DA adiaba ic local (spin) densi y app oxima ion in TDDFT
TDOEP ime-dependen op imized eec i e po en ial
TDSLA Sla e app oxima ion in TDDFT
LIST OF ABBREVIATIONS, FUNCTIONALS, AND METHODS
111
S sys em
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SIC sel -in e ac ion co ec ion
SIE sel -in e ac ion e o
sl semilocal
SLA Sla e app oxima ion
SRE sel -consis ency esidual e o
SSE s ochas ic Sch ödinge equa ion
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xc exchange-co ela ion
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Pub1
Publica ion 1
Ene gy ans e and Fö s e 's dipole coupling
app oxima ion in es iga ed in a eal- ime
Kohn-Sham scheme
D. Ho mann, T. Kö zdö e , and S. Kümmel
Physikalisches Ins i u , Uni e si ä Bay eu h, D-95440 Bay eu h, Ge many
Physical Re iew A
82
, 012509 (2010)
c

2010 The Ame ican Physical Socie y
DOI: 10.1103/PhysRe A.82.012509
a ailable a :
h p://link.aps.o g/doi/10.1103/PhysRe A.82.012509
ABSTRACT
We p esen a scheme o in es iga e ene gy ans e by eal- ime p opaga ion o he Kohn-
Sham equa ions. The scheme's pu pose is o check and go beyond he dipole coupling
app oxima ion unde lying a Fö s e - ype ene gy ans e , and o ob ain in o ma ion abou
he coupling on he g ounds o he densi y- unc ional heo y. We obse e de ia ions om
he dipole coupling app oxima ion o small molecules.
PHYSICAL REVIEW A 82, 012509 (2010)
Ene gy ans e and F¨
o s e ’s dipole coupling app oxima ion in es iga ed
in a eal- ime Kohn-Sham scheme
D. Ho mann, T. K¨
o zd¨
o e , and S. K¨
ummel
Physikalisches Ins i u , Uni e si ¨
a Bay eu h, D-95440 Bay eu h, Ge many
(Recei ed 19 Ap il 2010; published 23 July 2010)
We p esen a scheme o in es iga e ene gy ans e by eal- ime p opaga ion o he Kohn-Sham equa ions.
The scheme’s pu pose is o check and go beyond he dipole coupling app oxima ion unde lying a F¨
o s e - ype
ene gy ans e , and o ob ain in o ma ion abou he coupling on he g ounds o he densi y- unc ional heo y. We
obse e de ia ions om he dipole coupling app oxima ion o small molecules.
DOI: 10.1103/PhysRe A.82.012509 PACS numbe (s): 31.15.ee, 31.70.Hq, 82.20.Rp, 82.20.W
I. ENERGY TRANSFER AND TIME-DEPENDENT
DENSITY FUNCTIONAL THEORY
Ene gy ans e is one o he mos undamen al p ocesses on
he molecula scale, go e ning ligh -ha es ing in biological
sys ems [1,2] and ene gy con e sion in elec onic de ices such
as o ganic sola cells [3–5] o ligh -emi ing diodes [6]. The
design p inciples o na u al ligh -ha es ing complexes [7–9]
ha e ound conside able in e es , as hopes a e high ha he
p inciples ealized in na u e can be mimicked in he design o
a i icial o ganic de ices [3,10,11].
One s anda d me hod o in e p e expe imen al da a o
exci a ion ene gy ans e be ween a dono (D) and an accep o
(A) molecule sepa a ed by he dis ance Ris he so-called
F¨
o s e esonance ene gy ans e (FRET) heo y [9–21].
F¨
o s e heo y desc ibes he non adia i e ene gy ans e
media ed by a (quan um-mechanical) coupling be ween he
ansi ion dipoles o he dono and accep o molecules [13–15].
One o he cen al assump ions in FRET is ha he coupling
be ween D and A can be desc ibed by a (poin )-dipole-dipole
in e ac ion, alling as 1/R3. Fu he mo e, FRET heo y is
o mula ed o he weak coupling egime (i.e., he isola ed D
and A exci ed s a es do no change signi ican ly on coupling).
Based on hese assump ions F¨
o s e de i ed an exp ession o
he ene gy ans e a e showing a cha ac e is ic R−6dis ance
dependence, wi h he nuclea ib a ions being subsumed in o
a spec al o e lap ac o be ween dono emission and accep o
abso p ion spec a. The esul ing, a he simple exp ession o
he ene gy ans e a e, see Sec. II o de ails, allows o
de e mining he in e molecula dis ance Rby spec oscopy o
he coupled D-A sys em, and D as well as A indi idually.
Thus, FRET has gained emendous impo ance as i
es ablishes a spec oscopic ule on he nanoscale [22,23].
Typically i is applied in a ange o dis ances om abou 10
o 100 ˚
A[24,25]. Howe e , in ecen yea s he applicabili y
o he dipole coupling app oxima ion unde lying FRET has
been ques ioned in many applica ions [25–33] as equen ly
he in e molecula dis ance Ris compa able o he molecules’
ex ension, o he D and A molecules a e connec ed by b idging
uni s.
Theo e ical insigh in o he alidi y o he dipole coupling
app oxima ion is, he e o e, o g ea impo ance. As he
molecules o p ac ical ele ance o en con ain many elec-
ons, ( ime-dependen ) densi y unc ional heo y (TD)DFT
appea s as a na u al choice o s udy he p oblem on a
i s -p inciples scale a bea able compu a ional cos . Recen ly,
TDDFT has been applied o F¨
o s e - ype exci a ion ene gy
ans e ques ions [34–41] in he Casida- ype linea esponse
o malism [42]. As a complemen a y app oach o he Casida
o malism, eal- ime implemen a ions [43–52]o TDDFTa e
inding inc easing a en ion due o hei accu acy and a o able
scaling, which allows one o apply hem o la ge sys ems
[53–55]. In addi ion, hey do no equi e compu a ion o he
exchange-co ela ion ke nel, which can be ad an ageous when
ad anced unc ionals ha explici ly employ he o bi als a e
used [47,49,56–58].
In he ollowing we p esen a eal- ime TDDFT scheme
o in es iga ing ene gy ans e and he dipole coupling
app oxima ion. A e sho ly e iewing he pe inen concep s
o FRET heo y in Sec. II, we discuss in Sec. III how he dipole
coupling can be inco po a ed in o he eal- ime me hodology.
This leads o a e y gene al scheme o quali a i ely checking
he alidi y o he dipole coupling app oxima ion, as demon-
s a ed in Sec. IV. Unde ce ain condi ions ha a e explained
in Sec. Vone can also de e mine he coupling ma ix elemen
quan i a i ely. We s ess ha in all ins ances we delibe a ely
do no use he Kohn-Sham Sla e de e minan as an app ox-
ima ion o he ue wa e unc ion, s aying uly on TDDFT
g ounds.
II. THE DIPOLE COUPLING APPROXIMATION
In he ollowing we b ie ly e iew he aspec s o F¨
o s e
heo y ha a e c ucial o he u he conside a ions. S a ing
wi h Fe mi’s Golden Rule, he ene gy ans e a e kET can be
w i en as [15]
kET =2π|V|2∞
0
d J(),(1)
whe e J() is he spec al o e lap be ween he no malized
dono emission and accep o abso p ion spec a. Ha ee
a omic uni s a e used h oughou . Vis he elec onic coupling
ma ix elemen
V=DA∗|ˆ
VC|D∗A,(2)
whe e he Coulomb in e ac ion ˆ
VCmedia es be ween he
ini ial and inal wa e unc ions. F¨
o s e heo y is based on
he assump ion ha he wa e unc ion o he o al sys em can
be sepa a ed in o D and A pa s due o negligible elec onic
coupling be ween D and A. Ini ially, he accep o is in i s
g ound s a e deno ed by |Aand he dono is in an exci ed
1050-2947/2010/82(1)/012509(9) 012509-1 ©2010 The Ame ican Physical Socie y
D. HOFMANN, T. K ¨
ORZD ¨
ORFER, AND S. K ¨
UMMEL PHYSICAL REVIEW A 82, 012509 (2010)
s a e |D∗. The inal wa e unc ion co esponds o he in e se
si ua ion. Thus, he s a es in Eq. (2) a e w i en as
DA∗|= 1
√2(
DA∗|−
A∗D|),
(3)
|D∗A= 1
√2(|
D∗A−|
AD∗),
whe e |
DA∗abb e ia es he p oduc s a e |D|A∗, espec-
i ely. Acco dingly, he coupling ma ix elemen spli s in o
he Coulomb con ibu ion (i.e., Dand D∗ha ing he same
coo dina es)
VC=
DA∗|ˆ
VC|
D∗A(4)
and he so-called exchange con ibu ion
Vx=
A∗D|ˆ
VC|
D∗A(5)
(i.e., A∗and D∗ha ing he same coo dina es) [14,59,60]. As
he o e lap be ween s a es alls exponen ially wi h inc easing
R,Vxis neglec ed in FRET and only he Coulomb e m
con ibu es o he F¨
o s e a e-o -exci a ion ene gy ans e .
(The exchange con ibu ion gi es he so-called Dex e - ype
ene gy ans e [59].)
Unde he assump ion ha he ex ension o he D and
A molecules is small compa ed o R
| A|
|R|1 and | D|
|R|1,(6)
(see Fig. 1 o he no a ion), he Coulomb in e ac ion be ween
he elec ons o D and A is expanded in a mul ipole se ies.
Expanding up o powe s R−3, he in e ac ion is w i en as

j,k
1
R+ A
k− D
j
=
j,k 1
|R|− A
k− D
jR
|R|3+3
2 A
k− D
jR2
|R|5
− A
k− D
j2
2|R|3+···,(7)
whe e he ksum uns o e all elec ons o he accep o , and
he jsum uns o e all elec ons in he dono . In oducing he
o ien a ion ac o
κDA =ˆ
eDˆ
eA−3(ˆ
eDR)(ˆ
eAR),(8)
wi h he ec o s ˆ
ei= i
| i|, he ansi ion dipole momen s
µD=D|
j
D
j|D∗
(9)
µA=A|
k
A
k|A∗,
and assuming o hogonali y o D and A s a es, one inds he
coupling ma ix elemen o FRET heo y
VFRET =κDA |µA||µD|
|R|3.(10)
VFRET ea u es he cha ac e is ic R−3dependence o F¨
o s e
heo y. Toge he wi h Eq. (1) his yields he a e o he F¨
o s e -
ype exci a ion ene gy ans e .
III. A DIPOLE COUPLING SCHEME IN THE
TDDFT CONTEXT
The p e iously p esen ed concep elies on using he s a es
|Aand |D. Fo he ypical molecules o in e es in ol ing
ens o hund eds o elec ons, he compu a ional cos o
calcula ing hese many-pa icle s a es wi h ab ini io wa e
unc ion me hods is p ohibi i e. (TD)DFT allows one o
de e mine he elec onic s uc u e o sys ems o ha size, ye
again, he many-pa icle s a es a e no accessible: E en i he
ul ima e exchange-co ela ion unc ional we e known, he e
is no eason o belie e ha gene ally he Kohn-Sham Sla e
de e minan will be close o he ue co ela ed wa e unc ion.
The e o e, a TDDFT scheme ha is in ended o in es iga e
he dipole coupling app oxima ion ha is inhe en o F¨
o s e
heo y mus be based solely on he a iable ha is eliable in
(TD)DFT, namely he densi y. One way o how his can be
achie ed is p esen ed in he ollowing.
The eal- ime o malism o TDDFT ha we wan o apply
is based on he ime-dependen Kohn-Sham (KS) equa ions
[43,45,48,50]
i∂
∂ ϕj( , )=hKS( , )ϕj( , ),(11)
whe e hKS is he ime-dependen KS Hamil onian gi en by
hKS( , )=−∇2
2+ H( , )+ xc( , )+ ex ( , ).(12)
The po en ial ex ( , ) ep esen s all ex e nal con ibu ions
(e.g., nuclei and a lase ield). The elec on in e ac ion is aken
in o accoun ia he Ha ee po en ial
H[n]( , )=n( , )
| − |d3 (13)
and he exchange-co ela ion (xc) po en ial xc( , ). Whe eas
H[n]( , ) is known explici ly as he unc ional de i a i e o
he classical Ha ee ene gy
EH[n]=1
2n( , )n( , )
| − |d3 d3 ,(14)
he xc po en ial has o be app oxima ed as i s exac o m is
unknown.
The eal- ime KS equa ions lend hemsel es e y na u ally
o a gene al and s aigh o wa d scheme o checking he
dipole coupling app oxima ion ha is a he hea o FRET:
One can explici ly implemen he mul ipole expansion o
F¨
o s e in o he TDDFT equa ions and compa e he o bi al
p opaga ion wi h he mul ipole expansion o ano he one
wi hou (i.e., wi h he usual ull KS Hamil onian). The de ails
o his s a egy a e explained in he ollowing.
In acco dance wi h he assump ion o an app eciable spa ial
sepa a ion o he D and A sys ems ha is unde lying FRET
heo y, we di ide he ull densi y in o a D and an A pa . Fo
he sake o being as explici in ou no a ion as possible, we
assume in all o he ollowing ha he D and A densi ies
012509-2
ENERGY TRANSFER AND F ¨
ORSTER’S DIPOLE COUPLING ... PHYSICAL REVIEW A 82, 012509 (2010)
((()))
()()()
FIG. 1. Dono and accep o a e sepa a ed by he dis ance R=|R|
and can hus be localized in he hal -spaces wi h nega i e and posi i e
xcoo dina e, o he sake o an explici no a ion. Each sys em
is cha ac e ized by i s mul ipole momen s Ni,diand Qi
jk,whe e
i={D,A}labels he sys em, espec i ely. Dis ances wi h espec o
he cen e o D and A, espec i ely, a e deno ed by Dand A.
a e localized in he hal spaces o nega i e and posi i e
xcoo dina es, espec i ely, as shown in Fig. 1
nD( )=(−x)n( ),(15)
nA( )=(x)n( ),
whe e (x) is he Hea iside s ep unc ion.
As e iewed p e iously, F¨
o s e ’s concep is based on using
he exac s a es o he sepa a e molecules, bu only aking he
classical (Ha ee) in e ac ion be ween he ansi ion dipoles
o D and A in o accoun o he in e molecula coupling.
Consequen ly, a TDDFT analog o F¨
o s e ’s app oach should
ake all elec on in e ac ion e ec s in o accoun [i.e., use
Hand he (unknown) exac xc] wi hin each molecule, bu
use only he Ha ee po en ial o he coupling be ween D
and A, and he la e only o low o de in a mul ipole
expansion.
In he ollowing we desc ibe how his expansion can be
inco po a ed in o a TDDFT scheme in p ac ice, and we s a
by no ing ha he e a e wo possible pa hs o implemen he
dipole coupling app oxima ion o he Ha ee coupling. The
i s is o s a wi h he Ha ee ene gy o Eq. (14), pe o m
he mul ipole expansion, and hen ake he unc ional de i a i e
o ob ain he po en ial. The second is o s a om he Ha ee
po en ial i sel and pe o m he dipole app oxima ion on he
le el o Eq. (13).
Following he i s ou e, he Ha ee ene gy spli s in o h ee
componen s
EH[n]=EH[nD]+EH[nA]+nA( )nD( )
| − |d3 d3 ,(16)
whe e he i s wo con ibu ions sepa a e in D and A, whe eas
he hi d e m con ains he in e ac ion be ween he D and
A sys ems. The expansion (7)isusedin he hi d e mon
he igh -hand side ( hs) o Eq. (16) o ob ain he Ha ee
ene gy Edd
Hin he dipole coupling app oxima ion. Calcula ing
he unc ional de i a i e o Eq. (16) we use he unc ional
chain ule o ake he sepa a ion in D and A densi ies in o
accoun
H[n]( )=
i=D,AδEdd
H[nD,nA]
δni( )
δni( )
δn( )d3 .(17)
Consequen ly, he po en ial in he dipole coupling app ox-
ima ion (supe sc ip dd) as de i ed om he ene gy ( hus
supe sc ip index E) can be w i en as
ddE
H[nD,nA]= H[nD]+ ddE,D
H[nA](−x)
+ H[nA]+ ddE,A
H[nD](x),(18)
whe e H[nD(A)] is he Ha ee po en ial o D(A) and
ddE,D(A)
H[nA(D)] is he po en ial esul ing om he hi d e m o
Eq. (16) in he D(A) hal -space. Wi h he help o he mul ipole
momen s o he D(A) densi y,
Ni=ni( i)d3 i,di= ini( i)d3 i,
(19)
Qi
jk =ni( i)3 i
j i
k− i2δjkd3 i,
whe e i=D,A, he F¨
o s e po en ial ha he dono densi y
gene a es in he hal space o A (c . Fig. 1)is
ddE,A
H[nD]( A)=ND
|R|−(ND A−dD)R
|R|3+ AdD
|R|3
+ND
2
3

j,k=13 A
j A
k− A2δjkRjRk
|R|5
+
3

j,k=1
QD
jkRjRk
2|R|5−3(R A)(RdD)
|R|5.(20)
The co esponding po en ial ddE,D
H[nA]( D) is ob ained om
Eq. (20) by in e changing he D and A supe sc ip s and
eplacing Rby −R.
As an al e na i e o his de i a ion we can ake he second
ou e. To his end we s a di ec ly a he le el o he po en ial
and w i e he Ha ee po en ial on he A side as
A
H[nD,nA]=nA( )
| A− |d3 +nD( D)
| A+R− D|d3 D.
(21)
This exp ession leads o ano he na u al way o ansla ing
F¨
o s e ’s Ha ee dipole coupling concep in o a TDDFT
scheme: The con ibu ion om he A densi y o he Ha ee
po en ial in A’s own hal space is aken in o accoun ully,
whe eas he con ibu ion om he D densi y o he Ha ee
po en ial in A’s hal space [ he second e m on he hs o
Eq. (21)] is expanded in analogy o Eq. (7). The la e leads o
he po en ial
dd ,A
H[nD]=ND
| A+R|+dD( A+R)
| A+R|3+
3

j,k=1
QD
jkRjRk
2| A+R|5,
(22)
012509-3

D. HOFMANN, T. K ¨
ORZD ¨
ORFER, AND S. K ¨
UMMEL PHYSICAL REVIEW A 82, 012509 (2010)
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
-40 -30 -20 -10 0 10 20 30 40
Ha ee po en ial [Ry]
x coodina e [boh ]
FIG. 2. (Colo online) Compa ison o he usual Ha ee po en ial
H[n] (black solid line) wi h he Ha ee po en ials dd
H[nD,nA] ( ed
do ed line) and ddE
H[nD,nA] (blue dashed line). The po en ials a e
plo ed along he axis ha uns h ough he cen e s o he wo sodium
dime s, while he posi ion o he molecules is −10 and +10 boh .
on he A side (whe e he supe sc ip index indica es ha
we app oxima ed he po en ial di ec ly, no he ene gy as in
ou e 1). The co esponding po en ial on he D side is
dd ,D
H[nA]=NA
| D−R|+dA( D−R)
| D−R|3+
3

j,k=1
QA
jkRjRk
2| D−R|5.
(23)
Using hese exp essions ins ead o ddE,A
Hand ddE,D
Hin
Eq. (18) de ines he po en ial dd
H.
Le us pause o a momen o compa e he wo ou es. The
i s one, s a ing om he ene gy, appea s mo e na u al i one
has in mind F¨
o s e ’s o iginal wo k, which was based on he
ene gy. Howe e , he second one, di ec ly using he po en ial,
seems o be mo e na u al in he con ex o KS (TD)DFT in
which he po en ial is eadily a ailable. Howe e , bo h a e
alid ansla ions o F¨
o s e ’s concep in o a eal- ime TDDFT
con ex , and i is hus no clea ap io iwhich one should
be p e e ed. To ge a be e unde s anding o he di e ences
be ween ddE
Hand dd
Hwe plo hem in Fig. 2 o a anspa en
example, a sys em o wo sodium dime s [61] aligned in
pa allel and sepa a ed by R=20 a.u..
While dd
His a he simila o he ull Ha ee po en ial,
he po en ial ddE
His conside ably oo low in he cen al
egion be ween he dime s and ises unphysically a away
om he dime s. This beha io can be aced back o he
ac ha , in calcula ing ddE
H, we always di ide by powe s
o he ixed R[see denomina o s o Eq. (20)] while he
nume a o s ise wi h g owing dis ance o he co esponding
molecule. In con as , in po en ial dd
H he dis ance en e s
in he denomina o . The e o e, dd
Hdecays wi h he g owing
dis ance o he co esponding dime . As his is he beha io
one na u ally expec s om a po en ial, we used dd
Hin ou
calcula ions [62].
IV. REAL-TIME INVESTIGATION OF THE DIPOLE
COUPLING APPROXIMATION
Ou ocus so a was on he concep ual wo k o ansla ing
F¨
o s e ’s dipole coupling idea in o he (TD)DFT con ex . Fo
using he scheme in p ac ical calcula ions, we ha e o add ess
he ques ion o which in luences he app oxima ion ha is used
o he desc ip ion o he xc e ec s will ha e.
The exci a ion spec um o wo iden ical molecules a a
sepa a ion la ge enough o he indi idual molecula densi ies
o no o e lap can quali a i ely be desc ibed as being simila o
he spec um o a single molecule, bu wi h possibly an ene -
ge ic spli ing o he monome exci a ions (Da ydo spli ing,
see Sec. V), and wi h addi ional cha ge- ans e exci a ions
om one monome o he o he ha ca y p ac ically ze o
oscilla o s eng h.
I is a well known p oblem o many commonly used densi y
unc ionals, in pa icula (semi)local ones, ha hey se iously
unde es ima e he ene gy o cha ge- ans e exci a ions. In he
KS amewo k, his ailu e is closely ela ed o he absence
o s ep-like s uc u es in he xc po en ial ha esul om
discon inui ies [63]inTDDFT,seeRe .[64] o a de ailed
discussion. I has been shown ha xc app oxima ions using
a la ge ac ion o exac exchange [39], o ange-sepa a ed
hyb id unc ionals [65], o sel -in e ac ion co ec ions [66]
can desc ibe cha ge ans e well. Using a unc ional ha
accu a ely desc ibes cha ge ans e is manda o y o an
accu a e desc ip ion o wo monome s a close dis ance [67].
In he p opaga ion se up he mo emen o cha ge is eco ded
in eal- ime and his can be used o di ec ly moni o o cha ge
ans e om one molecule o he o he . This buil -in wa ning
allows one o check agains lea ing he “la ge sepa a ion”
si ua ion which was desc ibed in Sec. II and which is he ocus
o in e es he e [68]. A la ge sepa a ion, he ene gy ans e
is comple ely domina ed by he Ha ee coupling. I emains
manda o y in any case ha he xc unc ional app oxima ion one
uses desc ibes he exci a ion spec um o he single molecules
wi h easonable accu acy, as o he wise he ime-dependen
densi y and ansi ion dipoles will be inco ec .
S a ing om wo molecules a a la ge sepa a ion Rand
dec easing he dis ance, ou aim is o check when he dipole
coupling app oxima ion b eaks down. Ou calcula ions a e
pe o med wi h ou eal- ime TDDFT ex ension [51,52]o
he eal-space code PARSEC [69]. The wo molecules a e
placed in wo di e en hal -spaces o he eal-space g id. All
in es iga ions a e pe o med in he so-called supe molecula
app oach [70], whe e we conside D and A as a combined
sys em. To dis inguish be ween a coupling o exci ed s a es
ha ul ills he assump ions leading o expansion (7) and a
coupling ha does no , we compa e he ime-dependen dipole
momen s o D and A ob ained by a DFT plus TDDFT un wi h
he ull Ha ee po en ial, o he ones ob ained in a second un
using he po en ial dd
H. The dipole momen s a e calcula ed
using he D and A cen e s o densi y in he DFT g ound s a e
as e e ence poin s.
Ou p oceeding is as ollows: (1) de e mine he spec um
o one single molecule, (2) choose he exci a ion o in e es ,
(3) pe o m a eal- ime TDDFT calcula ion using he ull
Ha ee po en ial wi h a sho lase pulse as he ini ial
exci a ion, (4) epea s ep (3) wi h dd
H, (5) compa e he
012509-4
ENERGY TRANSFER AND F ¨
ORSTER’S DIPOLE COUPLING ... PHYSICAL REVIEW A 82, 012509 (2010)
ime-dependen dipole momen s ob ained in s eps (3) and (4).
The lase pulse is added as an ex e nal po en ial du ing he i s
s eps o he eal- ime p opaga ion only wi hin he hal -space
o D. The equency, leng h, and shape o he pulse a e uned
o exci e he sys em only in a na ow equency band a ound
he exci a ion chosen in s ep (2). (See Appendix A o de ails.)
In he ollowing we demons a e he use o his app oach
in calcula ions o dime s o Na2and C7H6O, espec i ely.
These sys ems a e anspa en enough o allow o a clea
explana ion and demons a ion o ou concep and hey a e
es ablished e e ence sys ems [36,71]. We employ he ime-
dependen local densi y app oxima ion (TDLDA) as i well
desc ibes he exci a ions o Na2and C7H6O[72,73] ha a e o
in e es in ou s udy. The possible unde es ima ion o cha ge-
ans e exci a ions ha was discussed a he beginning o his
sec ion is no o conce n he e as we a e speci ically in e es ed in
he majo exci a ions ca ying dipole oscilla o s eng h. These
a e desc ibed well by TDLDA o he sys ems ha we s udy.
We i s in es iga e he coupling be ween wo sodium
dime s wi h pa allel alignmen o he wo Na2axes as a
unc ion o he dis ance Rbe ween he wo Na2molecules.
As he exci a ion o in e es we choose he one ha is a
2.1eVin heNa
2spec um [71,72]. A ypical ime e olu ion o
he accep o dipole momen is shown in Fig. 3. A he chosen
dis ance, R=15 boh , he e a e ob ious de ia ions be ween
he ime-dependen dipole momen ha is ob ained using he
ull Ha ee po en ial and he one ha is ob ained using dd
H.
The de ia ions a e a mani es a ion o he b eakdown o
he dipole coupling app oxima ion a 15 boh . A sys ema ic
analysis o he de ia ions in he dipole momen a a ange o
dis ances om 45 o 15 boh is depic ed in Fig. 4. Quali a i ely,
weseeinFig.4 h ee egimes: Fo e y la ge dis ances (45
and 35 boh ) he di e ences anish and one is clea ly in he
dipole egime. A smalle dis ances (a ound 25 boh in he
FIG. 3. (Colo online) Accep o dipole momen along he
Na2molecula axis, eco ded as a unc ion o ime, o a sys em
o wo Na2wi h a sepa a ion o R=15 boh . The ime e olu ion was
calcula ed using he ull Ha ee po en ial (solid, ed) and he po en ial
dd
H(do ed, black). The coupling be ween D and A mani es s i sel
in he bea equency ωbea o an oscilla ion be ween D and A
(c . Sec. V). This bea equency can be ex ac ed om he A dipole
momen ia ωbea =2π/Tbea .
-0.001
-0.0008
-0.0006
-0.0004
-0.0002
0
0.0002
0.0004
0.0006
0.0008
0 5 10 15 20 25 30
di e ence o accep o dipole momen s [a.u.]
ime [ s]
FIG. 4. (Colo online) Di e ence be ween he accep o dipole
momen s ob ained in a calcula ion using he ull Ha ee po en ial
and ano he one using dd
H o an Na2dime , o di e en alues o
R: 15 boh (do dashed, ed), 25 boh (dashed, blue), 35 boh (do ed,
g een) and 45 boh (solid, black).
p esen example) one s a s o obse e de ia ions, and o ye
smalle dis ances (15 boh ) he dipole momen s di e e y
signi ican ly du ing hei ime e olu ion—al hough he densi y
o e lap is s ill small. He e he dipole app oxima ion clea ly
ails and highe mul ipoles a e ele an .
The same e ec s a e seen in he second example in which
we examine he coupling be ween wo C7H6O molecules ha
a e aligned in pa allel as depic ed in Fig. 5. We again ocus on
he lowes exci a ion wi h app eciable oscilla o s eng h. The
dipole momen ime e olu ion was calcula ed o a ange o
dis ances om R=40 boh o R=12 boh , again using he
wo di e en po en ials as discussed p e iously. Cha ac e is ic
esul s (dis ances o 24 o 12 boh ) a e shown in Fig. 6.The
di e ences a R=12 boh and R=16 boh signi ican ly
exceed he di e ences a he la ge sepa a ions. The e o e,
one can a gue quali a i ely ha up o 16 boh sepa a ion
one canno speak o a F¨
o s e - ype coupling. F om 20 boh
on, he di e ences in he dipole momen ime e olu ion a e
no ably smalle and he F¨
o s e - ype dipole app oxima ion can
be conside ed alid [74].
As ou analysis has so a ocused on he o al dipole
momen s, i is a he gene al and he concep can
FIG. 5. (Colo online) Sys em o wo C7H6O molecules, whe e
he dis ance Ris a ied om 40 o 12 boh .
012509-5
D. HOFMANN, T. K ¨
ORZD ¨
ORFER, AND S. K ¨
UMMEL PHYSICAL REVIEW A 82, 012509 (2010)
-6e-05
-4e-05
-2e-05
0
2e-05
4e-05
6e-05
0 5 10 15 20 25 30
di e ence o accep o dipole momen s [a.u.]
ime [ s]
FIG. 6. (Colo online) Di e ence o accep o dipole momen in
a sys em o wo C7H6O be ween a calcula ion using he ull Ha ee
po en ial and ano he one wi h he po en ial dd
H. The molecules
a e sepa a ed by 12 boh (do dashed, ed), 16 boh (dashed, blue),
20 boh (do ed, g een), and 24 boh (solid, black).
s aigh o wa dly be ex ended, e.g. o unequal dime s. Com-
pa ing cha ac e is ic ea u es o he dipole momen e olu ion
ob ained wi h ull coupling and wi h dipole coupling, one can
assess he us ange o he dipole app oxima ion.
I he exci a ions o in e es ha e he special ea u e o
being a he well sepa a ed om all o he exci a ions so ha
one can hink abou he coupling as a coupling in a wo-
le el sys em, hen one can also de e mine he coupling ma ix
elemen quan i a i ely om he TDDFT calcula ion—despi e
he ac ha one does no explici ly know he s a es in TDDFT.
This is he opic o he nex sec ion.
V. EXTRACTING THE COUPLING MATRIX ELEMENT
FROM A REAL-TIME PROPAGATION
A coupling ma ix elemen o ype (2) en e s he ene gy
ans e a e (1), and i is a close-lying ques ion whe he and
how i can be e alua ed. In p inciple, one could calcula e he
ini ial and inal s a es and e alua e Eq. (2) di ec ly. Howe e ,
as discussed p e iously, he s a es a e ha d o compu e om
i s p inciples and he e is no igo ous eason o iden i y he
KS Sla e de e minan wi h he ue wa e unc ion. The e o e,
i one wan s o s ay on he sa e o mal g ounds o (TD)DFT
by no using he KS Sla e de e minan as an app oxima ion o
he ue wa e unc ion, one has o hink abou al e na i e ways
o de e mine he coupling ma ix elemen .
The Da ydo spli ing [75–77] is such an al e na i e
[34–39,41]. In he case o wo equal molecules, he Da y-
do spli ing  equals he ene gy spli ing E o he
(nea ly) degene a e exci a ion ene gies o he monome s in he
supe molecule [34,36,37,39]. In he discussed si ua ion  is
p opo ional o he coupling ma ix elemen (2)
V=
2.(24)
This obse a ion can be exploi ed in TDDFT by calcula ing
he spec um [50,78–80] and deducing he coupling ma ix
elemen om he ene gy spli ing. In he eal- ime app oach
o TDDFT spec a a e usually de e mined by ini ially exci ing
he sys em wi h a momen um boos [45,81], p opaga ing he
ime-dependen KS equa ions [82] using he dipole momen
as obse able, and calcula ing he dipole spec um ia a
Fou ie ans o ma ion [81]. As he ene gy spli ing is ypically
much smalle han he exci a ion ene gies hemsel es, high
esolu ion and hus compa ably long p opaga ion imes a e
needed o esol e he spli ing.
The e o e, we conside an al e na i e way o ex ac ing he
ele an in o ma ion by obse ing he D and A dipole momen s
sepa a ely. I conside ably educes he compu a ional load. We
no e ha he Da ydo spli ing  and consequen ly also he
coupling ma ix elemen Vmani es s i sel as a equency
ωbea in he ime-dependen D (and A) dipole momen . This
equency can be ex ac ed as a bea equency ωbea o an
oscilla ion be ween D and A. (See Appendix B o de ails.)
F om i one ob ains
V=ωbea .(25)
Apa om he bea he dipole momen o he sys em oscilla es
only a equencies close o he lase equency ha was used
o he exci a ion. Hence he bea equency can be de e mined
easily om he D and A dipole momen s (see Fig. 3). The
essen ial obse a ion now is ha o doing so, he dipole
momen ime e olu ion only has o be eco ded long enough
o cap u e one hal cycle o he bea . This is a much sho e
ime han he ime needed o ob ain an accu a e spec um om
he Fou ie ans o ma ion. Thus, he coupling s eng h can be
de e mined wi h mode a e simula ion imes.
We used his app oach o de e mine he coupling o he
lowes exci a ion o he wo sodium dime s. Figu e 7shows
he hus ob ained dis ance dependence o he coupling ma ix
elemen . One obse es signi ican di e ences om he dipole
0.001
0.01
0.1
10 15 20 25 30 35 40
coupling ma ix elemen [eV]
dime sepa a ion [boh ]
FIG. 7. (Colo online) Coupling ma ix elemen be ween he
exci ed s a es a 2.1eVo woNa
2 e sus he dime sepa a ion.
Red c osses show he esul s om ou eal- ime and eal-space
implemen a ion. As a guide o he eye we plo ed a do ed line
wi h slope −3 which co esponds o a F¨
o s e - ype coupling. As
a con i ma ion o ou implemen a ion we also epo esul s ha we
ob ained using linea esponse TDDFT om a comme cial p og am
package [83–85] (black c osses).
012509-6
ENERGY TRANSFER AND F ¨
ORSTER’S DIPOLE COUPLING ... PHYSICAL REVIEW A 82, 012509 (2010)
coupling beha io o dis ances below 20 boh . Abo e 25 boh
he coupling alls wi h he expec ed 1/R−3dependence.
As a es o ou eal- ime and eal-space implemen a ion
we ollowed Re . [39] and compu ed linea esponse TDDFT
spec a o a couple o dime dis ances wi h he TURBOMOLE
package [83–85] using he local densi y app oxima ion and
Fully Op imized Con ac ed Gaussian Basis Se s o T iple
Ze a Valence Quali y (de 2-TZVP) [86]. The coupling ma ix
elemen s (see Fig. 7) a e calcula ed acco ding o Eq. (24)
exploi ing he Da ydo spli ing. The ag eemen be ween he
wo me hods is e y good and he obse ed small di e ences
can be a ibu ed o echnical di e ences such as he use o
pseudopo en ials and basis se s e sus eal space.
Fo pu ing ou indings in o pe spec i e one should
compa e he dis ance o 25 boh o he bond leng h o he
sodium dime , which is 5.78 boh . The a io o he ex ension o
he molecula sys ems o hei sepa a ion indica es ha one has
o be ca e ul in elying on he dipole coupling app oxima ion
o ex ended sys ems ha a e no e y a apa .
VI. CONCLUSION
We ha e p esen ed a quali a i e and a quan i a i e scheme
o in es iga ing he coupling beha io o wo molecules
wi hin he eal- ime app oach o TDDFT. The app oach allows
one o dis inguish be ween a F¨
o s e - ype dipole coupling and
a non-dipole coupling on e y gene al g ounds. I he coupled
exci a ions ha a e o in e es can be conside ed as a wo-le el
sys em (i.e., la ge ene ge ic sepa a ion om o he exci a ions),
hen he coupling ma ix elemen can be de e mined e icien ly
in he eal- ime scheme by e alua ing he Da ydo spli ing
om a bea equency.
Al eady he small molecules in es iga ed in his manusc ip
show no able de ia ions om he F¨
o s e - ype dipole coupling
a mode a e dis ances. Fo la ge molecules such as ypically
used o spec oscopic labeling o in molecula elec onics,
he ques ion a which dis ances FRET can be elied on is
hus o g ea signi icance. Theo e ical ools such as he one
discussed he e can play an impo an ole, as hey enable us o
check whe he he FRET dipole coupling app oxima ion can
be applied o no .
ACKNOWLEDGMENTS
Financial suppo by he DFG and he Eli es udienp o-
g amm “Mac omolecula Science” is g a e ully acknowl-
edged.
APPENDIX A: EXCITATION WITH A LASER PULSE
He e we b ie ly explain how we exci e he sys em by a
lase pulse in he eal- ime in es iga ion o he dipole coupling
app oxima ion in Sec. IV. The lase ac s as a ime-dependen
ex e nal po en ial in dipole app oxima ion
EL( )=Een ( )sin(ωL ),(A1)
whe e Een ( ) is he pulse en elope. The equency ωLo
he pulse was chosen o equal he exci a ion ene gy we we e
in e es ed in. We used he en elope Een ( )=Emax sin2(π
TL)
as his o m leads o a Fou ie spec um in which he side
maxima a e much lowe han he main peak ( ypically he
heigh o he i s side maximum is less han 3% o he
heigh o he main peak). The leng h TLo he pulse is
uned by compa ing he Fou ie spec um o he pulse wi h
he exci a ion spec um o he molecule: TLwas chosen such
ha he i s side maximum o he pulse’s spec um lay
close o he pulse’s main equency han he neighbo ing
peak in he molecule’s exci a ion spec um. In all cases we
in es iga ed, he exci a ions o he single molecules we e
su icien ly sepa a ed om each o he o dominan ly exci e
jus one exci ed s a e o he sys em. This could be e i ied
om he ime-dependen dipole signal.
APPENDIX B: BEAT OSCILLATION IN THE
TIME-DEPENDENT ACCEPTOR DIPOLE MOMENT
In his Appendix we b ie ly explain why ωbea can be
ex ac ed om he dipole momen . We s a by no ing ha
Eq. (24) is de i ed in a wo-s a e model [35,36], whe e
|
D∗Aand |
DA∗a e a pai o esonan s a es. The ime
e olu ion
|( )=a1( )|
D∗A+a2( )|
DA∗,(B1)
o he wo-s a e sys em wi h ini ial s a e |(0)=|
D∗Ais
gi en by he coe icien s a1( ) and a2( )[87]
|a1( )|2=cos2(V ),(B2)
|a2( )|2=sin2(V ).
The coe icien s oscilla e wi h he bea equency ωbea ,see
Eq. (25). The co esponding ime-dependen dipole momen
dA( )=( )| A|( )on he A side can be calcula ed as
dA( )=|a1( )|2A| A|A+|a2( )|2A∗| A|A∗,(B3)
whe e we exploi ed he o hogonali y o |Dand |D∗.I he
s a ic dipole momen A| A|Ao A anishes, (B3) simpli ies
o
dA( )=|a2( )|2A∗| A|A∗.(B4)
The e o e, he esonance oscilla ion o he coe icien s can
be obse ed in he ime e olu ion o he dipole momen dA( ).
Bo h Eqs. (B3) and (B4) can be used o de e mine he coupling
ma ix elemen V ia he bea equency (25).
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