Computational study of statically confined electron systems by means of quantum chemical and machine learning techniques

DOCTORAL DISSERTATION
Compu a ional s udy o s a ically con ined elec on
sys ems by means o quan um chemical and machine
lea ning echniques/ Es akikoki Kon ina u ako
Elek oi Sis emen Az e ke a Konpu azionala Machine
Lea ning e a Kimika Kuan ikoko Me odoak E abiliz
XABIER TELLERIA ALLIKA
Supe ised by
D . Jon Ma in Ma xain Be aza and D . Jose Ma ia Me ce o La aza
OCTOBER 2022
Compu a ional s udy o s a ically con ined
elec on sys ems by means o quan um chemical
and machine lea ning echniques/ Es akikoki
Kon ina u ako Elek oi Sis emen Az e ke a
Konpu azionala Machine Lea ning e a Kimika
Kuan ikoko Me odoak E abiliz
Disse a ion p esen ed in
Submi ed in ul illmen o he equi emen s o he
D O C T O R A L P RO G R A M I N T H E O R E T I C A L
C H E M I S T RY A N D C O M P U TAT I O N A L
M O D E L L I N G
P esen ed by
X A B I E R T E L L E R I A A L L I K A
Supe ised by
D R . J O N M AT T I N M AT X A I N B E R A Z A
and
D R . J O S E M A R I A M E R C E R O L A R R A Z A
Euskal He iko Unibe si a ea/Uni e sidad del Pa´ıs Vasco & Donos ia In e na ional
Physics Cen e
In DONOSTIA,OCTOBER 2022
(cc) 2022 Xabie Telle ia Allika (cc by-nc-nd 4.0)
CONTENTS
Con en s iii
Lis o Figu es i
Lis o Tables ix
1 in oduc ion 1
1.1 The elec onic s uc u e p oblem . . . . . . . . . . . . . . . . . . . . 1
1.2 Impo ance o elec on co ela ion . . . . . . . . . . . . . . . . . . . 3
1.2.1 Wigne c ys als and molecules . . . . . . . . . . . . . . . . . 3
1.2.2 Quan um Do s and Hooke a oms . . . . . . . . . . . . . . . . 4
1.3 Machine Lea ning assis ance . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.1 Supe ised lea ning: neu al ne wo ks o p edic ion . . . . . . 6
1.3.2 Semisupe ised lea ning . . . . . . . . . . . . . . . . . . . . . 7
1.4 Scopeo hiswo k ............................ 9
2 quasi-one dimensional sys ems 11
2.1 In oduc ion................................ 12
2.2 Compu a ional p o ocol . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.1 Quan um mechanical model . . . . . . . . . . . . . . . . . . . 12
2.2.2 Dis ibu ed Gaussian o bi als . . . . . . . . . . . . . . . . . . 13
2.3 Resul s and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.1 Two elec on sys em . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.2 Sys ems o h ee and ou elec ons . . . . . . . . . . . . . . . 20
2.4 Concluding ema ks ........................... 23
3 quasi- wo dimensional sys ems 25
3.1 In oduc ion................................ 26
3.2 Compu a ional p o ocol . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2.1 Dis ibu ed Gaussian o bi als . . . . . . . . . . . . . . . . . . 27
3.2.2 Quan um mechanical compu a ions . . . . . . . . . . . . . . . 28
3.3 Resul s and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3.1 Onepa iclesys ems....................... 29
3.3.2 Few pa icle sys ems . . . . . . . . . . . . . . . . . . . . . . . 32
3.4 Concluding ema ks ........................... 35
4 sphe ical hooke a oms 37
4.1 In oduc ion................................ 38
4.2 Compu a ionalme hods ......................... 40
4.3 Resul s and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.3.1 Two-elec on sys ems . . . . . . . . . . . . . . . . . . . . . . . 42
4.3.2 Fou -elec on sys ems . . . . . . . . . . . . . . . . . . . . . . 44
4.3.3 Six-elec on sys ems . . . . . . . . . . . . . . . . . . . . . . . 45
iii

i con en s
4.3.4 Eigh -elec on sys ems . . . . . . . . . . . . . . . . . . . . . . 45
4.3.5 Ten-elec on sys ems . . . . . . . . . . . . . . . . . . . . . . . 46
4.3.6 Sc eened Hooke A om . . . . . . . . . . . . . . . . . . . . . . 47
4.3.7 Decomposing he ene gy in o di e en con ibu ions . . . . . 48
4.3.8 CoulombHoles.......................... 51
4.4 Concluding ema ks ........................... 53
5 ml assis ed phase diag ams 61
5.1 In oduc ion................................ 62
5.2 Compu a ional Me hods . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.2.1 The modynamic model o eu ec ic composi ion . . . . . . . . 63
5.2.2 Quan um chemical me hods . . . . . . . . . . . . . . . . . . . 64
5.2.3 Semi-supe ised lea ning echniques . . . . . . . . . . . . . . 66
5.2.4 Wo k p ocedu e and calcula ion se up . . . . . . . . . . . . . 67
5.3 Resul s and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.3.1 Solid-liquid phase diag ams wi h a single eu ec ic poin . . . 69
5.3.2 G ound s a e spin mul iplici y o Hooke a oms . . . . . . . . 71
5.3.3
De ec ing co alen bonding in sphe ically con ined
He2
sys ems
73
5.4 ConcludingRema ks........................... 76
6 gaussian con inemen and connec ion o hooke a oms 77
6.1 In oduc ion................................ 78
6.2 Compu a ional Me hods . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.2.1 One-body in eg als conce ning Gaussian con inemen s . . . . 79
6.3 Resul s and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.3.1
Deeply con ined one cen e sys ems wi h e en numbe o elec ons
82
6.3.2
Loosely con ined wo elec on sys ems wi h sc eened Coulomb
in e ac ion ............................ 84
6.4 ConcludingRema ks........................... 88
7 summa y, main conclusions and u u e wo ks 89
7.1 Mainconclusions............................. 90
7.2 Fu he wo ks............................... 91
7.2.1 Some p elimina y esul s conce ning endohed al sys ems . . . 92
8 euska azko labu pena 95
8.1 Sa e a................................... 95
8.1.1 Egi u a elek onikoa en p oblema . . . . . . . . . . . . . . . . 95
8.1.2 Ko elazio elek onikoa en ga an zia . . . . . . . . . . . . . . 97
8.1.3 Ikaske a au oma ikoa en lagun za . . . . . . . . . . . . . . . 99
8.2 Lanhonenhelbu ua ........................... 102
8.3 Egindako lanen labu penak . . . . . . . . . . . . . . . . . . . . . . . 103
8.3.1 Sasi dimen sio ba eko sis emak . . . . . . . . . . . . . . . . . 103
8.3.2 Sasi bi dimen sio sis emak . . . . . . . . . . . . . . . . . . . . 106
8.3.3 Hi u dimen sioko Hooke-n a omoa . . . . . . . . . . . . . . . 108
8.3.4 Makina ikaskun za bidez lo u iko ase diag amak . . . . . . 113
con en s
8.3.5 Po en zial gauss a ak . . . . . . . . . . . . . . . . . . . . . . 117
8.4 Emai zanagusiak............................. 120
8.4.1
Kon inamendu ha monikoa ike zeko p o okolo konpu azionalen
ga apena ............................. 120
8.4.2
Ziu gabe asun laginke en me odoen inplemen azioa ase dia-
g amaklo zeko ......................... 120
8.4.3
Kon inamendu gauss a ei dagozkien go pu z ba eko in eg alak
inplemen a u e a au e iaz lo u iko emai zak heda u . . . . 121
8.5 Ondo ionagusiak............................. 123
8.6 E o kizune akolanak........................... 124
bibliog aphy 127
LIST OF FIGURES
Figu e 2.1
No malised densi y p o iles o single (blue) and iple (o -
ange) spin s a es o se e al con inemen s eng hs
k
. No ice
in he weak con inemen limi , bo h p o iles a e alike and
hey di e in he s ong con inemen limi . . . . . . . . . . . 14
Figu e 2.2
Va ia ion o he loca ion pa ame e wi h espec o he con-
inemen pa ame e
k
o single and iple s a es, he o ange
squa e ep esen s he asymp o ic limi o loca ion index o
he iple s a e.......................... 15
Figu e 2.3
Pa icle-hole en opy o wo elec on sys em in high spin
s a e o se e al alues o con inemen pa ame e
k
. Maximal
en opy alue is ound o a alue a ound k=1×10−3.3 . . 16
Figu e 2.4
Ene gies o single and iple spin s a es using minimal
gaussian basis (le ) and ene gy di e ence be ween he sin-
gle and he iple s a es o he wo elec on sys em using
minimal basis ( igh ). The ze o ene gy gap happens a ound
a alue close o k=5×10−4. ................. 20
Figu e 2.5
No malised densi y p o iles o h ee (blue) and ou (o -
ange) elec on sys ems in high spin con igu a ion o se e al
con inemen s eng hs k. .................... 21
Figu e 2.6
Compu ed elec on-hole en opies o
n=
3, 4 numbe o elec-
ons wi h high spin s a e o se e al alues o he con inemen
pa ame e k. .......................... 22
Figu e 3.1
Geome y o he dis ibu ed gaussian unc ions acco ding o
g
numbe . All dis ances be ween closes neighbou s is gi en
by he pa ame e δ........................ 27
Figu e 3.2
Loga i hm o absolu e e o s (a) and s anda d de ia ions
(b) o he en i s s a es and he i s h ee degene a e
g oups wi h (
nx+ny) = {
1, 2, 3
}
. Same colo s a e ep esen
analy ically degene a e s a es. . . . . . . . . . . . . . . . . . 30
Figu e 3.3
Ob ained es ima ed ela i e e o s o es ima ed
δop
using
a ious NN models consis ing o wo hidden laye s wi h se e al
numbe o neu ons pe laye . . . . . . . . . . . . . . . . . . . 32
Figu e 3.4
Densi y di e ence be ween he CASSCF(2,8) and HF single
spin wa e- unc ions as a unc ion o he dis ance wi h espec
o he po en ial minimum o se e al alues o k. ...... 34
i
LIST OF FIGURES ii
Figu e 3.5
Densi y di e ence be ween he CASSCF(3,8) and HF qua e
spin wa e- unc ions as a unc ion o he dis ance wi h espec
o he po en ial minimum o se e al alues o k. ...... 34
Figu e 4.1
T iple -single ene gy gap, in eV, calcula ed a he CAS(
Ne
,
No b
)
(dashed line) and MRMP2(
Ne
,
No b
) (con inuous line) le els
o heo y wi h he ETBS-6S basis se , as a unc ion o he
numbe o o bi als
No b
included in he ac i e space. All cases
co espond o a CASSCF wa e unc ion in which all elec ons
a e included in he ac i e space, excep o he cu es in
magen a o he 10 elec on sys em, ha co espond o wa e-
unc ions in which he 1s o bi al occupa ion is se o 2, and
he e o e, he ac i e space is composed o 8 elec ons and
No b-2o bi als. ......................... 49
Figu e 4.2
A) Le Figu e: T iple -single ene gy gap, in eV, calcula ed
a he MRMP2(
Ne
,10) le el o heo y (2-, 4-, 6-, and 8-
elec on sys ems) and MRMP2(8,14) (10-elec on sys em),
as a unc ion o he numbe o elec ons and o di e en
alues o
λ
. B) Righ Figu e: T iple -single ene gy gap, in eV,
calcula ed a CAS(
Ne
,10) (dashed line) and MRMP2(
Ne
,10)
(con inous line) le els o heo y o 2,4,6 and 8 elec ons, and
CAS(8,14) (dashed line) and MRMP2(8,14) (con inuous line)
le els o heo y o he 10-elec on sys em, as a unc ion o
he deg ee o sc eening (
λ
). All calcula ions we e done wi h
he ETBS-6S basis se . . . . . . . . . . . . . . . . . . . . . . 49
Figu e 4.3
Coulomb holes calcula ed a he CASSCF(
Ne
,13)/ETBS-
6S le el o heo y o bo h ull (solid lines) and
λee =
1.0
sc eened-coulombic (dasshed lines) Hooke sys ems wi h di -
e en numbe o elec ons (
Ne
=2,4,6,8 and 10) and o sin-
gle (black lines) and iple s a es ( ed lines). . . . . . . . . 52
Figu e 5.1
Decision ee used o labelling each poin in he
(x1
,
T)
plane
using eu ec ic poin coo dina es and mel ing cu es. . . . . . 64
Figu e 5.2
Flowcha o he gene al p ocedu e o phase diag am con-
s uc ion.............................. 68
Figu e 5.3
Ag/Si solid-liquid phase diag am. G een is he he e ogeneous
solid phase
α+β
, da k blue is he homogeneous ideal liquid
mix u e
L
, ligh blue is
β+L
and pu ple is
α+L
; he whi e
do s ep esen he sampled poin s. . . . . . . . . . . . . . . . 69
Figu e 5.4
KNO
3
/LiNO
3
solid-liquid phase diag am. G een is he he -
e ogeneous solid phase
α+β
, da k blue is he homogeneous
ideal liquid mix u e
L
, ligh blue is
β+L
and pu ple is
α+L
;
he whi e do s ep esen he sampled poin s. . . . . . . . . . 70
4in oduc ion
a subs a e o Wigne c ys als seem o be bilaye ma e ials in which low densi ies
a e ob ained by means o eme ging moi ´e pa e ns [28–31].
In he con ex o non-pe iodic ini e sys ems, localized elec onic s a es a e also
called Wigne molecules and hey eme ge when he con inemen o he elec ons
is a he weak [32–34]. Thus, in o de o desc ibe Wigne molecules o , in gene al,
low densi y elec on sys ems, elec on co ela ion e ec s on bo h, ene gy and wa e-
unc ion, mus be ea ed by means o high enough compu a ional me hods. Highly
accu a e densi ies o sys ems in low-co ela ion egimes as well as in high-co ela ion
egimes ha e been ob ained in he li e a u e [35–38]. I simila sys ems con aining
la ge numbe o elec ons (
n >
2) a e o be conside ed, a iche a ie y o elec onic
s a es consis ing on se e al g ound s a e spin mul iplici ies and mul i-de e minan al
ea u es a ise [39–41]. Al hough such sys ems can be employed o unde s and many-
body in e ac ions, he compu a ional cos inc eases wi h he size o he sys em. In
spi e o hese di icul ies, hese model sys ems ha e been epea edly used in he
calib a ion o elec onic s uc u e me hods, o hey p o ide e y a iable dynamic
and nondynamic elec on co ela ion egimes [42–44] ha pose a g ea challenge
o cu en compu a ional me hods. [37, 45–57] Such calib a ion has been possible
because o he ecen a ailabili y o highly accu a e analy ical and benchma k
da a.[39, 40, 58–64]
1.2.2 Quan um Do s and Hooke a oms
Quan um Do s (QDs) a e egions o space whe e, due o a gi en ex e nal po en ial,
pa icles such as elec ons a e con ined. As hese egions sh ink, a c i ical size
is eached a which p ope ies o he mic oscopic wo ld - as ene gy quan isa ion,
elec onic and magne ic s uc u e - eme ge. F om a physical chemical poin o
iew, a molecule o a nanopa icle ha con ines i s alence elec ons can also be
aken as QDs. Nowadays, in he dawn o quan um echnologies, QDs o m a as
and in e es ing ield o s udy. These sys ems cons i u e one o he mos elemen al
componen s o nanoscale de ices which ha e a wide ange o applica ions such as:
sola ene gy ha es ing, op oelec onics and quan um compu ing de ices among
many o he s [65]. Ve y ecen wo ks show u he ields o applica ions o QDs as
he moelec ici y [66], ca alysis [67–70] and, mos ema kably, quan um compu a ion
[71–75].
One o he simples and mos adequa e models used in heo e ical s udies
conce ning QDs a e he so called Ha monium o Hooke’s a oms in which elec ons
a e con ined in a sphe ical ha monic po en ial [76] o which he Hamil onian is gi en
as in equa ion
(1.4)
. Such models con ain pa ame e s ha may be uned in o de
o ep esen ea u es co esponding o eal QDs [77, 78]. Fo ins ance, wo k ca ied
ou in ou g oup using a Hookean exac h ee-body model o examine elec on
co ela ion in a wo-elec on sphe ical quan um do con i med ha iple -single
ansi ions ake place as he ex e nally applied magne ic ield inc eases[79]. Howe e ,

1.3 machine lea ning assis ance 5
he limi a ion o using an exac model es ic ed ou s udy o wo-elec on sys ems.
Tha is, he analy ical solu ions[80] o speci ic cu a u e pa ame e o he wo-
elec on Hooke a oms (
ω2=1
4
,
1
100 . . .
) a e well known, which can lead in p inciple
o highly accu a e densi ies o sys ems in low-co ela ion egimes (
ω2→ ∞
) as well
as in high-co ela ion egimes (
ω2→
0) [35–38]. I simila sys ems con aining la ge
numbe o elec ons (
n >
2) a e o be conside ed, a iche a ie y o elec onic s a es
consis ing on se e al g ound s a e spin mul iplici ies and non-dynamical elec on
co ela ion (mul i-de e minan al ea u es) a ise [39–41]. Al hough such sys ems can
be employed o unde s and many-body in e ac ions, he compu a ional cos inc eases
wi h he size o he sys em.
H=−1
2
n
X
i=1∇2
i+ω2
2
n
X
i=1
2
i+
n
X
j>i=1
1
ij
(1.4)
As in Hooke model a oms, he inco po a ion o elec on co ela ion e ec s has
been shown o be essen ial o an adequa e in e p e a ion o he expe imen al
spec a and anspo p ope ies in Quan um Do s [81–86]. In quan um do s, as
opposed o eal a oms, he e ec o elec on co ela ion may be a ied a will
h ough manipula ion o he dimension and shape o he nanoc ys al as well as
o he s eng h, bounda ies and symme ies o he con ining ields[87, 88]. Besides,
he elec on-elec on in e ac ion can be sc eened due o la ice, he doping o he
cha ges induced on he me al ga es[89]. This ac makes he quan um do many-body
p oblem mo e complex han he mo e amilia a omic case.
Finally, Hooke model sys ems ha e been epea edly used in he calib a ion
o elec onic s uc u e me hods, o hey p o ide e y a iable dynamic and non-
dynamic elec on co ela ion egimes [42–44] ha pose a g ea challenge o cu en
compu a ional me hods. [37, 45–57]. Such calib a ion has been possible because o
he ecen a ailabili y o highly accu a e analy ical and benchma k da a [39, 40,
58–63].
1.3 MACHINE LEARNING ASSISTANCE
IBM de ines Machine Lea ning (ML) in he ollowing way ”Machine lea ning is a
b anch o A i icial In elligence (AI) and compu e science which ocuses on he use
o da a and algo i hms o imi a e he way ha humans lea n, g adually imp o ing
i s accu acy”. Acco ding o his de ini ion, any ML solu ion is based on wo pilla s:
he da a (which a e empi ically ob ained abs ac numbe s) and an algo i hm which
ope a es upon hese da a in o de o gain some kind o knowledge. In addi ion (as
i is implici ly s a ed), employing a gi en algo i hm, he quali y o such knowledge
imp o es as he numbe o da a inc eases.
6in oduc ion
Machine lea ning me hods a e ecen ly being employed in a ious ields o chem-
is y [90–93] such as: elec onic s uc u e heo y de elopmen [94–98], e icien new
phase disco e y and phase diag am building [99–102], pa e ns in chemical p ope ies
[91, 103–106], ob aining Po en ial Ene gy Su aces [107–111] and eac ion pa hs and
ou comes [112–114] among o he s. Taking his b ie compila ion o examples in o
accoun , ML me hods a e and will be impo an ools in chemical sciences.
F om a wide pe spec i e, he e a e wo ’ adi ional’ machine lea ning disciplines,
he so called
unsupe ised
and
supe ised
lea ning pa adigms. Gi en a se o da a
X={x1. . . xn}
he o me aims o ind in e es ing s uc u e in he da a such as
clus e ing and dimension educ ion by echniques such as k-means Clus e ing, and
P incipal Componen analysis. Meanwhile, he la e also conside s one mo e se
Y={y1. . . yn}
and i s goal is o lea n a mapping which connec s bo h se s
X
and
Y
by echniques such as Linea and Logis ic Reg ession, Random Fo es s and Vec o
Suppo ed Machines. Then, semi-supe ised lea ning is hal way be ween hese wo;
while s ill conside ing bo h
X
and
Y
se s, he numbe o elemen s in
X
is a la ge
han he numbe o elemen s in
Y
and hus any ob ained mapping unc ion u ns
ou o be inaccu a e [115–117].
In his wo k wo supe ised lea ning algo i hms ha e been employed, i.e. Random
Fo es s and Neu al Ne wo ks as well as one semisupe ised algo i hms: Unce ain y
Sampling.
1.3.1 Supe ised lea ning: neu al ne wo ks o p edic ion
As i has been s a ed in he p e ious sec ion, supe ised lea ning echniques aim
o build a machine
:X→Y
which is able o map a ea u e se
X
in o a label
se
Y
such ha (bea ing in mind he gene al ML de ini ion) i ge s be e a doing
so by inc easing he size o bo h se s. Gi en a labelled da a se
D
, a p ope ML
algo i hm is able o in e he equi ed unc ion
o m aining da a con ained in
he se
T ⊂ D
and an e o o such in e ed unc ion can be ob ained by using he
complemen a y se
E ⊂ D
,
T ∩E =∅
. Among he housands o echniques designed
o such pu pose, in his wo k deep Neu al Ne wo ks (NN) ha e been employed o
in e op imal basis se pa ame e s as i is shown in chap e 2.
In he mid o he XX cen u y, wi h he aim o modelling how human b ains
p ocess in o ma ion, he wo ks o W. S. McCulloch, W. Pi s and F. Rosenbla
ga e ise o he i s implemen a ions o he pe cep on [118, 119] which by means o
non linea unc ions (as he sigmoid [120]) ope a ing upon weigh ed sums o inpu
ga e ce ain logical o con inuous ou pu s. As i has been p o ed, by a anging a
gi en numbe o hese neu ons in laye s and coupling hese se e al laye s such ha
he ou pu o he
i
- h laye becomes he inpu o he
(i+
1
)
- h one, hese machines
a e (in p inciple) uni e sal app oxima es [121–125]. Tha is, gi en enough da a and
la ge enough numbe o laye s and neu ons in each laye , hese ne wo ks can be
ained o app oxima e any a bi a y unc ion.
1.3 machine lea ning assis ance 7
1.3.2 Semisupe ised lea ning
Mos supe ised machine lea ning p edic ion and classi ica ion models wo k p o-
ided he ex ension o he lea ning da a se is la ge enough. Ne e heless, in many
eal-wo ld si ua ions, as in ou case, ob aining independen a iable alues (la-
bels) may be ime-demanding o compu a ionally/economically expensi e. Thus,
i is wise o gain u he expe ise based on al eady es ablished a ailable knowl-
edge and in es ou esou ces (will, ime and money) in enligh ening he da kes
co ne s o ou igno ance. Semi-supe ised and ac i e lea ning echniques can be
applied upon da a o which we ha e wo di e en se s: a labelled se
L
and an
unlabelled se
U
. Namely, he labeled se
L
is composed by poin s o med as an
o de ed Ca esian p oduc
(x1
,
y1). . . (xl
,
yl)
whe e
YL={y1. . . yl}∈{
1
. . . C}
is
he se o possible labels (ca ego ies). Then, we also ha e he unlabelled se
U
com-
posed by
(xl+1
,
yl+1). . . (xl+u
,
yl+u)
poin s whe e he labels
YU={yl+1. . . yl+u} ∈
{
1
. . . C}
a e unknown and he ca dinali y o his se is much la ge han he o me
one, l << u.
By doing so, we shall know whe e o map he nex expe imen o ob ain phase
diag ams as i has been done in e y ecen wo ks [126, 127]. We ha e been using
wo complemen a y echniques: label p opaga ion and unce ain y sampling.
Unde he ini ial assump ions, we ha e buil a g id in pa ame e space wi h
X={x1. . . xl+u}
coo dina es om which a numbe
l
will be labelled wi h jus one
label o m ca ego ies
Y={
1
. . . C}
and he emaining
u
coo dina es a e no labelled
(o cou se, all ca ego ies a e ep esen ed in he
l
labelled da a, so
ca d(l)≥ca d(C)
).
We also assume ha he numbe o unlabelled coo dina es is a la ge han he
labelled ones
l << u
. The poin he e is o use he in o ma ion (labels and coo dina es)
o he
l
labelled da a and in e he p obabili ies o unlabeled da a o belong o each
o he
C
possible ca ego ies and compu e he unce ain y based on hese p obabili ies.
1.3.2.1 Label P opaga ion Algo i hm
X. Zhu and Z. Ghah amani de eloped an algo i hm named ‘label p opaga ion
algo i hm’[128] in which in o ma ion om labelled da a is p opaga ed o unlabelled
da a in a s ochas ic p ocess by means o Ma ko chains. He e, one de ines a g aph
con aining all -labelled and unlabelled- da a and de ines he connec ion s eng h
(weigh ) by pai s. Using p ope ies ela ed o homogeneous Ma ko chains and
s ochas ic ma ices, i can be shown ha he algo i hm con e ges o a unique
s a iona y solu ion
(1.5)
o any which ini ial guess o he ma ix
Y(0)
we may
make.
Yu= (I−˜
Tuu)−1˜
TulYl(1.5)
8in oduc ion
The main d awback o his me hod is he ac ha o dense o la ge enough
g ids, he compu a ion o he ma ix
(I−˜
Tuu)−1
is expensi e o , in he wo s case,
he ma ix is singula . This p oblem may be sides epped by, ins ead o using he
closed o mula
(8.5)
aking an ini ial label ec o and i e a ing i . Ne e heless,
since con e gence is eached a e se e al i e a ions and la ge ma ices a e s ill
in ol ed, his algo i hm is s ill cos ly o dense enough g ids and his de iciency is
mo e ema kable as he numbe o dimensions o he ea u e space inc eases. As
an al e na i e, o he , a he cheap, classi ie s such as Random Fo es s (RF) can be
used o sol e he p oblem. Since RF do no ha e o e i ing p oblems, using a la ge
enough numbe o ees, he me hod is uni e sal o all sys ems s udied in his wo k.
The e o e, he ini ialisa ion se is employed o ain a RF model which will label he
unlabelled poin s assigning hem o each ca ego y wi h a gi en p obabili y, ha is,
an equi alen app oach as using label p opaga ion.
1.3.2.2 Unce ain y sampling
Lea ning is widening ou knowledge by gaining mo e in o ma ion whe e ou igno ance
o unce ain y is deepes . In o de o lea n in an e icien way, one should ocus
on one’s lack o expe ise a he han wha she/he al eady knows. Unce ain y
sampling is a way o sampling jus he poin s in which he unce ain y is he la ges ,
bu how can we measu e unce ain y? P o ided we ha e a p obabili y dis ibu ion
unc ion
PC(y|x)
such ha gi en a pa ame e ec o
x
gi es he p obabili ies o ha
poin o belong o each o he ca ego ies
y
in he se
{
1
. . . C}
, we may de ine some
unce ain y measu es as: Leas Con iden
(1.6)
, Ma gin
(1.7)
and Shannon En opy
(1.8)
. The i s one maximizes he leas likely ( he e o e he mos unce ain) o he
unlabelled da a, he second one is mo e lexible han he o me in he sense ha
i akes in o accoun he wo mos likelies p edic ion poin s. Finally, he Shannon
en opy is a measu e o in o ma ion con en ; he less in o ma ion a a iable con ains,
he mo e we know abou i s na u e [129].
uLC (x) = 1−max
CP(C|x)(1.6)
uMS (x) = 1−[P(C1|x)−P(C2|x)](1.7)
uSE (x) = −X
C
P(C|x)log P(C|x)(1.8)
1.4 scope o his wo k 9
1.4 SCOPE OF THIS WORK
Once he s a e o he a conce ning Wigne molecules, Hooke a oms and some
machine lea ning echniques has been desc ibed, in his sec ion he main goals o
he p esen wo k will be p o ided. We shall now lis he mains goals o his wo k:
1.
Op imize dis ibu ed gaussian basis unc ions o co ec ly desc ibing he
one- and wo-dimensionally con ined elec onic sys ems by means o machine
lea ning app oaches (Neu al Ne wo ks) o a iable con inemen pa ame e
k=ω2
. Besides, se an op imal wa e unc ion based elec onic s uc u e me hod
o ob aining accu a e wa e unc ions. Employ he se led p o ocole in o de o
s udy he Wigne loca ion o small numbe o elec ons
n={
2, 3, 4
}
in he
high spin s a e.
2.
Op imize one-cen e e en empe ed basis unc ions using classical op imiza-
ion echniques (simplex and New on-Raphson) o p ope ly desc ibing h ee
dimensional sphe ical Hooke a oms composed by ew numbe o elec ons
n={
2, 4, 6, 8, 10
}
and con inemen pa ame e
k=ω2=
1
/
4. Se le an op i-
mal wa e unc ion based me hod o ob aining accu a e ene gies o he lowes
laying single and iple s a es.
3.
Implemen and imp o e machine lea ning me hods based on semi-supe ised
lea ning and unce ain y sampling o ob aining phase diag ams e icien ly and
p o ide some chemically ele an sys ems.
4.
Calcula e analy ical one-body in eg als co esponding o gaussian con inemen
po en ials in e ms o gaussian basis unc ions and implemen hem in
GAMESS
US. P o ide some examples ela ed o he p e ious sec ions.

Chap e 2
QUASI-ONE DIMENSIONAL
SYSTEMS
In his chap e , localiza ion p ope ies o sys ems composed by ew elec ons con ined
in a quasi-one-dimensional ha monic po en ial ha e been s udied by semi-analy ical
p ocedu es o wo elec on sys ems, and by mo e obus quan um chemical ap-
p oaches conce ning wo, h ee and ou elec on sys ems in high spin s a e. By
using elec onic-s uc u e p ope ies ob ained by mul i- e e ence me hods such as he
one-body densi y and he pa icle-hole en opy, we ha e been able o de ine a pa h
ha connec s he co esponding Wigne molecules wi h he Fe mi liquids by a ying
he ha monic-po en ial con inemen pa ame e
k
. We conclude ha he pa icle-hole
en opy is a smoo h unc ion o he con inemen pa ame e and connec s he wo limi
cases, and shows a maximum alue which posi ion depends on he numbe o elec ons.
Submi ed o JCP
Uploaded o he a Xi h p://a xi .o g/abs/2207.12014
11
12 quasi-one dimensional sys ems
2.1 INTRODUCTION
In he p e ious chap e , some cha ac e is ics abou he na u e o he Wigne c ys als
and molecules ha e been in oduced. Due o he in insic na u e o such sys ems, in
o de o build models which ep esen hose phases elec on co ela ion e ec s mus
be p ope ly desc ibed. Some p e ious s udies ackled he issue o one-dimensional
sys ems composed by ew elec ons (
n=
2, 3
and
4) and he co ela ion e ec s
a ying he leng h o he con inemen box and he compu a ion me hod compa ing
ROHF wi h FCI [130]. Fa om being only heo e ical models, sys ems con aining
hese numbe o elec ons ha e been obse ed expe imen ally [131, 132].
In his chap e , we ha e s udied one-dimensional sys ems wi h (
n=
2, 3
and
4)
elec ons con ined in ha monic po en ials wi h se e al o de s o magni ude o he
con inemen pa ame e
k
. In a i s s ep, we ha e seen ha single and iple s a es
o wo elec on sys ems ha e simila densi y p o iles in he weak con inemen egime
(
k→
0) while o la ge con inemen egime (
k→ ∞
) he densi y o each spin s a e
con e ges o he ee pa icle densi y. Then, o high spin s a es, o all numbe o
elec ons, we ha e compu ed he elec on-hole en opy which is linked o co ela ion
e ec s [133–135] and we ha e been able o connec wo lowes en opy phases: he
Wigne molecule and he Fe mi liquid. These en opy cu es depend on he numbe
o elec ons and he po en ial pa ame e
k
and, he e o e, on he densi y o he
sys em. In all cases we ha e obse ed he e is a ce ain alue o
k
o which he
elec on-hole en opy is maximal, hence s a ic co ela ion e ec s play a big ole.
2.2 COMPUTATIONAL PROTOCOL
2.2.1 Quan um mechanical model
In his chap e , we ha e s udied sys ems composed by
n={
2, 3, 4
}
elec ons which
in e ac ia Coulomb po en ials among hem and a e con ined in a one-dimensional
ha monic well, hence he Hamil onian o he sys em can be exp essed as in equa ion
(2.1).
H=−1
2
n
X
i=1∇2
i+k
2
n
X
i=1
2
i+
n
X
j>i=1
1
ij
(2.1)
As i is well known [136, 137], in he pu e one-dimensional case he Coulomb
ope a o is singula ; in o de o sides ep his p oblem, we shall sol e he Sch ¨odinge
equa ion by using h ee dimensional basis unc ions dis ibu ed along one unique
di ec ion. The inal ene gy is hen ob ained by sub ac ing he ans e se componen s
o he kine ic ene gy o each elec on [138]. In his chap e , we will s udy he beha io
o such quasi-one-dimensional sys ems.
2.3 esul s and discussion 13
2.2.2 Dis ibu ed Gaussian o bi als
In o de o p ope ly desc ibe ou sys ems, we use a one dimensional g id o gaussian
unc ions gi en by equa ion
(2.2)
whe e he posi ion o each basis unc ion is
dis ibu ed along he xaxis Ri= (xi, 0, 0).
ϕi( ;α,Ri) = 2α
π3/4
exp(−α( −Ri)2)(2.2)
These gaussians ha e been se as ollows: gi en a numbe o elec ons
n
and a
ha monic po en ial wi h cu a u e pa ame e
k
, he semiclassical u ning poin s o
he highes ene gy le el a e gi en as
x0(n
,
k) = ±(2n+1)2
k1/4
. Using hese poin s
as well as he posi ion o he minimum o he po en ial well (
x=
0) we place a
basis unc ion in each o hem and we add
m
equidis an gaussians be ween he
semi-classical u ning poin s and 2
m
mo e o each side o hese poin s. In his se ing,
he e is a o al numbe o
M=
3
+
4
m
basis unc ions and he dis ance be ween
consecu i e basis is gi en by
δ=2|x0|
m+1
. Since all basis unc ions had he same exponen
α
, he o e lap be ween neighbou ing unc ions is gi en as
S(α
,
δ) = exp(−αδ2/
2
)
.
One may also ew i e he las o e lap unc ion in e ms o a dimensionless pa ame e
ξ=αδ2
as
S(ξ) = exp(−ξ/
2
)
. This dimensionless pa ame e cha ac e izes he
esolu ion o he basis se which in ou case has been chosen o be
ξ=
1.0 [130,
138–141] . By doing so, he exponen s a e dependen on he con inemen , he numbe
o elec ons and he numbe o basis unc ions as α=ξ(m+1)2
4x2
0(n,k).
2.3 RESULTS AND DISCUSSION
2.3.1 Two elec on sys em
As i has been in oduced in he p e ious sec ion, o each con inemen s eng h
pa ame e
k
, we ha e ixed he posi ion o h ee gaussians: one whe e he con inemen
po en ial has i s minimum (
x=
0) and one on each classical u ning poin s (
x=
±x0(k
,
n)
). Once hese poin s a e se led, we ha e added 100 equidis an gaussians
(
δ=x0
13
) wi h he same exponen (
α=δ−2
), 50 o hem in be ween he wo u ning
poin s and 25 a each side o hem, gi ing ise o a o al numbe o 103 basis
unc ions.
Using hese basis, we ha e compu ed he single and iple spin s a es a CASSCF
le el o heo y using se e al numbe s o ac i e o bi als o se e al alues o he
con inemen pa ame e
k
. A e ha ing un some ini ial es s, we ha e concluded
ha he op imal ac i e space is he one ob ained om CASSCF(2,5). By using his
20 quasi-one dimensional sys ems
k
as can be seen in igu e 2.4. Fo small alues o
k
, bo h s a es a e e y close
in ene gy while he iple s a e is a bi bellow he single s a e, bu o some
alue a ound
k=
5
×
10
−4
bo h lines c oss and hen he iple s a e is highe
in ene gy o all
k
. I is con enien o ecall ha o la ge
k
, he ene gy o he
iple s a e goes as
E0T=
2
k1/2
while he ene gy o he single s a e goes as
E0S=k1/2+ (2/π)1/2k1/4.
Figu e 2.4:
Ene gies o single and iple spin s a es using minimal gaussian basis (le )
and ene gy di e ence be ween he single and he iple s a es o he wo
elec on sys em using minimal basis ( igh ). The ze o ene gy gap happens
a ound a alue close o k=5×10−4.
Conside ing he minimal basis model, we ha e shown ha below a pa icula
small alue o
k
, he iple s a e (which ep esen he mo e localized sys em) is lowe
in ene gy han he single s a e (which ep esen s he less localized sys em) and
beyond
k≈
5
×
10
−4
we ind he opposi e si ua ion. Thanks o his simple model,
we ha e been able o de e mine he o de o magni ude o
k
o which localiza ion
may happen.
2.3.2 Sys ems o h ee and ou elec ons
Based on he esul s o he p e ious sec ion conce ning wo elec on sys ems, we
may b ie ly s udy sys ems composed by h ee and ou elec ons and see i we
ob ain simila beha iou s. Bea ing in mind he obse ed p ope ies o he wo
elec on sys em, we ha e cons ained ou sel es o he high spin s a e upon which
CASSCF(
n
, 2
n
) calcula ions (being
n
is he numbe o elec ons) wi h he g id
o gaussian basis unc ions s a ed in he p e ious sec ions ha e been pe o med.
The ob ained densi ies a e shown in igu e 2.5. Once again, we ha e compu ed
he pa icle-hole en opies using he occupa ion numbe s o each sys ems and he
ob ained esul s a e g aphically ep esen ed in igu e 2.6.

2.3 esul s and discussion 21
As i can be seen, e en hough bo h cu es ha e simila shape, he alue o
k
o which he en opy is maximum depends on he numbe o elec ons ( he la ge
he numbe o elec ons, he smalle he alue o
k
a which he en opy eaches i s
maximum). We shall in e ha he beha iou is simila o he wo elec on sys em
and hus he conclusions a e equi alen . In he ligh o hese esul s, we shall de ine
a kind o ansi ion s a e aking place a he maximal en opy poin
S(kmax
,
n)
such
ha connec s he Wigne molecule and Fe mi liquid s a es. The size and posi ion o
such ansi ion s a e depends, o cou se, on he numbe o elec ons.
Figu e 2.5:
No malised densi y p o iles o h ee (blue) and ou (o ange) elec on sys ems
in high spin con igu a ion o se e al con inemen s eng hs k.
22 quasi-one dimensional sys ems
Figu e 2.6:
Compu ed elec on-hole en opies o
n=
3, 4 numbe o elec ons wi h high
spin s a e o se e al alues o he con inemen pa ame e k.
2.4 concluding ema ks 23
2.4 CONCLUDING REMARKS
In his chap e , we ha e s udied sys ems composed by wo, h ee and ou elec-
ons con ined in a quasi-one-dimensional ha monic po en ial cha ac e ised by he
con inemen pa ame e k.
Fo he wo elec on sys em, we ha e ob ained analy ical o mula ions using a
minimal basis se and a one-de e minan wa e unc ion. In his way, we ha e been
able o model he Wigne -molecule egime as a iple wi h wo iden ical gaussians
basis unc ions and he Fe mi liquid egime as a single wi h a single gaussian basis
unc ion. Using his scheme, we ha e minimized he ene gy o each sys em o se e al
alues o he con inemen pa ame e
k
and ha e seen ha a c ossing be ween he
wo ypes o wa e unc ions happens a ound
k=
5
×
10
−4
. This indica es ha , a
his le el o heo y, some change in he na u e o he wa e unc ion mus ake place
a ound his alue.
Basis se s composed o a la ge numbe o o e lapping gaussian unc ions we e
also used in o de o app oach he limi o a comple e basis se . We pe o med
a scan o elec onic s uc u e de i ed p ope ies such as one-pa icle densi y and
pa icle-hole en opy and bo h indices coincide ha o
k
alues a ound 1
×
10
−4
and 1
×
10
−3
a ansi ion is aking place ei he because he loca ion index o he
single and he iple s a e s a o di e ge o because he pa icle-hole en opy
eaches a maximum. Anyhow, he o de o magni ude o
k
a which he ansi ion
s a s is he same as he one ob ained by he simpli ied model.
Ex ending he s udy o high-spin s a es conce ning h ee and ou elec ons
sys ems, we ha e obse ed he same beha iou . In he wo limi cases, he densi y
and he pa icle-hole en opies co espond o he ones expec ed o he Wigne
molecule
(k→
0
)
and he Fe mi liquid (
k→ ∞
). By a ying he con inemen
pa ame e
k
, we ha e compu ed a smoo h pa h connec ing hese ex eme s a es
such ha he pa icle-hole en opy eaches a maximum. Fo e y small alues o
he con inemen pa ame e , we ha e ob ained some nume ical ins abili ies and ha e
no ob ained eliable in o ma ion on how he hole-pa icle en opy beha es in his
egime. Fu he wo k conce ning he s udy o his egime can be done in a close
u u e.
Chap e 3
QUASI-TWO DIMENSIONAL
SYSTEMS
In his chap e , he p ocess o ob aining op imal gaussian basis se s dis ibu ed
in a hexagonal g id o modelling ha monically con ined wo dimensional sys ems
is desc ibed. As a i s s ep, hese basis unc ions a e op imized by means o he
a ia ional p inciple o one-body sys ems and he ob ained esul s a e compa ed
o analy ical ene gies. As a esul , he ob ained nume ical alues o he ene gies
a e no only accu a e wi h espec o he analy ical ones, bu also degene acy is
p ese ed. Using p ope ly op imized neu al ne wo ks, we ha e been able o ob ain
gene al op imal basis unc ions o a bi a y alues o he con inemen pa ame e
k
in he ange (1
×
10
−9
, 1
×
10
1
). The quali y o he ob ained basis unc ions inc eases
wi h he numbe o unc ions and pe o m be e o la ge alues o he con inemen
pa ame e k.
In p og ess
25

26 quasi- wo dimensional sys ems
3.1 INTRODUCTION
A as ly employed p ac ice in elec onic s uc u e simula ions is o expand he
co esponding many-body wa e unc ion in e ms o one-elec on basis unc ions.
In he pa icula case o quan um chemis y, he mos popula basis unc ions a e
gaussian unc ions si ing on each o he nuclei which compose he s udied chemical
sys em [142–144]. Besides, he use o loa ing gaussians includes some lexibili y o
be e desc ibe sp ead elec onic densi y wi hou he need o including an explici
nucleus [138, 145–149].
As i has been shown in p e ious wo ks, dis ibu ed
s
- ype gaussian o bi als a e
a he good basis unc ions o desc ibing many-body e mionic sys ems [130, 138–
140]. This app oach may be employed o s udy wo dimensional quan um sys ems
which ha e wide expe imen al and heo e ical in e es [150–153]. Howe e , he scale
o some sys ems is o he o de o he nanome e , o which i a single cen e is
conside ed, linea dependency p oblems will happen. In o de o se an example,
expe imen ally, Quan um Do s based on G aphene (GQD) and T ansi ion Me al
Dichalcogenides (TMDs) ha e been syn hesised in a ange o sizes in he in e al
1-7 nm [154–163]. Among hem, GQD wi h isible ligh pho o-luminescence ha e
diame e s a ound 1.0-2.5 nm [164, 165]. Wi h espec o moi ´e cells (as gene alising
Mo physics), he densi ies o such s a es a e o he magni ude o 1
×
10
16e−/m2
[166–171] and o smalle densi ies, some au ho indica e he p esence o Wigne
c ys allisa ion [172–175]. Since o such low densi ies (as i was i s ly p edic ed
by Wigne himsel [20] and expe imen ally con i med o 2D sys ems [176, 177])
he in e -elec onic in e ac ion ene gy is la ge han he kine ic one, i is i al o
desc ibe he elec on-elec on in e ac ion using accu a e echniques [37, 39, 40, 42–
64]. Bea ing his in mind, se ing a quan um chemical app oach o model such la ge
sys ems is c ucial. We shall es he alidi y o ou app oach o desc ibe Wigne
loca ion in many elec on sys ems.
The inal aim o his wo k is o ob ain an op imal wo-dimensional hexagonal
mesh composed by
s
- ype o bi als being
α
he exponen o such o bi als and
δ
he
dis ance be ween closes neighbou s. These wo pa ame e s a e selec ed such ha
gi en a ha monic con inemen s eng h
k
, he size o he mesh and a pa ame e
ξ
ha indica es he o e lap be ween neighbou ing o bi als, one ge s he minimal
one-pa icle ene gy. Wi h he aim o ge ing gene al unable basis se s, an A i icial
Neu al Ne wo k has been ained o p oduce he op imal mesh. Once he basis se is
op imised o gi en inpu pa ame e s, we ha e used quan um chemical me hods o
compu ing p ope ies conce ning he elec onic s uc u e o ha monically con ined
sys ems o elec ons. By doing so, all echnologies employed in his ield conce ning
high le el me hods such as he mul i- e e ence ones can be applied in la ge numbe
o elec ons.
3.2 compu a ional p o ocol 27
3.2 COMPUTATIONAL PROTOCOL
3.2.1 Dis ibu ed Gaussian o bi als
Hexagonal g ids composed by 3D
s
- ype Gaussian o bi als gi en as in equa ion
(3.1)
ha e been employed. The numbe o gaussian unc ions is gi en by
M=
3
g2−
5
g+
1
whe e
g≥
3 is ela ed o he numbe o shells in he hexagonal pa e n. In igu e
3.1 a schema ic ep esen a ion o he geome y o he basis se is gi en o se e al
alues o gnumbe .
ϕi( ;α,Ri) = 2α
π3/4
exp(−α( −Ri)2)(3.1)
Figu e 3.1:
Geome y o he dis ibu ed gaussian unc ions acco ding o
g
numbe . All
dis ances be ween closes neighbou s is gi en by he pa ame e δ.
28 quasi- wo dimensional sys ems
Using his se ing, all basis unc ions had he same exponen
α
and since he se
{Ri}M
gi es ise o a hexagonal pa e n he dis ance be ween closes neighbou s is he
same o any wo o hem. The o e lap be ween close-neighbou ing unc ions is gi en
as
S(α
,
δ) = exp(−αδ2/
2
)
whe e
δ
is he dis ance be ween he s a ed neighbou s.
One may also ew i e he las o e lap unc ion in e ms o a dimensionless pa ame e
gi en as
ξ=αδ2
as
S(ξ) = exp(−ξ/
2
)
; such dimensionless pa ame e cha ac e izes
he esolu ion o he basis se and con ols he alue o he en ies o he o e lap
ma ix. Due o he ac ha oo small alues o
ξ
gi e ise o linea dependency
p oblems ( he o e lap ma ix becomes singula ) while oo la ge alues imply ha
he o e lap is no good enough o desc ibe a con inuous space, he op imal alue o
ξ
has an uppe and lowe bound. P e ious wo ks ha e shown ha his pa ame e mus
belong in a na ow in e al a ound
ξ=
1.00; in his wo k, we ha e aken se e al
alues in he in e al ξ∈[0.85, 1.15][130, 138–140].
3.2.2 Quan um mechanical compu a ions
In a i s s ep, we ha e s udied one-body p oblems using iso opic ha monic con-
inemen po en ials. Fo such sys ems, he gene al Hamil onian is gi en by equa ion
(3.2)
which is sepa able (one e m o each coo dina e). The e o e, he g ound s a e
ene gy o he iso opic 2D ha monic oscilla o is gi en as E0=k1/2.
H(x,y) = 1
2−∂2
x+kx2−∂2
y+ky2=H(x) + H(y)(3.2)
Using he basis desc ibed in he p e ious sec ion, we can ew i e he co esponding
one-body Sch ¨odinge equa ion as a gene alised eigen alue p oblem
(3.3)
whe e
S
is
he o e lap ma ix,
T
is he kine ic ene gy ma ix,
V
is he po en ial ene gy ma ix
and
E
is he diagonal ma ix con aining he eigen alues o he Hamil onian while
C
con ains he eigen ec o s in i s columns.
(T+V)C=ESC (3.3)
The co esponding ma ix elemen s a e gi en by equa ions
(3.4)
,
(3.5)
and
(3.6)
espec i ely; no ice ha he elemen s o he kine ic and po en ial ene gy depend on
he elemen s o he o e lap, hus hey can be de ined in he same loop. Since he
basis unc ion a e h ee dimensional in na u e, by sol ing he gene al eigen alue
p oblem, we ob ain quasi-2D solu ions. In o de o co ec he ob ained ene gies,
since all gaussian ha e he same exponen , we may ac o ize ou he ans e se
componen o he wa e unc ion and sub ac he co esponding con ibu ion o he
kine ic ene gy, which in his case equals
α/
2 pe elec on [138]. By doing so, we
ob ain he co esponding co ec ed ene gies which co espond o he equi alen 2D
sys em.
3.3 esul s and discussion 29
Sij =exp −α
2d2
ij(3.4)
Tij =α
2(3−αd2
ij )Sij (3.5)
Vij =k
8α(2+α[(x2
i+x2
j) + (y2
i+y2
j)])Sij (3.6)
As i has been s a ed in he p e ious sec ion, i we se he alue o he exponen
α=ξ/δ2
o a ious alues o
ξ
, by means o he a ia ional p inciple, (gi en
(ξ
,
g
,
k)
) we may op imize he g ound s a e ene gy wi h espec o he dis ance
be ween he basis unc ions δ.
3.3 RESULTS AND DISCUSSION
3.3.1 One pa icle sys ems
In a i s s ep, we aim o gain some knowledge o how o build an app op ia e basis
g id exposed in p e ious sec ion o a gi en sys em. Tha is, o a gi en po en ial
cha ac e ized by he con inemen pa ame e
k
, a gi en eal numbe
ξ
which con ols
he size o he en ies in he o e lap ma ix and a gi en numbe o basis unc ions
ep esen ed by
g
, wha is he op imal dis ance be ween neighbo ing basis
δop
o
desc ibing ou sys em?
In o de o answe his ques ion, we ha e aken he ollowing s eps: we op imize
he one pa icle g ound s a e ene gy wi h espec o dis ance be ween neighbou ing
basis by sol ing he ma ix Sch ¨odinge equa ion o se e al alues o (
k
,
ξ
,
g
). We
aimed o gain expe ience on se e al o de s o magni ude o
k∈[
1
×
10
−10
, 9
]
and
ha e employed
ξ∈[
0.85, 1.15
]
wi h
g∈[
4, 10
]
numbe s. A o al numbe o 1190
op imized basis se s we e ob ained using he B en op imiza ion scheme o con ex
unc ions.
As a esul o hose op imiza ions, we ex ac ed he i s en eigen alues o he
Hamil onian and ha e compa ed hem o he analy ical alues such ha we ob ain
he ela i e e o gi en as
ϵ=Ei−E0,i
E0,i
whe e
Ei
is he ene gy o he
i
- h s a e and
E0,i
is he co esponding analy ical alue. Resul s o such e o s o op imized basis
se s a e gi en in igu e 3.2. As i can be seen, e en i his e o ge s la ge o highe
ene gy s a es, he maximal e o in all he samples is o he o de o 5 %. Ano he
ea u e o be conside ed is he ac ha no only he ela i e e o mus be aken
in o accoun , bu also he ac ha analy ically degene a e s a es should emain
degene a e in he nume ical app oach. In o de o measu e he di e ence o ene gy
in each ene gy le el o se e al s a es gi en (
k
,
ξ
,
g
) we ha e compu ed he ela i e
s anda d de ia ion o each se and hen pe o med s a is ics among all s anda d
de ia ions. As i is shown in igu e 3.2, he maximal ela i e s anda d de ia ion

Chap e 4
SPHERICAL HOOKE ATOMS
Single and iple spin s a e ene gies o h ee-dimensional Hooke a oms,
i
.
e
.elec-
ons in a quad a ic con inemen , wi h e en numbe o elec ons (2, 4, 6, 8, 10)
is discussed using Full-CI and CASSCF ype wa e unc ions wi h a a ie y o basis
se s and conside ing pe u ba i e co ec ions up o second o de . The e ec o he
sc eening o he elec on-elec on in e ac ion is also discussed by using a Yukawa- ype
po en ial wi h di e en alues o he Yukawa sc eening pa ame e (
λee
=0.2, 0.4,
0.6, 0.8, 1.0). Ou esul s show ha he single s a e is he g ound s a e o 2 and
8 elec on Hooke a oms, whe eas he iple is he g ound spin s a e o 4, 6 and
10 elec on sys ems. This sugges s he ollowing
Au bau
s uc u e 1
s <
1
p <
1
d
wi h single g ound spin s a es o sys ems in which he gene a ion o he iple
implies an in e -shell one elec on p omo ion, and iple g ound s a es in cases
when he e is a pa ial illing o elec ons o a gi en shell. I is also obse ed ha
he sc eening o elec on-elec on in e ac ions has a sizable quan i a i e e ec on
he ela i e ene gies o bo h spin s a es, specially in he case o 2 and 8 elec on
sys ems, a ou ing he single s a e o e he iple . Howe e , he sc eening o he
elec on-elec on in e ac ion does no p o oke a change in he na u e o he g ound
spin s a e o hese sys ems. By analyzing he di e en componen s o he ene gy, we
ha e gained a deepe unde s anding o he e ec s o he kine ic, con inemen and
elec on-elec on in e ac ion componen s o he ene gy.
Submi ed o IJQC
37
38 sphe ical hooke a oms
4.1 INTRODUCTION
Quan um do s ha e a ac ed conside able a en ion in he las yea s. The possibili y
o c ea ing a i icial a oms in which he elec ons a e con ined o a cen e h ough
a quad a ic ype po en ial opens he possibili y o designing new nanoelec onic
de ices wi h p ope ies a will, by p ecisely con olling he deg ee o con inemen .
Fo ins ance, ansi ions ne e obse ed in na u al a oms can be ob ained in he
a i icial ones, which could be o pa amoun impo ance in designing new lase s. [178]
Ano he p ope y ha has a ac ed conside able a en ion is he de e mina ion
o he iple -single gap in con ined sys ems, o hei use as s a es o a qubi ,
o o implemen logical ga es in quan um compu ing. The exci a ion spec um o
wo-elec on wo-dimensional (2D) quan um do s has been in es iga ed by unneling
spec oscopy, [179] and he heo e ical p edic ion o iple -single ansi ions wi h
inc easing magne ic ield has been expe imen ally co obo a ed. Al hough less s udied,
3D quan um do s a e also a subjec o in e es [86]. Examples o his a e magne ically
apped e mion apo s con ined by pa abolic po en ials[180, 181] o quan um de ec s
in diamond c ys als used as basic gadge s in quan um compu ing[182, 183].
One o he simples and mos adequa e models used in heo e ical s udies
conce ning QDs a e he so called Ha monium o Hooke’s a om in which elec ons
a e con ined in a sphe ical ha monic po en ial [76]. Such models con ain pa ame e s
ha may be uned in o de o ep esen ea u es co esponding o eal QDs [77, 78].
Fo ins ance, wo k ca ied ou in ou g oup using a Hookean exac h ee-body model
o examine elec on co ela ion in a wo-elec on sphe ical quan um do con i med
ha iple -single ansi ions ake place as he ex e nally applied magne ic ield
inc eases.[79] Howe e , he limi a ion o using an exac model es ic ed ou s udy
o wo-elec on sys ems. Tha is, he analy ical solu ions[80] o speci ic cu a u e
pa ame e o he wo-elec on Hooke a oms (
ω2=1
4
,
1
100 . . .
) a e well known, which
can lead in p inciple o highly accu a e densi ies o sys ems in low-co ela ion
egimes (
ω2→ ∞
) as well as in high-co ela ion egimes (
ω2→
0) [35–38]. I simila
sys ems con aining la ge numbe o elec ons (
Ne>
2) a e o be conside ed, a iche
a ie y o elec onic s a es consis ing on se e al g ound s a e spin mul iplici ies
and non-dynamical elec on co ela ion (mul i-de e minan al ea u es) a ise [39–41].
Al hough such sys ems can be employed o unde s and many-body in e ac ions, he
compu a ional cos inc eases wi h he size o he sys em.
As in Hooke model a oms, he inco po a ion o elec on co ela ion e ec s has
been shown o be essen ial o an adequa e in e p e a ion o he expe imen al spec a
and anspo p ope ies in Quan um Do s. [81, 82, 84–86] In quan um do s, as
opposed o eal a oms, he e ec o elec on co ela ion may be a ied a will h ough
manipula ion o he dimension and shape o he nanoc ys al as well as o he s eng h,
bounda ies and symme ies o he con ining ields[87]. Besides, he elec on-elec on
in e ac ion can be sc eened due o la ice, he doping o he cha ges induced on
he me al ga es[89]. This ac makes he quan um do many-body p oblem mo e
complex han he mo e amilia a omic case.
4.1 in oduc ion 39
Finally, Hooke model sys ems ha e been epea edly used in he calib a ion
o elec onic s uc u e me hods, o hey p o ide e y a iable dynamic and non-
dynamic elec on co ela ion egimes [42–44] ha pose a g ea challenge o cu en
compu a ional me hods. [37, 45–57] Such calib a ion has been possible because o
he ecen a ailabili y o highly accu a e analy ical and benchma k da a.[39, 40,
58–64]
Se e al well es ablished me hods o he elucida ion o a omic/molecula elec-
onic s uc u e ha e been applied o quan um do s. Salien among hese a e diag-
onaliza ions o la ge con igu a ion in e ac ion ep esen a ions o he Hamil onian
ma ix (usually e e ed as “exac ”diagonaliza ions)[184–189], Ha ee-Fock (HF)[187,
190–192], coupled-clus e [193], densi y unc ional heo y[194–197], and quan um
Mon e Ca lo calcula ions[198]. Le us emphasize, howe e , ha e en o wo-elec on
quan um do s,[199] i has been obse ed ha in o de o accoun p ope ly o he
elec on co ela ion e ec s, one mus go beyond pe u ba i e schemes based on he
independen -pa icle model o local spin-densi y unc ional heo y.[179] Las bu no
leas , some o us ha e ound ha he use o lexible basis se is c ucial o a co ec
desc ip ion o s ong co ela ion e ec s in ha monium.[40, 54, 199] Wo k ca ied
ou in ou g oup using a Hookean exac h ee-body model o examine elec on
co ela ion in a wo-elec on sphe ical quan um do con i med ha iple -single
ansi ions ake place as he ex e nally applied magne ic ield inc eases [79].
In he p esen pape , we add ess he o bi al occupa ion pa e n o es ablish
he Au bau p inciple o h ee-dimensional quan um do s wi h an e en numbe o
elec ons (
Ne
=2, 4, 6, 8, 10). Full con igu a ion in e ac ion (Full-CI) o
Ne
=2, and
comple e ac i e space sel consis en ield (CASSCF) ype wa e unc ions, o
Ne>
2,
a e employed o accoun o elec on co ela ion e ec s. On op o his, second-o de
pe u ba ion co ec ions o hese ene gies a e also conside ed. Bo h he single and
he iple s a es will be e alua ed in o de o asce ain whe he he g ound s a e
wa e unc ion is ei he spin unpola ized o spin pola ized. This will p o ide, by he
same oken, he es ima e o he iple -single ene gy gap as he numbe o elec ons
o he quan um do s inc eases.
Fi s , we ha e employed he Dunning’s amily o co ela ion consis en (CC)
basis se s up o sex uple ze a. One should no ice ha hese basis se s a e op imized
o Coulombic sys ems, and, he e o e, o he sake o consis ency, we ha e compa ed
hei accu acy wi h he a ailable benchma k da a. [39, 41, 62, 199] We epo he
ull da a as supplemen a y ma e ial. In summa y, al hough hese absolu e ene gies
do no each ull accu acy, he single - iple gaps we e in ull ag eemen wi h he
benchma k da a, excep o he en-elec on sys em o which quali a i e di e ences
we e obse ed depending on which CC basis se was used. In o de o u he imp o e
he pe o mance o he basis se , we ha e op imized a se o e en- empe ed basis se s
(ETBS he ea e ) o he Hooke po en ial conside ing di e en numbe o elec ons.
A e ca e ul inspec ion, he bes balance be ween accu acy and pe o mance was
ob ained o he basis se op imized wi h six elec ons in he single s a e, we call
his basis se ETBS-6S, and i is he one mainly used h oughou he pape .
40 sphe ical hooke a oms
Finally, we also in oduce a model o ake in o accoun he sc eening o elec on-
elec on in e ac ions h ough an elec on-elec on Yukawa ype po en ial (also known
as Debye-Yukawa po en ial as e e ence o he Debye-H¨uckel heo y so employed in
he s udy o elec oly es and plasmas), as in p e ious wo ks [200]. The sc eening in
elec on-elec on in e ac ion is o en included by in oducing an e ec i e dielec ic
media [201–203]. In he p esen wo k, we analyze he use o a Yukawa po en ial,
which a sho ange is simila o he Coulomb po en ial, o analyze he e ec o he
elec on-elec on sc eening in co ela ion e ec s. We obse e ha e en o an ex eme
sc eening, co ela ion e ec s a e s ill impo an , acco ding o he co esponding
Coulomb holes, highligh ing he impo ance o adequa ely ea elec on co ela ion
in hese sys ems.
4.2 COMPUTATIONAL METHODS
Le us conside he ollowing gene alized Hamil onian ope a o (in a omic uni s) o
ou Ne-elec on sys em:
b
H=−
Ne
X
i
1
2∇2
i+
Ne
X
i
1
2ω2 2
i+
Ne
X
j>i
e−λee ij
ij
(4.1)
whe e
i
is he dis ance ec o be ween he
i h
elec on and he cen e o he
ha monic po en ial, which o all he calcula ions o his pape is cen e ed a he
o igin. This Hamil onian ep esen s a ha monically con ined
Ne
-elec on sys em,
wi h a con inemen s eng h
ω2
, whose in e -elec onic in e ac ion has been sc eened
s a ically by a Yukawa-like a enua ed in e ac ion po en ial, ha ing a sc eening
leng h λ−1
ee .
Recall ha o
Ne
=2 and
λee
=0, he Hamil onian ope a o o Eq. 4.1 co e-
sponds o he wo-elec on Hooke a om, which can be sepa a ed in o i s in acula
coo dina es, namely, he elec on-elec on ela i e dis ance ec o =
1−
2
and he
cen e o mass coo dina e ec o R=1
2( 1+ 2) as[80, 204]
b
H=−∇2
R
4+ω2R2−∇2
+1
4ω2 2+1
(4.2)
being
=| |
and
R=|R|
, espec i ely. Eq. (2) un eils ha he cen e o mass o
he elec ons will beha e as a ha monic oscilla o wi h a sp ing cons an o 2
ω2
and
a g ound s a e ene gy o
ER=
3
√ω2
. Likewise, Eq. (2) indica es as well, ha he
elec ons will emain in he p oximi y o each o he o hey a e e ained wi hin
ini e in e -elec onic dis ances by he po en ial
V( ) = 1
4ω2 2+1
, (4.3)
4.2 compu a ional me hods 41
which is bes seen as an e ec i e con inemen po en ial. This model sys em is
commonly known as Hooke a om, Hookean o ha monium.[205]
In summa y, wo ypes o sys ems ha e been conside ed in his pape :
•
Coulombic Hooke A om: The Hamil onian includes an ha monic con inemen
e m (ω2=0.25) wi h Coulombic like elec on-elec on epulsion (λee =0.0).
•
Yukawa Hooke A om: The Hamil onian includes an ha monic con inemen e m
(
ω2=1
4
) and a Yukawa- ype sc eened elec on epulsion (
λee =
0.2, 0.4, 0.6, 0.8, 1.0).
The co esponding one-elec on con inemen in eg als and he wo-elec on
Yukawa- ype in eg als ha e been implemen ed by ou g oup in an in-house code and
he co esponding in eg al package in e aced wi h he
GAMESS-US
p og am [206,
207] o pe o m he calcula ions desc ibed in his wo k. HF, Full-CI, CASSCF and
mul i e e ence second-o de M¨olle -Plesse (MRMP2) me hods we e used along wi h
a ious basis se s o aug-cc ype, and op imized e en- empe ed basis se (ETBS) o
2, 4, 6 and 8 elec ons sys ems. Fo each o he sys ems, he HF ene gy o he single
s a e was calcula ed. The co esponding o bi als we e used o pe o m Full-CI (in
he case o 2-elec on sys ems) and CASSCF and MRMP2 calcula ions in he case
o 2, 4, 6, 8 and 10 elec ons) o he single and iple spin s a es.
As said in he in oduc ion, we ha e employed wo ypes o basis se s: i) s anda d
Dunning’s amily o co ela ion consis en (cc) basis se s up o sex uple ze a and ii)
e en- empe ed basis se s (ETBS) op imized o he Hooke a om. The d awback o
aug-cc-pVNZ basis se s is ha hey a e op imized o Coulombic sys ems. The e o e,
we ha e compa ed hei accu acy wi h he a ailable benchma k da a [39, 41, 62,
199] o Hookean sys ems. We epo he ull da a as supplemen a y ma e ial, bu in
summa y, he conclusion is ha , al hough absolu e ene gies wi h aug-cc-pVNZ basis
se s do no each ull accu acy, he single - iple gaps a e in ull ag eemen wi h he
benchma k da a. The only excep ion o his ule is he en-elec on sys em, which
shows a mo e e a ic beha iou wi h impo an quali a i e di e ences conce nig he
g ound s a e spin mul iplici y among he a ious aug-cc-pVNZ basis se s.
In o de o imp o e he absolu e ene gies, we ha e op imized an e en- empe ed
basis se in he p esence o a ha monic po en ial (
ω2=
0.25) conside ing di e en
numbe o elec ons. We ha e employed uncon ac ed ETBS wi h angula momen um
L=
0 o
L=
3 and he same numbe
N
o p imi i es pe shell. The
L
and
N
dependen exponen s o he p imi i es a e e en- empe ed ollowing he scheme:
ζk
LN (ω2) = ω2
2αL,N(ω2)βL,N(ω2)k−1, 1 ≤k≤N, (4.4)
whe e he pa ame e s
α
and
β
a e op imized by minimizing he CASSCF ( ull
elec on and 13 ac i e o bi als) ene gies. Simila s a egies ha e been ollowed in
p e ious publica ions. [56, 199, 208]

42 sphe ical hooke a oms
A e ca e ul inspec ion, he bes balance be ween accu acy and pe o mance
was ob ained o he basis se op imized wi h six elec ons in he single s a e,
we call o his basis se he ETBS-6S. This basis se can be cha ac e ized as an
uncon ac ed 4(SPDF) basis se wi h he ollowing exponen alues: 0.2404032,
0.3130329, 0.4076051, 0.5307491. The o al numbe o basis se s is 80, ou o which
68 a e linea ly independen and hese a e he only one used in he calcula ions. The
esul s o his basis se showed a be e ag eemen in bo h absolu e ene gies and
single - iple gaps wi h espec o he a ailable benchma k da a han he s anda d
aug-cc-pVNZ se ies, and he e o e, we ocus ou discussion on he esul s ob ained
wi h his basis se .
4.3 RESULTS AND DISCUSSION
This sec ion is o ganized in acco dance o he numbe o elec ons o he sys em.
Thus, we s a ou discussion wi h wo elec on sys ems, and hen 4, 6, 8, and 10
elec on sys ems ollow. In he case o he wo-elec on Coulombic Hooke a om,
he exac ene gy is known, [80, 204] and he e o e, he e is a benchma k alue o
calib a e he basis se s employed h oughou his wo k. In he o he cases, we will use
he benchma k da a a ailable in he li e a u e.[39, 41, 62, 199] Thus, he i s sec ion
is dedica ed o his calib a ion. Then, he discussion is cen e ed on he iple -single
gap o each o he sys ems, ocusing ou discussion on he ac o s ha a ec his
gap, such as he elec on epulsion sc eening and he numbe o elec ons in he
sys em. To unde s and hese ends, we analyze he co esponding na u al o bi als,
so ha we can ela e he solu ion o a speci ic elec onic con igu a ion. Based on
his in o ma ion, we come up wi h an
Au bau
s uc u e o desc ibe he elec on
illing pa e n in hese con ined sys ems.
4.3.1 Two-elec on sys ems
The esul s o he wo-elec on Hooke- ype a om wi h Coulombic elec on epulsion
a e shown in able 4.1. We emphasize ha o his sys em he exac ene gy o he
single s a e is known, [80, 204] namely 2.0 a.u., and, he e o e, we use his e e ence
alue o calib a e he accu acy o he a ious basis se s used h oughou his wo k
( able 4.1). The e is a subs an ial di e ence be ween aug-cc-pVDZ and he es o
he basis se s. Full-CI ene gy o he single s a e is 2.055213 a.u. wi h his basis se .
The use o aug-cc-pVTZ (25 basis unc ions including d o bi als) leads o a educ ion
o he ene gy e o o one o de o magni ude, leading o a Full-CI ene gy o 2.004107
a.u. To u he educe he ene gy e o by one o de o magni ude, one has o go
up o he aug-cc-pV5Z basis se (2.000476 a.u.) wi h 105 basis unc ions. A his
poin , i is wo h no icing ha ou esul s a e o compa able accu acy o he bes
unex apola ed esul epo ed by Ma i o e al. [199] o
ω2
=0.25 , namely, 2.0002965
a.u., ob ained using sys ema ic sequences o Gaussian p imi i es wi h e en- empe ed
4.3 esul s and discussion 43
exponen s. Finally, he use o he aug-cc-pV6Z basis se , which con ains 182 basis
unc ions, leads only o a mino imp o emen in he ene gy, 2.000196 a.u. Mo eo e ,
his basis se shows la ge linea dependencies, and in ac , he o al numbe o
molecula o bi als in he a ia ional space is educed o 139 upon elimina ion o
linea dependencies. This ac has led us no o conside his basis se any u he
in his cu en wo k. Besides, wo mo e basis se s ha e been es ed, which will be
e e ed as aug-cc-pV5Z* and aug-cc-pV6Z*. This basis se a e c ea ed by emo ing
he basis unc ions wi h angula momen um g ea e o equal o ou . The educ ion
in size o he basis se is subs an ial, om 105 o 75 o aug-cc-pV5Z, and om 182
o 95 o aug-cc-pV6Z. Mo eo e , he la e basis se upon emo al o
g
and
h
basis
unc ions does no show linea dependencies. The educ ion o basis se size has
only mino e ec s in he ene gy. In he case o aug-cc-pV5Z basis se , he ene gy
inc eases only om 2.000476 a.u. o 2.000685 a.u., whe eas o aug-cc-pV6Z, om
2.000196 o 2.000426 a.u. Conside ing he pe o mance o ou op imized ETBS-6S
basis se , i gi es a alue o he single o 2.000396 a.u., ha is, he second lowes
ene gy alue and only imp o ed by he conside ably highe aug-cc-pV6Z basis se .
The iple -single gap (
∆T−S
), in eV, o wo-elec on sys ems can be ound
in Table 4.1. I espec i e o he basis se employed in he calcula ion, he lowes
ene gy o bi al is o
s
- ype, ollowed by a shell o
p
o bi als .[80] We will call o hese
o bi als he 1
s
and 1
p
o bi als. Hence, he single s a e is o med by he double
occupa ion o he 1
s
o bi al and he iple s a e co esponds o he p omo ion o
one o he 1
s
elec ons o one 1
p
o bi al plus a one-spin lip. The iple -single
gap is la ge. A he Full-CI/aug-cc-pV6Z le el o heo y, he gap is 9.86 eV and
9.78 eV o Full-CI/ETBS-6S, a e y simila alue o ou e e ence alue o 9.79 eV,
which ully jus i ies he use o ou ETBS-6S basis se as a good comp omise be ween
accu acy and compu a ional cos . On he o he hand, he be e pe o mance o he
ETBS-6S basis se o e he Dunning ones is mo e e iden when mo e elec ons a e
conside ed (see below), and he e o e, we will discuss in he manusc ip he esul s
o he ETBS-6S basis se , in compa ison wi h some alues o he aug-cc-pV6Z*
basis se . The esul s o he es o he Dunning’s basis se s can be ound in Table
4.1 and 4.9.
Since he use o Full-CI is p ohibi i e as he numbe o elec ons inc eases,we ha e
analyzed he pe o mance o CASSCF and MRMP2 me hods, using he ETBS-6S
basis se s. In Table 4.2, we epo he ene gies ob ained a he CASSCF and MRMP2
le els o heo y using di e en ac i e spaces. The esul s can also be isualized in
Fig. 4.1, whe e we ha e analyzed he con e gence o CASSCF (dashed line) and
MRMP2 (con inuum line) as we conside mo e o bi als (
No b
in he ac i e window).
We conside om a minimal window o 2 o bi als up o 13 o bi als, which co espond
o he 1
s
, 1
p
, 1
d
, 2
s
and 2
p
shells. The con e gence in
∆T−S
is ob ained qui e as ,
specially o MRMP2 me hod, wi h an excellen ag eemen wi h Full-CI esul s.
44 sphe ical hooke a oms
4.3.2 Fou -elec on sys ems
The esul s o he ou -elec on sys ems can be ound in Tables 4.3 and 4.9, and in
Fig. 4.1. In his case, he use o Full-CI was compu a ionally p ohibi i e. On he
o he hand, mul ide e minan al wa e unc ions a e manda o y due o subs an ial
nea -degene acy e ec s. Consequen ly, we ha e decided o use mul icon igu a ional
wa e unc ions o he CASSCF ype. We ha e in es iga ed ac i e spaces ha span
om 4 o 13 o bi als, co esponding o he 1
s
, 1
p
, 1
d
, 2
s
and 2
p
shells. We ha e
included he ou elec ons in his ac i e o bi al space and conside all possible
exci a ions wi hin he window ha a e compa ible wi h he desi ed spin s a e (single
o iple ). The s a ing o bi als o he CASSCF calcula ions co espond o he HF
o bi als o he single s a e. The bigges calcula ion using 4 elec ons and 13 o bi als
o he CAS window leads o 2366 CSFs (Con igu a ion S a e Func ions) o he
single s a e, and 3003 CSFs o he iple s a e. We ha e also conside ed second-
o de pe u ba ion co ec ions based on hese CASSCF wa e unc ions (MRMP2,
he ea e ).
The elec onic s uc u e o he single s a e p esen s double occupa ion o he
1s o bi al and one o he 1p o bi als. The iple s a e co esponds mainly o a
con igu a ion in which he 1s o bi al is doubly occupied and wo 1p o bi als ha e one
elec on each. Due o he degene acy o he 1p shell, he esul an single and iple
CASSCF wa e unc ions show la ge nondynamical co ela ion o nea -degene acy
e ec s, and, he e o e, he need o use mul iple con igu a ions in he wa e unc ion.
As one can see in Tables 4.3 and 4.9, he alues o
∆T−S
a e nega i e o all
basis se s and me hods conside ed, indica ing ha in he case o 4 elec ons he
con inemen has led o a iple -spin g ound s a e. No ice om Fig. 4.1 ha he
con e gence o he esul s wi h he window size is qui e as . The alues o he
iple -single gap a he CAS(4,13) and MRMP2(4,13) le els o heo y wi h he
aug-cc-pV6Z* basis se a e -1.11 eV and -1.08 eV, espec i ely, in nice ag eemen
wi h he e e ence alue by Cioslowski e al. [41] o -1.00 eV. The use o he ETBS-6S
basis se s sligh ly imp o es hese esul s, gi ing alues o -1.06 and -1.04 eV a
he CAS(4,13) and MRMP2(4,13) le els o heo y. Howe e , we can see ha he
CAS(4,13)/ETBS-6S ene gies a e signi ican ly close o he e e ence alues han
he CAS(4,13)/aug-cc-pV6Z* ones. Fo ins ance, he ene gy alue o he single
and iple s a es a e 6.398720 and 6.359758 , espec i ely, in good ag eemen wi h
he alues o Cioslowski e al.[41] 6.385543 and 6.348830 a.u., espec i ely. The
in oduc ion o second-o de pe u ba ion co ec ions, albei non- a ia ional, u he
imp o es his ag eemen , gi ing ene gies, 6.390085 and 6.352024, ha a e e en close
o he ones by Cioslowski e al.[41] An ad an age o he MRMP2 ene gies (see Figu e
4.1) is ha hey a e less dependen on he ac i e space o he CASSCF wa e unc ion
and, hus, we can use a smalle ac i e space wi hou sac i icing he accu acy. The
single - iple gap ob ained wi h MRMP2(4,13) is -1.04 eV, in e y good ag eemen
wi h he one by Cioslowski e al., [41] namely, -1.00 eV.
4.3 esul s and discussion 45
4.3.3 Six-elec on sys ems
The esul s o he six elec on sys ems can be ound in Tables 4.4 and 4.9, and in
Fig. 4.1. As in he p e ious case, we use CASSCF ype wa e unc ions, wi h di e en
ac i e spaces ha go om a minimal 6 o bi al window up o 13 o bi als (1
s
, 1
p
, 1
d
,
2
s
, and 2
p
), and include all six elec ons. Ou la ges CAS window in ol es 26026
CSFs o he single s a e, and 39039 o he iple .
The inclusion o wo u he elec ons in he 1
p
-shell yields a e y simila iple -
single gap and ends close o he ones obse ed o 4-elec on sys ems. Again, we
ob ain nega i e alues o
∆T−S
, indica ing ha he g ound s a e is a iple . Fo
ins ance, he alue o
∆T−S
is -1.03 and -1.0 eV a he CAS(6,13)/ETBS-6S and
MRMP2(6,13)/ETBS-6S le els o heo y. A glance a Fig. 4.1 clea ly demons a es
ha he esul s a e e y well con e ged wi h espec o he window size, specially
o he MRMP2 le el o heo y. The alues o he aug-cc-pV6Z* basis se a e e y
simila , namely, -1.06 and -1.05 eV. Ou esul s o he single - iple gap ag ee
sa is ac o ily wi h he e e ence calcula ions o he 6-elec on Hookean a om, [62]
which yield a alue o -0.95 eV wi h an ene gy o 12.066294 a.u. o he single
and 12.031275 a.u. o he iple . In he case o he six-elec on sys em, ETBS-6S
absolu e ene gies again gi e a signi ican imp o emen o e he aug-cc-pV6Z* ones.
Howe e , now he di e ence be ween ou CASSCF/ETBS-6S and he e e ence
ene gies o S asbu ge [62] inc eases wi h espec o he 4-elec on case. Fo ins ance,
CAS(6,13)/ETBS-6S ene gies a e 12.115772 and 12.077925 a.u. o single and iple
s a es, espec i ely, whe eas he e e ence ene gies a e 12.066294 and 12.031275 a.u..
Howe e , he in oduc ion o pe u ba ion co ec ions lowe s he ene gies o 12.082805
and 12.046142 a.u a he MRMP2(6,13)/ETBS-6S le el o heo y, yielding a iple -
single gap only di e ing by 0.05 eV wi h espec o he -0.95 eV gap ob ained
om he da a o S asbu ge . [62] In summa y, al hough he quali y o ou absolu e
ene gies o each o he s a es dec eases wi h he inc easing numbe o elec ons, he
es ima ion o he iple -single gap emains co ec .
4.3.4 Eigh -elec on sys ems
Tables 4.5 and 4.9, and Fig. 4.1 summa ize he esul s o 8-elec on sys ems, using
CASSCF wa e unc ions wi h an ac i e space composed o 8 o 13 o bi als (1
s
, 1
p
,
1
d
, 2
s
, and 2
p
o bi als) and 8 elec ons. In he case o ou la ges window, he
CASSCF(8,13) wa e unc ion yields 143143 CSFs o he single s a e, and 234234
CSFs o he iple . The co esponding second-o de pe u ba i e co ec ions can
also be ound in Table 4.5.
In he case o he single s a e, he 8-elec on sys em co esponds o an elec onic
con igu a ion in which he 1s and 1p shells a e ul illed. In he case o he iple
s a e, based on he analysis o he occupancies o na u al o bi als, an elec onic
con igu a ion o 1
s2
1
p5
1
d1
- ype is obse ed wi h ETBS-6S and aug-cc-pV6Z* basis
52 sphe ical hooke a oms
sys ems, which leads o a educ ion o he con inemen ene gy. Ou esul s poin s o
a ela ionship o he in ashell elec on-elec on co ela ion (as i occu s o single
4-, 6- and 10-elec on and iple 8-elec on sys ems), wi h he p omo ion o a mo e
compac elec onic cloud.
−0.08
−0.06
−0.04
−0.02
0.00
0.02
0.04
0.06
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
N = 2
h(u)
u (Å)
ull, S = 0
ull, S = 1
sc eened, S = 0
sc eened, S = 1
−0.25
−0.20
−0.15
−0.10
−0.05
0.00
0.05
0.10
0.15
0.20
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
N = 4
h(u)
u (Å)
ull, S = 0
ull, S = 1
sc eened, S = 0
sc eened, S = 1
−0.30
−0.25
−0.20
−0.15
−0.10
−0.05
0.00
0.05
0.10
0.15
0.20
0.25
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
N = 6
h(u)
u (Å)
ull, S = 0
ull, S = 1
sc eened, S = 0
sc eened, S = 1
−0.30
−0.25
−0.20
−0.15
−0.10
−0.05
0.00
0.05
0.10
0.15
0.20
0.25
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
N = 8
h(u)
u (Å)
ull, S = 0
ull, S = 1
sc eened, S = 0
sc eened, S = 1
−0.30
−0.25
−0.20
−0.15
−0.10
−0.05
0.00
0.05
0.10
0.15
0.20
0.25
0.0 1.0 2.0 3.0 4.0 5.0
N = 10
h(u)
u (Å)
ull, S = 0
ull, S = 1
sc eened, S = 0
sc eened, S = 1
Figu e 4.3:
Coulomb holes calcula ed a he CASSCF(
Ne
,13)/ETBS-6S le el o heo y
o bo h ull (solid lines) and
λee =
1.0 sc eened-coulombic (dasshed lines)
Hooke sys ems wi h di e en numbe o elec ons (
Ne
=2,4,6,8 and 10) and o
single (black lines) and iple s a es ( ed lines).

4.4 concluding ema ks 53
4.4 CONCLUDING REMARKS
In he p esen chap e , we ha e p esen ed a ho ough s udy o 2-,4-,6-,8- and 10-
elec on sys ems con ined in a sphe ical quan um do in hei single and iple spin
s a es, wi h he aim o de e mining he iple -single gap o Hooke- ype sys ems.
The e ec o sc eening he elec on-elec on in e ac ion has also been aken in o
accoun e ec i ely by he in oduc ion o a Yukawa ype po en ial.
Ou esul s show an in e es ing pa e n in he iple -single gap as he numbe
o elec ons inc eases. Thus, single s a e is he g ound s a e o 2 and 8, whe eas
he iple s a e is he g ound spin s a e in 4 and 6 elec on sys ems. The si ua ion
o 10-elec on sys em is again a iple g ound s a e, bu wi h a gap smalle han
he one ound o 4 and 6 elec ons. Ou esul s can be eadily a ionalized in e ms
o he ollowing o bi al o de ing 1
s <
1
p <
1
d
, wi h a dec ease in he successi e
ene gy gaps.
We ha e also obse ed ha he sc eening o elec on-elec on in e ac ion has
a sizable e ec , no only on he absolu e ene gies o each s a e, bu also on he
iple -single gap. The iple -single gap o he 2 elec on and 8 elec on cases is
specially sensible o he sc eening e ec , a ou ing he single s a e o e he iple .
This can be ela ed o an inc ease o he co esponding gap be ween shells upon
sc eening. Howe e , he in luence o sc eening in he iple -single gap o 4-, 6- and
10-elec on cases was much mo e educed, a ou ing again he single s a e bu , in
no case, his sc eening p oduced a swi ch be ween spin s a es.
54 sphe ical hooke a oms
Table 4.1:
HF and Full-CI ene gies (a.u.) o he single and iple spin s a es o he
wo-elec on Hooke a om (ω2=0.25; λee =0.0)
Basis Size Con ac ion Single HF Single FCI T iple FCI ∆T−S(eV)
aug-cc-pVDZ 9 (5s2p/3s2p) 2.087904 2.055213 2.386080 9.00
aug-cc-pVTZ 25 (7s3p2d/4s3p2d) 2.038634 2.004107 2.373537 10.05
aug-cc-pVQZ 55 (8s4p3d2 /5s4p3d2 ) 2.038443 2.001484 2.383204 10.39
aug-cc-pV5Z* 75 (9s5p4d3 /6s5p4d3 ) 2.038408 2.000685 2.373767 10.15
aug-cc-pV6Z* 95 (11s6p5d4 /7s6p5d4 ) 2.038423 2.000426 2.362738 9.86
aug-cc-pV5Z 105 (9s5p4d3 2g/6s5p4d3 2g) 2.038400 2.000476 2.373701 10.16
aug-cc-pV6Z 182 (11s6p5d4 3g2h/7s6p5d4 3g2h) 2.038404 2.000196 2.362579 9.86
ETBS-6S 80 (4s4p4d4 /4s4p4d4 ) 2.038400 2.000396 2.359673 9.78
4.4 concluding ema ks 55
Table 4.2:
HF, CAS(2,
No b
) and MRMP2(2,
No b
) ene gies wi h he ETBS-6S basis se ,
in a omic uni s, o he single and iple spin s a es o he wo-elec on Hooke
a om. The RHF ene gy o he single s a e is 2.038400 and 2.038423 a.u. o
he ETBS-6S and aug-cc-pV6Z* basis se s, espec i ely.
Basis No b CAS(Ne,No b)∆T−SMRMP2(Ne,No b)∆T−S
Single T iple (eV) Single T iple (eV)
2 2.026787 2.364478 9.19 2.006140 2.360420 9.64
3 2.016225 2.364478 9.48 2.003759 2.360428 9.70
4 2.006551 2.364478 9.74 2.001197 2.360601 9.78
5 2.003410 2.362773 9.78 2.000701 2.360358 9.79
6 2.003056 2.361106 9.74 2.000670 2.359828 9.77
7 2.002705 2.361106 9.75 2.000637 2.359819 9.77
8 2.002358 2.361106 9.76 2.000601 2.359820 9.77
9 2.002019 2.361106 9.77 2.000559 2.359821 9.78
10 2.001681 2.361106 9.78 2.000516 2.359828 9.78
11 2.001630 2.359907 9.75 2.000513 2.359694 9.77
12 2.001280 2.359859 9.76 2.000481 2.359687 9.77
13 2.001082 2.359821 9.76 2.000461 2.359688 9.77
FCI 2.000396 2.359673 9.78
aug-cc-pV6Z* 13 2.001203 2.362886 9.84 2.000483 2.362749 9.86
aug-cc-pV6Z* FCI 2.000426 2.362738 9.86
Re e ence [80] 2.000000 2.359657 9.79
56 sphe ical hooke a oms
Table 4.3:
HF, CAS(4,
No b
) and MRMP2(4,
No b
) ene gies wi h ETBS-6S basis se , in
a omic uni s, o he single and iple spin s a es o he ou -elec on Hooke
a om. The RHF ene gy o he single s a e is 6.505162 and 6.517905 o he
EBTS-6S and aug-cc-pV6Z* basis se s, espec i ely.
Basis No b CAS(4, No b)∆T−SMRMP2(4,No b)∆T−S
Single T iple (eV) Single T iple (eV)
4 6.466497 6.420487 -1.25 6.393971 6.357443 -0.99
5 6.436891 6.409056 -0.76 6.395878 6.358352 -1.02
6 6.432908 6.395502 -1.02 6.395253 6.357424 -1.03
7 6.412845 6.387924 -0.68 6.391439 6.356035 -0.96
8 6.416955 6.379714 -1.01 6.392298 6.354322 -1.03
9 6.412845 6.371627 -1.12 6.391439 6.353215 -1.04
10 6.410052 6.368950 -1.12 6.391231 6.352928 -1.04
11 6.409228 6.365185 -1.20 6.391078 6.352579 -1.05
12 6.408315 6.361114 -1.28 6.390980 6.352180 -1.06
13 6.398720 6.359758 -1.06 6.390085 6.352024 -1.04
aug-cc-pV6Z* 13 6.412708 6.371770 -1.11 6.403958 6.364288 -1.08
Re e ence[41] 6.385543 6.348830 -1.00
Table 4.4:
HF, CAS(6,
No b
) and MRMP2(6,
No b
) ene gies wi h ETBS-6S basis se , in
a omic uni s, o he single and iple spin s a es o he ou -elec on Hooke
a om. The RHF ene gy o he single s a e is 12.253446 and 12.287010 o he
EBTS-6S and aug-cc-pV6Z* basis se s, espec i ely.
Basis No b CAS(6, No b)∆T−SMRMP2(6,No b)∆T−S
Single T iple (eV) Single T iple (eV)
6 12.177056 12.154962 -0.60 12.091183 12.055694 -0.97
7 12.171843 12.132887 -1.06 12.089361 12.051390 -1.03
8 12.147118 12.117039 -0.82 12.086838 12.050945 -0.98
9 12.137934 12.099985 -1.03 12.085660 12.048808 -1.00
10 12.132401 12.094292 -1.04 12.084591 12.047936 -1.00
11 12.126035 12.087898 -1.04 12.083819 12.047183 -1.00
12 12.120721 12.082893 -1.03 12.083437 12.046636 -1.00
13 12.114932 12.077925 -1.01 12.082804 12.046141 -1.00
aug-cc-pV6Z* 13 12.149632 12.110496 -1.06 12.120797 12.082391 -1.05
Re e ence[62] 12.066294 12.031275 -0.95
4.4 concluding ema ks 57
Table 4.5:
HF, CAS(8,
No b
) and MRMP2(8,
No b
) ene gies wi h ETBS-6S basis se , in
a omic uni s, o he single and iple spin s a es o he ou -elec on Hooke
a om. The RHF ene gy o he single s a e is 19.190980 and 19.247016 o he
EBTS-6S and aug-cc-pV6Z* basis se s, espec i ely.
Basis No b CAS(8, No b)∆T−SMRMP2(8,No b)∆T−S
Single T iple (eV) Single T iple (eV)
8 19.113340 19.395160 7.67 19.008248 19.272166 7.18
9 19.083985 19.371303 7.82 19.003255 19.270391 7.27
10 19.075987 19.362461 7.79 19.000736 19.268463 7.28
11 19.069697 19.353362 7.72 19.000079 19.269118 7.32
12 19.063989 19.343947 7.62 18.999332 19.269118 7.34
13 19.053174 19.334793 7.66 18.997973 19.268892 7.37
aug-cc-pV6Z* 13 19.110238 19.437774 8.91 19.065563 19.386165 8.72
Re e ence[39] 19.038 19.430 10.6
Table 4.6:
HF, CAS(10,
No b
) and MRMP2(10,
No b
) ene gies wi h ETBS-6S basis se , in
a omic uni s, o he single and iple spin s a es o he ou -elec on Hooke
a om. The RHF ene gy o he single s a e is 27.932821 and 28.174226 o he
EBTS-6S and aug-cc-pV6Z* basis se s, espec i ely.
Basis No b CAS(10, No b)∆T−SMRMP2(10,No b)∆T−S
Single T iple (eV) Single T iple (eV)
10 27.828015 27.807466 -0.56 27.675164 27.665330 -0.27
11 27.813009 27.790373 -0.62 27.678880 27.664692 -0.39
12 27.795515 27.773761 -0.59 27.688082 27.663346 -0.67
13 27.777241 27.757081 -0.55 27.689185 27.661990 -0.74
aug-cc-pV6Z* 13 28.020989 28.007019 -0.38 27.951539 27.937691 -0.38

58 sphe ical hooke a oms
Table 4.7:
HF, CAS(
Ne
,10) and MRMP2(
Ne
,10) ene gies, in a omic uni s, o he single
and iple spin s a es o he s-Hooke model a di e en alues o he
λ
sc eening
pa ame e .
λHF CAS(Ne, 10) ∆T−SMRMP2(Ne,10) ∆T−S
Single Single T iple (eV) Single T iple (eV)
2-elec on
0.0 2.038400 2.001681 2.361106 9.78 2.000516 2.359828 9.78
0.2 1.879433 1.845903 2.212892 9.99 1.844644 2.211584 9.98
0.4 1.777898 1.750206 2.131426 10.37 1.748859 2.130299 10.38
0.6 1.710108 1.688101 2.084166 10.78 1.686715 2.083500 10.80
0.8 1.663091 1.645841 2.055639 11.15 1.644470 2.055161 11.17
1.0 1.629427 1.615941 2.037805 11.48 1.614631 2.037480 11.51
4-elec on
0.0 6.505162 6.410052 6.368950 -1.12 6.391231 6.352928 -1.04
0.2 4.447821 5.522013 5.481478 -1.10 5.502540 5.464743 -1.03
0.4 5.098802 5.027175 4.988257 -1.06 5.008695 4.972340 -0.99
0.6 4.786937 4.729994 4.692890 -1.01 4.712556 4.678623 -0.92
0.8 4.584830 4.539335 4.505120 -0.93 4.524068 4.492776 -0.85
1.0 4.447821 4.411219 4.380074 -0.85 4.398160 4.369611 -0.78
6-elec on
0.0 12.253446 12.133679 12.094292 -1.07 12.084454 12.047936 -0.99
0.2 10.066994 9.960398 9.921242 -1.07 9.909188 9.873006 -0.98
0.4 8.877716 8.792459 8.754633 -1.03 8.745296 8.710196 -0.96
0.6 8.171402 8.104086 8.069290 -0.95 8.064118 8.030753 -0.91
0.8 7.723094 7.671069 7.638238 -0.89 7.636670 7.605815 -0.84
1.0 7.424057 7.382940 7.353003 -0.81 7.354441 7.326141 -0.77
8-elec on
0.0 19.190980 19.075987 19.362461 7.79 19.000736 19.268463 7.28
0.2 15.182763 15.084400 15.384014 8.15 15.005631 15.289623 7.73
0.4 13.068595 12.994991 13.317898 8.79 12.922972 13.235696 8.51
0.6 11.834935 11.781507 12.128228 9.43 11.719577 12.060148 9.27
0.8 11.060156 11.021423 11.389781 10.02 10.969653 11.334431 9.93
1.0 10.547028 10.518701 10.905855 10.53 10.475994 10.861138 10.48
10-elec on
0.0 27.932821 27.828015 27.807466 -0.56 27.675164 27.665330 -0.27
0.2 21.621750 21.528857 21.511339 -0.48 21.375745 21.363283 -0.34
0.4 18.408818 18.334295 18.317410 -0.46 18.203431 18.190047 -0.36
0.6 16.573893 16.515099 16.499172 -0.43 16.408780 16.395457 -0.36
0.8 15.437499 15.390866 15.376062 -0.40 15.305881 15.293178 -0.35
1.0 14.692340 14.654914 14.641306 -0.37 14.587232 14.575380 -0.32
4.4 concluding ema ks 59
Table 4.8:
Decomposi ion o he o al ene gy (a.u.) o he single and iple s a es in o
di e en con ibu ions, kine ic ene gy, con inemen ene gy ( ha is he one
co esponding o he monoelec onic ha monium con inemen ope a o ), and
elec on-elec on epulsion ene gy. Decomposi ion o he iple -single gap
∆T−S
in o kine ic (
∆K
T−S
), con inemen (
∆Vw
T−S
) and elec on-elec on epulsion e ms
(
∆Vee
T−S
). Two cases a e conside ed: i) Hooke a om wi h s anda d Coulombic
in e ac ions be ween elec ons and ii) s-Hooke in which he elec on-elec on in-
e ac ions a e sc eened by he Yukawa po en ial wi h
λee
=1.0.
∆s−u
co esponds
o he di e ences be ween he sc eened and unsc eened calcula ion.
NeModel Single Ene gy Componen s (a.u.) T iple Ene gy Componen s (a.u.) T iple -Single Gap (eV)
Kin. E. Vcon Vee To al E. Kin. E. Vcon Vee To al E. ∆K
T−S∆Vw
T−S∆Vee
T−S∆T−S
2 Hooke 0.664231 0.888647 0.448803 2.001681 0.920322 1.093808 0.346977 2.361106 6.97 5.58 -2.77 9.78
s-Hooke 0.702495 0.815055 0.098391 1.615941 0.974102 1.027525 0.036178 2.037805 7.39 5.78 -1.69 11.48
∆s−u0.038264 -0.073592 -0.350412 -0.385740 0.053780 -0.066283 -0.310799 -0.323301 0.42 0.20 1.08 1.70
%∆s−u5.8 -8.3 -78.1 -19.3 5.8 -6.06 -89.6 -13.7
4 Hooke 1.575643 2.661960 2.172449 6.410052 1.575087 2.648067 2.145797 6.368950 -0.02 -0.38 -0.73 -1.12
s-Hooke 1.802488 2.247689 0.361041 4.411219 1.810012 2.234690 0.335372 4.380074 0.20 -0.35 -0.70 -0.85
∆s−u0.226845 -0.414271 -1.811408 -1.998833 0.234925 -0.413377 -1.810425 -1.988876 0.22 0.03 0.03 0.27
%∆s−u14.4 -15.6 -83.4 -31.2 14.9 -15.61 -84.4 -31.2
6 Hooke 2.298682 4.810967 5.024030 12.133679 2.296848 4.797225 5.000219 12.094292 -0.05 -0.37 -0.65 -1.07
s-Hooke 2.815149 3.794034 0.773756 7.382940 2.821571 3.781367 0.750065 7.353003 0.17 -0.30 -0.64 -0.81
∆s−u0.516467 -1.016933 -4.250274 -4.750739 0.524723 -1.015858 -4.250154 -4.741289 0.22 0.03 0.01 0.26
%∆s−u22.5 -21.1 -84.6 -39.2 22.8 -21.18 -85.0 -39.2
8 Hooke 2.889288 7.320715 8.865985 19.075987 3.129535 7.489993 8.742933 19.362461 6.54 4.61 -3.35 7.79
s-Hooke 3.755694 5.445591 1.317417 10.518701 4.034318 5.638694 1.232843 10.905855 7.58 5.25 -2.30 10.53
∆s−u0.866406 -1.875124 -7.548568 -8.557286 0.904783 -1.851299 -7.510090 -8.456606 1.04 0.64 1.05 2.74
%∆s−u30.0 -25.6 -85.1 -44.9 28.9 -24.72 -85.9 -43.7
10 Hooke 3.857152 10.517423 13.453440 27.828015 3.857015 10.511402 13.439049 27.807466 0.00 -0.16 -0.39 -0.56
s-Hooke 5.168257 7.609567 1.877090 14.654914 5.171068 7.603958 1.866280 14.641306 0.08 -0.15 -0.29 -0.37
∆s−u1.311105 -2.907856 -11.576350 -13.173101 1.314053 -2.907444 -11.572769 -13.166160 0.08 0.01 0.10 0.19
%∆s−u34.0 -27.6 -86.0 -47.3 34.1 -27.66 -86.1 -47.3
60 sphe ical hooke a oms
Table 4.9:
Ha ee-Fock and CASSCF ene gies, in a omic uni s, o he single and iple
spin s a es o he 4, 6 , 8 and 10-elec on sys ems. Calcula ions done a he
CAS-SCF(Ne,13) le el o heo y.
Basis Ha ee-Fock CASSCF(Ne,13) ∆T−S
Single Single T iple (eV)
4 elec ons
aug-cc-pVTZ 6.508926 6.431129 [1s21p2] 6.378120 [1s21p2] -1.44
aug-cc-pVQZ 6.534156 6.439280 [1s21p2] 6.396548 [1s21p2] -1.16
aug-cc-pV5Z 6.539774 6.436668 [1s21p2] 6.394851 [1s21p2] -1.14
aug-cc-pV5Z* 6.539818 6.436879 [1s21p2] 6.394866 [1s21p2] -1.14
aug-cc-pV6Z* 6.517905 6.412708 [1s21p2] 6.371770 [1s21p2] -1.11
6 elec ons
aug-cc-pVTZ 12.285452 12.195292 [1s21p4] 12.147453 [1s21p4]-1.30
aug-cc-pVQZ 12.282169 12.167338 [1s21p4] 12.125916 [1s21p4] -1.13
aug-cc-pV5Z 12.315804 12.183503 [1s21p4] 12.144069 [1s21p4] -1.07
aug-cc-pV5Z* 12.315906 12.185233 [1s21p4] 12.145583 [1s21p4] -1.08
aug-cc-pV6Z* 12.287010 12.149632 [1s21p4] 12.110496 [1s21p4] -1.06
8 elec ons
aug-cc-pVTZ 19.337940 19.271538 [1s21p6] 19.714647 [1s21p52s1] 12.06
aug-cc-pVQZ 19.204578 19.104295 [1s21p6] 19.505586 [1s21p52s1] 10.92
aug-cc-pV5Z 19.261715 19.131805 [1s21p6] 19.493381 [1s21p51d1] 9.84
aug-cc-pV5Z* 19.261851 19.137723 [1s21p6] 19.528024 [1s21p52s1] 10.62
aug-cc-pV6Z* 19.247016 19.110238 [1s21p6] 19.437774 [1s21p51d1] 8.91
10 elec ons
aug-cc-pVTZ 28.699209 28.615431 [1s21p62s2] 29.258914 [1s21p62s11d1] 17.51
aug-cc-pVQZ 28.186450 28.066966 [1s21p62s2] 28.369080 [1s21p62s11d1] 8.22
aug-cc-pV5Z 28.197306 28.039504 [1s21p62s2] 28.063383 [1s21p62s11d1] 0.65
aug-cc-pV5Z* 28.197450 28.050011 [1s21p62s2] 28.157144 [1s21p62s11d1] 2.92
aug-cc-pV6Z* 28.174226 28.020989 [1s21p61d2] 28.007019 [1s21p61d2] -0.38
* Modi ied basis se emo ing basis unc ions wi h l≥4
Chap e 5
ML ASSISTED PHASE DIAGRAMS
In his chap e we p esen a sys ema ic p ocedu e o build phase diag ams o chemi-
cally ele an p ope ies by he use o a semi-supe ised machine lea ning echnique
called unce ain y sampling. Conc e ely, we ocus on g ound s a e spin mul iplici y
and chemical bonding p ope ies. As a i s s ep, we ha e ob ained single-eu ec ic-
poin -con aining solid-liquid sys ems which ha e been sui able o con as ing he
alidi y o his app oach. Once his was se led, on he one hand, we ha e buil
magne ic phase diag ams o se e al Hooke a oms con aining ew elec ons (4 and
6) apped in sphe oidal ha monic po en ials. Changing he pa ame e s o he con-
inemen po en ial such as cu a u e and aniso opy and in e elec onic in e ac ion
s eng h, we ha e been able o ob ain and a ionalise magne ic phase ansi ions
lipping he g ound s a e spin mul iplici y om single (non magne ic) o iple
(magne ic) s a es. On he o he hand, Bade ’s analysis is pe o med upon helium
dime s con ined by sphe ical ha monic po en ials. Co alency is s udied using desc ip-
o s as sign o
∆ρ( C)
and
H( C)
and he dependency on he deg ees o eedom
o he sys em is s udied i.e. po en ial cu a u e
ω2
and in e a omic dis ance
R
. As
a esul , we ha e obse ed ha he e may exis a co alen bond be ween He a oms
o sho enough dis ances and s ong enough con inemen . This machine lea ning
p ocedu e could, in p inciple, be applied o he s udy o o he chemically ele an
p ope ies in ol ing phase diag ams, sa ing a lo o compu a ional esou ces.
Published: Xabie Telle ia-Allika, Jose M. Me ce o, Xabie Lopez, and Jon M. Ma x-
ain, ”Building machine lea ning assis ed phase diag ams: Th ee chemically ele an
examples”, AIP Ad ances 12, 075206 (2022) h ps://doi.o g/10.1063/5.0088784
61
68 ml assis ed phase diag ams
Figu e 5.2: Flowcha o he gene al p ocedu e o phase diag am cons uc ion.
3375 g id poin s. Finally, in ela ion o He dime s, b oad anges in in e a omic dis ance
R∈[
1.0, 4.0
]˚
A and con inemen s eng h
ω2∈[
0.0, 4.0
]
whe e employed gi ing ise o a
o al numbe o 250000 g id poin s.
S ep 2: We selec he ini ialisa ion poin s in he g id buil in s ep 1. Fo he solid-liquid
sys em, a o al numbe o 50 uni o m dis ibu ed poin s whe e selec ed aking he Ca esian
p oduc o 5 equidis an poin s o
x
and 10 equidis an poin s o
T
. Wi h espec o
Hooke a oms, by hypo hesis, he edges o he pa ame e g id would con ain all di e en
phases; he e o e, since h ee pa ame e s we e employed, he ini ialisa ion se was ob ained
using he ex eme 8 poin s. Meanwhile, o he He dime s, he ex ema o he g id we e
conside ed along wi h 100 poin s selec ed a andom.
S ep 3: Once a gi en poin in he pa ame e g id is selec ed (ei he he ini ialisa ion
ones o a new one coming om s ep 5), we ca y ou so ing ee o equi ed elec onic
s uc u e calcula ions in
GAMESS US
and we assign a label o his poin depending on he
esul ob ained ei he by he so ing ee, he sign o he ene gy gap
∆T−S
o densi y
laplacian o o al ene gy densi y.
S ep 4: We p opaga e he label in o ma ion om he da a poin s o he unlabelled
ones using 200 ees o o m he RF.
S ep 5: Wi h he in o ma ion om he p e ious s ep, we compu e Shannon’s en opy
o all guessed poin s in he g id and selec he one wi h he highes sco e. We ake his
poin ou o he unlabelled poin s pool
U
, pe o m elec onic s uc u e calcula ions as in
s ep 3 and upda e he labelled da abase L.

5.3 esul s and discussion 69
5.3 RESULTS AND DISCUSSION
5.3.1
Solid-liquid phase diag ams wi h a single eu ec ic poin
In a i s s ep, in o de o es he alidi y and use ulness o his app oach, we ha e
s udied se e al solid-liquid phase diag ams o sys ems which include a single eu ec ic poin
conside ing he equi ed idealisa ions and app oxima ions. Thus, we ha e employed he
me hodology p esen ed in wo k [219] and ha e chosen some examples in ol ing bina y
mix u es: Ag/Si, KNO
3
/LiNO
3
and K
2
SO
4
/Li
2
SO
4
. The equi ed da a we e ob ained
om he same sou ce [219].
Fo all cases, as a nai e app oach, we ha e compu ed g ids in he
(x1
,
T)
plane aking 100
poin s o
x1
in he [0,1] ange and 500 poin s o
T
in he [0.25
×min(T ,1
,
T ,2)
, 1.25
×
max(T ,1
,
T ,2)
] ange such ha he o al numbe o g id poin s was 50000. Fo he
ini ialisa ion se , we ha e aken ano he g id wi h 5 poin s o
x1
and 10 poin s o
T
in he
same anges as he o me one; hen, we ha e sampled o he 100 poin s using unce ain y
sampling which gi es a o al numbe o 150 sampled poin s. In o he wo ds, in o de o
build a a he accu a e phase diag am we ha e used a 0.3% o poin s compa ed o he
nai e all g id app oach. The ob ained phase diag ams a e ep esen ed in igu es 5.3, 5.4
and 5.5.
Figu e 5.3:
Ag/Si solid-liquid phase diag am. G een is he he e ogeneous solid phase
α+β
,
da k blue is he homogeneous ideal liquid mix u e
L
, ligh blue is
β+L
and
pu ple is α+L; he whi e do s ep esen he sampled poin s.
As i has been obse ed in all h ee s udied cases, he unce ain y sampling me hod
samples mo e ho oughly he egion whe e he la ge numbe o phases coexis , i.e. nea he
eu ec ic poin . Finding eu ec ic composi ions is c ucial o some pha macological [228–230]
70 ml assis ed phase diag ams
Figu e 5.4:
KNO
3
/LiNO
3
solid-liquid phase diag am. G een is he he e ogeneous solid
phase
α+β
, da k blue is he homogeneous ideal liquid mix u e
L
, ligh blue is
β+Land pu ple is α+L; he whi e do s ep esen he sampled poin s.
Figu e 5.5:
K
2
SO
4
/Li
2
SO
4
solid-liquid phase diag am. G een is he he e ogeneous solid
phase
α+β
, da k blue is he homogeneous ideal liquid mix u e
L
, ligh blue is
β+Land pu ple is α+L; he whi e do s ep esen he sampled poin s.
5.3 esul s and discussion 71
and g een chemis y [231, 232] applica ions. The e o e, his me hod shall be in e es ing o
disco e ing no el eu ec ic o mula ions.
5.3.2 G ound s a e spin mul iplici y o Hooke a oms
Fo hese sys ems, we ha e employed a g id in he
(λ
,
ω2
xy
,
ω2
z)
space wi h 15 poin s o each
dimension in he domain
[
0.05, 2.55
]×[
0.20, 0.30
]×[
0.20, 0.30
]
, he e o e a o al numbe o
3375 g id poin s. In his case, we ha e aken jus 58 poin s in he g id o be compu ed,
which is he 1.72 % o all poin s.
We shall s a ou discussion by desc ibing he ob ained g ound s a e spin mul iplici y
phase diag ams. In igu e 5.6 we ind magne ic phase diag ams o 4 (le hand side) and
6 ( igh hand side) elec on sys ems. The e ical axis ep esen s he elec on-elec on
in e ac ion sc eening pa ame e
λ
while he ho izon al one ep esen s he a io be ween
he con inemen along he
x/y
axis and he
z
axis
ε=ω2
x,y
ω2
z
. We shall p oceed o analyse
he ob ained diag ams.
On he one hand, o 4 elec on sys ems, one shall obse e ha single s (non magne ic
s a es labelled in ed) appea in egions o which
ω2
x,y
is la ge,
ω2
z
is small and, o some
(ω2
xy
,
ω2
z)
poin s close o he bounda y, hey a e mo e abundan as
λ
inc eases. Unde he
aniso opic po en ial induced by he ac ha
ωx,y> ωz
he one body ene gies along he
z
axis a e smalle han in he
x
,
y
plane and in i ue o he
Au bau
p inciple, he o me
will be occupied i s ly gi ing ise o single s a es. Gi en an aniso opic enough po en ial
o which he g ound s a e is s ill a iple (as in he sphe ical case), we may s ill ob ain a
ansi ion o he single s a e by inc easing he sc eening po en ial
λ
as by doing so we a e
u ning o he exchange in e ac ion which does no compensa e he high spin s a e and
one body in e ac ions will impose single spin s a es.
On he o he hand, o 6 elec on sys ems i is ob ious ha some ea u es o he phase
diag am di e om he one ob ained o he 4 elec on sys em. Fi s , he g ound s a e spin
mul iplici y beha es in he opposi e way as compa ed o he case o 4 elec ons; in his
case, single s a es appea o small alues o
ω2
x,y
cu a u es and a he same ime la ge
alues o
ω2
z
. Besides, he beha iou wi h espec o he elec on in e ac ion sc eening
pa ame e
λ
is he same as in he p e ious case, he highe i is ( o a couple o sui able
cu a u es (
ω2
x,y< ω2
z
) he mos likely i is o ha po en ial o gi e ise o a single spin
mul iplici y o he elec onic g ound s a e.
In bo h cases, o la ge alues o he Yukawa-like sc eening pa ame e in he elec on-
elec on in e ac ion
λ
, quan um exchange in e ac ion becomes weake and bo h sys ems
show g ound s a e spin mul iplici y ansi ion in he close neighbou hood o he e ical
line
ε=
1. Besides, in he low
λ
alues egime whe e exchange in e ac ion a e no negligible,
in bo h cases we may obse e ha spin ansi ion does no happen as soon as we dis o
he sphe ical symme y o he po en ial; we obse e he e is an ine ia o hold he high
spin s a e ( iple
S
=1 agains single
S
=0). This is a na u al consequence o exchange
in e ac ion as i lowe s he o al ene gy o he sys em ia same spin pa icles; hus he
mo e pa icles wi h same spin he e a e, he mo e s able he sys em becomes.
Taking i up o he non-in e ac ing elec on sys em
(λ→ ∞)
, he Hamil onian is only
composed by one-pa icle ope a o s o which he o bi al ene gy eigen alues in a omic uni s
a e gi en by 5.8. In his case, depending on he asymme y pa ame e
ε
which is de ined
72 ml assis ed phase diag ams
Figu e 5.6:
Sampled poin s in he (
ϵ
,
λ
) plane. Red indica es single g ound spin s a e while
blue indica es iple g ound spin s a e
as he a io be ween he axial con inemen pa ame e s
ω2
x,y
and
ω2
z
, he ene gy o he
p
o bi als will spli as i is shown in Figu e 5.7. Taking in o accoun he Pauli’s exclusion
p inciple, we may i up o a couple o elec ons in each o bi al and, acco ding o Hund’s
ule, in case o degene acy hey will occupy degene a e o bi als such ha o al spin is
maximised. Keeping hese wo many-body quan um ules in mind, i is easy o see ha
he g ound s a e spin mul iplici y does no only depend on he con inemen po en ial bu
also in he numbe o elec ons.
ϵnx,ny,nz=ωx,y(nx+ny+1) + ωznz+1
2(5.8)
When he con inemen is sha pe along he
z
di ec ion han in he
x
and
y
di ec ions
(ω2
z> ω2
x,y)
( his is sphe ici y pa ame e
ε <
1), he degene acy o he
p
o bi al spli s in o
wo main g oups, he
pz
o bi al (highe in ene gy) and he plane composed by he s ill
degene a e
px
and
py
(lowe in ene gy). The e o e, when we dis o he sphe ical symme y
in his way no hing essen ial happens o 4 elec on sys ems as a as g ound s a e spin
mul iplici y is conce ned. Howe e , o six elec on sys ems, as
ε
ge s smalle , he ene gy o
he
pz
o bi al goes highe and we ob ain a poin in which all
px
and
py
o bi als a e doubly
occupied while
pz
o bi al is emp y. I we a e o i all 6 elec ons in his po en ial, we shall
see ha he g ound s a e spin mul iplici y is he single ; hus, magne ic p ope ies a e
al e ed in he p ocess o comp ising he sphe e.
On he con a y, ins ead o comp essing he sphe e, i we s e ch i along he
z
axis we
ob ain a smalle cu a u e along his di ec ion
(ω2
z< ω2
x,y)
and he sphe ici y pa ame e
becomes
ε >
1. Once again, acco ding o he one-body ene gy o mula
(5.8)
his implies he
5.3 esul s and discussion 73
Figu e 5.7: Schema ic a omic o bi al ene gy spli ing along asymme y pa ame e ϵ
o bi al ene gies, degene a e in he sphe ical case, spli in o wo b anches: he
pz
o bi al ( his
ime lowe in ene gy) and he plane composed by he s ill degene a e
px
and
py
(highe in
ene gy). Applying he occupancy ules, one shall see ha o 4 elec on sys ems he g ound
s a e mul iplici y is he single (all pai ed elec ons) while o 6 elec on sys ems he e is no
change as long as g ound s a e mul iplici y is conce ned. As opposed o he p e ious case,
he magne ic p ope ies o he 4 elec on sys em al e along his ans o ma ion while he
6 elec on sys em is able o keep i s magne ic p ope ies.
5.3.3
De ec ing co alen bonding in sphe ically con ined
He2
sys ems
In a i s s ep, we ha e buil he co alency phase diag ams shown in Figu e 5.8 a HF/aug-
cc-pVTZ le el using as label he sign ( ed o nega i e and blue o posi i e) o he
laplacian o he densi y a he bond c i ical poin ∆ρ( C)and he o al ene gy densi y a
he bond c i ical poin
H( C)
. In o al, we ha e compu ed 500
×
500 g id poin s in he
[
0.10, 4.00
]×[
1.00, 4.00
]
in e al in he
(ω2
,
R)
plane aking 104 poin s o ini ialisa ion
and 200 mo e o building he diag am he e o e using 0.1216% o all g id poin s.
In hese diag ams we ep esen he con inemen cu a u e
ω2
in he e ical axis and he
in e a omic dis ance
R
in he ho izon al one. As i can be seen o
∆ρ( C)
, nega i e alues
a e ob ained in egions whe e he in e a omic dis ance lays a ound 1.00 and 1.52
˚
A
and
con inemen cu a u es la ge han 2.00. On he o he hand, as a as
H( C)
is conce ned,
we ha e ound nega i e alues o all in e a omic dis ances sho e han 1.50
˚
A
. Beyond
his h eshold, we may obse e ha o la ge enough ha monic con inemen cu a u e,
he sign o he o al ene gy densi y a he BCP swi ches om posi i e o nega i e; he
cu a u e equi ed o make his swi ch happen is la ge as we inc ease he in e a omic
dis ance be ween he He a oms.
A his le el o heo y, o he ange in which he e a e co alency indica o s, ou model
comes in e ms wi h p e ious wo ks based on all-a om calcula ions on simila sys ems. Fo
example o
He2
@
C20H20
whe e he He-He dis ance is epo ed o be 1.265
˚
A
and posi i e

74 ml assis ed phase diag ams
Figu e 5.8:
Sign o o al ene gy densi y o He
2
sys em compu ed a HF/aug-cc-pVTZ le el,
ed is o nega i e and blue is o posi i e.
alue o
∆ρ( C)
[210] which is compa ible wi h ou calcula ions o
ω2<
2.00. Also o
in e a omic dis ances smalle han 1.60
˚
A
, o
He2
@
B12N12
and
He2
@
B16N16
sys ems 1.306
˚
A
and 1.456
˚
A
dis ances a e epo ed [216]. Fo hese wo sys ems, posi i e
∆ρ( C)
a e
epo ed while o al ene gy densi y is nega i e in he o me (a ound 1
×
10
−2
a.u.) and
ze o in he la e which comes along wi h ou esul s. Fo la ge dis ances, in a ecen wo k
1.520
˚
A
and 1.546
˚
A
dis ances a e epo ed o
He2
@
C36
and
He2
@
C40
espec i ely [212];
he epo ed sign o
∆ρ( C)
and
H( C)
a e posi i e (o he o de o 5
×
10
−3
a omic uni s).
Finally, He a om couples con ined in
B40
cages ha e been epo ed o be 1.672
−
1.640
˚
A
apa [217] and
∆ρ( C)
and
H( C)
bo h u ned ou o be posi i e which is compa ible
wi h ou esul s o small ω2 alues.
So a , esul s conce ning he na u e o bonding in helium dime ob ained by ou
ha monic con inemen model seem o ag ee wi h all a om models o se e al cages. Bea ing
in mind ha eal cages may induce a a he small po en ial cu a u e
ω2
and he ac
ha swi ch in he sign o
H( C)
happens somewhe e in he in e al
[
1.40, 2.60
]˚
A
o
in e nuclea dis ance, we shall ocus ou a en ion on s udying his egion and include
elec on-co ela ion e ec s in ou calcula ions. Hence, we ha e pe o med CASSCF(4,8)/aug-
cc-pVTZ calcula ions in his egion. To do so, we ha e aken a 20
×
20 g id poin s in he
(ω2
,
R)
plane wi h e enly sepa a ed poin s in he
(
0.00, 1.00
)×(
1.40, 2.6
)
domain. Using
hese poin s, we ha e compu ed le el maps as in Figu e 5.9. In he h ee maps, we indica e
he in e a omic dis ance in he ho izon al axis, he con inemen cu a u e in he e ical
axis and we ha e compu ed le el maps o h ee Bade desc ip o s ( om le o igh ):
∆ρ( C)
,
H( C)
and
−G( C)/V( C)
which ake alues in di e en anges acco ding o
each desc ip o labelled in ed o low alues and in blue o highe ones. As we can see o
∆ρ( C)
, we ha e ound posi i e alues in his domain and i inc eases as he in e a omic
5.3 esul s and discussion 75
dis ance is sho e an i is no highly dependen on
ω2
specially o a he sho in e a omic
dis ances. Fo
H( C)
we ha e ound nega i e alues which ge smalle (mo e nega i e)
as we sho en he in e nuclea dis ance and inc ease he con inemen cu a u e which is
compa ible wi h p io HF calcula ions as well as wi h all-a om calcula ions in which mo e
nega i e alues o his pa ame e appea when he size o he con ining cage is smalle .
Figu e 5.9:
Co alency indica o s o MMPT2(4,8)/aug-cc-pVTZ calcula ed densi ies o
sphe ical con inemen , om le o igh :
∆ρ( C)
,
H( C)
and
−G( C)/V( C)
.
Red shades indica e low alues, whi e indica es medium and blue shades indica e
high alues.
76 ml assis ed phase diag ams
5.4 CONCLUDING REMARKS
In his chap e we ha e been able o p oduce compu a ionally a o dable magne ic and
co alency phase diag ams o some ew elec on Hooke a oms and helium dime s con ined
in ha monic po en ials espec i ely by sampling he po en ial ea u e space by means
o semi-supe ised lea ning echnologies. Con a y o he con en ional way o explici ly
calcula ing e e y poin in a g id, we ha e aken se e al ini ialisa ion poin s and ha e
p opaga ed hei in o ma ion using classi ie s based on Random Fo es , which, compa ed o
label p opaga ion algo i hm, has enabled us o s udy dense g ids. By jus calcula ing he
poin s whe e in o ma ion is maximal, we ha e been able o educe he numbe o equi ed
compu a ions (in ou case he equi ed poin s o compu a ion we e below 2% o all g id
poin s).
The alida ion o he me hod employing RFs was pe o med by compu ing analy ical
ideal solid-liquid phase diag ams in ol ing a single eu ec ic poin . As i has been shown,
his me hod samples all in e phases, specially he neighbou hood o he eu ec ic poin in
which all ou phases coexis .
F om p e ious wo ks on 4 and 6 elec on sys ems con ined in sphe ical ha monic
po en ials wi h cu a u e
ω2=
0.25, i is known ha he g ound s a e spin mul iplici y
o hese sys ems is a iple (
S
=1) and he i s single (
S
=0) lays a ound 1 eV abo e
in ene gy. I we conside he elec ons do no in e ac among hem, he whole sys em
beha es as a sys em composed by
n
bodies which s ill mus ul ill basic many-body sys ems
ea u es as: Pauli’s exclusion p inciple,
Au bau
illing p inciple and Hund’s ule. Unde
hese assump ions, we s a e ha g ound s a e spin mul iplici y will be imposed by he
symme y o he con ining po en ial and, by al e ing he symme y o i , we may ob ain
g ound s a e mul iplici ies ei he single s o iple s depending on he po en ial pa ame e s,
he s eng h o in e elec onic in e ac ions and he numbe o elec ons.
On he o he hand, Bade analysis pe o med upon con ined helium dime s by means o
ha monic po en ial seems o be a a he good model which cap u es he main ea u es o all
a om app oaches. As a as sign o densi y laplacian and o al ene gy densi y is conce ned,
all ends ha e been co ec ly desc ibed by his simple model which shows ha e ec i e
con inemen po en ials can be employed o cap u e he essence o complex con ined sys ems
and a he simple models can be employed o desc ibe hem. This ac enabled he usage
o high heo e ical le el compu a ions upon hese sys ems.
As a i s s ep o many o he applica ions o chemical in e es , we conclude hese
machine lea ning echniques may be use ul o classi ica ion and explo a ion.
Chap e 6
GAUSSIAN CONFINEMENT AND
CONNECTION TO HOOKE ATOMS
In his chap e , we ha e compu ed and implemen ed one-body in eg als conce ning gaussian
con inemen po en ials o e gaussian basis unc ions. Then, we ha e se an equi alence
be ween gaussian and Hooke a oms and we ha e obse ed ha , acco ding o single and
iple s a e ene gies, bo h sys ems a e equi alen o la ge con inemen dep h o a se ies
o e en numbe o elec ons
n=
2, 4, 6, 8 and 10. Unlike wi h ha monic po en ials, gaussian
con inemen po en ials a e dissocia i e o small enough dep h pa ame e ; his ea u e is
c ucial in o de o model phenomena such as ioniza ion. In his case, in addi ion o co e-
sponding Taylo -se ies expansions, he i s diagonal and sub-diagonal Pad´e app oximan
we e also ob ained, use ul o compu e he uppe and lowe limi s o he dissocia ion dep h.
Hence, his me hod in oduces new ad an ages compa ed o o he s.
Submi ed o IJQC
77
84 gaussian con inemen and connec ion o hooke a oms
he eg ession. As a as i s anha monic e ms
g
a e conside ed, hey a e ob ained by
aking he slope o he linea eg ession which -in he wo s case scena io- has an e o o
4
×
10
−4au
. I we ake a deepe insigh o he
g
alues, we may immedia ely no ice ha , o
a gi en spin s a e ei he single o iple , does no d ama ically change wi h he size o he
ac i e space while i is highly dependen on he numbe o elec ons
n
. Besides he numbe
o elec ons, his anha monic e m also depends on he spin s a e aking he wo elec on
sys em as he mos no o ious one. F om he p e iously exposed heo y his beha iou was
expec ed since
g
ep esen s a sum o e elec ons o an a e aged alue o a qua ic po en ial
wi h espec o a many-body no malised wa e- unc ion; he e o e,
g
condenses a lo o
in o ma ion abou he sys em: he cu a u e, he numbe o elec ons and he spin s a e.
In he wo s case scena io - he one o CASSCF(8,10)(S)/ETBS-6S calcula ions- he
eg ession co ela ion pa ame e was
R2=
0.9623. Howe e , his is a p e y odd case
and he a e age alue o his s a is ic is
R2=
0.9991. I can also be seen ha e en he
Hooke a om ene gy is compa able o he ones ob ained by o he me hods, he anha monic
con ibu ion
g
is qui e di e en e en i we compa e i o he one ob ained by including
dynamical co ela ion e ec s ia pe u ba ion me hods a he same heo y le el. We ha e
also obse ed ha as soon as he ac i e space size is augmen ed, he co ela ion pa ame e
ge s apidly close o 1 app oaching pe ec linea dependency.
6.3.2
Loosely con ined wo elec on sys ems wi h sc eened
Coulomb in e ac ion
Based on he ac ha wo elec on sys ems wi h single spin s a e ha e a leas one occupied
bound s a e, we ha e been wonde ing a which poin o gaussian po en ial dep h he whole
sys ems dissocia es (
EG(Vd
0) =
0). On op o his, we ha e also conside ed elec on-elec on
in e ac ion o be sc eened and in wha measu e i a ec s he loosely bound sys em’s s abili y.
Hence, we ha e modelled hese sys ems using Hamil onians as in equa ion
(6.17)
whe e
he con inemen Gaussian po en ial has been de ined as in he p e ious sec ion and he
elec onic Coulomb in e ac ion is eplaced by a Yukawa-like po en ial wi h exponen
λ >
0.
We ha e aken 10 alues o
λ
pa ame e in he e enly sepa a ed ange
[
0.10, 1.00
]
and 20
alues o
V0
also in a e enly sepa a ed ange
[−
1.50,
−
0.50
]
a MRMP2(2,13)/ETBS-6S
le el o heo y o single s a es; he esul s o hese calcula ions can be ound in igu e 6.1.
Poin s wi h posi i e ene gies in his plo a e somehow meaningless since posi i e ene gies
belong o dissocia ed sys ems (sca e ing s a es); ne e heless, ene gies a e posi i e and
eal since he basis unc ion hemsel es c ea e he Di ichle bounda ies. A a i s glimpse,
one shall obse e ha he dissocia ion limi dep h is smalle as
λ
is la ge ( he e o e
elec on-elec on in e ac ion is weake ).
Le us y o make sense o he ob ained esul s. As in he p e ious sec ion, we ha e
expanded ou Gaussian sys ems ene gy now aking an addi ional e m as in equa ion
(6.18)
whe e he anha monic con ibu ions
g1
and
g2
a e gene ally aken as posi i e despi e
he ac he sign is al e na ing in he o iginal Taylo -like se ies. Now, we shall ob ain
EH
,
g1
and
g2
om da a using linea eg ession (omi ing all
EG>
0 da a) ia small

6.3 esul s and discussion 85
Figu e 6.1:
Ene gies o gaussian con inemen wi h wo elec ons in single spin s a e o
se e al sc eening pa ame e λ alues.
squa es minimisa ion; om he esidues, we may no ice ha hey ollow an expec ed cubic
polynomial end due o he ac ha we ha e immed he Taylo -like se ies a ha o de .
H=−1
2
2
X
i=1∇2
i−
2
X
i=1
V0e−ω2
2V0 2
i+e−λ 12
12
(6.17)
EG+2V0=EH+g1
V0
+g2
V2
0
+Oω8
V2
0(6.18)
Once we ha e ob ained he coe icien s we can sol e he equa ion
(6.18)
o
EG
as in
equa ion
(6.19)
whe e we s ill ha e a Taylo -like se ies. Now, we can exploi an in e es ing
p ope y o he ene gy unc ion: in he Taylo -like se ies signs a e al e na ing, he e o e i
is a S iljies unc ion. Tha means i we ob ain he main Pad´e sequence (i.e. he sequence
con aining he diagonal and lowe diagonal Pad´e app oximan s
PN
N1
V0
and
PN
N+11
V0
espec i ely), we shall ob ain physically ele an alues such as he dissocia ion limi o he
sys em. Since we only ha e ob ained he i s h ee coe icien s o he Taylo -like se ies, he
highes o de main Pad´e sequence we shall ob ain is he one composed by he diagonal
app oximan
P1
1(V−1
0)
and he subdiagonal app oximan
P1
2(V−1
0)
gi en in equa ions
(6.20)
and
(6.21)
espec i ely. I is known ha he sequence
P1
2(V−1
0)
will con e ge o
he igh ene gy om bellow while
P1
1(V−1
0)
will con e ge om abo e. Fo ob aining he
dissocia ion limi , we ake hese Pad´e app oximan s sol e hem o
V0
ob aining he lowe
and uppe limi o he dissocia ion dep hs
Vd−
0
and
Vd+
0
espec i ely in e ms o he
physical quan i ies
EH
,
g1
and
g2
o a gi en sc eening pa ame e
λ
. All ob ained esul s
a e condensed in able 6.2.
86 gaussian con inemen and connec ion o hooke a oms
EG
V0
=−2+EH
V0
+g1
V2
0
+g2
V3
0
+Oω8
V4
0(6.19)
P1
11
V0=−2g1+EH−2EHV0
g1−EHV0
(6.20)
P1
21
V0=(4g1+2EH)V2
0−(4g2+4g1EH)V0
−(E2
H+2g1)V2
0+ (g1+2g2+EH)V0+ (g2EH−g2
1)
(6.21)
Table 6.2:
Hooke a om ene gy (
EH
), i s anha monic e ms (
g1
,
g2
) and bound dissocia ion
limi s ob ained o se e al sc eening pa ame e alues o 2 elec on sys ems
wi h single spin s a e a CASSCF(2,13)/ETBS-6S and MRMP2(2,13)/ETBS-6S
le els. All alues a e gi en in a omic uni s.
EHg1g2Vd−
0Vd+
0
λCASSCF MRMP2 CASSCF MRMP2 CASSCF MRMP2 CASSCF MRMP2 CASSCF MRMP2
0.1 1.9379 1.9372 -0.3839 -0.3839 0.0163 0.0164 0.731 0.730 0.771 0.770
0.2 1.8719 1.8712 -0.3829 -0.3830 0.0200 0.0201 0.689 0.688 0.731 0.731
0.3 1.8204 1.8197 -0.3802 -0.3803 0.0232 0.0233 0.657 0.657 0.701 0.701
0.4 1.7770 1.7753 -0.3712 -0.3691 0.0232 0.0220 0.635 0.634 0.680 0.680
0.5 1.7397 1.7394 -0.3578 -0.3587 0.0203 0.0208 0.618 0.617 0.664 0.663
0.6 1.7143 1.7135 -0.3568 -0.3569 0.0233 0.0234 0.603 0.603 0.649 0.648
0.7 1.6902 1.6890 -0.3487 -0.3482 0.0223 0.0222 0.593 0.592 0.639 0.638
0.8 1.6702 1.6686 -0.3418 -0.3406 0.0215 0.0211 0.584 0.584 0.630 0.630
0.9 1.6532 1.6514 -0.3354 -0.3339 0.0206 0.0201 0.578 0.577 0.624 0.624
1.0 1.6376 1.6369 -0.3280 -0.3281 0.0192 0.0192 0.573 0.572 0.619 0.618
I we ocus ou a en ion upon a gi en
λ
alue and s udy a gi en es ima ed physical
p ope y, we shall no ice ha including dynamic co ela ion e ec s ia pe u ba ion me hods
does no qui e make a big di e ence wi h espec o he same quan i y ob ained by egula
CASSCF me hod.
Now, as a as
EH
is conce ned, his ene gy is smalle as
λ
inc eases which is o be
expec ed o Hooke a oms. In con as o he s a ed o me esul s, in his e y wo k,
we ob ained a Hooke wo elec on single a om ene gy o
λ=
0.2, 0.4, 0.8 and 1.0 o be
EH=
1.8459, 1.7502, 1.6881, 1.6458 and 1.6159 a.u. espec i ely a CASSCF/ETBS-6S
while in able 6.2 he ob ained ene gies a e in a e age 0.025 a.u. highe . As we ha e
discussed in he p e ious sec ion, accu a e Hooke ene gies a e ob ained o deep po en ials,
ne e heless, in his sec ion we ha e been dealing wi h loosely con ined sys ems. The e o e,
on he basis o his app oxima ion, we may s a e ha ou es ima ions a e a he easonable
and bo h egimes ha e p e y unique ea u es.
As o he anha monic e ms,
g1
we may ob e se i also ge s smalle as
λ
inc eases. We
may hypo hesise ha as elec on-elec on in e ac ion ge s weake , co ela ion e ec s a e
also u ned o and elec ons a e mo e likely ound in he cen e o he po en ial well, hus
his i s anha monic e m becomes smalle . On he o he hand, he second anha monic
e m g2ge s a maximum o λ=0.6 and hen dec eases.
6.3 esul s and discussion 87
Figu e 6.2: Limi dissocia ion po en ials o se e al λ alues.
Finally, he lowe and uppe bounds o he limi ioniza ion po en ials ha e in he
wo s case scena io a 0.046 a.u. ampli ude as we may obse e ha we ha e p edic ed hei
beha iou in e ms o physical cons an s by se ing he co esponding Pad´e app oximan s
(6.20)
and
(6.21)
o ze o and sol ing o
V0
. We shall see ha hese limi po en ials a e
shallowe and asymp o ic o a limi alue a which only one-body in e ac ions a e ele an .
Thus, we ge an ob ious conclusion, as elec on-elec on in e ac ion is u ned o he
po en ials does no need o do so much ”wo k” o con ine he in e ac ing pa icles and
shallowe po en ials a e s ill able o con ine hem. We may ind a isual summa y in igu e
6.2.
88 gaussian con inemen and connec ion o hooke a oms
6.4 CONCLUDING REMARKS
In his wo k we ha e compu ed and implemen ed he equi ed one-body in eg als o
quan um pa icles con ined in gaussian po en ial wells o which he cen e o he basis
unc ion and he cen e o he po en ials do no need o coincide. Such implemen a ion
has been in e aced o elec onic s uc u e so wa e
GAMESS-US
so ha we can make use o
i s quan um chemical compu a ion machine y o s udy sys ems o elec ons con ined in
dissocia i e po en ials.
Fi s ly, we ha e pe o med compu a ions on deeply con ined sys ems (la ge
V0
pa-
ame e ) wi h con olled wid h pa ame e such ha he cu a u e o he po en ial a he
minimum poin was
ω2=
0.25. Since p e ious esul s on ha monic po en ials ha e been
well es ablished o
n=
2, 4, 6, 8 and 10 elec ons, by means o Taylo se ies we ha e shown
ou calcula ions a e compa ible wi h he o me ones.
Finally, we ha e s udied dissocia i e sys ems composed by wo elec ons in which he
con en ional Coulomb ope a o was subs i u ed by Yukawa po en ials. In his case, we ha e
no only ob ained he co esponding Taylo -se ies expansion bu also he i s diagonal and
sub-diagonal Pad´e app oximan which we e use ul o compu e he uppe and lowe limi s
o he dissocia ion dep h o se e al sc eening pa ame e s λ.
Chap e 7
SUMMARY, MAIN CONCLUSIONS
AND FUTURE WORKS
In his las chap e , we ha e no only summa ized he main esul s and conclusions ob ained
in he p esen esea ch wo k, bu also ha e ske ched new esea ch lines based on he gained
knowledge.
In pa icula , we ha e ocused ou a en ion on he possibili y o applying gaussian
con inemen po en ials so as o model endohed ally doped cage clus e s. In his case, he
doping a om is explici ly ep esen ed in he model Hamil onian while he con inemen
po en ial can be adap ed so as o mimic p ope ies o he all a om sys em. Hence, he po en ial
pa ame e s, namely
(V0
,
β)
can be chosen such ha ele an p ope ies like ioniza ion
ene gies, elec on a ini ies and single - iple gaps ob ained om employing ull Hamil onian
app oaches can be eco e ed om he simpli ied models.
Wi h he aim o ob aining accu a e esul s wi hou comp omising he compu a ional
capaci ies, he de eloped machine lea ning me hods can be adap ed and ex ended o building
eliable simpli ied models. On he one hand, op imiza ion schemes simila o he ones
desc ibed in chap e s 3 and 4 can be employed o ob ained op imal basis unc ions gi en
some inpu po en ial pa ame e s and doping agen ’s a omic numbe . On he o he hand,
i some p ope ies such as lowes laying spin s a e mul iplici y, he sign o single - iple
gap o he sign o magne ic coupling cons an s change by a ying he ex e nal po en ial
pa ame e s, me hods p esen ed in chap e 5 can be employed o building phase diag ams
o such p ope ies e icien ly.
89

90 summa y, main conclusions and u u e wo ks
7.1 MAIN CONCLUSIONS
In his hesis wo k some ad ances conce ning sys ems o elec ons con ined by s a ic
po en ials ha e been p o ided. Fi s , compu a ional p o ocols o s udying sys ems o
ew elec ons wi h se e al o al spin numbe s con ined in one, wo and h ee dimensional
ha monic po en ials ha e been de eloped. In pa icula , on he one hand, adequa e ei he
dis ibu ed o one-cen e e en- empe ed gaussian basis unc ions ha e been ob ained by
means o classical con ex op imiza ion echniques. On he o he hand, hese basis unc ions
along wi h he employed mul i e e en ial CASSCF and MRPT2 me hods ha e been p o en
o accu a ely desc ibe Wigne loca ion phenomenon o weak con inemen po en ials, g ound
s a e spin mul iplici ies, Au bau s uc u e and single - iple ene gy gaps.
Fo sphe ical Hooke a oms, based on he Au bau s uc u e o he sys em, he spin
mul iplici y o he lowes ene gy con igu a ion pu ely depends on he numbe o elec ons.
Howe e , i sphe ical symme y is b oken such ha he ha monic con inemen along a
gi en axis di e s om he con inemen along he pe pe ndicula plane (i.e.
ω2
z=ω2
x,y
),
he g ound s a e spin mul iplici y does no only depend on he numbe o elec ons, bu
also on how sphe ical symme y has been b oken. Gi en a numbe o elec ons, a machine
lea ning me hod based on semi-supe ised lea ning and unce ain y sampling has been
de elop in o de o ob ain g ound s a e spin mul iplici y phase diag ams e icien ly in he
space composed by he s a ed con inemen pa ame e s and he elec on-elec on sc eening
pa ame e . In addi ion, his me hod has also be p o en o be pe o ming when applied
on u he chemically ele an sys ems such as solid-liquid bina y sys ems and chemical
bonding analysis.
As an ex ension o ha monic con inemen po en ials, gaussian con inemen po en ials
ha e been in oduced so as o p ope ly desc ibe many-cen e molecula ea u es, disso-
cia ion p ocesses and anha monic con ibu ions. The equi ed one-body in eg als ha e
been implemen ed in
GAMESS US
; he alidi y o such implemen a ion has been p o en by
compa ing he equi alen Hooke a om ene gies wi h he ones ob ained in p e ious wo ks in
he li e a u e and ou own esul s.
The esul s and me hodologies de eloped in he p esen wo k can be ex ended o u he
sys ems such as ones consis ing on di e en con inemen egimes, la ge numbe o elec ons
wi h iche numbe o g ound s a e spin mul iplici ies and employed he ob ained models
o mo e ealis ic sys ems by app op ia ely i ing he po en ial pa ame e s o all-a om
sys ems.
7.2 u he wo ks 91
7.2 FURTHER WORKS
In a nu shell, in his hesis we ha e de eloped compu a ional p o ocols which lead o:
ob aining op imal basis se s o ha monically con ined one-, wo-, and h ee-dimensional
sys ems; implemen ing machine lea ning algo i hms o op imizing some o he s a ed basis
unc ions ob aining phase diag ams e icien ly and implemen ing and p oo ing he alidi y
o gaussian con inemen po en ials. Wi h he aim o ex ending he gained knowledge o
o he sys ems and based on he expe ise o ou g oup, we ha e conside ed o apply he
de eloped models on he s udy o endohed ally doped cage clus e s. We shall now in oduce
he wo main opics conce ning his wo k: endohed ally doped cage clus e s (so ela ed o
a i icial a oms o supe a oms) and quan um sys ems con ined by s a ic po en ials. A e
discussing some wo ks and esul s ob ained in bo h ields, we will es ablish a b idge be ween
bo h disciplines and p esen ou app oach o do so.
On he one hand, clus e s a e somewhe e be ween bulk ma e ials and molecules, which
makes hem ha e in e es ing elec onic p ope ies. Among hese sys ems, we may ind he
sphe ical hollow nanoclus e s which may be doped as i was i s ly done o lan hanum in
ulle ene-60 La@C
60
[270] ollowed by o he simila sys ems [271, 272] and he so in e es ing
N@C
60
which po en ially can be used as an a omic clock [273]. Besides ulle ene-based-
species, we may also ind sys ems called ”supe a oms” [274–277] as Al
12
cages which
depending on he dopan can gi e ise o supe halogens (using B as dopan ), supe alkali
me als (using P as dopan ), supe chalcogen (using Ca as dopan ) o s able 40 elec on species
using Si as dopan [278]. O he in e es ing sys ems a e he ones based on semiconduc o
clus e s o which con inemen -induced p ope ies such as unable abso p ion, emission and
pho oluminescence ha e been obse ed [279–285]. In line wi h supe a om sys ems, ”a omic”
p ope ies a ise om collec i e beha iou o he whole elec onic s uc u e; ne e heless,
some species such as a oms con ined in semi-conduc ing clus e s [286–290] o which he
clus e ac s only as a con inemen , a omic p ope ies a e e y simila o he ones obse ed
by isola ed dopan elemen s.
On he o he hand, gaussian po en ial wells ha e been employed in a he heo e ical
elec onic s uc u e s udies [252–256] as well as in mo e ”applied” condensed ma e s udies
conce ning: quan um do s [233–237], he mo-magne ic p ope ies [239–242] and in e ac ions
wi h elec ic and lase ields [243–248]. These po en ials a e p e y simila o ha monic
po en ials so employed in benchma k o elec onic s uc u e me hods [36, 37, 41, 80,
261–266] wi h he sub le di e ence ha hey ha e a ini e numbe o bound s a es and can
gi e ise o molecula s uc u e (chap e 6).
Bea ing hese wo concep s in mind, as a i s s ep, we would like o model con ined
alkali me al and halogen a oms in gaussian po en ials and s udy hei beha iou conce ning
ioniza ion ene gy and elec onic a ini y in ma e ial clus e s. We ha e ob ained app oxima ed
es ima es o such ene gies based on ee a oms’ ioniza ion ene gy, hei highes occupied
a omic o bi al and he gaussian con inemen pa ame e s by using he Helmann-Feyman
heo em as well as i s o de pe u ba ion me hods and checked he alidi y o such
app oxima ions by means o explici calcula ions.
92 summa y, main conclusions and u u e wo ks
7.2.1
Some p elimina y esul s conce ning endohed al sys-
ems
The gene al Hamil onian o an a om wi h nuclea cha ge
Z
and
n
elec ons con ined in a
gaussian po en ial wi h dep h and wid h pa ame e s
V0
and
β
espec i ely can be w i en
as in equa ion (7.1).
H=−1
2
n
X
i=1∇2
i−Z
n
X
i=1
1
i
+
n
X
j>i
1
ij −V0
n
X
i=1
e−β 2
i(7.1)
I we conside monode e minan al wa e unc ions, apply he adiaba ic app oxima ion
(Koopmans’ heo em) and use he Helmann-Feynman heo em, we ob ain an app oxima e
a ia ion o he ioniza ion po en ial wi h espec o con inemen dep h in e ms o he
con inemen po en ial pa amen e s and he n h a omic o bi al:
∂I
∂V0
=−*Ψion 
n−1
X
i=1
e−β 2
iΨion++*Ψa om 
n
X
i=1
e−β 2
iΨa om+
≈Dϕ( n)e−β 2
nϕ( n)E
Thus, i we conside
V0
as a small pa ame e , we may in eg a e he esul ob ained by
Helmann-Feynman heo em in he ange
V′
0∈[
0,
V0]
such ha we ob ain an app oxima e
alue o he ioniza ion po en ial
I
in e ms o he uncon ined ioniza ion po en ial
I0
, he
con inemen po en ial and he
n h
a omic o bi al as in
(7.2)
. O cou se, in his app oxima ion
we a e aking o g an ed ha he shape o he a omic o bi al and, in gene al, he wa e
unc ion is uncoupled o he po en ial pa ame e s
V0
and
β
which is no ue acco ding
o elec onic s uc u e compu a ions conce ning Hooke a oms (chap e s 4 and 5). Thus,
he cu a u e o he po en ial de ined as
ω2=
2
βV0
may be an in e es ing pa ame e o
u u e discussions.
As an al e na i e o mula ion, one shall apply pe u ba ion me hods up o i s o de
co ec ion o ene gy and keep he unpe u bed wa e unc ion using
V0
as he equi ed small
pa ame e . S ill in a monode e minan al o mula ion and using he adiaba ic app oxima ion,
i can be shown ha one eaches he same esul o app oxima ed ioniza ion ene gies.
I≈I0+V0Dϕ( n)e−β 2
nϕ( n)E(7.2)
Using a simila s a egy, i we conside he neu al a om is composed by
n
elec ons and,
he e o e, he anion by
n+
1 elec ons, elec on a ini y
A
can be de ined as he nega i e
sign o ioniza ion ene gy o he anionic species and he e o e i is s aigh o wa d o see
ha i can app oxima ely be exp essed as in equa ion (7.3).
A≈A0−V0
nDϕ( n+1)e−β 2
n+1ϕ( n+1)E(7.3)
7.2 u he wo ks 93
As i can be seen om bo h equa ions
(7.2)
and
(7.3)
, conside ing
V0
is small enough,
i
β
is e y small i.e. a low cu a u e con inemen , he e ec o he po en ial on he sys em
is such ha i shi s he ioniza ion ene gies by a ac o o
V0
. On he con a y, i
β
is la ge
enough, he po en ial will ha e no e ec on he sys em since, in he limi , i app oaches a
null measu e index unc ion.
Now, he key poin is o compu e he in eg al in equa ions
(7.2)
and
(7.3)
in which
in o ma ion abou elec onic s uc u e and con inemen po en ial is encoded. As i is
commonly done in quan um chemis y, we expand basis unc ions in e ms o con ac ed
gaussian p imi i e unc ions and by elec onic s uc u e compu a ions we ob ain he
coe icien s o such basis o bi als o ob ain a omic (molecula ) o bi als. Hence, we shall
ew i e he equi ed in eg al as a bilinea o m as in
(7.4)
whe e he ma ix
V
con ains
in o ma ion abou he in e ac ion be ween he
p h
and
q h
p imi i e gaussian unc ions wi h
he con inemen po en ial ( hese in eg als a e gi en in chap e 6), he ma ix
C
con ains
he
q h
p imi i e expansion coe icien s o he
i h
basis unc ion and he ec o
Φ
con ains
he expansion coe icien s o he basis unc ions o he
n h
elec on ob ained by elec onic
s uc u e calcula ions.
Dϕ( n)e−β 2
nϕ( n)E=X
jipq
c†
jcic(j)†
pc(i)
qZg†
pgqe−β 2
nd3 = (CΦ)†VCΦ(7.4)
Wi h he aim o explo ing i hese app oxima ions a e alid a HF/6-311++G le el
o alkali me als M=Li,Na,K and MP2/6-311++G le el o halogens X=F,Cl in o de o
desc ibe ioniza ion po en ials and elec on a ini ies in con ined sys ems. We ha e also
ca ied ou calcula ions conce ning H a om a HF/6-311++G le el so ha we could gain
a be e pic u e conce ning ioniza ion po en ial. In o de o ob ain a gene al pic u e,
we ha e buil a 20
×
20 g id in he
(V0
,
β)∈[−
0.500,
−
0.001
]×[
0.10, 2.00
]
domain and
ha e compu ed he e o
ε
in each poin by compa ing he explici ly ob ained ioniza ion
ene gy
I
wi h he one ob ained by means o o mulas
(7.2)
and
(7.3)
, namely
I heo
as
ε=|I−I heo|
|I|×
100. As i can be seen in igu e 7.1, he ob ained e o does no only depend
on he con inemen po en ial pa ame e s bu also on he a omic numbe o he a om we
a e s udying. Fo all cases he ob ained e o is smalle o small alues o he po en ial
dep h pa ame e
V0
, which is expec ed om he pe u ba ion-me hod scheme. Ano he
sha ed ea u e is he ac ha , in gene al, he e o ge s smalle o la ge alues o he
wid h pa ame e
β
; we may also highligh he cases conce ning Na and K o which he
e o seems o ha e a minimum o some alues o he wid h pa ame e .
So a , we ha e aken he i s s eps o s udy his kind o sys ems. The e a e many ways in
which ou p edic ions can be imp o ed: ex end he heo y o mul i e e ence wa e unc ions,
ob ain op imized basis se s o each elemen gi en he con inemen pa ame e s
(V0
,
β)
in a
simila way as in chap e 3, compu e u he e ms in he pe u ba ion se ies and so on.
100 euska azko labu pena
8.1.3.2 E di-gainbegi a u ako ikaskun za
Gainbegi a u iko ikaskun za eknika gehienei eske lo u iko i aga penen e a sailkapen
e edu zeha zak iza en di a, baldin e a ikaskun za-da uen kopu ua nahiko handia bada. Hala
e a guz iz e e, e eali a ean age zen di en egoe a asko an, (gu e kasuan bezala) ez-menpeko
aldagaia en e ike ak lo zea denbo a alde ik edo ekonomikoki ga es ia izan dai eke. Be az,
ezagu za esku aga ian oina i u iko espe ien zia gehiago lo zea e a gu e baliabideak
(bo onda ea, denbo a e a di ua) gu e ezjakin asuna en zoko ik ilunenak a gi zen inbe i zea
zuhu a da. E di-gainbegi a u iko e a ikaske a ak iboko eknikak da uen a abe a aplika
dai ezke, e a ho e a ako bi mul zo desbe din di ugu.
E di-gainbe i a u iko ikaskun za e a ikaskun za ak iboko eknikak bi da u so a des-
be din di ugun kasue an aplika di zakegu: e ike a u iko so a ba
L
e a e ike a u gabeko
bes e so a ba
U
. Izenak adie azi bezala, ekike a u iko da a so a ezuga i e a e ike en
bide kadu a Ca esia o dena uz osa u iko
(x1
,
y1). . . (xl
,
yl)
i xu adun pun uek osa zen
du e, non
YL={y1. . . yl}∈{
1
. . . C}
ka ego ia posible guz ien so a den. Bes alde,
e ike a gabeko e a
(xl+1
,
yl+1). . . (xl+u
,
yl+u)
pun uez osa u iko so a
U
e e badugu, non
YU={yl+1. . . yl+u} ∈ {
1
. . . C}
e ike ak ezezagunak di en e a so a honen ka dinali a ea
au ekoa ena baino askoz handiagoa den l << u.
Ho ela, lan oso be ie an egin den moduan [126, 127], ase diag amak e aki zeko
asmoz, hu engo espe imen ua zein baldin zape an egin jaki eko da u so a haue an dagoen
in o mazioa e abil genezake.
Hasie ako hipo esiak kon uan izanda, pa ame o espazioa disk e iza uz
X={x1. . . xl+u}
i xu ako sa e ba e aiki dugu, non lehenengo
l
pun uak hasie ako konpu azioen bidez
Y={
1
. . . C}
so an age zen di en ka ego ia guz iak age zen di ela ik e ike a uko di-
ugun e a gainon zeko
u
pun uak (hasie a ba ean) e ike a u gabe ge a uko di en
l << u
.
Ja aian, lehen
l
pun ue ako e ike en in o mazioa gainon zeko pun ue a a heda uz, e ike-
a u gabeko pun uek
C
ka ego ia posible guz ie an ego eko du en p obabili a e banake ak
lo uko di ugu, adibidez, e ike en hedapen algo i moa e abiliz.
X. Zhu-k e a Z. Ghah amani-k ”e ike en hedapena” ga a u zu en algo i moa i eske
[128] e ike a u ako da uen in o mazioa e ike a u gabeko da ue a a Ma ko -en p ozesu ba en
bidez heda zen da. Ho , e ike a u ako e a e ike a ik gabeko da u guz iak bil zen di uen g a o
ba e a binakako elka ekin zak modula zen di uz en pisuak de ini zen di ugu. Ma ko -
en ka e homogeneoen e a ma ize es okas ikoen p opie a eak e abiliz, edozein hasie ako
baldin za
Y(0)
ha u a e e, algo i moak soluzio geldiko baka ba e a konbe gi zen duela
oga u zu en (8.5).
Yu= (I−˜
Tuu)−1˜
TulYl(8.5)
Me odo ho en desaban aila nagusia
(I−˜
Tuu)−1
ma izea en konpu azioan da za,
e abili ako sa eak nahiko handiak edo den soak badi a, objek u ho en konpu azioa ga es ia
edo ezinezkoa izan dai eke. A azo ho i saihes eko,
(8.5)
ekuazioan age zen den o mula
i xia e abili beha ean, hasie ako e ike a bek o e ba ha u e a Ma ko -en e eduan i e a uz
in o mazioa heda zea lo zen dugu. Hala e a guz iz e e, ma ize handiak e abil ze akoan
konbe gen zia lo zeko hainba i e azio egin beha di enez, algo i mo ho ek as una iza en
ja ai zen du; ba ez e e, sa een den si a ea edo a dimen sio kopu ua handi zen badi ugu.
Al e na iba gisa, konpu azionalki me keagoak di en klasi ika zaileak (esa e ako, ausazko

8.1 sa e a 101
basoak) p oblema ho i ebaz eko esna ap oposak di a. Gain-doi ze a azo ik iza en ez
du enez, zuhai z kopu u nahiko handia e abiliz lan hone an ike u ako sis ema guz iak
ikas eko me odo unibe salak di a. Gauzak ho ela, hasie ako da u so a ba e abiliz ausazko
basoe an oina i u iko modelo ba en ena zen dugu; ho i e abiliz, e ike a u gabeko pun uek
e a ka ego ia bakoi zean ego eko du en p obabili a eak esku a zen di ugu. Ho ela eginik,
e ike en hedapen algo i moa e abil zea en baliokidea den me odo ba eza i dugu.
Ikas ea hone an da za: gu e ezjakin asuna sakonena den alo e an in o mazio gehiago
lo zea alo ho i bu uz dugun ezagu za zabal zeko. Modu e aginko ean ikasi ahal iza eko,
no be ak ezagu za eza duen pun ue an a e a ja i beha du. Ziu gabe asun laginke a
ziu gabe asun handiena du en pun uen laginak ha zea ahalbide a zen digun me odoa
da, baina, nola kuan i ika genezake ziu gabe asuna? Demagun p obabili a e banake a
un zio ba dugula
PC(y|x)
zeinek, pa ame o bek o e ba
x
emanik,
{
1
. . . C}
ka ego ie an
ego eko
y
p obabili a eak ema en dizkigun. Ho ela, esku a u ako p opabili a eak e abiliz,
pun u jakin ba en ziu gabe asuna hainba e a an de ini dezakegu: kon ian za xikiena
(8.6)
, ziu gabe asun ma ginala
(8.7)
e a Shannon-en en opia
(8.8)
. Lehenak e ike a ik
gabeko da uen auke a ik xikiena (ho ega ik zalan zaga iena) maximiza zen du; biga ena
au ekoa baino malguagoa da, bi i aga penik p obableenak kon uan ha zen bai i u. Azkenik,
Shannon-en en opia in o mazio edukia en neu ia da; aldagai ba ek zenba e a in o mazio
gehiago izan, honen na u a i bu uz o duan e a gu xiago dakigu [129].
uLC (x) = 1−max
CP(C|x)(8.6)
uMS (x) = 1−[P(C1|x)−P(C2|x)](8.7)
uSE (x) = −X
C
P(C|x)log P(C|x)(8.8)
102 euska azko labu pena
8.2 LAN HONEN HELBURUA
Wigne -en molekulei, Hooke-n a omoei e a makinen ikaskun zako eknika ba zuei bu uzko
a ea en egoe a desk iba u ondo en, a al hone an o aingo lana en helbu u nagusiak jasoko
di ugu:
1.
Dimen sio ba ean e a bi an inda -pa ame o desbe dinen (
k=ω2
) bidez po en zial
ha monikoe an kon ina u ik dauden sis emak desk iba zeko e abil zen di en oina i
un zio gauss a bana uak op imiza zea ML eknikak (sa e neu onalak) e abiliz.
Ho e az gain, egi u a elek onikoa lo zeko ahalbide a zen du en me odo op imoak
esku a zea e a ho iek
n={
2, 3, 4
}
elek oidun spin ga aiko sis emak ike zeko
e abil zea.
2.
Zen u ba eko oina i un zioak eknika klasikoen bidez (simplex e a New on-Raphson)
op imiza uz hi u dimen sio ako
n={
2, 4, 6, 8, 10
}
elek oidun e a
k=ω2=
1
/
4
po en zial pa ame odun Hooke-n a omoen desk ibapen egokia esku a zea e a ho i
dagokion me odo op imoan oina i u a, oina izko single e e a iple ea en ene gia
zeha zak e dies ea.
3.
Makinen ikaskun za-me odoak hobe zea e a inplemen a zea. E di-gainbegi a u ako
ikaskun zan e a ziu gabe asun laginke a eknike an oina i u a, ase diag amak
e aginko asunez lo zea e a kimikoki esangu a suak di en sis ema ba zuk ho ni zea.
4.
Kon inamendu po en zial gauss a ei e a oina i un zio gauss a ei dagozkien go pu z
baka eko in eg al anali ikoak kalkula zea e a
GAMESS US
kodean inplemen a zea.
Au eko a ale an lo u iko emai zekin lo u a du en kalkuluak bu u zea.
8.3 egindako lanen labu penak 103
8.3 EGINDAKO LANEN LABURPENAK
8.3.1 Sasi dimen sio ba eko sis emak
Au eko lan ba zue an, po en zial esplizi u gabeko e a elek oi gu xidun sis emak az e u
zi uz en ku xa-luze a aldako ak e abiliz, e a ko elazio elek uak ROHF (ge uza i ekiko
Ha ee-Fock uhin un zio muga ua) e a FCI (kon igu azio-elka ekin za osozko uhin un zioa)
uhin un zio me odoen bi a zen ike u zi uz en [130]. Sis ema ho iek be ezko in e es
eo ikoa badu e e e, lan hone an ike u iko elek oi kopu u be be ak di uz en sis emak
espe imen alki beha u di uz e [131, 132].
Lan honen lehen kapi uluan, sasi dimen sio ba ean po en zial ha monikoen bidez
kon ina u ik dauden elek oi gu xidun (
n={
2, 3, 4
}
) sis emen lokalizazio p opie a eak
az e u di ugu. Ho ela, bada, sis ema ho iek modeliza zeko,
(8.9)
ekuazioan age zen den
e agile Hamil onda a e abili dugu.
H=−1
2
n
X
i=1∇2
i+k
2
n
X
i=1
2
i+
n
X
i=1,j>i
1
ij
(8.9)
Li e a u an jakina denez, dimen sio ba ean Coulomb-en e agilea singula a da [136,
137]; a azo ho i konpon zeko, hi u dimen sionalak di en e a le o ba en heda zen di en
s
mo ako oina i un zioak e abili di ugu. Ho i dela e a, dimen sio pu u ba izan beha ean,
sis ema sasi-unidimen sionala dela diogu. Au eko lane an o ga u du en moduan, oina i
ho iek e abiliz lo u iko ene gia i sobe a dauden zeha kako osagai bien eka pena kenduz,
sis ema en ene gia zuzendua lo zen dugu [138]. Kasu hone an, lo u ako uhin un zioak
e a egokian e ep esen a zeko asmoa ekin,
x=
0 pun uan zen a u iko e a
x
no anzkoan
heda zen den dimen sio ba eko a e e egula dun sa e ba ean zeha
s
mo ako un zio
gauss a no maliza uak koka u di ugu. Oina i un zio ho iek
(8.10)
ekuazioa en bidez
adie azi a daude; hemen,
Rµ= (xµ
, 0, 0
)T
bek o eak
µ
un zioa en zen u-koo dena uak
ema en di u.
ϕµ( ;α,Rµ) = 2α
π3/4
exp(−α( −Rµ)2)(8.10)
E a hone an de ini u iko segmen ua en luze a be e iko egoe a en ene gia ga aieneko
i zule a pun uak
x0(n
,
k) = ±(2n+1)2
k1/4
e abiliz de ini u dugu. Pun u ho ie a iko
bakoi zean e a
x=
0 pun uan un zio bana ja i ondo en, a e e egula e an bana u iko
bes e 4
m
un zio ja i di ugu e a i zule a pun u klasiko bien a ean 2
m
daude. Ho az,
guz i a
M=
3
+
4
m s
mo ako un zio ja i di ugu e a alboz alboko un zioen a eko
dis an zia
δ=2x0
m+1
izan da. Fun zio guz i ho iek
α
be e zaile be a du enez, ondoz ondoko
un zioen a eko gaineza pen in eg ala
S(α
,
δ) = exp(−αδ2/
2
) = exp(−ξ/
2
)
da. Lan
hone an, au eko hainba e an oina i uz [130, 138–141] , gaineza pen pa ame oa
ξ=
1.0
balioan inka u dugu. Ondo ioz,
α
be e zailea po en zial ha monikoa en na u a en (
k
) e a
guz i a e abili ako oina i un zioen kopu ua en menpekoa da
α=ξ(m+1)2
4x2
0(n,k)
. Lan hone an,
m=25 ja i dugu.
104 euska azko labu pena
Wigne -en lokalizazioa ka ak e iza zeko an, sis ema en p opie a e bi e abili di ugu:
go pu z ba eko den si a ea
(8.11)
e a pa ikula-hu sune en opia
(8.12)
. Lehenengoan,
φi( )
o bi al na u alak di a e a
γi
koe izien eak ko elazio elek onikoa en ga an zia ekin
lo zen di ugun okupazio zenbakiak di a [291, 292]. Biga enean, au ekoan bezala, okupazio
zenbakiak e abiliz pa ikula-hu sune en opia age zen da. De inizioz, adie azpen ho i
e abiliz lo u iko balioak absolu uki zenba e a handigoak izan, sis ema osoa en ka ak e e
mul ide e minan ala o duan e a handiagoa izango da.
ρ( ) = X
i
γiφ∗
i( )φi( )(8.11)
S=−X
i
γilog γi+ (1−γi)log(1−γi)(8.12)
Elek oi biz osa u iko sis emak modeliza zeko, CASSCF(2,5) me odoen bidez hainba
k
balioen za spin single e e a iple e egoe en egi u a elek onikoen kalkuluak bu u u
di ugu. Spin egoe a bakoi za en za go pu z ba eko den si a eak esku a u di ugu (2.1
i udia). Ikus dezakegunez, lo u iko den si a e p o ila
k
kon inamendu pa ame o e a spin
egoe a en menpekoa da. Espe o bezala,
k
balio handien za egoe a single ea i dagokion
den si a eak maximo baka a dauka, o dea, egoe a iple eak (Pauli en esklusio p in zipioa
dela e a) bi mu u di u.
k
balio xikien kasuan, aldiz, spin egoe a biek maximo bi di uz e
e a an zeko den si a e p o ila dauka e. Sis ema ho ien izae a mul ide e minan ala en
na u a az e zeko an, hainba
k
balio i dagokien egoe a iple ea en pa ikula-hu sune
en opiak kalkula u di ugu. 2.3 i udian ikus dezakegun moduan,
k
balio handien kasuan
pa ikula-hu sune en opia ze o an za doa. Ho ek kon inamendu sendoa en mugan pa ikula
independen ea en e edua sis ema en desk ibapen ona dela e a lo zen den Fe mi- en gasa en
desk iba zeko Sla e -en de e minan e baka nahikoa dela esan nahi du. Lo u iko ku ba en
i xu a au e iaz egindako lane an lo u iko emai zekin ba e aga ia dela [40, 41] adie azi
beha dugu. Kon inamendu pa ame oa en balioa xiki zen joan ahala, ka ak e e mul i-
de e minan ala i dagokion en opia handiko egoe a lo zen dela ikusi dugu, e a en opiak
be e balio maximoa k=5×10−4ingu uan izan ondo en, be iz jais en has en da.
Sis ema honi bu uzko ezagu za handiagoa iza eko an, oina i minimodun e edu anali iko
bi e abili di ugu. Alde ba e ik,
α
be e zailedun e a po en zial ha monikoa en minimoan
zen a u iko gauss a ba e abiliz, bi elek oiz okupa u iko o bi ala lo zen dugu, ho -
ela egoe a single ea en (Fe mi- en likidoa) e edua lo zen dugula ik
1Σg=σ↑↓
0
. Bes e
alde ik,
x0=±
1
/(
4
k)1/3
o eka pun u klasiko bakoi zean
α
be e zailedun oina i gauss a
ba koka u e a sime ia adap a uz sis ema i dagozkion
(2.6)
(bikoi ia, konbinazio sime ikoa)
e a
(2.7)
(bakoi ia, konbinzio an isime ikoa) o bi alak lo u di ugu. Azken ho iek e abi-
liz, egoe a iple ea en (Wigne -en k is ala) e edu sinpli ika ua lo u dugu
3Πu=σ↑
gσ↑
u
.
Spin egoe a bakoi za en za i dagokion oina iei dagozkien go pu z ba e a biko in eg alak
kalkula uz, egoe a bakoi za en ene gia esp esioak lo u di ugu. Kasu bie an ene gia
k
kon inamendu pa ame oa e a
α
oina ien be e zailea en un zioa dela oga u ondo en,
k
bakoi za en za ene gia minimiza u dugu
α
hu a pa ame o ba iazional gisa ha uz.
Ho ela eginik, egoe a bakoi za en ene gia op imoa en ku bak lo u di ugu (ikusi 2.4.
i udia). Lo u iko emai ze an oina i uz,
k
balio xikien za , spin egoe a bien ene giak
oso an zekoak izanda, iplea en ene gia singlea ena baino ze bai xikiagoa da. Os e a,
8.3 egindako lanen labu penak 105
k=
5
×
10
−4
pun ua en ingu uan single ea iple ea baino egonko agoa bilaka zen dela
ikusi dugu. Emai z hauek me odo au e a uak e abiliz e die si akoekin e ka uz, e edu
sinple hau lokalizazioa zein
k
balio ingu uan age uko den au esa eko e abilga ia dela
ondo ioz a u dugu.
Emai za hauek ha u a, hi u e a lau elek oiz oso u iko sis em an an zeko joe ak
age zen di en ike u nahi izan dugu. Asmo ho ekin, CASSCF(
n
, 2
n
) (non,
n
elek oi
kopu ua den), bakoi za i dagokion gauss a ez e a u iko sa ea e biliz, spin al uko kalkuluak
bu u u di ugu. Ho ien i ee a balioak e abiliz, hainba
k
- i dagozkien den si a eak (2.5
i udia) e a pa ikula-hu sune en opiak (2.6 i udia) kalkula u di ugu.
Ho ie an ikus dezakegunez, ku ba bien i xu a an zekoa badu e e e, en opia maximoa i
dagokion
k
- en balioa elek oi kopu ua en menpekoa da: zenba e a elek oi kopu ua
handiagoa izan, o duan e a xikiagoa izango da
k
ho en balioa. Az e u iko sis ema
guz iek an zeko joe a kuali a iboa du ela naba i u dugu; ondo ioz, Fe mi- en likidoa e a
Wigne -en k is ala en a eko an sizio an zeko p ozesu ba
S(kmax
,
n)
pun ua ekin de ini
dai ekelakoan gaude. Pun u ho en posizioa elek oi kopu ua en menpekoa izango da.
Ba u ako emai ze an oina i u a ze a ondo ioz a u dugu: kon inamendua eza zen
duen kanpo po en zial ha monikoa en
k
pa ame oa en balio bezain xikien za sis ema
e a zen du en elek oiek lokaliza zeko joe a du e; ho ela, Wigne -en molekula ba e a zen
da. A e gehiago, hainba kon inamendu pa ame o i dagozkien elek oi-hu sune en opiak
kalkula uz, egoe a lokaliza u e a ez-lokaliza uen a eko en opia maximodun an sizio
egoe ak au ki u di ugu.

106 euska azko labu pena
8.3.2 Sasi bi dimen sio sis emak
Dimen sio ba eko kasua bi dimen sio a a o oko uz, hainba elek oiz oso u iko sis emak
desk iba zeko, kimika kuan ikoan ohikoak di en nukleoe an zen a u iko [142–144] zein
espazioko edozein pun u a bi a io an zen a u iko [138, 145–149] un zio gauss a ak
e abil zen di ugu. Sis ema ho iek in e es eo iko zein espe imen ala du enez [150–153],
egi u a elek onikoa en eo ian oina i uko me odoak e abili ahal iza eko beha ezkoak
di en p o okolo konpu azioanalak ga a zea p emiazkoa da. Hala e e, sis ema ba zuen eskala
nanome oa en o denakoa da, e a ho e a ako zen o baka ba az e zen bada, menpeko-
asun linealeko a azoak ge a uko di a. Adibide ba ema ea en, espe imen alki lo u iko
g a enoan oina i u iko pun u kuan ikoek (GQD) e a an sizio me alen dikalkogenu oe an
(TMD) oina i u iko nanopa ikula ba zuek 1-7 nm-ko amaina du e [154–163]. Lan honen
biga en kapi uluan, bi dimen sioko po en zial ha moniko sime ikoen bidez kon ina u ik
dauden hainba elek oiz oso u iko sis emak desk iba zeko beha di en oina i gauss a
ba zuk p oposa u di ugu. Gaine a, lo u iko oina i un zioak po en zial ha monikoa en kon-
inamendu pa ame oa k, oina i un zio kopu u o ala e a gaineza pen pa ame oa ekiko
op imioak di a.
Ho ela,
(8.10)
ekuazioan age zen di en
s
mo ako o bi al gauss a ak sa e hexagonal
ba eko nodoe an koka u di ugu. Kasua en a abe a, guz i a
M=
3
g2−
5
g+
1 oina i
un zio e abili di ugu, non
g≥
3 zenbaki osoak 3.1 i udian age zen den pa oi hexagonalak
di uen ze enda kopu ua adie az en duen. Kon igu azio hone an ondoz ondoko un zioen
a eko gaineza pen in eg ala
S(α
,
δ) = exp(−αδ2/
2
) = exp(−ξ/
2
)
da, non
δ
ondoz ondoko
un zioen a eko dis an zia den. De ini u dugun
ξ=αδ2
gaineza pen pa ame oa en
balioa en a abe a, menpeko asun linealak (balo e oso xikiak) edo gune ez-ja ai uak
(balo e oso handiak) izan di zakegu. Au e iaz bu u u iko lane an oina i u a [130, 138–
140], lan hone an ξ∈[0.85, 1.15] a eko balioak az e u di ugu.
Lehen u a sean, po en zial ha moniko iso opiko bidimen sional ba ean kon ina u ik
dagoen pa ikula ba eko p oblemei e epa a u diegu. Ho ela, sis ema hao ei dagokien
Hamil onda a
(8.13)
ekuazioan age zen dela e a be e oina izko egoe a en ene gia zeha za
E0=k1/2dela a zeman dugu.
H(x,y) = 1
2−∂2
x+kx2−∂2
y+ky2=H(x) + H(y)(8.13)
Desk iba u iko oina i un zioak e abiliz, go pu z ba eko Sch ¨odinge -en ekuazioa
(8.14)
adie azpenean age zen den balo e p opio o oko uen p oblema moduan adie azi dugu.
Ho ,
T
ene gia zine ikoa,
V
ene gia po en ziala,
S
gaineza pen,
E
balio p opio e a
C
bek o e p opio ma izeak di a. Fo ga u dugun moduan, ene gia zine iko e a po en ziala en
ma ize ho iei dagozkien elemen uak gaineza pen ma izea en elemen uen menpekoak
di enez, guz iak bukle be ean de ini u di zakegu. Iza ez, e abili ako oina i un zioak
hi u dimen sionalak di enez, egi an Sch ¨odinge -en ekuazioa eba ziz, sasi bi dimen sioko
emai zak e dies en di ugu. Lo u iko ene giak (balo e p opioak) zuzen zeko, gauss a
guz iek be e zaile be a du enez, zeha kako osagaia en eka pena en ondo ioz lo zen dugun
gehiegizko ene gia zine ikoa kedu dugu. Kasu hone an elek oi bakoi zeko
α/
2 ene gia
eka pena kendu dugu.
8.3 egindako lanen labu penak 107
(T+V)C=ESC (8.14)
P in zipio ba iazionalean oina i u a,
(ξ
,
g
,
k)
balio jakin ba zuen za oina izko egoe -
a en ene gia ondoz ondoko unz zioen a eko dis an zia ekiko op imiza uz, hainba sa e az
osa u iko Dda u basea lo u dugu.
Enpi ikoki oga u dugun legez, e a ho e an lo u iko Hamil onda a en balio p opioak
ez di a baka ik zeha zak izan, hendeka u iko mailen na u a e ep oduzi zeko gai e e izan
ga a. Da u base hone an age zen di en sa e en e a emai za anali ikoen a eko e o e
e la ibo maximoa %5ekoa izan da e a hendeka u iko so en za kalkula u iko desbide a ze
es anda e la iboa %10−4koa izan da.
D
da u basea e abilik, osaga iak di en so a bi lo u di ugu: ikaske a so a ba
L
e a az e ze
T
so a ba (
D=L∪T
,
L∩T =∅
). Ho iek e abiliz, bi ge uza ezku uz e a
neu ona kopu u aldako dunez e a u iko hainba sa e neu onal op imiza u di ugu e a ho ien
un sa
(log k
,
ξ
,
g)
sa e a moduan emanik
log δop
i ee a balioa lo zea da. Lo u iko sa e
op imalak ak ibazio un zio sigmoideak, lehen ge uzan sei neu ona e a biga enean lau di u.
Op imizazioa 10 aldiz gu u za u iko balioz a zea en bidez egindako e o e kuad a ikoen
minimizazioa bu u uz lo u dugu. Emai za gisa, i ee ako balioen es ima u iko e o e
e la iboa %2.5ekoa dela ikusi dugu; ho en ondo ioz, oina i un zioen be e zaileen e o ea
∆αop =2ξ
δ2
op
∆log δop
moduan adie azi dugu. Oha u azken e o e hau xikiagoa bihu zen
dela kkon inamendu pa ame oa en balioa handi zen den heinean.
Lo u ako oina i un zio op imoek hainba elek oiz konposa u iko sis eme an age zen
di en ko elazio e ek uak desk iba zeko du en gai asuna oga zeko asmoz, bi e a hi u
elek oiz oso u iko e a spin baxu e a al uko sis emak az e u di ugu. Ho e a ako,
(8.15)
ekuazioan age zen den e agile Hamil onda a e abili dugu.
H=1
2
n
X
i=1
(k(x2
i+y2
i)−∇2
i) +
n
X
j>i=1
1
ij
(8.15)
E ek u ho ien neu i kuan i a iboa iza eko, koo dena u e adialean zeha CASSCF(
n
,8)
mailan e a UHF mailan kalkula u iko uhin un zioen a eko den si a e di e en zia kalkula u
dugu
k∈(
1
×
10
−5
, 1
×
10
−10)
balioak e abiliz. Koo dena u e adiala en balio xikien za
uhin ko ela ua en den si a ea ez ko ela ua ena baino xikiagoa dela ikusi dugu. Aldiz,
koo dena u e adiala en balio handien za uhin ko ela ua en den si a ea ez ko ela ua ena
baino handiagoa dela ikusi dugu. E ek u ho i naba iagoa da
k
pa ame oa en balioa
handi zen den heinean.
Labu bilduz, (20,100) uni a e a omikoko amaina du en sis emak desk iba zeko sa e
neu onalen bidez lo u iko sa e hexagonale an bana u iko oina i un zio gauss a ak
lo u di ugu. E o kizuneko aplikazio gisa, bi dimen sioko g a enoan oina i u iko pun u
kuan ikoen amainako sis emak desk iba zeko e abili ahal izango di ugu.
108 euska azko labu pena
8.3.3 Hi u dimen sioko Hooke-n a omoa
Pun u kuan ikoei bu uzko ike ke a eo ikoe an e abil zen di en e edu sinple e a egokiene ako
ba Ha monium edo Hooke-n a omoa de i zona da, non elek oiak po en zial es e iko
ha moniko ba en bidez kon ina u a dauden [76]. E edu ho iek sis ema e ealen p opie a eak
simula zeko doi di zakegun hainba pa ame o di uz e [77, 78]. Esa e ba e ako, gu e aldean
egindako lan ba ean kanpo e emu magne iko ba aplika uz, bi elek oidun Hooke-n a omo
ba ean ge a zen den single e- iple e an sizioa e a ho e an ko elazio elek onikoak
duen ga an zia az e u genuen [79]. Hala e e, e edu zeha zak e abil zeak gu e ike ke a
bi elek oi sis eme a a muga u zuen. Izan e e, bi elek oiz oso u iko Hooke-n a omoa en
kasuan, kon inamendu pa ame o zeha z ba zuen za baino (
ω2=1
4
,
1
100 . . .
) ez dago
soluzio anali iko ik[80]. Balio ho iek zeha zak badi a e e, ko elazio baxu (
ω2→ ∞
) e a
al uko (
ω2→
0) kon inamenduei bu uzko in o mazioa lo zeko e abilga iak di a [37, 39,
40, 42–64] [42–44].
Lauga en kapi uluan azaldu dugunez, Hooke-n a omoak desk iba zeko bezain egokiak
di en oina iak lo zea p emiazkoa da. Sis ema Coulomb-a en za op miza u iko aug-cc
oina i nahiko handiak e abiliz elek oi gu xiko sis emak e a egokian desk iba zeko gai
baga a e e, elek oi kopu ua handi zen dugun heinean (ha u 10 elek oidun sis emen
adibidea) menpeko asun linealen a azoez gain, ez ga a gauza izan oina izko egoe a en
spin anizkoi z asuna e a, ondo ioz, sis ema en be e ze egi u a e a egokian desk iba zeko.
ETBS-6S oina i un zioak e abiliz (e e e en ziazko balioekin alde a u a), beha u dugun
e o e handiena mHa ee o denekoa izan da.
Kapi ulu hone an egoe a single e e a iple ean dauden
n={
2, 4, 6, 8, 10
}
elek oiz
osa u iko
ω2=
0.25 kon inamendu pa ame oa du en Hooke-en a omoak desk iba zeko
e a single e- iple e ene gia a ea esku a zeko p ozedu a ba au kez u dugu. Gaine a,
Yukawa- en po en zialak e abiliz, elek oi-elek oi elka ekin ze an pan aila ze e ek uak
duen e agina az e u dugu, e a,
(8.16)
ekuazioan age zen den e agile Hamil onda a
e abiliz, sis ema hauen gaineko egi u a elek onikoa en kalkuluak bu u u di ugu.
H=−
n
X
i
1
2∇2
i+
n
X
i
1
2ω2 2
i+
n
X
i
n
X
j>i
e−λee ij
ij
(8.16)
Ene gia absolu uen za balio zeha zak lo zeko an,
ω2=
0.25 kon inamendu pa ame oa
e abiliz, hainba elek oi kopu u desbe din (
n={
2, 4, 6, 8, 10
}
) di uz en spin egoe a single e
e a iplea du en sis emen za op imoak di en oina i un zioak lo u di ugu. Ho ela,
kon a u ik gabeko
L=
0 e a
L=
3 momen u angelua en a eko e a
N
ge uzadun
oina iak e abili di ugu; ho ien be e zaileak
L
e a
N
zenbakiekiko du en menpeko asuna
(8.17) ekuazioan adie azi dugu.
ζk
LN (ω2) = ω2
2αL,N(ω2)βL,N(ω2)k−1, 1 ≤k≤N(8.17)
Azken ekuazio hone an age zen di en
αL,N(ω2)
e a
βL,N(ω2)
pa ame oak CASSCF(
n
, 13)
e a bi elek oien kasuan FCI mailan lo u iko ene giak minimiza uz lo u di ugu; op i-
mizazio p ozesu gisa simplex e a New on-Raphson me odoak e abili di ugu [56, 293, 294].
8.3 egindako lanen labu penak 109
A e az ikuska u ondo en, sei elek oi egoe a single ean den sis ema en zako lo u iko oina -
iek (izenez ETBS-6S) zehaz asuna en e a e endimendua en a eko o eka onena du enak
di ela ondo ioz a u dugu. Fun zio ho iek inko u gabeko 4(SPDF) oina iak di a e a
(8.17)
ekuazioan age zen di en pa ame o op imoak
αL,N=
0.240432 e a
βL,N=
1.3021162 izan
di a. Ho ela, lo zen den be e zaileen segida: 0.2404032, 0.3130329, 0.4076051, 0.5307491
da. Guz i a ETBS-6S so a 80 oina i un zioek osa zen du e e a ho ie a iko 68 linealki
askeak di a.
Bi elek o iz oso u iko sis ema Coulomb-ia en emai zak 4.1 aulan age zen di a; spin
egoe a single ea en ene gia zeha za 2.0 uni a e a omikoa (a.u.) dela kon uan izanda, sis ema
honen za lo u iko emai zak gainon zekoenen emai zen kali a ea kalib a zeko e abili di ugu.
Oina i un zioen segidan ikus en dugun moduan, FCI mailan: 2.055213 (aug-cc-pVDZ),
2.004107 (aug-cc-pVTZ), 2.000476 (aug-cc-pV5Z) e a 2.000196 (aug-cc-pV6Z) uni a e
a omiko izan di a. Emai za ho iek li e a uan age zen di enekin ba bada oz e e, single e-
iple e ene gia a ean e o ea 0.15 eV-koa da. Os e a, maila be an ETBS-6S oina iak
e abiliz, ene gia abosuluez gain, a e ho en e o ea 0.01 eV-koa izan da; ondo ioz, lan
hone an ga a u iko oina iak e abil zea jus i ika u ik dago. Elek oi kopu ua handi u ahala
FCI me odoak oso ga es iak bilaka zen di enez, CASSCF e a MRMP2 me odoek ETBS-6S
oina iekin lo zen du en zehaz asuna analisa u dugu; 4.2 aulan hainba espazio ak ibo
e abiliz lo u iko emai zak bildu a daude. Ikus en den moduan, e a hone an e die si ako
emai zak FCI me odoa en emai zen an zekoak di a.
Lau elek oiz oso u iko sis emen za , jada, FCI kalkuluak bu u zea oso ga es ia da;
bes alde iple ea en oina izko egoe an eka pen handiena duen kon igu azioan 1
p
o -
bi ale an spin be eko bi elek oi daudenez, uhin un zio mul ide e minan alak e abil zea
p emiazkoa da. 1
p
ge uza en hendekapena dela e a, spin egoe a bien uhin un zioak
desk iba zeko hainba kon igu azio bil zen di uz en uhin un zioak e abili beha di ugu.
Gu e kasuan, CASSCF(4,
m
) e a MRMP2(4,
m
) (
m=
4, 5
. . .
13) me odoei eske lo u iko
emai zak 4.3 aulan bil zen di a. CAS(4,13)/aug-cc-pV6Z* e a MRMP2(4,13)/aug-cc-
pV6Z* me odoak e abiliz, single e- iple e ene gia a ea 1.11 eV e a 1.08 eV di a hu enez
hu en; emai za hauek be ez li e a u an age zen di enen an zekoak(1.00 eV)[41] di en
a en, CAS(4,13)/ETBS-6S e a MRMPT2(4, 13)/ETBS-6S me odoekin lo u iko ene gia
absolu uen balioen e o eak (e e e en ziazko balioekin konpa a uz) xikiagoak izanda,
single e- iple ea en emai za e e hobea dela ikusi dugu. Alegia, single e- iple e ene gia
a ea en balioak 1.06 e a 1.04 eV izan di a.
Sei elek oiz oso u iko sis emen za (au eko kasua en an zeko e an), iple ea en oina -
izko egoe an eka pen handiena duen kon igu azioan 1
p
bi o bi ale an spin be eko bi
elek oi e a bes e o bi alean au kako spina du en bi elek oi daudenez, uhin un zio mul i-
de e minan alak e abil zea e e beha ezkoa da. Be az, CASSCF(6,
m
) e a MRMP2(6,
m
)
(
m=
6, 7
. . .
13) me odoei eske esku a u iko emai zak 4.4 aulan bil zen di a. Ikusi dogunez,
1
p
ge uzan bes e bi elek oi gehi u a en, single e- iple e ene gia a ea au eko kasua en
an zekoa da; kapi ulu hone an op imiza u iko oina iak e abiliz, 1.03 eV e a 1.00 eV-ko
balioak lo zen di ugu (hu enez hu en) CAS(6,13)/ETBS-6S e a MRMP2(6,13)/ETBS-6S
maile an. Gaine a, lo u ako emai zak li e a u an age zen den 0.95 eV-ko emai za ekin
[62] ba da oz. Ene gia absolu uen zehaz asuna elek oi kopu ua ekin xiki zen den a en,
es ima u iko single e- iple e a een balioak zuzenak iza en ja ai zen du ela ikusi dugu.
Oina izko egoe a elek onikoa en uhin un zio monode e minan ala ge uza i xikoa
bada e e, zo zi elek oidun sis emak e e CASSCF(8,
m
) e a MRMP2(8,
m
) (
m=
8, 9
. . .
13)
116 euska azko labu pena
na u a kobalen ea dela ondo ioz a u du e [214–216, 218]. 4. kapi uluan, lo u a kobalen ea en
desk iba zaileek kanpo po en zial es e iko ha monikoa en inda pa ame oa
ω2
e a nukleoen
a eko dis an zia ekiko
R
du en menpeko asuna az e u di ugu. Pa ame o hauek
(8.21)
Hamil onda ean age zen di a.
H=−1
2
4
X
i=1∇2
i−2
2
X
J=1
4
X
i=1
1
iJ
+
4
X
j>i=1
1
ij
+1
2
4
X
i=1
T
iW i+1
R(8.21)
Lehen pausuan, 5.8 i udian age zen di en kobalen zia ase diag amak HF/aug-cc-pVTZ
mailan esku a u di ugu. E ike a gisa, Laplace- a ak
∆ρ( C)
e a ene gia o ala en den si-
a eak
H( C)
lo u a pun u k i ikoan du en seinuak e abili di ugu. E a be ean, pa ame o
espazioa 500
×
500 sa e ba e abiliz disk e iza u dugu
[
0.10, 4.00
]×[
1.00, 4.00
]⊆(ω2
,
R)
domeinuan e a guz i a ho en 200 pun u e abiliz (0.1216%) kobelen zia ase diag amak
esku a u di ugu. Maila hone an lo ako
∆ρ( C)
- en kasuan ikus dezakegunez, balio nega -
iboak nukleoen a eko dis an zia 1.00 e a 1.52
˚
A
a ean dagoenean e a 2.00 ku ba u a
baino balio handiagoen za age zen da. Bes e alde ik,
H( C)
- i dagokionez, nukleoen
a eko dis an zia 1.50
˚
A
baino xikiagoa denean balo e nega iboak au ki u di ugu edozein
ku ba u a balio e abiliz. A a i balo e ho i gaindi uz, lo u a pun u k i ikoan ebalua u iko
ene gia o ala en den si a ea en seinua alda u egi en da po en zial ku ba u a handi zen
dugun heinean; seinu aldake a hu a ge a zeko beha den ku ba u a po en zial minimoa
nukleoen a eko dis an zia ekin go ako a da. Teo ia maila hone an, gu e e dua e abiliz
lo u iko emai zak hain lan au eka ie an lo u ikoenen an zekoak di a; esa e ba e ako,
He2
@
C20H20
sis eman He-He dis an zia 1.265
˚
A
izanik
∆ρ( C)
- en balio posi iboa ema en
du e [210], gu e kasua ekin e ka uz
ω2<
2.00. Helio a omoen a eko dis an zia 1.60
˚
A
baino xikiagoa den sis emen za (
He2
@
B12N12
e a
He2
@
B16N16
)[216]
∆ρ( C)
balio posi i-
boak e a
H( C)
balio nega iboak eman di uz e. In e eseko sis ema ho e i bu uz gehiago
ikas eko an, 20
×
20 sa e uni o me ba e aiki dugu
(
0.00, 1.00
)×(
1.40, 2.6
)⊆(ω2
,
R)
domeinuan e a pun u bakoi zean CASSCF(4,8)/aug-cc-pVTZ kalkuluak bu u u ondo en,
∆ρ( C)
e a
H( C)
esku a u di ugu; ho ela, 5.9 i udian age zen di en ses a ku bak
e aiki di ugu.
∆ρ( C)
- i e epa a uz, domeinu osoan balio posi iboak au ki u di ugu e a
be e magni udea handi u egi en da nukleoen a eko dis an zia xiki zen den heinean; e a
be ean ku ba u a pa ame oa ekiko ia askea da. Azkenik, domeinu ho e an
H( C)
- en
balio nega iboak esku a u di ugu e a eu en magni udea handi u egi en da nukleoen a eko
dis an zia xiki zen den heinean; azken behake a ho i HF kalkulue a ik e a a omo guz iak
e abiliz lo u iko emai zekin ba e aga ia dela naba i u dugu.
Labu bilduz, a al hone an ikuspun u kimiko ik in e esga iak di en hainba aplikazio
emanda, ML eknikak ase diag amak e aiki zeko e a ase be iak au ki zeko esna e abil-
ga iak di ela ondo ioz a u dugu.

8.3 egindako lanen labu penak 117
8.3.5 Po en zial gauss a ak
Kon inamendu po en zial gauss a ek komuni a e zien i ikoa en a e a e aka i du e. Alde
ba e ik, ma e ia konden sa ua en eo ian ma ize e die oalee an in eg a u iko pun u kuan-
ikoen e eduak lo zeko e abil zen di uz e [233–236]. Be iki bi e a hi u dimen sioko pun u
kuan ikoei bu uzko lane an hainba p opie a e en e eduak lo zeko e abili di uz e, bes eak
bes e: Aha ono -Bohm oszilazioak [237], kohe en zia gale a e ek uak [238], p opie a e
e momagne ikoak [239–242], kanpo e emuen elka ekin zak [243–248], p opie a een men-
peko asun opologikoak [249, 250], elka ze kuan ikoa [251], e edugin za ma ema ikoa [252–
254], elek oi gu xidun sis emak [255, 256] e a aba . Bes e alde ba e ik, isika nuklea ean
α
pa ikulen a eko elka ekin zak desk iba zen di uz en Ali-Bodme po en zialak [257]
o aindik e e egi u a nuklea eko e edue an e abil zen di uz e [258–260]. Azken ho ie an,
po en zial gauss a ak di en moduan e abil zen di en a en, ma e ia konden sa ua i bu-
uzko lane an egi u a elek onikoa en kalkuluak egi e akoan, po en zial gauss a en o dez
ho iei hu bil zen di en po en zial ha monikoak e abil zen di a. Ho iei, Hooke-n a omoan
de i ze. Ho iek kasu ba zue an hu bilke a ap oposak iza en badi a e e, pun u kuan ikoen
lo u iko egoe ak desk iba ze ako o duan hainba mugapen di uz e. Lehenik, po en zial
ho iek e abil zeak egi u a molekula a en gale a daka ; izan e e, hainba pun u an zen a-
u iko po en zial ha monien konbinazio lineal o o en emai za bes e zen u baka ba eko
po en zial ha monikoa da. Biga enez, po en zial ha monikoei in ini u lo u iko egoe a eslei
diezazkiekegu; ho az, ionizazioa e a disoziazioa bezalako p ozesuak desk iba zea ezinezkoa
da.
Lan honen 6. kapi uluan, hi u dimen sioko zen u baka eko po en zial gauss a e an
kon ina u ik dauden
n={
2, 4, 6, 8, 10
}
elek oidun sis emak
(8.22)
ekuazioan age zen den
Hamil onda a e abiliz az e u di ugu. Dagozkion bi go pu ze ako Yukawa elka ekin zei
dagozkien in eg alak gu e aldean au e iz egindako lane a ik be esku a u di ugu [226].
H=−1
2
n
X
i=1∇2
i−
n
X
i=1
V0e−β 2
i+
n
X
j>i=1
e−λ ij
ij
(8.22)
Po en zial gauss a en bidez kon ina u iko sis emak az e zeko, beha ezkoak di en
N
zen u di uen e a
(8.23)
ekuazioan age zen den kanpo po en ziala i dagozkion ma ize
elemen uak kalkula u beha di ugu. Gu e kasuan, kimika kuan ikoan egin ohi denez, ma ize
elemen uak oina i un zio gauss a ak e abiliz
(8.24)
ekuazioan age zen di en moduan
lo u di ugu. In eg al hauen kalkulua egi eko, un zio gauss a en bide kadu a eo ema
(ingelesez Gaussian p oduc heo em) bi aldiz aplika uz, balio anali iko i xiak esku a u
di ugu.
Vex ( ) = −
N
X
i=1
V0,ie−βi( −R0,i)2(8.23)
ZG1(α1,RA,l1,m1,n1)G2(α2,RB,l2,m2,n2)e−βi( 1−R0,i)2d 1(8.24)
118 euska azko labu pena
Kalkula u iko in eg alen emai zak egokiak di ela e a po en zial hauei bu uz gehiago
ikas ea en, lehendabiziko kalkulu ba zuk bu u u di ugu. Ho ien emai ze an oina i u a,
lo u iko in eg alen balioa e a ho ien inplemen azio konpu azionala zuzenak di ela oga u
dugu.
Lehenik,
{n=
2, 4, 6, 8, 10
}
elek oiz oso u iko spin egoe a single ea e a iple ean
dauden sis emen gaineko CASSCF e a MRMP2 kalkuluak
m={
10, 11, 12, 13
}
o bi al
e abiliz bu u u di ugu. Sis ema hauek 4. kapi uluan age zen di enekin lo u a e a zeko
asmoz,
V0
pa ame oa en balio handiak e a zabale a pa ame o ap oposdun po en zial
gauss a e an kon ina u iko sis emak az e u di ugu
ω2=
2
βV0=
1
/
4. Ho i dela e a,
kapi ulu ho an au kez u iko EBTS-6S oina iak e abil zea egokia izan da. Ildo be ean,
V0
pa ame oa en 30 balio auke a u di ugu uni o meki bana u iko
[
10.0, 300.0
]
a ean. Hain
balio handiak ha uz, un zio po en zial gauss a a en Taylo -en se ie moduan adie azi
dugu. Ga apen ho e an age zen di en lehen hi u gaiak po en zial ha moniko baliokideei
dagozkien elemen uak di a; gaine a, ba ezbes eko ene gia zine ikoak
T
e a elka ekin za
Coulomb-ia ek
Vee
po en zial sakone a pa ame oa ekiko menpeko asunik ez du ela ha zen
badugu, sis ema gauss a a en ene gia
EG
e a sis ema Hooke-a baliokidea en ene giak
EH
(6.14)
ekuazioa en bidez lo u ik daude. Adie azpen ho i e abiliz e a egi u a elek onikoa en
kalkulue a ik esku a u ako da uak e abiliz,
EH
e a lehen zuzenke a anha monikoa
g
e eg esio linealen bi a ez de e mina u di ugu (ikusi 6.1 aula). Ikusi dugunez, e eg esioen
bidez esku a u ako
EH
balioak 4. kapi uluan age zen di en an zekoak di a; lo u iko
e o e ik handiena 1
×
10
−5
Ha ee-koa izan da. Lehenengo gai anha monikoa i dagokionez,
e o e ik handiena 4
×
10
−4
uni a e a omikoduna izan da. A zeman dugunez, ho en balioa
ez da naba i alda zen espazio ak iboan dauden o bi al kopu ua ekin,
n
elek oi kopu ua e a
spin egoe a ekiko, o dea, menpeko asuna naba iagoa da (bi elek oiz koposa u iko sis emen
kasua naba mena da). Ko elazio koe izien ea i e epa a uz, kasu guz iak ha uz, be e
ba ezbes eko balioa
R2=
0.9991 izan da e a kasu ik oke ena CASSCF(8,10)(S)/ETBS-6S
kalkulue a ik e a o i akoa R2=0.9623.
Azkenik, po en zial gauss a en bidez kon ina u ik e a egoe a singlean dauden bi
elek oiz oso u iko sis emek behin za lo u iko egoe a ba du ela kon uan izanik, sis ema
ho iek zein po en zial sakone a balio en zako disozia zen di en az e u dugu (
EG(Vd
0) =
0).
Gaine a, elek oien a eko elka ekin zek disoziazio p ozesuan du en e agina ike zeko
xedea ekin, Yukawa- en po en zialean age zen di en
λ
pa ame oa en 10 balio ha u
di ugu
[
0.10, 1.00
]
a e uni o mean. E a be ean, sakone a pa ame oa en 20 balio ha u
di ugu
[
0.50, 1.50
]
uni o meki bana u iko a ean e a egi u a elek onikoa en kalkuluak
CASSCF(2,13)/ETBS-6S e a MRMP2(2,13)/ETBS-6S maile an bu u u di ugu (ikusi 6.1
i udia). Au eko kasuan egin dugun an zeko e an, po en zial gauss a a Taylo -en se ie
moduan ga a u dugu, baina kasu hone an biga en gai anha monikoa e e ha u dugu;
ho ela, adie azpen un zional egokia e abiliz,
EH
,
g1
e a
g2
balioak e eg esio linealen
bidez es ima u di ugu. 6.2 aulan age zen di en emai zak kon uan izanik, e eg esio ekniken
bidez es ima u iko balioek pe u bazio me odoek gehi zen du en ko elazio dinamikoa ekiko
askeak di a az e u iko
λ
balio guz ien za ; ho ela, CASSCF e a MRMP2 me odoen bidez
lo u iko es imazioak an zekoak di ela oha u ga a. Ho ela, esku a u iko sis ema Hooke-
a baliokidea en ene gia
EH
ge o e a xikiagoa da
λ
handi zen den heinean; hala e a
guz iz e e, 4. kapi uluan age zen di en CASSCF/ETBS-6S mailan esku a u iko emai zekin
alde a u a, e a hone an lo u iko ene giak ba ezbes e 0.025 a.u. al uagoak di a. An zeko
neu iko e o eak espe o di ugu, izan e e, Hooke-n a omo baliokideen ene giak e a zeha zean
8.3 egindako lanen labu penak 119
lo zeko, disoziazio limi ean baino, sakonki kon ina u iko sis emak ike u beha di ugu. Gai
ez ha monikoei dagokienez,
g1
xiki zen doa
λ
handi zen den heinean. Hipo esi nagusia:
elek oi-elek oi elka ekinz ak ahul zen di en heinean, ko elazio e ek uak e e i zal zen
joango di a e a elek oiak po en zial pu zua en minimo ik hu bil au ki zeko auke a handi u
egi en da; ho az sis ema osoa ”ha monikoagoa” da e a
g1
-en magni udea xiki u egi en
da. Bes alde,
g2
biga en gai anha monikoa hazi egi en da
λ=
0.6 pun u aino e a ge o
xiki zen has en da.
Disoziazio a a ien balioa ahalik e a zeha zenak esku a zeko an, Taylo -en ga apena en
gaien seinuak al e na uak di ela oga u ondo en, un zioa S iljies mo akoa dela oha u ga a.
Ho i ho ela, Pad´e- en sekuen zia nagusiak ha u a (
P1
1(V−1
0)
e a
P1
2(V−1
0)
) disoziazio
sakone a limi ea en goi e a behe mugak de e mina u di ugu. 6.2 i udian ikus dezakegunez,
pan aila ze pa ame oa zenba e a handiagoa izan, elek oi-elek oi elka ekin zak o duan
e a ahulagoak izango di a e a po en zialak ez du ze an hain sakona izan beha pa ikulak
kon ina u ik (ene gia nega ibodun egoe a) man en zeko.
120 euska azko labu pena
8.4 EMAITZA NAGUSIAK
8.4.1
Kon inamendu ha monikoa ike zeko p o okolo kon-
pu azionalen ga apena
Tesi hone ako 3., 4. e a 5. kapi ulue an, sasi ba , sasi bi e a hi u dimen siodun po en zial
Ha monikoe an kon ina u iko elek oi gu xidun sis emak az e zeko konpu azio p o okoloen
ga apena au kez u dugu.
Alde ba e ik, sasi-ba e a sasi-bi dimen sio ako sis emak az e zeko, hi u-dimen sioko
s
mo ako oina i un zio gauss a bana uak e abili di ugu. Lehenengoen za beha adina
un zio bana u zi en uni o meki bana u ako sa e ba ean zeha . Fun zio ho iei dagozkien
be e zaileak
α
e a zen uak
{ i}
,
k
kon inamendu pa ame oa en menpekoak izanik,
po en ziala en in o mazioa in eg a u ik du e; bes alde, ondoz-ondoko un zioen a eko
gaineza pen in eg ala en balioa
S(α
,
δ) = exp(−αδ2/
2
) = exp(−
1
/
2
)
eza i dugu. Biga -
ena en za , un zio gauss a ak sa e hexagonal ba ean bana u ik daudela, ondoz ondoko
oina i un zioen a eko dis an zia
δ
, gaineza pen pa ame oa
ξ
, kon inamendu pa ame oa
k
e a oina i- un zio konpu u o ala emanik dagozkien
α
be e zaileak ba iazionalki op-
imiza u di ugu. Zenbai op imizazio egin di ugu kon inamendu pa ame oen hedadu a
handia azal zeko. Pa ikula baka a en e edu ik lo u iko emai za anali ikoak e abiliz,
sa e neu onal e eduak op imiza u e a eba u geni uen emai za ho iek edozein balio a a
heda zeko helbu ua ekin. Kon igu azio ho iek e abiliz e edu ho ien za e o e xikiak lo u
ez ezik, me odo mul i-e e e en zialei eske kon inamendu xikiko sis eme an age zen den
Wigne -en lokalizazioa modu egokian e ep esen a u e e egin dugu.
Bes e alde ba e ik, lau ge uzaz oso u iko zen u baka eko oina i un zioak (izenez
4SPDF )
k=ω2=
1
/
4 kon inamendu pa ame odun Hooke-n hi u-dimen sioko a o-
moen za op imiza u di ugu. Hainba elek oi kopu u (
n={
2, 4, 6, 8, 10
}
), spin egoe a
(single e e a iple e) e a pan aila ze pa ame o en
(λ={
0.0, 0.2, 0.4, 0.6, 0.8, 1.0
})
op i-
mizazioak bu u u ondo en, oina i unibe sal op imoa sei elek oiz e a egoe a single eko
sis ema ako lo u akoa zela ondo ioz a u dugu (izenez ETBS-6S). Oina i ho iek balia u a,
sis ema guz ien single e e a iple e spin egoe en ene gia absolu uak lo uz, au e iz a -
gi a a u iko emai zekin ba da ozela ikusi dugu. Ho ez gain, lo u ako singele e- iple e
di e en zia balioak zeha zak e a sis ema en Au bau egi u a zuzena e e badela egiaz a u
dugu.
8.4.2
Ziu gabe asun laginke en me odoen inplemen azioa ase
diag amak lo zeko
Bosga en kapi uluan, ETBS-6S oina ia en op imizazio-p ozesua e a zehaz asuna desk i-
ba u di ugu. Aipa u bezala, ho iek medio di ela, Hooke-n a omo es e ikoen egi u a elek-
onikoko kalkuluen bidez lo zen den Au bau egi ua 1
s <
1
p <
1
d
dela ikusi dugu.
Be e ze-egi u a hone an oina i uz, lau e a sei elek oiz osa u iko Hooke-n a omo es e ikoen
oina izko egoe a en spin mul iplizi a ea iple ea dela be ehala ikusi dugu. Hala e a guz iz
e e, sime ia es e ikoa apu uz ge o,
p
o bi alak ez di a gehiago hendeka u ik egongo, e a
alde bi an be eiziko di a:
{px
,
py}
e a
{pz}
; ho ela, oina izko egoe a en spin mul ipliz-
8.4 emai za nagusiak 121
i a ea sime ia es e ikoa nola haus en den e a sis ema konposa zen duen elek oi kopu ua en
menpekoa izango da.
Hooke-n a omo ez-es e ikoen oina izko egoe a en spin aniz asuna (single ea edo
iple ea) zein baldin za an eza zen den ule zeko, 6. kapi uluan e di-gainbegi a u iko
ikaskun zan e a ziu gabe asun laginke a me odoe an oina i u ako makina-ikaskun za
p ozedu a ba p oposa u dugu.
Lehen pauso gisa, p oposa u iko p ozesua ilus a zeko, pun u eu ek iko baka ba du en
e eduzko solido-likido sis ema ba zuen ase diag amak lo u di ugu. Ikus dezakegunez,
ziu gabe asun laginke a eknikei eske , algo i moa i e ase guz iak lagin zeko gai da, be eziki,
pun u eu ek iko ik hu bil dauden pun uak.
Hooke-n a omo aniso opikoei dagokienez,
x
e a
y
no abidee an
ω2
x,y
kon inamendu
pa ame oa e a
z
no abidean
ω2
z
kon inamendu pa ame oa eza i di ugu. Gaine a, elek oi-
elek oi elka ekin za modula zen duen
λ
Yukawa pa ame oa ekin e e hainba oga
egin di ugu. Adie azi ako laginke a p ozesua gauza u ondo en, kasu es e iko guz ien za
(ω2
x,y=ω2
z
,
∀λ)
oina izko egoe a en spin mul iplizi a ea iple ea izan dela a zeman
dugu. Bes alde,
ω2
z> ω2
x,y
den kasue an, lau elek oidun sis emen za oina izko egoe a en
spin mul iplizi a ea single ea den bi a ean, sei elek oidun sis emen za iple ea iza en
ja ai zen du. Kon a a,
ω2
z< ω2
x,y
den kasue an, sei elek oidun sis emen za oina izko
egoe a en spin mul iplizi a ea single ea den bi a ean, lau elek oidun sis emen za iple ea
iza en ja ai zen du. Sis ema e eduga i ho iek e o kizunean e abili ahal izango di a elek oi
kopu u handiagoek osa u ako sis emak az e zeko.
Azkenik, me odoa en e abile a en ga an zia ilus a zen duen bes e adibide ba ema ea-
ga ik, po en zial es e iko ha monikoe an kon ina u iko helio-dime oen izae a kobalen ea
Bade -en eo ian age u ohi di en lo u a desk iba zaileak e abiliz az e u dugu. Kasu hone-
an, helio a omoen a eko lo u a kimikoa en izae a kobalen ea neu u dugu, lo u a pun u
k i ikoan kalkula u ako den si a ea en Laplace a a en e a guz izko ene gia-den si a ea en
zeinuak e abiliz. Gaine a, po en ziala en kon inamendu pa ame oa e a dis an zia in e nuk-
lea a, ase-diag ama e aiki zeko kanpoko pa ame o gisa e abili di ugu. Egoe a kobalen een
eskualde ezbe dinak i aga i a en, kuali a iboki desk iba zaile biek nukleoen a eko dis-
an zia xiki e a kon inamendu balio handie an egoe a kobalen een exis en zia au esa en
du e.
8.4.3
Kon inamendu gauss a ei dagozkien go pu z ba eko
in eg alak inplemen a u e a au e iaz lo u iko emai zak
heda u
O ain a e ike u ako sis eme an, elek oiak po en zial ha monikoen bidez kon ina u ik egon
di a. E edu ho iek pun u kuan ikoen e a a omo a i izialen lo u iko egoe ak desk iba zeko
lehen hu bilke a gisa e abil di zakegun a en, ez di a kapaz sis ema e eale an ga an z-
i suak di en hainba ˜naba du a desk iba zeko. Lehenik, hainba pun u an zen a u iko
po en zial ha monikoen konbinazio lineal ba o aindik po en zial ha moniko ba denez,
zen u ani ze a ik e a o i ako egi u a molekula a beha zea ezinezkoa da. Biga enez,
po en zial ha monikoek in ini u egoe a lo u di uz e; ho i dela e a, ionizazioa e a disoziazioa
bezalako p ozesuak desk iba zeko e abilezinak di a.

122 euska azko labu pena
Lan honen zazpiga en kapi uluan, kon inamendu gauss a en ga an zia e a p opie a e
o oko ak aipa zen di ugu; bes eak bes e (po en zial ha monikoak ez bezala) egi u a
molekula a e a lo u iko egoe a kopu u ini uak di uz ela. Ho e az gain, beha ezkoak
di en go pu z ba eko in eg alen kalkulua en o mulazioa adie azi dugu.
Po en zial ho iek e abiliz, lehendabiziko emai za ba zuk ema eko asmoa ekin, kasu ilus-
a zaile bi au kez u di ugu. Alde ba e ik, sakonki kon ina u iko (
V0
sakone a pa ame oa en
balio handiak) elek oi kopu u bikoi iz (
n={
2, 4, 6, 8, 10
}
) e a u iko sis emen single e-
iple e ene gia a ea CASSCF(
n
,
m
)/ETBS-6S e a MRPT2(
n
,
m
)/ETBS-6S maile an
m={
10, 11, 12, 13
}
o bi al ak ibo e abiliz kalkula u e a au e iaz Hooke-n a omoen za
lo u iko emai zekin alde a u di ugu. Taylo -en se ieak e a e eg esio linealak a eko,
Hooke-n a omo baliokidea en ene gia e a lehen gai anha monikoa e eg esio ik lo u iko
pa ame oak e abiliz lo u di ugu. E a ho e an lo u iko Hooke-n a omo baliokidea en
ene giak e a lauga en kapi uluan age zen di enak oso an zekoak di ela ikusi dugu. Bes e
alde ba e ik, Yukawa po en zialen bidez elka eki zen e a spin egoe a single ea du en
elek oi biz oso u iko sis ema en disoziazio mugak ike u di ugu. Kasu hone an, Taylo -en
se ieak e abili beha ean, konbe gen zia azka agoa du en Pad´e- en sekuen zia nagusie a a
jo dugu; ho ela, objek u ho ien p opie a ee an oina i u a, hainba pan aila ze pa ame o i
dagozkien goi e a behe disoziazio mugak lo u di ugu.
8.5 ondo io nagusiak 123
8.5 ONDORIO NAGUSIAK
Lan hone an po en zial es a ikoen bidez kon ina u iko elek oi sis emei bu uzko au e apen
ba zuk jaso di a. Lehenik, ba , bi e a hi u dimen sioko po en zial ha monikoei eske ,
hainba spin zenbaki di uz en elek oi gu xiz konposa u iko sis emak az e zeko konpu azio-
p o okoloak ga a u di ugu. Zehazki, ba e ik, op imizazio eknika klasikoen bidez, zen u
baka eko e a dimen sio ba en e a bi an bana u iko oina i un zio gauss a egoki ba zuk
lo u di ugu. Bes e alde ba e ik, oina i un zio ho iek CASSCF e a MRPT2 me odo mul i
e e e en zialak e abiliz: kon inamendu ahulean gauza zen den Wigne -en lokalizazioa e a
single e- iple e ene gia a eak modu egokian desk iba zeko gai izan ga a.
Hooke-n a omo es e ikoen kasuan, sis ema en Au bau egi u a ha u a, oina izko ego-
e a en spin anizkoi z asuna elek oi kopu ua en menpekoa da. Hale e, sime ia es e ikoa
apu u egi en da (
ω2
z=ω2
x,y
) oina izko egoe a en spin mul iplizi a ea elek oi kopu ua en
menpe ego eaz gain, sime ia ho i hau si izan dugun e a en menpekoa e e bada. Elek oi
kopu u jakin ba ha u a, e di gainbegi a u iko e a ziu gabe asun lagineke a eknikak
e abil zen di uen ga a u iko makina ikaske a me odo ba en bidez, kon inamendu e a pan-
aila ze pa ame oen menpekoak di en oina izko egoe a en spin mul iplizi a ea en ase
diag amak e aiki zea lo u dugu. Gaine a, me odo ho i kimikoki esangu a suak di en bes e
sis ema gehiago ike zeko (solido-likido nahas e bina ioak e a lo u a kimikoa en na u a en
analisia) e abilga ia dela oga u dugu.
Po en zial ha monikoen luzapen gisa, zen u ani ze a ik e a o i ako p opie a e moleku-
la ak, disoziazio p ozesuak e a eka pen anha monikoak bil zen di uz en kon inamendu
po en zial gauss a ei sa e a eman diegu. Beha ezkoak di en go pu z ba eko in eg alak
GAMESS-US so wea -ean inplemen a zeaz gain, baliokideak di en sis ema Hooke-onda ekin
alde a uz lo u iko emai zei e epa a uz po en zial bi ho ien a eko lo u a e e oga u dugu.
E o kizun hu bilean, lan hone an lo u ako emai zak e a ga a u iko me odoak kon ina-
mendu e egimen gehiago a a e a sis ema e ealis agoe a a heda uko di ugu.
124 euska azko labu pena
8.6 ETORKIZUNERAKO LANAK
Labu bilduz, lan hone an sasi-ba , sasi-bi e a hi u dimen sio an po en zial ha monikoen
bidez kon ina u iko elek oi sis emak desk iba zeko beha ezkoak di en p o okolo kon-
pu azioanalak ga a u di ugu. Ho e a ako, makina ikaskun zan oina i u iko hainba eknika
e abili di ugu beha ezkoak di en oina i un zioak op imiza zeko e a egi u a elek oniko ik
e a o i ako p opie a eei dagozkien ase diag amak i udika zeko. Azkenik, kon inamendu po-
en zial gauss a ei dagozkien go pu z baka eko in eg alak lo u e a inplemen a u ondo en,
po en zial ha monikoa ekin lo u iko emai zekin dauka en lo u a e e eza i dugu.
Lo u ako ezagu za bes e sis ema ba zue a a heda zeko helbu uz, e a gu e aldea en
espe ien zian oina i u a, lan hone an ga a u iko e eduak dopa u iko klus e endoed ikoe an
aplika zea kon side a u dugu. O ain, p oiek u honi bu uzko bi gai nagusiak au kez uko
di ugu: a omo a i izial e a supe a omoekin lo u ik dauden dopa u iko klus e endoed ikoak
e a po en zial es a ikoei eske kon ina u iko sis ema kuan ikoak. Bi alo e an esku a u ako
lan e a emai za ba zuk ez abaida u ondo en, bi diziplinen a eko zubi ba eza iko dugu
e a gu e ikuspegia au kez uko dugu.
Alde ba e ik, klus e ak ma e ia konden sa ua en e a molekulen a ean daude, e a
ho ek p opie a e elek oniko in e esga iak iza ea e agi en du. Sis ema ho ien a ean,
hu sik dauden nanoklus e es e ikoak au ki dai ezke, esa e ba e ako ain zinda ia izan
zen ule eno 60 lan ano ekin dopa u iko klus e a La@C
60
[270], an zekoak di en bes e
hainba [271, 272] e a po en zialki e loju a omiko gisa e abil dai ezkeen ule eno ni ogeno
klus e ak [273]. Fule enoan oina i u ako espeziez apa e, ”supe a omo” izendun hainba
sis ema e e badaude[274–277], bes eak bes e, Al
12
ku xen kasuan, dopa zailea en na u a en
a abe a, sis ema osoa supe halogeno (bo oa ekin dopa uz), supe alkalino ( os o oa ekn
dopa uz), supe kalkogeno (kal zioa ekin dopa uz) edo 40 elek oidun sis ema egonko
gisa (silizio ekin dopa uz) ha dezakegu [278]. Bes e sis ema in e esga i ba zuk klus e
e die oalee an oina i u akoak di a, zeine a ako hainba p opie a e doiko (xu gapena,
emisioa e a o olumineszen zia) hau eman di en [279–285]. Sis ema supe a omikoe an,
p opie a e ”a omikoak” egi u a elek oniko osoa en po ae a kolek ibo ik so zen di a;
haa ik, klus e e die oalee an kon ina u iko a omoen kasuan, sis ema osoa en p opie a eak
elemen u dopan e isola ue an beha u akoen oso an zekoak di a [286–290].
Bes e alde ba e ik, kon inamendu po en zial gauss a ak egi u a elek onikoa i bu-
uzko bai lan eo ikoe an [252–256], bai ma e ia konden sa ua i bu uzko lan aplika u-
agoe an e abili izan di uz e; hala nola: pun u kuan ikoak [233–237], p opie a e e mo-
magne ikoen ike ke a [239–242] e a LASER e a e emu elek ikoak e a ma e ia en a eko
elka ekin zak az e zeko [243–248]. Po en zial ho iek egi u a elek onikoko me odoe an
lo u iko emai zak kalib a zeko e a ho ien kali a ea neu zeko e abil zen di en po en zial
ha monikoen an zekoak di en a en [36, 37, 41, 80, 261–266], haien a eko desbe din asun
nagusia lehenengoa egi u a molekula ak e a lo u iko egoe a ini uak desk iba zeko gai dela
da (6. kapi ulua).
Bi kon zep u ho iek (dopa u iko klus e e die oaleak e a po en zial gauss a ak) kon-
uan inzanda, lehen u a s bezala, po en zial gauss a e an kon ina u a dauden me al
alkalinoen e a halogenoen ionizazio ene giak e a a ini a e elek onikoak ike zea gus a uko
li zaiguke. A omo askeen ionizazio ene gia, okupa u iko ene gia ga aiena en o bi ala e a
lehen o deneko pe u bazio me odoa (e a Helmann-Feyman eo ema aplika uz) e abiliz,
8.6 e o kizune ako lanak 125
ene gia ho ien hu bilezko balioak e die si di ugu e a eu en balioak kalkulu esplizi ue a ik
lo u iko emai zekin e ka u di ugu.
Sis ema endohed ikoen zako hasie ako emai za ba zuk
V0
sakone a e a
β
zabale a pa ame oak di uen po en zial gauss a ba en bidez
Z
zen-
baki a omikodun e a
n
elek oidun a omo ba en Hamil onda a
(8.25)
ekuazioa en bidez
adie aziko dugu.
H=−1
2
n
X
i=1∇2
i−Z
n
X
i=1
1
i
+
n
X
j>i
1
ij −V0
n
X
i=1
e−β 2
i(8.25)
Uhin un zio monode e minan alak ha u a, hu bilke a adiaba ikoa (Koopmans-en eo-
ema) e a Helmann-Feyman eo ema aplika uz, ionizazio po en zialak sakone a pa ame oa ekiko
duen menpeko asuna i dagokion e a
n
-ga en o bi al a omikoa en menpekoa den esp esio
hu bildu ba lo u dugu:
∂I
∂V0
=−*Ψion 
n−1
X
i=1
e−β 2
iΨion++*Ψa om 
n
X
i=1
e−β 2
iΨa om+
≈Dϕ( n)e−β 2
nϕ( n)E
Ho ela,
V0
pa ame oa en balioa xikia dela ha zen badugu, Helmann-Feynman
eo ema e abiliz esku a u iko esp esioa
V′
0∈[
0,
V0]
a e xikian in eg a uz ge o, ion-
izazio ene gia en esp esio hu bildua lo u dugu (ikusi
(8.26)
). Fo mulazio al e na iboa:
V0
pa ame oa en balio xikiak ha uz, lehen o deneko pe u bazio zuzenke a e abiliz emai za
be a esku a u dugu. Jakina, hu bilke a ho e an dagokion o bi al a omikoa en (e a e a
o oko ean uhin un zio osoa en) o ma po en zial pa a oekiko askea dela ain za ha u
dugu; kasu o oko ean ho i ez da zuzena Hooke-n a omoen kasuan ikusi dugun moduan
(4. e a 5. kapi uluak). Ondo ioz,
ω2=
2
βV0
moduan de ini u iko ku ba u a pa ame oa
e o kizuneko ike ke e an kon uan iza eko ezauga i adie azga ia izango dela us e dugu.
I≈I0+V0Dϕ( n)e−β 2
nϕ( n)E(8.26)
An zeko es a egia ja ai uz,
n
elek oidun a omo neu o ba e a honi dagokion
n+
1
elek oidun anioia ha uz,
A
a ini a e elek onikoa anioia en ionizazio ene gia baina seinu
nega iboa duen kan i a e bezala de ini uz, (8.27) ekuazioan age zen den hu bilke a lo u
dugu.
A≈A0−V0Dϕ( n+1)e−β 2
n+1ϕ( n+1)E(8.27)
Adie azpen bie an (
(8.26)
e a
(8.27)
) ikus dezakegun moduan,
V0
pa ame oa en balio
adin xikiak ha u a,
β
pa ame oa en balioa e e xikia bada, po en zial gauss a ak sis ema
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