Inhomogeneous mul iscale dynamics in ha monic la ices
Da id Cube o and Sophia N. Yali aki
Ci a ion: The Jou nal o Chemical Physics 122, 034108 (2005); doi: 10.1063/1.1829253
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Inhomogeneous mul iscale dynamics in ha monic la ices
Da id Cube o and Sophia N. Yali akia)
Depa men o Chemis y, Sou h Kensing on Campus, Impe ial College, London SW7 2AZ, Uni ed Kingdom
共Recei ed 10 June 2004; accep ed 14 Oc obe 2004; published online 3 Janua y 2005兲
We use p ojec ion ope a o s o add ess he coa se-g ained mul iscale p oblem in ha monic sys ems.
S ochas ic equa ions o mo ion o he coa se-g ained a iables, wi h an inhomogeneous le el o
coa se g aining in bo h ime and space, a e p esen ed. In con as o p e ious app oaches ha
ypically s a wi h he modynamic a e ages, he key elemen o ou app oach is he use o a
p ojec ion ma ix chosen bo h o i s physical appeal in analogy o mechanical s abili y heo y and
o i s algeb aic p ope ies. We show ha he modynamic equilib ium can be eco e ed and ob ain
he luc ua ion dissipa ion heo em a pos e io i. All sys em-speci ic in o ma ion can be compu ed
om a se ies o easible molecula dynamics simula ions. We eco e p e ious esul s in he
li e a u e and show how his app oach can be used o ex end he quasicon inuum app oach and
commen on implica ions o dissipa i e pa icle dynamics ype o me hods. Con a y o wha is
assumed in he la e models, he s ochas ic p ocess o all coa se-g ained a iables is no necessa ily
Ma ko ian, e en hough he a iables a e slow. Ou app oach is applicable o any sys em in which
he coa se-g ained egions a e linea . As an example, we apply i o he dynamics o a single
mesoscopic pa icle in he in ini e one-dimensional ha monic chain. © 2005 Ame ican Ins i u e o
Physics. 关DOI: 10.1063/1.1829253兴
I. INTRODUCTION
The p ocesses unde lying he p ope ies o ma e ials and
biological assemblies o en span a ange o leng h and ime
scales. In unde s anding and p edic ing hei beha io , i
would be desi able o s a om an a omis ic desc ip ion
which could be capable o exhibi ing he con inuum and
mac oscopic beha io o he sys em. Molecula dynamics
共MD兲alone is p esen ly incapable o b idging scales o such
o de s o magni ude. Coa se-g aining schemes ha ackle
di e en aspec s o his b oad p oblem a e cu en ly an ac-
i e a ea o esea ch. O pa icula in e es is he possibili y
o ollow he sys em dynamically and no jus ob ain i s he -
modynamic p ope ies.
A numbe o ‘‘mesoscale’’ dissipa i e pa icle dynamics
共DPD兲 ype me hods, o example, in oduce pa iclelike
a iables in con inuous equa ions and associa e wi h hem
conse a i e, dissipa i e, and andom o ces.1,2 The o igin
and app op ia e unc ional o m o hese o ces is s ill no
ully esol ed.3,4 A di e en app oach ha aims o a oid
hese equa ions al oge he elies on exploi ing h ough e i-
cien nume ical in eg a ion echniques a la ge numbe o
sho MD simula ions.5Ano he le el o complexi y is in o-
duced when he sys em needs o be inhomogeneously coa se
g ained. One o he mos success ul me hods unde his ca -
ego y, he quasicon inuum,6in ol es no explici ime no
ini e empe a u e. A gene aliza ion has been a emp ed bu
he dynamics ha e been in oduced ad hoc.7One o he main
emaining challenges in such me hods emains he e lec ions
a bounda ies be ween egions o di e en le el o
desc ip ion.8Coa se-g ained dynamics ha can sys ema i-
cally desc ibe he sys em a di e en scales in bo h ime and
space emain a challenge. A gene al scheme ha p esc ibes
he o m o he equa ions o be used based on an equi alen
oo ing o amewo k is cu en ly lacking.
We ha e used he p ojec ion ope a o app oach o Mo i
and Zwanzig9 o ob ain he dynamical equa ions o an inho-
mogeneous mul iscale sys em. To illus a e his, we sol e
exac ly he mul iscale inhomogeneous p oblem o ha monic
sys ems. In his ega d, ha monic sys ems, apa om hei
ob ious adequacy as a i s app oxima ion o solids, p o ide
an excellen amewo k o p obe hese ideas. The p ojec o
ope a o app oach has been used ex ensi ely in he pas . One
o he main d awbacks has been he e alua ion o he o mal
exp essions ob ained by he heo y. By using an algeb aic
p ojec ion and in analogy o mechanical s abili y, we ob ain
unc ional o ms o he o ces ha can be compu ed by a
se ies o sho molecula dynamics simula ions. The key
poin o ou app oach is ha we a oid al oge he he i s s ep
o a e aging and ob ain he he modynamic equilib ium
p ope ies a pos e io i. The mechanical analogy has been
i s in oduced in he li e a u e by Deu ch and Silbey o a
single pa icle in a la ice.10 In ha espec , ou esul s can be
conside ed a gene aliza ion o hei wo k. We also eco e as
a special case he esul s o Adelman and Doll o a om/solid
su ace sca e ing in ha monic la ices.11 Addi ionally, he
o malism p o ides a connec ion o he quasicon inuum
me hod wi h explici ime and ini e empe a u e. Finally, we
commen on he o m o he andom o ces and memo y
ke nels ha a e usually assumed in o he mesoscale dissipa-
i e dynamics ype o me hods.
The pape is o ganized as ollows: In Sec. II we in o-
duce he sys em and he coa se-g ained scheme. In Sec. III
we desc ibe he equa ions o he coa se-g ained a iables
using he Mo i o malism. In Sec. IV we in oduce a di e -
en p ojec ion ma ix and ede i e he heo y using ha inne
a兲Elec onic mail: [email p o ec ed]
THE JOURNAL OF CHEMICAL PHYSICS 122, 034108 共2005兲
122, 034108-10021-9606/2005/122(3)/034108/9/$22.50 © 2005 Ame ican Ins i u e o Physics
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p oduc , which allows us o ob ain closed exp essions o he
esul s in Sec. IV and a nume ical p ocedu e o compu e
hem in Sec. V. In Sec. VI we p opose a simula ion p oce-
du e o s udy he dynamics. We apply he heo y de eloped
in he p e ious sec ions o s udy he coa se-g ained dynamics
o a single mesoscopic pa icle in he in ini e one-
dimensional 共1D兲ha monic chain in Sec. VII. Finally, Sec.
VIII p o ides a sho summa y and conclusions.
II. SYSTEM DETAILS AND COARSE-GRAINING
SCHEME
We conside a ha monic sys em o dimension dconsis -
ing o Npa icles o mass mi. We will assume ha he in-
e ac ion be ween any wo pa icles only depends on he dis-
ance be ween hem. The Hamil onian o his sys em is
H⫽兺
␮
⫽1
d
冉
兺
i
1
2mi i,
␮
2⫹1
2兺
i,jxi,
␮
Aijxj,
␮
冊
,共1兲
whe e xi,
␮
and i,
␮
a e he
␮
componen s o he de ia ion
om he equilib ium posi ion and eloci y o he pa icle i,
espec i ely, and Aij is a symme ic ma ix ha sa is ies he
s abili y condi ion
兺
iAij⫽0. 共2兲
The a iables o in e es a e he coo dina es and eloci ies o
he cen e o mass o each o he coa se-g ained egions o
he sys em
Xk,
␮
⫽1
Mk兺
i
k
mixi,
␮
,Vk,
␮
⫽1
Mk兺
i
k
mi i,
␮
,共3兲
whe e Mk⫽兺i
kmiis he o al mass o he coa se-g ained e-
gion o pa icle k. The numbe o oscilla o s in each egion is
allowed o a y in acco dance wi h he desi ed coa se-
g aining le el. Fo example, i we a e in e es ed in keeping
he a omis ic le el in one pa o he sys em, he co espond-
ing egions con ain only one oscilla o and he coa se-
g ained a iables a e jus he coo dina es and eloci ies o
he o iginal oscilla o s. A he same ime, no e e y oscilla o
need o be ela ed o a coa se-g ained pa icle. We may be
in e es ed in coa se-g aining comple ely a pa o he sys em
comp ising many oscilla o s. In ha case no index kis asso-
cia ed wi h such egion. We p esen in Fig. 1 an example o
an inhomogeneous coa se-g aining scheme.
III. MORI THEORY
The equa ions o mo ion o hese a iables can be ob-
ained in a s aigh o wa d manne by using he Mo i p ojec-
ion ope a o o malism.9The equa ions a e ob ained by p o-
jec ing on o he subspace spanned by he ele an a iables.
The p ojec ion ope a o is de e mined by he choice o he
inne p oduc (B,C) in he Hilbe space o all unc ions
B(x, ) and C(x, ) o phase space coo dina es.9I is cus-
oma y o choose canonical equilib ium a e ages (B,C)
⫽
具
BC
典
eq , whe e
具
B共x, 兲
典
eq⫽
兰
dNxdN B共x, 兲e⫺H/kBT
兰
dNxdN e⫺H/kBT,共4兲
kBbeing he Bol zmann cons an and T he empe a u e. We
hen ob ain he ollowing linea gene alized Lange in equa-
ions:
dXk共 兲
d ⫽Vk共 兲,共5兲
Mk
dVk共 兲
d ⫽⫺兺
l
冋
⌳klXl共 兲⫹
冕
0
d
␶
⌽kl共
␶
兲Vl共 ⫺
␶
兲
册
⫹Rk共 兲,共6兲
whe e ⌳kl is a eno malized ma ix o ce, ⌽kl( ) a memo y
ke nel, and Rk( ) a o ce which con ains he in o ma ion o
he non ele an a iables we ha e le ou . We ha e d opped
he spa ial labels
␮
in hese exp essions o cla i y, because
he Hamil onian we a e using he e does no p o ide any cou-
pling be ween he di e en componen s.
The explici exp essions o he unknown quan i ies in
Eq. 共6兲in ol e complica ed unc ions con aining o mal p o-
jec ion ope a o s. Howe e , he ollowing in o ma ion can be
ob ained ela i ely easily 共see Chap. 8 o Re . 9兲: he o ce
Rk( ) sa is ies
具
Rk共 兲
典
⫽0共7兲
and
具
Rk共 兲Rl共0兲
典
⫽kBT⌽kl共 兲,共8兲
whe e 具¯典deno es a e ages o e a s a is ical ensemble o
he ini ial condi ions ha is close o equilib ium. Equa ion
共8兲is called he non-Ma ko ian luc ua ion-dissipa ion heo-
em. I gua an ees ha he sys em admi s he he modynamic
equilib ium solu ion.
The ad an age o his app oach is ha i p o ides us wi h
equa ions o mo ion whe e he o ces a e decomposed in
h ee pa s. The i s wo a e unc ions o he ele an a i-
ables X( ) and V( ) alone, and can be iden i ied wi h con-
se a i e and dissipa i e ype o ce e ms, espec i ely. The
o ce Rk( ) can be conside ed as a andom o ce. The e o e,
he equa ions can be cas inside he heo y o s ochas ic p o-
cesses. While he physical in e p e a ion o hese equa ions is
clea , he main p oblem esides in inding bo h closed o ms
o hese exp essions o ⌳kl and ⌽kl( ) as well as ways o
ac ually compu ing hem in p ac ice. This is no an easy ask
FIG. 1. Example o a coa se-g aining scheme in a 2D ha monic sys em.
034108-2 D. Cube o and S. N. Yali aki J. Chem. Phys. 122, 034108 (2005)
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e en in he case o linea sys ems. Fo example, wi h his
p ojec ion me hod, we ind he ma ix ⌳kl as he in e se o
he ma ix
具
Xl⬘Xl
典
eq /kBT, i.e.,
兺
l⬘
⌳kl⬘
具
Xl⬘Xl
典
eq
kBT⫽兺
l⬘
具
XkXl⬘
典
eq
kBT⌳l⬘l⫽
␦
kl .共9兲
This in e se ma ix is gene ally di icul o compu e. Special
ca e is equi ed as can be seen by no ing ha he indi idual
elemen s
具
Xl
2
典
eq do no exis in he he modynamic limi
(N→⬁) because o ansla ional in a iance.
IV. MORI THEORY WITH A SPECIAL INNER PRODUCT
A. The mechanical analog: Random o ces
The exp essions in he p eceding sec ion ha e been ob-
ained by using equilib ium a e ages as he inne p oduc .
Al hough wi h his app oach we ob ained s aigh away Eqs.
共5兲–共9兲, i is no clea how one can p oceed u he .
A be e insigh is gained i we use ins ead an inne
p oduc wi h he ollowing p ope ies:
共xi,
␮
,xj,
␯
兲⫽共 i,
␮
, j,
␯
兲⫽
␦
ij
␦
␮
␯
mi
⫺1,共xi,
␮
, j,
␯
兲⫽0.
共10兲
Using his inne p oduc , he p ojec ion ope a o o mal-
ism p oduces he equi alen equa ions o he p eceding sec-
ion. Speci ically, we eco e Eq. 共5兲and a modi ied equa ion
o he ime de i a i e o V( ), Eq. 共6兲 ha now in ol es a
memo y ke nel ela ed o he ime con olu ion o he posi-
ion X( ). In eg a ing by pa s his in eg al we a i e a
Mk
dVk共 兲
d ⫽⫺兺
l
冋
AklXl共 兲⫹
冕
0
d
␶␤
kl共
␶
兲Vl共 ⫺
␶
兲
⫹
␤
kl共 兲Xl共0兲
册
⫹R
ˆk共 兲,共11兲
whe e we ha e w i en Akl⫽兺i
k兺j
lAij. We now need o com-
pu e he andom o ce R
ˆk( ) and he memo y ke nel unc ion
␤
kl( ). The la e is connec ed o he o me by he p ojec-
ion ope a o s o malism as
␤
kl共 兲⫽⫺
冕
0
d
␶
关LR
ˆk共
␶
兲,Xl共0兲兴Ml,共12兲
whe e Lis he Liou ille9ope a o o he sys em. We la e
ela e each quan i y o hose o he p eceding sec ion 关see
Eq. 共6兲兴.
The andom o ce is in gene al a complica ed unc ion o
he ini ial coo dina es and eloci ies o he sys em. In ha -
monic sys ems, howe e , i can be exp essed as a linea unc-
ion o he ini ial condi ions 共see Sec. 8.5 o Re . 9兲. This will
u he allow us o es ablish a mechanical o ce analogy wi h
a e e ence mechanical sys em in which he coa se-g ained
a iables a e held ixed.
We s a by assuming
R
ˆk共 兲⫽兺
iqi
k共 兲mixi共0兲⫹ i
k共 兲mi i共0兲.共13兲
Consis ency equi es ha he ime dependen coe icien s a e
gi en by i
k( )⫽
兰
0
d
␶
qi
k(
␶
) and he se o equa ions
miq
¨i
k共 兲⫽⫺兺
jA
ˆijqj
k共 兲,共14兲
miqi
k共0兲⫽⫺兺
j
k
关共I⫺P
ˆ兲A兴ij,q
˙i
k共0兲⫽0, 共15兲
whe e he do deno es ime de i a i e,
A
ˆ⫽共1⫺P
ˆ兲A共1⫺P
ˆ 兲共16兲
and P
ˆis a p ojec ion ma ix de ined by
P
ˆij⫽mi
Mk共i兲
␦
k共i兲,k共j兲,共17兲
whe e k(i) deno es he index o he g oup in which he os-
cilla o ibelongs and P
ˆ he ansposed ma ix.
No e ha i x⫽(x1,x2,...) is he ec o o med by all
oscilla o s’ coo dina es, P
ˆ x⫽Xgi es a ec o whose com-
ponen s a e he cen e s o mass o he coa se-g ained pa -
icles. In addi ion, i io jbelongs o a egion ha has been
o ally coa se g ained, hen P
ˆij⫽0. Thus, i bo h iand j
belong o such egions, hen A
ˆij⫽Aij.
I is easy o show ha he new o ce ma ix A
ˆis sym-
me ic and sa is ies he s abili y condi ion 共2兲. The e o e,
qk( ), and hus he andom o ces R
ˆk( ), a e exp essed in
e ms o he mechanical p oblem 共14兲–共15兲. In ac , using he
algeb a we p esen below, we can show ha he andom
o ces can be u he w i en as
R
ˆk共 兲⫽⫺兺
i
k
冉
兺
jAij关x
ˆj共 兲⫹
ˆj共 兲兴
冊
,共18兲
whe e x
ˆ( ) and
ˆ( ) a e he solu ion o he mechanical p ob-
lem 共14兲wi h he ini ial condi ions x
ˆ(0)⫽(I⫺P
ˆ )x(0) and
ˆ(0)⫽(I⫺P
ˆ ) (0). This is a gene aliza ion o he esul s
p esen ed in Re . 10 o a single pa icle in a la ice.
The mechanical p oblem gene a ed by A
ˆis ela ed o he
o iginal mechanical p oblem o he unde lying la ice bu
ins ead he cen e o mass o he coa se-g ained pa icles a e
ixed, as illus a ed by he ac ha he ollowing p ope y is
obeyed a all imes:
P
ˆ qk共 兲⫽P
ˆMqk共 兲⫽0, 共19兲
whe e Mij⫽
␦
ijmiis he mass ma ix.
No e ha i ini ially he e is no diso de in he coa se-
g ained la ice, i.e., he e a e only collec i e ini ial de ia ions
om equilib ium, hen R
ˆk( )⫽0 a all imes. This is a c ucial
p ope y o linea sys ems ha will allow us o compu e he
ele an magni udes o he coa se-g ained heo y om
simple simula ions.
The solu ion o Eqs. 共14兲–共15兲can be o mally w i en
as
qk共 兲⫽Re共ei⍀ 兲qk共0兲,共20兲
whe e ⍀is he equency ma ix gi en by ⍀2⫽M⫺1A
ˆand i
is he imagina y uni . Equa ion 共19兲is a consequence o
A
ˆP
ˆ ⫽0, which implies ⍀P
ˆ ⫽0. Thus, he subspace gene -
a ed by P
ˆ co esponds o eigen ec o s o A
ˆwi h ze o eigen-
alues, each one co esponding o e e y coa se-g ained pa -
034108-3 Inhomogeneous mul iscale dynamics J. Chem. Phys. 122, 034108 (2005)
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icle. As a consequence, nei he ⍀no A
ˆpossess an in e se.
Howe e , since bo h ma ices a e diagonalizable (A
ˆis sym-
me ic and ⍀is He mi ian wi h he scala p oduc xy
⫽x My), we can de ine he pseudoin e se ⍀⫺1so ha
⍀⫺1⍀⫽⍀⍀⫺1⫽I⫺P
ˆ ,⍀⫺1P ⫽0. 共21兲
Then, he pseudoin e se A
ˆ⫺1⫽(⍀⫺1)2M⫺1 u ns ou o be
symme ic.
B. Memo y ke nel
We ha e ob ained he andom o ces h ough he ans-
o ma ion o an equi alen mechanical sys em. We can now
ackle he ke nel
␤
kl( ), which using Eqs. 共13兲and 共12兲, can
be w i en as
␤
kl共 兲⫽兺
j
冕
0
d
␶
冕
0
␶
d
␶
⬘qj
k共
␶
⬘兲Ai
l共22兲
whe e Ai
l⫽兺j
lAij. In he limi →⬁, each qi
k( ) decays o
ze o as he ini ial pe u ba ion 共15兲is p opaga ed o he
bounda ies o he sys em by elas ic wa es. Howe e , since
Eq. 共22兲in ol es a double ime in eg al, in gene al
␤
kl( )
does no decay o ze o wi h ime. A be e ep esen a ion is
ob ained by calcula ing he in eg als in Eq. 共22兲using Eq.
共20兲, which leads o
␤
kl共 兲⫽⌽kl共 兲⫺⌽kl共0兲,共23兲
whe e
⌽kl共 兲⫽⫺兺
jAj
l关A
ˆ⫺1Mqk共 兲兴j共24兲
wi h he ollowing p ope ies
兺
l⌽kl共 兲⫽0 and ⌽kl共 兲⫽⌽lk共 兲.共25兲
This ke nel is expec ed o anish as →⬁in he he mody-
namic limi (N→⬁), in which case he bounda ies a e e-
mo ed. No e ha we ha e used he same le e o name his
ke nel and he one we ob ained in Sec. III using equilib ium
a e ages as he inne p oduc . We will show in Sec. IVC ha
hey a e indeed he same.
We can now e u n o he gene alized Lange in equa ion
共11兲, which becomes
Mk
dVk共 兲
d ⫽⫺兺
l关Akl⫺⌽kl共0兲兴Xl共 兲
⫺兺
l
冕
0
d
␶
⌽kl共
␶
兲Vl共 ⫺
␶
兲
⫺兺
l⌽kl共 兲Xl共0兲⫹R
ˆk共 兲.共26兲
Equa ion 共26兲is he gene aliza ion o he p esen a bi a y
inhomogeneous coa se g aining o he esul s o Adelman
and Doll o a om/solid su ace sca e ing in ha monic
solids.11
No e ha wi h he iden i ica ions
⌳kl⫽Akl⫺⌽kl共0兲共27兲
and
Rk共 兲⫽⫺兺
l⌽kl共 兲Xl共0兲⫹R
ˆk共 兲,共28兲
Eq. 共26兲is jus Eq. 共6兲.
Finally, no e ha Eqs. 共25兲and 共2兲imply he ollowing
expec ed p ope y o he conse a i e o ce
兺
l⌳kl⫽0. 共29兲
C. Fluc ua ion-dissipa ion heo ems
Using he explici exp essions 共13兲and 共24兲, and he
in eg a ion o mula o Gaussian dis ibu ions, we eadily
ob ain
具
R
ˆk共 兲R
ˆl共0兲
典
A
ˆ⫽kBT⌽kl共 兲,共30兲
whe e
具
¯
典
A
ˆdeno es a e ages o e he ini ial condi ions
wi h he canonical dis ibu ion
␳
ˆ(x, )⬀exp(⫺H
ˆ/kBT),
whe e H
ˆhas he same o m as he Hamil onian o Eq. 共1兲,
bu wi h A
ˆins ead o A. This is he e sion o he luc ua ion
dissipa ion heo em ha appea s in Re . 11. We show in Ap-
pendix A ha he a e ages aken wi h
具
¯
典
A
ˆin Eq. 共30兲a e
he same as he cons ained a e ages
具
¯
典
X0⫽0de ined by
he canonical dis ibu ion wi h he cen e o mass o he
‘‘coa se-g ained’’ pa icles X0 ixed a hei equilib ium al-
ues (X0⫽0), p o iding a physical meaning o he
luc ua ion-dissipa ion equa ion 共30兲. Ne e heless, no e ha
Eq. 共30兲does no hold o gene al cons ained a e ages
具
B共x, 兲
典
X0⫽
兰
dNxdN B共x, 兲e⫺H/kBT
␦
共P
ˆ x⫺X0兲
兰
dNxdN e⫺H/kBT
␦
共P
ˆ x⫺X0兲,共31兲
wi h X0di e en om hei equilib ium alues.
Fo ou pu poses i is mo e con enien o in oke he
s anda d o m o he luc ua ion-dissipa ion heo em, in ol -
ing he andom o ce Rk( )
具
Rk共 兲Rl共0兲
典
eq⫽kBT⌽kl共 兲,共32兲
whe e he a e ages a e aken wi h he ull equilib ium dis i-
bu ion using A. We can p o e Eq. 共32兲s aigh away by using
he ollowing algeb aic iden i y:
A
ˆ⫺1⫽A
ˆ⫺1AP
ˆ A⫺1P
ˆAA
ˆ⫺1⫹A
ˆ⫺1AP
ˆ A⫺1共I⫺P
ˆ兲
⫹共I⫺P
ˆ 兲A
ˆ⫺1AP
ˆ A⫺1⫹共I⫺P
ˆ 兲A⫺1共I⫺P
ˆ兲,共33兲
whe e A⫺1is he pseudoin e se o Ade ined in simila ash-
ion o Eq. 共21兲by A⫺1A⫽AA⫺1⫽I⫺PT.(PT)ij⫽1/Nis he
p ojec o on he ansla ional mode 关gi en by Eq. 共2兲兴, he
only ze o equency mode in egula la ices.
Using a simila algeb a i can be eadily shown ha he
conse a i e o ce ⌳kl de ined by Eq. 共27兲is he same as he
o ce we ob ained in Sec. III, de ined by Eq. 共9兲. Al e na-
i ely, we can p o e his by aking he limi T→0 in Eqs. 共6兲
and 共26兲. Since Vk( )→0 in ha limi and bo h equa ions a e
exac o any ini ial condi ion, hey mus con ain necessa ily
he same conse a i e o ce.
Finally, no e ha i we w i e Eq. 共5兲as
034108-4 D. Cube o and S. N. Yali aki J. Chem. Phys. 122, 034108 (2005)
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冕
0
d
␶
Vl共 ⫺
␶
兲⫽Xl共 兲⫺Xl共0兲,共34兲
mul iply his equa ion by an a bi a y cons an
␥
kl and add i
o Eq. 共6兲we ob ain
Mk
dVk共 兲
d ⫽⫺兺
l
冋
⌳
¯
klXl共 兲⫹
冕
0
d
␶
⌽
¯
kl共
␶
兲Vl共 ⫺
␶
兲
册
⫹R
¯
k共 兲,共35兲
whe e ⌳
¯
kl⫽⌳kl⫹
␥
kl ,⌽
¯
kl⫽⌽kl⫺
␥
kl , and R
¯
k⫽兺l
␥
klXl
⫹Rk. The e o e, by changing
␥
kl we can ob ain an in ini e
numbe o di e en equa ions o mo ion, all wi h a andom
o ce R
¯
sa is ying he luc ua ion-dissipa ion o mula 共30兲
共 hough wi h ⌽ins ead o ⌽
¯
). Howe e , he co esponding
ke nel ⌽
¯
kl a long imes con e ges o ⫺
␥
kl , esul ing in an
a i icial nondecaying memo y e ec . Fu he mo e, R
¯
k共as
well as R
ˆk) does no sa is y he s anda d luc ua ion-
dissipa ion o mula 共32兲. In ac , in gene al he luc ua ions
o hose andom o ces di e ge in he uncons ained he mo-
dynamic equilib ium due o ansla ional in a iance. The e-
o e, we will e e in he ollowing o he Lange in equa ion
gi en by Eq. 共6兲o equi alen ly Eq. 共26兲.
V. A FEW IMPORTANT RESULTS
The exp ession 共24兲we ob ained o he memo y ke nel
is s ill oo o mal o mos p ac ical si ua ions. In his sec ion
we p esen a ew exac esul s ha will acili a e i s calcula-
ion bo h analy ically and nume ically.
A. Veloci y au oco ela ion unc ion
Le us conside he eloci y au oco ela ion ma ix
Ckl共 兲⫽
具
Vk共 兲Vl共0兲
典
eq .共36兲
Mul iplying Eq. 共6兲by Vl(0) and aking equilib ium a e -
ages we ob ain a se o di e en ial equa ions wi hou noise
Mk
dCkl共 兲
d ⫽⫺兺
l⬘
冕
0
d
␶
关⌳kl⬘⫹⌽kl⬘共 ⫺
␶
兲兴Cl⬘l共
␶
兲,
共37兲
which oge he wi h he ini ial condi ion Ckl(0)
⫽
␦
klMkkBTde e mine Ckl( ). Al e na i ely, i we know
Ckl( ) we can de i e ⌽kl( ) om hese equa ions.
In Appendix B we ake ad an age o his ela ionship o
show ha i Ckl( ) p esen s a discon inui y in i s de i a i e
a ime , hen he memo y ke nel p esen s a Di ac-del a sin-
gula i y a he same ime . To be mo e p ecise,
⌽kl共 兲⫽⫺
冋
C
˙kl共0兲
Cll共0兲
␦
共 兲⫹兺
⌬C
˙kl共 兲
Cll共0兲
␦
共 ⫺ 兲
册
⫹
␸
kl共 兲,共38兲
whe e ⌬C
˙kl( )⫽C
˙kl(
⫹)⫺C
˙kl(
⫺), deno es he imes
whe e C
˙kl( ) is discon inuous, and
␸
kl( ) is an o he wise
smoo h unc ion. This p ope y will become use ul when we
conside coa se g aining in he ime scale in Sec. VII.
The abo e esul s a e gene al and no pa icula o linea
sys ems. Howe e , in ha monic sys ems he e is a simple
way o compu e he au oco ela ion ma ix. In ac ,
Ckl共 兲⫽Vk共 兲Vl共0兲,共39兲
whe e Vk( ) is he eloci y o he coa se-g ained pa icle kin
he sys em s a ing om he ollowing ini ial condi ions:
xi共0兲⫽0 and i共0兲⫽
␦
k共i兲,lVl共0兲,共40兲
wi h Vl(0)2⫽MlkBT. This me hod exploi s he ac ha he
noise Rk( ) anishes wi h his pa icula choice o he ini ial
condi ions, as shown in Sec. IVA.
B. Nume ical calcula ion o he ke nel
and conse a i e o ces
No e ha Eqs. 共15兲and 共19兲imply ha i he g oup k
con ains only one oscilla o , and i is connec ed h ough he
o ce ma ix Awi h o he single-oscilla o g oups, hen
qi
k( )⫽0 o all iand hence ⌽kl( )⫽Rk( )⫽0 a all imes.
Thus, in a egion whe e we ha e kep he a omis ic desc ip-
ion, he gene alized Lange in equa ion 共6兲 educes o he
o iginal New on’s equa ion.
The e o e, he p oblem is educed o compu ing he con-
se a i e o ces ⌳kl and he ke nels ⌽kl( ) o hose coa se-
g ained pa icles ha belong o ac ual coa se-g ained egions
o in he p oximi ies o hem. Because o Eq. 共27兲, we jus
need o calcula e he memo y ke nel ⌽kl( ). This can be
done by sol ing Eq. 共14兲nume ically. Then, by using Eq.
共22兲we can ob ain
␤
kl( ). This in ol es a double nume ical
in eg a ion. The long ime limi o
␤
kl( ) gi es ⌽kl(0), om
which we can de e mine ⌳kl and ⌽kl( ).
Al e na i ely, we can use he mo e di ec me hod o Cai
e al.8In his app oach we un a molecula dynamics simu-
la ion o he ha monic sys em s a ing om he ollowing
ini ial condi ion:
xi共0兲⫽
⑀
␦
k共i兲land i共0兲⫽0. 共41兲
Nex , he cen e o mass o he coa se-g ained pa icles is
kep ixed by means o an ex e nal o ce on each oscilla o
Fk(i). The e o e, Eq. 共26兲becomes
兺
i
k
Fk共i兲共 兲⫺关⌳kl⫹⌽kl共 兲兴
⑀
⫽0. 共42兲
By de e mining he ex e nal o ce equi ed o keep he cen e
o mass ixed we can ob ain he conse a i e o ces and he
memo y ke nel. This me hod is also applicable o si ua ions
in which he ha monic cha ac e is a i s -o de app oxima-
ion o a mo e complex sys em.
The me hods abo e p o ide us wi h a nume ical ep e-
sen a ion o ⌽kl( ) a a high accu acy. This is nume ically
‘‘exac ’’ in he mic oscopic ime scale. Howe e , he ke nel
is likely o display a highly oscilla o y beha io when we
look a i on a coa se ime scale. In o de o ob ain a ea-
sonable smoo h unc ion in he mesoscopic ime scale we
need o coa se g ain in ime u he . We can do ha as ol-
lows: le us call
␦
and ⌬Ⰷ
␦
he basic ime s eps in he mi-
c oscopic and mesoscopic ime scales, espec i ely. One im-
034108-5 Inhomogeneous mul iscale dynamics J. Chem. Phys. 122, 034108 (2005)
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media e app oach would be o calcula e he coa se-g ained
ke nel ⌽
¯
kl( ) a he cen e o he mesoscopic in e al jas a
simple ime in eg al:
⌽
¯
共 j兲⫽1
⌬
冕
j⫺⌬/2
j⫹⌬/2d
␶
⌽共
␶
兲,共43兲
wi h j⫽⌬(j⫹1/2). Howe e , he se o da a ⌽
¯
( j) is going
o depend s ongly on he pa icula choice we ha e made o
he exac loca ion o he mesoscopic imes j. Fo example,
i we look a ⌽
¯
( j⫹
␦
), de ined as in Eq. 共43兲bu in eg a ing
om j⫺/⌬/2⫹
␦
o j⫹/⌬/2⫹
␦
ins ead, we may ob ain a
e y di e en alue due o he high oscilla o y beha io o
he aw da a. Ne e heless, we can de ine ins ead he coa se-
g ained ke nel a jas he a e age o all hese possible alues
inside ha ime s ep ⌬. This is equi alen o calcula ing he
ollowing con olu ion:
⌽
¯
共 兲⫽
冕
d
␶
⌽共
␶
兲
␪
共 ⫺
␶
兲,共44兲
whe e
␪
共 兲⫽2
⌬
冉
1⫺
兩
兩
2
⌬
冊
H
冉
1⫺
兩
兩
2
⌬
冊
,共45兲
and H(x) is he Hea iside uni -s ep unc ion. By applying
Eq. 共44兲 wo imes, we a e able o ep oduce he analy ical
esul s o he example we p esen in Sec. VII.
VI. AN ‘‘
AB INITIO
’’ METHOD FOR THE SIMULATION
OF THE DYNAMICS
Once we know ⌽kl( ) nume ically we can simula e he
dynamics by using a DPD-like algo i hm. I we assume ha
he s ochas ic p ocess Rk( ) is Gaussian 共which is jus i ied as
long as we conside small de ia ions om equilib ium兲, hen
i is de e mined by he i s momen s: 共7兲and 共8兲. In his
case, we can exploi he p ope ies 共25兲 o gene a e he se o
co ela ed a iables Rk( ) by using a se o independen
Gaussian a iables
␰
kl⫽⫺
␰
lk , so ha Rk( )⫽兺l⫽k
␰
kl and
具
␰
kl共 兲
␰
k⬘l⬘共0兲
典
eq⫽kBT⌽kl共 兲共
␦
kl⬘
␦
lk⬘⫺
␦
kk⬘
␦
ll⬘兲.共46兲
This is he essen ial ing edien used in DPD o p ese e mo-
men um conse a ion. In ac , since Eqs. 共29兲and 共25兲imply
⌳kk⫽⫺兺
l⫽k⌳kl and ⌽kk共 兲⫽⫺兺
l⫽k⌽kl共 兲,共47兲
espec i ely, we can w i e Eq. 共6兲in he o m
Mk
dVk共 兲
d ⫽兺
l⫽k
再
⫺⌳kl关Xl共 兲⫺Xk共 兲兴
⫺
冕
0
d
␶
⌽kl共 ⫺
␶
兲关Vl共
␶
兲⫺Vk共
␶
兲兴⫹
␰
kl共 兲
冎
,
共48兲
which e y much esembles he DPD equa ions o mo ion.
Howe e , he i s ques ion ha a ises is whe he we can
conside ⌽kl( ) as Di ac-del a unc ions, as cus oma y in
DPD. I is equen ly a gued ha hey can be conside ed so
because he coa se-g ained a iables a e slow compa ed o
he a omic ime scales. This would imply ha he ull s o-
chas ic p ocess
兵
X( ),V( )
其
would be Ma ko ian in he el-
e an ime scale. We will show nex ha his should no be
assumed.
VII. AN EXAMPLE: COARSE-GRAINING
IN THE 1D HARMONIC CHAIN
Le us illus a e he abo e esul s wi h a simple example:
he dynamics o a single coa se-g ained pa icle o blob
o med by nconsecu i e oscilla o s in he one-dimensional
in ini e ha monic chain. Fo simplici y we se all in insic
pa ame e s o he chain o uni y 共 he mass o each oscilla o ,
he elas ic cons an , and he equilib ium spacing be ween
hem兲. Le us deno e wi h Xand V he ele an a iables
共now scala magni udes兲o he single blob. Due o ansla-
ional in a iance, he conse a i e o ce ⌳on he blob an-
ishes. This can be shown by using
具
X2
典
→⬁in Eq. 共9兲.
The e o e, he gene alized Lange in equa ion 共6兲becomes
ndV共 兲
d ⫽⫺
冕
0
d
␶
⌽共
␶
兲V共 ⫺
␶
兲⫹R共 兲.共49兲
We now use he me hod p oposed in Sec. VA o com-
pu e he memo y ke nel. The au oco ela ion unc ion can be
ob ained by s udying he dissipa ion o an ini ial eloci y
pe u ba ion consis ing o all oscilla o s a es a he equilib-
ium posi ions excep he ones in he blob, which s a in-
s ead wi h eloci y V0. The ime dependen eloci y V( )o
he coa se-g ained pa icle can be compu ed di ec ly om he
exac solu ion o he in ini e 1D ha monic chain, which is
exp essed in e ms o Bessel unc ions,
V共 兲⫽V0兺
i⫽1
n
兺
j⫽1
n
J2
兩
i⫺j
兩
共2 兲,共50兲
o om he mac oscopic displacemen ield u(x, ),12 which
e i ies he wa e equa ion
⳵
2u
⳵
2⫽
⳵
2u
⳵
x2.共51兲
This equa ion can be sol ed easily by s anda d me hods, and
he eloci y is ob ained by in eg a ing
⳵
u(x, )/
⳵
o e he
blob’s egion. Bo h me hods p o ide he same esul o
la ge n,
V共 兲⬃V0共1⫺ /n兲H共1⫺ /n兲.共52兲
No e ha V( ) changes in a ime scale o o de ⬃n.We
al eady see om Eq. 共52兲 ha he eloci y au oco ela ion
decay is no exponen ial bu linea , which is an indica ion
ha he s ochas ic p ocess is non-Ma ko ian. We can sol e
Eq. 共49兲by using Laplace ans o ms.13 This is done in Ap-
pendix C. By changing o he p ope ime scale *⫽ /n,
wi h ⌽*( *)⫽n⌽(n *), we ob ain
冕
0
⬁d *⌽*共 *兲⫽2, 共53兲
and
034108-6 D. Cube o and S. N. Yali aki J. Chem. Phys. 122, 034108 (2005)
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⌽*共 *兲⫽
␦
共 *兲⫺
␦
共 *⫺1兲⫹e *⫺H共 *⫺1兲e *⫺1共 *⫹1兲
⫹兺
k⫽2
⬁
H共 *⫺k兲e *⫺k共⫺1兲k共 *⫺k兲k⫺2
共k⫺2兲!
⫻
冉
共 *⫺k兲2
k共k⫺1兲⫹2共 *⫺k兲
k⫺1⫹1
冊
.共54兲
No e ha he Di ac del as appea as a consequence o he
ime scale change. A he mic ocospic ime scale he au o-
co ela ion unc ion 共50兲does no p esen discon inui ies in
i s ime de i a i e.
We ha e plo ed in Fig. 2 he memo y ke nel 共wi hou
he Di ac-del a unc ions兲gi en by Eq. 共54兲. We ha e also
calcula ed nume ically his ke nel by using he me hod o
Sec. VB. Bo h esul s a e indis inguishable in he plo . I is
clea ha he ke nel has no decayed comple ely in he ime
scale o V( )( *⬃1), and hus we canno app oxima e i
wi h a single Di ac del a and he andom e m as whi e noise,
e en in he limi n→⬁.
This phenomenon is gene al and has implica ions be-
yond his pa icula example. The eason o his long-li ed
ke nels is ha he ime scale o Vk( ) is gi en by he dissi-
pa ion, which in ha monic sys ems is due o wa e p opaga-
ion, and he decay ime scale o he ke nels is on he same
ime scale. This is clea om Eqs. 共14兲–共15兲. In he mesos-
copic limi A
ˆ ends o A, and qi
k( ) ends o he solu ion o
he mechanical p oblem consis ing o an ini ial pe u ba ion
a ound he edge o he coa se-g ained egion k. Thus, ⌽kl( )
will no app eciably decay a leas un il he elas ic wa es
ha e emo ed he pe u ba ion, which will be abou he ime
i akes o he sound o c oss he mesoscopic egion, i.e., a
mesoscopic ime. This is a memo y e ec ha occu s inside
he coa se-g ained pa icles, causing he whole p ocess o be
non-Ma ko ian. The e o e, we canno conside he ke nels o
he au oco ela ion unc ion o he andom o ces Di ac-del a
unc ions, such as in DPD.
In Re . 14, Espan
˜
ol jus i ied he DPD me hod by s udy-
ing he equa ion o mo ion o an in ini e numbe o coa se-
g ained pa icles in he one-dimensional ha monic chain. In
con as o he example discussed abo e, now he in ini e
ha monic chain is pa i ioned in o an in ini e numbe o
blobs, each one o med by nconsecu i e oscilla o s, which is
mo e closely ela ed o he cus oma y coa se-g aining
scheme in DPD. Howe e , his wo k elies on he Ma ko ian
assump ion o he dynamics, which we ha e shown o be
unjus i ied. We p esen in Fig. 3 he scaled memo y ke nel
⌽3
*⫽n⌽3connec ing wo coa se-g ained pa icles kand l
sepa a ed by wo coa se-g ained pa icles, i.e.,
兩
k⫺l
兩
⫽3, as
compu ed om a molecula dynamics simula ion wi h blobs
o size n⫽10000. As be o e, he aw da a has been coa se
g ained in ime wi h a mesoscopic ime s ep ⌬*⫽⌬/n
⫽0.01. I is clea ly seen ha he ke nel is on he mesoscopic
ime scale *⫽ /n, as expec ed. In ac , by applying
he heo y o Sec. VA we ob ain ha his ke nel con ains
h ee Di ac-del a unc ions a di e en mesoscopic imes,
2
␦
( *⫺3)⫺
␦
( *⫺2)⫺
␦
( *⫺4) 共 ecognizable in Fig. 3 as
lines going ou o scale兲, a ound *⫽3, which is he ime i
akes he sound o p opaga e om one blob o he o he . We
epo he co esponding analy ical exp essions o his p ob-
lem and he genela iza ion o highe dimensions elsewhe e.
VIII. CONCLUSIONS
We ha e applied p ojec ion ope a o s o he coa se-
g ained mul iscale p oblem in ha monic sys ems wi h an in-
homogeneous le el o coa se g aining. The cus oma y ap-
p oach o using equilib ium a e ages as inne p oduc
p oduces he igh gene alized Lange in equa ion, bu he
explici exp essions a e di icul o calcula e. Using an al e -
na i e inne p oduc we ha e been able o p o ide explici
exp essions o he conse a i e o ces, he memo y ke nels
and he andom o ces, in e ms o a mechanical analog.
Based on hese exp essions, we ha e shown ha he memo y
ke nels and he eloci y au oco ela ion unc ions can be
compu ed in linea sys ems om a single molecula dynam-
ics simula ion. These esul s ep esen a gene aliza ion o
p e ious analy ical wo k on ha monic la ices in ol ing he
de i a ion o a se ies o algeb aic p ope ies.
In addi ion, a me hod ha esembles DPD has been p o-
posed and is applicable o any sys em in which he coa se-
g ained egion is linea . This can also be seen as a na u al
ex ension o he quasicon inuum me hod o accoun o ini e
empe a u e and dynamics. Mo eo e , we ha e shown ha
he memo y ke nels and also he au oco ela ion unc ion o
he andom o ces used in he simula ions should no be con-
side ed as Di ac-del a unc ions, as cus oma y in mos ap-
p oaches, wi hou jus i ica ion. Ins ead, he p oposed simula-
ions would need o include he compu a ion o he ime
FIG. 2. Scaled memo y ke nel ⌽*⫽n⌽as a unc ion o *⫽ /n共wi hou
he Di ac del as a *⫽0, 1兲 o a coa se-g ained pa icle in a 1D ha monic
chain.
FIG. 3. Memo y ke nel connec ing wo coa se-g ained pa icles sepa a ed
by wo blobs in he 1D ha monic chain.
034108-7 Inhomogeneous mul iscale dynamics J. Chem. Phys. 122, 034108 (2005)
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con olu ions in ol ing memo y ke nels. In o de o accoun
o he luc ua ion-dissipa ion heo em, he andom o ces
need o be gene a ed using a colo ed noise algo i hm. O he -
wise, one isks ha he sys em simula ed does no end o he
he modynamic equilib ium. Me hods add essing his issue
exis in he li e a u e,15,16 howe e , u he wo k is equi ed
o es hei e iciency in hese pa icula p oblems.
ACKNOWLEDGMENT
This wo k was unded in pa by he EU g ow h p ojec
SENTIMATS unde Con ac No. G5RD-CT-2001.
APPENDIX A: CONSTRAINED AVERAGES
S a ing om Eq. 共13兲we ob ain
具
R
ˆk共 兲R
ˆl共0兲
典
A
ˆ⫽qk共 兲 M
具
xx
典
A
ˆMql共0兲
⫽qk共 兲 M共1⫺P
ˆ 兲
具
xx
典
A
ˆ共1⫺P
ˆ兲Mql共0兲
⫽qk共 兲 M
具
共x⫺X兲共x⫺X兲
典
A
ˆMql共0兲.
共A1兲
Le us now conside he a iable change om x o cen e
mass coo dina es X⫽P
ˆ xplus a numbe o ela i e coo di-
na es which we will ep esen using ec o ial no a ion as
␰
. Each a iable
␰
ican be de ined as he di e ence be ween
he componen xio he oscilla o iand a ixed e e ence
oscilla o inside he same g oup. Then, he quan i y B(x)
⫽(x⫺X)(x⫺X) as a unc ion o
兵
X,
␰
其
does no depend
on he cen e o mass coo dina es X, i.e., i is a unc ion
o
␰
alone: B(x⫹␭X)⫽B(x) o all ␭and x. We show
nex ha o any unc ion wi h he same p ope y,
具
B(x)
典
A
ˆ
⫽
具
B(x)
典
X0⫽0.
Equa ion 共16兲implies
x Ax⫽x A
ˆx⫺X AX⫹2x AX,共A2兲
being x A
ˆxa unc ion o
␰
alone. The e o e, using his ex-
p ession and changing a iables we ob ain
具
B共x兲
典
X0⫽0⫽
兰
dNxB共x兲exp共⫺x Ax/2kBT兲
␦
共P
ˆ x兲
兰
dNxexp共⫺x Ax/2kBT兲
␦
共P
ˆ x兲
⫽
兰
dNxB共x兲exp共⫺x A
ˆx/2kBT兲
␦
共P
ˆ x兲
兰
dNxexp共⫺x A
ˆx/2kBT兲
␦
共P
ˆ x兲
⫽
兰
dXd
␰
B共x兲exp共⫺x A
ˆx/2kBT兲
␦
共X兲
兰
dXd
␰
exp共⫺x A
ˆx/2kBT兲
␦
共X兲
⫽
兰
d
␰
B共x兲exp共⫺x A
ˆx/2kBT兲
兰
d
␰
exp共⫺x A
ˆx/2kBT兲
⫽
兰
dXd
␰
B共x兲exp共⫺x A
ˆx/2kBT兲
兰
dXd
␰
exp共⫺x A
ˆx/2kBT兲
⫽
具
B共x兲
典
A
ˆ.共A3兲
APPENDIX B: SINGULARITIES
IN THE MEMORY KERNEL
Le us conside he Fou ie ans o m o he au oco ela-
ion unc ion
ckl共
␻
兲⫽
冕
0
⬁d e⫺i
␻
Ckl共 兲共B1兲
and analogously o he memo y ke nel
␾
kl(
␻
). In his ep-
esen a ion, Eq. 共37兲becomes
兺
l⬘
␾
kl⬘共
␻
兲cl⬘l共
␻
兲⫽Mk关Ckl共0兲⫺i
␻
ckl共
␻
兲兴
⫺兺
l⬘
⌳kl⬘
cl⬘l共
␻
兲
i
␻
.共B2兲
Assume C
˙kl( ) is discon inuous a he imes
⫽ 1, 2,.... Then, he asymp o ic beha io o ckl(
␻
) in he
limi
␻
→⬁can be ob ained by in eg a ion by pa s17
c共
␻
兲⬃C共0兲
i
␻
⫺1
␻
2
冋
C
˙共0兲⫹兺
⌬C
˙共 兲e⫺i
␻
册
,共B3兲
whe e ⌬C
˙( )⫽C
˙(
⫹)⫺C
˙(
⫺). Inse ing his exp ession in
Eq. 共B2兲we ob ain
兺
l⬘
␾
kl⬘共
␻
兲cl⬘l共
␻
兲⬃⫺Mk
i
␻
冋
C
˙共0兲⫹兺
⌬C
˙共 兲e⫺i
␻
册
.
共B4兲
On he o he hand, since
␾
共
␻
兲should be bounded in he limi
␻
→⬁, we also ha e
兺
l⬘
␾
kl⬘共
␻
兲cl⬘l共
␻
兲⬃
␾
kl共
␻
兲Mk
Cll共0兲
i
␻
.共B5兲
Combining bo h exp essions
␾
kl共
␻
兲⬃⫺1
Cll共0兲
冋
C
˙共0兲⫹兺
⌬C
˙共 兲e⫺i
␻
册
,共B6兲
in he limi
␻
→⬁. The in e se ans o m is gi en by
Eq. 共38兲.
APPENDIX C: CALCULATION
OF THE MEMORY KERNEL
The Laplace ans o m o he au oco ela ion unc ion
V(s) is ob ained om Eq. 共52兲as
034108-8 D. Cube o and S. N. Yali aki J. Chem. Phys. 122, 034108 (2005)
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