J. Chem. Phys. 110, 473 (1999); h ps://doi.o g/10.1063/1.478108 110, 473
© 1999 Ame ican Ins i u e o Physics.
Reo ganiza ion ene gies o cha ge ans e
eac ions in bina y mix u es o dipola ha d
sphe e sol en s: A Mon e Ca lo s udy
Ci e as: J. Chem. Phys. 110, 473 (1999); h ps://doi.o g/10.1063/1.478108
Submi ed: 07 July 1998 . Accep ed: 25 Sep embe 1998 . Published Online: 21 Decembe 1998
C. Denk, M. Mo illo, F. Sánchez-Bu gos, and An onio Sánchez
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Reo ganiza ion ene gies o cha ge ans e eac ions in bina y mix u es
o dipola ha d sphe e sol en s: A Mon e Ca lo s udy
C. Denka) and M. Mo illo
Uni e sidad de Se illa, Fı
´sica Teo
´ ica, Apa ado 1065, E-41080 Se illa, Spain
F. Sa
´nchez-Bu gos and An onio Sa
´nchez
Uni e sidad de Se illa, Quı
´mica Fı
´sica, Facul ad de Quı
´mica, C/ P o eso Ga cı
´a Gonza
´lez s/n,
E-41012 Se illa, Spain
~Recei ed 7 July 1998; accep ed 25 Sep embe 1998!
We s udy he beha io o he eo ganiza ion ene gy o simple cha ge ans e eac ions in mix u es
o dipola ha d sphe e luids by Mon e Ca lo simula ion. The s a ic dielec ic cons an s o he
sol en s a e also ob ained om he simula ion. They a e used as inpu in he eo ganiza ion ene gy
exp essions p o ided by he Ma cus heo y and he mean sphe ical app oxima ion. Thus, a
compa ison be ween he alues ob ained om he heo e ical exp essions and ou simula ion esul s
is possible. The dependence o he eo ganiza ion ene gy wi h he mix u e composi ion and he
in luence o p e e en ial sol a ion e ec s is also discussed. © 1999 Ame ican Ins i u e o Physics.
@S0021-9606~99!51401-3#
I. INTRODUCTION
Elec on ans e ~ET! eac ions in pola sol en s a e im-
po an in many biological and chemical p ocesses. Sol en
luc ua ions p o ide he ansi ion s a e con igu a ions o he
sol en –solu e sys em necessa y o he ET. Ma cus1–4 has
gi en a physical pic u e o ET eac ions, desc ibing he mul-
idimensional ee ene gy su aces o he p oduc and eac-
an s a es as pa abolic su aces in e ms o a sui able eac-
ion coo dina e. He ela ed he ac i a ion ee ene gy DG‡ o
he eac ion ee ene gy DGand he sol en eo ganiza ion
ene gy l. Ma cus also de i ed a simple exp ession o l,
based on a mac oscopic ea men o he sol en . The eo -
ganiza ion ene gy lis hen gi en in e ms o he s a ic and
op ical dielec ic cons an s o he sol en and a geome ical
ac o .
Al hough he dielec ic con inuum model ~Ma cus o -
mula!p o ides an adequa e quali a i e desc ip ion o he e-
o ganiza ion ene gy, i s quan i a i e p edic ions a e o en
imes a a iance wi h he expe imen al indings.5–7 Molecu-
la desc ip ions o he sol en a e, in p inciple, capable o
o e coming some o he di icul ies associa ed wi h a mac-
oscopic ea men o he sol en . The mean sphe ical ap-
p oxima ion ~MSA! ea men o he eo ganiza ion ene gy
ecognizes he sol en molecula i y by desc ibing he sol en
molecules as ha d sphe es wi h poin dipoles in hei cen e s.
Wi hin he MSA, an exp ession o lhas been de eloped
ha includes he ha d sphe e adius o he sol en
molecules.8,9
The need o a mic oscopic desc ip ion is o pa icula
in e es in mix u es o pola sol en s. Fo mos mix u es one
obse es a nonideal ~nonlinea !beha io o he sol a ion en-
e gy o an ionic o dipola solu e wi h espec o he mola
ac ions o he species p esen in he sol en .10 This beha -
io is usually e med as p e e en ial sol a ion. Fo ET eac-
ions in mix u es o pola sol en s, one migh also expec an
in luence o p e e en ial sol a ion e ec s on he sol en e-
o ganiza ion ene gy l. Bo h op ical11 and he mal12,13 ex-
pe imen al da a o ET eac ions in mix u es o wa e and
o ganic cosol en s show a beha io o l ha canno be ex-
plained wi h a con inuum model o he sol en . Analy ical
app oxima ions o he mic oscopic desc ip ion o he sol a-
ion ene gy in pola mix u es ha e been p esen ed in he
li e a u e.10,14
In his pape we will in es iga e he beha io o he e-
o ganiza ion ene gy lin pola sol en s by means o Mon e
Ca lo ~MC!simula ions. A quan i a i e p edic ion o expe i-
men al da a12,13 would equi e e y sophis ica ed simula ion
echniques. A his poin we a e a he in e es ed in he gen-
e al beha io o l. Fo his eason we ha e adop ed a simple
sol en model, wi h as ew adjus able pa ame e s as possible.
Ou sol en will be modeled ei he as a pu e sol en o as a
bina y mix u e. In bo h cases we will conside dipola , non-
pola izable ha d sphe e molecules. The solu e consis s o wo
cha ged ha d sphe es. We will s udy cha ge sepa a ion and
cha ge ecombina ion p ocesses o his sys em. As he sol-
en molecules a e conside ed nonpola izable, he op ical di-
elec ic cons an will be
e
op 51 o all sol en s. In o de o
compa e he nume ical esul s o lwi h he heo e ical p e-
dic ions, we need o e alua e he s a ic dielec ic cons an .
Ex ensi e simula ions ha e been ca ied ou in o de o de-
e mine
e
0 o ou di e en sol en models.
The ou line o he pape is as ollows: In Sec. II we
desc ibe ou sol en model and e iew he mos impo an
simula ion de ails. In Sec. III we discuss he e alua ion o
he dielec ic cons an s o he sol en models used in his
wo k. In Sec. IV, he applica ion o he MC echnique o he
calcula ion o ee ene gy su aces o cha ge ans e eac-
ions is summa ized. The esul s a e p esen ed and discussed
in Sec. V. A compa ison wi h he heo e ical esul s as ob-
a!Elec onic mail: [email p o ec ed]
JOURNAL OF CHEMICAL PHYSICS VOLUME 110, NUMBER 1 1 JANUARY 1999
4730021-9606/99/110(1)/473/11/$15.00 © 1999 Ame ican Ins i u e o Physics
ained o he con inuum sol en model ~Ma cus!and he
mic oscopic desc ip ion ~MSA!is made. Finally, Sec. VI
con ains some concluding ema ks.
II. MODEL AND MC SIMULATIONS
Ou sol en is modeled as a liquid o dipola , nonpola -
izable ha d sphe e molecules. Two sol en molecules iand k
in e ac ia a long anged dipole–dipole in e ac ion:
Vi,k5
m
i–
m
k
ik
323~
m
i– ik!~
m
k– ik!
ik
5 ik>2 s~1!
and a epulsi e sho anged in e ac ion ~ he ha d sphe e po-
en ial, Vi,k5` o ik5
u
ik
u
5
u
i2 k
u
,2 s). He e, sis he
adius o he ha d sphe es,
m
iis he dipole momen and iis
he posi ion o molecule i.
In his wo k we s udy pu e sol en s and bina y mix u es
o dipola ha d sphe e luids. The adius o he sol en mol-
ecules is aken o be he same o all sol en componen s. A
packing ac ion o
h
50.417 co esponding o a dense liquid
was used in all simula ions. The pola i y o a pu e sol en
will be exp essed in e ms o he dimensionless pa ame e
y54
p
m
2
9kBT,~2!
whe e
is he densi y o he liquid. A mix u e o wo sol-
en s will be composed o wo species, he less pola species
L, cha ac e ized by i s pola i y yL, and ano he species H
wi h a highe pola i y yH. The composi ion o he mix u es
will be desc ibed by he mola ac ions o he mo e pola
species H h oughou his wo k.
Pe iodic bounda y condi ions wi h he minimum image
con en ion15,16 we e applied o a cubic simula ion box o
side leng h L. This me hod equi es he use o a cu o o
he long anged dipola in e ac ions. In o de o accoun o
he con ibu ions beyond he cu o adius c<L/2 we ha e
adop ed he gene alized eac ion ield me hod.17 We conside
he mo ing bounda y dielec ic implemen a ion, i.e., he sub-
sys em inside he cu o sphe e a ound each molecule is
hough o be imme sed in a con inuum dielec ic cha ac e -
ized by a mac oscopic s a ic dielec ic cons an
e
RF . A mol-
ecule in e ac s di ec ly wi h all molecules inside i s cu o
sphe e ia Eq. ~1!and wi h he eac ion ield. The eac ion
ield a he cen e o molecule idue o he molecules inside
he cu o sphe e su ounding i is p opo ional o he o al
dipole momen inside i s cu o sphe e16
m
i–Ei
52~
e
RF21!
2
e
RF11
1
c
3
m
i•(
ik< c
m
k5
am
i•(
ik< c
m
k,~3!
whe e we ha e de ined a sc eening cons an
a
. Summing up
all e ms con ibu ing o he o al po en ial ene gy we ob ain
E o 51
2(
kÞi, ik< c
Vi,k
dd 21
2
a
(
i
m
i
2~4!
wi h an e ec i e in e ac ion ene gy
Vi,k
dd 5
m
i–
m
k~12
a
ik
3!
ik
323~
m
i– ik!~
m
k– ik!
ik
5.~5!
The las e m in Eq. ~4!desc ibes he cons an sel -
in e ac ion o he molecules wi h hei own eac ion ields.
The o m o he e ec i e in e ac ion in Eq. ~5!is e y con-
enien o compu a ional pu poses. The inclusion o he e-
ac ion ield only equi es wo ex a loa ing poin ope a ions
o each e alua ion o he in e ac ion ene gy, which ende s
his me hod compu a ionally easible, e en o ela i ely
la ge sys ems. I has been shown ha he compu a ionally
mo e expensi e Ewald summa ion echnique yields equi a-
len esul s o he dielec ic cons an when applied o simila
sys ems.18,19 Analogous conclusions ha e been d awn o
ee ene gy calcula ions o ionic hyd a ion.20
Sys em con igu a ions we e gene a ed in he ollowing
manne : s a ing om a gi en con igu a ion, a molecule was
selec ed andomly. I was displaced om i s ini ial posi ion
in a cube o side leng h Dxwi h a uni o m dis ibu ion. Now,
a o a ional axis (x,yo z) was selec ed a andom and he
dipole o ien a ion o he molecule was o a ed by a uni o mly
dis ibu ed angle in he ange 2D
<
<D
a ound his
axis. The pa ame e s Dxand D
we e adjus ed o each an
accep ance a io o app oxima ely 30%. The implemen a ion
o he simula ion is along he lines o he s anda d Mon e
Ca lo echniques.15,16
III. DIELECTRIC CONSTANT
The applica ion o he s a is ical mechanical heo y o
he dielec ic cons an 21,22 o ini e size simula ion sys ems
wi h bounda y condi ions equi es some modi ica ions. This
issue has been lucidly add essed by Neumann.18 F om his
analysis i ollows ha he s a ic dielec ic cons an
e
0in he
eac ion ield ~RF!geome y is gi en by
e
052
e
RF~11
z
!11
112
e
RF2
z
,~6!
wi h
z
54
p
3
b
^
M2
&
L353ygK.~7!
He e
^
M2
&
is he con igu a ional a e age o he squa ed o al
dipole momen o he sys em and
b
51/kBT. We ha e also
indica ed he ela ion o
z
wi h he pola i y yand he Ki k-
wood g ac o gK5
^
M2
&
/N
m
2. The dielec ic cons an
e
RF
ha cha ac e izes he eac ion ield was de e mined sel -
consis en ly in he simula ions, so ha
e
RF'
e
0 o all sol-
en s.
The sol en molecules we e ini ially p epa ed in an cc
s uc u e and he sys em was hen allowed o elax du ing
53106MC con igu a ions. Mean alues we e ob ained om
NMC553108subsequen MC con igu a ions, excep whe e
o he wise s a ed. In o de o ob ain an es ima e o he s a is-
ical e o s, we calcula ed mean alues o
z
o e blocks o
106con igu a ions. We ca ied ou a ious es s o check he
con e gence o he alues o he dielec ic cons an ob ained
in ou simula ions. In Fig. 1 we show he unning a e ages o
e
0 o wo sol en s wi h y52.18 and y52.90 o wo di e -
en alues o he maximum o a ional angle D
. In he case
o he sol en wi h he highe pola i y, he alues o
e
0con-
e ge e y slowly. E en o NMC553108con igu a ions we
474 J. Chem. Phys., Vol. 110, No. 1, 1 Janua y 1999 Denk
e al.
s ill ind a de ia ion o D
e
051.6 be ween he wo simula-
ions ~ heo e ically, he simula ion esul s should be indepen-
den o D
). The s a is ical e o s es ima ed using he block
a e ages we e D
e
050.4 o simula ion ~a!and D
e
050.5 o
simula ion ~b!. This indica es ha , e en wi h 53108MC
con igu a ions, he phase space o his highly pola sys em
has no been su icien ly explo ed o ob ain p ecise alues o
e
0. The de ia ion o he wo cu es will be used as an es i-
ma e o he e o o he dielec ic cons an ~app oxima ely
3%! o sol en s wi h high pola i y. Sol en s wi h lowe mo-
lecula dipole momen s show a much as e con e gence o
e
0. Fo sol en s wi h a dielec ic cons an
e
0<30 we ound
23108MC con igu a ions o be su icien o de e mine
e
0
wi h a ela i e e o o app oxima ely 3%.
We ha e also s udied he dependency o
e
0on he sys-
em size. Simula ion ~a!o Fig. 1 was epea ed o a sys em
wi h N5864 pa icles; he esul ing alue o
e
0is shown in
Table I @simula ion ~e!#. The esul s indica e ha
e
0is inde-
penden o he size o he sys em o N>256 wi hin he e o
limi s.
As al eady men ioned, he pa ame e
e
RF was adjus ed
sel -consis en ly by epea ing each simula ion, using he e-
sul
e
0o a simula ion as an inpu pa ame e
e
RF in he nex
simula ion. In all cases, i was su icien o epea each simu-
la ion only once, as
e
0depends e y weakly on
e
RF .In
simula ion ~ !~see Table I!we ha e epea ed simula ion ~b!
wi h
e
RF558. The esul s a e p ac ically iden ical.
The dielec ic cons an
e
0was hen de e mined o a
pu e sol en o e a wide ange o alues o he molecula
pola i y y. A sys em wi h N5256 pa icles was used in hese
simula ions and he mean alues o
e
0we e ob ained a e
53108MC con igu a ions. Fo he highes pola i y, y
53.0, we inc eased he numbe o MC con igu a ions o
NMC5109 o he easons men ioned abo e. The esul s a e
shown in Fig. 2.
The heo y o liquids p o ides a ious app oxima ions
o he s uc u e o he dipola ha d sphe e luid.23 O pa -
icula in e es is he so-called MSA, as i p o ides analy ical
exp essions o he co ela ion unc ions and he dielec ic
cons an .24 We ind i ins uc i e o compa e ou simula ion
esul s wi h he MSA heo e ical p edic ions. The pai dis i-
bu ion unc ion can be expanded as23
h~1,2!5hS~R!1hD~R!D~1,2!1hD~R!D~1,2!,~8!
whe e D~1,2!is he cosine o he angle o med by he dipole
o ien a ions o wo molecules and 2D(1,2)
m
2/R3is he
dipole–dipole in e ac ion as de ined in Eq. ~1!. The MSA
p o ides he unc ions hS(R),hD(R) and hD(R) in e ms o
he adial dis ibu ion unc ion ~ d !o he Pe cus–Ye ick
~PY!solu ion o ha d sphe es a di e en densi ies.
In Fig. 3 we show he adial dis ibu ion unc ion
gS(R)5hS(R)11 o a highly pola sol en (y53.00). Sol-
en s wi h a lowe pola i y ha e a e y simila d wi h a
sligh ly lowe main peak. The MSA esul o gS(R)~also
shown in Fig. 3!is jus he PY d o ha d sphe es a densi y
and does no depend on he molecula pola i y. Al hough
he ag eemen is globally good, he e a e de ia ions be ween
he MSA and he simula ions. In he egion close o con ac ,
he MSA subes ima es he alue o gS(R), which is o p i-
ma y impo ance o many he modynamic p ope ies o he
liquid. I also p edic s a somewha slowe decay om he
peak alue o he i s minimum when compa ed wi h he
simula ions.
The dielec ic p ope ies o he sol en a e ela ed o
hD(R), which desc ibes he angula co ela ion o wo mol-
ecules a a gi en dis ance R. The dielec ic cons an
e
0is
FIG. 1. Running a e ages o
e
0 o ~a!,~b!y52.90 and ~c!,~d!y52.18. The
sys ems we e simula ed wi h wo di e en alues o he maximum o a ional
angle D
5
p
/2 ~a!and ~c!and D
51.0 ~b!and ~d!.
FIG. 2. The s a ic dielec ic cons an
e
0 o a pu e sol en . The e o ba s o
he simula ion esul s indica e he es ima ed ela i e e o o 3% ~see main
ex !. The solid line is a ou h deg ee in e pola ion polynomial. The do ed
line co esponds o he heo e ical esul as ob ained om he MSA.
TABLE I. Dielec ic cons an s ob ained o sol en s wi h y52.90
~a!,~b!,~e!,~ !and y52.18 ~c!,~d!
e
0NMC ND
e
RF
~a!59.68 53108256
p
/2 70
~b!58.05 53108256 1.0 70
~c!28.29 53108256
p
/2 30
~d!29.05 53108256 1.0 30
~e!58.41 23108864
p
/2 70
~ !58.78 53108256 1.0 58
475J. Chem. Phys., Vol. 110, No. 1, 1 Janua y 1999 Denk
e al.
de e mined by he pola i y yand he Ki kwood g ac o gK
@see Eq. ~6!#. The Ki kwood g ac o gKcan be ob ained
om hD(R) by in eg a ion
gK5114
p
3
E
0
`hD~R!R2dR.~9!
In Fig. 4 we show hD(R) o a highly pola sol en (y
53.0) as ob ained om ou simula ions and he co espond-
ing MSA esul . The MSA does no p o ide a good app oxi-
ma ion o hD(R) as i unde es ima es he angula co ela ion.
This esul s in an unde es ima ion o gKand hus
e
0. This
de ia ion is mo e p onounced o sol en s wi h a high pola -
i y ~see Fig. 2!. Bu , e en o sol en s wi h he lowes pola -
i y unde conside a ion in his wo k, he MSA esul o
hD(R) does no ag ee wi h he simula ion da a. The dielec ic
cons an as ob ained by he MSA gi es a good app oxima-
ion o
e
0only o sol en s wi h gK'1. These limi a ions o
he MSA a e well known and mo e sophis ica ed heo ies
based on he hype ne ed chain ~HNC!app oxima ion p o-
ide much be e angula co ela ion unc ions and dielec ic
cons an s o dipola ha d sphe e luids. Compa isons o he
MSA and o he heo ies can be ound in he li e a u e.23,25–33
Le us now conside he case o bina y sol en mix u es.
We s udy wo di e en ypes o mix u es: ~A!mix u es wi h
wo componen s o simila pola i ies (yH53.0 and yL
52.18), and ~B!mix u es whose componen s a e o a he
di e en pola i y (yH53.0 and yL50.75). Fo each ype o
mix u e, he dielec ic p ope ies will only depend on i s mo-
la composi ion ~de ined by he mola ac ion o species H,
H), i he empe a u e is held ixed. This gi es us he pos-
sibili y o s udying wo qui e di e en mix u es, spanning a
wide ange o dielec ic cons an s. The simula ion p ocedu e
is he same as o pu e sol en s, and he dielec ic cons an is
e alua ed by using Eqs. ~6!and ~7!.
In Fig. 5 we show he dielec ic cons an s as ob ained o
he simula ed composi ions o mix u es o ype ~A!and ~B!.
The alues o
e
0 o he pu e sol en s ( H50 and H51)
we e aken om he simula ions desc ibed abo e ~see Fig.
2!. Fo bo h mix u es we obse e an almos quad a ic depen-
dence o he s a ic dielec ic cons an on he mola ac ion
H. This beha io can be unde s ood by no icing ha in he
case o a bina y mix u e,
^
M2
&
/Nis o a good app oxima ion
a quad a ic unc ion o he mola ac ion H. Fo
e
0
'
e
RF , i ollows om Eq. ~6! ha he dielec ic cons an is
essen ially a linea unc ion o
z
, which in u ns is p opo -
ional o
^
M2
&
/Nas can be seen in Eq. ~7!.
IV. REORGANIZATION ENERGIES
A. Calcula ion o he eo ganiza ion ene gy
om molecula simula ions
In Sec. III we ha e analyzed he dielec ic beha io o
dipola ha d sphe e sol en s. We will now s udy he si ua ion
when a solu e is imme sed in he sol en . Ou aim he e is o
s udy he ene ge ics o he mal cha ge ans e eac ions be-
ween wo solu e molecules in he p esence o a pola sol-
en . We ha e adop ed a simple model o he solu e: The
solu e consis s o wo ha d sphe e molecules ~dono and ac-
cep o !wi h gi en adii dand asepa a ed by a ixed dis-
FIG. 3. The adial dis ibu ion unc ion gS(R)(R5 /2 s) o a sol en wi h
y53.00 ~ illed ci cles!. The do ed line ep esen s gS(R) as ob ained om
he MSA.
FIG. 4. The angula co ela ion unc ion hD(R) o a highly pola sol en
(y53.0, illed ci cles!. The do ed line co esponds o hD(R) as p o ided by
he MSA.
FIG. 5. S a ic dielec ic cons an
e
0 o bina y mix u es o dipola ha d
sphe es wi h yL52.18 @mix u e ~A!, ci cles#and yL50.75 @mix u e ~B!,
squa es#in e ms o he mola ac ion H. In bo h cases yH53.0. The solid
lines ep esen quad a ic in e pola ion polynomials.
476 J. Chem. Phys., Vol. 110, No. 1, 1 Janua y 1999 Denk
e al.

ance d. The dono and accep o in he eac an s a e ca y a
cha ge qdand qa, espec i ely; a nega i e ne cha ge ans-
e s om he dono o he accep o . We will conside he
eac ion coo dina e o his p ocess in e ms o a cha ging
pa ame e
j
, so ha
qd~
j
!5qd1
j
e,~10!
qa~
j
!5qa2
j
e.~11!
He e, eis he elemen a y cha ge ~ ha we will ake as posi-
i e!. The eac an s a e o he solu e is gi en by
j
50, while
he p oduc s a e is ob ained by se ing
j
51. In his wo k we
ha e s udied wo ypical ans e eac ions: ~i!a cha ge e-
combina ion ~CR!p ocess qd52eand qa51eand ~ii! he
in e se p ocess, a cha ge sepa a ion ~CS!p ocess (qd5qa
50). The a es o hese p ocesses a e go e ned by an ac i-
a ion ee ene gy law. Ma cus1–4 ob ained o he nonadia-
ba ic elec on ans e a e
ke 5
k
A
4
p
l/
b
exp~2
b
DG‡!,~12!
whe e he ac i a ion ee ene gy DG‡is de ined as
DG‡5~DG1l!2
4l.~13!
He e,
k
is a ma ix elemen desc ibing he elec onic cou-
pling be ween eac an and p oduc s a e, and DGand l
deno e he ee-ene gy change o he eac ion and he eo -
ganiza ion ene gy, espec i ely. Equa ion ~12!was de i ed
unde he assump ion o a classical sol en ha esponds
linea ly o a edis ibu ion o cha ges.
The p ocedu e used o he e alua ion o eo ganiza ion
ene gies in cha ge ans e eac ions om simula ions is well
documen ed.34–36 He e we will b ie ly indica e he main
poin s. Le H
j
deno e he sol en –solu e in e ac ion ene gy
o a solu e s a e cha ac e ized by he cha ging pa ame e
j
and a ce ain ixed sol en con igu a ion. The ene gy gap
DV5H12H0desc ibes he ene gy di e ence be ween p od-
uc s and eac an s o a gi en con igu a ion o he sol en .
An elec on ans e will ake place only o sol en con igu-
a ions ha ul ill he condi ion DV52Ei~F ank–Condon
p inciple!, whe e Eiis he in insic ene gy di e ence
be ween he gas phase elec onic s uc u es o he ini ial
and inal s a e o he solu e. Le us de ine he andom a i-
able D5DV, wi h a p obabili y law gi en by p
j
(D)
5
^
d
(D2DV)
&
j
, whe e he angula b acke s indica e an
equilib ium a e age aken wi h a canonical dis ibu ion de-
sc ibing a sys em a empe a u e Tand Hamil onian H
j
. The
a e o he ET p ocess is p opo ional o he p obabili y den-
si y p0(2Ei) o he ene gy gap DVha ing a alue 2Ei.
This p obabili y dis ibu ion co esponds o a sol en in he -
modynamical equilib ium wi h he eac an s a e o he sol-
u e,
j
50.
The main p oblem in simula ions lies in he cons uc ion
o his p obabili y densi y as he en i e phase space o he
sol en deg ees o eedom has o be explo ed. This is an
imp ac ical ask, and one eso s o a ee ene gy pe u ba ion
me hod,34 which is based on he ollowing. Fo a gi en alue
o he ac ional cha ge pa ame e
j
,DVis sampled a ound
a alue D
j
wi h a dis ibu ion ha is app oxima ed e y well
by a Gaussian
p
j
~D!51
A
2
ps
j
2e2~D2D
j
!2/2
s
j
2.~14!
Va ious s a es
j
o he solu e a e simula ed, each simula ion
p o iding a p obabili y dis ibu ion p
j
(D) a ound D
j
. These
dis ibu ions can be pieced oge he 34 o yield he dis ibu ion
p0(D) o e a wide ange o alues D. Zhou and Szabo35
ha e p oposed a simpli ied me hod ha pe mi s ob aining
p0(D) by simula ing only wo s a es o he solu e:
j
50 and
j
51. In hei wo k, he eo ganiza ion ene gy is gi en by
l5D02DG10 ,~15!
wi h he ee-ene gy di e ence be ween s a e
j
and he eac-
an s a e o he solu e
DG
j
05
E
0
j
d
j
8D
j
8.~16!
The mean alue o he ene gy gap D
j
o a ce ain alue o
he cha ging pa ame e
j
can be ob ained by a cubic in e -
pola ion polynomial in
j
, whe e all coe icien s a e de e -
mined by he mean alues D0and D1and he dispe sions
b
s
0
2and
b
s
1
2in he eac an and p oduc s a e, espec i ely.
Inse ing he polynomial exp ession o Zhou and Szabo in
Eqs. ~15!and ~16!leads o
lCR51
2~D02D1!11
12 ~
b
s
0
22
b
s
1
2!,~17!
lCS51
2~D02D1!21
12 ~
b
s
0
22
b
s
1
2!.~18!
As poin ed ou by Zhou and Szabo, he accu acy o he in-
e pola ion polynomial app oxima ion o D
j
can be checked
by ca ying ou an addi ional simula ion o
j
51/2. The
alue o he ene gy gap D1/2 ob ained is hen compa ed o he
alue p o ided by he in e pola ion o mula. We ound a
de ia ion o less han 1% o all cases conside ed he e. F om
he in e pola ion polynomial, he p obabili y densi y p0(D)
can also be cons uc ed.35
B. Simula ion de ails
The inclusion o he cha ged solu e pa icles in he simu-
la ion in oduces a di icul y when dealing wi h sys ems o
ini e size: The o ien a ion o he dipole momen s o he sol-
en molecules is aniso opic and he pe iodic eplica ion o
he cen al simula ion cell does no ep esen a physical pic-
u e o he sol en . This issue has been widely discussed in
he li e a u e37 and a ious me hods ha e been de eloped o
a oid pe iodic bounda y condi ions ~pbc! o such sys ems.
Sphe ical simula ion cells in conjunc ion wi h a sui able
me hod o a oid sel -pola iza ion on he su ace o he cell
a e usually employed.38
We ha e conside ed simula ions wi h pe iodic bounda y
condi ions and wi hin a sphe ical geome y. In he pbc simu-
la ions, he cha ge–dipole in e ac ions can be desc ibed ia
an e ec i e in e ac ion po en ial ha includes he eac ion
ield in a simila manne o Eq. ~5!:
477J. Chem. Phys., Vol. 110, No. 1, 1 Janua y 1999 Denk
e al.
Vi,j
qd5qi
S
2
ij
2
a
D
m
j– ij,~19!
whe e
a
is he sc eening cons an as de ined in Eq. ~3!. The
solu e cha ges in e ac wi h all dipoles inside hei co e-
sponding cu o sphe es. The cu o adius o he cha ge–
dipole in e ac ions was se o be he same as o he dipole–
dipole in e ac ions.
In he sphe ical model, he simula ion cell is a sphe ical
essel wi h he solu e si ua ed a he cen e o he essel.
Each molecule in e ac s wi h all molecules inside he essel
and elec os a ic in e ac ions a e aken ully in o accoun @se
a
50 in Eqs. ~5!and ~19!#. The simula ion sphe e is di ided
in o wo egions: an inne sphe e, whe e he molecules a e
allowed o mo e and o a e and an ou e shell, whe e he
molecules a e kep ixed du ing he simula ion. The mol-
ecules cons i u ing he ou e shell a e si ua ed ini ially in a
cc s uc u e wi h andomly dis ibu ed dipole momen s. The
ixed dipoles in he ou e shell p e en an unphysical pola -
iza ion o he dipoles close o he su ace o he essel. A
co ec ion o he ene gy gap D0has o be applied in he case
o he sphe ical model. Con ibu ions om he sol en ou -
side he sphe e can be pa ially aken in o accoun by con-
side ing he sphe e o be imme sed in a con inuous dielec ic
medium wi h dielec ic cons an
e
ou . The in luence o his
dielec ic medium is no aken in o accoun du ing he simu-
la ion ~as i would equi e he ull solu ion o he Laplace
equa ion inside he sphe e o each con igu a ion!, bu i s
e ec on D0may be es ima ed by elec os a ic conside -
a ions. Fo a CR p ocess, he con ibu ions o D0 om ou -
side he simula ion cell wi h adius Ra e
D0
ou 52~
e
ou 21!
2
e
ou 11
~ed!2
R3,~20!
whe e dis he dis ance be ween he solu e cha ges.
We s udied he dependency o D0on he size o he
sys em o bo h models. A highly pola sol en (y52.90)
was used o sol a e wo solu e ions wi h equal adii d5 a
5a51.44Å, cha ges qd52e,qa51eand sepa a ed by a
dis ance o d56 Å. The sol en adius was se o be he
same as he solu e adius and a packing ac ion o
h
50.417 was used.
In he case o he pbc geome y we simula ed sys ems
wi h N5246 490 854 sol en molecules. The cu o adius
o he dipole–dipole and cha ge–dipole in e ac ions was se
o c5min(L/2,8 s) o each sys em, whe e Lis he side-
leng h o he cubic simula ion cell. Fo he sys em wi h N
5854 sol en molecules we conduc ed a simula ion wi h he
ull cu o c5L/2 in o de o check he in luence o he
educed cu o on D0.
In he sphe ical geome y we simula ed he same sys em
wi h a o al o N o 5673 sol en molecules. In his case, he
adius o he inne sphe e ~whe e he molecules a e allowed
o mo e!was a ied. The esul s a e combined in Table II.
Fo small sys em sizes, he wo geome ies p oduce qui e
di e en alues o he ene gy gap D0. The pbc simula ions
o e es ima e D0, whe eas he sphe ical model esul s in oo
small alues o he gap. As he sys em size inc eases, bo h
models yield simila esul s. As he size o he ou e shell in
he sphe ical model dec eases ~las alue in Table II!, he
pola iza ion o he cell su ace esul s in lowe alues o
D0. The educed cu o c58 s511.52Å o he pbc sys em
wi h N5854 sol en molecules gi es p ac ically he same
alue o D0as he sys em wi h c5L/2514.75Å.
Al hough he sphe ical geome y pe mi s he usage o
smalle sys em sizes, we ha e op ed o he pbc geome y
wi h N5854 sol en molecules and a educed cu o c
58 s o ou simula ions. Ou choice is based on he ac
ha he sphe ical model was no able o ep oduce he dielec-
ic cons an o he sol en . Thus he usage o a single ge-
ome y o he de e mina ion o bo h he dielec ic cons an
and he eo ganiza ion ene gy is clea ly p e e able. The pbc
geome y also allows us o calcula e he sol en adial dis i-
bu ion unc ions. Wi h ou choice o he sys em size and he
cu o adius, a molecule close o he simula ion cell bound-
a y only in e ac s wi h a small ac ion o he pe iodic im-
ages o he sol en molecules in he i s sol a ion shell.
When applying he con en ional Mon e Ca lo me hod
o c ea ing sys em con igu a ions ~ ansla ion and o a ion o
a molecule!in he case o mix u es, we no iced a e y slow
elaxa ion o he sys em. I a cha ged solu e is p esen , he
ini ial cc s uc u e o he sol en elaxes apidly o a si ua-
ion whe e he solu e is sol a ed by he molecules ha a e
ini ially in he icini y o he solu e. Thus, he mola compo-
si ion o he i s sol a ion shells depends on he ini ial po-
si ions o he wo species wi hin he cc g id. The high den-
si y o sol en molecules close o he solu e now ende s a
es uc u e o he sol a ion shell ex emely di icul . I he
wo componen s o he mix u e a e equally sized, one can
adop a much mo e e icien me hod o c ea ing con igu a-
ions: The con en ional ‘‘mo e’’ is al e na ed wi h a
‘‘swap’’ o molecules: wo molecules o di e en species a e
selec ed andomly. Now, he iden i ies o he molecules a e
in e changed, i.e., he dipole momen o he wo molecules is
changed. We only change he absolu e alues o he dipole
momen s; he dipole o ien a ions a e no al e ed. Each new
con igu a ion is gene a ed ei he by a mo e o by a swap and
he p obabili y o a swap was se o 10%.
V. RESULTS AND DISCUSSION
The eo ganiza ion ene gies lCR and lCS o a cha ge
ecombina ion and a cha ge sepa a ion p ocess ha e been
TABLE II. Values o D0 o pbc and he sphe ical model. The sol en and
solu e pa ame e s a e speci ied in he main ex . Fo he sphe ical model N
e e s o he numbe o molecules ha a e allowed o mo e. In his geome y
he o al numbe o molecules was held ixed a N o 5673.
ND0~kcal/mol! cGeome y
246 226.9 c5L/2 pbc
490 223.1 c58 spbc
854 221.9 c58 spbc
854 222.1 c5L/2 pbc
239 217.9 ¯sphe ical
371 222.4 ¯sphe ical
521 223.5 ¯sphe ical
593 218.6 ¯sphe ical
478 J. Chem. Phys., Vol. 110, No. 1, 1 Janua y 1999 Denk
e al.
calcula ed o a ious sol en s using he me hods desc ibed
abo e. The solu e consis s o wo ha d sphe es wi h adius
d5 a5a51.44Å and dis ance d56Å.
A. Pu e sol en s
We i s s udy he case o ET eac ions in pu e sol en s.
We ha e conside ed sol en s wi h pola i ies yin he ange o
0.75<y<3.0. The sol en adius was chosen o be he same
as he solu e adius. A packing ac ion o
h
50.417 and a
empe a u e o T5300K was used in all simula ions.
We will compa e ou simula ion da a wi h he eo gani-
za ion ene gy exp essions p o ided by he Ma cus heo y
and he MSA. In he Ma cus con inuum desc ip ion, he e-
o ganiza ion ene gy o a cha ge ans e o an elemen a y
cha ge be ween wo equally sized solu e molecules o adius
aa a dis ance din a nonpola izable sol en wi h s a ic di-
elec ic cons an
e
0is gi en by
l5e2
S
121
e
0
DS
1
a21
d
D
.~21!
The MSA heo y akes in o accoun he molecula aspec o
he sol en . In he case o in ini ely sepa a ed cha ge cen e s
(d→`), he eo ganiza ion ene gy is jus he sum o he
sol a ion ee ene gies o he wo ions8,9
l5e2
S
121
e
0
D
1
a~11
d
!.~22!
He e,
d
is a co ec ion o he sol a ed ion adius, which o a
e y good app oxima ion is gi en by39
d
53 s/a~1081/3
e
0
1/622!21.~23!
I he dis ance be ween he ions is no in ini e, he inclusion
o he ion–ion in e ac ion in Eq. ~22!wi hin he amewo k
o he MSA would equi e he knowledge o he mean ion–
ion po en ial. We will app oxima e his e m by he sc eened
in e ac ion in a con inuum dielec ic medium, as i has been
done by o he s.36 The inclusion o his e m in Eq. ~22! e-
sul s in
lMSA5e2
S
121
e
0
DS
1
a~11
d
!21
d
D
.~24!
We should keep in mind ha his app oxima e lMSA does no
eme ge om a pu ely mic oscopic pic u e o he sol en , as
i s in luence on he ion–ion in e ac ion is aken in o accoun
by a con inuum desc ip ion. This is expec ed o be a good
app oxima ion, as long as he dis ance o he ions is su i-
cien ly la ge.
When e alua ing Eqs. ~24!and ~23!, we ha e o p o ide
he solu e adius a, he ion dis ance d, he sol en adius s
and he s a ic dielec ic cons an o he sol en
e
0. The la e
may be ob ained di ec ly om he MSA, using he ollowing
ela ions:23
3y5~114
j
!2
~122
j
!42~122
j
!2
~11
j
!4,
~25!
e
05~114
j
!2~11
j
!4
~122
j
!6.
Fo compa ison wi h expe imen al da a i is o en mo e con-
enien o use he expe imen al alue o
e
0in Eqs. ~24!and
~23!.
We will hus compa e h ee heo e ical exp essions wi h
ou simula ion da a: ~a! he Ma cus exp ession Eq. ~21!,~b!
he ‘‘consis en ’’ MSA esul as gi en by Eq. ~24!using he
sol en pola i y yin o de o de e mine
e
0 om he MSA and
~c! he ‘‘expe imen al’’ MSA esul lMSA
ex , using he dielec-
ic cons an s as ob ained om he simula ions in Eq. ~24!.
The esul s a e shown in Fig. 6, whe e we ep esen he e-
o ganiza ion ene gy e sus he Peka ac o 121/
e
0. In his
ep esen a ion, he Ma cus esul @Eq. ~21!# is a s aigh line.
We ind ha lCS.lCR o all simula ed pu e sol en s.
Simila esul s ha e been ob ained by o he au ho s o com-
pa able sys ems.36 This is due o he combina ion o wo
e ec s:34,40 Fi s , he ee ene gy su aces a e no s ic ly
pa abolic, as i would be equi ed by linea esponse heo y.
Second, e en i he de ia ions om pa abolic su aces a e
small, he cu a u e o he pa abolas may be di e en in he
eac an and p oduc s a es. Bo h ea u es gi e ise o a di -
e en eo ganiza ion ene gy o he cha ge sepa a ion and
cha ge ecombina ion p ocesses.
The Ma cus exp ession o l esul s in an o e es ima ion
o he eo ganiza ion ene gy. Equa ion ~21!gi es l(y
53.0)5172.6 kcal/mole o he sol en wi h he highes po-
la i y. In he plo we ha e scaled he co esponding cu e o
coincide wi h lMSA
ex a his pola i y. This is equi alen o
using an e ec i e ion adius o a52.0Å in Eq. ~21!. When
he geome ic ac o is adjus ed in his way, he con inuum
desc ip ion s ill does no desc ibe well he o e all beha io
FIG. 6. Reo ganiza ion ene gies o a pu e sol en . The squa es ep esen
lCS, he ci cles co espond o lCR as ob ained om he simula ions. The
dashed line is he esul as ob ained by he con inuum desc ip ion ~Ma cus!
scaled o coincide wi h lMSA
ex o he highes pola i y. The do ed line co -
esponds o he heo e ical MSA esul o he simula ed sys em. The MSA
esul lMSA
ex , using
e
0as ob ained om he simula ions as an inpu pa am-
e e , is ep esen ed by he solid line.
479J. Chem. Phys., Vol. 110, No. 1, 1 Janua y 1999 Denk
e al.
o he eo ganiza ion ene gy. This ea u e is expec ed. The
exp ession o lin he con inuum desc ip ion, Eq. ~21!, in-
dica es ha he eo ganiza ion ene gy depends only on
e
0 o
ixed adii and sepa a ion dis ance and his dependence is
e y weak.
Fo he MSA esul s, we ind a much be e ag eemen
be ween he heo e ical exp essions and he simula ion da a.
The heo e ical MSA esul ~b!gi es good ag eemen o
small pola i ies, bu unde es ima es he eo ganiza ion en-
e gy o high pola i ies. This is no su p ising, as i is well
known ha he MSA unde es ima es he dielec ic cons an
o high sol en pola i ies. This esul s in an unde es ima ion
o he Peka ac o 121/
e
0on one hand and o he geome ic
ac o 1/a(11
d
) on he o he .
The MSA exp ession lMSA
ex wi h
e
0 aken om he simu-
la ion da a yields eo ganiza ion ene gies which a e close o
he simula ion esul s o he cha ge ecombina ion p ocess.
By cons uc ion, he MSA does no dis inguish be ween he
cha ge ecombina ion and cha ge sepa a ion p ocesses,
which as shown by he simula ion esul s, ha e di e en e-
o ganiza ion ene gies. Thus, he good ag eemen o lMSA
ex
wi h he esul s o he cha ge ecombina ion p ocess in he
case p esen ed is p obably o ui ous. Howe e , his exp es-
sion desc ibes he o e all shape o bo h cu es much be e
han he Ma cus app oxima ion. The heo e ical exp ession
o lwi hin he amewo k o MSA includes he geome ic
ac o 1/a(11
d
) which depends on
e
0 ia Eq. ~23!. The
abili y o lMSA
ex o ep oduce he end o he simula ion da a
o bo h p ocesses sugges s ha he inclusion o his geome -
ic ac o is an impo an co ec ion o he Ma cus exp ession
o he eo ganiza ion ene gy, Eq. ~21!. The geome ic ac o ,
as gi en by he MSA, can be in e p e ed as an e ec i e
solu e adius a(11
d
) and is a consequence o he sol en
molecula i y in he MSA pic u e. This emphasizes he need
o a molecula desc ip ion o he sol en , e en in he case o
pu e sol en s.
F om hese esul s one may be emp ed o conclude ha
he MSA is able o p edic he sol en s uc u e a ound he
solu es. We a e cu en ly examining his ques ion o single
ions in solu ion. P elimina y esul s indica e ha he pola -
iza ion densi y o he sol en a ound an ion is no e y well
ep esen ed by he MSA exp ession. Howe e , he ee en-
e gy o sol a ion, gi en as an in eg al o e he pola iza ion
densi y, is p edic ed qui e well by he MSA exp ession.
B. Mix u es
We will now p oceed o he case o bina y mix u es o
pola sol en s. We ha e calcula ed he eo ganiza ion ene gy
o he wo mix u es s udied in he p e ious sec ion @mix u e
~A!wi h simila pola i ies o he wo componen s: yH53.0
and yL52.18 and mix u e ~B!wi h a he di e en pola i ies
o he wo componen s: yH53.0 and yL50.75#. The esul s
o mix u e ~A!a e shown in Fig. 7 and o mix u e ~B!in
Fig. 8.
When he solu e is no p e e en ially sol a ed by one o
he wo componen s o he mix u e, one would expec a lin-
ea beha io o he sol a ion ene gy, as he mola ac ion o
he componen s is a ied.10 Fo nonpola izable sol en s, he
heo e ical exp essions o he eo ganiza ion ene gy educe
o a sol a ion ene gy in he limi d→`. Thus, a linea ~o
close o linea !dependence o lon he mola ac ion H
would indica e ha no p e e en ial sol a ion is p esen in he
p ocess. Fo bo h mix u es we obse e de ia ions om a
linea beha io in he eo ganiza ion ene gy. In mix u e ~A!
we obse e an almos linea beha io o lCS, while all al-
ues o lCR o he mix u es lie abo e a s aigh line connec -
ing he esul s o he pu e sol en s. Fo mix u e ~B! his
beha io is much mo e p onounced. When adding a small
mola ac ion o he componen wi h he highe pola i y, he
eo ganiza ion ene gy shows a d as ic inc ease, and e y
quickly he eo ganiza ion ene gy l eaches a alue close o
ha o he pu e sol en ( H51). As i was al eady obse ed
o mix u e ~A!, he eo ganiza ion ene gy o he cha ge
ecombina ion p ocess lCR de ia es mo e om he linea
FIG. 7. Reo ganiza ion ene gies o mix u e ~A!(yH53.0 and yL52.18).
The squa es ep esen lCS, he ci cles co espond o lCR. The linea depen-
dence, as expec ed in he limi d→` o ideal sol a ion, is ep esen ed as a
dashed line o each case. The xsymbols ep esen he MSA esul s lMSA
ex .
FIG. 8. Reo ganiza ion ene gies o mix u e ~B!(yH53.0 and yL50.75).
The squa es ep esen lCR, he ci cles co espond o lCS. The solid lines
connec ing he da a poin s ha e been ob ained by spline in e pola ion. The
linea dependence, as expec ed in he limi d→` o ideal sol a ion, is
ep esen ed as a dashed line o each case. The xsymbols ep esen he
MSA esul s lMSA
ex .
480 J. Chem. Phys., Vol. 110, No. 1, 1 Janua y 1999 Denk
e al.