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A time-evolution approach to intermolecular and thermodynamic potentials
Chemical Physics 7 (1975) 73-83 0 NorthHolland Publishing Company
A TIME-EVOLUTION
APPROACH TO INTERMOLECULAR
AND THERMODYTWlWC POTENTIALS*
Bruno LlNDER fiWtmen[ of fllemish): 7heF~ori&SIart! Univeniry. Tallahassee, FIcwi& 3.?306, USA Received 19 September 1974 A unified treatment is presented of ths interaction potentials (intermolecular forces) and the thermodynamic potentials (free enew~ functions) of molecular fluids. The treatment is based on a time-evolution approach to equilibrium which tt-accs the propagation of electrical and density fluctuations from the onset of the disturbance to the time equtiibrium IS rutty eswbk.+mi ‘Ihe ~=ow is sped;llired to a one-component. monntomic fluid interacting through geneked two-body intermoleculzu potentials (to infinite order). BY neglecting correlations between multiple excitations. II simpleexpressionis obtained for the free energy in terms of the molecule-density susceptibility of the unperturbed tluid and the intermolecular potential. The letter is formulated in terms of the charge-density susceptibilities of the noninterxting molecuies. and the static (Coulomb) potential. The theory takes account of translational quantum effects. It is shown that for a classicalfluid the free energy is consistent with formulations obtained from standard statistical mechanics within RPA.
1. Introduction In recent years, several new techniques have been proposed for treating the (equilibrium) thermodynamic properties of fluids. The most general theories are formulated in terms of the Coulomb interaction between the charged particles (electrons and nuclei) of the system and require the full complexity of quantum-statistical mechanics. The more standard techniques treat the quantum and statistical aspects separately. Thus, first the interaction energy between the charged particles are worked out quantum-mechanically for fvted intermolecular separations. Subsequently, the effects of the translational motion of the centers of mass of the molecules are brought in statisticaily. A unified treatment based on molecular- and charge-density correlation functions was recently proposed by Baer [l]. The intermolecular energies (or interaction potentials) are commonly referred to as intermolecular forces. The concept of an intermolecular force is predicated on the assumption that the electrons in a molecule move faster than the center of mass of the molecule. Indeed, it is this difference in time scale that makes it possible to analyze the complex dynamical motion of all the particles of the system as two separate events: one associated with the establishment of electronic correlations, the other with the establishment of translational molecular correlations. As long as the time scales are vastly different (and they generally are for molecular substances), then each type of correlation can be treated quantum-statistically and the resulting expression for the free energy of fluid will contain intermolecular potentials that may be temperature-dependent and translational parameters which may include quantum effects. The commonly employed procedure which treats the electronic fluctuations (i.e., the intermolecular potentials) as strictly quanta1 and the translational fluctuations as strictly classical imposes further restrictions not warranted by the dynamical behavior of the particles. In this paper we present a time-evolution approach to the thermodynamic properties of a molecular fluid, which treats the electronic and molecular translational fluctuations in a unified manner. The method is based on a generalization of the reaction potential technique used in the treatment of long-range forces [2,31_ The theory is first generalized to take account of the short-range interactions and then extended to include the effects of translational fluctuations. The present work is confined to the treatment of monatornic molecules and takes account only of two-body intermolecular interaction potentials (to all orders). The extension to arbitrary systems interacting through many-body potentials will be considered elsewhere. * Supported by a Grant from the National Science Foundation.
74
R Ltnderf A time-evolution
approach 10 inte?molecuuPr and thenmdpumtc
potentiais
2. Procedure 2.1. Electronic fluctuations The procedure for generating the long-range (or van der Waals) forces by means of the reaction potential technique was discussed earlier [2,3]. Here, we consider only the broad aspects of the theory and present a summary of the result for two-body interactions. In essence, one molecule, i, is singled out and treated explicitly and the remaining molecules with the radiation field are treated collectively as the “surrounding”. The reaction potential +(ri,t) is the scalar potential (retardation effects are here neglected) at the site rt of molecule i produced by the surrounding which was perturbed by the electrical fluctuations in i at all points r,! and at all times t’ < t. The polarization free energy for this process is obtained from the coupling between *(rj,f) and the charge-density p(ri,r). By taking account also of the polarization free energy arising from the fluctuations in the reaction potential parameters one obtains the mutual interaction free energy of i with the medium and, by summing over all i, the overall interaction free energy or interaction potential of the system. (To avoid possible confusion between this free energy and the true thermodynamic free energy of the fluid we shall henceforth refer to the former as interaction potential or intermolecular potential.) The interaction potential can be written as a function of two charge-density susceptibilities (to be discussed in more detail in the following paragraph): the susceptibility xi of molecule i and its conjugate reaction-susceptibihties Gi. The latter can be developed in a series involving the susceptibilities of the molecules of the surrounding and the static potentials of the radiation field. If i has only one neighboring molecule j, the reaction susceptibility is obtained by considering the hypothetical process whereby a fluctuation at r,! propagates to r,!, perturbsi at rj and acts back at ri. If there are two neighborsi and p. the reaction susceptibility will contain additional terms one of which represents a process whereby a charge-fluctuation in i first propagates to j, then top and finally returns to i, and also a term representing the process i+p + 1‘+i. Of course, the interaction path involving two molecules can also proceed via the path i+j+i+j+i, i.e., the disturbance is propagated more than once between the two molecules. The sequences i+j+i and i-j-i*j+i give rise respectively to two-body second order and two-body fourth order perturbation terms in the expansion of the interaction potential; the sequence i+ j+p+i gives rise to a three-body third order perturbation term. Higher order perturbation terms are generated in similar fashion. In this paper we consider only two-body interaction potentials. By neglecting correlations between multiple excitations (the deccrrelation approximation), we obtain the following long-range (or dispersion) part the generalized intermolecular potential*
of
xkk*(Ri$)
v&’ Xk;t(Rj;Q)V&
.
of
The xw’(Ri;P) and xK&(Rj:Q) are the charge-density susceptibilities i and j whose centers of mass are located respectively at the fixed positions Ri and Rj ;u& is the static potential; p = 1/kg T where kg is the Boltzmann constant and Tthe temperature. The prime on the first summation sign denotes that the Q=O term must be multiplied in eq. (1) refer to the susceptibilities in momentum space at the discrete frequencies by;. The susceptibilities ) along the imaginary axis of the complex frequency plane f=w - iy. These SUSyp=2nQ//S(whereQ=0,1,2,... ceptibilities are related to the real frequency susceptibilities in coordinate space X(rj, ri, W) by the following set of equations:
x We have in the past used the symbol AF or AF2 to denote the infmite order two-body interaction potential. We shall henceforth characterize the interaction potential bv the symbol VlRi ;Rj) and reserve the symbol F solely for the Helmholtz bee energy of the fluid.
B. Linder/A
XM’ (Ri ; w) = j-d+ =exP
time-evolun’on
i expWr&
approach to intermolecular
IiiIl n b-+0
c
-l
75
potentials
x(rj,ri’. w) exp(-ik’.rJ
[-i(k’-k)-Ri]SdgiSdSIexp(ik’Si)Xi(Si,Sl,w)exp
&(ri,r;,W)=
end thermodynamic
(2)
(-s’.$),
(Pm,k(fl)Pnnk(ri)
Pmn(fi)Pmn(fj)
0,m
-ib
w-Onm
m.n
-
W+o+r,-ib
(3)
).
Here, ri and ri are measured from a space-fixed origin, common to all molecules; & and &! are measured from the =fi-‘(En -Em). the E;being the unperturbed center of mass of molecule i, k.,gi=rt -Riv5~=r~-Ri;~nm of the density matrix electronic energies; the omnr are the diagonal components 6 = exp(-Mo)/tr
exp(-@I$,
in which Ho is the hamiltonian density operator p(ri): b(ri)=e
of the non-interacting
the omn (fi) are the matrix elements
system;
of the charge-
CS(ri-rj),
CZ,6(ri-rJ-e Y
i
where e is the absolute magnitude of the electronic charge, r, and rj are respectively the position coordinates of the nuclei and electrons and Z, is the atomic number of the uth atom. The susceptibility along the imaginary axis is related to the susceptibility along the real frequency axis, x(o) = x’(w) - ix”(w), by the equation x(P) E x(-i&
= (2,rr)Jdw
w x”(w)/(w2
+yp’).
(5)
0 The potential
uk is defined
as
where 52 is the volume of the container
and u(ri,rj)G
u(lri-ril)
E Irj-ri(-*is
the static potential.
Hence,
Vk =4ir/n2.
(6b)
It should be noted that the charge-density susceptibilities depend generaLly on the temperature and so does V,(Ri, Rj). In practice, however, only the ground electronic states of the molecules are populated to any appreciable extent and the effect of the temperature is negligible. For low temperatures (and room temperature can be considered low for electronic excitations), the spacing between the pointsye tends to zero and the summation over II in eq. (1) can be replaced by an integration ,z’
-d
= (fiplzn)~
dr 0
thus cancelling the 8-l. In this case the discrete frequencies appearing in the x(n) 3 x(-iye) replaced by the continuous frequencies -iy. The charge-density susceptibility can be gotten from the response function x(r,r’.
t, r’)=
Wi)([P(r’,
in which the angular brackets through the integration x(r,r’,
t’)i@.
denote
must, of
course,
r)l_)
(7)
a quantum-statistical
w) = (i/II) blimO i d7 tr O[j(r’. 0
be
average and may be most conveniently
t ’ ), p(r. r)]_ exp (-iwT -br).
obtained
r = t--t’.
(8) .
B. LinderiA lime-evolution approach to intermolecular and thermodynamic
76
pofentials
exists a simple relation between the Fourier transform of the commutator([b(r’. 0). c(r,r)]_) and the Fourier transform of the anticommutator ([p(r’,O), fi(r,~)] +) and thus also of the correlation function (fi(r’,O) ~(P,T)). The long-range forces can thus be interpreted as arising from the correlative motion of the charges (essentially the electrons) in neighboring molecules. In the addition to the diagrams embodied in eq. (1) there are others not included in the formula. These arise from fluctuations at say j which propagate through the potential ~(~j,~i) to i giving rise to a first order perturbaetc., giving rise to third and higher order pertion term and from propagations which follow the path j +i-j-i, turbation terms. In the absence of charge-overlap, such terms are eliminated either by symmetry or the decorrelation approximation. In the range of overlap, however, their contribution can be substantial. The first order perturbation term is particularly large at short distances and is mainly responsible for the short-range repulsion. This term can be written
There
(9) where $I, is the wave-function fik(Ri) =
exp
(-
ik.
of the unperturbed
Ri)Jdg
b (5)
eXp
molecules
i and j and
(-ik.k).
Higher order terms in this series can be generated in similar fashion. Thus, the third order perturbation similar to (9) except for a numerical factor and the replacement of uk by C e
F
term is
UkX~‘(Rj;P)Uk’Xk’k(Ri;P)Uk-
This term and all other higher order perturbation gives the overall intermolecular potential
V(RivRj)=
terms in this series will be neglected.
Combining
(9) and (1)
FUkCOn,m (@mIBk(Ri)P_k(Rj)I@m) In
The expression for the intermolecular pair potential given by eq. (10) has been written in its most general form so as to bring out the similarity between it and the thermodynamic potential, to be discussed later. For the kind of molecules we are considering here the spacing between the energy levels is such that the attractive potential is virtually temperature-independent. The leading term in the expression of the logarithm, when developed in a multipole series, yields -(6fil1?R$)J
dwi J dWj Q’rl(Wi)a,!‘(Wj)/(Oi + Oj), 0 0 which is a form of the London dispersion formula 121. The o”(o) are the imaginary parts of the ordinary (spatially-independent) polarizabilities of the isolated molecules. (Throughout this work the properties of the non-interacting system are assumed to be known or obtainable.) The repulsive part of the potential may in first approximation be gotten from the product wave-functions of the non-interacting molecules. Corrections for exchange can be included by using a symmetrized base. The major contribution, however, comes from the Coulomb penetration term which, for simpfi molecules, varies exponen-
B. LinderiA
limesvolution appmach lo in&nnolecubr
and thermodynamic
porentils
77
tially at short separations. Thus, in spite of the different form, formula (10) gives, for simplified models, the same dependence on Rjj as the standard semiempirical expressions. 2.2. ~~slationalIluctuariof2s The formulation developed so far is based on the assumption that the centers of mass of the moleccles are fixed in space. We now inquire what happens to the intermolecular potential when this restriction is relaxed. The way transiational, rotational and vibrational motion affect intermolecular forces was recently analyzed by Baer md Ben-Shaul [4]. We focus on the translational motion and examine the problem from a different point of view. There is nothing in the theory which led to the formulation of the interaction potential that requires the molecular centers to be fixed and, indeed, the effects of translational motion can be incorporated formally by averaging the susceptibilities not only over the electronic states but also over the translational ones. This requires the statistical operator in eq. (8) to be a product of two operators, one associated with the translational coordinates, otr. and one with the electronic coordinates, o,,. Although the recipe calls for integrating the response function in eq. (8) over the full range of r from 0 to wit is often unnecessary to do so. Actually, it suffices to carry out the integration over a time interval Ar which. though large compared to the correlation time 7, for electronic transitions, is small compared to the correlation time Ttfor translational motion. The correlation time for a particular mode of motion is the time beyond which the correlation function (p(r,~)p(r’, 0)) is effectively zero. Though this is not a precise definition it does suffice to identify the proper time scale for the transition. As a measure of 7e, we may take the time required to make a transition between two electronic states. Such transitions occur typically in the W region and are - 5 eV and thus roughly Te = (1 /mnm)W = 10-‘5s. As an estimate of Tt we take the time a molecule travels a distance of the order of its diameter, i.e., the collision time. For N2 (diameter - 3 A) the average speed at room temperature is -SXlO’*As-‘andso7 t - 10-12s. For larger molecules 7t is considerably longer. Thus, if we choose a time interval AT larger than 7, but less than rt then during this interval the electronic correlations, and therefore also the intermolecular interactions, will be fully established. Such a time interval is much too short for translational transitions to occur - the translational motion is negligible. To bc more explicit, if we let &(et . ..ep) represent the electronic wavefunction of molecule i and JIr(&) the translational wavefunction of its center of mass, then eq. (8) will contain terms like
During the time interval AT the contribution and the generalized susceptibility becomes
JbRiSdR:
C o,,I~t(R,)I*
from 0~1~ will be negligible.’
Summation
over t’ produces
S(Ri-RI)
s(Ri-RR:)X(ri,‘i,W)
where X(ri.‘i,W) is’the ordinary susceptibility for futed center of mass whose Fourier transform is given by eq. (2). Similar considerations apply to the higher order susceptibilities. These play a role in generating the multiple reflection potentials which are responsible for the higher order terms in the expansion of the logarithm in eq. (1). To obtain these one needs to evaluate products of quantities like
where the 2 are the response-function
operators defined by
ih-’ [P(r’, 0. P(r, r)l_. l
For terms t > As the !ranslational parameters contribute to the exponential. The electronic correlation function, howcvcr, &ops sharply to zero fort > 7e and thus when the integration ISextended to m the result remains unaltered.
B. LinderjA time.evolution approrrchto
78
intermolecular
and thermodynamic potenttalr
Again, if we include translational wave functions and pick a time interval AT larger than the correlation time for these multiple electronic transitions. we can factor out the translational parameters and obtain expressions Like
~dR,$dR;~dR;j.dR/‘...
XC m
~on~~r(Ri)~2G(Ri-R;)S(R;-Ri”)G(Ri”-R;~
‘mm(~ml~(fi,fi, t,f’)~(ff:fi’“,f”,t”‘)...l~r;r).
&I replacing the quantum statistical average of the product of the x’s by the product of their averages (the decorreMion approximation) we obtain an expression identical to eq. (1) except that each term in the expansion of the logarithm is averaged over the configuration space of Rt and Ri. The same applies to the leading (repulsive) term of eq. (10) (which is independent of any correlation times). The overall interaction potential becomes (V)=
c
where G1 (R;:
, *It 0rmf’(G, (Ri) $,a(Ri)l V(Ri,Ri)IJI,(Ri)
9,*(*i)),
S2-‘/ * exp (ik. R). (We ignore here possible symmetry
(V) = 52-*JdRiJdRi
(1I) effects.) Eq. (11) reduces to
V(Ri.Ri),
and for a system of N molecules
we have
F”)=$N(N-I)U’L
(12a)
We refer to this as the first order free energy. It is a correction to the free energy of the non-interacting system that arises from the electronic correlations (intermolecular forces). The time during which these correlations are established is-so short that the molecules are still randomly distributed. We my take. as is common practice, a reference state in which the molecules are partially correlated, say through the repulsive potential. Let VEf(Ri, Rj> represent the interaction potential of the unperturbed system and ~YC functions. The fust order free energy is then given by “translational” vhf(R1 , . . . , RN) the corresponding F(‘)=fN(N-1):
ohfhf (‘PA,
I V(Ri, Rj) I *hf),
(12b)
where P(Rt,Rj) = V(Rt, Rj) - V=f(Rt, Rj) is the perturbation potential. The zeroth order free energy FO will, in this case, contain contributions from V& as well as from the kinetic (and_uossibly electronic) energies of the molecules. The first order free energy corrects for the perturbation potential V. Here again the time interval during which F(l) is established is much too short for ihe molecules to redistribute themselves and the system retains the distribution characteristic of the unperturbed state. We now inquire what happens at subsequent times. It was already noted that for times greater than AT the electronic correlations are zero and the translational motion can have no direct influence on them. The translational motion causes disturbances, however, in the density of the fluid which propagate through the medium - this time not through the Coulomb potential but through the intermolecular potential. Indeed, the same procedure employed for generating the electronic correlations can be used almost without modification for generating the density correlations. One needs to replace the chargedensity susceptibilities of the molecules by molecule-density susceptibilities of the fluid and the static potential by the intermolecular potential. The density correlations are responsible for the generation of the higher order free energy terms of the fluid. The particle-density or molecule-density is defined as n‘(R) =&R-R,% with matrix elements
03)
B. Llnder/A rime-evolution approach lo inrermolecutar and rhermodynamic porenrials
..-, RN)‘PN(R,Rp,--.,R,,r),
nn~~(R)=NSdR,...dRN*n;(R.R,, and an average or equilibrium ii = c
79
density
anln, Q,,+,(R) = N/R.
We define also a molecule-density
(15)
susceptibility
i(R,s)=fi(R,~)-ii. X(R, R’,w) = (i/G) ;imo Jdr((i(R’,O). ~(R,T)] _) exp(-iwr-br). - 0 Since the system under consideration is a macroscopic fluid we can expect it to be translationally X(R. R’) = X( IR-R’I). Its Fourier transform will depend on only one variable K, XK(W) =Jd(R’--R)
exp [-iK.
(If-R)]
(16) invariant
and so
X(R, R’,w),
(17)
in contrast to the charge-density susceptibility which depends on two-variables k and k’. Also, in contrast io the charge-density susceptibility which is the property of a molecule, the molecule-density susceptibility is a property of the entire fluid and contains no labels characterizing the molecules. The free energy terms associated with the density fluctuations of the molecules can now be generated the same way as !he free energy terms associated with the charge fluctuations. The leading term in the reaction potential is &(2)(R, r)=JdR”‘j-dR’~dR’~dr’i+R,R”‘)~(R”’,R”, _w
t,f’)~(R”.R’)i(R’,t’),
(18)
where i!?(R,R’. t. t’)G (i/fi)[i(R’.
f’), fi(R, t)]_-
Equation (18) represents a process wh_ereby a fluctuation in the density at a point R’ and time I’ propagates CIUOU~~the intermole_cular potential V to a point R” at t’. produces a response at R’” at a later time t and propagates again through V producing the potential 4 (2) at R and t. (The points R and R’ lie within a correlation distance from each other and so do also R” and R”‘.) The corresponding element of work (free energy) taken over alI R is dti&
=+~dR[&(*)(R.
t), drj(R, t)],.
0%
(The symmetric form is I;sed becausethe 6 and i do not in general commute.) ‘The susceptibility X is proporby a linear parameter A tional to v2 and hence @(*) varies as q3 . Thus, if we let each IJ increase simultaneously from 0 to 1, we obtain
which
reduces
to
x F(R’,R”)x’(R”,R”‘,~J) This reduction theorem
is accomplished
F(R”,R)
c0thf
pfiiw.
by Fourier decomposing
the integrand
(21)
and by applying
the fluctuationdissipation
a0
fl. iinderfA
rime-evolrtrionapproach co inlermolccu&ar aad thermodynamic
porentiLF
Equation (21) is pan of the fluid free energy and is associated with the fluctuations arising in the molecule-denst’). To this we must add an an~+Io~ous expression. in which the prime and double prime on the X’s are interchanged, representing the contributions arising from fluctuations in the reaction potential parameters. The result is ity q(R’
X c(R',R")X(RI1,Rrrr,w) F(R"',R) ~0thf D&w.
(23)
We refer to this as the second order free energy; it is the leading term of a series associated tions. The higher order terms are generated
i%tl’l,R’V)X(RTV,RV,
W)
Switching to K space and imaginary
where X,
is defined
with density The third order term, for example, is
fluctua-
dRV x(R,R',W) V(R',R")X(R",R"', W)
Fc3)= -(R/12n)ReiJdR... X
in a similar fashion.
p(R”,R)_coth+
frequency
flhW.
(24)
axis yields the rather simple expression
for the perturbation
series
by (17) and
~K=CZ-'~d(R'-R)V"(R,R')exp[-iK-(R'-R)]. (Implicit in this derivation is the decorrelation approximation.) and (17); see also eq. (3)] :
(26) The susceptibility
X,(Q) can be written
[eqs. (16)
(W)hIN (‘)-KINhI_ (‘)-K)hlN(W)Nhf), -wNhf
(27)
WNhl- i 2nk! @fi)-l
-i 2ne (pri)-’
where fK=tiK-tiK
=JdRi(R)exp(-iK*R)
= Cexp(-iK*Rj)-N6K,+
(28)
i
It is noted that r&o = N so that this term is cancelled by EKzO. Thus X,(Q) can be written in terms of tiK+O with only off-diagonal elements (n~),+~~. By interchanging the labels N, M in the second term of eq. (27), one obtains
&@a,= fi-’
M%‘(&N(n-K)Nhf
where the prime on Z denotes
wNhf @hfM w&
ONA’)
+ [2aP(Bfl)-‘]2
(29)
’
that the terms M = N are to be deleted.
3. The fluid free energy
The first order free energy has so far been expressed
in terms of the spatid
coordinates.
It is convenient
to ex-
81
B. LinderfA rime-evolution approach 13 infermolecular and thermodynamic poren&ls
press F(‘) also in terms of K-dependent F(I)=;
s$ vK g
variables like the higher order terms. It is easy to see that
~~~(*~lti~ri_~
-NIeM),
where FK is given by (26). Putting eq. (25) in closed form and combining it with eq. (30) yields
This is the general expression for the free energy applicable to simple classical and quantum fluids interacting through two-body potentials. There is striking resemblance between this formula and the expression for the interaction potential given by eq. (10). It is seen that free energy shift can be obtained from the molecule- and chargedensity susceptibilities and correlation functions of the unperturbed system. The molecule-density susceptibility is related to the dynamic form factor of the unperturbed system
s(K’w)=M% o,dnK),tfh’ (“x)N,Sf”(~-~N,+f)z
(32)
by the expression w[l-exp(-(3fiw)lSW.w) , -gkJ c.s + [ZnP(j3fi)4]2 a
xK+&~w=
.
(33)
This follows from the definition of X,(E), the fluctuation-dissipation theorem [eq. (22)! and the transformation given by eq. (5). For K=O, X,=,(Q) =O. The first order free energy, F(l) can obtiously be related to the static form factor, SK NSK=z
o~~(*~lI”K”-KIQ~&
(34)
and thus the overall free energy shift can be expressed in terms of the static and dynamic form factors. It should be noted that if the ideal gas is taken as reference state then the K=O term will be the only surviving term in Fob. C7usticaZform. In the classical limit only the II= 0 contributes yielding F=Fi+$
$J 5$~r~n_~--N)~+(2P)-~
F
@n[l-xC,l(0)VK]+X&O)?K}.
(35)
The angular brackets here refer to classical averaging over configuration space with exp [- 0 V=f(R,-,RI)] as weighting factor. The classical value for X:(O) is most easily obtained from eq. (29) by replacing obfhf - oNN with a,+,,+¶ [ I- exp (- Ptrw~M)] Z=~AWNMwhich yields X&o
cl = - PlnKaK ,d = - SNSK,
XKd,o(O)= 0.
(36)
The classical form factor is related to the pair correlation function g(R) of the reference system S$ = 1 + (N/O)bR
[g(R) - I] exp (-i Km R).
(37)
The first order free energy term can, of course, be related also to SK and thus for the classical system the Free energy shift is entirely expressible in terms of the static form factor. The result obtained here for the cbssicd SYStem is equivalent to the one of Anderson and Chandler [S] within RPA. We conclude this section by comparing the classical expression for F given by eq. (35) with the standard statistical formula (38)
B. LinderiA rimc~evolution approach IO intermokcdar and thermodynamic porentds
a2 For this purpose
we take as reference
system
the completely
F=Fo+~(I-N-‘)(n~=O”K=0)V~=O+(28)-1
noninteracting
system and write
~Iln[l-X,d(o)VK]+~~(O)VK}.
(39)
(r!K-_O n_K=O) = IV2
(40)
Here. X&o(O)=
--P(nKn-K)
The angular brackets In[l--X:(O)
=-PN,
X&&O)
denote a non-weighted V,]= - In SgO [X:(O)
Vj]’
= - ln S$O(n~.“-~)f In reducing this expression ((“K n-,)9
space. The log term in (39) can be written
(KPO)
(--#V$K)= - ln(exp(-pviC(“Kn_K)]).
we have made use of the relation
(41)
[6] _
= s! (IIK ?LKY.
We next add the remaining X?(O)
= 0,
average over configuration
term
in the form
VK = In exp (-_PVKN)
and F(l)
= (2P)-‘In
exp [-@VK=~(nK=OnK=o
-IV)],
and obtain F-FO=-
(2~)-11n~~~(exp[-P~~(~~n_X-N)]).
The exact classical expression transformed to
F=-P-‘ln(exp[-P&J
for the free energy (based on pairwise adrlitivity) -+flKcO
VK(nKn_K-N)]).
Equation (43) reduces to (42) if the nK are taken to be statistically independent, way that within RPA the present and statistical formulations are equivalent.
(42)
given by (38) can readily be (43)
and this shows in a more direct
4. Discussion We have shown that the response theory formulation developed for treating intermolecular potentials can be generalized to treat thermodynamic potentials of fluids. Of prime consideration in this development are the different time scales for establishing electronic and translationa correlations. During the initial period of the perturbation the charge fluctuations propagate through the Coulomb potential giving rise to intermolecular forces. At subsequent times translational fluctuations propagate through the intermolecular force field and alter the molecular distribution functions. By employing the decorrelation approximation one obtains a simple expression for the free energy shift in terms of the molecule-density susceptibility and the intermolecular potential. The latter may, in turn, be expressed in terms of the charge-density susceptibilities of the molecules and the Coulomb potential. The decorrelation approximation is a convenient and simplifying approximation but not essential to the formalism. It amounts to replacing averages of products of response-function operators by the product of their averages and is, in some respect, equivalent to the timedependent Hartree or random phase approximation
B. Linderl.4 rime-evolurion
and thermodynamic poIenriaLF
approach IO intermolecular
83
used in many-body theories. By relaxing this restriction one can obtain a more accurate but also more complicated expression for the free energy, one which entails multiple transition frequencies and momentum variables. Although the treatment presented here was specialized to systems possessing electronic and translational motion, the same analysis c3n be employed when there are additional degrees of freedom. In particular, if the mole-
cules have permanent moments and the rotational correlation time is much shorter than the translational
one, then
one can use the Same formulae to treat this more general case. One needs to replace the electronic ckargedensity susceptibility in V(Ri,Rj> by a more general one which is averaged over both electronic and rotational states. This has the effect of introducing induc$on and orientation forces in the potential. Since the latter is strongly temperature-dependent SO is V(Ri,Rj) or V(Ri,Rj). (Th e reader is referred to ref. [2] and especially the second paper for a general treatment of dispersion, induction and orientation forces.) The opposite extreme case is one in which the rotational correlation time is very much greater than the electronic and comparable to the translational one. In this case, the rotational motion will have no effect on the intermolecular potential other than to introduce a fist order contribution arising from the interaction of the permanent moments. The molecule-density susceptibility however will have to be generalized to include both transhtional and rot3tional correlations. There are examples of both extreme cases discussed in this and the preceding paragmph but the majority of caSes will probably f3U in-between. The analysis in section 2.2 provides a proper framework for their description. t h-c h--n confined to molecules interacting through two-body intermolecular potentials The present 7’ 111terms in the expansion of the intermolecular potential are established prior to (infinite order). iL to the onset of translatiol. _. C~I. _ 1sand thus contribute to the fluid free energy even in lowest order of per. the (non-additive) many-body potentials. Generalization of the theory turbation. The same is expected tc .tempted in future publications. to include the non-additive potent1
Acknowledgement The author is grateful to the Department of Physical Chemistry of the Hebrew University, where part of this work was done, and especially to Professor S. Baer for his encouragement and hospitality. The author also wishes to thank Dr. T.B. MacRury for helpful suggestions.
References [ 11 S. Baer. I. Chem. Phyo. 60 (1974) 435. [21 B. Linder and D.A. Rabenold. Advan. Quantum Chem. 6 (1972) B. Linder. Advan. Chem. Phys. 12 (1967) 225. [3] T.B. MacRury and 8. finder, J. Chem. Phys. 58 (1973) [4 1 S. Baer and A. Ben-Shaul, hlol. Phys. 19 (1970) 33.
5388
203.
See also
and 5398.
[S] H.C. Anderson and D. Chandler, J. Chem. Phys. 53 (1970) 547; 54 (1971) 26. 161 R. Brout and P. Caruthen, Lectures on the many-electron problem (Interscience,
New York, 1963) p. 33.
A TIME-EVOLUTION
APPROACH TO INTERMOLECULAR
AND THERMODYTWlWC POTENTIALS*
Bruno LlNDER fiWtmen[ of fllemish): 7heF~ori&SIart! Univeniry. Tallahassee, FIcwi& 3.?306, USA Received 19 September 1974 A unified treatment is presented of ths interaction potentials (intermolecular forces) and the thermodynamic potentials (free enew~ functions) of molecular fluids. The treatment is based on a time-evolution approach to equilibrium which tt-accs the propagation of electrical and density fluctuations from the onset of the disturbance to the time equtiibrium IS rutty eswbk.+mi ‘Ihe ~=ow is sped;llired to a one-component. monntomic fluid interacting through geneked two-body intermoleculzu potentials (to infinite order). BY neglecting correlations between multiple excitations. II simpleexpressionis obtained for the free energy in terms of the molecule-density susceptibility of the unperturbed tluid and the intermolecular potential. The letter is formulated in terms of the charge-density susceptibilities of the noninterxting molecuies. and the static (Coulomb) potential. The theory takes account of translational quantum effects. It is shown that for a classicalfluid the free energy is consistent with formulations obtained from standard statistical mechanics within RPA.
1. Introduction In recent years, several new techniques have been proposed for treating the (equilibrium) thermodynamic properties of fluids. The most general theories are formulated in terms of the Coulomb interaction between the charged particles (electrons and nuclei) of the system and require the full complexity of quantum-statistical mechanics. The more standard techniques treat the quantum and statistical aspects separately. Thus, first the interaction energy between the charged particles are worked out quantum-mechanically for fvted intermolecular separations. Subsequently, the effects of the translational motion of the centers of mass of the molecules are brought in statisticaily. A unified treatment based on molecular- and charge-density correlation functions was recently proposed by Baer [l]. The intermolecular energies (or interaction potentials) are commonly referred to as intermolecular forces. The concept of an intermolecular force is predicated on the assumption that the electrons in a molecule move faster than the center of mass of the molecule. Indeed, it is this difference in time scale that makes it possible to analyze the complex dynamical motion of all the particles of the system as two separate events: one associated with the establishment of electronic correlations, the other with the establishment of translational molecular correlations. As long as the time scales are vastly different (and they generally are for molecular substances), then each type of correlation can be treated quantum-statistically and the resulting expression for the free energy of fluid will contain intermolecular potentials that may be temperature-dependent and translational parameters which may include quantum effects. The commonly employed procedure which treats the electronic fluctuations (i.e., the intermolecular potentials) as strictly quanta1 and the translational fluctuations as strictly classical imposes further restrictions not warranted by the dynamical behavior of the particles. In this paper we present a time-evolution approach to the thermodynamic properties of a molecular fluid, which treats the electronic and molecular translational fluctuations in a unified manner. The method is based on a generalization of the reaction potential technique used in the treatment of long-range forces [2,31_ The theory is first generalized to take account of the short-range interactions and then extended to include the effects of translational fluctuations. The present work is confined to the treatment of monatornic molecules and takes account only of two-body intermolecular interaction potentials (to all orders). The extension to arbitrary systems interacting through many-body potentials will be considered elsewhere. * Supported by a Grant from the National Science Foundation.
74
R Ltnderf A time-evolution
approach 10 inte?molecuuPr and thenmdpumtc
potentiais
2. Procedure 2.1. Electronic fluctuations The procedure for generating the long-range (or van der Waals) forces by means of the reaction potential technique was discussed earlier [2,3]. Here, we consider only the broad aspects of the theory and present a summary of the result for two-body interactions. In essence, one molecule, i, is singled out and treated explicitly and the remaining molecules with the radiation field are treated collectively as the “surrounding”. The reaction potential +(ri,t) is the scalar potential (retardation effects are here neglected) at the site rt of molecule i produced by the surrounding which was perturbed by the electrical fluctuations in i at all points r,! and at all times t’ < t. The polarization free energy for this process is obtained from the coupling between *(rj,f) and the charge-density p(ri,r). By taking account also of the polarization free energy arising from the fluctuations in the reaction potential parameters one obtains the mutual interaction free energy of i with the medium and, by summing over all i, the overall interaction free energy or interaction potential of the system. (To avoid possible confusion between this free energy and the true thermodynamic free energy of the fluid we shall henceforth refer to the former as interaction potential or intermolecular potential.) The interaction potential can be written as a function of two charge-density susceptibilities (to be discussed in more detail in the following paragraph): the susceptibility xi of molecule i and its conjugate reaction-susceptibihties Gi. The latter can be developed in a series involving the susceptibilities of the molecules of the surrounding and the static potentials of the radiation field. If i has only one neighboring molecule j, the reaction susceptibility is obtained by considering the hypothetical process whereby a fluctuation at r,! propagates to r,!, perturbsi at rj and acts back at ri. If there are two neighborsi and p. the reaction susceptibility will contain additional terms one of which represents a process whereby a charge-fluctuation in i first propagates to j, then top and finally returns to i, and also a term representing the process i+p + 1‘+i. Of course, the interaction path involving two molecules can also proceed via the path i+j+i+j+i, i.e., the disturbance is propagated more than once between the two molecules. The sequences i+j+i and i-j-i*j+i give rise respectively to two-body second order and two-body fourth order perturbation terms in the expansion of the interaction potential; the sequence i+ j+p+i gives rise to a three-body third order perturbation term. Higher order perturbation terms are generated in similar fashion. In this paper we consider only two-body interaction potentials. By neglecting correlations between multiple excitations (the deccrrelation approximation), we obtain the following long-range (or dispersion) part the generalized intermolecular potential*
of
xkk*(Ri$)
v&’ Xk;t(Rj;Q)V&
.
of
The xw’(Ri;P) and xK&(Rj:Q) are the charge-density susceptibilities i and j whose centers of mass are located respectively at the fixed positions Ri and Rj ;u& is the static potential; p = 1/kg T where kg is the Boltzmann constant and Tthe temperature. The prime on the first summation sign denotes that the Q=O term must be multiplied in eq. (1) refer to the susceptibilities in momentum space at the discrete frequencies by;. The susceptibilities ) along the imaginary axis of the complex frequency plane f=w - iy. These SUSyp=2nQ//S(whereQ=0,1,2,... ceptibilities are related to the real frequency susceptibilities in coordinate space X(rj, ri, W) by the following set of equations:
x We have in the past used the symbol AF or AF2 to denote the infmite order two-body interaction potential. We shall henceforth characterize the interaction potential bv the symbol VlRi ;Rj) and reserve the symbol F solely for the Helmholtz bee energy of the fluid.
B. Linder/A
XM’ (Ri ; w) = j-d+ =exP
time-evolun’on
i expWr&
approach to intermolecular
IiiIl n b-+0
c
-l
75
potentials
x(rj,ri’. w) exp(-ik’.rJ
[-i(k’-k)-Ri]SdgiSdSIexp(ik’Si)Xi(Si,Sl,w)exp
&(ri,r;,W)=
end thermodynamic
(2)
(-s’.$),
(Pm,k(fl)Pnnk(ri)
Pmn(fi)Pmn(fj)
0,m
-ib
w-Onm
m.n
-
W+o+r,-ib
(3)
).
Here, ri and ri are measured from a space-fixed origin, common to all molecules; & and &! are measured from the =fi-‘(En -Em). the E;being the unperturbed center of mass of molecule i, k.,gi=rt -Riv5~=r~-Ri;~nm of the density matrix electronic energies; the omnr are the diagonal components 6 = exp(-Mo)/tr
exp(-@I$,
in which Ho is the hamiltonian density operator p(ri): b(ri)=e
of the non-interacting
the omn (fi) are the matrix elements
system;
of the charge-
CS(ri-rj),
CZ,6(ri-rJ-e Y
i
where e is the absolute magnitude of the electronic charge, r, and rj are respectively the position coordinates of the nuclei and electrons and Z, is the atomic number of the uth atom. The susceptibility along the imaginary axis is related to the susceptibility along the real frequency axis, x(o) = x’(w) - ix”(w), by the equation x(P) E x(-i&
= (2,rr)Jdw
w x”(w)/(w2
+yp’).
(5)
0 The potential
uk is defined
as
where 52 is the volume of the container
and u(ri,rj)G
u(lri-ril)
E Irj-ri(-*is
the static potential.
Hence,
Vk =4ir/n2.
(6b)
It should be noted that the charge-density susceptibilities depend generaLly on the temperature and so does V,(Ri, Rj). In practice, however, only the ground electronic states of the molecules are populated to any appreciable extent and the effect of the temperature is negligible. For low temperatures (and room temperature can be considered low for electronic excitations), the spacing between the pointsye tends to zero and the summation over II in eq. (1) can be replaced by an integration ,z’
-d
= (fiplzn)~
dr 0
thus cancelling the 8-l. In this case the discrete frequencies appearing in the x(n) 3 x(-iye) replaced by the continuous frequencies -iy. The charge-density susceptibility can be gotten from the response function x(r,r’.
t, r’)=
Wi)([P(r’,
in which the angular brackets through the integration x(r,r’,
t’)i@.
denote
must, of
course,
r)l_)
(7)
a quantum-statistical
w) = (i/II) blimO i d7 tr O[j(r’. 0
be
average and may be most conveniently
t ’ ), p(r. r)]_ exp (-iwT -br).
obtained
r = t--t’.
(8) .
B. LinderiA lime-evolution approach to intermolecular and thermodynamic
76
pofentials
exists a simple relation between the Fourier transform of the commutator([b(r’. 0). c(r,r)]_) and the Fourier transform of the anticommutator ([p(r’,O), fi(r,~)] +) and thus also of the correlation function (fi(r’,O) ~(P,T)). The long-range forces can thus be interpreted as arising from the correlative motion of the charges (essentially the electrons) in neighboring molecules. In the addition to the diagrams embodied in eq. (1) there are others not included in the formula. These arise from fluctuations at say j which propagate through the potential ~(~j,~i) to i giving rise to a first order perturbaetc., giving rise to third and higher order pertion term and from propagations which follow the path j +i-j-i, turbation terms. In the absence of charge-overlap, such terms are eliminated either by symmetry or the decorrelation approximation. In the range of overlap, however, their contribution can be substantial. The first order perturbation term is particularly large at short distances and is mainly responsible for the short-range repulsion. This term can be written
There
(9) where $I, is the wave-function fik(Ri) =
exp
(-
ik.
of the unperturbed
Ri)Jdg
b (5)
eXp
molecules
i and j and
(-ik.k).
Higher order terms in this series can be generated in similar fashion. Thus, the third order perturbation similar to (9) except for a numerical factor and the replacement of uk by C e
F
term is
UkX~‘(Rj;P)Uk’Xk’k(Ri;P)Uk-
This term and all other higher order perturbation gives the overall intermolecular potential
V(RivRj)=
terms in this series will be neglected.
Combining
(9) and (1)
FUkCOn,m (@mIBk(Ri)P_k(Rj)I@m) In
The expression for the intermolecular pair potential given by eq. (10) has been written in its most general form so as to bring out the similarity between it and the thermodynamic potential, to be discussed later. For the kind of molecules we are considering here the spacing between the energy levels is such that the attractive potential is virtually temperature-independent. The leading term in the expression of the logarithm, when developed in a multipole series, yields -(6fil1?R$)J
dwi J dWj Q’rl(Wi)a,!‘(Wj)/(Oi + Oj), 0 0 which is a form of the London dispersion formula 121. The o”(o) are the imaginary parts of the ordinary (spatially-independent) polarizabilities of the isolated molecules. (Throughout this work the properties of the non-interacting system are assumed to be known or obtainable.) The repulsive part of the potential may in first approximation be gotten from the product wave-functions of the non-interacting molecules. Corrections for exchange can be included by using a symmetrized base. The major contribution, however, comes from the Coulomb penetration term which, for simpfi molecules, varies exponen-
B. LinderiA
limesvolution appmach lo in&nnolecubr
and thermodynamic
porentils
77
tially at short separations. Thus, in spite of the different form, formula (10) gives, for simplified models, the same dependence on Rjj as the standard semiempirical expressions. 2.2. ~~slationalIluctuariof2s The formulation developed so far is based on the assumption that the centers of mass of the moleccles are fixed in space. We now inquire what happens to the intermolecular potential when this restriction is relaxed. The way transiational, rotational and vibrational motion affect intermolecular forces was recently analyzed by Baer md Ben-Shaul [4]. We focus on the translational motion and examine the problem from a different point of view. There is nothing in the theory which led to the formulation of the interaction potential that requires the molecular centers to be fixed and, indeed, the effects of translational motion can be incorporated formally by averaging the susceptibilities not only over the electronic states but also over the translational ones. This requires the statistical operator in eq. (8) to be a product of two operators, one associated with the translational coordinates, otr. and one with the electronic coordinates, o,,. Although the recipe calls for integrating the response function in eq. (8) over the full range of r from 0 to wit is often unnecessary to do so. Actually, it suffices to carry out the integration over a time interval Ar which. though large compared to the correlation time 7, for electronic transitions, is small compared to the correlation time Ttfor translational motion. The correlation time for a particular mode of motion is the time beyond which the correlation function (p(r,~)p(r’, 0)) is effectively zero. Though this is not a precise definition it does suffice to identify the proper time scale for the transition. As a measure of 7e, we may take the time required to make a transition between two electronic states. Such transitions occur typically in the W region and are - 5 eV and thus roughly Te = (1 /mnm)W = 10-‘5s. As an estimate of Tt we take the time a molecule travels a distance of the order of its diameter, i.e., the collision time. For N2 (diameter - 3 A) the average speed at room temperature is -SXlO’*As-‘andso7 t - 10-12s. For larger molecules 7t is considerably longer. Thus, if we choose a time interval AT larger than 7, but less than rt then during this interval the electronic correlations, and therefore also the intermolecular interactions, will be fully established. Such a time interval is much too short for translational transitions to occur - the translational motion is negligible. To bc more explicit, if we let &(et . ..ep) represent the electronic wavefunction of molecule i and JIr(&) the translational wavefunction of its center of mass, then eq. (8) will contain terms like
During the time interval AT the contribution and the generalized susceptibility becomes
JbRiSdR:
C o,,I~t(R,)I*
from 0~1~ will be negligible.’
Summation
over t’ produces
S(Ri-RI)
s(Ri-RR:)X(ri,‘i,W)
where X(ri.‘i,W) is’the ordinary susceptibility for futed center of mass whose Fourier transform is given by eq. (2). Similar considerations apply to the higher order susceptibilities. These play a role in generating the multiple reflection potentials which are responsible for the higher order terms in the expansion of the logarithm in eq. (1). To obtain these one needs to evaluate products of quantities like
where the 2 are the response-function
operators defined by
ih-’ [P(r’, 0. P(r, r)l_. l
For terms t > As the !ranslational parameters contribute to the exponential. The electronic correlation function, howcvcr, &ops sharply to zero fort > 7e and thus when the integration ISextended to m the result remains unaltered.
B. LinderjA time.evolution approrrchto
78
intermolecular
and thermodynamic potenttalr
Again, if we include translational wave functions and pick a time interval AT larger than the correlation time for these multiple electronic transitions. we can factor out the translational parameters and obtain expressions Like
~dR,$dR;~dR;j.dR/‘...
XC m
~on~~r(Ri)~2G(Ri-R;)S(R;-Ri”)G(Ri”-R;~
‘mm(~ml~(fi,fi, t,f’)~(ff:fi’“,f”,t”‘)...l~r;r).
&I replacing the quantum statistical average of the product of the x’s by the product of their averages (the decorreMion approximation) we obtain an expression identical to eq. (1) except that each term in the expansion of the logarithm is averaged over the configuration space of Rt and Ri. The same applies to the leading (repulsive) term of eq. (10) (which is independent of any correlation times). The overall interaction potential becomes (V)=
c
where G1 (R;:
, *It 0rmf’(G, (Ri) $,a(Ri)l V(Ri,Ri)IJI,(Ri)
9,*(*i)),
S2-‘/ * exp (ik. R). (We ignore here possible symmetry
(V) = 52-*JdRiJdRi
(1I) effects.) Eq. (11) reduces to
V(Ri.Ri),
and for a system of N molecules
we have
F”)=$N(N-I)U’L
(12a)
We refer to this as the first order free energy. It is a correction to the free energy of the non-interacting system that arises from the electronic correlations (intermolecular forces). The time during which these correlations are established is-so short that the molecules are still randomly distributed. We my take. as is common practice, a reference state in which the molecules are partially correlated, say through the repulsive potential. Let VEf(Ri, Rj> represent the interaction potential of the unperturbed system and ~YC functions. The fust order free energy is then given by “translational” vhf(R1 , . . . , RN) the corresponding F(‘)=fN(N-1):
ohfhf (‘PA,
I V(Ri, Rj) I *hf),
(12b)
where P(Rt,Rj) = V(Rt, Rj) - V=f(Rt, Rj) is the perturbation potential. The zeroth order free energy FO will, in this case, contain contributions from V& as well as from the kinetic (and_uossibly electronic) energies of the molecules. The first order free energy corrects for the perturbation potential V. Here again the time interval during which F(l) is established is much too short for ihe molecules to redistribute themselves and the system retains the distribution characteristic of the unperturbed state. We now inquire what happens at subsequent times. It was already noted that for times greater than AT the electronic correlations are zero and the translational motion can have no direct influence on them. The translational motion causes disturbances, however, in the density of the fluid which propagate through the medium - this time not through the Coulomb potential but through the intermolecular potential. Indeed, the same procedure employed for generating the electronic correlations can be used almost without modification for generating the density correlations. One needs to replace the chargedensity susceptibilities of the molecules by molecule-density susceptibilities of the fluid and the static potential by the intermolecular potential. The density correlations are responsible for the generation of the higher order free energy terms of the fluid. The particle-density or molecule-density is defined as n‘(R) =&R-R,% with matrix elements
03)
B. Llnder/A rime-evolution approach lo inrermolecutar and rhermodynamic porenrials
..-, RN)‘PN(R,Rp,--.,R,,r),
nn~~(R)=NSdR,...dRN*n;(R.R,, and an average or equilibrium ii = c
79
density
anln, Q,,+,(R) = N/R.
We define also a molecule-density
(15)
susceptibility
i(R,s)=fi(R,~)-ii. X(R, R’,w) = (i/G) ;imo Jdr((i(R’,O). ~(R,T)] _) exp(-iwr-br). - 0 Since the system under consideration is a macroscopic fluid we can expect it to be translationally X(R. R’) = X( IR-R’I). Its Fourier transform will depend on only one variable K, XK(W) =Jd(R’--R)
exp [-iK.
(If-R)]
(16) invariant
and so
X(R, R’,w),
(17)
in contrast to the charge-density susceptibility which depends on two-variables k and k’. Also, in contrast io the charge-density susceptibility which is the property of a molecule, the molecule-density susceptibility is a property of the entire fluid and contains no labels characterizing the molecules. The free energy terms associated with the density fluctuations of the molecules can now be generated the same way as !he free energy terms associated with the charge fluctuations. The leading term in the reaction potential is &(2)(R, r)=JdR”‘j-dR’~dR’~dr’i+R,R”‘)~(R”’,R”, _w
t,f’)~(R”.R’)i(R’,t’),
(18)
where i!?(R,R’. t. t’)G (i/fi)[i(R’.
f’), fi(R, t)]_-
Equation (18) represents a process wh_ereby a fluctuation in the density at a point R’ and time I’ propagates CIUOU~~the intermole_cular potential V to a point R” at t’. produces a response at R’” at a later time t and propagates again through V producing the potential 4 (2) at R and t. (The points R and R’ lie within a correlation distance from each other and so do also R” and R”‘.) The corresponding element of work (free energy) taken over alI R is dti&
=+~dR[&(*)(R.
t), drj(R, t)],.
0%
(The symmetric form is I;sed becausethe 6 and i do not in general commute.) ‘The susceptibility X is proporby a linear parameter A tional to v2 and hence @(*) varies as q3 . Thus, if we let each IJ increase simultaneously from 0 to 1, we obtain
which
reduces
to
x F(R’,R”)x’(R”,R”‘,~J) This reduction theorem
is accomplished
F(R”,R)
c0thf
pfiiw.
by Fourier decomposing
the integrand
(21)
and by applying
the fluctuationdissipation
a0
fl. iinderfA
rime-evolrtrionapproach co inlermolccu&ar aad thermodynamic
porentiLF
Equation (21) is pan of the fluid free energy and is associated with the fluctuations arising in the molecule-denst’). To this we must add an an~+Io~ous expression. in which the prime and double prime on the X’s are interchanged, representing the contributions arising from fluctuations in the reaction potential parameters. The result is ity q(R’
X c(R',R")X(RI1,Rrrr,w) F(R"',R) ~0thf D&w.
(23)
We refer to this as the second order free energy; it is the leading term of a series associated tions. The higher order terms are generated
i%tl’l,R’V)X(RTV,RV,
W)
Switching to K space and imaginary
where X,
is defined
with density The third order term, for example, is
fluctua-
dRV x(R,R',W) V(R',R")X(R",R"', W)
Fc3)= -(R/12n)ReiJdR... X
in a similar fashion.
p(R”,R)_coth+
frequency
flhW.
(24)
axis yields the rather simple expression
for the perturbation
series
by (17) and
~K=CZ-'~d(R'-R)V"(R,R')exp[-iK-(R'-R)]. (Implicit in this derivation is the decorrelation approximation.) and (17); see also eq. (3)] :
(26) The susceptibility
X,(Q) can be written
[eqs. (16)
(W)hIN (‘)-KINhI_ (‘)-K)hlN(W)Nhf), -wNhf
(27)
WNhl- i 2nk! @fi)-l
-i 2ne (pri)-’
where fK=tiK-tiK
=JdRi(R)exp(-iK*R)
= Cexp(-iK*Rj)-N6K,+
(28)
i
It is noted that r&o = N so that this term is cancelled by EKzO. Thus X,(Q) can be written in terms of tiK+O with only off-diagonal elements (n~),+~~. By interchanging the labels N, M in the second term of eq. (27), one obtains
&@a,= fi-’
M%‘(&N(n-K)Nhf
where the prime on Z denotes
wNhf @hfM w&
ONA’)
+ [2aP(Bfl)-‘]2
(29)
’
that the terms M = N are to be deleted.
3. The fluid free energy
The first order free energy has so far been expressed
in terms of the spatid
coordinates.
It is convenient
to ex-
81
B. LinderfA rime-evolution approach 13 infermolecular and thermodynamic poren&ls
press F(‘) also in terms of K-dependent F(I)=;
s$ vK g
variables like the higher order terms. It is easy to see that
~~~(*~lti~ri_~
-NIeM),
where FK is given by (26). Putting eq. (25) in closed form and combining it with eq. (30) yields
This is the general expression for the free energy applicable to simple classical and quantum fluids interacting through two-body potentials. There is striking resemblance between this formula and the expression for the interaction potential given by eq. (10). It is seen that free energy shift can be obtained from the molecule- and chargedensity susceptibilities and correlation functions of the unperturbed system. The molecule-density susceptibility is related to the dynamic form factor of the unperturbed system
s(K’w)=M% o,dnK),tfh’ (“x)N,Sf”(~-~N,+f)z
(32)
by the expression w[l-exp(-(3fiw)lSW.w) , -gkJ c.s + [ZnP(j3fi)4]2 a
xK+&~w=
.
(33)
This follows from the definition of X,(E), the fluctuation-dissipation theorem [eq. (22)! and the transformation given by eq. (5). For K=O, X,=,(Q) =O. The first order free energy, F(l) can obtiously be related to the static form factor, SK NSK=z
o~~(*~lI”K”-KIQ~&
(34)
and thus the overall free energy shift can be expressed in terms of the static and dynamic form factors. It should be noted that if the ideal gas is taken as reference state then the K=O term will be the only surviving term in Fob. C7usticaZform. In the classical limit only the II= 0 contributes yielding F=Fi+$
$J 5$~r~n_~--N)~+(2P)-~
F
@n[l-xC,l(0)VK]+X&O)?K}.
(35)
The angular brackets here refer to classical averaging over configuration space with exp [- 0 V=f(R,-,RI)] as weighting factor. The classical value for X:(O) is most easily obtained from eq. (29) by replacing obfhf - oNN with a,+,,+¶ [ I- exp (- Ptrw~M)] Z=~AWNMwhich yields X&o
cl = - PlnKaK ,d = - SNSK,
XKd,o(O)= 0.
(36)
The classical form factor is related to the pair correlation function g(R) of the reference system S$ = 1 + (N/O)bR
[g(R) - I] exp (-i Km R).
(37)
The first order free energy term can, of course, be related also to SK and thus for the classical system the Free energy shift is entirely expressible in terms of the static form factor. The result obtained here for the cbssicd SYStem is equivalent to the one of Anderson and Chandler [S] within RPA. We conclude this section by comparing the classical expression for F given by eq. (35) with the standard statistical formula (38)
B. LinderiA rimc~evolution approach IO intermokcdar and thermodynamic porentds
a2 For this purpose
we take as reference
system
the completely
F=Fo+~(I-N-‘)(n~=O”K=0)V~=O+(28)-1
noninteracting
system and write
~Iln[l-X,d(o)VK]+~~(O)VK}.
(39)
(r!K-_O n_K=O) = IV2
(40)
Here. X&o(O)=
--P(nKn-K)
The angular brackets In[l--X:(O)
=-PN,
X&&O)
denote a non-weighted V,]= - In SgO [X:(O)
Vj]’
= - ln S$O(n~.“-~)f In reducing this expression ((“K n-,)9
space. The log term in (39) can be written
(KPO)
(--#V$K)= - ln(exp(-pviC(“Kn_K)]).
we have made use of the relation
(41)
[6] _
= s! (IIK ?LKY.
We next add the remaining X?(O)
= 0,
average over configuration
term
in the form
VK = In exp (-_PVKN)
and F(l)
= (2P)-‘In
exp [-@VK=~(nK=OnK=o
-IV)],
and obtain F-FO=-
(2~)-11n~~~(exp[-P~~(~~n_X-N)]).
The exact classical expression transformed to
F=-P-‘ln(exp[-P&J
for the free energy (based on pairwise adrlitivity) -+flKcO
VK(nKn_K-N)]).
Equation (43) reduces to (42) if the nK are taken to be statistically independent, way that within RPA the present and statistical formulations are equivalent.
(42)
given by (38) can readily be (43)
and this shows in a more direct
4. Discussion We have shown that the response theory formulation developed for treating intermolecular potentials can be generalized to treat thermodynamic potentials of fluids. Of prime consideration in this development are the different time scales for establishing electronic and translationa correlations. During the initial period of the perturbation the charge fluctuations propagate through the Coulomb potential giving rise to intermolecular forces. At subsequent times translational fluctuations propagate through the intermolecular force field and alter the molecular distribution functions. By employing the decorrelation approximation one obtains a simple expression for the free energy shift in terms of the molecule-density susceptibility and the intermolecular potential. The latter may, in turn, be expressed in terms of the charge-density susceptibilities of the molecules and the Coulomb potential. The decorrelation approximation is a convenient and simplifying approximation but not essential to the formalism. It amounts to replacing averages of products of response-function operators by the product of their averages and is, in some respect, equivalent to the timedependent Hartree or random phase approximation
B. Linderl.4 rime-evolurion
and thermodynamic poIenriaLF
approach IO intermolecular
83
used in many-body theories. By relaxing this restriction one can obtain a more accurate but also more complicated expression for the free energy, one which entails multiple transition frequencies and momentum variables. Although the treatment presented here was specialized to systems possessing electronic and translational motion, the same analysis c3n be employed when there are additional degrees of freedom. In particular, if the mole-
cules have permanent moments and the rotational correlation time is much shorter than the translational
one, then
one can use the Same formulae to treat this more general case. One needs to replace the electronic ckargedensity susceptibility in V(Ri,Rj> by a more general one which is averaged over both electronic and rotational states. This has the effect of introducing induc$on and orientation forces in the potential. Since the latter is strongly temperature-dependent SO is V(Ri,Rj) or V(Ri,Rj). (Th e reader is referred to ref. [2] and especially the second paper for a general treatment of dispersion, induction and orientation forces.) The opposite extreme case is one in which the rotational correlation time is very much greater than the electronic and comparable to the translational one. In this case, the rotational motion will have no effect on the intermolecular potential other than to introduce a fist order contribution arising from the interaction of the permanent moments. The molecule-density susceptibility however will have to be generalized to include both transhtional and rot3tional correlations. There are examples of both extreme cases discussed in this and the preceding paragmph but the majority of caSes will probably f3U in-between. The analysis in section 2.2 provides a proper framework for their description. t h-c h--n confined to molecules interacting through two-body intermolecular potentials The present 7’ 111terms in the expansion of the intermolecular potential are established prior to (infinite order). iL to the onset of translatiol. _. C~I. _ 1sand thus contribute to the fluid free energy even in lowest order of per. the (non-additive) many-body potentials. Generalization of the theory turbation. The same is expected tc .tempted in future publications. to include the non-additive potent1
Acknowledgement The author is grateful to the Department of Physical Chemistry of the Hebrew University, where part of this work was done, and especially to Professor S. Baer for his encouragement and hospitality. The author also wishes to thank Dr. T.B. MacRury for helpful suggestions.
References [ 11 S. Baer. I. Chem. Phyo. 60 (1974) 435. [21 B. Linder and D.A. Rabenold. Advan. Quantum Chem. 6 (1972) B. Linder. Advan. Chem. Phys. 12 (1967) 225. [3] T.B. MacRury and 8. finder, J. Chem. Phys. 58 (1973) [4 1 S. Baer and A. Ben-Shaul, hlol. Phys. 19 (1970) 33.
5388
203.
See also
and 5398.
[S] H.C. Anderson and D. Chandler, J. Chem. Phys. 53 (1970) 547; 54 (1971) 26. 161 R. Brout and P. Caruthen, Lectures on the many-electron problem (Interscience,
New York, 1963) p. 33.