2060
J. Phys. Chem. 1983, 87, 2060-2064
Theory of Orientational Pair Correlations in Molecular Fluids Davld Chandler' and Diane M. Richardson School of Chemical Sciences, University of Illinois, Urbana, Iillnols 6180 1 (Received: Februaty 8, 1983)
Spurred by new measurements of orientational pair correlation factors, we have reanalyzed the deficiencies of the RISM theory in predicting these long wavelength properties. Our approach is based upon the so-called proper integral equation theory of pair correlations in molecular fluids. The proper integral equation is formally exact and represents a generalization of the Ornstein-Zernike equation of simple fluids. An approximation yields the RISM equation, but this approximation demolishes all but the most trivial facets of long wavelength orientational pair correlations. Fortunately the approximation can be improved upon and predictions can be made about the pair correlation factors probed with dielectric measurements and light scattering. This paper discusses the general framework of the theory as it pertains to orientational pair correlations, especially the g, factor relevant to depolarized light scattering.
opposed to an atomic liquid, atoms can be strongly bound to others by chemical or intramolecular interactions. These forces tend to constrain the configuration space available
1. Introduction The theory of polyatomic molecular fluids has progressed to the stage where it is now possible to predict reliably the average local arrangements of atoms in fluids as complex as liquid benzene and its derivatives, and as important as water and aqueous solutions. Much of this progress is associated with the approach often referred to as the interaction site formalism, from which useful approximations such as the RISM equation have been produced.' In this article, we discuss an application of this formalism to study orientational pair correlation factors. The experimental determination of these long wavelength properties has long been plagued by ambiguities associated with local field corrections. The Flygare group played a significant role in unraveling some of these difficulties.2 This work together with more recent studies of Madden and co-workers3 which combine Cotton-Mouton experiments with Rayleigh depolarized light scattering have placed the experimental results on firmer ground. From the perspective of the interaction site formalism, the theory for orientational pair correlation factors has also been plagued with difficulties. These problems, however, seem to be resolved by the newly developed proper integral equation t h e ~ r y .We ~ focus on this development in this paper. In section 2, the physical ideas associated with the RISM equation are discussed. While plausible, these ideas fail to be accurate in the long wavelength regime.5 The extension of the RISM theory to a rigorous treatment, the proper integral equation, is discussed in section 3. The calculation of orientational pair correlation factors is then discussed in section 4. 2. RISM Theory
to the atoms, and it is these constraints that produce effects, such as orientational pair correlations, which distinguish molecular fluids from simple atomic systems. For example, one may juxtapose the structure of a monatomic fluid with that of a diatomic system. This is done in Figure 1. The illustrations show regions of a monatomic liquid and a homonuclear diatomic liquid (drawn schematically in two dimensions for artistic convenience), and their respective intermolecular radial distribution functions. In both the monatomic and diatomic systems, the atoms of neighboring molecules are close together. This packing is characteristic of typical liquid densities. Since the van der Waals radius, a/2, is the only characteristic length in the monoatomic system, the packing of atoms is very simple. The qualitative behavior of g ( r ) is thus easily understood by viewing the schematic picture of the monatomic fluid. In the diatomic case, the packing is more complicated as there are two length scales in this system: the chemical bond length, L , which is the sum of the covalent radii for the atoms, and a, which is twice the van der Waals radii. Notice from the figure that because the liquid is dense, it is very likely that a tagged atom (the one chosen to be at the origin) will be touching an atom associated with another molecule. Thus, just as in the monatomic case,g(r) should have a significant peak around r a. However, unlike the monatomic fluid, each atom in the diatomic system is attached to a second atom within the same molecule. Thus, for each atom within a distance Q of the tagged atom, there will be another atom within a distance u + L. This coupling of the intramolecular and the intermolecular lengths is manifested by a shoulder or a bump in g(r) at r = Q L . In addition, the interference between two lengths tends to broaden subsequent peaks in g(r) from what one expects to find for monatomic liquids. The radial distribution function drawn in Figure 1 for monatomic species is like that found for liquid argon.6 The diatomic distribution function is similar to that for liquid n i t r ~ g e n .For ~ most diatomics, the primary effects
+
Role of Various Length Scales. In a molecular fluid, as (1) For a review, see D. Chandler in 'The Liquid State of Matter: Fluids, Simple and Complex", E. W. Montroll and J. L. Lebowitz, Ed., North Holland, Amsterdam, 1982. (2)W. H. Flygare, Phil. Trans. R. SOC.London, Ser. A, 293,277(1979), and references cited therein. (3)M. R. Battaglia, T. I. Cox, and P. A. Madden, Mol. Phys., 37,1413 (1979);38,1539(1979). See also the molecular dynamics study of R. W. Impey, P. A. Madden, and D. J. Tildesley, ibid., 44, 1319 (1981). (4) D. Chandler, R. Silbey, and B. M. Ladanyi, Mol. Phys., 46, 1335 (1982). (5)D. Chandler, Faraday Dkcuss. Chem. SOC.,66,71 (1978);D.E. Sullivan and C. G. Gray, Mol. Phys., 42,443 (1981). 1
(6)L. Verlet, Phys. Reu., 166, 201 (1968). (7)J. Barojas, d. Levesque, and B. Quentrec, Phys. Reu. A , 7,1092 (1973);C. S. Hsu, D. Chandler, and L. J. Lowden, Chem. Phys., 14, 213 (1976).
0 1983 American Chemical Society
The Journal of Physlcal Chemlstty, Vol. 87, No. 12, 1983 2061
Orientational Pair Correlations in Molecular Fluids
gir)
where @(r,r’) = .si?(lr - r’l) is the intramolecular pair distribution function between two different sites a and y within the same molecule; c,,(r,r’) = c,,(lr - r’l) is a site-site direct correlation function; the parameter p is the bulk particle density (we assume a uniform one-component system, generalizations are Btraightforward); and we employ the matrix multiplication rule
g ir)
(AB),,(r,r’) =
S d r ” Aul(r,r”)Bey(r”,r’) (2.5) 1
-
0
20
Flgure 1. Radial distribution functions and reglons of a dense fluid for an atomic fluid, left, and a homonuclear diatomic fluid, right.
of the interference due to chemical bond lengths are to broaden peaks and to produce the shoulder at r = u + L. But for liquid Br2, which has a much larger bond length than N2, the shoulder is separated from the peak at r = u, and the experiments by Stanton et alSsshow a second pronounced peak at r = u + L. In some circumstances involving polyatomic species, the coupling of intramolecular and intermolecular lengths is associated with interlocking of molecules. This phenomenon gives rise to significant peaks in the appropriate radial distribution functions. An example is found in liquid carbon tetra~hloride.~ We have stressed how certain features in g(r) are evidence of the plurality of length scales which characterize a molecular fluid. These features are also evidence of orientational pair correlations. For example, if all orientations of neighboring N2 molecules were equally likely, g(r) would not look as pictured in Figure 1. Rather than having discernable features, the main peak would be diffuse indicating a simple uniform distribution of distances between u and u + L. Phenomenology. One way of incorporating the effects of chemical bonding (i.e., intramolecular arrangements of atoms) on intermolecular correlations is based upon the following generalization of the Ornstein-Zernike equation:
h = wco + poch = wc(1 - poc)-lo
(2.1)
or
x = pw + p2h = (1 - poc)-lpw
(2.2)
Here
is the equilibrium correlation function between site a at r and site y in another molecule at r’; Xay(r,r’)is the a-y density-density correlation function; its intramolecular part is given by
o,,(r,r’) = 6,,6(r - r’) + .s$+,r’)
(2.4)
(8)G.W.Stanton, J. H. Clarke, and J. C. Dore, Mol. Phys., 34, 823 (1977). , 5228 (1974); (9)L. J. Lowden and D. Chandler, J. Chem. P h y ~ .61, A. H. Narten, ibid., 65,573 (1976).
Equation 2.1 or 2.2 can serve as a definition of the direct correlation functions cay(r).This can be verified as follows: One may introduce the spacial Fourier transforms defined, for example, as
= (exp[ik.(rp) - riy))])
(2.6)
where rl’) refers to the position of the a t h site (or atom) in molecule i, and the brackets (-) denote the equilibrium ensemble average. These transforms diagonalize the space integrations in (2.1) aQd (2.2) and allow one to solve for 8(k) in terms of G ( k ) ,h(k),and 6-l (k). This last matrix is ill-defined at precisely k = 0, but away from this point everything is well-behaved. As a result, 8(k) and thus cay(r) can be defined in terms of (2.1) or (2.2). Such a definition, however, is little more than a tautology. The utility of the direct correlation function arises from its physical interpretation as some sort of solvent mediated effective pair interaction between sites in different molecules. The usual graphical representation of eq 1.1given in Figure 2 shows that this equation describes intermolecular siteaite coi relations as oeing “propagated” by singly connected chains of intramolecular pair correlations and intermolecular site-site direct correlations. We might expect, therefore, that c&) should have the same range or spacial extent as the actual intermolecular pair potential since cay(r)should embody all those mechanisms for correlations that are more than singly connected. A detailed analysis of the graphical series for h&) shows, however, that this expectation is only an approximation! Indeed, except for the linear perturbative regime, it is impossible to derive eq 2.1 from a t~pologicalreduction of the exact cluster expansion for h, (r).l0 This fact, proved in ref 4, is evident from consiaeration of graphs depicting nonlinear couplings as illustrated in Figure 3. Nevertheless, useful theories can be constructed by assuming that c&) is short ranged when the intermolecular potential is short ranged. We call such theories RISM (for reference interaction site model) theories. Included among these are the RISM equation’ and Rossky’s extended RISM equation.” The former was designed to treat nonassociated fluids while the latter has been successfully applied to polar associated liquids including water. In all these applications, one focuses attention on the qualitative nature of the local arrangements of pairs of atoms in the fluid, and it is in this regard that the theory appears successful. With respect to orientational pair correlation factors, however, the theory is not correct. These properties are associated with the long wavelength (IO) B. M. Ladanyi and D. Chandler, J. Chem. Phys., 62,4308(1975); D. Chandler and L. R. Pratt, ibid., 65,2925 (1976). (11) F.Hirata and P. J. Rossky, Chem. Phys. Lett., 83,329(1981);F. Hirata, B.M. Pettitt, and P. J. Rossky, J. Chem. Phys., 77,509 (1982); B. M.Psttitt and P. J. Rossky, ibid.,77, 1451 (1982).
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The Journal of Physical Chemistry, Vol. 87,No. 12, 1983
c-h+ Q
= Y
- + A + &
errors in the RISM theory are associated with assumption that c?,(r) is short ranged or, more precisely, the assumption that
- c@@+cilZ#?&+a Y
+
a
Y
O
Chandler and Richardson
Y
...
Figure 2. The RISM equation. The unlabeled bonds represent the cay functions, the wavy lines represent st: functions, and the two point hypervertex function represents sf: plus a single vertex. See ref 4 and references cited therein for more details.
4
where (kBp)-lis the temperature, and uay(r)is the site-site intermolecular pair potential. That being the case, one might seek corrections to the RISM theory by adding a long-ranged correction to eq 2.11 and adjusting the form of this correction in such a manner as to guarantee consistency with the exact value of the orientational pair correlation factor. Such an idea has been advocated by Cummings and Stell.13 With this approach, Cummings and Stell and, more recently, Cummings and S ~ l l i v a nshow ' ~ that in order to recover correct orientational pair correlation factors from eq 2.1, c,&) must be long ranged or even diverge at large distances. For example, when considering the case of nonpolar linear triatomic molecules (e.g., CSJ where the intermolecular interactions are believed to be short ranged, a correct prediction of g, from eq 2.1 requires
a
cay(r)
Y Figure 3. Interaction site graph that cannot be treated properly within the context of eq 2.1 with short-ranged direct correlation functions. I t depicts an atom or site a on one molecule pushing slmultaneously on two sites of another molecule (one of them is y) via a solventmediated effective interaction v a Y ( r )[i.e., vpy(r) represents a sum of (perhaps) a number of two particle irreducible graphs].
contributions to the spacial correlations. For example, consider the g, factor defined for linear symmetric molecules as
where ui is the unit vector along the axis of the ith molecule, Pz(...) refers to the second Legendre polynomial, and N refers to the total number of molecules in the system. It can be shown12 (see eq 4.2) that g, is determined by a linear combination of the moments h$ = (1/5!) l d r r4ha,(r)
(2.8)
where hi$ is the coefficient of k4 in the small wavevector expansion
-
Dayr
large r
(2.12)
where the matrix of constants D,, depends upon the value of g, and equals zero only when g, = 1. Unfortunately, without further information, it is not possible to utilize this approach to predict values of g,. Results like eq 2.12 provide only the criteria necessary for consistency between eq 2.1 and the correct long wavelength pair structure of a molecular fluid. It is perhaps unsettling to conceive of a theory for neutral nonpolar fluids for which effective interactions diverge at large distances. These strange interactions can be viewed as a consequence of the inability of eq 2.1, without approximation, to account naturally for the nonlinear couplings between molecules. This point is established in the next section with the proper integral equation theorya4The considerations lead both to an understanding of results like eq 2.12 plus the ability to predict the values of orientational pair correlation factors. 3. Proper Integral Equation By considering the topologies of graphs contributing to hay(r), one may show that it is possible to arrange the cluster series into the following form:4
where In general, the g, factor should differ from the ideal gas result, g, = 1. When the appropriate linear combination is formed in the RISM approximation, however, one finds5 (see eq 4.5) that k2IRISM
=
(2.10)
This result, while not implying that the RISM theory neglects local orientational pair correlations, does show that, in the long Wavelength regime, the RISM phenomenology fails to be correct. Cummings-Stell Approach. If we retain eq 2.1, these
(12) J. S. Herye and G. Stell, J. Chem. Phys., 66, 795 (1977); D. E. Sullivan and C. G. Gray, ref 5.
and co, cl, c,, and c b are the four topologically distinct matrices of site-site direct correlation functions. In particular, co is the sum of all the nodeless graphs for which the two roots are not s circles; c1is the sum of all nodeless graphs for which the left root is an s circle but the right is not; C? = c,; and c b is the sum of nodeless graphs for which both roots are s circles. Some of these graphs are shown in Figure 4. See ref 4 and references cited therein for more details. (13) P. T. Cummings and G . Stell, Mol. Phys., 46, 383 (1982). (14) P. T. Cummings and D. E. Sullivan, Mol. Phys., 46, 665 (1982).
The Journal of Physical Chemistry, Vol. 87, No. 12, 1983 2083
Orientational Pair Correlations in Molecular Fluids
"
Y
a
"
Y
"
We may suppose that co,cl,cr,and cb are all short ranged since this is to be expected from the graphical definitions and the physical interpretations of these functions. However, even with this assumption, eq 3.5 implies that the RISM c is long ranged. To show this, note that due to normalization, at k = 0, the elements of both w and x are independent of the indices cy and y. Further, from eq 3.1, one may deduce
Y
PQ =
a
Y
"
Y
Figure 4. Interaction site graphs for co,c,, c, and cbwith f bonds and s vertices. See ref 4 and references therein for definitions.
Equation 3.1 is called the uproper integral equation" because it follows from an exact topological reduction of the cluster series for h&). I t is in this sense that the RISM eq 2.1 is not "proper". In addition to the graphical definitions, an alternative set of definitions can be constructed by inverting the coupled Ornstein-Zernike-like equations discussed in ref 4. That is, one notes that
is the matrix of pair correlation functions for free atoms dissolved a t infinite dilution in the fluid: further
is the matrix of pair correlation functions between an atom that is part of a molecule and a free atom at infinite dilution in the fluid. Equations 3.3 and 3.4, the transpose of eq 3.4, and eq 3.1 suffice to define these direct correlation functions since by inverting these four matrix equations we can express the four direct correlation function matrices in terms of the well-defined pair correlation functions, b,hl + h,, ho + h,, and h. Note that eq 3.1 bears a striking resemblance to the RISM eq 2.1. One obtains the RISM theory by replacing fl with its low density limit Q=w
and regarding the remaining direct correlation function, co, as the RISM c. It is the correction to Q w , namely AQ, which contains cl, c, and cb and therefore describes the nonlinear couplings referred to in the previous section (see Figures 3 and 4). We shall see that AQ is responsible for the correct long wavelength description of the orientational pair correlations. This fact was recently illustrated in a study of dielectric constant of polar fluids.15 In the next section, we focus on the g2factor for nonpolar substances. Before turning to that study, we note that we already have an answer to why the RISM c must be long ranged if eq 2.1 is to be correct at small k. By comparing eq 2.1 and 3.1, one finds (eq 4.3 of ref 4) c = co - p-'(Q-' - w-')
(3.6)
from which we see that, at k = 0, the elements of Q are also independent of the indices. Thus, at k = 0, the matrices o and Q are noninvertible. That is, i2-l and 0-l both diverge as k 0. This small k divergence implies that c is long ranged unless Q = w. In other words, if we insist on retaining eq 2.1 as an exact definition of c, the nonzero AQ is responsible for c being long ranged.
-
4. Orientational Pair Correlation Factor g2
The g, factor defined in eq 2.7 plays a central role in the determination of the depolarized Rayleigh ratio, the Kerr constant, and the Cotton-Mouton constant.16 The measurement of just one of these properties, however, is not sufficient to determine g,. For example, the depolarized Rayleigh ratio, R , depends on both g, and a solvent-renormalized polarizability anisotropy. Without invoking a theory for the latter quantity, the determination of R does not fix g,. As discussed by Battaglia et al.,3however, a judicious combination of experiments can determine g, for many liquids, and they have performed such experiments. The connection between g2and the spacial pair correlations in a molecular fluid is found by considering the Fourier transforms p&Jk)
= ( ( N - 1) exp[ik.(rp) - rhy))])
(4.1)
If WE assume the intermolecular interactions are short ranged (and the system is away from a critical point), the coefficients of the small k expansion, eq 2.9, are well-behaved at least through the k4 term.17J8 One may analyze the form of these coefficients by noting that the length of the intramolecular vectors rp) - rp) are bounded, hence the exponential in eq 4.1 can be expanded in powers of these vectors dotted into k. Such expansions are straightforward though somewhat simplified in the case where molecules are assumed to be rigid. In that case one finds for linear symmetric molecules12
where 1 is the intramolecular distance between nonequivalent sites 1 t n d 2 within a molecule, and h(0) = h,,(O). [Recall that h J k ) is independent of a and y at k = 0.1 Analogous expressions for flexible nonpolar molecules can be obtained though l4 is then (lr!') - ri2)14). The linear combination of correlation functions in eq 4.2 is similar in form to that encountered in the study of the dielectric constant."*18 Since there exists15J7a compact expression relating the dielectric constant to the hyper-
(3.5)
(15) D. Chandler, C. G. Joslin, and J. M. Deutch, Mol. Phys., 47,871
(1982).
x(1 - cox)-'
Y
(16) A. D. Buckingham, Discuss. Faraday SOC.,43, 205 (1967), and references cited therein. (17) D. Chandler, J. Chem. Phys., 67,1113 (1977). e G. Stell, J. Chem. Phys., 65,18 (1976). (18) J. S. H ~ y and
J. Phys. Chem. 1903, 87, 2064-2072
2064
vertex function, 9,we seek a similar expression for g,. This is done by casting eq 4.2 in the form gz -- 1 = ( l / p ) T r
*d4)- ( 5 / 4 p ) g ( O ) + 2
(4.3)
g,
1 = Tr
- (5/4)Ah(O)
(4.5)
is defined through the expansion
where
A6
-
=
6 ( k ) - &(k)
= Ah(0)
+ k2A9(*)+ k4A9(4)+ ... (4.6)
where
* =(45,141[;l
R
--1 0
..]
i.e. 14.1)
is analogous to the @ matrix employed in the dielectrics study,15J7x ( ~=) p d 4 ) p2h(4)is the k 4 coefficient in the small k expansion of x, and g(0) = p + p2h(0). One may express x ( ~and ) ?(O) in terms of Q though eq 3.1, and several symmetry properties of the x matrix can be used to deduce analogous properties of 9 through the use of eq 3.6. From these manipulations, and the assumption that the site-site direct correlation functions are short ranged, one findsIg
+
AQL?,! = (1/5!)1dr r4AR,,(r)
(4.7)
and A8,,(0) = Ah(0). [Recall that &,,(k) and 8,,(k) are independent of a and y a t k = 0.1 Equation 4.5 is a new result for the g2 factor. From this formula, it is obvious why the RISM theory with A 9 = 0 yields the ideal gas result g, = 1. Further, the formula provides a straightforward route to calculations of g2 from the interaction site formalism. Such calculations will be reported (19)D. M. Richardson and D. Chandler, to be published.
Rotational Spectroscopy and the Properties of Hydrogen-Bonded Dimers B H A A. C. Legon Christopher Ingold Laboratories, Department of Chemistry, University College London, London WC 1H OAJ, United Kingdom (Received: November 5, 1982)
The properties of hydrogen-bonded dimers B-HA that are available from investigations of their rotational spectra are discussed by reference to selected examples. The rotational spectra referred to have been observed by using two techniques: Stark-modulation microwave spectroscopy and pulsed-nozzle,Fourier-transform microwave spectroscopy. The second of these techniques has been recently introduced by Flygare and is described briefly. Among the dimer properties considered are molecular geometry and symmetry, various details of the hydrogen-bond stretching and bending potential energy functions (such as vibrational energy separations, barriers to inversion, and dissociation energy), and indicators of the electric charge redistribution that accompanies formation of B-HA (such as the electric dipole moment and the electric field gradients at certain nuclei). Included among examples of species Ba-HA referred to are (PH,.HX), (H,O.HF), (N2.HF)( H m H - H C l )and (HCN-HF).
1. Introduction
The hydrogen bond is ubiquitous. Its properties have profound consequences throughout the physical and biological sciences. It is, for example, a commonplace that the nature of the physical world would be very different in the absence of a hydrogen bond between water molecules. To determine the properties of the hydrogen bond interaction is, therefore, a matter of importance. But this interaction is generally a weak one and can be greatly perturbed by lattice or solvent effects in solid or liquid phases, respectively. As a consequence, a detailed investigation of the intrinsic properties of this weak binding refers preferably to the gas phase at low pressure. Two types of hydrogen bond can be recognized: intraand intermolecular. The advantage of considering the second type in isolation is that the properties of the separated component molecules are often well-known and then the extent to which they become modified on dimer formation can be appraised. This article is accordingly 0022-3654/83/2087-2064$0 1.50/0
TABLE I : Properties of Hydrogen-Bonded Dimers B...HA Available from Rotational Spectroscopy molecular s y m m e t r y a n d geometry details of t h e hydrogen-bond potential energy function: stretching (e.g., force constants, dissociation energy) bending (e.g., barriers to inversion) details of t h e electric charge redistribution o n formation of B . . . H A (e.g., electric dipole m o m e n t s , electric field gradients)
restricted to a discussion of intermolecular hydrogenbonded dimers B--HA at low pressure in the gas phase. The properties that we seek for dimers B-HA are summarized in Table I. The molecular symmetry and geometry obviously tell of the site in B to which the hydrogen atom of HA is bound. The details of the hydrogen bond stretching and bending potential energy functions give, through force constants, the energy required for small displacements along the coordinates that describe stretching and bending of the bond or, through the dis0 1983 American Chemical Society
