A molecular site-site integral equation that yields the dielectric constant

Abstract

Our recent derivation [K. M. Dyer et al., J. Chem. Phys. 127, 194506 (2007)] of a diagrammatically proper, site-site, integral equation theory using molecular angular expansions is extended to polar fluids. With the addition of atomic site charges we take advantage of the formal long-ranged potential field cancellations before renormalization to generate a set of numerically stable equations. Results for calculations in a minimal (spherical) angular basis set are presented for the radial distribution function, the first dipolar (110) projection, and the dielectric constant for two model diatomic systems. All results, when compared to experiment and simulation, are a significant quantitative and qualitative improvement over previous site-site theories. More importantly, the dielectric constant is not trivial and close to simulation and experiment.

INTRODUCTION

A significant challenge for any theory of the liquid state is to correctly describe the equilibrium structure and thermodynamics of polar fluids, especially the static dielectric constant.^{1, 2, 3} The situation for site-site model fluids is particularly intriguing,^{4, 5, 6} especially considering that the challenge has, in general, only been met successfully for model fluids that can be described by single-center multipole expansions.⁷ In contrast, the application of the interaction site model^{8, 9} (ISM) class of integral equation theories has been, to date, at best only modestly successful in comparison,^{10, 11} many modifications being only ad hoc patches to provide improvement in the asymptotic behavior. Further, as with the structural results,^{12, 13, 14} although the diagrammatically proper ISM theory is known to be formally capable of predicting the dielectric constant,¹⁵ a general quantitative implementation for polar fluids has not been demonstrated yet.^{13, 16}

Recently, for apolar fluids,^{17, 18} we have shown that a new diagrammatically proper⁹ class of theories can be derived from the molecular Ornstein–Zernike(OZ) theory^{3, 19} in full angular generality. The resulting molecular integral equation retains full mutual angular information about the chemical bonds of the intramolecular geometry in the pair correlations. The result is both a formally complete theory for site-site model integral equations, in the sense of the molecular OZ formalism of Blum,²⁰ and a means of formulating approximate numerical solutions that are a significant improvement over earlier efforts. The essential step¹⁷ comes from recognizing that the diagrammatically proper chains, including the intramolecular distribution function, can be used in an Allnatt-style renormalization²¹ of the intermolecular potential in the closure after certain long-ranged terms are canceled from the angular average of the potential. Earlier efforts with a similar goal have been successful at extracting some thermodynamic integrals^{22, 23} and reduced-basis information from reference ISM (RISM) type theories.^{24, 25, 26} The present site-site molecular integral-equation theory is constructed to be equivalent in the complete basis to a full molecular theory.¹⁹ Here we extend the method to polar diatomic model fluids.

The long-ranged nature of the Coulomb potential terms necessary for describing polar fluids analytically changes the nature of most theories of the liquid state. While a simple resummation for RISM theories results in numerically tractable equations, the dielectric constant remains trivial.^{4, 5} Similarly, the proper ISM theories allow a solution when a subset of diagrams are renormalized¹⁵ but that class of theories is found to be practically nonrenormalizable when considering nontrivial dielectric properties.^{13, 16}

In extending our angularly expanded, site-site molecular integral equation theory to polar systems, we find that the Coulomb potential for charge-neutral systems can be treated by considering long-ranged cancellations of the full site-site potential in the resulting molecular correlation functions. We derive the resulting equations by first using charge neutrality or more generally the spherical harmonic addition theorem to cancel the site-site Coulomb functions at long range, and then find a remainder that may be conveniently resummed in a subset of diagrams.^{21, 27, 28} The resulting equations for the molecular pair correlation functions, g(r_ij,Ω₁,Ω₂), retain the full angular as well as radial dependence and may be reduced to lesser dimensionality readily by integration.

Here, we present results using a minimal (spherical) basis-set approximation for the intermediate angular average in the convolution of the site-site radial distribution functions. We obtain nontrivial results for the first dipolar projection of the correlation functions and the predicted dielectric constant in the liquid state for the polar site-site diatomic model systems considered. All results are shown to be significant improvements over previous theories. We demonstrate conclusively that this molecular site-site integral equation theory, even using only the minimal angular basis set, is essentially quantitatively predictive of the dielectric constant of polar systems.

THEORY

We review the equations for our molecular liquid state theory in the case of an arbitrary diatomic fluid¹⁷ and then extend it to long-ranged potentials. The fundamental equations, though quite involved in detail, can be derived in a quite simple manner. We combine the two separate integral equation routes used in the study of molecular fluids, those of Andersen and Chandler²⁹ and Blum and Torruella,²⁰ through the diagrammatically proper integral equations of Chandler et al.⁹

By recognizing that the diagrammatic series³⁰ of the ISM integral equation theories are also fully expandable in the full set of molecular coordinates necessary to the Blum–Torruella formalism, the requirement that these systems be formally equivalent leads us immediately to the realization¹⁷ that the proper interaction site OZ integral equation, when combined with a molecular convolution term which is a simple bridge function in the basic site-site theory,⁸ is simply a particular diagrammatic resummation of the full molecular series.³ Resummation or renormalization in studies of the liquid state is a subject due originally to Allnat,²¹ who introduced the method in order to deal more or less generally with potential functions, which introduce mathematical poles into the functions of the liquid state, whether in real space or Fourier space. Once we recognize the particular route to achieve equivalence of the ISM and molecular fluid theories, the resulting expansions are both formally complete and quantitatively predictive to this point of simple test cases,¹⁷ including application to intramolecular conformations.¹⁸

For site-site models, if we choose an origin for arbitrary sites i and j on molecules 1 and 2, respectively, then the four distinct terms in the molecular potential between diatomic molecules are given by

$$\beta U({\mathbf{r}}_{ij},{\Omega }_1,{\Omega }_2)=\beta U_{ij}\left(r_{ij}\right)+\beta U_{jj}\left(\mid {\mathbf{r}}_{ij}-{\mathbf{l}}_1\mid \right)+\beta U_{ii}\left(\mid {\mathbf{r}}_{ij}+{\mathbf{l}}_2\mid \right)+\beta U_{ji}\left(\mid {\mathbf{r}}_{ij}-{\mathbf{l}}_1+{\mathbf{l}}_2\mid \right)=\beta U^o\left(r_{ij}\right)+\beta U^l({\mathbf{r}}_{ij},{\Omega }_1)+\beta U^r({\mathbf{r}}_{ij},{\Omega }_2)+\beta U^b({\mathbf{r}}_{ij},{\Omega }_1,{\Omega }_2),$$ (1)

where β=1∕k_BT, k_B is the Boltzmann constant, T is the absolute temperature, l₁ and l₂ are the bond displacement vectors between sites on molecules 1 and 2, respectively, r_ij is the displacement vector between site i on molecule 1 and site j on molecule 2, and Ω₁ and Ω₂ are the orientations of molecules 1 and 2, respectively. For example, the form U_ji(∣r_ij−l₁+l₂∣), illustrated in Fig. 1, refers to the potential between sites j,i on molecules 1, 2 as measured in the coordinate system defined by sites i,j on molecules 1 and 2, respectively. In the last line, we follow our previous practice^{17, 18} and label the different functions in βUnone, left, right, both (o,l,r,b) in our extension of the convention of Chandleret al.,⁹ according to the spatial and angular dependence of the different functional forms in the sum.

Figure 1.

A graphic illustrating the coordinate system used in this work, for the r_ji=r_ij−l₁+l₂ vector displacement needed for the both potential term, U(r_ij,Ω₁,Ω₂)=U_ji(∣r_ij−l₁+l₂∣). The dashed line is the vector displacement r_ij between sites i,j; the solid line is r_ji, and the axis denote an arbitrary origin coincident with site i, which the l vectors are rotated with respect to. Please note the sign convention implicit in the subscript order. That is, i,j has the opposite directionality to j,i.

Given the potential form, the molecular pair distribution function can be defined^{3, 19} by the exact closure relation,

$$g({\mathbf{r}}_{ij},{\Omega }_1,{\Omega }_2)=\exp (-\beta U({\mathbf{r}}_{ij},{\Omega }_1,{\Omega }_2)+t({\mathbf{r}}_{ij},{\Omega }_1,{\Omega }_2)+B({\mathbf{r}}_{ij},{\Omega }_1,{\Omega }_2)),$$ (2)

where B(r_ij,Ω₁,Ω₂) is the molecular bridge function,^{3, 31}t(r_ij,Ω₁,Ω₂)=g(r_ij,Ω₁,Ω₂)−c(r_ij,Ω₁,Ω₂)−1=h−c is the indirect correlation function, h=g−1 is the pair correlation function, and c is the direct correlation function, defined for an arbitrary coordinate system by the molecular (OZ) (Ref. ³²) equation in Fourier space,

$$h(\mathbf{k},{\Omega }_1,{\Omega }_2)=c(\mathbf{k},{\Omega }_1,{\Omega }_2)+{⟨c(\mathbf{k},{\Omega }_1,{\Omega }_3)\rho h(\mathbf{k},{\Omega }_3,{\Omega }_2)⟩}_{{\Omega }_3},$$ (3)

where h(k,Ω₁,Ω₂) is the Fourier transform of h(r,Ω₁,Ω₂) similarly for c, and the brackets refer to the spherical average over the orientation of molecule 3. If for each site-site coordinate pair i and j in the potential function, we expand g(r_ij,Ω₁,Ω₂) in the rotationally invariant basis set of Blum and Torruella,²⁰ then, as shown previously,¹⁷ the closure is an exponential of a site-site pairwise sum. Since the exponential of a sum is the product of exponentials, then using a Mayer expansion³³ of the site-site functions, the closure can be topologically resummed such that in the minimal (000) basis set the radially symmetric pair functions are defined as

$$c^o\left(r_{ij}\right)=\exp (-\beta U^o\left(r_{ij}\right)+t^o\left(r_{ij}\right)+B^o\left(r_{ij}\right))-t^o\left(r_{ij}\right)-1=g^o\left(r_{ij}\right)-t^o\left(r_{ij}\right)-1,$$ (4)

$${\overline{c}}^{l;000}\left(r_{ij}\right)=g^o\left(r_{ij}\right){⟨\exp (-\beta {\overline{U}}^l({\mathbf{r}}_{ij},{\Omega }_1)+B^l({\mathbf{r}}_{ij},{\Omega }_1))⟩}_{{\Omega }_1}\exp \left({\tau }^{l;000}\left(r_{ij}\right)\right)-{\tau }^{l;000}\left(r_{ij}\right),$$

$${\overline{c}}^{r;000}\left(r_{ij}\right)=g^o\left(r_{ij}\right){⟨\exp (-\beta {\overline{U}}^r({\mathbf{r}}_{ij},{\Omega }_2)+B^r({\mathbf{r}}_{ij},{\Omega }_2))⟩}_{{\Omega }_2}\exp \left({\tau }^{r;000}\left(r_{ij}\right)\right)-{\tau }^{r;000}\left(r_{ij}\right),$$

$${\overline{c}}^{b;000}\left(r_{ij}\right)=g^o\left(r_{ij}\right)[{⟨\exp (-\beta {\overline{U}}^b({\mathbf{r}}_{ij},{\Omega }_1,{\Omega }_2)-\beta {\overline{U}}^l({\mathbf{r}}_{ij},{\Omega }_1)-\beta {\overline{U}}^r({\mathbf{r}}_{ij},{\Omega }_2)+B^b({\mathbf{r}}_{ij},{\Omega }_1,{\Omega }_2)+B^l({\mathbf{r}}_{ij},{\Omega }_1)+B^r({\mathbf{r}}_{ij},{\Omega }_2))⟩}_{{\Omega }_1,{\Omega }_2}\exp ({\tau }^{b;000}\left(r_{ij}\right)+{\tau }^{l;000}\left(r_{ij}\right)+{\tau }^{r;000}\left(r_{ij}\right))-{⟨\exp (-\beta {\overline{U}}^l({\mathbf{r}}_{ij},{\Omega }_1)+B^l({\mathbf{r}}_{ij},{\Omega }_1))⟩}_{{\Omega }_1}\exp \left({\tau }^{l;000}\left(r_{ij}\right)\right)-{⟨\exp (-\beta {\overline{U}}^r({\mathbf{r}}_{ij},{\Omega }_2)+B^r({\mathbf{r}}_{ij},{\Omega }_2))⟩}_{{\Omega }_2}\exp \left({\tau }^{r;000}\left(r_{ij}\right)\right)+1]-{\tau }^{b;000}\left(r_{ij}\right),$$

where there is no ambiguity by dropping the angular indices for the none radial functions such as c^o(r_ij). The $\overline{U}$ functions and the τ functions are renormalized²¹ analogs of the potential U and indirect correlation functions t, defined by the short-ranged radial and angular resummations

$$\beta {\overline{U}}^l({\mathbf{r}}_{ij},{\Omega }_1)=\beta U^l({\mathbf{r}}_{ij},{\Omega }_1)-t_{jj}^o\left(\mid {\mathbf{r}}_{ij}-{\mathbf{l}}_1\mid \right),$$ (5)

$$\beta {\overline{U}}^r({\mathbf{r}}_{ij},{\Omega }_2)=\beta U^r({\mathbf{r}}_{ij},{\Omega }_2)-t_{ii}^o\left(\mid {\mathbf{r}}_{ij}+{\mathbf{l}}_2\mid \right),$$

$$\beta {\overline{U}}^b({\mathbf{r}}_{ij},{\Omega }_1,{\Omega }_2)=\beta U^b({\mathbf{r}}_{ij},{\Omega }_1,{\Omega }_2)-t_{ji}^o\left(\mid {\mathbf{r}}_{ij}-{\mathbf{l}}_1+{\mathbf{l}}_2\mid \right)-{\tau }_{jj}^l(\mid {\mathbf{r}}_{ij}-{\mathbf{l}}_1\mid ,{\Omega }_1,{\Omega }_2)-{\tau }_{ii}^r(\mid {\mathbf{r}}_{ij}+{\mathbf{l}}_2\mid ,{\Omega }_1,{\Omega }_2),$$

where

$${\tau }^l({\mathbf{r}}_{ij},{\Omega }_1)=t^l({\mathbf{r}}_{ij},{\Omega }_1)-{\varphi }^l({\mathbf{r}}_{ij},{\Omega }_1),$$ (6)

$${\tau }^r({\mathbf{r}}_{ij},{\Omega }_2)=t^r({\mathbf{r}}_{ij},{\Omega }_2)-{\varphi }^r({\mathbf{r}}_{ij},{\Omega }_2),$$

$${\tau }^b({\mathbf{r}}_{ij},{\Omega }_1,{\Omega }_2)=t^b({\mathbf{r}}_{ij},{\Omega }_1,{\Omega }_2)-{\varphi }^b({\mathbf{r}}_{ij},{\Omega }_1,{\Omega }_2),$$

and the ϕ(r,Ω₁,Ω₂) functions are defined by

$${\varphi }^l(r_{ij},{\Omega }_1)\equiv \exp (-\beta {\overline{U}}^l({\mathbf{r}}_{ij},{\Omega }_1))-1,$$ (7)

$${\varphi }^r(r_{ij},{\Omega }_2)\equiv \exp (-\beta {\overline{U}}^r({\mathbf{r}}_{ij},{\Omega }_2))-1,$$

$${\varphi }^b(r_{ij},{\Omega }_1,{\Omega }_2)\equiv \exp (-\beta {\overline{U}}^l({\mathbf{r}}_{ij},{\Omega }_1)-\beta {\overline{U}}^r({\mathbf{r}}_{ij},{\Omega }_2)-\beta {\overline{U}}^b({\mathbf{r}}_{ij},{\Omega }_1,{\Omega }_2))-\exp (-\beta {\overline{U}}^l({\mathbf{r}}_{ij},{\Omega }_1)-\beta {\overline{U}}^r({\mathbf{r}}_{ij},{\Omega }_2))-\left[(\exp (-\beta {\overline{U}}^l({\mathbf{r}}_{ij},{\Omega }_1))-1)(\exp (-\beta {\overline{U}}^r({\mathbf{r}}_{ij},{\Omega }_2))-1)(\exp (-\beta {\overline{U}}^b({\mathbf{r}}_{ij},{\Omega }_1,{\Omega }_2))-1)\right].$$

Thus, the ϕ functions are like the Mayer f-bond for the effective potential constructed site pairwise from the bare molecular potential less the contributions from those terms in the indirect correlation functions, which are simple coordinate transformations of the corresponding none, left, or right term. Note that, as with the closure as explicitly given in Eq. 4, the first order τ⁰⁰⁰(r) and ϕ⁰⁰⁰(r) functions are simply the normalized angular averages of the preceding equations.

Finally, to first order in the rotationally invariant basis set, the generating integral equation is a diagrammatically proper OZ equation,

$$\mathbf{H}=\mathbf{\chi }+\mathbf{S}+(\mathbf{\chi }+\mathbf{S})\mathbf{\rho }\mathbf{H},$$ (8)

$$\mathbf{\chi }=\overline{\mathbf{c}}-\mathbf{\Phi }+(\overline{\mathbf{c}}-\mathbf{\Phi }){\mathbf{\rho }}^{\prime }\mathbf{\chi },$$

where χ is the set of full molecular convolution products to first order in density, all matrices are the familiar species labeled o,l,r,b block matrices;¹² the elements of $\overline{\mathbf{c}}$ and Φ are the respective Fourier transforms of the ${\overline{c}}^{000}\left(r_{ij}\right)$ and ϕ⁰⁰⁰(r_ij) functions; the elements of S are the usual intramolecular distribution functions taken here to be simply rigid constraints;^{8, 34} and the ρ and ρ^′ matrices are block diagonal,

$${\mathbf{\rho }}_{ii}=\left(\begin{array}{cc}\rho & \rho \\ \rho & 0\end{array}\right),$$ (9)

$${\mathbf{\rho }}_{ii}^{\prime }=\left(\begin{array}{cc}0& 0\\ 0& \rho ∕2\end{array}\right),$$

zero otherwise. The scalar ρ is the molecular number density of the fluid, and the factor of 1∕2 in ρ^′ is the coordinate system degeneracy of diatomic models.¹⁷ The generalization to reactive fluids is straightforward,¹⁷ though not considered here. The reader will note that Eq. (8) appears in Eq. (39) of our previous paper¹⁷ in its algebraic solved form; here we write it in the unsolved form.

The equations to this point have been derived without reference to the form of the site-site potential. We now deal with the atomic partial charge potential for polar (and charged) systems which has historically been a nontrivial issue^{13, 16, 35, 36} due to the long-ranged behavior of the 1∕r functional form. In this molecular system of equations, however, as is the case with multipolar expansions,^{7, 37, 38} dealing numerically with the Coulomb potential of site-site polar molecules can be made straightforward. We take advantage of the formal cancellation of the aggregate sitewise sum of the long-ranged tails of the charge-charge functions in a neutral system. The interested reader will note that, due to our use of diatomic molecules, this is a greatly simplified version of the more general discussion in Ref. 19, particularly Secs. II C and II D therein, and is discussed as well for the point-charge∕dipole system in Chap. 12 of Ref. 39. For example, for the Coulomb potential U^C of a polar diatomic system, we have

$$\beta U^c({\mathbf{r}}_{ij},{\Omega }_1,{\Omega }_2)=\frac{\beta q_i{q}_j}{r_{ij}}+\frac{\beta q_j{q}_j}{\left(\mid {\mathbf{r}}_{ij}-{\mathbf{l}}_1\mid \right)}+\frac{\beta q_i{q}_i}{\left(\mid {\mathbf{r}}_{ij}+{\mathbf{l}}_2\mid \right)}+\frac{\beta q_j{q}_i}{\left(\mid {\mathbf{r}}_{ij}-{\mathbf{l}}_1+{\mathbf{l}}_2\mid \right)},$$ (10)

where for the immediate discussion, we adopt the sign convention that q_i=−q_j for i≠j.

Following the standard Legendre expansions^{19, 39} with r_ij>l₁+l₂, for charge-neutral molecules, ∑_iq_i=0, the left, right, and both terms of the Coulomb potential expand as

$$\frac{\beta q_j{q}_j}{\left(\mid {\mathbf{r}}_{ij}-{\mathbf{l}}_1\mid \right)}=\frac{\beta q_j{q}_j}{r_{ij}}\sum _{n=0}^{\infty }{\left(\frac{l_1}{r_{ij}}\right)}^n{P}_n\left(\cos {\theta }_1\right),$$ (11)

$$\frac{\beta q_i{q}_i}{\left(\mid {\mathbf{r}}_{ij}+{\mathbf{l}}_2\mid \right)}=\frac{\beta q_i{q}_i}{r_{ij}}\sum _{n=0}^{\infty }{(-1)}^n{\left(\frac{l_2}{r_{ij}}\right)}^n{P}_n\left(\cos {\theta }_2\right),$$

$$\frac{\beta q_j{q}_i}{\left(\mid {\mathbf{r}}_{ij}-{\mathbf{l}}_1+{\mathbf{l}}_2\mid \right)}=\frac{\beta q_j{q}_i}{r_{ij}}\sum _{n=0}^{\infty }\sum _{m=0}^{\infty }\sum _l\left(\frac{4\pi }{(2n+1)(2m+1)}\right){(-1)}^m{\left(\frac{l_1}{r_{ij}}\right)}^n{\left(\frac{l_2}{r_{ij}}\right)}^m{Y}_n^l(\cos {\theta }_1,{\varphi }_1)Y_m^{-l}(\cos {\theta }_2,{\varphi }_2),$$

respectively, where the sum over l in the last equation is restricted to the lesser of −m⩽l⩽m or −n⩽l⩽n in the usual way,^{19, 39} the P_n(cos θ) are the Legendre polynomials, and the $Y_m^l(\cos \theta ,\varphi )={(-1)}^m\sqrt{(2m+1)∕4\pi (m-l)\text{!}∕(m+l)\text{!}}{P}_m^l\left(\cos \theta \right)\exp \left(il\varphi \right)$ are the standard spherical harmonics for $P_m^l\left(\cos \theta \right)$ the associated Legendre polynomials. Here, (θ₁,θ₂,ϕ₁₂)=(Ω₁,Ω₂) are the relative molecular coordinates, with θ₁ the angle between l₁ and r_ij, θ₂ the angle between l₂ and r_ij, and ϕ₁₂=ϕ₁−ϕ₂ the dihedral angle between l₁ and l₂. This is, of course, the general Legendre expansion from which the familiar multipole expansion can be derived.^{19, 39, 40}

As usually seen term-by-term in the derivation of the multipole expansion, the general expansions, when added together, cancel the none, left, and right terms completely. The cancellation of the none terms is simply the charge-neutral sum of the first terms in each series,

$$\frac{\beta q_i{q}_i}{r_{ij}}+\frac{\beta q_i{q}_j}{r_{ij}}+\frac{\beta q_j{q}_j}{r_{ij}}+\frac{\beta q_j{q}_i}{r_{ij}}=0,$$ (12)

due to q_i=−q_j. To see the cancellation for the left and right series, we first take separately the right and left averages of the both term above. That is, due to orthogonality, by alternately setting m and n to 0, we average separately over the orientations of the right and left molecules in the both potential, which, combined with the standard normalization of the spherical harmonics, reduces to

$${⟨\frac{\beta q_j{q}_i}{\left(\mid {\mathbf{r}}_{ij}-{\mathbf{l}}_1+{\mathbf{l}}_2\mid \right)}⟩}_{{\Omega }_1}=\frac{\beta q_j{q}_i}{r_{ij}}\sum _{m=0}^{\infty }{(-1)}^m{\left(\frac{l_2}{r_{ij}}\right)}^m{P}_m\left(\cos {\theta }_2\right),$$ (13)

$${⟨\frac{\beta q_j{q}_i}{\left(\mid {\mathbf{r}}_{ij}-{\mathbf{l}}_1+{\mathbf{l}}_2\mid \right)}⟩}_{{\Omega }_2}=\frac{\beta q_j{q}_i}{r_{ij}}\sum _{n=0}^{\infty }{\left(\frac{l_1}{r_{ij}}\right)}^n{P}_n\left(\cos {\theta }_1\right).$$

Notice that, since q_i=−q_j, these series are equal and opposite in sign to the right and left Coulomb potential terms, respectively. If we now pull these separate partial sums through the complete both series,

$$\frac{\beta q_j{q}_i}{\left(\mid {\mathbf{r}}_{ij}-{\mathbf{l}}_1+{\mathbf{l}}_2\mid \right)}=\frac{\beta q_j{q}_i}{r_{ij}}+\frac{\beta q_j{q}_i}{r_{ij}}\sum _{m=1}^{\infty }{(-1)}^m{\left(\frac{l_2}{r_{ij}}\right)}^m{P}_m\left(\cos {\theta }_2\right)+\frac{\beta q_j{q}_i}{r_{ij}}\sum _{n=1}^{\infty }{\left(\frac{l_1}{r_{ij}}\right)}^n{P}_n\left(\cos {\theta }_1\right)+\sum _{n=1}^{\infty }\sum _{m=1}^{\infty }\sum _l\left(\frac{4\pi }{(2n+1)(2m+1)}\right){(-1)}^m{\left(\frac{l_1}{r_{ij}}\right)}^n{\left(\frac{l_2}{r_{ij}}\right)}^m{Y}_n^l(\cos {\theta }_1,{\varphi }_1)Y_m^{-l}(\cos {\theta }_2,{\varphi }_2),$$ (14)

where the series here begin at m, n=1, the cancellation of each term with the corresponding oppositely charged left and right terms is obvious, and we have that

$$\beta U^c({\mathbf{r}}_{ij},{\Omega }_1,{\Omega }_2)=\frac{\beta q_j{q}_i}{r_{ij}}\sum _{n=1}^{\infty }\sum _{m=1}^{\infty }\sum _l\left(\frac{4\pi }{(2n+1)(2m+1)}\right){(-1)}^m{\left(\frac{l_1}{r_{ij}}\right)}^n{\left(\frac{l_2}{r_{ij}}\right)}^m{Y}_n^l(\cos {\theta }_1,{\varphi }_1)Y_m^{-l}(\cos {\theta }_2,{\varphi }_2),$$ (15)

where m,n both start now at 1. This result has two practical consequences that we exploit formally.

First, in the none, left, right, both conventions, the complete molecular Coulomb potential is only, and identically, a both term. Thus, taken as a sum, there is no need to separate the Coulomb term sitewise as we do the short-ranged potential terms. The practical result is that the Coulomb potential appears uncanceled in only the both terms, in a single part of the closure, such that the closure equations for the none, left, and right radial site-site direct correlation functions contain only the short-ranged or non-Coulomb potential terms, and thus remain unchanged from the above. The only term in which the molecular Coulomb potential then still remains is both term of the site-renormalized direct correlation function. The ${\overline{c}}^{b;000}$ term becomes [for the molecular hypernetted-chain (HNC) approximation]

$${\overline{c}}_{ij}^{b;000}\left(r\right)=g_{ij}^o\left(r\right)[{⟨\exp (-\beta {\overline{U}}^b({\mathbf{r}}_{ij},{\Omega }_1,{\Omega }_2)-\beta U^c({\mathbf{r}}_{ij},{\Omega }_1,{\Omega }_2)-\beta {\overline{U}}^l({\mathbf{r}}_{ij},{\Omega }_1)-\beta {\overline{U}}^r({\mathbf{r}}_{ij},{\Omega }_2)+{\tau }^{b;000}\left(r_{ij}\right)+{\tau }^{l;000}\left(r_{ij}\right)+{\tau }^{r;000}\left(r_{ij}\right))⟩}_{{\Omega }_1,{\Omega }_2}-{⟨\exp (-\beta {\overline{U}}^l({\mathbf{r}}_{ij},{\Omega }_1)+{\tau }^{l;000}\left(r_{ij}\right))⟩}_{{\Omega }_1}-{⟨\exp (-\beta {\overline{U}}^r({\mathbf{r}}_{ij},{\Omega }_2)+{\tau }^{r;000}\left(r_{ij}\right))⟩}_{{\Omega }_2}+1]-{\tau }^{b;000}\left(r_{ij}\right),$$ (16)

where we use the closed form, vector function of βU^c given in Eq. 10 and, since there are no uncanceled charge-charge potential terms in the none, left, and right closures, ϕ^b(r_ij,Ω₁,Ω₂) remains unchanged from Eq. 7, and does not include the full Coulomb divergence.

Second, we may now follow standard methods for charged systems^{21, 27, 28} and resum the asymptotic form of the remaining site-site Coulomb potential

$$\beta U_s^c({\mathbf{r}}_{ij},{\Omega }_1,{\Omega }_2)=\frac{\beta q_i{q}_j(1-\mathrm{erf}\left(\alpha r_{ij}\right))}{r_{ij}}+\frac{\beta q_j{q}_j(1-\mathrm{erf}\left(\alpha \mid {\mathbf{r}}_{ij}-{\mathbf{l}}_1\mid \right))}{\left(\mid {\mathbf{r}}_{ij}-{\mathbf{l}}_1\mid \right)}+\frac{\beta q_i{q}_i(1-\mathrm{erf}\left(\alpha \mid {\mathbf{r}}_{ij}+{\mathbf{l}}_2\mid \right))}{\left(\mid {\mathbf{r}}_{ij}+{\mathbf{l}}_2\mid \right)}+\frac{\beta q_j{q}_i(1-\mathrm{erf}\left(\alpha \mid {\mathbf{r}}_{ij}-{\mathbf{l}}_1+{\mathbf{l}}_2\mid \right))}{\left(\mid {\mathbf{r}}_{ij}-{\mathbf{l}}_1+{\mathbf{l}}_2\mid \right)}=\beta U^c({\mathbf{r}}_{ij},{\Omega }_1,{\Omega }_2)-\psi ({\mathbf{r}}_{ij},{\Omega }_1,{\Omega }_2),$$ (17)

where erf(αr) is the error function and ψ(r_ij,Ω₁,Ω₂) are the Coulomb resummation functions with Fourier transforms, ψ(k_ij,Ω₁,Ω₂), given by

$$\psi ({\mathbf{k}}_{ij},{\Omega }_1,{\Omega }_2)=4\pi \frac{q_i{q}_j{e}^{-k^2∕\left(4{\alpha }^2\right)}}{k^2}(1-e^{-i{\mathbf{k}}_{ij}\cdot {\mathbf{l}}_1})(1-e^{-i{\mathbf{k}}_{ij}\cdot {\mathbf{l}}_2}).$$ (18)

For the Fourier transform functions, k_ij here does not refer to numerical elements of k. Rather, just as r_ij refers to the particular coordinate system between sites i and j in real, r-space, k_ij refers to the equivalent coordinate system in reciprocal, k-space.¹⁹

We note here that the chemical bonds (formally infinitely long ranged) have a direct interplay in this resummation and have not relinquished their directionality in the above equation. As an example, truncating our intermediate convolutions at the 000 basis set level, the only terms (which are both functions) required of the Fourier transform of ψ are

$${\psi }^{b;000}\left(k\right)=4\pi \frac{q_i{q}_j{e}^{-k^2∕\left(4{\alpha }^2\right)}{(1-\sin \left(kl\right)∕\left(kl\right))}^2}{k^2}.$$ (19)

Finally, the χ term in the OZ equations becomes

$$\mathbf{\chi }=\overline{\mathbf{c}}-\mathbf{\Phi }-\mathbf{\psi }+(\overline{\mathbf{c}}-\mathbf{\Phi }-\mathbf{\psi }){\mathbf{\rho }}^{\prime }\mathbf{\chi },$$ (20)

the first radial term in the full molecular convolution chain sum⁴¹ to all orders in the m,n,l projections.

There are two final considerations. First, the convergence parameter in the error functions α has been found to be best for atomic fluids²⁷ when α≈1. For molecular fluids, this value would actually scale the functions inside the molecular contact distance, r≈2l. Since we require only that the long-ranged tails of the function be resummed, we use instead the value α≈1∕2l. This value leads to stable numerical behavior of the equations for all models and phase points investigated in this work. We note that the grid truncations required for finite numerical work must include considerations of the molecular geometry (or bond length) to avoid apparent imbalances in the molecular cancellation. This is clearly related to the questions of potential truncation by site or by molecule seen in integral equations and simulations some time ago.⁴² Finally, as for the apolar fluids, by restraining the Coulomb potential exclusively to the both terms as above, we completely avoid the question of auxiliary sites with short-ranged shields on various charges¹⁶ by construction, and thus there is no need to introduce false molecular potential terms^{13, 36} to avoid catastrophic collapse due to the pole in the potential functions. Below, we demonstrate the results of the theory for two polar diatomic systems.

RESULTS

We investigate two polar diatomic models here: the polar version of the HCl model used previously^{17, 36} and a polar model^{43, 44} with the geometry of N₂. HCl is of interest as it is a hydrogen bonding fluid, where as the “N₂” model has explicit nonspherical packing effects. For the HCl model, with the theory presented here, there is no need to add a repulsive core to the hydrogen site, and thus the H–H and H–Cl Lennard-Jones terms are all zero as originally designed. The charges for the H and Cl sites are q_i=±0.2e, respectively, the bond length is l_HCl=1.3 Å, at T=313 K, 0⩽ρ⩽0.018 molecules∕ų. The Cl–Cl Lennard-Jones parameters are σ_ClCl=3.353 Å and ϵ_ClCl∕k_B=259 K. The model geometry is thus equivalent to a finite-length dipole embedded asymmetrically in a Lennard-Jones sphere. For the N₂ model, we use σ_NN=3.720 Å, ϵ_NN∕k_B=37.3 K, and q_i=±0.2e values, with l=1.10 Å, T=562 K, and ρ=0.014 molecule∕ų. All calculations are performed in the molecular HNC approximation, B^o=B^l=B^r=B^b=0, using Eqs. 4, 7, 8 modified by Eqs. 16, 17, 19, 20. Finally, we stress that while these particular models are not strongly dielectric materials, as for water and other more complicated molecular systems, no site-site theory has, to date, successfully described even weakly nontrivial dielectric properties. More complex, and thus, stronger dielectric materials are beyond the scope of the present work, where we concentrate on the fundamental results unobtainable by previous methods.

All numerical calculations were performed using the standard methods described previously¹⁷ on a radial grid of 2048 points with a range of 60 Å, while the angular integrations were performed with 16 and 32 points in the polar θ and azimuthal ϕ integration grids, respectively, as before. Although in total the equations of the theory are quite detailed, we point out that the actual calculations are reasonably straightforward. The system of equations is simply a molecular OZ system, using an HNC closure, in which the complete initial set of functions, ${\overline{c}}^{000}$ and ϕ⁰⁰⁰, are simply subsets of the angularly averaged molecular f-bond. As before, it is simplest to hold the angularly dependent functions constant, solve the OZ system for the τ⁰⁰⁰ functions, recompute the new angular averages, and iterate further to convergence. Using this system, the phase points investigated typically required ten such full angular iterations, or approximately 30 min of calculation time on a typical desktop workstation. With the error function method implemented, with α=1.08∕2l, the numerical behavior of the standard Picard method was extremely stable for the thermodynamic phase points investigated.

The results are summarized in Figs. 23456. The structural results for this model have been recently examined²⁶ using a reduced-basis theory derived in a similar fashion as we use, but from the original RISM integral equation. Their g(r) results for this particular model appear similar to those presented below, although no thermodynamic information or higher order projections beyond the g(r) results were presented in that work.²⁶ In Fig. 2, we show our results for the site-site radial distribution functions for liquid HCL at T=313 K and ρ=0.018 molecules∕ų versus the results for simulation and compared with the XRISM theory (we use the standard short-ranged shield for the hydrogen site in the XRISM calculations only). As with the apolar case, all functions are a clear and significant improvement over previous RISM-type theories, which are well-summarized for diatomic models in Ref. 13.

Figure 2.

The site-site radial distribution functions for HCl at T=313 K and ρ=0.018 molecules∕ų, as predicted in this work. The dashed line is the result of the theory of this work, the solid line is the result of simulation, and the dotted line is the result of the XRISM theory (Refs. ³⁵, ³⁶).

Figure 3.

The site-site $h_{ij}^{110}\left(r\right)$ projections for HCl. The line types are as for Fig. 2 (Note: XRISM has no 110 angular projection, by construction).

Figure 4.

The site-site g_ij(r)’s for the charged model N₂ at ρ=0.014 and T=562 K. The line types are as for Fig. 2.

Figure 5.

The site-site $h_{ij}^{110}\left(r\right)$ projections for the charged model N₂. The line types are as for Fig. 2.

Figure 6.

The predicted dielectric constant ϵ values for the HCl mode at T=313 K. The solid line is the result of this work and the dashed line is for ϵ=1+3y which is the XRISM result. The circle and square are for ab initio simulation and experimental results (Ref. ⁴⁷), respectively, and the diamond is our simulation result reported in this work.

In Fig. 3, we demonstrate the angular information available from the theory, even in the minimal intermediate angular averaging basis set, by plotting the site-site $h_{ij}^{110}\left(r\right)$ projections for the HCl model versus simulation. The h¹¹⁰ projections are defined^{19, 20} as

$$h^{110}\left(r_{ij}\right)=\frac{1}{\int {\left({\Phi }^{110}({\Omega }_1,{\Omega }_2)\right)}^{2}d{\Omega }_{1}d{\Omega }_2}\int h({\mathbf{r}}_{ij},{\Omega }_1,{\Omega }_2){\Phi }^{110}({\Omega }_1,{\Omega }_2)d{\Omega }_{1}d{\Omega }_2$$ (21)

and are easily measured from simulation using methods extended from g(r) calculations.⁴⁵ As in the previously studied case for the two-dimensional g(r,θ) functions¹⁷ and the dihedral distribution functions of butane,¹⁸ this is simply an extension of the angular integrations necessary for the closure as defined in Eq. 4. Since the angular basis-set truncation here is severe, the resulting functions cannot be considered to be fully quantitative. However, the plots in Fig. 3 show that the predicted dipolar correlations are nontrivial even for the minimal-basis-set approximation and provide a satisfactory qualitative description of the angular correlations of the dipoles.

The structural and angular predictions for the N₂ model are presented in Figs. 45. There, we have the radial and 110 projections, respectively. First, we note that, unlike the apolar model of N₂ examined previously, the smaller Lennard-Jones well of this polar model results in a stable solution without the need to resort to an effective density∕variational method.¹⁷ The radial distributions are a significant improvement, especially at longer ranges, on par with what we expect from the results for the apolar version of the model,¹⁷ while the 110 projections are again qualitative only. There are two major effects here. First, the truncation of the basis set is severe as mentioned. Second, physically, we see the principle steric effects. The bond length of the N₂ model combined with nontrivial volume effects due to the Lennard-Jones potential on both sites of the model introduces well-known angular behavior that cannot be completely described by using only the linear diagrams in the intermediate average. In effect, this also highlights the quality of the angular results for the HCl case in Fig. 3.

In Fig. 6 we show the predicted dielectric constant of HCl as a function of density at T=313 K. We calculate the dielectric constant ϵ directly using the Kirkwood g-factor definition⁴⁶

$$\frac{(ϵ-1)(2ϵ+1)}{9ϵ}=g_{K}y,$$ (22)

$$g_K=1+\frac{4\pi \rho }{3}{\int }_0^{\infty }{h}_{ij}^{110}\left(r\right)r^{2}dr,$$

$$y=\frac{4\pi {\mu }^2\rho }{9k_{B}T},$$

where μ is the dipole moment for the model. We note that, since g_K∝⟨μ_i⋅μ_j⟩∕⟨μ⟩², the ensemble-averaged mutual orientation of different dipoles, there is a sign dependence introduced by calculating g_K in the site-site coordinate systems, and, if we use $I_{ij}=\int h_{ij}^{110}\left(r\right)r^{2}dr$, then

$$I_{11}=-I_{12}=-I_{21}=I_{22},$$ (23)

due to the particular conventions for μ_i⋅μ_j in each site-site coordinate pair. This allows a measure of thermodynamic consistency for the calculation and, indeed, we find that the dielectric constants and g_K factors calculated from the different site-site functions agree satisfactorily in these calculations. The density dependence of ϵ is plotted in comparison to the well-known XRISM “ideal gas” result, ϵ=1+3y, the experimental and ab initio simulation results⁴⁷ for HCl at ρ=0.014, and our own simulation results for ρ=0.018 calculated from simulation using standard methods for the Ewald sum.⁴⁸ The agreement is remarkable, with a slight overestimation of the value of ϵ over the density range investigated. This particular deviation from experiment is entirely consistent with the work on single-center model molecular fluids, especially in the various linearized (LHNC) and quadratic (QHNC) hypernetted-chain approximations (see Ref. 7 in particular for a summary) and can reasonably be attributed to the basis set truncation.

For N₂, the dielectric constant is also close to simulation. At 562 K and ρ=0.014, 1+3y=1.84, and we find from simulation that ϵ=2.2, which we essentially duplicate from the theory, ϵ=2.1, although as a practical matter the results are probably within the error of the simulation method.^{48, 49} Overall, the site-site theory presented above essentially quantitatively predicts the radial structure and dielectric constant of polar diatomic fluids, in agreement with both experiment and simulation for these models.

CONCLUSIONS

A long standing goal in the study of the theory of liquids has been a reliable and quantitative method of predicting the equilibrium structure and thermodynamics of polar liquids without resorting to simulation. Without such a theory, experiment has had to rely on simulations for model calculations. While extremely useful as such, a simulation is not a theory amenable to asymptotic analysis. They can be computationally quite tedious, especially for bulk properties such as the free energy or dielectric constant.

The authors have recently focused on extending exact simple fluid bridge diagram methods^{31, 50, 51, 52, 53} to the study of site-site model integral equation theories. By examining the diagrammatically proper ISM theory in the context of the full molecular theory, we have found that recognition of the diagrammatic similarity of the site-site model fluid expansions^{29, 30} to the h-bonded bridge diagram series of simple fluids, leads to molecularly proper integral equations for the inter- and intramolecular structures and thermodynamics of simple diatomic liquids.^{17, 18} Here, we have also shown that, properly constructed, these first-principles methods extend quite readily to polar diatomic fluids. The central result of the current work is that, for the first time, a site-site model theory using a low order angular expansion has been shown to be quantitatively predictive of the radial structure of polar liquids and the static dielectric constant. Further, the theory also provides a strong qualitative description of the angular correlations of the model fluids with minimal effort in the intermediate angular averages.

Overall, these equations result in a convincing practical advance for the case of dipolar liquids. The dielectric constant of these systems, important for both technological and fundamental reasons, has traditionally served as a rigorous test of any liquid state theory, a test which has yielded almost uniformly disappointing results to this point for common site-site models theories and their variants such as RISM and PISM. We were successfully able to consider a hydrogen bonding liquid such as HCL and our next goal is clearly water, where efforts focused on extending the molecular OZ methods that were successful for diatomic systems^{54, 55} to water site-site model systems have shown somewhat disappointing results.^{56, 57} In that context, the work presented here is only a necessary first step. There are tedious but straightforward steps to extend the method to higher angular order for arbitrary model fluid solvents. These are due to the obvious increase in the number and dimensionality of the required angular integrals. In the future, we will examine this directly, with the goal of using methods related to the spherical harmonics addition and integration theorems to reduce the complexity of the calculations.^{50, 58}