We present the dispersion-induced dipole moments of coupled Drude oscillators obtained from two approaches. The first approach evaluates the dipole moment using the second-order Rayleigh-Schrödinger perturbation theory wave function allowing for dipole-dipole and dipole-quadrupole coupling. The second approach, based on response theory, employs an integral of the dipole-dipole polarizability of one oscillator and the dipole-dipole-quadrupole hyperpolarizability of the other oscillator over imaginary frequencies. The resulting dispersion dipoles exhibit an *R*^{−7} dependence on the separation between the two oscillators and are connected to the leading-order *C*_{6}/*R*^{6} dispersion energy through the electrostatic Hellmann-Feynman theorem.

## I. INTRODUCTION

Long-range dispersion interactions between atoms or molecules are generally explained in terms of correlated charge fluctuations of the two atoms (or molecules).^{1} The simplest wave function approach that accounts for dispersion is second-order Möller-Plesset (MP2) perturbation theory.^{2} In this picture, the dispersion interaction results from the interaction of fluctuating dipoles of the two atoms or molecules. In contrast, in the last paragraph of Feynman’s classic 1939 paper, he observed that the long-range dispersion interaction between two spherical atoms causes each atom to acquire a permanent dipole moment with the negative ends of the two dipoles pointed toward each other.^{3} The dipole on each atom and the force on the nuclei resulting from the charge distortion display an *R*^{−7} dependence on the separation of the atoms.^{3–5} Feynman further observed that one can obtain the dispersion energy from a classical electrostatic calculation involving the altered charge distributions of the atoms. Eliason and Hirschfelder^{6} confirmed numerically the existence of the *R*^{−7} dispersion force for two interacting H atoms in their *S* ground states, while Byers Brown and Whisnant^{7} confirmed the existence of dispersion dipoles. Hunt showed, through a non-linear response approach, that the dispersion-induced dipole on one atom (or molecule) results from the interaction of its dipole-dipole-quadrupole hyperpolarizability with the instantaneous dipole on the other atom (or molecule).^{8} This approach was exploited in Refs. 9–11. Hunt also presented an analytical proof of Feynman’s conjecture about the dispersion force for interacting atoms in *S* states, and generalized it to molecules of arbitrary symmetry.^{12} She has also been able to reconcile the two seemingly different views of dispersion.^{12}

In the present article, we analyze the dispersion-induced dipole for the case of two interacting quantum Drude oscillators.^{13} In doing so, we draw on the work of Linder and Kromhout^{10} and Galatry and Gharbi.^{11} The latter authors have already published the expression for the dispersion dipole for two interacting 3D Drude oscillators, which they obtained using a response-function approach. In the present study, we report the second-order wave function that gives the dispersion dipoles of the interacting oscillators and use it to obtain the dispersion-induced changes in the charge distribution. We also confirm that one can obtain the *C*_{6} coefficient for the interacting Drude oscillators from the electrostatic force resulting from the altered charge distributions.

## II. THEORY

A Drude oscillator consists of two fictitious charges +*q* and −*q* coupled harmonically through the force constant *k*.^{13,14} In general, the +*q* charge is assumed to be fixed, while the −*q* charge is displaceable. The problem of two quantum Drude oscillators with dipole-dipole coupling is a textbook model for illustrating the origin of the long-range *C*_{6}/*R*^{6} dispersion interaction.^{14} Here, *R* is the distance between the two oscillators, and *C*_{6} is the dispersion coefficient associated with the *R*^{−6} contribution. However, when only dipole-dipole coupling is allowed, this model does not give rise to the permanent dipoles predicted by Feynman: To recover the permanent dipoles induced by dispersion, it is necessary to include dipole-quadrupole coupling.

In this work, we assume that the two oscillators lie on the z-axis, and, for simplicity, we initially assume that oscillator *A* only has charge fluctuations in the z-direction and that the coupling with oscillator *B* is through the −2*q*^{2}*z*_{1}*z*_{2}/*R*^{3} and $2 q 2 z 1 \theta 2 zz / R 4 $ terms, where *θ ^{zz}* = ½(3z

^{2}− r

^{2}). In this case, the relevant wave function through second-order (in atomic units) becomes

where in configuration $ i j $, *i* and *j* specify the states of oscillators *A* and *B*, respectively, and *ω*_{0} is the frequency of the Drude oscillator. $ 0 $, $ 1 $, and $ 2 $ denote the ground and first two excited states of the one-dimensional Drude oscillator. In Eq. (1), we have left out terms that do not contribute to the dipole on oscillator *B*. Figure 1 shows the dispersion-induced change in the charge density for the wave function in Eq. (1) extended to also include the distortion on oscillator *A* caused by the fluctuation in the z-direction of the charge distribution on oscillator *B*. The oscillators are separated by *R* = 12 *a*_{0}, and the parameters are chosen to be *q* = 0.5 a.u., the mass *m* = 1 a.u., and *k* = 0.16 a.u., resulting in *α* = 20 a.u.^{3} and *ω*_{0} = 0.40 a.u., values roughly appropriate for Kr. As seen from this figure, each oscillator acquires a permanent dipole with the negative ends of the dipoles pointing towards each other just as Feynman predicted for interacting atoms.

Using the wave function in Eq. (1), the dipole moment of oscillator *B* is calculated to be (retaining terms through *R*^{−7})

where ^{A}*α* and ^{B}*α* are the static dipole polarizabilities of *A* and *B*, which are equal to *q*^{2}/*k*, and ^{B}*B ^{zzzz}* is the

*zzzz*component of the static dipole-dipole-quadrupole hyperpolarizability of

*B*, which is equal to $2 q 3 / m 2 \omega o 4 $ for a Drude oscillator. This latter result was derived using Eq. (A4) of Ref. 10. The charge displacements depicted in Figure 1 have both dipolar and quadrupolar contributions, the latter of which results from the dipole-dipole coupling. The distortion due to dipole-dipole coupling alone is shown in Figure 2.

The induced dipole on *B* due to the dipole fluctuations on *A* can also be expressed in terms of the integral of the product of the frequency-dependent dipole polarizability of *A* and the frequency-dependent dipole-dipole-quadrupole hyperpolarizability of *B* evaluated at imaginary frequency,^{9–12}

where *T*_{αδ} and *T*_{ε,βγ} are the standard dipole-dipole and dipole-quadrupole coupling tensors, respectively. Bishop and Pipin have reported accurate values of *α*(*iω*) and *B*(−*iω*, *iω*) as a function of *ω* for H, He, and H_{2}.^{15} For the simplified 1D model described above, one need only to consider *α ^{zz}*(

*iω*) and

*B*(−

^{zzzz}*iω*,

*iω*) which are given by

and

where the latter result was obtained using Eq. (A4) of Linder and Kromhout.^{10} Interestingly, the frequency dependence of *B ^{zzzz}* is even simpler for the Drude oscillator than obtained using closure relations as used by these authors. Using Eqs. (4) and (5) to evaluate the integral in Eq. (3) gives the same expression for the dipole moment as was obtained from the perturbation theory wave function (Eq. (2)).

For the simplified 1D model considered above, the force on the fixed charge of oscillator *B*, obtained by use of the electrostatic Hellmann-Feynman theorem, is

which is also equal to −6*C*_{6}/*R*^{7}. This gives the result $ C 6 = 1 3 \alpha 2 \omega 0 $, compared to the exact result of $ 1 2 \alpha 2 \omega 0 $ for the *C*_{6} value for 1D Drude oscillators.^{11} This discrepancy is simply a consequence of our neglect in the simplified treatment of contributions of ^{B}*B ^{zzxx}* and

^{B}

*B*in response to the fluctuation of the charge density of

^{zzyy}*A*in the z-direction. When these are included, the correct value of

*C*

_{6}is obtained for the 1D case.

In three dimensions, the Hamiltonian for two interacting Drude oscillators, with dipole-dipole and dipole-quadrupole coupling, has the form

where $ H A ( 0 ) $ and $ H B ( 0 ) $ are the Hamiltonians for the non-interacting oscillators, $ \mu \alpha I $ and $ \theta \alpha \beta I $ are the dipole and quadrupole moment operators for oscillator *I*, and the standard Einstein summation notation is used. The resulting wave function through second-order contains three *R*^{−3}, ten *R*^{−4}, and 34 *R*^{−7} terms. Using this to evaluate ^{B}*μ ^{z}* gives

In the evaluation of Eq. (8), the product of the *R*^{−3} and *R*^{−4} terms in the wave function contributes 1/6 of the dipole moment while the *R*^{−7} terms contribute 5/6. In comparison, Hirschfelder and Eliason found that 93% of the contributions to the dipole moment arise from the *R*^{−7} terms in the wave function for the case of two hydrogen atoms.^{6} We note that Levine also considered the dispersion dipole for interacting Drude oscillators.^{16} However, due to the approximations that he made in averaging the interactions, the resulting dispersion dipole is 10/3 times larger than that given by Eq. (8).

In treating the interacting 3D Drude oscillators using the response function approach it is necessary to consider seven components of *B*, *B ^{zzzz}*,

*B*,

^{zzxx}*B*,

^{zzyy}*B*,

^{zxxz}*B*,

^{zyyz}*B*, and

^{zxzx}*B*, which are related as follows:

^{zyzy}^{8,10}

Allowing for these relations, the net dispersion dipole on oscillator *B* can be expressed as

where *α ^{zz}* and

*B*are defined above in Eqs. (4) and (5). Evaluation of the integral gives the same result for the dipole induced on oscillator

^{zzzz}*B*as obtained from the perturbation theory approach (Eq. (8)). Moreover, for the 3D Drude oscillator case, the value of

*C*

_{6}evaluated using the electrostatic Hellmann-Feynman theorem is $ 3 4 \alpha 2 \omega 0 $ which is the exact value of the

*C*

_{6}coefficient for two interacting 3D Drude oscillators as evaluated using second-order perturbation theory. In Figure 3, we display a contour plot of the dispersion-induced changes in the charge distribution of two interacting 3D Drude oscillators separated by

*R*= 12

*a*

_{0}, and with the same parameters as used above for the 1D case.

## III. CONCLUSIONS

In this work we derive, using both perturbation theory and a response function approach, the expression for the dispersion-induced dipoles of interacting Drude oscillators. We also derive the C_{6} dispersion coefficient from the electrostatic Hellmann-Feynman theorem. As noted in previous studies, examining H_{2} and He_{2} at interatomic distances beyond the overlap region,^{17–19} the permanent dipoles resulting from the dispersion interactions are very small. Due to the small values of these dipoles, it has been concluded by some researchers that it is essential to use variational (or, at least, coupled-cluster) approaches to accurately predict dispersion dipoles. Although one has to be careful in extrapolating from the Drude model to “real” atoms or molecules, our results suggest that even the second-order Rayleigh-Schrödinger perturbation theory wave function should suffice for estimating the dispersion-induced dipoles.

In recent years, several methods of correcting density-functional theory for dispersion that involve distortions of the atomic charges have been introduced. These include the dispersion-corrected atom-centered potential (DCACP) approach,^{20–22} the density displacement model of Hesselmann,^{23} and the self-consistent Tkatchenko-Scheffler^{24} model of Ferri and co-workers.^{25} However, the success of these models at describing dispersion interactions appears to be unrelated to the electrostatic Hellmann-Feynman approach for obtaining the *C*_{6} coefficient from the charge distortion.

## Acknowledgments

This research was supported by a grant from the National Science Foundation No. CHE1362334 and a Fellowship from the Pittsburgh Quantum Institute. We acknowledge valuable discussions with Dr. V. Voora.