Intermolecular forces from asymptotically corrected density functional description of monomers

doi:10.1016/S0009-2614(02)00533-X

Chemical Physics Letters

Volume 357, Issues 3–4, 10 May 2002, Pages 301-306

https://doi.org/10.1016/S0009-2614(02)00533-X Get rights and content

Abstract

Symmetry-adapted perturbation theory based on Kohn–Sham determinants, SAPT(KS), was shown before to perform poorly for the electrostatic energy which is potentially exact in this approach. We demonstrate that some deficiencies of SAPT(KS) result from wrong asymptotics of exchange-correlation potentials. On applying an asymptotic correction, we were not only able to recover the electrostatics, but also the first-order exchange and second-order induction and exchange-induction energies fairly accurate. Dispersion is still reproduced poorly but can be computed reasonably accurately from the damped asymptotic expansion.

Introduction

Density functional theory (DFT), one of the most popular methods for predicting properties of molecular systems, fails when used in the supermolecular approach to compute the interaction potentials for systems with a significant dispersion component. Numerous papers have been devoted to this problem, see [1] for references and a quantitative analysis. Recently, Williams and Chabalowski (WC) [2] proposed a perturbational approach where only the unperturbed monomers are described by DFT, whereas the interaction energies are obtained from symmetry-adapted perturbation theory (SAPT) [3] expressions utilizing Kohn–Sham (KS) orbitals and orbital energies. The WC approach was to replace the Hamiltonian partitioning of the standard SAPT [3]: H=F+ξW+λV by the partitioning H=K+ξW^KS+λV, where F=F_A+F_B is the sum of the Fock operators for monomers A and B,W=W_A+W_B is the sum of the Moller–Plesset fluctuation operators, V is the intermolecular interaction operator, K=K_A+K_B is the sum of the KS operators, and W^KS may be formally defined as W^KS=H_A+H_B−K, where H_X is the Hamiltonian for the unperturbed monomer X. In the KS-based perturbation theory, WC included only terms of zeroth order in ξ, effectively neglecting the operator W^KS and using the Hamiltonian H^KS=K+λV. We denote by SAPT(KS) this simplified, KS-operator-based single-perturbation theory. In order to achieve interaction energies accurate to a few percent, one has to include in the standard SAPT fairly high-order terms in W, thereby making SAPT computer resource intensive. SAPT(KS), lacking the intramonomer correlation operator, scales extremely well with system size which manifests itself for large systems like CH₃CN–CO₂ for which WC reported a speed up by a factor of 128 of SAPT(KS) over SAPT. The question therefore is whether or not SAPT(KS) can achieve significantly better accuracy than the regular SAPT at the same cost, i.e., in the zeroth order in ξ [denoted by SAPT(0)].

The exchange, induction, and dispersion components of interaction energy are defined in terms of ground- and excited-state wave functions, therefore we cannot a priori conjecture that SAPT(KS) will recover these terms accurately. In contrast, the electrostatic energy, E⁽¹⁾_elst, depends on the unperturbed monomer charge densities only, which should be well reproduced by DFT, and we can expect SAPT(KS) to provide accurate values for this term. WC, however, observed that for all the systems they studied E_elst⁽¹⁾ was consistently too negative. In particular, for He₂ at the near-minimum separation the intramonomer correlation effect on the electrostatic energy was a factor of about 10 too large in magnitude, indicating a wrong asymptotic density behavior resulting from a deficiency in continuum exchange-correlation (XC) potentials in the asymptotic region [4], [5]. WC did point out that an asymptotically corrected XC potential might yield a more accurate value of E_elst,⁽¹⁾ and the initial goal of this project was to check this conjecture.

Section snippets

Asymptotic correction

The inability of continuum functionals to yield asymptotically correct potentials lies in their failure to account for the derivative discontinuity (DD) [4], [5], [6], [7] in the energy E(N) at integer particle number N. In [4], [5] it is argued that a DD-free potential should behave as $v^{DDF}_{xc,σ} (r)→−1/r+(I_{σ} +ϵ_{HOMO,σ})$ , where I_σ is the ionization potential of electrons of spin σ and ϵ_HOMO,σ is the highest occupied molecular orbital eigenvalue. However, most existing continuum functionals yield XC

Method

The SAPT [10] suite of codes interfaced with CADPAC [11] has been used for all our calculations. The SAPT interaction energy components have been computed at the highest [3] level of intramonomer correlation possible with the current codes [10]. The δ^HF_int,resp term is not included in comparisons since it is of the third and higher order in V, whereas the SAPT(KS) expansion in V is truncated at the second order. SAPT interaction energy components will be computed with response [3], [12]

Helium dimer

We discuss calculations for He₂ in detail as nearly exact benchmark results for the interaction components are available for this system. To reduce basis set dependent effects, we used a fairly large basis of 177 contracted Cartesian Gaussian functions from [19] where it is denoted Dc147 since it contains 147 spherical functions. Calculations in two smaller basis sets indicate that both the asymptotically corrected and uncorrected SAPT(KS) results are well converged in the basis Dc147 except

Theoretical justification for SAPT(KS)

The fairly accurate results obtained in this work for the electrostatic, induction, and exchange energies in the SAPT(KS) approach call for some explanation. As already discussed, such an explanation is straightforward for E⁽¹⁾_elst. Now consider the first-order exchange energy, E⁽¹⁾_exch, which in the single-exchange approximation is expressible by an integral of the interaction density matrix defined as $ρ_{int} (ij)=N_{A} N_{B} ∫ψ_{A}^{*} ψ_{B}^{*} Pψ_{A} ψ_{B} d τ^{′}_{ij},$ where i and j refer to space and spin coordinates of N_A and N_B

SAPT(KS) + damped dispersion

SAPT(KS) is not able to yield an accurate enough dispersion energy, and most of the disagreement with SAPT lies in this energy component. Thus, it would be advantageous if the dispersion energy could be obtained reliably and at low cost using some other approach. One such way is to use the damped asymptotic expansion for dispersion as in the HFD method [1], [22]. We have checked this possibility for He₂ using asymptotic dispersion constants from [19]. The damping coefficient was obtained from a

Conclusions

In summary, we have demonstrated that a symmetry-adapted perturbation theory based on KS orbitals and orbital energies is capable of recovering, to a good accuracy, the electrostatic, exchange, and induction components of the interaction energy provided that the asymptotic correction is applied to the XC potentials. The accuracy is significantly better than that achieved by the computationally equivalent SAPT(0) approach. A theoretical justification for the success of SAPT(KS) for E⁽¹⁾_elst and E

Acknowledgements

This research was supported by the NSF grant CHE-9982134. We thank Dr. Cary Chabalowski for providing a direction for this work, Prof. Bogumil Jeziorski for many insights, and both of them for helpful discussions.

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Intermolecular forces from asymptotically corrected density functional description of monomers

Abstract

Introduction

Section snippets

Asymptotic correction

Method

Helium dimer

Theoretical justification for SAPT(KS)

SAPT(KS) + damped dispersion

Conclusions

Acknowledgements

Chem. Phys. Lett.

J. Chem. Phys.

J. Phys. Chem. A

J. Chem. Phys.

J. Chem. Phys.

Phys. Rev. Lett.

Phys. Rev. Lett.

Men. R. Acad. Italia