Second-order Dispersion Energy Research Articles

How Much Dispersion Energy Is Included in the Multiconfigurational Interaction Energy?

We demonstrate how to quantify the amount of dispersion interaction recovered by supermolecular calculations with the multiconfigurational self-consistent field (MCSCF) wave functions. For this purpose, we present a rigorous derivation which connects the portion of dispersion interaction captured by the assumed wave function model—the residual dispersion interaction—with the size of the active space. Based on the obtained expression for the residual dispersion contribution, we propose a dispersion correction for the MCSCF that avoids correlation double counting. Numerical demonstration for model four-electron dimers in both ground and excited states described with the complete active space self-consistent field (CASSCF) reference serves as a proof-of-concept for the method. Accurate results, largely independent of the size of the active space, are obtained. For many-electron systems, routine CASSCF interaction energy calculations recover a tiny fraction of the full second-order dispersion energy. We found that the residual dispersion is non-negligible only for purely dispersion-bound complexes.

Second-Order Dispersion Energy Based on Multireference Description of Monomers.

We propose a method for calculating a second-order dispersion energy for weakly interacting multireference systems in arbitrary electronic states. It is based on response properties obtained from extended random phase approximation equations. The introduced formalism is general and requires only one- and two-particle reduced density matrices of monomers. We combine the new method with either generalized valence bond perfect pairing (GVB) or complete active space (CAS) self-consistent field description of the interacting systems. In addition to a general scheme, three approximations, leading to significant reduction of the computational cost, are developed by exploiting Dyall partitioning of the monomer Hamiltonians. For model multireference systems (H2···H2 and Be···Be) the method is accurate, unlike its single-reference-based counterpart. Neither GVB nor CAS description of single-reference monomers improves the dispersion energy with respect to the Hartree-Fock-based results.

Dispersion Energy of Symmetry-Adapted Perturbation Theory from the Explicitly Correlated F12 Approach.

Methods of the explicitly correlated F12 approach are applied to the problem of calculating the uncoupled second-order dispersion energy in symmetry-adapted perturbation theory. The accuracy of the new method is tested for noncovalently bound complexes from the A24 data set [J. Řezáč and P. Hobza, J. Chem. Theory Comput. 2013, 9, 2151] using standard orbital basis sets aug-cc-pV XZ supplemented with auxiliary aug-cc-pV XZ_OPTRI sets. For near equilibrium geometries, it is possible to recover the dispersion energy with average relative errors consistently smaller than 0.1% (with respect to the CBS extrapolated limit estimated from regular orbital calculations). This level of accuracy is achieved already in the basis set of a triple-ζ quality, when a Slater-type correlation factor exp(-0.9 r12) is combined with variant C of the F12 approach. The explicitly correlated approach clearly outperforms regular orbital calculations in the basis set of quintuple-ζ quality (average relative errors of 1%).

Dispersion energy from density-fitted density susceptibilities of singles and doubles coupled cluster theory

A new method of calculation of the second-order dispersion energy is proposed. It is based on the Longuet-Higgins formula [Faraday Discuss. Chem. Soc. 40, 7 (1965)], which describes the dispersion interaction in terms of frequency-dependent density susceptibilities of monomers. In this study, the density susceptibilities are obtained from the coupled cluster theory at the singles and doubles level. Density fitting is applied in order to reduce the computational effort for the evaluation of density susceptibilities. It is shown that density fitting improves the scaling of the computational resources with molecular size by one order of magnitude without affecting the accuracy of the resulting dispersion energy. Numerical results are presented for several van der Waals molecules to illustrate the performance of the new approach.

Intermolecular potentials from supermolecule and monomer calculations

The use of monomer properties to improve supermolecule calculations of intermolecular potentials is reviewed. For Van der Waals dimers, the MP2 supermolecule method is too inaccurate for most purposes, and the CCSD(T) supermolecule method requires too much computer time for large molecules. Using perturbation theory to analyse the MP2 supermolecule energy shows that the second-order dispersion energy is the main source of the inaccuracy. It is shown that the dispersion energy can be improved by using more accurate dispersion energy coefficients which can be obtained from monomer frequency-dependent polarizabilities. The supermolecule MP2 electrostatic and exchange-repulsion interaction energies can also be recalculated or scaled to a higher level of theory, using monomer charge densities. Applying these corrections to the MP2 supermolecule energy does not require much additional computer time, and gives potential energy surfaces with comparable accuracy to supermolecule CCSD(T) calculations. Possible extensions of the method to different supermolecule methods and to larger molecules are discussed.

Intermolecular interaction energies by topologically partitioned electric properties II. Dispersion energies in one-centre and multicentre multipole expansions

Multicentre multipole expansions in principle should allow one to solve the ‘shape’ convergence problem arising in the calculation of long-range interaction energies between large non-spherical molecules via point-multipole expansions. In part I of this series (1996, Molec. Phys., 88, 69) it was shown that this indeed is the case for first-order electrostatic and secondorder induction energies when employing distributed multipole moments and static polarizabilities generated from the topological partitioning of the molecular volume as provided by Bader's ‘atoms-in-molecules’ theory. Their topologically partitioned polarizabilities is used in the present contribution to compare the convergence behaviour of one-centre and multicentre multipole expansions of the secondorder dispersion energy for homomolecular dimers of the water, carbon monoxide, cyanogen and urea molecules. The findings are similar to those for the induction energy; the radial ‘extension’ convergence problem, which already exists for point-multipole expanded interaction energies between atoms, necessarily persists, but the angular convergence problems linked to the shape of the interacting molecules can successfully be treated by multicentre multipole expansions of the dispersion energy. generalization to frequency-dependent,

Dispersion energy in the coupled pair approximation with noniterative inclusion of single and triple excitations

The second-order dispersion energy in the coupled-pair (coupled-cluster doubles) approximation has been derived. The coupled-pair amplitudes are subsequently used in a perturbation theory type expression to account for the effects of single and triple excitations. This approach selectively sums to infinite order important classes of intramonomer correlation diagrams resulting in a better theoretical description of the dispersion interaction compared to a finite-order perturbation treatment. Numerical results have been obtained for He2, Ar–H2, Ar–HF, (HF)2, (H2O)2, and He–F− in various geometries and basis sets to illustrate the performance of the nonperturbative versus perturbative treatments of the intramonomer correlation contributions to the energy of the dispersion interaction.

Representations of dispersion energy damping functions for interactions of closed shell atoms and molecules

Abstract Analytical expressions are developed for the individual partial wave components of the non-expanded second-order dispersion energy for the H(1s)H(1s) interaction and the results are used to obtain analytical expressions for the related damping functions f ( l a , l b , R ). The average or overall damping functions f 2 n ( R ), for each R −2 n expanded dispersion energy, are given as a weighted average of the f ( l a , l b , R ) with n = l a + l b +1. The analytical results, and related numerical calculations, are used to discuss the nature of the average and the partial wave damping functions as a function of l a , l b , n and R . For example it is shown that the individual partial wave components of the f 2 n ( R ) are essentially independent of l a and l b for all relevant values of the interatomic distance R . Reliable representations of the average damping functions are of considerable importance in the construction of potential energy models and the various literature representations are compared and contrasted with each other and with a new representation of the f 2 n ( R ) given in this paper. The discussion includes general comments on the use of the H(1s)H(1s) damping functions in the construction of potential energy models for other interactions through the use of interspecies distance scaling methods.

The Spherical Expansion of H2Dimer Interaction Energy by Double Symmetry-Adapted Perturbation Theory

Abstract The weak interaction energy of H2 dimer is studied by double symmetry-adapted perturbation theory (SAPT) within second-order of intermolecular and intramonomer perturbation for molecular simulations. The assumed orientations of H2 dimer are linear, parallel, T type and X type. Among four orientations T orientation is the most stable, while linear orientation is the most repulsive. The second-order dispersion energy E disp (2) is the most attractive contribution in all orientations. The interaction energy has the anisotropy, so we expressed our total interaction energy by the spherical expansion to compare with the experimental value. The isotropic interaction energy is about 85% of the experimental value.

Accuracy of the Boys and Bernardi function counterpoise method

The performance of the Boys and Bernardi function counterpoise (FCP) method in eliminating the basis set superposition error (BSSE) is studied for He2, at R=5.6 a.u., within the supermolecular coupled electron pair approximation (CEPA-1) method. A series of one-electron Gaussian basis sets is designed that allows a systematic approach to the basis set limit value of the interaction energy. Every basis set contains a part suitable to reproduce the atomic correlation energy and a second part optimized for the dispersion interaction in He2. BSSE-free correlated first-order interaction energies [E(1)], calculated using perturbation theory, are reported for each of these sets. Extrapolation to the basis set limit yields a new value of 33.60±0.02 μH for E(1) at R=5.6 a.u. Extending previous work, the supermolecular CEPA-1 interaction energies for each set are then compared to the total of E(1) and the BSSE-free Mo/ller–Plesset second-order dispersion energy reported previously. While for some basis sets the uncorrected ΔE values deviate up to 43 K from the perturbation estimate, the FCP-corrected results always agree within 0.4 K. A virtuals-only counterpoise procedure is considered as well, but fails badly. The remaining discrepancies in the FCP results are ascribed to a failure of the Mo/ller–Plesset approach to precisely model the dispersion energy at the CEPA level. This problem is removed in a further, more stringent test where supermolecular EintCEPA-intra results, in which only the intra-atomic correlation (at the CEPA-1 level) is taken into account, are directly compared to the BSSE-free E(1) values. In this test the FCP-corrected supermolecular results agree, for the larger sets, to within 0.001 K with the results expected on the basis of E(1). These findings demonstrate, for the first time, that at least in He2 the FCP recipe yields interaction energies that correspond precisely (to machine precision) to the basis set and correlation method at hand.

Analysis of the potential energy surface of Ar–NH3

The combination of supermolecular Mo/ller–Plesset treatment with the perturbation theory of intermolecular forces is applied in the analysis of the potential energy surface of Ar–NH3. Anisotropy of the self-consistent field (SCF) potential is determined by the first-order exchange repulsion. Second-order dispersion energy, the dominating attractive contribution, is anisotropic in the reciprocal sense to the first-order exchange, i.e., minima in one nearly coincide with maxima in the other. The estimated second-order correlation correction to the exchange effect is nearly as large as a half ΔESCF in the minimum and has a ‘‘smoothing’’ effect on the anisotropy of ε(20)disp. The model which combines ΔESCF with dispersion energy (SCF+D) is not accurate enough to quantitatively describe both radial and angular dependence of interaction energy. Comparison is also made between Ar–NH3 and Ar–PH3, as well as with the Ar dimer.

The impact of higher polarization functions of second-order dispersion energy. Partial wave expansion and damping phenomenon for He 2

Convergence properties of the Møller—Plesset dispersion energy with respect to saturation of polarization basis set have been investigated, by means of the partial wave expansion through “i” terms, for the He dimer. The expansion proved to be slowly convergent and ineffective if accurate calculations are aimed at. The damping functions for individual terms have been evaluated and shown to be in good agreement with the semi-empirical ones. The basis set extension effect, occurring when occupied and virtual orbitals of monomers are evaluated with the basis set of a whole dimer, has been found to improve upon the slow convergence of the partial wave expansion. Additional improvement is observed on adding bond-centered functions.

Correlated calculation of the interaction in the nitromethane dimer

The interaction energy of the nitromethane dimer at several separations between the monomers was calculated using fourth-order many-body perturbation theory (MBPT), with single, double, and quadruple (SDQ) excitations included. The self-consistent field (SCF) counterpoise (CP) correction, and second-order dispersion energy were also computed. A double-zeta plus polarization basis was used. The monomers were oriented so that a hydrogen atom on each monomer could form a hydrogen bond with an oxygen on the other monomer. An interaction energy of 3.57 and 5.04 kcal/mol was found at the SCF and SDQ-MBPT(4) levels of theory, respectively. The CP-corrected SCF energy added to the second-order dispersion energy gives an interaction energy of 5.62 kcal/mol.

Non-expanded dispersion energies and damping functions for Ar2 and Li2

The non-expanded second-order dispersion energies and damping functions associated with the long-range dispersion energies varying as R−6, R−8and R−10 have been calculated for Ar2 and Li2 with the time-dependent Hartree-Fock method, using extended Gaussian basis sets. These results are used to discuss the difficulties associated with ab initio computations of these quantities.

Long-range interactions of Li( n = 2) states with each other and the long-range interaction of Li(2s 2S) with Li(3s 2S)

We compute and analyze the potential energy for the twenty-six lowest-lying states of Li 2 which correspond asymptotically to the interaction of Li(2s 2S) with Li(2s 2S). Li(2p 2P) or Li(3s 2S), and the interaction Li(2p 2P) with Li(2p 2P) to obtain the leading terms of the first-order electrostatic energies, and the second-order dispersion energies. Ion-pair perturbations are found to dominate the potential curves of several states. The polarizabilities of Li 2s, 2p and 3s in various fields are calculated.

Long range induction and dispersion interactions between Hartree-Fock subsystems

A consistent perturbation theory of long range induction interactions between two subsystems described in the Hartree-Fock approximation is presented. For the wavefunction of the interacting system represented by the Hartree product of the subsystem Hartree-Fock wavefunctions (the HHF approximation) the resulting perturbation equations are shown to be precisely equivalent to those of the coupled Hartree-Fock perturbation theory. In contrast to some earlier proposals concerning the calculation of induction energies the present treatment provides the HF level of accuracy for each of the two interacting subsystems. The non-perturbative solutions of the induction energy problem are then used to build a variation scheme for the calculation of intra- and inter-subsystem correlation effects. By using different partition schemes for the total hamiltonian the functionals derived for the second-order inter-subsystem correlation energy represent either the uncoupled or the coupled pair schemes. In the lowest order with respect to induction interactions they are equivalent to the corresponding ordinary second-order dispersion energy functionals. The structure of the perturbed wavefunctions is analysed and it is shown that in order to obtain the whole of the induction effects one has to take into account some portion of intra- and inter-subsystem double excitations which can be expressed in the ‘unlinked’ form. Also some computational aspects of the theory presented in this paper are discussed.

The interaction between first excited state hydrogen atoms and a different non-degenerate atom: an example of second-order resonance forces

The interaction of a first excited state hydrogen atom with a different non-degenerate atom is considered in the R -1 multiple approximation. The interaction energies for the Π states arising from this interaction have the form usually associated with second-order dispersion energies while those for the Σ states behave quite differently as a function of R. Because of a second-order ‘resonance within one molecule effect’ the second-order Σ-state energies do not possess single R -1 expansions that are formally valid for all values of R consistent with the multipole treatment. The expansions of these Σ-state energies, that are valid for ‘small’ R, contain terms varying as both even and odd powers of R -1; the lead R -7 ‘odd’ term competes strongly with the usual R -6 dispersion energy. The expansions that are valid for ‘large’ R contain terms varying as even powers of R -1 only and usually have little physical meaning. The He(1S)-H(n=2) interaction is considered as a specific example of these results and the interaction energies for this system are evaluated through O(R -8) by using one-centre pseudostate methods. The general significance of the results is also discussed briefly.

Charge overlap effects dependence on the nature of the interaction

The He-He, He-H(1s), H(1s)-H(1s) and H(1s)-H(2s) interactions are considered as model systems for studying how charge overlap effects in second-order dispersion energies vary as a function of the nature of the interacting species. The non-expanded second-order energy and the corresponding multipole R -1 expansions, through all powers of R -1, are obtained for each interaction using Unsold's average energy approximation. The results are used to discuss the limitations of the usefulness of the R -1 expansions.

Inequalities for Multipole Dispersion Interactions

Some theoretical relations are obtained for various multipole dispersion force constants and for the total second-order dispersion energy in the long-range interactions between atoms.