SCF theory of intermolecular interactions without basis set superposition error

Special SCF LCAO MO type equations are derived, permitting “supermolecule” calculations for intermolecular interactions, excluding basis set superposition error (BSSE) from the beginning on the basis of the “chemical Hamiltonian approach”. (No additional “monomer” calculations are necessary to correct for BSSE.) The formalism excluding the BSSE results in a non-Hermitean Fock matrix; an algorithm is proposed to obtain the required molecular orbitals, in which no integral transformation is needed.     

A BSSE-free SCF algorithm for intermolecular interactions

A very simple modification of the usual (～N⁴) SCF procedure is proposed, permitting the exclusion of basis set superposition errors (BSSE ) in problems of intermolecular interactions. No a posteriori corrections are required. The results of this “CHA /F method” are numerically close to those of the Boys–Bernardi correction scheme but are free from the “overcompensation” characteristic of the latter at smaller distances.     

SCF perturbation theory for unimolecular reactions

A CNDO/2 SCF perturbation theory is presented for interpreting the form of CNDO/2 potential energy surfaces of unimolecular reactions. The analysis is performed by calculating the energy change E arising from a distortion of the molecular geometry along the reaction coordinate. E is decomposed into different perturbational contributions which are appropriate for an interpretation of the perturbation energy E. Moreover, E is resolved into energy parts arising from a single occupied orbital and contributions due to pairwise orbital interactions. In this way one evaluates numerically how the form of the occupied and unoccupied orbitals determines the magnitude of E. If the distortion occurs along a definite symmetry coordinate, group-theoretical arguments can be applied to discuss the magnitude of characteristic components of the perturbation energy. The SCF perturbation theory is used to analyze the isomerization of ethylene, cis-2-butene and cis-2-butenenitrile.This work was partially supported by Nato-Grant No. 1072     

Improved intermolecular SCF theory and the BSSE problem

Analytical and numerical studies are performed concerning the exclusion of the basis set superposition error (BSSE ) from the SCF calculations of intermolecular interactions. Based on these studies a new procedure is proposed, which consists of the following steps: (1) determine the orbitals by the SCF scheme based on the recent “chemical Hamiltonian approach” (CHA-SCF method), i.e., excluding the delocalization effects caused by BSSE , and then (2) calculate the usual energy expectation value. (This gives results superior to those obtained by the previous nonsymmetric CHA energy formula.) The actual numerical calculations performed for different simple systems (He₂, water dimer) by using various basis sets indicate that the CHA/CE (CHA with “conventional energy” formula) potential curves are well-balanced and are close to those obtained by the Boys–Bernardi (BB ) method and usually (but not necessarily) go slightly beyond the latter. So our method gives results better than (or close to) those given by the BB method by performing only a single ～N⁴ calculation at each geometrical arrangement of the system.     

SCF theory of molecular interactions

Perturbation theory is used to analyze the interactions between two closed-shell systems in the SCF approximation. The effects of orbital overlap are included. The dominant terms through second order are identified, including electrostatic, penetration, polarization, charge transfer, and SCF dispersion effects. The basis set limit is studied so that basis set superposition contributions may be identified. Calculations are presented for He? He and Na⁺? H₂O.     

Application of SCF perturbation theory to molecular calculations

A method for solving Roothaan's molecular orbital equations by means of SCF perturbation theory is presented. An estimate of the accuracy of the third order expansion is made for the CNDO/2 approximation from a comparison of the results from direct calculations. It is found that the third order theory is sufficiently accurate for quantitative studies.     

Calculation of molecular geometries by SCF perturbation theory

An SCF perturbation method for investigating the effects of an arbitrary change in nuclear configuration on the electronic structure of a molecule is reported. Illustrative calculations for CO₂ and CH₃OH are presented. It is found that, when taken to second order, the error in the calculated change in energy which results from a change in nuclear configuration is typically in the range of 0.1 to 7%.     

Second-order energy components in basis-set-superposition-error-free intermolecular perturbation theory

 Recently a basis-set-superposition-error-free second-order perturbation theory was introduced based on the “chemical Hamiltonian approach” providing the full antisymmetry of all wave functions by using second quantization. Subsequently, the “Heitler–London” interaction energy corresponding to the sum of the zero- and first-order perturbational energy terms was decomposed into different physically meaningful components, like electrostatics, exchange and overlap effects. The first-order wave function obtained in the framework of this perturbation theory also consists of terms having clear physical significance: intramolecular correlation, polarization, charge transfer, dispersion and combined polarization–charge transfer excitations. The second-order energy, however, does not represent a simple sum of the respective contributions, owing to the intermolecular overlap. Here we propose an approximate energy decomposition scheme by defining some “partial Hylleraas functionals” corresponding to the different physically meaningful terms of the first-order wave functions. The sample calculations show that at large and intermediate intermolecular distances the total second-order intermolecular interaction energy contribution is practically equal to the sum of these “physical” terms, while at shorter distances the overlap-caused interferences become of increasing importance. Received: 18 June 2001 / Accepted: 28 August 2001 / Published online: 16 November 2001     

Analytic techniques for the coupled multiconfiguration SCF perturbation theory

The perturbation theory based on the paired excitation multiconfiguration self-consistent field approach of Clementi and Veillard is considered. The coupled first-order perturbed orbital equations are analysed and an appropriate computational scheme for their solution is discussed. The proposed computational scheme is analogous to the technique employed for the solution of the coupled Hartree–Fock equations in the one-configuration approximation. However, because of the presence ofnondiagonal Lagrangian multipliers and the use of different one-electron operators for different orbitals, the present scheme raises some new computational problems. In this context a new technique for the solution of the unperturbed multiconfiguration self-consistent field equations is proposed. A simple illustration of the superiority of the multiconfiguration perturbation approach with respect to the ordinary coupled Hartree–Fock scheme is given. Also the validity of the variation formulation of the presented scheme and its relation to the finite-field approach are discussed.     

Rapid computation of intermolecular interactions in molecular and ionic clusters: self-consistent polarization plus symmetry-adapted perturbation theory

A method that we have recently introduced for rapid computation of intermolecular interaction energies is reformulated and subjected to further tests. The method employs monomer-based self-consistent field calculations with an electrostatic embedding designed to capture many-body polarization (the "XPol" procedure), augmented by pairwise symmetry-adapted perturbation theory (SAPT) to capture dispersion and exchange interactions along with any remaining induction effects. A rigorous derivation of the XPol+SAPT methodology is presented here, which demonstrates that the method is systematically improvable, and moreover introduces some additional intermolecular interactions as compared to the more heuristic derivation that was presented previously. Applications to various non-covalent complexes and clusters are presented, including geometry optimizations and one-dimensional potential energy scans. The performance of the XPol+SAPT methodology in its present form (based on second-order intermolecular perturbation theory and neglecting intramolecular electron correlation) is qualitatively acceptable across a wide variety of systems-and quantitatively quite good in certain cases-but the quality of the results is rather sensitive to the choice of one-particle basis set. Basis sets that work well for dispersion-bound systems offer less-than-optimal performance for clusters dominated by induction and electrostatic interactions, and vice versa. A compromise basis set is identified that affords good results for both induction and dispersion interactions, although this favorable performance ultimately relies on error cancellation, as in traditional low-order SAPT. Suggestions for future improvements to the methodology are discussed.     

Coupled multiconfiguration self-consistent field (MC SCF) perturbation theory

The analytical form of the perturbation theory for the MC SCF method of Veillard and Clementi is presented. The appropriate second-order energy functional which takes into account the self-consistency requirements, leads to a set of coupled first-order perturbed equations determining the perturbed configuration coefficients and orbitals. The second-order energy formula derived from this functional can be given a clear physical interpretation. The present analytical approach is compared with the finite perturbation MC SCF scheme. The possibility of the approximate solution of the coupled MC SCF perturbation equations is also discussed and the so-called uncoupled procedures are devised. In the limit of the single determinant wave function the present formulae are shown to be equivalent to the appropriate Hartree-Fock perturbation results. The differences between the one-configuration SCF and the MC SCF approach are illustrated by the calculation of the electric dipole polarizability of. HZ in the CNDO/2 approximation. It is shown that the one-configuration SCF approaches cannot account for the correct asymptotic properties of the second-order energy for large internuclear distances. This feature of the SCF perturbation theories does not depend on the specific approximations of the CNDO/2 scheme and is corrected by using the MC SCF perturbation theory.     

Calculation of frequency-dependent polarizabilities by means of SCF perturbation theory

The SCF perturbative method developed for the calculation of static polarizabilities is extended to the frequency-dependent case. Calculations are reported for helium and methane. The helium results are compared with those of other workers.     

Third-order interactions in symmetry-adapted perturbation theory

We present an extension of many-body symmetry-adapted perturbation theory (SAPT) by including all third-order polarization and exchange contributions obtained with the neglect of intramonomer correlation effects. The third-order polarization energy, which naturally decomposes into the induction, dispersion, and mixed, induction-dispersion components, is significantly quenched at short range by electron exchange effects. We propose a decomposition of the total third-order exchange energy into the exchange-induction, exchange-dispersion, and exchange-induction-dispersion contributions which provide the quenching for the corresponding individual polarization contributions. All components of the third-order energy have been expressed in terms of molecular integrals and orbital energies. The obtained formulas, valid for both dimer- and monomer-centered basis sets, have been implemented within the general closed-shell many-electron SAPT program. Test calculations for several small dimers have been performed and their results are presented. For dispersion-bound dimers, the inclusion of the third-order effects eliminates the need for a hybrid SAPT approach, involving supermolecular Hartree-Fock calculations. For dimers consisting of strongly polar monomers, the hybrid approach remains more accurate. It is shown that, due to the extent of the quenching, the third-order polarization effects should be included only together with their exchange counterparts. Furthermore, the latter have to be calculated exactly, rather than estimated by scaling the second-order values.     

Calculation of the energies of interatomic and intermolecular van der Waals interactions in terms of variational perturbation theory

A new quantum mechanical method is suggested for calculating separate contributions to the energies of interatomic and intermolecular van der Waals interactions. The method involves the multipole expansion of the electrostatic potential of two interacting charge systems, which is treated as a perturbation. Equations for the induction and dispersion contributions are obtained by varying the second-order correction to the interaction energy. The wave functions of isolated systems in the ground state are taken in single determinatal form. The excited-state wave functions are expressed as superpositions of singly excited configurations (the Tamm-Dankov approximation). The density matrix formalism allowed us to obtain the equations in compact matrix form. The method is applied to calculations of the dispersion energies of interactions between two- and four-electron atoms and between two benzene molecules. M. E. Evsev'ev State Pedagogical Institute, Mordoviya. Translated fromZhurnal Strukturnoi Khimii, Vol. 36, No. 4, pp. 600–607, July–August, 1995. Translated by I. Izvekova     

SCF perturbation calculations on metalloporphyrins

The influence of metal ion on the oxidation and ionisation potentials of metalloporphyrins is investigated by the simple electrostatic model using SCF perturbation theory. The zero order wavefunctions are obtained from PPP and CNDO/2 methods. The wide variations in redox potentials with metal and the relative insensitivity of the optical transitions with metal are very well accounted for by the perturbation approach.     

A BSSE-free SCF algorithm for intermolecular interactions. IV. Generalization for open-shell systems

The simple and efficient a priori BSSE-free SCF method (CHA/F) proposed in the previous parts of this series is extended to the case of open-shell systems treated at the single-determinant UHF level of theory. The appropriate equations were derived and sample calculations are presented on three different (H₃⁺(DOTTED BOND)H, H₂O(DOTTED BOND)H, and CH₄(DOTTED BOND)H) systems. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 68: 151–158, 1998     

Application of second-order SCF perturbation theory to the calculation of mixed-frequency hyperpolarizabilities from time-dependent Hartree-Fock theory

Second-order SCF perturbation theory is used to solve the TDHF equations of Dalgarno and Victor through the introduction of frequency dependent density matrices. Exploratory calculations are reported for the frequency dependent polarizability and hyper-polarizability of LiH.     

Application of SCF perturbation theory to the study of tetrahedrally bonded valence crystals

The bond interaction method has been applied to the study of hexagonal and cubic BeO, BN and diamond. It is found that the CNDO method yields the cubic structure as the more stable forms of BN and diamond; the BeO calculation did not converge well. It is concluded that the method is limited to the study of relatively covalent systems.     

A BSSE-free SCF algorithm for intermolecular interactions. II. Sample calculations on hydrogen-bonded complexes

The usefulness and reliability of the recent BSSE -free SCF algorithm based on the “chemical Hamiltonian approach” (CHA /F ) is demonstrated by calculating potential curves for several hydrogen-bonded complexes with 4-31G , 6-31G , and 6-31G ** basis sets. It is concluded that the CHA /F scheme gives results that are numerically close to those of the Boys–Bernardi a posteriori correction scheme but are free from the “overcompensation” characteristic of the latter at smaller distances and given basis sets. © 1992 John Wiley & Sons, Inc.