Variation-perturbation Approach Research Articles

Procedure supplementing SCF interaction energies by dispersion term evaluated in dimer basis set within variation-perturbation approach

A variation-perturbation procedure for the evaluation of dispersion interaction, originally proposed by Jeziorski and van Hemert, has been reformulated to include basis set extension effects on an equal footing with the SCF interaction energy, corrected for basis set superposition error (BSSE). This approach has been tested for He2, (H2)2, (H2O)2, and (C2H4)2 complexes.

The systematic cancellation of errors in a variation-perturbation approach to the calculation of intermolecular potentials

We give a description of a variation-perturbation approach to calculating intermolecular potentials that possess a systematic cancellation of errors. An important consequence is the possibility of using integrals over AOs that are quite approximate, while still obtaining satisfactory values for the potential.

Interlaminar stress analysis for laminated plates containing a curvilinear hole

This study presents a variational-perturbation approach for the three-dimensional stress analysis around a curvilinear cutout in composite laminates. The solution is based on a composite expansion and assumed stress finite element methods. Stress field solutions for angleply and cross-ply laminates containing an elliptical hole have been presented. The effects of the ellipse aspect ratio and the fiber orientation on the interlaminar shear stress magnitude have been investigated. A failure criterion based on the interlaminar distortional energy function has been suggested. According to this criterion, it was found that, for angle-ply laminates containing an elliptical hole, a delamination failure initiation occurs at a point along the perpendicular to the direction of loading.

A new variation-perturbation approach: The fast configuration interaction method

A new variational method which has been successfully applied in quantum chemistry is presented. The method allows a good approximation for the correlation energy of a finite Fermi system to be obtained with less computer time than the existing methods.

Variation-perturbation approach to complex eigenvalues of resonance states

ADVERTISEMENT RETURN TO ISSUEPREVArticleNEXTVariation-perturbation approach to complex eigenvalues of resonance statesFrank. WeinholdCite this: J. Phys. Chem. 1979, 83, 11, 1517–1520Publication Date (Print):May 1, 1979Publication History Published online1 May 2002Published inissue 1 May 1979https://doi.org/10.1021/j100474a030RIGHTS & PERMISSIONSArticle Views66Altmetric-Citations4LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InReddit PDF (540 KB) Get e-Alerts

Electron affinities by a variation-perturbation approach

A simple variation-perturbation method recently developed is applied to the calculation of electron affinities. In this method a common basis is used for both the ion and the neutral system. The resulting secular equations are of a very small dimension. The matrix elements entering these equations can be evaluated by diagrammatic methods. The formalism is compared with the one-body Green's function method. The electron affinities of Li, Na, F, Cl, and OH are calculated.

A simple variational-perturbational approach to the correlation problem in atoms and molecules

Abstract A simple ansatz for the ground state of atoms and molecules is proposed which is also suitable for large internuclear distances where perturbation theory breaks down. In its simplest form, where only doubly-excited configurations are included, the ansatz leads to 97.3% of the exact correlation energy of H 2O calculated in the doubly-excited configurational space with the same basis set. The calculated correlation energy for N 2 shows the correct behaviour as a function of internuclear distances.

Reduced Partitioning Procedure in Configuration Interaction Studies. I. Ground States

A variation-perturbation approach to configuration interaction (CI) studies of atoms and molecules has been derived which yields a series of converging upper bounds to the CI energy. As a consequence, the method permits an adequate representation of a large n-particle wavefunction in terms of a very low order nonlinear perturbation summation. This ``reduction'' procedure is found to have simplifications for full CI studies as well as suggesting certain systematic improvements over the standard Hartree-Fock (HF) and ``truncated'' CI calculations. The first order solution, which is equivalent to the application of a geometric sumrule to the perturbation expansion is also shown to be derivable from a ``steepest descent'' argument. In preliminary calculations on H2 and the HeH+ molecular ion with a HF function as the unperturbed state, the first order result gives 60% to 70% of the energy difference between the full CI energy and the HF value, while only a third order treatment accounts for better than 90%. Potential curves for the ground state of HeH+ for the eight lowest orders of solution and their spectroscopic parameters are also obtained, with the eighth order energies differing from the full CI values by less than 10−4 a.u. at all internuclear separations.

Ground-State Energy of a Many-Particle Boson System

The ground state of a many-particle boson system is studied for two closely related limits: the uniform limit and the weak coupling limit. The former is defined by $\ensuremath{\alpha}=1\ensuremath{-}g(0)\ensuremath{\ll}1$ and the latter by $\ensuremath{\beta}=\frac{(N\ensuremath{-}{N}_{0})}{N}\ensuremath{\ll}1$, where $g(r)$ is the radial distribution function and ${N}_{0}$ is the mean occupation number of the zero-momentum state. In the uniform limit the variation-perturbation approach based on (a) the method of correlated basis functions and (b) the series expansion in powers of $\ensuremath{\alpha}$ is found to be equivalent to the field-theoretic treatment given by Brueckner (for the charged-boson gas) in the weak coupling limit. In particular, it is shown that the variation-perturbation energy obtained for the uniform limit in the momentum representation is identical through second order to the ground-state energy evaluated for $\ensuremath{\beta}\ensuremath{\ll}1$ by summing one- and two-ring diagrams in the Bogoliubov occupation-number representation. The charged-boson gas and the one-dimensional boson system with a $\ensuremath{\delta}$-function interaction are considered to examine some of the interesting features of the uniform-limit procedure.

Analysis of the Karplus-Kolker approximation to Hartree-Fock perturbation theory

The Karplus-Kolker approximate uncoupled Hartree-Fock perturbation scheme is compared with the corresponding coupled variation-perturbation approach. The first-order perturbation-like corrections to the Karplus-Kolker second-order energies are defined and their meaning and relative importance are discussed. Also a geometric approximation to the coupled Hartree-Fock perturbation theory, based on the Karplus-Kolker method, is examined. To account properly for the coupling effects an alternative approach, called the coupled geometric approximation, is proposed. Appropriate numerical calculations were performed for the electric dipole polarizabilities and dipole shielding factors of two-, four- and ten-electron atomic systems. The first-order corrected Karplus-Kolker second-order energies are usually far from the corresponding coupled Hartree-Fock results. When the coupling effects are negligible, the geometric approximation gives quite satisfactory results. In the case of large coupling terms, the coupled geometric approach leads to an essential improvement of the calculated second-order atomic properties. These final results are very close to the corresponding coupled Hartree-Fock values.

Approximate variation-perturbation calculations of the second-order molecular properties

The variation-perturbation technique is applied to the calculation of the electric polarizability tensor of diatomic molecules within the CNDO/2 approximation. The results obtained in this paper are similar to those found in the non-empirical variation-perturbation approach within the minimal STO basis set. They are also compared with other CNDO/2 calculations and with existing experimental data. It follows from this comparison that the CNDO/2 variation-perturbation method does not allow for the calculation of the polarizability anisotropies. On the other hand, the rotational average of the polarizability tensor can be predicted in agreement with both the experimental data and nonempirical results.

Nuclear Corrections to Electronic Expectation Values: Zero-Point Vibrational Effects in the Water Molecule

A variation–perturbation approach is used to determine nuclear corrections to electronic properties of polyatomic molecules. To illustrate the technique, general expressions for the first-order zero-point vibrational corrections are derived and applied to various one-electron properties of the water molecule. These include the electric dipole and molecular quadrupole moments, the diamagnetic susceptibility, the diamagnetic shielding constants, the quadrupole coupling constants, and the Hellmann–Feynman forces. The numerical results, which depend upon the combined use of the LCAO MO CI wavefunctions of Reeves and Boys and of the experimental data of Papous̆ek and Pliva, show that electronic expectation values need to be vibrationally corrected for quantitative comparisons of theory with experiment. For the water molecule, these corrections are typically about 1%, but can be as large as 10%, of the equilibrium values. The net sign of each correction is determined by the symmetric stretching mode in H2O and D2O, and the O–H and/or O–D stretch in HDO. Of particular importance to the proper treatment of the vibrational average is a potential surface containing the proper number of appropriately chosen symmetric and asymmetric configurations of nuclei.

Studies in Raman Intensity Theory

Raman intensity theory is investigated with the Kramers–Heisenberg dispersion theory as a basis. The sum over excited states is transformed into an infinite series in ground-state matrix elements involving an average energy. Particular choice of this average energy and neglect of all terms N ≥ 2 amount to the same approximations as in a variation–perturbation approach used by others. This truncated series, after a transformation through a quantum mechanical identity, assumes a particularly simple form. Calculations using this simple expression are presented for both the H2+ ion and the H2 molecule for several approximate wavefunctions. In several additional calculations for the H2 molecule, a more drastic approximation is employed where it is assumed that the major contribution to the derived polarizability is from the normalization factor of the wavefunction rather than from the detail of the wavefunction itself. The results of these calculations are compared with other calculations and, in the case of H2, also with experimental data. The qualitative success is encouraging. Finally, tentative explanations, based on the present approximation, are offered for some common empirical rules in Raman spectroscopy. These concern (1) the additivity properties for various bond types, (2) the fact that highly ionic bonds tend to exhibit weak Raman intensity while covalent bonds exhibit strong intensity, and (3) that totally symmetric vibrations almost always exhibit significantly greater intensity than do nontotally symmetric vibrations.

Many-Electron Theory of Nonclosed-Shell Atoms and Molecules. II. Variational Theory

The theory developed in I is generalized into a variational many-electron theory by making the first-order correlation processes in the wavefunction ``exact to all orders'' and by including the effects of unlinked clusters. Equations for obtaining the significant correlation effects are derived by the variation—perturbation approach. Symmetry properties of the correlation functions and a semiempirical theory based on the form of the variational energy are also discussed. The examples of lithium, 1S-carbon, and C2H4+ illustrate some aspects of the theory.

A Variation-Perturbation Approach to the Interaction of Radiation with Atoms and Molecules

A variation-perturbation formulation for time-dependent interactions is presented and applied to the interaction between atoms or molecules and a semiclassical radiation field. Details of the treatment through second order are given for the steady-state problem and are compared with the standard perturbation-theory results. It is demonstrated that the usual expression for the time-dependent perturbation solution, which consists of a sum over an infinite set of excited states (including any continuum), can be replaced by a small number of frequency-dependent functions that satisfy certain time-independent differential equations. Since an exact solution to these equations is very difficult for all but the simplest cases, an equivalent variational formulation of more general utility is introduced. The induced time-dependent electric and magnetic moments of atoms and molecules in an oscillating electromagnetic field are evaluated in terms of the frequency-dependent functions. With the usual continuum medium assumptions, the macroscopic polarizations are determined from these moments, and simple formulas for the index of refraction and optical rotatory power are obtained. The method is illustrated by a simple application to the dynamic polarizability of the harmonic oscillator and by a more detailed examination of the dynamic polarizability and index of refraction of the nonrelativistic hydrogen atom. A comparison is made between an exact solution and a variational approximation for this problem. The excellent agreement between the two methods provides an indication of the power of the variational formulation. An indication of its utility for more general applications is given.

Variation-Perturbation Approach to the Interaction of Radiation with Matter

A variation-perturbation technique is outlined as an alternative to the ordinary time-dependent perturbation method. The essential point of the technique is to factor out the effective time dependence of the perturbed wavefunction and to reduce the remaining problem to the solution of time- dependent inhomogeneous differential equations. For simple cases a direct integration is possible, while a variational approach is applicable to more complex equations. (P.C.H.)