Calculation of molecular systems via coordinate wave functions

Expressions are derived for the matrix elements of the Coulomb interaction of a system of charges consisting of two subsystems, each in a state with a given spin. It is found that the interaction energy of the two subsystems is independent of the spin S₂ of the second subsystem to quantities of the order of the square of the overlap integral for the one-electron orbitals if the first subsystem has spin S[₁=0. Examples are given of calculations by this method. An appendix gives tables of the matrices needed to calculate the interaction energy of two subsystems with a total number of particles from 3 to 6.     

Computation of molecular systems via coordinate wave functions

The total wave function is represented as the product of the coordinate and spin wave functions in order to formulate quantum chemistry problems in a form free from spin wave functions. The permutation symmetry of the coordinate function remembers the spin. Coordinate functions of a given symmetry are constructed via Young operators. The relation of Young's schemes to quantum-chemical structures is discussed.Read at the Symposium on Quantum Chemistry, Palanga, June 1965.     

Computation of molecular systems via coordinate wave functions

A method is presented for constructing the basis functions of the irreducible representations of the molecular point-group symmetry in terms of coordinate wave functions corresponding to a given total spin S of the molecule. The resulting functions are the eigenfunctions of the 2S+1 molecular multiplets and lead to quasi-diagonalization of the secular equation. Examples are given of calculations for the energy levels of a system of four electrons in a tetrahedral field and for those of the six electrons of benzene in the state with S=0.     

Considerations on Gaussian expanded generator coordinate wave functions

The recently developed integral discretization technique for the generator coordinate method is applied to discuss some features of Gaussian expanded wave functions. Using the hydrogen atom ground state, it is shown that the nuclear cusp problem has an interesting relation with the generator coordinator weight function, which allows the production of accurate wave functions by an integration criterion.     

A minimum principle for molecular systems allowing the use of discontinuous wave functions

In this paper the minimum principle proposed for atomic systems by Hall, Hyslop and Rees [1] is generalized to molecules. It is shown that this generalization retains the advantage of admitting the use of a larger class of trial wave functions, for example those with discontinuities, than is possible in the usual minimum energy principle. The further advantage that the upper bounds obtained by this treatment are always at least as good as those of the Rayleigh-Ritz method is also preserved. The theory is applied to the H ion, potential energy curves are obtained for various “cut-off” wave functions, and the equilibrium internuclear distance is calculated. The optimization of the “cut-off” region so that the upper bound is minimized is also discussed.     

Calculation scheme for optimizing multiconfigurational wave functions

Theoretical and Experimental Chemistry -     

Second order coalescence conditions of molecular wave functions

Kato's cusp condition gives the exact first order dependence of molecular wave functions on interparticle separation near the coalescence of two charged particles. We derive conditions correct to second order in interparticle separation, which concern second order derivatives of the wave function at the coalescence point. For identical particle coalescence, we give equations correct to third order. In addition to a universal, particle dependent term, a system and state dependent term arises in the higher order conditions, which we interpret as an effect of Coulombic screening. We apply our analysis to the standard orbital-based methods of quantum chemistry and discuss the implications for Jastrow- and R12-type correlation factors.     

Highly compact wave functions for He-like systems

Wave functions which are compact, but still quite accurate, are extremely valuable as tools for gaining understanding of quantum systems. This paper investigates the use for that purpose of functions that depend exponentially on all the interparticle distances of a few-body system, illustrated by a study of the ground electronic states of the He isoelectronic series (Z from 1 to 10). Using as few as 4 exponential basis functions, it is found that nonrelativistic energies are reproduced to within 38 microhartrees of the exact values, an error far less than for previously reported compact wave functions. Other properties are also well-represented.     

Long-range molecular interaction coefficients computed from frost-model wave functions

The average long-range interaction energy between two molecules can be written as an inverse asymptotic series in the intermolecular separation distanceR. Using Frost-model wave functions, the dispersion coefficients of the first three (R ^–6,R ^–8,R ^–10 terms in the series are obtained. Coefficients of three- and four-body non-additive interaction energies are also calculated and the form of the dispersion interaction when retardation effects are included is examined.     

Analysis of molecular orbital wave functions in terms of valence bond functions for molecular fragments. I. Theory

A method of expansion of molecular orbital wave functions into valence bond (VB ) functions is extended to molecular fragments. The wave function is projected onto a basis of mixed determinants, involving molecular orbitals as well as fragment atomic orbitals, and is further expressed as a linear combination of VB functions, characteristic of structural formulas of the fragment but whose remaining bonds are frozen. Structural weights for the fragment are deduced from this expression. Delocalized molecular orbitals are used as a startpoint, as they are after an ordinary SCF calculation. Wave functions of medium-sized molecules may be analyzed with reasonable storage requirements in a computer.     

Evaluation on multicenter integrals for the hartree-fock-roothaan equations in molecular coordinate systems

Bakinskii State University. Translated from Zhurnal Strukturnoi Khimii, Vol. 32, No. 5, pp. 135–139, September–October, 1991.     

On the molecular electrostatic potentials obtained from CNDO wave functions

The various approximations proposed for computing molecular potentials with CNDO wave functions are tested on the case of guanine and shown to be unable to reproduce correctly the essential fine features of theab initio potential.     

Asymptotic separability of three-body continuum wave functions for Coulomb systems

We consider a general three-body Coulomb system above the threshold for total break-up. For large particle separations the Schrödinger equation for this system is shown to be separable in terms of new suitably defined 6-dimensional parabolic coordinates. Although these coordinates are not orthogonal, mixed second derivative terms in the expression for the kinetic energy do not contribute at large particle separations to the Coulomb modification of free particle states. Rejection of mixed second derivatives at all particle configurations yields a three-body continuum wave function that satisfies exact asymptotic boundary conditions for Coulomb systems and treats all two-body interactions fully symmetric. This analysis splits the wave equation into an ‘unperturbed’ part plus a perturbation where the unperturbed part takes into account all long-range correlations.     

Calculations of molecular wave functions in terms of modulated plane waves

The possibility of representing molecular orbitals in terms of a many centre expansion of modulated plane waves is discussed. All integrals appearing in the usual SCF procedure for the calculation of the energy are obtained in an analytical form. The resulting formulae are rather simple to be evaluated.

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Zusammenfassung Es wird die Möglichkeit, Molekülorbitale mittels einer Mehrzentrenentwicklung von modulierten ebenen Wellen darzustellen, diskutiert. Alle Integrale, die in einer SCF-Rechnung üblichen Typs (z. B. CNDO) erscheinen, werden in einer einfach auszuwertenden analytischen Form angegeben.

     

Calculation of the Stark effect of neon I usingjl coupled wave functions

A method to calculate the Stark shifts and splittings in noble gases is described and results for neon I are presented. The energy values in a static homogeneous electric field are found by diagonalizing the energy matrix numerically. This matrix consists of the energy values of the free atom taken from experiment and of off-diagonal matrix elements of the electric field operator. The latter are computed using wave functions consisting of a radial function of the excited electron found by numerical integration, and of a two-particle (core+electron) spin-orbital part represented by rigorousjl coupling. As an example, the splitting of the levels 6s to 6p is shown and is explained in terms of atomic level positions, the relative size of matrix elements and of selection rules. A nomenclature for the high field Stark effect is developed in accord with group theory.     

Supplementary d and f functions in molecular wave functions: Optimum and nonoptimum exponents

Energy optimization calculations have been carried out to determine the variability of optimum p, d, and f polarization function exponents in molecules containing first- and second-row elements and in normal valency and hypercoordinate species. Optimum exponents were determined for single sets of higher-order functions at both Hartree–Fock and correlated (Moller–Plesset) levels of theory using the Dunning–Hay double zeta and the McLean–Chandler triple zeta basis sets. More detailed calculations were used to test the response to nonoptimum d and f function exponents of the total energy, the optimum geometry, and harmonic stretching frequencies. The variability in optimum exponents and the size of the energy penalties incurred by adopting nonoptimum values reduces the utility of standard exponents for p, d, and f polarization functions. © 1993 John Wiley & Sons, Inc.