Variation-perturbation approach

Because of the difficulty in evaluating the infinite sums needed to find Ea E°. .., one sometimes uses a variation-perturbation approach, in which E 2 E. .. are evaluated by looking for functions that minimize certain integrals involving H see Hameka, p. 223, for details. 

To evaluate ot theoretically, one must use perturbation theory. Ramsey did so (see Murrell and Harget, pp. 121-123), and found that o is the sum of two terms, a positive term od (called the diamagnetic contribution, since it decreases the applied field), and a negative term op (the paramagnetic contribution). The term op involves the usual perturbation-theory sum over excited states, and therefore is difficult to calculate however, one can use a variation-perturbation approach (Section 1.10) in op calculations. For molecular protons, od exceeds op, and aH is positive. 

Because of the difficulty in evaluating the paramagnetic contribution, ab initio calculations of NMR shielding constants are few in number and limited to rather small molecules. Jaszunski and Sadlej used a variational perturbation approach with an ab initio SCF wave function of H20 and found aH = 28.3 ppm. [M. Jaszunski and A. J. Sadlej, Theor. Chim. Acta, 27, 135 (1972).] The known proton screening constant in H2 is 26.6 ppm,6 and gaseous H20 shows a proton chemical shift of 3.6 ppm relative to gaseous H2 hence the experimental H20 proton shielding constant is 30.2 ppm. 

Interatomic Force Constants (IFCs) are the proportionality coefficients between the displacements of atoms from their equilibrium positions and the forces they induce on other atoms (or themselves). Their knowledge allows to build vibrational eigenfrequencies and eigenvectors of solids. This paper describes IFCs for different solids (SiC>2-quartz, SiC>2-stishovite, BaTiC>3, Si) obtained within the Local-Density Approximation to Density-Functional Theory. An efficient variation-perturbation approach has been used to extract the linear response of wavefunctions and density to atomic displacements. In mixed ionic-covalent solids, like SiC>2 or BaTiC>3, the careful treatment of the long-range IFCs is mandatory for a correct description of the eigenfrequencies. 

The knowledge of these quantities, coupled with the equation of motion, allows to find the vibrational eigenfunctions and eigenvalues, in the harmonic approximation. In order to compute the IFCs, a variation-perturbation approach to the DFT has been used. 

We wish to end this section by saying that the variation-perturbation approach as discussed above, introduces a natural hierarchy of gradually more and more sophisticated models starting from the crude evaluation of the electrostatic energy in the distributed multipole approximation, and ending with the inclusion of the intramolecular and dispersion contributions at the MP2 or even more correlated level. 

Nunes and Gonze [153] have recently extended DFPT to static responses of insulating ciystals for any order of perturbation theory by combining the variation perturbation approach with the modern theory of polarization [154]. There are evident similarities between this formalism and (a) the developments of Sipe and collaborators [117,121,123] within the independent particle approximation and (b) the recent work of Bishop, Gu and Kirtman [24, 155,156] at the time-dependent Hartree Fock level for one-dimensional periodic systems. 

Astrand, P.-O., Ruud, K., Sundholm, D. A modified variation-perturbation approach to zero-point vibrational motion. Theor. Chem. Acc. 103, 365-373 (2000)... 

A variation perturbation approach has recently been given by Gallup and Gerratt We begin from the secular equation for the lowest eigenvalue Eq of... 

These effects are examined in Sections XVIII, XIX, and XX after we have listed which ones are involved in various theories. These theories can be derived from the exact x and E by a general variation-perturbation approach. This approach, which also allows one to estimate the errors left after a certain approximation has been made, is described first. 

The variation-perturbation approach then briefly consists of (a) minimizing a large portion (which has a stationary or minimum point) of E, Eq. (58), to obtain a trial function and (b) substituting this back into the entire E to get an upper limit to E. 

The complete form of the Many-Electron Theory , which is the main topic of this article, however, is not a perturbation theory. Both the many-electron theory and Brueckner-type theories are now derived from the exact % and E by the general variation-perturbation approach. The approach which we call variation-perturbation for lack of a better name should not be confused with perturbation theory. 

By far, the largest effects in %, Eq. (20), are the pair correlations, Ofj. They should therefore be obtained as accurately as possible. TTie remaining effects need only be estimated, provided they are indeed small. In the forthcoming sections we shall obtain the pairs exactly (i.e. to all "orders ) from the major portions of the exact x and the energy according to the variation-perturbation approach, and then examine and estimate the / and the many-electron effects (Sections XIX and XVIII). Both in concept and practice, this leads to very simple results. 

The M,/s in it will be obtained (Section XXII) by minimizing a large portion of E according to the variation-perturbation approach. They will then also give us estimates of the correction... 

This paper is a tribute to Professor Osvaldo Goscinski and his scientific work. Our scientific roads have crossed only occasionally, but there is a seminal paper that had a strong influence on our work on the direct Cl method. This is the paper from 1970 on the variational-perturbation approach [25]. Per Siegbahn and 1 used this approach in the early development of the direct Cl method and the convergence rate of the secular problem was increased by an... 

O. Goscinski and O. Sinanoglu Upper and Lower Boimds and the Generalized Variation-Perturbation Approach to Many Electron Theory Int. J. Quantum Chem. 2, 397 (1968). 

The response functions theory the PCM method [1] is an extension of the response theory for molecules in the gas phase [2, 3], This latter is based on a variational-perturbation approach for the description of the variations of the electronic wave function and of the changes of the observables properties at the various orders of perturbation with respect to the perturbing fields, and no restrictions are posed on the nature of the observables and on the nature of the perturbing fields, and the theory gives also access to a direct determination of the transition properties (i.e. transition energies and transition probabilities) associated with transitions between the stationary states of the molecular systems. The PCM response theory adds to this framework several new elements. 

In Tables 1 and we 2 compare some components obtained in variation-perturbation approach [4] and other variational decompositions ]22-23] implemented in popular GAMESS system [24] with the most accurate benchmark... 

To develop the variation-perturbation approach we may suppose that 1 is an approximation to the required eigenfunction and that we wish to improve this approximation by adding functions 02> and solving... 

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