A variation-perturbation approach

For the purposes of this section, we write the Hamiltonian in the form (11.3.9), namely 

There are many ways of dealing with the small terms, most of them perturbational in character and presupposing knowledge (at least in principle) of exact eigenfunctions of Hq. We develop only one approach, based on the variation-perturbation theory of Section 2.5, which has the merits of simplicity, flexibility and generality and does not require knowledge of exact unperturbed wavefunctions. This approach also lends itself well to the physical interpretation of the mode of interaction of the small terms. 

Generally speaking, we shall be interested in some particular set of degenerate eigenstates of Hq, and the effect on these states of the perturbation H. In the language of Section 2.5, these states form the A set , containing, say, approximate wavefunctions the fi set  

As a first approximation, E in (11.4.4) is given the value E appropriate to the unperturbed A set, and the inverse matrix is expanded in order to obtain an explicit, order-by-order form of Heff. 

It will now be assumed that the functions 0., although not exact eigenfunctions of Hq, are variationally determined approximations, in which case the matrix of Hq will be diagonal  

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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. 

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. 

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

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 this paper a method [11], which allows for an a priori BSSE removal at the SCF level, is for the first time applied to interaction densities studies. This computational protocol which has been called SCF-MI (Self-Consistent Field for Molecular Interactions) to highlight its relationship to the standard Roothaan equations and its special usefulness in the evaluation of molecular interactions, has recently been successfully used [11-13] for evaluating Eint in a number of intermolecular complexes. Comparison of standard SCF interaction densities with those obtained from the SCF-MI approach should shed light on the effects of BSSE removal. Such effects may then be compared with those deriving from the introduction of Coulomb correlation corrections. To this aim, we adopt a variational perturbative valence bond (VB) approach that uses orbitals derived from the SCF-MI step and thus maintains a BSSE-free picture. Finally, no bias should be introduced in our study by the particular approach chosen to analyze the observed charge density rearrangements. Therefore, not a model but a theory which is firmly rooted in Quantum Mechanics, applied directly to the electron density p and giving quantitative answers, is to be adopted. Bader s Quantum Theory of Atoms in Molecules (QTAM) [14, 15] meets nicely all these requirements. Such a theory has also been recently applied to molecular crystals as a valid tool to rationalize and quantitatively detect crystal field effects on the molecular densities [16-18]. 

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. 

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. 

Banerjee and Harbola [69] have worked out a variation perturbation method within the hydrodynamic approach to the time-dependent density functional theory (TDDFT) in order to evaluate the linear and nonlinear responses of alkali metal clusters. They employed the spherical jellium background model to determine the static and degenerate four-wave mixing (DFWM) y and showed that y evolves almost linearly with the number of atoms in the cluster. 

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)... 

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. 

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... 

The second-order perturbation expressions that Stacey derived for ratios of linear and bilinear functionals are not unique to the variational approach. In this section these expressions were obtained with a straightforward perturbation approach (50). Moreover, Seki (63) has shown that the Usachev-Gandini approach can be used to derive a GPT expression for the static reactivity for perturbations that do not retain criticality. This does not detract from the value of the variational approach. The latter provides stationary functionals for all kinds of integral parameters from which one can obtain exact expressions for these parameters [e.g., Eq. (140) for the static reactivity] as well as a consistent derivation of the equations for the distribution functions and of constraints imposed on them. 

Alternatively, there are perturbation methods to estimate Ecorreiation- Briefly, in these methods, you take the HF wavefunction and add a correction—a perturbation—that better mimics a multi-body problem. Moller-Plesset theory is a common perturbative approach. It is called MP2 when perturbations up to second order are considered, MP3 for third order, MP4, etc. MP2 calculations are commonly used. Like CISD, MP2 allows single and double excitations, but the effects of their inclusion are evaluated using second-order perturbation theory rather than variationally as in CISD. An even more accurate type of perturbation theory is called coupled-cluster theory. CCSD (coupled-cluster theory, singles and doubles) includes single and double excitations, but their effects are evaluated at a much higher level of perturbation theory than in an MP2 calculation. 

MNDOC is a correlated version of MNDO. Unlike all previously discussed methods, MNDOC includes electron correlation explicitly and thus differs from MNDO at the level of the underlying quantum chemical approach (a) while being completely analogous to MNDO in all other aspects (b)-(d) except for the actual values of the parameters. In MNDOC electron correlation is treated conceptually by full configuration interaction, and practically by second-order perturbation theory in simple cases (e.g., closed-shell ground states) and by a variation-perturbation treatment in more complicated cases (e.g., electronically excited states).The MNDOC parameters have been determined at the correlated level and should thus be appropriate in all MNDO-type applications which require an explicit correlation treatment for a qualitatively suitable zero-order description. In closed-shell ground states... 

Truncation at the first-order temi is justified when the higher-order tenns can be neglected. Wlien pe higher-order tenns small. One choice exploits the fact that a, which is the mean value of the perturbation over the reference system, provides a strict upper bound for the free energy. This is the basis of a variational approach [78, 79] in which the reference system is approximated as hard spheres, whose diameters are chosen to minimize the upper bound for the free energy. The diameter depends on the temperature as well as the density. The method was applied successfiilly to Lennard-Jones fluids, and a small correction for the softness of the repulsive part of the interaction, which differs from hard spheres, was added to improve the results. 

The second aspect is more fundamental. It is related to the very nature of chemistry (quantum chemistry is physics). Chemistry deals with fuzzy objects, like solvent or substituent effects, that are of paramount importance in tautomerism. These effects can be modeled using LFER (Linear Free Energy Relationships), like the famous Hammett and Taft equations, with considerable success. Quantum calculations apply to individual molecules and perturbations remain relatively difficult to consider (an exception is general solvation using an Onsager-type approach). However, preliminary attempts have been made to treat families of compounds in a variational way [81AQ(C)105]. 

The key element in London s approach is the expansion of the electrical potential energy in multipole series. Since neutral molecules or portions of molecules are involved, the leading term is that for dipole-dipole interaction. While attention has been given to higher-order terms, these are usually small, and the greater need seems to be for improved treatment of the dipole-dipole terms. London used second order perturbation theory in his treatment, but Slater and Kirkwood38,21 soon followed with a variation method treatment which yielded similar results. Other individual papers will be mentioned later, but the excellent review of Mar-genau26 should not be overlooked. 

The methodology presented here expands the recent CASPT2 approach to more flexible zeroth-order variational spaces for a multireference perturbation, either in the Moller-Plesset scheme or in Epsein-Nesbet approach [70-72]. Furthermore, it allows for the use of a wide set of possible correlated orbitals. These two last points were discussed elsewhere [34]. 

A more realistic approach to quantify the pressure field is to consider the effect of turbulence [6]. For a pipe flow, the turbulent pressure fluctuations are due to velocity perturbations as a result of the formation of eddies. The instantaneous turbulent velocity can be calculated by assuming a sinusoidal velocity variation in... 

For a reaction of the type shown in (74) with hydroxide ion in excess, the expected variation of the time constant (t-1) for the first-order approach to equilibrium after a temperature perturbation is given by (75). Thus a plot of reciprocal relaxation time (t 1) against hydroxide ion concentration is... 

The exact approach to the problem of dynamic (linear) stability is based on the solution of the equations for small perturbations, and finding eigenvalues and eigenfunctions of these equations. In a conservative system a variational principle may be derived, which determines the exact value of eigenfrequency... 

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