Variation-perturbation method

As an alternative to the Ritz variational method, a variational perturbation method based on a perturbation expansion can be utilized. Here, the Hamiltonian is separated into an unperturbed Hamiltonian and a perturbing potential,... 

Various semi-empirical methods have been compared for all properties in an important review by Klopman and O Leary, whilst Adams et < /. have compared the finite field, the variation, and the second-order infinite sum methods for the calculation of a in DNA bases. They find that the variation-perturbation method gives the most reliable results, but as their calculations were at the iterative extended Hiickel level there is no guarantee as to the generality of their conclusions. 

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. 

Initial results obtained for TPA and for photoelectron spectra of small systems, show that anharmonicity must be included in the calculation of EC factors to reproduce experiment [54, 77, 104]. However, it is difficult to treat larger anharmonic systems by means of perturbation theory. Such systems can be handled by applying the variation/perturbation methods of electronic structure theory that have been, and continue to be, extended to the vibrational Schrodinger equation as discussed earlier. The EC integrals tliat appear in the equations for resonant (hyper)polarizabilities may be calculated employing approaches like VSCF, VMP2, VCI and VCC, That will allow us to Include anharmonic contributions to all orders and thereby remove the intrinsic limitations of the perturbation expansion in terms of normal coordinates. 

The Variation-Perturbation Method. The variation-perturbation method allows one to accurately estimate EP and higher-order perturbation-theory energy corrections for the ground state of a system without evduating the infinite sum in (9.36). The method is based on the inequality... 

Extended articles on the most common electron correlation methods such as limited configuration interaction (Cl see Configuration Interaction), M0ller-Plesset many-body perturbation theory (MBPT see M0ller Plesset Perturbation Theory), variation-perturbation methods (such as PCILO see Complete Active Space Self-consistent Field (CASSCF) Second-order Perturbation Theory (CASPT2) and Configuration Interaction), and coupled cluster theory (CC see Coupled-cluster Theory), as well as on explicitly ri2-dependent wave functions (see rxi Dependent Wave functions), can be found elsewhere. 

The next step towards increasing the accuracy in estimating molecular properties is to use different contributions for atoms in different hybridi2ation states. This simple extension is sufficient to reproduce mean molecular polarizabilities to within 1-3 % of the experimental value. The estimation of mean molecular polarizabilities from atomic refractions has a long history, dating back to around 1911 [7], Miller and Sav-chik were the first to propose a method that considered atom hybridization in which each atom is characterized by its state of atomic hybridization [8]. They derived a formula for calculating these contributions on the basis of a theoretical interpretation of variational perturbation results and on the basis of molecular orbital theory. 

I have elected to include a discussion of the variational principle and perturbational methods, although these are often covered in courses in elementary quantum mechanics. The properties of angular momentum coupling are used at the level of knowing the difference between a singlet and a triplet state. 1 do not believe that it is necessary to understand the details of vector coupling to understand the implications. 

The idea of coupling variational and perturbational methods is nowadays gaining wider and wider acceptance in the quantum chemistry community. The background philosophy is to realize the best blend of a well-defined theoretical plateau provided by the application of the variational principle coupled to the computational efficiency of the perturbation techniques. [29-34]. In that sense, the aim of these approaches is to improve a limited Configuration Interaction (Cl) wavefunction by a perturbation treatment. 

In order to overcome the optimization process of the (hyper) polarizabilities calculations, we have been led to deeply study the perturbational and variational methods and in particular the variation-perturbation treatment introduced by Hylleras (20) since 1930. We will not develop here the theoretical framework of the recent study of N. El Bakali Kassimi (21). We propose criteria for generating adequate sets of polarization functions necessary to calculate (hyper) polarizabilities. 

We have first been concerned with the computational point of view. Through the calculation of the dynamic polarizability of CO, we have developed a method based on the conventional SCF-Cl method, using the variational- perturbation techniques the first-order wavefunction includes two parts (i) the traditional one, developed over the excited states and (ii) additional terms obtained by multiplying the zeroth—order function by a polynomial of first-order in the electronic coordinates. This dipolar... 

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]. 

A different approach to obtaining approximation solutions to quantum mechanical problems is provided by the variation method. It is particularly useful when there is no closely related problem that yields exact solutions. The perturbation method is not applicable in such a case. 

Wong K-Y, Gao J (2007) An automated integration-free path-integral method based on Kleinert s variational perturbation theory. J Chem Phys 127(21) 211103... 

Although we cannot solve the wave equation for the helium atom exactly, the approaches described provide some insight in regard to how we might proceed in cases where approximations must be made. The two major approximation methods are known as the variation and perturbation methods. For details of these methods as applied to the wave equation for the helium atom, see the quantum... 

An important advantage of MP2 and higher-order perturbation methods is their size-consistency at every order. This is in contrast to many variational Cl methods, for which the calculated energy of two identical non-interacting systems might not be equal to twice that of an individual system. Size-consistent scaling is also characteristic of QCI and CC methods, which are therefore preferable to standard Cl-type variational methods for many applications. 

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