Variation-perturbation procedure

The variation-perturbation procedure is capable of yielding a lower bound to the exact polarizability provided that the exact is known. 

For non-empirical calculations it is not necessary that R be entirely negligible. It only has to be smaller than [2i>i ] The variation-perturbation procedure then applies. The larger XSlf part gives the iJ[Pg.358]

Sections XVI through XXI show that after degeneracies are removed the main effects are the pair correlations, Eqs. (77) and (82) or (85). These are now to be obtained according to the variation-perturbation procedure by minimizing a large portion of E, Eq. (82). Once determined they will also give estimates of ft and R. Properties of pair functions, their relation to the states of actual, free two-electron systems and methods for their evaluation have been given in detail previously. The results are summarized here. 

According to the variation-perturbation procedure, one would now minimize each with trial functions subject to... 

The coupling terms A are easily estimated. They are much smaller than the (2 >p ip ) term so that the variation-perturbation procedure applies (Section XIV). The calculated from Eqs. (157b) alone can be used to calculate the A as well. 

Equation (3.17) is then solved by a variational-perturbative procedure, where the coupled-cluster T and A amplitudes are expanded with respect to the components of the perturbation V(t) ... 

Many-body calculations which go beyond the Hartree-Fock model can be performed in two ways, i.e. using either a variational or a perturbational procedure. There are a number of variational methods which account for correlation effects superposition-of-configurations (or configuration interaction (Cl)), random phase approximation with exchange, method of incomplete separation of variables, multi-configuration Hartree-Fock (MCHF) approach, etc. However, to date only Cl and MCHF methods and some simple versions of perturbation theory are practically exploited for theoretical studies of many-electron atoms and ions. 

Quantum mechanical calculations are carried out using the Variational theorem and the Har-tree-Fock-Roothaan equations.t - Solution of the Hartree-Fock-Roothaan equations must be carried out in an iterative fashion. This procedure has been called self-consistent field (SCF) theory, because each electron is calculated as interacting with a general field of all the other electrons. This process underestimates the electron correlation. In nature, electronic motion is correlated such that electrons avoid one another. There are perturbation procedures whereby one may carry out post-Hartree-Fock calculations to take electron correlation effects into account. " It is generally agreed that electron correlation gives more accurate results, particularly in terms of energy. 

The frequency dependence of the reaction field factor is calculated by using a frequency dependent dielectric permittivity r,(ro), which is an experimental quantity related to the solvent. Computation of the frequency dependent multipole polarizabilities is feasible, in principle, by perturbation techniques. Nevertheless tlris procedure is tedious and one generally prefers some variation-perturbation scheme [20]. In addition, such a computation is still limited to small systems and can scarcely be extended economically to molecules of chemical interest. Hence a further simplification has been proposed. It consists in assuming that the quantitiesand are... 

For more substantial structural differences between conformers involving multiple dihedral angles, importance sampling or the perturbation procedure are probably the best approaches using a one-dimensional reaction path with concerted variations in the angles. Obtaining even a complete two-dimensional map for the solvent effect, e.g. is computationally too demanding. Adequate simulations... 

We now follow the familiar procedure of perturbation theory to extract from eqn 1 the first order expression of the variations indieated by A. Let us start from the normalization condition, and denote by M.j the j-th eolumn of any given matrix M. Since both C,j and Cj are normalized to unity, to first order in AC, as has been mentioned, N may be taken to be unity, and we must have ... 

In this section we examine the ground-state energy of the helium atom by means of both perturbation theory and the variation method. We may then compare the accuracy of the two procedures. 

Chapters 7 and 8 discuss spin and identical particles, respectively, and each chapter introduces an additional postulate. The treatment in Chapter 7 is limited to spin one-half particles, since these are the particles of interest to chemists. Chapter 8 provides the link between quantum mechanics and statistical mechanics. To emphasize that link, the ffee-electron gas and Bose-Einstein condensation are discussed. Chapter 9 presents two approximation procedures, the variation method and perturbation theory, while Chapter 10 treats molecular structure and nuclear motion. 

Quantum-mechanical approximation methods can be classified into three generic types (1) variational, (2) perturbative, and (3) density functional. The first two can be systematically improved toward exactness, but a systematic correction procedure is generally lacking in the third case. 

The Breit-Pauli Hamiltonian is an approximation up to 1/c2 to the Dirac-Coulomb-Breit Hamiltonian obtained from a free-particle Foldy-Wouthuysen transformation. Because of the convergence issues mentioned in the preceding section, the Breit-Pauli Hamiltonian may only be employed in perturbation theory and not in a variational procedure. The derivation of the Breit-Pauli Hamiltonian is tedious (21). 

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