Perturbation type approximations

Algorithmic details of the NRT variational minimization (5.61) differ somewhat for dominant reference structures (where all elements of the density operators are considered) and weaker secondary structures (which are treated by a simpler perturbative-type approximation involving diagonal density operator elements only). In each case, the variational minimization of (5.61) can be equivalently expressed as the majdmization of a corresponding fractional improvement y(vv) (for secondary structures) or F(W) (for reference structures) that expresses the percentage reduction of the multiresonance error in (5.61) from its initial single-resonance value. Of course, there is no assurance that this error can be reduced to zero [i.e., that or F(1V) can achieve 100% accuracy], because the... 

Next, we present some observations concerning the connection between the reconstruction process and the iterative solution of either CSE(p) or ICSE(p). The perturbative reconstruction functionals mentioned earlier each constitute a finite-order ladder-type approximation to the 3- and 4-RDMCs [46, 69] examples of the lowest-order corrections of this type are shown in Fig. 3. The hatched squares in these diagrams can be thought of as arising from the 2-RDM, which serves as an effective pair interaction for a form of many-body perturbation theory. Ordinarily, ladder-type perturbation expansions neglect three-electron (and higher) correlations, even when extended to infinite order in the effective pair interaction [46, 69], but iterative solution of the CSEs (or ICSEs) helps to... 

The main approximation methods belong to two categories variation and perturbation type. 

The main problems involved with the Levich-Dogonadze model are those of agreement with experiment and of the fundamental hypotheses used. They are probably correct in assuming that the transition between the different electronic terms should, in principle, be calculated (if this is possible) using the perturbation theory approximation for a quantum transition probability (w) from a process of Landau-Zener type, i.e.. 

If the predissociation line is sharp, indicating only a small probability of the resonance state breaking up, then a perturbation type approach may be used. This approach is very clearly described in Shapiro s paper of 1972 where he introduces and tests out numerical procedures for evaluating the bound-continuum integrals needed in both this approximate perturbation (or so called golden rule) approach and also in the exact theory of photodissociation processes. The basic theory of the golden rule expressions has been presented by Levine.It has also been carefully derived by Beswick and Jortner who have used it in a pioneering study of vibrational predissociation. 

Most of the techniques described in this Chapter are of the ab initio type. This means that they attempt to compute electronic state energies and other physical properties, as functions of the positions of the nuclei, from first principles without the use or knowledge of experimental input. Although perturbation theory or the variational method may be used to generate the working equations of a particular method, and although finite atomic orbital basis sets are nearly always utilized, these approximations do not involve fitting to known experimental data. They represent approximations that can be systematically improved as the level of treatment is enhanced. 

These compounds have been the subject of several theoretical [7,11,13,20)] and experimental[21] studies. Ward and Elliott [20] measured the dynamic y hyperpolarizability of butadiene and hexatriene in the vapour phase by means of the dc-SHG technique. Waite and Papadopoulos[7,ll] computed static y values, using a Mac Weeny type Coupled Hartree-Fock Perturbation Theory (CHFPT) in the CNDO approximation, and an extended basis set. Kurtz [15] evaluated by means of a finite perturbation technique at the MNDO level [17] and using the AMI [22] and PM3[23] parametrizations, the mean y values of a series of polyenes containing from 2 to 11 unit cells. At the ab initio level, Hurst et al. [13] and Chopra et al. [20] studied basis sets effects on and y. It appeared that diffuse orbitals must be included in the basis set in order to describe correctly the external part of the molecules which is the most sensitive to the electrical perturbation and to ensure the obtention of accurate values of the calculated properties. 

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. 

We applied the Liouville-von Neumann (LvN) method, a canonical method, to nonequilibrium quantum phase transitions. The essential idea of the LvN method is first to solve the LvN equation and then to find exact wave functionals of time-dependent quantum systems. The LvN method has several advantages that it can easily incorporate thermal theory in terms of density operators and that it can also be extended to thermofield dynamics (TFD) by using the time-dependent creation and annihilation operators, invariant operators. Combined with the oscillator representation, the LvN method provides the Fock space of a Hartree-Fock type quadratic part of the Hamiltonian, and further allows to improve wave functionals systematically either by the Green function or perturbation technique. In this sense the LvN method goes beyond the Hartree-Fock approximation. 

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