Variation-perturbation approach many-electron theory

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. 

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). 

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. 

An alternative approach to the perturbation theory in treating many-electron systems is the configuration-interaction (Cl) method which is based on the variational principle. Nonrelativistic Cl techniques have been used extensively in atomic and molecular calculations. The generalization to relativistic configuration-interaction (RCI) calculations, however, presents theoretical as well as technical challenges. The problem originates from the many-electron Dirac Hamiltonian commonly used in RCI calculations ... 

More appropriate than perturbation approaches for improving on the energy are variational approaches (under the specific caveats discussed in chapter 8 with respect to the negative-energy states), because the total electronic energies obtained are much better controlled, an essential property since the exact reference is not known for any interesting many-electron molecule. In particular, we shall address the second-generation MCSCF methods mentioned in chapter 8. For reference to molecular Cl and CC theory, please consult sections 8.5.2 and 8.9, respectively. 

The technique for dealing with this problem is well known from nonrelativistic calculations on many-electron systems. One-particle basis sets are developed by considering the behavior of the single electron in the mean field of all the other electrons, and while this neglects a smaller part of the interaction energy, the electron correlation, it provides a suitable starting point for further variational or perturbational treatments to recover more of the electron-electron interaction. It is only natural to pursue the same approach for the relativistic case. Thus one may proceed to construct a mean-field method that can be used as a basis for the perturbation theory of QED. In particular, the inclusion of the Breit interaction in the mean-field calculations ensures that the terms of O(a ) are included to infinite order in QED. 

The Moller-Plesset method uses perturbation theory to correct for the electron correlation in a many-electron system. The Moller-Plesset method has the advantage that it is a computationally faster approach than Cl computations however, the disadvantage is that it is not Variational. A non-Variational result is not, in general, an upper bound of the trae ground-state energy. In the MoUer-Plesset method, the zero-order Hamiltonian is defined as the sum of all the N one-electron Hartree-Fock Hamiltonians, H", as given in Equation 9-30. 

Recent calculations of the band structure of polyethylene have employed variations of the ab initio method incorporating electron correlation. Sun and Bartlett (1996) utilised many-body perturbation theory to encompass electron correlation in the ab initio framework. Siile et al. (2000) and Serra et al. (2000) have employed variants of DFT. These calculations involved the optimisation of local effective potentials and a local-density approximation respectively. Figure 4.13 shows a comparison of the band structure obtained by Siile et al., Fig. 4.13(d), with those obtained by other ab initio DFT calculations using the Hartree-Fock (HF), Fig 4.13(a), and Slater approaches, Figs. 4.13(b) and (c). 

One seemingly sensible approach to the relativistic electronic structure theory is to employ perturbation theory. This has the apparent advantage of representing supposedly small relativistic effects as corrections to a familiar non-relativistic problem. In Appendix 4 of Methods of molecular quantum mechanics, we find the terms which arise in the reduction of the Dirac-Coulomb-Breit operator to Breit-Pauli form by use of the Foldy-Wouthuysen transformation, broken into electronic, nuclear, and electron-nuclear effects. FVom a purely aesthetic point of view, this approach immediately looks rather unattractive because of the proliferation of terms at the first order of perturbation theory. To make matters worse, many of the terms listed are singular, and it is presumably the variational divergences introduced by these operators which are referred to in [2]. Worse still, higher-order terms in the Foldy-Wouthuysen transformation used in this way yield a mathematically invalid expansion. 

In order to improve the mean field description of the electronic structure one has to go beyond the single-configuration approach. [12, 13] IWo main strategies have been developed to introduce correlation effects. In the first case, one employs methods based on many-body perturbation theory (MBPT). [12, 21] They allow the treatment of so-called dynamical correlation effects in cases where the HF method already provides a reasonable description of the ground state. However, these perturbation theoretical methods are not variational, that is the calculated value for the energy does not provide an upper bound to the true energy of the system. 

Other methods which go beyond the Hartree-Fock level of approximation include Cluster Methods and Many-Body Perturbation Theory (Wilson 1984). These approaches involve the introduction of repulsion effects due to simultaneous interactions between three, four, and even more electrons in the expansion of the wavefunction. One important drawback of cluster methods and many-body perturbation theory is that they are not variational. That is to say, the calculated energies no longer represent upper bounds and it is possible to obtain predictions in excess of 100% of the experimental values. Nevertheless, their use is capable of reducing the error in the calculation of the energy of the helium atom to something of the order of lO %. 

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