Wave functions, single-particle, variational

Writing the Euler-Lagrange equations in terms of the single-particle wave functions (tpi) the variation principle finally leads to the effective singleelectron equation, well-known as the Kohn-Sham (KS) equation ... 

In the next step, which is numerically the most demanding, the differential equations (3) are solved. Two possible strategies using a variational expansion of the single particle wave functions, /., are described below. After the eigenvalues and eigenfunctions have been found, a new ("output") charge density can be... 

Earlier studies of positron-molecule elastic scattering did not involve such detailed descriptions of the scattering process as do the variational and R-matrix formulations. Instead, the interaction between the positron and the molecule was represented by a relatively simple model potential, and the positron wave function F(r 1) was assumed to satisfy the equivalent single-particle Schrodinger equation... 

One of the earliest approximations for W is due to Hartree, who considered independent el trons moving in the field of the other electrons in the system. The variational wave function has the form of a product of single-particle functions, i.e.. 

The static and dynamic mean-field equations are derived, for a given energy-density functional, by variation with respect to the single-particle wave functions Ritz variational principle of minimal energy yields the static mean-field equations,... 

Considering die bosonic nature of the particles (the wave function has to be symmetric see Chapter 1), we will use = (/>( )(/) (2) as a variational function, where is a normalized spinorbital. This represents a restriction analogous to taking a single Slater determinant because an exact wave function should not be =

[Pg.399]

But, since the hamiltonian (Eq. 1) is not a sum of single-particle hamiltonians, the true wave function can not be written in the product form of Eq. 2, which furthermore does not have the required antisymmetry property. Nevertheless, a product wave function of any form, and thus also of the form of Eq. 2, can be used in the variational principle of quantum mechanics ... 

We have already pointed out that the situations in which a single-determinant wave function can be used as a reference for a correlated method are much fewer in relativistic theory than in nonrelativistic theory. We must therefore resort to methods that do not assume a single-determinant reference. Whether we treat dynamical correlation perturbationally or variationally, it is usually the case that we need to obtain eigenfunctions of some A-particle Hamiltonian, and so we turn to Cl methods. 

It is interesting to note that in the extreme Kondo limit, i.e. nf-> 1, the pole of the resonance level disappears from the f level spectrum. The existence of a pole in the unoccupied part of the spectrum gives hint to an excited state of the system. A suitable trial wave function for the existed state is defined in eq. (49), because it has the correct variational energy. In this state, which is doubly degenerate, the singlet correlation between f and band electrons is broken. One can think of the resonance level as a single-particle state (Liu 1989a), and the operator which puts an electron in this state is... 

MOs first appear in the framework of the Hartree-Fock (HF) method, which is a mean-field treatment [17,22]. The basic idea is to start from an A-particle wave-function that is appropriate for a system of non-interacting electrons. Having fixed the Ansatz for the A-particle wavefunetion in this way, the variational principle is used in order to obtain the best possible approximation for the fully interacting system. Such independent particle wavefunctions are Slater-determinants, which consist of antisymmetrized products of single-particle wavefunctions (x)J (the antisymmetry brought about by the determinantal form is essential in order to satisfy die Pauh principle). Thus, the Slater-determinant is written as... 

In closing this section we emphasize that the MCTDH equations of motion conserve the norm of the wave function and, for time-independent Hamiltonians, the mean energy. This follows directly from the variational principle. Moreover, the MCTDH wave function converges toward the numerically exact wave function with increasing numbers of single-particle functions. 

For the correlation energy no general explicit expression is known, neither in terms of orbitals nor in terms of densities. A simple way to understand the origin of correlation is to recall that the Hartree energy is obtained in a variational calculation in which the many-body wave function is approximated as a product of single-particle orbitals. Use of an antisymmetrized product (a Slater determinant) produces the Hartree energy and the exchange... 

---