Variational principles orbital functional theory

Since the exact ground-state electronic wave function and density can only be approximated for most A-electron systems, a variational theory is needed for the practical case exemplified by an orbital functional theory. As shown in Section 5.1, any rule T 4> defines an orbital functional theory that in principle is exact for ground states. The reference state for any A-electron wave function T determines an orbital energy functional E = Eq + Ec,in which E0 = T + Eh + Ex + V is a sum of explicit orbital functionals, and If is aresidual correlation energy functional. In practice, the combination of exchange and correlation energy is approximated by an orbital functional Exc. 

The problem has now become how to solve for the set of molecular orbital expansion coefficients, c. . Hartree-Fock theory takes advantage of the variational principle, which says that for the ground state of any antisymmetric normalized function of the electronic coordinates, which we will denote H, then the expectation value for the energy corresponding to E will always be greater than the energy for the exact wave function ... 

HF (HF) theory is based on the idea that one takes an antisymmetrized product wavefunction and uses the variational principle to obtain the best possible approximation to the A -particle wavefunction that cannot be represented by such a single determinant. Thus, one inserts the single determinant into the Rayleigh-Ritz functional and performs a constraint variation of the orbitals. The results of the variational process are the famous HF equations that are satisfied by each of the orbitals ... 

Applying the variational principle to the energy given by Eq. 1, Kohn and Sham reformulated the density functional theory by deriving a set of one-electron Hartree-like equations leading to the Kohn-Sham orbitals v().(r) involved in the calculation of p(r)15. The Kohn-Sham (KS) equations are written as follows ... 

It is clear from the preceding sections that the powerful A-representable constraints from the orbital representation do not extend to the spatial representation. This suggests reformulating the variational principle in g-density functional theory in the orbital representation. 

An early example implementing the general approach to take into account first the intrabond correlation, is presented by the PCILO - perturbational configuration interaction of localized orbitals method [121,122], As one of its authors, J.-P. Malrieu mentions in [122], the PCILO method opposes the majority of the QC methods in all the fundamental concepts. In contrast to the majority of the methods based on the variational principle, the PCILO method is based on estimating the energy by perturbation theory. Also, the majority of the QC methods use one-electron HFR approximation, at least as an intermediate construct, whereas the PCILO is claimed to addresses directly the V-electron wave function and takes into account all surviving matrix elements of the electron-electron interactions. In contrast with other QC... 

Within MO theory, the problem of determining the electronic structure of a molecule is then reduced to the determination of the coefficients of the expansions represented by Eqs. (1) and (2). On the basis of the variational principle, the best coefficients (and, correspondingly, the best orbitals and functions) for the ground state are those that lead to a minimum energy. 

The central field approximation and the simplifications which result from it allow one to construct a highly successful quantum-mechanical model for the AT-electron atom, by using Hartree s principle of the self-consistent field (SCF). In this method, one equation is obtained for each radial function, and the system is solved iteratively until convergence is obtained, which leaves the total energy stationary with respect to variations of all the functions (the variational principle ). The Hartree-Fock equations for an AT-electron system are equivalent to several one electron radial Schrodinger equations (see equation (2.2)), with terms which make the solution for one orbital dependent on all the others. In essence, the full AT-electron problem is approximated by a smaller number of coupled one-electron problems. This scheme is sometimes (somewhat inappropriately) referred to as a one-electron model in fact, the Hartree-Fock equations are a genuine AT-electron theory, but describe an independent particle system. 

In advanced Slater theory, more than one Slater function is taken in a linear combination to generate the best approximation to particular atomic orbitals and we have seen that this best standard could be based on the degree of fit to the numerical radial functions or the linear combinations that returned the variation principle best eigenvalue. In such cases, these coefficients are undetermined until the best eigenvalues have been calculated and the overall requirement of normalization is imposed. This is a general problem, which leads us to the theory of the self-consistent field (57,58,61,62, 42,47,53) developed by Hartree in his early calculations (1) and to Chapter 5. 

The variational principle of the energy density functional theory based on the definition (33) is a straightforward consequence of the quantum mechanical variational principle (8) and the functional mapping (13). It is clearly orbit-dependent... 

Kohn and Sham proposed that the kinetic energy of the electrons could be calculated from a set of orbitals, Xr which are expressed with a basis set of functions for which the individual orbital coefficients are determined in a manner somewhat similar to that used to determine the coefficients of the orbitals in HF theory. DFT thus becomes a self-consistent procedure in which one starts with a hypothetical system of noninteracting electrons that have the same electron density as the system of interest, determines the corresponding wave functions, and uses the variational principle to minimize the energy of the system and produce a new electron density. That density serves as a basis for a new iteration of the procedure, and the process is repeated until convergence is achieved. 

The Coulomb repulsion between the two electrons is l/r,2, where r,2 is the distance between the electrons. We are going to use the variation principle on a linear expansion. All basis functions contain one (or more) exponential function where the effective nuclear charge is obtained from screening theory (Section 2.2.6). Since there is one other electron in the same shell, the screening factor is equal to S = 0.35. A rather good function to approximate the Is orbital is thus exp[-(2 - 0.35)r] = exp(-1.65r). 

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