Eigenvalues orbitals-self-consistent field

It is well known that the PES cannot be given analytically in quantum chemistry, but can be calculated point by point through iterative solution of matrix eigenvalue problems arising from the application of LCAO-MO SCF Cl (Linear Combination of Atomic Orbitals - Molecular Orbital Self-Consistent-Field Configuration Interaction) methods. If... 

At the energy minimum, each electron moves in an average field due to the Other electrons and the nuclei. Small variations in the form of the orbitals at this point do not change the energy or the electric field, and so we speak of a self-consistent field (SCF). Many authors use the acronyms HF and SCF interchangeably, and I will do so from time to time. These HF orbitals are found as solutions of the HF eigenvalue problem... 

The Hartree-Fock equations form a set of pseudo-eigenvalue equations, as the Fock operator depends on all the occupied MOs (via the Coulomb and Exchange operators, eqs. (3.36) and (3.33)). A specific Fock orbital can only be determined if all the other occupied orbitals are known, and iterative methods must therefore be employed for determining the orbitals. A set of functions which is a solution to eq. (3.41) are called Self-Consistent Field (SCF) orbitals. 

Since the Fock matrix is dependent on the orbital coefficients, the Roothaan equations have to be repeatedly solved in an iterative process, the self-consistent field (SCF) procedure. One important step in the SCF procedure is the conversion of the general eigenvalue equation (7) into an ordinary one by an orthogonalization transformation... 

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 occupied spin orbitals included in the operators J and K have to be the solutions of Equation 2.30. An iterative method is used, where the successive solutions of Eqnation 2.30 define J and K. This procedure is repeated until the energy eigenvalues are within a stipnlated convergence limit. This solution is called the self-consistent field (SCE) solution. Physically, SCF means that screening and penetration effects are taken into account in the best possible way within the one-electron approximation. 

As a matter of fact, as in the Hartree-Fock (HF) scheme, the KS equation is a pseudo-eigenvalue problem and has to be solved iteratively through a self-consistent field procedure to determine the charge density p(r) that corresponds to the lowest energy. The self-consistent solutions 4>ia resemble those of the HF equations. Still, one should keep in mind that these orbitals have no physical significance other than in allowing one to constitute the charge density. We want to stress that the DFT wavefunction is not a Slater determinant of spin orbitals. In fact, in a strict sense there is no A -electron wavefunction available in DFT. ... 

The basis for all wave function based ab initio methods is the Hartree-Fock (HF) approach. [11, 12] It makes use of a single-determinant ansatz constructed from one-electron spin orbitals. These orbitals describe the motion of each electron within the field of the nuclei and the mean field of the remaining n-1 electrons. The mean field is not known a priori, but depends on the orbitals which are determined self-consistently from the eigenvalue problem of the Fock operator. 

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