Slater determinants total spin

If there are 4 electrons in the molecule, then yr, and y/2 are occupied (and yr3and y/4 are virtual oibitals). The occupied orbitals are used to construct the total wavefunction, as a Slater determinant of spin orbitals. 

In these methods, each electron is moving in a mean-field potential due to the other electrons. The total wavefunction T is written as a determinant, the Slater determinant, of spin-orbitals. 

V i hen the Slater determinant is expanded, a total of N1 terms results. This is because there are N different permutations of N eleefrons. For example, for a three-electron system with spin orbitals X2 and xs the determinant is... 

In the ordinary Hartree-Fock scheme, the total wave function is approximated by a single Slater determinant and, if the system possesses certain symmetry properties, they may impose rather severe restrictions on the occupied spin orbitals see, e.g., Eq. 11.61. These restrictions may be removed and the total energy correspondingly decreased, if instead we approximate the total wave function by means of the first term in the symmetry adapted set, i.e., by the projection of a single determinant. Since in both cases,... 

The basis of the expansion, ifra, are configuration state functions (CSF), which are linear combinations of Slater determinants that are eigenfunctions of the spin operator and have the correct spatial symmetry and total spin of the electronic state under investigation [42],... 

The Slater determinant for the total wavefunction T of a 2 -electron atom or molecule is a 2 x 2 determinant with 2 rows due to the 2 electrons and 2 columns due to the 2 spin orbitals (you can interchange the row/column format) since these are closed-shell species, the number of spatial orbitals i// is half the number of electrons. We use the lowest n occupied spatial orbitals (the lowest 2n spin orbitals) to make the determinant. 

The functions 4> are the spatial molecular (or atomic) orbitals or wavefunctions that (along with the spin functions) make up the overall or total molecular (or atomic) wavefunction ijj, which can be written as a Slater determinant (Eq. 5.12). Concerning the energies from the fact that... 

We have restricted ourselves to the case of complexes for which the excitation energies in the /-system are much higher than the excitation energy in the (/-shell of the metal. In these complexes the number of valence electrons in the ligand subsystem is even and thus the ground state of the /-system can be approximated by a single Slater determinant (I> with zero total spin. Then the wave function I acquires the form of eq. (1.245) ... 

The total, antisymmetric function for a closed-shell configuration is expressed as a Slater determinant built-up from the spin-orbitals. In the case of open-shell configurations, a linear combination of Slater determinants may be needed in order to obtain a function with the same symmetry and multiplicity characteristics as the state under consideration. 

The electronic structure methods are based primarily on two basic approximations (1) Born-Oppenheimer approximation that separates the nuclear motion from the electronic motion, and (2) Independent Particle approximation that allows one to describe the total electronic wavefunction in the form of one electron wavefunc-tions i.e. a Slater determinant [26], Together with electron spin, this is known as the Hartree-Fock (HF) approximation. The HF method can be of three types restricted Hartree-Fock (RHF), unrestricted Hartree-Fock (UHF) and restricted open Hartree-Fock (ROHF). In the RHF method, which is used for the singlet spin system, the same orbital spatial function is used for both electronic spins (a and (3). In the UHF method, electrons with a and (3 spins have different orbital spatial functions. However, this kind of wavefunction treatment yields an error known as spin contamination. In the case of ROHF method, for an open shell system paired electron spins have the same orbital spatial function. One of the shortcomings of the HF method is neglect of explicit electron correlation. Electron correlation is mainly caused by the instantaneous interaction between electrons which is not treated in an explicit way in the HF method. Therefore, several physical phenomena can not be explained using the HF method, for example, the dissociation of molecules. The deficiency of the HF method (RHF) at the dissociation limit of molecules can be partly overcome in the UHF method. However, for a satisfactory result, a method with electron correlation is necessary. 

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