How to use HF and RHF depending on the application for energy minimization using quantum chemistry software.
Thank You.
Dear Iresh,
Hartree–Fock (HF) method is a method of approximation for the determination of the wave function and the energy of a quantum many-body system in a stationary state. There are two kinds of calculations when using the HF method: (1) RHF (Restricted Hartree-Fock and (2) UHF (Unrestricted Hartree-Fock).
Unrestricted Hartree–Fock (UHF) theory is the most common molecular orbital method for open shell molecules where the number of electrons of each spin are not equal. While restricted Hartree–Fock theory uses a single molecular orbital twice, one multiplied by the α spin function and the other multiplied by the β spin function in the Slater determinant, unrestricted Hartree–Fock theory uses different molecular orbitals for the α and β electrons. This has been called a different orbitals for different spins (DODS) method. The result is a pair of coupled Roothaan equations, known as the Pople–Nesbet–Berthier equations.
Approximations of HF:
The Hartree–Fock method makes five major simplifications in order to deal with this task:
1-The Born–Oppenheimer approximation is inherently assumed. The full molecular wave function is actually a function of the coordinates of each of the nuclei, in addition to those of the electrons.
2-Typically, relativistic effects are completely neglected. The momentum operator is assumed to be completely non-relativistic.
3-The variational solution is assumed to be a linear combination of a finite number of basis functions, which are usually (but not always) chosen to be orthogonal. The finite basis set is assumed to be approximately complete.
4-Each energy eigenfunction is assumed to be describable by a single Slater determinant, an antisymmetrized product of one-electron wave functions (i.e., orbitals).
5-The mean field approximation is implied. Effects arising from deviations from this assumption, known as electron correlation, are completely neglected for the electrons of opposite spin, but are taken into account for electrons of parallel spin. (Electron correlation should not be confused with electron exchange, which is fully accounted for in the Hartree–Fock method.).
Relaxation of the last two approximations give rise to many so-called post-Hartree–Fock methods.
References:
Berthier, Gaston (1954). "Extension de la methode du champ moleculaire self-consistent a l'etude des couches incompletes" [Extension of the method of molecular self-consistent field to the study of incomplete layers]. Comptes rendus hebdomadaires des séances de l'Académie des sciences (in French) 238: 91–93.
Pople, J. A.; Nesbet, R. K. (1954). "Self-Consistent Orbitals for Radicals". The Journal of Chemical Physics 22 (3): 571.
Szabo, A.; Ostlund, N. S. (1996). Modern Quantum Chemistry. Mineola, New York: Dover Publishing.
Hoping this will be helpful,
Rafik
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