Self-consistent field energy

TABLE 1. Self-Consistent Field energies (in hartree) of the He atom for some excited states as a function of the size, M, of the even-tempered basis set used to parametrize the orbitals. In the column headed (a) the same even-tempered basis set optimized for the ground state, is used for aU states ... 

The self-consistent-field energies are given in hartree the triple-excitation energy, Ei t, the quadruple-excitation energy,... 

Maslen, P.E., Jayatilaka, D., Colwell, S.M., Amos, R.D., Handy, N.C. Higher analytical derivatives. 2. The 4 derivative of self-consistent-field energy, J. Chem. Phys. 1991, 95, 7409-17. 

The next step was sustained by the assumption that the correlation energy can be seen as the perturbation of the self-consistent-field energy, which is associated with a wave function derived for a single electronic configuration. At this point the basic methods of approximation used in quantum chemistry, namely the perturbation and variational, can be considered. 

Escf is the self-consistent field energy, f is the electric field, a is a nuclear coordinate, is the one-electron atomic orbital integral, If is related to the derivative of the molecular orbital coefficients with respect to a by... 

Escf The self-consistent field energy in hartrees. 

Table 11-2 provides information on energies for a number of atoms in their ground states. Self-consistent-field energies are presented for three levels of basis set complexity. In the STO single-f level, a miiumal basis set of one STO per occupied AO is used, and the energy is minimized with respect to independent variation of every orbital exponent f. The STO double- basis set is similar except that there are two STOs for each AO, the only restriction being that the STOs have the same spherical harmonics as the AOs to which they correspond. 

Bowman J (1978) Self-consistent field energies and wavefunctions for coupled oscillators. J Chem Phys 68 608... 

For the case of intramolecular energy transfer from excited vibrational states, a mixed quantum-classical treatment was given by Gerber et al. already in 1982 [101]. These authors used a time-dependent self-consistent field (TDSCF) approximation. In the classical limit of TDSCF averages over wave functions are replaced by averages over bundles of trajectories, each obtained by SCF methods. 

Gerber, R.B., Buch, V., Ratner, M.A. Time-dependent self-consistent field approximation for intramolecular energy transfer. I. Formulation and application to dissociation of van der Waals molecules. J. Chem. Phys. 77 (1982) 3022-3030. 

Using as many methods as are available to you for comparison (Mathcad, QBASIC, and TRUE BASIC), determine the self-consistent field (SCF) energies of the He atom and of the ions Li+, Be " ", and B +. Fill in the SCF column of Table 8-1. 

Cl calculations can be used to improve the quality of the wave-function and state energies. Self-consistent field (SCF) level calculations are based on the one-electron model, wherein each electron moves in the average field created by the other n-1 electrons in the molecule. Actually, electrons interact instantaneously and therefore have a natural tendency to avoid each other beyond the requirements of the Exclusion Principle. This correlation results in a lower average interelectronic repulsion and thus a lower state energy. The difference between electronic energies calculated at the SCF level versus the exact nonrelativistic energies is the correlation energy. 

The second step determines the LCAO coefficients by standard methods for matrix diagonalization. In an Extended Hiickel calculation, this results in molecular orbital coefficients and orbital energies. Ab initio and NDO calculations repeat these two steps iteratively because, in addition to the integrals over atomic orbitals, the elements of the energy matrix depend upon the coefficients of the occupied orbitals. HyperChem ends the iterations when the coefficients or the computed energy no longer change the solution is then self-consistent. The method is known as Self-Consistent Field (SCF) calculation. 

Ab initio calculations are iterative procedures based on self-consistent field (SCF) methods. Normally, calculations are approached by the Hartree-Fock closed-shell approximation, which treats a single electron at a time interacting with an aggregate of all the other electrons. Self-consistency is achieved by a procedure in which a set of orbitals is assumed, and the electron-electron repulsion is calculated this energy is then used to calculate a new set of orbitals, which in turn are used to calculate a new repulsive energy. The process is continued until convergence occurs and self-consistency is achieved." ... 

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 self-consistent field function for atoms with 2 to 36 electrons are computed with a minimum basis set of Slater-type orbitals. The orbital exponents of the atomic orbitals are optimized so as to ensure the energy minimum. The analysis of the optimized orbital exponents allows us to obtain simple and accurate rules for the 1 s, 2s, 3s, 4s, 2p, 3p, 4p and 3d electronic screening constants. These rules are compared with those proposed by Slater and reveal the need for the screening due to the outside electrons. The analysis of the screening constants (and orbital exponents) is extended to the excited states of the ground state configuration and the positive ions. 

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