Self-consistent field procedure

SCF (self-consistent field) procedure for solving the Hartree-Fock equations SCI-PCM (self-consistent isosurface-polarized continuum method) an ah initio solvation method... 

A completely different route to the A-electron problem is provided by DPT. On an operational level it can be thought of as an attempt to improve on the HE method by including correlation effects into the self-consistent field procedure. 

The physical reasoning for why these densities were frequently employed in the earlier days of density functional theory was that in this way the degeneracy of the partially filled d-orbitals could be retained. A technical reason why these densities still have to be employed in some recent investigations is that calculations with integral orbital occupations simply do not converge in the self consistent field procedure (see, e. g., Blanchet, Duarte, and Salahub, 1997). Such densities correspond to a representation of a particular state 2S+1L with Mg = S and a spherical averaging over ML. 

Heavy atoms exhibit large relativistic effects, often too large to be treated perturba-tively. The Schrodinger equation must be supplanted by an appropriate relativistic wave equation such as Dirac-Coulomb or Dirac-Coulomb-Breit. Approximate one-electron solutions to these equations may be obtained by the self-consistent-field procedure. The resulting Dirac-Fock or Dirac-Fock-Breit functions are conceptually similar to the familiar Hartree-Fock functions the Hartree-Fock orbitals are replaced, however, by four-component spinors. Correlation is no less important in the relativistic regime than it is for the lighter elements, and may be included in a similar manner. 

Now we have written down a wave function appropriate for use in the case where H = h(i). In HF theory, we make some simplifications so many-electron atoms and molecules can be treated this way. By tacitly assuming that each electron moves in a percieved electric field generated by the stationary nuclei and the average spatial distribution of all the other electrons, it essentially becomes an independant-electron problem. The HF Self Consistent Field procedure (SCF) will be bent on constructing each x(x) to give the lowest energy. 

Figure 8.20 An energy level diagram for ferrocene. The MO energies are those calculated by Shustorovich and Dyatkina, using a self-consistent field procedure. The positions of the ring and iron orbitals on this diagram are only approximate.

Hiickel and extended Huckel methods are termed semi-empirical because they rely on experimental data for the quantification of parameters. There are other semi-empirical methods, such as CNDO, MINDO, INDO, in which experimental data are still used, but more care is taken in evaluating the Htj. These methods are self-consistent field procedures based on 3 SCF. They are discussed in various works on molecular orbital theory.4... 

The relativistic or non-relativistic random-phase approximation (RRPA or RPA)t is a generalized self-consistent field procedure which may be derived making the Dirac/Hartree-Fock equations time-dependent. Therefore, the approach is often called time-dependent Dirac/Hartree-Fock. The name random phase comes from the original application of this method to very large systems where it was argued that terms due to interactions between many alternative pairs of excited particles, so-called two-particle-two-hole interactions ((2p-2h) see below) tend to... 

The process is continued for k cycles till we have a wavefunction and/or an energy calculated from that are essentially the same (according to some reasonable criterion) as the wavefunction and/or energy from the previous cycle. This happens when the functions i/ (l), i//(2),. .., j/(n) are changing so little from one cycle to the next that the smeared-out electrostatic field used for the electron-electron potential has (essentially) ceased to change. At this stage the field of cycle k is essentially the same as that of cycle k — 1, i.e. it is consistent with this previous field, and so the Hartree procedure is called the self-consistent-field-procedure, which is usually abbreviated as the SCF procedure. 

Now, in the Hartree-Fock method (the Roothaan-Hall equations represent one implementation of the Hartree-Fock method) each electron moves in an average field due to all the other electrons (see the discussion in connection with Fig. 53, Section 5.23.2). As the c s are refined the MO wavefunctions improve and so this average field that each electron feels improves (since J and K, although not explicitly calculated (Section 5.2.3.63) improve with the i// s ). When the c s no longer change the field represented by this last set of c s is (practically) the same as that of the previous cycle, i.e. the two fields are consistent with one another, i.e. self-consistent . This Roothaan-Hall-Hartree-Fock iterative process (initial guess, first F, first-cycle c s, second F, second-cycle c s, third F, etc.) is therefore a self-consistent-field procedure or SCF procedure, like the Hartree-Fock procedure... 

Although wave equations are readily composed for more-electron atoms, they are impossible to solve in closed form. Approximate solutions for many-electron atoms are all based on the assumption that the same set of hydrogen-atom quantum numbers regulates their electronic configurations, subject to the effects of interelectronic repulsions. The wave functions are likewise assumed to be hydrogen-like, but modified by the increased nuclear charge. The method of solution is known as the self-consistent-field procedure. 

Roothaan s Self-Consistent-Field Procedure.—While numerical integration techniques may be used to solve the Hartree-Fock equations in the case of atoms by the iterative method described above, the lower symmetry of the nuclear field present in molecules necessitates the use of an expansion for the determination of the molecular orbitals by a method developed by Roothaan.81 In Roothaan s approach, it is assumed that each molecular orbital may be adequately represented by a linear expansion in terms of some (simpler) set of basis functions xj, i.e. 

Ab initio MO computer programmes use the quantum-chemical Hartree-Fock self-consistent-field procedure in Roothaan s LCAO-MO formalism188 and apply Gaussian-type basis functions instead of Slater-type atomic functions. To correct for the deficiencies of Gaussian functions, which are, for s-electrons, curved at the nucleus and fall off too fast with exp( —ar2), at least three different Gaussian functions are needed to approximate one atomic Slater s-function, which has a cusp at the nucleus and falls off with exp(— r). But the evaluation of two-electron repulsion integrals between atomic functions located at one to four different centres is mathematically much simpler for Gaussian functions than for Slater functions. 

At the HF-SCF level of theory, the wave function is determined completely by the molecular orbitals. In the vast majority of cases, these are given by linear combinations of atom-centered basis functions, and these MO coefficients are obtained by the self-consistent field procedure. The first-order change to the wave function is therefore governed by the first-order change in the MO coefficients. It is not difficult to work out expressions for the derivatives of the MO coefficients,22 and one obtains... 

The fastest scaling for an ab initio method is Hartree Fock theory utilizing the self-consistent-field procedure (HF/SCF), which for a given basis set scales as the number of electrons N4. In other words, double the size of the calculation and it will take... 

We can also formulate this in a different manner and say that the self-consistent field procedure plays a crucial role in 4-component theory because it serves to define the spinors that isolate the n-electron subspaces from the rest of the Fock space. In this manner it determines in effect the precise form of the electron-electron interaction used in the calculations. Both aspects are a consequence of the renormalization procedure that was followed when fixing the energy scale and interpretation of the vacuum. The experience with different realizations of the no-pair procedure has learned that the differences in calculated chemical properties (that depend on energy differences and not on absolute energies) are usually small and that other sources of errors (truncation errors in the basis set expansion, approximations in the evaluation of the integrals) prevail in actual calculations. 

The self-consistent field procedure in Kohn-Sham DFT is very similar to that of the conventional Hartree-Fock method [269]. The main difference is that the functional Exc[p] and matrix elements of Vxc(r) have to be evaluated in Kohn-Sham DFT numerically, whereas the Hartree-Fock method is entirely analytic. Efficient formulas for computing matrix elements of Vxc(r) in finite basis sets have been developed [270, 271], along with accurate numerical integration grids [272-277] and techniques for real-space grid integration [278,279]. 

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