Slater potential

The main difficulty in solving the Hartree-Fock equation is caused by the non-local character of the potential in which an electron is orbiting. This causes, in turn, a complicated dependence of the potential, particularly of its exchange part, on the wave functions of electronic shells. There have been a number of attempts to replace it by a local potential, often having an analytical expression (e.g. universal Gaspar potential, Slater approximation for its exchange part, etc.). These forms of potential are usually employed to find wave functions when the requirements for their accuracy are not high or when they serve as the initial functions. 

Here we use the self-consistent-charge (SCC) approximationto construct the coulomb potential Vc. The effective atomic charges are estimated by Mulliken population analysis and are spherically averaged around each nucleus. Then, they are superimposed to construct the molecular coulomb potential. For the exchange potential. Slater s approximation ... 

Experimental data for L-shell ionization potentials Slater, J. C. In Quantum Theory of Atomic Structure, Vol. I, p. 206. New York McGraw 1960, and valence state ionization potentials Hinze, J., Jaffd, H. H. J. Am. Chem. Soc. 84, 540 (1962). 

There are several theoretical studies on LaO and related lanthanide oxides. We have already mentioned the ligand-field theory model calculations of Field (1982) as well as Carette and Hocquet (1988). More recently, Kotzian et al. (1991a,b) have applied the INDO technique (Pople et al. (1967) extended to include spin-orbit coupling [see also Kotzian et al. (1989a, b)] to lanthanide oxides (LaO, CeO, GdO and LuO). The authors call it INDO/S-CI method. The INDO parameters were derived from atomic spectra, model Dirac Fock calculations on lanthanide atoms and ions to derive ionization potentials, Slater-Condon factors and basis sets. The spin-orbit parameter is derived from atomic spectra in this method. 

Table 3.1 Determination of static equilibrium in GaAs - test of convergence with number of plane waves. The equilibrium lattice constant a is found from the condition p(a ) = 0, the bulk modulus is B = -V (dp/dV). (Local potential. Slater exchange a=0.8, 2 special points, B from a linear fit, a Srom a quadratic one.)... 

Slater and Kirkwood s idea [121] of an exponential repulsion plus dispersion needs only one concept, damping fiinctions, see section Al.5.3.3. to lead to a working template for contemporary work. Buckingham and Comer [126] suggested such a potential with an empirical damping fiinction more than 50 years ago ... 

Slater P C and Koster G F 1954 Simplified LCAO method for the periodic potential problem Phys. Rev. 94 1498-524... 

One more quantum number, that relating to the inversion (i) symmetry operator ean be used in atomie eases beeause the total potential energy V is unehanged when all of the eleetrons have their position veetors subjeeted to inversion (i r = -r). This quantum number is straightforward to determine. Beeause eaeh L, S, Ml, Ms, H state diseussed above eonsist of a few (or, in the ease of eonfiguration interaetion several) symmetry adapted eombinations of Slater determinant funetions, the effeet of the inversion operator on sueh a wavefunetion P ean be determined by ... 

The single Slater determinant wavefunction (properly spin and symmetry adapted) is the starting point of the most common mean field potential. It is also the origin of the molecular orbital concept. 

This result was rediscovered by Slater (1951) with a slightly different numerical coefficient of C. Authors often refer to a term Vx which is proportional to the one-third power of the electron density as a Slater-Dirac exchange potential. 

Just to remind you, the electron density and therefore the exchange potential are both scalar fields they vary depending on the position in space r. We often refer to models that make use of such exchange potentials as local density models. The disagreement between Slater s and Dirac s numerical coefficients was quickly resolved, and authors began to write the exchange potential as... 

So far there have not been any restrictions on the MOs used to build the determinantal trial wave function. The Slater determinant has been written in terms of spinorbitals, eq. (3.20), being products of a spatial orbital times a spin function (a or /3). If there are no restrictions on the form of the spatial orbitals, the trial function is an Unrestricted Hartree-Fock (UHF) wave function. The term Different Orbitals for Different Spins (DODS) is also sometimes used. If the interest is in systems with an even number of electrons and a singlet type of wave function (a closed shell system), the restriction that each spatial orbital should have two electrons, one with a and one with /3 spin, is normally made. Such wave functions are known as Restricted Hartree-Fock (RHF). Open-shell systems may also be described by restricted type wave functions, where the spatial part of the doubly occupied orbitals is forced to be the same this is known as Restricted Open-shell Hartree-Fock (ROHF). For open-shell species a UHF treatment leads to well-defined orbital energies, which may be interpreted as ionization potentials. Section 3.4. For an ROHF wave function it is not possible to chose a unitary transformation which makes the matrix of Lagrange multipliers in eq. (3.40) diagonal, and orbital energies from an ROHF wave function are consequently not uniquely defined, and cannot be equated to ionization potentials by a Koopman type argument. 

The Vext operator is equal to Vne for A = 1, for intermediate A values, however, it is assumed that the external potential Vext(A) is adjusted so that the same density is obtained for both A = 1 (the real system) and A = 0 (a hypothetical system with noninteracting electrons). For the A = 0 case the exact solution to the Schrddinger equation is given as a Slater determinant composed of (molecular) orbitals, for which the... 

The key element in London s approach is the expansion of the electrical potential energy in multipole series. Since neutral molecules or portions of molecules are involved, the leading term is that for dipole-dipole interaction. While attention has been given to higher-order terms, these are usually small, and the greater need seems to be for improved treatment of the dipole-dipole terms. London used second order perturbation theory in his treatment, but Slater and Kirkwood38,21 soon followed with a variation method treatment which yielded similar results. Other individual papers will be mentioned later, but the excellent review of Mar-genau26 should not be overlooked. 

The discussion in the previous section indicated that Nett values would be expected to exceed substantially the actual number of outer shell electrons. Table IV amply confirms this conclusion. Since N appears to the power in the Slater-Kirkwood equation, the deviations are exaggerated. Thus, in making a very similar treatment a few years ago in which slightly different empirical potentials were used, the writer31 found substantially smaller effective N values. For the same reason a relatively crude effective... 

In view of the complications of the intermolecular potential (as compared to the interatomic potential of the rare gas atoms) the comparisons for molecules in Tables II, III, and IV should be judged with caution. The apparent discrepancies from the theories for single atoms can be misleading. An example is the calculation for CH4 on the Slater-Kirkwood theory where Table IV shows the absurd value of 24 for the effective number of electrons. Pitzer and Catalano32 have applied the Slater-Kirkwood equation to the intermolecular potential of CH4 by addition of all the individual atom interactions and, with N = 4 for carbon and 1 for hydrogen, obtained agreement within 5 per cent for the London energy at the potential minimum. 

Douglas, A. S., Proc. Cambridge Phil. Soc. 52, 687, "A method for improving energy-level calculations for series electrons." Inclusion of a polarization potential in the Hartree-Slater-Fock equation. 

Slater, E.C., Rosing, J., Mol, A. (1973). The phosphorylation potential generated by respiring mitochondria. Biochim. Biophys. Acta 292, 534-553. 

Ward JAM, JME Ahad, G Lacrampe-Couloume, GF Slater, EA Edwards, BS Lollar (2000) Hydrogen isotope fractionation of toluene potential for direct verification of bioremediation. Environ Sci Technol 34 4577-4581. 

Since the Fock operator is a effective one-electron operator, equation (1-29) describes a system of N electrons which do not interact among themselves but experience an effective potential VHF. In other words, the Slater determinant is the exact wave function of N noninteracting particles moving in the field of the effective potential VHF.5 It will not take long before we will meet again the idea of non-interacting systems in the discussion of the Kohn-Sham approach to density functional theory. 

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