Density functional theory basis Slater determinants

The particular choice of the equidensity orthmormal orbitals defining the Slater determinant that yields a prescribed electrcm density p(r) has been proposed by Harriman [18] on the basis of the pioneering works by Macke [19] and Gilbert [20]. Alternative constructions and extensions have also been suggested [21, 22]. in die density functional theory such A-electron wavefiinctions are mvolved in the formal density constrained search of Levy [17]. 

In principle, either method should be open to us to use in solid-state calculations. However, in practice the choice of a plane-wave basis set rather limits the use of any method within the framework of Fock theory. The problem is that the Slater determinant (Eq. 3.5) set up within the formalism of a plane-wave basis set is very large, and so impedes the performance of the calculations. Therefore, use of a plane-wave basis set, for the time being at least, limits the approximations made to H to those formalized in density functional theory. 

All the computational results here used density functional theory (B3LYP [74-77]) and the Gaussian program (G03 in Sect. 4.2 G09 elsewhere) to find a Slater-determinant wavefunction [78, 79]. The electron density, stress tensor, and other properties were evaluated using AIMALL [80]. In all calculations, large polarized basis sets were used. Further computational details may be found in our other papers on this topic [31, 32]. 

The basis set is the set of madiematical functions from which the wave function is constructed. As detailed in Chapter 4, each MO in HF theory is expressed as a linear combination of basis functions, the coefficients for which are determined from the iterative solution of the HF SCF equations (as flow-charted in Figure 4.3). The full HF wave function is expressed as a Slater determinant formed from the individual occupied MOs. In the abstract, the HF limit is achieved by use of an infinite basis set, which necessarily permits an optimal description of the electron probability density. In practice, however, one cannot make use of an infinite basis set. Thus, much work has gone into identifying mathematical functions that allow wave functions to approach the HF limit arbitrarily closely in as efficient a manner as possible. 

For the very small systems in Table 7.1, it is possible to approach the exact solution of the Schrodinger equation, but, as a rule, convergence of the correlation energy is depressingly slow. Mathematically, this derives from the poor ability of products of one-electron basis functions, which is what Slater determinants are, to describe the cusps in two-electron densities that characterize electronic structure. For the MP2 level of theory, Schwartz (1962)... 

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