Density matrices single Slater determinant

The origins of density functional theory (DFT) are to be found in the statistical theory of atoms proposed independently by Thomas in 1926 [1] and Fermi in 1928 [2]. The inclusion of exchange in this theory was proposed by Dirac in 1930 [3]. In his paper, Dirac introduced the idempotent first-order density matrix which now carries his name and is the result of a total wave function which is approximated by a single Slater determinant. The total energy underlying the Thomas-Fermi-Dirac (TFD) theory can be written (see, e.g. March [4], [5]) as... 

The density matrix description is useful when discussing the electron correlations. The statement that the motion of electrons is correlated can be given an exact sense only if the two-electron density matrices eqs. (1.199) and (1.200) are used. In terms of the wave function, the statement of the correlated character of electron motions sounds like a negative statement the non-correlated (Hartree-Fock) wave function is one which is represented by a single Slater determinant, and the correlated one... 

Though we can compare electron densities directly, there is often a need for more condensed information. The missing link in the experimental sequence are the steps from the electron density to the one-particle density matrix f(1,1 ) to the wavefunction. Essentially the difficulty is that the wavefunction is a function of the 3n space coordinates of the electrons (and the n spin coordinates), while the electron density is only a three-dimensional function. Drastic assumptions must be introduced, such as the description of the molecular orbitals by a limited basis set, and the representation of the density by a single Slater-determinant, in which case the idempotency constraint reduces the number of unknowns... 

That Eq. (44) and Eq. (34) are different is due to the fact that GHF exchange is quadratic in the first-order density matrix, whereas Slater s approximation depends on the 4/3 power of the density. Only for the special case of a single determinantal wavefunction that has constant direction of S (i.e. for conventional but not generalized single determinants) can we regroup terms in Eq. (43) to yield Eq. (34). For this case N and S can be simplified as... 

For a single Slater determinant all information is contained in its one particle density matrix [96]. For this special case we use q instead of 7. 

The ground-state wavefunction corresponding to this noninteracting system is then a single Slater determinant i(x) of the lowest occupied orbitals i(x) of the Kohn-Sham differential equation. The Dirac [15] single-particle density matrix y,(r, r ) that results from this Slater determinant is... 

In the case where is a single Slater determinant, the eigenvalues n, of the first-order density matrix reduce to occupation numbers, with values either 0 or 1. In the more general case, where is a sum of Slater determinants, the eigenvalues rii can be shown to obey the relationships... 

In fact in this resides the power of the density matrix formalism reducing a many-body problem to the single particle density matrix, abstracted from the single Slater determinant of Eq. (4.190) called also as Fock-Dirac matrix... 

Both TDHF and the TDDFT follow the dynamics of a similar quantity a single Slater determinant that can be uniquely described by an idempotent single-electron density matrix p (with p = p).62,63,77,78 However, they yield different equations of motion for p(t). stemming from the different interpretation of p tj. In the TDHF, p(t) is viewed as an approximation for the actual single-electron density matrix, whereas in TDDFT p(t) is an auxiliary quantity constrained... 

Since the many-electron wave function is represented by a single Slater determinant, the total density matrix p t) must be a projector at all times ... 

The first-order reduced density matrix of the exact wave function also has a number of weakly occupied orbitals for which i > N. These orbitals never appear in a single Slater determinant method, but are important for a correct description of the correlation hole. 

L5wdin showed that Ynso is the best possible approximation to the exact density (Equation 1.103). The one-matrix YoCEIO of the trial wave function corresponding to a single Slater determinant with orbitals <[) may be expanded as follows ... 

A number of expectation values cannot be obtained from the density, but require the one-matrix or the two-matrix. To obtain the one-matrix or two-matrix, one has to first define a wave function. Normally, the Slater determinant for the N lowest energy orbitals is nsed, bnt a single Slater determinant cannot possibly be the correct wave function. As has been shown at the end of Chapter 1, the correct one-matrix contains weakly occupied NSOs, because the correct wave function is a snperposition of many Slater determinants. It is unthinkable that the DPT orbitals would give correct results for all expectation valnes, when the nonzero occupation numbers of the one-matrix are incorrectly equal to unity. 

In the special case that the wave function T is a Slater determinant, i.e., the wave function of N noninteracting fermions, the single-particle density matrix can be written in terms of the orbitals comprising the determinant,... 

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