Slater determinant definition

A very different approach has been followed by Zhao et al. [54-57] who based their method on the constrained search definition of the Kohn-Sham kinetic energy. It follows from this definition that, from all Slater determinants which yield a given density, the Kohn-Sham determinant will minimize the kinetic energy. Suppose we have an exact density po- If one minimizes the Kohn-Sham kinetic energy under the constraint... 

This definition is consistent with the definition of overlap between two Slater-determinants having the same number of electrons. The overlap between Slater determinants having a different number of electrons is not defined. The extension to have a well-defined, but zero, overlap between two occupation number vectors with different numbers of electrons is a special feature of the Fock-space formulation that allows a unified description of systems with a different number of electrons. As a special case of Eq. (1.3), the vacuum state is defined to be normalized... 

Here h are the one-electron integrals including the electron kinetic energy and the electron-nuclear attraction terms, and gjjkl are the two-electron repulsion integrals defmed by (3 19). The summations in (3 24) are over the molecular orbital basis, and the definition is, of course, only valid as long as we work in this basis. Notice that the number of electrons does not appear in the defmition of the Hamiltonian. All such information is found in the Slater determinant basis. This is true for all operators in the second quantization formalism. 

Most of the formalism to be developed in the coming sections of these lecture notes will be independent of the specific definition of the configurational basis, in which we expand the wave function. We therefore do not have to be very explicit about the exact nature of the basis states hn>. They can be either Slater determinants or spin-adapted Configuration State Functions (CSF s). For a long time it was assumed that CSF s were to be preferred for MCSCF calculations, since it gives a much shorter Cl expansion. Efficient methods like GUGA had also been developed for the solution of the Cl problem. Recent... 

To understand the spin properties of Slater determinants one first has to have an expression for the S2 operator. From the definition of S2 and by using the step up and step down spin operators it is easy to show that... 

The MO concept is directly related to an approximate wavefunction consisting of a Slater determinant of occupied one-particle wavefunctions, or molecular orbitals. The Hartree-Fock orbitals are by definition the ones that minimize the expectation value of the Hamiltonian for this Slater determinant. They are usually considered to be the best orbitals, although it should not be forgotten that they are only optimal in the sense of energy minimization. 

From this definition it is evident that application of Yj to the Fermi vacuum is equivalent to annihilation of a particle (or creation of a hole) in 14>0 >. The effect of YA on the Fermi vacuum state is the creation of a particle (or annihilation of a hole) in I 0>. The effect of YA" on the Fermi vacuum is the creation of a particle in the virtual spin-orbitals and finally, the effect of YA" is the annihilation of a particle in virtual spin-orbitals. Thus e.g., a singly excited Slater determinant I ) can be described as... 

For a single Slater determinant, the exact behaviour of the modulating function is attained when we use Hartree-Fock orbitals as the generating ones. The behavior of this modulating function F/fsF can be easily determined. In fact, by inspection of Eqs. (8) and (9), we see that it is a positive function (the density is positive definite and tends to zero at infinity, the multiplicative... 

Just as for polyelectronic atoms, the electronic wavefunction for a molecule must be antisymmetric the Pauli principle. Thus, electron spin correlation is accommodated by the definition of the wavefunction as a Slater determinant whose elements are the occupied molecular orbitals. 

These definitions may be extended to any basis of orthonormal r-electron states. Thus, if one has a basis of Slater determinants (A), then any state /) is ... 

By exchange, we mean the density functional definition of exchange, in which the wavefunction is a Slater determinant whose density is the exact density of the interacting system, and which minimizes the energy of the non-interacting system in the Kohn-Sham external potential, iv,a=o- Another useful concept is the pair distribution function, defined as[14]... 

The Hartree-Fock and the Kohn-Sham Slater determinants are not identical, since they are composed of different single-particle orbitals, and thus the definition of exchange and correlation energy in DFT and in conventional quantum chemistry is slightly different [52]. 

The N-particle function (Pq e lA y ) is given, in general as a linear combination of Slater determinants constructed from plane waves, thus extending the treatment of both Macke [53, 54] and of March and Young [55]. Thus, we have < o = IxC/cXx. where Xk is the Slater determinant y = (JV) det [( li,..., and < ij,(r) is the plane wave defined in Eq. (1), with momentum e Z. Clearly, in view of this definition of o 6 the resulting density is... 

It is sometimes forgotten that, although Eqs. (18) and (19) hold for any wave function, Eqs. (20) and (21) are true only if the wave function is a Slater determinant. For physically realistic wave functions, electrons with different spin are correlated as well and in some cases this correlation can be of exactly the same type as the Fermi correlation. (As the definition of Fermi correlation states that it occurs only for electrons with the same spin, it must not be called Fermi correlation). 

We may conclude that the definition of the creation and annihilation operators and the simple anticommutation relations are equivalent to the Slater-Condon rules. This opens up for us the space spanned by the Slater determinants i.e., all the integrals involving Slater determinants can be easily transformed into the one- and two-electron integrals involving spinorbitals. 

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