Energy of a Slater Determinant

The variational problem is to minimize the energy of a single Slater determinant by choosing suitable values for the MO coefficients, under the constraint that the MOs remain orthonormal. With cj) being an MO written as a linear combination of the basis functions (atomic orbitals) /, this leads to a set of secular equations, F being the Fock matrix, S the overlap matrix and C containing the MO coefficients (Section 3.5). 

In this case the basis functions (coordinate system) are non-orthogonal, the overlaps are contained in the S matrix. By multiplying from the left by S and inserting a unit matrix written in the form (13.19) may be reformulated as 

The latter equation is now in a standard form for determining the eigenvalues of the F matrix. The eigenvectors contained in C can then be backtransformed to the original coordinate system (C = S C )- 

Equation (13.20) corresponds to a symmetrical orthogonalization of the basis. The initial coordinate system, (the basis functions %) is non-orthogonal, but by multiplying with a matrix such as S the new coordinate system has orthogonal axes. 

There are choice other than S for such an orthogonalizing transformation, any Inafrix which has the property X SX = I can be used. 

The variational problem may again be formulated as a secular equation, where the coordinate axes are many-electron functions (Slater determinants),, which are orthogonal (Section 4.2). 

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Derivative Techniques 240 10.4 Lagrangian Techniques 242 10.5 Coupled Perturbed Hartree-Fock 244 10.6 Electric Field Perturbation 247 10.7 Magnetic Field Perturbation 248 10.7.1 External Magnetic Field 248 13.1 Vibrational Normal Coordinates 312 13.2 Energy of a Slater Determinant 314 13.3 Energy of a Cl Wave Function 315 Reference 315 14 Optimization Techniques 316... 

If one works out this expression one obtains equations that are identical to equations (316) and (317). These equations were first derived by Talman and Shadwick [45]. Since in our procedure we optimized the energy of a Slater determinant wavefunction under the constraint that the orbitals in the Slater determinant come from a local potential, the method is also known as the optimized potential method (OPM). We have therefore obtained the result that the OPM and the expansion to order e2 are equivalent procedures. The OPM has many similarities to the Hartree-Fock approach. Within the Hartree-Fock approximation one minimizes the energy of a Slater determinant wavefunction under the constraint that the orbitals are orthonormal. One then obtains one-particle equations for the orbitals that contain a nonlocal potential. Within the OPM, on the other hand, one adds the additional requirement that the orbitals must satisfy single-particle equations with a local potential. Due to this constraint the OPM total energy ) will in general be higher than the Hartree-Fock energy Fhf, i.e., Ex > E. We refer to Refs. [46,47] for an application of the OPM method for molecules. 

Modern density functional methods, that can be traced back to a paper by Kohn and Sham [231], avoid the evaluation of the kinetic energy as afunctional of the density. One rather introduces an artificial non-interacting system in a modified external potential - with the same density as the considered system and one evaluates the kinetic energy of this system as the kinetic energy of a Slater determinant. So the density functional methods in current use, are strictly speaking not genuine density functional methods. 

Let us first briefly recall how these GSMO s are obtained The energy of a Slater determinant (where a is the usual antisymmetrizer)... 

The calculation of the energy of a Slater determinant and the interaction between two different Slater determinants may seem a rather complicated task given the large number of terms (A ) when the determinant is written in its explicit form. However, the Slater-Condon rules given in Table 1.1 establish a few simple relations to calculate matrix elements between two Slater determinants. 

SCF Self-consistent field. Calculation in which (->) AO coefficients in an (-+) LCAO wavefunction are variationally optimized, thereby minimizing the electronic energy of a Slater determinant, composed of the resulting MOs. 

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