Wave functions Slater determinants

The energy of a wave function containing variational parameters, like an HF (one Slater determinant) or MCSCF (many Slater determinants) wave function. Parameters are typically the MO and state coefficients, but may also be for example basis function exponents. Usually only minima are desired, although in some cases saddle points may also be of interest (excited states). 

Interchange of two rows of the Slater determinant changes the sign of the wave function, which is therefore antisymmetric with respect to interchange of electrons. When two rows or columns are identical, the determinant is zero. The Slater determinant wave function therefore obeys the Pauli exclusion principle for fermions. 

The single-Slater determinant includes correlation between the motion of two electrons with parallel spins that avoid each other because of the exclusion principle (Szabo and Ostlund 1989), but correlation between the motion of electrons with opposite spin is neglected. The wave function of Eq. (3.2) does not prevent the two electrons from being at the same point in space, which is physically impossible. The Slater determinant wave function is therefore described as uncorrelated. 

For a multi-Slater determinant wave function, orbitals which satisfy Eq. (3.6), and therefore Eq. (3.7), can still be defined. For these orbitals, referred to as the natural spin orbitals, the coefficients nt are not necessarily integers, but have the boundaries 0 n, 1. 

We now prove that any unitary transformation of the orbitals leaves a closed-shell SCF Slater-determinant wave function unchanged, thereby showing the validity of transforming to localized MOs. Let i//loc be the SCF wave function written using localized orbitals. We wish to prove that if/loc equals can, where pc n uses the canonical MOs. In the notation of (1.260), we have... 

In these methods, the ground state must be of a type that can be described by a single Slater determinant of orthonormal spin orbitals. One such type, which is the most common ground state, is a closed-shell system (i.e. all occupied MOs are doubly occupied). We let or 0 > denote the Slater determinant wave function for the ground state, which will usually be made up of HF MOs, although in fact... 

The orbital exponent a, and the location of the orbital center Ri are the variational parameters. These Gaussians are positioned in the molecule in the same way as in Lewis s concept of valence theory. A single Slater determinant wave function for the non-orthogonal basis set of doubly occupied orbitals, Gj is constructed, and the total electronic energy is given by the well-known formula... 

The extension of the basis can improve wave functions and energies up to the Hartree-Fock limit, that is, a sufficiently extended basis can circumvent the LCAO approximation and lead to the best molecular orbitals for ground states. However, this is still in the realm of the independent-particle approximation 175>, and the use of single Slater-determinant wave functions in the study of potential surfaces implies the assumption that correlation energy remains approximately constant on that part of the surface where reaction pathways develop. In cases when this assumption cannot be accepted, extensive configuration interaction (Cl) must be included. A detailed comparison of SCF and Cl results is available for the potential energy surface for the reaction F + H2-FH+H 47 ). 

This state vector, which is the second-quantized counterpart of a configuration space Slater determinant wave function, is a solution of the equation... 

Jz obtained from anti-symmetrized Slater determinant wave functions ai,a2, , njv) such that... 

In fact, this is Kohn-Sham exchange energy [see Eq. (11.72)], because the Slater determinant wave function used to calculate it is the Kohn-Sham determinant not the Hartree-Fock one. 

The Hartree wave function lacks the anti-symmetry of electron-wave function exchange. Thereafter Fock uses the Slater determinant wave function like... 

The total wave function can be much improved if it is written with the help of two or many configurations, each corresponding to a Slater determinant wave function. The most general way of writing a wave function of this type is... 

Orbitals that are the solutions of Equation 2.30 are called canonical Hartree-Eock orbitals. Please remember that a unitary transformation can be applied to the orbitals of a Slater determinant without changing the total (Slater determinant) wave function. 

According to Section 1.6.2, we should make a linear combination of aU possible Slater determinants, calculate the matrix elements, and determine the coefficients in an eigenvalue problem. We will choose a simpler way, however. It is sufficient to note that certain combinations of (Ml,Ms) quantum numbers correspond to one, two, or three Slater determinants. Wave functions with different quantum numbers cannot be mixed. For a given linear combination, the number of states in the boxes in Table 2.2 are the same. Each (L,S)-multiplet must be represented by a single Slater determinant for a given (Ml,Ms) - quantum number. Thus, the following scheme must be possible ... 

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