Normalization Slater determinants

The values of cion and c ,v determine the weights of the respective components, and reflect the relative stabilization of the VB states in solution e g. a polar solvent is expected to stabilize the ionic TBu+Cl-) relative to the covalent x Bu-Cl). By contrast, the HF wave function for BuCl is the (normalized) Slater determinant[22]... 

If one could solve Eq. (203) exactly for exact energy— provided that the reference function is n-representable (e.g., is a normalized Slater determinant). The unitary transformation preserves the n-representability. Equation (203) is an infinite-order nonlinear set of equations and not easy to solve. However, the perturbation expansion terminates at any finite order. We have [6,12]... 

The next step is to find orbitals that minimize the expectation value of /complete in Eq. (13.1), given Eq. (13.3) for Hqm/mm. If we take as our wave function a standard normalized Slater determinant, we have... 

The starting point is a shell consisting of a collection of spin-orbitals. Occupation of these orbitals is represented by a normalized Slater determinant. An... 

Here the sequence of spin-orbitals between vertical bars represents a normalized Slater determinant. The af operators are so-called creation operators and 0> is the vacuum state. We may undo the action of the creation operators by introducing so-called annihilation operators, as exemplified below ... 

Slater determinants are usually constructed from molecular spinorbitals. If, instead, we use atomic spinorbitals and the Ritz variational method (Slater determinants as the expansion functions), we would get the most general formulation of the valence bond (VB) method. The beginning of VB theory goes back to papers by Heisenbeig, the first application was made by Heitler and London, and later theory was generalized by Hurley, Lennard-Jones, and Pople. The essence of the VB method can be explained by an example. Let us take the hydrogen molecule with atomic spinorbitals of type liaO and Vst (abbreviated as aa and b ) centered at two nuclei. Let us construct from them several (non-normalized) Slater determinants, for instance ... 

If the function 4 is taken as a normalized Slater determinant built of N spinorbitals 4>i, from the I rule of Slater-Condon (see Appendix M available at http //booksite.elsevier. com/978-0-444-59436-5, we replace there h with b) for (4> — r)) ). 

In order to show how we ealeulate the dipole moment in praetiee, let us use the Hartree-Fock approximation. Using the normalized Slater determinant we have as the Haitree-Fock approximation to the dipole moment ... 

For example, the s)mibol 4> (001000100000...) means a normalized Slater determinant of dimension 2, containing the spinorbitals 3 and 7. The symbol (001000...) does not make sense because the number of digits 1 has to equal 2, etc. 

The essence of the VB method can be explained by an example. Let us take the hydrogen molecule with atomic spinorbitals of type Ha a and si /3 denoted shortly as a a and fc/3 centred at two nuclei. Let us construct from them several (non-normalized) Slater determinants, for instance ... 

Electronic structure calculations in atoms with more than two electrons require properly antisymmetrized trial functions. For a closed-shell atom with p electrons having paired spins, such a function can be written in the form of a normalized Slater determinant ... 

For ease of exposition we shall for the moment assume that the orbitals r are orthonormal so that the spin-orbitals are also orthonormal. If the are then chosen as normalized Slater determinants or proper linear combinations of them, they too may be chosen to be an orthonormal set so that M i = dt/. The secular problem then simplifies to... 

Another possibility to avoid the quasi-independence in the definition of the DAFH and DI offers the decomposition of the pair density into symmetric and antisymmetric parts, which can be also seen as a partitioning into the singlet and triplet components of the pair density." For wavefunctions given by single normalized Slater determinant based on N orthonormal orbitals J/ the pair density of the singlet coupled electrons can be written as" ... 

Let ( />(x) be a basis of M orthonormal spin orbitals, where the coordinates x represent collectively the spatial coordinates r and the spin coordinate a of the electron. A Slater determinant is an antisymmetrized product of one or more spin orbitals. For example, a normalized Slater determinant for N electrons may be written as... 

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