Orthonormality Slater determinants

The Hilbert space corresponding to N electrons is the sum of the orthogonal subspaces S2 , n = 0,1,2,... IV, which are spanned by the n-tuply excited (orthonormal) Slater determinants. Elements of the space Q are all linear combinations of n-tuply excited Slater determinants. It does not mean, of course, that each element of this space is an n-tuply excited Slater determinant. For example, the sum of two doubly excited Slater determinants is a doubly excited Slater determinant only when one of the excitations is common to both determinants. 

The functions Pa, P t and P are well behaved. In particular, the functions Pa and Pjt, respectively, may be expanded in terms of orthonormal Slater determinants Dak and D i built from some orthonormal set of one-electron functions zpi, ip2 (mo-... 

Additionally, orthonormal Slater determinants result in orthonormal CSFs,... 

In faet, the Slater determinants themselves also are orthonormal funetions of N eleetrons whenever orthonormal spin-orbitals are used to form the determinants. 

A Hartree-Fock wave function can be written as a single Slater determinant, composed of a set of orthonormal MOs (eq. (3.20)). 

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). 

Molecular rearrangement resulting from molecular collisions or excitation by light can be described with time-dependent many-electron density operators. The initial density operator can be constructed from the collection of initially (or asymptotically) accessible electronic states, with populations wj. In many cases these states can be chosen as single Slater determinants formed from a set of orthonormal molecular spin orbitals (MSOs) im as / =... 

Averages of properties require integrals over CSFs which can readily be written for one- and two-electron operators, insofar the Slater determinants and the MSOs are orthonormal by construction, in terms of one- and two-electron density matrices. 

The calculation of the cross matrix elements (6) is somewhat more difficult, because the Slater Determinants involved in them are constructed with two sets of non-orthonormal spinorbitals. This calculation, however, may be greatly simplified, if the two sets are assumed to be corresponding, that is, if they fulfill the following condition [14] ... 

Now that we have decided on the form of the wave function the next step is to use the variational principle in order to find the best Slater determinant, i. e., that one particular Osd which yields the lowest energy. The only flexibility in a Slater determinant is provided by the spin orbitals. In the Hartree-Fock approach the spin orbitals (Xi 1 are now varied under the constraint that they remain orthonormal such that the energy obtained from the corresponding Slater determinant is minimal... 

The one-electron integral in Eq. (2.19) can then be evaluated by substitution of the MO expressions obtained from the relevant Slater determinant [Eq. (2.15)] and the application of the requirements for MO orthonormality in the resulting integral expressions [Eq. (2.3)]. After a substantial amount of algebraic manipulation, it can be shown that the relevant integral can be expressed as the sum of simple one-electron integrals ... 

In a well known practical but approximate method to solve the GS problem, known as the Hartree-Fock (HF) approximation (see e.g. [10]), the domain of variational functions P in Eq. (9) is narrowed to those that are a single Slater determinant (D) 9 d, constructed out of orthonormal spin orbitals tj/iix) ... 

Here, the minimization is over all sets of orthonormal orbitals, subject to the requirement that satisfies the (g, K) conditions. Because an ensemble of Slater determinants is incapable of describing electron correlation, one must... 

Let be Bm = electron function basis. Slater-determinants constructed over Bm span an orthonormal, jx = ( ) dimensional subspace of the N-electron Hilbert space. The projection of the exact wave function in this subspace ( ) can be given as a linear... 

Most Cl calculations involve configurations formed from a common set of orthonormal orbitals by spin and symmetry adaptation of Slater determinants. In this case S is a unit matrix and the formation of H is greatly simplified. 

Let 0i(r,a), i = 1, m denote a basis of m orthonormal spin orbitals where r are the spatial coordinates and o is the spin coordinate. A Slater determinant is an antisymmetric linear combination of one or more of these spin orbitals. The occupation of a given Slater determinant can be written as an occupation number vector, ln>, where nj is one if spin orbital j is occupied in the Slater determinant and nj is zero if spin orbital (j> is unoccupied. 

As was shown in chapter three we can compute the transition densities from the Cl coefficients of the two states and the Cl coupling coefficients. Matrix elements of two-electron operators can be obtained using similar expresssions involving the second order transition density matrix. This is the simple formalism we use when the two electronic states are given in terms of a common orthonormal MO basis. But what happens if the two states are represented in two different MO bases, which are then in general not oithonormal We can understand that if we realize that equation (5 8) can be derived from the Slater-Lowdin rules for matrix elements between Slater determinants. In order to be a little more specific we expand the states i and j ... 

In the relativistic DVME method, the interactions among the states represented by the Slater determinants, i.e., the Cl can be analyzed using explicit many-electron wave functions expressed by eq. (23). From the orthonormality of the Slater determinants, the inner product of the /th many-electron wave function with itself can be expanded as... 

Although a Slater-determinant reference state 4> cannot describe such electronic correlation effects as the wave-function modification required by the interelec -tronic Coulomb singularity, a variationally based choice of an optimal reference state can greatly simplify the -electron formalism. 4> defines an orthonormal set of N occupied orbital functions occupation numbers = 1. While () = 1 by construction, for any full A-electron wave function T that is to be modelled by it is convenient to adjust (T T) > 1 to the unsymmetrical... 

The reference state T is a Slater determinant constructed as a normalized antisymmetrized product of N orthonormal spin-indexed orbital functions orbital energy functional E = To + Ec is to be made stationary, subject to the orbital orthonormality constraint (i j) = StJ, imposed by introducing a matrix of Lagrange multipliers. The variational condition is... 

It can be proven [31] that all possible Slater determinants of N particles constructed from a complete system of orthonormalized spin-orbitals 4>k form a complete basis in the space of normalized antisymmetric (satisfying the Pauli principle) functions, of N electrons i.e. for any antisymmetric and normalizable (K one can find expansion amplitudes so that ... 

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... 

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