Determinantal Expansion

The effectiveness of the method is exhibited by Figure 2 in which the energy errors of truncated expansions are plotted versus the numbers of determinants in these expansions. For each of the four systems shown, one curve displays this relationship for the expansions generated by the just discussed a priori truncations, whereas the other curve is obtained a posteriori by starting with the full SDTQ calculation in the same orbital basis and, then, simply truncating the determinantal expansion based on the ordering established by the exact coefficients of the determinants. There is practically no difference in the number of determinants needed to achieve an accuracy of 1 mh. 

Because subsystems A and B do not interact, it must be that T a consists of a determinantal expansion in functions taken solely from the set Ha, and similarly uses only those spin orbitals in Br. It follows that T a and are strongly orthogonal [53]. Two antisymmetric functions f x, ..., Xp) and g yi,..., yg) are said to be strongly orthogonal if... 

We now specialize the discussion to the ligand field theory situation and define the orthonormal set of spin-orbitals we shall use in the determinantal expansion of the many-electron functions Vyy for the groups M and L. First we suppose that we have a set of k orbitals describing the one-electron states in the metal atom these will be orthonormal solutions of a Schrodinger equation for a spherically symmetric potential, V<,(r), which may be thought of as the average potential about the metal atom which an electron experiences ... 

As discussed in detail in Refs. 77 and 82, for example, this expansion is not N-fold (where N is the number of electrons in the system) for the lower perturbational orders, but truncates to include only modest excitation levels. For example, the first-order wavefunction, which may be used to compute both the second- and third-order energies, contains contributions from doubly excited determinants only, whereas the second-order wavefunction, which contributes to the fourth- and fifth-order perturbed energies, contains contributions from singly, doubly, triply, and quadruply excited determinants. Furthermore, the sum of the zeroth- and first order energies is equal to the SCF energy. This determinantal expansion of the perturbed wavefunctions suggests that we may also decompose the cluster operators, T , by orders of perturbation theory ... 

Next are methods based on zeroth-order wave functions that are linear combinations of Slater determinants that again may or may not include a treatment of residual electron correlation effects. In these multiconfigurational (MC) approaches, the starting wave function is presupposed to be given by a determinantal expansion... 

It is important to note, however, that there are fundamental differences between FSCC and SRCC with respect to the nature of their excitation operators. For a given truncation of the cluster operators beyond simple double excitations, the determinantal expansion space available in an FSCC calculation is smaller than those of SRCC calculations for the various model space determinants. A class of excitations called spectator triple excitations must be added to the FSCCSD method to achieve an expansion space that is in some sense equivalent to that of the SRCC. But even then, the FSCC amplitudes are restricted by the necessity to represent several ionized states simultaneously. Thus, we should not expect the FSCCSD to produce results identical to a single reference CCSD, nor should we expect triple excitation corrections to behave in the same way. The differences between FSCC and SRCC shown in Table I and others, below, should be interpreted as a manifestation of these differences. 

If the rows of a matrix become linearly dependent, the determinant of that matrix vanishes. This is also true of the determinantal expansion terms. The spin orbitals in the determinant must be linearly independent for the function to have non-zero values for some set of electron coordinates. For an orthonormal spin-orbital basis, this means that no spin orbital may be occupied more than once. The determinantal form of the expansion basis shows that a particular electron cannot be associated with a particular orbital or, accordingly, with a particular region of space. It is still convenient, however, to refer to the electrons in a particular orbital as if the wavefunction were a simple orbital product. The strict interpretation of these references is to the orbital, which is occupied in the determinant by all the electrons, and not to the individual electron which happens to occupy the orbital in some of the Nl terms of the determinant. 

The calculation of expectation values of operators over the wavefunction, expanded in terms of these determinants, involves the expansion of each determinant in terms of the N expansion terms followed by the spatial coordinate and spin integrations. This procedure is simplified when the spatial orbitals are chosen to be orthonormal. This results in the set of Slater Condon rules for the evaluation of one- and two-electron operators. A particularly compact representation of the algebra associated with the manipulation of determinantal expansions is the method of second quantization or the occupation number representation . This is discussed in detail in several textbooks and review articles - - , to which the reader is referred for more detail. An especially entertaining presentation of second quantization is given by Mattuck . The usefulness of this approach is that it allows quite general algebraic manipulations to be performed on operator expressions. These formal manipulations are more cumbersome to perform in the wavefunction approach. It should be stressed, however, that these approaches are equivalent in content, if not in style, and lead to identical results and computational procedures. 

The coefficient of a particular determinant within a CSF is given by the product of the factors / given in Table II for all the levels. The phase factors that are unique to the unitary group approach are those determined by b. These factors are determined by the CSF coupling and not by the individual determinants. Thus these phase factors result in the multiplication of the total CSF by some overall sign factor. Table III shows the determinantal expansion for the set of doublet CSFs consisting of five singly occupied orbitals, 01,02.04.05. nd one doubly occupied orbital, 0j. The sparseness of the... 

As we shall see, the most common use of the variation method is not to find a set of linear parameters in the determinantal expansion of the wavefunction but to model the electronic structure and optimise the parameters contained in the mathematical formulation of that model. 

Slater determinants) in terms of these one-electron functions. We then consider the Hartree-Fock approximation in which the exact wave function of the system is approximated by a single Slater determinant and describe its qualitative features. At this point, we introduce a simple system, the minimal basis (Is orbital on each atom) ab initio model of the hydrogen molecule. We shall use this model throughout the book as a pedagogical tool to illustrate and illuminate the essential features of a variety of formalisms that at first glance appear to be rather formidable. Finally, we discuss the multi-determinantal expansion of the exact wave function of an N-electron system. 

In the limit of a complete set of orbitals, we may arrive at the exact solution to the Schrodinger equation in the form (4.1.2) but the determinantal expansion then becomes infinite. In practice, we must resort to truncated expansions and thus be satisfied with approximate solutions to the electronic SchrOdinger equation. 

More generally, we may impose the correct Cbulomb-cusp behaviour on any determinant-based wave function by multiplying the determinantal expansion 4> by some correlating Junction y of the form [7]... 

---