Representation determinantal

Density functional theory, direct molecular dynamics, complete active space self-consistent field (CASSCF) technique, non-adiabatic systems, 404-411 Density operator, direct molecular dynamics, adiabatic systems, 375-377 Derivative couplings conical intersections, 569-570 direct molecular dynamics, vibronic coupling, conical intersections, 386-389 Determinantal wave function, electron nuclear dynamics (END), molecular systems, final-state analysis, 342-349 Diabatic representation ... 

Unfortunately, the A-representability constraints from the orbital representation are not readily generalized to the spatial representation. A first clue that the A-representability problem is more complicated for the spatial basis is that while every A-representable Q-density can be written as a weighted average of Slater determinantal Q-densities in the orbital resolution (cf. Eq. (54)), this is clearly not true in the spatially resolved formulation. For example, the pair density (Q = 2) of any real electronic system will have a cusp where electrons of opposite spin coincide but a weighted average of Slater determinantal pair densities,... 

The variational procedure in Eq. (100) is in the spirit of the Kohn-Sham ansatz. Since satisfies the (g, K) conditions, it is A-representable. In general, Pij ig corresponds to many different A-electron ensembles and one of them,, corresponds to the ground state of interest. However, for computational expediency in computing the energy, a Slater determinantal density matrix,... 

The simplest approach, of course, is to maintain the minimum-determinantal description and reoptimize all of the orbitals. In practice, however, such an approach is practical only in instances where die ground-state and the excited-state wave functions belong to different incduciblc representations of die molecular point group (cf. Section 6.3.3). Otherwise, the variational soludon for die excited-state wave function is simply to collapse back to the ground-state wave function And, even if the two states do differ in symmetry, the desired excited state may not be the lowest energy such state widiiii its irrep, to which variational optimization will nearly always lead. 

The application of HF and KS-DFT is fundamentally limited to wave functions that can be expressed as single Slater determinants. This restricts their utility in dealing with states like the A2 state of phenylnitrene, which is two-determinantal in character (Figure 14.3). One can apply HF or KS-DFT to the single determinant that, restricted to representation of the singly occupied orbitals and with normalization implicit, is written... 

Thus, there are two Au MOs, two B2g MOs, three Blu MOs, and three B2g MOs. By constructing SALCs corresponding to these representations, the well-nigh hopeless problem of solving a 10 x 10 determinantal equation is reduced to the tractable task of solving two quadratic and two cubic equations. This has already been illustrated in Section 7.2. 

A localized molecular orbital representation is the closest approach that can be achieved, for a given determinantal wavefunction, to an electrostatic model of a molecule 44>. With truly exclusive orbitals, electron domains interact with each other through purely classical Coulombic forces and the wavefunction reduces, for all values of the electronic coordinates, to a single term, a simple Hartree product. 

The determinantal functions must be linearly independent and eigenfunctions of the spin operators S2 and Sz, and preferably they belong to a specified row of a specified irreducible representation of the symmetry group of the molecule [10, 11]. Definite spin states can be obtained by applying a spin projection operator to the spin-orbital product defining a configuration [12]. Suppose d>0 to be the solution of the Hartree-Fock equation. From functions of the same symmetry as d>0 one can build a wave function d>,... 

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. 

Determinantal representation of CSFs. Expansion of CSFs in the unitary group approach in terms of spin-orbital determinants. The coefficients are determined as products of factors,/k, determined from the Clebsch-Gordan coefficients and phase factors of Table II. The coefficient sparseness of the determinants is predictable and corresponds to the allowed area principle. 

For small wavefunction expansions where an explicit determinantal representation of the wavefunction may be constructed, it is straightforward to determine the solutions to Eq. (241) by constructing the overlap matrix of the first-order variational space. This is required in the solution of the variational super-CI equations for which these linear dependences must be explicitly identified and eliminated For larger CSF expansion lengths or large orbital basis sets, however, this step could easily dominate the entire iterative procedure, so alternate methods must be examined. The solutions to Eq. (241) may be determinedby first operating from the left with a CSF... 

If we define three vectors a, b and c, as in Problem 5.9, the expression a (6 X c), known as the scalar triple product, yields a scalar quantity, the magnitude of which provides the formula for the volume, F, of a parallelepiped with adjacent edges defined by vectors a, b and c (an example in chemistry being a crystalline unit cell). If the determinantal representation of A x c is used, then, on expanding the determinant from the first row, and evaluating the three scalar products, we obtain ... 

A determinantal wave function expressed in this form is a coherent state. The associated Lie group is the unitary group U K) and the reference state Eq. (127) is the lowest weight state of the irreducible representation [1 0 ] of Lf(K). The stability group is U N) X U(K — N). The norm in an orthonormal basis of spin orbitals is... 

The rules given by Slater must be used when evaluating matrix elements of any operator in a Slater determinantal representation (Slater, 1960, p. 291). These rules reduce the matrix elements of one- and two-electron operators between many-electron Slater determinantal functions to simple sums over spatial orbital... 

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