Determinantal product states

We note that similar selection rules have been derived on the basis of determinantal product states, using the expansion theorem of Laplace [15]. The relationship between both formalisms is still under study [16],... 

The decade ended with the publication of the two-volume work of Slater (1960) on atomic structure. A cautious treatment is given of tensor operators and fractional parentage. Expansions in determinantal product states are still resorted to, and there is no discussion of Racah s use of groups. A little later, the writer attempted to remedy that deficiency (Judd 1963). His prefatory assurance that the needs of the experimentalist were borne in mind was sourly commented on, in a private conversation, by Edien. The recollection that the book of Condon and Shortley (1935) gave similar problems to Russell (its dedicatee), provided some solace. 

The procedure for finding the first-order energy corrections, through the use of formulae (1.11), (1.12) etc., can be greatly simplified by finding in advance linear combinations of the determinantal product states, such that... 

On adding the two quantum numbers of the parent, the total number of quantum numbers characterizing the states of the three non-equivalent electron configuration is 12, which is the same as the number required for characterizing the determinantal product states. 

The Pauli principle can be included in a more general symmetry principle, proposed by Heisenberg and Dirac, which states that electrons and other particles with half-integer spin must be represented by wavefunctions which are totally antisymmetric with respect to an interchange of the spin and coordinates of any two electrons. Slater showed that this requirement could be satisfied by a normalized determinantal product wavefunction of the form... 

Electron Nuclear Dynamics (48) departs from a variational form where the state vector is both explicitly and implicitly time-dependent. A coherent state formulation for electron and nuclear motion is given and the relevant parameters are determined as functions of time from the Euler equations that define the stationary point of the functional. Yngve and his group have currently implemented the method for a determinantal electronic wave function and products of wave packets for the nuclei in the limit of zero width, a "classical" limit. Results are coming forth protons on methane (49), diatoms in laser fields (50), protons on water (51), and charge transfer (52) between oxygen and protons. 

In the independent particle picture, the ground state of helium is given by Is2 xSo. For this two-electron system it is always possible to write the Slater determinantal wavefunction as a product of space- and spin-functions with certain symmetries. In the present case of a singlet state, the spin function has to be... 

It is generally found that if one increases the flexibility of a single-determinantal wavefunction by allowing each space orbital to assume an independent form (rather than insisting on double occupation by an and a (1 electron for those orbitals which would otherwise be so occupied as dictated by the electronic configuration) that the asymptotic difficulties of the wavefunction are removed. Thus, the unrestricted Hartree-Fock method usually predicts the correct dissociation products of a molecular system.140 The symmetrical (C2 ,) insertion of Of3P) into Ha yields the 33i state of the HaO system. The electronic configuration of this state expressed in terms of the unrestricted set of orbitals is... 

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 construction of a basis of this space by sums and products of Slater determinantal basis sets of the component spaces is trivial. Obviously the dimension of the composite Hilbert space Y is considerably enlarged in comparison with the physical N particle space H, which contains all particle-number conserving excitations. The expression (6), however, only allows a rather rough estimate of the actual dimension of the relevant space for which it gives an upper bound. The minimal space needed is certainly smaller because symmetries may reduce the relevant spaces drastically. E. g. the fact that both reference states were chosen equal in Eq. (5) allows to replace the third component of the extended states by 1 4 ( 0 1 s which spares the component from... 

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