Determinantal product

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... 

The representation in which L and S are good quantum numbers is obtained by forming suitable linear combinations of the determinantal product wavefunctions, equation (3.72). The first-order shift in the energy produced by the electrostatic interaction is given by... 

VVc can now see why the normalisation factor of the Slater determinantal wavefunction is I v/N . If each determinant contains N terms then the product of two Slater determinants, ldeU rminant][determinant], contains (N ) terms. However, if the spin orbitals form an oi lhonormal set then oidy products of identical terms from the determinant will be nonzero when integrated over all space. We Ccm illustrate this with the three-electron example, k ljiiiidering just the first two terms in the expansion we obtain the following ... 

So far there have not been any restrictions on the MOs used to build the determinantal trial wave function. The Slater determinant has been written in terms of spinorbitals, eq. (3.20), being products of a spatial orbital times a spin function (a or /3). If there are no restrictions on the form of the spatial orbitals, the trial function is an Unrestricted Hartree-Fock (UHF) wave function. The term Different Orbitals for Different Spins (DODS) is also sometimes used. If the interest is in systems with an even number of electrons and a singlet type of wave function (a closed shell system), the restriction that each spatial orbital should have two electrons, one with a and one with /3 spin, is normally made. Such wave functions are known as Restricted Hartree-Fock (RHF). Open-shell systems may also be described by restricted type wave functions, where the spatial part of the doubly occupied orbitals is forced to be the same this is known as Restricted Open-shell Hartree-Fock (ROHF). For open-shell species a UHF treatment leads to well-defined orbital energies, which may be interpreted as ionization potentials. Section 3.4. For an ROHF wave function it is not possible to chose a unitary transformation which makes the matrix of Lagrange multipliers in eq. (3.40) diagonal, and orbital energies from an ROHF wave function are consequently not uniquely defined, and cannot be equated to ionization potentials by a Koopman type argument. 

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. 

For small determinantal wavefunctions these statements are easily verified by explicit expansion the general proof rests on the fact that the determinant of a matrix product is equal to the product of the determinants of the matrices. 

Answer. Orbitals are one-electron wave functions, ). The fact that electrons are fermions requires that each electron be described by a different orbital. The simplest form of a many-electron wave function, T(l, 2,..., Ne), is a simple product of orbitals (a Hartree product), 1(1) 2(2) 3(3) NfNe). However, the fact that electrons are fermions also imposes the requirement that the many-electron wave function be antisymmetric toward the exchange of any two electrons. All of the physical requirements, including the indistinguishability of electrons, are met by a determinantal wave function, that is, an antisymmetrized sum of Hartree products, ( 1,2,3,..., Ne) = 1(1) 2(2) 3(3) ( ). If (1,2,3,...,Ne) is taken as an approximation of (1,2,..., Ne), i.e., the Hartree-Fock approximation, and the orbitals varied so as to minimize the energy expectation value,... 

The wavefunction F° then follows as an antisymmetrized product built from the single-particle functions q>i(r, ms) for the Z electrons (Slater determinantal wave-function, see below and Section 7.2), where r is the spatial vector and ms the spin magnetic quantum number. 

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... 

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. 

HF Surfaces for Ss2 Reactions.—We now return to the review of the reactions which have been studied at the single-determinantal SCF level of approximation. The majority of these studies have concentrated on mapping out the minimum energy pathways between reactants and products. Such reaction pathways for a number of Ss2 displacement reactions have been determined. The systems which... 

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... 

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