Single determinant wavefunctions

The calculation mixes all single determinant wavefunctions that can be obtained from the ground state by exciting electrons from a subset of the occupied orbitals (of the ground state) to a subset of the unoccupied orbitals. The subsets are specified as a fixed number (highest occupied or lowest unoccupied) or by an energy criterion associated with the energy difference between the occupied orbital and the unoccupied orbital. 

However, if this is not the case, the perturbations are large and perturbation theory is no longer appropriate. In other words, perturbation methods based on single-determinant wavefunctions cannot be used to recover non-dynamic correlation effects in cases where more than one configuration is needed to obtain a reasonable approximation to the true many-electron wavefunction. This represents a serious impediment to the calculation of well-correlated wavefunctions for excited states which is only possible by means of cumbersome and computationally expensive multi-reference Cl methods. 

Suppose there are In electrons in the system, half with a-spin and half with -spin, and we form a single-determinant wavefunction with the spatial functions and so that... 

Hartree-FockWavefunction. The simplest quantum-mechanically correct representation of the many-electron wavefimction. Electrons are treated as independent particles and are assigned in pairs to functions termed Molecular Orbitals. Also known as Single-Determinant Wavefunction. 

Pople-Nesbet Equations. The set of equations describing the best Unrestricted Single Determinant Wavefunction within the LCAO Approximation. These reduce the Roothaan-Hall Equations for Closed Shell (paired electron) systems. 

Single-Determinant Wavefunction See Hartree-Fock Wavefunction. 

The Kohn-Sham (KS) scheme43 introduces a single-determinant wavefunction in terms of the KS orbitals and partitions the HKUEDF into three main pieces ... 

Excited states, and those unusual molecules with electrons of opposite spin singly occupying different spatial MO s (open-shell singlets) cannot be properly treated with a single-determinant wavefunction. They must be handled with approaches beyond the Hartree-Fock level, such as configuration interaction (Section 5.4). 

In the Bom interpretation (Section 4.2.6) the square of a one-electron wavefunction ij/ at any point X is the probability density (with units of volume-1) for the wavefunction at that point, and j/ 2dxdydz is the probability (a pure number) at any moment of finding the electron in an infinitesimal volume dxdydz around the point (the probability of finding the electron at a mathematical point is zero). For a multielectron wavefunction T the relationship between the wavefunction T and the electron density p is more complicated, being the number of electrons in the molecule times the sum over all their spins of the integral of the square of the molecular wavefunction integrated over the coordinates of all but one of the electrons (Section 5.5.4.5, AIM discussion). It can be shown [9] that p(x, y, z) is related to the component one-electron spatial wavefunctions ij/t (the molecular orbitals) of a single-determinant wavefunction T (recall from Section 5.2.3.1 that the Hartree-Fock T can be approximated as a Slater determinant of spin orbitals i/qoc and i// /i) by... 

Given the HF single determinant wavefunction of an iV-electron system, the Koopmans theorem29 states that the energy required to produce an (2V-l)-single determinant wavefunction by removing an... 

When the orbitals are determined in this manner, with the only restriction being the orthonormality constraint, equation (12), they yield the best possible antisymmetrized product wavefunction (i.e., a single determinant wavefunction) for the system in question since the resulting fP(x R) yields the lowest possible value for E(R). The heart of the Hartree-Fock model is to replace the detailed and accurate description of the repulsions between every pair of electrons in the system by the average field that each electron exerts on every other. This is a consequence of the one-electron or orbital basis of the model which leads to a product wavefunction. The probability, Pa(ri,r2)dridr2, that electron 1 be in some small volume element about ri when electron 2 is simultaneously in some small volume element about ra is, in a simple product-type wavefunction, given by the product of the two singleparticle probabilities,... 

When the second of the equivalence restrictions is removed, a single determinant wavefunction of lower energy is usually obtained. In fact, it is possible for a wave-function obtained in this way, a so-called unrestricted Hartree-Fock (UHF) wavefunction191 (perhaps more properly called a spin-unrestricted Hartree-Fock wavefunction) to go beyond the Hartree-Fock approximation and thus include some of the correlation energy. Lowdin192 describes this as a method for introducing a Coulomb hole to supplement the Fermi hole already accounted for in the RHF wavefunction. 

The obvious advantage of this method is the retention of the single-determinant wavefunction. The drawback is that the resulting wavefunction is no longer an eigenfunction of the spin operator . There are cases where the eigenvalue equation... 

It has been customary to classify methods by the nature of the approximations made. In this sense CNDO, INDO (or MINDO), and NDDO (Neglect of Diatomic Differential Overlap) form a natural progression in which the neglect of differential overlap is applied less and less fully. It is now clearer that there is a deeper division between methods, related to their objectives. On the one hand are approximate methods which set out to mimic the ab initio molecular orbital results. The objective here is simply to find a more economical method. On the other hand, some workers, recognizing the defects of the MO scheme, aim to produce more accurate results by the extensive use of parameters obtained from experimental data. This latter approach appears to be theoretically unsound since the formalism of the single-determinant wavefunction and the Hartree-Fock equations is retained. It can be argued that the use of the single-determinant wavefunction prevents the consistent achievement of predictions better than those obtained by the ab initio scheme where no further... 

Downward-directed lines represent hole states (orbitals occupied in the reference) and upward directed lines represent particle states (orbitals unoccupied in the reference). Hence, one may interpret Figure 1(d) as a single-determinant wavefunction that differs from the reference by a single excitation from orbital (j), to orbital Furthermore, this convention implies that the reference wave-function itself is represented by empty space, as indicated in Figure 1(c). 

Almost as a corollary to the discussion of the strategy explicated in the previous section, we may note that essentially the same procedure can be invoked for calculating the SCF or MC-SCF wavef unctions. Let the closed-shell ground state of a 2n-electron system be represented by a single determinant wavefunction constructed from 2n-spin orbitals. 

The sum is over n the occupied MOs rj/i and for a closed-shell molecule each n,- = 2, for a total of 2n electrons. Equation (7.1) applies strictly only to a single-determinant wavefunction 4, but for multideterminant wavefunctions arising from configuration interaction treatments (section 5.4) there are similar equations [8]. A shorthand for p(x, y, z)dxdydz is p r)dr, where r is the position vector of the point with coordinates (x, y, z). 

We have shown in this introductory study that the two-particle density matrix can be used as a basis for indices of bond order or strength, and of Diels-Alder reactivity. In the approximation used there was a close parallel between the indices thus obtained and those developed within the one-particle matrix. To some extent this may be a little fortuitous and depend on the fact that for single-determinant wavefunctions there is a simple formula relating the elements of the two matrices ... 

While the uncoupled Hartree-Fock method and the single transition approximation have the merit of computational simplicity, they suffer, however, in particular from the usually unsatisfactory description of electronically excited states with a single-determinant wavefunction. 

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