Multi-determinant wavefunction

Let us summarise this discussion of NOs with one simple fact the shortest expansion of a closed-shell multi-determinant wavefunction can be written in terms of a set of orbitals which are doubly occupied in the expansion of the electron density. 

We begin by considering a multi-determinant wavefunction of the form eqn ( 22.3) and try to find approximations to the orbitals which will validate our particular choice of ... 

Use the determinants formed by occupying arbitrary selections of the occupied and virtual orbitals as a basis for perturbation theory. That is, allow the perturbation to mix the single-determinant ground state with various excited single determinants to give a multi-determinant wavefunction in the presence of the perturbation. 

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. 

More accurate multi-determinant configuration-interaction (Cl) wavefunctions are described by specifying the types of substitutions ( excitations ) from the starting HF... 

UHF Methods. A major drawback of closed-shell SCF orbitals is that whilst electrons of the same spin are kept apart by the Pauli principle, those of opposite spin are not accounted for properly. The repulsion between paired electrons in spin orbitals with the same spatial function is underestimated and this leads to the correlation error which multi-determinant methods seek to rectify. Some improvement could be obtained by using a wavefunction where electrons of different spins are placed in orbitals with different spatial parts. This is the basis of the UHF method,40 where two sets of singly occupied orbitals are constructed instead of the doubly occupied set. The drawback is of course that the UHF wavefunction is not a spin eigenfunction, and so does not represent a true spectroscopic state. There are two ways around the problem one can apply spin projection operators either before minimization or after. Both have their disadvantages, and the most common procedure is to apply a single spin annihilator after minimization,41 arguing that the most serious spin contaminant is the one of next higher multiplicity to the one of interest. 

Making the central assumption that the best possible single-determinant function formed from a given basis is also the best possible starting point from which to make a multi-determinant expansion" we take the first term in the expansion eqn ( 20.1) to be o, the HF wavefunction so that eqn ( 20.1) becomes... 

Taking into account the electronic correlation is mandatory if a quantitative description of the electronic stmcture and energy of the system of interest is required. In addition, in some cases, the inclusion of the electronic correlation effects are necessary to obtain even a qualitatively correct description of the electronic structure of the system. By definition, the mean-field approximation resulting from the approximation of the multi-electron wavefunction by a single Slater determinant is unable to account for the electronic correlation. A correlated electronic wavefunction must then be written as a linear combination of several Slater determinants... 

Since the exact solution of Schrodinger s equation for multi-electron, multi-nucleus systems turned out to be impossible, efforts have been directed towards the determination of approximate solutions. Most modern approaches rely on the implementation of the Born-Oppenheimer (BO) approximation, which is based on the large difference in the masses of the electrons and the nuclei. Under the BO approximation, the total wave-function can be expressed as the product of the electronic il/) and nuclear (tj) wavefunctions, leading to the following electronic and nuclear Schrodinger s equations ... 

Continuum wavefunctions required for the calculation of bound-free matrix elements of the type (3.1) can be determined by several methods. In principle, all techniques which have been developed in the field of atomic and molecular collisions [see Thomas et al. (1981) for a comprehensive overview] can be employed with only slight modifications and extensions. Since many readers are probably not familiar with the calculation of multi-dimensional continuum wavefunctions, we shall briefly describe in the following one particular method, which is rather universal. 

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