Multi-electron wavefunction

Ceulemans considers a dn electron state, split by an octahedral field into the e and t2 levels, so that all the n electrons are in the t2 subshell. In the notation of Sugano et al. [27], rjj(t2SrMsMr) is the multi-electronic wavefunction, with SMs irrep labels for the total spin and rMr irrep labels in the octahedral group for the orbital state. We use a real orbital basis in which all njm factors take their simplest possible forms and suppress S, r and Mr below. It takes six electrons (three pairs each of opposed spin) to fill this t2 subshell. Ceulemans [7] particle-hole conjugation operator 0() has the effect of conjugating the occupancies within this subshell, and of... 

That is, the multi-electron wavefunction is formed from products of molecular spin orbitals. Here, ( )f(rj) means the j electron with co-ordinate rj is in the i molecular orbital with P spin. The spin should be thought of as an additional... 

On the other hand the permutation symmetry of multi-electron wavefunctions is restricted by the Pauli principle. 

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... 

Our individual one-electron HF or KS wavefunctions represent the individual molecular orbitals, and the square of the wavefunction gives us the probability distribution of each electron within the molecule. We do not know the form of the real multi-electron wavefunction a priori, nor the individual one-electron HF or KS functions, but we can use the mathematical principle that any unknown function can be modeled by a linear combination of known functions. A natural choice for chemists would be to use a set of functions that are similar in shape to individual atomic orbitals. To do this, we need to consider atomic radial distribution functions, such as the ones shown in Figure 3.2 for hydrogen. These are plots of how the electron density varies at any given distance away from the nucleus. 

Martinez T J, BenNun M and Levine R D 1996 Multi-electronic-state molecular dynamics—a wavefunction approach with applications J. Phys. Chem. 100 7884... 

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. 

Nevertheless, the one-electron approach does have its deHciencies, and we believe that a major theoretical effort must now be devoted to improving on it. This is not only in order to obtain better quantitative results but, perhaps more importantly, to develop a framework which can encompass all types of charge-transfer processes, including Auger and quasi-resonant ones. To do so is likely to require the use of many-electron multi-configurational wavefunctions. There have been some attempts along these lines and we have indicated, in detail, how such a theory might be developed. The few many-electron calculations which have been made do differ qualitatively from the one-electron results for the same systems and, clearly, further calculations on other systems are required. 

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 ... 

The above studies all involved only one and two electron systems. And with the exception of the Be high spin excited states (61) none required the use of "Fermi statistics" (wavefunction antisymmetry) in the Monte Carlo simulations. This is of course a prerequisite for multi-electron systems. We have recently carried out REP-QMC simulations on some three-electron systems. Aluminum is probably the simplest. In Table III we show energies for two states of Al and also for Al. ... 

The formalism described here to derive energy-consistent pseudopotentials can be used for one-, two- and also four-component pseudopotentials at any desired level of relativity (nonrelativistic Schrbdinger, or relativistic Wood-Boring, Douglas-Kroll-Hess, Dirac-Coulomb or Dirac-Coulomb-Breit Hamiltonian implicit or explicit treatment of relativity in the valence shell) and electron correlation (single- or multi-configurational wavefunctions. The freedom... 

MCSCF multi-configurational SCF. Electronic wavefunction is a linear combina-... 

Further developments [3] lead naturally to improved solutions of the Schrodinger equation, at least at the Hartree-Fock limit (which approximates the multi-electron problem as a one-electron problem where each electron experiences an average potential due to the presence of the other electrons.) The authors apply a continuous wavelet mother. v (x), to both sides of the Hartree-Fock equation, integrate and iteratively solve for the transform rather than for the wavefunction itself. In an application to the hydrogen atom, they demonstrate that this novel approach can lead to the correct solution within one iteration. For example, when one separates out the radial (one-dimensional) component of the wavefunction, the Hartree-Fock approximation as applied to the hydrogen atom s doubly occupied orbitals is, in spherical coordinates. 

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. 

The main focus in this chapter is on two themes On the one hand, the discussion aims at presenting, without many formulae, certain critical characteristics and properties of unstable states in general, and, in particular, of states that are normally represented by multi-particle wavefunctions, such as the ones that are found in the spectra of atoms and small molecules. On the other hand, the discussion aims at demonstrating why and how it is possible to tackle, from first principles and from a many-electron point of view, a large number of problems that have to do with field-free and field-induced unstable states, for which reliable electronic structure calculations are, in principle, feasible. 

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