Many-electron molecular orbital

The development of localized-orbital aspects of molecular orbital theory can be regarded as a successful attempt to deal with the two kinds of comparisons from a unified theoretical standpoint. It is based on a characteristic flexibility of the molecular orbital wavefunction as regards the choice of the molecular orbitals themselves the same many-electron Slater determinant can be expressed in terms of various sets of molecular orbitals. In the classical spectroscopic approach one particular set, the canonical set, is used. On the other hand, for the same wavefunction an alternative set can be found which is especially suited for comparing corresponding states of structurally related molecules. This is the set of localized molecular orbitals. Thus, it is possible to cast one many-electron molecular-orbital wavefunction into several forms, which are adapted for use in different comparisons fora comparison of the ground state of a molecule with its excited states the canonical representation is most effective for a comparison of a particular state of a molecule with corresponding states in related molecules, the localized representation is most effective. In this way the molecular orbital theory provides a unified approach to both types of problems. 

Spin-Orbit Matrix and Electric Field Gradient Tensor Derived from Many-Electron Molecular Orbital Wave Functions... 

Note that while in Fig. 1.1 the so called many-electronic molecular orbitals are about, in Eq. 1.50 the uni- (or i-) electronic molecular orbitals are assumed, each of them moving with its Hamiltonian that is effective since it contains also the influence of the all other electrons in the system upon it. Even more, these molecular mono-orbitals are further decomposed in mathematical atomic orbitals, see 4> s in Eq. 1.53 that are essentially not so different from the individual (p s wave-functions of the N-electrons considered in Eig. 1.1. The actual atomic orbitals are mathematical rather physical objects (for which reason they are often called as basis set) and reflect the atomic participation in the molecular or bonding system rather that the total or valence number of electrons in the system. As such, viewed as the linear combination over the atomic orbitals, the resulted MO-LCAO wave-function... 

The problem of evaluating matrix elements of the interelectron repulsion part of the potential between many-electron molecular Sturmian basis functions has the degree of difficulty which is familiar in quantum chemistry. It is not more difficult than usual, but neither is it less difficult. Both in the present method and in the usual SCF-CI approach, the calculations refer to exponential-type orbitals, but for the purpose of calculating many-center Coulomb and exchange integrals, it is convenient to expand the ETO s in terms of a Cartesian Gaussian basis set. Work to implement this procedure is in progress in our laboratory. 

One approach to the treatment of electron correlation is referred to as density functional theory. Density functional models have at their heart the electron density, p(r), as opposed to the many-electron wavefimction, F(ri, r2,...). There are both distinct similarities and distinct differences between traditional wavefunction-based approaches (see following two sections) and electron-density-based methodologies. First, the essential building blocks of a many-electron wavefunction are single-electron (molecular) orbitals, which are directly analogous to the orbitals used in density functional methodologies. Second, both the electron density and the many-electron wavefunction are constructed from an SCF approach which requires nearly identical matrix elements. 

Tho author has repeatedly [6j pointed out that the valency conception of non-localized bonds (resonance among several valence-bond structures, many centre molecular orbitals) lias no reality and only arises from a neglect of electronic repulsions. It is noteworthy that on this basis the possibility of a contribution of structures IIIc and d (IVc %od d) to Ufa and b (IVa and b) and vice versa would not enter and, thus, this objection against the discussed interpretation of the hydrogen bond would be removed ... 

These investigations have reaffirmed earlier conclusions [6] that the observations related to the electronic spectra of organic compounds cannot be accounted for by the widely accepted theoretical interpretations based on molecular structures involving non-localized bonds (many-centre molecular orbitals) and of the simplifying assumption that only the n electrons are important or need to be considered for the electronic transitions. They can be understood with the help of classical structures involving only localized bonds and inductive electron displacements. 

Many of the principles and techniques for calculations on atoms, described in section 6.2 of this chapter, can be applied to molecules. In atoms the electronic wave function was written as a determinant of one-electron atomic orbitals which contain the electrons these atomic orbitals could be represented by a range of different analytical expressions. We showed how the Hartree-Fock self-consistent-field methods could be applied to calculate the single determinantal best energy, and how configuration interaction calculations of the mixing of different determinantal wave functions could be performed to calculate the correlation energy. We will now see that these technques can be applied to the calculation of molecular wave functions, the atomic orbitals of section 6.2 being replaced by one-electron molecular orbitals, constructed as linear combinations of atomic orbitals (l.c.a.o. method). 

In computations with the RIP, it is found that many relativistic molecular orbitals (RMOs) are almost identical to the DFAOs of the constituent atoms in that the coefficient of one DFAO is very close to unity while the remaining coefficients are small. These orbitals and the electrons they contain are termed the core, and the electrons not included in the core are accommodated in the... 

Since the number of integrals to be evaluated increases rapidly with molecular size, semi-empirical calculations are more practical at present for larger molecules. The most widely used semi-empirical approach is based upon Pople s theory (22, 23) of molecular diamagnetism within the independent electron framework. All explicit two electron terms become zero in this method. Thus the necessity of evaluating many integrals is avoided and the remainder can be approximated by the methods implicit in all valence electron molecular orbital calculations. (24)... 

Buckingham suggested that one-electron molecular orbitals do not give an adequate description for calculation of spin densities and that many-electron wave functions are needed. He also did not see why the hyperfine tensor should be symmetric. McGarvey replied that this is only an approximation and that in practice the antisymmetric terms are usually not detectable since they enter in higher order. 

Many of the colors of vegetation are due to electronic transitions in conjugated 7c-electron systems. In thefree-electron molecular orbital (FEMO) theory, the electrons in a conjugated molecule are treated as independent particles in a box of length L. Sketch the form of the two occupied orbitals in butadiene predicted by this model and predict the minimum excitation energy of the molecule. The tetraene CH2=CHCH=CHCH=CHCH=CH2 can be treated as a box of length 8R, where R = 140 pm (as in this case,... 

Although orbital wave functions, such as Hartree-Fock, generalized valence bond, or valence-orbital complete active space self-consistent field wave functions, provide a semi-quantitative description of the electronic structure of molecules, accurate predictions of molecular properties cannot be made without explicit inclusion of the effects of dynamical electron correlation. The accuracy of correlated molecular wave functions is determined by two inter-related expansions the many-electron expansion in terms of antisymmetrized products of molecular orbitals that defines the form of the wave function, and the basis set used to expand the one-electron molecular orbitals. The error associated with the first expansion is the electronic structure method error the error associated with the second expansion is the basis set error. Only by eliminating the basis set error, i.e., by approaching the complete basis set (CBS) limit, can the intrinsic accuracy of the electronic structure method be determined. 

The natural orbital concept, as originally formulated by Per-Olov Lowdin, refers to a mathematical algorithm by which bestpossible orbitals (optimal in a certain maximum-density sense) are determined from the system wavefunction itself, with no auxiliary as sumptions or input. Such orbitals inherently provide the most compact and efficient numerical description of the many-electron molecular wavefunction, but they harbor a type of residual multicenter indeterminacy (akin to that of Hartree-Fock molecular orbitals) that somewhat detracts from their chemical usefulness. 

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