Self-consistent field wave functions molecules

A key development in quantum chemistry has been the computation of accurate self-consistent-field wave functions for many diatomic and polyatomic molecules. The principles of molecular SCF calculations are essentially the same as for atomic SCF calculations (Section 11.1). We shall restrict ourselves to closed-shell configurations. For open shells, the formulas are more complicated. 

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. 

T. A. Halgren and W. N. Lipscomb, Self-consistent-field wave-functions for complex molecules. The approximation of partial retention of diatomic differential overlap, J. Chem. Phys. 58 1569 (1973). 

Cl calculations can be used to improve the quality of the wave-function and state energies. Self-consistent field (SCF) level calculations are based on the one-electron model, wherein each electron moves in the average field created by the other n-1 electrons in the molecule. Actually, electrons interact instantaneously and therefore have a natural tendency to avoid each other beyond the requirements of the Exclusion Principle. This correlation results in a lower average interelectronic repulsion and thus a lower state energy. The difference between electronic energies calculated at the SCF level versus the exact nonrelativistic energies is the correlation energy. 

How does a rigorously calculated electrostatic potential depend upon the computational level at which was obtained p(r) Most ab initio calculations of V(r) for reasonably sized molecules are based on self-consistent field (SCF) or near Hartree-Fock wavefunctions and therefore do not reflect electron correlation in the computation of p(r). It is true that the availability of supercomputers and high-powered work stations has made post-Hartree-Fock calculations of V(r) (which include electron correlation) a realistic possibility even for molecules with 5 to 10 first-row atoms however, there is reason to believe that such computational levels are usually not necessary and not warranted. The Mpller-Plesset theorem states that properties computed from Hartree-Fock wave functions using one-electron operators, as is T(r), are correct through first order (Mpller and Plesset 1934) any errors are no more than second-order effects. 

The development of ab-initio self-consistent field (SCF) wave functions for clusters by Bagus et al. provides a powerful basis for interpreting the experimental observations related to the localized bonding of adsorbed molecules (43-47). In these... 

Now we have written down a wave function appropriate for use in the case where H = h(i). In HF theory, we make some simplifications so many-electron atoms and molecules can be treated this way. By tacitly assuming that each electron moves in a percieved electric field generated by the stationary nuclei and the average spatial distribution of all the other electrons, it essentially becomes an independant-electron problem. The HF Self Consistent Field procedure (SCF) will be bent on constructing each x(x) to give the lowest energy. 

Analytic, exact solutions cannot be obtained except for the simplest systems, i.e. hydrogen-like atoms with just one electron and one nucleus. Good approximate solutions can be found by means of the self-consistent field (SCF) method, the details of which need not concern us. If all the electrons have been explicitly considered in the Hamiltonian, the wave functions V, will be many-electron functions V, will contain the coordinates of all the electrons, and a complete electron density map can be obtained by plotting Vf. The associated energies E, are the energy states of the molecule (see Section 2.6) the lowest will be the ground state , and the calculated energy differences En — El should match the spectroscopic transitions in the electronic spectrum. 

Now we are ready to start the derivation of the intermediate scheme bridging quantum and classical descriptions of molecular PES. The basic idea underlying the whole derivation is that the experimental fact that the numerous MM models of molecular PES and the VSEPR model of stereochemistry are that successful, as reported in the literature, must have a theoretical explanation [21], The only way to obtain such an explanation is to perform a derivation departing from a certain form of the trial wave function of electrons in a molecule. QM methods employing the trial wave function of the self consistent field (or equivalently Hartree-Fock-Roothaan) approximation can hardly be used to base such a derivation upon, as these methods result in an inherently delocalized and therefore nontransferable description of the molecular electronic structure in terms of canonical MOs. Subsequent a posteriori localization... 

Calculations by the self-consistent field LCAO-MO method for the ground state wave function of the pyrazine molecule indicate that the lone pairs are quite different. The lower lone pair is little delocalized (1.88 electrons on nitrogen), but the second lone pair is as delocalized as the lone pair in pyridine with 1.37 electrons on nitrogen, 0.22 electrons on hydrogen, and 0.40 electrons on carbon.63... 

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 calculating a theoretical photoelectron spectmm, the atomic ionization cross section a. is usually taken so far from the theoretical values calculated for a neutral free atom in the ground state. However, the MO calculation by DV-Xa method is carried out self consistently and provides Q. by Mulliken population analysis using the SCF MO wave function calculated. In the present calculations, the atomic orbital Xj used for the basis function flexibly expands or contracts according to reorganization of the charge density on the atom in molecule in the self-consistent field. Furthermore, excited state atomic orbitals are sometimes added to extend the basis set. In such a case, the estimation of peak intensity of the photoionization using the data of Oj previously published is not adequate. Thus a calculation of the photoionization cross section is required for the atomic orbital used in the SCF calculation in order... 

Each of these methods is based on the AFDF approach. Within the framework of the conventional Hartree-Fock-Roothaan-Hall self-consistent field linear combination of atomic orbitals (LCAO) ab initio representation of molecular wave functions built from molecular orbitals (MOs), the AFDF principle can be formulated using fragment density matrices. For a complete molecule M of some nuclear configuration K, using an atomic orbital (AO) basis of a set of n AOs density matrix P can be determined using the coefficients of AOs in the occupied MOs. The electronic density p(r) of the molecule M, a function of the three-dimensional position variable r, can be written as... 

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