Calculation of molecular electronic wave functions and energies

Calculation of molecular electronic wave functions and energies 

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). 

Before discussing the problems of many-electron wave functions for molecules, it is instructive to consider the special cases of the Hj and H2 molecules, containing one and two electrons respectively. The electronic wave function for H2 can actually be calculated exactly within the Bom-Oppenheimer approximation, not analytically but by using series expansion methods an excellent description of the calculation has been given by Teller and Sahlin [17], and we give a summary of the method in appendix 6.1. We will, however, use the II2 and H2 molecules to illustrate the l.c.a.o. method in the next two subsections. 

The hydrogen molecular ion has long been a test bed for quantum theoretical methods, and continues so to be. We now present first the simplest quantitative treatment, leading to calculations of the electronic energy as a function of the intemuclear distance, R. The Hamiltonian for the 112 molecule may be written in the form 

In order to determine the ground state energy of the molecule at a fixed value of R we use the Variation Theorem, that is, 

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Calculation of molecular electronic wave functions and energies [1 207... 

Using the Born-Oppenheimer approximation, electronic structure calculations are performed at a fixed set of nuclear coordinates, from which the electronic wave functions and energies at that geometry can be obtained. The first and second derivatives of the electronic energies at a series of molecular geometries can be computed and used to find energy minima and to locate TSs on a PES. 

A recent comparison (33.) of quantities calculated from molecular electronic wave functions for pyrrole and pyrazole indicated that ARCANA values compared more favorably with large-scale ab initio values than those calculated by other non-rigorous methods in the case of orbital energies of occupied molecular orbitals, gross atomic populations of heteroatoms, and total overlap populations including negative overlap populations between nonbonded atoms. 

In this expression, ge is the g factor for an isolated spin (2.0023), X is the spin-orbit coupling constant, C is a proportionality constant calculated from the electronic wave functions, and A is the energy difference between the ground state and the first excited state. Values of X have been obtained for a number of atoms and ions from atomic spectra, but the particular value to be used in a molecular system can only be approximated by this. In general, X values increase with atomic number. The values of AE can sometimes be deduced from electronic spectra. Thus the g anisotropy is related to the electronic wave function, and if sufficient information is known about the electronic wave function the principal g components can be calcu-... 

In this chapter, we look at the techniques known as direct, or on-the-fly, molecular dynamics and their application to non-adiabatic processes in photochemistry. In contrast to standard techniques that require a predefined potential energy surface (PES) over which the nuclei move, the PES is provided here by explicit evaluation of the electronic wave function for the states of interest. This makes the method very general and powerful, particularly for the study of polyatomic systems where the calculation of a multidimensional potential function is an impossible task. For a recent review of standard non-adiabatic dynamics methods using analytical PES functions see [1]. 

There is no unequivocal answer to the question as to which is the better method. Calculations by the VB method are likely to be more reliable than those by the MO method, but in practice are much more difficult to carry out. For many-electron molecules the MO procedure is simpler to visualize because we combine atomic orbitals into molecular orbitals and then populate the lower-energy orbitals with electrons. In the VB method, atomic orbitals are occupied, but the electrons of different atoms are paired to form bonds, a process that requires explicit consideration of many-electron wave functions. To put it another way, it is easier to visualize a system of molecular orbitals containing N electrons than it is to visualize a hybrid wave function of N electrons. 

In molecular orbital (MO) theory, which is the most common implementation of QM used by chemists, electrons are distributed around the atomic nuclei until they reach a so-called self-consistent field (SCF), that is, until the attractive and repulsive forces between all the particles (electrons and nuclei) are in a steady state, and the energy is at a minimum. An SCF calculation yields the electronic wave function 4C (the electronic motion being separable from nuclear motion thanks to the Born-Oppenheimer approximation). This is the type of wave function usually referred to in the literature and in the rest of this chapter. 

For molecules of chemical interest it is not possible to calculate an exact many-electron wave function. As a result, we have to make certain approximations. The most commonly made approximation is the molecular orbital approximation, which is outlined in the next section. Within such a framework, it is useful to define various levels of computational method, each of which can be applied to give a unique wave function and energy for any set of nuclear positions and number of electrons. If such a model is clearly specified and if it is sufficiently simple to apply repeatedly, it can be used to generate molecular potential energy surfaces, equilibrium geometries, and other physical properties. Each such theoretical model can then be explored and the results compared in detail with experiment. If there is sufficient consistent success, some confidence can then be acquired in its predictive power. Each such level of theory therefore should be thoroughly tested and characterized before the significance of its prediction is assessed. 

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