Electronic wave function for molecule

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 goal of this section is to introduce the first of the two main methods for generating approximate electronic wave functions for molecules. The methods we describe work well in qualitative descriptions of bonding and are used for that purpose in all branches of chemistry. They also serve as the starting point for sophisticated computer calculations of molecular electronic structure with readily available molecular modeling software packages. 

Aerts, P. J. C. (1986) Towards relativistic quantum chemistry—on the ab initio calculation of relativistic electron wave functions for molecules in the Hartree-Fock-Dirac approximation. PhD thesis, Rijksuniversiteit te Groningen, Netherlands. 

We decided to investigate corrections to the B. O. approximation for a number of diatomic hydrides and deuterides in order to obtain information about electronic isotope effects on the gas phase isotopic exchange reactions (1). Such calculations require a knowledge of electronic wave functions for molecules and the evaluation of a number of integrals. While such calculations would at one time have been very difficult, the availability of a large digital computer makes such calculations feasible now. The results of some of these calculations will be reported in the following sections they are reported in more detail elsewhere (2, 4),... 

The passage from wave function (3) to (4) gives us a new perspective on the role of the ionic stmctures and suggests an entirely novel direction for constmcting electronic wave functions for molecules, to which we now turn. 

In the preceding chapters we have discussed the energy levels and electronic wave functions for molecules in which the nuclei were assumed to be at rest. For a diatomic molecule the electronic energy levels were given by the solution of the equation... 

To use direct dynamics for the study of non-adiabatic systems it is necessary to be able to efficiently and accurately calculate electronic wave functions for excited states. In recent years, density functional theory (DFT) has been gaining ground over traditional Hartree-Fock based SCF calculations for the treatment of the ground state of large molecules. Recent advances mean that so-called time-dependent DFT methods are now also being applied to excited states. Even so, at present, the best general methods for the treatment of the photochemistry of polyatomic organic molecules are MCSCF methods, of which the CASSCF method is particularly powerful. 

Until now we have implicitly assumed that our problem is formulated in a space-fixed coordinate system. However, electronic wave functions are naturally expressed in the system bound to the molecule otherwise they generally also depend on the rotational coordinate 4>. (This is not the case for E electronic states, for which the wave functions are invariant with respect to (j> ) The eigenfunctions of the electronic Hamiltonian, v / and v , computed in the framework of the BO approximation ( adiabatic electronic wave functions) for two electronic states into which a spatially degenerate state of linear molecule splits upon bending. 

I. 33 A. The electronic wave function for the normal benzene molecule can be composed of terms corresponding to the Kekute structures I and... 

In his valuable paper Molecular Energy Levels and Valence Bonds Slater developed a method of formulating approximate wave functions for molecules and constructing the corresponding secular equations.1 Let a,b, repreamt atomic orbitals, each occupied by one valence electron, and a and 0 represent the electron spin functions for spin orientation -f i and — J, respectively. Slater showed that the following function corresponds to a valence-bond structure with bonds a-----b, c---d, and so forth ... 

All results are based upon master eq. (16). One of the chief deficiencies of many discussions of chemical transitions of excited molecules is made apparent by the formalism. Considerable effort has been devoted to development of electronic wave functions for A and B. Transition probabilities are then discussed in terms of superficial examination of the relationships between the wave functions. In discussions of the subject, considerable bickering may arise because of divergence of opinion as to the goodness of electronic wave functions. While discussion of the quality of approximate wave functions has real significance in structural chemistry, it seems to be a matter of secondary importance in treatment of the dynamic problem at the present time. Almost any kind of electronic wave function is likely to be of better quality than any available perturbation operators (// ). A secondary problem arises from the fact that the vibrational part of ifi1, is likely to be relatively unknown.)- At the present time our best approach to the problem appears to be use of experiments to read back the nature of the perturbation. This leads to an iterative procedure in which the implications of relationships between wave functions are examined experimentally to lead to tentative generalizations that are, in turn, used to predict results of more experiments. The procedure is essentially that used by Zimmerman and his group,7 by Woodward and Hoffman,25 and, in one form or another, by various other authors. 

Almost all approaches to many-electron wave functions, for both atoms and molecules, involve their formulation as products of one-electron orbitals. If the interelectronic repulsion term in (6.23) is small compared with the other terms, the Hamiltonian is approximately separable into independent operators for each electron and the two-electron wave function, f (1, 2), can be written as a simple product of one-electron functions,... 

Our earlier discussion of electronic wave functions for many-electron atoms drew attention to the main inadequacy of the Hartree-Fock single determinant treatment it does not take account of the correlation between the motions of electrons with opposite spins. In molecules this can even lead to qualitative deficiencies in the description of electronic structure, such as the failure to describe dissociation correctly. For example, the correct wave function for the singlet state of the hydrogen molecule at large... 

Both of these expressions are defined in the molecule-fixed (q = 0) coordinate system, and are expectation values over the electronic wave function for the vibronic state r. The contributions of (8.214) are included in the expressions for the first-order energies of the rotational levels given in table 8.6. There are, of course, many non-zero matrix elements in the case (b) basis, all of which are listed by Chiu [40]. 

Any method representing the many-electron wave function for a molecule as an antisymmetrized product of one-electron orbitals... 

The first term (kinetic energy) is a summation over all the particles in the molecule. The second term (potential enctgy) uses Coulomb s law to calculate the interaction between every pair of particles in the molecule, where e, and cj ate the charges on particles i and j. For electrons, the charge -e, while the charge for a nucleus is Ze, where Z Is the atomic number. The summation nutation ipairwise interaction terms in the summation (e.g., eg / = e e, and should only appear in the potential energy term once). The denominator r in the. second term is the distance between particles i anil j. J. i is understood to be the electronic wave function for a many-atom system. 

The LCAO method extends to molecules the description developed for many-electron atoms in Section 5.2. Just as the wave function for a many-electron atom is written as a product of single-particle AOs, here the electronic wave function for a molecule is written as a product of single-particle MOs. This form is called the orbital approximation for molecules. We construct MOs, and we place electrons in them according to the Pauli exclusion principle to assign molecular electron configurations. 

Computer calculations of molecular electronic structure use the orbital approximation in exactly the same way. Approximate MOs are initially generated by starting with trial functions selected by symmetry and chemical intuition. The electronic wave function for the molecule is written in terms of trial functions, and then optimized through self-consistent field (SCF) calculations to produce the best values of the adjustable parameters in the trial functions. With these best values, the trial functions then become the optimized MOs and are ready for use in subsequent applications. Throughout this chapter, we provide glimpses of how the SCF calculations are carried out and how the optimized results are interpreted and applied. 

Both electrons occupy this bonding orbital, satisfying the condition of indistin-guishability and the Pauli principle. Recall from Section 6.2 that the electronic wave function for the entire molecule in the LCAO approximation is the product of all of the occupied MOs, just as an atomic wave function is the product of all occupied Hartree orbitals of an atom. Thus, we get... 

Now we can compare the LCAO and VB versions of the electronic wave functions for the molecule directly by multiplying out iAmo rearranging terms to obtain... 

The theory of the chemical bond is one of the clearest and most informative examples of an explanatory phenomenon that probably occurs in some form or other in many sciences (psychology comes to mind) the semiautonomous, nonfundamental, fundamentally based, approximate theory (S ANFFBAT for short). Chemical bonding is fundamentally a quantum mechanical phenomenon, yet for all but the simplest chemical systems, a purely quantum mechanical treatment of the molecule is infeasible especially prior to recent computational developments, one could not write down the correct Hamiltonian and solve the Schrodinger equation, even with numerical methods. Immediately after the introduction of the quantum theory, systems of approximation began to appear. The Born Oppenheimer approximation assumed that nuclei are fixed in position the LCAO method assumed that the position wave functions for electrons in molecules are linear combinations of electronic wave functions for the component atoms in isolation. Molecular orbital theory assumed a characteristic set of position wave functions for the several electrons in a molecule, systematically related to corresponding atomic wave functions. 

Although electronic wave functions for diatomic molecules containing unlike nuclei cannot be rigorously classified as even or odd, they often approach members of these classes rather closely, and obey an approximate selection rule of the above type.)... 

Group theory is a branch of mathematics that involves elements with defined properties and a single method to combine two elements called multiplication. The symmetry operators belonging to any symmetrical object form a group. The theorems of group theory can provide useful information about electronic wave functions for symmetrical molecules, spectroscopic transitions, and so forth. 

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