Covalent wavefunction theory

Interatomic Force Constants (IFCs) are the proportionality coefficients between the displacements of atoms from their equilibrium positions and the forces they induce on other atoms (or themselves). Their knowledge allows to build vibrational eigenfrequencies and eigenvectors of solids. This paper describes IFCs for different solids (SiC>2-quartz, SiC>2-stishovite, BaTiC>3, Si) obtained within the Local-Density Approximation to Density-Functional Theory. An efficient variation-perturbation approach has been used to extract the linear response of wavefunctions and density to atomic displacements. In mixed ionic-covalent solids, like SiC>2 or BaTiC>3, the careful treatment of the long-range IFCs is mandatory for a correct description of the eigenfrequencies. 

It can be seen that the first two terms are the same as the valence bond wavefunction, and there are an additional two terms. The first two are commonly called the covalent terms because each has the electrons associated with both centers. The final two are known as ionic terms because each places two electrons at one center (i.e. H+ H and H H ). Valence bond theory generally neglects ionic terms of this type, whereas in MO theory the covalent and ionic terms are treated equally. 

The HF wavefunction takes the form of a single Slater determinant, constructed of spin-orbitals, the spatial parts of which are molecular orbitals (MOs). Each MO is a linear combination of atomic orbitals (LCAOs), contributed by all atoms in the molecule. The wavefunction in classical VB theory is a linear combination of covalent and ionic configurations (or structures), each of which can be represented as an antis5nnmetrised product of a string of atomic orbitals (AOs) and a spin eigenfunction. The covalent structures recreate the different ways in which the electrons in the AOs on the atoms in the molecule can be engaged in bonding or lone pairs. An ionic structure contains one or more doubly-occupied AOs. Each of the structures within the classical VB wavefunction can be expanded in terms of several Slater determinants constructed from atomic spin orbitals. 

D11.1 Our comparison of the two theories will focus on the manner of construction of the trial wavefunctions for the hydrogen molecule in the simplest versions of both theories. In the valence bond method, the trial function is a linear combination of two simple product wavefunctions, in which one electron resides totally in an atomic orbital on atom A. and the other totally in an orbital on atom B. See eqns I l.l and 11.2, as well as Fig. 11.2. There is no contribution to the wavefunction from products in which both electrons reside on either atom A or B. So the valence bond approach undervalues, by totally neglecting, any ionic contribution to the trial function. It is a totally covalent function. The molecular oibital function for the hydrogen molecule is a product of two functions of the form of eqn 11.8, one for each electron, that is. 

From the quantitative point of view the calculation of the energy (and other properties) associated to VB wavefunctions was hampered by the non-orthogonality of the atomic orbitals centered on the different atoms of the molecule. In addition, and quite disappointing, was the fact that, in most cases, to achieve the same accuracy as calculations based on the Molecular Orbital (MO) model, several ionic structures had to be considered in the VB wavefunction. For example, even considering 20 covalent and 39 ionic structures in the VB wave-function for the water molecule, the energy of the VB wavefunction is still not comparable to the results based on the HF model using the same basis set. Thus, besides the poor performance when compared to the MO calculations, the consideration of the ionic structures caused the VB theory to loose its most important characteristics chemical interpretability and compactness of the wavefunction. 

Throughout our discussion, we refer to ionic lattices, suggesting the presence of discrete ions. Although a spherical ion model is used to describe the structures, we shall see in Section 6.13 that this picture is unsatisfactory for some compounds in which covalent contributions to the bonding are significant. Useful as the hard sphere model is in describing common crystal structure types, it must be understood that it is at odds with modem quantum theory. As we saw in Chapter 1, the wavefunction of an electron does not suddenly drop to zero with increasing distance from the nucleus, and in a close-packed or any other crystal, there is a finite electron density everywhere. Thus all treatments of the solid state based upon the hard sphere model are approximations. 

An ab initio formulation of VB theory is possible but it is more cumbersome than its MO or DFT equivalents. Instead, the strategy adopted by Warshel has been to define the principal resonance structures for the reaction process of interest and, for each, parametrize an appropriate empirical function that describes the energy of the structure as a function of the relevant geometrical parameters. The ground state potential energy surface is then obtained by solving the secular equation for the resonance structures. Let us take H-F again as our example [21]. In this case, there are two resonance structures, the covalent and the ionic, with wavefimctions V i and respectively. The total wavefunction for the system, is a linear combination of these wavefimctions ... 

The problems are not limited to H2. A classic foible is that of F2 in which a molecule with a weak covalent bond is calculated to be unbound by Hartree-Fock theory. While the difference in the calculated energy of the covalent/ionic combination and that of the molecule at equilibrium has occasionally been taken to be the bond energy, the energy difference between that of the neutral separated products and of the molecule is more formally correct. The calculated energy of the F2 molecule at equilibrium is less stable than that of the separated atoms. Calculated results are usually not that wrong. However, problems with dissociation are not limited to homonuclear bonds or even to covalent bonds. For example, in the gas phase, hence in the absence of solvent, the ionic NaCl homolytically dissociates into Na-l-CI and not Na+ - -Cr the electron affinity of Cl is less than the ionization energy of Na. Furthermore, the problem is not merely at infinite separation - even at the equilibrium distance there is an incorrect contribution of the ionic component to the molecular wavefunction. Any dissociation for which the number of unpaired electrons in the molecule and its components differ is problematic. We note the problem... 

As noted above (equations 9 and 10), each pair of valence NHOs /ia, /ib leads to a complementary pair of valence bond (/)ab) and antibond ( ab) orbitals. Although the latter orbitals play no role in the elementary Lewis picture, their importance was emphasized by Lennard-Jones and Mulliken in the treatment of homonuclear diatomic molecules. Since valence antibonds represent the residual atomic valence-shell capacity that is not saturated by covalent bond formation, they are generally found to play the leading role in noncovalent interactions and delocalization effects beyond the Lewis structure picture. Indeed, it may be said that the NBO treatment of bond-antibond interactions constitutes its most unique and characteristic contribution toward extending the Lewis structure concepts of valence theory. Although the NBO hybrids and polarization coefficients are chosen to minimize the role of antibonds, the final non-zero weighting of non-Lewis orbitals reflects their essential contribution to wavefunction delocalization. 

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