Wave functions, hydrogen bonds

Valence bond and molecular orbital theory both incorporate the wave description of an atom s electrons into this picture of H2 but m somewhat different ways Both assume that electron waves behave like more familiar waves such as sound and light waves One important property of waves is called interference m physics Constructive interference occurs when two waves combine so as to reinforce each other (m phase) destructive interference occurs when they oppose each other (out of phase) (Figure 2 2) Recall from Section 1 1 that electron waves m atoms are characterized by their wave function which is the same as an orbital For an electron m the most stable state of a hydrogen atom for example this state is defined by the Is wave function and is often called the Is orbital The valence bond model bases the connection between two atoms on the overlap between half filled orbifals of fhe fwo afoms The molecular orbital model assembles a sef of molecular orbifals by combining fhe afomic orbifals of all of fhe atoms m fhe molecule... 

If a covalent bond is broken, as in the simple case of dissociation of the hydrogen molecule into atoms, then theRHFwave function without the Configuration Interaction option (see Extending the Wave Function Calculation on page 37) is inappropriate. This is because the doubly occupied RHFmolecular orbital includes spurious terms that place both electrons on the same hydrogen atom, even when they are separated by an infinite distance. 

The optimum value of c is determined by the variational principle. If c = 1, the UHF wave function is identical to RHF. This will normally be the case near the equilibrium distance. As the bond is stretched, the UHF wave function allows each of the electrons to localize on a nucleus c goes towards 0. The point where the RHF and UHF descriptions start to differ is often referred to as the RHF/UHF instability point. This is an example of symmetry breaking, as discussed in Section 3.8.3. The UHF wave function correctly dissociates into two hydrogen atoms, however, the symmetry breaking of the MOs has two other, closely connected, consequences introduction of electron correlation and spin contamination. To illustrate these concepts, we need to look at the 4 o UHF determinant, and the six RHF determinants in eqs. (4.15) and (4.16) in more detail. We will again ignore all normalization constants. 

In order to include the spin of the two electrons in the wave function, it is assumed that the spin and spatial parts of the wave function can be separated so that the total wave function is the product of a spin and a spatial wave function F — iAspace sp n Since our Hamiltonian for the H2 molecule does not contain any spin-dependent terms, this is a good approximation (NB—the complete Hamiltonian does contain spin-dependent terms, but for hydrogen they are rather small and do not appreciably affect the energetics of chemical bonding). For a two-electron system it turns out that there are four possible spin wave functions they are ... 

Because electrons have wave-like properties, orbital interactions involve similar addition or subtraction of wave functions. When two orbitals are superimposed, one result is a new orbital that is a composite of the originals, as shown for molecular hydrogen in Figure 10-2. This interaction is called orbital overlap, and it is the foundation of the bonding models described in this chapter. 

For the hydrogen bonded complexes (FH)2, (HC1)2, (H20)2, FH-CO, FH -OC, FH -NH3, CIH- NH3, OH2-NH3, and H30+—H20, with binding energies ranging from 1.7 kcal/mol (FH-CO) to 32.9 kcal/mol (H30+—H20), Tuma, Boese, and Handy, 1999, compared results of several density functional methods with high level conventional wave function based data. With the exception of the HCTH functional, all functionals were found to over-... 

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