Bond dissociation wave function differences

It is a well-known fact that the Hartree-Fock model does not describe bond dissociation correctly. For example, the H2 molecule will dissociate to an H+ and an atom rather than two H atoms as the bond length is increased. Other methods will dissociate to the correct products however, the difference in energy between the molecule and its dissociated parts will not be correct. There are several different reasons for these problems size-consistency, size-extensivity, wave function construction, and basis set superposition error. 

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. 

We will now look at how different types of wave functions behave when the O-H bond is stretched. The basis set used in all cases is the aug-cc-pVTZ, and the reference curve is taken as the [8, 8J-CASSCF result, which is slightly larger than a full-valence Cl. As mentioned in Section 4.6, this allows a correct dissociation, and since all the valence electrons are correlated, it will generate a curve close to the full Cl limit. The bond dissociation energy calculated at this level is 122.1 kcaPmol, which is comparable to the experimental value of 125.9 kcal/mol. 

The approach used first, historically, and the one this book is about, is called the valence bond (VB) method today. Heitler and London[8], in their treatment of the H2 molecule, used a trial wave function that was appropriate for two H atoms at long distances and proceeded to use it for all distances. The ideal here is called the separated atom limit . The results were qualitatively correct, but did not give a particularly accurate value for the dissociation energy of the H—H bond. After the initial work, others made adjustments and corrections that improved the accuracy. This is discussed folly in Chapter 2. A cmcial characteristic of the VB method is that the orbitals of different atoms must be considered as nonorthogonal. 

Relativistic Methods 204 8.1 Connection Between the Dirac and Schrodinger Equations 207 8.2 Many-particle Systems 210 8.3 Four-component Calculations 213 11.4.1 Ab Initio Methods 272 11.4.2 DFT Methods 273 11.5 Bond Dissociation Curve 274 11.5.1 Basis Set Effect at the HF Level 274 11.5.2 Performance of Different Types of Wave Function 276... 

For simple covalent bond breaking reactions, a bound state in the R-BO scheme correlates to a diradical asymptotic state. This latter state represents in the laboratory world a collision pair. In a-space we can define an intermolecular distance. For all values of such distance, the system cannot change its electronic state in an adiabatic process. The asymptotic state must be orthogonal to the bound state. It is therefore necessary that the electronic wave function of the collision pair show one node more than the bounded system. The energy expectation values as a function of the intermolecular distance for the two states would cross above the dissociation energy limit. The corresponding FC factor can hence be different from zero. Experimentally, it is well known that most of the bond-forming processes may have a small barrier (about 1 Kcal/mol) [22]. 

What have we learned from the above exercise First of aU that in order to obtain a correct potential curve for the dissociation of a covalent chemical bond with a single determinant wave function, we have to sacrifice both the spin and space symmetry. Secondly, we have seen that the closed shell HF wave function is stable around the equilibrium geometry. Even if we allow different orbitals for different spin, the wave function will converge to a solution where the electrons arrange themselves in closed shells. This is quite general and occurs for most molecules around their equilibrium geometry. 

Valence bond (VB) theory. In VB theory one starts with the occupied atomic orbitals of the atoms and constructs a many-electron wave function to describe bonding directly in terms of these atomic orbitals. Although similar to MO, the differences will become transparent (below). VB theory is most useful for describing reactions and bond dissociations because the many-electron states of the atoms are built into VB. However, VB is computationally much more complicated than MO, and it is much less obvious how to describe excited states in terms of VB. Important chemical concepts such as resonance are based on VB concepts. 

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