Molecular electronic energies, analytical results

We consider in this section some approximate analytical solutions to the electronic Schrodinger equation, in order to provide some basic insight into the energetics of the making and breaking of chemical bonds. Since most of the results are well known from quantum mechanics/chemistry, we will only present the key points and sometimes omit detailed proofs. 

In order to obtain the potential energy surfaces associated with chemical reactions we, typically, need the lowest eigenvalue of the electronic Hamiltonian. Unlike systems such as a harmonic oscillator and the hydrogen atom, most problems in quantum mechanics cannot be solved exactly. There are, however, approximate methods that can be used to obtain solutions to almost any degree of accuracy. One such method is the variational method. This method is based on the variational principle, which says 

3Several commercial quantum chemical programs like Gaussian (by Gaussian, Inc., www.gaussian.com) and Spartan (by Wavefunction, Inc., www.wavefun.com) are available for the solution of the electronic Schrodinger equation. 

The exact ground-state wave function tpo and the associated energy Eq satisfy the Schrodinger equation 

if we use a trial function (cf ) that depends on some parameters, we can vary these parameters in order to minimize and we will always obtain an energy that is larger than or equal to (if we happen to obtain the exact ground-state wave function) the exact ground-state energy. 

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This effect can be illustrated by Fig. 14.2. The effective range of local modification of the sample states is determined by the effective lateral dimension 4ff of the tip wavefunction, which also determines the lateral resolution. In analogy with the analytic result for the hydrogen molecular ion problem, the local modification makes the amplitude of the sample wavefunction increase by a factor exp( — Vi) 1.213, which is equivalent to inducing a localized state of radius r 4tf/2 superimposed on the unperturbed state of the solid surface. The local density of that state is about (4/e — 1) 0.47 times the local electron density of the original stale in the middle of the gap. This superimposed local state cannot be formed by Bloch states with the same energy eigenvalue. Because of dispersion (that is, the finite value of dEldk and... 

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