Unperturbed energy eigenvalue

Note that both the numerator and denominator in the final expression are always positive expressions in the case of the denominator, we know this because is the lowest energy eigenvalue of the unperturbed system. (The denominator reduces to a difference in orbital energies.)... 

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... 

The solution, namely the complex eigenvalue of the field-induced resonance state, is a function of the frequency and strength of the field. For normal cases where there are no serious field-induced near-degenaracies, it is connected smoothly to the unperturbed energy Eq ... 

The first-order correction to the wave function and the second-order correction to the energy eigenvalue involve sums over all of the unperturbed wave functions and energy eigenvalues. We do not derive these formulas, but present them here. The formula for the coefficient a j in Eq. (G-7) is... 

Equations (5.3), (5.7), and (5.8) form the starting point for a systematic perturbation theory analysis, whose deeper details need not concern us here (see V B, p. 16ff). In this approach, the NLS model //op is regarded as the unperturbed Hamiltonian, with known eigenfunction and energy eigenvalue eP" that are assumed to be well understood. The resonance-type corrections to energy (EP ), density (pnl), or other properties can then be expressed (analyzed or evaluated) in orderly fashion from the known properties of the model Lewis system. The NBO... 

The unperturbed wavefunction in Equation 8-24 can now be applied to the unperturbed Hamiltonian in Equation 8-20. This yields the unperturbed energy that consists of nothing more than a sum of two hydrogen-like eigenvalues as found previously in Equation 8-13. 

By way of illustration it will be assumed that Ek is non-degenerate and AH is small enough to ensure that the perturbed energy level k is closer to Ek than to any other unperturbed level. The new eigenvalue problem is... 

The form of Hmuit having been obtained, it now remains to discuss how to solve the subsequent eigenvalue equation. For radiation-matter coupling that is small relative to intramolecular Coulomb potential energies, the interaction Hamiltonian may be considered as a perturbation on the particle-field sysfem. A perfurbafion fheory solufion is then the most obvious choice. The first two terms of Eq. (6) are faken fo constitute the unperturbed Hamiltonian Ho, so that... 

In this formulation the zeroth-order level energy is just the Dirac eigenvalue Enj summed over the electrons in the unperturbed state. In particular, = (N Ho N), where the unperturbed states and the action of Ho are explicitly given by... 

This is true regardless of the perturbation mechanism (J-dependent or J-independent matrix elements) because the trace of a matrix is representation invariant the sum of the basis function (i.e., deperturbed) energies is equal to the sum of the eigenvalues. Since the approximately unperturbed Bmain and Emain(O) values are usually known from a relatively perturbation-free portion of the band, the constants for the perturbing state can be inferred from Eqs. (4.3.10) and (4.3.11). If the average energy plot shows any deviation from linearity, this implies either an incorrect line identification or an additional perturber. 

For a p-fold degenerate energy level, the eigenvalue problem of the perturbation operator V is first solved in the basis of the eigenfunctions of the unperturbed Hamiltonian IP that span the degenerate space associated with the energy level... 

---