Approximate excitation level

Table 11 Calculated Excitation Energie.s (in eV) for CH in Comparison with SCF, CISD, and FCI Results. The Last Row Gives the Approximate Excitation level (AEL), which is a Measure for the Numbers of Electrons Involved in the Electronic Transition. For Computational Details, see Refs, 35, 41, and 110...

Thus, we can use the approximate quantum number m to label such levels. Moreover, it may be shown [11] that (1) 3/m is one-half of an integer for the case with consideration of the GP effect, while it is an integer or zero for the case without consideration of the GP effect (2) the lowest level must have m = 0 and be a singlet with Ai symmetry in 53 when the GP effect is not taken into consideration, while the first excited level has m = 1 and corresponds to a doublet E conversely, with consideration of the GP effect, the lowest level must have m = j and be a doublet with E symmetry in S, while the first excited level corresponds to m = and is a singlet Ai. Note that such a reversal in the ordering of the levels was discovered previously by Hancock et al. [59]. Note further thatj = 3/m has a meaning similar to thej quantum numbers described after Eq. (59). The full set of quantum numbers would then be... 

So far everything is exact. A complete manifold of excitation operators, however, means that all excited states are considered, i.e. a full Cl approach. Approximate versions of propagator methods may be generated by restricting the excitation level, i.e. tmncating h. A complete specification furthermore requires a selection of the reference, normally taken as either an HF or MCSCF wave function. 

Figure 4 Conduction band levels and excitation levels of infinite periodic hydrogen chains by using different approximations of the polarization propagator. The left part refers to the crystalline orbital energy differences, namely, the Hartree-Fock excitation energies the right part refers to the random phase approximation results obtained by using 41 k-points in half the first Brillouin zone.

Figure 8.1 The harmonic potential and the Morse potential, together with vibrational energy levels. The harmonic potential is an acceptable approximation for molecular separations close to the equilibrium distance and vibrations up to the first excited level, but fails for higher excitations. The Morse potential is more realistic. Note that the separation between the vibrational levels decreases with increasing quantum number, implying, for example, that the second overtone occurs at a frequency slightly less than twice that of the fundamental vibration.

The excitation level matches that for Tp. The simplest possible approximation to Ec is given by MBPT(2) or MP2,... 

Eqs (46) and (49) are the basic equations of the ECC theory described in ref 124. The approximate ECC methods, such as ECCSD, are obtained by truncating the many-body expansions of cluster operators T and Z at some excitation level < N. so that T is replaced by eq (4), and Z is replaced by... 

If one of these errors is vastly larger than the other, it is the only one to consider. The problem changes only if the bath-induced and the non-RWA errors are similar. In this case, we find that dephasing of the doubly excited state caused by 5 in (4.185) is a fourth-order effect and hence can be ignored in the present second-order treatment. The result of this approximation is that the system can be split into two completely separate subsystems O and , the former suffering only from dephasing and the latter only from unwanted population of the doubly excited level 1 162)- Th Hamiltonians of these systems are... 

---