Single-configuration approximation

Actually, two bands of quite different intensity separated by nearly 2 eV are observed Whereas the positions of the two bands could possibly be accommodated by appropriate parametrization, this is not possible for the band intensities which reveals the limitations inherent in the single-configuration approximation. 

Indeed it is easy to see that, in general, the symmetry of the model will not be recovered by the variational solution since, if any one of the R departs from the symmetry of H, then the coupling operators Vrs will destroy the symmetry of the other departures from symmetry will quickly propogate throughout the model solution of the form (9) will have rather complicated behaviour in the variational process, for example each single-configuration approximation should show characteristic saddle point behaviour when variations 5 R are admitted. The minimum in the variational expression when the S R are constrained to have the correct symmetry should also be a local maximum with respect to symmetry-breaking variations S R. 

A very special complication can arise from the breakdown of the single configuration approximation. This is most likely to be a problem in systems where the d-orbital energy ordering alters drastically during a reaction an example may be the butadiene coupling reaction discussed below. 

In single-configurational approximation yj(0) f 0 only for s electrons. That is why in (22.44) the symbol xp has subscript s. The whole dependence of the shift under investigation on the pecularities of the electronic shells of an atom is contained only in the multiplier y s(0). Unfortunately, formula (22.44) does not account for the deviations of the shape of the nucleus from spherical symmetry. Therefore it is unfit for non-spherical nuclei. The accuracy of the determination of all these quantities may be improved by accounting for both the correlation and relativistic effects [157, 158]. A universal program to compute isotope shifts in atomic spectra is described in [159]. 

However, due to the admixture of weak interactions it may occur that the parity is no longer a completely exact quantum number. The same is true for J if we account for hyperfine interactions. Fortunately, due to the weakness of the above-mentioned interactions, the parity and total momentum are the most accurate quantum numbers. In many cases a single-configuration approximation describes fairly accurately atomic characteristics, then the configuration may also be treated as an exact quantum number. However, quite often one has to account for the admixtures (superposition) of other configurations. 

The total moment of a complex of configurations may be presented as a sum of two terms nl(K) representing the moment of the total spectrum in single-configuration approximation, and the second, An (K), describing the correction due to superposition of configurations, i.e. 

An interesting recent study of the perturbations of the 3II and. 4 11 states of CO due to spin-orbit and rotational-orbit interactions has appeared.320 By including Cl functions built from HF orbitals optimized for each state, good agreement was obtained between theory and experiment in most cases, but the single-configuration approximation is seriously in error. 

Unfortunately, even with an incomplete one-electron basis, a full Cl is computationally intractable for any but the smallest systems, due to the vast number of. V-electron basis functions required (the size of the Cl space is discussed in section 2.4.1). The Cl space must be reduced, hopefully in such a way that the approximate Cl wavefunction and energy are as close as possible to the exact values. By far the most common approximation is to begin with the Hartree-Fock procedure, which determines the best single-configuration approximation to the wavefunction that can be formed from a given basis set of one-electron orbitals (usually atom centered and hence called atomic orbitals, or AOs). This yields a set of molecular orbitals (MOs) which are linear combinations of the AOs ... 

The molecular orbital theory of complexes (6), developed out of crystal field theory (7) by the inclusion of both a- and sr-covalent interactions, has been successfully applied 8—13) to the octahedral (14. 15) carbonyls of the 4 metals (V i, Cr , Mn+ etc.). The qualitative discussion of this octahedral case is particularly simple because of a clean-cut separation (within the single-configuration approximation) of a- and ji-systems. 

In cases (1) and (2), the HE single-configuration approximation (SCA) is, in general, inadequate as a zero-order model, either for purposes of efficient high-level computation or for purposes of semiquanfitative understanding of the main features of a process or a properfy. 

The main N-electron matrix element of the one-electron dipole operator in the single configuration approximation would be (lag lag )(lo- 2cr ) (2ag 3ap(2ag D l r ). This is not zero if the orbitals are obtained self-consistently for each configuration separately. 

There exist several SCF codes for the solution of radial equations the Hartree-Fock [16] equations are only one example, and the case described above is that of the single configuration approximation, in which each electron has well-defined values of n and l. There exist several other possibilities as stressed above, in Hartree s original method, the exchange term was left out in the Hartree-Slater method [17], an approximate expression is used for the form of the exchange term. The Cowan code [20] is a pseudorelativistic SCF method, which avoids the complete four-component wavefunctions by simulating relativistic effects. 

Another desirable aspect of using the TDA and RPA approaches is that they both use a common set of molecular orbitals, which aids both in developing qualitative interpretations of the excitation process and also in calculating properties such as transition moments. The latter depends on (i j r i i )p, where r = is the dipole operator. It is easy to evaluate such a one-electron property provided i / and are described in terms of the same orthonormal orbital set. When different orbitals are used in and l —typically to get the best possible solution for both states—the resultant nonorthogonality causes a number of complications. This is particularly true when an entire spectrum of electronic states is the objective and all transition moments are required. Nevertheless, all the methods discussed so far neglect electron correlation effects, and one must go beyond the single configuration approximation if quantitative accuracy is to be achieved. 

The single-configuration approximation is an approximate representation for the diabatic electronic function. Such an approximation is expected to result in... 

Since J+L = J+ 17) the L-uncoupling operator is a one-electron operator, and consequently, in the single-configuration approximation, the configurations describing the two interacting states can differ by no more than one spin-orbital. The electronic part of the perturbation matrix element is then proportional to the same (7r+ l+ spin-electronic perturbation. However, owing to the presence of the J+ operator, the total matrix element of the B(.R)J+L operator between (fi — 1 and ft) is proportional to [J(J + 1) — 0(0 — 1)]1 2-... 

The B2n state is the lowest of three 2n valence states arising from the 5a2 1-7T3 2-7T2 configuration (Section 3.2.3). The configurations of the B2n and D2E+ states differ by two orbitals, which means that the B D heterogeneous interaction is forbidden in the single configuration approximation. However, the electrostatic B2n C2n interaction is configurationally allowed (Section 3.3.2), and it is the admixture of C2n character into both B2n and D2E+ states that is responsible for the nominal B D perturbation. This is an indirect effect of the form... 

Until this point, the single-configuration approximation for each electronic state has been assumed/ The C2 b3E X E+ interaction is one example where it is necessary to represent one of the interacting states by a mixture of two configurations. Another example involving predissociation is discussed in Section 7.11.1. 

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