One-configuration approximation

Thus, the use of wave function in the form (23.67) and operators in the form (23.62)—(23.66) makes it possible to separate the dependence of multi-configuration matrix elements on the total number of electrons using the Wigner-Eckart theorem, and to regard this form of superposition-of-configuration approximation itself as a one-configuration approximation in the space of total quasispin angular momentum. 

There are many molecules, in particular some radicals, that are naturally described in terms of two or more resonance structures, and for which the one-configuration approximation is not appropriate. Such molecules are generally subject to the well-known broken-symmetry artifact, whereby a wave function calculated at an insufficient level of theory is of lower symmetry than the nuclear framework, which results in erroneous energetics and discontinuities of the calculated potential surface. The formyloxyl radical (42 43 in Scheme 20) is a typical example, but the problem is very general and includes... 

Flexible rotor balancing must be performed with the rotor whirl configuration approximating the mode in question. The operating speed(s) is in the vicinity of a major flexible mode resonance (damped critical speed). As these two speeds approach one another, a tighter... 

Fig. 7.7 Schematic diagrams for common electron configurations of Ni " complexes in the one-electron approximation. The resulting valence electron contributions V z are obtained from Table 4.2...

However, since and -5 asymptote to the same function, one might approximate (U) = S dJ) in (3.57) so that the acceptance probability is a constant.3 The procedure allows trial swaps to be accepted with 100% probability. This general parallel processing scheme, in which the macrostate range is divided into windows and configuration swaps are permitted, is not limited to density-of-states simulations or the WL algorithm in particular. Alternate partition functions can be calculated in this way, such as from previous discussions, and the parallel implementation is also feasible for the multicanonical approach [34] and transition-matrix calculations [35],... 

However, if this is not the case, the perturbations are large and perturbation theory is no longer appropriate. In other words, perturbation methods based on single-determinant wavefunctions cannot be used to recover non-dynamic correlation effects in cases where more than one configuration is needed to obtain a reasonable approximation to the true many-electron wavefunction. This represents a serious impediment to the calculation of well-correlated wavefunctions for excited states which is only possible by means of cumbersome and computationally expensive multi-reference Cl methods. 

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. 

Having chosen to work within a particular one of our one-configuration family of models, there are some important decisions to be taken about specific computations within that model. Most important of these is the question of numerical approximation whether numerical approximations are to be used and, if so, how. 

In quantum mechanics, as we have already seen, one can approximately describe the hydrogen molecular ion as consisting of Ha+ and Hb, or Hb+ and Ha. Some combination of wave functions representing these two configurations is needed as an approximation of the actual state of affairs. The state of H2+ can then be thought of as a resonance hybrid of the two. 

Fig. 2. This figure shows the electronic energy of the ground state of H2 molecule, calculated in a crude approximation using only one configuration. The benchmark calculation of Kolos and Wolniewicz is exhibited for comparison. Accuracy can be seen to be improved by using more atomic orbitals even when a rough approximation is used for the interelectron repulsion matrix element.

For example for a box with four cells containing four balls, there is only one configuration and VF = 1, S = 0. When the box expands further to 64 cells, W = 64 /(60 4 ) = 635,376 and S/k = 13.36. When the box expands further to a very large number m, we use Stirling s approximation to calculate the number of combinations ... 

Transition state theory can be used to test reaction dynamics on a molecular scale. Thus one can hypothesize a spatial configuration of the atoms in the transition state and from this calculate A S° the predicted rate constant can then be compared to that observed. If the agreement is not acceptable, the molecular configuration of the transition state can be adjusted until such agreement is obtained. Assuming this molecular configuration approximates the actual form of the intermediate in the reaction, one can learn something about the chemical dynamics of the reaction. 

Unfortunately, if a single configuration is used to approximate the many-electron wave function, electrons of opposite spin remain uncorrelated. The tacit assumption that electrons of opposite spin move independently of each other is, of course, physically incorrect, because, in order to minimize their mutual Coulombic repulsion energy, electrons of opposite spin do certainly tend to avoid each other. Therefore, a wave function, T, that consists of only one configuration will overestimate the Coulombic repulsion energy between electrons of opposite spin. 

It seems not unlikely that HCoCCO) has a configuration approximating the trigonal bipyramid and H2Fe(CO)4 one approximating the octahedron. 

The four valence-bond structures or configurations, 4a-d, are combined mathematically to give four hybrid states, and of these, the lowest-energy one corresponds approximately to the normal state of the molecule. The calculation shows that the structures 4a and 4b, which have one electron in each p orbital, are the major contributors to the hybrid of ethene. The valence-bond structures, 4c and 4d, are ionic structures, which correspond to the conventional formulas, 4e and 4f ... 

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. 

Here we consider an optical transition between Aj and E electronic states of a center of a trigonal symmetry. To describe the vibrations of the center we use the collinear-configurational approximation [27] in which only the central forces are taken into account in the optical center (taking account of deviations from this approximation, see later). If one restricts oneself to the linear vibronic coupling in the e state, then in this approximation the potential energy operators in the Ai and E electronic states can be presented in the form ... 

Above the collinear-configurational approximation (1) was used. If one wants to take into account the deviations from this approximation then in equation (3) the term V1 = Y.mwnQni

[Pg.139]

Improvement of the crystal-field splitting calculation has been achieved by two different approaches. On one hand the basis set of wavefunctions was extended to include also excited configurations. This approach will be dealt with in sect. 4.4.6. On the other hand, the one-electron approximation has been relaxed to take into account electron correlation effects. The original formulation of the correlation crystal-field parameterization has been proposed by Bishton and Newman (1968). Judd (1977) and Reid (1987) redefined the operators to ensure their mutual orthogonality ... 

One approach to the approximate representation of molecular bodies is based on molecular isodensity contours, MIDCOs, defined with respect to some fixed nuclear configuration K and some electron density threshold a. A MIDCO G(a,K) is defined (in the fixed nuclear configuration approximation) as the collection of all those points r of the three-dimensional space where the electronic density is equal to the threshold a ... 

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