Truncated expansions

This Legendre expansion converges rapidly only for weakly anisotropic potentials. Nonetheless, truncated expansions of this sort are used more often than justified because of their computational advantages. 

Figure 1. Rate of convergence of SDTQ-CI truncated expansions for FORS1 and FORS2 active spaces for HNO and NCCN molecules (a) Open-circles correspond to Cl configurations generated from SD-CI natural orbitals ...

An important result, found for the SDTQ[N/N] wavefiinctions of all molecules considered, is that the split-localized molecular orbitals yield a considerably faster convergence for truncated expansions than the natural orbitals. For example, for NCCN SDTQ[18/18], millihartree accuracy is achieved by about 50,000 determinants of the ordering based on split-localized orbitals whereas about 150,000 determinants are needed for the natural-orbital-based ordering. This observation calls for the revision of a widely held bias in favor of natural orbitals. 

Considering now the calculated energies of the truncated expansions as functions of the estimated normalization deficiencies, one finds that, in all cases, the energies approach the flill-SDTQ value for the limit Ac (Ntr)-- 0 from above along a linear or weakly quadratic curve. By means of an extrapolation of this curve one can then determine the degree of truncation necessary for the error not to exceed the desired threshold, say 1 mh, without having to calculate a Cl wavefimctions larger than the truncated one. 

The effectiveness of the method is exhibited by Figure 2 in which the energy errors of truncated expansions are plotted versus the numbers of determinants in these expansions. For each of the four systems shown, one curve displays this relationship for the expansions generated by the just discussed a priori truncations, whereas the other curve is obtained a posteriori by starting with the full SDTQ calculation in the same orbital basis and, then, simply truncating the determinantal expansion based on the ordering established by the exact coefficients of the determinants. There is practically no difference in the number of determinants needed to achieve an accuracy of 1 mh. 

The truncation procedure for fiill-valence-space and N-electrons-in-N-orbitals SDTQ MCSCF waveflmctions is based on choosing split-localized molecular orbitals as configuration generators since they lead to the greatest number of deadwood configurations that can be deleted. A quite accurate estimation method of identifying the latter has been developed so that the truncation can be performed a priori. The method has been shown to be effective in applications to the molecules HNO, OCO and NCCN where, for instance, the energies of the full SDTQ[N/N] calculations are recovered to better than 1 mh by truncated expansions that require only 11.8%, 10.9% and 6.3%, respectively, of the number of determinants in the full calculations. Similar trends are observed for the FORS 1 model. 

Equation (2.18) is a linear variation function. (The summation indices prevent double-counting of excited configurations.) The expansion coefficients cq, c, c%, and so on are varied to minimize the variational integral. o) is a better approximation than l o)- In principle, if the basis were complete. Cl would provide an exact solution. Here we use a truncated expansion retaining only determinants D that differ from I Tq) by at most two spin orbitals this is a singly-doubly excited Cl (SDCI). 

Of course the truncated expansion in equation (1) is meaningful provided... 

Usually this method is used on an H-depleted molecular graph, truncated expansions being obtained considering only fragments up to a user-defined size. Some methods for -> log P estimations are based on cluster expansion. Moreover, a new method for the calculation of embedding frequencies for acyclic trees based on spectral moments of iterated line graph sequence was proposed recently [Estrada, 1999]. 

The key elements of the MWR are the expansion functions (also called the trail-, basis- or approximating functions) and the weight functions (also known as test functions). The trial functions are used as the basis functions for a truncated series expansion of the solution, which, when substituted into the differential equation, produces the residual. The test functions are used to ensure that the differential equation is satisfied as closely as possible by the truncated series expansion. This is achieved by minimizing the residual, i.e., the error in the differential equation produced by using the truncated expansion instead of the exact solution, with respect to a suitable norm. An equivalent requirement is that the residual satisfy a suitable orthogonality condition with respect to each of the test functions. 

This shows that the effect of an orbital transformation, as in Eq. (76), on any expansion ket is exactly represented by the action of the exponential operator exp( - iA) on that ket. The important feature of the operator representaton is that it gives the effect of the orbital transformation expressed in the original orbital basis. Since Eq. (99) is valid for any ket, it also holds for any linear combination of kets and, therefore, for the MCSCF wavefunction. It may be noted that when A = 0, this unitary operator reduces to the identity operator. For small values of the parameters A , it is useful to consider the truncated expansion of the exponential operator... 

When the particle Stokes number is not small, the truncated expansion for the particle velocity mean particle velocity must be calculated from the disperse-phase momentum equation described in Section 4.3.7. Let us for the time being consider a very dilute population of identical particles. The mean velocity of these particles can be found by solving Eq. (4.91). For small particle... 

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