Wave function expansions

For the systematic approach based on so(4, 2) it is more convenient to use the second choice and calculate E(k). Thus, using the wave function expansion... 

In 2003, Aquino, Flores-Riveros and Rivas-Silva [115] approached the problem via a wave function expansion in terms of generalized Hylleraas (GH) basis sets, where the Hamiltonian is expressed in Hylleraas coordinates... 

Normal mode labels correspond to leading component in wave function expansion. D = E - E , where E is result of nth-order CVPT. 

These terms include the ft)-component of M 1 and M 2 thus, applying the perpendicular field Hamiltonian leads to a mixture of different Ms for the wave function of the next FC order. Thus, the dimensions in the perpendicular case are larger than those in the parallel case see Tables 13.1 and 13.2. Note that the M expansion is naturally introduced by the Hamiltonian, and we did not take any consideration of the initial or g functions. This feature comes from the exact structure of the FC wave function and shows an important merit of the FC method the correct wave function expansion is always achieved, even when we start from a simple initial function. 

So far, we focused on conventional quantum chemical approaches that approximate the FCI wave function by truncating the complete N-particle Hilbert space based on predefined configuration selection procedures. In a different approach, the number of independent Cf coefficients can be reduced without pruning the FCI space. This is equivalent to seeking a more efficient parameterization of the wave function expansion, where the Cl coeflBcients are approximated by a smaller set of variational parameters that allow for an optimal representation of the quantum state of interest. Different approaches, which we will call modern solely to distinguish them from the standard quantum chemical methods, have emerged from solid-state physics. 

From Eq. (8.14), we can immediately read off the Cl coefficients of the wave function expansion. If we write the summation over the elements of the eigenvector of Hgi explicitly, we obtain for the Cl coefficient of one Slater determinant n... 

In order to reconstruct a CASCI-type wave function expansion, which contains the most important configurations, a Monte Carlo sampling routine can then be employed to sample the complete N-particle Hilbert space [92]. 

However, although, starting from this point, many sophisticated methods for wave function expansion, for example, the coupled cluster approach, multi-configuration self-consistent-field method or multi-reference Cl methods, have been developed, the correlation problem faced many computational limitation, some of them almost insurmountable, due to the immense number of integrals to be evaluated. 

For a GCF expansion, the density contributions can be determined directly from the partial overly matrices and the arc factors. This allows graph density analysis of wave function expansions that are much larger than could be considered with conventional approaches that rely on explicit summations of CSF contributions. For a normalized GCF wave function, the node density Dk may be computed as [30,32]... 

It is obvious from the denominator in the second term that the matrix will become non-Hermitian when the zeroth-order energies of the determinants in So are not the same. Therefore, this recipe only works for (nearly-)degenerate states. One advantage of this "perturb-and-then-diagonalize approach is that the length of the wave function expansion remains of the dimension of the model space, and hence, especially suitable for analysis purposes. Multideterminantal perturbation schemes that follow the diagonalize-and-then-perturb approach are described in Sect.4.3.3. 

This article has been organized as follows. Section 2 summarizes the notation and formulation of the main concepts used in this work it also reports the formulation of the Shannon entropy indices in terms of the seniority numbers of the Slater determinants. In Sect. 3, we present numerical values of those indices for wave functions of selected atomic and molecular systems these values allow one to characterize the compactness of the wave function expansions. The calculation level and the computational details are also indicated in this section. An analysis and discussion of these results are reported in Sect. 4. Finally, in the last section, we highlight the main conclusions and perspectives of this work. 

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