Full configuration-interaction wave functions

For a given one-electron expansion, the exact solution to the Schrodinger equation may be written as a linear combination of all determinants that can be constructed from this one-electron basis in the A-electron Fock space  

The expansion coefficients may be obtained from the variation principle as described in Section 4.2.3 and the solution is called the full configuration-interaction (FCI) wave function. The number of determinants in an FCI wave function is given by the expression 

The factorial dependence of the number of Slater determinants on the number of spin orbitals and electrons (5.3.2) makes FCI wave functions intractable for all but the smallest systems. However, in those cases where FCI calculations can be carried out, the results are often very useful since their solutions are exact within the chosen one-electron basis. In comparison with FCI, the errors introduced in the treatment of the / -electron expansion by less accurate wave functions can be identified and examined in isolation from the errors in the one-electron space. In this way, the FCI calculations may serve as useful benchmarks for the A -electron treatments of the standard models, exhibiting their strengths and deficiencies in a transparent manner. Still, for the FCI benchmark calculations to be worthwhile, the orbital basis must be sufficiently flexible to provide a reasonably faithful representation of the electronic system - otherwise our conclusions will be based on a distorted representation of the true system, with little relevance for accurate calculations carried out in larger basis sets. 

In this section, we shall present two examples of FCI wave functions that, in the subsequent sections, will be used in our discussion of the standard computational models the hydrogen molecule in the large cc-pVQZ basis and the water molecule in the small cc-pVDZ basis. For a description of these basis sets, see Section 8.3.3. 

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This series could be continued to include all possible substituted determinants, in which case it would correspond to a full configuration interaction wave function,... 

The exponential operator T creates excitations from 4>o according to T = l + 72 + 73 + , where the subscript indicates the excitation level (single, double, triple, etc.). This excitation level can be truncated. If excitations up to Tn (where N is the number of electrons) were included, vPcc would become equivalent to the full configuration interaction wave function. One does not normally approach this limit, but higher excitations are included at lower levels of coupled-cluster calculations, so that convergence towards the full Cl limit is faster than for MP calculations. 

Eq. (8.94) represents an exact expression for the quantum mechanical state of a many-electron system. Note that the expansion coefficients Cj of the N-particle basis states are directly related to the expansion coefficients of the one-particle states. It is sufficient to know either of them (and this fact is related to the observation made below that a full configuration interaction wave function does not require the optimization of orbitals see the next section). 

Abstract This work reports the formulation of Shannon entropy indices in terms of seniority numbers of the Slater determinants expanding an A-electron wave fimc-tion. Numerical determinations of those indices prove that they provide a suitable quantitative procedure to evaluate compactness of wave functions and to describe their configurational structures. An analysis of the results, calculated for full configuration interaction wave functions in selected atomic and molecular systems, allows one to compare and to discuss the behavior of several types of molecular orbital basis sets in order to achieve more compact wave... 

We can follow the procedure adopted in Section 1 for the exact Schrodinger equation and conjecture that the full configuration interaction wave function for the supersystem can be written... 

At this point we are sufficiently equipped to consider briefly the methods used to approximate the wave functions constructed in the restricted subspace of orbitals. So far the only approximation was to restrict the orbital basis set. It is convenient to establish something that might be considered to be the exact solution of the electronic structure problem in this setting. This is the full configuration interaction (FCI) solution. In order to find one it is necessary to construct all possible Slater determinants for N electrons allowed in the basis of 2M spin-orbitals. In this context each Slater determinant bears the name of a basis configuration and constructing them all means that we have their full set. Then the matrix representation of the Hamiltonian in the basis of the configurations ( >K is constructed ... 

In principle, the theory reviewed in Sections 4-6 can be applied to interactions of arbitrary systems if the full configuration interaction (FCI) wave functions of the monomers are available, and if the matrix elements of H0 and V can be constructed in the space spanned by the products of the configuration state functions of the monomers. For the interactions of many-electron monomers the resulting perturbation equations are difficult to solve, however. A many-electron version of SAPT, which systematically treat the intramonomer correlation effects, offers a solution to this problem. 

In practice, therefore, we must work with finite-dimensional orbital spaces, properly optimized so as to yield the best representation of the w-electron wave function. In addition, we must, for a given finite orbital basis, find a way to determine the coefficients in the expansion (Eq. (14)). The best wave function (i.e., the wave function with the lowest energy) is obtained by optimizing all expansion coefficients Cp variationally, as done in the full configuration interaction (FCI) method [18-20] ... 

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