Full CI

For /2(Ar)i7, an extensive CI-CSP simulation was carried out, and the results were compared with those of the simple CSP approximation. Both calculations are for the ultrafast dynamics following excitation of the I2 into the B state. We found that the CI-CSP calculation, including doubly excited configurations , is close to converged for times up to t 500 fs, when 1500 configurations are included. Fig. 2 shows co(t)p, the coefficient of the CSP term and the doubly excited terms in the full CI-CSP wavefunction,... 

A seventh degree polynomial fit to the theoretical curves has been used (columns A and B). In column A use is made of the variational MR SD-Cl energies while in column B estimated full-CI energies 111] are utilised... 

The implementation of weights for functions in the full Cl space differs from the above, since it is impractical to form the full overlap matrix, at least for very large cases. Instead we must rely on one or more full-CI transformations, making use of Eq. (45), and the relations... 

In the implementation of Eq. (48) within CASVB, we use the fact that the effect of an orbital permutation is very straightforward to realize in the determinant basis. Just as for more general transformations, the permutation may be decomposed into separate a and P parts, and the transformation P x pP carried out either in two steps, or as a single pass through all the determinants. This procedure is quite inexpensive, even for a Cl vector in the complete CASSCF space. In our implementation of the full-CI stracture transformation (described in Section 2.1), we have employed a decomposition of O with full pivoting, in order to improve numerical accuracy. 

The Hamiltonian in normal order with respect to its own exact eigenfunction (of full-CI type) is... 

Daul, S., Ciofini, I., Daul, C., White, S.R. Full-CI quantum chemistry using the density matrix renormalization group. Int. J. Quantum Chem. 2000, 79(6), 331. 

The Hy-CI function used for molecular systems is based on the MO theory, in which molecular orbitals are many-center linear combinations of one-center Cartesian Gaussians. These combinations are the solutions of Hartree-Fock equations. An alternative way is to employ directly in Cl and Hylleraas-CI expansions simple one-center basis functions instead of producing first the molecular orbitals. This is a subject of the valence bond theory (VB). This type of approach, called Hy-CIVB, has been proposed by Cencek et al. (Cencek et.al. 1991). In the full-CI or full-Hy-CI limit (all possible CSF-s generated from the given one-center basis set), MO and VB wave functions become identical each term in a MO-expansion is simply a linear combination of all terms from a VB-expansion. Due to the non-orthogonality of one-center functions the mathematical formalism of the VB theory for many-electron systems is rather cumbersome. However, for two-electron systems this drawback is not important and, moreover, the VB function seems in this case more natural. 

Abstract. The elements of the second-order reduced density matrix are pointed out to be written exactly as scalar products of specially defined vectors. Our considerations work in an arbitrarily large, but finite orthonormal basis, and the underlying wave function is a full-CI type wave function. Using basic rules of vector operations, inequalities are formulated without the use of wave function, including only elements of density matrix. 

This is the well-known full-CI scheme. The two-electron density matrix is defined by the formula... 

The components of vectors D k.=i..m are completely defined by the parameters of the underlying full-CI type wave function, and the index sets of Slater-determinants and their subdeterminants according to (13). The number of vectors D is ( ), and this is of course equal to the number of geminals g constructed over the M-dimensional one-particle basis Bm-... 

The most interesting corollary of the results of the previous chapter is, that using basic vector operations and features of vectors, inequalities relating the elements of density matrix can be formulated. Vectors D are completely determined by the configurational coefficients of the underlying full-CI type wave function, but we do not need the knowledge of these coefficients when deriving the inequalities. 

Having discussed ways to reduce the scope of the MCSCF problem, it is appropriate to consider the other limiting case. What if we carry out a CASSCF calculation for all electrons including all orbitals in the complete active space Such a calculation is called full configuration interaction or full CF. Witliin the choice of basis set, it is the best possible calculation that can be done, because it considers the contribution of every possible CSF. Thus, a full CI with an infinite basis set is an exact solution of the (non-relativistic, Bom-Oppenheimer, time-independent) Schrodinger equation. 

Note that no reoptimization of HF orbitals is required, since the set of all possible CSFs is complete . However, tliat is not much help in a computational efficiency sense, since the number of CSFs in a full CI can be staggeringly large. The trouble is not the number of electrons, which is a constant, but the number of basis functions. Returning to our methanol example above, if we were to use Hie 6-31G(d) basis set, the total number of basis functions would be 38. Using Eq. (7.9) to determine the number of CSFs in our (14,38) full CI we find that we must optunize 2.4 x 10 expansion coefficients ( ), and this is really a rather small basis set for chemical purposes. 

Thus, full CI calculations with large basis sets are usually carried out for only the smallest of molecules (it is partly as a result of such calculations that the relative contributions to basis-set quality of polarization functions vs. decontraction of valence functions, as discussed in Chapter 6, were discovered). In larger systems, the practical restriction to smaller basis sets makes full CI calculations less chemically interesting, but such calculations remain useful to the extent that, as an optimal limit, they permit an evaluation of the quality of other methodologies for including electron correlation using the same basis set. We turn now to a consideration of such other methods. 

If we consider all possible excited configurations that can be generated from the HF determinant, we have a full CI, but such a calculation is typically too demanding to accomplish. However, just as we reduced the scope of CAS calculations by using RAS spaces, what if we were to reduce the CI problem by allowing only a limited number of excitations How many should we include To proceed in evaluating this question, it is helpful to rewrite Eq. (7.1) using a more descriptive notation, i.e.. 

If we assume that we do not have any problem with non-dynamical correlation, we may assume that there is little need to reoptimize the MOs even if we do not plan to carry out the expansion in Eq. (7.10) to its full CI limit. In that case, the problem is reduced to determining the expansion coefficients for each excited CSF that is included. The energies E of N different CI wave functions (i.e., corresponding to different variationally determined sets of coefficients) can be determined from the N roots of the CI secular equation... 

The central tenet of CC theory is that the full-CI wave function (i.e., the exact one within the basis set approximation) can be described as... 

Of course, operating on the HF wave function with T is, in essence, full CI (more accurately, in full CI one applies 1 T), so one may legitimately ask what advantage is afforded... 

Table 7.1 Basis set convergence for HF and full CI energies of CO and O, respectively...

Table 7.4 Average errors in correlation energies (kcal mol ) compared to full CI for various methods applied to HB, H2O, and HF at both equilibrium and bond-stretched geometries...

In molecular orbital theory, there is a clear and well defined path to the exact solution of the Schrodinger equation. All we need do is express our wave function as a Unear combination of all possible configurations (full CI) and choose a basis set that is infinite in size, and we have arrived. While such a goal is essentially never practicable, at least the path to it can be followed unambiguously until computational resources fail. 

As is well known, CASSCF wavefunctions are invariant to general (i.e. nonunitary) linear transformations of the active orbitals. As such, we may seek alternative, but equivalent, representations in which a small number of configurations are dominant. This is achieved in our case by means of efficient computational schemes for carrying out exactly the transformations of full-CI spaces induced by nonunitary transformations of orbital spaces [9]. 

Figure 1. Comparison between VB and full-CI results with the same basis-set for the LiH system as a function of projectile-target distance. The r(LiH) distance is fixed at the equilibrium distance of the diatomic molecule and the Jacobi angle (the projectile-LiH centre of mass-target H angle) is fixed at the value of 169°.

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