Direct Cl Methods

As illustrated above, even quite small systems at the CISD level results in millions of CSFs. The variational problem is to extract one or possibly a few of the lowest eigenvalues and -veetors of a matrix the size of millions squared. This cannot be done by standard diagonalization methods where all the eigenvalues are found. There are, however, iterative methods for extraeting one, or a few, eigenvalues and -veetors of a large matrix. The Cl problem eq. (4.6) may be written as 

---

In direct CL methods, the target analyte can be the substrate, the oxidant, or the catalyst. In some cases, the analyte can also be an inhibitor that decreases the intensity of the CL signal. This has helped expand the scope of direct methodologies, which was formerly restricted to the few available CL reactions. In indirect CL methods, the analyte is usually the fluorophore. These methods have a broader scope than their direct counterparts as a result of the wide variety of— mainly organic—substances that can act as fluorophore, either as such or following derivatization into fluorescent molecules. 

The final step in the generalization of the direct Cl method was the realization that the full list of coupling coefficients Afj rs can be generated from a small set of prototype coupling coefficients B ra, initially termed internal coupling coefficients. 

P.E.M. Siegbahn, The Direct Cl Method, in Methods in Computational Molecular Physics, (G.H.F. Diercksen and S. Wilson, eds.) D. Reidel Publishing Company, Dordrecht (1983). 

P.E.M. Siegbahn, A New Direct Cl Method for Large Cl Expansions in a Small Orbital Space, Chem. Phys. Letters 109, 417 (1984). 

Older versions of SOCI programs are very I/O intensive because they used to store the Hamiltonian matrix on disk and read it in every iteration step.52,141 Integral-driven direct methods for spin-orbit coupling came up in the mid 1980s123,142 following the original fomulation of direct Cl methods for spin-independent Hamiltonians.143-146 Modern direct SOCI programs can easily handle several million determinants.108,147-151... 

Roos BO, Siegbahn PEM (1980) A direct Cl method with a multiconfigurational reference state. Int J Quantum Chem 17 485-500... 

A possible way to solve the convergence problems consists in using high order energy derivatives in the Taylor energy expansion. The drawback is that a higher derivative calculation is really expensive. Consequently some authors had included these terms in some approximate way / 12,1 4, 28/. This, combined, for instance, with the Direct Cl method of Knowles and Handy, using Slater determinants instead of CSF s, overcomes some problems. 

In typical applications, only the ground electronic state or perhaps a few of the low lying excited states are of interest. Hence methods which obtain only the lowest few roots of the Cl matrix are greatly preferred over methods which compute the entire spectrum. Furthermore, storage requirements are greatly reduced if H is not stored at all direct Cl methods, discussed in section 4, form products He = a directly from the MO integrals. 

Direct Cl methods often require an index vector which points to a list of all allowed excitations from a given iV-electron basis function. Using alpha and beta strings, the index vector need not be the length of the Cl vector—its size is dictated by the number of alpha or beta strings, which (for a full Cl) is approximately the square root of the number of determinants. This results from the fact that in determinant-based Cl, electrons in alpha spin-orbitals can be excited only to other alpha spin-orbitals, and electrons in beta spin-orbitals can be excited only to other beta spin-orbitals (because of the restriction to a single value of Ms). 

The direct Cl method was proposed in 1972 by Roos. The idea of this method is to avoid the explicit construction and storage of the large Hamilton matrix. Instead, the eigenvectors are found iteratively. The basic operation in each iteration is to form the vector g = H c directly from the molecular integrals and the trial vector c. The optimum algorithm to form this product... 

The second possible contraction scheme was first proposed by Meyer , and discussed in the context of the direct Cl method by Siegbahn ". In this case all configurations which have the same external but different internal parts are contracted, and the scheme is therefore called the internal contraction. The internally contracted configurations are generated by applying pair excitation operators to the complete MCSCF reference function. Therefore, the number of contracted configurations and variational parameters is independent of the number of reference configurations. It only depends on the number of correlated internal orbitals and the size of the basis set. ... 

Like any general direct Cl method, the shape-driven graphical unitary group approach can also be used to perform FCI calculations for some cases, but as in Handy s determinantal scheme, matrix elements involving higher than double excitations from the reference CSF are processed using scalar algorithms. Like the Handy determinantal scheme, this approach has been used to provide several benchmarks. 

Saunders V R and Van Lenthe J H 1983 The direct Cl method a detailed analysis Mol. Phys. 48 923-54... 

---