Size of the Cl Matrix

What is a typical size of the CI matrix Consider a small system, H2O with a 6-31G(d) basis. For the purpose of illustration, let us for a moment return to the spinorbital description. There are 10 electrons and 38 spin-MOs, of which 10 are occupied and 28 are empty. There are possible ways of selecting n electrons out of the 10 occupied orbitals, and K2s,n ways of distributing them in the 28 empty orbitals. The number of excited states for a given excitation level is thus Kiq. K2s,n, and the total number of 

Excitation level n Number of nth excited CSFs Total number of CSFs 

The number of excited determinants thus grows factorially with the size of the basis set. Many of these excited determinants will of course have different spin multiplicity (triplet, quintet etc. states for a singlet HF determinant), and can therefore be left out in the calculation. Generating only the singlet CSFs, the number of configurations at each excitation level is shown in Table 4.1. 

In the general case of N electrons and M basis functions the total number of singlet CSFs that can be generated is given by 

Since only doubly excited determinants have non-zero matrix elements with the HF state, these are the most important. This may be illustrated by considering a full Cl 

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In large systems there can be many orbitals in a small energy range, and the size of the Cl matrix can be very sensitive to the value of the maximum excitation if you use Biergy Criterion. Since calculation time depends heavily on the size of the Cl matrix, you can end up with very long calculations, especially if you use the ab initio methods or the MNDO, AMI, or PM3 semi-empirical methods. This could exhaust the memory of your system. Again, inspecting the results of an RHF (no Cl) calculation will help you avoid these pitfalls. 

The disappearance of matrix elements between the HF reference and singly excited Wliat IS a typical size of the Cl matrix Consider a small system, H2O with a 6-31G(d)... 

Figure 7.4 Structure of the Cl matrix as blocked by classes of determinants. The HF block is the (1,1) position, the matrix elements between the HF and singly excited determinants are zero by Brillouin s theorem, and between the HF and triply excited determinants are zero by the Condon-Slater rules. In a system of reasonable size, remaining regions of the matrix become increasingly sparse, but the number of determinants in each block grows to be extremely large. Thus, the (1,1) eigenvalue is most affected by the doubles, then by the singles, then by the triples, etc...

Finally, a few words on the size of system that can be treated. The limiting parameters using the (n-l-l)th eigenvalue from the diagonalization of the Cl matrix as a ... 

The comparison of the calculated spectra of the free ions and the ones in the crystal is not straightforward. Indeed, in the crystal, the presence of the first coordination shell increases the number of electrons and basis functions in the calculations, resulting in a blow-up of the Cl expansion, mainly due to the generated doubly-excited configurations. One should bare in mind that this increase is about six time as fast in double group symmetries as in the non-relativistic symmetry. In a non effective Hamiltonian method, the only way to keep the size of the DGCI matrix to an affordable size of few million configurations, is to cut down the number of correlated electrons. This may essentially deteriorate the quality of electron correlation as the contributions of the spin-orbit interaction... 

This elementary fact enables us to reduce the size of the Cl matrices by choosing linear combinations of the double excitations that have a particular spin symmetry. In particular, since we are normally concerned with ground-state singlets, we can use linear combinations of double-excitation determinants that are singlet spin functions. The only difference which this will make to our method is the detailed form of the Hkl in the matrix H (the singlets are still orthonormal). 

It was not long before it was realized that even evaluating and storing the Hamiltonian placed too severe limits on the size of the Cl expansion. For a million determinants, the number of Hamiltonian matrix elements is 10 —far too large to store on disk and read in each iteration, even if the matrix was sparse and the sparsity could be exploited. The Hamiltonian is, however, composed of integrals over the molecular... 

The immediate consequence of this increased number of determinants is an increase of the size of the Hamiltonian matrix. Eor the = 0 case, the relativistic Hamiltonian matrix has nine times as many elements as in the nonrelativistic Hamiltonian matrix. We would expect the work in multiplying the Hamiltonian matrix by a vector (the time-consuming step in modern Cl procedures, as discussed in the previous section) to increase by a factor of 36 9 from the size of the Hamiltonian matrix and 4 from the complex arithmetic. 

Due to the size of the variational problem, a large Cl is usually not a practicable method for recovering dynamic correlation. Instead, one usually resorts to some form of treatment based on many-body perturbation theory where an explicit calculation of all off-diagonal Cl matrix elements (and the diagonalization of the matrix) are avoided. For a detailed description of such methods, which is beyond the scope of this review, the reader is referred to appropriate textbooks295. For the present purpose, it suffices to mention two important aspects. 

The first conclusion we may draw is that the minimal dressing of a Cl matrix (CAS-SDCI in our study) not only ensures the importanty property of size-extensivity, but also improves the absolute values of the yielded energies. This can be stated in all studied cases single bond breaking, two single bond breaking and triple bond breaking. 

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