Coupled Cluster Methods
Full CI and the Dimension of the CI Space
In the previous section's discussion of the CI method, it was tacitly assumed that one could find a complete basis into which the problem might be expanded; however, in reality, a basis set must contain an infinite number of functions to be truly complete. Even with the best computers man can hope to construct, the problem of constructing and diagonalizing an infinite Hamiltonian matrix will never be tractable. Despite this drawback, such an infinite basis set CI serves as a idealistic reference and has been been denoted as ``complete CI.'' While the ultimate goal of complete CI will never be obtained, it is possible to select a finite basis of K one-electron functions which span as much of the chemically significant regions of the infinite space as possible and to construct a finite set of 2K spin orbitals which may in turn be used to generate Slater determinants or CSFs which can then be used as N-electron basis functions in the CI expansion. Typically the set of spin orbitals is generated from the one-electron or atomic basis functions through the SCF procedure of the previous chapter, although it is possible to employ other methods such as the multi-configuration SCF method described in the next chapter. The designation ``full CI'' is used to describe CI calculations which include all determinants or CSFs generated from a finite basis set. Even using a finite basis set, the size of the Hamiltonian matrix grows rapidly with increasing basis set size, and in practice full CI calculations have only been performed using adequate basis sets for a few small molecules. The very fact that theorists have been able to perform as many full CI calculations as they have is due to a theorem of quantum mechanics which states that if a Hermitian operator commutes with the Hamiltonian, then eigenfunctions of that operator which possess different eigenvalues do not contribute to the same block of the Hamiltonian. This theorem allows the Hamiltonian matrix to be subdivided into non-interacting blocks which may be diagonalized separately. Moreover, if the desired wave function, typically that of the ground state, is known a priori to be an eigenfunction of some Hermitian operator with a particular eigenvalue, then any N-electron basis functions which are also eigenfunctions of that Hermitian operator but have different eigenvalues may be eliminated from the CI procedure.
One of the most common Hermitian operators used to narrow down the CI
space is the spin angular momentum operator S2:
 |
(44) |
If the desired electronic state is a triplet (S=1), then any N-electron basis function which is not a triplet can safely be excluded from the full CI calculation. This reduction of the CI space is particularly feasible if the N-electron basis functions are CSFs, i. e. small linear combinations of Slater determinants which are constructed to be eigenfunctions of S2. Taking into account spin symmetry, the size of the full CI space in terms of CSFs is given by Weyl's dimension formula[4]:
 |
(45) |
where N is the number of electrons, n is the number of orbitals, and S is the total spin. If the CI calculation is performed in a determinant basis instead of a CSF basis, then one must be extremely careful to include all determinants which would contribute to CSFs with the correct spin quantum number. In terms of determinants, the size of the CI space when only spin symmetry is considered is given by
 |
(46) |
Next to S2, the most common set of Hermitian operators used to reduce the size of the CI space are spatial symmetry operators such as rotation or reflection operators. The exact benefits of employing spatial symmetry restrictions on the CI space are difficult to judge in general because they depend upon the symmetry point group to which the molecule belongs and the nature of the basis functions. Consider, for example, the DZP full CI wave function of NH2, a molecule which belongs to the C2v point group. The DZP basis set for NH2 contains 26 basis functions, yielding 13 A1, 2 A2, 4 B1, and 7 B2 symmetry adapted atomic orbitals. With this basis, the full CI wave function describing the X 2B1state of NH2 contains 96,089,928 CSFs when symmetry restrictions are employed, compared to 384,813,000 CSFs when no such restrictions are used. Thus, in this instance, symmetry restrictions reduced the number of CSFs by a factor of about 4.