Configuration Interaction and
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Excitation Based Selection Schemes

Despite the drastic reduction in the CI space which can be achieved using both spin and spatial symmetry considerations, the size of a full CI wave function depends factorially upon the size of the basis set and, as such, remains practical only for small molecules with at most two or three heavy atoms or for systems with extremely high symmetry. It is important to point out the distinction between practical and feasible in this context. While it would be feasible to compute a full CI wave function for a larger system with a miniscule basis set, it is not practical because the basis set would not sufficiently span the chemically important space and the results would be almost worthless. Given that full CI calculations are not practical for many of the systems chemists wish to study, it becomes necessary to find a means of truncating the CI space without greatly sacrificing accuracy.

Perhaps the most common method of reducing the CI space is to select a single dominant configuration, i. e. a Slater determinant or CSF, as a reference function and include other configurations based upon their relation to the reference. The question of exactly how to determine a priori which configurations will be dominant in the CI wave function without performing a full CI calculation itself is still an active area of research, but in most cases theorists resort to using the SCF wave function as the reference. For the vast majority of molecules, the SCF wave function, which is the best single determinant wave function, serves as an adequate reference near the molecule's equilibrium geometry. Having selected a reference configuration, all of the remaining configurations can be classified according to excitation level, where an excitation is defined as the replacement of a spin orbital in the reference by some other spin orbital. In this scheme the expression for the full CI may be written as

\begin{displaymath}\vert \Psi \rangle = C_0 \vert \Phi_0 \rangle + \sum_{ra} c_a... ...<b<c} c_{abc}^{rst} \vert \Phi_{abc}^{rst} \rangle + \cdots, \end{displaymath} (47)

where $\vert \Phi_a^r \rangle$ indicates a configuration in which spin orbital a has been replace by spin orbital b. The simplest method of truncating this full CI expansion is to select a excitation level and remove all terms containing higher excitations. In terms of excitation levels, where $\vert S \rangle$, $\vert D \rangle$, $\vert T \rangle$ and $\vert Q \rangle$ represent all single, double, triple, and quadruple excitations, the structure of the Hamiltonian matrix may be written as

\begin{displaymath}{\bf H} = \begin{array}{c} \langle \Phi_0 \vert \\ \langle... ...vdots & \vdots & \vdots & \vdots & \vdots \end{array}\right] . \end{displaymath} (48)

Only the lower diagonal of the Hamiltonian was provided because if the set of molecular orbitals are real valued functions, as is typically the case, then the Hamiltonian is a symmetric matrix. Notice that matrix elements of the Hamiltonian between the reference, assumed to be the SCF wave function, and all singly excited configurations are rigidly zero, a condition not mandated by Slater's rules. Instead, these matrix elements are zero due to Brillouin's Theorem[5] which shows that the SCF procedure is equivalent to guaranteeing that the SCF wave function does not mix with any singly excited configurations.

The most commonly employed excitation truncated CI method is the CI singles and doubles wave function (CISD or SDCI) in which all configurations that differ from the reference by more that two spin orbitals are removed from the CI space. The motivation for truncating the CI expansion given in (2.47) to include only single and double excitations derives directly from Slater's rules, which state that matrix elements of the Hamiltonian between any two configurations which differ by more than two spin orbitals are identically zero. Therefore, only singly and doubly excited configurations may interact directly with the reference. This does not mean that higher excitations are not important, it just implies that more highly excited configurations must interact indirectly with the reference through their matrix elements with the singly and doubly excited configurations. Moreover, if an SCF reference is employed, then all of the singly excited configurations would also be restricted to an indirect interaction with the reference due to Brillouin's Theorem. Despite this difficulty, singly excited configurations are almost always included in any CI calculation because they are comparatively small in number and are important for determining one-electron properties. As a whole, the CISD method has been very reliable, and in well behaved systems which contain 10 or fewer electrons, it has been observed to recover roughly 95% of the basis set correlation energy.[6]

After the CISD method, the next most commonly employed excitation truncated CI method is the CISDTQ wave function which adds configurations including triple and quadruple excitations into the CISD CI space. Once again the motivation for including these configurations may be traced back to Slater's rules. In this case, only triply and quadrupily excited configurations can interact directly with the singly and doubly excited configurations which in turn interact directly with the reference. For small systems containing 10 or fewer electrons, the CISDTQ method has been observed to recover more than 99% of the basis set correlation energy.[6] Furthermore, the CISDTQ wave function has been found to perform well for molecules displaced far from their equilibrium geometries, recovering 98.6% of the correlation energy for a water molecule with its O-H bonds stretched to twice their equilibrium bond lengths. In conclusion, the CISD and CISDTQ methods are both common truncated CI schemes which perform well for small systems, but many other methods which are based on the concept of excitation based selection may be found in the literature.

This page maintained by Brian C. Hoffman
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