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Next: Restricted Hartree-Fock References Up: Configuration Interaction Singles Previous: Introduction

CIS Energy Equations

Since there are only two types of determinants according to excitation level (i.e., the reference and single excitations), there are only two relevant types of matrix elements, $\langle \Phi_0 \vert {\hat H}
\vert \Phi_i^a \rangle$ and $\langle \Phi_i^a \vert {\hat H} \vert \Phi_j^b \rangle$. Assuming that the determinants are made up of a common set of orthonormal spin orbitals, these matrix elements may be evaluated using Slater's rules. The first is given by

\begin{displaymath}\langle \Phi_0 \vert {\hat H} \vert \Phi_i^a \rangle =
...um_{k \in \Phi_0} \langle ik \vert\vert ak \rangle =
\end{displaymath} (2)

where the Fock matrix element Fpq is defined as

\begin{displaymath}F_{pq} = h_{pq} + \sum_{k \in \Phi_0} \langle pk \vert\vert qk \rangle.
\end{displaymath} (3)

The other relevant matrix elements are of the form $\langle \Phi_i^a \vert {\hat H} \vert \Phi_j^b \rangle$. The singly excited determinants may differ from each other by two spin orbitals if $i \neq j$ and $a \neq b$. If so, the determinants are already in maximum coincidence and the matrix element is of the form

\begin{displaymath}\langle \cdots a \cdots j \cdots \vert {\hat H} \vert \cdots i \cdots b \cdots \rangle
\end{displaymath} (4)


\begin{displaymath}\langle \cdots j \cdots a \cdots \vert {\hat H} \vert \cdots b \cdots i \cdots \rangle
\end{displaymath} (5)

The matrix elements are $\langle aj \vert\vert ib \rangle$ and $\langle ja \vert\vert bi \rangle$, respectively, and these antisymmetrized integrals are equal to each other (and also to $-\langle ja \vert\vert ib \rangle$ and $-\langle aj \vert\vert bi \rangle$).

For the case i=j, $a \neq b$, the matrix elements are

\begin{displaymath}\langle \Phi_i^a \vert {\hat H} \vert \Phi_i^b \rangle =
...t\vert bk \rangle =
F_{ab} - \langle ai \vert\vert bi \rangle.
\end{displaymath} (6)

Likewise, for the case $i \neq j$, a=b, the matrix elements are

\begin{displaymath}\langle \Phi_i^a \vert {\hat H} \vert \Phi_j^a \rangle =
...\vert jk \rangle =
-F_{ij} - \langle ia \vert\vert ja \rangle.
\end{displaymath} (7)

Finally, when i=j and a=b,
$\displaystyle \langle \Phi_i^a \vert {\hat H} \vert \Phi_i^a \rangle$ = $\displaystyle \sum_{k \in \Phi_0} h_{kk} + \frac{1}{2} \sum_{k,l \in \Phi_0}
...\in \Phi_0} \langle ka \vert\vert ka \rangle
- \langle ia \vert\vert ia \rangle$ (8)
  = $\displaystyle E_{0} - F_{ii} + F_{aa} - \langle ia \vert\vert ia \rangle,$  

where $E_0 = \langle \Phi_0 \vert {\hat H} \vert \Phi_0 \rangle$; if $\vert \Phi_0 \rangle$ is obtained by an SCF procedure, then this is the SCF energy.

Using the permutational symmetries of the antisymmetrized two-electron integrals, the two-electron terms for the preceding four cases can be combined (rearranging the integral $\langle ia \vert\vert ja \rangle$ requires the assumption that the orbitals are real). This yields the final, compact result

\begin{displaymath}\langle \Phi_i^a \vert {\hat H} \vert \Phi_j^b \rangle =
...{ij} - F_{ij} \delta_{ab}
+ \langle aj \vert\vert ib \rangle.
\end{displaymath} (9)

This is equation (11) of Maurice and Head-Gordon [2], who extended the CIS method to the case of restricted open-shell (ROHF) and unrestricted (UHF) reference determinants. Note that E0occurs along the diagonal of the entire matrix H; this means that we can subtract E0 before diagonalizing and add it back later to each of the eigenvalues. If all matrix elements Fia = 0, as they often are, then the reference determinant does not mix with any of the excited determinants and $\vert \Phi_0 \rangle$ is already an eigenfunction of the CIS Hamiltonian with eigenvalue E0; furthermore, the eigenvalues of the CIS Hamiltonian less the E0diagonal terms represent excitation energies. From this point onward, E0 will be subtracted from the Hamiltonian.

Given the above matrix elements, it remains to write down the CIS energy expression. Recall that the CIS wavefunction is expanded as

\begin{displaymath}\vert \Psi \rangle = c_0 \vert \Phi_0 \rangle + \sum_{ia} c_i^a \vert \Phi_i^a \rangle.
\end{displaymath} (10)

Assuming real CI coefficients, the energy is given by

\begin{displaymath}E_{\rm CIS} =
+ 2 \sum_{ia} c_0 c_i^a F_{ia}
+ \sum...
+ \sum_{ijab} c_i^a c_j^b \langle aj \vert\vert ib \rangle.
\end{displaymath} (11)

For a closed-shell SCF reference $\vert \Phi_0 \rangle$, off-diagonal terms of the Fock matrix vanish, and the expression becomes

\begin{displaymath}E_{\rm CIS} =
E_{\rm SCF}
+ \sum_{ia} (c_i^a)^2 (\epsilon...
+ \sum_{ijab} c_i^a c_j^b \langle aj \vert\vert ib \rangle,
\end{displaymath} (12)

which matches equation (2.15) of Foresman et al. [1] once the two-electron integral is rearranged. Of course, this equation is only useful once the CI coefficients are known. In general, the lowest several eigenvectors are of interest in a CIS study; these can be obtained by iteratively diagonalizing the CIS Hamiltonian using Davidson's method [3] or the Davidson-Liu Simultaneous Expansion Method [4]. Iterative solution calls for diagonalization of the Hamiltonian in a small subspace of trial vectors, with the set of vectors being expanded every iteration until convergence. This requires calculation of the following quantities, usually called the $\sigma$ vectors:

\begin{displaymath}\sigma_{I} = \sum_{J} H_{IJ} c_{J}.
\end{displaymath} (13)

One ${\mathbf \sigma}$ vector must be computed for each cin the set of trial vectors. For CIS, ${\mathbf \sigma}$ can be written
$\displaystyle \sigma_0$ = $\displaystyle \sum_{jb} \langle \Phi_0 \vert \overline{H} \vert \Phi_j^b \rangle c_j^b$ (14)
$\displaystyle \sigma_i^a$ = $\displaystyle \langle \Phi_{ia} \vert \overline{H} \vert \Phi_0 \rangle c_0
+ \sum_{jb} \langle \Phi_i^a \vert \overline{H} \vert \Phi_j^b \rangle c_j^b,$ (15)

where the bar over H is a reminder that E0 has been subtracted from the Hamiltonian. These expressions can be expanded to
$\displaystyle \sigma_0$ = $\displaystyle \sum_{jb} c_j^b F_{jb}$ (16)
$\displaystyle \sigma_{ia}$ = $\displaystyle c_0 F_{ia}
+ \sum_{jb} c_j^b
F_{ab} \delta_{ij}
- F_{ij} \delta_{ab}
+ \langle aj \vert\vert ib \rangle
\right].$ (17)

The above equations make it clear that $\sigma$ can be computed directly from the one- and two-electron integrals without the need to explicitly compute or store the one- and two-electron coupling coefficients $\gamma^{IJ}_{pq}$ and $\Gamma^{IJ}_{pqrs}$ as separate quantities; this makes the CIS method a direct CI procedure. As noted by Foresman et al. [1], the CIS iterations can actually be performed in a ``double-direct'' fashion; i.e., the integrals can also be computed on-the-fly as needed. As shown by Maurice and Head-Gordon [2], the contribution of the two-electron integrals to $\sigma_i^a$ can be written as a Fock-like matrix,
$\displaystyle {\tilde F}_{ia}$ = $\displaystyle \sum_{jb} c_j^b \langle aj \vert\vert ib \rangle$ (18)
  = $\displaystyle \sum_{\mu \nu} C_{\mu a}^* C_{\nu i}
\sum_{\lambda \sigma} \langl...
...bda \vert\vert \nu \sigma \rangle
\sum_{jb} C_{\lambda j}^* c_j^b C_{\sigma b},$ (19)

where $C_{\mu i}$ are the coefficients defining the transformation from atomic orbitals (AOs) to molecular orbitals (MOs). Evaluation of ${\tilde F}_{ia}$ can be carried out as a series of matrix multiplies. The pseudodensity matrix,

{\tilde P}_{\lambda \sigma} = \sum_{jb} C_{\lambda j}^* c_j^b C_{\sigma b},
\end{displaymath} (20)

can be multiplied by the two-electron integrals as they are formed in the atomic orbital basis to yield the AO Fock-like matrix,

{\tilde F}_{\mu \nu} = \sum_{\lambda \sigma} \langle \mu \lambda \vert\vert \nu \sigma \rangle
{\tilde P}_{\lambda \sigma},
\end{displaymath} (21)

which is transformed back into the MO basis by

{\tilde F}_{ia} = \sum_{\mu \nu} C_{\mu a}^* {\tilde F}_{\mu \nu}
C_{\nu i}.
\end{displaymath} (22)

next up previous
Next: Restricted Hartree-Fock References Up: Configuration Interaction Singles Previous: Introduction
C. David Sherrill