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

$$\langle \Phi_0 \vert {\hat H} \vert \Phi_i^a \rangle = h_{ia... ...um_{k \in \Phi_0} \langle ik \vert\vert ak \rangle = F_{ia},$$ (2)

where the Fock matrix element F_pq is defined as

$$F_{pq} = h_{pq} + \sum_{k \in \Phi_0} \langle pk \vert\vert qk \rangle.$$ (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

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

or

$$\langle \cdots j \cdots a \cdots \vert {\hat H} \vert \cdots b \cdots i \cdots \rangle$$ (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

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

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

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

Finally, when i=j and a=b,

$$\langle \Phi_i^a \vert {\hat H} \vert \Phi_i^a \rangle \sum_{k \in \Phi_0} h_{kk} + \frac{1}{2} \sum_{k,l \in \Phi_0} \l... ...\in \Phi_0} \langle ka \vert\vert ka \rangle - \langle ia \vert\vert ia \rangle$$ = (8)

$$E_{0} - F_{ii} + F_{aa} - \langle ia \vert\vert ia \rangle,$$ =

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

$$\langle \Phi_i^a \vert {\hat H} \vert \Phi_j^b \rangle = E_0... ...{ij} - F_{ij} \delta_{ab} + \langle aj \vert\vert ib \rangle.$$ (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 F_ia = 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

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

Assuming real CI coefficients, the energy is given by

$$E_{\rm CIS} = E_0 + 2 \sum_{ia} c_0 c_i^a F_{ia} + \sum... ... + \sum_{ijab} c_i^a c_j^b \langle aj \vert\vert ib \rangle.$$ (11)

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

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

$$\sigma_{I} = \sum_{J} H_{IJ} c_{J}.$$ (13)

One ${\mathbf \sigma}$ vector must be computed for each cin the set of trial vectors. For CIS, ${\mathbf \sigma}$ can be written

$$\sigma_0 \sum_{jb} \langle \Phi_0 \vert \overline{H} \vert \Phi_j^b \rangle c_j^b$$ = (14)

$$\sigma_i^a \langle \Phi_{ia} \vert \overline{H} \vert \Phi_0 \rangle c_0 + \sum_{jb} \langle \Phi_i^a \vert \overline{H} \vert \Phi_j^b \rangle c_j^b,$$ = (15)

where the bar over H is a reminder that E₀ has been subtracted from the Hamiltonian. These expressions can be expanded to

$$\sigma_0 \sum_{jb} c_j^b F_{jb}$$ = (16)

$$\sigma_{ia} c_0 F_{ia} + \sum_{jb} c_j^b \left[ F_{ab} \delta_{ij} - F_{ij} \delta_{ab} + \langle aj \vert\vert ib \rangle \right].$$ = (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,

$${\tilde F}_{ia} \sum_{jb} c_j^b \langle aj \vert\vert ib \rangle$$ = (18)

$$\sum_{\mu \nu} C_{\mu a}^* C_{\nu i} \sum_{\lambda \sigma} \langl... ...bda \vert\vert \nu \sigma \rangle \sum_{jb} C_{\lambda j}^* c_j^b C_{\sigma b},$$ = (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,

$${\tilde P}_{\lambda \sigma} = \sum_{jb} C_{\lambda j}^* c_j^b C_{\sigma b},$$ (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,

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

which is transformed back into the MO basis by

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