Configuration Interaction Singles

Adapted from my CHEM 221B final project at Berkeley, following Szabo & Ostlund.

Hartree-Fock hands us one state, the ground state, and stops there. What if we want the color of a molecule, the energy of its first excited state, the fate of an absorbed photon? Configuration interaction singles (CIS) is the smallest possible step beyond Hartree-Fock that answers these questions, and it is worth understanding well, because nearly every affordable excited-state method in use today inherits its structure.

The ansatz

Recall that a Hartree-Fock calculation leaves us with a determinant $\ket{\Psi_0}$ built from occupied spin orbitals, plus a set of unoccupied (virtual) orbitals it never used. The cheapest way to build new states is to promote one electron, replacing occupied orbital $i$ with virtual orbital $a$ to form the singly excited determinant $\ket{\Psi_i^a}$. CIS takes its wavefunctions to be linear combinations of these,

$$\ket{\Phi} = \sum_{ia} c_i^a \ket{\Psi_i^a}$$

Why should such a small basis describe an excited state? Notice first that it cannot improve the ground state. The Brillouin theorem tells us that singly excited determinants do not couple to the Hartree-Fock ground state through the Hamiltonian, $\bra{\Psi_i^a} \hat H \ket{\Psi_0} = 0$. The singles block of the Hamiltonian is therefore decoupled from the ground state, and diagonalizing it yields states that are orthogonal to $\ket{\Psi_0}$ and can be interpreted as excitations rather than corrections.

The eigenvalue problem

Within the singles space, the Hamiltonian (measured from the Hartree-Fock energy) has matrix elements

$$A_{ia,jb} = (\epsilon_a - \epsilon_i)\,\delta_{ij}\delta_{ab} + \langle aj \,|\, ib\rangle - \langle aj \,|\, bi\rangle$$

The first term is the orbital-energy gap you would guess from a molecular-orbital diagram. The two-electron integrals correct that guess, and the correction matters. The gap $\epsilon_a - \epsilon_i$ alone overestimates most excitation energies badly, because it ignores the interaction between the excited electron and the hole it left behind. Diagonalizing,

$$\pmb{A}\pmb{X} = \omega \pmb{X}$$

the eigenvalues $\omega$ are the CIS excitation energies and the eigenvectors are the amplitudes $c_i^a$ of each excitation.

CIS stays in use because of a short list of properties that are rarely found together in one cheap method. The energies are variational upper bounds to the exact ones. The cost is comparable to Hartree-Fock itself. The total energies are size consistent, meaning $E(A+B) = E(A) + E(B)$ for noninteracting fragments. And CIS gets the $-1/r$ asymptotic attraction between a separated electron and hole right, which standard time-dependent DFT famously does not. When a charge-transfer excitation energy from TD-DFT looks suspiciously low, a CIS calculation is the classic sanity check.

From CIS to TDHF

Time-dependent Hartree-Fock reuses the matrix $\pmb A$ and couples it to a second block,

$$\begin{pmatrix} \pmb{A} & \pmb{B} \\ \pmb{B} & \pmb{A} \end{pmatrix} \begin{pmatrix} \pmb{X} \\ \pmb{Y} \end{pmatrix} = \omega \begin{pmatrix} \pmb{1} & 0 \\ 0 & -\pmb{1} \end{pmatrix} \begin{pmatrix} \pmb{X} \\ \pmb{Y} \end{pmatrix}, \qquad B_{ia,jb} = \langle aj \,|\, bi\rangle - \langle aj \,|\, ib\rangle$$

The $\pmb Y$ amplitudes describe de-excitations, and the spectrum comes in $\pm\omega$ pairs, of which only the positive half is physical since the reference is already the ground state. Setting $\pmb B = 0$ recovers CIS, which is why CIS is also called the Tamm-Dancoff approximation to TDHF. TDHF energies come out slightly lower than CIS, but the improvement is modest against the doubled dimension of the problem, and in practice the Tamm-Dancoff form is often preferred for its stability.

Toward TD-DFT

CIS excitation energies are qualitative, typically off by an eV or so, because a single determinant per excitation has no room for the orbital relaxation and correlation that follow a real excitation. The structure, however, is the durable part. Swap the Hartree-Fock reference for a Kohn-Sham one and the $\pmb A$ and $\pmb B$ blocks for their DFT counterparts, and the TDHF equations above become TD-DFT, the workhorse of practical excited-state calculations.

Feel free to contact me if there is an error or confusion!