The derivation of the CI-singles energy and wave function^{216, 270} begins by selecting the HF single-determinant wave function as reference for the ground state of the system:
$${\mathrm{\Psi}}_{\mathrm{HF}}=\frac{1}{\sqrt{n!}}det\left\{{\chi}_{1}\chi {}_{2}\mathrm{\cdots}{\chi}_{i}\chi {}_{j}\mathrm{\cdots}{\chi}_{n}\right\}$$ | (7.1) |
where $n$ is the number of electrons, and the spin orbitals
$${\chi}_{i}=\sum _{\mu}^{N}{c}_{\mu i}{\varphi}_{\mu}$$ | (7.2) |
are expanded in a finite basis of $N$ atomic orbital basis functions. Molecular orbital coefficients $\{{c}_{\mu i}\}$ are usually found by SCF procedures which solve the Hartree-Fock equations
$$\mathrm{\mathbf{F}\mathbf{C}}=\bm{\epsilon}\mathrm{\mathbf{S}\mathbf{C}},$$ | (7.3) |
where S is the overlap matrix, C is the matrix of molecular orbital coefficients, $\epsilon $ is a diagonal matrix of orbital eigenvalues and F is the Fock matrix with elements
$${F}_{\mu \upsilon}={H}_{\mu \upsilon}+\sum _{\lambda \sigma}\sum _{i}{c}_{\mu i}{c}_{\upsilon i}(\mu \lambda ||\upsilon \sigma )$$ | (7.4) |
involving the core Hamiltonian and the anti-symmetrized two-electron integrals
$$(\mu \mu ||\lambda \sigma )=\int \int {\varphi}_{\mu}({\mathbf{r}}_{1}){\varphi}_{\nu}({\mathbf{r}}_{2})\left(\frac{1}{{r}_{12}}\right)[{\varphi}_{\lambda}({\mathbf{r}}_{1}){\varphi}_{\sigma}({\mathbf{r}}_{2})-{\varphi}_{\sigma}({\mathbf{r}}_{1}){\varphi}_{\lambda}({\mathbf{r}}_{2})]d{\mathbf{r}}_{1}d{\mathbf{r}}_{2}$$ | (7.5) |
On solving Eq. (7.3), the total energy of the ground state single determinant can be expressed as
$${E}_{\mathrm{HF}}=\sum _{\mu \upsilon}{P}_{\mu \upsilon}^{\mathrm{HF}}{H}_{\mu \upsilon}+\frac{1}{2}\sum _{\mu \upsilon \lambda \sigma}{P}_{\mu \upsilon}^{\mathrm{HF}}{P}_{\lambda \sigma}^{\mathrm{HF}}(\mu \lambda ||\upsilon \sigma )+{V}_{\mathrm{nuc}}$$ | (7.6) |
where ${P}^{\mathrm{HF}}$ is the HF density matrix and ${V}_{\mathrm{nuc}}$ is the nuclear repulsion energy.
Equation (7.1) represents only one of many possible determinants made from orbitals of the system; there are in fact $n(N-n)$ possible singly substituted determinants constructed by replacing an orbital occupied in the ground state ($i$, $j$, $k,\mathrm{\dots}$) with an orbital unoccupied in the ground state ($a$, $b$, $c,\mathrm{\dots}$). Such wave functions and energies can be written
$${\mathrm{\Psi}}_{i}^{a}=\frac{1}{\sqrt{n!}}det\left\{{\chi}_{1}\chi {}_{2}\mathrm{\cdots}{\chi}_{a}\chi {}_{j}\mathrm{\cdots}{\chi}_{n}\right\}$$ | (7.7) |
$${E}_{ia}={E}_{\mathrm{HF}}+{\epsilon}_{a}-{\epsilon}_{i}-(ia||ia)$$ | (7.8) |
where we have introduced the anti-symmetrized two-electron integrals in the molecular orbital basis
$$(pq||rs)=\sum _{\mu \upsilon \lambda \sigma}{c}_{\mu p}{c}_{\upsilon q}{c}_{\lambda r}{c}_{\sigma s}(\mu \lambda ||\upsilon \sigma )$$ | (7.9) |
These singly excited wave functions and energies could be considered crude approximations to the excited states of the system. However, determinants of the form Eq. (7.7) are deficient in that they:
do not yield pure spin states
resemble more closely ionization rather than excitation
are not appropriate for excitation into degenerate states
These deficiencies can be partially overcome by representing the excited state wave function as a linear combination of all possible singly excited determinants,
$${\mathrm{\Psi}}_{\mathrm{CIS}}=\sum _{ia}{a}_{i}^{a}{\mathrm{\Psi}}_{i}^{a}$$ | (7.10) |
where the coefficients $\{{a}_{ia}\}$ can be obtained by diagonalizing the many-electron Hamiltonian, A, in the space of all single substitutions. The appropriate matrix elements are:
$${A}_{ia,jb}=\u27e8{\mathrm{\Psi}}_{i}^{a}\left|H\right|{\mathrm{\Psi}}_{j}^{b}\u27e9=({\epsilon}_{a}-{\epsilon}_{j}){\delta}_{ij}{\delta}_{ab}-(ja||ib)$$ | (7.11) |
According to Brillouin’s, theorem single substitutions do not interact directly with a reference HF determinant, so the resulting eigenvectors from the CIS excited state represent a treatment roughly comparable to that of the HF ground state. The excitation energy is simply the difference between HF ground state energy and CIS excited state energies, and the eigenvectors of A correspond to the amplitudes of the single-electron promotions.
CIS calculations can be performed in Q-Chem using restricted (RCIS),^{216, 270} unrestricted (UCIS), or restricted open-shell^{618} (ROCIS) spin orbitals.
$molecule 0 1 C O 1 CO H 1 CH 2 A H 1 CH 2 A 3 D CO = 1.2 CH = 1.0 A = 120.0 D = 180.0 $end $rem JOBTYPE = opt EXCHANGE = hf BASIS = 6-31G* $end @@@ $molecule read $end $rem EXCHANGE = hf BASIS = 6-311(2+)G* CIS_N_ROOTS = 15 Do 15 states CIS_SINGLETS = true Do do singlets CIS_TRIPLETS = false Don’t do Triplets $end