Configuration interaction theory (CI)

Warning: Configuration interaction is a new feature of the QMol-grid package, and has not been fully tested and benchmarked.

Consider an atomic or molecular system with $N$ active electrons in the Born-Oppenheimer approximation (fixed nuclei). In atomic units, the Hamiltonian operator describing the system is

$$ \hat{\mathcal{H}} (x_{1} ,x_{2} ,\ldots ,x_{N} ) = \sum_{k=1}^{N}{ \hat{h} (r_k )} + \sum_{1\le k < k' \le N} V_{\textrm{ee}} (r_k -r_{k^{\prime } } ) ~~~~ \textrm{with} ~~~~ \hat{h} (r)=-\frac{\Delta }{2}+V_{\textrm{ne}} (r), ~~~~~~ (1) $$

where $x_k =(r_k ,\omega_k )$ are the electronic coordinates including the spatial $r_k$ and spin $\omega_k$ information. The operator $\hat{h}$ is the core Hamiltonian, while $V_{\textrm{ee}}$ and $V_{\textrm{ne}}$ are the electron-electron and total electron-nucleus interaction potentials, respectively. Configuration interaction (CI) expands the expression for the Hamiltonian operator of equation (1), and its associated wave functions, in a basis of configuration states formed as Slater determinants of orthonormal spin orbitals. The CI algorithm implemented in QMol-grid is discussed in [Visentin 2025]. Briefly, for a given set of $K\ge N$ spin orbitals

$$ \chi_1 ,\chi_2 ,\ldots,\chi_K ~~ \textrm{such} ~ \textrm{that} ~~ \int \chi_k (x)^* \chi_l (x)~\textrm{d}x=\delta_{k,l} ~~ \forall 1\le k\le l\le K, ~~~~~~ (2) $$

a configuration state is formed by the antisymmetrized product (Slater determinant) of $N$ unique spin orbitals of the form

$$ |\chi_{\vec{k} } \rangle =|\chi_{k_1 } \chi_{k_2 } \ldots\chi_{k_N } \rangle =\frac{1}{\sqrt{N!}}\sum_{\sigma \in S_N } \textrm{sgn}(\sigma )\prod_{l=1}^N \chi_{\sigma (k_l )} (x_l ), ~~~~~~ (3) $$

where $S_N$ is the group of permutations in $[1,N]$ , $\textrm{sgn}(\sigma )$ is the signature of the permutation $\sigma$ , and we have introduced the short-hand notation ${\vec{k}}$ $ {= (k_{1} ,k_{2}, \ldots, k_{N} )}$ . The CI matrix is then built by calculating the matrix elements ${CI}_{{\vec{k}} {\vec{l}} } $ $=\langle$ $\chi_{{\vec{k}} }$ $ |\hat{\mathcal{H}} |$ $\chi_{{\vec{l}} }$ $ \rangle$ for all the configuration states $\lbrace $ $\vec{k}$ $ \rbrace$ in the expansion basis. Upon diagonalization of the CI matrix, one gets the expressions for the ground and excited states of the Hamiltonian operator of equation (1) in the vector space spanned by the configuration-state basis.

The QMol-grid package uses a restricted basis of spin orbitals $\chi_k (x)=\phi_{k^{\prime } } (r)\gamma (\omega )$ , where up and down spin channels share the same spatial-orbital basis set $\lbrace \phi_k \rbrace_k$ and $\gamma$ is the spin function. An arbitrary set of orthonormal spatial orbitals can be used (i.e., the CI calculations do not rely on Hartree-Fock orbitals). It also provides support for calculating the dipole-coupling matrix elements $\langle $ $\chi_{{\vec{k}} }$ $ |\sum_{k=1}^N r_k |$ $\chi_{{\vec{l}} } $ $ \rangle$ , e.g., used for laser-driven time-dependent CI calculations.

• QMol_CI_conv performs CI calculations using an explicit convolution scheme to calculate the electron-electron repulsion integrals.

A preliminary step to CI calculations in the QMol-grid package typically involve running a density-functional theory (DFT) or Hartree-Fock (HF) calculation to determine the spatial-orbital basis to be used in the CI.

References

[Visentin 2025] G. Visentin and F. Mauger, "Configuration-interaction calculations with density-functional theory molecular orbitals for modeling valence- and core-excited states in molecules," arXiv:2509.08245 (2025).