Double Configuration Interaction Singles: Scalable and size-intensive approach for orbital relaxation in excited states and bond-dissociation

We present a novel theoretical scheme for orbital relaxation in configuration interaction singles (CIS) based on a perturbative treatment of its electronic Hessian, whose analytical derivation is also established in this work. The proposed method, which can be interpreted as a “CIS-then-CIS” scheme, variationally accounts for orbital relaxation in excited states, thus significantly reducing the overestimation of charge-transfer excitation energies commonly associated with standard CIS. Additionally, by incorporating de-excitation effects from CIS, we demonstrate that our approach effectively describes single bond dissociation. Notably, all these improvements are achieved at a mean-field cost, with the pre-factor further reduced with the efficient algorithm introduced here, while preserving the size-intensive property of CIS.

💡 Research Summary

The authors introduce a new method, Double Configuration Interaction Singles (DCIS), to address two long‑standing deficiencies of the conventional Configuration Interaction Singles (CIS) approach: the lack of orbital relaxation for excited states and the inability to describe bond‑breaking processes. The key idea is to treat the orbital rotation operator κ̂, which generates a unitary transformation of the reference determinant, only to first order (1 – κ̂) and then to apply a second CIS calculation on top of the already‑CIS‑optimized wavefunction. In this “CIS‑then‑CIS” picture the original CIS state is used as a reference, the orbital rotations are approximated linearly, and the resulting wavefunction contains both the original single excitations and the double excitations generated by the product of two single excitations (|Φ_ab^ij⟩).

To make the approach variational, the authors derive the full electronic Hessian of CIS with respect to both orbital rotation parameters (κ_ai) and CI coefficients (c_ai). The Hessian consists of three blocks: oo (orbital‑orbital), oc (orbital‑CI), and cc (CI‑CI). The oo block reduces to the familiar Hartree–Fock Hessian plus additional de‑excitation terms that are zero in HF but non‑zero in CIS, thereby incorporating a modest amount of electron correlation. The oc and cc blocks contain the coupling between orbital rotations and CI residuals. By truncating the Hessian to first order, the authors obtain a set of linear equations that can be solved without iterative non‑linear optimization, thus avoiding the convergence problems typical of full orbital‑optimized CIS.

The resulting DCIS equations can be cast as a generalized eigenvalue problem