Orbital Optimization and Neural-Network-Assisted Configuration Interaction Calculations of Rydberg States

TL;DR

Rydberg states challenge traditional electronic-structure methods due to their diffuse electron density. The authors combine state-specific orbital optimization in a plane-wave Hartree–Fock framework with neural-network–assisted selective CI (NNCI) to enable accurate and scalable excited-state calculations. They demonstrate near-$TBE$ accuracy for the $2s$ Rydberg state of H$_2$ in full CI and show that NNCI yields excitation energies for NH$_3$ and H$_2$O that closely match experimental values and high-level benchmarks, using far fewer determinants. The approach leverages diffuse-tail–friendly plane-wave orbitals and targeted determinant selection to extend high-accuracy calculations to challenging Rydberg states and potentially other long-range excitations in larger systems.

Abstract

Rydberg excited states of molecules pose a challenge for electronic structure calculations because of their highly diffuse electron distribution. Even large and elaborate atomic basis sets tend to underrepresent the long-range tail, overly confining the Rydberg state. An approach is presented where the molecular orbitals are variationally optimized for the excited state using a plane wave basis set in Hartree-Fock calculations, followed by configuration interaction calculations on the resulting reference. Using excited state optimized plane wave orbitals greatly enhances the convergence of the many-body calculation, as illustrated by a full configuration interaction calculation of the 2s Rydberg state of H$_2$. A neural-network-based selective configuration interaction approach is then applied to calculations of the 3s, 3p$_x$ and 3p$_y$ states of H$_2$O and the 3s and 3p$_z$ states of NH$_3$. The obtained values of excitation energy are in close agreement with experimental measurements as well as previous many-body calculations based on sufficiently diffuse atomic basis sets. Previously reported high-level calculations limited to atomic basis sets lacking extra diffuse functions, such as aug-cc-pVTZ, give significantly higher estimates due to confinement of the Rydberg states.

Figures (6)

• Figure 1: Top left: Orbital occupation in the spin restricted Hartree-Fock calculation of the ground and $2s$ Rydberg excited states of the H2 molecule. Bottom left and right: Comparison of the $2s$ Rydberg orbital obtained in three different calculations, displayed as contour maps (red and blue lines corresponding to regions of positive and negative amplitude, respctively) and amplitude along the molecular axis. In a ground state Hartree-Fock calculation initialized using the cc-pVTZ (blue solid line) or aug-cc-pVTZ (blue dashed line) basis set, the $2s$ orbital has shorter tails than in an excited state calculation where the occupied orbitals are variationally optimized for the $2s$ Rydberg state (orange curve). In each case, the occupied orbitals are represented by a plane wave basis, but the unoccupied orbitals are largely determined by the atomic basis set used in the initialization. Circles mark the positions of the protons.
• Figure 2: Energy of $2s$ Rydberg state of H2 as a function of H-H distance from full CI calculations using different sets of orbitals, with respect to the ground state energy minimum. The occupied orbitals are variationally optimized in Hartree-Fock plane wave (PW) calculations for the ground ($(1\sigma_g)^2$, downward triangles), or $2s$ state ($1\sigma_g 2s$, upward triangles), while the unoccupied orbitals are represented with the cc-pVTZ or aug-cc-pVTZ atomic basis sets used in the initialization. The size of triangles reflects the overlap of the full CI many-body state with the $1\sigma_g 2s$ two Slater determinant wave function. Inset: Expanded scale around the minimum. Theoretical best estimates (TBE) are from Refs.Kolos1969 (KW, lower branch) and Wolniewicz1993 (WD, upper branch).
• Figure 3: Excitation energy for $3s$ and $3p_z$ Rydberg states of the NH3 molecule calculated with NNCI using 52 MOs, formed initially from the aug-cc-pVTZ basis set, but then optimized variationally for the occupied ones in plane wave based Hartree-Fock calculations, either for the ground state (NNCI aug-cc-pVTZ) or for the target Rydberg state (NNCI PW opt). For comparison, experimental estimates from Refs.Skerbele65 and Arfa91 are shown as well as extrapolated full CI calculations (exFCI) from Ref.Mountaineering2018 based on the aug-cc-pVQZ basis set.
• Figure 4: Convergence of the excitation energy for the $3s$ and $3p_z$ Rydberg states of NH3 with respect to the number of Slater determinants included in the NNCI calculation, in all cases generated from 52 MOs formed initially from the aug-cc-pVTZ basis set. The occupied orbitals are optimized in plane wave based Hartree-Fock calculations either for the ground state (orange) or the target excited state (green).
• Figure 5: Comparison of NNCI calculated excitation energy for the $3s$, $3p_y$ and $3p_x$ Rydberg states of H2O, based on 52 MOs initially formed from the aug-cc-pVTZ basis set but then optimized variationally for the target Rydberg state (NNCI PW opt), with experimentally measured Chutjian75 values, illustrating good agreement. Results of several other theoretical calculations are also shown: extrapolated full CI calculations (exFCI) based on the aug-cc-pVQZ basis set Mountaineering2018, EOM-CCSDTQ calculations in the complete basis set (CBS) limit Mountaineering2021, available only for two of the states, as well as results of multireference GMS SU CCSD calculations H2O_cc_2006 based on either a cc-pVTZ+diff basis set that includes extra diffuse functions, or the standard aug-cc-pVTZ basis set. The latter overly confines the $3p_x$ leading to a large overestimate of the excitation energy, by more than 1 eV.