Open-shell frozen natural orbital approach for quantum eigensolvers

TL;DR

This work introduces a spin-restricted open-shell frozen natural orbital (FNO) workflow based on ZAPT2, designated ZAPT-FNO, to dramatically reduce the size of the virtual space in quantum eigensolvers while preserving accuracy. By constructing the open-shell second-order density matrix $P^{(2)}$, diagonalizing the virtual-virtual block to obtain natural occupations, and applying a $ riangle E_{ ext{FNO}}$ correction, the method achieves systematic convergence of total energies and singlet-triplet gaps with smaller active spaces, even in large augmented basis sets. Benchmarks on H$_2$O$_2$, O$_2$, CH$_2$, and the Ir(ppy)$_3$ complex demonstrate smoother, more reliable convergence than canonical MO selection, and accurate T$_1$–S$_0$ gaps approaching experimental values when combined with iQCC, illustrating a practical path to accurate, resource-efficient open-shell quantum simulations. The approach is particularly advantageous for large, diffuse basis sets, enabling chemically accurate predictions for sizable open-shell systems and excited states with realistic active spaces suitable for quantum hardware. Limitations arise in stretched geometries where perturbation theory breaks down, guiding when to rely on or avoid the $ riangle E_{ ext{FNO}}$ correction. Overall, ZAPT-FNO offers a robust framework for preparing compact, accurate active spaces for quantum simulations of complex open-shell and excited-state chemistry.

Abstract

We present an open-shell frozen natural orbital (FNO) approach, which utilizes the second-order Z-averaged perturbation theory (ZAPT2), to reduce the restricted opten-shell Hartree-Fock virtual space size with controllable accuracy. Our ZAPT2 frozen natural orbital (ZAPT-FNO) selection scheme significantly outperforms the canonical molecular orbital virtual space truncation scheme based on Hartree-Fock orbital energies, especially when using large multiple-polarized and augmented basis sets. We demonstrate that the ZAPT-FNO-selected virtual orbitals lead to a systematic convergence of the correlation energies, but more importantly to the singlet-triplet T$_1$-S$_ 0$ energy gaps with respect to the complete active space (CAS) [occupied + virtual] size. We confirm our findings by simulating T$_1$-S$_ 0$ gaps in H$_2$O$_2$ and O$_2$ molecules using the traditional complete active space configuration interaction (CASCI) approach, as well as in stretched CH$_2$, for which we also employed the iterative qubit coupled cluster (iQCC) method as a quantum eigensolver. Finally, we applied the iQCC method with ZAPT-FNO-selected active space to the phosphorescent Ir(ppy)$_3$ complex with 260 electrons, where extended basis sets are required to achieve chemical (ca. 1 m$E_h$) accuracy. In this case, CASCI results are not available; however, the iQCC-computed T$_1$-S$_ 0$ gaps show robust convergence with enlarging basis set and CAS size, approaching the experimental value. Thus, the ZAPT-FNO method is very promising for improving the accuracy of quantum chemical modelling in a resource-efficient manner, and opens the door to simulating open-shell states of large materials within realistic active space sizes and without compromising on basis-set quality.

Figures (6)

• Figure 1: The total correlation energy, $E_{\text{ZAPT}}^{(2)}$ computed at an increasing number of frozen virtuals for both the and approaches using the aug-cc-pVTZ basis set on the high spin triplet state of H2O2. The total correlation energy at an increasing number of frozen virtuals decreases rapidly for the orbitals, while a large amount of the correlation energy is recovered using the approach.
• Figure 2: The T$_1$-S$_0$ energy gap for H2O2 with the aug-cc-pVTZ basis set using the approach (green) and the approach (red dashed). The grey box indicates "chemical accuracy" of 1m. The $E^{\text{gap}}$ is within 1m at 20% frozen virtuals, while the approach does not converge to within 1m until the full set of MOs is used.
• Figure 3: (a) for the virtual orbitals of O2 with the aug-cc-pVTZ basis set, for the S$_0$ and T$_1$ states. The orbitals used for sizes of (8,8), (8,16), (8,22), and (8,30) are shown by horizontal dashed lines, indicating that all orbitals to the left of that line are included in the given active space using the approach. (b) Convergence of the T$_1$-S$_0$ energy gap as a function of the active space size for O2 with the aug-cc-pVTZ basis set.
• Figure 4: (a) Bond dissociation energy path of a breaking of a single C-H bond in the CH2 molecule (left). T$_1$-S$_0$ energy gap across the dissociation pathway for breaking a single C-H bond in CH2 (right). (b) Bond dissociation energy path of the symmetric stretching of both C-H bonds in the CH2 molecule (left). T$_1$-S$_0$ energy gap across the dissociation pathway for symmetric stretching of both C-H bonds in CH2 (right). The calculations are performed on the selected orbitals also used to produce the qubit Hamiltonian for the optimization. Chemical accuracy of 1m relative to the calculations is indicated by the grey box in the right panel for both (a) and (b). Dashed lines are shown to guide the eye.
• Figure 5: (a) Heatmap showing the orbital overlap $S_{\text{MO}}$ matrix between the coefficients and semicanonicalized coefficients for the first 20.0 virtual orbitals of the Ir(ppy)3 complex (labelled orbitals [range-phrase= --]2039 as these are the last 20.0 orbitals in the CAS(40,40) set). This matrix captures how much each orbital from the set overlaps with the corresponding index in the set. The $S_{\text{MO}}$ for orbital 21.0 is highlighted in a red box. (b) Orbital 21.0 from the set which is a Rydberg-like orbital, compared with orbital 21.0 from the set which is localized, and shows $s$ and $p$-like character on the carbon rings.