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Efficient Random Phase Approximation for Diradicals

Reza G. Shirazi, Vladimir V. Rybkin, Michael Marthaler, Dmitry S. Golubev

TL;DR

This work introduces a static direct random phase approximation (RPA) treatment to renormalize the parameters of a simple two-orbital model for diradicals, capturing the screening effect of doubly occupied and empty environment orbitals on the diradical pair. The authors derive analytical expressions for the singlet-triplet splitting $\Delta E_{ST}$ within a renormalized six-state two-orbital framework and define how RPA yields screened Coulomb integrals $\tilde{h}_{psqr}$ that feed into the model. Benchmarking against high-level CASSCF/NEVPT2 results for ten molecules shows that the RPA-corrected model significantly improves predictions for diradicals with small gaps, with the triplet-density implementation performing best and reducing the average error to around a few kcal/mol after excluding outliers. The approach offers a cheap, physically transparent way to extend the two-orbital diradical model, with potential applicability to other strongly correlated systems such as double quantum dots.

Abstract

We apply the analytically solvable model of two electrons in two orbitals to diradical molecules, characterized by two unpaired electrons. The effect of the doubly occupied and empty orbitals is taken into account by means of random phase approximation (RPA). We show that in the static limit the direct RPA leads to the renormalization of the parameters of the two-orbital model. We test our model by comparing its predictions for the singlet-triplet splitting with the results from multi-reference CASSCF and NEVPT2 simulations for a set of ten molecules. We find that, for the whole set, the average relative difference between the singlet-triplet gaps predicted by the RPA-corrected two-orbital model and by NEVPT2 is about 40%. For the five molecules with the smallest singlet-triplet splitting the accuracy is better than 20%.

Efficient Random Phase Approximation for Diradicals

TL;DR

This work introduces a static direct random phase approximation (RPA) treatment to renormalize the parameters of a simple two-orbital model for diradicals, capturing the screening effect of doubly occupied and empty environment orbitals on the diradical pair. The authors derive analytical expressions for the singlet-triplet splitting within a renormalized six-state two-orbital framework and define how RPA yields screened Coulomb integrals that feed into the model. Benchmarking against high-level CASSCF/NEVPT2 results for ten molecules shows that the RPA-corrected model significantly improves predictions for diradicals with small gaps, with the triplet-density implementation performing best and reducing the average error to around a few kcal/mol after excluding outliers. The approach offers a cheap, physically transparent way to extend the two-orbital diradical model, with potential applicability to other strongly correlated systems such as double quantum dots.

Abstract

We apply the analytically solvable model of two electrons in two orbitals to diradical molecules, characterized by two unpaired electrons. The effect of the doubly occupied and empty orbitals is taken into account by means of random phase approximation (RPA). We show that in the static limit the direct RPA leads to the renormalization of the parameters of the two-orbital model. We test our model by comparing its predictions for the singlet-triplet splitting with the results from multi-reference CASSCF and NEVPT2 simulations for a set of ten molecules. We find that, for the whole set, the average relative difference between the singlet-triplet gaps predicted by the RPA-corrected two-orbital model and by NEVPT2 is about 40%. For the five molecules with the smallest singlet-triplet splitting the accuracy is better than 20%.
Paper Structure (13 sections, 78 equations, 12 figures, 1 table)

This paper contains 13 sections, 78 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. Theory
  3. System Hamiltonian
  4. Simple two-orbital model
  5. Direct RPA approximation
  6. Computational details
  7. Molecular test set
  8. Quantum-chemical calculations
  9. Results and discussion
  10. Conclusion
  11. ACKNOWLEDGMENTS
  12. RPA Hamiltonian (\ref{['HRPA']}) and RPA in quantum chemistry
  13. Static limit of direct RPA

Figures (12)

  • Figure 1: Ten molecules considered in our study: (1) $p-$benzyne, (2) $m-$benzyne, (3) $o-$benzyne, (4) 2,5-didehydropyridinium cation (DDP-1) Sheng, (5) 6-cyano-2,5-didehydropyridinium cation (DDP2-2) Sheng, (6) trimethylethylene (TME), (7) cyclopentane cation (CPC), (8) phenylnitrene (PN), (9) phenylcarbene (PC), (10) trimethylenemethane (TMM).
  • Figure 2: Errors of considered methods in $E_{ST}$ with respect to the NEVPT2 reference for the molecular test set.
  • Figure 3: $o$-benzyne.
  • Figure 4: $m$-benzyne.
  • Figure 5: $p$-benzyne.
  • ...and 7 more figures