2601002829
  • Open Access
  • Article

Excitation Energies in a Polarizable Environment: A Comparison of State-Averaged and Linear-Response CASSCF/AMOEBA Formulations

  • Tommaso Nottoli,   
  • Lorenzo Lapi,   
  • Giacomo Londi,   
  • Filippo Lipparini,   
  • Benedetta Mennucci *

Received: 10 Dec 2025 | Revised: 09 Jan 2026 | Accepted: 13 Jan 2026 | Published: 05 Mar 2026

Abstract

Accurate modeling of excitation energies in complex environments is essential for interpreting spectroscopic signatures. A central challenge which is relevant to many systems, is achieving a balanced description of excited states with distinct electronic characters, such as locally excited (LE) and charge transfer (CT) states, while consistently incorporating environmental effects. Here, we compare the linear-response (LR) and state-averaged (SA) formulations of CASSCF when coupled to the polarizable AMOEBA force field. Using the LE and CT excitations of the substrate–cofactor pair in the photoenzyme CvFAP as a test case, we show that LR- and SA-CASSCF offer complementary strengths. LR-CASSCF provides an unbiased mapping of the excitation manifold, whereas SA-CASSCF can refine selected states to obtain a more quantitative description. This combined strategy offers a robust framework for interpreting spectroscopic features of embedded chromophores, especially in systems where CT character and environmental polarization play a decisive role.

References 

  • 1.

    Guido, C.A.; Rosa, M.; Cammi, R.; et al. An open quantum system theory for polarizable continuum models. J. Chem. Phys. 2020, 152, 174114.

  • 2.

    Corni, S.; Cammi, R.; Mennucci, B.; et al. Electronic excitation energies of molecules in solution within continuum solvation models: Investigating the discrepancy between state-specific and linear-response methods. J. Chem. Phys. 2005, 123, 134512.

  • 3.

    Lunkenheimer, B.; K¨ohn, A. Solvent Effects on Electronically Excited States Using the Conductor-Like Screening Model and the Second-Order Correlated Method ADC(2). J. Chem. Theory Comput. 2013, 9, 977–994.

  • 4.

    Guido, C.A.; Jacquemin, D.; Adamo, C.; et al. Electronic Excitations in Solution: The Interplay between State Specific Approaches and a TD-DFT Description. J. Chem. Theory Comput. 2015, 11, 5782–5790.

  • 5.

    Schwabe, T. General theory for environmental effects on (vertical) electronic excitation energies. J. Chem. Phys. 2016, 145, 154105.

  • 6.

    Guareschi, R.; Valsson, O.; Curutchet, C.; et al. Electrostatic versus Resonance Interactions in Photoreceptor Proteins: The Case of Rhodopsin. J. Phys. Chem. Lett. 2016, 7, 4547–4553.

  • 7.

    Nifosı, R.; Mennucci, B.; Filippi, C. The key to the yellow-to-cyan tuning in the green fluorescent protein family is polarisation. Phys. Chem. Chem. Phys. 2019, 21, 18988–18998.

  • 8.

    Roos, B.O.; Taylor, P.R.; Siegbahn, P.E.M. A complete active space SCF method (CASSCF) using a density matrix formulated super-CI approach. Chem. Phys. 1980, 48, 157–173.

  • 9.

    Roos, B.O. The complete active space self-consistent field method and its applications in electronic structure calculations. Adv. Chem. Phys. 1987, 69, 399–445.

  • 10.

    Werner, H.J. Matrix-formulated direct multiconfiguration self-consistent field and multiconfiguration reference configurationinteraction methods. Adv. Chem. Phys. 2009, 69, 1–62.

  • 11.

    Olsen, J. The CASSCF method: A perspective and commentary. Int. J. Quant. Chem. 2011, 111, 3267–3272.

  • 12.

    Lischka, H.; Nachtigallova, D.; Aquino, A.J.; et al. Multireference approaches for excited states of molecules. Chem. Rev. 2018, 118, 7293–7361.

  • 13.

    Beebe, N.H.; Linderberg, J. Simplifications in the generation and transformation of two-electron integrals in molecular calculations. Int. J. Quant. Chem. 1977, 12, 683–705.

  • 14.

    Koch, H.; Sanchez de Meras, A.; Pedersen, T.B.; et al. Reduced scaling in electronic structure calculations using Cholesky decompositions. J. Chem. Phys. 2003, 118, 9481–9484.

  • 15.

    Pedersen, T.B.; Lehtola, S.; Fdez. Galv´an, I.; et al. The versatility of the Cholesky decomposition in electronic structure theory. Wiley Interdiscip. Rev. Comput. Mol. Sci. 2024, 14, e1692.

  • 16.

    Aquilante, F.; Boman, L.; Bostr¨om, J.; et al. Cholesky decomposition techniques in electronic structure theory. In Linear-Scaling Techniques in Computational Chemistry and Physics: Methods and Applications; Springer: Dordrecht, The
    Netherlands, 2011; pp. 301–343.

  • 17.

    Folkestad, S.D.; Kjønstad, E.F.; Koch, H. An efficient algorithm for Cholesky decomposition of electron repulsion integrals. J. Chem. Phys. 2019, 150, 194112.

  • 18.

    Zhang, T.; Liu, X.; Valeev, E.F.; et al. Toward the minimal floating operation count Cholesky decomposition of electron repulsion integrals. J. Phys. Chem. A 2021, 125, 4258–4265.

  • 19.

    Stanton, J.F.; Gauss, J.; Cheng, L.; et al. CFOUR, Coupled-Cluster Techniques for Computational Chemistry, a Quantum-Chemical Program Package. Available online: http://www.cfour.de (accessed on 2 February 2026).

  • 20.

    Matthews, D.A.; Cheng, L.; Harding, M.E.; et al. Coupled-cluster techniques for computational chemistry: The CFOUR program package. J. Chem. Phys. 2020, 152, 214108.

  • 21.

    Lipparini, F.; Gauss, J. Cost-effective treatment of scalar relativistic effects for multireference systems: A CASSCF implementation based on the spin-free Dirac–Coulomb Hamiltonian. J. Chem. Theor. Comput. 2016, 12, 4284–4295.

  • 22.

    Nottoli, T.; Gauss, J.; Lipparini, F. Second-order CASSCF algorithm with the Cholesky decomposition of the two-electron integrals. J. Chem. Theory Comput. 2021, 17, 6819–6831.

  • 23.

    Nottoli, T.; Gauss, J.; Lipparini, F. A black-box, general purpose quadratic self-consistent field code with and without Cholesky decomposition of the two-electron integrals. Mol. Phys. 2021, 119, e1974590.

  • 24.

    Nottoli, T.; Burger, S.; Stopkowicz, S.; et al. Computation of NMR shieldings at the CASSCF level using gauge-including atomic orbitals and Cholesky decomposition. J. Chem. Phys. 2022, 157, 084122.

  • 25.

    Nottoli, T.; Lapi, L.; Alessandro, R.; et al. An Efficient and Robust Implementation of CASSCF Linear Response Theory. J. Phys. Chem. A 2025, 129, 8441–8452.

  • 26.

    Nottoli, T.; Giann`ı, I.; Levitt, A.; et al. A robust, open-source implementation of the locally optimal block preconditioned conjugate gradient for large eigenvalue problems in quantum chemistry. Theor. Chem. Acc. 2023, 142, 69.

  • 27.

    Alessandro, R.; Giann`ı, I.; Pes, F.; et al. Linear response equations revisited: A simple and efficient iterative algorithm. J. Chem. Theory Comput. 2023, 19, 9025–9031.

  • 28.

    Cossi, M.; Barone, V.; Robb, M.A. A direct procedure for the evaluation of solvent effects in MC-SCF calculations. J. Chem. Phys. 1999, 111, 5295–5302.

  • 29.

    Hedeg˚ard, E.D.; List, N.H.; Jensen, H.J.A.; et al. The multi-configuration self-consistent field method within a polarizable embedded framework. J. Chem. Phys. 2013, 139, 044101.

  • 30.

    Li, Q.; Mennucci, B.; Robb, M.A.; et al. Polarizable QM/MM multiconfiguration self-consistent field approach with statespecific corrections: Environment effects on cytosine absorption spectrum. J. Chem. Theor. Comput. 2015, 11, 1674–1682.

  • 31.

    Liu, H.; Jenkins, A.J.; Wildman, A.; et al. Time-Dependent Complete Active Space Embedded in a Polarizable Force Field. J. Chem. Theor. Comput. 2019, 15, 1633–1641.

  • 32.

    Song, C. State averaged CASSCF in AMOEBA polarizable water model for simulating nonadiabatic molecular dynamics with nonequilibrium solvation effects. J. Chem. Phys. 2023, 158, 014101.

  • 33.

    Song, C.; Wang, L.P. A Polarizable QM/MM Model That Combines the State-Averaged CASSCF and AMOEBA Force Field for Photoreactions in Proteins. J. Chem. Theory Comput. 2024, 20, 6632–6651.

  • 34.

    Sepali, C.; Goletto, L.; Lafiosca, P.; et al. Fully Polarizable Multiconfigurational Self-Consistent Field/Fluctuating Charges Approach. J. Chem. Theory Comput. 2024, 20, 9954–9967.

  • 35.

    Nottoli, T.; Bondanza, M.; Lipparini, F.; et al. A Polarizable CASSCF/MM Approach Using the Interface Between OpenMMPol Library and Cfour. J. Comput. Chem. 2025, 46, e27550.

  • 36.

    Mazza, F.; Trinari, M.; Sepali, C.; et al. Analytical Nuclear Gradients for the Multiconfigurational Self-Consistent Field Method Coupled with the Polarizable Fluctuating Charges Model. arXiv 2025, arXiv:2511.10295.

  • 37.

    Ponder, J.W.; Wu, C.; Ren, P.; et al. Current status of the AMOEBA polarizable force field. J. Phys. Chem. B 2010, 114, 2549–2564.

  • 38.

    Bondanza, M.; Nottoli, T.; Nottoli, M.; et al. The OpenMMPol library for polarizable QM/MM calculations of properties and dynamics. J. Chem. Phys. 2024, 160, 134106.

  • 39.

    Sorigue, D.; Legeret, B.; Cuine, S.; et al. An algal photoenzyme converts fatty acids to hydrocarbons. Science 2017, 357, 903–907.

  • 40.

    Sorigue, D.; Hadjidemetriou, K.; Blangy, S.; et al. Mechanism and dynamics of fatty acid photodecarboxylase. Science 2021, 372, eabd5687.

  • 41.

    Nottoli, M.; Lipparini, F. General formulation of polarizable embedding models and of their coupling. J. Chem. Phys. 2020, 153, 224108.

  • 42.

    Jensen, H.J.; Jørgensen, P. A direct approach to second-order MCSCF calculations using a norm extended optimization scheme. J. Chem. Phys. 1984, 80, 1204–1214.

  • 43.

    Lipparini, F.; Lagard`ere, L.; Stamm, B.; et al. Scalable Evaluation of Polarization Energy and Associated Forces in Polarizable Molecular Dynamics: I. Toward Massively Parallel Direct Space Computations. J. Chem. Theory Comput. 2014, 10, 1638–1651.

  • 44.

    Greengard, L.; Rokhlin, V. A fast algorithm for particle simulations. J. Comput. Phys. 1987, 73, 325–348.

  • 45.

    Caprasecca, S.; Jurinovich, S.; Lagard`ere, L.; et al. Achieving Linear Scaling in Computational Cost for a Fully Polarizable MM/Continuum Embedding. J. Chem. Theory Comput. 2015, 11, 694–704.

  • 46.

    Lipparini, F. General Linear Scaling Implementation of Polarizable Embedding Schemes. J. Chem. Theory Comput. 2019, 15, 4312–4317.

  • 47.

    Londi, G.; Salvadori, G.; Mazzeo, P.; et al. Protein-Driven Electron-Transfer Process in a Fatty Acid Photodecarboxylase. JACS Au 2025, 5, 158–168.

  • 48.

    Martin, R.L. Natural transition orbitals. J. Chem. Phys. 2003, 118, 4775–4777.

Share this article:
How to Cite
Nottoli, T.; Lapi, L.; Londi, G.; Lipparini, F.; Mennucci, B. Excitation Energies in a Polarizable Environment: A Comparison of State-Averaged and Linear-Response CASSCF/AMOEBA Formulations. Photochemistry and Spectroscopy 2026, 2 (1), 10. https://doi.org/10.53941/ps.2026.100010.
RIS
BibTex
Copyright & License
article copyright Image
Copyright (c) 2026 by the authors.