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Vertical Excitation Energies of Embedded Systems: The Vertical Excitation Model (VEM) within Polarizable QM/MM

Chiara Sepali, Piero Lafiosca, Linda Goletto, Tommaso Giovannini, Chiara Cappelli

Abstract

Polarizable Quantum Mechanics/Molecular Mechanics (QM/MM) approaches based on fluctuating charges and dipoles (QM/FQ(F$μ$)) are formulated within the state-specific Vertical Excitation Model (VEM) to compute vertical excitation energies of solvated systems. This methodology overcomes the limitations of the widely used Linear Response (LR) approach. While LR can capture the dynamic response of the solvent to the QM transition density, it neglects the solvent reorganization that follows solute relaxation upon electronic excitation. In contrast, the VEM framework explicitly accounts for this effect. Benchmark calculations of vertical excitation energies using QM/FQ(F$μ$) are reported for a representative set of solutes - acrolein, acetone, caffeine, p-nitroaniline, coumarin 153, doxorubicin, and betaine-30 - comparing VEM with LR, corrected LR (cLR), and cLR 2 schemes. The results reveal notable variations in solvent response depending on the character of the electronic transition and demonstrate that optimal accuracy can be achieved by selecting the most appropriate model for each specific system and excitation.

Vertical Excitation Energies of Embedded Systems: The Vertical Excitation Model (VEM) within Polarizable QM/MM

Abstract

Polarizable Quantum Mechanics/Molecular Mechanics (QM/MM) approaches based on fluctuating charges and dipoles (QM/FQ(F)) are formulated within the state-specific Vertical Excitation Model (VEM) to compute vertical excitation energies of solvated systems. This methodology overcomes the limitations of the widely used Linear Response (LR) approach. While LR can capture the dynamic response of the solvent to the QM transition density, it neglects the solvent reorganization that follows solute relaxation upon electronic excitation. In contrast, the VEM framework explicitly accounts for this effect. Benchmark calculations of vertical excitation energies using QM/FQ(F) are reported for a representative set of solutes - acrolein, acetone, caffeine, p-nitroaniline, coumarin 153, doxorubicin, and betaine-30 - comparing VEM with LR, corrected LR (cLR), and cLR 2 schemes. The results reveal notable variations in solvent response depending on the character of the electronic transition and demonstrate that optimal accuracy can be achieved by selecting the most appropriate model for each specific system and excitation.

Paper Structure

This paper contains 10 sections, 19 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Theory
  3. QM/FQ and QM/FQFmu polarizable embedding approaches
  4. QM/FQ and QM/FQFmu in the LR, cLR and cLR2 regimes
  5. Vertical Excitation Model formulation of QM/FQ and QM/FQFmu
  6. Computational details
  7. Results and Discussion
  8. Analysis of electronic transitions
  9. Excitation energies and solvatochromic shifts
  10. Conclusion and future perspectives

Figures (5)

  • Figure 2: HOMO (red and blue) and LUMO (yellow and green) orbitals involved in the studied electronic transitions of acrolein (ACRO), acetone (ACE), caffeine (CAFF), doxorubicin (DOXO), betaine-30 (BET), para-nitroaniline (PNA), and coumarin 153 (C153).
  • Figure 3: QM/FQF$\mu$ vertical excitation energies of (a) ACRO and (b) ACE, averaged over 200 snapshots. Data refer to the HOMO–LUMO $n \to \pi^*$ transition, under different solvent regimes (GSRF, LR, cLR, cLR$^2$, and VEM). The dashed line indicates the correction to the GSRF vertical excitation energy.
  • Figure 4: QM/FQF$\mu$ vertical excitation energies of (a) CAFF and (b) DOXO, averaged over 200 snapshots. Data refer to the HOMO–LUMO $\pi \to \pi^*$ transition, under different solvent regimes (GSRF, LR, cLR, cLR$^2$, and VEM). The dashed line indicates the correction to the GSRF vertical excitation energy.
  • Figure 5: QM/FQF$\mu$ vertical excitation energies of (a) BET, (b) PNA, and (c) C153 averaged over 200 snapshots for the HOMO–LUMO CT transitions. Data refer to different solvent regimes (GSRF, LR, cLR, cLR$^2$, and VEM). The dashed line indicates the correction to the GSRF vertical excitation energy.
  • Figure 6: QM/FQ$^{a,b}$ and QM/FQF$\mu$ solvatochromic shifts ($\omega_{model} - \omega_{vac}$) for the investigated molecules (ACRO, CAFF, ACE, DOXO, BET, PNA, C153) computed using different solvent models (GSRF, LR, cLR, cLR$^2$, VEM). Experimental solvatochromic shifts in water are taken from Refs. hayes1965solventbayliss1954solventbayliss1968solventcatalan2011solvatochromismrenge2009solvent for ACE, moskvin1966experimental for ACRO, reichardt2011solvents for BET, kovalenko2000femtosecond for PNA, and horng1995subpicosecond for C153 (in DMSO). Computational data are taken from Ref. rijal2022quantum for CAFF and Ref. nicoli2022assessing for DOXO.