Systematic improvement of molecular excited state calculations by inclusion of nuclear quantum motion: A mode-resolved picture and the effect of molecular size

Abstract

The energies of molecular excited states arise as solutions to the electronic Schrödinger equation and are often compared to experiment. At the same time, nuclear quantum motion is known to be important and to induce a redshift of excited state energies. However, it is thus far unclear whether incorporating nuclear quantum motion in molecular excited state calculations leads to a systematic improvement of their predictive accuracy, making further investigation necessary. Here, we present such an investigation by employing two first-principles methods for capturing the effect of quantum fluctuations on excited state energies, which we apply to the Thiel set of organic molecules. We show that accounting for zero-point motion leads to much improved agreement with experiment, compared to "static" calculations that only account for electronic effects, and the magnitude of the redshift can become as large as 1.36 eV. Moreover, we show that the effect of nuclear quantum motion on excited state energies largely depends on the molecular size, with smaller molecules exhibiting larger redshifts. Our methodology also makes it possible to analyze the contribution of individual vibrational normal modes to the redshift of excited state energies, and in several molecules, we identify a limited number of modes dominating this effect. Overall, our study provides a foundation for systematically quantifying the shift of excited state energies due to nuclear quantum motion and for understanding this effect at a microscopic level.

Mapping Charge-Transfer Excitations in Bacteriochlorophyll Dimers from First Principles

“…Excitations of different spatial localization and character are known to be affected in different ways by molecular vibrations [79]. Our goal here is to provide a mode-resolved picture of excitation energy renormalization in a BCL dimer due to thermally-activated vibrations, following earlier work by Hele et al [80]. For this purpose we started from the crystal structure of the special pair dimer SP and performed a full geometry optimization using the def2-TZVP basis set and B3LYP exchange-correlation functional.…”

Cyclo[18]carbon including zero-point motion: ground state, first singlet and triplet excitations, and hole transfer

“…The anharmonic effects that arise at room temperature can be captured by approaches such as classical Molecular Dynamics, however these would entirely miss the impact of nuclear quantum fluctuations, which is significant and even dominant for smaller molecules. 20,22 Ab initio Molecular Dynamics at finite temperatures have been implemented attempting to describe excited state properties, 43 optical spectroscopy 44 and excited state dynamics, 45 while, path integral molecular dynamics have been implemented attempting to describe nuclear quantum and anharmonic effects. 46,47 Finally, while our technique focuses on describing the effect of equilibrium nuclear configurations at a chosen temperature, excited state spectroscopy simulations have been implemented to provide information on the effect of vibrations that have been displaced due to electronic excitation, 48 or even on peculiarly displaced vibrational configurations that can lead to nonadiabatic transitions.…”

“…In our case of cyclo [18]carbon, the computational cost of the quadratic approximation is comparable to the Monte Carlo method described above (a comparison of the two methods in terms of computational cost can be found in ref. 22). Although the quadratic method is less accurate than the Monte Carlo method, since it involves an expansion of the observable and a finitedifference estimation of its second derivative, it has a significant advantage: in eqn (10), the correction to the observable is expressed as a sum over the vibrational modes, the contribution of each of which can be isolated, allowing for additional microscopic insights.…”

“…Although the quadratic method is less accurate than the Monte Carlo method, since it involves an expansion of the observable and a finite-difference estimation of its second derivative, it has a significant advantage: in eqn (10), the correction to the observable is expressed as a sum over the vibrational modes, the contribution of each of which can be isolated, allowing for additional microscopic insights. Moreover, it has been shown that, for the specific case of computing excited state energies at a temperature T , the quadratic approximation may be written in the equivalent form 22,56 where ω Eν denotes the frequency of vibrational mode ν on the excited state. In organic molecules, the excited state surface generally has a smaller curvature than the ground state one, due to contributions from anti-bonding molecular orbitals, giving ω ν > ω Eν in most cases.…”

“…19 The second one is a less exact, yet mode-resolved, method, called below the quadratic method. This approach has been successfully applied recently to calculate the exciton energies of solid state organic semiconductors, 20 and the Thiel 21 set of molecules plus anthracene, tetracene and pentacene, 22 leading to improved agreement with experiment. Also, some of us have recently employed RT-TDDFT to study hole transfer in linear cumulenic and polyynic chains.…”