Nonequilibrium solvent polarization

The free energy E( E) invested in creation of a nonequilibrium solvent polarization P can be expressed as a series in even powers of P with the two first terms as follows ... 

Anharmonic higher order terms gain importance for stronger solute-solvent couplings requiring 0 in Eq. [121]. The nonequilibrium solvent polarization can be considered as an ET reaction coordinate. The curvature of the corresponding free energy surface is... 

Classical and semi-classical theories of electron transfer provide quantitative models for determining the reaction pathway. Of particular importance is the theory of nonequilibrium solvent polarization based on the dielectric continuum model.5 From these theories Eqs. 

Figure 1. Nonequilibrium free energy as a function of instantaneous solvent polarization for the ground electronic state S0 and the excited state Si of an ideal probe. In this example the equilibrium solvent polarization in S, (X q) is larger than in S0 (Ar q) because the dipole moment is larger in Si than S0. Reprinted from Ref. 31 with permission, from J. Chem. Phys. 88, 2372 (1988). Copyright 1988, American Physical Society.

Here Fe(t) and Fg(t) are the time-dependent nonequilibrium Helmholtz free energies of the e and g states, respectively. The energy difference A U(t) can be replaced by a free energy difference due to the fact that the entropy is unchanged in a Franck-Condon transition [51]. Free energies in Eq. (3) can be represented [54] by a sum of an equilibrium value Fcq and an additional contribution related to nonequilibrium orientational polarization in the solvent. Thus for the free energy in the excited state Fe(t) we have... 

In the simplest picture of the nonequilibrium state, only a fraction of the solvent degrees of freedom is able to follow the quick change in the electronic structure of the solute, while the slow degrees of freedom take a longer time to equilibrate with the new state of the solute. More detailed descriptions of the time evolution of the solvent polarization have been reported [15] and similar results have also been recently achieved in the context of the PCM [13,14],... 

This analysis shows that in order to account properly for solvent polarity effects, a solvation model has to be characterized by a larger flexibility with respect to the same model for ground state phenomena. In particular, it should be possible to shift easily from an equilibrium to a nonequilibrium regime according to the specific phenomenon under scrutiny. In the following section, we will show that such a flexibility can be obtained in continuum models and generalized to QM descriptions of the electronic excitations. 

The motions associated with the degrees of freedom of the solvent molecules involve different time scales. In particular, typical vibration times being of the order of 10-14-10-12s, it is clear that the orientational component of the solvent polarization cannot instantaneously readjust to follow the oscillating solute , so that a nonequilibrium solute-solvent system has to be considered. 

The solvent polarization can be formally decomposed into different contributions each related to the various degrees of freedom of the solvent molecules. In common practice such contributions are grouped into two terms only [41,52] one term accounts for all the motions which are slower than those involved in the physical phenomenon under examination (the slow polarization), the other includes the faster contributions (the fast polarization). The next assumption usually exploited is that only the slow motions are instantaneously equilibrated to the momentary molecule charge distribution whereas the fast cannot readjust, giving rise to a nonequilibrium solvent-solute system. 

Of course, there is more to a chemical reaction than its rate constant the reaction path or mechanism is also of central interest. Once again, nonequilibrium solvation is crucial in describing this path. In an equilibrium solvation picture, the solvent polarization would remain equilibrated throughout the reaction course, but this assumption is rarely satisfied for an actual reaction path, because of the same considerations noted above for the rate constant. Indeed these nonequilibrium solvation effects can qualitatively change the character of the reaction path as compared with an equilibrium solvation image. Dielectric continuum dynamic descriptions thus have an important role to play here as well. Indeed, we will employ in this contribution the reaction path Hamiltonian formulation previously developed [48,49], which can be used to generate a reaction path which is the analog in solution of the well-known Fukui reaction path in the gas phase [50], The reaction path will be discussed for both reaction topics in this contribution. 

We have already mentioned in the Introduction (Section 3.7.1) the importance of conical intersections (CIs) in connection with excited electronic state dynamics of a photoexcited chromophore. Briefly, CIs act as photochemical funnels in the passage from the first excited S, state to the ground electronic state S0, allowing often ultrafast transition dynamics for this process. (They can also be involved in transitions between excited electronic states, not discussed here.) While most theoretical studies have focused on CIs for a chromophore in the gas phase (for a representative selection, see refs [16, 83-89], here our focus is on the influence of a condensed phase environment [4-9], In particular, as discussed below, there are important nonequilibrium solvation effects due to the lack of solvent polarization equilibration to the evolving charge distribution of the chromophore. 

Solvation Regime When dealing with solvent effects, it is to be kept in mind that absorption and emission are time-dependent processes. In this case (as ours), the time dependency of the spectroscopic phenomenon is not explicitly accounted for in the computational methodology, that is, time-independent methods are exploited. Moreover, the motions associated to the degrees of freedom of the solvent molecules involve different time scales. In the particular case of vibrational spectroscopy, typical vibration times being on the order of lO -lO s, it is clear that the orientational component of the solvent polarization cannot instantaneously readjust to follow the oscillating solute, so that a nonequilibrium solute-solvent system should in principle be considered. 

A second complication is the presence of two alternative approaches for computing the solvent response state-specific (SS) and linear-response (LR). Cammi et al. and Corni et alf examined the difference between the two for exact solute states in equilibrium and nonequilibrium regimes. In brief, the difference between the two formalisms stems from the Hartree partition of the solute-solvent wave function. In SS, the solvent polarization is evaluated through the interaction with the density of the solute in each particular electronic state (ground or... 

On the other hand, the election of the equilibrium solvent polarization to obtain the nonequilibrium values is a convenient way to represent the effect of the solvent delay, but clearly other solvent polarizations (associated with thermal fluctuations, for instance) that affect the solute dynamics are not taken into account. 

We conclude this discussion with some data demonstrating the importance of equilibrium versus nonequilibrium solvation for aqueous-phase ionization. Table 7 shows VDEs for several aqueous-phase nucleobases, nucleosides, and nucleotides, computed at the MP2 level using both equilibrium and nonequilibrium PCMs. In the former case, the parent and the ionized species are separately equilibrated to a solvent whose dielectric constant is = 78, whereas in the latter case the nonequilibrium version of TD-DFT -I- PCM is used to estimate a correction on electron detachment such that only = 1.78 is used to relax the solvent polarization. (See Ref. 309 for details.)... 

In this contribution, we describe and illustrate the latest generalizations and developments[1]-[3] of a theory of recent formulation[4]-[6] for the study of chemical reactions in solution. This theory combines the powerful interpretive framework of Valence Bond (VB) theory [7] — so well known to chemists — with a dielectric continuum description of the solvent. The latter includes the quantization of the solvent electronic polarization[5, 6] and also accounts for nonequilibrium solvation effects. Compared to earlier, related efforts[4]-[6], [8]-[10], the theory [l]-[3] includes the boundary conditions on the solute cavity in a fashion related to that of Tomasi[ll] for equilibrium problems, and can be applied to reaction systems which require more than two VB states for their description, namely bimolecular Sjy2 reactions ],[8](b),[12],[13] X + RY XR + Y, acid ionizations[8](a),[14] HA +B —> A + HB+, and Menschutkin reactions[7](b), among other reactions. Compared to the various reaction field theories in use[ll],[15]-[21] (some of which are discussed in the present volume), the theory is distinguished by its quantization of the solvent electronic polarization (which in general leads to deviations from a Self-consistent limiting behavior), the inclusion of nonequilibrium solvation — so important for chemical reactions, and the VB perspective. Further historical perspective and discussion of connections to other work may be found in Ref.[l],... 

Finally, the last two terms in G(2.12) account for the effects of the solvent orientational polarization in the nonequilibrium solvation. The matrix K0 is the inverse of the solvent orientational polarization interaction energy matrix I0 whose elements are defined, analogously to Ienm, by... 

Most of the theoretical works concerning dynamical aspects of chemical reactions are treated within the adiabatic approximation, which is based on the assumption that the solvent instantaneously adjusts itself to any change in the solute charge distribution. However, in certain conditions, such as sudden perturbations or long solvent relaxation times, the total polarization of the solvent is no longer equilibrated with the actual solute charge distribution and cannot be properly described by the adiabatic approximation. In such a case, the reacting system is better described by nonequilibrium dynamics. 

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