This large solvent effect is in good accord with the results obtained previously by Gao and Alhambra,33 even if in their work a completely different model is used to represent the solvent molecules i.
In that paper, the authors find, for the first term of series I, a decrease of the single CC bond length of the order of 0. The same conclusions can be reached from a different way, by taking into account the changes on the net charges of the various chemical groups present in the molecule, when passing from gas phase to solution. In Table 1. T This behavior once again shows that in solution the cyanine-like structure assumes a much more important role and becomes the main one for the series II.
The electron-migration towards the A group is reflected also in the net charges of the intermediate CH groups which all increase in magnitude passing from vacuum to aqueous solution 1. IR spectra: intensities and frequencies The definition of vibrational intensities for molecules in solution requires some modifications with respect to the isolated system.
The presence of the solvent in fact modifies not only the solute charge distribution but also the probing electric field acting on the molecule. As we shall see in the following sections, this is a problem of general occurrence when an external field interacts with a molecule in a condensed phase historically it is known as 'local field effect'.
Assuming the harmonic approximation as valid, vibrational frequencies for a solvated molecule can be expressed in terms of the second derivatives of the proper energy functional, the free energy Q, with respect to the 30 R. Tomasi normal coordinates i. Starting from eq. In the previous sections we have resumed the procedures to get the all the required free energy derivatives appearing in eqs. The basic point is how to describe the solvent response to solute vibrations, or more precisely, if, and how, to introduce solvent polarization under non-equilibrium conditions.
The concept of solute-solvent non-equilibrium system involves many different aspects related to the real nature of the solvent polarization. This latter can be decomposed into different contributions related to the various degrees of freedom of the solvent molecules. The motions associated to these degrees of freedom involve different timescales: suffice it to recall the difference orders of magnitude in the relaxation times typical of the translational and electronic motions.
In the Computational Modelling of the Solvent Effects on Molecular Properties 31 common practice, such contributions are grouped in two terms only: one term accounts for all the motions which are slower than those involved in the physical phenomenon under examination, the other includes the faster contributions.
The next assumption usually exploited is that only the latter are instantaneously equilibrated to the momentary molecule charge distribution whereas the former cannot readjust, giving rise to a non-equilibrium solvent-solute system. This partition and the following non-equilibrium approach has been originally formulated and commonly applied to electronic processes for example solute electronic transitions as well as to the evaluation of solute response to external oscillating fields see next sections.
In the case of vibrations the slow term will contain the contributions arising from the motions of the solvent molecules as a whole translations and rotations , whereas the fast term will take into account the internal molecular motions electronic and vibrational.
The solvents of which we have modeled the effect are: cyclohexane eye , carbon tetrachloride CCI4 , benzene CeHe , 1,2-dichloroethane 1,2-dce and acetonitrile acn. In order to evaluate the differences in the approaches, we have performed calculations exploiting both equilibrium and non-equilibrium models within the PCM-IEF framework.
All the computed frequencies and IR intensities we will report in the following are obtained in the harmonic approximation, no anharmonic effects have been considered. The geometries of all the systems have been re-optimized in each phase. Tomasi Table 1. Let us see what happens to the frequency values in passing from gas phase to solution, as evaluated by equilibrium and non-equilibrium models and experimentally.
To make the comparison easy, we report in Table 1. Equilibrium and non-equilibrium S values are instead very similar in the case of non-polar solvents: this is not surprising since static and optical dielectric constants are very similar in such cases. Passing to IR intensities, we first note that both equilibrium and nonequilibrium models give the correct trend in passing from apolax to polar solvents; both the two sets of computed results and the experimental data show a net increment of the intensities from vacuo to polar solvents.
For a more detailed analysis we prefer to shift from absolute values to differences as done in the previous analysis on frequencies; in Table 1. Table 1. In particular, the nonequilibrium results present a by far better agreement with experiments, leading the absolute value to be within the experimental range.
On the contrary, very large discrepancies are obtained between equilibrium and experimental data. Once again, in apolar solvents the two models give very similar results and both well agree with experiments.
The results we have presented are sufficient to show that the computational procedure gives frequencies and intensities comparable with the experimental ones. Tomasi does not significantly change the frequency: on the contrary a better behavior of the non-equilibrium model turns out evident when looking at intensities in polar solvents.
Electric dipole polarizabilities The dipole hyper polarizabilities of a molecule in vacuo are defined in terms of a Taylor expansion of the dipole moment when the molecule is subjected to an external homogeneous electric field; alternatively, they may be expressed in terms of the corresponding Taylor expansion of the energy of the molecule.
In this case the external field corresponds to the perturbing field acting locally on the molecular solute and the expansion of the energy is substituted by the expansion of the free energy function G. These elements constitute the essential part of the linear and nonlinear optics, a subject for which there is a remarkable interest to know the influence of solvation effects.
The molecular response to an electric field regards its whole charge distribution, electron and nuclei. We may introduce also for molecules in solution the usual partition of the theoretical chemistry into electric and nuclear parts. Electronic contribution The electronic contribution can be computed using two derivative schemes involving quantum mechanical calculations of the free energy or, alternatively, of the dipole moment followed by derivatives with respect to the perturbing external field, computed at zero intensity.
To obtain time-dependent properties, we have to pass from the basic model to an extended version in which the solute is described thorough a time-dependent Schrodinger equations.
In this extended version of the model we have also to introduce the time-dependence of the solvent polarization, which is expressed in terms of a Fourier expansion and requires the whole frequency-spectrum of the dielectric permittivity e w of the solvent. Tomasi of the components of the external field. Electronic hyperpolarizabilities of solvated system: urea in water In this section, we shall summarize a study on dynamical polarizability and hyperpolarizability tensors a, ft, and j of urea in vacuo and in water we have published on the Journal of Molecular Structure Theochem.
The solvent model is the Polarizable Continuum Model PCM whereas vibrational averaging of the optical properties along the C-0 stretching coordinate has been obtained by the DiNa package44 both in vacuo and in solution.
Tomasi The geometry of the urea molecule has been optimized at the Restricted Hartree-Fock level using the standard polarized triple zeta basis set TZP. The geometry optimization has been performed both in vacuo and in aqueous solution. The prediction of hyper polarizabilities depends upon products of matrix elements of the electron position operator f. Consequently, unlike the molecule's energy, which primarily depends upon inverse power of f, any study of hyper polarizabilities must allow for an adequate description of the more diffuse regions of the molecule's wave function.
Basis sets that are usually found to be adequate for a dipole moment require further extension to adequately account for the polarizability tensor, a, and even further extensions for the first hyperpolarizability, 0, or for the second hyperpolarizability, 7.
There are also other factors to be considered. Vibrational corrections originating from the coupling between the electronic and the nuclear motions are sometimes important in obtaining predictive accuracy for molecular hyper polarizabilities. Such corrections are actually of two kinds: the first one simply involves averaging of the observables over the accessible vibrational states, the second one involves contributions to the hyper polarizability from vibronic intermediate states.
The latter is described in the next section; here we shall consider the former contribution only. The averaging has been obtained by a variational procedure using spline basis functions.
It is worth noting that in aqueous solution the vibrational correction to the j3z value is markedly greater than in gas phase. Thus it is advisable to consider both the solvent and the vibrational effects, and their possible coupling, to get reliable values of molecular nonlinear properties. Without the inclusion of the frequency dependence, the various experimentally distinct nonlinear processes would coincide in the static limit.
Moreover, dispersion effects can have a relatively large effect on the observed hyper polarizability, and predictive calculations will frequently require more accurate dispersion values than those estimated.
This would be particularly true if the exciting frequency ui becomes close to a resonance. A knowledge of the frequency dependence is also important in identifying possible resonance enhanced effects that might make a particular molecule suitable for a specific application.
As said before, in the presence of a dielectric medium the analysis is further complicated by effects related to the solvent dynamical response to the external oscillating field. One way which allows to take into account the latter aspect is to use a dielectric permittivity varying with the exciting frequency. In the case of water as solvent the dielectric permittivity e w may be described with the help of the Debye formula 1. The dispersion behavior of the single electric properties will be separately analyzed in the following paragraphs, a Polarizability: a.
This effect can be related to the shape of the e w function. By applying Debye formula it is easy to see that, at the frequency considered in our calculations, the value of e w is practically equal to e oo.
If one considers the 'sum-over-states' method for the calculation of polarizabilities with these values in minds, then it is easy to give a qualitative explanation of the behavior indicated above. In fact, when the solvent response function is described by e oo the actual stabilization of the excited states will be less than in a situation where the same response depends on e 0 , hence smaller it will be the correspondent a value.
The nuclear relaxation is the dominant contribution and can be computed in two ways: by perturbation theory or by finite field approximation. We shall limit ourselves to the perturbation theory.
In the BO approximation, the perturbation theory implies the evaluation of the first terms of the expansion of the free energy function and of the properties with respect to the nuclear coordinates and to the external field. At level of double harmonicity electric and mechanical only linear 44 R. Tomasi term are considered in the expansion of the properties and only quadratic terms are considered in the expansion of the vibrational potential.
Qa denotes the normal mode associated to ua and each partial derivative is evaluated at the equilibrium geometry of the solvated system.
The vibrational hyperpolarizabilities of push-pull systems in solution In this section we report a second extract of the study we have published on the Journal of the American Chemical Society32 about solvent effects on electronic and vibrational components of linear and nonlinear optical properties of Donor-Acceptor polyenes.
On the contrary, no comparisons with data from literature can be made for solvated systems, as never studied before. Anyway, our results show that the solvent effects induce either Computational Modelling of the Solvent Effects on Molecular Properties 45 Table 1.
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