Polarizable Embedding: Modelling of Solvent Effects in UV/Visual Spectroscopy
Particularly in the field of spectroscopy there have always been close links between experimental and theoretical work. The latest developments in computational chemistry allow now for an increasing number of theoretical investigations to simulate the important aspect of solvent effects very accurately.
Everything started with the investigation of the hydrogen atom. The intensive efforts to understand the experimentally observed spectral lines led to the essential fundamentals of quantum mechanics. Bohr's model of the atom which resulted from this was a milestone on the way to our present understanding of atomic and molecular systems. Since then, there has always been a very fruitful exchange of information between experimentalists and theorists. Many experiments are impossible to interpret without a well-founded theoretical model. On the other hand, new and increasingly more precise measurements result in deeper insights into the atomic world.
For a long time now, the knowledge obtained from theoretical chemistry has not only enabled a qualitative understanding but has also enabled quantitative predictions. The rapid progress in computational chemistry in the fields of methodology and hardware has primarily contributed to this. In spite of this, the demands for computing power are high for present-day computer models. To keep the problems calculable approximations to the experimental reality have to introduced often. One of the most frequent simplifications is to neglect the solvent in which the measurement was made.
Of course, this assumption is only permissible to a limited extent. In general, a change in the spectroscopic properties due to the solvent is to be expected. If this phenomenon relates to the absorption of visible light, this is referred to as solvatochromism . An example of this is illustrated in Figure 1, which shows various solutions of the molecule betaine-30. Depending on the polarity of the solvent, completely different colour impessions are perceived. This effect can only be modelled if the influence of the medium is taken into account. Therefore, for the comprehensive understanding of spectroscopic experiments it is essential to consider the environment.
This does not only apply to basic questions of purely theoretical interest. It can also be decisive for the examination of applications. An example is the simulation of new dyes for polymer-based solar cells. New dyes are necessary in order to ensure the best possible utilization of white sunlight for the generation of electricity. Computers are an obvious choice as economical tools for a rational design in this context. However, in order to make reliable predictions of the colour characteristics, the remaining environment in the solar cell must be considered.
Quantum Mechanical Modelling of Solvents
A model which is as complete as possible consists of a quantum mechanical description of the total solvent/dye system. Naturally, the dynamic character of the system must be considered. It is not sufficient to simply calculate an arrangement of all the atoms (referred to as a snapshot of the atomic positions), but rather several thousand different arrangements must be generated. If such calculations are preformed as a matter of routine, a purely quantum mechanical approach cannot be used. The computational effort is simply too great.
To describe the effect of solvents in spite of this, there are two basic approaches for efficient computer models. The first method merely implicitly describes the solvent environment as a dielectric continuum which can be polarized. That is, in the model, the molecule of interest is enclosed in an cavity, which represents an interface. The induced charge depending on several solvent parameters is determined for this interface. The outer potential which is calculated in this way is then the resulting influence on the absorption characteristics of the molecule. This implicit approach significantly speeds up the calculations. However the great simplification does not allow solvent-specific influences to be taken into account. Above all, the formation of hydrogen bonds cannot be described with sufficient accuracy. However, these often have a great effect on the dye molecules.
The alternative approach starts by describing the molecule in solution with classical moleculardynamics (MDs) and the extraction of several representative snapshots from the simulation. In a subsequent step, a model from a quantum mechanical (QM) and a classical molecular mechanical (MM) point of view is created for each of these snapshots. This is referred to as a QM/MM approach. In this, the atoms of the solvent a reprimarily depicted as point charges (corresponding to their partial charges in the solvent molecule). These charges create an electrostatic potential, which also influences the absorption of light in the dye molecule. As the positions of the atoms vary for each snapshot, the dynamic character of the solvent is accounted for. In addition, due to the explicit treatment of solvent atoms, the effect of hydrogen bonds can be depicted. The price for this is a great computational effort, as first of all the MD simulation must be performed and then about one hundred further QM/MM calculations are necessary, instead of a single calculation as in the implicit approaches.
In addition to the differences between the approaches, which have already been described, there is a further aspect which is relevant for the description of solvatochromism.
In the standard QM/MM approaches, the partial charges of the solvent molecules are fixed. Because of this, the electrostatic potential does not change even if the charge density in the QM part changes, i.e. no feedback between the two systems is described. Due to the absorption of light and the excitation of an electron to a higher energy level there is often a change in the electron density, which often has a significant effect on the environment of the dye molecule. Hence, due to the interaction with the solvent, a stabilisation or destabilisation of the excited state may occur. This in turn means that light of higher (or lower) energy is required in order to achieve the electronic excitation. The change in the absorption energy has a direct influence on the solvatochromic effect. While this circumstance is not described in conventional QM/MM approaches, in principle this can be achieved by the implicit solvent model.
To achieve an explicit description of the solvent while at the same time obtaining a mutual coupling between the QM and MM sub-system, the method of polarizable embedding (PE) has been developed [2-4]. Here, dipole moments are induceddue to the polarizable nature of the classic alatomic centres, which provide the necessary feedback. PE is therefore an extension of the standard QM/MM approaches. A further advantage is that in the classical region, the atoms are not only described by their point charges and their polarizable nature, but also by higher multipole moments. The values for this are not derived from empirical data. Rather, they can be obtained directly from quantum mechanical calculations. This greater independence from experiments enables a more flexible modelling of different solvents or other environments. The method can therefore be combined with both the approach of density functional theory , which is now the standard method of computational chemistry, as well as with the more precise coupled-cluster method [3,4]. In an initial study it could be demonstrated that with this approach the experimental data for the solvatochromism of acetone in the solvents water, methanol, acetonitrile and carbon tetrachloride could be reproduced with very high accuracy . This is clearly shown in comparison with other methods illustrated in Figure 2. Both the implicit description of the solvent (indicated here with PCM) as well as the conventional QM/MM approach cannot satisfactorily depict the correct trend.
Furthermore, the method also allows the examination and better understanding of the individual contributions of various effects of the solvent.  A further interesting aspect is that from the point of view of a computer model, there is little difference between a solvent environment and a protein scaffold. Therefore the PE method can also be applied to questions concerning the spectroscopy of biological molecules. For example it could be demonstrated that the experimentally confirmed low influence on the absorption of the chromophor in Photoactive Yellow Protein (PYP) is not due to a negligible interaction with the protein, but rather is caused by mutual negation of physical effects. An essential component of this is the different induced polarization in the ground and excited state between chromophor and protein so that the PE method was essential for a correct calculation .
At present, the PE method is in an active stage of development, so that it will certainly take some time before it can be routinely used by the users of computer simulations. However, the potential of the method has already been demonstrated in several applications . The PE method is therefore a valuable tool for the better simulation and understanding of environmental effects on UV/visual spectroscopy.
 Reichardt C. and Welton T.: Solvents and Solvent Effects in Organic Chemistry, Wiley-VCH, Weinheim2011
 Olsen J. M. et al.: J. Chem. Theory Comput. 6, 3721-3734 (2010)
 Sneskov K. et al.: J. Chem. Phys. 134, 104108 (2011)
 Schwabe T. et al.: J. Chem. Theory Comput. 8, 3274-3283 (2012)
 Schwabe T. et al.: J. Chem. Theory Comput. 7, 2209-2217 (2011)
 Sneskov K. et al.: Phys. Chem. Chem. Phys. 13, 18551-18560 (2011)
 Olsen J. M. H. and Kongsted J.: Adv. Quantum Chem. 61, 107-143 (2011)