PMC:4264129 / 35124-50965
Annnotations
{"target":"https://pubannotation.org/docs/sourcedb/PMC/sourceid/4264129","sourcedb":"PMC","sourceid":"4264129","source_url":"http://www.ncbi.nlm.nih.gov/pmc/4264129","text":"2.2.2. Special Problems\nIn general, for the past twenty years optimized molecular geometries in solution have been obtained by applying a continuum solvent model. The idea was introduced and subsequently developed by the Tomasi group as the PCM model [51,52]. Since the 1990’s, different continuum solvent models [53,54,55,56,57,58,59] and extension beyond the dielectric approximation [60] have been developed to account for the solvent effects on the geometry and energy/free energy of dissolved molecules. Several reviews summarize these models and compare the results obtained from different approaches [52,61,62,63].\nThe basic idea in the widely used PCM method [51] is that the solute is placed in a cavity carved in the continuum dielectric solvent, and the solute and the solvent mutually polarize each other. As a consequence, the solute’s geometry changes slightly and its internal energy increases when compared to its optimized gas-phase energy. The energy-increase is balanced by the developing solute-solvent electrostatic interaction energy. The final results are obtained through an iterative self-consistent-field (SCF) process that finds the total energy minimum and its related geometry. For the geometry optimization and energy/free energy calculations, all methods can be utilized, which were mentioned in relation to gas-phase calculations [33,34,35,36,37,38]. Thus, geometry optimizations can be performed by means of the HF, MP2 and DFT methods, and higher level energy calculations can be performed up to the CCSD(T)/CBS level. The customary basis set for geometry optimization and frequency analysis is 6-311++G**, but even the aug-cc-pvtz set has been applied [15,64].\nWhen a molecule dissolves, a close molecular environment is encountered that is in contrast to the most frequently applied ideal-gas model, where no potential energy interaction is considered even through the collisions of the molecules. Although the solute-solvent interactions are substantial, the effect of a non-polar or only slightly polar solvent (CCl4, CHCl3) on the molecular geometry is generally small [15,64]. The geometric effect could be, however, large when a solute with an intramolecular H-bond in the gas phase dissolves in a protic solvent such as water or methanol, which have both proton donor and acceptor sites. In this case, the X–H…Y intramolecular H-bond may collapse while solute-solvent H-bonds are formed using the free XH and Y sites.\nThe weakest point of the continuum dielectric solvent model is that the above solute-solvent H-bond(s) are only implicitly mimicked by polarization of the solvent and concomitant appearance of surface charges on the inner surface of the cavity: Negative surface charges opposite to a polar hydrogen and positive ones in the lone-pair regions of the solute’s oxygens and nitrogens. Although this response is qualitatively correct, the calculated solute-solvent stabilization energy is underestimated [65,66]. Thus, for proper calculation of the free energy changes when a polar solute with or without internal H-bond(s) dissolves in a protic solvent, explicit consideration of the solute-solvent intermolecular H-bonds becomes necessary.\nThis requirement can be largely satisfied by adopting the supermolecule + continuum approach, where the solute is surrounded by a number of explicit solvent molecules. The solute and the explicit solvent molecules mimic the H-bonds in the first solvation shell within the cavity carved in the continuum solvent. The critical question then becomes, how many explicit solvent molecules are to be considered.\nFor constructing the starting geometry of a supermolecule, knowledge of microsolvated solute structures is very helpful. In these systems, the central, polar molecule with or without an intramolecular H-bond is solvated by a few solvent molecules. Locations of the solvent molecules (water, methanol) indicate the most preferable solvation sites of the solute with a hydrogen donor/acceptor solvent. Useful information can be obtained from experimental gas-phase hydration/solvation studies augmented with theoretical calculations [12,13,14,16,23,67,68,69,70] or specific theoretical calculations for hydrated amino acid side chains, nucleotid base and sugar models [71,72,73,74,75].\nRecent calculations proved [15,64] that application of at least the aug-cc-pvtz basis set is required for reliable estimation of the relative solute free energies. If the solute has 6–10 C, N, O atoms and connected hydrogens, 500 basis functions could easily be required. If such a solute has to be surrounded by at least 5–6 water molecules, the number of basis functions increases to about 1000. The number of basis functions could be somewhat reduced by considering the solvent molecules with a lower basis set, with, e.g., 6-31+G**. While the supermolecule + continuum approach can be useful theoretically, it suffers from several technical challenges.\n(1) The geometry optimization for a system with 500–1000 basis functions is very slow in solution. If one wants to prove the local-energy-minimum character of the supermolecule and calculate thermal corrections, very small remaining forces should be allowed only at the end of the optimization. It is almost unreachable for a number of systems (or only by the application of the very time-consuming analytical second-derivative methods), in cases when torsion or intermolecular vibration frequencies could be as low as a few cm−1.\n(2) The number of explicit solvent molecules to be considered can become be critical. In a real, dilute solution the solute is surrounded by solvent molecules all around. Except for the simplest modeling cases like the partial solvation of γ-amino-butyric acid with 2–5 water molecules [76] or consideration of 3–8 water molecules during the HOCl catalyzed tautomerization of β-cyclopentadione [77], a considerably larger number of water molecules is generally required for reasonable modeling of the solvation sphere even for a small organic molecule. An impressive example was provided by Lu [78], who optimized the geometry of the Al(H2O)63+·12 H2O hydrate at the B3LYP/6-31+G(d,p) level in a water continuum by the PCM method. The resulting structure was of nearly spherical symmetry, easily allowing for the formation of the water network. Consideration of eighteen solvent molecules was necessary for mimicking the first and second hydration shells in a dilute solution.\n(3) The results of Lu and coworkers call attention to the need for the supermolecule to reasonably mimic the immediate in-solution environment of the solute. With a relatively small number of explicit solvent molecules within the supermolecule (for example, 3–4 water molecules, originally each of them facing a polar site), the water-water interactions may dominate over the solute-water interactions. Instead of forming 3–4 solute-water hydrogen bonds, a water cluster is then formed on some side of the solute and the number of water-solute hydrogen bonds would be smaller than expected in a water box with hundreds of water molecules. A successful tetrahydrate model in a continuum solvent was developed by Nagy for the syn-anti transformation of the acetic acid carboxylic group [79], whereas three waters in hydrogen bonds to the solute were not enough for modeling the immediate solvation environment of the transition state for 2F-phenol [15].\n(4) In general, only the first solvation shell around the polar sites can be modeled. Moreover, even in these cases, the explicit-solvent/continuum interface suffers from neglecting the consideration of the solvent-solvent hydrogen bonds. For methanol or acetonitrile solvents, the problem is not dramatic since the polar site of the solvent molecules should point toward the solute while the methyl group is located mainly on the outer surface of the supermolecule. Then the first-sphere solvent molecules can create a non-polar surface toward its continuum representation. This is surely not the case for explicit water molecules and is likewise questionable for a solvent like acetic acid with two stericly separated polar sites.\n(5) Geometry optimization for a supermolecule leads to an overly ordered structure, which is not maintained due to thermal disordering in a real solution.\n(6) If one wants to study the structure of a dilute or moderately concentrated (1 molar) solution as well as solute dimerization, boxes of a large number of explicit solvent molecules should be considered. These studies typically then require Monte Carlo (MC) [80] or molecular dynamics (MD) [81] simulations.\nDuring MC calculations, the solution model is a large solvent box with hundreds of solvent molecules and one or a few solute molecules embedded in the solvent. Atoms are represented by point-like centers characterized with atomic charges and assigned van der Waals parameters. The interaction energy of the atoms in different molecules is calculated by pair potentials and the total energy is the sum of these pair-interaction energies. Macroscopic thermodynamic quantities are estimated by averaging the individual values calculated for a large number of consecutive configurations. A configuration means a specific geometric arrangement of the elements in the solution box. The method is a probability method, meaning that a new configuration with modified geometric arrangement of the elements is considered for the above averaging upon the probability of the acceptance of the total energy change. If the new configuration is rejected, the old one is considered one more time in the averaging process. The most frequently applied sampling procedures are the Metropolis procedure [82] or some suitable procedure [83,84], which can accelerate the convergence of the calculated averages for thermodynamic quantities or help to more quickly reach an equilibrium solution structure by applying a biased energy calculation and probability for the acceptance of a new configuration. The goal is to generate a series of configurations corresponding to the Boltzmann distribution. For constant temperature simulations, the temperature is a parameter of the expression determining the acceptance probability. After having generated the required set of configurations, the average energy, enthalpy, volume, etc., can be calculated as an arithmetic mean of the individual values obtained with each configuration. If a biasing sampling was used, the probability of the acceptance has to be corrected before calculating averages.\nThe MD simulation is a deterministic process. The solution box is established as described above, but the atomic masses are also considered. A force field is used for calculating the total energy of the system with a given geometric arrangement (with Cartesian coordinates for each atom) at a reference time “t”. The force field contains terms accounting for the energy contributions by atoms bound along a 1–2–3–4 path, as well as for interactions of more remote atoms within the molecule and with all atoms in other molecules. The system generally is not in energy minimum, thus there are forces acting on the atoms. Using the gradient of the total energy, the forces acting on each atom can be determined. Applying Newton’s law, the position of the atoms at t + Δt can be calculated by means of the determined instantaneous velocities. Δt must be small, generally being chosen between 0.5 and 2 fs. In the latter case, the X–H distances are kept at a constant value. The temperature is related to the sum of the atomic kinetic energies. The simplest way to keep the temperature at a constant value is by scaling the determined atomic velocities or by coupling a thermostat to the system. Letting the simulation run long enough, sometimes for tens of nanoseconds, the average of the thermodynamic quantities can be obtained for a simulation period, where some structural characteristics, e.g., the solution density, have reached an equilibrium value already. By examining the trajectories calculated for geometric parameters of the solute, structural changes can be followed.\nUsing intermolecular pair-potentials such as OPLS-AA [85,86], Amber [87] or CHARMM22 [88], the largest problem is the development of the relevant atomic charges for the molecule. For example, Amber was originally parameterized for biopolymers and DNA, and no special charge parameters were available for, e.g., the HO–CHx–CHx–Y (Y = OH, NH2, NH3+, x = 1, 2) substructures. Furthermore, the OH and Y charges (and the CHx values, as well) should be conformation dependent, since, e.g., there is an intramolecular H-bond for the tG+g− conformer of 1,2-ethanediol, which is missing in the tTt form (Figure 2). Also, conformation dependent charges have to be used with Y = NH2, and NH3+ for the OCCN gauche and trans structures [89]. Recent developers of force fields suggest using molecular electrostatic potential (MEP) fitted charges, where the MEP should be obtained for the in-solution optimized solute. Since the solute and the continuum solvent mutually polarize each other, the MEP obtained at the end of the SCF procedure for the geometry optimization and energy minimization, reflects the electrostatic potential of a polarized solute in a polarized solvent environment. Charge fitting is a working tool, although different fitting methods (ES [90], RESP, [91], CHELPG [92]) lead to different results. Nonetheless, the problem is again of how to optimize the solute in a solvent environment. The MEP for the supermolecule is not relevant, since the solvent molecules are not so strictly bound to the solute due to thermal disorientation in a large solvent box as would follow from the structure of an optimized supermolecule. Furthermore, there is generally some charge transfer between the elements of the supermolecule. Although the programs force the charge of the total supermolecule to zero or some +/− integer for ionic solutes, the individual charges for the solute and the surrounding solvent molecules generally differ from 0, +/−1 etc. Thus, the MEP- fitted charges for a supermolecule should not be directly accepted for the atoms of the solute and applied in the MC of MD intermolecular pair-potentials. A possibility is that the solute geometry is accepted from the optimized supermolecule, and the MEP is fitted for the pure solute in a single-point calculation. This is, however, not a consistent procedure. The author has not found a good solution for this problem when surveying the literature.\nDespite the listed potential problems, the continuum solvent approach has been one of the most frequently used theoretical methods for characterizing the geometry and the energy/free energy for dissolved molecules. Since chemical equilibria depend on relative rather than absolute free energies, the problems mentioned above may not emerge in every case so harshly, and the errors could be partially cancelled. For example, the energy minimization in the supermolecule approach leads to too tightly bound water molecules. In a model, where the thermal disordering effect is also taken into consideration (MC and MD), a more loosely bound first solvation shell is expected. Nonetheless, since relative energy data are to be compared for the supermolecules with different solute conformations, the error is probably decreased. Also a more or less cancelled error may be expected regarding the interaction energy between the explicit solvent molecules at the outer surface of the supermolecules and the continuum. In a study for a series of compounds, it is a good practice to compare the computational results with available experimental values. Unfortunately, however, good-quality experimental results generally not are available in the literature for equilibria, where a number of conformers have been detected in 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