Solvation calculations

It can be shown (see, e.g., Hansen and McDonald, 1986, section 2.5) that aU thermodynamic quantities can be calculated from the radial distribution function, g(r). This function, which can be obtained experimentally from X-ray or neutron diffraction, can also be calculated, provided the intermolecular potential function is known. The aim of the integral equation theories is to calculate the radial distribution function from the intermolecular potential V(r). There are several versions that differ according to the choice of simplifying assumptions made. They each result in an equation that contains g(r) associated with the Boltzmann factor exp(-/ v(r)). For example, the PY equation (Hansen and McDonald, 1986, p. 119 Percus and Yevick, 1958) 

p is 1/A r. The hypemetted chain (HNC) equation (van Leeuwen et al, 1959) and the equation of Bom and Green (1949) involve similar terms, though they are in logarithmic form. The Bom-Green equation also contains the first derivative with respect to r, so it is an integrodifferential equation for g(r) in terms of V(r). 

Going from pure liquids to solutions, by either MC or MD methods, the pure solvent is first simulated, and then one of the solvent molecules in the model is replaced by a solute molecule. After a number of cycles to allow the system to relax to accommodate the intruder, the chain of calculations continues until convergence of the 

It arises from the reaction field, that is, the field within the cavity due to the charges induced in the wall of the cavity by the dipole, acting back on the dipole. 

If the Hamiltonian of the molecule in vacuo is the Hamiltonian of the same molecule in the cavity may be written where is given by Equation 5.10 

---

Molecular volumes are usually computed by a nonquantum mechanical method, which integrates the area inside a van der Waals or Connolly surface of some sort. Alternatively, molecular volume can be determined by choosing an isosurface of the electron density and determining the volume inside of that surface. Thus, one could find the isosurface that contains a certain percentage of the electron density. These properties are important due to their relationship to certain applications, such as determining whether a molecule will fit in the active site of an enzyme, predicting liquid densities, and determining the cavity size for solvation calculations. 

Transition Structures and Ab Initio and Semiempirical Gas-Phase AND Solvation Calculations ... 

A comparison of the second-order rate coefficients for nitration of 2,4,6-tri-methylpyridine and 1,2,4,6-tetramethylpyridinium ion (both at the 3-position) shows similarity of profile in the common acidity region and a rapidly increasing rate with acidity for the trimethyl compound at acidities below 90 wt. % (where the usual maximum is obtained). These two pieces of evidence show reaction to occur on the conjugate acid as also indicated by the large negative entropy of activation. Surprisingly, the tetramethyl compound is less reactive than the trimethyl compound so maybe this is an example of steric hindrance to solvation. Calculation of the encounter rate also showed that reaction on the free base was unlikely. 

FIGURE 3.25. Potential energy profiles (from B3LYP/6-13G calculations) for the clevage of 3- and 4-nitrobenzyl chloride anion radicals (a and b, respectively) in the gas phase (top) and in a solvent (middle and bottom) (from COSMO solvation calculations with a dielectric constant of 36.6 and 78.4, respectively). Dotted and solid lines best-fitting Morse and dissociative Morse curves, respectively. Adapted from Figure 3 of reference 43, with permission from the American Chemical Society. 

FIGURE 3.31. Reaction of alkyl chlorides with NO as a function of steric hindrance at carbon. Central line reactant state Left-hand lines ET transition and product states. Right-hand lines 5 2 transition and product states. Numbers above and below the lines are the standard free energies (in eV) numbers in parentheses are the entropies in meV/K. The numbers on the structures are bond lengths in A. From MP2/6-31G //UHF/6-31G and COSMO (solvation) calculations. 

Note that it is fairly common in the literature for continuum solvation calculations to be reported as having been carried out using Poisson-Boltzmann electrostatics even when no electrolyte concentration is being considered, i.e., the Poisson equation is considered a special case of the PB equation and not named separately. 

Obviously, intermediates, which are formed in the gaseous phase, arise as a result of the strong electrostatic attraction of species with no solvation. Calculations predict that mixing of the nitronium ion with benzene causes an interaction of frontal molecular orbitals. As a result, their energetic levels are changed. At a distance between the reagents of from 0.025 to 0.015 nm, an electron transfer is possible, even from benzene to the nitron-... 

Sampling the conformational space of solute(s) by MC or MD algorithms requires many intramolecular and solvation calculations and accordingly simplicity in the solute Hamiltonian and computer efficiency in the continuum method used to compute solvation are key requirements. This implies that, with some exceptions [1], MD/MC algorithms are always coupled to purely classical descriptions of solvation, which in order to gain computer efficiency adopt severe approximations, such as the neglect of explicit electronic polarization contributions to solvation (for a discussion see ref. [1]). In the following we will summarize the major approaches used to couple MD/MC with continuum representations of solvation. 

After performing ab initio and solvation calculations to examine the decarboxylation reaction in water, the free energy surface of the enzyme-catalyzed reaction was explored. An initial ODCase-OMP complex was constructed from the structure of the ODCase-6-azaUMP complex reported by Pai and coworkers,22... 

Solvation effects were neglected in the Kubicki and Apitz study, in part, because of limited computer power.65 As a matter of general practice, however, one would probably run these types of gas-phase calculations prior to running simulations within a solvent even with unlimited computer resources. This is a common strategy to evaluate the effects of solvation on structure and one that provides an initial guess for the solvation calculations. One advantage of the molecular modeling approach is the ability to add and subtract components at will in order to assess the effects they have on the behavior of the system. 

In the previous Sections, we have paid attention to the definition of the cavity in the different computational procedures we have examined. A proper definition of the cavity plays an important role in assessing the quality of continuum solvation calculations. It is convenient to collect under a specific heading some remarks on this point. 

The standard state of AGsoi H+ must also be taken into account to produce reliable results. Free energies may be calculated using an ideal gas at 1 atm as a reference for gas-phase calculations or with an ideal gas at 1 mol/L, which is used with free energy of solvation calculations. The value for AGsoi and AGgas depends on which standard state is used in their determination. Furthermore, a homogenous equilibrium, where all of the species are in the same standard state, is necessary to obtain reliable pK results. The conversion of the 1 atm standard state to the 1 mol/L standard state can be derived from the relationship between the... 

The change in free energy of solvation calculation for the reaction is the largest source of error in pKa calculations. To determine the most accurate method we must look both at the type of solvation model used, implicit, explicit, or cluster continuum method (likewise described as implicit-explicit), and the specific level of theory. As previously mentioned, ionic species, in particular, are extremely difficult to calculate because of their strong electrostatic effects and large free energy of solvation values [8,14,23,25]. 

Namazian and Halvani studied pKa calculations with an explicit water using the B3LYP/6-31+G(d,p) level of theory for free energy calculations in the gas phase and PCM/B3LYP/6-31+G(d,p) with the UAO radius for solvation calculations. Using a data set of 66 acids, they found the method accurate within an average of 0.58 pKa units. The thermodynamic cycles used an explicit water, as in Eq. (9). Although the method produced pKa values within 0.6 pKa units, there is... 

Gao and colleagues studied several methods of solvation calculations [29] ... 

For the free energy of solvation calculation, however, it is difficult to discern the most accurate method. Recently, there have been numerous publications exploring the use of the cluster continuum method with anions. With regard to implicit solvation, there are no definite conclusions to the most accurate method, yet for the PB models the conductor-like models (COSMO CPCM) appear to be the most robust over the widest range of circumstances [23]. At this writing, the SMVLE method seems to be the most versatile, as it can be used by itself, or with the implicit-explicit model, and the error bars for bare and clustered ions are the smallest of any continuum solvation method. The ability to add explicit water molecules to anions and then use the implicit method (making it an implicit-explicit model) improves the results more often than the other implicit methods that have been used in the literature to date. 

Due to the numerous potential cycles using explicit molecules, levels of theory, basis sets, and types of molecules, it is impossible to determine one specific method that produces the most accurate pKa values. Rather, this review serves to summarize the current literature and illustrate various schemes that have been successful. Accurate attention to detail and the use of benchmark calculations or experimental values to assist in determination of the correct method to use for a particular system is highly recommended. Further research on thermodynamic cycles using explicit cycles, clustered water structures, conformational effects, and advances in continuum solvation calculations will continue to advance this field. 

The structure of the solute in vacuum was optimized (energy minimized) using the Hartree-Fock (HF) method with the DZPsp(df) basis set (Lin and Sandler, 1999a). The same geometry was used in subsequent solvation calculations without further optimization. 

An application of continuum solvation calculations that has not been extensively studied is the effect of temperature. A straightforward way to determine the solvation free energy at different temperatures is to use the known temperature dependence of the solvent properties (dielectric constant, ionization potential, refractive index, and density of the solvent) and do an ab initio solvation calculation at each temperature. Elcock and McCammon (1997) studied the solvation of amino acids in water from 5 to 100°C and found that the scale factor a should increase with temperature to describe correctly the temperature dependence of the solvation free energy. Tawa and Pratt (1995) examined the equilibrium ionization of liquid water and drew similar conclusions. An alternative way to study temperature effect is through the enthalpy of solvation. The temperature dependence of is related to the partial molar excess enthalpy at infinite dilution,... 

---