Continuum calculations

While TD-DFT continuum calculations for molecules, such as camphor, are not yet quite practicable, efforts to create highly parallel computer codes capable of tackling this scale of problem are expected to be fruitful soon. In the meantime TD-DFT studies for computahonaUy less demanding small molecules [66-68] or highly symmetric molecules, such as SFg [79], have provided indicahons of the general value of the inclusion of electron response effects. 

Jean-Charles, A. Nicholls, A. Sharp, K. Honig, B. Tempzyck, A. Hendrickson, T. Still, W.C., Electrostatic contributions to solvation energies comparison of free energy perturbation and continuum calculations., J. Am. Chem. Soc. 1991,113, 1454-1455... 

COSMO models [27-30] were compared with the SM approach [22] by Klamt [43] and by Cramer and Truhlar [44]. A very recent paper by Klamt and coworkers [45] shows that improved calculated pKa values are obtained for the limited domain of strong to moderately weak acids by a cluster-continuum method in which the acid and conjugate base are each associated with one or a few solvent molecules and this cluster is then continuum-calculated with COSMO-RS. The authors point out, however, that for the calculation of pKa a consistent and generally applicable method is still lacking . This paper clarifies the problem raised in [41]. The matter is under study.1... 

I cite three papers to show that standard continuum calculations can give satisfactory first-principles pKa values Shields and coworkers used a thermodynamic cycle with gas phase and continuum calculations to obtain satisfactory results for six simple carboxylic acids [46]. These were absolute calculations in the sense that no acid was used as a reference point, although the experimental gas phase free energy and aqueous solvation energy of the proton were resorted to. Not quite as esthetically satisfying perhaps, were relative calculations in which acetic acid was used as a reference compound [47]. Similar to the absolute acid calculations was work with phenols that was said to be among the most accurate of any such calculations for any group of compounds [48]. 

Hybrid solvation Implicit solvation plus Explicit solvation microsolvation subjected to the continuum method. Here the solute molecule is associated with explicit solvent molecules, usually no more than a few and sometimes as few as one, and with its bound (usually hydrogen-bonded) solvent molecule(s) is subjected to a continuum calculation. Such hybrid calculations have been used in attempts to improve values of solvation free energies in connection with pKp. [42], and also [45] and references therein. Other examples of the use of hybrid solvation are the hydration of the environmentally important hydroxyl radical [52] and of the ubiquitous alkali metal and halide ions [53]. Hybrid solvation has been surveyed in a review oriented toward biomolecular applications [54]. 

What physical properties of solvents have been used to parameterize them for continuum calculations ... 

Persistent interactions are not limited to hydrogen bonds. We mention for example those appearing in solutions of molecules with a terminal C=0 or C=N group dissolved in liquids such as acetone or dimethylsulfoxide. These solvents prefer at short distances an antiparallel orientation which changes at greater distances to a head-to-tail preferred orientation. The local antiparallel orientation is somewhat reinforced by the interaction with the terminal solute group and it is detected by the PCM calculation of nuclear shielding and vibrational properties. Recent experimental correlation studies [25] have confirmed the orientational behaviour of these solvents found in an indirect way from continuum calculations. The physical effect found in this class of solvent-solute pairs seems to be due to dispersion forces. 

Ref. (234) reported a theoretical study of the solvent effects on various isomers of the palladium hydride complex PdH3Cl(NH3)2 in dichloromethane. The influence of the solvent was investigated by discrete MP2 and SAPT, and continuum SCRF calculations. The theoretical relation between SCRF and SAPT, Eq. (1-177), was fully confirmed by the numerical results from the discrete SAPT and continuum SCRF calculations, cf. Table 1-7 and Figure 1-4. Interestingly, both the discrete MP2 and continuum SCRF models predicted the same relative stabilities for the isomers of PdH3Cl(NH3)2 in dichloromethane. Small energetic differences between the results of the discrete and continuum calculations could be explained by the entropy effects, neglected in the discrete model. 

The atomic radii may be further refined to improve the agreement between experimental and theoretical solvation free energies. Work on this direction has been done by Luque and Orozco (see [66] and references cited therein) while Barone et al. [67] defined a set of rules to estimate atomic radii. Further discussion on this point can be found in the review by Tomasi and co-workers [15], It must be noted that the parameterization of atomic radii on the basis of a good experiment-theory agreement of solvation energies is problematic because of the difficulty to separate electrostatic and non-electrostatic terms. The comparison of continuum calculations with statistical simulations provides another way to check the validity of cavity definition. A comparison between continuum and classical Monte Carlo simulations was reported by Costa-Cabral et al. [68] in the early 1980s and more recently, molecular dynamics simulations using combined quantum mechanics and molecular mechanics (QM/MM) force-fields have been carried out to analyze the case of water molecule in liquid water [69],... 

A. Jean-Charles, A. Nicholls, K. Sharp, B. Honig, A. Tempczyk, T. F. Hendrickson, and W. C. Still,/. Am. Chem. Soc., 113, 1454 (1991). Electrostatic Contributions to Solvation Energies Comparison of Free Energy Perturbation and Continuum Calculations. 

A number of basis functions were generated and utilized for two steps of the continuum calculations. First, an extended basis set with continuum wavefunctions of limited spatial extent and the minimal set of atomic orbitals corresponding to the states filled with electrons were used to obtain self-consistent charge density. Then a further extended basis set was used to represent the states in the continuum through the orthogonalization to the wavefunctions for the electron charge density. To reduce the number of matrix elements, the basis functions were transformed to the symmetry-adapted orbitals before the orthogonalization. 

Still, J. Am. Chem. Soc., 113, 145 (1991). Electrostatic Contributions to Solvation Free Energy Perturbation and Continuum Calculations. 

Models of this configuration are more difficult to set up and are more challenging to compute than are reaction path models. In addition, the reaction distributions predicted by continuum models primarily reflect the assumed hydrologic properties and kinetic rate constants, which normally comprise the most poorly known parameters in a natural system. Since a valid reaction model is a prerequisite for a continuum model, the first step in any case is to construct a successful reaction path model for the problem of interest. The reaction path model provides the modeler with an understanding of the nature of the chemical process before he undertakes more complex modeling. The scope of this book is restricted to reaction models like those already described we will not consider the added complexities of continuum calculations. 

Finally, accurate relative pK values can be obtained by combining the results of high-level calculations in the gas phase with CPCM continuum calculations of solvation energies. 

It is important to mention that the displacement of the interfacial nodes, U/, are not degrees of freedom in the continuum calculation Because the interfacial nodes coincide with interface atoms, r/, they are moved as atoms and appear as fixed nodes with prescribed displacements to the other elements of the continuum region. Lastly, it must be remarked that this formulation of the continuum/atomistic coupling does not allow for the use of a unique energy functional because E includes the elastic energy of the pad atoms, whose energy is already implicitly contained in the continuum energy, E. ... 

To bypass the limitations of the Cauchy-Born rule, in 2006, Lu et al. proposed a more involved scheme to couple standard DFT to quasi-continuum calculations. In their method, the part of the system far away from the zone of interest is described using a classical (nonquantum) quasi-continumn approach (see discussion above on QC for details), i.e., considering both local (continuum) and nonlocal (atomistic) terms. Classical potentials (EAM in the applications presented) are used to evaluate the energy within the QC calculations. A third region is considered as well, covering the part of the system that needs a more detailed description. It is in this region that density functional theory is used. 

Metal-Ceramic Interfaces, Pergamon, Oxford, 1990, p. 63. Coupled Atomistic-Continuum Calculations of Near Interface Cracking in Metal/Ceramic Composites. 

In this section an approximate form of the chemical potential of a species in solution is derived. This approximate chemical potential is then used to derive an equation for the relative binding free energy that is suitable for dielectric continuum calculations. Steps are then taken to refine the relative binding free energy such that the errors due to the approximations invoked are be minimized. The refined relative binding free energy is then cast at various levels of detail in order highlight its important contributions. 

Another interesting aspect of continuum solvent methods is their low computational cost. For this reason continuum calculations may replace more expeasive computer simulations for the evaluation of simple properties as the solvation free energy (the numerical results are of comparable accuracy with the two approaches) and nowadays represent the only option open for calculations requiring a good QM level. However, the complementarity between computer simulations and continuum solvent calculatioas mast be stre.ssed. Specific,... 

---