From owner-chemistry@ccl.net Sun Sep 5 12:51:01 2010 From: "Michael K. Gilson mgilson*o*ucsd.edu" To: CCL Subject: CCL: Gsolv - pKa - Frequency Message-Id: <-42698-100905124804-8284-JtNA8j9XeB1A/V7pffb/hA]|[server.ccl.net> X-Original-From: "Michael K. Gilson" Content-Transfer-Encoding: 8bit Content-Type: text/plain; charset=ISO-8859-1; format=flowed Date: Sun, 05 Sep 2010 09:47:53 -0700 MIME-Version: 1.0 Sent to CCL by: "Michael K. Gilson" [mgilson:ucsd.edu] When one speaks of "the entropy of a molecule", whether in solution or gas phase, one is usually implicitly referring to the partial molar entropy of that molecular species, \partial S \over \partial n, where S is the total entropy of the system and n is the number of moles the molecule of interest. This is a well-defined quantity, both in gas-phase and solution, and it is usefully related to the chemical potential of the species \mu. Thus, the chemical potential (often loosely referred to as the free energy of the species) is its partial molar free energy \partial G \over \partial n. One can also write the partial molar enthalpy as \partial H \over \partial n, and then one has: \mu \equiv \partial G \over \partial n = \partial H \over \partial n - T \partial S \over \partial n This is obviously analogous to G=H-TS, which applies to the system as a whole. When we refer to the thermodynamics of chemical reactions, we are implicitly referring to partial molar quantities. With this as background, the answer is yes, the (partial molar) entropy of a molecule in gas phase is typically differently from its (partial molar) entropy in solution. Regards, Mike On 9/5/2010 8:45 AM, Christopher Cramer cramer..umn.edu wrote: > Sent to CCL by: Christopher Cramer [cramer+*+umn.edu] > In response to: > >> Sent to CCL by: "W Flak" [williamflak]![yahoo.com] >> >> * Is the entropy of a molecule in gas phase different from that in a solution? >> if yes, please direct me to the undergraduate book I should read >> >> * I read many posts on pKa calculation and found some people (as Andreas >> Klamt,2006) recommended not to include frequency in pKa calculation, and the >> others include it. What do you recommend? >> > > These questions must be answered from both a fundamental standpoint and a practical standpoint. From a fundamental standpoint, it is perhaps worth emphasizing that there is no formal way to DEFINE the entropy of a solute in solution. Entropy is an extensive property. We can speak of the entropy of the entire solution (assuming that the solution is itself an isolated system), but the only way to define the entropy of the SOLUTE would be to invoke an approximation whereby there are no interactions between the solute and the solvent (as in the ideal-gas approximation, which does indeed allow us, in all undergraduate stat-mech textbooks, to define system partition functions as products of molecular partition functions and thus talk about individual molecular entropies). > > So, for people who do explicit simulations, this doesn't pose any real ambiguity. One can evaluate certain expectation properties of an ensemble (like a solution) in order to estimate entropy and one is doing perfectly rigorous thermodynamics. And, most people in that community recognize the dangers of attempting to partition system properties (like the entropy) into solute and solvent components, although discussions on this point continue to surface from time to time. > > The continuum solvation approximation, however, lends itself to the pretense that the solute continues to be an isolated molecule, so that we may imagine that we are somehow being rigorous if we construct molecular and ensemble partition functions based solely on solute properties (i.e., the usual electronic, translational, rotational, and vibrational components). Thus, it SEEMS to be sensible to compute frequencies in both the gas and solution phases, and to compute the thermal contributions to free energy therefrom, and to take the DIFFERENCE in the gas and solution phases as part of what contributes to the free energy of transfer when going from the gas phase to solution (free energy of transfer is synonymous with free energy of solvation). > > However, there are several issues to address. As already noted above, it's not really consistent with the fundamental thermodynamics of the "real" system, it only seems rigorous because one has replaced the full solvent with a continuum, but it is a continuum that DOES interact with the solute, so the solute doesn't really satisfy the "isolated" criterion anymore. Moreover, there are certain paradoxes. For instance, what does one think about the computed rotations and the associated rotational partition function? A molecule surrounded by solvent clearly is not a quantum mechanical rigid rotator (which is the model that we use in ideal gas statistical mechanics). Coming back to simulations, for the moment, if one were to compute the frequencies present in the condensed phase (by Fourier transform of the dipole-dipole autocorrelation function), one would find that in the very low frequency region of the spectrum, certain regions would be associated with moderate to large am! > plitude solute "rotational-like movements" that are coupled to solvent molcules also moving in the first solvation shell. Such modes are typically called librations. An important aspect of the librations is that they absolutely CANNOT be separated into solute and solvent specific modes -- they are intimately coupled. Put differently, one cannot take the 3N-6 vibrational degrees of freedom of the solute-solvent supersystem and cleanly identify 3m-6 that belong to the m solute atoms (where there are also n solvent atoms and (m+n)=N). > > Instead, one relies on the semiempirical nature of continuum solvation models to avoid (or finesse, depending on your point of view...) this problem. Insofar as the models themselves tend to have parameters that are optimized against experimental data, they clearly already incorporate the free energy effects arising from partition function changes, even though it is rather hard to define those changes other than operationally within the continuum approximation. So, in some sense attempting to approximate them by recomputing frequencies in solution is double counting. BUT, parameterization of continuum models has been accomplished primarily using data for rather "simple" molecules. So, it IS conceivable that in some much more complicated case, where perhaps there are very low frequency motions that will be strongly perturbed by solvation, an attempt to adjust the statistical mechanics by considering the partition functions would be useful. Having said that, I can't say tha! > t I've ever seen a convincing case for this, nor that it seems likely that a continuum model is the ideal way in which to address the issue in the first place, but I accept it as being possible. > > So, coming finally to practical concerns, within the context of third-law entropies, we have a good way to compute "absolute" free energies for ideal gases, and theorists do this routinely by adding thermal contributions, derived from frequency calculations and the usual particle in a box, rigid rotator, harmonic oscillator approximations, to very high level electronic energies. At that point, the "best" (in my opinion) way to define the free energy in solution is to do a continuum calculation and add the derived free energy of solvation to the gas phase free energy. One can certainly re-optimize the geometry in solution, in order to assess the impact of that on the free energy change. And, if one is dying to know, one can recompute frequencies too, just to see if some large effect has been introduced. When there IS a large effect in the partition function, however, there is no simple answer to the question of "what is the best way to incorporate this into the free energy! > change?" > > To wrap this discussion up, let us address one or two other practical points. What if your solute geometry in solution is not stationary in the gas phase? This might be true for a zwitterion, for example. Well, perforce you are going to have a hard time getting a gas-phase partition function and associated thermal contributions for that structure. You are pretty much stuck with doing the frequencies in solution. And, to be consistent, should you do all of the other frequencies in solution, too? Good question. I would say, "probably", but one would want to check and hope that it makes little difference for the other species that DO have gas-phase stationary points. > > In addition, for those still pondering the issue of the entropy of the "system", it is worth noting that the continuum approximation nearly always invokes linear response in the electrostatics. Put simply, that implies that one half of the favorable electrostatic interaction energy between the polarized solute and solvent is consumed as the work required to polarize the continuum in the first place. In practice, this quantity is often assumed to be dominated by enthalpy, but this is just an assumption -- only through the definition of a temperature-dependent continuum model is it possible to in principle define an enthalpic and entropic component to the electrostatic part of the solvation free energy. (Note, again, that even that phrase: "electrostatic part of the free energy", is not really rigorous -- free energy is also extensive, so imagining that one can divide it up into electrostatics and non-electrostatics is conceptually convenient, but not rigorously valid as t! > he two cannot be decoupled in a real system -- one can only try to design model cases where one or the other might be expected to heavily dominate and hope that these will be useful in the construction of theoretical models). > > Finally, a plug for a recent paper on pKa prediction that will likely be a useful read for those interested in the activity: Ho, J. M.; Coote, M. L. "A universal approach for continuum solvent pK(a) calculations: are we there yet?" Theor. Chem. Acc. 2010, 125, 3-21. > > Chris > -- > > Christopher J. Cramer > Elmore H. Northey Professor > University of Minnesota > Department of Chemistry > 207 Pleasant St. SE > Minneapolis, MN 55455-0431 > -------------------------- > Phone: (612) 624-0859 || FAX: (612) 626-7541 > Mobile: (952) 297-2575 > email: cramer:+:umn.edu > jabber: cramer:+:jabber.umn.edu > http://pollux.chem.umn.edu/~cramer > (website includes information about the textbook "Essentials > of Computational Chemistry: Theories and Models, 2nd Edition")> > -- Michael K. Gilson, M.D., Ph.D. Professor Skaggs School of Pharm. and Pharm. Sci. University of California San Diego 9500 Gilman Drive, MC 0736 La Jolla, California, 92093-0736 Office: 858-822-0622 http://pharmacy.ucsd.edu/faculty/Gilson.shtml