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
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