CCL:G: Thermodynamic Data & Solvation - Calculation Questions:



 Sent to CCL by: MIELCZAREK Detlev Conrad [detlev-conrad.mielczarek * ifpen.fr]
 Dear Uwe, many thanks for the explanation and the reference/paper.
 Dear Eric, many thanks for your very comprehensive response - with a detailed
 explanation how and why the terms are chosen - possibly the best I have seen
 yet.
 Hopefully I won't have any further questions after sitting down to properly read
 your response and the linked paper.
 Once again, many thanks for your two comprehensive responses.
 Kind regards
 Detlev
 Detlev Conrad Mielczarek
 Scientific Visitor/Post Doctorant
 IFP Energies nouvelles
 France
 detlev-conrad.mielczarek -  - ifpen.fr
 -----Original Message-----
 > From: owner-chemistry+detlev-conrad.mielczarek==ifpen.fr|ccl.net [mailto:owner-chemistry+detlev-conrad.mielczarek==ifpen.fr|ccl.net] On
 Behalf Of Eric Hermes erichermes**gmail.com
 Sent: 06 April 2017 18:57
 To: MIELCZAREK Detlev Conrad
 Subject: CCL:G: Thermodynamic Data & Solvation - Calculation Questions:
 Sent to CCL by: Eric Hermes [erichermes!^!gmail.com] Dr. Mielczarek,
 As always, the free energy of a species is defined with respect to some  choice
 of reference state, such that for a reaction of the form:
 (1) A + B -> C
 with a standard state free energy of reaction ΔG°, one can write the
 equilibrium constant as:
 (2) Keq = exp[-ΔG°/(kB T)] = a_C / (a_A a_C)
 Where a is the *activity* of a species. In the case of an ideal gas, the
 activity is simply the density divided by the *standard state*
 density:
 (3) a = ρ/ρ°
 The numerical value of Keq then depends on choice of reference state, but the
 equilibrium densities do not. Eq. 2 satisfies this requirement.
 One can show that the free energy of a species at an *arbitrary* reference state
 is given by:
 (4) G = G° + kB T ln[ρ/ρ°]
 This is *general*, but you can show that for an ideal gas the expression derives
 from the translational entropy.
 Now, when we are discussing free energy of solvation, the reaction
 becomes:
 (5) A_g -> A_soln
 Which of course brings with it changes to both the enthalpy and the entropy.
 These issues are *orthogonal* to the reference state issue, though. The standard
 state for species in the gas phase is usually taken to be 1 bar (occasionally 1
 atm is chosen instead), but the standard state for species in *solution* is
 typically chosen to be 1 M, which is very different! With these choices of
 reference state, the free energy of solvation ΔG°solv can be used to
 calculate the equilibrium constant like this (using P for pressure to visually
 distinguish from the density ρ):
 (6) Keq = exp[-ΔG°solv/(kB T)] = (ρ_A/(1 M)) / (P_A / (1 bar))
 As before, the numerical value of Keq depends on the choice of reference state,
 but the equilibrium density and pressure do not.
 So, if your goal is to determine the equilibrium properties of a solvation
 process, then you do not need to be concerned about the choice of reference
 state -- you need to be *aware* of it in order to calculate the properties
 directly, but whichever choice you make you will get the same answer. On the
 other hand, if you wish to compare your free energies of solvation to
 experimental values, you should most definitely ensure that the value you are
 calculating uses a reference state of 1 M for solution-phase species and 1 bar
 for gas-phase species.
 In the case of SMD, by default it will use the *same* reference state for the
 gas-phase and solution-phase species. The actual value of the reference state is
 irrelevant, as ΔGsolv has the same value for any choice of reference state
 so long as the reactants and products have the *same* reference state (if you
 are not convinced, stare at eqs 2, 3, and 4 in the context of eq 5 until you
 believe me :) ).
 This means you can arbitrarily say what reference state ΔGsolv is at, say
 1 bar, for both reactants and products. Then, if you want to calculate
 ΔG°solv (the *experimental* standard state value), you just need to
 use eq 4:
 (7) ΔGsolv = Gsoln - G°gas (we choice a reference state of 1 bar)
 (8) Gsoln = G°soln + kB T ln[(1 bar / (kB T))/1 M] (ideal gas law)
 (9) ΔG°solv = G°soln - G°gas = ΔGsolv - kB T ln[(1
 bar / (kB T))/1 M]
 Note that you would have gotten the *exact same result* if you had arbitrarily
 chosen a reference state of 1 M instead for ΔGsolv (if you don't believe
 me, try it yourself -- start with ΔGsolv = G°soln - Ggas).
 ---
 Now, as to your point about the entropy of solution phase species -- I'm not
 sure the paper you are linking is making the claim you are saying it does. It is
 an *unarguable fact* that (at the typical choice of standard states) a species
 in solution has less entropy than the same species in the gas phase. This *must*
 be the case because the molecules in an ideal gas are non-interacting,
 uncorrelated, and undergoing ballistic motion, whereas in solution molecules are
 *caged* by the solvent and undergoing diffusive motion, which means they have
 significantly less freedom of motion.
 The paper you link talks about *vibrational* motion, which is going to be
 *significantly* less perturbed by the presence of solvent. The
 *vast* majority of entropy for gas-phase species comes from translational
 motion, not vibrational motion.
 ---
 Finally, I'm not sure SMD is well-suited to give a *breakdown* of enthalpic and
 entropic terms to the free energy of solvation. The procedure you discuss will
 get *some* of that breakdown, but ultimately several contributions to
 ΔGsolv are all entangled with one another. 
 What I mean by that is if you simply perform two *single point* calculations on
 the same species, one with and one without SMD correction, the difference in the
 potential energies between those two calculations will include a mixture of
 enthalpic and entropic effects. 
 Doing the full thermodynamic calculations (i.e. doing a frequency calculation
 and reading the thermodynamic data printed in the Gaussian output file) will
 give you a *full* estimate for ΔGsolv. But, in both calculations (with and
 without SMD) the calculated enthalpic and entropic contributions are arising
 from the same set of approximations
 -- harmonic oscillator for vibration, rigid rotor for rotation, and *ideal gas*
 for translation.
 This works because the *potential energy* difference between the systems is
 parameterized such that the free energy difference calculated by Gaussian is a
 good approximatino of the total free energy of solvation. Since the enthalpy and
 entropy calculated by Gaussian for both calculations are using the same
 approximations, they will have very similar (but not identical!) values. All of
 those complicated factors such as the loss of translational entropy due to
 solvation are baked into the *electronic energy* that is calculated by the SMD
 method.
 The most *robust* way of separating enthalpic and entropic contributions would
 be to calculate the full value of the free energy at a variety of different
 temperatures, then numerically evaluating dG/dT to find the entropy. However,
 SMD as implemented in Gaussian is not temperature-dependent, so you cannot
 actually do this. There are other continuum solvation models developed by Cramer
 and Truhlar which do have temperature-dependence, which would allow you to do
 this, but as far as I can tell they are not included in Gaussian.
 ---
 I hope this answers all of the questions that you had. Please let me know if
 anything I said was unclear or you want additional assistance.
 Eric Hermes
 On Thu, 2017-04-06 at 06:24 +0000, MIELCZAREK Detlev Conrad detlev-
 conrad.mielczarek-.-ifpen.fr wrote:
 > Sent to CCL by: MIELCZAREK Detlev Conrad [detlev-
 > conrad.mielczarek]^[ifpen.fr] Dear CCL, a question on thermodynamic
 > data & solvation from me, maybe you can help me.
 >
 > So, the basic problem for me is, that I am calculating/want to
 > calculate thermodynamic data (Hf, S - hence also dG) in solvation,
 > using implicit solvation models, SMD with a COSMO cavity to be
 > specific. For my application, these should be accurate enough. (So no
 > molecular dynamics simulations etc.)
 >
 > Solvation models are normally parametrised for dGsolv - so this value
 > can be extracted from the quantum chemistry calculation as the
 > difference in the calculated Gibbs Free Enthalpy.
 > Hf can calculated easily in the gas phase, and a re-optimisation of
 > the structure with solvation should capture the majority of the impact
 > of solvation on the enthalpy. (Which is dominated by molecular
 > structure.)
 > (I guess there is the case of stabilisation and complexes, such as are
 > reported for water. However this is currently beyond the scope of my
 > work.)
 >
 > The topic of solvation has been discussed previously on the CCL here:
 > http://www.ccl.net/chemistry/resources/messages/2011/12/01.001-dir/
 > http://www.ccl.net/chemistry/resources/messages/2011/10/06.005-dir/
 > http://www.ccl.net/chemistry/resources/messages/2014/05/01.004-dir/
 > And there is the book "Essentials of Computational Chemistry Theories
 > and Models" from Professor Cramer with a section on phase change (the
 > source of my confusion).
 >
 > Specifically, the discussion concerning the energy change related to
 > the state conversion causes me some grief.
 >
 > On the one hand, the CCL responses read as if this should be applied
 > in the case of any phase change, but then others suggest this is
 > applicable only if the process is a second order reaction and thus the
 > total number of mols changes? - The latter view seems to agree with
 > the book...
 >
 > So if I have compound A in both the gas and liquid phase (from a
 > quantum chemistry calculation), do I need to account for the phase
 > change/change of state or not? Or is it something that can be included
 > in the parametrisation of the solvation model/the quantum chemistry
 > code already?
 >
 > Just to add more confusion to the topic: I have trialled a commercial
 > product which gives the Gibbs Enthalpy of Solvation in kcal/mol for
 > mol/L concentrations and using a very low end/fast functional, it
 > gives values similar to when a correction term is added... on the
 > other hand, where available, the calculated values without correction
 > agree with the published values in the SMD paper: http://pubs.acs.org
 > /doi/abs/10.1021/jp810292n (Supplementary Data)
 >
 > In addition, a regular computational chemistry calculation sees very
 > little (virtually no) difference in the entropy between the gaseous
 > and solvated phase. This would agree with the CCL-linked paper here h
 > ttp://pubs.acs.org/doi/abs/10.1021/jp205508z . But this would clash
 > with the common expectation that entropy in the liquid phase is
 > reduced...
 >
 > Hence, I would highly appreciate if someone knowledgeable in the field
 > of solvation could guide me onto the correct track.
 >
 > Detlev Conrad Mielczarek
 > Scientific Visitor/Post Doctorant
 > IFP Energies nouvelles
 > France
 >
 > www.ifpenergiesnouvelles.fr
 >
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 environment before printing !
 Ce message et toutes ses pièces jointes sont confidentiels et établis
 à l'intention exclusive de ses destinataires. Toute utilisation non
 conforme à sa destination, toute diffusion ou toute publication, totale ou
 partielle, est interdite, sauf autorisation expresse. IFP Energies nouvelles
 décline toute responsabilité au titre de ce message. This message and
 any attachments are confidential and intended solely for the addressees. Any
 unauthorised use or dissemination is prohibited. IFP Energies nouvelles should
 not be liable for this message.
 __________________________