Chris<= /div>

On Dec 2, 2011, at 2:23 AM, Andreas Klamt klamt = ~ cosmologic.de wrote:

Sent to CCL by: Andreas Klamt [klamt++cosmologic.de] =20 =20
Dear CCLers,

as Chris Cramer has shifted the thread from a special Gaussian to a general continuum solvation context, and since he suggests his post "as a useful archival reference for CCL users present and future" I need to make a few remarks on Chris' comprehensive entry:

- Chris is starting his entry with a reference to a paper by Ribero, Marenich, Cramer and Truhlar (RMCT), which he considers as a dispositive paper. Unfortunately that paper already in the abstract explicitly contradicts to our recent paper (Ho, Klamt, Coote, Comment on the correct use of continuum solvent models, J. Phys. Chem. A 2010, 114, 13442). This clearly shows that apparently some aspects are under controversial discussion between the experts, while Chris is describing things as if their view would be generally accepted. Without any question, I could agree with 70% of his statements, but I disagree with at least 20%.

- It should be especially noted that in the theoretical derivation of the expression for the effective vibrational potential of the solute given in the RMCT paper there is an important either mathematical or conceptional error: They are starting from a correct, classical formulation of the partition function, incl. integrals over the solute coordinates R and their momenta P, and over the external (solvent) coordinates r and momenta p. Then they separate the integral over R and define the remainder to be an entity W(R), and in the next sentence they identify W(R) with the potential of mean force (PMF(R)). But the remainder of the partition function includes the integration over the momenta P of the solute, while the PMF(R) at a given set of internal coordinates R is just the integral over all states of the outer system at these fixed coordinates R, i.e. PMF(R) does not include an integration over the kinetic states P of the solute. Hence W(R) and PMF(R) cannot be the same. By assuming that they are the same RMCT are tacitly omitting the P integral, but it is just this integral which is the problematic part in the partition function of a solute in a solvent. Hence all the further statement in the RMCT paper are based on a wrong derivation.

- Also the statements of Chris regarding the liberation free energy are only partly right: Yes, the volume integral gives the same (almost trivial) RTln(V/V0) result as in the gas phase, but Ben Naim has no expression for the integral over the kinetic states. Hence nobody, neither Ben Naim, nor Chris Cramer, nor me has a formally correct theory for the complete partition function of a solute in a solvent with respect to the translational, stiff-rotational, and vibration/rotation contributions. The solvation free energy contribution resulting from the differences of partition function compared to the ideal gas partition function are all effectively parameterized into the non-electrostatic contributions of our solvation models, e.g. in to the cavitation free energy, etc. Lukily, these contributions seem to be sufficiently systematic (most likely to first order surface proportional) in order to be subsumed in the functional forms of the non-electrostatic contributions. Otherwise all of our continuum solvation models would not work.

- But as a consequence of this - if we do not know how to formulate the partition function in solution, and since the differences are already empirically included in the solvation free energy of the continuum solvation mode - it is worthless, yes even counterproductive, to try to calculate the vibrational part of this partition function from the frequencies in the continuum solvent, as RMCT are suggesting in their paper.

I am absolute convinced, and I have stated this several times before in the CCL, and we worked it out in more detail in the Ho, Klamt, Coote paper) that the most consistent way to calculate the free energy in solution is
- to first calculate the total free energy in the gas phase as accurately as we can
- and then add the effective free energy of solvation of the solute as calculated with a good continuum solvation model (electrostatic + non-electrostatic, or the dG_solv in COSMO-RS)

Stay out of the calculation of the frequencies in solution for th purpose of a free energy calculation, unless you first write down a consistent theory for the partition function in solution, which would be highly appreciated, but seems to be out of reach.

Best regards

Andreas

Am 01.12.2011 06:38, schrieb Christopher Cramer cramer[A]umn.edu:
This question, in = various forms, does just keep coming up...

While I'm going to take the opportunity to refer to a detailed, formal, and I hope dispositive paper just published by the Minnesota group (doi 10.1021/jp205508z), I'll also make = some effort here to be more cookbooky in an explanation here. Pedagogy in front! Not just for Gaussian, but more = general.

The free energy of solvation is, clearly, defined as the DIFFERENCE in the free energy of a species in the gas phase and in solution. Free energy is an ensemble property, not a molecular property, so we are immediately faced with the need to make some approximations in order to render the modeling tractable.

In the gas phase, those approximations are by now fairly standard and nearly universal. We make the ideal-gas approximation (so that the partition function of a mole of molecules is simply the product of Avogadro's number of molecular partition functions), and we make the rigid rotator and harmonic oscillator approximations to simplify the rovibrational partition functions, and, voila, we have a means to compute a standard state free energy (Gibbs free energy -- free enthalpy for my colleagues in (far more logical) German-speaking countries). Pretty much every electronic structure program on earth will do this for you when you run a vibrational frequency calculation in the gas phase. Poof -- you get E, H, S (third law), and G (and Cv too, if you care).

Note that you should be careful to recognize that (i) unless you overrode it, the program chose some default standard state concentration (1 bar? 1 atm? you should know...) and temperature (nearly always 298 K) and scale factor for vibrational frequencies (typically 1, but that might not be the best value for a given level of theory) and (ii) the harmonic oscillator approximation is catastrophically bad for super-low frequency vibrations (below, say, 50 wavenumbers, to pick an arbitrary value). There are fixes for the latter problem, but I'll let someone else post about that.

Now, what about G of a solute in solution? Well, to begin, since we're not dealing with a pure component anymore (at least, not if the solvent is different than the solute), we need to assume that we have an ideal solution, and we should recognize that we're talking about a partial molar quantity (more often referred to in the rigorous literature as a chemical potential than a free energy, but that's a matter of tradition). In any case, we need to assemble a free energy as a sum of

electronic energy (i.e., potential and kinetic energy of electrons for a fixed set of nuclear positions)

coupling to medium (which includes electrostatic and non-electrostatic components, although, it being chemistry, EVERYTHING is electrostatic... -- in practice, however, with continuum models, electrostatic means what you get by assuming the molecule is a charge distribution in a cavity embedded in a classical dielectric medium in which case one can apply the Poisson equation -- non-electrostatic is everything else -- dispersion, cavitation, covalent components of hydrogen bonding, hydrophobic effects, you name it)

temperature dependent translational (?), rotational (?), vibrational, and conformational contributions -- the question marks indicate conceptual issues

So, a few points to bear in mind. The optimal geometry in solution is unlikely to be the same as that in the gas phase -- but it might be close. You just have to decide for yourself if you want to reoptimize or not.

Same for the vibrational and conformational contributions to free energy in solution -- they might be very, very close to those in the gas phase -- or they might not. If you assume that they ARE the same, you avoid having to do a frequency calculation in solution (and you avoid wondering what it means to do vibrational frequencies in a continuum, which in principle means a surrounding that is in equilibrium with the solute -- but how can a medium composed of molecules be fully in equilibrium with a molecular solute on the timescale of the solute's vibrations, since the solvent vibrations are on the same timescale?)

Note that if I assume no change in the various parts of the free energy EXCEPT for the electronic energy and solute-solvent coupling (more on that momentarily), life is pretty easy. The free energy of solvation is the difference in the self-consistent reaction-field (SCRF) energy INCLUDING non-electrostatic effects, and the electronic energy in the gas phase. That is, I look at the expectation value of <H+(1/2)V> for the solvated wave function, where V is the reaction field operator, add non-electrostatic effects (typically NOT dependent on the quantum wave function, so added post facto, although there are a few exceptions in the literature), subtract the expectation value of <H> for the gas-phase wave function (note that you might have done the two expectation values at different geometries, or you might have used the gas-phase geometry for both -- your choice -- the former is more "physical", certainly, but the latter is a useful approximation in many instances), and you are done. You've got the free energy of solvation FOR IDENTICAL STANDARD-STATE CONCENTRATIONS. That is, the number you have in hand assumes no change in standard-state concentration. However, many experimental solvation free energies are tabulated for, say, 1 atm gaseous standard states and 1 M solution standard states. To compare the computed value to the tabulated value, one needs to correct for the standard-state concentration difference.

In the interest of the cookbook, let me be more practical. Thus, let's say that I compute a gas-phase G value, including all contributions, electronic and otherwise of

-3.000 00 a.u.

and, let's say that the electronic energy alone in the gas phase is

-3.020 00 a.u. (so ZPVE and thermal contributions to G = are +0.020 00 a.u.)

and, finally, let's say that my SCRF calculation provides an electronic energy INCLUDING non-electrostatic effects of

-3.030 00 a.u.

In that case, my free energy of solvation is -0.010 00 a.u. (difference of -3.030 00 and -3.020 00 a.u.) And, if I want to think about my free energy in solution, I can make the assumption that there is no change in the ZPVE and thermal contributions, in which case I would have G in solution equals -3.010 00 a.u. (which is gas-phase G of -3.000 00 a.u. plus free energy of solvation -0.010 00 a.u.)

But, just to be clear, if my gas phase G referred to a 1 atm standard state concentration, for an ideal gas at 298 K and 1 atm, that implies a molarity of 1/24.5 M. If I want my G in solution to be for a 1 M standard state, I need to pay the entropy penalty to compress my concentration from 1/24.5 M to 1 M, which is about 1.9 kcal/mol (the proof is left to the reader...) So, my 1 M free energy in solution is not -3.010 00 but rather about -3.006 99 a.u.

The above is an example of how almost all free energies of solvation and free energies in solution are computed in the literature using continuum solvation models (at least if they're done properly!)

Lots of important details glossed over a bit above (in the interests of clarity, I demur). But, to be more thorough, let's note:

1) Why was it (1/2)V in the SCRF calculation? -- the = 1/2 comes from linear response theory and assumes that you spend precisely half of the favorable coupling energy organizing the medium so that it provides a favorable reaction field.

2) How can there be a translational partition function = for a solute in solution? There isn't one -- but there is something called a liberational free energy associated with accessible volume, and Ben-Naim showed some time ago that the value is identical to that for the a particle-in-a-box having the same standard-state concentration -- i.e., there is no change on going from gas-phase translational partition function to liberational partition function for the same standard-state concentration for an ideal solution. When there are issues with non-accessible volume, however, account must be taked (cf. Flory-Huggins theory).

3) How can there be a rotational partition function for = a solute in solution? There isn't one -- solute rotations become librations that are almost certainly intimately coupled with first-solvation-shell motions. In essence, assuming no change in "rotational partition function" implies assuming no free energetic consequence associated with moving from rotations to librations. This remains a poorly resolved question, but, in practice, since most continuum solvation models are semiempirical in nature (having been parameterized against experimental data) any actual changes in free energy have been absorbed in the parameterization as best as possible. If you find that unsatisfying, hey, feel free not to use continuum solvent models -- it's certainly ok by me...

4) Where did those non-electrostatic effects come from? Every model is different in that regard, and I won't attempt to summarize a review's-worth of material in an email. Lots of nice Chem. Rev. articles over the years on continuum models if you want to catch up.

Finally, what is described above is a popular approach for computing solvation free energies and free energies in solution, but by no means the only approach out there. A non-exhaustive list to compute either or both solvation free energies or free energies in solution includes free-energy perturbation from explicit simulations, RISM-based models, fragment-based models derived > from a statistical mechanical approach (including COSMO-RS and variations on that theme), fragment-based models from expert learning, and models relying on alternative physicochemical approaches to computing interaction energies (e.g., SPARC). These alternative models can be quantal, classical, or SMILESal (which is to say, more in the realm of chemoinformatics than physical chemistry). Let a thousand flowers bloom.

I hope that this post serves as a useful archival reference for CCL users present and future. Best wishes to all for a peaceful winter solstice (or summer, for my antipodeal colleagues).

Chris

On Nov 30, 2011, at 1:46 PM, Close, David M. CLOSED#,#mail.etsu.edu wrote:

--

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
<= div style=3D"margin-top: 0px; margin-right: 0px; margin-bottom: 0px; = margin-left: 0px; ">http://pollux.chem.umn.edu=
(website includes information about the textbook "Essentials
= of Computational Chemistry: Theories and Models, 2nd Edition")
--=20
PD. Dr. Andreas Klamt
CEO / Gesch=E4ftsf=FChrer
COSMOlogic GmbH & Co. KG
Burscheider Strasse 515
D-51381 Leverkusen, Germany

phone  	+49-2171-731681
fax    	+49-2171-731689
e-mail 	klamt^cosmologic.de
web    	www.cosmologic.de

HRA 20653 Amtsgericht Koeln, GF: Dr. Andreas Klamt
Komplementaer: COSMOlogic Verwaltungs GmbH
HRB 49501 Amtsgericht Koeln, GF: Dr. Andreas Klamt
-=3D This is automatically added to each message by the mailing script = =3D-E-mail to subscribers: CHEMISTRY:_:ccl.net or use: http://www.ccl.ne= t/cgi-bin/ccl/send_ccl_message E-mail to administrators: CHEMISTRY-REQUEST:_:ccl.net = or use http://www.ccl.ne= t/cgi-bin/ccl/send_ccl_message Subscribe/Unsubscribe:=20 http://www.ccl.net/c= hemistry/sub_unsub.shtml Before posting, check wait time at: http://www.ccl.net Job: http://www.ccl.net/jobs=20 Conferences: http:/= /server.ccl.net/chemistry/announcements/conferences/ Search Messages: http://www.ccl= .net/chemistry/searchccl/index.shtmlhttp://www.ccl.net/spammers.txt RTFI: http://www.cc= l.net/chemistry/aboutccl/instructions/

Christopher J. Cramer

Elmore H. Northey Professor

University of Minnesota

Department of = Chemistry

Minneapolis, MN 55455-0431

Phone: (612) 624-0859 || = FAX: (612) = 626-7541

email: cramer:_:umn.edu

jabber: cramer:_:jabber.umn.edu

http://pollux.chem.umn.edu<= /p>

(website includes = information about the textbook "Essentials

of Computational = Chemistry: Theories and Models, = 2nd Edition")

= --Apple-Mail-2-929885713-- From owner-chemistry@ccl.net Fri Dec 2 12:27:00 2011 From: "=?ISO-8859-1?Q?Jo=E3o_Henriques?= jmhenriques_._fc.ul.pt" To: CCL Subject: CCL:G: wb97XD in g09 Message-Id: <-45970-111202122432-7097-vz28mc8t3INLdLqPHTa/4w^server.ccl.net> X-Original-From: =?ISO-8859-1?Q?Jo=E3o_Henriques?= Content-Transfer-Encoding: 8bit Content-Type: text/plain; charset=ISO-8859-1 Date: Fri, 2 Dec 2011 17:24:25 +0000 MIME-Version: 1.0 Sent to CCL by: =?ISO-8859-1?Q?Jo=E3o_Henriques?= [jmhenriques()fc.ul.pt] Hi, I assume you are doing some population computations, so here is what you need to do: 1) On the command route, add the readradii flag under pop, e.g Pop(bla,bla,bla,readradii). 2) At the end of your input type the atom name and radius, e.g. Lu XX. 3) Do some research regarding the radius value. I never used Lu on my calculations. That's all, Best regards, On Thu, Dec 1, 2011 at 10:40 AM, quartarolo|a|unical.it wrote: > > Sent to CCL by: quartarolo _ unical.it > Dear all, > I'm running a calculation with Gaussian09 for a Lutetium complex using the > wB97XD functional. > The program stops with the following message: > "R6DR0: No vdW radius available for IA= 71". > Is there a way to define this parameter manually in the input file and > eventually where to search for vdW radius of lutetium. > > best regards > > Quartarolo Domenico > > ---------------------------------------------------------------- > This message was sent using IMP, the Internet Messaging Program. > > > > > > **** Riservatezza / Confidentiality **** > In ottemperanza al D.Lgs. n. 196 del 30/6/2003 in materia di protezione dei > dati personali, le informazioni contenute in questo messaggio sono > strettamente riservate ed esclusivamente indirizzate al destinatario > indicato (oppure alla persona responsabile di rimetterlo al destinatario). > Vogliate tener presente che qualsiasi uso, riproduzione o divulgazione di > questo messaggio e' vietato. Nel caso in cui aveste ricevuto questo > messaggio per errore, vogliate cortesemente avvertire il mittente e > distruggere il presente messaggio.> > > > Job: http://www.ccl.net/jobsConferences: > http://server.ccl.net/chemistry/announcements/conferences/> > > -- João Henriques, MSc in Biochemistry Faculty of Science of the University of Lisbon Department of Chemistry and Biochemistry C8 Building, Room 8.5.47 Campo Grande, 1749-016 Lisbon, Portugal E-mail: joao.henriques.32353__gmail.com / jmhenriques__fc.ul.pt http://intheochem.fc.ul.pt/members/joaoh.html From owner-chemistry@ccl.net Fri Dec 2 18:12:01 2011 From: "David A Mannock dmannock(~)ualberta.ca" To: CCL Subject: CCL:G: Delta G of solution calculation? Message-Id: <-45971-111202155103-3962-h+Qq0spj//HnsoJ5P2AkOQ^server.ccl.net> X-Original-From: David A Mannock Content-Type: multipart/alternative; boundary=0016e6d9772ae23eb804b3221f1f Date: Fri, 2 Dec 2011 13:50:51 -0700 MIME-Version: 1.0 Sent to CCL by: David A Mannock [dmannock..ualberta.ca] --0016e6d9772ae23eb804b3221f1f Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Andreas, That is an excellent and informative response to this question. In my own work the variation between stable and metastable hydration states in solution and at surfaces is a key feature that we have struggled to understand. Are there clearly defined relationships for the molecular structure and hydration number other than those defined in terms of the number of CH2/CH3/OH, etc. groups and an explanation of the changing hydration number with solute concentration? Also, if I recall the works of Stephen White, John Nagle et al. At membrane surfaces, the volume of the water molecule in the cavity space in the interfacial region is smaller than the 30Ang^3 value. This has been attributed to entropic effects, but I have not seen a more extensive explanation of this phenomenon. Dave On Fri, Dec 2, 2011 at 1:23 AM, Andreas Klamt klamt ~ cosmologic.de < owner-chemistry#,#ccl.net> wrote: > Sent to CCL by: Andreas Klamt [klamt++cosmologic.de] > Dear CCLers, > > as Chris Cramer has shifted the thread from a special Gaussian to a > general continuum solvation context, and since he suggests his post "as a > useful archival reference for CCL users present and future" I need to mak= e > a few remarks on Chris' comprehensive entry: > > - Chris is starting his entry with a reference to a paper by Ribero, > Marenich, Cramer and Truhlar (RMCT), which he considers as a dispositive > paper. Unfortunately that paper already in the abstract explicitly > contradicts to our recent paper (Ho, Klamt, Coote, Comment on the correct > use of continuum solvent models, J. Phys. Chem. A 2010, *114*, 13442). > This clearly shows that apparently some aspects are under controversial > discussion between the experts, while Chris is describing things as if > their view would be generally accepted. Without any question, I could agr= ee > with 70% of his statements, but I disagree with at least 20%. > > - It should be especially noted that in the theoretical derivation of the > expression for the effective vibrational potential of the solute given in > the RMCT paper there is an important either mathematical or conceptional > error: They are starting from a correct, classical formulation of the > partition function, incl. integrals over the solute coordinates R and the= ir > momenta P, and over the external (solvent) coordinates r and momenta p. > Then they separate the integral over R and define the remainder to be an > entity W(R), and in the next sentence they identify W(R) with the potenti= al > of mean force (PMF(R)). But the remainder of the partition function > includes the integration over the momenta P of the solute, while the PMF(= R) > at a given set of internal coordinates R is just the integral over all > states of the outer system at these fixed coordinates R, i.e. PMF(R) does > not include an integration over the kinetic states P of the solute. Hence > W(R) and PMF(R) cannot be the same. By assuming that they are the same RM= CT > are tacitly omitting the P integral, but it is just this integral which i= s > the problematic part in the partition function of a solute in a solvent. > Hence all the further statement in the RMCT paper are based on a wrong > derivation. > > - Also the statements of Chris regarding the liberation free energy are > only partly right: Yes, the volume integral gives the same (almost trivia= l) > RTln(V/V0) result as in the gas phase, but Ben Naim has no expression for > the integral over the kinetic states. Hence nobody, neither Ben Naim, nor > Chris Cramer, nor me has a formally correct theory for the complete > partition function of a solute in a solvent with respect to the > translational, stiff-rotational, and vibration/rotation contributions. Th= e > solvation free energy contribution resulting from the differences of > partition function compared to the ideal gas partition function are all > effectively parameterized into the non-electrostatic contributions of our > solvation models, e.g. in to the cavitation free energy, etc. Lukily, > these contributions seem to be sufficiently systematic (most likely to > first order surface proportional) in order to be subsumed in the function= al > forms of the non-electrostatic contributions. Otherwise all of our > continuum solvation models would not work. > > - But as a consequence of this - if we do not know how to formulate the > partition function in solution, and since the differences are already > empirically included in the solvation free energy of the continuum > solvation mode - it is worthless, yes even counterproductive, to try to > calculate the vibrational part of this partition function from the > frequencies in the continuum solvent, as RMCT are suggesting in their pap= er. > > I am absolute convinced, and I have stated this several times before in > the CCL, and we worked it out in more detail in the Ho, Klamt, Coote pape= r) > that the most consistent way to calculate the free energy in solution is > - to first calculate the total free energy in the gas phase as accurately > as we can > - and then add the effective free energy of solvation of the solute as > calculated with a good continuum solvation model (electrostatic + > non-electrostatic, or the dG_solv in COSMO-RS) > > Stay out of the calculation of the frequencies in solution for th purpose > of a free energy calculation, unless you first write down a consistent > theory for the partition function in solution, which would be highly > appreciated, but seems to be out of reach. > > Best regards > > Andreas > > > Am 01.12.2011 06:38, schrieb Christopher Cramer cramer[A]umn.edu: > > This question, in various forms, does just keep coming up... > > While I'm going to take the opportunity to refer to a detailed, formal, > and I hope dispositive paper just published by the Minnesota group > (doi 10.1021/jp205508z), I'll also make some effort here to be more > cookbooky in an explanation here. Pedagogy in front! Not just for Gaussia= n, > but more general. > > The free energy of solvation is, clearly, defined as the DIFFERENCE in > the free energy of a species in the gas phase and in solution. Free energ= y > is an ensemble property, not a molecular property, so we are immediately > faced with the need to make some approximations in order to render the > modeling tractable. > > In the gas phase, those approximations are by now fairly standard and > nearly universal. We make the ideal-gas approximation (so that the > partition function of a mole of molecules is simply the product of > Avogadro's number of molecular partition functions), and we make the rigi= d > rotator and harmonic oscillator approximations to simplify the > rovibrational partition functions, and, voila, we have a means to compute= a > standard state free energy (Gibbs free energy -- free enthalpy for my > colleagues in (far more logical) German-speaking countries). Pretty much > every electronic structure program on earth will do this for you when you > run a vibrational frequency calculation in the gas phase. Poof -- you get > E, H, S (third law), and G (and Cv too, if you care). > > Note that you should be careful to recognize that (i) unless you > overrode it, the program chose some default standard state concentration = (1 > bar? 1 atm? you should know...) and temperature (nearly always 298 K) and > scale factor for vibrational frequencies (typically 1, but that might not > be the best value for a given level of theory) and (ii) the harmonic > oscillator approximation is catastrophically bad for super-low frequency > vibrations (below, say, 50 wavenumbers, to pick an arbitrary value). Ther= e > are fixes for the latter problem, but I'll let someone else post about th= at. > > Now, what about G of a solute in solution? Well, to begin, since we're > not dealing with a pure component anymore (at least, not if the solvent i= s > different than the solute), we need to assume that we have an ideal > solution, and we should recognize that we're talking about a partial mola= r > quantity (more often referred to in the rigorous literature as a chemical > potential than a free energy, but that's a matter of tradition). In any > case, we need to assemble a free energy as a sum of > > electronic energy (i.e., potential and kinetic energy of electrons for a > fixed set of nuclear positions) > > coupling to medium (which includes electrostatic and non-electrostatic > components, although, it being chemistry, EVERYTHING is electrostatic... = -- > in practice, however, with continuum models, electrostatic means what you > get by assuming the molecule is a charge distribution in a cavity embedde= d > in a classical dielectric medium in which case one can apply the Poisson > equation -- non-electrostatic is everything else -- dispersion, cavitatio= n, > covalent components of hydrogen bonding, hydrophobic effects, you name it= ) > > temperature dependent translational (?), rotational (?), vibrational, > and conformational contributions -- the question marks indicate conceptua= l > issues > > So, a few points to bear in mind. The optimal geometry in solution is > unlikely to be the same as that in the gas phase -- but it might be close= . > You just have to decide for yourself if you want to reoptimize or not. > > Same for the vibrational and conformational contributions to free energy > in solution -- they might be very, very close to those in the gas phase -= - > or they might not. If you assume that they ARE the same, you avoid having > to do a frequency calculation in solution (and you avoid wondering what i= t > means to do vibrational frequencies in a continuum, which in principle > means a surrounding that is in equilibrium with the solute -- but how can= a > medium composed of molecules be fully in equilibrium with a molecular > solute on the timescale of the solute's vibrations, since the solvent > vibrations are on the same timescale?) > > Note that if I assume no change in the various parts of the free energy > EXCEPT for the electronic energy and solute-solvent coupling (more on tha= t > momentarily), life is pretty easy. The free energy of solvation is the > difference in the self-consistent reaction-field (SCRF) energy INCLUDING > non-electrostatic effects, and the electronic energy in the gas phase. Th= at > is, I look at the expectation value of for the solvated wave > function, where V is the reaction field operator, add non-electrostatic > effects (typically NOT dependent on the quantum wave function, so added > post facto, although there are a few exceptions in the literature), > subtract the expectation value of for the gas-phase wave function (no= te > that you might have done the two expectation values at different > geometries, or you might have used the gas-phase geometry for both -- you= r > choice -- the former is more "physical", certainly, but the latter is a > useful approximation in many instances), and you are done. You've got the > free energy of solvation FOR IDENTICAL STANDARD-STATE CONCENTRATIONS. Tha= t > is, the number you have in hand assumes no change in standard-state > concentration. However, many experimental solvation free energies are > tabulated for, say, 1 atm gaseous standard states and 1 M solution standa= rd > states. To compare the computed value to the tabulated value, one needs t= o > correct for the standard-state concentration difference. > > In the interest of the cookbook, let me be more practical. Thus, let's > say that I compute a gas-phase G value, including all contributions, > electronic and otherwise of > > -3.000 00 a.u. > > and, let's say that the electronic energy alone in the gas phase is > > -3.020 00 a.u. (so ZPVE and thermal contributions to G are +0.020 00 > a.u.) > > and, finally, let's say that my SCRF calculation provides an electronic > energy INCLUDING non-electrostatic effects of > > -3.030 00 a.u. > > In that case, my free energy of solvation is -0.010 00 a.u. (difference > of -3.030 00 and -3.020 00 a.u.) And, if I want to think about my free > energy in solution, I can make the assumption that there is no change in > the ZPVE and thermal contributions, in which case I would have G in > solution equals -3.010 00 a.u. (which is gas-phase G of -3.000 00 a.u. pl= us > free energy of solvation -0.010 00 a.u.) > > But, just to be clear, if my gas phase G referred to a 1 atm standard > state concentration, for an ideal gas at 298 K and 1 atm, that implies a > molarity of 1/24.5 M. If I want my G in solution to be for a 1 M standard > state, I need to pay the entropy penalty to compress my concentration fro= m > 1/24.5 M to 1 M, which is about 1.9 kcal/mol (the proof is left to the > reader...) So, my 1 M free energy in solution is not -3.010 00 but rather > about -3.006 99 a.u. > > The above is an example of how almost all free energies of solvation and > free energies in solution are computed in the literature using continuum > solvation models (at least if they're done properly!) > > Lots of important details glossed over a bit above (in the interests of > clarity, I demur). But, to be more thorough, let's note: > > 1) Why was it (1/2)V in the SCRF calculation? -- the 1/2 comes from > linear response theory and assumes that you spend precisely half of the > favorable coupling energy organizing the medium so that it provides a > favorable reaction field. > > 2) How can there be a translational partition function for a solute in > solution? There isn't one -- but there is something called a liberational > free energy associated with accessible volume, and Ben-Naim showed some > time ago that the value is identical to that for the a particle-in-a-box > having the same standard-state concentration -- i.e., there is no change = on > going from gas-phase translational partition function to liberational > partition function for the same standard-state concentration for an ideal > solution. When there are issues with non-accessible volume, however, > account must be taked (cf. Flory-Huggins theory). > > 3) How can there be a rotational partition function for a solute in > solution? There isn't one -- solute rotations become librations that are > almost certainly intimately coupled with first-solvation-shell motions. I= n > essence, assuming no change in "rotational partition function" implies > assuming no free energetic consequence associated with moving from > rotations to librations. This remains a poorly resolved question, but, in > practice, since most continuum solvation models are semiempirical in natu= re > (having been parameterized against experimental data) any actual changes = in > free energy have been absorbed in the parameterization as best as possibl= e. > If you find that unsatisfying, hey, feel free not to use continuum solven= t > models -- it's certainly ok by me... > > 4) Where did those non-electrostatic effects come from? Every model is > different in that regard, and I won't attempt to summarize a review's-wor= th > of material in an email. Lots of nice Chem. Rev. articles over the years = on > continuum models if you want to catch up. > > Finally, what is described above is a popular approach for computing > solvation free energies and free energies in solution, but by no means th= e > only approach out there. A non-exhaustive list to compute either or both > solvation free energies or free energies in solution includes free-energy > perturbation from explicit simulations, RISM-based models, fragment-based > models derived > from a statistical mechanical approach (including COSMO-= RS > and variations on that theme), fragment-based models from expert learning= , > and models relying on alternative physicochemical approaches to computing > interaction energies (e.g., SPARC). These alternative models can be > quantal, classical, or SMILESal (which is to say, more in the realm of > chemoinformatics than physical chemistry). Let a thousand flowers bloom. > > I hope that this post serves as a useful archival reference for CCL > users present and future. Best wishes to all for a peaceful winter solsti= ce > (or summer, for my antipodeal colleagues). > > Chris > > > > On Nov 30, 2011, at 1:46 PM, Close, David M. CLOSED#,#mail.etsu.eduwrote= : > > Does anyone know how Gaussian calculates deltaG of solvation? This > was in G98 and was automatic. In G03 one had to add SCFVAC in the > scrf(CPCM,read) read list to get the delta G result. I have several > questions about the methods used. I presume that during the SCRF > calculation the program has to have a separate SCF calculation on the inp= ut > molecule in the gas phase, along with a frequency calculation to know th= e > free energy of the gas phase structure. Is this correct? If so, are the > ZPE and free energy correction terms multiplied by a scale factor (0.92 f= or > example?). Also to do solvation energy calculations one need to convert > the gas phase reference state from 1 atmos. to 1 M. Does the program > report this corrected delta G value? **** > My reason for asking is that I have done these actual calculations > separately with energy/frequency calculations on the neutral and anion, a= nd > I get slightly difference answers from those answers using SCFVAC, and I > need to know why this is?**** > Regards, Dave Close.**** > > > -- > > > 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 > > (website includes information about the textbook "Essentials > > of Computational Chemistry: Theories and Models, 2nd Edition") > > > > > > > > > > -- > PD. Dr. Andreas Klamt > CEO / Gesch=E4ftsf=FChrer > COSMOlogic GmbH & Co. KG > Burscheider Strasse 515 > D-51381 Leverkusen, Germany > > phone +49-2171-731681 > fax +49-2171-731689 > e-mail klamt^cosmologic.de > web www.cosmologic.de > > HRA 20653 Amtsgericht Koeln, GF: Dr. Andreas Klamt > Komplementaer: COSMOlogic Verwaltungs GmbH > HRB 49501 Amtsgericht Koeln, GF: Dr. Andreas Klamt > > -=3D This is automatically added to each message by the mailing script = =3D-look u= p > the X-Original-From: line in the mail header. E-mail to subscribers: > CHEMISTRY#,#ccl.net or use:= E-mail to administrators: > CHEMISTRY-REQUEST#,#ccl.net or useBefore posting, check wait > time at: http://www.ccl.netConferences: > http://server.ccl.net/chemistry/announcements/conferences/ Search > Messages: http://www.ccl.net/chemistry/searchccl/index.shtml If your mail > bounces from CCL with 5.7.1 error, check:= RTFI: > http://www.ccl.net/chemistry/aboutccl/instructions/ --0016e6d9772ae23eb804b3221f1f Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Andreas, That is an excellent and informative response to this question. In= my own work the variation between stable and metastable hydration states i= n solution and at surfaces is a key feature that we have struggled to under= stand. Are there clearly defined relationships for the molecular structure = and hydration number other than those defined in terms of the number of CH2= /CH3/OH, etc. groups and an explanation of the changing hydration number wi= th solute concentration? Also, if I recall the works of Stephen White, John= Nagle et al. At membrane surfaces, the volume of the water molecule in the= cavity space in the interfacial region is smaller than the 30Ang^3 value. = This has been attributed to entropic effects, but I have not seen a more ex= tensive explanation of this phenomenon. Dave

On Fri, Dec 2, 2011 at 1:23 AM, Andreas Klam= t klamt ~ cosmologic.de <owner-chemistry#,#ccl.net<= /a>> wrote:

Sent to CCL by: Andreas Klamt [klamt++cosmologic.de] =20 =20 =20
Dear CCLers,

as Chris Cramer has shifted the thread from a special Gaussian to a general continuum solvation context, and since he suggests his post "as a useful archival reference for CCL users present and future&q= uot; I need to make a few remarks on Chris' comprehensive entry:

- Chris is starting his entry with a reference to a paper by Ribero, Marenich, Cramer and Truhlar (RMCT), which he considers as a dispositive paper. Unfortunately that paper already in the abstract explicitly contradicts to our recent paper (Ho, Klamt, Coote, Comment on the correct use of continuum solvent models, J. Phys. Chem. A 2010, 114, 13442). This clearly shows that apparently some aspects are under controversial discussion between the experts, while Chris is describing things as if their view would be generally accepted. Without any question, I could agree with 70% of his statements, but I disagree with at least 20%.

- It should be especially noted that in the theoretical derivation of the expression for the effective vibrational potential of the solute given in the RMCT paper there is an important either mathematical or conceptional error: They are starting from a correct, classical formulation of the partition function, incl. integrals over the solute coordinates R and their momenta P, and over the external (solvent) coordinates r and momenta p. Then they separate the integral over R and define the remainder to be an entity W(R), and in the next sentence they identify W(R) with the potential of mean force (PMF(R)). But the remainder of the partition function includes the integration over the momenta P of the solute, while the PMF(R) at a given set of internal coordinates R is just the integral over all states of the outer system at these fixed coordinates R, i.e. PMF(R) does not include an integration over the kinetic states P of the solute. Hence W(R) and PMF(R) cannot be the same. By assuming that they are the same RMCT are tacitly omitting the P integral, but it is just this integral which is the problematic part in the partition function of a solute in a solvent. Hence all the further statement in the RMCT paper are based on a wrong derivation.

- Also the statements of Chris regarding the liberation free energy are only partly right: Yes, the volume integral gives the same (almost trivial) RTln(V/V0) result as in the gas phase, but Ben Naim has no expression for the integral over the kinetic states. Hence nobody, neither Ben Naim, nor Chris Cramer, nor me has a formally correct theory for the complete partition function of a solute in a solvent with respect to the translational, stiff-rotational, and vibration/rotation contributions. The solvation free energy contribution resulting from the differences of partition function compared to the ideal gas partition function are all effectively parameterized into the non-electrostatic contributions of our solvation models, e.g. in to the cavitation free energy, etc.=A0 Lukily, these contributions seem to be sufficiently systematic (most likely to first order surface proportional) in order to be subsumed in the functional forms of the non-electrostatic contributions. Otherwise all of our continuum solvation models would not work.

- But as a consequence of this - if we do not know how to formulate the partition function in solution, and since the differences are already empirically included in the solvation free energy of the continuum solvation mode - it is worthless, yes even counterproductive, to try to calculate the vibrational part of this partition function from the frequencies in the continuum solvent, as RMCT are suggesting in their paper.

I am absolute convinced, and I have stated this several times before in the CCL, and we worked it out in more detail in the Ho, Klamt, Coote paper) that the most consistent way to calculate the free energy in solution is
- to first calculate the total free energy in the gas phase as accurately as we can
- and then add the effective free energy of solvation of the solute as calculated with a good continuum solvation model (electrostatic + non-electrostatic, or the dG_solv in COSMO-RS)

Stay out of the calculation of the frequencies in solution for th purpose of a free energy calculation, unless you first write down a consistent theory for the partition function in solution, which would be highly appreciated, but seems to be out of reach.

Best regards

Andreas

Am 01.12.2011 06:38, schrieb Christopher Cramer cramer[A]umn.edu:
This question, in various forms, does just keep coming up...

While I'm going to take the opportunity to refer to a detailed, formal, and I hope dispositive paper just published by the Minnesota group (doi=A010.1021/jp205508z), I'll also make s= ome effort here to be more cookbooky in an explanation here. Pedagogy in front! Not just for Gaussian, but more general.

The free energy of solvation is, clearly, defined as the DIFFERENCE in the free energy of a species in the gas phase and in solution. Free energy is an ensemble property, not a molecular property, so we are immediately faced with the need to make some approximations in order to render the modeling tractable.

In the gas phase, those approximations are by now fairly standard and nearly universal. We make the ideal-gas approximation (so that the partition function of a mole of molecules is simply the product of Avogadro's number of molecular partition functions), and we make the rigid rotator and harmonic oscillator approximations to simplify the rovibrational partition functions, and, voila, we have a means to compute a standard state free energy (Gibbs free energy -- free enthalpy for my colleagues in (far more logical) German-speaking countries). Pretty much every electronic structure program on earth will do this for you when you run a vibrational frequency calculation in the gas phase. Poof -- you get E, H, S (third law), and G (and Cv too, if you care).

Note that you should be careful to recognize that (i) unless you overrode it, the program chose some default standard state concentration (1 bar? 1 atm? you should know...) and temperature (nearly always 298 K) and scale factor for vibrational frequencies (typically 1, but that might not be the best value for a given level of theory) and (ii) the harmonic oscillator approximation is catastrophically bad for super-low frequency vibrations (below, say, 50 wavenumbers, to pick an arbitrary value). There are fixes for the latter problem, but I'll let someone else post about that.

Now, what about G of a solute in solution? Well, to begin, since we're not dealing with a pure component anymore (at least= , not if the solvent is different than the solute), we need to assume that we have an ideal solution, and we should recognize that we're talking about a partial molar quantity (more often referred to in the rigorous literature as a chemical potential than a free energy, but that's a matter of tradition). In any case, we need to assemble a free energy as a sum of

electronic energy (i.e., potential and kinetic energy of electrons for a fixed set of nuclear positions)

coupling to medium (which includes electrostatic and non-electrostatic components, although, it being chemistry, EVERYTHING is electrostatic... -- in practice, however, with continuum models, electrostatic means what you get by assuming the molecule is a charge distribution in a cavity embedded in a classical dielectric medium in which case one can apply the Poisson equation -- non-electrostatic is everything else -- dispersion, cavitation, covalent components of hydrogen bonding, hydrophobic effects, you name it)

temperature dependent translational (?), rotational (?), vibrational, and conformational contributions -- the question marks indicate conceptual issues

So, a few points to bear in mind. The optimal geometry in solution is unlikely to be the same as that in the gas phase -- but it might be close. You just have to decide for yourself if you want to reoptimize or not.

Same for the vibrational and conformational contributions to free energy in solution -- they might be very, very close to those in the gas phase -- or they might not. If you assume that they ARE the same, you avoid having to do a frequency calculation in solution (and you avoid wondering what it means to do vibrational frequencies in a continuum, which in principle means a surrounding that is in equilibrium with the solute -- but how can a medium composed of molecules be fully in equilibrium with a molecular solute on the timescale of the solute's vibrations, since the solvent vibrations are on the same timescale?)

Note that if I assume no change in the various parts of the free energy EXCEPT for the electronic energy and solute-solvent coupling (more on that momentarily), life is pretty easy. The free energy of solvation is the difference in the self-consistent reaction-field (SCRF) energy INCLUDING non-electrostatic effects, and the electronic energy in the gas phase. That is, I look at the expectation value of <H+(1/2)V> for the solvated wave function, where V is the reaction field operator, add non-electrostatic effects (typically NOT dependent on the quantum wave function, so added post facto, although there are a few exceptions in the literature), subtract the expectation value of <H> for the gas-phase wave function (note that you might have done the two expectation values at different geometries, or you might have used the gas-phase geometry for both -- your choice -- the former is more "physical", certainly, but the latter is a= useful approximation in many instances), and you are done. You've got the free energy of solvation FOR IDENTICAL STANDARD-STATE CONCENTRATIONS. That is, the number you have in hand assumes no change in standard-state concentration. However, many experimental solvation free energies are tabulated for, say, 1 atm gaseous standard states and 1 M solution standard states. To compare the computed value to the tabulated value, one needs to correct for the standard-state concentration difference.

In the interest of the cookbook, let me be more practical. Thus, let's say that I compute a gas-phase G value, including all contributions, electronic and otherwise of

-3.000 00 a.u.

and, let's say that the electronic energy alone in the gas phase is

-3.020 00 a.u. =A0(so ZPVE and thermal contributions to G are +0.020 00 a.u.)

and, finally, let's say that my SCRF calculation provides an electronic energy INCLUDING non-electrostatic effects of

-3.030 00 a.u.

In that case, my free energy of solvation is -0.010 00 a.u. (difference of -3.030 00 and -3.020 00 a.u.) And, if I want to think about my free energy in solution, I can make the assumption that there is no change in the ZPVE and thermal contributions, in which case I would have G in solution equals -3.010 00 a.u. (which is gas-phase G of -3.000 00 a.u. plus free energy of solvation -0.010 00 a.u.)

But, just to be clear, if my gas phase G referred to a 1 atm standard state concentration, for an ideal gas at 298 K and 1 atm, that implies a molarity of 1/24.5 M. If I want my G in solution to be for a 1 M standard state, I need to pay the entropy penalty to compress my concentration from 1/24.5 M to 1 M, which is about 1.9 kcal/mol (the proof is left to the reader...) So, my 1 M free energy in solution is not -3.010 00 but rather about -3.006 99 a.u.

The above is an example of how almost all free energies of solvation and free energies in solution are computed in the literature using continuum solvation models (at least if they'r= e done properly!)

Lots of important details glossed over a bit above (in the interests of clarity, I demur). But, to be more thorough, let's note:

1) =A0Why was it (1/2)V in the SCRF calculation? -- the 1/2 comes from linear response theory and assumes that you spend precisely half of the favorable coupling energy organizing the medium so that it provides a favorable reaction field.

2) =A0How can there be a translational partition function for a solute in solution? There isn't one -- but there is something called a liberational free energy associated with accessible volume, and Ben-Naim showed some time ago that the value is identical to that for the a particle-in-a-box having the same standard-state concentration -- i.e., there is no change on going from gas-phase translational partition function to liberational partition function for the same standard-state concentration for an ideal solution. When there are issues with non-accessible volume, however, account must be taked (cf. Flory-Huggins theory).

3) =A0How can there be a rotational partition function for a solute in solution? There isn't one -- solute rotations become librations that are almost certainly intimately coupled with first-solvation-shell motions. In essence, assuming no change in "rotational partition function" implies assuming no free energetic consequence associated with moving from rotations to librations. This remains a poorly resolved question, but, in practice, since most continuum solvation models are semiempirical in nature (having been parameterized against experimental data) any actual changes in free energy have been absorbed in the parameterization as best as possible. If you find that unsatisfying, hey, feel free not to use continuum solvent models -- it's certainly ok by me...

4) =A0Where did those non-electrostatic effects come from? Every model is different in that regard, and I won't attempt to summarize a review's-worth of material in an email. Lots of nic= e Chem. Rev. articles over the years on continuum models if you want to catch up.

Finally, what is described above is a popular approach for computing solvation free energies and free energies in solution, but by no means the only approach out there. A non-exhaustive list to compute either or both solvation free energies or free energies in solution includes free-energy perturbation from explicit simulations, RISM-based models, fragment-based models derived > from a statistical mechanical approach (including COSMO-RS and variations on that theme), fragment-based models from expert learning, and models relying on alternative physicochemical approaches to computing interaction energies (e.g., SPARC). These alternative models can be quantal, classical, or SMILESal (which is to say, more in the realm of chemoinformatics than physical chemistry). Let a thousand flowers bloom.

I hope that this post serves as a useful archival reference for CCL users present and future. Best wishes to all for a peaceful winter solstice (or summer, for my antipodeal colleagues).

Chris

On Nov 30, 2011, at 1:46 PM, Close, David M. CLOSED#,#mai= l.etsu.edu wrote:

= =A0 Does anyone know how Gaussian calculates deltaG of solvation?=A0 This was in G98 and was automatic.=A0 In G03 one had to add SCFVAC in the scrf(CPCM,read) read list to get the delta G result.=A0 I have several questions about the methods used.=A0 I presume that during the SCRF calculation the program has to have a separate SCF calculation on the input molecule =A0in the gas phase, along with a frequency calculation to know the free energy of the gas phase structure.=A0 Is this correct?=A0 If so, are the ZPE and free energy correction terms multiplied by a scale factor (0.92 for example?).=A0 Also to do solvation energy calculations one need to convert the gas phase reference state from 1 atmos. to 1 M.=A0 Does the program report this corrected delta G value?=A0<= u>

= =A0=A0My reason for asking is that I have done these actual calculations separately with energy/frequency calculations on the neutral and anion, and I get slightly difference answers from those answers using SCFVAC, and I need to know why this is?
= =A0 Regards, Dave Close.

--

Christopher J. Cramer

Elmore H. Northey Professor

University of Minnesota

Department of Chemistry

207 Pleasant St. SE

Minneapolis, MN 55455-0431

----------------------= ----

Phone:=A0= =A0(612) 624-0859 || FAX:=A0=A0= (612) 626-7541

Mobile: (952) 297-2575

email: =A0cramer:_:umn.edu

jabber: =A0cramer:_:jabber.umn.edu

http://pollux.chem.umn.edu
(website includes information about the textbook "Essentials

=A0 =A0=A0= of Computational Chemistry:=A0= =A0Theories and Models, 2nd Edition")
--=20
PD. Dr. Andreas Klamt
CEO / Gesch=E4ftsf=FChrer
COSMOlogic GmbH & Co. KG
Burscheider Strasse 515
D-51381 Leverkusen, Germany

phone  	+49-2171-731681
fax    	+49-2171-731689
e-mail 	klamt^co=
smologic.de
web    	www.cosmolog=
ic.de

HRA 20653 Amtsgericht Koeln, GF: Dr. Andreas Klamt
Komplementaer: COSMOlogic Verwaltungs GmbH
HRB 49501 Amtsgericht Koeln, GF: Dr. Andreas Klamt
-=3D This is automatically added to each message by the mailing script =3D-E-mail to subscribers: CHEMISTRY#,#ccl.net or use: http://www.ccl.net/cgi-bin/ccl/send_ccl_message E-mail to administrators: CHEMISTRY-REQUEST#,#ccl.net or use http://www.ccl.net/cgi-bin/ccl/send_ccl_message Subscribe/Unsubscribe:=20 http://www.ccl.net/chemistry/sub_unsub.shtml Before posting, check wait time at: http://www.ccl.net Job: http://www.ccl.n= et/jobs=20 Conferences: http://server.ccl.net/chemistry/announcements/co= nferences/ Search Messages: http://www.ccl.net/chemistry/searchccl/index.shtmlhttp://= www.ccl.net/spammers.txt RTFI: http://www.ccl.net/chemistry/aboutccl/instructions/

--0016e6d9772ae23eb804b3221f1f-- From owner-chemistry@ccl.net Fri Dec 2 20:13:01 2011 From: "Jun Zhang coolrainbow * yahoo.cn" To: CCL Subject: CCL: How to calculate the independent degrees of freedom of a molecule? Message-Id: <-45972-111202201132-2659-lky+hEg5/08k3IUVy/1BDw[a]server.ccl.net> X-Original-From: Jun Zhang Content-Type: multipart/alternative; boundary="-849746693-977869748-1322874681=:77500" Date: Sat, 3 Dec 2011 09:11:21 +0800 (CST) MIME-Version: 1.0 Sent to CCL by: Jun Zhang [coolrainbow%a%yahoo.cn] ---849746693-977869748-1322874681=:77500 Content-Type: text/plain; charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable Hello everyone:=0A=A0=0AFor a nonlinear N-atom molecule there are 3N-6 inte= rnal degrees of freedom, of course.=A0 However, for a point group-constrain= t molecule, such as C2v H2O, there are only 2 independent degrees of freedo= m. So, is=A0there a systematic way to determine the number and the contents= of independent degrees of freedom of an arbitarily symmetry-constraint mol= ecule? I heard that some knowledge like Polya theorem are requied, but I am= not sure. It will be better if some reviews can be reccommended.=0A=A0=0AT= hank you in advance. Any suggestions will be appreciated.=0A=A0=0ABest rega= rds!=0A=A0=0A--------------------------------------------------------------= --=0AJun Zhang (coolrainbow=-=yahoo.cn)=0AComputational Chemistry Group=0ANo.= 94, Weijinlu=0ANankai University =0ATianjin, China ---849746693-977869748-1322874681=:77500 Content-Type: text/html; charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable

Hello ever= yone:

For a nonlinear = N-atom molecule there are 3N-6 internal degrees of freedom, of course. = ; However, for a point group-constraint mol= ecule, such as C2v H2O, there are only 2 independent degrees o= f freedom. So, is there a systematic way to determine the number and t= he contents of independent degrees of freedom of an arbitarily symmetry-con= straint molecule? I heard that some knowledge like Polya theorem are requie= d, but I am not sure. It will be better if some reviews can be reccommended= .

Thank you in advance= . Any suggestions will be appreciated.

Best regards!

------------------= ----------------------------------------------
Jun Zhang (coolrainbow=-=ya= hoo.cn)
Computational Chemistry Group
No.94, Weijinlu
Nankai Unive= rsity
Tianjin, China

---849746693-977869748-1322874681=:77500--