Subject: mixing entropies for a molecule with multiple minima (corr.)

From: Patrick Bultinck <[email-masked]>
Date: Sat, 5 Aug 2000 14:16:07 +0100 (NFT)

Dear all,

There has been a little confusion on the question I raised, because I 
jumped to DS° calculations too rapidly (some DX° notations should have 
been X°). So here is the corrected version :

>I would like to calculate thermodynamical parameters (H°, S°
>and G°) for an ensemble of conformations of a molecule. H° etc. can
>easily be calculated for each conformation by calculating the hessian. G°
>is also easily calculated of course. Using Boltzmann's law, one can
>calculate the mole fractions of each conformation you would have for one
>mole of the molecule. H° is easily calculated for the ensemble using the
>individual mole fractions and H° values for each conformation. S° also in
>first approximation, but... should S° not be augmented with a mixing
>entropy term because you are mixing different conformations, which you all
>treat separately as ideal gases ? I think you would need to add a term
>S°(mix)=sum(x_i*ln(x_i)), where x_i is the mole fraction for conformation
>i. Since you treat them as ideal gases, nothing changes in H° for the
>ensemble (they do not interact). Using the mixing-corrected S°, a new G°
>can be calculated.
>
>So I am using the following steps to calculate G° for an ensemble of
>conformations of the same molecule :
>- Calculate H°, S° and G° for each conformation separately
>- Correct where necessary for chiral degeneracy of each conformation, and
>obtain (for some conformations) chiral corrected values (not every ab
>initio program does this on the fly).
>- Calculate Boltzmann determined occupations for all conformations.
>- Calculate H° for the ensemble of conformations, as well as S° from
>individual H° and S° values of all conformations
>- Correct S° with a mixing term, accounting for the fact that each
>conformation is an ideal gas, and that mixing ideal gases brings about a
>mixing entropy.
>- Calculate a new G° value from H° and the corrected S° value.
>
>In my opinion, only then can I calculate the DG° value in a reaction :
>metal + ligand -> complex.
>
>I would very much appreciate your opinions on this. I am not ignorant in
>thermodynamics, but I would like to have more experienced people to comment
>on my way to calculate G° for an ensemble of conformations of the same
>molecule. Especially for the fact that Boltzmann is used to calculate mole
>fractions for each conformation, based on individual values for each
>conformation, and that then later on, a term is added to S° which can only
>be calculated for an ensemble (but to calculate S_mix, you need the mole
>fractions first)...
>
>I think the method I suggest might be correct, since not including 
S(mix) terms for the ligand in two reactions sometimes yields crazy 
results, e.g. entropically more favourable formation of 6-membered ring 
complex than a 5-membered ring complex starting from two different 
chelating ligands (ethylenediamine and 1,3-propanediamine).

>Thanks,
>Patrick Bultinck
>Ghent University
>Belgium