Subject: Summary: correlating in silico TS energies with ee values

From: Joe Harriman <s808o/at/unb.ca>
Date: Thu, 10 Jun 2004 10:43:23 -0300

Thanks for everyone for their replies.  The following is a summary of the responses
given to my initial question.  In short I had asked if there were any methods available
to correlate experimental ee values with TS energies obtained in silico.  Hope this can
help others as well.

Hi Joe,

The critical number you would need is the activation energy, i.e. the difference between
the transition state energy and the reactants ground state energy (ideally free
energies including the entropic term). This should correlate with the energy of
activation (Ea) from the Arrhenius equation. 

http://www.shodor.org/UNChem/advanced/kin/arrhenius.html

Then, you would need to test the correlation of the calculated activation energies with
some experimentally-derived numbers to:
 1) establish that the model and theory-level does reasonably reproduce your
experimentally observed results
 2) to create a 'calibration' equation to more accurately estimate 
 activation energies for new reaction paths

If kinetically controlled, then the product ratios should be related to the respective
reaction rates. If thermodynamically controlled, then the product energies (energies of
reaction) should control the product ratios. This approach has been applied in a paper
by: Malwitz, N., Reaction Kinetic  Modeling from PM3 Transition State Calculations, J.
Phys. Chem., Vol 99, 
 No. 15, 1995 p. 5291

 Regards,
 David Gallagher
 CAChe Group, Fujitsu
 Portland, Oregon

Hi,

This comes out of the Eyring or Arrhenius equations, which 
you find in any basic physical chemistry text book.  Arrhenius: k = A  exp(-Ea/RT); I
prefer Eyring, but they give the same results when you're looking at relative rates. 
As a first approximation assume that the constant(s) are equal for both paths (usually
including the entropy, a fairly strong approximation...).  I assume that you're looking
at an irreversible step, then the ratio of products, r, can be obtained simply as the
ratio of rate constants, r = exp(DEa/RT), where DEa is the difference in transition
state energy (all the constants disappear in the division).  When you have the ratio,
the ee is easily obtained from ee = (r-1)/(r+1), which is the excess divided by the
total, the definition of ee.

         /Per-Ola

Joe,

The enantiomeric excess can be expressed as a ratio which means that you should be able
to predict the ration from using the dG values at the dtationary point (TS).  At a
minimum, you need to do frequency calculations in  order to obtain dG values. You know
that one can consider that a pair of diastereomeric transition states (and they have to
be to have different TS energies as enantiomeric TS have equivalent energies) can be
considered to be an equilibrium reaction so K# = exp[-DG/RT]. So you can compare TS
Equilibrium constants K#(1)/K#(2) = exp[(DG(2)-DG(1))/RT]

> From this it is easy to see how you could predict ration of optical isomers.

 I hope this helps...

 Mark

Hi,

All other things equal, the energies can be treated as classical  
barriers relative to the ground-state for the start of the reaction. I would calculate
that as the end of an IRC run (or DRC with some handwaving), depending on the program
you are using. Then use the apparent DeltaEdagger in the Arrhenius equation.   Or use
the delta(deltaEdagger) as a measure of the relative ratios of the isomers.This should
lead (at worst) to a prediction of the predominant isomer, assuming kinetic control of
the reaction. Delta(deltaE) (from the two ends of the IRC) can be used as a  prediction
of the predominant isomer, assuming thermodynamic control.

 Sb

Hi,

First, I am not an expert in this area, and I will be very much 
interested in the summary of all the answers that
you will get.

That said, I think that it depends on wether the two enantiomeric  products are obtained
under a kinetic control or under thermodynamic control. If the experimental conditions
are such that you obtain the thermal equilibrium for your products then you should use
their energies to calculate their proportions. (Bearing in mind that enantiomers have
the same energy, then you should get ee=0.) Assuming that you are under kinetic
control, I would suggest using the  Transition State Theory that states that the rate
constant is proportional to exp(-Delta_G(TS)/RT). Then you can see that the
concentration of each product is proportionnal to k and thus to this exponential term :
[A]/[B]=exp(-(Delta_G(TS-A)-Delta_G(TS-B))/RT) If you think that the entropic
contribution is the same for both TS, you can write:
[A]/[B]=exp(-(Delta_E(TS-A)-Delta_E(TS-B))/RT)

 hope this helps,
 Paul.

Joe, 

        Please take a look at the following publication from Ken Houk's group.   

        I did the calculations for stereoselective hydroborating agents - mono and
di-isopinanylcampheylboranes.  We used a hybrid QM/MM method at that time due to the
difficulty of using a complete QM approach for the system. If you read the paper and
references carefully, you should see some formulas showing how to convert relative
energies into enantiomeric excess values. 

        There is also an earlier publication (of which I am a co-author) - should be
Tetrahedron, or something similar listed in the Science paper.  Sorry, I do not a copy
of the paper with me at the moment. 

        Contact me or Ken Houk (UCLA) if you have further questions. 

        Regards, 
        Jim Metz