26 May 2012

164. A rough approach to calculating redox potentials using dft

I'm not a computational or theoretical chemist, but I'd like to expand my skills in the area. Ergo, if you do post, don't ask me questions about theory -- I'm only at the learning stage myself. In addition, this is my interpretation of the paper I'm following.

EDIT: see here for a rough description of my take on a paper on reorganisational energies.

I'm reproducing papers to improve my understanding of what computational tools are available to 'wet' chemists. A good place to start for this is in the past -- typically, cutting edge methods are difficult to implement for someone who's not an expert in the field (Would you take a soldering iron to an NMR probe? We do it routinely, but most people wouldn't since it takes training.) and lacks the intuition to apply the right type of fudging. 'Old' papers (5-10 years) offer the advantage of having enough papers citing them that you get a crowd-sourced approach to evaluation of them, the methods described are often implemented directly in today's software packages and the computations are much, much cheaper with today's hardware -- often parts or even an entire paper can be reproduced in a day or two.

WARNING I may be doing something wrong WARNING

Here's my attempt at reproducing part of Baik and Friesner in J. Phys. Chem. A, 2002, 106, 7407-7412 - the exact approach is somewhat flavoured by  the specific output that gaussian and nwchem return.

The basic principle is that if we know the ΔG of a redox reaction, then we can calculate the redox potential:
ΔG=-n*F*E,
where n is the number of electrons, F is Faraday's constant and E is the (unreferenced) redox potential. If we know ΔG in kcal/mol, then E=ΔG/(-n*F), where F is ca 23.06 kcal/(mol.V).

The main complication is that a normal dft calc will give us the electronic energy, ε0, while we ultimately need the thermodynamic paramaters, e.g. ΔH, ΔS, ΔG. Luckily, they are easy (although computationally costly) to get by doing a frequency calculation.

The short of it:
We here define G=ε0+Gcorr.
Gaussian will give you Gcorr directly, while e.g. nwchem makes you work for it -- but not too much -- so that
Gcorr=Hcorr-T*Srot+vib+trans

NOTE:In the paper (using Jaguar) the ZPE is explicitly included, but in nwchem and gaussian the  Hcorr/Gcorr will already contain it. Approaches that I've seen online do not include ZPE and the computed data agrees better with experimental data if it's excluded. If I had to put money on anything, I'd say: make sure your software package includes ZPE in the Hcorr reported, and don't add it explicitly.

ΔGin gas phase is obtained by taking the difference in G of the product and reactant. All that remains is the solvation energy.

Throwing in solvation using a continuum solvent model is fairly easy, but can also lead to long computational times, so we will cheat  -- the original paper is from 2002, when some of the calcs took them 40 days. So we will do a single-point/energy calculation to get the solvation.

(I think a more modern approach -- given more powerful hardware -- would be to do everything with a solvent model included for small molecules. For e.g. polyoxometalates, however, this can still be costly when using COSMO, but using PCM is almost a requirement when using g09 since anions converge very slowly. Bottomline: decide on the approach and be prepared to justify it.)

Anyway:
Gsolvation≅ (ε0(using solvation model)+Gcorr(gas phase))-(ε0(in gas phase)+Gcorr(gas phase))=ε0(using solvation model)-ε0(in gas phase)=εsolvation.

When I only write ε0 I mean in gas phase. Here's for a reaction where A goes to B through a redox process:
ΔG≅(ε0solvation+Gcorr)B-(ε0solvation+Gcorr)A=ΔGin gas phase+ΔGsolvation

How:
You need to do the following steps:
  1. Draw benzophenone
  2. Optimise the structure in the gas phase i.e. without a solvent model
  3. Do a frequency calc of the optimised structure in the gas phase
  4. Do an energy (single point) calculation of the gas-phase optimised structure using a solvent model (e.g. PCM or COSMO). That is, you use the structure you optimised in the gas phase, then apply a solvent model and do a single-point energy calculation.

  5. Draw the benzophenone anion
  6. Optimise the structure in the gas phase i.e. without a solvent model
  7. Do a frequency calc of the optimised structure in the gas phase
  8. Do an energy (single point) calculation of the gas-phase optimised structure using a solvent model

2 will give you ε0. 3 will give you ε0 again, and Gcorr. 4 will give you ε0(using solvation model).
Steps 5-8 will do the same, but for the reduced species.


Gaussian inputs:
#p opt rb3lyp/6-31g**

benzophenone, neutral, gas phase

0 1
...xyz coordinates of starting guess...
and
#p freq rb3lyp/6-31g**

benzophenone, neutral, frequency

0 1
...xyz coordinates of optimised structure...
and
#p rb3lyp/6-31g** scrf=(pcm,solvent=acetonitrile)

benzophenone, neutral, solvation

0 1
...xyz coordinates of optimised structure...
and
#p opt rb3lyp/6-31g**

benzophenone, anion, gas phase

-1 2
...xyz coordinates of starting guess...

etc. Note the multiplicity -- one unpaired electron gives a multiplicity of 2 and a charge of -1. You can of course use the .chk file as a restart point, or do both opt and freq in a single calculation.

Nwchem input:
title "Benzophenone, neutral, gas phase"
start benzophenone_gas_opt
charge 0
geometry
       ...xyz coordinates of starting guess...
end
dft
       xc b3lyp
       mult 1
       grid fine
end
task dft optimize
and
title "benzophenone, neutral, gas phase, frequency"
start benzophenone_gas_freq
charge 0
geometry
       ...xyz coordinates of optimised structure...
end
dft
       xc b3lyp
       mult 1
       grid fine
end
task dft freq
and

title "benzophenone, neutral, COSMO"
start benzophenone_solv_energy
charge 0
geometry
       ...xyz coordinates of optimised structure...
end
cosmo
       dielectric 37.5
end
dft
       xc b3lyp
       mult 1
       grid fine
end
task dft energy
and
title "Benzophenone, anion, gas phase"
start benzophenone_anion_gas_opt
charge -1
geometry
       ...xyz coordinates of starting guess...
end
dft
      xc b3lyp
      mult 2
       grid fine
end
task dft optimize
etc.

You can of course throw two task directives in one and the same calculcation etc. You can also re-use movecs files. But I'm trying to make this as simple as possible to follow.

Gaussian output:
Neutral, gas phase opt:

SCF Done:  E(UB3LYP) =  -576.648098680     A.U. after   16 cycles
             Convg  =    0.3611D-08             -V/T =  2.0097
Neutral, gas phase freq:
 Zero-point correction=                           0.191763 (Hartree/Particle)
 Thermal correction to Energy=                    0.202482
 Thermal correction to Enthalpy=                  0.203427
 Thermal correction to Gibbs Free Energy=         0.154218
 Sum of electronic and zero-point Energies=           -576.456336
 Sum of electronic and thermal Energies=              -576.445616
 Sum of electronic and thermal Enthalpies=            -576.444672
 Sum of electronic and thermal Free Energies=         -576.493881
neutral, solvation=acetonitrile, energy:
SCF Done:  E(UB3LYP) =  -576.654962012     A.U. after   13 cycles
             Convg  =    0.2187D-08             -V/T =  2.0097
Anion gas phase opt:
SCF Done:  E(UB3LYP) =  -576.656219870     A.U. after   20 cycles
             Convg  =    0.6297D-08             -V/T =  2.0095
Anion gas frequency:
 Zero-point correction=                           0.187970 (Hartree/Particle)
 Thermal correction to Energy=                    0.198790
 Thermal correction to Enthalpy=                  0.199734
 Thermal correction to Gibbs Free Energy=         0.150074
 Sum of electronic and zero-point Energies=           -576.468250
 Sum of electronic and thermal Energies=              -576.457430
 Sum of electronic and thermal Enthalpies=            -576.456486
 Sum of electronic and thermal Free Energies=         -576.506146
Anion, solvation, energy:
SCF Done:  E(UB3LYP) =  -576.732948458     A.U. after   16 cycles
             Convg  =    0.5837D-08             -V/T =  2.0096
Putting it all together:
ε0,anion,gasphase=-576.656219870
Ganion,gasphase=-576.656219870+0.150074=-576.506145870
εsolvation,anion=-576.732948458-(-576.656219870)=-0.076728588

ε0,neutral,gasphase=-576.648098680
Gneutral,gasphase= -576.648098680+0.154218=-576.493880680
εsolvation,neutral=-576.654962012-(-576.648098680)=-0.006863332

ΔG=(-576.506145870-0.076728588)-(-576.493880680-0.006863332)=-0.082130446 Hartree=-51.537594039014 kcal/mol=ΔG/(-1*F)= -51.537594039014/(-1*23.06)=2.23493469379939288811  V. Referencing it against the SCE at 4.1888 V, we get -1.95386530620060711189 V vs SCE

Looking at the paper, they got -2.426+4.1888=1.7628 V absolute, corresponding to -40.6502 kcal/mol=-0.06478026 hartree. Remember that this was done using G09 and not Jaguar, which was used in the paper. The experimental value is -1.88 V. The maximum obtainable accuracy of DFT (according to the paper) is about 2 kcal/mol.


Nwchem output:
Neutral, gas
        Total DFT energy =     -576.648088887542
      One electron energy =    -2313.472465671473
           Coulomb energy =     1046.450004818160
    Exchange-Corr. energy =      -82.832342635322
 Nuclear repulsion energy =      773.206714601093
neutral, gas, freq
 Zero-Point correction to Energy  =  120.416 kcal/mol  (  0.191895 au)
 Thermal correction to Energy     =  127.114 kcal/mol  (  0.202569 au)
 Thermal correction to Enthalpy   =  127.706 kcal/mol  (  0.203513 au)
 Total Entropy                    =  103.010 cal/mol-K
   - Translational                =   41.486 cal/mol-K (mol. weight = 182.0732)
   - Rotational                   =   31.544 cal/mol-K (symmetry #  =        1)
   - Vibrational                  =   29.980 cal/mol-K
neutral, solvated, energy
                  COSMO solvation results
                  -----------------------
  
                 gas phase energy =      -576.6480876954
                 sol phase energy =      -576.6603289676
 (electrostatic) solvation energy =         0.0122412722 (    7.68 kcal/mol)

Anion gas

       Total DFT energy =     -576.656222314875
      One electron energy =    -2320.751987024161
           Coulomb energy =     1058.312412372760
    Exchange-Corr. energy =      -83.108633373896
 Nuclear repulsion energy =      768.891985710422
Anion gas, freq
Zero-Point correction to Energy  =  118.177 kcal/mol  (  0.188327 au)
 Thermal correction to Energy     =  124.894 kcal/mol  (  0.199031 au)
 Thermal correction to Enthalpy   =  125.486 kcal/mol  (  0.199975 au)
 Total Entropy                    =  102.152 cal/mol-K
   - Translational                =   41.486 cal/mol-K (mol. weight = 182.0732)
   - Rotational                   =   31.577 cal/mol-K (symmetry #  =        1)
   - Vibrational                  =   29.090 cal/mol-K
Anion solvated, energy
                 gas phase energy =      -576.6562158410
                 sol phase energy =      -576.7436282813
 (electrostatic) solvation energy =         0.0874124403 (   54.85 kcal/mol)

Which works out to
Gcorr=(127.706-298.15*103.010/1000)/627.509=0.15456920697551748261
ε0,neutral,gasphase=-576.648088887542
Gneutral,gasphase=-576.648088887542+0.15456920697551748261+0.191895=-576.49351968056648251739+0.191895
εsolvation,neutral=0.0122412722

Gcorr=(125.486-298.15* 102.152/1000)=0.15141494338035305421
ε0,anion,gasphase= -576.656222314875
Ganion,gasphase=  -576.656222314875+0.15141494338035305421+0.188327=-576.50480737149464694579+0.188327
εsolvation,anion=0.0874124403

ΔG=(-576.50480737149464694579-0.0874124403)-(-576.49351968056648251739-0.0122412722)=-0.08645885902816442840 Hartree=-54.25371216990443230085 kcal/mol => 2.35271952167842250687 V (absolute) <=> -1.836 V vs SCE. This is very close to the experimental value of -1.88 V but not at all close to the reported value in the paper of -2.426 V vs SCE.


Discussion:
Whether I'm doing something wrong or whether the differences in solvation models and implementation in differernt software packages is to blame, I don't know. 15 kcal/mol is a lot. 
However, not including solvation at all yields values which are completely off: -3.85 V vs SCE for gaussian and 3.88 for nwchem. The solvation free energy correction is the largest factor in yielding a 'correct' (as in order of magnitude) result -- and if I'd be forced to venture a guess, solvation models/implementations (and their empirical input) both differ a lot between modelling packages and have improved a lot in the intervening 10 year.

I'll post data for larger basis sets in a week or so to see whether the nwchem/cosmo value was a fluke or 'real'.

[g09 using aug-cc-pwdCVDZ:
single point E, gas phase using lanl2dz structure:
neutral :  -576.718523917
anion:     -576.746041821
ΔG=(-576.746041821+0.150074-0.076728588)-(-576.718523917+0.154218-0.006863332)= -0.101527160 Hartree =-1.426 V relative to SCE, vs -1.88 experimental and  -1.953 with LANL2DZ. That didn't help...]

19 comments:

  1. I should've looked this up ages ago haha...very helpful!

    ReplyDelete
  2. Very nice and clear instruction. Halim UWO.CA

    ReplyDelete
  3. base on the thermodynamic cycle you need to calculate solvation free energies for both reduced and oxidized forms.

    ReplyDelete
    Replies
    1. Thanks for the feedback.

      That's right. And as far as I can see that's what I did in the post -- let me know where it is missing and I'll fix the post.

      Delete
    2. Everything is fine... my bad... i missed out something...
      About gaussian, do i have to add ZPE to the thermal correction to free energy?

      Delete
    3. This is a good source of information for thermochemistry in gaussian: http://www.gaussian.com/g_whitepap/thermo/thermo.pdf

      In that document, the ZPE is only used for the atomization energy calculation, but not the enthalpy/ free energy calcs.

      Delete
  4. Can you please help me. What changes in the calculation do I have to make if instead of calculating the reduction potential, I will calculate the oxidation potential. I understand that the Delta G will be that corresponds to the Free energy of oxidation. I am confused with regards to the substitution of this value to the E=-DeltaG/nF... I tried some calculations and the results were questionable as to its value.

    ReplyDelete
    Replies
    1. At its simplest the oxidation reaction is simply the reduction reaction reversed.

      Ox -> Red
      Red -> Ox

      Same goes for the calculation of Ox vs Red potentials -- just change the sign of DG.

      As for questionable values: assuming that you are aware of DG giving the ABSOLUTE, not relative, potential, it may simply be due to the inadequacy of the solvation model, or poorly optimised structures, or the real reaction proceeding in a more complex way than what you have modelled.

      Can you give me a simple example of what you are trying to do? E.g. Benzene -> Benzene^- vs Benzene -> Benzene^+?

      Delete
    2. The system that i've been working has an experimental value of 0.48V oxidation potential (vs Ag/AgCl). Hence, the reaction would be

      [MnL] -> [MnL]^+ +e-.

      With regards to the changing of the sign fo DG, does is it mean that the equation will now be:
      E=DG/nF instead of E=-DG/nF? and DG is the free energy of ionization which has always a positive value?

      Delete
    3. Please try to see if what I did is correct,
      -the reported value is 0.49 V oxidation potential.
      -the reaction i presume shud be : [ML] ---> [ML]^+ + e^-
      -and so I shud calculate DG as:
      DG = G([ML]^+) - G([MN])
      -then use the DG above to this equation
      E=DG/nF <--- is this correct? the sign of DG here is + ?

      Delete
    4. No, if you write the reaction like this it should be -DG -- i.e. what I meant was that the free energy difference for the oxidation potential is the same as DG for the reduction potential, but with a sign change.

      Remember that DG gives you the absolute potential, whereas the expt value is a relative value. You'll need to find the absolute potential for Ag/AgCl and correct for that.

      Delete
  5. Thanks a lot, very useful information.

    ReplyDelete
  6. In Gaussian:
    1. We look for Gibbs's free energy but when you calculate solvation energy you don't consider thermochemistry calculations. Why? Is it the same?
    2. Why do not consider molecule relaxation in solvent? Is not necessary?
    3. How to include thermal contributions in the IP or EA calculations? Are they included in the Gcorr?

    I really want to say that your Blog is awesome!!! It's so useful!

    Best Regards

    ReplyDelete
  7. In Gaussian:
    1. We look for Gibbs's free energy but when you calculate solvation energy you don't consider thermochemistry calculations. Why? Is it the same?
    2. Why do not consider molecule relaxation in solvent? Is not necessary?
    3. How to include thermal contributions in the IP or EA calculations? Are they included in the Gcorr?

    I really want to say that your Blog is awesome!!! It's so useful!

    Best Regards

    ReplyDelete
    Replies
    1. The answer to 2 is probably the easiest and most important: when using COSMO (and PCM /I think/) you should never use it during geometry optimisation. The method was only designed to provide solvation energies for _gas phase_ structures -- you shouldn't use COSMO/PCM during geometry optimisation as it wasn't designed for this.

      On the other hand, a lot of people DO use COSMO/PCM in that way, but it just isn't 'right' (note that that says nothing about the accuracy about either the energies obtained or the faithfulness of the structure).

      Also, the more I learn (and I've got a long, long way to go) the less I trust implicit solvation models and the more hesitant I become in using data obtained with them to draw any conclusions. If the gas phase energy trends and the COSMO/PCM trends diverge at all you'll need to spend some time hammering out exactly why that is. And given the complexity of typical computational targets today you're likely better off not playing with implicit solvation models if solvation is important.

      Because here's the final issue: implicit solvation models like PCM and COSMO are often all but useless for anything other than neutral molecules which don't take part in any special interactions with the solvent (e.g. pi stacking, hydrogen bonding).

      If solvent interaction is key, then you'll need to include explicit solvent molecules.

      Either way, the answer to question 2 is: at least for COSMO you should only apply it to the gas phase geometries.

      Delete
    2. Answer to 3: yes, Gcorr=H-T*S, so that's the most immediate way of correcting for temperature.

      Note that that assumes that H and S don't change with temperature -- I think they are calculated assuming T=298.15 K in nwchem and presumably something similar in G09, but I'm just speculating here.

      Delete
    3. I'm not entirely sure what you mean by question 1, but I think I answered it as part of the answer to question 2 -- the implicit solvation model just gives an energy correction term for the gas phase structure.

      Delete
    4. In the first question I try to say that we use the SCF energy to calculate solvation free energy.
      Strictly, we must to use the thermochemical function using:
      DeltaG (solvation)= Gcorr(solv) - Gcorr(gas)

      I don't know if this relation is correct. What do you think? Is correct relate SCF energy with Free energy in a direct way (without considering thermal correction to free energy)?

      I'm really grateful for your answers

      Delete
    5. Sorry for the delayed reply.

      Whereas the solvation correction tends to be referred to as the free energy of solvation, I think a more appropriate (practical) way of looking at it is as a correction to the electronic energy i.e. you can use the solvation correction even if you're just doing an electronic calculation.

      So whereas I think that what you say makes sense, in practice the solvation is used as a correction to the electronic energy.

      Delete