Solvation SCF cycle
2.1 Maximizing accuracy in calculations
2.2.3 Deprotonation reactions—pK a calculations
barriers and enthalpic relative energies are usually on par with free energy barriers and free energy relative energies shows that the entropic contribution intrinsically treated in PBF solvation is minor.
In summary, calculations on associative ligand exchange reactions can be done ad- equately by DFT methods with appropriate free energy corrections and experimental solvation energies when available. Although by far the most popular of DFT meth- ods, B3LYP does not display itself in an outstanding light in this study. The overall errors in B3LYP can fairly easily be attributed to its larger errors in cationic transi- tion states. It is the versatility of B3LYP that still makes it a desirable choice when using DFT calculations, and still arguably superior to PBE and BP86 in multipurpose calculations.
The benefit to BP86 and PBE functionals is their faster performance in addi- tion to their mathematical purity. B3LYP and mPW1PW91 hybrid functionals are roughly equivalent to mPW1PW91, winning out over B3LYP slightly. There are no apparent gains in using kinetic operator gradients, as in the cases with M06 and SB98 functionals. Lastly, the M06 functional appears to be unreliable in these classes of calculations despite its heavy parameterization, though its unreliability could be due to its heavy parameterization.
Figure 2.3: Investigated deprotonation reactions
reactions add an additional hurdle from the standpoint of simulation, and that is treatment of a proton solvated in solution. The difficulty in treating protons is two- fold: (1) it is impossible to calculate E orEsolv from a species with no electrons, and (2) the compact charge of a proton in solvent is very difficult to model with implicit solvation.1 We present three different approaches to this problem:
1. All-QM simulations involving explicit H3O+: To calculate deprotonation reactions entirely with QM, the most straightforward approach would be to include an explicit H2O to the left side and an explicit hydronium ion (H3O+) to the right side of the balanced reaction. This allows all solvation energies to be calculated from QM.
Sample calculation:
The following reaction (which has a experimental ∆G = 9.6 kcal/mol) [Pt(Cl)3(H2O)]−+ H2O→[Pt(Cl)3(OH)]2−+ H3O+ has the following computed values:
[Pt(Cl)3(H2O)]−
Egas =−1576.385745 Eh =−989197.0 kcal/mol Esolv =−0.10808 Eh =−67.8 kcal/mol
ZPVE = 18.1 kcal/mol Hvib298K+ 3∗RT = 5.4 kcal/mol
1Recent studies within our group indicate that implicit solvation energies converge after adding 18 explicit waters to a QM simulation of proton, however most of this solvation energy is captured with 4 explicit waters.
−T∗Svib298K =−7.0 kcal/mol RT∗ln(24.5) = 1.9 kcal/mol
G=−989246.5kcal/mol [Pt(Cl)3(OH)]2−
Egas =−1575.727198 Eh =−988783.8 kcal/mol Esolv=−0.31737 Eh =−199.2 kcal/mol
ZPVE = 10.2 kcal/mol Hvib298K+ 3∗RT = 5.1 kcal/mol
−T∗Svib298K =−6.4 kcal/mol RT∗ln(24.5) = 1.9 kcal/mol
G=−988972.1 kcal/mol H2O
G=−47962.6 kcal/mol H3O+
Egas=−76.70581677 Eh =−48133.6 kcal/mol Esolv=−0.16034881 Eh=−100.6 kcal/mol
ZPVE = 22.0 kcal/mol Hvib298K+ 3∗RT = 1.8 kcal/mol
−T∗S =−3.7 kcal/mol (S obtained from (45.9 cal/mol∗K)[26] + (−33.6 cal/mol∗K)[27]) RT∗ln(24.5) = 1.9 kcal/mol
G=−48212.2 kcal/mol.
Thus, the relative free energy is:
∆G = [(−988972.1 kcal/mol) + (−48212.2 kcal/mol)]
−[(−989246.5 kcal/mol) + (−47962.6 kcal/mol)]
= 24.8 kcal/mol!
This result thus has an error of 15.6 kcal/mol, an obviously grave deficiency in this type of calculation, and likely the result of a poor description of the solvation energy of H3O+.
2. Simulations using experimental H3O+solvation energies: As we believe the description of H3O+ solvation is poor, we examined whether using the experimental solvation energy of H3O+ would improve the accuracy of the test calculations. For the same deprotonation reaction as above we calculate the relative energies:
[Pt(Cl)3(H2O)]− G=−989246.5 kcal/mol
[Pt(Cl)3(OH)]2− G=−988972.1 kcal/mol
H2O
G=−47962.6 kcal/mol H3O+
Egas=−76.70581677 Eh =−48133.6 kcal/mol ZPVE = 21.6 kcal/mol
Hvib298K + 3∗RT + pV = 2.4 kcal/mol
−T∗S =−14.4 kcal/mol Gsolv =−110.2 kcal/mol G=−48234.2 kcal/mol.
In this approach, calculating the deprotonation of [Pt(Cl)3(H2O)]− leads to a free energy difference of
∆G = [(−988972.1 kcal/mol) + (−48234.2 kcal/mol)]
−[(−989246.5 kcal/mol) + (−47962.6 kcal/mol)]
= 2.8 kcal/mol.
While this is a noticeable improvement over the QM solvation energy calculations, this ∼ 7 kcal/mol error is not satisfactory for the purposes of mechanistic investiga- tions.
3. Simulations using experimental H+ solvation energies An alternate ap- proach would be to use the experimental energies for H+ instead of the explicit hy- dronium. For the deprotonation reaction:
[Pt(Cl)3(H2O)]− →[Pt(Cl)3(OH)]2−+ H+
we calculate the relative energies:
Sample calculation:
[Pt(Cl)3(H2O)]− G=−989246.5 kcal/mol
[Pt(Cl)3(OH)]2−
G=−988972.1 kcal/mol H+
Hgas= 3∗RT = 1.8 kcal/mol
−T∗Sgas =−7.82 kcal/mol Gsolv=−254.0 kcal/mol[27]
GH+ =−270.3 kcal/mol.
In this approach, calculating the deprotonation of [Pt(Cl)3(H2O)]− leads to a free energy difference of
G = [(−988972.1 kcal/mol) + (−270.3 kcal/mol)]
−(−989246.5 kcal/mol)
= 4.1 kcal/mol
which is thus far the best agreement with the experimental value, 9.6 kcal/mol.
We now present data on calculations of the various DFT methods. At first glance, it should be clear how difficult deprotonation reactions can be in simulations. Errors in these cases are far above expected errors for stable intermediates, and even with treatments using experimental solvation energies, all calculations are underestimated by at least 5 kcal/mol. Interestingly, across the three deprotonation reactions, the absolute value of the mean error and the RMS error are essentially identical. This suggests that while an large error is present, a single empirical correction could be used to reduce the large errors in these specific calculations. The empirical correction would simply be the value of the mean error for that specific DFT method. Employing such a correction reduces errors to an RMS error of∼1 kcal/mol. While this is not a long- term solution, this result suggests that a critical shortcoming of DFT simulations can
metals are strongly warranted.
Barrier Heights (kcal/mol) Reaction ref. Thermo. Exp. Density Functionals
B3LYP BP86 M06 mPW PBE SB98 Pd-1 [28] ∆H‡ 12 12.9 12.7 10.7 13.7 10.8 12.7
∆G‡ 16 17.7 17.6 15.0 18.2 15.8 17.3 Pd-2 [28] ∆H‡ 10 13.0 11.4 7.1 12.5 9.8 11.8
∆G‡ 14 15.2 13.6 9.3 14.7 12.0 14.0 Pd-3 [29] ∆H‡ – 13.1 11.9 10.9 12.9 10.5 12.3
∆G‡ 19 18.5 17.5 15.9 18.0 16.3 17.6
Pd-4 [29] ∆H‡ – 12.3 9.9 5.8 10.8 8.4 10.7
∆G‡ 14 16.2 13.8 9.6 14.6 12.3 14.5 Pd-5 [29] ∆H‡ – 16.7 15.9 14.3 17.4 14.4 16.4
∆G‡ 20 21.1 20.4 18.2 21.4 19.1 20.6 Pd-6 [29] ∆H‡ – 15.9 13.4 8.4 15.0 11.8 14.7
∆G‡ 15 17.8 15.3 10.3 16.9 13.6 16.5
Pd-7 – ∆H‡ – 15.8 15.4 14.1 16.3 14.0 15.4
∆G‡ – 22.3 22.0 20.1 22.4 20.8 21.7
Pd-8 – ∆H‡ – 15.1 12.5 8.0 13.6 11.0 13.7
∆G‡ – 18.8 16.3 11.7 17.3 14.7 17.4
Pd-9 – ∆H‡ – 14.5 14.1 13.2 14.4 12.7 14.0
∆G‡ – 20.3 20.0 18.5 19.9 18.8 19.6
Pd-10 – ∆H‡ – 13.9 11.8 7.8 12.0 10.2 12.4
∆G‡ – 17.8 15.7 11.7 15.9 14.1 16.3 Pd-11 [28] ∆H‡ 14 16.1 15.7 15.2 16.8 14.5 16.0
∆G‡ 18 22.5 22.2 21.1 22.8 21.1 22.2 Pd-12 [28] ∆H‡ 10 15.0 12.5 8.4 14.0 11.1 14.1
∆G‡ 12 18.7 16.3 12.2 17.8 14.9 17.8
Relative thermodynamics (kcal/mol) Reaction ref. Thermo. Exp. Density Functionals
B3LYP BP86 M06 mPW PBE SB98
Pd-1!Pd-2 [28] ∆H 2 -0.1 1.3 3.6 1.2 1.0 0.9
∆G 2 2.5 4.0 5.7 3.4 3.8 3.3
Pd-3!Pd-4 [29] ∆H – 0.7 2.0 5.1 2.1 2.2 1.7
∆G 5 2.3 3.7 6.3 3.4 4.0 3.1
Pd-5!Pd-6 [29] ∆H – 0.8 2.5 5.9 2.4 2.7 1.7
∆G 5 3.3 5.2 8.0 4.6 5.5 4.0
Pd-7!Pd-8 – ∆H – 0.7 2.9 6.1 2.8 3.1 1.6
∆G – 3.5 5.8 8.4 5.1 6.1 4.2
Pd-9!Pd-10 – ∆H – 0.6 2.3 5.4 2.4 2.5 1.6
∆G – 2.5 4.3 6.7 3.9 4.7 3.3
Pd-11!Pd-12 [28] ∆H 4 1.2 3.2 6.8 2.7 3.3 2.0
∆G 6 3.7 5.9 8.9 5.0 6.2 4.3
Table 2.1: Calculated thermochemical data for associative ligand exchange reactions involving Pd(II)
Barrier Heights (kcal/mol) Reaction ref. Thermo. Exp. Density Functionals
B3LYP BP86 M06 mPW PBE SB98 Pt-1 [30] ∆H‡ 21 18.6 18.4 15.8 19.7 16.9 18.8
∆G‡ 23 23.1 23.0 19.7 23.9 21.6 23.0 Pt-2 [30] ∆H‡ 18.4 17.4 15.5 11.0 17.0 13.9 16.5
∆G‡ 21 19.4 17.4 12.9 18.9 15.9 18.4 Pt-3 [30] ∆H‡ 20 18.8 17.9 16.6 18.9 16.3 18.2
∆G‡ 23 24.0 23.2 21.3 23.8 21.8 23.2 Pt-4 [30] ∆H‡ 17.7 17.2 14.3 10.9 15.3 12.7 15.5
∆G‡ 19 19.5 16.7 13.3 17.7 15.1 17.8 Pt-5 [30] ∆H‡ 24 23.2 22.7 19.9 23.8 21.4 23.9
∆G‡ 28 27.6 27.3 23.8 28.0 26.2 28.2 Pt-6 [30] ∆H‡ 22.8 21.6 18.8 13.7 21.0 17.2 21.6
∆G‡ 23 23.3 20.6 15.4 22.7 19.0 23.4 Pt-7 [30] ∆H‡ – 22.8 21.7 19.9 23.1 20.8 22.8
∆G‡ 26 27.8 27.0 24.6 27.9 26.2 27.7 Pt-8 [30] ∆H‡ – 22.3 18.9 14.9 20.7 17.6 21.0
∆G‡ 26 25.8 22.4 18.4 24.2 21.1 24.5 Pt-9 [30] ∆H‡ – 19.5 18.7 17.7 20.3 17.3 19.9
∆G‡ 21 24.7 24.1 22.6 25.3 22.9 25.0 Pt-10 [30] ∆H‡ – 18.9 16.1 13.2 17.2 14.7 17.7
∆G‡ 18 22.5 19.7 16.8 20.8 18.3 21.3 Pt-11 [30] ∆H‡ – 24.0 22.8 21.3 24.3 21.8 23.8
∆G‡ 26 29.7 28.7 26.6 29.7 27.9 29.4 Pt-12 [30] ∆H‡ – 23.8 20.4 16.8 22.3 19.1 22.6
∆G‡ 20 26.8 23.5 19.9 25.3 22.2 25.7
Relative thermodynamics (kcal/mol) Reaction ref. Thermo. Exp. Density Functionals
B3LYP BP86 M06 mPW PBE SB98 Pt-1!Pt-2 [30] ∆H 2.6 1.2 2.9 4.8 2.7 2.9 2.3
∆G 2 3.7 5.6 6.9 4.9 5.8 4.6
Pt-3!Pt-4 [30] ∆H 2.3 1.7 3.6 5.7 3.6 3.6 2.7
∆G 4 4.4 6.5 8.1 6.1 6.7 5.4
Pt-5!Pt-6 [30] ∆H 1.2 1.6 3.9 6.2 2.9 4.2 2.3
∆G 5 4.2 6.7 8.5 5.3 7.2 4.8
Pt-7!Pt-8 [30] ∆H – 0.5 2.8 5.0 2.4 3.2 1.8
∆G 5 2.1 4.6 6.2 3.8 5.1 3.2
Pt-9 !Pt-10 [30] ∆H – 0.6 2.6 4.5 3.2 2.6 2.2
∆G 5 2.3 4.4 5.8 4.6 4.6 3.7
Pt-11!Pt-12 [30] ∆H – 0.3 2.4 4.5 2.0 2.7 1.2
∆G 6 2.9 5.2 6.7 4.4 5.6 3.7
Table 2.2: Calculated thermochemical data for associative ligand exchange reactions involving Pt
RMS errors (kcal/mol) Reaction Thermo. No. of Density Functionals
Reactions B3LYP BP86 M06 mPW PBE SB98
Pd reactions ∆H‡ 4 3.1 1.7 1.9 2.9 0.9 2.5
∆G‡ 8 3.3 2.5 3.6 3.1 2.4 2.9
Pt reactions ∆H‡ 6 1.3 2.9 6.3 1.5 4.3 1.8
∆G‡ 12 3.1 2.1 4.1 2.6 2.5 2.7
anionic reactions ∆H‡ 8 1.6 2.5 5.6 1.7 3.8 1.7
∆G‡ 12 1.2 1.6 4.9 1.2 2.7 1.1
cationic reactions ∆H‡ 2 3.8 2.1 1.4 3.5 0.9 3.2
∆G‡ 8 4.9 3.0 1.8 4.1 2.0 4.1
all reactions ∆H‡ 10 2.2 2.5 5.1 2.2 3.4 2.1
∆G‡ 20 3.2 2.3 3.9 2.8 2.5 2.7
RMS errors (kcal/mol) Reaction Thermo. No. of Density Functionals
Reactions B3LYP BP86 M06 mPW PBE SB98
Pd reactions ∆H 2 2.5 0.8 2.3 1.1 0.9 1.6
∆G 4 1.9 1.3 3.0 1.2 1.2 1.4
Pt reactions ∆H 3 0.9 1.7 3.7 1.2 1.9 0.7
∆G 6 2.4 2.0 2.9 1.7 2.1 1.9
anionic reactions ∆H 4 1.3 1.5 3.3 1.1 1.7 0.8
∆G 6 1.3 2.1 3.6 1.6 2.2 1.4
cationic reactions ∆H 1 2.8 0.8 2.8 1.3 0.7 2.0
∆G 4 3.1 0.9 1.6 1.5 0.6 2.1
all reactions ∆H 5 1.7 1.4 3.2 1.2 1.6 1.2
∆G 10 2.2 1.7 3.0 1.6 1.8 1.7
Table 2.3: RMS errors for associative ligand exchange reactions
Deprotonation Energies (kcal/mol)
Reaction Thermo. Exp. Density Functionals
B3LYP BP86 M06 mPW PBE SB98
Prot-1 ∆G 9.6 1.9 -0.6 3.5 4.3 -0.5 3.9
Prot-2 ∆G 7.1 0.9 -1.1 1.9 2.8 -1.1 2.7
Prot-3 ∆G 11.3 3.4 0.2 4.9 6.0 0.3 5.4
Mean Errors ∆G – -7.3 -9.8 -5.9 -5.0 -9.8 -5.3
RMS Errors ∆G – 7.3 9.9 6.0 5.0 9.8 5.4
RMS Errors with correction ∆G – 0.7 1.2 0.5 0.5 1.2 0.7 Table 2.4: Calculation results for deprotonation reactions