First-Principles
Prediction of Acidities in the
Gas and Solution Phase
Prof. Michelle Coote and Dr Junming Ho
1
Why are Chemists
interested in pKas?
2
1
Chemical Speciation
•  Oxicams (polyprotic acids) are non-steroidal antiinflammatory compounds
CH3
S
OH O
N
S
O O
CATION
N
H
CH3
N
H
K1ZN
K1ZO
Ka1
K1N
O
O
N
H
N
S
CH3
O O
zwitterion (O)
CH3
CH3
CH3
OH
S
OH O
S
KT2
N
N
H
N
S
CH3
neutral product
O O
KT1
N
H
O
S
N
N
H
N
S
CH3
O O
zwitterion (N)
NEUTRAL
Ka2
K2N
K2ZO
K2ZN
CH3
O
O
N
S
O O
S
N
H
CH3
ANION
N
3
Predicting Protonation State
•  Polyamines are potential ion channel blockers
•  Electrostatic interactions to anionic residues in
channel.
• 
http://sf.anu.edu.au/~rph900/dan/bsp_animation_long.mov
H2N
N
NH2
1+
2+
3+
4+ ???
NH2
fA =
1
[H+ ]
[H+ ]2
[H+ ]3
[H+ ]4
; fHA+ =
; fHA2+ =
; fHA3+ =
; fHA 4+ =
2
3
4
Do
K1Do
K1K 2Do
K1K 2K 3Do
K1K 2K 3K 4Do
4
2
Thermochemical cycles
pK a =
HA (aq)
!Gsolv(HA)
!Gsoln
*
!Gsoln
RT ln(10)
H+(aq) + A-(aq)
!Gsolv(H+)
!Ggas
HA (g)
!Gsolv(A-)
H+(g) + A-(g)
!Gsoln = !Ggas + !Gsolv (H + ) + !Gsolv (A" ) " !Gsolv (HA)
5
Thermochemical cycles
!Gsoln = !Ggas + " ni !GS (Product i ) - " n j !GS (Reactant j )
i
j
•  The solution phase reaction energy of a reaction in
any solvent can be calculated in this manner
provided that
‒  ΔGs of reactants and products are available either from
experimental measurements or calculations
6
3
Standard States
HA (aq)
!G*soln
!G*solv(HA)
!G*solv(H+)
!Gogas
HA (g)
• 
• 
• 
H+(aq) + A-(aq)
!G*solv(A-)
H+(g) + A-(g)
Standard state for solutions (*) is 1 mol L-1 (or 1 molal)
Standard state for gas phase (o) is 1 atm (or 1 bar)
A correction is needed to convert ΔGogas to standard state of 1 mol L-1
%
!G*gas = !G°gas + !nRTln(RT)
R=8.314 J K -1mol-1; R=0.0821 L atm K -1mol-1
• 
Correction term arises from changes in translational entropy as the
pressure (or concentration) of the ideal gas changes
•  Δn is the number of moles of products less reactants
7
Example 1. Construct a thermodynamic cycle and use it to
calculate the pKa (at 298 K) of hydrofluoric acid using the following
data provided
!GS* (HF) = "31.7 kJ/mol
!GS* (F" ) = "439.3 kJ/mol
!GS* (H+ ) = "1112.5 kJ/mol
pK a =
*
!Gsoln
RT ln(10)
HF(g) # H+ (g)+F- (g); !Go =1529 kJ/mol
8
4
Example 1. Construct a thermodynamic cycle and use it to
calculate the pKa (at 298 K) of hydrofluoric acid using the following
data provided
!GS* (HF) = "31.7 kJ/mol
!GS* (F" ) = "439.3 kJ/mol
!GS* (H+ ) = "1112.5 kJ/mol
pK a =
*
!Gsoln
RT ln(10)
HF(g) # H+ (g)+F- (g); !Go =1529 kJ/mol
*
o
% ) = 17.1 kJ mol-1
!Gsoln
= !Ggas
+ !GS* (F " ) + !GS" (H + ) " !GS* (HF) + RT ln(RT
pK a =
*
!Gsoln
= 2.9
RT ln(10)
9
The key ingredients
•  ΔGgas from molecular orbital calculations or density functional methods
• 
ΔGS from continuum solvent models or MD simulations
10
5
Gas phase acidities
o
!Gacid
= G o (H + ) + G o (A- ) - G o (HA)
= G o (H + ) + E(A- ) - E(HA) + Gcorr (A" ) " Gcorr (HA)
-26.3 kJ mol-1 @ 298 K
Ideal gas partition
functions
Thermal corrections
(including ZPVE)
based on the harmonic
oscillator rigid rotor
model
(opt + freq; low level)
Electronic energies
From MOT or DFT
methods
(Single-point; high level)
11
Gas phase acidities
60.0
MAD (neutrals)
ADmax (neutrals)
50.0
MAD (cations)
ADmax (cations)
40.0
30.0
20.0
10.0
0.0
BP86
B971
B3LYP
BMK
M05-2X
HF
MP2
G3MP2+ CBS-QB3
Calculated gas phase acidities using different levels of theory for electronic energies.
Calculations were based on B3LYP/6-31+G(d) geometries and corresponding
thermal corrections
12
6
!Gogas
HF(g)
3.
F-(g) + H+(g)
Solvation free energy and Continuum solvent models
Clearly, an essential ingredient for the computation of !G*soln is the solvation free energy of each
species that appear in the chemical reaction, !G*s. This is typically calculated in Gaussian 03
using a continuum solvent model, where it is evaluated as the interaction energy between the
solute and the solvent which is treated as a bulk dielectric continuum. A schematic drawing
illustrating the difference between an explicit and continuum solvent model is shown below:
Solvation Gibbs Energies
Molecular dynamics simulations are expensive
Continuum models are much more cost effective and can deliver comparable, if not
better accuracy.
10
13
Continuum model solvation energies
!Gs* = !GES + !Gcav + !Gdisp"rep
Main parameters:
(1)  Level of theory – HF? DFT? MP2? Basis set?
- Affects ΔGES
(2)  Choice of radii {rH, rC, rN, rO etc}
-  Affects ΔGES , ΔGcav and ΔGdisp-rep
Almost all continuum models contain parameters, e.g. {ri}, which have been
optimized at a particular level of theory to reproduce experimental solvation energies.
Semi-empirical nature of these models means that it is important to adhere to
parameterization protocol for best accuracy.
14
7
The CPCM-UAHF model
The Conductor-like Polarisable Continuum Model.
(1)  Level of theory: HF/6-31G(d) for neutrals and HF/6-31+G(d) for ions
(2)  UAHF: The radius of each atom contains additional parameters which takes into
account the formal charge and hybridisation of the atom.
(3)  Accuracy: ~ 4 kJ mol-1 for neutrals and 15 kJ mol-1 for ions.
15
Gaussian03. Input for Gaussian09 is different. See Appendix
16
8
Variational PCM results
=======================
<psi(0)| H |psi(0)>
(a.u.) =
<psi(0)|H+V(0)/2|psi(0)>
(a.u.) =
<psi(0)|H+V(f)/2|psi(0)>
(a.u.) =
<psi(f)| H |psi(f)>
(a.u.) =
<psi(f)|H+V(f)/2|psi(f)>
(a.u.) =
Total free energy in solution:
with all non electrostatic terms
(a.u.) =
-------------------------------------------------------------------Electrostatic contributions to solvation free energy.
-76.010631
-76.021150
-76.022089
-76.009608
-76.022097
(Unpolarized solute)-Solvent
(kcal/mol) =
(Polarized solute)-Solvent
(kcal/mol) =
Solute polarization
(kcal/mol) =
Total electrostatic
(kcal/mol) =
-------------------------------------------------------------------“Non-electrostatic” contributions to solvation free energy.
-6.60
-7.84
0.64
-7.19
Cavitation energy
Dispersion energy
Repulsion energy
Total non electrostatic
What we want!
ΔG*s
4.45
-5.15
1.40
0.70
(kcal/mol) =
(kcal/mol) =
(kcal/mol) =
(kcal/mol) =
DeltaG (solv)
(kcal/mol) =
-6.50
--------------------------------------------------------------------
Beware! This is not the Gsoln that we seek.
-76.020988
This is ΔG*s(H2O)
17
Effects of geometrical relaxation
The above example calculates the solvation free energy by performing BOTH solvent
and gas phase calculation on the solution-optimised geometry.
Provided that solution and gas phase geometries are very similar, this is a
reasonable approximation.
For conformationally flexible molecules, where solution phase and gas phase
geometries differ significantly, effects of geometrical relaxation needs to be added
into ΔGs.
*
*
!Gsolv
" !Gsolv
(soln geom) + !Erelax
*
= !Gsolv
(soln geom) + (Egas //soln - Egas //gas)
18
9
Example 2. Given the experimental gas phase acidity acetic acid
(CH3COOH) and acetone (CH3COCH3) are 1427 and 1514 kJ mol-1
respectively, calculate the aqueous pKa values of each acid using solvation
energies obtained from the CPCM-UAHF model at the HF/6-31G(d) level
of theory on solution phase optimized geometry.
Tip: Start with the gas phase optimized geometry and use it as the input
geometry for your solvation calculation.
pK a =
*
!Gsoln
RTln(10)
*
!Gsolv
(H+ ) = "1112.5 kJ / mol
19
M
OOOOLLLLDDDDEEEENNNN
M
M
M
M
M
MOOOLLLDDDEEENNN
M
OOOOLLLLDDDDEEEENNNN
M
M
M
M
M
MOOOLLLDDDEEENNN
defaults used
defaults used
O
O
last point
last point
H
C
H
C
H
H
C
C
O
C
H
H
H
H
H
H
M
OOOOLLLLDDDDEEEENNNN
M
M
M
M
M
MOOOLLLDDDEEENNN
M
OOOOLLLLDDDDEEEENNNN
M
M
M
M
M
MOOOLLLDDDEEENNN
defaults used
defaults used
O
O
last point
last point
H
H
H
C
C
C
H
C
H
C
O
H
H
H
20
10
Example 2. Given the experimental gas phase acidity acetic acid
(CH3COOH) and acetone (CH3COCH3) are 1427 and 1514 kJ mol-1
respectively, calculate the aqueous pKa values of each acid using solvation
energies obtained from the CPCM-UAHF model at the HF/6-31G(d) level
of theory on solution phase optimized geometry.
ΔG*s(acetone) = -3.8 kcal mol-1or -15.8 kJ mol-1
ΔG*s(enolate) = -63.9kcal mol-1or -267.4 kJ mol-1
ΔG*s(acetic) = -7.6 kcal mol-1or -31.6 kJ mol-1
ΔG*s(acetate) = -77.2 kcal mol-1or -323.1 kJ mol-1
ΔG*soln(acetone) = 1514 + (-267.4) + (-1112.5) - (-15.8) + 7.9 = 157.8 kJ mol-1
ΔG*soln(acetic) = 1427 + (-323.1) + (-1112.5) – (-31.6) + 7.9 = 30.9 kJ mol-1
pKa(acetone) = 27.6 cf. expt (19.2)
pKa(acetic acid) = 5.4 cf. expt (4.8)
21
The Direct/Absolute Method
• 
• 
• 
ΔGgas from high level ab initio method (error ~ 5 kJ mol-1)
ΔGsolv(H+ has uncertainty of no less than 10 kJ mol-1.
ΔGsolv from continuum solvent models (e.g. CPCM-UAHF)
–  Neutrals (error ~ 5 kJ mol-1)
–  Ions (Error >= 15 kJ mol-1)
• 
An error of 5.7 kJ mol-1 at r.t. corresponds to 1 unit error in pKa
• 
Acetic/Acetate is in the parameterisation dataset of the the PCM-UAHF model,
therefore the good agreement is not surprising.
• 
Acetone is a carbon acid (not considered in parameterisation) so errors are
much larger.
22
11
Proton exchange scheme
Isodesmic proton exchange method
–  Relies on structural similarity with HRef to maximise error
cancellation in ΔGgas and ΔΔGsolv.
–  No need for ΔGs(H+)
!G*soln
HA(aq, 1M) + Ref-(aq, 1M)
"!G*solv(HA)
HA(g, 1M)
+
"!G*solv(Ref-)
Ref-(g, 1M)
pK a (HA) =
HRef(aq, 1M) + A-(aq, 1M)
!G*solv(HRef)
!G*gas
HRef(g, 1M)
+
!G*solv(A-)
A-(g, 1M)
*
!Gsoln
+ pK a (HRef)
RTln(10)
23
Example 3. Using a proton exchange scheme to improve the accuracy
of a pKa calculation.
Provided below are some organic molecules and their experimental gas
phase and aqueous acidities (acidic proton in bold).
Acid
CH3COOH
HCN
CH3COOCH3
CH3NO2
CH3OH
!Goacid(kJ/mol)
1427
1433
1528
1467
1569
pKa
4.8
9.4
25
10.3
15.5
(a) From the above Table, identify the acid that you would use to set up an
isodesmic proton exchange reaction for calculating the pKa of acetone.
(b) Construct a thermodynamic cycle for the proton exchange reaction and
use it to calculate the pKa of acetone. You should use the CPCM-UAHF model
for your solvation calculations (at the HF/6-31G(d) level of theory) and the gas
phase acidity of acetone provided in the previous example.
24
12
Example 3. Using a proton exchange scheme to improve the accuracy of a
pKa calculation.
Pick methylacetate!
!G*soln
O
O
O
(aq)
(aq)
O
!Gogas
O
O
(aq)
(aq)
!G*solv
!G*solv
!G*solv
!G*solv
O
O
O
O
(g)
(g)
O
O
(g)
(g)
ΔG*s(acetone) = -3.8 kcal mol-1or -15.8 kJ mol-1
ΔG*s(enolate) = -63.9kcal mol-1or -267.4 kJ mol-1
*
!Gsoln
+ pK a (HRef)
RTln(10)
= 21.4
pK a (HA) =
ΔG*s(methylacetate) = -3.5 kcal mol-1or -14.8 kJ mol-1
ΔG*s(enolate) = -62.1kcal mol-1or -259.9 kJ mol-1
ΔG*soln = -20.5 kJ mol-1
25
For further details, see attached
references in hand-out.
26
13
ONIOM approximation
N
NH
R2
R1
NH
R3
R2
+
N
R4
R2
R1
R2
R4
R3
+
NH
!EEX (MP2/L)
R4
R1
N
NH
R3
!ECORE (G3)
H
R1
N
R3
+
+
H+
!E (G3)
R4
ONIOM APPROXIMATION
!E(G3) " !ECORE (G3)+!EEX (MP2/L)
27
14