First-Principles Prediction of Acidities in the Gas and Solution - - PDF document

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First-Principles Prediction of Acidities in the Gas and Solution Phase

• Prof. Michelle Coote and Dr Junming Ho

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Why are Chemists interested in pKas?

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• Oxicams (polyprotic acids) are non-steroidal anti-

inflammatory compounds

Chemical Speciation

S N O O CH3 OH N H O S N O O CH3 O N H O S N O O CH3 OH N H O zwitterion (O) neutral product S N O O CH3 OH N O zwitterion (N) N H S N S N H S N H S S N O O CH3 O N H O N S CATION NEUTRAL ANION Ka1 Ka2 K1ZO K1N K1ZN K2ZO K2N K2ZN KT1 KT2 CH3 CH3 CH3 CH3 CH3

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• Polyamines are potential ion channel blockers
• Electrostatic interactions to anionic residues in

channel.

• http://sf.anu.edu.au/~rph900/dan/bsp_animation_long.mov

Predicting Protonation State

N NH2 NH2 H2N 1+ 2+ 3+ 4+ ???

fA = 1 Do ; fHA+ = [H+] K1Do ; fHA2

2+ = [H+]2

K1K2Do ; fHA3

3+ =

[H+]3 K1K2K3Do ; fHA4

4+ =

[H+]4 K1K2K3K4Do

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Thermochemical cycles

pKa = !Gsoln

*

RT ln(10)

HA (aq) H+(aq) + A-(aq)

!Gsoln

!Gsoln = !Ggas + !Gsolv(H +) + !Gsolv(A") " !Gsolv(HA)

HA (g)

!Gsolv(HA) !Gsolv(H+) !Gsolv(A-) !Ggas

H+(g) + A-(g)

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Thermochemical cycles

!Gsoln = !Ggas + ni!GS(Producti) - nj

j

"

!GS(Reactant j)

i

"

• 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

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Standard States

• 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 ΔGo

gas to standard state of 1 mol L-1

!Ggas

*

= !Ggas

°

+ !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

HA (aq) H+(aq) + A-(aq)

!G*soln

HA (g)

!G*solv(HA) !G*solv(H+) !G*solv(A-) !Gogas

H+(g) + A-(g)

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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

HF(g) # H+(g)+F-(g); !Go=1529 kJ/mol

pKa = !Gsoln

*

RT ln(10)

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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

HF(g) # H+(g)+F-(g); !Go=1529 kJ/mol

pKa = !Gsoln

*

RT ln(10) !Gsoln

*

= !Ggas

• + !GS

* (F ") + !GS "(H +) " !GS * (HF) +RT ln( %

RT) =17.1 kJ mol-1 pKa = !Gsoln

*

RT ln(10) = 2.9

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The key ingredients

• ΔGgas from molecular orbital calculations or density functional methods
• ΔGS from continuum solvent models or MD simulations

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Gas phase acidities

!Gacid

• = Go(H +) + Go(A-) - Go(HA)

= Go(H +) + E(A-) - E(HA) + Gcorr (A") "Gcorr (HA)

• 26.3 kJ mol-1 @ 298 K

Ideal gas partition functions Electronic energies From MOT or DFT methods (Single-point; high level) Thermal corrections (including ZPVE) based on the harmonic

• scillator rigid rotor

model (opt + freq; low level)

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Gas phase acidities

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

0.0 10.0 20.0 30.0 40.0 50.0 60.0 BP86 B971 B3LYP BMK M05-2X HF MP2 G3MP2+ CBS-QB3

MAD (neutrals) ADmax (neutrals) MAD (cations) ADmax (cations)

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Solvation Gibbs Energies

Molecular dynamics simulations are expensive Continuum models are much more cost effective and can deliver comparable, if not better accuracy.

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Continuum model solvation energies

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

!Gs

* = !GES + !Gcav + !Gdisp"rep

Almost all continuum models contain parameters, e.g. {ri}, which have been

• ptimized 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.

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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.

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• Gaussian03. Input for Gaussian09 is different. See Appendix

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Variational PCM results ======================= <psi(0)| H |psi(0)> (a.u.) = -76.010631 <psi(0)|H+V(0)/2|psi(0)> (a.u.) = -76.021150 <psi(0)|H+V(f)/2|psi(0)> (a.u.) = -76.022089 <psi(f)| H |psi(f)> (a.u.) = -76.009608 <psi(f)|H+V(f)/2|psi(f)> (a.u.) = -76.022097 Total free energy in solution: with all non electrostatic terms (a.u.) = -76.020988

• Electrostatic contributions to solvation free energy.

(Unpolarized solute)-Solvent (kcal/mol) = -6.60 (Polarized solute)-Solvent (kcal/mol) = -7.84 Solute polarization (kcal/mol) = 0.64 Total electrostatic (kcal/mol) = -7.19

• “Non-electrostatic” contributions to solvation free energy.

Cavitation energy (kcal/mol) = 4.45 Dispersion energy (kcal/mol) = -5.15 Repulsion energy (kcal/mol) = 1.40 Total non electrostatic (kcal/mol) = 0.70 What we want! ΔG*s DeltaG (solv) (kcal/mol) = -6.50

• Beware! This is not the Gsoln that we seek.

This is ΔG*s(H2O)

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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)

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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

• f 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.

19 pKa = !Gsoln

*

RTln(10) !Gsolv

*

(H+) = "1112.5 kJ / mol 20

MOLDEN MOLDEN MOLDEN MOLDEN MOLDEN MOLDEN MOLDEN MOLDEN MOLDEN MOLDEN MOLDEN MOLDEN MOLDEN MOLDEN MOLDEN MOLDEN

MOLDEN MOLDEN MOLDEN

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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

• f 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) 22

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.

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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+)

HA(aq, 1M) + Ref-(aq, 1M) HRef(aq, 1M) + A-(aq, 1M)

!G*soln !G*gas "!G*solv(HA) !G*solv(HRef) !G*solv(A-)

HA(g, 1M) + Ref-(g, 1M) HRef(g, 1M) + A-(g, 1M)

"!G*solv(Ref-)

pKa(HA) = !Gsoln

*

RTln(10) + pKa(HRef)

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Example 3. Using a proton exchange scheme to improve the accuracy

• f a pKa calculation.

Provided below are some organic molecules and their experimental gas phase and aqueous acidities (acidic proton in bold).

Acid !Go

acid(kJ/mol)

pKa CH3COOH 1427 4.8 HCN 1433 9.4 CH3COOCH3 1528 25 CH3NO2 1467 10.3 CH3OH 1569 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.

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Example 3. Using a proton exchange scheme to improve the accuracy of a pKa calculation. Pick methylacetate!

O O O O O O (aq) (aq) (aq) (aq)

!G*soln !G*solv !G*solv !G*solv !G*solv

O O O O O O (g) (g) (g) (g)

!Gogas

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(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

pKa(HA) = !Gsoln

*

RTln(10) + pKa(HRef) = 21.4 26

For further details, see attached references in hand-out.

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ONIOM approximation

NH R1 R2 R3 R4 N R1 R2 R3 R4 + H+ !E (G3)

ONIOM APPROXIMATION !E(G3) " !ECORE(G3)+!EEX (MP2/L)

NH N + H !ECORE (G3) NH R1 R2 R3 R4 N + NH N R1 R2 R3 R4 + !EEX (MP2/L)