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π electron density and pKa maps

Vinegar is sour and wine is not, yet acetic acid and ethanol both carry an O–H group. When the proton leaves, its two bonding electrons stay behind. Where that pair can go, through the superposition of p orbitals, decides how willingly the proton leaves. The maps compute it for twenty validated molecules or any SMILES you enter; the last section tests the idea against measured pKa values.

Any molecule

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The server builds one RDKit conformer and runs the same orbital layer as the curated set; your browser draws the maps.

Solution mode titrates every ionisable site of the molecule together: each charged group makes it harder for its neighbours to lose a proton (or easier, for opposite charges). Below the map: every step from pH 0 to 14.

Gut feeling

A proton leaves; its electrons stay

Brønsted and Lowry (1923) defined an acid as a proton donor. The proton leaves as a bare nucleus, so the bonding pair stays on the acid. The pKa then measures how comfortable the remaining base is with that extra charge.

A gut feeling follows. The further the pair can spread, the weaker it pulls the proton back, and the lower the pKa. Ethoxide keeps its charge on one oxygen (pKa 15.9). Acetate shares it between two (4.76).

Spreading happens through the π system. The p orbital on oxygen overlaps with the p orbitals of its neighbours, and the electrons occupy superpositions of all of them. To make the gut feeling testable, the spreading needs numbers.

Forecast

Superposition, written out

Each π molecular orbital is a superposition of atomic p orbitals with Hückel coefficients c. Squaring a sum produces two kinds of terms, and that split is what the maps show.

ψk = Σr crk λrone molecular orbital: a weighted sum of atomic p orbitals λr
ρ = Σr Prr λr2 + Σr≠s Prs λr λsdensity = atomic part + interference part, with Prs = Σk nk crk csk
Qr = corer − Prrnet charge on atom r (Coulson 1939); core = π electrons the atom brings + its formal charge
V(r) = ke Σj Qj / |r − rj|the electric potential those charges create (ke = 14.40 V·Å per e)

The λr are Slater 2p orbitals made orthogonal with Löwdin's symmetric procedure. In that basis Hückel's zero-overlap assumption is exact: the populations Prr add up to the electron count, and the interference part integrates to zero. Superposition moves charge from atoms into bonds; the total stays the same. Ruedenberg (1962) identified this interference term as the physical origin of the chemical bond. The readout beside the map integrates both parts for the molecule on screen.

From these fields come three forecasts, each a different guess about which part of the electricity the proton feels. Their parameters were fixed from Streitwieser's 1961 table before any pKa was compared.

Charge left on the site

Q of the atom that held the proton, in the base form. Less charge left behind means a weaker pull on the proton and a lower pKa.

Qsite in e

Potential at the proton

V at the spot where the proton sat, from every charge in the molecule, including charge that sits nearby without being conjugated to the site.

Vprobe in volts

Extra delocalization

How much more π energy the base gains than the acid by letting the site's p orbital mix with its neighbours.

ΔEdeloc in units of |β|

Test

Forecast against measured pKa

O–H acids. The charge left on oxygen ranks the twelve acids best: R² 0.92 and a leave-one-out error of 1.3 pKa units over a span of 15.5 units. The potential at the proton reaches R² 0.76 (2.3 units), the delocalization energy 0.82 (1.9 units).

N bases. Here the potential at the proton does best: R² 0.97 with a leave-one-out error of 1.4 units. Five molecules make this a hint rather than a result.

The instructive miss. In 3-nitrophenol the charge left on oxygen is the same as in phenol (−0.821 against −0.820). In the phenoxide HOMO the meta carbon that carries the nitro group has a coefficient of 0.10, against 0.37 to 0.43 at the ortho and para carbons, so it gets less than a tenth of their weight. Deprotonation moves 0.002 e onto a meta nitro group and 0.11 e onto a para one. For the hydrocarbon analogue, the benzyl anion, Longuet-Higgins's rule makes the meta coefficients exactly zero. Open Orbitals on 3-nitrophenol to see them. The measured pKa still drops by 1.6 units. That part of the effect reaches the proton through space and through σ bonds. The potential descriptor counts the nitro group's charges wherever they sit, and it does register the change (−12.43 V against −12.85 V).

The charge descriptor also cannot tell charge shared by two oxygens (acetate) from charge spread over carbons and nitro oxygens (picrate). Both leave about −0.67 e on the site, yet their pKa differ by 4.4 units. The three hollow points were declared outside the model in advance, because their substituents act through σ bonds. The π model gives trifluoroethanol and formic acid exactly the descriptor of their parents and gives 4-chlorophenol a slightly less acidic one; the measured values move the other way.

All values as a table

Limits

What the model cannot see

  • σ effects. Inductive pull (CF₃, Cl) and hyperconjugation (CH₃ against H) travel through σ bonds, which this model leaves out. A σ layer in the spirit of Bent's rule is the next step.
  • Water. Every pKa here is measured in water, which screens charge and hydrogen-bonds to the anion. The fits therefore test rankings, and the slopes are not physical constants. Solvent can even reverse an order: in the gas phase tert-butanol is a stronger acid than methanol, the opposite of water (Brauman & Blair 1970).
  • Geometry. Bond lengths are idealised and every molecule is flat, with acid and base sharing one geometry so that difference maps show electrons only. Real picrate twists its ortho nitro groups out of the ring, which weakens their conjugation. The anilinium proton sits on top of the nitrogen p lobe for a like-for-like comparison.
  • In-plane lone pairs. A lone pair in the molecular plane (pyridine nitrogen, carbonyl oxygen) does not belong to the π system, so pyridine-type bases are left out.
  • Parameters. Heteroatom values are Streitwieser's textbook set; those for the nitro group are the least certain. None were adjusted to pKa data.

Origins

Where the ideas come from

  • Acids as proton donors. J. N. Brønsted, Recl. Trav. Chim. Pays-Bas 42, 718 (1923); T. M. Lowry, Chem. Ind. 42, 43 (1923).
  • The p-scale. S. P. L. Sørensen, Biochem. Z. 21, 131 (1909). pKa uses the same negative logarithm as his pH.
  • The π-electron model. E. Hückel, Z. Phys. 70, 204 (1931).
  • Orbital exponents. J. C. Slater, Phys. Rev. 36, 57 (1930).
  • π charges and fractional bond orders. C. A. Coulson, Proc. R. Soc. A 169, 413 (1939).
  • Pairing and the meta nodes. C. A. Coulson & G. S. Rushbrooke, Proc. Camb. Phil. Soc. 36, 193 (1940); H. C. Longuet-Higgins, J. Chem. Phys. 18, 265 (1950).
  • Symmetric orthogonalisation. P.-O. Löwdin, J. Chem. Phys. 18, 365 (1950).
  • Heteroatom parameters. A. Streitwieser, Molecular Orbital Theory for Organic Chemists (Wiley, 1961).
  • Interference as the origin of the bond. K. Ruedenberg, Rev. Mod. Phys. 34, 326 (1962).
  • Field and resonance, measured. L. P. Hammett, J. Am. Chem. Soc. 59, 96 (1937); C. G. Swain & E. C. Lupton, J. Am. Chem. Soc. 90, 4328 (1968). Their empirical split of substituent effects mirrors forecasts 2 and 3.
  • Solvent can reverse acidity. J. I. Brauman & L. K. Blair, J. Am. Chem. Soc. 92, 5986 (1970).
  • Measured pKa values. E. P. Serjeant & B. Dempsey, Ionisation Constants of Organic Acids in Aqueous Solution (IUPAC, Pergamon 1979); P. Ballinger & F. A. Long, J. Am. Chem. Soc. 82, 795 (1960) for ethanol. Compilations agree to about ±0.05 units.