Solvation (Condensed Phase) Models

Chemical Phenomena

Video VI.i What Condensed Phases are Important?

• Homogeneous liquid soluons are the most common condensed phases found in experimental chemistry. We’ll focus on these, but make connecons to others • Solids • Surfaces • Liquid crystal soluons • Supercrical ﬂuids • Membranes • Note, in certain instances, the ﬁne line between a condensed phase and a supermolecular complex Why is Solvaon Important?

• Condensed-phase properties depend on the condensed-

phase wave function, and <Ψgas|A|Ψgas> may be very

different from <Ψsol’n|A|Ψsol’n> • Interactions between two (or more) molecules in solution depend on partial desolvation of each • Potential energy hypersurfaces (and hence kinetics and equilibria) may be quantitatively and qualitatively different in solution by comparison to the gas phase

Solvatochromism of Dye ET30 (S1 – S0)

N +

O –

Solvent Color λmax, nm anisole yellow 769 acetone green 677 2-pentanol blue 608 ethanol violet 550 methanol red 515 Enzyme–Substrate Interactions

o ∆G g E(g) + S(g) E•S(g)

o o o ∆G S(E) ∆G S(S) ∆G S(E•S)

E(s) + S(s) E•S(s) o ∆G aq Related Potenal Energy Surfaces % ( A o ' [ ]sol * ΔGS (A) = lim ' −RT ln * A 0 A [ ]→ ' [ ]gas * gas-phase & eq ) surface

€

o ΔG S(x,y) E solvated surface Equilibrium constants from differences in energy of local minima

Rate constants from differences in energy of (x,y) connected local minima and saddle points PES Slice — The Menschutkin Reacon

gas phase

H NH + CH Cl H NCH + + Cl– 3 3 H3N C Cl 3 3 H H

G ∆Gsolv(reactants) ∆Gsolv(products)

aqueous solution The Generic Free-Energy Cycle

o,‡ ΔGgas o ΔGS (‡)

Go,rxn E Δ gas

ΔG o(R) o S ΔGS (P) o,‡ ΔGsol

o,rxn ΔGsol R ‡ P

arbitrary coordinate Explicit and Implicit Solvent Modeling: Two Alternave Approaches

• Explicit solvent modeling is conceptually obvious: add lots of solvent molecules and compute blue curve on last slide • Implicit solvent modeling is more subtle: forget about molecular representaon of solvent; compute gas-phase curve by one method, then compute free energies of solvaon at points of interest by a diﬀerent method (this can actually be done in a blue-curve fashion too, but the solvent remains implicit) (Equilibrium) Free Energy of Solvation

o o ΔG = ΔG + G S ENP CDS

SOLVENT DEPENDENCE

E ⇒ Electronic Energy $ N ⇒ Nuclear Repulsion # Solvent Dielectric P ⇒ Solute-Solvent Polarization "

C ⇒ Cavitation $ D ⇒ Dispersion # Other Solvent Properties S ⇒ Structural etc. " (also called “non-electrostac” component) Solvation (Condensed Phase) Models

Explicit Solvent—Atomistic Analysis

Video VI.ii Some Rules for Explicit Solvent Modeling

• Rule 1: It takes a lot of solvent molecules to look like a solution. Put differently, clusters only tell one about clusters. • Consequence: Quantum mechanics is very, very expensive (although use of Car-Parinello approach is ongoing). Instead, molecular mechanics (i.e., force ﬁeld) approaches tend to be used for at least some of the system. • Tools: Periodic boxes or Ewald sums to limit system size. QM/MM implementations. But It Is There! Solvent Density Analysis

Nagan, et al. J. Am. Chem. Soc. 1999, 121, 7310. But It Is There! Solvent Density Analysis

Nagan, et al. J. Am. Chem. Soc. 1999, 121, 7310. Some Rules for Explicit Solvent Modeling

• Rule 2: Equilibrium properties (e.g., free energies) require averaging over phase space. • Consequence: Sampling phase space becomes a key issue. Brute force is impossible in a real system. Free energy convergence can be very slow. • Tools: Monte Carlo or Molecular Dynamics sampling until apparent (ergodic) convergence. More robust if multiple trajectories are run from different starting points. Integrang over Phase Space

∫ Ξ(r)P(r)dr Ξ = PS P r dr ∫PS ( )

Expectation values (whether quantum or classical) are dictated by the relative probabilities of being in different regions of phase space € P r = e−E(q,p) / kBT Q = P r dr ( ) ∫PS ( )

Key point: Don’t waste time evaluating Ξ(r) if P(r) is zero.

Difﬁculty:€ Phase space is 6N-dimensional. If you only want to sample all possible combinations of either positive or negative values for each coordinate (i.e., hit every “hyperoctant” in phase space once), you need 26N points! MC/MD Provides Ready Access to Many Properes in Soluon

• Average structures (with standard deviaons); note that these structures may be determined with constraints imposed (e.g., from NMR NOE measurements) • All kinds of quantum mechanical properes (albeit usually averaged over fewer snapshots than the enre trajectory) • Structural details associated with the solvaon shell (typically NOT available from implicit models) Radial Distribuon Funcon g 1 1 NA NB g r = δ r − r AB( ) ∑∑ [ AiBj ] V N A • N B i =1 j =1

1 €

r Solvation (Condensed Phase) Models

Explicit Solvent—Free Energy Perturbation

Video VI.iii Integrang over Phase Space

∫ Ξ(r)P(r)dr Ξ = PS P r dr ∫PS ( )

Expectation values are dictated by the relative probabilities of being in different regions of phase space € P r = e−E(q,p) / kBT Q = P r dr ( ) ∫PS ( )

Key point: Don’t waste time evaluating Ξ(r) if P(r) is zero.

Difﬁculty:€ Phase space is 6N-dimensional. If you only want to sample all possible combinations of either positive or negative values for each coordinate (i.e., hit every “hyperoctant” in phase space once), you need 26N points! What About Free Energy as the Average Property? 1 1 A = k T ln B Q

$ E(q,p) / kBT −E(q,p) / kBT ' & ∫∫ e e dqdp) = kBT ln & −E(q,p) / kBT ) % ∫∫ e dqdp (

$ E(q,p) / kBT ' Q = kBT ln ∫∫ e P q,p dqdp %& ( ) ()

E / kBT = kBT ln e

EB / kBT EA / kBT A − A = kBT ln e − kBT ln e B A B A € $ M ' $ M ' 1 B 1 A & Ei / kBT ) & Ei / kBT ) = kBT ln ∑ e − kBT ln ∑ e & MB ) & MA ) % i ( % i (

Horrifyingly slowly convergent for two structures A and B because sampling procedures are designed to sample predominantly low-energy points, but high-energy points contribute exponenally more to sum.€ A Trick With the Integral

$ −EB(q,p) / kBT ' QA QB & ∫∫ e dqdp ) AB − AA = kBT ln = − kBT ln = −kBT ln Q Q & −EA(q,p) / kBT ) B A % ∫∫ e dqdp( $ E q,p / k T $ E q,p / k T E q,p / k T ' ' ∫∫ e− B( ) B e A ( ) B e− A ( ) B dqdp & %& () ) = −k T ln& ) B E q,p / k T & ∫∫ e− A( ) B dqdp ) %& () $ $ E q,p / k T E q,p / k T ' E q,p / k T ' ∫∫ e− B( ) B e A( ) B e− A( ) B dqdp & %& () ) = −k T ln& ) B E q,p / k T & ∫∫ e− A ( ) B dqdp ) %& () $ ' −[EB(q,p)−EA (q,p)] / kBT −(EB−EA) / kBT = −kBT ln& ∫∫ e P(q,p)dqdp) = −kBT e % ( A

Thus, sampling over coordinates for A, determine exponenal of energy diﬀerence € between A and B.

Simpler because only a single ensemble, but what if sample over A is not ergodic for B? How Can We Ensure That A and B Are “Similar”?

e.g., HCN vs. HNC % a b q q ( % a b q q ( ΔE = ' HH − HH + H B H D * − ' HH − HH + H A H D * H D ' r12 r 6 εr * ' r12 r 6 εr * & H B H D H B H D H B H D ) & H AH D H AH D H AH D )

HC €

−(EB−EA) / kBT A − A = −k T ln e r HD B A B HAHD A r HBHD

HA HB € E λ = λE + 1− λ E ( ) B ( ) A Free Energy Perturbaon

€ How Can We Ensure That A and B Are “Similar”?

e.g., HCN vs. HNC

% a b q q ( % a b q q ( ΔE = 0.05' HH − HH + H B H D * + 0.95' HH − HH + H A H D * H D ' r12 r 6 εr * ' r12 r 6 εr * & H B H D H B H D H B H D ) & H AH D H AH D H AH D ) % a b q q ( − ' HH − HH + H A H D * ' r12 r 6 εr * & H AH D H AH D H AH D ) HC +% a b q q ( % a b q q (. = 0.05-' HH − HH + H B H D * − ' HH − HH + H A H D *0 ' r12 r 6 εr * ' r12 r 6 εr * ,-& H B H D H B H D H B H D ) & H AH D H AH D H AH D )/0

1 −(Eλ +dλ −Eλ ) / kBT € A − A = −kBT ∑ ln e r HD B A HAHD λ=0 λ r HBHD

H H € A B E λ = λE + 1− λ E ( ) B ( ) A Free Energy Perturbaon

€ Monitoring the Alchemical Change

ΔG

Diﬀerence between forward and reverse is one Hysteresis is another measure of error

0 λ 1 Computaonal Alchemy o ΔG g E(g) + S(g) E•S(g)

o o o ΔG S(E) ΔG S(S) ΔG S(E•S)

E(s) + S(s) E•S(s) o ΔG aq(1) o o 0 ΔΔG S ΔΔG S

E(s) + S´(s) E•S´(s) o ΔG aq(2) Computaonal Alchemy

Note that perturbaon to nothing (annihilaon) is allowed, although some care must be taken for so drasc a perturbaon. Solvation (Condensed Phase) Models

Implicit Solvent—Electrostatics

Video VI.iv Some Rules for Implicit Solvent Modeling

• Rule 1: If one replaces the solvent molecules with a continuum dielectric (an equilibrium averaging, in essence) one is left with a system no “larger” than the solute. • Consequence: All solvent-structural information is lost, but if one can afford QM for gas phase, one can afford QM for solution. Polarization now arises from ﬁrst principles (self-consistent reaction ﬁeld— SCRF—induced in the medium). • Tools: Numerous methods to solve or approximate the Poisson equation for continuous or discrete charge representations. Some Rules for Implicit Solvent Modeling

• Rule 1: If one replaces the solvent molecules with a continuum dielectric (an equilibrium averaging, in essence) one is 1 G = − ρ(r)φ(r)dr left with a system no “larger” than the 2 ∫ solute. • Consequence: All solvent-structural information is lost, but if one can afford QM for gas phase, one can afford QM for solution. Polarization now arises from €ﬁrst principles (self-consistent reaction ﬁeld— SCRF—induced in the medium). • Tools: Numerous methods to solve or approximate the Poisson equation for ∇ε (r) • ∇φ(r) = −4πρ(r) continuous or discrete charge representations.

€ Solvent-induced Polarizaon

1 ΔGENP = ΨS H + V (ΨS ) ΨS 2

SCRF € operator Polarizaon Example—Nitroaromac Radical Anion

NH 2 NH2 Br NO 2 [ H ] Br NH2

NO 2 NO2 Conducng Sphere (Ion) Example

Sphere carries charge q and has radius α. q Charge distribuon on a conducng sphere is ρ(s) = , s on surface 4πα2 q Potenal outside sphere is φ(r) = − ε r Thus, solving for the work of charging requires integrang only € over the surface of the sphere: € 1 G = − ρ s φ s ds ∫S ( ) ( ) 2 and the free energy of polarizaon GP is the 1 q q diﬀerence in the work of charging in soluon (ε ≠ 1) = ds 2 ∫S 4πα 2 εα and the gas phase (ε = 1) 2 1 q 1$ 1' q2 = the Born 2 εα GP = − &1 − ) 2% ε( α equaon

€ Dipole in a Sphere Example (Kirkwood-Onsager)

Sphere carries dipole µ at center and has radius α. 2 1$ 2(ε −1)' µ Analogous analysis leads to GP = − & ) 2& 2ε +1 ) 3 % ( )( α

., 1 $ 2(ε −1)' ΨµΨ .0 This leads to the Schrödinger equaon: - H − & ) µ1Ψ = EΨ € . 2 & 2ε +1 ) 3 . / % ( ) ( α 2 which is variaonally minimized by a Slater determinant formed from solvated orbitals that are eigenfuncons of: € , 2 0 . $ 2(ε −1)' ΨµΨ . - F − & ) 1ψ = ε ψ i 3 i i i SCRF . %& (2ε +1) () α . / 2 Note that the solvated orbitals and associated eigenvalues will be diﬀerent than their gas-phase counterparts € Some Rules for Implicit Solvent Modeling

• Rule 2: A continuum dielectric is a ﬁction, so it is best not to get too caught up in theoretical rigor (ΔGENP is not even a physical observable…)

• Consequence: Construction of the dielectric cavity and/or methods for solving or approximating the Poisson equation can vary signiﬁcantly from one model to the next.

• Tools: Parameterization just as important as for force-ﬁeld models (even if occasionally it is stealth-parameterization). Cavies and Their Fillings Ideal Cavies

Spheres and ellipsoids: Permit analytic solution of Poisson equation for interior charge expressed as multipole expansion

Charge in a sphere: Born equaon

1% 1( q2 ΔG = − '1 − * Reaction Field ε P 2& ε) α

solute q is charge and α is radius of cavity sphere €

Cavies and Their Fillings Ideal Cavies

Spheres and ellipsoids: Permit analytic solution of Poisson equation for interior charge expressed as multipole expansion

Dipole in a sphere: Kirkwood-Onsager equaon

1% 2(ε −1)( µ2 ΔGP = − ' * 3 Reaction Field ε 2& (2ε +1)) α

solute µ is dipole moment and α is radius of € cavity sphere; requires SCRF since relaxaon of Ψ aﬀects µ

Cavies and Their Fillings Ideal Cavies

Spheres and ellipsoids: Permit analytic solution of Poisson equation for interior charge expressed as multipole expansion

Any number of mulpoles in a sphere or ellipse: Rivail and Rinaldi

1 L l L l # G = − M m f mm # M m # P ∑ ∑ ∑ ∑ l ll # l # Reaction Field ε 2 l =0 m =−l l # =0m # =−l #

solute € M is solute or reacon ﬁeld mulpole moment (order l component m) and f is a “reacon-ﬁeld factor”

Cavies and Their Fillings Ideal Cavies

Spheres and ellipsoids: Permit analytic solution of Poisson equation for interior charge expressed as multipole expansion

But such cavies tend to be very unrealisc for arbitrarily shaped molecules Cavies and Their Fillings Arbitrary Cavies

Arbitrary cavities require non-analytic or approximate solutions of the Poisson equation for interior charge expressed as either a continuous charge distribution or a single- or multicenter multipole expansion

Can Integrate through Volume ε Connuous distribuon: PCM solute Can use (MST) of Tomasi and approximate many coworkers (also form for COSMO) Poisson equation

Can Integrate at Surface

Cavies and Their Fillings Arbitrary Cavies

Can Integrate through Volume ε

Mulpole expansion: solute Can use Rivail and many approximate coworkers form for Poisson equation

Can Integrate at Surface

Cavies and Their Fillings Arbitrary Cavies

1 $ 1' atoms GP = − &1 − ) ∑qkqk *γ kk * Generalized Born: GB 2% ε ( k, k * (Sll and subsequent others, classical) and SMx (Cramer & 2 −1/ 2 γ = r 2 +α α e−rkk # / dkk #α kα k # Truhlar, quantum kk # ( kk # k k # ) SCRF) €

€ Bulk Electrostac Eﬀects Generalized Born (GB) equaon

atoms 1$ 1' ε GP = − &1 − ) ∑qkγ kk'qk' 2% ε( kk'

−1/ 2 2 −r 2 / dα α kk ' k k ' q2 r e q2 γ kk' = ( kk' + α kα k' ) q1 q q1 q3 € 3

Liming behaviors… € Avoids charge penetraon rkk' >> 0 rkk' = 0 and can be put into pairwise form (advantages) but 2 1 & 1 # qk qk' 1$ 1' qk requires paral atomic GP = − $1− ! GP = − &1 − ) charges (disadvantage). Now 2 ε r 2% ε( α € % " kk' € k heavily used for classical Coulomb’s Law Born’s Equaon simulaons, less for quantum 1/2(–gas + soluon) monatomic ion calculaons. € Solvation (Condensed Phase) Models

Implicit Solvent—Non-electrostatics

Video VI.v Some Rules for Implicit Solvent Modeling

• Rule 3: Electrostatics are only part of the free energy of solvation.

• Consequence: One needs to somehow account for cavitation, dispersion, solvent- structural changes, etc., if one wants to make contact with the experimental observable.

• Tools: It’s nice if you can also ﬁx up your electrostatic approximations at the same time How to Account for Nonelectrostac Terms?

• Ignore them completely (potenally valid for polyelectrolytes, where electrostac eﬀects will be expected to dominate in any case) • Aempt to compute separately using, e.g., scaled-parcle theory to esmate cavitaon costs, dispersion from models employing atomic or group polarizabilies, others from…? • Assume proporonality to solvent-accessible surface area of atoms or groups and parameterize microscopic surface tensions (or surface tension funconals) • Connuum solvaon is semiempirical from the outset, so parameterizaon is no sin (CJC personal opinion…) First-Solvaon-Shell Contribuons

Cavitaon, Dispersion, Structural rearrangement of solvent

atomic-number-dependent atoms $ atoms ' parameters GCDS = ∑ Ak &σ k + ∑σ kk' (R)) k % k' (

atomic surface tension designed to account for the eﬀect of solute geometry

€ 0.4 Å Switching funcons (T ) 0.8 k,k’

0.6

(Å)

- 0.4

0.2

0.0 1 1.1 1.2 1.3 1.4 Solvent Accessible Surface Area RC-O (Å) (deﬁnes the ﬁrst solvaon shell) Microscopic Surface Tensions

GCDS = ∑Akσk k

Example: The SMx universal solvaon models

€ descr (ξ j ) σ = σˆ ξ Z is atomic number i ∑ Z i j j Some Descriptors ξ: n is solvent index of refracon γ is solvent macroscopic surface tension α is solvent hydrogen-bonding acidity (Abraham) € β is solvent hydrogen-bonding basicity (Abraham) Microscopic Surface Tensions

o ΔGS,expt − ΔGENP = GCDS = ∑ Akσ k k

Example: The SMx universal solvaon models

Aer parameterizaon (about 72 parameters for 2500 data {H,C,N,O,F,S,P,Cl,Br- compounds} in 91 solvents including water) SM8 has a mean unsigned error of approximately 0.6 kcal mol–1 for neutrals and 3-6 kcal mol–1 for ions, depending on solvent Examples of Solvent Descriptors

H2O C6H6 CH2Cl2 dielectric constant 78.36 2.27 8.93 Abraham’s hydrogen bond acidity 0.82 0.00 0.10 Abraham’s hydrogen bond basicity 0.38 0.14 0.05 refractive index 1.33 1.50 1.42 surface tension (cal mol-1Å-2) 104.71 40.62 39.15 carbon aromaticity 0.00 1.00 0.00 electronegative halogenicity 0.00 0.00 0.67 SM8 Performance

Mean unsigned errors (kcal/mol) for SM8 and some other popular connuum solvaon models

Solute class Data SM8 IEF-PCM C-PCM PB All equal

N G03/UA0 GAMESS Jaguar to mean

aqueous neutrals 274 0.5 4.9 1.6 0.9 2.7

nonaq. neutrals 666 0.6 6.0 2.8 2.3 1.5 aqueous ions 112 3.2 12.4 8.4 4.0 8.6 nonaqueous ions 220 4.9 8.4 8.4 8.1 8.6

Cramer, C. J.; Truhlar, D. G. Acc. Chem. Res. 2008, 41, 760

Marenich, A. V.; Cramer, C. J.; Truhlar, D. G. “Performance of SM6, SM8, and SMD on the SAMPL1 Test Set for the Prediction of Small-Molecule Solvation Free Energies” J. Phys. Chem. B 2009, 113, 4538. Ribeiro, R. F.; Marenich, A. V.; Cramer, C. J.; Truhlar, D. G. “Prediction of SAMPL2 Aqueous Solvation Free Energies and Tautomeric Ratios Using the SM8, SM8AD, and SMD Solvation Models” J. Comput.-Aid. Mol. Des. 2010, 24, 317. SMD Example from G09

TS structure for water- Gas phase assisted tautomerizaon SCF Done: E(RM06) = -510.336253045 A.U. aer 1 cycles of 1-methylthymine Dipole moment (Debye): Tot = 5.3667

Aqueous SCF Done: E(RM06) = -510.358270799 A.U. aer 12 cycles Convg = 0.4387D-08 -V/T = 2.0099 SMD-CDS (non-electrostac) energy (kcal/mol) = 4.81 (included in total energy above)

Dipole moment (Debye): Tot = 7.5777

Free energy of solvaon = –510.358271 – (–510.336253) = –0.022018 a.u. = –13.8 kcal/mol

Note (i) signiﬁcant non-electrostac component and (ii) signiﬁcant polarizaon as judged by increased dipole moment. Introducon of Temperature Dependence into GCDS Funconal Form for GCDS

atoms $ atoms ' SM6 GCDS = ∑ Ak &σ k + ∑σ kk' (R)) k % k' (

contribuon at 298 K GCDS = GCDS(T0)+ SM6T from SM6 ! € ΔSS (T −T0)+ ! Δ CP [(T −T0) −T ln(T T0)] temperature dependence of the non-electrostac contribuons to € the free energy of solvaon relave to the value at T0 (298 K) Introducon of Temperature Dependence into GCDS Funconal Form for GCDS

atoms $ atoms ' SM6 GCDS = ∑ Ak &σ k + ∑σ kk' (R)) k % k' (

entropy-like GCDS = GCDS(T0)+ SM6T component % atoms % atoms (( € ΔS ΔS ' ∑ Ak 'σ k + ∑ σ kk' (R)** ( T −T0)+ & k & k' )) % atoms % atoms (( ΔCP ΔCP ' ∑ Ak 'σ k + ∑ σ kk' (R)** [ (T −T0) −T ln(T T0)] & k & k' )) temperature dependence of the non-electrostac heat capacity-like contribuons to the free energy of solvaon relave component € to the value at T0 (298 K) Quality of Parameterizaon for Benzene

Variaon of the free energy of hydraon of benzene relave to 298 K n Experiment n Linear 1.2 n Unrestricted - 1 14 parameters n Final - 0.8 7 parameters 0.6 (kcal/mol) 0.4 0.2 0 -0.2 -0.4 Free energy -0.6 270 290 310 330 350 370 Temperature (K) Solvation (Condensed Phase) Models

Implicit Solvent—Phase-phase Partitioning

Video VI.vi What Do We Predict with SMx Solvaon Models?

gas-phase gas-phase solvent A

o o o ΔGS ΔGS (self ) ΔGS (A → B)

pure solution liquid solution of solute solvent B Absolute free energy of solvation Free energy of self-solvation Transfer free energy of solvation Solvation free energy — Vapor pressure Partition coefﬁcient all solvents, no types

By combining these, we also calculate solubility. Have also extended to: • Interface adsorption • Membrane permeability Free Energies of Solvation and Partition Coefficients

Agas

o o ∆GS (1) ∆GS (2)

Asol’n (1) o Asol’n (2) ∆G(1)↔(2) Natural and Unnatural Nucleic Acid Bases

CH3 NH2 NH2 NH2 HN 4 6 7 3 H Br 1 5 N N N N N N 5 8 6 N N O 2 N O N H 2 N 4 9 H N 1 3 CH3 CH3 CH3 CH3 9-methyladenine 9,N6-dimethyladenine 1-methylcytosine 5-bromo-1-methylcytosine

O NH2 H NH2 H CH3 CH3 N N N N N N

N N O N O N H2N N H2N N CH3 CH3 CH3 CH3 2-amino-9-methylpurine 2,6-diamino-9-methylpurine 1-methylthymine 1,5-dimethylcytosine

O O O O H H H Br H N H N N N N N

N N O N O N H2N N H N CH3 CH3 CH3 CH3

9-methylguanine 9-methylhypoxanthine 1-methyluracil 5-bromo-1-methyluracil

Giesen et al. J. Phys. Chem. B 1997, 101, 5084. Calculated Chloroform/Water Partition Coefficients

Solute SM5.4/A Experiment

logKCHCl3/H2O logKCHCl3/H2O 9-Methyladenine –1.6 –0.8 * = predicted 9,N6-Dimethyladenine –0.3 0.4* before measurement 2-Amino-9-methylpurine –1.9 –0.5*

2,6-Diamino-9-methylpurine –2.4

9-Methylguanine –4.1 –3.5

9-Methylhypoxanthine –3.5 –2.5

1-Methylcytosine –4.3 –3.0

5-Bromo-1-methylcytosine –2.4

1-Methylthymine –0.3 –0.5

1,5-Dimethylcytosine –3.1

1-Methyluracil –1.2 –1.2

5-Bromo-1-methyluracil –0.3 –0.7*

Mean unsigned error: 0.7 Solubility from Solvation Free Energy and Vapor Pressure

Agas

o o P ∆GS

Apure o Asol’n S Mean-unsigned error (MUE) in predicted log P•

SM5.42R Solute class No. data HF B3LYP AM1 hydrocarbons 11 0.3 0.3 0.2 aromatics 6 0.2 0.3 0.2 alcohols/phenols 9 0.3 0.2 0.3 ethers 4 0.2 0.2 0.2 carbonyls 11 0.7 0.5 0.8 esters 7 0.3 0.2 0.6 CHN compounds 7 0.2 0.2 0.4 nitro compounds 5 0.1 0.2 0.3 HCNO compounds 60 0.3 0.3 0.4 halocarbons 15 0.4 0.4 0.5 all liquid solutes 75 0.3 0.3 0.4

Winget et al. J. Phys. Chem. B 2000, 104, 4726 Mean-unsigned error (MUE) in predicted log S

SM5.42R Solute class No. data HF B3LYP AM1 UNIFAC hydrocarbons 11 0.5 0.4 0.4 1.4 aromatics 6 0.1 0.0 0.1 0.2 alcohols/phenols 9 0.3 0.2 0.3 0.6 ethers 4 0.5 0.5 0.5 0.5 carbonyls 11 0.4 0.4 0.5 0.3 esters 7 0.3 0.3 0.2 0.1 CHN compounds 7 0.7 0.5 0.6 0.5 nitro compounds 5 0.4 0.4 0.2 0.2 HCNO compounds 60 0.4 0.4 0.4 0.6 halocarbons 10 0.2 0.3 0.2 0.4 all liquid solutes 70 0.4 0.3 0.3 0.5 solid solutes 13 0.3 0.4 0.5 0.8

Thompson et al. J. Chem. Phys. 2003, 119, 1661 Analysis of Membrane/Water Partitioning

* * * * * * * * * * * * * * * * * * * * * * * * *

= Solute * Known starting concentrations Microsyringe to sample aqueous phase Interior vesicle volume negligible “Solvent” Model for Phosphatidyl Choline

O O

O O NMe3 O P O O

SM5.4 requires: ε n α β γ

Best guess: 5.0 1.37 0.0 0.9 27 (Const.)

Regression fit: (5.0) 1.40 (0.0) 1.15 25 0.59 Water/Phosphatidyl Choline Partitioning

OH OH OH OH

OH CH3

Cl nC5H11

NH2 NH2 NMe2

OH NH2 Cl Cl

NO2

Cl CH3 Cl Cl Cl Cl Cl OH

Cl Cl

NO2 CH3

OH O N

NO2 Predicted vs Experimental log K (m/w) 5 y = - 5.2182e-2 + 0.90812x R^2 = 0.795

2 Predicted

0 0 1 2 3 4 5 Experiment

Chambers et al. in Rational Drug Design, Truhlar, et al. Eds.; Springer: New York, 1999, p. 51. Soil/Water Partitioning

Important factor controlling the persistence of environmental contaminants

carbamates, phosphonothioates, polyhalogenated aromatics, ureas, horrible molecules Dirt

[X]soil / %OC KOC = [X]aqueous

organic remarkably constant from clay to loam to peat carbon “Solvent” Model for Soil

"Dirt/MIDI!"

SM5.42R requires: ε n α β γ

Best guess: ? ? ? ? ?

Regression fit: (15.0) 1.379 0.61 0.60 46.0

Mean unsigned error over 387 compounds = 1 log unit Correlation of a Subset of Chlorinated Biphenyls

5.5

5 OC K 4.5 y = 1.05x - 0.99

3.5

Experimental log 3

2.5

2 2 2.5 3 3.5 4 4.5 5 5.5

Calculated log K OC

Clm Cln Theory allows for an atom-by-atom decomposition of the partitioning energies to better understand factors affecting them Solvation (Condensed Phase) Models

Hybrid Explicit/Implicit Solvent “it was the best of models, it was the worst of models”

Video VI.vii Equilibria: Thema mit Veränderungen

pKa (Born-Haber cycle)

o ∆G g – + o $ AH A + H −ΔGaq / RT (g) (g) (g) pK = −log e a ( ) ∆Go (AH) ∆Go (A–) ∆Go (H+) S S S ΔGo$ = aq – + 2.303RT AH(s) A (s) + H (s) o´ ∆G aq 1) Need diﬀuse funcons in basis set and good theory to get accurate gas-phase deprotonaon free energy € 2) Can’t compute E for H+ (no electrons!) so electronic structure programs are reluctant to compute thermal contribuons to G (but a good spreadsheet will) o –1 3) ΔG S of proton is an experimental quanty (–264.0 kcal mol ) 4) Standard-state concentraon-change free energy must be included –1 5) Each non-cancelled error of 1.4 kcal mol in any step will lead to an error in pKa of 1 pK unit — errors in ionic solvaon free energies are potenally much larger than this… 6) Can correct for funconal-group systemac errors, see: Klicic et al. J. Phys. Chem. A 2002, 106, 1327. Free Energy Cycles and Ions

No explicit water molecules

! ΔGg(AH) AH(g) A−(g) + H+(g)

* * - ΔG∗(H+) € ΔG S (AH) ΔG S (A ) S € € € € AH(aq) A−(aq) + H+(aq) 2.303RTpKa €

€ € 2.303RTpK = ΔG!€(AH) − ΔG∗(AH)€+ ΔG∗(A−) +ΔG∗(H+) € a g S S S

€ Free Energy Cycles and Ionic Clusters Treat the ion as a cluster

ΔG (AH) g − + AH (g) + H2O (g) H2O • A (g) + H (g) * ∗ * - ∗ + Δ G S (AH) ΔGS(H2O) Δ GS (H 2 O • A ) ΔGS(H ) € € € € € H O (aq) − + AH (aq) + 2 H2O • A (aq) + H (aq)

€ 2.303RTpKa € ! ∗ ∗ ∗ − ∗ + 2.303RTpKa = ΔGg(AH) − ΔGS(AH) − ΔGS(H2O) +ΔGS(H2O • A ) +ΔGS(H ) € € € € € For addional details on cluster solvaon free energies and their use to set the € absolute solvaon free energy of the proton and the absolute potenal of the normal hydrogen electrode (NHE), see Kelly, C. P.; Cramer, C. J.; Truhlar, D. G. J. Phys. Chem. B 2006, 110, 16066. Example: pKa of Methanol (Expt. 15.5) − − MeOH / MeO MeOH / H2O •MeO (Cycle 1) (Cycle 2)

€ Experimental data: € Experimental data: ! ! ΔGg = + 375.0 kcal/mol ΔGg = + 358.0 kcal/mol ∗ + ∗ + ΔGS(H ) = − 265.9 kcal/mol ΔGS(H ) = − 265.9 kcal/mol ∗ ∗ Δ GS(MeOH) = − 5.11kcal/mol ΔGS(MeOH) = − 5.11kcal/mol ∗ Δ GS(H2O) = − 6.32 kcal/mol

€ Calculated (SM6) data: Calculated (SM6) data: * - € ΔG (MeO ) = - 88. 3 kcal/mol * - S ΔG S (H 2O • MeO ) = - 81. 8 kcal/mol

pKa = 20.4 pKa =16.0

€ € Adding More Waters

Experimental pKa’s - 2- H2CO3 HCO3 CO3 pK 6.4 pK 10.3 a1 = a2 =

Calculated pK ’s a 2- € € H H No. H O pK pK O O O 2 a1 a2

0 -0.6 1.6 H C H 1 1.3 5.0 O O

2€ € 2.3 7.8 H H 3 4.2 9.0 O

Adding explicit water molecules improves the accuracy of the calculaon Clustering Other Ions

25 a K

15 Experimental p

-5 -5 5 15 25 35

Calculated pKa

, : bare ion used : clustered ion used Kelly, C. P.; Cramer, C. J.; Truhlar, D. G. J. Phys. Chem. A. 2006, 110, 2493 Ru(bpy)(damp) Catalyst N N NMe2 RuIV Me2N O N

crical chemical step

Vigara et al. Chem. Sci. 2012, 3, 2576. Photo courtesy Natalie Lucier of Flickr under Creave Commons license