1.Solvation Models and 2. Combined QM / MM Methods See review article on Solvation by Cramer and Truhlar: Chem. Rev. 99, (1999)

Part 1. Solvation Models §Some describe explicit solvent molecules §Some treat solvent as a continuum §Some are hybrids of the above two: l Treat first solvation sphere explicitly while treating surrounding solvent by a continuum model l These usually treat inner solvation shell quantum mechanically, outer solvation shell classically Each of these models can be further subdivided according to the theory involved: classical (MM) or quantum mechanical

Explicit QM Water Models §Sometimes as few as 3 explicit water molecules can be used to model a reaction adequately: Could use HF, DFT, MP2, CISD(T) or other theory.

Explicit Force Field Water Models §3 types: l rigid model, with interactions described by pairwise Coulombic and Lennard-Jones potentials l flexible models l polarizable models §TIP3P, TIP4P, and TIP5P rigid water models: (transferable intermolecular potentials, three parameter)

Continuum (Reaction Field) Models  Consider solvent as a uniform polarizable medium of fixed dielectric constant  having a solute molecule M placed in a suitably shaped cavity. 

Continuum Models… §Creation of the cavity costs energy (i.e., it is destabilizing), whereas dispersion interactions between solute and solvent molecules add stabilization. The electronic charge of solute M polarizes the medium, inducing charge moments, which add electrostatic stabilization:  G solvation =  G cavity +  G dispersion +  G electrostatic 

Models Differ in 5 Aspects 1.Size and shape of the solute cavity 2.Method of calculating the cavity creation and the dispersion contributions 3.How the charge distribution of solute M is represented 4.Whether the solute M is described classically or quantum mechanically 5.How the dielectric medium  is described. (these 5 aspects will be considered in turn on the following slides) 

1a. Solute Cavity Size and Shape Spherical Ellipsoidalvan der Waals (Born) (Onsager) (Kirkwood) r r

1b. Solvent-Accessible Surface §A solvent-accessible surface is made by connecting the center of a “rolling” a sphere (e.g., 1.5 Å radius) around the vdW (electron density iso-) surface of the molecule. §This method excludes pockets that are inaccessible even to small solvent molecules.

2. Cavity & Dispersion Energy §The energy required to create the cavity (entropy factors and loss of solvent-solvent vdW interactions) and the stabilization due to dispersion (vdW interactions, including some repulsion) are usually assumed to be proportional to the surface area of the solute M and the surface tension of the solvent. §These contributions may be treated as one term (proportional to the entire molecular volume) or as a sum of terms with each atom type having a different proportionality constant, such parameters derived by best fit to experimental solvation energies.

3a. Charge Distribution of Solute §Some use atom-centered charges, such as Mulliken charges, considered to be at the center of a sphere representing each atom as in the vdW model §Other approaches involve a dipole or multipole expansion (the simplest of these for a neutral molecule involves only the dipole moment). §Multipole expansions methods often need several orders (dipole, quadrupole, hexapole, octupole, decapole, etc.) for best results.

3b. Charge Distribution of Solute r (charge = q) (dipole =  ) (dipole + polarizability) (assuming vdW sized spheres for each atom) (this calculation is summed over all atoms)

4. Description of Solute M Solute molecule M may be described by: l classical molecular mechanics (MM) l semi-empirical quantum mechanics (SEQM), l ab initio quantum mechanics (QM) l density functional theory (DFT), or l post Hartree-Fock electron correlation methods (MP2 or CISDT).

5. Describing the Dielectric Medium  Usually taken to be a homogeneous static medium of constant dielectric constant  §May be allowed to have a dependence on the distance from the solute molecule M. §In some models, such as those used to model dynamic processes, the dielectric may depend on the rate of the process (e.g., the response of the solvent is different for a “fast” process such as an electronic excitation than for a “slow” process such as a molecular rearrangement.)

Example: SM5.4/A in Titan’s AM1 §Employs a generalized Born approximation with semiempirical parameter sets to represent solute- solvent interaction. §Cavity is made of interlocking spheres (~ vdW surface) §Charge distribution of the solute is represented by atom-centered Mulliken charges Born: (Chris Cramer, U. Minn)

Hybrid Solvation Model Red = highest level of theory (MP2, CISDT) Blue = intermed. level of theory (HF, AM1, PM3) Black = lowest level of theory (MM2, MMFF), or Continuum

How Good are Solvation Models? §For neutral solutes, experimental free energies of solvation between the range of +5 to -15 kcal/mol are measurable to an accuracy of ± 0.1 kcal/mol. §Continuum models of solvation can calculate energies of solvation to within 0.7 kcal/mol on a large data set of neutral molecules. §The solvation energy of charged species can be measured only to accuracies of ± 5 kcal/mol. Computed solvation energies have similar errors.

Part 2. Hybrid QM/MM Methods §For problems such as modeling the mechanism of an enzyme, MM is not good enough and QM is too costly. A hybrid approach offers the best solution. §This method is essentially the same approach as is used in the hybrid model of solvation.

Hybrid QM/MM Methods §Hybrid QM/MM methods may employ any combination of high level theory (HF, MP2, DFT) to model a small, select part of the molecule, and any type of MM (MM3, MMFF) to model the rest of the structure. §the ONIUM method (next slide) in Gaussian 03 allows several layers: e.g., CISD(T), then HF or AM1, then MM for outside.

ONIUM (layering) Method Red = highest level of theory (MP2, CISDT) Blue = intermed. level of theory (HF, AM1, PM3) Black = lowest level of theory (MM2, MMFF) catalytic triad of carboxypeptidase

Problems of Hybrid Approaches §The biggest problem is how to adequately model the interface or boundary between the QM-modeled region and the MM-modeled region. §In some respects it is the same problem faced in using explicit solvent molecules for the modeling the first solvation shell and a continuum model for more distant solvent molecules in the hybrid model.

Problems of Hybrid Approaches §One promising approach is to “cleave” bonds that occur at the interface between the “layers” and “cap” each end of the bond with a hypothetical hydrogen atom. §Hybrid QM/MM methodology such as the ONIUM method is experiencing increasing use and remarkable success in the solution of complex biochemical problems.