Solvation Models and 2. Combined QM / MM Methods

1. Solvation Models and2. Combined QM / MM Methods See review article on Solvation by Cramer and Truhlar: Chem. Rev. 99, 2161-2200 (1999)

2. Part 1. Solvation Models • Some describe explicit solvent molecules • Some treat solvent as a continuum • Some are hybrids of the above two: • Treat first solvation sphere explicitly while treating surrounding solvent by a continuum model • 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

3. ExplicitQM 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.

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

5. Continuum (Reaction Field) Models • Consider solvent as a uniform polarizable mediumof fixed dielectric constant e having a solute molecule M placed in a suitably shaped cavity. e

6. Continuum Models… e • 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 polarizesthe medium, inducing charge moments, which add electrostatic stabilization: DGsolvation = DGcavity + DGdispersion + DGelectrostatic

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

8. 1a. Solute Cavity Size and Shape Spherical Ellipsoidal van der Waals (Born) (Onsager) (Kirkwood) r r

9. 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.

10. 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.

11. 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.

12. 3b. Charge Distribution of Solute (assuming vdW sized spheres for each atom) r (dipole + polarizability) (charge = q) (dipole = m) (this calculation is summed over all atoms)

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

14. 5. Describing the Dielectric Medium • Usually taken to be a homogeneous static medium of constant dielectric constant e • 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.)

15. Example: SM5.4/A in Titan’s AM1 (Chris Cramer, U. Minn) • 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:

16. 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

17. 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.

18. 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 hybridapproach offers the best solution. • This method is essentially the same approach as is used in the hybrid model of solvation.

19. 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 ONIUMmethod (next slide) in Gaussian 03 allows several layers: e.g., CISD(T), then HF or AM1, then MM for outside.

20. 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

21. 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.

22. 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.