Solvation Models and 2. Combined QM / MM Methods

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Solvation Models and 2. Combined QM / MM
Methods

• See review article on Solvation by Cramer and
  Truhlar Chem. Rev. 99, 2161-2200 (1999)

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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
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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.
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Explicit Force 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)
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Continuum (Reaction Field) Models

• Consider solvent as a uniform polarizable medium
  of fixed dielectric constant e having a solute
  molecule M placed in a suitably shaped cavity.

e
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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
  polarizes the medium, inducing charge moments,
  which add electrostatic stabilization
• DGsolvation DGcavity DGdispersion
  DGelectrostatic

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Models Differ in 5 Aspects
e

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 e is described.

(these 5 aspects will be considered in turn on
the following slides)
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1a. Solute Cavity Size and Shape

• Spherical Ellipsoidal van der
  Waals
• (Born) (Onsager)
  (Kirkwood)

r
r
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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.
  

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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.

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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.

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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)
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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).

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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.)

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Example SM5.4/A in Titans 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
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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
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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.

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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.

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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.

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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
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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.

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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.