Computational Chemistry Molecular Mechanics/Dynamics F = Ma Quantum Chemistry Schr Ö dinger Equation H  = E 

Molecular Mechanics Force Field Bond Stretching Term Bond Angle Term Torsional Term Non-Bonding Terms: Electrostatic Interaction & van der Waals Interaction

Bond Stretching Potential E b = 1/2 k b (  l) 2 where, k b : stretch force constant  l : difference between equilibrium & actual bond length Two-body interaction

Bond Angle Deformation Potential E a = 1/2 k a (  ) 2 where, k a : angle force constant   : difference between equilibrium & actual bond angle  Three-body interaction

Periodic Torsional Barrier Potential E t = (V/2) (1+ cosn  ) where, V : rotational barrier  : torsion angle n : rotational degeneracy Four-body interaction

Non-bonding interaction van der Waals interaction for pairs of non-bonded atoms Coulomb potential for all pairs of charged atoms

MM Force Field Types MM2Small molecules AMBERPolymers CHAMMPolymers BIOPolymers OPLSSolvent Effects

CHAMM FORCE FIELD FILE

/A o /(kcal/mol)

/(kcal/mol/A o2 ) /Ao/Ao

/(kcal/mol/rad 2 ) /deg

/(kcal/mol)/deg

Algorithms for Molecular Dynamics Runge-Kutta methods: x(t+  t) = x(t) + (dx/dt)  t Fourth-order Runge-Kutta x(t+  t) = x(t) + (1/6) (s 1 +2s 2 +2s 3 +s 4 )  t +O(  t 5 ) s 1 = dx/dt s 2 = dx/dt [w/ t=t+  t/2, x = x(t)+s 1  t/2] s 3 = dx/dt [w/ t=t+  t/2, x = x(t)+s 2  t/2] s 4 = dx/dt [w/ t=t+  t, x = x(t)+s 3  t] Very accurate but slow!

Algorithms for Molecular Dynamics Verlet Algorithm: x(t+  t) = x(t) + (dx/dt)  t + (1/2) d 2 x/dt 2  t x(t -  t) = x(t) - (dx/dt)  t + (1/2) d 2 x/dt 2  t x(t+  t) = 2x(t) - x(t -  t) + d 2 x/dt 2  t 2 + O(  t 4 ) Efficient & Commonly Used!

General QM/MM scheme QM MM Combined QM/MM method 1.QM is used to describe the site where reactions occur, including those atoms make important and direct interactions to atoms undergoing valence change in the reactions process. 2.MM is used to describe the rest of the system. Presumably atoms in these regions contribute to the reaction moieties through a static and classical electrostatic fashion. Hao Hu, HKU

A simple approach: ONIOM method Our owN n-layered Integrated molecular Orbital + molecular mechanics Method += Mechanical embedding model Hao Hu, HKU

Electrostatic embedding model QM MM When MM atoms are represented as point charges The forces on MM atoms Hao Hu, HKU

Dirty details: QM/MM boundary Dangling bond Linked hydrogen Local orbital Pseudo atom Hao Hu, HKU

Monte Carlo Metropolis sampling v v’ v’’ Hao Hu, HKU

Free energy, enthalpy, & Entropy Partition function of canonical ensemble In classical mechanics, potential energy is independent to kinetic energy C(N) is a constant for the same N Hao Hu, HKU

Free energy (Helmholtz) Internal energy Entropy <>: Ensemble average Hao Hu, HKU Free energy, enthalpy, & Entropy

Difficulty for absolute Free energy simulation Absolute free energy requires converged integration on 3N dimensions No experimental absolute free energies available Relative free energies are what really matters kBTkBT Hao Hu, HKU

Free energy perturbation Free energy difference between two states State 1; energy E1State 2; energy E2 Hao Hu, HKU

Free energy perturbation Zwanzig, R. W., J. Chem. Phys. 1954, 22: Free energy difference between two states Hao Hu, HKU

Free energy perturbation More details for the solvation free energy case State 1; energy E1State 2; energy E2 Hao Hu, HKU

Complex system Hao Hu, HKU