Lecture: Classical Modeling Classical Modeling: Deterministic Methods

Molecular Mechanics

This is the determination of minimum energy configuration using various techniques to compute the global minimum of a function with many degrees of freedom. The function used represents the internal energy of a collection of atoms that form a molecule and is based on the coordinates of the particles (atoms) of the system. A system of N atoms has 3N-5 degrees of freedom. There are short range interaction that describe atoms bonded to each other as well as long range interactions (called non-bonded interactions) describing the electrostatic and Van der Waals forces. In addition, the surface is highly convoluted and a multitude of local minima exist.

The target is to obtain a 3-D structure with the lowest energy as determined from the potential function. The computational schemes scale as N2/2 in compute complexity and storage schemes scale as O(N), where N is the number of atoms in the system.

In molecular mechanics calculations, the evolution of the system is neglected and one typically obtains a minimum energy configuration. The procedure is to find a geometric configuration that minimizes the energy of the system with the constraints that atoms cannot bump into each other and that the atoms are bonded to each other.

Consider an energy function tex2html_wrap_inline250

Let tex2html_wrap_inline252 be the initial geometry.

Geometric Rearrangement is

tex2html_wrap_inline254 where X1 is the new geometric configuration.

A Taylor series of the energy function using this rearrangement is;

tex2html_wrap_inline262
where

tex2html_wrap_inline264

tex2html_wrap_inline266 is called the gradient

tex2html_wrap_inline268 is called the Hessian

Some molecular mechanics methods:

Systematic search
Valid for systems with few degrees of freedom
Minimization techniques (see the discussion Introduction to Geometry Optimization )

Molecular Dynamics Calculations

The target is to obtain a time evolution of the 3-D structure. Exploration of conformation space. As mentioned previously (see the Classical Modeling: Introduction ) the initial positions and velocities need to be provided, in additon to other parameters of the system, such as the mass, charge, etc. The forces are the gradient of the potential function (see the Classical Modeling: Energy Function )

The molecular dynamics formulation starts with Newton's equation of motion
tex2html_wrap_inline250

The solution of this second order differenential equation is obtained by integrating twice to obtain velocities and position. The initial position and velocities are determined by the specified geometry, while the velocities are set according to the simulation temperature, using random assignments of velocity based upon a statistical distribution. The simulation temperature is taken to be room temperature of 300o Kelvin. This procedure is based on numerical integration along a constant energy surface within the framework of a particular statistical ensemble.

Microcanonical ensemble
NVE Molecular Dynamics (constant particle number N, constant volume V and constant energy E). Numerical integration along an isotherm.
Canonical ensemble
NVT Molecular Dynamics (constant particle number N, constant volume V and constant temperature T). Numerical integration along an isotherm.

The MD simulation is advanced in time using a modified Verlet algorithm, namely, the leap frog formulation. The updated velocity is therefore derived from the acceleration and the current velocity. This is expressed numerically as,
displaymath66
Where the force, at time t, can be determined from
displaymath66
Where the change in temperature is obtained from the simulation temperature. Then the actual velocity, for the ith atom is determined to be:
displaymath66
where the velocities V(i) are the average velocity for each incremental time step in the simulation. One specifies initial positions, initial velocities and the orther necessary parameters and then computes positions at half time step n+1/2 and velocities at time step n+1

Derivation of Verlet Leapfrog Algorithm

The following 3 equations are used in the MD simulation:
displaymath66
displaymath68
displaymath70
F(t) is found by computing tex2html_wrap_inline72 and summing over all interparticle pairs.

Derivation of Equations

Expanding in a Taylor series about a=t with the variable tex2html_wrap_inline74
displaymath76
displaymath78
eqnarray21

This becomes, by definition of tex2html_wrap_inline80
eqnarray25

This implies
displaymath82
eqnarray30

Now
eqnarray34
displaymath84

This completes the derivation of the equations used in the algorithm. To understand why it's called the Verlet leapfrog algorithm consider the following Taylor series expansion:
displaymath86
displaymath88
displaymath90
displaymath92
displaymath94

So we see the half-step velocities are related to the position increments. This allows us to rewrite our algorithm as
displaymath96
displaymath98
displaymath100

So the half-step velocties are used to calculate the advanced positons tex2html_wrap_inline102 which are themselves used to find F and in turn the next half step velocity as the cycle repeats itself. Hence the term "leapfrog". Note that tex2html_wrap_inline104 is not required to step particles forward.

For a 1000 degree temperature difference the thermal energy lost to the environment amounts to about 0.01% of the atom velocity per iteration or cycle. This effect can be cumulative, i.e., as the simulation time increases there is a substantial impact on the simulation results. The relationship is not linear with time, but does increase with time.

Free Energy Perturbation

Free Energy changes relates to small perturbations of a system which are determined from a molecular dynamic simulation in which the potential energy function is slowly changed such that the system reversibly changes from state A to state B.

Thermodynamics says that there is a relation between free energy and work. tex2html_wrap_inline38 work done (at a constant volume) in a quasi-static (isothermal) process. Reversible (quasi-static) means that an equilibrium distribution (Maxwell-Boltzmann) is maintained all the time during the process. tex2html_wrap_inline40 The free-energy dynamics samples the probability distribution P.

There are two variants:

  1. slow growth
    tex2html_wrap_inline42
    The value of tex2html_wrap_inline44 is changed by tex2html_wrap_inline46 after each timestep tex2html_wrap_inline48
  2. Multi-Configuration Thermodynamic Integration (MCTI)
    tex2html_wrap_inline50
    The mean value tex2html_wrap_inline52 is accumulated at a series of values of tex2html_wrap_inline44 separated by tex2html_wrap_inline56 where tex2html_wrap_inline48

This can be described by the thermodynamic cycle
where;

In this cycle, tex2html_wrap_inline68
tex2html_wrap_inline70
tex2html_wrap_inline72

Thus, to compute the change in free energy differences for tex2html_wrap_inline74 one need know tex2html_wrap_inline76.



Author: Ken Flurchick