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
Let
be the initial geometry.
Geometric Rearrangement is
where X1 is the new geometric
configuration.
A Taylor series of the energy function using this rearrangement is;
where
is called the gradient
is called the Hessian
Some molecular mechanics methods:
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
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.
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,
Where the force, at time t, can be determined from
Where the change in temperature is obtained from the simulation temperature.
Then the actual velocity, for the ith atom is determined to be:
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
The following 3 equations are used in the MD simulation:
F(t) is found by computing
and summing over all interparticle pairs.
Derivation of Equations
Expanding in a Taylor series about a=t with the variable
This becomes, by definition of
This implies
Now
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:
So we see the half-step velocities are related to the position increments.
This allows us to rewrite our algorithm as
So the half-step velocties are used to calculate the advanced positons
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
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 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.
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.
The free-energy dynamics samples the probability
distribution P.
There are two variants:
This can be described by the thermodynamic cycle
where;
In this cycle,
Thus, to compute the change in free energy differences for
one need know
.