1549380323-Statistical Mechanics Theory and Molecular Simulation

364 Free energy calculations

e. Develop a Monte Carlo approach for sampling the distribution function

P(q) =

1

Z

e−βφ(q)

from part d.

f. Derive molecular dynamics equations of motion, including full expressions

for the force onqusing the chain rule on the derivatives (dU/dx)(dx/dq)

and (dU/ ̃ dx)(dx/dq), and develop a numerical procedure for obtaining

these forces.

Hint: Consider expanding exp[−βU ̃(x)] in a set of orthogonal polyno-

mials such as Legendre polynomialsPl(α(x)) withα(x)∈[− 1 ,1]. What

should the functionα(x) be?

8.14. It has been suggested (Peterset al., 2007) that the committor probability

pB(r) for a single reaction coordinateq(r) can be approximated by a function

πB(q(r)) that depends onronly throughq(r).

a. What are the advantages and disadvantages of such an approximation?

b. Suppose thatπB(q(r)) can be accurately fit to the following functional

form

πB(q(r)) =

1 + tanh(q(r))

2

.

Isq(r) a good reaction coordinate? Why or why not?