1549380323-Statistical Mechanics Theory and Molecular Simulation

354 Free energy calculations

A(x) =D 0

(

x^2 −a^2

) 2

−

λ^2

2 k

x^2. (8.10.22)

The bare double well has minima atx=±aand a barrier height ofD 0 a^4 while the free
energy in eqn. (8.10.22) has minima atx=±

√

a^2 +λ^2 /(4D 0 k) and a barrier height of
D 0 a^4 +λ^2 a^2 /(2k) +λ^4 /(16D 0 k^2 ). Thus, in order to ensure sufficient barrier-crossing
in an adiabatic dynamics simulation, the temperature of the “reaction coordinate”
xshould satisfykTx> D 0 a^4. Now we only need to choose a massmxsuch thatx
is adiabatically decoupled fromy. Consider the specific example ofD 0 = 5,a= 1,
k= 1,λ= 2.878,my= 1, andkTy= 1. To see how the choice of the massmxaffects
the final result, we plot, in Fig. 8.7, the free energy profile obtainedin a simulation
of length 10^8 steps forTx= 10Tyand two different choices ofmx. The bare double-
well potential is also shown on the plot for reference. We see that asmxincreases

z

x

y

φ

θ

r

1 2

(^34)
1
2
3
4
Fig. 8.8 Schematic showing a coordinate system that can be used to obtain a dihedral angle
as an explicit coordinate from the positions of four atoms.
from 10myto 300my, the free energy profile obtained approaches the analytical result
in eqn. (8.10.22). Fig. 10 of Rossoet al.(2002) gives an excellent illustration of the
expected dynamical behavior of adiabatically decoupled variables using the technique
of time correlation functions (to be discussed in Chapter 13).
The adiabatic free energy dynamics approach can be used to generate the two-
dimensional free energy surface of the alanine dipeptide in Fig. 8.5. Its use requires a
transformation to a coordinate system that includes the backbone dihedral angles as
explicit coordinates (however, see Abrams and Tuckerman, 2008 ). Fig. 8.8 illustrates
how the transformation can be carried out. Each set of four neighboring atoms along
the backbone of a polymer or biomolecule define a dihedral angle. Fig.8.8 shows that
when any four atoms with positionsrk+1,...,rk+4labeled as 1, 2, 3, and 4 in the figure
are arranged in a coordinate frame such that the vectorr 3 −r 2 lies along thez-axis
and the vectorr 2 −r 1 is parallel to thex-axis, then when the vectorr 4 −r 3 is resolved
into spherical-polar coordinates, the azimuthal angle is the dihedral angle denotedφin
the figure. Since the transformation can be applied anywhere in thechain, let the four