1549380323-Statistical Mechanics Theory and Molecular Simulation

Metadynamics 357

is a time interval. The purpose of this bias potential is to add Gaussians of heightW
and width ∆sat intervalsτGto the potential energy so that as time increases, these
Gaussians accumulate. If the system starts in a deep basin on the potential energy
surface, then this basin will be “filled in” by the Gaussians, thereby lifting the system
up toward the barrier until it is able to cross into the next basin, which is subsequently
filled by Gaussians until the system can escape into the next basin, and so forth.
Our analysis of the adiabatic dynamics approach shows that if the reaction co-
ordinates move relatively slowly, then they move instantaneously not on the bare
potential energy surface but on the potential of mean force surfaceA(q 1 ,...,qn). Thus,
if Gaussians are added slowly enough, then as time increases,UGtakes on the shape of
−A(q 1 ,...,qn), since it has maxima whereAhas minima, and vice versa. Thus, given a
long trajectoryrG(t) generated using the bias potential, the free energy hypersurface
is constructed using

A(q 1 ,...,qn)≈−W

∑

t=τG, 2 τG,...,

exp

[

−

∑n

α=1

(qα−fα(rG(t)))^2

2∆s^2

]

. (8.11.7)

A proposed proof that eqn. (8.11.7) generates the free energy profile is beyond the
scope of this book; the reader is referred to the work of Laioet al.(2005) for an
analysis based on the Langevin equation (see Chapter 15). It has also been proposed
that the efficiency of metadynamics can be improved by feeding information about the
accumulated histogram into the procedure for adding the Gaussians (Barducciet al.,
2008).
Before closing this section, we note briefly that some of the ideas from metady-
namics have been shown by Maragliano and Vanden-Eijnden (2006) and by Abrams
and Tuckerman (2008) to greatly simplify the adiabatic free energydynamics ap-
proach, eliminating the need of explicit variable transformations, asdiscussed in Sec-
tion 8.10 (Maragliano and Vanden-Eijnden, 2006; Abrams and Tuckerman, 2008). In
order to derive this scheme, we start by writing theδ-functions in eqn. (8.6.4) as the
limit of a product of Gaussians

P(s 1 ,...,sn) =

CN

Q(N,V,T)

lim

κ→∞

(

βκ

2 π

)n/ 2 ∫

dNpdNre−βH(r,p)

×

∏n

α=1

exp

[

−

1

2

βκ(fα(r 1 ,...,rN)−sα)^2

]

. (8.11.8)

The product of Gaussians can be added to the potentialU(r) as a set of harmonic
oscillators with force constantκ. If, in addition, we multiple eqn. (8.11.8) by a set of
nadditional uncoupled Gaussian integrals

∏n

α=1

∫

dpαe−βp

(^2) α/ 2 mα
,
then we can define an extended phase-space Hamiltonian of the form of the following
form