adaptive biasing force Overcoming free energy barriers in molecular simulations using an average force 

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Theoretical foundations Strictly speaking, a potential
of mean
force (PMF) is the reversible work supplied to the system to bring two solvated particles,
or
ensembles of particles, from an infinite separation to some contact distance:
w(r) = −1/β ln g(r) (1) Here, g(r) is the pair
correlation
function of the two particles, or ensembles thereof. The vocabulary “PMF” has,
however, been extended to a
wide range of order parameters, or reaction coordinate surrogates, that
go far beyond simple interatomic or
intermolecular
distances. In this perspective, generalization of equation (1) is not straightforward. This
explains
why it may be desirable to turn to a definition suitable for any type of parameter, ξ:
A(ξ) = 1/β ln P(ξ) + A_{0 }(2) A(ξ) is the free energy of the state defined by a particular value of ξ, which corresponds to an isohypersurface in phase space. A_{0} is a constant and P(ξ) is the probability density to find the chemical system of interest at ξ. The connection between the derivative of the free energy with respect to the order parameter, dA(ξ)/dξ, and the forces exerted along the latter may be written as: dA(ξ)/dξ = 〈∂V(x) / ∂ξ〉_{ξ}  1/β 〈∂ ln J / ∂ξ〉_{ξ }=_{ }〈F_{ξ}〉_{ξ }_{ }(3) where J is the determinant of the Jacobian for the transformation from generalized to Cartesian coordinates. The first term of the ensemble average corresponds to physical forces exerted on the system, derived from the potential energy function, V(x). The second contribution is a purely geometric correction that accounts for difference in phase space availability as the order parameter varies. It is worth noting, that, contrary to its instantaneous component, F_{ξ}, only the average force, 〈F_{ξ}〉_{ξ }is physically meaningful. In the framework of the average
biasing
force (ABF) approach, F_{ξ} is accumulated in small windows or bins of finite
size, δξ, thereby providing an estimate
of the
derivative dA(ξ)/dξ defined in equation (3). The force
applied along
the order parameter to overcome free energy barriers is defined as:
F^{ABF} = 〈F_{ξ}〉_{ξ }∇_{ξ} ξ (4) As sampling of the phase space
proceeds,
the estimate of dA(ξ)/dξ is progressively refined. The
biasing force, F^{ABF}, introduced in the equations
of motion
guarantees that in the bin centered about ξ, the force acting along the
order
parameter averages to zero over time. Evolution of the system along ξ is, therefore, governed mainly
by its self diffusion properties.
