This section describes the -dependence of the potentials
used for free energy calculations (see sec. Free energy calculations). All common
types of potentials and constraints can be interpolated smoothly from
state A () to state B () and vice
versa. All bonded interactions are interpolated by linear interpolation
of the interaction parameters. Non-bonded interactions can be
interpolated linearly or via soft-core interactions.
Starting in GROMACS 4.6, is a vector, allowing different
components of the free energy transformation to be carried out at
different rates. Coulomb, Lennard-Jones, bonded, and restraint terms can
all be controlled independently, as described in the
mdp options.
The example given here is for the bond potential, which is harmonic in
GROMACS. However, these equations apply to the angle potential and the
improper dihedral potential as well.
Fourth-power bond stretching and cosine-based angle potentials are
interpolated by linear interpolation of the force constant and the
equilibrium position. Formulas are not given here.
It should be noted that it is also possible to express a pathway from
state A to state B using and (see
(144)). It may seem to make sense physically to vary the
force field parameters and rather than
the derived parameters and . However, the
difference between the pathways in parameter space is not large, and the
free energy itself does not depend on the pathway, so we use the simple
formulation presented above.
When the mass of a particle changes, there is also a contribution of the
kinetic energy to the free energy (note that we can not write the
momentum as
, since that would result in the
sign of being
incorrect 99):
The constraints are formally part of the Hamiltonian, and therefore they
give a contribution to the free energy. In GROMACS this can be
calculated using the LINCS or the SHAKE algorithm. If we have
constraint equations for LINCS, then
where is the displacement vector
between two particles and is the constraint distance between
the two particles. We can express the fact that the constraint distance
has a dependency by
In a free-energy calculation where particles grow out of nothing, or
particles disappear, using the simple linear interpolation of the
Lennard-Jones and Coulomb potentials as described in
(256) and (255) may lead to poor
convergence. When the particles have nearly disappeared, or are close to
appearing (at close to 0 or 1), the interaction energy
will be weak enough for particles to get very close to each other,
leading to large fluctuations in the measured values of
(which, because of the simple
linear interpolation, depends on the potentials at both the endpoints of
).
To circumvent these problems, the singularities in the potentials need
to be removed. This can be done by modifying the regular Lennard-Jones
and Coulomb potentials with “soft-core” potentials that limit the
energies and forces involved at values between 0 and
1, but not at or 1.
In GROMACS the soft-core potentials are shifted versions
of the regular potentials, so that the singularity in the potential and
its derivatives at is never reached. This formulation was
introduced by Beutler et al.100:
where and are the normal “hard core” Van der
Waals or electrostatic potentials in state A () and
state B () respectively, is the
soft-core parameter (set with sc_alpha in the
mdp file), is the soft-core
power (set with sc_power), is the radius
of the interaction, which is or an input
parameter (sc_sigma) when or
is zero. Beutler et al.100 probed various
combinations of the power values for the Lennard-Jones
and Coulombic interactions. GROMACS uses for both,
van der Waals and electrostatic interactions.
For intermediate , and alter
the interactions very little for and
quickly switch the soft-core interaction to an almost constant value for
smaller (Fig. 32). The force is:
The original GROMOS Lennard-Jones soft-core
function100 uses , but gives a smoother
curve. Another issue that should be
considered is the soft-core effect of hydrogens without Lennard-Jones
interaction. Their soft-core is set with
sc_sigma in the mdp file. These
hydrogens produce peaks in at
is 0 and/or 1 for and close to 0 and/or 1
with . Lowering sc_sigma
will decrease this effect, but it will also increase the interactions
with hydrogens relative to the other interactions in the soft-core
state.
When soft-core potentials are selected (by setting
sc_alpha>0), and the Coulomb and Lennard-Jones
potentials are turned on or off sequentially, then the Coulombic
interaction is turned off linearly, rather than using soft-core
interactions, which should be less statistically noisy in most cases.
This behavior can be overwritten by using the mdp option
sc-coul to yes. Note that the
sc-coul is only taken into account when lambda states
are used, not with couple-lambda0 /
couple-lambda1, and you can still turn off soft-core
interactions by setting sc-alpha=0. Additionally, the
soft-core interaction potential is only applied when either the A or B
state has zero interaction potential. If both A and B states have
nonzero interaction potential, default linear scaling described above is
used. When both Coulombic and Lennard-Jones interactions are turned off
simultaneously, a soft-core potential is used, and a hydrogen is being
introduced or deleted, the sigma is set to sc-sigma-min,
which itself defaults to sc-sigma-default.
In this section we describe the functional form and parameters for
the soft-cored non-bonded interactions using the formalism by Gapsys et al.185.
The Gapsys et al. soft-core is formulated to act on the level of van der Waals and electrostatic forces:
the non-bonded interactions are linearized at a point defined as, or , respectively.
The linearization point depends on the state of the system as controlled by the parameter and
two parameters (set with sc-gapsys-scale-linpoint-q) and (set with sc-gapsys-scale-linpoint-lj).
The dependence on guarantees that the end-states are properly represented by their hard-core potentials.
Fig. 33 illustrates the behaviour of the linearization point, forces and integrated potential energies with respect
to the parameters and . The optimal choices of the parameter values have been systematically explored in 185. These recommended values are set by default when sc-function=gapsys is selected: sc-gapsys-scale-linpoint-q=0.3 and sc-gapsys-scale-linpoint-lj=0.85.
The parameter is a unitless scaling factor in the range .
It scales the position of the point from which the van der Waals force will be linearized.
The linearization of the force is allowed in the range ,
where setting results in a standard hard-core van der Waals interaction.
Setting closer to 1 brings the force linearization point towards
the minimum in the Lennard-Jones force curve ().
This construct allows retaining the repulsion between two particles with non-zero C12 parameter at any value.
The parameter has a unit of and is defined in the range .
It scales the position of the point from which the Coulombic force will be linearized.
Even though in theory can be set to an arbitrarily large value,
algorithmically the linearization point for the force is bound in the range ,
where setting results in a standard hard-core Coulombic interaction.
Setting to a larger value softens the Coulombic force.
In all the notations below, for simplicity, the distance between two atoms and is noted as , i.e. .
where the switching point between the soft and hard-core Lennard-Jones forces
for state A, and
for state B.
In analogy to the Beutler et al. soft core version, is the radius of the interaction, which is
or an input parameter (set with sc-sigma-LJ-gapsys) when C6 or C12 is zero. The default value for this parameter is sc-sigma-LJ-gapsys=0.3.
where the switching point between the soft and hard-core electrostatic forces is
for state A, and
for state B.
The dependence of the linearization point for both van der Waals and Coulombic interactions is of the same power .