Free energy interactions#
This section describes the
Starting in GROMACS 4.6,
Harmonic potentials#
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.
GROMOS-96 bonds and angles#
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.
Proper dihedrals#
For the proper dihedrals, the equations are somewhat more complicated:
Note: that the multiplicity
Tabulated bonded interactions#
For tabulated bonded interactions only the force constant can interpolated:
Coulomb interaction#
The Coulomb interaction between two particles of which the charge varies
with
where
Coulomb interaction with reaction field#
The Coulomb interaction including a reaction field, between two
particles of which the charge varies with
Note that the constants
Lennard-Jones interaction#
For the Lennard-Jones interaction between two particles of which the
atom type varies with
It should be noted that it is also possible to express a pathway from
state A to state B using
Kinetic Energy#
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
after taking the derivative, we can insert
Constraints#
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
where
Thus the
The (zero) contribution
For SHAKE, the constraint equations are
with
Soft-core interactions: Beutler et al.#
![../../_images/softcore.png](../../_images/softcore.png)
Fig. 32 Soft-core interactions at
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
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
In GROMACS the soft-core potentials
where sc_alpha
in the
mdp file), sc_power
), sc_sigma
) when
For intermediate
where
The original GROMOS Lennard-Jones soft-core
function100 uses sc_sigma
in the mdp file. These
hydrogens produce peaks in 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
.
Soft-core interactions: Gapsys et al.#
In this section we describe the functional form and parameters for the soft-cored non-bonded interactions using the formalism by Gapsys et al.183.
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, sc-gapsys-scale-linpoint-q
) and sc-gapsys-scale-linpoint-lj
).
The dependence on sc-function=gapsys
is selected: sc-gapsys-scale-linpoint-q=0.3
and sc-gapsys-scale-linpoint-lj=0.85
.
![../../_images/gapsys-sc.png](../../_images/gapsys-sc.png)
Fig. 33 Illustration of the soft-core parameter influence on the linearization point (top row),
forces (middle row) and energies (bottom row)
for van der Waals (left column) and electrostatic interactions (right column).
The case of two interacting atoms is considered.
In state A both atoms have charges of 0.5 and
The parameter
The parameter
In all the notations below, for simplicity, the distance between two atoms
Forces: van der Waals interactions#
where the switching point between the soft and hard-core Lennard-Jones forces
sc-sigma-LJ-gapsys
) when C6 or C12 is zero. The default value for this parameter is sc-sigma-LJ-gapsys=0.3
.
Explicit expression:
Forces: Coulomb interactions#
where the switching point
Explicit expression:
Energies: van der Waals interactions#
Explicition definition of energies:
Energies: Coulomb interactions#
: van der Waals interactions#
Here we provide the explicit expressions of
for Coulomb interactions#
Here we provide the explicit expressions of