.. _feia:
Free energy interactions
------------------------
This section describes the :math:`\lambda`-dependence of the potentials
used for free energy calculations (see sec. :ref:`fecalc`). All common
types of potentials and constraints can be interpolated smoothly from
state A (:math:`\lambda=0`) to state B (:math:`\lambda=1`) 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, :math:`\lambda` 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
:ref:`mdp` options.
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.
.. math:: \begin{aligned}
V_b &=&{\frac{1}{2}}\left[{(1-{\lambda})}k_b^A +
{\lambda}k_b^B\right] \left[b - {(1-{\lambda})}b_0^A - {\lambda}b_0^B\right]^2 \\
{\frac{\partial V_b}{\partial {\lambda}}}&=&{\frac{1}{2}}(k_b^B-k_b^A)
\left[b - {(1-{\lambda})}b_0^A + {\lambda}b_0^B\right]^2 +
\nonumber\\
& & \phantom{{\frac{1}{2}}}(b_0^A-b_0^B) \left[b - {(1-{\lambda})}b_0^A -{\lambda}b_0^B\right]
\left[{(1-{\lambda})}k_b^A + {\lambda}k_b^B \right]\end{aligned}
:label: eqnfepharmpot
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:
.. math:: \begin{aligned}
V_d &=&\left[{(1-{\lambda})}k_d^A + {\lambda}k_d^B \right]
\left( 1+ \cos\left[n_{\phi} \phi -
{(1-{\lambda})}\phi_s^A - {\lambda}\phi_s^B
\right]\right)\\
{\frac{\partial V_d}{\partial {\lambda}}}&=&(k_d^B-k_d^A)
\left( 1+ \cos
\left[
n_{\phi} \phi- {(1-{\lambda})}\phi_s^A - {\lambda}\phi_s^B
\right]
\right) +
\nonumber\\
&&(\phi_s^B - \phi_s^A) \left[{(1-{\lambda})}k_d^A - {\lambda}k_d^B\right]
\sin\left[ n_{\phi}\phi - {(1-{\lambda})}\phi_s^A - {\lambda}\phi_s^B \right]\end{aligned}
:label: eqnfeppropdihedral
**Note:** that the multiplicity :math:`n_{\phi}` can not be
parameterized because the function should remain periodic on the
interval :math:`[0,2\pi]`.
Tabulated bonded interactions
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
For tabulated bonded interactions only the force constant can
interpolated:
.. math:: \begin{aligned}
V &=& ({(1-{\lambda})}k^A + {\lambda}k^B) \, f \\
{\frac{\partial V}{\partial {\lambda}}} &=& (k^B - k^A) \, f\end{aligned}
:label: eqnfeptabbonded
Coulomb interaction
~~~~~~~~~~~~~~~~~~~
The Coulomb interaction between two particles of which the charge varies
with :math:`{\lambda}` is:
.. math:: \begin{aligned}
V_c &=& \frac{f}{{\varepsilon_{rf}}{r_{ij}}}\left[{(1-{\lambda})}q_i^A q_j^A + {\lambda}\, q_i^B q_j^B\right] \\
{\frac{\partial V_c}{\partial {\lambda}}}&=& \frac{f}{{\varepsilon_{rf}}{r_{ij}}}\left[- q_i^A q_j^A + q_i^B q_j^B\right]\end{aligned}
:label: eqnfepcoloumb
where :math:`f = \frac{1}{4\pi \varepsilon_0} = {138.935\,458}` (see
chapter :ref:`defunits`).
Coulomb interaction with reaction field
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The Coulomb interaction including a reaction field, between two
particles of which the charge varies with :math:`{\lambda}` is:
.. math:: \begin{aligned}
V_c &=& f\left[\frac{1}{{r_{ij}}} + k_{rf}~ {r_{ij}}^2 -c_{rf}\right]
\left[{(1-{\lambda})}q_i^A q_j^A + {\lambda}\, q_i^B q_j^B\right] \\
{\frac{\partial V_c}{\partial {\lambda}}}&=& f\left[\frac{1}{{r_{ij}}} + k_{rf}~ {r_{ij}}^2 -c_{rf}\right]
\left[- q_i^A q_j^A + q_i^B q_j^B\right]
\end{aligned}
:label: eqdVcoulombdlambda
**Note** that the constants :math:`k_{rf}` and :math:`c_{rf}` are
defined using the dielectric constant :math:`{\varepsilon_{rf}}` of the
medium (see sec. :ref:`coulrf`).
Lennard-Jones interaction
~~~~~~~~~~~~~~~~~~~~~~~~~
For the Lennard-Jones interaction between two particles of which the
*atom type* varies with :math:`{\lambda}` we can write:
.. math:: \begin{aligned}
V_{LJ} &=& \frac{{(1-{\lambda})}C_{12}^A + {\lambda}\, C_{12}^B}{{r_{ij}}^{12}} -
\frac{{(1-{\lambda})}C_6^A + {\lambda}\, C_6^B}{{r_{ij}}^6} \\
{\frac{\partial V_{LJ}}{\partial {\lambda}}}&=&\frac{C_{12}^B - C_{12}^A}{{r_{ij}}^{12}} -
\frac{C_6^B - C_6^A}{{r_{ij}}^6}
\end{aligned}
:label: eqdVljdlambda
It should be noted that it is also possible to express a pathway from
state A to state B using :math:`\sigma` and :math:`\epsilon` (see
:eq:`eqn. %s <eqnsigeps>`). It may seem to make sense physically to vary the
force field parameters :math:`\sigma` and :math:`\epsilon` rather than
the derived parameters :math:`C_{12}` and :math:`C_{6}`. 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.
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 :math:`\mathbf{p}` as
m :math:`\mathbf{v}`, since that would result in the
sign of :math:`{\frac{\partial E_k}{\partial {\lambda}}}` being
incorrect \ :ref:`99 <refGunsteren98a>`):
.. math:: \begin{aligned}
E_k &=& {\frac{1}{2}}\frac{\mathbf{p}^2}{{(1-{\lambda})}m^A + {\lambda}m^B} \\
{\frac{\partial E_k}{\partial {\lambda}}}&=& -{\frac{1}{2}}\frac{\mathbf{p}^2(m^B-m^A)}{({(1-{\lambda})}m^A + {\lambda}m^B)^2}\end{aligned}
:label: eqnfepekin
after taking the derivative, we *can* insert
:math:`\mathbf{p}` = m :math:`\mathbf{v}`, such that:
.. math:: {\frac{\partial E_k}{\partial {\lambda}}}~=~ -{\frac{1}{2}}\mathbf{v}^2(m^B-m^A)
:label: eqnfepekinderivative
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
:math:`k = 1 \ldots K` constraint equations :math:`g_k` for LINCS, then
.. math:: g_k = | \mathbf{r}_{k} | - d_{k}
:label: eqnfepconstr
where :math:`\mathbf{r}_k` is the displacement vector
between two particles and :math:`d_k` is the constraint distance between
the two particles. We can express the fact that the constraint distance
has a :math:`{\lambda}` dependency by
.. math:: d_k = {(1-{\lambda})}d_{k}^A + {\lambda}d_k^B
:label: eqnfepconstrdistdep
Thus the :math:`{\lambda}`-dependent constraint equation is
.. math:: g_k = | \mathbf{r}_{k} | - \left({(1-{\lambda})}d_{k}^A + {\lambda}d_k^B\right).
:label: eqnfepconstrlambda
The (zero) contribution :math:`G` to the Hamiltonian from the
constraints (using Lagrange multipliers :math:`\lambda_k`, which are
logically distinct from the free-energy :math:`{\lambda}`) is
.. math:: \begin{aligned}
G &=& \sum^K_k \lambda_k g_k \\
{\frac{\partial G}{\partial {\lambda}}} &=& \frac{\partial G}{\partial d_k} {\frac{\partial d_k}{\partial {\lambda}}} \\
&=& - \sum^K_k \lambda_k \left(d_k^B-d_k^A\right)\end{aligned}
:label: eqnconstrfreeenergy
For SHAKE, the constraint equations are
.. math:: g_k = \mathbf{r}_{k}^2 - d_{k}^2
:label: eqnfepshakeconstr
with :math:`d_k` as before, so
.. math:: \begin{aligned}
{\frac{\partial G}{\partial {\lambda}}} &=& -2 \sum^K_k \lambda_k \left(d_k^B-d_k^A\right)\end{aligned}
:label: eqnfepshakeconstr2
Soft-core interactions
~~~~~~~~~~~~~~~~~~~~~~
.. _fig-softcore:
.. figure:: plots/softcore.*
:height: 6.00000cm
Soft-core interactions at :math:`{\lambda}=0.5`, with :math:`p=2` and
:math:`C_6^A=C_{12}^A=C_6^B=C_{12}^B=1`.
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
:eq:`Equations %s <eqdVljdlambda>` and :eq:`%s <eqdVcoulombdlambda>` may lead to poor
convergence. When the particles have nearly disappeared, or are close to
appearing (at :math:`{\lambda}` 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
:math:`\partial V/\partial {\lambda}` (which, because of the simple
linear interpolation, depends on the potentials at both the endpoints of
:math:`{\lambda}`).
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 :math:`{\lambda}` values between 0 and
1, but not *at* :math:`{\lambda}=0` or 1.
In |Gromacs| the soft-core potentials :math:`V_{sc}` are shifted versions
of the regular potentials, so that the singularity in the potential and
its derivatives at :math:`r=0` is never reached:
.. math:: \begin{aligned}
V_{sc}(r) &=& {(1-{\lambda})}V^A(r_A) + {\lambda}V^B(r_B)
\\
r_A &=& \left(\alpha \sigma_A^6 {\lambda}^p + r^6 \right)^\frac{1}{6}
\\
r_B &=& \left(\alpha \sigma_B^6 {(1-{\lambda})}^p + r^6 \right)^\frac{1}{6}\end{aligned}
:label: eqnfepsoftcore
where :math:`V^A` and :math:`V^B` are the normal “hard core†Van der
Waals or electrostatic potentials in state A (:math:`{\lambda}=0`) and
state B (:math:`{\lambda}=1`) respectively, :math:`\alpha` is the
soft-core parameter (set with ``sc_alpha`` in the
:ref:`mdp` file), :math:`p` is the soft-core :math:`{\lambda}`
power (set with ``sc_power``), :math:`\sigma` is the radius
of the interaction, which is :math:`(C_{12}/C_6)^{1/6}` or an input
parameter (``sc_sigma``) when :math:`C_6` or :math:`C_{12}`
is zero.
For intermediate :math:`{\lambda}`, :math:`r_A` and :math:`r_B` alter
the interactions very little for :math:`r > \alpha^{1/6} \sigma` and
quickly switch the soft-core interaction to an almost constant value for
smaller :math:`r` (:numref:`Fig. %s <fig-softcore>`). The force is:
.. math:: F_{sc}(r) = -\frac{\partial V_{sc}(r)}{\partial r} =
{(1-{\lambda})}F^A(r_A) \left(\frac{r}{r_A}\right)^5 +
{\lambda}F^B(r_B) \left(\frac{r}{r_B}\right)^5
:label: eqnfepsoftcoreforce
where :math:`F^A` and :math:`F^B` are the “hard core†forces. The
contribution to the derivative of the free energy is:
.. math:: \begin{aligned}
{\frac{\partial V_{sc}(r)}{\partial {\lambda}}} & = &
V^B(r_B) -V^A(r_A) +
{(1-{\lambda})}\frac{\partial V^A(r_A)}{\partial r_A}
\frac{\partial r_A}{\partial {\lambda}} +
{\lambda}\frac{\partial V^B(r_B)}{\partial r_B}
\frac{\partial r_B}{\partial {\lambda}}
\nonumber\\
&=&
V^B(r_B) -V^A(r_A) + \nonumber \\
& &
\frac{p \alpha}{6}
\left[ {\lambda}F^B(r_B) r^{-5}_B \sigma_B^6 {(1-{\lambda})}^{p-1} -
{(1-{\lambda})}F^A(r_A) r^{-5}_A \sigma_A^6 {\lambda}^{p-1} \right]\end{aligned}
:label: eqnfepsoftcorederivative
The original GROMOS Lennard-Jones soft-core
function\ :ref:`100 <refBeutler94>` uses :math:`p=2`, but :math:`p=1` gives a smoother
:math:`\partial H/\partial{\lambda}` curve. Another issue that should be
considered is the soft-core effect of hydrogens without Lennard-Jones
interaction. Their soft-core :math:`\sigma` is set with
``sc_sigma`` in the :ref:`mdp` file. These
hydrogens produce peaks in :math:`\partial H/\partial{\lambda}` at
:math:`{\lambda}` is 0 and/or 1 for :math:`p=1` and close to 0 and/or 1
with :math:`p=2`. 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 :ref:`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`` Â /
``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``.
Recently, a new formulation of the soft-core approach has been derived
that in most cases gives lower and more even statistical variance than
the standard soft-core path described above \ :ref:`101 <refPham2011>`,
:ref:`102 <refPham2012>`. Specifically, we have:
.. math:: \begin{aligned}
V_{sc}(r) &=& {(1-{\lambda})}V^A(r_A) + {\lambda}V^B(r_B)
\\
r_A &=& \left(\alpha \sigma_A^{48} {\lambda}^p + r^{48} \right)^\frac{1}{48}
\\
r_B &=& \left(\alpha \sigma_B^{48} {(1-{\lambda})}^p + r^{48} \right)^\frac{1}{48}\end{aligned}
:label: eqnnewsoftcore
This “1-1-48†path is also implemented in |Gromacs|. Note that for this
path the soft core :math:`\alpha` should satisfy
:math:`0.001 < \alpha < 0.003`, rather than :math:`\alpha \approx
0.5`.