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doc/html/_sources/compute_fep.txt
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@ -61,15 +61,15 @@ thermodynamic integration (FDTI) or Bennet's acceptance ratio method
The potential energy of the system is decomposed in three terms: a
background term corresponding to interaction sites whose parameters
remain constant, a reference term *U*\ <sub>0</sub> corresponding to the
remain constant, a reference term :math:`U_0` corresponding to the
initial interactions of the atoms that will undergo perturbation, and
a term *U*\ <sub>1</sub> corresponding to the final interactions of
a term :math:`U_1` corresponding to the final interactions of
these atoms:
.. image:: Eqs/compute_fep_u.jpg
:align: center
A coupling parameter &lambda; varying from 0 to 1 connects the
A coupling parameter :math:`\lambda` varying from 0 to 1 connects the
reference and perturbed systems:
.. image:: Eqs/compute_fep_lambda.jpg
@ -79,7 +79,7 @@ It is possible but not necessary that the coupling parameter (or a
function thereof) appears as a multiplication factor of the potential
energy. Therefore, this compute can apply perturbations to interaction
parameters that are not directly proportional to the potential energy
(e.g. &sigma; in Lennard-Jones potentials).
(e.g. :math:`\sigma` in Lennard-Jones potentials).
This command can be combined with :doc:`fix adapt <fix_adapt>` to
perform multistage free-energy perturbation calculations along
@ -91,26 +91,26 @@ stepwise alchemical transformations during a simulation run:
This compute is suitable for the finite-difference thermodynamic
integration (FDTI) method :ref:`(Mezei) <Mezei>`, which is based on an
evaluation of the numerical derivative of the free energy by a
perturbation method using a very small &delta;:
perturbation method using a very small :math:`\delta`:
.. image:: Eqs/compute_fep_fdti.jpg
:align: center
where *w*\ <sub>i</sub> are weights of a numerical quadrature. The :doc:`fix adapt <fix_adapt>` command can be used to define the stages of
&lambda; at which the derivative is calculated and averaged.
where :math:`w_i` are weights of a numerical quadrature. The :doc:`fix adapt <fix_adapt>` command can be used to define the stages of
:math:`\lambda` at which the derivative is calculated and averaged.
The compute fep calculates the exponential Boltzmann term and also the
potential energy difference *U*\ <sub>1</sub>-\ *U*\ <sub>0</sub>. By
choosing a very small perturbation &delta; the thermodynamic
potential energy difference :math:`U_1 -U_0`. By
choosing a very small perturbation :math:`\delta` the thermodynamic
integration method can be implemented using a numerical evaluation of
the derivative of the potential energy with respect to &lambda;:
the derivative of the potential energy with respect to :math:`\lambda`:
.. image:: Eqs/compute_fep_ti.jpg
:align: center
Another technique to calculate free energy differences is the
acceptance ratio method :ref:`(Bennet) <Bennet>`, which can be implemented
by calculating the potential energy differences with &delta; = 1.0 on
by calculating the potential energy differences with :math:`\delta` = 1.0 on
both the forward and reverse routes:
.. image:: Eqs/compute_fep_bar.jpg
@ -241,12 +241,12 @@ trajectories during which the volume fluctuates or changes :ref:`(Allen and Tild
**Output info:**
This compute calculates a global vector of length 3 which contains the
energy difference (\ *U*\ <sub>1</sub>-\ *U*\ <sub>0</sub>) as c_ID[1], the
Boltzmann factor exp(-(\ *U*\ <sub>1</sub>-\ *U*\ <sub>0</sub>)/\ *kT*\ ), or
*V*\ exp(-(\ *U*\ <sub>1</sub>-\ *U*\ <sub>0</sub>)/\ *kT*\ ), as c_ID[2] and the
volume of the simulation box *V* as c_ID[3]. *U*\ <sub>1</sub> is the
energy difference ( :math:`U_1-U_0` ) as c_ID[1], the
Boltzmann factor :math:`\exp(-(U_1-U_0)/kT)`, or
:math:`V \exp(-(U_1-U_0)/kT)`, as c_ID[2] and the
volume of the simulation box :math:`V` as c_ID[3]. :math:`U_1` is the
pair potential energy obtained with the perturbed parameters and
*U*\ <sub>0</sub> is the pair potential energy obtained with the
:math:`U_0` is the pair potential energy obtained with the
unperturbed parameters. The energies include kspace terms if these
are used in the simulation.