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update fmass part of fix pimd/langevin document

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This commit is contained in:
Yifan Li
2023-03-18 10:17:01 -04:00
parent 76c19410c7
commit fd21a584bf
1 changed files with 26 additions and 2 deletions
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doc/src/fix_pimd.rst
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@ -30,10 +30,11 @@ Syntax
* keywords for style *pimd/langevin*
.. parsed-literal::
*keywords* = *method* or *integrator* or *ensemble* or *fmmode* or *scale* or *temp* or *thermostat* or *tau* or or *iso* or *aniso* or *barostat* or *taup* or *fixcom* or *lj*
*keywords* = *method* or *integrator* or *ensemble* or *fmmode* or *fmass* or *scale* or *temp* or *thermostat* or *tau* or or *iso* or *aniso* or *barostat* or *taup* or *fixcom* or *lj*
*method* value = *nmpimd*
*integrator* value = *obabo* or *baoab*
*fmmode* value = *physical* or *normal*
*fmass* value = scaling factor on mass
*temp* value = Temperature (temperarate unit)
Temperature = target temperarate of the thermostat
*thermostat* values = style seed
@ -81,9 +82,11 @@ by the following equations:
.. math::
Z = & \int d{\bf q} d{\bf p} \cdot \textrm{exp} [ -\beta H_{eff} ] \\
H_{eff} = & \bigg(\sum_{i=1}^P \frac{p_i^2}{2m_i}\bigg) + V_{eff} \\
H_{eff} = & \bigg(\sum_{i=1}^P \frac{p_i^2}{2M_i}\bigg) + V_{eff} \\
V_{eff} = & \sum_{i=1}^P \bigg[ \frac{mP}{2\beta^2 \hbar^2} (q_i - q_{i+1})^2 + \frac{1}{P} V(q_i)\bigg]
$M_i$ is the fictitious mass of the $i$-th mode, and m is the actual mass of the atoms.
The interested user is referred to any of the numerous references on
this methodology, but briefly, each quantum particle in a path integral
simulation is represented by a ring-polymer of P quasi-beads, labeled
@ -152,6 +155,27 @@ masses of beads, which can be used for the Partial Adiabatic CMD
:ref:`(Hone) <Hone>`, or to be set as P, which results in the fictitious
masses to be equal to the real particle masses.
The keyword *fmmode* of *fix pimd/langevin* determines the mode of fictitious
mass preconditioning. There are two options: *physical* and *normal*. If *fmmode* is
*physical*, then the physical mass of the particles are used (and then multiplied by
*fmass*). If *fmmode* is *normal*, then the physical mass is first multiplied by the
eigenvalue of each normal mode, and then multiplied by *fmass*. More precisely, the
fictitious mass of *fix pimd/langevin* is determined by two factors: *fmmode* and *fmass*.
If *fmmode* is *physical*, then the fictitious mass is
.. math::
$M_i = \mathrm{fmass} \times m$
If *fmmode* is *normal*, then the fictitious mass is
.. math::
$M_i = \mathrm{fmass} \times \lambda_i \times m$
where $\lambda_i$ is the eigenvalue of the $i$-th normal mode.
.. note::
Fictitious mass is only used in the momentum of the equation of motion
($\bf{p}_i=M_i\bf{v}_i$), and not used in the spring elastic energy
($\sum_{i=1}^P \frac{1}{2}m\omega_P^2(q_i - q_{i+1})^2$, $m$ is always the
actual mass of the particles).
The keyword *sp* is a scaling factor on Planck's constant, which can
be useful for debugging or other purposes. The default value of 1.0
is appropriate for most situations.