# 9. Exploring the phase space¶

## 9.1. Introduction¶

This section assumes that one has defined a force-field model as explained in the previous section, Force-field models. The tools discussed in this section allow one to explore the phase space of a system (and derive its thermodynamic properties) using a force field model.

All algorithms are implemented such that they assume very little about the internals of the force field models. The force field takes atomic positions and cell vectors as input, and returns the energy (and optionally forces and a virial tensor). All algorithms below are only relying on this basic interface.

Most of the algorithms are extensible through so-called hooks. These hooks are pieces of code that can be plugged into a basic algorithm (like a Verlet integrator) to add functionality like writing trajectory files, sampling other ensembles or computing statistical properties on the fly.

One important aspect of yaff.analysis is that that trajectory data can be written to an HDF5 file. In short, HDF5 is a cross-platform format to store efficiently any type of binary array data. A HDF5 file stores arrays in a tree sturcture, which is similar to files and directories in a regular file system. More details about HDF5 can be found on wikipedia and on the non-profit HDF Group website. This format is designed to handle huge amounts of binary data and it greatly facilitates post-processing analysis of the trajectory data. By convention, Yaff stores all data in HDF5 files in atomic units.

## 9.2. Molecular Dynacmis¶

### 9.2.1. Overview of the Verlet algorithms¶

The equations of motion in the NVE ensemble can be integrated as follows:

verlet = VerletIntegrator(ff, 1*femtosecond, temp0=300)
verlet.run(5000)


This example just propagates the system with 5000 steps of 1 fs, but does nearly nothing else. After calling the run method, one can inspect atomic positions and velocities of the final time step:

print verlet.vel
print verlet.pos
print ff.system.pos     # equivalent to the previous line
print verlet.ekin/kjmol # the kinetic energy in kJ/mol.


By default all information from past steps is discarded. If one is interested in writing a trajectory file, one must add a hook to do so. The following example writes a HDF5 trajectory file:

hdf5_writer = HDF5Writer(h5.File('output.h5', mode='w'))
verlet = VerletIntegrator(ff, 1*femtosecond, hooks=hdf5_writer, temp0=300)
verlet.run(5000)


The parameters of the integrator can be tuned with several optional arguments of the VerletIntegrator constructor. See yaff.sampling.verlet.VerletIntegrator for more details. The exact contents of the HDF5 file depends on the integrator used and the optional arguments of the integrator and the yaff.sampling.io.HDF5Writer. The typical tree structure of a trajectory HDF5 file is as follows. (Comments were added manually to the output of h5dump to describe all the arrays.):

\$ h5dump -n production.h5
HDF5 "production.h5" {
FILE_CONTENTS {
group      /
group      /system                          # The 'system' group contains most attributes of the System class.
dataset    /system/bonds
dataset    /system/charges
dataset    /system/ffatype_ids
dataset    /system/ffatypes
dataset    /system/masses
dataset    /system/numbers
dataset    /system/pos
dataset    /system/rvecs
group      /trajectory                      # The 'trajectory' group contains the time-dependent data.
dataset    /trajectory/cell                 # cell vectors
dataset    /trajectory/cons_err             # the root of the ratio of the variance on the conserved quantity
#     and the variance on the kinetic energy
dataset    /trajectory/counter              # an integer counter for the integrator steps
dataset    /trajectory/dipole               # the dipole moment
dataset    /trajectory/dipole_vel           # the time derivative of the dipole moment
dataset    /trajectory/econs                # the conserved quantity
dataset    /trajectory/ekin                 # the kinetic energy
dataset    /trajectory/epot                 # the potential energy
dataset    /trajectory/epot_contribs        # the contributions to the potential energy from the force field parts.
dataset    /trajectory/etot                 # the total energy (kinetic + potential)
dataset    /trajectory/pos                  # the atomic positions
dataset    /trajectory/rmsd_delta           # the RMSD change of the atomic positions
dataset    /trajectory/rmsd_gpos            # the RMSD value of the Cartesian energy gradient (forces if you like)
dataset    /trajectory/temp                 # the instantaneous temperature
dataset    /trajectory/time                 # the time
dataset    /trajectory/vel                  # the atomic velocities
dataset    /trajectory/volume               # the (generalized) volume of the unit cell
}
}


The hooks argument may also be a list of hook objects. For example, one may include the yaff.sampling.nvt.AndersenThermostat to reset the velocities every 200 steps. The yaff.sampling.io.XYZWriter can be added to write a trajectory of the atomic positions in XYZ format:

hooks=[
HDF5Writer(h5.File('output.h5', mode='w')),
AndersenThermostat(temp=300, step=200),
XYZWriter('trajectory.xyz'),
]


By default a screen logging hook is added (if not yet present) to print one line per iteration with some critical integrator parameters. The output of the VerletIntegrator is as follows:

VERLET ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
VERLET Cons.Err. = the root of the ratio of the variance on the conserved
VERLET             quantity and the variance on the kinetic energy.
VERLET d-rmsd    = the root-mean-square displacement of the atoms.
VERLET g-rmsd    = the root-mean-square gradient of the energy.
VERLET counter  Cons.Err.       Temp     d-RMSD     g-RMSD   Walltime
VERLET ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
VERLET       0    0.00000      299.5     0.0000       93.7        0.0
VERLET       1    0.15231      286.4     0.0133      100.1        0.0
VERLET       2    0.17392      297.8     0.0132       90.6        0.0
VERLET       3    0.19803      306.8     0.0137       82.1        0.0


The screen output is geared towards detecting simulation errors. The parameters Cons.Err., Temp, d-RMSD, g-RMSD should exhibit only minor fluctuations in a proper MD run, except when the system only consists of just a few atoms. The wall time should increase at a somewhat constant rate.

It is often desirable to control the amount of data processed by the hooks, e.g. to limit the size of the trajectory files and the amount of screen output. Most hooks have start and step arguments for this purpose. Consider the following example:

hooks=[
VerletScreenLog(step=100)
HDF5Writer(h5.File('output.h5', mode='w'), start=5000, step=10),
XYZWriter('trajectory.xyz', step=50),
AndersenThermostat(temp=300, step=1000),
]


In this example, the screen output contains only one line per 100 NVE iterations. The HDF5 trajectory only contains trajectory data starting from step 5000 with intervals of 10 steps. The XYZwriter only contains the positions of the atoms every 50 steps. The Andersen thermostat only resets the atomic velocities every 1000 steps.

For a detailed description of all options of the VerletIntegrator and the supported hooks, we refer to the reference documentation:

### 9.2.2. Initial atomic velocities¶

When no initial velocities are given to the constructor of the VerletIntegrator constructor, these velocities are randomly sampled from a Poisson-Boltzmann distribution. The temperature of the distribution is controlled by the temp0 argument and if needed, the velocities can be rescaled by using the scalevel0=True argument.

The default behavior is to not remove center-of-mass and global angular momenta. However, for the Nose-Hoover thermostat, this is mandatory and done automatically. For the computation of the instantanuous temperature, one must know the number of degrees of freedom (ndof) in which the kinetic energy is distributed. The default value for ndof is in line with the default initial velocities. ndof is always set to 3N, except for the Nose-Hoover thermostat, where ndof is set to the number of internal degrees of freedom.

One may specify custom initial velocities and ndof by using the vel0 and ndof arguments of the VerletIntegrator constructor. The module yaff.samplling.utils contains various functions to set up initial velocities.

## 9.3. Geometry optimization¶

A basic geometry optimization (with trajectory output in an HDF5 file) is implemented as follows:

hdf5 = HDF5Writer(h5.File('output.h5', mode='w'))
opt = CGOptimizer(CartesianDOF(ff), hooks=hdf5)
opt.run(5000)


The CartesianDOF() argument indicates that only the positions of the nuclei will be optimized. The convergence criteria are controlled through optional arguments of the yaff.sampling.dof.CartesianDOF class. The run method has the maximum number of iterations as the only optional argument. If run is called without arguments, the optimization continues until convergence is reached.

One may also perform an optimization of the nuclei and the cell parameters as follows:

hdf5 = HDF5Writer(h5.File('output.h5', mode='w'))
opt = CGOptimizer(FullCellDOF(ff), hooks=hdf5)
opt.run(5000)


This will transform the degrees of freedom (DOFs) of the system (cell vectors and Cartesian coordinates) into a new set of DOF’s (scaled cell vectors and reduced coordinates) to allow an efficient optimization of both cell parameters atomic positions. One may replace yaff.sampling.dof.FullCellDOF by any of the following:

The optional arguments of any CellDOF variant includes convergence criteria for the cell parameters and the do_frozen option to freeze the fractional coordinates of the atoms.

## 9.4. Harmonic approximations¶

Yaff can compute matrices of second order derivatives of the energy based on symmetric finite differences of analytic gradients for an arbitrary DOF object. This is the most general approach to compute such a generic Hessian:

hessian = estimate_hessian(dof)


where dof is a DOF object like CellDOF and others discussed in the previous section. The routines discussed in the following subsections are based on this generic Hessian routine. See yaff.sampling.harmonic for a description of the harmonic approximation routines.

### 9.4.1. Vibrational analysis¶

The Cartesian Hessian is computed as follows:

hessian = estimate_cart_hessian(ff)


This function uses the symmetric finite difference approximation to estimate the Hessian using many analytic gradient computations. Further vibrational analysis based on this Hessian can be carried out with TAMkin:

hessian = estimate_cart_hessian(ff)
gpos = np.zeros(ff.system.pos.shape, float)
epot = ff.compute(gpos)

import tamkin
mol = tamkin.Molecule(system.numbers, system.pos, system.masses, epot, gpos, hessian)
nma = tamkin.NMA(mol)
invcm = lightspeed/centimeter
print nma.freqs/invcm


One may also compute the Hessian of a subsystem, e.g. for the first three atoms, as follows:

hessian = estimate_cart_hessian(ff, select=[0, 1, 2])


### 9.4.2. Elastic constants¶

Yaff can estimate the elastic constants of a system at zero Kelvin. Just like the computation of the Hessian, the elastic constants are obtained from symmetric finite differences of analytic gradient computations. The standard approach is:

elastic = estimate_elastic(ff)


where elastic is a symmetric 6 by 6 matrix with the elastic constants stored in Voight notation. If the system under scrutiny does not change its relative coordinates when the cell is deformed, one may use a faster approach:

elastic = estimate_elastic(ff, do_frozen=True)

A detailed description of this routine can be found here: yaff.sampling.harmonic.estimate_elastic().