# Trajectory Analysis¶

analysis is a standalone program that performs analysis of molecular dynamics trajectories created with AMS. It can produce histograms and radial distribution functions. It is also used under the hood in AMSmovie (MD Properties menu bar).

This is an example showing how to compute the oxygen-oxygen radial distribution function of a MD simulation using the analysis utility program:

Task RadialDistribution

TrajectoryInfo
Trajectory
KFFilename ams.results/ams.rkf
Range 1 1000 2
End
End

NBins 1000
AtomsFrom
Element O
End
AtomsTo
Element O
End
End
eor


The analysis program reads one or more trajectory files (filename.rkf) from an AMS molecular dynamics (MD) or a Grand Canonical Monte Carlo (GCMC) simulation. The file information is supplied in the TrajectoryInfo input block. In this block, a separate Trajectory subblock needs to be supplied for each trajectory file. The Trajectory subblock contains a mandatory keyword KFFilename, and an optional keyword Range. The latter contains the initial frame to be read, the final frame to be read, and optionally the stepsize. By default all frames on the trajectory file are read.

TrajectoryInfo
NBlocksToCompare integer
Trajectory
KFFilename string
Range integer_list
End
End

TrajectoryInfo
Type: Block All the info regarding the reading of the trajectory files.
NBlocksToCompare
Type: Integer 1 Get an error estimate by comparing histograms for NBLocks time blocks of the trajectory.
Trajectory
Type: Block True All info regarding the reading of a single trajectory file.
KFFilename
Type: String ams.rkf The name of the AMS trajectory file.
Range
Type: Integer List Two or three values: start frame, end frame, step size.

All tools in the analysis program provide an option to obtain information on the equilibration of the simulation. If the optional keyword NBlocksToCompare in the TrajectoryInfo block is set to a value $$N$$ higher than 1, the trajectory is divided into $$N$$ blocks, and the analysis results for each block are compared. The variation in the analysis result is provided as a standard deviation.

The Analysis tool computes radial distribution functions $$g(r)$$ if the Task keyword is set to RadialDistribution.

Task [RadialDistribution | Histogram | AutoCorrelation | MeanSquareDisplacement | AverageBinPlot]

Task

Further details on the radial distribution functions are then set in the RadialDistribution block. If more than one RadialDistribution block is present in the input, more than one radial distribution function will be computed. The result is printed to output as text, as well as stored in a binary file (plot.kf).

### Description¶

A radial distribution function $$g(r)$$, or pair correlation function, is a density of distances between particles, relative to the average distance density. The x-axis variable represents a distance $$r$$, while the y-axis represents the relative density of that distance. For a complete homogeneous system of particles the $$g(r)$$ values for the distances between all particles equals 1 everywhere.

Two sets of atoms $$\mathbb{S}_{\textrm{from}}$$ and $$\mathbb{S}_{\textrm{to}}$$, of length $$n_{\textrm{from}}$$ and $$n_{\textrm{to}}$$ respectively, are specified with the keywords AtomsFrom and AtomsTo in the RadialDistribution block. As a result the program computes $$n_{\textrm{from}}*n_{\textrm{to}}$$ distances $$r_{ij}^s$$ between atom $$i$$ in $$\mathbb{S}_{\textrm{from}}$$ and atom $$j$$ in $$\mathbb{S}_{\textrm{to}}$$ for each trajectory frame $$s$$ out of a total of $$n_{\textrm{frames}}$$ frames.

A normalized histogram is then computed from these distances, resulting in a function $$N(r)$$.

$$N(r)=\frac{1}{n_{\textrm{frames}}} \sum_{s=1}^{n_{\textrm{frames}}} \sum_{i=1}^{n_{\textrm{from}}}\sum_{j=1}^{n_{\textrm{to}}} \delta(r_{ij}^s-r)$$.

This histogram is converted to a density, by dividing all values $$N(r)$$ with the volume $$V(r)= 4 \pi r^2 dr$$ of a sphere-slice at radius $$r$$ with thickness $$dr$$.

The density is further converted to a relative density by dividing with the total density of the system $$\rho_{\textrm{tot}} = \frac{n_{\textrm{from}}*n_{\textrm{to}}}{V_{\textrm{tot}}}$$, yielding the final radial distribution function $$g(r)$$.

$$g(r) = \frac{N(r)}{V(r)*\rho_{\textrm{tot}}}$$

### Options¶

Non-periodic systems The above equation assumes that the volume $$V_{\textrm{tot}}$$ of the system is a well-defined quantity. This assumption is correct for systems with 3D periodicity, where the $$V_{\textrm{tot}}$$ is defined as the volume of the periodic cell. In such a system the value of $$r$$ can be no larger than $$r_{\textrm{max}}$$, the radius of the largest sphere that can be placed inside the periodic cell.

If a system is non-periodic in one or more direction, then the program still computes a $$g(r)$$, only if the radius $$r_{max}$$ is supplied by the user with the Range keyword in the RadialDistribution block. The radius is the second value supplied.

RadialDistribution
Range float_list
End

RadialDistribution
Type: Block True All input related to radial distribution functions.
Range
Type: Float List Either one, two, or three real values. If one it is the stepsize. If two, it is the minimum value and the maximum value. If three, it is the minimum value, the maximum value, and the stepsize. The stepsize overrides NBins.

In this case the volume $$V_{\textrm{tot}}$$ is assumed to be the volume of a sphere with radius $$r_{\textrm{max}}$$.

NPT simulations The above equation further assumes that the volume $$V_{\textrm{tot}}$$ is constant throughout the simulation. The $$g(r)$$ of the trajectory from an NPT simulation can still be computed, and in this case $$V_{\textrm{tot}}$$ is the average value of the volume of the periodic cell.

Simulations with varying numbers of atoms The above equation also assumes that $$n_{\textrm{from}}$$ and $$n_{\textrm{to}}$$ remain constant throughout the simulation. However, in a Molecular Gun simulation particles can be added to the system, and in a GCMC simulation particles can be both added and removed from the system. Nonetheless, the program still computes a $$g(r)$$ in these situations.

If the AtomsFrom and AtomsTo blocks contain element names (supplied with the recurring Element keyword), then every time atoms are added to or removed from the system, the sets of atoms $$\mathbb{S}_{\textrm{from}}$$ and $$\mathbb{S}_{\textrm{to}}$$ are re-evaluated.

If the AtomsFrom and AtomsTo blocks contain atom numbers (supplied with the recurring Atom keyword), these numbers are updated in the sets $$\mathbb{S}_{from}$$ and $$\mathbb{S}_{to}$$ every time atoms are added to or removed from the system. If one of the atoms from the set disappears, the number of distances contributing to the $$g(r)$$ decreases.

Note: Currently, the values of $$n_{from}$$ and $$n_{to}$$ in the normalization factor are taken from the last frame of the simulation.

Warning: If multiple trajectories are supplied, and the number of atoms changes between the end of one trajectory and the beginning of another, this may result in an error in the atom numbers used by the program internally.

## Histogram¶

The Analysis program computes histograms if the Task keyword is set to Histogram.

Task [RadialDistribution | Histogram | AutoCorrelation | MeanSquareDisplacement | AverageBinPlot]

Task

Further details on the histogram need to be specified in the Histogram block. If more than one Histogram block is present in the input, more than one histogram will be computed. The result is printed to output as text, as well as stored in a binary file (plot.kf). By default the histogram contains the number of occurrences of a certain value, but the normalized occurrence is provided if the keyword Normalized in the Histogram block is specified.

Histogram
Normalized Yes/No
End

Histogram
Normalized
Type: Bool No Give the normalized histogram.

Histograms can be computed for every quantity stored on the molecular dynamics trajectory file (ams.rkf) in the section History. Example quantities are PotentialEnergy, KineticEnergy, TotalEnergy, Temperature. In the histogram block, this quantity is selected with the keyword Variable in the Axis subblock. If more than one Axis subblock is present, the dimensionality of the histogram is increased: Three Axis subblocks result in a 3D histogram.

For each histogram axis, the number of bins can be selected with the NBins keyword in the Axis block, in which case the range of values along each axis is automatically determined. The default NBins value is 100.

Alternatively, a range and a stepsize can be selected with the keyword Range in the Axis subblock. The keyword Range can contain one, two, or three values: 1: Only a stepsize. 2: A smallest value and a largest value. 3: A smallest value, a largest value, and the stepsize.

Histogram
Axes
Axis
NBins integer
Range float_list
Variable string
End
End
End

Histogram
Type: Block True All input related to histograms.
Axes
Type: Block Specifications for the histogram axes.
Axis
Type: Block True Specifications for a single histogram axis.
NBins
Type: Integer 100 The number of bins along the histogram axis.
Range
Type: Float List Either one, two, or three real values. If one it is the stepsize. If two, it is the minimum value and the maximum value. If three, it is the minimum value, the maximum value, and the stepsize. The stepsize overrides NBins.
Variable
Type: String The quantity along the histogram axis.

## Autocorrelation Functions¶

The Analysis program computes autocorrelation functions (ACF) if the Task keyword is set to AutoCorrelation.

Task [RadialDistribution | Histogram | AutoCorrelation | MeanSquareDisplacement | AverageBinPlot]

Task

Further details need to be specified in the AutoCorrelation block. If more than one AutoCorrelation block is present in the input, more than one ACF will be computed. The result is printed to output as text, as well as stored in a binary file (plot.kf).

AutoCorrelation
Atoms
Atom integer
Element string
End
DataReading [Auto | AtOnce | BlockWise]
InputValues
Values float_list
End
MaxStep integer
NPointsHighestFreq integer
Property [Velocities | DipoleMomentFromCharges | InputValues | DiffusionCoefficient |
DipoleDerivativeFromCharges | PressureTensor | Viscosity]
TimeStep float
UseAllValues Yes/No
VecElements
Index integer
End
End

AutoCorrelation
Atoms
Type: Block Relevant if Property is set to Velocities, DipoleMomentFromCharges, DipoleDerivativeFromCharges, or DiffusionCoefficient. Atom numbers or elements for the set of atoms for which the property is read/computed. By default all atoms are used.
Atom
Type: Integer True Atom number.
Element
Type: String True Element Symbol Atom.
DataReading
Type: Multiple Choice Auto [Auto, AtOnce, BlockWise] The KF data can be read in and handled once, or blockwise. The former is memory intensive, but mostly faster. If Auto is selected, the data is read at once if it is less than 1 GB, and blockwise if it is more.
InputValues
Type: Block Relevant if Property is set to InputValues. All input values (a vector on each line) need to be provided in this block, using the keyword Values (possibly multiple times).
Values
Type: Float List True The values at each step (on a single line)
MaxStep
Type: Integer The maximum number of time steps for which the autocorrelation function will be computed. The default is half of the number of provided frames.
NPointsHighestFreq
Type: Integer 4 The number of points (timesteps) used for the highest frequency displayed in spectrum. This determines up to which frequency the spectrum is displayed. If the spacing between time-steps used for the ACF is 1 fs, then by default the maximum frequency displayed is 0.25 fs-1 (or 8339 cm-1). This corresponds to a (default) value of NPointsHighestFreq of 4. A higher number selected here, will result in a lower maximum frequency returned by the program. The lowest possible value (spectrum up to highest possible frequency) is 2.
Property
Type: Multiple Choice DipoleDerivativeFromCharges [Velocities, DipoleMomentFromCharges, InputValues, DiffusionCoefficient, DipoleDerivativeFromCharges, PressureTensor, Viscosity] Compute the ACF either from velocities (from rkf), the dipole moment (from coordinates and atomic charges in rkf), the dipole moment derivative (from velocities and atomic charges in rkf), from the pressure tensor (from rkf), or from values specified in input. Selecting DiffusionCoefficient is equivalent to selecting Velocities. The default, DipoleDerivativeFromCharges, results in the computation of an IR spectrum.
TimeStep
Type: Float Relevant if Property is set to InputValues. The time separating the entries (in fs). If Property is set to any of the other quantities, it can be read from an RKF file, and the timestep is read from the RKF file as well. The read value then overrides this keyword.
UseAllValues
Type: Bool No By default the same number of values are used for each t-step in the ACF. This has the advantage that all values in the ACF are equally reliable, but it does mean that for the smaller timesteps much of the data is not used. To switch this off and use all data, UseAllValues can be set to true
VecElements
Type: Block A set of indices referring to a subset of the property vector. Works in combination with the atoms block. For example, in combination with the property Velocities, the Atoms block allows the selection of a subset of atoms, while the VecElelements block allows the selection of a subset of vector elements (e.g. 1 and 2 for the elements x and y). Currently not implemented with InputValues.
Index
Type: Integer True Element of the property vector.

### Description¶

An autocorrelation function $$C(t)$$ describes the average correlation (overlap) of a (vector) property $$\textbf{A}$$ with itself as a function of time.

$$C(t) = \langle \textbf{A}(0) \cdot \textbf{A}(t)) \rangle$$

The average runs over all time-intervals $$\left( t_{0}, t_{0}+t \right),\left( t_{1}, t_{1}+t \right),...,\left( t_{N}, t_{N}+t \right)$$, with $$t_{N} = t_{n} - t_{m}$$. Here $$n$$ is the total number of simulation steps in the trajectory, and $$m$$ is the number of discrete $$t$$ values for which $$C(t)$$ is computed. The value $$m$$ can be set with the keyword MaxStep, and defaults to half the total number of simulation steps. As stated above, the default average runs over the same number of time intervals for each value of $$t$$. If UseAllValues is selected, the average runs over all available time intervals for each value of $$t$$, which is large for small $$t$$, and smallest ($$N$$) for $$t_{m}$$.

The normalized autocorrelation function $$c(t)$$ describes the decorrelation of the property with time, and always starts at 1.0 at $$t=0$$.

$$c(t) = \frac{\langle \textbf{A}(0) \cdot \textbf{A}(t)) \rangle}{\langle \textbf{A}(0) \cdot \textbf{A}(0)) \rangle}$$

In most cases short timescale fluctuations are important, so frequent storage of the desired property is required (when preparing the molecular dynamics simulation, set the Frequency keyword in the Trajectory block of the MolecularDynanimcs settings low, preferably to 1).

A power spectrum is automatically computed by Fourier transform of the autocorrelation function, and provides information on the frequencies of the signal. When the selected property is the dipole moment or the dipole moment derivative, the power spectrum matches the IR spectrum.

### Options¶

Autocorrelation functions can be computed for different simulation properties: 1) Dipole moments from coordinates and atomic charges 2) Dipole moment derivatives from velocities and atomic charges 3) Velocities 4) The pressure tensor 5) User provided values. Selecting 6) Diffusion coefficient is equivalent to selecting Velocities.

AutoCorrelation
Property [Velocities | DipoleMomentFromCharges | InputValues | DiffusionCoefficient |
DipoleDerivativeFromCharges | PressureTensor | Viscosity]
End

AutoCorrelation
Type: Block True All input related to auto correlation functions.
Property
Type: Multiple Choice DipoleDerivativeFromCharges [Velocities, DipoleMomentFromCharges, InputValues, DiffusionCoefficient, DipoleDerivativeFromCharges, PressureTensor, Viscosity] Compute the ACF either from velocities (from rkf), the dipole moment (from coordinates and atomic charges in rkf), the dipole moment derivative (from velocities and atomic charges in rkf), from the pressure tensor (from rkf), or from values specified in input. Selecting DiffusionCoefficient is equivalent to selecting Velocities. The default, DipoleDerivativeFromCharges, results in the computation of an IR spectrum.

With the keyword MaxStep the number of values $$m$$ in the autocorrelation function ($$t = [0,t_{1},t_{2},....,t_{m}]$$) can be set. The default value is half of the total number of simulation steps $$n$$ used.

A subset of atoms for which the property $$\textbf{A}$$ should be selected/computed can be provided in the block Atoms. The block can contain element names (recurring keyword Element), or individual atom numbers (recurring keyword Atom).

AutoCorrelation
Atoms
Atom integer
Element string
End
End

AutoCorrelation
Type: Block True All input related to auto correlation functions.
Atoms
Type: Block Relevant if Property is set to Velocities, DipoleMomentFromCharges, DipoleDerivativeFromCharges, or DiffusionCoefficient. Atom numbers or elements for the set of atoms for which the property is read/computed. By default all atoms are used.
Atom
Type: Integer True Atom number.
Element
Type: String True Element Symbol Atom.

A subset of vector elements can be provided with the subblock VecElements. By default all vector elements found on the RKF file will be used.

AutoCorrelation
VecElements
Index integer
End
End

AutoCorrelation
Type: Block True All input related to auto correlation functions.
VecElements
Type: Block A set of indices referring to a subset of the property vector. Works in combination with the atoms block. For example, in combination with the property Velocities, the Atoms block allows the selection of a subset of atoms, while the VecElelements block allows the selection of a subset of vector elements (e.g. 1 and 2 for the elements x and y). Currently not implemented with InputValues.
Index
Type: Integer True Element of the property vector.

## Mean Square Displacement¶

The Analysis program computes mean square displacements (MSD) if the Task keyword is set to MeansSquareDisplacement.

Task [RadialDistribution | Histogram | AutoCorrelation | MeanSquareDisplacement | AverageBinPlot]

Task

Further details need to be specified in the MeanSquareDisplacement block. If more than one MeanSquareDisplacement block is present in the input, more than one MSD will be computed. The result is printed to output as text, as well as stored in a binary file (plot.kf).

MeanSquareDisplacement
Atoms
Atom integer
Element string
End
DataReading [Auto | AtOnce | BlockWise]
InputValues
Values float_list
End
MaxStep integer
Property [Coords | InputValues | DiffusionCoefficient]
StartTimeSlope float
TimeStep float
UseAllValues Yes/No
VecElements
Index integer
End
End

MeanSquareDisplacement
Atoms
Type: Block Relevant if Property is set to any quantity that is available per atom (Coords, DiffusionCoefficient). Atom numbers or elements for the set of atoms for which the property is read/computed are provided here. By default all atoms are used.
Atom
Type: Integer True Atom number.
Element
Type: String True Element Symbol Atom.
DataReading
Type: Multiple Choice Auto [Auto, AtOnce, BlockWise] The KF data can be read in and handled once, or blockwise. The former is memory intensive, but mostly faster. If Auto is selected, the data is read at once if it is less than 1 GB, and blockwise if it is more.
InputValues
Type: Block Relevant if Property is set to InputValues. All input values (a vector on each line) need to be provided in this block, using the keyword Values (possibly multiple times).
Values
Type: Float List True The values at each step (on a single line)
MaxStep
Type: Integer The maximum number of time steps for which the mean square displacement function will be computed. The default is half of the number of provided frames.
Property
Type: Multiple Choice Coords [Coords, InputValues, DiffusionCoefficient] Compute the MSD from the property selected here (from rkf). Selecting DiffusionCoefficient is equivalent to selecting the property Coords.
StartTimeSlope
Type: Float 0.0 The MSD has a nonlinear regime at short timescales, and a linear regime at long timescales. To determine the slope, the starting point for the linear regime has to be determined. This keyword sets the starting time in fs. If set to zero, the starttime will be automatically determined.
TimeStep
Type: Float Relevant if Property is set to InputValues. The time separating the entries (in fs). If Property is set to any of the other quantities, it can be read from an RKF file, and the timestep is read from the RKF file as well. The read value then overrides this keyword.
UseAllValues
Type: Bool No By default the same number of values are used for each t-step in the MSD. This has the advantage that all values in the MSD are equally reliable, but it does mean that for the smaller timesteps much of the data is not used. To switch this off and use all data, UseAllValues can be set to true
VecElements
Type: Block A set of indices referring to a subset of the property vector. Works in combination with the atoms block. For example, in combination with the property Coords, the Atoms block allows the selection of a subset of atoms, while the VecElelements block allows the selection of a subset of vector elements (e.g. 1 and 2 for the elements x and y). Currently not implemented with InputValues.
Index
Type: Integer True Element of the property vector.

### Description¶

The mean square displacement $$MSD(t)$$ describes the average change of a (vector) property $$\textbf{A}$$ over time. This property is usually a set of atom coordinate vectors, but the implementation is entirely general.

$$MSD(t) = \langle [\textbf{A}(0) - \textbf{A}(t)]^2 \rangle$$

The average runs over all time-intervals $$\left( t_{0}, t_{0}+t \right),\left( t_{1}, t_{1}+t \right),...,\left( t_{N}, t_{N}+t \right)$$, with $$t_{N} = t_{n} - t_{m}$$. Here $$n$$ is the total number of simulation steps in the trajectory, and $$m$$ is the number of discrete $$t$$ values for which $$MSD(t)$$ is computed. The value $$m$$ can be set with the keyword MaxStep, and defaults to half the total number of simulation steps. As stated above, the default average runs over the same number of time intervals for each value of $$t$$. If UseAllValues is selected, the average runs over all available time intervals for each value of $$t$$, which is large for small $$t$$, and smallest ($$N$$) for $$t_{m}$$.

The most common use of the mean square displacement is for the computation of the diffusion coefficient, in which case the selected property is the set of atom coordinates (Coords). The diffusion coefficient is proportionate to the slope of this mean square displacement function, and therefore this slope is automatically computed. The function $$MSD(t)$$ becomes linear only after an initial time interval, and the user can set this initial time with the keyword StartTimeSlope. If not provided, this start time is automatically determined. To allow the user to determine if the linear regime has been sufficiently sampled, the slope of $$MSD(t)$$ as a function of $$t$$ is computed as well. If the slope has not converged to a stable value, the user should select a larger value of $$t_m$$ or continue the molecular dynamics simulation for a longer time.

### Options¶

Autocorrelation functions can be computed for different simulation properties: 1) Coordinates 2) User provided values. Option 3) Diffusion coefficient is equivalent to option 1).

MeanSquareDisplacement
Property [Coords | InputValues | DiffusionCoefficient]
End

MeanSquareDisplacement
Type: Block True All input related to auto correlation functions.
Property
Type: Multiple Choice Coords [Coords, InputValues, DiffusionCoefficient] Compute the MSD from the property selected here (from rkf). Selecting DiffusionCoefficient is equivalent to selecting the property Coords.

A subset of atoms for which the property $$\textbf{A}$$ should be selected/computed can be provided in the block Atoms. The block can contain element names (recurring keyword Element), or individual atom numbers (recurring keyword Atom).

MeanSquareDisplacement
Atoms
Atom integer
Element string
End
End

MeanSquareDisplacement
Type: Block True All input related to auto correlation functions.
Atoms
Type: Block Relevant if Property is set to any quantity that is available per atom (Coords, DiffusionCoefficient). Atom numbers or elements for the set of atoms for which the property is read/computed are provided here. By default all atoms are used.
Atom
Type: Integer True Atom number.
Element
Type: String True Element Symbol Atom.

A subset of vector elements can be provided with the subblock VecElements. By default all vector elements found on the RKF file will be used.

MeanSquareDisplacement
VecElements
Index integer
End
End

MeanSquareDisplacement
Type: Block True All input related to auto correlation functions.
VecElements
Type: Block A set of indices referring to a subset of the property vector. Works in combination with the atoms block. For example, in combination with the property Coords, the Atoms block allows the selection of a subset of atoms, while the VecElelements block allows the selection of a subset of vector elements (e.g. 1 and 2 for the elements x and y). Currently not implemented with InputValues.
Index
Type: Integer True Element of the property vector.

## AverageBinPlot¶

The Analysis program can plot two arbitrary properties, present on the RKF file, against each other averaged over each bin if the Task keyword is set to AverageBinPlot.

Task [RadialDistribution | Histogram | AutoCorrelation | MeanSquareDisplacement | AverageBinPlot]

Task

Further details need to be specified in the AverageBinPlot block. If more than one AverageBinPlot block is present in the input, more than one AverageBinPlot will be computed. The result is printed to output as text, as well as stored in a binary file (analysis.kf).

AverageBinPlot
Nbins integer
Property
Axis float_list
Name string
Region string
Vector string
End
Timestep integer
End

AverageBinPlot
Nbins
Type: Integer 10 Number of bins that are plotted
Property
Type: Block True Property 1
Axis
Type: Float List If defined the dot_product along this axis will be taken
Name
Type: String Name of the property
Region
Type: String Name of the atom region among which the forces will be averaged
Vector
Type: String is it a vector? yes/no
Timestep
Type: Integer 1 Timestep used for the plotting

In the case of plotting the viscosity or friction coefficient, it is sufficient to only specify one property(meaning either Viscosity or FrictionCoefficient), which will be plotted automatically against the timestep.

## Viscosity¶

The viscosity can be computed from an equilibrated Molecular Dynamics run as the integral over the pressure tensor autocorrelation function.

$$\eta = \frac{1}{10T} \int_{t=0}^{t=t_{max}} \langle \P_ij(0) \cdot \P_ij{v}(t)) \rangle dt$$

The viscosity is computed if the task AutoCorrelation is selected, and if in the AutoCorrelation block Viscosity is selected as the Property.

$AMSBIN/analysis <<eor Task AutoCorrelation AutoCorrelation Property Viscosity End eor  AutoCorrelation Type: Block True All input related to auto correlation functions. Property Type: Multiple Choice DipoleDerivativeFromCharges [Velocities, DipoleMomentFromCharges, InputValues, DiffusionCoefficient, DipoleDerivativeFromCharges, PressureTensor, Viscosity] Compute the ACF either from velocities (from rkf), the dipole moment (from coordinates and atomic charges in rkf), the dipole moment derivative (from velocities and atomic charges in rkf), from the pressure tensor (from rkf), or from values specified in input. Selecting DiffusionCoefficient is equivalent to selecting Velocities. The default, DipoleDerivativeFromCharges, results in the computation of an IR spectrum. Again, a subset of atoms can be selected with the sublock Atoms. The value of the viscosity is written to the output, as well as to the KF file. ## Diffusion Coefficient¶ The diffusion coefficient can be computed in two ways. 1. As the integral over the velocity autocorrelation function. 2. As the slope of the mean square displacement of the atomic coordinates. The latter is more commonly used, as the former requires trajectory information to be stored at very short time intervals. ### From Velocity Autocorrelation¶ The diffusion coefficient can be defined as an integral over the velocity autocorrelation function. $$D = \frac{1}{d} \int_{t=0}^{t=t_{max}} \langle \textbf{v}(0) \cdot \textbf{v}(t)) \rangle dt$$ The factor $$\frac{1}{d}$$ corrects for the dimension of the system, which we assume to equal the length of the provided vector $$\textbf{v}$$. The dimension $$d$$ equals 3, unless specified otherwise in the subblock VecElements. In a system that is periodic in less than 3 dimensions, it may make sense to provide only the vector elements along the periodic dimensions. By default, however, all vector elements provided are used. The diffusion coefficient is computed if the task AutoCorrelation is selected, and if in the AutoCorrelation block DiffusionCoefficient is selected as the Property. The result is completely equivalent to selecting the task AutoCorrelation with Velocities as the Property keyword. $AMSBIN/analysis <<eor
AutoCorrelation
Property DiffusionCoefficient
End
eor

AutoCorrelation
Type: Block True All input related to auto correlation functions.
Property
Type: Multiple Choice DipoleDerivativeFromCharges [Velocities, DipoleMomentFromCharges, InputValues, DiffusionCoefficient, DipoleDerivativeFromCharges, PressureTensor, Viscosity] Compute the ACF either from velocities (from rkf), the dipole moment (from coordinates and atomic charges in rkf), the dipole moment derivative (from velocities and atomic charges in rkf), from the pressure tensor (from rkf), or from values specified in input. Selecting DiffusionCoefficient is equivalent to selecting Velocities. The default, DipoleDerivativeFromCharges, results in the computation of an IR spectrum.

### From Mean Square Displacement¶

The mean square displacement becomes linear with time, after an initial time interval. We can therefore define the linear part of the function as follows.

$$MSD(t) = {\langle \textbf{r}(0) - \textbf{r}(t)) \rangle} = at + b$$,

with $$a$$ as the slope of the function. The diffusion coefficient is proportional to the slope $$a$$.

$$D = \frac{1}{2d} a$$

Here, $$d$$ is the dimensionality of the system, or the length of the provided vector $$\textbf{r}$$. The dimension $$d$$ equals 3, unless specified otherwise in the subblock VecElements. In a system that is periodic in less than 3 dimensions, it may make sense to provide only the vector elements along the periodic dimensions. By default, however, all vector elements provided are used.

The diffusion coefficient is computed if the task MeanSquareDisplacement is selected, and if in the MeanSquareDisplacement block DiffusionCoefficient is selected as the Property. The result is completely equivalent to selecting the task MeanSquareDisplacement with Coords as the Property keyword.

\$AMSBIN/analysis <<eor

MeanSquareDisplacement
Property
In both cases, a subset of atoms can be selected with the sublock Atoms, and a subset of vector elements (in this case elements 1=X, 2=Y, 3=Z for the Cartesian coordinates) can be selected with the subblock VecElements.