Cerius²·Forcefield Engines

5       Analyzing Results

The C²·Analysis module is used to analyze trajectory files generated by simulations (both minimization and dynamics, as well as other modules such as quantum mechanics, which are documented separately). Several types of analyses can be performed, depending on the run conditions and the type of data generated. These include statistical analysis of properties, calculation of diffusion constants from mean-square displacements, calculation of the dipole-dipole autocorrelation function and power spectrum, and analysis of property fluctuations.

Velocity data can be analyzed to obtain the velocity autocorrelation function and power spectrum. Structural analysis can also be performed, where the radial distribution function is calculated and a Fourier transform is applied to obtain the structure factor.

This section explains

This section includes information on:

Preparing trajectories

Output from analysis

Property statistics

Table 5 . Finding information about analyzing results

If you want to know about:	Read:
Finding the Analysis module.	Accessing the tools.
Simulation theory and uses.	Forcefield-Based Simulations.
Specifying output of minimizatins.	Specifying output.
Performing minimizations.	Minimization.
Chemically realistic dynamics simulations.	Dynamics conditions--the thermodynamics ensemble.
Controlling kinetic or thermodynamic temperature.	Temperature control.
Applying constant stress and/or pressure in dynamics.	Pressure and stress control.
Specifying output of dynamics runs.	Specifying output.
Performing dynamics simulations.	Dynamics Simulation.
Loading a trajectory file.	Loading a trajectory file.
Format of trajectory file.	File Formats.
Analyzing part of a trajectory file.	Selecting frames from a loaded trajectory.
Using a set of models as a "trajectory".	Using a set of models to create a "trajectory".
Graphing various properties.	Basic statistics of various properties.
Displacement analysis.	Mean-square displacement, Self-diffusion constant.
Velocity results.	Velocity autocorrelation, Power spectrum.
Structural distribution.	Radial distribution function, Structure factor.
Dipole properties and functions.	Dipole-dipole interactions.
Mactoscopic response properties.	Fluctuation analysis.
Display of graphical and printed output from analyses.	Output as graphs and text.
Aniimating models.	Displaying model conformations.
Displaying individual model conformations.	Displaying model conformations.
Converting trajectory data to tables.	Output in tables format.

Related information

Several other modules (e.g., C²·Conformer Analysis, C²·Polymer Properties), which are documented separately, also contain analysis utilities.

You should already know...

Forcefield-Based Simulations, Molecular Dynamics, contains information on the types of information that can be obtained from trajectory files output by dynamics simulations run under various conditions.

Accessing the tools

Most tools for analyzing results of minimization and dynamics simulations are accessed from one of the decks of cards in the main Visualizer control panel, the OFF METHODS card deck.

To access the card in the C²·Analysis module, click the deck selector in the main control panel and choose OFF METHODS from the list that appears. Then click the title of the ANALYSIS card to bring it to the front. The deck of cards menu area should now look like this:

Preparing trajectories

Finding information

This section includes information on:

Loading or creating a trajectory

Selecting frames from a loaded trajectory

You should already know...

Information on trajectories is contained in Forcefield-Based Simulations under Dynamics trajectories.

Loading or creating a trajectory

You should already know...

Using Cerius² file browser controls is explained in Cerius2 Modeling Environment under Loading model structure files.

Accessing the tools

Select the Input menu item from the ANALYSIS card to access the Analysis Input control panel.

Loading a trajectory file

Use the browser box in the Analysis Input control panel to find a trajectory file.

Trajectory file names usually end in .trj, .atrj, .qtrj, .his, or .arc.

To load the desired file, either double-click its name in the file list box or select its name and then click the SELECT pushbutton.

Using a set of models to create a "trajectory"

To use selected models as, in effect, a trajectory, select SELECTED MODELS under Choose Data Source in the Analysis Input control panel. It is possible, for some purposes (noted under Property statistics where relevant), to analyze only one model; however, more reliable results are obtained when a set of conformations (from selected models or from a trajectory file) is analyzed.

Additional information

Please see the on-screen help for details on the functioning of each control in the Analysis Input control panel.

File formats

File formats are documented in File Formats.

Selecting frames from a loaded trajectory

• The structure was not fully equilibrated for the first few steps saved in the file.

• You have a very long trajectory file and do not need to analyze all of it.

• You want to compare various portions of the trajectory with one another.

• You saved data more frequently than was needed for your present analytical needs.

First you load a trajectory file (which automatically informs you of the number of frames in the file, via the Last entry box in the Analysis Input control panel), and then you specify the frame or time values as desired.

Accessing the tools

Select the Input menu item from the ANALYSIS card to access the Analysis Input control panel.

Selecting the frames

First load the desired trajectory file (Loading or creating a trajectory). Then, if you want to analyze only some of the frames in the file, select those frames by changing the First frame to extract, the Last frame to extract, and/or the Step interval entry boxes in the Analysis Input control panel. You can change either the Frame or the equivalent Time, whichever is more convenient.

Tip

Additional information

Please see the on-screen help for details on the functioning of each control in the Analysis Input control panel.

Output from analysis

Finding information

This section includes information on:

Displaying model conformations

Output as graphs and text

Output in tables format

You should already know...

Loading trajectories (Loading or creating a trajectory) and selecting frames (Selecting frames from a loaded trajectory) are covered above.

Displaying model conformations

Related information

The C²·Visualizer has a more limited, independent utility for animating models with a trajectory file loaded within that utility. Please see Cerius2 Modeling Environment under Animating models.

Accessing the tools

To access the Analysis Show Frames control panel, select the Show Frames menu item from the ANALYSIS card.

To access the Analysis Input control panel, select the Input menu item from the ANALYSIS card.

Displaying single conformations

To display the model in the conformation of a single frame in the loaded trajectory file, you can use the Current controls in either the Analysis Input or Analysis Show Frames control panel.

To display a single frame, enter a frame number in the left entry box or a time in the right entry box.

To step through frames until you get to the desired frame (or conformation), use the up and down arrows between the two entry boxes.

To display a conformation by picking a point from a trajectory graph (Output as graphs and text), assure that the Pick frame from graph tool in the Analysis Show Frames control panel is highlighted, then click any point on any graph that was output by a Profile calculation (see Calculations).

Animating the model

You can also drag the mouse in the graph window to step through conformations equivalent to the graphed points.

To animate the model at a constant speed and proceeding once through the trajectory frames, set the popup in the Analysis Show Frames control panel to STOP AT END and click the forward or reverse video-control arrow.

To animate the model continuously, set the popup in the Analysis Show Frames control panel to REVERSE AT END or LOOP AT END and click the forward or reverse video-control arrow.

To stop a continuous animation, click the square control that is located between the two video-control arrows.

Information on the current frame

To obtain information on the energy components, time step, and other properties of the model in an individual frame, select the frame by any of the methods described for displaying a single conformation (Displaying single conformations). Click the Information on current frame action button in the Analysis Show Frames control panel.

The information is printed in the text window.

Saving an individual conformation

To save any single frame (model conformation), select the frame by any of the methods described for displaying a single conformation (Displaying single conformations). Select the File/Save Model... menu item from the menu bar in the main control panel. Save the model with the Save Model control panel, as described in Cerius2 Modeling Environment, under Saving model structure files.

Additional information

Please see the on-screen help for details on the functioning of each control in the Analysis Show Frames and Analysis Input control panels.

Output as graphs and text

Related information

Cerius2 Modeling Environment, Working with Graphs, gives information on managing graphs (Managing graphs) and controlling aspects of their display (Displaying and editing graphs). Printing graphs is discussed on Printing models and graphs.

Accessing the tools

Select the Analyze/Statistics menu item from the ANALYSIS card to access the Analysis Statistics control panel. Then click its Output... pushbutton to access the Analysis Output control panel.

Sending output to graphs and/or the text window

Check or uncheck the Plot check box to send output to a graph or not.

Check or uncheck the Text Port check box to send additional output to the text window or not. (Some information is always sent to the text window.)

Additional information

Please see the on-screen help for details on the functioning of each control in the Analysis Statistics and Analysis Output control panels.

Output in tables format

Related information

Cerius2 Modeling Environment contains complete documentation on working with tables in the C²·Tables module, in Working with
Tables. In addition, some other modules (which are documented separately) include other tables utilities.

Accessing the tools

Select the Analyze/Statistics menu item from the ANALYSIS card to access the Analysis Statistics control panel. Then click its Output... pushbutton to access the Analysis Output control panel.

Saving results as table files

To send the output to a file that can be viewed, analyzed, and manipulated with the C²·Tables module, check the Output to Tables File check box in the Analysis Output control panel.

Use the file browser controls to specify the directory in which to store the file, as well as the file's name.

To not send output to a Cerius² table file, uncheck the Output to Tables File check box.

Additional information

Please see the on-screen help for details on the functioning of each control in the Analysis Output control panel.

Property statistics

Finding information

This section includes information on:

Basic statistics of various properties

Mean-square displacement

Self-diffusion constant

Velocity autocorrelation

Power spectrum

Radial distribution function

Structure factor

Dipole-dipole interactions

Fluctuation analysis

Related information

Cerius2 Modeling Environment, Working with Graphs, gives information on managing graphs (Managing graphs) and controlling aspects of their display (Displaying and editing graphs). Printing graphs is discussed on Printing models and graphs.

Forcefield-Based Simulations, Molecular Dynamics, contains full information on the types of information that can be obtained from trajectory files output by dynamics simulations run under various conditions.

Basic statistics of various properties

You should already know...

Loading trajectories (Loading or creating a trajectory), selecting frames (Selecting frames from a loaded trajectory), and specifying output (Output from analysis) are covered above.

Technical notes

The exact properties listed in the Select Properties list box in the Analysis Statistics control panel depend on the contents of the loaded trajectory.

Accessing the tools

Select the Analyze/Statistics menu item from the ANALYSIS card to access the Analysis Statistics control panel.

Selecting properties

To select a property to analyze, highlight its name in the Select Properties list box in the Analysis Statistics control panel by clicking it.

To select more than one property at once, <Shift>-click to select several adjacent properties or <Ctrl> -click to select properties whose names are not continuous.

To deselect a highlighted property, <Ctrl> click it. To deselect all highlighted properties and select another one, simply click the desired property.

Properties listed within angle brackets (e.g., <Dipole Magnitude>) cannot be selected or graphed until they have been calculated.

To select measurements, you need to first create them. To do this, select the Geometry/Measurements... menu item from the menu bar in the main control panel to access the Measurements control panel. Create measurements as described in Cerius2 Modeling Environment, under Measuring models.

Calculations

For the selected property or properties, you can use the Analysis Statistics control panel to calculate:

• A time-dependent profile (i.e., a history) of the property values, by clicking the Profile action button.

The value of the property is plotted vs. time for trajectories that contain time information (such as those output by dynamics simulations) and vs. frame number for trajectories that do not contain time information (such as those output by minimization runs).

• A histogram showing the frequency distribution of the property, by clicking the Distribution statistics action button.

You can change the number of bins into which the histogram is divided or request that output be sent only to the text window (several types of information are always sent to the text window) with the Distribution Preferences control panel. Access this control panel by clicking the Preferences... pushbutton to the right of the Distribution statistics action button in the Analysis Statistics control panel.

• The running or block average of the property, by clicking the Running Average or Block Average action button.

The average is plotted vs. time for trajectories that contain time information (such as those output by dynamics simulations) and vs. frame number for trajectories that do not contain time information (such as those output by minimization runs).

To changed the width or interval used in block averaging, click the Preferences... button to the right of the Block Average action button to access the Block Average Preferences control panel.

To search the trajectory file for a number of frames containing the highest or lowest values of a selected property, use the Search for controls in the Analysis Statistics control panel.

The results are printed in the text window

Additional information

Please see the on-screen help for details on the functioning of each control in the Analysis Statistics, Distribution Preferences, and Block Average Preferences control panels.

Mean-square displacement

You should already know...

Loading trajectories (Loading or creating a trajectory), selecting frames (Selecting frames from a loaded trajectory), and specifying output (Output from analysis) are covered above.

Concepts

The mean-square displacement (MSD) is calculated as follows:

Eq. 2

0 < m + n = k
m = Maximum number of points allowed for the MSD calculation.
n = Number of data points used for averaging.
i = Step counter (increment).
k = Total number of snapshots read in.

The value for k is set when specifying how many frames to analyze (Selecting frames from a loaded trajectory). The default value for m is k/2, but this, as well as the step counter i, can be changed using the options on the Mean Squared Displacement control panel (Accessing the tools).

You can calculate the MSD for all or selected atoms in the model. A plot of the MSD vs. time is displayed and updated as the calculation proceeds.

Accessing the tools

Select the Analyze/MSD menu item from the ANALYSIS card to access the Mean Squared Displacement control panel.

Prerequisites

You need to load a trajectory (Loading or creating a trajectory) and select what frames to use (Selecting frames from a loaded trajectory) in the MSD calculation.

Setting up an MSD calculation

Specify whether the calculation should include all or only selected atoms by setting the Atoms popup in the Mean Squared Displacement control panel.

To specify the values of parameters in Eq. 2, set these controls:

m = Max. Time (as number of frames or in picoseconds).
i = Origin Step (as number of frames or in picoseconds).

To perform the MSD calculations assuming the system is anisotropic, check the All Anisotropic Components check box. Six MSD plots (one for each of the anisotropic components) are calculated and displayed. The isotropic MSD plot is also calculated and displayed.

Graph output

To display a plot that is continuously updated as the MSD calculation proceeds, check the Monitor Calculation check box.

To display the finished plot only at the end of the MSD calculation, uncheck the Monitor Calculation check box.

Performing the MSD calculation

Click the Calculate MSD action button in the Mean Squared Displacement control panel.

Additional information

Please see the on-screen help for details on the functioning of each control in the Mean Squared Displacement control panel.

Self-diffusion constant

You should already know...

Loading trajectories (Loading or creating a trajectory), selecting frames (Selecting frames from a loaded trajectory), specifying output (Output from analysis), and calculating the MSD (Mean-square displacement) are covered above.

Concepts

The self-diffusion constant is obtained using the Einstein relation:

Eq. 3

N = Number of atoms.
r = Position of the particle.
t = Time.

How it works

A least-squares linear fit of the MSD can be done, and the self-diffusion constant can be determined from the slope of this line.

Often a better linear fit can be produced by discarding the initial quadratic segment of the MSD curve.

Technical notes

The linear fit is always done using the most recently calculated MSD. When all anisotropic components are calculated and displayed, this corresponds to the last plot displayed.

Accessing the tools

Select the Analyze/MSD menu item from the ANALYSIS card to access the Mean Squared Displacement control panel.

Calculating the self-diffusion constant

After calculating the MSD (Mean-square displacement), examine the graph of MSD vs. time to determine whether the entire curve or a segment of it is to be used for linear fitting.

To specify that only part of the MSD curve by used in the self-diffusion calculation, change the values for the beginning and end of the included range in the MSD fit range entry boxes in the Mean Squared Displacement control panel. Otherwise, the entire MSD-vs.-time plot is used.

Start the calculation by clicking the Calculate Line Fit action button in the control panel.

The resulting diffusion constant D is displayed in the text window, in units of both cm² s^-1 atom^-1 and Ų ps^-1 atom^-1.

Additional information

Please see the on-screen help for details on the functioning of each control in the Mean Squared Displacement control panel.

Velocity autocorrelation

This type of information is useful in predicting the dynamic and vibrational properties of materials at elevated temperatures and in predicting thermal effects on infrared or Raman spectra.

You should already know...

Loading trajectories (Loading or creating a trajectory), selecting frames (Selecting frames from a loaded trajectory), and specifying output (Output from analysis) are covered above.

Concepts

The autocorrelation function C is defined as follows:

Eq. 4

0 < m + n = k
m = Maximum number of points allowed for the autocorrelation function calculation.
n = Number of data points used for averaging.
i = Step counter (increment).
V = Velocity.
k = Total number of snapshots read in.

The value for k is set when specifying how many frames to analyze (Selecting frames from a loaded trajectory). The default value for m is k/2, but this, as well as the step counter i, can be changed using the options on the Velocity Autocorrelation control panel (Accessing the tools).

You can calculate the VACF for all or selected atoms in the model. A plot of C vs. time is displayed and updated as the calculation proceeds.

Accessing the tools

Select the Analyze/VACF menu item from the ANALYSIS card to access the Velocity Autocorrelation control panel.

Prerequisites

You need to load a trajectory (Loading or creating a trajectory) and select what frames to use (Selecting frames from a loaded trajectory) in the VACF calculation.

Setting up a VACF calculation

Specify whether the calculation should include all or only selected atoms by setting the Atoms popup in the Velocity Autocorrelation control panel.

To specify the values of parameters in Eq. 4, set these controls:

m = Max. Time (as number of frames or in picoseconds).
i = Origin Step (as number of frames or in picoseconds).

To perform the VACF calculations assuming the system is anisotropic, check the All Anisotropic Components check box. Six VACF plots (one for each of the anisotropic components) are calculated and displayed. The isotropic VACF plot is also calculated and displayed.

Graph output

To display a plot that is continuously updated as the VACF calculation proceeds, check the Monitor Calculation check box.

To display the finished plot only at the end of the VACF calculation, uncheck the Monitor Calculation check box.

Performing the VACF calculation

Click the Calculate VACF action button in the Velocity Autocorrelation control panel.

Additional information

Please see the on-screen help for details on the functioning of each control in the Velocity Autocorrelation control panel.

Power spectrum

This type of information is useful in predicting the dynamic and vibrational properties of materials at elevated temperatures and in predicting thermal effects on infrared or Raman spectra.

You should already know...

Loading trajectories (Loading or creating a trajectory), selecting frames (Selecting frames from a loaded trajectory), specifying output (Output from analysis), and calculating the VACF (Velocity autocorrelation) are covered above.

Concepts

The frequency resolution of the power spectrum, , is given by:

Eq. 5

t_max = Time length of the analyzed data.

Because the time interval between two successive frames t must be fixed for a velocity autocorrelation calculation, the maximum frequency point in the power spectrum curve, f_max, is also fixed, as seen by the following relationship:

Eq. 6

Thus, decreasing the value of makes t_max larger and effectively increases the resolution of the power spectrum curve.

Decreasing t_max results in a less detailed curve. Options are available that allow you to specify values for both and t_max.

Technical notes

The fast version of the Fourier transform is used.

The Fourier transform is always done using the most recently calculated set of VACF data. When all anisotropic components are calculated and displayed, this corresponds to the last plot displayed.

Accessing the tools

Select the Analyze/VACF menu item from the ANALYSIS card to access the Velocity Autocorrelation control panel.

Calculating the power spectrum

After calculating the VACF (Velocity autocorrelation), specify the Frequency Interval ( in Eq. 5) and Maximum Time (t_max in Eq. 5) allowed, in the Calculate Power Spectrum section of the Velocity Autocorrelation control panel.

Start the calculation by clicking the Calculate Power Spectrum action button in the control panel.

A graph of the power spectrum is displayed.

Additional information

Please see the on-screen help for details on the functioning of each control in the Velocity Autocorrelation control panel.

Radial distribution function

This type of information is useful in revealing overall structural properties such as packing, ordering, compressibility, and phase transitions.

You should already know...

Loading trajectories (Loading or creating a trajectory), selecting frames (Selecting frames from a loaded trajectory), and specifying output (Output from analysis) are covered above.

Concepts

Because the system is assumed to be isotropic, the radial distribution function is centrosymmetric. The following expression is used to calculate the radial distribution function G_AB(r) between two selected groups, A and B:

Eq. 7

N_A = Number of atoms in group A.
N_B = Number of atoms in group B.
N_AB = Number of atoms common to both groups A and B.
V = Unit cell volume (for nonperiodic systems V = 1).

You can also specify that all atoms be included in the calculation (this is the default). In this case, N_A = N_B = N, where N is the total number of atoms in the model (nonperiodic systems) or in the unit cell (periodic systems). Eq. 7 then becomes:

Eq. 8

Eq. 9

r_Ai = Position of the ith atom in group A.
r_Bj = Position of the jth atom in group B.

Eq. 10

N_bins = Number of bins.
r_cutoff = Cutoff distance.

Distance interval and cutoff distance

You can specify values for dr and r_cutoff using the Distance Interval and Cut-Off Distance controls in the RDF control panel. For periodic systems, the computation time is proportional to r²cutoff. Thus, increasing the cutoff distance significantly increases the time required to calculate G_AB(r). This does not apply to nonperiodic systems, for which a value large enough to include all possible pair interactions should be chosen to assure accurate calculation of the structure factor.

For a given cutoff distance, decreasing the value of the Distance Interval results in a larger number of bins and effectively increases the resolution of the curve. If the value is too small, however, the curve can become noisy.

Technical notes

Calculations can be done using trajectory file data, a set of conformations (Using a set of models to create a "trajectory"), or the coordinates of the current model. Because the use of trajectory file data allows averaging over multiple frames, the results obtained are statistically more reliable than those obtained from the single frame data of the current model.

Accessing the tools

Select the Analyze/RDF menu item from the ANALYSIS card to access the RDF control panel.

Prerequisites

Unless you want to use the coordinates of the current model or a set of models (Using a set of models to create a "trajectory"), you need to load a trajectory (Loading or creating a trajectory) and select what frames to use (Selecting frames from a loaded trajectory) in the RDF calculation.

Setting up an RDF calculation

Specify whether the calculation should include all atoms, only selected atoms, or only certain atoms by setting the popup in the RDF control panel.

If you set the popup to ELEMENTS or FF TYPES, then specify the elements or forcefield atom types to use for groups A and B (Eq. 7), using the two popups that appear.

To specify the values of parameters in Eq. 10, set these controls:

r_cutoff = Cut-Off Distance (Å).
dr = Distance Interval (Å).

To display a plot that is continuously updated as the RDF calculation proceeds, check the Monitor Calculation check box.

To display the finished plot only at the end of the RDF calculation, uncheck the Monitor Calculation check box.

Performing the RDF calculation

To start the RDF calculation, click the Calculate g(r) action button in the RDF control panel.

Additional information

Please see the on-screen help for details on the functioning of each control in the RDF control panel.

Structure factor

This type of information is useful in revealing overall structural properties such as packing, ordering, compressibility, and phase transitions. The structure factor is particularly useful in that it can be directly compared with X-ray diffraction data.

You should already know...

Loading trajectories (Loading or creating a trajectory), selecting frames (Selecting frames from a loaded trajectory), specifying output (Output from analysis), and calculating the RDF (Radial distribution function) are covered above.

Concepts

The structure factor, S_AB(k) between two selected groups, A and B, is related to the radial distribution function G_AB(r) through a 3D Fourier transform:

Eq. 11

k = Wave vector.

Eq. 12

Eq. 13

= Atomic density, N/V.

For periodic systems, however, G_AB(r) = 1 when r . Therefore, Eq. 13 should be rewritten as follows:

Eq. 14

Eq. 15

Because the delta function (k) = 0 when k 0, it can be removed analytically, resulting in the following:

Eq. 16

Two controls are available in the RDF control panel that can be used to alter the resolution of the S_AB(k) curve: k-Space Interval and Maximum Distance. The k-Space Interval specifies the value for the k-space interval k, which corresponds to the bin width in the S_AB(k) curve:

Eq. 17

r = Distance interval used in the G_AB(r) calculations.
M_bins = Number of bins in the S_AB(k) curve.

For a given distance interval, decreasing the value of k results in a greater number of bins and effectively increases the resolution of the S_AB(k) curve.

The Maximum Distance specifies the cutoff distance rmax used in calculating the structure factor:

Eq. 18 rmax = Mbins r

The Fourier transform is always done using the most recently calculated set of RDF data.

Accessing the tools

Select the Analyze/RDF menu item from the ANALYSIS card to access the RDF control panel.

Calculating the structure factor

After calculating the RDF (Radial distribution function), specify the k-Space Interval (dk in Eq. 17) and the Maximum Distance (r_max in Eq. 18) allowed, in the RDF control panel.

Start the calculation by clicking the Calculate Structure Factor action button in the control panel.

A graph of structure factor vs. k is displayed.

Additional information

Please see the on-screen help for details on the functioning of each control in the RDF control panel.

Dipole-dipole interactions

The power spectrum of the dipole correlation function can be related to the infrared spectrum of the system you are studying.

You should already know...

Loading trajectories (Loading or creating a trajectory), selecting frames (Selecting frames from a loaded trajectory), and specifying output (Output from analysis) are covered above.

Concepts

The dipole moment is defined as the magnitude of the dipole vector. It is origin-independent only if there is no net charge on the system. By convention, the origin used for the dipole moment is halfway between the centers of the positive and negative charge.

For a collection of point charges, the dipole vector is defined as:

Eq. 19

4.802 = Factor necessary to convert the value to Debyes from angstroms and atomic changes.
Q_{min =} The smaller absolute value of the total positive charge and the total negative charge.
= Center of charge (positive or negative) calculated by:

Eq. 20

Where the sums run over all positive or negative charges as appropriate, and:

q_i = charge on atom i.
r_i,a = x, y, or z component of the coordinates of charge i.

Calculation of the dipole autocorrelation and power spectrum is analogous to similar calculations using velocities (Velocity autocorrelation and Power spectrum).

Technical notes

Calculations can be done using trajectory file data, a set of models (Using a set of models to create a "trajectory"), or the coordinates of the current model. Because the use of trajectory file data allows averaging over multiple frames, the results obtained are statistically more reliable than those obtained from the single frame data of the current model.

The fast version of the Fourier transform is used for calculating the dipole-dipole power spectrum.

Accessing the tools

Select the Analyze/Dipole menu item from the ANALYSIS card to access the Dipole control panel.

Prerequisites

Unless you want to use a set of models (Using a set of models to create a "trajectory") or the coordinates of the current model, you need to load a trajectory (Loading or creating a trajectory) and select what frames to use (Selecting frames from a loaded trajectory) in the calculation of the dipole vector and moment (magnitude).

Calculating the dipole-dipole correlation function requires that the dipole vector and moment have been calculated.

Calculating a dipole-dipole power spectrum requires that the dipole-dipole correlation function has been calculated.

Dipole vector and dipole moment

Specify whether the calculation should include all or only selected atoms by setting the atoms popup in the Dipole control panel.

Click the Calculate Dipole action button to perform the calculation.

Once the dipole vector and moment have been calculated, their values can be graphed with the commands in the Analysis Statistics control panel (Basic statistics of various properties).

Dipole-dipole correlation function

First calculate the dipole vector and moment (Dipole vector and dipole moment), and make sure the trajectory contains constant time steps.

To specify the values of parameters in Eq. 4, set these controls:

m = Max. Time (as number of frames or in picoseconds).
i = Origin Step (as number of frames or in picoseconds).

To display a plot that is continuously updated as the calculation of the dipole-dipole correlation function proceeds, check the Monitor Calculation check box.

To display the finished plot only at the end of the calculation, uncheck the Monitor Calculation check box.

To start the calculation, click the Calculate Dipole Autocorrelation action button in the Dipole control panel.

Dipole-dipole power spectrum

First calculate the dipole-dipole correlation function (Dipole-dipole correlation function).

Specify the Frequency Interval ( in Eq. 5) and Maximum Time (tmax in Eq. 5) allowed, in the Dipole-Dipole Power Spectrum section of the Dipole control panel.

Start the calculation by clicking the Calculate Power Spectrum action button in the control panel.

A graph of the power spectrum is displayed.

Additional information

Please see the on-screen help for details on the functioning of each control in the Dipole control panel.

Fluctuation analysis

You should already know...

Loading trajectories (Loading or creating a trajectory), selecting frames (Selecting frames from a loaded trajectory), and specifying output (Output from analysis) are covered above.

Thermodynamic properties

You can calculate the fluctuations in the following properties:

• Adiabatic compressibility--Response of system volume to a reversible change in pressure.

• Isothermal compressibility--Response of system volume to changes in pressure at constant temperature.

• Thermal pressure coefficient--Response of pressure to changes in temperature at constant volume.

• Gruneisen parameter--Response of pressure to changes in internal energy at constant volume.

• Isobaric heat capacity--Response of internal energy to changes in temperature at constant volume.

• Isometric heat capacity--Response of internal energy to changes in temperature at constant pressure.

• Volume expansivity--Change in volume, per-unit volume, with temperature at a constant pressure.

• Isoenthalpic Joule-Thomson coefficient--Change in pressure with temperature at constant enthalpy.

• Isothermal Joule-Thomson coefficient--Change in enthalpy with pressure at a constant temperature.

• Velocity of sound--The speed of sound waves in the system.

Dielectric constant--How well the system screens the interaction between charged sites within it. Calculated using fluctuations in the dipole moment, which must be calculated first (Dipole-dipole interactions). The calculation uses the Kirkwood-Frohlich equation:

Eq. 21

e = Dielectric constant.
M = Dipole moment of sample.
V = Volume of sample.
T = Temperature.

Technical notes

Some trajectory files do not contain enough information to generate all possible fluctuation properties. The software automatically lists only the properties that can be calculated.

Accessing the tools

Select the Analyze/Fluctuation Properties menu item from the ANALYSIS card to access the Fluctuations control panel.

Prerequisites

You need to load a trajectory (Loading or creating a trajectory) and select what frames to use (Selecting frames from a loaded trajectory) in the fluctuation calculations.

Selecting properties

To select a property to analyze, highlight its name in the Select Properties list box in the Fluctuations control panel by clicking it.

To select more than one property at once, <Shift>-click to select several adjacent properties or <Ctrl>-click to select properties whose names are not continuous.

To deselect a highlighted property, <Ctrl> click it. To deselect all highlighted properties and select another one, simply click the desired property.

Calculations

For the selected property or properties, you can use the Fluctuations control panel to calculate:

• The value of the property, by clicking the Calculate action button.

The value of the property is printed in the text window.

• The running or block average of the property, by clicking the Running Average or Block Average action button.

The average is plotted vs. time.

To changed the width or interval used in block averaging, click the Preferences... button to the right of the Block Average action button to access the Block Average Preferences control panel.

Please see the on-screen help for details on the functioning of each control in the Fluctuations control panel.