Step 9c - Centre-of-Mass Spatial Density

Radial distribution functions provide a spherically-averaged picture of structural correlations between sites, so when there are strongly directional interactions (such as hydrogen bonds, for instance) or simply where the central molecule of interest lacks symmetry, understanding the distribution of other sites in 3D space around a given molecule type is much more informative. A spatial distribution function or spatial density function (SDF) tells us, in a manner analogous to the radial distribution function, the densities of some object in 3D space relative to the bulk density. In other words, given a central molecule, the SDF tells us where other molecules or sites prefer to locate themselves.

The SDF is an “orientation-dependent” function - rather than averaging spherically around a molecule as in the RDF, the SDF preserves spatial information relative to some reference set of axes related to the local geometry of the central molecule. So, our first task is to define a set of reference axes for our water molecule.

Set Up Site Axes

We will define axes on our centre-of-mass site. In general, a system of reference axes can be constructed quite easily from the following steps:

1. Define a coordinate that will represent the origin of the axes (i.e. the local coordinate {0,0,0}). In the present example this will be the centre-of-mass of the molecule.
2. Select a group of one or more atoms whose coordinates, when averaged, will define absolutely the direction of the x axis.
3. Select a group of one or more atoms whose coordinates, when averaged, will define the rough direction of the y axis (and more importantly the XY plane), and which will be orthogonalised with respect to x (i.e. made to form a 90° angle).
4. Define the z axis from the cross product of the x and y axes.

When defining a site within Dissolve it is your responsibility to provide atom indices for the origin and the representative x and y directions - Dissolve will do the rest. Thus:

Go to the Water species tab and open the Analysis Sites section

If you don’t already have the COM site, click-drag with the left mouse button in the species viewer and draw a box around the whole molecule to select it, then press the Create button in the toolbar, and rename the new site to COM.

Make sure the COM site is selected in the list on the left

Select the oxygen atom and click the X Axis button in the viewer toolbar

Select either one of the hydrogen atoms and click the Y Axis button in the viewer toolbar

You’ll note that, as soon as you pressed the Y Axis button a set of axes appeared in the site viewer, letting you know that your definition is complete and showing you how your system of axes are oriented.

Create the Analysis Layer

We will add another of Dissolve’s predefined analysis layers to get the modules that we want:

Layer ⇨ Create ⇨ Analysis ⇨ Average Molecule & SDF

Along with the calculation of the SDF, it is useful to have the actual average geometry of the species around the specified site, and so the layer provides the following modules:

Module	Purpose
CalculateAvgMol	Calculates the average geometry of the parent species around a specified reference site
CalculateSDF	Calculates the spatial density function of a particular site around a central reference site

As the name implies, the CalculateAvgMol ) module will provide us with the average species geometry which we can use as a reference when we plot the spatial density function calculated from the CalculateSDF module. The CalculateSDF module takes two sites as input - the central site A about which the distribution of site B will be calculated. For both modules, the principal (central) site must have a system of axes defined.

Let’s proceed and set up the two modules in the layer. First, CalculateAvgMol :

Select the CalculateAvgMol module to display its options

Open the Control settings group

Set the target Site for the average molecule calculation to be COM (it will be the only one available, since it is the only one which has a set of axes defined)

And now CalculateSDF :

Select the CalculateSDF module to display its options

Open the Control settings group

Set the central SiteA to COM (again, the only one available)

Set the surrounding SiteB to O

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