Shortcut: SPHERICAL_HARMONIC

Module	symfunc
Description	Usage
Calculate the values of all the spherical harmonic funtions for a particular value of l.

Output components

This action calculates the values in the following table. These values can be referenced elsewhere in the input by using this Action's label followed by a dot and the name of the value required from the list below.

Input

The arguments that serve as the input for this action are specified using one or more of the keywords in the following table.

Further details and examples

Calculate the values of all the spherical harmonic funtions for a particular value of l.

This action allows you to the all the spherical harmonic functions for a particular input $L$ value. As discussed in more detail in the article provided above the spherical harmonics this action calculates have the following general form:

$$Y_l^m(\theta,\phi) = wNe^{im\phi} P_l^m(\cos\theta)$$

where $N$ is a normalisation constant, $P_l^m$ is tn associated Legendre Polynomial and $w$ is an optional weight that can be passed in an input argument or simply set equal to one. $e^{i\phi}$ and $\cos(\theta) are computed from the other input arguments,$x, y$and$z$ as follows:

$$e^{i\phi} = \frac{x}{r} + i\frac{y}{r} \qquad \textrm{and} \qquad \cos(\theta) = \frac{z}{r} \qquad \textrm{where} \qquad r = \sqrt{x^2 + y^2 + z^2}$$

At present, the arguments for this action must be matrices. However, it would be easy to add functionality that would allow you to compute this function for scalar or vector input. The arguments must all have the same shape as the two output components will also be matrices that are calculated by applying the function above to each of the elements of the input matrix in turn. The number of components output will be equal to $2(2L+1)$ and will contain the real and imaginary parts of the $Y_l^m$ functions with the the $2l+1$ possible $m$ values.

The following intput provides an example that demonstrates how this function is used:

Click on the labels of the actions for more information on what each action computes

dThe DISTANCE_MATRIX action with label d calculates the following quantities:
 Quantity   
 Type   
 Description  
d.w
matrix
a matrix containing the weights for the bonds between each pair of atoms
d.x
matrix
the projection of the bond on the x axis
d.y
matrix
the projection of the bond on the y axis
d.z
matrix
the projection of the bond on the z axis: DISTANCE_MATRIXCalculate a matrix of distances between atoms. This action has hidden defaults. More details GROUPthe atoms for which you would like to calculate the adjacency matrix=1-10 COMPONENTS also calculate the components of the vectors connecting the atoms in the contact matrix
d: DISTANCE_MATRIXCalculate a matrix of distances between atoms. This action uses the defaults shown here. More details GROUPthe atoms for which you would like to calculate the adjacency matrix=1-10 COMPONENTS also calculate the components of the vectors connecting the atoms in the contact matrix  NL_CUTOFF The cutoff for the neighbor list=0.0 NL_STRIDE The frequency with which we are updating the atoms in the neighbor list=1 CUTOFF ignore distances that have a value larger than this cutoff=-1
cThe SPHERICAL_HARMONIC action with label c calculates the following quantities:
 Quantity   
 Type   
 Description  
c.rm-n1
matrix
the real parts of the spherical harmonic values with the m value given  This is the n1th of these quantities
c.rm-0
matrix
the real parts of the spherical harmonic values with the m value given  This is the 0th of these quantities
c.rm-p1
matrix
the real parts of the spherical harmonic values with the m value given  This is the p1th of these quantities
c.im-n1
matrix
the real parts of the spherical harmonic values with the m value given  This is the n1th of these quantities
c.im-0
matrix
the real parts of the spherical harmonic values with the m value given  This is the 0th of these quantities
c.im-p1
matrix
the real parts of the spherical harmonic values with the m value given  This is the p1th of these quantities: SPHERICAL_HARMONICCalculate the values of all the spherical harmonic funtions for a particular value of l. More details Lthe value of the angular momentum=1 ARGthe values input to this function=d.x,d.y,d.z
PRINTPrint quantities to a file. More details ARGthe labels of the values that you would like to print to the file=c.rm-n1 FILEthe name of the file on which to output these quantities=real_part_m-1
The PRINT action with label  calculates somethingPRINTPrint quantities to a file. More details ARGthe labels of the values that you would like to print to the file=c.im-n1 FILEthe name of the file on which to output these quantities=imaginary_part_m-1
PRINTPrint quantities to a file. More details ARGthe labels of the values that you would like to print to the file=c.rm-0 FILEthe name of the file on which to output these quantities=real_part_m0
PRINTPrint quantities to a file. More details ARGthe labels of the values that you would like to print to the file=c.im-0 FILEthe name of the file on which to output these quantities=imaginary_part_m0
PRINTPrint quantities to a file. More details ARGthe labels of the values that you would like to print to the file=c.rm-p1 FILEthe name of the file on which to output these quantities=real_part_m+1
PRINTPrint quantities to a file. More details ARGthe labels of the values that you would like to print to the file=c.im-p1 FILEthe name of the file on which to output these quantities=imaginary_part_m+1

The DISTANCE_MATRIX command in the above input computes 3 $10\times10$ matrices. These 3 $10\times10$ matrices are used in the input to the sphierical harmonic command, which in turn outputs 6 $10\times10$ matrices that contain the real and imaginary parts when the three spherical harmonic functions with $l=1$ are applied element-wise to the above input. These six $10\times10$ matrices are then output to six separate files.

In the above example the weights for every distance is set equal to one. The following example shows how an argument can be used to set the $w$ values to use when computing the function above.

Click on the labels of the actions for more information on what each action computes

sThe CONTACT_MATRIX action with label s calculates the following quantities:
 Quantity   
 Type   
 Description  
s
matrix
a matrix containing the weights for the bonds between each pair of atoms: CONTACT_MATRIXAdjacency matrix in which two atoms are adjacent if they are within a certain cutoff. More details GROUPspecifies the list of atoms that should be assumed indistinguishable=1-10 SWITCHthe input for the switching function that acts upon the distance between each pair of atoms. Options for this keyword are explained in the documentation for LESS_THAN.={RATIONAL R_0=1.0}
scThe CONTACT_MATRIX action with label sc calculates the following quantities:
 Quantity   
 Type   
 Description  
sc.w
matrix
a matrix containing the weights for the bonds between each pair of atoms
sc.x
matrix
the projection of the bond on the x axis
sc.y
matrix
the projection of the bond on the y axis
sc.z
matrix
the projection of the bond on the z axis: CONTACT_MATRIXAdjacency matrix in which two atoms are adjacent if they are within a certain cutoff. More details GROUPspecifies the list of atoms that should be assumed indistinguishable=1-10 SWITCHthe input for the switching function that acts upon the distance between each pair of atoms. Options for this keyword are explained in the documentation for LESS_THAN.={RATIONAL R_0=1.0} COMPONENTS also calculate the components of the vectors connecting the atoms in the contact matrix
cThe SPHERICAL_HARMONIC action with label c calculates the following quantities:
 Quantity   
 Type   
 Description  
c.rm-n1
matrix
the real parts of the spherical harmonic values with the m value given  This is the n1th of these quantities
c.rm-0
matrix
the real parts of the spherical harmonic values with the m value given  This is the 0th of these quantities
c.rm-p1
matrix
the real parts of the spherical harmonic values with the m value given  This is the p1th of these quantities
c.im-n1
matrix
the real parts of the spherical harmonic values with the m value given  This is the n1th of these quantities
c.im-0
matrix
the real parts of the spherical harmonic values with the m value given  This is the 0th of these quantities
c.im-p1
matrix
the real parts of the spherical harmonic values with the m value given  This is the p1th of these quantities: SPHERICAL_HARMONICCalculate the values of all the spherical harmonic funtions for a particular value of l. More details Lthe value of the angular momentum=1 ARGthe values input to this function=sc.x,sc.y,sc.z,s
PRINTPrint quantities to a file. More details ARGthe labels of the values that you would like to print to the file=c.rm-n1 FILEthe name of the file on which to output these quantities=real_part_m-1
The PRINT action with label  calculates somethingPRINTPrint quantities to a file. More details ARGthe labels of the values that you would like to print to the file=c.im-n1 FILEthe name of the file on which to output these quantities=imaginary_part_m-1
PRINTPrint quantities to a file. More details ARGthe labels of the values that you would like to print to the file=c.rm-0 FILEthe name of the file on which to output these quantities=real_part_m0
PRINTPrint quantities to a file. More details ARGthe labels of the values that you would like to print to the file=c.im-0 FILEthe name of the file on which to output these quantities=imaginary_part_m0
PRINTPrint quantities to a file. More details ARGthe labels of the values that you would like to print to the file=c.rm-p1 FILEthe name of the file on which to output these quantities=real_part_m+1
PRINTPrint quantities to a file. More details ARGthe labels of the values that you would like to print to the file=c.im-p1 FILEthe name of the file on which to output these quantities=imaginary_part_m+1

This function is used in the calculation of the Steinhardt order parameters, which are described in detail here.

Syntax

The following table describes the keywords and options that can be used with this action