Q4

	This is part of the symfunc module
	It is only available if you configure PLUMED with ./configure –enable-modules=symfunc . Furthermore, this feature is still being developed so take care when using it and report any problems on the mailing list.

Calculate fourth order Steinhardt parameters.

The fourth order Steinhardt parameters allow us to measure the degree to which the first coordination shell around an atom is ordered. The Steinhardt parameter for atom, \(i\) is complex vector whose components are calculated using the following formula:

\[q_{4m}(i) = \frac{\sum_j \sigma( r_{ij} ) Y_{4m}(\mathbf{r}_{ij}) }{\sum_j \sigma( r_{ij} ) } \]

where \(Y_{4m}\) is one of the fourth order spherical harmonics so \(m\) is a number that runs from \(-4\) to \(+4\). The function \(\sigma( r_{ij} )\) is a switchingfunction that acts on the distance between atoms \(i\) and \(j\). The parameters of this function should be set so that it the function is equal to one when atom \(j\) is in the first coordination sphere of atom \(i\) and is zero otherwise.

The Steinhardt parameters can be used to measure the degree of order in the system in a variety of different ways. The simplest way of measuring whether or not the coordination sphere is ordered is to simply take the norm of the above vector i.e.

\[Q_4(i) = \sqrt{ \sum_{m=-4}^4 q_{4m}(i)^{*} q_{4m}(i) } \]

This norm is small when the coordination shell is disordered and larger when the coordination shell is ordered. Furthermore, when the keywords LESS_THAN, MIN, MAX, HISTOGRAM, MEAN and so on are used with this colvar it is the distribution of these normed quantities that is investigated.

Other measures of order can be taken by averaging the components of the individual \(q_4\) vectors individually or by taking dot products of the \(q_{4}\) vectors on adjacent atoms. More information on these variables can be found in the documentation for LOCAL_Q4, LOCAL_AVERAGE and NLINKS.

Examples

The following command calculates the average Q4 parameter for the 64 atoms in a box of Lennard Jones and prints this quantity to a file called colvar:

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

contract shortcuts

q4: Q4 SPECIESthis keyword is used for colvars such as coordination number. =1-64 D_0 could not find this keyword =1.3 R_0 could not find this keyword =0.2 MEAN( default=off ) calculate the mean of all the quantities.  The Q4 action with label q4 calculates a single scalar value
PRINT ARGthe input for this action is the scalar output from one or more other actions. =q4.mean FILEthe name of the file on which to output these quantities =colvar The PRINT action with label 

q4_grp: GROUP ATOMSthe numerical indexes for the set of atoms in the group. =1-64 The GROUP action with label q4_grp defines a group of atoms so that they can be referred to later in the input
q4_mat: CONTACT_MATRIX GROUPspecifies the list of atoms that should be assumed indistinguishable. =1-64 R_0 could not find this keyword =0.2 D_0 could not find this keyword =1.3 NNcompulsory keyword ( default=6 )
The n parameter of the switching function =6 MMcompulsory keyword ( default=0 )
The m parameter of the switching function; 0 implies 2*NN =0 COMPONENTS( default=off ) also calculate the components of the vector connecting the atoms
in the contact matrix  The CONTACT_MATRIX action with label q4_mat calculates the following quantities:

 Quantity   
 Description  
q4_mat.w
a matrix containing the weights for the bonds between each pair of atoms
q4_mat.x
the projection of the bond on the x axis
q4_mat.y
the projection of the bond on the y axis
q4_mat.z
the projection of the bond on the z axis


q4_sh: SPHERICAL_HARMONIC ARGthe input to this function. =q4_mat.x,q4_mat.y,q4_mat.z,q4_mat.w Lcompulsory keyword 
the value of the angular momentum =4 The SPHERICAL_HARMONIC action with label q4_sh calculates the following quantities:

 Quantity   
 Description  
q4_sh.rm-n4
the real parts of the spherical harmonic values with the m value given  This is the n4th of these quantities
q4_sh.rm-n3
the real parts of the spherical harmonic values with the m value given  This is the n3th of these quantities
q4_sh.rm-n2
the real parts of the spherical harmonic values with the m value given  This is the n2th of these quantities
q4_sh.rm-n1
the real parts of the spherical harmonic values with the m value given  This is the n1th of these quantities
q4_sh.rm-0
the real parts of the spherical harmonic values with the m value given  This is the 0th of these quantities
q4_sh.rm-p1
the real parts of the spherical harmonic values with the m value given  This is the p1th of these quantities
q4_sh.rm-p2
the real parts of the spherical harmonic values with the m value given  This is the p2th of these quantities
q4_sh.rm-p3
the real parts of the spherical harmonic values with the m value given  This is the p3th of these quantities
q4_sh.rm-p4
the real parts of the spherical harmonic values with the m value given  This is the p4th of these quantities
q4_sh.im-n4
the real parts of the spherical harmonic values with the m value given  This is the n4th of these quantities
q4_sh.im-n3
the real parts of the spherical harmonic values with the m value given  This is the n3th of these quantities
q4_sh.im-n2
the real parts of the spherical harmonic values with the m value given  This is the n2th of these quantities
q4_sh.im-n1
the real parts of the spherical harmonic values with the m value given  This is the n1th of these quantities
q4_sh.im-0
the real parts of the spherical harmonic values with the m value given  This is the 0th of these quantities
q4_sh.im-p1
the real parts of the spherical harmonic values with the m value given  This is the p1th of these quantities
q4_sh.im-p2
the real parts of the spherical harmonic values with the m value given  This is the p2th of these quantities
q4_sh.im-p3
the real parts of the spherical harmonic values with the m value given  This is the p3th of these quantities
q4_sh.im-p4
the real parts of the spherical harmonic values with the m value given  This is the p4th of these quantities


q4_denom_ones: ONES SIZEcompulsory keyword 
the number of ones that you would like to create =64 The ONES action with label q4_denom_ones calculates a single scalar value
q4_denom: MATRIX_VECTOR_PRODUCT ARGthe input for this action is the scalar output from one or more other actions. =q4_mat.w,q4_denom_ones The MATRIX_VECTOR_PRODUCT action with label q4_denom calculates a single scalar value
q4_sp: MATRIX_VECTOR_PRODUCT ARGthe input for this action is the scalar output from one or more other actions. =q4_sh.*,q4_denom_ones The MATRIX_VECTOR_PRODUCT action with label q4_sp calculates the following quantities:

 Quantity   
 Description  
q4_sp.rm-n4
the product of the matrix q4_sh.rm-n4 and the vector q4_denom_ones
q4_sp.rm-n3
the product of the matrix q4_sh.rm-n3 and the vector q4_denom_ones
q4_sp.rm-n2
the product of the matrix q4_sh.rm-n2 and the vector q4_denom_ones
q4_sp.rm-n1
the product of the matrix q4_sh.rm-n1 and the vector q4_denom_ones
q4_sp.rm-0
the product of the matrix q4_sh.rm-0 and the vector q4_denom_ones
q4_sp.rm-p1
the product of the matrix q4_sh.rm-p1 and the vector q4_denom_ones
q4_sp.rm-p2
the product of the matrix q4_sh.rm-p2 and the vector q4_denom_ones
q4_sp.rm-p3
the product of the matrix q4_sh.rm-p3 and the vector q4_denom_ones
q4_sp.rm-p4
the product of the matrix q4_sh.rm-p4 and the vector q4_denom_ones
q4_sp.im-n4
the product of the matrix q4_sh.im-n4 and the vector q4_denom_ones
q4_sp.im-n3
the product of the matrix q4_sh.im-n3 and the vector q4_denom_ones
q4_sp.im-n2
the product of the matrix q4_sh.im-n2 and the vector q4_denom_ones
q4_sp.im-n1
the product of the matrix q4_sh.im-n1 and the vector q4_denom_ones
q4_sp.im-0
the product of the matrix q4_sh.im-0 and the vector q4_denom_ones
q4_sp.im-p1
the product of the matrix q4_sh.im-p1 and the vector q4_denom_ones
q4_sp.im-p2
the product of the matrix q4_sh.im-p2 and the vector q4_denom_ones
q4_sp.im-p3
the product of the matrix q4_sh.im-p3 and the vector q4_denom_ones
q4_sp.im-p4
the product of the matrix q4_sh.im-p4 and the vector q4_denom_ones


q4_norm2: COMBINE PERIODICcompulsory keyword 
if the output of your function is periodic then you should specify the periodicity
of the function. =NO POWERScompulsory keyword ( default=1.0 )
the powers to which you are raising each of the arguments in your function =2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2 ARGthe input to this function. =q4_sp.rm-n4,q4_sp.im-n4,q4_sp.rm-n3,q4_sp.im-n3,q4_sp.rm-n2,q4_sp.im-n2,q4_sp.rm-n1,q4_sp.im-n1,q4_sp.rm-0,q4_sp.im-0,q4_sp.rm-p1,q4_sp.im-p1,q4_sp.rm-p2,q4_sp.im-p2,q4_sp.rm-p3,q4_sp.im-p3,q4_sp.rm-p4,q4_sp.im-p4 The COMBINE action with label q4_norm2 calculates a single scalar value
q4_norm: CUSTOM ARGthe input to this function. =q4_norm2 FUNCcompulsory keyword 
the function you wish to evaluate =sqrt(x) PERIODICcompulsory keyword 
if the output of your function is periodic then you should specify the periodicity
of the function. =NO The CUSTOM action with label q4_norm calculates a single scalar value
q4: CUSTOM ARGthe input to this function. =q4_norm,q4_denom FUNCcompulsory keyword 
the function you wish to evaluate =x/y PERIODICcompulsory keyword 
if the output of your function is periodic then you should specify the periodicity
of the function. =NO The CUSTOM action with label q4 calculates a single scalar value
q4_mean: MEAN ARGthe input to this function. =q4 PERIODICcompulsory keyword 
if the output of your function is periodic then you should specify the periodicity
of the function. =NO The MEAN action with label q4_mean calculates a single scalar value
PRINT ARGthe input for this action is the scalar output from one or more other actions. =q4.mean FILEthe name of the file on which to output these quantities =colvar The PRINT action with label 

The following command calculates the histogram of Q4 parameters for the 64 atoms in a box of Lennard Jones and prints these quantities to a file called colvar:

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

contract shortcuts

q4: Q4 SPECIESthis keyword is used for colvars such as coordination number. =1-64 D_0 could not find this keyword =1.3 R_0 could not find this keyword =0.2 HISTOGRAMcalculate a discretized histogram of the distribution of values. ={GAUSSIAN LOWER=0.0 UPPER=1.0 NBINS=20 SMEAR=0.1}  The Q4 action with label q4 calculates a single scalar value
PRINT ARGthe input for this action is the scalar output from one or more other actions. =q4.* FILEthe name of the file on which to output these quantities =colvar The PRINT action with label 

q4_grp: GROUP ATOMSthe numerical indexes for the set of atoms in the group. =1-64 The GROUP action with label q4_grp defines a group of atoms so that they can be referred to later in the input
q4_mat: CONTACT_MATRIX GROUPspecifies the list of atoms that should be assumed indistinguishable. =1-64 R_0 could not find this keyword =0.2 D_0 could not find this keyword =1.3 NNcompulsory keyword ( default=6 )
The n parameter of the switching function =6 MMcompulsory keyword ( default=0 )
The m parameter of the switching function; 0 implies 2*NN =0 COMPONENTS( default=off ) also calculate the components of the vector connecting the atoms
in the contact matrix  The CONTACT_MATRIX action with label q4_mat calculates the following quantities:

 Quantity   
 Description  
q4_mat.w
a matrix containing the weights for the bonds between each pair of atoms
q4_mat.x
the projection of the bond on the x axis
q4_mat.y
the projection of the bond on the y axis
q4_mat.z
the projection of the bond on the z axis


q4_sh: SPHERICAL_HARMONIC ARGthe input to this function. =q4_mat.x,q4_mat.y,q4_mat.z,q4_mat.w Lcompulsory keyword 
the value of the angular momentum =4 The SPHERICAL_HARMONIC action with label q4_sh calculates the following quantities:

 Quantity   
 Description  
q4_sh.rm-n4
the real parts of the spherical harmonic values with the m value given  This is the n4th of these quantities
q4_sh.rm-n3
the real parts of the spherical harmonic values with the m value given  This is the n3th of these quantities
q4_sh.rm-n2
the real parts of the spherical harmonic values with the m value given  This is the n2th of these quantities
q4_sh.rm-n1
the real parts of the spherical harmonic values with the m value given  This is the n1th of these quantities
q4_sh.rm-0
the real parts of the spherical harmonic values with the m value given  This is the 0th of these quantities
q4_sh.rm-p1
the real parts of the spherical harmonic values with the m value given  This is the p1th of these quantities
q4_sh.rm-p2
the real parts of the spherical harmonic values with the m value given  This is the p2th of these quantities
q4_sh.rm-p3
the real parts of the spherical harmonic values with the m value given  This is the p3th of these quantities
q4_sh.rm-p4
the real parts of the spherical harmonic values with the m value given  This is the p4th of these quantities
q4_sh.im-n4
the real parts of the spherical harmonic values with the m value given  This is the n4th of these quantities
q4_sh.im-n3
the real parts of the spherical harmonic values with the m value given  This is the n3th of these quantities
q4_sh.im-n2
the real parts of the spherical harmonic values with the m value given  This is the n2th of these quantities
q4_sh.im-n1
the real parts of the spherical harmonic values with the m value given  This is the n1th of these quantities
q4_sh.im-0
the real parts of the spherical harmonic values with the m value given  This is the 0th of these quantities
q4_sh.im-p1
the real parts of the spherical harmonic values with the m value given  This is the p1th of these quantities
q4_sh.im-p2
the real parts of the spherical harmonic values with the m value given  This is the p2th of these quantities
q4_sh.im-p3
the real parts of the spherical harmonic values with the m value given  This is the p3th of these quantities
q4_sh.im-p4
the real parts of the spherical harmonic values with the m value given  This is the p4th of these quantities


q4_denom_ones: ONES SIZEcompulsory keyword 
the number of ones that you would like to create =64 The ONES action with label q4_denom_ones calculates a single scalar value
q4_denom: MATRIX_VECTOR_PRODUCT ARGthe input for this action is the scalar output from one or more other actions. =q4_mat.w,q4_denom_ones The MATRIX_VECTOR_PRODUCT action with label q4_denom calculates a single scalar value
q4_sp: MATRIX_VECTOR_PRODUCT ARGthe input for this action is the scalar output from one or more other actions. =q4_sh.*,q4_denom_ones The MATRIX_VECTOR_PRODUCT action with label q4_sp calculates the following quantities:

 Quantity   
 Description  
q4_sp.rm-n4
the product of the matrix q4_sh.rm-n4 and the vector q4_denom_ones
q4_sp.rm-n3
the product of the matrix q4_sh.rm-n3 and the vector q4_denom_ones
q4_sp.rm-n2
the product of the matrix q4_sh.rm-n2 and the vector q4_denom_ones
q4_sp.rm-n1
the product of the matrix q4_sh.rm-n1 and the vector q4_denom_ones
q4_sp.rm-0
the product of the matrix q4_sh.rm-0 and the vector q4_denom_ones
q4_sp.rm-p1
the product of the matrix q4_sh.rm-p1 and the vector q4_denom_ones
q4_sp.rm-p2
the product of the matrix q4_sh.rm-p2 and the vector q4_denom_ones
q4_sp.rm-p3
the product of the matrix q4_sh.rm-p3 and the vector q4_denom_ones
q4_sp.rm-p4
the product of the matrix q4_sh.rm-p4 and the vector q4_denom_ones
q4_sp.im-n4
the product of the matrix q4_sh.im-n4 and the vector q4_denom_ones
q4_sp.im-n3
the product of the matrix q4_sh.im-n3 and the vector q4_denom_ones
q4_sp.im-n2
the product of the matrix q4_sh.im-n2 and the vector q4_denom_ones
q4_sp.im-n1
the product of the matrix q4_sh.im-n1 and the vector q4_denom_ones
q4_sp.im-0
the product of the matrix q4_sh.im-0 and the vector q4_denom_ones
q4_sp.im-p1
the product of the matrix q4_sh.im-p1 and the vector q4_denom_ones
q4_sp.im-p2
the product of the matrix q4_sh.im-p2 and the vector q4_denom_ones
q4_sp.im-p3
the product of the matrix q4_sh.im-p3 and the vector q4_denom_ones
q4_sp.im-p4
the product of the matrix q4_sh.im-p4 and the vector q4_denom_ones


q4_norm2: COMBINE PERIODICcompulsory keyword 
if the output of your function is periodic then you should specify the periodicity
of the function. =NO POWERScompulsory keyword ( default=1.0 )
the powers to which you are raising each of the arguments in your function =2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2 ARGthe input to this function. =q4_sp.rm-n4,q4_sp.im-n4,q4_sp.rm-n3,q4_sp.im-n3,q4_sp.rm-n2,q4_sp.im-n2,q4_sp.rm-n1,q4_sp.im-n1,q4_sp.rm-0,q4_sp.im-0,q4_sp.rm-p1,q4_sp.im-p1,q4_sp.rm-p2,q4_sp.im-p2,q4_sp.rm-p3,q4_sp.im-p3,q4_sp.rm-p4,q4_sp.im-p4 The COMBINE action with label q4_norm2 calculates a single scalar value
q4_norm: CUSTOM ARGthe input to this function. =q4_norm2 FUNCcompulsory keyword 
the function you wish to evaluate =sqrt(x) PERIODICcompulsory keyword 
if the output of your function is periodic then you should specify the periodicity
of the function. =NO The CUSTOM action with label q4_norm calculates a single scalar value
q4: CUSTOM ARGthe input to this function. =q4_norm,q4_denom FUNCcompulsory keyword 
the function you wish to evaluate =x/y PERIODICcompulsory keyword 
if the output of your function is periodic then you should specify the periodicity
of the function. =NO The CUSTOM action with label q4 calculates a single scalar value
q4_bt1: BETWEEN ARGthe input to this function. =q4 SWITCHThis keyword is used if you want to employ an alternative to the continuous function
defined above. =GAUSSIAN,LOWER=0,UPPER=0.05,SMEAR=0.1 The BETWEEN action with label q4_bt1
The following command could be used to measure the Q4 parameters that describe the arrangement of chlorine ions around the sodium atoms in sodium chloride. The imagined system here is composed of 64 NaCl formula units and the atoms are arranged in the input with the 64 Na \(^+\) ions followed by the 64 Cl \(-\) ions. Once again the average Q4 parameter is calculated and output to a file called colvar

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




q4: Q4 SPECIESAthis keyword is used for colvars such as the coordination number. =1-64 SPECIESBthis keyword is used for colvars such as the coordination number. =65-128 D_0 could not find this keyword =1.3 R_0 could not find this keyword =0.2 MEAN( default=off ) calculate the mean of all the quantities.  The Q4 action with label q4 calculates a single scalar value
PRINT ARGthe input for this action is the scalar output from one or more other actions. =q4.mean FILEthe name of the file on which to output these quantities =colvar The PRINT action with label 


q4_grp: GROUP ATOMSthe numerical indexes for the set of atoms in the group. =1-64 The GROUP action with label q4_grp defines a group of atoms so that they can be referred to later in the input
q4_mat: CONTACT_MATRIX R_0 could not find this keyword =0.2 D_0 could not find this keyword =1.3 NNcompulsory keyword ( default=6 )
The n parameter of the switching function =6 MMcompulsory keyword ( default=0 )
The m parameter of the switching function; 0 implies 2*NN =0 COMPONENTS( default=off ) also calculate the components of the vector connecting the atoms
in the contact matrix  The CONTACT_MATRIX action with label q4_mat
If you simply want to examine the values of the Q4 parameters for each of the atoms in your system you can do so by exploiting the command DUMPATOMS as shown in the example below. The following output file will output a file in an extended xyz format called q$.xyz for each frame of the analyzed MD trajectory. The first column in this file will contain a dummy name for each of the atoms, columns 2-4 will then contain the x, y and z positions of the atoms, column 5 will contain the value of the Q4 parameter, columns 6-15 will contain the real parts of the director of the \(q_{6m}\) vector while columns 15-24 will contain the imaginary parts of this director.

 Click on the labels of the actions for more information on what each action computes 
q4: Q4 SPECIESAthis keyword is used for colvars such as the coordination number. =1-64 SPECIESBthis keyword is used for colvars such as the coordination number. =65-128 D_0 could not find this keyword =1.3 R_0 could not find this keyword =0.2 MEAN( default=off ) calculate the mean of all the quantities.   You cannot view the components that are calculated by each action for this input file. Sorry 
DUMPATOMS ATOMSthe atom indices whose positions you would like to print out. =q4 ARGthe input for this action is the scalar output from one or more other actions. =q4_anorm FILEcompulsory keyword 
file on which to output coordinates; extension is automatically detected =q4.xyz  You cannot view the components that are calculated by each action for this input file. Sorry 

Glossary of keywords and components

Description of components



 Quantity   
 Keyword   
 Description    

 lessthan  
 LESS_THAN   
the number of colvars that have a value less than a threshold   

 morethan  
 MORE_THAN   
the number of colvars that have a value more than a threshold   

 altmin  
 ALT_MIN   
the minimum value of the cv   

 min  
 MIN   
the minimum colvar   

 max  
 MAX   
the maximum colvar   

 between  
 BETWEEN   
the number of colvars that have a value that lies in a particular interval   

 highest  
 HIGHEST   
the largest of the colvars   

 lowest  
 LOWEST   
the smallest of the colvars   

 sum  
 SUM   
the sum of the colvars   

 mean  
 MEAN   
the mean of the colvars   

 _vmean  
 VMEAN   
the norm of the mean vector   

 _vsum  
 VSUM   
the norm of the mean vector   

The atoms involved can be specified using



 SPECIES  
this keyword is used for colvars such as coordination number. In that context it specifies that plumed should calculate one coordination number for each of the atoms specified. Each of these coordination numbers specifies how many of the other specified atoms are within a certain cutoff of the central atom. You can specify the atoms here as another multicolvar action or using a MultiColvarFilter or ActionVolume action. When you do so the quantity is calculated for those atoms specified in the previous multicolvar. This is useful if you would like to calculate the Steinhardt parameter for those atoms that have a coordination number more than four for example   

Or alternatively by using



 SPECIESA  
this keyword is used for colvars such as the coordination number. In that context it species that plumed should calculate one coordination number for each of the atoms specified in SPECIESA. Each of these cooordination numbers specifies how many of the atoms specifies using SPECIESB is within the specified cutoff. As with the species keyword the input can also be specified using the label of another multicolvar   

 SPECIESB  
this keyword is used for colvars such as the coordination number. It must appear with SPECIESA. For a full explanation see the documentation for that keyword   

Compulsory keywords



 NN  
( default=6 ) The n parameter of the switching function   

 MM  
( default=0 ) The m parameter of the switching function; 0 implies 2*NN   

 D_0  
( default=0.0 ) The d_0 parameter of the switching function   

 R_0  
The r_0 parameter of the switching function   

Options



 HIGHEST  
( default=off ) this flag allows you to recover the highest of these variables.   

 LOWEST  
( default=off ) this flag allows you to recover the lowest of these variables.   

 SUM  
( default=off ) calculate the sum of all the quantities.   

 MEAN  
( default=off ) calculate the mean of all the quantities.   

 LOWMEM  
( default=off ) this flag does nothing and is present only to ensure back-compatibility   

 VMEAN  
( default=off ) calculate the norm of the mean vector.   

 VSUM  
( default=off ) calculate the norm of the sum of all the vectors  



 SWITCH  
the switching function that it used in the construction of the contact matrix   

 LESS_THAN  
calculate the number of variables that are less than a certain target value. This quantity is calculated using \(\sum_i \sigma(s_i)\), where \(\sigma(s)\) is a switchingfunction.. You can use multiple instances of this keyword i.e. LESS_THAN1, LESS_THAN2, LESS_THAN3...   

 MORE_THAN  
calculate the number of variables that are more than a certain target value. This quantity is calculated using \(\sum_i 1 - \sigma(s_i)\), where \(\sigma(s)\) is a switchingfunction.. You can use multiple instances of this keyword i.e. MORE_THAN1, MORE_THAN2, MORE_THAN3...   

 ALT_MIN  
calculate the minimum value. To make this quantity continuous the minimum is calculated using \( \textrm{min} = -\frac{1}{\beta} \log \sum_i \exp\left( -\beta s_i \right)  \) The value of \(\beta\) in this function is specified using (BETA= \(\beta\)).   

 MIN  
calculate the minimum value. To make this quantity continuous the minimum is calculated using \( \textrm{min} = \frac{\beta}{ \log \sum_i \exp\left( \frac{\beta}{s_i} \right) } \) The value of \(\beta\) in this function is specified using (BETA= \(\beta\))   

 MAX  
calculate the maximum value. To make this quantity continuous the maximum is calculated using \( \textrm{max} = \beta \log \sum_i \exp\left( \frac{s_i}{\beta}\right) \) The value of \(\beta\) in this function is specified using (BETA= \(\beta\))   

 BETWEEN  
calculate the number of values that are within a certain range. These quantities are calculated using kernel density estimation as described on histogrambead.. You can use multiple instances of this keyword i.e. BETWEEN1, BETWEEN2, BETWEEN3...   

 HISTOGRAM  
calculate a discretized histogram of the distribution of values. This shortcut allows you to calculates NBIN quantites like BETWEEN.