LOCAL_Q4

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

Calculate the local degree of order around an atoms by taking the average dot product between the \(q_4\) vector on the central atom and the \(q_4\) vector on the atoms in the first coordination sphere.

The Q4 command allows one to calculate one complex vectors for each of the atoms in your system that describe the degree of order in the coordination sphere around a particular atom. The difficulty with these vectors comes when combining the order parameters from all of the individual atoms/molecules so as to get a measure of the global degree of order for the system. The simplest way of doing this - calculating the average Steinhardt parameter - can be problematic. If one is examining nucleation say only the order parameters for those atoms in the nucleus will change significantly when the nucleus forms. The order parameters for the atoms in the surrounding liquid will remain pretty much the same. As such if one models a small nucleus embedded in a very large amount of solution/melt any change in the average order parameter will be negligible. Substantial changes in the value of this average can be observed in simulations of nucleation but only because the number of atoms is relatively small.

When the average Q4 parameter is used to bias the dynamics a problems can occur. These averaged coordinates cannot distinguish between the correct, single-nucleus pathway and a concerted pathway in which all the atoms rearrange themselves into their solid-like configuration simultaneously. This second type of pathway would be impossible in reality because there is a large entropic barrier that prevents concerted processes like this from happening. However, in the finite sized systems that are commonly simulated this barrier is reduced substantially. As a result in simulations where average Steinhardt parameters are biased there are often quite dramatic system size effects

If one wants to simulate nucleation using some form on biased dynamics what is really required is an order parameter that measures:

• Whether or not the coordination spheres around atoms are ordered
• Whether or not the atoms that are ordered are clustered together in a crystalline nucleus

LOCAL_AVERAGE and NLINKS are variables that can be combined with the Steinhardt parameters allow to calculate variables that satisfy these requirements. LOCAL_Q4 is another variable that can be used in these sorts of calculations. The LOCAL_Q4 parameter for a particular atom is a number that measures the extent to which the orientation of the atoms in the first coordination sphere of an atom match the orientation of the central atom. It does this by calculating the following quantity for each of the atoms in the system:

\[ s_i = \frac{ \sum_j \sigma( r_{ij} ) \sum_{m=-4}^4 q_{4m}^{*}(i)q_{4m}(j) }{ \sum_j \sigma( r_{ij} ) } \]

where \(q_{4m}(i)\) and \(q_{4m}(j)\) are the fourth order Steinhardt vectors calculated for atom \(i\) and atom \(j\) respectively and the asterisk denotes complex conjugation. 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 sum in the numerator of this expression is the dot product of the Steinhardt parameters for atoms \(i\) and \(j\) and thus measures the degree to which the orientations of these adjacent atoms is correlated.

Examples

The following command calculates the average value of the LOCAL_Q4 parameter for the 64 Lennard Jones atoms in the system under study and prints this quantity to a file called colvar.

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

q4: Q4 
SPECIES
this 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 The Q4 action with label q4
lq4: LOCAL_Q4 
SPECIES
this keyword is used for colvars such as coordination number. 
=q4 
SWITCH
This keyword is used if you want to employ an alternative to the continuous switching
function defined above. 
={RATIONAL D_0=1.3 R_0=0.2}  
MEAN
take the mean of these variables. 
 The LOCAL_Q4 action with label lq4 calculates a single scalar value
PRINT 
ARG
the input for this action is the scalar output from one or more other actions. 
=lq4.mean 
FILE
the name of the file on which to output these quantities 
=colvar The PRINT action with label 

The following input calculates the distribution of LOCAL_Q4 parameters at any given time and outputs this information to a file.

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

q4: Q4 
SPECIES
this 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 The Q4 action with label q4
lq4: LOCAL_Q4 
SPECIES
this keyword is used for colvars such as coordination number. 
=q4 
SWITCH
This keyword is used if you want to employ an alternative to the continuous switching
function defined above. 
={RATIONAL D_0=1.3 R_0=0.2}  
HISTOGRAM
calculate how many of the values fall in each of the bins of a histogram. 
={GAUSSIAN LOWER=0.0 UPPER=1.0 NBINS=20 SMEAR=0.1}  The LOCAL_Q4 action with label lq4 calculates the following quantities:

 Quantity   
 Description  
lq4.between-1
the number/fraction of values within a certain range. This is calculated using one of the formula described in the description of the keyword so as to make it continuous. You can calculate this quantity multiple times using different parameters.  This is the 1th of these quantities
lq4.between-2
the number/fraction of values within a certain range. This is calculated using one of the formula described in the description of the keyword so as to make it continuous. You can calculate this quantity multiple times using different parameters.  This is the 2th of these quantities
lq4.between-3
the number/fraction of values within a certain range. This is calculated using one of the formula described in the description of the keyword so as to make it continuous. You can calculate this quantity multiple times using different parameters.  This is the 3th of these quantities
lq4.between-4
the number/fraction of values within a certain range. This is calculated using one of the formula described in the description of the keyword so as to make it continuous. You can calculate this quantity multiple times using different parameters.  This is the 4th of these quantities
lq4.between-5
the number/fraction of values within a certain range. This is calculated using one of the formula described in the description of the keyword so as to make it continuous. You can calculate this quantity multiple times using different parameters.  This is the 5th of these quantities
lq4.between-6
the number/fraction of values within a certain range. This is calculated using one of the formula described in the description of the keyword so as to make it continuous. You can calculate this quantity multiple times using different parameters.  This is the 6th of these quantities
lq4.between-7
the number/fraction of values within a certain range. This is calculated using one of the formula described in the description of the keyword so as to make it continuous. You can calculate this quantity multiple times using different parameters.  This is the 7th of these quantities
lq4.between-8
the number/fraction of values within a certain range. This is calculated using one of the formula described in the description of the keyword so as to make it continuous. You can calculate this quantity multiple times using different parameters.  This is the 8th of these quantities
lq4.between-9
the number/fraction of values within a certain range. This is calculated using one of the formula described in the description of the keyword so as to make it continuous. You can calculate this quantity multiple times using different parameters.  This is the 9th of these quantities
lq4.between-10
the number/fraction of values within a certain range. This is calculated using one of the formula described in the description of the keyword so as to make it continuous. You can calculate this quantity multiple times using different parameters.  This is the 10th of these quantities
lq4.between-11
the number/fraction of values within a certain range. This is calculated using one of the formula described in the description of the keyword so as to make it continuous. You can calculate this quantity multiple times using different parameters.  This is the 11th of these quantities
lq4.between-12
the number/fraction of values within a certain range. This is calculated using one of the formula described in the description of the keyword so as to make it continuous. You can calculate this quantity multiple times using different parameters.  This is the 12th of these quantities
lq4.between-13
the number/fraction of values within a certain range. This is calculated using one of the formula described in the description of the keyword so as to make it continuous. You can calculate this quantity multiple times using different parameters.  This is the 13th of these quantities
lq4.between-14
the number/fraction of values within a certain range. This is calculated using one of the formula described in the description of the keyword so as to make it continuous. You can calculate this quantity multiple times using different parameters.  This is the 14th of these quantities
lq4.between-15
the number/fraction of values within a certain range. This is calculated using one of the formula described in the description of the keyword so as to make it continuous. You can calculate this quantity multiple times using different parameters.  This is the 15th of these quantities
lq4.between-16
the number/fraction of values within a certain range. This is calculated using one of the formula described in the description of the keyword so as to make it continuous. You can calculate this quantity multiple times using different parameters.  This is the 16th of these quantities
lq4.between-17
the number/fraction of values within a certain range. This is calculated using one of the formula described in the description of the keyword so as to make it continuous. You can calculate this quantity multiple times using different parameters.  This is the 17th of these quantities
lq4.between-18
the number/fraction of values within a certain range. This is calculated using one of the formula described in the description of the keyword so as to make it continuous. You can calculate this quantity multiple times using different parameters.  This is the 18th of these quantities
lq4.between-19
the number/fraction of values within a certain range. This is calculated using one of the formula described in the description of the keyword so as to make it continuous. You can calculate this quantity multiple times using different parameters.  This is the 19th of these quantities
lq4.between-20
the number/fraction of values within a certain range. This is calculated using one of the formula described in the description of the keyword so as to make it continuous. You can calculate this quantity multiple times using different parameters.  This is the 20th of these quantities


PRINT 
ARG
the input for this action is the scalar output from one or more other actions. 
=lq4.* 
FILE
the name of the file on which to output these quantities 
=colvar The PRINT action with label 

The following calculates the LOCAL_Q4 parameters for atoms 1-5 only. For each of these atoms comparisons of the geometry of the coordination sphere are done with those of all the other atoms in the system. The final quantity is the average and is outputted to a file

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

q4a: Q4 
SPECIESA
this keyword is used for colvars such as the coordination number. 
=1-5 
SPECIESB
this keyword is used for colvars such as the coordination number. 
=1-64 
D_0
 could not find this keyword 
=1.3 
R_0
 could not find this keyword 
=0.2 The Q4 action with label q4a
q4b: Q4 
SPECIESA
this keyword is used for colvars such as the coordination number. 
=6-64 
SPECIESB
this keyword is used for colvars such as the coordination number. 
=1-64 
D_0
 could not find this keyword 
=1.3 
R_0
 could not find this keyword 
=0.2 The Q4 action with label q4b
w4: LOCAL_Q4 
SPECIES
this keyword is used for colvars such as coordination number. 
=q4a,q4b 
SWITCH
This keyword is used if you want to employ an alternative to the continuous switching
function defined above. 
={RATIONAL D_0=1.3 R_0=0.2}  
MEAN
take the mean of these variables. 
 
LOWMEM
( default=off ) lower the memory requirements 
 The LOCAL_Q4 action with label w4 calculates a single scalar value
PRINT 
ARG
the input for this action is the scalar output from one or more other actions. 
=w4.* 
FILE
the name of the file on which to output these quantities 
=colvar The PRINT action with label 

Glossary of keywords and components

Description of components

When the label of this action is used as the input for a second you are not referring to a scalar quantity as you are in regular collective variables. The label is used to reference the full set of quantities calculated by the action. This is usual when using MultiColvar functions. Generally when doing this the previously calculated multicolvar will be referenced using the DATA keyword rather than ARG.

This Action can be used to calculate the following scalar quantities directly. These quantities are calculated by employing the keywords listed below. These quantities can then be referenced elsewhere in the input file by using this Action's label followed by a dot and the name of the quantity. Some of them can be calculated multiple times with different parameters. In this case the quantities calculated can be referenced elsewhere in the input by using the name of the quantity followed by a numerical identifier e.g. label.lessthan-1, label.lessthan-2 etc. When doing this and, for clarity we have made it so that the user can set a particular label for each of the components. As such by using the LABEL keyword in the description of the keyword input you can customize the component name

Quantity	Keyword	Description
between	BETWEEN	the number/fraction of values within a certain range. This is calculated using one of the formula described in the description of the keyword so as to make it continuous. You can calculate this quantity multiple times using different parameters.
highest	HIGHEST	the highest of the quantities calculated by this action
lessthan	LESS_THAN	the number of values less than a target value. This is calculated using one of the formula described in the description of the keyword so as to make it continuous. You can calculate this quantity multiple times using different parameters.
lowest	LOWEST	the lowest of the quantities calculated by this action
mean	MEAN	the mean value. The output component can be referred to elsewhere in the input file by using the label.mean
min	MIN	the minimum value. This is calculated using the formula described in the description of the keyword so as to make it continuous.
moment	MOMENTS	the central moments of the distribution of values. The second moment would be referenced elsewhere in the input file using label.moment-2, the third as label.moment-3, etc.
morethan	MORE_THAN	the number of values more than a target value. This is calculated using one of the formula described in the description of the keyword so as to make it continuous. You can calculate this quantity multiple times using different parameters.

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 coordination 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

NUMERICAL_DERIVATIVES	( default=off ) calculate the derivatives for these quantities numerically
NOPBC	( default=off ) ignore the periodic boundary conditions when calculating distances
SERIAL	( default=off ) do the calculation in serial. Do not use MPI
LOWMEM	( default=off ) lower the memory requirements
TIMINGS	( default=off ) output information on the timings of the various parts of the calculation
SWITCH	This keyword is used if you want to employ an alternative to the continuous switching function defined above. The following provides information on the switchingfunction that are available. When this keyword is present you no longer need the NN, MM, D_0 and R_0 keywords.
MEAN	take the mean of these variables. The final value can be referenced using label.mean. You can use multiple instances of this keyword i.e. MEAN1, MEAN2, MEAN3... The corresponding values are then referenced using label.mean-1, label.mean-2, label.mean-3...
MORE_THAN	calculate the number of variables more than a certain target value. This quantity is calculated using \(\sum_i 1.0 - \sigma(s_i)\), where \(\sigma(s)\) is a switchingfunction. The final value can be referenced using label.morethan. You can use multiple instances of this keyword i.e. MORE_THAN1, MORE_THAN2, MORE_THAN3... The corresponding values are then referenced using label.morethan-1, label.morethan-2, label.morethan-3...
LESS_THAN	calculate the number of variables less than a certain target value. This quantity is calculated using \(\sum_i \sigma(s_i)\), where \(\sigma(s)\) is a switchingfunction. The final value can be referenced using label.lessthan. You can use multiple instances of this keyword i.e. LESS_THAN1, LESS_THAN2, LESS_THAN3... The corresponding values are then referenced using label.lessthan-1, label.lessthan-2, label.lessthan-3...
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= \(\beta\)) The final value can be referenced using label.min. You can use multiple instances of this keyword i.e. MIN1, MIN2, MIN3... The corresponding values are then referenced using label.min-1, label.min-2, label.min-3...
BETWEEN	calculate the number of values that are within a certain range. These quantities are calculated using kernel density estimation as described on histogrambead. The final value can be referenced using label.between. You can use multiple instances of this keyword i.e. BETWEEN1, BETWEEN2, BETWEEN3... The corresponding values are then referenced using label.between-1, label.between-2, label.between-3...
HISTOGRAM	calculate how many of the values fall in each of the bins of a histogram. This shortcut allows you to calculates NBIN quantities like BETWEEN. The final value can be referenced using label.histogram. You can use multiple instances of this keyword i.e. HISTOGRAM1, HISTOGRAM2, HISTOGRAM3... The corresponding values are then referenced using label.histogram-1, label.histogram-2, label.histogram-3...
MOMENTS	calculate the moments of the distribution of collective variables. The mth moment of a distribution is calculated using \(\frac{1}{N} \sum_{i=1}^N ( s_i - \overline{s} )^m \), where \(\overline{s}\) is the average for the distribution. The moments keyword takes a lists of integers as input or a range. Each integer is a value of \(m\). The final calculated values can be referenced using moment- \(m\). You can use the COMPONENT keyword in this action but the syntax is slightly different. If you would like the second and third moments of the third component you would use MOMENTS={COMPONENT=3 MOMENTS=2-3}. The moments would then be referred to using the labels moment-3-2 and moment-3-3. This syntax is also required if you are using numbered MOMENT keywords i.e. MOMENTS1, MOMENTS2...
LOWEST	this flag allows you to recover the lowest of these variables. The final value can be referenced using label.lowest
HIGHEST	this flag allows you to recover the highest of these variables. The final value can be referenced using label.highest