LOCAL_Q3

This is part of the crystallization module |

Calculate the local degree of order around an atoms by taking the average dot product between the \(q_3\) vector on the central atom and the \(q_3\) vector on the atoms in the first coordination sphere.

The Q3 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 Q3 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 parameteters allow to calculate variables that satisfy these requirements. LOCAL_Q3 is another variable that can be used in these sorts of calculations. The LOCAL_Q3 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:

\[ s_i = \frac{ \sum_j \sigma( r_{ij} ) \sum_{m=-3}^3 q_{3m}^{*}(i)q_{3m}(j) }{ \sum_j \sigma( r_{ij} ) } \]

where \(q_{3m}(i)\) and \(q_{3m}(j)\) are the 3rd order Steinhardt vectors calculated for atom \(i\) and atom \(j\) respectively and the asterix 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.

- 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 amongst 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 the label of the components customizable. 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. |

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. |

mean | MEAN | the mean value. The output component can be refererred 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. |

- Compulsory keywords

DATA | the labels of the action that calculates the multicolvars we are interested in |

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

MM | ( default=12 ) The m parameter of the switching function |

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

R_0 | The r_0 parameter of the switching function |

- Options

NOPBC | ( default=off ) ignore the periodic boundary conditions when calculating distances |

SERIAL | ( default=off ) do the calculation in serial. Do not parallelize |

LOWMEM | ( default=off ) lower the memory requirements |

MEAN | ( default=off ) take the mean of these variables. The final value can be referenced using |

TOL | this keyword can be used to speed up your calculation. When accumulating sums in which the individual terms are numbers inbetween zero and one it is assumed that terms less than a certain tolerance make only a small contribution to the sum. They can thus be safely ignored as can the the derivatives wrt these small quantities. |

SWITCH | This keyword is used if you want to employ an alternative to the continuous swiching 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. |

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.more_than. You can use multiple instances of this keyword i.e. MORE_THAN1, MORE_THAN2, MORE_THAN3... The corresponding values are then referenced using label.more_than-1, label.more_than-2, label.more_than-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.less_than. You can use multiple instances of this keyword i.e. LESS_THAN1, LESS_THAN2, LESS_THAN3... The corresponding values are then referenced using label.less_than-1, label.less_than-2, label.less_than-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. |

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

MOMENTS | calculate the moments of the distribution of collective variables. The \(m\)th 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- \(m\). |

- Examples

The following command calculates the average value of the LOCAL_Q3 parameter for the 64 Lennard Jones atoms in the system under study and prints this quantity to a file called colvar.

Q3 SPECIES=1-64 D_0=1.3 R_0=0.2 LABEL=q3 LOCAL_Q3 ARG=q3 SWITCH={RATIONAL D_0=1.3 R_0=0.2} MEAN LABEL=lq3 PRINT ARG=lq3.mean FILE=colvar

The following input calculates the distribution of LOCAL_Q3 parameters at any given time and outputs this information to a file.

Q3 SPECIES=1-64 D_0=1.3 R_0=0.2 LABEL=q3 LOCAL_Q3 ARG=q3 SWITCH={RATIONAL D_0=1.3 R_0=0.2} HISTOGRAM={GAUSSIAN LOWER=0.0 UPPER=1.0 NBINS=20 SMEAR=0.1} LABEL=lq3 PRINT ARG=lq3.* FILE=colvar

The following calculates the LOCAL_Q3 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

Q3 SPECIESA=1-5 SPECIESB=1-64 D_0=1.3 R_0=0.2 LABEL=q3a Q3 SPECIESA=6-64 SPECIESB=1-64 D_0=1.3 R_0=0.2 LABEL=q3b LOCAL_Q3 ARG=q3a,q3b SWITCH={RATIONAL D_0=1.3 R_0=0.2} MEAN LOWMEM LABEL=w3 PRINT ARG=w3.* FILE=colvar