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PLUMED manual

Shortcut: GYRATION_TENSOR

Shortcut: GYRATION_TENSOR

Module	colvar
Description	Usage
Calculate the gyration tensor using a user specified vector of weights
output value	type
the gyration tensor that was computed from the weights	matrix

Details and examples

Calculate the gyration tensor using a user specified vector of weights

The elements of the $3 \times 3$ gyration tensor are defined as:

$G_{\alpha\beta} = \frac{\sum_i^{n} w_i (\alpha_i-\alpha_{\rm COM})(\beta_i - \beta_{\rm COM})}{\sum_i^{n} w_i}$

where $\alpha_i$ and $\beta_i$ can be the $x$ , $y$ or $z$ coordinates of atom $i$ and $\alpha_{\rm COM}$ and $\beta_{\rm COM}$ can be the $x$ , $y$ or $z$ components of the center, which is calculated using:

${r}_{\rm COM}=\frac{\sum_i^{n} {r}_i\ w_i }{\sum_i^{n} w_i}$

The following example shows how you can calculate and print the gyration tensor from the positions of the fist 10 atoms using PLUMED

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

gThe GYRATION_TENSOR action with label g calculates the following quantities: Quantity    Type    Description  
g matrix the gyration tensor that was computed from the weights: GYRATION_TENSORCalculate the gyration tensor using a user specified vector of weights This action is a shortcut. More details ATOMSthe group of atoms that you are calculating the Gyration Tensor for=1-10
# PLUMED interprets the command:
# g: GYRATION_TENSOR ATOMS=1-10
# as follows (Click the red comment above to revert to the short version of the input):
g_centThe CENTER_FAST action with label g_cent calculates the following quantities: Quantity    Type    Description  
g_cent atoms virtual atom calculated by CENTER_FAST action: CENTERCalculate the center for a group of atoms, with arbitrary weights. More details ATOMSthe group of atoms that appear in the definition of this center=1,2,3,4,5,6,7,8,9,10
g_wThe CONSTANT action with label g_w calculates the following quantities: Quantity    Type    Description  
g_w vector the constant value that was read from the plumed input: ONESCreate a constant vector with all elements equal to one More details SIZEthe number of ones that you would like to create=10
g_distsThe DISTANCE action with label g_dists calculates the following quantities: Quantity    Type    Description  
g_dists.x vector the x-component of the vector connecting the two atoms
g_dists.y vector the y-component of the vector connecting the two atoms
g_dists.z vector the z-component of the vector connecting the two atoms: DISTANCECalculate the distance/s between pairs of atoms. More details COMPONENTS calculate the x, y and z components of the distance separately and store them as label ATOMS1the pair of atom that we are calculating the distance between=g_cent,1 ATOMS2the pair of atom that we are calculating the distance between=g_cent,2 ATOMS3the pair of atom that we are calculating the distance between=g_cent,3 ATOMS4the pair of atom that we are calculating the distance between=g_cent,4 ATOMS5the pair of atom that we are calculating the distance between=g_cent,5     # Action input conctinues with 5 further ATOMSn keywords, 
gThe MATRIX_PRODUCT action with label g calculates the following quantities: Quantity    Type    Description  
g matrix the product of the two input matrices: COVARIANCE_MATRIXCalculate a covariance matix More details ARGthe vectors of data from which we are calculating the covariance=g_dists.x,g_dists.y,g_dists.z WEIGHTSthis keyword takes the label of an action that calculates a vector of values=g_w
# --- End of included input --- PRINTPrint quantities to a file. More details ARGthe labels of the values that you would like to print to the file=g FILEthe name of the file on which to output these quantities=colvar

In this example input the weights of all the atoms are set equal to one. The 9 elements of the resulting gyration matrix will be output to the file colvar here. If you want the weights of the atoms to be set equal to the masses you can use either of the following equivalent commands:

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

gThe GYRATION_TENSOR action with label g calculates the following quantities: Quantity    Type    Description  
g matrix the gyration tensor that was computed from the weights: GYRATION_TENSORCalculate the gyration tensor using a user specified vector of weights This action is a shortcut. More details ATOMSthe group of atoms that you are calculating the Gyration Tensor for=1-10 MASS calculate the center of mass
# PLUMED interprets the command:
# g: GYRATION_TENSOR ATOMS=1-10 MASS
# as follows (Click the red comment above to revert to the short version of the input):
g_centThe CENTER_FAST action with label g_cent calculates the following quantities: Quantity    Type    Description  
g_cent atoms virtual atom calculated by CENTER_FAST action: CENTERCalculate the center for a group of atoms, with arbitrary weights. More details ATOMSthe group of atoms that appear in the definition of this center=1,2,3,4,5,6,7,8,9,10 WEIGHTSwhat weights should be used when calculating the center=@Masses
g_mThe MASS action with label g_m calculates the following quantities: Quantity    Type    Description  
g_m vector the MASS of the atom/s: MASSGet the mass of one or multiple atoms More details ATOMSthe atom numbers that you would like to store the masses and charges of=1,2,3,4,5,6,7,8,9,10
g_distsThe DISTANCE action with label g_dists calculates the following quantities: Quantity    Type    Description  
g_dists.x vector the x-component of the vector connecting the two atoms
g_dists.y vector the y-component of the vector connecting the two atoms
g_dists.z vector the z-component of the vector connecting the two atoms: DISTANCECalculate the distance/s between pairs of atoms. More details COMPONENTS calculate the x, y and z components of the distance separately and store them as label ATOMS1the pair of atom that we are calculating the distance between=g_cent,1 ATOMS2the pair of atom that we are calculating the distance between=g_cent,2 ATOMS3the pair of atom that we are calculating the distance between=g_cent,3 ATOMS4the pair of atom that we are calculating the distance between=g_cent,4 ATOMS5the pair of atom that we are calculating the distance between=g_cent,5     # Action input conctinues with 5 further ATOMSn keywords, 
gThe MATRIX_PRODUCT action with label g calculates the following quantities: Quantity    Type    Description  
g matrix the product of the two input matrices: COVARIANCE_MATRIXCalculate a covariance matix More details ARGthe vectors of data from which we are calculating the covariance=g_dists.x,g_dists.y,g_dists.z WEIGHTSthis keyword takes the label of an action that calculates a vector of values=g_m
# --- End of included input --- g2The GYRATION_TENSOR action with label g2 calculates the following quantities: Quantity    Type    Description  
g2 matrix the gyration tensor that was computed from the weights: GYRATION_TENSORCalculate the gyration tensor using a user specified vector of weights This action is a shortcut. More details ATOMSthe group of atoms that you are calculating the Gyration Tensor for=1-10 WEIGHTSwhat weights should be used when calculating the center=@Masses
# PLUMED interprets the command:
# g2: GYRATION_TENSOR ATOMS=1-10 WEIGHTS=@Masses
# as follows (Click the red comment above to revert to the short version of the input):
g2_centThe CENTER_FAST action with label g2_cent calculates the following quantities: Quantity    Type    Description  
g2_cent atoms virtual atom calculated by CENTER_FAST action: CENTERCalculate the center for a group of atoms, with arbitrary weights. More details ATOMSthe group of atoms that appear in the definition of this center=1,2,3,4,5,6,7,8,9,10 WEIGHTSwhat weights should be used when calculating the center=@Masses
g2_mThe MASS action with label g2_m calculates the following quantities: Quantity    Type    Description  
g2_m vector the MASS of the atom/s: MASSGet the mass of one or multiple atoms More details ATOMSthe atom numbers that you would like to store the masses and charges of=1,2,3,4,5,6,7,8,9,10
g2_distsThe DISTANCE action with label g2_dists calculates the following quantities: Quantity    Type    Description  
g2_dists.x vector the x-component of the vector connecting the two atoms
g2_dists.y vector the y-component of the vector connecting the two atoms
g2_dists.z vector the z-component of the vector connecting the two atoms: DISTANCECalculate the distance/s between pairs of atoms. More details COMPONENTS calculate the x, y and z components of the distance separately and store them as label ATOMS1the pair of atom that we are calculating the distance between=g2_cent,1 ATOMS2the pair of atom that we are calculating the distance between=g2_cent,2 ATOMS3the pair of atom that we are calculating the distance between=g2_cent,3 ATOMS4the pair of atom that we are calculating the distance between=g2_cent,4 ATOMS5the pair of atom that we are calculating the distance between=g2_cent,5     # Action input conctinues with 5 further ATOMSn keywords, 
g2The MATRIX_PRODUCT action with label g2 calculates the following quantities: Quantity    Type    Description  
g2 matrix the product of the two input matrices: COVARIANCE_MATRIXCalculate a covariance matix More details ARGthe vectors of data from which we are calculating the covariance=g2_dists.x,g2_dists.y,g2_dists.z WEIGHTSthis keyword takes the label of an action that calculates a vector of values=g2_m
# --- End of included input --- g3The GYRATION_TENSOR action with label g3 calculates the following quantities: Quantity    Type    Description  
g3 matrix the gyration tensor that was computed from the weights: GYRATION_TENSORCalculate the gyration tensor using a user specified vector of weights This action is a shortcut. More details ATOMSthe group of atoms that you are calculating the Gyration Tensor for=1-10 MASS_WEIGHTED set the masses of all the atoms equal to one
# PLUMED interprets the command:
# g3: GYRATION_TENSOR ATOMS=1-10 MASS_WEIGHTED
# as follows (Click the red comment above to revert to the short version of the input):
g3_centThe CENTER_FAST action with label g3_cent calculates the following quantities: Quantity    Type    Description  
g3_cent atoms virtual atom calculated by CENTER_FAST action: CENTERCalculate the center for a group of atoms, with arbitrary weights. More details ATOMSthe group of atoms that appear in the definition of this center=1,2,3,4,5,6,7,8,9,10 WEIGHTSwhat weights should be used when calculating the center=@Masses
g3_mThe MASS action with label g3_m calculates the following quantities: Quantity    Type    Description  
g3_m vector the MASS of the atom/s: MASSGet the mass of one or multiple atoms More details ATOMSthe atom numbers that you would like to store the masses and charges of=1,2,3,4,5,6,7,8,9,10
g3_distsThe DISTANCE action with label g3_dists calculates the following quantities: Quantity    Type    Description  
g3_dists.x vector the x-component of the vector connecting the two atoms
g3_dists.y vector the y-component of the vector connecting the two atoms
g3_dists.z vector the z-component of the vector connecting the two atoms: DISTANCECalculate the distance/s between pairs of atoms. More details COMPONENTS calculate the x, y and z components of the distance separately and store them as label ATOMS1the pair of atom that we are calculating the distance between=g3_cent,1 ATOMS2the pair of atom that we are calculating the distance between=g3_cent,2 ATOMS3the pair of atom that we are calculating the distance between=g3_cent,3 ATOMS4the pair of atom that we are calculating the distance between=g3_cent,4 ATOMS5the pair of atom that we are calculating the distance between=g3_cent,5     # Action input conctinues with 5 further ATOMSn keywords, 
g3The MATRIX_PRODUCT action with label g3 calculates the following quantities: Quantity    Type    Description  
g3 matrix the product of the two input matrices: COVARIANCE_MATRIXCalculate a covariance matix More details ARGthe vectors of data from which we are calculating the covariance=g3_dists.x,g3_dists.y,g3_dists.z WEIGHTSthis keyword takes the label of an action that calculates a vector of values=g3_m
# --- End of included input --- PRINTPrint quantities to a file. More details ARGthe labels of the values that you would like to print to the file=g,g2,g3 FILEthe name of the file on which to output these quantities=colvar

This input above should output three identical $3\times 3$ matrices.

If you want to use an arbitrary set of weights for the atoms you can use the following syntax.

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

cThe CONSTANT action with label c calculates the following quantities: Quantity    Type    Description  
c vector the constant value that was read from the plumed input: CONSTANTCreate a constant value that can be passed to actions This action has hidden defaults. More details VALUESthe numbers that are in your constant value=1,3,2,4
c: CONSTANTCreate a constant value that can be passed to actions This action uses the defaults shown here. More details VALUESthe numbers that are in your constant value=1,3,2,4  NROWS the number of rows in your input matrix=0 NCOLS the number of columns in your matrix=0
gThe GYRATION_TENSOR action with label g calculates the following quantities: Quantity    Type    Description  
g matrix the gyration tensor that was computed from the weights: GYRATION_TENSORCalculate the gyration tensor using a user specified vector of weights This action is a shortcut. More details ATOMSthe group of atoms that you are calculating the Gyration Tensor for=1-4 WEIGHTSwhat weights should be used when calculating the center=c UNORMALIZED do not divide by the sum of the weights
# PLUMED interprets the command:
# g: GYRATION_TENSOR ATOMS=1-4 WEIGHTS=c UNORMALIZED
# as follows (Click the red comment above to revert to the short version of the input):
g_centThe ARGS2VATOM action with label g_cent calculates the following quantities: Quantity    Type    Description  
g_cent.x scalar the x coordinate of the virtual atom
g_cent.y scalar the y coordinate of the virtual atom
g_cent.z scalar the z coordinate of the virtual atom
g_cent.mass scalar the mass of the virtual atom
g_cent.charge scalar the charge of the virtual atom: CENTERCalculate the center for a group of atoms, with arbitrary weights. More details ATOMSthe group of atoms that appear in the definition of this center=1,2,3,4 WEIGHTSwhat weights should be used when calculating the center=c
g_distsThe DISTANCE action with label g_dists calculates the following quantities: Quantity    Type    Description  
g_dists.x vector the x-component of the vector connecting the two atoms
g_dists.y vector the y-component of the vector connecting the two atoms
g_dists.z vector the z-component of the vector connecting the two atoms: DISTANCECalculate the distance/s between pairs of atoms. More details COMPONENTS calculate the x, y and z components of the distance separately and store them as label ATOMS1the pair of atom that we are calculating the distance between=g_cent,1 ATOMS2the pair of atom that we are calculating the distance between=g_cent,2 ATOMS3the pair of atom that we are calculating the distance between=g_cent,3 ATOMS4the pair of atom that we are calculating the distance between=g_cent,4
gThe MATRIX_PRODUCT action with label g calculates the following quantities: Quantity    Type    Description  
g matrix the product of the two input matrices: COVARIANCE_MATRIXCalculate a covariance matix More details ARGthe vectors of data from which we are calculating the covariance=g_dists.x,g_dists.y,g_dists.z WEIGHTSthis keyword takes the label of an action that calculates a vector of values=c UNORMALIZED do not divide by the sum of the weights
# --- End of included input --- PRINTPrint quantities to a file. More details ARGthe labels of the values that you would like to print to the file=g FILEthe name of the file on which to output these quantities=colvar

Although the input weights are CONSTANT here that is not a requirement. You can use a vector of weights from any action that outputs a vector in the input for the WEIGHTS keyword here. Notice, also that the denominator in the expression for $G_{\alpha\beta}$ is set equal to one rather than the sum of the weights as we used the UNORMALIZED flag.

Similar functionality to the functionality in the examples above is used in the GYRATION shortcut. There is, however, no fast version of the GYRATION_TENSOR command in the way that there is a fast version of the GYRATION command that is used when the weights are all one or when the masses are used as the weights.

A note on periodic boundary conditions

Calculating the gyration tensor is normally used to determine the shape of a molecule so all the specified atoms would normally be part of the same molecule. When computing gyration tensors it is important to ensure that the periodic boundaries are calculated correctly. There are two ways that you can manage periodic boundary conditions when using this action. The first and simplest is to reconstruct the molecule similarly to the way that WHOLEMOLECULES operates. This reconstruction of molecules has been done automatically since PLUMED 2.2. If for some reason you want to turn it off you can use the NOPBC flag as shown below:

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

gThe GYRATION_TENSOR action with label g calculates the following quantities: Quantity    Type    Description  
g matrix the gyration tensor that was computed from the weights: GYRATION_TENSORCalculate the gyration tensor using a user specified vector of weights This action is a shortcut. More details ATOMSthe group of atoms that you are calculating the Gyration Tensor for=1-5 NOPBC ignore the periodic boundary conditions when calculating distances
# PLUMED interprets the command:
# g: GYRATION_TENSOR ATOMS=1-5 NOPBC
# as follows (Click the red comment above to revert to the short version of the input):
g_centThe CENTER_FAST action with label g_cent calculates the following quantities: Quantity    Type    Description  
g_cent atoms virtual atom calculated by CENTER_FAST action: CENTERCalculate the center for a group of atoms, with arbitrary weights. More details ATOMSthe group of atoms that appear in the definition of this center=1,2,3,4,5 NOPBC ignore the periodic boundary conditions when calculating distances
g_wThe CONSTANT action with label g_w calculates the following quantities: Quantity    Type    Description  
g_w vector the constant value that was read from the plumed input: ONESCreate a constant vector with all elements equal to one More details SIZEthe number of ones that you would like to create=5
g_distsThe DISTANCE action with label g_dists calculates the following quantities: Quantity    Type    Description  
g_dists.x vector the x-component of the vector connecting the two atoms
g_dists.y vector the y-component of the vector connecting the two atoms
g_dists.z vector the z-component of the vector connecting the two atoms: DISTANCECalculate the distance/s between pairs of atoms. More details COMPONENTS calculate the x, y and z components of the distance separately and store them as label NOPBC ignore the periodic boundary conditions when calculating distances ATOMS1the pair of atom that we are calculating the distance between=g_cent,1 ATOMS2the pair of atom that we are calculating the distance between=g_cent,2 ATOMS3the pair of atom that we are calculating the distance between=g_cent,3 ATOMS4the pair of atom that we are calculating the distance between=g_cent,4 ATOMS5the pair of atom that we are calculating the distance between=g_cent,5
gThe MATRIX_PRODUCT action with label g calculates the following quantities: Quantity    Type    Description  
g matrix the product of the two input matrices: COVARIANCE_MATRIXCalculate a covariance matix More details ARGthe vectors of data from which we are calculating the covariance=g_dists.x,g_dists.y,g_dists.z WEIGHTSthis keyword takes the label of an action that calculates a vector of values=g_w
# --- End of included input --- PRINTPrint quantities to a file. More details ARGthe labels of the values that you would like to print to the file=g FILEthe name of the file on which to output these quantities=colvar

An alternative approach to handling PBC is to use the PHASES keyword. This keyword instructs PLUMED to use the PHASES option when computing the position of the center using the CENTER command. Distances of atoms from this center are then computed using PBC as usual. The example shown below shows you how to use this option

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

gThe GYRATION_TENSOR action with label g calculates the following quantities: Quantity    Type    Description  
g matrix the gyration tensor that was computed from the weights: GYRATION_TENSORCalculate the gyration tensor using a user specified vector of weights This action is a shortcut. More details ATOMSthe group of atoms that you are calculating the Gyration Tensor for=1-5 PHASES use trigonometric phases when computing position of center of mass
# PLUMED interprets the command:
# g: GYRATION_TENSOR ATOMS=1-5 PHASES
# as follows (Click the red comment above to revert to the short version of the input):
g_centThe CENTER_FAST action with label g_cent calculates the following quantities: Quantity    Type    Description  
g_cent atoms virtual atom calculated by CENTER_FAST action: CENTERCalculate the center for a group of atoms, with arbitrary weights. More details ATOMSthe group of atoms that appear in the definition of this center=1,2,3,4,5 PHASES use trigonometric phases when computing position of center
g_wThe CONSTANT action with label g_w calculates the following quantities: Quantity    Type    Description  
g_w vector the constant value that was read from the plumed input: ONESCreate a constant vector with all elements equal to one More details SIZEthe number of ones that you would like to create=5
g_distsThe DISTANCE action with label g_dists calculates the following quantities: Quantity    Type    Description  
g_dists.x vector the x-component of the vector connecting the two atoms
g_dists.y vector the y-component of the vector connecting the two atoms
g_dists.z vector the z-component of the vector connecting the two atoms: DISTANCECalculate the distance/s between pairs of atoms. More details COMPONENTS calculate the x, y and z components of the distance separately and store them as label ATOMS1the pair of atom that we are calculating the distance between=g_cent,1 ATOMS2the pair of atom that we are calculating the distance between=g_cent,2 ATOMS3the pair of atom that we are calculating the distance between=g_cent,3 ATOMS4the pair of atom that we are calculating the distance between=g_cent,4 ATOMS5the pair of atom that we are calculating the distance between=g_cent,5
gThe MATRIX_PRODUCT action with label g calculates the following quantities: Quantity    Type    Description  
g matrix the product of the two input matrices: COVARIANCE_MATRIXCalculate a covariance matix More details ARGthe vectors of data from which we are calculating the covariance=g_dists.x,g_dists.y,g_dists.z WEIGHTSthis keyword takes the label of an action that calculates a vector of values=g_w
# --- End of included input --- PRINTPrint quantities to a file. More details ARGthe labels of the values that you would like to print to the file=g FILEthe name of the file on which to output these quantities=colvar

Input

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

Keyword	Type	Description
ATOMS	atoms	the group of atoms that you are calculating the Gyration Tensor for

Full list of keywords

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

Keyword	Type	Default	Description
ATOMS	input	none	the group of atoms that you are calculating the Gyration Tensor for
NOPBC	optional	false	ignore the periodic boundary conditions when calculating distances
WEIGHTS	optional	not used	what weights should be used when calculating the center
PHASES	optional	false	use trigonometric phases when computing position of center of mass
MASS	optional	false	calculate the center of mass
MASS_WEIGHTED	optional	false	set the masses of all the atoms equal to one
UNORMALIZED	optional	false	do not divide by the sum of the weights

References

More information about how this action can be used is available in the following articles:

J. Vymětal, J. Vondrášek, Gyration- and Inertia-Tensor-Based Collective Coordinates for Metadynamics. Application on the Conformational Behavior of Polyalanine Peptides and Trp-Cage Folding. The Journal of Physical Chemistry A. 115, 11455–11465 (2011)