Action: CONTACT_MATRIX

Module	adjmat
Description	Usage
Adjacency matrix in which two atoms are adjacent if they are within a certain cutoff.
output value	type
a matrix containing the weights for the bonds between each pair of atoms	matrix

Output components

This action can calculate the values in the following table when the associated keyword is included in the input for the action. 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.

Name	Type	Keyword	Description
w	matrix	COMPONENTS	a matrix containing the weights for the bonds between each pair of atoms
x	matrix	COMPONENTS	the projection of the bond on the x axis
y	matrix	COMPONENTS	the projection of the bond on the y axis
z	matrix	COMPONENTS	the projection of the bond on the z axis

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
GROUPA	atoms	when you are calculating the adjacency matrix between two sets of atoms this keyword is used to specify the atoms along with the keyword GROUPB
GROUPB	atoms	when you are calculating the adjacency matrix between two sets of atoms this keyword is used to specify the atoms along with the keyword GROUPA
ATOMS	atoms	the atoms for which you would like to calculate the adjacency matrix

Further details and examples

Adjacency matrix in which two atoms are adjacent if they are within a certain cutoff.

A useful tool for developing complex collective variables is the notion of the so called adjacency matrix. An adjacency matrix can be an $N \times N$ matrix in which the $i$ th, $j$ th element tells you whether or not the $i$ th and $j$ th atoms/molecules from a set of $N$ atoms/molecules are adjacent or not. Alternatively, you can calculate an adjacency matrix between a set of $N$ atoms and a second set of $M$ atoms. For this type of matrix the $i$ th, $j$ th element tells you whether the whether the $i$ th atom in the first group and the $j$ th atom in the second group are adjacent or not. The adjacency matrix in this case is thus $N \times M$ .

The simplest type of adjacency matrix is a contact matrix. The elements of a contact matrix are calculated using:

$a_{ij} = \sigma( r_{ij} )$

where $r_{ij}$ is the magnitude of the vector connecting atoms $i$ and $j$ and where $\sigma$ is a switching function. If you want to calculate a contact matrix for one group of atoms the input would look something like this:

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

c1The CONTACT_MATRIX action with label c1 calculates the following quantities: Quantity    Type    Description  
c1 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=1-7 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=2.6 NN=6 MM=12}

Alternatively, if you want to calculate the contact matrix between two groups of atoms you would use an input like following:

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

c2The CONTACT_MATRIX action with label c2 calculates the following quantities: Quantity    Type    Description  
c2 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 GROUPAwhen you are calculating the adjacency matrix between two sets of atoms this keyword is used to specify the atoms along with the keyword GROUPB=1-7 GROUPBwhen you are calculating the adjacency matrix between two sets of atoms this keyword is used to specify the atoms along with the keyword GROUPA=8-20 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=2.6 NN=6 MM=12}

Once you have calculated a contact matrix you can perform various matrix operations by using the tools in the matrixtools or clusters modules.

Coordination numbers

If a contact matrix, $\mathbf{C}$ , is multiplied from the back by a vector of ones the $i$ th element of the resulting matrix tells you the number of atoms that are within $r_c$ of atom $i$ . In other words, the coordination numbers of the atoms can be calculated from the contact matrix by doing matrix vector multiplication.

This realisation about the relationship between the contact map and the coordination number is heavily used in PLUMED. For example, to calculate and print the coordination numbers of the first 7 atoms in the system with themselves you would use an input something like this:

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

c1The CONTACT_MATRIX action with label c1 calculates the following quantities: Quantity    Type    Description  
c1 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=1-7 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=2.6 NN=6 MM=12}
onesThe ONES action with label ones calculates the following quantities: Quantity    Type    Description  
ones vector a vector of ones with the required number of elements: ONESCreate a constant vector with all elements equal to one This action is a shortcut. More details SIZEthe number of ones that you would like to create=7
# PLUMED interprets the command:
# ones: ONES SIZE=7
# as follows (Click the red comment above to revert to the short version of the input):
onesThe CONSTANT action with label ones calculates the following quantities: Quantity    Type    Description  
ones vector the constant value that was read from the plumed input: CONSTANTCreate a constant value that can be passed to actions More details NOLOG do not report all the read in scalars in the log VALUESthe numbers that are in your constant value=1,1,1,1,1,1,1
# --- End of included input --- ccThe MATRIX_VECTOR_PRODUCT action with label cc calculates the following quantities: Quantity    Type    Description  
cc vector the vector that is obtained by taking the product between the matrix and the vector that were input: MATRIX_VECTOR_PRODUCTCalculate the product of the matrix and the vector More details ARGthe label for the matrix and the vector/scalar that are being multiplied=c1,ones
PRINTPrint quantities to a file. More details ARGthe labels of the values that you would like to print to the file=cc FILEthe name of the file on which to output these quantities=colvar

Implmenting the coordination number this way is useful as there are many different ways to define whether two atoms/molecules and to construct a "contact" matrix based on the result. For example:

You could say that two molecules are connected if they are within a certain distance of each other and if they have the same orientation (see TORSIONS_MATRIX).
You could say that two water molecules are connected if they are hydrogen bonded to each other (see HBOND_MATRIX).
You could say that two atoms are connected if they are within a certain distance of each other and if they have similar values for a CV (see OUTER_PRODUCT).

When the coordination numbers is implemented in the way described above (by doing the matrix-vector multiplication) you have the flexibility to define the contact matrix that is used in the multiplication in whatever way you choose. In other words, this implementation of the coordination number is much more flexible. For example, suppose you want to calculate the number of atoms that have a coordination is greater than 3.0. You can do this with PLUMED using the following input:

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

# Calculate the contact matrix for the first seven atoms in the system
c1The CONTACT_MATRIX action with label c1 calculates the following quantities: Quantity    Type    Description  
c1 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=1-7 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=2.6 NN=6 MM=12}
# Calculate the coordination numbers for the first seven atoms in the system
onesThe ONES action with label ones calculates the following quantities: Quantity    Type    Description  
ones vector a vector of ones with the required number of elements: ONESCreate a constant vector with all elements equal to one This action is a shortcut. More details SIZEthe number of ones that you would like to create=7
# PLUMED interprets the command:
# ones: ONES SIZE=7
# as follows (Click the red comment above to revert to the short version of the input):
onesThe CONSTANT action with label ones calculates the following quantities: Quantity    Type    Description  
ones vector the constant value that was read from the plumed input: CONSTANTCreate a constant value that can be passed to actions More details NOLOG do not report all the read in scalars in the log VALUESthe numbers that are in your constant value=1,1,1,1,1,1,1
# --- End of included input --- ccThe MATRIX_VECTOR_PRODUCT action with label cc calculates the following quantities: Quantity    Type    Description  
cc vector the vector that is obtained by taking the product between the matrix and the vector that were input: MATRIX_VECTOR_PRODUCTCalculate the product of the matrix and the vector More details ARGthe label for the matrix and the vector/scalar that are being multiplied=c1,ones
# Set the ith element of the vector mtc equal to one if the coordination number of atom i is greater than 3.
mtcThe MORE_THAN action with label mtc calculates the following quantities: Quantity    Type    Description  
mtc vector the vector obtained by doing an element-wise application of a function that is one if the if the input is more than a threshold to the input vectors: MORE_THANUse a switching function to determine how many of the input variables are more than a certain cutoff. More details ARGthe values input to this function=cc SWITCHThis keyword is used if you want to employ an alternative to the continuous swiching function defined above={RATIONAL D_0=3 R_0=1}
# Calculate the number of atoms with a coordination number greater than 3.
sThe SUM action with label s calculates the following quantities: Quantity    Type    Description  
s scalar the sum of all the elements in the input vector: SUMCalculate the sum of the arguments More details ARGthe values input to this function=mtc PERIODICif the output of your function is periodic then you should specify the periodicity of the function=NO
PRINTPrint quantities to a file. More details ARGthe labels of the values that you would like to print to the file=s FILEthe name of the file on which to output these quantities=colvar

Alternatively, consider the CV that was used to study perovskite nucleation in this paper. The CV here measures the number of methylamonium molecules that are attached to at least 5 other methylamoniusm molecules, at least 7 lead atoms and at least 11 ionide ions. We can calculate something akin to this CV and apply a restraint on the resulting quantity by using the following input file:

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

# Lead ions
PbThe GROUP action with label Pb calculates the following quantities: Quantity    Type    Description  
Pb atoms indices of atoms specified in GROUP: GROUPDefine a group of atoms so that a particular list of atoms can be referenced with a single label in definitions of CVs or virtual atoms. More details ATOMSthe numerical indexes for the set of atoms in the group=1-64
# Iodide atoms
IThe GROUP action with label I calculates the following quantities: Quantity    Type    Description  
I atoms indices of atoms specified in GROUP: GROUPDefine a group of atoms so that a particular list of atoms can be referenced with a single label in definitions of CVs or virtual atoms. More details ATOMSthe numerical indexes for the set of atoms in the group=65-256
# Methylamonium "atoms" -- in the real CV these are centers of mass rather than single atoms
cnThe GROUP action with label cn calculates the following quantities: Quantity    Type    Description  
cn atoms indices of atoms specified in GROUP: GROUPDefine a group of atoms so that a particular list of atoms can be referenced with a single label in definitions of CVs or virtual atoms. More details ATOMSthe numerical indexes for the set of atoms in the group=257-320

ones64The ONES action with label ones64 calculates the following quantities: Quantity    Type    Description  
ones64 vector a vector of ones with the required number of elements: ONESCreate a constant vector with all elements equal to one This action is a shortcut. More details SIZEthe number of ones that you would like to create=64
# PLUMED interprets the command:
# ones64: ONES SIZE=64
# as follows (Click the red comment above to revert to the short version of the input):
ones64The CONSTANT action with label ones64 calculates the following quantities: Quantity    Type    Description  
ones64 vector the constant value that was read from the plumed input: CONSTANTCreate a constant value that can be passed to actions More details NOLOG do not report all the read in scalars in the log VALUESthe numbers that are in your constant value=1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
# --- End of included input --- # Contact matrix that determines if methylamonium molecules are within 8 A of each other
cm_cncnThe CONTACT_MATRIX action with label cm_cncn calculates the following quantities: Quantity    Type    Description  
cm_cncn 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=cn 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=0.8}
# Coordination number of methylamounium with methylamonium
cc_cncnThe MATRIX_VECTOR_PRODUCT action with label cc_cncn calculates the following quantities: Quantity    Type    Description  
cc_cncn vector the vector that is obtained by taking the product between the matrix and the vector that were input: MATRIX_VECTOR_PRODUCTCalculate the product of the matrix and the vector More details ARGthe label for the matrix and the vector/scalar that are being multiplied=cm_cncn,ones64
# Vector with elements that are one if coordiantion of methylamonium with methylamonium >5
mt_cncnThe MORE_THAN action with label mt_cncn calculates the following quantities: Quantity    Type    Description  
mt_cncn vector the vector obtained by doing an element-wise application of a function that is one if the if the input is more than a threshold to the input vectors: MORE_THANUse a switching function to determine how many of the input variables are more than a certain cutoff. More details ARGthe values input to this function=cc_cncn SWITCHThis keyword is used if you want to employ an alternative to the continuous swiching function defined above={RATIONAL R_0=5 NN=12 MM=24}

# Contact matrix that determines if methylamoinium moleulcule and Pb atom are within 7.5 A of each other
cm_cnpbThe CONTACT_MATRIX action with label cm_cnpb calculates the following quantities: Quantity    Type    Description  
cm_cnpb 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 GROUPAwhen you are calculating the adjacency matrix between two sets of atoms this keyword is used to specify the atoms along with the keyword GROUPB=cn GROUPBwhen you are calculating the adjacency matrix between two sets of atoms this keyword is used to specify the atoms along with the keyword GROUPA=Pb 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=0.75}
# Coordination number of methylamonium with Pb
cc_cnpbThe MATRIX_VECTOR_PRODUCT action with label cc_cnpb calculates the following quantities: Quantity    Type    Description  
cc_cnpb vector the vector that is obtained by taking the product between the matrix and the vector that were input: MATRIX_VECTOR_PRODUCTCalculate the product of the matrix and the vector More details ARGthe label for the matrix and the vector/scalar that are being multiplied=cm_cnpb,ones64
# Vector with elements that are one if coordination of methylamounium with lead is >7
mt_cnpbThe MORE_THAN action with label mt_cnpb calculates the following quantities: Quantity    Type    Description  
mt_cnpb vector the vector obtained by doing an element-wise application of a function that is one if the if the input is more than a threshold to the input vectors: MORE_THANUse a switching function to determine how many of the input variables are more than a certain cutoff. More details ARGthe values input to this function=cc_cnpb SWITCHThis keyword is used if you want to employ an alternative to the continuous swiching function defined above={RATIONAL R_0=7 NN=12 MM=24}

ones192The ONES action with label ones192 calculates the following quantities: Quantity    Type    Description  
ones192 vector a vector of ones with the required number of elements: ONESCreate a constant vector with all elements equal to one This action is a shortcut. More details SIZEthe number of ones that you would like to create=192
# PLUMED interprets the command:
# ones192: ONES SIZE=192
# as follows (Click the red comment above to revert to the short version of the input):
ones192The CONSTANT action with label ones192 calculates the following quantities: Quantity    Type    Description  
ones192 vector the constant value that was read from the plumed input: CONSTANTCreate a constant value that can be passed to actions More details NOLOG do not report all the read in scalars in the log VALUESthe numbers that are in your constant value=1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
# --- End of included input --- # Contact matrix that determines if methylamoinium moleulcule and I atom are within 6.5 A of each other
cm_cnIThe CONTACT_MATRIX action with label cm_cnI calculates the following quantities: Quantity    Type    Description  
cm_cnI 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 GROUPAwhen you are calculating the adjacency matrix between two sets of atoms this keyword is used to specify the atoms along with the keyword GROUPB=cn GROUPBwhen you are calculating the adjacency matrix between two sets of atoms this keyword is used to specify the atoms along with the keyword GROUPA=I 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=0.65}
# Coordination number of methylamonium with I
cc_cnIThe MATRIX_VECTOR_PRODUCT action with label cc_cnI calculates the following quantities: Quantity    Type    Description  
cc_cnI vector the vector that is obtained by taking the product between the matrix and the vector that were input: MATRIX_VECTOR_PRODUCTCalculate the product of the matrix and the vector More details ARGthe label for the matrix and the vector/scalar that are being multiplied=cm_cnI,ones192
# Vector with elements that are one if coordination of methylamounium with lead is >11
mt_cnIThe MORE_THAN action with label mt_cnI calculates the following quantities: Quantity    Type    Description  
mt_cnI vector the vector obtained by doing an element-wise application of a function that is one if the if the input is more than a threshold to the input vectors: MORE_THANUse a switching function to determine how many of the input variables are more than a certain cutoff. More details ARGthe values input to this function=cc_cnI SWITCHThis keyword is used if you want to employ an alternative to the continuous swiching function defined above={RATIONAL R_0=11 NN=12 MM=24}

# Element wise product of these three input vectors.
# mm[i]==1 if coordination number of corrsponding methylamounium with methylamonium is >5
# and if coordination of methylamounium with Pb is >7 and if coordination of methylamounium with I > 11
mmThe CUSTOM action with label mm calculates the following quantities: Quantity    Type    Description  
mm vector the vector obtained by doing an element-wise application of an arbitrary function to the input vectors: CUSTOMCalculate a combination of variables using a custom expression. More details ARGthe values input to this function=mt_cncn,mt_cnpb,mt_cnI FUNCthe function you wish to evaluate=x*y*z PERIODICif the output of your function is periodic then you should specify the periodicity of the function=NO

# Sum of coordination numbers and thus equal to number of methylamoniums with desired coordination numbers
ffThe SUM action with label ff calculates the following quantities: Quantity    Type    Description  
ff scalar the sum of all the elements in the input vector: SUMCalculate the sum of the arguments More details ARGthe values input to this function=mm PERIODICif the output of your function is periodic then you should specify the periodicity of the function=NO

rrThe RESTRAINT action with label rr calculates the following quantities: Quantity    Type    Description  
rr.bias scalar the instantaneous value of the bias potential
rr.force2 scalar the instantaneous value of the squared force due to this bias potential: RESTRAINTAdds harmonic and/or linear restraints on one or more variables. This action has hidden defaults. More details ARGthe values the harmonic restraint acts upon=ff ATthe position of the restraint=62 KAPPA specifies that the restraint is harmonic and what the values of the force constants on each of the variables are=10
rr: RESTRAINTAdds harmonic and/or linear restraints on one or more variables. This action uses the defaults shown here. More details ARGthe values the harmonic restraint acts upon=ff ATthe position of the restraint=62 KAPPA specifies that the restraint is harmonic and what the values of the force constants on each of the variables are=10  SLOPE specifies that the restraint is linear and what the values of the force constants on each of the variables are=0.0

COMPONENTS flag

If you add the flag COMPONENTS to the input as shown below:

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

c4The CONTACT_MATRIX action with label c4 calculates the following quantities: Quantity    Type    Description  
c4.w matrix a matrix containing the weights for the bonds between each pair of atoms
c4.x matrix the projection of the bond on the x axis
c4.y matrix the projection of the bond on the y axis
c4.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=1-7 COMPONENTS also calculate the components of the vectors connecting the atoms in the contact matrix 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=2.6 NN=6 MM=12}

then four matrices with the labels c4.w, c4.x, c4.y and c4.z are output by the action. The matrix with the label c4.w is the adjacency matrix that would be output if you had not added the COMPONENTS flag. The $i,j$ component of the matrices c4.x, c4.y and c4.z contain the $x$ , $y$ and $z$ components of the vector connecting atoms $i$ and $j$ . Importantly, however, the components of these vectors are only stored in c4.x, c4.y and c4.z if the elements of c4.w are non-zero.

Optimisation details

Adjacency matrices are sparse. Each atom is only be connected to a small number of neighbours and the vast majority of the elements of the contact matrix are thus zero. To reduce the amount of memory that PLUMED requires PLUMED uses sparse matrix storage. Consequently, whenever you calculate and store a contact matrix only the elements of the matrix that are non-zero are stored. The same thing holds for the additional values that are created when you use the COMPONENTS flag. The components of the vectors connecting atoms are only stored when the elements of c4.w are non-zero.

We can also use the sparsity of the adjacency matrix to make the time required to compute a contact matrix scale linearly rather than quadratically with the number of atoms. Element $i,j$ of the contact matrix is only non-zero if two atoms are within a cutoff, $r_c$ . We can determine that many pairs of atoms are further appart than $r_c$ without computing the distance between these atoms by using divide and conquer strategies such as linked lists and neighbour lists. To turn on these features you need to set the D_MAX parameter in the switching functions. The value you pass to the D_MAX keyword is used as the cutoff in the link cell algorithm.

References

More information about how this action can be used is available in the following articles: - G. A. Tribello, F. Giberti, G. C. Sosso, M. Salvalaglio, M. Parrinello, Analyzing and Driving Cluster Formation in Atomistic Simulations. Journal of Chemical Theory and Computation. 13, 1317–1327 (2017)

Syntax

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

Keyword	Type	Default	Description
GROUPA	input	none	when you are calculating the adjacency matrix between two sets of atoms this keyword is used to specify the atoms along with the keyword GROUPB
GROUPB	input	none	when you are calculating the adjacency matrix between two sets of atoms this keyword is used to specify the atoms along with the keyword GROUPA
ATOMS	input	none	the atoms for which you would like to calculate the adjacency matrix
NL_CUTOFF	compulsory	0.0	The cutoff for the neighbor list
NL_STRIDE	compulsory	1	The frequency with which we are updating the atoms in the neighbor list
NN	compulsory	6	The n parameter of the switching function
MM	compulsory	0	The m parameter of the switching function; 0 implies 2*NN
D_0	compulsory	0.0	The d_0 parameter of the switching function
R_0	compulsory	none	The r_0 parameter of the switching function
SERIAL	optional	false	do the calculation in serial
COMPONENTS	optional	false	also calculate the components of the vectors connecting the atoms in the contact matrix
NOPBC	optional	false	don't use pbc
GROUP	optional	not used	specifies the list of atoms that should be assumed indistinguishable
SWITCH	optional	not used	the 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.

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Quantity	Type	Description
c1	matrix	a matrix containing the weights for the bonds between each pair of atoms