Coordination Numbers

Count of covalent bonds for an atom in a molecule.

Introduction

The concept of atomic coordination numbers (CNs) has been introduced by Grimme and co-workers. CNs represent hybridisation conditions for atoms inside a molecular environment that agrees quite well with chemical intuition. Within the kallisto program CNs are calculated in a pairwise sum that incorporates atomic covalent radii as introduced by Pyykkö. Furthermore, the differences in atomic electronegativities have been introduced for each pair as shown in its definition below.

$$CN_i = \sum\limits_i^N\sum\limits_{j \ne i} \frac{\delta_{AB}^{EN}}{2}\left( 1 + \text{erf}\left(-k_0\left(\frac{R_{AB}-R_{AB}^{cov}}{R_{AB}}\right)\right) \right) \\ \text{where} \quad \delta_{AB}^{EN} = \left( k_1 \exp\left(|EN_A - EN_B| + k_2 \right)^2\right)/k_3$$

The parameters used within the above definition are as follows:

Parameter	Value
k0	4.1
k1	19.1
k2	254.6
k3	254.6

Those parameters have been obtained by a least-squared fit to Wiberg bond orders of different di-atomic molecules. Here, Pauling electronegativities (EN), the internuclear distance of pair AB (RAB), and covalent atomic radii (RcovAB = RcovA + RcovB) are used.

Define the Subcommands

cns

> kallisto cns options arguments

options

arguments

Applications

To calculate coordination numbers, I call the subcommand cns

> kallisto cns alanine-glycine.xyz
3.98207181
3.00785834 
2.99800682 
3.98095768 
2.993322   
1.00025168
2.99979217 
1.00140949 
1.98927904 
0.99678598 
0.9964672  
4.0016192
0.9932187  
0.99711911 
0.99333844 
0.99320143 
0.99209619 
0.99397073
0.99397068 
0.99397075
# Save output to file 'cns'
> kallisto cns --out cns alanine-glycine.xyz
> cat cns
3.98207181
3.00785834 
2.99800682 
3.98095768 
2.993322   
1.00025168
2.99979217 
1.00140949 
1.98927904 
0.99678598 
0.9964672  
4.0016192
0.9932187  
0.99711911 
0.99333844 
0.99320143 
0.99209619 
0.99397073
0.99397068 
0.99397075

Now we obtain a list of atomic coordination numbers, which is in agreement with chemical intuition.