A simple mathematical model for prediction of nitrogen dioxide molecular structure

Gianluca Argentini

gianluca.argentini@rielloburners.com
gianluca.argentini@gmail.com

Where it is shown that one can numerically compute the angle between atomic bonds and length bonds in N(Subscript[O, 2]) molecule with a good agreement with experimental data.

In this paper we assume that nitrogen and oxygen atoms have valence electronic configuration 2(s^2)2(p^3) and 2(s^2)2(p^4) respectively. In N(Subscript[O, 2]) nitrogen dioxide molecule, N atom and each O atom share two pairs of electrons while N preserves a single unpaired electron (see (^1)ChEBI). The pairs of electrons shared by N and O are bonding pairs, according to VSEPR theory terminology (see (^(1,2))Gillespie).
   From VSEPR theory, in this molecule the three atoms have a planar configuration. Let be α = ONO the angle with vertex at N nucleus, that is the angle between atomic bonds. Also, we consider the circle (Subscript[C, N ]) centred on N nucleus and having radius (Subscript[R, N ]) equal to Van der Waals radius of N (see (^1)Gillespie), and the two circles (Subscript[C, O ])centred on O nuclei and having radii (Subscript[R, O ]) equal to Van der Waals radius of O, positioned along the line NO with O nucleus at a distance, from N, equal to a value d to be defined. Let be A the point of intersection between (Subscript[C, N ])and the vertical line traced from N nucleus, and B and C the two intersections between (Subscript[C, O ])and the lines NO respectively (see Fig.1). Finally, denote with (Subscript[χ, N]) and (Subscript[χ, O]) the nitrogen and oxygen electronegativities.

Fig. 1 The geometrical representation of nitrogen dioxide molecule.

In the most simple representation, assuming (Subscript[cr, N ]) and (Subscript[cr, O ]) the covalent radius of N and O respectively, the length d of the bonds N-O is assumed to be the sum (Subscript[cr, N ]) + (Subscript[cr, O]). But nitrogen and oxygen are two chemical elements having great electronegativity values, so the real length bond d is lower than the sum of the two covalent radii (see (^2)Gillespie). Also, (Subscript[χ, O]) > (Subscript[χ, N]) , so O atom attracts valence electrons and the extension of the bond electronic pair domain decreases along the NO direction. We can think that d : ((Subscript[cr, N ])+(Subscript[cr, O])) = (Subscript[χ, N]) : (Subscript[χ, O]) , therefore the length bond is

d=((Subscript[cr, N ])+(Subscript[cr, O]))((Subscript[χ, N])/(Subscript[χ, O]))

If Q is the charge concentration at point A of the nitrogen unpaired electron, then for the double bonds N-O the charge concentrations at points B and C, due to two valence electrons of oxygen, are 2Q. The Coulomb electrostatic energy among the three electronic pairs domains is

W=Q  2Q(1/(Subscript[d, NO]))+Q  2Q(1/(Subscript[d, NO]))+2Q 2Q(1/(Subscript[d, OO]))

where is (Subscript[d, NO]) the distance between N and O atoms, and (Subscript[d, OO]) is the distance between O atoms in the molecule. If we consider a cartesian system with origin at the middle point of the segment BC, then the cartesian coordinates of A, B and C are A = {0, (Subscript[R, N ])+ (d + (Subscript[R, O])) cos a/2}, B = {-(d + (Subscript[R, O])) sin a/2, 0} and C = {(d + (Subscript[R, O])) sin a/2, 0} respectively. The geometric configuration of the molecule in its ground state is determinated by atoms positioning so that W is minimum.
   We can find numerically, using a mathematical software, the value of α for which W has its minimum (see (^1)Trott). Note that, for this purpose, it is not restrictive to suppose Q = 1. These are the values of geometric parameters used for computation (see (^1)CCCBDB and (^1)Gillespie): (Subscript[R, N ])=150 pm, (Subscript[R, O ])=140 pm, (Subscript[cr, N ])= 70 pm, (Subscript[cr, O ])= 60 pm, (Subscript[χ, N]) = 3.1, (Subscript[χ, O]) = 3.5 .
   In Fig. 2 there is a plot of the Coulomb energy as function of the angle a the N-O bonds. Minimum is obtained for a value of 2.35 rad, that is α = 134.65°. Note that this value is in good agreement with experimental values reported in technical literature, e.i. α = 134.1° in (^1)CCCBDB.
   Note that the d length bond, computed using previous formula, is 120.5 pm, to be compared with the value of 119.4 pm from experimental data (see (^1)CCCBDB).

Fig. 2 Graph of the Coulomb energy vs the angle α = ONO, due to electronic pairs in nitrogen dioxide molecule.

Summary Table			
			Experimental	This model
ONO angle [°]		134.1		134.65	
bond length [pm]	119.4		120.5

References
(^1) CCCBDB, Computational Chemistry Comparison and Benchmark DataBase, http://cccbdb.nist.gov/, 2013
(^1) ChEBI, Chemical Entities of Biological Interest, https://www.ebi.ac.uk/chebi, 2014
(^1) R. J. Gillespie and I. Hargittai, The VESPR Model of Molecular Geometry, Boston, Allyn and Bacon, 1991
(^2) R. J. Gillespie and P. L. A. Popelier, Chemical Bonding and Molecular Geometry, Oxford & New York, Oxford University Press, 2001
(^1) M. Trott, The Mathematica Guidebook for Numerics, New York, Springer, 2006