The torsional potential energy surfaces of 1,2dinitrobenzene, 1,3dinitrobenzene, and 1,4dinitrobenzene were calculated using the B3LYP functional with 631G(d) basis sets. Threedimensional energy surfaces were created, allowing each of the two CN bonds to rotate through 64 positions. Dinitrobenzene was chosen for the study because each of the three different isomers has widely varying steric hindrances and bond hybridization, which affect the energy of each conformation of the isomers as the nitro functional groups rotate. The accuracy of the method is determined by comparison with previous theoretical and experimental results. The surfaces provide valuable insight into the mechanics of conjugated molecules. The computation of potential energy surfaces has powerful application in modeling molecular structures, making the determination of the lowest energy conformations of complex molecules far more computationally accessible.
Torsional potential energy calculations provide conformational information and allow finding the barriers to the rotations of bonds. Early applications were performed in organic compounds phosphates, 1,3butadiene, polypeptides, and dimethyl groups [
The calculations by Bongini and Bottoni compared HartreeFock and secondorder MøllerPlesset methods [
In this manuscript, we explore the use of the torsional PES to determine the geometry of the lowest energy state for molecules, a methodology which might be extended to larger molecules such as proteins. We computationally determine the energy correlation between pairs of rotors on the molecule. These correlations can be reduced to a matrix differential equation, and the eigenvalues and eigenvectors of the equation will specify the lowest energy conformation of the molecule being studied. Since the energy of the system is represented by a differential equation in matrix form, it can be iterated computationally to a convergence at the lowest energy state of the molecule. The knowledge of the correlations between different rotors allows local energy minima distinct from the global minima to be easily identified.
We report the torsional PESs for dinitrobenzene isomers. The twodimensional PES plots show the potential energy of the molecule as a function of the angles each NO_{2} group makes with respect to the ring. The molecular structure of dinitrobenzene allows for three isomers and thus our calculations explore torsional PESs for modeling conjugated systems with and without steric effects. As shown in Figure
Molecular structure of orthodinitrobenzene and definition of the coordinate system. The twodimensional PESs depend on the angles
The geometry of the nitro groups in the dinitrobenzene isomers has been explored before. Freed et al., studied the variations in the linewidths of electron spin resonance spectra of the carbonnitrogen anion for each isomer of the dinitrobenzene anions, where they found alternation in the linewidths of the metaisomer, which arise from rotation of the nitro groups out of the planar conformation [
The behavior of the nitro groups in dinitrobenzene is not without controversy. Dinitrobenzene substituent reactions have been studied before [
All the molecular modeling and singlepoint energy computations were performed using the Gaussian 09 package [
Many individual calculations can be combined to yield the torsional potential energy surface. To plot the surface, a
The torsional PES of each isomer, expressed as a function of the torsional angles, can be expanded in terms of Legendre polynomials:
The coefficients for the expansions of the potential energy (in units of Ha) for each isomer are presented in Table
Coefficients of the Legendre polynomial expansion. For each expansion, we truncate the polynomials once data at the lower energies fit well. The terms marked with “
Calculated coefficients (mHa)  Paraisomer  Metaisomer  Orthoisomer 


9.8006  10.7201  10.0467 

−5.6462  −6.0068  0.0000 

0.2811  0.2856  4.8531 

0.0155  0.0136  0.0000 



2.1196 

−5.6462  −6.0068  0.000 

0.2811  0.2856  4.8531 

0.0155  0.0136  0.0000 



2.1196 

0.2850  0.0664  19.2731 



0.0000 



1.0792 



18.4973 
The Pearson correlation coefficient,
In the above equation,
We have constructed the torsional PES for the three dinitrobenzene isomers using firstprinciples methods. The surface is symmetric, with a global minimum corresponding to (
The ground state conformation of paradinitrobenzene.
The torsional potential energy surface of paradinitrobenzene is plotted in Figure
Torsional PES of paradinitrobenzene. The conformational energies (in Hartree) of the minima are zero, while the barriers around (
As would be expected from the opposition of the NO_{2} groups, no effects of the electronic repulsion can be seen, since the molecule is completely planar in its lowest energy state. A slice of the potential corresponding to symmetric rotations, where both angles vary simultaneously, or in mathematical terms,
Plot of the electronic energy above the minimum for the paraisomer as a function of the torsional angle of the NO_{2} groups, using the computed Legendre polynomial expansion.
The torsional energy is seen to have a behavior very similar to the cosine squared dependence on the angle that would be expected if no steric effects were present.
The minimum energy conformation of metadinitrobenzene is displayed in Figure
The ground state conformation of metadinitrobenzene.
As for the paraisomer, the ground state conformation of the metaisomer is clearly planar. The torsional PES of metadinitrobenzene is plotted in Figure
The torsional PES of metadinitrobenzene. The conformational energies (in Hartree) of the minima are zero, while the barriers around (0,
The global energy minimum of the surface is seen to be located at (
The Legendre expanded electrostatic energy above the minimum as a function of the CN bond angles is plotted for the metaisomer.
The torsional PES of orthodinitrobenzene, presented in Figure
Torsional PES of orthodinitrobenzene. The conformational energies (in Hartree) of the minima are set to zero. The barriers around (
Here, eight minima are observed to be present in the surface. The global energy minima are not observed at (
The lowest energy conformation of orthodinitrobenzene.
It can also be seen that each of the CN bonds is bent away from the other NO_{2} group, increasing the spacing of the oxygen atoms from each other. The torsional stresses are seen to be great enough to force the molecule out of the planar conformation preferred by bond hybridization, because the repulsion between the NO_{2} groups is very strong. Thus, the orthoisomer is the only one of the three isomers studied which is greatly influenced by steric repulsion. The slice of the torsional PES of the orthoisomer pertaining to symmetric rotations is illustrated in Figure
A slice of the Legendre polynomial expansion of orthodinitrobenzene, considering symmetric rotations of the nitro groups, with both nitro groups rotating together.
To verify that the strain on the orthoisomer was not improperly considered by the constraints of the scan, all the bond length and angle constraints were relaxed for the conformation associated with the global energy minimum, and the strains on each atom were minimized. Table
The effect of relaxing the scan constraints at the orthodinitrobenzene minimum energy conformation.
Orthodinitrobenzene bonds  Average original length (Å)  Average final length (Å) 

CC  1.395  1.391 
CH  1.085  1.086 
CN  1.475  1.453 
NO  1.227  1.226 
We have constructed the torsional PES for dinitrobenzene isomers using firstprinciples methods. The paraisomer and metaisomer are found to have weak interactions between nitro groups, so the torsional potentials can be accurately modeled using very few terms in an expansion. For the orthoisomer, however, the strong steric repulsion between the adjacent NO_{2} groups makes the theoretical form of the potential much more complex. However, the fourthorder expansion for the potentials only differs from the computed potentials towards the energy maximum. This difference is not important as regards the goal of this research, which is to enable the computation of lowest energy structures of molecules. Since the goal is to determine the location in angle space of the energy minimum, the exact characteristics of an energy maximum have no relevance to the usefulness of the potential. The result is the confirmation that the NO_{2} groups of orthodinitrobenzene do not lie in the plane of the benzene ring. A previous report stated that the NO_{2} groups are not planar for the orthoisomer [
The modeling of the properties of tworotor molecules, an important first step to allow the reduction of the problem of a molecular geometry to a system of linear differential equations, can be accomplished to great accuracy by the computation of torsional potentials. This method can be tailored to available computational resources by adjusting the spacing between the points in the torsional potential, as well as by altering the region around the torsional minima for which the potential will be computed. The set of correlations between combinations of tworotor systems in a molecule can be computed far more easily than a normal geometry optimization, because the number of molecular degrees of freedom is limited to two at a time.
Some of the results in this article were presented at the 2015 APS March Meeting [
The authors declare that they have no conflicts of interest.
This work used the OSU High Performance Computing Center at Oklahoma State University supported in part through the National Science Foundation (Grant no. OCI1126330). The authors thank S. Dai, A. Adams, and C. Fennell for useful discussions.