Molecular orbital calculations (on dinitrogen tetroxide and related species)
Degree GrantorUniversity of Canterbury
Degree NameDoctor of Philosophy
The rapid increase over the last few years in the application of theoretical methods to the study of chemical problems is well known. Although new numerical techniques are being developed and theoretical advances are slowly being made, the major cause of this expansion is a continuing series of improvements in the speed and memory capabilities of digital computers. These advances, as well as improving the accuracy of wavefunctions calculated for small molecules, permit the application of well-established methods to larger molecular systems. Molecular orbital methods which do not use any experimental parameters other than molecular geometry are usually referred to as ab initio methods. This thesis is mainly concerned with the application of two such methods, which incorporate different approximations, to a study of the molecule dinitrogen tetroxide, N₂O₄. Chapter One outlines the molecular orbital theory necessary for an understanding of these two methods. Most of the calculations were ultimately done using the second method, known as the Multi-Gaussian Expansion technique. In Chapter Two the structure and bonding of N₂O₄ and of nitrogen dioxide, N₂, the monomer into which N₂O₄ readily dissociates, are reviewed. Particular emphasis is given to the results of previous molecular orbital calculations on these molecules. An understanding of the nature of the N-N bonding in N₂O₄ is a long-sought goal. The most stable isomer has a planar structure in which the two nitrogen dioxide moieties are joined by a N-N bond that is very much longer than the normal N-N single bond, such as occurs, for example, in hydrazine, N₂O₄. If O-O interactions were responsible for the long bond the molecule might be expected to adopt a staggered rather than a planar structure. The geometry of the N₂ moiety is also remarkably similar to that in free N₂. Several explanations for these features have been advanced, and the N-N bond has been variously described as (i) a normal sigma bond plus partial pi bond; (ii) a '"pi-only" bond in which there is no sigma bond at all between the nitrogen atoms; (iii) a "splayed" single bond which, although of sigma type, requires the molecule to be planar in order to achieve maximum overlap; (iv) a charge transfer configuration; (v) a sigma bond weakened by delocalization of oxygen lone pair electrons into a N-N antibonding orbital with a little pi bonding to account for planarity; and (vi) a normal sigma bond plus a partial pi bond with destabilization coming from 2p orbitals on the nitrogen atoms. These theories were based on calculations which did not include all of the electrons in the molecule and which were semi-empirical to the extent that some experimental parameters were required; To the author's knowledge the calculation described here represents the first non-empirical all-electron treatment of N₂O₄. A brief discussion is given of a type of increased valence formula which may be applied to systems having four electrons in three overlapping atomic orbitals on three atoms. When N₂O₄ is described in terms of these formulae the weak N-N bond results from the fact that the molecule is represented by a resonance between several valence bond structures, some of which lack an N-N bond. In particular, this approach suggests that stability should arise from reduced net charges on the nitrogen atoms, three centre O-N-N interactions, and long range interactions between the nitrogen atom of one moiety and the oxygen atoms of the other moiety. These suggestions are qualitatively consistent with the present results. Several problems arose in the application to N₂O₄ of the original versions of the computer programs embodying the methods of Chapter One. These problems are discussed in Chapter Three. Because all forty-six electrons were considered the amount of computer time required for the calculations was very much greater than for smaller systems. The equations must be solved by an iterative procedure, which normally involves the diagonalization of a matrix to find its eigenvalues and eigenvectors. For a large matrix, especially if some of these eigenvalues are very close together, this procedure is a primary source of truncation errors which build up on each iteration. Alternative methods were therefore sought for solving the equations of Chapter One. Most successful of these was a direct minimization procedure called the conjugate gradients method. A much improved rate of convergence to the solution was obtained when this method was used in conjunction with the matrix diagonalization method. Another major improvement was brought about by making maximum use of the molecule's symmetry. In the approximate molecular orbital theory used here the overall wavefunction is an antisymmetrized product of molecular orbitals, each of which is expressed as a linear combination of basis functions located on the atoms. In exploiting the molecule's symmetry these basis functions are initially combined into symmetry orbitals, whose use in the final iterative procedure results in less trouble from rounding errors. The effect is that the 30 x 30 matrix which is to be diagonalized is transformed to block diagonal form, each smaller block of which can be diagonalized separately. A further advantage of this method is that it facilitates the selection of the particular electronic state for which the wave function is to be calculated. For the multi-Gaussian expansion method a large number of integrals need to be calculated. Because of the high symmetry of N₂O₄ many small groups of these are equal in magnitude and therefore only one member of each group needs to be calculated. This leads to a drastic reduction in computer time. The method used for organizing the selection and storage of these integrals is discussed in Section 3.3. It should be mentioned here that because dinitrogen trioxide, N₂O₃, for which a similar type of weak N-N bonding has been suggested, has much less symmetry a similar calculation would be correspondingly more difficult. In order to compare the N₂O₄ wave function with that of its monomer, calculations to the same levels of approximation were required for nitrogen dioxide. Because this species has an unpaired electron the methods of Chapter One are not suitable and wavefunctions were obtained by the two different methods which are described in Chapter Four. Convergence difficulties with these methods are well known. It has been suggested that these result from incorrect choice of eigenvectors on each iteration. This problem was investigated, and a suitable method of choosing the eigenvectors was incorporated into the computer program. Once wave functions had been obtained it became necessary to interpret them. Methods for doing this are discussed in Chapter Five. The Mulliken population analysis partitions the density into various atom populations and overlap populations between atoms. The magnitude and sign of this overlap give an indication of the amount of bonding between two atoms. A recently proposed bond energy analysis was also employed. This partitions the total energy of the molecule into contributions from single atoms and from groups of two, three and four atoms. An excellent correlation was found between the two-centre bond energies thus calculated and the Mulliken overlap populations. The population analysis is, however, sensitive to the choice of basis functions. A more complete description of the electron density distribution is therefore obtained from contour plots through various planes in the molecule. For the wave functions computed by the methods of Chapter One the molecular orbitals are not unique, in that particular kinds of transformations of the functions amongst themselves leave the total wavefunction unchanged. One form of transformation leads to localized orbitals which more closely resemble traditional chemical ideas of bonds, inner shells and lone pairs. The method used for this localization is discussed in Section 5.4 and Appendix V. The results of the calculations are given and discussed in Chapter Six. Results for the nitrite ion, N₂- obtained by the methods of Chapter One are compared with those of an exact ab initio calculation with the same basis functions and molecular geometry, taken from the literature. When the multi Gaussian expansion wave function for N₂O₄ was examined the following features were apparent. In comparison with N₂ and its ions N₂⁺and N₂- all covalent bonding was reduced. The results indicated that the weakness of the N-N bond was due to the fact that the N-N antibonding orbital was filled and the expected N-N sigma bonding orbital was unoccupied. The major interpretive problem was therefore to understand why the molecule was stable at all since there was very little N-N pi bonding. From the bond energy analysis the stability of this state was found to be due to a lowering of the energy associated with the electrons close to the N atoms, long range N-O interactions, and surprisingly large and negative three centre O-N-N energies. As noted above these three features are important in the increased valence description of N₂O₄. An investigation of the rotational barrier was not attempted because this would require the calculation of a wave function at the perpendicular configuration. This has lower symmetry and in addition to problems from rounding errors would require a much greater amount of computer time to calculate the larger number of unique integrals. Reported calculations with similar size basis sets have. produced results for rotational barrier values which are not very good estimates. Throughout most of this thesis distances are given in atomic units (Bohr radii). For comparison with literature values, however, in Chapter Two distances are given in nanometers. 1 a.u. of length = 5.29167 x 10-¹¹m Energies are given either in atomic units (Hartrees) or in the SI units of kilojoules per mole: 1 a.u. of energy = 1 Hartree = 27.2107 eV where 1 eV per particle = 23.061 kcal mole-¹ = 96.487 kJ mole-¹. These values are taken from the tables of Cohen & Du Mond. In the equations all expressions for operators are in corresponding atomic units.