Strain Energies in Homoatomic Nitrogen Clusters N(4), N(6), and N(8)

Strain energies and resonance energies can be obtained as the energy changes for appropriate homodesmotic reactions using ab initio calculated total energies as the energies of the reactants and products involved. Homodesmotic reactions conserve bond types and preserve valence environments at all atoms, requirements that favor the cancellation of basis set and electron correlation errors in the ab initio energies. In this paper we calculate strain energies and resonance energies for N(4), N(6), and N(8) clusters in a number of chemically significant but, for nitrogen, hypothetical structural forms. The nitrogen cluster strain energies are generally of the same order of magnitude as those of isostructural hydrocarbon clusters, and individual differences can be explained by using the ring strain additivity rule and recognizing the effect of the presence of lone pairs of electrons on nitrogen clusters but not on the hydrocarbons. Resonance energies of the nitrogen clusters are much smaller than those of the comparable aromatic hydrocarbons. The differences can be rationalized by considering the relative strengths of CC and NN single and double bonds. Strain and resonance energies of nitrogen clusters are compared with those previously reported for homoatomic clusters of phosphorus and arsenic. Trends through the series are remarkably similar, but strain energies for clusters from lower periods are progressively smaller. Strain and resonance have been important organizing concepts in organic chemistry for many years. Estimates of corresponding parameters for inorganic analogs are only now becoming available.     

Maximum hardness in P6 isomers

Parr and Chattaraj proposed a principle of maximum hardness for stable molecular structures. Pearson and Palke used ab initio SCF MO calculations for ammonia and ethane to demonstrate the operation of the principle. In this paper, we present ab initio SCF MO results for five isomeric forms of the homoatomic P⁶ cluster as further support for the principle of maximum hardness. © 1994 John Wiley & Sons, Inc.     

Three-Dimensional Hückel Theory for closo-Carboranes

We have recently developed a 3-dimensional Hückel method for cluster compounds. The method uses a set of approximations for Coulomb, resonance, and overlap integrals very similar to those employed in the familiar 2-dimensional Hückel theory for the pi electrons of planar conjugated hydrocarbons. The method can be adapted to heteroatomic clusters by introducing heteroatomic Coulomb integrals, alpha(Y) = alpha(X) + hbeta, whereh is a parameter for heteroatom Y. In this paper, we use the 3-dimensional Hückel method to study the properties of the closo-carboranes, C(2)B(n)()(-)(2)H(n)(). We calibrate the method by choosing a value of the heteroatomic parameter h that distinguishes positional isomers by energy and gives them relative energies in rough agreement with those established by observation and ab initio calculations. We obtain modest improvement in matching ab initio relative energies of isomers by means of a three-parameter, first-order perturbation treatment. We use the calibrated method to evaluate various mechanisms proposed for the isomerizations of C(2)B(4)H(6), C(2)B(5)H(7), and C(2)B(6)H(8), all of which have been observed to undergo intramolecular isomerizations. Rearrangements of C(2)B(6)H(8) have been satisfactorily explained by a single-DSD (diamond-square-diamond) process. Those for C(2)B(5)H(7) require at least two DSD processes, concerted, consecutive, or overlapping. Several different mechanisms have been proposed for the rearrangement of C(2)B(4)H(6). In evaluating intermediate and transition state structures, the 3-dimensional Hückel method gives higher energies to those structures with a larger number of nontriangular faces, a plausible conclusion except that occasionally it is wrong. In comparison with ab initio results, the 3-dimensional Hückel method fails to give low energies for classical structures.     

Energies of the lowest singlet S and triplet S states of helium by the local energy method

The least squares local energy method is applied to the helium atom in greater detail in a formulation which can easily be extended to more complicated atoms. The energies of the 1 ¹ S and 2³ S states are calculated to be – 2.9025 H and – 2.1753 H respectively. These values are in excellent agreement with the non-relativistic values of – 2.9037 H and – 2.1752 H calculated by Pekeris.

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Zusammenfassung Die Methode, die Varianz der lokalen Energie zu minimisieren, wird in einer leicht auf kompliziertere Atome zu erweiternden Form ausführlicher auf das Heliumatom angewandt. Die Energien des 1 ¹ S – und des 2 ³ S–Zustandes werden zu – 2,9025 bzw. – 2,1753 at. E. berechnet. Diese Werte stimmen ausgezeichnet mit den von Pekeris berechneten nichtrelativistischen Werten von – 2,9037 bzw. – 2,1752 at. E. überein.

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Résumé Le calcul de l'énergie locale par la méthode des moindres carrés est appliqué plus en détail à l'atome d'hélium, dans une forme qui s'étend aisément à des atomes plus compliqués. Les énergies des états 1 ¹ S et 2 ³ S se calculent à – 2,9025 et – 2,1753 u. a., respectivement. Ces valeurs sont en excellent accord avec les valeurs de – 2,9037 et – 2,1752 u. a. (sans relativité) calculées par Pekeris.

     

The discretized flow on domains of attraction: a structural stability result

It is well known that, for stepsize sufficiently small, compactattractors of ordinary differential equations persist underdiscretization. The present paper describes the structure ofthe discrete-time dynamical system obtained via discretizationon A(M_h)\M_h where M_h is the approximate attractor and A(M_h)is its domain of attraction. The existence of a smooth embeddinginto a continuous-time parallelizable flow is proved. The constructioncan be used to define sections for discretizations and can beinterpreted as a justification of the method of modified equations.     

Optimized Structures and Relative Stabilities of the Carboranes from Ab Initio Calculations

Ab initio calculations at the STO-3G level were performed on almost all of the possible isomers for the entire series of closo-carboranes, C₂B_n-2H_n, 5 ? n ? 12. Geometry optimizations using the gradient method were also included in all calculations. We report here the relative energies obtained for the various isomers as well as the optimized structures. These calculations confirm our previous predictions of relative stabilities obtained from topological charge stabilization. Comparisons of our structures with those from experimental data provide us with a measure of reliability for bond distances obtained using ab initio SCF MO calculations at the STO-3G level. Results from the geometry optimization substantiated the experimentally known fluxional behavior of the 8 and 11 atom polyhedra.     

Details of the reaction graphs for intramolecular isomerizations of the carboranes

From proposed mechanisms for framework reorganizations of the carboranes C₂B _n-2H _n ,n = 5–12, we present reaction graphs in which points or vertices represent individual carborane isomers, while edges or arcs correspond to the various intramolecular rearrangement processes that carry the pair of carbon heteroatoms to different positions within the same polyhedral form. Because they contain both loops and multiple edges, these graphs are actually pseudographs. Loops and multiple edges have chemical significance in several cases. Enantiomeric pairs occur among carborane isomers and among the transition state structures on pathways linking the isomers. For a carborane polyhedral structure withn vertices, each graph hasn(n -1)/2 graph edges. The degree of each graph vertex and the sum of degrees of all graph vertices are independent of the details of the isomerization mechanism. The degree of each vertex is equal to twice the number of rotationally equivalent forms of the corresponding isomer. The total of all vertex degrees is just twice the number of edges orn(n - 1). The degree of each graph vertex is related to the symmetry point group of the structure of the corresponding isomer. Enantiomeric isomer pairs are usually connected in the graph by a single edge and never by more than two edges.     

Ab initio SCF MO results for the carborane isomers 3,5-C2B6H8, 1, 7-C2B7H9, and 1,2-C2B7H9

We report the geometry-optimized total energies and bond distances for the closo-carborane isomers 3,5-C₂B₆H₈, 1,7-C₂B₇H₉, and 1,2-C₂B₇H₉ calculated by the ab initio SCF MO method using the STO -3G basis set. Relative energies are compared with those of the other carborane isomers in the 8- and 9- atom classes. These results complete the set of calculations at the same level of theory for all deltahedral carborane isomers except for those of the 11-atom class.