Low-order macroelements for two- and three-dimensional Stokes flow

The use of low-order elements for approximating fluid flow is attractive because all the elemental contributions can be quickly and easily obtained. One of the drawbacks is that low-order elements often give rise to spurious pressure modes or incompatible velocity and pressure approximations. In this paper linear velocity and linear pressure elements are described for both two- and three-dimensional flow that always produce stable solutions provided the elements are assembled into simple macroelements following easily used rules. Some examples of this idea are given for Stokes flow and compared with another popular low-order method. © 1993 John Wiley & Sons, Inc.     

The use of infinite grid refinements at singularities in the solution of Laplace's equation

Summary The approximate solution of Laplace's equation using the finite element method is considered. Particular emphasis is given to problems in which there are boundary singularities and the use of infinite refinements in the grid of triangles in the neighbourhood of these singularities is analysed. A particular type of infinite grid refinement is proposed and some examples are given.     

β‐Carboline (norharman)

The structure of β‐carboline, also called norharman (systematic name: 9H‐pyrido[3,4‐b]indole), C₁₁H₈N₂, has been determined at 110 K. Norharman is prevalent in the environment and the human body and is of wide biological interest. The structure exhibits intermolecular N—H...N hydrogen bonding, which results in a one‐dimensional herringbone motif. The three rings of the norharman molecule collectively result in a C‐shaped curvature of 3.19 (13)° parallel to the long axis. The diffraction data show shorter pyridyl C—C bonds than those reported at the STO‐3G level of theory.     

The synergy between qualitative theory, quantitative calculations, and direct experiments in understanding, calculating, and measuring the energy differences between the lowest singlet and triplet states of organic diradicals

This perspective describes research, carried out in the authors' labs over the past forty years, aimed at understanding, predicting, and measuring the singlet-triplet energy differences (ΔE(ST)) in diradicals. A theory for qualitatively predicting the ground states of diradicals and the use of Negative Ion Photoelectron Spectroscopy (NIPES) for measuring ΔE(ST) are described. The application of this theory, ab initio calculations, and NIPES to the prediction and measurement of ΔE(ST) in a wide variety of organic diradicals is detailed. Among the diradicals that are discussed in this perspective are HN, CH(3)N, PhN, CH(2), trimethylenemethane (TMM), oxyallyl (OXA), meta-benzoquinodimethane (MBQDM), meta-benzoquinone (MBQ), tetramethyleneethane (TME), 1,2,4,5-tetramethylenebenzene (TMB), and D(8 h) cyclooctatetraene (COT). All of these diradicals have been studied in one and, in most cases, in both of the authors' laboratories. The studies of OXA and D(8h) COT were, in fact, collaborations between the research groups of the authors. These two projects both took advantage of the ability of NIPES to provide information about transition states. Transition-state spectroscopy was used to measure the carbonyl stretching frequency in the singlet state of OXA and to establish that D(8h) COT violates the strictest version of Hund's rule.     

A perimidine-derived non-Kekulé triplet diradical

6,9-Di(tert-butyl)-1-methyltetrazolo[1,5-a]perimidine (1) has been synthesized from naphthalene in seven steps. The EPR spectra, recorded after irradiation of 1 in a butyronitrile matrix at 77 K (lambda = 351 nm) and in Ar and Xe matrixes at 4.6 K (lambda > or = 345 nm), showed a six-line, high-field signal (Delta m(S) = +/- 1), centered at 3350 G in butyronitrile, along with a half-field signal (Delta m(S) = +/- 2), which is characteristic for triplets. Simulation of the observed EPR spectra gave values for the zero-field splitting parameters of |D/hc|/cm(-1) = 0.0105, |E/hc|/cm(-1) = 0.0014 in butyronitrile and |D/hc|/cm(-1) = 0.0107, |E/hc|/cm(-1) = 0.0016 in Ar. These EPR parameters are consistent with the diradical 5,8-di(tert-butyl)-2-(N-methylimino)perimidine-1,3-diyl ((3)2) as source of the EPR spectra. Linearity of the Curie-Weiss plot and UB3LYP and (14/14)CASPT2 calculations of the singlet-triplet energy difference (DeltaE(ST) approximately 8-10 kcal/mol) indicate that the triplet is the ground state of 2, as predicted for such a nondisjoint diradical.     

Molecular orbitals of the oxocarbons (CO)n, n = 2-6. Why does (CO)4 have a triplet ground state?

Cyclobutane-1,2,3,4-tetrone has been both predicted and found to have a triplet ground state, in which a b(2g) σ MO and an a(2u) π MO are each singly occupied. The nearly identical energies of these two orbitals of (CO)(4) can be attributed to the fact that both of these MOs are formed from a bonding combination of C-O π* orbitals in four CO molecules. The intrinsically stronger bonding between neighboring carbons in the b(2g) σ MO compared to the a(2u) π MO is balanced by the fact that the non-nearest-neighbor, C-C interactions in (CO)(4) are antibonding in b(2g), but bonding in a(2u). Crossing between an antibonding, b(1g) combination of carbon lone-pair orbitals in four CO molecules and the b(2g) and a(2u) bonding combinations of π* MOs is responsible for the occupation of the b(2g) and a(2u) MOs in (CO)(4). A similar orbital crossing occurs on going from two CO molecules to (CO)(2), and this crossing is responsible for the triplet ground state that is predicted for (CO)(2). However, such an orbital crossing does not occur on formation of (CO)(2n+1) from 2n + 1 CO molecules, which is why (CO)(3) and (CO)(5) are both calculated to have singlet ground states. Orbital crossings, involving an antibonding, b(1), combination of lone-pair MOs, occur in forming all (CO)(2n) molecules from 2n CO molecules. Nevertheless, (CO)(6) is predicted to have a singlet ground state, in which the b(2u) σ MO is doubly occupied and the a(2u) π MO is left empty. The main reason for the difference between the ground states of (CO)(4) and (CO)(6) is that interactions between 2p AOs on non-nearest-neighbor carbons, which stabilize the a(2u) π MO in (CO)(4), are much weaker in (CO)(6), due to the much larger distances between non-nearest-neighbor carbons in (CO)(6) than in (CO)(4).     

Evidence for the formation of the (Ph3P)2Pt complex of 3,7-dimethyltricyclo[3.3.0.03,7]oct-1(5)-ene, the most highly pyramidalized alkene in a homologous series. Isolation and X-ray structure of the product of the ethanol addition to the complex

[reaction: see text] Attempts to isolate the (Ph(3)P)(2)Pt complex of the highly pyramidalized olefin 3,7-dimethyltricyclo[3.3.0.0(3,7)]oct-1(5)-ene 2 by generation of 2 in the presence of (Ph(3)P)(2)PtC(2)H(4), followed by crystallization of the complex (2-Pt) from THF-ethanol, resulted in the isolation of the adduct of 2-Pt with ethanol (5). Calculations confirm that addition of alcohol across the C1-C5 bond is more favorable in 2-Pt than in the corresponding (Ph(3)P)(2)Pt complexes of less pyramidalized olefins, despite the stronger Pt-C bonds in 2-Pt.