Stereodynamics of 9,11-Diphenyl-10-azatetracyclo[6.3.0.0.(4,11)0.(5,9)]undecanes. Highly Restricted Nitrogen Inversion and Isolated Phenyl Rotation. X-ray Crystallographic, Dynamic NMR, and Molecular Mechanics Studies

The (1)H NMR spectra of 10-benzyl-9,11-diphenyl-10-azatetracyclo[6.3.0.0.(4,11)0.(5,9)]undecane (BnPh(2)()) and 10-methyl-9,11-diphenyl-10-azatetracyclo[6.3.0.0.(4,11)0.(5,9)]undecane (MePh(2)()) decoalesce due to slowing inversion at nitrogen and to slowing isolated bridgehead phenyl rotation. The high nitrogen inversion barriers in MePh(2)() (DeltaG() = 12.2 +/- 0.1 kcal/mol at 250 K) and BnPh(2)() (DeltaG() = 10.6 +/- 0.1 kcal/mol at 215 K) are typical of tertiary amines in which at least one C-N-C bond angle is constrained to a small value. Compared to the minuscule rotation barriers about sp(2)-sp(3) carbon-carbon bonds in simple molecular systems, the bridgehead phenyl rotation barriers in MePh(2)() (DeltaG() = 9.8 +/- 0.1 kcal/mol at 210 K) and BnPh(2)() (DeltaG() = 9.8 +/- 0.1 kcal/mol at 210 K) are unusually high. Molecular mechanics calculations (MMX force field) suggest that the origin of the high phenyl rotation barriers lies in the close passage of an o-phenyl proton and a methyl (or benzylmethylene) proton in the transition state. BnPh(2)() crystallized from hexane as white needles in the monoclinic system Pn. Unit cell dimensions are as follows: a = 12.198(1) ?, b = 6.1399(6) ?, c = 14.938(2) ?, beta = 107.470(4) degrees, V = 1067.1(2) ?(3), Z = 2. In the crystal molecular structure, the imine bridge CNC bond angle in BnPh(2)() is constrained to a small value (96 degrees ). The benzylic phenyl group is oriented gauche to the nitrogen lone pair.     

Conservative methods for the Toda lattice equations

We are concerned with the numerical integration of the Todalattice equations by using different conservative methods. Numericalexperiments suggest that the global error for isospectral schemesdecreases exponentially with time but it is almost constantfor either symplectic or more general integrators. We providea theoretical explanation for these experimental findings.