On the periodic points of a two-parameter family of maps of the plane

We study the dynamics of a two-parameter family of noninvertible maps of the plane, derived from a model in population dynamics. We prove that, as one parameter varies with the other held fixed, the nonwandering set changes from the empty set to an unstable Cantor set on which the map is topologically equivalent to the shift endomorphism on two symbols. With the help of some numerical work, we trace the genealogies of the periodic points of the family of period 5, and describe their stability types and bifurcations. Among our results we find that the family has a fixed point which undergoes fold, flip and Hopf bifurcations, and that certain families of period five points are interconnected through a codimension-two cusp bifurcation.     

Racemization barriers of 1,1'-binaphthyl and 1,1'-binaphthalene-2,2'-diol: a DFT study

Density functional theory has been applied to the study of various pathways and transition states for the configurational inversion of 1,1'-binaphthyl (1) and 1,1'-binaphthalene-2,2'-diol (2). The preferred pathway is found to be anti with centrosymmetric transition state. Whereas the reaction path of 1 goes downhill from transition to ground state, in the case of 2 it contains one unexpected local minimum. Very satisfactory agreement with available experimental values of activation Gibbs energies is achieved.     

Three-photon near-threshold photoionization dynamics of isooctane

The electron survival probability following three-photon (9.3 eV total) near-threshold photoionization of neat isooctane is measured with sub-50 fs time resolution. The measured dynamics are nonexponential in time and are well described by a diffusion-controlled electron-cation recombination model. Excitation-power-dependent studies indicate that the unperturbed three-photon threshold ionization is only observed for pump irradiance below 0.5 TW cm2. At excitation fields above this level, the signal is no longer cubic in the excitation irradiance, and the observed electron survival probability dramatically changes, decaying as a single exponential in time.     

Rotationally resolved electronic spectra of 1,2-dimethoxybenzene and the 1,2-dimethoxybenzene-water complex

Rotationally resolved S(1) <-- S(0) electronic spectra of 1,2-dimethoxybenzene (DMB) and its water complex have been observed and assigned. The derived values of the rotational constants show that the bare molecule has a planar heavy-atom structure with trans-disposed methoxy groups in its ground and excited electronic states. The transition of DMB is polarized along the b-axis bisecting the methoxy groups, demonstrating that its S(1) state is an (1)L(b) state. Higher energy bands of DMB are also polarized along the b-axis and have been tentatively assigned to different vibrational modes of the (1)L(b) state. The water complex origin appears 127 cm(-1) to the blue of the bare molecule origin. Analyses of the high resolution spectra of DMB/H(2)O and DMB/D(2)O suggest that the water molecule is attached via two O-H...O hydrogen bonds to the methoxy groups in both electronic states. A tunneling motion of the attached water molecule is revealed by a splitting of these spectra into two subbands. Potential barriers to this motion have been determined.     

Two-photon absorption in quadrupolar pi-conjugated molecules: influence of the nature of the conjugated bridge and the donor-acceptor separation

Quadrupolar-type substitution of pi-conjugated chromophores with donor and acceptor groups has been shown to increase their two-photon absorption (TPA) response by up to two orders of magnitude. Here, we apply highly correlated quantum-chemical calculations to evaluate the impact of the nature of conjugated bridge and the charge-transfer distance on that enhancement. We compare chromophores with phenylenevinylene-, thienylenevinylene-, polyene-, and indenofluorene-type backbones substituted by dimethylamino and cyano groups. In all compounds, we find a strongly TPA-active A(g) state (either 2A(g) or 3A(g)) in the low-energy region, as well as a higher lying TPA-active state (mA(g)) at close to twice the energy of the lowest lying one-photon allowed state; the smaller energy detuning in the mA(g) states results in very large TPA cross sections delta. We also investigate the influence of the degree of ground-state polarization on TPA. Independent of the nature of the backbone and the donor-acceptor separation, delta displays the same qualitative evolution with a maximum before the cyanine-like limit; the highest TPA cross sections are calculated for distirylbenzene- and polyene-based systems.     

Beyond the benzene dimer: an investigation of the additivity of pi-pi interactions

The benzene dimer is the simplest prototype of pi-pi interactions and has been used to understand the fundamental physics of these interactions as they are observed in more complex systems. In biological systems, however, aromatic rings are rarely found in isolated pairs; thus, it is important to understand whether aromatic pairs remain a good model of pi-pi interactions in clusters. In this study, ab initio methods are used to compute the binding energies of several benzene trimers and tetramers, most of them in 1D stacked configurations. The two-body terms change only slightly relative to the dimer, and except for the cyclic trimer, the three- and four-body terms are negligible. This indicates that aromatic clusters do not feature any large nonadditive effects in their binding energies, and polarization effects in benzene clusters do not greatly change the binding that would be anticipated from unperturbed benzene-benzene interactions, at least for the 1D stacked systems considered. Three-body effects are larger for the cyclic trimer, but for all systems considered, the computed binding energies are within 10% of what would be estimated from benzene dimer energies at the same geometries.     

Calculation of Franck-Condon factors including anharmonicity: simulation of the C2H4+X2B3u<--C2H4X1A(g) band in the photoelectron spectrum of ethylene

Our new simple method for calculating accurate Franck-Condon factors including nondiagonal (i.e., mode-mode) anharmonic coupling is used to simulate the C2H4+X2B3u<--C2H4X1A(g) band in the photoelectron spectrum. An improved vibrational basis set truncation algorithm, which permits very efficient computations, is employed. Because the torsional mode is highly anharmonic it is separated from the other modes and treated exactly. All other modes are treated through the second-order perturbation theory. The perturbation-theory corrections are significant and lead to a good agreement with experiment, although the separability assumption for torsion causes the C2D4 results to be not as good as those for C2H4. A variational formulation to overcome this circumstance, and deal with large anharmonicities in general, is suggested.     

Kinetic studies of some substituted hexarhodium carbonyl clusters

Reactions of the halides X- (X- = chloride, bromide or iodide) with the substituted cluster Rh6(CO)15(PPh3) in oxygen-free chloroform lead to [Rh5(CO)14(PPh3)]-, Rh(CO)2(PPh3)2X and [Rh(CO)2X2]- in the molar ratios 2:1: approximately 13. Oxidation by the solvent is assumed to lead to most of the Rh(I) product, and the stoichiometry for reactions with I- can be defined as 4Rh6(CO)15(PPh3) + 27I- + 12CHCl3 --> 2[Rh5(CO)14(PPh3)]- + Rh(CO)2(PPh3)2I + 13[Rh(CO)2I2]- + 6C2H2Cl4 + 4CO + 12Cl-. This can be rationalized quite simply with the aid of a few generally justifiable assumptions. Rate constants for reactions with bromide increase to a limiting value with increasing [Br-] in a way that shows that breaking of one Rh-Rh bond, with an unusual closo to nido structural change, is rate determining. This opening of the cluster might be spontaneous or solvent induced. To complete the reaction, the bromide has to compete with the reverse nido to closo change. The same closo to nido change is also a major rate determining step for reactions with P(OPh)3 in oxygen-free solutions, and for reactions with bromide in oxygenated solutions in the presence of trifluoroacetic and some other acids. The limiting rates increase slightly with increasing basicity of the ligands P(p-XC6H4)3 along the series X = F3C, Cl, F, H and MeO. Activation parameters for these reactions are reported.     

syn-2,3-Disubstituted-2,3-dihydro-1,4-benzoxathiin rings: preparation from quinone ketals and 2-mercaptoethanols

A general method for the preparation of syn-2,3-disubstituted-2,3-dihydro-1,4-benzoxathiin rings from 2-mercaptoethanols and quinone ketals is presented. This ring system is produced by Michael addition of a 2-mercaptoethanol to a quinone ketal, followed by cyclization of the initial Michael adduct, and subsequent aromatization to afford a syn-2,3-disubstituted-1,4-benzoxathiin in fair to good chemical yield. Several chiral syn-2,3-disubstituted-2,3-dihydro-1,4-benzoxathiin rings were prepared with this method from enantioenriched 2-mercaptoethanols. No loss of enantiopurity was observed.     

Reactions of Molecules with Two Equivalent Functional Groups. 5. Anomalous Reactivity of 1,10-Cyclooctadecanedione. Crystal and Molecular Structures of cis,cis-1,10-Diphenyl-1,10-cyclooctadiene and the Stereoisomers of 1,10-Diphenyl-1,10-cyclooctadecanediol(1)

The reaction of the title diketone (3) with phenyl Grignard produces (with rate constant k(1)) the conjugate base (6-M) of 10-hydroxy-10-phenylcyclooctadecanone (6), which is subsequently converted (with rate constant k(2)) to the conjugate base of the title diol, as a mixture of the cis (7, 55%) and trans (8, 45%) isomers. The ratio k(2)/k(1), 2.2 +/- 0.4, indicates that the carbonyl group in 6-M is 4.4 times as reactive as each carbonyl in 3. Competition experiments further demonstrate that the relative rates (per carbonyl) for addition of phenylmagnesium bromide to 3, 10-methylenecyclooctadecanone (11), and cyclopentadecanone (12) are 1.0:0.60:1.92. Possible reasons for this order of reactivity are discussed. Diols 7 and 8 undergo facile double dehydration to form the title diene 13, which is predicted by molecular mechanics calculations to be the most stable of the four possible symmetrical diene isomers. The structures of 7, 8, and 13 were secured by single-crystal X-ray studies.