Understanding highly excited states via parametric variations

Highly excited vibrational states of an isolated molecule encode the vibrational energy flow pathways in the molecule. Recent studies have had spectacular success in understanding the nature of the excited states mainly due to the extensive studies of the classical phase space structures and their bifurcations. Such detailed classical-quantum correspondence studies are presently limited to two or quasi two dimensional systems. One of the main reasons for such a constraint has to do with the problem of visualization of relevant objects like surface of sections and Wigner or Husimi distributions associated with an eigenstate. This neccesiates various alternative techniques which are more algebraic than geometric in nature. In this work we introduce one such method based on parametric variation of the eigenvalues of a Hamiltonian. It is shown that the level velocities are correlated with the phase space nature of the corresponding eigenstates. A semiclassical expression for the level velocities of a single resonance Hamiltonian is derived which provides theoretical support for the correlation. We use the level velocities to dynamically assign the highly excited states of a model spectroscopic Hamiltonian in the mixed phase space regime. The effect of bifurcations on the level velocities is briefly discussed using a recently proposed spectroscopic Hamiltonian for the HCP molecule.Comment: 12 pages, 9 figures, submitted to J. Chem. Phy

A New Approach toward Transition State Spectroscopy

Chirped-Pulse millimetre-Wave (CPmmW) rotational spectroscopy provides a new class of information about photolysis transition state(s). Measured intensities in rotational spectra determine species-isomer-vibrational populations, provided that rotational populations can be thermalized. The formation and detection of S0 vinylidene is discussed in the limits of low and high initial rotational excitation. CPmmW spectra of 193 nm photolysis of Vinyl Cyanide (Acrylonitrile) contain J=0-1 transitions in more than 20 vibrational levels of HCN, HNC, but no transitions in vinylidene or highly excited local-bender vibrational levels of acetylene. Reasons for the non-observation of the vinylidene co-product of HCN are discussed.Comment: Accepted by Faraday Discussion

Resonance dynamics of DCO (X~ 2A′\widetilde{X}\,{}^2A') simulated with the dynamically pruned discrete variable representation (DP-DVR)

Selected resonance states of the deuterated formyl radical in the electronic ground state ($\widetilde{X}\,{}^2A'$) are computed using our recently introduced dynamically pruned discrete variable representation (DP-DVR) [H. R. Larsson, B. Hartke and D. J. Tannor, J. Chem. Phys., 145, 204108 (2016)]. Their decay and asymptotic distributions are analyzed and, for selected resonances, compared to experimental results obtained by a combination of stimulated emission pumping (SEP) and velocity-map imaging of the product D atoms. The theoretical results show good agreement with the experimental kinetic energy distributions. The intramolecular vibrational energy redistribution (IVR) is analyzed and compared with previous results from an effective polyad Hamiltonian. Specifically, we analyzed the part of the wavefunction that remains in the interaction region during the decay. The results from the polyad Hamiltonian could mainly be confirmed. The C=O stretch quantum number is typically conserved, while the D-C=O bend quantum number decreases. Differences are due to strong anharmonic coupling such that all resonances have major contributions from several zero-order states. For some of the resonances, the coupling is so strong that no further zero-order states appear during the dynamics in the interaction region, even after propagating for 300 ps.Comment: in pres

Topologically coupled energy bands in molecules

We propose a concrete application of the Atiyah-Singer index formula in molecular physics, giving the exact number of levels in energy bands, in terms of vector bundles topology. The formation of topologically coupled bands is demonstrated. This phenomenon is expected to be present in many quantum systems.Comment: 16 pages, 3 figure

Embeddings and Ramsey numbers of sparse k-uniform hypergraphs

Chvatal, Roedl, Szemeredi and Trotter proved that the Ramsey numbers of graphs of bounded maximum degree are linear in their order. In previous work, we proved the same result for 3-uniform hypergraphs. Here we extend this result to k-uniform hypergraphs, for any integer k > 3. As in the 3-uniform case, the main new tool which we prove and use is an embedding lemma for k-uniform hypergraphs of bounded maximum degree into suitable k-uniform `quasi-random' hypergraphs.Comment: 24 pages, 2 figures. To appear in Combinatoric

Spectroscopic Interpretation: The High Vibrations of CDBrClF

We extract the dynamics implicit in an algebraic fitted model Hamiltonian for the deuterium chromophore's vibrational motion in the molecule CDBrClF. The original model has 4 degrees of freedom, three positions and one representing interbond couplings. A conserved polyad allows in a semiclassical approach the reduction to 3 degrees of freedom. For most quantum states we can identify the underlying motion that when quantized gives the said state. Most of the classifications, identifications and assignments are done by visual inspection of the already available wave function semiclassically transformed from the number representation to a representation on the reduced dimension toroidal configuration space corresponding to the classical action and angle variables. The concentration of the wave function density to lower dimensional subsets centered on idealized simple lower dimensional organizing structures and the behavior of the phase along such organizing centers already reveals the atomic motion. Extremely little computational work is needed.Comment: 23 pages, 6 figures. Accepted for publication in J. Chem. Phy

A sharp threshold for random graphs with a monochromatic triangle in every edge coloring

Let $\R$ be the set of all finite graphs $G$ with the Ramsey property that every coloring of the edges of $G$ by two colors yields a monochromatic triangle. In this paper we establish a sharp threshold for random graphs with this property. Let $G(n,p)$ be the random graph on $n$ vertices with edge probability $p$. We prove that there exists a function $\hat c=\hat c(n)$ with $0 0$, as $n$ tends to infinity Pr[G(n,(1-\eps)\hat c/\sqrt{n}) \in \R ] \to 0 and Pr [ G(n,(1+\eps)\hat c/\sqrt{n}) \in \R ] \to 1. A crucial tool that is used in the proof and is of independent interest is a generalization of Szemer\'edi's Regularity Lemma to a certain hypergraph setting.Comment: 101 pages, Final version - to appear in Memoirs of the A.M.