Theory of anharmonically modified Coriolis coupling in the S1 state of benzene and relation to experiment

Avoided crossings between quasidegenerate rovibrational states in the Doppler-free two-photon excitation of the 141 mode in the S1 excited state of benzene are treated theoretically. Two sets of avoided crossings in plots of spectral line frequency vs J at a given K and DeltaK have been reported experimentally between an initially prepared "light" state (141 in zeroth order) and dark states, namely, one which in zeroth order is a 51101161 state, the other being in zeroth order a 62111 and/or possibly a 31161 state, implicated earlier by Neusser et al. The identification of these states makes the phenomenon an excellent candidate for treatment of the avoided crossing via a Van Vleck transformation, no other basis set states being needed for the diagonalization in order to extract the important features. Two successive transformations are used for handling direct coupling and coupling via virtual states. The dominant calculated contribution to the coupling is, jointly, Coriolis plus cubic–cubic anharmonic interactions between vibrational modes.Playing less of a role are Coriolis terms in which the inverse moment of inertia tensor is expanded up to quadratic terms in the coordinates. There results a 5×5 (for coupling to 51101161 ) and a 3×3 (for coupling to 62111 or 31161 ) matrix of the transformed Hamiltonian, each of which can also be described, if desired, to a very good approximation by a 2×2 matrix. The coupling element V0 and the difference of the rotational constants for the light and dark states (DeltaB) are obtained from the plots of line position vs J(J+1) obtained. For the 141 to 51101161 and for the 141 to 62111 couplings the theoretical results are in reasonable agreement with the experimental results, no adjustable parameters being employed. For a coupling of 141 to 31161 the calculated V0 would be much too high compared with experiment (a factor of 10), the coupling involving the exchange of only three instead of four vibrational quanta. A situation in which the 141 state is coupled to the 62111 state to yield an avoided crossing and off-resonantly coupled to the 31161 state would be consistent with some experimental results and not affect the reasonable agreement of the slope difference and splitting for the avoided crossing plots

A rectangular additive convolution for polynomials

We define the rectangular additive convolution of polynomials with nonnegative real roots as a generalization of the asymmetric additive convolution introduced by Marcus, Spielman and Srivastava. We then prove a sliding bound on the largest root of this convolution. The main tool used in the analysis is a differential operator derived from the "rectangular Cauchy transform" introduced by Benaych-Georges. The proof is inductive, with the base case requiring a new nonasymptotic bound on the Cauchy transform of Gegenbauer polynomials which may be of independent interest

Crystallization of random matrix orbits

Three operations on eigenvalues of real/complex/quaternion (corresponding to $\beta=1,2,4$) matrices, obtained from cutting out principal corners, adding, and multiplying matrices can be extrapolated to general values of $\beta>0$ through associated special functions. We show that $\beta\to\infty$ limit for these operations leads to the finite free projection, additive convolution, and multiplicative convolution, respectively. The limit is the most transparent for cutting out the corners, where the joint distribution of the eigenvalues of principal corners of a uniformly-random general $\beta$ self-adjoint matrix with fixed eigenvalues is known as $\beta$-corners process. We show that as $\beta\to\infty$ these eigenvalues crystallize on the irregular lattice of all the roots of derivatives of a single polynomial. In the second order, we observe a version of the discrete Gaussian Free Field (dGFF) put on top of this lattice, which provides a new explanation of why the (continuous) Gaussian Free Field governs the global asymptotics of random matrix ensembles.Comment: 25 pages. v2: misprints corrected, to appear in IMR

Theory of fluorescence excitation spectra using anharmonic-coriolis coupling in S1 and internal conversion to S0. I. General formalism

A treatment of one- or two-photon fluorescence excitation spectra is described using the vibration–rotation coupling of zeroth order states in the excited electronic state and nonadiabatic coupling to the ground state. Using perturbation theory, experimental harmonic frequencies, an anharmonic force field, and various theoretical Coriolis coupling constants, a quasistationary molecular eigenstate in an excited electronic state S1 is first calculated. The S1 eigenstate is then coupled via the nonadiabatic nuclear kinetic energy operator (internal conversion) to rovibronic states in the ground state manifold, the latter states approximated in a simple manner. A search algorithm is used to select the S1 dark states and the S0 states. Both the perturbation theory coefficient and the Franck–Condon factors are employed in the evaluation function used in the search. The results are applied in part II to the channel three problem in benzene

Interlacing Families II: Mixed Characteristic Polynomials and the Kadison-Singer Problem

We use the method of interlacing families of polynomials introduced to prove two theorems known to imply a positive solution to the Kadison--Singer problem. The first is Weaver's conjecture $KS_{2}$ \cite{weaver}, which is known to imply Kadison--Singer via a projection paving conjecture of Akemann and Anderson. The second is a formulation due to Casazza, et al., of Anderson's original paving conjecture(s), for which we are able to compute explicit paving bounds. The proof involves an analysis of the largest roots of a family of polynomials that we call the "mixed characteristic polynomials" of a collection of matrices.Comment: This is the version that has been submitte