An effective-hamiltonian approach to CH5+, using ideas from atomic spectroscopy

In this talk we present the first steps in the design of an effective Hamiltonian for the vibration-rotation energy levels of CH$_{5}^{+}$. Such a Hamiltonian would allow calculation of energy level patterns anywhere along the path travelled by a hypothetical CH$_{5}^{+}$ (or CD$_{5}^{+}$) molecule as it passes through various coupling cases, and might thus provide some hints for assigning the observed high-resolution spectra. The steps discussed here, which have not yet addressed computational problems, focus on mapping the vibration-rotation problem in CH$_{5}^{+}$ onto the five-electron problem in the boron atom, using ideas and mathematical machinery from Condon and Shortley’s book on atomic spectroscopy. The mapping ideas are divided into: (i) a mapping of particles, (ii) a mapping of coordinates (i.e., mathematical degrees of freedom), and (iii) a mapping of quantum mechanical interaction terms. The various coupling cases along the path correspond conceptually to: (i) the analog of a free-rotor limit, where the H atoms see the central C atom but do not see each other, (ii) the low-barrier and high-barrier tunneling regimes, and (iii) the rigid-molecule limit, where the H atoms remain locked in some fixed molecular geometry. Since the mappings considered here often involve significant changes in mathematics, a number of interesting qualitative changes occur in the basic ideas when passing from B to CH$_{5}^{+}$, particularly in discussions of: (i) antisymmetrization and symmetrization ideas, (ii) $n,l,m_{l},m_{s}$ or $n,l,j,m_{j}$ quantum numbers, and (iii) Russell-Saunders computations and energy level patterns. Some of the mappings from B to CH$_{5}^{+}$ to be discussed are as follows. Particles: the atomic nucleus is replaced by the C atom, the electrons are replaced by protons, and the empty space between particles is replaced by an “electron soup.” Coordinates: the radial coordinates of the electrons map onto the five local C-H stretching modes, the angular coordinates of the electrons map onto three rotational degrees of freedom and seven bending vibrational degrees of freedom. The half-integral electron spins map onto half-integral proton spins or onto integral deuterium spins (for CD$_{5}^{+}$). Interactions: the Coulomb attraction between nucleus and electrons maps onto a Morse-oscillator C-H stretching potential, spin-orbit interaction maps onto proton-spin-overall-rotation interaction, and Coulomb repulsion between electrons maps onto some kind of proton repulsion that leads to the equilibrium geometry

FURTHER PROGRESS IN FITTING 13000 TORSION-WAGGING-ROTATIONAL MW AND IR vt = 0,1 TRANSITIONS IN CH3NH2 USING THE HYBRID (TUNNELLING + INTERNAL ROTATION) PROGRAM

A few years ago, the authors wrote a hybrid program to fit rotational levels in molecules with one CH$_{3}$ internal-rotation large-amplitude motion, one NH$_{2}$ inversion large-amplitude motion, and symmetry described by the G$_{12}$ PI group. This program was applied with success to the MW spectrum of 2-methylmalonaldehyde, but the rather small data set for this molecule did not provide a stringent test of the model. More challenging is the application of the hybrid program to CH$_{3}$NH$_{2}$, since this molecule has a much larger data set, containing both MW and IR transitions, as well as having a more extensive $v_{t}$, $J$, and $K$ quantum number coverage. In our ISMS talk this year we will first give an overview of our best least-squares fit to date: The data set contains slightly more than 2500 MW and 11000 IR transitions with $J \le 32$ and $K \le 14$, which are fit to a weighted standard deviation of 1.64 using 71 parameters. Next, we present an assessment of this fit’s strong points (e.g., significantly less parameters, ability to predict spectra in higher torsional states) and weak points (e.g., somewhat larger standard deviation, greater parameter correlation) when compared to the best all-tunneling-model fit in the literature. Based on this assessment, we believe that our fit, as well as the predictive abilities of the program, are sufficiently good that we can now begin considering collaborations with measurement and assignment campaigns of $v_{t}$ = 1 MW data and $v_{t}$ = 2, 3 IR data already underway in other laboratories. Finally, we will present a slightly modified ordering scheme for the operators in this hybrid program, and describe the need for devising a contact transformation treatment to specify determinable parameters in the hybrid Hamiltonian, in order to reduce parameter-correlation problems during the trial-and-error fitting process. A knowledge of determinable parameters would be particularly useful here, since there is little previous experience to guide the choice of “a good set” of higher-order constants to float when carrying out large fits having over 10000 lines, $J_{max}$ = 40, $K_{max}$ = 15, $v_{t}$ = 0 and 1 torsional states, A$_{1}$, A$_{2}$, B$_{1}$, B$_{2}$, E$_{1}$, E$_{2}$ symmetry species, and nearly 100 parameters

UNUSUAL INTERNAL ROTATION COUPLING IN THE MICROWAVE SPECTRUM OF PINACOLONE

The molecular-beam Fourier-transform microwave spectrum of pinacolone (methyl textit{tert}-butyl ketone) has been measured in several regions between 2 and 40 GHz. Assignments of a large number of A and E transitions were confirmed by combination differences, but fits of the assigned spectrum using several torsion-rotation computer programs based on different models led to the unexpected conclusion that no existing program correctly captures the internal dynamics of this molecule. A second puzzle arose when it became clear that roughly half of the spectrum remained unassigned even after all predicted transitions were added to the assignment list. Quantum chemical calculations carried out at the MP2/6-311++G(d,p) level indicate that this molecule does not have a plane of symmetry at equilibrium, and that internal rotation of the light methyl group induces a large oscillatory motion of the heavy textit{tert}-butyl group from one side of the $C_s$ saddle point to the other. The effect of this non-$C_s$ equilibrium structure was modeled for $J = 0$ levels by a simple two-top torsional Hamiltonian, where magnitudes of the strong top-top coupling terms were determined directly from the textit{ab initio} two-dimensional potential surface. A plot of the resultant torsional levels on the same scale as a one-dimensional potential curve along the zig-zag path connecting the six (unequally spaced) minima bears a striking resemblance to the 1:2:1 splitting pattern of levels in an internal rotation problem with a six-fold barrier. A plot of the six minima closely resembles the potential surface for methylamine. This talk will focus on implications of these resemblances for future work

CROSS-CONTAMINATION OF FITTING PARAMETERS IN MULTIDIMENSIONAL TUNNELING TREATMENTS

In this talk we examine the two-dimensional tunneling formalism used previously to fit the hydrogen-transfer and internal-rotation splittings in the microwave spectrum of 2-methylmalonaldehyde in an effort to determine the origin of various counterintuitive results concerning the isotopic dependence of the internal-rotation splittings in that molecule. We find that the cause of the problem lies in a “parameter contamination” phenomenon, where some of the numerical magnitude of splitting parameters from modes with large tunneling splittings “leaks into” the parameters of modes with smaller tunneling splittings. We further find that such parameter contamination, which greatly complicates the determination of barrier heights from the least-squares-fitted splitting parameters, will be a general problem in spectral fits using the multi-dimensional tunneling formalism, since it arises from subtle mathematical features of the non-orthogonal framework functions used to set up the tunneling Hamiltonian. Transforming to a physically less intuitive orthonormal set of basis functions allows us to give an approximate numerical estimate of the contamination of tunneling parameters for 2-methylmalonaldehyde by combining a dominant tunneling path hypothesis with results recently given for the hydrogen-transfer--internal-rotation potential function for this molecule._x000d

K-scrambling in a near-symmetric top molecule containing an excited noncoaxial internal rotor

9 pages, 8 figures, 2 tables.-- PACS: 34.30.+h; 33.15.Mt; 33.15.HpClassical trajectories on rotational energy surfaces and coherent-state quantum projections have been used to study an asymmetric-top molecule containing a freely rotating internal symmetric top whose symmetry axis is not coincident with a principal axis of the molecule. Stationary points on the rotational energy surface, which strongly influence the trajectories, increase in number from two to four to six as J/n increases from zero to infinity (where J is the total and n is the free-internal-rotor angular momentum). For some J/n values trajectories can arise that sample a large fraction of K values (where K is the z-axis projection of J), corresponding in quantum wave functions to extensive K mixing in the symmetric-top basis set |J,K. When such mixing cannot be made small for any choice of z axis, we call it K scrambling. For typical values of the torsion–rotation coupling parameter , rotational eigenfunctions for a given J and torsional state turn out to be quite different from eigenfunctions for the same J in some other torsional state. Nonzero rotational overlap integrals are then distributed among many rotational functions for each (n,n) pair, which may, in turn, contribute to internal rotation enhancement of intramolecular vibrational energy redistribution. We have also examined near-free-rotor levels of our test molecule acetaldehyde, which arise for excitation of ten or more quanta of methyl group torsion, and find that barrier effects do not change the qualitative picture obtained from the free-rotor treatment.This work was supported in part by the Spanish DGES (Project No. PB96-0881) and the Division of Chemical Sciences, Office of Basic Energy Sciences, Office of Energy Research, U.S. Department of Energy.Peer reviewe

Rotational energy surfaces of molecules exhibiting internal rotation

10 pags., 6 figs., 3 tabs.Rotational energy surfaces [W. G. Harter and C. W. Patterson, J. Chem. Phys. 80, 4241 (1984)] for a molecule with internal rotation are constructed. The study is limited to torsional states at or below the top of the barrier to internal rotation, where the extra (torsional) degree of freedom can be eliminated by expanding eigenvalues of the torsion-K-rotation Hamiltonian as a Fourier series in the rotational degree of freedom. For acetaldehyde, considered as an example, this corresponds to considering vt=0, 1, and 2 (below the barrier) and vt=3 (just above the barrier): The rotational energy surfaces are characterized by locating their stationary points (maxima, minima, and saddles) and separatrices. Rather complicated catastrophe histories describing the creation and annihilation of pairs of stationary points as a function of 7 are found at moderate J for given torsional quantum number (v t) and symmetry species (A,E). Trajectories on the rotational energy surface which quantize the action are examined, and changes from rotational to vibrational trajectories caused by changes in the separatrix structure are found as a function of J for vt=2. The concept of a "best" quantization axis for the molecule-fixed component of the total angular momentum is examined from a classical point of view, and it is shown that labeling ambiguities encountered in the literature for torsion-rotation energy levels, calculated numerically in the rho-axis system, can be eliminated by reprojecting basis-set K values onto an axis passing through an appropriate stationary point on the rotational energy surface. © 1994 American Institute of Physics.This work was supported in part by the Division of Chemical Sciences, Office of Basic Energy Sciences, Office of Energy Research, U.S. Department of Energy, and by the Consejo Superior de Investigaciones Cientificas of Spain

APPROXIMATE THEORETICAL MODEL FOR THE FIVE ELECTRONIC STATES (Ω\Omega = 5/2, 3/2, 3/2, 1/2, 1/2) ARISING FROM THE GROUND 3d93d^{9} CONFIGURATION IN NICKEL HALIDE MOLECULES AND FOR ROTATIONAL LEVELS OF THE TWO Ω\Omega = 1/2 STATES IN THAT MANIFOLD

Author Institution: Optical Technology Division, NIST, Gaithersburg, MD 20899-8441, USAAn effective Hamiltonian for a non-rotating diatomic molecule containing only crystal-field and spin-orbit operators has been set up to describe the energies of the five spin-orbit components that arise in the ground electronic configuration of the nickel monohalides. The model assumes that bonding in the nickel halides has the approximate form Ni$^{+}$X$^{-}$, with an electronic $3d^{9}$ configuration plus closed shells on the Ni$^{+}$ moiety and a closed shell configuration on the X$^{-}$ moiety. Least-squares fits of the observed five spin-orbit components of the three lowest electronic states in NiF and NiCl are then carried out in terms of the three crystal field parameters $C_{0}, C_{2}, C_{4}$ and the spin-orbit coupling constant $A$. Following this, the usual effective Hamiltonian $B(\textbf{J-L-S})^2$ for a rotating diatomic molecule is used to derive expressions for the unusually large $\Omega$-type doubling parameter $p$ in the two $\Omega$ = 1/2 states in the $3d^{9}$ manifold. These expressions show (for certain sign conventions) that the sum of the two $p$ values should be $-2B$, but that their difference can vary between $-10B$ and $+10B$. The theoretical magnitudes for $p$ are in good agreement with the two observed $p$ values for both NiF and NiCl, but the signs are not. The experimental signs can be brought into agreement with the theoretical signs by a fairly massive change in +/- parity assignments in the NiF and NiCl literature. The last part of the talk will focus on the theoretical and experimental implications of these parity changes

National Bureau of Standards Reports

Report describing procedures for making quantum mechanical calculations of rotational energy levels and rotational line intensities in diatomic molecules. The procedures are illustrated by sample calculations. A familiarity with the material of this report should enable a practicing electronic spectroscropist to carry out, though in a rather mechanical way, his own theoretical calculations for molecules under experimental investigation