The Multiresonant Hamiltonian Model and Polyad Quantum Numbers

Chapter 4

Downloaded by NORTH CAROLINA STATE UNIV on January 11, 2013 | http://pubs.acs.org Publication Date: June 10, 1997 | doi: 10.1021/bk-1997-0678.ch004

The Multiresonant Hamiltonian Model and Polyad Quantum Numbers for Highly Excited Vibrational States William F. Polik and J. Ruud van Ommen Department of Chemistry, Hope College, 35 East 12th Street, Holland, MI 49423

The multi-resonant Hamiltonian model is a spectroscopic Hamiltonian capable of fitting highly excited vibrational states in polyatomic molecules with a minimum number of parameters. It treats nonresonant interactions among harmonic oscillator product basis states using second-order perturbation theory and resonant interactions explicitly using harmonic oscillator matrix elements. The pure vibrational spectrum of formaldehyde (H CO) is analyzed up to 10000 cm with several multi-resonant Hamiltonian models, resulting in fits with less than 3 cm standard deviation. The identified vibrational resonances imply three polyad quantum numbers for formaldehyde: N = v , N = v + v + v + v , and N = 2v + v + v + v + 2v + v . The goodness of these vibrational polyad quantum numbers is discussed, and generalizations to other chemical systems are offered. 2

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A fundamental understanding of chemical reactivity requires a detailed understanding of the reactant molecules. Molecular structure has been studied in great detail for many molecules at or near the equilibrium geometry. However, far fewer studies have been carried out on molecules in excited vibrational states. For a reaction with an activation energy barrier, only molecules in excited vibrational states can sample phase space far from the equilibrium geometry and ultimately undergo chemical reaction. While molecules reacting on a single potential energy surface possess both rotational and vibrational energy, the significant geometry change required to reach the transition state geometry typically requires large amounts of vibrational energy. Thus, a model for chemical reactivity must account for the nature of highly excited vibrational states in reactant molecules. Spectroscopic characterization of excited vibrational states offer direct insight into the nature of energetic molecules. At low energies in small molecules, interactions between specific vibrational states can cause level shifting. Such © 1997 American Chemical Society In Highly Excited Molecules; Mullin, A., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1997.

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interactions are characterized as spectroscopic perturbations and are typically analyzed by using nondegenerate perturbation theory or diagonalizing small matrices. At higher energies, the onset of extensive state mixing can lead to the apparent destruction of the regular patterns of energy levels which existed at lower energies. At still higher energies or in larger molecules, fractionation of energy levels may be observed due to mixing of the spectroscopically active zero-order state with unspecified background states. Ultimately, complete state mixing leads to a statistical description of energy levels. The mixing of vibrational states is commonly referred to as Intramolecular Vibrational Redistribution (IVR) (7). The extent of IVR is directly measurable in a spectrum, assuming that the evidence for state mixing is not obscured by another effect, e.g., rotational congestion. Accurate experimental measurements of excited state properties also assist the theoretical description of excited molecules. The dynamics of a chemical reaction is governed by the underlying potential energy surface (PES). While quantum chemical calculations are generally reliable for obtaining the correct PES topology and are becoming increasingly accurate, they have not yet achieved "spectroscopic accuracy." Thus, one of the best ways to obtain an accurate PES is to start with a calculated PES, which has the correct functional form and good initial values for the potential parameters, and refine this surface using accurate experimental parameters. A second way in which experiments assist theory is by serving as a rigorous test for quantum chemistry methodologies. If general methods can be developed that accurately calculate experimental results for several prototypical systems, then those methods can be more trusted to make accurate predictions for similar, unmeasured chemical systems. The determination of high quality data sets for excited molecular states is therefore essential to the continued development of quantum chemistry methodologies. In this paper, several methods for obtaining spectra of excited vibrational levels are reviewed and contrasted. One method for obtaining vibrational spectra which are free from rotational congestion is described, and results are presented for formaldehyde. The multi-resonant Hamiltonian model is developed and used to fit the assigned formaldehyde spectrum. The power of this model is that it accounts for the apparent destruction of regular spectroscopic sequences with a minimum number of parameters, each of which has a simple physical interpretation. An analysis is performed on the resonances used in the Hamiltonian model, revealing conserved dynamical quantities known as polyad quantum numbers. Conclusions are drawn regarding the dynamics of formaldehyde, and generalizations to other molecular systems are offered. Spectroscopy of Excited Vibrational States Spectra of excited states in molecules are more difficult to obtain than spectra of states near equilibrium. The most straightforward method of accessing excited states is high overtone absorption spectroscopy. In this method, a laser is scanned through the energy region of interest and absorption is directly or indirectly monitored. A particularly sensitive implementation is intracavity optoacoutistic spectroscopy, in which the sample cell is placed inside a dye laser cavity, the excitation source is

In Highly Excited Molecules; Mullin, A., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1997.

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POLIK & VAN O M M E N

Multiresonant Hamiltonian Model

chopped, and absorbance is monitored with an acoustic microphone at the chopping frequency through a lock-in amplifier (2). Because the transitions being measured are vibrational overtones, which are forbidden in the harmonic oscillator approximation, they usually have extremely weak absorbances. The absorbance strength of a transition depends on the anharmonicity and the quantum number change of the vibration. High overtone spectroscopy is therefore best suited for studying C - H or O-H stretching vibrations, as stretches are typically more anharmonic than bends and fewer quanta of excitation are required to reach a given energy due to the relatively high frequency of the hydrogen stretch. High overtone spectroscopy suffers from broad spectroscopic linewidths, typically 50 c m or greater, due to inhomogeneous broadening from rotational congestion at room temperature. Such broad linewidths can prevent detection of weaker transitions or nearby states, as well as obscure the homogeneous linewidth which is the measure of the IVR rate. The recently developed method of infrared laser assisted photofragment spectroscopy (IRLAPS) allows infrared absorption spectra to be recorded in a supersonic expansion, thereby reducing the problems associated with rotational congestion (3). A tremendous breakthrough in the study of excited states came with the development of Stimulated Emission Pumping (SEP) (4). In this method, a fixed "pump" laser populates a single rovibronic level in an excited electronic state, and fluorescence from the laser populated level is monitored. A scanning "dump" laser stimulates emission from the level populated by the pump laser to the excited states of interest. Transitions to excited states are detected as dips in the fluorescence from the laser populated level. Since SEP involves an electronic transition, transition intensities depend on Franck-Condon factors. SEP spectroscopy therefore complements high overtone spectroscopy by accessing different excited states. SEP has two main advantages over high overtone spectroscopy. SEP is a double-resonance method which allows the use of rotational selection rules to simplify a spectrum or to confirm assignments. Also, spectroscopic linewidths are typically laser or Doppler limited, resulting in linewidths on the order of 0.1 cm" with pulsed lasers. The chief limitation of SEP is that signals are detected as small changes on top of a large, potentially noisy, fluorescence background. In practice, the dynamic range of SEP is limited by saturation effects or upward transitions from high "dump" laser intensities at one end and by the extent to which fluctuations in the background fluorescence level can be reduced at the other end. A variation of SEP which circumvents the problem of a fluctuating background is degenerate four-wave mixing spectroscopy (SEP-DFWM) which offers significantly increased sensitivity at the expense of signal linearity (5). Dispersed Fluorescence (DF) spectroscopy has enjoyed a recent resurgence as a method for obtaining excited state spectra (6). In this method, a "pump" laser populates a single rovibronic level in an excited electronic state, just as with SEP. Instead of using a second laser to stimulate emission down to excited vibrational states, however, a monochromator is used to disperse the fluorescence. Transitions to excited states are detected as peaks in the fluorescence intensity spectrum. As with SEP, DF spectroscopic intensities depend on Franck-Condon factors and rotational selection rules can be used to simplify a spectrum. The principal advantages of DF over SEP are that DF is a zero background technique, is not subject to saturation effects, and can be recorded much more quickly. Spectroscopic transitions observed -1


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by DF are not obscured by background fluorescence fluctuations, although in practice one is ultimately limited by detector readout noise. Intensities are much more reliable in DF because there is no danger of saturation. A 10000 c m DF spectrum can be recorded in a day, whereas a comparable SEP spectrum would require months or years of effort along with multiple dye and optics changes. The principle disadvantage of DF relative to SEP is resolution, which is typically monochromator limited resulting in linewidths of several cm . DF and SEP methods are complementary in that DF permits rapid acquisition of medium resolution survey spectra while SEP is useful for higher resolution studies of more limited frequency ranges. 1


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Figure 1. Techniques for recording spectra of excited vibrational states, a) Overtone spectroscopy, b) Stimulated Emission Pumping (SEP), c) Dispersed Fluorescence (DF). Pure Vibrational Spectroscopy of Formaldehyde Formaldehyde (H CO) is one of the simplest and most studied polyatomic molecules. Excited vibrational levels in formaldehyde have been studied by all three aforementioned methods: overtone, SEP, and DF spectroscopy. Prior to 1984, 24 vibrational states had been assigned primarily by analysis of Fourier transform infrared (FTIR) overtone and combination band spectra. In 1984 Reisner et al. assigned 57 new vibrational states using SEP spectroscopy and obtained the first complete set of harmonic and anharmonic spectroscopic constants, ω,° and χ , for a tetra-atomic molecule (7). In 1996 Bouwens et al. assigned 198 new vibrational states using DF spectroscopy in a supersonic expansion, bringing the total number of assigned vibrational states in So formaldehyde to 279 (6). Bouwens et al. also corrected several misassignments in the literature and determined an improved set of vibrational spectroscopic constants. Formaldehyde has also been the subject of many theoretical studies. The most accurate ab initio calculation of the PES to date is the 1993 study by Martin, Lee, and Taylor (8). Burleigh, McCoy, and Sibert have used the data of Bouwens et al. to refine this surface to 7,600 cm" , resulting in an average mean deviation between experimental and calculated energies of 1.5 cm" (9). The tremendous increase in the number of assigned states for formaldehyde was due to the recently developed technique of pure vibrational spectroscopy, in which selection rules are used to eliminate rotational congestion entirely (6). Formaldehyde belongs to the point group, and the So and S\ electronic states have 2

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A\ and A electronic symmetry, respectively. Vibrational states for which both v and v +v are even have Αι vibrational symmetry, states for which both v and v +v are odd have A vibrational symmetry, states for which v is odd and v +v is even have B\ vibrational symmetry, and states for which v is even and v +v is odd have B vibrational symmetry. Evaluation of the electric dipole transition integral by symmetry arguments reveals that in the 5ι

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Off-diagonal matrix elements are of the form ( . . . , ν , . , ν ^ ν ^ . . ^ . · . ^ , +l,v . +l,v* -1,...) ;

= ^ ( v . k h + l X v , | i , | v , + l)(v,|

The Multiresonant Hamiltonian Model and Polyad Quantum Numbers

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