Photofragment angular momentum distributions in the molecular frame. III. Coherent effects in the photodissociation of polyatomic molecules with circularly polarized light

We extend the a(q) (k)(s) polarization parameter model [T. P. Rakitzis and A. J. Alexander, J. Chem. Phys. 132, 224310 (2010)] to describe the components of the product angular momentum polarization that arise from the one-photon photodissociation of asymmetric top molecules with circularly polarized photolysis light, and provide a general equation for fitting experimental signals. We show that the only polarization parameters that depend on the helicity of the circularly polarized photolysis light are the A(0) (k) and Re[A(1) (k)] (with odd k) and the Im[A(1) (k)] (with even k); in addition, for the unique recoil destination (URD) approximation [for which the photofragment recoil v arises from a unique parent molecule geometry], we show that these parameters arise only as a result the interference between at least two dissociative electronic states. Furthermore, we show that in the breakdown of the URD approximation (for which the photofragment recoil v arises from a distribution of parent molecule geometries), these parameters can also arise for dissociation via a single dissociative electronic state. In both cases, the A(0) (k) and Re[A(1) (k)] parameters (with odd k) are proportional to cosΔφ, and the Im[A(1) (k)] parameters (with even k) are proportional to sinΔφ, where Δφ is the phase shift (or average phase shift) between the interfering paths so that Δφ can be determined directly from the A(q) (k), or from ratios of these A(q) (k) parameters. Therefore, the determination of these A(q) (k) parameters with circularly polarized photolysis light allows the unambiguous measurement of coherent effects in polyatomic-molecule photodissociation.     

Maximum-entropy analysis of momentum densities in diatomic molecules

The one-particle density in momentum space γ(p) is studied for diatomic molecules by using the maximum-entropy technique. The knowledge of one or more momentum expectation values <p"> provides approximations on the density γ(p) for any value of the momentum, which are convergent when increasing the number of known moments. Other unknown expectation values are estimated in terms of the constructed maximum-entropy densities. A numerical study of the quality of the approximations is carried out by means of experimental and theoretical data for the momentum expectation values involved. Experimental errors are also taken into account to have an idea of the sensibility of the results to the information from which they are obtained. © 1997 John Wiley & Sons, Inc.     

On the use of NDO approximate wavefunctions in the evaluation of momentum density and radial momentum density distributions in polyatomic molecules

The use of single determinantal approximate molecular wavefunctions of the LCAO MO NDO type for the calculation of the momentum densityρ(p) and the radial momentum density distributionJ(p) is discussed. In each case, these expressions should be orientationally invariant and the momentum density should be normalized. Combining these two requirements, it is shown that only two approximations are physically significant:

1. NDO wavefunctions are used andρ(p) andI(p) are approximated respectively up to an INDO and a CNDO level;
2. Overlap integrals are explicitly taken into account when solving the Roothaan SCF equations or deorthogonalized NDO functions are employed, together with the unapproximated expressions.

     

Electron momentum density distribution in homonuclear diatomic molecules

Individual orbital contributions to the electron momentum densities of first-row homonuclear diatomic molecules are discussed. It is shown that the nodal surfaces in the orbital EMDs arise from a diffraction factor with both geometric and electronic components. The positions of the nodal surfaces convey information on the electronic structure. The results are illustrated with a Hartree-Fock-Slater calculation of the F₂(X¹Σ_g⁺) molecule.     

The rotational structure of three-photon resonances of polyatomic molecules

A general formulation of the selection rules and line strengths of three-photon excitation spectra is presented for molecules of arbitrary symmetry. For symmetric-top molecules the paper extends the discussion by Nieman in the context of rovibronic transitions of NH₃. For asymmetric-top molecules it is shown that the rotational line strengths can be sensitive to quantum-mechanical interference between contributions from the various components of the three-photon vibronic transition tensor. This transition tensor is discussed within a perturbation-theory framework in terms of several models of intermediate-state participation.     

Two-photon state selection and angular momentum polarization probed by velocity map imaging: application to H atom photofragment angular distributions from the photodissociation of two-photon state selected HCl and HBr

A formalism for calculating the angular momentum polarization of an atom or a molecule following two-photon excitation of a J-selected state is presented. This formalism is used to interpret the H atom photofragment angular distributions from single-photon dissociation of two-photon rovibronically state selected HCl and HBr prepared via a Q-branch transition. By comparison of the angular distributions measured using the velocity map imaging technique with the theoretical model it is shown that single-photon dissociation of two-photon prepared states can be used for pathway identification, allowing for the identification of the virtual state symmetry in the two-photon absorption and/or the symmetry of the dissociative state. It is also shown that under conditions of excitation with circularly polarized light, or for excitation via non-Q-branch transitions with linearly polarized light the angular momentum polarization is independent of the dynamics of the two-photon transition and analytically computable.     

A correlation of excited state rotational constants for diatomic molecules

The concepts of “orbital stress” and “transition stress” are defined and applied to N₂, N⁺₂, CO, and CO⁺. The bond lengths and rotational constants of excited electronic states are related to the transition stress, and the response of the electrons and nuclei to the transition stress is shown to be a molecular property, essentially independent of the electronic configuration or state.     

Vibrational—rotational structure in the angular distribution and intensity of photoelectrons from diatomic molecules. III. Effects of higher partial

Total and differential cross sections for the photoionization of H₂ are calculated for each vibrational—rotational transition induced in the molecule. The transition moment is evaluated in the two-center spheroidal coordinates with varying internuclear distance. The wavefunction for the ejected electron is calculated with partial-wave coupling taken into account. It is found that the calculation with only the lowest partial wave retained gives a sufficiently reliable result for most of the cross sections for 584 A and 736 A photons. Only the exception is the anisotropy parameter for the photoelectron angular distribution in the case where rational transition |ΔJ| = 2 occurs. The parameter, especially for fast photoelectrons, increases very much when higher partial waves are included in the calculation. Some preliminary results for the incidence of 304 A line are also shown.     

Formaldehyde photodissociation: dependence on total angular momentum and rotational alignment of the CO product

Quasiclassical trajectory calculations are reported to investigate the effects of rotational excitation of formaldehyde on the branching ratios of the fragmentation products, H2+CO and H+HCO. The results of tens of thousands of trajectories show that increased rotational excitation causes suppression of the radical channel and enhancement of the molecular channel. Decomposing the molecular channel into "direct" and "roaming" channels shows that increased rotation switches from suppressing to enhancing the roaming products across our chosen energy range. However, decomposition into these pathways is difficult because the difference between them does not appear to have a distinct boundary. A vector correlation investigation of the CO rotation shows different characteristics in the roaming versus direct channels and this difference is a potentially useful signature of the roaming mechanism, as first speculated by Kable and Houston in their experimental study of photodissociation of acetaldehyde [P. L. Houston and S. H. Kable, Proc. Nat. Acad. Sci. 103, 16079 (2006)].     

Experimental and theoretical investigations of angular distributions of fragments from dissociations of polyatomic ions

The possibility of correcting relative intensities of fragment ions obtained by angle-resolved mass spectrometry, for the effects of translational energy release, was investigated. The computational procedure is essentially an extension of that described by Todd et al. [10] to take account of a distribution function for translational energy release. The theoretical model permits derivation of a condition under which the deconvolution should be least ambiguous. In practice, the limiting features of the procedure appear to be lack of knowledge of the translational energy release distributions, and the very poor inter-laboratory reproducibility of the experimental data. Some experimental data are reported here for the case of the molecular ions of n-butylbenzene, obtained under conditions of high angular resolution (±0.02°). Floating the collision cell electrically, to separate unimolecular and collision-induced reactions, is shown to have a marked effect upon both relative intensities and values of translational energy release. Reasonable agreement was obtained between the present experimental data and theoretical predictions.     

Rovibrational momentum densities of diatomic molecules in the rigid rotor-harmonic oscillator approximation

The rigid rotor-harmonic oscillator model in the momentum representation is used to examine the rovibrational momentum density of a diatomic molecule. The effects on this function of rotational and vibrational excitation, thermal averaging, and nuclear spin statistics are exhibited in calculations on N₂ and thehydrogen halides, and a brief comparison with electron momentum densities is made.     

The nodal structure of the momentum distributions of molecules

The nodal structure of molecular momentum distributions is studied by considering the simplest case of the ground state of the hydrogen molecular ion. By examining the exact expansion of the H₂⁺ momentum distribution, it is shown that an infinite sequence of nodes does exist along the p_z axis (z axis parallel to the bond axis) but not nodal planes perpendicular to the p_z axis (as is found for the simplest LCAO function). The nodes are those points where nonplanar nodal surfaces cross the p_z axis. It is also shown that molecular systems with more than one electron cannot, in the ground state, have nodal surfaces in their momentum distributions. Implications for the directional Compton profiles J( q ) are discussed.     

The parity-adapted basis set in the formulation of the photofragment angular momentum polarization problem: the role of the Coriolis interaction

We present a theoretical framework for calculating the recoil-angle dependence of the photofragment angular momentum polarization taking into account both radial and Coriolis nonadiabatic interactions in the diatomic/linear photodissociating molecules. The parity-adapted representation of the total molecular wave function has been used throughout the paper. The obtained full quantum-mechanical expressions for the photofragment state multipoles have been simplified by using the semiclassical approximation in the high-J limit and then analyzed for the cases of direct photodissociation and slow predissociation in terms of the anisotropy parameters. In both cases, each anisotropy parameter can be presented as a linear combination of the generalized dynamical functions fK(q,q',q,q') of the rank K representing contribution from different dissociation mechanisms including possible radial and Coriolis nonadiabatic transitions, coherent effects, and the rotation of the recoil axis. In the absence of the Coriolis interactions, the obtained results are equivalent to the earlier published ones. The angle-recoil dependence of the photofragment state multipoles for an arbitrary photolysis reaction is derived. As shown, the polarization of the photofragments in the photolysis of a diatomic or a polyatomic molecule can be described in terms of the anisotropy parameters irrespective of the photodissociation mechanism.     

Effects of angular momentum recoupling on depolarization spectra in alkali-rare gas optical half collisions

The depolarization mechanisms of excited alkali atoms interacting with ground state rare gas atoms are investigated using the method of far wing spectroscopy of collision pairs under single collision conditions. From a semiclassical theory explicite expressions for the spectra of alkali multipole components σ _k ^(?) (ω _L ) are derived assuming quasistatic excitation at a localized internuclear separationR _L(ω _L ) related to the laserfrequency ω _L as well as realistic models for the half collision following excitation. The collision models are characterized by different Hund's coupling regions, where excitation takes place and which are traversed on the collision path. Due to selection rules for excitation of populations and coherences and for support of multipoles the σ _k ^(?) (ω _L ) are shown to depend strongly on the collision model. From the spectra thus a labelling of the initial molecular states and mapping of the change in coupling case is possible. Estimations of the contributions of the various angular momentum recoupling effects are given.     

Static charge distributions which generate exact potential curves for diatomic molecules

For a diatomic molecule, there exists a charge density which can be used to generate the exact potential energy curve for the molecule, by integration of the classical Poisson equation. This density is displayed for H⁺₂, H₂ and LiH, and it is shown to be similar in shape to the united-atom electron density.     

Finite electric field valence shell calculations of polarizability gradients and raman depolarization ratios for diatomic molecules

Calculation of polarizability gradients have been made for a number of diatomic molecules using the Finite Field CNDO/II approximate SCF method. Comparison with experimental results suggests that the method will be generally useful for the prediction and interpretation of Raman intensities.     

The discrete representation correspondence between quantum and classical spatial distributions of angular momentum vectors

This work demonstrates that the quantum mechanical moments of a state described by the density matrix correspond to discrete spherical harmonic moments of the classical multipole expansion of the spatial distribution of the angular momentum vectors. For the diagonal density matrix elements, this work exploits the fact that the quantum mechanical vector coupling (Clebsch-Gordan) coefficients become increasingly accurate discrete representations of spherical harmonics as j increases. A Schwinger-type basis accounts for nonaxially symmetric angular distributions, which result in nonzero off-diagonal elements of the density matrix. The resulting discrete minimum uncertainty picture of the classical moments has a stringent equivalence with the quantum mechanical one for all j and provides an unambiguous connection for the classical and quantum moments in the large j limit. The equivalence is numerically tested for simple models, and there is a satisfying equivalence even for small j. Applications, implications, and extensions are indicated, and the relevance of this work for the interpretation of classical mechanical simulations of inelastic and reactive molecular collisions will be documented elsewhere.     

Semiclassical IVR approach to rotational excitation of non-polar diatomic molecules by non-resonant laser pulses

We apply the ‘classic’ semiclassical initial value representation (SC-IVR) approach to describe rotational excitation of non-polar diatomic molecules by intense short non-resonant laser pulses. We also investigate the applicability of the quantum mechanical sudden approximation. It is found that the SC-IVR approach gives accurate rotational excitation probabilities in regimes where the sudden approximation fails. Primitive semiclassical wavefunction propagation is used to illustrate the phenomenon of angular focussing of rotation states by strong pulses.