The subduction of vibrational states of molecules from the momentum space of simple harmonic motion

A single-particle model of molecular vibrational states is proposed in which the normal modes are projected out of the body vibrations of an infinite simple harmonic sphere. This model assigns the spurious change of mass or centre of mass and leads to removal of mass monopoles and dipoles from the system. These conservation conditions impose strict boundary conditions on the potential and basis functions. On incorporation into the model they result in a set of loop equations in which the potential is proportional to the fundamental vibration. The simplest solutions to these equations strongly resemble the Poschl-Teller generalization of the Morse potential. The solutions have been extended to incorporate the repulsive states and generate the set of net attractive states appropriate to the anharmonic potential.The basis functions of this potential display both angular and radial node structures. The degeneracies between radial and angular mode patterns can be studied by transformation into an angular coordinate space. In this way coupling to other phenomena described in similar angular momentum space can be performed directly before subduction to real displacement space.On leave from the Department of Chemistry, Catholic University of Leuven, Celestijnenlaan 200F, B-3030 Leuven-Belgium     

High-resolution millimeter wave spectroscopy and multichannel quantum defect theory of the hyperfine structure in high Rydberg states of molecular hydrogen H2

Experimental and theoretical methodologies have been developed to determine the hyperfine structure of molecular ions from detailed studies of the Rydberg spectrum and have been tested on molecular hydrogen. The hyperfine structure in l=0-3 Rydberg states of H2 located below the X 2Sigmag+(v+=0,N+=1) ground state of ortho H2+ has been measured in the range of principal quantum number n=50-65 at sub-MHz resolution by millimeter wave spectroscopy following laser excitation to np and nd Rydberg states using a variety of single-photon and multiphoton excitation sequences. The np1(1), nd1(1), and the nf1(0-3) Rydberg states were found to be metastable and to have lifetimes of more than 5 micros beyond n=50. Members of other series, such as the nd1(2), nd1(3), and the np1(0) series, were found to have lifetimes of more than 1 mus. Local perturbations induced by low-n Rydberg states belonging to series converging on rovibrationally excited levels of H2+ reduce the lifetimes in narrow ranges of n values. The hyperfine structure is strongly dependent on the value of the orbital angular momentum l. In the penetrating s and p states at n approximately 50 the exchange interaction dominates over the hyperfine interaction and the levels can be labeled by the total electron spin angular momentum quantum number S (S=0 or 1). In the less penetrating d and f Rydberg states, the hyperfine interaction between the core nuclear and electron spins is larger than the exchange interaction and the Rydberg states are of mixed singlet and triplet character. A procedure based on the Stark effect and on the systematic analysis of selection rules and combination differences was developed to determine the orbital and the total angular momentum quantum numbers l and F and to construct an energy map of p and f Rydberg levels between n=54 and 64 with relative positions of an accuracy of better than 1 MHz. Multichannel quantum defect theory (MQDT) was extended to treat the hyperfine structure in molecular Rydberg states and was used to analyze the observed hyperfine structure of the p and f Rydberg states of H2. The frame transformation between the Born-Oppenheimer channels described by the angular momentum coupling scheme (abetaJ) and the asymptotic channels described by the (e[bbetaS+]) coupling scheme was derived and enables an elegant treatment of all intermediate coupling cases. Purely ab initio quantum defect theory reproduced the experimentally determined positions to within 40 MHz for the p levels and 13 MHz for the f levels. By slight adjustments of the quantum defect functions and their energy dependences and by consideration of the p-f interaction, of the singlet-triplet splittings of the f levels, and of the departure of the ionic levels from pure coupling case (bbetaS+), the agreement between theory and experiment could be improved to 600 kHz. By comparing the results of MQDT calculations of the hyperfine structure of f Rydberg levels with those of coupled equations calculations, the frame transformation approximation of MQDT was shown to be accurate to within 300 kHz. The extrapolated ionic hyperfine structure of the X 2Sigmag+(v+=0,N+=1) ionic level corresponds to the ab initio prediciton of Babb and Dalgarno [Phys. Rev. A 46, R5317 (1992)] within the experimental error.     

Speed dependent rotational angular momentum polarization of the O2 (a 1Deltag) fragment following ozone photolysis in the wavelength range 248-265 nm

The translational anisotropy and rotational angular momentum polarization of a selection of rotational states of the O2 (a 1Deltag; v=0) photofragment formed from ozone photolysis at 248, 260, and 265 nm have been determined using the technique of resonance enhanced multiphoton ionization in combination with time of flight mass spectrometry. At 248 nm, the dissociation is well described as impulsive in nature with all rotational states exhibiting similarly large, near-limiting values for the bipolar moments describing their angular momentum alignment and orientation. At 265 nm, however, the angular momentum polarization parameters determined for consecutive odd and even rotational states exhibit clear differences. Studies at the intermediate wavelength of 260 nm strongly suggest that such a difference in the angular momentum polarization is speed dependent and this proposal is consistent with the angular momentum polarization parameters extracted and reported previously for longer photolysis wavelengths [G. Hancock et al., Phys. Chem. Chem. Phys. 5, 5386 (2003); S. J. Horrocks et al., J. Chem. Phys. 126, 044308 (2007)]. The alternation of angular momentum polarization for successive odd and even J states may be a consequence of the different mechanisms leading to the formation of the two O2 (a 1Deltag) Lambda doublets. Specifically, the involvement of out of plane parent rotational motion is proposed as the origin for the observed depolarization for the Delta- relative to the Delta+ state.     

Dyson orbitals for ionization from the ground and electronically excited states within equation-of-motion coupled-cluster formalism: theory, implementation, and examples

Implementation of Dyson orbitals for coupled-cluster and equation-of-motion coupled-cluster wave functions with single and double substitutions is described and demonstrated by examples. Both ionizations from the ground and electronically excited states are considered. Dyson orbitals are necessary for calculating electronic factors of angular distributions of photoelectrons, Compton profiles, electron momentum spectra, etc, and can be interpreted as states of the leaving electron. Formally, Dyson orbitals represent the overlap between an initial N-electron wave function and the N-1 electron wave function of the corresponding ionized system. For the ground state ionization, Dyson orbitals are often similar to the corresponding Hartree-Fock molecular orbitals (MOs); however, for ionization from electronically excited states Dyson orbitals include contributions from several MOs and their shapes are more complex. The theory is applied to calculating the Dyson orbitals for ionization of formaldehyde from the ground and electronically excited states. Partial-wave analysis is employed to compute the probabilities to find the ejected electron in different angular momentum states using the freestanding and Coulomb wave representations of the ionized electron. Rydberg states are shown to yield higher angular momentum electrons, as compared to valence states of the same symmetry. Likewise, faster photoelectrons are most likely to have higher angular momentum.     

Calculation of argon trimer rovibrational spectrum

Rovibrational spectra of Ar3 are computed for total angular momenta up to J=6 using row-orthonormal hyperspherical coordinates and an expansion of the wave function on hyperspherical harmonics. The sensitivity of the spectra to the two-body potential and to the three-body corrections is analyzed. First, the best available semiempirical pair potential (HFDID1) is compared with our recent ab initio two-body potential. The ab initio vibrational energies are typically 1-2 cm-1 higher than the semiempirical ones, which is related to the slightly larger dissociation energy of the semiempirical potential. Then, the Axilrod-Teller asymptotic expansion of the three-body correction is compared with our newly developed ab initio three-body potential. The difference is found smaller than 0.3 cm-1. In addition, we define approximate quantum numbers to describe the vibration and rotation of the system. The vibration is represented by a hyper-radial mode and a two-degree-of-freedom hyperangular mode, including a vibrational angular momentum defined in an Eckart frame. The rotation is described by the total angular momentum quantum number, its projection on the axis perpendicular to the molecular plane, and a hyperangular internal momentum quantum number, related to the vibrational angular momentum by a transformation between Eckart and principal-axes-of-inertia frames. These quantum numbers provide a qualitative understanding of the spectra and, in particular, of the impact of the nuclear permutational symmetry of the system (bosonic with zero nuclear spin). Rotational constants are extracted from the spectra and are shown to be accurate only for the ground hyperangular mode.     

Propensity rules for orientation by atom impact: II. Many-state description of direct transitions

This paper contains a theoretical analysis of orientation and alignment created in direct, collision-induced transitions among atomic states with arbitrary angular momentum. Using the natural coordinate frame, general propensity rules are derived in the velocity region of maximum transition probability and their range of validity is investigated. The predictions are tested and illustrated by nine-state calculations for Li(n=2, 3) transitions in Li-He collisions.     

Photofragment angular momentum distribution beyond the axial recoil approximation: predissociation

We present the quantum mechanical expressions for the angular momentum distribution of the photofragments produced in slow predissociation. The paper is based on our recent theoretical treatment [J. Chem. Phys. 123, 034307 (2005)] of the recoil angle dependence of the photofragment multipole moments which explicitly treat the role of molecular axis rotation on the electronic angular momentum polarization of the fragments. The electronic wave function of the molecule was used in the adiabatic body frame representation. The rigorous expressions for the fragment state multipoles which have been explicitly derived from the scattering wave function formalism have been used for the case of slow predissociation where a molecule lives in the excited quasibound state much longer than a rotation period. Possible radial nonadiabatic interactions were taken into consideration. The optical excitation of a single rotational branch and the broadband incoherent excitation of all possible rotational branches have been analyzed in detail. The angular momentum polarization of the photofragments has been treated in the high-J limit. The polarization of the photofragment angular momenta predicted by the theory depends on photodissociation mechanism and can in many cases be significant.     

A new method for the derivation of exact vibration-rotational kinetic energy operator in internal coordinates

An efficient angular momentum method is presented and used to derive analytic expressions for the vibration-rotational kinetic energy operator of polyatomic molecules.The vibration-rotational kinetic energy operator is expressed in terms of the total angular momentum operator J,the angular momentum operator J and the momentum operator p conjugate to Z in the molecule-fixed frame Not only the method of derivation is simpler than that in the previous work,but also the expressions ot the kinetic energy operators arc more compact.Particularly,the operator is easily applied to different vibrational or rovibrational problems of the polyatomic molecules by variations of matrix elements Gn of a mass-dependent constant symmetric matrix     

Quantum theory of molecular collisions in a magnetic field: efficient calculations based on the total angular momentum representation

An efficient method is presented for rigorous quantum calculations of atom-molecule and molecule-molecule collisions in a magnetic field. The method is based on the expansion of the wave function of the collision complex in basis functions with well-defined total angular momentum in the body-fixed coordinate frame. We outline the general theory of the method for collisions of diatomic molecules in the (2)Σ and (3)Σ electronic states with structureless atoms and with unlike (2)Σ and (3)Σ molecules. The cross sections for elastic scattering and Zeeman relaxation in low-temperature collisions of CaH((2)Σ(+)) and NH((3)Σ(-)) molecules with (3)He atoms converge quickly with respect to the number of total angular momentum states included in the basis set, leading to a dramatic (>10-fold) enhancement in computational efficiency compared to the previously used methods [A. Volpi and J. L. Bohn, Phys. Rev. A 65, 052712 (2002); R. V. Krems and A. Dalgarno, J. Chem. Phys. 120, 2296 (2004)]. Our approach is thus well suited for theoretical studies of strongly anisotropic molecular collisions in the presence of external electromagnetic fields.     

Two-step resonant photoionization spectroscopy of Co I and Ni I: continuum structure near the first ionization limit

We explored the continuum structure of Co I and Ni I near the first ionization limit using one-color and two-color two-step photoionization spectroscopy. The ionization spectra of both elements show a large number of autoionizing features. The use of different excitation schemes involving intermediate states with different angular momentum allowed the assignment of the angular momentum for part of the detected autoionizing states. Several of these states could be identified as members of different Rydberg series.     

Theoretical study of the He-HF+ complex. II. Rovibronic states from coupled diabatic potential energy surfaces

The bound rovibronic levels of the He-HF+ complex were calculated for total angular momentum J=1/2, 3/2, 5/2, 7/2, and 9/2 with the use of ab initio diabatic intermolecular potentials presented in Paper I and the inclusion of spin-orbit coupling. The character of the rovibronic states was interpreted by a series of calculations with the intermolecular distance R fixed at values ranging from 1.5 to 8.5 A and by analysis of the wave functions. In this analysis we used approximate angular momentum quantum numbers defined with respect to a dimer body-fixed (BF) frame with its z axis parallel to the intermolecular vector R and with respect to a molecule-fixed (MF) frame with its z axis parallel to the HF+ bond. The linear equilibrium geometry makes the He-HF+ complex a Renner-Teller system. We found both sets of quantum numbers, BF and MF, useful to understand the characteristics of the Renner-Teller effect in this system. In addition to the properties of a "normal" semirigid molecule Renner-Teller system it shows typical features caused by large-amplitude internal (bending) motion. We also present spectroscopic data: stretch and bend frequencies, spin-orbit splittings, parity splittings, and rotational constants.     

Quantum-dynamical study in three-dimensional space of the exchange reaction H+ClH'→HCl+H' with zero total angular momentum $ \left( {\overrightarrow J = 0} \right) $

The 3D quantum dynamics of the exchange reaction H + ClH′→HCl+H′ with zero angular momentum was studied. The nonstationary Schr?dinger equation was solved by expansion into a basis set using discrete variable representation. The probabilities of the reaction were determined in relation to the total energy for the ground and first excited vibrational states.     

Algebraic approach to the asymmetric rigid rotor

A pure algebraic treatment of the eigenvalue equation corresponding to the asymmetric top is presented. The algebraic method employs the Holstein–Primakoff bosonic realization of the angular momentum algebra. Explicit determination of the linear boson transformation coefficients of the eigenstates is carried out by means of the coherent states formalism. No reference to special functions is needed and a completely algebraic approach is achieved. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 77: 704–709, 2000     

REMPI-TOF studies of the translational anisotropy and the polarization of the O ((1)D2) photofragment angular momentum following ozone photolysis at 298 nm

The translational anisotropy and the polarization of the electronic angular momentum of the O ((1)D2) fragment produced from the 298 nm photodissociation of ozone have been determined using resonance enhanced multiphoton ionization (REMPI) in conjunction with time-of-flight mass spectrometry (TOFMS). The translational anisotropy parameter beta, which is necessarily averaged over the O2 co-fragment rotational distribution, is measured to be 1.08 +/- 0.04. This is consistent with that expected for the (1)B2 <-- (1)A1 transition within an impulsive model if the tangential velocity associated with the zero point motion of the bend is constricted to opening the bond angle. Molecular frame polarization parameters of rank up to k = 4 have been extracted for the O ((1)D2) fragment and the calculated m(J) populations show a strong preference for the absolute value(m(J)) = 1 states. A small coherence term is also observed, a manifestation of the nuclear geometry of the dissociating molecule and the existence of possible non-adiabatic processes in the exit channel. The orientation associated with the mapping of the photon helicity onto the O ((1)D2) electronic angular momentum distribution was observed to have been quenched. However, the parameter gamma1', which describes the contribution to the orientation from a coherent superposition of a parallel and perpendicular excitation where the photofragment angular momentum lies perpendicular to both the recoil velocity and to the transition dipole moment, was determined to be -0.06.     

Vector correlations in photodissociation of polarized polyatomic molecules beyond the axial recoil limit

We present the full quantum mechanical theory of the angular momentum distributions of photofragments produced in photolysis of oriented/aligned parent polyatomic molecules beyond the axial recoil limit. This paper generalizes the results of Underwood and Powis(28,29) to the case of non-axial recoil photodissociation of an arbitrary polyatomic molecule. The spherical tensor approach is used throughout this paper. We show that the recoil angular distribution of the angular momentum polarization of each of the photofragments can be presented in a universal spherical tensor form valid for photolysis in diatomic or polyatomic molecules, irrespective of the reaction mechanism. The angular distribution can be written as an expansion over the Wigner D-functions in terms of the set of the anisotropy-transforming coefficients c(K(i)q(i))(K) (k(d), K(0)) which contain all of the information about the photodissociation dynamics and can be either determined from experiment, or computed from quantum mechanical theory. An important new conservation rule is revealed through the analysis, namely that the component q(i) of the initial reagent polarization rank K(i) and the photofragment polarization rank K onto the photofragment recoil direction k is preserved in any photolysis reaction. Both laboratory and body frame expressions for the recoil angle dependence of the photofragment angular momentum polarization are presented. The parent molecule polarization is shown to lead to new terms in the obtained photofragment angular distributions compared with the isotropic case. In particular, the terms with |q(i)| > 2 can appear which are shown to manifest angular momentum helicity non-conservation in the reaction. The expressions for the coefficients c(K(i)q(i))(K) (k(d), K(0)) have been simplified using the quasiclassical approximation in the high-J limit which allows for introducing the dynamical functions and the rotation factors which describe the decreasing of the photofragment angular momentum orientation and alignment due to the rotation of the molecular axis during photodissociation. In this case, the resultant recoil angle dependence is also presented in a form where the anisotropy of the parent molecular ensemble is expressed in terms of the molecular axis distribution, rather than in terms of the molecular density matrix.     

Representing and selecting vibrational angular momentum states for quasiclassical trajectory chemical dynamics simulations

Linear molecules with degenerate bending modes have states, which may be represented by the quantum numbers N and L. The former gives the total energy for these modes and the latter identifies their vibrational angular momentum jz. In this work, the classical mechanical analog of the N,L-quantum states is reviewed, and an algorithm is presented for selecting initial conditions for these states in quasiclassical trajectory chemical dynamics simulations. The algorithm is illustrated by choosing initial conditions for the N = 3 and L = 3 and 1 states of CO2. Applications of this algorithm are considered for initial conditions without and with zero-point energy (zpe) included in the vibrational angular momentum states and the C-O stretching modes. The O-atom motions in the x,y-plane are determined for these states from classical trajectories in Cartesian coordinates and are compared with the motion predicted by the normal-mode model. They are only in agreement for the N = L = 3 state without vibrational angular momentum zpe. For the remaining states, the Cartesian O-atom motions are considerably different from the elliptical motion predicted by the normal-mode model. This arises from bend-stretch coupling, including centrifugal distortion, in the Cartesian trajectories, which results in tubular instead of elliptical motion. Including zpe in the C-O stretch modes introduces considerable complexity into the O-atom motions for the vibrational angular momentum states. The short-time O-atom motions for these trajectories are highly irregular and do not appear to have any identifiable characteristics. However, the O-atom motions for trajectories integrated for substantially longer period of times acquire unique properties. With C-O stretch zpe included, the long-time O-atom motion becomes tubular for trajectories integrated to approximately 14 ps for the L = 3 states and to approximately 44 ps for the L = 1 states.     

Coordinate transformation methods to calculate state-to-state reaction probabilities with wave packet treatments

A procedure for the transformation from reactant to product Jacobi coordinates is proposed, which is designed for the extraction of state-to-state reaction probabilities using a time-dependent method in a body-fixed frame. The method consists of several steps which involve a negligible extra computational time as compared with the propagation. Several intermediate coordinates are used, in which the efficiency depends on the masses of the atoms involved in the reaction. A detailed study of the relative efficiency of using reactant and product Jacobi coordinates is presented for several systems, and simple arguments are found depending on the masses of the atoms involved in the reaction. It is found that the proposed method is, in general, more efficient than the use of product Jacobi coordinates, specially for nonzero total angular momentum. State-to-state reaction probabilities are obtained for Li+FH-->LiF+H and F+HO-->FH+O collisions for several total angular momenta.     

Even-parity auto-ionizing states of iridium I observed by resonance laser ionization spectroscopy

Auto-ionizing states of neutral iridium were observed in the continuum structure near the first ionization limit using one-color and two-color two-step resonance laser ionization spectroscopy. The total angular momentum of 20 even-parity auto-ionizing states could be determined from a combined analysis of the two-color spectra obtained with ionization schemes using intermediate states with different total angular momentum. Double-resonant ionization schemes were evaluated by fluence-dependence measurements, and photo-ionization cross-sections for resonant ionization transitions were determined. We could also identify several high-lying members of ns, np and nd Rydberg series converging to the first ionization limit of the atom.     

Magnetic circular dichroism measurements of the magnetisation properties of matrix-isolated species having spin angular momentum only

Magnetic circular dichroism has been used for the first time to study the magnetisation properties of some matrix-isolated species. The samples chosen for measurement have one, three, five and six unpaired electrons in their ground electronic states and, nominally, no orbital angular momentum. The experimental results follow the theoretical predictions very closely, but the nature of the magnetisation function is such that it is not easy to distinguish between different combinations of g-value and total spin angular momentum.