Parent-molecule rotational depolarization of photofragment angular momentum distributions: diatomic and polyatomic molecules

We extend the A(q)(k) polarization-parameter model, which describes product angular momentum polarization from one photon photodissociation of polyatomic molecules in the molecular frame [J. Chem. Phys., 2010, 132, 224310], to the case of rotating parent molecules. The depolarization of the A(q)(k) is described by a set of rotational depolarization factors that depend on the angle of rotation of the molecular axis γ. We evaluate these rotational depolarization factors for the case of dissociating diatomic molecules and demonstrate that they are in complete agreement with the results of Kuznetsov and Vasyutinskii [J. Chem. Phys., 2005, 123, 034307] obtained from a fully quantum mechanical approach of the same problem, showing the effective equivalence of the two approaches. We further evaluate the set of rotational depolarization factors for the case of dissociating polyatomic molecules that have three (near) equal moments of inertia, thus extending these calculations to polyatomic systems. This ideal case yields insights for the dissociation of polyatomic molecules of various symmetries when we compare the long lifetime limit with the results obtained for the diatomic case. In particular, in the long lifetime limit the depolarization factors of the A(0)(k) (odd k), Re(A(1)(k)) (even k) and Im(A(1)(k)) (odd k) for diatomic molecules vanish; in contrast, for polyatomic molecules the depolarization factors for the A(0)(k) (odd k) reduce to a value of 1/3, whereas for the Re(A(1)(k)) (even k) and Im(A(1)(k)) (odd k) they reduce to 1/5.     

Rotational angular momentum polarization: the influence of stray magnetic fields

We show that weak residual magnetic fields can significantly affect the preparation and measurement of molecular rotational angular momentum alignment in a typical gas-phase stereodynamics apparatus. Specifically, polarization spectroscopy, a third-order nonlinear spectroscopic technique, is used to prepare and probe the collisional and noncollisional losses of rotational angular momentum alignment of OH X (2)Pi. Residual magnetic fields of the order of the geomagnetic field are shown to have a significant effect on the prepared polarization on a submicrosecond timescale. This can be expected to be a significant effect for many gas-phase free radicals, such as those of interest in combustion, atmospheric chemistry, and the burgeoning field of cold molecules. We demonstrate a simple experimental remedy for this problem.     

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.     

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.     

The role of angular momentum in collision-induced vibration-rotation relaxation in polyatomics

Vibrational relaxation of the 6(1) level of S(1)((1)B(2u)) benzene is analyzed using the angular momentum model of inelastic processes. Momentum-(rotational) angular momentum diagrams illustrate energetic and angular momentum constraints on the disposal of released energy and the effect of collision partner on resultant benzene rotational excitation. A kinematic "equivalent rotor" model is introduced that allows quantitative prediction of rotational distributions from inelastic collisions in polyatomic molecules. The method was tested by predicting K-state distributions in glyoxal-Ne as well as J-state distributions in rotationally inelastic acetylene-He collisions before being used to predict J and K distributions from vibrational relaxation of 6(1) benzene by H(2), D(2), and CH(4). Diagrammatic methods and calculations illustrate changes resulting from simultaneous collision partner excitation, a particularly effective mechanism in p-H(2) where some 70% of the available 6(1)-->0(0) energy may be disposed into 0-->2 rotation. These results support the explanation for branching ratios in 6(1)-->0(0) relaxation given by Waclawik and Lawrance and the absence of this pathway for monatomic partners. Collision-induced vibrational relaxation in molecules represents competition between the magnitude of the energy gap of a potential transition and the ability of the colliding species to generate the angular momentum (rotational and orbital) needed for the transition to proceed. Transition probability falls rapidly as DeltaJ increases and for a given molecule-collision partner pair will provide a limit to the gap that may be bridged. Energy constraints increase as collision partner mass increases, an effect that is amplified when J(i)>0. Large energy gaps are most effectively bridged using light collision partners. For efficient vibrational relaxation in polyatomics an additional requirement is that the molecular motion of the mode must be capable of generating molecular rotation on contact with the collision partner in order to meet the angular momentum requirements. We postulate that this may account for some of the striking propensities that characterize polyatomic energy transfer.     

CNDO calculation of polarizabilities including polarization functions in the basis set

Electric polarizabilities are calculated by solving the first- and second-order perturbation equations through a variational procedure. Satisfactory numerical results are obtained for a number of molecules using the CNDO approximation with an extended basis set.     

The photodissociation dynamics of O2 at 193 nm: an O3PJ angular momentum polarization study

In the following paper we present translational anisotropy and angular momentum polarization data for O((3)P(1)) and O((3)P(2)) products of the photodissociation of molecular oxygen at 193 nm. The data were obtained using polarized laser photodissociation coupled with resonantly enhanced multiphoton ionization and velocity-map ion imaging. Under the jet-cooled conditions employed, absorption is believed to be dominated by excitation into the Herzberg continuum. The experimental data are compared with previous experiments and theoretical calculations at this and other wavelengths. Semi-classical calculations performed by Groenenboom and van Vroonhoven [J. Chem. Phys, 2002, 116, 1965] are used to estimate the alignment parameters arising from incoherent excitation and dissociation and these are shown to agree qualitatively well with the available experimental data. Following the work of Alexander et al. [J. Chem. Phys, 2003, 118, 10566], orientation and alignment parameters arising from coherent excitation and dissociation are modelled more approximately by estimating phase differences generated subsequent to dissociation via competing adiabatic pathways leading to the same asymptotic products. These calculations lend support to the view that large values of the coherent alignment moments, but small values of the corresponding orientation moments, could arise from coherent excitation of (and subsequent dissociation via) parallel and perpendicular components of the Herzberg I, II and III transitions.     

The decomposition of the SCF interaction energy in hydrogen bonded dimers corrected for basis set superposition errors: An examination of the basis set dependence

A systematic examination of the components of the interaction energy, obtained with the Kitaura and Morokuma method, for nine H-bonded dimers without and with counterpoise corrections (CP ) is presented. The nine dimers H_n A …? HBH_m correspond to all the possible combinations of HF, H₂O, and NH₃ as electron donors and electron acceptors. The interaction energy and the corresponding components have been computed over a sizable interval of intermolecular distances with five basis sets (STO -3G, MINI -1, 3-21G, 4-31G, 6-31G**) selected among those most extensively used to study interactions in larger systems. The CP corrections to the ΔE components have been obtained with a method, implemented in our group, which permits assignment to the pertinent components of ΔE of a physically reasonable portion of the CP correction even though different CP corrections are adopted. We examine here three versions of the CP correction, namely, the full CP correction (i.e., the original version of Boys and Bernardi) and CP corrections limited to the virtual space of the partner or to the electron donor only. The resulting data are employed to assess the basis set dependence of several models of hydrogen bonding (the electrostatic model, the semiclassical model, etc.) both with and without CP corrections.     

The photodissociation dynamics of ozone at 193 nm: an O(1D2) angular momentum polarization study

Polarized laser photolysis, coupled with resonantly enhanced multiphoton ionization detection of O(1D2) and velocity-map ion imaging, has been used to investigate the photodissociation dynamics of ozone at 193 nm. The use of multiple pump and probe laser polarization geometries and probe transitions has enabled a comprehensive characterization of the angular momentum polarization of the O(1D2) photofragments, in addition to providing high-resolution information about their speed and angular distributions. Images obtained at the probe laser wavelength of around 205 nm indicate dissociation primarily via the Hartley band, involving absorption to, and diabatic dissociation on, the B 1B2(3 1A1) potential energy surface. Rather different O(1D2) speed and electronic angular momentum spatial distributions are observed at 193 nm, suggesting that the dominant excitation at these photon energies is to a state of different symmetry from that giving rise to the Hartley band and also indicating the participation of at least one other state in the dissociation process. Evidence for a contribution from absorption into the tail of the Hartley band at 193 nm is also presented. A particularly surprising result is the observation of nonzero, albeit small values for all three rank K = 1 orientation moments of the angular momentum distribution. The polarization results obtained at 193 and 205 nm, together with those observed previously at longer wavelengths, are interpreted using an analysis of the long range quadrupole-quadrupole interaction between the O(1D2) and O2(1Deltag) species.     

The photodissociation dynamics of ozone at 226 and 248 nm: O(3PJ) atomic angular momentum polarization

Speed distributions, and spatial anisotropy and atomic angular momentum polarization parameters have been determined for the O((3)P(J)) products following the photodissociation of ozone at 248 and 226 nm using velocity map ion imaging. The data have been interpreted in terms of two dissociation mechanisms that give rise to fast and slow products. In both cases, excitation is believed to occur to the B state. Consistent with previous interpretations, the speed distributions, translational anisotropy parameters, and angular momentum polarization moments support the assignment of the major pathway to curve crossing from the B to the repulsive R surface, generating fast fragments in a wide range of vibrational states. For the slow fragments, it is proposed that following excitation to the B state, the system crosses onto the A state. The crossing seam is only accessible to molecules that are highly vibrationally excited and therefore possess modest recoil speeds. Once on the A state, the wavepacket is thought to funnel through a conical intersection to the ground state. The velocity distributions, spatial anisotropy parameters, spin-orbit populations and polarization data each lend support to this mechanism.     

The effect of basis set superposition error on the convergence of interaction energies

 For the intermolecular interaction energies of ion-water clusters [OH⁻(H₂O) _n (n=1,2), F⁻(H₂O), Cl⁻(H₂O), H₃O⁺(H₂O) _n (n=1,2), and NH₄ ⁺(H₂O) _n (n=1,2)] calculated with correlation-consistent basis sets at MP2, MP4, QCISD(T), and CCSD(T) levels, the basis set superposition error is nearly zero in the complete basis set (CBS) limit. That is, the counterpoise-uncorrected intermolecular interaction energies are nearly equal to the counterpoise-corrected intermolecular interaction energies in the CBS limit. When the basis set is smaller, the counterpoise-uncorrected intermolecular interaction energies are more reliable than the counterpoise-corrected intermolecular interaction energies. The counterpoise-uncorrected intermolecular interaction energies evaluated using the MP2/aug-cc-pVDZ level is reliable. Received: 14 March 2001 / Accepted: 25 April 2001 / Published online: 9 August 2001     

Photofragment angular momentum distribution beyond the axial recoil approximation: the role of molecular axis rotation

We present the quantum-mechanical expressions for 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 paper generalizes the result of Siebbeles et al. [J. Chem. Phys. 100, 3610 (1994)] to the case of dissociation of rotating molecules. The electronic wave function of the molecule was used in the adiabatic body-frame representation. The obtained rigorous expressions for the fragment state multipoles have been explicitly derived from the scattering wave-function formalism and then simplified using the quasiclassical approximation in the high-J limit. Possible radial and Coriolis nonadiabatic interactions have been taken into consideration. It is shown that the rotation of the molecular axis is described by a number of rotation factors which depend on the rank of the incident-photon polarization matrix, on the dissociation mechanism, and on the classical angle of rotation of the molecular axis gamma.     

The photodissociation dynamics of OCS at 248 nm: the S((3)P(J)) atomic angular momentum polarization

The dissociation of OCS has been investigated subsequent to excitation at 248 nm using velocity map ion imaging. Speed distributions, speed dependent translational anisotropy parameters, and the atomic angular momentum orientation and alignment are reported for the channel leading to S((3)P(J)). The speed distributions and beta parameters are in broad agreement with previous work and show behavior that is highly sensitive to the S-atom spin-orbit state. The data are shown to be consistent with the operation of at least two triplet production mechanisms. Interpretation of the angular momentum polarization data in terms of an adiabatic picture has been used to help identify a likely dissociation pathway for the majority of the S((3)P(J)) products, which strongly favors production of J=2 fragment atoms, correlated, it is proposed, with rotationally hot and vibrationally cold CO cofragments. For these fragments, optical excitation to the 2 (1)A(') surface is thought to constitute the first step, as for the singlet dissociation channel. This is followed by crossing, via a conical intersection, to the ground 1 (1)A(') state, from where intersystem crossing occurs, populating the 1 (3)A(')1 (3)A(")((3)Pi) states. The proposed mechanism provides a qualitative rationale for the observed spin-orbit populations, as well as the S((3)P(J)) quantum yield and angular momentum polarization. At least one other production mechanism, leading to a more statistical S-atom spin-orbit state distribution and rotationally cold, vibrationally hot CO cofragments, is thought to involve direct excitation to either the (3)Sigma(-) or (3)Pi states.     

Photodissociation dynamics of OCS at 248 nm: the S((1)D(2)) atomic angular momentum polarization

The dissociation of OCS has been investigated subsequent to excitation at 248 nm. Speed distributions, speed dependent translational anisotropy parameters, angular momentum alignment, and orientation are reported for the channel leading to S((1)D(2)). In agreement with previous experiments, two product speed regimes have been identified, correlating with differing degrees of rotational excitation in the CO coproducts. The velocity dependence of the translational anisotropy is also shown to be in agreement with previous work. However, contrary to previous interpretations, the speed dependence is shown to primarily reflect the effects of nonaxial recoil and to be consistent with predominant excitation to the 2 (1)A(') electronic state. It is proposed that the associated electronic transition moment is polarized in the molecular plane, at an angle greater than approximately 60 degrees to the initial linear OCS axis. The atomic angular momentum polarization data are interpreted in terms of a simple long-range interaction model to help identify likely surfaces populated during dissociation. Although the model neglects coherence between surfaces, the polarization data are shown to be consistent with the proposed dissociation mechanisms for the two product speed regimes. Large values for the low and high rank in-plane orientation parameters are reported. These are believed to be the first example of a polyatomic system where these effects are found to be of the same order of magnitude as the angular momentum alignment.     

The 6-31G++ basis set: An economical basis set for correlated wavefunctions

The 6-31G ⁺⁺ basis set is described. This basis set is very similar to the existing 6-31G ^** set but is somewhat smaller through the use of five (rather than six) second-order Gaussians (d functions) and has polarization function exponents optimized for correlated rather than Hartree–Fock wavefunctions. The performance of 6-31G ⁺⁺ is compared with that of the 6-31G ^** and 6-31G ^** basis sets through calculation of the geometries and atomization energies for the set of molecules LiH, FH, H₂O, NH₃, CH₄, N₂, CO, HCN, and HCCH.     

On the floating basis set model for studying the vibronic interaction in polyatomic molecules

The floating basis set model by Johnson and Weigang is reinvestigated using a general picture of the molecular hamiltonian. It is found that Johnson and Weigang, though considering explicitly that electrons move with the nuclei, ignored a contribution to the electronic wavefunction from the coupling of the vibrational and electronic motions of the electrons. The vibrational wavefunctions and energies of the molecule are also modified.     

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.     

Symmetrization of eigenfunctions of angular momentum in point groups

The eigenfunctions |jm〉 of angular momentum can combine linearly to make basis functions of irreducible representations of point groups. We surmount the projection operator and find a new method to calculate the combination coefficients. It is proven that these coefficients are components of eigenvectors of some hermitian matrices, and that for all pure rotation point groups, the coefficients can be made real numbers by properly choosing the azimuth angles of symmetry elements of point groups in the coordinate system. We apply the coupling theory of angular momentum to obtain the general formulas of the basis functions of point groups. By use of our formulas, we have calculated the basis functions with half‐integers j from 1/2 to 13/2 of double‐valued irreducible representations for the icosahedral group. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 83: 286–302, 2001     

The effect of basis set superposition error on the convergence of intermolecular interaction energies for deprotonated complexes

The intermolecular interaction energies of the deprotonated hydrogen-bonded complexes F(-)(HF), F(-)(H(2)O), F(-)(NH(3)), Cl(-)(HF), SH(-)(HF), H(2)P(-)(HF), OH(-)(H(2)O), OH(-)(H(2)O)(2), OH(-)(NH(3)), Cl(-)(H(2)O), SH(-)(H(2)O), H(2)P(-)(H(2)O), Cl(-)(NH(3)), SH(-)(NH(3)), H(2)P(-)(NH(3)), Cl(-)(HCl), Cl(-)(H(2)S), Cl(-)(PH(3)), SH(-)(H(2)S), SH(-)(PH(3)), and H(2)P(-)(PH(3)) were calculated with correlation consistent basis sets at the MP2, MP4, QCISD(T), and CCSD(T) levels. When the basis set is smaller, the counterpoise-uncorrected intermolecular interaction energies are closer to the complete basis set limit than the counterpoise-corrected intermolecular interaction energies. The counterpoise-uncorrected intermolecular interaction energies obtained at the MP2/aug-cc-pVDZ level of theory are close to the interaction energies obtained at the extrapolated complete basis set limit in most of the complexes. Also, we investigate the accuracy of the other levels.