Bent Triatomic Molecules Research Articles

Angular distributions from photodissociations. V. The bent triatomic molecule

The generalized Franck–Condon theory of bent triatomic molecule dissociations is extended to permit the calculation of state-to-state photofragment angular distributions. The axial recoil limiting forms of Zare and Bersohn are obtained for the state-averaged angular anisotropies β̄ with corrections for the effects of bending vibrations. For certain orientations of the transition dipole within the molecule, βfi is found to be nearly independent of fragment rotational angular momentum j, while for other orientations, it is found to be a strongly oscillatory function of j. In some cases, calculated photofragment rotational distributions at fixed detection angles are strong functions of the detection angle, suggesting the possibility of partial state selection by spatial resolution. Approximate analytical expressions for βfi are presented for initially bound molecular states that are either nonrotating or are well described by a prolate symmetric top wave function with k=0. The distributions are linear combinations of parallel and perpendicular-type distributions. Numerical applications are provided for direct photodissociations from the bent X̃ HOC1 and C̃ HCN states.

Rotational distributions from photodissociations. IV. The bent triatomic molecule

The generalized Franck–Condon theory for the photodissociations of bent triatomic molecules is formulated, including bending and rotational degrees of freedom. The nonseparable multidimensional Franck–Condon amplitudes are reduced to a rapidly convergent series of products of one-dimensional integrals. The Franck–Condon factors provide the photofragment quantum state distributions when inelastic scattering on the unbound surface is negligible. When the scattering is important, they provide the nascent amplitudes which are the driving terms in the full scattering treatment. Calculations of the expected photofragment rotational distributions are given under dipole and scalar couplings which are, respectively, appropriate to direct photodissociation and predissociations. Semiquantitative closed form WKB approximations to the Franck–Condon amplitudes are derived to exhibit the essential physical features of these amplitudes. The results are readily interpreted in terms of the Heisenberg uncertainty principle, the orientational constraints imposed by the bond angle of the bent triatomic molecule, and the conservation of angular momentum. A simple classical approximation is also used and results are in qualitative agreement to the quantal calculations. Comparisons are also provided with recent rotational distributions observed in H2S photodissociation and illustrative calculations are presented for predissociation of HCN from the C̃ state and direct photodissociation of HOCl from its ground state.

Photodissociation of triatomic molecules: Rotational scattering effectsa)

An exact quantum method for calculating the state-to-state direct photodissociation probabilities for arbitrary (linear or bent) triatomic molecules is presented. It is shown that this method, which requires the accumulation of Franck–Condon overlap integrals between the initial state and a scattering state satisfying arbitrary scattering boundary conditions, is equivalent to the solution of the driven equations of Band, Freed, and Kouri. For both HCN and ICN the effects of variation of the final state potential on the vibrational and rotational distributions of the CN fragments are examined. Agreement of our exact calculations with the approximate calculations of Morse and Freed is very good.

The Renner effect in a bent triatomic molecule using the adiabatic approach

The consequences of the Renner-Teller effect in molecules executing large amplitude bending motion have been treated using an adiabatic approach (Barrow, Dixon and Duxbury [8]) and using a perturbed linear molecule approach (Jungen and Merer [7]). This paper makes detailed numerical comparisons of these two methods in calculations of the vibronic levels of H2S+, NH2 and CO2 +. It is shown that the two methods are completely equivalent. For molecules in which at least one component of a Renner-Split degenerate state is bent the adiabatic approach is computationally considerably the less demanding of the two. However, for linear molecules the Jungen and Merer approach is more rapidly convergent.

Quantitative energy level correlations for linear, bent and internally rotating triatomic molecules

This paper is concerned with the study of the quantitative correlation of the rotation-bending energy levels of a linear or bent triatomic molecule with the energy levels of the molecule when there is free internal rotation of a diatomic fragment [as in, for example, Ar(HCl)]. The LiNC and KCN molecules are used as model systems. A correlation parameter γ r is introduced to quantify the position a molecule occupies in these correlation diagrams; this parameter has the value +1 for an ideal bent molecule, −1 for an ideal linear molecule, and −3 for an ideal free internal rotor triatomic molecule. The rotation-bending energy levels of the HCN-HNC isomerization system, and of the double minimum HO 2 system, are also studied.

Rotational distributions in photodissociation: the bent triatomic molecule

The full three-dimensional quantum theory of the photodissociation (both pre- and direct) of a bent triatomic molecule is presented. Franck-Condon amplitudes are evaluated for rotational and orbital angular momentum distributions for predissociation from the C̃ state of HCN and for direct photodissociation from the ground state of HOCl. Results agree with a simple classical model.

Two large-amplitude motions in triatomic molecules. Force field of the 1B2 (1A′) state of SO2

A program has been developed to calculate the energy levels associated with the two large-amplitude stretching vibrations ν1 and ν3 of a bent triatomic molecule in which the ν3 oscillation occurs in a double minimum potential. Employing the two large-amplitude Hamiltonian H0s(ρ1,ρ3) obtained earlier by Brand and Rao. [J. Mol. Spectrosc., 61, 360 (1976)], the vibrational energy levels (v1,v3even/odd) of SO2 molecule in its 1B2 (1A′) excited state are calculated. The nine parameters of the potential function V0(ρ1,ρ3) are then adjusted to give a least-square fit to the 12 observed vibrational term values corresponding to the levels (v1,v3even) of S16O2 and S18O2. A three-dimensional picture of the potential surface V0(ρ1,ρ3) using the final set of force constants is also presented. The saddle point of this surface is at (ρO1=1.5525 Å, ρO3=0.0 Å) and the absolute minima occur at (ρe1=1.5644 Å, ρe3=±0.0745 Å). Barrier height, i.e., the height of the saddle point above the absolute minima, is 140 cm−1.

Polarized photofluorescence excitation spectroscopy

When a triatomic molecule ABC is photolysed by a beam of light to produce the fragments AB and C, the directional nature of the photo-dissociation process manifests itself in a number of ways; in particular, the angular distribution of each fragment is anisotropic, and the angular momentum vector J of the AB photofragment points in space with a preferential distribution with respect to the direction of the light beam. If the AB diatomic fragment is electronically excited, then the emission from AB* will, in the absence of collisions, be polarized, and a measurement of the degree of polarization, p, will provide information about the symmetry nature and dissociation dynamics of the photoexcited state (ABC)*. A number of limiting cases are presented for the sign and magnitude of p, calculated in the classical high J limit, following dissociation of both linear and bent triatomic molecules.

Simple electrostatic models for vibrating unsymmetrical triatomic molecules and triatomic ions

Bond charge, point-dipole models are used to derive simple universal relations,Kθθ=0.0435 (K11K22)1/2 −0.0086 (K 11 + K22), andKθθ = 0.11 (K11K22)1/2 −0.0055 (K11 +K22), between the bond stretching force constantsK11 andK22 and bond bending force constantKθθ for linear and bent unsymmetrical triatomic molecules respectively. The relations are shown to be approximately valid for a number of molecules. An extension of the models to include charged species (ions) in triatomic molecules is also presented and tested, giving good results.

Large Amplitude Hamiltonian of a Triatomic Molecule

Abstract An expression for the Hamiltonian H(ρ1, ρ2, ρ3) of a vibrating-rotating triatomic molecule is derived using three curvilinear coordinates gi, Q 3 in such a way that the Hamiltonian obtained is applicable to any bent triatomic molecule and allows for large displacements in all the three modes of motion. A variational technique is then used to calculate the low lying vibrational energy levels (υ1, υ2, υ3) of the H2O molecule in its X̃1A1 ground state. The kinetic energy of the Hamiltonian T(ρ1, ρ2, ρ3) takes into account the large amplitude character of the three modes together with their interaction. But in order to minimize the formidable amount of computation, a simple quadratic potential F(ρ1, ρ2, ρ3) is assumed for all the three modes which only serves to illustrate the method of calculation.

A spline-fitted potential surface for bent triatomic molecules

The rotated-Morse curve cubic spline method developed previously is extended to bent triatomic molecules by using 2D cubic spline. The Yates—Lester potential for bent H 3 is shown to be accurately fitted over the entire range of the three internal coordinates, with a standard deviation of less than 1 kcal mol −1. The spline method compares favorably in computational speed with the analytical potential. 1 Associated with the CNRS (ERAno. 549).

The vibration-rotation problem in triatomic molecules allowing for two large-amplitude stretching vibrations

An expression for the Hamiltonian of a vibration-rotating triatomic molecule is derived, using two curvilinear stretching coordinates ϱ 1 and ϱ 3 and one rectilinear bending coordinate S 2, in such a way that the Hamiltonian obtained is applicable to any bent triatomic molecule and allows for large displacements along the stretching coordinates. From this, a zeroth-order Hamiltonian H s 0 ( ϱ 1, ϱ 3) is obtained, describing the energy levels associated with the two stretching vibrations ν 1 and ν 3. The vibrational energy levels ( v 1, v 3 even) of an XY 2 molecule having unequal bond lengths at equilibrium are then calculated. The kinetic energy T 0 ( ϱ 1, ϱ 3) of the Hamiltonian effectively takes into account the two large-amplitude motions in ν 1 and ν 3 together with their interaction. A model calculation is described for a bent XY 2 molecule (SO 2 in its 1A′ ( 1B 2) excited state) in which the ν 3 oscillation occurs in a double-minimum potential. Coupling by kinetic energy terms in the Hamiltonian turns out to be very small in this example.

Hartree-Fock calculations and photoelectron spectra of SSO and NSF

Results from ab initio Hartree-Fock calculations on the ground states of SSO and NSF are reported. The calculations employ large basis sets of Gaussian functions of essentially double zeta quality. The photoelectron spectra of the bent triatomic molecules with 18 valence electrons, i.e. SSO, NSF, O3, SO2, NSCl are compared. Further experimental details of the photoelectron spectrum of SSO are presented and an assignment of the observed ionic states of SSO⊕ is attempted. The calculated dissociation energies, dipole moments and the population analyses are given. The correlation effect is qualitatively discussed in connection with the applicability of Koopmans' theorem for the bent tri-atomic molecules.

The Renner effect in a bent triatomic molecule executing a large amplitude bending vibration

The theory is given for a model hamiltonian which combines the effects of large amplitude bending for a bent triatomic molecule with those arising from the coupling of vibrational, electronic and spin angular momenta characteristic of linear molecules. Two numerical examples are given for the application of the theory. (i) The vibronic and spin-orbit coupling between the A, 2 A 1 and [Xtilde], 2 B 1 states of PH2 is discussed in detail in terms of the 2Π u state of linear PH2 with which they correlate. The v 2 and K dependence of the doublet splitting in the A state is well represented in terms of a single spin coupling parameter for the 2Π u state. (ii) The calculations give a quantitative account of the inversion for high v 2 of the K = 0 and K = 1 levels of the bent B 2 component of the a, 3 A 2 state of CS2 arising from its correlation with a 3Δ2u state of linear CS2.

Anharmonic bend-stretch interaction force constants in bent triatomic molecules

The anharmonicity in the bending mode of a symmetric bent triatomic molecule is modeled by a simple double-minimum potential and, by assuming a dependence of the force constants of this potential on the symmetric stretching co-ordinate, a model is developed for the anharmonic bend-stretch interaction force constants. In this way the 19-parameter quartic force field of the bent XY 2 molecule may be represented by a 9-parameter model. The parameters in the proposed potential function have been adjusted to the experimental spectroscopic data for SO 2 and NO 2.

Theoretical Approach to Potential Energy Functions for Linear AB2 and ABC and Bent AB2 Triatomic Molecules

Let the coordinate system for a linear AB2 triatomic molecule have its origin on the A nucleus with R1 the distance to one B nucleus, R2 the distance to the other, and θ the apex angle. Then a working formula for the Born-Oppenheimer potential energy near equilibrium, W(R1, R2, θ), is W(R1, R2, θ) =WD(R1)+WD(R2)+A/|R3|N−B/(R1+R2)N,where W D(R1) and W D(R2) are potential functions for the ground state diatomic molecule AB, R3 is the vector sum R1+R2, A and B are constants and N is an integer. The potential energy for linear ABC tri-atomic molecules is given by Eq. (i) with R2 [or R1] scaled: R2→ η R2. Equation (i) is tested for CO2, CS2, OCS, HCN, and N2O by predicting all force constants up to fourth order except for the harmonic bending constant which is used in the parameterization. For bent AB2 molecules the working formula for the potential energy is W(R1, R2, θ) = WD(R1)+WD(R2)+A/|R3|N−B/(R1+R2)N+C/|R3|M−Aθ cos(θ − θe), (M≠ N),where θe is the equilibrium angle. Equation (ii) is tested for H2O, SO2, and O3 by predicting all force constants up to fourth order except for the harmonic bending and stretch-bend interaction constants which are used in the parameterization. A theory is presented for the presence of the W D(R) components in Eqs. (i) and (ii). It is suggested that the inverse |R3| terms partially represent averaged multipole interaction energies while the inverse (R1+R2) terms and angular term in Eq. (ii) partially represent valence orbital effects. The formulas for the parameters in Eqs. (i) and (ii) are given. Suggestions for writing down potential functions for larger molecules are included.

Anharmonic force constants of water

The observed vibrational energies and rotational constants have been used in a least-squares determination of the force constants in the general quartic force field of the isolated water molecule. The force constants have been examined in the formalism proposed by Machida and Overend and attempts have been made to simplify the potential function in order to develop a model with general applicability to bent triatomic molecules.

Higher vibrational levels of a bent triatomic molecule

Higher vibrational levels of a bent triatomic molecule

Infra-red dichroism and protein structure

We have been considering the direction of the transition moment of the carbonyl band of the peptide group at ca . 1650 cm -1 and have concluded that it is inclined considerably to the carbonyl bond and lies somewhere between the OC and the ON directions. This arises from two main causes, the first of which is the mechanical interaction of the CO and CN bond vibrations. As a result of the resonance, which has been proved by the X-ray evidence to exist in the amide group, the force constant of the CN bond must be increased to a value sufficient to interact appreciably with that in the CO bond, the force constant of which is corre­spondingly reduced. This enables us to consider the NCO part of the group in first approximation as a bent triatomic molecule, the bond vibrations interacting ‘out of’ and ‘in’ phase to give vibration modes ( a ) and ( b ) respectively (figure 28). The vibration ( a ) corresponds to the amide band at 1650 cm -1 , and from the directions of the motions of the atoms it may be inferred that the transition moment is inclined at an angle to the CO bond and veers towards the ON direction. The ‘ b ’ vibration occurs at a lower frequency and is complicated by resonance with an NH deformation vibration. We hope to publish fuller details of this shortly.

Contribution to the Study of Electronic Spectra of Bent Triatomic Molecules

The electronic absorption spectra of the SO2, SeO2, and TeO2 molecules have been studied under widely varied experimental conditions. The vibrational analysis of the ultraviolet absorption region of SO2 with center of gravity near 2000A reveals that it is formed by at least two electronic systems with the following vibrational constants: System α1···ν0=42,170 cm−1, ν1′=963 cm−1, ν2′=379 cm−1, x′ν2′=2 cm−1.System α2···ν0=44,236 cm−1, ν1′=775 cm−1, ν2′=375 cm−1.The presence in the same region of a third electronic system, with ν2′=360 cm−1 and ν1′=845 cm−1, seems highly probable. The character of the vibrational structure of the band systems and a preliminary analysis of the rotational structure of several bands have enabled us to obtain some information about the molecular structure of the excited states. The vibrational analysis of the c-system of SeO2 in the visible, and the TeO2 bands in the visible and ultraviolet, have enabled us to correlate the electronic states of the three similar molecules.