Transition Moment Matrix Elements Research Articles

Ab Initio rovibrational spectrum of the NaH2 + ion–quadrupole complex

The electronic and rovibrational structure of (1A1) NaH2 + has been investigated using a relativistically-corrected, all-electron coupled-cluster with singles, doubles and perturbative triples (CCSD(T)) ansatz. For the electronic ground state this ansatz yielded equilibrium Na–H bond lengths (R e ) of 2.4208 A and an equilibrium H–Na–H bond angle (θe) of 17.8°. An analytical potential energy surface (PES) was embedded in the rovibrational Hamiltonian. The PES was constructed using 118 CCSD(T) points and exhibited a residual error of 1.2 cm−1. The rovibrational Hamiltonian was diagonalised using variational techniques. The vibrational and rovibrational eigenvectors were assigned using a configuration weight scheme in terms of normal modes and the Mulliken assignment scheme, respectively. For the ground vibrational state of (1A1) NaH2 +, the vibration-averaged bond lengths 〈R〉 and angle 〈θ〉 were 2.4995 A and 17.1°, respectively. The ab initio (1A1) NaH2 + PES yielded a dissociation energy (D 0) value of 10.3 kJ mol−1, which is in excellent agreement with the experimental value of 10.3 ± 0.8 kJ mol−1 (Bushnell et al. in J Phys Chem 98:2044, 1994). An analytical dipole moment surface was constructed using 90 CCSD(T) points. Rovibrational spectra of (1A1) NaH2 +, (1A′) NaHD+ and (1A1) NaD2 + for v ≤ 10, J ≤ 5 were constructed using rovibrational transition moment matrix elements calculated in a novel manner that employs the analytical dipole moment surface (DMS). The rovibrational structure of the Na+–H2 v HH = 1 ← v HH = 0 band was calculated and compared to that of Li+–H2.

Direct Investigation of the Validity of Vertical Approximation in the Calculation of Transition Moment Matrix Elements: n → π* Transition in Methyl Formate

The absolute optical oscillator strength for the n → π* transition in methyl formate is calculated, within a procedure that has recently been used by our group to determine the optical oscillator strength of symmetry-forbidden transitions and which consists of expanding the (squared) transition moment along the normal coordinates of vibration. In the present work, we extend its application to a symmetry-allowed transition and discuss the applicability of the vertical approximation. To validate the present results, the UV photoabsorption cross-section was measured, and the result is compared with the one previously reported.

Selective vibrational excitations in the OX (X=F, Cl, Br, I) molecules

Using ab initio calculated potential energy and electric dipole moment functions for the X2Π states of OF, OCl, OBr and OI, two models have been tested to selectively populate their vibrational modes by ultrashort coherent light pulses. For a given form of the pulses either a perturbative approach using discrete vibrational eigenstates and electric dipole transition moment matrix elements or a wavepacket propagation technique were used. The optimisation of the pulse parameters and the mechanisms of the multiphoton processes are discussed. For the target level v=10 populations of more than 60% have been achieved in all four molecules. For OCl it was possible to populate selectively all target levels between v=2 and v=15 using chirped pulses.

Computerized simulation and fitting of singlet–triplet spectra of orthorhombic asymmetric tops: Theory and extensions to molecules with large multiplet splittings

Motivated by our recent finding that the singlet–triplet bands of selenoformaldehyde involve an upper state with large zero field splittings, we have extended the theory and written a program for predicting and fitting such rotationally resolved spectra. Triplet state matrix elements for a case (A) basis have been developed, including corrections for centrifugal and spin–centrifugal distortion. The full Hamiltonian matrix has been symmetry adapted, simplifying the problem to four individual matrices of approximately equal size for molecules of orthorhombic symmetry. Diagonalization of these matrices yields triplet state energies that are in agreement with previous treatments using a basis in which the spin splittings are small relative to the rotational intervals. Methods have been developed for sorting the eigenvalues and assigning quantum labels regardless of the magnitude of the spin splittings. The calculation of the relative intensities of the rotational lines within a band has been programmed using transition moment matrix elements from the literature. The selection rules for various upper state symmetries have been developed in a form useful for the analysis of spectra. Band contour predictions of spectra for various coupling cases have been presented.

Absolute Rovibrational Line Intensities in the ν 5 Band System of Cyanogen, NCCN

The far-infrared spectrum of the lowest-lying rovibrational band system ν 5 of cyanogen, NCCN, was measured in the wavenumber range from 180 to 280 cm −1. A Fourier transform interferometer was used. Absolute rovibrational line intensities of the fundamental and the first hot band were evaluated. The pure vibrational transition moment matrix elements for the fundamental and the first hot band, |〈μ (00),(01) vib〉| and |〈μ (01),(02) vib〉|, were determined to be 0.1812(19) and 0.1790(30) D, respectively. Furthermore, four nitrogen-broadened spectra of this band system were measured in the pressure range between p N 2 = 8 and 30 mbar (0.1 mbar cyanogen), yielding a nitrogen-broadening coefficient of 0.12 cm −1/atm decreasing slightly with increasing rotational quantum number J. No pressure-induced shifts of line positions were observed in this pressure range. The effects of instrumental distortion on line intensities are discussed.

Comparison of the coefficients of the transition matrix elements calculated by approximation methods and morse-potential wavefunctions

Coefficients for transition moment matrix elements of the OH group of methanol in different solvents were calculated using approximation methods and Morse-potential wavefunctions. A comparison of the obtained results shows that both methods produce nearly the same values except for three coefficients which are needed to calculate the third dipole moment derivative.

Rotational energy levels and line intensities for 2S+1Λ-2S+1Λ and 2S+1(Λ ± 1)-2S+1Λ transitions in a diatomic molecule van der Waals bonded to a closed shell partner

Abstract Hamiltonian matrix elements needed for calculating rotational energy levels are derived for a planar complex consisting of an open-shell diatomic molecule and a closed-shell partner. These matrix elements take account of spin-orbit interaction and a Renner-Teller-like splitting term, but not of the effects of large-amplitude internal rotation of the diatomic fragment within the complex. Rotational levels obtained by numerically diagonalizing this Hamiltonian matrix for given J are, as expected, strongly influenced by the degree of quantization of projections of the electron orbital, electron spin, and total angular momentum along the two natural axes in the problem, i.e., by the degree of quantization along the internuclear axis of the open-shell diatomic fragment, or along the inertial a axis of the near-symmetric-top planar complex. The degree of quantization of these varioss projections is in turn determined by the relative sizes of the spinorbit interaction, the Renner-Teller interaction, and products of the rotational constants and quantum numbers of the form BJ , CJ , and AK , as well as by the angle between the diatomic internuclear axis and the inertial a axis of the complex. Transition-moment matrix elements needed for calculating intensities in spin and orbitally allowed transitions in the open-shell diatomic fragment are also derived. Such transitions have the form 2 S +1 Λ- 2 S +1 Λ and 2 S +1 (Λ ± 1)- 2 S +1 Λ, where Λ = Σ, Π, Δ, Φ, etc. A brief discussion of how to use the Hamiltonian and transition-moment matrix elements in a computer program is given.

Variation of Dipole Moment Function of O-H Bond of t-Butanol with the Nature of Solvent and Its Temperature

Abstract The intermolecular influence on dipole moment function is evaluated for O-H bond of t-butanol in different nonpolar solvents at temperatures ranging from 10° to 60°C employing IR band intensities of fundamental and first overtone bands. Two sets of dipole moment derivatives have been calculated corresponding to – and +- combinations of the transition moment matrix elements R10 and R20, the values for ++ and -+ combinations are equal in magnitude to those for - and +- combinations, respectively and opposite in signs. In general the dipole moment derivatives increase on lowering the temperature as well as with increasing molecular interactions with the solvent molecules. Dipole moment plots with dimensionless coordinate ξ [=(r-re)/re, where r and re are internuclear distances during vibration and at equilibrium, respectively] are reported for various systems considered. It is found that for +- combination the dipole moment maximum shifts to higher internuclear distances when polarisation of the sol...

Infrared band intensities: Transition moment matrix elements for fundamentals and overtones: Part III. Isotope effect

A theoretical analysis of the isotope effect on IR band intensities of the first threa harmonics of a diatomic oscillator is presented. The coefficients a ij for the dipole moment derivatives appearing in the expressions for transition moment matrix elements R iu for the transitions i → 0, where i = 1, 2 and 3 are calculated for OH, OD, HCl and DCl species and their dependence on the reduced mass of the oscillator is evaluated. From these studies it is concluded that, within the limits of logical approximations, the intensities of the fundamental, first overtone and second overtone are proportional to μ −1, μ − 3 2 and μ −2, respectively, where μ is reduced mass. The intensities for these three bands for OD and DCl are predicted to be approximately 1 2 , 1 3 and 1 4 times their values for OH and HCl oscillators.

The ν2 + ν4 and 2 ν2 isotropic Raman bands of 12CH 4

The purely isotropic Raman spectrum of the ν 1 band, the ν 2 + ν 4 band (enhanced through interaction with ν 1), and the 2 ν 2 band of 12CH 4 was obtained with a spectral resolution of 0.30–0.35 cm −1 from exposures with different orientations of the linearly polarized exciting light. The ν 2 + ν 4 and 2 ν 2 bands show partially resolved rotational structure. The spectra are interpreted in terms of a model which takes explicitly into account vibrational and rovibrational interactions with other vibrational states, using molecular constants determined primarily from infrared spectra. The computed contours are in excellent agreement with the experimental ones and the observed and calculated peak wavenumbers agree within one tenth of the spectral resolution limit, except for a small region near the ν 1 band. The good overall agreement represents an independent check on the overall correctness of the previously reported molecular constants. A detailed discussion is given of the contributions to the intensities of individual transitions from the three transition moment matrix elements, which in an isolated-band model are the intensity parameters of the ν 1, 2 ν 4, and 2 ν 2 isotropic bands, respectively.

Analysis of the ν1 and ν2 + ν4 bands of 12CH 4

Line assignments, positions, strengths, and experimentally determined upper states are reported for the ν 1 and ν 2 + ν 4 infrared bands of 12CH 4. The bands have been analyzed using infrared spectra recorded with different optical densities and temperatures at 0.02- and 0.01-cm −1 resolution using a four-passed grating spectrometer at Florida State University and a Fourier transform spectrometer at Kitt Peak National Observatory. Both ν 1 and ν 2 + ν 4 lines are assigned through J′ = 14. For J′ ≥ 10, the upper state of the vibrationally infrared inactive ν 1 band interacts strongly with ν 2 + ν 4 and is observed in the infrared spectrum with line strengths on the order of 10 −3 cm −2 atm −1. The upper-state energies and transition intensities are calculated from the molecular constants and transition moment matrix elements obtained through a simultaneous analysis of ν 1, ν 3, ν 2 + ν 4, 2 ν 4, and 2 ν 2.

The 2ν4 isotropic and anisotropic Raman bands of CH4

AbstractThe Raman spectrum of the very weak 2ν4 band of CH4 has been recorded in the region 2460–2675 cm−1 with a spectral resolution of about 0.30 cm−1 and with two different orientations of the linearly polarized exciting light relative to the direction of observation. The two exposures are used to obtain the purely isotropic and the purely anisotropic Raman spectrum of the band. The two spectra are interpreted in terms of a model which takes explicitly into account vibrational and rotation‐vibrational interactions with other vibrational states. Using molecular constants determined primarily from infrared spectroscopy, theoretical contours are calculated which agree very satisfactorily with the experimental ones. For the isotropic spectrum it is found that two different values of the ratio between the 2ν4(A1) and the ν1 transition moment matrix elements reproduce the spectrum equally well. It is shown that the major part of the intensity in the isotropic and anisotropic Raman spectrum is borrowed from the ν1(A1) and the ν3(F2) fundamentals, respectively, through Fermi‐type interactions, and it is also shown that the 2ν4 isotropic Raman band would have been 4–6 times more intense, if the 2ν4(A1) transition moment matrix element had been negligible.

ChemInform Abstract: A THEORETICAL MODEL FOR THE INTERACTING UPPER STATES OF THE ν1, ν3, 2ν2, ν2 + ν4, AND 2ν4 BANDS IN METHANE

Abstract The effective vibration-rotation Hamiltonians complete to fourth order in the Amat-Nielsen scheme for the upper states of the ν1, ν3, 2ν2, ν2 + ν4, and 2ν4 bands in methane are reviewed, and the major vibration-rotation interactions (H30, H 40 , H 21 , H 31 , H 22 ) connecting the different vibrational states are discussed. Explicit matrix elements in a basis of harmonic oscillator-symmetric rotor basis functions are given for the purely vibrational terms and for the vibration-rotation interactions. Expressions for spectral intensities of infrared and Raman spectra are presented, and the selection rules and transition moment matrix elements are stated. A computer program is described which, incorporating all these features, can be used for prediction of infrared and Raman spectra and for determination of molecular constants from observed spectra by a least-squares routine. As an example the program is applied to the 2ν4 isotropic Raman spectrum of 12CH4, leading to a very good agreement between the experimental and calculated spectra.

A theoretical model for the interacting upper states of the ν1, ν3, 2 ν2, ν2 + ν4, and 2 ν4 bands in methane

The effective vibration-rotation Hamiltonians complete to fourth order in the Amat-Nielsen scheme for the upper states of the ν 1, ν 3, 2 ν 2, ν 2 + ν 4, and 2 ν 4 bands in methane are reviewed, and the major vibration-rotation interactions ( H 30, H ̃ 40 , H ̃ 21 , H 31 , H ̃ 22 ) connecting the different vibrational states are discussed. Explicit matrix elements in a basis of harmonic oscillator-symmetric rotor basis functions are given for the purely vibrational terms and for the vibration-rotation interactions. Expressions for spectral intensities of infrared and Raman spectra are presented, and the selection rules and transition moment matrix elements are stated. A computer program is described which, incorporating all these features, can be used for prediction of infrared and Raman spectra and for determination of molecular constants from observed spectra by a least-squares routine. As an example the program is applied to the 2 ν 4 isotropic Raman spectrum of 12CH 4, leading to a very good agreement between the experimental and calculated spectra.

Infrared band intensities: a comparative study of the transition moment matrix elements for the fundamental and overtones: Part II . Application to CO, HCl and OH

The various expressions considered in Part I for the transition moment matrix elements of fundamental and first two overtones are applied to carbon monoxide. The coefficients a ij in the expressions R io = Σ a ij p j (where R io is the transition moment integral for the O → i vibrational transition and p j is the dipole moment derivative ∂ j P /∂XXX j , XXX = ( r — r e)/ r e, r e is equilibrium bond distance) are reported for i, j = 1, 2, 3. It is found that these coefficients do not vary by more than 5% when compared for the same i, j values in various expressions irrespective of the most exhaustive treatments used in deriving the original expressions. On the basis of the values of the coefficients obtained for CO, generalisations have been suggested on the effects of inclusion of mechanical and electrical anharmonicity to the intensities of fundamental and first two overtones. It is generally observed that the contribution of p' 1, is about 100 fold more than the contribution of p' 2, for R 10. On the other hand the contributions of p' 1, and p, for R 20 and R 30 are of nearly equal magnitude but opposite in sign. The contribution of p' 1 to R 10 is much higher than its contribution to R 20 and R 20. The various observations lead us to conclude that, whereas the effect of inclusion of mechanical anharmonicity on the intensity of the fundamental band is negligible, this effect is almost comparable to the effect of inclusion of electrical anharmonicity for the first two overtones. Simple forms of the a ij expressions are applied to HC1 and OH to demonstrate the effect of variation of molecular constants on the a ij values. On the basis of the observed trend in the values of these coefficients for CO, HCl and OH general remarks on the effects of hydrogen bonding on IR band intensities are given.

Infrared band intensities: a comparative study of the transition moment matrix elements for fundamentals and overtones: Part I. General

A complete survey of the various expressions reported by different authors for transition moment matrix elements for infrared transitions of diatomic molecules has been made. The different expressions for fundamentals and overtones are presented in uniform coordinates. Although the expressions look different when compared in their original forms it is found that with the uniform coordinates, several of them are similar in the first few terms. Expressions obtained from the consideration of Morse potential as well as those obtained from inclusion of anharmonic potential are discussed. From the various expressions presented in uniform coordinates general remarks about the effects of inclusion of mechanical and electrical anharmonicity on the intensities of the fundamental and first two overtones are made. Since the effects of inclusion of mechanical and electrical anharmonicity are opposite in sign for the first overtone, it is further discussed that an increase in the intensity of the fundamental band on hydrogen bonding and a slight weakening of the intensity of the first overtone or no change in its intensity can be explained on the basis of an increase in the first and second derivatives of dipole moments on hydrogen bonding. Some general remarks are made regarding the dipole expansions and intensity expressions for polyatomic molecules.

Dipole Moment Derivatives of Free and Hydrogen Bonded O-H Bond of Methanol

Abstract Infrared intensities of fundamental and first overtone bands of O-H stretching mode of methanol are reported in a few hydrogen bonding solvents. From these intensity data the first and second dipole moment derivatives have been computed. It is noticed that the intensity of the fundamental band increases appreciably on hydrogen bonding whereas the first overtone does not show much change in intensity. Two sets of dipole moment derivatives obtained from ++ and -+ combinations of the transition moment matrix elements R10 and R20 are considered, the values for – and -+ combinations are equal in magnitude to those of ++ and +- combinations respectively and opposite in signs. It is observed that in all the case studied both the dipole moment derivatives of O-H bond increase on hydrogen bonding. It is shown that with an increase in the values of both the dipole moment derivatives not only one can explain an appreciable increase in

Comments on theF-factor approximation for HF(1∑)

Using a linear dipole moment function, the analytical vibrational-rotational transition moment matrix elements of Herman and Rubin, and their simplified formulation, the so-called F-factor approximation, of Herman, Rothery and Rubin are compared for the HF molecule in its 1∑ electronic ground state. The comparison reveals that, both for the v′ = 1, J′ →v″ = 0, J″ and the v′ = 2, J′ →v″ = 1, J″ transitions, the F-factor approximation gives a better than 5 per cent agreement with the HR theory for J″ up to 25, but becomes rapidly worse thereafter.

Rotational line intensities for singlet-triplet transitions in molecules belonging to the point groups D2 h, C2 v, and D2

Explicit expressions are presented for transition moment matrix elements and Hamiltonian matrix elements which can be used directly in computer programs for calculating rigid rotor line intensities for singlet-triplet transitions in molecules belonging to the point groups D 2 h , C 2 v , and D 2. The calculational procedure outlined and the various matrix elements presented can be used for triplet states exhibiting large or small spin splittings in the nonrotating molecule.

Electronic Transition Moment Integrals for First Ionization of CO and the A - X Transition in CO+. Some Limitations on the use of the r-Centroid Approximation.

Integrals necessary for the determination of transition moment matrix elements from experimental data have been evaluated numerically by use of vibrational wave functions derived from RKR potentials. A power series expansion for the electronic transition moment has been assumed. The significant quantities which can be related to an arbitrary center of expansion are vibrational overlap integrals and quantities of the form ʃψ v' r n ψ v″ dr. Experimental band intensities and relative populations for vibrational levels of the initial electronic state are needed to determine the expansion coefficients. Transition moment integrals have been calculated for first ionization from the ground electronic state of CO and for the A 2Π i - X 2Σ+ transition of CO+. Comparison of these integrals with previous calculations based on Morse functions has shown them to be rather sensitive to the wave-functions [potentials] used. Characteristics generally attributed to the r-centroid and related integrals are examined, and some limitations on the use of the r-centroid approximation are discussed, following a review of assumptions made in the use of that approximation.