Fourth-order contributions to the rotational-vibrational matrix elements for diatomic molecules; application to the Fς′v(m) functions for CO

Using the rotational potential function expanded through the quartic term as a perturbation, and considering successively a quartic and a quintic dipole moment function, general expressions are obtained for the vibration-rotation matrix elements corresponding to the transitions vJ→?′J′ with ?′?v+5.The vibration-rotation interaction functions F^?′_v(m) are deduced for the transitions v→?′ (v=0, 10, 20 and?′?v+4) of CO from calculations including third then fourth-order contributions to the matrix elements. The results obtained for the coefficients of the quartic polynomials fitting these functions are compared and their validity is discussed.     

Radial matrix elements and dipole moment function for the ground state of CO

Radial matrix elements 〈vJ|x^k|v′J′〉 for k = 0–5, v = 0–12, |v′- v| = 0–4, and J up to 150 have been calculated for CO using accurate wavefunctions obtained from the numerical solution of the Schrödinger equation with a second-order RKR potential curve. These are used in conjunction with a model dipole moment function (a Padé approximant which has the correct united and separated atom limits and R^?4 long-range behavior) to analyze the experimental intensity data. For all the levels considered, we conclude that an adequate representation of the dipole moment function is provided by a five-term power series expansion. This simplifies the computation of dipole moment matrix elements, typical results of which are presented to illustrate the dependence on the rotational and vibrational quantum numbers.     

High order vibrational matrix elements for diatomic molecules; dipole moment function and transition moments of CO

The fifth and sixth-order contributions to the vibrational matrix elements are obtained for the transitions v→v'(v'?v+4) using an eight power internuclear Dunham potential and a quartic power series expansion of the dipole moment function. The results for the dipole moment coefficients M₀ to M₄ of CO and the transition moments R_v^v' (with v=5, 10, 20) deduced from a calculation including succesively third, fourth, fifth and sixth-order perturbation theory are compared. The validity of these results is discussed.Using a quintic dipole moment function, general expressions for the vibrational matrix elements corresponding to the transitions v→v'(v'? v+5) are also presented and the influence of these contributions on the calculation of the dipole moment coefficients as well as the hot band transition moments of CO is shown.     

Theoretical infrared transition matrix elements for the water molecule

Using the Hartree-Fock-Roothaan method, symmetric potential energy and dipole moment functions have been obtained for the ground state of the water molecule. These functions were transformed and analyzed in terms of the normal symmetric stretching (v₁) and bending (v₂) coordinates for the calculation of i.r. transition matrix elements. There is satisfactory agreement between this work and the experimental values for the transition matrix element of the v₂ fundamental. However, a discrepancy of a factor of 2.5 is observed between calculated and experimental matrix elements for the v₁ fundamental.     

Nonempirical and quasi-classical approaches to the calculation of the electronic transition dipole moments

The dependence of the electronic transition dipole moment on the internuclear separation is calculated nonempirically for the Meinel band of the N ₂ ⁺ molecular ion using the Gaussian 98 software package with different basis sets. The results are compared with the dependences obtainedby the semiempirical methods. The basic relationships of the R-centroid method are derived, and the quasi-classical nature of this method is demonstrated. It is confirmed that the dependence of the electronic transition dipole moment on the internuclear separation is described most reliably by using the semiempirical methods. The vibronic transition dipole moment is shown to be an oscillating function of the vibrational quantum number v1 (with v2=const) or v2 (with v1=const).     

Vibration-rotation matrix elements for diatomic molecules; vibration-rotation interaction functions Fv′v (m) for CO

General expressions for the fourth-order vibrational matrix elements are obtained for the transitions v → v′ (v ? v′ v + 4) using a sixth-power internuclear potential and a quartic dipole moment function. The results for the dipole moment coefficients M₀ to M₄ of CO and for some transition moments R^v′_v deduced from a calculation including successively second, third and fourth order perturbation theory are compared.Using the rotational potential function expanded through the cubic term as a perturbation, we have also obtained general expressions for the vibration-rotation matrix elements. The vibration-rotation interaction functions F^v′_v (m) are calculated for the transitions v → v′ (v = 0, 10, 20 and v ? v′ ? v + 4) of CO and the coefficients C, D, E and G of the quartic polynomials fitting these functions are deduced. Taking account of uncertainties in the matrix elements R⁰₀ to R⁴₀, an error estimate for the coefficients of F²₀(m) and F³₀(m) is given.     

Vibration-rotational and rotational intensities for CO isotopes

Dipole moment matrix elements have been computed for the five most abundant isotopes of CO. The wave functions utilized were obtained from a direct solution of the Schrödinger equation with an accurate RKR potential. The dipole moment function, in the form of a Padé approximant, was chosen to reproduce the experimental measurements near equilibrium, to have the proper united and separated atom limits, and to have the correct long-range asymptotic functional dependence on internuclear separation. Because of the large number of transitions involved, and to facilitate applications, the squares of the dipole moment matrix elements were fitted by a least-squares procedure to polynomials in v and J. Predictions for the 5-0 and 6-0 rotationless matrix elements and Herman-Wallis coefficients are given, and their dependence on the isotopic reduced mass is discussed. For the pure rotational band, v = v′ = 0, explicit Einstein A values and transition frequencies were calculated for the three most abundant isotopes for J up to 55. The corresponding dipole moment matrix elements were also fitted to simple polynomials in m and the dependence of the coefficients on the reduced mass given. The present results incorporate the most accurate and extensive intensity measurements and theoretical dipole moment function data for any heteronuclear diatomic molecule. In view of this, because of the importance of the CO laser, the accuracy of the spectral frequencies, and the ubiquity of the CO molecule, it is reasonable to expect that some of these lines, in particular, in the fundamental band of ¹²C¹⁶O, can serve as laboratory standards for intensity measurements.     

Theoretical dipole moment function of the X1Σ+ state of HI

Multiconfiguration self-consistent-field calculation has yielded the dipole moment function for the X¹Σ⁺ state of HI, which qualitatively confirms the experimental finding that the dipole derivative at R_e is negative. The calculated dipole moment for the v = 0 vibrational level is 0.665 D for both HI and DI as compared with the experimental values of 0.38 and 0.445 ± 0.02 D, respectively. The calculated potential curve yields values of R_e, D₀, and ΔG_v+1/2, in good agreement with spectroscopic data. A simple valence bond explanation has been provided for the qualitative difference between the dipole moment function of HI and those of the smaller hydrogen halides.     

Infrared spectral line parameters of HBr and DBr at elevated temperatures

The electric dipole matrix elements <v'J'μvJ> for pure rotation and vibration-rotation transitions, with |;m|;<-40 and v<-v'<-6, have been derived for HBr and DBR by using the Rydberg-Klein-Rees (RKR) potentials and numerical solutions of the Schrödinger equation. An improved dipole-moment expansion was determined by fitting these matrix elements to the available experimental data on line intensities. A least squares analysis of the available line position constants gave an improved set of Dunham coefficients good for spectral lines with both large and small quantum numbers v and J. The results were then used to generate a compilation of individual line parameters for the Δv = 1 bands of HBr and DBr at temperatures up to 3000K. These parameters, together with previously compiled line parameters for HCl, HF, CO and NO, are being used for line-by-line calculations of radiance from a hot source as seen through an atmospheric path.     

Adequate representation of the dipole moment function for diatomic molecules and the matrices of vibration-rotation transition moments

An exponential representation of perturbations is used as a basis of the perturbed Morse oscillator approach which is applied, in a matrix form, for calculating the radial matrix elements for diatomic molecules. An analytic procedure is developed to deduce an exponential-power series expansion for the dipole moment function M(r) from experimental spectral intensities. It is shown that for real anharmonic molecules, the series expansion in powers of e^ar (α being the Morse parameter) is an adequate form for representing transition operators, just as the usual series expansion in powers of internuclear distance r is adequate for the case of a harmonic oscillator, and it is equivalent to a series expansion in vibrational wavefunctions. An exponential-power series expansion is derived as well for a model dipole moment function which has a correct long-range dependence and limit. To exemplify the accuracy and efficiency of the technique proposed, the (40 × 40) matrices of vibration-rotation transition moments 〈vJ|M(r)|v′J′〉(v, v′ = 0, 1, …, 39) have been calculated for the ground state of CO. Typical results of these computations are presented (up to v = 35, J = 100, and v′ ? v = 1–4) to illustrate the dependence of vibration-rotation interaction functions on the vibrational and rotational quantum numbers.     

The dipole moment of the FO radical

The electric dipole moment of the FO radical (${}^2\text{Π}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$) has been directly measured by observing its saturated absorption laser magnetic resonance spectrum (v = 1 ← 0) in the presence of a moderate electric field. The resulting dipole moment [μ(v = 0) = 0.0043(4) D, μ(v = 1) = 0.0267(9) D] is extremely small, and shows a large relative change with vibrational state. The sign of the dipole moment is the same for the two vibrational states. Previous failures to detect microwave or gas phase EPR spectra of FO are fully explained by this small dipole moment.     

Calculated vibrational transition probabilities of OH(X2Π)

The theoretically derived dipole moment function of OH(X²Π) obtained by Stevens Das, Wahl, Neumann, and Krauss is used to calculate the absolute intensities of the vibrational-rotational transitions of the OH Meinel bands. The calculations take full account of the spin uncoupling and vibration-rotation coupling which markedly influence the radiative transition probabilities. The effect of lambda-doubling on the vibrational transitions is analyzed and generally found to be negligible. Results are tabulated for Δv = v′ – v″ ranging from the fundamental transitions Δv = 1 to the Δv = 5 overtone, for v′ = 1–9 and J′ = 0.5 – 15.5. A comparison is made with available data, and various features of the OH spectrum are examined that are of aeronomical and experimental interest. Thermally averaged emission rates are presented for Δv = 1–5, and the validity of the rotational temperatures commonly derived from experimental intensity distributions is questioned.     

Investigation of surface self-diffusion of Ni atoms on the Ni(100) plane by means of work function measurements and Monte Carlo calculations

A novel method is presented to study migration of adsorbed metal atoms on a clean (hkl) surface of a metal substrate. The system used was Ni on Ni(100). We make use of the assumption that each single step of random walk motion of an individual adatom on the metal substrate refers to an activation energy ΔE_if (indices i and f giving the number of occupied nearest neighbour sites in the initial and final position respectively). Furthermore, we assume that the dipole moment p_v of an adatom is determined by the number v of occupied nearest neighbour sites. Via their dipole moments the adsorbed atoms induce a change of work function ΔΦ of the substrate metal, ΔΦ being related to the average dipole moment per unit area by the Helmholtz equation. To measure ΔΦ we use a pendulum device. At fixed low coverage θ the variation of ΔΦ with temperature T has been calculated numerically within the framework of a Monte Carlo simulation (MCS) model, using various activation energies ΔE_if and dipole moments p_v as input parameters. By fitting these data to the experimental curves we could derive the following dipole moments and activation energies: p₀ = 0.45 ± 0.05 D, p₁ = 0.36 ± 0.05 D, p₂ = 0.27 ± 0.05 D, p₃ = 0.18 ± 0.05 D, ΔE₀₀ 0.60 ± 0.02 eV, ΔE₁₀ = 0.70 ± 0.025 eV, ΔE₂₁ = 0.80 ± 0.05 eV, ΔE₂₀ = 0.90 ± 0.05 eV. We compare these results with those of other workers.     

Zum Dipolmoment zweiatomiger Moleküle

The Stark effect of a rotating anharmonic oscillator is calculated including terms up to 10^?7 relative accuracy for comparison with recent high precision experiments. A series expansion is used for the internuclear potential and the instantaneous dipole moment as a function ofr, the nuclear distance. The measurable effects are

1. 1.
   the change of the dipole moment with the vibrational quantum numberv up to terms proportional to (v + 1/2)²;     
   
   

Einstein A coefficients and absolute line intensities for the E2Π–X2Σ+ transition of CaH

Einstein A coefficients and absolute line intensities have been calculated for the E²Π–X²Σ⁺ transition of CaH. Using wavefunctions derived from the Rydberg–Klein–Rees (RKR) method and electronic transition dipole moment functions obtained from high-level ab initio calculations, rotationless transition dipole moment matrix elements have been calculated for all 10 bands involving v′=0,1 of the E²Π state and v″=0,1,2,3,4 of the X²Σ state. The rotational line strength factors (Hönl–London factors) are derived for the intermediate coupling case between Hund's case (a) and (b) for the E²Π–X²Σ⁺ transition. The computed transition dipole moments and the spectroscopic constants from a recent study [Ram et al., Journal of Molecular Spectroscopy 2011;266:86–91] have been combined to generate line lists containing Einstein A coefficients and absolute line intensities for 10 bands of the E²Π–X²Σ⁺ transition of CaH for J-values up to 50.5. The absolute line intensities have been used to determine a rotational temperature of 778±3 °C for the CaH sample in the recent study.     

Stark effect measurements on the NaK molecule

In a newly built molecular beam apparatus we have measured permanent electric dipole moments of the B ¹ Π and X ¹ Σ ⁺ states of NaK to test the vibrational dependence of the dipole moment function as predicted by ab initio calculations by Aymar and Dulieu [J. Chem. Phys. 122, 204302 (2005)]. Four vibrational bands were studied, the dipole moment for the B ¹ Π decreases from 2.73(2) D for v = 3 to 1.99(2) D for v = 13. The vibrational dependence shows a fairly good agreement with theory, but the absolute values are lower than those from the calculation by 13%. The ground state dipole moment of 2.72(6) confirms the earlier result by Wormsbecher et al. [J. Chem. Phys. 74, 6983 (1981)].     

Application of the contact transformation method to torsional problems in methyl silane

The effect of the torsional degree of freedom on redundancies in the Hamiltonian and on the dipole operator has been investigated for methyl silane-like molecules. By applying a rotational contact transformation to the torsion-rotation Hamiltonian H_TR for the ground vibrational state, a systematic method is demonstrated for treating the redundancies that relate different terms in H_TR. In general, with this method, the experimentally accessible molecular parameters in the reduced Hamiltonian can be related to the physically significant molecular parameters in the untransformed Hamiltonian. It is shown that H_TR contains a new term which has matrix elements with selection rules (ΔK = ±3), (Δσ = 0), and Δv_T arbitrary, where v_T and σ label the torsional levels and sublevels, respectively. As a result of this term, the distortion dipole constant μ_D which characterizes (ΔK = ±3) matrix elements in C_3v molecules cannot, in systems like CH₃SiH₃, be ascribed entirely to centrifugal distortion but can contain a significant contribution from torsional effects. Furthermore, new transitions can appear in the pure torsional bands which may be strong enough to observe in low barrier molecules. By applying a vibrational contact transformation, the form is derived of the leading torsional terms in the dipole moment expansion. The four dipole distortion constants μ₀^T, μ₂^T, μ_|;^T, and μ_Λ^T which characterize these terms are related to the molecular parameters that enter the Coriolis, centrifugal distortion, and anharmonicity contributions to the vibration-torsion-rotation Hamiltonian.     

Frequency and intensity measurements in the fundamental infrared band of HD

The fundamental band of HD has been studied at room temperature and 0.6 atm pressure with a 48-m absorption path and a resolution of 0.01 cm^?1. The frequencies of nine electric dipole and three electric quadrupole transitions were measured with an accuracy of 0.001 cm^?1, and their analysis gives improved molecular parameters for the v = 0 and 1 states of HD. The intensities of the dipole transitions were measured in order to determine the v = 1-0 electric dipole transition moment. These measurements extend earlier experiments to higher J values, thus refining the determination of the rotational dependence of the transition moment.     

Systematics of vibrational intensities for “diatomic” molecules: Approximate formulas

Several approximations are discussed which enable one to estimate electric dipole moment transition matrix elements for diatomic molecules based on a very limited amount of experimental data. For strongly polar molecules, a simplistic point-charge dipole model in which there is no vibrational charge flux predicts a simple relation between the permanent moment, M₀, and its derivative with respect to internuclear separation, M₁. This is found to be approximately valid for a surprisingly large number of polar molecules. Next, approximate matrix elements for the fundamental and overtone bands are presented for a linear dipole moment function. Finally, systematic trends for molecules having similar electronic structures are investigated and combined with the approximations discussed above in order to estimate the transition moments for several molecules of astrophysical interest. An extension to the ν₃ fundamental and overtone bands of CH₄ is suggestive that some of these ideas may be applicable to certain polyatomic molecules as well.     

Third-order theory of the line intensities in the allowed and forbidden vibrational-rotational bands of C3v molecules

A third-order theory of the intensities of the allowed and “forbidden” (perturbation-allowed) transitions to the fundamental vibrational levels of C_3v semirigid molecules has been worked out by using the method of contact transformations applied to the electric dipole moment operator. Explicit expressions have been obtained for the linestrengths of the allowed (Δk = 0, ±1) as well as forbidden (Δk = ±2, ±3, ±4) transitions from the ground vibronic state to the fundamental vibrational levels of C_3v molecules. The treatment takes into account all the important Coriolis and anharmonic interactions in a C_3v molecule, including the effect of the “2, 2” and “2, −1” l-type interactions and the Δk = ±3 interactions on the intensities of the allowed and forbidden vibrational-rotational transitions. The expressions for the linestrengths of the allowed and forbidden transitions are given here in a form suitable to fit the experimental data on the intensities in the vibrational-rotational spectra of C_3v molecules.