Rydberg-Klein-Rees Method Research Articles

Calculating macroscopic gas molar heat capacity of SO molecule based on rovibrational energy level

<sec>Sulfur oxide (SO) is a kind of well-known diatomic molecule which becomes one of the major pollutants in the atmosphere. Control of the heat capacity of SO molecule is of great significance for elucidating its macroscopic evolution process. In the research of macroscopic systems composed of many particles as well as several matters, it is an important approach to obtain macroscopic thermodynamic quantities of the system by constructing a partition function from the microscopic information of molecule. For diatomic molecules in a certain electronic state, the partition function can directly be obtained by calculating the rovibrational energy of the system to acquire the macroscopic molar heat capacities.</sec><sec>In this work, the contribution of rotational behavior to molar heat capacity is further considered. The potential energy function for the ground electronic state of SO is constructed by the variational algebraic method (VAM) and RKR (Rydberg-Klein-Rees) method, in which the former one can determine the complete vibrational energy levels of an electronic state of a molecule. The rovibrational energy level of the system is obtained by analytical solution, and then the molar heat capacity of SO macroscopic gas in the temperature range of 300–6000 K is calculated by quantum statistical ensemble theory The above calculation depends only on the experimental vibrational energy, experimental rotational spectral constant and the dissociation energy of SO molecule. Fortunately, through comparison between theoretical calculation results and experimental data, we find that the molar heat capacity of gaseous SO molecule can be well predicted by employing the full set of rovibrational energy to describe the internal vibration and rotation of SO molecule. The idea of calculating the molar heat capacity by using the full set of rovibrational energy makes up for the shortcomings of previous work where molar heat capacity is calculated by using the approximate model characterizing the molecular rotational behavior, and also provides a new research paradigm for solving macro thermodynamic quantities based on micro statistical processes .</sec>

Potential energy curves, turning points, Franck–Condon factors and $r$-centroids for the astrophysically interesting $\mbox{S}_{2}$ molecule

We have theoretically studied the seven lowest triplet states of the disulfur species. For such investigations, analytical potential energy curves (APECs) for $X^{3}\varSigma _{g}^{-}$ ground and $A^{\prime\,3} \Delta _{u}$, $A^{3}\varSigma _{u}^{+}$, $B^{\prime\prime\,3}\varPi _{u}$, and $B^{3}\varSigma _{u}^{-}$ excited electronic states were constructed within the extended Hartree–Fock approximate correlation energy (EHFACE) model by Varandas. Once that these analytical representations are obtained, nuclear properties, such as vibrational energies, classical turning points, and spectroscopic constants were calculated. Particularly, comparisons between these vibrational levels with those obtained via the Rydberg–Klein–Rees (RKR) methodology as implemented in Le Roy’s RKR1 code are reported. The impact of tight $d$ augmented correlation consistent basis on the energies and frequencies is shown. We also re-examined the vibronic (vibration-electronic) transition parameters as Franck–Condon (FC) factors and $r$-centroids for the bands of the $B^{\prime\prime\,3}\varPi _{u} $–$X^{3}\varSigma _{g}^{-}$, $B^{3}\varSigma _{u}^{-} $–$X^{3}\varSigma _{g}^{-}$, $C^{3}\varSigma _{u}^{-} $–$X ^{3}\varSigma _{g}^{-}$, and $D^{3}\varPi _{u} - X^{3}\varSigma _{g}^{-}$ systems of the $\mbox{S}_{2}$ molecule. The vibrational levels and turning points of the two Rydberg states $C^{3}\varSigma _{u}^{-}$ and $D^{3}\varPi _{u}$ were computed only with the RKR method. Our results can be employed in rationalizations of astrochemical and astrophysical observations.

Observation and assignment of the [formula omitted] state of [formula omitted

The 21∏g state of Rb2 has been observed using the infrared infrared double resonance method. The low part of the potential curve of the state is characterized for the first time. The absolute vibrational numbers are determined with isotope effect. The molecular constants of this state are obtained from a Dunham fit and the potential curve of the state is constructed using the Rydberg–Klein–Rees method.

The [formula omitted] electronic state of KLi molecule

The A1Σ+←X1Σ+ band system in the 39K7Li molecule is investigated by the polarisation labelling spectroscopy technique. The excited state is characterised by a set of Dunham coefficients describing rovibrational levels v=0–34,J=9–73 and the corresponding potential energy curve is constructed with the Rydberg–Klein–Rees method. The main equilibrium constants are Te=12097.11(3)cm-1, (with respect to the minimum of the electronic ground state), ωe=137.0821(131)cm-1,Re=3.9466(3)Å.

Vibrational distributions in N 2 with an improved calculation of energy levels using the RKR method

The potential curve of the N 2 ground state ( X 1 Σ g + ) has been accurately reconstructed using the Rydberg–Klein–Rees (RKR) method extrapolated by a Hulburt and Hirschfelder potential for longer internuclear distances. Solving the corresponding radial Schrödinger equation yields 60 bound levels above v = 0, whose energies may be fitted by the following polynomial expansion: G ( v ) = ∑ i = 1 5 a i ( v + 1 / 2 ) i , with the coefficients a i in cm −1 equal to 2378.1, −18.516, 0.26662, −6.2127 × 10 −3, and 3.4215 × 10 −5, respectively, for i = 1–5. The effects of considering this more accurate potential on the vibrational distribution function of N 2(X, v) molecules obtained in nitrogen microwave discharges and post-discharges are compared with previous calculations, using a smaller vibrational manifold.

First observation of the 3 1Π and 4 1Π states of NaLi molecule

Two previously unknown electronic states of NaLi molecule have been observed by polarisation labelling spectroscopy in the energy region of 29 900–34 100 cm −1 above the bottom of the molecular ground state potential well. The states are identified as 3 1Π(Na(3d) + Li(2s)) and 4 1Π(Na(4p) + Li(2s)). The Dunham coefficients are derived for both states and the potential energy curves constructed by the Rydberg–Klein–Rees method.

Arrays of radiative transition probabilities for [formula omitted] plasmas

Franck–Condon factors, Einstein coefficients, and band oscillator strengths have been calculated for the most prominent diatomic radiative transitions found in high-temperature CO 2 – N 2 gases and plasmas. A bibliographic review has allowed selecting the most up-to-date spectroscopic constants, which were used for rebuilding the electronic states potential curves through the Rydberg–Klein–Rees method. Vibrational wavefunctions have been determined for such potential curves and combined with the most up-to-date electronic transition moments found in the literature, in order to determine the transition radiative properties. The proposed set should provide an useful and accurate spectroscopic database for the determination of the radiative properties of CO 2 – N 2 gases and plasmas.

Estimation of dissociation energy of NiC molecule

The experimental potential-energy curve for the electronic ground state of an astrophysically important NiC molecule is constructed by applying the Rydberg–Klein–Rees method as modified by Vanderslice et al. The ground state dissociation energy is estimated by a curve-fitting technique using the modified Lippincott potential function, and this estimated value of D 0 = 3.454 eV is in good agreement with the value of D 0 ⩾ 3.34 eV reported by Brugh and Morse.

Spectroscopic studies of atmospheric interest on NO and NO +

In recent years, high-resolution photoelectron spectroscopy and ab initio calculations have considerably revised and enlarged the understanding of the electronic structure of the NO and NO + molecules. The experimental potential energy curves for the different electronic states of atmospheric interest molecules like NO and NO + are constructed by using the Rydberg–Klein–Rees method as modified by Vanderslice et al. The ground state dissociation energies are determined by curve fitting technique using the five parameter Hulburt–Hirschfelder (H–H) function. The estimated dissociation energies are 6.381 and 10.693 eV for NO and NO +, respectively. These values are in good agreement with the literature values. The r-centroids and Franck–Condon factors (FC Factors) for the band system of B 2Π r –X 2Π of NO and a 3Σ +–X 1Σ +, A 1Π–X 1Σ + of NO + molecules have been calculated employing an approximate analytical methods of Jarmain and Fraser, and Nicholls and Jarmain. The absence of the bands in these systems is explained.

Spectroscopic investigations on cometary molecules CO +, CH and CH +

The experimental potential energy curves for the electronic ground states of molecules like CO +, CH and CH + observed in comets are constructed by using the Rydberg–Klein–Rees method as modified by Vanderslice et al. The ground state dissociation energies are determined by curve fitting technique using the five parameter Hulburt–Hirschfelder (H–H) function. The estimated dissociation energies are 8.33±0.19, 3.48±0.903 and 3.89±0.102 eV for CO +, CH and CH +, respectively. These values are in good agreement with the literature values. Estimated dissociation energies of CO +, CH and CH + are used in the relation given by Gaydon and ionization potentials are evaluated for CO and CH molecules. The estimated ionization potentials are 14.03 and 10.85 eV , respectively for CO and CH. The Franck–Condon factors and r-centroids for the band system of A 2 Π – X 2 Σ + of CO + molecule have been calculated employing an approximate analytical methods of Jarmain and Fraser. The absence of the bands in this system is explained.

Spin–orbit interactions, new spectral data, and deperturbation of the coupled b 3∏u and A 1∑u+ states of K2

We report calculations of the spin–orbit energy as a function of internuclear distance R within the b 3∏u state of K2, and between the b 3∏0u and A 1∑u+ states, together with new spectroscopic data on the b state and previously unpublished data on the A state. Both the new data and previous data are fitted to Hamiltonian parameters using the discrete variable representation (DVR) method. The DVR matrix includes nonrelativistic Born–Oppenheimer potentials and spin–orbit interactions, which are scaled to match the known asymptotic limits and to best fit the experimental data. We report fitted Dunham coefficients that yield the A and b state potentials by means of the Rydberg–Klein–Rees method. These parameters thus take into account second-order spin–orbit perturbation shifts (from the vibrational levels of these two states) which are normally not considered in band-by-band fits to spectroscopic data.

Spectroscopic Studies on Astrophysically Interesting TaO TaS, ZrS and SiO+ Molecules

The true experimental potential energy curves for the electronic ground states of astrophysically important TaO, TaS, ZrS and SiO+molecules are constructed by using the Rydberg–Klein–Rees method as modified by Vanderslice et al. The ground state dissociation energies are determined by curve fitting techniques using the five parameters Hulburt-Hirschfelder (H-H) function. The estimated dissociation energies are 8.19 ± 0.17, 6.9 ±0.14, 5.89 ± 0.12 and 5.75 ± 0.12 eVfor TaO, TaS, ZrS and SiO+ respectively. These values are in good agreement with the literature values. The r-centriods and Franck–Condon factors (FC Factors) for the bands of K2φ5/2- X2Δ3/2(K-X) system of TaO, A-X2 Δ (A-X) and B-X2 Δ (B-X)systems of TaS, B1 Π - X1 Σ+(B-X) system of ZrS and B2Σ+ - X2 Σ+(B-X) and A2 Π - X2 Σ+(A-X)systems of SiO+ molecules have been calculated. The Franck–Condon factors (FC factors) are evaluated by the approximate analytical methods of Jarmain and Fraser. The absence of the bands of these systems is explained.

Optical–optical double resonance polarization spectroscopy of highly excited states of 23Na39K

The Doppler-free high resolution spectrum of the transitions to highly excited states of the 23Na39K molecule was measured by using the technique of the optical–optical double resonance polarization spectroscopy. Several highly excited states 1Σ+, 1Π, and 1Δ were found, and the molecular constants for these states were determined. The vibrational numbering of the 1Δ state (G 1Δ state) was estimated, and the potential energy curve was calculated by the Rydberg–Klein–Rees method. Many perturbed lines were observed. The indirect perturbation between 1Σ+ and 1Δ levels through the L-uncoupling interaction with a distant 1Π state was identified. The hyperfine splittings were observed in transition lines to a 3Π state, which is perturbed by a 1Σ+ state.

First observation of the quasibound levels and tunneling line broadening in the 3 1Πg state of Na2 using an ultrasensitive ionization detector

All quasibound vibrational energy levels v=27 and 28 of the Na2 3 1Πg state were accessed by using optical–optical double resonance and detected with a shielded ionization detector. The line broadening due to the tunneling effect was observed to be larger than 1 cm−1 at the rovibrational level v=28, J=31. The potential curve has been determined by the Rydberg–Klein–Rees method and modified using Jeung’s calculation for the large distance barrier and comparison of the calculated and the measured line widths. The barrier height was found to be ∼126 cm−1 above its asymptotic limit (3s+4p) at 5.98 Å. There also exists an unobserved shallow outer well with depth ∼35 cm−1 at 9.5 Å according to our modification of Jeung’s calculated potential. One way to possibly probe this outer well is also discussed.

A two-isotope higher-order RKR-type inversion procedure

The Rydberg–Klein–Rees (RKR) method is a valuable everyday tool for determining diatom potential energy curves. However, the RKR method is based upon the first-order WKB approximation, and while in most cases the potential curves it yields are adequate, there exist situations in which their deficiencies are unacceptable. This work addresses the above problem by presenting and testing a new higher-order, RKR-like inversion procedure which is exact within the third-order WKB approximation. The new method relies upon the existence of data for two isotopes. The turning point expressions obtained in the new method have exactly the same structure as the ordinary RKR expressions, and the procedure requires no iteration. Tests for two well-defined model problems show that the present method can yield potentials which are more accurate than those obtained from first-order RKR calculations, independent of whether or not the latter includes the widely used Kaiser correction.

Electronic states and molecular constants of Bi2

Potential energy curves of several electronic states of Bi2 have been calculated by the Rydberg–Klein–Rees method using molecular constants obtained from laser-excited photoluminescence of Bi2. These potential curves are used to compute Franck–Condon factors. Franck–Condon factors for the A–X system are compared with measured intensities of photoluminescence series extending over 50 vibrational levels (v″= 0–49) and including both Franck–Condon maxima. It is shown that more accurate molecular constants can be derived by using the information contained in these intensity data. Another important result is achieved by deriving previously unknown vibrational and rotational constants of the upper electronic state of the E–B doublet photoluminescence series using only line intensity measurements. Relative energy positions of the electronic states have been obtained by perturbation considerations and by measuring the temperature dependent changes of relative intensities of photoluminescence lines belonging to different electronic transitions. These experiments suggest that there is an electronic state X ′, which is lower in energy than the X state. A potential energy diagram for Bi2 is derived including the re-evaluated published spectroscopic data of Bi2.

Effects of the Langer Transformation on the Calculation of Internuclear Potential Curves

The Langer transformation is applied to the radial equation for internuclear motion. The quantization integral is obtained and applied to the calculation of the effective potential curve by the Rydberg–Klein–Rees method. First- and second-order RKR expressions for the turning points are derived. It is shown that the Langer transformation affects the RKR method in all orders.

Spectroscopy of Na2 Using Laser-Induced Fluorescence

The argon-ion laser lines at 4658, 4727, 4765, 4880, 4965, 5017, and 5145 Å are each found to excite one or more resonance fluorescence series of the Na2 (X 1Σg+ − B 1Πu) blue-green band system. Altogether, 19 different fluorescence progressions have been identified and assigned υ, J quantum numbers. The absolute wavenumbers of many of these emission lines have been measured interferometrically using a Fabry–Perot etalon crossed with a Bass–Kessler type spectrograph. A revised set of spectroscopic constants has been determined for both the upper and lower states which reproduce the observed term energies to better than 0.1 cm−1 on the average. Using these improved spectroscopic data, potential curves have been constructed for the B and X states of Na2 by the Rydberg–Klein–Rees method. These potential curves are used in turn to compute Franck–Condon factors for the Na2 B–X band system.

Potential-Energy Curves for CO and CO+

Potential-energy curves for the X 1Σ+, A 1Π, I 1Σ−, a 3Π, a′ 3Σ+, d 3Δi, and e 3Σ− electronic states of CO and the X 2Σ+, A 2Πi, and B 2Σ+ electronic states of CO+ have been calculated using a modification of the Rydberg—Klein—Rees method. The curves for the a 3Π, d 3Δi, and A 2Πi states, which are intermediate between Hund's Cases (a) and (b), represent fictitious rotationless states calculated using the true Bv. Benesch and co-workers have shown theoretically that this representation is correct for such intermediate cases. The numerical results for the A 2Πi state presented here confirm their conclusions.

Higher Approximations in the Rydberg—Klein—Rees Method

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