Quantum Defect Functions Research Articles

Calculations of the 1s^2ns High Rydberg States Spectrum of the Lithium Atom

Ionization potentials and quantum defects of 1s^2 ns bound states and their adjacent continuum states for the lithium atom are calculated with the R-matrix theory, and then the quantum defect function of the 1s^2ns channel is obtained, which varies slowly with the energy based on the quantum defect theory. The initial quantum defect function is calibrated from the most accurate theoretical value of the higher excited state and is used to calculate the quantum defects, ionization potentials, and term energies of the 1s^2ns high Rydberg states of the lithium atom. The present calculations may help in performing measurements of the 1s^2ns high Rydberg states for the lithium atom.

Term energies of 1s2nf high Rydberg states for the lithium atom

Ionization potentials and quantum defects of 1s2nf bound states and their adjacent continuum states for the lithium atom are calculated with the R-matrix theory; then the quantum defect function (QDF) of the 1s2nf channel is obtained, which varies smoothly with the energy based on the quantum defect theory. The original QDF is calibrated from a more accurate literature value and used to calculate quantum defects, ionization potentials and term energies (relative to the 1s22s ground state) of 1s2nf high Rydberg states of the lithium atom. The term energies in this work are, on the whole, in good agreement with the experimental data in the literature.

Energies of the 1s2nd Rydberg states for the lithium atom

In this work, energy levels and quantum defects of 1s2nd bound states and their adjacent continuum states for the lithium atom are calculated with the R-matrix theory, and then the quantum defect function of the 1s2nd channel is obtained, which vary smoothly with the energy based on the quantum defect theory. The obtained quantum defect function is calibrated from more accurate experimental value of higher Rydberg state and used to calculate quantum defects, ionization potentials and transition energies (relative to the 1s22s ground state) of 1s2nd high Rydberg states of the lithium atom. Present calculations may help in performing measurements of yet unmeasured 1s2nd Rydberg states of this system in future.

Ionization Potentials and Quantum Defects of 1s2np2P Rydberg States of Lithium Atom

In this work, ionization potentials and quantum effects of 1s2np2P Rydberg states of lithium are calculated based on the calibrated quantum defect function. Energy levels and quantum defects for 1s2np2P bound states and their adjacent continuum states are calculated with the R-matrix theory, and then the quantum defect function of the 1s2np (n ≥ 7) channel is obtained, which varies smoothly with the energy based on the quantum defect theory. The accurate quantum defect of the 1s27p2P state derived from the experimental data is used to calibrate the original quantum defect function. The new function is used to calculate ionization potentials and quantum effects of 1s2np2P (n ≥ 7) Rydberg states. Present calculations are in agreement with recent experimental data in whole.

Dissociation and predissociation in the ns 1B1 and np1A2 series of H2O

It is shown how a carefully devised quantum defect function can be used to unify direct dissociation of the lowest member of a Rydberg series, with predissociation of the higher vibronic states. Results, for a one-dimensional model, obtained by the techniques of multichannel quantum defect theory (MQDT) are compared with those of a simpler Fano golden rule approximation. A simplified model calculation for the ns1B1 series of H2O and D2O yields results in good accord with experiment, and a dramatic isotope effect is explained. A similar application to the elusive np1A2 series leads to the prediction of sharp rotational structure for the np1A2(000) bands with n≥4, despite the evidence of a very broad Franck–Condon profile for the dissociative 3p1A2 state. A band at 90 051 cm−1 in the 2+1 REMPI spectrum is tentatively assigned to the 4p member of this 1A2 series.

Multireference configuration interaction and quantum defect calculations on the Rydberg states of the BH molecule†

Configuration interaction calculations have been carried out on electronic states of BH and BH+. The calculated spectroscopic constants are in good agreement with experimental data, where available. Quantum defect functions have been calculated from the ab initio potential energy curves of 4s(1Σ+), 4s(3Σ+), 4p(1Σ+), 4p(3Σ+), 4p(1Π), 3d(1Π), 3d(1Δ), 4d(3Δ), 4f(1Π), 4f(1Φ), 4f(1Δ) and 4f(3Δ) Rydberg states of BH, which may be used to generate all the Rydberg states of each series up to the ionization limit. Calculated vibrational autoionization widths for the Rydberg states are smaller than 0.5 cm−1.

Ab initio and quantum-defect calculations for the Rydberg states of ArH

Potential energy curves were evaluated for the ground and thirteen low-lying excited electronic states of the ArH molecule over a wide range of internuclear distances by the multi-reference averaged quadratic coupled cluster method. The ab initio energy differences and transition dipole moments were used to estimate Einstein emission coefficients, absorption oscillator strengths and radiative lifetimes. Diagonal and off-diagonal quantum defects, as functions of internuclear distance, were extracted from ab initio potentials of the lowest Rydberg states of the neutral ArH molecule by taking account of configuration interaction between Rydberg series converging to the ground and two electronic excited states of the ArH(+) cation. The derived quantum-defect functions were used to generate manifolds of higher excited Rydberg states. The agreement between experimental and calculated energies and radiative transition probabilities was found to be as good as or better than that obtained by earlier calculations.

Calculations of high Rydberg states for the lithium atom

We present calculations of high Rydberg states for the lithium atom by two different theoretical methods. The first method is based on the $R$-matrix theory to obtain the energy levels and the quantum defects for the $1{s}^{2}nd\phantom{\rule{0.2em}{0ex}}^{2}D$ and $1{s}^{2}nf\phantom{\rule{0.2em}{0ex}}^{2}F$ states and their adjacent continuum states. Then the quantum defect functions of the two channels are obtained which vary smoothly with energy based on quantum defect theory. The second method is based on the full core plus correlation method. Our results are compared with data in the literatures; especially the results of the $R$-matrix method are in excellent agreement with microwave measurements for the lithium atom.

Resonance calculations of d − f intervals for the lithium Rydberg states

By using the energies of the 1s2nd (n=3–5) and 1s2nf (n=4–6) states for a lithium atom, calculated with the full core plus correlation method, quantum defect functions are obtained which vary smoothly with energy based on quantum defect theory. Then these functions are separately used to construct a system equation to calculate the energies and quantum defects of the 1s2nd and 1s2nf (n=7–11) states for the lithium atom. Separations between the centres of gravity of the nd and nf levels (n=7–11) are also predicted and compared with the experimental and theoretical data available in the literature.

Theoretical study of the ArH+ electronic states

Potential energy curves, permanent multipole and transition dipole moments were evaluated for the ground and low-lying excited electronic states of the ArH+ cation over a wide range of internuclear distance by the multireference averaged quadratic coupled cluster method (MR-AQCC). The electric dipole polarisability of the ground X 1sigma state was evaluated by the finite-field method. The permanent multipole moments and dipole polarisabilities corresponding to the ArH+ X 1sigma+ state were used to estimate quantum defect functions of the nonpenetrating d- and f-complex Rydberg states of neutral ArH molecule. The ground state dipole function and potential were tested by a simulation of intensity distributions in the rovibrational deltav = 1 bands, radiative lifetimes and rotational g-factors for the X 1sigma+ state.

High-resolution millimeter wave spectroscopy and multichannel quantum defect theory of the hyperfine structure in high Rydberg states of molecular hydrogen H2

Experimental and theoretical methodologies have been developed to determine the hyperfine structure of molecular ions from detailed studies of the Rydberg spectrum and have been tested on molecular hydrogen. The hyperfine structure in l=0-3 Rydberg states of H2 located below the X 2Sigmag+(v+=0,N+=1) ground state of ortho H2+ has been measured in the range of principal quantum number n=50-65 at sub-MHz resolution by millimeter wave spectroscopy following laser excitation to np and nd Rydberg states using a variety of single-photon and multiphoton excitation sequences. The np1(1), nd1(1), and the nf1(0-3) Rydberg states were found to be metastable and to have lifetimes of more than 5 micros beyond n=50. Members of other series, such as the nd1(2), nd1(3), and the np1(0) series, were found to have lifetimes of more than 1 mus. Local perturbations induced by low-n Rydberg states belonging to series converging on rovibrationally excited levels of H2+ reduce the lifetimes in narrow ranges of n values. The hyperfine structure is strongly dependent on the value of the orbital angular momentum l. In the penetrating s and p states at n approximately 50 the exchange interaction dominates over the hyperfine interaction and the levels can be labeled by the total electron spin angular momentum quantum number S (S=0 or 1). In the less penetrating d and f Rydberg states, the hyperfine interaction between the core nuclear and electron spins is larger than the exchange interaction and the Rydberg states are of mixed singlet and triplet character. A procedure based on the Stark effect and on the systematic analysis of selection rules and combination differences was developed to determine the orbital and the total angular momentum quantum numbers l and F and to construct an energy map of p and f Rydberg levels between n=54 and 64 with relative positions of an accuracy of better than 1 MHz. Multichannel quantum defect theory (MQDT) was extended to treat the hyperfine structure in molecular Rydberg states and was used to analyze the observed hyperfine structure of the p and f Rydberg states of H2. The frame transformation between the Born-Oppenheimer channels described by the angular momentum coupling scheme (abetaJ) and the asymptotic channels described by the (e[bbetaS+]) coupling scheme was derived and enables an elegant treatment of all intermediate coupling cases. Purely ab initio quantum defect theory reproduced the experimentally determined positions to within 40 MHz for the p levels and 13 MHz for the f levels. By slight adjustments of the quantum defect functions and their energy dependences and by consideration of the p-f interaction, of the singlet-triplet splittings of the f levels, and of the departure of the ionic levels from pure coupling case (bbetaS+), the agreement between theory and experiment could be improved to 600 kHz. By comparing the results of MQDT calculations of the hyperfine structure of f Rydberg levels with those of coupled equations calculations, the frame transformation approximation of MQDT was shown to be accurate to within 300 kHz. The extrapolated ionic hyperfine structure of the X 2Sigmag+(v+=0,N+=1) ionic level corresponds to the ab initio prediciton of Babb and Dalgarno [Phys. Rev. A 46, R5317 (1992)] within the experimental error.

Semiclassical theory for the quantum defect function of diatomic molecules

The simple analytical expression for the quantum defect function, Δlm(R),of a diatomic molecule is derived using the uniform semiclassicalapproach including the effects of hidden crossing. Δlm(R) is a function ofthe internuclear distance Rand the effective charge at each nucleus is also a function ofR. Thus,in the case of homonuclear molecules this expression gives universal dependenceon R.The derived expression is applied to H2molecules. The quantum defects are found to be in good agreement with the abinitio numerical data by Wolniewicz and Dressler (1994 J. Chem. Phys. 100 444).

Theoretical ab initio study of the electronic states of KrH and KrH+: Quantum defect and complex coordinate calculations on the Rydberg states of KrH

Potential energy curves have been calculated for the ground and excited electronic states of KrH and the cation KrH+ by ab initio configuration interaction calculations using effective core potentials for Kr. Quantum defect functions have been determined from the ab initio potentials of the low-lying Rydberg states of KrH and potential energy curves have been generated for higher n (s,p,d) Rydberg states. The resulting bound–bound transition energies are in excellent agreement with experimental data. The interaction of the 5p B 2Π state with the 5s and 5p A 2Σ and C 2Σ+ states and their predissociation by X 2Σ+ has been treated by multistate complex scaling calculations for both KrH and KrD. Much larger predissociation widths are obtained in KrH than in KrD, in agreement with experimental observations.

Lifetime measurements and quantum-defect theory treatment of the k 3Πu− state of hydrogen molecule

The experimental and theoretical lifetimes for rovibronic k 3Πu− states of H2 and D2 isotopomers have been investigated over a wide range of vibrational v′ and rotational N′ quantum numbers. Lifetimes have been measured by a delayed coincidence method, combined with direct electron-impact excitation of the ground state molecules and single photon counting techniques to detect induced fluorescence to the a 3Σg+ state. Pronounced pressure-dependence of the experimental lifetimes was observed and properly taken into account. The pure radiative lifetimes of the k 3Πu− states were estimated using theoretical transition dipole moments responsible for the visible k 3Πu→a 3Σg+ transition plus infrared emission on the higher-lying Λg3 states belonging to the 3s,d 3Λg complex. Both the predissociative and autoionization decay rates were predicted by the Fermi-Golden rule based on radial coupling matrix elements for the k 3Πu∼c 3Πu and k 3Πu∼X2Σg+(H2+) pairs of interacting states, respectively. The required electronic matrix elements as a function of internuclear distance R were derived in the framework of quantum-defect theory modified to allow explicit consideration of regular radial coupling effects. The relevant quantum-defect functions of all states treated were extracted from published highly accurate Born–Oppenheimer potential curves. Both the total theoretical radiative, predissociative plus autoionization rates and the calculated rovibronic term values agree well with their experimental counterparts. The N′-dependence of the experimental and theoretical lifetimes is found to be negligible for both isotopomers while a pronounced v′-dependence is observed for a H2 isotopomer. The vibrational predissociation is very weak comparing with radiative decay for both isotopomers whereas the autoionization rate is comparable with the radiative ones for the H2 v′⩾4 levels though it is still negligible for the D2 v′⩽6 levels.

Ab initio R-matrix/multichannel quantum defecttheory study of nitric oxide

Ab initio R-matrix techniques are combined withmulti-channel quantum defect theory to determine theproperties of arbitrarily highly excited electronic states ofnitric oxide. Results are obtained for l = 0-4 channelsassociated with the four lowest NO+ target channels, namely1Σ+, 3Σ+, 3Π and3Δ. Energy and bond length variations of the quantumdefect functions for all target states are reported. Certainsmall surface amplitude R-matrix poles also serve to determinethe energies of valence states. Diabatic potential curves forRydberg states converging to the 1Σ+ and3Σ+ target states and for the valence states arepresented. Systematic jumps in the pσ quantum defectsfor all target states as the bond length increases beyondR = 2.5 au are attributed to Rydbergization of the antibonding2pσ* molecular orbital.

The Rydberg states of NO2: Vibrational autoionization of the ndσ states

Ab initio configuration interaction calculations were carried out on the potential-energy surfaces of the ground and Rydberg excited electronic states of NO2. The results show that potential-energy curves with typically Rydberg form are obtained for most of the excited states at linear geometries, similar to the ground-state potential of NO2+. At nonlinear geometries valence-Rydberg interactions complicate the potential-energy surfaces of the excited states. Quantum defect functions have been determined from the ab initio results on the 3dσ Rydberg state and vibrational autoionization widths have been calculated for excited vibrational levels of members of the ndσ series.

Theoretical study on the Rydberg states of NeH: Ab initio quantum defect and complex coordinate calculations

Ab initio calculations have been carried out on the potential energy curves of the Rydberg states of NeH up to 3d. Quantum defect functions have been calculated from the ab initio potentials and potential energy curves and vibrational levels for higher n (s,p,d) Rydberg states have been generated. The interaction of the 2p B 2Π state with the 2s and 2p, A 2Σ+ and C 2Σ+ states and their predissociation by X 2Σ+ has been treated by multi-state complex coordinate scaling calculations for both NeH and NeD. The results are consistent with previous 2×2 calculations on the predissociation of the A 2Σ+ and C 2Σ+ states. Finally, a calculation of the interaction between the Π2, Σ+2, and Δ2 3d states in NeH and NeD shows appreciable mixing between the states at high values of the rotational quantum number.

The Rydberg states of H 2 O

Empirical quantum defect functions are fit to the vibrational origins of the known rotationally resolved bands of H2O and D2O, including new analyses for bands in the region of the 3d (100) cluster...

On the construction and use of ab initio quantum defect functions for

Ab initio adiabatic and diabatic symmetric stretch/bend potentials of Rydberg states of have been employed for calculations of corresponding quantum-defect functions. These are in turn employed for the generation of the potential-energy surfaces of higher-lying Rydberg states, for the calculation of vibrational levels of the linear state and for the calculation of autoionization line widths in the series. The results are generally in good agreement with existing experimental information.

Secular perturbation theory of long-range interactions.

Solutions of Schroedinger's equation for systems interacting with long-range power-law potentials, [ital V][sub int]([ital r])[proportional to][ital r][sup [minus][ital n]] with [ital n][ge]2, are cast in the form of a power series in [ital V][sub int]. These perturbations lead to secular divergences that are eliminated by renormalizing the angular-momentum quantum number. Long-known perturbation techniques in classical mechanics and quantum-field theory yield modified effective-range formulas, quantum-defect functions, and solutions of close-coupling equations for wave propagation in long-range fields. As an example, we extract in first-order perturbation theory a modified effective-range expansion for the phase shift of an electron interacting with an atomic 1/[ital r][sup 4] polarization potential. Near thresholds, the method is applicable to all power-law potentials with [ital n][ge]2, and to their combinations, as well as to multichannel problems involving anisotropic potentials.