Variational calculations on the hydrogen molecular ion

We present high-precision non-relativistic variational calculations of bound vibrational—rotational state energies for the H⁺ ₂ and D⁺ ₂ molecular ions in each of the lowest electronic states of Σ _g , Σ _u , and Π _u symmetry. The calculations are carried out including coupling between Σ and Π states but without using the Born—Oppenheimer or any adiabatic approximation. Convergence studies are presented which indicate that the resulting energies for low-lying levels are accurate to about 10^?13. Our procedure accounts naturally for the lambda-doubling of the Π _u state.     

Spectroscopic and molecular properties of 14 selected electronic states of Si2 molecule

The potential energy curves (PECs) of the X³Σ_g⁻, D³Π_u, a¹Δ_g, b¹Π_u, H′³Σ_u⁻, K³Σ_u⁻, 1³Σ_u⁺, 1³Π_g, 2³Σ_u⁺, 2³Π_g, 3³Π_g, 3³Σ_u⁺, 2³Π_u and 2³Σ_g⁻ electronic states of the Si₂ molecule are investigated using the complete active space self-consistent field (CASSCF) method followed by the valence internally contracted multireference configuration interaction (MRCI) approach with the correlation-consistent basis sets of Dunning and co-workers. The effects on the PECs by the core-valence correlation and relativistic corrections are included. The way to consider the relativistic correction is to use the third-order Douglas-Kroll Hamiltonian approximation. The core-valence correlation correction is made with the aug-cc-pCV5Z basis set. And the relativistic correction is performed at the level of cc-pV5Z basis set. To obtain more reliable results, the PECs determined by the MRCI calculations are also corrected for size-extensivity errors by means of the Davidson modification (MRCI+Q). The PECs of all these electronic states are extrapolated to the complete basis set limit by the total-energy extrapolation scheme. Using the PECs, the spectroscopic parameters are determined and compared with those reported in the literature. With these PECs determined by the MRCI+Q/CV+DK+56 calculations, the vibrational levels and inertial rotation constants of the first 20 vibrational states are evaluated and compared with the RKR data for these electronic states when the rotational quantum number J equals zero. On the whole, as expected, the most accurate spectroscopic parameters and molecular constants of the Si₂ molecule are determined by the MRCI+Q/CV+DK+56 calculations. And the spectroscopic parameters of the 1³Σ_u⁺, 1³Π_g, 2³Σ_u⁺, 2³Π_g, 3³Π_g, 3³Σ_u⁺, 2³Π_u and 2³Σ_g⁻ electronic states obtained by the MRCI+Q/CV+DK+56 calculations should be good prediction for future laboratory experiment.     

Comparison of semiempirical and ab initio absolute probabilities of rovibronic transitions for the I 1Π g − , J 1Δ g − →C 1Π u ± system of bands of the H2 molecule

The absolute values of probabilities of the I ¹Π _g ^? , v′, J′; J ¹ Δ _g ^? , v′, J′→C ¹Π _u ^± , v″, J″ spontaneous transitions in the H₂ molecule (for the vibrational and rotational quantum numbers v′=v″=0–3, J′=1–6, and J″=J′, J′ ±1) are calculated by using ab initio and semiempirical data on the dipole moments of the 3dπ ¹Π_g, 3dδ¹Δ_g→2pπ¹Π_u electronic transitions. In both cases, the calculations are performed both in the adiabatic approximation and with an allowance for the nonadiabatic effect of electronic-rotational interaction. The coefficients of expansion of the wave functions of perturbed rovibronic states in the Born-Oppenheimer basis functions used in the calculations were obtained in the approximation of pure precession from experimental values of the terms. It was found that the values of transition probabilities based on the ab initio calculations systematically exceed the corresponding semiempirical data by a factor of 1.2–1.9 for the I ¹Π_g→C ¹Π _u ^± transition and by a factor of 1.4–1.6 for the J ¹Δ _g ^? →C ¹Π _u ^± transition. It was established that the difference between the ab initio and semiempirical values of electronic transition moments virtually has no effect on the dependence of the transition probabilities on the vibrational quantum numbers. The discrepancies between the results of adiabatic and nonadiabatic calculations are significant and reach two orders of magnitude, which is indicative of the important role of perturbations in the probabilities of the transitions considered.     

Ab initio potential energy curves and vibrational levels for the c, l, and i states of the hydrogen molecule

Potential energy curves for the hydrogen molecule in the c³Π_u, I¹Π_g, and i³Π_g state have been calculated in the Born-Oppenheimer approximation. Highly flexible wavefunctions have been used, and for each state the calculations have been carried out for 40 different internuclear distances in the region 1 ≤ R ≤ 12 a.u. For the Π_g states the wavefunctions are known to change their character around R = 5 a.u. The effect of this change on various components of the energy has been analyzed. The vibrational Schrödinger equation for all states has been solved for H₂, HD, and D₂. For H₂ the resulting energies are compared with the experimental values and it is shown that the adiabatic effects are likely to be responsible for the existing discrepancy between the theoretical and experimental values.     

Accurate theoretical spectroscopic investigations of the 20 Λ-S and 58 Ω states of O2+ cation including the spin–orbit coupling effect

The potential energy curves (PECs) are calculated for the 20 Λ-S states (X²Π_g, A²Π_u, B²Σ^?_g, a⁴Π_u, b⁴Σ^?_g, b′⁴Π_g, c⁴Σ^?_u, 1²Σ⁺_g, 1²Σ⁺_u, 1²Σ^?_u, 1⁴Σ⁺_g, 1⁴Σ⁺_u, 1⁴Δ_g, 1⁴Δ_u, 1⁶Σ⁺_g, 1⁶Σ⁺_u, 1⁶Π_g, 1⁶Π_u, 2⁴Π_g and 2⁴Π_u) of O₂⁺ cation and their corresponding 58 Ω states. Of these 20 Λ-S states, the 1⁶Π_u state is found to be repulsive. The 1²Σ⁺_g, 1⁴Σ⁺_u, c⁴Σ^?_u and 1⁴Δ_u states are found to possess the double well. The b⁴Σ^?_g, 1⁶Σ⁺_g, 1⁴Σ⁺_u, a⁴Π_u, A²Π_u, 1⁶Π_g and 2⁴Π_g states are found to be inverted with the spin–orbit coupling effect included. The b′⁴Π_g, 1⁶Π_g, 1⁶Σ⁺_g, 1⁴Σ⁺_u and 1⁴Δ_u states, and the second well of the 1²Σ⁺_g state are found to be the weakly bound states. The b′⁴Π_g state is found to possess one well with one barrier. The PECs are calculated by the complete active space self-consistent field method, which is followed by the internally contracted multireference configuration interaction approach with the Davidson correction in combination with the aug-cc-pV6Z basis set. The core–valence correlation and scalar relativistic corrections are included. The convergent behaviour of present calculations is discussed with respect to the basis set and theoretical level. The spin–orbit coupling effect is accounted for. The PECs are extrapolated to the complete basis set limit. The spectroscopic parameters are evaluated, and compared with available measurements. It demonstrates that the spectroscopic parameters reported here can be expected to be reliably predicted ones.     

Observation of perturbations in the rovibronic transition probabilities for the (4d)r 3Π g − , (4d) s 3 Δ g − → (2p) c 3Π u ± band systems of the H2 molecule

The ratios of the radiative transition probabilities for the lines of the P, Q, and R branches of the (4d)r ³Π _g ^? , (4d)s ³Δ _g ^? → (2p) c ³Π _u ^± band systems of the H₂ molecule have been measured for the first time. Significant (to two orders of magnitudes) differences are found between the experimental values and the adiabatic theory predictions. It is established that the results of the nonadiabatic calculation performed by us in the pure-precession approximation taking into account the electronic-rotational interaction of the 4d ³Π_g and 4d ³Δ_g states are in agreement with the experimental data. The optimal energies of rovibronic levels of the r ³Π _g ^? , s ³Δ _g ^? , c ³Π _u ^? , and c ³Π _u ⁺ states have been found and reidentification of 11 from 54 spectral lines, assigned previously to the (0-0) and (1-1) bands, was performed.     

Ab initio calculation of transition dipole moments for transitions between valence states in oxygen molecules

Results of ab initio calculations of potential-energy curves for 20 singlet and 20 triplet valence states of oxygen with configuration interaction taken into account in the 6-31G basis are presented. Transition dipole moments of triplet-triplet (1³Π_g ← B ³Σ _u ^? , 1³Π_g ← A ³Σ _u ⁺ , 1³Π_g ← A′ ³Δ_u, B ³Σ _u ^? ← X ³Σ _g ^? , 2³Π_u ← 1³Π _g, 2³Σ _g ^? ← B ³Σ _u ^? , 1³Π_u ← X ³Σ _g ^? , 2³Π_u ← X ³Σ _g ^? , 2³Π _g ← A′³Δ_u, 3³Π_g ← A′ ³Δ _u, 2³Δ_u ← 2³Π_g, 3³Π_g ← B ³Σ _u ^? , and 2³Π_g ← A ³Σ _u ⁺ ) and singlet-singlet (2¹Σ _g ⁺ ← 2¹Π_u, 2¹Π_u ← 1¹Π ^g, ¹Π_u ← 2¹Δ_g, 1¹Π_g ← c ¹Σ _u ^? , ¹Π_u ← b ¹Σ _g ⁺ , 1¹Δ _u ← a ¹Δ_g, 2¹Π_u ← a ¹Δ_g, 2¹Δ_g ← 1¹Δ_u, ¹Π _u ← a ¹Δ _g, 1¹Π_u ← b ¹Σ _g ⁺ , 2¹Π_g ← 1¹Π_u, 2¹Π _g ← c ¹Σ _u ^? , 1¹Δ _u ← 1¹Π _g, f′Σ _u ⁺ ← b ¹Σ _g ⁺ , 2¹Σ _g ⁺ ← f ′¹Σ _u ⁺ , 3¹Π_g ← 1¹Δ_u) radiative transitions are calculated as functions of internuclear separation. The possibility of observing these transitions under experimental conditions is discussed.     

Adiabatic Energies of Excited Σu States of the Hydrogen Molecule

Accurate Born-Oppenheimer potential energy curves and adiabatic corrections are computed for the six lowest ¹Σ_u states of the hydrogen molecule. For the 4¹Σ_u-6¹Σ_u states adiabatic term values of the vibrational levels supported by the potentials and the corresponding rotational constants are given. For the outer potential well of the 6¹Σ_u state, Franck-Condon factors are listed for the possible transitions to the HH?¹Σ_g state.     

The radiative characteristics of the rovibronic states of the hydrogen molecule: IV. Relative probabilities of the r 3Π g ? , s 3Δ g ? → c 3Π u ± spontaneous transitions of the H2 molecule

A statistical analysis of the available literature data and data obtained in this work on the wave numbers of the lines that appear in triplet-triplet rovibronic transitions of the H₂ molecule was performed. This allowed us to verify and refine the controversial identification of spectral lines and find the optimum rovibronic level energy values for the c ³Π _u ^±, r ³Π _g ⁻, and s ³Δ _g ⁻ states. The ratios between the line strengths of the P, Q, and R branches of the (4dπ)r ³Π _g ⁻, (4dδ)s ³Δ _g ⁻ → (2pπ)c ³Π _u ^± band systems of the H₂ molecule were measured systematically. The calculation results obtained in the Frank-Condon approximation differed substantially (by up to two orders of magnitude) from the experimental data. The dependences of the ratios between rovibronic line strengths of the r ³Π _g ⁻ → c ³Π _u ^± and s ³Δ _g ⁻ → c ³Π _u ^± transitions on the rotational quantum number N′ of the upper level were found to correlate with each other. The deviations of adiabatic theory increase as N′ grows, which is evidence of an important role played by electronic-rotational interactions in the perturbation of transition probabilities. The experimental ratios between the probabilities of rovibronic transitions satisfactorily agree with the results of calculations within the framework of the simple nonadiabatic model taking into account electronic-rotational interaction of radiating adiabatic states in the approximation of pure precession. The dependences of transition probabilities on N′ were obtained for the first time for the first four diagonal bands of the r ³Π _g ⁻, s ³Δ _g ⁻ → c ³Π _u ^± transitions. Original Russian Text ? S.A. Astashkevich, B.P. Lavrov, A.V. Modin, I.S. Umrikhin, 2008, published in Khimicheskaya Fizika, 2008, Vol. 27, No. 2, pp. 22–38.     

Electronic structure of lanthanide dimers

Relativistic energy-consistent small-core lanthanide pseudopotentials of the Stuttgart-Bonn variety and extended valence basis sets have been used for the investigation of some selected lanthanide dimers with open 4f shell, that is Ce₂, Pr₂ and Gd₂. Comparison is made with results of corresponding previous studies of La₂ and Lu₂ as well as to available experimental data. The trends in the molecular constants of the dimers of the lanthanide series are discussed. The ground state candidates of Ce₂(4f¹4f¹σ² _gπ⁴ _u ¹Σ⁺ _g, ¹Σ^? _u, ³Σ^? _g, ³Σ⁺ _u, ¹6_g, ³6_u) and Pr₂(4f²4f²σ² _gπ⁴ _u ⁵Σ⁺ _g, ⁵Σ^? _u, ⁵10_g) are degenerate within 20cm^?1 and have the same valence subconfiguration σ² _g π⁴ _u, which was previously found to give rise to the La₂(4f⁰4f⁰σ² _gπ⁴ _u ¹Σ⁺ _g) ground state. In the case of Gd₂ the 4f⁷4f⁷σ² _gσ¹ _uσ² _u; ¹⁹Σ^? _g ground state found previously is confirmed. The derived molecular constants are the best theoretical estimates available at present and show a satisfactory agreement with experimental data. Discrepancies in the vibrational constants of La², Ce² and Pr² are shown to be probably related to quite large Ar-matrix shifts.     

Non-adiabatic coupling between three states

Non-adiabatic transition probabilities in systems of three adiabatic molecular states of the same symmetry are calculated by the semiclassical/adiabatic technique and numerically. The terms considered are three 1 _g (or 1 _u ) molecular states of alkali diatom M ₂; in the limit of three atoms these terms correlate with the ground ² S _1/2 and excited ² P_j (j = ½, 3/2) atomic states. The results are discussed in connection with the corresponding Massey parameter and can be used to estimate non-adiabatic effects in photodissociation of M ₂*(B ¹Π _u ) and in collisional excitation of M.     

用能量自洽法研究碱金属双原子分子的势能曲线

用能量自洽法(ECM)研究了碱金属双原子分子一些电子激发态的势能曲线：Na₂ 分子的2¹Π_g,4³Π_g和b³Π< sub>u电子激发态,K₂分子的a³Σ^＋_{u,2¹Πg,B¹Πu和A^{关键词： 能量自洽 双原子分子 势能 碱金属}}     

Electronic transition dipole moment functions and difference potentials for transitions among low-lying states of Li2 and Na2

Electronic transition dipole moment functions based on ab initio multiconfiguration self-consistent field wavefunctions are computed for the transitions ¹Σ_u⁺-¹Σ_g⁺, ³Σ_g⁺-³Σ_u⁺, ¹Π_u-¹Σ_g⁺, ³Π_g-³Σ_u⁺, ¹Σ_u⁺-¹Π_g, ³Σ_g⁺-³Π_u, ¹Π_u-¹Π_g, and ³Π_g-³Π_u in Li₂ and Na₂. (In each case the states are the lowest lying of their symmetry.) We also calculate the matrix element 〈³Σ_u⁺|i(L_x - iL_y)|³Π_u〉 for the predissociation of the ³Π_u state by the ³Σ_u⁺ state. Several unobserved spectral features are predicted.     

Comparative study of the semiempirical three-parameter potential energy functions for diatomic molecules

A comparative study is made of four three-parameter semiempirical potential energy functions for 32 electronic states of diatomic molecules and their ions:n ₂:X¹gS _g ⁺ ,B ³π_g,A ³ gS_u,C ³ _u,B′ ³ gS_u.a ¹ π_g, a′gS _u ^? ,Ω ¹δ_u N ₂ ⁺ :X ² gS _g ^gS +A ² π,C ² gS _u ⁺ ,B ² gS _u ⁺ CO:X¹gS⁺,a ³ π, a′³ gS_u,e ³ gS^?,d ³gD₁,A ¹π CO⁺:X²gS⁺,A ² π,B ²gS⁺ O₂:X³gS _g ^? ,B ³ gS_u,c ¹ gS _u ^? ,b ¹gS _g ^s ,a ¹ δ_g,c ³ δ_u O ₂ ⁺ :X ²π_g,A ² π_g, a¹ π_g,b ⁴ gS _g ^? A program for numerically integrating the radial Schrödinger equation by the Cooley method is worked out. Certain additional units are introduced to conserve computer time. The resulting vibrational levels are compared with the experimental levels for all the electronic states studied. It is concluded on the basis of this analysis that it is not possible to describe equally well all the electronic states of various molecules on the basis of any single three-parameter potential function. A method for choosing a potential function for describing some particular electronic state of a diatomic molecule is proposed.     

Experimental determination of the potential energy curves of the I(3/2u) and I(3/2g) states of Kr+ 2

The I(3/2u) and I(3/2g) states of Kr⁺ ₂ have been investigated by pulsed-field-ionization zero-kinetic-energy (PFI-ZEKE) photoelectron spectroscopy following (2 + 1′) resonance-enhanced multiphoton excitation via the 0⁺ _g Rydberg state located below the Kr?([4p]⁵5p[1/2]₀) + Kr(¹S₀) dissociation limit of Kr₂. From the positions of a large number of vibrational bands in the spectra of the ⁸⁴Kr₂ and ⁸⁴Kr-⁸⁶Kr isotopomers, the adiabatic ionization potentials (IP(I(3/2u)) = 112672.4 ± 0.8cm^?1, IP(I(3/2g)) = 111 395.0 ± 1.4cm^?1), the dissociation energies (D ⁺ ₀(I(3/2u)) = 368.8 ± 2.0cm^?1, D ⁺ ₀(I(3/2g)) = 1646.2 ± 2.3cm^?1) and vibrational constants for both ionic states have been determined. Potential energy curves have been extracted which perfectly reproduce all experimental observations and are accurate over a wide range of energies and internuclear distances. The equilibrium internuclear distances (R ⁺ _e(I(3/2u)) = 4.11 ± 0.04 Å, R ⁺ _e(I(3/2g)) = 3.35 ± 0.10 Å) have been derived by comparing the intensity distribution in the PFI-ZEKE photoelectron spectra to calculated Franck-Condon factors. The dissociation energy of the I(3/2g) state and the equilibrium internuclear distance of the I(3/2u) state differ markedly from previously reported values.     

Long range adiabatic potentials and scattering lengths for the EF,e and h states of the hydrogen molecule

Accurate EF ¹Σ _g , e ³Σ _u and h ³Σ _g state adiabatic potentials are computed for internuclear distances up to 44 atomic units. All the energies are more accurate by a fraction of a wave number from existing results. Scattering lengths for collisions of 1S and 2S hydrogen atoms are evaluated. These differ significantly from earlier results. This is because for some states the scattering length is very sensitive to even small changes in the potential.     

Kinetic ion-acoustic solitary waves in collisional plasmas

In this study, a bipolar high-voltage pulse with 20 ns rising time is employed to generate diffuse dielectric barrier discharge plasma using wire-plate electrode configuration in nitrogen at atmospheric pressure. The gas temperature of the plasma is determined by comparing the experimental and the best fitted optical emission spectra of the second positive bands of N₂(C³Π_u → B³ Π_g, 0-2) and the first negative bands of N₂ ⁺ (B² Σ_u ⁺ → X² Σ_g ⁺, 0-0). The effects of the concentration of argon and oxygen on the emission intensities of N₂ (C³Π_u → B³Π_g, 0-0, 337.1 nm), OH?(A ²Σ → X²Π, 0-0) and N₂ ⁺ (B² Σ_u ⁺ → X² Σ_g ⁺, 0-0, 391.4 nm) are investigated. It is shown that the plasma gas temperature keeps almost constant with the pulse repetition rate and pulse peak voltage increasing. The emission intensities of N₂ (C³Π_u → B³Π_g, 0-0, 337.1 nm), OH(A²Σ → X²Π, 0-0) and N₂ ⁺ (B² Σ_u ⁺ → X² Σ_g ⁺, 0-0, 391.4 nm) rise with increasing the concentration of argon, but decrease with increasing the concentration of oxygen, and the influences of oxygen concentration on the emission intensities of N₂(C³Π_u → B³Π_g, 0-0, 337.1 nm) and OH (A²Σ → X²Π, 0-0) are more greater than that on the emission intensity of N₂ ⁺ (B² Σ_u ⁺ → X² Σ_g ⁺, 0-0, 391.4 nm).     

Calculation of adiabatic potentials of Li<Subscript>2</Subscript><Superscript>+</Superscript>

We report adiabatic potential energy curves of the Li₂ ⁺ molecule. Our curves are tabulated according to internuclear distance from 2 a₀ to 100 a₀. We compare our theoretical results with the ones calculated by other authors and potential energy curves derived from experiments. For the ground state and 17 excited states we calculate spectroscopic parameters and compare them with parameters obtained by other authors. For the first time we present three new minima for 3²Σ_u ⁺, 4²Σ_u ⁺ and 2²Π_g excited states. In our approach we use the configuration interaction method where only the valence electrons of Li atoms are treated explicitly. The core electrons are represented by pseudopotential. All calculations are performed by means of MOLPRO program package.     

Rovibronic Levels for the (1-3)Πg Manifold of B2: Exterior Complex Scaling Finite Element Calculations Based on an Adiabatic and a Strictly Diabatic Basis

A one-dimensional complex scaled finite element method was applied on an adiabatic basis of B₂ in order to find rovibronic term energy values for the (1)³Π_g; (v, N)=(0-8, 0-25) and (2)³Π_g; (v, N)=(0-10, 0-25) levels. The method was also applied to the (1)³Π_g; (v, N)=(0-8, 0-25) and (2)³Π_g; (v, N)=(0-1, 0-25) levels in a strictly diabatic framework. Adiabatic single-channel and diabatic coupled-channel total wavefunctions were obtained and used in order to identify the vibrational levels. Comparing levels for the interacting two-state (1-2)³Π_g and three-state (1-3)³Π_g system, a constant energy shift of about 1.7 cm⁻¹ is found. Comparisons between the adiabatic Born-Oppenheimer (BO) and the diabatic (1)³Π_g; (v, N)=(0-8, 0-25) levels show differences between −20 and 7 cm⁻¹, while the corresponding shifts for the (2)³Π_g; (v, N)=(0, 0-25) and (1, 0-25) levels are about 50 and 60 cm⁻¹, respectively. A comparison between our three-state approximation and experimental observations of the (2)³Π_g-A³Π_u electronic transition shows a difference in the line positions of about 665 cm⁻¹. The calculated widths for all but the (1)³Π_g; (v, N)=(7-8, 0-25), as well as the (2)³Π_g; (v, N)=(0, 0-25) BO and diabatic rovibronic levels, have small but with N increasing predissociation rates. The (1)³Π_g; (v>8, N=0-25) BO and the (2)³Π_g; (v, N)=(1, 0-25) diabatic levels are strongly predissociated with widths ≥16 cm⁻¹.