Ab initio potential energy curves and vibrational levels for the C and D 1Πu states of the hydrogen molecule

Two lowest eigenvalues of the Hamiltonian for the hydrogen molecule corresponding to the ¹Π_u symmetry have been calculated in the Born-Oppenheimer approximation. They represent the C and D states, respectively. Different highly flexible wavefunctions were used for both states and the calculations were performed for a wide range of internuclear distances: 1 ≤ R ≤ 12 a.u. for the C state, and 1 ≤ R ≤ 25 a.u. for the D state. The calculated potential energy curves are more accurate than any previous ab initio results for the above states. The vibrational Schrödinger equation for both states has been solved for H₂, HD, and D₂ using the Numerov method. The resulting energies and rotational constants are compared with the experimental values.     

Electronic transition moments between excited singlet states of the H2 molecule

The twelve transition moments which connect each of the three ¹Σ_g⁺ states EF, GK, and $\text{H}\text{H}$ with the three ¹Σ_u⁺ states B,B′, $\text{B″}\text{B}$ and with the state C¹Π_u were computed in the range of internuclear distances 1.25 ≤ R ≤ 15 a.u. using accurate electronic wavefunctions. The relative phases of the wavefunctions, which determine the signs of the transition moments, were consistently defined at all R values. The strong R dependences of the transition moments reflect the R-dependent changes in electronic character of the states involved in these transitions. At large R values the ¹Σ_g⁺-¹Σ_u⁺ transition moments are dominated by the two-electron charge resonance in the ionic configuration H⁺ + H^? whose transition moment is $\text{M(R) ? R}$.     

Transition Moments betweenΔg,Πg, andΠuStates of the H2Molecule

Electronic transition moments are computed for dipole transitions between theCandD¹Π_u,IandR¹Π_gandJandS¹Δ_gstates of the hydrogen molecule for internuclear distancesR∈ [1, 15] bohr. The character of the wavefunctions at large internuclear distances is investigated. It is shown that in theDandRstates the asymptotic configurations are strongly mixed forR≤ 15 bohrs. TheSstate has mixed character even atR= 40 bohr.     

Improved potential energy curve and vibrational energies for the electronic ground state of the hydrogen molecule

The potential energy curve for the electronic ground state of the hydrogen molecule has been recomputed for intermediate and large internuclear separations. for 2.4 ≤ R ≤ 8.0a.u. the previous potential energy curve has been improved. The largest improvement amounts to 5.5 cm^?1, and was obtained in the vicinity of R = 4.4a.u.. Using the new potential energy curve, and the adiabatic and relativistic corrections, the vibrational and rotational energy levels have been calculated for H₂, HD, and D₂. The deviations of the calculated energy levels G(v) of H₂ and D₂ from the observed values follow very closely the nonadiabatic corrections resulting from the Van Vleck formula.     

A theoretical study of the electronic transition moment for the C2 swan band system

Self-consistent-field plus configuration-interaction calculations have been performed on the a³Π_u and d³Π_g states of C₂. Good agreement is obtained between the theoretical potential energy curves and these obtained by a Klein-Dunham analysis of measured molecular constants. The sum of the squares of the theoretical transition moments Σ|R_e|² between these states at 2.44 bohr is 4.12, a.u. which is in good agreement with the range of values of 3.3–3.6 a.u. obtained from shock tube measurements. The computed variation of the Σ|R_e|² with internuclear distance is in remarkably good agreement with the relative measurements by Danylewych and Nicholls.     

High-resolution Fourier spectrometry of 14N2 infrared emission spectrum: Extensive analysis of the B3Πg-A3Σu+ system

The B³Π_g-A³Σ_u⁺ system of N₂ excited in a microwave discharge was recorded between 3 000 and 18 000 cm^?1 with a high-resolution Fourier spectrometer. There are no observed irregularities in the levels of the A³Σ_u⁺ and B³Π_g states at least for the values of v and J considered here, except the predissociation in the B³Π_g state for v = 12 and J higher than 33. This predissociation will be checked with more complete data in another article. Thirty three bands of the first positive system with 0 ≤ v′ ≤ 12 and 0 ≤ v″ ≤ 8 are analyzed. The molecular parameters of the B³Π_g and A³Σ_u⁺ are obtained by a complete fitting procedure. Derived values of equilibrium constants are deduced; RKR potential energy curves for the two states are constructed, and the Franck-Condon factors calculated for the A-B system.     

Nonadiabatic effects in the B, C, B′, and D states of H2

The Kolos-Wolniewicz potentials for the H₂$\text{B}{}^1\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{+}}$ and C¹Π_u states were used together with the hypothesis of pure precession for the rotation-electronic interaction, to calculate the nonadiabatic energy levels of these states for J = 1 to 5. The complete coupling matrix was computed using accurate numerical vibrational wavefunctions. The calculated Λ-doubling of the v = 0 to 12 C vibrational levels generally agrees well with experimental values, and the nonadiabatic shifts in the B rotational constants qualitatively explain the difference between the theoretical and RKR potentials for this state.The interaction of the B′${}^1\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{+}}$ and D¹Π_u states was also investigated, but only qualitatively since adiabatic potentials of sufficient accuracy do not exist for these states. The Λ-doubling of the Dv = 0 rotational levels agrees well with the experimental values. An appreciable “background” nonadiabatic shift in the B′ rotational constants was found. This shift, which is nearly 5 percent of B_v, is in addition to that of strong local two-level interactions and was not “deperturbed” in constructing the B′ RKR potential. The result is that the RKR turning points differ by about 0.04 au from the “true” adiabatic turning points. This conclusion is supported by a Hartree-Fock calculation of the B′ potential to the left of R_e.     

Observation of the 5pΠu Rydberg state of Li2

Some weak, collisionally induced transitions in ⁷Li₂ have been recorded by Fourier transform spectrometry in the near infrared, following excitation of the 5d¹Π_g state by optical-optical double resonance. They have been assigned as transitions to the 1 ¹Δ_g state from levels v=0 and 1 of a new ungerade Rydberg state, 5p¹Π_u. Quantum defect considerations indicate that the principal quantum number for this new state is 5, and that the assignment to 5p is compatible with a Rydberg series of which the lowest members would be the B¹Π_u and C¹Π_u states.     

Rotational analysis of the A2Πu-X2Πg Second Negative band system of O2+

Existing high-resolution data for the $\text{O}{}_2{}^{\mathrm{+}}\text{A}{}^2\text{Π}{}_{\mathrm{u}}\text{- X}{}^2\text{Π}{}_{\mathrm{g}}$ Second Negative band system have been analyzed using a nonlinear least-squares fit that employs numerically diagonalized Hamiltonians. Values for the full set of molecular constants of the A²Π_u and X²Π_g states are obtained for the first time. In addition to values for ν₀(v′, v″), B_v, and D_v, the values for the spin-orbit coupling constants A_v are determined for both states. For the X²Π_g state, which is near Hund's case (a), the agreement between these A_v″ values and those predicted by theory is good. However, for the A²Π_u state, which is much nearer to case (b), these A_v′ values and theory disagree both in magnitude and in variation with vibrational level. The A²Π_u state is an inverted state for vibrational levels v′ ≤ 5 and is a regular state for levels v′ = 6–8 (the upper limit of present high-resolution data). Λ-doubling parameters are determined for the X²Π_g state, the only state where Λ-doubling is statistically significant. Spin-rotation interaction is not statistically significant for either state. Dunham Y_i0 and Y_i1 expansion coefficients are determined for each state. Theoretical D_v values calculated from RKR potentials are used to improve the B_v values in the reduction of the data.     

Theoretical electronic transition moments for the Ballik-Ramsay,Fox-Herzberg,and Swan systems OF C2

Potential energy curves and spectroscopic constants T_e, r_e, ω_e, and ω_eχ_e have been calculated for the a³Π_u, b³Σ^-_g, d³Π_g and e³Π_g electronic states of C₂ using self-consistent-field plus configuration-interaction techniques. These results are in excellent agreement with those obtained from experiments. The variation of the electronic transition moment with internuclear separation has been calculated for the Ballik-Ramsay system (b³Σ^-_g-a³Π_u), Fox-Herzberg (e³Π_g-a³Π_u), and Swan (d³Π_g-a³Π_u) band systems. These results are in good agreement with existing experimental and theoretical data.     

The electronic spectrum of AgBr2: Ab initio benchmark vs. DFT calculations on the lowest ligand-field states including spin-orbit effects

The X²Π_g, ²Σ_g⁺ and ²Δ_g states of AgBr₂ have been studied through benchmark ab initio CASSCF + Averaged Coupled Pair Functional (ACPF) and DFT calculations using especially developed valence basis sets to study the transition energies, geometries, vibrational frequencies, Mulliken charges and spin densities. The spin-orbit (SO) effects were included through the effective hamiltonian formalism using the |ΛSΣ〉 ACPF energies as diagonal elements. At the ACPF level, the ground state is ²Π_g, in contradiction with ligand-field theory and Hartree-Fock results. The ACPF adiabatic excitation energies of the ²Σ_g⁺ and ²Δ_g states are 3825 and 20 152 cm⁻¹, respectively. The inclusion of the SO effects leads to a pure Ω = 3/2 (²Π_g) ground state, a Ω = 1/2 (97% ²Π_g + 3% ²Σ_g⁺) A state, a Ω = 1/2 (3% ²Π_g + 97% ²Σ_g⁺) B state, a Ω = 5/2 (²Δ_g) C state and a Ω = 3/2 (99% ²Δ_g) D state. The B97, B3LYP and PBE0 functionals, which were shown to yield accurate transition energies for CuCl₂, overestimate the X²Π_g-²Σ_g⁺ T_e by around 25% but provide a qualitative energetic ordering in agreement with CASSCF and ACPF results. The nature of the bonding in the X²Π_g ground state is different from that of AgCl₂ since the Mulliken charge on the metal is 0.95 while the spin density is only 0.39. DFT strongly delocalizes the spin density providing even smaller values of around 0.13 on Ag not only for the ground state, but also for the ²Σ_g⁺ state.     

Molecular constants for the I1Πg and J1Δg states of the 3d complex in H2, HD,and D2

The experimental rovibronic energies of the 3dπI¹Π_g^? and 3dδJ¹Δ_g^? states of H₂ and D₂ (v = 0–4 and N = 1–11) have been fitted by rovibronic constants, including L-uncoupling through constants B_v^ΠΔ and D_v^ΠΔ. Comparison of the constants obtained for H₂ and D₂ yields information on the Born-Oppenheimer and adiabatic electronic energies T_e^BO and T_e^AD, and on the diagonal corrections for nuclear motion. T_e^BO derived from experiment for the I¹Π_g state lies 2 cm^?1 below the ab initio calculation of Ko?os and Rychlewski (J. Mol. Spectrosc., 66, 428–440 (1977). The nuclear mass dependence of the ω_e and B_e values in H₂ and D₂ deviates from simple isotope relationships but agrees with expectations based on the R-dependence of the diagonal corrections for nuclear motion through the term $\text{〈L}{}^2\text{?2}\text{Λ}{}^2\text{〉}\text{2μR}{}^2$, i.e., $\text{+2}\text{μR}{}^2$ for 3dπ and $\text{?1}\text{μR}{}^2$ for 3dδ.     

Doppler-limited dye laser excitation spectrum of the C2 Swan band (v′ − v″ = 1 − 0)

The dye laser excitation spectrum of the Swan band (v′ ? v″ = 1 ? 0) has been observed with Doppler-limited resolution. The C₂ molecule was generated by the reaction of microwave discharge products of CF₄ with CH₄. The high sensitivity of laser excitation spectroscopy has enabled us to observe not only $\text{ΔΩ}\text{= 0}$ transitions, but also $\text{ΔΩ}\text{= ± 1}$ transitions and to determine molecular constants including the spin-orbit coupling constant for both the d³Π_g and a³Π_u states. The parameters thus obtained for a³Π_u were favorably compared with those previously obtained from the Ballik-Ramsay band. The present results on the d³Π_gv = 1 state were combined with those of an earlier Fourier spectroscopic study on the Swan band (v′ ? v″ = 0 ? 0) to derive equilibrium molecular constants for the d³Π_g state. The contributions of the spin-rotation interaction and the centrifugal correction for the spin-orbit coupling to the energy levels in a ³Π state have been discussed in detail.     

The HH1Σg+ state of hydrogen: Adiabatic calculation of vibronic states in H2, HD,and D2

The nuclear-mass-dependent diagonal corrections to the electronic energies of the $\text{H}\text{H}{}^1\text{Σ}{}_{\mathrm{g}}{}^{\mathrm{+}}$ state of hydrogen are computed in the range 1.25 ≤ R ≤ 11 a.u. The correction energy goes through a pronounced peak near R = 3 a.u., which is discussed in terms of the character of the electronic wavefunction. The rovibrational structures of H₂, HD, and D₂ in the adiabatically corrected double-minimum potential curves of these isotopes in the $\text{H}\text{H}$ state are presented. Comparison with experimental data indicates the presence of appreciable nonadiabatic perturbations.     

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.     

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.     

S2分子B″3Πu态的势能函数和光谱常数的理论研究

推导了S₂分子B″³Π_u态的合理离解极限.用Gaussian 94 QCISD(T)方法和6-311++G^**基组计算了S₂分子B″³Π_u以及X³Σ^-_g态的势能曲线.给出了S₂分子B″³Π_u态的Murrell-Sorbie势能函数和光谱常数.B″³Π_u与B³Σ^-_u态在排斥支重叠范围大；同时，B″³Π_u与X³Σ^-_g态有相同离解极限，因而，在吸引支有重叠.讨论了B″³Π_u与B³Σ^-_u和X³Σ^-_g态相互作用的特征. 关键词：     

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.     

j3Δg state of the hydrogen molecule

The potential energy curve for the j³Δ_g state of the hydrogen molecule has been calculated in the Born-Oppenheimer approximation. Highly flexible wavefunctions depending explicitly on the interelectronic distance have been used. The vibrational Schrödinger equation for the j state has been solved for H₂, HD, and D₂. The results are compared with the available experimental data, and it is shown that the adiabatic and nonadiabatic effects are mainly responsible for the existing discrepancy. The singlet-triplet separation for the j-J pair of states is also discussed.     

Potential energy curves for the 48 lowest-lying molecular states of LiH+

The electronic energies of the 20 lowest lying ²Σ⁺ states, the 14 lowest lying ²Π states, and the 14 lowest lying ²Δ states of LiH⁺ have been calculated in the range 2 ≤ R ≤ 20 a.u. from a model potential approach and using truncated diatomic orbitals (TDO) as basis set. Results in very good agreement with the more recent literature were obtained for the spectroscopic constants, R_e and D_e, for the ground state X²Σ⁺.