Nuclear Motion Corrections Research Articles

The E F, G K, and H H̄ 1Σg+ states of hydrogen. Improved ab initio calculation of vibrational states in the adiabatic approximation

Electronic energies and diagonal energy corrections for nuclear motion are calculated for the first three excited 1Σg+ states of H2 at some 70 internuclear separations in the interval R(a.u.) ∈{1,15}. More accurate electronic wave functions comprising, respectively, 129, 118, and 110 terms for the EF, GK, and HH̄ states, and a new method for the evaluation of the relevant integrals are used. The maximum values of the adiabatic energy corrections are found to be larger than the previously published values by 60 cm−1 (EF at R=3.2 a.u.), 55 cm−1 (GK at R=2.85 a.u.), and 38 cm−1 (HH̄ at R=2.95 a.u.), while the electronic energies at the same values of R are now lower by −6, −26, and −7 cm−1. The adiabatic ab initio vibrational energies of the EF, v=0, levels lie 1.9 (H2) and 1.4 cm−1 (HD and D2) above the experimental values. All higher vibrational levels of the three electronic states are appreciably affected by nonadiabatic energy shifts.

The I1Πg state of hydrogen: adiabatic corrections, energy levels, L uncoupling, and electronic transition moments

Employing Kołos and Rychlewski's ab initio wave functions for the electronic I1Πg. state of molecular hydrogen (Dieke's 3dπ 3E) we calculated the diagonal corrections for nuclear motion to the electronic energies over the internuelear-distance range R = 1 – 12 au. The vibration–rotation energy levels calculated in the adiabatic approximation are compared with spectroscopic results in H2, HD, and D2 and it is concluded that the convergence error of the electronic energy in the inner well [Formula: see text] of the double-minimum potential curve is 1.0 ± 0.5 cm−1. The electronic matrix elements of angular-momentum coupling with the [Formula: see text] state and of the dipole transition moments to the (B, B′) [Formula: see text] and C1Πu states are also presented.

The properties of LiH in its ground and first excited electronic state

With the aid of recently produced potential curves for the X 1Σ+ and A 1Σ+ states of lithium hydride, the rovibronic energies of 7LiH have been calculated. From these energies the standard spectroscopic constants have been evaluated. The effect on these constants of inclusion of adiabatic corrections (for diagonal nuclear motion) in the potential curves is discussed. The expectation values of certain powers of the electronic coordinates are also tabulated.

An adiabatic study of the A 1Σ+ state of LiH

Using a 250-term configurational-interaction wave function with a 33-term elliptical orbital basis, the potential curve for the first excited state (A 1Σ+) of lithium hydride has been determined. With the corresponding wave functions, the adiabatic (diagonal nuclear motion) corrections have been calculated. These are found to be, in part, in qualitative agreement with the values extracted from an extensive analysis of the spectroscopic data for the A 1Σ+–X 1Σ+ system for the four LiH isotopes.

Molecular constants for the I1Π g and J1Δ g states of the 3 d complex in H 2, HD, and D 2

The experimental rovibronic energies of the 3 dπ I 1Π g − and 3 dδ J 1Δ g − states of H 2 and D 2 ( 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 H 2 and D 2 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 I 1Π g state lies 2 cm −1 below the ab initio calculation of Kołos and Rychlewski ( J. Mol. Spectrosc., 66, 428–440 (1977). The nuclear mass dependence of the ω e and B e values in H 2 and D 2 deviates from simple isotope relationships but agrees with expectations based on the R-dependence of the diagonal corrections for nuclear motion through the term 〈L 2−2 Λ 2〉 2μR 2 , i.e., +2 μR 2 for 3 dπ and −1 μR 2 for 3 dδ.

Improved adiabatic energy for the 3Σ +g state of the hydrogen molecule

Diagonal corrections for nuclear motion have been calculated for the equilibrium internuclear separation in the a 3Σ + g state of the hydrogen molecule. Their new value is by 2.133 cm −1 smaller than the previous less accurate result. If this correction is applied to the theoretical value of the dissociation energy it removes the existing small discrepancy with the experimental results.

The scattering states of HD +

Elastic and charge-transfer cross-sections for collisions of protons (or deuterons) with deuterium (or hydrogen) atoms have been calculated for 31 values of the energy from 0.001 to 7.5 eV. They were obtained by a fully quantum mechanical method. The internuclear potentials obtained from the Born-Oppenheimer separation were improved by the addition of the first order corrections for nuclear motion, and accurate phase shifts were obtained by numerical solution of a pair of coupled radial Schrödinger equations. The cross-sections are compared with experimental and theoretical values obtained previously.

The EF and GK[formula omitted] states of hydrogen : Adiabatic calculation of vibronic states in H 2, HD, and D 2

The Born-Oppenheimer energies of the first two excited 1 Σ g + states of hydrogen, EF and GK, are computed with higher accuracy (70-term wavefunctions) and over larger R ranges (1 a.u. < R ≤ 15 a.u.) than previously available. Less accurate results over the same large R range are also presented for the third excited 1 Σ g + state, H, whose potential curve is found to exhibit a broad and flat outer minimum, H , near R = 11 a.u., similar to the one in the third 1 Σ u + state, B . The diagonal corrections for nuclear motion in the GK state and the energies and B values of the vibrational states in the EF and GK potentials are computed in the adiabatic approximation up to the dissociation limit. The GK potential curve has two minima, each of which accommodates one single vibrational level, while all other levels lie above the potential barrier. These ab initio results, in which the effects of nuclear-momentum coupling between different electronic states are neglected, agree surprisingly well with experimental data although the electronic GK wavefunction goes through two marked changes of character within the R range of the lowest vibrational levels, viz., in the regions of the potential barrier and of the outer minimum. All of the experimentally established and irregular singlet- g levels of H 2, which had been given electronic labels Q, 3 K, 3 D, L, M, U, and N by Richardson and by Dieke, can now be attributed to the two electronic Born-Oppenheimer states EF and GK, both of whose vibration-rotation structures are characterized by double-minimum potential curves. The experimental energy levels reveal perturbations whose quantitative explanation requires nonadiabatic computations of rovibronic states; results of such a treatment will be presented separately. In this first paper we include a preliminary survey of the experimental evidence regarding the magnitudes of these vibronic interactions within the 1 Σ g + states and of the coupling between electronic motion and nuclear rotation, i.e., 1 Σ g +- 1 Π g- 1 Δ g interactions ( l-uncoupling).

The computation of nuclear motion and mass polarization adiabatic energy corrections for several states of the hydrogen molecule

This paper reports the computation of the diagonal matrix elements of the nuclear motion and mass polarization operators (often referred to as adiabatic corrections) for several electronic states of the hydrogen molecule. It is observed that reasonably simple wavefunctions will yield accurate expectation values for the nuclear motion operators. The present calculation covers a greater range of R for the X 1Σ+g and B 1Σ+u states than is covered in the very accurate work of Kolos and Wolniewicz, and includes computations for the C 1Πu and D 1Πu states for which no theoretical results have been previously reported. For the latter states the present diagonal corrections have been used in conjunction with the very recent Born–Oppenheimer potential energy curves of Kolos and Rychlewski [J. Mol. Spectrosc. 62, 109 (1976)] to compute vibrational level energies for both H2 and D2. The diagonal corrections largely remove the mass-dependent part of the discrepancy between theory and experiment for these vibrational level energies.

The dissociation of an orbiting resonance : calculation of eigen-energies and widths of the quasi-bound states of4HeH+

Born-Oppenheimer eigenergies (EnJ ) of the ro-vibrational (n, J) quasibound levels of 4HeH+ in the X1Σ+ electronic state are calculated using an improved outer boundary condition and their values are compared with those of three other methods, namely, the uniform JWKB quadratic approximation, the application of an outer boundary condition pertaining to the irregular Bessel function and the maximization of the time-delay function. The widths (Γ nJ ) of these states obtained by semiclassical and quantum techniques are also compared. The boundary condition may be applied to other diatomic ionic orbiting complexes (i.e. H2 +) which dissociate via the decay of quasibound states to an ion and a neutral atom. For the broader resonances, the differences between various estimates of a particular EnJ can exceed the diagonal nuclear-motion correction.

Fried-Martin corrections for mesonic Auger transitions

Using excited circular orbits of απ −e −, αK −e − and α p e − atoms as examples, it is shown that nuclear motion corrections to computed Auger rates can depend sensitively on the interpretation of the wave functions actually employed and also on the multipole order of the transition.

Nuclear motion corrections to some electric and magnetic properties of diatomic molecules

It is commonly assumed that the nuclei are at a fixed separation when relating the electric quadrupole moment of a diatomic molecule in a 1Σ state to the rotational g value and the magnetizability anisotropy. This assumption is examined with particular reference to the hydrogen isotopes and is found to lead to an error of approximately 1 per cent in the quadrupole moment. The paramagnetic magnetizability and rotational g value function for hydrogen are extracted from published expérimental data and combination of the latter with an accurate calculation of the quadrupole moment leads to precise values of the magnetizability anisotropy for the lower vibration-rotation states of hydrogen. An experimental measurement of the rotational g value for just one further state or isotope of hydrogen is required to make the analysis more accurate.

Corrections to the Born-Oppenheimer approximation and electronic effects on isotopic exchange equilibria. II

If nonadiabatic effects are neglected (the adiabatic approximation), the potential energy for the motions of the nuclei of a molecule is the sum of the Born-Oppenheimer electronic energy and the adiabatic correction (diagonal nuclear motion correction). This adiabatic correction has been evaluated at the experimental equilibrium internuclear separation for a number of first- and second-row diatomic hydrides and deuterides. The electronic wavefunctions which have been used in these calculations are LCAO MO SCF functions which range in quality from so-called minimum basis set functions to so-called near-Hartree-Fock functions. The explicit R dependence of the LCAO coefficients has been taken into account. The adiabatic corrections to the electronic energy are functions of the nuclear masses. With the assumption of the distance independence of the adiabatic correction, the calculations here can be used to evaluate the effect of the adiabatic correction on the equilibrium constants for the isotopic exchange reactions HX+HD=DX+H2. For the HX systems on which calculations were done here, these correction factors range, with good single configuration wavefunctions, from 0.963 to 1.113 at 300°K.

Nuclear-Motion Corrections to the Fine Structure of Helium

The motion of the nucleus yields numerically important order $\frac{m}{{M}_{\mathrm{He}}}$ corrections to the fine structure of the helium atom. We present here the derivation of such corrections applied to the $2^{3}P$ fine-structure splittings in ${\mathrm{He}}^{4}$.

Some accurate results for three-particle systems

Results of a variational calculation of the nonadiabatic ground state energy of H 2 + are presented. Diagonal corrections for nuclear motion have also been calculated for the electronic ground state of H 2 + . The adiabatic potential energy curve has been employed to calculate the rotational and vibrational levels for the H 2 + ion and for muonic molecules. Nonadiabatic energy corrections are discussed. Forpμd the adiabatic wave function is compared with the corresponding nonadiabatic result.

Lower Bound for the Dissociation Energy of the Hydrogen Molecule

An asymptotic formula for the diagonal nuclear motion corrections is derived for the hydrogen molecule, and a discussion is given of the errors which arise in the adiabatic dissociation energies from the failure of the adiabatic approximation to yield the exact energy of the dissociation products.

Theoretical Investigation of the Lowest Double-Minimum State E, F 1Σg+ of the Hydrogen Molecule

Accurate potential-energy curve for the first excited ∑1g+ state of the hydrogen molecule has been calculated for 1 ≤ R ≤ 12 a.u. The curve is known to have a double minimum, and for the outer minimum a significantly lower energy has been obtained than all previously reported results. The diagonal corrections for nuclear motion have also been computed and found to have a very sharp peak in the vicinity of the potential maximum. Vibrational and rotational levels have been calculated for all isotopes of the hydrogen molecule, and some results for H2, D2and T2 are shown and compared with experimental data. For the outer minimum, two, three, and four vibrational levels for H2, D2, and T2, respectively, are reported which lie below the lowest levels observed experimentally. For the observed levels good agreement with experimental data is obtained; however, the agreement is significantly worse when only the clamped nuclei potential without the diagonal corrections is used in the calculation of the vibrational energies. The wavefunction is analyzed in terms of some simple functions and appears to drastically change its character in the vicinity of the potential maximum.

Vibration–Rotation Interaction Effects in Calculated Franck–Condon Factors. I. The Ionization of H2 and D2

Computed Franck–Condon factors for the ionization of H2 and D2 are reported which include previously neglected vibration–rotation interaction effects. Eigenfunctions were obtained by direct solution of the Schrödinger equation in which the exact centrifugal potential is explicitly included. Only υ″ = 0 and small K″ states of the neutral molecule have been considered and attention is confined to those transitions for which |K″ − K| = 0, 2. The results show a clear dependence of the computed Franck–Condon factor on rotational quantum number. For K″ = K, this effect is small except for transitions to the uppermost vibrational states of the ion. For |K″ − K| = 2, the effect is substantial even for transitions to low-lying vibrational states of the ion. By using a sum rule, it is shown that the probability of dissociative ionization exhibits a similar dependence on rotational state. Comparison of the present results for the case K″ = K = 0 with previous computation shows that adiabatic corrections for nuclear motion, explicitly included in this work, have a wholly negligible influence on computed Franck–Condon factors.

Improved Theoretical Ground-State Energy of the Hydrogen Molecule

The potential-energy curve for the electronic ground state of the hydrogen molecule has been calculated for 1 ≤ R ≤ 3.2 a.u. in double precision and using a 100-term expansion for the electronic wavefunction. Accuracy of the previously computed diagonal corrections for nuclear motion has been tested. The vibrational equation has been solved for all isotopes of the hydrogen molecule and for the rotational quantum number J ≤ 10. The calculated adiabatic dissociation energy of H2, corrected for relativistic and radiative effects, is by 3.8 cm−1 larger than the experimental value, hence the theoretical total energy is by the same amount lower than the experimental value. The calculated vibrational quanta for H2 are by 0.5–0.9 cm−1 larger than the experimental ones.

Vibrational and Rotational Energies for the B 1Σu + , C 1Πu, and a 3Σg + States of the Hydrogen Molecule

An accurate potential-energy curve, including the diagonal corrections for nuclear motion, has been computed for the a 3Σg + state of H2, and the previously computed potential-energy curve for the B 1Σu + state has been supplemented with additional energy values for several internuclear distances. For both states, as well as for the C 1Πu state, the Schrödinger equation for nuclear motion has been integrated to yield the vibrational and rotational energies, as well as the rotational constants for all six isotopes of the hydrogen molecule. The reported theoretical results are in a good agreement with experimental values.