Adiabatic Corrections Research Articles

On spin evolution in a time-dependent magnetic field: Post-adiabatic corrections and geometric phases

We examine both quantum and classical versions of the problem of spin evolution in a slowly varying magnetic field. Main attention is given to the first- and second-order adiabatic corrections in the case of in-plane variations of the magnetic field. While the first-order correction relates to the usual adiabatic Berry phase and Coriolis-type lateral deflection of the spin, the second-order correction is shown to be responsible for the next-order geometric phase and in-plain deflection. A comparison between different approaches, including the exact (non-adiabatic) geometric phase, is presented.

Direct perturbation theory for the dark soliton solution to the nonlinear Schrödinger equation with normal dispersion

After finding the basic solutions of the linearized nonlinear Schrödinger equation by the method of separation of variables, the perturbation theory for the dark soliton solution is constructed by linear Green's function theory. In application to the self-induced Raman scattering, the adiabatic corrections to the soliton's parameters are obtained and the remaining correction term is given as a pure integral with respect to the continuous spectral parameter.

Numerical Calculation of Adiabatic Corrections with the Born-Handy Method*

A modified algorithm of the numerical calculation of adiabatic corrections is proposed. It is based on earlier approaches, introduced by Cencek and Kutzelnigg and by the present author. The adiabatic correction is approximated in terms of overlap integrals between electronic wave functions for a given geometry and for a single nucleus shifted by h. The leading term of error is proportional to the square of h in the original methods. The new approach, while requires the same wave functions as those methods, shrinks the error, so that it becomes proportional to h 4 in atomic cases. Test calculations show, that similar behavior is retained also for two-atom molecules and additional stable decimal digit of the adiabatic correction can be obtained.

Adiabatic and non-adiabatic corrections to rovibrational energies of diatomic molecules: variational calculations with experimental accuracy

Using the variational technique, eigensolutions of the radial Herman-Asgharian equation accounting for non-adiabatic terms are determined within the experimental accuracy of the high-resolution spectroscopy. This method, which is independent of the algebraic and numerical approaches currently used in the literature for a "direct-potential-fit" of diatomic rovibrational spectra, is shown to be useful for validation of available calculations and for resolving some controversial issues. Comparative discussions are reported in this paper for a dozen diatomic molecules.

On a possible variation of the proton-to-electron mass ratio: H 2 spectra in the line of sight of high-redshift quasars and in the laboratory

Recently the finding of an indication for a decrease of the proton-to-electron mass ratio μ = m p/ m e by 0.002% in the past 12 billion years was reported in the form of a Letter [E. Reinhold, R. Buning, U. Hollenstein, P. Petitjean, A. Ivanchik, W. Ubachs, Phys. Rev. Lett. 96 (2006) 151101]. Here we will further detail the methods that led to that result and put it in perspective. Laser spectroscopy on molecular hydrogen, using a narrow-band and tunable extreme ultraviolet laser system at the Laser Centre Vrije Universiteit Amsterdam, results in transition wavelengths of spectral lines in the B 1 Σ u + - X 1 Σ g + Lyman and C 1 Π u - X 1 Σ g + Werner band systems at an accuracy of (4–11) × 10 −8, depending on the wavelength region. This corresponds to an absolute accuracy of 0.000004–0.000010 nm. A database of 233 accurately calibrated H 2 lines is presented here for future reference and comparison with astronomical observations. Recent observations of the same spectroscopic features in cold hydrogen clouds at redshifts z = 2.5947325 and z = 3.0248970 in the line of sight of two quasar light sources (Q 0405−443 and Q 0347−383) resulted in 76 reliably determined transition wavelengths of H 2 lines at accuracies in the range 2 × 10 −7 to 1 × 10 −6. Those observations were performed with the Ultraviolet and Visible Echelle Spectrograph at the Very Large Telescope of the European Southern Observatory at Paranal, Chile. A third ingredient in the analysis is the calculation of an improved set of sensitivity coefficients K i , a parameter associated with each spectral line, representing the dependence of the transition wavelength on a possible variation of the proton-to-electron mass ratio μ. The new model for calculation of the K i sensitivity coefficients is based on a Dunham representation of ground state and excited state level energies, derived from the most accurate data available in literature for the X 1 Σ g + ground electronic state and the presently determined level energies in the B 1 Σ u + and C 1Π u states. Moreover, the model includes adiabatic corrections to electronic energies as well as local perturbation effects between B and C levels. The full analysis and a tabulation of the resulting K i coefficients is given in this paper. A statistical analysis of the data yields an indication for a variation of the proton-to-electron mass ratio of Δ μ/ μ = (2.45 ± 0.59) × 10 −5 for a weighted fit and Δ μ/ μ = (1.99 ± 0.58) × 10 −5 for an unweighted fit. This result, indicating the decrease of μ, has a statistical significance of 3.5 σ. Mass-variations as discussed relate to inertial or kinematic masses, rather than gravitational masses. Separate treatment of the data gives a similar positive result for each of the quasars Q 0405−443 and Q 0347−383. The statistical analysis is further documented and possible systematic shifts underlying the data, with the possibility of mimicking a non-zero Δ μ/ μ value, are discussed. The observed decrease in μ corresponds to a rate of change of d ln μ/d t = −2 × 10 −15 per year, if a linear variation with time is assumed. Experiments for detecting a possible variation of μ in the modern epoch via ultra-precision experiments on H 2 quadrupole transitions are proposed.

Explicitly correlated potential energy surface of , including relativistic and adiabatic corrections

After a short historical account of the theory of the H3+ ion, two ab initio methods are reviewed that allow the computation of the ground-state potential energy surface (PES) of H3+ in the Born-Oppenheimer (BO) approximation, with microhartree or even sub-microhartree accuracy, namely the R12 method and the method of explicitly correlated Gaussians. The BO-PES is improved by the inclusion of relativistic effects and adiabatic corrections. It is discussed how non-adiabatic effects on rotation and vibration can be simulated by corrections to the moving nuclear masses. The importance of the appropriate analytic fit to the computed points of the PES for the subsequent computation of the rovibronic spectrum is addressed. Some recent extensions of the computed PES in the energy region above the barrier to linearity are reviewed. This involves a large set of input geometries and the correct treatment of the dissociation asymptotics, including the coupling with the first excited singlet state. Some comments on this state as well as on the lowest triplet state of H3+ are made. The paper ends with a few remarks on the ion H5+.

Rotation–vibrational states of and the adiabatic approximation

We discuss recent progress in the calculation and identification of rotation-vibrational states of H3+ at intermediate energies up to 13,000 cm(-1). Our calculations are based on the potential energy surface of Cencek et al. which is of sub-microhartree accuracy. As this surface includes diagonal adiabatic and relativistic corrections to the fixed nuclei electronic energies, the remaining discrepancies between our calculated and experimental data should be due to the neglect of non-adiabatic coupling to excited electronic states in the calculations. To account for this, our calculated energy values were adjusted empirically by a simple correction formula. Based on our understanding of the adiabatic approximation, we suggest two new approaches to account for the off-diagonal adiabatic correction, which should work; however, they have not been tested yet for H3+. Theoretical predictions made for the above-barrier energy region of recent experimental interest are accurate to 0.35 cm(-1) or better.

Near-infrared spectroscopy of above the barrier to linearity

Since the Royal Society Discussion Meeting on H3+ in 2000, the laboratory spectroscopy of H3+ has entered a new regime. For the first time, transitions of H3+ above the barrier to linearity have been observed. A highly sensitive near-infrared spectrometer based on a titanium:sapphire laser and incorporating a dual-beam, double-modulation technique with bidirectional optical multi-passing has been developed in order to detect these transitions, which are more than 4600 times weaker than the fundamental band. We discuss our recent work on the 2v1 + 2v2(2) <-- 0, 3v1 + v2(1) <-- 0, v1 + 4v2(2) <-- 0, v1 + 4V2(4) <-- 0 and 2v1 + 3v2(1) <-- 0 combination bands and the 5v2(1) <-- 0, 5v2(3) <-- 0, 52(5) <-- 0 and 6v2(2) <-- 0 overtone bands. Experimentally determined energy levels provide a critical test of ab initio calculations in this challenging energy regime (greater than 10,000 cm(-1)). By comparing the experimental energy levels and theoretical energy levels from ab initio calculations in which the adiabatic and relativistic corrections are incorporated, the extent of higher-order effects such as non-adiabatic and radiative corrections is revealed.

Potential Energy Curves for Excited States of the Hydrogen-Antihydrogen System

The potential energy curves for the hydrogen-antihydrogen (HH) system in states with a leptonic orbital angular momentum projection Lambda=0, 1, 2, 6, and 30 are presented. Within the framework of the adiabatic picture, explicitly correlated Gaussians are used as basis functions which describe accurately the hydrogen-antihydrogen interaction. The critical internuclear distances where the system transforms into positronium and protonium atoms are found. Adiabatic corrections to the potential energy curves are also estimated.

Isotopically invariant analysis of vibration–rotation transitions of HBr and its isotopologues

The vibration–rotation transition frequencies of HBr and its isotopologues reported in the literature were simultaneously analyzed in a least-squares fit to determine isotopically invariant molecular parameters U kl and Born–Oppenheimer breakdown parameters Δ kl . A total of 692 lines was fitted to Watson’s expression involving 25 adjustable parameters to a weighted unitless standard deviation of σ ≈ 0.91. From new data in the far-infrared region and those of improved precision in the mid-infrared and the near-infrared region, the breakdown parameters for the bromine atom were determined; the parameter Δ 10 Br for the harmonic frequency improved the fit and Δ 01 Br for the rotational constant enabled us to obtain the information on the adiabatic correction to the Born–Oppenheimer approximation.

Evaluation of the adiabatic correction for LiH, RbH and CsH molecules

The evaluation of the first and second derivative of the non adiabatic coupling between the several 1Σ + adiabatic states of LiH, RbH and CsH molecules, is considered from accurate diabatic and adiabatic data. Such derivatives of the electronic wave function are determined through a numerical differentiation of the rotational matrix connecting the diabatic and adiabatic representations. The first as well as the second derivative presents many peaks, related to ionic–neutral and neutral–neutral coupling between the 1Σ + states. Such radial coupling has been exploited to calculate the first adiabatic correction, which corresponds to the diagonal term of the second derivative divided by the reduced mass, for the ground and some excited states of the three molecules. The second adiabatic correction has been determined using the Virial theorem. The first adiabatic correction was added to the adiabatic potential energy curves to re-determine the corrected spectroscopic constants and vibrational energy levels. The vibrational shift, which is the difference between the corrected and the adiabatic levels, has been calculated for X, A, C and D 1Σ + states of the three molecules. A shift of some 10 cm −1 is observed for some vibrational levels showing the breakdown of the Born–Oppenheimer approximation.

Theoretical study of the A state of helium dimer†

Microhartree accuracy wave functions composed of exponentially correlated Gaussians (ECG) are employed to calculate the electronic energy curve of the A state of the helium dimer. The Born–Oppenheimer energy is augmented by adiabatic and relativistic corrections. Calculations are performed for a wide range of internuclear separations R including the separated atoms limit. Computed vibrational energy levels and other spectroscopic parameters are compared with the experimental data. The electric dipole transition moments and the Einstein spontaneous emission coefficients are reported for the X system. †Dedicated to Professor Andrzej Sadlej on the occasion of his 65th birthday.

On the relativistic generalization of Keldysh ionization theory

Relativistic generalization of the semiclassical theory of tunnel and multiphoton ionization of atoms and ions in the field of a high-intensity electromagnetic wave (Keldysh theory) with linear and elliptic polarizations is developed. The exponential factor in the ionization rate of relativistic bound state is calculated for arbitrary values of the adiabaticity parameter γ. In the case of low-frequency laser radiation, γ ≪ 1 , the asymptotically exact formula is derived for the ionization rate of atomic s-level, including the Coulomb, spin and adiabatic corrections and the pre-exponential factor. The ionization rate of the ground state of a hydrogen-like atoms (ions) with Z ≲ 100 is calculated, depending on laser intensity and ellipticity of radiation.

Relativistic theory of tunnel and multiphoton ionization of atoms in a strong laser field

Relativistic generalization is developed for the semiclassical theory of tunnel and multiphoton ionization of atoms and ions in the field of an intense electromagnetic wave (Keldysh theory). The cases of linear, circular, and elliptic polarizations of radiation are considered. For arbitrary values of the adiabaticity parameter γ, the exponential factor in the ionization rate for a relativistic bound state is calculated. For low-frequency laser radiation , an asymptotically exact formula for the tunnel ionization rate for the atomic s level is obtained including the Coulomb, spin, and adiabatic corrections and the preexponential factor. The ionization rate for the ground level of a hydrogen-like atom (ion) with Z ≲ 100 is calculated as a function of the laser radiation intensity. The range of applicability is determined for nonrelativistic ionization theory. The imaginary time method is used in the calculations.

Indication of a Cosmological Variation of the Proton-Electron Mass Ratio Based on Laboratory Measurement and Reanalysis ofH2Spectra

Based on highly accurate laboratory measurements of Lyman bands of H2 and an updated representation of the structure of the ground X 1sigma(g)+ and excited B 1sigma(u)+ and C 1pi(u) states, a new set of sensitivity coefficients K(i) is derived for all lines in the H2 spectrum, representing the dependence of their transition wavelengths on a possible variation of the proton-electron mass ratio mu = m(p)/m(e). Included are local perturbation effects between B and C levels and adiabatic corrections. The new wavelengths and K(i) factors are used to compare with a recent set of highly accurate H2 spectral lines observed in the Q 0347-383 and Q 0405-443 quasars, yielding a fractional change in the mass ratio of deltamu/mu = (2.4 +/- 0.6) x 10(-5) for a weighted fit and deltamu/mu = (2.0 +/- 0.6) x 10(-5) for an unweighted fit. This result indicates, at a 3.5sigma confidence level, that mu could have decreased in the past 12 Gyr.

Bounds for the scattering length of spin-polarized helium from high-accuracy electronic structure calculations

We developed a series of correlation-consistent, polarized multiple zeta basis sets optimized specifically for the energy of the 2 3S state of helium atom. These basis sets were subsequently augmented with diffuse functions optimized for the van der Waals constants C6 through C14 which determine the asymptotic behavior of the second-order dispersion interaction between 2 3S helium atoms at large interatomic separation R. The resulting bases were applied to compute the Born-Oppenheimer (BO) potential for the lowest 5Sigmag+ state of the helium dimer. The coupled cluster and the full configuration-interaction techniques were employed to account for the electron correlation effects. The cardinal number extrapolation technique was used to obtain the complete-basis-set limit V(R) for the interaction potential and to find its lower VL(R) and upper VU(R) bounds. The resulting potentials were fitted to an analytical function containing accurate van der Waals constants C6 through C12 (including C11). We found that the complete-basis-set BO potential has a well depth De=1048.24+/-0.36 cm-1. The highest rotationless vibrational level is bound by D14=90.2+/-4.7 MHz, much stronger than the previous most accurate estimation of 15.2 MHz. The error bounds for De and D14 were obtained using the VL(R) and VU(R) potentials. The S-wave scattering length computed using the VL(R), V(R), and VU(R) potentials (assuming atomic masses) is aL=7.41 nm, a=7.54 nm, and aU=7.69 nm, respectively. We also computed the adiabatic, relativistic, and quantum electrodynamics (QED) corrections to the BO potential. When these corrections are taken into account the values of D14 and of a (both computed assuming nuclear masses) are 87.4+/-6.7 MHz and 7.64+/-0.20 nm; the error bounds reflect now also the uncertainty of the included adiabatic, relativistic, and QED corrections. The value of the scattering length resulting from our investigation lies outside the error bounds of all experimental determinations based on the properties of Bose-Einstein condensate of spin-polarized helium atoms.

Adiabatic potentials for the interaction of atomic antihydrogen with He and He+

Potential energy curves for interactions of antihydrogen atoms with singlet and triplet states of helium atom and helium cation have been calculated within the adiabatic approximation. The leptonic Schrödinger equation was solved variationally with the basis of explicitly correlated Gaussian functions. Adiabatic corrections have been calculated with the Born–Handy method. For the and systems, critical distances have been found to be below 1.7 bohr and 1.0 bohr and the rearrangement cross sections have been computed with the semiclassical method. The possible existence and structure of complex atomcules is discussed.

Analysis of Pure Rotational and Vibration-rotational Spectra of NaCl X1Σ+and Quantum-chemical Calculation of Related Molecular Properties

We fitted published frequency and wave number data for pure rotational and vibration-rotational spectra, respectively, of 23Na35Cl and 23Na37Cl to derive parameters related to potential energy and to the rotational g factor. For comparison with these experimental data we undertook quantum-chemical computation of adiabatic corrections, rotational and vibrational g factors, electric dipolar moment and its derivative as a function of internuclear distance in a range near Re as a test of an algebraic approach to spectral analysis; experimental, 0.0287 ± 0.0014, and calculated, 0.02149, values of gr at Re are in moderate agreement. The combined results are discussed from a point of view of computational spectrometry.

Adiabatic corrections to the energies of 50 typical medium size molecules containing H atoms

The adiabatic correction is calculated for a series of 50 typical medium size molecules containing H atoms, most of them for the first time. The effect increases with the size of organic compounds. For this kind of molecules it is found, as a general rule, that the adiabatic correction per electron is an almost constant value, amounting approximately 0.27 × 10 −3 au.

The vibrational g-factor of dihydrogen from theoretical calculation and analysis of vibration-rotational spectra

We present the first results from quantum-chemical calculation of a vibrational g-factor; the calculations were performed at the level of full configuration interaction using a basis set of aug-cc-pVQZ quality. The theoretical results are consistent with experimental results from analysis of pure rotational and vibration-rotational spectra of dihydrogen in six isotopic variants, in which calculated results for either the rotational g-factor or adiabatic corrections are employed to constrain fits of coefficients of radial functions from wave numbers of transitions. When fits are constrained with data for the rotational g-factor, we reproduce also the radial dependence of adiabatic corrections relative to their value at equilibrium internuclear separation.