Diagonal Correction Research Articles

Rank-Augmented LU-Algorithm for Computing Generalized Matrix Inverses

A rank-augmnented LU-algorithm is suggested for computing a generalized inverse of a matrix. Initially suitable diagonal corrections are introduced in (the symmetrized form of) the given matrix to facilitate decomposition; a backward-correction scheme then yields a desired generalized inverse.

Improved digital aperture corrector

A high-speed digital system for the aperture correction of television images is described in which the amount of diagonal correction applied can be varied with respect to that in the horizontal and vertical directions. It is shown that this facility is desirable in order to correct for different forms of spread function, and that the resolution of diagonal detail in the image is improved by using the appropriate amount of diagonal correction. Noise increase in the system is examined and the errors introduced by rounding off the digitised video and correction signals are also considered.

Adiabatic Corrections in a Simple Model of Two Interacting Electronic-Potential Curves

Accurate numerical methods are applied to the vibronic problem resulting from two interacting intersecting harmonic curves. Two examples are considered, corresponding to either degenerate or nondegenerate electronic states. The energies are obtained with (a) the zeroth-order Born-Oppenheimer approximation; (b) the so-called adiabatic approximation, which includes the diagonal corrections to (a); (c) the full vibronic procedure. Several features in the correlation diagrams giving the energies as a function of the configuration-interaction matrix element are given an explanation. We also note that, in the model treated here, inclusion of the diagonal corrections does not always lead to an improvement of the energies.

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.

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.

Effets stark et zeeman combines sur les transitions hydrogenoïdes de l'helium neutre

The perturbation theory of Waller and Foster (WF) for hydrogenic lines of neutral Helium is generalized in order to take into account an external magnetic field H having an arbitrary angle with respect to an external constant electric field F . The diagonal correction has been evaluated numerically, taking into account recent experimental data. A FORTRAN IV program written for the CDC 3600 computer allows the calculation of the displacements and the intensities for any hydrogenic transition. Special attention is given to the n' = 2 → n = 4 and n' = 2 → n = 5 transitions in neutral helium. The corresponding spectral components were studied carefully for parallel and perpendicular fields. The calculated displacements are in excellent agreement with those determined experimentally. Concerning the intensities of the components, the calculations confirm Foster's observations, namely, that the intensities become asymmetric if magnetic and electric interactions are of comparable strengths, and that there exist a strong mutual interaction between the components for perpendicular fields. In the latter case, the asymmetry increases with increasing orbital quantum number l. The shift and intensity of some spectral components belonging to the He I-transitions {2–4} ortho, {2–5} ortho, {2–4} para, and {2–5} para are given in graphical form for the three constant electric field strengths: 5, 50 and 100 kV/cm, with the magnetic field strength ranging from 0,2 kΓ to 80 kΓ. Due to the large number of possible spectral components, the graphical representation is restricted to components being polarized perpendicular to both the electric and magnetic fields. In two tables, experimental and theoretical shifts of the components belonging to the transitions 2 3 P → 5 3 S, 5 3 P, 5 3 D, 5 3 F and 5 3 G, and 2 1 P → 5 1 S, 5 1 D , 5 1 F, and 5 1 G have been listed, thus permitting a numerical comparison of our theoretical calculations with the experimental results of Foster and Pounder ( H = 25.8 kV, F = 64 kV/cm, H ⊥ F ) and Lebowsky and Steubing ( H = 32 kΓ, F = 13.9 kV /cm, H ⊥ F ). In graphs and tables, all shifts are counted from the unperturbed transitions 2 P–4 D or 2 P–5 D, respectively. These numerical results will be used later to calculate the profiles of Stark broadened lines in the presence of magnetic fields in plasmas.

Potential-Energy Curve for the B 1Σu+ State of the Hydrogen Molecule

An accurate potential-energy curve for the B 1Σu+ state of H2 has been computed using variational wavefunction in the form of an expansion in elliptic coordinates and depending explicitly on the interelectronic distance. The theoretical clamped-nuclei binding energy (De=28 896.3 cm−1) with diagonal corrections for nuclear motion (ΔD=−46.1 cm−1) gives the adiabatic binding energy De=28 850.2 cm−1 in a good agreement with the experimental value De=28 852.8 cm−1. However, for small and large values of the internuclear distance, the theoretical potential-energy curve very significantly differs from the corresponding RKR curve. The computed wavefunctions are analyzed in terms of some simple functions and it is shown that for 3<R<7 a.u. the B state is predominantly ionic in character.

Accurate Adiabatic Treatment of the Ground State of the Hydrogen Molecule

Accurate ground-state energies of the hydrogen molecule have been computed using wavefunctions in the form of expansions in elliptic coordinates and including explicitly the interelectronic distance. The computations have been made with 54-term expansions (0.4≤R≤3.7) and with 80-term expansions (0.5≤R≤2.0). For the equilibrium internuclear distance, the best total energies obtained in the two cases are —1.1744701 a.u. and —1.1744746 a.u., respectively, the corresponding binding energies being 38 291.8 and 38 292.7 cm—1. Employing the 54-term wavefunctions, the relativistic corrections and the diagonal corrections for nuclear motion have been computed for several internuclear distances. For equilibrium their contributions to the binding energy have been found to be —0.526 and 4.947 cm—1, respectively. Thus the final theoretical binding energy for H2 amounts to 38 297.1 cm—1 and is a little larger than the experimental value 38 292.9±0.5 cm—1. The discrepancy may be due to the adiabatic approximation.