Corrections For Relativistic Effects Research Articles

Coupled cluster study of the X̃ 2Π and à 2Σ+ electronic states of the HCGe radical: Renner–Teller splitting and the effects of relativistic corrections

The X̃ 2Π and à 2Σ+ states of the germanium methylidyne radical (HCGe) have been investigated at the SCF, CISD, CCSD, and CCSD(T) levels of theory. The total energies, equilibrium geometries, dipole moments, harmonic vibrational frequencies, infrared intensities, and Renner–Teller splitting are reported. The relativistic one-electron Darwin and mass-velocity terms are calculated using first-order perturbation theory and the effects of these corrections on energetics, harmonic vibrational frequencies, and Renner–Teller splitting are discussed. At our highest level of theory [CCSD(T)/cc-pVQZ], the ground electronic state (X̃ 2Π) has a linear geometry with re(CH)=1.079 Å and re(CGe)=1.769 Å, in good agreement with the experimental values of r0(CH)=1.067 Å and r0(CGe)=1.776 Å. In the electronically excited à 2Σ+ state, HCGe is also found to have a linear geometry with re(CH)=1.074 Å and a much shorter re(CGe)=1.669 Å at the [CCSD(T)/cc-pVQZ] level, in agreement with experimental values of r0(CH)=1.059 Å, r0(CGe)=1.674 Å, and the observation of a C–Ge triple bond character. The à 2Σ+ state C–Ge stretching vibrational frequency is determined to be ω3=990.2 cm−1 at the CCSD(T)/cc-pVTZ level with the inclusion of relativistic effects, which is in essentially perfect agreement with the experimental value of ν3=990 cm−1. With the same method, the X̃ 2Π state harmonic vibrational frequencies are predicted to be 846 cm−1 for the C–Ge stretching, 443 and 506 cm−1 for the two nondegenerate bending Renner–Teller components, and 3249 cm−1 for the C–H stretching modes for which experimental values are not available. The quantum mechanical splitting (T0 value) was determined to be 38.0 kcal/mol at the nonrelativistic CCSD(T)/TZ3P(2 f,2d)+2diff level, while it is found to be 39.9 kcal/mol with the inclusion of relativity, in very good agreement with the experimental value of 39.8 kcal/mol. The Renner parameter (ε) is determined to be −0.1386, and the effects of relativity were seen to produce a smaller Renner parameter of −0.1329.

All-Electron Ab Initio Investigation of the Electronic States of the PdC Molecule

The electronic structure of transition metal containing molecules are extremely complicated and extensive calculations are required for reliable descriptions. In spite of this the results can often be interpreted in simple terms. The electronic structure of PdC is consistent with the molecular orbital diagram, as shown. Corrections for relativistic effects are shown to be extremely important for PdC.

Triplet–singlet energy gaps in iodo-carbenes (I–C–X): Remarkable discrepancy between theory and experiment

The triplet–singlet energy gaps in iodo-carbenes, I–C–X with X = H, F, Cl, Br and I, were computed using the MP2, CCSD(T), CASSCF/CASPT2, MR-SDCI, MR-ACPF and B3LYP methods, with basis sets up to 6-311++G(3df,2p) and aug-cc-pvTZ and effective core potentials. Corrections for relativistic effects were also incorporated. Our results indicate that diiodo-carbene (CI2) is likely to possess a singlet ground state lying around 30 kJ mol−1 below the lowest triplet state, thus at variance with that of a recent negative ion photoelectron spectroscopic study (R. L. Schwartz, G. E. Davico, T. M. Ramont and W. C. Lineberger, J. Phys. Chem. A, 1999, 103, 8213). In addition, the present study confirms the singlet character of other iodo-carbenes with substantial triplet–singlet gaps and also points out the remarkably large fluctuations and discrepancies in the absolute values of the energy splittings, not only between experiment and theory, but also between ab initio quantum chemical methods.

Binding of aluminium to coinage metals: electron correlation and relativistic effects

Relativistic coupled cluster calculations have been carried out for diatomic molecules formed by aluminium and the coinage metal atoms (Me). The relativistic effects are accounted for by using the spin-free Douglas–Kroll formalism while the electron correlation contribution is calculated at the level of the CCSD(T) method. The theoretical data are in satisfactory agreement with the known experimental values and available results of other calculations. The AlAu molecule with the calculated relativistic dissociation energy of 3·41 eV is the most stable species in the series. The spectroscopic parameters determined are analysed in terms of the electron correlation and relativistic contributions to their values. These results parallel those obtained recently for a similar series of boron compounds. Dipole and quadrupole moments also were investigated as was the parallel component of the dipole polarizability of all three AlMe molecules, with particular attention given to their decomposition into electron correlation and relativistic contributions. Within the one electron non-relativistic approximation all these molecules turn out to have negative dipole moments (polarity Al(-) Me(-)) whose sign is changed already by taking into account the electron correlation. The relativistic effects enhance the positive dipole moment values and make AlAu the most polar molecule in the series, with a dipole moment of about 1·58 D. Similarly, due to both electron correlation and relativistic contributions, the quadrupole moment of AlAu is positive and differs by more than 7 au from the value calculated in the one-electron non-relativistic approach. The present results obtained without the spin-free Douglas–Kroll formalism are compared with those calculated by using the spin-free (MVD) Pauli correction for relativistic effects. It follows that for systems as heavy as AlAu the MVD correction to non-relativistic results becomes insufficient. This is partly because of the violation of the low-order perturbation treatment of relativistic effects and partly due to the increasing importance of non-additive relativistic-correlation contributions.

Towards the accurate computation of properties of transition metal compounds: the binding energy of ferrocene

Recently published ab initio computations on ferrocene in the framework of coupled-cluster as well as multireference perturbation theory are exploited to obtain a theoretical best estimate of 655 ± 15 kcal/mol for the heterolytic bond disruption enthalpy. This best estimate was obtained by establishing the complete atomic orbitals basis set limit at the level of single-reference second-order perturbation theory, and computing corrections for relativistic effects, semicore (3s/3p) electron correlation effects, vibrational zero-point energy corrections, and structural relaxation energies of the fragments. Agreement with the experimental value of 635 ± 6 kcal/mol is unsatisfactory.

Study of the importance of relativistic, correlation, and relaxation effects on ionization energy of atoms by a relativistic and correlated local density method

The effect of relativistic corrections, correlation, and relaxation on the ionization energy of inner and outer orbitals of light and heavy atoms has been studied using the ‘‘fully’’ correlated relativistic local density method—called the RCΞ method. The validity of Koopmans’ theorem is also discussed in this paper.

Relativistic corrections to the atomic electron affinities.

The effect of relativistic corrections in the electron affinities (${\mathit{A}}_{\mathit{e}}$) of the atoms (Li through I) has been determined. The relativistic contributions to the ${\mathit{A}}_{\mathit{e}}$ are negative, except for Cu, Rh, and Ag atoms and are large for some transition-metal atoms. The values of the corrected ${\mathit{A}}_{\mathit{e}}$ have been used in order to estimate the contribution of the correlation energies to the ${\mathit{A}}_{\mathit{e}}$.

Unitary theory of the deuteron photodisintegration.

The deuteron photodisintegration in the \ensuremath{\Delta}-resonance region is calculated within a unitary theory of the NN-N\ensuremath{\Delta}-\ensuremath{\pi}NN system coupled to the photon. The photoexcitation amplitude of the \ensuremath{\Delta} is deduced from the ${M}_{1+}$((3/2) pion photoproduction amplitude consistently with its background term and the \ensuremath{\pi}N interaction. The NN-N\ensuremath{\Delta} transition amplitude is obtained from the coupled equations for the NN-N\ensuremath{\Delta}-\ensuremath{\pi}NN system. The one-pion-exchange currents as well as the normal single-nucleon current are included. The polarization parameters are reproduced quantitatively, but the cross section is slightly underestimated. Comparison of our results with those of the NN-N\ensuremath{\Delta} coupled-channel model is made. Effects of relativistic corrections and gauge-invariance constraints are discussed.

Electron affinities of alkaline-earth and actinide elements calculated with the local-spin-density-functional theory.

The generalized exchange local-spin-density-functional theory with self-interaction correction and correction of the statistical exchange potential was used to calculate the electron affinities of alkaline-earth and actinide elements. The effect of relativistic correction by the mass velocity and Darwin terms and correlation correction on the electron affinities is discussed. The calculation of the negative ions shows that, although the correlation correction to the potential is very small, it is essential for obtaining converged values for most of the negative ions. The calculated results predict stable negative ions for the alkaline-earth elements Ca, Sr, Ba, and Ra, supporting the calculations of Vosko et al. [Phys. Rev. A 39, 446 (1989)] and Fischer [Phys. Rev. Lett. 59, 2263 (1987); Phys. Rev. A 39, 963 (1989)]. Estimated electron affinities for the actinide elements are given.

Energy spectrum of the dirac equation for the Scharzschild and Kerr fields

We consider the effect of relativistic corrections and rotation of the central body on the structure of the energy spectrum of a particle with spin in the Schwarzchild and Kerr fields. A splitting of levels is obtained, which corresponds to the classical shift of the perihelion of the orbit and precession of the plane of the gravitational spin-orbit interaction and several nonlinear spin effects are calculated.

Accurate a b i n i t i o calculations which demonstrate a 3Πu ground state for Al2

The spectroscopic parameters and separations between the three low-lying X 3Πu, A 3Σ−g, and a 1Σ+g states of Al2 are studied as a function of both the one-particle and n-particle basis sets. Approximate correlation treatments are calibrated against full CI calculations correlating the six valence electrons in a double-zeta plus two d-function basis set. Since the CASSCF/MRCI 3Πu–3Σ−g separation is in excellent agreement with the FCI value, the MRCI calculations were carried out in an extended (20s13p6d4f)/[6s5p3d2f] Gaussian basis. Including a small correction for relativistic effects, our best estimate is that the 3Σ−g state lies 174 cm−1 above the 3Πu ground state. The 1Σ+g state lies at least 2000 cm−1 higher in energy. At the CPF level, inclusion of 2s and 2p correlation has little effect on De, reduces Te by only 26 cm−1, and shortens the bond lengths by about 0.02 a0. Further strong support for a 3Πu ground state comes from the experimental absorption spectra, since both observed transitions can be convincingly assigned as 3Πu→3Πg. The (2)3Πg state is observed to be sensitive to the level of correlation treatment, and to have its minimum shifted to shorter r values, such that the strongest experimental absorption peak probably corresponds to the 0→2 transition.

The 2p-3s transitions in atomic oxygen

The 2p4-2p33s transitions in atomic oxygen were investigated by the MCHF method with Breit-Pauli corrections for relativistic effects. Particular attention was given to the spin-forbidden 2p4 1D-2p33s 3So transition recently identified in the satellite dayglow spectrum. A somewhat smaller branching ratio for the decay of the 3So state to the 1D state relative to 3P than that derived by Meier and Conway (1985) from modelling the 3P-3So transition was found.

The Knight shift of francium in cesium

Using the PAD technique the Knight shift for Fr in Cs at 273 K has been measured with the 15− isomer in212Fr. The result 4.9(3)% may be well understood in terms of the conventional analysis. A dominant correction for relativistic effects is the prime source for the extraordinarily large value.

Calculation of lower electronic states of the GaO and InO molecules by the SCF-X? scattered wave method

In the overlapping atomic spheres approximation of the SCF-Xα SW method in different variants (spin-restricted, spin-unrestricted, and quasirelativistic), we have calculated fragments of the potential curves for the electronic ground state (X 2∑) and the lower excited states (1 2π, 2 2∑, 2 2π, 3 2∑) of the GaO and InO molecules. We have calculated the spectroscopic constants (Te, ωe, ωexe, Be, αe, Re, Ae) for the molecules in the indicated states, the matrix elements for electronic transitions from the electronic ground state to the excited states, the Franck-Condon factors for the transitions 3 2∑, 2 2π, 2 2∑ ↔ 1 2π, X 2∑ (v′, v″ ≦10). For the InO molecule, we discuss the effect of relativistic corrections on the calculated molecular constants.

The effect of electron correlation on the metal—ligand bond in ferrocene

Large-scale MC SCF and externally contracted CI calculations have been performed to determine the metal—ligand distance in ferrocene. With a basis set of 174 contracted functions, A CI expansion of more than a million configurations. and a correction for relativistic effects, the computed distance is still 0.07 Å longer than experiment.

Ionisation cross sections of Cd, Sn, Te, I and Ba for protons and alpha particles

K-shell X-ray production cross sections of 48Cd, 50Sn, 52Te, 53I and 56Ba have been measured for alpha particles with energies between 2.2 and 2.8 MeV. In addition, production cross sections for 52Te have been measured for protons between 1.6 and 3 MeV. The values have been compared with the earlier experiments and with the theoretical predictions given by four different approaches: the perturbed stationary state model corrected for relativistic, projectile Coulomb deflection, and energy-loss effects, the plane-wave Born approximation (PWBA) with corrections for relativistic effects, effective binding and Coulomb deflection, the semiclassical approximation corrected for the same effects as the PWBA in its most recent formulation and, lastly, the binary model corrected for effective energy and relativistic effects. The different degrees of agreement for each theory in the cases examined are discussed.

Bethe-salpeter equation with instantaneous harmonic oscillator exchange

The Bethe-Salpeter equation in the form due to Cung et al. (Ann. Phys. (N.Y.) 98 (1976), 516) is investigated for the special case of instantaneous harmonic oseillator exchange, An exact reduction to a pair of coupled ordinary differential equations for the radial excitations of the 3( J ± 1) J modes is achieved. The equations in the mass zero case are brought to a form which is quite close to Whittaker's equation. This similarity to Whittaker's equation is exploited in a computer study of the level structure as a function of the quark mass. This study covers the region from a highly relativistic spectrum depending only upon J to the nonrelativistic regime where the spectrum depends only upon L. An expression for the leptonic width of a 3 S 1 state in terms of the Bethe-Salpeter wave function is derived and applied to the ψ-family. The effect of relativistic corrections is to reduce the predicted value of the leptonic width compared to the value calculated by assuming nonrelativistic kinematics. It is also shown that the relativistic treatment allows a 3 D 1 state to couple directly to a virtual photon.

Bonding in Metal Atom Cluster Compounds from the d‐Orbital Overlap Model to SCF‐Xα‐SW Calculations

AbstractThis paper gives a survey of results pertaining to the electronic structures of metal atom cluster compounds. The development of progressively more complete and accurate calculations, from the simple d‐orbital overlap model of Cotton and Haas (1964) through approximate Hartree–Fock (Fenske–Hall) calculations to recent SCF‐Xα‐SW calculations incorporating corrections for relativistic effects. It is shown that the qualitative essentials of the bonding are apparent even in the simplest treatment but that the more sophisticated methods are required to interpret spectra and other experimental data quantitatively.

K-Shell X-Ray Production of Na, Mg, Al, P, S, and Cl by 0.4-2.4 MeV 11H and 42He Ions

K-shell x-ray cross sections for thin solid targets of 11Na, 12Mg, 13A1, 15P, 16S, and L.17Cl have been measured for 0.4-2.4 MeV H and He ions. Ratios of ?(He)/4?(H) are determined as a function of energy/ mass to reduce the experimental uncertainties. The absolute cross sections and the cross section ratios are compared to theoretical predictions of the binary encounter approximation, the plane wave Born approximation, and the perturbed stationary state theory with Coulomb deflection (CPSS). It is shown that the CPSS theory with an approximate correction for relativistic effects gives good estimates of the experimental data.

Gamma polarisation in the n + p → γ + d reaction due to weak parity violating effects

We study the effect of parity violating weak interactions on the polarisation of the photon in the thermal neutron capture np → γd. The effect of various strong potentials is analysed in detail in the non-relativistic calculation. It is shown that, due to the short range of the weak interactions, the high momenta tail of the deuteron wave functions plays an important role. The effects of relativistic corrections are then considered. They are shown to be important for a precise quantitative calculation, but unable to explain the present disagreement between theory and experiment.