Relativistic Energy Corrections Research Articles

A closed-shell coupled-cluster treatment of the Breit-Pauli first-order relativistic energy correction.

First-order relativistic corrections to the energy of closed-shell molecular systems are calculated, using all terms in the two-component Breit-Pauli Hamiltonian. In particular, we present the first implementation of the two-electron Breit orbit-orbit integrals, thus completing the first-order relativistic corrections within the two-component Pauli approximation. Calculations of these corrections are presented for a series of small and light molecules, at the Hartree-Fock and coupled-cluster levels of theory. Comparisons with four-component Dirac-Coulomb-Breit calculations demonstrate that the full Breit-Pauli energy corrections represent an accurate approximation to a fully relativistic treatment of such systems. The Breit interaction is dominated by the spin-spin interaction, the orbit-orbit interaction contributing only about 10% to the total two-electron relativistic correction in molecules consisting of light atoms. However, the relative importance of the orbit-orbit interaction increases with increasing nuclear charge, contributing more than 20% in H(2)S.

Perturbation energy expansions based on two-component relativistic Hamiltonians

The approximate elimination of the small-component approach provides ansatze for the relativistic wave function. The assumed form of the small component of the wave function in combination with the Dirac equation define transformed but exact Dirac equations. The present derivation yields a family of two-component relativistic Hamiltonians which can be used as zeroth-order approximation to the Dirac equation. The operator difference between the Dirac and the two-component relativistic Hamiltonians can be used as a perturbation operator. The first-order perturbation energy corrections have been obtained from a direct perturbation theory scheme based on these two-component relativistic Hamiltonians. At the two-component relativistic level, the errors of the relativistic correction to the energies are proportional to α4Z4, whereas for the relativistic energy corrections including the first-order perturbation theory contributions, the errors are of the order of α6Z6–α8Z8 depending on the zeroth-order Hamiltonian.

Anatomy of relativistic energy corrections in light molecular systems

Relativistic energy corrections which arise from the use of the Dirac-Coulomb Hamiltonian, and the Gaunt and Breit interaction operators, plus Lamb-shift effects have been determined for the global minima of the ground electronic states of C2H6, NH3, H2O, [H,C,N], HNCO, HCOOH, SiC2, SiH− 3, and H2S, and for barrier characteristics for these molecular systems (inversion barrier of NH3 and SiH− 3, barrier to linearity of H2O, H2S, and HNCO, rotational barrier of C2H6, difference between conformations of HCOOH (Z/E) and SiC2 (linear/T-shaped), and isomerization barrier of HCN/HNC). The relativistic calculations performed at the Hartree-Fock and the highly correlated CCSD(T) levels employed a wide variety of basis sets. Comparison of the perturbational and the four-component fully variational results indicate that the Coulomb-Pauli Hamiltonian and the lowest order Hamiltonian of direct perturbation theory (DPT(2)) are highly successful for treating the relativistic energy effects in light molecular systems both at a single point on the potential energy hypersurface and along the surface. Electron correlation contributions to the relativistic corrections are relatively small for the systems studied, and are comparable with the 2-electron Darwin correction. Corrections beyond the Dirac-Coulomb treatment are usually rather small, but may become important for high accuracy ab initio calculations.

Theoretical study for the Pt2Au- and PtAu2- ethylene interaction

A theoretical study was done for the on-top coordination mode of the ethylene molecule chemisorbed over the Pt (100) surface, which was modeled by the Pt2Au and PtAu2 systems. In this on-top reaction mode, the CC axis is parallel to the PtPt, PtAu, or AuAu axis, respectively. Calculations were done with the Turbomole program, which is a density functional theory based method. Relativistic energy corrections were included and electron-core potentials were used. Orbital basis sets of DZP quality were employed for all the atomic species. The gradient-dependent BLYP functional was used for the description of the exchange–correlation energy. It was found that the ethylene moiety is severely perturbed by this metallic surfaces. We have found that in Pt3C2H4, which contains two Pt surface atoms and one Pt bulk atom, the moiety is chemisorbed with a binding energy (BE) of 50.3 kcal/mol. Substitution of the bulk Pt atom by an Au atom reduces the BE to 37.6 kcal/mol, but it also produces strong structural changes on ethylene. Substitution of one surface Pt atom by one Au atom produces a PtPtAu system, which is able to chemisorb the ethylene moiety with a BE of 13.31 kcal/mol. Similarly, replacement of one surface Pt atom and one bulk Pt atom by Au atoms, respectively, gives a Pt–Au2 system where the BE is 14.2 kcal/mol. Finally, replacement of the two surface Pt atoms by Au atoms produces a PtAu2 system which is unable to chemisorb the C2H4 molecule, since here there is an energy barrier of 12.72 kcal/mol. Thus, the substitution of the Pt bulk (surface) atoms by Au particles enhances (diminishes) the chemisorption of C2H4. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 75: 699–707, 1999

Resolution of some mathematical problems arising in the relativistic treatment of the S states of three-electron systems

Some of the mathematical difficulties that arise in the evaluation of the Breit–Pauli relativistic energy corrections for the S states of three-electron systems are resolved. Evaluation of the expectation value of the Breit–Pauli Hamiltonian using explicitly correlated wave functions leads to sets of integrals that diverge individually. By appropriately combining these integrals, and using some judicious series expansions, all the integration problems are resolved in terms of well-known auxiliary functions.

Effective Hamiltonian for near-degenerate states in relativistic direct perturbation theory. II. H2+-like systems

The recently developed effective Schrödinger-like Hamiltonian equation (EHA) in a model space of near-degenerate nonrelativistic two-component spinors is applied to the relativistic energy corrections at first, second, and third order of c−2 within the framework of relativistic direct perturbation theory (DPT). The dominant singular part of the total relativistic correction is already recovered by the lowest-order effective Hamiltonian in the spirit of degenerate perturbation theory, while the perturbative expansion needs to account for only the small remaining part. Numerical results for groups of excited potential curves of the one-electron H2+-like quasimolecule Sn299+ are presented and discussed. In general the most efficient approach is first-order EHA-DPT for the set of states, followed by single-state DPT of higher orders.

Relativistic and binding energy corrections to heavy quark fragmentation to polarised quarkonium

Abstract We calculate the fragmentation function for a charm quark to decay inclusively into transversely and longitudinally polarised J / ψ , including relativistic and binding energy corrections in powers of the quark relative velocity v . We use these fragmentation functions to calculate the asymmetry in the angular distribution of leptons produced in the subsequent decay of J / ψ . We point out that the inclusion of the said corrections reduces this asymmetry from 5% to about 2%.

Relativistic and binding energy corrections to heavy quark fragmentation functions

We calculate the fragmentation function for a charm quark to decay inclusively into $S$-wave charmonium states, including relativistic and binding energy corrections in powers of the quark relative velocity $v$. We also use these fragmentation functions to estimate their contribution to the production rate of ${\ensuremath{\eta}}_{c}$ and $J/\ensuremath{\psi}$ in ${Z}^{0}$ decay. These corrections contribute about $38$% to the integrated $c\ensuremath{\rightarrow}J/\ensuremath{\psi}+X$ fragmentation. For ${\ensuremath{\eta}}_{c},$ these corrections are found to be small.

Relativistic and binding energy corrections to direct photon production in Upsilon decay.

A systematic gauge-invariant method, which starts directly from QCD, is used to calculate the rate for an $\ensuremath{\Upsilon}$ meson to decay inclusively into a prompt photon. An expansion is made in the quark relative velocity $v$, which is a small natural parameter for heavy quark systems. Inclusion of these $O({v}^{2})$ corrections tends to increase the photon rate in the middle $z$ range and to lower it for larger $z$, a feature supported by the data.

Variational solution of the two-center Dirac-Coulomb problem in the free-particle representation

The problem of the hydrogen molecular ion H 2 + is considered in the free-particle momentum-space representation of the Dirac equation. This representation permits a straightforward interpretation of the relativistic energy corrections arising from the kinetic and the potential energy expansion as well as the contribution from virtual electron-positron pairs.

Finite nucleus effects on relativistic energy corrections

The effect of using a finite nucleus model in quantum-chemical calculations is examined. Relativistic corrections from the first-order Foldy—Wouthuysen terms are affected indirectly by the change in wavefunction, but also directly as a result of revised expressions for the Darwin and spin—orbit terms due to the change in nuclear potential. A calculation for the Rn atom indicates that the mass—velocity and Darwin corrections are much more sensitive to the finite nucleus than the non-relativistic total energy, but that the total contribution for these two terms is quite stable provided the revised form of the Darwin term is used. The spin—orbit interaction is not greatly affected by the choice of nuclear model.

Determination of mean inner potential and thickness in wedge shaped specimens by interferometry and electron holography

Knowlege of a specimens' mean innner potential and thickness is important for many purposes including holography, analytical (AEM) and microstructural analysis. This paper shows that both electron interferometry and holography can easily be used to determine the mean inner potential, as well as, the thickness at every point within a TEM specimen. The relative fringe shift, Δs/s , in a hologram by an object is linearly proportional to the object's mean inner potential, V, and thickness, d, and is given by,where C is considered a constant and described by the relativistic energy correction divided by the acceleration energy, E, and wavelength, λ, i.e. [(1 + eE/mc2)/(l + eE/2mc2)/2Eλ]. Inorder to separate d and V another holographic image is required to provide another image equation. This image equation is easily provided by rotating the specimen a known angle, Δθ, such that d becomes a function of Δθ.

Convenient parametrization for deep inelastic structure functions of the deuteron.

The spin, as well as spin-averaged, twist-two structure functions of the deuteron are calculated in a version of the convolution model that incorporates relativistic and binding energy corrections. A simple parametrization of these structure functions is given in terms of a few deuteron wave-function parameters and the free nucleon structure functions. This allows for an easy comparison of structure functions calculated using different deuteron models.

An accurate solution of the two-centre Dirac equation for H +2 by the finite-element method

A two-dimensional, fully numerical approach to the four-component Dirac equation using the finite-element method (FEM) is employed for a diatomic system. For H + 2, an absolute accuracy of about 10 −10 au for the 1σ g orbital energy is obtained by using only 2601 grid points. Using a difference method, the relativistic energy correction can be calculated with an accuracy of about 10 −13 au.

A relativistic C.I. study on theJ=1/2 even spectrum of Sr II

We present a method for performing relativistic CI calculations in ground and excited atomic and ionic levels. An electron occupying a relativistic shellnκ in a given electronic configuration is described by a single numerical four-component Dirac-Fock orbital having the samen and κ quantum numbers to those of the shellnκ. Application of this method yields estimates for the I.P. (88 741 cm−1) and the core correlation energy (−30916 cm−1) for Sr II and for the total correlation energy in Sr III (−30916 cm−1). Core-valence correlation energies for the |core 5s〉 (−4379 cm−1), |core 6s〉 (−1191 cm−1) and |core 13s〉 (−32 cm−1) levels have been calculated for Sr II. Estimates for the total relativistic, Breit, vacuum polarization and self energy corrections for these levels are also reported. Configurations in which the core is not fully occupied have been found to yield significant contributions to the correlation energies of both ground and excited levels. Our results show that full scale relativistic CI calculations using numerical four-component Dirac-Fock orbitals are feasible and provide a useful ab-initio tool for the investigation of atomic properties within the framework of fully relativistic theories.

The origin of relativistic effects of atomic orbitals

a(R)= integral R0 ( integral psi a psi d Omega )r2 dr curves are presented for different contributions a to the energy of atomic orbitals. While all radial shells contribute about equally to the non-relativistic kinetic and potential orbital energies, there is almost perfect cancellation of these energies in the inner shells, and the total energy of an orbital is almost solely determined by its outermost shell. In contrast to this, the first-order relativistic mass-velocity, Darwin and spin-orbit energies originate from the innermost shells only, while all radial shells contribute to the so-called indirect relativistic orbital energy correction,. The indirect effect is important also for s atomic orbitals except for the central columns of the periodic system, where the indirect destabilisation is compensated by indirect stabilisation. This explains the 'gold maximum' of relativistic corrections.

The relativistic energy correction of the H2+ molecule

From variational solutions of the Dirac and Schrödinger equation using extended Gaussian basis sets the relativistic energy correction ΔEr of H2+ at R = 2.0 [Bohr] is estimated to be − 7.366 [μHa]. The leading coefficients in a series expansion of ΔEr in powers of the fine structure constant are obtained numerically. It is concluded that the value of ΔEr was slightly underestimated in earlier calculations based on the Dirac equation, whereas it was slightly overestimated in those based on perturbation theory.

Relativistic perturbation theory: II. One-electron variational perturbation calculations

For pt.I see ibid., vol.19, p.149 (1986). The effect of the approximate character of the non-relativistic wavefunction on the second-order relativistic energy correction for the ground state of hydrogen-like atoms is illustrated by means of prototype calculations using Gaussian-type basis functions. Generalisation of the concept of even-tempered basis sets for use in relativistic calculations is proposed. Applications using Gaussian-type and exponential-type functions are presented and the convergence of the second-order relativistic correction with increasing size of basis set is examined. Some attention is paid to the possibility of computation of the third-order energy correction using the first-order correction to the wavefunction.

Quasirelativistic methods

AbstractA comparative study of quasirelativistic equations used in atomic structure calculations has been performed. A uniform derivation of all the equations is presented, and some of their specific features are discussed in detail. Electron density distributions, orbital energies, and expectation values of rn obtained with different methods are compared with the ones resulting from the Schrödinger and Dirac equations. The most accurate are found to be the equations of Wood and Boring and of Barthelat, Pelissier, and Durand. (They reproduce almost exactly the Dirac electron densities and expectation values.) The simplest, though least accurate, equation is proposed by us. It gives the relativistic energy corrections with about 6% accuracy and retains exactly the form of the nonrelativistic Schródinger equation. Consequently, its application in analytical SCF‐CI calculations does not require any additional integral calculation.

Validity of first-order perturbation theory for relativistic energy corrections

First-order relativistic calculations give a relative error in the estimate of the total correction which increases like Z 2 from 0.26% for neon to 8% for xenon.