Relativistic Propagator Research Articles

Probing the Planck scale: the modification of the time evolution operator due to the quantum structure of spacetime

The propagator which evolves the wave-function in non-relativistic quantum mechanics, can be expressed as a matrix element of a time evolution operator: i.e. GNR(x) = 〈x2|UNR(t)|x1〉 in terms of the orthonormal eigenkets |x〉 of the position operator. In quantum field theory, it is not possible to define a conceptually useful single-particle position operator or its eigenkets. It is also not possible to interpret the relativistic (Feynman) propagator GR(x) as evolving any kind of single-particle wave-functions. In spite of all these, it is indeed possible to express the propagator of a free spinless particle, in quantum field theory, as a matrix element 〈x2|UR(t)|x1〉 for a suitably defined time evolution operator and (non-orthonormal) kets |x〉 labeled by spatial coordinates. At mesoscopic scales, which are close but not too close to Planck scale, one can incorporate quantum gravitational corrections to the propagator by introducing a zero-point-length. It turns out that even this quantum-gravity-corrected propagator can be expressed as a matrix element 〈x2|UQG(t)|x1〉. I describe these results and explore several consequences. It turns out that the evolution operator UQG(t) becomes non-unitary for sub-Planckian time intervals while remaining unitary for time interval is larger than Planck time. The results can be generalized to any ultrastatic curved spacetime.

Four-component relativistic calculations of electronic excitations in tris-(allyl)-iridium complex: Influence of spin-orbit coupling on the electronic structure and excitation spectrum

Large-scale four-component relativistic calculations have been carried out for excited states of the tris-(allyl)-iridium complex, Ir(C3H5)3, using the relativistic polarization propagator method. The main focus was on providing insight into the lowest excited states of the three different structural forms of the complex, C3, C3h, and C1, including prediction of the excitation spectra and elucidation of spin-orbit coupling (SOC) effects for these forms. To unravel SOC effects, results of spin-orbit (SO) calculations were contrasted to results of spin-free (SF) calculations. Both the SO and SF calculations predict the lowest excited states of all three forms are of Rydberg type. We identified several Rydberg series, characterized by different hole configurations, that constitute the low-energy parts of the SO and SF excitation spectra. The SO spectra, however, differ notably from the corresponding SF spectra, pointing out to the influence of SOC. The most pronounced differences are observed for the SO and SF spectra of the C3h form. Our analysis indicates that SOC in this form causes strong alterations of the frontier occupied orbitals, participating in the excitations. As a result, excited states belonging to the first two Rydberg series are characterized by different electronic structure (hole configurations) in the SO and SF cases. Besides the orbital alteration effect, two other SOC effects, zero-field splitting and singlet-triplet mixing were identified in the SO spectra of all three forms. These effects result in new lines in the SO spectra and notable redistribution of the spectral intensity.

Influence of the nuclear charge distribution and electron correlation effects on magnetic shieldings and spin-rotation tensors of linear molecules.

The nuclear charge distribution effects (NChDE) on two response properties, the NMR magnetic shielding (σ) and the nuclear spin-rotation (SR) constants (M), are analyzed. We do it employing point-like and Gaussian-like models for describing the nuclear charge density of three linear molecules: HBr, HI and HAt. According to our results, both properties are sensitive to the NChDE. We show that the NChDE are almost completely relativistic, i.e., they nearly vanish in the non-relativistic limit of both properties. We calculated the NChDE on σ and M, and analyzed the differences between them in terms of a relativistic relation between these two properties. Using that relation we found that the electronic core mechanisms are the main ones for the NChDE on the shielding of nuclei of both, molecules and free atoms. The NChDE are smaller on SR constants than on shieldings. Nevertheless, within the relativistic polarization propagator formalism at the RPA level of approach they are very important for SR constants of nuclei in heavy-atom-containing compounds. Astatine in HAt has the largest influence: MAt = −9.95 kHz for a point-like model and −50.10 kHz for a Gaussian-like model. Correlation effects must be included and we do it using different DFT schemes. The PBE0 functional gives results that are closest to experiments for Br and I, though the LDA gives the closest for hydrogen. The value of the SR constant of At is reduced among 350 kHz and 500 kHz from its RPA value, when different and usual functionals are applied. Given that the NChDE on M and σ are mostly relativistic in their origin, these effects are also dependent on electron correlation. They have also a nonvanishing dependence with the Gaunt electron–electron interactions.

Propagator of Dirac oscillator in 2D with the presence of the minimal length uncertainty

In order to emphasize the role of quantum correction in two dimensions with the presence of the minimal length uncertainty relation, let us consider the case of relativistic Dirac oscillator in $ (2+1)$ -dimensional momentum space. At modified midpoint discretization interval $ (\bar{p}_{n} = (p_{n}+p_{n-1})/2)$ we can obtain the exact relativistic propagators, also the energy eigenvalues and their corresponding normalized radial momentum space eignefunctions are deduced.

Calculation of the covariant matrix elements, and cross-sections of Compton diffusion on a bound electron

This paper completes a previous published work that calculated analytically the relativistic wavefunctions for bound electron in a Compton diffusion process. This work calculates the relativistic propagator and the Wronskian of the two associated Feynman diagrams of Compton diffusion (emission first and absorption first). Then find an explicit expression for the covariant matrix elements separated into two parts: spin-angular part and radial part. Using these explicit expressions, the effective cross-section for Compton diffusion in the most general form is obtained in terms of basic dynamical and static quantities, like electron’s and photon’s 4-momenta and atomic number. The form of the cross-section is put ready for numerical calculations.

Relativistic and electron correlation effects on NMR J-coupling of Sn and Pb containing molecules

We studied the influence of relativistic and electron correlation effects on NMR J-couplings in the following set of heavy-atom containing molecules: \(XY_4\) and H\(_3XX\)H\(_3\) (X = Sn, Pb; Y = H, F, Cl, Br, I). We applied two formalisms, the relativistic polarization propagator approach at random phase level of approach (RelPPA-RPA) and density functional theory (DFT) with functionals as implemented in the DIRAC code. We have chosen four functionals that have different amount of HF exchange (PBE0, B3LYP, BLYP, BP86). For those molecular systems, results of calculations with BLYP functional have the best performance as compared with available experimental data. As was previously found for magnetic shieldings in other molecular systems we are able to show here that DFT functionals must be modified in order to obtain reliable results of NMR J-coupling within the relativistic regime. We can state that there is a non-linear dependence among both, electron correlation and relativistic effects that should be introduced in the functionals. The functionals implemented in the DIRAC code are standard nonrelativistic ones which were parameterized with data taken from light-atom containing molecules. This explains why they are not able to properly introduce relativistic effects on NMR parameters, like J-coupling constant. Lastly we show that in the analysis of J-couplings for the family of compounds mentioned above, one must consider the effects of a third heavy-atom that is close to the J-coupled atoms of the same molecule, specially for \(^n\)J(H–H). This kind of effect is similar to the newest and so called heavy-atom effect on vicinal heavy atoms, HAVHA, proposed for the NMR-shielding constant. Such effects are among the most important relativistic effects in the family of compounds studied in this work.

Absolute value of the nuclear magnetic shielding of silicon and germanium atoms in Si/Ge(CH3)4

The reference value of NMR magnetic shieldings, σref(X), of a nucleus X is of highest importance when theoretical analysis of chemical shifts are envisaged. For light atoms the absolute scale can be obtained using an old relationship among spin–rotation constants and magnetic shieldings, though for heavy-atom-containing molecules such nonrelativistic relationship is not valid any longer. Then, for such molecules the search for σref needs new strategies to be followed. We present here new values of σref(Si) and σref(Ge) for tetramethyl silane and tetramethyl germanium that were obtained applying a simple procedure which mix accurate experimental chemical shifts and theoretically obtained magnetic shieldings in a set of selected molecular systems. We considered that in experiments one usually measures chemical shifts, (δ). We calculated σref(Si) and σref(Ge) from a set of seventeen and sixteen heavy-halogen-containing molecules, respectively. We found out that σref[Si; Si(CH3)4] in gas phase should be close to 410.49±6.77ppm and σref[Ge; Ge(CH3)4] should be close to 1705.29±19.51ppm. Such theoretical values were obtained by performing calculations within the relativistic polarization propagator method, RelPPA, at RPA level of approach. Based on those values of σref, we recalculated the chemical shifts of the whole set of molecular systems obtaining a good reproduction of known experimental results. We also obtained a highly correlated relationship among the absolute values of the nuclear magnetic shieldings of Si and Ge atoms in halogen substituted tetrahedral compounds.

Calculation of the lowest electronic excitations of the alkaline earth metals using the relativistic polarization propagator

In this work we use the recently implemented four-component polarization propagator for accurate single excitation calculations of alkaline earth metals and compare our results to experimental data. Various approximations to the Dirac–Coulomb Hamiltonian are additionally tested. In Ca spin–orbit coupling already leads to noticeable zero field splitting, which gradually increases for the heavier homologs finally invalidating the singlet and triplet state characterizations. For all systems we observe a very good agreement with experimental transition energies in the considered energy range. For Sr, Ba and Ra non-relativistic approaches already exhibit unacceptable deviations in the reproduction of transition energies and spectral structure. The obtained excited final states are analyzed in terms of atomic donor and acceptor orbital contributions. Our results stress the necessity to use relativistic implementations of the polarization propagator for an accurate description of both electron correlation and relativistic effects contributing to excitation spectra of heavy systems.

Theoretical analysis of NMR shieldings of group-11 metal halides on MX (M = Cu, Ag, Au; X = H, F, Cl, Br, I) molecular systems, and the appearance of quasi instabilities on AuF.

Accurate calculations of nuclear magnetic shieldings of group-11 metal halides, σ(M; MX) (M = Cu, Ag, Au; X = H, F, Cl, Br, I), were performed with relativistic and nonrelativistic theoretical schemes in order to learn more about the importance of the involved electronic mechanisms that underlie such shieldings. We applied state of the art schemes: polarization propagators at a random phase level of approach (PP-RPA); spin-free Hamiltonian (SF); linear response elimination of small component (LRESC) and density functional theory (DFT) with two different functionals: B3LYP and PBE0. The results from DFT calculations are not close to those from the relativistic polarization propagator calculations at the RPA level of approach (RelPP-RPA), in line with previous results. The spin-orbit (SO) contribution to a shielding constant is important only for MF molecules (M = Cu, Ag, Au). Different electronic mechanisms are considered within the LRESC method, bunched into two groups: core- and ligand-dependent. For the analysed shieldings the core-dependent electronic mechanisms are the most important ones; the ligand-dependent being only important for MF molecules. An out of range value for σ(Au) is found in AuF. It was previously reported in the literature, either originated in the large fluorine electronegativity together with large spin-orbit coupling contributions; or, due to Fermi-contact contributions. We argue here that such an unexpected large value is an artifact originated in the appearance of quasi instabilities, and show how to handle this apparent problem.

Relativistic and electron-correlation effects on the nuclear magnetic resonance shieldings of molecules containing tin and lead atoms.

The reference values for NMR magnetic shieldings, σ(ref), are of the highest importance when theoretical analysis of chemical shifts are envisaged. The fact that the nonrelativistically valid relationship among spin-rotation constants and magnetic shieldings is not any longer valid for heavy atoms requires that the search for σ(ref) for such atoms needs new strategies to follow. We present here results of σ(ref) that were obtained by applying our own simple procedure which mixes accurate experimental chemical shifts (δ) and theoretical magnetic shieldings (σ). We calculated σ(Sn) and σ(Pb) in a family of heavy-halogen-containing molecules. We found out that σ(ref)[Sn;Sn(CH3)4] in gas phase should be close to 3864.11 ± 20.05 ppm (0.5%). For Pb atom, σ(ref)[Pb;Pb(CH3)4] should be close to 14475.1 ± 500.7 ppm. Such theoretical values correspond to calculations with the relativistic polarization propagator method, RelPPA, at the RPA level of approach. They are closer to experimental values as compared to those obtained applying few different functionals such as PBE0, B3LYP, BLYP, BP86, KT2, and KT3 of the density functional theory, DFT. We studied tin and lead shieldings of the XY(4-n)Z(n) (X = Sn, Pb; Y, Z = H, F, Cl, Br, I) and PbH(4-n)I(n) (n = 0, 1, 2, 3, 4) family of compounds with four-component functionals as implemented in the DIRAC code. For these systems results of calculations with RelPPA-RPA are more reliable than DFT ones. We argue about why those DFT functionals must be modified in order to obtain more accurate results of NMR magnetic shieldings within the relativistic regime: first, there is a dependence among both electron-correlation and relativistic effects that should be introduced in some way in the functionals; and second, the DIRAC code uses standard nonrelativistic functionals and the functionals B3LYP and PBE0 were parametrized only with data taken from light elements. It can explain why they are not able to properly introduce relativistic effects on nuclear magnetic shieldings. We finally show that in the analysis of magnetic shieldings for the family of compounds mentioned above, one must consider the newest and so-called heavy-atom effect on vicinal heavy atoms, HAVHA. Such effects are among the most important relativistic effects in these kind of compounds.

The relativistic polarization propagator for the calculation of electronic excitations in heavy systems

In this work, we present a new four-component implementation of the polarization propagator for accurate calculations of excited states in heavy systems. Differences to existing nonrelativistic realizations are detailed and the energetically lowest final states of the ns(2)np(6) → ns(2)np(5)(n + 1)s(1) and ns(2)np(6) → ns(2)np(5)(n + 1)p(1) transitions in noble gases are calculated and compared with experimental data. Already for the light atoms Ne and Ar spin-orbit coupling leads to noticeable zero field splitting that gradually increases in the heavier homologues and eventually invalidates the LS-based description of singlet and triplet excited states. For all four noble gases Ne through Xe, we observe a very good agreement with experimental transition energies in the considered energetic range where the extended version of the propagator implementation in general yields better excitation energy differences than the strict variant. In the extended version, off-diagonal first-order contributions in the two-particle-two-hole block are included that are not present in the strict variant. In case of Kr and Xe, nonrelativistic approaches already exhibit unacceptable deviations in the reproduction of transition energies and the spectral structure. The obtained excited final states are analyzed in terms of atomic contributions to the donor and acceptor orbitals constituting the corresponding wave functions. The relativistic polarization propagator provides a consistent description of electron correlation and relativistic effects especially relevant for the heavier systems where these two contributions are no longer separable.

Spin–orbit effects, electronic decay and breakdown phenomena in the photoelectron spectra of iodomethane

In this work we discuss the theoretical photo-double ionization spectrum of iodomethane which was obtained by the newly developed relativistic two-particle propagator and compare it to experimental data. For a correct interpretation of the spectral features in the outer valence region, spin–orbit effects play a major role and induce new features to the spectra. Additionally, wide-energy range single and double ionization spectra of iodomethane were calculated revealing possible electronic decay reactions after high-energy primary ionization. In the low energy region of the single ionization spectrum pronounced breakdown was observed resulting in a highly correlated manifold which was characterized in detail via population analysis.

Strong configuration interaction in the double ionization spectra of noble gases studied by the relativistic propagator method

In this work, the four-component two-particle propagator technique is employed for the calculation of double ionization spectra of the noble gas atoms Ne through Rn. For a correct assignment of the individual final states, inclusion of spin-orbit coupling and electron correlation is mandatory and is accounted for in the framework of the relativistic propagator. It was observed that the $n{s}^{2}n{p}^{4}$(${}^{3}\phantom{\rule{-0.16em}{0ex}}{P}_{2,1,0}$, ${}^{1}\phantom{\rule{-0.16em}{0ex}}{D}_{2}$, ${}^{1}\phantom{\rule{-0.16em}{0ex}}{S}_{0}$) manifolds of all investigated noble gas dications exhibit a clear main-state character with only small admixture from other configurations. This also refers to the $2{s}^{1}2\phantom{\rule{-0.16em}{0ex}}{p}^{5}$(${}^{3}\phantom{\rule{-0.16em}{0ex}}{P}_{2,1,0}^{o}$, ${}^{1}\phantom{\rule{-0.16em}{0ex}}{P}_{1}^{o}$) states of Ne${}^{2+}$. In the argon, krypton, and xenon dications, the $n{s}^{1}n{p}^{5}$(${}^{3}\phantom{\rule{-0.16em}{0ex}}{P}_{2,1,0}^{o}$) states, and especially the $n{s}^{1}\phantom{\rule{-0.16em}{0ex}}n{p}^{5}$(${}^{1}\phantom{\rule{-0.16em}{0ex}}{P}_{1}^{o}$) ones, lose intensity due to pronounced configuration interaction. These states experience strong mixings with ground-state shake-up satellites, which occupy the same energy region. The composition of the $5{s}^{1}5{p}^{5}$(${}^{1}\phantom{\rule{-0.16em}{0ex}}{P}_{1}^{o}$) singlet state of Xe${}^{2+}$ is studied in detail by analyzing the corresponding eigenvector. As long as a $LS$ coupling picture can be approximately maintained, the amount of singlet-triplet splitting decreases in the sequence from neon to xenon. In the $6{s}^{1}6{p}^{5}$ manifold of Rn${}^{2+}$, a complete disappearance of well-defined main states takes place leading to a dense and complicated spectrum governed by very strong multiconfiguration effects. Relativistic corrections to the Coulomb interaction are accounted for by inclusion of the Gaunt (magnetic) term.

Unitarized Chiral Perturbation Theory in a finite volume: Scalar meson sector

We develop a scheme for the extraction of the properties of the scalar mesons f0(600), f0(980), and a0(980) from lattice QCD data. This scheme is based on a two-channel chiral unitary approach with fully relativistic propagators in a finite volume. In order to discuss the feasibility of finding the mass and width of the scalar resonances, we analyze synthetic lattice data with a fixed error assigned, and show that the framework can be indeed used for an accurate determination of resonance pole positions in the multi-channel scattering.

ρ self-energy at finite temperature and density in the real-time formalism

The ρ meson self-energy in nuclear matter from baryonic loops is analysed in the real-time formulation of field theory at finite temperature and density. The discontinuities across the branch cuts of the self-energy function are evaluated for an exhaustive set of resonances in the loops considering the fully relativistic thermal baryon propagator including anti-baryons. Numerical calculations show a significant broadening of the ρ spectral function coming from the Landau cut. Adding the contribution from mesonic loops, the full spectral function of the ρ in a thermal gas of mesons, baryons and anti-baryons in equilibrium is evaluated at various values of temperature and baryonic chemical potential.

Relativistic effects on the shielding of SnH2XY and PbH2XY (X, Y = F, Cl, Br and I) heavy atom–containing molecules

The relativistic polarization propagator approach is one of the most reliable methods available today for the calculation of NMR spectroscopic parameters on heavy atom–containing molecules, though its implementation is still at RPA or FOPPA (first-order) level of approach. Two-component methods like the LR-ESC method make possible the analysis of the electronic origin of relativistic effects due to its splitting in several mechanisms which are (or not) sensitive to the molecular structure or the nature of the chemical environment of the atom under study. In this article we present the study of some nuclear magnetic shieldings on the heavy atom for the following systems: SnXH3 (X = H, F, Cl, Br, I), SnXYH2 (X, Y = F, Cl, Br, I) and PbXH3 (X = H, F, Br, I). Total LR-ESC calculations are confronted to benchmark RPA calculations and then analyzed in order to get the main trends and discuss the electronic origin of the shielding of two kinds of atoms involved in such systems: central and substituent atoms. The electronic origin of the heavy atom effects on vicinal heavy atoms (HAVHA), recently proposed, is analyzed. It is shown that the passive third-order Spin orbit mechanism does not explain the total pattern though is still the most important. There are two other mechanisms involved: the so called here PSO-OZ and the L-PSO-K. Both mechanisms do contain the PSO perturbative Hamiltonian (which also include kinetic energy correcting terms). In the case of \(\hbox{SnH}_2\hbox{I}_2\), the HAVHA effect on σ(Sn) is of the order of 16%. When the central atom is as heavy as Sn, the active SO contribution on the shielding of such atom becomes larger than the passive SO, which is small in this case. This would mean that the HALA-type effect is strongly diminished when applied on a vicinal heavy atom. Quite a similar pattern though with larger relativistic effects is observed for the central lead atom.

The UKB prescription and the heavy atom effects on the nuclear magnetic shielding of vicinal heavy atoms

Fully relativistic calculations of NMR magnetic shielding on XYH3 (X = C, Si, Ge and Sn; Y = Br, I), XHn (n = 1-4) molecular systems and noble gases performed with a fully relativistic polarization propagator formalism at the RPA level of approach are presented. The rate of convergence (size of basis set and time involved) for calculations with both kinetic balance prescriptions, RKB and UKB, were investigated. Calculations with UKB makes it feasible to obtain reliable results for two or more heavy-atom-containing molecules. For such XYH3 systems, the influence of heavy vicinal halogen atoms on sigma(X) is such that heavy atom effects on heavy atoms (vicinal plus their own effects or HAVHA + HAHA effects) amount to 30.50% for X = Sn and Y = I; being the HAHA effect of the order of 25%. So the vicinal effect alone is of the order of 5.5%. The vicinal heavy atom effect on light atoms (HALA effect) is of the order of 28% for X = C and Y = I. A similar behaviour, but of opposite sign, is observed for sigma(Y) for which sigmaR-NR (I; X = C) (HAHA effect) is around 27% and sigmaR-NR(I; X = Sn) (HAVHA + HAHA effects) is close to 21%. Its electronic origin is paramagnetic for halogen atoms but both dia- and paramagnetic for central atoms. The effect on two bond distant hydrogen atoms is such that the largest variation of sigma(H) within the same family of XYH3 molecules appears for X = Si and Y = I: around 20%. In this case sigma(H; X = Sn, Y = I) = 33.45 ppm and sigma(H; X = Sn, Y = H) = 27.82 ppm.

The appearance of an interval of energies that contain the whole diamagnetic contribution to NMR magnetic shieldings

Working within relativistic polarization propagator approach, it was shown in a previous article that the electronic origin of diamagnetic contributions to NMR nuclear magnetic shielding, sigmad, are mostly excitations that fit in a well defined interval of energies such that 2mc2<or==(epsiloni-epsilons)<4mc2. That interval of energies does not have, in principle, any physical reason to be so well defined, and gives a large amount of the total contribution to sigmad, e.g., close to 98% of it. Then a further study is given in this article, where we show some of the main characteristics of that interval of energy, such as its universal appearance and basis set independence. Our main result is the finding that sigmad is completely described by that interval of excitation energies, i.e., there is no contribution arising from outside of it. Most of the contributions belonging to that interval arise from virtual electronic energies larger than -3mc2. For heavier atoms, there are few contributions from states with virtual negative energies smaller than -3mc2. The model systems under study were noble gases, XH (X=Br, I, and At), XH2 (X=O, S, Se, Te, and Po), XH3 (X=N, P, As, Sb, and Bi); XH4 (X=Sn and Pb), and SnXH3 (X=Br and I). The pattern of contributions of occupied molecular orbitals (MOs) is also shown, where the 1s1/2 is the most important for excitations ending in the bottom half part of the above mentioned interval. On the other hand, the contribution of the other occupied MOs are more important than that of 1s1/2 for the other part of such interval. We also show that sigmad is electron correlation independent within both relativistic and nonrelativistic domain. In the case of sigmap, we find out a clear dependence of electron correlation effects with relativistic effects, which is of the order of 30% for Pb in PbH4.

Relativistic diffusion processes and random walk models

The nonrelativistic standard model for a continuous, one-parameter diffusion process in position space is the Wiener process. As is well known, the Gaussian transition probability density function (PDF) of this process is in conflict with special relativity, as it permits particles to propagate faster than the speed of light. A frequently considered alternative is provided by the telegraph equation, whose solutions avoid superluminal propagation speeds but suffer from singular (noncontinuous) diffusion fronts on the light cone, which are unlikely to exist for massive particles. It is therefore advisable to explore other alternatives as well. In this paper, a generalized Wiener process is proposed that is continuous, avoids superluminal propagation, and reduces to the standard Wiener process in the nonrelativistic limit. The corresponding relativistic diffusion propagator is obtained directly from the nonrelativistic Wiener propagator, by rewriting the latter in terms of an integral over actions. The resulting relativistic process is non-Markovian, in accordance with the known fact that nontrivial continuous, relativistic Markov processes in position space cannot exist. Hence, the proposed process defines a consistent relativistic diffusion model for massive particles and provides a viable alternative to the solutions of the telegraph equation.

Relativistic effects on the nuclear magnetic shieldings of rare-gas atoms and halogen in hydrogen halides within relativistic polarization propagator theory

In this work an analysis of the electronic origin of relativistic effects on the isotropic dia- and paramagnetic contributions to the nuclear magnetic shielding sigma(X) for noble gases and heavy atoms of hydrogen halides is presented. All results were obtained within the 4-component polarization propagator formalism at different level of approach [random-phase approximation (RPA) and pure zeroth-order approximation (PZOA)], by using a local version of the DIRAC code. From the fact that calculations of diamagnetic contributions to sigma within RPA and PZOA approaches for HX(X=Br,I,At) and rare-gas atoms are quite close each to other and the finding that the diamagnetic part of the principal propagator at the PZOA level can be developed as a series [S(Delta)], it was found that there is a branch of negative-energy "virtual" excitations that contribute with more than 98% of the total diamagnetic value even for the heavier elements, namely, Xe, Rn, I, and At. It contains virtual negative-energy molecular-orbital states with energies between -2 mc2 and -4 mc2. This fact can explain the excellent performance of the linear response elimination of small component (LR-ESC) scheme for elements up to the fifth row in the Periodic Table. An analysis of the convergency of S(Delta) and its physical implications is given. It is also shown that the total contribution to relativistic effects of the innermost orbital (1s1/2) is by far the largest. For the paramagnetic contributions results at the RPA and PZOA approximations are similar only for rare-gas atoms. On the other hand, if the mass-correction contributions to sigma(p) are expressed in terms of atomic orbitals, a different pattern is found for 1s1/2 orbital contributions compared with all other s-type orbitals when the whole set of rare-gas atoms is considered.