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![Fig. 2.1. Mendeleev Periodic Table as published in 1869, Ref. [31].](profile/Miroslav-Urban/publication/228631594/figure/fig1/AS:669967544500230@1536744118100/Mendeleev-Periodic-Table-as-published-in-1869-Ref-31_Q320.jpg)
Relativistic effects in atomic and molecular properties
We present an overview of basic principles and methods of the relativistic quantum chemistry. Practical aspects of different methods will be discussed stressing their capability of providing accurate predictions of molecular properties, particularly in species containing a heavy metal element....
Citations
... Coinage metal group is a classic example to denote and learn how relativistic effects affect the bonding properties, as summarized by Pyykkö, Schmidbauer, Schwerdtfeger, Iliaš, Reiher, and Wolf, among others. [40][41][42][43][44][45][46][47][48] In this sense, the [(NHC-M) 2 pyz] 2+ aggregate allows to evaluate further the role of relativistic effects in a molecular motor complex acting over the accepting N → M bond from pyrazine towards the metal contacts. In this report, we explore the nature of the pyrazinecoinage metal bond as a mechanical bond as a prototypical molecular rotor unit. ...
... But in relativistic framework, the Dirac equation of bound muon can be solved exactly or using relativistic perturbation (Niri and Anjami, 2018, Firew Meka, 2020, Adamu and Ngadda, 2017, Deck et al., 2005and Eshetu Diriba and Gashaw Bekele, 2021. It is worth noting that the relativistic effects play key role in the studies of the molecular properties such as electron affinities, ionization potentials, reaction, dissociation energies, spectroscopic and other properties (Iliaš et al., 2010 andPyper, 2020). The relativistic effects can also be used to investigate the rotation curve of disk galaxies which is significant at large radii (Deur, 2021). ...
Hydrogen is the simplest atom in nature which helps us to study fundamental properties and structure of atoms. In this work, the perturbed Schrodinger equation is used to estimate relativistic mass correction with potentials from finite size sources. This study is done with the assumption that the changes in both energy eigenvalues and eigenfunctions are negligible when considering the finite size of the nuclei. The relativistic mass corrections to 1s 1/2 and 2s_1/2 states using potential from finite size source are obtained and compared with corrections using potential from point-like source. The results show that, for hydrogen-like atoms with light nuclei the relativistic mass corrections due to the finite size source roughly coincides with that of point-like source. However, for atoms with heavy nuclei the two corrections display strong disagreement in which the corrections with finite size nuclei are significantly smaller than that of point-like nuclei. Thus, this study offers the first attempt at including relativistic mass correction to energy eigenvalues of electronic and muonic hydrogen-like atoms with finite-size nuclei. When experimental data become available, comparison of these theoretical predictions may give a better insight into how relativistic corrections are affected by finite-size nuclei.
... In particular, relativistic effects in atomic densities and related functionals for many-electron systems [1]. As both the number of electrons (N) in the atom and its nuclear charge (Z) increase, relativistic effects become more important and have to be taken into account to understand atomic or molecular properties [2][3][4][5]. In this order of ideas, these developments in relativistic quantum theory have made it possible to obtain accurate electronic properties for heavy elements with the aim of predicting their chemical and physical behaviors. ...
The relativistic density functional calculations including scalar and spin–orbit effects via the ZORA approximation and including solvent effects were carried out on the Tungsten intermediates [W6I14]2− clusters. In addition, these considerations were supported by molecular quantum similarity studies using four similarity descriptors such as overlap and coulomb indices, and their euclidean distances. The current calculations also indicate that the electronic similarities of the lowest excited states of the intermediates clusters and molecular quantum similarity with the strongly luminescent W6I14 cluster, suggest that these intermediates metal clusters (W3I9, W3I9_I1, W4I11, W4I11_I2, W5I13, W6I14) could be luminescent. This would imply that the luminescence property is evident from the W3I9 unit, this unit being the minimum necessary to present this property.
... | www.nature.com/naturecommunications very few heavy element molecules relied on spectroscopy techniques [28][29][30][31] and, thus, the relativistic effects were mainly discussed in theoretical aspects 32,33 . Although a Pb dimer has a much longer bond length (7 Å) than other relativistic molecules [33][34][35] and there is a substantial adatom-substrate interaction, the molecular splitting mainly arises from the direct overlap of relativistic p orbitals. ...
We fabricate artificial molecules composed of heavy atom lead on a van der Waals crystal. Pb atoms templated on a honeycomb charge-order superstructure of IrTe2 form clusters ranging from dimers to heptamers including benzene-shaped ring hexamers. Tunneling spectroscopy and electronic structure calculations reveal the formation of unusual relativistic molecular orbitals within the clusters. The spin–orbit coupling is essential both in forming such Dirac electronic states and stabilizing the artificial molecules by reducing the adatom–substrate interaction. Lead atoms are found to be ideally suited for a maximized relativistic effect. This work initiates the use of novel two-dimensional orderings to guide the fabrication of artificial molecules of unprecedented properties. Artificial molecules supported on templated surfaces attract enormous interest due to their tunable electronic properties. Here the authors use STM experiments and DFT calculations to show the formation of Pb artificial clusters on a IrTe2 honeycomb template that are maximally stabilized by relativistic effects.
... The direct and real space observation of relativistic molecular orbitals are exceptional, since most of previous experiments for very few heavy element molecules relied on spectroscopy techniques [28][29][30][31] and, thus, the relativistic effects were mainly discussed in theoretical aspects [32,33]. Although a Pb dimer has a much longer bond length (7Å) than other relativistic molecules [33][34][35] and there is a substantial adatom-substrate interaction, the molecular splitting mainly arises from the direct overlap of relativistic p orbitals. ...
We fabricate artificial molecules composed of heavy atom lead on a van der Waals crystal. Pb atoms templated on a honeycomb charge-order superstructure of IrTe2 form clusters ranging from dimers to heptamers including benzene-shaped ring hexamers. Tunneling spectroscopy and electronic structure calculations reveal the formation of unusual relativistic molecular orbitals within the clusters. The spin-orbit coupling is essential both in forming such Dirac electronic states and stabilizing the artificial molecules by reducing the adatom-substrate interaction. Lead atoms are found to be ideally suited for a maximized relativistic effect. This work initiates the use of novel two dimensional orderings to guide the fabrication of artificial molecules of unprecedented properties.
... 16 In the case of large (typically nonsymmetric) closed-shell systems, such as the ones studied in this work, relativity treatment can be reduced to so-called scalar relativistic effects (SRE)s. 17,18 This is a substantial simplification as it allows us to neglect e. g. spin-orbit coupling and use a relativistic one-component Hamiltonian (instead of four-or two-component) approach with practically negligible overhead compared to nonrelativistic one. ...
... If we decompose a noncovalent interaction into electrostatic, induction and dispersion forces, all of these contributions may be affected by SREs through dipole moments and dipole polarizabilities. 18 Among common halogens, this applies to bromine and iodine, which possess high polarizabilities. 22 In fact, X-bonding may be dispersion-driven as indicated by symmetry-adapted perturbation studies. ...
Halogen bond (X-bond) is a noncovalent interaction between a halogen atom and an electron donor. It is often rationalized by a region of the positive electrostatic potential on the halogen atom, so-called $\sigma$-hole. The X-bond strength increases with the atomic number of the halogen involved, thus for heavier halogens, relativistic effects become of concern. This complicates the quantum chemical description of X-bonded complexes and impedes a detailed understanding of the nature of the attraction. To quantify scalar relativistic effects (SREs) on the interaction energies and $\sigma$-hole properties, we have performed highly accurate quantum chemical calculations at the complete basis set limit of several X-bonded complexes and their halogenated monomers. The SREs turned to be comparable in magnitude to the effect of basis set. The nonrelativistic calculations typically underestimate the attraction by up to 5% or 23% for brominated and iodinated complexes, respectively. SREs yield smaller, more flatten and more positive $\sigma$-holes, although the differences to nonrelativistic values are small. Further for X-bonds, we highlight the importance of diffuse functions in the basis sets and provide support for using basis sets with pseudopotentials as an affordable alternative to a more rigorous Douglas-Kroll-Hess relativistic theory.
... 358−360 Twocomponent methods such as the recently introduced X2C approximation are less computer time-consuming than fourcomponent procedures but are still computationally expensive especially when electron correlation is included. [346][347][348][349][350][351][352][353][354]361,362 Therefore, the presently most commonly used quantum chemical approaches including relativistic effects in chemistry are based on one-component (scalar relativistic) approximations, such as the Douglas−Kroll method modified by Hess 363 or the Zeroth Order Regular Approximation (ZORA). 364 A very popular time-efficient and quite accurate alternative is the use of relativistic effective core potentials (ECPs), also called pseudopotentials. ...
The focus of this review is the presentation of the most important aspect
of chemical bonding in molecules of the main group atoms according to the current state of knowledge. Special attention is given to the difference between the physical mechanism of covalent bond formation and its description with chemical bonding models, which are often confused. This is partly due to historical reasons, since until the development of quantum theory there was no physical basis for understanding the chemical bond. In the absence of such a basis, chemists developed heuristic models that
proved extremely valuable for understanding and predicting experimental studies. The great success of these simple models and the associated rules led to the fact that the model conceptions were regarded as real images of physical reality. The complicated world of quantum theory, which eludes human imagination, made it difficult to link heuristic models of chemical bonding with quantum chemical knowledge. In the early days of quantum chemistry, some suggestions were made which have since proved untenable. In recent decades, there has been a stormy development of quantum chemical methods, which are not limited to the quantitative accuracy of the calculated
properties. Also, methods have been developed where the experimentally developed models can be quantitatively expressed and visually represented using mathematically well-defined terms that are derived from quantum chemical calculations. The calculated numbers may however not be measurable values. Nevertheless, as orientation data for the interpretation and classification of experimental findings as well as a guideline for new experiments, they form a coordinate system that defines the
multidimensional world of chemistry, which corresponds to the Hilbert space formalism of physics. The nonmeasurability of model values is not a weakness of chemistry but a characteristic by which the infinite complexity of the material world becomes scientifically accessible and very useful for chemical research. This review examines the basis of the commonly used quantum chemical methods for calculating molecules and for analyzing their electronic structure. The bonding situation in selected representative molecules of main-group atoms is discussed. The results are compared with textbook knowledge of common chemistry.
... [3][4][5] Recently, advances in electron correlation theories have made the accurate prediction of Xray ionization and excitation spectra viable. [6][7][8][9] A high level correlation theory should, in principle, take care of correlation and orbital relaxation in a generic situation involving energy differences of the ground and ionized, electron attached, or excited states. However, for core IP in particular, a proper correlation theory must also accurately capture not only a partial cancellation of dynamic correlation between the states but also take account of the large orbital relaxation in an explicit manner. ...
The orbital relaxation attendant on ionization is particularly important for the core electron ionization potential (core IP) of molecules. The Unitary Group Adapted State Universal Coupled Cluster (UGA-SUMRCC) theory, recently formulated and implemented by Sen et al. [J. Chem. Phys. 137, 074104 (2012)], is very effective in capturing orbital relaxation accompanying ionization or excitation of both the core and the valence electrons [S. Sen et al., Mol. Phys. 111, 2625 (2013); A. Shee et al., J. Chem. Theory Comput. 9, 2573 (2013)] while preserving the spin-symmetry of the target states and using the neutral closed-shell spatial orbitals of the ground state. Our Ansatz invokes a normal-ordered exponential representation of spin-free cluster-operators. The orbital relaxation induced by a specific set of cluster operators in our Ansatz is good enough to eliminate the need for different sets of orbitals for the ground and the core-ionized states. We call the single configuration state function (CSF) limit of this theory the Unitary Group Adapted Open-Shell Coupled Cluster (UGA-OSCC) theory. The aim of this paper is to comprehensively explore the efficacy of our Ansatz to describe orbital relaxation, using both theoretical analysis and numerical performance. Whenever warranted, we also make appropriate comparisons with other coupled-cluster theories. A physically motivated truncation of the chains of spin-free T-operators is also made possible by the normal-ordering, and the operational resemblance to single reference coupled-cluster theory allows easy implementation. Our test case is the prediction of the 1s core IP of molecules containing a single light- to medium-heavy nucleus and thus, in addition to demonstrating the orbital relaxation, we have addressed the scalar relativistic effects on the accuracy of the IPs by using a hierarchy of spin-free Hamiltonians in conjunction with our theory. Additionally, the contribution of the spin-free component of the two-electron Gaunt term, not usually taken into consideration, has been estimated at the Self-Consistent Field (ΔSCF) level and is found to become increasingly important and eventually quite prominent for molecules with third period atoms and below. The accuracies of the IPs computed using UGA-OSCC are found to be of the same order as the Coupled Cluster Singles Doubles (ΔCCSD) values while being free from spin contamination. Since the UGA-OSCC uses a common set of orbitals for the ground state and the ion, it obviates the need of two N⁵ AO to MO transformation in contrast to the ΔCCSD method.
... The Dirac theory of the electron was first applied to manyelectron atomic systems by Swirles [1], who used the Hartree-Fock formalism in conjunction with the Dirac equation. Many further works have since appeared [2][3][4][5][6][7][8][9][10][11], including textbooks [12]. Autschbach [13] has discussed electron correlation effects in relativistic calculations. ...
... Works published up to January 2016 were summarized by Pyykkӧ in the database 'RTAM' [14]. In particular, Rose, Grant, and Pyper (RGP) [5] discussed the effects of relativity on the 71 Lu (4f) 14 (5d) 1 (6s) 2 , 79 Au (5d) 10 (6s) 1 , 81 Tl (5d) 10 (6s) 2 (6p) 1 states, all of which involve a single d, s, or p electron. RGP introduced the terms, direct and indirect relativistic effects, which will be used after explanation in the present article. ...
... Works published up to January 2016 were summarized by Pyykkӧ in the database 'RTAM' [14]. In particular, Rose, Grant, and Pyper (RGP) [5] discussed the effects of relativity on the 71 Lu (4f) 14 (5d) 1 (6s) 2 , 79 Au (5d) 10 (6s) 1 , 81 Tl (5d) 10 (6s) 2 (6p) 1 states, all of which involve a single d, s, or p electron. RGP introduced the terms, direct and indirect relativistic effects, which will be used after explanation in the present article. ...
This work examines the relativistic and nonrelativistic effective charges (values of Z <sub>eff</sub>) for valence-shell electrons from <sub>1</sub>H to <sub>103</sub>Lr. Differences between relativistic and nonrelativistic Z <sub>eff</sub> values are investigated in detail. Except for <sub>46</sub>Pd (4 d )<sup>10</sup> (5 s )<sup>0</sup>, all atoms have n s or n p spinors/orbitals as their outermost shell. Apart from <sub>24</sub>Cr (3 d )5 (4s)<sup>1</sup>, <sub>41</sub>Nb (4 d )<sup>4</sup> (5 s )<sup>1</sup>, and <sub>42</sub>Mo (4 d )<sup>5</sup> (5 s )<sup>1</sup>, the relativistic n s <sub>+</sub> Z <sub>eff</sub> values are always larger than the corresponding nonrelativistic n s values. For all atoms having n p <sup>+</sup> spinors as their outermost shell, in contrast, the n p <sup><sub>+</sub></sup> Z <sub>eff</sub> values are smaller than the corresponding n p Z <sub>eff</sub> values. In the valence n d and n f shells of transition metal atoms, the lanthanoid and actinoid atoms, the relativistic Z <sub>eff</sub> values are always smaller than the corresponding nonrelativistic n d and n f Z <sub>eff </sub>values except for <sub>46</sub>Pd (4 d )<sup>10</sup> (5 s )<sup>0</sup>. There are considerable differences between values of Z <sub>eff</sub> for the two series, but the relativistic Z <sub>eff</sub> values plotted against the atomic number ( Z ) recognizably retain the shape of the nonrelativistic Z <sub>eff</sub> versus Z curve. This fact implies that the nonrelativistic equation of motion for electrons largely determines the electronic configurations of atoms.
... We now discuss relativistic effects in the valence ns shells in more detail. Relativistic effects are negligibly small for 1 H, 2 He, 3 Li, 4 Be, 11 Na, 12 Mg; then, they increase gradually for the series 19 K, 20 Ca, the first transition-metal atoms ( 21 Sc− 29 Cu), 30 Zn, 37 Rb, 38 Sr, the second transition-metal atoms ( 39 Y− 47 Ag), 48 Cd, 55 Cs, 56 Ba, the lanthanoid atoms ( 57 La− 71 Lu), the third transition-metal atoms ( 72 Hf− 79 Au), 80 Hg, 87 Fr, 88 Ra, and the actinoid atoms ( 89 Ac− 103 Lr). We, however, note that between the alkali-metal atoms and alkaline-earth metal atoms (between 19 K and 20 Ca, 37 Rb and 38 Sr, 55 Cs and 56 Ba, 87 Fr and 88 Ra), ratio(ε) is almost constant, indicating that shielding effects from the inner core of the alkali-metal atoms and alkaline-earth metal atoms are similar and the shielding effects from another ns electron are quite small. ...
Periodic trends in relativistic effects are investigated from 1H through 103Lr using Dirac–Hartree–Fock and nonrelativistic Hartree–Fock calculations. Except for 46Pd (4d¹⁰) (5s⁰), all atoms have as outermost shell the ns or n’p spinors/orbitals. We have compared the relativistic spinor energies with the corresponding nonrelativistic orbital energies. Apart from 24Cr (3d⁵) (4s¹), 41Nb (4d⁴) (5s¹), and 42Mo (4d⁵) (5s¹), the ns+ spinor energies are lower than the corresponding ns orbital energies for all atoms having ns spinor (ns+) as the outermost shell, as some preceding works suggested. This indicates that kinematical effects are larger than indirect relativistic effects (the shielding effects of the ionic core plus those due to electron–electron interactions among the valence electrons). For all atoms having np+ spinors as their outermost shell, in contrast, the np+ spinor energies are higher than the corresponding np orbital energies as again the preceding workers suggested. This implies that indirect relativistic effects are greater than kinematical effects. In the neutral light atoms, the np– spinor energies are close to the np+ spinor energies, but for the neutral heavy atoms, the np– spinor energies are considerably lower than the np+ spinor energies (similarly, the np– spinors are considerably tighter than the np+ spinors), indicating the importance of the direct relativistic effects in np–. In the valence nd and nf shells, the spinor energies are always higher than the corresponding orbital energies, except for 46Pd (4d¹⁰) (5s⁰). Correspondingly, the nd and nf spinors are more diffuse than the nd and nf orbitals, except for 46Pd.
