Article Cite This: J. Chem. Theory Comput. 2018, 14, 3025−3039 pubs.acs.org/JCTC Relativistic Spin−Orbit Heavy Atom on the Light Atom NMR Chemical Shifts: General Trends Across the Periodic Table Explained Jan Vícha,*,† Stanislav Komorovsky,∥ Michal Repisky,Δ Radek Marek,‡ and Michal Straka*,§ † Center of Polymer Systems, University Institute, Tomas Bata University in Zlín, Třída T. Bati, 5678, CZ-76001, Zlín, Czech Republic ∥ Institute of Inorganic Chemistry, Slovak Academy of Science, Dúbravská cesta 9, SK-84536 Bratislava, Slovakia Δ Center for Theoretical and Computational Chemistry, Department of Chemistry, UiT − The Arctic University of Norway, N-9037 Tromsø, Norway ‡ CEITEC - Central European Institute of Technology, Masaryk University, Kamenice 5/A4, CZ-62500 Brno, Czech Republic § Institute of Organic Chemistry and Biochemistry of the Czech Academy of Sciences, Flemingovo nám. 2, CZ-16610, Prague, Czech Republic * S Supporting Information ABSTRACT: The importance of relativistic effects on the NMR parameters in heavy-atom (HA) compounds, particularly the SO- HALA (Spin−Orbit Heavy Atom on the Light Atom) effect on NMR chemical shifts, has been known for about 40 years. Yet, a general correlation between the electronic structure and SO-HALA effect has been missing. By analyzing 1H NMR chemical shifts of the sixth-period hydrides (Cs−At), we discovered general electronic-structure principles and mechanisms that dictate the size and sign of the SO-HALA NMR chemical shifts. In brief, partially occupied HA valence shells induce relativistic shielding at the light atom (LA) nuclei, while empty HA valence shells induce relativistic deshielding. In particular, the LA nucleus is relativistically shielded in 5d2−5d8 and 6p4 HA hydrides and deshielded in 4f0, 5d0, 6s0, and 6p0 HA hydrides. This general and intuitive concept explains periodic trends in the 1H NMR chemical shifts along the sixth-period hydrides (Cs−At) studied in this work. We present substantial evidence that the introduced principles have a general validity across the periodic table and can be extended to nonhydride LAs. The decades-old question of why compounds with occupied frontier π molecular orbitals (MOs) cause SO-HALA shielding at the LA nuclei, while the frontier σ MOs cause deshielding is answered. We further derive connection between the SO-HALA NMR chemical shifts and Spin−Orbit- induced Electron Deformation Density (SO-EDD), a property that can be obtained easily from differential electron densities and can be represented graphically. SO-EDD provides an intuitive understanding of the SO-HALA effect in terms of the depletion/ concentration of the electron density at LA nuclei caused by spin−orbit coupling due to HA in the presence of a magnetic field. Using an analogy between the SO-EDD concept and arguments from classic NMR theory, the complex question of the SO- HALA NMR chemical shifts becomes easily understandable for a wide chemical audience. 1. INTRODUCTION standard experimental ranges and thus considerably hindering The importance of relativistic effects in understanding NMR the experimental detection of new HA compounds.12,13 An chemical shifts in heavy-element compounds was recognized example is the recent preparation of the first stable PbII over 40 years ago, when Nomura, Takeuchi, and Nagakawa hydride,13 inspired by predictions of relativistic 1H NMR proposed the hitherto unknown “spin-polarization shifts.”1 chemical shifts in PbII compounds as high as +90 ppm.11 These shifts were introduced to explain the enigmatic normal Due to its nonlocal character,14 δSO(LA) encodes informa- halogen dependence2 of the Light-Atom (LA) NMR chemical tion about the electronic structure of the heavy atom and its shifts in a series of organic compounds featuring different surroundings.3,15−20 Interest in this phenomenon is rapidly halogen atoms. Because the relativistically induced NMR shift growing as an increasing number of connections between at the LA originates mainly from Spin−Orbit (SO) Coupling δSO(LA) and the structural/electronic properties of the HA (SOC) at the Heavy Atom (HA),3−6 this effect is termed Spin− complexes are discovered. For instance, δSO(1H) reflects the Orbit Heavy Atom on the Light Atom (SO-HALA) effect,3,7 coordination environment of the HA center in SnII and PbII and induced shifts are called SO-HALA NMR chemical shifts,3,7 hydrides.11 The polar/covalent character of the HA-LA bond δSO(LA). The δSO(LA) value may reach several hundreds of parts per Received: February 9, 2018 million (ppm),8−11 pushing the LA chemical shifts beyond the Published: April 20, 2018 © 2018 American Chemical Society 3025 DOI: 10.1021/acs.jctc.8b00144 J. Chem. Theory Comput. 2018, 14, 3025−3039 Journal of Chemical Theory and Computation Article Figure 1. Molecular structures and 1H NMR chemical shifts for hydrides of sixth-period elements. The corresponding hydrogen atoms are highlighted in blue. Experimental, δexp(1H), calculated total, δcalc(1H), and calculated SO-HALA, δSO(1H), chemical shifts are shown. Details and references to the experimental studies are given in Table S1. Chemical shifts are reported with respect to TMS. can be judged from the size of δSO(LA),20,21 and this concept energies of the σ and π frontier orbitals, which lead to shielding can be further extended to addressing the stability and reactivity or deshielding effect at LA.14,35 A general electronic structure of the HA complexes.22,23 This approach is particularly origin of the positive/negative sign of the SO-HALA effect has interesting for d0-element catalysts of olefin metathesis,24 not been satisfactorily explained. where reactivity can be predicted from δ(LA).25,26 In this study, we provide a simple and general understanding The mechanism of the SO-HALA nuclear shielding involves of how the SO-HALA chemical shifts depend on the electronic the interaction of the SOC at HA with magnetic dipole at LA structure of HA molecules across the periodic table. The 1H nuclei via the orbital angular momentum operator.3,7 In the NMR chemical shifts in a series of hydrides of the sixth period presence of an external magnetic field, SOC produces nonzero (Cs−At) are calculated and analyzed. Experimentally known spin density, even in a closed-shell molecule. The induced spin compounds are used where possible. Periodic trends in the 1H density, which predominantly propagates to LA nuclei via the chemical shifts are observed in the sixth period hydrides, with Fermi-contact (FC) mechanism, contributes to magnetic the sign of δSO switching several times along the sixth period. interactions and the LA NMR chemical shift, δ(LA). The Using the recently introduced MO analysis of the δSO(LA) in SO/FC mechanism requires the involvement of the LA atomic the framework of third-order perturbation theory (PT3),14 we ns orbitals to bring the SO-induced spin-polarization into derive a generally valid connection between the sign of δSO(LA) contact with the LA nucleus.7,27 Hence, particularly strong SO- and the electronic structure (molecular orbital types) of the HA HALA effects are observed for the 1H NMR chemical shifts of compounds. We show that the suggested concepts are H nuclei bonded directly to HA.6,10,11,28 The δSO(LA) value is extendable across the periodic table by demonstrating examples further influenced by the SOC of the 5d, 6p, and 5f HA atomic of hydrides of fourth to seventh period elements. The same orbitals involved in the frontier MOs19−21 as well as by the principles are also applicable to non-hydrogen light atoms, as covalence/polarity of the HA-LA bond.14,18,20,21 shown on 13C NMR chemical shifts of sixth-period compounds The SO-HALA effect can be either shielding or deshielding, with HA-carbon bonds. depending on the character of frontier orbitals, as was for the Linking the above-mentioned orbital-based mechanisms with first time noted by Pyykö et al.3 and repeatedly confirmed by newly introduced graphical concepts of Spin−Orbit and others.15,19,20,27−31 The deshielding SO effect has been Magnetically Induced Spin Density (SOM-ISD) and Spin− associated with occupied σ-type HA-LA bonding MOs, while Orbit Electron Deformation Density (SO-EDD)14 results in a the π-type MOs lead to a shielding δSO(LA).3,15,27,32,33 For simple and general explanation of δSO(LA) using classical instance, Os, Ir, and Pt complexes with several occupied 5d π- concepts of NMR theory, valid across the periodic table. type MOs are associated with a negative (shielding) value of δSO(LA),15,19,33−35 while clearly deshielding δSO(LA) values are 2. RESULTS AND DISCUSSION reported in Hg complexes with σ-type frontier MOs derived 2.1. 1H NMR Shifts of Sixth-Period Hydrides (Cs−At). from Hg 6s and 6p AOs.14,20,33,35,36 A strong dependence of We first show how the SO-HALA effect evolves along the sixth δSO(LA) on the structural trans effect (i.e., on the substituent in period using hydrides of the sixth-period elements. The the trans position to the LA)19,34 correlates with the relative experimental δexp(1H) (where available), calculated δcalc(1H), 3026 DOI: 10.1021/acs.jctc.8b00144 J. Chem. Theory Comput. 2018, 14, 3025−3039 Journal of Chemical Theory and Computation Article Figure 2. Trends in 1H NMR chemical shifts of the hydrides of sixth-period elements. The δexp(1H) values are in black. δcalc(1H) values are in green, and δSO(1H) values are in magenta. Schematic representations of the valence-shell electronic configurations of the HAs in their corresponding oxidation states are shown. Occupied and vacant lone pair orbitals of HAs (LPHA and LPHA*) are shown in dark and light blue, respectively. For clarity, ligand field effects and orbital degeneracy are not considered. Chemical shifts are reported with respect to TMS. and δSO(1H) for the hydrides of CsI to AtI are shown in Figure A clear correlation between the electronic configuration at 1. Detailed information about the selected compounds and the HA and the δSO(1H) is seen in Figure 2. In hydrides with references to experimental studies are collected in Table S1. formally empty HA valence shells, a deshielding δSO(1H) is found. Distant bulky ligands in some experimental systems were in This case is observed in 6s05d0 (CsI and BaII), 4f05d0 (LaIII and calculations substituted by methyl groups and/or hydrogen CeIV), 4f145d0 (LuIII), 5d0 (HfIV and TaV), 5d106s0 (HgII and atoms to save computational resources. Such structural AuI), and 6s26p0 (TlI and PbII) hydrides. On the other hand, modifications have marginal effects on the calculated δcalc(1H) hydrides with partially f illed HA valence shells (i.e., those with and δSO(1H) values.28 some occupied LPHA MOs) exhibit shielding δSO(1H). This Calculated δcalc(1H) of model hydride compounds were used effect occurs in the 5d2 (WIV) through 5d8 (AuIII) and 5p2 for analysis where experimental NMR data were not available, (PoII) through 5p4 (AtI) hydrides. Maximum shielding in 5d i.e., for Cs, Ce, Tl, Pb, Po, and At. This substitution is possible transition-metal complexes is observed in the 5d6 IrIII system due to the excellent accuracy of the Spin−Orbit Zeroth-Order with 3xLPIr. In 6p elements, the shielding scales with the Regular Approximation (SO-ZORA) computational ap- number of LPHA MOs reaching a maximum at the 6p4 AtI proach, 11,37 as seen from a comparison between the hydride with 2xLPAt. A similar correlation between the number experimental and calculated δ(1H) values in Figures 1 and 2. of LPHA and δSO(1H) with respect to the changes in the Note that δSO is obtained as a difference between the SO- oxidation state of HA was reported for iodine compounds by ZORA (with SOC) and ZORA (without SOC) calculations; Kaupp et al.32 details are given in the Methods section. Though the changes of the sign of δSO(LA) have been often The trends in δexp(1H), δcalc(1H), and δSO(1H) in the sixth- discussed in the literature, particularly along the 5d Pt−Au−Hg period hydrides together with schematic valence electronic series,14,19,20,35 the periodic trends of δSO(LA) in closed-shell configurations at HAs are shown in Figure 2. Regarding the complexes along the whole sixth period (except Rn) are electronic structure at HA, we will refer to electron lone pair reported for the first time here. We will now focus on orbitals of HA (orbitals composed entirely of HA atomic orbitals that do not contribute to bonding) as LPHA. For clearer explanation of the “oscillating” behavior of δSO(1H) in Figure 2. discussion and analysis, we will use an analogical description of 2.2. The Electronic Structure Origin of the Sign of empty atomic orbitals located at HAs that are not part of any δSO(1H). Comprehending the connection between the δSO(1H) antibonding MOs, referring to them as virtual or vacant lone values and electronic structure of the HA in these systems is pair orbitals, LPHA*. crucial to understand the periodic trends in Figure 2. This In Figure 2, the trend in the δcalc(1H) values is strongly connection can be explained using the recently developed third- dictated by the δSO(1H), which accounts for a major part of the order perturbation theory-based approach (PT3) in which the total δcalc(1H) in most of the compounds. The somewhat larger SO-induced nuclear shielding constant, σSO (δSO = −σSO) is differences between the δcalc(1H) and δSO(1H) in 5dn (n = 2−8) obtained in a perturbative manner using nonrelativistic complexes are related to the nonrelativistic paramagnetic MOs.2,3,14,40,41 The MO analysis in the framework of the shielding induced at the H atoms by magnetic currents from PT3 approach enables us to see how individual MOs contribute the unfilled 5d shell at the central HA, known as the to the total σSO and hence how electronic structure factors Buckingham-Stephens effect.15,28,38,39 Note also the significant influence the σSO. We limit using the PT3 method to the Fermi- δSO(1H) in the 6s0 BaII hydride, which is similar in size to that contact mechanism of SO-induced NMR shielding (σSO/FC) of 5d0 complexes. This is due to involvement of 5d Ba orbitals which covers typically 95% of the total perturbative σSO in bonding, see section 2.5. Another point to note in Figure 2 is (negative δSO).5,6 We note that PT3 MO analysis provides the large SO-HALA deshielding predicted for the CeIV hydride qualitative results needed for understanding of the δSO(LA). also discussed in section 2.5. Due to introduced approximations, the results are somewhat 3027 DOI: 10.1021/acs.jctc.8b00144 J. Chem. Theory Comput. 2018, 14, 3025−3039 Journal of Chemical Theory and Computation Article Figure 3. Schematic triangle representation14 of SO/FCΔ and SO/FC∇ MO couplings in eqs 1 and 2 with respect to different orbital types (σ, dark purple; σ*, light purple; LP, dark blue; LP*, light blue). Schematic electronic configuration of the HA is shown on the left. Signs of the individual coupling contributions to σSO/FC(1H) are color-coded according to their naturered for deshielding and blue for shielding. Couplings that do not contribute in a given electronic configuration of the HA are crossed out. less accurate than, e.g., SO-ZORA used throughout this work; give rise to positive (shielding) or negative (deshielding) for details, see below. contributions to σSO/FC(LA), depending on the MOs involved. The PT3 expressions for σSO/FC in eqs 1 and 2 show that the We now proceed to the analysis of the sign of σSO/FC. SO/FC mechanism couples the occupied (φi,φj) and virtual Because of the energy denominators in eqs 1 and 2, the crucial (φa,φb) MOs through three operatorsthe SOC operator, MO couplings that contribute to the total σSO/FC(LA) arise r−3 ̂ HAl , the FC operator, δ , and the angular momentum HA LA from the frontier MOs (small denominators). We shall consider operator, l ̂ . The one-electron MO energies, εi, εj, εa, and εb, HA four basic frontier orbital types: bonding σHA‑LA, antibonding enter in the energy denominators. All quantities are in atomic σ*HA‑LA, LPHA, and LPHA* MOs; in a briefer notation, σ, σ*, LP, units (au). and LP*, respectively. At this moment, we do not include the Δ frontier occupied/virtual MOs corresponding to the bonding of σSO/FC (LA) ≈ HAs with other ligands, L (HA-L/HA-L*). Their role is HA −3 ̂ HA clarified below. 1 occ occ vac ⟨φa|rHA l |φi⟩⟨φi|δ LA|φj⟩⟨φj|l ̂ |φa⟩ The two limiting cases of electronic configuration of HA − 4 ∑∑∑ together with MO couplings between the considered frontier c i=1 j=1 a=1 (εi − εa)(εj − εa) MO types are visualized in Figure 3. Figure 3a depicts the + permutations (1) situation in the absence of occupied LPHA MOs, while Figure 3b illustrates the situation with all LPHA occupied, i.e., in the ∇ absence of LPHA* MOs. σSO/FC (LA) ≈ Eqs 1 and 2 give rise to six different PT3 MO coupling HA HA 1 occ vac vac −3 ̂ ⟨φi|rHA l |φa⟩⟨φa|δ LA|φb⟩⟨φb|l ̂ |φi⟩ contributions to σSO/FC(LA) (three for each SO/FC type). + ∑∑∑ These couplings are visualized by triangles with corresponding c4 i=1 a=1 b=1 (εi − εa)(εi − εb) MOs and operators, Figure 3. The sign of each MO coupling + permutations (2) contribution (represented by a triangle) to σSO/FC(1H) can be deduced as follows. Because the final values of the integrals in Eqs 1 and 2 show two distinct PT3 contributions to σSO/FC with eqs 1 and 2 are phase (sign) invariant, we can fix the sign of the opposite signs.14 We refer to them as SO/FCΔ (eq 1) and SO/ heavy-atom AO contributing to the bonding (σ) and FC∇ (eq 2) to retain compatibility with our previous work.14 antibonding (σ*) HA-LA MOs. This choice has two They differ in that they describe the coupling of either two consequences. First, the hydrogen 1s AO will contribute with occupied MOs with one vacant MO (SO/FCΔ) or two vacant opposite signs to the σ HA‑LA and σ*HA‑LA MOs. The MOs with one occupied MO (SO/FC∇). The particular corresponding FC integral will thus be positive only if the combination is expressed by the orientation of the triangle in two σ orbitals are of the same type (either bonding or the superscript. Note that the two occupied (vacant) MOs antibonding), i.e., ⟨σ(*)|δLA|σ(*)⟩ > 0. Similarly, it will be involved in SO/FCΔ (SO/FC∇) coupling correspond to negative if their types differ, i.e., ⟨σ|δLA|σ*⟩ < 0. Second, if we identical MOs when i = j (a = b). Both contributions generally neglect small interatomic contributions, all four integrals 3028 DOI: 10.1021/acs.jctc.8b00144 J. Chem. Theory Comput. 2018, 14, 3025−3039 Journal of Chemical Theory and Computation Article Figure 4. Visualization of the PT3 MO couplings, δSO/FC, in (a) TlIH and (b) AtIH. The arrows represent the following operators: The SO coupling (r−3 ̂ HA LA ̂ HA HAl ), the Fermi contact (δ ), and the angular momentum (l ) operators. The schematic frontier MO energy diagrams are given on the left. The contributions to δSO/FC(1H) from each coupling are given inside the triangles. ̂ |σ(*)⟩ and ⟨X|r−3 ⟨X|lHA ̂ ( ) HAl |σ * ⟩, where X is either LP or LP*, HA In a general system, the δSO(LA) values will result from a will have the same sign. Note that in the final account, the signs combination of these two coupling types, with their overall of the scalar product is irrelevant, since the integrals are coming value determined by the energy separation between frontier in pairs, and the corresponding contributions from SOC and MOs and the magnitude of the integrals. Good examples of angular momentum integrals are always positive. Finally, the such combination are 5d8 and 5d10 systems, where the FC operator, δLA (do not mix with chemical shift δ(LA)) properties of the ligand trans to LA can modulate the sign of couples only σ-type orbitals (LP(*) does not have contribution δSO(LA),14,35 see below. from the s orbital on the LA), and in contrast, the SO operator, The metal−ligand MOs, HA-L and HA-L*, are responsible r−3 ̂ HAl , does not couple σ-type orbitals with each other. HA for bonding between HAs and ligands other than the LA but Combining this reasoning and the prefactor sign of eqs 1 and 2, were omitted for clarity in the derivation of the sign of each respective contribution to σSO/FC(1H) for the sign of each σSO/FC(LA) above. In fact, these MOs are composed of the combination of σHA‑LA and σ*HA‑LA and of LPHA and LPHA* in same HA AOs that form LPHA and LPHA* MOs in the absence eqs 1 and 2 can be explicitly determined. of other ligands. Hence, they must contribute to the The results are illustrated graphically using the orbital σSO/FC(LA) in a similar manner as the LPHA and LPHA* coupling triangles in Figure 3. The system in Figure 3a has no MOs. However, their influence is smaller because HA-L/HA- occupied LPHA MOs, so the three couplings involving LPHA do L* MOs have less HA character than the corresponding pure not contribute, as indicated by the red crosses. The three LPHA/LPHA* MOs due to admixture with other ligand AOs. remaining couplings in Figure 3a, which involve LPHA*, are all Also, the bonding HA-L MOs are usually lower in energy than the corresponding LPHA orbitals, which leads to larger energy deshielding. Because the occupied σHA‑LA and the vacant denominators in eqs 1 and 2, i.e., smaller contributions from σ*HA‑LA MOs are coupled with the vacant LP* MO in different particular couplings. This effect is nicely illustrated for the pathways, this coupling can be described as σ(*)HA‑LA ↔ LPHA* example of BiIII hydride given below. coupling. To maintain the compatibility with previous works, Note that, due to the structural trans ef fect (influence of the which were based mostly on PT2 analysis,14,20,35 we shall refer ligand in trans position to LA on the bond between HA and to the couplings involving LPHA* as σHA‑LA ↔ LPHA* or SO- LA), the ligand that is trans to LA plays a special role for HALA deshielding, or as the SO+ mechanism. The best δSO(LA).14,19,34,35 It strongly affects the composition and conditions for the SO+ deshielding mechanism are found in rrelative energy of frontier MOs involved in couplings (Figure complexes with an empty HA valence shell, e.g., in the 5d0 and 3) as reported recently by Greif et al.35 and Novotný at al.14 6p0 hydrides (σHA‑LA near the HOMO, several LPHA* MOs This affects the SO+ deshielding mechanism, Figure 3a. A available). Clearly, the hydrogen nuclei in these compounds are strong trans ligand will raise the energy of σHA‑LA, thus deshielded (Figure 2). enhancing its coupling with LPHA* or HA-L* and increasing Figure 3b depicts the situation where the frontier LPHA MOs the contribution of the SO-HALA deshielding mechanism are all filled and LPHA* MOs are not available. The deshielding (Figure 3a). The sign of δSO(LA) can even revert in 5d8 and LPHA*-based combinations then vanish, and all the remaining 5d10 complexes, changing from a shielding to a deshielding MO couplings are shielding. These combinations all include value at LA, if sufficiently strong trans ligands are used.14,20,35 coupling between occupied LPHA and σ(*)HA‑LA MOs. We shall The data here also reveal a connection to the long-known refer to the shielding couplings involving LPHA as LPHA ↔ observation that frontier occupied σ MOs give rise to σ*HA‑LA or SO-HALA shielding, or as the SO− mechanism. The deshielding values of δSO(LA), while occupied π MOs give SO− mechanism is the main reason why shielding δSO(1H) rise to shielding values of δSO(LA)8,15,19,32,42,3,27 (see the values are observed in 5d2−5d8 and 5p2 and 5p4 hydrides in Introduction). The SO-active π/π* MOs are usually those with Figure 2. a large admixture of HA AO/AO* character, similar to LPHA. 3029 DOI: 10.1021/acs.jctc.8b00144 J. Chem. Theory Comput. 2018, 14, 3025−3039 Journal of Chemical Theory and Computation Article When these MOs are populated, the SO-HALA shielding mechanism occurs. When these MOs are empty, the σHA‑LA is usually HOMO, and the SO-HALA deshielding mechanism dominates. The validity of these described concepts will be further demonstrated on 6p hydrides. 2.3. The δSO(1H) Values along the Series of 6p Hydrides from TlH to AtH. The 1H NMR chemical shifts in the 6p hydride complexes (Figure 2) are clearly dominated by the δSO(1H) term. The enormous deshielding demonstrated by δSO(1H) = +130 ppm11 in TlI−H steadily decreases toward ∼1 ppm in the BiIII−H complex and becomes a shielding, with δSO(1H) = −23 ppm in AtI−H. Note that we switched back to the chemical shift (δ) scale (instead of nuclear magnetic shielding, σ) as it is more natural for experimentalists. Figure 5. Selected SO-ZORA orbital magnetic couplings for δSO(1H) In the following, the δSO/FC(1H) is used to describe total in (a) TlH and (b) AtH. Scalar relativistic ZORA (without SO), on the PT3-calculated values, and δSO/FC is used for individual orbital left, gives purely Ramsey-type coupling (l,̂ paramagnetic spin orbit coupling contributions. The valence MOs and dominant (PSO) term). The leading-order spin−orbit contribution to NMR coupling contributions δSO/FC to the total PT3 δSO/FC(1H) in shifts in the two-component SO-ZORA framework originates from the TlH and AtH are shown in Figure 4. The valence space of TlH interplay of angular momentum, l,̂ and the Fermi-contact (δLAσ̂) operator. consists of σTl−H, 2xLP*Tl, and σ*Tl−H MOs, and that of AtH consists of σAt−H, 2xLPAt, and σ*At−H MOs. The metal 6s orbital is not considered as it does not directly contribute to the SOC Figure 5 are compared. For instance, in TlH (Figure 5a), the SOC mixes the H 1s character from σTl−H and σ*Tl−H bonds effects due to its almost spherical symmetry.20 (MO41 and MO44) into LP*Tl orbitals (MO42* and MO43*) A key difference between TlH and AtH is in the 6p orbitals and thus “activates” this coupling to propagate the spin density MO42 and MO43, Figure 4 on the left. They are vacant in TlH to the LA via the FC operator (δLAσ̂).14,35 The corresponding and populated in AtH. The coupling of vacant LP*Tl MOs with coupling gives δpara(1H) = +5 ppm at the scalar relativistic σTl−H in TlH (Figure 4a) induces SO-HALA deshielding. The ZORA level and δpara (1H) = +73 ppm at the SO-ZORA level, total PT3 δSO/FC(1H) is +58 ppm, while the leading δSO/FC resulting in a δSO of +68 ppm for this particular coupling contributions sum up to ΣδSO/FC = +48 ppm (Figure 4a). In (Figure 5a). In AtH (Figure 5b), SOC mixes LPAt (MO42) AtH, the occupied LPAt MOs and their coupling with σ*At−H with the H 1s orbital from σTl−H and σ*Tl−H (MO41 and induces SO-HALA shielding (Figure 4b), with a total PT3 MO44*), producing approximately 4 ppm greater shielding at δSO/FC(1H) of −29 ppm (ΣδSO/FC = −25 ppm in Figure 4b). the SO-ZORA level than the ZORA level, i.e., the δSO = −4 The results in Figure 4 are well in line with general ppm for this particular coupling (Figure 5b). hypothesis derived in section 2.2. However, an observant reader The SO-ZORA MO analysis of all investigated sixth-period may note that the PT3-calculated δSO(1H) in TlH of 58 ppm is hydrides is given in Figures S1−S22 and S27 in the Supporting approximately half of the SO-ZORA value (130 ppm) from Information. A detailed comparison of PT3 and SO-ZORA Figure 2. This difference is due to the use of different analyses of TlH, BiH, and AtH complexes is shown in Figure approximations and less accurate computational levels that are S27. required for transparent MO PT3 analysis (see Methods BiIII(CH3)2H in Figure 6 has both occupied and virtual Bi Section). Although the approximations used in PT3 enable us 6p-based MOs involved in the metal−ligand HA-L and HA-L* to understand the sign of the δSO(1H) from the first-principles, MOs. Both SO-HALA shielding and SO-HALA deshielding which is unachievable at the SO-ZORA level, SO-ZORA occur hereall six types of SO/FC PT3 orbital couplings are calculations can provide more accurate results. From a theoretical point of view, the ground-state SO-ZORA MOs are already SO-coupled, and δSO is thus described by the Ramsey-type second-order PT (PT2) couplings. Instead of rather complicated SO-ZORA operators, the two-component FC δLAσ̂, where σ̂ corresponds to Pauli matrices, and angular momentum (lHA ̂ ) operators are responsible for the leading- order contribution to δSO at the PT2 level. To obtain PT3 expressions like in eqs 1 and 2, one could start from PT2 expressions and insert the SO changes to MOs while discarding all but the first-order terms in the SOC. The SO-ZORA approach and SO-ZORA MO analysis (Figure 5) produce results that are more quantitative and provide a link with previous studies of the SO-HALA effect, which were typically based on MO analysis at the SO-ZORA relativistic level.18−20,35,42,43 The two-orbital σHA‑LA ↔ LPHA* Figure 6. Visualization of the PT3 orbital couplings in BiIII(CH3)2H. deshielding (TlH) and LPHA ↔ σ*HA‑LA shielding (AtH) The arrows represent the following operators: The SO coupling mechanisms are clearly seen in Figure 5. The SOC, which is (r−3 ̂ HA LA HAl ), the Fermi-contact (δ ), and the angular momentum (l ) ̂ HA included in the wave function at the SO-ZORA level, manifests operators. The schematic energy diagram of the MOs is shown on the itself by mixing the σ and LPHA-type orbitals, which is clearly left. The contributions to δSO from each coupling are given inside the seen when the ZORA (left) and SO-ZORA orbitals (right) in triangles. 3030 DOI: 10.1021/acs.jctc.8b00144 J. Chem. Theory Comput. 2018, 14, 3025−3039 Journal of Chemical Theory and Computation Article active (Figure 6). The contributions of the shielding and the deshielding mechanisms nearly cancel each other, which explains the smaller value of total δSO(LA) in BiIII compounds than in other 6p0 complexes.10 The absolute values of particular coupling contributions are also notably smaller than those in the isoelectronic Tl hydride because of the larger energy differences between occupied and vacant MOs, i.e., larger denominators in eqs 1 and 2, see above. In addition, the Bi AO character of the HA-L/HA-L* MOs is considerably smaller than in the LPAt or LP*Tl MOs, which are nearly 100% 6p- or 6p*-based in AtH or TlH (Table S2). This leads overall to smaller individual coupling contributions.10,11 PT3 MO analysis gives a δSO(1H) of −4 ppm in BiIII hydride, while the value of +1 ppm was obtained using SO-ZORA (Figure 1). This Figure 7. Visualization of the PT3 orbital couplings in (a) 5d0 HfIV difference is due to the use of different computational levels, see and (b) 5d6 IrIII. The arrows represent the following operators: The the Methods section. A detailed SO-ZORA analysis of the BiIII SO coupling (r−3 ̂ HA LA HAl ), the Fermi-contact operator (δ ), and the hydride is shown in Figure S27. applied external magnetic field is represented by the angular In summary, the overall trend in hydride chemical shifts in 6p momentum operator (lHA ̂ ). The contributions to δSO/FC from each hydrides (Figure 2) can be rationalized by an interplay of the coupling are given inside the triangles. predicted SO+ and SO− mechanisms (Figures 3−6). The importance of the SO+ mechanism grows with an increasing number of LP HA * MOs. The importance of the SO − couplings contained in the σHA‑LA ↔ LPHA* mechanism are mechanism then rises with an increasing number of LPHA actually forbidden; i.e., the corresponding matrix elements in MOs (Figure 2). The trend is actually semiquantitative at the eqs 1 and 2 vanish.43,44 For example, the product of magnetic SO-ZORA level. The δSO(1H) value decreases with the coupling between the iridium 5dz2 AO (mixed in the σIr−H MO) decreasing number of virtual LPHA* MOs from TlI to BiIII at and the Ir 5dx2−y2 AO (vacant LP*Ir MO) is identically zero.43,44 a rate of ∼60 ppm per LPHA* MO (130 ppm/2 LPHA*, 62 Because all π-type 5d orbitals of Ir are occupied in 5d6 square- ppm/1 LPHA*, and 1 ppm/0 LPHA*; Figure 2 and Table S2). pyramidal complexes, the small calculated deshielding con- The trend continues with the increasing number of occupied tributions originate mostly from couplings between the Ir 5dz2 LPHA MOs from BiIII to AtI (increasing importance of the SO− component of the σIr−H orbital and the upper-lying MO*s with mechanism) by approximately 11 ppm per LPHA MO (1 ppm/0 Ir 5dxz and 5dyz admixture. LPHA, −11 ppm/1 LPHA, and −23 ppm/2 LPHA). The fact that maximum shielding is reached for the 5d6 HA 2.4. The δSO(1H) Values along the Series of 5d electronic configuration is caused by a combination of several Hydrides (Hf−Hg). Examples of HfIV and IrIII Hydrides. factors. First, octahedral and square-pyramidal ligand fields, The δSO(1H) in 5d hydrides shows distinct trends in Figure 2. A common in 5d4−5d8 complexes, lead to a preference for LPHA deshielding (positive) δSO(1H) is observed in 5d0 HfIV and TaV ↔ σ*HA‑LA couplings because the π-type metal 5d orbitals are hydrides, whereas the presence of even a single occupied LPHA populated first due to stabilizing crystal-field effects.19 Second, MO in the 5d2 WII hydride system inverts its sign. The δSO(1H) the stabilization of the occupied 5d LPHA MOs due to the value then decreases nearly linearly with the number of increasing nuclear charge along the IrIII−HgII series increases occupied LPHA orbitals from 5d2 to 5d6 hydrides, where it the HA 5d-5d* orbital energy differences, which reduces the reaches a minimum (maximum shielding) in the IrIII complex. coupling efficiency (increasing the energy denominators in eqs The δSO(1H) value then increases (shielding decreases) as the 1 and 2). Third, the involvement of the HA 6p orbitals in the series progresses toward 5d10 AuI and HgII complexes. σHA‑LA bond, which leads to the deshielding 6p ↔ 6p* SO+ In Figure 7, we compare two 5d hydrides, a 5d0 HfIV hydride mechanism, see above, begins to play a role in 5d8 complexes with δSO(1H) = +8 ppm and a 5d6 IrIII hydride with δSO(1H) = and is particularly important in 5d10 complexes.14,20,35 −29.4 ppm. Their structures can be found in Figure 1. Note The importance of the SO-HALA deshielding mechanism that these two compounds do not have trans substituents. The thus generally decreases from 5d2 to 5d8 complexes due to the 5d0 HfIV hydride has no occupied LPHA, so its δSO(1H) is decreasing number of available LPHA* MOs, the stabilization of dictated mainly by the σHA‑LA ↔ LPHA* deshielding (Figure the σHA‑LA MO energy,14,35 and the vanishing symmetry- 7a). Though the shielding HA-L ↔ σ*HA‑LA couplings are forbidden 5d ↔ 5d* σHA‑LA↔LPHA* couplings (see above). possible in the HfIV hydride due to an admixture of π-type Hf The population of even a single LPHA MO in a 5d2 WIV 5d orbitals in the HA-L bonding MO, their contributions to complex is thus sufficient to establish the predominance of the δSO(1H) are negligible owing to the large energy denominators LPHA↔ σ*HA‑LA shielding mechanism, leading to a negative (see above). δSO(1H) value although small nonvanishing σHA‑LA↔LPHA* In the 5d6 IrIII hydride, the presence of the three LPHA MOs deshielding couplings still contribute to δSO(1H). For details, leads to strong LPHA ↔ σ*HA‑LA shielding at the hydrogen see the graphical PT3 MO analysis of the 5d2 WIV complex in nuclei (Figure 7b). Interestingly, deshielding contributions are Figure S28. still present as the complex formally has one LPHA* and several Note that the sign change of the δSO(1H) and δ(1H) that HA-L* MOs. However, they are negligible (<1 ppm in total) as occurs between 5d0 and 5d2 transition-metal complexes could the corresponding σHA‑LA MO needed for the σHA‑LA ↔ LPHA* be a viable tool for determining the metal oxidation state during deshielding coupling is a low-lying MO,35 which leads to large various reactions and catalytic processes, where heavy atoms energy denominators in eqs 1 and 2, making the contribution often experience changes in oxidation states, e.g., from WIV to from this MO small. Furthermore, certain 5d ↔ 5d* AO WVI. The presence of distinct values of δ(1H) are advantageous 3031 DOI: 10.1021/acs.jctc.8b00144 J. Chem. Theory Comput. 2018, 14, 3025−3039 Journal of Chemical Theory and Computation Article because these signals should be observable without difficulty in filled 4f shell restricts the NMR investigations of diamagnetic complex NMR spectra. complexes of 4f elements to 4f0 (LaIII and CeIV) and 4f14 (LuIII) 2.5. Examples of 6s and 4f Hydrides and General- species. Hydride complexes of all three metals have been ization of SO-HALA Mechanism Across the Periodic experimentally studied,49,50 though the δexp(1H) value for the Table. The examples discussed in sections 2.3 and 2.4 have CeIV complex has not been reported.51 The LaIII complex provided solid evidence that the δSO(1H) value is dictated by studied here is actually a μ-H-bonded dimer,50,52 while our the two simple orbital coupling mechanisms. The full analysis analysis was performed on its monomeric unit. of δSO(1H) for the whole series of sixth-period hydrides The experimental and calculated δ(1H) values for LaIII and (Figures 1 and 2) at the SO-ZORA relativistic level is given in LuIII hydrides are approximately +10 ppm, analogous to those Tables S1 and S2 and Figures S1−S22 and S27. The results in the 5d0 hydrides of HfIV and TaV (Figure 2). Detailed MO confirm that ideas outlined above are valid generally along the analysis of LaIII and LuIII hydrides reveals that the SO+ sixth period. The deshielding σHA‑LA↔ LPHA* (HA-L*) mechanism is dominant in these compounds (Table S1 and couplings of the SO+ mechanism are confirmed in 6s0, 6p0, Figures S3 and S5). 5d0, and 4f0 HA hydrides, while the shielding σ*HA‑LA ↔ LPHA The CeIV hydride is remarkably different. Its predicted (HA-L) couplings of the SO− mechanism are dominant in δ ( H) and δSO(1H) values of +31 and of +21.1 ppm are calc 1 5d2−5d8, 6p2, and 6p4 complexes. In the following, we unique among the diamagnetic 4f-element hydrides and beyond demonstrate these general principles on further examples. the standard 1H NMR range (<20 ppm). We are presently Because individual spinors can have rather complex and trying to repeat the experimental synthesis51 and obtain an nonintuitive shapes (compare ZORA and SO-ZORA MOs in NMR spectrum of this compound to confirm our theoretical Figure 5), MO couplings are illustrated using scalar ZORA prediction. The CeIV hydride differs from its analogues because MOs, as is customary.10,16,19,20,42 For clarity, we only show of the availability of the Ce 4f AOs in chemical bonding. The examples of MO−MO* couplings, where δSO(OMC) (OMC = σCe−H bonding MO has 6% of the metal 4f AO character, unlike orbital magnetic coupling) stands for contribution from the σLa−H and σLu−H bonding MOs (<1%). This contribution coupling between two particular MOs. allows for 4f ↔ 4f* couplings in the σHA‑LA ↔ LPHA* 6s0 CsI and BaII Hydrides. Despite the fact that the HA 6s deshielding mechanism, which considerably increases the size orbital itself does not contribute to the SOC, a deshielding of δSO(1H) in CeIV hydride, see Figure 9 for details. The value of δSO(1H) is obtained for CsH and BaH2 (Figure 2 and Table S1). The primary reason for deshielding δSO(1H) is mixing of HA 5d AOs into the bonding σHA‑LA MOs (which is also reflected in bent rather than linear structure of BaH2),45,46 see Figure 8. The availability of the low-lying 5d LPHA* MOs Figure 8. Example of the σHA‑LA ↔ LPHA* deshielding coupling in 6s0 BaIIH2. (MO35* of BaH2 in Figure 8) then allows for 5d ↔ 5d* σHA‑LA↔ LPHA* deshielding. The example in Figure 8 shows Figure 9. Example of σHA‑LA ↔ LPHA* deshielding coupling between coupling between σBa−H (MO29) with Ba 5dxz character and σCe−H (MO138) and LP*Ce (MO156*), both with Ce 5d and Ce 4f LPHA* (MO35*) with Ba 5dyz character. The deshielding value AO contributions. Due to the complex composition of the presented of δSO(1H) thus provides evidence that Ba employs 5d MOs in MOs, the couplings of particular Ce 4f and 5d AOs, based on a bonding and behaves partially an early transition metal.45,46 detailed MO composition analysis (Table S2), are schematically The total δSO(1H) value of 3.8 ppm found in BaH2 is not illustrated at the bottom. negligible, particularly when compared with the δexp(1H) value of the hydrides of other alkaline-earth metals. The difference differences between La and Ce complexes are analogous to between the experimental δexp(1H) values of CaH2 (4.5 ppm) those between ThIV and UVI species, where ThIV behaves like a and BaH2 (8.7 ppm)47 is 4.2 ppm, which is very similar to the 6d0 element (5f orbitals are too diffuse to play an important calculated ΔδSO(1H) value of 3.8 ppm. The presence of σHA‑LA role), while in UVI (and PaV) complexes, the 5f0 LP* MOs are ↔ LPHA* SO-HALA deshielding thus naturally explains the responsible for a large deshielding δ(1H), see ref 8 and below. experimental trends in 1H NMR chemical shifts in the alkaline- The SO-HALA Mechanism Across the Periodic Table. The earth group hydrides (Ca, Sr, and Ba).47,48 The SO-HALA investigation of heavy-element hydrides along the sixth-period effect also explains the reported deshielding of 1H signals in above indicates that the two MO coupling mechanisms hydride-doped strontium mayenite in comparison with contributing to δSO(1H), i.e., the σHA‑LA ↔ LPHA* deshielding corresponding resonances in its calcium analog. Such differ- and LPHA ↔ σ*HA‑LA shielding, apply generally for 6s, 6p, 5d, ences could not be reproduced by scalar relativistic and 4f hydrides. The rules should thus apply generally across calculations.48 the periodic table, e.g., for 4d or 5f HA hydrides. One has to Diamagnetic 4f0 and 4f14 Hydrides. The open-shell keep in mind, however, that SOC is correspondingly smaller in electronic structure of lanthanide compounds with a partially lighter HA, as it formally scales with ∼ Z2.5 HA (ref 41) where Z is 3032 DOI: 10.1021/acs.jctc.8b00144 J. Chem. Theory Comput. 2018, 14, 3025−3039 Journal of Chemical Theory and Computation Article Figure 10. Structures of selected compounds with δexp(1H), δcalc(1H), and δSO(1H) for highlighted (in blue) hydride ligands. Because of enforced closed-shell configuration of the UIV 5f2 system, only the δSO(1H) is given to illustrate the SO-HALA mechanism. the atomic number of HA. We note, though, that SO-HALA as the reduced importance of the SO+ deshielding mechanism induced chemical shifts do not always follow this trend.53−55 (energy denominators and symmetry restrictions),43,44 similar For instance, the δSO(1H) may be occasionally larger in the 3d to those discussed for transition-metal hydrides in section 2.4. metal complexes than in their 4d analogues due to lower energy Beyond the Hydrides. Hydrides represent ideal model denominators, which partly counteract SO matrix elements, as compounds for studying δSO(LA) because of the simple well as due to the crystal field effects, which reflect the bonding electronic structure of the hydrogen atom. Nevertheless, the overlap between metal and ligand orbitals.55,56 trends and ideas presented above for hydrides can be in fact Results for the fourth to seventh period hydride complexes transferred to heavier LAs, such as 13C. This fact is that can be found in the literature comply with the two SO- demonstrated on a series of sixth-period ethyl compounds, HALA mechanisms described above.25,28,31,55,56 To further listed and analyzed in Table S4. The trends in their calculated illustrate the general validity of the concepts introduced above, δSO(13C) values (Figure 12) are rather similar to those of the we put forward a few more examples in Figure 10. hydrides in Figure 2, except for much larger absolute values of An example of a fourth period hydride is a 3d8 CoI hydride δSO(13C). (Figure 10).57 Approximately 1/3 of its total δ(1H) = −18.7 ppm is due to the SO-HALA effect, while the remainder is caused by a strong Buckingham−Stephens effect.28 As expected, the major orbital couplings contributing to δSO(1H) are the 3d ↔ 3d* LPCo ↔ σCo−H* shielding couplings (Figure S23 and Table S3). The recent observation of large SO-HALA effects in SnII complexes10,11 provides an example of a fifth-period SnII hydride58 with calculated δSO(1H) of +13 ppm (Figure 10) caused predominantly by 5p ↔ 5p* σSn−H ↔ LPSn* deshielding couplings (Figure S24 and Table S3). The seventh period 5f0 UVI complexes are particularly interesting due to the enormous deshielding δSO(1H) values recently predicted by Greif et al.8,9,59 This deshielding is due to 5f σU−H ↔ 5f LPU* couplings of the SO+ mechanism as illustrated in Figure 11a. What occurs if the empty 5f0 shell Figure 12. Trends in δcalc(13C) and δSO(13C) values of an ethyl carbon LA bound to a HA in model sixth period ethyl complexes. The SO-ZORA MO couplings were analyzed for model complexes of ethylthallium and ethylastatine (Table S4). The MO couplings in Figure 13 parallel those discussed in detail for Figure 11. Examples of SO-ZORA orbital couplings in UVI (a) and UIV (b) species. becomes populated? No diamagnetic closed-shell actinide complexes beyond 5f0 are known so far (recently prepared ThII compounds are 6d2 systems60,61), but such compounds may be prepared in the future. In a computational experiment, we enforced a closed-shell 5f2 electronic structure on existing uranium 5f2 open-shell triplet hydride (Figure 10) to estimate the effect of the LPHA ↔ σ*HA‑LA shielding mechanism on δSO(1H) in 5f-element compounds. In analogy to 5d0 vs 5d2 systems (Figure 2), the occupation of a single 5f LPU MO inverts the sign of δSO(1H). δSO(1H) decreases by more than 100 ppm, when going from the 5f0 UVI (+65 ppm) to 5f2 UIV (−37 ppm) complex. This difference is due to the 5f LPU ↔ 5f Figure 13. Selected MO plots and SO-ZORA couplings for (a) σU−H* shielding couplings (see example in Figure 11b) as well ethylthallium and (b) ethylastatine. 3033 DOI: 10.1021/acs.jctc.8b00144 J. Chem. Theory Comput. 2018, 14, 3025−3039 Journal of Chemical Theory and Computation Article TlH and AtH (Figures 4 and 5). A σTl−C ↔ LPTl* deshielding coupling in ethylthallium contributes +102 ppm to the total δSO(13C) = 218.3 ppm, while an LPAt ↔ σAt−C* shielding coupling is responsible for a −18 ppm contribution to the total calculated δSO(13C) = −44.1 ppm in ethylastatine. We note that HA-LA π-bonding can also occur in nonhydride ligands. It is known that π-bonding MOs cause SO shielding effects at LA (see Introduction).14 We assume that the contribution of π-bonding MOs shared by HAs and LAs are qualitatively the same as those from LPHA as the π- bonds between HA and LA utilize the same HA AOs as LPHA MOs. Thus, the mechanism should be somewhat similar to that of LPHA ↔ σ*HA‑LA shielding, as shown in Figure 13b. A Figure 14. Comparison of SOM-ISD and SO-EDD calculated at two confirmation of this hypothesis requires a more detailed levels of theory (fully relativistic DKS and SO-ZORA). Red/blue color represents SO-induced depletion/concentration in electron or spin analysis of nonhydride ligands, which is beyond the scope of density. Figures are plotted using isovalues of 0.00005 au for SO-EDD this work. and 0.00000005 au for SOM-ISD. 2.6. Visualization of the SO-HALA effect. Connecting δSO(LA) with Spin−Orbit Electron Deformation Density by a decrease (increase) of electron density at the LA, e.g., by a (SO-EDD) through Spin−Orbit and Magnetically In- more/less electronegative substituent. For instance, in TlH, duced Spin Density (SOM-ISD). Is there a way to visualize where the SO+ mechanism derived from σHA‑LA ↔ LPHA* the modulation of the electronic structure of a molecule by orbital couplings is observed, SOC “pulls” the electron density SOC, and how it is related to δSO? We have recently introduced from LA (depletion of electron density represented in red) to a ground state property termed Spin−Orbit Electron the LP* π-space of the HA (concentration of electron density Deformation Density (SO-EDD), which is obtained as a shown in blue). The opposite effect is seen in AtH, where the difference between the electron densities calculated with and shielding LPHA ↔ σ*HA‑LA mechanism “pushes” the electron without the SOC. Its value at LA nuclei correlates with the sign density from the LPAt MO toward the LA σ space. of δSO(LA).14 Here, we establish a firmer connection between Numerous examples of the correlation between SO-EDD δSO(LA) values and the SO-EDD by elaborating the relation- and the sign of δSO(LA) are demonstrated in Figures S1−S26 ship’s theoretical foundations and presenting further examples and S29 in the Supporting Information. The SO-EDD correctly of SO-EDD. reflects the sign of δSO(LA) for each studied compound. Although the PT3 expressions in eqs 1 and 2 were derived Selected examples of SO-EDD plots for sixth-period hydrides only for the SO/FC contribution to the NMR, σFC/SO, they are shown in Figure 15. The SO-EDD plots are consistent with represent the leading linear contribution of SOC to the Spin− Orbit Magnetically Induced Spin Density (SOM-ISD) at LA, see Theoretical Background. Shortly, in the presence of both SOC and a magnetic field (measured by the angular momentum operator, lHA ̂ ), the SOC and angular momentum operators couple the σHA‑LA and σ*HA‑LA MOs with the HA- based orbitals. This action “generates” an induced spin density at LA and also in orbitals that would have no density at the LA nucleus in calculations at a nonrelativistic level, i.e., in LPHA/ LPHA* or HA-L/HA-L* MOs. The spin density felt by the LA nucleus is represented by the FC operator in the coupling triangles (δLA in Figure 3). The δFC/SO (1H) value is proportional to the functional value of the SOM-ISD components at the LA nucleus via eq 7 in the Theoretical Background section. If only one heavy nucleus is present in the system, the SOM- Figure 15. SO-EDD plots obtained as differences between the electron density calculated at the SO-ZORA and at the ZORA levels for ISDand thus also the δSO(LA)are connected on a selected sixth period hydrides, along with the hydride δSO(1H) values. qualitative level to SO-EDD14 (see eqs 6−9 in the Theoretical Red and blue colors represent SO-induced depletion and concen- Background section). SO-EDD can be easily obtained as the tration of electron density, respectively. Values of δSO(1H) are color- difference between the electron density calculated with and coded accordingly. Isovalues of 0.00005 au are used for SO-EDD. without SOC. Underlying theory indicates that both SOM-ISD and SO-EDD have a similar topology (although different scaling) in the vicinity of the LA, while their behavior around the HA is in principle different. This relation is graphically deshielding values of δSO(1H) in 6s0, 4f0, 5d0, and 5d10 illustrated for TlH and AtH compounds in Figure 14. Note that complexes (decrease of electron density at the LA) and SO-EDD plots calculated at four-component DKS and two- shielding ones in 5d2, 5d6, and 5d8 complexes. They even component SO-ZORA levels are similar to each other. encompass the sign change of δSO(1H) between the 5d0 and 5d2 The correlation between δSO(LA) values and SO-EDD is and between 5d8 and 5d10 complexes. The correlation between particularly intriguing, as it offers an interesting parallel to the SO-EDD and the sign of δSO(1H) extends beyond the sixth classical arguments regarding NMR shielding in diamagnetic period hydrides (Figures S23−S26) and to nonhydride ligands molecules, i.e., that the (de)shielding effect at the LA is caused (Figure S29). 3034 DOI: 10.1021/acs.jctc.8b00144 J. Chem. Theory Comput. 2018, 14, 3025−3039 Journal of Chemical Theory and Computation Article Despite rather complex underlying theory, the trends in the already rising interest in linking NMR parameters with the δSO(LA) sign can thus be explained to a general audience by structure/chemical properties of heavy-element compounds. using an easily understandable analogy to classical NMR theory arguments based on changes of electron density around the LA 4. METHODS nuclei. Molecular Structures. Molecular geometries are based on published X-ray data where applicable and otherwise prepared 3. CONCLUSIONS in silico. All structures were optimized using the PBE0 functional62,63 and def-TZVPP (f-elements) or def2-TZVPP The 1H NMR chemical shifts for hydrides of the sixth period basis sets (other atoms),64 with corresponding relativistic (Cs−At) were calculated and analyzed to understand the SO- effective core potentials (ECPs)65 for the heavy-metal atoms HALA (Spin−Orbit Heavy Atom on the Light Atom) NMR (ECP replacing 28 core electrons for fifth row elements, 46 chemical shifts across the periodic table. The total 1H NMR core electrons for Cs-Ba and La-Lu, and 60 core electrons for shifts show distinct periodic trends along the sixth-period Hf-At and seventh row elements). The dispersion correction hydrides, which are determined by the oscillating SO-HALA (D3) by Grimme was used in the optimizations,66 with the chemical shifts, δSO(LA). The hydrogen nucleus is relativisti- exception of compounds with 4f and 5f elements (D3 not cally deshielded in complexes with a formally empty valence defined). Use of this level of theory has been justified by our shell at the HA, namely, in the 6s0 (CsI and BaII), 4f05d0 (LaIII previous studies of metal complexes.34,37,67−69 An implicit and CeIV), 4f145d0 (LuIII), 5d0 (HfIV and TaV), 5d106s0 (HgII conductor-like screening solvent model (COSMO)70 with and AuI), and 6s26p0 (TlI and PbII) hydrides. The hydrogen solvent parameters corresponding to the experimental solvents nucleus is shielded in hydrides where the HA has a partially was used in the geometry optimizations of the experimentally occupied valence shell (5d2 WIV to 5d8 AuIII, 6p2 Po, and 6p4 known compounds; model complex geometries were optimized At). The maximum shielding is observed for 5d6 IrIII and 6p4 in vacuo. AtI hydrides. The SO-HALA effect typically scales with the Calculation of NMR Chemical Shifts (SO-ZORA). NMR number of available lone pair orbitals at the heavy atom, LPHA. chemical shifts and δSO(1H) were calculated using the SO- Third-order perturbation theory (PT3) molecular orbital ZORA with the PBE40 functional, i.e., the standard PBE0 (MO) analysis allowed us to explain the electronic structure functional with the exact-exchange admixture set to 40% in the origin of the δSO(LA) sign and its oscillation along the sixth ADF 2016 package.71,72 This level provided excellent results in period hydrides from the first principles. We have identified our previous studies17,34,37,68 and is referred to as the SO- two general mechanisms related to the electronic structure at ZORA level here. Calculated magnetic shielding values were the HA, which control the sign and size of the δSO(LA). The referenced to the trimethylsilane (TMS) frequency. The mean SO-HALA deshielding, SO+, arises from the coupling of the absolute deviation between the δcalc(1H) value calculated at the occupied σHA‑LA bonding MO with the vacant MOs located at PBE40/TZP level with SO-ZORA and δexp(1H) is 2.3 ppm. the HA, i.e., LPHA* (empty “lone pair” orbitals at HA) or HA- This result indicates a rather good agreement between theory L* (antibonding orbitals of other ligands). The effect is and experiment with respect to the wide range of 1H NMR strongest when the σHA‑LA and LPHA* MOs are frontier MOs chemical shifts (>70 ppm) in the studied hydrides. (HOMO and LUMO, respectively), as is the case in most s0, p0, Analysis of NMR Chemical Shifts and Electronic d0, and f0 HA compounds. The SO-HALA shielding, SO−, arises Structure (SO-ZORA). The SO-ZORA MO analysis of the from the coupling of the occupied LPHA (HA-L) orbitals with NMR chemical shifts as implemented in the ADF 2016 package the σ*HA‑LA antibonding MO. The effect is strongest when the was used.30,44 To analyze the contributions of individual MOs frontier occupied orbitals are formed from pure LP MOs at the to SO NMR chemical shifts (δSO), we subtracted the HA and the virtual σ*HA‑LA MO is the LUMO or nearby. The paramagnetic contributions (δpara) of individual MOs at the results from the PT3 MO analysis are consistent with those of SO-ZORA and ZORA levels. The δSO value was then calculated the SO-ZORA MO analysis, which gives more accurate results as10,42 for NMR chemical shifts, comparable to experimental data. We provide solid evidence that the described two mechanisms δ SO = δ para[SO‐ZORA] − δ para[ZORA] (3) apply generally for δSO(1H) across the periodic table and also NMR chemical shift contributions of degenerate spin− for δSO(LA) of nonhydride LA. orbitals (spinors) were summed up and reported as As the δSO(LA) value is directly linked to the Spin−Orbit contributions of the parental scalar relativistic MOs for clarity. and Magnetically Induced Spin Density (SOM-ISD) at the LA The atomic-orbital composition of M−H bonds was nucleus, which is qualitatively connected to the Spin−Orbit determined from the Natural Localized Molecular Orbitals Electron Deformation Density (SO-EDD), the trends in (NLMOs) obtained from a Natural Bond Orbital (NBO) δSO(LA) can be understood on a qualitative level using the analysis73 as implemented in the NBO 6.0 module (Gennbo)74 classical arguments related to the NMR chemical shift. Namely, interfaced to ADF code. the relativistic deshielding effect is accompanied by spin−orbit- Analysis of SO/FC mechanism using PT3 approach. coupling (SOC)-induced withdrawal of electron density from The calculations and analysis of SO contributions to the NMR the LA, while the shielding at the LA is connected to SOC shielding constants were performed using the developers’ pushing electron density toward the LA. This simplification was version of the ReSpect 5.0.0 program package75 within the valid in all tested systems. framework of the Kohn−Sham (DFT) level of theory as By comprehending the nature of the SO-HALA effect and its described previously.14 The PT3 implementation in the relation to the electronic structure of the HA, we have opened a ReSpect program was based on the static (frequency-free) new and simple way for understanding the often enigmatic case of the third-order response theory presented in ref 76. The trends in the NMR chemical shifts of LAs in heavy-element implementation was identical to that in refs 40 and 41, while compounds. These findings are believed to further increase the the notation SO/FC used in this work corresponds to FC-I 3035 DOI: 10.1021/acs.jctc.8b00144 J. Chem. Theory Comput. 2018, 14, 3025−3039 Journal of Chemical Theory and Computation Article terms in ref 41 and is part of the so-called SO term in ref 38. Magnetically Induced Spin Density SOM-ISD. In other words, The spin-dipolar (SD) term was not discussed for hydride- both magnetic field and SOC are necessary for spin density to based ligands due to its minimal contribution, which is due to be nonvanishing. To take SOC into consideration, we the hydrogen 1s orbital having a nonvanishing density at the employed both SO-ZORA and four-component (DKS) frame- nucleus and the resulting dominance of the FC term. The works. Interestingly, eq 5 is more general than eq 4 in both limitation of the tested systems to only one HA center in the approaches since it contains more responses to SOC than the complex allows the summation in the SO operator to be linear response. restricted to ∑nuc N −3 N̂ ̂ HA −3 HA N=1Z rN l ≈ Z rHAl , as the nuclear charge of For a qualitative analysis, we will rewrite eq 5 in a more the HA is much larger than that of the LA. This approach was suitable form in the two-component framework used to simplify the discussion of eqs 1 and 2 and PT3 ⎧ occ vac HA ⎫ diagrams; however, the full summation in the SO operator is 4π ⎪ ⟨φa|l ̂ |φi⟩ ⎪ used in the calculations of PT3 MO contributions (see, for σ (LA) = 2 Re⎨∑ ∑ φi (LA)σφ SO † ̂ a(LA) ⎬ 9c ⎪ ⎩ i=1 a=1 (εi − εa) ⎭ ⎪ example, Figure 4). To make the text and particularly the figures more concise, we wrote the SO operator in the (6) simplified form of r−3 HAl ̂ in the text. HA Here, φi and φa correspond to occupied and vacant molecular For the transparent analysis of MO contributions in PT3, we spinors, respectively, ε represents one-electron energies, c is the used the common gauge origin methodology and a relatively speed of light, σ̂ is a vector composed of the three Pauli small basis set (Dyall’s valence double-ζ basis).77−79 Further, all ̂ is the angular momentum operator centered matrices, and lHA DFT kernels (both first and second order) are omitted in eqs 1 on the HA. In eq 6 Cartesian components of sigma matrix and and 2 and in the calculations of the PT3 MO contributions. of the angular momentum operator are summed. Note that the This approximation can only be made in the framework of the MOs in eq 6 are two-component spinors and contain a pure DFT functionals (PBE in this work), where the kernel contribution from SOC, while MOs in eqs 1 and 2 are simpler, contributions are generally not dominant. For more informa- nonrelativistic MOs. In analogy to the PT3 analysis in ref 14, tion on the details of the PT3 implementation and PT3 we neglect all the contributions from the exchange correlation analysis, see ref 14. and Hartree−Fock kernels in eq 6. To obtain feasible results Calculations of SOM-ISD and SO-EDD. The SOM-ISD from analysis based on eq 6, the gauge origin of the angular was obtained as the difference between the values calculated momentum operator, lHA ̂ , must be placed on the metal center using a full four-component relativistic Dirac−Kohn−Sham (for a more detailed discussion, see ref 14). For a practical (DKS) formalism based on the Dirac−Coulomb Hamiltonian interpretation of eq 6, see Figures 5 and S27 and the and the restricted magnetically balanced basis for the small corresponding discussion. component,80,81 as implemented in the developer version of the Eqs 4 and 5 represent the direct link between the SO- ReSpect 5.0.0 code75 (noted as DKS approach), and a scalar induced chemical shift, δSO, andρB. When taking only the linear relativistic calculation with the same setup (SO integrals were response with respect to the SOC into consideration, we can neglected in four-component calculations, denoted sc-DKS). write The PBE functional82,83 and uncontracted Dyall’s valence double-ζ basis set for all atoms were used in both cases.77−79 8π Bx ,SO σ SO/FC(LA) = [ρx (LA) + ρyBy ,SO (LA) The SO-EDD was calculated as the difference in electron 9c densities calculated at the SO-ZORA/DKS and ZORA/sc-DKS + ρzBz ,SO (LA)] levels. (7) where ρ B,SO denotes the quadratic response of the spin density 5. THEORETICAL BACKGROUND with respect to the magnetic field and SOC. If the system has Magnetically Induced Spin Density (MISD) and Its only one heavy atom, PT3 theory can be used to show that on a Connection to the δ SO and Spin−Orbit Electron qualitative level, SOM-ISD and SO-EDD (ρ0SO,SO) are Deformation Density (SO-EDD). The leading cause of the connected by the simple expression SO-induced changes in the isotropic nuclear magnetic shielding ZHA Bx ,SO at the LA is the FC mechanism.5,6 (ρx (LA) + ρyBy ,SO (LA) + ρzBz ,SO (LA)) c δ SO(LA) = −σ SO(LA) ≅ −σ SO/FC(LA) (4) ≈ ρ0SO,SO (LA) (8) The connection between the δSO value and the FC For closed-shell singlet systems, ρSO = 0, so the first nonzero 0 mechanism can be rewritten using the theory connecting the response of the electron density, ρ0, to SOC is quadratic. Eq 8 magnetically induced current and the NMR shielding tensor. As is valid under the assumption that the matrix elements of SO recently discussed,84 the first-order SO effects on NMR operator r−3 ̂ and angular momentum operator lHA HA ̂ have the HAl shielding originate exclusively from spin currents. The under- same sign. Even with this assumption, the connection lying theory is beyond the scope of this work, but we readily represented by eq 8 will break down when close to the heavy write the result here: metal, where many different spinors have nonzero contribu- 8π Bx tions and different magnitudes of r−3 HAl ̂ and lHA HA ̂ matrix σ SO(LA) = [ρ (LA) + ρyBy (LA) + ρzBz (LA)] elements will therefore cause differences in SOM-ISD and SO- 9c x (5) EDD. In eq 5, ρ (LA) represents the MISD on atom LA. An B However, the SOM-ISD and SO-EDD have the same interesting result from eq 5 is that for closed-shell singlet qualitative effect on the ligand atom in practice, as shown in systems, MISD is nonzero only if the SOC is included in the Figures 14 and 15 of the main text. As a result, the sign of underlying theory, which is then referred to as Spin−Orbit and δSO(LA) can be predicted using just the sign of the SO-EDD on 3036 DOI: 10.1021/acs.jctc.8b00144 J. Chem. Theory Comput. 2018, 14, 3025−3039 Journal of Chemical Theory and Computation Article the ligand atom, ρSO,SO 0 (LA). On the basis of these observations were provided by the CESNET (LM2015042) and the CERIT and using eqs 4, 7, and 8, the connection between δSO(LA) and Scientific Clouds (LM2015085). ■ the SO-EDD ρSO,SO 0 (LA) can be expressed as 8π Bx ,SO REFERENCES δ SO(LA) = − [ρ (LA) + ρyBy ,SO (LA) + ρzBz ,SO (LA)] 9c x (1) Nomura, Y.; Takeuchi, Y.; Nakagawa, N. Substituent Effects in 8π Aromatic Proton NMR Spectra. 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Giant Spin− Figure S29: SO-EDD plots at the SO-ZORA level for orbit Effects on 1H and 13C NMR Shifts for uranium(VI) Complexes EtTl and EtAt (PDF) Revisited: Role of the Exchange−correlation Response Kernel, ■ AUTHOR INFORMATION Corresponding Authors Bonding Analyses, and New Predictions. Phys. Chem. Chem. Phys. 2016, 18, 30462−30474. (10) Vícha, J.; Marek, R.; Straka, M. High-Frequency 13C and 29Si NMR Chemical Shifts in Diamagnetic Low-Valence Compounds of *E-mail: [email protected]. TlI and PbII: Decisive Role of Relativistic Effects. Inorg. Chem. 2016, *E-mail: [email protected]. 55, 1770−1781. ORCID (11) Vícha, J.; Marek, R.; Straka, M. High-Frequency 1H NMR Jan Vícha: 0000-0003-3698-8236 Chemical Shifts of SnII and PbII Hydrides Induced by Relativistic Effects: Quest for PbII Hydrides. Inorg. Chem. 2016, 55, 10302− Stanislav Komorovsky: 0000-0002-5317-7200 10309. Radek Marek: 0000-0002-3668-3523 (12) Rivard, E.; Power, P. P. Recent Developments in the Chemistry Michal Straka: 0000-0002-7857-4990 of Low Valent Group 14 Hydrides. Dalton Trans. 2008, 33, 4336− Notes 4343. The authors declare no competing financial interest. (13) Schneider, J.; Sindlinger, C. P.; Eichele, K.; Schubert, H.; ■ Wesemann, L. Low-Valent Lead Hydride and Its Extreme Low-Field 1H NMR Chemical Shift. J. Am. Chem. Soc. 2017, 139, 6542−6545. ACKNOWLEDGMENTS (14) Novotný, J.; Vícha, J.; Bora, P. L.; Repisky, M.; Straka, M.; This work was supported by the Czech Science Foundation Komorovsky, S.; Marek, R. Linking the Character of the Metal−Ligand (16-05961S) and by the Ministry of Education, Youth and Bond to the Ligand NMR Shielding in Transition-Metal Complexes: Sports of the Czech Republic under the National Sustainability NMR Contributions from Spin−Orbit Coupling. J. Chem. Theory Program I (LO1504) and II (LQ1601) and the Multilateral Comput. 2017, 13, 3586−3601. Scientific and Technological Cooperation Project in the (15) Wolff, S. K.; Ziegler, T. Calculation of DFT-GIAO NMR Shifts Danube Region (8X17009). Institutional support was provided with the Inclusion of Spin-Orbit Coupling. J. Chem. Phys. 1998, 109, 895−905. by the Czech Academy of Sciences, project RVO-61388963. (16) Sutter, K.; Autschbach, J. Computational Study and Molecular Financial support from the SASPRO Program (contract no. Orbital Analysis of NMR Shielding, Spin−Spin Coupling, and Electric 1563/03/02), cofinanced by the European Union, the Slovak Field Gradients of Azido Platinum Complexes. J. Am. Chem. Soc. 2012, Academy of Sciences, and the Slovak Grant Agencies VEGA 134, 13374−13385. and APVV (contract no. 2/0116/17, APVV-15-0726 and DS- (17) Pawlak, T.; Munzarová, M. L.; Pazderski, L.; Marek, R. 2016-0009), is duly acknowledged. Computational resources Validation of Relativistic DFT Approaches to the Calculation of NMR 3037 DOI: 10.1021/acs.jctc.8b00144 J. Chem. 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