Temperature dependence of contact and dipolar NMR chemical shifts in paramagnetic molecules Bob Martin and Jochen Autschbach Citation: The Journal of Chemical Physics 142, 054108 (2015); doi: 10.1063/1.4906318 View online: http://dx.doi.org/10.1063/1.4906318 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/142/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in NMR chemical shift as analytical derivative of the Helmholtz free energy J. Chem. Phys. 138, 054113 (2013); 10.1063/1.4789398 Ab initio calculations of NMR chemical shifts J. Chem. Phys. 128, 052201 (2008); 10.1063/1.2816784 Transition metal NMR chemical shifts from optimized effective potentials J. Chem. Phys. 126, 074101 (2007); 10.1063/1.2436876 Density functional calculations of NMR shielding tensors for paramagnetic systems with arbitrary spin multiplicity: Validation on 3 d metallocenes J. Chem. Phys. 126, 024107 (2007); 10.1063/1.2423003 Non-Abelian point group symmetry in direct second-order many-body perturbation theory calculations of NMR chemical shifts J. Chem. Phys. 108, 8295 (1998); 10.1063/1.476258

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THE JOURNAL OF CHEMICAL PHYSICS 142, 054108 (2015)

Temperature dependence of contact and dipolar NMR chemical shifts in paramagnetic molecules Bob Martin and Jochen Autschbacha) Department of Chemistry, University at Buffalo, State University of New York, Buffalo, New York 14260-3000, USA

(Received 13 November 2014; accepted 6 January 2015; published online 6 February 2015) Using a recently proposed equation for NMR nuclear magnetic shielding for molecules with unpaired electrons [A. Soncini and W. Van den Heuvel, J. Chem. Phys. 138, 021103 (2013)], equations for the temperature (T) dependent isotropic shielding for multiplets with an effective spin S equal to 1/2, 1, 3/2, 2, and 5/2 in terms of electron paramagnetic resonance spin Hamiltonian parameters are derived and then expanded in powers of 1/T. One simplifying assumption used is that a matrix derived from the zero-field splitting (ZFS) tensor and the Zeeman coupling matrix (g-tensor) share the same principal axis system. The influence of the rhombic ZFS parameter E is only investigated for S = 1. Expressions for paramagnetic contact shielding (from the isotropic part of the hyperfine coupling matrix) and pseudo-contact or dipolar shielding (from the anisotropic part of the hyperfine coupling matrix) are considered separately. The leading order is always 1/T. A temperature dependence of the contact shielding as 1/T and of the dipolar shielding as 1/T 2, which is sometimes assumed in the assignment of paramagnetic chemical shifts, is shown to arise only if S ≥ 1 and zero-field splitting is appreciable, and only if the Zeeman coupling matrix is nearly isotropic (∆g = 0). In such situations, an assignment of contact versus dipolar shifts may be possible based only on linear and quadratic fits of measured variable-temperature chemical shifts versus 1/T. Numerical data are provided for nickelocene (S = 1). Even under the assumption of ∆g = 0, a different leading order of contact and dipolar shifts in powers of 1/T is not obtained for S = 3/2. When ∆g is not very small, dipolar and contact shifts both depend in leading order in 1/T in all cases, with sizable contributions in order 1/T n with n = 2 and higher. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4906318] I. BACKGROUND AND MOTIVATION

Chemical shifts in paramagnetic molecules (herein referred to as pNMR shifts) shifts have, for a long time, been analyzed by separating the total isotropic nuclear magnetic shielding into three separate contributions1–7 σ = σ orb + σ c + σ pc. (1) orb Here, σ is the “orbital” shielding, defined to be the only nonvanishing contribution in the case of a diamagnetic molecule. To avoid confusion, we do not further discuss the separation of σ orb into a “paramagnetic” and a “diamagnetic” contribution. Explicit expressions for this and the other shielding terms are given below. In most cases, the shielding values are small and the isotropic chemical shift with respect to the same isotope in a reference (ref) compound is then simply δ = σ ref − σ. We focus here on the shielding of the probe nucleus, for example, of a ligand atom in a paramagnetic metal complex, rather than the chemical shift. The NMR shielding contribution specific to the electron paramagnetism is typically split into a σ c contact (c) term, and a σ pc pseudo-contact (pc) term.3 The electron-nucleus magnetic hyperfine coupling (hfc) plays a central role, as does the electron magnetic moment quantified in terms of electron g-factors. Both effects are also underlying electron paramagnetic resonance (EPR), along with zero-field splitting a)Email: [email protected]

0021-9606/2015/142(5)/054108/10/$30.00

for an effective spin S ≥ 1. In a nonrelativistic quantum theoretical framework, and for the simple case of a spin-doublet (S = 1/2), the pseudo-contact shielding arises exclusively from the anisotropic dipolar “spin-dipole” (SD) mechanism of hfc. Hence, pc NMR shifts are also commonly referred to as “dipolar” shifts. We use both labels interchangeably. The contact shieldings are caused by a non-zero spin density at the nucleus of interest and they are associated with the isotropic “Fermi contact” (FC) mechanism of hfc. For point nuclei, the nonrelativistic FC operator affords a Dirac delta-distribution term centered on the nucleus of interest. In a relativistic quantum theoretical framework8–11 and/or with finite nuclei, the distinction between contact and noncontact operators becomes somewhat blurred. Spin-orbit (SO) coupling further complicates the analysis, as it blurs the distinction between electron spin and orbital angular momentum. For the purpose of the analysis presented herein, as a working definition, we refer to pseudo-contact, or dipolar, isotropic shielding as the electron paramagnetism contribution to σ in Eq. (1) arising from the anisotropic part of the hfc tensor. Contact shielding is associated with the isotropic component of the hfc tensor. A similar distinction has also recently been proposed in Ref. 7. In the nonrelativistic limit, this definition recovers the usual distinction between contact and purely dipolar terms, and it remains largely intact in relativistic calculations of hfc on heavy elements systems except for the most extreme cases of SO coupling.12,13

142, 054108-1

© 2015 AIP Publishing LLC

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For paramagnetic ligand NMR shifts caused by an openshell metal center at some distance, contact shifts would indicate covalent interactions by which spin density “leaks” from the metal to the ligands (or vice versa13), whereas pc shifts may occur “through space” without covalent metal-ligand interactions. It is no surprise, then, that experimentalists have found the contact versus pc separation very useful. According to an often-cited14–27 analysis by Bleaney1 of pNMR ligand shifts induced by lanthanoid ions using crystal-field theory, the σ c and σ pc contributions vary with temperature (T) in leading order as T −1 and T −2, respectively. This would allow to assess the relative magnitudes of contact versus dipolar paramagnetic shift contributions by variable temperature (VT) NMR experiments,21,28–34 and thereby give insight about the extent of metal-ligand covalent interactions.19–21,23,25,26,35 For instance, Desreux and Reilley utilized this separation, while also acknowledging that it was somewhat controversial. It is sometimes assumed that the presence of large paramagnetic effects on NMR shifts indicates a dominance of contact terms. However, in Ref. 36, we recently showed that for plutonyl(VI)carbonate more than half of the experimentally observed increased shielding of the carbonate carbons by 376 ppm—relative to diamagnetic uranyl(VI)carbonate—is potentially attributable to dipolar terms. It has been pointed out that the approximations used in Bleaney’s pseudo-contact shift formula are quite restrictive, and large deviations were found between theoretical and experimental data.37,38 In 1970, Kurland and McGarvey2 derived a formula for the pseudo-contact shift in terms of the magnetic susceptibility that does not demonstrate a T −2 behavior but a more complex exponential dependence resembling well-known related expressions put forth later by Bertini et al.39 These expressions do not obviously lead to different T-dependencies for contact and dipolar shifts. Pennanen and Vaara4 (PV) proposed in 2008 an expression to calculate pNMR chemical shifts from EPR spin Hamiltonian parameters, including effects from zero-field splitting (ZFS). The PV expression affords an overall T −1 factor but also an additional temperature-dependence from Boltzmann-averaging over the eigenstates of the ZFS tensor which may lead to a more intricate T-dependence of the shielding contributions. Recently, a new general theoretical framework for the pNMR shift has been proposed by Soncini and van den Heuvel (SvH),40 who made a case that the treatment of ZFS in the context of paramagnetic NMR by PV was not correct. The temperature-dependence of the various shielding contributions in the SvH expression is more intricate than in the PV framework. However, some results presented by SvH already provided hints that a simple T −1 versus T −2 dependence of contact and dipolar terms in the pNMR shielding is unlikely. Our group made use of Bertini’s equations in previous work to estimate pNMR dipolar shifts.36,41 In one of the Bertini expressions, the dipolar NMR shifts arise from the anisotropy of the magnetic susceptibility combined with the through-space dipolar part of the hfc tensor in which the spin density is approximated as a point magnetic dipole located at the metal center. The T-dependence enters via the susceptibility whose theoretical expression is closely related to the SvH pNMR shielding expression (vide infra). Curiously, Bleaney noted

J. Chem. Phys. 142, 054108 (2015)

that the 1/T 2-dependent pseudo-contact shifts observed for lanthanoid complexes require a lack of magnetic anisotropy, while in Bertini’s susceptibility expression, magnetic anisotropy is required for non-zero dipolar shieldings. In the present work, we analyze the SvH NMR shielding expression with a particular focus on the temperaturedependence of the various terms. To keep the analysis simple, the pNMR shift is considered for an axial system, which covers many experimentally relevant cases. Pseudo-spins S = 1/2, S = 1, S = 3/2, S = 2, and S = 5/2 are considered explicitly, and the isotropic shielding is expanded in powers of 1/T. For S = 1, we also investigate the changes when going from an axial to a non-axial system. A numerical example is provided by nickelocene, an axial S = 1 system with experimentally and theoretically well characterized pNMR shifts and large paramagnetic effects thereupon. ZFS is described in terms of second-rank spin operators, and Zeeman and hyperfine (HF) coupling are described in terms of first-rank spin operators. Inclusion of higher rank operators is possible,42 which would then lead to the appearance of higher-rank g and hfc tensors in the weight of each term in the temperature expansion. However, this is unlikely to alter the main conclusions. The T-dependence of the shielding is shown to be more intricate than a simple 1/T behavior for contact and a simple 1/T 2 behavior for pseudo-contact shifts, respectively. For S = 1, 2, and 5/2 with appreciable zero-field splitting but very small g-tensor anisotropy (∆g), the leading paramagnetic term in the isotropic shielding is, indeed, proportional to 1/T with negligible 1/T 2 behavior, while the leading dipolar term has a 1/T 2 dependence proportional to the ZFS parameter D. For S = 3/2, and for all considered spins if ∆g is not small, a clean separation is not possible. We discuss the SvH NMR expression in Sec. II and derive the separation of contact vs. dipolar isotropic shielding. Explicit expressions for different S along with expansions in powers of 1/T are derived and discussed in Sec. III. The influence of the rhombic ZFS parameter E is investigated in Sec. IV for S = 1. Nickelocene results are presented in Sec. V. This work concludes with a brief summary. II. THEORETICAL DETAILS A. The pNMR shielding expression by SvH

In Ref. 43, SvH derived a general formula for the Tdependent shielding tensor elements of a nucleus in a paramagnetic molecule as the bi-linear derivative of the Helmholtz free energy F as σiNj (T) =

∂ 2F . ∂ Bi ∂ µ Nj

(2)

Here, i, j ∈ {x, y, z}, N is the nucleus of interest, µiN is a component of its spin magnetic moment vector µ N , and Bi is a component of the external magnetic field vector B. Here and elsewhere in this work, the derivatives are understood to be taken at B = 0, µ N = 0. We adopt a similar notation as SvH and follow their lines of derivation in this section in order to render the present article reasonably self-contained.

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In a quantum mechanical framework, calculating a bilinear derivative (a double-perturbation molecular property44) requires the linear and bi-linear derivatives of the Hamiltonian Hˆ ∂ Hˆ HF , (3a) Hˆ Ni = ∂ µiN ∂ Hˆ Hˆ iZ = , (3b) ∂Bi ∂ 2 Hˆ DS . (3c) Hˆ N ij = ∂Bi ∂ µ Nj The term in (3a) is the HF operator component i for nucleus N describing the perturbation of the molecule by the presence of the nuclear spin magnetic moment, (3b) is the Zeeman (Z) operator component i describing the perturbation by a static external magnetic field, and (3c) is the diamagnetic shielding (DS) operator component i, j for nucleus N. The field-free Hamiltonian may be expressed as  Eλ|λa⟩⟨λa|. (4) Hˆ 0 = λ,a

The normalized eigenfunctions |λa⟩ corresponding to the eigenvalues Eλ have an additional index a to count the components of degenerate states. For convenience, we assume the latter to be mutually orthogonal. SvH obtained the temperaturedependent shielding tensor from the Helmholtz energy as σiNj (T) =

∂ 2F ∂Bi ∂ µ Nj

DS = ⟨ Hˆ N i j ⟩0 + ⟨



β

ˆ ˆ HF e w H0 Hˆ iZe−w H0 Hˆ N j dw⟩0.

(5)

0

In the previous equation, ⟨···⟩0 indicates a thermal average in the canonical ensemble corresponding to Hˆ 0. Further, β = 1/(k BT) with k B being the Boltzmann constant and T the absolute temperature. SvH carried out the ensemble average before the integration over w, assuming knowledge of the exact eigenfunctions of Hˆ 0. Subsequent integration over w led to a sum-over-states (SOS) like expression for the shielding tensor elements,  1  − kEλT  DS σiNj (T) = e B  ⟨λa| Hˆ N i j |λa⟩ Q λ  a 1  HF + ⟨λa| Hˆ iZ|λa ′⟩⟨λa ′| Hˆ N j |λa⟩ k BT a,a′  HF   ⟨λa| Hˆ iZ|λ′a ′⟩⟨λ′a ′| Hˆ N j |λa⟩   . + 2 Re (6) Eλ′ − Eλ  λ′,λ a,a ′  In Eq. (6), Q = λ,a exp(− βEλ) is the partition function. Several notable features of the expression for σiNj (T) were already pointed out by SvH but are worth reiterating here, with additional comments: HF 2 ˆ ˆZ ˆ DS ˆ DM (i) Replacing Hˆ N j with H j and H N i j with Hi j = (∂ H) /(∂Bi ∂B j ), respectively, gives the van Vleck equation for the magnetizability tensor components χi j (T) = (∂ 2F)/ (∂Bi ∂B j ). The operator Hˆ iDM j is the diamagnetic magneHF tizability operator. In a similar vein, replacing Hˆ iZ by Hˆ M i DS DSO DSO 2 ˆ and Hˆ N i j by Hˆ M i N j , where Hˆ M i N j = (∂ H)/(∂ µM i∂ µN j)

is the diamagnetic spin-orbital operator, would give an expression for the indirect NMR nuclear spin-spin coupling (J-coupling) between nuclei M and N. It is reassuring that the T-dependent shielding tensor has the same formal structure as the van Vleck magnetizability, since this is also the case for the shielding and magnetizability expressions that apply to systems without electron paramagnetism. The van Vleck equation is derived in some textbooks (e.g., Ref. 45) without explicitly invoking the Helmholtz energy route used by SvH, from a perturbation expansion of Boltzmann average of the magnetic moment induced by the external and the hyperfine field. (ii) For non-degenerate states, the term on the right hand side of Eq. (6) proportional to β = 1/k BT vanishes because HF the time-odd operators Hˆ iZ, Hˆ N j have no diagonal matrix elements. If one further assumes a non-relativistic theoretical framework, then each term in the sum over λ in Eq. (6) is identical to the Ramsey SOS shielding tensor expression for state λ. With the substitutions of operators suitable for J-coupling, the corresponding Ramsey J-coupling expression is obtained. Within a two- or four-component relativistic framework, Eq. (6) includes all the well-known heavy atom effects on the shielding, for instance, from SO coupling. We shall not discuss the various available relativistic frameworks for calculating NMR chemical shifts for closed-shell molecules as this topic has been reviewed several times in recent years.8,9,46–49 (iii) The term in σiNj (T) being unique to the electron paramagnetism is the Boltzmann average of  HF β a,a′⟨λa| Hˆ iZ|λa ′⟩⟨λa ′| Hˆ N j |λa⟩. For good reasons, SvH referred to it as the Curie term. In a relativistic framework, sizable relativistic effects can be expected for the Curie shielding term. For instance, the matrix elements HF of Hˆ N j are known to be sensitive to relativistic effects (both scalar and spin-orbit), and the matrix elements of Hˆ iZ are sensitive to SO coupling. For a degenerate magnetic ground state with excited states that are energetically well separated from the ground state, the shielding tensor components then have an intrinsic leading 1/T dependence from this term. If the ground state is a spin multiplet, it may be split by ZFS and a more intricate T-dependence may result. In a relativistic framework where the wavefunctions explicitly include effects from SO coupling and the dipolar spin-spin interaction, a calculation would already include the ZFS in Hˆ 0 and its eigenfunctions.

B. The SvH pNMR expression for a pseudo-spin Hamiltonian and its eigenfunctions

We continue to follow the derivation by SvH, by invoking ˆ N , B) rather than the EPR pseudo-spin Hamiltonian for H(µ working with many-electron wavefunctions and an actual molecular Hamiltonian. If desired, the spin Hamiltonian parameters can be calculated from first principles before applying the subsequent equations. Instead of taking the detour via the EPR spin Hamiltonian, Eq. (6) may be utilized within a

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quantum chemical framework—either directly as written or in a suitable formulation using linear response theory. Using I N to denote the nuclear spin vector (µ N = g N β N I N) and Sˆ for the electronic pseudo-spin vector operator in Cartesian representation, the EPR spin Hamiltonian reads Hˆ = Sˆ · D · Sˆ + βe B · g · Sˆ + Sˆ · A N · I N

(7)

with ZFS, the Zeeman interaction, and the hyperfine interaction, respectively, from left to right on the right-hand side. Further, g N is the nuclear g-factor, β N is the nuclear magneton, D is a 3 × 3 matrix representing the ZFS tensor, βe is the Bohr magneton ( βe = 1/2 in atomic units), gik is an element of the N 3 × 3 Zeeman coupling matrix g (“g-tensor”), and Aik is an element of the electron-nucleus hyperfine coupling matrix A N for nucleus N. We use upright-bold notation for matrices and italic-bold for vectors. The multiplication dots indicate matrixˆ The fieldvector contractions to yield a scalar Hamiltonian H. free Hamiltonian and two of the perturbation operators of Eqs. (3) based on the Hamiltonian (7) are given as ˆ Hˆ 0 = Sˆ · D · S,  Hˆ iZ = − βe gik Sˆk ,

(8a) (8b)

k HF Hˆ N j

1 ˆ N Sk A j k . = gN β N k

(8c)

Indices N indicating a specific nucleus are implied from here on and not explicitly written. Consider a multiplet for spin S that is split by the zerofield interaction. Let the sets of eigenfunctions of Hˆ 0 and the eigenvalues be |Sλa⟩ and Eλ, respectively. For this multiplet, the shielding tensor elements of Eq. (6) are given by  − Eλ 1 βe gik Al j σiPj = − e k BT gN β N k BT Q k l λ  ×  ⟨Sλa| Sˆk |Sλa ′⟩⟨Sλa ′| Sˆl |Sλa⟩ a,a′   ⟨Sλa| Sˆk |Sλ′a ′⟩⟨Sλ′a ′| Sˆl |Sλa⟩   . (9) +2k BT Re  ′ E − E λ λ λ′,λ a,a ′  The EPR pseudo-spin Hamiltonian only describes the electron paramagnetism, and, therefore, there is no information about the nuclear shielding originating from paramagnetic and diamagnetic orbital current densities. We use a superscript “P” to indicate the shielding contribution resulting from just the electron paramagnetism. A factor of 1/(k BT) has been extracted from the λ-summation to emphasize the overall 1/T pre-factor of the paramagnetic shielding. The shielding tensor can be written in a compact notation using matrix multiplications as σP = −

βe g Z A. g N β N k BT

The elements of the 3 × 3 matrix Z are defined as  1  − kEλT  e B  ⟨Sλa| Sˆk |Sλa ′⟩⟨Sλa ′| Sˆl |Sλa⟩ Zk l = Q λ a,a′

(10)

  ⟨Sλa| Sˆk |Sλ′a ′⟩⟨Sλ′a ′| Sˆl |Sλa⟩   . + 2k BT Re  ′ − Eλ E λ ′ ′ λ ,λ a,a  (11) If the hyperfine spin Hamiltonian is written as I·A· Sˆ instead of the reverse order used in Eq. (7), then Eq. (10) affords AT , the transpose of A. Previously, PV4 proposed a different approach to incorporate effects from ZFS in calculated pNMR shielding. In our notation, the PV approach leads to the following expression for the elements of Z: 1  − kEλT  e B ⟨Sλa| Sˆk Sˆl |Sλa⟩. (12) Zk′ l = Q λ a SvH argued that the PV expression leads to an unphysical Tdependence of one of the unique shielding tensor elements of an S = 1 axial model system. In the absence of ZFS (i.e., for a pseudo-spin doublet, or D = 0 for cases with S > 1/2), the eigenstates of Hˆ 0 are pure pseudo-spin eigenfunctions. In this case, both the SvH and the PV expressions for Z give identical results, S(S + 1) δk l . (13) 3 Here, δ k l is the Kronecker delta. For S = 1/2, one obtains the shielding expression for a doublet previously derived by Patchkovskii and Moon.3 The expression for the pNMR shielding tensor can be easily adapted for use in first-principles calculations. Missing in the spin Hamiltonian of Eq. (7) are the effects related to the orbital current densities. One may pragmatically absorb all those effects into the orbital shielding of Eq. (1) and calculate this component of the shielding tensor with readily available methodology, for instance, for a diamagnetic analog of a paramagnetic system of interest.3 The shielding terms related to the electron paramagnetism can then be accessed via Eq. (10) if calculations of the ZFS tensor, the g-matrix, and the hyperfine coupling matrix are feasible. Within such a theoretical framework, the complete shielding tensor is then given by no ZFS: Zk0l =

βe 1 g Z A. (14) g N β N k BT Comparison with Eq. (1) shows that isotropic contact and dipolar shifts originate from the isotropic part of g ZA. For doublets, or for spin higher than 1/2 but vanishing ZFS, the above expression simplifies to σ = σ orb −

βe S(S + 1) g A, (15) g N β N 3k BT which has been used frequently in computational studies of pNMR chemical shifts.50–55 If one assumes that g and A are diagonal in the same coordinate system, the contact shielding is proportional to giso Aiso and the dipolar shielding is proportional to 2∆g∆A, with ∆g, ∆A being the anisotropic parts of the Zeeman and hyperfine interaction, and the constant of proportionality being in both cases −( βe S(S + 1))/(3g N β N k BT). Therefore, Eq. (15) plainly gives the same T −1 temperature dependence for the contact and dipolar shieldings. Equation (14) has been used less frequently, and apart from work by Soncini and coworkers it has been used in conjunction with Z′ of Eq. (12), typically showing a weak influence from ZFS. σ = σ orb −

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C. Separation of contact and dipolar terms

Based on Eqs. (11) and (13), a 1/T 2 dependence can only enter the pNMR shielding in the presence of non-zero ZFS, or if excited electronic multiplets are thermally populated. The latter scenario was considered in Bleaney’s analysis, but because the excited states for most of the considered lanthanoid ions were at comparatively high energy, the effects were often rather minor. We ignore other electronic multiplets and ask the question whether ZFS for a given multiplet is capable of rendering the dipolar shielding proportional to 1/T 2 while preserving a 1/T dependence of the contact shielding. Doublets are not considered in detail since we already know that in this case dipolar and contact shielding must have the same 1/T dependence. The reader is reminded that the working definition used in this study for dipolar and contact shielding rests on the separation of the hfc tensor into an anisotropic and an isotropic part. The ZFS removes the degeneracy of the spin microstates in the absence of an external magnetic field.56 A ZFS tensor D arises from the assumption that a system behaves as a single effective spin and the Hamiltonian has the form of Eq. (8a).57 When the molecule is oriented to diagonalize D, the tensor is characterized by two values, D and E,58 as shown in Eq. (16), where D is the axial parameter and E is the rhombic parameter, −D 0 0 + *. 3 + E // .. −D (16) D=. 0 −E 0 // . // .. 3 2D 0 0 , 3 As written, D is traceless. This choice can be made since any global shift in the energies of the microstates does not affect their relative weights.59 We consider an axial system, with the principal magnetic axis along the z direction. In this case, the rhombic parameter can be ignored. Further, the energy of the multiplet gets globally shifted by +D/3 such that D simplifies to *0 0 0 + D = ... 0 0 0 /// . (17) ,0 0 D We then have Hˆ 0 = D Sˆz2. The energies are integer multiples of D. The eigenstates of Hˆ 0 are |SMs ⟩ spin eigenfunctions, which causes the off-diagonal elements of Z in Eq. (11) to vanish. The resulting diagonal matrix Z may be written in terms of parallel (along the principal magnetic axis) and perpendicular components, or alternatively in terms of an isotropic component Ziso = 13 Tr(Z) and an anisotropic (dipolar) tensor Zdip defined in terms of ∆Z = 31 (Z ∥ − Z⊥) as * Z⊥ Z = ... 0 ,0 * Ziso = ... 0 , 0

0 Z⊥ 0 0 Ziso 0

0+ 0 /// = Ziso + Zdip Z∥0 + * −∆Z 0 /// + ... 0 Ziso - , 0

In the most general case, the principal axes of g and D do not have to be collinear. However, the eigenvalues for the system cannot be solved easily in this case.59 Therefore, we assume that for the axial system under consideration, g and D share the same principal axis system, i.e., the Zeeman coupling matrix g can be partitioned in the same way as Z in Eqs. (18). In Ref. 40, SvH demonstrated that, for the simple case of a spin triplet with non-vanishing axial ZFS and simultaneously axial g and A tensors (i.e., a system with D∞h symmetry) their shielding expression produced an expected behavior σ⊥P → const. in the low temperature limit (where D ≫ k BT), while the PV expression (Eq. (12)) was shown to produce an undesired Curie-law type temperature dependence. A distinction between contact and dipolar shielding was not considered. In the following, we consider spin states with S up to 5/2 analytically, we extend the analysis to an axial system with a general hfc tensor, and in particular we consider the isotropic shielding dependence on powers of 1/T separated into contributions from the isotropic and dipolar part of the hfc tensor, respectively. The possibility of g and A sharing the same principal axis system is not a suitable general assumption even for an axial complex. For instance, while the hfc tensor for a metal center or another atom lying directly on the principal magnetic axis is expected to be diagonal just like g and D, a ligand atom in a different position is not likely to afford a diagonal hfc matrix. However, this does not change the fact that in the absence of ZFS all shielding tensor contributions resulting from the electron paramagnetism are strictly proportional to 1/T, save for temperature-dependent changes in the Boltzmann populations of different electronic spin multiplets. With diagonal matrices for g and Z of the form shown in Eq. (18) but without further assumptions about the orientation of the principal axis system of the hfc tensor, we have * g⊥ g Z A = ... 0 ,0

0 g⊥ 0

0 + * Z⊥ 0 /// ... 0 g∥ - , 0

0 Z⊥ 0

0 + * A11 0 /// ... A21 Z ∥ - , A31

A12 A22 A32

A13 + A23 /// A33 -

*. g⊥ Z⊥ A11 g⊥ Z⊥ A12 g⊥ Z⊥ A13 +/ = .. g⊥ Z⊥ A21 g⊥ Z⊥ A22 g⊥ Z⊥ A23 // . (19) g Z A g Z A g Z A ∥ ∥ 32 ∥ ∥ 33 , ∥ ∥ 31 We define, as previously, Aiso = (1/3)Tr(A). Further, a hfc tensor anisotropy ∆A can be defined as ∆A = 31 (A33 − A11+2 A22 ). With these definitions, the isotropic average of the shielding tensor of Eq. (10) is given by  βe P σiso =− Aiso(2g⊥ Z⊥ + g ∥ Z ∥ ) 3k BT g N β N  + 2∆A(g ∥ Z ∥ − g⊥ Z⊥)   βe =− Aiso giso(Z ∥ + 2Z⊥) + 2∆g(Z ∥ − Z⊥) 3k BT g N β N   + ∆A 2giso(Z ∥ − Z⊥) + 2∆g(2Z ∥ + Z⊥) . (20)

0 −∆Z 0

0 + 0 /// . 2∆Z -

(18)

One may also substitute Z ∥ + 2Z⊥ = 3Ziso, Z ∥ − Z⊥ = 3∆Z, and 2Z ∥ + Z⊥ = 3(Ziso + ∆Z) in the last expression. Equation (20) results in terms depending upon Aiso and upon ∆A. Per our working definition, these constitute the contact and dipolar

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J. Chem. Phys. 142, 054108 (2015)

(pseudo-contact) contributions to the shielding, respectively. Both are discussed in Sec. III. For the simple case of A being diagonal as well, the supplementary material collects expressions for the parallel and perpendicular components of the resulting diagonal shielding tensor and plots of their temperature dependence.60 It is clear already from Eq. (20) that ZFS would have to lead to a cancellation of the zeroth-order terms of 1/T expansions of Z ∥ and Z⊥ in very specific ways in order to lead to a vanishing 1/T contributions to the dipolar shielding. III. ANALYSIS OF THE T -DEPENDENCE OF THE PNMR SHIELDING FOR S = 1, 3/2, 2, AND 5/2

Equation (11) was considered for systems with spin 1, 1.5, 2, and 2.5 based on the eigenfunctions and eigenstates of the ZFS tensor of Eq. (17). The corresponding expressions for Z ∥ and Z⊥ of Eq. (18) are read as follows: S=1:

S=

3 : 2

Z∥ =

2e−β D , 1 + 2e−β D

(21a)

Z⊥ =

2 −β D ) β D (1 − e , −β D 1 + 2e

(21b)

Z∥ =

1 + 9e−2β D , 4(1 + e−2β D )

(21c)

Z⊥ =

1 + 4β3D (1 − e−2β D ) 1 + e−2β D

1 S= : 2

S=1:

S=

3 : 2

S=2:

S=

5 : 2

,

(21d)

S=2:

Z∥ =

Z⊥ =

S=

5 : 2

Z∥ =

Z⊥ =

2e−β D + 8e−4β D , 1 + 2e−β D + 2e−4β D 6 βD

− 3β14D (e−β D ) − 3β4D (e−4β D ) 1 + 2e−β D + 2e−4β D

(21e)

,

(21f)

1 + 9e−2β D + 25e−6β D , 4(1 + e−2β D + e−6β D ) 9 + β8D − 2β11D (e−2β D ) − 2β5D (e−6β D ) 4(1 + e−2β D + e−6β D )

(21g)

. (21h)

Substituting these expressions in Eq. (20) for Z ∥ and Z⊥ and defining a common pre-factor Ciso as Ciso = −

βe 3g N β N

(22)

gives the expressions for the isotropic nuclear shielding and the contact (c) and dipolar (pseudo-contact, pc) contributions. We next examine the temperature dependence of the contact and dipolar contributions to the isotropic shieldings for these spin states by expansion of the relevant terms in Eq. (20) in powers of 1/T, truncated at the third order term. For comparison, we also show the S = 1/2 case.

 3giso , 4k BT   3∆g pc σiso = Ciso∆A , 4k BT   2giso 2D∆g D 2giso  c σiso = Ciso Aiso  − 2 2 − 3 3,  k BT 3k BT 9k BT    4∆g 2D(giso + ∆g) 2D 2∆g  pc σiso = Ciso∆A  − − ,  k BT 3k 2BT 2 9k 3BT 3    5(7giso + 2∆g) D(29∆g − 2giso) 2D 2(giso − ∆g)  c σiso = Ciso Aiso  − + , 12k BT  9k 2BT 2 27k 3BT 3    5(giso + 8∆g) D(29giso + 25∆g) 2D 2(giso − ∆g)  pc  σiso = Ciso∆A  − − , 6k BT  9k 2BT 2 27k 3BT 3    6giso 42D∆g D 2(16∆g − 7giso)  c σiso = Ciso Aiso  − +  ,  k BT 5k 2BT 2 5k 3BT 3    2 12∆g 42D(giso + ∆g) 2D (8giso + ∆g)  pc − σiso = Ciso∆A  −  ,  k BT 5k 2BT 2 5k 3BT 3    2 35giso 56D∆g 4D (27∆g − 7giso)  c σiso = Ciso Aiso  − +  ,  4k BT 3k 2BT 2 9k 3BT 3    2 35∆g 56D(giso + ∆g) 4D (27giso + 13∆g)  pc σiso = Ciso∆A  − +  .  2k BT 3k 2BT 2 9k 3BT 3  

c σiso = Ciso Aiso

(23a) (23b) (23c)

(23d)

(23e)

(23f)

(23g)

(23h)

(23i)

(23j)

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054108-7

B. Martin and J. Autschbach

Consider a situation where the anisotropy of the Zeeman coupling matrix, ∆g, is very small. ∆g = 0 does not necessarily imply that giso is close to the free-electron g-value. For instance, the octahedral 5 f 1 complex NpF6 has an isotropic g matrix, but because of the strong SO coupling in the 5 f shell of the metal giso = −0.6 is very far from ge . For orbitally non-degenerate spin multiplets and weak SO coupling, i.e., for many systems containing relatively light elements, the gfactors do not deviate much from ge , and ∆g is then also small simply because of the overall small g-shifts (deviations of the g-factors from ge ). In the event that ∆g is indeed vanishingly small, examination of Eqs. (23) reveals that for S = 1, 2, and 5/2, the term proportional to 1/T in the dipolar nuclear shielding vanishes, while the contribution proportional to 1/T 2 is non-zero as long as D and giso are non-zero. In turn, if ∆g = 0, the contribution proportional to 1/T in the contact nuclear shielding is non-zero for S = 1, 2, and 5/2 as long as giso is non-zero, while the 1/T 2 contribution vanishes if ∆g = 0. In this case, a separation of contact and dipolar shifts from experimental data via plotting the shifts versus 1/T and extracting linear and quadratic terms from a fit would indeed be possible. It would be important that giso D 2/(k BT)2 is sufficiently small such that the third order terms can be neglected. The 1/T n contributions to the pNMR shielding are proportional to D n−1. This means, for instance, that a 1/T 2 dependence of dipolar shifts can only be noticed if D is sizable. It also means, as stated already, that all shielding contributions other than those proportional to 1/T vanish if D = 0. The case of S = 3/2 shows that for non-zero D, there is at least one exception to the approximate 1/T versus 1/T 2 dependence of contact versus dipolar shifts. Here, even for ∆g = 0, the contact shielding has a non-zero 1/T 2 contribution and the dipolar shift has a non-zero 1/T contribution. If Aiso and ∆A are of equal magnitude and ∆g = 0, the 1/T 2 term in the dipolar shielding has a pre-factor of 29/2 compared to the 1/T 2 term in the contact shielding, and therefore an observed non-linear behavior of the chemical shift versus 1/T would still indicate the likely presence of dipolar shifts. However, the dipolar shielding also has a 1/T contribution that would carry a factor of 2/7 compared to the 1/T term in the contact shielding if Aiso and ∆A are of equal magnitude, i.e., a significant portion of the 1/T dependence would then be caused by the dipolar shielding. If ∆g is not small, Eqs. (23) show that a clean separation of contact and dipolar chemical shifts via temperature dependent NMR measurements and simple fits to 1/T and 1/T 2 is not possible for any of the pseudo-spin values that we have investigated. In fact, Bleaney already made a point that a dipolar shift proportional to 1/T 2 requires ∆g = 0. He assumed the presence of a multiplet with a given total angular momentum, the components of which were then split by a weak crystal field such that the populations were approximately equal, resulting in an effective ∆g ≈ 0. In the present theoretical framework, such a cancellation of 1/T terms in the dipolar shielding would result from averaging equivalent shielding expressions for different components of the multiplet, assuming their ∆g average to zero. The Bertini expression mentioned in the introduction, which requires magnetic anisotropy, corresponds to a leading term proportional to ∆g/T in the dipolar shielding seen for the various spin states in Eqs. (23).

J. Chem. Phys. 142, 054108 (2015)

FIG. 1. Comparison of contact (left) and dipolar (right) contributions to σ iso for a spin triplet, versus 1/T , using a D value of 50 cm−1 and ∆g /g iso values of 0.1 (upper panel) and 2.0 (lower panel).

An example is provided in Figure 1, where we assume S = 1, a D parameter of 50 cm−1 and compare the isotropic shielding versus 1/T over a range of 200 K for the case of very small ∆g/giso and a rather large ∆g/giso. In the former case, the plot for the contact term appears linear while the plot for the dipolar term exhibits curvature. With a large ∆g/giso that may be encountered with a heavy transition metal complex or an actinide complex, the situation is completely changed; now the plot for the contact shielding exhibits significant curvature while a large contribution in order 1/T appears to dominate the dipolar shielding.

IV. INFLUENCE OF THE RHOMBIC ZFS PARAMETER FOR S = 1

If the rhombic parameter E in Eq. (16) is nonzero but the energy of the multiplet is still globally shifted by +D/3, D is given by *. E D = .. 0 ,0

0 −E 0

0+ 0 /// . D-

(24)

The corresponding ZFS Hamiltonian Hˆ 0 reads in the basis of |S,MS ⟩ spin eigenfunctions for S = 1 *D ˆS · D · Sˆ = .. 0 . ,E

0 0 0

E+ 0 /// . D-

(25)

For E = 0, there is a non-degenerate eigenstate at zero energy, and a doubly degenerate level at energy +D. A non-zero rhombic ZFS parameter E breaks the degeneracy of the latter level. The eigenvectors and eigenvalues of (25) are |0⟩ = (0,1,0) with Eλ = 0, |−⟩ = √12 (−1,0,1) with Eλ = D − E, and |+⟩ = √12 · (1,0,1) with Eλ = D + E. Evaluating Eq. (11) with this set of eigenvectors and energies gives a diagonal matrix Z with the following principal values:

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054108-8

B. Martin and J. Autschbach

J. Chem. Phys. 142, 054108 (2015)

Z11 =

2 −β(D+E) ) β(D+E) (1 − e , −β(D−E) −β(D+E) (1 + e +e )

(26a)

Z22 =

2 −β(D−E) ) β(D−E) (1 − e , −β(D−E) −β(D+E) (1 + e +e )

(26b)

Z33 =

1 −β(D−E) − e−β(D+E)) β E (e . (1 + e−β(D−E) + e−β(D+E))

(26c)

The resulting contact and dipolar isotropic shielding expressions read

   2 2 (giso − ∆g) β(D+E) (1 − e−β(D+E)) + β(D−E) (1 − e−β(D−E)) Ciso  Aiso  = k BT 1 + e−β(D−E) + e−β(D+E)    (giso + 2∆g) β1E (e−β(D−E) − e−β(D+E))   , + 1 + e−β(D−E) + e−β(D+E)     2 2 −β(D+E) −β(D−E) (∆g − g ) (1 − e ) + (1 − e ) iso C β(D+E) β(D−E) iso pc ∆A  σiso = k BT 1 + e−β(D−E) + e−β(D+E)    (2giso + 4∆g) β1E (e−β(D−E) − e−β(D+E))   . + 1 + e−β(D−E) + e−β(D+E) 

c σiso

(27a)

(27b)

Expansion in powers of 1/T finally gives   2giso 2D∆g (D 2 + 3E 2)giso  c − 2 2− σiso = Ciso Aiso  (28a)  ,  k BT 3k BT 9k 3BT 3    4∆g 2D(giso + ∆g) 2(D2 + 3E 2)∆g  pc − σiso = Ciso∆A  −  .  k BT 3k 2BT 2 9k 3BT 3  (28b)

elocene has the structure shown in Fig. 2 and affords a spintriplet ground state. pNMR shifts were calculated at 298 K. The complex was also considered by PV, and we have included

We find that the parameter E does not explicitly enter the shielding expression until order 1/T 3. Lifting the degeneracy of the ZFS eigenstates of energy D leaves the terms in order 1/T and 1/T 2 unaffected such that in the case of ∆g = 0 it is still possible to separate the two pNMR shielding mechanisms by their dependence of inverse powers of T. Equations (28a) and (28b) exactly reduce to Eqs. (23c) and (23d) for vanishing E.

PBE0a PVb Expt.c

TABLE I. Calculated EPR g -factors and ZFS for nickelocene (S = 1). g iso

∆g

D (cm−1)

2.0620 2.0454 2.04

−0.0164 −0.0214 −0.02

1.62 104.4 25.6

a Present

calculations using the ADF code with Slater-type basis functions and the PBE0 hybrid functional for g -factors, hfc, and orbital shielding. The ORCA code was used to calculate the ZFS parameter. Details are provided in the supplementary material.60 b PV Gaussian03 calculations with ZFS obtained from ORCA calculations, using the PBE0 functional with the IGLO-III basis set for the ligand atoms and the basis set constructed by Munzarová and Kaupp61 for the metal center.4 c Solid state measurements, Ref. 62.

V. AN APPLICATION: NICKELOCENE

This section reports NMR and EPR parameters for Nickelocene, calculated with density functional theory (DFT). Nick-

TABLE II. Calculated hfc, NMR shielding, and chemical shifts for nickelocene, from Eq. (14) based on the experimental value of D.

PBE0a PVc Expt.

H C H C H C

orb (ppm) σ iso

A iso (MHz)

∆A (MHz)

δ (ppm)

26.44 95.25 26.30 90.31

−3.61 4.85 −3.69 5.33

0.14 4.16 0.02 3.96

−258.15b 1427.46b −261.20b 1557.22b −257d 1514e

a Present

FIG. 2. Optimized structure of nickelocene.

calculations using the ADF code with Slater-type basis functions and the PBE0 hybrid functional for g -factors, hfc, and orbital shielding. Chemical shifts were computed at 298 K with reference to Tetramethylsilane (TMS) calculated at the same level of theory (1H 31.45 ppm 13C 187.95 ppm). b For comparison, if the equation of PV is used instead of Eq. (14), the calculated shifts are (1H −258.38 ppm 13C 1427.63 ppm) and (1H −261.44 ppm 13C 1557.57 ppm) for PBE0 and PV, respectively. c Data from Pennanen and Vaara.4 d In solution, Ref. 63. e In solution, Ref. 64.

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054108-9

B. Martin and J. Autschbach

J. Chem. Phys. 142, 054108 (2015)

FIG. 3. Nickelocene: Variation in the total isotropic shielding and the contact and dipolar contributions versus 1/T , based on the data presented in Table II.

FIG. 4. Nickelocene: Variation in the total isotropic shielding and the contact and dipolar contributions versus 1/T in the range 200–400 K, based on the data presented in Table II. The absolute temperatures (K) are also indicated.

the basic parameters reported in their article4 for comparison (Table I). However, the pNMR shifts were calculated based on Eqs. (9) and (1) in both cases rather than using the PV shielding expression together with the PV data. Due to the large differences in the calculated values of D, the shielding values in Table II are based on the experimentally determined value of 25.6 cm−1.62 Corresponding shielding values using the PV Equation (12) for the elements of Z can be found in the footnotes of Table II. As one might expect from the small ZFS, at ambient temperatures, the numerical differences between the SvH and PV expressions are very small. The temperature dependence of the calculated shieldings is graphically displayed in Figs. 3 and 4. The calculated carbon and proton chemical shifts for nickelocene are in reasonable agreement with experimental data, given the known sensitivity of hfc tensors to approximations in density functionals and to the fraction of exact exchange in hybrid functionals (25% for PBE0). Also, the signs are correctly obtained by the calculations. As far as the T-dependence is concerned, due to this system having S = 1 and axial symmetry with E = 0, and ∆g being very small, the calculated shielding components do vary in leading order as 1/T for contact and 1/T 2 for dipolar at not too low temperatures. For temperatures below ca. 50 K (1/T = 0.02 K−1), higher order terms in the 1/T expansion become important and the temperature dependence becomes much more intricate. VI. CONCLUSIONS

A separation of pNMR contact and dipolar chemical shifts in terms of their dependence on different leading powers of 1/T is approximately possible under limited circumstances (i) S ≥ 1 and D being reasonably large is a requirement. (ii) Some values for S appear to be excluded from the list of spin multiplets for which a separation is possible, with S = 3/2 being a verified example. (iii) ∆g being very small compared to D/(k BT) is a requirement for a possible separation in case of the other spin values considered herein. Based on these findings, an assignment of contact and dipolar shifts based on the slope

and curvature of the dependence of VT pNMR shifts when plotted versus 1/T would be unreliable unless it is also verified by theoretical considerations and/or additional measurements of EPR parameters that the separation is valid and that higher order terms can be neglected. In situations where a separation is possible, VT NMR measurements would give access to contact and dipolar shifts separately. Taking the example of a complex with a paramagnetic metal center, knowledge of the relative contributions from contact versus dipolar mechanisms would then give information about the extent of metal-ligand covalent interactions. ACKNOWLEDGMENTS

This research was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Heavy Element Chemistry program, under Grant No. DE-FG02-09ER16066. We further acknowledge computer time and technical support by the Center for Computational Research (CCR) at the University at Buffalo. 1B.

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