NMR Shifts. III computing Knight shifts and chemical shifts:

Transcription

1 NMR Shifts. III cmputing Knight shifts and chemical shifts: ) Shift mechanisms, hyperfine fields, Susceptibility: While K and d are frmally identical, typically K refers t shifts in metals r due t cnductin electrns (nrmally electrn spinrelated), while d refers t effects in insulatrs and due t nn-cnductin electrns (rbitalrelated). Here I discuss mstly the main terms: the cntact interactin, which causes Knight shifts in simple metals (giving K p r the Pauli Knight shift); the diamagnetic shielding, a chemical shift term imprtant fr insulatrs, ntatin d dia (ften negligible, in metals), and the rbital shift (K rb ) r paramagnetic chemical shift (d para ). Smetimes this term can be defined as a Knight shift r chemical shift, depending n whether we have an insulatr r metal; they are really ne and the same. Shift mechanisms mitted frm this list include: ntably fr transitin metals the cre plarizatin fr this see fr example the Carter et al. review and als the small spin-diplar shift (K dip ) and the nuclear-nuclear shift (d n ); these are mentined briefly belw. Furthermre nte that spin rbit interactins are neglected here, allwing us t make the apprximatin that the spin and rbital susceptibilities are cmpletely separate. Characterizing the shifts in slids where spin-rbit cupling is significant (as might be the case with heavy atms present) can be difficult, and this is a tpic f current research. Understanding f spin-rbit effects is smewhat mre develped fr mlecular systems, as fr example discussed by Pyykkö, Ther. Chem. Acc. 0, 4-6 (000). Generically, the magnetic shifts can be expressed in the general frm, K i χ i r δ i χ i, [6] where c is the electrn susceptibility (per atm), fr spins r rbital mments f a particular lcalized rbital, and H HF represents a hyperfine field fr the specific interactin. H HF is the effective magnetic field at the nucleus per electrn mment. Nte that we are in CGS units; c per atm has units cm, and H HF measures the field per mment (G/µ has units cm - ). Hence K and d btained frm [6] are dimensinless as wuld be expected. Fr anistrpic interactins, the spin prtin f H HF is typically replaced by the spin hyperfine tensr, represented by A. A appears in the sectin belw and nte that it has units f energy (hyperfine field multiplied by γ ). ) Hyperfine fields; Fermi cntact and ther spin shifts: Cnsider first the Hamiltnian representing lw-rder interactins f the nucleus (I) with electrn spin (S) and rbital (L) mments: H = I i J I j + I i A S j + γ e Ii L mmc r j + A term. [7] The last term is nt always included, but as we see belw it cntributes t diamagnetism and the diamagnetic shift. The first term includes direct diple-diple interactins between nuclei as well as indirect (electrn-mediated) parts, but it des nt nrmally lead t net shifts s it will be disregarded belw. [T the extent that neighbring nuclei have nnzer plarizatin this term may give an additinal shift (d n ), hwever its cntributin is almst always verwhelmed by larger shifts.] A is the hyperfine spin tensr described abve, nt the same as the vectr ptential A appearing in the last term. A generally cnsists f the usual diple field plus the delta-functin Fermi cntact term:

2 (I H IS = γ gµ i ˆr)(S j ˆr) I i S j 8πδ (r) B I r i S j. [8] The sum extends ver the N nuclear sites i as well as all f the electrns j; fr energy per site we shuld divide by N. Hwever since the nuclei act independently I dispensed with the i summatin belw. [8] lks like a Zeeman interactin and can be expressed frmally via an effective lcal field: H γ I H lc. [9] The first term in [8] is nrmally small but it can be imprtant fr transitin metals with partially filled d bands, where it results in the diplar Knight shift (K dip ). The cntact term in [8] cnnects the nucleus t the wavefunctin at r = 0 (r within the vlume f the nucleus). Fr practical purpses this is relevant fr s-electrns nly. Since electrn spin dynamics are much faster than NMR time scales, the shift crrespnding t an energy difference between I z m-states is prprtinal t the average s-rbital spin S z : 8π ϕ s () E m = ψ H ψ m = γ gµ FS B m S z. The FS subscript designates an average ver participating states n the Fermi surface. In terms f the s-partial Pauli susceptibility (given as susceptibility per atm), S z = χ p ( ) s H / (gµ B ), and the energy average becmes ( χ p ) s H. Finally, ne can see that the definitin [] crrespnds t E m = γ m 8π ϕ () s FS H lc H K, and cmbining with [9] fr m = we can btain the psitive shift, K p = 8π ϕ s() FS ( χ p ) s. [0] Psitive means that f is greater than zer ( dwnfield shift ). The curly-bracket term in [0] is the same as H HF. Fr a given species, atmic-based values f the Fermi-cntact H HF ften may suffice as a reasnable apprximatin, and a table f such fields are listed fr example in Carter et al. (e.g. H HF = 0 MG is fund fr valence s electrns in indium, which is the effective field at the nucleus if a single unpaired electrn ppulated this rbital). K p is istrpic since the spin susceptibility is directin-independent. Diplar spin cntributins t K are in general anistrpic due t the directinal dependence f the diple interactin, hwever nrmally the presence f shift anistrpy indicates the cntributin f a paramagnetic rbital shift, discussed later. Recnstructing the lgic frm the last paragraph, we see that K (r d) can be btained by cmputing ΔE nucl / (γ H ) prvided E crrespnds t a term linear in γ and in H ; such linearity was assumed abve. Mre generally, this can be expressed at least fr istrpic cases, K = E µ H r δ = E µ H, [] where µ = γ is the nuclear mment. The average energy refers t that f a particular nuclear state. T cmpare, it is useful t recall that the susceptibility is als be btained by a similar derivative, χ = M H E, [] V H

3 where the thermdynamic relatin in [] gives the vlume susceptibility (unitless in cgs). The magnetizatin is frmally related t the derivative f the free energy, but in this case it suffices t use the average energy. Other spin-shift terms aside frm Fermi cntact may appear when, fr example, unpaired spins lead t a Curie-type paramagnetic susceptibility. In such cases [0] may be replaced by, K para χ Curie, where H HF here refers t the relevant lcal interactin between the nucleus and paramagnetic spin. This leads t an bvius Curie-like dependence f the shift and linewidths. In cases with n direct verlap, the diple interactin may prvide the spin-nuclear cupling t the paramagnetic spins, althugh in such cases because f the symmetry f the diplar field, in istrpic systems the net shift will be zer. Returning t the Knight shift [0], nte that χ p represents the Pauli susceptibility, χ p µ B g(ε F ) in the limit f weak interactins in simple metals, hwever the derivatin is mre general and χ p in [0] can be replaced by the exact slutin as lng as it is limited t the n-site s-partial term fr the site f interest. Fr example with e-e interactins the Stner-enhanced susceptibility may apply, χ χ ( Ug(ε F )). Krringa relaxatin thery applies in these cases, giving / T T, and this can be used as a means t separate the spin parts f K, since the rbital shift mechanisms defined belw d nt lead t relaxatin prcesses in slids. Hwever, nte that the simple Krringa-prduct K T T = γ e (4πk B γ n ) is numerically exact nly in the limit f weak interactins. See als the review f van der Klink and Brm, Prg. Nucl. Magn. Resn. Spectrsc. 6 (000) 89 0 fr mre infrmatin. ) Orbital shifts and perturbatin thery: Hamiltnian: Cnsider the magnetic field H el felt by the electrn (at r) due t the nuclear diple µ plus external field H : its the vectr ptential is, A = ( H r) / + ( µ r) / r. [] [This is in the symmetric gauge centered at the nucleus.] The vectr ptential augments the electrn mmentum peratr: p = ( ih + ea / c). Squaring t btain the Hamiltnian, H = p / m, yields terms linear and bi-linear in A: H = H 0 ie A + e mc mc A H 0 + H + H. [4] We will see that H yields diamagnetic shifts, and H yields paramagnetic shifts. (a) Electrn susceptibilities: It is instructive t derive these terms first. Nte that in this case we cnsider nly the H term in A, disregarding the nucleus. Frm [] we see that terms in the energy bilinear in H are needed. The diamagnetic term cmes frm H = e mc A = e 8mc (H rsinθ), and the crrespnding average energy is E = ψ H ψ. The derivative [] yields the expressin, χ dia = e 4mc ψ x + y ψ, [5]

4 where ψ represents the grund state wavefunctin (all-electrn, includes cre as well as valence states). The result is the per-atm susceptibility. Fr the paramagnetic shift the relevant term is, H = ie A = ie mc mc ( r H ) = ie mc ( r) H = µ BL H. [6] Because f quenching f angular mmentum, the grund state fr nn-magnetic materials nrmally has zer expectatin value fr. S, the first-rder perturbatin term ΔE = 0 H 0 vanishes, and we need secnd-rder perturbatin thery, where the crss term bilinear in H is: ( ) ψ ΔE = µ B H. [7] i E E i The sum is ver all excited states. Frm the derivative [] we btain the Van Vleck susceptibility: ψ i ψ χ VV = µ B. [8] i E i E The term is always psitive (ntice that I reversed the denminatr, and that E i > E ). It is smetimes called temperature independent paramagnetism, r TIP. It is imprtant if the filled and unfilled valence states cme frm the same L 0 shell, s that the matrix element is nn-zer, and largest when there is a small gap between these states. (b) Diamagnetic chemical shift ( Shielding ): Cmparing t the derivatin f χ dia abve, the relevant crss term in H linear in bth µ and H (& see []) is ΔH = e H µ (rsinθ). As befre ΔE = ψ 8mc r H ψ is the firstrder energy difference, and the derivative [] yields δ dia = e mc ψ x + y ψ r. [9] The diamagnetic nuclear shielding depends nly n the shape and density f the electrn clud, and all electrns cntribute, including cre states, s states, etc. Fr filled shells, r a pwder average, we btain the istrpic shift, δ dia = e mc ψ r ψ. The cntributin f a single rbital with knwn r wuld be δ dia = e mc r. Cmparing [6], ntice that this can be put in the standard frm δ i χ i, with H HF = r r. Nte als that deep cre states wuld likely require a Dirac-equatin treatment, hence sme numerical crrectins, althugh in NMR we measure shifts relative t a reference cmpund, fr which the cre cntributin is presumably nearly unchanged. (c) Paramagnetic term: 4

5 Cmpare [6], but with the nuclear diple term als included in A ([]), we can see that the relevant term is ΔH = e H L mc + µ r, and the energy linear in bth µ and H is: H ΔE = µ µ { ψ 0 / r ψ + c.c. } B [0] i E E i The derivative [] yields: { ψ 0 / r ψ + c.c. } δ para = µ B. [] i E i E This term is psitive definite, hence paramagnetic. Sum is ver all excited states, which makes the matrix elements challenging t calculate. Only p r d (r f) states cntribute because f the angular mmentum matrix elements. δ para may be large in transitin metals where filled and empty d states are clse in energy. It is als in general anistrpic, since the superpsitin f angular mmentum sub-states cntained in the lcal rbitals in will change upn change f crystal rientatin. Ntice als that as lng as a single rbital dminates the matrix elements we can write δ para χ VV, and fr this case H HF =. Estimates f the /r average can be r fund in tables f atmic functins (example: Fraga et al., Handbk f Atmic Data is in the TAMU library) and used fr estimating the shift vs. susceptibility. (d) Gauge prblem and extended crystal systems: The derivatins abve fr rbital susceptibilities and chemical shifts may have additinal challenges when extended t slid-state systems, when the vectr ptential [] is defined in a gauge centered arund a particular nucleus. In neighbring cells the basic relatinships are in principle still valid, but the cntributins pick up large additinal terms that cancel but make calculatins smewhat impractical. A number f imprved methds have been intrduced in recent years t address these and related issues 4,5,6. The methd intrduced fr FLAPW cmputatins, and implemented in the WIENk cde 5 uses the alternative current density frmulatin f the shifts, as described belw. Cnceptually, it may be helpful t examine the matrix elements invlved in paramagnetic shielding fr slids frm [] by cnsidering that the wave functins ψ and are cnstructed f lcalized rbitals with relatively small verlap = ϕ(r j ) e i k R j / N (tight binding limit). One methd t address the gauge prblem is t re-define the center f the gauge in each unit cell, 5 in which case the Hamiltnian appears unchanged cmpared t that f the islated atm. Since the peratr acts nly upn the lcal rbitals, the integral ψ 0 will be nnzer nly fr a vertical (virtual) transitin in k-space acrss the band-gap. This is similar t the (real) interband transitins familiar in ptics (see figure) and such states are als the nes cupled by the paramagnetic shift matrix elements in weakly interacting systems. 5

6 Vertical transitin (e) Current density: In QM the current density is derived by expanding the time derivative: t ψ ψ =ψ * t + ψ* t ψ. [] Since ψ ρ is the prbability density, we can frmulate a cntinuity equatin, ρ t = j, where j is a prbability current. Derivatives n the right f [] are btained frm the Schrödinger equatin and frm [4], t ψ = i i Hψ = + m + ie c A ψ i (U + µ BH S)ψ. [] With a little algebra we btain, i j = m ψ * ψ ( ψ * )ψ + e Aψ * ψ. [4] mc Defining the vectr ptential as in [], it turns ut that the first term results in the paramagnetic susceptibility, and the secnd the diamagnetic part. Diamagnetic susceptibility: Cnsider fr simplicity a spherically symmetric atm. With H alng +z, the vectr ptential [] circulates in the +f directin. The electric current density is e times [4], giving current lps that cntribute t an induced mment ppsing the applied field. A lp at (r, q) with crss-sectin r dr dq (see sketch belw) has radius r sin q, and magnetic mment (cgs units): d µ = I area ( ẑ) = e c mc Aψ * ψ (r dr dθ)πr sin θ( ẑ), [5] taking the last term frm [4]. Using A = H rsinθ /, and integrating, µ = ( ẑ) e H 4mc e πr dr dθ sinθ r sin θψ * ( ψ ) = ( ẑ) 4mc H r sin θ which is equivalent t [5] multiplied by H. Thus χ dia per atm is the same as derived befre. The first term in [4] similarly leads t the paramagnetic term, althugh nte that the grund state ψ has n net circulatin, s the first-rder perturbed wavefunctin must be used in this case. [6] 6

7 Current density and vectr ptential fr χ dia calculatin. The chemical shifts can be btained by using the current [4] plus the Bit-Savart law t btain the magnetic field acting n the nucleus. This is the basis fr the methd used in refs. 6. Nte that the abve-mentined gauge prblems must still be addressed fr extended slids; Laskwski and Blaha address this 6 using a methd f Pickard and Mauri 4 t refrmulate the tw cntributins t the induced current int a single expressin that is gauge invariant. The resulting DFT-based current density depends entirely upn btaining an accurate grund-state electrn density and its respnse t a magnetic field. The secnd paper 6 addresses the results btained frm different DFT functinal chices in the FLAPW methd, and it appears frm these reprts that the PBE functinal yields results within abut 80% f the crrect results, with hybrid functinals surprisingly giving results that are nt particularly imprved. The tentative cnclusin is that these discrepancies can be attributed in mst part t difficulty in crrectly estimating energies f excited states (which affects the results even thugh the specific methd used invlves directly calculating the field-dependence f the grund-state wavefunctins, rather than calculating unccupied states as in intermediate step fr perturbatin thery, as in the methd leading up t eqn. []). Numbered references: C. P. Slichter Principles f Magnetic Resnance, (Springer, 996); G. C. Carter, L. H. Bennett, and D. J. Kahan, Metallic Shifts in NMR (Pergamn, New Yrk, 977). Shastry and Abrahams, Phys. Rev. Lett. 7, 9 (94); Ishikagi and Mriya, J. Phys. Sc. Japan 65, 40 (96); R. E. Walstedt, The NMR Prbe f High-T c Materials (Springer, 008). Fr chemical shift thery in general, J. D. Memry, Quantum Thery f Magnetic Resnance Parameters (968); And & Webb, Thery f NMR Parameters (98); Calculatin f NMR and EPR Parameters, ed. Kaup, Bühl, Malkin (004). Fr verview f Knight shifts & metals specifically, gd places t start are Winter, Magnetic Resnance in Metals (97); A. Narath in Hyperfine Interactins (ed. Freeman & Frankel, Academic, 967) p. 87; als references () abve r the Walstedt bk cited abve in ref. (). 4 F Mauri, B G Pfrmmer, S G Luie, Phys. Rev. Lett. 77, 500 (996); C J Pickard, F Mauri, Phys. Rev. B 6, 450 (00). 5 J R Cheeseman, G W Trucks, T A Kieth, M J Frisch, J. Chem Phys. 04, 5497 (996). 6 R Laskwski, P Blaha, Phys. Rev. B 85, 05 (0); R Laskwski, P Blaha, F Tran, Phys. Rev. B 87, 950 (0). 7