Prediction of NMR Parameters in the Solid-State

1 JPO

v. close to axial symmetry about Prediction of NMR parameters in the the P-O bond solid-state Jonathan Yates

Materials Modelling Laboratory, Oxford Materials 2 JPSi

1 ‘z’ component slightly tilted from the internuclear vectors. Some deviation from axial symmetry

Wednesday, 14 April 2010 Nuclear Magnetic Resonance

Zero Field Applied Field

Population differences NMR signal is proportional to population difference •! For 1H in a 9.4 T field at 298 K For 1H in 9.4T at 298K

"E = 2.65 # 10–25 J ω = γ B

nupper/nlower = exp (–"E / kT) magnetogyric ratio nupper/nlower = 0.999935 fundamental nuclear constant Sensitivity is an issue! Need Large γ and/or B

•! Populations differences in NMR are very small – sensitivity is always a concern •! Sensitivity is helped by higher magnetic field strengths and higher ! nuclei

•! Need to generate a strong magnetic field (~2 to 28 T) •! Use a superconducting wire (~100 km for 850 MHz)

NbTi 10 K 15 T ~£10 / m

Nb3Sn 18.3 K 30 T ~£50 / m Common Isotopes

ϒ Freq Abundance Isotope Spin 107 T-1 rad s-1 MHz @ 9.4T %

1H 1/2 26.75 400 100

13C 1/2 6.73 100.6 1.1

29Si 1/2 -5.32 79.6 4.7

31P 1/2 -10.84 162.1 100 Common Isotopes

ϒ Freq Abundance Isotope Spin 107 T-1 rad s-1 MHz @ 9.4T %

1H 1/2 26.75 400 100

13C 1/2 6.73 100.6 1.1

29Si 1/2 -5.32 79.6 4.7

31P 1/2 -10.84 162.1 100

17O 5/2 -3.63 54.1 0.04

49Ti 7/2 -1.51 22.6 5.41 Solid-StateSuperconducting NMR Spectrometers magnets

600 MHz 14.1 T 400 MHz 9.4 T ~£800,000 ~£300,000 Image courtesy Sharon Ashbrook (St Andrews) Magnetic Shielding

Flurbiprofen non-steriodal anti-inﬂamatory

6 (ω ωref ) 10 δiso = − × ωref

chemical shift (ppm)

δiso = σref σiso Each distinct C atom experiences a different magnetic field and − resonates at a unique frequency. Measure the change wrt a standard (for 13C this is liquid magnetic shielding tetramethylsilane) (from calc) 2 equivalently and by convention, within the 1st Brillouin planewave/pseudopotential formalism - it allows individ- Zone. ual operations to be performed in the most efficient basis, The problem of solving for an infinite number of elec- e.g. the kinetic energy operator in Fourier space, and a trons has become one of calculating for a finite number local potential in real space. of bands at an infinite number of k-points. However, physical properties are expected to be smoothly varying functions of k and hence many integrals can be well ap- D. The pseudopotential approximation proximated by a finite sampling of k. A common scheme for Brillouin Zone integration are the sets of regular in- The electrons in an atom can be divided into two types tegration grids introduced by Monkhorst and Pack. — core electrons and valence electrons. The core electrons are tightly bound to the nucleus, while the valence electrons are more extended. A working definition for 1. The supercell approximation core electrons is that they are the ones which play no part in the interactions between atoms, while the valence elec- The application of periodic boundary conditions forces trons dictate most of the properties of the material. It periodicity on the system studied. To address systems is common to make the frozen core approximation. The which do not have full three dimensional translational core electrons are constrained not to differ from their symmetry — for example the study of defects, impuri- free atomic nature when placed in the solid state envi- ties or even the interaction of molecules and surfaces, ronment. This reduces the number of electronic degrees the supercell approximation can be used. Aperiodic sys- of freedom in an all electron calculation. It is a very tems are approximated by enclosing the region of in- good approximation. A different, but physically related, terest in either bulk material (for a defect) or vacuum approach is taken in the pseudopotential approximation. (for a molecule) and then periodically repeating this cell Since, in an all electron calculation, the valence elec- throughout space (Figure??) The supercell must be large tron wavefunctions must be orthogonal to the core wave- enough for the fictitious interactions between neighboring functions they necessarily have strong oscillations in the cells to be negligible. region near the nucleus (see the all electron wavefunction in Figure 3). Given that a planewave basis set is to be used to describe the wavefunctions, these strong oscilla- C. The planewave basis set tions are undesirable — requiring many plane waves for an accurate description. Further, these oscillations are of In order to solve the eigenvalue problem of Equation 2 very little consequence for the electronic structure in the numerically the eigenstates must be represented by some solid, since they occur close to the nucleus, and interact basis set. While there are many possible choices, the one little with the neighboring atoms. In the pseudopotential made here is to use planewaves as the basis. There are approach only the valence electrons are explicitly consid- many advantages in the use of planewaves: they form ered, the effects of the core electrons being integrated a mathematically simple basis, they naturally incorpo- within a new ionic potential. The valence wavefunctions rate periodic boundary conditions, and, perhaps most need no longer be orthogonal to the core states, and so importantly planewave calculations can be taken system- the orthogonality oscillations disappear, hence far fewer atically to convergence as a function of the size of the plane waves are required to describe the valence wave- basis. The Kohn-Sham eigenstates are expressed as, functions. n n i(k+G).r Ψk(r) = ck(G)e , (5) !G IV. CHEMICAL SHIELDING IN A PERIODIC SYSTEM where the sum is over all reciprocal lattice vectors G. To truncate the basis set the sum is limited to a set of When a sample of matter is placed in a uniform, exter- reciprocal lattice vectors contained within a sphere with Induced Current a radius defined by the cutoff energy, E : nal magnetic field electronic currents flow throughout the cut material. For an insulating, non-magnetic material only !2|k + G|2 the orbitalTom computeotio nthe ochemicalf the shiftselect we justro needns toco calculatentribut e to this cur- ≤ E . (6) the current induced by the external magnetic field 2m cut rent. The current j(r), produces a non-uniform induced magnetic field in the material, which is given by the Biot- Hence, the basis set is defined by the maximum kinetic Savart law as energy component it contains. Physical quantities can be Biot-Savart r r! converged systematically by increasing Ecut. (figure of 1 3 ! ! − Bin(r) = d r j(r ) × . (7) shielding convergence) A Fast-Fourier Transform (FFT) c " |r − r!|3 can be used to change the representation of the eigen- The chemical shielding tensor is defined as the ratio be- states from a sum of Fourier components to to a uniform tween this induced field, and the external applied field grid of points in the real-space unit cell. The use of nu- Obtain current within perturbation merically efficient FFTs is one key to the success of the Bin(r) = −!σ(r)Bext. (8) theory (linear response)

(0) (1) 2 O = O + O + (B ) Bin = -σ B0 O note: σ is a rank 2 tensor Planewaves and Pseudopotentials

To simulate periodic systems planewaves are a convenient choice: however describing the tightly bound core states, and oscillatory part of the valence states close to the nucleus are prohibitively expensive. We must approximate...

Frozen Core Approximation • “Core” electrons taken from free atom ﬁxed during calculation

Pseudopotential Approximation • Valence electrons experience weak effective potential in the core region

Note: Typically these two approximations are used together. But this does not have to be the case. Some codes can employ ʻself-consistentʼ pseudopotentials which allow the core states to ʻrelaxʼ to their speciﬁc environment. Atomic States

Si Approximations

Overcoming the previous approximations

Frozen Core Approximation • Contribution of “core” electrons to shielding is not chemically sensitive 1s states in Carbon contribute ~200ppm in diamond, benzene, proteins ie core states contribute to shielding - but not shift.

Pseudopotential Approximation • Use PAW method to ﬁx-up valence wavefunction in the core region COMPUTATIONS OF MAGNETIC RESONANCE PARAMETERS FOR CRYSTALLINE SYSTEMS: PRINCIPLES 5 COMPUTATIONS OF MAGNETIC RESONANCE PARAMETERS FOR CRYSTALLINE SYSTEMS: PRINCIPLES 5

The shielding of the reference standard, σ iso, could be iso ref 800 35 The shielding of the referenceobtained standard, by performingσref , could a be calculation on800 the35 standard com- obtained by performing a calculation on the standard com- 30 pound itself, but this is often not convenient—for30 example, if 600 pound itself, but this is often not convenient—for example, if 600 the reference is a liquid or solution. For this reason, most stud- 25 H the reference is a liquid or solution. For this reason,iso most stud- 25 H C ies have obtained σref as the intercept of a line of unit gradient 400 ies have obtained σ iso as the intercept of a line of unit gradient 400 20 C F ref ﬁtted to a graph of calculated shieldings against20 experimental 20 25F 30 35 Si ﬁtted to a graph of calculated shieldingsshifts. against experimental 20 25 30 35 200 Si P shifts. 200 P 0 0 GIPAW-USP (ppm)

3.2 Core States and PseudopotentialsGIPAW-USP (ppm) COMPUTATIONS3.2 Core States OF MAGNETIC and Pseudopotentials RESONANCE PARAMETERS FOR CRYSTALLINE SYSTEMS: PRINCIPLES 5 −200 COMPUTATIONS OF MAGNETIC RESONANCE PARAMETERS FOR CRYSTALLINE SYSTEMS: PRINCIPLES 5 In the planewave-pseudopotential approach,−200 it is implicit that In the planewave-pseudopotentialiso approach, it is implicit that 400 The shielding of the reference standard, σ , couldthe be core electrons35 can be treated separately from the valence − ref 800 iso 400 −400 −200 0 200 400 600 800 obtained by performingthe core a electrons calculation can on the beThe standard treated shielding com- separately of the reference from standard, the valenceσref , could be − 800 35 obtained bystates performing in an a calculation atomic30 code. on the Historically, standard com- the validity−400 of−200 the frozen0 200 400 600 800 pound itself, butstates this is in often an not atomic convenient—for code. Historically, example, if the validity600 of the frozen 30 (a) All electron shielding (ppm) pound itself,core but this approximation is often not convenient—for for NMR example, properties if (a) had600 been doubted,All but electron shielding (ppm) the reference is a liquid or solution. For this reason, most stud- 25 H coreiso approximation forthe NMR reference properties is a liquid or had solution. been For doubted, this reason, but most stud- 1125 H ies have obtained σ as the intercept of a line of unit gradienta carefuliso study400 by Gregor, Mauri, and Car Cshowed that, ref ies have obtained σ as the intercept11 20 of a line of unit gradient 400 F C ﬁtted to a grapha of careful calculated study shieldings by againstGregor, experimental Mauri,ref and Car showed20 25 that,30 35 20 800 F35 ﬁtted to a graphif the of calculated core and shieldings valence against experimental states are partitioned35 20Si in a25 gauge-30 35 shifts. if the core and valence states are partitioned200 in a gauge- 800 P Si shifts. invariant way, the shielding of the core electrons200 is chemically P30 invariant way, the shielding of the core electrons0 is chemically 30 600

insensitive.GIPAW-USP (ppm) The contribution of the core electrons600 0 to chemical 3.2 Core Statesinsensitive. and Pseudopotentials The contribution of the core electrons to chemical GIPAW-USP (ppm) 25 3.2 Core Statesshifts and can Pseudopotentials hence be neglected. If absolute25 shieldings are H −200 −200 400 H C shifts can hence be neglected. If absolute shieldings are 20 In the planewave-pseudopotential approach,In it is the implicit planewave-pseudopotentialrequired, that the core approach, contribution it is implicit is that most conveniently400 calculated C F required, the core contribution is most conveniently calculated 20 20F 25 30 35 the core electrons can be treated separatelythe from core the electrons valenceby setting can be treated the−400 separately gauge from origin the valence at the atomic−400 center,20 25 so that 30 35 Si −400 −200 0 200 400 600−400 800−200 0 200 400 600200 800 Si states in an atomicby code. setting Historically, the gauge the validitystates origin ofin an the atatomic frozen the code. atomic Historically, center, the validity so that of the frozen P the shielding(a) is purely diamagneticAll electron shielding and (ppm)(a)200 given by theAll Lamb electron shielding (ppm) P core approximationthe for shielding NMR properties is purely hadcore been diamagnetic approximation doubted, but for and NMR given properties by had the been Lamb doubted, but a careful study by Gregor, Mauri, and Cara11 carefulshowed studyformula, that, by Gregor, Mauri, and Car11 showed that, 0

formula, 2 35 GIPAW-USP (ppm) if the core and valence states are partitionedif the in core a gauge- and valence states800 are35 partitioned inµ ae gauge- n(r)0800 µ e2 n(r) 0 3 GIPAW-USP (ppm) invariant way, the shielding of the core electronsinvariant is chemically0 way, the shielding3 of the core30σ electronsij δij is chemicallyd r 30 (23) σ δ d r (23) 600 insensitive. The contribution of the coreij electronsinsensitive.ij to chemical The contribution of600 the core electrons= 4 toπ chemicalme r −200 = 4πme r 25 ! −200 25 shifts can hence be neglected. If absoluteshifts shieldings can hence! are be neglected. If absolute shieldings are H H As shown in400 Figure 2, the use of pseudopotentials400 C implies C required, the core contribution is most20 conveniently calculated 20 F required, the coreAs contribution shown is in most Figure conveniently 2, the calculated use of pseudopotentials implies F20 25 30 35 −400 by setting thea gauge nonphysical origin at the form20 atomic for 25 center, the 30 so wavefunction that 35 −400 inSi the region 400Si 200 by setting the gauge origin at the atomic center, so that 200 − P − 0 200 400 600 800 a nonphysical formthe for shielding the wavefunction is purely diamagnetic200 in and the given region by the Lamb −400 P −200 0 200 400 600 800 the shielding is purely diamagnetic and given by theclose Lamb to the nucleus. For this reason, a formalism based on (b) All electron shielding (ppm) formula, close to the nucleus. Forformula, this reason, a formalism based on (b) 0 All electron shielding (ppm) pseudopotentials2 0 might appear a poor choiceGIPAW-USP (ppm) for the calculation 2 µGIPAW-USP (ppm) 0e 3 n(r) pseudopotentialsµ0e might3 n(r) appear a poorσij choiceδij for thed r calculation (23) σij δij d r of(23)nuclear= 4πmagneticme r resonance parameters.−200 The upper part of Figure 4 Isotropic magnetic shielding for nuclear sites in a range of of nuclear= 4πmagneticme r resonance parameters. The−200! upper part of Figure 4 Isotropic magnetic shielding for nuclear sites in a range of ! As shownFigure in Figure 4 2, shows the use of shieldings pseudopotentials computed implies for a range of isolated molecules. The graphs show shieldings obtained by the GIPAW method As shown inFigure Figure 2, 4 the shows use of shieldings pseudopotentials computed implies for a range of isolated molecules.− The400 graphs show shieldings obtained by the GIPAW method a nonphysical form for the−400 wavefunction in the region −400 −200 0 200and 400 ultrasoft 600 pseudopotentials 800 plotted against all-electron shielding. The a nonphysical form for the wavefunction in the regionmolecules by−400 using−200 only0 the 200and pseudowavefunctions ultrasoft 400 600 pseudopotentials 800 (i.e., plotted against all-electron shielding. The molecules by usingclose only to the the nucleus. pseudowavefunctions For this reason, a formalism (i.e., based on (b) All electron shieldingstraight (ppm) line represents perfect agreement. The upper figure (a) shows close to the nucleus. For this reason, a formalism basedneglecting on (b) the fact that theAll electron pseudowavefunction shieldingstraight (ppm) line represents differs perfect from agreement. The upper figure (a) shows neglecting the fact thatpseudopotentials the pseudowavefunction might appear a poor choicediffers for from the calculation the contribution without GIPAW augmentation, the lower figure (b) plots pseudopotentials might appear a poor choiceof fornuclear the calculationmagneticthe true resonance all-electron parameters. wavefunction The upper part close ofthe contributionFigure to the 4 nucleus),Isotropic without magnetic and GIPAW shielding the augmen for nucleartation, sites the in a lower range of figure (b) plots of nuclear magneticthe true resonance all-electron parameters. wavefunction The upper part close of toFigure the nucleus), 4 Isotropic magnetic and the shielding for nuclear sites in a range of the total contribution. (Reproduced from Ref. 13. American Physical Figure 4 showscorresponding shieldings computed large for basis a range set of isolated quantum-chemicalthe totalmolecules. contribution. The graphs calculations. (Reproduced show shieldings obtained from Ref. by the 13. GIPAW American method Physical  Figure 4 showscorresponding shieldings computed large for basis amolecules range set of quantum-chemical isolated by using onlymolecules. the pseudowavefunctionsThe calculations. graphs show shieldings (i.e., obtainedand by ultrasoft the GIPAW pseudopotentials method plotted againstSociety, all-electron 2007) shielding. The and ultrasoft pseudopotentials plotted againstSociety, all-electron 2007) shielding. The 13 molecules by using only the pseudowavefunctionsneglecting theFor (i.e., fact proton that the pseudowavefunction shieldings, the13 differs agreement from isstraight reasonable. line represents For perfect agreement.C, The upper figure (a) shows For proton shieldings, the agreement isstraight reasonable. line represents For perfectC, agreement. The upper figure (a) shows neglecting the fact that the pseudowavefunctionthe true differs all-electronthe from pseudopotential wavefunction close to the results nucleus), reproduce and the the the contribution general without trend GIPAW augmen of tation, the lower figure (b) plots the true all-electronthe wavefunction pseudopotential close to the results nucleus), reproduce and the the the contribution general without trend GIPAW of augmentation,the the total lower contribution. figure (b) plots (Reproduced from Ref. 13.  American Physical corresponding large basis set quantum-chemical calculations. Society, 2007) and n is a composite index that accounts for the angular the all-electronthe total contribution. shieldings (Reproduced although from13 and Ref. the 13.n is slopeAmerican a composite is Physical somewhat index that accounts for the angular correspondingthe large all-electron basis set quantum-chemical shieldingsFor proton calculations. although shieldings, the theSociety, slope agreement 2007) is is somewhat reasonable. For C, For proton shieldings, the agreement is reasonable.the pseudopotential Forless13C, than results unity. reproduce Uncorrected the general trend pseudopotential ofmomentum and calculations the number for of projectors.momentum In simple and the terms, number the of projectors. In simple terms, the less than unity. Uncorrected pseudopotential calculations for and n is a composite index that accounts for the angular the pseudopotential results reproduce thethe general all-electron trendproton shieldings of and although first row the elementsslope is somewhat havePAW been transformation presented (see works e.g., by computingPAW transformation the component works of a by computing the component of a the all-electronproton shieldings and although first row the slopeelementsless than is somewhat unity. have Uncorrected beenand presentedn pseudopotentialis a composite (see calculations e.g., index that for accountsmomentum for the and angular the number of projectors.certain In simple atomic-like terms, the state (say 2p) in a pseudowavefunction, less than unity.Ref. Uncorrected 12); the pseudopotential agreementproton calculations can and first beRef. for row improved 12); elementsmomentum the have by agreement and beenusing the presented number rather can (see of be projectors. e.g.,certain improvedPAW In atomic-like simple transformation by terms, using the state works rather by (say computing 2p) in the a component pseudowavefunction, of a proton and first row elements have been presentedRef. 12); (see thehard e.g., agreement pseudopotentialsPAW can transformation be improved (i.e., by works using a by small rather computingand matchingcertain replacing the component atomic-like radius). the of statepseudized Fora (say the 2p) component inand a pseudowavefunction, replacing by its the all-electron pseudized component by its all-electron hard pseudopotentials (i.e., a small matching radius). For the and replacing the pseudized component by its all-electron Ref. 12); the agreement can be improvedhard by pseudopotentials usingsecond rather (i.e., rowcertain a elements, small atomic-like matching Si state radius). and (say P, For 2p) Figure the in a pseudowavefunction, 4 shows the impact of form. This may seem like a rather approximate procedure, second row elements,second Si and row P, elements, Figure Si 4and and shows replacing P, Figure the 4 the impact shows pseudized the of impact component ofform.form. This by This its may all-electron may seem seem like like a rather a rather approximate approximate procedure, procedure, hard pseudopotentials (i.e., a small matching radius). For the but because the atomic states formbut a good because basis for the the atomic states form a good basis for the second row elements,the pseudopotential Si and P, Figure 4 approximation showsthe pseudopotential the impactthe of to pseudopotential approximation beform. dramatic This to may bewith approximation dramatic seem almost like with a almost ratherbut to approximate be because dramatic procedure, the with atomic almost states form a good basis for the all of the chemical sensitivitybut because lost. In the order atomic to make states direct form awavefunction good basis in for the the region close towavefunction the nucleus, it can in be the region close to the nucleus, it can be the pseudopotentialall of approximation the chemical to be dramatic sensitivity with almostall lost. of In the order chemical to make sensitivity direct lost.wavefunction Inmade order highly to accuratein make the by region direct using multiple close projectors. to the nucleus, it can be all of the chemical sensitivity lost. In ordercomparison to make to direct experiment,wavefunction it is therefore in the vital region to correctly close to the nucleus, it can be made highly accurate by using multiple projectors. account for thecomparison use of pseudopotentials to experiment, and to obtain shieldings it ismade therefore highlyWithin vital the accurate to PAW correctly scheme, by using for multiple an all-electron projectors. local or comparison tocomparison experiment, it to is therefore experiment, vital to it correctly is thereforemade vital highly to accurate correctly by using multiple projectors. with the accuracy of all-electron calculations. semilocal operator O, the correspondingWithin pseudooperator, theO PAW, scheme, for an all-electron local or account for theaccount use of pseudopotentials for the use and of pseudopotentials to obtain shieldingsaccount and forWithin to the obtain use the of shieldings PAW pseudopotentials scheme, for anWithin and all-electron to the obtain local PAW shieldings or scheme, for an all-electron local or The basis for such an approach was provided by projector is given by with the accuracy of all-electron calculations. with thesemilocal accuracy operator of all-electronO, the corresponding calculations.semilocal pseudooperator, operator OO,, the correspondingsemilocal pseudooperator, operator O, theO corresponding, pseudooperator, O, with the accuracy of all-electronaugmented wave calculations. (PAW) method introduced by van de Walle # The basis for such an approach was provided by projector14 The basisis given for by such an approach was provided by projector is given by The basis for such anand approach Blochl.¨ In this was scheme, provided a linear by transformation projector mapsis given by O O pR,n φR,n O φR,m augmented wave (PAW) method introducedthe by valence van de pseudo Walle wavefunctions $ onto the correspondingT = +# | ! # | | ! 14 augmented wave (PAW)Projector method Augmentedaugmented introduced Waves wave by van˜ (PAW) de Walle method introduced by van deR Walle,n,m # # and Blochl.¨ In this scheme, a linear transformationall-electron wavefunctions,maps $14 | O$! , O pR,n φR,n O φR,m " $ 14 T and Blochl.¨ In this˜ = scheme,+ | a linear! # transformation| | ! # maps# the valence pseudoand wavefunctions Blochl.¨ In$ thisonto scheme, the corresponding a linear transformation| ! = T | ! mapsR,n,m O O φR,n O φR,mp pR,m φ O φO(25) O pR,n φR,n O φR,m | ˜ ! " −#T | | !R#,n | R,n R,m all-electron wavefunctions, $ $ , the valence pseudo wavefunctionsT $ $ onto the corresponding= + | ! # | | =! + | ! # | | ! the valence pseudo˜ wavefunctions $1 onto[ φR the,n correspondingφ#R,n ] pR,n # (24) R,n,m % R,n,m | ! = T | ! T = ˜ + | !−| ˜ ! # ˜ φ|R,n O φR,m | ˜p!R,m As constructed(25) in# equation# (25), the pseudooperator O " | ! R,n −# | | ! # | " # $ $ all-electronpseudo wavefunctions,all-electron $ all-electron$", wavefunctions, $ $acting, on pseudowavefunctions will give the same matrix# 1 [ φR,n φR,n ] pR,n (24) ˜ | ! = T% | ˜ ! # # φ ,n O#φ ,m p ,m (25) T = + | !−| ˜ ! # ˜ φR| ,n| are! = all-electronT | ! As atomiclike constructed states in# obtained equation# from (25), theelements pseudooperator as the all-electronOφR,n operatorO φR,mO actingpR on,m all-electron R (25) R R R,n | ! # # −# | | ! # | a calculation on anacting isolated on atom pseudowavefunctions and φR,n are the will givewavefunctions. the same matrix−# | | ! # | " | ! φR,n are all-electron atomiclike1 statescorresponding obtained[ φR,n from pseudizedφR,nelements states.] pR,np asR,n the1are all-electron a set(24)[ ofφ projectorsR operator,n φORacting,nFor] p a onR system,n all-electron under a uniform(24) magnetic field, PAW alone % | ! ˜ T# ˜= | + | !−|As˜ constructed! # ˜ | in# equation (25),%As the constructed pseudooperator in# equationO # (25), the pseudooperator O a calculation on an isolatedT atom= and+ suchφR that|,n arepR!−,n theφR| ,m wavefunctions.!δR#,R˜δn,m|. R labelsR,n the# atomic site is not a computationally# realistic# solution.# In using a set of # R,n| ! # ˜ | ˜ $ ! = $ acting on pseudowavefunctions will give the same matrix corresponding pseudized states. pR,n are a" set of projectors For a system under" a uniformacting magnetic on field, pseudowavefunctions PAW alone will give the same matrix # ˜ | - φ are+ all-electron atomiclike states obtained from such that pR,n φRφ,mR,n δRare,R δn,m all-electron. R labels the# atomiclike atomic siteR,n statesis not a obtained computationally from realisticelements solution. In as using the a all-electron set of operatorelementsO acting as the on all-electron all-electron operator O acting on all-electron# # ˜ | ˜| $ ! =! $ | ! # a calculationpseudo on an isolateda atom calculation and φ onR,n anare isolated the atomwavefunctions. and φR,n are the wavefunctions. | ! | ! corresponding pseudized states.correspondingpR,n are a set pseudized of projectors states. pR,nForare a system a set of under projectors a uniform magneticFor a system field, under PAW alone a uniform magnetic field, PAW alone # ˜ | # ˜ | such that pR,n φR ,m δR,R δsuchn,m. R thatlabelspR,n the#φR atomic,m δ siteR,R δn,mis. R notlabels a computationally the# atomic site realisticis solution. not a computationally In using a set realistic of solution. In using a set of # ˜ | ˜ $ ! = $ # ˜ | ˜ $ ! = $ GIPAW = Gauge-Including Projector Augmented Waves Modification of PAW by Pickard and Mauri for systems in an external magnetic field - plays a role all-electron similar to GIAO in quantum chemistry techniques GIPAW CALCULATION OF NMR CHEMICAL SHIFTS FOR… PHYSICAL REVIEW B 76, 024401 ͑2007͒ A theory for solid-state NMR TABLE I. Chemical Shielding parameters for a O-͑SiH ͒ clus- 800 35 3 2 (a) ter derived from the ␣-quartz structure. The all-electron calculations 30 GIPAW vs Gaussian 600 ͑Ref. 15͒ use the IGAIM method with cc-pCVxZ basis sets for O 25 H and Si. The GIPAW-NCP calculations ͑Ref. 15͒ use Troullier- test on small molecules (ppm) 400 C 20 F Martins pseudopotentials with a 120 Ry plane-wave cutoff. The 20 25 30 35 n.b. v. big Gaussian basis sets USP Si plane-wave cutoff for the GIPAW-USP calculation is 60 Ry. - 200 P

JRY, C. Pickard, F. Mauri PRB 76, 024401 (2007) 0 USP NCP All-electron GIPAW

-200 O ␴iso 317.62 315.62 316.49

Convergence of shielding with #planewaves valence uncorrected using -400 ⌬aniso −109.75 −111.12 −109.25 -400 -200 0 200 400 600 800 All Electron Shielding (ppm) ␩ 0.02 0.02 0.03

800 35 (b) Si͑1͒ ␴iso 340.55 345.23 340.98 30 ⌬aniso 150.61 147.89 151.07 600 25 H ␩ 0.08 0.09 0.08

(ppm) 400 C 20 F 20 25 30 35 Si͑2͒ ␴iso 339.28 343.84 339.60

USP Si

- 200 P ⌬aniso 149.53 147.01 150.24 ␩ 0.08 0.08 0.08 with GIPAW with 0 GIPAW

-200 the convergence of the isotropic O chemical shielding -400 -400 -200 0 200 400 600 800 against maximum plane-wave energy for NCPs and USPs All Electron Shielding (ppm) with augmentation regions of both 1.3 and 1.5 bohr. The faster convergence rate of USPs is apparent. We also note FIG. 1. Color online Isotropic chemical shielding for nuclear ͑ ͒ that for both NCPs and USPs, the final converged result is sites in a range of molecules ͑Ref. 35͒. The graphs show shieldings obtained with the GIPAW-USP method plotted against all-electron independent of the size of augmentation region, demonstrat- ing the stability of the GIPAW approach. shielding ͑Ref. 30͒. The straight line represents perfect agreement. The upper figure ͑a͒ shows the contribution without GIPAW aug- Finally, we consider truly crystalline systems, for which mentation ͑␴ +␴ ͒; the lower figure ͑b͒ plots the total we compare to existing GIPAW calculations using NCP, and bare core 36 contribution. experiment. We now use the PBE density functional which has been shown to give chemical shifts in good agreement 15,19 17 the GIPAW augmentations are included, the agreement with with experiment. In Table II, we present O NMR pa- the all-electron results is excellent. As for the norm- rameters calculated using USPs for three silicate materials, conserving case,11 the largest absolute deviations are for together with results from Ref. 15 using NCP and experi- phosphorus. However, these represent a small fraction of the mental results. The structures of the materials and the details total range of phosphorus chemical shieldings. of the Brillouin zone integration are the same as used in Ref. We next consider silicate compounds. In Ref. 15, O and Si 15. Plane waves up to a maximum energy of 40 Ry are used. chemical shielding tensors were computed for a small cluster The agreement of the parameters computed with USPs, both derived from the ␣-quartz structure, both using the GIPAW with the existing NCP results and experiment, is excellent. In approach with NCP and with the IGAIM approach with a particular, the assignment of the oxygen sites in coesite is large ͑pentuple zeta͒ Gaussian basis. In Table I, we compare consistent between the two sets of computational results. For isotropic chemical shieldings, chemical shielding anisotropy these materials, we find that the use of USPs gives a 50% ⌬ , and chemical shielding asymmetry ␩ computed with aniso 380 the three approaches. The agreement between the approaches All-electron is good. We also use this silicate cluster to examine the con- USP rcut=1.3 USP rcut=1.5 vergence of the chemical shielding with the size of the plane- (ppm) 360 NCP rcut=1.3 wave basis. Clearly, the rate of convergence depends on the NCP rcut=1.5 size of the pseudopotential augmentation region; a larger

Shielding 340 augmentation region allows for a softer pseudopotential and hence fewer plane waves are required for numerical convergence. However, the augmentation region should not be so Isotropic 320 O 7 large that neighboring augmentation regions have signiﬁcant 1

overlap; otherwise, errors will be introduced. The relatively 300 0 20 40 60 80 100 120 large Si–O bond lengths in silicates allow for a large aug- Maximum Planewave Energy (Ry) mentation region for an oxygen pseudopotential and we consider values of 1.3 and 1.5 bohr. We note that for the shorter FIG. 2. Convergence with plan-ewave cutoff energy of the 17O oxygen bonds found in organic materials, only the smaller isotropic chemical shielding in a O-͑SiH3͒2 cluster derived from the augmentation region will be appropriate. In Fig. 2, we plot ␣-quartz structure. The all-electron result is taken from Ref. 15.

024401-7 First-principles NMR

Crystal Structure X-ray, Neutron diffraction Cambridge Structural Database Calculate forces C1 -0.602 -0.537 6.384 refine structure C2 0.135 0.622 5.689 H1 0.257 0.361 4.639 ...

311.0ppm

Calculate chemical shifts

316.9ppm Examples Steven Brown (Warwick)

13C axis Maltose sugar used in brewing 1 Haxis

MAS-J-HMQC Examples

Maltose 13C sugar used in brewing 1 Haxis

x xx x xx x

xx x x x

x MAS-J-HMQC

x - ﬁrst principles Examples

13C axis 1 Haxis

Molecule to solid variation due to intermolecular interactions (weak hydrogen bonds) x - ﬁrst principles

J. Am. Chem. Soc. 127 10216 (2005) Figure 5

NMR Interactions Figure 5

Chemical Shift

orbital currents

Quadrupolar coupling (EFG) nuclei I>1/2 interact with local electric ﬁeld gradients Function of charge density spin-spin coupling (eg J-coupling) 36 splitting of resonance due to nucleus-nucleus interaction hard to observe in solids....

36 6 COMPUTATIONS OF MAGNETIC RESONANCE PARAMETERS FOR CRYSTALLINE SYSTEMS: PRINCIPLES

localized functions, the gauge-origin problem, well known in The quadrupolar coupling constant, CQ and the asymmetry quantum chemical approaches (see Shielding: Overview of parameter, ηQ can be obtained from the diagonalized electric Theoretical Methods), has been introduced. In short, equation field gradient tensor whose eigenvalues are labeled VXX,VYY, (25) will require an infinitely large number of projectors VZZ, such that V ZZ > V XX > V YY : in order for the computed shieldings to be translationally | | | | | | invariant (i.e., independent of the choice of gauge origin). To eVZZQ CQ (30) address this problem, Pickard and Mauri8 introduced a field- = h dependent transformation operator B, which, by construction, imposes the translational invarianceT exactly: and VYY VXX η − (31) Q V ie r R B ie r R B = ZZ B 1 e 2h · × [ φR,n φR,n ] pR,n e− 2h · × (26) T = + | "−| ˜ " $ ˜ | R,n ! Within the planewave-pseudopotential approach, the charge density is expressed as the sum of three terms,15 and there The resulting approach is known as the gauge including are correspondingly three distinct contributions to the EFG. projector augmented wave (GIPAW ) method. Although orig- First, there is a contribution arising from the ionic charges inally formulated for norm-conserving pseudopotentials, the (sum of the nuclear and core-electron charge). From the site extension to the more computationally efficient ultrasoft pseu- under consideration, this appears as an infinite lattice of point dopotentials has been presented by Yates et al.13 Figure 4 charges whose EFG contribution can be obtained using an shows shieldings computed using ultrasoft pseudopotentials Ewald summation. Secondly, there is the contribution of the and the GIPAW scheme, together with large basis set quantum- pseudized valence charge density, which is evaluated by using chemical calculations. For the shieldings in these isolated a reciprocal space form of equation (29). Finally, there is a molecules, the agreement is essentially perfect. For crystalline PAW contribution to account for the difference between the systems, validation of the technique comes only from compar- pseudo and all-electron charge densities on the atomic site ison to NMR experiments; numerous studies have been made under consideration. in recent years, just two are mentioned in section “Examples”.

4.2 Indirect Coupling

4 OTHER MAGNETIC RESONANCE PARAMETERS The indirect (or J ) coupling manifests itself in splittings of the NMR resonance, or as a modulation of the spin-echo signal. 4.1 Electric Field Gradient Physically, the coupling arises from the interaction of two nuclei, K and L, with magnetic moments, µK and µL, mediated For a quadrupolar nucleus (spin I > 1/2), the observed by the electrons in the system. The first complete analysis NMR response includes an interaction between the quadrupole of this indirect coupling was provided by Ramsey16,17 who moment of the nucleus Q, and the electric field gradient (EFG) showed that the J -coupling tensor, ←→J KL, is obtained from generated by its surroundings. The EFG is a second rank, the magnetic field induced at nucleus K due to the perturbative symmetric, traceless tensor V (r) given by effect of nucleus L,

∂Eα(r) 1 ∂Eγ (r) (1) !γKγL Vαβ (r) δαβ (27) B (R ) J µ (32) ∂r 3 ∂r in K ←→KL L = β − γ γ = 2π · ! where γ K and γ L are the gyromagnetic ratios of nuclei K where α,β,γ denote the Cartesian coordinates x,y,z and E α(r) and L. At the nonrelativistic level, two distinct mechanisms is the local electric field at the position r, which can be (1) contribute to B (R ). First, the nuclei interact via the calculated from the charge density n(r): in K Electric Field Gradients electronic charge; the nucleus L induces an orbital current j(1)(r) which, in turn, creates a magnetic field at nucleus K. n(r) E ( ) 3r (r r ) The nuclei also interact via the electronic spin; the nucleus L α r d 3 α α% (28) Function of= the charger densityr - ie ground-state− property. (1) " | − %| inducing a polarization of the electronic spin, m (r), which Also computed by all-electron codes such as Wien2k, Crystal (1) also contributes to the magnetic field at nucleus K. Bin (RK) The EFGcharge tensor is then equal to is given by density 2 (r r )(r r ) (1) µ0 (1) 3rKrK rK 3 3 n(r) α α% β β% B (R ) m (r) d r EFG V (r) d r δ 3 − − (29) in K −5| | αβ 3 αβ 2 = 4π · rK = r r% # − r r% $ " % | | & Eigenvalues " | − | | − | µ0 8π (1) 3 m (r)δ(rK) d r Vxx, Vyy, Vzz V > V > V + 4π 3 To compute| zz| | theyy| EFG| xx tensor| in a periodic system is less " demanding than calculating either the shielding or indirect µ0 rK (1)( ) 3 j r 3 d r (33) Quadrupolar Couplingcoupling tensors, as it requires only knowledge of the ground- + 4π × rK state charge density, ground-state wavefunctions and the " | | eQVzz CQ =position of the ions in the unit cell—no linear response rN r RN with RN the position of nucleus N; δ is the Dirac calculationh is required. delta= function.− Asymmetry Vxx Vyy ηQ = − Vzz

Note: The quadrupolar moment, Q, is a nuclear property. Most recent values given in “Year-2008 Standard Values of Nuclear Quadrupole Moments”: P. Pyykkö, Mol. Phys. 106, 1965-1974 (2008) 17 But note Q appears as a simple scaling factor O MAS Glutamic Acid . HCl J-coupling

Electron-mediated interaction of nuclear spins

Nucleus A causes a magnetic ﬁeld at nucleus B (and vice versa) A B

No J-coupling

In solids J is rarely revealed in splitting with J-coupling J of peaks in 1D spectra (anisotropic interactions broaden peaks)

Increasing use of techniques, eg those using spin-echo modulation, to anti-parallel parallel measure J in solids. J-coupling

Orbital Dipole induces orbital motion of electrons (due to electron charge) This creates a localised magnetic ﬁeld Orbital Current similar mechanism to NMR shielding

Spin Dipole induces spin density (relative displacement of ↑and ↓ electrons) This creates induced ﬁeld via:

Fermi-Contact (ʻsʼ like density at nucleus) Induced Spin density: C2H2 Spin-dipolar (non-spherical distribution of spin) similar mechanism to EPR-hyperﬁne & Knight-shift H C C H Total J = Orbital + Spin -ve +ve -ve

+ve FULL PAPER G. Gottarelli, P. Mariani et al.

Figure 1. Hydrogen-bond pattern of the twoJ-couplingribbonlike assemblies of guanosine derivatives.

Guanosine ribbons seem particularly interesting as their ordered molecular layers, which are obtained on solid surfaces by evapo- self-assembles into ribbons [15] ration of the LC phase in CHCl3 , display rectifying and molecular electronicsphotoc (FET)onductive properties.[16] In a preliminary communication[17] we have described a new lyomesophase formed by derivative 1 in hexadecane; more a [18] 2hJN7a,N1b recently, Kato and co-workers have reported thermotropic aand lyotropic[19] phases formed by folic acid derivatives due to similar hydrogen-bond patterns. In this paper, we report the structure of ribbon I, obtained from single-crystal X-ray diffraction of deoxyguanosine derivative 3, and a study of b the gel-like liquid crystals formed in different solvents by derivatives 1 and 2 and by the two other guanosines 4 and 5, 2h 2h JN7b,N1a = 6.2±0.4 Hz (expt) b JwhichN7b,N1abelong to the ribo series and contain the acetonide 6.5 Hz (calc) group. We believe that the variations of the gel-like structures with the chemical constitution of the gelators and with the NMR spectrum that slowly transforms into a different 2hJN7a,N1b = 7.4±0.4 Hz (expt) solvent yield a good understanding of the self-assembly spectrum corresponding to the ribbon stable in CHCl3 process leading to the gel-like phases,[20] and that the presence solution (ribbo7.7n II)Hz; the trans (calc)ition from ribbon I to ribbon II [12] of the two different ribbons in the different lyomesophases is is J.faster Am. Chem.in the Soc.presence 130, 12663of moisture (2008) in the solvent. These remarkable.

Abstract in Italian: I derivati lipofili della guanosina allo stato cristallino e in soluzione sono autoassociati in aggregati a Results and Discussion nastro. La struttura dei nastri e¡ stata caratterizzata mediante diffrazione dei raggi X da cristallo singolo e, in soluzione, X-ray single-crystal study: Compound 3 crystallises in the mediante NMR e ESI-MS. Allo stato solido e in soluzione chiral group P1, with two independent molecules in the cloroformica sono presenti due diversi nastri con un diverso asymmetric unit. The two molecules differ in the conforma- schema di legami a idrogeno. Le fasi di tipo gel ottenute in tion of the lateral chains attached to the guanosine skeleton, esadecano, toluene e cloroformio sono state studiate con la as can be seen in Figure 2. microscopia ottica e la diffrazione dei raggi X a basso angolo: The two types of molecules interact in the solid state by il tipo di fase osservata e¡ in relazione con la struttura means of hydrogen bonding between the guanine residues molecolare dei composti e dipende fortemente dal solvente (Figure 3). The NH group of the six-membered ring interacts presente. Vengono discusse le strutture delle fasi tenendo in exclusively with the N atom of the five-membered ring considerazione la presenza dei due diversi nastri. (N(H) ± N 2.827(2) and 2.909(2) ä). The two H atoms on the NH2 groups interact with the oxygen atom on the six-

2144 ¹ WILEY-VCH Verlag GmbH, 69451 Weinheim, Germany, 2002 0947-6539/02/0809-2144 $ 20.00+.50/0 Chem. Eur. J. 2002, 8, No. 9 Practical Details

*.param ﬁle task : magres chemical shift/shielding magres_task : shielding electric ﬁeld gradient efg nmr both

Must use on-the-ﬂy pseudopotentials

Highly sensitive to geometry (optimise H X-ray positions) CONVERGE (basis cut-off & k-points) *.castep ﬁle

======| Chemical Shielding Tensor | |------| | Nucleus Shielding tensor | | Species Ion Iso(ppm) Aniso(ppm) Asym | | H 1 23.81 5.27 0.40 | | H 2 24.75 -3.35 0.85 | | H 3 27.30 -5.79 0.90 |

| O 5 -43.73 504.95 0.47 | | O 6 -63.53 620.75 0.53 | | O 7 -43.73 504.95 0.47 | | O 8 -63.53 620.75 0.53 | ======

Anisotropy Asymmetry

σ = σ 1/2(σ σ ) η = 3(σyy σxx)/2σaniso aniso zz − xx − yy − *.magres ﬁle

======Atom: O 1 ======O 1 Coordinates 1.641 1.522 5.785 A

TOTAL Shielding Tensor

218.1858 12.1357 -25.7690 13.4699 191.6972 -7.2419 -25.9178 -6.5205 216.3180

O 1 Eigenvalue sigma_xx 185.6127 (ppm) O 1 Eigenvector sigma_xx 0.5250 -0.8103 0.2603 O 1 Eigenvalue sigma_yy 193.8979 (ppm) O 1 Eigenvector sigma_yy 0.4702 0.5310 0.7049 O 1 Eigenvalue sigma_zz 246.6904 (ppm) O 1 Eigenvector sigma_zz -0.7094 -0.2477 0.6598

O 1 Isotropic: 208.7337 (ppm) O 1 Anisotropy: 56.9351 (ppm) O 1 Asymmetry: 0.2183

Note: shielding tensor has a symmetric and an antisymmetric component. Typical NMR experiments are only sensitive to the symmetric part. Therefore we only diagonalise the symmetric part of the shielding tensor Practical Things

Some practical things to keep in mind

Structure NMR is very sensitive to structure! XRD atom positions are often not sufﬁcient for accurate simulation of NMR parameters

Dynamics Atoms move - spectra are recorded at ﬁnite T. How to account for that in (static) calculations?

Accuracy How good are our present (approximate) density functionals? Structure

Anomeric Carbon

Crystallises as: 80% α-Maltose 20% β-Maltose Compute spectra for both structures separately and superimpose Structure

Anomeric Carbon

After relaxing neighbouring atom O1ʼ which had large force / thermal ellipsoid Dynamics

MLF peptide 13C rms error

Static relaxed structure 4.2ppm 2 pm ab-initio MD 3.3ppm 5 μm classical MD 3.0ppm

Ab-initio MD Captures large (20ppm) ﬂuctuations on fs timescale

Classical MD Samples rotomeric states

half of sites have errors <1ppm - limitations of PBE

J. Am. Chem. Soc. 132, 3993 (2010) Magnetic Resonance File Format

Aim Provide a complete archival format for a ﬁrst principles calculation of NMR parameters

Visualisation

Ab-initio Abinit, CASTEP ﬁle-format Spin-simulation QE, Wien2k etc

Database Jmol viewer

Open-source Crystal-viewer Multi-platform (Java) Can embed in website

New developments • display selected values on structure • convert shielding to chemical shift • switch isotopes • visualization of tensors (D, J, σ, EFG) • relative orientations (Euler angles) • select spin system for export Database

• Ab-initio calculations provide NMR parameters for all atoms in the system

• Papers often report only selected results

• Store full information as “ab-initio nmr ﬁle-format” in a database

• Allows later retrieval and processing of data

• Establish structural trends Consultation

Draft speciﬁcation and examples www.ccpnc.ac.uk

Feedback and Comments to [email protected]

by 19th October 2012 Getting more information

NMR Books Good Introduction Nuclear Magnetic Resonance (Oxford Chemistry Primers) P. J. Hore

More advanced Spin Dynamics: Basics of Nuclear Magnetic Resonance Malcolm H. Levitt

Solid state NMR Solid-State NMR: Basic Principles and Practice David Apperley, Robin Harris, Paul Hodgkinson

Useful survey of applications Multinuclear Solid-State Nuclear Magnetic Resonance of Inorganic Materials Kenneth J.D. MacKenzie, M.E. Smith Recent Review Articles Recent advances in solid-state NMR spectroscopy of spin I = 1/2 nuclei Anne Lesage, Phys. Chem. Chem. Phys., 2009, 11, 6876 Recent advances in solid-state NMR spectroscopy of quadrupolar nuclei Sharon E. Ashbrook, Phys. Chem. Chem. Phys., 2009, 11, 6892 Getting more information

GIPAW Theory A more in depth introduction to the theory (email JRY for a copy) Computations of Magnetic Resonance Parameters for Crystalline Systems: Principles Jonathan R. Yates, Chris J. Pickard Encyclopedia of Magnetic Resonance (2008) doi:10.1002/9780470034590.emrstm1009

Exhaustive (and exhausting!) review of applications and theory (email JRY for a copy) First-Principle Calculation of NMR Parameters Using the GIPAW (Gauge Including Projector Augmented Wave) Method: a Chemist's Point of View Bonhomme, Gervais, Babonneau, Coelho, Pourpoint, Azais, Ashbrook, Grifﬁn, Yates, Mauri, Pickard, Chemical Reviews (in press 2012)

Original Theory Papers: All-electron magnetic response with pseudopotentials: NMR chemical shifts, Chris J. Pickard, and Francesco Mauri. Phys. Rev. B, 63, 245101 (2001)

Calculation of NMR Chemical Shifts for extended systems using Ultrasoft Pseudopotentials Jonathan R. Yates, Chris J. Pickard, and Francesco Mauri. Physical Review B 76, 024401 (2007)

A First Principles Theory of Nuclear Magnetic Resonance J-Coupling in solid-state systems Sian A. Joyce, Jonathan R. Yates, Chris J. Pickard, Francesco Mauri J. Chem. Phys. 127, 204107 (2007) www.gipaw.net