Solid-state NMR Winter School 2016 Problem Set 1

1. Show that threefold or greater symmetry results in a chemical shift with η=0.

2. Prove that U exp[A]U + = exp[UAU + ] for U unitary.

3. Show that R A (α, β,γ ) = az (α )ay (β )az (γ ), where all rotations are given in the space-fixed frame. Hint, use the fact that rotations about an arbitrary axis can be accomplished by first rotating that axis to one of the coordinate axes, rotating about the coordinate axis, and then rotating back to the original axis location.

4. Write out the propagator for the MREV-8 sequence and use it to help find the average Hamiltonian, H(0). What is the chemical shift scaling for this sequence?

5. In their classic paper on sideband intensities under magic-angle spinning, Herzfeld and Berger (J. Herzfeld and A. E. Berger, Journal of Chemical , 1980, 73, 6021-6030) write both the lab frame magnetic field/spin operators and the chemical shift tensor in the MAS rotor frame, H cs = −γ Tr{σrotor (B ⊗ I )rotor }

rotor PAS −1 rotor rotor lab −1 rotor = −γ Tr{R(ΩPAS )σ R (ΩPAS )R(Ωlab )(B ⊗ I ) R (Ωlab )}

Provide an alternate derivation of the Hamiltonian used as the starting point in the Herzfeld- Berger analysis (corresponding to the frequency given in equation [16] of their paper) by writing the components in the laboratory frame and using a two-step transformation of the spatial tensor from the PAS to rotor frame and then from the rotor to lab frames. Use spherical and the passive rotation convention.

Downloaded 15 Nov 2007 to 138.23.132.34. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp Downloaded 15 Nov 2007 to 138.23.132.34. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp Downloaded 15 Nov 2007 to 138.23.132.34. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp Downloaded 15 Nov 2007 to 138.23.132.34. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp Downloaded 15 Nov 2007 to 138.23.132.34. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp Downloaded 15 Nov 2007 to 138.23.132.34. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp Downloaded 15 Nov 2007 to 138.23.132.34. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp Downloaded 15 Nov 2007 to 138.23.132.34. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp Downloaded 15 Nov 2007 to 138.23.132.34. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp Downloaded 15 Nov 2007 to 138.23.132.34. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp Tensors and Rotations in NMR

LEONARD J. MUELLER Department of Chemistry, University of California, Riverside, CA 92521

ABSTRACT: The transformation of second-rank Cartesian tensors under rotation plays a fundamental role in the theoretical description of nuclear magnetic resonance experi- ments, providing the framework for describing anisotropic phenomena such as single crystal rotation patterns, tensor powder patterns, sideband intensities under magic-angle sample spinning, and as input for relaxation theory. Here, two equivalent procedures for effecting this transformation—direct rotation in Cartesian space and the decomposition of the Cartesian tensor into irreducible spherical tensors that rotate in subgroups of rank 0, 1, and 2—are reviewed. In a departure from the standard formulation, the explicit use of the spherical tensor for the decomposition of a spatial Cartesian tensor is intro- duced, helping to delineate the rotational properties of the basis states from those of the elements. The result is a uniform approach to the rotation of a physical system and the corresponding transformation of the spatial components of the NMR Hamilto- nian, expressed as either Cartesian or spherical tensors. This clears up an apparent inconsistency in the NMR literature, where the rotation of a spatial tensor in spherical tensor form has typically been partnered with the inverse rotation in Cartesian form to produce equivalent transformations. Ó 2011 Wiley Periodicals, Inc. Concepts Magn Reson Part A 38: 221–235, 2011.

KEY WORDS: irreducible spherical tensor; Cartesian tensor; rotation matrices; Wigner rotation matrix elements

INTRODUCTION periodic under rotation, while powdered samples of randomly oriented crystallites give broad, but typi- The NMR Hamiltonian is anisotropic, giving rise to cally structured, resonances. Time-dependent modu- spectroscopic lines that depend not only on local mo- lation of the orientation through sample spinning can lecular and chemical structure, but also on orientation average the position and breadth of these peaks, but within the static magnetic field (1–7). For example, single crystals have resonance frequencies that are even for samples in rapid isotropic motion, vestiges of anisotropic interactions remain as routes to relaxa- tion (1, 2, 5, 8). Received 3 June 2011; revised 31 July 2011; Anisotropic interactions in the NMR Hamiltonian accepted 1 August 2011 are represented by second-rank Cartesian tensors Correspondence to: Leonard J. Mueller. E-mail: Leonard.Mueller@ ucr.edu whose transformational properties under rotation play a fundamental role in the theoretical description of Concepts in Magnetic Resonance Part A, Vol. 38A(5) 221–235 (2011) NMR experiments (1–7). Two approaches to treating Published online in Wiley Online Library (wileyonlinelibrary.com). DOI 10.1002/cmr.a.20224 these rotations are the direct transformation of the Ó 2011 Wiley Periodicals, Inc. second-rank spatial tensor in Cartesian form and the

221 222 MUELLER decomposition of the Cartesian tensor into irreduci- coupling), the static magnetic field (chemical shield- ble spherical tensor components that rotate in sub- ing interaction), or the molecular l groups of rank 0, 1, and 2 (9–13). While these two vector (spin-rotation interaction). A is a second-rank approaches must give equivalent results, there is an Cartesian tensor and is a molecular level property apparent and curious need in the NMR literature to that depends on local geometry, electronic structure, partner the rotation in one representation with its and molecular orientation. For example, the Hamilto- inverse rotation in the other to find consistency in the nian for the chemical shielding interaction has the final transformed tensor. Here, the transformation of form, second-rank tensors in Cartesian and spherical forms are reviewed and it is shown that discrepancies in HCS ¼ g I s B 0 1 0 1 their sense of rotation can be reconciled by explicitly sxx sxy sxz 0 writing the Cartesian tensor as an expansion in the ir- B C B C I I I @ A @ A reducible spherical tensor basis and taking care to ¼ g ðÞx y z syx syy syz 0 szx szy szz Bz distinguish the rotational properties of the underlying spherical tensor basis components from those of the ¼ g IxsxzBz þ IysyzBz þ IzszzBz ; ½2 expansion coefficients. The result is a uniform and consistent approach to the rotation of the physical for the static, laboratory-frame magnetic field aligned system and the corresponding transformation of the along the z-axis. The chemical shielding interaction spatial components of the NMR Hamiltonian, results from currents in the electron density induced expressed as either Cartesian or spherical tensors. by the external magnetic field, which in turn produce This review begins with a brief introduction to an additional magnetic field that interacts with the second-rank Cartesian tensors in the NMR Hamilto- nuclear spin. Although the induced field is not neces- nian and rotation matrices and operators from both sarily aligned along the lab-frame magnetic field, it is the active and passive points of view. Next, the truncated under the secular approximation (3), in spherical tensor basis is introduced and explicit rela- which terms that do not commute with the Zeeman tions for the transformation of spherical tensor matrix interaction are discarded, to give elements under rotation of the physical system are derived. The resulting coefficient equation differs CS;secular H ¼ g IzszzBz: [3] from the customary equation used in the theoretical description of NMR experiments, and the relation- The theoretical description of NMR spectroscopy ship between the two is shown, highlighting the error relies on the transformation of spatial tensors under in the sense of rotation for the latter. A worked rotation. Two equivalent ways to treat this transfor- example for the transformation of an ab initio chemi- mation are the direct rotation of the second-rank cal shielding tensor is then presented to illustrate the spatial tensor in Cartesian form and the decomposi- consistency of this approach, before final comments tion of the Cartesian tensor into irreducible spherical on the Hamiltonian in spherical tensor form and the tensor components that are independently rotated and choice of reference frame. recombined to form the transformed Cartesian tensor. For the direct rotation, the initial (A) and transformed 0 (A ) tensors are related by (14)

THE NMR HAMILTONIAN AND 0 1; ROTATIONS A ¼ RA R [4] where R is the corresponding rotation matrix. For the The general form of a term in the NMR Hamiltonian transformation in irreducible spherical tensor form, (3, 4, 6, 7)is the individual spherical tensor components Akq trans- form in an analogous fashion to the angular momen- Hl cl I l Sl; ¼ A [1] tum basis kets, with the qth tensor component of a rank-k spherical tensor rotating into a sum of the cl where is a constant specific to a given interaction, other tensor components of the same rank, I is a spin angular momentum vector operator, and Sl is another vector, which, depending on the particular interaction, may be the same spin angular momentum Xk R DðkÞ ; operator (quadrupolar interaction), a different spin Akq ¼ pqðÞOR Akp [5] angular momentum operator (J coupling or dipolar p¼k

Concepts in Magnetic Resonance Part A(Bridging Education and Research) DOI 10.1002/cmr.a TENSORS AND ROTATIONS IN NMR 223 with weightings determined by the Wigner rotation define the Wigner rotation matrices from both the ðkÞ matrix elements, Dpq (VR)(9–13). As will be dis- active and passive points of view. Bouten’s expres- cussed in detail below, Eq. [5] is not completely sions agree with those given by Fano and Racah (9), R th analogous to Eq. [4]; Akq describes what the q -com- Wigner (11), and the revised version of Edmonds ponent of the rank-k tensor, Akq, transforms into and (16). As these matrices are essential to this discus- should not be confused with the qth-component of the sion, they are presented below. transformed tensor. The precise meanings of Eqs. [4] and [5] depend on whether the transformation is considered to be the Rotation Matrices in Cartesian Space result of a rotation of the object (the active point of Rotations in three-dimensional (3D) space can be view) or a rotation of the coordinate frame of refer- parameterized by three Euler angles, V ={a,b,g}, ence (the passive point of view) (14, 15), which are that relate the orientation of a stationary set of axes, reviewed in the following sections. For a physical Oxyz, to a rotatable set, OXYZ. Following Bouten rotation of the object, Eq. [4] expresses the Cartesian 15 0 ( ), if the two frames are initially coincident, then tensor after the rotation, A , in terms of the tensor for the Euler angles describe the following successive the object in its original orientation, A, while Eq. [5] rotations as the OXYZ axes are transformed to their describes what the initial Akq spherical-tensor com- final orientation: first, a rotation about the z-axis by ponent rotates into. For a coordinate frame transfor- an angle a (using a right-hand rule) that reorients the mation, Eq. [4] describes the tensor in the rotated X Y Y 0 and axes, with the transformed axis defining an frame, A , in terms of the elements of the tensor in intermediate axis u; second, a rotation about the u the original frame, A; Eq. [5] expresses what a single axis by an angle b, placing the Z axis in its final ori- tensor component of the original frame, Akq, looks entation; and third, a rotation about the transformed like in the rotated frame. Z axis by an angle g to place the OXYZ axes in their To compare the Cartesian and spherical tensor final orientation. These are illustrated in Figure 1. representations of a rotational transformation first Two points of view exist for how these Euler requires explicit expressions for the rotation matrices angles describe the orientation of an object, defined and the corresponding Wigner rotation matrix ele- by a body-fixed set of axes, relative to an observer- ments. Significant confusion over this correspon- fixed reference frame in which the object is viewed. dence exists in the literature due in part to mistakes Under the active rotation convention, different orien- in two standard texts on angular momentum, the first tations of the object are the result of rotations of the by Rose (10) and the second by Edmonds (12) object in the observer-fixed frame. In this case, the (although the text by Edmonds was subsequently re- body-fixed axes are associated with the rotatable vised (16)). Bouten (15) points out these well-hidden OXYZ frame above, and the reference axes with the errors and explicitly writes out the form of the rota- stationary Oxyz frame. Such a transformation is tion matrices and quantum mechanical operators that defined by the active rotation matrix in 3D space,

0 1 cos a cos b cos g sin a sin g cos a cos b sin g sin a cos g cos a sin b @ A RAðÞ¼a; b; g sin a cos b cos g þ cos a sin g sin a cos b sin g þ cos a cos g sin a sin b : [6] sin b cos g sin b sin g cos b

As expected, an active p/2 rotation about the z axis associated with the stationary Oxyz frame and the ref- takes an initial vector0 1 aligned0 1 along x to one along y, erence axes with the rotatable OXYZ frame. Different 1 0 orientations of the object are effected by different p ; ; @ A @ A i.e., RA 2 0 0 0 ¼ 1 . sets of Euler angles that transform the observer-fixed 0 0 frame from initial coincidence with the body-fixed From the standpoint of passive rotation it is the frame to its new perspective on the object. The trans- observer-fixed frame that rotates relative to the body- formation is defined by the passive rotation matrix in fixed frame. In this case, the body-fixed axes are 3D space,

0 1 cos a cos b cos g sin a sin g sin a cos b cos g þ cos a sin g sin b cos g @ A RPðÞ¼a; b; g cos a cos b sin g sin a cos g sin a cos b sin g þ cos a cos g sin b sin g : [7] cos a sin b sin a sin b cos b

Concepts in Magnetic Resonance Part A(Bridging Education and Research) DOI 10.1002/cmr.a 224 MUELLER

Figure 1 Rotation in three-dimensional space parameterized by the Euler angles, V ={a,b,g}, that relate the orientation of a rotatable set of axes, OXYZ, to a stationary set, Oxyz.

In the passive convention, a rotation0 1 of p0/2 about1 Z must have the same numerical values for its rotation 1 0 matrix elements regardless of whether it is considered X Y @ A @ A takes an -vector to – , R p; ; 0 ¼ 1 . to be the result of an active or a passive transformation; PðÞ2 0 0 0 0 it is only the values ascribed to the specific Euler angles In both the active and passive contexts, the coordi- that are different. In this sense, there is considerable lat- nates of the object are written in the observer-fixed itude in choosing a convention when setting up trans- reference frame, which is Oxyz for an active rotation formations in the laboratory frame; problems involving and OXYZ for passive, and so any transformation that coordinate frame transformations can often be recast as ~a b~ takes an initial vector to a final vector , physical rotations and vice versa (see Question 1 for an example of this). It is also worth noting that if one were 0 1 0 1 to perform an active rotation followed by a passive ax bx rotation with the same three Euler angles, the object @ a A @ b A; R y ¼ y [8] would be restored to its original orientation in the a b z z observer-fixed frame, a consequence of the fact that

Concepts in Magnetic Resonance Part A(Bridging Education and Research) DOI 10.1002/cmr.a TENSORS AND ROTATIONS IN NMR 225

Table 1 Wigner Rotation Matrix Elements (17)

ðkÞ -i a p -i g q ðkÞ ðkÞ i g p i a q ðkÞ Active: Dpq (a,b,g)=e e dpq (b) Passive: Dpq (a,b,g)=e e dpq (-b)

ð0Þ d ð2Þ ð2Þ b 00ðÞ¼b 1 d ðÞ¼b d ðÞ¼b cos4 22 2 2 2 dð2Þ dð2Þ dð2Þ dð2Þ 21ðÞ¼b 12ðÞ¼b 2 1ðÞ¼b 1 2ðÞb 1 ¼ sin b ðÞ1 þ cos b 2 ð2Þ ð2Þ ð2Þ ð2Þ ð1Þ ð1Þ b d d d d d ðÞ¼b d ðÞ¼b cos2 20ðÞ¼b 02ðÞ¼b 20ðÞ¼b 0 2ðÞb 11 1 1 2 rffiffiffi 3 ¼ sin2 b 8

ð2Þ ð2Þ ð2Þ ð2Þ ð1Þ ð1Þ b d d d d d d 2 2 1ðÞ¼b 1 2ðÞ¼b 21ðÞ¼b 12ðÞb 1 1ðÞ¼b 11ðÞ¼b sin 2 1 ¼ sin b ðÞ1 þ cos b 2 ð1Þ ð1Þ ð1Þ d d d ð2Þ ð2Þ b 01ðÞ¼b 10ðÞ¼b 0 1ðÞb d d 4 2 2ðÞ¼b 22ðÞ¼b sin 1 2 ¼dð1ÞðÞ¼b pffiffiffi sin b 10 2

dð1Þ ð2Þ ð2Þ 1 00ðÞ¼b cos b d ðÞ¼b d ðÞ¼b ðÞ2 cos b 1 ðÞcos b þ 1 11 1 1 2 1 dð2Þ ðÞ¼b dð2Þ ðÞ¼b ðÞ2 cos b þ 1 ðÞ1 cos b 1 1 11 2

dð2ÞðÞ¼b dð2Þ ðÞ¼b dð2ÞðÞ¼b dð2Þ ðÞb 10 0r1ffiffiffi 01 10 3 ¼ sin b cos b 2 1 dð2ÞðÞ¼b 3 cos2 b 1 00 2

The active and passive representations of this opera- a; b; g 1 a; b; g : RPðÞ¼RA ðÞ [9] tor are again derived by Bouten (15) respectively as,

O a; ; eia Iz eib Iy eig Iz ; Wigner Rotation Matrices AðÞ¼b g [11]

To calculate the Wigner matrix elements for the rota- and, tion of the spherical tensor components requires an expression for the quantum mechanical rotation ig Iz ib Iy ia Iz OPðÞ¼a; b; g e e e : [12] operator. For every rotation of a physical system in real space, there corresponds a unitary quantum me- The former is familiar from NMR as the form of the chanical operator in the state space of the physical active rotations used in Liouville space transforma- system that transforms a ket according to tions (5). Note that both are defined in terms of the angular momentum operators in the observer-fixed 0 O a; ; jic ¼ ðÞb g jic [10] frame of reference, taken to be Oxyz in both cases

Concepts in Magnetic Resonance Part A(Bridging Education and Research) DOI 10.1002/cmr.a 226 MUELLER now. As well, both are unitary and each other’s duce the irreducible spherical tensor basis (18, 19), inverse which forms a matrix basis for the decomposition of Cartesian tensors. The spherical tensor basis (STB) is O a; ; O1 a; ; : nd AðÞ¼b g P ðÞb g [13] related to the 2 -rank Cartesian tensor basis (CTB) by The Wigner rotation matrix elements are defined in terms of the matrix elements of these operators in Âà 1ffiffi kq T00 ¼p Txxþ Tyyþ Tzz the generalized angular momentum basis | i, 3Âà i T ¼pffiffi Txy Tyx 10 2 DðkÞ kpO kq; Âà pqðÞ¼OR hjRji [14] 1 T 6 Tzx Txz6i Tzy Tyz 1 1 ¼2 Âà 1ffiffi T20 ¼ p 3Tzz Txxþ Tyyþ Tzz which arise naturally under the application of closure 6Âà 1 to the rotation of an initial angular momentum ket T 6 Txz Tzx6i Tyz Tzy 2 1 ¼ÂÃ2 þ þ 1 T2 62 ¼ Txx Tyy6i Txyþ Tyx ½16 X Xk0 2 R 0 0 jikq ¼ ORjikq ¼ jik p hjk p ORjikq k0 p k0 in keeping with the standard definition of spherical ¼ 3 4 6 20 21 Xk Xk tensors used in NMR ( , , , , ). Explicitly the ðkÞ Cartesian basis is written ¼ jikphjkpORjikq ¼ DpqðÞOR jikp: ½15 p¼k p¼k 0 1 0 1 100 010 B C B C Table 1 summarizes the Wigner rotation matrices for Txx ¼ @ 000A Txy ¼ @ 000A both the active and passive rotation operators. Note 000 000 that it is the first index of the Wigner rotation matrix 0 1 001 that is summed over in Eq. [15], just as in Eq. [5]. B C Also, by definition, Eq. [15] describes the rotation of Txz ¼ @ 000A th th the q -basis or vector component, not the q -basis 000 component of the rotated vector in state space. This 0 1 0 1 000 000 is similarly true for the rotation of the spherical ten- B C B C @ A @ A sor components in Eq. [5], a point that is often con- Tyx ¼ 100 Tyy ¼ 010 000 000 fused in the NMR literature (although by no means 0 1 unique to NMR) with Eq. [5] frequently taken to 000 mean the qth component of the transformed tensor. B C Tyz ¼ @ 001A To avoid confusion, the transformed tensor will be 0 1 000 referred to as A = RA R and its tensor components 0 1 0 1 0 as Akq, while a rotated tensor component will be des- 000 000 R B C B C ignated as Akq. Tzx ¼ @ 000A Tzy ¼ @ 000A It is also important to distinguish the tensor com- 100 010 ponents, which are themselves tensors, from matrix 0 1 000 elements of the tensors, which are scalars associated B C with the magnitude of a specific tensor component. Tzz ¼ @ 000A ½17 The latter will be denoted by lower case letters, e.g., 001 axx. The analogy is to vectors in Cartesian space, ~ v ^ v ^ v ^ with the vector v ¼ xx þ yy þ zz having an and the spherical tensor basis is x-vector component vxx^ and an x-scalar component (or matrix element/coefficient) vx. 0 1 0 1 100 010 1 B C i B C T00 ¼pffiffiffi@ 010A T10 ¼pffiffiffi@ 100A 3 2 001 000 THE SPHERICAL TENSOR BASIS AND THE 0 1 ROTATION OF SPHERICAL TENSORS 001 1B C T1 61 ¼ @ 00i A To derive equations for the transformation in the 2 6i spherical tensor representation, it is useful to intro- 1 0

Concepts in Magnetic Resonance Part A(Bridging Education and Research) DOI 10.1002/cmr.a TENSORS AND ROTATIONS IN NMR 227

0 1 0 1 100 00 1 ÂÃ 1 B C 1B C a 1 a a ia a ffiffiffi@ A @ 6i A 2 61 ¼ xzþ zx yzþ zy T20 ¼ p 0 10T2 61 ¼ 00 2 6 2 002 1 6i 0 1ÂÃ 0 1 a2 62 ¼ axx ayy iaxyþ ayx ½22 1 6i 0 2 1B C T 6 ¼ @ 6i 10A: ½18 2 2 2 can readily be shown to satisfy Eq. [20] by direct 000 substitution. The spherical tensor basis states transform under Both the CTB and STB are complete and satisfy the rotation analogously to the angular momentum basis orthonormalization conditions kets, with the qth basis component of a rank-k spheri- no cal tensor transforming into a combination of basis y Tr TklTmn ¼ dkmdln: [19] states of the same rank (9–13), Xk R 1 ðkÞ The coefficients, akj, for the expansion of an arbitrary Tkq ¼ RTkqR ¼ DjqðÞOR Tkj: [23] Cartesian tensor, A, in the spherical basis, j¼k

This equation, which will be referred to as the tensor X2 Xk component equation (or component equation for short), A ¼ akjTkj [20] can be verified by direct substitution of the Wigner rota- k¼0 j¼k tion matrix elements and the Cartesian representations R of R and Tkq. It again highlights that Tkq is the transfor- can be obtained by mation of the qth-component of the rank-k tensor, not no the qth-component of the transformed tensor. Equation y akj ¼ Tr TkjA : [21] [23] also clarifies the meaning of equation [5], which should be interpreted as a scalar multiple of equation [23]. To transform the full tensor, A is first expanded As the STB components are not Hermitian, the ma- and then rotated in the spherical tensor basis trix trace is taken with the conjugate of the corresponding basis component. The coefficients A0 ¼ RAR1 obtained by [21] ! X2 Xk 1 ¼ R akjTkj R 1 ÂÃ k¼0 j¼k a00 ¼pffiffiffi axxþ ayyþ azz X2 Xk Xk 3 ðkÞ akj D OR Tkq i ÂÃ ¼ qjðÞ k¼0 j¼k q¼k a10 ¼ pffiffiffi axy ayx 2 X2 Xk Xk ÂÃ ðkÞ 1 ¼ DqjðÞOR akjTkq ½24 a 6 ¼ azx axz iazy ayz 1 1 2 k¼0 q¼k j¼k 1 ÂÃ a20 ¼ pffiffiffi 3azz axxþ ayyþ azz 6 or explicitly in matrix form as

0 ð0Þ a ¼ D ðÞOR a 00 00 0 00 1 0 1 0 1 ð1Þ ð1Þ ð1Þ a a0 D R D R D R 1 1 1 1 1 1ðÞO 10ðÞO 11ðÞO @ A B C B C a10 @ a0 A ¼ B Dð1Þ Dð1Þ Dð1Þ C 10 @ 0 1ðÞOR 00ðÞOR 01ðÞOR A a11 a0 ð1Þ ð1Þ ð1Þ 11 D ðÞOR D ðÞOR D ðÞOR 0 1 1 10 11 1 0 1 ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ 0 1 a0 D ðÞOR D ðÞOR D ðÞOR D ðÞOR D ðÞOR a B 2 2 2 1 20 21 22 C 2 2 2 2 ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ B a0 C B D R D R D R D R D R CB a C B 2 1 C B 1 2ðÞO 1 1ðÞO 10ðÞO 11ðÞO 12ðÞO CB 2 1 C B a0 C B Dð2Þ Dð2Þ Dð2Þ Dð2Þ Dð2Þ CB a C: B 20 C ¼ B 0 2ðÞOR 0 1ðÞOR 00ðÞOR 01ðÞOR 02ðÞOR CB 20 C @ a0 A B ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ C@ a A 21 @ D ðÞOR D ðÞOR D ðÞOR D ðÞOR D ðÞOR A 21 a0 1 2 1 1 10 11 12 a 22 Dð2Þ Dð2Þ Dð2Þ Dð2Þ Dð2Þ 22 2 2ðÞOR 2 1ðÞOR 20ðÞOR 21ðÞOR 22ðÞOR

Concepts in Magnetic Resonance Part A(Bridging Education and Research) DOI 10.1002/cmr.a 228 MUELLER

While Eqs. [24] and [25] describe the rotation of the These two expressions seem to confuse the rotation full tensor, the theoretical treatment of NMR interac- of the coefficients and the rotation of the basis set tions often focuses on particular matrix elements of components and, in light of the coefficient equation, the Cartesian tensor for different orientations of the should not be valid. In fact Eq. [27] is not correct in physical system (3, 4). For example, due to truncation that it is inconsistent with the Cartesian transforma- at high magnetic field, it is the matrix element associ- tion under rotation as written in Eq. [4]. Rather, Eq. ated with the secular component of a tensor that prin- [27] should be written with the inverse of the rotation cipally determines its spectrum, such as szz (in the matrix in the Wigner rotation elements, laboratory frame) of the chemical shielding tensor. From Eq. [24], the coefficient of the qth-component Xk b0 DðkÞ b : of the rank–k, transformed tensor is kq ¼ jqðÞOR1 kj [29] j¼k Xk 0 ðkÞ akq ¼ DqjðÞOR akj: [26] This can be shown by noting that the appropriate j¼k pairing of the bkj coefficients with the basis compo- nents to form the full tensor is We will refer to this as the coefficient equation to dis- tinguish it from the component equation (Eq. [23]). X2 Xk b ; The coefficient equation is strikingly similar in form A ¼ kjTkj [30] to the tensor component equation, with just a switch k¼0 j¼k in the order of the indices for the Wigner coefficient. b Great care needs to be taken to distinguish these two and so the kj must satisfy the coefficient equation, equations as they express different physical situa- Xk tions. The component equation applies to the tensor 0 ðkÞ bkq ¼ DqjðÞOR bkj; [31] basis components; it describes how a single tensor j¼k component transforms into a sum of the other compo- nents under rotation. The coefficient equation, how- which can be rewritten as Eq. [29] by taking the ever, gives the coefficient of a specific tensor compo- complex-conjugate of both sides and making use of nent after the transformation of an arbitrary tensor. ðkÞ ðkÞ the property that Djq (VR1)=Dqj (VR). Yet in the NMR literature, equations for the trans- The distinction between Eqs. [27] and [29] is fun- formation of spherical tensor coefficients are ubiqui- damental and clarifies the apparent inconsistency tously written as if they were the same as the tensor between the rotation of second-rank Cartesian tensors 3 4 20 21 component equation ( , , , ), in Cartesian and spherical tensor forms found in the NMR literature, where the rotation using Eq. [27] ? k must be partnered with its inverse rotation in Carte- P k 0 Dð Þ b [27] bkq ¼ jqðÞOR kj sian form to produce equivalent transformations. One j¼k prominent example of this mispairing is Appendix B of Mehring’s text (4), which couples the Cartesian with coefficients, bkj, defined to be the complex-con- rotation matrix from the passive perspective (the jugates of those in [22], inverse of the active perspective) with the Wigner rotation matrix elements in the active form, an error ÂÃ that can be traced back to Rose’s text (10). Others (6, 1ffiffiffi b00 ¼ a ¼p axxþ ayyþ azz 20 21 00 3 , ) have noted that this mispairing is necessary i ÂÃ and directly link the representation of the rotation in b10 ¼ a ¼pffiffiffi axy ayx spherical form with its inverse rotation in the Carte- 10 2 sian representation, but fail to note that the inverse 1ÂÃ b a a a 6ia a rotation should be associated with the Wigner rota- 1 61 ¼ 1 61 ¼ zx xz zy yz 2 tion matrix elements. The correct correspondence 1 ÂÃ b a ffiffiffi a a a a between the physical rotation of the system and its 20 ¼ 20 ¼ p 3 zz xxþ yyþ zz 6 Cartesian and spherical tensor transformations is sig- ÂÃ 1 nificant, particularly in structural studies where con- b 6 ¼ a 6 ¼ axzþ azx6iayzþ azy 2 1 2 1 2 nections back to a molecular frame are made. As the ÂÃ 1 spatial components of the NMR Hamiltonian are b 6 ¼ a ¼ axx ayy6iaxyþ ayx ½28 2 2 2 62 2 tied to the molecular frame of reference, they rotate

Concepts in Magnetic Resonance Part A(Bridging Education and Research) DOI 10.1002/cmr.a TENSORS AND ROTATIONS IN NMR 229

Figure 2 Two orientations of glycine in the laboratory frame related by an active rotation p; p; p RA 4 6 3 of the molecular coordinates. The standard CPK scheme is used to designate the atom colors (H, white; C, gray; N, blue; O, red).

0 1 0 1 with the molecule. The result is that in many cases xb xa @ b A p p p @ a A the transformation effected by the application of Eq. y ¼ RA ; ; y [27] has been the opposite of what was intended or zb 4 6 3 za expected. For example, Haeberlen (3) uses Eq. [27] 0 qffiffi qffiffi qffiffi 1 0 1 with the Wigner matrix elements from the passive 1 3 5 1 1 1 a B 4 2 4 2 2 2 C x perspective, so is actually performing an active rota- B qffiffi qffiffi qffiffi C @ ya A; 4 ¼ B 3 3 1 1 1 1 C [32] tion of the tensors (and molecules). Mehring ( ), as @ 4 2 4 ffiffi 2 2 ffiffi 2 A a 20 p p z well as Spiess and Schmidt-Rohr ( ), use the active 1 3 3 Wigner rotation matrix elements and Eq. [27], so are 4 4 2 actually performing a passive rotation. where the superscripts a and b refer to the initial and final molecular orientations, respectively, shown in the corresponding figures. In this expression, the EXAMPLE: THE TRANSFORMATION OF A new molecular orientation is considered to be the CHEMICAL SHIELDING TENSOR resultÈÉ of an active rotation with Euler angles a p; p; p OA ¼ ¼ 4 b ¼ 6 g ¼ 3 . However, the orienta- As a worked example, the transformation of the ab ini- tion in 2(b) could equally be considered theÈ result of a p; tio chemical shielding tensor for the alpha carbon of a passive rotation with Euler angles OP ¼ ¼3 p; p glycine in the gas phase is considered. This is done in b ¼6 g ¼4g; either way, the numerical form of three ways. In method I, the ab initio chemical shield- the rotation matrix and the orientations shown in Fig- ing tensor is calculated directly using Gaussian03 (22) ure 2 would be the same. for an initial and rotated orientation. In method II, the For the alpha carbon, the calculated ab initio transformed tensor is calculated from the initial tensor chemical shielding tensors (Gaussian03, B3LYP, and the rotation matrix in Cartesian form (Eq. [4]). In 6311þþG**) corresponding to the molecular orien- method III, the transformation is effected using spheri- tations in Figure 2(a) and 2(b) are cal tensors and the coefficient equation (Eq. [26]). To 0 1 be consistent, all three of these must agree. 151:5 8:71:5 a Figure 2(a) shows the geometry optimized struc- s ¼ @ 6:9 131:0 5:0 A [33] ture of glycine in the gas phase calculated using 5:1 2:8119:4 Gaussian03 (B3LYP, 6311þþG**), in which the molecular orientation in the laboratory (observer)- and fixed coordinate frame is chosen by the program dur- 0 1 ing refinement to be the standard nuclear orientation, 129:01:20:1 b;I @ A with the principal axes of the ten- s ¼ 0:6 152:0 10:9 [34] sor aligned along the three Cartesian axes. Figure 3:1 7:7 120:9 2(b) shows the same molecule in which the atomic Cartesian coordinates from Figure 2(a) have been (in ppm). The additional superscript on the latter is rotated according to meant to signify that it was calculated using method I.

Concepts in Magnetic Resonance Part A(Bridging Education and Research) DOI 10.1002/cmr.a 230 MUELLER

The chemical shielding tensor for orientation b jic0 ¼ Ojic [39] can also be directly calculated from the tensor for the initial orientation and the rotation matrix using and operator tied to the molecular frame method II, direct matrix multiplication in the Carte- sian representation, A0 ¼ OAOy [40]   b; p p p a p p p s II ¼ R ; ; s R1 ; ; having an expectation value that is invariant under A 4 6 3 A 4 6 3 0 1 molecular rotation 129:01:20:2 B C : : : : 0 0 ¼ @ 0 6 152 0 10 9 A ½35 hjc A0jic ¼ hjc Ajic : [41] 3:1 7:8 120:9

This agrees with method I within the expected com- DYADIC PRODUCTS, THE NMR putational accuracy. HAMILTONIAN, AND REFERENCE Finally, method III can be used, which employs FRAMES the use of spherical tensors. First, the initial chemical shielding tensor is decomposed into its spherical The advantage of irreducible spherical tensors is that components by calculating the coefficients (Eq. [21]) they isolate elements of second-rank Cartesian ten- no sors that transform together under rotation. To con- a y a struct the Hamiltonian directly in terms of spherical s ¼ Tr T s ; [36] kj kj tensor components and take full advantage of these transformational properties, the Cartesian space, spin for k = {0,1,2} and j ={k,...,k}. These transform operator-containing components of the NMR Hamil- according to the coefficient equation (Eq. [26]) to tonian are first written as a dyadic product (3, 4, 6) give the coefficients for orientation b, 0 1 l l l Sx Ix Sx Iy Sx Iz k  X l Sl I @ SlI SlI SlI A: b DðkÞ a : S ¼ ¼ y x y y y z [42] skq ¼ qj OR p;p;p skj [37] l l l AðÞ4 6 3 S Ix S Iy S Iz j¼k z z z

Here again, the Wigner rotation matrix elements will and the NMR Hamiltonian as a trace over the product of the spatial and spin-containing Cartesian tensors have the same numerical values whether we considerÈ this an active rotation with Euler angles O ¼ a ¼ p; A 4 ÈÉ b p; g p or a passive rotation with Euler angles l l l l l l l ¼ 6 Èɼ 3g H ¼ c I A S ¼ c Tr A S : [43] a p; p; p OP ¼ ¼3 b ¼6 g ¼4 . Once calculated, the full chemical shielding tensor for orientation b An analogous decomposition into the spherical tensor can be reconstituted according to Eq. [20] as basis can be applied to the dyadic product with coef- X2 Xk ficients b;III b s ¼ skjTkj k j k Âà ¼0 ¼ 0 1 1 s00 ¼pffiffiffi SxIx þ SyIy þ SzIz 129:01:20:2 3 B C Âà ¼ @ 0:6152:0 10:9 A; ½38 i s10 ¼ pffiffiffi SxIy SyIx 3:1 7:8 120:9 2 1Âà s1 61 ¼ SzIx SxIz iSzIy SyIz in agreement with both methods I and II. 2 Âà Note that the molecular coordinates and the tensor 1 s20 ¼ pffiffiffi 3SzIz SxIx þ SyIy þ SzIz transform together. This is reasonable, given that 6 they both correspond to quantum mechanical observ- 1Âà s 6 ¼ SxIz þ SzIx iSyIz þ SzIy ables that are tied to the molecular frame of refer- 2 1 2 ence. Molecular frame observables also rotate in the 1Âà s 6 ¼ SxIx SyIy iSxIy þ SyIx ; ½44 same sense as kets in state space, as Schmidt-Rohr 2 2 2 and Spiess point out (20), with a transformed molec- ular ket allowing the Hamiltonian to be written

Concepts in Magnetic Resonance Part A(Bridging Education and Research) DOI 10.1002/cmr.a TENSORS AND ROTATIONS IN NMR 231

ÈÉ l l l l H ¼ c Tr A S The Hamiltonian in Eqs. [47] and [48] is written () X2 Xk X2 Xk0 from the perspective of the laboratory frame, with the l ¼ c Tr akjTkj sk0 j0 Tk0 j0 spatial tensor transformed to that frame before being k¼0 j¼k k0¼0 j0¼k0 combined with the spin components (also in the labora- X2 Xk tory frame) to form the Hamiltonian. While the external l j ¼ c ðÞ1 akjsk j: ½45 magnetic field and spin operators associated with NMR k¼0 j¼k observables are tied to the laboratory frame, making this an obvious choice for writing the tensor compo- Here, the identity nents of the Hamiltonian, the trace in Eq. [43] is invari- ant to a change in basis, so the spatial and spin tensors j0 y in the Hamiltonian may in fact be written in any arbi- Tk0 j0 ¼ðÞ1 Tk0 j0 [46] trary frame. For example, in some cases it may be con- venient to consider the tensor components of the Ham- has been used to simplify the trace. Note that the iltonian from within the spatial tensor PAS frame, coefficients in the Cartesian dyadic product contain ÈÉ l l spin operators, which themselves may be treated H ¼ c Tr APAS SPAS : [49] using spherical tensors in spin space (2, 23–25), a related but distinct issue from the representation of The in the PAS must ultimately be related Cartesian tensors in real space treated here. back to the spin tensor in the laboratory frame, as the The transformation of the Hamiltonian under rota- latter contains the lab-frame angular momentum tion in real space can now be written directly in operators that direct the spin dynamics, but again the terms of the rotation of the Cartesian tensor in Eq. tensors are simply related by [43] or through the transformation of the spherical PAS PAS lab 1 PAS ; tensor coefficients, akj, in Eq. [45]. For example, if S ¼ R Olab S R Olab [50] the spatial tensor has known components in its prin- cipal axis system (the in which the where now the Euler angles parameterize the orienta- symmetric part of the spatial tensor is diagonal), then tion of the laboratory frame as seen by an observer in the tensor components in the laboratory frame can be the PAS frame. calculated and combined with the spin-containing This leads to a potential source of confusion, as components to give the Hamiltonian as the Euler angles are defined relative to observers in different frames in Eqs. [47] and [49]. They are ÈÉ l l related, however, and by comparing Eqs. [47]–[50], H ¼ c Tr Alab Slab and making use of the cyclic property of the trace, ÈÉ cl Tr Olab PAS 1 Olab lab ÈÉ ¼ R PAS A R PAS S [47] l l H ¼ c Tr R Olab APAS R1 Olab Slab ÈÉPAS PAS l and ¼ c Tr APAS R1 Olab Slab R Olab ÈÉPAS PAS l ¼ c Tr APAS R OPAS Slab R1 OPAS X2 Xk X2 Xk ÈÉlab lab Hl cl jalabslab cl j cl Tr 1 PAS PAS PAS lab ; ¼ ðÞ1 kj k j ¼ ðÞ1 ¼ R Olab A R Olab S ½51 k¼0 j¼k k¼0 j¼k Xk it is seen that ðkÞ lab PAS lab D 0 O a 0 s ; ½48 jj PAS kj k j j0¼k lab 1 PAS R OPAS ¼ R Olab Þ [52] respectively, in Cartesian and spherical tensor forms. Vlab or Here PAS is the set of Euler angles that parameter- ize the orientation of the principal axis system (PAS) R alab ; blab ; llab ¼ R1 aPAS; bPAS; gPAS as seen by an observer in the laboratory frame. The PAS PAS PAS lab lab lab PAS; PAS; aPAS : superscripts ‘‘lab’’ and ‘‘PAS’’ have been added to ¼ R glab blab lab ½53 the tensors to indicate that the components are defined relative to an observer in those frames (and In other words, two observers in the lab and PAS the l dropped from the tensors for notational conven- frames have opposite views of the rotation that takes ience). Both Eqs. [47] and [48] can be simplified their frames from initial coincidence to their final ori- considerably under the secular approximation (see entation relative to each other, exactly as one would Question 2). expect. Equation [51] also highlights that even for

Concepts in Magnetic Resonance Part A(Bridging Education and Research) DOI 10.1002/cmr.a 232 MUELLER tensors written in the lab frame, the transformation of spatial components of the NMR Hamiltonian repre- the spatial components of the Hamiltonian to that sented by either Cartesian or spherical tensors. frame can be replaced by the application of the inverse transformation to the spin containing dyadic Questions and Answers terms. Finally, it is noted that in some cases a frame dis- Question 1. There is considerable latitude in choosing tinct from the lab or PAS may be more convenient to either the active or passive convention for represent- work in. For example, in their classic paper on side- ing a transformation in NMR. For example, consider band intensities under magic-angle spinning, Herz- the transformation of a chemical shielding tensor feld and Berger (26) choose the MAS rotor as the under a change in coordinate frame of reference from frame of reference and then write both the lab frame the tensor PAS to the laboratory frame. Describe this magnetic field/spin operators (which are now time- transformation using the passive point of view and then recast the problem using the active point of view. dependent) and the chemical shift tensor in the rotor Answer: frame. Question 3 below presents an alternate deriva- The use of the passive point of view in this con- tion of the Hamiltonian used as the starting point in text is straight-forward. Letting the Euler angles the Herzfeld-Berger analysis, writing the components PVlab parameterize the rotation that would take the in the laboratory frame and using a two-step transfor- PAS PAS frame into coincidence with the laboratory mation of the spatial tensor from the PAS to rotor frame, the chemical shielding tensor in the lab frame and then rotor to lab frames. would be written: s lab R POlab s PAS R1 POlab : CONCLUSIONS ¼ P PAS P PAS Alternatively, the coordinate frame transformation The motivation behind the use of irreducible spheri- could be considered to result from an active rotation cal tensors in NMR is that they allow the components that takes the tensor PAS from initial alignment with of second-rank Cartesian tensors to be grouped the laboratory frame to its final orientation in the labo- according to their underlying rotational symmetry. In ratory frame, parameterized by the Euler angles AVlab , principal, this should simplify the treatment of aniso- PAS tropic interactions in the NMR Hamiltonian. In prac- lab lab s lab ¼ R AO s PAS R1 AO : tice, the inconsistent use of active and passive con- A PAS A PAS ventions and errors in several of the classic texts on In this case, sPAS is both the initial tensor in the labo- angular momentum have kept these benefits from ratory frame and the tensor in the PAS frame. PVlab being fully realized and have led in many cases to PAS and AVlab are of course related; if PVlab ={a,b,g} transformations that correspond to the inverse of the PAS PAS then AVlab ={g,b,a}. stated rotation. In situations where spectroscopic PAS observables are averaged over the full set of Euler Question 2. Write out the chemical shielding Hamilto- angles or no connection is made back to a molecular nian under the secular approximation in terms of the frame of reference, this confusion in rotation is of lit- PAS components, sXX, sYY,andsZZ,andtheEuler tle consequence. The calculated CSA powder line Vlab angles, PAS, that relate the PAS and laboratory shape, for example, would be the same regardless of frames. Use the active rotation convention. whether the molecular rotation was parameterized Answer: correctly or with an inverted sense. In cases where (a) Using Cartesian Tensors: absolute connections between the spatial tensor and The starting point for this problem is Eq. [47]. molecular or crystalline frames are desired, however, First, the following associations are made for the these discrepancies must be reconciled. chemical shielding interaction: Here a consistent approach to the transformation l of second-rank Cartesian tensors in Cartesian and c ¼ g; spherical tensor forms has been shown. The introduc- tion of an explicit spherical tensor basis for the 0 1 decomposition of second-rank Cartesian tensors helps sXX 00 PAS PAS @ A delineate the rotational properties of the basis states A ¼ s ¼ 0 sYY 0 from those of the matrix elements. This provides a 00sZZ uniform approach to the rotation of the physical ; system and the corresponding transformation of the ðignoring the antisymmetric termsÞ

Concepts in Magnetic Resonance Part A(Bridging Education and Research) DOI 10.1002/cmr.a TENSORS AND ROTATIONS IN NMR 233

rffiffiffi and PAS 1 3 a ¼ pffiffiffi½¼3sZZ ðÞsXX þ sYY þ sZZ d 0 1 20 6 2 000 PAS lab a ¼ 0 Slab ¼ ðÞB I ¼ @ 000A; 2 61 BzIx BzIy BzIz PAS 1 1 a ¼ ½¼sXX sYY Zd 2 62 2 2 for the static magnetic field aligned along the lab- frame z-axis. Under the secular approximation, Slab and can be further simplified by dropping the Ix and Iy spin slab 1ffiffiffiB I angular momentum terms, which do not commute 00 ¼p z z Zeeman 3 with the Zeeman interaction, H = gBzIz,allow- slab ing the chemical shielding Hamiltonian to be written 10 ¼ 0 ÂÃ ÈÉ slab 1 B I iB I secular HCS lab s PAS 1 lab B I : 1 61 ¼ z x z y 0 ¼ g R OPAS R OPAS zz z z 2 rffiffiffi lab 2 s ¼ BzIz Using the active convention for writing the rotation 20 3 matrix, ÂÃ lab 1 secular s ¼ BzIx iBzIy 0  2 61 2 HCS B I 2 2 2 ¼ g z z sZZ cos b þ sXX sin b cos g slab  2 62 ¼ 0 2 2 þsYY sin b sin g ; Equation [48] can then be written where Vlab ={a,b,g} are the Euler angles that HCS alabslab alabslab PAS ¼ g 00 00þ g 20 20 would take the PAS frame from initial coincidence X2 with the lab frame to its final orientation in the lab Dð0Þ lab aPASslab Dð2Þ lab aPASslab ¼g 00 OPAS 00 00þ g 0 j0 OPAS 2 j0 20 frame. j0¼2 The Hamiltonian can be also be written in terms n rffiffiffi 1 of the isotropic shift, B I B I Dð2Þ lab ¼ g z zs þ g z z 0 2 OPAS Zd rffiffiffi 6 1 s ¼ ðÞsXX þ sYY þ sZZ ; Dð2Þ lab Dð2Þ lab 1 3 þ 00 OPAS d 02 OPAS Zdg  6 anisotropy, 3 cos2 b 1 1 ¼ g B I s þ d Z sin2 b cos 2gÞ : z z 2 2 d ¼ sZZ s; Question 3. In their classic paper on sideband inten- and asymmetry, sities under magic-angle spinning, Herzfeld and sYY sXX Berger (26) write both the lab frame magnetic field/ Z ¼ ; d spin operators and the chemical shift tensor in the MAS rotor frame, as, ÈÉ Hcs Tr rotor B I rotor  ¼g s ðÞ 2 n HCS B I 3 cos b 1 1 2 : Tr rotor s PAS 1 rotor rotor ¼ g z z s þ d Z sin b cos 2gÞ ¼g R OPAS R OPAS R Olab 2 2 o B I lab 1 rotor ðÞ R Olab (b) Using Spherical Tensors: The starting point here is Eq. [48], with the same Provide an alternate derivation of the Hamiltonian associations given above and noting that used as the starting point in the Herzfeld-Berger anal- ysis (corresponding to the frequency given in Eq. pffiffiffi PAS 1 [16] of their paper) by writing the components in the a ¼pffiffiffi½¼sXX þ sYY þ sZZ 3 s 00 3 laboratory frame and using a two-step transformation aPAS ¼ 0 of the spatial tensor from the PAS to rotor frame and 10 then from the rotor to lab frames. Use spherical ten- aPAS 1 61 ¼ 0 sors and the passive rotation convention.

Concepts in Magnetic Resonance Part A(Bridging Education and Research) DOI 10.1002/cmr.a 234 MUELLER

Answer: Under the secular approximation, the where the Euler angles Vrotor ={a,b,g} parameterize chemical shift Hamiltonian in spherical tensor form is PAS the rotation that would take the PAS to the rotor frame. With some minor algebraic and trigonometric Hcs ¼g alabslab g alabslab 0000 20 20 rearrangement, this can be reduced to Eq. [16] of Dð0Þ lab arotorslab their article. ¼g 00 Orotor 00 00 X2 Dð2Þ lab arotorslab g 0 j0 Orotor 2 j0 20 j0¼2 ACKNOWLEDGMENTS Dð0Þ lab Dð0Þ rotor aPASslab ¼g 00 Orotor 00 OPAS 00 00 The author would like to thank Dr. Moreno Lelli for X2 X2 Dð2Þ lab Dð2Þ rotor aPASslab lively discussions on rotation matrices and Professor g 0 j0 Orotor j0 j00 OPAS 2 j00 20 j0¼2 j00¼2 Alex Bain, Mr. Lingchao Zhu, and Mr. Chen Yang X2 X2 for a careful reading and comments on the text. This B I Dð2Þ lab work was supported by NSF grant CHE0848607. ¼g z zs g 0 j0 Orotor j0¼2 j00¼2 Dð2Þ rotor aPASslab j0 j00 O j00 PAS 2 20 REFERENCES X2 B I B I Dð2Þ lab ¼g z zs g z z 0 j0 Orotor j0¼2 1. Abragam A. 1961. Principles of Nuclear Magnetism.  rffiffiffi Oxford: Clarendon Press. Dð2Þ rotor 1 Dð2Þ rotor 2. Slichter CP. 1980. Principles of Magnetic Resonance. j0 O Zd þ j0 O 2 PAS 6 0 PAS Berlin; New York: Springer-Verlag. rffiffiffi  3. Haeberlen U. 1976. High Resolution NMR in Solids: Dð2Þ rotor 1 d j0 2 OPAS Zd Selective Averaging. New York: Academic Press. 6 4. Mehring M. 1983. Principles of High Resolution NMR in Solids. Berlin; New York: Springer-Verlag. Using the passive convention, the Euler angles that 5. Ernst RR, Bodenhausen G, Wokaun A. 1987. Princi- would take the rotor-fixed frame to the laboratory ples of Nuclear Magnetic Resonance in One and Two . Oxford: Oxford University Press. frame under MAS are 6. Smith SA, Palke WE, Gerig JT. 1992. The hamilto- lab nians of NMR. Part I. Concepts Magn Reson 4:107– O ¼ fgort; ym; 0 ; rotor 144. where 7. Smith SA, Palke WE, Gerig JT. 1992. The hamilto- nians of NMR. Part II. Concepts Magn Reson 4:181– 1 1 ym ¼ cos pffiffiffi : 204. 3 8. Smith SA, Palke WE, Gerig JT. 1994. The hamilto- nians of NMR. Part IV: NMR relaxation. Concepts (N.B., this is just one of the possible ways in which Magn Reson 6:137–162. MAS can be parameterized). Substituting above and 9. Fano U, Racah G. 1959. Irreducible Tensorial Sets. simplifying the expression gives New York: Academic Press. 10. Rose ME. 1957. Elementary Theory of Angular Mo-  mentum. New York: Wiley. 11. Wigner EP. 1959. Group Theory and Its Application Hcs B I : ¼g z z s to the of Atomic Spectra. New  York: Academic Press. 1 1 1 þ d sin2 b Z cos 2a 1 þ cos 2b 12. Edmonds AR. 1957. Angular Momentum in Quantum 2 4 3 Mechanics. Princeton, NJ: Princeton University Press. cosðÞ 2ort þ 2g 13. Zare RN. 1988. Angular Momentum: Understanding Spatial Aspects in Chemistry and Physics. New York: 1 þ Zd sin 2a cos b sinðÞ 2ort þ 2g Wiley. 3 14. Goldstein H. 1980. Classical Mechanics. Reading, 1 1 MA: Addison-Wesley Pub. Co. pffiffiffid 1 þ Z cos 2a sin 2b cosðÞort þ g 2 3 15. Bouten M. 1969. On rotation operators in quantum pffiffiffi  2 mechanics. Physica 42:572–580. þ Zd sin 2a sin b sinðÞort þ g 16. Edmonds AR. 1996. Angular Momentum in Quantum 3 Mechanics. Princeton, NJ: Princeton University Press.

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17. Brink DM, Satchler GR. 1962. Angular Momentum. Gaussian 03. Revision B.05. Gaussian, Inc., Wallingford, Oxford: Clarendon Press. CT, 2004. 18. Cook RL, Delucia FC. 1971. Application of theory of 23. Bowden GJ, Hutchison WD. 1986. - irreducible tensor operators to molecular hyperfine formalism for multiple-quantum NMR. 1. Spin-1 structure. Am J Phys 39:1433–1454. nuclei. J Magn Reson 67:403–414. 19. Nielsen RD, Robinson BH. 2006. The spherical ten- 24. Bowden GJ, Hutchison WD, Khachan J. 1986. Tensor sor formalism applied to relaxation in magnetic reso- operator-formalism for multiple-quantum NMR. 2. nance. Concepts Magn Reson Part A 28A:270–290. Spins-3/2, spin-2, and spin-5/2 and general-I. J Magn 20. Schmidt-Rohr K, Spiess HW. 1994. Multidimensional Reson 67:415–437. Solid-State NMR and Polymers. London; San Diego: 25. Bain AD, Berno B. 2011. Liouvillians in NMR: the Academic Press. direct method revisited. Prog Nucl Magn Reson Spec- 21. Shekar SC, Jerschow A. 2008. Tensors in NMR. trosc, in press; doi: 10.1016/j.pnmrs.2010.12.002. In:Harris RK,Wasylishen RE, eds. Encyclopedia of 26. Herzfeld J, Berger AE. 1980. Sideband intensities in Magnetic Resonance. Chichester: Wiley. NMR-spectra of samples spinning at the magic angle. 22. Frisch MJ, Trucks GW, Schlegel HB, Scuseria GE, J Chem Phys 73:6021–6030. Robb MA, Cheeseman JR, J. A.Montgomery Jr., Vreven T, Kudin KN, Burant JC, Millam JM, Iyengar SS, Tomasi J, Barone V, Mennucci B, Cossi M, Scal- BIOGRAPHY mani G, Rega N, Petersson GA, Nakatsuji H, Hada M, Ehara M, Toyota K, Fukuda R, Hasegawa J, Ish- ida M, Nakajima T, Honda Y, Kitao O, Nakai H, Len Mueller is an Associate Professor in Klene M, Li X, Knox JE, Hratchian HP, Cross JB, the Department of Chemistry at the Uni- Adamo C, Jaramillo J, Gomperts R, Stratmann RE, versity of California, Riverside. He Yazyev O, Austin AJ, Cammi R, Pomelli C, Ochter- received his B.S. in Chemistry from the ski JW, Ayala PY, Morokuma K, Voth GA, Salvador University of Rochester (1988), C.P.G.S. P, Dannenberg JJ, Zakrzewski VG, Dapprich S, Dan- in Natural Science (Chemistry) from the iels AD, Strain MC, Farkas O, Malick DK, Rabuck University of Cambridge (1989), and Ph.D. in Chemistry from the California AD, Raghavachari K, Foresman JB, Ortiz JV, Cui Q, Institute of Technology (1997). From Baboul AG, Clifford S, Cioslowski J, Stefanov BB, 1996–1998 he was an American Cancer Society Postdoctoral Liu G, Liashenko A, Piskorz P, Komaromi I, Martin Fellow at the Massachusetts Institute of Technology. Len’s RL, Fox DJ, Keith T, Al-Laham MA, Peng CY, research interests include nuclear magnetic resonance Nanayakkara A, Challacombe M, Gill PMW, Johnson spectroscopy as a probe of molecular and biological structure B, Chen W, Wong MW, Gonzalez C, Pople JA. and dynamics.

Concepts in Magnetic Resonance Part A(Bridging Education and Research) DOI 10.1002/cmr.a Physica 42 (1969) 572-580 o North-Holland Publishing Co., Amsterdam

ON THE ROTATION OPERATORS IN QlJANTlJM MECHANICS

1’I. ROIJTEN

A fdeling uooY theoretiscke Physica, S.C.K. ,VTol, Belqit?

Received 20 September 1968

synopsis The confusion, existing about the exact form of the rotation operators in quantum mechanics, is cleared up by indicating some well hidden errors in the two standard books on angular momentum.

1. Introduction. The consequences of the rotational symmetry of a Hamiltonian in quantum mechanics are described in many booksi-5). To every rotation in real space, there corresponds a in the Hilbert space of the physical states. In some cases, an explicit expression for these operators is needed. Several different expressions are used in the literature which manifestly cannot all be exact. The existing confusion about the exact form of the rotation operators can partly be ascribed to small differences in the choice of the Euler angles and to the existence of an active and a passive point of view, which are often not clearly distinguished. However, the main cause of the confusion appears to be due to some well hidden errors in the derivation of the rotation operators in the two standard booksr~s) on angular momentum. In this paper, we want to point out these errors and to indicate the exact expression for the rotation operators, both in the active and in the passive point of view. For simplicity, we consider only the case of a scalar field. In particular, we are thinking of the wave function of a single spinless particle. This simple case contains already the whole difficulty. Generalization to more complicated systems is straightforward.

2. Choice of the Euler angles. Let us consider a “fixed” system of axes Oxyz and a “rotated” system of axes OXYZ (fig. 1). The orientation of the rotated system relative to the fixed system is determined by the three Euler angles @y. We follow Messiahs) for the definition of the Euler angles. We choose one of the half axes orthogonal to the zOZ plane and call it Ou. The Euler angles are defined as

01 = (Oy, Ou); #8 = (Oz, 02); y = (Ou, OY).

572 ON THE ROTATION OPERATORS IN QUANTUM MECHANICS 573

Fig. 1. Definition of the Euler angles.

If one considers the system of axes OXYZ to coincide originally with Oxyz, it will be transformed to its rotated position by three successive rotations:

(i) a rotation about the axis Oz through an angle (Y; (ii) a rotation about the axis Ou through an angle p; (iii) a rotation about the axis OZ through an angle y. The angles 01,p, y are algebraic quantities which are taken as positive or negative according to whether the corresponding rotations about the three axes are positive or negative*.

3. Definition of the rotation matrices. We consider an object (or a physical system), whose position in space, relative to a fixed origin 0, is determined by the position vectors of all its points. For definiteness, we consider one

point P of this abject with position vector 0”. We choose the system of axes Oxyz as a frame of reference. The position of the point P is now determined, relative to this frame of reference, by the

three Cartesian coordinates xyz of the vector o”p. One can think of rotations in two different ways. 3.1. Active point of view. In the active point of view, the object is rotated about the fixed or&in 0. Every point P of the object will acquire

a new position P’ in space. The coordinates of the new position vector &’ in the fixed frame of reference Qxyq will be denoted x’y’z’. It is possible to characterize aa active rotation by three Euler angles in the following way. We introduce! a system of axes OXYZ which are fixed in the object and which are chosen, so as to coincide originally (i.e. before the rotation of the object is carried out), with the axes Oxyz. The rotated

* A positive (negative) rotation about a given aXis is one which would carry a right handed screw in the positive (negative) direction along that axis. 574 M.BouTEN

position of the object is now determined by the final orientation of the ax OXYZ. The three Euler angles OL,8, y which, from section 2, determine tl final orientation of OXYZ relative to the fixed system of axes Oxyz, c; thus be taken to characterize the active rotation. The relation between the coordinates x, y, x of 6%’ and x’, y’, Z’ of 6 is easily obtained:

x’ cosa: co@ cosy - sinol siny -cosol cosg siny - sinol cosy cosa sir@ x y' = sina cos/3cosy+coscu. siny - sinol co@ siny + coso( cosy sincu.sir@ y O(2’ - sir43 cosy sin/? siny cosg >( z

We abbreviate this as

y3 = A(a, @, y) ; The set of matrices A(ol, 8, y) form a matrix group which we denote by G The matrices A (OL,p, y) have the following interesting property. Introducir- the two special matrices

case -sin8 0

M(0) = sin8 COST 0; ( 0 0 1> C c0se 0 sin e

iv(e) = 0 I 0 , - ( sin e 0 COST >

which are respectively equal to A(800) = A (000) and A (OeO), it is possibl to write every A(cY, @, y) in a standard from as a product of these speci; matrices M(0) and N(8) :

A (01,BP Y) = Mb) N(B) M(Y). The physical meaning of this property is, that the rotated position of th object will also be obtained by three successive rotations about the fixe axes : (i) a rotation about the axis Oz through an angle y ; (ii) a rotation about the axis Oy through an angle ,!I; (iii) a rotation about the axis Oz through an angle 01. It is also important to note the trivial sounding fact that there exists a; isomorphism between the matrices of GA and the active rotations about th different axes in space.

3.2. Passive point of view. In the passive point of view, we keep thl object fixed in space but rotate the frame of reference about the origin 0 The new orientation of the frame of reference will be called OXYZ, tc distinguish it from its original orientation Oxyz. The passive rotation car ON THE ROTATION OPERATORS IN QUANTUM MECHANICS

immediately be characterized by the three Euler angles 01, b, y WI determine the orientation of the system of axes OXYZ relative to the sys of axes Oxyz. Every position vector o”p of a point of the object will be representet its three coordinates x, y, z in the original frame of reference Oxyz, am its three coordinates X, Y, Z in the rotated frame of reference OXYZ. relation between the original and the new coordinates is given by X ~0s~ co$ cosy -sinar siny sinor co+ cosy + cosa siny -sir@ cosy \ Y = -cosol co+? siny-sinol cosy -sinol co@ siny+socar cosy siq3 siny O(Z cosol sir@ sinor sir@ cosg I We abbreviate this as

i; = P(ol, /9, y) ;. The set of matrices P(ol, /?, y) form a matrix group which we denote by Using the same matrices M(B) and N(8) defined in (2), it is also pas: to write P(ol, /?, y) in a standard form P(% B, y) = M(-_y) N(-_B) W--or). The physical meaning of this relation is obvious. Contrary to the situation in the active point of view, there does not f an isomorphism between the matrices of Gp and the passive rotations al the different axes in space *. Whereas, for example, the matrix N( -8) a corresponds to a passive rotation through /3 about the Oy-axis, v occurring in the product N(-_B) M(--ol), i t corresponds to a passive rota through fl about the Ou axis, which forms an angle 01with the Oy axi will be seen, further on, that the faulty expressions for the rotation oper; in the literature arise from supposing such an isomorphism between matrix group Gp and the passive rotations about the different axes in ST From (3) and (5), or directly from inspecting the matrices in (1) and one obtains the following relation between the matrices A(ol, /?, y) ; P(% B, y): P(% /% y) = A(-_y, -/A -4 = A+/$4 = A-l (CyBy). Here At is the transposed matrix of A. From (6), it follows that the m: groups GA and Gp are anti-isomorphica). This means that there exists a to-one correspondence between the elements A(ol, 8, y) of GA and P(ol, of Gp such that, if

A(% Bl, Yl) A(% I%?,yz) = A(a3, 83, y3), one then has

J-+2, B2, y2) w% #h, Yl) = +a, 83, y3), * There exists an isomorphism between the matrices P(u @ y) and the pz rotations about the different axes which are fixed in the frame of reference. 576 M. BOUTEX

where the order of the matrices in the product has been reversed. In particular, one must be careful in writing down the Euler angles corresponding to a product of two active or two passive rotations. For example, whereas

one has

4. Rotation operators in quantum mechanics. 4.1. AC t i v e p o i n t of view. We consider a spinless particle, described by a wave function @(x, y, z) in the frame of reference Oxyz. An active rotation with Euler angles 01, is, y will result in a new state of this particle, described by a new wave function @‘(x, y, z) in the fixed frame of reference Oxyz. Since @(x, y, z) is a scalar field, the new function CD’will have the same value in the point P’ as the original function CDhas in the point P: @‘(x’, y’, 2’) = @(X, y, 2)

or, using (I),

@[A(a, /!I, y) i] = d,(r). (9)

For a given matrix A (CL,j3, y), one obtains in this way a mapping of every vector CDof the Hilbert space of one particle, onto an, in general, different vector @’ of the Hilbert space. This mapping is represented by a unitary operators) which we denote by O$, and which is called the rotation operator in the active point of view. From (9), it is defined by

0&, &) = @[A-l (01, p, y) Y]. (10) Wigner5) has stressed the importance of putting the inverse of the matrix A(ol, ,8, y) in the r.h.s. of the defining equation (10). Only in this case will the operators 0’l;k,, constitute a representation of the group GA of matrices A(cq /I, y). This means that there exists a one to one correspondence between the matrices A(a, /?, y) and the unitary operators S&,

A (01, B> Y) ti a$,, such that, if

In view of (3), it is then sufficient to obtain the operators corresponding to the matrices M(8) and N(B). These are easily obtained (see e.g. p. 445 of ON THE ROTATION OPERATORS IN QUANTUM MECHANICS 577

ref. 3) :

M(e) t+ episLz; (11) N(8) t+ eCieLv.

It then follows immediately that

&IA = e-iaLz e-iSLy e-iyLz orsv (12) This agrees with the expression, obtained by Messiah, in a slightly different way.

4.2. Passive point of view. We consider the same spinless particle with wave function @(x, y, z) in the frame of reference Oxyz. In the rotated frame of reference OXYZ, the same state will be described by another wave function @“(X, Y, 2). Since @(x, y, z) is a scalar field, the new function @” will have the same value as the original function CDin the same physical point P:

@“(X, Y, Z) = @(x, y, z)

or, using eq. (4),

W[P(a, p, y) ; = CD& (13) For a given matrix P(a, fi, y), one obtains a mapping of every vector @ of the Hilbert space of one particle onto an, in general, different vector CD”of the Hilbert space. This mapping is represented by a unitary operator5) which we denote by 0$, and which is called the rotation operator in the passive point of view. From (13), it is defined by

O$, @(; = @[P-l(a, /!I,y) v;. (14) The operators c?&,, defined in this way, constitute a representation of the group Gp of matrices P(a, /I,y). From (5), and using the expression (11) for the operators corresponding to the matrices M(0) and N(8), one immediately obtains

ocO, = eivLz eVLv eioiLz (15)

This expression is given by Fano and Racaha), and is also implicitly con- tained in Wigner’s book (note that Wigner interchanges the definition of the Euler angles 01 and y !). Both these authors state clearly that they are con- sidering rotations in the passive point of view. It is to be noticed that the operators (12) and (15) are related by

as they should be, in view of eq. (6). 578 M. HOUTEN

5. Comparisorz with the standard books 0% angular momentum. 5.1. Ed- mondsi). Using the same definition for the Euler angles a, 8, y and for the rotation operator O$,, Edmonds obtains the following expression @& (Edmonds) = eiaL2 eiPLy eiyLZS (17) One can show in several ways that this expression is incorrect. Let us make a special choice for a scalar field, e.g. @(x, y, z) = z. In view of the defining equation (14) one must have

o&z = P-i(ol, /3, y) 31 x + p-1(% j3, y)aa y + p-1(% B, y)33 2 =

= --?G sin /3 cos y + y sin /3 sin y + z cos 8.

In the last line, we have introduced the explicit matrix elements (4) of the matrix P(B, @, y), using the property P-i = Pt. The r.h.s. of this relation depends only on the angles p and y, but not on 01.The 1.h.s. on the other hand becomes using Edmonds’s operator (17)

ice, ei/jLv eiyLz z = eiaLz eii(L, z e , which depends only on the angles o( and p, and not on the angle y. Another way of seeing that Edmonds’s expression (17) for the operator O&, cannot be correct, is by showing that these operators do not form a representation of the matrix group Gp, as they should do from the definition (14). Consider for example the product (8) of matrices of GP. If the operators (17) formed a representation of Gp one should have

iaL, . eiSLv e~~Lz = eiBL, eiC:,+ a) Lz e , which is obviously wrong. The error made in the derivation of Edmonds, can be indicated in the following way. Having obtained the exact result which can be expressed by saying that the operator corresponding to a rotation about the z-axis is given by eiaLZ, Edmonds generalizes this result to conclude : The operator corresponding to a rotation through p about the ZLaxis is iPL, e , the operator corresponding to a rotation through y about the Z axis is eivLz (18) It then follows that

O$, (Edmonds) = eiyLz eiBLUeiaLZ, (19) which can be reduced to eq. (17) by expressing L, and LZ in terms of L,, L, and L,. ON THE ROTATION OPERATORS IN QUANTUM MECHANICS 579

The error is made in (la), in associating an operator with every passive rotation about the different axes in space. From the definition (14), it follows that there exists an isomorphism between the operators O&, and the matrices P(cx, p, y) of the group GP. On the other hand, as we have seen in section 3, there does not exist an isomorphism between the matrices of the group Gp and the passive rotations about the different axes in space. Consequently, there does not exist an isomorphism between the passive rotations about the axes in space and the rotation operators O$,, as is assumed in Edmonds’s derivation. In the active point of view, on the contrary, there does exist an isomorphism between the active rotations about the axes in space and the matrices A (01,8, y) of GA, and consequently also with the rotation operarors @$,,. As shown by Messiahs), one has the correspondence

an active rotation through (Y about the z axis H e-iorLz, an active rotation through p about the u axis e, eeisLU, an active rotation through y about the 2 axis ++ emivLz, so that

oit = e-iyLz e-iPLu e-iorLz c&J Expressing L, and LZ in terms of Lg, L, and L, reduces this expression to eq. (12).

5.2. Roses). From pages 16 and 18 of Rose’s book, it follows that Rose is also considering the passive point of view. However, his rotation operator, which we will denote gaav, is defined in a different way:

*asv@(; = @[P(% B, y) ;. (20) As stressed by Wigners), the operators &?‘aoydefined in this way do not constitute a representation of the matrix group Gp. The operators 9’a~v are, however, anti-isomorphic with the matrices P(a, /I, y). It then follows, in view of eq. (5) and using (1 I), that

-iaL e - iSL, e -iyL, 9 a~Y = e (21)

The exactness of this expression is also obvious from eqs. (6), (10) and (12). Rose obtains this same result by making two errors which cancel each other. He writes

-iPL, e-iaLz w a#v L- e-‘uLz. e , (22) and expressing L, and LZ in terms of Lz, L, and Lz, he obtains (21). In writing down (22), he asssociates a rotation operator with every passive rotation about the different axes in space, making the same error as Edmonds. Moreover, he has taken the same order in the product of the rotation 580 ON THE KOTATION OPEKATORS IN QUANTUM MECHANICS

operators as in the product of the rotations whereas, from the definition (20), the order should be reversed. These two errors turn out to cancel each other, yielding the exact operator (2 1). However, it remains to be remembered that the operators (21) do not constitute a representation of the matrix group Gp but rather of the group GA. The same remark applies obviously also to the derived representation matrices.

Acknowledgement. I am indebted to Dr. P. J. Brussaard for a discussion of the manuscript.

REFERENCES

1) Edmonds, A. R., Angular Momentum in Quantum Mechanics, l’rinceton University Press (Princeton, 1957). 2) Rose, M. E., Elementary Theory of Angular Momentum, Wiley (New York, 1957). 3) Messiah, A., Mecanique Quantique, Dunod (Paris, 1960). 4) Fano, U and Racah, G., Irreducible Tensorial Sets, Academic Press (New York, 1959). 5) Wigner, E. I’., Group Theory and its Applications to the Quantum Mechanics of Atomic Spectra, Academic Press (New York, 1959). 6) Zassenhaus, H. J., The Theory of Groups, Chelsea Publishing Company (New York, 1956).