Module II, Lecture 03: Formal Theory of Rotations

Rotation operators in physics It stands to reason that in the absence of external fields and perturbations the result of an experiment on a physical system should not depend on the choice of coordinates. In other words, it is reasonable to assume that space itself is uniform and isotropic. In particular, the energy of a physical system should not be changed by static coordinate translations and rotations:   TTHTHˆˆˆˆˆ †  (1)   RRHRHˆˆˆˆˆ †  where Tˆ is a unitary operator that performs coordinate system translation and Rˆ is the rotation oper‐ ator, which is also unitary because it preserves norms and angles. Because the relations above hold for any wavefunction, they must hold for the corresponding operators. Therefore, both Tˆ and Rˆ com‐ mute with the Hamiltonian: ˆˆˆˆ†1 ˆˆˆˆ  ˆˆ THTH THTH   HT ,  0 (2) ˆˆˆˆ†1 ˆˆˆˆ  ˆˆ RHRH RHRH   HR ,  0 This leads to the conservation of the corresponding observables: d TiHTˆˆˆ...  ,   0 dt  (3) d RiHRˆˆˆ...  ,   0 dt  We will now find out what these observables are. Let us derive the opera‐ tor performing a rotation by a small angle  in the XY plane:

Rxxˆ : cos y sin  Emmy Noether proved in 1915  Ryxˆ :sincos y that any differentiable symmetry  (4) of a physical system leads to the Rzzˆ  :  conservation law for the genera‐  tor of that symmetry. Differentia‐ Rˆ xyz, ,  x cos  y sin  , x sin   y cos  , z ble symmetries are the subject of    Lie group theory. Because the angle  is small, we can use a Taylor expansion to second term around   0 :

ˆˆ ˆ 2 RR 0  R   O (5)   0 The derivative in brackets is computed using the composite function differentiation rule:

ˆ   Rxyxyzxy cos sin , sin cos ,  ...   (6)  0  0  yx The angular momentum operator is instantly recognizable. We therefore find the following expression for the operator performing a rotation by an infinitesimal angle d around the Z axis:   Rdˆˆˆ1 iLd  L  i x  y (7)  ZZ yx Spin Dynamics, Module II, Lecture 03 – Dr Ilya Kuprov, University of Southampton, 2013 (for all lecture notes and video records see http://spindynamics.org) ˆ Similar treatment can demonstrate that small rotations in the YZ and XZ planes are performed by LX ˆ and LY respectively. In the case of infinitesimal translation, linear momentum operators and the asso‐ ciated conservation laws appear in a similar fashion. Infinitesimal time evolution treatment leads to the conservation of energy. These are very profound results – all conservation laws stem from symmetries.

We can now obtain a differential equation for the rotation operator itself and solve it. The result ex‐ plains the “rotating frame” name that is commonly given to the interaction representation:

RdRdRˆˆˆˆˆ   1 iLdR    Z  ˆˆ ˆ (8) RdR  dR  ˆ iLˆˆ R   iL ˆˆ R  R ˆ   eiLZ ddZZ The conclusion is that all rotation operators are exponentials of the angular momentum operators. Simi‐ lar calculations for the other two axes lead to the very important result that the transformation of any continuous function under a coordinate system rotation in 3 is given by:

ˆˆˆ Rfxyzeeefxyzˆ ,,  ,, iLXYZ iL iL  ,, (9) ˆˆˆ    LiyzXYZˆˆˆˆ ˆˆ; Lizx   ; Lixy    zy  xz   yx  that is, the angular momentum operators are the basis of the Lie algebra corresponding to the Lie group of rotations. Our extensive knowledge of the properties of the angular momentum operators is now presented in a different light – the commutation relations between angular momentum operators ˆˆ ˆ ˆˆ ˆ ˆˆ ˆ [LLXY , ] iL Z [ LL YZ , ] iL X [ LL ZX , ] iL Y (10) are now identified as structure relations of a Lie algebra. Their central role in the angular momentum theory is explained by the fact that they generate the rotation group. Note that the expressions for the angular momentum operators as well as the conservation laws for the corresponding observables are consequences of just one assumption – that physical space is isotropic.

Rotation group The rotation group, known formally as SO3,  , the special orthogonal group in three dimensions, consists of all three‐dimensional rotations – continuous linear transformations of 3 that leave the sca‐ lar product invariant and have a determinant of +1. It is a subgroup of the orthogonal group O3,  , which also includes inversions. The real field is often implicitly assumed and the group is referred to simply as SO3 . Elements of SO3 are related by the exponential map aia exp to their infini‐ tesimal generators in so3 , which is a Lie algebra of imaginary Hermitian 33 matrices. In the math‐ ematics literature the exponential map is defined as aa exp  and so3 is the algebra of real an‐ tisymmetric 33 matrices. Because any three‐dimensional rotation can be decomposed into a combi‐ nation of three rotations around the coordinate axes, SO3 is a triparametric Lie group. Strictly speaking, only the commutator and the linear combination with real coefficients are defined in ˆˆ ˆ ˆ2 so3 , and therefore operators like LLiL XY and LX do not belong there – their exponentials do not correspond to rotations. They do, however, occur in physical reality and for that reason we should make space for them. A universal enveloping algebra U a of a Lie algebra a is (for physics purposes, mathematicians again beg to differ) the associative algebra obtained from a by allowing plain products Spin Dynamics, Module II, Lecture 03 – Dr Ilya Kuprov, University of Southampton, 2013 (for all lecture notes and video records see http://spindynamics.org) and complex coefficients. It allows us to move from a non‐ associative structure of the original Lie algebra to an associative structure while preserving the representations. It also often contains many physically significant operators that are missing from a . ˆˆˆˆ2222 The total momentum operator LLLLXYZ, other powers of momentum projection operators as well as the raising and lowering operators belong to U so3 . It is therefore clear that we are in most cases working with U so3  when we build angular momentum Hamiltonians. The total momentum operator Lˆ2 is also the Casimir operator (the sum of squares of all generators) of so3 . It commutes with all elements of the algebra, but does not belong to it. Irreducible representations of Lˆ2 are always proportional to the unit matrix. Group theory was introduced into quantum mechanics by Eugene Wigner (above) and Hermann Weyl. Irreducible matrix represen‐ Parameterization of rotations tations of the rotation group are presently It is very important to understand that when a molecule rotates known as Wigner matrices. or undergoes rotational diffusion in three‐dimensional space, spins do not rotate. They get translated in space, but their projections on the lab frame axes stay the same. The anisotropic interactions, however, do rotate, because in most cases they are determined by the electronic structure and the angles that the distance vectors make with the applied magnetic field.

It is easy to verify by direct inspection that the exponentials of the following complex Hermitian matrices give the rotation matrices around the three coordinate axes: 000  1 0 0     Ji0 0 exp iJ 0 cos sin 11      00i  0sincos  00i  cos0 sin     JiJ0 0 0 exp 0 1 0 (11) 22      i 00  sin0 cos  00i  cossin0   Ji0 0  exp iJ  sincos 0 33    000 001 These matrices are therefore a basis set of a representation of so3 in gl3,  and a generator set for a representation of SO3 in GL3,  with the rotation angles ,, as continuous parame‐ ters. The generators can be recovered from their exponentials by a tangent space transformation:  J iexp iJ J  i  exp iJ  J  i  exp iJ (12) 112233000     An infinite number of other generator sets and therefore parameterizations exist. Importantly, the three generators should not commute, otherwise the tangent space transformation becomes ill defined. The most popular parameterizations in the current use include: Spin Dynamics, Module II, Lecture 03 – Dr Ilya Kuprov, University of Southampton, 2013 (for all lecture notes and video records see http://spindynamics.org)  Euler angles convention A 1. Rotate about the Z axis through an angle  02   2. Rotate about the new Y axis through an angle  0  3. Rotate about the new Z axis through an angle  02   This is a very inconvenient sequence because the reference frame drifts with the object. It is only useful when the frame of reference is object‐centred (e.g. for an aeroplane).

 Euler angles convention B 1. Rotate about the Z axis through an angle  02   2. Rotate about the Y axis through an angle  0  3. Rotate about the Z axis through an angle  02   This sequence is more convenient and easier to visualize in the spin dynamics context because the frame of reference remains the same throughout the transformation. The angles involved are the same and the two conventions A and B accomplish identical rotations.

Euler angles are notoriously difficult to convert into because they are not a valid parameterization of SO3 , since the generators corresponding to the three Euler angles do not obey the required commutation relations – it is easy to see that they do not commute in the right way: 00iii  00  00    Ji0 0 , J  0 0 0 , Ji  0 0 (13) 123      000 i 00  000 The commutator of the first and the last generator yields zero instead of the second generator. In practice this leads to nasty singularities:   0 and    points are singular, meaning that the dif‐ ferential equations involving Euler angles run into analytical and numerical difficulties. In general, while it is always possible to translate Euler angles into more regular conventions (angle‐axis, DCM, quaternions etc.), it is not easy to go back.

 Angle‐axis parameterization Any rotation may be defined in terms of a unit vector and an angle of rotation around that vector.  Given a vector n of unit length and a rotation angle  , the rotation matrix is:

2 cos nnnnnnnXXYZXZY 1 cos 1  cos sin 1  cos sin 2 RnnYX1 cos n Z sin cos  n Y  1 cos  nn YZ  1  cos  n X sin  (14) 2 nnZ XYZYX1 cos n sin nn  1  cos  n sin cos  n Z  1 cos  This may be shown to be a valid parameterization of the rotation group and all singularities associat‐ ed with Euler angles disappear in this approach. Angle‐axis parameterization also has the benefit of being easy to visualize.

 Quaternions Another valid parameterization of SO3 that has no singularities and in which a rotation is defined by four parameters {,,,abcd }, such that abcd222  2 1 and adibci  (15) bciadi 

Spin Dynamics, Module II, Lecture 03 – Dr Ilya Kuprov, University of Southampton, 2013 (for all lecture notes and video records see http://spindynamics.org) is a family of generators of SU 2 . The quaternion representation is most easily obtained from an‐ gle‐axis parameters:

anXYsin / 2 bn sin / 2 (16) cnZ sin / 2 d cos  / 2 Given a unit quaternion wxyz,,, , the rotation matrix is obtained as: 12y22 2 z 2 xy  2 zw 2 xz  2 yw 22 R 22xy zw 122  x z 22 yz  xw (17) 22 22xzywyzxwx 22 122 y

 Directional cosine matrix An explicit 3x3 matrix performing the rotation in question, i.e. R1 AR . Usually results from the in‐ strumental readout or from the diagonalization of a tensor specified explicitly as a 3x3 matrix. In the latter case the DCM is the matrix of eigenvectors of the tensor, and its application to any vector or matrix would rotate them into the eigenframe of the tensor.

It should be noted that the matrix of eigenvectors of an interaction tensor, if taken straight from the diagonalization procedure, is often a reflection away from the DCM due to randomness associated with eigenvector phases and labels. For this reason it is usually inadvisable to use diagonalization as a source of the directional cosine matrix.

In simple practical calculations, the rotation matrix treatment of spin interaction tensor rotations:     ˆ ˆˆˆˆ1 Rˆ LAS  L R ARS  (18)  does often suffice. R is defined as a superposition of rotations around the three laboratory frame axes: cos sin 0 cos  0 sin  1 0 0  R, , sin  cos  0 0 1 0 0 cos  sin  (19)  0 0 1 sin 0 cos 0 sin cos In complex spin systems, however, this approach quickly becomes cumbersome. In particular, it com‐ pletely blocks all attempts at performing relaxation theory treatment. We do therefore need a more regular way of describing rotations using representation theory of SO3 .

Irreducible representations of the rotation group We will now consider representations of SO3 with operators acting on the Hilbert space of all well‐ behaved functions on 3 . The complete basis set of that space is infinite and therefore any faithful ma‐ trix representation of SO3 would be infinite‐dimensional – our task is to reduce it. SO3 is gener‐ ated by the three angular momentum operators, and so the task of finding irreducible representations ˆ ˆ ˆ amounts to finding a transformation that simultaneously partitions LX , LY and LZ into the smallest possible blocks. Because all three momentum projection operators commute with the Casimir operator Lˆ2 , they share the invariant subspaces with it and it would therefore suffice to determine which fami‐ lies of functions are invariant under the total angular momentum operator. Skipping the well‐known derivation, we will simply note here that eigenfunctions of Lˆ2 are known as spherical harmonics:

Spin Dynamics, Module II, Lecture 03 – Dr Ilya Kuprov, University of Southampton, 2013 (for all lecture notes and video records see http://spindynamics.org) 1/2 mm  21l  lm ! mim YPelmlll, 12  cos   ,  , 1,..., lm4 lm ! l  (20) m l m 22ml/2 dd1  Pxlll1 x Px Px  ll x  1 dx m 2!ldx Sets of spherical harmonics of a given rank span the invariant subspaces of the total angular momentum operator as well as the three projection operators: ˆˆ2 LYlm l l1 Y lm LYZ lm  mY lm 1 LYˆ  l l11 m m  Y l l 11 m m  Y (21) X,1,1lm2  l m l m 1 LYˆ  l l 11 m m  Y l l 11 m m  Y Y,1,1lm2i  lm lm Therefore the spherical harmonics of different ranks form basis sets for different irreducible representa‐ tions of SO3 – a spherical harmonic is transformed by any rotation into a linear combination of spherical harmonics of the same rank. Matrix elements of irreducible representations: l ˆ Dmlm,mlm ,,  YY,,R  ,,  (22) are known as Wigner functions and the corresponding matrices as Wigner rotation matrices. They will occur very often from now on, particularly in the context of spin relaxation theory.

Spherical harmonics may also be written via Cartesian coordinates: ˆ l m Ylm , Yxyzlm  ,, TLlm  1 1 0 0 Eˆ 4 4 3 3 x  iy 1 1 1  ei sin   Lˆ 8 8 r 2  3 3 z 1 0 cos Lˆ 4 4 r Z 3 3 x  iy 1 1 –1 ei sin Lˆ 8 8 r 2  2 5 22i 15 x  iy 1 ˆ2 2 2 e sin  L 32 32 r 2 2 5 5 x  iy z 1 i   LLˆˆ LL ˆˆ 2 1  e sin cos  2  ZZ 8 8 r 2 5 52zxy222  21 2 0 2 LLLLLˆˆˆˆˆ2  3cos  1 2 Z   16 16 r 34 5 15 x  iy z 1 i   LLˆˆ LL ˆˆ 2 –1 e sin cos 2 ZZ 8 8 r 2 2 5 22i 15 x  iy 1 ˆ2 2 –2 e sin  L 32 32 r 2 2

Spin Dynamics, Module II, Lecture 03 – Dr Ilya Kuprov, University of Southampton, 2013 (for all lecture notes and video records see http://spindynamics.org) The operators obtained by replacing the Cartesian coordinates with Carte‐ sian spin operators are known as irreducible spherical tensor operators (IST). The table above gives single‐spin irreducible spherical tensor opera‐ tors. Their two‐spin analogues: 1 TLSˆˆ2 ,  LSˆ 2 2 1 TLSˆˆˆ2 ,  LSLSˆˆ (23) 1ZZ2 

ˆˆˆˆ2 21ˆˆˆ TLS0ZZ,  LSLSLS    34 will be useful in setting up coupling tensor rotations. For linear interactions ˆ Spherical tensors were intro‐ (such as Zeeman interaction) the L operator vector is replaced by the duced into NMR by Jack Freed magnetic field vector. For quadratic couplings both sets of operators refer (above) and Bryan Sanctuary. to the same spin. Because of their common group‐theoretical heritage, spherical harmonics and irreduc‐ ible spherical tensors have common rotation properties:

l l ˆˆˆˆˆlll   ˆ  l RTT, ,m m',DD m m  , , RYY   , , lm lm ', m m   , , (24) ml ml  The second‐rank Wigner matrix, which occurs particularly often, is given in the table below.

2 2 2 2 2 dm,2 dm,1 dm,0 dm,1 dm,2

2 2 2 1cos  1cossin    3 2 1cossin    1cos  d  sin   2,n 4 2 8 2 4

2 1cossin    cos  1 2 3 cos  1 2 1cossin    d  cos   sin 2  cos   1,n 2 2 8 2 2 2 3 3 3cos2   1 3 3 d sin2  sin 2  sin 2 sin2  0,n 8 8 2 8 8

2 1cossin    cos  1 2 3 cos  1 2 1cossin    d cos  sin 2  cos   1,n 2 2 8 2 2

2 2 2 1cos  1cossin    3 2 1cossin    1cos  d sin  2,n 4 2 8 2 4

The full Wigner functions are defined in terms of the reduced functions given above as:

22 im   in Dmn,, ed mn  e (25) Because all spin interactions are at most second‐rank, the following expansion in terms irreducible spherical tensor operators holds in all cases (extended as noted above to linear and quadratic cases):   12 ˆ ˆ ˆˆˆ 00   11 ˆ   22 ˆ  LSA  aLSaTkn k n 00  aT m m  aT m m (26) kn,12 m m {X,Y,Z} The first rank terms will henceforth be ignored – they are suspected to exist, but have never been ob‐ served because they are very small. The following table gives the coefficients required for the transfor‐ mation between the standard 3x3 matrix representation and irreducible spherical tensors:

Spin Dynamics, Module II, Lecture 03 – Dr Ilya Kuprov, University of Southampton, 2013 (for all lecture notes and video records see http://spindynamics.org) l ˆ l lm, am Tm 100 1 1 ˆ ˆ 0, 0 aaa   XX YY ZZ  LS010 3 3  001 00 1 1 1 ˆ ˆ 1,1  LiS00  aaiaaZX XZ ZY YZ  2 2  10i 00i i 1 ˆ ˆ 1, 0 aa Li00 S  XY YX   2 2  000 00 1 1 1 ˆ ˆ 1, 1 aaiaa  LiS00   ZX XZ ZY YZ  2 2  10i 10i 1 1 ˆ ˆ 2, 2 aaiaa  Li10  S  XX YY XY YX  2 2  000 001 1 1 ˆ ˆ 2,1 aaiaa  LiS00  XZ ZX YZ ZY  2 2  10i 100 1 1 ˆ ˆ 2, 0 2aaa  ZZ XX YY  LS010 6 6  002 00 1 1 1 ˆ ˆ 2, 1 aaiaa  LiS00  XZ ZX YZ ZY  2 2  10i 10i 1 1 ˆ ˆ 2, 2 aaiaa  Li 10  S  XX YY XY YX  2 2  000

The ISTs in this table have been written out explicitly in matrix form to expose their symmetry. Coeffi‐ cients and signs are determined by the requirement to preserve the rotation and commutation proper‐ ties of the spherical tensor operators. If the interaction tensor is diagonal in the current reference frame, the transformation becomes particularly simple:   ˆ ˆ ˆˆˆˆˆˆ LSaLSaLSaLSA XXXX  YYYY  ZZZZ  (27) aaa02222aaa aa  XX YY ZZTTTTˆˆˆˆ ZZ XX YY   XX YY   3600222 

Spin Dynamics, Module II, Lecture 03 – Dr Ilya Kuprov, University of Southampton, 2013 (for all lecture notes and video records see http://spindynamics.org) Setting up rotations for complex spin systems In rigid molecules, the simple rotational transformation rule derived above for the irreducible spherical tensors is inherited by the full spin system Hamiltonian: 2 ˆˆ2 ˆ HH,,iso  Dkm  ,, Q km (28) km,2 ˆ where Qkm (called rotational basis operators) are linear combinations of irreducible spherical tensors corresponding to the interactions within the spin system. No matter how large and complicated the spin system, there are always just 25 rotational basis operators. To derive the expressions for Qˆ , we would ˆ km note that for a multi‐spin system in a rigid molecule, rotated by Rˆ , we have: 2 ˆˆ ˆ22 ˆ HHiso R amm LTL  Lm2 22 (29) ˆˆˆˆˆˆ22  22 RaLSTLSRmm,,  aSSTSS mm ,, LSm,2 S m  2 where Hˆ is the isotropic part of the Hamiltonian and the three terms in brackets correspond to line‐ iso ˆ ar, bilinear and quadratic couplings within the spin system. After we apply the rotation Rˆ , we get: 2 ˆˆ22 ˆ  2 HHiso  amk LTLD km  Lkm,2 (30) 22 22ˆˆ  2  22  2 aLSTLSm,, kDD km  aSSTSS m ,, k km LSkm,, 2 S km ,  2 Reordering the terms and taking Wigner functions out of the brackets yields:

2 ˆˆ222  ˆ  2 ˆ 2 2 ˆ 2 HHiso Dkm aLTL m k   aLSTLS m,, k   aSSTSS m ,, k  (31) km,2  L LS , S We can now define the terms in brackets as the rotational basis and conclude that all information about the amplitudes and internal orientations of all interactions has been packaged into just 25 operators: ˆ 2222 ˆˆ       22 ˆ  QaLTLaLSTLSaSSTSSkmmk  mk ,,     mk ,,  (32) LLSS, All required expressions for ISTs, spherical tensor coefficients and Wigner functions are tabulated above.

Setting up spin system rotations – a summary:

1. Get all interactions into 3x3 Cartesian matrix form. l 2. Translate the interaction matrices into spherical tensor parameters am using the rela‐ tions given in the table above. Ignore the first‐rank components. ˆ ˆ 3. Compute the isotropic Hamiltonian Hiso and the 25 rotational basis operators Qkm . If at all possible, avoid using Euler angles to parameterize rotations. 4. The spin system Hamiltonian at any orientation relative to the frame of reference in which the original tensors were specified is given by Equation (28).

This is in practice the most consistent and straightforward way of setting up rotations in complicated spin systems. In particular, it avoids re‐creating all coupling operators at every orientation when powder patterns are computed and also facilitates relaxation theory treatment.

Spin Dynamics, Module II, Lecture 03 – Dr Ilya Kuprov, University of Southampton, 2013 (for all lecture notes and video records see http://spindynamics.org)