Residual dipolar coupling constants: An elementary derivation of key equations

Transcription

1 Conceps in Magneic Resonance 1A, Residual dipola coupling consans: An elemena deivaion of ke equaions F. Kame, M. V. Deshmukh, H. Kessle and S. J. Glase* Insiu fü Oganische Chemie und Biochemie, Technische Univesiä München, Lichenegsaße 4, 85747, Gaching, Geman * Coespondence o: Pof. D. Seffen Glase Phone: Fa: , Glase@ch.um.de

2 ABSTRACT Residual dipola couplings RDC have ecenl found a wide ange of applicaions in high esoluion NMR of iomolecules in he liquid sae. A non-isoopic oienaional disiuion of a molecule of inees esuls in non-eo aveage dipola coupling consans. Hee, we pesen an inuiive inoducion o he alignmen enso and an elemena deivaion of ke equaions. KEYWORDS alignmen enso, coupling consan, dipola coupling, dipola Hamilonian, ode mai, poaili mai, esidual dipola coupling, Saupe mai.

3 INTRODUCTION While dipola couplings ae he dominan ineacions in solid sae NMR of spin 1/ nuclei, he ae aveaged o eo fo isoopicall eoiening molecules in he liquid sae. This makes i possile o achieve high-esoluion speca wih elaive ease in liquid sae NMR. On he ohe hand, a wealh of sucual infomaion is los if dipola couplings vanish. Howeve, even in liquid sae NMR, molecules can e paiall aligned, e. g. eenal fields magneic o elecic o anisoopic solvens 1-8. Fo eample in liquid csalline solvens, he dissolved molecules ae paiall aligned hough seic and anisoopic ineacions wih he solven molecules, and dipola couplings can e oseved 9, 10. The ecen success and wide use of such esidual dipola couplings is due o he developmen and chaaceiaion of seveal new alignmen media such as icelles 5, filamenous phage Pf1 11, and polaclamide gels 1, 1, which make i possile o ceae a elaivel small, unale degee of alignmen. This allows he specoscopis o adjus he alignmen in such a wa, ha he sie of he aveage dipola coupling is in he ode of he J couplings. In his case, he esuling speca ae sill simple, and dipola coupling consans can e measued elaivel eas compaing line spliings in isoopic and in aligned samples. Techniques o measue esidual dipola couplings and a wide ange of applicaions have een discussed in a nume of aicles and eviews -8. Hee, we evisi he fundamenal quesion of how o calculae he epeced esidual dipola coupling consan fo a homonuclea e.g. 1 H- 1 H o heeonuclea e.g. 15 N- 1 H spin pai. This uns ou o e a supisingl simple calculaion if we know he oienaion and he hee pincipal componens of he so-called alignmen enso. This alignmen enso is a ke concep, ha is cucial o undesand esidual dipola couplings. Howeve, in ou epeience, man sudens and even seasoned paciiones in he field of NMR have concepual 1

4 difficulies o full undesand he phsical meaning of he alignmen enso which someimes lead o seious misconcepions vide infa. This ma esul in pa fom he common pacice in lieaue, o deive he alignmen enso using mahemaicall elegan, u no ve inuiive appoaches ased on spheical hamonics, hei addiion heoems, Legende polnomials, Wigne oaion maices and a confusing nume of angles eween vaious aes. In conas, we hee use a seamlined geomeic appoach simila o he oiginal deivaion Saupe 9, 10, which is ased on he Caesian epesenaion of vecos. Ecep fo he mos asic ules of mai and veco muliplicaion, onl elemena mahemaics is needed o deive he alignmen enso. As a didacical aid on he wa o undesanding he alignmen enso, we discuss he elaed poaili enso. Numeical eamples and illusaing figues ae used o conve he phsical meaning of hese ensos. In he Appendi, vaious epessions fo he esidual dipola coupling consans commonl found in lieaue ae deived fom he pesened ke esuls. STATIC DIPOLAR COUPLING HAMILTONIAN We conside wo spins I and S wih an inenuclea veco R v see Fig. 1. This veco can e epessed in he fom v v R R, [1] whee R is he disance eween he wo nuclei and v is a uni veco poining in he diecion of R v. Similal, he veco epesening he eenal magneic field B v can e epessed in he fom

5 v v B B, [] whee B is he magniude of he saic magneic field, and v is a uni veco poining in he L diecion of he magneic field. In he la fame, L, L, whee convenion he magneic field poins along he L ais, he uncaed dipola coupling Hamilonian has he fom H D D% I L S L ' I L S L ' I L S L. [] If he spins I and S ae heeonuclea, he second and hid em in he acke can e negleced, esuling in he simple weak dipola coupling Hamilonian H [4] D DI L S L which has he same fom as he weak heeonuclea J coupling Hamilonian. In oh cases, he dipola coupling consan which in he weak coupling limi coesponds diecl o he epeimenall oseved line spliings in unis of H is given 14: 1 cos D, ' [5] R % whee is he angle eween he inenuclea veco and he magneic field see Fig. 1. The em

6 µ I 0h [6] 8 S onl depends on phsical consans 14: he gomagneic aios I and S of spin I and S especivel, he Planck consan h h / and he pemeaili of vacuum µ 0. Fo eample, fo 1 H- 1 H, 1 C- 1 H and 15 N- 1 H spin pais, kh Å, kh Å and 6.5 kh Å, especivel. The maimum possile value of cos is 1 fo 0 o, and hence, accoding o Eq. [5], he maimum possile dipola coupling consan is D ma / R 1 1/ / / R, [7] which coesponds e. g. o 1.7 kh fo a 15 N- 1 H spin pai wih disance R 1.04 Å. Rememe ha he scala poduc eween wo uni vecos is idenical o he cosine of he angle eween he wo vecos. Hence, he em cos in Eq. [5] can alwas e epessed in he fom v cos T v. [8] Hee, T v is a ow veco he anspose of he column veco v which allows us o wie he scala poduc of he wo vecos as a usual mai poduc eween he 1 mai v T and he 1 mai v vide infa. 4

7 TIME-DEPENDENT AND AVERAGE DIPOLAR COUPLING HAMILTONIAN Now we conside he wo spins I and S o e pa of a molecule in soluion. In he la fame, he magneic field veco B v is consan poining along he L ais, u he inenuclea veco R v is now ime-dependen see Fig. A. Fo simplici, we assume ha he molecule is igid no inenal dnamics and consan disance R, such ha he imedependence of R v is solel due o he oaional umling moion of he molecule. Hence, he em cos and as a esul also he dipola coupling consan D and he dipola coupling Hamilonian is ime-dependen. Fo poeins, he oaional coelaion ime is in he ode of nanoseconds and on he ime-scale of he NMR epeimen, onl he imeaveaged dipola Hamilonian H D gives ise o spliings in he specum elaaion effecs caused he flucuaions of he dipola Hamilonian will no e consideed hee. The ime-aveaged dipola coupling consan 1 D cos ' [9] R % epesens he so-called esidual dipola coupling consan, which depends on he aveage alignmen of he molecule. OUTLINE AND KEY RESULTS The goal of his manuscip is o deive a geneal appoach fo he calculaion of D fo an pai of spins if he alignmen popeies of he molecule ae known. Befoe we go ino he fomal deivaion, we give a ief ouline of he seps and sae he final esul. Fis, we L move fom he la fame, L, L c.f. Fig. A o a fame of efeence,, ha is 5

8 fied o he molecule. In his fame of efeence, he em cos can e convenienl epessed wih he help of a poaili enso P, which is a second ode appoimaion of he oienaional poaili disiuion of he diecion of he eenal magneic field in he molecula fied fame of efeence 6, 15. This poaili enso P can e epesened an ellipsoid c.f. Fig. A wih a fied oienaion in he chosen molecula fame,,. The pincipal values P~, P~ and P~ of he poaili enso i.e. he lenghs of he half aes of he poaili ellipsoid ae he poailiies o find he magneic field along he coesponding pincipal aes of he poaili ellipsoid, and hence P~ P~ P~ 1. Fo eample, fo an isoopicall eoiening molecule, P~ P~ P~ 1/, and he poaili ellipsoid is educed o a sphee see Fig. 4 C. On he ohe hand, if a molecule is full aligned, P~ P~ 0 and P~ 1 convenion, he pincipal elemens ae odeed wih inceasing magniude, i.e. he poaili enso is educed o a single line in he diecion of he magneic field. In geneal, he pincipal aes of he poaili ellipsoid define a special molecula fied ais ssem ~, ~, ~, in which he calculaion of esidual dipola coupling consans is especiall simple see Fig. B: If we know he hee Caesian componens ~, ~ and ~ of an given inenuclea uni veco v in his pincipal ais ssem, he em cos in Eq. [8] is simpl given cos ~ ~ ~ ~ P P P. [10] ~ ~ 6

9 If his simple equaion deived elow is inseed ino Eq. [9], he esidual coupling consan can e pediced fo an aia spin pai in a molecule, as long as he oienaion and pincipal values of he poaili enso ae known. Wih his ke esul, we can calculae evehing, and we could sop hee, ecep ha esidual dipola coupling consans ae commonl no epessed in ems of he inoduced poaili enso P coesponding in geneal o a eal smmeic mai wih ace 1 u in ems of is aceless pa is esolven P 1/ 1, which is called he alignmen enso A 5: 1 A P 1. [11] The hee pincipal componens A~, A~ and A~ of he alignmen enso A ae simpl given 1 A ~ P~, 1 A ~ P~, 1 A ~ P~, [1] and he pincipal aes of A and P ae idenical. Noe ha in conas o he poaili enso P see Figs. and 4, he alignmen enso A canno e epesened as an ellipsoid, ecause one o wo of he pincipal componens A~, A~, and A~ of he alignmen enso ae negaive if an of he hee componens is noneo due o A ~ A~ A~ 0. Alenaive gaphical epesenaions of he effecs of he alignmen enso ae shown in Figs. 5 and 6 vide infa. 7

10 In ems of he pincipal componens of he alignmen enso, he em cos 1/ in he equaion fo he esidual dipola coupling consan Eq. [9] can e epessed as 1 cos ' A ~ ~ A~ ~ A~ ~. [1] % If his equaion is inseed ino Eq. [9], i is again possile o pedic he esidual coupling consan fo an aia spin pai in a molecule, povided ha he oienaion and pincipal values of he alignmen enso ae known. Convesel, he alignmen enso A o he poaili enso P can e deemined if a sufficien nume of epeimenal dipola coupling consans ae measued fo a given molecule 16. As will e shown elow, he alignmen enso A and he poaili enso P is chaaceied five independen paamees. Theefoe, a leas five dipola coupling consans need o e measued in ode o deemine he five unknown paamees 16. In man cases, i is also possile o accuael pedic he alignmen enso A 17 o he poaili enso P fo a given molecule in a given liquid csalline solven, and hence o pedic he epeced dipola coupling consans fo a poposed molecula sucue fom fis pinciples. 8

11 9 DERIVATION OF THE PROBABILITY AND ALIGNMENT TENSORS In an aiail chosen molecula fame wih aes,, see Fig. B, a given inenuclea veco R v is consan sill assuming a igid molecule wihou inenal dnamics:. R % R R v v [14] Howeve, in his fame of efeence, he diecion of he magneic field veco B v is imedependen if he molecule umles in soluion:. B % B B v v [15] The definiion of cos via he scala poduc of he uni vecos v and v c.f. Eq. [8] is valid in an fame of efeence. Hence, we can epess cos in he molecula fame as a funcion of he componens of he uni vecos v and v, which poin in he vaing diecion of he magneic field B v and of he consan inenuclea veco R v, especivel: % ' cos v T v [16], and

12 10. cos [17] Noe ha Eq. [17] can also e epessed in he fom % % cos '. [18] Hence, he ime aveage of cos is given. cos T v P v % % ' [19] We call he mai % P [0] he poaili mai. Fo a known poaili mai P, he esidual dipola coupling consan Eq. [9] is given

13 v 1 T v D ' P. [1] R % The mai P is eal, smmeic, and has a ace of 1 ecause { } P P P 1 P [] since definiion, v is a uni veco, and hence, 1 fo all imes. Theefoe, P is full specified onl five independen paamees. The mai P can e epesened gaphicall as an ellipsoid see Fig. and 4. The hee pincipal aes ~, ~ and ~ of his ellipsoid ae defined he hee eigenvecos of he mai P and he lenghs of he hee half aes ae defined he eigenvalues P~, P~ and P~ see Fig. A. In he special fame of efeence defined his pincipal ais ssem see Fig. B, he mai P is diagonal: P~ 0 0 P 0 P~ 0. [] % 0 0 P~ In his case he eigenvalues pincipal values P, ~ ~ ~ ~ P and P ~ ~ ae he poailiies o find he magneic field along he pincipal aes ~, ~ and ~, especivel. Theefoe we call P simpl he poaili enso. Rigoousl, P coesponds o he sum of he eo and second ode em of a spheical hamonics epansion of he poaili 11

14 1 disiuion funcion desciing he oienaion of a efeence veco elaive o a igid od 4, 15. In he pincipal ais ssem, Eq. [1] fo he calculaion of he esidual dipola coupling educes simpl o % ' 1 ~ ~ ~ ~ ~ ~ P P P R D. [4] Fo eample, in he saic case, % v is consan, and hence,. % P [5] The mai has a much simple fom in he pincipal ais fame ~, ~, ~ whee he ~ ais is paallel o he veco v. In his efeence fame, % % ~ ~ ~ v and % P [6] In his case, he poaili ellipsoid is educed o a line along he ~ ais and he dipola coupling consan is

15 D D ~ R % 1 '. [7] Fo a compleel isoopicall eoiening molecule, he aveages, ae eo, and P ~ P~ P~ 1/, i. e. he poaili mai, P 0 0 [8] % is diagonal in an molecula fied fame of efeence. Hence, hee is an equal poaili of 1/ fo he magneic field diecion o poin along all hee aes of efeence. The coesponding poaili ellipsoid is a sphee wih adius 1/ see Fig. 4 C, and he esidual dipola coupling consan is D R % 1 ~ ~ ~ ' 0. 1 [9] Fig. 4 A shows an eample of an aiall smmeic poaili ellipsoid wih he pincipal values P ~ P~ 0. 5 and P ~ Fig. 4 B shows an eample wihou aial smme whee P ~ 0., P ~ 0. and P ~ Noe ha he lack of aial smme simpl means ha hee ae wo diffeen poailiies P P~ ~ fo he magneic field o poin along he pincipal aes ~ and ~ of he molecula-fied poaili enso. Howeve, his does no means impl ha in he la fame hee ae diffeen poailiies fo he molecule o e 1

16 aligned along he L o P, L diecion. Fo eample in he case shown in Fig. 4 B, ~ 0. P ~ 0. and P ~ 0. 5 ae he poailiies ha he ~, ~ and ~ aes ae aligned paallel o B 0. In he NMR lieaue, i is no cusoma o conside he poaili enso P which can e nicel depiced as an ellipsoid, u o use is aceless pa which is called he alignmen enso 1 A P 1. [0] If we mulipl A fom he lef wih he uni ow veco vt and fom he igh wih he column veco v and using Eq. [19] and Eq. [0], we ge v T v vt' 1 v A % P- 1 vt v 1 v P - 1 cos, T v [1] which can also e used o calculae he esidual dipola coupling consan in Eq. [1]: v T v A D. [] R 14

17 P and A have he same pincipal ais ssem ~, ~, ~ ecep fo a possile eodeing of he ais laels if he convenion is used ha P ~ P~ P~ and A ~ A~ A~, and he pincipal values ae elaed 1 A ~ P~, 1 A ~ P~, and 1 A ~ P~, [] wih A ~ A~ A~ 0. In he pincipal ais ssem 1 cos ' A ~ ~ A~ ~ A~ ~, [4] % and hence, he esidual dipola coupling consan is given A~ ~ ~ ~ A ~ A ~ D. [5] R The alignmen enso canno e epesened as an ellipsoid, ecause a leas one of he pincipal values is alwas negaive if A 0. 15

18 In Fig. 5, we show a gaphical epesenaion of he A ensos which coespond o he P v T v A ensos shown in Fig. 4. The plos show he sufaces whee he em is consan. R Hence, if spin I is assumed o e locaed in he oigin, he plos show he possile locaions of spin S fo which he esidual dipola coupling consan has he same magniude. Fo he case of an isoopicall eoiening molecule spheical poaili enso, he esidual dipola coupling is alwas eo, and no such suface eiss. The dependence of he scaling faco cos 1/ on he oienaion of he inenuclea veco is someimes shown he colo of a uni sphee. Fo he hee cases shown in Fig. 4 and 5, he coesponding gascale coded suface epesenaions of he alignmen ensos ae shown in Fig. 6. The gascale inensi epesens he scaling faco of a esidual dipola coupling consan if spin I is locaed a he oigin and spin S is moved ove he suface, i. e. assuming a consan inenuclea disance. Fo eample, in he aiall smmeic case shown in Fig. 6 A wih A ~ A~ 1/ 1 and A ~ 1/ 6, he scaling faco cos 1/ is eo if he ~ -componen of he inenuclea veco is ~ 1/, which is saighfowad o see if Eq. [4] is se o eo and using. This coesponds o an angle of accos 1/ he magic ~ ~ 1 ~ angle eween he inenuclea veco and he ~ -ais. Fo he case shown in Fig. 6 B wih A ~ / 15, A ~ 1/ 0 and A ~ 1/ 6, he pola angle, when he scaling faco is eo, depends also on he aimuhal angle eween he ~ -ais and he pojecion of v on he ~ / ~ plane. Fo eample, in he ~ / ~ plane, he scaling faco is eo if ~ / 16

19 coesponding o accos / , and in he ~ / ~ plane, he scaling faco is eo if ~ 1/ 6 accos 1/ In he isoopic case shown in Fig. 6 C, he scaling faco cos 1/ is eo fo all oienaions of he inenuclea veco R v. APPENDIX In he appendi, he ke equaions Eq. [4] and Eq. [5] fo he calculaion of he esidual dipola coupling consan D ae eepessed in vaious foms found in lieaue. If he uni veco v is defined in ems of he pola coodinaes and in he pincipal ais ssem of he alignmen enso A, hen v % ~ ~ ~ sin' cos sin' sin % cos' [6] and hence accoding o Eq. [4]: ' cos 1 * A ~ sin cos A~ sin sin A~ cos. [7] % This can e simplified noing ha cos 1 cos / and sin 1 cos / : * cos 1 ' % A~ ~ A~ A sin sin cos sin A ~ A ~ A sin ~ A ~ A~ sin sin cos A cos ~ cos A cos ~ [8] 17

20 Since A is a aceless mai, A ~ A~ A~, and we can ewie Eq. [8] as * 1 ' * sin ' A~ A~ cos % A~ cos sin cos. % [9] The pe-faco of A % can e fuhe simplified using he elaion sin 1 cos : cos 1 cos sin cos 1 cos 1. [40] Thus, we aive a * 1 ' A ~ A~ A~ cos % cos 1 sin cos. [41] Eq. [41] can alenaivel e epessed in ems of he pincipal values S %, S % and S % of he Saupe mai o ode mai S, which is simpl he alignmen mai A scaled a faco of /, if he opical ais of he liquid csal is collinea wih he diecion of he magneic field 9, 10: S / A. [4] Hence, 18

21 * cos { S ~ cos 1 S~ S~ sin cos }. 1 ' 1 % [4] Ofen, he aial componen A a of he alignmen enso is defined as 5 A a A~ S~, [44] and he homic componen A of he alignmen enso is defined as A A ~ ~ A S~ S~. [45] Wih hese definiions, we can epess Eqs. [41] and [4] as - cos, cos / ' 1 A sin / cos, 1 * 1 ' % A a. [46] 0 which in un is ofen wien as 1 / cos 0 1. Aa %, * - Aa cos % 1 Rsin cos ' { cos % 1 sin cos } [47] whee 19

22 A A R [48] a is called he homici of he alignmen enso and A~ A~ S~ S~ R [49] A~ S~ is called he asmme paamee which descies he deviaion fom aiall smmeic odeing 6. So fa, we have assumed a igid molecule ha umles in soluion. In he pesence of inenal moions he deivaion of esidual dipola couplings ecomes moe complicaed 6, 18, 19. Povided he alignmen pocess is no affeced inamolecula moion, he analsis is elaivel saighfowad. If he inenal moion of he inenuclea veco v is aiall v smmeic wih espec o he aveage oienaion av, he dipola coupling epeced fo his aveage oienaion is scaled a faco, which is idenical o a genealied ode paamee S 0 S The lae coesponds mahemaicall o he spin elaaion ode paamee 19, 0, u ehiis a sensiivi o moions eending o he millisecond ime scale 6, 18. This leads o he following equaion of he esidual dipola coupling consan: { cos % 1 sin cos }. Aa D S [50] R 0

23 This epession is ofen ewien using he maimum dipola coupling D R ma / / c. f. Eq. [7] o he so called magniude of he esidual dipola coupling enso D D A / 7: a ma a D S D D S SD a ma { cos ' 1 * sin cos } A { cos ' 1 * sin cos } ma a Aa % P * cos sin cos, [51] whee P cos 1 / is he second-ode Legende polnomial. Finall, we use he esuls deived in his manuscip o inoduce he conceps of he genealied degee of ode GDO of a given alignmen enso A and he genealied angle eween wo diffeen alignmen ensos 1 A and A. In complee analog o he scala poduc eween wo eal vecos, he scala poduc e- 1 ween wo eal maices e. g. wo alignmen maices A and A is defined as 1 1 A A A ij A ij [5] i, j and he nom A of he eal mai A is given A A A. [5] i, j A ij 1

24 The maimum ode is found fo he saic case, whee he poaili enso P ma is given Eq. [6] in he pincipal ais ssem. The coesponding maimum alignmen enso A P 1/ 1 has he fom ma ma ' 1/ 0 0 A ma 0 ' 1/ 0. [54] 0 0 / % The nom of A ma is given A ma. [55] The genealied degee of ode GDO of a given ode mai A can e defined as A GDO A. [56] A ma In ems of he Saupe mai S / A c. f. Eq. [4], his can e wien as 6, GDO S. [57] In lieaue, he smol is ofen used fo he GDO u we do no use he smol hee in ode o avoid confusion wih he pola angle defined in Eq. [6].

25 The GDO is independen of he molecula-fied fame, in which he alignmen enso A is epessed. In he pincipal ais ssem onl he diagonal elemens of A ae noneo and Eq. [56] simplifies o GDO A ~ ~. ~ A A [58] Fo aiall smmeic alignmen ensos A A~ A~ / his simplifies fuhe o : ~ GDO 1 A % 4 A ~ ~ S ~. 1 4 A ~ A ~ A ~ [59] Wih he help of he scala poduc, we can also define he genealied angle eween wo alignmen ensos 1 A and A which coespond e. g. o wo diffeen alignmen media. If he mai epesenaions of 1 A and A ae given in a common molecula fame of efeence, he cosine of he genealied angle eween hese alignmen ensos can e defined as he nomalied scala poduc eween hem : 1 A A cos. [60] 1 A A

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28 0. Lipai G and Sao A. Model-fee appoach o he inepeaion of Nuclea Magneic Resonance Relaaion in Macomolecules.. Analsis of Epeimenal Resuls. J Am Chem Soc 198; 104: Meile J, Pompes JJ, Pei W, Giesinge C, Büschweile R. Model-fee Appoach o he Dnamic Inepeaion of Residual Dipola Couplings in Gloula Poeins. J Am Chem Soc 001; 1: Tolman JR, Al-Hashimi HM, Ka LE, Pesegad JH. Sucual and Dnamic Analsis of Residual Dipola Coupling Daa fo Poeins. J Am Chem Soc 001; 1: Sass J, Codie F, Hoffmann A, Rogowski M, Cousin A, Omichinski JG, Löwen H, Gesiek S. Puple Memane Induced Alignmen of Biological Macomolecules in he Magneic Field. J Am Chem Soc 1999; 11:

29 FIGURE CAPTIONS Figue 1: Definiion of he angle eween he inenuclea veco R v connecing spins I Figue : Figue : Figue 4: and S and he magneic field veco B v. The uni vecos v and v poin in he diecion of R v and B v, especivel. Effec of molecula umling of a igid molecule as seen Panel A fom he la fame of efeence wih aes L, L, L and Panel B fom an aia molecula fame of efeence wih aes,,. In he la fame Panel A, he magneic field B v is consan and poins definiion along he L ais, wheeas he inenuclea veco R v keeps changing is diecion. In a molecula fame Panel B, he siuaion is evesed: hee, an given inenuclea veco is consan, wheeas he oienaion of he magneic field is ime-dependen. The molecule, a given inenuclea veco R v and he poaili ellipsoid a gaphical epesenaion of he poaili enso P, c.f. Eq. [5] ae shown Panel A in an aiail chosen molecula fame c.f. Fig. B and Panel B in he special coodinae ssem wih aes ~, ~, ~ defined he pincipal aes of he poaili ellipsoid. Eamples of hee chaaceisic poaili ellipsoids gaphical epesenaions of he poaili enso P, c.f. Eq. [5] as seen fom he pincipal ais ssem wih aes ~, ~, ~ c.f. Fig. B. Panel A shows an aiall smmeic poaili ellipsoid wih P % P % 0.5 and P % 0.5 Panel A. Panel B depics a homic po- 7

30 aili ellipsoid wih P % 0., P % 0. and P % 0.5. Panel C shows an isoopic poaili ellipsoid wih P % P % P % 1/. Figue 5: Gaphical epesenaions of he alignmen ensos Panel A which coespond o he hee poaili ensos shown in Fig. 4 A-C. The pincipal componens of he alignmen enso ae A A ~ A~ 0.5 1/ 1/ 1, A ~ 0.5 1/ 1/ 6, B A ~ 0. 1/ /15, A ~ 0. 1/ 1/ 0, A ~ 0.5 1/ 1/ 6 and C vt v - A ~ A~ A~ 1/ 1/ 0. The plos show he sufaces whee A / R 1Å ligh ga o -1 Å - dak ga if he ~, ~ and ~ aes ae laelled in unis of Å. Figue 6: Fo he hee cases shown in Fig. 4 and 5 wih A A ~ A~ 1/ 1, A ~ 1/ 6, B A ~ / 15, A ~ 1/ 0, A ~ 1/ 6 and C A ~ A~ A~ 0 he magniude of he scaling faco cos 1/ is coded on a uni sphee as a funcion of he oienaion of he inenuclea veco R v whie: vanishing scaling faco. Posiive and negaive scaling facos ae denoed he especive sign. 8

31 Figue 1

32 Figue

33 Figue

34 Figue 4 Figue 5 Figue 6

35 Figue 4 Figue 5 colo - - Figue 6 colo