Supporting Information for. Strongly Enhanced Raman Optical Activity in Molecules by. magnetic response of nanoparticles

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1 upportig Iformtio for trogly Ehcd m Opticl Activity i Molculs by mgtic rspos of oprticls Tog Wu 1 Xiuhui Zhg ogyo Wg 1 Xigdog Zhg 1* 1 chool of Physics Bijig Istitut of Tchology Bijig 181 Chi chool of Chmistry Bijig Istitut of Tchology Bijig 181 Chi *Corrspodig Authors: zhgxd@bit.du.c 1. Thorticl drivtio for m d Followig th formlism of Jsko d cusri 1 w giv th xprssios for th totl iducd multipols i this prt. I cotrst to prvious thory th mgtic polrizility of th NP is cosidrd. Furthrmor istd of usig qusi-sttic pproximtio th EM filds r writt i th ll wv zos icludig th r-fild itrmdit d fr-fild rgios. W strt by oticig th EM fild d th lctric fild grdit t th positio of th molcul c b writt s E E E B (1) I i g i m i b bc b c b B B E E () I i m i mg i b b bc b c E E E B (3) g i gg i gm i t b c c cd c d c c Accordig to qs 6 d 7 i th mi txt th xprssios for tsor lmts mrkd by i qs 1 d c b rdily gott which r giv by C P C C (4) 1 z bz m zb 1 P (5) c (6) g bc 1

2 P C1 C zbz (7) c m zbc1p c (8) (9) mg (1) gg cd c bd g c c C P E1 cb z E cz E bz c E3 z bz cz E4 bc z E5 z bc bz cz (11) gm 1 ' z ' z E 6 c P b ' c ik ' zz ' zz c r cr (1) with E 1 C P s (13) 4 ikr xp ikr ik (14) s m (15) 4 k i 3c 3 4 (16) r r r E E E k 3i 3c 3 4 r r r 6k 15i 15c r r r ik i 4 3 r r k E5 i cr E i c 6 r r (17) (18) (19) () (1)

3 I vlutig qs 4-1 should b qul with which is th frqucy of th icidt wv. ubstitutig qs 1-3 ito qs 1-3 i th mi txt th multipols iducd o th molcul c b rdily gott. Th rditio fild t th m shiftd frqucy p origitd from th multipol xcittio of th molcul c iduc dipol rsposs o th sphricl NP ccordig to qs 4 d 5 i th mi txt. Th totl multipols of th molcul-np hybrid systm r show i qs i th mi txt. Isrtig qs 4 d 5 ito ths qutios d usig th xprssios for th multipol rditio filds th totl multipols c lso b xprssd s m () m g b b bc bc m m (3) m mg b b bc bc m (4) g gg gm c c cd cd c c Th cocrt xprssios for tsor lmts mrkd by i qs -4 c b rdily gott which r giv by 1 z bz C C C (5) C F F (6) g bc 1 cz z bz cz r m C (7) c 1 zb m 1 1 zb i p z ' r C C1 ' b C 'z zb c (8) i p C1 Cbzz C1 r z ' 'z b c c (9) mg i p bc G1 zbcz z ' C F1 ' b cz F ' zbzcz 3 (3) 3

4 g 3 3 c Cr bzc zbc ' z ' c C1 cz 3 zbz ' z C 3 r C c 3 gm c 1 zc bz bzc z ' zc ' z 3 C F F 3 C F F C F F gg cd c dz 1 z cz dz bz bc dz 1 bz cz dz z zc dz 1 zc dz (31) (3) (33) whr F 1 F d G 1 i ths qutios r giv by 3 F 1 ic c i r 3 r 3 r 3 c F G i 5ic 5c c r r r 3 i 3 3i 1 3 c r cr r (34) (35) (36) I vlutig qs 5-33 should b st to b qul with p which is th m shiftd frqucy. (1). Exprssios for () Erd t rd t E G () d Erd t A Usig th fr fild pproximtio for rditio filds of multipols E i q 16 c b rd t xprssd s: 1 E C m c d d b d rd t b c b c d d d i p d b c b bc 3c (37) E (1) ubstitutig qs 1-3 d qs -4 ito this qutio xprssios for rd t 4

5 () Erd t G () d rd t E A r giv by E C α E + α B - α E + α B 1 d m i 1 d m m i εcb cb' α b'c' c'b" Eb" - εcb cb' α b'c' c'b" Bb" c c d d (1) rd t c' c'd' d'b' b' c' c'd' d'b' b' bc' c'd' d'b' b' bc' c'd' d'b' b' () rd t i m i b i m i () rd t iωp d b g i g m i c' α c'b" b"c" E c" + c' α c'b" b"c" Bc" 3c d d d iω b c + α E + α B 3c E G C 3G E 3G B p g i g m i bcc' c'b" b"c" c" bcc' c'b" b"c" c" m i i c' d' d' b' b' c' d' d' b' b' d d d d d 1 b 11 d m i d g b c g * c ' bc' c bcc ' b c' bc c' 3 3 c 9 c d d d d d m b m 1 d i d gm b c gm i m i c ' bc' c bcc ' b c' bc c' G c'd' d'b' Eb ' + d'b' Bb ' c 3 c E A C A E A E A B g i gg i gm i c'd'' d''b' b' c''f' 'f'b'd' b' d' c'd'' d''b' b' d d d d d 1 b 11 d m i d g b c g * c ' bc' c bcc ' b c' bc c' 3 3 c 9 c d d d d d g b g 1 d mg i d gg b c gg i m i c'd' bc'd' c bcc'b' b c'd' bc c'd' A 'c'd' ' b' Eb ' ' b' Bb ' c 3 c (38) (39) (4) whr C p ik d d d ( ) ( ) 4 / b d b d with d d big th positio vctor of th dtctor. (1) 3. Ordr lysis for trms i E A () rd t E G d () E rd t rd t I this prt ordrs of mgitud for trms occur i q 38 q 39 d q 4 r vlutd usig th followig ssumptios d coditios: 5

6 Th molculr rspos tsors stisfy A O O d A G 1 / [ d ] [1 ] / O 1 whr d dots lgth o th ordr of molculr dimsios. Th chirl molcul is clos to th surfc of NP ( r s ). Th rdius of NP is much smllr th th wvlgth ( s ). isprsio of th NP is cosidrd to b o. Bsd o this ssumptio th pproximtio p is usd. For covic w lso dfi th dimsiolss d m s hcmt fctors rltd with th lctric d mgtic dipol rsposs of th NP which c b xprssd s c s 3 (41) 4 r s m 3 m 4 r (4) Bsd o th ruls mtiod ov trms i qs 38-4 c b substitutd by som simpl vrils with sm ordrs of mgitud. Th substitutios r giv by s r (43) 1 (43b) A d (43c) G cka (43d) 1 (43) 1 (43f) d E 1 i (43g) i i B E / c (43h) E ke (43i) i i 6

7 Mkig us of th substitutios giv i q 43 ordrs of mgitud for trms occurrd i (1) () E rd t rd t E G () d rd t E A r listd i Tl 1 d Tl -c. Not th blu fot trms (Color Oli) r vishd if diffrt phss btw th filds scttrd by th molcul d NP r glctd. (1) Tl 1. Ordrs of mgitud for trms occur i E r vlutd by xprssios prstd rd t i this tl. Th blu fot trms r vishd if w igor phs diffrcs btw th fild scttrd by th NP d molcul. C i q 38 is omittd for cocisio. + 1 ( + 1) ( ) ( + 1) ( ) ( + 1) ( ) ( + ) ( ) ( ) () Tl. Ordrs of mgitud for trms occur i Erd t G () d rd t E A r vlutd by xprssios prstd i this tl. Th blu fot trms hv th sm mig with thos i Tl 1. Nottios r V d / r kr ; U kd kr Tl.1 C is omittd for cocisio. ; '= 1; = 1. imilr with m ' m () / ( + ) ( + ) ( + ) ( + ) / ( ) 7

8 (b) / ( + ) / ( + 1) ( + ) ( + 1) (c) / / ( + 1) If w prsum O m ordrs of mgitud c b dotd by th umbr of strisks (*) or strs ( ) i Tl 1 d Tl. Trms mrkd with thr strisks (strs) r / r ( or / s ) tims lrgr th thos with two d th squc gos o. Thos trms without strisk or str r trivil d c b igord. Accordig to q 17 i th mi txt ( p ) G I EICP OA d ( p ) A I EICP OA r built by products of trms listd i Tl 1 d trms i Tl. Thus ( p ) G I EICP OA d ( p ) A I EICP OA scl s ( p) G ( p) A 3 IEICP OA O IEIC POA KpO kd (1 m) (44) if th rtrdtio of th pl wv d th phs diffrc btw th fild scttrd by th NP d molcul r igord. A discussio o ths two ffcts is providd i th xt prt. Although both ( p ) G I EICP OA d ( p ) A I EICP OA r proportiol to th mgtic dipol rspos of th NP diffrt mchisms should b ssigd i itrprtig thir hcmts. 8

9 imilr with E th EOA c b modld s two-stp procss. Firstly multipols r iducd o th molcul by th icidtl wv th th molcul rdits fild t m shiftd frqucy. Ths two stps r wll dscribd by qs 1-3 d qs -4 rspctivly. Mchisms of th EOA c b udrstood by lyzig th migs of d tsors i ths qutios. From th tls ov w fid tht ( p ) G I EICP OA is domitd by trms which r proportiol to d. Accordig to q 7 q 15 d q 9 d r both fuctios of s m th mgtic dipol polrizility of th NP. Assumig m is o th sm ordr of mgitud with th followig rltio holds B E E. (45) i m i mg i b b bc b c Thus th totl mgtic fild t th positio of th molcul c b pproximtd s B B. (46) I i b O th othr hd udr EOA coditios th molcul dos ot rdit i th fr spc. Wh th molcul is clos to th NP mtiod ov lctric d mgtic dipol rditios of th molcul r both strogly modifid. Accordig to qs -4 th rditio fild of th mgtic dipol of th molcul is giv by E E m E m E m (47) ( m) mb d b m qbc gm rd t bc c bc c bc d d whr m b E d d b E r vcuum rditio filds of uit lctric d mgtic dipols prlllig with b xis. q bc E is th lctric fild of uit qudrupol with bc compot. From th xprssios of th giv i q 7 q 9 d q 3 E is domitd by th first ( m) rd t trm d c b pproximtd s E E m. (48) ( m) mb rd t bc c 9

10 From q 46 c b dotd s th hcmt fctor of th mgtic fild t th positio of th molcul. Accordig to q 48 bc c b usd to chrctriz th mplifictio of th molculr mgtic dipol rditio. Bsd o th migs of d bc hcmt of ( p ) G I EICP OA c b modld s sum of two ffcts: lrg mgtic fild coctrtio t th positio of th molcul d rditio hcmt of th mgtic dipol momt of th molcul. Ehcmt mchism for ( p ) A I EICP OA c b giv i similr wy. From tl 1 d tl w fid tht ( p ) A I EICP OA is domitd by trms which r proportiol to gm c m m d mg bc. imilr with d ll of thm r fuctios of th mgtic dipol polrizility ( ). Howvr s m gm c m mg d bc r vishd withi th qusi-sttic m pproximtio. To clcult ( p ) A I EICP OA xprssios for lctric d mgtic filds crtd by multipol sourcs i ll wv zos icludig th r-fild itrmdit d fr-fild rgios r rquird Accordig to q 1 d q 3 gm i B d cb c rprst lctric fild d lctric fild m i b grdit iducd by th mgitud fild of th icidt wv. Thy r much smllr th E i b g i d ce c which r fild d grdit iducd by th lctric fild compot of th icidtl wv. O th othr hd th rditio filds of th lctric dipol d th lctric qudrupol of th molcul should b xprssd s E E E E (49) ( ) d b mb m qbc g rd t bc c bc c bc d d E E E E. (5) ( q) d b g mb mg qbc gg rd t b cd cd b cd cd bc d d Th scod trm o th right sid of q 49 is much smllr th th othr two trms. imilrly th 1

11 mb mg cotributio of E b cd cd to th molculr qudrupol rditio is trivil comprig with th first d third trms i q 5. Bcus of this th rditios of th molculr lctric dipol d lctric qudrupol r oly slightly ltrd by trms with mg or bc. m Bsd o th lysis ov o c fid th lctric fild d lctric fild grdit r oly slightly ltrd by th mgtic dipol rspos of th NP. imilr th rditios of molculr lctric dipol d lctric qudrupol r ot strogly iflucd by th mgtic rspos of th NP. Thus th hcmt of ( p ) A I EICP OA by th mgtic dipol rspos of th NP should b udrstood s th sytry brkig of () th lctric fild d lctric fild grdit impigd o th molcul d (b) fild rditd by th lctric dipol d th lctric qudrupol of th molcul. 4. trdtio d Phs diffrc ffcts I this prt w ivstigt th trdtio d Phs diffrc hcmt mchisms of th EOA. From th tls i th prvious prt o c fid tht th products of g i E d trms lik ' A g i E ' or ' A E ' (1) i rd t i () E rd t A; G i ' α E ' giv th ( ) ldig-ordr trm of I p E. Jsko t.l 1 hv climd tht ths trms should b ICP OA vishd bcus th orittio vrgig of th dirctios of th pl wv. This coclusio is drw bsd o th qusi-sttic pproximtio whr th rtrdtio ffct of th icidt wv is i igord. If w xpd th icidt pl wv s: i E k r this prt of sigl mrgs 1 oc gi du to th scod trm i this xpsio. Trms itroducd by phs diffrc btw th filds scttrd by th NP d th chirl ω / c A E molcul lso ply pivotl rol i hcig th OA sigl. For xmpl g g i () i E rd t tims (1) i E i α E rd t givs EICP-OA sigl which scls s 11

12 ( p) 4 IEICP OA Kp kd O. (51) u to th xistc of rtrdtio d phs diffrc ffcts th circulr itsity diffrc E c b i pricipl s lrg s its uhcd coutrprt v wh th NP possss mrly th lctric dipol rspos. This is i violt cotrst with prvious prdictios 1 which clim tht lctric dipolr NPs giv ouiform hcmt of m d OA with E sigifictly smllr th. Howvr it is worth to ot tht oly ( p ) A I EICP OA is strogly hcd by ths two mchisms whil ( p ) G I EICP OA scls oly s cub of. Numricl rsults hv b prformd for odsorbd d dsorbd css. For th odsorbd cs th chirl molcul my bid i y dirctio with rspct to th surfc of th NP with qul probility of ll orittios. A vrg ovr ll molculr orittios is tk. I cotrst th orittio of th molcul is fixd for th dsorbd cs. Hr w mploy th mtrix polriztio thory (MPT) dvlopd by Bour. 34 Furthrmor to gurt th rliility of th umricl rsults th lctric qudrupol rspos of th NP which c b xprssd s ˆs s E E Q T (5) is lso icludd i th Bour s formul. E is th lctric fild icidt o th NP d T dots th trspos oprtio. Hr th tsor qudrupol momts r dtrmid by th sytricl prts of th lctricl fild grdits E. Th s is th sclr polrizility which c b Q obtid by th mthods dscribd lswhr. 5 1

13 Figur 1. () urfc hcd m scttrig itsitis I. (b) G ( p ) E compots of EOA spctr ( I ( p ) G E ICPOA ). (c) A compots of EOA spctr ( I ( p ) A E ). (d) Circulr itsity ICPOA diffrc E. ()-(h) r rrgd sm with ()-(d) but for th molculr orittio vrg css. W cosidr systm whr chirl molcul ()-(-)-CHFClBr is st to b 11 m wy from th ctr of 1m Au NP s show i th ist of figur 1(c). Th rspos tsors of th molcul r clcultd by th GAUIAN softwr t th B3LYP/6-31+G** lvl. I ( p ) E I ( p ) G ( p ) A E I ICP OA E d ICP OA E s fuctios of wvlgth r prstd i Figur 1-d d Figur 1-h for th dsorbd d th odsorbd css rspctivly. Th rsults (blu lis) prstd i Figur 1 r sigls s fuctio of wvlgth for th molculr vibrtio mod t th frqucy of 5 cm -1. I dditio to xct umricl rsults (blu lis) w hv lso prformd umricl rsults by usig th followig kids of pproximtios: igor MP: Th diffrt phss of filds scttrd by th molculr multipols d th Au NP r igord. pcificlly withi this pproximtio th totl multipols of th molcul-np combid systm r giv by (53) s 13

14 m m m (54) s (55) s istd of tht providd by qs i th mi txt. igor PW: Th rtrdtio ffct of th icidt pl wv is igord. I prformig this i pproximtio th icidt wv is st s E by droppig th phs trm. is costt vctor which dots th polriztio of th pl wv. igor All: Igor both th rtrdtio d th phs diffrc. From Figurs 1 b d f w fid tht thr is o ffct o th clcultd rsults of ( p) I E ( p ) d I G E ICP OA wh th rtrdtio ffct or th phs diffrc hs b igord. Howvr from Figur 1c d Figur 1g w hv obsrvd tht omissio of y o of ths two ffcts ( p ) lds to smllr I A E ICP OA compots of OA. By compriso th rd li with th blu li i Figur 1c o c fid tht th ifluc of th rtrdtio ffct of th pl wv is much strogr th th phs diffrc ffct wh th molcul is fixd with rspct to th surfc of th NP. pcificlly it c b foud tht wh th rtrdtio of th pl wv is igord th A compot of sigl is oly roud 1/4 of th xct rsult t y giv wvlgth of th icidt wv. Howvr from Figur 1g iflucs of ths two ffcts r comprl upo orittiol vrgig of th molculs. As show i Figur 1g th A compot of OA is 3/4 of th xct vlu t y giv wvlgth wh glctig ithr th rtrdtio or th phs diffrc ffct. Comprig Figur 1b with Figur 1c o c fid tht th OA for th dsorbd cs is ( p ) domitd by I A E ICP OA ( p ). u to th xistc of th rtrdtio ffct I A E ICP OA is sigifictly hcd by th Au NP. Thus th EOA sigl for this cs c b s lrg s 14

15 4 KpO kd. I cotrst from Figur 1f d Figur 1g th EOA for th ( p ) odsorbd cs is still domitd by I G E ICP OA which is ot iflucd by th ffcts discussd i this prt. Thus th EOA sigl for this cs scls s K p 3 kd O. 5. Elctric qudrupol missio r th NP Figur. Th corrspodig fr fild rditio pttrs lim r Q re r for lctric qudrupols big put 1m wy from th surfc of i sphrs with s=65m (th scod list) d corrspodig rsults for th Au sphrs (th third list). 15

16 I this prt w show fr fild rditio pttrs for vrious kids of qudrupol. Th mgituds of th qudrupols r giv by th first list. Th scod list of th figur shows th corrspodig rsults for th qudrupol big put 1m wy from i NP. Th lctric qudrupols rdit t th wvlgth of 549m. Th third list is rrgd similr to th scod list but th i sphrs r substitutd by Au sphrs d th lctric qudrupols rdit t th wvlgth of 537.6m. From ths figurs o c s tht th qudrupol rditio hcmt fctors for i NP r ot lrgr th tht for Au NP. frcs [1] Jsko B. G.; cusri G. E. urfc Ehcd m Opticl Activity of Molculs o Orittiolly Avrgd ubstrts: Thory of Elctromgtic Effcts. J. Chm. Phys [] u E. C. L.; Etchgoi P. G. Pricipls of urfc-ehcd m pctroscopy d ltd Plsmoic Effcts; Elsvir: Amstrdm 9. [3] Bouř P. Mtrix Formultio of th urfc-hcd m Opticl Activity Thory. J. Chm. Phys [4] Novák V.; Šbstík J.; Bouř P. Thorticl Modlig of th urfc-ehcd m Opticl Activity. J. Chm. Thory Comput [5] Evlyukhi A. B.; ihrdt C.; Zywitz U.; Chichkov B. N. Collctiv rsocs i mtl oprticl rrys with dipol-qudrupol itrctios. Phys. v. B