Molecular QED of coherent and incoherent sumfrequency and second-harmonic generation in chiral liquids in the presence of a static electric field

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1 Moecuar Physcs ISSN: (Prnt) (Onne) Journa homepage: Moecuar QED of coherent and ncoherent sumfrequency and second-harmonc generaton n chra quds n the presence of a statc eectrc ed P. Fscher & A. Saam To cte ths artce: P. Fscher & A. Saam (2010) Moecuar QED of coherent and ncoherent sum-frequency and second-harmonc generaton n chra quds n the presence of a statc eectrc ed, Moecuar Physcs, 108:14, , DOI: / To n to ths artce: Pubshed onne: 21 Ju Submt your artce to ths ourna Artce vews: 52 Vew reated artces Ctng artces: 4 Vew ctng artces Fu Terms & Condtons of access and use can be found at Downoad by: [Harvard Lbrary] Date: 16 June 2016, At: 06:43

2 Moecuar Physcs Vo. 108, No. 14, 20 Juy 2010, RESEARCH ARTICLE Moecuar QED of coherent and ncoherent sum-frequency and second-harmonc generaton n chra quds n the presence of a statc eectrc ed P. Fscher and A. Saam* The Rowand Insttute at Harvard, Harvard Unversty, Cambrdge, MA 02142, USA; Department of Chemstry, Wae Forest Unversty, Wnston-Saem, NC 27109, USA (Receved 2 Apr 2010; na verson receved 12 May 2010) Coherent second-order nonnear optca processes are symmetry forbdden n centrosymmetrc envroents n the eectrc-dpoe approxmaton. In quds that contan chra moecues, however, and whch therefore ac mrror mage symmetry, coherent sum-frequency generaton s possbe, whereas second-harmonc generaton remans forbdden. Here we appy the theory of moecuar quantum eectrodynamcs to the cacuaton of the matrx eement, transton rate, and ntegrated sa ntensty for sum-frequency and second-harmonc generaton tang pace n a chra qud n the presence and absence of a statc eectrc ed, to examne whch coherent and ncoherent processes exst n the eectrc-dpoe approxmaton n quds. Thrd- and fourth-order tme-dependent perturbaton theory s empoyed n combnaton wth snge-sded Feyan dagrams to evauate two contrbutons arsng from statc ed-free and ed-nduced processes. It s found that, n addton to the coherent term, an ncoherent process exsts for sum-frequency generaton n quds. Surprsngy, n the case of dc-ed-nduced second-harmonc generaton, the ncoherent contrbuton s found to aways vansh for sotropc chra quds even though hyper-rayegh second-harmonc generaton and eectrc-ed-nduced secondharmonc generaton are both ndependenty symmetry aowed n any qud. Keywords: moecuar quantum eectrodynamcs; sum-frequency generaton; second-harmonc generaton; statc eectrc ed; chraty 1. Introducton Motvated by observatons of second-harmonc generaton (SHG) sas emanatng from crystas acng a centre of nverson, n whch the eectrc poarzaton s second order n the apped oscatng eectrc ed, ed Gordmane [1] n 1965 to predct that sum- and dfference-frequency generaton (S/DFG) can tae pace n an sotropc non-centrosymmetrc medum, thereby enabng the structure of chra quds to be probed spectroscopcay. Interest n ths phenomenon has resurfaced n recent years as a resut of expermenta measurements of the quadratc susceptbty of optcay actve moecues n souton va three-wave mxng processes nvovng nfra-red and vsbe radaton [2 4]. In partcuar, the coherent superposton of two ncomng optca eds on non-racemc soutons of 1,1 0 -b-2-naptho and monene produced eectromaetc radaton at the sum-frequency, aowng the chraty of the qud to be arrved at drecty. Further, the sum- and dfference-frequency emsson sa ntensty from a randoy orented sampe of chra moecues when subect to two ncdent tme-varyng eectrc eds, E ð! 1 Þ and E ð! 2 Þ, of dfferng frequency! 1 and! 2 where Latn subscrpts denote Cartesan tensor components, has been examned theoretcay [5 14]. Generay, the quadratc poarzaton of the nonnear medum, P ð2þ ð! s=d Þ, s evauated usng the formasm of sem-cassca radaton theory va P ð2þ ð! s=d Þ¼ ð2þ ð! s=dþe ð! 1 ÞE ð! 2 Þ: ð1:1þ It s seen from the prevous equaton that the poarzaton P ð2þ ð! s=d Þ s expressbe n terms of the second-order nonnear optca susceptbty tensor ð2þ ð! s=dþ, where! s=d s the sum- and dfferencefrequency,! s ¼! 1! 2 and! d ¼! 2! 1, respectvey. In an sotropc medum, (2) s a pseudoscaar that s proportona to the rst hyperpoarzabty, commony denoted by, and whch therefore offers the possbty by whch the chraty of the souton may be probed, snce, and hence (2), are of opposte s for the enantomers of a chra souton and the correspondng bu susceptbty s zero for a racemc *Correspondng author. Ema: saama@wfu.edu ISSN prnt/issn onne ß 2010 Tayor & Francs DOI: /

3 1858 P. Fscher and A. Saam mxture. However, the technque does not aow for enantomers to be dstngushed between as the expermenta sa s proportona to the matude square of these quanttes. A nove off-shoot of the process descrbed above occurs when an addtona statc eectrc ed s swtched on, eadng to a contrbuton n the ntensty that s proportona to the cross-term formed by the SFG and the eectrc ed-nduced SFG (ESFG) susceptbty, namey an SFG-ESFG contrbuton, and thereby to a nonnear chra eectro-optc effect [15 17]. Wth ths technque, the chraty and absoute conguraton of optcay actve moecues n souton may be ascertaned by expotng the fact that the nterference term between the dc-ed and statc edfree SFG contrbutons depends on the handedness of the enantomer. Expermenta conrmaton has been provded by measurements reported on 1,1 0 -b- 2-naptho dssoved n tetrahydrofuran [16]. In ths paper we derve the theory of coherent and ncoherent SFG-ESFG scatterng n quds and as under what conguratons t s senstve to chraty. Ths aows us to aso consder the case of degenerate ncdent frequences, correspondng to second harmonc generaton (SHG) processes. It s nown that ncoherent Rayegh scatterng (ncoherent SHG) s senstve to chraty f contrbutons beyond the eectrc dpoe approxmaton are consdered [18]. Snce ncoherent SHG, whch requres a non-zero and s forbdden f the moecues are centrosymmetrc, and eectrc-ed-nduced SHG (EFISH), whch s aowed n any qud, both occur, one mght expect that ther cross-term, descrbed by a party-odd seventh ran observabe, coud mae SHG senstve to chraty. It s found that ths contrbuton vanshes. We rst empoy the theory of moecuar quantum eectrodynamcs (QED) [19 22] to compute the matrx eement, transton rate and ntegrated ntensty for SFG from chra speces, both n the presence and absence of an apped statc eectrc ed. The contrbuton arsng when the dc eectrc ed acts s compared wth the term obtaned wthout the statc ed, as we as wth the nterference term occurrng between these two contrbutons. In ths ast such term, the ntegrated ntensty s neary proportona to the strength of the statc eectrc ed, and depends on the product of the second- and thrd-order dpoar moecuar susceptbty. Both the coherent and the ncoherent contrbutons are evauated and examned. We then consder the case of SHG. A dstnct advantage ganed by the use of the moecuar QED formasm s that the radaton ed obeys the postuates of quantum mechancs. Ths ensures that eectron photon nteractons are descrbed from a fundamenta pont of vew, n contrast to the sem-cassca radaton theory formuaton n whch ony matter s treated quantum mechancay, wth the externa eectromaetc ed(s) beng consdered as a cassca externa perturbaton on the matera system. Cacuaton of the matrx eements for SFG n the absence and presence of a statc eectrc ed proceeds n the foowng two sectons, respectvey. Rate and sa ntensty expressons are then presented n Secton 4. Secton 5 examnes the ncoherent eectrc ed-nduced SFG and SHG processes. The wor s summarzed n the sxth and na secton. 2. Matrx eement n the absence of a statc eectrc ed Consder a chra moecue, ntay n the ground eectronc state g, wth energy E g, nteractng wth two monochromatc aser eds contanng n 1 photons of mode ð 1, 1 Þ and n 2 photons of mode ð 2, 2 Þ, where denotes the wavevector n the drecton of propagaton, and the poarzaton ndex of the photon. Let the optcay actve speces absorb a snge photon from each of the two asers, and emt a snge photon at the sum-frequency, whose mode s denoted by ð s, s Þ, and return to the ground eectronc state. The nta and na states of the tota radaton matter system may be represented as ¼E g ; n 1 ð 1, 1 Þ, n 2 ð 2, 2 Þ, ð2:1aþ f ¼E g ; ðn 1 1Þð 1, 1 Þ, ðn 2 1Þð 2, 2 Þ; 1ð s, s Þ, ð2:1bþ wth the state of the radaton ed ncuded expcty snce t forms an ntrnsc part of the dynamca system when treated by the technques of moecuar QED. The tota Hatonan for the nteracton of eectromaetc eds wth atoms and moecues s therefore a sum of three terms, H tota ¼ X H mo ðþh rad X H nt ðþ: ð2:2þ The rst contrbuton s the faar moecuar Hatonan encountered n non-reatvstc quantum theory of matter. The second term s the Hatonan operator for the eectromaetc ed. It s commony expressed n second quantzed form through the annhaton and creaton operators for a (, )-mode photon, a ðþ ðþ and a yðþ ðþ, respectvey. For the overwhemng maorty of probems of nterest to chemca physcsts, the mutpoar form of the coupng Hatonan between radaton and matter s the

4 Moecuar Physcs 1859 most amenabe [23,24]. Invong the eectrc dpoe approxmaton yeds the recozabe form for the nteracton Hatonan H nt ðþ ¼" 1 0 ðþd? ðr Þ, ð2:3þ where ðþ s the eectrc dpoe moment operator of speces, ocated at R, and d? ðrþ s the second quantzed transverse eectrc dspacement ed operator at poston r whose mode expanson s d? ðrþ ¼ X hc" 1=2 0 e ðþ ðþa ðþ ðþe r 2V, e ðþ ðþa yðþ ðþe r, ð2:4þ n whch e ðþ ðþ s a compex unt eectrc poarzaton vector for a photon of mode ð, Þ, the overbar denotes the compex conugate, and V s the quantzed mode voume. It shoud be ponted out that the eectrc dpoe approxmaton s not sufcent to descrbe near chroptca effects such as optca rotaton, crcuar dchrosm, and crcuar dfferenta scatterng of ht, n sotropc soutons. For these, and severa other processes, hgher-order mutpoe contrbutons need to be ncuded. In vrtuay a cases ths may be satsed by accountng for maetc dpoe and eectrc quadrupoe coupng terms. For the probem of nterest n the present wor, however, n whch a nonnear optca effect s domnant, the eectrc dpoe nteracton term aone s adequate. As a consequence, n many appcatons the eectrc dpoar form of the secondorder response functon, (2) can ead to observabes such as energy shfts, transton rates and sa ntenstes that are arger n matude than that arsng from near optca actvty phenomena that depend on the rotatory strength tensor or the dynamc mxed eectrc maetc dpoe poarzabty, the ast two quanttes dependng on hgher mutpoe moment contrbutons n addton to the eectrc dpoe coupng term. Snce the coupng between radaton and matter s sma, the nteracton Hatonan s consdered as a perturbaton on the nteractng system, enabng a perturbaton theory souton to be effected. SFG s a three-photon process, and the matrx eement s cacuated usng the thrd-order tme-dependent perturbaton theory formua M ð3þ ¼ X fh nt II h II Hnt I IH nt, ð2:5þ ðe I,II I E ÞðE II E Þ where the sum s carred out over a ntermedate states I and II that n to f, and the denomnators denote dfferences n energy between ntermedate and nta states. Evauaton of the matrx eement s factated by the drawng of tme-ordered dagrams, an adaptaton of the faar snge-sded Feyan graphs, and expcated by Ward [25]. An extenson to doubesded Feyan dagrams woud aow the descrpton of pure dephasng phenomena va a densty matrx formasm to be undertaen wthout the need to ncude bath-states n the bass set. The effects descrbed here, however, are fuy captured by the smper formasm due to Ward [25], where one nds that sx topoogcay dstnct dagrams contrbute to the thrd-order matrx eement for SFG. They are depcted n Fgure 1. In ths vsua representaton, tme proceeds from eft to rght. Incomng arrows denote ed actons correspondng to photon absorpton, whe an outgong arrow desates the correspondng emsson nteracton. Intermedate eectronc states are abeed m and n. Note that, overa, energy s conserved subect to E ¼ hc s hc 1 hc 2 ¼ 0, that s at the sum-frequency h! s ¼ hð! 1! 2 Þ: After addng the contrbutons from the sx tme-orderngs, and summng over a moecues,, the tota SFG matrx eement s M sfg hc 3=2 ¼ ð 1 2 s Þ 1=2 ðn 1 n 2 Þ 1=2 e ð 1Þ ð 1 Þ 2" 0 V e ð 2Þ ð 2 Þe ð sþ ð s Þ X e ð 1 2 s ÞR ðþ, ð2:6þ where R s the poston vector of moecue n the space-xed frame, and the moecuar rst hyperpoarzabty tensor,, s gven by ¼ X ( mg ðe m,n mg hc 1 ÞðE ng hc 1 hc 2 Þ mg ðe mg hc 2 ÞðE ng hc 1 hc 2 Þ mg ðe mg hc 1 ÞðE ng hc 1 hc s Þ mg ðe mg hc 2 ÞðE ng hc 2 hc s Þ mg ðe mg hc s ÞðE ng hc 1 hc s Þ ) mg, ð2:7þ ðe mg hc s ÞðE ng hc 2 hc s Þ where the summaton excudes the nta state g.

5 1860 P. Fscher and A. Saam proportona to the square of the number of scatterers. It s evauated n Secton 4. Thus from (2.6), the matrx eement reduces to M sfg ¼N hc 3=2 ð 1 2 s Þ 1=2 ðn 1 n 2 Þ 1=2 e ð 1Þ ð 1 Þ 2" 0 V e ð 2Þ ð 2 Þe ð sþ ð s Þ : ð2:8þ The effect of an externa dc ed on chra SFG s studed n the foowng secton. 3. Matrx eement n the presence of a statc eectrc ed The matrx eement for statc eectrc-ed-nduced SFG from chra moecues s now cacuated. In addton to the two aser beams actng on the moecue, consdered n the prevous secton, a statc eectrc ed s apped, and a photon at the sum-frequency s agan emtted. The nta and na states of the system are agan represented by the ets (2.1). To account for coupng of matter to the statc eectrc ed, whose strength s E, an extra term s added to the nteracton Hatonan (2.3). Ths s of the form H statc nt ðþ ¼ðÞE, ð3:1þ where E ¼ ^EE, wth ^E the unt vector n the drecton of the apped statc eectrc ed. The eadng contrbuton to the matrx eement s now of fourth order. It s computed usng the tmedependent perturbaton theory formua M ð4þ ¼ X f Hnt III hiii H nt II hii H nt I hh I nt : ðe I,II,III I E ÞðE II E ÞðE III E Þ ð3:2þ Fgure 1. Snge-sded Feyan dagrams for sum-frequency generaton. Consder forward scatterng, for whch 1 2 ¼ s, and the photon momenta are conserved. Moreover, the matrx eement s poston ndependent and s N tmes the snge moecue matrx eement when there are N moecues present. The scatterng amptudes from dfferent centers nterfere constructvey, and the forward scatterng s sad to be coherent. Ceary, n ths case the scattered ntensty s At ths order, 24 dagrams contrbute to the matrx eement. They are ustrated n Fgure 2, wth, m and n denotng ntermedate eectronc states of moecue. In ths pctora representaton, the statc eectrc ed s depcted by an ncomng arrow wth zero frequency. Summng over the contrbuton from each of the tme-orderngs yeds, for the statc eectrc ednduced SFG matrx eement, the expresson M esfg ¼N hc 3=2 ð 1 2 s Þ 1=2 ðn 1 n 2 Þ 1=2 2" 0 V Ee ð 1Þ ð 1 Þe ð 2Þ ð 2 Þe ð sþ ð s Þ ^E, ð3:3þ where, as n the contrbuton arsng n the absence of a dc-ed (2.6), the case of forward scatterng

6 Moecuar Physcs 1861 Fgure 2. Snge-sded Feyan dagrams for sum-frequency generaton n the presence of an eectro-statc ed.

7 1862 P. Fscher and A. Saam has been consdered. The 24-term, non-ndex symmetrc, fourth-ran susceptbty tensor s gven by Rate expressons and ntegrated sa ntenstes are computed n the next secton. ¼ X,m,n where once agan the nta state s excuded from the summaton. The optca phase-matchng condtons of the forward scattered statc eectrc ed-nduced contrbuton are dentca to those eadng to the dc-ed free sum-frequency matrx eement (2.8), and both can therefore contrbute to the sa ed. The tota matrx eement s gven by the sum of (2.8) and (3.3), namey the contrbutons to SFG n chra moecues arsng from the statc eectrc ed ndependent and dependent terms, respectvey. Thus M ¼ M sfg ( M esfg ¼N hc 3=2 ð 1 2 s Þ 1=2 2" 0 V ðn 1 n 2 Þ 1=2 e ð 1Þ ½ E ^E Š: ð 1 Þe ð 2Þ ðe hc 1 ÞðE mg hc 1 hc 2 ÞðE ng hc 1 hc 2 Þ ðe hc 1 ÞðE mg hc 1 ÞðE ng hc 1 hc 2 Þ ð 2 Þe ð sþ ð s Þ ð3:5þ 4. Sum-frequency generaton sa ntenstes The transton rate for SFG n chra moecues n the presence or absence of a statc eectrc ed s ready cacuated from the Ferm Goden Rue formua ¼ 2 f h M 2, ð4:1þ where f s the densty of na states of the system. From the matrx eement (3.5), the probabty and the rate are seen to be a sum of three contrbutons: M 2 ¼M sfg ðe hc 2 ÞðE mg hc 1 hc 2 ÞðE ng hc 1 hc 2 Þ ðe hc 2 ÞðE mg hc 2 ÞðE ng hc 1 hc 2 Þ E ðe mg hc 1 ÞðE ng hc 1 hc 2 Þ E ðe mg hc 2 ÞðE ng hc 1 hc 2 Þ ðe hc 1 ÞðE mg hc 1 hc 2 ÞE ng ðe hc 1 ÞðE mg hc 1 ÞðE ng hc 2 Þ ðe hc 2 ÞðE mg hc 1 hc 2 ÞE ng ðe hc 2 ÞðE mg hc 2 ÞðE ng hc 1 Þ E ðe mg hc 1 ÞðE ng hc 2 Þ E ðe mg hc 2 ÞðE ng hc 1 Þ ðe hc 1 ÞðE mg hc 2 ÞE ng ðe hc 1 ÞðE mg hc 2 ÞðE ng hc 2 Þ ðe hc 2 ÞðE mg hc 1 ÞE ng ðe hc 2 ÞðE mg hc 1 ÞðE ng hc 1 Þ E ðe mg hc 1 hc 2 ÞðE ng hc 2 Þ E ðe mg hc 1 hc 2 ÞðE ng hc 1 Þ ðe hc 1 hc 2 ÞðE mg hc 2 ÞE ng ðe hc 1 hc 2 ÞðE mg hc 2 ÞðE ng hc 2 Þ ðe hc 1 hc 2 ÞðE mg hc 1 ÞE ng ðe hc 1 hc 2 ÞðE mg hc 1 ÞðE ng hc 1 Þ ) ðe hc 1 hc 2 ÞðE mg hc 1 hc 2 ÞðE ng hc 2 Þ, ðe hc 1 hc 2 ÞðE mg hc 1 hc 2 ÞðE ng hc 1 Þ ð3:4þ M esfg 2 ¼M sfg 2 M esfg 2 ð4:2þ 2Re M sfg M esfg, where the overbar ses the compex conugate. The rst term corresponds to the probabty for SFG, the second represents the anaogous probabty nduced by the statc eectrc ed, whe the thrd term

8 Moecuar Physcs 1863 s the nterference term between these two aforementoned processes. The contrbuton to the rate from each of these three terms s now evauated separatey. For conventona SFG, n whch emsson occurs n a cone, whose eement of sod ange s d, the nntesma rate s gven by d sfg d ¼ hc " V2 s n 1n 2 N 2 e ð 1Þ ð 1 Þe ð 2Þ ð 2 Þ e ð sþ ð s Þ 2, ð4:3þ on substtutng for the densty of states f ¼ 2 s V d= ð2þ 3 hc, where the anguar bracets denote an orentatona average of the moecuar response tensor (2.7). Denng the mean ntensty of the ncomng beam as I ¼ hn hc2, ¼ 1, 2, ð4:4þ V the SFG ntegrated ntensty s then found to be I sfg d sfg ¼ hc s d ¼ I 1I 2 4 s N " 3 0 c eð 1Þ ð 1 Þe ð 2Þ ð 2 Þ e ð sþ ð s Þ 2 : ð4:5þ As expected, the resut (4.5) reduces to the ntensty for SHG when the two ncomng photons are of dentca mode, and whch then vanshes on orentatona averagng. An sotropc orentatona average s performed on the moecuar tensor n (4.5), as s approprate for a qud, usng the resut [26] 1 ¼ 6 " ", ð4:6þ where " s the Lev Cvta ant-symmetrc tensor, Latn subscrpts refer to Cartesan tensor components n the space-xed frame of reference as before, and Gree ndces denote components n the moecuexed coordnate axes. Thus the SFG ntegrated ntensty s [12] I sfg ¼ I 1I 2 4 s N " 3 0 c " 2 P e ð1þ ð 1 Þ, e ð2þ ð 2 Þ, e ð sþ ð s Þ, ð4:7þ where P s a functon of the poarzaton vectors of the ncomng and emtted radaton, whch f taen to be pane poarzed, s gven expcty by P e ð1þ ð 1 Þ, e ð2þ ð 2 Þ, e ðsþ ð s Þ ¼e ð1þ ð 1 Þ 2 e ð2þ ð 2 Þ 2 e ðsþ ð s Þ 2 e ð sþ ð s Þ 2 e ð 2Þ ð 2 Þe ð 1Þ ð 1 Þ 2 e ð 1Þ ð 1 Þ 2 e ð sþ ð s Þe ð 2Þ ð 2 Þ 2 e ðsþ ð s Þe ð2þ ð 2 Þ e ð 2 Þ ð 2 Þe ð1þ ð 1 Þ e ðsþ ð s Þe ð1þ ð 1 Þ e ð s Þ ð s Þe ð1þ ð 1 Þ e ðsþ ð s Þe ð2þ ð 2 Þ e ð 2 Þ ð 2 Þe ð1þ ð 1 Þ e ð2þ ð 2 Þ 2 e ðsþ ð s Þe ð1þ ð 1 Þ 2 : ð4:8þ The requrement of mutuay orthogona poarzatons cannot be satsed for conear beams, and a nonconear beam geometry s therefore necessary. Dsperson causes s even for conear beams, and a non-conear beam conguraton w further ncrease the phase msmatch,, between the emtted and ncomng waves. The optmum expermenta geometry therefore needs to stre a baance between mzng the phase-msmatch whst aowng for mutuay orthogona poarzaton states [27]. If the drecton of poarzaton of the emtted photon s perpendcuar to ether of the two absorbed photons, then P has an especay smpe form. That s, f e ðsþ ð s Þ?e ð1þ ð 1 Þ and e ð2þ ð 2 Þ, then ony the rst two terms of Equaton (4.8) reman, and P e ð1þ ð 1 Þ, e ð2þ ð 2 Þ, e ðsþ ð s Þ ¼e ðsþ ð s Þ 2 e ð1þ ð 1 Þ 2 e ð2þ ð 2 Þ 2 e ð2þ ð 2 Þe ð1þ ð 1 Þ 2 ð4:9þ Secondy, f e ð 1Þ ð 1 Þ?e ð 2Þ ð 2 Þ then the second, fourth and fth terms of Equaton (4.8) dsappear, and P for ths stuaton s P ¼e ð 1Þ ð 1 Þ 2 e ð 2Þ ð 2 Þ 2 e ð sþ ð s Þ 2 e ð 1Þ ð 1 Þ 2 e ð sþ ð s Þe ð 2Þ ð 2 Þ 2 e ð 2Þ ð 2 Þ 2 e ð sþ ð s Þe ð 1Þ ð 1 Þ 2 : ð4:10þ Thrdy, f the poarzaton vector of the emtted photon s orthogona to e ð1þ ð 1 Þ, whch n turn s perpendcuar to e ð2þ ð 2 Þ, then ony the rst term of the factor (4.8) remans. Now the second term contrbutng to the rate s examned, namey that arsng from the statc eectrc ed-nduced process, whose matrx eement M esfg was gven by (3.3). Anaogousy to (4.5), the ntegrated ntensty s gven by I esfg d esfg ¼ hc s d ¼ I 1I 2 4 s E2 N " 3 0 c e ð 1Þ ð 1 Þe ð 2Þ ð 2 Þ e ð sþ ð s Þ ^E 2, ð4:11þ after substtutng for the densty of states, and the mean ntenstes of the ncomng tme-dependent eds (4.4). Acton of a statc eectrc ed w exert a torque on the moecuar ground state permanent dpoe moment 00, and therefore gve rse to addtona

9 1864 P. Fscher and A. Saam temperature-dependent terms n the orentatona dstrbuton of the sampe moecues. Hence for poar moecues, the tensor Equaton (3.4) and subsequent equatons w aso contan terms of the form ¼ 00 =ðt Þ: It s seen that (4.11) reduces to the ntensty for statc eectrc ed-nduced SHG on substtutng 1 ¼ 2 ¼, and s ¼ 2, usng the formua for the mean ntensty due to two beams of dentca mode from reaton (4.4), I 1 I 2 ¼ I 2 0 ¼ hn 2 ðh 2 c 4 2 =V 2 Þ, and nsertng the second-order coherence factor g ð2þ ¼ nðn 1Þ = hn 2 : Ths produces I eshg ¼ I2 0 4 E 2 N 2 g ð2þ 2 2 " 3 0 c e ðþ ðþe ðþ ðþe ð sþ ð s Þ ^E 2, ð4:12þ n agreement wth earer moecuar QED cacuatons [28] (and Ref. [19], Secton 9.2), where the 24-term,, -ndex symmetrc fourth-ran moecuar response tensor s obtaned from (3.4) on nsertng! 1 ¼! 2 ¼!: Returnng to resut (4.11), a rotatona average s now performed on, whch acs ndex symmetry. Usng standard technques [26], ths s found to be h ¼ 1 30 ð4 Þ ð 4 Þ ð 4 Þ, ð4:13þ where Gree subscrpts refer, once agan, to Cartesan components n the moecue-xed frame. Contractng (4.13) wth the statc and eectromaetc poarzaton vectors n (4.11) resuts n the rotatonay averaged ntensty I esfg ¼ I 1I 2 4 s E2 N " 3 0 c 1 30 f½4ð~e 1 ~e 2 Þð~e s ^EÞð~e 1 ~e s Þð~e 2 ^EÞ ð~e 1 ^EÞð~e 2 ~e s ÞŠ ½ð~e 1 ~e 2 Þð~e s ^EÞ4ð~e 1 ~e s Þð~e 2 ^EÞ ð~e 1 ^EÞð~e 2 ~e s ÞŠ ½ð~e 1 ~e 2 Þð~e s ^EÞð~e 1 ~e s Þð~e 2 ^EÞ 4ð~e 1 ^EÞð~e 2 ~e s ÞŠ g 2 ; ð4:14þ where the subscrpts 1, 2 and s have been used to abe photon modes. The radant ntensty s quadratc n the statc eectrc ed strength, s proportona to the ntenstes of the ncdent eds, and s proportona to the fourth power of the sum-frequency wavevector. From the matrx eement (3.5), the cross-term between SFG wth and wthout the presence of a statc eectrc ed, contrbutng to the thrd term of the probabty (4.2), may be obtaned. It comprses two terms, a coherent and an ncoherent contrbuton. The former s gven by 2Re M sfg M esfg ¼ 2N 2 hc s n 1 n 2 e ð 1Þ ð 1 Þ 2" 0 V e ð 1Þ 0 e ð sþ ð 1Þe ð 2Þ ð s Þe ð sþ ð 0 s Þ E ^E : ð 2 Þe ð 2Þ ð 2Þ 0 ð4:15þ Substtutng for the densty of states, and the mean ntenstes of the two ncdent beams (4.4), the ntegrated sa ntensty for the cross-term s found to be d I coh ¼ hc s d ¼ I 1I 2 4 s N2 E 16 2 " 3 0 c eð 1Þ ð 1 Þe ð 1Þ ð 1Þ 0 e ð 2Þ ð 2 Þe ð 2Þ ð 2Þe ð sþ 0 ð s Þe ð sþ ð 0 s Þ ^E 0 0 0, ð4:16þ whch appes to the moecue n xed orentaton reatve to the two oscatng eds, as we as to the drecton of the statc eectrc ed. The rate and sa ntensty for sotropc moecues s computed from (4.16) usng the resuts for the orentatonay averaged moecuar response tensors (4.6) and (4.13), and s hd I coh ¼ hc s d ¼ I 1I 2 4 s N2 E " 3 0 c " 0 0 0eð 1Þ ð 1Þ 0 e ð 2Þ ð 2Þe ð sþ 0 ð 0 s Þ" nh 4ðe 1 e 2 Þðe s ^E Þðe 1 e s Þðe 2 ^E Þ h ðe 1 ^E Þðe 2 e s Þ ðe 1 e 2 Þðe s ^E Þ 4ðe 1 e s Þðe 2 ^E Þðe 1 ^E Þðe 2 e s Þ h ðe 1 e 2 Þðe s ^E Þðe 1 e s Þðe 2 ^E Þ o 4ðe 1 ^E Þðe 2 e s Þ : ð4:17þ The ntegrated ntensty s proportona to the ntenstes of each aser beam, and to the fourth power of the wavevector of the sum-frequency, but s ony neary proportona to the strength of the statc eectrc ed.

10 Moecuar Physcs Incoherent eectrc-ed-nduced sum-frequency and second-harmonc generaton It s nterestng to note that, n addton to the coherent optca processes consdered thus far, there aso exsts an ncoherent nonnear scatterng ntensty that depends neary on an eectro-statc ed, and that has not yet been observed. The coherent cross-term between the chray senstve SFG term and the achra eectrc ed-nduced sum-frequency generaton (ESFG) contrbuton s, n a qud, proportona to the product of the correspondng rotatonay averaged tensors, and, as mped by the ast term on the rght of Equaton (4.2), and gven expcty by resut (4.17). Ths eads to the predcton that a correspondng ncoherent term w arse, consstng of the rotatonay averaged seventh-ran tensor formed by the product of the thrd-ran sum-frequency tensor, and the fourth-ran eectrc-ed-nduced sum-frequency response tensor. In the eectrc-dpoe approxmaton, ths tensor s odd under party and may as such gve rse to a pseudoscaar, characterstc of a chra observabe quantty. Due to the engthy nature of the ensung expressons n the evauaton of the sotropc ncoherent term, ony an outne of ts dervaton w be gven, wth an expct form beng presented for two partcuar conguratons of ncomng oscatng and statc, and emtted eds. For moecues n xed reatve orentaton to the two aser eds, the statc eectrc ed and the scattered radaton, the ncoherent contrbuton to the statc ed-nduced sum-frequency ntensty s proportona to the number of scatterers, and s gven by I ncoh ¼ I 1I 2 4 s NE 16 2 " 3 0 c eð 1Þ ð 1 Þe ð 1Þ ð 1Þe ð 2Þ 0 ð 2 Þ e ð 2Þ ð 2Þe ð sþ ð 0 s Þe ð sþ ð s Þ ^E : 0 ð5:1þ To obtan the sa for sotropc moecues requres performng an orentatona average over the seventh-ran Cartesan tensor product of moecuar response functons ¼ I ð7þ, ð5:2þ where I (7) expresses neary ndependent combnatons of sotropc seventh-ran somers n the space- and moecue-xed axes. It taes the form 0 1T 0 1 " " : : I ð7þ ¼ B : C M ð7þ : B : : A : " " ð5:3þ The rst and thrd factors st n row and coumn format the 36 neary ndependent sotropc somers of ran seven n the two frames of reference. They may be obtaned from Tabe III of Ref. [26]. The mdde factor n the ast formua s a matrx of coefcents. It s gven expcty by Equaton (34) of the aforementoned Ref. [26]. Evauatng the matrx products n (5.2) and contractng wth the poarzaton factors n (5.1) yeds the ncoherent contrbuton to the scattered ntensty. Sxteen of the 36 terms arsng from contracton of tensor products n the space-xed coordnate system are found to vansh on notng the ndex symmetry propertes of the poarzaton product, and the ant-symmetry property of the Lev Cvta tensor. Whe ths greaty reduces the overa number of terms appearng n the na answer, t s st too ong to gve here or as Suppementary Matera. The fu expresson may be obtaned from ether of the authors. Instead, a few speca conguratons of dynamc and statc eds are consdered, whch resut n consderaby reduced formuae. An especay smpe answer resuts when the two ncomng oscatng eds are perpendcuar to each other, and both are mutuay orthogona to the poarzaton vector of the photon emtted at the sum-frequency. In ths case, the ntegrated sa ntensty vanshes. The resut for two other conguratons of nterest, correspondng to ether one of the ncomng beams beng poarzed perpendcuar to the emtted radaton, whch n turn are both orthogona to the drecton of the statc eectrc ed, s gven n the appendx. Returnng to expresson (5.1), t s straghtforward to cacuate the ncoherent sa ntensty for the cross-term arsng due to statc ed-free and statc ed-nduced SHG n chra moecues by settng e ð 1Þ ð 1 Þ¼e ð 2Þ ð 2 Þ¼e ðþ ðþ, yedng I shg ncoh ¼ I2 0 4 g ð2þ NE 2 " 3 0 c e ðþ ðþe ðþ 0 ðþeðþ ðþe ðþ ðþ 0 e ð sþ ð s Þe ð sþ ð 0 s Þ ^E 0 0 0, ð5:4þ where s ¼ 2 has been empoyed agan. Interestngy, on orentatona averagng subect to mutuay perpendcuar ncomng, emtted and statc eds, the above resut vanshes. A seventh-ran average s necessary, as t was for the correspondng SFG nterference term outned earer. Vanshng of ndvdua contrbutons to the average s due to ndex symmetry assocated wth dentca nput beam poarzatons and ndex asymmetry of the Lev Cvta tensor, or from consderatons of expermenta ed geometry. That (5.4) vanshes on averagng s n contrast to hyper-rayegh scatterng, whch exsts for SHG n any qud, as we as ts statc eectrc-ed-nduced anaogue. It s aso worth

11 1866 P. Fscher and A. Saam pontng out that chra hyper-rayegh scatterng can be observed by a sxth-ran observabe formed by the product of two SHG hyperpoarzabtes, where, however, one of the transtons s maetc-dpoe aowed [18]. 6. Concusons Wthn the framewor of non-reatvstc quantum eectrodynamcs, the matrx eement has been cacuated for SFG from chra moecues both wth and wthout the presence of an apped statc eectrc ed. These contrbutons arose from thrd- and fourth-order perturbaton theory, wth evauaton of the probabty amptude factated by the drawng of tme-ordered dagrams. Integrated sa ntenstes were then computed va the Ferm Goden rue transton rate and apped to stuatons n whch the optcay actve speces was n a xed orentaton reatve to ncomng and emtted radaton, or was competey randoy orented. SFG from a chra moecue n souton depends on the rst hyperpoarzabty, a pseudoscaar quantty that changes s when one enantomer s repaced by ts mrror mage form. The ntegrated ntensty aso depends on the ntensty of each mpngng beam, and on the fourth power of the sumfrequency wavevector [6]. Appcaton of a statc eectrc ed to a chra souton undergong SFG was aso studed. In ths case, the sa ntensty s found to depend on the moecuar second-order hyperpoarzabty tensor, a quantty that s proportona to the thrd-order poarzaton. In addton to dependng neary on the product of aser ntenstes, as we as on the fourth power of the sum-frequency wavevector, the ntegrated ntensty n ths case s proportona to the square of the strength of the statc eectrc ed. A usefu spectroscopc probe of optcay actve quds s provded by the cross-term of the dc-ed-free and statc-ed-dependent terms. It s composed of a coherent contrbuton and an ncoherent term. The former survves random orentatona averagng, s neary proportona to the statc eectrc ed strength, and depends on the product of the moecuar secondand thrd-order susceptbtes. It permts the determnaton of the absoute conguraton of a chra souton va a purey eectrc dpoar process, as has been observed expermentay. A concomtant chray senstve contrbuton to the ntensty of scattered SFG s predcted to occur wth ncoherent ht scatterng n the presence of an eectro-statc ed, athough the anaogous SHG process vanshes on sotropc averagng and s therefore unobservabe for chra quds. References [1] J.A. Gordmane, Phys. Rev. 138 A, 1599 (1965). [2] M.A. Ben, T.A. Kuaov, K.-H. Ernst, L. Yan and Y.R. Shen, Phys. Rev. Lett. 85, 4474 (2000). [3] M.A. Ben, S.H. Han, X. We and Y.R. Shen, Phys. Rev. Lett. 87, (2001). [4] P. Fscher, K. Becwtt, F.W. Wse and A.C. Abrecht, Chem. Phys. Lett. 352, 463 (2002). [5] P.K. Yang and J.Y. Huang, J. Opt. Soc. Am. B 15, 1698 (1998). [6] D.L. Andrews and I.D. Hands, J. Phys. B 31, 2809 (1998). [7] P. Fscher, A.D. Bucngham and A.C. Abrecht, Phys. Rev. A 64, (2001). [8] P. Fscher, F.W. Wse and A.C. Abrecht, J. Phys. Chem. A 107, 8232 (2003). [9] A.J. Moad and G.J. Smpson, J. Phys. Chem. B 108, 3548 (2004). [10] G.J. Smpson, Chem. Phys. Chem. 5, 1301 (2004). [11] M.A. Ben, Y.R. Shen and R.A. Harrs, J. Chem. Phys. 120, (2004). [12] S. Cheon and M. Cho, Phys. Rev. A 71, (2005). [13] S. Cheon and M. Cho, J. Phys. Chem. A 113, 2438 (2009). [14] O. Rosya and S. Muame, Mo. Phys. 107, 265 (2009). [15] A.D. Bucngham and P. Fscher, Chem. Phys. Lett. 297, 239 (1998). [16] P. Fscher, A.D. Bucngham, K. Becwtt, D.S. Wersma and F.A. Wse, Phys. Rev. Lett. 91, (2003). [17] P. Fscher and F. Hache, Chraty 17, 421 (2005). [18] V. Ostroverhov, R.G. Petsche, K.D. Snger, L. Suhonova, R.J. Tweg, S.-X. Wang and L.C. Chen, J. Opt. Soc. Am. B 17, 1531 (2000). [19] D.P. Crag and T. Thrunamachandran, Moecuar Quantum Eectrodynamcs (Dover, New Yor, 1998). [20] D.L. Andrews and P. Acoc, Optca Harmoncs n Moecuar Systems (Wey-VCH, Wenhem, 2002). [21] A. Saam, Int. Rev. Phys. Chem. 28, 405 (2008). [22] A. Saam, Moecuar Quantum Eectrodynamcs (Wey, New Jersey, 2010). [23] E.A. Power and S. Zenau, Ph. Trans. R. Soc. London A 251, 427 (1959). [24] R.G. Wooey, Proc. R. Soc. London A 321, 557 (1971). [25] J.F. Ward, Rev. Mod. Phys. 37, 1 (1965). [26] D.L. Andrews and T. Thrunamachandran, J. Chem. Phys. 67, 5026 (1977). [27] P. Fscher, D.S. Wersma, R. Rghn, B. Champae and A.D. Bucngham, Phys. Rev. Lett. 85, 4253 (2000). [28] Y.T. Lam and T. Thrunamachandran, J. Chem. Phys. 77, 3810 (1982).

12 Appendx A: Incoherent contrbuton to the ed-free statc ed-nduced SFG sa ntensty Conguraton n whch e ð1þ, e ðsþ and ^E are mutuay perpendcuar The sotropc sa ntensty for the ncoherent contrbuton to the cross-term between statc ed-free and statc ednduced sum-frequency generaton when e ð1þ ð 1 Þ? e ðsþ ð s Þ? ^E s gven by Moecuar Physcs 1867 I 1 I 2 4 s NE " 3 0 cð" e ð1þ e m ð1þ eð2þ e ð2þ m ^E f½" ð Þ" ð Þ " ð Þ " ð Þ" ð5 5 ÞŠ ½3" " ð3 9 ÞŠ ½" " " ÞŠ ½" " ð 7 Þ2ð" " ÞŠ g " e ð1þ e ð1þ e ð2þ e ð2þ m ðsþ e eðsþ m ^E f½" ð Þ" ð Þ " ð Þ" ð 5ð ÞÞŠ ½" " ð ÞŠ ½9" 3" ð ÞŠ ½7" " ð Þ2" ð ÞŠ g " e ð1þ e ð1þ ðsþ e eðsþ ^E m eð2þ e ð2þ m f½" ð Þ" ð Þ" ð Þ " ð ÞŠ ½5" " ð5 9 ÞŠ ½9" " ð9 19 ÞŠ ½19" " ð19 37 Þ14ð" " ÞŠ g " e ð1þ e ð1þ ðsþ e ^E m eð2þ e ð2þ m e ðsþ f½" ð Þ" ð Þ" ð Þ " ð ÞŠ ½7" " ð7 17 ÞŠ ½7" " ð7 17 ÞŠ ½21" " ð21 51 Þ18ð" " ÞŠ gþ:

13 1868 P. Fscher and A. Saam Conguraton n whch e ð2þ, e ðsþ and ^E are mutuay perpendcuar The sotropc sa ntensty for the ncoherent contrbuton to the cross-term between statc ed-free and statc ed-nduced sum-frequency generaton when e ð2þ ð 2 Þ? e ðsþ ð s Þ? ^E s gven by I 1 I 2 4 s NE " c " e ð1þ e m ð1þeð2þ e m ð2þ ^E 1 " " " " " 5 5 3" " ð3 9 Þ " " " Þ " " ð 7 Þ2 " " " e ð1þ e m ð1þ ðsþ e eðsþ ^E m " " " " " " " 14" ð Þ 14" " ð22 34 Þ" ð2 12 Þ14" " m e ð1þ e ð1þ ðsþ e eðsþ ^E m " " ð " " " " ð5 11 Þ 5" " ð5 11 Þ C A 15" " ð15 33 Þ12ð" "