Module III, Lecture 02: Correlation Functions and Spectral Densities

Transcription

1 Modue III, Lecture : orreaton Functons and Spectra enstes The ony part of Boch Redfed Wangsness spn reaxaton theory that we coud say nothng about n the prevous ecture were the correaton functons. They are, n genera, unknown, but the smpe and popuar cases of sotropc and ansotropc rotatona dffuson w be treated n ths ecture. orreaton functons sotropc rotatona dffuson It was demonstrated n the prevous ectures that, n the case of rotatonay moduated spn nteractons n rgd moecues, the ansotropc part of the spn Hamtonan aways has the foowng form: Hˆ t t Qˆ (), where t are second rank Wgner functons that depend on moecuar orentaton (and therefore tme) and Q ˆ are rotatona bass operators. Under the assumpton that the nose n the system s statonary, the most genera form of a correaton functon between two Wgner functons s: * * G t t () Spn ynamcs, Modue III, Lecture r Iya Kuprov, Unversty of Southampton, 3 (for a ecture notes and vdeo records see where the anguar brackets denote the ensembe average, e.g. the average over a the dfferenty orented moecues n a qud sampe. Such ensembe averages may be recast n terms of probabty dstrbutons, for exampe: x xp x dx (3) p x s the ensembe probabty densty functon for the vaues of x. In ths way, the ensembe where average of the product of dfferenty tmed Wgner functons n Equaton () can be wrtten n terms of ntegras of probabtes of encounterng systems n a partcuar orentaton: G * * P P,,d d (4) s the probabty densty of moecues havng an orentaton at tme, havng had an orentaton at tme zero. Probabty denstes P and P,, may be thought of as fractona concentratons they gve the concentraton of moecues n a partcuar ocaton n the orentaton space. where P s the probabty densty of nta orentatons and P,, We w assume n ths secton that the moecues n our sampe are spherca partces undergong sotropc rotatona dffuson. That means that the nta probabty dstrbuton P s unform, meanng that P /, and so the ntegra smpfes to: G P d d There w be dffuson from one orentaton nto another, and we w assume that P obeys the sotropc rotatona dffuson equaton: *,, (5) P, ˆ ˆ ˆ ˆ, LX LY LZ P,, L P,, (6) where s the rotatona dffuson constant and L ˆ X, Lˆ ˆ Y, L Z are the anguar momentum operators. Because Wgner functons are egenfunctons of the tota momentum operator ˆL :

2 ˆ L (7) t s reasonabe to seek a souton to Equaton (6) n terms of Wgner functons: m A () P,,, If we substtute ths souton back nto Equaton (6), we get the expresson for the tme dependent coeffcents A,: A, A, m m (9) A, A, A, e A The genera souton therefore acqures the form: () P,, e m where are the coeffcents determned by the nta condton. The competeness condton for Wgner functons of nteger rank s: * () m and therefore the nta condton (a systems pontng at at tme zero) yeds: m * P,, * e m () We can now use the orthogonaty condton for Wgner functons: to compute the ntegra n Equaton (5): * d, k m, m (3) * Gabcd ab cd P,,d d e d d * * ab cd m e d d * * ab cd m e, a m, b, c m, d m 5 e ac, bd, 6 And so the fna expresson for the correaton functon n the case of sotropc rotatona dffuson s: (4) Spn ynamcs, Modue III, Lecture r Iya Kuprov, Unversty of Southampton, 3 (for a ecture notes and vdeo records see

3 * ac, bd, 6,, acbd Gabcd ab cd e e kt 4r 3 r 6 3kT (5) where s known as the rotatona correaton tme, s the dynamc vscosty of the sovent and r s the hydrodynamc radus of the moecue, whch s assumed to be spherca. The equaton connectng the rotatona dffuson constant to the partce radus, sovent vscosty and temperature s derved from the Stokes aw. Equaton (5) s the most common rotatona correaton functon n current use. orreaton functons rotatona dffuson of a symmetrc top For a symmetrc top moecue, the rotatona dffuson equaton acqures the foowng form: P, ˆ ˆ ˆ, LZ LX LY P,, (6) where and are the axa and equatora rotatona dffuson constants of what s now assumed to be an epsoda partce, and L ˆ X, Lˆ ˆ Y, L Z are the same anguar momentum operators as above. Wgner functons are st a convenent choce, because they are aso egenfunctons of the dffuson operator n the square brackets n Equaton (6): ˆ ˆ ˆ ˆ ˆ ˆ ˆ LZ LX L Y LZ LX LY LZ (7) m, m m Ths beng the ony dfference from the entre dervaton gven above (everythng ese fows through dentcay), we can concude that n the case of the axay ansotropc rotatona dffuson: * ac, bd, 6b Gabcd ab cd e () 5 where the rotatona dffuson constants are reated to the two rad of the epsod n a way that drecty foows from Equaton (5): kt kt, (9) r 3 3 r Importanty, Equaton (6) mpcty assumes the moecuar frame of reference to be the egenframe of the rotatona dffuson tensor ths condton must be observed durng the spn system setup for the reaxaton theory cacuatons. orreaton functons ansotropc rotatona dffuson In the fuy ansotropc case we have: P,, ˆ ˆ ˆ XXLX YYLY ZZL Z P,, () n whch XX, YY, ZZ are now the three (assumed dfferent) egenvaues of the rotatona dffuson tensor. The Wgner functons are no onger the egenfunctons of the dfferenta operator n the square brackets, but they are not very far from them, because: Spn ynamcs, Modue III, Lecture r Iya Kuprov, Unversty of Southampton, 3 (for a ecture notes and vdeo records see

4 Lˆ Z m ˆ ˆ ˆ L L L m m X Y k, m () Wth some effort (verfabe by back substtuton foowed by appcaton of Equatons ()), we can obtan the second rank egenfunctons of the fuy ansotropc rotatona dffuson operator n the bass of second rank Wgner functons: m 4XX YY ZZ XX YY ZZ 4 3 XX YY ZZ 4 4 XX YY ZZ 5 XX YY ZZ where the m ndex now runs pany from to 5 projecton ranks have been mxed n the new egenfunctons, and therefore m[,] ndexng s no onger approprate. The new shorthand symbos are defned as foows: XX YY ZZ 3 XX YY ZZ XX YY XX ZZ YY ZZ Other ranks w make an appearance n the equatons beow, but woud not eventuay be necessary. Smary to the treatment before, the genera souton to Equaton () s: XX YY () P,, e m k Because the egenfunctons n the tabe above have been normazed, the competeness reaton remans the same as Equaton () and for the case where a systems are ntay ponted at we get: * P,, e (3) m k Keepng n mnd the Wgner functon orthogonaty reatons n Equaton (3), we can now take the ntegra n Equaton (5): Spn ynamcs, Modue III, Lecture r Iya Kuprov, Unversty of Southampton, 3 (for a ecture notes and vdeo records see

5 G e d d * * abcd ab cd k m * * * e ab d cd d k m 5 * * achmbhmd e,kahmb,kchmd e k m k 5 where the m ndex on the egenvaues has been dropped (they dd not depend on t anyway) and the superscrpt ndcatng second rank has been omtted because t s no onger agebracay reevant. The new constants h mb may be computed by brute force and depend on ther ndces n the foowng way: m \b 3 (4) 4 5 Spectra densty functons Let us now ook at a specfc eement of the reaxaton superoperator n Hbert space n a representaton where the statc Hamtonan s dagona (ths s a sgnfcant assumpton t s not aways feasbe, for purey numerca reasons, to dagonaze a Hamtonan): ˆ Rˆ ˆ G ˆ Qˆ e Qˆ e ˆ d H ˆ Hˆ Tr [,[, ]] (5) a ab b a b Usng the cycc permutaton rue, we can reorder the commutators under the trace: ˆ ˆ ˆ ˆ ˆ H ˆ H H ˆ H ˆ ˆ ˆ ˆ ˆ a Q m e Qe b e Qe a b Q Hˆ ˆ (, ) (, ) =Tr ˆ H ˆ ab, ˆ ab e Q ˆ ˆ ˆ e P Pk, m [ a,[ b, Q]] Tr [,[, ]] Tr [,[, ]] The trace of a product of four matrces can be computed expcty n the ndexed notaton: Tr e Qˆ e Pˆ e Qˆ e Pˆ Hˆ Hˆ ˆ ˆ ( ab, ) H H ( ab, ) r rs sj rsj ˆ (, ) (, ) j ˆ ab j ˆ ˆ ab j ˆ ˆ( ab, ) e Q e P e Q P e Q P j j j j j j j j j j (6) (7) and the ntegra becomes a coecton of Fourer transforms: Spn ynamcs, Modue III, Lecture r Iya Kuprov, Unversty of Southampton, 3 (for a ecture notes and vdeo records see

6 a ˆ ˆ ˆ( ab, ) (, ) R ˆ ˆ ˆ ab b Q P G e d Q P j j J j j j j j J G e d, j j j ab ˆ The Fourer transform of the correaton functon s caed the spectra densty functon. It has a physca meanng of energy densty of the stochastc perturbaton at a gven frequency. In the sotropc tumbng approxmaton we have: / J e e d (9) / The rea part of ths functon s a Lorentzan curve. It determnes the reaxaton rate (t woud acqure a mnus when t appears n the Louve von Neumann equaton). The (usuay much smaer) magnary part contrbutes to the frequency part of the LvN equaton and for that reason s known as dynamc frequency shft. The FS s usuay gnored n practca smuatons. Spectra denstes n non rgd moecues: the Lpar Szabo mode Anaytca smpcty s a rare vrtue n reaxaton theory, and t s probaby the ony vrtue of the Lpar Szabo restrcted oca moton mode. We can observe that, n a spherca moecue where the overa rotatona dffuson s ndependent from the restrcted oca dffuson of a partcuar group, the correaton functon can be factored as goba oca () G G G (3) Lpar Szabo approxmaton assumes that both functons are exponenta and the resutng spectra densty functon therefore s: S S J, (3) nt where S s caed order parameter. It can be nterpreted as a fracton of the fu body ange that s spanned by the restrcton cone of the nterna moton. Ths form for the spectra densty functon s wdey used n proten NMR spectroscopy. Transatona dffuson The reaxaton theory derved n the prevous ecture specfcay assumed rotatonay moduated nteractons because a compete and eegant reaxaton theory s ony avaabe for the rotatonay moduated case. Speca cases for transatonay moduated nteractons are aso anaytca and we w consder a smpe case of a transatonay moduated scaar nteracton n ths secton. onsder a correaton functon of the amptude mutper taken at two dfferent tmes n a system governed by sotropc transatona dffuson: f t of a dstance dependent nteracton G f t f t f f (3) From the same probabty argument as we used for the rotatona dffuson n Equaton (4), we get: G f r f r Pr Pr, r, drdr 3 3 (33) Spn ynamcs, Modue III, Lecture r Iya Kuprov, Unversty of Southampton, 3 (for a ecture notes and vdeo records see

7 s the probabty densty of a system havng a dstance r at tme, havng had a dstance r at tme zero. We w assume that the where Pr s the probabty densty of dstances and Pr, r, probabty dstrbuton P s unform, and so the ntegra smpfes to 3 3 G fr frpr, r,drdr V (34) where V s the sampe voume. The probabty densty Pr, r, may be thought of as fractona concentraton t gves the concentraton of systems wth a partcuar vaue of the dstance. There w be dffuson from one dstance nto another, and we w assume that P obeys the dffuson equaton: P r r P r r P r r r r,,,,,,, where s the reatve dffuson constant and the hard sphere boundary condton: P (36) r r d where d s the dstance of cosest approach. There are no smpe anaytca soutons for the correaton functon n Equaton (34), but varous mts and approxmatons are avaabe n the terature. (35) Spn ynamcs, Modue III, Lecture r Iya Kuprov, Unversty of Southampton, 3 (for a ecture notes and vdeo records see