Molecular Spectroscopy and Group Theory Chemstry 630 Bruce Johnson I. Interacton of Lght and Matter Molecular Spectroscopy uses lght to probe structure and dynamcs of molecules. Most spectroscopy focuses on structure, e.g., electronc state quantum numbers, vbratonal force constants, equlbrum geometres and moments of nerta. Increasngly, experments and theory are also probng dynamcs,.e., tme development of quantum probablty ampltudes generated by photolytc means. Most of ths course wll deal wth the structure aspects. Quantzed feld treatment: The most general treatment of the nteracton of an assembly of charged partcles, e.g., electrons and nucle, wth an external electromagnetc feld requres quantum electrodynamcs. The full Hamltonan for the system s schematcally H H matter + H feld + H nteracton (I.) H matter s the feld-free Hamltonan of the materal system. H feld s the Hamltonan of the free radaton feld n the absence of the matter. Transtons are caused by H nteracton. Absorpton corresponds to the external feld gvng up a quantum of energy (photon) n a partcular mode to the matter. Emsson s the opposte process. Both stmulated and spontaneous emsson are allowed. There are cases, e.g., spontaneous emsson and multphoton exctaton, where t s more mportant to consder the radaton feld quantum mechancally. Semclasscal feld treatment: The external feld s treated classcally, so the Hamltonan becomes explctly tme-dependent. The Hamltonan for a system of charged partcles (electrons and nucle) n an external electromagnetc feld becomes H " p # q A r,t m [ ] +, j + "V Coulomb r # r j " q $ r,t H matter + H nteracton (I.) Here the vector potental A and the scalar potental ϕ satsfy Maxwell's Equatons E " #$ " %A %t, B # & A (I.3)
For many spectroscopc applcatons, t s suffcent to consder the felds semclasscally. Example: Sngle mode electromagnetc feld (Propagaton along z-axs, lnear polarzaton along x-axs) Sutable vector and scalar potentals n MKS unts are A( r,t) " z ˆ c E 0 x x cos (# t " k z) (I.4) "( r,t) # E 0 x x cos ( $ t # k z) (I.5) Inserton nto Maxwell's equatons leads to E( r,t) x ˆ E 0 x cos (" t # k z) (I.6) B( r,t) y ˆ c E 0 x cos (" t # k z) (I.7) ω : angular frequency (radans / sec) ν ω / π : crcular frequency (cycles / sec, Hertz) k : wave vector λ : wavelength ( k k π / λ ω / c) " ν / c : wave number [ / λ (cm) ] In MKS unts, the magntudes of the electrc and magnetc felds dffer by c. x E y B z Fgure I-
General electromagnetc felds: We can also have crcularly- or elptcally-polarzed EM felds usng approprate lnear combnatons of exp[+(ω t k. r)] and exp[ (ω t-k. r)]. In addton, we may Fourer-analyze more general, non-monochromatc, felds as sums or ntegrals over many (k,ω) modes. Energy: The classcal electromagnetc feld transports energy. For the monochromatc EM wave above, the ntensty (energy flow per sec per unt area) s gven by the Poyntng vector S(r, t) S µ 0 E " B, S S µ 0 c E 0 x cos (# t $ k z) (I.8) whch "poynts" n the drecton of propagaton (postve z n Fg. I-). Here µ 0 s the magnetc permeablty of free space, 4π/07. The cycle-averaged ntensty s S µ 0 c E 0 x (I,9) Force: The Lorentz force on a partcle (charge q and velocty v) due to the EM feld s F qe + q v " B (I.0) The two terms dffer n magntude by v /c. The lghtest charged partcles we worry about are electrons, and the rato v /c for an electron n the lowest Bohr orbt of the hydrogen atom s α /37. Thus, the electrc feld provdes the domnant force n EM radaton felds. Multpole Expansons The nteracton of an external electromagnetc feld wth an assembly of charged partcles, e.g., an atom or molecule, s frequently descrbed by terms n the Hamltonan contanng multpole moments and powers of the felds. Frst, let us ntroduce the center of mass of the molecule, R " m M r, M " m (I.) Then expand the vector and scalar potentals n the Hamltonan of Eq. (I.) about R, H " p # q A R,t m [ # q ( r.$)a ( R,t) #...] + "V Coulomb r # r j, j + " q # R,t + " q ( r.$)# R,t +... (I.3) 3
At ths pont, t s convenent to perform a untary (gauge) transformaton ntally consdered by Mara Göppert-Meyer n the 930's to express everythng n terms of the E and B felds at the center of mass. Later extended by others, t s now called the Power-Zenau-Woolley transformaton, and expresses the Hamltonan n the Multpolar Gauge as H " p m + "V Coulomb r # r j, j +q tot "( R,t) #µ $E( R,t) # ( ' Q %& %,&x,y,z (% E & ( R,t) +... " $ µ ( m) + M µ e % & #P ' ( ) * B R,t " B( R,t) * µ m $ + M µ ' # P % & ( ) (I.4) where the felds are evaluated at R and the multpole moments are moments of charge and current n the molecule. Electrc monopole moment (E0 -- total charge) q tot " q (I.5) Electrc dpole moments (E) µ " q ( r # R) µ x " q ( x # X), µ y " q ( y #Y), µ z " q ( z # Z) (I.6) Electrc quadrupole moments (E) Q x x " q ( x # X), Q x y " q ( x # X) ( y #Y),... (I.7) Magnetc dpole moments (M) µ ( m) " q ( r # R)$ % p # P ( ' * (I.8) & m M ) 4
Here P s the momentum operator for moton of the center of mass. In the absence of magnetc felds, one can separate center of mass moton from the other coordnates n H. Otherwse, couplng terms n the last lne of Eq. (I.4) prevent complete separaton. The multpole moments depend only on degrees of freedom ndependent of the center of mass and are approxmately calculable by ordnary quantum chemstry methods. Magnetc monopoles (M0) have so far not been observed n nature. Spn: Snce the electrons and nucle also have spn, extra terms appear n the molecular Hamltonan and multpole moments. For example, n addton to the magnetc dpole terms n (3) correspondng to orbtal moton of the charges, there are also ntrnsc contrbutons arsng from the spn s of each partcle µ ( m) " q ( r # R)$ % p # P ( ' * + " q g s & m M ) m (I.9) where g s a constant whch, for electrons, s very close to. In a related ven, the nucle n the molecule may possess permanent electrc quadrupole moments whch must be added to (I.7). We wll not worry further about these terms for now. Magntudes of laboratory felds: For unform DC electrc felds, E 0 8 Volts / m s a large feld (near typcal thresholds for gas breakdown due to onzaton). The atomc unt of electrc feld strength s the feld experenced by the electron of the hydrogen atom n ts frst Bohr orbt due to the proton -- 5 0 Volts / m. For unform magnetc felds, 0 5 Gauss s large for macroscopc felds, whereas an atomc-scale feld s 0 9 Gauss. Laboratory felds are mnor perturbatons n these cases. A smlar comparson holds for CW lasers nstead of DC felds. However, fast-pulse lasers are now capable of delverng very hgh peak ntenstes, at least for short perods of tme. Not surprsngly, the chef effect seems to be to tear molecules apart va onzaton. Electrc dpole approxmaton: The magnetc dpole (M) nteractons tend to be reduced by a factor of α ~ /37 relatve to the elctrc dpole (E) nteractons. For E nteractons, we expect reducton n sze by k r where r s some average dstance of partcles from the center of mass,.e., usually only a few Angstroms. Even for wave numbers n the ultravolet regon of the spectrum, k r s small. For example, at a wavelength λ 60 nm, k π / λ 0.004 Å. Thus the fact that optcal wavelengths are usually much larger than molecular dmensons means that optcal spectroscopy s domnated by the E term unless the molecular electrc dpole happens to be very small. The electrc dpole approxmaton neglects all nteractons above E, resultng n H el"dp # p m + #V Coulomb r " r j, j + q tot $ R,t "µ e %E R,t (I.0) 5
-- The second-to-last term also vanshes f the molecule s neutral (q tot 0), leavng only the dpole moment term. -- The latter term s frequently smplfed by statng that the orgn of the coordnate system s the molecular center of mass. Ths gves the common smplfed Hamltonan for a neutral molecule H el"dp "µ e # p m + #V Coulomb r " r j, j $E 0,t H 0 + H' (I.) Here H 0 s the feld-free molecular Hamltonan and H' s the term couplng the molecular coordnates to the external feld. Ths s the startng pont for most semclasscal treatments of nteracton of a molecule or atom wth an EM feld, and t s what we focus on. If we need to nclude translatonal moton, we go back a step. Tme-dependent perturbaton theory How does one solve Eq. (I.) n general? The tme-dependent Schrödnger equaton for ths system s $ H 0 + H'"h # ' & )* r % #t(,r,...,r N,t 0 (I.) H 0 s assumed to have orthonormal egenfunctons ψ k (0) and egenenerges E k (0), ( 0) [ H 0 " E k ] # ( 0) k r,r,...,r N 0, (I.3) provdng a sutable bass for expanson of the full wave functon: "( r,r,...,r N,t) # c k t k [ t /h ] " ( 0) k r,r,...,r N exp $ E k 0 (I.4) All the tme dependence of the wave functon appears n the coeffcents, whch must now be determned. Ths s the Drac varaton of constants procedure. The exponentals are explctly ncluded for convenence (they could be ncluded n the c k ). Insertng ths Ansatz nto Eq. (I.) and makng use of Eq. (I.3), 6
$ H 0 + H'"h # ' & )* r % #t(,r,...,r N,t ( 0) " exp # E k t /h k k [ ] c k ( t 0 { )[ H 0 + H'#E k ] # hc k( t )} $ k [ ] c k t ( 0) " exp # E k t /h [ H'# hc k ( t) ] 0 $ k r,r,...,r N ( 0) r,r,...,r N 0 (I.5) Ths can be multpled by ψ m (0)* and ntegrated over all coordnates (usng the orthonormalty of the wave functons) to obtan a system of tme-dependent equatons for the c k, ( 0) " exp # E k t /h k [ ] c k t [ m H' k # hc k( t)$ m k] 0 (I.6) hc k ( t) " c k ( t)exp # m k t k m H' k, # m k E ( 0) m $ ( 0) Ek h (I.7) Now let us assume the monochromatc feld used before (wth E x 0 " F x 0 to avod confuson), neglectng the spatal dependence of the cosne term n the long wavelength lmt E( r,t) " x ˆ F 0 x cos (# t) (.8) H' "µ # E( r,t) $ " µ 0 x Fx cos (% t) (I.9) If the molecule starts n state n, then c k (0) δ kn. Then the sum reduces to a sngle term (k n) and we fnd to a frst approxmaton that the tme-ntegral of (I.7) between 0 and t yelds c m ( t) " m n + t # exp( $ h 0 m n t' ) m H' n dt' " # m n $ h F 0 x m µ x n cos & t' t % exp & 0 m n t' dt' " m n # h F 0 x m µ x n e % t' #% t' ( + e ) dt' t $ exp % 0 m n t' " m n # h F 0 x m µ x n # $ m n + $ % exp $ m n t + $ t ' ' & # + exp $ m n t # $ t $ m n #$ ( * * ) (I.30) 7
Suppose ω mn > 0 [E (0) m > E (0) n ]. If ω ω mn, the last term, whch comes from the exp(-ω t) term of the cosne, domnates because the denomnator becomes small. The frst exponental s called the antresonant term and s usually neglgble by comparson for ω ω mn ; droppng ths term s called the Rotatng Wave Approxmaton. The resonant behavor of the transton ampltude corresponds to photon absorpton (we need a quantzed feld treatment to really prove ths). When squared, we get the nstantaneous transton probablty c m ( t) " 0 F x h $ exp # m n t $ # t m µ x n # m n $# 0 F x h m µ x n sn [(" m n #") t /] " m n #" (I.3) If we start from state m and calculate the probablty of emsson to n, ths corresponds to the negatve frequency regon ω ω mn, so that the frst bracketed term n Eq. (I.30) domnates. It turns out that we end up wth exactly the same result except that <m µ x (e) n> becomes <n µ x (e) m>. Thus, snce the squares of these ampltudes are the same, the probablty of absorpton from n to m s the same as for stmulated emsson from m to n. Spontaneous emsson s not part of ths semclasscal descrpton. Range of valdty Upper bound on t: Eq. (I.30) neglects second and hgher orders n tme-dependent perturbaton theory. It s only vald provded the transton probablty does not grow too large,.e., c m <<. From Eq. (I.3) we can see that, for tmes short relatve to the nverse of the detunng ω ω mn, the transton probablty grows as t, c m ( t) " 0 F x h m µ x n t (I.3) and relablty of the frst-order treatment requres that t << F x 0 h (I.33) m µ x n 8
Lower bound on t: For a fxed value of t, the probablty n Eq. (I.3) vares wth ω as the square of the snc [or sn(x)/x] functon 0.8 0.6 [Sn(x)/x]^ 0.4 0. -0-5 5 0 Fgure I- x The nodes mmedately around the man lobe fall at ω ω mn ± π / t, so that the effectve wdth for whch the probablty s sgnfcant s gven by Δω 4π / t. Our neglect of the antresonant term above ammounted to assumng that ths snc functon does not overlap an analogous one centered around ω ω mn,.e., that " m n >> #" 4 $ t (I.34) Therefore, for the Rotatng Wave Approxmaton to be vald, t >> " # m n (I.35) Consstency of the two bounds (I.33) and (I.35) therefore requres that " F x 0 m µ x n << h # m n (I.36).e., that the matrx element of the perturbaton between states m and n be much smaller than ther energy splttng. Thus, Eq. (I.3) s also lmted to felds weak enough to satsfy Eq. (I.36). 9
Rab Oscllatons N. F. Ramsey, "Molecular Beams", p. 9 J. I. Stenfeld, "Molecules and Radaton", p. 39 Suppose there are only two states and undergong a resonant transton wth unperturbed wave functons ψ, ψ and energes E h" and E h". In ths two-state case, we do not have to use perturbaton theory. [Consequently, we drop the superscrpt (0).] Instead, we attempt to drectly solve for the tme-dependent coeffcents c (t) and c (t) n "( t) c ( t)" + c ( t)" (I.37) Inserted n the tme-dependent Scrödnger equaton, we get $ H 0 + H'" h # ' & ) c % #t ( ( t)* + c ( t)* [ ] 0 (I.38) One can take projectons separately wth ψ * and ψ * to obtan equatons for the c (t): hc ( t) c ( t) H 0 + H' + c ( t) H 0 + H' c ( t) E + c ( t) H' (I.39) hc ( t) c ( t) H 0 + H' + c ( t) H 0 + H' c ( t) H' + c ( t) E (I.40) Here t s assumed that the egenstates are orthonormal and that the perturbaton H' has no dagonal elements between them (usually the case). From the dscusson above, we throw away the antresonant term n each case,.e., keep the e ωt exponental for absorpton ( ) and the e ωt term for stmulated emsson ( ), leadng to the equatons hc ( t) c ( t) E " c ( t) µ x F 0 x e # t (I.4) hc ( t) " c ( t) µ x F 0 x e "# t + c ( t) E (I.4) If the egenstates are real, the matrx elements of the dpole moment are also real. We defne the Rab frequency to be " R h µ x 0 Fx h µ x 0 Fx (I.43) 0
n terms of whch hc ( t) c ( t) E " c ( t) h# R e# t (I.44) hc ( t) " c ( t) h# R e"# t + c ( t) E (I.45) The solutons to these equatons are cos % "t c t + -, c ( t) " R # # $ & * #"t &. + #* sn % ( 0 exp t ' " $ '/ ) E &. - % ( 0 (I.46), $ h '/ $ #t ', $ + sn & ) exp * t % ( + E '/. & ) (I.47) - % h ( 0 where Δ s the detunng of the feld frequency from the transton frequency of the atom or molecule $ " # $# # $ h E (I.48) $ E and " # + $ R & + ( h µ x $ %$ ' 0 Fx ) + * (I.49) The probablty of beng n the excted state at tme t s then P ( t) c ( t) " R $ #t ' # sn & ) (I.50) % (.e., t oscllates wth a frequency Ω/ whch depends on the amount of detunng, the transton dpole moment, and the feld ntensty. Cases: Ωt π, 3π, 5π,... -- P maxes out at ω R / Ω,.e., populaton nverson If on-resonance, Δ 0, Ω ω R, P maxes out at,.e., complete nverson
P (t)!0 0.8 0.6!>0 0.4 0. t Fgure I-3 Note: Ths has assumed that the feld ampltude F x 0 s constant. It turns out that, on resonance, smlar results hold f F x 0 F x 0 (t) vares slowly n tme relatve to the optcal cycle (.e., as n pulsed lasers). The feld can be slowly turned on and off, and the relevant quantty s the pulse area at tme t h "# A t t e $ µ x Fx 0 t' dt' (I.5) wth P sn [A(t)/]. If A(t) π, 3π,..., we get populaton nverson and call ths a π pulse. Example: For the partcular case of the calcum 4 S 4 P resonance transton at λ 4.7 nm, the transton dpole matrx element s.4 x 0 9 C m [R. C. Hlborn, Am. J. Phys. 50, 98 (98)]. If we take an electrc feld strength of 0 4 Volts/m, ths gves us ω R.4 x 0 9 x 0 4 /.05 x 0 34.3 x 0 9 Hz. So a sngle complete Rab oscllaton for ths feld strength takes about 0.43 ns. Varaton: Some experments can also look at the behavor as functon of detunng for fxed tme P ( t) $ " R & (" #" ) sn & + " R & % (" #" ) + " R ' ) t) ) ( (I.5) The envelope of the oscllatons has a Lorentzan falloff wth detunng. See T. R. Dyke, et. al., J. Chem. Phys. 57, 77 (97). Tme lmts: Rab oscllatons wll not contnue forever. Eventually other processes such as spontaneous emsson to other states, collsons and loss of phase coherence among the atoms or molecules n the system wll nterfere.
Short-tme and weak-feld lmt of Rab oscllaton: If we look at small Ωt, Eq. (I.50) gves -- P ω R t / 4, the same result as frst-order perturbaton theory, Eq. (I.3). Ths s only vald at the very begnnng of the frst cycle n the graph above. Averagng Over Fnte Frequency Ranges In practce the lght source always has a fnte bandwdth and measurements of photon absorpton ntegrate over some fnte range n energy or frequency (no truly monochromatc sources!). To do ths, we want to use the radaton energy densty. Frst, for the monochromatc plane wave wth electrc feld n drecton x, the cycle-averaged energy flow per second per unt area (ntensty) s S µ 0 c F 0 x (I.53) The energy of the feld passng through a cross-sectonal area A n tme t s therefore gven by (F x 0 ) A t / µ 0 c. Ths energy sweeps out a volume A c t, so the radaton energy densty s the rato of these two quanttes " x µ 0 c F 0 x # 0 F 0 x (I.54) where ε 0 ( c/µ 0 ) s the permttvty of free space. For monochromatc lght, the number of photons s smply " x /h#. Let us go back to the weak-feld perturbaton theory expressons for the transton probablty so that we are not n prncple lmted to two states. The probablty of an absorpton transton from n to m ( n) s then gven by (I.3) rewrtten as c m ( t) " x h # 0 m µ x n [ t] sn $ % &% m n $ % &% m n (I.55) "Whte" lght: For non-monochromatc lght sources, we must average ths over the frequences. If there s a contnuous range, we replace ρ x by u x (ν) dν, where u x (ν) s an energy per unt volume per unt frequency nterval, and ntegrate: c m ( t) h " 0 m µ x n $ dv u x (# ) [ t] sn % # &# m n % # &# m n (I.56) The ntegraton s llustrated n Fg. I-4. 3
Fgure I-4. From Atkns, "Molecular Quantum Mechancs" As t grows, the snc-squared functon n the ntegrand becomes more and more sharply peaked at the center frequency ω mn πν mn, the heght growng as t and the wdth dmnshng as /t. The area under ths curve then grows as t. In ths lmt u x (ν) may be approxmated by u x (ν mn ) and the ntegral extended to (-, ) wthout major error. Usng - [sn(x)/x] dx π, Eq. (I.56) becomes c m ( t) m µ h x " 0 n u x (# mn ) t (I.57) Ths s Ferm's Golden Rule, the probablty that a sngle molecule n state n makes a transton to state m after llumnaton for a tme t. We then say that the number of molecules makng ths transton per second (the transton rate) s gven by d dt N n c m t " h # 0 m µ x n N n u x ( $ mn ) (I.58) where N n s the number of molecules ntally n state n. It s mportant to remember that the lnear t-dependence of c m n Eq. (I.57), or the constancy of the rate n Eq. (I.58), s the result of havng averaged over a fnte bandwdth. 4
Isotropc (unpolarzed) lght: If the radaton s sotropc nstead of plane-polarzed, we have u x u y u z u/3, where u s the total radaton densty per unt frequency. In ths case, c m ( t) " m µ n u ( $ 6 h mn ) t (I.59) # 0 where <m µ (e) n> s the transton dpole moment (a vector), that s m µ n m µ x n + m µ y n + m µ z n (I.60) Watch for factors of (/3) n the lterature. In ths case, t s due to the unpolarzed lght. In others, t s due to an averagng over random orentatons of the molecule. Drac delta functons: Another thng one commonly fnds n the lterature for these expressons s a Drac delta functon n frequency. Ths s easly understood from the above. One of the standard representatons of the delta functon [Cohen-Tannoudj, et. al., Quantum Mechancs, p. 470] tells us that, usng standard scalng propertes of delta functons, lm t "large [ t] # ( $ %$ m n ) sn # $ %$ m n # t &[# ( $ %$ m n )] t &( $ %$ m n ) (I.6) Therefore Eq. (I.56) for x-polarzed lght leads to lm t "large c m ( t) t h # 0 m µ x n u x ( $ mn ) % d$ &( $ '$ m n ) (I.6) and ths s frequently abbrevated wthout the # d". Whenever such a delta functon appears t must be remembered that an ntegraton over frequency must be performed, even f t s only a tny regon of ntegraton. Actually, the transton may be broadened by means dscussed later, n whch case the δ functon wll be replaced by a less sngular lneshape functon also normalzed to unt area. In practce the fnte bandwdth of the lght source may allow transtons to several or many states m. In ths case, the total probablty of photon absorpton s P( t) " c m ( t) (I.63) m n band and the total rate of absorpton s, as before, N n dp/dt. 5
Macroscopc Pcture of Absorpton and Emsson Assume a monochromatc collmated beam mpngng on a sample Absorpton k,! k,! Sample Stmulated Emsson k,! k,! Sample Spontaneous Emsson k,! Sample k,! k 5,! k 4,! Fgure I-5 k 3,! Absorpton and stmulated emsson correspond to subtracton and addton, respectvely, of photons to the beam. In both cases, the departng photons have the same frequency and momentum as those enterng. Spontaneous emsson corresponds to photons havng the same frequency ω and k k ω/c, but wth roughly an sotropc dstrbuton of drectons of k. 6
Momentum s conserved. Absorpton of a photon wth momentum p hk (p h/λ) means the sample acqures ths momentum. Ensten Rate Equatons Ensten provded an argument by whch the spontaneous emsson probablty could be obtaned from the semclasscal theory, even though t does not normally appear there. (Ths s motvated by the fact that all excted states expermentally have fnte lfetmes before decay by photon radaton or some other process even f an external feld s not appled.) Consder a system of matter and radaton n equlbrum at temperature T. In the case of unpolarzed whte lght or a contnuous range of lght, we calculate from Eq. (.59) the rate of absorpton to be B n"m N n u (# mn ), where B n"m s the Ensten coeffcent for absorpton, B n "m 6h # 0 m µ n (I.64) The rate of stmulated emsson from m to n s smlarly B m"n N m u (# mn ), where we have ntroduced a new B constant. However, we know from the dscusson after Eq. (I.3) that B n"m B m"n # B (I.65) To accommodate spontaneous emsson (present even when u 0), we must ntroduce another rate constant; the rate of spontaneous emsson from m to n s A m"n N m. We abbrevate ths as well, rememberng that there s only one A constant. Fgure I-6 For smplcty, let us now consder that there are only two states avalable, and. The rates of change of the populatons are then gven by 7
dn dt " dn dt " B # N u ( $ ) + B # N u ( $ ) + A # N " B N u + B N u + A N (I.66) B " B " # B, A " # A, u ( $ ) # u (I.67) N + N N const (I.68) At steady state, the rates of upward and downward transtons balance, so B N u B N u + A N (I.69) N N B u B u + A (I.70) If the system s also n thermal equlbrum, the equlbrum values of N and N are gven by a Boltzmann dstrbuton, so that (k B Boltzmann constant) and N $ exp " h# ' & ) (I.7) N % k B T ( * # A B u exp h" & -, % / (I.7) + $ k B T '. For a cavty at room temp (300 K), hν k B T for ν 6 0 Hz or λ 50 µ (far nfrared). Longer wavelengths (mcrowave, RF) correspond to hν << k B T or A << Bu,.e. thermally stmulated emsson domnates. For shorter wavelengths (near-ir, vs, UV), spontaneous emsson domnates. Now Planck's famous law for blackbody radaton n equlbrum wth matter gves u (" ) 3 8# h" c 3 $ exp h" ' & ) * % k B T ( (I.73) so evaluatng ths at ν and nsertng nto (I.7) leads to 8
A B 8" h# 3 6" 4 3 # µ (I.74) c 3 3$ 0 hc 3 (These relatons are slghtly modfed f ether level or s degenerate -- See Stenfeld's text.) It s possble to ntegrate exactly the rate equatons for the two level system, leadng to an excted state populaton N t # % $ N ( 0) " N B u & ( exp " B u + A B u + A' [ t] + N B u B u + A (I.75) wth N N N, N const. Alternatvely, nstead of usng the ensemble descrpton, we can use the probablty densty P (t) N (t) / N, P ( 0) " P t # % $ B u & ( exp " B u + A B u + A' [ t] + B u B u + A (I.76) Steady state populaton: P (") Specal case: No radaton feld (u 0) B u B u + A (I.77) P ( t) P ( 0) exp ("A t) (I.78) The spontaneous rate relates to the natural lfetme of the excted state,.e., the populaton decreases by /e after a tme /A. Specal case: Weak radaton feld (B u << A) P ( t) P ( 0) exp ("A t) + B u A [ " exp ("A t) ] (I.79) Exponental change to weak steady state populaton. Specal case: Intense radaton feld (B u >> A) P ( 0) " $ % P t # & ' ( exp ( " B ut ) + (I.80) Exponental rse (or fall, f there s ntal populaton nverson) to steady state populaton of /. Fully saturated transton. Only /, at most, of the populaton can be n level at long tmes. Then populaton nverson (P > /) can only occur f some pumpng mechansm other than radaton contrbutes, or f more than two levels are consdered. 9
Beer's Law and Ensten Coeffcents In practce, lght s sent nto macroscopc samples of matter and attenuaton s measured as a functon of lght frequency ("absorpton spectrum"). Fgure I-7. Absorbng quantum system We take A' cross-sectonal area (cm ) here, not to be confused wth the Ensten coeffcent. The lght-source ntensty I (W/cm -Hz) s related to the energy densty per unt frequency nterval (sometmes called spectral energy densty) by I (" ) c u (" ) (I.8) Also, the ntensty s related to the photon flux Φ (photons/cm -sec) mpngng on the sample by dvdng by the photon energy, I ( # ) " # h# c u ( # ) h# (I.8) The number of photons passng through area A' n tme t s therefore A't I ( # ) " # A't (I.83) h# Now assume the optcally-thn lmt of small fractonal absorpton, I I 0. Almost all the molecules are n the ground state and P (t) <<,.e., we can gnore stmulated and spontaneous emsson. In ths lmt, P (t) s [see Eq. (I.76)] 0
P ( 0) " P t # % $ B u & ( exp " B u + A B u + A' [ t] + B u B u + A " [ # exp (# B ut) ] " B u t B I t c (I.84) The number of lower-state molecules n an nfntesmal volume element A' dl s N n A' dl (I.85) where n s the densty of molecules n state (N N). produced n ths volume element durng tme t s therefore N N P ( t) n A' dl B I t c The number of excted molecules (I.86) and ths must be equal to the number of photons that have been absorbed, correspondng to a decrease n lght ntensty di < 0: n A' dl B I t c " di A't (I.87) h# or di I " n dl B h# c (I.88) Integraton of Eq. (.85) gves us $ I I 0 exp &"n B h# % c L ' ) I ( 0 exp "* L (I.89) The absorbance or optcal depth α L s the fractonal decrease ln(i 0 /I), where the absorpton coeffcent α can be expressed usng our form for the Ensten B-coeffcent (unpolarzed lght) as " n B h# c h# n µ (I.90) c 6$ 0 h Eq. (I.89) s Beer's Law, and Eq. (I.90) provdes a drect lnk between the macroscopcally measured α and the mcroscopcally calculated transton dpole matrx elements.
The absorpton cross secton s the contrbuton per molecule to the absorpton coeffcent, $ ( # ) " # B h# n c (I.9) and has unts area/molecule. Ths represents an effectve area each molecule presents to ncomng photons. Beer's Law then s expressed as I I 0 exp ("n # L) (I.89') Note: In the lterature, you wll frequently fnd that Beer's Law stated as I /I 0 0 "# C L (I.89'') where C s the concentraton n moles/lter and ε s the extncton coeffcent n lters/mole-cm. Ths s mostly just a change of unts, but there s also a change from the natural base of logarthms to base 0. Watch out for numercal factors of ln(0).303 n the value of α. Caveat: In many dscussons, the rate of photon absorpton s expressed n terms of the cross secton: R dp dt B I c " I h# " $ (I.9) In practcal applcatons, the cross secton always has some wdth n frequency. If ν 0 s the center frequency, then one ntegrates over ths bandwdth (assumed for the moment to be much narrower than the bandwdth of u(ν)), B c $ #(" ) d" (I.93) h" 0 Oscllator strength: Ths s also a commonly used quantty for descrpton of electronc transtons, f " # 0 mc 3 " $ 0 e A 4 " # 0 m h$ 0 " e B mc & %( $ ) d$ (I.94) " e For sngle-electron transtons from state to j n the atom or molecule, the value of ths quantty s that the sum over all states j equals unty (sum rule), " f j (I.95) j Here f j gves the relatve transton strength nto each level j startng from level.
Lne Broadenng Homogeneous broadenng: Each element of an ensemble has an dentcal envronment and resonant frequences Lfetme (or natural) broadenng Spontaneous emsson s a prmary example. All excted levels radate. The excted state populatons decay exponentally n tme, leadng to Lorentzan lneshapes, "(# ) " 0 g (# ), g (# ) $# HWHM /% (# &# 0 ) + $# HWHM, ' g (# ) d# (I.96) Here g(ν) s the lneshape functon and σ 0 s the ntegrated absorpton cross secton (unts cm /sec, not cm ) " 0 # "( $ ) d$ (I.97) $ 0 / % "# HWHM m>! m n>! n "# HWHM Lneshape Fgure I-8. Spontaneous emsson and Lorentzan lneshape Wesskopf and Wgner result for spontaneous emsson transton from m to n: "# HWHM $ ave %, $ ave ' + * ), (I.98) ( & n + If n> s the ground state, there can be no emsson. Then & m " n #, $% HWHM 4& " m (Ι.99) Spectral lnes are unsharp to ths degree. 3
Pressure (or collson) broadenng: Collsons shorten excted state lfetmes. The lneshapes are stll Lorentzan except that there s a collsonal contrbuton to "# HWHM. Inhomogeneous broadenng: Each molecule n an ensemble has dfferent resonant frequences. Doppler broadenng If an atom or molecule has velocty component v z along the propagaton drecton of the EM feld, ths changes the number of cycles t encounters per second. The frequency s changed from the lab frequency ν 0 to " " 0 ( ± v z /c). There s usually a dstrbuton of veloctes, e.g., the probablty for a gven v z s proportonal to a Boltamann factor exp(-mv z /kt). Ths produces a gaussan lneshape, "(# ) " 0 g (# ), g (# ) ln( ' ) exp )&ln $ %# HWHM ) ( ( # &# 0) %# HWHM *,, + (I.00) "(# ) " 0 g (# ), $# HWHM # c k B T ln M (I.0) Fgure I-9. (a) Homogeneously and (b) nhomogeneously broadened lneshapes. From Stenfeld, Molecules and Radaton Thermal populatons of nternal vbratonal and rotatonal states of molecules: To the extent that these are unresolved n spectroscopc experments, ths s nhomogeneous broadenng. Power (saturaton) broadenng: If the radaton s too ntense, frst order perturbaton theory cannot be used because of the substantal depleton of lower level populaton. Local envronments may provde further broadenng. External felds for gas phase. Amorphous solds 3. Lqud phase 4. Adsorpton on surfaces 4
Cross secton revsted The dervaton of the absorpton cross-secton above used broadband radaton. Relaxng ths restrcton, the rate of upward transtons can be wrtten as a frequency ntegraton over the crosssecton tmes the photon flux R dp dt # d" $ (" )%(" ) (I.0) In the broadband exctaton lmt, t s assumed that the molecular cross-secton s sharply peaked wth respect to the spectral energy densty u(ν) (hν/c) Φ(ν), whch can then be approxmated by ts value at lne center and taken out of the ntegral R broadband " #( $ 0 ) % d$ &( $ ) #( $ 0 ) h$ 0 c B (I.03) However, n the narrowband lmt (often the case usng lasers!), ths rate s nstead approxmated by replacng the lneshape of the transton by t's value at the band center of the lght source R narrowband " #( $ 0 ) % d$ &( $ ) (I.04) In other words, we chew a hole out of the spectral profle of the transton. Maxmum value of cross secton: It s generally true that the cross secton for a transton s lmted to a mnmum wdth determned by natural broadenng. If ths s the only broadenng mechansm present, t can be proven that exactly on resonance σ takes the value " max # $ (I.05) and ths s true no matter what the system s. For an optcal wavelength around 600 nm, ths s 6 x 0 0 cm. However, the presence of any other broadenng mechansms can reduce the actual value of σ enormously (several orders of magntude). Selecton Rules One-electron: In the case of a one-electron atom, the egenfunctons are " nlm ( r,#,$ ) R n l ( r) Y l m (#,$) (I.06) 5
where the R n l are the famlar radal hydrogenc wave functons and the Y l m are sphercal harmoncs, defned n terms of the assocated Legendre polynomals by m Y l m (",#) N l m P l ( " ) $ e m# (I.07) Gven an arbtrarly-orented external feld Ε, the perturbng Hamltonan s proportonal to E" r r ( E x sn# cos$ + E y sn# sn$ + E z cos# ) (I.08) In transton matrx elements between two states " n'l'm' and " nlm, the orentatonal varables can be explctly ntegrated out for each of these terms (see, e.g., Condon and Shortley, The Theory of Atomc Spectra): " " # d$ sn$ 0 0 * # d% Y l' m' $,% cos$ Y l m ( $,% ) l +# m " l',l+ " m',m l + ( l ++ m) ( l + 3) l # m + " l',l# " m',m l # ( l + m) ( l +) (I.09) " " # d$ sn$ 0 0 * # d% Y l' m' $,% sn$ cos% Y l m ( $,% ) " # l',l+ # m',m+ ( l + m +) l + m + l + ( l + 3) + # l',l" # m',m+ ( l " m) l " m " l " ( l +) + " l',l+ " m',m# ( l # m +) l # m + l + ( l + 3) # " l',l# " m',m# ( l + m) l + m # l # ( l +) (I.0) " " # d$ sn$ 0 0 * # d% Y l' m' $,% sn$ sn% Y l m ( $,% ) " # l',l+ # m',m+ ( l + m +) l + m + l + ( l + 3) + # l',l" # m',m+ ( l " m) l " m " l " ( l +) " # l',l+ # m',m" ( l " m +) l " m + l + ( l + 3) + # l',l" # m',m" ( l + m) l + m " l " ( l +) (I.) 6
Thus we have the selecton rules for electrc dpole transtons z-polarzaton: Δl ± Δm 0 x-,y-polarzaton: Δl ± Δm ± Changes n n, governed by the radal matrx elements, are not restrcted. As long as there are no spn-dependent terms n the Hamltonan or dpole operator, there are no spn transtons (Δm s 0). Party: One can show that the party of the wave functon, the factor by whch t changes under reflecton of all the coordnates through the orgn, s ( ) l. (Ths comes from the behavor of the sphercal harmonc functons under the reflecton.) The electrc dpole operator s of odd party; In order for the matrx element to not vansh,.e., for the total ntegrand have even party, the party of the two wave functons must be dfferent (Laporte rule). Asde: In actualty, the hydrogen atom s a pathologcal example because of a very specal extra degeneracy. Most sphercally symmetrc potentals are not degenerate n the l quantum number, but a hgher symmetry of the hydrogen atom allows ths to happen, e.g., the s levels are degenerate wth the p levels n the absence of spn-orbt nteractons. As a drect consequence of ths, the hydrogen atom allows for a separaton of varables n ether parabolc or sphercal coordnates, but the parabolc egenfunctons are not all egenfunctons of party. As an example (see Paulng and Wlson, Introducton to Quantum Mechancs), the most convenent zeroth-order wave functons for descrbng the hydrogenc Stark effect nvolve lnear combnatons of the s and p wave functons, and the latter have dfferent partes. In general, however, we can classfy atomc wave functons by party. Hgher multpoles: Electrc quadrupole (E) and magnetc dpole (M) operators have even party. Therefore the selecton rules requre that ntal and fnal states have the same party. 7