Molecular Quantum Mechanics and Molecular Spectra, Molecular Symmetry, and Interaction of Matter with Radiation Fre´de´ric Merkt and Martin Quack Laboratorium fu¨r Physikalische Chemie,ETH Zu¨rich, Zu¨rich, Switzerland “The language of spectra, a true atomic music of the of the spectral lines of the hydrogen atom was found by spheres” Balmer (1885a,b). The key to modern spectroscopy arises, Sommerfeld (1919), as translated and cited by Pais (1991) however,fromquantumtheory,startingwithanunderstand- ing of the intensity distribution of blackbody radiation on the basis of quantization by Planck (1900a,b), the under- 1 INTRODUCTION standingofthephotoelectriceffectwiththephotonconcept duetoEinstein(1905),andmostdirectlytheunderstanding The aim of this article is to introduce the basic experi- of the spectrum of the hydrogen atom on the basis of the mentalandtheoreticalconceptsunderlyingmolecularspec- “old” quantum theory as introduced by Bohr, (1913a,b,c,d) troscopy. At a purely empirical level, human color vision including also Bohr’s condition for spectroscopic transi- can be considered to be a form of spectroscopy, although tions between energy levels E and E connected by the i f the relation between the observed “color” of light and the absorption or emission of photons with frequency ν and fi electromagneticspectrumisnotsimple.Inthatsense,audi- Planck’s constant h: tory perception can be considered to be a better frequency (cid:1) (cid:1) (cid:1) (cid:1) analyzer,althoughthiswouldbeconsideredaspectroscopy (cid:1)∆E (cid:1)=(cid:1)E −E (cid:1)=hν (1) ofsoundratherthanofelectromagneticradiation.Therela- fi f i fi tionbetweenspectroscopyandmusichasbeenaddressedin Thisequation, together with Bohr’stheoryof the hydrogen the words of Sommerfeld, which themselves refer implic- atom as a dynamical system having a quantized energy- itly to the earliest days of human science, as cited at the level structure, forms the basis of the use of line-resolved beginning of this article. high-resolution spectroscopy in understanding the dynam- Early spectroscopic experiments, leading later to appli- icsofatomicandmolecular—later,alsonuclear—systems. cationsinchemistry,canbefollowedinhistoryfromNew- The underlying quantum dynamical theory of such micro- ton’sspectrum,theobservationofthe“Fraunhoferlines”in scopic systems was then completed by introducing “mod- the solar spectrum as appearing on the cover of this hand- ern” quantum theory in the work of Heisenberg (1925), book,infraredspectraobservedbytheHerschels(fatherand Schro¨dinger, (1926a,b,c,d,e), and Dirac (1927, 1929). In son), ultraviolet (UV) photochemistry observed by Ritter, essence, current high-resolution molecular spectroscopy and,finally,thestartofchemical,analytical,andastronom- reliesontheuseofthistheoreticalframeworkinitsapplica- ical spectroscopy with the work of Bunsen and Kirchhoff tiontofurtheringourknowledgeonthefundamentalaspects in the nineteenth century. The “mathematical harmony” of molecular quantum dynamics as well as in its numerous practical applications in science and technology. HandbookofHigh-resolutionSpectroscopy.EditedbyMartinQuack and Fre´de´ricMerkt.2011JohnWiley& Sons,Ltd. The outline of this introductory article to this handbook ISBN:978-0-470-74959-3. is thus briefly as follows. In Section 2, we introduce 2 ConceptsinMolecular Spectroscopy some basic equations of quantum mechanics as needed for 2002). Following Hamilton, one obtains the canonical spectroscopy. Section 3 treats the quantum dynamics of Hamiltonian differential equations of motion accordingly: coherent radiative transitions. Section 4 discusses the basic (cid:2) (cid:3) dq ∂H concepts underlying spectroscopic experiments. Section 5 k =q˙ = (3) k givesabriefintroductiontothecharacteristicsofmolecular dt ∂pk (cid:2) (cid:3) energy levels. Section 6 discusses molecular symmetry dp ∂H (MS) and basic group theory as relevant to spectroscopy. k =p˙k =− (4) dt ∂q k Section7dealswithradiationlesstransitionsandlineshapes for high-resolution spectroscopy. The dynamics of the classicalsystem is thus obtained from the solution of 6N coupled differentialequations. Provided 2 QUANTUM MECHANICS AND that one knows some exact initial condition for one point in phase space, all future and past states of the system in SPECTROSCOPY terms of the set {q (t),p (t)} can be calculated exactly. k k Further considerations arise if the initial state is not known Quantum mechanics provides the underlying theory for exactly, but we do not pursue this further. molecular spectroscopy. It is dealt with in much detail in One approach to quantum dynamics replaces the func- relevant text books by Messiah (1961), Cohen-Tannoudji tions H, p , q by the corresponding quantum mechanical et al. (1973), Sakurai (1985), and Landau and Lifshitz operators(Hkˆ,pkˆ ,qˆ )ortheirmatrixrepresentations(H,p , (1985). For the historical background, one can also consult k k k q ) resulting in the Heisenberg equations of motion: Dirac (1958) and Heisenberg (1930), which remain of k (cid:4) (cid:5) ipnrteesreenstt sfeocrtiotnheisfutondinamtroednutacle csoomnceepotfs.thTehbeasaicimcoonfcetphtes dqˆk = 2π qˆk,Hˆ (5) dt ih and definitions. This can obviously not replace a more (cid:4) (cid:5) detailed introduction to quantum mechanics, which can be dpˆk = 2π pˆ ,Hˆ (6) k dt ih found in the textbooks cited as well as in numerous other books on this topic. whichinvol√venowPlanck’squantumofaction(orconstant) h, and i= −1. Following Dirac (1958), these equations 2.1 Classical Mechanics and Quantum are the quantum mechanical equivalent of the Poisson- Mechanics bracket formulation of classical mechanics and one can, in fact, derive the corresponding classical equations of Manysystemsinbothclassicalandquantummechanicscan motion from the Heisenberg equations of motion if one be described by the motion of interacting point particles, uses quantum mechanics as the more fundamental starting where the physical “particles” are replaced by points of point (see Sakurai (1985), for instance). The classical limit mass m with position at the centerof mass of the particle. k of quantum mechanics has also found interest in molecular For planetary systems, the “particles” would be the sun reaction dynamics in a different framework (Miller 1974, and planets with their moons (plus planetoids and artificial 1975).Equations(5)and(6)containthecommutatoroftwo satellites,etc.).Foratomicandmolecularsystemsthe“point ˆ ˆ operators A and B in general notation: particles” can be taken to be the nuclei and electrons to (cid:4) (cid:5) within a good approximation. Aˆ,Bˆ =AˆBˆ −BˆAˆ (7) In classical dynamics, one describes such an N-particle system by a point in the mathematical phase space, which As quantum mechanical operators and their matrix repre- has dimension 6N with 3N coordinates (for instance sentations do not, in general, commute, this introduces a Cartesian coordinates x , y , z for each particle “k”) k k k new element into quantum mechanics as compared to clas- and 3N momenta p , p , p . Such a point in phase xk yk zk sical mechanics. For instance, in Cartesian coordinates the space moving in time contains all mechanically relevant coordinate operator xˆ is simply multiplicative, while the k information of the dynamical system. In the nineteenth- momentumoperatorpˆ isgivenbythedifferentialoperator century Hamiltonian formulation of classical mechanics, xk one writes the Hamiltonian function H as the sum of the h ∂ pˆ = (8) kinetic (T) and potential V energies: xk 2πi∂x k H =T +V (2) leading to the commutator (cid:6) (cid:7) h in terms of generalized coordinates qk and their conjugate xˆ , pˆ =i (9) momentap (LandauandLifshitz1966,Goldstein1980,Iro k xk (2π) k Conceptsin Molecular Spectroscopy 3 and the corresponding Heisenberg uncertainty relation Ψ (x ,y ,z , . . ., x ,y ,z ,t) depending on the par- 1 1 1 N N N (Messiah 1961) ticle coordinates and time and satisfying the differential equation (time-dependent Schro¨dinger equation) h ∆x ∆p ≥ (10) k xk (4π) h ∂Ψ(x ,y ,z ,. . .,x ,y ,z ,t) i 1 1 1 N N N 2π ∂t where ∆x and ∆p are defined as the root mean square deviationskof the coxrkresponding ideal measurement results =Hˆ Ψ(x1,y1,z1,. . .,xN,yN,zN,t) (14) for the coordinates x and momenta p . Similar equations k xk applytoy ,z withp ,p ,etc.forallparticleslabeledby The physical significance of the wavefunction Ψ (also k k yk zk their index k. It is thus impossible in quantum mechanical called state function) can be visualized by the probability systems to know experimentally the position of the “point density in phase space” to a better accuracy than allowed by the P(x ,y ,z ,. . .,x ,y ,z ,t) Heisenberg uncertainty relation in a quantum mechanical 1 1 1 N N N state. In classical mechanics, on the other hand, the xk and =Ψ(x1. . .zN,t)Ψ∗(x1. . .zN,t) p , etc., commute and the point in phase space can, in xk =|Ψ(x . . .z ,t)|2 (15) principle,bedefinedandmeasuredwitharbitraryaccuracy. 1 N A somewhat more complex reasoning leads to a similar P is real, positive, or zero, whereas Ψ is, in general, a “fourth” uncertainty relation for energy E and time t: complex-valued function. h P(x ,y ,z ,. . .,z ,t)dx dy dz . . .dz gives the ∆E∆t ≥ (11) 1 1 1 N 1 1 1 N (4π) probability of finding the quantum mechanical system of point particles in the volume element (dx . . . dz ) at 1 N We note that equations (10) and (11) are strictly inequal- position (x ,. . .,z ) at time t. 1 N ities, not equations in the proper sense. Depending on the The differential operator in equation (14) is sometimes system considered,the uncertaintycanbe larger than what called the energy operator Eˆ would be given by the strict equation. If the equal sign in equations(10)and(11)applies,onespeaksofa“minimum- Eˆ =i h ∂ (16) uncertaintystateorwavepacket”(seebelow).Thecommu- 2π∂t tatorsinequations(5)and(6)arereadilyobtainedfromthe Thus one can write formofthekineticenergyoperatorinCartesiancoordinates: Tˆ = 1(cid:8)N (cid:9)pˆx2k + pˆy2k + pˆz2k(cid:10) (12) Eˆ Ψ(r,t)=Hˆ Ψ(r,t) (17) 2 m m m k=1 k k k where we introduce the convention that r represents, in general,a complete set of space (andspin) coordinatesand and includesthespecialcaseofsystemsdependingonlyonone Hˆ =Tˆ +Vˆ (13) coordinate, which then can be called r. The solution of equation (14) has the form ˆ if the potential energy V is a multiplicative function of the Ψ(r,t)=Uˆ (t,t )Ψ(r,t ) (18) 0 0 coordinatesoftheparticles(forinstance,withtheCoulomb potential for charged particles). ˆ The time-evolution operator U(t,t ) operating on Ψ(r,t ) 0 0 While this so-called Heisenberg representation of quan- ˆ produces the function Ψ(r,t). U satisfies the differential tum mechanics is of use for some formal aspects and also equation certain calculations, frequently, the “Schro¨dinger represen- tation” turns out to be useful in spectroscopy. ˆ i h ∂U(t,t0) =HˆUˆ (t,t ) (19) 2π ∂t 0 2.2 Time-dependent and Time-independent Thus, in general, one has to solve this differential equation Schro¨dinger Equation ˆ ˆ inordertoobtainU(t,t ).If,however,H doesnotdepend 0 ˆ upon time, U(t,t ) is given by the equation 0 2.2.1 Time-dependent Schro¨dinger Equation (cid:11) (cid:12) In the Schro¨dinger formulation of quantum mechanics Uˆ (t,t )=exp −2πiHˆ ·(t −t ) (20) 0 0 (“wave mechanics”), one introduces the “wavefunction” h 4 ConceptsinMolecular Spectroscopy ˆ ˆ The exponential function of an operator Q as a matrix The eigenfunctions of H are called stationary states representation of this operator is given by equation (21): (cid:2) (cid:3) E t exp(Qˆ)=(cid:8)∞ Qˆn (21) Ψk(r,t)=ψk(r)exp −2πi hk (26) n! n=0 The name for stationary states is related to the time independence of the probability density One of the most important properties of Ψ is that it (cid:1) (cid:1) satisfiestheprincipleoflinearsuperposition.IfΨ1(r,t)and P(r,t)=Ψ (r,t)Ψ∗ (r,t)=|Ψ (r,t)|2 =(cid:1)ψ (r)(cid:1)2 (27) Ψ (r,t) satisfy equation (14) as possible representations k k k k 2 of the dynamical state of the system, then the linear The time-independent Schro¨dinger equation (25) is superposition thus derived as a special case from the time-dependent Schro¨dinger equation. Ψ(r,t)=c Ψ (r,t)+c Ψ (r,t) (22) 1 1 2 2 is also a possible dynamical state satisfying equation (14), 2.2.3 General Time-dependent States ˆ asisreadilyshown,giventhatH isalinearoperatorandc , 1 Making use of the superposition principle (equation 22), c are complex coefficients. However, Ψ(r,t), in general, 2 ˆ the general solution of the Schro¨dinger equation results as is not an eigenstate of H. follows: (cid:2) (cid:3) (cid:8) (cid:8) E t Ψ(r,t)= c ψ (r)exp −2πi k = c Ψ (r,t) 2.2.2 Special Case of Stationary States and k k h k k Time-independent Schro¨dinger Equation k k (28) We assume that Hˆ does not depend on time. We consider IfHˆ doesnotdependontime,suchasinthecaseofisolated ˆ the special case where Ψk(r,t) is an eigenfunction of H atomic and molecular systems, the coefficients ck are time with eigenvalue Ek. Thus independent, generally complex coefficients. According to the principle of spectral decomposition, the probability of Hˆ Ψ (r,t)=EˆΨ (r,t)=E Ψ (r,t) (23) k k k k measuring an energy E in the time-dependent state given k by equation (28) is Thesolutionforthisspecialcaseisgivenbyequation (24): p (E )=|c |2 =c c∗ (29) h ∂Ψ (r,t) k k k k k i k =E Ψ (r,t) 2π ∂t k k ˆ (cid:2) (cid:3) Thus, with time-independent H, the p are independent of k E t =E ψ (r)exp −2πi k (24) time as is also the expectation value of the energy k k h (cid:8) (cid:5)E(t)(cid:6)= |c |2E (30) ˆ k k H beingindependent of time, one candivide equation (23) by exp(−2πiEkt/h) and obtain Figure 1illustratesthespectraldecompositionfortwotypes of spectra. With high-resolution spectroscopy providing Hˆ ψ (r)=E ψ (r) (25) k k k the Ek and ψk, equation (28) provides the basis for E)k pk p( E 0 1 2 3 4 (a) (b) k Figure 1 Spectral decomposition schemes: illustration of spectral decomposition of a time-dependent state. p (E )=|c |2 is the k k k probability of measuring the eigenvalue E in the time-dependent state given by Ψ(r,t). (a) Irregular spectrum and distribution. (b) k Harmonic oscillatorwith a Poisson distribution. Conceptsin Molecular Spectroscopy 5 constructing a general time-dependent state from high- where, consistent with our definitions above, the notation resolution spectroscopic results. The energy in a time- r stands either for the relevant coordinate in a one- dependentstateisthereforenotawell-definedquantity,but dimensional system and the integration is carried out itisonlydefinedbymeansofastatisticaldistributiongiven over the range of definition of r, or else r represents by p . The distribution satisfies the uncertainty relation the set of all coordinates and the integration corresponds k given by equation (11). to a multiple integral over the space of all coordinates. The equations given above for operators remain valid for the corresponding matrix representations replacing the ˆ 2.3 Time-evolution Operator Formulation of operators Q by the corresponding matrices Q. Quantum Mechanics 2.4 Time-dependent Perturbation Theory and Equation (19) can be made the starting point for a general Matrix Representation of the Schro¨dinger ˆ formulationofquantummechanics.U providesnotonlythe Equation solution of the Schro¨dinger equation according to equation (18)butalsoofthetimedependenceofthedensityoperator WeconsideradecompositionoftheHamiltonianHˆ accord- (see, for instance, Messiah (1961) and Sakurai (1985)) ingtoazero-orderHamiltonianHˆ (whichmightbeuseful, 0 (cid:8) iftheSchro¨dingerequationwithHˆ hasasimpleanalytical 0 ρˆ(t)= pn|Ψn(cid:6) (cid:5)Ψn| (31) solution such as a collection of harmonic oscillators) and ˆ ˆ n a “perturbation” operator V needed to complement H in 0 order to describe the complete Hamiltonian satisfying the Liouville–von Neumann equation Hˆ =Hˆ +Vˆ (36) (cid:4) (cid:5) 0 i h dρˆ(t) = Hˆ,ρˆ(t) (32) 2π dt We assume Hˆ and Vˆ to be time independent, although 0 many of the following steps can be carried out similarly by means of the solution with time-dependent Hamiltonians and “perturbations” Vˆ. The perturbation might sometimes be small, but this is ρˆ(t)=Uˆ(t,t0)ρˆ(t0)Uˆ†(t,t0) (33) not necessary. We assume the solution of the Schro¨dinger ˆ equation to be known with H 0 This equation is of particular importance for statistical mechanics. Hˆ0ϕk(r)=Ekϕk(r) (37) Furthermore,theHeisenbergequationsofmotion(equat- ions 5 and 6) for the operators pˆk and qˆk, as for any Theϕk formacompletebasisandthegeneralwavefunction ˆ is given by generalized operator Q corresponding to the dynamical (cid:2) (cid:3) observable Q, are solved by means of the equation (cid:8) E t Ψ(r,t)= c (t)ϕ (r)exp −2πi k (38) Qˆ(t)=Uˆ†(t,t )Qˆ(t )Uˆ(t,t ) (34) k k h 0 0 0 Here the coefficients c (t) depend explicitly upon time k ˆ This equation is of importance for time-resolved spec- because the ϕ are not eigenfunctions of H. If the k troscopy, where one might, for instance, observe the time- ϕ (r) and E are known, the time dependence of the k k dependent electric dipole moment M(t) described by the c (t) provides, in essence, the solution of the Schro¨dinger k dipole operator µˆ. We note the difference in sign of equation with the complete Hamiltonian including the ˆ equation(32)andtheHeisenbergequationsofmotion,with perturbation V. Inserting Ψ(r,t) into the time-dependent resulting differences for the solutions given by equations Schro¨dinger equation (14) with Hˆ =Hˆ +Vˆ and simpli- 0 (33) and (34). This is no contradiction, as the density fying the equations by means of the matrix representation operator ρˆ does not correspond to a dynamical variable of the operator Vˆ (equation 35) (observable) of the quantum system. (cid:13) (cid:1) (cid:1) (cid:14) (cid:1) (cid:1) Given a complete orthonormal basis ϕ , which can V = ϕ (cid:1)Vˆ(cid:1)ϕ (39) k jk j k also be the eigenstate basis ψ , one can define matrix k representations of the various operators one obtains a set of coupled differential equations Qjk =(cid:13)ϕj(cid:1)(cid:1)(cid:1)Qˆ(cid:1)(cid:1)(cid:1)ϕk(cid:14)=(cid:15) +∞ϕ∗jQˆϕkdr (35) i2hπdcdjt(t) =(cid:8)exp(iωjkt)Vjkck(t) (40) −∞ k 6 ConceptsinMolecular Spectroscopy where we use the abbreviations can be computed accordingly. Equations (48) and (49) are equivalent to the original Schro¨dinger equation. Approxi- 2πE 2πE ω =ω −ω = j − k (41) mations arise from the truncation of the generally infinite jk j k h h matricesatfinitesize,fromerrorsintroducedbythenumer- Defining a matrix element of some kind of Hamiltonian icalalgorithmsusedforthecalculationsofmatrixelements, matrix, and in the calculations of the various matrix operations. H˜ =exp(iω t)V (42) jk jk jk 3 QUANTUM DYNAMICS OF thesetofequationsdefinedbyequation(40)canbewritten SPECTROSCOPIC TRANSITIONS in matrix notation: UNDER EXCITATION WITH i h dc(t) =H˜(t)c(t) (43) COHERENT MONOCHROMATIC 2π dt RADIATION ˜ In this matrix representation, H(t) depends on time. How- ever, one can make the substitution 3.1 General Aspects a =exp(−iω t)c (44) While traditional spectroscopy in the optical domain has k k k used weak, incoherent (quasi-thermal) light sources (see and obtain also Section 4), present-day spectroscopy frequently uses (cid:9) (cid:10) high-power coherent laser light sources allowing for a (cid:8) i h daj = V a + h ω a (45) variety of phenomena ranging from coherent single-photon 2π dt jk k 2π j j transitions to multiphoton transitions of different types. k Figure 2 provides a summary of mechanisms for such Defining the diagonal matrix transitions. (cid:16) (cid:17) While excitation with incoherent light can be based on hω E = E = j (46) a statistical treatment (Section 4), excitation with coher- Diag j (2π) ent light can be handled by means of quantum dynamics asoutlinedinSection 2.Intense,coherentlaserradiationas one obtains alsoelectromagneticradiationintheradiofrequencydomain h da(t) (cid:18) (cid:19) used in nuclear magnetic resonance (NMR) spectroscopy i = E +V a=H(a)a(t) (47) 2π dt Diag (Ernst etal. 1987) canbe treatedasa classicalelectromag- netic wave satisfying the general wave equations (50) and whereaisthecolumnmatrixofcoefficientsa=(a1,a2,. . . (51) resulting from Maxwell’s theory: a ,. . .a )T and H(a) is a time-independent matrix repre- k n sentation of the Hamiltonian as defined above. ∂2E ∇2E=µµ εε (50) Equation (47) is thus a matrix representation of the 0 0 ∂t2 original Schro¨dinger equation, which makes the influence ∂2B of the perturbation Vˆ explicit. The corresponding time- ∇2B=µµ εε (51) 0 0 ∂t2 independent Schro¨dinger equation is obtained following Section 2.2.2 and we do not repeat the corresponding E is the electric field vector and B the magnetic field equations. The solution of equation (47) is given by the vector (magnetic induction), µ, µ , ε, ε are the normal 0 0 matrix representations of equations (18)–(20). Thus fieldconstants(see StohnerandQuack2011:Conventions, Symbols, Quantities, Units and Constants for High- a(t)=U(a)(t,t0)a(t0) (48) resolution Molecular Spectroscopy, this handbook), with ε =µ=1 in vacuo. The nabla operator ∇ is defined by with equation (52): U(a)(t,t )=exp(−2πiH(a)(t −t )/h) (49) ∂ ∂ ∂ 0 0 ∇ =e +e +e (52) x y z ∂x ∂y ∂z These equations are generally useful for numerical compu- tation, provided that the basis functions ϕ and the E are where e , e , e are the unit vectors in a right-handed k k x y z knownandthematrixelementsV neededinequation(47) Cartesian coordinate system. The classical electromagnetic jk Conceptsin Molecular Spectroscopy 7 nw Direct We consider here, for simplicity, the special case of a classical z-polarized electromagnetic wave propagating in vacuo in the y-direction with slowly varying (or constant) 0 field amplitudes E (t) and B (t) (see Figure 3): 0 0 Goeppert-Mayer E (y,t)=|E (t)|cos(ωt +η(cid:8)−k y) (53) z 0 ω B (y,t)=|B (t)|cos(ωt +η(cid:8)−k y) (54) 2w x 0 ω ω =2πν is the angular frequency, k =2π/λ the angular ω wavenumber, ν =c/λ the ordinary frequency and λ the 0 (cid:8) wavelength. At a given position y the phase η can be combined with the phase −k y to an overall phase (η = ω nw Quasiresonant stepwise k y−η(cid:8)). excitation ω The extension to more general cases is straightforward 2w (see also Quack (1998)). The intensity of the radiation is, in general, (cid:20) w εε I(y,t)=|E (y,t)|2 0 (55) z µµ 0 (cid:21) (cid:22) 0 and averaging over time with cos2x =1/2, one has from equations (53) and (55) Incoherent stepwise (cid:20) 2(w ± ∆w) excitation I(t)= 1|E (t)|2 εε0 (56) 0 2 µµ 0 1(w ± ∆w) For the speed of light, one has in some medium with refractive index n m 0 c c =(µµ εε )−1/2 = (57) m 0 0 n m Figure 2 Mechanisms for radiative excitation. [After Quack and in vacuo (µ=ε =1) (1998).] Dotted lines give the transitions, curved full lines, the dipolecoupling. c =(µ ε )−1/2 (58) 0 0 wavecanbeunderstoodasthecoherentstatedescriptionof Wecanmentionheresomepracticalequationsforcalculat- thequantumfieldinthelimitofverylargeaveragenumber ing electric and magnetic field strengths when irradiating (cid:5)n(cid:6) of quanta per field mode (Glauber 1963a,b, Perelomov with monochromatic radiation of given intensity I: 1986). Coherent laser radiation and also radiofrequency (cid:1) (cid:1) (cid:20) radiation are frequently characterized by (cid:5)n(cid:6)>1010. Thus (cid:1)(cid:1)(cid:1) E0 (cid:1)(cid:1)(cid:1)(cid:9)27.44924 I (59) the classical approximation to the electromagnetic field is V cm−1 Wcm−2 excellent. The situation of weak thermal light sources in (cid:1) (cid:1) (cid:20) the optical domain is very different ((cid:5)n(cid:6)<1), requiring a (cid:1)(cid:1)(cid:1)B0(cid:1)(cid:1)(cid:1)(cid:9)9.156×10−6 I (60) quantum statistical treatment (Section 4). T Wcm−2 z E(y) B(y) y 1 2 l l x Figure 3 Schematic representation of a z-polarized monochromatic wave. 8 ConceptsinMolecular Spectroscopy A further quantity characterizing the irradiation over Hˆ ϕ =E ϕ =(cid:1)ω ϕ (67) Mol k k k k k some period of time t is the fluence F(t) defined by equation (61): and write the solution of the time-dependent Schro¨dinger (cid:15) t equationinthebasisϕ ofmoleculareigenstateswithtime- F(t)= I(t(cid:8))dt(cid:8) (61) dependent coefficients:k 0 (cid:8) For wavelengths λ>100nm, one can assume E and B to Ψ(r,t)= b (t)ϕ (r) (68) k k be constant over the extension of the atomic or molecular k system at any given time (∆y >1nm), which leads to the dipole approximation for the interaction energy between Insertingthisintoequation(14),weobtainasetofcoupled molecule and field: differential equations: Vˆel.dipole =−µel·E (62) i(cid:1)dbj =(cid:8)H b (t) (69) jk k dt where µ is the electric dipole vector given by equa- k el tion (63), with charges q for the particles with position i or in matrix notation, vector r : i (cid:8) db(t) µ = q r (63) i(cid:1) =H(t)b(t) (70) el i i dt i This is again, in essence, a matrix representation of the Similarly, one has the interaction energy with a magnetic original Schro¨dinger equation (see Section 2). Assuming dipole µ magn molecularstatesofwell-definedparity,thediagonalelectric Vˆ =−µ ·B (64) dipole matrix elements vanish and we have the diagonal magn.dipole magn elements of H(t): For the present quantum dynamical treatment of coher- (cid:13) (cid:1) (cid:1) (cid:14) (cid:1) (cid:1) ent excitation, we restrict our attention to electric dipole H =E = ϕ (cid:1)Hˆ (cid:1)ϕ ≡(cid:1)ω (71) ii i i Mol i i transitions in a field given by equation (53), and there- fore we can write, with the z-component µz of the electric For other situations such as for chiral molecules or dipole operator (and abbreviating η =kωy−η(cid:8)), as fol- if parity violation were important (see Quack 2011: lows: Fundamental Symmetries and Symmetry Violations Vˆ =−µˆ E (y,t)=−µˆ |E (t)|cos(ωt −η) (65) from High-resolution Spectroscopy, this handbook), one el.dipole z z z 0 would have also a diagonal contribution from the elec- tricdipoleinteractionenergy.Disregardingsuchcaseshere, The extension to magnetic dipole transitions is straightfor- the electric dipole interaction energy leads to off-diagonal ward.WegivehereonlyabriefsummaryandrefertoQuack matrix elements: (1978, 1982, 1998) for more detail. (cid:13) (cid:1) (cid:1) (cid:14) (cid:1) (cid:1) H = ϕ (cid:1)Vˆ (t)(cid:1)ϕ (72) kj k el.dipole j 3.2 Time-dependent Quantum Dynamics in an Oscillatory Electromagnetic Field Dividing H by (cid:1)cos(ωt −η) we obtain a matrix element kj V ,whichisindependentoftime,ifwecanassume|E (t)| kj 0 We consider now the time-dependent Schro¨dinger equation tobesufficientlyslowlyvaryingintimethatitcanbetaken (14) with a time-dependent Hamiltonian constantforthetimeperiodunderconsideration,asweshall Hˆ(t)=Hˆ −µ |E (t)|cos(ωt −η) (66) do, replacing E0(t) by E0, Mol z 0 which is of the form of equation (36), with Hˆ being the V = Hkj =−(cid:21)ϕ (cid:1)(cid:1)µˆ (cid:1)(cid:1)ϕ (cid:22)|E0| =V∗ (73) Mol kj [(cid:1)cos(ωt −η)] k z j (cid:1) jk time-independent Hamiltonian for the isolated molecule in ˆ the absence of fields and the interaction Hamiltonian V We then obtain a set of coupled differential equations in is now a time-dependent, oscillatory function. We assume matrix notation: the solution of the time-independent Schro¨dinger equation for the isolated molecule to be given by equation (67) d i b(t)={W+Vcos(ωt −η)}b(t) (74) ((cid:1)=h/2π): dt Conceptsin Molecular Spectroscopy 9 wherewehavedefinedthediagonalmatrixW bythematrix U(t,t )=F(t,t )exp(A(t −t )) (81) 0 0 0 elements W ≡ω . kk k This isstill a practicallyexactrepresentationof the orig- F(t0,t0)=1 (82) inal time-dependent Schro¨dinger equation for the physical F(t +nτ)=F(t) (83) situation considered here. Because of the essential time dependence in Vcos(ωt −η), there is no simple closed A(t(cid:8))=A(t(cid:8)(cid:8)) (all t(cid:8),t(cid:8)(cid:8)) (84) expression in the form of the exponential function anal- ogous to equations (18), (20) or (48), (49). Apart from Itisthensufficienttointegratenumericallyoveroneperiod numerical, stepwise solutions discussed in Quack (1998), τ andthenobtaintheevolutionforalltimesbymatrixmul- one can make use of series expansions such as the Mag- tiplications according to equations (81)–(84). In particular, nus expansion. This solves equation (70) by means of the at multiples of the period τ one finds (with t =0), 0 following series for U(t,t ): 0 U(τ)=exp(Aτ) (85) b(t)=U(t,t )b(t ) (75) 0 0 U(nτ)=[U(τ)]n (86) U(t ,t )=1 (76) 0 0 (cid:9) (cid:10) There has been considerable literature making use of Flo- (cid:8)∞ quet’s theorem for the treatment of coherent excitation and U(t,t )=exp C (77) 0 n there also exist computer program packages (see Quack n=0 (1998)). We discuss here a further useful approximation. The firsttwo terms aregiven bythe following expressions: (cid:15) 3.4 Weak-field Quasi-resonant Approximation t i(cid:1)C = H(t(cid:8))dt(cid:8) (78) (WF-QRA) for Coherent Monochromatic 0 t0 Excitation (cid:23) (cid:24) (cid:15) (cid:15) 1 t t(cid:8)(cid:8)(cid:6) (cid:7) i(cid:1)C =− H(t(cid:8)),H(t(cid:8)(cid:8)) dt(cid:8) dt(cid:8)(cid:8) (79) We consider a level scheme for coherent excitation with 1 2 t0 t0 levels near the resonance as shown in Figure 4. One can then associate with each molecular level of energy Higher terms contain(cid:6)more complex(cid:7) combinations of com- E =(cid:1)ω an integer photon number n for near-resonant k k k mutators of the type H(t(cid:8)),H(t(cid:8)(cid:8)) . From this one recog- excitation such that nizes that the series terminates after the first term given by equation (78), if H(t(cid:8)) and H(t(cid:8)(cid:8)) commute at all t(cid:8), ωk =nkω+xk (87) (cid:8)(cid:8) t , which is true if H does not depend on time, result- ing in the exponential solutions already discussed. There where xk is a frequency mismatch for exact resonance at a(cid:6)re other (rar(cid:7)e) cases of time-dependent H(t), but with the best choice of nk. H(t(cid:8)),H(t(cid:8)(cid:8)) =0. One can, however, also make use of Under the conditions that (i) there is a sequential near- the periodicity of the field using Floquet’s theorem (Quack resonantexcitationpath,(ii(cid:1))onl(cid:1)ylevelswithageneralreso- nancemismatchsatisfying(cid:1)D (cid:1)(cid:11)ωcontributeeffectively 1978, 1998). kj to excitation (quasi-resonant co(cid:1)ndi(cid:1)tion), and (iii) the cou- pling matrix elements satisfy (cid:1)V (cid:1)(cid:11)ω (weak-field con- kj dition), one can approximately derive a set of coupled 3.3 Floquet Solution for Hamiltonians with equations with an effective Hamiltonian that does not Strict Periodicity depend upon time. For this purpose, one makes the simple substitution (Quack 1978, 1998) With H =(cid:1){W+Vcos(ωt −η)} from equation (74), one has obviously a =exp(in ωt)b (88) k k k H(t +τ)=H(t) (80) resulting in the set of differential equations (cid:8) with period τ =2π/ω. idak =x a + 1 V a (89) k k kj j Making use of the Floquet theorem (or Floquet– dt 2 j(cid:12)=k Liapounoff theorem, see Quack, (1978, 1998) for the his- torical references), one has the following form for the or in matrix form (with the diagonal matrix X and time-evolution matrix (with some integer n): Xkk =xk) 10 Conceptsin MolecularSpectroscopy i g i k xk nkw ency 2 c i gi j u 1 i gi e freq 0 b xk k v w– 0 w Effecti a ij gi k w = y) / h c Fenigerugrye-l5eveElfsfcehcetimvee-forfeqFuigeunrcey4.scheme corresponding to the g ener b (nk – 1)w = naw cular a use V(cid:8)/2 rather than V/2 for the general matrix. This is e graphically shown in Figure 5 for the same level scheme ol M as in Figure 4, but with effective energies that are “on the ( nw a same energy shell” and thus effective couplings between levelsofsimilareffectiveenergy.Wenotethecloseanalogy to the dressed atom (dressed molecule) picture by Cohen- 2 Tannoudji, which uses, however, a different derivation 1 (Cohen-Tannoudji et al. 1992). We note that the quasi- 0 resonant transformation as given in Quack (1978, 1998) can be written in matrix notation: Figure 4 Energy-level scheme [(After Quack 1982).] The molecular energy levels are marked as horizontal full lines. The a =Sb (94) horizontal dashed lines correspond to the energies E +n (cid:1)ω of 0 k the ground state (E ) with n photons. It is sometimes useful to 0 k with the diagonal matrix describe decay phenomena by adding an imaginary energy con- tribution,for instance E =Re(E)−iγ /2 as indicated. l l (cid:13) S =exp(in ωt) (95) kk k (cid:16) (cid:17) da 1 i = X + V(cid:8) a (90) Similarly, a transformation for the density matrix P(a) dt 2 from P(b) can be derived in this approximation, resulting in the solution of the Liouville–von Neumann equationfor One can interpret this equation by means of an “effective P(t) by Hamiltonian” (cid:16) (cid:17) H(a) =(cid:1) X + 1V(cid:8) (91) P(a)(t)=SP(b)S† (96) eff 2 P(a)(t)=U(a)(t,t )P(a)(t )U(a)†(t,t ) (97) eff 0 0 eff 0 and the corresponding effective time-evolution matrix (cid:25) (cid:26) For details, we referto Quack(1978, 1982, 1998). We turn H(a)(t −t ) now to a simple application to the special case of coherent U(a)(t,t )=exp −2πi eff 0 (92) eff 0 h radiative excitation connecting just two quantum states. (cid:11) (cid:2) (cid:3) (cid:12) 1 =exp −i X + V(cid:8) (t −t ) 0 2 3.5 Coherent Monochromatic Excitation a(t)=U(a)(t,t )a(t ) (93) between Two Quantum States eff 0 0 This is quite a remarkable result as it corresponds to If only two quantum states are considered, one obtains a replacing the molecular energies E by new effective schemeforthecoherentmonochromaticradiativeexcitation k energies (cid:1)X and the couplings V by new effective as shown in Figure 6. kk kj couplings (V /2) for near-resonant levels (and implicitly Equation (70) simplifies to the set of just two coupled kj by zero for far off-resonant levels). We can therefore differential equations: