Fundamentals of -Vibration Spectra

S. Albert, K. Keppler Albert, H. Hollenstein, C. Manca Tanner, M. Quack

ETH Zürich, Laboratory of Physical Chemistry, Wolfgang-Pauli-Str. 10, CH-8093 Zürich, Switzerland, Email: [email protected]

reprinted from

“Handbook of High-Resolution ”,

Vol. 1, chapter 3, pages 117–173, M. Quack, and F. Merkt, Eds. Wiley Chichester, 2011, ISBN-13: 978-0-470-06653-9. Online ISBN: 9780470749593, DOI: 10.1002/9780470749593

with compliments from Professor Martin Quack, ETH Zürich Abstract

We provide a survey of fundamental aspects of rotation–vibration spectra. A basic understanding of the concepts is obtained from a detailed discussion of rotation–vibration spectra of diatomic with only one vibrational degree of freedom. This includes approximate and exact separation of rotation and vibration, effective spectroscopic constants, the effects of nuclear spin and statistics, and transition probabilities derived from the form of the function. The underlying assumptions and accuracy of the determination of molecular structure from spectra are discussed. Polyatomic molecules show many interacting vibrational degrees of freedom. Energy levels and spectra are discussed on the basis of normal coordinates and effective Hamiltonians of interacting levels in Fermi resonance, and in more complex resonance polyads arising from anharmonic potential functions. The resulting time-dependent dynamics of intramolecular energy flow is introduced as well. Effective Hamiltonians for interacting rotation–vibration levels are derived and applied to the practical treatment of complex spectra. Currently available computer programs aiding assignment and analysis are outlined.

Keywords: vibration-rotation spectra; intramolecular vibrational energy flow; fermi resonance; time-dependent vibrational dynamics; effective Hamiltonians; diatomic molecules; nuclear spin and statistics; electric dipole moment function; line intensities; band strengths

Fundamentals of Rotation–Vibration Spectra

Sieghard Albert, Karen Keppler Albert, Hans Hollenstein, Carine Manca Tanner and Martin Quack Laboratorium f¨ur Physikalische Chemie, ETH Z¨urich, Z¨urich,Switzerland

1 INTRODUCTION Jets, Sigrist 2011: High-resolution Infrared Laser Spec- troscopy and Sensing Applications, Weber 2011: High-resolution infrared (IR) spectroscopy is the key to High-resolution of , Havenith the quantum-state-resolved analysis of molecular rota- and Birer 2011: High-resolution IR-laser Jet Spec- tion–vibration spectra and the consequent understanding troscopy of Formic Acid Dimer, Hippler et al. 2011: Mass of the quantum dynamics of molecules. Such quantum state and Isotope-selective ,Amano resolution has been common in pure rotational spectra from 2011: High-resolution and Infrared Spec- the very beginning of this field after about 1945, on the troscopy of Molecular Cations,Freyet al. 2011: High- basis of microwave (MW) technology (see Bauder 2011: resolution Rotational Raman Coherence Spectroscopy Fundamentals of , this hand- with Femtosecond Pulses,Demtroder¨ 2011: Doppler-free book). In the IR range of the spectrum, however, which is Laser Spectroscopy, Stanca-Kaposta and Simons 2011: important for the study of combined rotational–vibrational High-resolution Infrared–Ultraviolet (IR–UV) Double- excitation, experimental resolution was sufficient in the past resonance Spectroscopy of Biological Molecules,Flaud only for the simpler diatomic and polyatomic molecules, and Orphal 2011: Spectroscopy of the Earth’s Atmo- largely limited by the technology of grating or more sphere and Herman 2011: High-resolution Infrared Spec- generally dispersive (Herzberg 1945, 1950, troscopy of : Theoretical Background and Nielsen 1962, Stoicheff 1962). There has been substan- Research Trends, among others, in this handbook). tial development of IR and Raman spectroscopy at much In parallel to the experimental developments, there higher resolution in recent decades, based on laser tech- has been great progress in the theoretical understanding nology, on one hand, and interferometric Fourier trans- of the of molecules on the basis of form infrared (FTIR) spectroscopy, on the other hand. quantum chemical ab initio theory as well as quantum This has made the rotation–vibration spectra of much dynamics in general (see Yamaguchi and Schaefer 2011: more complex molecules accessible to the full analysis Analytic Derivative Methods in Molecular Electronic of the rotational–vibrational fine structure and the fron- Structure Theory: A New Dimension to Quantum tier in this area of research is moving toward even larger Chemistry and its Applications to Spectroscopy,Tew polyatomic molecules. Several articles in this handbook dis- et al. 2011: Ab Initio Theory for Accurate Spectroscopic cuss these experimental developments (see,forinstance, Constants and Molecular Properties, Breidung and Thiel Albert et al. 2011: High-resolution Fourier Transform 2011: Prediction of Vibrational Spectra from Ab Initio Infrared Spectroscopy,Snelset al. 2011: High-resolution Theory, Mastalerz and Reiher 2011: Relativistic Elec- FTIR and Diode Laser Spectroscopy of Supersonic tronic Structure Theory for Molecular Spectroscopy, Marquardt and Quack 2011: Global Analytical Poten- Handbook of High-resolution Spectroscopy. Edited by Martin Quack tial Energy Surfaces for High-resolution Molecular and Fred´ eric´ Merkt.  2011 John Wiley & Sons, Ltd. Spectroscopy and Reaction Dynamics, Watson 2011: ISBN: 978-0-470-74959-3. Indeterminacies of Fitting Parameters in Molecular 118 Fundamentals of Rotation–Vibration Spectra

Spectroscopy, Carrington 2011: Using Iterative Methods the situation is much more complex. We start from the to Compute Vibrational Spectra, Tennyson 2011: High Born–Oppenheimer approximation (see Bauder 2011: Fun- Accuracy Rotation–Vibration Calculations on Small damentals of Rotational Spectroscopy,Worner¨ and Merkt Molecules, Boudon et al. 2011: Spherical Top Theory 2011: Fundamentals of Electronic Spectroscopy and and Molecular Spectra,Koppel¨ et al. 2011: Theory of Marquardt and Quack 2011: Global Analytical Potential the Jahn–Teller Effect and Field et al. 2011: Effective Energy Surfaces for High-resolution Molecular Spec- Hamiltonians for Electronic Fine Structure and Poly- troscopy and Reaction Dynamics, this handbook), where atomic Vibrations, this handbook). Thus, many aspects of the or nuclei move in an effective potential V (R) rotation–vibration spectroscopy are covered in great detail defined by the solution of the electronic Schrodinger¨ in the individual articles of this handbook. equation. R is the distance between the nuclei. We con- The goal of this article is to provide a brief, largely sider molecules in their electronic ground state (see Worner¨ didactic discussion of some of the basic concepts, where and Merkt 2011: Fundamentals of Electronic Spec- one considers vibrational motion in addition to the rota- troscopy, this handbook, for a discussion pertaining to tional motion treated in the preceding article by Bauder molecules in excited electronic states). Indeed, one can 2011: Fundamentals of Rotational Spectroscopy,this consider “best” effective potentials V (R) going beyond handbook. This also helps in defining some of the basic the Born–Oppenheimer approximation, still retaining the concepts in this field (see also Stohner and Quack 2011: concept that the explicit motion of the electrons can be Conventions, Symbols, Quantities, Units and Constants neglected and, thus, one has to consider only the dynamics for High-resolution Molecular Spectroscopy, this hand- of nuclear motion in this effective potential (Section 2.1). book). Further basic concepts related to the present arti- Because this effective potential, in general, has no simple cle can be found in Merkt and Quack 2011: Molecular form, there are consequently no simple analytical expres- Quantum Mechanics and Molecular Spectra, Molecular sions available for the energy levels or rovibrational term Symmetry, and Interaction of Matter with Radiation, values except for model potentials. Thus, in practical spec- this handbook. We also discuss some aspects arising in the troscopy, one uses approximations leading to simple formu- practical analysis of complex rotation–vibration spectra of lae for the term values, which can be expressed by tables of polyatomic molecules, because this topic is not well cov- constants summarizing these term value expressions (Huber ered in the traditional textbooks and has developed in more and Herzberg 1979). The approaches, starting either from recent years through the development of new experimental “exact” solutions of the Schrodinger¨ equation for nuclear possibilities. The structure arising from electronic excita- motion in an effective potential or using simple term formu- tion is covered in Worner¨ and Merkt 2011: Fundamentals lae for energy levels, are conceptually different, and similar of Electronic Spectroscopy, this handbook. conceptual differences also exist in the case of polyatomic This article is organized as follows. We start with a molecules. Symmetry and nuclear spin introduce further detailed discussion of diatomic molecules in Section 2. effects in homonuclear diatomic molecules (Section 2.3). Section 3 extends the treatment to polyatomic molecules To describe spectroscopic transitions between energy lev- with several interacting vibrations including anharmonic els, one has to furthermore consider the radiative transition resonance. The time-dependent picture of intramolecuar probabilities, for example, in the electric dipole approx- motion and energy flow is presented in Section 4. Section imation for absorption spectra or related approximations 5 discusses the analysis of interaction of rotation and for Raman spectra (Section 2.4). We also discuss concepts vibration, and Section 6 deals with pattern recognition and related to the determination of molecular structure from assignment aids for complex molecular spectra. high-resolution IR spectra.

2 ENERGY LEVELS AND IR SPECTRA 2.1 Quasi-exact Treatment of OF DIATOMIC MOLECULES: Rotation–Vibration States of Diatomic Molecules in the Framework of the INTERACTION OF ROTATION AND Born–Oppenheimer Approximation or More VIBRATION AND EFFECTS OF General Effective Potentials NUCLEAR SPIN Figure 1 schematically shows a diatomic with This section considers some basic concepts of spectra variable bond length R. For example, the reader may con- of vibrating and rotating molecules, using diatomic sider 12C16O (with no nuclear spin) or 1,2H19F (neglect- molecules for illustration. These concepts also find their ing nuclear spin effects for now), both in their 1Σ correspondence in polyatomic molecules, where, however, closed shell, nondegenerate electronic ground state in the Fundamentals of Rotation–Vibration Spectra 119

V(R)

z Harmonic z2 z (E) 1 De 0 s m 0 1 D0 (E) De m 2 Morse R

Figure 1 scheme. The origin of the E0 molecule-fixed coordinate system is located at the center of mass 0 Re R (S) of the molecule. Figure 2 V (R), full line, and harmonic poten- Born–Oppenheimer approximation (see Bauder 2011: Fun- tial, dotted line. The relevant parameters are the zero-point energy (E) damentals of Rotational Spectroscopy and Worner¨ and E0, the electronic dissociation energy De , and the thermo- 0 = 0 = Merkt 2011: Fundamentals of Electronic Spectroscopy, dynamic bond dissociation energy at 0 K D0 ∆dissH0 /NA (E) this handbook). Restricting the motion to one dimension De − E0. (along the z axis), one has a pure vibrational and trans- lational problem; separating the center of mass (S) trans- with the reduced coordinate lational motion in a standard way (Cohen-Tannoudji et al. = − 1977, Demtroder¨ 2003) and using momentum conservation Q (R Re) f/hν (8) for the molecule as a whole, one obtains the Hamiltonian operator for the internal (vibrational) motion: Because of close analogies with the classical mechanics of oscillation, one uses the concept of the force constant ˆ2 ˆ = ˆ + ˆ = p + Hv Tv Vv V(R) (1) ∂2V(R) 2µ f = (9) ∂R2 R=Re with the momentum operator pˆ and the reduced mass µ, and the classical oscillation ν = cω m m e µ = 1 2 (2) m + m 1 2 1 f ν = (10) where m1 and m2 are the masses of the two nuclei (or 2π µ atoms, see below). Figure 2 shows two typical model potentials, as an The normalization factors Nv in equation (7) are obtained illustration, for which the time-independent vibrational from Schrodinger¨ equation +∞ ∗ = ˆ Ψ v (Q)Ψ v(Q) dQ 1 (11) HvΨ v(R) = EvΨ v(R) (3) −∞ √ v −1/2 Nv = πv!2 (12) has simple solutions. With the harmonic potential Vh(R),

1 and the Hermite polynomials Hv (Q) V (R) = f(R− R )2 (4) h 2 e dv H (Q) = (−1)vexp(Q2) exp(−Q2) (13) one has v dQv 1 E = hν v + , v = 0, 1, 2 ... (5) Similarly, with the “anharmonic” Morse potential v 2 = (E) { − − − }2 VM(R) De 1 exp[ a(R Re)] (14) or equivalently with vibrational ωe one obtains E 1 v = + ωe v (6) 2 hc 2 Ev 1 1 = ωe v + − ωexe v + (15) 2 Ψ v(Q) = NvHv(Q)exp(−Q /2) (7) hc 2 2 120 Fundamentals of Rotation–Vibration Spectra

with the usual spectroscopic constants ωe and ωexe in units oscillator. The conventional spectroscopic constants of the of wavenumbers (m−1 or cm−1, see Stohner and Quack anharmonic Morse oscillator are related to the properties of 2011: Conventions, Symbols, Quantities, Units and Con- the potential stants for High-resolution Molecular Spectroscopy,this handbook) 1 f D h 1/2 ω = = a eM (18) e 2 2 2πc µ 2π cµ ωe ωexe = (16) 4DeM with the force constant f being defined by equation (9). (E) One can also write De DeM = (17) hc ha2 ω x = (19) e e 8π 2cµ Equation (15) is valid for discrete energies Ev≤vmax , where (E) v is given by the highest bound level with E

Harm. osc. Morse osc. where the last approximate equation is useful for practical ) ) − 1 36 − 1 36 − − calculations with constants given in spectroscopic units.

cm v = 0 5 cm v = 0 5

hc 27 hc 27 Rewriting the energy levels as a difference measured 18 18 from the zero point level, one finds

)/(1000 9 )/(1000 9

R R ( ( (E − E ) V 0 V 0 v 0 = − − 2 4 4 (ωe ωexe)v ωexev (22) 2 2

| v =0 | v =0

v v hc 2 2 (E − E ) 0 0 v 0 = − 2 and | y and | y ω0v ω0x0v (23) v −2 v −2

y y hc −4 −4

4 4 = − = 2 v =1 2 v =1 with the new constants ω0 ωe ωexe and ω0x0 ωexe | |

v 2 v 2 for the Morse oscillator. 0 0 The energy difference of adjacent energy levels is a linear and | y and | y

v −2 v −2 y y −4 −4 function of v:

4 4 2 v =2 2 v =2 − | | ∆E(v) (E + E )

v 1 v v 2 v 2 = = (ω0 − ω0x0) − 2ω0x0v (24) 0 0 hc hc and | y and | y

v −2 v −2 y y This relation can be used for practical determination of the −4 −4 spectroscopic constants from a linear graph (Figure 4). One 4 4 2 v =3 2 v =3 | |

v 2 v 2 readily also sees that there is a finite number of discrete 0 0 levels with

and | y and | y 2DeM v −2 v −2  y y vmax (25) −4 −4 ωe 4 v =4 4 v =4 2 2 | |

v 2 v 2 Thus one has about twice as many levels as the number 0 0 of harmonic levels up to dissociation. The first interval and | y and | y v −2 v −2 =˜

y y ∆E(0)/hc νfM is also called the anharmonic funda- −4 −4 mental of the Morse oscillator (y-intercept, 4 4 2 2

| v =5 | v =5 Figure 4). v 2 v 2 0 0 The one-dimensional motion can be extended to three and | y and | y

v −2 v −2 dimensions by using conservation in y y −4 −4 isotropic space (Cohen-Tannoudji et al. 1977, Zare 1988, 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 Demtroder¨ 2003). This allows for an exact separation of R/(100 pm) R/(100 pm) the wavefunction in polar coordinates (R, ϑ, ϕ) according Figure 3 Eigenfunctions (full line) and probability densities to equation (26): (dotted line) of the harmonic oscillator (left) and the anharmonic = Morse oscillator (right). Ψ(R,ϑ,ϕ) Φv,J (R)YJ,MJ (ϑ, ϕ) (26) Fundamentals of Rotation–Vibration Spectra 121

∆E specifically designed for diatomic molecule problems hc (Le Roy 2007). In general, there are no simple w −w x intercept 0 0 0 formulae available. However, we mention the close analogy to the hydrogen , where V (R) is the Coulomb poten- tial (V(R)∝−e2/R). Indeed, the problem at hand is the − general problem of two point masses separated by R and 2w0x0 slope moving in free space under the influence of an interaction potential V (R). Figure 5 shows the lowest “rotation–vibration energies” for the Coulomb potential in the corresponding effective 0 v 2DeM potentials Veff(R, l), where the “rotational” quantum num- −1 we ber is labeled “l” by convention for the orbital angular momentum. Because of the special of the Coulomb = Figure 4 Term values as differences f(v) ∆E(v)/(hc);see potential, the “vibrational ground state” for each effective equation (24). potential Veff(R, l) is degenerate with excited states aris- ing from lower values of l (l − 1,l− 2,...0), leading YJ,MJ (ϑ, ϕ) are the spherical harmonics; here we use the to the well-known “Coulomb degeneracy” (slightly lifted conventional symbols for the angular momentum quantum in reality). For diatomic molecules, no such degeneracy number J and the quantum number MJ for the z component arises because V (R) is qualitatively very different from the of the angular momentum. The angular dependence of the Coulomb potential. Indeed, in general, no simple exact term “rotational” wavefunctions is, thus, the same as for the formulae arise. hydrogen atom, with the conventional notation Yl,ml (ϑ, ϕ). In this respect an exception results for the Morse The “vibrational” wavefunctions Φv,J (R) depend upon potential. As shown by Pekeris (1934), and Nielsen (1951, the double index v,J because they are solutions of a 1959), one finds one-dimensional Schrodinger¨ equation with an effective J -dependent potential 2 Ev,J 1 1 = ωe v + − ωexe v + 2 hc 2 2 = + h + Veff(R, J ) V(R) J(J 1) (27) + + − 2 + 2 8π 2µR2 BvJ(J 1) DvJ (J 1) (32)

Veff(R, J ) = V(R)+ Vcent(J ) (28) 15 000 where Vcent(J ) is interpreted as a centrifugal potential. The effectively one-dimensional differential equation to be 0 solved is thus −15 000 3s, 3p, 3d HΦˆ (R) = E Φ (R) v,J v,J v,J (29) 2s, 2p −30 000 )

with − 1 −45 000 2 ˆ h ∂ 2 ∂ )/(cm H =− R + Veff(R, J ) (30) R ( − 8π 2µR2 ∂R ∂R 60 000 = 0 eff ∼ V = 1 or rewriting the differential equation in the practical form −75 000 = 2 (Demtroder¨ 2003) −90 000 1 d dΦ (R) R2 v,J R2 dR dR −105 000 1s 8π 2µ + E − V (R, J ) Φ (R) = 0 (31) −120 000 h2 v,J eff v,J 0 3.0 6.0 9.0 12.0 15.0 R/(100 pm) The one-dimensional differential equation is readily solved by numerical methods, for instance, using the Numerov– Figure 5 Potential functions with vibrational eigenvalues for the Cooley algorithm, and programs are available that are hydrogen atom (for l = 0, 1, 2). 122 Fundamentals of Rotation–Vibration Spectra

50 000 means obvious that this is the best choice. In calculating J = 0 effective rotational energies following equation (27), one 40 000 = J 50 would assume that the indeed includes 30 000 J = 100 J = 200 the masses of the electrons which are, however, not incor- ) 20 000 − 1 porated at the position of the two nuclei separated by R. 10 000 cm A better assumption would be to compute the moments hc 0 of inertia by means of the actual probability distributions ) /(

R −10 000 ( of the electrons arising from the ground-state electronic eff

~ − V 20 000 wavefunction. Nevertheless, in the first approximation, the −30 000 use of atomic masses might be plausible for the calculation −40 000 of the effective rotational reduced mass µr to be used in −50 000 equation (27). On the other hand, for the purely vibrational 0123456problem, the most logical interpretation in the framework R/(100 pm) of the straight Born–Oppenheimer (BO) approximation would be a motion of the bare nuclei in the effective ˜ Figure 6 Morse potential Veff(R) of the HF molecule from the potential field generated by the electronic motion (see SO-3 potential (Klopper et al. 1998) and effective potential for Bauder 2011: Fundamentals of Rotational Spectroscopy, various values of J . this handbook). In principle, for very simple molecules + such as H2 or H2 one could treat the quantum mechanical problem directly as a three or four particle system, without 1 need for the BO approximation. Bv = Be − αe v + (33) 2 However, particularly for the heavier atoms, it seems h physically obvious that the inner shell electrons are so B = (34) e 2 2 tightly bound to the nuclei that they effectively move with 8π cµRe the nuclei and thus contribute to the effective vibrational 3h2ω 1 1 α = e − reduced masses µv. One might, thus, include some useful e 2 2 2 2 (35) 16π µR De aRe a R fraction of the total number of electrons of the “atoms” e e 1 with masses “mi” in the calculation of the reduced masses Dv = De + β v + (36) µ in equation (2). The situation is by no means trivial and e 2 v is related to replacing the true many-body Hamiltonian by 2 8ωexe 5αe α ωe an effective few-body Hamiltonian (two-body Hamiltonian β = D − − e e e 3 (37) ωe ωe 24Be for diatomic molecules). The uncertainties in these masses are thus directly linked to the Born-Oppenheimer or other 4B3 D = e related effective Hamiltonian approximations. There has e 2 (38) ωe been some debate on which is the “best” choice for effective reduced masses µv and µr; for further discussion, we The comparatively simple formula (equation (32)) for refer to the review by Tennyson 2011: High Accuracy the rotation–vibration energy levels of a Morse oscillator Rotation–Vibration Calculations on Small Molecules, suggests a generalization in terms of simple power series this handbook and the papers by Kutzelnigg (2007), Bunker expansions as a function of v and J . Before addressing such and Moss (1977), Schwartz and Leroy (1987) and Coxon equations, we show the effect of the centrifugal potential and Hajigeorgiou (1999). At ultrahigh precision, the mass for a typical example (Figure 6). For low J , there is little distribution within the extended nuclei must also be taken change in the effective potential, very different from the into account. hydrogen atom. For J = 1 and 2, the effects would not be visible in the figure. Large effects arise only for J>100, which is a further reason to look for another simple treat- 2.2 Effective Treatment in Terms of ment of the rotation–vibration problem. Spectroscopic Constants Here, it is necessary to make a remark on the calcula- tion of the effective reduced masses µ in these equations. Equation (32) for the Morse oscillator lends itself to In the practical analysis of spectra, the most commonly a simple interpretation. The purely v-dependent terms used assumption is to introduce the atomic masses mi in are the results of the anharmonic vibrational motion equation (2), i.e., including the masses of the electrons alone. Neglecting the small centrifugal distortion term 2 2 in the atoms with the masses mi. It is, however, by no DvJ (J + 1) , the rotational term BvJ(J + 1) corresponds Fundamentals of Rotation–Vibration Spectra 123

to the energy levels, with an effective rotational E 1 j v,J = y v + [J(J + 1)]k (45) constant Bv. hc jk 2 Separating the fast vibrational motion from the rotational j k motion in a Born–Oppenheimer-like adiabatic manner, B v One can use the following relations (Herzberg 1950, is interpreted as the expectation value of the rotational Demtroder¨ 2003) constant over the vibrational wavefunction: h y10  ωe (46) Bv  Ψ v Ψ v (39) 8π 2cµR2 y20 −ωexe (47) It is furthermore common to allow for a representation y30  ωeye (48) of energy levels by simply extending the power series of equation (32) to higher powers by adding further empir- y01  Be (49) ically adjustable constants (Huber and Herzberg 1979). y02 −De (50) Thus, one can write the vibrational term values as y −α (51) 2 11 e Ev 1 1 = G(v) = ωe v + − ωexe v + y −β (52) hc 2 2 12 e y  γ (53) 1 3 1 4 21 e + ω y v + + ω z v + +··· (40) e e 2 e e 2 B − ω x α ω α2ω2 y  e e e + e e + e e (54) 00 4 12B 144B2 and add to this the rotational term values defined by e e

For the Morse potential, only y10, y20, y01, y02, y11, EJ(v) = F (J ) = B J(J + 1) − D J 2(J + 1)2 and y are different from zero and one can establish hc v v v 12 exact relationships to the more conventional spectroscopic + 3 + 3 +··· HvJ (J 1) (41) parameters. However, in general, equations (46)–(54) are approximate. to give the total rotational–vibrational energy Also, in a real molecule, one might absorb some effects E from higher terms in the lower terms by setting all terms v,J = G(v) + F (J ) (42) hc v after some point in the expansion to zero. These facts have to be observed when comparing results arising from Furthermore, one uses the expansions different fits to experimental data in the literature. 2 = − + 1 + + 1 + ··· Bv Be αe v γ e v (43) 2 2 2.3 Symmetry, Nuclear Spin, and Statistics in 1 1 2 Homonuclear Diatomic Molecules D = D + β v + + δ v + + ··· (44) v e e 2 e 2 Other important effects in the energy level structures One notes the slight inconsistencies in the sign conven- arise for homonuclear diatomic molecules from symmetry, tion: to have commonly positive ωexe,αe,andDe,the nuclear spin, and statistics. The generalized Pauli principle corresponding terms in the power series are defined with is one of the fundamental laws of atomic and molecular sci- a negative sign, whereas all further terms have a positive ence, going beyond the mere laws of quantum mechanics sign. (see Quack 2011: Fundamental Symmetries and Sym- These equations are used in the least squares adjustment metry Violations from High-resolution Spectroscopy, of the corresponding “effective spectroscopic constants” this handbook). The consequences of the Pauli principle (ωe, ωexe, Be, αe, De, βe,etc.) to the measured transition on the electronic structure due to the indistinguishability (Albritton et al. 1976). One should not over- of electrons are discussed in an article by Worner¨ and interpret the physical significance of such parameters, but Merkt 2011: Fundamentals of Electronic Spectroscopy, they can be considered as resulting from the diagonaliza- this handbook. There are, however, further important con- tion of an effective Hamiltonian matrix to appropriate order sequences on rotation–vibration spectra arising from the to obtain its diagonal (eigenvalue) form. identity of nuclei. Indeed, we know that the solution of the A slightly different development was proposed by Schrodinger¨ equation in the Born–Oppenheimer approxi- Dunham (1932): mation may generally be written as 124 Fundamentals of Rotation–Vibration Spectra

(e) total spin wavefunction can be written as a product of the Ψ tot(re,Rn ...)= Ψ e(re,Rn ...)Ψ (Rn) v individual wavefunctions, at least as far as its symmetry · (ev) (evr) Ψ r (ϑ,ϕ,χ)Ψns (σ ) (55) properties are concerned, which results in a total of four possibilities with appropriate symmetry with respect to E = E + E(e) + E(ev) + E(evr) (56) tot e v r ns permutation of the indices 1 and 2 and satisfying the triangle condition for the total nuclear spin I Here, the upper indices indicate the implicit dependence upon electronic (e), vibrational (v), and rotational (r) states |I − I |≤I ≤ I + I (59) by computing the appropriate expectation values in the 1 2 1 2 given state. For the diatomic molecule, we obviously have 1. The three normalized symmetrical functions have total R = R. n nuclear spin I = 1: The wavefunction of equation (55) is a solution of the Schrodinger¨ equation within appropriate approximations, = √1 + = but it is still incomplete because it does not satisfy ϕs [α(1)β(2) α(2)β(1)] with MI 0 1 2 the symmetry requirements imposed by the generalized (60) Pauli principle. A proper Ψ tot must be antisymmetric with = = ϕs α(1)α(2) with MI 1 (61) respect to the permutation of two fermions (particles with 2 = =− half odd integer spin) and symmetric with respect to the ϕs3 β(1)β(2) with MI 1 (62) permutation of two bosons (particles with integer spin). We shall assume that electrons have been dealt with by an 2. The one antisymmetrical wavefunction has I = 0(and = appropriate electronic structure treatment for the electronic MI 0): ground state (see Worner¨ and Merkt 2011: Fundamentals = √1 − of Electronic Spectroscopy, this handbook), resulting in ϕa [α(1)β(2) α(2)β(1)] (63) allowed electronic ground-state terms. We now consider 2 the effect on rotational–vibrational structure arising from Because of the generalized Pauli principle, the total wave- identical nuclei. function Ψ must be antisymmetric with respect to the We start with the simple and historically important case tot permutation (12) of the indices of the two protons; thus, of H2 (see also Quack 2011: Fundamental Symmetries and Symmetry Violations from High-resolution Spec- (12)Ψ =−Ψ (64) troscopy, this handbook). The ground-state electronic term tot tot is 1Σ+, the electronic wavefunction, thereby satisfying the g Considering the properties of the product wavefunction Pauli principle for electrons. This totally symmetric term in (equation (55)), one can thus either combine symmetric the point group D∞ is also symmetrical upon the permuta- h rotational functions with even J with the antisymmetric tion of the two protons in H . The vibrational wavefunction 2 nuclear spin function (equation (63)) or else combine the Ψ (e)(R) or Φ (R) depends on the magnitude of separa- v v,J antisymmetric rotational wavefunctions with odd J with tion R of the two nuclei and, thus, is also symmetric with the three symmetric nuclear spin functions with an extra respect to a permutation of the coordinates of the two pro- degeneracy due to M = 0, ±1, giving an extra nuclear tons. For a diatomic molecule, the rotational wavefunction I spin statistical weight 3 for all odd J (in addition to the Ψ (ev)(ϑ,ϕ,χ) reduces to the spherical harmonics Y r J,MJ (2J + 1) degeneracy resulting from M ). These two sets of (equation (26)). These are symmetric under permutation of J states of hydrogen do not interconvert easily because of the the coordinates of the two nuclei for even J and antisym- dynamical principle of approximate nuclear spin symmetry metric for odd J as can be seen by the inspection of the conservation (see Quack 2011: Fundamental Symmetries spherical harmonics. We discuss next the symmetry of the and Symmetry Violations from High-resolution Spec- nuclear spin functions Ψ (evr)(σ ). Each proton may exist ns troscopy, this handbook). They behave like different iso- in two spin states, depending on the magnetic quantum mers of hydrogen H (called nuclear spin isomers), which number m for the projection of the proton spin on the 2 p can be separated and can be stable for months at room z-axis, which we define as temperature in the absence of catalysts for interconversion. =+ ≡ = The more abundant nuclear spin isomer with the higher ψspin, p (mp 1/2) α(j 1, 2) (57) 1,2 nuclear spin statistical weight 3 (and odd J for H2) is called =− ≡ = ortho-hydrogen, while the less abundant nuclear spin iso- ψspin, p1,2 (mp 1/2) β(j 1, 2) (58) mer with nuclear spin statistical weight 1 (and even J for Depending on whether we consider the first or the second H2) is called para-hydrogen. In ordinary hydrogen gas, the proton (j = 1, 2), we write α(1), α(2), or β(1), β(2). The two forms coexist as a mixture. Because of the extra nuclear Fundamentals of Rotation–Vibration Spectra 125 spin statistical weights (3 and 1), in samples at thermal will be lifted slightly for instance) and more generally equilibrium with many J populated, there will be an inten- lead to further deviations from the expression for Etot in sity alternation (3 : 1) in spectral lines associated with odd equation (56). or even J in the ground state of the relevant transition A second consideration concerns the symmetry properties (for instance, in the rotation–vibration Raman spectrum of electronic terms under permutation of identical nuclei. 3 − of H2; see Weber 2011: High-resolution Raman Spec- We illustrate this with the example of the O2( Σg ) ground troscopy of Gases, this handbook). Similar effects arise state, which has led to some confusion not only in the for all homonuclear diatomic molecules with nuclei of spin textbook literature (for instance, Demtroder¨ 2003, but also 15 1/2 (for instance, N2). For nuclear spins Ii > 1/2, one can in a number of other cases). This is because of the readily show, by inspection of the functions generated fol- understanding of the symmetry property of the electronic 3 − lowing the triangle condition (equation (59)), that the total term Σg (in D∞h) as related to the symmetry under the number of symmetrical nuclear spin functions is permutation (12) of the two nuclei. It is, thus, necessary to consider the permutation of the two nuclei arising from a (2I + 1)2 + 2I + 1 N = i i (65) sequence of point group operations as follows: s 2 ◦ 1. Rotate the molecule by 180 . and the total number of antisymmetrical functions is 2. Invert the electrons back through the center of symme- 2 try. (2Ii + 1) − 2Ii − 1 Na = (66) 3. Reflect the electrons at the plane containing the molec- 2 ular symmetry axis 1–2 perpendicular to the axis of

Obviously, with Ii > 1/2, one has more than two possible rotation involved in operation 1. values of I in equation (59). The deuterium nucleus, for Operation 1 changes the sign of the rotational wave- instance, has ID = 1 and thus I = 0, 1, 2 with Ns = 6and function for odd rotational angular momentum quantum Na = 3, in the case of D2. Because the deuteron is a boson, the total wavefunction numbers J as discussed above, or N, in conventional nota- tion, if J contains an electronic spin contribution (as in the Ψ tot must be symmetrical; thus, even J (for the electronic − + 3 | − |≤ ≤ + ground state 1Σ ) combine with the six symmetrical triplet ground state of O2( Σg ), N S J N S g = nuclear spin wavefunctions and are called ortho-deuterium with total electronic spin S 1 for a triplet). Operation 2 (because of the higher abundance, see above) and odd changes the sign of the electronic wavefunction for unger- J combine with the three antisymmetrical wavefunctions ade terms (Σu, Π u, ∆u,. . .). For gerade terms such as Σg, the sign remains unchanged. Operation 3 changes the sign (para-deuterium). − − Several further considerations are now in order. Equation of the electronic wavefunction for X terms (here Σ ). (55) is an approximation, and, in particular, nuclear For degenerate electronic terms Π, ∆,. . . the twofold =±| | spin–rotation interactions can couple states of different degeneracy with Λ Λ is slightly lifted by molecular nuclear spin as non-Born–Oppenheimer interactions can rotation (Λ-doubling, see below and see also Worner¨ and couple different electronic states. Nevertheless, it is possible Merkt 2011: Fundamentals of Electronic Spectroscopy, to obtain the proper symmetry relations from the approx- this handbook) containing a symmetric and an antisymmet- imation (equation (55)), as can be seen from the exact ric function under this reflection. The total wavefunction wavefunction, which in general will be a sum of infinitely must, thus, be symmetric including the sequence of the many terms of the form of equation (55): three operations above for bosons, and antisymmetric for fermions. j 16 3 − = Applying this to O2( Σ ), one thus finds that the elec- Ψ tot,exact cj Ψ tot(re,Rn,...) (67) g j tronic term is antisymmetric with respect to the permutation of the nuclei. The total wavefunction must be symmetric, Our discussion following equation (55) concerned only the because the 16O nucleus with nuclear spin I = 0 is a boson. dominant leading term with the largest cj in equation (67). Thus, only odd rotational angular momenta N = 1, 3, 5 j 3 − However, all further terms Ψ tot must satisfy the generalized result in allowed states for O2( Σg ) (with nuclear spin sta- 18 Pauli principle in the same way as the leading term, even tistical weight 1). The same is true for O2 with the boson though they may mix individual terms in the products 18OhavingI = 0. Thus, these molecules have a “zero point 0 (vibronic mixing, nuclear spin rotation mixing, etc.). It is rotation”, which affects the dissociation energy ∆dissH0 of 16 18 16 18 thus sufficient to consider only the leading term, if one is O2 and O2, but not of O O, which has all rotational only interested in the symmetry properties. Mixing may, angular momenta allowed (N = 0, 1, 2,. . .), because there however, decrease degeneracies (nuclear spin degeneracy are no symmetry restrictions. 126 Fundamentals of Rotation–Vibration Spectra

For 17O, one has I = 5/2 and thus this nucleus is anticipated the neutron as a component of the nucleus. a fermion. The total wavefunction Ψ tot must be anti- Intensity alternation in spectra was discovered by Mecke 3 − (1925) and discussed by Hund (1927) (see also Rasetti symmetric. Therefore, the antisymmetric Σg electronic ground state can combine with antisymmetric (odd N) 1930, Heitler and Herzberg 1929). Some of the interest- rotational wavefunctions and antisymmetric nuclear spin ing history is covered by Herzberg (1939, 1950) (see also 17 wavefunctions (Na = 15 from equation (66), para- O2) Quack 2011: Fundamental Symmetries and Symmetry or with symmetric (even N) rotational wavefunctions and Violations from High-resolution Spectroscopy, this hand- symmetric nuclear spin wavefunctions (Ns = 21 accord- book). 17 ing to equation (65), ortho- O2). Figure 19 in the arti- The fact that the neutron (free or bound in the nucleus) cle by Worner¨ and Merkt 2011: Fundamentals of Elec- cannot be an electron bound to a proton is also clear from tronic Spectroscopy, this handbook, gives a summary of the spin and statistical properties: the neutron has spin 1/2 the symmetry properties of a number of electronic terms and is a fermion, whereas the electron binding proton would − of diatomic molecules. We have introduced these basis be a boson with spin 0 or 1. Thus, the radioactive β decay concepts of symmetry, nuclear spin, and statistics for the also + − simplest case of diatomic molecules. The general concept n → p + e (incomplete) (68) is of relevance for more complex symmetric molecules with identical nuclei, in which the use of group theory cannot be complete, because of the statistics and spin, and becomes necessary (see Oka 2011: Orders of Magnitude now we know that the symmetry properties are saved by and Symmetry in Molecular Spectroscopy and Quack the electron antineutrino, a fermion of spin 1/2 2011: Fundamental Symmetries and Symmetry Viola- + − tions from High-resolution Spectroscopy, this handbook). n → p + e + νe (correct) (69) The intensity alternations resulting from nuclear spin sta- tistical weights in spectral line patterns are among the Thus, while the neutrino was postulated historically by more powerful “practical” instruments in the symmetry Pauli in order to preserve the symmetry law of energy assignment of molecular states. Good examples of diatomic conservation in β-decay, a second route to this hypothesis molecule spectra with intensity alternations are given in would be spin and statistics (see also Quack 2011: Fun- Weber 2011: High-resolution Raman Spectroscopy of damental Symmetries and Symmetry Violations from Gases, this handbook. For spectra of polyatomic molecules, High-resolution Spectroscopy, this handbook). see Albert et al. 2011: High-resolution Fourier Trans- A note is also relevant concerning the nature of nuclear form Infrared Spectroscopy, this handbook. “spin”: this property would better be called nuclear angu- Apart from the practical importance in the assignment lar momentum, because in general it contains contributions of high-resolution spectra, the role of nuclear spin, sym- from orbital angular momentum in current nuclear mod- metry, and statistics has been of fundamental importance els. This fact is actually also important for nuclear parity for the understanding of nuclear structure in a historical and quadrupole moments. Table 1 lists a small selection context. Before the , one hypoth- of properties of particularly important nuclei in molecu- esis of nuclear structure assumed that the heavier nuclei lar spectroscopy; a more detailed table is given by Hippler contained electrons, bound to protons within the nucleus et al. 2011: Mass and Isotope-selective Infrared Spec- by a special force, in order to neutralize part of the pro- troscopy, this handbook and Cohen et al. (2007). This ton charges. In this case, the deuteron would consist of table is instructive because it shows the apparent com- two protons and an electron tightly bound together. Such plexity of the patterns of nuclear angular momenta and a “deuteron” would have half odd integer spin and be a parities given by I Π as understood on the basis of the fermion, leading to a quite different nuclear spin statis- relatively simple nuclear shell model of Goeppert–Mayer tics than observed in reality. The historical example was and Jensen including orbital angular momenta. Some sim- 14 14 actually N2, where the real N nucleus is a boson of ple (and obvious) rules include the absence of nuclear spin 1 (similar to the deuteron), whereas the hypotheti- spin for gg nuclei (gerade–gerade for even numbers 14 cal “electron binding” 7N would consist of seven elec- of protons and even numbers of neutrons). Nuclei with trons and 14 protons, resulting again in a “fermion 14N” I ≥ 1 have quadrupole moments, leading, in general, to with half-odd integer spin. The intensity alternation actu- more easily visible hyperfine structure in rotation–vibration 14 ally observed in the Raman spectra of N2 demonstrated spectra. Magnetic moments mN are of obvious relevance the boson (spin 1) nature of 14N (resulting from seven for magnetic resonance spectroscopy (Ernst et al. 1987). protons and seven neutrons), thus excluding the “elec- We note that the masses m given in Table 1 are atomic tron binding” model of the nucleus. Early high-resolution masses (including the electron masses). Table 2 gives spectroscopy thus helped to shape nuclear theory and for further illustration some ground state properties of Fundamentals of Rotation–Vibration Spectra 127

Table 1 Ground-state properties of some selected light nuclei: charge number Z, element symbol, mass number A,atomicmassm, natural abundance as mole fraction x in terms of percent, nuclear “spin” I and parity Π, magnetic moment mN (in units of the Bohr magneton µN), and quadrupole moment Q.

Π 2 Z Symbol Am/Da 100xI mN/µN Q/fm

1H 11.007825032 99.9885 (1/2)+ +2.7928473 0 D22.014101777 0.0115 1+ +0.8574382 +0.286 5 B 10 10.012937 19.93+ +1.800644 +8.47 11 11.009305 80.1 (3/2)− +2.688648 +4.07 6 C 12 12.0(defined) 98.93 0+ 00 13 13.003354837 1.07 (1/2)− +0.702411 0 14 14.003241989 − 0+ 00 7 N 14 14.003074004 99.636 1+ +0.403761 +2.001 15 15.000108898 0.364 (1/2)− −0.2831888 0 8 O 16 15.994914619 99.757 0+ 00 17 16.99913170 0.038 (5/2)+ −1.89379 −2.578 18 17.999161 0.205 0+ 00 9 F 19 18.9984032 100 (1/2)+ +2.628868 0

Table 2 Ground-state properties of some molecules: electronic emission and the spectra of homonuclear diatomic mole- 2S+1 ground-state term symbol Γ x , the ground-state N value, cules are observable by Raman spectroscopy (see Merkt angular momentum quantum number J , total nuclear spin I,and and Quack 2011: Molecular Quantum Mechanics and total angular momentum F with total parity Π in the ground state. Molecular Spectra, , and Interac- tion of Matter with Radiation, this handbook). To discuss 2S+1 Π Molecule Γ x NJ I F the structure of actually observed line spectra, we have to address the consequences of selection rules and rela- 1 + + H2 Σg 00 0 0 tive and absolute intensities of the lines. For electric dipole 12 16 1 + + C O Σ 00 0 0 transitions, the integrated line strength G (or integrated 13 16 1 + − fi C O Σ 0 0 1/2 (1/2) absorption cross section) is proportional to the absolute 14C16O 1Σ+ 00 0 0+ 12C17O 1Σ+ 0 0 5/2 (5/2)+ square of the matrix element of the electric dipole operator: 15 1 + + N2 Σg 00 0 0 + − 3 14N15N 1Σ 0 0 1/2 (1/2) 8π 2 − Gfi = |Mfi| (70) 0 0 3/2 (3/2) 3hc0(4π0) 16 3 − + O2 Σg 10 0 0 10 1+ with G being given by the appropriate integration over the + fi 20 2 molecular absorption cross section: 19 1 + + F2 Σg 00 0 0 −1 Gfi = σ fi(ν)˜ ν˜ dν˜ (71) molecules, the complete ground state having well defined line “particle properties” (including nuclear parity) and might and be considered similar to a fundamental particle of some relevance for applications in fundamental physics (see 2 |M |2 = f µ i (72) Quack 2011: Fundamental Symmetries and Symmetry fi ρ ρ Violations from High-resolution Spectroscopy, this hand- book). where the sum is extended over the space-fixed Cartesian The total angular momentum quantum number |I − J |≤ coordinates with the components µρ of the electric dipole F ≤ I + J arises from the combination of the J (= N if operator (ρ = x,y,z) = S 0) and nuclear spin (I) angular momentum. = = µ µρeρ riqi (73) 2.4 Radiative Transitions and Spectra of ρ i Diatomic Molecules qi are the charges of the quasi-point particles in the The rotation–vibration spectra of heteronuclear diatomic molecule (electrons and nuclei), ri the corresponding posi- molecules are readily observed by IR absorption and tion vectors, and ex , ey,andez the unit vectors. Analysis of 128 Fundamentals of Rotation–Vibration Spectra

the rotational wavefunctions (see Merkt and Quack 2011: J′ Molecular Quantum Mechanics and Molecular Spectra, 3 Molecular Symmetry, and Interaction of Matter with Radiation, this handbook) results in the angular momentum 2 for an allowed rotational–vibrational transi- R(1)–P(1) 1 tion in a diatomic molecule in a 1Σ state such as CO 0

∆J =±1 (74)

The group of lines corresponding to ∆J =+1 is called the P(3) R(2) R-branch with (initially) increasing wavenumber and the P(2) R(1) group of lines with ∆J =−1 the P-branch, with (initially) P(1) R(0) decreasing wavenumber. Figure 7 shows a CO spectrum J′′ with assignment of the observed transitions given as P(J ) and R(J ), where J is the angular momentum quantum 3 number of the lower level, as illustrated by the level scheme 2 of Figure 8. R(0)–P(2) 1 The figures also clearly illustrate how energy inter- 0 vals in the vibrational ground state can be obtained by “ground-state combination differences” (GSCDs), i.e., dif- Figure 8 Rotational levels in the lower vibrational state (referred to as J ) and in the upper vibrational state (referred ferences of spectral line frequencies. For example, the difference ν˜[R(0)] −˜ν[P(2)] corresponds to [E(J = 2) − to as J ). Some transitions of the P- and R-branch as well as combination differences are also indicated. E(J = 0)]/(hc) in the vibrational ground state. Thus, IR rotation–vibration spectroscopy at high resolution pro- vides an alternative to pure rotational (MW of far IR) spectroscopy for determining rotational energies in the Linear approximation vibrational ground state. Similarly, excited state rotational (tangent at Re) energies can be isolated by appropriate combination differ- ences, for example, ν˜[R(1)] −˜ν[P(1)] in Figure 8. ) R To obtain exact absolute transition intensities for rota- ( tion–vibration transitions, one has to evaluate the inte- gral in equation (72) using the exact wavefunction from equation (26), where Φv,J (R) is in general not simple. However, because it depends upon R,wehavetoconsider the R-dependent dipole moment function µ(R). The qual- 0 Re R itative behavior and even the semiquantitative behavior of Figure 9 Dipole moment µ(R) of a diatomic molecule as a function of the distance between the nuclei R.

R(0) R(0) 2 2 these electric dipole functions are often well described by a FWHM function proposed by Mecke (1950) and shown graphically −1 R(2) 1 0.0068 cm P(1) in Figure 9:

0 n 1 µ(R) = bR exp(−αR) (75) 2147.05 .10 Absorbance P(3) where b, n, and α are adjustable parameters to describe R(4) P(5) the behavior of µ(R). It is obvious that homonuclear 0 diatomic molecules do not show electric dipole pure rota- 2100 2120 2140 2160 2180 tion–vibration spectra, because µ(R) vanishes for symme- ∼ − try reasons at all values of R. n /cm 1 To derive some further simple approximate rules, one ≡ Figure 7 CO spectrum measured at low temperature sometimes considers pure vibrational wavefunctions Ψ v (T ≈ 13 K) in a supersonic jet. [Reproduced with permission Φv,J =0(R) as an approximation for the vibrational part of from Amrein et al. (1988b).] the wavefunction, also for higher J , and one describes the Fundamentals of Rotation–Vibration Spectra 129 dipole function with a Taylor series expansion: ∂µ −1 µ(R) = µ(Re) + (R − Re) ∂R R e 1 ∂2µ − + (R − R )2 +··· (76) 2 2 e

2 ∂R ]

Re 2

Thus, one can define a purely vibrational transition moment /(pm) −3

G [

g ∂µ l M = µ(R ) Ψ |Ψ + Ψ |(R − R )| Ψ  vv e v v v e v −4 ∂R CHCl3 exp. Re CHCl3 Fit 1 ∂2µ 2 CHF3 exp. + Ψ v (R − Re) Ψ +··· (77) 2 ∂R2 v −5 CHF Fit Re 3

The first term in equation (77) vanishes for v = v , because of the orthogonality of the vibrational eigenfunctions. For −6 v = v , it results in the “permanent electric dipole moment” 1234 of the molecule, leading to pure rotational transitions. (a) N For harmonic oscillator functions, one has a selection 2.0 rule ∆v =±1 for the second term and ∆v = 0, ±2forthe third term. This nonlinear term thus provides intensity for overtone transitions ∆v = 2 even in the harmonic oscillator 1.5 CHF3 (one refers figuratively to “electrical ” due to ) e this term). On the other hand, if one has anharmonic (Morse R 1.0 oscillator-like) wavefunctions Ψ v, Ψ v , then even the linear )/ ( R term in the dipole moment (∂µ/∂R)R (R − Re) leads to a ( e CHCl3 nonvanishing electric dipole matrix element for |∆v| > 0. Thus, one refers to the intensity of the overtone transition 0.5 being due to “mechanical anharmonicity”. In general, both effects are important and for diatomic molecules, it is in 0 fact not very difficult to carry out the integration exactly, 0.5 1.5 2.5 (b) provided that µ(R) is known. For polyatomic molecules, R/ Å the situation is more complex and, therefore, many calcula- tions of vibrational intensities are carried out for separable Figure 10 (a) Overtone intensities for the CH chromophore normal vibrations in the so-called double harmonic approx- (quantum number N)0→ N transition (after Lewerenz and imation, which assumes harmonic oscillator wavefunctions Quack 1986). Experimental and theoretical fit for CHF3 (R /R = 6.0, i.e.,  1) and for CHCl (R /R = 1.05). (b) and retains only the linear term in the Taylor series expan- m e 3 m e Empirical dipole functions, approximately valid around Re ± 0.3 sion of the dipole moment function. Obviously, this can at A˚ and extrapolated with the Mecke function outside this range. best give an approximation for “fundamental” vibrational For CHF3 and CDF3 µ(Re) would be 1.01 and 1.3 D, for CHCl3 transitions (i.e., ∆v = 1 from v = 0) as overtone transitions 3.8 D, and for CDCl3 1.5 D, when adjusted to the fundamental band strength with the Mecke function (note that permanent dipole vanish in this approximation. ± ± moments µ0 are for CHF3 1.6 0.1 D and for CHCl3 1.4 0.5D Figure 10 illustrates electric dipole functions for the in qualitative agreement). See Lewerenz and Quack (1986) for rather localized CH-stretching normal vibrations in CHF3 precise definitions and parameters and further discussion. and CHCl3 (“quasi-diatomic vibration” but including some motion of the heavy atoms). Although a linear approximation might be acceptable for CHF3, the dipole larger than the contribution from the term (∂µ/∂R)Re .Such functions of CHCl3 show a maximum in µ(R) near phenomena must be considered when using quantum chem- ≈ ˚ Re 1 A; thus, (∂µ/∂R)Re is zero or very small, which ical program packages computing approximate (“double is reflected by the overtone intensity (∆v = 2) for CH- harmonic”) vibrational band intensities, but they are obvi- stretching in CHCl3, which is larger than that for the fun- ously not of fundamental importance, because more accu- damental transition (∆v = 1), because the contribution to rate calculations are possible. For intensities and selection 2 2 the transition intensity from the term with (∂ µ/∂R )Re is rules in Raman spectra, we refer the reader to Merkt 130 Fundamentals of Rotation–Vibration Spectra

and Quack 2011: Molecular Quantum Mechanics and 2Π 2Π 3/2 1/2 g Molecular Spectra, Molecular Symmetry, and Interac- 300 tion of Matter with Radiation and Weber 2011: High- J = 5/2 24 resolution Raman Spectroscopy of Gases, this hand- J = 7/2 32

book. − 1 200 J = 3/2 16 cm /

T J = 1/2 8 2.5 Diatomic Radicals in Degenerate Electronic 100 J = 5/2 24 Ground States

J = 3/2 16 This topic is closely related to the electronic structure and 0 spectra discussed in detail by Worner¨ and Merkt 2011: Fundamentals of Electronic Spectroscopy, this hand- Figure 11 Level scheme for the lowest states of the OH radical. book. We give here just a short account of some of the most important consequences in rotation–vibration spectra 2 2 thus, Π 1/2 is lower than Π 3/2 in energy, whereas for observed in the IR and illustrate these with two promi- the OH radical ground state, one has A =−139 cm−1,and nent examples. For degenerate electronic ground states of 2 2 Π 3/2 is lower in energy than Π 1/2. There is a contri- diatomic radicals, one uses the following nomenclature bution to the from the electronic angular for the electronic angular momentum quantum number Λ, momentum with the quantum number Ω = |Σ + Λ|: which gives the absolute magnitude of the projection of the electronic angular momentum on the molecular axis T (J, Ω) = B [J(J + 1) − Ω2] with J ≥ Ω (83) el =  rot v Jz Λ The resulting term values are thus approximately Λ = 0, 1, 2, 3,... (78) 2 nomenclature Σ,Π,∆,Φ T = Te + Trot = T0 + AΛΣ + Bv[J(J + 1) − Ω ] (84)

The analogy to the nomenclature S, P, D, and F for Figure 11 shows the level scheme for OH relative to the atomic terms with L = 0, 1, 2, 3 is readily seen, although lowest level with J = 3/2. the physical significances of L and Λ are quite different. The degeneracies in these levels are partly lifted by Similarly, the S for the electronic Λ-doubling and hyperfine splittings. The two Λ compo- term is given by the upper left exponent in the electronic nents correspond to eigenfunctions of slightly different term symbol: energy and angular dependence

(2S+1) ΛΣ+Λ (79) 1 ΨΛ± = √ exp(−iΛϕ) ± exp(iΛϕ) (85) 2 The quantum number Σ for the projection of the electron spin onto the molecular axis can take 2S + 1values: with ϕ being the angle of rotation with respect to the molecular axis. This “Λ-doubling” in OH is in the gigahertz =− − + − + − Σ S, S 1, S 2,. . .,S 1,S (80) range. Finally, hyperfine coupling of the molecular angular momentum J with nuclear spin I leads to levels with total For instance, in the case of the doublet ground states of the angular momentum quantum numbers: important NO and OH radicals, one has Λ = 1 and thus a 2Π term with the two possibilities |J − I| ≤ F ≤ J + I (86) 2 2 Π 1/2 and Π 3/2 (81) For the ground state of OH with J = 3/2andI = 1/2for = In addition to the rotational energy given by the term the proton, thus one has two possibilities with F 1or2. formulae in Section 2.2, one has the extra “electronic” term The corresponding level scheme is shown in Figure 12. for the rotation vibration structure: We have provided this discussion to give an idea of the orders of magnitude expected for high-resolution spectra. Te = T0 + AΛΣ (82) For much more detailed discussion of the corresponding spectral structures, we refer the reader to Zare (1988), The spin–orbit coupling constant A can be positive or neg- Brown and Carrington (2003) and the article by Worner¨ and ative. For the NO ground state, one has A = 123.1cm−1; Merkt 2011: Fundamentals of Electronic Spectroscopy, Fundamentals of Rotation–Vibration Spectra 131

Λ Hyperfine F g γN 1 Splitting T (N) = BN(N + 1) + for J = N + (91) splitting 1 2 2 2 5 γ(N + 1) 1 50 MHz T (N) = BN(N + 1) − for J = N − (92) 13 2 2 2 The spectroscopic constant γ is frequently very small. For 3Σ states, one has three sublevels for each rotational quantum number N: J = 3/2 1700 MHz 2Π 3/2 T1(N) = BN(N + 1) − λ + B(2N + 3) 1/2 − B a2 − 2a + (2N + 3)2 γ 2 5 + γ(N + 1) + for J = N + 1 (93) 50 MHz 2 13 T2(N) = BN(N + 1) for J = N (94) Figure 12 Level scheme for the OH radical including hyperfine T3(N) = BN(N + 1) − λ − B(2N − 1) splittings (schematic, not to scale). 1/2 + B a2 − 2a + (2N − 1)2 this handbook (see also Western 2011: Introduction to γ − γN − for J = N − 1 (95) Modeling High-resolution Spectra, this handbook). 2 In practice, one can use the following approximate term = formulae: with a λ/B. The coupling constant λ is frequently on the order of rotational constants. 2 A 2 2 If one has an electronic angular momentum Λ>0, the T( Π ) = + B + (J + 1/2) − Λ (87) 3/2 2 eff selection rules for IR rotation vibration transitions are − 2 A 2 2 T( Π ) = + B − (J + 1/2) − Λ (88) ∆J = 0, ±1 (96) 1/2 2 eff A = A − 2B (89) where the ∆J = 0 transitions correspond to a Q-branch. 1 ± B Figure 13 shows as an example the low-temperature vibra- Beff± = B (90) AΛ tional spectrum of NO with one Q-branch because at the low temperature of about 17 K in a supersonic jet 2 If one has Λ = 0, one uses the term formulae for the expansion, only the Π 1/2 ground state is populated. At sublevels of a 2Σ state: higher temperatures, one finds a second Q-branch arising

= NO jet-FTIR, Trot 17 K

0.2

0.1 Absorbance

0

1850 1860 1870 1880 1890 1900 ∼ n /cm−1

Figure 13 Spectrum of NO at low temperature (17 K) in a supersonic jet. Here, the Napierian absorbance ln(I0/I) is shown, see also Amrein et al. (1988b). 132 Fundamentals of Rotation–Vibration Spectra

Nitric oxide T = 300 K R(7/2) 1.5 R(5/2) 1/2 1/2 R(7/2)3/2 R(5/2) Q(3/2) R(3/2) 3/2 1.0 P(5/2)1/2 3/2 P(5/2) Q(5/2)3/2 3/2 Q(3/2) R(1/2)1/2 P(3/2)1/2 1/2 0.5 Q(1/2)1/2 Absorbance Q(5/2)1/2 0

0.15 jet - FTIR

0.10

0.05

0

= Trot 17 K

1867.2 1867.6 .7 1871.0 .1 1875.7 .8 .9 6.0 .1 1881.0 .1 1884.3 1887.5.6 1890.7 .8 .9 ∼ − n /cm 1

Figure 14 High-resolution (0.004 cm−1 FWHM) FTIR spectra of neat . Only small regions of the spectra containing spectral lines are displayed on an expanded scale (0.05 cm−1 per division). Top: spectrum recorded in a static cell at room temperature (T = 300 K). Center: jet-FTIR spectrum. (Nozzle diameter 100 µm, distance from the nozzle z = 6 mm, backing pressure 650 kPa, 2 background pressure 0,6 Pa.) Note the absence of lines of the excited electronic state Π 3/2. Bottom: computed spectrum for a rotational temperature of 17 K. [Reproduced with permission from Amrein et al. (1988b).] The Napierian absorbance ln(I0/I) is shown.

Table 3 Summary of the conventions for the various angular Fundamentals of Electronic Spectroscopy, this hand- momentum quantum numbers. book).

Quantum number Description

Λ Projection of electronic angular 2.6 The Determination of Molecular momentum Structures from Infrared and Raman Σ Projection of electronic spin Rotation–Vibration Spectra Ω = |Λ + Σ| Projection as given by formula . (sometimes Ω = Λ + Σ) N Angular momentum from The article by Bauder 2011: Fundamentals of Rotational rotation excluding spins Spectroscopy, this handbook, has already provided an S Electronic spin extensive discussion of the practice of determination of | − | ≤ ≤ + N S J N S Angular momentum including structure from pure rotation (MW) spectra. We complement electronic spin this here with some additional aspects arising in the Ii Nuclear spin of nucleus i I Total nuclear spin analysis of rotation–vibration spectra in terms of molecular |I − J | ≤ F ≤ I + J Total angular momentum structure. We use diatomic molecules for illustrating the basic concepts because of their relative simplicity. Most of the concepts can be readily transferred to the much more 2 from Π 3/2 and also the corresponding P- and R-branch complex situation of polyatomic molecules. transitions; Figure 14 shows expanded sections of the spec- In Section 2.4, we have already discussed the deter- trum, in which the effects of Λ-doubling are visible. mination of the vibrational ground-state rotational level Table 3 provides a brief didactic summary of the conven- structures from GSCDs in rotation–vibration spectra. Thus, tions for the various angular momentum quantum numbers with such data, essentially all the methods discussed in (see also Stohner and Quack 2011: Conventions, Symbols, Bauder 2011: Fundamentals of Rotational Spectroscopy, Quantities, Units and Constants for High-resolution this handbook, for the analysis of pure rotational MW spec- Molecular Spectroscopy and Worner¨ and Merkt 2011: tra can be applied to determine the molecular structure. Fundamentals of Rotation–Vibration Spectra 133

Usually, the MW data are much more precise, whereas the the ground-state probability distributions derived from the IR GSCDs frequently cover a much larger range of rota- vibrational wavefunctions, and for a typical X–H bond tional levels up to very high J for the heavier polyatomic length, one obtains relative uncertainties in the range 5–8%, molecules. If both sets of data are available, a combined for heavier molecules, thus, in the percent range. Moving analysis using a weighted least squares adjustment is fre- beyond this, one could argue that certain parameters of quently in order. these distributions can be derived more accurately (for The new aspect of rotation–vibration data consists in instance, the expectation value of the distribution, or obtaining results for excited vibrational states in terms the maximum in the distribution). These will be isotope of Bv, Dv, etc. Adjustment of the appropriate constants dependent, but still would be empirically well defined. One in equations (43) and (44) to the observed spectroscopic could argue that the physical sizes, say, of HF and DF data results, among other constants in values for the are actually different, for instance, when interacting with “equilibrium rotational constant Be”, is frequently possible other molecules. With very accurate data, one might expect with a precision of five to six significant digits. This then to get again relative uncertainties in the 10−4 –10−5 can be translated into an equilibrium bond length Re of range. similar accuracy by means of equation (34) provided that However, one probably must be extremely cautious when the uncertainty in the reduced masses µ is negligible or interpreting structural data at a higher relative accuracy. disregarded and that the physical significance of Re as While the frequencies in spectra can, of course, be mea- the position of the minimum of the potential function can sured to much higher accuracy, their interpretation in terms be justified within the framework of an accurate effective of structure should probably in general not claim accu- Hamiltonian model. Both these aspects leave room for racies corresponding to relative uncertainties better than some doubt. We have already discussed uncertainties in 10−4 –10−5 at best, in terms of their real physical signifi- the definitions of reduced masses in Section 2.3. The first cance, regardless of which concept or definition of structure uncertainty results from the point mass assumption for is used. nuclei, which have extensions in the femtometer range. For typical bond lengths, in the 100 pm range (1 A),˚ the relative uncertainty due to this approximation is about 3 POLYATOMIC MOLECULES 10−5, similar to common experimental uncertainties. The relative uncertainties related to the treatment of electronic 3.1 General Aspects masses when calculating µ can be similarly estimated − − to be about 10 5 to about 10 4 (given the electron: The treatment of the rotation–vibration dynamics and spec- − proton mass ratio of 5 × 10 4). An even more serious tra of polyatomic molecules is conceptually similar but uncertainty might arise from the concept of an effective more complex than that for diatomic molecules because potential. Using slightly different effective (“BO-like”) of the larger number of internal degrees of freedom. Hamiltonians, different definitions of these potentials lead Indeed, in the adiabatic or Born–Oppenheimer approxi- to different minima. This effect is not easily estimated. mation and related approximations the potential function However, different Re obtained for different isotopomers V (R) in equation (1) now is to be replaced by a potential can give an indication of the actual uncertainties. For energy hypersurface V(q1,q2,q3,. . .,q3N−6) depending 1 19 H = − example, for H F, one finds Re 91.6808 pm and for on 3N 6 degrees of freedom, where N is the number 19 D = D F one has Re 91.694 pm (Huber and Herzberg 1979). of atoms or nuclei in the molecule. At present we take −4 The relative difference is on the order of 10 and can the qi to be some kind of generalized coordinate (see be considered to arise from a combination of several of below). the uncertainties discussed. Thus, while it is possible to In principle, any consistent set of coordinates could determine “apparent” Re in a self-consistent way from be chosen to formulate the Hamiltonian for the quantum spectroscopic data to within uncertainties of 10−5 or dynamics of the molecule given an appropriate effective 10−6 (and even better), one should be cautious about potential energy hypersurface. A conceptually and dynami- claiming relative uncertainties of less than about 10−4, cally simple choice consists of simply taking the Cartesian when the underlying physical assumptions and significance coordinates of the atoms (or nuclei) x1,y1,z1,...,xi,yi, are considered. zi ...,xN ,yN ,zN . In practice, it is, however, desirable to From another point of view, one could argue that the reduce the number of degrees of freedom by appropri- actual uncertainty in molecular structures (here, the bond ate (exact) separation of the center of mass motion and lengths of diatomic molecules) is related to the root- approximate separation of rotational motion. Often, there mean-square deviation of distances measured in actual are special physical aspects of the internal molecular motion scattering experiments. This can be readily calculated from that suggest the use of other special sets of coordinates, for 134 Fundamentals of Rotation–Vibration Spectra instance, for low-frequency internal degrees and other large With this simplified classical Hamiltonian, the dynamics amplitude motions (see Bauder 2011: Fundamentals of are separable. One can show that there is an orthogonal Rotational Spectroscopy, this handbook). There is, how- transformation to the set of normal coordinates Qk (Wilson ever, one set of coordinates that finds, by far, the widest et al. 1955): range of applications for ordinary rigid molecules: normal 3N coordinates make the vibrational problem separable in the Q = L q (102) limit of the harmonic (multidimensional) potential function. k ik i i=1 The vibrational Hamiltonian can be written as a sum of Hamiltonians of harmonic oscillators. We introduce here After this orthogonal transformation with L−1 = LT, one the basic concepts of the normal coordinate treatment, just can write the kinetic energy T as in order to define the main features, nomenclature, and con- ventions. A fairly complete treatment from the point of view 1 3N dQ 2 T = i (103) of molecular spectroscopy can be found in Wilson et al. 2 dt (1955) and numerous other papers. i=1 The potential energy V is given by

3.2 Normal Coordinates and Normal Vibrations 1 3N V = λ Q2 (104) 2 k k We start by writing the classical Hamiltonian for the k=1 motion of N atoms in Cartesian coordinates x ,x ,. . .,x , 1 2 i with λ being functions of the force constants F .This ...,x : k ij 3N allows us now to write the classical Hamiltonian for 3N 2 every separate vibrational mode and the total vibrational m dx H = i i + V(x ,. . .,x ) (97) Hamiltonian as a sum over all the modes: 2 dt 1 3N i=1 3N dQ H = H Q , i (105) We consider the potential function with a well-defined min- i i dt imum at the equilibrium geometry of the molecule defined i=1 e by xi . It is then convenient to introduce mass-weighted Translating this to quantum mechanics and solving the displacement coordinates qi defined by appropriate Schrodinger¨ equation (See Merkt and Quack √ √ 2011: Molecular Quantum Mechanics and Molecular q = m ∆x = m (x − xe) (98) i i i i i i Spectra, Molecular Symmetry, and Interaction of Mat- ter with Radiation, this handbook, Wilson et al. 1955) The kinetic energy remains exactly separable in these coor- one obtains energy levels E and wave functions dinates. One uses a Taylor expansion for the potential v1,v2,v3... = e Ψ v1,v2,v3...: around the minimum (xi xi ): 3N 3N 3N 3N ∂V 1 ∂2V E = E (106) V =V + q + q q +··· v1,v2,v3... vi 0 i i j = ∂qi 2 ∂qi∂qj i 1 i=1 0 i=1 j=1 0 (99) 3N Ψ (Q ,Q ,Q ,. . .)= Ψ (Q ) (107) where V0 is a constant, which can be set to zero. The first v1,v2,v3... 1 2 3 vi i derivatives are zero at the minimum of the potential. The i=1 second derivatives are the force constants Fij . We now As in classical mechanics, one can describe the vibrational neglect higher terms of the expansion. The classical Hamil- molecular motion as a superposition of 3N harmonic vibra- tonian takes the following form: tions. For the quantum energy levels of the ith mode, one has 1 3N 3N H = T + Fij qiqj (100) 1 2 E = v + hν (108) i=1 j=1 vi i 2 i with with 3N 2 1 dqi 1 1 f T = (101) ν = λ = i (109) 2 dt i i i=1 2π 2π µi Fundamentals of Rotation–Vibration Spectra 135

−

n1 n2 n3

Figure 15 Graphical representation of the classical harmonic vibrations of .

n1 n2 n3

Figure 16 Graphical representation of the classical harmonic vibrations of SO2. where fi are generalized force constants (depending on the Fij and the masses) and µi are similarly generalized reduced masses. For the discussion of the modes of zero n2a frequency (translation and rotation) as well as the mathe- − matical treatment in terms of F and G matrix formalism, 667 cm 1 we refer to Wilson et al. (1955). n2b In general, the normal coordinates Qi involve an in-phase motion of all the atoms in a molecule. This is usually graphically represented by arrows indicating the relative size of the displacement of the atoms in the classical −1 n1 1388 cm vibrational motion associated with each mode. As an example, we show in Figure 15 the graphical representation of the classical harmonic vibrations of the water molecule with nine degrees of freedom and three n 2349 cm−1 vibrational modes (six zero frequency modes correspond- 3 ing to three translational and three rotational degrees of Figure 17 Graphical representation of the classical harmonic freedom). Because of the relatively light hydrogen atoms, vibrations of CO2. all three vibrational modes involve mostly a motion of the two hydrogen atoms with the heavy atom moving only slightly. In contrast, the more even mass distribution Table 4 Fundamental wavenumbers of CO2. in the SO molecule leads to a substantial displacement 2 Mode ν˜/cm−1 Description of all three atoms in the three harmonic normal vibrations (Figure 16). ν2a 667 Bending Figure 17 shows the normal vibrations of CO2,and ν2b 667 Bending the fundamental transition wavenumbers are given in ν1 1388 Symmetric stretching Table 4. This is an example of a linear molecule with ν3 2349 Asymmetric stretching two degenerate bending vibrations, leading to a vibrational angular momentum. We use here the notation ν2a and ν2b for the two degenerate bending vibrations. Another summarized in Table 5. Finally, Figure 19 shows the nor- common notation uses the vibrational angular momen- mal modes of CHD2I, with three light and one very heavy l tum quantum number l (v1, v2 , v3), see equation (112). substituent at the C atom. Table 6 summarizes harmonic AspectrumofCO2 can be found in Albert et al. normal vibrational frequencies as well as the fundamental 2011: High-resolution Fourier Transform Infrared Spec- transition frequencies. We discuss some additional aspects troscopy, this handbook. Figure 18 shows the normal of the vibrational overtone spectra for these molecules vibrations of CHClF2, and the fundamental frequencies are in the following subsection. Many other examples of 136 Fundamentals of Rotation–Vibration Spectra

n2 n3 n1

n4 n5 n6

n7 n8 n9

Figure 18 Graphical representation of the classical harmonic vibrations of CHClF2.

n1 n2 n3

n4 n5 n6

n7 n8 n9

Figure 19 Graphical representation of the classical harmonic vibrations of CHD2I. normal vibrations and spectra can be found in various available, including didactic examples (see, for instance articles of this handbook (see Albert et al. 2011: High- McIntosh and Michaelian 1979a,b,c). In the separable resolution Fourier Transform Infrared Spectroscopy, normal-mode picture of molecular vibrations, the molec- Weber 2011: High-resolution Raman Spectroscopy of ular vibrational motion is described by a collection of Gases, Herman 2011: High-resolution Infrared Spec- independent harmonic oscillators, not too different from troscopy of Acetylene: Theoretical Background and the harmonic oscillator picture of a diatomic molecule. Research Trends and Hippler et al. 2011: Mass and Each normal vibration has its own vibrational spectrum. Isotope-selective Infrared Spectroscopy, this handbook). It turns out, however, that anharmonicity is very important Programs for normal-mode calculations are also readily in reality. Fundamentals of Rotation–Vibration Spectra 137

35 −1 Table 5 Fundamentals of CH ClF2 (band centers in cm ). 3.3 Anharmonic Resonances and Effective Hamiltonians (a) −1 Mode Γ ν˜ i /cm References Approximate Description 3.3.1 General Aspects (c)–(e) ν1 A 3021 63351(4) s-(CH) As discussed in Quack 2011: Fundamental Symmetries (c),(f) ν2 A 1313 09551(2) b-(CH) (g)–(i) and Symmetry Violations from High-resolution Spec- ν3 A 1108 72738(2) s-(CF) 1108 72704(2) (j) troscopy, this handbook, the approxima- (b)(k),(l) tion introduces a far too high symmetry and too many ν4 A 812 9300(45) s-(CCl) (m),(n) ν5 A 596 371399(5) b-(CF) constants of the motion into the vibrational dynamics of (o)–(q) ν6 A 412 928543(5) b-(CCl) polyatomic molecules. In reality, anharmonic terms in the (c),(f) ν7 A 1351 70198(2) b-(CH) potential break this high symmetry, resulting in much (g)–(i) ν8 A 1127 28175(2) s-(CF) more complex spectra and dynamics. One can, in princi- (j) 1127 28149(2) ple, retain a description in normal coordinates but intro- ν A 366 197216(5) (o)–(q) b-(CF) 9 duce anharmonic terms in the potential (cubic, quartic, etc.) In the given approximate assignments the following abbreviations are (Nielsen 1951, 1959, Wilson et al. 1955, Mills 1974, Cal- used: s refers to a stretching and b to a bending mode (after Albert et al. ifano 1976, Papousekˇ and Aliev 1982, Aliev and Watson (2006)). (a) Γ gives the vibrational symmetry species in the Cs point group. 1985): (b)deperturbed value. (c) Amrein et al. (1985). 1 2 1 (d) V(Q)= λ Q + Φ Q Q Q Amrein et al. (1988a). 2 k k 6 k,l,m k l m (e)Fraser et al. (1992). k k,l,m (f)Thompson et al. (2003). (g) Albert et al. (2004b). 1 (h)Luckhaus and Quack (1989). + Φk,l,m,nQkQlQmQn +··· (110) (i) 24 Snels and D’Amico (2001). k,l,m,n (j)Thompson et al. (2004). (k) Albert et al. (2006). This type of expansion is useful if there is a unique deep (l)Ross et al. (1989a). (m)Klatt et al. (1996). minimum on the potential hypersurface. The correspond- (n)Gambi et al. (1991). ing equilibrium geometry is taken as the reference con- (o)Kisiel et al. (1997). figuration for the expansion. The normal coordinates Qk (p)Kisiel et al. (1995). (q)Merke et al. (1995). are orthogonal, and with this choice of coordinates, the quadratic terms in the potential are diagonal, the first mixed terms being cubic as shown in equation (110). The mini- mum of the potential energy is taken as zero. Because the ˜ Table 6 Harmonic ωi and experimental νi band centers of the normal coordinates depend on masses, the potential con- nine fundamental vibrational modes of CHD2I. References of stants Φ are different for different isotopomers even within high-resolution analyses are also indicated when available. the Born–Oppenheimer approximation. However, because −1(a) −1 one usually starts with a formulation of the potential in Mode Symmetry Description ωi /cm ν˜ i /cm terms of a set of coordinates in which the potential is inde- (b) ν1 A CH stretching 3199.4 3029.6790 pendent of the masses, such as internal coordinates, the (c)–(e),(a) ν2 A CD stretching 2280.1 2194 anharmonic potential constants for each isotopomer can be (c)–(e),(a) ν3 A CH bending 1216.8 1170 derived to satisfy the conditions of the Born–Oppenheimer (c)–(e),(a) ν4 A CD bending 1047.7 1018 (g) approximation or any other effective potential, which is ν5 A CD-bending 780.0 754.4738 (f) independent of the atomic masses (see Marquardt and ν6 A CI-stretching 543.9 508.7898 (d)–(e),(a) Quack 2011: Global Analytical Potential Energy Sur- ν7 A CD-stretching 2401.7 2313 (d)–(e),(a) ν8 A CH-bending 1332.3 1287 faces for High-resolution Molecular Spectroscopy and (g) ν9 A CI-bending 680.1 659.8692 Reaction Dynamics, this handbook). For ab initio calcula- tions of such “force fields” or potential functions, we refer (a)Horka´ et al. (2008). (b)Hodges and Butcher (1996). to Yamaguchi and Schaefer 2011: Analytic Derivative (c)Riter and Eggers (1966). Methods in Molecular Electronic Structure Theory: A (d)Duncan and Mallinson (1971). New Dimension to Quantum Chemistry and its Applica- (e) Santos and Orza (1986). tions to Spectroscopy,Tewet al. 2011: Ab Initio Theory (f)Kyllonen¨ et al. (2004). (g)Kyllonen¨ et al. (2006). for Accurate Spectroscopic Constants and Molecular Properties and Breidung and Thiel 2011: Prediction of 138 Fundamentals of Rotation–Vibration Spectra

Vibrational Spectra from Ab Initio Theory, this hand- E v1v2... = ; book. In this article, we do not go into the details of the G(v1, v2,. . . li,li+1,. . .) hc normal coordinate treatment including anharmonic poten- 1 tials (see the extensive literature cited above) but rather = ωs vs + + ωt (vt + 1) 2 s t restrict our discussion to the basic concepts. In a man- ner very similar to the case of diatomic molecules, one 1 1 + x v + v + can now either handle the quantum mechanical problem of ss s s ≥ 2 2 molecular vibrations with an exact variational (numerical) s s s technique or can use perturbation theory, leading to sim- 1 + x v + (v + 1) ple term formulae and “effective Hamiltonians”. The latter st s 2 t s t treatment is most commonly used in spectroscopy and first + + + we discuss it briefly, with the emphasis on anharmonic res- xtt (vt 1)(vt 1) onance. We then summarize some recent developments on t t≥t the relation of the relevant parameters in these two treat- + gtt lt lt +··· (112) ments. This relation has been shown to be more complex t t≥t than previously anticipated with some unexpected conse- quences for spectra and dynamics (Lewerenz and Quack where s and s denote non degenerate and t and t twofold 1988, Quack 1990, Marquardt and Quack 1991, 2001, degenerate vibrational normal modes. Here, the l quantum see also Marquardt and Quack 2011: Global Analytical number is related to the projection of the vibrational angular Potential Energy Surfaces for High-resolution Molec- momentum of the twofold degenerate vibration on the ular Spectroscopy and Reaction Dynamics, this hand- symmetry axis. Analogous expressions hold if there are book). We then briefly discuss local vibrations and group modes with higher degeneracy dt = 3, 4 ..., replacing the frequencies in IR spectra, as well as some aspects of tunnel- expressions in parentheses (vt + 1) by appropriate terms ing problems arising in “nonrigid” molecules with multiple with (vt + dt /2) in equation (112). minima. Alternatively, one can relate the level energies to the energy E0 of the vibrational ground state as being set to zero. Then, one obtains 3.3.2 Term Formulae from Anharmonic − Ev1v2... E0 Perturbation Theory for Polyatomic Molecules = G0(v1, v2,. . .) hc = ˜ + If the Taylor expansion of equation (110) converges rapidly νj vj xij vivj j i j≥i and if there are no “resonances” between close-lying states (see below), excluding degenerate vibrations, one can write + + gij lilj ... (113) the anharmonic term formula for a polyatomic molecule i j≥i on the basis of perturbation theory in analogy to diatomic molecules as ˜ Again the corresponding “spectroscopic constants” νj , xij ,

gij ...can be expressed in terms of the potential param- E eters and other molecular properties. These expressions v1v2... = G(v , v ,. . .) hc 1 2 are useful in describing vibrational spectra of polyatomic molecules. They frequently fail, however, if two or more 1 = ω v + vibrational states are close in energy, which may lead to an k k 2 k anharmonic resonance interaction. 1 1 + xkl vk + vl + +··· (111) 3.3.3 Elementary Description of a Two-level 2 2 k≥l Anharmonic Resonance (Fermi Resonances, Darling–Dennison Resonances, etc.) The harmonic frequencies and anharmonic constants can be We give here a brief summary of the two-level resonance, related to the more fundamental force constants and other closely retaining the original notation from Herzberg (1945) properties of the molecules (see for example Papousekˇ and for didactic reasons. Let us assume that the harmonic Aliev 1982). For symmetric top molecules with twofold energies from equation (106) in the normal-mode picture denegerate vibrations, one has similarly are labeled as zero-order energies for one energy level Fundamentals of Rotation–Vibration Spectra 139

0 = with the eigenfunctions Ei Ev1v2v3... and also for a second energy level “acciden- = 0 − 0 tally” very close in energy in the normal-mode approxima- Ψ i aΨ i bΨ n (119) 0 = tion En Ev v v .... The corresponding zero-order wave- 0 0 1 2 3 Ψ n = bΨ + aΨ (120) 0 = i n functions according to equation (107) are Ψ i Ψ v1v2v3... and Ψ 0 = Ψ . The effect of including the higher 2  2 n v1v2v3... If one has δ 4W , one has approximately order terms beyond the first sum in equation (110) can be taken into account locally for these two close-lying levels δ W 2 E  E ± + (121) by computing the matrix element i,n ni 2 δ = 0∗ ˆ 0 → 0 → 0 0 − 0 = → Win ... Ψ i WΨndQ1dQ2dQ3 ... (114) or Ei Ei and En En with En Ei δ and Ψ i 0 → 0 Ψ i , Ψ n Ψ n and thus a very small effect of the “non- where the integration is understood to cover the complete resonant” perturbation, which provides a justification for neglecting non-resonant levels in isolating just the two-level space of the coordinates. Win is supposed to include all terms in the anharmonic Hamiltonian contributing to the resonance problem. For the coefficients in equations (119) integral. The effective Hamiltonian matrix for coupling and (120), one has just the two levels in “resonance”, with “resonance” being 1/2 defined as a similar magnitude of the coupling |W | and 4W 2 + δ2 + δ in a = (122) the zero-order energy separation of the two levels 2 4W 2 + δ2 | |≈| 0 − 0|=| | 1/2 Win Ei En δ (115) 4W 2 + δ2 − δ b = (123) takes the simple matrix form 2 4W 2 + δ2 0 In the classic case of a Fermi resonance (Fermi 1931), = Ei Win = H11 H12 H eff 0 (116) Ψ corresponds to a fundamental level of one vibration Wni En H21 H22 i (v1 = 1, v2 = 0, and all other vk = 0) and Ψ n to an over- ∗ = = = This matrix is Hermitian (Wni Win) and in many tone of a second vibration (v1 0, v2 2, and all other cases real and symmetric (Wni = Win = W), which we vk = 0). If one assumes that the electric dipole transition assume here, without loss of generality, because we need strength for IR absorption is large for the fundamental | |2 =| |2 = 2 | 0|ˆ| 0 |2 =| 0 |2 only Wni Win W . The situation is schematically ( Ψ o µ Ψ i Moi ) and essentially zero for the over- | 0|ˆ| 0 |2 = drawn in Figure 20. The eigenvalues of the two-level tone ( Ψ o µ Ψ n 0), then one can readily see that the Hamiltonian are transition strengths for the observable transitions to the eigenstates Ψ and Ψ in the high-resolution spectrum are 1 i n E = E ± 4W 2 + δ2 (117) proportional to the squares of a and b, respectively. This i,n ni 2 is a common assumption in the analysis of spectra, allow- (E0 + E0) (E + E ) E = i n = i n (118) ing one to determine all parameters of the Fermi resonance ni 2 2 equations from the observed line frequencies and intensi- ties. The classic case of the Fermi resonance actually con- cerned the Raman spectrum of CO2, in which one assumed Yn,En that the Raman cross section for the symmetric stretch- 0 = ing fundamental Ψ i Ψ 100 is large, but small for the 0 = Y 0,E 0 bending overtone Ψ n Ψ 0200. The zero-order energies are d n n 0 ≈ −1 ≈ 0 E Ei /(hc) 1335 cm En/(hc), whereas the observed ni 0 0 −1 Yi ,Ei W levels are at about 1388 and 1285 cm , both giving a + strong Raman signal. Furthermore, only the Σg compo- 0 Yi,Ei nent Ψ 02 0 (with vibrational angular momentum quantum number l = 0, indicated by the superscript (0,20,0)) can give a nonzero matrix element W,whereastheΠ compo- nent Ψ 0220 has a vanishing W because of the symmetry 0 0 Y0 , E0 selection rule on the integral in equation (114). There- fore, E(0, 22, 0) is not shifted by the resonance and the Figure 20 Energy diagram of a Fermi resonance between two level separation |E(0, 20, 0) − E(0, 22, 0)|/(hc) is about energy levels. 50 cm−1, in the observed eigenstate spectrum, where the 140 Fundamentals of Rotation–Vibration Spectra harmonic level labels have obviously no exact meaning not necessary in a more general treatment, of which we for the E(0, 20, 0) level, which is heavily mixed. A more outline some aspects with some pertinent examples. detailed discussion of the Fermi resonance in CO2 can be found in Herzberg (1945) and Califano (1976). Numerous resonances of similar type have been observed in many 3.4 Many-level Anharmonic Resonances in other molecules since the first discussion of the CO2 reso- Molecules Containing an Isolated CH nance by Fermi (1931). Another type of “named” resonance Chromophore is the so-called Darling–Dennison resonance in which the two modes of different symmetry but similar harmonic fre- 3.4.1 Overtone Spectra of CHX3 Molecules and 0 = 0 = quency exchange two quanta (thus Ψ i Ψ 20... and Ψ n Polyad Structure Ψ ). This type of resonance was discussed by Darling 02... Figure 21 shows a survey spectrum of the CH-stretching and Dennison (1940) for H O and frequently occurs in not 2 overtone absorption in CHF (second overtone correspond- only dihydrides but also in many other molecules. More 3 ing to three quanta of CH-stretching, Dubal¨ and Quack general types of resonances are simply called anharmonic 1984a). As the CH-stretching wavenumber is about twice resonances. If rotational levels are involved, one speaks the CH-bending wavenumber, one expects a strong Fermi of rovibrational resonances. We have given this introduc- resonance. There are, indeed, two bands of almost equal tory discussion to provide some basis for a commonly used strength, each showing the characteristic P, Q, R structure analysis for two-level resonances. Several questions deserve attention in this context: of a parallel band, as for CH-stretching overtones in this symmetric top molecule. 1. Is the assumption that only one state results in a Thus, one might be tempted to analyze this observed zero-order line strength (either Raman or IR, and also spectrum using a two-level Fermi resonance model, and others such as fluorescence) justified? Such a state is this was, indeed, the basis of the first analysis by Bernstein sometimes called the bright state or, perhaps to be and Herzberg (1948). Although the other bands are quite preferred, the chromophore state. weak, nevertheless, they are important for the multistate 2. Is the neglect of further interacting levels justified? Fermi resonance. The effective Hamiltonian is shown in 3. Is the common assumption that the first low-order term Figure 22. The Fermi resonance system is, in fact, described in the Taylor expansion of the potential equation (110) by a polyad (here tetrad) of four strongly coupled zero-order is sufficient to describe the resonance using the matrix levels. In the effective Hamiltonian, one takes the term element in equation (114) justified? For a Fermi reso- formula in equation (113) as a diagonal structure, and one 2 nance, this would be a Φkl2 QkQl term and for a Dar- assumes a general form similar to the one from perturbation 2 2 ling–Dennison resonance the Φk2l2 QkQl term, which theory on a harmonic oscillator model as the off-diagonal might thus then be directly evaluated from the reso- structure. The effective Hamiltonian is assumed to be block- nance analysis (Mills 1974). This is furthermore fre- diagonal in the polyad quantum number N quently complemented by the assumption that low- order terms are generally larger than higher order terms 1 N = vs + vb (124) in the Taylor expansion. 2

It turns out that all of these assumptions must be ques- where vs is the CH-stretching quantum number and vb tioned under many circumstances. They are also obviously the quantum number for the degenerate CH-bending mode

2.0 ABX8 ) I CHF / 3 0 I 1.0

In ( 7880 7900 8260 8310 B A 0 8000 8200 8400 8600 8800 ∼ n /cm−1

Figure 21 Survey spectrum of the N = 3 Fermi resonance polyad in CHF3. The inserts are magnified portions from the same survey spectrum (P = 0.5 × 105 Pa, resolution 1 cm−1). [Reproduced with permission from Dubal¨ and Quack (1984a).] Fundamentals of Rotation–Vibration Spectra 141

|3, 0, 0〉 |2, 2, 0〉 |1, 4, 0〉 |0, 6, 0〉 subspace of the relevant normal coordinates and the effective Hamiltonian. For the variational treatment, one H 3 − 3/2k 0 0 | 〉 11 sbb 3, 0, 0 uses reduced dimensionless normal coordinates and a Tay- lor expansion of the potential as follows: − 3/2k 3 −2k | sbb H22 sbb 0 2, 2, 0〉 H 3 = 1 1 − 3 − 3 = 2 + 2 + 2 0 2k sbb H k sbb |1, 4, 0〉 V(Qs,Qb) ωsy ωbQb CsbbyQb 33 2 2 2 + 2 2 + 4 + 4 0 0 − 3 k H 3 |0, 6, 0〉 Cssbby Qb CbbbbQb CsbbbbyQb 2 sbb 44 + 3 2 +··· Csssbby Qb (126) 8684 +130 0 0 Here, we use the “Morse coordinate” + + 130 8526 213 0 [1 − exp (−aQ )] y = s (127) 0 +213 8310 +226 a with an effective Morse parameter a to describe the one- 00+226 8036 dimensional, very anharmonic C–H stretching potential 2 (D = ωs/(2a ) is the corresponding dissociation wavenum- Figure 22 Structure of the Fermi resonance Hamiltonian for the ber) and about 30 terms in the expression of equation (126) N = 3 polyad in CHF3. Top: General form. Bottom: Numerical − are generally needed for an accurate representation of values in cm 1 units from the best fit. [Reproduced with permis- sion from Dubal¨ and Quack (1984a).] either ab initio potentials or empirically derived poten- tials. The anharmonic vibrational problem is solved numer- ically either by basis set representation methods (Lewerenz (l is the corresponding vibrational angular momentum b and Quack 1988, Marquardt and Quack 1991) or by dis- quantum number). The structure of the Hamiltonian takes crete variable techniques (Luckhaus and Quack 1992, Beil the following form for nonzero matrix elements: et al. 1994, 1997). As both the effective (experimental) N and the vibrational variational spectra are well described Hv ,v ,l ;(v −1),(v +2),l ˆ s b b s b b by the simple effective Hamiltonian Heff of the previ- 2 ous subsections, one can derive the following transforma- = vs, vb,lb|k Q Q |vs − 1, vb + 2,lb (125) sbb s b tions: 1/2 1 1 =− k v (v − l + 2)(v + l + 2) ZTH Z = Diag(E ,E ...E) (128) 2 sbb 2 s b b b b eff 1 2 n T V H varV = Diag(E1,E2 ...En) (129) where we introduce primed quantities for Qs and Qb for effective dimensionless reduced normal coordinates and where Z and V are the corresponding eigenvector matrices of the effective and variational Hamiltonians. Thus, one can ksbb for the effective anharmonic coupling constant, which is related to the structure defined by the expansion in derive the matrix representation of the effective Hamilto- equation (110) but not identical to the corresponding force nian using the similarity transformation: constant, as we shall see. Such effective Hamiltonians have T T T H eff = (V Z ) H varVZ (130) been shown to describe the overtone spectra of many CHX3 molecules reliably. Such success would be interpreted as a If we arrange the eigenfunctions ψ of eigenvalues E as success of low-order perturbation theory to describe the n n the column vector many-level Fermi resonance. A more careful investigation shows, however, that some caution is necessary with such ψ = (ψ ,ψ ,ψ ...ψ )T (131) a conclusion. 1 2 3 n

and similarly the basis functions φn of the variational ˆ 3.4.2 Variational Treatment and Effective Fermi Hamiltonian Hvar as Resonance Hamiltonian = T φ (φ1,φ2,φ3 ...φn) (132) Lewerenz and Quack (1988), Dubal¨ et al. (1989), and ˆ Marquardt and Quack (1991) have investigated the rela- and the aprioriunknown basis functions χ n of Heff as tion between a full vibrational variational treatment of = T the CH-stretching and bending Fermi resonance in the χ (χ 1,χ2,χ3 ...χn) (133) 142 Fundamentals of Rotation–Vibration Spectra we can write in matrix notation only be the case if the expansion of the actual potential were to stop after very few terms, which is not the case ψ = V Tφ = ZTχ (134) for the real molecular systems. Rather, the few effective Hamiltonian parameters are complicated combinations and obtain an explicit expression for χ in terms of the of very many potential parameters appearing in a long functions φn, which are known in real coordinate space expansion of the type of equation (110) or (126). 3. The vibrational variational treatment combined with χ = ZV Tφ (135) the effective Hamiltonian treatment allows a compact representation of spectroscopic data on the one hand. Because of the block diagonal structure of Z , one can At the same time, it establishes the nontrivial relation restrict the expansion of χ i to the Nith block with explicit between the effective Hamiltonian parameters and notation as follows: the true potential parameters and it allows for a representation of molecular eigenfunctions and of basis = χ i Zik Vjkφj (136) functions of the effective Hamiltonians in terms of k(Ni ) j wavefunctions in real coordinate space by means of transformations given by equations (128)–(136). These relations are, in fact, generally useful, when relating 4. In particular, the long-believed dogma of the simple a treatment using an effective “spectroscopic” Hamiltonian perturbation theoretical relation between the anhar- defined by a small number of spectroscopic parameters monic potential constants and the effective Hamiltonian such as the Hamiltonian of Section 3.4.1 with a full constants cannot be maintained. variational Hamiltonian defined in real coordinate space. These investigations during the decade following 1980 have We illustrate this last point with some results for proto- resulted in the following, perhaps surprising, conclusions: typical molecules in Table 7. One can readily see the fairly similar magnitude of the various “spectroscopic” effec- 1. Often anharmonic resonance spectra can be described tive Hamiltonian parameters and their difference from the using effective Hamiltonians similar in structure to corresponding potential constants. Clearly, the traditional normal coordinate expansions of very low order in relation due to perturbation theory (Nielsen 1951, 1959, equation (110) with very few parameters. Mills 1974) 2. These parameters may not be interpreted, in general, as potential parameters in real coordinate space, even = though equation (110) might suggest this. This would ksbb Csbb (137)

Table 7 Spectroscopic constants and force constants for the Fermi resonance in CHX3 molecules (after Quack 1990).

(a) (b) (c) (d) (e) CHD3 CHF3 CHCl3 CHBr3 CH(CF3)3

Constant Exp. Ab initio Exp. Ab initio Exp. Exp. Exp. −1 ν˜ s/(cm ) 3048 3148 3086 3126 3096 3110 305 −1 ν˜ b/(cm ) 1292 1326 1370 1432 1221 1148 1353 −1 x ss/(cm ) −58 −67 −64 −67 −65 −66 −68 −1 x bb/(cm ) −4.5 −5.0 −5.6 −7.5 −6.5 −5.0 1.2 −1 x ab/(cm ) −22 −34 −29 −34 −26 −21 −22 −1 g bb/(cm ) 2.6 3.8 9.7 11.5 7.8 3.2 1.9 −1 |k sbb|/(cm )30± 15 31 100 ± 10 99 85 ± 15 75 ± 30 70 ± 15 −1 (f) Csbb/(cm )97± 50 140 ± 15 187 187 265 245 115 −1 Cssbb/(cm ) −50 −78 −92 −85 −116 −94 −50 a)Experiments: Campargue and Stoeckel (1986), Ben Kraiem et al. (1989), Campargue et al. (1989a,b), Lewerenz and Quack (1988), Lewerenz (1987), Ha et al. (1987), Peyerimhoff et al. (1984). Theory: Dubal¨ et al. (1989), Lewerenz and Quack (1988), Lewerenz (1987), Ha et al. (1987), Peyerimhoff et al. (1984), Segall et al. (1987). (b)Experiments: Dubal¨ and Quack (1984a), von Puttkamer and Quack (1989), Quack (1982). Theory: Dubal¨ et al. (1989), Lewerenz and Quack 13 (1988), Lewerenz (1987), Ha et al. (1987). For CHF3, see Hollenstein et al. (1990). (c) 13 Dubal¨ et al. (1989), Lewerenz and Quack (1986), Baggott et al. (1986), Green et al. (1987), Wong et al. (1987). For CHCl3, see Hollenstein et al. (1990). (d)Ross et al. (1989a), Hollenstein et al. (1990), Davidsson et al. (1991), Manzanares et al. (1988). (e)Dubal¨ et al. (1989), von Puttkamer et al. (1983b), Baggott et al. (1985). (f) −1 An ab initio calculation (Amos et al. 1988) gives Csbb = 278 cm in surprisingly good agreement with experiment (Dubal¨ et al. 1989). Fundamentals of Rotation–Vibration Spectra 143 does not hold very well. Marquardt and Quack (1991) have effective Hamiltonians can be found in the article by Field derived improved approximations: et al. 2011: Effective Hamiltonians for Electronic Fine Structure and Polyatomic Vibrations, this handbook. 8ω2 k = b C sbb + sbb (138) ωs(2ωb ωs) 3.4.3 Concluding Remarks on Multistate and Anharmonic Resonances Many spectra of polyatomic molecules are analyzed with 8ω2 = − 3 2 b the effective Hamiltonian as illustrated in Section 3.4.2, ksbb 1 a N 4 ωs(2ωb + ωs) and, even more frequently, only local resonances are iden- tified, if any, as discussed in Section 3.3. The connection to 4ωb + ωs + a2N C + aNC + ∆k (139) the full variational treatment as discussed in Section 3.4.2 2ω + ω sbb ssbb sbb b s is carried out only rarely, but it is very important for the understanding of the underlying dynamics, in particular, where a is the Morse parameter as defined above, N an also time-dependent wavepacket dynamics as discussed in averaged polyad quantum number for the polyads retained Section 4. in the description, and ∆k a correction term defined as: sbb The very extensive work of Amat et al. (1971) in terms C2 of various contact transformations should be seen in the ≈− × sbb ∆ksbb aN 2.6 (140) ωs tradition of effective Hamiltonian analyses. Beyond the dis- cussion of Field et al. 2011: Effective Hamiltonians for We do not discuss in detail additional efforts extended Electronic Fine Structure and Polyatomic Vibrations, to treatments in curvilinear coordinates (Carrington et al. this handbook, already mentioned, the article by Herman 1987, Green et al. 1987, Wong et al. 1987, Voth et al. 2011: High-resolution Infrared Spectroscopy of Acety- 1984) including contributions by Sibert and coworkers lene: Theoretical Background and Research Trends, (Sibert 1990) and other kinds of perturbation theoreti- this handbook, discusses effective Hamiltonian analyses. cal approaches. Table 8 summarizes the results from the Polyad dynamics of effective Hamiltonians has been dis- approximations discussed here. In general, full accuracy cussed by a number of authors from various points of can be achieved only with a complete vibrational varia- view (Jung et al. 2001, Kellman and Lynch 1986). Reviews tional treatment. The general Fermi resonance treatment covering effective Hamiltonian analyses include those by has been extended to the CH-chromophore in Cs and C1 Quack (1990), Lehmann et al. (1994), Nesbitt and Field symmetrical environments; in addition to the Fermi reso- (1996), Gruebele and Bigwood (1998), Perry et al. (1996), nances, Darling–Dennison resonances (of a “quartic” type) and Herman et al. (1999). In addition, more extensive ref- involving the two CH-bending modes and “cubic” anhar- erences can be found in the article by Hippler et al. 2011: monic resonances coupling the two bending modes a and Mass and Isotope-selective Infrared Spectroscopy,this b with the stretching mode s by an effective coupling con- handbook. stant ksab are included. We refer to Horka´ et al. (2008) as The question of treating the anharmonic vibrational prob- a paper which also includes a summary of results for many lem in small polyatomic molecules variationally, including molecules. An extensive treatment of various aspects of all degrees of freedom, is dealt with in the articles by Ten- nyson 2011: High Accuracy Rotation–Vibration Calcu- Table 8 Fermi resonance coupling constants; values in cm−1 lations on Small Molecules, this handbook and Carrington (after Marquardt and Quack 1991). 2011: Using Iterative Methods to Compute Vibrational Spectra, this handbook. So far this is restricted to small

(a) (a) (b) (c) ksbb Csbb ksbb ksbb molecules. Our discussion of treating only a subspace vari- ationally (Lewerenz and Quack 1988) finds its justification ± CHF3 100 10 187 118 113 in the substantial separation of timescales and coupling ± CHD3 30 15 97 75 54 strengths, when one considers additional degrees of free- CHCl 89 265 129 115 3 dom. We discuss this question in Section 4. CH(CF3)3 65 115 82 74 ±55 The final topic concerns the polyad structure, when an CHBr3 55 15 181 114 81 increasing number of degrees of freedom are coupled. (a) From direct fit to experimental data (Segall et al. 1987, Lewerenz and Figure 23 shows the zero-order polyad levels for CHF3 for Quack 1988, Dubal¨ et al. 1989, Ross et al. 1989b, Davidsson et al. 1991). = (b) Prediction from equation (138). A1 polyads from N 1 to 7. The number of coupled levels (c) Prediction from equation (139); constants from Dubal¨ et al. (1989), in each polyad is N + 1 ∝ N, because the vibrational angu- Ross et al. (1989b), Davidsson et al. (1991); N = 3.5 assumed. lar momentum l for the bending vibration is approximately 144 Fundamentals of Rotation–Vibration Spectra

+600 vibrational variational calculations would be much more difficult, although perhaps not impossible (see Marquardt CHF3 and Quack 2011: Global Analytical Potential Energy Sur- faces for High-resolution Molecular Spectroscopy and +400 A1-polyads Reaction Dynamics, this handbook). (Zero-order states) > +200 |6,2,0 |5,4,0> |4,6,0> 3.5 Local Modes, Group Frequencies, and IR |4,4,0> > > Chromophores ) |5,2,0 |3,8,0 − 1 0 |4,2,0> |3,6,0> cm > |2,10,0> When discussing the CH stretching overtone spectra, one |3,2,0 > hc |3,4,0 implicitly uses the notion of a fairly localized normal vibra- /( |2,2,0> 0 |2,8,0> E − |1,2,0> tion of the isolated CH chromophore. Indeed, it is the ∆ 200 |2,4,0> > > |0,2,0> |2,6,0 |1,12,0 “infrared chromophore” nature of this localized vibration that results in a simple appearance of overtone spectra, > |1,4,0 |1,10,0> − where we have a very dense set of vibrational states that 400 > |1,6,0 |1,8,0> are “dark” and do not appear in spectra except by cou- |0,4,0> |0,14,0> pling to the chromophore levels. Indeed, the chromophore concept can be made the basis for a description of such −600 |0,6,0> |0,12,0> spectra (Quack 1990). Another closely related concept is |0,8,0> |0,10,0> that of group frequencies, which are characteristic for cer- tain types of functional groups. For high-frequency modes 1234567 such as O–H stretching in alkanols and N–H stretching N in amines, the relationship to fairly localized normal vibra- tions is clear and sometimes a quasi-diatomic treatment of Figure 23 Reduced energy plot for the zero-order states the local vibrations is possible. However, there are also |v , v ,l  of CHF . The zero of energy is redefined at each N to s b b 3 functional groups leading to more delocalized vibrations be the energy of the |vs, 0, 0 state, which is not shown explicitly. The coupled states for one polyad appear as one column with a such as the C=O stretching in organic amides and others, given N. [Reproduced with permission from Dubal¨ and Quack on which there is extensive literature available. A more (1984a).] interesting variant of the local mode theory has been devel- oped over many decades for hydrides with two or more conserved on short timescales in C3v molecules (Luckhaus high-frequency modes (XH2,XH3,etc.)andH2Opro- and Quack 1993). Figure 24 shows a coupling scheme just vides a simple example. Here, the normal modes are, by for one (N = 4) CH-stretching chromophore polyad in a definition, delocalized over both X–H coordinates (sym- C1 or Cs symmetrical environment. Then, the number of metric and antisymmetric stretching). It has been argued, coupled levels in each polyad increases approximately pro- however, that at high excitations a zero-order Hamilto- portional to N 2. Finally, if all s vibrational degrees of nian starting from localized X–H stretching modes pro- freedom are included, the number of levels coupled in vides a better approach than the normal-mode approach each polyad would increase approximately as N (s−1),pro- (Mecke 1936, 1950, Henry and Siebrand 1968, Child and portional to the vibrational density of states. Ultimately, Lawton 1981). We refer to Quack (1990) for a sum- this leads to a sequential scheme for intramolecular vibra- mary of the historical development of this approach. On tional and finally rotational–vibrational redistribution in the other hand, in large polyatomic molecules, the “high- polyatomic molecules and the concept of the global vibra- resolution” eigenstates at high densities of states may be tional state, which would be delocalized over all degrees of assumed to be very delocalized, lending themselves per- freedom (Quack 1981, Quack and Kutzelnigg 1995). haps to a statistical treatment in terms of global vibrational Given the possibility of variational vibrational calcu- states (Quack 1981, 1990, Lehmann et al. 1994). These lations and analysis, one might also wonder whether an concepts go beyond an introductory survey and we refer effective Hamiltonian analysis has any justification at all. to some relevant articles in this handbook (see Herman It turns out that, in practice, a fit to experimental spectra 2011: High-resolution Infrared Spectroscopy of Acety- requires a compact description in terms of few parameters. lene: Theoretical Background and Research Trends and For this purpose, fits of effective Hamiltonian parameters Quack 2011: Fundamental Symmetries and Symmetry are an ideal intermediate step in spectroscopic analyses. A Violations from High-resolution Spectroscopy, this hand- direct approach starting with a potential hypersurface and book). Fundamentals of Rotation–Vibration Spectra 145

Polyad N = 4 400

sab 320 311 302

240 231 222 213 204

160 151 142 133 124 115 106

080 071 062 053 044 035 026 017 008

sa2 sab sb2 a2b2

k g saa k sab k sbb

Fermi Darling–Dennison

Figure 24 The coupling scheme for polyad N = 4ofCHD2I, where the states are labeled in the circles by the quantum numbers vsvavb in the zero-order picture, where vs is the quantum number for stretching mode and va and vb are quantum numbers for the bending modes. The underlined states have A symmetryinaCs symmetric molecule. [Reproduced with permission from Horka´ et al. (2008).]

3.6 Large Amplitude Motion in Nonrigid High-resolution Microwave and Infrared Spectroscopy Molecules Including Several Equivalent of Molecular Cations and Havenith and Birer 2011: Minima High-resolution IR-laser Jet Spectroscopy of Formic Acid Dimer, this handbook). If one has very floppy systems A straightforward treatment of large-amplitude motions such as clusters bound by van der Waals or hydrogen bonds, of very nonrigid molecules with several equivalent or in general, the best approach is a variational treatment, for nonequivalent potential minima of very similar energy instance, using discrete variable representations (DVR) (see clearly is not possible. In such cases, one may sometimes Baciˇ c´ and Light 1989, Tennyson 2011: High Accuracy define a reaction path connecting the minima and define Rotation–Vibration Calculations on Small Molecules Using Iterative Methods to Com- normal modes along this path, for instance, inversion in and Carrington 2011: pute Vibrational Spectra, this handbook). In some cases, ammonia and internal rotation in or methanol. The quantum Monte-Carlo techniques are useful (Quack and dynamics and spectra of such nonrigid molecules have been Suhm 1991). discussed in the article by Bauder 2011: Fundamentals of Rotational Spectroscopy, this handbook. Coupling to vibrational degrees of freedom leads to mode selective tun- 4 MOLECULAR IR SPECTRA AND neling, studied for instance in ammonia isotopomers (see Snels et al. 2011: High-resolution FTIR and Diode Laser MOLECULAR MOTION Spectroscopy of Supersonic Jets, this handbook), aniline 4.1 General Aspects (Fehrensen et al. 1998, 1999), or H2O2 (Fehrensen et al. 2007, see Marquardt and Quack 2011: Global Analytical Deriving molecular motion from IR rotation–vibration Potential Energy Surfaces for High-resolution Molec- spectra has a long history and can be discussed at several ular Spectroscopy and Reaction Dynamics, this hand- levels, leading to slightly differing pictures of molecular book). For more details on this topic, we refer to these arti- motion. cles and also to others present in this handbook (see Bauder 2011: Fundamentals of Rotational Spectroscopy,Cami- 1. The historically oldest and still very widely used nati 2011: of Large Molecules picture is derived from the classical normal-mode and Molecular Complexes, Hippler et al. 2011: Mass and treatment. In this approach, spectroscopy is used to Isotope-selective Infrared Spectroscopy, Amano 2011: derive the normal vibrations and the motion of the 146 Fundamentals of Rotation–Vibration Spectra

atoms is then described by the classical mechanical Gaussian probability following the classical mechani- normal vibrations. If one considers the arrows in cal trajectory (Schrodinger¨ 1926, Quack and Sutcliffe Figures 15–19 literally as describing atomic motions 1985, Marquardt and Quack 1989). This picture would (not just relative displacements along a normal coor- then remain valid for all harmonic degrees of freedom dinate), then these figures give a schematic view and thus the classical picture of molecular vibrations of these classical vibrations. Didactic movies using would be qualitatively valid also for the true quantum classical dynamics showing these normal vibrations motion, if the harmonic normal-mode description were exist and are widely distributed. The overall motion adequate to describe vibrational spectra and dynamics when several normal vibrations are excited would cor- (Quack 1995). respond to a more complicated classical Lissajous 2. It turns out, as we have seen, that anharmonic con- motion. It turns out that the coherent-state descrip- tributions to the spectra are very important. One might tion of the quantum harmonic oscillator also provides then think of deriving anharmonic potentials from spec- some justification for this picture for the true quan- tra and describing molecular motion quasi-classically tum motion. The coherent quantum state shown for a by computing trajectories using the anharmonic poten- typical “molecular” harmonic oscillator in Figure 25 tials. Indeed, this approach is widely used. However, (with a fundamental wavenumber of 1000 cm−1)isa it turns out that anharmonicity (when large) leads to

−1 Figure 25 Quasi-classical oscillation of the harmonic oscillator with resonant, coherent excitation (ν˜ =˜νL = 1000 cm ) (after Quack and Sutcliffe 1985). The images show the axis t toward the back (the time step separating two lines is 1 fs). The abscissa from left to right shows the spatial coordinate q and the ordinate the probability density |Ψ |2. Top: from t = 0 to 0.1 ps (practically unperturbed ground state). Bottom: from 16.5 to 16.6 ps (practically free oscillation after resonant excitation). Fundamentals of Rotation–Vibration Spectra 147

dynamical consequences, making a quantum descrip- of a Lorentzian line shape with the coupling strength V 2 to tion necessary. a quasi-continuum of state density ρ. It can be derived from 3. A quantum description, which has come into wide an (almost) exactly solvable model and is not a consequence use more recently, uses effective Hamiltonians, such of first-order perturbation theory. The application of this as the Fermi resonance Hamiltonians as discussed idea to an exponential decay of a highly excited vibrational in Section 3, and computes anharmonic intramolecu- state into a quasi-continuum of vibrational states by IVR lar vibrational redistribution (IVR) in terms of time- and an exponential line shape analysis was done by dependent quantum basis state populations. This Bray and Berry (1979), deriving very short (100 fs) decay provides some insights into quantum redistribution pro- times for highly excited CH-stretching states in benzene. cesses. However, it does not provide a complete picture Numerous analyses of this kind have appeared since then. of the true quantum motion, because the wavefunctions Of course, one can also find exponential decay through corresponding to the basis states are either completely IVR by vibrational predissociation into true continua, for unknown or arbitrary, or they might be known as instance, for (HF)2 (Hippler et al. 2007, Manca et al. approximations, such as normal-mode wavefunctions. 2008). They still carry a large uncertainty, related to the uncer- More generally, one can go beyond these simple coupling tainties originating from the basis states. schemes (Quack 1981) and finally derive state populations 4. If the transformation to a true variational Hamilto- pk for some spectroscopic effective Hamiltonian. The basic nian is known along the lines of Section 3.4.2, then equations, given some Hamiltonian matrix H (“effective” a complete quantum description in terms of quantum or “true”), are given by the propagator mechanical wavepackets and time-dependent probabil- ity densities is possible, the accuracy of which is only U(t, t0) = exp (−2πiH (t − t0)/h) (142) limited by the uncertainties of the primary spectro- = scopic data and the accuracy of the corresponding spec- b(t) U(t, t0)b(t0) (143) troscopic analysis. We provide here some examples for 2 pk =|bk| (144) analyses along the lines of the last two approaches. For the two-level Fermi resonance problem, the full time evolution matrix can be easily written (see Merkt and 4.2 Time-dependent Population Dynamics from Quack 2011: Molecular Quantum Mechanics and Molec- Simple Coupling Models and Effective ular Spectra, Molecular Symmetry, and Interaction of Hamiltonians Matter with Radiation, this handbook, for the mathemat- ically identical Rabi problem, which is explicitly solved Simple coupling models relating spectra and time-depen- in terms of the U matrix therein). The numerical calcu- dent population dynamics can be traced to the early model lation for more complicated examples is straightforward. of Bixon and Jortner (1968). This model for an exponen- Figure 26 shows the result for basis state populations of the tial decay of an initial state into a dense set of back- ground states was motivated by the need to understand the irreversible relaxation of excited electronic states by 1.0 internal conversion and intersystem crossing processes (see 1 Merkt and Quack 2011: Molecular Quantum Mechan- 2 ics and Molecular Spectra, Molecular Symmetry, and Interaction of Matter with Radiation, this handbook). N 0.5 The essence of the result of Bixon and Jortner (1968) is P the realization that exponential decay of an initial state 3 can occur not only with truly continuous spectra, such as in predissociation (Herzberg 1966), but also with quasi- continua, such as in internal conversion processes. In addi- 4 0 tion, the Bixon–Jortner model provides an important result: 0 0.1 0.2 0.3 0.4 0.5 0.6 the “Golden Rule” t /ps 2πΓ 4π 2V 2 k = = ρ (141) Figure 26 Time evolution for the populations of states inter- h h acting in the N = 3 polyad of CHF3 calculated with the spec- troscopic Fermi resonance Hamiltonian and the initial condition This provides a relation between the exponential decay rate p(|3, 0, 0) = 1. [Reproduced with permission from Dubal¨ and constant k and the full width at half maximum Γ (FWHM) Quack (1984a).] 148 Fundamentals of Rotation–Vibration Spectra

Table 9 Typical IVR lifetimes of initial CH-stretching excitation in alkyl and acetylene compounds estimated from spectroscopic data. See reviews (Quack 1990, 1991, Lehmann et al. 1994) for further tables (after Quack and Kutzelnigg 1995).

Example τ IVR/ps Method and reference

(a) Alkyl CH <0.1 Effective Hamiltonian Heff dynamics X3CH (Quack 1990, 1991, Baggott et al. 1985) (X=D,F,Cl,Br,CF3) or real molecular Hamiltonian dynamics Hmol in subspace (Quack 1991, Marquardt et al. 1986, Marquardt and Quack 1991) (a) XF2CH <0.1 Heff (Dubal¨ and Quack 1984a,b (a) X2FCH <0.1 Amrein et al. 1985, Quack 1991) or Hmol (X=D,Cl,(CF3)) (Luckhaus and Quack, 1992, 1993, Quack 1991, Quack and Stohner 1993) Acetylene CH Temperature-dependent lineshape (CX3)3CC≡CH(X=F) ≥10–20 (von Puttkamer et al. 1983a) (v=1) ≥60 Molecular beam linewidth, bolometric (X=F) (Gambogi et al. 1993, Lehmann et al. 1994) X=H >200 Molecular beam, bolometric (Lehmann et al. 1994) linewidth X=D ≥40 Molecular beam, bolometric (Lehmann et al. 1994) linewidth (CH3)3Si–C≡C–H Inhomogeneous FTIR, temperature dependent (v=1) ≥2000 (von Puttkamer, K. and Quack, M. (1983) unpublished work cited in Quack 1991) bolometric (Kerstel et al. 1991) CH3CH2C≡C–H ≥269 Statistical Heff (band structure (McIllroy and Nesbitt 1990, Bethardy and Perry 1993)).

(a) Detailed dynamics (nonexponential) obtained from 10 to 1000 fs. effective Hamiltonian represented for CHF3 in Figure 22 4.3 Time-dependent Wavepacket Analysis (spectra in Figure 21). An interesting observation beyond the very short time for IVR (<100 fs for a complete initial The state population analyses contain an element of arbi- = decay of the vs 3 CH-stretching state) is the very sub- trariness, because the time evolution of populations depends stantial population of two other states (achieving maximum upon the choice of basis states (Beil et al. 1997). For populations of about 80 and 40%), although the spectrum in instance, if one chooses eigenstates as the basis there is Figure 21 shows only two prominent bands. Nevertheless, no time evolution of populations, and if one chooses some a two-level Fermi resonance analysis would be dynamically very highly localized states, one might get evolution on a quite misleading as it would lead to a periodic beating of 1-fs or even shorter timescale. While often the physical sit- just two states instead of the correct involvement of three uation may suggest some reasonable effective Hamiltonian, as shown in Figure 26. the ambiguities can be removed altogether if a transforma- Quite a few examples of this type have been analyzed in tion to a “true” variational Hamiltonian can be carried out terms of IVR along those lines and Table 9 gives a summary along the lines of Section 3.4.2. of some results (Quack and Kutzelnigg 1995). Figure 27 shows such a result for CHF3,whichwas, One of the more interesting results concerns the very in fact, the first example where such a wavepacket anal- different initial redistribution times for the isolated alkyl ysis was carried out (Marquardt et al. 1986, Marquardt and CH chromophore states (τ<100 fs) and the acetylenic Quack 1991). The new aspect beyond the short timescale CH chromophore (τ>1 ps) originally derived by von Put- is now the evolution of probability density in space. As tkamer et al. (1983b) and since confirmed by many exper- can be seen, this evolution is not simply a quasi-classical iments, now also including time-resolved femtosecond- oscillation between two coupled pendula, but rather one pump probe experiments (Kushnarenko et al. 2008a,b). finds a nonclassical delocalization. It has been found Rather than discussing more such examples, we now con- that with artificially very small one has sider evolution of the quantum mechanical wavepacket a quasi-classical coherent state propagation, very similar derived from high-resolution IR spectra, which provides an to the behavior of two classical anharmonically coupled additional level of understanding. oscillators. On the other hand, with large anharmonicity (as Fundamentals of Rotation–Vibration Spectra 149

Laser off |y|2 Q b

0 fs Q b

Qs

80 fs

20 fs

125 fs

30 fs

220 fs

45 fs

380 fs

Figure 27 Wavepacket motion for the two strongly coupled CH-stretching (Qs) and bending (Qb) vibrations in the CHF3 molecule. 2 |Ψ(Qs,Qb,t)| is the probability distribution on the femtosecond timescale after the initial CH-stretching excitation at t = 0. [Reproduced from Marquardt and Quack (2001).]

applicable to CHF3), one finds nonclassical delocalization of thought, one can also analyze the dynamics in a very on the timescale of IVR. This introduces the new concept coarse-grained way by studying time-dependent entropy DIVR (delocalization IVR) as opposed to CIVR (classical (Quack 1990, Luckhaus et al. 1993, Quack 2011: Fun- IVR) and the tuning of this phenomenon by anharmonic- damental Symmetries and Symmetry Violations from ity constitutes a fundamental finding for intramolecular High-resolution Spectroscopy, this handbook). dynamics (Marquardt and Quack 1991, 2001, Quack 1993, 1995). These phenomena can be investigated both for isolated 5 ROTATION–VIBRATION SPECTRA OF molecules as shown in Figure 27 and for molecules under POLYATOMIC MOLECULES coherent excitation (Quack and Stohner 1993). Figure 28 shows such an example for CH-stretching and bending 5.1 General Aspects wavepackets in CHD2F (Luckhaus et al. 1993). One can recognize an initial almost coherent state oscillation for The basic concepts of the interaction of vibration and rota- the first 100 fs, followed by IVR and delocalization on tion in polyatomic molecules are similar to the ones dis- timescales <1 ps. In addition, with wavepacket analysis cussed in Section 2 for diatomic molecules. However, the under coherent excitation, one finds the pronounced differ- exact separation of rotational and vibrational motion pos- ence in timescales for the alkyl–CH chromophore and the sible for diatomic molecules cannot be directly translated acetylenic–CH chromophore as derived from the spectrum to polyatomic molecules. Nevertheless, one can distinguish (Quack and Stohner 1993). Taking a slightly different line between the two approaches. 150 Fundamentals of Rotation–Vibration Spectra

50–150 fs (exc. 2865 cm−1) 400– 450 fs (exc. 2865 cm−1) 950–1050 fs (exc. 2865 cm−1)

(a) (b) (c)

−2 Figure 28 (a) Time-dependent probability density P(qv,t) of CHD2F under IR-multiphoton excitation with I = 20 W cm , ν˜ FL = −1 2865 cm . The abscissa shows a dimensionless CH-stretching normal coordinate range −4.5 ≤ qs ≤+8.5, the timestep separating two lines toward the back is 0.5 fs; total time range shown from 50 to 150 fs after the start of radiative excitation. (b) As (a) but from 400 to 450 fs with time step 0.25 fs. (c) As (a) but showing probability density P(qi,t), integrated over qs, and the dimensionless CH-bending out-of-plane qo, time range 950–1050 fs, time step 0.5 fs, range of the dimensionless CH-bending in-plane −8 ≤ qi ≤+8. [Reproduced with permission from Luckhaus et al. (1993).]

1. Exact (numerical) treatment of the rotation–vibration with emphasis on the rotational level structure. Here, problem given an effective multidimensional poten- we complement this with a brief summary of the tial hypersurface (the lowest dimension, for a triatomic basic theory and a short section on the practice molecule, would be a three-dimensional hypersurface of rotation–vibration analysis including a discussion in a four-dimensional space–with the fourth dimen- of an efficient general program available for rota- sion being the potential energy coordinate). Such tion–vibration analysis of asymmetric tops. approaches are discussed in this handbook by Ten- nyson 2011: High Accuracy Rotation–Vibration Calculations on Small Molecules, and Carrington 2011: Using Iterative Methods to Compute Vibra- 5.2 The Rotation–Vibration Hamiltonian tional Spectra, this handbook. Until today, these approaches have been limited to triatomic, four atom, The rotation–vibration Hamiltonian of polyatomic mole- and at most five atom molecules and very few anal- cules is discussed in the article by Bauder 2011: Fun- yses of rotation–vibration spectra along these lines damentals of Rotational Spectroscopy, this handbook, have been carried out, quite in contrast to diatomic and has a long history. We may refer here to the clas- molecules, where this type of “exact” approach is sic papers by Nielsen (1951, 1959), the work by Wilson more common. et al. (1955) (particularly Chapter 11 therein) and Wat- 2. Effective Hamiltonian analyses leading to sets of son (1968), with important subsequent work by Howard spectroscopic parameters. Different from diatomic and Moss (1971), Louck (1976), Makushkin and Ulenikov molecules, the effective rotation–vibration parameters, (1977), Meyer (1979), and Islampour and Kasha (1983). however, concern matrix representations of appropri- We introduce here the basic concepts, with the aim of ate Hamiltonians prior to diagonalization; thus, rota- defining notations. The usual starting point is the classi- tion–vibration energies cannot, in general, be given cal Hamiltonian of equation (97) in Cartesian coordinates as simple formulae for the Hamiltonian in diagonal for all particles written in quantum mechanical operator structure. For the purely vibrational problem of anhar- form: monic resonances, an example has already been given in Section 3, where we introduced diagonal term for- h2 3N 1 ∂2 mulae for a zero-order Hamiltonian and other param- Hˆ =− + V(xˆ ,. . .,x ) 2 2 1 3N (145) 4π 2mi ∂x eters defining the off-diagonal structures. The rota- i=1 i tion–vibration problem has been addressed already 3N ˆ2 in the preceding article by Bauder 2011: Funda- pi ˆ = + V(x1,. . .,x3N ) 2mi mentals of Rotational Spectroscopy, this handbook, i=1 Fundamentals of Rotation–Vibration Spectra 151

q ˆ Transformation to a system of arbitrary coordinates k fol- Jα are the angular momentum operators in the molecule- lowing Podolsky (1928) gives fixed system, and pf the total linear momentum in space- fixed direction f .   −1 ˆ 1 −1/4 1/2 −1/4 ˆ H = |g| pˆj |g| Gjkpˆk|g| + V(q1,. . .,q3N ) 2 µ = I − ζ α ζ β q q  j,k αβ αβ ik jk i j (155) (146) i,j,k

Iαβ are the components of the moments of inertia , with the momentum operators pˆk given by = = − α with α,β x,y,z,andi, j, k 1,. . .,3N 6, and ζ ik ˆ the common Coriolis coupling parameters. Π α are the h ∂ components of vibrational angular momentum: pˆk =−i (147) 2π ∂qk ˆ = α ˆ Π α ζ ij qipj (156) and i,j N α =− α = − 1 ∂qj ∂qk ζ ij ζ ji (lnβ,ilnγ ,j lnγ ,ilnβ,j ) (157) Gjk = (148) m ∂x ∂x n=1 i i i i where the matrix l defines the transformation from normal = ∂xi ∂xi gjk mi (149) coordinates to mass-weighted Cartesian coordinates in the ∂qj ∂qk i molecule-fixed system. |g|=Det(g) (150)

This can be simplified to 5.3 Effective Hamiltonian Analysis for Rotation–Vibration Spectra of 1 Asymmetric Tops Hˆ = pˆ G pˆ +ˆu(q ,. . .,q ) 2 j jk k 1 3N j,k The general Hamiltonian of the preceding section can be applied in several ways to rotation–vibration spectra. One + V(qˆ ,. . .,q ) (151) 1 3N can distinguish three cases depending on the moments of h2 1 ∂ ∂ ln (|g|) inertia I , I , I after transformation to the principal axis u(qˆ ,. . .,q ) = G a b c 1 3N 8π 2 4 ∂q jk ∂q system (see Bauder 2011: Fundamentals of Rotational j,k j k Spectroscopy, this handbook) or the corresponding rota- 1 ∂ ln (|g|) ∂ ln (|g|) tional constants ordered by convention A ≥ B ≥ C and + Gjk (152) 16 ∂qj ∂qk with X = A, B, C: h Using center of mass separation, Euler angles for rotation, X = (158) 2 Eckart conditions, and transformation to normal coordi- 8π cIX nates, Watson (1968) has derived a simplified general effec- For rigid rotors, apparently simple forms can be derived for tive rotation–vibration Hamiltonian, which we write here the rotational energy levels when some of the A, B, C are in the notation of Aliev and Watson (1985) identical (see Bauder 2011: Fundamentals of Rotational Spectroscopy, this handbook). For A = B = C, one has a 1 h2 spherical rotor (or spherical top); for A = B = C, one has Hˆ = p2 + (Jˆ − Πˆ )µ (Jˆ − Πˆ ) 2M f 8π 2 α α αβ β β a prolate symmetric top and for A = B = C, one has an α,β oblate symmetric top. We refer to Bauder 2011: Funda- 1 + pˆ2 +ˆu + V(qˆ ,. . .,q ) (153) mentals of Rotational Spectroscopy, this handbook and 2 k 1 3N k Herzberg (1945, 1966) for rotational–vibrational energy level structures for these cases. It turns out, however, that where the rotation–vibration problem for spherical and symmet- ric rotors is actually complicated by the occurrence of h2 degeneracies, which are lifted at higher order. In general, uˆ =− µ (154) spherical and symmetric rotors possess degenerate vibra- 32π 2 αα α tions, if the identity of moments of inertia arises from the 152 Fundamentals of Rotation–Vibration Spectra symmetry of the molecule (Herzberg 1945). Here, we do number v not discuss these cases in more detail, referring to Bauder ˆ vv(S) = (S) ˆ2 + (S) ˆ2 + (S) ˆ2 − ˆ4 2011: Fundamentals of Rotational Spectroscopy,this Hrot A Jz B Jx C Jy DJ J handbook, and in particular to Boudon et al. 2011: Spher- − D Jˆ2Jˆ2 − D Jˆ4 ical Top Theory and Molecular Spectra, this handbook, JK z K z ˆ2 ˆ2 ˆ2 ˆ4 ˆ4 ˆ6 for spherical rotors. + d1J (J+ + J−) + d2(J+ + J−) + HJ J Asymmetric rotors have no simple term value formulae; + ˆ4 ˆ2 + ˆ2 ˆ4 + ˆ6 + ˆ4 ˆ2 however, they have the advantage that degenerate vibra- HJKJ Jz HKJJ Jz HK Jz h1J (J+ tions do not occur, in general. This allows one to deal ˆ2 ˆ2 ˆ4 ˆ4 ˆ6 ˆ6 + J−) + h2J (J+ + J−) + h3(J+ + J−) (159) with the rotation–vibration problem in a fairly general way. We discuss here an implementation, which exists also in where the form of a computer program (Luckhaus and Quack ˆ2 = ˆ2 + ˆ2 + ˆ2 ˆ = ˆ ± ˆ 1989), which was designed to allow for treating all kinds J Jx Jy Jz and J± Jx iJy of interaction of rotation and vibration in a very general way. A similar approach was later also used by Pickett Note that in Equation (159) as well as in the following (1991). Following Watson, one can, on the basis of the equations in this section we do not explicitly write the general Hamiltonian discussed in Section 5.2, define effec- v-dependence of the rotational constants to simplify the tive Hamiltonians that form blocks of “polyads” similar to expressions. r those discussed in Section 3. However, now both anhar- We also give the A-reduced Hamiltonian in the I monic and Coriolis-type rotation–vibration couplings are representation as another example with terms diagonal in included (see the scheme in Figure 29). If the molecule has the vibrational quantum number v: some symmetry, anharmonic interactions are allowed only ˆ vv(A) = (A) ˆ2 + (A) ˆ2 + (A) ˆ2 between states of the same vibrational symmetry species Hrot A Jz B Jx C Jy (as indicated by vertical arrows in Figure 29), whereas rovi- − ˆ4 − ˆ2 ˆ2 − ˆ4 ∆J J ∆JKJ Jz ∆K Jz brational Coriolis interactions also couple states of different vibrational symmetry (but with the same total, rovibrational − 1 ˆ2 + ˆ2 ˆ2 + ˆ2 [δJ J δK Jz , J+ J−]+ symmetry species, of course). Figure 29 shows a case of 2 + ˆ6 + ˆ4 ˆ2 + ˆ2 ˆ4 + ˆ6 a Cs-symmetrical molecule (for example, CHClF2 (Luck- φJ J φJKJ Jz φKJJ Jz φK Jz haus and Quack 1989, Albert et al. 2006)). As an example 1 ˆ4 ˆ2 ˆ2 ˆ4 ˆ2 ˆ2 with C symmetry, we mention CH D (Ulenikov et al. + [η J + η J J + η J , J+ + J−]+ 2v 2 2 2 J JK z K z 2006). Watson (1968) has given several representations for such effective Hamiltonians and we give here the S- (160) reduced effective Hamiltonian in the I r representation as Note that in the following discussion the choice of the rep- an example, with terms diagonal in the vibrational quantum resentation will not be explicitly mentioned in the notation of the rotational constants to simplify the expressions. For the off-diagonal terms representing a α-Coriolis coupling E (α = x,y,z), one has: ˆ v v = v v ˆ + v v ˆ ˆ Hrot iξ α Jα ηβγ [Jβ , Jγ ]+ (161)

where α,β,γ are all different. The ξ α constants are related to the common Coriolis parameters ζ α by ω ω 1/2 v v = α n + m + vn ξ α 2Bαζ m,n (vm 1) ωm ωn 4 (162) in which ωm and ωn are the harmonic frequencies of the fundamentals, the coupling being between the (vm, vn, vk) A′ A′′ and the (vm+1, vn−1, vk) states. For anharmonic coupling, which can be of the Fermi resonance, Darling–Dennison type, or more complicated, one has

Figure 29 Illustration of a resonance polyad of an effective ˆ v v = + ˆ2 + ˆ2 + ˆ2 − ˆ2 Hamiltonian including various types of interactions. Hrot F FJ J FzzJz Fxy(Jx Jy ) (163) Fundamentals of Rotation–Vibration Spectra 153

This introduces a rotational dependence of the anharmonic For the parameter refinement, the program uses the resonance. In the program WANG of Luckhaus and Quack Marquardt (1961) algorithm (Press et al. 1986), a nonlinear (1989) all these couplings (and higher terms, if needed) least squares fit procedure. The required first derivatives are included. The symmetry of Hamiltonian and coupling of χ 2 (the sum of squared deviations) with respect to schemes can be specified in a very simple and general the spectroscopic parameters pα are calculated analytically, way as described in the Appendix of Luckhaus and Quack using the Hellmann–Feynman theorem, i.e., the derivatives (1989) which we almost literally recall here. of the eigenvalues εi are obtained according to The number of diagonal blocks into which the Hamilto- nian is split has to be specified. For each of these “symmetry ∂ = † ∂ εi ci H ci (164) blocks”, a small matrix determines the mechanism that cou- ∂pα ∂pα ples the different vibrational states and the Wang blocks that appear in the symmetry block. By the numbers contained where εi is the ith eigenvalue, ci is the ith eigenvector, in these matrices (“coupling schemes”), the program iden- H the Hamiltonian matrix, and pα the αth spectroscopic tifies the subroutines to be called for setting up the various parameter. diagonal (Wang) and off-diagonal (coupling) blocks. The As long as there is no coupling between vibrational diagonal numbers consist of two digits: the first one specify- states, the usual set of rotational quantum numbers will ing the vibrational state and the second one the Wang block generally provide an unambiguous assignment of a transi- (1, 2, 3, 4 = E+,E−,O+,O−). The off-diagonal numbers tion to a pair of eigenstates (the vibrational states involved stand for the type of coupling (0, 1, 2, 3 = none, x-, y-, have, of course, also to be specified). This is not true in the z-Coriolis coupling). The corresponding block structure of case of coupled vibrational states, as the energetical order- the program is very easily extended to include other types ing of eigenstates belonging to different Wang blocks varies of coupling simply by assigning a new number to another with the spectroscopic parameters. On the other hand, the subroutine containing the necessary formulae for the matrix ordering of rotational quantum numbers within the same elements. Wang block is normally independent of the actual parame- As an example we give the coupling scheme used ters. One, therefore, first assigns the eigenvectors of a given for the a-Coriolis interaction of the two CF-stretching symmetry block (of a given J value) to the various Wang 35 fundamentals in CH ClF2 (Luckhaus and Quack 1989): blocks, where care must be taken that the number of eigen- the lower state is identical with the vibrational ground state vectors assigned to a Wang block is equal to its dimension. (state no. 1). There is no coupling and, therefore, each of The sum over the squared coefficients of the corresponding the four Wang blocks forms a symmetry block by itself basis functions serves as a criterion. Within these sets, the 11 12 13 14 same correspondence between rotational quantum numbers and the eigenvalue-ordered eigenvectors as in the absence The upper state consists of the two CF-stretching funda- of coupling is assumed. This method guarantees a smooth mentals (a : ν3 = state no. 2; a : ν8 = state no. 3) coupled transition from the cases of appreciable to those of van- by a-Coriolis interaction (a-axis corresponding to z-axis in ishing coupling. The assignments have generally proved I r representation). Neglecting the (symmetry allowed) c- stable, i.e., there is no “switching” (or crossing) of assign- Coriolis coupling (code:2) leads to four symmetry blocks ments over a reasonable range of values of the coupling containing two Wang blocks each parameters. Only assignments for which the deviations between a :213 243 a :223 233 observed and calculated wavenumbers are less than a 332 333 331 334 given tolerance are considered for the calculation of χ 2 and its derivatives. The number of assignments consid- If c-Coriolis coupling were included, the Hamiltonian for ered changes, therefore, from one refinement cycle to the the upper state would only split into two symmetry blocks: next (floating data base). Convergence is assumed for relative changes of the weighted mean square deviation a :213 0 2 a :22302(χ2/Σ[statistical weights]) of less than 0.5 × 10−4.This 332 2 0 331 2 0 not only helps cope with typing errors in large sets of 02 243 02 233assignments but is also necessary when combining the pro- 20 333 20 334gram with a simple automatic assignment procedure. The advantage of this method over simply setting the c- In this case, a stick spectrum is calculated with the refined Coriolis coupling blocks equal to zero is obvious since parameters and the experimental spectrum is searched for the operations count for matrix diagonalization is roughly peaks coinciding within a given tolerance, with calculated quadratic in the matrix dimension. transitions not yet assigned. An intensity limit helps avoid 154 Fundamentals of Rotation–Vibration Spectra

−1 Table 10 Spectroscopic parameters (in cm )fortheν3-andν8- Hamiltonians. Line intensity information complements this, 35 fundamentals (C–F stretching) of CH ClF2. Standard deviations usually by comparing simulated and experimental data, are given in parentheses as uncertainties in the last digits (after although least squares analyses of intensities are also pos- Luckhaus and Quack 1989). sible and sometimes carried out. These effective Hamilto- nians can then be used for analyses of time-dependent state ν3 ν8 populations. One can also use the rotational constants in the −1 (a) (a,b) ν˜ 0 /cm 1108.7292(1) 1127.2854(1) various vibrational states to determine Ae, Be,andCe at the A /cm−1 0.341271(39) 0.338478(38) equilibrium geometry, very similar to diatomic molecules − B /cm 1 0.16226587(82) 0.16195055(61) discussed in Section 2 but with added complexities. We −1 C /cm 0.11616297(59) 0.11735914(52) refer here to the critical discussion of equilibrium struc- −6 −1 DJ /(10 cm ) 0.02937(37) 0.05887(26) −6 −1 tures of derived from spectra of CH2D2 (Ulenikov DJK /(10 cm ) 0.0607(22) 0.2872(11) −6 −1 et al. 2006). Another approach to equilibrium structures of DK /(10 cm ) 0.2237(71) 0.1318(83) −6 −1 d1 /(10 cm ) −0.02107(26) −0.01081(21) polyatomic molecules is to adjust parameters of a potential −6 −1 d2 /(10 cm ) −0.01010(10) −0.001183(92) hypersurface in the calculation of vibrational ground-state −12 −1 HJ /(10 cm ) 0.227 0.227 properties and compare these with the experimental con- −12 −1 HK /(10 cm ) 1.958 1.958 stants A0, B0, C0 of the vibrational ground state (Hollen- −1 ξ z /cm 0.38370(94) stein et al. 1994). Such calculations have been carried out rms error /cm−1 0.0018 even for high-dimensional (CH ) and very floppy molecules Assignments 1424 2748 4 Max. J 40 51 (such as (HF)2) (Quack and Suhm 1991). Along fairly simi- Range/cm−1 1095–1158 lar lines, Gauss and Stanton (1999) have proposed comput- ing the difference between Ae, Be, Ce and A0, B0, C0 from ( ) − a Corrected by a small average calibration shift of +0.0018 cm 1. ab initio calculations combined with perturbation theory to (b) Here we correct a misprint of the original paper. derive semi-experimental equilibrium structures.

“assignment” of noise. A new fit is then performed and the whole procedure is repeated until no new assignments 6 PATTERN RECOGNITION FOR are found. As extrapolations beyond the range of quantum SPECTROSCOPIC ASSIGNMENTS AND numbers for which a set of spectroscopic parameters has been optimized are rather unreliable, the simulation in each ANALYSIS OF ROVIBRATIONAL step is limited to quantum numbers only slightly higher SPECTRA WITH LOOMIS–WOOD than those already assigned. The range of assigned quantum DIAGRAMS numbers is thus extended stepwise. As the program can be run interactively, the necessary calculations can be carried 6.1 General Aspects out efficiently. As an example, Table 10 gives the results for param- The high-resolution rotation–vibration spectra of poly- eters obtained for the two Coriolis-coupled CF-stretching atomic molecules often show an enormous complexity, 35 fundamentals in CH ClF2 (a “dyad”). A more compli- which makes assignment of the rotation–vibration states cated example is provided by the tetrad of levels 2ν9(A ), connected in a transition, the first step in any analysis,

2ν6(A ), ν6 + ν9(A ), and the fundamental ν4(A ) in that sometimes very difficult. There are several techniques that same molecule analyzed by Albert et al. (2006). Here, ν4 help to establish and confirm such quantum number assign- and 2ν6 are coupled by a Fermi resonance, whereas 2ν9 ments: and 2ν are coupled to ν + ν by Coriolis interactions. 6 6 9 1. One can check the assignments by ground-state (or The set of parameters obtained is given in Table 11. See excited-state) combination differences, which must also Snels et al. 2011: High-resolution FTIR and Diode match to within the experimental accuracy (Section 2). Laser Spectroscopy of Supersonic Jets, this handbook, for 2. One can establish the symmetry assignment by means an extended treatment of ν /ν Coriolis resonance band of 3 8 of nuclear spin statistical weights and the resulting line CHClF . 2 intensity alternations in high-resolution spectra. When considering the relatively large numbers of param- 3. One can simulate intensities in spectra obtained as eters in the effective Hamiltonians of this kind, one should a function of temperature and thereby gain effective also see the much larger number of line data analyzed in information on the lower state of the observed line. such spectra. Frequently, many thousands of lines are accu- rately measured and assigned and thus an enormous data All these methods are quite widely used and the combina- set is described in a very compact form by such effective tion difference technique has even provided the basis for a Fundamentals of Rotation–Vibration Spectra 155

−1 35 Table 11 Spectroscopic constants (in cm )oftheν4, 2ν6,ν6 + ν9,and2ν9 tetrad of CH ClF2 (after Albert et al. 2006).

(a) (b) (b) Ground state ν4(A ) 2ν6(A )ν6 + ν9(A ) 2ν9(A )

ν˜ 0 0.0 812.9300(45) 825.4091(45) 778.61 732.41 A 0.341392948 0.34103816(137) 0.34151924(138) 0.34218295 0.34302622 B 0.162153959 0.16119383(30) 0.16194755(30) 0.16195124 0.16197450 C 0.116995535 0.11651661(52) 0.11671129(40) 0.11658536 0.11648176 −8 ∆J /10 5.2247979 5.0155(35) 5.5275(30) 5.224797 5.2247979 −8 ∆JK/10 15.3149617 13.497(27) 10.741(22) 15.3149617 15.3149617 −8 ∆K /10 16.4159900 18.651(34) 20.507(37) 16.4159900 16.4159900 −8 δJ /10 1.4746802 1.4383(11) 1.4998(10) 1.4746802 1.4746802 −8 δK /10 16.7271720 16.342(11) 13.4991(87) 16.7271720 16.7271720 −12 ΦJ /10 0.02341953 0.02341953 0.02341953 0.02341953 0.02341953 −12 ΦJK/10 0.33456479 0.33456479 0.33456479 0.33456479 0.33456479 −12 ΦKJ/10 −0.03422368 −0.03422368 −0.03422368 −0.03422368 −0.03422368 −12 ΦK /10 0.09860155 0.09860155 0.09860155 0.09860155 0.09860155 −12 φJ /10 0.01093423 0.01093423 0.01093423 0.01093423 0.01093423 −12 φJK/10 0.18072503 0.18072503 0.18072503 0.18072503 0.18072503 −12 φK /10 3.05377929 3.05377929 3.05377929 3.05377929 3.05377929 F −7.6839(36) −3 Fzz/10 1.4715(12) ξ z 0.023 0.023 − − ξ y 0.218 0.218 −8 (c) ηKxz/10 7.593(94) ndata 2284 1590 max J 78 78 max Ka 30 30 −3 drms/10 0.308 0.288

(a) Blanco et al. (1996). (b) Rotational constants and Coriolis interaction constants transferred from Kisiel et al. (1997). (c) Interaction between ν4 and ν6 + ν9. computer program for automatic assignments in the Zurich¨ the development of personal computers. Scott and Rao group. Nevertheless, often the initial assignment is difficult (1966) studied the 13.7 µm region of the acetylene spec- and this cannot be readily carried out automatically. Here, trum with the first program able to generate Loomis–Wood interactive visual aids for pattern recognition can be help- diagrams. Later, several interactive Loomis–Wood assign- ful and we describe as an important example the method ment programs were developed, including the pioneering originally published by Loomis and Wood (1928) and now work of Winnewisser et al. (1989). The so-called Giessen used in some laboratories. Loomis–Wood program was originally designed only for The Loomis–Wood algorithm uses the periodicity of a linear molecules (Haas et al. 1994, Schulze et al. 2000, recurring pattern of lines in the spectrum and organizes Albert et al. 1996, 1997a,b, 1998, 2001) but is also useful in the lines in such a way that the periodicity is translated the analysis of spectra of asymmetric top molecules (Albert into emerging, recognizable, continuous patterns in the et al. 2003, 2004a,b, Albert and Quack 2007). The first Loomis–Wood diagram. The concept of the Loomis–Wood generation of Loomis–Wood programs uses line positions diagram arose in 1928 (Loomis and Wood 1928): to assign and intensities to recognize spectral patterns (McNaughton the rotational structure of the blue-green bands of the et al. 1991, Winther et al. 1992, Stroh et al. 1992). Most of diatomic molecule Na2, Loomis and Wood plotted ν(J)˜ − the programs were based on a DOS interface, which limits ν˜[Q(J )], the difference between the observed transition and the resolution. Some programs already provided approaches that corresponding to the Q-branch, as a function of the to assign spectra of symmetric top molecules (Moruzzi J quantum number for four measured bands. P- and R- et al. 1994, Brotherus 1999) and distinguish asymmetry branches always appear as smooth curves, nearly straight splitting components (Moruzzi et al. 1998). Nowadays, the lines with different slopes. new programs take advantage of graphic tools and mouse- These diagrams seem helpful but were not often used in interactive approaches in order to make the assignment the beginning since they are time consuming when built process easier. Here, we refer to the Giessen Loomis–Wood manually. The first significant improvement came with program adapted for Igor Pro (Neese 2001), the program 156 Fundamentals of Rotation–Vibration Spectra written by Gottselig (2004) for the Zurich¨ group, the to rewrite equation (165) in powers of (−J ): CAAARS package (Medvedev et al. 2005) and its alter- 2 native AABS (Kisiel et al. 2005) optimized for asymmetric ν˜[P(J )] −˜ν0 =(2Bv + ∆B)(−J)+ (∆B − ∆D)(−J) top rotors, LWW (Lodyga et al. 2007) for symmetric rotors −2(2D + ∆D)(−J)3 −∆D(−J)4 (168) developed for Windows, PGOPHER (Western 2010), or v ATIRS (Tasinato et al. 2007). Besides Loomis–Wood plots, Equation (168) is a polynomial of fourth degree in (−J ). some of these programs provide further help such as the The same treatment can be done for transitions of the calculation of combination differences or simulation of the R-branch: experimental spectrum following a fit procedure. In our group, we have recently developed a new pack- ν˜[R(J )] =˜ν + F (J + 1) − F (J ) (169) age called LOOM4WANG (Albert et al. 2011) to plot 0 v v Loomis–Wood diagrams, which has the advantage of work- Analogous to Equation (168), ν˜[R(J )] −˜ν can then be ing on Unix, Windows, and Mac platforms. The pro- 0 written as a polynomial, in (J + 1) this time, with the same gram was developed under Linux using the GTK+/Gnome coefficients: Application Development suite (Version 2) packagea for the graphical interface. For the optimization and fit pro- ν˜[R(J )]−˜ν = (2B +∆B)(J + 1)+(∆B−∆D)(J + 1)2 cedures, the code uses the Fortran C-MINUIT package 0 v 3 4 from the CERN program library (James 1998). All the C- −2(2Dv +∆D)(J + 1) −∆D(J + 1) MINUIT fitting facilities for the different expressions of (170) the Hamiltonian operator can be used. The present ver- In general, the transitions for P- and R-branches can sion of the program can fit spectroscopic parameters of be written as a polynomial of fourth degree in m with a linear molecule, taking the centrifugal distortion cor- m =−J for the P-branch and m = J + 1fortheR- rections up to the second-order into account, and the branch: upcoming version will also be able to treat the case of symmetric top molecules. Furthermore, LOOM4WANG ν˜ LW (m) =˜ν[P(−m)] −˜ν0 for m<0 (171) provides an interface for the external fit program WANG =˜ − −˜ ≥ (Luckhaus and Quack 1989). All the Loomis–Wood ν[R(m 1)] ν0 for m 1 (172) diagrams shown in the following were generated with LOOM4WANG. with

2 ν˜ LW (m) = (2Bv + ∆B)m + (∆B − ∆D)m 6.2 Linear Molecules 3 4 −2(2Dv + ∆D)m − ∆Dm (173) The concept of Loomis–Wood diagrams is based on the expression of the rovibrational transitions for linear Usually, ∆B, Dv ,and∆D are several orders of magni- molecules. Here, we detail the case of transitions for tude smaller than Bv , which means that, in a first approx- the P-branch with correction of the centrifugal distortion imation, all the terms in m of degree higher than one are + constants up to the first order as an example: negligible compared to 2Bv ∆B in equation (173), at least for m values that are not very large. For example, ν˜[P(J )] =˜ν0 + Fv (J − 1) − Fv (J ) in the case of carbonyl sulfide (OCS), the vibrational −1 2 2 band at ν˜ 0 = 3095.55442(9) cm has been assigned to the =˜ν0 + Bv J(J − 1) − Dv (J − 1) J 1200–0000 transition of the main isotopomer 16O12C32S 2 2 − Bv J(J + 1) − Dv J (J + 1) (165) (Maki and Wells 1991); the analysis of the rotational

transitions results in the following rotational constants: where ν˜ = G(v ) − G(v ) is the band center of the vibra- −1 −4 −1 0 Bv = 0.202849 cm , ∆B = 6.23025 10 cm , Dv = − − − − tional transition, Bv is the rotational constant for the lower 4.24954 10 8 cm 1,and∆D =−7.14171 10 9 cm 1;in vibrational state and Bv for the upper vibrational state, other words, terms with order higher than one are negligible Dv is the correction of the centrifugal distortion to the up to J = 64. first order for the lower vibrational state, and Dv that The almost linear relationship between m and the tran- for the upper vibrational state. We use the following rela- sition (the higher order terms being negligible at first tions approximation) is exploited as follows: in the modern form of the Loomis–Wood diagram, the rovibrational Bv = Bv + ∆B (166) spectrum is cut in segments of width ∼2Bv (actually Dv = Dv + ∆D (167) 2Bv + ∆B) and stacked one beneath the other in order to Fundamentals of Rotation–Vibration Spectra 157

0.8

0.6

0.4 Absorbance

0.2

0.0 3060 3065 3070 3075 30803085 3090 3095 3100 3105 3110 3115 3120 3125 3130 Wavenumber /cm−1

Figure 30 FTIR spectrum of OCS recorded with a White type cell of 19.2 m length with the Bruker IFS125 prototype ZP 2001 in the 3055–3130 cm−1 region. The box indicates the region illustrated in Figure 31 (pressure of OCS is ca 0.2 mbar, decadic absorbance lg(I0/I) is shown). highlight lines belonging to the same group of transitions. For larger molecules, a more exact description of the tran- Figure 30 shows the FTIR spectrum of the OCS molecule sition, taking into account the correction of the centrifugal in the 3055–3130 cm−1 region and Figure 31 shows how distortion constants up to the second or even higher order, −1 the ∼2Bv -wide slices of the 3100–3104 cm region of may be required; it leads to a polynomial of at least the the spectrum are arranged one beneath the other in order to sixth degree in m: reveal patterns in the spectrum. ˜ = + + − 2 The corresponding Loomis–Wood diagram constructed νLW (m) (2Bv ∆B)m (∆B ∆D)m 3 with the LOOM4WANG program is shown in Figure 32: − 2(2Dv + ∆D − Hv − ∆H/2)m the lines of the spectrum are represented by bars with + − 4 + + 5 lengths proportional to the intensity of the transitions (3∆H ∆D)m 3(2Hv ∆H )m 6 in the spectrum. If a correct value of Bv is used, the + ∆Hm bars corresponding to the same group of rovibrational +··· (174) transitions appear aligned in the Loomis–Wood diagram. The pattern at the center of Figure 32 is composed of the Hv is the correction of the centrifugal distortion to the most intense lines of the spectrum and corresponds to the second order in the lower vibrational state, and Hv is that transitions of the 1200–0000 band of the main isotopomer in the upper vibrational state with ∆H = Hv − Hv .These 16O12C32S. terms are smaller than Dv and ∆D, which does not change The top of the diagram corresponds to the lower end of the interpretation of the Loomis–Wood diagram. the spectral range shown in Figure 30, and as the wavenum- Plotting the same Loomis–Wood diagram as in Figure 34, ber increases from left to right in Figure 30, it increases one beside the other three times, allows the user to from top to bottom by row in Figure 32. Therefore, the recognize the continuity of the patterns. They are of weaker top half of the diagram illustrates transitions lower than the intensity and with different slopes: these patterns have ˜ band center ν0, i.e., the P-branch, and the bottom half of been assigned to the splitting of the 131–010 transition the diagram shows transitions higher than the band center, (see Maki and Wells 1991 and references therein). Since i.e., the R-branch. The closer the transition to the center of v2 > 0, l>0, the state is split into e and f components of the diagram, the lower is the J value. Figures 33(a) and (b) l, corresponding to the B and C branches on the diagram illustrate the effect of neglecting the correction of the cen- (Figure 34). trifugal distortion: the deviation from the vertical straight Obviously, the interpretation of the spectrum of a linear line for large J values (tails of the plot) in Figure 33(a) molecule such as OCS is one of the simplest cases for indicates the limitations of the linear approximation of the a Loomis–Wood plot and may not require such graphical polynomial because Dv as well as ∆D were set to zero tools to get the proper assignment. This section has in Figure 33(a), whereas they were optimized in 33(b), all illustrated the key points of the Loomis–Wood diagram, other parameters being kept the same. which can be useful for more complex spectra. 158 Fundamentals of Rotation–Vibration Spectra am. 3104 3104 dicated box is retained 3104 3104 3104 3104 3104 3104 Figure 30 in order to obtain the patterns aligned for a Loomis–Wood diagr * * 3103 ed on each slice to make the displacement of the spectrum clear. Only the in * 0.8 0.6 0.4 0.2 0.0

1 − * 0.8 0.6 0.4 0.2 0.0 3102 * 0.8 0.6 0.4 0.2 0.0 Wavenumber / cm * 0.8 0.6 0.4 0.2 0.0 * 0.8 0.6 0.4 0.2 0.0 3101 * 0.8 0.6 0.4 0.2 0.0 Superposition of slices of a section of the FTIR spectrum of OCS as indicated in 0.8 0.6 0.4 0.2 0.0 3100 0.8 0.6 0.4 0.2 0.0 Figure 31 The star indicates thefor first the line Loomis–Wood considered in diagram this shown pattern in here Figure and 32. is repeat Fundamentals of Rotation–Vibration Spectra 159

P-branch values

m

R-branch

− + B 0.0 B

Figure 32 Loomis–Wood diagram of the FTIR spectrum of OCS shown in Figure 30. The box indicates the spectral region −1 −1 selected in Figure 30. The spectroscopic parameters are as follows: ν˜ 0 = 3095.55442 cm , B = 0.2028567408 cm ,and∆B = −0.0005455008 cm−1.

−B 0.0 +B −B 0.0 +B (a) (b)

Figure 33 Loomis–Wood diagram of the FTIR spectrum of OCS shown in Figure 30. The spectroscopic parameters are as −1 −1 −1 −1 −1 follows: ν˜ 0 = 3095.55442 cm , B = 0.2028567408 cm ,and∆B =−0.0005455008 cm .(a)D = 0cm and ∆D = 0cm ; (b) D = 4.409752 10−8 cm−1,and∆D =−5.7191468 10−9 cm−1.

6.3 Nonlinear Molecules 6.3.1 Symmetric Top Molecules

Each type of molecule has its own selection rules and, therefore, displays different patterns in Loomis–Wood Here we discuss the case of the prolate symmetric top diagrams. In this section, we describe how we can anticipate (A>B= C), and that for the oblate symmetric top (A = the Loomis–Wood plot patterns of nonlinear molecules, on B>C) can be easily obtained by analogy (see below). the basis of that observed for linear molecules. Two kinds of bands are observable: parallel (∆K = 0and 160 Fundamentals of Rotation–Vibration Spectra

B-branch

P-branch

C-branch values

m

R-branch

−3B −B 0.0 +B +3B

Figure 34 Loomis–Wood diagram of the FTIR spectrum of OCS shown in Figure 30 reproduced three times. The spectroscopic −1 −1 −1 −8 −1 parameters are as follows: ν˜ 0 = 3095.55442 cm , B = 0.2028567408 cm , ∆B =−0.0005455008 cm , D = 4.409752 10 cm , and ∆D =−5.7191468 10−9 cm−1.

−1 Table 12 Rotational constants (in cm ) of the ground state and the ν1 state of CH3I (after Paso and Antilla 1990).

= − Rotational constant X Ground state ν1 state ∆X Xν1 X

A 5.173931 5.121724 −52.207 × 10−3 B 0.250215625 0.250167425 −0.0482 × 10−3 −6 −6 −9 DJ 0.2103983 × 10 0.2118883 × 10 1.49 × 10 −6 −6 −6 DJK 3.294545 × 10 3.254545 × 10 −0.04 × 10 −6 −6 −6 DK 87.34 × 10 88.06 × 10 0.72 × 10

∆J =±1forK = 0, or ∆J = 0, ±1forK = 0), and For K = 0, the equation is analogous to equation (173) perpendicular (∆K =±1, ∆J = 0, ±1). obtained in the case of the linear molecule as discussed The expression of the rotational term value is in Section 6.2 with DJ = Dv , and therefore we expect a pattern similar to that shown in Figure 32 for the = + + − 2 − 2 Fv(J, K)prolate BvJ(J 1) (Av Bv)K DJv J OCS molecule. Regarding K = 0, for each K value, we × + 2 − + 2 − 4 get a new pattern in the Loomis–Wood diagram, shifted (J 1) DJKv J(J 1)K DKv K − 2 − 4 = (175) from (∆A ∆B)K ∆DK K compared to the K 0 − where we have considered the centrifugal distortion cor- pattern. Since (∆A ∆B) is generally much larger than = = rection up to the first order to simplify the mathematical ∆DK , as soon as the K 0andK 1 patterns are treatment. identified, the other are shifted by roughly K2 times the difference between the K = 0andK = 1 patterns. For Parallel Band instance, Table 12 lists the spectroscopic constants of the In the case of parallel bands, the transition for the ν band of the symmetric top molecule CH I. In this Loomis–Wood diagram can be written as follows for both 1 3 case, ∆DJK is seven orders of magnitude smaller than P- and R-branches: 2 2Bv + ∆B, and up to K = 300, ∆DJKK is negligible 2 compared to 2B + ∆B: the patterns appear with almost ν˜ LW (m) = (2Bv + ∆B + ∆DJKK )m + (∆B − ∆DJ v the same slope as that for K = 0. The deviation from − 2 2 ∆DJKK )m the “straight line”, however, starts at lower J values 3 4 with increasing K value since ∆DJ is only two orders − 2(2DJ + ∆DJ )m − ∆DJ m of magnitude smaller than 2DJ (third power of m in 2 4 + (∆A − ∆B)K − ∆DK K (176) equation (176)). Fundamentals of Rotation–Vibration Spectra 161

Regarding the oblate symmetric top molecules, one has Perpendicular band to replace (Av − Bv) in equation (175) by (Cv − Bv), which Here, we discuss the characteristics of the Loomis–Wood leads to plot by comparing it to that corresponding to a parallel band. In the case of a perpendicular band, the transition = + + − 2 − 2 + 2 Fv(J, K)oblate BvJ(J 1) (Cv Bv)K DJv J (J 1) can be written as follows: 2 4 − DJK J(J + 1)K − DK K (177) 2 v v ν˜ LW (m) = [2Bv + ∆B + ∆DJK(K ± 1) − DJK × ± + − − and the corresponding transition for the Loomis–Wood (2K 1)]m [∆B ∆DJ (∆DJK diagram: 2 2 × (K ± 1) − DJK (2K ± 1))]m − 2(2DJ 3 4 2 + ∆DJ )m − ∆DJ m + (∆A − ∆B) ν˜ LW (m) = (2Bv + ∆B + ∆DJKK )m + (∆B − ∆DJ × ± 2 − − ± 2 2 3 4 (K 1) (Av Bv )(2K 1) − ∆DJKK )m −2(2DJ +∆DJ )m −∆DJ m − ± 4 − ± 2 2 4 ∆DK (K 1) DK (2K 1)[K + (∆C − ∆B)K − ∆DK K (178) + (K ± 1)2] (181) The treatment of the Loomis–Wood diagram is, there- fore, exactly the same as for prolate symmetric top with ∆K =±1, m =−J for the P-branch and m = (J + molecules. 1) for the R-branch. Since the ∆J = 0 transitions are allowed for K = 0, Given the rotational constant X in equation (176) for 2 2 one could wonder how the Q-branches are handled in a a parallel band, ∆XK is replaced by ∆X(K ± 1) − Loomis–Wood diagram: X(2K ± 1) in equation (181) for a perpendicular band. In this case, the added correction −X(2K ± 1) may be larger than the term ∆X(K ± 1)2 itself. Moreover, this correction ν˜[Q(J )] −˜ν0 = Fv (J, K) − Fv (J, K) may be nonnegligible for the low powers of m in the 2 = Bv J(J + 1) + (Av − Bv )K polynomial. Therefore, no more straight line–like patterns 2 2 2 can be expected, as seen in the Loomis–Wood plot for a − D J (J + 1) − D J(J + 1)K Jv JKv parallel band, although patterns should still be recognizable. 4 2 The main component of the distance to the band center ν˜ − D K −[B J(J +1)+(A −B )K 0 Kv v v v 3 is proportional to DK K : patterns with different values of 2 2 2 − + − + K start at different slices of the spectrum in the diagram, DJv J (J 1) DJKv J(J 1)K and are apparently similar to those at different m values 4 − DKv K ] (179) when the parameters of the diagram are optimized for a parallel band. For an oblate symmetric top molecule, the 2 ν˜[Q(J )] −˜ν0 = (∆B − ∆DJKK )J + (∆B − ∆DJ transition can be written as follows: 2 2 − ∆DJKK )J 2 ν˜ LW (m) = [2Bv + ∆B + ∆DJK(K ± 1) 3 4 − (2∆DJ )J − (∆DJ )J + [(∆A − ∆B) − DJK (2K ± 1)]m + [∆B − ∆DJ 4 −∆DK ]K (180) 2 2 − (∆DJK(K ± 1) − DJK (2K ± 1))]m 3 4 Here again, the Loomis–Wood term can be considered to − 2(2DJ + ∆DJ )m − ∆DJ m be a polynomial of fourth degree in m with m = J , but + − ± 2 − − ± the coefficients of the first and third powers are different (∆C ∆B)(K 1) (Cv Bv )(2K 1) than those in equation (176). Since for the first power of 4 − ∆DK (K ± 1) − DK (2K ± 1) the polynomial, ∆B is much smaller than the largest term × 2 + ± 2 2Bv found in the other expressions of ν˜[P(J )] −˜ν0 and [K (K 1) ] (182) ν˜[R(J )] −˜ν0, the pattern of the Q-branch appears as a horizontal line in the center of the diagram and cannot 6.3.2 Spherical Top Molecules be as easily assigned with the help of the Loomis–Wood diagram as the P- and R-branches. Other programs have The similarity in the expressions for the rotational energy been developed to analyze this part of the spectrum, for of a spherical top and that of a linear molecule makes the example, QBRASS (Stroh 1991). analysis of the spherical top’s Loomis–Wood plot easier. 162 Fundamentals of Rotation–Vibration Spectra

Indeed, a spherical top molecule can be considered to a-axis (z) be the limiting case of a symmetric top molecule with 2 I Ia = Ib = Ic (i.e., A = B = C) with all coefficients of K in the expression of the term value equal to zero: c-axis (y) = + − 2 + 2 Fv(J, K)sph. top BvJ(J 1) DJv J (J 1) (183) 213 pm 108° In equation (183), we have once again considered the C centrifugal distortion correction up to the first order to simplify the mathematical treatment. This equation is the 108 pm D same as that used in the case of linear molecules. The H Loomis–Wood diagram of a spherical top molecule should, D therefore, look similar to that of a linear molecule as described in Section 6.2. Figure 35 Illustration of the CHD2I molecule and its principal axis system. The ac plane defines the Cs symmetry plane in the plane of the page, the b axis being perpendicular to the ac 6.3.3 Asymmetric Top Molecules plane. The equilibrium geometry parameters are from ab initio calculations by Horka´ et al. (2008). [Reproduced with permission The case of asymmetric top molecules may seem to be from Albert et al. 2010.] the most complicated because the K degeneracy of the symmetric top is removed. The three kinds of allowed for J = 0 − 6 in Herzberg (1945) and citations therein. transitions are defined with J , Ka,andKc: Nevertheless, it is sometimes possible, in the first approxi- mation, for the analysis (and not for the determination of the

∆J =±1,∆Ka = 0, ±2, ±4 ..., rotational constants) to consider the molecule as a nearly symmetric top. In this case, the Loomis–Wood diagram =± ± ± = ∆Kc 1, 3, 5 ..., for Ka 0 and (184) is quite similar to that of a symmetric top molecule; a− − − ∆J = 0, ±1,∆Ka = 0, ±2, ±4 ..., (respectively b and c ) type bands of an asymmetric top exhibit similar patterns to those of a parallel (respec- =± ± ± = ∆Kc 1, 3, 5 ..., for Ka 0 (185) tively perpendicular) band of a symmetric top molecule. To illustrate this point, we discuss the case of CHD2I, a for a-type transitions, molecule belonging to the Cs point group. Figure 35 shows the structure of the molecule with the principal axes a, b, =± =± ± ± ∆J 1,∆Ka 1, 3, 5 ... and c.

∆Kc =±1, ±3, ±5 ... (186) The asymmetry parameter κ of CHD2Idefinedby equation (190) for b-type transitions, and 2B − A − C κ = (190) A − C ∆J =±1,∆Ka =±1, ±3, ±5 ..., is close to −1(−0.9978; Riter and Eggers 1966); therefore, ∆Kc = 0, ±2, ±4 ... for Kc = 0, and (187) the molecule can be considered to be a nearly prolate ∆J = 0, ±1,∆K=±1, ±3, ±5 ..., a symmetric top. Figure 36 shows an overview of the ν1 band of CHD I corresponding to the CH-stretching component ∆Kc = 0, ±2, ±4 ... for Kc = 0 (188) 2 (Albert et al. 2010). Typical features of an a/c-type hybrid for c-type transitions. band are visible: the more intense a-type transitions form The second reason for the complexity of the treatment is the central Q-branch and the almost symmetric P- and R-branches are well resolved. The c-type transitions are the term Wτ in the expression of the rotational energy: ∆K =±1 detectable due to the strong a Q-branches spaced by (B + C) (B + C) more than 5 cm−1. The most intense region of the spectrum F(J ) = J(J + 1) + A − W (189) τ 2 2 τ has already been assigned (Hodges and Butcher 1996) and data presented here include a larger spectral range. where τ is the subscript added to take the K-type dou- The Loomis–Wood diagram of the a-type transitions of bling into account (−J ≤ τ ≤ J )andWτ is a function the ν1 band is shown in Figure 37 in the approximation of of the rotational constants A, B,andC and the quantum a linear molecule Hamiltonian. In this case, each almost number J . For more details, see the algebraic expressions straight pattern corresponds to the P-(top half of the Fundamentals of Rotation–Vibration Spectra 163

0.8

0.6

0.4 Absorbance

0.2

0.0 3000 3020 3040 3060 Wavenumber / cm−1

Figure 36 Overview of the FTIR spectrum of CHD2Iintheν1 region recorded in a White-type cell of 9.6 m length with the Bruker IFS125 prototype ZP 2001. [Reproduced with permission from Albert et al. (2010).]

= Ka 1

P-branch = Ka 5

= = = = = Q-branch Ka 3 Ka 2 Ka 0 Ka 7 Ka 4 values

m

R-branch

−(B +C)/2 0.0 +(B +C)/2

Figure 37 Loomis–Wood diagram of the a-type transitions of the ν1 band of CHD2I. The following parameters were used: −1 −1 −6 −1 −1 −1 ν˜ 0 = 3029.68 cm , (B + C)/2 = 0.215 cm , ∆[(B + C)/2] = 2.006 10 cm , D = 0cm ,and∆D = 0cm . [Reproduced with permission from Albert et al. 2010.]

diagram) and R-(bottom half of the diagram) branches (Ka = 1, 2): each P- and R-branch pattern is split into two of a-type transitions with the same Ka value; they are components. similar to those expected for a parallel band of a prolate Figure 38 shows a larger view of the Loomis–Wood dia- symmetric top with the spacing between a given Ka gram of Figure 37, in which weaker patterns corresponding pattern and the Ka = 0 pattern roughly proportional to to the P-and R-branches of the c-type transitions are also 2 Ka . The difference from a Loomis–Wood diagram of a visible. The middle part of the diagram corresponds to prolate symmetric top is due to the asymmetry splittings the a-type transitions discussed above. The curved pat- (Ka + Kc = J or J + 1) observed for low Ka values terns observed at the top of the diagram correspond to the 164 Fundamentals of Rotation–Vibration Spectra

= ← Ka 9 10 = ← Ka 8 9 = ← Ka 7 8 P-branch, = ← Ka 5 6 c-type K = 2←3 a K = 3←4 = ← a Ka 4 5

a-type values

m = ← = ← Ka 4 3 Ka 5 4 = ← K = 7←6 = ← Ka 3 2 a Ka 8 7 R-branch c-type

−(B+C )/2 0.0 +(B +C)/2

Figure 38 Loomis–Wood diagram of the a- (in the middle of the diagram) and c-type transitions (above the a-type transitions for the P- −1 branch and below for the R-branch, respectively) of the ν1 band of CHD2I. The following parameters were used: ν˜ 0 = 3029.68 cm , (B + C)/2 = 0.215 cm−1, ∆[(B + C)/2] = 2.006 10−6 cm−1, D = 0cm−1,and∆D = 0cm−1. [Reproduced with permission from Albert et al. 2010.]

P-branches of the c-type transitions, while those on the bot- expression of the transitions due to ∆Ka = 0 compared to tom of the diagram correspond to the R-branches. They are the linear molecule Hamiltonian approximation used to plot similar to those expected for a perpendicular band of a pro- the Loomis–Wood diagram. late symmetric top. The lines in a given pattern have the Figure 39 shows the same Loomis–Wood diagram as ← same Ka Ka value but not the same Kc (and therefore Figure 37, without taking the intensity of the spectral lines

Kc) value; the c-type patterns are no longer almost straight into account. This makes it possible to highlight small like the a-type patterns because of the correction in the deviations observed from the smoothed patterns: these are

P-branch

Q-branch values

m

R-branch

−(B+C ) 0.0 +(B+C )

Figure 39 Loomis–Wood diagram of the a-type transitions of the ν1 fundamental of CHD2I. The diagram is obtained with the same parameters as in Figure 37. Circles highlight some of the weak resonances observed. Fundamentals of Rotation–Vibration Spectra 165 local resonances that can be revealed by Loomis–Wood Albert, S., Albert, K., and Quack, M. (2004a) Rovibrational anal- + diagrams. They prove the existence of at least one other ysis of the ν4 and ν5 ν9 bands of CHCl2F. Journal of Molec- ular Structure, 695–696, 385–394. vibrational state interacting with the ν1 state, which must also be considered during the fitting procedure as discussed Albert, S., Albert, K.K., and Quack, M. (2011) High-resolution in Section 5.3. Fourier transform infrared spectroscopy, in Handbook of High- resolution Spectroscopy, Quack, M. and Merkt, F. (eds), John Wiley & Sons, Ltd., Chichester, UK. Albert, S., Albert, K.K., Winnewisser, M., and Winnewisser, B.P. 7 CONCLUDING REMARKS (1998) The rovibrational overtone spectrum of H13CNO up to 3600 cm−1: a network of resonance systems. Berichte At high resolution, molecular rotation–vibration spectra der Bunsengesellschaft f¨ur Physikalische Chemie, 102(10), contain an enormous amount of information on molecular 1428–1448. structures and vibrational and rotational quantum motions. Albert, S., Albert, K.K., Winnewisser, M., and Winnewisser, B.P. 13 15 This information is encoded in spectral line intensities (2001) The FT-IR spectrum of H C NO in the ranges 1800–3600 and 6300–7000 cm−1. Journal of Molecular Struc- and positions, which can frequently be measured with ture, 599(1–3), 347–369. stunning precision. In addition, the amount of information Albert, S., Hollenstein, H., Quack, M., and Willeke, M. (2004b) can be quantitatively enormous and involves the necessity Doppler-limited FTIR spectrum of the ν3(a )/ν8(a ) Cori- of analyzing thousands and tens of thousands of ro- olis resonance dyad of CHClF2: analysis and comparison vibrational transitions. We have discussed here some of with ab initio calculations. Molecular Physics, 102(14–15), the fundamental aspects, ranging from the basic concepts 1671–1686. for the relatively simple case of diatomic molecules to Albert, S., Hollenstein, H., Quack, M., and Willeke, M. (2006) the tools available for analyzing the great complexity Rovibrational analysis of the ν4,2ν6 Fermi resonance band 35 of the spectra of polyatomic molecules. Many of the of CH ClF2 by means of a polyad Hamiltonian involving the + other articles in this handbook provide further insight vibrational levels ν4,2ν6, ν6 ν9 and 2ν9, and comparison with ab initio calculations. Molecular Physics, 104(16–17), at the current level of research in this exciting area of 2719–2735. spectroscopy. Albert, S., Keppler Albert, K.K., Manca Tanner, C., Mariotti, F., and Quack, M. (2011) in preparation. Albert, S., Manca Tanner, C., and Quack, M. (2010) High res- ACKNOWLEDGMENTS olution spectrum and rovibrational analysis of the ν1CH- stretching fundamental in CHD2I. Molecular Physics, 108(18), Our work is supported by ETH Zurich¨ and the Schweiz- 2403–2426. erischer Nationalfonds. Albert, S. and Quack, M. (2007) High resolution rovibrational spectroscopy of pyrimidine. Analysis of the B1 modes ν10b and ν4 and the B2 mode ν6b. Journal of Molecular Spectroscopy, END NOTE 243(2), 280–291. Albert, S., Winnewisser, M., and Winnewisser, B.P. (1996) Net- a.http://developer.gnome.org/. works of anharmonic resonance systems in the rovibrational overtone spectra of the quasilinear molecule HCNO. 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