Appendix IV: Spectroscopy of polyatomic molecules. Determination of the. rotational constants.

1 Appendix IV: Spectroscopy of polyatomic molecules. Determination of the rotational constants. IV.1. Glossary of expressions and notations used in spectroscopy IV.2. Selection rules IV.2.1 Electric dipole transitions IV.2.2 Raman transitions IV.3. Centrifugal distortion correction and planarity relations IV.4. Coriolis contributions to α β k constants IV.5. Equilibrium structure of HCO + IV.6. Determination of the axial constant of a symmetric molecule IV.6.1. forbidden transitions (or perturbation allowed transitions) IV.6.2. loop method IV.7. Example of assignment of an isolated infrared band IV.8. Equilibrium rotational constants of sulfur dioxide IV.9. Equilibrium structure of HCP References IV.1. Glossary of expressions and notations used in spectroscopy Name Definition Notation asymmetry parameter κ = 2B A C κ A C band center Position of the fictitious ν i center of the band (J' = 0 J" = 0 transition). branch notation a the value ΔJ = J" J' = 2, 1, 0, 1,2 is indicated by the ΔK a ΔK c ΔJ letter O, P, Q, R, S, respectively. The changes in K a, K c is " " Ka K (J ") c indicated by adding letters as a left superscript according the same rule. for a symmetric or near-symmetric tops the usual notation is ΔK ΔJ K ( J ) bright band Very strong band centrifugal distortion constants in S reduction D J, D JK, D K, d 1, d 2 in A reduction Δ J, Δ JK, Δ K, δ J, δ K

2 combination band Rovibrational transitions between the ground state and a combination of excited state ν i +ν j notation for combination state For a state with v i =n i, v j =n j i n i j n j dark band dark state difference band Extremely weak band which only becomes observable in particular cases Only involved in dark bands Rovibrational transitions between ν j and ν i direction cosines The orientation of the molecular system relative to the laboratory system is described by three Euler angles, θ, φ and χ. The laboratory F = (X,Y,Z) and molecular ξ = (x,y,z) coordinates systems are related by the transformation F=Φ.ξ or F = ξ Φ Fξ ξ doubly degenerate state fundamental band fundamental state harmonic frequency hot band oblate top overtone band overtone band polyad prolate top rotational constants in a vibrational state v For linear molecules with more than two atoms, symmetric top and spherical top molecules, some modes are doubly degenerate Rovibrational transitions between the ground state and the first excited state First excited state of a vibrational state ν j ν i Φ F ξ d i = 2 ν v s =1 for a non degenerate state v t =1, =1 for a degenerate state. Rovibrational transitions ν i +ν j ν i between ν i +ν j and ν i κ > 0, see asymmetry parameter Rovibrational transitions 2ν, 3ν, between the ground state and the second or higher excited state Rovibrational transitions 2ν between the ground state and the second or higher excited state set of several interacting states κ < 0, see asymmetry parameter A v, B v, C v or B ξ v with ξ = a, b, c ω i

3 rotational quantum numbers total angular momentum component of the angula r momentum in the direction of the symmetry axis for a symmetric top pseudo-quantum numbers triply degenerate state K a is the quantum number of the prolate top limiting case and K c the quantum number of the oblate top limiting case For spherical molecules some modes are triply degenerate J J K +J K a, K c d i = 3 vibrational anharmonicity x rs, g tt' constant b vibrational quantum numbers with t only for degenerate v r, t states: t = v t, v t 2, v t 4,, v t vibrational states for linear The v 1, v 2 and v 3 are the triatomic molecules vibrational quanta for the ν 1 ( v 1,v 2 2,v 3 ) ν 2 and ν 3 modes, 2 v 2 is associated with the ν 2 doubled degenerated mode vibrational states for nonlinear triatomic molecules vibrational quanta for the ν 1 The v 1, v 2 and v 3 are the (v 1,v 2,v 3 ) ν 2 and ν 3 modes The upper and lower levels of a transition are indicated by aprime ' and double prime " respectively. The vibrational term formula is G(v) = ω r ( v r + d r 2) + x rs ( v r + d r 2) ( v s + d s 2) + g t t t t r r s The index r is usually assigned in descending frequency order, symmetry species by symmetry species. The index t is kept for degenerate modes. t t IV.2. Selection rules In a rotational spectrum the intensity of a transition is proportional to the square of the permanent electric dipole moment, whereas in an infrared spectrum the intensity of an absorption band is proportional to the square of the derivative of the dipole moment with respect to the normal coordinate q. The intensity of Raman bands depends on polarizability. As the selection rules of

4 these three types of transitions are different, they furnish complementary information. IV.2.1 Electric dipole transitions The following selection rules hold in the infrared or in the microwave regions: ΔJ = J ' J " = 0, ±1 (J = 0 J = 0, forbidden). (IV.1) For an electric dipole transition, the line intensity is proportional to t ϕ' µ Z ϕ" 2 t where µ Z is the Z- laboratory fixed component of the transition moment operator, and ϕ'> and ϕ"> the upper and lower state wavefunctions. A transition is observable provided that: t ϕ' µ Z ϕ" 0 (IV.2) t Z This implies that the symmetry type (Γ in short hand notation) of the product Γ( ϕ'>) Γ( µ ) Γ( ϕ">) of the upper and lower wavefunctions and of the transition moment operator must contain the completely symmetric representation of the molecular group. To go further in details on the selection rules, we will assume that that the nuclei are performing low amplitude motion and therefore that the harmonic approximation can be applied. Also at a first order the vibrational and rotational part of the wavefunctions ( vib> and R>) in short hand notation) can be separated. In these conditions, the upper and lower states wavefunctions are written as: ϕ'> vib'> R'>, ϕ"> vib"> R"> (IV.3) t Also µ Z is developed in tensorial products of vibrational and rotational operators: t 0 i i j µ Z = Φ Zξ µ ξ + qi µ ξ + qiq j µ ξ... ξ i i, j (IV.5) In this expression the µ 0 i ξ, µ ξ and i j ξ µ parameters are the ξ components of the permanent

5 dipole moment, of the first derivative relative the q i dimensionless coordinate and the second derivative relative to q i and q j, of the dipole moment, respectively. Pure rotational or infrared bands for which both vibrational and rotational selection rules are fulfilled are said as infrared active. Vibrational selection rules: The vibrational selection rules state that: vib'> = vib">, µ ξ 0 0, (IV.6) for a pure rotational transition, and i <vib' q i vib"> 0, µ ξ 0, (IV.7) <vib' q i q j vib"> 0, i j ξ µ 0, (IV.8) for the ν i fundamental band and its associated hot bands ν i + ν j ν j ; for the combination ν i + ν j or ν i ν j difference bands, respectively. Rotational selection rules The rotational selection rules state that: <R' Φ F ξ R"> 0 (IV.9) Table IV.1 gives a detailed description of the selection rules for linear and symmetric top molecules which may be classified, as parallel (//) or perpendicular ( ), according to the direction of the ξ axis which is either // or to the molecular axis of the molecule. Accordingly, the variation of, for linear molecules are Δ = 0 and Δ = ±1, (resp. of K for symmetric tops, ΔK = 0 or Δ K ±1), for // and transitions, respectively. Finally, Table IV.2 gives the rotational contribution, in term of ( <R' Φ F ξ R"> 2 ), to the intensity for symmetric top, linear molecules (for

6 linear molecules, use instead of K) or spherical top molecules. For asymmetric top molecules three types of transitions (A, B or C) are observable according to the variation of the K a and K c quantum numbers (see Table IV.3). The detailed line intensity calculations can be found in (Aliev and Watson 1985). Let us notice that hybrid bands, for example both A- and B- type bands for planar C s type molecules, are observable, depending of the symmetry of the molecules and of the relative values of the a and b components of the transitions moment parameters. IV.2.2 Raman transitions Raman transitions involve the polarizability tensor of the considered molecules. This means that the vibrational and rotational selection rules differ significantly from those observed for electric dipole transitions. For example, for a Raman spectrum, the rotational selection rules are ΔJ = J ' J " = 0, ±1, ±2 (IV.10) and therefore differ from those observed in infrared (ΔJ=2 transitions are forbidden in infrared). The reader is referred to Herzberg (1966), Wilson et al. (1980) Hollas (1998) for a more detailed description of Raman selection rules. Raman and infrared spectroscopy can give complementary information to the structure of a molecule. This was the case for deuterated germane, 70 GeD 4, (Pierre et al. 2002). For this spherical top molecule (T d symmetry), the four fundamental bands, were investigated by infrared and Raman spectroscopies leading to the determination of the equilibrium bond length r e = 1.5173(1) Å. Indeed, the ν 1 band (GeD symmetric stretch, a 1 symmetry) at 1510.5 cm -1 which is a dark band in infrared, was investigated by Raman spectroscopy. On the other hand, infrared spectra were used to study the bending dyad ν 2 /ν 4 (symmetric bend, a 1 symmetry, and

7 antisymmetric, t 2 symmetry, at 660.8 and 599.9 cm -1, respectively) and the ν 3 (antisymmetric bend, t 2 symmetry, at 1523.4 cm -1 ). In principle only the ν 3 and ν 4 bands are infrared active bands, but due to strong interactions between the upper vibrational energy levels involved in the ν 2 and ν 4 bands, forbidden lines appear in the bending dyad infrared spectrum, which allowed also to fit some of the ground state parameters. IV.3. Centrifugal distortion correction and planarity relations From the experimental quartic centrifugal distortion constants, Δ in the A-reduction or D in the S- reduction, the determinable constants T (or τ) are derived (Watson 1977) T xx = 1 4 τ xxxx = Δ J 2δ J = D J + 2d 1 + 2d 2 (IV.11a) T yy = 1 4 τ yyyy = Δ J + 2δ J = D J 2d 1 + 2d 2 (IV.11b) T zz = 1 4 τ zzzz = Δ J Δ JK Δ K = D J D JK D K (IV.11c) T 1 = T yz + T xz + T xy = 1 4 τ 1 = 3Δ J Δ JK = 3D J D JK 6d 2 (IV.11d) T 2 = B x T yz + B y T xz + B z T xy = 1 4 τ 2 = ( B x + B y + B z )Δ J 1 ( 2 B + B x y )Δ JK + ( B x B y )( δ J + δ K ) (IV.11e) = ( B x + B y + B z ) D J 1 ( 2 B + B x y ) D JK + ( B x B y )d 1 6B z d 2 with

8 T xy = 1 4 τ = 1 xxyy ( 4 τ + 2τ xxyy xyxy ) (IV.12) and where (x, y, z) = (b, c, a) or (a, b, c) in the I r or in the III r representations, respectively. The experimental information is obviously not sufficient to obtain all the τ αβγδ constants. However, in the particular case of a planar molecule, with a b being the molecular plane, Dowling (1961) has demonstrated that the following relations hold for the equilibrium constants τ bcbc = τ acac = 0 (IV.13a) τ cccc = C 4 A τ aaaa + 2 C 4 A 2 B τ + C 2 aabb B 4 τ bbbb (IV.13b) 2 τ bbcc = C B τ bbbb + C A 2 τ aabb (IV.13c) 2 τ aacc = C A τ aaaa + C B 2 τ aabb (IV.13d) Using these relations, there are only four independent τ αβγδ constants left, and the centrifugal distortion correction, Eq. (4.51) may be easily calculated. In the particular case of a bent XY 2 molecule, the following relation may also be used to calculate τ abab τ abab = 16 ABC ω 3 2 (IV.14) IV.4. Coriolis contributions to α k β constants The Coriolis operator is H 21 operator and therefore it can have the matrix elements with the total

9 Δv = 2, 2, or 0. For example in Eq. (67) of chapter 8 in Computational Molecular spectroscopy (Sarka and Demaison 2000) the first term gives the matrix elements with Δv = 2, 2 elements while the second term gives the Δv = 0 matrix elements. The Δv = 2, 2 elements do not cause resonance problems because they connect the states differing in the energy by ω k + ω l. The Δv = 0 matrix elements may cause problems because they connect the states differing in the energy by ω k ω l. We can separate the Coriolis contributions of both terms to α k β constants as follows: Δv= ±2 Δv=0 β β ( α k ) = α Cor ( k ) + α β Cor ( k ) Cor (IV.15) where Δv= ±2 β ( α k ) = 2 Bβ Cor Δv=0 β ( α k ) = 2 Bβ Cor m m ( ) 2 ( ω m + ω k ) β ( ζ mk ) 2 ω ω m k ω m ω k ( ) 2 ( ω m ω k ) β ( ζ mk ) 2 ω + ω m k ω m ω k (IV.16) (IV.17) The Eqs. (IV.15, IV.16, IV.17) can be easily derived e.g. from Eq. (101) in (Aliev and Watson 1985) and the following text, where Eq. (IV.17) follows from the last line in Eq. (101) and Eq. (IV.16) from the last but one line in Eq. (101). Eq. (IV.16) is identical with Eq. (17.8.5) in (Papousek and Aliev 1982), but there is a misprint in their equation and B should be squared. If we take into account explicitly the Coriolis resonance between the two states, say ω k,ω n, by introducing corresponding matrix element explicitly then the Coriolis contribution to the constant α β k will consist of the complete Eq. (IV.16) and the summation in Eq. (IV.17) is

10 restricted to all m n. The same applies also to α n β. If we sum the Coriolis contributions we find that β ( α k ) Cor k Δv= ±2 0 while Δv=0 β ( α k ) Cor k = 0 so that β ( α k ) = Cor k k Δv= ±2 β ( α k ) Cor (IV.18) Thus, while the Δv = 0 cancel out the Δv = ±2 contributions do not and should be taken into account. IV.5. Equilibrium structure of HCO + The experimental equilibrium structure of the formyl ion, HCO +, was first obtained by Woods (1988) using the equilibrium rotational constants of HCO +, H 13 CO +, and HC 18 O +. DCO + was not used because a Fermi interaction was suspected between the states v 1 = 1 and v 2 = 4 0 (Kawaguchi et al. 1986; Hirota and Endo 1988). Furthermore, to obtain a good fit, it was necessary to include a set of γ-constants, some of them rather large. However, several high-level ab initio studies (Sebald 1990, Martin et al. 1993) found a much shorter CH bond length, see Table IV.4. An ab initio calculation of the anharmonic force field (Puzzarini et al. 1996a) invalidates the Fermi resonance hypothesis and the small value of α 1 (DCO + ) is attributed to a Coriolis intraction between 10 0 0 and 01 1 1 states. This Coriolis interaction also affects HCO +, albeit to a lesser extent. This was confirmed by a recent experimental study of the rotational spectra in the vibrational states 020, 030, 040, 01 1 0; and 100 (Dore et al. 2003). This example shows how difficult it is to obtain a reliable structure using only experimental information. In this particular case, ab initio calculations first indicated the existence of a problem and were then able to propose an explanation which was finally substantiated experimentally.

11 IV.6. Determination of the axial constant of a symmetric molecule There is an advantage of infrared over microwave in the particular case of symmetric top molecules. Due to the selection rule ΔK = 0, it is not possible to determine the axial rotational constant (A for a prolate top, C for an oblate top). In principle, the simplest solution is to use Raman spectroscopy since the selection rules are different, e.g. ΔK = ±2 transitions are possible but, in practice, its lower resolving power limits the accuracy. However, there are two powerful methods which allow us to obtain this constant with precision IV.6.1. forbidden transitions (or perturbation allowed transitions) (Papousek 1989, Nakagawa 1984) Suppose that two nearby rovibrational levels (k = +1) and (k = -2) with different values of k are mixed through a resonance ( is the angular momentum quantum number which occurs in degenerate vibrations: = vt, vt - 2, -vt). The corresponding wavefunctions are also mixed and the normally forbidden transitions (broken arrows) become allowed. The frequency difference between allowed transitions and forbidden ones permits to obtain the axial rotational constant, see figure IV.1. This method was first applied to CH3I by Maki and Hexter (1970) and later to many light molecules where perturbation effects are rather strong. It is even not necessary to observe forbidden transitions because the resonance may sufficiently affect the spectrum so that the axial constant may be determined from a fit of the allowed transitions. For instance, in the analysis of the ν 5 band of FClO 3, systematic displacements of observed P P6 and P P9 lines from their calculated values were noticed. These

12 displacements could be explained by a Δk = ±3 resonance and the introduction of the corresponding off-diagonal element enabled the determination of A0 with accuracy (Burczyk et al. 1991). Other more approximate methods are described by Nemes (1984). IV.6.2. loop method It is possible to calculate energy differences in the ground state with two different K values through the analysis of a fundamental band νt', a hot band (νt + νt') ±2 - νt', and the corresponding combination band (νt + νt') ±2, see figure IV.2. This method was first proposed by Nakagawa (1984) and was recently reviewed by Graner and Bürger (1997) who applied it to many molecules for which hot bands are easy to observe (i.e. rather heavy molecules). IV.7. Example of assignment of an isolated infrared band The assignment of an infrared spectrum usually is done iteratively. The first assignments are performed by searching for regularities in the spectrum. For example, Fig. IV.3, (Brizzi et al. 2005) shows an example of such regularities observed in the Q-branch during the analysis of the ν 1 band of the 79 Br 14 N 18 O isotopic species of nitrosyl bromide near 1751.3 cm -1. For this planar C s molecule the ν 1 band (N-O stretch) which should be hybrid (i.e. with both A-type and B-type transitions) is, in fact, an almost a pure A- type band. In addition, for BrNO, the upper (v 1 = 1, in short hand 1 1 ) and lower (v = 0) state rotation constants (A 1 = 81794.9, B 1 = 3530.7 and C 1 = 3381.2 MHz and A 0 = 82500.3,

13 B 0 = 3516.42 and C 0 = 3369.3 MHz) are close to the prolate symmetric top limit. In these conditions, the position of the upper and lower state energy levels can be estimated Q (approximately) using Eq. (4.41). The K ( J ) subbands correspond to the J K a, K c J K a, Q a clustered transitions. For these lines, an approximate position is given by the following expression: K c Q Q K a (J) E 1 + ( B 1 + C 1 ) 2 ( ) 2 2 ( ) J(J +1) + ( A 1 ( B 1 + C 1 ) 2) K a 2 ( ) J(J +1) + ( A 0 ( B 0 + C 0 ) 2) K a B 0 + C 0 (IV.19) In Eq. ( 4.19), the subscripts 1 and 0 stands for the 1 1 and ground state rotational constants, respectively, and E 1 is the vibrational energy for state 1 1. From Fig. IV.3 a first guess on some of the α 1 ξ can be obtained examining the Q branch. Indeed Q from the J=K a = 7 and J=K a = 6 ( J ) branch structures (at 1750.152 and 1750.454 cm -1, QK a respectively), and the distance between the first J lines within these of the two structures ( Q Q 6 (7) Q Q 6 (6) ~ 0.00058 cm -1, and Q Q 7 (8) Q Q 7 (7) ~ 0.0067 cm -1 ), one get (A 0 (B 0 +C 0 )/2) (A 1 (B 1 +C 1 )/2)) ~ 0.0024 cm -1, ((B 0 +C 0 )/2 - (B 1 +C 1 )/2) ~ 0.00044 cm -1, and therefore α 1 A ~ 705 MHz. In this way it is possible to have reasonable starting values for the 1 1 upper state rotational constants, and to perform the analysis, once the constants of the lower state are known. To continue the assignment, it is necessary to use calculated predictions both for line positions and relative line intensities. For this calculation, it is necessary to use a theoretical model which accounts for the symmetry properties of the molecule, and, if necessary, for the resonances. The upper state levels are inserted in a least squares fit to get a new set of upper states parameters allowing better predictions and hence more assignments to be made. This process is repeated

14 iteratively until the complete assignment of the spectrum. When the analysis is progressing, it may be necessary to use a more sophisticated theoretical model in order to account for a resonance which has been detected during the assignment procedure. The assignment of an isolated band is usually straightforward. Indeed, the line predictions are reliable since, in the absence of perturbations, the upper state energy levels are modeled accurately using a standard rotational Hamiltonian. However, assignments may be difficult, for example, in congested Q branches. One should be cautious to avoid incorrect assignments since the energy levels calculation might appear as satisfactory when considering the residuals of the least squares fit, although the rotational parameters are incorrect. To ensure the correctness of the analysis, it is necessary to compare in detail the observed and calculated spectra (line positions and intensities). This will be demonstrated using the example of the assignment of the ν 6 band of nitric acid, HNO 3. This band corresponds to the O-NO 2 stretching mode at 646.964 cm -1 for H 15 NO 3 and it is, in principle a hybrid band but, as for BrNO, it behaves as a pure A-type band. To perform its assignment it is necessary to consider the structure and the relative intensities of the P, Q and R branches which are characteristic of a planar oblate molecule, (A ~ 0.43, B ~ 0.40 and C ~ 0.21 cm -1 ). Indeed for A-type bands of nitric acid, the strongest lines in the P and R branches involve J Ka K c rotational levels with low K a values (K a << K c ~ J). These P and R lines are grouped in stacks involving the same value of (2J K c ), see Fig. IV.4. Each P and R cluster is separated from the next one by about (B + C)/2 ~ 0.43 cm -1, and the assignment performed by searching for these regularities can easily be confirmed by ground state combination differences calculated using accurate ground state rotational constants. Using preliminary parameters

15 deduced from these P and R branches attributions, tentative assignments can be performed in the very congested Q branch, see Fig. IV.5. Unfortunately Q branches for A-type bands of nitric acid are more difficult to analyze than for a prolate molecule like BrNO. Indeed in the A-type Q branches the strongest lines of nitric acid only involve very high K a values (K c << K a ~ J). This means that the rotational energy levels involved in the P and R transitions are different from those involved in the Q branch, and that extrapolations are uneasy. The first assignments in the Q branch should concern the Q Q (J) series (corresponding to K K a =J c = 0 or 1 clustered transitions), which are the strongest and most regular ones. However each J-term of this Q series is separated from the following one by only ~0.01 cm -1. It is clear that erroneous attributions can easily be done. To test the effects of incorrect assignments in the Q branch, we have shifted the J value by one Q (J (J 1)) for all the Q ( J ) lines assigned by Keller (Keller et al.1998), see the bottom K a = J trace of Fig. IV.5. Surprisingly, with this incorrect Q branch assignment, the results of the energy level calculation for the 6 1 state appear as satisfactory as when the assignment is correct. Indeed as in the original study, 95% of the 6 1 energy levels are reproduced within 0.001 cm -1. In order to demonstrate the bad quality of the parameters resulting from this incorrect fit, it is necessary to compare carefully the observed and calculated spectra in each detailed spectral region of the ν 6 band, and for these comparisons, two lists of ν 6 line positions and intensities were generated using, for the first one, the correct parameters and for the second one, those obtained by shifting all Q-branch assignments. As shown in Fig. IV.4, for both calculations the comparison with the observed spectrum is excellent in the whole range of the P and R branches. However, Fig. IV.5, which shows the similar modeling in the central part of the Q branch, indicates that the calculation performed

16 when shifting the Q assignment leads to strong disagreements between the observed and calculated spectra. As a conclusion it is clear that one should not be confident on the quality of a line positions calculation (only) which may appear as satisfactory even if several assignments are incorrect. A more realistic test is performed by a careful comparison of the observed and calculated spectra in term of positions and intensities, in each spectral range of the investigated infrared band. Table IV.5 lists the rotational constants and the quartic centrifugal distortion constants for H 15 ξ NO 3 in the ground vibrational state v = 0, together with the variation of these parameters α 6 in the 6 1 excited state. For the A, B and C rotational constants, both the "correct" and "incorrect" α 6 ξ values are in reasonable agreement. It is clear that performing incorrect assignments in the Q branch of the ν 6 band has a visible error impact for the centrifugal distortion constants mainly for the Δ K and Δ JK centrifugal distortion constants. IV.8. Equilibrium rotational constants of sulfur dioxide Sulfur dioxide, SO 2, has three vibrational modes: the SO symmetric stretch (ν 1 ), the OSO bending (ν 2 ), and the SO antisymmetric stretch (ν 3 ), located at 1151.712, 517.872, and 1362.060 cm -1, respectively for 32 S 16 O 2 (Lafferty et al. 1993). The resonances are weak in general, but not completely negligible if high accuracy is requested for the rotational constants. Table IV.6 details the form of the block diagonalized Hamiltonian matrix which was used for the detailed investigation of the ν 1, ν 3, ν 1 +ν 3 (2499.871 cm -1 ), 2ν 3 (2713.383 cm -1 ), and 3ν 3 bands (2713.383

17 cm -1 ). Indeed, it was necessary to account for the 1 1 2 3 3 2 and 1 1 2 3 3 1 3 3 anharmonic resonances, for the 2 1 1 1 and 2 1 3 1 1 1 3 1 Fermi resonances, and for the 1 1 3 1 C-type Coriolis (Flaud et al. 1993a, Lafferty et al. 1993, Flaud and Lafferty 1993b). It is important to underline that data coming from seven infrared or microwave experimental studies performed from 1969 to 1993 were collected (Flaud and Lafferty 1993b) to obtain the equilibrium structure. Heterodyne frequency measurements performed in the ν 1 and ν 3 bands of 32 SO 2 (Vanek et al. 1990) are satisfactorily reproduced when accounting for the resonances coupling the 1 1, 2 1 and 3 1 interacting energy levels (Flaud et al. 1993a). Such an agreement is impossible for the 1 1 and 3 1 rotational parameters obtained by neglecting these resonances (Guelachvili et al. 1984). This is shown in Table 4.7 which gives a short sample of the comparisons of these two calculations. From the set of A v, B v and C v rotational constants obtained for several excited states of 32 SO 2 it was possible to derive through a least squares fit calculation the A e, B e and C e equilibrium rotational constants together with theα i ξ and γ ij ξ vibrational-rotational interaction constants (ξ = A, B and C) involved in Eq. (4.43): these parameters are collected in Table IV.8. For the v = 1 1, 2 2 and 3 1 interacting vibrational states, Table IV.9 compares the excited state rotational constants calculated from Eq. (4.43), with the empirical values when accounting or neglecting for the 1 1 3 1 and 1 1 2 2 resonances. It is clear that the quality of the agreement between the determined A v B v and C v and the calculated ones is much better when the resonances are accounted for. As a matter of conclusion, accounting for the resonances improves the accuracy achieved for the equilibrium rotational parameters, and therefore the quality of the structural parameters obtained experimentally for the analyzed molecular species.

18 IV.9. Equilibrium structure of HCP The equilibrium structure of the linear molecule methylidynephosphine, HCP, was first determined in 1973 (Strey and Mills 1973) by correcting the ground state rotational constants of HCP and DCP using an experimental anharmonic force field, see Table IV.10. The experimental equilibrium structure was obtained later (Lavigne et al. 1984) using the equilibrium rotational constants of HCP and DCP and taking into account the Fermi resonance between the v 3 = 1 and v 2 = 2 0 states. The new r e (CH) bond length was found smaller by 0.0032 Å than the previous value. The isolated CH bond stretching frequency (Dréan et al. 1996) as well as ab initio calculations (Botschwina and Sebald 1983) indicate that the value of Lavigne et al. (1984) is too small. This was attributed to a Fermi resonance between the levels v 1 = 1 and v 3 = 2 (Pépin and Cabana 1986). Later, Puzzarini et al. (1996b) calculated an ab initio potential energy surface and were able to point out the importance of this second Fermi resonance in DCP affecting the states v 1 = 1 and v 3 = 2. This resonance lowers B 1 by as much as ten MHz. A new equilibrium structure was also estimated. The accuracy of this structure was finally confirmed by ab initio calculations at the CCSD(T)/V5Z level of theory (Koput 1996). The example shows again that it is not easy to determine a reliable experimental equilibrium structure.

19 References Botschwina, P. and Sebald, P. 1983. Vibrational frequencies from anharmonic ab initio/empirical potential energy functions: Stretching vibrations of hydroisocyanic acid, phosphaethyne, isocyanoacetylene, and phosphabutadiyne. J. Mol. Spectrosc. 100: 1-23. Brizzi, G., Puzzarini, C., Perrin, A., Orphal, J., Willner, H., and Garcia, P. 2005. High resolution Fourier transform infrared spectrum of 79 Br 14 N 18 O: analysis of the ν 1 band. J. Mol. Struct. 742:37 41. Burczyk, K., Bürger, H., Le Guennec, M., Wlodarczak, G., and Demaison, J. 1991. High- Resolution FTIR and Millimeter-Wave Study of FClO 3 : Ground State Rotational Constants Including A 0, Structure, and the ν 2, ν 3, ν 5 and ν 6 Excited States J. Mol. Spectrosc. 148: 65-79. Coudert, L.H., Maki, A.G., and Olson, W.B. 1987. High-resolution measurements of the ν 2 and 2ν 2 -ν 2 bands of SO 2. J. Mol. Spectrosc. 124:437-442. Dore, L., Beninati, S., Puzzarini, C., and Cazzoli, G. 2003. Study of the vibrational interactions in DCO + by millimeter-wave spectroscopy and determination of the equilibrium structure of formyl ion. J. Chem. Phys. 118: 7857-7862. Dowling J.M. 1961. Centrifugal distortion in planar molecules. J. Mol. Spectrosc. 6: 550-553. Dréan, P., Demaison, J., Poteau, L., and Denis, J.-M. 1996. Rotational Spectrum and Structure of HCP. J. Mol. Spectrosc. 176: 139-145. Drouin B.J., Miller C.E., Fry J.L., Petkie D.T., Helminger P., Medvedev Y.R. 2006. Submillimeter measurements of isotopes of nitric acid. J. Mol. Spectrosc. 236:29 34. Flaud, J.-M., and Lafferty, W.J. 1993a. 32 S 16 O 2 : a refined analysis of the 3ν 3 band and

20 determination of equilibrium rotational constants, J. Mol. Spectrosc. 161:396-402. Flaud, J.-M., Perrin, A., Salah, L.M., Lafferty, W.J, and Guelachvili, G., 1993b. A reanalysis of the (010) (020) (100) and (001) rotational levels of 32 S 16 O 2. J. Mol. Spectrosc. 160:272-278. Graner, G., Demaison, J., Wlodarczak, G., Anttila, R., Hillman, J. J., and Jennings, D. E. 1988. A preliminary determination of the A 0 rotational constant of propyne. Mol. Phys. 64: 921-932. Graner, G. and Bürger, H.1997. Hot Bands in Infrared Spectra of Symmetric Top and Some Other Molecules. A useful Tool to Reach Hidden Information. in Vibration-Rotational Spectroscopy and Molecular Dynamics, ed. P. Papousek, 239-297. Singapore: World Scientific. Guelachvili, G., Ulenikov, O.N., and Ushakova, G.A. 1984. Analysis of the ν 1 and ν 3 Absorption Bands of 32 S 16 0 2. J. Mol. Spectrosc. 108: 1-5. Hirota, E. and Endo, Y. 1988. Microwave Spectroscopy of HCO + and DCO + in Excited Vibrational States. J. Mol. Spectrosc. 127: 527-534. Kawaguchi, K., McKellar, A.R.W., and Hirota, E. 1986. Magnetic field modulated infrared laser spectroscopy of molecular ions: The ν 1 band of DCO +. J. Chem. Phys. 84: 1146-1148. Keller, F., Perrin, A. Flaud, J.-M., Johns, J.W.C., Lu, Z., and Looi, E.C. 1998. High resolution analysis of the ν 6, ν 7, ν 8 and ν 9 bands of H 15 NO 3 measured by Fourier transform Spectroscopy. J. Mol. Spectrosc. 191;306-310. Koput, J. 1996. The equilibrium structure and spectroscopic constants of HCP an ab initio study. Chem. Phys. Lett. 263: 401-406. Lafferty, W.J, Pine A.S., Flaud, J.-M. and Camy-Peyret C. 1993. The 2ν 3 band of 32 S 16 O 2 : line

21 positions and intensities. J. Mol. Spectrosc. 157: 499-511. Lavigne, J., Pépin, C., and Cabana, A. 1984. The vibration-rotation spectrum of D 12 CP in the region of the ν 2 band: The spectroscopic constants for the states 00 0 O, 02 0 0, and 02 2 O and the bond lengths of the molecule. J. Mol. Spectrosc. 104: 49-58. Maki, A. G. and Hexter, R. M. 1970. Resonance Interactions with ν 5 in CH 3 I; A Method of Determining A 0. J. Chem. Phys. 53: 453-454. Martin, J.M.L., Taylor, P.R., and Lee, T.J. 1993. Accurate ab initio quartic force fields for the ions HCO + and HOC +. J. Chem. Phys. 99: 286-292. Nakagawa, T. 1984. Experimental Methods for Determining the Rotational Constant A 0 of a Symmetric Top Molecule. J. Mol. Spectrosc. 104: 402-404. Nemes, L. 1984. Vibrational effects in spectroscopic geometries. In Vibrational Spectra and Structure; ed. J.R. Durig, Vol. 13, 161-221. Amsterdam: Elsevier. Papousek, D. 1989. "Forbidden" transitions in molecular vibrational-rotational spectroscopy. Coll. Czech. Chem. Commun. 54: 2555-2630. Papousek, D, Papouskova, Z., Hsu, Y.-C., Chen, P., Pracna, P., Klee, S., Winnewisser, M., and Demaison, J. 1993. Determination of A 0 and D K of 13 CH 3 F from the k = ±2 Forbidden Transitions to the v 5 = 1 Vibrational Level. J. Mol. Spectrosc. 159: 62-68. Pépin, C. and Cabana, A. 1986. The ν 1 + ν 2 vibration-rotation band of D 12 CP and evidence for Fermi resonance in the [100, 002] diad. J. Mol. Spectrosc. 119: 101-106. Pierre, G., Boudon, V., MKadmi, E.B., Bürger, H., Bermejo, D., and R. Martınez, R. 2002. Study of the Fundamental Bands of 70 GeD 4 by High-Resolution Raman and Infrared Spectroscopy: First Experimental Determination of the Equilibrium Bond Length of Germane, J. Mol. Spectrosc. 216: 408 418.

22 Puzzarini, C., Tarroni, R., Palmieri, P., Carter, S., and Dore, L. 1996a. Accurate ab initio prediction of the equilibrium geometry of HCO + and of rovibrational energy levels of DCO +. Mol. Phys. 87: 879-898. Puzzarini, C., Tarroni, R., Palmieri, P., Demaison, J., and Senent, M. L. 1996b. Ro-vibrational energy levels and equilibrium geometry of HCP. J. Chem. Phys. 105: 3132-3139. Sebald, P. 1990. PhD Thesis, Kaiserslautern. Strey, G. and Mills, I.M. 1973. The anharmonic force field and equilibrium structure of HCN and HCP. Mol. Phys. 26: 129-138. Vanek, M.D., Wells, J.S., and Maki, A.G. 1990. Heterodyne frequency measurements on SO 2 near 41 THz (1370 cm -1 ). J. Mol. Spectrosc. 141: 346-347. Woods, R.C. 1988. Microwave spectroscopy of molecular ions in the laboratory and in space. Phil. Trans. R. Soc. Lond. A 324: 141-146.

23 Figure IV.1. Scheme of the energy level crossings in the states k = +1 and k = -2 of the level v5 = 1 of CH3F (J" = 13) (Papousek et al. 1993)

24 Figure IV.2. Scheme of levels in propyne, CH3C CH used to obtain Ground State Combination Difference between K and K + 3 (K = k ) (Graner et al. 1988).

25 Fig IV.3. Portion of spectrum of the ν 1 band of 79 Br 14 N 18 O for the Q-branch head. The observed and the synthetic spectra for the Q Q 6 and Q Q 7 branches are presented together with assignments of J" values. The experimental (a) and calculated spectra (b) are displaced for visual clarity. Reprinted from (Brizzi, 2005),copyright (2005) with permission from Elsevier.

Fig IV.4. Portion of the R branch of the ν 6 band of H 15 NO 3. 26

Fig IV.5. Portion of the Q branch of H 15 NO 3. 27

28 Table IV.1. Selection rules for infrared transitions in linear and symmetric (with a 3-fold symmetry axis molecules (in all cases ΔJ = 0, ±1) Δ = 0 Linear molecules Parallel transition (//) Molecules with a 3-fold symmetry axis Δ = 0, ±3, ±6, ΔJ = 0, ±1 (for 0) ΔJ = ±1 (for = 0) ΔΚ = 0 ΔJ = 0, ±1 (for K 0) Perpendicular transition ( ) ΔJ = ± 1 (for K = 0) Δl = ±1 ΔJ = 0, ±1 Δl = ±1, ±2, ±4, ΔΚ = ±1 ΔJ = 0, ±1 Δ ΔΚ = 0, ±3, ±6,

29 Table IV.2. Rotational contribution ( <R' Φ F ξ R"> 2 ) to the intensity for symmetric top, linear molecules (for linear molecules, use instead of K) or spherical top molecules. Symmetric top or linear molecules Spherical top // (Parallel transitions) (Perpendicular transitions) K = K = 0 K = K 0 K = K±1 2 2 J = J 1 J J K J 1 2 2 J = J 0 2K( 2 1 ) J = J+1 (J+1) + ( 2J + 1)( J K)( J + 1± K) J J ( J + 1 ) 1 2 ( J K ) ( J 1 K ( ) J ) (4J 2 1) ( ) J ( J + 1) ( ) ( ) 2 J + 1 2 K 2 1 ( J + 1± K)( J + 2± K) J + 1 2 J + 1 (2J+1) 2 (2J+1)(2J+3)

30 Table IV.3. Selection rules for vibration-rotation transitions for asymmetric top molecules Selections rules Approximate selection rules ΔJ = 0, ±1 A-type µ a 0 ΔK a = even, ΔK c = odd ΔK a = 0, ΔK c = ±1 B-type µ b 0 ΔK a = odd, ΔK c = odd ΔK a = ±1, ΔK c = ±1 C-type µ c 0 ΔK a = odd, ΔK c = even ΔK a = ±1, ΔK c = 0 hybrid µ ξ &µ ξ' 0 ξ- & ξ'-types with ξ = A, B or,c Forbidden ΔK a = even, ΔK c = even

31 Table IV.4. Equilibrium structures of HCO + (in Å). r(hc) r(co) method reference 1.097247(38) 1.104738(23) experimental Woods 1988 1.0919(5) 1.1058(2) ab initio Sebald 1990 1.09215 1.10545 ab initio Martin et al. 1993 1.0919(9) 1.1055(3) semi-experimental Puzzarini et al. 1996a 1.09204 1.10558 experimental Dore et al. 2003

32 Table IV.5. Rotational constants for the ground state of H 15 NO 3 and vibrational dependence in the 6 1 vibrational state. H 15 NO 3 v = 0 6 1 6 1 a Correct Incorrect assignment b assignment A (MHz) 13012.263 13007.422(6) 13006.652(20) B (MHz) 12096.924 12054.180(6) 12054.762(18) C (MHz) 6260.137 6279.824(1) 6279.843(3) Δ K (khz) 7.33846 9.492(17) 12.63(14) Δ JK (khz) -4.49804-7.630(19) -11.62(17) Δ J (khz) 8.901231 9.6891(38) 10.32(27) δ K (khz) 7.49446 7.7449(50) 6.805(43) δ J (khz) 3.782612 3.8119(19) 4.124(14) a Ground state parameters from (Drouin et al. 2006). b Data from (Keller et al. 1998).

33 Table IV.6. Form of the block diagonalized Hamiltonian matrix used for the detailed investigation of the ν 1 (1151.712 cm -1 ), ν 3 (1362.061 cm -1 ), ν 1 +ν 3 (2499.871 cm -1 ), 2ν 3 (2713.383 cm -1 ), and 3ν 3 bands (2713.383 cm -1 ) of 32 SO 2 (Flaud et al. 1993a, 1993b). The letters W, C, F and Anh stand for Watson s type rotational operators, Eqs. (4.11) and (4.15), C-type Coriolis, Eq. (4.30), Fermi and anharmonic resonances, Eq. (4.35), respectively. v 0 2 1 2 2 1 1 3 1 2 3 1 1 2 1 2 2 3 1 1 1 3 1 3 2 1 1 2 3 3 3 1 1 2 3 3 1 0 W 2 1 W 2 2 W F C 1 1 F W 3 1 C W 2 3 W 1 1 2 1 W 2 2 3 1 W F 1 1 3 1 F W 3 2 W Anh 1 1 2 3 Anh W 3 3 W Anh 1 1 2 3 3 1 Anh W

34 Table IV.7. Heterodyne frequency measurements (in MHz) for the ν 3 band of 32 SO 2. Obs Frequency a J Ka, Kc J Ka, Kc o c b o c c 40458986.4(50) 19 0, 19 20 0, 20 9.6 0.0 40740829.0(50) 16 12, 5 16 12, 4 7.9 0.9 40741496.8(50) 25 11, 14 25 11, 15 10.7 0.7 40741887.7(40) 15 12, 3 15 12, 4 11.1 0.9 40743809.8(60) 13 12, 1 13 12, 2 13.8 0.8 41792579.5(40) 57 8, 49 56 8, 48 13.5 0.1 41793377.8(50) 59 10, 49 58 10, 48 31.3 5.8 41794759.0(40) 60 11, 50 59 11, 49 13.5 0.1 a (Vanek et al. 1990). b (Guelachvili et al. 1984). c (Flaud et al. 1993b).

35 Table IV.8. 32 SO 2 Equilibrium rotational constants and rotation-vibration interacting parameters (Flaud, 1993a). ξ A B C B ξ e 60502.212(260) 10359.2338(380) 8845.1048(400) α 1 ξ 33.330(450) 50.0229(900) 42.3281(910) α 2 ξ 1130.360(550) 2.6235(330) 15.6847(470) α 3 ξ 616.557(230) 34.9537(390) 32.4054 ξ γ 0.599(220) 0.0486(430) 0.1037(440) 11 ξ γ 6.862(280) 0.0873(550) 0.4035(560) 12 ξ γ 6.252(400) 0.1365(730) 0.8505(800) 13 ξ γ 24.766(380) 0.17280(590) 0.07912(590) 22 ξ γ 17.845(400) 23 ξ γ 3.8780(160) 0.03510(310) 0.07671(310) 33 α i / 2 273.567 41.177 45.209 i α i B e ξ (%) 0.45 0.40 0.51 γ ij / 4 2.70 0.05 0.09 i, j γ ij 2 α i (%) 0.99 0.13 0.19

36 Table IV.9. Experimentally determined (Obs) and calculated (Calc) rotational constants for the 1 1, 2 1, and 3 1 interacting states of 32 SO 2 (all values in khz). State Calc a Obs Calc b resonances taken into account Obs Calc resonances neglected 1 1 A 60 810 989 0.009 385 c B 10 268 260 0.12 40.8 c C 8 757 360 0.18 2.88 c 2 2 A 63 185 731 0.84 26.7 d B 10 322 371 0.32 7.58 d C 8 767 842 3.71 67.2 d 3 1 A 60 157 703 10.0 677 c B 10 283 118 1.56 62.2 c C 8 766 816 0.45 22.8 c a Calculated with Eq. (4.20). b (Flaud et al. 1993a, 1993b). c (Guelachvili 1984). d (Coudert 1987).

37 Table IV.10. Equilibrium structures of HCP (in Å). r(hc) r(cp) method reference 1.0692(7) 1.5398(2) experimental a Strey and Mills 1973 1.06596(11) 1.540452(18) experimental Lavigne et al. 1984 1.070(1) 1.540(1) mixed method b Dréan et al. 1996 1.0702(10) 1.5399(2) semi-experimental Puzzarini et al. 1996b 1.0706 1.5399 ab initio Koput 1996 a Using an experimental anharmonci force field. b Combination of isolated stretching frequency and scaled ab initio calculations.