Ionization from a double bond: Rovibronic

JOURNAL OF CHEMICAL PHYSICS

VOLUME 120, NUMBER 4

22 JANUARY 2004

Ionization from a double bond: Rovibronic photoionization dynamics of ethylene, large amplitude torsional motion and vibronic coupling in the ground state of C2 H4¿ S. Willitsch, U. Hollenstein, and F. Merkt Physical Chemistry, ETH Zurich, CH-8093 Zurich, Switzerland

共Received 3 September 2003; accepted 30 October 2003兲 Rotationally resolved pulsed-field-ionization zero-kinetic-energy photoelectron spectra of the ˜X ˜ ⫹ transition in ethylene and ethylene-d4 have been recorded at a resolution of 0.09 cm⫺1 . The →X spectra provide new information on the large amplitude torsional motion in the cationic ground state. An effective one-dimensional torsional potential was determined from the experimental data. ⫹ Both C2 H⫹ 4 and C2 D4 exhibit a twisted geometry, and the lowest two levels of the torsional potential form a tunneling pair with a tunneling splitting of 83.7(5) cm⫺1 in C2 H⫹ 4 and of . A model was developed to quantitatively analyze the rotational structure 37.1(5) cm⫺1 in C2 D⫹ 4 of the photoelectron spectra by generalizing the model of Buckingham, Orr, and Sichel 关Philos. Trans. R. Soc. London, Ser. A 268, 147 共1970兲兴 to treat asymmetric top molecules. The quantitative analysis of the rotational intensity distributions of allowed as well as forbidden vibrational bands enabled the identification of strong vibronic mixing between the ˜X ⫹ and Ã ⫹ states mediated by the torsional mode ␯ 4 and a weaker mixing between the ˜X ⫹ and ˜B ⫹ states mediated by the symmetric CH2 out-of-plane bending mode ␯ 7 . The vibrational intensities could be accounted for quantitatively using a Herzberg–Teller-type model for vibronic intensity borrowing. The adiabatic ionization energies of C2 H4 and C2 D4 were determined to be 84 790.42(23) cm⫺1 and 84 913.3(14) cm⫺1 , respectively. © 2004 American Institute of Physics. 关DOI: 10.1063/1.1635815兴

I. INTRODUCTION

finding that was rationalized by invoking the effects of hyperconjugation.5,6 This prediction was corroborated by the recording of the spectrum of the first Rydberg state of ethylene 共the R state兲 in the vacuum ultraviolet by Merer and Schoonveld.7 The spectrum was interpreted in terms of an ionic core undergoing a large amplitude motion between twisted structures with torsional angles of ⫾25° through a potential barrier of 290 cm⫺1 at the planar geometry. The advent of photoelectron spectroscopy in the 1960s enabled a direct investigation of the cation, and since the early work of Turner and co-workers8 numerous classical photoelectron spectroscopic studies of C2 H4 and its fully deuterated isotopomer C2 D4 have been reported.9–17 Striking aspects of the high-resolution photoelectron spectra are first, the observation of an extended progression in the torsional mode of C2 H⫹ 4 which provides evidence for the twisted geometry of the ethylene cation and second, the observation of ⫹ a forbidden transition to cationic states of C2 H⫹ 4 and C2 D4 with three quanta ( v 4 ⫽3) in this mode. Although this weak 共and forbidden兲 band was unambiguously observed in three studies,15–17 the reasons for its observation have not been discussed. The torsional motion in the electronic ground state was modelled by Pollard et al.15 in terms of a double-minimum potential with a twisting angle of 27⫾2° and a torsional barrier at the planar geometry of 270⫾150 cm⫺1 . A subsequent re-evaluation of these results by Chau18 yielded a twisting angle of 28⫾1° and a barrier of 280⫾60 cm⫺1 .

CvC double bonds represent key elements that define structural and dynamical properties of larger polyatomic molecules. The role of the electron density on their function as structural elements is of fundamental interest in chemistry. Photoelectron spectroscopy offers the means to selectively modify the electron density in a CvC double bond and observe the structural and dynamical consequences, as is illustrated here in a benchmark study of the photoionization dynamics of ethylene. Ethylene C2 H4 is an important molecule in the theory of chemical bonding and in chemistry in general because it is the simplest molecular system that exhibits a double bond. Within the framework of molecular orbital theory, the double bond character of the CvC bond of ethylene is explained by its highest occupied molecular orbital, the well-known ␲-type molecular orbital, whose rigidity with respect to twisting about the C–C axis is also responsible for the planar geometry of ethylene in its electronic ground state. Ethylene thus represents a prototype system to study the electronic structure of molecules with double bonds 共see, e.g., Refs. 1–3, and references cited therein兲 and to study by photoelectron spectroscopy the effects of the removal of an electron out of the bonding ␲-type orbital on the molecular geometry, the dynamics and the energy level structure. Already very early semiempirical electronic structure calculations predicted a twisted equilibrium geometry for the ethylene radical cation in its electronic ground state,4,5 a 0021-9606/2004/120(4)/1761/14/$22.00

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© 2004 American Institute of Physics

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J. Chem. Phys., Vol. 120, No. 4, 22 January 2004

However, these two studies differ as to whether the lowest vibrational state is located below or above the barrier 共compare Fig. 4 of Ref. 15 with Table 2 of Ref. 18兲. This issue could have been resolved from an observation of the first excited torsional level ( v 4 ⫽1). Unfortunately, this level has not been observed experimentally so far either because of insufficient resolution or sensitivity. An alternative experimental estimate for the twisting angle in the cationic electronic ground state was performed in an electron paramagnetic resonance 共EPR兲 spectroscopic study19 which led to a value between 8 and 23 degrees based on a comparison between the experimentally determined proton hyperfine coupling constant and its prediction by a semiempirical calculation. From the results of the photoelectron spectroscopic studies it was soon realized that nonadiabatic couplings between the electronic states of the ethylene radical cation are pervasive. Theoretical investigations of these interactions20–24 showed that nonadiabatic effects must be included in the interpretation of the second photoelectron band, corresponding to the Ã ⫹ state which is connected to the ˜X ⫹ state by a ˜ ⫹ states exconical intersection.21,22 Similarly, the ˜B ⫹ and C 24 ⫹ ˜ hibit a conical intersection. The C state converts adiabatically into the ˜X ⫹ state by a twisting of the molecule, and fast radiationless decay has been observed in the photoelectron bands corresponding to these states.15 Ko¨ppel et al.20,22 showed that the deviation from planarity in the ˜X ⫹ state is caused by vibronic coupling with the Ã ⫹ state. By comparison of model calculations with the classical photoelectron spectra, it was suggested in the same studies that the effect of vibronic coupling on the intensity distribution of the first photoelectron band is small and that the band can be adequately described within the framework of the Franck– Condon approximation. Several ab initio studies on the ground state of C2 H⫹ 4 have also been reported. Whereas early Hartree–Fock calculations yielded a planar equilibrium geometry for the cationic ground state,25,26 more sophisticated methods that take into account electron correlation all predict a D 2 equilibrium geometry.20,24,27–32 Still, the results for the twisting angle and the height of the torsional barrier at the planar geometry vary widely depending on the method and basis set used, and the challenges in accurately calculating the equilibrium geometry have been discussed in Refs. 27 and 32. In the most recent investigation, a twisting angle of 21.0° and a barrier at the planar geometry of 116 cm⫺1 were determined, and the positions of the first torsional levels were computed.32 This ab initio study, however, did not include the zero-point vibrational motion. Although the twisted geometry of the cationic ground state is now well established, there is still a considerable degree of disagreement concerning the twisting angle, the shape of the torsional potential in the electronic ground state and whether large-amplitude motion via tunneling takes place or not. The desires to achieve a more precise characterization of this important molecular cation, to investigate the dynamics associated with ionization from a double bond, and to apply the methods of very high resolution photoelec-

Willitsch, Hollenstein, and Merkt

tron spectroscopy recently developed in our laboratory33,34 to larger polyatomic molecules incited us to start a pulsed-fieldionization zero-kinetic-energy 共PFI-ZEKE兲 photoelectron ˜ ⫹ photoelectronic transition spectroscopic study of the ˜X →X of C2 H4 and C2 D4 . The resolution of better than 0.1 cm⫺1 achieved in the present study is two to three orders of magnitude higher than in the previous photoelectron spectroscopic investigations and enabled a more accurate determination of the geometry and torsional potential, a detailed investigation of rotationally resolved photoionization dynamics and a re-evaluation of the question of vibronic coupling in the first photoelectron band of ethylene. II. EXPERIMENT

The PFI-ZEKE photoelectron spectrometers used in this work have been described in detail previously35,36 and only aspects relevant to the present study are summarized here. Tunable vacuum ultraviolet 共VUV兲 radiation was generated by resonance-enhanced difference-frequency mixing (˜␯ VUV ⫽2˜␯ 1 ⫺˜␯ 2 ) in a krypton gas beam using the (4p) 5 ( 2 P 1/2) 5p ⬘ 关 1/2兴 (J⫽0)←(4p) 6 1 S 0 two-photon resonance at 2˜␯ 1 ⫽98 855.071 cm⫺1 . Mixtures of 10% ethylene 共Pangas, purity 99.95%兲 and ethylene-d4 共Cambridge Isotope Laboratories, isotopic purity 98%兲 in argon were introduced into the measurement chamber in a pulsed supersonic expansion in vacuum from a nozzle stagnation pressure of 3.0 bar which resulted in a rotational temperature T rot⬇8 K. The supersonic molecular beam crossed the VUV laser beam behind a skimmer in the photoionization region of the spectrometer where the charged particles were produced either by photo- or pulsedfield ionization and accelerated towards a microchannel plates detector by a pulsed electric field. PFI-ZEKE photoelectron spectra were recorded by exciting ethylene with a single VUV photon from rotational levels of the ground vibronic state to very high Rydberg states located immediately below the ionization thresholds and monitoring the wave number dependent electron signal produced by pulsed-field ionization.37 Overview spectra of the ⫹ ˜X ⫹ 2 B 3u ground electronic state of C2 H⫹ 4 and C2 D4 were ⫺1 recorded at moderate resolution (0.7 cm ) and provide information on the vibrational structure up to 2000 cm⫺1 above the ground vibrational state of the cations. In these experiments, the VUV radiation was generated using the output of two pulsed dye lasers 共Lambda Physik Scanmate 2兲. The fixed-frequency laser in the difference-frequency mixing process ( ␯ 1 ) was operated with an intracavity e´talon. High Rydberg states were field-ionized using a two-pulse electric field sequence with amplitudes of ⫹275 mV/cm and ⫺415 mV/cm, respectively. The two lowest vibronic bands were measured using our narrow-bandwidth of C2 H⫹ 4 (0.008 cm⫺1 ) near Fourier-transform limited VUV laser36 and the procedure for recording high-resolution PFI-ZEKE photoelectron spectra outlined in Ref. 33. To achieve maximal selectivity in the pulsed-field ionization of high Rydberg states, a multipulse electric field sequence was employed consisting of successive pulses with amplitudes of ⫹140, ⫺85, ⫺130, ⫺175, and ⫺220 mV/cm. The field-ionization



Ionization from a double bond C2 H4⫹

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FIG. 1. Schematic representation of the most important molecular orbitals of the ethylene radical cation in terms of a linear combination of atomic orbitals. The correlation and symmetry labels in the suitable symmetry groups D 2h , D 2 , D 2d and in a semi-united atom basis are given for torsional angles between 0 and 90 degrees. ᐉ ⬙ and ␭ ⬙ denote the quantum numbers of the orbital angular momentum and its projection on the molecular axis in the semi-united atom basis. The dominant electronic configurations of the first three electronic states of the cation are depicted on the left of the figure. The arrows indicate the relative orientation of the electron spins.

yield was recorded by setting detection windows in the electron time-of-flight spectrum after each field pulse of negative polarity. The spectrum with the best resolution was obtained by collecting the electron signal in the second window. The linewidth of the narrowest lines amounted to ⌫ FWHM ⫽0.09 cm⫺1 . The wave number calibration of the overview spectra was achieved by simultaneously recording optogalvanic spectra of neon and making reference to tabulated transition wave numbers.38 In the high-resolution measurements, the VUV wave number was calibrated with an accuracy of better than 0.015 cm⫺1 using the procedure described in Ref. 39. III. ELECTRONIC STRUCTURE AND MOLECULAR SYMMETRY

Ethylene in its vibronic ground state is a planar nearprolate asymmetric top of D 2h point group symmetry. The a-inertial axis coincides with the CvC-bond, the b-axis lies in the molecular plane and the c-axis perpendicular to it. Following common practice,6 the a, b, and c axes are identified with the z, y, and x axes, respectively. The most important molecular orbitals are displayed schematically in Fig. 1 which also qualitatively depicts their dependence on the torsional angle. The highest occupied molecular orbital 共HOMO兲 is the 1b 3u ␲-type orbital that is responsible for the double bond character of the CvC bond. Ionization from this orbital gives rise to the ˜X ⫹ 2 B 3u ( 2 B 3 in D 2 ) ground electronic state of the ethylene radical cation. If the two CH2 moieties of the molecule are twisted against each other by a torsional angle ␶, the molecular point group symmetry is lowered to D 2 until the torsional angle reaches 90° at which point the CH2 moieties lie perpendicular to

each other and the molecule assumes a configuration of D 2d point group symmetry. With increasing torsional angle the energy of the 1b 3u orbital is raised, whereas the energy of the 1b 2g orbital 共the lowest unoccupied molecular orbital, LUMO兲 is lowered. These orbitals form a degenerate pair of orbitals of e symmetry at the perpendicular configuration. The same holds true for the 1b 2u and 1b 3g orbitals which also become degenerate at a twisting angle of 90°. In the electronic ground state of neutral ethylene, the perpendicular geometry corresponds to a saddle point on the potential energy surface, giving rise to a large torsional barrier of about 21 000 cm⫺1 . 40 In the cation, the E state which is the ground state at the perpendicular geometry is unstable with respect to Jahn–Teller distortion along the torsional coordinate.6,7,24 Upon twisting, orbitals of the same symmetry (b 3 or b 2 in D 2 ) mix. This effect is especially pronounced for the b 3 orbitals which lie close in energy. The mixing of these orbitals results in a net stabilization at slightly twisted geometries and leads to the occurrence of a double-minimum potential along the torsional coordinate in the cationic ground state.6,22 Depending on the depth of these minima, the molecular symmetry 共MS兲 group is D 2h (M ) if the energy splittings resulting from the tunneling motion are observable at the resolution of the experiment or D 2 (M ) if it is not.41 The effects of tunneling through the large barrier at the perpendicular geometry, the height of which was estimated to be between 10 000 and 12 000 cm⫺1 , 15,24,32 are not expected to be observable at the experimental resolution achieved in the present study. Hence, we restrict our discussion to the large-amplitude tunneling motion through the potential barrier at the planar geometry within the D 2h (M ) and D 2 (M ) MS groups.


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TABLE I. Rotational selection rules for photoionization transitions from the ˜X ( 1 A g ) ground state of neutral ethylene to the X ˜ ⫹ ( 2 B 3u ) ground electronic state of the ethylene cation in terms of allowed changes for the asymmetric ⫹ ⫹ top quantum numbers ⌬K a ⫽K ⫹ a ⫺K a⬙ and ⌬K c ⫽K c ⫺K c⬙ . ⌫ v and ᐉ denote the upper state vibrational symmetry and the orbital angular momentum quantum number of the photoelectron, respectively. The assignment of the twelve normal modes ␯ i of the ethylene cation to the irreducible representations of D 2h (M ) is given in the last column. ⌫⫹ v

⌬K a

⌬K c

ᐉ

Ag Au B 1g B 1u B 2g B 2u B 3g B 3u

odd odd odd odd even even even even

even even odd odd odd odd even even

even odd even odd even odd even odd

constant is much larger than the B and C constants, giving rise to a concentration of many lines in a relatively small wave number interval. In contrast, bands of vibrational symmetry ⌫ ⫹ v ⫽A g , A u , B 1g , and B 1u can be expected to show a broad rotational structure because 兩 ⌬K a 兩 ⬎0.

B. Simulation of rotational transition intensities ␯1 , ␯2 , ␯3 ␯4 ␯ 11 , ␯ 12 ␯8 ␯ 9 , ␯ 10 ␯5 , ␯6 ␯7

IV. THEORY A. Photoionization selection rules

When applied to photoionization transitions from rotational levels of the vibronic ground state of neutral ethylene ⬙ ⫽A g ) to selected rovibrational levels 共vibronic symmetry ⌫ ve ⫹2 ˜ of the X B 3u ground electronic state of the cation, the general rovibrational symmetry selection rule for photoionization transitions42

To gain insight into the rotational fine structure of the observed vibronic bands, the model developed by Buckingham, Orr, and Sichel43 for the calculation of rotationally resolved photoionization cross sections for diatomic molecules was extended to the treatment of asymmetric tops. A similar model has also been reported by McKoy and co-workers 共see Ref. 44, and references cited therein兲. In the present study, however, we aim at deriving expressions for relative rotational line intensities in photoelectron spectra that are easy to implement numerically and do not require elaborate electronic structure calculations, but nevertheless provide insights into the dominant features of the photoionization dynamics. In Ref. 43, the total photoionization cross section between two rovibronic states of a diatomic molecule that are treated within the orbital approximation and in the limit of Hund’s angular momentum coupling case 共b兲 is given as ⬁

⫹ ⬙ 丢 ⌫ e ⫺ 傻⌫ * ⌫ rve 丢 ⌫ rve

共1兲

can be expressed as ⫹ ⌫ r⬙ 丢 B 3u 丢 ⌫ ⫹ v 丢 ⌫ r 丢 ⌫ e ⫺ 傻A u .

␴ tot⬀q 2v

Q共 ᐉ⬙兲

E,ᐉ ⫺1 兩 C ␣ ⬙ ᐉ ⬙ 兩 2 关 ᐉ ⬙ 兩 F ␣ ⬙ ᐉ⬙⬙ 兩 2 兺 2ᐉ ⫹1 ⬙ ᐉ ⫽兩␭ 兩

⬙

⬙

⫹ 共 ᐉ ⬙ ⫹1 兲兩 F ␣ ⬙ ᐉ⬙⬙ 兩 2 兴 , E,ᐉ ⫹1

共2兲

⫹ ⬙ represent the rovibronic symmetries and ⌫ rve In Eq. 共1兲, ⌫ rve of the ionic and neutral states, respectively, ⌫ e ⫺ is the symmetry of the photoelectron and ⌫ * is the dipole moment representation of the molecular symmetry group.41 The symmetry ⌫ e ⫺ is either A g or A u , depending on whether the orbital angular momentum quantum number ᐉ of the photoelectron is even or odd. The rotational symmetries ⌫ r⫹ and ⌫ r⬙ depend on the asymmetric top quantum numbers K a and K c . For K a K c ⫽ee, eo, oe and oo ⌫ r ⫽A g , B 1g , B 3g , and B 2g , 41 respectively, whereby e and o represent even and odd values of the quantum numbers. Table I lists the rotational selection rules in terms of the changes of the asymmetric top ⫹ quantum numbers ⌬K a ⫽K ⫹ a ⫺K ⬙ a and ⌬K c ⫽K c ⫺K c⬙ which were derived for different cationic vibrational symmetries ⌫⫹ v using Eq. 共2兲. The symmetry labels within D 2 (M ) can be obtained from the D 2h (M ) results by suppressing the g/u subscripts. Transitions to vibrational states with symmetries ⌫ ⫹ v differing only in their g/u symmetry exhibit the same rotational selection rules, but these require that the photoelectron be ejected as a partial wave of different g/u symmetry. Moreover, the rotational structures of the photoelectron bands are strongly dependent on the symmetry of the cationic vibrational state: When ⌫ ⫹ v ⫽B 2g , B 2u , B 3g , and B 3u , observation of sharp Q-type branches can be expected because transitions with ⌬K a ⫽0 are allowed and the rotational A

共3兲

where q 2v represents the Franck–Condon factor and ᐉ ⬙ and ␭ ⬙ are the quantum numbers of an orbital angular momentum component and its projection along the molecular axis in a single-center expansion of the molecular orbital ␾ ␣ ⬙ ,␭ ⬙ from which ionization occurs: ⬁

␾ ␣ ⬙ ,␭ ⬙ ⫽

兺

ᐉ ⬙⫽兩␭ ⬙兩

C ␣ ⬙ ᐉ ⬙ R ␣ ⬙ ᐉ ⬙ 共 r 兲 Y ᐉ ⬙ ,␭ ⬙ 共 ␪ , ␸ 兲 .

共4兲

C ␣ ⬙ ᐉ ⬙ are the expansion coefficients, R ␣ ⬙ ᐉ ⬙ (r) stand for functions that depend only on the radial coordinate r, ␣ ⬙ represents all other quantum numbers required to define the state of the molecule, and Y ᐉ ⬙ ,␭ ⬙ ( ␪ , ␸ ) are spherical harmon-

ics. The functions F ␣ ⬙ ᐉ⬙⬙ in Eq. 共3兲 are radial transition integrals as defined in Eq. 共25兲 of Ref. 43. The factor Q(ᐉ ⬙ ) incorporates the dependence of the photoionization cross section on the angular 共especially the rotational兲 coordinates. The extension of this model to asymmetric tops relies on the following approximations: E,ᐉ ⫾1

共1兲 The effects of the electron spin are neglected. This assumption is perfectly valid for singlet electronic states. For doublet electronic states, the approximation is justified as long as the fine structure induced by the electron spin cannot be resolved experimentally. In asymmetric tops, the electron spin usually manifests itself by a spin-




rotation coupling that is generally small in larger polyatomic molecules.45 In the present study, spin-rotation splittings could not be resolved in the spectra. 共2兲 The quantum number of the total angular momentum without spin N is identified with the rotational angular momentum. 共3兲 As in Ref. 43, the electronic wave function is represented by a single Slater determinant and the photoelectron is assumed to be ejected from a definite molecular orbital. Moreover, it is assumed that ionization occurs vertically and that the molecular orbital structure does not change upon the removal of an electron. This treatment entails the neglect of electron correlation, i.e., of configuration interaction. However, it will be demonstrated in Sec. V B that the present formalism can also be applied to the case where several configurations contribute to the electronic state under consideration provided that one of these dominantly accounts for the intensity of a given vibronic band. This is typically the case when a band gains intensity by vibronic coupling 共see Sec. V B兲. The advantage of using the orbital approximation is— besides a considerable simplification of the calculation—that it provides a simple and appealing physical picture for the angular momentum change of the molecular core upon ionization and captures the dominant aspects of the photoionization dynamics: the angular momentum transferred between the initial (N⬙ ) and final (N⫹ ) states can be identified with the electronic orbital angular momentum 艎⬙ of the orbital out of which ionization occurs, i.e., the electronic angular momentum ‘‘hole’’ left behind in the molecule after ejection of the photoelectron 共see also the discussion in Ref. 46兲. Physical intuition and a qualitative picture of the molecular orbitals of the neutral state can often be used to identify the dominant contributions to the sum of Eq. 共4兲. In the extension of Eq. 共3兲 to treat asymmetric top molecules, the molecular wave function of a diatomic molecule 兩 ␣ N⌳M N 典共here, the spin quantum numbers are disregarded兲 which is a special case of a symmetric top wave function41 is substituted by the corresponding wave function of an asymmetric top 兩 ␣ NK a K c M N 典 , where M N denotes the quantum number associated with the projection of N on the spacefixed z-axis. The asymmetric top eigenfunctions 兩 ␣ NK a K c M N 典 are expanded in a basis of symmetric top functions 兩 ␣ NKM N 典 ,

Q 共 ᐉ ⬙ 兲 ⫽ 共 2N ⫹ ⫹1 兲 ⫻

⫻

冋兺冉

⫹ ⫹ ⫹ N ⫹ ,K ,K a c N ⬙ ,K ⬙ a ,K c⬙

K ⫹ ,K ⬙

共 ⫺1 兲 K c K ⫹

N⫹

ᐉ⬙

N⬙

⫺K ⫹

␭⬙

K⬙

N

兺

K⫽⫺N

N,K a ,K c

cK

兩 ␣ NKM N 典 ,

共5兲

where K represents the projection of the rotational angular N,K ,K momentum on the symmetric top axis and c K a c are expansion coefficients. For ethylene which is a near-prolate symmetric top, this axis is conveniently identified with the a-axis. This substitution leads to a new expression of the Q(ᐉ ⬙ )-factor:

冊册

cK

⬙

2

共6兲

.

In the 3 j-symbol of Eq. 共6兲, N ⬙ (N ⫹ ) and K ⬙ (K ⫹ ) stand for the quantum numbers associated with the total angular momentum without spin and its projection on the molecular axis of the neutral 共ionic兲 states. Moreover, the factor q 2v 关see Eq. 共3兲兴 now contains the dependence on all vibrational variables and, in a first approximation, q 2v can be expressed as a product of Franck–Condon factors. The other terms in Eq. 共3兲 remain formally unchanged, and the total photoionization cross section thus becomes ⬁

␴ tot⬀ ␳ ⬙ q 2v ⫻

⫻

冋

冉

2N ⫹ ⫹1 ⬙ 2ᐉ ⬙ ⫹1

兺 ᐉ ⫽兩␭ 兩 ⬙

兺 K ,K ⫹

⫹ ⫹ ⫹ N ⫹ ,K ,K a c N ⬙ ,K a⬙ ,K c⬙

⬙

共 ⫺1 兲 K c K ⫹

N⫹

ᐉ⬙

N⬙

⫺K ⫹

␭⬙

K⬙

冊册

cK

⬙

2

⫻ 兩 C ␣ ⬙ ᐉ ⬙ 兩 2 关 ᐉ ⬙ 兩 F ␣ ⬙ ᐉ⬙⬙ 兩 2 ⫹ 共 ᐉ ⬙ ⫹1 兲兩 F ␣ ⬙ ᐉ⬙⬙ 兩 2 兴 . E,ᐉ ⫺1

E,ᐉ ⫹1

共7兲 In Eq. 共7兲, an additional weighting with the population of the rotational state of the neutral species

⬙ /kT rot其 ␳ ⬙ ⫽ ␹ ⬙ 共 2N ⬙ ⫹1 兲 exp兵 ⫺E rot

共8兲

is introduced that is proportional to the spin statistical weight ␹ ⬙ and the magnetic degeneracy factor (2N ⬙ ⫹1). The rota⬙ in the Boltzmann factor of Eq. 共8兲 is meational energy E rot sured relative to the state of lowest energy having the same nuclear spin symmetry. For ethylene in its vibronic ground state, the nuclear spin statistical weight is ␹ ⬙ ⫽7 for states with K a⬙ K c⬙ ⫽ee and ␹ ⬙ ⫽3 otherwise.41 In the analysis of experimental spectra, Eq. 共7兲 can be used in several ways. If relative rotational line intensities within a given vibronic band are of interest, the factor q 2v can be treated as a scaling factor and the parameters

B ␣ ⬙ ,ᐉ ⬙ ª 兩 C ␣ ⬙ ᐉ ⬙ 兩 2 关 ᐉ ⬙ 兩 F ␣ ⬙ ᐉ⬙⬙ 兩 2 ⫹(ᐉ ⬙ ⫹1) 兩 F ␣ ⬙ ᐉ⬙⬙ 兩 2 兴 can be fitted to the experimental spectrum. Alternatively, if estimates for the coefficients C ␣ ⬙ ᐉ ⬙ are available, the factors E,ᐉ ⫺1 E,ᐉ ⫹1 b ␣ ⬙ ,ᐉ ⬙ ªᐉ ⬙ 兩 F ␣ ⬙ ᐉ⬙⬙ 兩 2 ⫹(ᐉ ⬙ ⫹1) 兩 F ␣ ⬙ ᐉ⬙⬙ 兩 2 can be treated as adjustable parameters within the model. In this way, Eq. 共7兲 provides a quantitative model for relative rotational transition intensities that is easy to implement numerically without the need of any vibronic structure calculations, and this approach will be followed in Sec. V C. Alternatively, relative vibrational intensities can be determined and are proportional to the vibrational factors q 2v because the outer sum on the right hand side of Eq. 共7兲 can be treated as a common scaling factor. Equation 共7兲 will be used in this way in Sec. V D to E,ᐉ ⫺1

兩 ␣ NK a K c M N 典 ⫽

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E,ᐉ ⫹1


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˜ 1 A g 00 →X ˜ ⫹ 2 B 3u ␯ v i transition of C2 H4 . At the experimental resolution of 0.7 cm⫺1 , the FIG. 2. Survey of the PFI-ZEKE photoelectron spectrum of the X i rotational structure is resolved in all bands. The upper trace in the upper panel represents the photoionization spectrum recorded by monitoring the v ⫹ wave-number-dependent C2 H4 ion yield. The assignment bars indicate the upper vibrational state ␯ i i where v i represents the number of quanta in the mode ␯ i . The diamonds and squares show experimental and calculated relative intensities for the torsional ( ␯ 4 ) bands, respectively. See text for details.

analyze the intensity distribution of the vibrational progression in the torsional mode. Finally, when combined with a theoretical calculation of the radial transition integrals F nᐉ ⬙⬙ and the vibrational factors q 2v , Eq. 共7兲 can be used for a full ab initio prediction of relative rovibronic line intensities in the photoelectron spectra of asymmetric tops. E,ᐉ ⫾1

V. RESULTS AND DISCUSSION A. Vibrational structure of the electronic ground state of C2 H4¿ and C2 D4¿

˜ ⫹ 2 B 3u ␯ v i viOverview spectra of the ˜X 1 A g 00 →X i ⫹ ⫹ bronic transitions in C2 H4 and C2 D4 in the wave number

˜ 1 A g 00 →X ˜ ⫹ 2 B 3u ␯ v i transition of C2 D4 . The upper trace in the upper panel represents the FIG. 3. Survey of the PFI-ZEKE photoelectron spectrum of the X i vi photoionization spectrum recorded by monitoring the wave-number-dependent C2 D⫹ 4 ion yield. The assignment bars indicate the upper vibrational state ␯ i where v i represents the number of quanta in the mode ␯ i . The assignment of the bands marked with an asterisk is tentative. The diamonds and squares show experimental and calculated relative intensities for the torsional ( ␯ 4 ) bands, respectively. See text for details.



range 84 600– 86 800 cm⫺1 are presented in Figs. 2 and 3, v respectively. In the following, we use the short notation ␯ i i vi to label a transition to a cationic vibrational state ␯ i where v i denotes the number of quanta in the mode ␯ i . The labeling of normal modes adopted in this study follows the convention used for neutral ethylene 共see e.g., Fig. 1 of Ref. 47兲. At the experimental resolution of 0.7 cm⫺1 , the rotational structure could be sufficiently well resolved in all vibrational bands in Figs. 2 and 3 for a rotational analysis to be performed. The top trace in the upper panel of both figures shows the photoionization 共PI兲 spectrum which was obtained ⫹ by monitoring the wave-number-dependent C2 H⫹ 4 /C2 D4 -ion yield, and the lower traces correspond to the PFI-ZEKE photoelectron spectra. The PI spectra display a sharp rise from a signal level of zero over the first band of the PFI-ZEKE spectra which indicates that the first bands in Figs. 2 and 3 correspond to the vibrationless origin bands 0 0 . The assignments of the remaining vibrational bands were established after identification of the vibrational symmetry of the upper state by its rotational structure using the rotational photoionization selection rules in Table I and model calculations of the relative rotational transition intensities as outlined in Sec. V C. These assignments are supported by comparison with previous experimental studies13–17 and recent ab initio results32 on the cation as well as with the fundamental wave numbers of neutral ethylene.47 Upon photoionization from the 1b 3u molecular orbital, the CvC bond length increases from 1.34 to 1.40 Å 共Ref. 32兲 and the molecule is twisted away from the planar geometry of the neutral ground state. Vibrational progressions in the CvC stretching and torsional coordinates are thus expected to be dominant in the photoelectron spectrum. This is indeed observed and the main contributions to the spectrum correspond to totally symmetric excitations in the ␯ 4 torsional, ␯ 3 CH2 bending, and ␯ 2 CvC stretching vibrations. The low wave number region of both spectra shows several bands of irregular spacing and intensity that were already partially resolved in previous photoelectron spectroscopic studies13–17 and assigned to a progression in the torsional mode ␯ 4 . The high resolution of our measurement, however, enabled the first observation of the 4 1 , 4 4 , and 4 5 1 3 5 6 bands of C2 H⫹ 4 and of the 4 , 4 , 4 , and 4 bands of ⫹ C2 D4 . The assignment of this progression will be further justified by the results of the modelling of the torsional motion in Sec. V D. The rotational structure of bands with an even number of quanta in the torsional mode differs markedly from that of bands with an odd number of quanta in this mode 共highresolution measurements of the rotational structure of the 0 0 and 4 1 bands are compared in Fig. 4 below兲, indicating that the two groups of bands differ in the vibrational symmetry of the upper state. Hence the adequate effective molecular symmetry group is D 2h (M ), because the torsional coordinate is of a u symmetry in D 2h (M ), whereas it is totally symmetric in D 2 (M ). In Fig. 2, the totally symmetric vibrational bands at 86 047 and 86 276 cm⫺1 are assigned to the fundamental bands of the symmetric CH2 bending ( ␯ 3 ) and CvC stretch-


1767

ing ( ␯ 2 ) vibrations which have also been observed in the previous classical photoelectron spectroscopic studies. Each of them forms the origin of a progression of combination bands with excitations of the torsional mode ␯ 4 which can readily be identified by their rotational structure and by comparison of their relative spacings with those of the bands of the fundamental torsional progression 4 v 4 . The 3 1 4 1 and 2 1 4 1 bands are observed here for the first time and, together with the 3 1 4 2 and 2 1 4 2 bands, offer the possibility to quantify the effects of the ␯ 2 and ␯ 3 modes on the torsional motion 共see below兲. The weak band at 85 690 cm⫺1 exhibits an intense sharp feature with weaker lines on its high wave number side. This rotational structure is consistent with an upper state of B 3u vibrational symmetry and the prominent line is identified as a Q-type rotational branch which consists of unresolved transitions with ⌬N⫽N ⫹ ⫺N ⬙ ⫽0, ⌬K a ⫽0 and ⌬K c ⫽0. Consequently, this feature is assigned to the 7 1 fundamental band. Our experimentally derived value for the ␯ 7 fundamental wave number of 901 cm⫺1 is also consistent with the ab initio predictions that range from 887 to 955 cm⫺1 共Ref. 32兲 and can be compared to ˜␯ 7 ⫽948.8 cm⫺1 in neutral ethylene.48 The spectrum of C2 D⫹ 4 in Fig. 3 also shows the fundamental excitations in the totally symmetric CD2 bending ( ␯ 3 ) and CvC stretching ( ␯ 2 ) vibrations in combination with excitations of the torsional mode. The combination bands 3 1 4 1 and 2 1 4 1 are observed here for the first time. The sharp line at 85 594 cm⫺1 is identified as the 7 1 band because its characteristic rotational structure indicates an upper state of B 3u vibrational symmetry. Moreover, its fundamental wave number of 683 cm⫺1 is similar to that of neutral C2 D4 关 ˜␯ 7 ⫽719.8 cm⫺1 共Ref. 49兲兴. The totally symmetric band at 86 270 cm⫺1 close to the 2 1 band is assigned to the first overtone band 7 2 . The totally symmetric band at 86 533 that partially overlaps with the 2 1 4 2 band is identified with the combination band 7 2 4 2 . The weak feature at 85 853 cm⫺1 which consists of a single sharp line is assigned to the Q-type branch of the 7 1 4 2 combination band because of its characteristic shape and its wave number offset from the 7 1 band. However, this assignment must remain tentative as it solely relies on a single line that is interpreted as a Q-type branch. Similarly, the weak feature at 85 494 cm⫺1 , which also appears as a single sharp line, is tentatively assigned to the Q-type branch of the 101 band with an upper state vibrational symmetry of B 2u . Its wave number of 583 cm⫺1 can be compared to ˜␯ 10⫽593.7 cm⫺1 in neutral C2 D4 . 49 ⫹ The spectra of C2 H⫹ 4 and C2 D4 reveal several vibra⫹ 1 1 tional interactions. In C2 H4 , the 2 4 band is perturbed by the almost degenerate 4 5 band of the same upper state vibrational symmetry A u . The perturbation results in an abnormally small wave number offset of 71 cm⫺1 of the 2 1 4 1 band from the 2 1 band which can be compared to 84 cm⫺1 between the 0 0 and 4 1 bands. A similar interaction is found 1 1 5 1 1 in C2 D⫹ 4 between the 3 4 and 4 bands. Here, the 3 4 band is slightly shifted to higher wave numbers compared to what can be expected from the splitting of the 0 0 and 4 1 bands. The 7 2 band is likely to gain intensity from the neighboring 2 1 band by a Fermi resonance, an effect that is even more pronounced between the 7 2 4 2 and 2 1 4 2 bands which


1768



TABLE II. Assignments and origins of the vibrational bands observed in the PFI-ZEKE photoelectron spectrum ˜ ⫹ transition of C2 H4 . The notation ␯ v i denotes the number of quanta v i in the mode ␯ i of the of the ˜X →X i cationic state. ⌫ ⫹ v represents the vibrational symmetry of the cationic state. Band

⌫⫹ v

˜␯ a/cm⫺1

Expt. 共Ref. 17兲b

˜␯ 0 c/cm⫺1

00 41 42 43 71 44 31 3 14 1 21 2 14 1 45 3 14 2 2 14 2

Ag Au Ag Au B 3u Ag Ag Au Ag Au Au Ag Ag

84 788.4共3兲 84 872.1共4兲 85 226.6共3兲 85 553.0共4兲 85 689.7共8兲 85 950.5共3兲 86 047.1共3兲 86 128.8共4兲 86 276.1共3兲 86 347.0共4兲f 86 382.0共4兲f 86 482.5共3兲 86 720.0共3兲

84 787

0.0 83.7共5兲 438.2共4兲 764.6共5兲 901.3共8兲 1162.1共5兲 1258.7共4兲 1340.4共5兲 1487.7共4兲 1558.6共5兲f 1593.6共5兲f 1694.1共4兲 1931.6共4兲

85 224 85 559

86 036 86 277

86 478 86 713

˜␯ 0,calcd/cm⫺1

˜␯ 0,calce/cm⫺1

83.8 438.2 764.5

122 324 688 955 1093 1290

1162.1

1557 1592.4

1545

Ionization energies not corrected for the field-induced lowering of the ionization threshold of ⌬˜␯ ⫽(2.3 ⫾1.3) cm⫺1 ; numbers in parentheses denote an uncertainty of one ␴ in the last digit. b The uncertainty of the line positions was reported to be 8 cm⫺1 . c Wave number relative to vibronic ground state. d Calculated term values using the one-dimensional model for the torsional motion, see Sec. V D. e Reference 32. CCSD共T兲/cc-pVTZ harmonic frequencies. For the ␯ 4 -mode, the torsional Schro¨dinger equation was solved explicitly in a pure electronic ab initio potential without considering the zero-point-energy vibrational motion. f Perturbed band, see text for details. a

are so close that their rotational structures overlap and that their intensities are almost equal. All assignments are summarized in Tables II and III which also list the origins of the bands determined from the analysis of the rotational structure as discussed in Sec. V C.

B. Vibronic coupling

The overview spectra shown in Figs. 2 and 3 reveal several bands with nontotally symmetric upper state vibrational symmetry. Assuming in a first approximation that the vi-

TABLE III. Assignments and origins of the vibrational bands observed in the PFI-ZEKE photoelectron spec˜ →X ˜ ⫹ transition of C2 D4 . The notation ␯ v i denotes the number of quanta v i in the mode ␯ i of the trum of the X i cationic state. ⌫ ⫹ v represents the vibrational symmetry of the cationic state. Band

⌫⫹ v

˜␯ a/cm⫺1

Expt. 共Ref. 17兲b

˜␯ 0 c/cm⫺1

00 41 42 43 101e 71 44 7 1 4 2e 31 45 3 14 1 3 14 2 46 21 72 2 14 1 2 14 2 7 24 2

Ag Au Ag Au B 2u B 3u Ag B 3u Ag Au Au Ag Ag Ag Ag Au Ag Ag

84 911.0共3兲 84 948.1共5兲 85 186.5共4兲 85 380.0共3兲 85 494.2共12兲 85 593.9共9兲 85 631.4共4兲 85 852.8共10兲 85 871.5共4兲 85 893.2共10兲f 85 912.1共4兲f 86 146.0共4兲 86 202.5共4兲 86 248.4共4兲f 86 270.3共4兲f 86 288.0共4兲 86 521.5共4兲f 86 532.9共5兲f

84 908

0.0 37.1共5兲 275.5共4兲 469.0共4兲 583.2共13兲 682.9共9兲 720.44共4兲 941.8共10兲 960.5共4兲 982.2共10兲f 1001.1共5兲f 1235.0共4兲 1291.5共4兲 1337.4共4兲f 1359.3共5兲f 1377.0共5兲f 1610.5共5兲f 1621.9共5兲f

85 185

85 628 85 874

86 164 86 229

86 522

˜␯ 0,calcd/cm⫺1 37.1 275.4 468.8

720.7

995.1

1291.5

Ionization energies not corrected for the field-induced lowering of the ionization threshold of ⌬˜␯ ⫽(2.3 ⫾1.3) cm⫺1 ; numbers in parentheses denote an uncertainty of one ␴ in the last digit. b The uncertainty of the line positions was reported to be 8 cm⫺1 in Ref. 17. c Wave number relative to vibronic ground state. d Calculated term values using the one-dimensional model for the torsional motion, see Sec. V D. e Assignment tentative, see text for details. f Perturbed band, see text for details.

a




bronic transition moment is only weakly dependent on the nuclear coordinates, it can be factored into a vibrational and an electronic part: ⫹ ⫹ ⬙ 典兩2⬇兩具 ␸ ⫹ 兩具 ␸ e ⫺ 兩具 ␸ ev 兩␮ ˆ ␣ 兩 ␸ ev ˆ ␣ 兩 ␸ e⬙ 典兩 2 , v 兩␸⬙ v 典具 ␸ e ⫺ 兩具 ␸ e 兩 ␮

共9兲

where ␸ e and ␸ v represent electronic and vibrational wave functions, respectively, ␸ e ⫺ the wave function of the photoˆ ␣ ( ␣ ⫽x,y,z) is a component of the electric electron and ␮ dipole moment operator. When considering transitions from the vibronic ground state of the neutral molecule 共vibrational symmetry ⌫ ⬙v ⫽A g ), Eq. 共9兲 implies that transitions to nontotally symmetric vibrational levels are vibronically forbidden 2 because the Franck–Condon factor 兩具 ␸ ⫹ v 4兩 ␸ ⬙ v 4 典兩 in the prod⫹ 2 2 vanishes. Hence, these uct 兩具 ␸ ⫹ v 兩␸⬙ v 典兩 ⫽ 兿 i兩具 ␸ v j兩 ␸ ⬙ v j典兩 bands must gain intensity by a different mechanism than those with totally symmetric upper state vibrational symmetry. As has been discussed in Sec. III, the 1b 3u and 1b 3g molecular orbitals are mixed when the molecule is twisted. Equivalently, this mixing can be viewed, in the realm of the orbital approximation discussed in Sec. IV B, as a mixing between the electronic states that arise from the configurations in which these orbitals are singly occupied. The observation of bands with an odd number of torsional quanta in the ˜X ⫹ 2 B 3u state of C2 H⫹ 4 thus has its origin in intensity borrowing from the Ã ⫹ 2 B 3g state which is formed by ionization from the 1b 3g orbital 共this intensity borrowing effect is known as the Herzberg–Teller effect50兲. Indeed, the symmetry requirement for vibronic coupling ˜⫹

⌫ Xe

Ã ⫹

丢 ⌫e

傻⌫ ⫹ v

共10兲

is fulfilled because B 3u 丢 B 3g ⫽A u . As a result of the mixing between the ˜X ⫹ and Ã ⫹ states, the electronic wave function of the cationic ground state ␸ ⫹ e can be represented as a su˜⫹

˜⫹

perposition of zero-order wave functions ␸ Xe and ␸ Ae of the ˜X ⫹ and Ã ⫹ states,20,41 respectively, ˜⫹

˜⫹

X A ␸⫹ e ⫽c ˜X ⫹ ␸ e ⫹c Ã ⫹ ␸ e ,

共11兲

where the coefficient c Ã ⫹ is a function of the torsional coordinate ␶ which induces the mixing and c ˜X ⫹ ⬇1 if only small deviations from the planar geometry are considered. Taking only the linear term of a power series expansion in ␶, c Ã ⫹ can be approximated as c Ã ⫹ ⬇c Ã⬘ ⫹ ␶ because c Ã ⫹ vanishes at the planar geometry. The vibronic transition moment becomes ⫹ ⬙ 典兩2 兩具 ␸ e ⫺ 兩具 ␸ ev 兩␮ ˆ ␣ 兩 ␸ ev ˜⫹

X ⬇ 兩 c ˜X ⫹ 具 ␸ ⫹ ˆ ␣ 兩 ␸ e⬙ 典 v 兩␸⬙ v 典具 ␸ e ⫺ 兩具 ␸ e 兩 ␮ ˜⫹

A ⫹c Ã⬘ ⫹ 具 ␸ ⫹ ˆ ␣ 兩 ␸ ⬙e 典兩 2 . v 兩␶兩␸⬙ v 典具 ␸ e ⫺ 兩具 ␸ e 兩 ␮

共12兲

For transitions between totally symmetric vibrational states, only the first term on the right-hand side of Eq. 共12兲 is nonvanishing. Within our model for the photoionization transition intensities, the contribution of the torsional mode to the vibrational factor q 2v in Eq. 共7兲 thus corresponds to the Franck–Condon factor 具 ␸ ⫹ v 兩␸⬙ v 典 for bands with an even 4

4

1769

number of torsional quanta in the ionic state and the electronic factor contains the zero-order wave function of the ˜X ⫹ state. Bands with an odd number of torsional quanta 共upper state vibrational symmetry A u ) gain intensity only because of the second term in Eq. 共12兲. Hence, the dependence of q 2v on the torsional motion is accounted for by the factor 具␸⫹ v 4兩 ␶ 兩 ␸ ⬙ v 4 典 for these bands, and the electronic factor contains the zero-order wave function of the Ã ⫹ state. In this way, the electronic transition moments can be evaluated using the orbital approximation because only one of the configurations of Eq. 共11兲 dominantly contributes to the intensi˜ ⫹ ) and (A ˜ ⫹ ) configurations for the bands ties, namely the (X with an even and odd number of torsional quanta, respectively. Equation 共12兲 expresses the fact that the vibronic coupling that leads to a twisted equilibrium geometry in the electronic ground state by configuration interaction also gives rise to the observation of bands with an odd number of torsional quanta in the ionic state. A calculation of the relative intensities of the successive torsional bands based on the evaluation of the vibrational factors in Eq. 共12兲 and a one-dimensional torsional potential gives good agreement with the experimental results as will be demonstrated in Sec. V D 共see also Figs. 2 and 3兲. A similar mechanism of intensity borrowing accounts for the observation of the 7 1 band. In this case, the upper state ⫹ is B 3u 丢 B 3u ⫽A g which enables couvibronic symmetry ⌫ ev ⫹2 ˜ pling to the B A g electronic state. This state arises by ionization from the 3a g molecular orbital. However, the vibronic coupling can be expected to be less efficient than for the odd torsional bands, because the energy separation between the ˜X ⫹ and the ˜B ⫹ state is considerably larger than the one between the ˜X ⫹ and the Ã ⫹ state.15 This is reflected by the low intensity of the 7 1 band in both isotopomers. Thus, the conclusion drawn by Ko¨ppel et al.20,22 that vibronic couplings are small in the first photoelectron band and that the intensity distribution is well described by the Franck–Condon principle has to be refined in the sense that intensity borrowing effects contribute only little to the total intensity of the band but give rise to the appearance of the bands with an odd number of torsional quanta and the 7 1 band.

C. Rotational structure

The upper traces of Figs. 4共a兲 and 4共b兲 show highresolution PFI-ZEKE photoelectron spectra of the 0 0 and 4 1 bands of C2 H⫹ 4 , respectively. The linewidth of the narrowest lines amounts to 0.09 cm⫺1 , which constitutes the best resolution achieved to date by PFI-ZEKE photoelectron spectroscopy of a polyatomic molecule, and is only slightly larger than the linewidth of 0.06 cm⫺1 obtained in the PFI-ZEKE photoelectron spectra of Ar and N2 . 33 At this resolution, the rotational structure could be fully resolved. A slight broadening of several rotational lines is attributed to an unresolved spin-rotational splitting. The assignment bars in Fig. 4 denote the dominant rotational transitions using the notation N K⬙ ⬙ K ⬙ a c


1770


Willitsch, Hollenstein, and Merkt ˜ TABLE IV. Results of the analysis of the rotational structure of the X ˜ ⫹ 0 0 bands of C2 H4 and C2 D4 . A ⫹ , B ⫹ , and C ⫹ represent the rota→X tional constants of the ionic state, IE the adiabatic ionization energy, and rms the root-mean-square error of the fit. Numbers in parentheses represent one standard deviation.

a

FIG. 4. Rotational structure of 共a兲 the 0 0 , and 共b兲 the 4 1 bands of the ˜ ⫹ transition high-resolution PFI-ZEKE photoelectron spectrum of the ˜X →X of C2 H4 . The narrowest lines exhibit a full width at half maximum of 0.09 cm⫺1 . 共c兲 Rotational structure of the 7 1 band of the PFI-ZEKE photoelectron spectrum of C2 H4 recorded at a resolution of 0.7 cm⫺1 . The assign⫹ ment bars indicate rotational transitions N K⬙ ⬙ K ⬙ →N K ⫹ K ⫹ , where N ⬙ , K a⬙ , a c

a

c

⫹ K c⬙ and N ⫹ , K ⫹ a , K c denote the asymmetric top quantum numbers in the neutral and ionic state, respectively. The intensity of the line marked with an asterisk is enhanced by a rovibrational channel interaction. In each panel, the upper and lower traces represent experimental and simulated spectra, respectively, whereby the rotational line intensities were calculated using the model for the rotational line intensities outlined in Secs. IV B and V C. The difference in the rotational structure of the 0 0 and 4 1 bands is caused by different ionization mechanisms for the two bands 共see text for details兲.

⫹

→NK⫹K⫹ . The rotational line positions were analyzed using a

c

rigid rotor rotational Hamiltonians for both the neutral and ionic states H⫽A ⬙ (⫹) N a⬙ (⫹)2 ⫹B ⬙ (⫹) N b⬙ (⫹)2 ⫹C ⬙ (⫹) N c⬙ (⫹)2 , ⫹

⫹

⫹

共13兲

where A ⬙ , B ⬙ , C ⬙ (A ,B ,C ) denote the rotational constants of the neutral 共ionic兲 state. When refining the molecular constants, the rotational constants of the ethylene ground state were held fixed at the values reported in Ref. 48 and only the ionization energy 共corresponding to the 0 00→0 00 transition兲 and the ionic rotational constants were fitted. Inclusion of centrifugal distortion terms in the rotational Hamiltonian did not yield a significant improvement of the fit, indicating that centrifugal distortion effects can be neglected at the accuracy of the measurements and the low degree of rotational excitation of the states involved in the

Band

0 0 C2 H⫹ 4

0 0 C2 D⫹ 4

A ⫹ /cm⫺1 B ⫹ /cm⫺1 C ⫹ /cm⫺1 IEa/cm⫺1 rms

4.770共16兲 0.9252共49兲 0.7832共58兲 84790.42共23兲 0.253

2.399共63兲 0.674共59兲 0.554共136兲 84913.3共14兲 0.150

Corrected for the field-induced lowering of the ionization energy.

observed transitions 共see Fig. 4兲. The results of the fit for the 1 0 0 band of C2 H⫹ 4 are listed in Table IV. The fit for the 4 band led to identical rotational constants within the uncertainty limits. The results of an analogous analysis of the rotational structure of the origin band 0 0 of C2 D⫹ 4 共see Fig. 3兲 is summarized in Table IV. In this fit, the lower state rotational constants were held fixed at the values reported in Ref. 51. The fits also yielded precise values for the adiabatic ionization energies of C2 H4 and C2 D4 , which, after correction for the field-induced lowering of the ionization thresholds, were determined to be 84 790.42(23) and 84 913.3(14) cm⫺1 , respectively. The lower traces in Figs. 4共a兲 and 4共b兲 show a simulation of the rotational structure using the constants derived from the fits, the model for the rotational transition intensities described in Sec. IV B and the selection rules of Table I. A rotational temperature T rot⫽8 K was assumed in the simulations. By inspection of Fig. 1, it can be seen that the 1b 3u molecular orbital correlates with a 2p ␲ x orbital in a semiunited atom basis. Thus, the main angular momentum contribution (ᐉ ⬙ ,␭ ⬙ ) in a single center expansion of this orbital is 共1, ⫾1). The simulation of the rotational structure of the 0 0 band in Fig. 4共a兲 using Eq. 共7兲 only takes into account this leading component. The agreement between the experimental and calculated spectra indicates that this approximation is adequate and that the photoelectron must represent a superposition of s and d partial waves. As discussed in the previous subsection, the odd v ⫹ 4 torsional bands with ⌫ ⫹ v ⫽A u gain their intensity by intensity borrowing from the Ã ⫹ state which arises by ionization from the 1b 3g molecular orbital. The 1b 3g orbital correlates with a 3d ␲ y orbital in a semi-united atom basis. The simulation in Fig. 4共b兲 only takes into account contributions from this main angular momentum component (ᐉ ⬙ , ␭ ⬙ )⫽(2,⫾1). The overall agreement between experimental and simulated rotational structures is very good for both bands. Only on the high-wave-number side can several weak lines be found in the experimental spectrum that are not accounted for by the simulation, e.g., the 1 11→4 23 transition of the 0 0 band or the 2 02→6 16 transition of the 4 1 band. These small contributions to the spectrum can be explained by higher angular momentum components in the single-center expansions of the respective orbitals. Within our model for the rotational transition intensities, the marked differences between the rotational structures of




the 0 0 and 4 1 bands can be rationalized by the different angular momentum transfer ᐉ ⬙ 共which corresponds to the electronic angular momentum ‘‘hole’’ left behind in the molecular core after photoionization兲 which leads to different values of the 3 j-symbol in Eq. 共7兲 for ᐉ ⬙ ⫽1 (0 0 band兲 and ᐉ ⬙ ⫽2 (4 1 band兲. This difference manifests itself in the fact that the spectrum of the 0 0 band is dominated by transitions with 兩 ⌬N 兩 ⫽0,1 whereas transitions with 兩 ⌬N 兩 ⫽0,2 account for the most intense lines in the spectrum of the 4 1 band. Thus, the different rotational structures of the bands are caused by a propensity rule which originates from the different angular momentum character of the orbitals from which the photoelectron is ejected. In other words, the vibronic coupling by which the odd torsional bands gain intensity is also ‘‘echoed’’ in their rotational structure, which, in turn, can serve as a means to identify the electronic state to which coupling occurs. The orbital approximation indeed implies that the rotational structure of the 4 1 band should be identical ˜ ⫹ transition, if small to that of the 0 0 band of the ˜X →A differences in the rotational constants are disregarded. Similarly, the rotational structure of the 7 1 band 关see upper trace of Fig. 4共c兲兴 could be simulated by assuming an intensity borrowing mechanism with the ˜B ⫹ state which arises by ionization from the 3a g molecular orbital. The 3a g molecular orbital correlates with a 3d ␴ orbital in a semiunited atom basis 共see Fig. 1兲 and its dominant angular momentum component (ᐉ ⬙ ,␭ ⬙ ) is 共2,0兲. The rotational structure of the 7 1 band could indeed be well reproduced assuming a dominant angular momentum component (ᐉ ⬙ ,␭ ⬙ )⫽(2,0) with a small admixture of 共0,0兲 character, as is illustrated by the simulation in the bottom trace of Fig. 4共c兲. The band consists of transitions with ⌬N⫽2 and ⌬K a ⫽0, in line with the dominant electronic angular momentum character (ᐉ ⬙ ,␭ ⬙ )⫽(2,0) of the orbital from which ionization occurs. This example again nicely illustrates how vibronic coupling imposes a propensity rule on the rotational transition intensities, and how the dominant angular momentum components can be extracted using the orbital approximation. The differences in the rotational structure of the bands with an even and odd number of torsional quanta are only clearly revealed in the very high resolution spectra in Fig. 4. Unambiguous rotational assignment would not have been possible at the 0.7 cm⫺1 resolution of the overview spectra. This nicely demonstrates the usefulness, and indeed the necessity of high-resolution techniques for recording PFIZEKE photoelectron spectra of larger polyatomic molecules. D. Large amplitude torsional motion

A striking property of the vibrational structure of the ˜X ⫹ state, also observed by Pollard et al.,15 is that the torsional term values are insensitive, at the low energies investigated here, to the excitation of the ␯ 2 and ␯ 3 modes 共see Tables II and III兲. This observation is all the more surprising as the C–C stretching and CH2 bending modes would have been expected, on the basis of the behavior of the ground electronic state of ethylene40,52 to be the modes most strongly coupled to the torsional motion. The apparent independence of the torsional motion from the ␯ 2 and ␯ 3 modes suggests

1771

that the torsional motion is separable, at low excitation energies, from these modes and can be treated in a first, rather coarse approximation by a one-dimensional torsional model, without explicitly considering anharmonic couplings. Such couplings must be considered in molecules such as H2 O2 , 53 H2 S2 , 54 HSOH,55 and their isotopomers that exhibit a similar torsional motion and also display chiral equilibrium structures. The torsional motion is assumed to take place in an effective double-minimum potential of the form V 共 ␪ 兲 ⫽2k ␪ 2 ⫹ ⑀ 共 exp兵 ⫺4 ␣␪ 2 其 ⫺1 兲

共14兲

proposed in Refs. 15 and 18. The angle ␪ is defined as half the torsional angle ␪ ⫽ ␶ /2 and k, ⑀, and ␣ are adjustable parameters. The potential given by Eq. 共14兲 corresponds to the sum of a harmonic potential and a Gaussian hump centered at the planar configuration. Its suitability to describe the torsional motion was tested by fitting the potential parameters to a set of 11 ab initio points in the interval 0° ⭐ ␶ ⭐60°. At each point, the Born–Oppenheimer potential energy was obtained after fully optimizing the D 2 geometry at the CCSD共T兲/cc-pVTZ level of theory. Although this calculation does not contain the zero-point-energy correction to the effective torsional potential, it is expected to capture the main features of the potential function. The three potential parameters could be refined so that the maximum deviation to the ab initio potential points did not exceed 1 cm⫺1 . A similar agreement was also obtained by fitting the potential parameters to a set of ab initio points calculated by solely varying the torsional angle and leaving all other internal coordinates fixed at the values corresponding to the fully optimized equilibrium geometry of the ˜X ⫹ state. Given that the maximum deviation between the two sets of potential points amounts to less than 60 cm⫺1 in the range ⫺40°⭐ ␶ ⭐40°, it appears that 共1兲 the torsional coordinate is only weakly coupled to other modes as was already suggested for the ␯ 2 and ␯ 3 modes by the experimental results 共see above兲 and 共2兲 the effective coordinate in the 1D model corresponds in first approximation to the torsional angle at low excitation energies. However, the torsional angle at the potential minimum and the height of the barrier at the planar geometry are not expected to exactly match the values obtained from a fit to experimental data because the zero point energy corrections, even if they are small, are likely to be of the same order of magnitude as the barrier in the Born–Oppenheimer potential. Indeed, the barrier height represents less than 5% of the zero point energy. The kinetic energy operator T( ␪ ) for the onedimensional torsional motion was derived using the method of Meyer and Gu¨nthard,56 T 共 ␪ 兲 ⫽⫺

d2 1 h , 8 ␲ 2 c 4m Hr⬜2 d ␪ 2

共15兲

where h represents Planck’s constant, c the speed of light, m H the mass of the hydrogen atom, and r⬜ the perpendicular distance of the H-atoms from the C–C-axis. The factor h/32␲ 2 c m Hr⬜2 corresponds to the rotational constant A ⫹ in cm⫺1 . The torsional Hamiltonian thus becomes


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TABLE V. Results of the analysis of the experimental positions of the ⫹ torsional levels of C2 H⫹ 4 and C2 D4 . The parameters k, ␣, and ⑀ of the torsional model potential 关Eq. 共14兲兴 were optimized to reproduce the experimental results. rms represents the root-mean-square error of the fit. The torsional angle ␶ min at the potential minimum and the height of the barrier at the planar geometry H B were determined from the model potential. Numbers in parentheses represent one standard deviation. C2 H⫹ 4 This work Expt. 共Ref. 15兲

C2 D⫹ 4 This work

k/cm⫺1 deg⫺2 ⑀ /cm⫺1 ␣ /10⫺4 cm⫺1 deg⫺2 rms

3.2270共232兲 4816.7共589兲 5.2293共436兲 0.365

4.687 7667 4.103

3.4800共285兲 5176.5共745兲 4.9183共463兲 0.506

H B /cm⫺1 ␶ min /deg

357共26兲 29.2共5兲

270共150兲 27共2兲

292共24兲 27.8共5兲

H 共 ␪ 兲 ⫽⫺A ⫹

d2 ⫹2k ␪ 2 ⫹ ⑀ 共 exp兵 ⫺4 ␣␪ 2 其 ⫺1 兲 . d␪2

共16兲

The corresponding Schro¨dinger equation was solved numerically using the discrete variable representation method of Meyer.57,58 The calculations were performed on an equidistant grid of 201 points in the interval ⫺45°⭐ ␪ ⭐⫹45°. At this grid resolution, the first ten torsional eigenvalues were converged to better than 10⫺4 cm⫺1 . The potential parameters were fitted to the vibrational term energies ˜␯ 0 共see Tables II and III兲. The A ⫹ rotational

FIG. 5. Torsional potentials and lowest five torsional eigenfunctions of the ⫹ ˜X ⫹ ground state of C2 H⫹ 4 共a兲 and C2 D4 共b兲 determined by a least-squares fit of the parameters of the model potential 关Eq. 共14兲兴 to the experimental positions of the observed torsional bands. The origin of the amplitude scale of each eigenfunction was placed at its corresponding eigenvalue. The full 共dashed兲 lines represent vibrational states of A g (A u ) symmetry. In both isotopomers, the first two torsional levels lie below the potential maximum at a torsional angle ␶ ⫽0° and constitute a tunneling pair. The barrier height H B and torsional angle at the potential minimum ␶ min amount to ⫺1 and 27.8(5)° for 357(26) cm⫺1 and 29.2(5)° for C2 H⫹ 4 and 292(24) cm ⫹ C2 D4 , respectively.

constant for the ground vibronic state obtained from the fit described in Sec. V C was used in Eq. 共16兲 because its value remains constant for all observed torsional bands within the limit of our experimental accuracy. Using a value for A ⫹ corresponding to the ab initio geometry32 did not change the results of the fit significantly. The 4 5 band was omitted from the fit because it is subject to a vibrational perturbation in both isotopomers 共see Sec. V A兲. The potential parameters obtained from the fits are listed in Table V and the calculated torsional term values are compared to the experimental results in Tables II and III. Figures 5共a兲 and 5共b兲 show the fitted torsional potentials and the first five torsional eigen⫹ functions for C2 H⫹ 4 and C2 D4 , respectively. The origin of the amplitude scale of each wave function was placed at its corresponding eigenvalue. Table V also gives the torsional angle ␶ min where the effective potential function is minimal and the value of the potential barrier at the planar geometry H B ⫽V( ␶ ⫽0°)⫺V( ␶ min). The three lowest eigenvalues were found to react very sensitively to small changes in the barrier height and width. In particular, it was not possible to reproduce the eigenvalue spectrum for C2 D⫹ 4 with the potential parameters determined for C2 H⫹ 4 . This result demonstrates the importance of the contribution of the zero-point vibrational motion to the effective 1D potential 共see also discussion above兲. From Fig. 5 it can be seen that in both isotopomers the 0 0 and 4 1 states lie below the potential maximum, effectively constituting a tunneling pair with a tunneling splitting of 83.7(5) cm⫺1 for C2 H⫹ and 4 37.1(5) cm⫺1 for C2 D⫹ 4 . The fitted torsional potentials of ⫹ C2 H⫹ 4 and C2 D4 exhibit effective barriers of 357(26) and ⫺1 292(24) cm , respectively. Table VI compares previous experimental and theoretical estimates for the barrier height H B and the torsional angle ␶ min in C2 H⫹ 4 with a short description of the method employed. Our values for ␶ min and H B are in agreement with previous estimates from classical photoelectron spectroscopic investigations. However, the resolution in these studies was two to three orders of magnitude lower than to the one achieved in the present work and did not allow one to observe the tunneling splitting which constitutes a very sensitive probe of the potential in the barrier region. The other experimental estimate for the minimum twisting angle stems from an electron paramagnetic resonance 共EPR兲 study of the ethylene cation in a SF6 matrix.19 A value between 8 and 23 degrees was suggested. The discrepancy between this study and our results might originate from the inaccuracy of the semiempirical calculation or from the influence of the matrix, both of which is reflected in the large error boundary of ␶ min given in Ref. 19. A similar theoretical evaluation of an EPR study at the SDCI level of theory yielded a minimum torsional angle of 28°. 59 The barrier heights calculated ab initio all lie significantly lower than our experimental results and are comparable to the value derived in our own ab initio calculation. It must however be emphasized that these barrier heights cannot be compared directly with the experimental results because none of the ab initio calculations considered the role of the zero point energy vibrational motion. For the same reason, the vibrational term values calculated directly from a




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TABLE VI. Comparison of the torsional angle ␶ min at the potential minimum and the height of the barrier at the planar geometry H B determined in the present study for the electronic ground state of C2 H⫹ 4 with the results of previous investigations. The upper part of the table lists the results of experimental studies that implicitly contain the zero-point-energy contribution to the effective torsional potential. The lower part lists the results of ab initio calculations that do not consider the zero-point-energy vibrational motion. Abbreviations: PES ⫽photoelectron spectroscopy, EPR⫽electron paramagnetic resonance. Ref.

␶ min /deg

H B /cm⫺1

20 15 18 59 19 This work

25 27 28 28 8-23 29.2

235 270 280

27 28 29 30 31 32

20.1 14 33 20.2 28.4 21.0

62 21

357

104 116

Method 共year兲 Classical PES 共1978兲 Classical PES 共1984兲 Re-evaluation of classical PES 共1985兲 Calculation of EPR hyperfine coupling constants 共1990兲 Calculation of EPR hyperfine coupling constants 共1996兲 PFI-ZEKE photoelectron spectroscopy MP2/3-21G, CEPA-1/6-311G(d f ,p) 共1984兲 SDCI with quadruples correction 共1991兲 BP/DZP 共1993兲 QCISD共T兲/6-311G(d,p) 共1998兲 B3LYP/6-311G(d,p) 共2001兲 CCSD共T兲/cc-pVTZ and extrapolation 共2002兲

purely electronic one-dimensional potential in Ref. 32 differ markedly from our results 共see last column of Table II兲. To make better contact with experiment, future ab initio calculations will need at least to include zero point energy corrections. Also needed would be specific studies of the importance of anharmonic couplings and of the exact tunneling coordinate. Using the wave functions in Fig. 5, an estimate of the relative vibronic transition moments was determined by calculating the contribution of the vibrational overlap integrals involving the torsional mode to the factor q 2v in Eq. 共12兲. For the torsional bands of A g 共and A u ) upper state vibrational symmetry, the vibrational integrals 兩具 4 i⫹ 兩 0 0 ⬙ 典兩 2 共and 兩具 4 i⫹ 兩 ␶ 兩 0 0 ⬙ 典兩 2 ) were computed with the lower state wave function determined using the model torsional potential of Wallace40 for neutral ethylene. To enable a comparison with the experimental results, both the calculated 共represented as squares in Figs. 2 and 3兲 and the integrated experimental 共indicated as diamonds in Figs. 2 and 3兲 band intensities were normalized to the experimental value of the 4 2 (4 3 ) band for the bands of A g (A u ) upper state vibrational symmetry. The good agreement between calculated and measured relative torsional band intensities supports the validity of our 1D model. Moreover, it shows that the intensities of the even torsional bands are well described within the Franck–Condon approximation, and that the intensities of the odd torsional bands can well be reproduced by the model for the intensity borrowing outlined in Sec. V B. VI. SUMMARY AND CONCLUSIONS

The ground electronic state of the ethylene radical cat⫹ ions C2 H⫹ 4 and C2 D4 was studied by PFI-ZEKE photoelectron spectroscopy. The improvement in resolution by two to three orders of magnitude compared to previous classical photoelectron spectroscopic studies enabled the complete resolution of the rotational structure and the observation of several hitherto unreported vibronic bands, including several members of the progression in the torsional mode. The new

data allowed for a more accurate characterization of the large amplitude motion of the ethylene cation along the torsional coordinate; in particular, the first two torsional states were found to be situated below the barrier maximum at the planar geometry, effectively constituting a tunneling pair with a tun⫺1 neling splitting of 83.7 cm⫺1 in C2 H⫹ in 4 and of 37.1 cm ⫹ C2 D4 . The zero point energy contribution to the effective torsional potential was found to be substantial, and its neglect is likely to account for the underestimation of the barrier height by all previous ab initio studies. The large amplitude torsional motion is accompanied by configuration interaction between the ˜X ⫹ and Ã ⫹ states which leads to the observation of ‘‘forbidden’’ torsional bands that gain intensity by vibronic coupling. Using a Herzberg–Teller-type model for this vibronic interaction and a quantitative model for rotationally resolved photoionization cross sections, it could be demonstrated that vibronic coupling has a profound influence on the rotational structure of the bands. Interpreting the angular momentum transfer accompanying the photoionization as the angular momentum of the orbital hole created upon ionization enabled, via photoionization selection and propensity rules, the identification of the dominant configuration interactions and thus the ionic electronic states to which vibronic coupling occurs. The conclusions drawn here from the detailed study of the photoionization dynamics of ethylene can be extended to a wider range of molecular systems: The loss of rigidity upon ionization out of a double bond can result in a substantial deviation from planarity and the associated configurational mixing. The configurational mixing leads, in turn, to the observation of ‘‘forbidden’’ vibrational bands with characteristic rotational structures. The occurrence of forbidden bands in the photoelectron spectra thus represents a general dynamical pattern in the photoionization of polyatomic molecules with CvC double bonds as has already been recognized by Ko¨ppel et al.21–23 A complete and consistent picture of the photoionization dynamics in these molecules can, however, only be gained from high-resolution photoelectron


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spectroscopic studies in which not only the vibrational structure but also the rotational structure is observed. ACKNOWLEDGMENTS

We thank Dr. M. Willeke 共ETH Zu¨rich兲 for useful discussions. This work is supported financially by the ETH Zu¨rich and the Swiss National Science Foundation under Project No. 200020-100050. R. S. Mulliken, J. Chem. Phys. 66, 2448 共1977兲. K. B. Wiberg, C. M. Hadad, J. B. Foresman, and W. A. Chupka, J. Phys. Chem. 96, 10756 共1992兲. 3 R. P. Krawczyk, A. Viel, U. Manthe, and W. Domcke, J. Chem. Phys. 119, 1397 共2003兲. 4 R. S. Mulliken and C. C. J. Roothaan, Chem. Rev. 41, 219 共1947兲. 5 R. S. Mulliken, Tetrahedron 5, 253 共1959兲. 6 A. J. Merer and R. S. Mulliken, Chem. Rev. 69, 639 共1969兲. 7 A. J. Merer and L. Schoonveld, Can. J. Phys. 47, 1371 共1969兲. 8 A. D. Baker, C. Baker, C. R. Brundle, and D. W. Turner, Int. J. Mass Spectrom. Ion Phys. 1, 285 共1968兲. 9 J. H. D. Eland, Int. J. Mass Spectrom. Ion Phys. 2, 471 共1969兲. 10 G. R. Branton, D. C. Frost, T. Makita, C. A. McDowell, and I. A. Stenhouse, J. Chem. Phys. 52, 802 共1970兲. 11 C. R. Brundle and D. B. Brown, Spectrochim. Acta, Part A 27A, 2491 共1971兲. 12 R. Stockbauer and M. G. Inghram, J. Electron Spectrosc. Relat. Phenom. 7, 492 共1975兲. 13 T. Cvitasˇ, H. Gu¨sten, and L. Klasinc, J. Chem. Phys. 70, 57 共1979兲. 14 P. M. Dehmer and J. L. Dehmer, J. Chem. Phys. 70, 4574 共1979兲. 15 J. E. Pollard, D. J. Trevor, J. E. Reutt, Y. T. Lee, and D. A. Shirley, J. Chem. Phys. 81, 5302 共1984兲. 16 L. Wang, J. E. Pollard, Y. T. Lee, and D. A. Shirley, J. Chem. Phys. 86, 3216 共1987兲. 17 D. M. P. Holland, D. A. Shaw, M. A. Hayes, L. G. Shpinkova, E. E. Rennie, L. Karlsson, P. Baltzer, and B. Wannberg, Chem. Phys. 219, 91 共1997兲. 18 F. T. Chau, J. Mol. Struct. 131, 383 共1985兲. 19 K. Toriyama and M. Okazaki, Appl. Magn. Reson. 11, 47 共1996兲. 20 H. Ko¨ppel, W. Domcke, L. S. Cederbaum, and W. von Niessen, J. Chem. Phys. 69, 4252 共1978兲. 21 H. Ko¨ppel, L. S. Cederbaum, and W. Domcke, J. Chem. Phys. 77, 2014 共1982兲. 22 H. Ko¨ppel, L. S. Cederbaum, W. Domcke, and S. S. Shaik, Angew. Chem., Int. Ed. Engl. 22, 210 共1983兲. 23 H. Ko¨ppel, L. S. Cederbaum, and W. Domcke, Chem. Phys. Lett. 110, 469 共1984兲. 24 C. Sannen, G. Ras¸eev, C. Galloy, G. Fauville, and J. C. Lorquet, J. Chem. Phys. 74, 2402 共1981兲. 25 W. A. Lathan, W. J. Hehre, and J. A. Pople, J. Am. Chem. Soc. 93, 808 共1971兲. 26 W. R. Rodwell, M. F. Guest, D. T. Clark, and D. Shuttleworth, Chem. Phys. Lett. 45, 50 共1977兲. 1 2

Willitsch, Hollenstein, and Merkt 27

N. C. Handy, R. H. Nobes, and H.-J. Werner, Chem. Phys. Lett. 110, 459 共1984兲. 28 K. Takeshita, J. Chem. Phys. 95, 1838 共1991兲. 29 L. A. Eriksson, S. Lunell, and R. J. Boyd, J. Am. Chem. Soc. 115, 6896 共1993兲. 30 N. Salhi-Benachenhou, B. Engels, M.-B. Huang, and S. Lunell, Chem. Phys. 236, 53 共1998兲. 31 Y.-J. Liu and M.-B. Huang, THEOCHEM 536, 133 共2001兲. 32 M. L. Abrams, E. F. Valeev, C. D. Sherrill, and T. D. Crawford, J. Phys. Chem. A 106, 2671 共2002兲. 33 U. Hollenstein, R. Seiler, H. Schmutz, M. Andrist, and F. Merkt, J. Chem. Phys. 115, 5461 共2001兲. 34 U. Hollenstein, R. Seiler, A. Osterwalder, M. Sommavilla, A. Wuëst, P. Rupper, S. Willitsch, G. M. Greetham, B. Brupacher-Gatehouse, and F. Merkt, Chimia 55, 759 共2001兲. 35 F. Merkt, A. Osterwalder, R. Seiler, R. Signorell, H. Palm, H. Schmutz, and R. Gunzinger, J. Phys. B 31, 1705 共1998兲. 36 U. Hollenstein, H. Palm, and F. Merkt, Rev. Sci. Instrum. 71, 4023 共2000兲. 37 G. Reiser, W. Habenicht, K. Mu¨ller-Dethlefs, and E. W. Schlag, Chem. Phys. Lett. 152, 119 共1988兲. 38 F. M. Phelps, M.I.T. Wavelength Tables 共MIT Press, Cambridge, MA, 1982兲, Vol. 2. 39 U. Hollenstein, R. Seiler, and F. Merkt, J. Phys. B 36, 893 共2003兲. 40 R. Wallace, Chem. Phys. Lett. 159, 35 共1989兲. 41 P. R. Bunker and P. Jensen, Molecular Symmetry and Spectroscopy, 2nd ed. 共NRC Research Press, Ottawa, 1998兲. 42 R. Signorell and F. Merkt, Mol. Phys. 92, 793 共1997兲. 43 A. D. Buckingham, B. J. Orr, and J. M. Sichel, Philos. Trans. R. Soc. London, Ser. A 268, 147 共1970兲. 44 K. Wang and V. McKoy, High Resolution Laser Photoionization and Photoelectron Studies 共Wiley, Chichester, 1995兲, pp. 281–329. 45 G. Herzberg, Molecular Spectra and Molecular Structure, Volume III, Electronic Spectra and Electronic Structure of Polyatomic Molecules 共Krieger, Malabar, 1991兲. 46 F. Merkt and T. P. Softley, Int. Rev. Phys. Chem. 12, 205 共1993兲. 47 R. Georges, M. Bach, and M. Herman, Mol. Phys. 97, 279 共1999兲. 48 E. Rusinek, H. Fichoux, M. Khelkhal, F. Herlemont, J. Legrand, and A. Fayt, J. Mol. Spectrosc. 189, 64 共1998兲. 49 A.-K. Mose, F. Hegelund, and F. M. Nicolaisen, J. Mol. Spectrosc. 137, 286 共1989兲. 50 G. Herzberg and E. Teller, Z. Phys. Chem. Abt. B 21, 410 共1933兲. 51 T. L. Tan, K. L. Goh, P. P. Ong, and H. H. Teo, Chem. Phys. Lett. 315, 82 共1999兲. 52 R. Wallace and J. P. Leroy, Chem. Phys. 144, 371 共1990兲. 53 B. Fehrensen, D. Luckhaus, and M. Quack, Chem. Phys. Lett. 300, 312 共1999兲. 54 M. Gottselig, D. Luckhaus, M. Quack, J. Stohner, and M. Willeke, Helv. Chim. Acta 84, 1846 共2001兲. 55 M. Quack and M. Willeke, Helv. Chim. Acta 86, 1641 共2003兲. 56 R. Meyer and H. H. Gu¨nthard, J. Chem. Phys. 49, 1510 共1968兲. 57 R. Meyer, J. Chem. Phys. 52, 2053 共1970兲. 58 R. Meyer, J. Mol. Spectrosc. 76, 266 共1979兲. 59 S. Lunell and M.-B. Huang, Chem. Phys. Lett. 168, 63 共1990兲.
