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Oct 1, 2004 - u←X4 g forbidden transitions. The parameters of a model Hamiltonian were fit to the bands and their corresponding forbidden transitions. Line.
JOURNAL OF CHEMICAL PHYSICS

VOLUME 121, NUMBER 13

1 OCTOBER 2004

High-resolution spectroscopy of the 2 2 ⌸ u ] X 4 ⌺ À g forbidden ¿ transitions of C2 Christopher G. Tarsitano, Christopher F. Neese, and Takeshi Okaa) Department of Chemistry, University of Chicago, Chicago, Illinois 60637; Department of Astronomy and Astrophysics, University of Chicago, Chicago, Illinois 60637; and The Enrico Fermi Institute, University of Chicago, Chicago, Illinois 60637

共Received 25 May 2004; accepted 9 July 2004兲 4 ⫺ The electronic absorption spectrum of the 共0,2兲, 共1,3兲, and 共6,9兲 bands of the B 4 ⌺ ⫺ u ⫺X ⌺ g system ⫹ of C2 was obtained using the velocity modulation technique in conjunction with heterodyne detection. The rotationally resolved spectrum shows perturbations, which are attributed to the 2 2 ⌸ u 2 state. The mixing between the B 4 ⌺ ⫺ u state and the 2 ⌸ u state for nearly degenerate levels generated 2 enough intensity borrowing to observe twenty 2 ⌸ u ←X 4 ⌺ ⫺ g forbidden transitions. The parameters of a model Hamiltonian were fit to the bands and their corresponding forbidden transitions. Line position measurements, line strength factors, and expectation values for the orbital angular momentum 具 ⌳ ⬘ 典 for the forbidden transitions are reported. Molecular parameters from the global fit of each band, including their corresponding forbidden transitions, are reported. © 2004 American Institute of Physics. 关DOI: 10.1063/1.1787493兴

jarane, Carre´, and Larzillie`re15 who analyzed the effect of the perturbation on the levels of the perturbed B 4 ⌺ ⫺ u state. In this paper, we present the direct observations of the ⫹ 2 2 ⌸ u ←X 4 ⌺ ⫺ g forbidden transitions of C2 . From a rotational analysis of the perturbations in the 共0,2兲, 共1,3兲, and 共6,9兲 bands of the B⫺X system of C⫹ 2 , we were able to predict forbidden transitions from the X 4 ⌺ ⫺ g state to the 2 2 ⌸ u state and their relative intensities. Twenty forbidden lines were measured and included in a least-squares fit of each band. In addition to the observed forbidden lines, we predict others whose intensities lied just below our detection limit or were overlapped with other lines. The expectation value of the orbital angular momentum 具 ⌳ ⬘ 典 was experimentally determined and presented to characterize the amount of ⌸ character in the perturbing levels.

I. INTRODUCTION

The molecular ion C⫹ 2 is conjectured to be an important species in the extraterrestrial medium. Using a satellite-born mass spectrometer, C⫹ 2 has been detected in the comet Halley1 and proposed to be present in the comet Giacobini-Zinner.2 This ion is predicted to play a role in the chemistry of the interstellar medium.3–5 In 1972, Meinel obtained a rotationally resolved spec2 trum at 249 nm and assigned it to a 2 ⌺ ⫺ g ⫺ ⌸ u transition of ⫹ 6 C2 . While the spectroscopic evidence suggests that the species is C⫹ 2 , several theoretical treatments disagree with this assignment.7,8 Later, O’Keefe and co-workers obtained a low using translational energy resolution spectrum of C⫹ 2 spectroscopy.9 Using the calculations of Petrongolo et al.,7 they were able to assign several electronic transitions. This work was followed by the definitive work of Maier and co-workers who obtained the matrix spectrum of C⫹ 2 and later obtained many bands of the B⫺X system in the gas phase.10–13 In their high-resolution gas-phase work, Maier and co-workers observed perturbations in the quartet splitting of the spectrum. While their experiment had the resolution to do a rotational analysis, the resolution was not sufficient for a detailed analysis of the spin-spin interaction or the perturbation. However, Maier and co-workers did predict that the perturbation was the result of a spin–orbit interaction with the 2 2 ⌸ u state, so named by Petrongolo et al.7 since the richness of electronic states of C⫹ 2 defies the use of ordinary alphabetic nomenclature. Maier’s prediction was based on the calculations of Petrongolo et al., which showed that the 2 2 ⌸ u state lies ⬇0.5 eV below the minimum of the B 4 ⌺ ⫺ u state.7,12 This prediction was validated in two spectroscopic analyses that followed by Zackrisson and Royen14 and Boud-

II. EXPERIMENT

The experimental setup is shown schematically in Fig. 1. We employed a double-modulation technique, velocity modulation with heterodyne detection, to improve the sensitivity of the spectrometer. The first technique, velocity modulation, developed by Gudeman, Saykally et al., provides the ion/neutral discrimination needed in the analysis of the spectrum.16,17 The second technique, heterodyne detection, developed by Bjorklund, permits the laser radiation to be modulated at radio/microwave frequencies and thus further improves the sensitivity of the spectrometer.18 The doublemodulation technique reduces the problem of residual amplitude modulation, which degrades the sensitivity enhancement of heterodyne detection alone. In this technique the spectral lines associated with ions have a line shape resembling a second-derivative Gaussian, whereas the transitions from neutral species have a line shape resembling a first-

a兲

Electronic mail: [email protected]

0021-9606/2004/121(13)/6290/8/$22.00

6290

© 2004 American Institute of Physics

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Spectroscopy of C2⫹

J. Chem. Phys., Vol. 121, No. 13, 1 October 2004

6291

FIG. 1. A schematic diagram of the absorption spectrometer using velocity modulation with heterodyne detection. The radiation from an Ar⫹ -pumped dye laser passes through a polarizer 共P兲, an electro-optic phase modulator 共EOM兲, and another polarizer before traversing the plasma cell and impinging upon a fast photoreceiver 共Det兲. The signal from the detector is demodulated by a double balanced mixer 共DBM兲, whose local oscillator source is the signal generator driving the EOM passed through a phase shifter 共␾兲. A pair of phase sensitive detectors 共PSDs兲 then demodulates this signal at 1 f and 2 f , where f is the discharge frequency of the plasma. Demodulation produces the characteristic second-derivative line shape for ions at 1 f and first-derivative line shape for ions and neutrals at 2 f . Shown on the computer screen is the R共9兲 quartet of the 共1,3兲 band from the B⫺X system of C⫹ 2 . Beamsplitters before and after the EOM send diagnostic beams through an I2 cell and spectrum analyzer, respectively. The spectrum analyzer is used to monitor the depth of the modulation which is observed on an oscilloscope. The absorption from the I2 cell is recorded along with the experimental spectrum and is compared against the wavelength meter to ensure proper wavelength meter behavior.

derivative Gaussian. A variant of this technique was developed by Wang et al., who also use the magnetic rotation effect.19 The radiation source consisted of a Coherent 899-29 Autoscan ring laser that was pumped by a Coherent Innova 200 Ar⫹ laser. The dye, Rhodamine 6G, provided an output power of more than several hundred milliwatts in the region of 17 085–17 450 cm⫺1 with a linewidth of 500 kHz. Data collection and control of the laser was accomplished via Coherent autoscan software. The absolute uncertainty in the frequency determined by the wavelength meter of the Coherent ring laser was 0.007 cm⫺1. An iodine spectrum was recorded to ensure proper wavelength meter behavior.20,21 The laser output was passed through a polarizer and a MgO:LiNbO3 electro-optical phase modulator 共EOM兲 driven at 166 MHz. An etalon was used to monitor the depth of the modulation. By monitoring the signal strength of a C⫹ 2 transition and adjusting the rf voltage across the MgO:LiNbO3 crystal, the optimal modulation index was experimentally determined to be ⬇2.3 or a carrier-to-sideband ratio, 0th:1st:2nd:3rd:4th⬇0.05:1.0:0.58:0.13:0.02. Then, the sidebanded beam was passed through the discharge cell and the absorption signal was detected using a New Focus fast photoreceiver. The ac signal from the detector was first mixed down using a double balance mixer and then demodulated at 1 f and 2 f using a pair of Stanford Research System SR510 lock-in amplifiers. The use of 1 f and 2 f detection helps further discriminate the signals of ionic and neutral molecules. The laser was step-scanned in 25 MHz increments. For each step, 500 data points per channel were acquired and averaged. The time constant of the lock-in amplifiers and the scanning rate of the laser were optimized by repeatedly scanning over a C⫹ 2 transition and monitoring the spectral line so that there were no distortions to the line shape or shifts in the rest frequency. It was determined that the optimal settings for the scanning rate and time constant were 167 MHz/s and 300 ms, respectively. For the weaker transitions, an average of 15 000 data points per step were used with a scan rate and time constant of 17 MHz/s and 3 s, respectively.

The C⫹ 2 molecular ions were produced in a water-cooled positive column discharge. In brief, the reagent gases 共5.1 Torr He and 10 mTorr of CO兲 flowed into a discharge cell through 18 inlets and pumped out by a mechanical pump through 9 outlets. The bore of the discharge had a length of 1.3 m and an inner diameter of 12 mm. The plasma was produced by applying a potential of several kilovolts at 8 kHz across the electrodes. The resulting discharge current was ⬇125 mA rms. Carbon monoxide was used as a reagent gas in preference to C2 H2 because C2 H2 produced C⫺ 2 effiinterfered ciently and the bands of the B⫺X system of C⫺ 2 with the spectrum of C⫹ 2 . However, the 共1,2兲 band of the A⫺X system of CO⫹ was detected within the 共1,3兲 band of C⫹ 2 , and C2 H2 was used to differentiate the species.

III. THEORY A. Effective Hamiltonian and matrix elements

An effective Hamiltonian is derived from the fundamental molecular Hamiltonian via an electronic contact transformation followed by a vibrational contact transformation. We use the derivation of Brown et al.22 except that we allow the 2 2 ⌸ u and B 4 ⌺ ⫺ u states to remain coupled through the electronic contact transformation. Thus, in addition to the effective Hamiltonians describing the 2 2 ⌸ u and B 4 ⌺ ⫺ u states, there is an effective Hamiltonian describing the interaction of the two states. The derivation of the effective Hamiltonian for the unperturbed X 4 ⌺ ⫺ g state follows the derivation of Brown et al. exactly. The Hamiltonians for the 4 ⌺ states, H ⌺ , and the 2 ⌸ state, H ⌸ , are given in Eqs. 共1兲 and 共2兲, respectively,

H ⌺ ⫽T ⌺ ⫹B ⌺ N2 ⫺D ⌺ N4 ⫹ ␥ ⌺ N•S⫹ ␥ ⌺D N2 N•S 2 ⫹ ␭ ⌺ 共 3S z2 ⫺S2 兲 , 3

共1兲

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J. Chem. Phys., Vol. 121, No. 13, 1 October 2004

Tarsitano, Neese, and Oka

H ⌸ ⫽T ⌸ ⫹B ⌸ N2 ⫺D ⌸ N4 ⫹A ⌸ L z S z

兩 n 2S⫹1 ⌳ ⍀ v J ef 典 ⫽

⫹ A ⌸D 共 L z S z N ⫹N L z S z 兲 1 2

2

2

1 &

兩 v 典 共 兩 n⌳ 典 兩 S⌺ 典 兩 J⍀ 典

⫿ 共 ⫺1 兲 s 兩 n ⫺⌳ 典 兩 S ⫺⌺ 典 兩 J ⫺⍀ 典 ).

2 2 ⫹ 12 p ⌸ 共 ⌳ ⫹ S ⫺ N ⫺ ⫹⌳ ⫺ S ⫹N ⫹ 兲 2 2 2 2 ⫺ 12 q ⌸ 共 ⌳ ⫹ N ⫺ ⫹⌳ ⫺ N⫹兲.

共2兲

The above Hamiltonians are written in terms of N⫽J⫺S instead of R⫽J⫺S⫺L. While R appears in the fundamental molecular Hamiltonian, N does not contain terms coupling different electronic states and is more appropriate for the 2 are defined effective Hamiltonians. The ladder operators ⌳ ⫾ 22 by Brown et al. The interaction Hamiltonian H ⬘ for the coupling between the 2 2 ⌸ u and B 4 ⌺ ⫺ u states is not readily written in operator notation, but the matrix elements of H ⬘ are derived in Sec. III B. Table I lists the matrix elements of the effective Hamiltonians. Several papers give these matrix elements in differing detail.14,15,23 All matrix elements are evaluated using a Wang-transformed Hund’s case a basis,

共3兲

In Eq. 共3兲, n is an index specifying the particular electronic state, ef is the rotationless parity,24 and s⫽1 for ⌺ ⫺ states and s⫽0 for all other states. The off-diagonal matrix elements are evaluated using the phase conventions defined by

具 S⌺ 兩 ␴ xz 兩 S⫺⌺ 典 ⫽ 共 ⫺1 兲 S⫺⌺ , 具 ⌳ 兩 ␴ xz 兩 ⫺⌳ 典 ⫽ 共 ⫺1 兲

⫺⌳

共4a兲 共4b兲

,

where ␴ xz is the symmetry operator corresponding to reflection in the xz plane of the molecule. B. Effective interaction Hamiltonian

The term in the fundamental molecular Hamiltonian responsible for coupling the 2 2 ⌸ u and B 4 ⌺ ⫺ u states is the spin–orbit term, H SO . The matrix elements of the interaction Hamiltonian H ⬘ , prior to the vibrational contact transformation, are given to the second order by

⫺ ⫺ ;r 典 ⫽ 具 2 2 ⌸ u⍀ ;r 兩 H SO 兩 B 4 ⌺ u⍀ ;r 典 具 2 2 ⌸ u⍀ ;r 兩 H ⬘ 兩 B 4 ⌺ u⍀



兺n

⫺ ;r 典 关 12 V ⌺ 共 r 兲 ⫹ 21 V ⌸ 共 r 兲 ⫺V n 共 r 兲兴 具 2 2 ⌸ u⍀ ;r 兩 H SO 兩 n;r 典具 n;r 兩 H SO 兩 B 4 ⌺ u⍀

关 V ⌺ 共 r 兲 ⫺V n 共 r 兲兴关 V ⌸ 共 r 兲 ⫺V n 共 r 兲兴

.

共5兲

TABLE I. Matrix elements for the 2 ⌸ and 4 ⌺ ⫺ states of a diatomic molecule (x⫽J⫹1/2). f f 4 ⫺ 2 4 2 2 具 n 4⌺ ⫺ 3/2v J e 兩 H 兩 n ⌺ 3/2v J e 典 ⫽T ⌺ ⫹2␭ ⌺ ⫹B ⌺ (x ⫺1)⫺D ⌺ (x ⫹x ⫺2)⫺ 2 ␥ ⌺ ⫺3 ␥ ⌺D (x ⫺1)

3

f f 4 ⫺ 2 4 2 3 具 n 4⌺ ⫺ 1/2v J e 兩 H 兩 n ⌺ 1/2v J e 典 ⫽T ⌺ ⫺2␭ ⌺ ⫹B ⌺ (x ⫾2x⫹3)⫺D ⌺ 关 (x ⫹13x ⫹6)⫾(4x ⫹12x) 兴

7

⫺ ␥ ⌺ (⫾x⫹ 2 )⫺ ␥ ⌺D 关 7x 2 ⫹9⫾(x 3 ⫹10x) 兴



f f 4 ⫺ 2 具n4⌺⫺ 3/2v J e 兩 H 兩 n ⌺ 1/2v J e 典 ⫽⫺ B ⌺ ⫺2D ⌺ 共 x ⫾x⫹1 兲 ⫺

冉 冉

冊 冊 册

␥ ⌺ ␥ ⌺D 2 ⫺ 共 x ⫾2x⫹6 兲 2 2

册冑

3x 2 ⫺3

A⌸ A ⌸D ⫹ B ⌸⫹ 共 x 2 ⫺1 兲 ⫺D ⌸ 共 x 4 ⫺x 2 兲 2 2 p⌸ A⌸ A ⌸D ⫹ B ⌸⫺ ⫹q ⌸ x 共 x 2 ⫹1 兲 ⫺D ⌸ 共 x 4 ⫹3x 2 兲 ⫿ 具 n 2 ⌸ 1/2v J ef 兩 H 兩 n 2 ⌸ 1/2v J ef 典 ⫽T ⌸ ⫺ 2 2 2 q⌸ 具 n 2 ⌸ 3/2v J ef 兩 H 兩 n 2 ⌸ 1/2v J ef 典 ⫽⫺ B ⌸ ⫺2D ⌸ x 2 ⫿ x 冑x 2 ⫺1 2 1 1 f f 4 ⫺ 2 具 n ⬘ ⌺ 3/2v ⬘ J e 兩 H 兩 n ⌸ 3/2v J e 典 ⫽⫺ ␰ 3/2⫺ ␰ D 共 x 2 ⫺1 兲 2 2

具 n 2 ⌸ 3/2v J ef 兩 H 兩 n 2 ⌸ 3/2v J ef 典 ⫽T ⌸ ⫹



f f 2 具 n ⬘4⌺ ⫺ 1/2v ⬘ J e 兩 H 兩 n ⌸ 1/2v J e 典 ⫽⫺

1 2)

␰ 1/2⫺

1 2)





␰ D 共 x 2 ⫾x⫹2 兲

1 2

f f 2 ␰ D 冑x 2 ⫺1 具 n ⬘4⌺ ⫺ 3/2v ⬘ J e 兩 H 兩 n ⌸ 1/2v J e 典 ⫽ f f 2 具 n ⬘4⌺ ⫺ 1/2v ⬘ J e 兩 H 兩 n ⌸ 3/2v J e 典 ⫽

1 )

␰ D 冑x 2 ⫺1

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Spectroscopy of C2⫹

J. Chem. Phys., Vol. 121, No. 13, 1 October 2004

具 S ⬘ ,⌺ ⬘ ,⌳ ⬘ ,⍀ 兩 H SO 兩 S,⌺,⌳,⍀ 典 ⫽ 共 ⫺1 兲 S ⬘ ⫺⌺ ⬘



S⬘

1

S

⌺⬘

⫺⌬⌺

⫺⌺



6293

具 S ⬘ ,⌳ ⬘ 储 H SO 储 S,⌳ 典 , 共7兲

where ⌬⍀⫽0 and ⌬⌳⫽⫺⌬⌺. There is an additional symmetry to the reduced matrix elements given by Eq. 共7兲, due to parity,

具 S ⬘ ,⌳ ⬘ ⫽⫺1 储 H SO 储 S,⌳⫽0 ⫾ 典 ⫽⫾ 具 S ⬘ ,⌳ ⬘ ⫽1 储 H SO 储 S,⌳⫽0 ⫾ 典 .

共8兲

This symmetry is needed to Wang transform the basis set. Using Eqs. 共5兲–共8兲, the interaction between the B 4 ⌺ ⫺ u and 2 2 ⌸ u states after the electronic contact transformation is ⫺ ;r 典 ⫽ 具 2 2 ⌸ u3/2 ;r 兩 H ⬘ 兩 B 4 ⌺ u3/2

⫺ ;r 典 ⫽ 具 2 2 ⌸ u1/2 ;r 兩 H ⬘ 兩 B 4 ⌺ u1/2

FIG. 2. A flowchart describing the analysis of a band in the spectrum. An iterative approach was used to fit the perturbations in the bands. From this fit, the line positions and relative intensities of forbidden transitions were predicted. The forbidden transitions were included in a final least-squares fit of each band and molecular parameters were obtained.

In Eq. 共5兲, n indexes all electronic states except for the 2 2 ⌸ u ⫺ 30 and B 4 ⌺ ⫺ The 兩 B 4 ⌺ u⍀ ;r 典 , 兩 2 2 ⌸ u⍀ ;r 典 , and 兩 n;r 典 u states. are electronic state vectors solving the electronic Schro¨dinger equation as a function of the internuclear distance r, and the V ⌺ (r), V ⌸ (r), V n (r) are the corresponding eigenvalues, i.e., ⫺ ⫺ ;r 典 ⫽V ⌺ 共 r 兲 兩 B 4 ⌺ u⍀ ;r 典 , H elec 兩 B 4 ⌺ u⍀

共6a兲

H elec 兩 2 2 ⌸ u⍀ ;r 典 ⫽V ⌸ 共 r 兲 兩 2 2 ⌸ u⍀ ;r 典 ,

共6b兲

H elec 兩 n;r 典 ⫽V n 共 r 兲 兩 n;r 典 .

共6c兲

The Wigner-Eckart theorem leads to a general expression for the spin–orbit matrix elements appearing in Eq. 共5兲.25,26

⫺ ␰ 3/2共 r 兲 , 2 ⫺ ␰ 1/2共 r 兲 2)

.

共9a兲

共9b兲

Equations 共9a兲 and 共9b兲 define the molecular parameters ␰ 3/2(r) and ␰ 1/2(r). It is evident from Eq. 共5兲 that ␰ 3/2(r) and ␰ 1/2(r) are each the sum of a first-order and second-order contribution. The first-order contribution is the reduced matrix element, 具 S ⬘ ,⌳ ⬘ 储 H SO 储 S,⌳ 典 , from Eq. 共7兲 and is the same for both ␰ 3/2(r) and ␰ 1/2(r). We expect the first-order contribution to be more significant than the second-order contribution; therefore, ␰ 3/2(r)⬇ ␰ 1/2(r). The vibrational contact transformation of the terms defined above is straightforward. After the second-order vibrational contact transformation, there are three molecular constants describing the perturbation, ␰ 3/2 , ␰ 1/2 , and ␰ D . The centrifugal distortion terms are easily derived by taking the anticommutator of the matrix defined by Eqs. 共9a兲 and 共9b兲 and the matrix of the N2 operator. In deriving the centrifugal distortion terms, the second-order contribution to ␰ 3/2(r) and ␰ 1/2(r) is ignored. The matrix elements of H ⬘ after the vibrational contact transformation are presented in Table I.

C. Line strength factors and expectation values

In addition to the transition energies calculated from the eigenvalues of the above Hamiltonian, there are several important properties that are calculated from the eigenvectors that aid in understanding the experimental spectrum. These properties include the rotational line strength factors, which are used to understand intensity borrowing in the forbidden transitions, and the expectation value of ⌳ ⬘ , which is an 2 index of the amount of mixing between the B 4 ⌺ ⫺ u and 2 ⌸ u states.

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N

17 097.6699共6兲a 17 093.0493共⫺1兲 17 089.7164共2兲 17 087.6490共8兲 17 086.8416共⫺2兲 17 087.2953共⫺2兲 17 089.0193* 17 091.9564* 17 096.1786共⫺14兲 17 101.5824共⫺1兲 17 107.8448共13兲 17 117.8680共9兲 17 126.4783共⫺17兲 17 136.9378共⫺16兲 17 148.7548共⫺12兲 P

R13(1) Q12(1)

P

1 3 5 7 9 11 13 15 17 19 21 23

17 301.5590共⫺9兲 17 296.9925共⫺23兲 17 293.7274共⫺6兲 17 291.7341共39兲 17 290.9967共⫺2兲 17 291.5283共11兲 17 293.3213共⫺1兲 17 296.3690* 17 300.6990共⫺3兲 17 306.2760共5兲 17 313.0486共⫺18兲 17 321.4792共⫺14兲 P

R13(1)

a

17 172.2362* 17 169.6979共4兲 17 168.5089* 17 168.0324共14兲 17 170.7198共⫺10兲 17 173.7212共5兲 17 178.1774共23兲 17 184.0434共⫺12兲 17 191.3156共⫺17兲 17 199.9888共⫺5兲 17 210.0604共35兲

P3

P4 共0,2兲 Band of the B

R1 4

4 ⫺ ⌺⫺ u ⫺X ⌺ g

system of 17 106.8722共11兲 17 114.5686共17兲 17 123.5312共12兲 17 133.7493共5兲 17 145.2182共4兲 17 157.9436*b 17 171.8732* 17 187.0619共⫺10兲 17 203.4159共5兲 17 220.6076共38兲 17 241.5309共13兲 17 261.0153共⫺20兲 17 282.3168* 17 304.9512共6兲

17 093.2651共5兲 17 089.9258共5兲 17 087.8476共⫺3兲 17 087.0256共⫺8兲 17 087.4531共⫺7兲 17 089.1121共⫺12兲 17 091.9564* 17 095.3789* 17 102.9960共25兲 17 109.0225共⫺17兲 17 117.6524共⫺12兲 17 126.6688共5兲 17 137.1779* 17 149.0106*

17 093.1380共10兲 17 089.8538共1兲 17 087.7941共⫺8兲 17 086.9803共1兲 17 087.4026共⫺21兲 17 089.0193* 17 092.3241共⫺17兲 17 095.3789* 17 103.3065共37兲 17 109.4102共6兲 17 117.3121共⫺1兲 17 126.5772共4兲 17 137.1779* 17 149.0106*

17 092.9128共2兲 17 089.6262共⫺3兲 17 087.5629共6兲 17 086.7332共⫺13兲 17 087.1112共⫺11兲 17 088.8109共10兲 17 094.5402共⫺12兲 17 096.7834共4兲 17 102.1298共11兲 17 108.8620共⫺5兲 17 116.8843* 17 126.2159共1兲 17 136.8115共6兲 17 148.6820共13兲

17 097.5651共6兲 17 097.7995共16兲

R

⌬J⫽⌬N Transitions R 17 106.6912共0兲 Q21(1)

Q43(1)

17 297.2619共⫺1兲 17 293.9941共⫺4兲 17 291.9965共7兲 17 291.2605共⫺6兲 17 291.7911共20兲 17 293.5657* 17 296.6162* 17 300.9181* 17 305.6037共⫺25兲 17 313.4870共⫺14兲 17 321.7253共8兲

17 297.1038共8兲 17 293.9096共⫺2兲 17 291.9377共8兲 17 291.2153共⫺3兲 17 291.7354* 17 293.5657* 17 296.6162* 17 300.9181* 17 305.8323共⫺2兲 17 313.5663共3兲 17 321.6363共⫺3兲

17 301.4600共⫺1兲

P

17 172.4048* 17 169.3861共9兲 17 169.0215共21兲 17 168.6266共6兲 17 170.9303共7兲 17 173.9506共⫺11兲 17 178.4063共⫺11兲 17 184.2730共⫺13兲 17 191.5444共⫺16兲 17 200.2184共⫺1兲 17 210.2896共13兲

Q12(1)

R2

R3

R4

17 107.0861共2兲 17 114.7764共5兲 17 123.7293共⫺4兲 17 133.9325共⫺9兲 17 145.3741共⫺21兲 17 158.0368共⫺17兲 17 171.8732* 17 186.2609* 17 204.8277共14兲 17 221.7777* 17 241.3148共⫺13兲 17 261.2058共2兲 17 282.5758* 17 305.1936*

17 106.7875共8兲 17 114.6798共0兲 17 123.6656共⫺8兲 17 133.8795共⫺19兲 17 145.3230共⫺4兲 17 157.9436* 17 172.2351共⫺17兲 17 186.2609* 17 205.1370共26兲 17 222.1707共18兲 17 240.9741共2兲 17 261.1148共13兲 17 282.5238* 17 305.1936*

17 114.4521共⫺10兲 17 123.4338共⫺1兲 17 133.6341共⫺16兲 17 145.0308共⫺1兲 17 157.7331共6兲 17 174.4515共⫺8兲 17 187.6629共⫺15兲 17 203.9596共⫺8兲 17 221.6214共⫺4兲 17 240.5582共⫺13兲 17 260.7523共⫺2兲 17 282.1915共⫺10兲 17 304.8747共⫺1兲

17 106.9579共7兲

R

17 310.9076共1兲 17 318.5287共⫺13兲 17 327.4224共⫺3兲 17 337.5720共⫺11兲 17 348.9770共8兲 17 361.6114* 17 375.5151* 17 390.6437* 17 406.1419共9兲 17 424.8089共⫺3兲 17 443.8018共⫺43兲

17 310.5863共12兲 17 318.4195共⫺23兲 17 327.3545共6兲 17 337.5217共⫺3兲 17 348.9358共7兲 17 361.6114* 17 375.5151* 17 390.6437* 17 406.3672共9兲 17 424.8860共1兲 17 443.7182共9兲

17 310.4347共⫺40兲

R

17 184.9560共⫺18兲 17 191.9535共17兲 17 201.6106共23兲 17 211.2311共6兲 17 223.5396共⫺9兲 17 236.5590共17兲 17 250.9936共⫺2兲 17 266.8258共5兲 17 284.0432共1兲 17 302.6390共⫺17兲 17 322.6137共10兲 17 343.9545共10兲

17 184.7088共6兲 17 191.8214共⫺42兲 17 201.6551共12兲 17 211.7789共⫺7兲 17 223.4094共2兲 17 236.4534共⫺5兲 17 250.8984共16兲 17 266.7290共⫺15兲 17 283.9510共16兲 17 302.5460共⫺26兲 17 322.5228共⫺5兲 17 343.8672共⫺10兲

C⫹ 2

⫹ 4 ⫺ 共1,3兲 Band of the B 4 ⌺ ⫺ u ⫺X ⌺ g system of C2 17 310.6406共⫺2兲 17 296.8344共5兲 17 318.2653共17兲 17 293.6431共10兲 17 327.1592共22兲 17 291.6677共⫺15兲 17 337.3116共28兲 17 290.9478共2兲 17 348.7159共15兲 17 291.4812共⫺9兲 17 361.3696共⫺15兲 17 293.2704共2兲 17 375.2770共12兲 17 296.3690* 17 390.4260共8兲 17 301.2517共⫺6兲 17 406.8121共18兲 17 306.3717共⫺11兲 17 424.3711共⫺1兲 17 313.2112共24兲 17 443.5622共1兲 17 321.3350共31兲

⌬J⫽⌬N Transitions R 17 301.6839共14兲 Q43(1)

⫹ 4 ⫺ 共6,9兲 Band of the B 4 ⌺ ⫺ u ⫺X ⌺ g system of C2 17 184.8022共⫺19兲 17 172.2362* 17 192.2569共⫺6兲 17 169.2668* 17 169.7823共⫺8兲 17 201.0809共29兲 17 169.0673共43兲 17 168.5089* 17 210.6300共⫺1兲 17 169.1729共9兲 17 168.7719共⫺6兲 17 223.3256共⫺17兲 17 170.7948共⫺4兲 17 170.4199共3兲 17 236.3237共6兲 17 173.8448共⫺10兲 17 173.4760共⫺7兲 17 250.7638共43兲 17 178.3100共10兲 17 177.9397共⫺16兲 17 266.5952共1兲 17 184.1779共⫺15兲 17 183.8124共5兲 17 283.8133共⫺22兲 17 191.4537共⫺1兲 17 191.0867共0兲 17 302.4136共⫺11兲 17 200.1283共⫺13兲 17 199.7647共7兲 17 322.3858共⫺7兲 17 210.2033共⫺7兲 17 209.8537* 17 343.7310*

Q32(1)

Q21(1)

The number in parenthesis denotes observed⫺calculated line position expressed in terms of the last digits. The asterisk denotes observed lines that were excluded from the fit as a result of overlapping.

b

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17 107.0210共8兲

17 318.1536共⫺11兲 17 327.0834共⫺30兲 17 337.2554共13兲 17 348.6653共⫺5兲 17 361.3151共⫺22兲 17 375.2375共⫺12兲 17 390.9757共⫺11兲 17 406.9051共⫺14兲 17 424.5294共7兲 17 443.4143共17兲

17 310.7862共13兲

17 192.3493共⫺43兲 17 201.1398共5兲 17 211.3822共⫺35兲 17 223.0391共8兲 17 236.0875共⫺10兲 17 250.5337共21兲 17 266.3665共25兲 17 283.5850共33兲 17 302.1797共⫺9兲 17 322.1573共11兲 17 343.5036共0兲

Tarsitano, Neese, and Oka

1 3 5 7 9 11 13 15 17 19 21 23

P2

J. Chem. Phys., Vol. 121, No. 13, 1 October 2004

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

P1

6294

⫹ 4 ⫺ TABLE II. Wave numbers of the observed lines in the B 4 ⌺ ⫺ u ⫺X ⌺ g system of C2 .

Spectroscopy of C2⫹

J. Chem. Phys., Vol. 121, No. 13, 1 October 2004

6295

4 ⫺ FIG. 3. A portion of the spectrum of the 共6,9兲 band of the B 4 ⌺ ⫺ u ←X ⌺ g transition is shown. The R共7兲 and P共9兲 共not shown兲 quartets are perturbed by both 2 the ⍀⫽3/2 and ⍀⫽1/2 spin–orbit states of the 2 ⌸ u state. In the B 4 ⌺ ⫺ u state, the F1 component of the N⫽8 level is perturbed by the J⫽9.5 level of the 2 2 ⌸ 1/2u spin–orbit state. The mixing of these two states allows for the QP11(9) and SR11(7) transitions. Similarly, the F2 component is mixing with the J ⫽8.5 level of the 2 2 ⌸ 3/2u state giving rise to the QP22(9) and SR22(7) transitions. The P共11兲 quartet happens to appear near the QP11(9) and QP22(9) spectral lines.

The eigenvectors are written as ⫺ ⫺ 兩 ⌿ ⬙ 典 ⫽ 具 X 4 ⌺ g3/2 J ⬙ ef 兩 ⌿ ⬙ 典 兩 X 4 ⌺ g3/2 J ⬙ ef 典 ⫺ ⫺ ⫹ 具 X 4 ⌺ g1/2 J ⬙ ef 兩 ⌿ ⬙ 典 兩 X 4 ⌺ g1/2 J ⬙ ef 典 ,

共10兲

⫺ ⫺ 兩 ⌿ ⬘ 典 ⫽ 具 B 4 ⌺ u3/2 J ⬘ ef 兩 ⌿ ⬘ 典 兩 B 4 ⌺ u3/2 J ⬘ ef 典

⫹具B

4

具 ⌳ ⬘ 典 ⫽ 兩 具 2 2 ⌸ u3/2J ⬘ ef 兩 ⌿ ⬘ 典 兩 2 ⫹ 兩 具 2 2 ⌸ u1/2J ⬘ ef 兩 ⌿ ⬘ 典 兩 2 .

⫺ ⫺ ⌺ u1/2 J ⬘ ef 兩 ⌿ ⬘ 典 兩 B 4 ⌺ u1/2 J ⬘ ef 典

⫹ 具 2 2 ⌸ u3/2J ⬘ ef 兩 ⌿ ⬘ 典 兩 2 2 ⌸ u3/2J ⬘ ef 典 共11兲

The scalar products appearing in Eqs. 共10兲 and 共11兲, e.g., ⫺ J ⬙ ef 兩 ⌿ ⬙ 典 , are the numerical elements of the eigen具 X 4 ⌺ g3/2 vectors calculated by the fitting routine. R The rotational line strength factors, S J ⬘ J ⬙ are calculated as

冋冉

J⬘

1

3 2

0

J⬙ ⫺

3 2



⫺ ⫺ ⫻具 ⌿ ⬘ 兩 B 4 ⌺ u3/2 J ⬘ ef 典具 X 4 ⌺ g3/2 J ⬙ ef 兩 ⌿ ⬙ 典





J⬘

1

1 2

0

J⬙ ⫺



⫺ 1 具 ⌿ ⬘ 兩 B 4 ⌺ u1/2 J ⬘ ef 典 2

⫺ ⫻具 X 4 ⌺ g1/2 J ⬙ ef 兩 ⌿ ⬙ 典



2

,

共13兲

IV. RESULTS AND DISCUSSION

⫹ 具 2 2 ⌸ u1/2J ⬘ ef 兩 ⌿ ⬘ 典 兩 2 2 ⌸ u1/2J ⬘ ef 典 .

R S J ⬘ J ⬙ ⫽ 共 2J ⬘ ⫹1 兲共 2J ⬙ ⫹1 兲

in the derivation. The intensity of a transition is the product R of S J ⬘ J ⬙ , an oscillator strength factor, a Franck-Condon factor, and a ground-state Boltzmann factor. The expectation value of ⌳ ⬘ is given by

共12兲

where 兩 ⌿ ⬙ 典 and 兩 ⌿ ⬘ 典 are the ground state and excited state eigenvectors, respectively. Equation 共12兲 is straightforward to derive 共see Zare, Sec. 6.5兲,27 although the half-integral electronic spin requires careful attention to the phase factors

4 À A. The B 4 ⌺ À u À X ⌺ g system

Determination of the line positions of the 2 2 ⌸ u ←X 4 ⌺ ⫺ g forbidden transitions required the analysis of the vibronic bands of the B⫺X system of C⫹ 2 . The analysis is represented as a flowchart in Fig. 2. The analysis of the Doppler-limited absorption spectrum of the 共0,2兲, 共1,3兲, and 共6,9兲 bands of the B⫺X system was straightforward. Figure 3 shows a portion of the spectrum. The absorption lines were fit to second-derivative Gaussian line shapes. Overlapped lines were fit simultaneously to a pair of second-derivative Gaussian line shapes. When the resulting rest frequencies of a pair of lines differed by less than 0.02 cm⫺1, the absorptions were excluded from the data set because the lines were not adequately resolved. The regions outside the perturbed portion of each band were assigned using the relative intensities from Kova´cs.28,29 Then these lines were fit in an initial least-squares fit using the Hamiltonian described in Sec. III and the matrix elements presented in Table I. An iterative approach was used to assign the perturbed portion of the spectrum. The iterative approach consisted of starting outside the perturbed region and gradually working toward the center of the perturbation by adding two to four lines at a time to the initial leastsquares fit. Analysis of the 共6,9兲 band was more difficult due to the perturbation occurring at low N. As a result, analysis

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6296

J. Chem. Phys., Vol. 121, No. 13, 1 October 2004

Tarsitano, Neese, and Oka

⫹ 4 ⫺ 2 TABLE III. Molecular parameters 共in cm⫺1兲 for the B 4 ⌺ ⫺ u , X ⌺ g , and 2 ⌸ u states of C2 .

共0,2兲 Band X 4⌺ ⫺ g

B 4⌺ ⫺ u

2 2⌸ u

v ⫽2 1.380 482共12兲a 6.369共11兲 ¯ ⫺6.470共37兲 ¯ v ⫽0 1.537 738共12兲 6.479共12兲 ⫺3.117共89兲 6.973共40兲

Bu D v ⫻106 ␥ v ⫻104 ␭ v ⫻102 ␥ D v ⫻106 Bv D v ⫻106 ␥ v ⫻104 ␭ v ⫻102

Bv D v ⫻105 Av A D v ⫻102 p v ⫻102 q v ⫻103 Coupling constants ␰ 3/2 ␰ 1/2 ␰ D ⫻104 Band origins 4 ⫺ T v ⬘ v ⬙ (B 4 ⌺ ⫺ u ⫺X ⌺ g ) 2 4 ⫺ T v ⬘ v ⬙ (2 ⌸ u ⫺X ⌺ g ) rms error

1.181 78共28兲 ⫺3.378共46兲 12.707共65兲 ⫺0.922共26兲 ⫺0.155共40兲 ⫺0.366共34兲 8.8672共88兲 9.9617共75兲 3.78共20兲 17 100.584 44共34兲 17 204.350共39兲 0.001 54

共1,3兲 Band v ⫽3 1.362 996共17兲 6.448共22兲 ¯ ⫺6.166共42兲 ¯ v ⫽1 1.520 179共17兲 6.554共24兲 ⫺1.10共14兲 7.570共45兲

1.108 18共21兲 ¯ 9.87共45兲 1.59共13兲 ¯ ¯ 4.0017共58兲 4.550共13兲 ¯ 17 304.331 24共34兲 17 447.014共78兲 0.001 58

共6,9兲 Band v ⫽9 1.253 992共17兲 5.882共23兲 ⫺36.0共19兲 ⫺2.612共94兲 3.955共99兲 v ⫽6 1.429 539共17兲 6.834共23兲 13.0共18兲 12.167共95兲

1.0178共15兲 ⫺21.1共19兲 6.720共23兲 ⫺4.23共12兲 ⫺3.830共89兲 ⫺3.6共11兲 3.1670共51兲 2.970共10兲 ¯ 17 178.903 74共56兲 17 188.767共17兲 0.001 87

a

The number in the parenthesis denotes one standard deviation expressed in terms of the last digits.

of this band started at higher values of N and iteratively approached lower values of N. In the preliminary fits of the unperturbed B⫺X lines, it ⬙ v , p v , q v , D ⌸ v , and ␰ D v to zero was necessary to fix ␥ ⬙v , ␥ D for the 共0,2兲 and 共1,3兲 bands. For the 共6,9兲 band, it was per⬙ v to get a reasonable fit. As tinent to include both ␥ ⬙v and ␥ D more lines of the perturbed region were included in the fits, the excluded parameters mentioned above were added as necessary. In the least-squares fits of the 共0,2兲 and 共1,3兲 bands, ␥ ⬘v and ␥ ⬙v cannot be determined independently; because of the selection rules for a 4 ⌺⫺ 4 ⌺ system, ␥ ⬘v and ␥ ⬙v are highly correlated. The consequence in setting ␥ ⬙v ⫽0 is that ␥ ⬘v becomes an effective constant approximately equal to ␥ ⬘v ⫺ ␥ ⬙v . The entire data set for the B⫺X system includes the following: 94 lines for the 共0,2兲 band with a maximum J of 30.5, of which five lines were transitions where ⌬J⫽⌬N; 77 lines for the 共1,3兲 band with a maximum J of 24.5, of which four lines were transitions where ⌬J⫽⌬N; and 82 lines for the 共6,9兲 band with a maximum J of 24.5. This data set is tabulated in Table II. While spectral lines with higher J values were observed, they were not used in the fit since their F 1 and F 4 components and F 2 and F 3 components were overlapped and difficult to deconvolute. B. The 2 2 ⌸ u À X 4 ⌺ À g forbidden transitions

From the fit of each band, the forbidden transitions and their relative intensities were predicted. In most cases the forbidden transitions were predicted to be within 0.05 cm⫺1 of the actual line position measurement. Each band was refit with its corresponding forbidden transitions; any previously

excluded parameters mentioned above were added as necessary. In the least-squares fits: seven forbidden transitions were added to the 共0,2兲 band, four forbidden transitions were added to the 共1,3兲 band, and nine forbidden transitions were added to the 共6,9兲 band. The molecular parameters from the least-squares fits are presented in Table III. The assignments of the forbidden transitions, their freR quencies, and line strength factors, S J ⬘ J ⬙ , are presented in Table IV. In addition, Table IV includes the calculated relative intensities for a rotational temperature of 400 K. The intensities presented are relative to the strongest transition in each band, which is the R 1 (9) transition in all three bands. The experimentally determined expectation value for the orbital angular momentum 具 ⌳ ⬘ 典 has been included to characterize the amount of ⌸ character in the perturbing state. Unfortunately, many of the frequencies of the forbidden transitions could not be measured as a result of overlapping with other transitions. In this case, their predicted frequencies and uncertainties at one standard deviation are presented in Table IV. Additionally, Table IV presents predicted forbidden transitions, whose intensities were below our detection limit, along with their calculated uncertainties at one standard deviation. If the interaction was only between the B 4 ⌺ ⫺ u and the 2 2 ⌸ u states, then the spin–orbit interaction parameters, ␰ 1/2 and ␰ 3/2 , would be equal as described in Sec. III B. That is, the ratio, ␰ 1/2 / ␰ 3/2 , would be equal to one. A reasonable least-squares fit of the data could not be obtained with the constraint ␰ 1/2⫽ ␰ 3/2 . From the parameters in Table III, the ratio, ␰ 1/2 / ␰ 3/2 , for the 共0,2兲, 共1,3兲, and 共6,9兲 bands is 1.1234共14兲, 1.1370共36兲, and 0.9378共35兲, respectively; there-

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Spectroscopy of C2⫹

J. Chem. Phys., Vol. 121, No. 13, 1 October 2004 ⫹ TABLE IV. The 2 2 ⌸ u ←X 4 ⌺ ⫺ g forbidden transitions of C2 .

Assign Q

R24(13) O P24(15) Q R13(13) O P13(15) R R12(15) R R23(15) P P12(17) P P23(17) R R23(17) R R12(17) S R11(21) S R11(19) P P23(19) P P12(19) Q P11(21) R

R12(17) P12(19) R R23(17) Q R24(15) P P23(19) O P24(17) P R14(13) N P14(15) Q R13(15) S R11(19) S R11(21) O P13(17) Q P11(21) Q P11(23) S R22(21) P

S

R11(7) P11(9) S R22(7) Q P22(9) P P12(5) R R12(3) P P23(5) R R23(3) O P24(3) Q Q23(1) P P23(7) P P12(7) R R23(5) R R12(5) Q P11(7) Q

Wave number/cm⫺1

Int.

2 2 ⌸ u ←X 4 ⌺ ⫺ g ( v ⫽2) 17 167.7237共19兲a 0.245 17 087.8108共31兲*b 0.199 17 168.8353共101兲* 0.090 17 088.9243共101兲* 0.073 17 198.2497共30兲 0.068 17 199.0662共28兲* 0.067 17 107.3648共10兲 0.052 17 108.1804共⫺44兲 0.052 17 189.8328共29兲* 0.037 17 189.5082共⫺41兲 0.033 17 229.4345共295兲* 0.029 17 234.6391共⫺5兲 0.028 17 088.0048共37兲 0.027 17 087.6794共29兲* 0.024 17 121.8793共32兲* 0.020 2 2 ⌸ u ←X 4 ⌺ ⫺ g ( v ⫽3) 17 409.5644共5兲 0.159 17 309.0293共1兲 0.117 17 411.3166共1兲 0.081 17 386.5186共1兲 0.073 17 310.7828共33兲* 0.060 17 296.7939共33兲* 0.056 17 371.0942共1651兲* 0.007 17 292.1993共1650兲* 0.005 17 385.3575共864兲* 0.005 17 440.9091共987兲* 0.004 17 428.9072共1732兲* 0.004 17 295.6330共863兲* 0.004 17 329.5883共987兲* 0.003 17 306.8257共1733兲* 0.002 17 431.5674共164兲* 0.002 2 2 ⌸ u ←X 4 ⌺ ⫺ g ( v ⫽9) 17 212.9053共⫺5兲 0.379 17 170.3064共⫺3兲 0.360 17 212.7390共⫺19兲 0.353 17 170.1375共11兲 0.340 17 172.0806共⫺5兲 0.134 17 194.6468共⫺9兲 0.115 17 172.6149共39兲 0.100 17 195.1719共⫺28兲 0.079 17 173.1290共6兲 0.064 17 185.6695共40兲* 0.037 17 161.4639共443兲* 0.021 17 161.5097共382兲* 0.020 17 194.0548共442兲* 0.019 17 194.0985共382兲* 0.019 17 176.4301共599兲* 0.017

S J⬘J⬙

具 ⌳ ⬘典

4.42 4.77 1.63 1.75 1.63 1.61 1.73 1.71 1.22 1.10 2.07 1.36 1.29 1.16 1.42

0.644 0.644 0.877 0.877 0.901 0.895 0.901 0.895 0.930 0.940 0.912 0.936 0.930 0.940 0.936

5.24 5.53 2.67 1.73 2.82 1.85 0.12 0.13 0.11 0.18 0.25 0.12 0.19 0.27 0.16

0.715 0.715 0.847 0.880 0.847 0.880 0.990 0.990 0.993 0.991 0.989 0.993 0.991 0.989 0.993

3.66 4.06 3.41 3.83 1.15 0.91 0.86 0.63 0.51 0.28 0.20 0.19 0.17 0.16 0.17

0.611 0.611 0.588 0.588 0.776 0.776 0.793 0.793 0.580 0.580 0.968 0.974 0.968 0.974 0.980

R

a

The number in parenthesis denotes observed⫺calculated line position expressed in terms of the last digits. b The asterisk denotes predicted transitions with one standard deviation uncertainty expressed in terms of the last digits.

fore the effect of the second-order term in Eq. 共5兲 is experimentally observable. V. CONCLUSION

The high-resolution absorption spectrum of the 共0,2兲, ⫹ 4 ⫺ 共1,3兲, and 共6,9兲 bands of the B 4 ⌺ ⫺ u ←X ⌺ g transition of C2 in the gas phase was obtained using velocity modulation in conjunction with heterodyne detection. The spectrum shows

6297

perturbations which are attributed to an interaction between 2 the B 4 ⌺ ⫺ u and the 2 ⌸ u states. The mixing between the 4 ⫺ 2 B ⌺ u state and the 2 ⌸ u state for nearly degenerate levels generated enough intensity borrowing to observe twenty 2 2 ⌸ u ←X 4 ⌺ ⫺ g forbidden transitions. The forbidden transitions have been included in a global least-squares fit of each band to obtain improved molecular parameters. ACKNOWLEDGMENTS

The authors wish to thank Dr. K. Freed for the insightful discussion of C⫹ 2 . We would also like to thank Dr. D. Hullah ´ da´mkovics for their earlier work on heterodyne and Ma´te´ A detection. This work was supported by NSF Grant. No. PHY0099442. D. Krankowsky, P. La¨mmerzahl, I. Herrwerth et al., Nature 共London兲 321, 326 共1986兲. 2 M. A. Coplan, K. W. Ogilvie, M. F. A’Hearn, P. Bochsler, and J. Geiss, J. Geophys. Res. 92, 39 共1987兲. 3 E. F. van Dishoeck and J. H. Black, Astrophys. J., Suppl. Ser. 62, 109 共1986兲. 4 T. Hasegawa, K. Volk, and S. Kwok, Astrophys. J. 532, 994 共2000兲. 5 T. Oka, J. A. Thorburn, B. J. McCall, S. D. Friedman, L. M. Hobbs, P. Sonnentrucker, D. E. Welty, and D. G. York, Astrophys. J. 582, 823 共2003兲. 6 H. Meinel, Can. J. Phys. 50, 158 共1972兲. 7 C. Petrongolo, P. J. Bruna, S. D. Peyerimhoff, and R. J. Buenker, J. Chem. Phys. 74, 4594 共1981兲. 8 P. Rosmus, H.-J. Werner, E.-A. Reinsch, and M. Larsson, J. Electron Spectrosc. Relat. Phenom. 41, 289 共1986兲. 9 A. O’Keefe, R. Derai, and M. T. Bowers, Chem. Phys. 91, 161 共1984兲. 10 D. Forney, H. Althaus, and J. P. Maier, J. Phys. Chem. 91, 6458 共1987兲. 11 M. Ro¨sslein, M. Wyttenbach, and J. P. Maier, J. Chem. Phys. 87, 6770 共1988兲. 12 J. P. Maier and M. Ro¨sslein, J. Chem. Phys. 88, 4614 共1988兲. 13 F. G. Celii and J. P. Maier, Chem. Phys. Lett. 166, 517 共1990兲. 14 M. Zackrisson and P. Royen, J. Mol. Spectrosc. 161, 1 共1993兲. 15 K. Boudjarane, M. Carre´, and M. Larzillie`re, Chem. Phys. Lett. 243, 571 共1995兲. 16 C. S. Gudeman, M. H. Begemann, J. Pfaff, and R. J. Saykally, Phys. Rev. Lett. 50, 727 共1983兲. 17 C. S. Gudeman and R. J. Saykally, Annu. Rev. Phys. Chem. 35, 387 共1984兲. 18 G. C. Bjorklund, Opt. Lett. 5, 15 共1980兲. 19 R. Wang, Y. Chen, P. Cai, J. Lu, Z. Bi, X. Yang, and L. Ma, Chem. Phys. Lett. 307, 339 共1999兲. 20 S. Gerstenkorn and P. Luc, Atlas du Spectroscopie d’absorption de la Molecule d’iode 共CNRS, Paris, 1978兲. 21 S. Gerstenkorn and P. Luc, Rev. Phys. Appl. 14, 791 共1979兲. 22 J. M. Brown, E. A. Colbourn, J. K. G. Watson, and F. D. Wayne, J. Mol. Spectrosc. 74, 294 共1979兲. 23 K. Kawaguchi and T. Amano, J. Chem. Phys. 88, 4584 共1988兲. 24 J. M. Brown, J. T. Hougen, K. P. Huber et al., J. Mol. Spectrosc. 55, 500 共1975兲. 25 K. F. Freed, J. Chem. Phys. 45, 4214 共1966兲. 26 H. Lefebvre-Brion and R. W. Field, Perturbations in the Spectra of Diatomic Molecules 共Academic, Orlando, 1986兲. 27 R. N. Zare, Angular Momentum 共Wiley, New York, 1988兲. 28 I. Kova´cs, Rotational Structure in the Spectra of Diatomic Molecules 共Adam Hilger Ltd., London, 1969兲. 29 E. E. Whiting, J. A. Paterson, I. Kova´cs, and R. W. Nicholls, J. Mol. Spectrosc. 47, 84 共1973兲. 30 Because the interaction between the 2 2 ⌸ u and B 4 ⌺ ⫺ u states is explicit throughout the electronic contact transformation. All of the sums relating the constants in H ⌺ and H ⌸ to the fundamental molecular Hamiltonian are over all electronic states except for the 2 2 ⌸ u and B 4 ⌺ ⫺ u states as well. This is the only modification to the relations given by Brown et al. 1

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