Energy level structure and potential energy curve of the C0u+ state of Ne2 by high-resolution VUV laser spectroscopy

Chemical Physics Letters 397 (2004) 344–353 www.elsevier.com/locate/cplett

Energy level structure and potential energy curve of the C 0þ u state of Ne2 by high-resolution VUV laser spectroscopy A. Wu¨est, F. Merkt

*

Physical Chemistry Laboratory, ETH Zu¨rich, CH-8093 Zu¨rich, Switzerland Received 23 July 2004; in final form 18 August 2004 Available online 25 September 2004

Abstract The C 0þ u state of Ne2 has been investigated by high-resolution vacuum ultraviolet laser spectroscopy. Rotationally resolved spec20 0 00 tra of the C 0þ X 0þ Ne2, 20Ne–22Ne and 20Ne–21Ne u ðv ¼ 0
12Þ g ðv ¼ 0Þ transitions have been recorded for the isotopomers þ and used to extract the potential energy curve of the C 0u state in a nonlinear, least-squares fitting procedure. The resulting curve ˚ located
167.7 and
77.5 possesses the correct long-range behaviour and two minima at internuclear distances of 2.95 and 4.09 A cm
1 below the Ne(1S0) + Ne*(3s 0 [1/2](J = 1)) dissociation limit, respectively. The C 0þ state is found to be predissociative and fragu ments to Ne(1S0) + Ne*(3s[3/2](J = 1)). Ó 2004 Elsevier B.V. All rights reserved.

1. Introduction Rare gas dimers belong to the category of molecules called Rydberg molecules [1]: Their ground neutral state is repulsive at short internuclear distances and only bound by weak van der Waals forces. Their excited electronic states are all predominantly of Rydberg character. Configuration and spin–orbit interactions strongly perturb their electronic structure, particularly at low nvalues. Low-lying electronically excited states possess mixed electronic configurations as a consequence of the existence of several closely spaced cationic electronic states, two in the case of Heþ 2 and six in the case of the other homonuclear rare gas dimer ions. Rare gas dimers and their ions play an important role in vacuum ultraviolet lamps [2,3], excimer laser systems [4], and rare gas ion lasers. The rare gas dimers also belong to the small class of molecules in which the first excited electronic states lie energetically so high above the ground electronic state that the corresponding single-photon transitions lie in *

Corresponding author. Fax: +41 1 632 10 21. E-mail address: [email protected] (F. Merkt).

0009-2614/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2004.08.087

the vacuum ultraviolet (VUV) range of the electromagnetic spectrum. The VUV spectrum of the homonuclear rare gas dimers is progressively shifted toward shorter wavelengths in the series Xe2, Kr2, Ar2 and Ne2. Consequently, high-resolution spectroscopic investigations are not straightforward, particularly for Ne2 for which all transitions lie below the LiF cut-off wavelength. Compared to the heavier dimers Ar2, Kr2 and Xe2 which have been extensively studied (see, e.g. [5–9]), relatively little experimental data are available on the excited electronic states of Ne2. Information on the electronic structure of Ne2 stems from studies by VUV absorption and emission spectroscopy [10,11], scattering experiments [12] and investigations of the metastable electronic states which possess predominantly triplet character [13–15]. In addition, several ab initio quantum chemical and semiempirical calculations have been performed on the electronically excited states of Ne2 [16–22]. The only high-resolution spectroscopic information available on the VUV spectrum of Ne2 stems from measurement of the absorption and emission spectrum by Tanaka, Yoshino and coworkers [10,11]. As part of their pioneering work on the VUV spectra of the rare gas dimers they obtained information on several band systems

A. Wu¨est, F. Merkt / Chemical Physics Letters 397 (2004) 344–353

connecting the ground and lowest electronically excited states and recorded four vibrational bands of the C 0þ X 0þ u g transition of Ne2 showing a partially resolved rotational structure [10,11]. We report here on a new measurement of the C 0þ X 0þ u g electronic transition of Ne2 by VUV laser spectroscopy at a resolution which enabled the full resolution of the rotational structure and the derivation of a potential function for the C 0þ u state. The double-minimum potential and the resulting irregular rovibrational energy level structure of the excited C 0þ u state prevented a simultaneous analysis of both electronic states, and a two-step procedure was followed in the analysis. In the first step, the potential energy curve of the X 0þ g ground state was determined from rovibrational combination differences [23]. In a second step, presented here, the ground state potential was used together with the experimental transition wave numbers to extract the potential energy curve of the upper C 0þ u state. The C 0þ u state belongs to the manifold of molecular states arising from the interaction of a ground state Ne(1S0) atom and an excited Ne* atom in one of the states corresponding to the excitation of a 2p valence electron into a 3s-orbital, i.e., in order of increasing energy, Ne*(3s[3/2](J = 2)), Ne*(3s[3/2](J = 1)), Ne*(3s 0 [1/ 2](J = 0)) and Ne*(3s 0 [l/2](J = 1)) [24]. The molecular states correlating to each of the four dissociation limits are best described in HundÕs case (c) nomenclature and can be derived with the rules given by Mulliken in [25]. Labelling the excited and ground state Ne atoms 1 and 2, respectively (J1 = J, J2 = 0), the quantum number X of the projection of the total electronic angular momentum J onto the internuclear axis takes the values X ¼ J ; J
1; . . . ; 0:

ð1Þ

When X = 0, the molecular orbital is further classified according to its symmetry with respect to reflection through a plane containing the internuclear axis. The corresponding character, denoted +/
, is obtained by the rule [25] X X þ=
for J 1 þ J 2 þ li þ lj ¼ even=odd; ð2Þ i

j

where li and lj are the orbital angular momentum quantum numbers of the individual electrons P of atoms 1 and 2, respectively. Noting that J = 0, li = odd and 2 P lj = even, it follows that 0+/0
correspond to J1 = odd/even. As a consequence of the g/u-symmetry of orbitals of homonuclear diatomic molecules, the following molecular orbitals correlate to the four dissociation limits: 1

þ Neð S0 Þ þ Ne ð3s0 ½1=2ðJ ¼ 1ÞÞ $ 1g ; 1u ; 0þ g ; 0u ; 1


Neð S0 Þ þ Ne ð3s0 ½1=2ðJ ¼ 0ÞÞ $ 0
g ; 0u ; 1

þ Neð S0 Þ þ Ne ð3s½3=2ðJ ¼ 1ÞÞ $ 1g ; 1u ; 0þ g ; 0u ; 1


Neð S0 Þ þ Ne ð3s½3=2ðJ ¼ 2ÞÞ $ 2g ; 2u ; 1g ; 1u ; 0
g ; 0u :

345

The C 0þ u state of Ne2 is weakly bound and correlates with the energetically highest of the four dissociation limits, i.e. Ne(1S0) + Ne*(3s 0 [1/2](J = 1)). The notation C 0þ u originates from the fact that it is the third electronically excited state accessible in a one-photon transition þ from the X 0þ g ground state (A 0u and B 1u being the first two) [6]. The C 0þ u state exhibits interesting physical properties: The dominant term in the long-range interaction series scales as R
3 (R designates the internuclear distance) because the atomic fragments Ne(2p6,1S0) and Ne*(2p53s 0 [1/2](J = 1)) can be brought into resonance by an electric-dipole transition [26]. Moreover, the nature of the correlation diagrams [27,28] and the spin–orbit interaction [16] lead to a double-minimum potential energy curve. In this work, we report a complete set of rovibrational transition wave numbers for the C 0þ X 0þ u g transition from which we derived a potential energy curve of spectroscopic accuracy for the C 0þ u state of Ne2. The measurements were carried out by (1 + 1 0 ) resonance-enhanced multiphoton ionisation (REMPI) spectroscopy of the three isotopomers 20Ne2, 20Ne–21Ne and 20Ne–22Ne and also revealed new information on the decay dynamics of the C 0þ u state, in particular the predissociation to the Ne(1S0) + Ne*(3s[3/2](J = 1)) fragments.

2. Experiment The experimental procedure used in the present study has been described in our previous report on the determination of the potential energy curve of the X 0þ g ground state [23] and is only briefly summarised here. The experimental setup consists of a tandem electron/ ion time-of-flight (TOF) spectrometer and a broadly tunable narrow-bandwidth VUV laser system. The VUV radiation was generated by resonance-enhanced sum-frequency mixing ð~mVUV ¼ 2~m1 þ ~m2 Þ in a supersonic krypton gas jet using the 5p[1/2](J = 0) 4p6 two-pho
1 ton resonance at 2~m1 ¼ 94092:867 cm [29]. Two dye lasers pumped by the 532 or 355 nm output of a Nd:YAG laser generated the necessary narrowbandwidth radiation, the wave number of which was doubled or tripled in b-barium-borate crystals. When operating the dye lasers with intracavity e´talons, the bandwidth of the VUV radiation amounted to 0.12 cm
1. The VUV wave number was calibrated by recording optogalvanic spectra of neon and e´talon spectra of the fundamental dye laser outputs and building the corresponding sums. The neon dimers were produced in a pulsed supersonic jet of pure neon. The nozzle stagnation pressure amounted to 6 bar and the timing of nozzle opening with respect to the laser pulse was chosen so as to

A. Wu¨est, F. Merkt / Chemical Physics Letters 397 (2004) 344–353

~mVUV

þ  ðiiÞ Ne2 ðX 0þ g Þ ! Ne2 ðC 0u Þ predissociation

1






! Neð S0 Þ þ Ne ð3s½3=2ðJ ¼ 1ÞÞ;

Ne ð3s½3=2ðJ ¼ 1ÞÞ 2~mVIS

~mVIS



! Ne ð4d½3=2ðJ ¼ 2ÞÞ ! Neþ þ e
: In scheme (i), (1VUV + 1 0 ) REMPI spectra are recorded by monitoring the Neþ 2 signal as a function of the VUV radiation wave number. Detecting the dimer ion signal mass-selectively by TOF mass spectrometry allows to record spectra of the different isotopomers of Ne2. However, predissociation of the C 0þ u state reduces the ionisation signal which can only be observed with good signal-to-noise (S/N) ratio when ~m3 is chosen so as to correspond to a transition from the C 0þ u state to an autoionising level of Ne2. This disadvantage is converted into an advantage in scheme (ii), which relies on monitoring the atomic Ne+ signal produced by (2VIS + 1VIS) REMPI of the Ne*(3s[3/2](J = 1)) fragment as a function of ~mVUV . Scheme (ii) is much more sensitive than scheme (i), but one must take into account the contributions of the different isotopomers to a given atomic Ne+ signal and carefully extract the spectrum of each Ne2 isotopomer on the basis of their natural abundances [23]. The radiation used in the ionisation step (~m3 or ~mVIS in scheme (i)/(ii)) was produced by a third dye laser and introduced into the spectrometer at an angle of 45° relative to the VUV radiation and the neon gas jet. The polarisation of the VUV beam ð~mVUV Þ was either parallel ð~m3 ¼ ~mVIS Þ or perpendicular ð~m3 ¼ 2~mVIS Þ to that of the ionising beam. In the case of scheme (ii), the visible radiation had to be focused into the spectrometer to enable efficient ionisation of the excited atomic fragment and was delayed by several nanoseconds relative to the VUV radiation to avoid AC Stark shifts and broadening 0 00 0 00 of the C 0þ X 0þ u ðv ; J Þ g ðv ; J Þ transitions of Ne2. 3. Results 0 00 0 00 Rovibronic transitions C 0þ X 0þ u ðv ; J Þ g ðv ; J Þ 00 0 for v = 0, 1 and several v could be recorded with both excitation–detection schemes for the isotopomers 20Ne2 and 20Ne–22Ne. 20Ne–21Ne was only detectable with scheme (ii) because of its small natural abundance (0.49%). The vibrational bands all consist only of an R- and a P-branch, and lack a Q-branch. They can thus þ be assigned to a 0þ g $ 0u one-photon transition in

Ne2+ signal (arb. units)

~m3

Ne2+ signal (arb. units)

~mVUV

þ  þ
ðiÞ Ne2 ðX 0þ g Þ ! Ne2 ðC 0u Þ ! Ne2 þ e ;

accord with the previous conclusion of Tanaka and Yoshino [10]. Because the Neþ 2 ionisation signal was in general very weak, spectra of good S/N ratio of selected bands were obtained by setting the wave number ~m3 so as to correspond to a strong transition to an autoionising Rydberg state of Ne2. Several examples are displayed in Figs. 1a– c which depict overview spectra of 20Ne2 recorded applying scheme (i). Fig. 1d shows an overview spectrum of 20Ne–22Ne obtained with scheme (ii). The higher sensitivity of this detection scheme is apparent from the much higher S/N ratio. To rationalise the irregular intensity distributions in the spectra shown in Fig. 1a–c, it is important to note that a Neþ 2 dimer ion can be created either by direct ionisation of the C 0þ u state or by autoionisation of a Rydberg state excited from the C 0þ u state. Moreover, the

Ne2+ signal (arb. units)

maximise the formation of Ne2 and minimise that of larger clusters. Two excitation–detection schemes were employed to measure the C 0þ X 0þ u g transition:

Ne+ signal (arb. units)

346

(a) 0.4

Scheme (i) ~ νUV = ~ν2

0.2 0.0 v’ =

0

1

2 3

4

5 6 7 89

2 3

4

5 6 7 89

2 3

4

5 6 7 89

(b) 0.4

Scheme (i) -1 -1 ~ νUV = 32733.2 cm , ~ νVIS = 16366.6 cm

0.2 0.0 v’ =

0

1

(c) 0.4

Scheme (i) -1 -1 ~ νVIS = 16226.6 cm νUV = 32453.2 cm , ~

0.2 0.0 v’ =

0

1

(d) 0.4

Atomic transition

Scheme (ii) -1 ~ νVIS = 16276.2 cm

0.2 0.0 v’ =

135750

0

1

135800

2 3

4

135850

5 6 7 89

135900

Wave number / cm-1 Fig. 1. Survey spectra (bandwidth of VUV radiation 0.4 cm
1) of the 20 0 00 C 0þ X 0þ Ne2 recorded with excitation– u ðv Þ g ðv Þ transition of detection scheme (i) (panels (a) to (c)) and of 20Ne–22Ne recorded with scheme (ii) (panel (d)). The spectra in panels (a) to (c) were obtained at different wave numbers of the ionising laser radiation indicated in each panel in the top left corner. The value of ~m2 indicated in panel (a) corresponds to one of the frequencies used in the four-wave mixing process. In panel (d), the atomic transition Ne*(3s 0 [1/ 2](J = 1)) Ne(1S0) is marked with an arrow.

A. Wu¨est, F. Merkt / Chemical Physics Letters 397 (2004) 344–353

ionisation or excitation can take place in a single- or a multi-photon process. The measured relative intensities thus do not correspond to the Franck–Condon factors 0 00 of the C 0þ X 0þ u ðv Þ g ðv Þ transitions but reflect the product of transition moments of all steps involved and may be further modulated by predissociation if the predissociation rates of the C 0þ u state depend on the vibrational quantum number. An atomic Ne+ ion can only be generated from the dimer by scheme (ii), i.e., predissociation of the C 0þ u state followed by (2 + 1) REMPI of the excited atomic Ne*(3s[3/2](J = 1)) fragment. The much better S/N ratio compared to the spectra obtained by monitoring the dimer ion indicate that the C 0þ u state predissociates within a few nanoseconds or less. Spectra of the region immediately below the dissociation limit recorded by scheme (ii) are displayed in Fig. 2 for 20Ne2 (upper trace) and 20Ne–22Ne (lower trace). They consist of vibrational bands with v 0 up to 12. An enlargement of the (v 0 = 0, v00 = 0) band of both isotopomers showing the completely resolved rotational structure is depicted in Fig. 1 of [23] where the R-branch and the P-bandhead are clearly recognizable. To document the sensitivity of scheme (ii) Fig. 3 depicts the transitions [(v 0 ,v00 ) = (2.0) and (3,0)] of the 20Ne–21Ne isotopomer. Table 1 lists all measured rovibronic transition wave numbers.

347

To determine the absolute value of the vibrational quantum number v 0 of the C 0þ u state, a two-step procedure was followed. In the first step, the vibrational band origins ~mv0 v00 and the rotational constants Bv 0 were derived in a least-squares fit using the observed transition wave 20 numbers ~mobs Ne2 and the relation v0 J 0 ;v00 J 00 of ~mobs mv0 v00 þ Bv0 J 0 ðJ 0 þ 1Þ
Bv00 J 00 ðJ 00 þ 1Þ; v0 J 0 ;v00 J 00 ¼ ~

ð3Þ

where J00 and J 0 designate the total angular momentum quantum numbers of the lower and upper electronic states, respectively. Transitions corresponding to values of J00 and J 0 up to 4 were considered for which the effects of the centrifugal distortion are negligible. The ground state rotational constants Bv00 = 0,1 were held fixed at the values given in [23] and were not fitted simultaneously because the double minimum nature of the potential energy curve of the C 0þ state prevents a u simultaneous analysis of both states as already discussed in [23]. Table 2 summarises the band centres and rotational constants of the successive bands seen in Figs. 1 and 2. Striking features of the energy level structure of the 0 C 0þ u state are that (a) the levels with v = 0 and to a lesser extent v 0 = 2 and 3 have rotational constants that are significantly larger than that of level v 0 = 1, and (b) the vibrational spacings are very irregular. Both observations strongly suggest that the potential energy curve

Ne*(3s’[1/2](J=1)) 20 1

Ne( S0) +

20

v’=10

v’=11 v’=12

dissociation limit

v’=8

v’=6

v’=9

20

Ne2

0

3 135860

135870

135880

135890 135900 -1 Wave number / cm

20

22

Ne- Ne

dissociation limit

Ne*(3s’[1/2](J=1)) 22 1

Ne( S0) +

20

v’=11 v’=12

v’=7

v’=6

v’=5

2

v’=8

1

v’=10

0

v’=9

Ion signal (arb. units)

1

v’=5

2

v’=4

v’=7

3

135910

135920

20 00 Fig. 2. Spectrum of the C 0þ X 0þ Ne2 (upper trace) and 20Ne–22Ne (lower trace) in the energy region close to the u g ðv ¼ 0Þ transitions of þ dissociation limit of the C 0u state recorded with excitation–detection scheme (ii). The broad structure in both spectra designated with a vertical, dashed line corresponds to the atomic transition Ne*(3s 0 [1/2/](J = 1)) Ne(1S0) at 135888.72 cm
1 [24].

348

A. Wu¨est, F. Merkt / Chemical Physics Letters 397 (2004) 344–353

(v’,v’’)

(2,0)

(3,0)

Ion signal (arb. units)

2

1

135844

135846

135848

135850

R(BH)

P(1)

P(2)

R(BH)

P(1)

P(2)

0

135852

135854

-1

Wave number / cm

20 0 00 Fig. 3. Spectrum of the C 0þ X 0þ Ne–21Ne recorded with excitation–detection scheme (ii). The band head of the u ðv ¼ 2; 3Þ g ðv ¼ 0Þ transition of R-branch (designated with R(BH)) is not resolved in either of the vibronic bands.

of the C 0þ u state possesses more than one well, in accord with previous theoretical predictions [16,19]. That the C 0þ u state does indeed have more than one potential well will be demonstrated in Section 4. In the second step, the absolute value of v 0 was determined by analysing the isotopic shifts of the levels v 0 = 0, 2 and 3 which have a vibrational wave function localised primarily in the inner potential well (see Fig. 4). The analysis resulted in an unambiguous assignment of the lowest observed level to v 0 = 0. We therefore conclude that the transitions observed at 135575 and 135345 cm
1 by Tanaka and Walker [11] and attributed to the C 0þ u state must correspond to another electronic state.

4. Potential energy curve of the C 0þ u state The potential energy curve of the C 0þ u state was determined in a weighted nonlinear least-squares fitting procedure [30–32] by adjusting the parameters of an analytical potential function used to calculate the rovibronic transition wave numbers ðiÞ ðiÞ ~mðiÞ v0 J 0 ;v00 J 00 ¼ Ev0 J 0
E v00 J 00 ;

ð4Þ ðiÞ

ðiÞ

of isotopomer (i) of Ne2. In Eq. (4), Ev0 J 0 and Ev00 J 00 designate the rovibronic term values (in cm
1) of the levels þ of the C 0þ u and X 0g states with respect to a common origin taken as the dissociation limit Ne(1S0) + Ne(1S0) ðiÞ of the ground electronic state. The values of Ev00 J 00 were

calculated from the potential energy curve of the ground state reported in [23] and agree with the experimental ðiÞ values to better than 0.1 cm
1. Ev0 J 0 corresponds to the eigenvalues of the Schro¨dinger equation 

 h
2 d2 ðiÞ ðiÞ ðiÞ ðiÞ þ hcV J 0 ðRÞ wv0 J 0 ðRÞ ¼ hcEv0 J 0 wv0 J 0 ðRÞ
2li dR2

ð5Þ

describing the internal motion of the nuclei in the C 0þ u state. Eq. (5) was solved numerically employing a discrete variable representation on an equidistant grid with ˚. 751 points between 1.5 and 20 A In Eq. (5), li designates the reduced mass (calculated ðiÞ using the masses of the atoms), wv0 J 0 the vibrational wave ðiÞ 0 0 function of the level (v J ), and V J 0 (R) the effective potential of isotopomer (i), which includes the centrifugal potential and has the form ðiÞ

V J 0 ðRÞ ¼ V ðRÞ þ

h J 0 ðJ 0 þ 1Þ : 8p2 li c R2

ð6Þ

The isotopomer-independent potential V(R) was described independently of other electronic states. If there were sufficient experimental information on the other states of u symmetry of the manifold correlating to the first four excited dissociation limits of Ne2 (see Section 1) one could model the coupling of the electronic states with the spin–orbit interaction matrix given in [16], as was, e.g., performed for the first six electronic states of Arþ 2 in [32]. The Born–Oppenheimer potential V(R) was represented by the following analytical function:

A. Wu¨est, F. Merkt / Chemical Physics Letters 397 (2004) 344–353

349

Table 1 0 00 0 00 Observed wave numbers of the rovibronic transitions C 0þ X 0þ mobs ) and differences between calculated and u ðv ; J Þ g ðv ; J Þ (designated as ~ measured transition wave numbers ðD~m ¼ ~mcalc
~mobs Þ for the three isotopomers 20Ne2, 20Ne–21Ne and 20Ne–22Ne 20

(v 0 ,v00 )

~mobs (0,0)

20

Ne2

P(1) P(2) P(4) P(6) P(8) P(10)

D~m
0.10
0.18
0.09
0.06
0.01

R(0) R(1) R(2) R(3) R(4) R(5) R(6) R(7) R(8) R(10)

135777.64(6)


0.13

135778.78(7)


0.12

135780.37(7)


0.13

135782.36(9)


0.07

135784.82(8) 135787.95(14)

0.02 0.04

(0,1)

P(2) P(4) R(0) R(2) R(4)

135763.27(16) 135763.97(15) 135763.88(14) 135765.32(14) 135767.70(16)


0.13
0.10
0.14
0.10
0.09

(1,0)

P(1) P(2) P(3) P(4) P(5) P(6)

135834.65(5)


0.23

135833.34(6)


0.07

135831.55(9)

0.15

(1,1)

(2,0)

135835.57(20)


0.22

135835.57(20)


0.15

135834.90(9) 135833.78(14)

0.15 0.45

P(2) P(4)

135821.18(17) 135820.75(18)


0.20
0.11

R(0) R(2) R(4)

135821.79(17) 135822.06(14) 135822.28(17)


0.21
0.08
0.14

135844.72(5)


0.06

135843.69(4)

0.22

135842.33(7)

0.60

135840.45(13)

1.05

135845.65(20)


0.06

P(1) P(2) P(3) P(4) P(5) P(6) P(7) P(8) R(0) R(1) R(2) R(3) R(4)

~mobs

20

Ne–22Ne

D~m

~mobs

D~m a

135776.68(22) 135776.68(22) 135776.98(9) 135777.85(10) 135779.32(10)

R(0) R(1) R(2) R(3) R(4) R(6)

Ne–21Ne

135845.86(4)

0.21

135845.65(20)

0.64

135775.53(60)

0.64

135776.40(13) 135776.89(13) 135777.47(16) 135778.20(19) 135778.96(23) 135779.81(36) 135780.84(20) 135781.91(16) 135783.24(32)

0.47 0.48 0.50 0.47 0.51 0.57 0.57 0.64 0.59

135835.17(5) 135834.74(6) 135834.22(11)


0.25
0.22
0.20

135832.66(17) 135830.75(15)

0.05 1.16

135835.63(32)a


0.25

135835.70(25)a


0.29

135844.93(24) 135844.49(13)


0.36
0.25

135844.68(6) 135844.32(7) 135843.86(7) 135843.41(8) 135842.78(21)


0.57
0.51
0.35
0.22 0.06

135845.51(37)a


0.36

135845.34(51)a


0.64

(continued on next page)

350

A. Wu¨est, F. Merkt / Chemical Physics Letters 397 (2004) 344–353

Table 1 (continued) 20

(v 0 ,v00 )

(3,0)

P(1) P(2) P(3) P(4) P(5) P(6) P(8) R(0) R(1) R(2) R(3) R(4)

(4,0)

(R,0)

(5,0)

P(1) P(2) R(0)

(6,0)

(8,0) (9,0) (10.0)

(11,0) (12,0)

20

Ne–22Ne

~mobs

D~m

~mobs

D~m

~mobs

D~m

135851.29(5)


0.10

135851.15(27) 135850.80(30)


0.04
0.07

135850.26(7)


0.02

135850.80(8) 135850.44(8) 135849.98(12) 135849.54(13)


0.09
0.11
0.10
0.18

135849.03(7) 135847.63(8)

0.16 0.58 135851.82(36)a


0.18

135851.40(43)a
0.18

135852.22(20)


0.10

135852.43(5)


0.03

135852.22(20) a

135861.80(32)

135871.25(24)a

0.32 0.65

0.76

P(1) P(2) R(0)

(7,0)

Ne–21Ne

20

Ne2

135879.42(24)a

0.62

P(1) P(2)

135861.02(23)a

0.74

135869.87(17) 135869.45(16)

0.97 0.94

135870.26(15)a

1.02

135878.03(22) 135877.60(14)

0.84 0.80

135878.49(16)a

0.80

135884.99(19) 135884.48(20)

0.46 0.50

R(0)

135886.27(43)a

0.30

135885.33(19)a

0.53

P(1) R(0)

135891.59(20)a

0.17

135890.43(15) 135890.79(30)a

0.31 0.34

P(1) R(0)

135895.75(25)a

0.01

135894.75(22) 135895.06(29)a

0.12 0.19

135898.02(31) 135897.49(25)

0.01 0.01


0.10

135898.35(29)a

0.04


0.08

a

0.12

a

0.14

P(1) P(2) R(0)

135898.88(30)a

R(0)

a

135901.07(30)

135900.60(49)

R(0)

135902.27(39)
1

The experimental uncertainties correspond to one standard deviation. ~mobs and D~m are both given in cm . a Band head.

V ðRÞ ¼ ½1
sðRÞV 1 ðRÞ þ sðRÞV 2 ðRÞ þ Ediss ð7Þ   ¼ ½1
sðRÞ A1 e
b1 R
B1 e
b1 R=b1    C3 C6
b2 R
f3 ðR; b2 Þ 3 þ f6 ðR; b2 Þ 6 þ sðRÞ A2 e R R þ Ediss ;

ð8Þ

where    1 R
Rs sðRÞ ¼ 1 þ tanh ; 2 Ws

ð9Þ

fn ðR; b2 Þ ¼ 1
e
b2 R and

k n X ðb2 RÞ ; k! k¼0

 Ediss ¼ ~m Ne ð3s0 ½1=2ðJ ¼ 1ÞÞ

ð10Þ 1 Neð S0 Þ :

ð11Þ

Because the potential energy curve of the C 0þ u state possesses two wells, the function V(R) is composed of two contributions V1(R) and V2(R) describing the inner and outer potential wells, respectively. The switch function s(R) with centre Rs and width Ws ensures a smooth transition from V1(R) to V2(R). Ediss represents the energetic position (in cm
1) of the dissociation limit of the

A. Wu¨est, F. Merkt / Chemical Physics Letters 397 (2004) 344–353 Table 2 0 00 Band centres ~mv0 0 of the C 0þ X 0þ u ðv Þ g ðv ¼ 0Þ transition and rotational constants B0v of the vibrational levels of the C 0þ u state of 20 Ne2 v0

~mv0 0 =cm
1

Bv 0 /cm
1a

Bv 0 /cm
1b

0 1 2 3

135777.22(7) 135835.40(6) 135845.43(5) 135851.99(6)

0.207(10) 0.092(21) 0.115(5) 0.114(7)

0.208 0.101 0.144 0.119

a

Determined with Eq. (3). R  1
h2 Calculated as Bv0 ¼ hc wv0 0 ðRÞ 2lR 2 wv0 0 ðRÞdR with the vibrational wave functions wv 0 0(R) determined from Eq. (5). b

be adequate to reproduce the experimental results with an absolute error smaller than 1 cm
1. In the same manner as for the potential energy curves of the lowest electronic states of Arþ 2 [32], the parameters A1 and B1 have been substituted by Re,1 and De,1 through the relations

dV 1 ðRÞ

¼0 ð12Þ dR R¼Re;1 and V 1 ðRe;1 Þ ¼ Ediss
De;1 :

ð13Þ

Similarly, A2 was replaced by Re,2 using

dV 2 ðRÞ

¼ 0: dR

Table 3 lists the parameters of the best fit (rootmean-square deviation (rms) = 3.1). The value Ediss = (135888.72 ± 0.04) cm
1 [24] was held fixed during the fitting procedure.

5. Discussion and conclusions Fig. 4 shows the potential energy curve of the C 0þ u state with the vibrational wave functions wv 0 ,J 0 = 1 (R), v 0 = 0
5 for 20Ne2. The effects of the double minimum potential, namely the irregular vibrational spacings and the localisation of the v 0 = 0 (1) level mainly in the inner (outer) well, are clearly visible. The long-range

v’=5

135860

v’=4 v’=3 v’=2 v’=1

135840

135820

1

1

135800

135780

v’=0 135760

135740

135720

135700 2.0

4.0

ð14Þ

R¼Re;2

-1

Energy relative to Ne( S0) + Ne( S0) dissociation limit / (hc cm )

þ C 0þ u state relative to the corresponding one of the X 0g state and is approximated to be isotopomer-independent because the energetic difference of the atomic transitions of 20Ne and 22Ne (0.003(3) cm
1 [33]) is considerably smaller than the experimental uncertainty. V1(R) has the form of a generalised Morse potential (see, e.g. [32]) and V2(R) is very similar to a Tang–Toennies potential [34], the only difference being the occurence of the R
3 term in the inverse power series. This term, which is absent in the electronic ground state of the rare gas dimers, arises here because of a resonant dipole interaction [26,35]. Values for the coefficients C3 and C6 were taken from [16] ðC 3 ¼ 0:26Eh a30 ; C 6 ¼
57:4Eh a60 Þ and not adjusted in the fitting procedure. Only terms up to R
6 were retained in the inverse power series because (a) no higher coeffecients Cn were known to us and (b) the potential given by Eq. (8) turned out to

351

6.0

8.0

Internuclear distance R / Å Fig. 4. Potential energy curve of the C 0þ u state of Ne2 and wave functions of the first few vibrational levels of

20

Ne2 for J 0 = 1.

352

A. Wu¨est, F. Merkt / Chemical Physics Letters 397 (2004) 344–353

Table 3 Parameters of the analytical function describing the potential energy curve of the C 0þ u state of Ne2 Parameter ˚ Re,1/A

Value 2.954 ± 0.008 185.0 2.0 2.776 ± 0.043 4.09 1.556 ± 0.003 8456 276632 3.145 ± 0.007 0.2 654978 22016 33577

De,1a/cm
1 b1a ˚
1 b1/A ˚ Re,2a/A ˚
1 b2/A

˚ 3) C3a/(cm
1 A
1 ˚ 6 a C6 /(cm A ) ˚ Rs/A ˚ Wsa/A

A1b/cm
1 B1b/cm
1 A2c/cm
1

The uncertainties correspond to 95% confidence intervals. a This parameter was kept

fixed during the fitting procedure. b 1 ðRÞ

Determined from dVdR ¼ 0 and V1(Re,1) = Ediss
De,1.

R¼Re;1

c 2 ðRÞ Determined from dVdR ¼ 0.

R¼Re;2

resonant dipole interaction proportional to R
3 is responsible for the slowly increasing potential energy curve at large internuclear distances and the relatively large number of vibrational states. 5.1. Comparison with previous experimental and theoretical results Tanaka, Yoshino and coworkers have investigated the C 0þ u state of Ne2 by absorption [10] and emission spectroscopy [11]. Four vibrational bands corresponding to transitions to v 0 = 0 and 1 from v00 = 0 and 1 were observed in absorption. Detection of higher lying vibrational levels of the C 0þ u state was hindered by the energetic proximity of the atomic Ne*(3s 0 [1/2](J = 1)) Ne (1S0) resonance. From the experimental data, a dissociation energy D0 of 126 cm
1 was estimated [10]. The emission spectra contained two additional lines at lower energies than the (v 0 = 0,v00 = 0) transition which led to the conclusion that the dissociation energy
1 of the C 0þ u state should be larger (D0  544 cm ) [11] than found in the analysis of the absorption spectrum. The analysis of our spectra yields a value of De = 167.7 cm
1 in better agreement with the analysis of the absorption than the emission spectrum (see Table 4). The ab initio quantum chemical [16,18,19] and semiempirical [21] potentials of the C 0þ u state of Ne2 reported in the literature agree qualitatively with the potential derived here in that they exhibit a shallow well and a local maximum. Cohen and Schneider [16] and Peyerimhoff and coworkers [18,19] performed ab initio configuration-interaction calculations treating the spin–orbit interaction by the semiempirical method pro-

Table 4 Comparison of the dissociation energies (De), the equilibrium internuclear distances (Re) of the inner and outer wells and of the position of the local maximum Vmax(Rmax) relative to the dissociation limit of the potential energy curve of the C 0þ u state of Ne2 Inner well De

Outer well Re

Vmax

Rmax

De

Reference

Re

[16]a [19]a [10]c [11]c 2.95
54.2 3.36 77.5 4.09 This workc
1 ˚. The energies are given in cm , the internuclear distances in A a Ab initio values. b D0 value. c Experimental values. 24 126b 544b 167.7

2.7 3.8

484 40

2.9 4.5

posed in [16]. The dissociation energy, equilibrium internuclear distance and the position of the potential maximum relative to the dissociation limit calculated in this way are also listed for comparison in Table 4. Both calculations underestimate the dissociation energy and yield a local maximum lying just above the dissociation limit. However, as pointed out by the authors of [19] Ôthe uncertainty in the calculated energies is estimated to be 0.2 mhartree ( 0.005 eV), values in the De and Vmax columns of this order are uncertain, and indeed it is not always clear whether such extrema exist at allÕ. Our analysis clearly confirms the existence of the maximum in the potential energy curve of the C 0þ u state but suggests that it is located below the dissociation limit. It also gives a slightly different equilibrium internuclear distance and has two minima. 5.2. Potential model The normalised rms value of 3.1 (a definition of this quantity can be found, for instance, in Eq. (18) of [32]) obtained in the nonlinear fit of the potential parameters to the transition wave numbers implies that the potential model used is not flexible enough. However, because the absolute errors are less than 1 cm
1 (see Table 1) we chose not to introduce additional parameters into the analytical potential function. To check the validity of our potential energy curve, rotational constants Bv0 ¼ 8ph2 lc hv0 j R
2 v0 i were calculated for the vibrational levels v 0 = 0
3 of 20Ne2. They are compared in Table 2 with the experimental results (see Eq. (3)). The values of Bv 0 are in good agreement except for v 0 = 2, which is the level with the wave function having the largest amplitude at the position of the potential barrier (see Fig. 4). This may indicate that the potential energy curve could be improved by introducing additional parameters describing the local barrier separating the two potential wells. Our model describes the potential energy curve of the C 0þ u state independently of other electronic states and

A. Wu¨est, F. Merkt / Chemical Physics Letters 397 (2004) 344–353

relies on the validity of the Born–Oppenheimer approximation. The fact that the calculated values of the transition wave numbers of the low J lines of the (0,0) band of 20Ne2 are systematically too small (by  0.1 cm
1), whereas they are too large (by  0.5 cm
1) in the case of 20Ne–22Ne (see Table 1) may be a manifestation of possible non-Born–Oppenheimer effects. Indeed, this behaviour is not caused by an erroneous absolute assignment of the vibrational quantum number v 0 but could originate in an isotopomer-specific coupling of the vibrational level v 0 = 0 to a vibrational level of a different electronic state such as, for instance, the A 0þ u state. Considering the large number of possible perturbations (spin–orbit and configuration interaction, nonadiabatic effects), it is not surprising that a fully satisfactory fit of a single potential energy curve could not be achieved. A more precise treatment of the C 0þ u state would have to include other electronic states which at present are not known with sufficient accuracy. However, the absolute accuracy of  1 cm
1 of the potential energy curve determined here for the C 0þ u state of Ne2 is in any case sufficient to serve as a very stringent test for future ab initio quantum chemical calculations of the excited electronic states of Ne2.

Acknowledgements This work is supported financially by the Swiss National Science Foundation and by ETH Zu¨rich.

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