Electronic spectrum (2 p σ u -1 s σ g ) of the D +2 ion

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Electronic spectrum (2pσ u -1sσ g ) of

the D+ 2 ionAlan Carrington a , Iain R. McNab a , Christine A.Montgomerie a & Richard A. Kennedy ba Department of Chemistry , University of Southampton ,Hampshire, SO9 5NH, Englandb Department of Chemistry , University of Birmingham ,Birmingham, B15 2TT, EnglandPublished online: 23 Aug 2006.

To cite this article: Alan Carrington , Iain R. McNab , Christine A. Montgomerie & RichardA. Kennedy (1989) Electronic spectrum (2pσ u -1sσ g ) of the D+ 2 ion, Molecular Physics: AnInternational Journal at the Interface Between Chemistry and Physics, 67:4, 711-738, DOI:10.1080/00268978900101401

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MOLECULA~ PHYSICS, 1989, VOL. 67, NO. 4, 711-738

Electronic spectrum (2pa~--ls~rg) of the D~" ion

by A L A N C A R R I N G T O N , l A I N R. M c N A B and C H R I S T I N E A. M O N T G O M E R I E

D e p a r t m e n t of Chemis t ry , Univers i ty of S o u t h a m p t o n , H a m p s h i r e SO9 5NH, Eng land

and R I C H A R D A. K E N N E D Y

D e p a r t m e n t of Chemis t ry , Univers i ty of Bi rmingham, B i rmingham B15 2TT, England

(Received 13 February 1989; accepted 3 March 1989)

We describe the first measurements of an electronic spectrum of the hydrogen molecular ion, in its perdeutero form, D~. The lower states involved in the spectrum are high-lying vibration-rotation levels of the lsag ground state, and the upper states are the vibration-rotation levels of the 2ptr u long-range van der Waals state. Most of the observed spectral lines, which involve v = 21 in the ground state, are in the infrared region spanned by the carbon dioxide laser, but two microwave electronic transitions, involving v = 26 and 27, have also been observed. We use a D~ ion beam, in which the high-lying vibration- rotation levels of the ground state are populated by the initial electron impact ionization process. The infrared transitions are induced by Doppler tuning the ion beam into resonance with an appropriate laser line, and excitation is detected by electric field dissociation of the upper levels, the resulting D § fragments being separated from other fragment ions and the parent ions with a magnetic analyser, and detected with an electron multiplier. The two microwave transitions were detected initially by microwave-infrared double resonance, but one of the transitions has also been observed by pure microwave spectroscopy, the phase of the signal showing that the 2ptr u van der Waals levels are already populated in the ion beam before laser excitation. Theoretical calculations predict that the 2pau long-range minimum, arising from the charge/induced-dipole interaction, supports seven vibration-rotation levels. We have detected transitions to all of them, determining the vibrational spacing and rotational constants which characterize the van der Waals state. The experimental results are compared with the predictions of Born-Oppenheimer and adiabatic calculations. The two theoretical methods give energies for the vibration-rotation levels of the 2ptr u state which are in close agreement with each other, but for the v = 21 level of the ground state the two methods differ in their predictions by almost 2cm-1. The nonadiabatic correction to the energy of the ground state v = 21 level is found to be close to - 0 - 5 cm-~. The 2pa~-Is% transition moments are also calculated for different values of the internuclear distance. All of the spectroscopic lines observed are unsplit, showing that the nuclear hyperfine interactions for the upper and lower states are closely similar in all cases studied. This result shows that the inversion symmetry of the electronic wavefunction is preserved even in the highest bound level of the 2pa u van der Waals state, which has a dissociation energy of 0.147 cm- 1.

1. Introduction

The hyd rogen molecu la r ion is famil iar to all s tudents of molecu la r s t ructure theory. I t is the only molecule for which the Schr6dinger equa t ion can be solved

0026-8976/89 $3.00 �9 1989 Taylor & Francis Ltd

712 A. Carrington et al.

exactly within the Born-Oppenheimer (fixed nuclei) approximation, and it provides the simplest examples of bonding and antibonding molecular orbitals. Theoretical investigations are, however, much more numerous than experimental studies of its spectra. The only direct spectroscopic investigation of H~ itself is the radiofrequency spectrum measured by Dehmelt and Jefferts [1], who combined quadrupole ion trapping with molecular alignment arising from photodissociation with polarized white light. They measured, with extremely high precision, the frequencies of nuclear hyperfine transitions within vibration-rotation levels of the ground electronic state, with v = 4 to 8 and N = 1. There have been no direct spectroscopic studies of the D~" ion, but a photoelectron spectrum of D 2 showing well resolved vibrational structure has been described I-2].

The homonuclear molecules H~ and D~ do not have electric dipole moments, so that dipole vibrational and rotational spectra are not possible. The heteronuclear species, HD +, does have a dipole moment because of the separation between the centres of charge and mass. Wing, Ruff, Lamb and Spezeski [3] used ion beam techniques to detect vibration-rotation transitions involving the lowest vibrational levels of HD § v = 0 to 3, and Carrington and his colleagues [4, 5, 6] have studied the vibration-rotation spectra of HD + which involve high-lying levels close to the dissociation limit.

Although the ls% ground electronic state of H~" has a binding energy D O of 2.65 eV, the first excited state (2ptr,) is usually considered to be repulsive. These two states become degenerate at the lowest dissociation limit. Consequently an electronic spectrum of H~- (and its isotopes) has neither been expected nor observed. This conclusion applies also to HD § the main difference between HD § and the homonuclear molecules being that the dissociation limits, H § + D and H + D § are separated by 29.8cm -1, H § + D being the lower. The origin of this difference appears in the solution of the SchriSdinger equations for the H and D atoms, the reduced mass being slightly different in the two cases.

The purpose of this paper is to show that H~- and D~ can have resolved electronic spectra, and to describe the measurement and analysis of such a spectrum for D~-; preliminary observations of this spectrum have been described elsewhere [7]. Our earlier statement that the lowest excited electronic state (2ptr,) is repulsive is not true for all values of the internuclear distance. It has been appreciated for many years that when a proton approaches a hydrogen atom, the charge distribution of the atom is polarized, resulting in an electrostatic interaction with the proton. This interaction was first examined theoretically by Coulson [8], who expressed the interaction as an inverse power series in the internuclear separation R. The leading term in this interaction has the form -~t/(8rte o R4), representing the charge/induced dipole interaction, where ~t is the polarizability of the hydrogen atom. Consequently the potential energy of the H + H + system decreases as R decreases from infinite nuclear separation, until the chemical or valence forces become dominant. The result is that the 2ptr, state is predicted to have a shallow van der Waals minimum at R = 6-64 A, with a Born-Oppenheimer well depth of 13.34cm -1. In both H~- and D~" this potential well is expected to support a small number of vibration-rotation levels, so that a discrete electronic spectrum arising through transitions from the lsag ground state is possible. It should be emphasized that the long-range van der Waals minimum also arises naturally from the solution of the electronic Schr6dinger equation to give the Born-Oppenheimer potential.

Apart from the observation and characterisation of the D ... D § van tier Waals

Electronic spectrum (2pau-lsag) of the D~ ion 713

state, we are also interested in determining the symmetry of the molecule in vibration-rotation levels close to the dissociation limit. In the ground electronic state of HD § we have shown [5, 6] that the electron distribution becomes increasingly asymmetric for vibration-rotation levels which approach the H § + D dissociation limit. We were able to resolve the proton and deuterium nuclear hyperfine structure in the infrared spectra, and for the last bound level (v = 22, N = 1), lying only 0.123 cm -1 below the dissociation limit, the deuterium Fermi contact interaction is essentially that of the free deuterium atom, the complementary proton interaction being almost zero. The electron is therefore virtually localized at the deuteron nucleus. In the homonuclear molecules, however, the nuclei are indistin- guishable and one would expect the inversion symmetry of the electronic states to be preserved at all internuclear distances. There is, however, an apparent contra- diction in the observation that, at infinite nuclear separation, the electron is localized at one nucleus. We return to this aspect in w 9, but point out here (and in more detail in w 7) that breakdown of the symmetry of the homonuclear molecule would be revealed by the observation of nuclear hyperfine splitting in the vibration- rotation components of the electronic spectrum.

We feel we should comment on our use of the name of van der Waals in describing the D . . . D § long-range state. We think it is useful to distinguish between molecular complexes bound by conventional chemical valence forces and those involving electrostatic interatomic or intermolecular forces which do not involve electron transfer and sharing. Complexes which are bound by dispersion forces (R-6) are commonly described as van der Waals complexes, but not everyone would agree that this term should be applied to ionic species of the type described in this paper, where the leading term in the multipole expansion of the electrostatic interaction describes the charge/induced-dipole interaction (R-4). However the H ... H § complex was described by Coulson [8], in 1941, as a van der Waals complex, and this description was also used subsequently by Herzberg [9]. On balance we feel that these authoritative precedents should be followed.

2. Principles The only published values of the van der Waals vibration-rotation levels for H~-

are those of Peek [10], using the Born-Oppenheimer potential. Peek commented on the existence of the corresponding levels for D~', with the prediction that v = 0, N = 0 to 4 and v = 1, N = 0 and 1 would be bound. He did not, however, calculate the values of the vibration-rotation energies. Our adiabatic calculations for the 2ptr u state, described in w 8, confirm the expectation of seven bound levels for D r , the lowest level (v = 0, N = 0) having a predicted binding energy of 5-513 cm-1, and the highest (v = 1, N = 1) having a binding energy of 0.147 cm-1. The first theoretical vibration-rotation energies for H~- and D f in their ground states were published by Hunter, Yau and Pritchard [11], using the adiabatic approximation. Subsequently Wolniewicz and Poll [12] have calculated nonadiabatic vibration-rotation energies for H~- and HD § including relativistic and radiative corrections, but unfortunately they did not extend their calculations to D~-.

From the adiabatic calculations of Hunter, Yau and Pritchard it was apparent to us that the 0-21 and 1-21 bands of the 2pau-lsag electronic spectrum of D~" would lie in the infrared region spanned by the carbon dioxide laser (874 to 1094 cm-1). There are, unfortunately, no corresponding band systems for H~- which

- - #z i - - - - i

21.0 1000

Figure 1. Adiabatic potential energy curves for the D~ lsag and 2pa= states in the region of the dissociation limit, showing some of the vibration-rotation levels studied in the present work.

are accessible with a carbon dioxide laser. In figure 1 we show the adiabatic 2ptr= potential curve for D~ with the predicted positions of the v = 0, N = 0 to 4 levels, and also the relevant port ion of the lsag potential curve showing the v = 21, 26 and 27 levels. The information presented in figure 1 suggests that the F ranck-Condon factors for the 0-21 and 1-21 bands are likely to be extremely small; they are, nevertheless, sufficiently large for the transitions to be driven with a carbon dioxide laser, as we shall show. In w 8 we present ab initio calculations of the transition dipole moments using adiabatic potential curves. Another uncertainty in the plan- ning of our experiments was the population of the v = 21 levels in D~. It is now well known that electron impact ionisation of H~ and its deuterium isotopes results in significant populations of all bound vibrational levels of the ground state. In our ion beam experiments the ions are accelerated out of the ion source into a high vacuum environment, and provided the source pressure is not too high, the vibra-

Electronic spectrum (2pa~- l str g) of the D ~ ion 715

tional populations produced in the ionisation process are preserved in the ion beam. Calculations of the Franck-Condon factors for ionisation [13] predict that the population factor of v = 21 in D~ should be approximately 0-002, which is sufficient for our purposes.

Detection of the infrared transitions must necessarily be accomplished by an indirect method, and our early attempts to detect excitation from the ground state were based upon monitoring the subsequent photodissociation of the upper state. However, our attention was drawn to the possibility of using an electric field to selectively dissociate the van der Waals levels. Bjerre and Keiding [14] have shown that visible laser excitation of O~ ions can be detected by subsequent electric field dissociation, and Hiskes [15] and Wind [16] have discussed the theory of the electric field dissociation of H~. An electric field couples the lstrg and 2pau states in D~-, and lowers the energy of the D + D § dissociation limit. Consequently vibration-rotation levels of the 2pau van der Waals state which are bound in zero field become quasibound with respect to the lowered dissociation limit, and may be expected to predissociate. Laser excitations to the 2pa~ levels are therefore detected by monitoring D § fragment ions subsequently produced in the region of an applied electric field.

3. Experimental methods The ion beam apparatus used in this work has undergone a number of changes

since it was originally described, so it is appropriate to review the main features here. A block diagram of the complete instrument is shown in figure 2. The molecular ions are produced by electron bombardment in an ion source located inside a chamber which is pumped by a 160mm Edwards oil diffusion pump. Electron energies of 40 to l l 0 e V are used, the electron emission current being l mA. The source contains an ion repeller plate to which low voltages may be applied. The source body is maintained at a positive potential of up to 5 kV maximum, the remainder of the apparatus being essentially at earth potential. Gas pressures measured outside the source during an experiment are typically 5 x 10-6torr , and we estimate the pressure inside the source to be 10-3 to 10 -4 torr.

The ions are accelerated through an exit slit (1 mm x 5 mm) and focused into a 90 ~ magnetic sector of 6 cm radius which allows identification and separation of ions of the desired charge to mass ratio. The ion beam then enters the main chamber which is separately pumped by a second 160mm oil diffusion pump. The constricted flight tube through the magnetic sector provides differential pumping between the source and main chambers, the pressure in the main chamber being typically 2 • 10-7 torr when the ion beam is present. After the magnetic sector the ion beam passes through a lens stack which provides focusing and deflection in the y and z directions. The electron beam in the source, and the magnetic field in the sector are in the z direction, the propagation of the ion beam being in (by convention) the x direction.

Following the first lens stack the ion beam passes through a perforated steel tube of 26 cm length and 2 cm diameter. This tube, which we call the drift tube, is electrically isolated from the remainder of the apparatus. The geometry of the ion beam machine enables a laser beam to be aligned parallel or antiparallel to and collinear with the ion beam; zinc selenide windows permit transmission of the laser beam through the apparatus, and spectroscopic transitions occur inside the drift

1st MAGNET

WINDOW ~ LENS ONE o os.o LASER SeAM . . . . . -, -" ~ I II I

ION SOURCE

DRIFT TUBE

2nd MAGNET LENS TWO

DRIFT T U B E ~ i

VOLTAGE �9

MODULATION ~ P VOLTAGE AMP

I' '.P A OO+ I MICROCOMPUTER

Figure 2.

WINDOW

MULTIPLIER

AMPLIFIER J

Block diagram of the ion beam apparatus, electronic units and computer control. The lens 2 assembly includes the electric field dissociation lens.

tube. After this region the ion beam encounters an electric field dissociation lens, consisting of four plates with 2 m m diameter holes to permit passage of the ion and laser beams. The two inner plates are at an appropriate positive potential, depending on the required strength of the electric field, and the two outer plates are at earth potential. We have used various plate separations, ranging from 1 to 10mm, depending on the dissociation energy of the vibrat ion-rotat ion level under investigation. The first earth plate is also used as an ion current monitor, and in the present work ion beam currents up to 10 -6 A are obtained; the D~- ion beam fluxes are therefore up to 10 t3 S -1. An upper limit to the positive potential which can be applied to the field dissociation lens is set by the source potential used; the value of the latter is determined by Doppler resonance requirements as discussed below.

Fragment D § ions formed in the field dissociation lens are steered and focused by a second lens stack identical with that preceding the drift tube. The ions are collected and analysed by a second 6 cm 90 ~ magnetic sector and detected with an electron multiplier. The separation of those D § fragments produced in the field lens from all other D § fragments is crucial to the success of the experiment, and is discussed in more detail in the following section.

The infrared laser used is an Edinburgh Instruments PL4 CW carbon dioxide laser; with 12CO 2 the laser is operated in a flowing gas mode and maximum output

Electronic spectrum (2pau-lstr g) of the D ~ ion 717

powers range from 50 to 10W. Alternatively the laser is operated in a sealed gas mode with 13CO2 and output powers of 20 to 1 W are available. The short and long term stabilities of the laser are sufficiently good to not limit the observed spectroscopic line widths. Laser lines are identified by means of a small grating monochro- mator; they are spaced at approximately 2cm -~ intervals and range from 874 to 1094cm -1. For spectroscopic searches we choose an appropriate laser line and source potential. We use the Doppler shift for spectroscopic tuning, and for D~- beam potentials from 1 to 5 kV the corresponding Doppler tuning range is approximately ___0.7 to __+ 1.6cm -~, the positive sign corresponding to antiparallel alignment of the ion and laser beams. For spectroscopic scanning an auxiliary sweep voltage is applied to the drift tube. A scan voltage of 200 volts corresponds to a wavenumber scan of about 0.07 cm- ~ at a source potential of 1 kV, or 0.03 cm- ~ at a source potential of 5 kV. Larger drift tube voltage scans are not feasible if maximum sensitivity is required, since the ion optics need to be retuned in order to maximise the ion beam intensity.

When searching for spectra we monitor the D § fragments originating in the field dissociation lens as a function of the effective laser wavenumber, and resonant vibration-rotation transitions result in an increased D § ion current. We add a sinusoidal modulation voltage to the d.c. scanning voltage applied to the drift tube, and detect the electron multiplier output current with an E G & G Model 5207 lock-in amplifier referenced at the Doppler (velocity) modulation frequency. Conse- quently first-derivative lineshapes are obtained. At modulation frequencies greater than a few kHz the limiting noise is the ion beam statistical noise.

We retain manual control of the ion source variables, the magnetic sector fields and the laser. The Doppler scans, recording, processing and storage of spectra are all under the control of an HP A600+ microcomputer through an HPIB IEEE instrument bus, as shown in figure 2. The spectra displayed later in this paper are recorded digitally, stored on a hard disc, and plotted directly with an H P 7550A graphics plotter.

4. Experimental results

The initial difficulty in recording the D~- spectrum described in this paper was, of course, locating the first resonance. In relation to the practical range and rate of scanning, an uncertainty of 0.5 cm-1 in the predicted resonance wavenumber results in a very time consuming search. Once the first resonance was located, the remainder were observed more readily. For the 2ptr u state theory predicts five rotational levels for v = 0, so that the 0-21 band should exhibit nine rotational components (five P-branch and four R-branch). We have observed seven of these components, the remaining two being at frequencies which are not accessible with our carbon dioxide laser. The 2pt L v = 1 level is expected to have two rotational levels, so that the 1-21 band should possess three rotational components; we have observed all three. Of the total of ten observed transitions we have been able to make independent parallel and antiparaUel measurements of seven, using different laser lines. This has enabled us to determine the earth field penetration factor I-4] for the ion source (0.9939), and therefore to correct the measured resonant ion beam potentials appro- priately. Parallel and antiparallel measurements of the same transition are then

3200 31'94 3188 3182 3176 31'70 BEAM POTENTIAL / V

ae'oo 35~ 3s~ 3~e ~e4 3~o BEAM POTENTIAL/V

FREQUENCY / GHz

Figure 3. Top: recording of the 2ptr. 0, 3-1strg 21, 2 infrared line; Centre: recording of the 2ptr, 1, l-lstrg 21, 0 infrared line; Bottom: recording of the ls% 26, 4-2ptr~, 0, 3 pure microwave line, obtained by monitoring the electric field dissociation of the 0, 3 level. The higher frequency component is induced by microwave radiation propagated parallel to the ion beam, whilst the lower frequency component arises from back- reflected (antiparaUel) microwave radiation. They are separated because of the Doppler effect, and the transition frequencies given in table 1 are extrapolated rest frequencies.

consistent to within 0-001 c m - ~. Figure 3 shows the two infrared lines with the best and the worst signal-to-noise ratio, both measured with a I s output time constant. All of the lines in the 0-21 band can be recorded with signal-to-noise ratios greater than 500 to 1 if high laser powers are used, but the lines are still easily observable at laser powers of 1 W. In no case is any hyperfine splitting of the vibrat ion-rotat ion lines observed; the line widths at high laser powers are between 9 and 18 MHz, and they decrease only slightly when the laser power is decreased. The full experimental results are listed in table 1.

We have already commented on the necessity of tuning the apparatus so that the D § fragments formed in the electric field dissociation lens are separated from other D § fragments formed elsewhere. The electric field, which is probably fairly inhomo- geneous, is established between the earth plates and the inner plates of the lens to which a positive potential is applied. The potential within the lens changes from

Table 1.

Experimental transition wavenumbers (cm-1) for the D~ infrared and microwave 2ptru-lstrg electronic spectrum.

Laser Upper Lower

state state Transition Ion Expt. 2pa u lsag Line wavenumber Dir. energy wavenumber

vN v"N" Isop. N /cm- 1 P/A /eV /cm - 1

2pau, O-lsag, 21 band 0,2-21,3 12 P(22) 942.3833 P 2039

13 R(38 ) 939-9499 A 4470 ~1-21,2 12R(6) 966.2504 P 4019

12 R(2) 963"2631 A 5009 ~0-21,1 12 R(30 ) 982.0955 P 3455

12 R(26 ) 979-7054 A 2182 ~1-21,0 13P(30) 991.0715 P 4106

12 R(42) 988.6466 A 1767 0,2-21,1 12 R(28) 980.9132 A 4427 0,3-21,2 12 R(10) 969"1395 P 5340

12 R ( 6 ) 966.2504 A 3167 0,4-21,3 12 P(18) 945-9802 P 2455

12 P(20) 944.1940 A 1039

1-21 band 1,0-21,1 12 R(40 ) 987-6202 P 5261

12 R(34 ) 984-3832 A 4859 1,1-21,0 13P(28) 993"0426 A 3565 1,1-21,2 12 R(12 ) 970.5472 P 1630

0-26 band 26,4--0,3 Microwave

1-27 band 1,0-27,1 Microwave

941.401

964-838

980.763

989.606

982.421 967.506

944.898

985.968

994.413 969-643

0.39791

0.37176

earth to the applied potential, so that the energy of the predissociating D~ ions will depend upon the position within the lens at which dissociation occurs. This results in fragment D § ions which have a characteristic momentum, so that they are separated from other D § ions which have different momenta by the second magnetic sector [17]. Electric field inhomogeneities lead to a spread in the momenta of the D § fragments, but they can still be separated satisfactorily. The momentum of the desired D § fragments is a sensitive function of the D~ dissociation energy and the applied electric field, which determines the predissociation lifetime. It is difficult to choose the correct setting of the magnetic sector field in advance, but easy to optimize its value once the resonance has been located. Equally, the tuning of the second lens is critical in optimising the separation and detection of the signal D § ions. We encounter particular problems in attempting to detect D~- ions in levels which have very small dissociation energies. The main reason for this is that predissociation occurs rapidly once the ions enter the electric field; consequently they dissociate in a region whose potential is only slightly positive with respect to earth. The resulting D § fragment ions therefore have a momentum close to that of fragment ions formed elsewhere in the apparatus, by photodissociation or collision- induced dissociation, and hence must be detected against the large background noise. This is the main reason for the poorer signal-to-noise ratio obtained for the

720 A. C a r r i n g t o n et al.

1, 1-21,0 t rans i t ion shown in figure 3. I t will be apprec ia ted , therefore, that spect ro- scopic searches are difficult because of the near - imposs ib i l i ty of choosing, in advance, the correct ad ju s tmen t of the second magne t ic sector, d issocia t ing po ten- tial, and focusing and deflect ion poten t ia l s in the lens. We hope these p rob lems will d iminish with fur ther experience in using the field d issoc ia t ion technique.

5. Rotational and vibrational analysis

The infrared exper imenta l da t a presented in table 1 can be ana lysed to p rov ide values of some of the molecu la r constants . However we canno t make full use of the conven t iona l me thods of ana lys ing electronic spec t ra [9-1, s imply because the van der W a a l s state possesses an insufficient n u m b e r of v i b r a t i o n - r o t a t i o n levels. In this sect ion we a d o p t the s t anda rd a n h a r m o n i c osc i l la tor model , and confine our a t ten- t ion to those cons tan t s which can be de te rmined f rom the exper imenta l da t a alone. In w 8 we compare the exper imenta l ly de te rmined molecu la r cons tan t s with those ob ta ined f rom ab initio theoret ica l ca lcula t ions of the po ten t ia l energy curves. We also explore the extent to which exper iment and theory can be c o m b i n e d legiti- mate ly to p rov ide fur ther values of the cons tan t s which charac te r i se the van der W a a l s state.

F o r the l sag , v = 21 and 2pau, v = 0 s tates we assume the s t anda rd three- te rm express ion for the ro t a t iona l energies,

Ev(N ) = Bv N ( N + 1) - D v N2(N + 1) 2 + Hv Na(N + 1) 3

F r o m the measurement s of seven ro t a t i ona l c o m p o n e n t s of the 0-21 band, l isted in table 1, we can de te rmine the three ro t a t iona l cons tan t s (Bo, D~, H~) for each v ib ra t iona l level, and also the band origin, Vo,21. The resul t ing values are presented in table 2.

Since the v = 1 level of the van der W a a l s s ta te possesses only two ro t a t iona l levels, we are able to measure only three ro t a t i ona l c o m p o n e n t s of the 1-21 band, as shown in table 1. C o m b i n i n g these measurement s with those of the 0-21 band we de te rmine direct ly the v = 0 to 1 v ib ra t iona l spacing in the long- range state, AG~/2 . Using the ro ta t iona l cons tan t s de te rmined for v = 21, we are also able to de te rmine the band origin, vl ,2t and the spacing be tween the N = 0 and 1 ro ta t iona l levels of the van der W a a l s state. The values a re given in table 2; we also list the values of ( l / R 2 ) - ~/2 for v = 21 of the g r o u n d s ta te a n d v = 0 of the van tier W a a l s state. N o o ther in fo rmat ion can be der ived f rom the exper imenta l measurement s t aken alone.

Table 2. Molecular constants for the 2pa u van der Waals state and the lstrg v = 21 level of D~. All values are in cm- 1, except for bond lengths which are in A.

1/0,21 = 9 8 9 " 0 4 5 Y1,21 = 9 9 4 ' 2 5 0

2pa u state AG1/2 = 5-205 B 0 = 0.28249 D O = 0-9353 x 10-3

v = 1, N = 0-1 separation = 0'162

l s%, v = 21 B21 = 4-1474 D21 = 3-223 x 10 - 3

H 0 = -0"1508 x 10 -4

H21 = -0"1587 x 10 -5

R 0 = 7"70

R21 = 2"0093

Electronic spectrum (2ptr~-lsag) of the D~ ion 721

6. Microwave spectra The highest bound vibration-rotation levels of the lsag ground state, with

v = 26 and 27, lie very close in energy to the 2pau levels. It was apparent to us that electronic transitions between the two states would lie in the microwave region, and that the dipole moments of these transitions should be very large. Intensity calculations presented in the next section confirm these expectations, and we have suc- ceeded in observing two microwave transitions, 26,4--0,3 and 1,0-27,1. The apparatus used has been described in detail elsewhere [18], so we present here only a brief summary. The drift tube shown in figure 2 is replaced by a much shorter tube, so that there is sufficient space for a waveguide cell to be placed between the drift tube and the electric field dissociation lens. We use the drift tube scanning and modulation scheme described earlier to locate a suitable infrared electronic transition (for example, 0, 3-21, 2) and adjust the source potential so that the ion beam is in resonance with the laser radiation with the drift tube at earth potential. The electric field dissociation lens and magnetic sector are adjusted so that D § ions from the fragmentation of 0, 3 are detected. Microwave radiation is amplitude-modulated and propagated through the waveguide cell; the microwave frequency is scanned and the 26, 4-0, 3 transition detected by measuring the change in D § ion current with a lock-in amplifier referenced at the modulation frequency. The phase of the observed microwave line corresponds to a decreased population of the 0, 3 level at resonance, but the microwave transition can also be detected by monitoring D § fragments from the field dissociation of 26, 4 (the higher of the two levels), in which case the phase of the microwave line corresponds to an increase in the fragment current. Most significantly, the microwave transition can be detected without infrared pumping, and the phase of the pure microwave line, shown in figure 3 (bottom), proves that at the point of entry of the ion beam into the waveguide cell the population of 0, 3 is higher than that of 26, 4. This is the first direct demonstration that the very highest vibration-rotation levels, including those of the 2pa~ van der Waals state, are populated in the ion beam. The 1, 0-27, 1 microwave transition has also been detected by microwave-infrared double resonance, the population of the 1,0 level being enhanced by infrared pumping from the 21, 1 level of the ground state. In this case, however, we are unable to observe the pure microwave spectrum; the microwave double resonance line has a relatively poor signal-to-noise ratio because of the difficulty of separating D § ions which arise from the fragmentation of 1,0 or 27, 1 from other background fragment ions. The measured microwave transition wavenumbers are presented in table 1. As in the case of the infrared lines, the microwave lines, which have a width of 6 MHz, show no evidence of nuclear hyperfine splitting.

7. Nuclear hyperfine splittings We have not observed any nuclear hyperfine splitting of the infrared or micro-

wave lines. However the vibration-rotation levels will be extensively split by the nuclear hyperfine interactions. In this section we describe briefly the origin and form of the splittings, and their spectroscopic consequences.

In the lsag ground state, levels of even N will have even values of the total nuclear spin (I = 2, 0), whilst odd N is associated with I = 1. The inverse rules hold for the rotational levels of the 2per u van der Waals state. The most important

interaction will be the Fermi contact interaction, so that we adopt the coupling scheme

S + I = G , G + N = F

The effective fine and hyperfine hamiltonian may be written in the form

~ e f f = bF S . I + t(2Sz 12 - Sx Ix - Sy Iy) + TS �9 N

where the three terms represent the Fermi contact interaction (bF) , the axial component of the dipolar interaction (t), and the spin-rotation interaction (y). A b initio theoretical values of these constants are presented in the next section. There are many other terms which could contribute to the effective hamiltonian, including the electric quadrupole interaction, the non-axial components of the dipolar interaction, and the purely nuclear interaction terms. These are too small to contribute significantly at the present level of resolution, but their possible effects should not be forgotten.

The allowed values of the quantum numbers are as follows:

I = 0 , G = 1/2, F = N + 1/2,

I = 1 , G = 3 / 2 , F = N + 3 / 2 , N + 1 / 2 ,

G = 1/2, F = N 4- I/2,

I = 2 , G = 5 / 2 , F = N 4 - 5 / 2 , N_+3/2, N4- 1/2,

G = 3/2, F = N 4- 3/2, N ___ 1/2,

except that not all the states exist for N = 0, 1 and 2. The matrix elements of the effective hamiltonian in the above basis are given in the Appendix. Calculation of the hyperfine energies involves, at the most, diagonalizing a set of 2 x 2 matrices, with appropriate values of the three constants inserted. As an example, figure 4 presents theoretical energy level diagrams for the vibration-rotation level 21,2 which has I = 2 and 0, and for 21, 3 with I = 1. The general forms of the hyperfine levels are similar for any v, N levels of the ground and first excited states, except that the splitting of the F states becomes very small close to the dissociation limit because the dipolar and spin-rotation constants are so small.

The electric dipole transition probability of an electronic vibration-rotation component is necessarily diagonal in the quantum number G. Consequently there are only two possible circumstances which could result in the rovibronic lines exhibiting a hyperfine splitting. The first is if the hyperfine constants for the upper and lower levels were sufficiently different. We have observed this situation in our studies of the vibration-rotation spectrum of the H D § ion, because of the increasing asymmetry in the electron distribution for levels near the dissociation limit. However, the calculations for D~- presented in the next section show that the hyperfine constants are likely to be very similar for the lower and upper levels involved in all of the transitions described in this paper. This conclusion depends, of course, on the inversion symmetry of the molecule being preserved. The second situation in which hyperfine splitting could be observed is if G is a poor quantum number; in other words, the off-diagonal matrix elements listed in the Appendix would have to

Electronic spectrum (2paa-lstrg) of the D~ ion

I--2 ,' G = 512

F N + 5 / 2

/ N + 3 / 2 , J ~ - - N + 1 / 2

~ N - 1 / 2 N - 3 / 2

N + 1 / 2

G --1/2

1=2 G =3/2

~ - N - 1 / 2

N - 3 / 2 < f 2 - �9 N - 1 / 2

\ ~ - N + 1 / 2 N + 3 / 2

Figure 4.

21,3 ,

G --3/2

G =1/2

N + 3 / 2 / / ,~. N + 1 / 2 r N - 1 / 2

x N - 3 / 2

N - 1 / 2 N - 1 / 2

Calculated hyperfine structure diagrams for the D~ lsag v = 21, N = 2 and v = 21, N = 3 vibration-rotation levels.

be large, because of large dipolar and spin-rotation interactions. Our ab initio calculations show, however, that only the Fermi contact interaction is likely to be significant. The fact that all the observed lines are unsplit is therefore significant in showing that the inversion symmetry is preserved, but we are, in consequence, unable to measure the hyperfine interaction in the present work. We return to this aspect in w 9. It may be possible to observe radiofrequency/infrared double resonance spectra similar to those we have described for H D § [19], but the situation for D~- is less favourable because the infrared laser drives all hyperfine levels within a vibration-rotation level.

8. Adiabatic calculations

The methods used in our calculations have been described in detail elsewhere [5, 20], and only a brief outline is presented here. In atomic units, the non-relativistic hamiltonian for the internal mot ion of the hydrogen molecular ion is:

~ V20 V2 ~20 Vr V R 1 1 1 = - - + O)

2 2/~ 8# 2/~ a R rle r2e" The position of the electron (rg) is referred to the centre of the internuclear vector (R -- r 2 - rl) , /,t is the reduced mass of the two nuclei, 1//~ a = l/m1 - 1/m 2 ( = 0 for

D~), and rl~ is the distance of the electron from nucleus 1. Exact solutions to the hamiltonian cannot be found; however the electronic Born-Oppenheimer equation,

~ o ( R , rg) = Eo(R)Oo(R, r 0) V 2 1 1 )

2 r l e r2e )

can be solved exactly to give the Born-Oppenheimer potential,

UB~ = Eo(R ) + 1/R.

The solution of (2) is performed in prolate spheroidal coordinates,

(r~ + r2o) 1 ~< ~ < ;] , ] R

(rle - r2e) - 1 ~ rl ~< t l - R '

using the truncated series expansions

(I)o(R, to) = L(R, ~)M(R, rl), (5)

L(R, 3) = exp [--p(~ -- 1)] on(R)L~[2p(~ -- 1)] , (6)

where p2 = _R2E0(R)/2

M(R, rl) = ~ fs(R)P~(rl) (7) s=O

where L, is a Laguerre polynomial and Ps is a Legendre polynomial. For the ground state (ls%) the Legendre expansion of M(R, rl) is restricted to s even, while for the first excited state (2ptru) only terms with s odd appear in the expansion.

More accurate vibration-rotation energy levels may be obtained by averaging the full hamiltonian (1) over the Born-Oppenheimer electronic wavefunction to give an effective, adiabatic potential for the nuclear motion

uAD(R) = Eo(R) + ~ -- *~ ~ ~ *o dr,. (8)

Once a potential has been calculated, the vibration-rotation energy levels are found by solving the radial Schr6dinger equation

{ d2 2# N(N + 1)} - ~ + ~-~ [U(R) - evN] + R 2 ZvN(R) = 0 (9)

where U(R) is given by (3) for a Born-Oppenheimer calculation, and (8) for an adiabatic calculation. N is the rotational quantum number and XvN(R) gives the radial dependence of the vibration-rotation wavefunction. Equation (9) was solved by the Numerov-Cooley algorithm [21], as implemented by LeRoy 1-22]. Since we are interested in energy levels which are close to a dissociation limit, particular attention was paid to the convergence of the energy levels with respect to both the grid interval and the grid range. The calculations for v = 21 of the lstrg state used a grid interval of 0.0004 ~, and a grid range of 0-1 to 32.0 hit. For the 2pa u state and for v = 26 and 27 of the lscr o state the grid range was increased (0.1/~ to 320.0/~,),

which required a corresponding increase in the grid interval to 0-004 A. This latter grid was also used for the calculation of the vibrational overlap integrals and transition dipole moments. The initial electronic calculations were performed on a rather coarser grid 0-1 (0-05) 50.0 (0.5) 160-0 (5.0) 620.0 ao, and interpolated by cubic splines to the grids described above.

The electronic transition dipole moment,

M(R) = fr R)erg r R) drg (10)

connecting the lsag and 2pa. states has also been calculated. This Z-Z transition only has a parallel component of M. The 2pa.-lsag transition is of the charge transfer type, and at large R we have

Mz(R ) "~ eR/2. (11)

From our radial wavefunctions we have computed the overlap integrals,

f Zv,N,(2pa~, R)~v,,N,,(lsag, R) (12) dR

and transition dipole moments,

f zv,N,(2pa~, R)Mz(R)~v,,N,,(lsag, R) (13) dR.

Figure 5 shows the vibrational wave functions for the ground state v = 21, N = 2 and v = 26, N = 4 levels, and the excited state v = 0, N = 3 level. This figure illus- trates the reasons why the transition dipole moment is so small for the 0-21 band, and so large for the 0-26 microwave resonance. Our infrared experiments are successful only because of the intense radiation source and the sensitive detection method.

The electronic wavefunction ~o can be used to calculate the radial dependence of the three hyperfine parameters b~, t and 7 [5] for both the lstrg and 2ptru states. Note that only the first-order contribution to y is calculated, since for the hydrogen molecular ion this is the dominant contribution 1-23, 24]. These results are averaged over X,N to obtain the hyperfine parameters for a particular vibration-rotation energy level, so that, for example,

bF(v, N) = f Zou(R)bF(R)z~s(R) dR, (14)

Table 3 lists the values of the fundamental constants [25] used in our calculations. Tabulations of the Born-Oppenheimer potentials and the adiabatic corrections have previously been given for HD + [5] in both the lsa and 2pa states, and values for D~- are readily obtained from this data. Similarly, values of the R dependence of bF and t (for both nuclei) and 7 for the lsa state of H D + have been presented elsewhere [5], and D~- values may be obtained by simply scaling the results. Hyperfine constants for the 2pa, state of D~- do not appear to have been presented previously; table 4 shows the R dependence of bF, t and ), for this state, and values of M~(R) are given in table 5.

The results of the vibration-rotation energy level calculations are presented in table 6. The dissociation energies are given relative to the appropriate D + D +

726 A. C a r r i n g t o n et al.

,' x , 0 , 3 - 2 6 , 4

I i , J I

0,3- 21,2

I I I I I 0 0 5 10 1 5 2

az~, Figure 5. Top: vibrational wave functions for the lsag v = 26, N = 4 and 2ptr= v = 0, N = 3

levels; Bot tom: vibrational wave functions for the lstrg o = 21, N = 2 and 2pa= v = 0, N = 3 levels.

d i s soc i a t i on l imit , wh ich is - 0 - 5 E n for the B o r n - O p p e n h e i m e r ca l cu l a t i ons and

- 0 . 5 ( 1 - me/mD) E n for the a d i a b a t i c c a l cu l a t i ons (E n be ing the Har t ree) . As t ab le 6

shows, the 2ptr u s ta te s u p p o r t s f ive r o t a t i o n a l levels for v = 0 a n d t w o for v = 1, in

a g r e e m e n t w i t h the p r ed i c t i ons o f P e e k [10] , a l t h o u g h he d id n o t p re sen t c a l cu l a t ed

energies .

Table 3. Values of the fundamental constants used in the adiabatic calculations 1-25].

Deuteron-electron mass ratio (roD~me) Deuteron mass (mD) Planck's constant (h) Bohr radius (ao) Hartree (E H = 2 x R~) Bohr magneton (#B) Nuclear magneton (#N) (#0/4n) = 1/(4ne o c 2) Free electron g factor (ge) Deuteron gyromagnetic ratio (gD)

3670.4830 3.343586 x 10-27kg 6.6260755 x 10-34 js 0.529177249 x 1 0 - t ~ 219474.6307 cm - 1 9.2740154 x 10 -24 jT -1 5-0507866 x 10- 27 J T - 1 10-7 N A - 2 2-002319304 0"85743823

e o is the permittivity of a vacuum, c is the velocity of light.

Table 4. Adiabatic hyperfine parameters for the 2pa~ state of Dr .

R/a o bF/MHz t/MHz ~/MHz

1.0 63.863 6.491 -22-246 2.0 141.021 0-852 -6.515 3.0 142.499 0.081 --0.477 4.0 130.460 0.122 0-709 5.0 121.138 0-154 0-768 6.0 115.450 0.144 0.617 7.0 112.281 0-119 0.461 8-0 110.626 0.093 0.339 9-0 109-809 0.072 0.250

10-0 109-429 0.055 0-187 11-0 109-264 0.042 0.142 12.0 109.199 0.033 0.110 13.0 109.178 0.026 0.087 14-0 109-174 0.021 0-070 15.0 109.177 0.017 0.057 16.0 109.181 0.014 0.047 17.0 109.185 0.011 0-039 18.0 109.189 0-010 0.033 19-0 109.192 0.008 0-028 20.0 109.194 0-007 0.024 30-0 109.203 0.002 0.007

Table 5. Parallel electronic transition dipole moment between the 2pa u and lsag states of D~.

R/a o M Jeao

1"0 0-6748985 2.0 1"0499426 3-0 1-4326868 4"0 1"8709098 5"0 2"3606948 6"0 2"8790850 7-0 3-4043259 8"0 3"9255941 9"0 4"4419782

10"0 4"9535077 11"0 5"4619056 12.0 5-9681850 13.0 6.4730064 14-0 6-9767947 15"0 7"4798295 16"0 7"9823006 17"0 8"4843408 18.0 8"9860456 19-0 9-4874852 20"0 9'9887121 30"0 14"9949953

1 ea o = 8'47836 x 10-3~ [-25]. = 2"54175 D [26].

728 A. Carrington e t al.

Table 6. Born-Oppenheimer and adiabatic energy levels and adiabatic hyperfine parameters for v = 21, 26 and 27 of the D~ 1sag state, and for v = 0 and 1 of the 2pa u state.

Born-Oppenheimer Adiabatic v, N (EoN -- ED) /cm- 1 (EoN -- ED)/cm- 1 bF/MHz t/MHz y/MHz

lstrg state 21,0 -992.336 -994.120 110-407 1.486 5.364 21,1 -984.058 -985.838 110.387 1.481 5.344 21,2 -967.581 -969.354 110.348 1.471 5.304 21, 3 - 943.063 - 944.825 110.290 1.455 5.243 21,4 -910.740 -912.487 110.215 1.435 5-163 21,5 -870.925 -872-653 110-123 1.410 5.063

26,4 -1-797 -1.867 109.182 0-065 0.231

27,1 -0.660 -0.676 109.192 0"017 0-059

2ptr ,, state 0,0 --5-510 --5-513 109.189 0"019 0"065 0, 1 --4.949 --4.952 109.188 0.019 0.063 0,2 --3.853 --3.856 109.188 0'018 0.060 0,3 --2.280 --2.283 109.188 0"017 0"056 0,4 --0.356 --0-359 109.189 0"014 0-047

1,0 --0.308 --0.308 109.202 0.004 0.013 1, 1 --0.147 --0"147 109.202 0"003 0"010

The calculated hyperfine constants are consistent with the qualitative remarks made in w 7. For the v = 21 and higher levels of the ground state and also for both vibrational levels of the van der Waals state the constants t and ~ are extremely small, while b E is close to half the value for a D atom with an infinitely heavy nucleus (218.410MHz), and changes very little between the v = 21 ground state levels and the excited state levels.

The calculated vibrat ion-rotat ion energies may be compared with the experimental wavenumbers for the infrared 0-21 and 1-21 bands, and also the two microwave transitions, as shown in Table 7. The adiabatic calculations give much better agreement with experiment than do the Born-Oppenheimer calculations, but even at the adiabatic level there is an obvious difference between experiment and theory for the 0-21 and 1-21 bands of about 0 .44cm-1. This discrepancy arises because a number of small but significant corrections have been omitted, namely, nonadiabatic, relativistic and radiative effects. In their calculations on H D +, Wolnie- wicz and Poll [12] have shown that when these corrections are included, the resulting energy levels give transition wavenumbers which agree with experiment to within 0.01 c m - 1 or closer. We have made estimates of the corresponding corrections for D~- in an at tempt to improve the level of agreement between experiment and theory.

(i) R a d i a t i v e e f f e c t s

The most extensive study of the radiative corrections for the hydrogen molecular ion (H~) is due to Bishop and Cheung [27]. They calculated the R-dependent radiative correction (in c m - 1) for the lsag state from the expression

AErad(R) = 0.227419.781 - In ( k o / E n ) ] ( p ( R ) / a o 3), (15)

Table 7. Comparison with experiment of the Born-Oppenheimer and adiabatic predictions for the 0-21 and 1-21 bands, and two components of the 0-26 and 1-27 bands of the 2ptru-lsag electronic spectrum of D~.

Transition Experiment B.O. theory Exp.-B.O. Adiabatic Exp.-Ad. v, N-v, N /cm- 1 /cm- 1 /cm- 1 /cm- 1 /cm- 1

2ptr,, 0-1s%, 21 band 0, 4-21, 5 - - 870.569 - - 872.294 - - 0, 3-21, 4 - - 908.460 - - 910.204 - - 0, 2-21, 3 941.401 939-210 2.191 940-969 0.432 0, 1-21, 2 964.838 962.632 2.206 964.402 0.436 0, 0-21, 1 980-763 978.548 2-215 980.325 0-438 0, 1-21,0 989"606 987.387 2.219 989.168 0.438 0, 2-21, 1 982.421 980.205 2.216 981.982 0.439 0, 3-21, 2 967'506 965.301 2.205 967.071 0.435 0,4-21, 3 944-898 942.707 2-191 944.466 0-432

1-21 band 1, 1-21, 2 969.643 967.434 2-209 969.207 0.436 1, 0-21, 1 985-968 983.750 2.218 985.530 0.438 1, 1-21,0 994.413 992-189 2.224 993-973 0.440

0-26 band 0,3-26,4 0"3979 0.483 -0"085 0.416 -0.018

1-27 band 1,0-27, 1 0-3718 0-352 0.020 0.368 0.004

where In (ko/EH) is the Bethe logarithm, and p(R) is the electron density at a nucleus. The evaluat ion of the Bethe logari thm is very difficult. Close to equilibrium (R ~- 2ao), Bishop and Cheung recommended a value of 2.35 for the Beth6 logarithm, but were unable to provide reliable values at much larger internuclear separations. Our interest is in the highest vibrational energy levels, and it will be better to use the H a tom value for the Bethe logarithm, 2.291 [28], so that

AErao(R) = 0"227419"781 - 2"291](p(R)/ao 3). (16)

AErad(R) is then averaged over the radial wavefunction to give the correct ion (in c m - 1) for a part icular v ibra t ion- ro ta t ion energy level

AEraa(v, N) = 0"227419"781 -- 2"291](p(g)/aoS)v,N. (17)

The value of (p(R))v. N is easily extracted from the radially-averaged Fermi contact parameter. At the dissociation limit AEr~d(oo) = 0.271 c m - 1 , so that inclusion of the radiative corrections gives a correct ion to the energy relative to the dissociation limit of:

(AErial(v, N) -- 0-271) cm - 1. (18)

The resulting corrections for v = 21 of the g round state are given in table 8; they are insignificant for the 2ptr u state.

(ii) Relativistic corrections

In order to evaluate the relativistic corrections, we have averaged the R- dependent relativistic corrections reported by Bishop for the lsag state [29] over the

Table 8.

A. Carrington et al.

Radiative, relativistic and nonadiabatic corrections to the adiabatic dissociation energies of the D~ ls% energy levels.

Level Radiative Relativistic Nonadiab. Corrected (EvN -- ED) (v, N) /cm- 1 /cm- 1 /cm- l /cm- 1

21,0 0-003 0.059 -0.515 -994.573 21, 1 0-003 0.059 -0.514 -986.290 21,2 0.003 0.059 -0.512 -969.804 21, 3 0.003 0.059 -0.509 -945.272 21,4 0.003 0"059 -0.505 -912.930 21, 5 0.002 0-059 - 0-499 - 873-091

26, 4 0.000 0-002 -0-018 - 1-883

27, 1 0-000 0.001 -0.005 -0.680 i

radial wavefunction:

N) = f zo (g)aEr~ dR. (19) AErel(v,

At the dissociation limit the relativistic correction is -1 .461 cm-1 , so the net correction is,

(AEre~(v, N) + 1.461)cm- 1, (20)

No tabulations of AEre~(R) are available for the 2pau state, but we anticipate that the values will be very close to the D atom value ( - 1-461 cm-1), and will therefore have very little effect on the calculated dissociation energies. The relativistic corrections for v = 21, 26 and 27 of the lsa~ state are given in table 8.

(iii) N onadiabatic corrections

Wolniewicz and Poll 1-12] have calculated the nonadiabatic corrections to the vibrat ion-rotat ion energy levels of the ground states of H~ and H D + by a variat ion-perturbat ion method. They presented their results in the form of an expansion:

AE"aa(v, N ) = A v + B v N ( N + 1) + C v N 2 ( N + 1) 2. (21)

The origin of these corrections can be understood classically. In an adiabatic calculation it is assumed that the electron can follow the motions of the nuclei exactly, so that the electronic wavefunction and energy depends only on the positions of the nuclei, and not on how they are moving. An adiabat ic potential function (8) can then be calculated. The electron is, however, unable to follow the motions of the nuclei exactly; the separation of nuclear and electronic motions is therefore not complete, and a potential function can no longer be defined. For the hydrogen molecular ion the effects of this electron slippage are quite small, and may be introduced as small corrections to the energy levels. The electron slippage may be described quantum mechanically by the mixing of excited electronic states into the ground state. The mixing occurs as a result of the kinetic energy operators in the hamiltonian (1) which are proport ional to 1/# (and also an additional operator proportional to 1/#a for HD+). As the dissociation limit is approached, the correc-

Electronic spectrum (2pau-lstrg) of the D~ ion 731

tion A o gets smaller. Recalling the interpretation of the nonadiabatic corrections given above, this is because the nuclei are most likely to be found at large internuclear separations where they are moving very slowly, and the electron slippage will therefore be very small.

These ideas suggest that Ao, the reduced mass #, and ED -- Eo0 (the energy of the level (v, 0) below the dissociation limit) should satisfy a relationship of the form:

A o ~- # - m f ( E D - - Eoo ) (22)

where f(ED -- Evo) is an unspecified #-independent function. Using Wolniewicz and Poll's values of A o for H~- and HD § it is indeed found that a relationship of the form (22) with m = 1 is satisfied quite well. From a plot of #Ao against (Eo - Eoo) we use the reduced mass of D~- to obtain an estimated A o value of -0 .523 cm-1 for the v = 21 level. In a similar manner it was discovered that the nonadiabatic parameter Bo is given by

B o _~ #-2g(E D - - Evo ) (23)

(although here the HD + and H f values agree less well), and a plot of#2Bv against (ED -- Eoo) gave an estimate for B21(D~) of 0.00052 cm-1. Co is extremely small and was therefore neglected. The correct dissociation energy for a nonadiabatic calculation of D~- is,

- -0 .5( - m D ~E u (24) \me + mDJ

which lies 0.008 c m - t below the adiabatic limit given previously, so that the effects of nonadiabatic corrections on the energies of v = 21 relative to the dissociation limit are to add

--0.523 + 0-008 + 0-00052 N(N + 1)cm -1 (25)

to the tabulated adiabatic energies. Estimates of the nonadiabatic corrections for v = 26 and 27 have been made in a similar fashion, and the results are given in table 8. Nonadiabatic calculations have not been reported for the 2ptr u state, but for the bound levels the nuclei will be moving slowly, and the nonadiabatic corrections are therefore likely to be very small.

When all three of the corrections described above are applied to the adiabatic energy levels, and the resulting new energy levels used to calculate the transition wavenumbers, the level of agreement with experiment for the 0-21 and 1-21 bands improves to --- 0.014 c m - t (see table 9). The remaining discrepancy between theory and experiment is the same for both the 1-21 and 0-21 bands and is independent of N. This is consistent with a slight overestimation of AZl, and experiment and theory would be brought into agreement by taking A 2 1 = --0"509 cm-1.

In table 10 we list the calculated values of the vibrational overlap integrals (12) and the transition dipole moments (13). For the infrared transitions the vibrational overlap integrals are very small for the 0-21 and 1-21 bands, as the radial wave functions plotted in figure 5 demonstrate. However the small magnitude of the overlap integral is partially compensated for by the large value of the charge transfer electronic transition moment, which is close to 11 D in the region where the two wavefunctions overlap (R -~ 4.65 A). In contrast the vibrational overlap integrals for

732 A. Ca r r i ng ton et al.

Table 9. Comparison of experimental transition wavenumbers with adiabatic predictions including nonadiabatic, radiative and relativistic corrections to the lstrg vibration- rotation levels.

Transition Experiment Corrected adiabatic Expt.-C. Ad. v, N-v, N / cm- 1 /cm - 1 /cm- 1

2pau, 0-1s%, 21 band 0,4-21, 5 - - 872-732 - - 0, 3-21,4 - - 910-647 - - 0, 2-21, 3 941.401 941-416 -0-015 0, 1-21, 2 964.838 964.852 -0-014 0, 0-21, 1 980-763 980.777 -0-014 0, 1-21, 0 989.606 989-621 -0 .015 0, 2-21, 1 982.421 982.434 -0-013 0, 3-21, 2 967.506 967.521 -0 .015 0, 4-21, 3 944.898 944-913 -0-015

1-21 band 1, 1-21, 2 969.643 969.657 -0-014 1, 0-21, 1 985-968 985.982 - 0.014 1, 1-21,0 994.413 994.426 -0 .013

0-26 band 0, 3-26, 4 0-3979 0.400 - 0.002

1-27 band 1, 0-27, 1 0-3718 0.372 0.000

Table 10. Vibrational overlap integrals and transition dipole moments for the components of the 2ptr~-ls% spectrum measured in this work. (The numbers given in brackets refer to the two transitions which we are unable to measure).

Transition Vibrational overlap Transition dipole v, N-v, N integral/10-3 moment /10-2D

2pau, O-lstrg, 21 band 0,4-21, 5 (0-49251) (0-50553) 0, 3-21,4 (0-57170) (0.58305) 0, 2-21, 3 0.60645 0"61533 0, 1-21, 2 0"61800 0"62471 0,0-21, 1 0.61410 0-61930 0, 1-21,0 0-57104 0"57538 0, 2-21, 1 0'53139 0-53626 0, 3-21, 2 0.47474 0.48044 0,4-21,3 0"38718 0-39344

1-21 band 1, 1-21, 2 0-22365 0-22568 1, 0-21, 1 0.25617 0.25789 1, 1-21,0 0-20674 0-20794

0-26 band 0, 3-26, 4 928-2 1907.2

1-27 band 1, 0-27, 1 860-65 3025-2

Electronic spectrum (2pou-lsao) of the D~ ion 733

the microwave transitions are very large, as is made clear in figure 5. Coupl ing the large vibrational overlap integrals with the strong charge-transfer electronic transition momen t gives the microwave transitions massive transit ion dipole moments.

We have used the Born -Oppenhe imer and adiabatic potential curves to calculate the energy levels directly, but it is also possible to obtain many of the molecular constants (such as Re, oge, By , . . . ) directly f rom the potential curves, wi thout first calculating the v ibra t ion- ro ta t ion energy levels. Fo r the 2pa u state, R e was found by using a 5-point numerical differentiation formula [30] to give values of dU/dR in the region about R = 6 . 6 4 / ~ ; f rom these data R e and D e were obtained, since dU/dR = 0 at R = R e . o~ e can also be calculated by using a 5-point differentiation formula [30] to give

d2U) (26) k = dR2,]a=R ~

so that

1 ( ~ ) 1/2 ~ = ~nc (27)

The resulting values are as follows:

Born -Oppenhe imer Adiabat ic Re/~ 6.640 6.640 Ddcm - x 13.342 13.346 ~odc m - 1 20"07 20-09

The rotat ional constants are obtained by treating the centrifugal energy term h2N(N + 1)/2#R 2 as a per turbat ion to the vibrational (N = 0) energy levels; they are listed in table 11. The method of Hu t son [31] was used to obtain B, D, H, L and M for v = 21 of the lsag state, and for v = 0 and 1 of the 2pau state. The series

Table 11. Rotational constants for v = 21 of the lscrg state of D~ and for v = 0 and 1 of the 2pG state, derived directly from the ab initio potentials by the method of Hutson [31].

B/cm- 1 Diem - 1 H/cm - 1 L/cm - 1 M/em- 1

ls%, v = 21 B.O. 4.14576 0"33809(- 2) 0"6568( - 6) -- 0" 1270( - 8) - 0" 1469( - 15) Adiab. 4"14755 0.33070(-2) 0"6577(-6) -0-1267(-8) 0.2777(-14) Exp. 4.14740 0-32230(- 2) - 1.5873(- 6) 0"0 0-0

2per u , v = 0 B.O. 0"28247 0"9903(- 3) --0"7731(- 5) - 0'1225(-- 6) --0-2527(- 8) Adiab. 0.28243 0.9894(- 3) - 0"7717( - 5) - 0.1222( - 6) - 0.2519( - 8) Exp. 0.28249 0-9353(- 3) - 1.5079(-- 5) 0.0 0-0

2pau, v = 1 B.O. 0.08649 0.2785(-- 2) -0 ' 9753( -4 ) --0-7894(- 5) -0 .8811( - 6) Adiab. 0.08654 0-2782(-2) -0 .9726( -4) -0-7859(-5) -0 .8758( -6)

converges quite rapidly for v = 21, and the values obtained from the adiabatic potential are in good agreement with those derived from the experimental data. For the 2pa~ state the agreement between the experimental and ab initio values of Bo and D O is also quite good, but the convergence of the series, even for v = 0, is slow. Using constants up to and including M the calculated energy difference between (0, 0) and (0, 4) agrees with the ab initio value to no better than 0.009 cm-1. For v = 1 the convergence is even slower, but the calculated separation between N = 0 and 1 using the constants listed in table 11 is 0.161 cm-1, in excellent agreement with the measured value. The ab initio adiabatic rotational constants given for v = 1 can therefore be regarded as satisfactory, and the calculated value of R~ is 13.9/~. However, because of the extreme anharmonicity and shallow depth of the van der Waals state potential, attempts to derive equilibrium values of vibrational or rotational constants are unjustified.

9. Discussion

The main purposes of the work described in this paper were to measure, for the first time, the electronic spectrum of the hydrogen molecular ion (in its perdeutero form, D~), and to characterize by spectroscopy the long-range 2ptr u van der Waals state, which was predicted theoretically many years ago, but had never been studied experimentally. Other purposes of the work were to determine the symmetry of the homonuclear molecule at its dissociation limit, to test the accuracy of different types of ab initio calculations, and to investigate the high-lying energy level populations which result from the ionisation of the neutral molecule. The experimental techniques which have been developed are also of interest, and are likely to be applic- able to other systems.

Both the Born-Oppenheimer and adiabatic calculations predict that the D~ 2pry u long-range minimum supports seven bound levels: v = 0 has five rotational levels and v = 1 has two. We have detected and measured transitions involving all of these levels. Even so, there is insufficient experimental data accessible for us to be able to perform a rigorous vibrational and rotational analysis, and the anharmonic nature of the potential means that our truncated series expansions for the vibrational and rotational energies are inadequate. However our experimental results provide confidence in the theoretical predictions, particularly those of the adiabatic calculations, and we feel that the constants given in table 2 are reasonably accurate. Born-Oppenheimer calculations for the higher excited electronic states of H~" [32] show that, in addition to the 3dcrg and 2pnu states which are genuinely bound, deeper and much longer-range van der Waals minima exist in a number of states which are otherwise repulsive, because the polarizability of the hydrogen atom increases with electronic excitation. Experimental studies of these minima would be of great interest.

Table 6 shows that the Born-Oppenheimer and adiabatic methods give remark- ably close vibration-rotation energies for the 2ptr~ state, and there seems little doubt that nonadiabatic contributions to the energies will be extremely small. For the v = 21 levels of the lsag state, however, the two ab initio methods differ by about 1.8 cm-1. If the Born-Oppenheimer calculations alone had been available to guide our experiments, we might well have failed to observe the D~" spectrum. As it is, our

uncertainty concerning the size of the nonadiabatic correction (which we now know to be about 0.5 cm- 1) resulted in long searches for the first spectroscopic line.

We chose to study D~ because calculations predicted that the 0-21 and 1-21 bands of the electronic spectrum would lie in the infrared region spanned by the carbon dioxide laser. We have no doubt that the experimental methods described in this paper would be equally successful in detecting the analogous spectrum of H~-, but unfortunately the predicted vibronic bands fall outside the carbon dioxide laser region. The 0-15 band origin in H~- is predicted to occur at 847cm -1, compared with the lowest carbon dioxide laser wavenumber available to us which is 874 cm- 1. The next band in H~', 0-14, is expected to occur at 1397 cm-1, compared with our upper experimental limit of 1094cm-1. Similar problems arise if one considers the possibility of using a carbon monoxide laser. The 0-13 band origin should occur at 2074 cm-1, a difficult region for the carbon monoxide laser although not impossible. However the vibronic bands at higher wavenumbers are predicted to have smaller transition dipole moments, and the laser power available from carbon monoxide is much less than that available from carbon dioxide. Our best chance of detecting the H~ 2pa u state would seem to lie with microwave spectroscopy, and we return to this point later.

In the case of HD § the 2ptr(v = 0)--lsa(v = 17) band is predicted to occur in the region 950 to 1050 cm-1, which is certainly conveniently in the range of the carbon dioxide laser. However the 2pa electronic state correlates with the H + D § dissociation limit, and its vibration-rotation levels therefore lie above the H § + D limit. Davis and Thorson [33] have calculated their predissociation lifetimes, and spectroscopic linewidths of several cm- 1 are predicted; this spectrum is therefore probably not accessible to us.

One of the most interesting questions which arises is whether, under all circumstances, the inversion symmetry of the electronic states is preserved in the homonuclear molecules, H f and D~-. We have established that in H D § the electron distribution becomes increasingly asymmetric in energy levels which approach the dissociation limit, H + + D. In the highest bound vibrational level, v = 22, the nuclear hyperfine interaction reveals that the electronic wavefunction is essentially D(ls), with almost no contribution from H(ls). These results are understandable, however, since the H + + D dissociation limit lies 29.8cm -1 below the H + D § dissociation limit. In the homonuclear molecules the dissociation limits are doubly degenerate, provided we ignore effects arising from nuclear spin. The infrared and microwave lines we have observed show no evidence of nuclear hyperfine splitting, for Which we can provide two different explanations. The first is that the electron distribution is equally asymmetric in all the levels studied, so that the hyperfine splittings are the same in both levels involved in each transition. This is extremely unlikely; we have, for example, studied the 1,0-27, 1 transition by microwave spectroscopy and the 1,0-21, 1 transition in the infrared, but both transitions exhibit single lines. We would, therefore, require any asymmetry in the electron distribution to be the same for all three levels. The second explanation, which must be correct, is that the electron distribution is symmetric in all of the lstr 9 and 2pa, levels studied, including those closest to the dissociation limit. This conservation of the inversion symmetry arises because of the indistinguishability of the two nuclei [34], irrespec- tive of the distance between them.

A different situation arises when we take the nuclear hyperfine interaction into account. The hydrogen atom, for example, exists in two hyperfine states with J = 0

and 1, which are separated by 1420.41 MHz. In H ; , therefore, there are actually four dissociation limits involving hydrogen atoms in their ground electronic state, one degenerate pair which involves J = 0 hydrogen atoms being separated by 1420.41 MHz from the other degenerate pair with J = 1 hydrogen atoms. It would therefore be extremely interesting to study a vibration-rotation level of H~- which had a dissociation energy small compared with the hyperfine splitting. Theoretical calculations for H~- actually predict that the highest 2ptr u level, with v = 1, N = 0, has a dissociation energy of only a few MHz. The N = 0 level in the 2ptru state has total nuclear spin I = 1, and in the coupling scheme described in w 7, the quantum number G has values of 3/2 and 1/2. Near to the dissociation limit we normally expect the proton Fermi contact interaction constant by to have a value close to 710MHz, with all other constants being very small. The hyperfine levels in the N = 0 state should therefore be split by about 1065 MHz (i.e. (3/2)bF) whereas at dissociation the hydrogen atom hyperfine splitting will be 1420.41 MHz. We do not expect the H~ ion to show unequal hyperfine splittings for the two protons, but the transition from atomic to molecular state is likely to reveal an unusual size of hyperfine splitting. In D~" there are also four lowest dissociation limits, the deuterium atom having J = 1/2 and 3/2 states separated by 218-05 MHz (0-0073 cm-1). However, the highest bound levels of the 2ptr. and lscrg states have dissociation energies of 0-147 and 0.136cm -1, which are very large compared with the deuterium atom hyperfine splitting.

The observation of the pure microwave transition, described in w 6, is of particular interest since it establishes that the vibration-rotation levels of the 2pa. state in D~- are populated by the formation processes occurring in the ion source. The fact that we find the population of the 2ptr. 0, 3 level to be higher than that of the l sa o 26, 4 level in the ion beam may be due to preferential electric field dissociation of ions in 26, 4 during the process of acceleration from the source, where the ion beam necessarily passes through electric fields with strengths up to 10 kV cm- 1. We would expect the initial electron impact ionization of D 2 (in v = 0) to populate all of the ground state vibrational levels of the ion, including v = 26 and 27. However, even the inner (classical) turning point of the 2ptr. potential is at 5.6 A, and direct population by the electron impact process seems unlikely. Under the conditions of our experiment, however, D~ ions are formed in high yield, and it may be that the observed 2pau D~- ions are formed by the dissociation (perhaps through gas collisions) of D~ in the ion source. It is also possible that passage of the ion beam through the accelerating electric field results in some population transfer from the lstr o to the 2ptr~ levels.

We have already noted that the infrared electronic spectrum of H~ is not accessible with the carbon dioxide laser, but the 2ptr, 0 , 2 - 1 s a o 19, 1 and 2ptr. 1 ,O- l sa o

19, 1 pure microwave transitions are predicted to occur at 17.748 and 6.386GHz. The levels involved in these transitions lie extremely close to the dissociation limit, and although they may be populated in the ion source, their populations are likely to be depleted during the process of acceleration from the ion source. It remains to be seen whether these microwave transitions will be detectable.

A.C. thanks the Royal Society for a Research Professorship, I.R.M. thanks the University of Southampton for a Research Fellowship, and C.A.M. thanks the British Petroleum Company plc for a Research Studentship. We are indebted to the SERC for grants towards the purchase of equipment.

Electronic spectrum (2p%-lsa~) of the D~ ion

Appendix

I = 0 F = N + 1/2, F = N - - 1/2,

I = 1 F = N + 3 / 2 , F = N - - 3 / 2 , F = N + 1/2,

F = N - 1/2

I = 2 F = N + 5 / 2 , F = N - - 5 / 2 , F = N + 3 / 2 ,

F = N - 3/2,

F = N + 1/2,

F = N - 1/2,

D~ matrix elements of the effective hamiltonian

G = 1/2, E = ~ N / 2 , G = 1/2, E = - 7 ( N + 1)/2.

G = 3 / 2 , E G = 3 / 2 , E M(3/2 , 3/2) M(1/2 , 1/2) M(3/2 , 1/2) M(3/2 , 3/2) M(1/2 , 1/2)

M(3/2 , 1/2)

= bF/2 + 7N/2 - tN/(2N + 3), = bF/2 -- ~(N + 1)/2 -- t(N + 1)/(2N - l), = b J 2 + ~(N - 3)/6 + t(N + 3)/(2N + 3),

= - - b F - - 7N/6, = --(y/3)[N(2N + 3)] x/2 - t[N/4(2N + 3)] ~/2,

= bE~2 -- y(N + 4)/6 + t(N -- 2)/(2N - 1), = - b y + 7(N + 1)/6, = - ( 7 / 3 ) [ ( N + 1X2N - 1)] ~/2 + t[(N + 1)/4(2N -- 1)] x/2

G = 5/2, E = b F -I- ~N/2 -- 2tN/(2N + 3), G = 5/2, E = bF -- 7(N + 1)/2 -- t(N + 1)/(2N -- 1), M(5/2 , 5/2) = bF + ~,(3N -- 5)/10 + 2t(N + 15)/5(2N + 3), M(3/2 , 3/2) = - - 3 b F / 2 -- 3) ,N/10 + 3tN/5(2N + 3), M(5/2 , 3/2) = - ( ~ / 5 ) [ 2 N ( 2 N + 5)] x/2

- - 9t[2N(2N + 5)]1/2/10(2N + 3), M(5/2 , 5/2) = bF -- 7(3N + 8)/10 + 2t(N -- 14)/5(2N - 1), M(3/2 , 3/2) = - -3 b F /2 + 37(N + 1)/10 + 3t(N + 1)/5(2N -- 1), M(5/2 , 3/2) = -- (~/5)[2(N + 1)(2N - 3)] 1/2 + (Dt/lO)[(2N + 2)(2N -- 3)]X/E/(2N - 1), M(5/2 , 5/2) = by + 7(N - 8)/10 + 4 t (4N 2 + I 0 N - 21) /5(2N - 1)(2N + 3), M(3/2 , 3/2) -- - -3bE/2 -- ~(N -- 3)/10 - 3t(N + 3) /5(2N + 3), M(5/2 , 3/2) = - - (7 /5 ) [3 (N + 2)(2N -- 1)] x/2 -- 3 t (2N - 9 ) [3 (N + 2)]~/2/10(2N + 3)(2N - 1) ~/2, M(5/2 , 5/2) = bF -- 7(N + 9)/10 + 4 t (4N 2 - 2 N -- 27) /5(2N -- 1)(2N + 3), M(3/2 , 3/2) = - - 3 b F / 2 + 7(N + 4)/10 -- 3t(N -- 2) /5(2N - 1), M(5/2 , 3/2) = -- (7 /5)[3(2N + 3)(N - 1)] ~/2 + (3t/lO)(2N + l l ) [ 3 ( N - 1)]~/2/(2N -- 1X2N + 3) ~/2.

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chem. Phys., 77, 34. [3] WING, W. H., RUFF, G. A., LAMa, W. E., and SPEZESKI, J. J., 1976, Phys. Rev. Lett., 36,

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[8] COULSON, C. A., 1941, Proc. R. Soc. Edinb. A, 6I, 20. [9] HERZBERG, G., 1950, Spectra of Diatomic Molecules (D. van Nostrand Co. Inc.).

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