- Email: [email protected]
Effects of polarization in the field ionization spectrum of H2O
Volume 287, number I ,2
29 November1991
CHEMICAL PHYSICS LETTERS
Effects of polarization in the field ionization spectrum of HZ0 R.D. Gilbert and M.S. Child Department ofTheoretical Chemistry, University of
Oxford, 5 South Pa& Road, O&d
OX1 3UB, UK
Received 15August 1991
Anomalous linesinthe ZEKE field-ionization spectrum of Hz0 are explained via a mechanism of energy redistribution between core rotation and the Rydbergelectron,mediatedby the Rydberg-core-dipoleinteraction.A quantum defectmodelof the interaction, involving the coupling of two closed channels, is discussed in relation to predictions ofthe intensities of the observed lines. Experimental implications of this polarization mechanism are suggested.
1. Introduction The rotationally resolved ZEKE field-ionization spectrum of Hz0 recently reported by Tonkyn et al. [ 11 is of added interest due to the appearance of certain anomalous lines. It is known in diatomic molecules that radiative transitions to high Rydberg states, since they involve the change from Hund’s case (b) to case (d), are governed by quite restrictive propensity rules which depend on the angular momentum properties of the initial state. Extension of these rules to polyatomic molecules [ 21 requires that the strong ndt lb, series in Hz0 show overall type C rotational structure, since lb, is nearly a pure p orbital. Most of the strong lines in the ZEKE spectrum conform to this prediction, but a few almost equally strong lines can be confidently assigned to type A transitions with Kg’=Kz = 0.Some weak type A lines are to be expected as limits of the weaker npt lb, series, but for these K: = K: =O is not allowed, since the propensity rules include 1KI: -td" I= 1K,+1,and the nodal plane of the b, orbital excludes contribution of atomic orbitals with J”=O. The aim of this paper is to suggest a mechanism to account for these anomalous lines. In the experiment, water is irradiated in a time-of-flight spectrometer with VLJVlight in a frequency range just below the ionization continuum and with a band-width of about 1 cm-‘, After a short delay on the order of a us, a small ( -0.3 to -0.5 V/cm) electric field is pulsed, sufficient to ionize the near-threshold Rydberg states. The resulting flux of near-zero-velocity electrons is recorded as a function of the VUV excitation frequency. Throughout the experiment, a small dc voltage ($0.05V) is maintained across the extraction plates to sweep out any electrons produced by direct ionization. The ZEKE spectrum therefore shows peaks at frequencies corresponding to excitation to Rydberg levels lying a cm-’ or two below the ionization limits associated with different rotational states of the positive ion. The rotational origin of the spectrum depends on the magnitude of the ionizing field. For the published spectrum [ 11, the origin is found to be 101764.5? I cm-‘. Tonkyn et al. measured the red shift due to the field and deduced an ionization potential for Hz0 of 101766-t 1 cm-‘, which is to be compared with the ionization limit of 101772 cm- ’ deduced from extrapolation of the Rydberg series [ 21. The observed spectra are reproduced in figs. 1 and 2 along with simulations based on H&l-London formulae [ 21. The observed spectrum in the center of figs. 1 and 2 is to be compared with the calculated intensity profile corresponding to a pure nd (case (d) )cpb, (case (b) ) transition, shown at the bottom, and the profile corresponding to a pure np (case (d))+db, (case (b)) transition, shown at the top. The peaks in figs. 1 and 2 are keyed to the frequencies tabulated in tables 1 and 2 for cross-reference. All the intensities are scaled to aid 0009-2614/91/S 03.50 0 1991 Elsevier Science Publishers B.V. All rights reserved.
153
Volume 287, number I,2
29 November 1991
CHEMICAL PHYSICS LETTERS
d’
b’ I
rw
3170d
b 10:iEO[ FREOL JENI
:Y
3 ICM-11
Fig. 1. Pulsed field ionization spectrum of HZO, x25 K. Top: calculated spectrum using HGnl-London intensities for np (case (d ) ) cdb, (case (b) ) Middle: experimental trace from ref. [ I]. Bottom: calculated spectrum using Hb;nl-London intensities for nd (case (d)) +pb, (case (b)). Calculated spectra are convolved with Lorcntzians of width 1cm-‘, and all intensities are scaled to aid comparison. Peaks are keyed for cross-reference with table I.
FREQUENCY
(CM-11
Fig. 2. Pulsed field ionization spectrum of DZO, x 50 K. Top: calculated spectrum using HiSnl-London intensities for np (case (d))+-db, (case (b)).Middle:experimental trace from ref. [I]. Bottom; calculated spectrum using Hiinl-London intensities for nd (case (d))+pb, (case (b)). Calculated spectra are convolved with Lorentzians of width 1cm-‘, and all intensities are scaled to aid comparison. Peaks are keyed for cross-reference with table 2.
comparison. In tables 1 and 2, all the experimental lines above a given threshold of intensity are tabulated with proposed assignments. In the H,O spectrum, 8 of the 16 tabulated lines are seen in the nd+ lb, calculation. Although the relative intensities are only approximately simulated, the calculation seems reliable as a guide in suggesting likely assignments. Only two of the observed lines are seen in the npc- lb, calculation, and the observed intensity of the one labelled a’ is much higher than calculated. The peaks marked with arrows in figs. 1 and 2 are not fully assignable to either of the calculated spectra, These are the “anomalous” lines which are to be interpreted by the mechanism to be proposed in what follows. The most obvious anomalies are the transitions with K:: = Kb = 0. These cannot be due to any direct one-photon process, since they are forbidden by selection rules which depend only on the symmetry of the ground state. Because of experimental constraints, the DzO spectrum of Tonkyn et al. is both hotter and less fully resolved than that of HzO. The density of lines makes interpretation of the spectrum somewhat less reliable. Yet, it is possible to assign several lines to the direct npt lb, excitation and almost all of the others to the direct ndt lb, process. The most obvious “anomalies” in the DzO spectrum are the lines at 101849.5, 101887.7, and 101924.3 cm-‘, and the shoulders at 101901.9 and 101934.3 cm-‘. In the case of the “allowed” lines, the series in question are optically accessible fmm the ground state, but 154
29 November 199I
CHEMICAL PHYSICS LETTERS
Volume 287, number 1,2
Table 1 Pulsed field ionization spectrum of H20, z 25 K [I]: all lines above 6.0 in relative intensity. uul is based on H,O+ rotational [ 111 and rotational origin of 101764.5 cm-’ “obr (cm-‘)
I
101722.2 101740.4 101747.9 101759.5 101778.5 101785.2
21.0 41.9 9.5 9.2 85.8 35.2
101803.4
23.4
101805.9 101828.3 101853.2 101858.2 101859.8 101870.6 101872.3 101878.1 101883.1
31.6 35.1 9.2 6.9 9.5 10.5 21.8 7.6
Assignment
Key
“ohs- “GUI (em-‘)
a
to.1 -0.3 -0.4 -0.1 to.3 -0.2 $1.0 +0.7 +0.2 0.0 +0.4 -0.6
: l
C : 11 e f *
constants
to.5 +0.8
b
to.2 t1.4
8.3
Table 2 Pulsed field ionization spectrum of D20, s 50 K [ I 1: all lines above 20.0 in relative intensity. Y,, is based on D,O+ rotational constants [12]androtationa1originof101914.5cm-’ “Oh, (cm-‘)
I
Assignment
Key
(cm-‘)
101848.7 101849.5 101860.3 101868.6 101870.3 101874.4
21.0 39.3 27.7 20.9 22.0 25.5
101878.6 101887.7 101891.3 101901.9
23.1 48.0 39.2 81.6
aw+110 %v-101
93.9
30x-31, lo,el,,
loI+ 2 20
t
1,1*2x
a
101911.0
33.5
101917.6
43.4
101922.6 101924.3
88.5 140.1
-0.5 -0.1
-0.4 :
101903.5
“ob. -
lOI+-
* C *
7-OP-212
d cl
lll~llcl
a’
313c303 212t202 lIo+l,, lll~l0I lO,~OLxl
f b g l
-0.4 -0.6 +0.3 to.5 +0.2 -0.4 +0.4 -0.3 - 1.0 0.0 +2.0 to.7 +0.1
ucal
VOba
I
Assignment
59.3 127.2 49.8 23.3 33.1 29.1
2,2+
Key
(cm-‘) 101934.3 101936.8 101947.6 101953.4 101957.5 101965.9 101967.5 101971.7 101976.7 101988.3
49.1 34.3 37.0 32.2
101990.0 101995.0 101997.5
20.9 21.7 24.3
102000.0 102009. I 102013.3 102025.8 102038.3 102041.6
34.6 24.6 22.1 23.6 21.2 20.1
lo,
1 ,o+~oo
;
V&l- “cdl (cm-‘)
t1.2 t1.0
2,,+10,
i
221’202
C’
312+202
j
220+110
k
+0.1
&,+l,,
I
-0.5
321c211
m d f 0
-0.3 +0.3 0.0 +0.3
P g’
to.3 +1.1
h’
-0.9
220+
lo,
22,400 ‘h-313
4,,+2,,
+0.8 +1.0
0.0
155
Volume 287, number 1,2
CHEMICAL PHYSICS LETTERS
29 November 1991
Fig. 3. Interaction between two Rydberg channels. Unperturbed energylevels are shown to the right and left. In the center are
plotted the components of series I (left) in each of the mixed states. The arrow and the bracket indicate mixed states excited by a laserwithfrequency 101764.5cm-’ and bandwidth 1cm-‘.
the anomalous lines may be induced by the state-mixing mechanism illustrated in fig. 3. This diagram was calculated using a two-channel quantum defect model of the interaction between the nd series terminating on the 1, , rotational state of HzO+ (series I) and the np series converging on the 00,,limit (series II). Transitions to series I are optically allowed from the lo, ground state, and give rise to the strong lines labelled c in fig. 1 and g in fig. 2. The anomalous lines assigned to OoO + lo, are to be explained by the proposed mechanism. The relevant equations are given in the Appendix. The two unperturbed series are shown to the right and left of the diagram. The center represents intensity borrowing due to the interaction, which results in a normal autoionization profile above the series II limit and a corresponding mixed packet below the limit. The mixed packet is subject to ionization by the pulsed field. The plotted intensity of a member of such a mixed packet is 1c, I* from eq. (8), multiplied by the series II state density. The appearance of anomalous lines via this state-mixing mechanism depends on the occurrence of degeneracies between states of series I and states of series II. This depends, in turn, on two considerations. First, if the level spacing in the allowed series I is less than the laser bandwidth, it will always be possible to excite at least one packet of mixed states very near the series II limit. Otherwise, the state mixing depends on a rare accidental degeneracy. In the present case, with the bandwidth about 1 cm-‘, transitions between series whose limits differ by more than about 40 cm-’ will be rare. This consideration alone (along with the selection rules discussed in the next section) is both necessary and sufficient to account for all the observed lines. The second consideration is that significant state mixing will only occur if the width rof the mixed packet is at least comparable to the series II level separation. Finally, it is also required that the lifetime r- l/rfor relaxation from series I to series II should be short compared with the time-delay in the experiment ( 10P6 s). An estimate of the width is given in section 2.
2. Matrix elements of the ion core dipole moment The observed selection rules are consistent with the interaction of the Rydberg states with the dipole moment of the ion core. The dipole moment of the water ion is about 0.95 au [3] and is therefore not insignificant with respect to the Coulomb potential over most of the orbits of even the high Rydberg electrons. A recent paper [ 41 reported field-induced autoionization in the Rydberg spectrum of Li2, mediated by the quadrupole moment of the ion core, an effect similar to the one considered here. 156
Volume 287, dilihbel; i,i
29 November 199 I
CHEMICAL PHYSICS LETTERS
The near-threshold Rydberg elect&%W&&S what may be approximated as a phase-shifted hydrogen-like orbital coupled to a symmetrized rotational state of the i&t ~orreto form an overall state of definite J=N+ +L, 1JiN+K+p) I vlm). Near threshold, the radial functions for tkff bound Coulomb system are approximately 2’/4
r-“4cos[JL(A+#t-~tX] )
R(c, I, r) z
with e= 1/2v2, For the pure Coulomb problem, the parameters v and h take their integral values n and 1,but the effect of the short-range interactions with the core may be modelled simply by the change in the value of n by the quantum d&M & [IRQin the phase of the long-range part of the wavefunction by the value xp,. Thus we take the valuesu= n-p/ and h I=&: C@mpericlon of eq. ( 1) with an asymptotic form of the ordinary Bessel function now gives
(2) Eq. (2) is familiar as the near-threshold approximation of the hydrogen atom wavefunction [ $61; it is generalized here to non-integral values of v and 1. Using standard integrals for the Bessel functions, we find
The expression vanishes identically for’ fnt&grrtll I’+& Thus, for non-penetrating Rydberg orbitals, where the quantum defects are quite small, the dipole coupling BBr&j#$iffsrent channels should be small as well. The coupling among the lower angular momentum channels, s, p, arid di however, will be relatively much larger. The angle-dependent factor in the Rydberg-core-dipole ifitetaction is
II
(J’I’N +‘K+‘p‘ y =
JliV +K+p)
C-1) AN++K+‘+l’+l~~l,(2N+‘+1)(2~++1) [2(l+s,+,)(l+s,+,,)]“”
X
N+’
[(
* )I
_K+' 1j(+ l N+
(4)
Here p is the expectation value of the dipole moment in the given electronic-vibrational state of the core, and y and t are coordinates of the Rydberg electron with respect to the molecule-fixed principal axes. Selection rules implied in this expression include the following: K+ (even)uK+‘(odd), Aiv+<1,
K+ (odd)++K+‘(even),
[.I-/(
IJ-1’1 dV+'
Kz(even)HK:‘(odd),
K: (odd)wKz’( even),
.
Combination of eqs. (3) and (4) gives an estimate of the matrix element coupling series I and series 11via the core dipole interaction. Average quantum defects may be estimated form the known p and d Rydberg series [2,7]: ~,=0.70 and ~~~0.1 1.Substitution into eq. (3) gives the approximate magnitude in au of the matrix element, O.~~(YV’)-‘j2. With v= 50 and v’ =330, as in the example of fig. 3, this is about 0.01 cm-‘. With a series II state density of about 150/cm-‘, and a series I state density of about OS/cm-‘, the interaction strength is sufficient to mix states from each series with a bandwidth of 1 cm-‘. It is interesting to note that the interaction element between the s series, with ps= 1.5 [ 8 1, and the p series with &,=0.7 is almost an order of magnitude bigger than the one between p and d. This shows that any Stark 157
Volume 287, number I,2
CHEMICAL PHYSICS LETTERS
29
November199I
mixing of the angular momentum channels due to the field will enhance the coupling between series 1 and series II levels. The autoionizing linewidth is given by the Fermi golden rule expression,
The same expression determines the range of the quasi-autoionizing interactions just below threshold, as indicated in fig. 3. Here the whole set of angular momentum quantum numbers is represented by the single symbol (Yor B. The ket with Lis the energy-normalized equivalent of the ket with Y,v here being the series I effective quantum number and E’the energy of the series II state. Substituting p and d quantum defects, and taking an angular integral of unity gives r of the order of 18000/u3 cm-’ and a lifetime of relaxation z=T-’ of the order 3X lo-l6 y3 s. For ~~50, this comes to rzO.14 cm-’ and 7~4~ lo-” s. Thus, for this example, the quasiautoionizing linewidth is an order of magnitude larger than the series II level spacing of 0.006 cm-‘, and the relaxation time is well within the experimental delay time of 10e6 s. Specific predictions of relative intensities would require a more detailed knowledge of the Rydberg series than is at present available. Leaving aside the possibility of mixing with other angular momentum channels, the p and d series alone comprise not two, but eight distinct Rydberg series. Since the dipole interaction is most heavily weighted near the core, the different C,, symmetry characters of the series will significantly affect their degree of mixing. A more accurate treatment of the pertinent interactions seems, however, for the time being, out of reach. In any case, there are a number of indications that the influence of the ionizing field should not be ignored in any detailed calculation. One effect of the ionizing field is to mix the various I-components of a given n(v) manifold, and eq. (5) shows that the resulting admixture of s and p character in the d series could dramatically increase the magnitude of r through reduction of A. Another important consideration is that, for rotational energy transfers of less than about 14 cm-‘, the relevant series I levels will he above the Inglis-Teller limit (n = 90 for a field of about Ex 0.3 V/cm), above which the Stark splitting 3n (n - 1 )E exceeds the Rydberg level separation n -3. Hence the polarization interaction in the presence of the field might be better viewed as an interaction between continua than between discrete series. These points remain to be addressed in a later more sophisticated calculation.
3. Conclusion Anomalous type A transitions with K; = K: = 0 in the ZEKE field-ionization spectrum of Hz0 [ 1] have been attributed to polarization-induced quasi-autoionizing state mixing between the optically allowed nd+ 1bl series (series I) and members of the npt 1bl series (series II), which are forbidden under case (b) to case (d) selection rules. The conditions of the experiment require the presence of a series 1 line within the laser bandwidth ( z 1 cm-‘) at a fixed (but field-dependent) energy below the series II limit, a condition that is necessarily satisfied provided the rotational energy separation between the two limits is less than roughly 40 cm-‘. All the observed anomalous lines meet this criterion and satisfy the necessary polarization selection rules. An estimate of the autoionization (or state-mixing) linewidth derived from the known dipole moment of H20+ [ 31 gives an order of magnitude comparable to the series II level spacing (as required for significant state mixing) and implies a Iifetime of the order of 10-“-lO-‘” s. Several aspects of the theory are open to experimental test. One prediction is that autoionization should be observed in the nd 1l 1+- 1b, 1Ol series immediately above the O,,,,+-101 limit. Secondly, only rare accidental anomalous lines might be found for transitions involving rotational energy transfer in excess of about 40 cm-‘. The necessary search over high J” and I#‘+levels at higher temperatures would presumably require some double-resonance technique to isolate the lines of interest. A third suggestion is that an increase in the ionizing 158
Volume287,number 1,2
CHEMICALPHYSICSLETTERS
29 November 1991
voltage would bring the exciting frequency into a sparser region of series II, thus diminishing the admixture with series I. The effect might be to decrease the anomalous line intensities. The picture might well be complicated, however, by the effects of level-crossings arising from non-adiabatic switching of the field. Finally, it is suggested that a sufficiently short time delay might attenuate the anomalous lines by allowing insufficient time for relaxation from series II to series I. Our proposed interpretation suggests that the ZEKE spectrum could provide valuable information on the strength of the p/d polarization relevant to aspects of the lower-frequency regions of the spectrum. For example, a recent analysis of the 3d+ lb, complex 193 in the approximation of “pure precession” yielded unreasonably large inertial defectsin the effective rotational constants, but there are insufficient data on the npc 1b1 series to assess whether these defects can be traced to polarization. The intensities of the np+ lb, series could also arise partly from polarization mixing with nd and ns. More detailed analysis of the lower frequency bands is clearly required, but there is little evidence of the npa, series beyond n=4, and none for the other np components. The present suggestion that the observation of anomalous lines provides a measure of the strength of d/p polarization means that the experiment of Tonkyn et al. could bear directly on an important but elusive interaction throughout the Rydberg series.
Appendix The quantum defect theory for two interacting channels takes as parameters: two ionization limits, Et and
EG, corresponding to two different states of the positive ion; two quantum defects, j& and &; and a mixing angle, 8, to parameterize the 2 x 2 orthogonal frame transformation U. The energies and compositions of mixed states below the lower ionization limit, EG , are determined by equations of the form [ lo] cos0sin[rc~,+/3,(E)] ( sinOsin[rrfiU,+/J,(E)]
-sinBsin[rCpb tB(E)] cos8sin[rt&+~,r(E)]
A, =. ’ )( Ab>
(A.1)
where
(A.21 Ry being the Rydberg constant. The energy levels correspond to zeroes of the determinant derived from eq. (A. 1) and the coefkientsd, and A, determine the admixture of non-normalized channel functions. The proper normalization is set in terms of coefficients 112 c,=({c0sec0s[sr~,t~,(E)]}A,-{sinBc0s[~~~++,(E)]}Ab)
c,,=((sinecos[nl4ta,(8)1:A.+(ForBcns[nlr”*,
,
g
(
)
(A.3)
subject to c:+c:,=1.
(A.4)
The quantities c: were verified, for small 6’values to sum to unity over a packet of mixed states localized around one unperturbed series I level; the quantity plotted in fig. 3 is (d/&r/dE) c:, which gives a probability density per unit energy spread over series II. At energies above the lower ionization limit, channel II becomes open, and eq. (A. 1) becomes 159
CHEMICAL PHYSICS LETTERS
Volume 287, number I,2
cosf?sin[x~~+~,(E)] sin 8 sin [ 7cpa- m(E)]
A, =O ’ K Ab1
-sinBsin[npbt+i(E)] cosBsin[n~~--r(E)]
29 November1991
(Al51
with the energy normalization of the mixed state provided by T={~~~~COS[~C~~-IT~(E)]}A,~{COS~COS[~C~~-T~(E)]}A~.
Here
7(E) is the
(A.61
channel II phase shift, and IFI-1= 1. Whenever B,(E) is approximated by (A.7)
with (A.8) solutions to eqs. (A.5) and (A.6) are given by
&(E)=L Ry cos2er(+-p,)3
r (E-E,
-4)2+r2/4
9
(A.91
where (A.lO) and (A.1 1) Comparison of eq. (A.lO) with eq. (5) encourages the identification tan t9=fip
WY/W (A+A’+ I ) (/I’-A) .
(A.12)
Acknowledgement We would like to acknowledge useful discussions with Dr. M. Glass-Maujean, Dr. T.P. Softley, and Mr. F. Merkt, and to thank Dr. R.G. Tonkyn and Dr. M.G. White for making their data available to us before its publication and for their comments and suggestions. [ 1 ] R.G. Tonkyn, R. Wiedmann, E.R. Grant and M.G. White, J. Chem. Phys., in press. [2] MS. Child and Ch. Jungen, .J. Chem. Phys. 93 (1990) 7756. [ 31 B. Weis, S. Carter, P. Rosmus, H.-J. Werner and P.J. Knowles, J. Chem. Phys. 91 (1989) 2818. [ 41 CR. Mahon, G.R. Janik and T.F. Gallagher, Phys. Rev. A 41 ( 1990) 3746. [ 51H.A. Bethe and E.E. Salpeter, Quantum mechanics of one- and two-electron atoms (Springer, Berlin, 1957) [6] U. Fano and A.R.P. Rau, Atomic collisions and spectra (Academic Press, New York, 1986). [7] E. Ishiguro, M. Sasanuma, H. Masuko, Y. Morioka and M. Nakamura, J. Phys. B I1 ( 1978) 993. [S] M.N.R. Ashfold, J.M. Bayley and R.N. Dixon, Can. J. Phys. 62 ( 1984) 1806. [9] R.D. Gilbert, M.S. Child and J.W.C. Johns, Mol. Phys., in press. [ 101C.H. Greene and Ch. Jungen, Advan. At. Mol. Phys. 21 (1985)5I. [I l] B.M. Dinelli, M.W. Crofton and T. Oka, J. Mol. Spectry. 127 (1988) 1. [ 121H. Lew and R, Groleau, Can. J. Phys. 65 (1987) 739.
160
29 November1991
CHEMICAL PHYSICS LETTERS
Effects of polarization in the field ionization spectrum of HZ0 R.D. Gilbert and M.S. Child Department ofTheoretical Chemistry, University of
Oxford, 5 South Pa& Road, O&d
OX1 3UB, UK
Received 15August 1991
Anomalous linesinthe ZEKE field-ionization spectrum of Hz0 are explained via a mechanism of energy redistribution between core rotation and the Rydbergelectron,mediatedby the Rydberg-core-dipoleinteraction.A quantum defectmodelof the interaction, involving the coupling of two closed channels, is discussed in relation to predictions ofthe intensities of the observed lines. Experimental implications of this polarization mechanism are suggested.
1. Introduction The rotationally resolved ZEKE field-ionization spectrum of Hz0 recently reported by Tonkyn et al. [ 11 is of added interest due to the appearance of certain anomalous lines. It is known in diatomic molecules that radiative transitions to high Rydberg states, since they involve the change from Hund’s case (b) to case (d), are governed by quite restrictive propensity rules which depend on the angular momentum properties of the initial state. Extension of these rules to polyatomic molecules [ 21 requires that the strong ndt lb, series in Hz0 show overall type C rotational structure, since lb, is nearly a pure p orbital. Most of the strong lines in the ZEKE spectrum conform to this prediction, but a few almost equally strong lines can be confidently assigned to type A transitions with Kg’=Kz = 0.Some weak type A lines are to be expected as limits of the weaker npt lb, series, but for these K: = K: =O is not allowed, since the propensity rules include 1KI: -td" I= 1K,+1,and the nodal plane of the b, orbital excludes contribution of atomic orbitals with J”=O. The aim of this paper is to suggest a mechanism to account for these anomalous lines. In the experiment, water is irradiated in a time-of-flight spectrometer with VLJVlight in a frequency range just below the ionization continuum and with a band-width of about 1 cm-‘, After a short delay on the order of a us, a small ( -0.3 to -0.5 V/cm) electric field is pulsed, sufficient to ionize the near-threshold Rydberg states. The resulting flux of near-zero-velocity electrons is recorded as a function of the VUV excitation frequency. Throughout the experiment, a small dc voltage ($0.05V) is maintained across the extraction plates to sweep out any electrons produced by direct ionization. The ZEKE spectrum therefore shows peaks at frequencies corresponding to excitation to Rydberg levels lying a cm-’ or two below the ionization limits associated with different rotational states of the positive ion. The rotational origin of the spectrum depends on the magnitude of the ionizing field. For the published spectrum [ 11, the origin is found to be 101764.5? I cm-‘. Tonkyn et al. measured the red shift due to the field and deduced an ionization potential for Hz0 of 101766-t 1 cm-‘, which is to be compared with the ionization limit of 101772 cm- ’ deduced from extrapolation of the Rydberg series [ 21. The observed spectra are reproduced in figs. 1 and 2 along with simulations based on H&l-London formulae [ 21. The observed spectrum in the center of figs. 1 and 2 is to be compared with the calculated intensity profile corresponding to a pure nd (case (d) )cpb, (case (b) ) transition, shown at the bottom, and the profile corresponding to a pure np (case (d))+db, (case (b)) transition, shown at the top. The peaks in figs. 1 and 2 are keyed to the frequencies tabulated in tables 1 and 2 for cross-reference. All the intensities are scaled to aid 0009-2614/91/S 03.50 0 1991 Elsevier Science Publishers B.V. All rights reserved.
153
Volume 287, number I,2
29 November 1991
CHEMICAL PHYSICS LETTERS
d’
b’ I
rw
3170d
b 10:iEO[ FREOL JENI
:Y
3 ICM-11
Fig. 1. Pulsed field ionization spectrum of HZO, x25 K. Top: calculated spectrum using HGnl-London intensities for np (case (d ) ) cdb, (case (b) ) Middle: experimental trace from ref. [ I]. Bottom: calculated spectrum using Hb;nl-London intensities for nd (case (d)) +pb, (case (b)). Calculated spectra are convolved with Lorcntzians of width 1cm-‘, and all intensities are scaled to aid comparison. Peaks are keyed for cross-reference with table I.
FREQUENCY
(CM-11
Fig. 2. Pulsed field ionization spectrum of DZO, x 50 K. Top: calculated spectrum using HiSnl-London intensities for np (case (d))+-db, (case (b)).Middle:experimental trace from ref. [I]. Bottom; calculated spectrum using Hiinl-London intensities for nd (case (d))+pb, (case (b)). Calculated spectra are convolved with Lorentzians of width 1cm-‘, and all intensities are scaled to aid comparison. Peaks are keyed for cross-reference with table 2.
comparison. In tables 1 and 2, all the experimental lines above a given threshold of intensity are tabulated with proposed assignments. In the H,O spectrum, 8 of the 16 tabulated lines are seen in the nd+ lb, calculation. Although the relative intensities are only approximately simulated, the calculation seems reliable as a guide in suggesting likely assignments. Only two of the observed lines are seen in the npc- lb, calculation, and the observed intensity of the one labelled a’ is much higher than calculated. The peaks marked with arrows in figs. 1 and 2 are not fully assignable to either of the calculated spectra, These are the “anomalous” lines which are to be interpreted by the mechanism to be proposed in what follows. The most obvious anomalies are the transitions with K:: = Kb = 0. These cannot be due to any direct one-photon process, since they are forbidden by selection rules which depend only on the symmetry of the ground state. Because of experimental constraints, the DzO spectrum of Tonkyn et al. is both hotter and less fully resolved than that of HzO. The density of lines makes interpretation of the spectrum somewhat less reliable. Yet, it is possible to assign several lines to the direct npt lb, excitation and almost all of the others to the direct ndt lb, process. The most obvious “anomalies” in the DzO spectrum are the lines at 101849.5, 101887.7, and 101924.3 cm-‘, and the shoulders at 101901.9 and 101934.3 cm-‘. In the case of the “allowed” lines, the series in question are optically accessible fmm the ground state, but 154
29 November 199I
CHEMICAL PHYSICS LETTERS
Volume 287, number 1,2
Table 1 Pulsed field ionization spectrum of H20, z 25 K [I]: all lines above 6.0 in relative intensity. uul is based on H,O+ rotational [ 111 and rotational origin of 101764.5 cm-’ “obr (cm-‘)
I
101722.2 101740.4 101747.9 101759.5 101778.5 101785.2
21.0 41.9 9.5 9.2 85.8 35.2
101803.4
23.4
101805.9 101828.3 101853.2 101858.2 101859.8 101870.6 101872.3 101878.1 101883.1
31.6 35.1 9.2 6.9 9.5 10.5 21.8 7.6
Assignment
Key
“ohs- “GUI (em-‘)
a
to.1 -0.3 -0.4 -0.1 to.3 -0.2 $1.0 +0.7 +0.2 0.0 +0.4 -0.6
: l
C : 11 e f *
constants
to.5 +0.8
b
to.2 t1.4
8.3
Table 2 Pulsed field ionization spectrum of D20, s 50 K [ I 1: all lines above 20.0 in relative intensity. Y,, is based on D,O+ rotational constants [12]androtationa1originof101914.5cm-’ “Oh, (cm-‘)
I
Assignment
Key
(cm-‘)
101848.7 101849.5 101860.3 101868.6 101870.3 101874.4
21.0 39.3 27.7 20.9 22.0 25.5
101878.6 101887.7 101891.3 101901.9
23.1 48.0 39.2 81.6
aw+110 %v-101
93.9
30x-31, lo,el,,
loI+ 2 20
t
1,1*2x
a
101911.0
33.5
101917.6
43.4
101922.6 101924.3
88.5 140.1
-0.5 -0.1
-0.4 :
101903.5
“ob. -
lOI+-
* C *
7-OP-212
d cl
lll~llcl
a’
313c303 212t202 lIo+l,, lll~l0I lO,~OLxl
f b g l
-0.4 -0.6 +0.3 to.5 +0.2 -0.4 +0.4 -0.3 - 1.0 0.0 +2.0 to.7 +0.1
ucal
VOba
I
Assignment
59.3 127.2 49.8 23.3 33.1 29.1
2,2+
Key
(cm-‘) 101934.3 101936.8 101947.6 101953.4 101957.5 101965.9 101967.5 101971.7 101976.7 101988.3
49.1 34.3 37.0 32.2
101990.0 101995.0 101997.5
20.9 21.7 24.3
102000.0 102009. I 102013.3 102025.8 102038.3 102041.6
34.6 24.6 22.1 23.6 21.2 20.1
lo,
1 ,o+~oo
;
V&l- “cdl (cm-‘)
t1.2 t1.0
2,,+10,
i
221’202
C’
312+202
j
220+110
k
+0.1
&,+l,,
I
-0.5
321c211
m d f 0
-0.3 +0.3 0.0 +0.3
P g’
to.3 +1.1
h’
-0.9
220+
lo,
22,400 ‘h-313
4,,+2,,
+0.8 +1.0
0.0
155
Volume 287, number 1,2
CHEMICAL PHYSICS LETTERS
29 November 1991
Fig. 3. Interaction between two Rydberg channels. Unperturbed energylevels are shown to the right and left. In the center are
plotted the components of series I (left) in each of the mixed states. The arrow and the bracket indicate mixed states excited by a laserwithfrequency 101764.5cm-’ and bandwidth 1cm-‘.
the anomalous lines may be induced by the state-mixing mechanism illustrated in fig. 3. This diagram was calculated using a two-channel quantum defect model of the interaction between the nd series terminating on the 1, , rotational state of HzO+ (series I) and the np series converging on the 00,,limit (series II). Transitions to series I are optically allowed from the lo, ground state, and give rise to the strong lines labelled c in fig. 1 and g in fig. 2. The anomalous lines assigned to OoO + lo, are to be explained by the proposed mechanism. The relevant equations are given in the Appendix. The two unperturbed series are shown to the right and left of the diagram. The center represents intensity borrowing due to the interaction, which results in a normal autoionization profile above the series II limit and a corresponding mixed packet below the limit. The mixed packet is subject to ionization by the pulsed field. The plotted intensity of a member of such a mixed packet is 1c, I* from eq. (8), multiplied by the series II state density. The appearance of anomalous lines via this state-mixing mechanism depends on the occurrence of degeneracies between states of series I and states of series II. This depends, in turn, on two considerations. First, if the level spacing in the allowed series I is less than the laser bandwidth, it will always be possible to excite at least one packet of mixed states very near the series II limit. Otherwise, the state mixing depends on a rare accidental degeneracy. In the present case, with the bandwidth about 1 cm-‘, transitions between series whose limits differ by more than about 40 cm-’ will be rare. This consideration alone (along with the selection rules discussed in the next section) is both necessary and sufficient to account for all the observed lines. The second consideration is that significant state mixing will only occur if the width rof the mixed packet is at least comparable to the series II level separation. Finally, it is also required that the lifetime r- l/rfor relaxation from series I to series II should be short compared with the time-delay in the experiment ( 10P6 s). An estimate of the width is given in section 2.
2. Matrix elements of the ion core dipole moment The observed selection rules are consistent with the interaction of the Rydberg states with the dipole moment of the ion core. The dipole moment of the water ion is about 0.95 au [3] and is therefore not insignificant with respect to the Coulomb potential over most of the orbits of even the high Rydberg electrons. A recent paper [ 41 reported field-induced autoionization in the Rydberg spectrum of Li2, mediated by the quadrupole moment of the ion core, an effect similar to the one considered here. 156
Volume 287, dilihbel; i,i
29 November 199 I
CHEMICAL PHYSICS LETTERS
The near-threshold Rydberg elect&%W&&S what may be approximated as a phase-shifted hydrogen-like orbital coupled to a symmetrized rotational state of the i&t ~orreto form an overall state of definite J=N+ +L, 1JiN+K+p) I vlm). Near threshold, the radial functions for tkff bound Coulomb system are approximately 2’/4
r-“4cos[JL(A+#t-~tX] )
R(c, I, r) z
with e= 1/2v2, For the pure Coulomb problem, the parameters v and h take their integral values n and 1,but the effect of the short-range interactions with the core may be modelled simply by the change in the value of n by the quantum d&M & [IRQin the phase of the long-range part of the wavefunction by the value xp,. Thus we take the valuesu= n-p/ and h I=&: C@mpericlon of eq. ( 1) with an asymptotic form of the ordinary Bessel function now gives
(2) Eq. (2) is familiar as the near-threshold approximation of the hydrogen atom wavefunction [ $61; it is generalized here to non-integral values of v and 1. Using standard integrals for the Bessel functions, we find
The expression vanishes identically for’ fnt&grrtll I’+& Thus, for non-penetrating Rydberg orbitals, where the quantum defects are quite small, the dipole coupling BBr&j#$iffsrent channels should be small as well. The coupling among the lower angular momentum channels, s, p, arid di however, will be relatively much larger. The angle-dependent factor in the Rydberg-core-dipole ifitetaction is
II
(J’I’N +‘K+‘p‘ y =
JliV +K+p)
C-1) AN++K+‘+l’+l~~l,(2N+‘+1)(2~++1) [2(l+s,+,)(l+s,+,,)]“”
X
N+’
[(
* )I
_K+' 1j(+ l N+
(4)
Here p is the expectation value of the dipole moment in the given electronic-vibrational state of the core, and y and t are coordinates of the Rydberg electron with respect to the molecule-fixed principal axes. Selection rules implied in this expression include the following: K+ (even)uK+‘(odd), Aiv+<1,
K+ (odd)++K+‘(even),
[.I-/(
IJ-1’1 dV+'
Kz(even)HK:‘(odd),
K: (odd)wKz’( even),
.
Combination of eqs. (3) and (4) gives an estimate of the matrix element coupling series I and series 11via the core dipole interaction. Average quantum defects may be estimated form the known p and d Rydberg series [2,7]: ~,=0.70 and ~~~0.1 1.Substitution into eq. (3) gives the approximate magnitude in au of the matrix element, O.~~(YV’)-‘j2. With v= 50 and v’ =330, as in the example of fig. 3, this is about 0.01 cm-‘. With a series II state density of about 150/cm-‘, and a series I state density of about OS/cm-‘, the interaction strength is sufficient to mix states from each series with a bandwidth of 1 cm-‘. It is interesting to note that the interaction element between the s series, with ps= 1.5 [ 8 1, and the p series with &,=0.7 is almost an order of magnitude bigger than the one between p and d. This shows that any Stark 157
Volume 287, number I,2
CHEMICAL PHYSICS LETTERS
29
November199I
mixing of the angular momentum channels due to the field will enhance the coupling between series 1 and series II levels. The autoionizing linewidth is given by the Fermi golden rule expression,
The same expression determines the range of the quasi-autoionizing interactions just below threshold, as indicated in fig. 3. Here the whole set of angular momentum quantum numbers is represented by the single symbol (Yor B. The ket with Lis the energy-normalized equivalent of the ket with Y,v here being the series I effective quantum number and E’the energy of the series II state. Substituting p and d quantum defects, and taking an angular integral of unity gives r of the order of 18000/u3 cm-’ and a lifetime of relaxation z=T-’ of the order 3X lo-l6 y3 s. For ~~50, this comes to rzO.14 cm-’ and 7~4~ lo-” s. Thus, for this example, the quasiautoionizing linewidth is an order of magnitude larger than the series II level spacing of 0.006 cm-‘, and the relaxation time is well within the experimental delay time of 10e6 s. Specific predictions of relative intensities would require a more detailed knowledge of the Rydberg series than is at present available. Leaving aside the possibility of mixing with other angular momentum channels, the p and d series alone comprise not two, but eight distinct Rydberg series. Since the dipole interaction is most heavily weighted near the core, the different C,, symmetry characters of the series will significantly affect their degree of mixing. A more accurate treatment of the pertinent interactions seems, however, for the time being, out of reach. In any case, there are a number of indications that the influence of the ionizing field should not be ignored in any detailed calculation. One effect of the ionizing field is to mix the various I-components of a given n(v) manifold, and eq. (5) shows that the resulting admixture of s and p character in the d series could dramatically increase the magnitude of r through reduction of A. Another important consideration is that, for rotational energy transfers of less than about 14 cm-‘, the relevant series I levels will he above the Inglis-Teller limit (n = 90 for a field of about Ex 0.3 V/cm), above which the Stark splitting 3n (n - 1 )E exceeds the Rydberg level separation n -3. Hence the polarization interaction in the presence of the field might be better viewed as an interaction between continua than between discrete series. These points remain to be addressed in a later more sophisticated calculation.
3. Conclusion Anomalous type A transitions with K; = K: = 0 in the ZEKE field-ionization spectrum of Hz0 [ 1] have been attributed to polarization-induced quasi-autoionizing state mixing between the optically allowed nd+ 1bl series (series I) and members of the npt 1bl series (series II), which are forbidden under case (b) to case (d) selection rules. The conditions of the experiment require the presence of a series 1 line within the laser bandwidth ( z 1 cm-‘) at a fixed (but field-dependent) energy below the series II limit, a condition that is necessarily satisfied provided the rotational energy separation between the two limits is less than roughly 40 cm-‘. All the observed anomalous lines meet this criterion and satisfy the necessary polarization selection rules. An estimate of the autoionization (or state-mixing) linewidth derived from the known dipole moment of H20+ [ 31 gives an order of magnitude comparable to the series II level spacing (as required for significant state mixing) and implies a Iifetime of the order of 10-“-lO-‘” s. Several aspects of the theory are open to experimental test. One prediction is that autoionization should be observed in the nd 1l 1+- 1b, 1Ol series immediately above the O,,,,+-101 limit. Secondly, only rare accidental anomalous lines might be found for transitions involving rotational energy transfer in excess of about 40 cm-‘. The necessary search over high J” and I#‘+levels at higher temperatures would presumably require some double-resonance technique to isolate the lines of interest. A third suggestion is that an increase in the ionizing 158
Volume287,number 1,2
CHEMICALPHYSICSLETTERS
29 November 1991
voltage would bring the exciting frequency into a sparser region of series II, thus diminishing the admixture with series I. The effect might be to decrease the anomalous line intensities. The picture might well be complicated, however, by the effects of level-crossings arising from non-adiabatic switching of the field. Finally, it is suggested that a sufficiently short time delay might attenuate the anomalous lines by allowing insufficient time for relaxation from series II to series I. Our proposed interpretation suggests that the ZEKE spectrum could provide valuable information on the strength of the p/d polarization relevant to aspects of the lower-frequency regions of the spectrum. For example, a recent analysis of the 3d+ lb, complex 193 in the approximation of “pure precession” yielded unreasonably large inertial defectsin the effective rotational constants, but there are insufficient data on the npc 1b1 series to assess whether these defects can be traced to polarization. The intensities of the np+ lb, series could also arise partly from polarization mixing with nd and ns. More detailed analysis of the lower frequency bands is clearly required, but there is little evidence of the npa, series beyond n=4, and none for the other np components. The present suggestion that the observation of anomalous lines provides a measure of the strength of d/p polarization means that the experiment of Tonkyn et al. could bear directly on an important but elusive interaction throughout the Rydberg series.
Appendix The quantum defect theory for two interacting channels takes as parameters: two ionization limits, Et and
EG, corresponding to two different states of the positive ion; two quantum defects, j& and &; and a mixing angle, 8, to parameterize the 2 x 2 orthogonal frame transformation U. The energies and compositions of mixed states below the lower ionization limit, EG , are determined by equations of the form [ lo] cos0sin[rc~,+/3,(E)] ( sinOsin[rrfiU,+/J,(E)]
-sinBsin[rCpb tB(E)] cos8sin[rt&+~,r(E)]
A, =. ’ )( Ab>
(A.1)
where
(A.21 Ry being the Rydberg constant. The energy levels correspond to zeroes of the determinant derived from eq. (A. 1) and the coefkientsd, and A, determine the admixture of non-normalized channel functions. The proper normalization is set in terms of coefficients 112 c,=({c0sec0s[sr~,t~,(E)]}A,-{sinBc0s[~~~++,(E)]}Ab)
c,,=((sinecos[nl4ta,(8)1:A.+(ForBcns[nlr”*,
,
g
(
)
(A.3)
subject to c:+c:,=1.
(A.4)
The quantities c: were verified, for small 6’values to sum to unity over a packet of mixed states localized around one unperturbed series I level; the quantity plotted in fig. 3 is (d/&r/dE) c:, which gives a probability density per unit energy spread over series II. At energies above the lower ionization limit, channel II becomes open, and eq. (A. 1) becomes 159
CHEMICAL PHYSICS LETTERS
Volume 287, number I,2
cosf?sin[x~~+~,(E)] sin 8 sin [ 7cpa- m(E)]
A, =O ’ K Ab1
-sinBsin[npbt+i(E)] cosBsin[n~~--r(E)]
29 November1991
(Al51
with the energy normalization of the mixed state provided by T={~~~~COS[~C~~-IT~(E)]}A,~{COS~COS[~C~~-T~(E)]}A~.
Here
7(E) is the
(A.61
channel II phase shift, and IFI-1= 1. Whenever B,(E) is approximated by (A.7)
with (A.8) solutions to eqs. (A.5) and (A.6) are given by
&(E)=L Ry cos2er(+-p,)3
r (E-E,
-4)2+r2/4
9
(A.91
where (A.lO) and (A.1 1) Comparison of eq. (A.lO) with eq. (5) encourages the identification tan t9=fip
WY/W (A+A’+ I ) (/I’-A) .
(A.12)
Acknowledgement We would like to acknowledge useful discussions with Dr. M. Glass-Maujean, Dr. T.P. Softley, and Mr. F. Merkt, and to thank Dr. R.G. Tonkyn and Dr. M.G. White for making their data available to us before its publication and for their comments and suggestions. [ 1 ] R.G. Tonkyn, R. Wiedmann, E.R. Grant and M.G. White, J. Chem. Phys., in press. [2] MS. Child and Ch. Jungen, .J. Chem. Phys. 93 (1990) 7756. [ 31 B. Weis, S. Carter, P. Rosmus, H.-J. Werner and P.J. Knowles, J. Chem. Phys. 91 (1989) 2818. [ 41 CR. Mahon, G.R. Janik and T.F. Gallagher, Phys. Rev. A 41 ( 1990) 3746. [ 51H.A. Bethe and E.E. Salpeter, Quantum mechanics of one- and two-electron atoms (Springer, Berlin, 1957) [6] U. Fano and A.R.P. Rau, Atomic collisions and spectra (Academic Press, New York, 1986). [7] E. Ishiguro, M. Sasanuma, H. Masuko, Y. Morioka and M. Nakamura, J. Phys. B I1 ( 1978) 993. [S] M.N.R. Ashfold, J.M. Bayley and R.N. Dixon, Can. J. Phys. 62 ( 1984) 1806. [9] R.D. Gilbert, M.S. Child and J.W.C. Johns, Mol. Phys., in press. [ 101C.H. Greene and Ch. Jungen, Advan. At. Mol. Phys. 21 (1985)5I. [I l] B.M. Dinelli, M.W. Crofton and T. Oka, J. Mol. Spectry. 127 (1988) 1. [ 121H. Lew and R, Groleau, Can. J. Phys. 65 (1987) 739.
160