Theoretical studies of ZEKE spectroscopy and dynamics of high Rydberg states

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Theoretical studies of ZEKE spectroscopy and dynamics of high Rydberg states

Yi-Hsieh Wang

a,e

, Y. Teranishi

b,c

, H. Mineo

a

, S.D. Chao

a,*

, H.L. Selzle

d

, H.J. Neusser

d

,

E.W. Schlag

d

, S.H. Lin

b,c,e a

Institute of Applied Mechanics, National Taiwan University, Taipei 106, Taiwan b

Institute of Atomic and Molecular Sciences, Academia Sinica, Taipei 106, Taiwan

c_{Institute of Applied Chemistry, Institute of Molecular Science, Chiao-Tung University, Hsin-Chu, Taiwan} d

Institut für Physikalische und Theoretische Chemie, Technische Universität München, Lichtenbergstr. 4, D-85748 Garching, Germany e

Research Center for Applied Sciences, Academia Sinica, Taipei 115, Taiwan

a r t i c l e

i n f o

Article history:

Received 22 September 2009 In ﬁnal form 29 December 2009 Available online 11 January 2010

a b s t r a c t

A main purpose of this Letter is to show how to employ the inverse Born–Oppenheimer approximation as a basis set to study zero kinetic energy (ZEKE) spectroscopy and the autoionization dynamics of the ZEKE states. The calculations of channel couplings, quantum defects, intensity borrowing, vibrational and rota-tional autotionizations will be demonstrated by using a homonuclear diatomic molecule as an example. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction

Since its invention by Müller–Dethlefs, Sander and Schlag in

1984[1,2], the zero kinetic energy (ZEKE) spectroscopy has seen

the explosive activity in high resolution spectroscopy of the ions. Using this technique, the ﬁrst examples of full rotational resolution in photoelectron spectra (PES) have been obtained. Consequently, there has been a considerable impetus given to the understanding of the behavior of molecules very close to their ionization thresholds.

However, while the power of the ZEKE technique to determine ionization potential (IP) of molecules and ionic rovibronic energies with high accuracy is widely recognized, and has been illustrated in many systems, the interpretation of rotational line intensities and dynamics of ZEKE states still poses some problems[3,4]. For example, do the ZEKE line intensities reﬂect direct ionization cross-sections? Are ZEKE-PES intensities consistent with conven-tional PES intensities? Can they be predicted by ab initio calcula-tions? Is there a systematic procedure to extract meaningful information from ZEKE line intensities? Rotational and vibrational autoionizations are believed to play important roles in ZEKE spec-troscopy. How can they be evaluated credibly? Anomalous intense peaks are often observed in ZEKE spectroscopy and they are usu-ally attributed to some resonance effects. How can we treat these resonance phenomena? In this Letter, we shall attempt to answer these questions by employing the theory of ZEKE spectroscopy based on the use of inverse Born–Oppenheimer approximation (IBOA)[5–8].

For simplicity of demonstration, we shall use a homonuclear diatomic molecule as an example to demonstrate how to calculate channel couplings, quantum defects, intensity borrowing in ZEKE spectroscopy, and dynamics of rotational and vibrational autoion-ization. In our opinion, the observed ZEKE band-shapes should de-pend on optical pumping of ZEKE levels, l mixing induced by a stray field, field-induced lowering of ionization thresholds, electric field-induced ionization, tunneling ionization, and autoionization. The ZEKE band-shapes will not be studied in this Letter, but will be reported in a future paper.

The present Letter will be organized as follows. In Section2, a typical ZEKE spectroscopy experiment will be presented and the IBOA will be described. In Sections3–5, we shall show how to use the IBOA to study quantum defects, and rotational and vibra-tional autoionizations, respectively. These will be followed by dis-cussion which is presented in Section6.

2. General consideration

In treating ZEKE spectroscopy and dynamics of the ZEKE Rydberg states, the multi-channel quantum defect theory (MQDT) is com-monly used[3,4,9–11], which is combined with the photo-electron spectroscopy model[12–17]. In this Letter, we shall use the inverse Born–Oppenheimer approximation (IBOA) to study these phenom-ena. Some preliminary results have been reported[5–7]. Several types of ZEKE experiments have been proposed[1–4]. For conve-nience of discussion, one of these experiments will be brieﬂy described, which is based on the one-photon absorption[5,6], and shown inFig. 1.

Consider a photoexcitation from state Kða

v

Þ to state MðcwmÞ, where a and c denote electronic states and

v

and w denote

*Corresponding author. Fax: +886 2 2363 9290. E-mail address:sdchao@iam.ntu.edu.tw(S.D. Chao).

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rovibrational states, and m denotes the Rydberg electron state. In the dipole approximation, the interaction Hamiltonian is

b

H0_¼

_l

*_E*

0cos

x

t ð2:1Þ

where

l

*is the dipole moment, E*0and

x

are the electric ﬁeld and frequency of the laser ﬁeld, respectively, and the absorption rate constant Wð1Þ K!Mis given by Wð1Þ_K!M¼

p

2h2j

l

* MK E * 0j2Dð

x

MK

x

Þ ð2:2Þ

where

l

*MK denotes the transition dipole moment and Dð

x

MK

x

Þ represents the line-shape function. The latter can be Lorentzian or Fano-type of line-shape depending on whether the M-level is iso-lated or coupled to a continuum. The thermal average rate constant Wð1Þ_{is given by} Wð1Þ ¼X K X M PKWð1Þ_K!M ¼

p

2h2 X K X M PKj

l

* MK E * 0j 2 Dð

x

MK

x

Þ ð2:3Þ

where PKdenotes the Boltzmann distribution.

It should be noted that the absorption rate constants given by

Eqs.(2.2) and (2.3)can easily be converted into the corresponding

absorption coefﬁcients or cross sections. However, the measure-ments of ZEKE signals involve not only the optical absorption from the transition of, for example, ða

v

Þ ! ðcwmÞ but also the extraction of ZEKE electrons from ZEKE states ðcwmÞ by a discriminating field and an extraction field. This is the reason why the absorption rate constants, instead of the absorption coefficients, are being used in treating ZEKE spectroscopy in this Letter.

In this Letter, we shall choose the inverse Born–Oppenheimer approximation[5–7]as a basis set for the ZEKE experiment. The molecular Hamiltonian in this case can be expressed as

b

H ¼ bHionþ bTe ð2:4Þ

where bHionis the Hamiltonian of the ion and bTedenotes the kinetic energy operator of the ZEKE electron. It follows that

b H

W

cwm¼ Ecwm

W

cwm ð2:5Þ

W

cwm¼

U

cwm

H

cw ð2:6Þ b Hion

H

cw¼ Ucw

H

cw ð2:7Þ and ðbTeþ UcwÞ

U

cwm¼ Ecwm

U

cwm ð2:8Þ

This shows that the ZEKE electron is moving in the potential energy surface provided by the ion core described by ðcwÞ. The energy lev-els within IBOA are shown graphically inFig. 2. In the ﬁgure, ZEKE states ðcwmÞ below the rovibronic state ðcwÞ of the ion are discrete while those above ðcwÞ are continuous ðcwkÞ. Here we refer to the

high Rydberg states near but below the ionization continuum as ‘ZEKE Rydberg’ states.

Based on the IBOA model for ZEKE spectroscopy, the ab initio calculations can be applied to calculate transition dipole moments involved in absorption rate constants shown in Eqs.(2.1)–(2.3). It should be noted that nearly all ZEKE experiments involve pulse-ﬁeld ionization of very high Rydberg states just below threshold, rather than extraction of true ZEKE electrons at, or above thresh-old. The Rydberg states form a pseudo-continuum, and as a conse-quence of the continuity of the transition probability on either side of an ionization limit, it is often assumed that the ZEKE transition intensities can be treated in the same manner as true photoioniza-tion intensities[12–17].

It should be noted that in the BOA model, the kinetic energy operator of nuclear motion bTncan be used to treat the excited state dynamics like internal conversion[18]and autoionization of low Rydberg states[19]. In a similar manner, the kinetic energy opera-tor of the ZEKE electron bTecan be employed to treat rotational and vibrational autoionization, as will be shown below.

Based on the use of the IBOA, the model of the ZEKE spectros-copy and the dynamics of its Rydberg states can be shown in

Fig. 2. Here the Born–Oppenheimer approximation (BOA) is used

to describe the neutral molecule. From this ﬁgure we can see that the ZEKE spectroscopy is related to the transition from the rovib-ronic state of the neutral molecule ða

v

Þ or Wav. In the BOA, Wav¼UaHav where Ua and Hav represent the wavefunctions of

the electronic motion and nuclear motion, respectively. We can also see that in the IBOA, the channel coupling is due to the cou-pling between ðcwmÞ and ðcw0_m0_{Þ while the autoionization is due} to the coupling of the ZEKE states ðcwmÞ with the ZEKE continuum states ðcw0_k0

Þ. 3. Quantum defect

Due to the channel couplings, each ZEKE state is not pure and thus we can calculate the quantum defect for a ZEKE state and the intensity borrowing in the ZEKE spectroscopy. For example, we can calculate the quantum defect as follows. Suppose that we let n denote the ZEKE state ðcwmÞ under consideration which will be coupled to other lower Rydberg states like n1, n2, . . . by the channel couplings as shown inFig. 2. Then we have

W

n¼ Cn;n

W

0nþ Cn;n1

W

0 n1þ Cn;n2

W

0 n2þ ð3:1Þ and b H

W

n¼ En

W

n ð3:2Þ Cn;nðHn;n EÞ þ Cn;n1Hn;n1þ Cn;n2Hn;n2þ ¼ 0 ð3:3Þ Cn1;nHn1;nþ Cn1;n1ðHn1;n1 EÞ ¼ 0 ð3:4Þ Cn2;nHn2;nþ Cn2;n2ðHn2;n2 EÞ ¼ 0 ð3:5Þ . . .

Here we assume that the couplings like Hni;njði – jÞ can be

ne-glected. It follows that

E ¼ Hn;nþ X ni jHn;nij 2 E Hni;ni ð3:6Þ or approximately E ¼ Hn;nþ X ni jHn;nij 2 Hn;n Hni;ni ð3:7Þ

In this Letter, for demonstration we shall consider the applications of the IBOA model to the dynamics and spectroscopy of ZEKE states of homonuclear diatomic molecules. In the following we consider a simple model. Notice that Ucwin the Schrödinger equation of the

Fig. 1. A schematic plot showing a one-photon ZEKE spectroscopy. The transition by a laser of frequencyxis from state KðavÞ to state MðcwmÞ. See text for more details.

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molecular ion core, Eq.(2.7), consists of the rovibronic energy levels Ecwof the molecular ion plus the potential energy Veof the ZEKE electron. That is,

ðbTeþ VeÞ

U

cwm¼

e

cwm

U

cwm ð3:8Þ

where

e

cwm¼ Ecwm Ecw; Ve¼ Ucw Ecw ð3:9Þ

For Vewe shall for simplicity use the multipole expansion

Ve¼ e2 r þ V 0 q ð3:10Þ V0 q¼ e Q ðRÞ r3 4

p

5 X2 m¼2 Y2 mðh; /ÞY2 mð

H

;

U

Þ " # ð3:11Þ

where Q ðRÞ denotes the core quadrupole moment.

For our purpose, we shall use the notations;

v

þ_{and N}þ_denote the vibrational and rotational quantum numbers of the ion, while ðnlmÞ represents the quantum numbers of high Rydberg states.

The quantum defect of N2has been experimentally determined by Merkt and Softley[10,20–22]. In the following, we demonstrate how to calculate it by choosing the state f

v

þ_{¼ 0; N}þ_{¼ 0ðnlmÞg as} an example. Notice that

v

0þ_{¼ 0; N}0þ ¼ 2ðn0_l0 Þ V0q

v

þ_{¼ 0; N}þ ¼ 0ðnlÞ D E ¼ eh

v

_v0þ_¼0jQðRÞj

v

_vþ_¼0ihRn0_l0jr3jR_nli 4

p

5 X mMþ X m0M0þ X m hYl0m0jY2 mjYlmi Y N0þ_¼2;M0þY₂_mY_Nþ_¼0;Mþ " # ð3:12Þ and hRn0;l0jrpjR_n;li ¼ ½ðn þ lÞ!ðn l 1Þ! 1 2_½ðn0_{þ l}0_Þ!ðn0_l0_1Þ!12 X nl1 k¼0 X n0_l0₁ k0¼0 ð1Þkþk0 2lþl0þkþk0_þ2ðl þ l0þ k þ k0 p þ 2Þ! ðn1_{þ n}01_Þlþl0þkþk0pþ3 " Fðnl; kÞFðn0_l0 ;k0Þ ð3:13Þ where Fðnl; kÞ ¼ ðn1_Þlþkþ2_ðk!Þ1 ½ðn l 1 kÞ!1½ð2l þ 1 kÞ!1 ð3:14Þ

and a similar expression for Fðn0_l0 ;k0Þ.

Therefore, we can calculate the quantum defect as the energy correction in Eq. (3.7) and evaluate the matrix element by Eq.

(3.12). Our preliminary results show that there seems no deﬁnite

correlation between the quantum defects and the rotational quan-tum number. For example, for n = 200 we obtain the quanquan-tum de-fects of 0.158, 0.322, 0.825 for N+_{= 1, 2, 3, respectively. On the} other hand from the formulation shown in the above, it is clear the vibrational quantum number dependence of the quantum defect is roughly linear. This has been found to be consistent with the experimental observation[7].

Merkt and Softley reported the rotationally resolved ZEKE spec-tra of N2 for the bands X2Rþgð

v

þ¼ 0; 1Þ X

0_Rþ

gð

v

¼ 0Þ[10]. For ð

v

þ_{¼ 0Þ ð}

_v

_{¼ 0Þ band, they observed that the Q-branch is strong} and the O-branch is weak, and pointed out that the intensity bor-rowing takes place. This intensity borbor-rowing can be treated by cal-culating the corresponding matrix elements similar to Eq.(3.12). Our calculations show that the transition 00! 2þof the S-branch, due to the channel coupling, can borrow the 00

! 0þ of the Q-branch, due to the large gap E2þ E₀þ (that is, off-resonance),

the 00! 2þ transition is weak as experimentally observed. The O-branch 20

! 0þ can borrow the intensity from 20

! 2þ. For the ð

v

0_{¼ 0 !}

_v

þ_{¼ 0Þ of N}

2, the spectral intensities of various bands of S, O, M, . . . branches are determined by the intensity borrowing from the Q-band and are determined by the energy differences like E4þ E₀þ; E₀þ E₂þ, etc. For O-branch, M-branch, etc., accidental

resonances can happen to these energy gaps and consequently, some anomalous peaks can be observed.

4. Rotational autoionization

We start with the Fermi golden rule expression for rotational autoionization using the IBOA as a basis set[5–7]

Wðcwm!cw0_kÞ¼ 2

p

h

W

cw0kHb 0 IBO

W

cwm D E 2

q

ðEkÞ ð4:1Þ

where

q

ðEkÞ is the density of state around energy Ek. Notice that

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W

cw0_k Hb0_IBO

W

cwm D E ¼ h 2 meh

U

cw0_kjh

H

cw0j

r

ej

H

cwi j

r

e

U

cwmi ð4:2Þ

For a homonuclear diatomic molecule, the interaction between the ZEKE electron and the molecular ion can be expressed as shown in

Eq.(3.11). In Eqs.(4.1) and (4.2)UcwmandUcw0_kdenote the bound

state and continuum state of the ZEKE electron, respectively, while

HcwandHcw0represent the wavefunctions the remaining degrees of

freedom of the ion.

Carrying out the time-independent perturbation to the ionic wavefunction, we obtain h

H

cw0j

r

ej

H

cwi ¼

H

0_cw0

r

eV0q

H

0_cw D E E0 cw E 0 cw0 ð4:3Þ and

W

cw0_k Hb0_IBO

W

cwm D E ¼ h 2 me 1 E0 cw E 0 cw0

U

cw0k

H

0cw0

r

eV0q

H

0_cw D E

r

e

U

cwm D E ð4:4Þ

For spherical harmonics, we obtain

Yl0_m0 @Y2 m @h @Y@hlm þ Yl0_m0 1 sin h @Y2 m @/ _{sin h}1 @Y@/lm ¼ Dl0_;lhY_l0_m0jY2 mjYlmi ð4:5Þ where Dl;l¼ 3 Dlþ2;l¼ 2l Dl2;l¼ 2ðl þ 1Þ 8 > < > : ð4:6Þ

Therefore, we can obtain the ﬁnal expression of transition matrix element as

v

0þ_N0þ JMJðkl 0 Þ bH0 IBO

v

þ_Nþ JMJðnlÞ D E ¼ h 2 me ðeÞ E0 vþNþE0_v0þN0þ h

v

_v0þjQ ðRÞj

v

_vþi 4

p

5 X mMþ X m0_M0þ X m CJMJ mMþC JMJ m0M0þhYl0m0jY2 mjYlmi Y N0þ_M0þY₂_mY_Nþ_Mþ " # Dl0;lhRkl0jr5jRnli3hRkl0jr4 dRnl dr ð4:7Þ

where Q ðRÞ denotes the core quadrupole moment,

v

vþ denotes the

wavefunction of vibrational state

_v

þ_{, and C}JMJ

mM represents the Clebsch–Gordan coefﬁcient for the coupling between the electronic orbital angular momentum and the core rotational motion; J and MJ denote the total angular momentum and its projection on the space-ﬁxed z axis. Besides, the radial integrals in Eq. (4.7) were performed numerically.

In the case of rotational autoionization of H2, the numerical re-sults are given in Table 1 for the transition f

v

þ_{¼ 0; N}þ

¼ 2; JMJðnlÞg ! f

v

0þ¼ 0; N0þ¼ 0; JMJðklÞg. For a homonuclear mol-ecule like H2, only transitions l0¼ l and l0¼ l 2 are allowed. In our calculation, the largest contribution comes from the transition l0¼ l, which is few orders of magnitude larger than those from l0¼ l 2 mainly due to its larger radial matrix element. Further-more, fromTables 1 and 2, we can see that the rotational autoion-ization rates decrease with increasing l and decrease with increasing n for a given l value. From the tables we can see that the n3scaling law holds well and gives a very consistent estimation of the autoionization rate. This dependence is easily veriﬁed as a

result of the factor of n3/2_{in the asymptotic bound-state} wave-function with high n.

5. Vibrational autoionization

We consider the vibrational autoionization described by the transition

f

v

þ_{¼ 1; N}þ

¼ 0; JMJðnlÞg ! f

v

þ¼ 0; N þ

¼ 2; JMJðklÞg

The calculated vibrational autoionization rates of H2are shown in

Table 2. FromTable 2, we can see that the behaviors of vibrational

autoionization are similar to those of rotational autoionization. With the same rotational transition, the discrepancy between them is mainly attributed to the vibrational matrix elements.

From Tables 1 and 2, we see that the autoionization rates are

quite sensitive to n; l. In addition, their energy dependence is given as follows. We have calculated the rates by changing the energy gap and k in Eq.(4.7). For example, for the transitions inTable 1 with n ¼ 150 and l ¼ 3, we obtain the rates of 4:842 106, 4:757 106, 4:647 106s1 _{for the energies of 100, 500,} 1000 cm1_{, respectively. We can see that the variation on rates is}

Table 1

Rotational autoionization for the transition of fvþ_{¼ 0; N}þ

¼ 2; JMJðnlÞg ! fv0þ_{¼ 0; N}0þ

¼ 0; JMJðklÞg. The energy of the continuum state is 156 cm1.

n l J Rate (s1_{) IBOA} Rate (s1_{) MI} 150 2 2 4.076E+07 4.076E+07 150 3 3 4.831E+06 4.830E+06 150 4 4 1.016E+06 1.016E+06 150 5 5 2.940E+05 2.940E+05 150 6 6 1.042E+05 1.042E+05 150 7 7 4.240E+04 4.239E+04 150 8 8 1.900E+04 1.900E+04 150 9 9 9.125E+03 9.124E+03 150 10 10 4.603E+03 4.602E+03 250 2 2 8.805E+06 8.804E+06 250 3 3 1.044E+06 1.043E+06 250 4 4 2.195E+05 2.195E+05 250 5 5 6.356E+04 6.355E+04 250 6 6 2.254E+04 2.254E+04 250 7 7 9.178E+03 9.177E+03 250 8 8 4.118E+03 4.118E+03 250 9 9 1.981E+03 1.981E+03 250 10 10 1.001E+03 1.001E+03 Table 2

Vibrational autoionization for the transition of fvþ_{¼ 1; N}þ

¼ 0; JMJðnlÞg ! fv0þ_{¼ 0; N}0þ

¼ 2; JMJðklÞg. The energy of the continuum state is 2035 cm1.

n l J Rate (s1 ) IBOA Rate (s1 ) MI 150 2 2 1.710E+06 1.710E+06 150 3 3 1.906E+05 1.905E+05 150 4 4 3.534E+04 3.534E+04 150 5 5 8.270E+03 8.269E+03 150 6 6 2.135E+03 2.135E+03 150 7 7 5.638E+02 5.637E+02 150 8 8 1.455E+02 1.455E+02 150 9 9 3.569E+01 3.569E+01 150 10 10 8.177E+00 8.178E+00 250 2 2 3.695E+05 3.694E+05 250 3 3 4.117E+04 4.116E+04 250 4 4 7.638E+03 7.637E+03 250 5 5 1.788E+03 1.788E+03 250 6 6 4.621E+02 4.621E+02 250 7 7 1.222E+02 1.222E+02 250 8 8 3.159E+01 3.158E+01 250 9 9 7.765E+00 7.764E+00 250 10 10 1.784E+00 1.786E+00

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small. In fact, our calculation shows that, in Eq.(4.7), the parenthe-sis containing the radial matrix element is approximately propor-tional to the energy difference which is roughly canceled out by the energy gap in the coefﬁcient.

In ZEKE experiments, the influence of Stark-mixing due to the stray residue field needs to be included in our discussion of autoi-onization rates. Chao et al. have proposed that the optically acces-sible populations in low-l states would transfer to high-l states, resulting in reduction of core penetration[7]. That is, the low-l states contributing significantly near the ion core experience a stronger interaction with the core. Hence, electrons in low-l states would quickly disappear through various relaxation processes.

FromTables 1 and 2, we can see that the transition rate decreases

signiﬁcantly as l increases for both rotational and vibrational autoi-onization. Consequently, after a long period of delay time, only the states with higher l would survive and be observed in ZEKE exper-iments. In the case of l ¼ 3, the lifetime of rotational autoionization from high Rydberg state is about 1

l

s, and that of vibrational autoi-onization is about one order of magnitude smaller. The autoioniza-tion in this range might be observed in the ZEKE experiments with comparable delay time.

6. Discussion

It should be noted that in Section2in discussing the theory of ZEKE spectroscopy we calculate the optical absorption rate constant Wðav!cwmÞfor various transitions like ða

v

Þ ! ðcwmÞ as a function of laser intensity, pulse-duration, laser-polarization and laser-wavelength. However, in ZEKE spectroscopy, the ZEKE signal is obtained by, in addition to this optical pumping (or absorption), the extraction of ZEKE electrons by a discriminating field and an extraction electric field, and thus the signal can often be interfered by autoionizations. In field extraction of ZEKE electrons, tunneling ionization might play some role.

The rotational and the vibrational autoionization have often been evaluated based on the multipole interaction (MI) model

[23–27], which was ﬁrst pointed out by Russek et al.[23]and

fol-lowed by Jungen and Miescher[24]and Eyler[25]. For example, for the Fermi golden rule expression of autoionization cwm ! cw0_k due to V0

qthe rate can be expressed as

Wðcwm!cw0_kÞ¼2

p

h

W

cw0kV 0 q

W

cwm D E 2

q

ðEkÞ ð6:1Þ It follows that

v

0þ_N0þ_JM Jðkl 0 Þ V 0q

v

þN þ_JM JðnlÞ D E ¼ eh

v

_v0þjQ ðRÞj

v

_vþihRkl0jr3jR_nli 4

p

5 X mMþ X m0M0þ X m CJMJ mMþC JMJ m0M0þhYl0_m0jY2 mjYlmi Y N0þ_M0þY₂_mY_Nþ_Mþ " # ð6:2Þ

Numerical calculations have been performed to compare the above expression with ours based on the use of IBOA. The results are given

inTables 1 and 2denoted as MI. We can see that the agreement for

both vibrational and rotational autoionization rates is excellent. Next, we shall compare the formulations between the two models as discussed by Russek et al.[23].

Referring to Eqs.(2.7) and (2.8), we can obtain the unperturbed Hamiltonian by neglecting the higher order interactions:

b H0 ion

H

0 cw¼ U 0 cw

H

0 cw ð6:3Þ bTe e2 r

U

0m¼ E 0 m

U

0 m ð6:4Þ

We note that to the zeroth-order, this representation is the same as the traditional Hund’s case (d). From Eq.(6.4), we can derive that

U

0f V 0 q bTe

U

0i D E bTe

U

0f V 0 q

U

0i D E ¼ E0i E 0 f

U

0f V 0 q

U

0i D E ð6:5Þ

Also notice that

h 2 2me

r

2 e

U

0 f V 0 q

U

0_i D E ¼ h 2 2me

U

0_f V0 q

r

2 e

U

0 i D E þ 2

U

0_f

r

eV0q

r

e

U

0i D E þ

U

0_f

r

2 eV 0 q

U

0_i D E h i ð6:6Þ

To derive Eq.(6.6), we need to perform integration by parts two times to pass re to the right side. Substituting Eq. (6.6) to Eq.

(6.5), we obtain

U

0f V 0 q

U

0i D E ¼ h 2 2me 1 E0 f E 0 i 2

U

0 f

r

eV0q

r

e

U

0i D E þ

U

0 f

r

2 eV 0 q

U

0 i D E h i ð6:7Þ

Because the quadrupole interaction involves the core dependence, we shall multiply both sides of Eq.(6.7)by the ionic wavefunctions and integrate over the ionic coordinates. Then, we obtain the autoi-onization matrix element used in the MI model

H

0cw0

U

0 f V 0 q

U

0i D E

H

0cw D E ¼ h 2 2me 1 E0 f E 0 i 2

H

0 cw0

U

0_f

r

eV0q

r

e

U

0i D E

H

0 cw D E h þ

H

0 cw0

U

0_f

r

2 eV 0 q

U

0 i D E

H

0 cw D Ei ð6:8Þ

Switching the sequence of the two integrals, the ﬁrst term on the right-hand side is in accordance with the matrix element in Eq.

(4.2)in the formulation of IBOA, while the second term is evidently

zero for the quadrupole term is a solution of Laplace equation in the electronic coordinate. Here we shall stress that this derivation is based on the use of multipole expansion, in which we employed the unperturbed wavefunction and its ﬁrst-order correction. The generality of Eq.(6.8)should be examined for other assumptions and conditions. In fact, the origins of the two formulations (Eqs.

(4.2) and (6.1)) are quite different and the results are not necessarily

equivalent. At any rate, we see that the IBOA approach is more ver-satile and can be systematically generalized to include other higher order interactions. As emphasized previously[23,25], the present model works mainly for non-penetrating states. For penetrating states, higher order terms need to be included.

It is also interesting to compare our IBOA to the MQDT[3,4,9– 11], which has been employed to study high Rydberg states. The MQDT is a scattering theory while our IBOA is based on eigenfunc-tion expansion; i.e., a bound state descripeigenfunc-tion. Using the IBOA, the interactions between the Rydberg electron and the ion core can be systematically included and calculated. Therefore, the IBOA in this sense is useful in providing an alternative approach to the MQDT to better understand the many faces of high Rydberg states. Acknowledgements

This work has been supported by National Science Council of ROC and Academia Sinica. The works of Yi-Hsieh Wang and S.D. Chao are also partly supported by the CQSE National Taiwan Uni-versity through 97R0066-66.

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