Fine-And Hyperfine-Structure Effects in Molecular Photoionization: I

Fine- and hyperfine-structure effects in molecular photoionization: I. General theory and direct photoionization Matthias Germann1 and Stefan Willitsch1, a) Department of Chemistry, University of Basel, Klingelbergstrasse 80, 4056 Basel, Switzerland (Dated: 2 October 2018) We develop a model for predicting fine- and hyperfine intensities in the direct photoionization of molecules based on the separability of electron and nuclear spin states from vibrational-electronic states. Using spherical tensor algebra, we derive highly symmetrized forms of the squared photoionization dipole matrix elements from which which we derive the salient selection and propensity rules for fine- and hyperfine resolved photoionizing transitions. Our theoretical results are validated by the analysis of the fine-structure resolved photoelectron spectrum of O2 (reported by H. Palm and F. Merkt, Phys. Rev. Lett. 81, 1385 (1998)) and are used for predicting hyperfine populations of molecular ions produced by photoionization.

I. INTRODUCTION dynamics.19–21 Thus, there is a growing need for theoretical models capable of describing fine- and hyperfine Photoionization and photoelectron spectroscopy are effects in molecular photoionization. Whereas the hy- among the eminent experimental techniques to gain infor- perfine structure in Rydberg spectra has previously been mation on the electronic structure of molecules, on their treated within the framework of multichannel quantum- 14–18 photoionization dynamics and the structure and dynam- defect theory (MQDT), we are not aware of any pre- ics of molecular cations.1–3 Since the first photoelectron vious treatments of hyperfine intensities in direct pho- spectroscopic studies of molecules in the 1960s, the reso- toionization which can be applied to the interpretation lution of the technique has steadily been improved, such of line intensities in high-resolution photoelectron spec- that vibrational and thereon rotational structure were tra and to the prediction of level populations of molec- resolved during the following decades.3,4 Along with the ular cations produced by photoionization. The present experimental progress, the theoretical understanding of work aims at filling this gap by developing closed expres- molecular photoionization has been successively refined. sions for fine- and hyperfine-structure resolved intensities Vibrational structure in the spectra can often be modeled in molecular photoionization. The theory is developed in terms of the Franck-Condon principle (see, e.g., Ref. here for diatomic molecules in Hund’s coupling case (b), 2). Concerning rotational structure, Buckingham, Orr but can readily be extended to other coupling cases by 22 and Sichel (BOS) presented in a seminal paper5 in 1970 a suitable basis transformation of our final result or to a model to describe rotational line intensities in photo- symmetric- and asymmetric-top molecules by a suitable 4 electron spectra of diatomic molecules. Subsequently, ex- modification of the rotational basis functions. tended models, e.g., for resonance-enhanced multiphoton In the present paper, we develop the general theory for ionization6 or to describe the angular distribution of the fine- and hyperfine-resolved photoionization intensities photoelectrons,7,8 have been developed. Also, the BOS and apply our model to the analysis of the fine structure model has been rephrased in terms of spherical tensor of the photoelectron spectrum of O2 from Ref. 13 and of algebra9 and extended to asymmetric rotors.4 Moreover, hyperfine propensities in the photoionization of N2. In 23 dipole selection rules for photoionizing transitions have the subsequent companion paper, we extend our model been developed based on these intensity models10 as well to resonance-enhanced multiphoton-ionization processes as on general symmetry considerations.11,12 and address the problem of hyperfine-preparation of Over the last decades, fine (spin-rotational) struc- molecular cations. ture has been resolved in high-resolution photoelectron spectra.13 Hyperfine structure has been resolved in millimeter-wave spectra of high Rydberg states of rare II. GENERAL CONSIDERATIONS gas atoms such as Kr and diatomics such as H2 and its isotopomers, see, e.g., Refs. 14–18. Fine- and hyperfine- We consider the ionization of a molecule M yielding structure effects in molecular photoionization have also the molecular ion M+ by ejection of a photoelectron e− become of importance in precision spectroscopy and dy- through interaction with electromagnetic radiation via namics experiments with molecular ions produced by arXiv:1606.02990v1 [physics.chem-ph] 9 Jun 2016 the electric-dipole operator µ: photoionization in which the ionic hyperfine populations µ are governed by the underlying hyperfine photoionization M −−−−→ M+ + e−. (1)

The relevant transition matrix element is given by

a)Electronic mail: [email protected] (hψM+ | hψe− |) µ |ψMi , (2) 2 with |ψMi standing for the internal quantum state of the III. FINE-STRUCTURE RESOLVED PHOTOIONIZATION neutral molecule and |ψM+ i|ψe− i the product of the in- INTENSITIES ternal state of the molecular ion M+ and the state of the photoelectron e−. We start by developing the theory for fine-structure- resolved photoionization intensities which will form the basis for the subsequent inclusion of hyperfine struc- The squared magnitude of the transition matrix ele- ture in Sec. IV. We note that a similar result for fine- ment summed over all the quantum states contributing structure resolved photoionization has previously been to the observed ionization rate, derived by McKoy and co-workers.6,24 In Hund’s case (b), fine structure manifests itself in the coupling of the orbital-rotational angular momentum N (corresponding to the mechanical rotation of the molecule in Σ states) with the electron spin S to form J: J = N + S. Therefore, we evaluate Eq. (3) for a diatomic + X X X 2 P (M → M ) = (hψM+ | hψe− |) µ |ψMi , molecule in Hund’s case (b) by expressing the quan- ψ ψ ψ M M+ e− tum states of the neutral molecule |ψMi and the molec- (3) ular ion |ψM+ i in the basis |nΛ, v, NΛSJMJ i and is proportional to the ionization probability per unit + + + + + + + + n Λ , v ,N Λ S J MJ , respectively. The defini- time. Here, the sums over ψM and ψM+ include all de- tions of the relevant angular-momentum quantum num- generate (or spectroscopically unresolved) states of the bers are summarized in Tab. I. n (n+) and v (v+) denote neutral molecule and the molecular ion, respectively, in- the electronic and the vibrational quantum number in volved in the photoionizing transition. The sum over the neutral molecule (molecular ion). The state of the ψe− includes the orbital angular momentum and the spin photoelectron |ψe− i is expressed as a tensor product of state of the emitted photoelectron. We suppose that nei- its spin state and its partial wave |s, msi |l, mli. ther the energy nor the angular distribution nor the spin The quantity P (J, J +) which is proportional to the polarization of the photoelectron is detected in the pho- ionization probability on the photoionizing transition toionization experiment. J → J + may hence be written as

+ ∞ l s J J 2 + X X X X X + + + + + + + + P (J, J ) = n Λ , v ,N Λ S J MJ hs, ms| hl, ml| µ |nΛ, v, NΛSJMJ i . + l=0 ml=−l ms=−s MJ =−J + MJ =−J (4)

We follow the approach of Xie and Zare9 and identify the state of the photon. Cjm stands for a Clebsch- j1m1j2m2 electric-dipole operator µ and the photoelectron partial 25–27 k Gordan coeﬃcient. The spherical tensor operator Tp wave |l, m i with the spherical tensors T1 and Tl , l µ0 −ml of Eq. (5) describes the combined eﬀect of absorbing respectively. We then contract the product of these two electromagnetic radiation via the electric dipole opera- 22,25,26 spherical tensors according to tor and ejecting a photoelectron in the state |l, mli (with l = k ± 1). l+1 X Tl ⊗ T1 = Ckp Tk, (5) The term with k = l does not contribute to the sum −ml µ0 l−ml1µ0 p k=|l−1| in Eq. (4) because of parity selection rules and may be omitted.9,11 Ignoring proportionality constants, the ma- where p = −ml + µ0 and µ0 denotes the polarization trix element in Eq. (4) is thus expressed as

where negative values for k are to be excluded. To proceed, we decouple spin and orbital-rotational 3

TABLE I. Angular momentum quantum numbers relevant to the photoionization of diatomic molecules Magnitude Mol.-ﬁxed Space-ﬁxed quant. num. projection projection Description

N Λ MN Orbital-rotational angular momentum of the neutral molecule S – MS Total electron spin of the neutral molecule J – MJ Total angular momentum of the neutral molecule excluding nuclear spin

I – MI Nuclear spin of the neutral molecule F – MF Total angular momentum of the neutral molecule + + + N Λ MN Orbital-rotational angular momentum of the molecular ion + + S – MS Total electron spin of the molecular ion + + J – MJ Total angular momentum of the molecular ion excluding nuclear spin + + I – MI Nuclear spin of the molecular ion + + F – MF Total angular momentum of the molecular ion l – ml Orbital angular momentum (partial wave) of the photoelectron s – ms Spin of the photoelectron (s = 1/2) 1 – µ0 Angular momentum due to the electric-dipolar interaction with the electromagnetic ﬁeld k q p Total orbital angular momentum transferred to/from the molecule in the ionization process (p = −ml + µ0) u – w Total angular momentum transferred to/from the molecule in the ionization process (w = −ms + p) angular momenta in the neutral state according to coordinates using Wigner rotation matrix elements

k X k |nΛ, v, NΛSJMJ i Tk = [Dk ]∗T0 , (10) X p pq q = CJMJ |nΛ, v, NΛM ,SM i , (7) q=−k NMN SMS N S MN ,MS where the 0 denotes operators in molecule-fixed coordi- and analogously in the ionic state. The transition matrix nates. element in Eq. (6) then reads, The Wigner rotation matrix elements only act on the 0k angular coordinates, whereas the tensor T q only oper- + + + + + + + + ates on the vibronic state. Therefore, we can write: n Λ , v ,N Λ S J MJ hs, ms| Tk |nΛ, v, NΛSJM i + + + + + + k p J n Λ , v ,N Λ MN Tp nΛ, v, NΛMN X X J +M + = CJMJ C J k NMN SMS + + + + X D + + + 0k E N MN S MS + + = n Λ , v T q nΛ, v MN ,MS M ,M N S (8) q=−k + + + + + + + + × n Λ , v ,N Λ MN ,S MS hs, ms| + + + k ∗ × N Λ MN [Dpq] NΛMN . (11) k × Tp |nΛ, v, NΛMN ,SMSi . Upon substituting the rotational states of the neutral k molecule by Wigner rotation matrices Since the tensor operator Tp does not operate on the spin functions, the matrix element on the last and next- r 2N + 1 h i∗ to-last line above can be separated into a rotational- hφ θ χ | NΛM i = D(N) (φ, θ, χ) , (12) N 2 MN Λ vibronic and a pure spin factor: 8π and analogously for the ion, we obtain for the last line in + + + + + + + + n Λ , v ,N Λ MN ,S MS hs, ms| Eq. (11) an integral over a product of three Wigner rotation matrices over the Euler angles φ, θ, χ which accounts × Tk |nΛ, v, NΛM ,SM i = p N S for:22,25,27 + + + + + + k n Λ , v ,N Λ MN Tp nΛ, v, NΛMN + + + k ∗ × S+M + hs, m | |SM i . (9) N Λ MN [Dpq] NΛMN S s S √ √ M +−Λ+ = 2N + + 1 2N + 1(−1) N In order to calculate the rotational-vibronic factor on N + k N N + k N the second last line in Eq. (9), we transform the spheri- × + + . (13) k −M p MN −Λ q Λ cal tensor operator Tp from space-fixed to molecule-fixed N 4

Because of the second Wigner 3j-symbol, this expres- electronic spin after ionization: sion vanishes for all values of q but q = Λ+ − Λ =: ∆Λ. Hence, only this value contributes to the sum in Eq. (11) + + S MS , sms and we may write + S +s Stot X X StotMStot = C + + hStotMStot | . (15) S MS sms + Stot=|S −s| MStot =−Stot

+ + + + + + k Assuming orthonormal spin states, we thus obtain for the n Λ , v ,N Λ MN Tp nΛ, v, NΛMN D E √ √ spin factor in Eq. (9): = n+Λ+, v+ T0k nΛ, v 2N + + 1 2N + 1 ∆Λ + + S MS , sms SMS + + + + M −Λ N k N N k N + × (−1) N + + . S +s Stot −MN p MN −Λ ∆Λ Λ X X StotMStot = C + + hStotMStot | SMSi (14) S MS sms + Stot=|S −s| MStot =−Stot (16)

SMS = C + + . (17) S MS sms Collecting these results and substituting them into To compute the spin part of Eq. (9), we couple the Eq. (6), we obtain the matrix element for spin-rotation- spin of the ion and the photoelectron to get the total resolved photoionization dipole transitions as:

+ + + + + + + + n Λ , v ,N Λ S J MJ hs, ms| hl, ml| µ |nΛ, v, NΛSJMJ i √ √ √ √ √ l−1+p+N+N +−Λ+−S−s+M ++M = 2N + + 1 2N + 1 2S + 1 2J + + 1 2J + 1(−1) J J √ + X l 1 k D + + + 0k E N k N × 2k + 1 n Λ , v T ∆Λ nΛ, v + −ml µ0 −p −Λ ∆Λ Λ k=l±1 + + + + (18) X X M +MS N S J NSJ × (−1) N + + + M M −M MN MS −MJ + + N S J MS ,MS MN ,MN N + k N S+ s S × + + , −MN p MN MS ms −MS

where the Clebsch-Gordan coeﬃcients have been re- avoided and the matrix element becomes,28 placed by Wigner 3j-symbols. + + + + + + + + n Λ , v ,N Λ S J MJ hs, ms| hl, ml|

µ |nΛ, v, NΛSJMJ i In principle, Eq. (18) completely describes the matrix √ √ √ √ √ = 2N + + 1 2N + 1 2S + 1 2J + + 1 2J + 1 element for fine-structure-resolved photoionization tran- √ l−1−Λ++N+J−s+2M +M + X k sitions and could—when substituted into Eq. (4)—be × (−1) J J (−1) 2k + 1 used for analyzing measured photoionization and photo- k=l±1 electron spectra and predicting relative photoionization l 1 k D k E × n+Λ+, v+ T0 nΛ, v intensities. However, the complexity of this expression −m µ −p ∆Λ complicates a deeper insight into the physics of the pho- l 0 k+s toionization process and the multiple sums render the N + k N X J + u J evaluation of this expression computationally expensive. × + (2u + 1) + −Λ ∆Λ Λ −MJ w MJ u=|k−s|  J + u J  u k s   However, using the properties of the Wigner 3j- × N + k N . w −p m symbols, the terms on the last and next-to-last line in s S+ s S  Eq. (18), may be expressed in from of a Wigner 9j- (19) symbol25–27 (the expression in curly brackets below). In + + this way, the sums over MN , MN , MS and MS are Here, the angular momentum quantum number u with 5 the associated space-fixed projection w (given by w = scribed by the electric-dipole operator. The angular mo- −ms + p) has been introduced. u represents the resul- menta associated with the photoelectron partial wave and tant of the coupling of k and s. Its physical meaning is the dipole excitation are connected with k, see Eq. (5). described further below. Since the dipole operator is a spherical tensor of first Substituting this matrix element into Eq. (4) and sim- rank, k is constrained to the values k = l − 1 or k = l + 1 plifying the result, we obtain the quantity P (J, J +) as: with k = l forbidden because of parity selection rules.9,11 We thus have k = 0, 2, 4,... for ± ↔ ± transitions and + + + P (J, J ) = (2N + 1)(2N + 1)(2S + 1)(2J + 1)(2J + 1) k = 1, 3, 5,... for ± ↔ ∓ transitions.9 2 X X N + k N The interplay between the different angular momenta × (2k + 1) −Λ+ ∆Λ Λ is expressed by the 9j-symbol in Eq. (21) in a compact l k=l±1 way. All rows and columns have to fulfill the triangle D E 2 25,26 × n+Λ+, v+ T0k nΛ, v inequality for the coupling of angular momenta. The ∆Λ couplings in the ionic and the neutral state are expressed  2 k+s J + u J by the first and the last column, respectively. The cou- X   × (2u + 1) N + k N pling of the angular momenta transferred to or from the + u=|k−s| S s S  molecule is described by the middle column: the spin of 2 2 the photoelectron s is coupled to k (the angular momen- X l 1 k X u k s × . tum associated with the photoelectron and the interac- −ml µ0 −p w −p ms tion with the electromagnetic field) to form u. The rows ml ms (20) of the 9j-symbol represent the angular momentum trans- fer during photoionization: according to the first row, Owing to the orthogonality properties of the Wigner 3j- u determines the change in the total angular momen- symbols,25–27 the cross terms in the above expression tum (excluding nuclear spin), whereas k decides on the vanish when summed over all possible values for MJ and change in the orbital-rotational angular momentum (sec- + ond row). Finally, s accounts for the electron-spin change MJ resulting in a particularly simple form for Eq. (20). For linear polarized radiation, as is used in most ex- in the molecule resulting from the ionization process, as periments, we have µ0 = 0 in a suitably chosen coordi- seen from the last row in the 9j-symbol. nate system. The sums over ml and ms then account for The value of k determines the maximum change of the 1/(3(2k + 1)) and we get as a final result: orbital-rotational angular momentum in the photoionization process, i.e., |∆N| = |N + − N| ≤ k, as may be 1 P (J, J +) = (2N + + 1)(2N + 1)(2S + 1)(2J + + 1) seen from the 3j-symbol or the middle row of the 9j- 3 symbol in Eq. (21). Transitions with k = 0 do not allow 2 X X N + k N any change in the orbital-rotational angular momentum, × (2J + 1) −Λ+ ∆Λ Λ i.e., N + = N. For transitions with k = 2, the values l k=l±1 ∆N = 0, ±1 and ±2 are possible (with ∆N = ±1 for- 2 D + + + 0k E bidden for Σ-Σ-transitions). The permissible values of k × n Λ , v T ∆Λ nΛ, v (and therefore the range of photoelectron partial waves  2 k+s + l) and their weighting factors are determined by the vi-  J u J  X + D E 2 × (2u + 1) N k N . (21) + + + 0k + bronic coefficients Ck := n Λ , v T ∆Λ nΛ, v . u=|k−s| S s S  To calculate these coefficients, the electronic structure The highly symmetrized form of Eq. (21) allows a de- of the molecular ion and its neutral precursor must be tailed and insightful physical interpretation and discus- known. Thus, the coefficients Ck may either be obtained sion of photoionization selection rules. The photoelectron from an ab-initio calculation or by fitting Eq. (21) to 4,9 ejected in the ionization process is described by a partial measured photoionization intensities. In many cases, wave l. The probability of a transition from a specific only one or a few Ck contribute substantially to the total neutral to an ionic state is obtained by summing over all transition probability. If so, only a few free parameters partial waves. The photoelectron partial waves allowed are needed to describe the intensities in a rotationally for a particular electronic state of the ion and the neutral resolved photoelectron spectrum. precursor molecule are constrained by the parity of these Within the orbital ionization model4,5 which assumes states.9,11,12 If neutral and ionic states have the same that the photoelectron is ejected from a single molecular parity (± ↔ ± transitions), only odd values of l occur: orbital and that the orbital structure does not change l = 1, 3, 5,... In the case of unequal parities of neutral upon ionization, k assumes the intuitive physical inter- and ionic state (± ↔ ∓), only even l values are allowed: pretation as the quantum number of the electronic orbital l = 0, 2, 4,... angular momentum left behind in the molecule after pho- In addition to the angular momentum carried by the toionization. In this approximation, the permissible val- departing photoelectron, the molecule also exchanges an- ues of k and the vibronic coefficients Ck can be obtained gular momentum with the electromagnetic wave as de- from a single-center expansion of the molecular orbital 6 from which the photoelectron is ejected. and our result may be simplified further. The 9j-symbol Moreover, the photoelectron spin s = 1/2 is coupled then equals a 6j-symbol,25,26 to k to form u. The possible values for u are thus u =  2 |k − 1/2| and u = k + 1/2. Similar to k determining J + u N 2   1 J + u N the change in the orbital-rotational angular momentum, N + k N = , (22) k N + s u determines the change in the total angular momentum S+ s 0  2(2N + 1) (excluding nuclear spin) according to the selection rule and Eq. (21) becomes, |∆J| = |J + − J| ≤ u, as inferred from the first row of the 9j-symbol in Eq. (21). For, e.g., k = 0 only u = 1/2 1 P (N,J +) = (2N + 1)(2N + + 1)(2J + + 1) is possible and thus only transitions with |∆J| = 1/2 are S=0 6 allowed. For k = 2, on the other hand, u = 3/2 and 2 X X N + k N u = 5/2 are possible, allowing values of |∆J| up to 5/2. × −Λ+ ∆Λ Λ Since the values of u are determined by the values of k, l k=l±1 the vibronic coefficients Ck describing the relative inten- D E 2 + + + 0k sities of different rotational lines in a photoelectron spec- × n Λ , v T ∆Λ nΛ, v trum also determine the relative intensities of transitions k+s 2 X J + u N connecting different fine structure components. Hence, × (2u + 1) . (23) the present model describes the spin-rotational effects in k N + s u=|k−s| photoionization without additional free parameters. Eq. (21) also provides a substantially more efficient way for numerical calculations of the transition probabilities as compared to direct evaluation of the transition IV. HYPERFINE-STRUCTURE RESOLVED matrix element in Eq. (18) and summation over angular PHOTOIONIZATION INTENSITIES momentum projection quantum numbers (as in Eq. (4)), since the number of computationally demanding multiple The above treatment is now extended to photoion- sums is minimized. izing transitions connecting hyperfine levels. To that Eq. (21) may be compared with the similar result given end, we need to consider the role of the nuclear spin in Eq. (6) of Dixit et al..29 The equivalence of the two in the neutral and ionic levels. We assume that both results can be seen by taking the squared absolute mag- of these levels may be described with the Hund’s case 22,30,31 nitude of Eq. (6) in Ref. 29, integrating over the entire (bβJ ) angular momentum coupling scheme. In this unit sphere and making use of orthogonality properties scheme, the total nuclear spin I is coupled to J yielding of Wigner symbols and spherical harmonics. the total angular momentum F: F = J + I. The ba- We note that Eq. (21) can also be directly applied to sis functions are denoted by |nΛ, v, NΛSJIFMF i and + + + + + + + + + + symmetric top-molecules exhibiting a Hund’s case (b)- n Λ , v ,N Λ S J I F MF for the neutral and type coupling hierarchy by substituting the quantum ionic states, respectively. Here, I and F (I+ and F +) numbers Λ, Λ+ by the symmetric-top quantum numbers denote the nuclear spin and the total angular momen- K,K+. Asymmetric top molecules can be treated by a tum quantum number, respectively, of the neutral (ionic) + suitable substitution of the rotational wavefunction as state. MF and MF denote the projection angular mo- shown in Ref. 4. Moreover, other Hund’s cases can mentum quantum numbers with respect to the space- be treated by an appropriate frame transformation of fixed z-axis associated with F and F +. All other quan- Eqs. (20) and (21), see, e.g., Ref. 22. tum numbers are defined as before, see Tab. I. The pho- Often, the neutral molecule is in a singlet state, i.e, toionization transition probability is then proportional to S = 0. In this case, we have J = N and S+ = s = 1/2 the quantity

+ ∞ l s F F + X X X X X + + + + + + + + + + P (F,F ) = n Λ , v ,N Λ S J I F MF hs, ms| hl, ml| + l=0 ml=−l ms=−s MF =−F + MF =−F 2

µ |nΛ, v, NΛSJIFMF i . (24)

The coupled angular momentum states are expressed rotational-vibronic and the nuclear spin states as in the decoupled tensor-product basis of the spin- |nΛ, v, NΛSJIFMF i = X X CFMF |nΛ, v, NΛSJM ,IM i , (25) JMJ IMI J I MI MJ 7 and equivalently for the ionic state. In this basis, the In essence, we have reproduced in Eq. (29) the result transition matrix element appearing in Eq. (24) accounts from Eq. (21) with an additional Wigner 6j-symbol de- for scribing the influence of the nuclear spin, i.e., the hyperfine structure effects. Since the photoelectron does n+Λ+, v+,N +Λ+S+J +I+F +M + hs, m | hl, m | F s l not carry nuclear spin, the same selection rule as for ∆J × µ |nΛ, v, NΛSJIFMF i = applies also for ∆F , namely |∆F | = |F + − F | ≤ u. X X F +M + Again due to the separability of the transition matrix C F CFMF + + + + JMJ IMI J MJ I MI element, the relative intensities of photoionizing transi- M +,M M +,M I I J J tions between particular hyperfine levels are determined + + + + + + + + + + × (hn Λ , v ,N Λ S J MJ ,I MI | hs, ms| by the magnitude of the vibronic transition matrix el- × hl, m |)µ |nΛ, v, NΛSJM ,IM i . (26) ements which also determine the intensities of different l J I rotational lines in a photoelectron spectrum. For vanish- Since the nuclear spin is neither affected by the absorp- ing nuclear spin, Eq. (29) reduces to Eq. (21). tion of electromagnetic radiation, nor by the ejection of the photoelectron,12 we may separate the nuclear spin states from the remaining transition matrix element ob- V. APPLICATIONS taining + + + + + + + + + + n Λ , v ,N Λ S J MJ ,I MI hs, ms| hl, ml| A. Example 1: Fine-structure resolved photoionization of × µ |nΛ, v, NΛSJMJ ,IMI i = molecular oxygen + + + + + + + + δI+I δ + n Λ , v ,N Λ S J M MI MI J In order to validate the present results, Eq. (21) is × hs, ms| hl, ml| µ |nΛ, v, NΛSJMJ i . (27) applied to the analysis of the spin-rotation-resolved pho- 3 − 4 − The transition matrix element on the last and next-to- toelectron spectrum of the X Σg → b Σg photoioniz- last line of Eq. (27) is the same as in Eq. (19). Substi- ing transition in molecular oxygen reported by Palm and 13 tuting Eq. (19) into (27), replacing Clebsch-Gordan coef- Merkt. 28 ficients by 3j-symbols and simplifying the result yields The energy level structure of neutral O2 in the elec- 3 − + the following expression for the transition probability: tronic ground state X Σg and of the O2 ion in the b4Σ− state is shown in Fig. 1. We are interested in the P (F,F +) = g + + + Q(1) line of the v = 0 → v = 0 band, i.e., in the tran- (2N + 1)(2N + 1)(2S + 1)(2J + 1)(2J + 1) sition v = 0,N = 1 → v+ = 0,N + = 1, which has + X X been measured with the highest resolution. Neutral O × (2F + 1)(2F + 1)δ + (2k + 1) 2 II 3 − l k=l±1 exhibits a total electron spin S = 1 in the X Σg state, 2 such that there are three spin-rotation components for 2 + + + + + 0k N k N N = 1: J = 0, 1 and 2. For O in the b4Σ− state, × n Λ , v T ∆Λ |nΛ, vi + 2 g −Λ ∆Λ Λ the total electronic spin is S+ = 3/2 giving rise to three  + 2 spin-rotation components with J + = 1/2, 3/2 and 5/2. k+s  J u J  +2 X + u J J Hence, there are in total nine different transitions be- × (2u + 1) N k N + + IF F tween the fine-structure components of the neutral and u=|k−s| S s S  ionic state involved in the Q(1) line. 2 2 X l 1 k X u k s × . (28) The experimental photoelectron spectrum of the fine- −ml µ0 −p w −p ms structure-resolved Q(1) line of Palm and Merkt13 is re- ml ms produced in Fig. 2 (a). The spectrum shows three well- The terms on the last line above again account for separated peaks spaced by about 2 cm−1 reflecting the 1/(3(2k + 1)) for linear polarized radiation (µ0 = 0), i.e., 3 − spin-rotation splitting in the X Σg state of neutral + P (F,F ) = O2. Every peak is composed of three partially over- 1 lapping lines which stem from transitions to different + + 4 − (2N + 1)(2N + 1)(2S + 1)(2J + 1)(2J + 1) spin-rotation components of the ionic b Σg state which 3 −1 + X X are spaced by about 0.4 cm , totaling to the nine spin- × (2F + 1)(2F + 1)δII+ rotation transitions of the Q(1) line. l k=l±1 To compare the measured spectrum with our theoret- + 2 D k E 2 N k N ical predictions, the experimental line intensities have × n+Λ+, v+ T0 nΛ, v ∆Λ −Λ+ ∆Λ Λ been extracted from the measured spectrum by fitting  2 Gaussian functions to the spectral features. The empiri- k+s J + u J 2 X   u J J + cal intensities found this way are shown as blue circles in × (2u + 1) N + k N . IF + F the stick spectrum of Fig. 2 (b). S+ s S  u=|k−s| From the analysis of rotationally, but not fine- (29) structure-resolved photoelectron spectra of Hsu et al.,32 8

J+! (E – IP) / (hc) [cm-1]! a J 2 4 3/2! 0.8! J 1 2 5 2 3 2 + 4 - ++ - O b 4! N 2 =g 1 O2 b !g ! 5/2! 0.4! + 1/2! 0.0! v = 0! 3 - J 0 3 3 - O2 X !g

O X ! units

2 g . J 1 J 1 2 5 2 3 2 arb 2 J 1 2 5 2 3 2 intensity 1 + 4 - -1 O+ b 4 - 2 !g J! E /(hc) [cm ]! O2 b !g 1! 4.0! 0 3 - O X 3!- N 2 = g 1 O2 X !g ! 4 2 0 2 4 6 2! 2.1! relative wavenumber cm1 v = 0! 0! 0.0 b ! 1.0 4 − FIG. 1. Spin-rotational energy-level structure of the b Σg 0.8 3 − + and the X Σg states of the O2 ion and of neutral O2, respectively. Levels are labeled by their total angular momentum intensity 0.6 + quantum numbers J and J for the neutral and ionic state, 0.4 respectively, as well as their relative term value (adapted from normalized Ref. 13). 0.2 0.0 4 2 0 2 4 6 relative wavenumber cm1 it has been established that the photoionization of O 2 mainly involves the coefficient C and to a smaller extent FIG. 2. (a) Measured fine-structure-resolved photoioniza- 0 3 − + 4 − tion spectrum of the O2 X Σg (v = 0,N = 1) → O2 b Σg C2. For k > 2, the vibronic matrix elements essentially + + vanish. (v = 0,N = 1) transition recorded by Palm and Merkt, digitized from Fig. 4 of Ref. 13. The assignment bars indi- According to our model, for k = 0 we have u = 1/2 cate transitions between levels of total angular momentum in Eq. (21), giving rise to transitions with ∆J = ±1/2 quantum number J and J + in the neutral and ionic states, with high intensities. Indeed, the lines with the highest respectively. (b) Stick spectrum showing the normalized in- intensity within each of three peaks for J = 0, 1 and 2 in tensities of individual spin-rotation-resolved transitions. Blue Fig. 2 (a) obey this criterion. On the contrary, transitions circles: experimental intensities extracted from the spectrum with |∆J| > 1/2 are only possible via the k = 2 vibronic in (a). Green diamonds: fit of our photoionization model matrix element with a considerably reduced magnitude. Eq. (23) to the measured intensities with two vibronic coef- Indeed, such lines show only low to medium intensities ficients C0 and C2 treated as free parameters. Red squares: in the experimental spectrum. spin-rotation-resolved intensities calculated with Eq. (23) using the relative values for the vibronic coefficients C0 = 0.6 For a quantitative analysis, we fitted the normalized and C2 = 0.4 determined from a rotationally-resolved photo- experimental intensities shown as blue circles in the stick electron spectrum in Ref. 32. All intensities are normalized spectrum of Fig. 2 (b) to the relative ionization rates to unity for the most intense transition.) as given by the transition probabilities from Eq. (21) weighted by the neutral-state level populations. The lat- ter were calculated from a Boltzmann distribution with a rotational temperature of 7 K as reported in Ref. 13. B. Example 2: Hyperfine propensities in the photoionization The two vibronic coefficients C and C were treated as 0 2 of molecular nitrogen free parameters. The line intensities obtained from this procedure are shown as green diamonds in Fig. 2 (b). As can be seen, the present model reproduces the mea- We are not aware of any hyperfine-resolved photoelec- sured photoionization intensities well. The ratio of the tron spectra reported in the literature that could serve two vibronic coefficients which accounts for the relative as a benchmark to test our photoionization model. In or- intensities of the k = 0 vs. k = 2 line obtained from the der to illustrate the implications of hyperfine structure in fit amounts to C2/C0 = 0.3/0.7. This result is in agree- direct photoionization, we calculated the relative inten- ment with the values of C2 = 0.4 ± 0.1, C0 = 0.6 ± 0.1 sities of hyperfine-resolved photoionization transitions of found in Ref. 32 in the analysis of a rotationally—instead molecular nitrogen which govern the hyperfine popula- of fine-structure—resolved photoelectron spectrum. For tions of the resulting cations.21 As an illustrative exam- comparison, the red squares in Fig. 2 (b) show the pre- ple, we studied the intensities of hyperfine components of 1 + + 2 + + dicted intensities using the values of the vibronic coeffi- the transition N2 X Σg N = 2 → N2 X Σg N = 4.A cients from Ref. 32. In this approach, no free parameters level scheme is shown in Fig. 3. As the neutral electronic except a global intensity normalization factor enter our ground state of N2 is a singlet state, the total electron modeling. Also in this case, the agreement with experi- spin vanishes and we have N = J. The 14N isotope has a ment is very good. nuclear spin of 1. For N2, a total nuclear spin of I = 0 or 2 9 2 13

is possible for N = 2 according to the Pauli principle. We

+ F 2

1.0 concentrate on the case I = 2. The N ion exhibits ﬁne a J 9 2 2 11 0.8 2 F and hyperﬁne structure. The rotational levels are split by 2 2 9 2 intensity

0.6 2

9 2 7 11 2 7 5 F 2 2 5 0.4 2 the spin-rotation interaction into two spin-rotation com- F 2 2 F 7 5 F 9 F F 5 7 F

0.2 F F normalized ponents labeled by the quantum number J, which may F F + + + + F take the values J = N + 1/2 and J = N − 1/2. F0 F1 F2 F3 F4

The spin-rotation levels are split further into hyperﬁne 2 11

+ levels associated with the quantum number F with the F 1.0 2 + + + + + + + b J 7 2 9

values F = J + I ,J + I − 1,..., |J − I |. 0.8 2 F 1 + 7 intensity 2

0.6 2 Similar to the previous example of O , the N X Σ 2 2 2 2 7

g 2 5 5 2 9 F 2 3 2 2

0.4 3 2 2 2 2 2 2 + F 2 + 5 3 11 F F 9 7 7 F 5 3 5 F

→ N X Σ photoionization transition is dominated by F

2 g 0.2 F normalized F F F F F F F 37,38 F the vibronic coefficients C0 and C2. Coefficients with k > 2 are considerably smaller and are thus neglected F0 F1 F2 F3 F4 + here. For the N = 2 → N = 4 transitions, we have FIG. 4. Relative intensities of hyperfine components of the + J = N = 2 and J = 9/2 or 7/2. Since for both of these 1 + + 2 + + + N2 X Σg N = J = 2 → N2 X Σg N = 4,J = 9/2 transitions we have |∆J| > 1/2, they may solely occur via (a) and J + = 7/2 (b) photoionization transition calculated the vibronic transition matrix element associated with by Eq. (29). The hyperfine levels in the neutral and ionic C2. state are labeled by their total angular momentum quantum The intensities of the hyperfine components calculated number F and F +, respectively. Transitions with relative from Eq. (29) are shown in Fig. 4. The relative intensi- intensities < 10−3 have been suppressed. ties of the hyperfine transitions are governed by the 6j- symbol on the last line of Eq. (29). A propensity towards transitions obeying the relation ∆J = ∆F is observed, similar as has previously been observed in bound-bound closed, symmetrized expressions of the relevant squared hyperfine-resolved transitions.33,39 photoionization-transition matrix elements which lend themselves to a straightforward derivation of photoionization selection rules and to an efficient computational VI. SUMMARY AND CONCLUSIONS implementation. The present results can be used in the analysis of fine- and hyperfine resolved photoelectron In the present paper, we have developed a general spectra and the prediction of cationic level populations framework for fine- and hyperfine intensities in the di- upon photoionization. rect photoionization of molecules. We have derived The present treatment does not, however, contain the effects of interactions between different ionization channels.40 Whereas these effects often play only a minor F+! role in direct photoionization, they may be observed in 13/2! 11/2! Rydberg and pulsed-field-ionization zero-kinetic-energy J+ = 9/2 ! 9/2! (PFI-ZEKE) photoelectron spectra. In this case, a 7/2! more detailed treatment of the scattering problem of 5/2! + N+ X 2!+ + 2 + N =2 4g! the photoelectron within the framework of MQDT is N X ! + 2 g F ! warranted.14–18 3/2! N a'' 1!+ 1 + 2 g + 5/2! N2 a'' !g J = 7/2 ! 7/2!

N X 1!+ 2 g 9/2! N X 1 + 2 !g 11/2! ACKNOWLEDGMENTS

N+ X 2!+ + 2 + 2 g N2 X !g This work has been supported by the Swiss National

+ Science Foundation as part of the National Centre of N a'' 1! + 2 g F! N a'' 1! 2 g 0! Competence in Research, Quantum Science & Technol-

+ 1! N X 1 2 !g ogy (NCCR-QSIT), the European Commission under the 1 + N = 2 J = N = 2 N X ! ! ! 4! 2 g Seventh Framework Programme FP7 GA 607491 COMIQ 2! 3! and the University of Basel

FIG. 3. Energy-level structure of the nitrogen molecular ion33 1D. W. Turner, C. Baker, A. D. Baker, and C. R. Brundle, and the neutral nitrogen molecule. The energetic order of the Molecular Photoelectron Spectroscopy. (Wiley-Interscience, Lon- 1 + hyperﬁne levels in the neutral N2 Σ state has been esti- don, 1970). g 2 mated using electric-quadrupole coupling constants extrapo- A. M. Ellis, M. Feher, and T. G. Wright, Electronic and Photo- lated from spectroscopic data on the neutral N A3Σ+ state electron Spectroscopy (Cambridge University Press, Cambridge, 2 u 2005). and from N complexes.34–36 2 3F. Merkt, S. Willitsch, and U. Hollenstein, “High-resolution Pho- toelectron Spectroscopy,” in Handbook of High-resolution Spec- 10

troscopy, Vol. 3, edited by M. Quack and F. Merkt (John Wiley 23M. Germann and S. Willitsch, (to be published). & Sons, Hoboken, 2011) p. 1617. 24M. Braunstein, V. McKoy, and S. N. Dixit, J. Chem. Phys. 96, 4S. Willitsch and F. Merkt, Int. J. Mass Spectrom. 245, 14 (2005). 5726 (1992). 5A. D. Buckingham, B. J. Orr, and J. M. Sichel, Phil. Trans. Roy. 25R. N. Zare, Angular Momentum (John Wiley & Sons, New York, Soc. Lond. A 268, 147 (1970). 1988). 6S. N. Dixit and V. McKoy, J. Chem. Phys. 82, 3546 (1985). 26D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, 7S. W. Allendorf, D. J. Leahy, D. C. Jacobs, and R. N. Zare, J. Quantum Theory of Angular Momentum (World Scientiﬁc, Sin- Chem. Phys. 91, 2216 (1989). gapore, 1989). 8K. Wang and V. McKoy, J. Chem. Phys. 95, 4977 (1991). 27A. R. Edmonds, Drehimpulse in der Quantenmechanik (Bibli- 9J. Xie and R. N. Zare, J. Chem. Phys. 97, 2891 (1992). ographisches Institut, Mannheim, 1964). 10S. N. Dixit and V. McKoy, Chem. Phys. Lett. 128, 49 (1986). 28M. Germann, Ph.D. thesis, University of Basel (2016). 11J. Xie and R. N. Zare, J. Chem. Phys. 93, 3033 (1990). 29S. N. Dixit, D. L. Lynch, V. McKoy, and W. M. Huo, Phys. 12R. Signorell and F. Merkt, Mol. Phys. 92, 793 (1997). Rev. A 32, 1267 (1985). 13H. Palm and F. Merkt, Phys. Rev. Lett. 81, 1385 (1998). 30R. A. Frosch and H. M. Foley, Phys. Rev. 88, 1337 (1952). 14H. J. Wörner, U. Hollenstein, and F. Merkt, Phys. Rev. A 68, 31T. M. Dunn, “Nuclear Hyperﬁne Structure in the Electronic 032510 (2003). Spectra of Diatomic Molecules,” in Molecular Spectroscopy: Mod- 15A. Osterwalder, A. Wüest, F. Merkt, and Ch. Jungen, J. Chem. ern Research, edited by K. Narahari Rao and C. Weldon Mathews Phys. 121, 11810 (2004). (Academic Press, New York, 1972). 16H. A. Cruse, Ch. Jungen, and F. Merkt, Phys. Rev. A 77, 042502 32C.-W. Hsu, M. Evans, S. Stimson, C. Y. Ng, and P. Heimann, (2008). Chem. Phys. 231, 121 (1998). 17Ch. Jungen, “Elements of Quantum Defect Theory,” in Handbook 33S. D. Rosner, T. D. Gaily, and R. A. Holt, J. Mol. Spectrosc. of High-resolution Spectroscopy, Vol. 1, edited by M. Quack and 109, 73 (1985). F. Merkt (John Wiley & Sons, Hoboken, 2011) p. 471. 34R. S. Freund, T. A. Miller, D. De Santis, and A. Lurio, J. Chem. 18D. Sprecher, Ch. Jungen, and F. Merkt, J. Chem. Phys. 140, Phys. 53, 2290 (1970). 104303 (2014). 35D. De Santis, A. Lurio, T. A. Miller, and R. S. Freund, J. Chem. 19X. Tong, A. H. Winney, and S. Willitsch, Phys. Rev. Lett. 105, Phys. 58, 4625 (1973). 143001 (2010). 36A. C. Legon and P. W. Fowler, Z. Naturforsch. 47a, 367 (1992). 20X. Tong, T. Nagy, J. Yosa Reyes, M. Germann, M. Meuwly, and 37P. Baltzer, L. Karlsson, and B. Wannberg, Phys. Rev. A 46, 315 S. Willitsch, Chem. Phys. Lett. 547, 1 (2012). (1992). 21M. Germann, X. Tong, and S. Willitsch, Nat. Phys. 10, 820 38G. Öhrwall, P. Baltzer, and J. Bozek, Phys. Rev. A 59, 1903 (2014). (1999). 22J. M. Brown and A. Carrington, Rotational Spectroscopy of 39M. Germann and S. Willitsch, Mol. Phys. 114, 769 (2016). Diatomic Molecules (Cambridge University Press, Cambridge, 40F. Merkt and T. P. Softley, Int. Rev. Phys. Chem. 12, 205 (1993). 2003).