is the scattering phase shift associated with the th continuum molecular orbital, and M

Transcription

1 Moecuar-orbita decomposition of the ionization continuum for a diatomic moecue by ange- and energy-resoved photoeectron spectroscopy. I. Formaism Hongkun Park and Richard. Zare Department of Chemistry, Stanford University, Stanford, Caifornia 9435 Received 9 September 995; accepted 2 December 995 A theoretica formaism is deveoped for the quantum-state-specific photoeectron anguar distributions PADs from the direct photoionization of a diatomic moecue in which both the ionizing state and the state of the ion foow Hund s case b couping. The formaism is based on the moecuar-orbita decomposition of the ionization continuum and therefore fuy incorporates the moecuar nature of the photoeectron ion scattering within the independent eectron approximation. The resuting expression for the quantum-state-specific PADs is dependent on two distinct types of dynamica quantities, one that pertains ony to the ionization continuum and the other that depends both on the ionizing state and the ionization continuum. Specificay, the eectronic dipoe-moment matrix eement r exp(i ) for the ejection of a photoeectron with orbita anguar momentum quantum number making a projection on the internucear axis is expressed as Ū exp(i )M, where Ū is the eectronic transformation matrix, is the scattering phase shift associated with the th continuum moecuar orbita, and M is the rea eectronic dipoe-moment matrix eement that connects the ionizing orbita to the th continuum moecuar orbita. Because Ū and depend ony on the dynamics in the ionization continuum, this formaism aows maxima expoitation of the commonaity between photoionization processes from different ionizing states. It aso makes possibe the direct experimenta investigation of scattering matrices for the photoeectron ion scattering and thus the dynamics in the ionization continuum by studying the quantum-state-specific PADs, as iustrated in the companion artice on the photoionization of O. 996 American Institute of Physics. S I. ITRODUCTIO Recent advances in theoretica and experimenta techniques have enabed the dynamica study of moecuar photoionization processes in unprecedented detai. Differentia photoionization cross sections can now be cacuated for sma moecues amost quantitativey using ab initio methods.,2 The deveopment of experimenta techniques has aowed for the resoution of individua quantum eves of the moecuar ion 3 7 as we as the photoeectron anguar distributions PADs associated with them. 8 3 The synergy between these theoretica and experimenta deveopments has significanty increased our knowedge about the photoionization of moecues and has narrowed the gap between our understanding of moecuar and atomic photoionization. Perhaps the best exampe that iustrates the experimenta progress in the dynamica study of moecuar photoionization is the previous report of a compete quantum mechanica description for the photoionization of the O A 2 state using ange- and energy-resoved photoeectron spectroscopy. 2 By ionizing opticay aigned O A 2 moecues in a specific quantum eve using resonance-enhanced mutiphoton ionization REMPI and by measuring the PADs associated with the production of each rotationa eve of the O X ion, investigators have been abe to deduce a the dynamica parameters needed to describe the photoionization process from the O A 2 state. The dynamica parameters, which are the eectronic dipoe-moment matrix eements that connect the O A 2 state to each partia wave in the ionization continuum that yieds the O X ion, provide a most detaied description of this process and can aso be directy compared with the corresponding quantities that are computed in ab initio cacuations. 4 Athough the above procedure has been appied so far ony to the photoionization of the O A 2 state, it shoud be generay appicabe to the study of photoionization dynamics of other sma diatomic moecues. In the one-photon ionization of a diatomic moecue, the ionization continuum that consists of a photoeectron and the moecuar ion in a given rovibronic state is couped to an ionizing state through the dipoar interaction between the moecue and the ionizing photon in the imit of ow ionization intensity. Therefore, the same fina state is reached in the photoionization from different ionizing states when the photoeectron energy and the rovibronic state of the moecuar ion are the same. Despite this commonaity between different photoionization pathways, the dipoe-moment matrix eements formuated based on the partia-wave decomposition of the moecuar ionization continuum pertain ony to a photoionization event from a given ionizing state and do not give detaied insight into the dynamics of the ionization continuum. This situation is in contrast to that in atomic photoionization, in which the reationship between dipoe-moment matrix eements for distinct photoionization processes has 4554 J. Chem. Phys. 4 (2), 22 March /96/4(2)/4554/4/$. 996 American Institute of Physics Downoaded 6 Jan 2 to Redistribution subject to AIP icense or copyright; see

2 H. Park and R.. Zare: Photoeectron spectroscopy. I. Formaism 4555 FIG.. A schematic iustration of the photoionization of a diatomic moecue. In the figure r designates a radia distance from the center of mass of the ion core to a fictitious surface that divides the ion-core region and the asymptotic region. been used to provide a unified description of the ionization continuum of an atom Because the ionic potentia under which the photoeectron moves is a centra fied within the independent eectron approximation, the partia-wave channes, each denoted by a definite eectronic orbita anguar momentum, form a compete set of independent ionization channes in atomic photoionization. Consequenty, the phase difference between dipoe-moment matrix eements for different partia-wave channes remains the same for photoionization processes from different eectronic states of an atom. The phase difference is in essence a characteristic of the ionization continuum of an atom, not the atomic energy eve from which photoionization occurs. When the photoeectron is far from the ion core, whether the ion core is atomic or moecuar, it moves principay under the infuence of the Couombic potentia because of the charge on the ion core see Fig.. 23 This region of the configuration space where the Couombic potentia dominates the photoeectron motion is designated as the asymptotic region. Because of the centra nature of the Couombic potentia, the partia-wave description of the ionization continuum is appropriate in the asymptotic region. On the other hand, the photoeectron fees the nature of the ionic core when it is near the core, which we ca the ion-core region. 24 The dynamics of the photoeectron in this region is what differentiates the atomic and the moecuar ionization continua. When the ion core is atomic, the photoeectron moves under the centra potentia within the independent eectron approximation, athough the potentia is different from the Couombic one because of the shieding caused by the presence of other eectrons. When the ion core is moecuar, the potentia under which the photoeectron moves is noncentra even in the independent eectron approximation, and the description of the ionization continuum based on partia waves with definite is not appropriate in the ion-core region. The separation of the eectronic configuration space into the ion-core region and the asymptotic region is justified because any mutipoar potentia that the photoeectron experiences fas much faster with distance than the Couombic potentia. The mutipoar interaction can thus be considered to be short range as ong as the ion-core region is arge enough to encompass a region of the photoeectron configuration space where the buk of mutipoar interactions occur. Because the ion-core region shoud be arge, however, the interaction between the photoeectron and the ion-core is not uniform even inside the ion-core region, and the interaction can be cassified into two distinct types depending on the distance between the photoeectron and the ion core. 25,26 When the photoeectron is inside the ion-core region but is outside the perimeter of the bound moecuar orbitas of the ion core, the interaction between the photoeectron and the ion core can be viewed roughy as that between the photoeectron and the mutipoes ocated at the center of mass of the ion. We ca this region of the configuration space the mutipoe-moment-interaction region. When the photoeectron is inside the perimeter of the bound moecuar orbitas, on the other hand, the interaction can no onger be viewed as that of the photoeectron and the mutipoe. Instead, the exchange interaction between the photoeectron and the bound eectrons in the core dominates the motion of the photoeectron. This region of the configuration space is designated as the eectron-exchange-interaction region. In this paper, we present a theoretica formaism for the quantum-state-specific PADs from the direct photoionization of a diatomic moecue based on the quantum scattering theory formaism and anguar momentum couping agebra. Unike the partia-wave formaism that was the basis of our previous work,,27 the present formaism treats expicity the mixing between different partia waves in the ion-core region, and it thus fuy incorporates the moecuar nature of the probem within the independent eectron approximation. The resuting expressions ceary show the commonaity between different photoionization processes and provide the reationship between the dipoe-moment matrix eements that coupes different ionizing states to the same ionization continuum. We note that most of the theoretica eements that form the basis of the present artice have aready appeared in the iterature. The scattering theory formaism used in this artice has been extensivey deveoped to expain atomic Rydberg spectroscopy and photoionization by Seaton and co-workers 28 in the framework of the mutichanne quantum defect theory MQDT. MQDT has subsequenty been adapted to moecuar probems through the pioneering efforts of Fano, Jungen, Greene, and co-workers. 23,24,26,29 The anguar momentum couping expressions that appear in this artice are, on the other hand, mosty adapted from our previous work,27 and that of McKoy and co-workers. 3 The present formaism is unique in its couping of these two theoretica machineries that aows for a unified description of the direct photoionization of a moecue within the independent eectron approximation. Because many of the dynamica parameters that appear in this formaism pertain directy to the ionization continuum J. Chem. Phys., Vo. 4, o. 2, 22 March 996 Downoaded 6 Jan 2 to Redistribution subject to AIP icense or copyright; see

3 4556 H. Park and R.. Zare: Photoeectron spectroscopy. I. Formaism without reference to the ionizing states, they shoud be common to photoionization processes from different ionizing states of a moecue as ong as the same fina state is reached, that is, as ong as the rovibronic state of the ion and the energy of the photoeectron are the same. Emphasis is given to the carification of basic physica ideas behind the theoretica formaism within the independent eectron framework. The paraes between the ideas in this formaism and those used to describe atomic photoionization are aso emphasized. The expressions given in this artice can be used by an experimentaist to extract dynamica information on the moecuar ionization continuum from the quantum-statespecific PADs; the procedure is iustrated in the companion artice on the photoionization of O. 3 This formaism can aso be empoyed to predict experimenta PADs based upon dynamica quantities obtained from theoretica cacuations or from the spectroscopy of high-ying Rydberg states. The organization of this artice is as foows. In Sec. II, we outine the construction of the one-eectron continuum moecuar orbita for a photoeectron moving in the fied of a diatomic ion core. Section II begins with a brief discussion about the one-eectron atomic continuum wave function to introduce the basic physica ideas and the machinery of scattering theory. In Sec. III, we present an expression for the quantum-state-specific PADs of a diatomic moecue based on the continuum-moecuar-orbita decomposition deveoped in Sec. II. The spin part of various wave functions is not expicity considered in Secs. II and III because we restrict ourseves to states of diatomic moecues that foow Hund s case b couping. In Sec. IV, we discuss the appicabiity of the present formaism in retrieving information about the ionization continuum from experimenta PADs. We aso discuss briefy in Sec. IV the possibe extension of the formaism to more genera moecuar photoionization probems. Appendix A contains a brief discussion about various matrices that appear in the main text; in Appendix B, we show the forma equivaence of the expressions in this artice and those in the open-continuum MQDT. II. COTIUUM MOLECULAR ORBITALS Photoionization can be considered as an absorption of a photon foowed by a haf coision between the photoeectron and the ion. The fina state reached by the photoabsorption is thus the coision state that describes the scattering event between the photoeectron and the ion. In this section, we construct the one-eectron continuum wave function for a photoeectron moving in a diatomic ion fied based on the scattering theory formaism. 29 First, we briefy discuss the one-eectron continuum wave function for a photoeectron moving in an atomic ion fied to ay the groundwork for the discussion of the moecuar continuum wave functions. Because a photoeectron aways moves under a centra potentia in the independent eectron approximation in the case of atomic photoionization, the orbita anguar momentum quantum number for the photoeectron is a good quantum number. Consequenty, the scattering between the photoeectron and the ion becomes a so-caed singe-channe phenomenon, 29 and the discussion becomes particuary transparent. As wi be seen, many of the quantities and ideas defined for the atomic continuum wave function can be directy carried over to the discussion of the moecuar continuum wave function. Because of the centra nature of the ionic potentia under which the photoeectron moves, the fina state for the photoeectron in atomic photoionization can be expanded in partia waves 29,32 that conform to the incoming-wave boundary condition 29 k m i e i Y m * kˆ m r;k. In Eq., r is the position vector of the photoeectron in the aboratory LAB frame whose origin is ocated at the atomic nuceus, and kkkˆ, where kˆ is the direction aong which the photoeectron is ejected in the LAB frame. The quantity k is the magnitude of the asymptotic inear momentum of the photoeectron, which is reated to the asymptotic photoeectron energy,, by k2, 2 2m e where m e designates the reduced mass of the eectron. Atomic units are used throughout this artice, and the meanings of anguar momentum quantum numbers that appear in this artice are isted in Tabe I. denotes the Couomb phase shift, which is given by karg im e Z/k, where Z is the net charge on the ion core. In photoionization of neutra species, Z, but we retain the symbo Z in the foowing expressions because many of the ideas deveoped in this section can be used to describe the ionization continuum for mutipy charged ions as we. Athough is an energy-dependent quantity, as is cear in Eq. 3 through its dependence on k, we do not expicity indicate its energy dependence to make the notation more compact. In Eq., m r;k is the energy-normaized wave function for the continuum eectron referenced to the scattering ampitude S. The asymptotic form of m r;k is given by 29,32 r m r;k m e where /2 k 2ir e ix S *ke ix Y m rˆ, x kr 2 m ez n2kr k. 5 We aso designate m r;k as the partia-wave basis function because it is the basis function that appears in the partia wave decomposition of the continuum state in Eq.. The coefficients on the right-hand side of Eq. 4 are chosen to ensure the normaization of m r;k per unit Rydberg energy interva. The scattering ampitude S designates the ampitude ratio of the outgoing and incoming wave components 3 4 J. Chem. Phys., Vo. 4, o. 2, 22 March 996 Downoaded 6 Jan 2 to Redistribution subject to AIP icense or copyright; see

4 H. Park and R.. Zare: Photoeectron spectroscopy. I. Formaism 4557 TABLE I. Anguar momentum quantum numbers used in the text. Anguar momentum Laboratory fixed projection Moecue-fixed projection Description M ucear rotation pus orbita anguar momentum for the ionizing state M ucear rotation pus orbita anguar momentum for the ion t M t t Anguar momentum transferred between the photoeectron and the ion J M J J Tota anguar momentum excuding nucear and eectronic spins m Photoeectron anguar momentum Photon anguar momentum for each vaue of, 29 and it can be reated to the scattering phase shift, (k) moduo, by the reation S ke 2i. 6 The physica meaning of wi be discussed beow. Because the photoeectron moves under the Couombic potentia in the asymptotic region, the one-eectron continuum wave function in that region can be exacty represented as a inear combination of the reguar and irreguar Couomb basis functions, f and g, respectivey. Here, f and g are two ineary independent soutions of the radia Schrödinger equation for the Couombic potentia whose anaytica properties are we documented 28 and whose asymptotic forms are given by 24 r f r;k 2m /2 e k sin x and r g r;k 2m /2 e k cos x. We thus introduce a continuum wave function that is the inear combination of f and g in the asymptotic region m r;k &r f r;kk kg r;ky m rˆ rr. 8 Here, r designates a radia distance from the atomic nuceus to a fictitious surface that divides the ion-core region and the asymptotic region. In Eq. 8, K is caed the reaction ampitude and is given by K ktan. 9 The reaction ampitude K represents the ratio of g with respect to f in m r;k. ote that the radia part of m r;k is rea, whereas that of m r;k is compex. The fact that g appears in m r;k is a direct manifestation of the presence of other eectrons in the ion-core region When the ion is hydrogenic, that is, when there are no other eectrons except the eectron being ionized in the ion-core region, the potentia under which the photoeectron moves is aways Couombic. In this case, the irreguar Couomb basis function, g, cannot be a part of the acceptabe continuum wave function because of the boundary condition at the origin. When there are other eectrons, however, g can be a part of m r;k outside the ion-core region because the boundary condition at the origin is ifted; the ion-core region is effectivey treated as a back box, and the boundary is moved to rr. The compex interaction between the photoeectron and the other eectrons in the ion-core region is nevertheess whoy represented by the scattering phase shift or equivaenty the scattering ampitude S and the reaction ampitude K outside the ion-core region. 29 The reationship between m r;k and m r;k is given by m r;kik k m r;k, as can be shown easiy from Eqs. 4 through 8. We note that m r;k is not normaized per unit Rydberg energy interva and thus introduce another one-eectron wave function m r;k that is propery normaized: m r;kcos m r;k. By substituting Eqs. 9 to into Eq., we obtain the photoeectron fina-state wave function k expressed in terms of m r;k. Before we proceed to a discussion of the fina oneeectron state for the moecuar photoionization, a comment on the physica meanings of quantities that have been introduced so far is appropriate. The scattering phase shift carries information on how the ion-core short-range interaction affects each partia-wave channe in the asymptotic region. It is aso an anaytic continuation of the quantum defect defined for the Rydberg states of the atom. 28 Because the photoeectron moves in the deep Couomb we near the ion core with a arge associated kinetic energy, the dynamics inside the ion-core region and thus are not very sensitive to the asymptotic photoeectron energy. 34 The sensitivity of to the J. Chem. Phys., Vo. 4, o. 2, 22 March 996 Downoaded 6 Jan 2 to Redistribution subject to AIP icense or copyright; see

5 4558 H. Park and R.. Zare: Photoeectron spectroscopy. I. Formaism photoeectron energy is roughy proportiona to the ratio of the energy variation to the ionization potentia of the atom. When the variation of the asymptotic photoeectron energy is sma, can therefore be approximated to be independent of energy and can be compared with the quantum defect for the high-ying Rydberg states of the atom. The compicated energy dependence of the one-eectron continuum wave functions m r;k and m r;k that stems from the eectronic motion in the asymptotic region is compacty represented by the we-known properties of the reguar and irreguar Couomb basis functions f and g and the Couomb phase shift. 24,29 The wave function m r;kand m r;k is the oneeectron eigenfunction of both the orbita anguar momentum operator and the atomic Hamitonian. Therefore, it can be identified as the continuum atomic orbita, setting aside the asymptotic boundary condition that shoud be appied to it through Eqs. and. Severa differences exist, however, between m r;k and bound atomic orbitas that are commony encountered in eectronic structure cacuations. Whereas a bound atomic orbita is defined ony at a specific orbita energy, m r;k is defined at every energy. For a given photoeectron energy, an infinite number of m r;k exist that are degenerate. Being a continuum wave function, m r;k is normaized to the Dirac deta function, that is, m r;k is an improper vector in the physica Hibert space. 35 For this reason, we cannot interpret m * (r;k) m (r;k) as representing the probabiity density of finding a photoeectron at a specific point in the configuration space, unike the case for bound orbitas. These differences are manifestations of the fact that m r;k is defined for the continuum eectron. The expressions for the fina state in moecuar photoionization can be obtained foowing a simiar procedure. The fact that the moecuar ion has a structure associated with it, however, introduces two important modifications in the resuting expressions. First, any one-eectron wave function for the moecuar ionization continuum shoud be referenced to the moecue-fixed MF frame. Second, the potentia under which the photoeectron moves in the ion-core region is noncentra, and the scattering between the photoeectron and the ion core becomes a mutichanne probem. 29 The photoeectron motion in the asymptotic region is nevertheess governed by the Couombic potentia as discussed in Sec. I, and the one-eectron fina-state wave function for the photoeectron in the photoionization of a diatomic moecue can be expanded in partia waves 3,36 that conform to the incoming-wave boundary condition, 37 just as for the one-eectron fina-state wave function for the photoeectron in the atomic photoionization k;r m i e i Y m * kˆd m * Rˆ r;k,r. 2 Here, k and are defined in the same way as in the atomic photoionization see Eqs. 2 and 3, and R is the internucear distance of the moecuar ion. In Eq. 2, r designates the position vector of the photoeectron in the moecue-fixed MF frame. The coordinate systems are defined such that the z axis of the MF frame is aong the internucear axis of the moecuar ion and the origin of the MF frame coincides with that of the LAB frame at the center of mass CM of the ion. Thus the two frames are reated to each other through a frame rotation by a set of three Euer anges Rˆ. Whenever there is no ambiguity, we adopt a convention that a vector with a prime is expressed in the LAB frame, whereas a vector without a prime is expressed in the MF frame. Comparison of Eqs. and 2 shows that the rotation matrix eement D m * (Rˆ ) appears in Eq. 2 because the one-eectron wave function r;k,r is expressed in the MF frame. Here, r;k,r is the energy-normaized one-eectron wave function for the continuum eectron referenced to the MFframe scattering matrix S matrix. The asymptotic form of r;k,r is given by 38 r r;k,r m /2 e k 2ir e ix * S k,re ix Y rˆ. 3 The moecuar wave function r;k,r is aso designated as the partia-wave basis function, just as m r;k is in atomic photoionization. S is the (,) eement of the S matrix for a given vaue of, which is the z-axis projection of in the MF frame. The S matrix is the mutichanne anaog of the scattering ampitude, S, defined in Eq. 4. Together with the incoming-wave boundary condition presented in Eq. 2, the form of Eq. 3 ensures that for arge r the continuum wave function is composed of one outgoing Couomb wave with and the incoming Couomb waves with a, consistent with the time-dependent representation of the photoionization process. 37,39 Inspection of r;k,r in Eq. 3 ceary reveas the moecuar nature of the continuum wave function despite its asymptotic form. Athough the photoeectron experiences the spherica Couombic potentia when it is in the asymptotic region, it moves under the nonspherica moecuar potentia when it is inside the ion-core region. Thus is not a good quantum number in this region, and the -mixing between different partia waves, or equivaenty the -changing coision between the photoeectron and the ion core, can occur when the photoeectron emerges from the ion core. In the independent eectron approximation, however, is sti a good quantum number because of the cyindrica nature of the diatomic ion-core potentia. The possibiity of -changing coisions within the same manifod is indicated in Eq. 3 by off-diagona eements of the S matrix, S ( ). The S matrix designates a portion of the fu S matrix that is bock diagona in. S gives the ampitude ratio of the outgoing Couomb wave with and incoming Couomb wave with for each vaue of. It is important to note that S being a continuum property, is a function of the asymptotic energy of the photoeectron and the internucear distance ony. The S matrix is a unitary matrix, and it com- J. Chem. Phys., Vo. 4, o. 2, 22 March 996 Downoaded 6 Jan 2 to Redistribution subject to AIP icense or copyright; see

6 H. Park and R.. Zare: Photoeectron spectroscopy. I. Formaism 4559 TABLE II. Gossary of terms used in the text. Term Atomic case Spherica potentia Diatomic Cyindrica potentia Description a (k) (k) Couomb phase shift Eectronic eigenchanne index; Orbita index S (k) S (k,r), eement of the scattering matrix for a given vaue of K (k) K (k,r), eement of the reaction matrix for a given vaue of U (k,r), eement of the eectronic transformation matrix, eement for a given vaue of (k) Eectronic eigenphase shift k k;r Photoeectron fina-state wave function m r;k r;k,r Continuum one-eectron wave function referenced to the scattering matrix; Partia-wave basis function m r;k r;k,r Continuum one-eectron wave function referenced to the reaction matrix m r;k (r;k,r) a The descriptions in parentheses appy ony to diatomic moecues. Eectronic eigenchanne wave function; Continuum orbita petey specifies the asymptotic outcome of the coision between the photoeectron and the ion core for a given. For the same reason as for the atomic continuum wave function, the one-eectron continuum wave function for the moecue can be expressed as a inear combination of f and g in the asymptotic region. We thus introduce the continuum wave function that is referenced to the MF-frame reaction matrix K matrix, 38 r;k,r, which is the mutichanne anaog of m r;k for an atom r;k,r &r f r;k K k,rg r;ky rˆ rr. 4 Once again, r is a radia distance from the CM of the ion core to a fictitious surface that divides the ion-core region and the asymptotic region, I is the unit matrix, and K is the MF-frame reaction matrix for a given. The K matrix is the mutichanne anaog of the reaction ampitude K that appears in Eq. 8. The fu K matrix is bock diagona in for the same reason the S matrix is. The matrix eement K signifies the ratio of g with respect to f in r;k,r, and it is a function of the photoeectron energy and internucear distance ony, just as S is. The reation between the S matrix and the K matrix is given by 4 S ik ik, 5a which can be rewritten in matrix notation as S IiK IiK. 5b Whereas the S matrix is a compex unitary matrix, the K matrix is a rea symmetric matrix a rea Hermitian matrix. Thus r;k,r has a rea radia part whereas the radia part of r;k,r is compex. Using Eqs. 3 through 5, it can be shown that r;k,r and r;k,r are reated to each other by the foowing equaity: 38,4 r;k,r IiK r;k,r. 6 Just as S and K defined for an atom do, both the S and K matrices competey specify the asymptotic outcome of the eectron scattering from the inear moecuar ion core for a given. The effect of the compex interactions that the photoeectron experiences inside the ion-core region is thus whoy represented by the K matrix outside the ion-core region. We emphasize that the K matrix defined in Eqs. 4 and 5, ike K that appears in the atomic continuum wave function, characterizes ony the short-range coision dynamics between the photoeectron and the ion core because of the anisotropic moecuar-ion-core potentia. 24,28 Because the eectron moves in the deep Couomb we near the ion core, the short-range dynamics and thus the K matrix are rather insensitive to the asymptotic photoeectron energy, just as K is insensitive. In many appications, the K matrix can be treated to be approximatey independent of energy. To this point, the discussion reveas the neary one-toone correspondence between various quantities defined for moecuar and atomic photoeectrons; S, K, r;k,r, and r;k,r are the mutichanne anaogs of S, K, m r;k, and m r;k, respectivey. This correspondence becomes more evident when we consider the scattering ampitude S and the reaction ampitude K defined for the atomic photoeectron as the diagona eements of the scatter- J. Chem. Phys., Vo. 4, o. 2, 22 March 996 Downoaded 6 Jan 2 to Redistribution subject to AIP icense or copyright; see

7 456 H. Park and R.. Zare: Photoeectron spectroscopy. I. Formaism ing matrix and the reaction matrix, respectivey see Tabe II. Because of the centra nature of the atomic potentia, the off-diagona eements of these matrices are a zero in the atomic case. ote, however, that the one-eectron wave functions r;k,r and r;k,r defined for the moecuar photoeectron are not the eigenfunctions of the moecuar Hamitonian in the ion-core region, as can be seen from the off-diagona eements of the S and K matrices. The atomic counterparts of them, m r;k and m r;k, are eigenfunctions of the atomic Hamitonian throughout the configuration space for the photoeectron in the independent eectron approximation. Indeed r;k,r and r;k,r can be rewritten in terms of another set of one-eectron wave functions that are eigenfunctions of the short-range moecuar Hamitonian based on the eigenchanne formuation of scattering theory. 23,24,42,43 The derivation is briefy outined beow. Because the K matrix is rea and symmetric, it can be diagonaized by the unitary transformation K U tan T U U tan U. 7a Equation 7a can be written in the matrix form K U tan U T, 7b where U is an eectronic transformation matrix that diagonaizes the K matrix, U T is the transpose of U, and tan is a diagona matrix with the eement tan on the th row. Because K is a rea matrix, we may choose U to be a rea orthogona matrix. We designate as the eectronic eigenchanne index or the moecuar orbita index and as the eectronic eigenphase shift. Using Eq. 7 and the rea unitary property of U,itcan be easiy shown that IiK U e i cos U T. 8 Here, e i cos is a diagona matrix with e i cos on the th row. Equation 8 can be inserted into Eq. 6 to obtain r;k,r U e i cos U r;k,r. Using Eq. 4 and the normaization properties of U, U U, U U, we can rewrite Eq. 9 as r;k,r where (r;k,r) is given by U e i r;k,r, r;k,r &r U f r;kcos g r;ksin Y rˆ rr. 22 The wave function (r;k,r) is designated as the eectronic eigenchanne wave function. 24,43 Finay, by inserting Eq. 2 into Eq., we can write the one-eectron continuum state that conforms to the incoming-wave boundary condition as k;r i e i Y m * kˆd * mrˆ m U e i r;k,r. 23 Before we proceed to the next section, we shoud investigate the physica meanings of, U, and (r;k,r), which were designated as the eectronic eigenphase shift, the eectronic transformation matrix eement, and the eectronic eigenchanne wave function, respectivey. (r;k,r) defined in Eq. 22 is an admixture of f and g with various just ike r;k,r in Eq. 4. Mathematicay, the (r;k,r) and the r;k,r are independent basis sets that are reated to each other by the orthogona transformation matrix U aside from the factor cos. The f and g that appear in (r;k,r) are, however, weighted regardess of with the same quantities, cos and sin, respectivey, unike those in r;k,r. This observation indicates that (r;k,r) is an eigenbasis that diagonaizes the K matrix. 24,43 Because the K matrix characterizes the scattering process between the photoeectron and the ion core for a given, (r;k,r) is the eigenfunction of the coision process described by the K matrix. In addition, because (r;k,r) and (r;k,r) ( ) are not mixed by the scattering of the eectron from the ion core, there shoud be no short-range interaction that coupes (r;k,r) and (r;k,r). 24 This ack of short-range mixing means that (r;k,r) is aso an eigenfunction of the short-range MF-frame eectronic Hamitonian at a given asymptotic energy of the photoeectron and the internucear distance R. Quantum mechanicay, any inear combination of various Couomb basis functions, f and g, with the same k and R can be considered an eigenfunction of the MF-frame eectronic Hamitonian as ong as it conforms to the appropriate boundary conditions because a of the inear combinations are defined to be degenerate. They are reated to each other by an appropriate unitary transformation. Among these eigenfunctions, however, the partia-wave basis function r;k,r and the eectronic eigenchanne wave function (r;k,r) stand out because of their physica significance. r;k,r is the basis function for the partia-wave decom- J. Chem. Phys., Vo. 4, o. 2, 22 March 996 Downoaded 6 Jan 2 to Redistribution subject to AIP icense or copyright; see

8 H. Park and R.. Zare: Photoeectron spectroscopy. I. Formaism 456 position of the continuum wave function, k;r. Because asymptoticay it is composed of the incoming and outgoing Couomb waves, the physica boundary condition at infinity that is appropriate for the photoionization process can be easiy appied to r;k,r. On the other hand, (r;k,r) is the eigenbasis for the coision between the photoeectron and the moecuar-ion core that is described by the eectronic K matrix. 24,43 It is aso the one-eectron eigenfunction of the short-range eectronic Hamitonian at a fixed nucear geometry. Considering that the short-range Hamitonian is what characterizes the moecuar nature of the scattering process, (r;k,r) can be appropriatey designated as the continuum moecuar orbita. Just ike the continuum atomic orbita, the continuum moecuar orbitas have the foowing properties: (r;k,r) is defined at every energy,. For a given photoeectron energy, there are infinite number of (r,k,r) that are degenerate. Because (r;k,r) is an improper vector in the physica Hibert space, we cannot interpret * (r;k,r) (r;k,r) as representing the probabiity density of finding the eectron at a specific point of the configuration space. With the above interpretations of (r;k,r) in mind, it is straightforward to assess the physica meanings of and U. The eectronic eigenphase-shift is a scattering phase shift associated with the eectronic eigenchanne wave function (r;k,r) at a given k and R. It carries information about how the ion-core short-range potentia acts on each eectronic eigenchanne. is aso an anaytic continuation of the eectronic quantum defect defined for the Rydberg states of moecues. 28 Because of the insensitivity of the short-range scattering dynamics to the energy of the photoeectron, can be compared with the corresponding eectronic quantum defect of high-ying Rydberg states obtained from MQDT anaysis of Rydberg spectra. 44 The correspon- dence between defined for moecues and defined for atoms can be recognized immediatey see Tabe II. U, which is defined to be the eement of the eectronic transformation matrix that diagonaizes the K matrix, can be interpreted as the projection of a partia-wave wave function onto the eigenstate of the coision or of the short-range Hamitonian for a given k and R. As for, U can be compared with the Rydberg-series mixing coefficient that can be determined from the spectroscopy of Rydberg states. 44 III. DIPOLE-MOMET MATRIX ELEMETS AD PHOTOELECTRO AGULAR DISTRIBUTIOS The dynamics of one-photon ionization in the weak-fied imit is governed by the eectric dipoe-moment matrix eements that connect the ionizing state to the ionization continuum. Thus an expression for the quantum-state-specific PADs can be obtained from expressions for these dipoe matrix eements. In this section, we derive an expression for the PAD when both the ionizing state and the state of the ion are described by Hund s case b couping. The derivation given in this artice can be easiy extended to other Hund s couping cases. The approximations made in the derivation are aso discussed in detai in this section. We note that the same approximations were used in the earier iterature on this subject without expicit justification.,27 The wave function for an ionizing state that foows Hund s case b couping is given by the Born Oppenheimer product pm 2 /2 8 2 p RD * M Rˆ. 24 Here, is the vibrationa quantum number for the ionizing state, and p represents quantum numbers needed to specify the ionizing state competey. A the anguar momentum quantum numbers are isted in Tabe I. (R) is a vibrationa wave function in the ionizing state that we set to be rea. The wave function for the composite state of the photoeectron and the ion that foows Hund s case b couping can aso be written as a simiar Born Oppenheimer type product using the one-eectron continuum wave functions given in Eq. 2 p M ;k /2 RD * M Rˆ i e i Y m * kˆd * mrˆ m à r;k,rp ]. 25 Here, represents the vibrationa quantum number for the ion, p represents the quantum numbers needed to specify the state of the ion competey, and (R) represents a vibrationa wave function in the ion eectronic state. à designates the antisymmetrization operator. Severa approximations are impicit in writing the wave functions as in Eqs. 24 and 25. The spin parts of both wave functions are not expicity written because the spins are decouped from the dynamics when we assume Hund s case b couping. This choice is equivaent to assuming that the spins remain couped throughout the ionization process to the spin vaue of the ionizing state. 24 The wave functions given for both the ionizing state and the fina state are aso not parity adapted. Because we do not consider the interactions that remove the degeneracy between different parity states, the fina resuts of our derivation remain the same when we use parity adapted wave functions, provided that the incoherent sum over each parity component is performed at the appropriate stage. Finay, by writing the fina-state wave function as Born Oppenheimer products as in Eq. 25, we are ignoring the couping between nucear vibrationa motion and the eectron being ionized when it is near the ion core. When the photoeectron is inside the ion-core region, the vibrationa wave function for the composite state of the photoeectron and the ion can be sighty different from that of the isoated ion. 26 The transition eectric-dipoe-moment matrix eement that connects the ionizing state and the fina state in our formaism is given by J. Chem. Phys., Vo. 4, o. 2, 22 March 996 Downoaded 6 Jan 2 to Redistribution subject to AIP icense or copyright; see

9 4562 H. Park and R.. Zare: Photoeectron spectroscopy. I. Formaism p M ;k D pm p M ;k s 4 /2 3 r s ˆ pm p M ;k r s Y rˆ s s pm, 26 where r s is the eectronic coordinate expressed in the LAB frame, ˆ is the unit poarization vector of ight in the LAB frame, and s is an index that runs over a eectrons in the moecue. D stands for the eectric dipoe-ength operator, where for ineary poarized ight with its eectric vector pointing aong the z axis in the LAB frame, and for circuary poarized ight that propagates aong the z axis in the LAB frame. We can express the dipoe-ength operator that appears in Eq. 26 in terms of the spherica harmonics defined in the MF frame r s Y rˆ s s r s s D * Rˆ Y rˆ s. 27 Combining expressions in Eqs. 24 through 27, we obtain p M ;k D pm 4 / /2 2 /2 i e i Y m m kˆ d D M Rˆ D m Rˆ D * Rˆ DM * Rˆ dr R[ r;k,rp à s r s Y rˆsp R. 28 The first integra in Eq. 28 can be evauated readiy using standard anguar momentum agebra, 45 and the resut can be found in the iterature,27 d D M Rˆ D m Rˆ D * Rˆ D * M Rˆ t t M M t m M t t t. M t M t t t 29 Here, t designates the anguar momentum transferred between the photoeectron and the ion with M t and t denoting its projections aong the LAB frame and the MF-frame z axes, respectivey, t. 3 The ast integra in Eq. 28 represents the vibrationay averaged eectronic dipoe-moment matrix eement that connects the ionizing state to the continuum partia wave with anguar momentum quantum number and its projection on the internucear axis.,27,3 We designate it as r e i r e i dr R[ r;k,rp à s r s Y rˆsp R. 3 Under the independent eectron approximation, Eq. 3 is simpified to a one-eectron integra r e i dr R r;k,rry rˆr R. 32 In Eq. 32, r is the coordinate of the eectron being ionized, and r is a one-eectron orbita of that eectron in the ionizing state. By inserting Eq. 9 into Eq. 32, we can express r e i in terms of the continuum eigenchanne quantities r e i dr RU e i r;k,rr Y rˆr R. 33 As noted in the previous section, the continuum quantities U and are dependent on the internucear distance R and formay cannot be taken outside the integra. When the R dependence of continuum quantities U and is sma and smooth over the range of R where the overap between (R) and (R) is significant, Eq. 33 can be written as where r e i Ū e i M, 34 M dr R r;k,rry rˆr R. 35 J. Chem. Phys., Vo. 4, o. 2, 22 March 996 Downoaded 6 Jan 2 to Redistribution subject to AIP icense or copyright; see

10 H. Park and R.. Zare: Photoeectron spectroscopy. I. Formaism 4563 Here, M represents the vibrationay averaged eectronic dipoe-moment matrix eement that connects the ionizing eectronic orbita to the th continuum moecuar orbita for each. It is important to note that M is rea because the radia parts of eectronic wave functions are rea and aso because the vibrationa wave functions are chosen to be rea. In Eq. 34, Ū and are the representative vaues of U and, respectivey, over that range of R where the vibrationa overap is significant. The equaity in Eq. 34 is exact if U and e i are monotonicay varying functions in this range of R, in which case Ū and can be chosen by the mean-vaue theorem. 46 Otherwise, the equaity shoud be considered an approximation reminiscent of the Franck Condon approximation for bound bound transitions, and Ū and can be taken as the vibrationay averaged vaues of U and, respectivey. The approximation that the variations of U and with R are sma and smooth is physicay reasonabe when we consider the direct photoionization of a bound state with sma vibrationa quanta. This approximation is ikey to break down, however, when some dynamica phenomena, such as a shape resonance, occurs in moecuar photoionization. It can be seen from Eq. 34 that the matrix Ū, which is the vibrationay averaged version of the eectronic transformation matrix U, acts as a unitary matrix that causes the rotation between two compex dipoe engths, r r e i and M M e i. Equations 34 and 35 are the key resuts of this anaysis. These equations are important because they express the separation of the dipoe-moment matrix eements that govern the photoionization dynamics into two parts: the quantities that are the properties of the continuum wave functions ony, and the quantities that depend on both the ionizing state and the ionization continuum. In atomic and moecuar physics, extensive experimenta work has been performed to determine either continuum-reated quantities such as quantum defects determined from Rydberg spectroscopy or quantities that depend both on the ionizing state and the continuum such as eectric dipoe-moment matrix eements determined from photoeectron spectroscopy. For atomic photoionization, the reationship between these two types of quantities has been used extensivey to interpret various experimenta findings in a unified fashion. 5,6,8 22 It has aso been shown that both the dipoe-moment matrix eements and the scattering matrix eements can be extracted by combining various experimenta data. 7,34,47,48 In moecuar photoionization, however, systematic attempts to reate experimenta findings in photoionization with coisiona parameters are rare despite the cose reationship between eectron ion coision and photoionization, which has ong been reaized in the MQDT framework. This rarity stems from experimenta and theoretica compexities inherent to the moecuar probem. For instance, whereas the partia waves are the coision eigenchannes in atomic photoionization in the centra-fied approximation, this situation is not the case in moecues as discussed in the previous section. Equation 34 permits us to extract detaied dynamica information pertaining to the ionization continuum from moecuar photoionization experiments. By inserting Eqs. 29 and 34 into Eq. 28, we obtain an expression for the transition eectric-dipoe matrix eement p M ;k D pm 4 /2 3 2 /2 2 /2 m i e i Y m kˆ Cm t M, where we define Ū e i M t Cm t M M 2 t M t m M t t t. M 36 t M t t t 37 The 3- j symbos given in Eq. 37 determine the seection rues for the bound free transitions when both the intermediate and the fina states are described by Hund s case b couping. Because photoionization seection rues for various Hund s couping cases have aready been discussed in the iterature, 49,5 we do not discuss them expicity here. ote that in Eq. 36, the summations over and can be truncated at some finite vaue see Appendix A. The summations over m and, which are quantum numbers reated to, are aso truncated accordingy. The corresponding summations in the partia-wave expansion of the continuum wave function are not restricted in Eq. 2. Because of the ight mass of the eectron, the centrifuga barrier associated with the eectronic motion near the ion core is very arge, and the penetration of the photoeectron into the ion-core region is negigiby sma when is arge. On the other hand, the eectron density of the bound eectronic orbita is concentrated in the ion-core region. In addition, the singe-center expansion of tighty bound moecuar orbitas usuay shows rapid convergence with. Consequenty, the integras in Eq. 32 become negigibe when is very arge. Because the (r;k,r) and the r;k,r are reated to each other by the unitary transformation U, is aso restricted when is restricted. The intensity of photoeectrons, or the rate of ejection of photoeectrons, associated with the ionization event p p and detected in the LAB-frame soid ange eement dsin d d is given by 27,3 J. Chem. Phys., Vo. 4, o. 2, 22 March 996 Downoaded 6 Jan 2 to Redistribution subject to AIP icense or copyright; see

11 4564 H. Park and R.. Zare: Photoeectron spectroscopy. I. Formaism I,2 fs F M D pm M M p M ;k pm D p M ;k M M. 38 Here, fs is the fine structure constant, F represents the photon fux of ight with frequency, and M M designates eements of the density matrix for the ionizing state 5 that specifies not ony the popuations of the different M subeves of a given eve in the ionizing state but aso the coherences between these eves. Because we consider the photoionization from a specific quantum eve in the ionizing state, the summation over is omitted in Eq. 38. For an isotropic ensembe, M M is simpy given by M M 2 M M. 39 When n REMPI is empoyed to study the photoionization from a specific quantum eve in the intermediate state, the specific form of M M for the intermediate state depends on the experimenta geometry. Various forms of M M can be found in the iterature for different methods by which the ionizing state is reached.,27,3 Inserting Eq. 36 into Eq. 38 we obtain where I, 82 fs 3 F m Y m kˆy * kˆ m M e i mm m Ū, Ū mm 22 i M M M t t Cm t M Cm t M M M M 4. 4 In Eq. 4, I, can be recast in terms of the spherica harmonics using the Cebsch Gordan series 27,45 where I, Leven M LM Y LM,, 42 F m m m 222L 4 L mm LM 82 fs 3 M e i. /2 m Ū m Ū L M M 43 In Eq. 43, LM depends not ony on the magnitudes of eectronic dipoe-moment matrix eements, M, but aso on the scattering phase shifts, and. Whereas U and can be assumed to be independent of energy, as discussed in the previous section, the Couomb basis functions and have a nontrivia dependence on the asymptotic photoeectron energy. The vaue of LM is, however, much ess sensitive to for two reasons. In contrast to their behavior at the asymptotic region, f and g are quite insensitive to near the ion core. 28 Thus M, whose magnitude stems mosty from the ion-core region, is insensitive to in most cases. Aso, whereas itsef is strongy dependent on, does not vary rapidy with. Using the properties of the gamma function and Stiring s approximation, 28 it can be shown that 2 k 2 Ok2. 44 For most sma diatomic ions, k varies typicay ess than. a.u. over the energy span of a few rotationa eves within the same vibrationa manifod. Hence, M and remain approximatey the same for different ion rotationa eves within the same vibrationa manifod. The form of the PAD for a given quantum eve of the ion, which is described by LM, can aso be approximated to be independent of energy over an energy range on the order of mev. IV. DISCUSSIO We have derived in this artice an expression for the quantum-state-specific PADs from direct photoionization of a diatomic moecue based on the moecuar-orbita decomposition of the ionization continuum. The resuting expression for the quantum-state-specific PADs is dependent on two distinct types of dynamica quantities, one that pertains to the ionization continuum and the other that depends both on the ionizing state and the ionization continuum; Ū and can be cassified as the former, whereas M ceary beongs to the atter. Because Ū and describe ony the dynamics in the ionization continuum, they shoud be common in photoionization processes from different ionizing states when the asymptotic photoeectron energy and the rovibronic state of the ion are the same. Hence, the present formaism aows for the maxima expoitation of the com- J. Chem. Phys., Vo. 4, o. 2, 22 March 996 Downoaded 6 Jan 2 to Redistribution subject to AIP icense or copyright; see