Molecular Applications of Quantum Defect Theory

ADVANCES IN ATOMIC A N D MOLECULAR PHYSICS, VOL. 21

MOLECULAR APPLZCATIONS OF QUANTUM DEFECT THEORY CHRIS H . GREENE Departmeni if Physics and Astronomy Louisiana State Cniver.riry Baton Rouge. Louisiana

Ch. JUNGEN Lahoratoire de Photophysique Molkciilaire du CNRS L'nibtersitP de Paris-Sud Orsay, France 1. Introduction . . . . . . . . . . . . . . . . . . . . 11. Quantum Defect Concepts and Formalism . . . . . .

111.

. . . . . . . . . . . . . . A. Origin of the Rydberg Formula . . . . . . . . . . . . . . . . . B. Multichannel Rearrangement Processes . . . . . . . . . . . . . C. The Eigenchannel Representation. . . . . . . . . . . . . . . . D. Photofragmentation Cross Sections . . . . . . . . . . . . . . . E. Physical Significance of the Eigenchannels . . . . . . . . . . . . Rovibrational Channel Interactions. . . . . . . . . . . . , , . . . A. Adaptation of the Quantum Defect Formalism to Molecular

Problems. . . . . . . . . . . . . . . . . . . . . . . . . B. Channel Interactions Involving Highly Excited Bound Levels . C . Channel Interactions Involving Continua . . . . . . . . . . D. Treatment of a Class of Non-Born-Oppenheimer Phenomena. IV. Electronic Interactions at Short Range . . . . . . . . . . . . . A. Theoretical Developments . . . . . . . . . . . . . . . . . B. Electronic Preionization in Molecular Nitrogen. . . . . . . . C. Photodissociation and Dissociative Recombination . . . . . , V. Discussion and Conclusions . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .

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51 54 54 56

58

62 63 66 66 69 16 87 91 91 102 106 I I5 1 I8

I. Introduction Rydberg states of diatomic molecules, in which one electron usually roams far from the nuclei and other electrons, differ dramatically in most 51 Copynghl 0 1985 by Academic Press, Inc All nghls of reproduction in any form reserved

52

Chris H . Greene and Ch. Jungen

respects from valence states. The dominant simplifying element of the valence-state physics is the slow time scale of nuclear vibration and nuclear rotation compared with the time scale of electronic motion. As higher and higher Rydberg electronic states are excited a point is reached at which these time scales become comparable, and in fact the electronic motion eventually becomes the slowest. At this stage the interpretation of molecular spectra cannot be handled by a conventional analysis based on the BornOppenheimer approximation and on the coupling of electronic angular momenta to the internuclear axis. As realized by Mulliken ( 1964), an understanding of this portion of the spectrum would require a radically different method of theoretical analysis. The first element of this analysis was the development of quantum defect theory by Seaton ( 1966)to handle the corresponding multichannel Rydberg spectrum in atoms and in electron-ion scattering. A comprehensive review of this approach has been presented recently by Seaton (1 983). High-resolution spectroscopic studies of H2photoabsorption by Chupka and Berkowitz ( 1969), Herzberg ( 1969), and Herzberg and Jungen ( 1972) provided an experimental impetus which sparked an extension of Seaton’s formulation to molecular Rydberg spectra by Fano (1 970). Fano’s article emphasized two new concepts, namely, the physical significance of eigenchannels of the electron -ion interaction and the necessity for a rotational frame transformation between the eigenchannels at small electronic distances r and the fragmentation channels at large r. Systematic exploitation of these concepts, including their extension to vibrational channel interactions, has permitted a nearly complete accounting of the experimental H2photoabsorption spectrum (Dehmer and Chupka, 1976) up to photon energies of 133,000 cm-’ (Jungen and Raoult, 1981). Mulliken (1976) has referred to Rydberg states as the “stepping stones toward ionization.” He thereby stressed the basic unity which exists between high Rydberg states and the adjacent ionization continua: the short-range physics is common to both while they differ only at long range. Quantum defect theory provides a unified description of discrete and continuous spectra in terms of the same parameters pertaining to the physics at short range. Thereby it accounts for the Rydberg structures as well as for their decay into the continuum. This decay is called preionization (or synonymously autoionization), or predissociation if the continuum is associated with nuclear rather than with electronic motion. Figure 1 displays two molecular spectra which exhibit striking effects due to both preionization and predissociation. Both spectra result from photoexcitation of nitric oxide and they cover the same wavelength region. The upper spectrum is a recording of the ion current resulting after excitation just above the ionization threshold: The small peaks superimposed on the flat continuum are Rydberg structures whose

+

, I

+

4

I

v*= 1

77,000

7 ~ 0 ENERGY / cm”

75,000

I

a

I 1.P vf 0

74,000

FIG.1 Medium-resolution photoionization(a) and photoabsorptionspectrum(b) of NO nearthe first ionization threshold.The spectra are taken with comparablespectral resolution (= 8 and 13an-’,respectively),but they correspondto Merent temperatures(20 and 78 K, respectively). The assignmentsof nplr, u Rydberg bands are those of Miescher and Alberti (1976) and are based on the high-resolutionabsorption spectrum. Sections marked *-* are reproduced in Fig. 21. [(a) From Ono et ul., 1980; (b) From Miescher et al., 1978.1

54

Chris H. Greene and Ch. Jungen

presence reveals that preionization occurs. The lower trace is an absorption spectrum taken under comparable experimental resolution. The strikingly different appearance of the two spectra reveals that a second decay channel, corresponding to production of neutral atomic fragments, must also interfere. Moreover, the different appearance of the Rydberg peaks in the two spectra indicates that predissociation is strong and competes with preionization. The aim of the present article is, first of all, to review the current state of molecular quantum defect theory. Second, emphasis will be placed on describing its applications to typical problems of molecular physics such as the interpretation of the spectra of Fig. 1. Although we do not explicitly discuss electron scattering by neutral molecules in this review, theoretical methods closely related to quantum defect theory have been applied to this class of problems as well. A thorough and fairly recent review of the current status of electron - molecule scattering theory has been presented by Lane ( 1980).

11. Quantum Defect Concepts and Formalism A. ORIGINOF THE RYDBERG FORMULA

The various approaches called “quantum defect theory” are essentially designed to take advantage of one simple fact: When one molecular electron moves sufficiently far from the remaining electrons and nuclei (say at r > r,), it experiences a purely Coulombic attraction. At any energy two independent solutions exist to the radial Coulomb Schrodinger equation, one of which is well behaved and regular at r = 0 while the other diverges and is accordingly irregular at r = 0. The radial wave function of the outermost electron must then reduce to a linear superposition of regular and irregular Coulomb wave functions (f;g),

r > r, y ( r ) = f ( r ) cos I I ~- g(r) sin np, (1) The constant p is called the quantum defect and is the usual scattering phase shift 6divided by II.In scattering theory one is usually interested in the phase shift at positive energies (E > 0) only, but in quantum defect theory the expression [Eq. ( 1 )] is meant to apply equally at E < 0, wherep then provides information about the bound-state spectrum. The underlying physical element which makes the quantum defect approach useful is the near energy independence of any property of the system (such as the quantum defect p ) determined at small radial distances. This

QUANTUM DEFECT THEORY

55

follows immediately from the strong attraction prevalent at small distances; that is, the local kinetic energy at r 4 ro is large even at and below the ionization threshold because of the deep Coulombic well. At large distances instead the wave function changes dramatically from an oscillatory character just above E = 0 to an exponential decay just below E = 0. Fortunately this energy dependence stemming from large radii is given in closed analytical form in terms of well-known, tabulated properties of Coulomb wave functions (Seaton, 1966). For example, the asymptotic form of the energynormalized regular and irregular solutions is

+ (zrn/k)In r + q] - ( 2 r n / ~ k ) ' /cos[kr ~ + (zrn/k)In r + q] ( 2 r n / ~ k ) sin[kr '/~

f(~, r) g(E,

r)

at positive energies E

f(~, r)

= k2/2rn

> 0, and

(rn/nK)1/2[sin/3 (D-lrrrnlKeKr ) - cos p (DrzmlKe-w 11

- (rn/n~)'/*[~os p (D-lr-zm/KeKr ) + sin p (Dr*mlKe-Kr)] at negative energies E = - ~ ~ / 2
r)

(3)

tron mass for the system of interest to the electron rest mass. In these expressions z is the charge of the ion, while the phase parameters q and /3 are constants with respect to r : V ( E ) = (zrn/k)In P(E) = K(U

+

+

2k - 1 ~ / 2 arg r(l 1 - izrn/k)

- 1)

(4)

in terms of the "effective quantum number" u = Z ~ / and K the orbital angular momentum quantum number 1. The constant D is given by Seaton (1 966) or by Greene et al. ( 1983), but it will not be needed here. These asymptotic forms of (fl g) will permit a straightforward application of the large-r boundary conditions relevant to scattering or photoionization processes. [In some applications, especially far below threshold, an alternative analytic pair of solutions (fo,go)should be used in place of (f;g).] The bound-level positions follow immediately from requiring the outer field solution [Eq. ( l)] to be finite at r + m. With the large-r forms (3) for E < 0 the constraint obtained for integer I is

+

sin n(u p ) = 0

(5)

equivalent to the empirical formula of Rydberg:

En = - rnz2/2(n- p ) 2

(6) Here the quantum defect p is nearly independent of energy, whereby Eq. (6) compactly parametrizes an infinite number of states.

56

Chris H . Greene and Ch. Jungen

In multichannel situations, of course, both Eqs. ( 5 ) and (6) must be generalized, as discussed below. Likewise Eqs. ( 1)-(3) remain valid for non-Coulombic long-range fields as well if the parameters q, /Iand , D appropriate to that long-range field are used in place of Eq. (4). Applications of this flexibility of quantum defect theory have expanded in recent years (Greene et al., 1979,1983; Watanabe and Greene, 1980), as evidenced, for example, by the treatments of molecular dissociation described in Sections II1,D and IV,A below (Giusti 1980; Colle, 1981; Nakamura, 1983; Mies, 1984; Mies and Julienne, 1984). B. MULTICHANNEL REARRANGEMENT PROCESSES

as

The quantum-mechanical amplitude for a rearrangement collision such

e-

+ H,+(i’)

+

e-

+ H2+(i)

(7)

is given by one element of a scattering matrix Sii,. It is defined mathematically through the asymptotic (outgoing wave) form of the stationary-state wave function having an incoming wave component in channel i f only and outgoing scattered waves in all channels i, The coordinate r in Eq. (8) represents the separation between the outermost electron and the H2+ center of mass, while o denotes all other independent coordinates of the system, including spin and angular coordinates. The factor @Lo) thus represents the complete wave function of H2+in the state i together with the spin and orbital angular momentum wave functions of the outermost electron with appropriate angular momentum coupling factors. The A operator indicates the Fermi antisymmetrization needed, but it satisfies only a formal requirement here since exchange is negligible asymptotically. Lastly f f ( r )represent outgoing/incoming wave solutions of the outer field (electron - ion) radial Schrodinger equation for channel i. Asymptotically these are essentially exp(+ ikir) aside from an additional logarithmic phase factor associated with the Coulomb tail. The normalization v ~ involves the relative velocity of the receding fragments; it ensures that lSii@)12 is the ratio of the outgoing radial probability flux in channel i to the incoming flux in channel i f . Alternative channels i can have different internal energies Ei of the fragments and possibly different reduced masses mi,so that the wave vector ki in f f ( r )at a total energy E is given in atomic units (Bethe and Salpeter, 1957) by

k: = 2rni(E - Ei)

(9)

~

/

~

57

QUANTUM DEFECT THEORY

The dimension of the square scattering matrix is No, the number of open channels [for which kf > 0 in Eq. (9)]; this is also the number of linearly independent solutions Yi. of the Schrodinger equation at the total energy E of interest obeying regularity boundary conditions at the origin. But while these No solutions and especially their asymptotic forms completely characterize the scattering information, they depend strongly on the energy for two reasons: 1. The scattering matrix has pole-type structures in the vicinity of autoionizing and predissociating resonances. 2. Even away from resonances the eigenphase shifts of S ( E ) are highly energy dependent, particularly close to fragmentation thresholds, because of the phase shift contributed by the long-range field [see Eq. (4)].

The main result of Seaton’s quantum defect theory (1966, 1983) is the parametrization of the complicated, energy-dependent S matrix in terms of a smooth, short-range reaction matrix K and standard parametersB(E) and ME) characteristic of the long-range potential. In many applications K can even be taken independent oftheenergy,and for this reason it lends itselfto a direct physical interpretation far more readily than does the scattering matrix. The reaction-matrix representation of the N independent solutions for an N-channel problem utilizes (A,gi),two independent solutions to the outerfield Schrodinger equation in channel i. Specially, the independent solutions outside a “reaction surface” at r = ro must be the generalization of Eq. (1):

Ti@)= A

N i- 1

Qi(w)[f;(r) Sii, - gi(r)K,,.],

r > ro

(10)

Note that the independent solutions of Eq. (10) are quite different from the S-matrix representation, Eq. (8). For one thing they are real, with Kii.symmetric. For another, there are more independent solutions (a total ofN) than the number of open channels (No).This reflects a key (and initially surprising) feature of the reaction-matrix representation: The boundary conditions at r + m have not yet been applied, whereby the N, = N - Noclosed channel components ( i = No 1, . . . , N) are exponentially divergent. The N X Nsmooth reaction matrix K contains the scattering information in slightly disguised form. It characterizes the large-r form of Nindependent solutions having an unacceptable divergence. In constructing physically acceptable solutions we utilize the fact that the form of this divergence is specified precisely by Eqs. (3) and (10). This allows us to find a new set of solutions Y j ( E ) ,withj = 1, . . . ,No, which are linear combinations of the Y,(E) in Eq. (10) that remain well behaved at r - m. This procedure is usually referred to as the “elimination” of the closed channels from the wave

+

58

Chris H . Greene and Ch. Jungen

function (see footnote 1 for details). The result is an No X No open-channel reaction matrix

K ( E ) = Kw - K"[K"

+ tan B(E)]-lKCo

( 1 1)

in terms of the parts of the original smooth reaction matrix referring to open and closed channels at a given energy E,

The B(E) in Eq. ( 1 1) is a diagonal N, X N, matrix whose elements are the negative energy phase parameters [Eq. (4)] in each closed channel, which diverge at each ionization threshold. This simple result gives in algebraic form all of the complicated energy dependences which are associated with closed-channel resonances. From the point of view of ab initio calculations, this is of considerable practical importance since the calculation can be confined to within a small volume of configuration space and to a coarse energy mesh, typically E 2 1 eV, yet it is capable of representing sharp resonance features in a scattering (or photoabsorption) experiment on an energy scale of cm-' units or less. It is also of practical importance for semiempirical attempts to account for experimental observations, which can thus be accomplished by fitting to (nearly) energy-independent quantities such as K. It may be useful to give the correspondence between our notations and the matrices introduced by Seaton (1 983). Thus our No X No reaction matrix K ( E ) in Eq. (1 1) is denoted R by Seaton, who called it the reactance matrix. The short-range N X N reaction matrix K is denoted 3" by Seaton. Our N X N frame transformation matrix U is denoted X by Seaton, and for our "short-range scattering matrix" (see footnote 2, p. 101), Seaton uses the symbol x. Lastly, Seaton's No X No matrix T coincides with our open-channel eigenvector matrix given in Eq. (20).

C. THEEIGENCHANNEL REPRESENTATION Fano ( 1970, 1975) has singled out the eigenvalues and eigenvectors of the reaction matrix as having special physical significance. These, denoted tan qua and U,, respectively, are found by diagonalizing K:

K = U tan(np)UT (13) where a superscript T has been used to denote the transpose of U. The eigenchannel wave functions yl,(E),a! = 1, . . . ,N, have a common phase shift npa, often called an "eigenquantum defect," in each of the fragmenta-

59

QUANTUM DEFECT THEORY

tion channels i. [In this article Greek indices (a,. . .) always refer to the eigenchannels, while italic indices (i, j , . . .) refer to the fragmentation channels.] The independent eigenchannel solutions have the following form outside the reaction volume:

In matrix notation these eigenchannels are related to the solutions ‘Pi@) in the reaction-matrix representation by

W=wCOS R/l

(15)

Here w and Y are square matrices whose rows represent the different fragmentation channel components i, and whose columns represent separate independent solutions of the Schrodinger equation at r > ro. The elimination of divergent closed-channel components is quite analogous to the procedure leading to Eq. ( 1 1) above. Any allowed solution must be a linear combination of the eigenchannels,

with the coefficientsA,(E) to be determined by large-r boundary conditions. The superposition [Eq. ( I6)] must decay exponentially in every closed channel (i E Q), while each open-channel component ( i E P)will be required to have a common eigenphase shift RZ. These two conditions when combined with Eqs. (3) and (4) imply the linear equations [see Lu (1971)l N

where

{U,

uia W P i

Fia(E) =

+~ a ) ,

sin n(- z +pa),

+

iEQ iEP

sin picia cos PisM, - sin w7Cia cos nr§ia,

+

iEQ

(18)

iEP The second set of equations, for example, with 43, = Viacos npa, will be

used below. This homogeneous system of equations has a nontrivial solution only if det(Fia) = 0, which can be satisfied by No values of the “collision eigenphase shift” n.s,,(E) and No solution vectors A,@) of Eq. (17), with p = 1, . . . , N o . The resulting “collision eigenchannel” solutions have the following form in the fragmentation zone ( r > ro),

Chris H. Greene and Ch. Jungen

60

where the orthogonal matrix T(E) is given in terms of the normalized A, by N

Tip(E)=

a- 1

UIa cos n(-

Tp

+ PaMw

and their normalization is now specified according to

The matrix T is unitary as can be seen as follows. By forming the complex superpositions of the collision eigenchannel solutions y/,(E),

we find after some manipulation a wave function with the asymptotic form of Eq. (8), with the elements of the scattering matrix given by

From this expression we see that the matrix T is in fact the matrix of eigenvectors of the scattering matrix and therefore is unitary, and in fact if a real normalization is adopted, it is a real orthogonal matrix (i-e., T-l = TT).' In applications to date using the eigenchannel formulation of quantum defect theory, the linear system of Eqs. ( 17) and (1 8) is solved by iteration. That is, a search is executed for values of T at which det{Fia) = 0. This The orthogonality of T can also be verified directly from the expressions of Eq. (20). We present the considerations here for the reader interested in details because they permit at the same time a verification ofthe form ofthe open-channel reaction matrix K(E)given in Eq. ( 1 1). First, we reexpand the channel coefficientsA,, in terms of new coefficients

Z, =

I: U,,,cos np,,A,, N

a- I

The linear system [Eqs. (17), (18)] then takes the form (tan

[!AT}

rani L

a,,.

+ Kii,)Z,, = 0,

for each i E Q, P

where the reaction matrix K is related to U andp by Eq. (13). Second, we partition the system into two parts referring to closed and open channels, respectively,

Km

tan(- ;T:

+ Km] [E:]

The "closed" portion of the linear system is now used to express the N, coefficientsZ c in terms of the No coefficients Z o according to Zc = -(tan

B + Km)-lKmZo

QUANTUM DEFECT THEORY

61

iterative calculation is needed because the “eigenvalue” z enters F, nonlinearly through a trigonometric function. One drawback of this iterative calculation is that the linear algebra cannot be solved “automatically,” using standard computer programs. It is possible neverthelessto put this system of equations into the standard form of ageneralized eigenvalue problem for tan atpwhich is convenient for numerical applications; this has not been used in the previous work, e.g., of Jungen and Raoult (198 1):

TA = tan at AA

(23)

with

and

A,

=

{O’U ,

iEQ iEP

cos ap,, This eigenproblem (Wilkinson, 1965) has No nontrivial solutions also since the rank of the A matrix is No. Standard routines are available for its efficient solution. Equations (23)-(25) apply also when all channels are closed, except that the homogeneous system, Eq. (23), then possesses a nontrivial solution only at certain discrete energy levelsEn.These must be determined in general by a numerical search. The concept of “energy-normalized” wave functions is no longer appropriate in this purely discrete regime, where the wave functions should be normalized to unity. In terms of an unnormalized solution vector A, of Eq. (23), the normalization integral is (Lee and Lu, 1973; Greene, 1980): N: = a-’

c [A, uj, i,a

COS(P,

+ ap,)]

(26) d X - AanU,, sin(P, ap,,) IdE 1E-E“ The main energy dependence contributing to the derivative in Eq. (26)

+

(I’

Insertion of this expression into the “open” portion ofthe system leads immediately to the form of Eq. ( 1 I ) for the No X No open-channel reaction matrix K(E),which is symmetric. Finally, we verify from the linear system, Eqs. (17), (18), using the expression, Eq. (20), for T and the definition of the coefficients Zi, that

Z & = Ti, cos mp The orthogonality of T now follows immediately from the orthogonality of Zo.

62

Chris H . Greene and Ch. Jungen

comes from the long-range phase parameter pi defined by Eq. (4)for a long-range attractive Coulomb field. To a good approximation then, particularly for a high Rydberg level, this result simplifies to

-

D. PHOTOFRAGMENTATION CROSSSECTIONS Application of the finitenessboundary condition at r 03 has reduced the number of physically relevant stationary state solutions wPat any energy E to No, the number of open channels. Various photoabsorption experiments performed using photons at a definite energy access specific linear combinations of these independent solutions. Scattering experiments can be used to study different linear combinations. Here we summarize the formulas which describe photoabsorption from a single bound state having total angular momentum Jo. In the electric dipole approximation the information required to calculate a photoabsorption cross section consists of matrix elements of the molecular dipole operator i * r , with E^ the photon polarization vector. We assume the incident light to be linearly polarized. Other possibilities are easily treated by the same methods. The initial and final stationary states, yo and w, will be assumed to have definite angular momenta Jo and J, respectively, which is permissible since J commutes with the field-free molecular Hamiltonian. Consequently the dynamics is isolated in Nreduced dipole matrix elements = (aJII r(l)llJo),using notation and conventions ofSobel’man for each J, D‘,“) (1972). Because yois typically limited to a small spatial region where ty, is a slowly varying function ofenergy, these matrix elements are likewise smooth in E. (Photoabsorption by an initial Rydberg state as occurs, for example, in a multiphoton experiment, represents a notable exception.) A smaller subset of No matrix elements D$J)is obtained using the eigenvector A, from Eq. (23h N

DLJ)(E)=

I]DLJ)Aw(E) a- I

(28)

Of course these “collision eigenchannel” matrix elements can vary rapidly with energy, since they incorporate resonance effects. The incoming wave boundary condition, described thoroughly by Starace ( 1982), must be satisfied when photofragments are observed in one channel i. In particular the real wP must be superposed to eliminate all outgoing

QUANTUM DEFECT THEORY

63

spherical wave components in channels i’ # i. This is accomplished by the complex superposition

which is thus connected to the initial state by a reduced dipole matrix element

The partial cross section (in a.u.) for photofragmentation into channel i is then an incoherent summation over all final-state angular momenta:

with a the fine-structure constant and o the photon energy in a.u. The total cross section is a sum over all of the No a, at each energy, and reduces to the simpler form involving the real DbJ),

Note that these expressions are applicable to either photoionization or photodissociation, or to both competing processes at once. In addition the dipole matrix elements of Eq. (30) serve as input into theoretical formulations of other observable properties such as the photofragment angular distributions (Dill and Fano, 1972; Fano and Dill, 1972), fragment spin polarization (Lee, 1974b), or the alignment and orientation of photofragments (Klar, 1979, 1980; Greene and Zare, 1982). The calculation of various observables in the photoionization of rare gas atoms has been reviewed by Johnson et al. ( 1980). E. PHYSICAL SIGNIFICANCE OF THE EIGENCHANNELS Strictly speaking, the reaction matrix representation, Eq. (lo), and the eigenchannel representation, Eq. ( 14), of independent solutions are equivalent, and related linearly by Eq. (1 5 ) . Yet the eigenchannels often have a simple physical significance,whose systematic exploitation can be extremely helpful. In particular, if the short-range Hamiltonian is approximately diagonal in some simple, standard representation, then the short-range reaction matrix ought to likewise be nearly diagonal in that representation. This follows from the (heuristic) quantum defect view of a scattering process

64

Chris H . Greene and Ch. Jungen

which is depicted schematically in Fig. 2. Consider a specific transition which results after an electron collides with the ionic core, causing a transition from a rovibrational state i of H2+ to a final state i’. Because r, was chosen such that this transition cannot take place at radii r > r,, it occurs as follows: First, an electron comes in toward the H2+ ion along the long-range potential Vi(r) which convergesat r 00 to a corresponding level Ei of H2+.It remains on this potential until it reaches r = r,, the edge ofthe reaction zone. Within the reaction zone the motion is complicated and not described by any local potential, but the net result of this collision with the core at r < r, is that the electron can be scattered and re-emerge from the core in a diflerenf long-range potential Vit(r).It can then move outward to the detectors at infinity along this potential. Ifwithin the core there is no interaction between wave functions with the channel structures i and i’ (i.e., if the Hamiltonian matrix element Hiit= 0), then clearly there should be no scattering from i to i’ within the core (i.e., Kii.= 0). Thus diagonality of the short-range Hamiltonian in a standard representation implies that K should also be diagonal in that representation. This simple concept can be developed considerably further. Thus, since the scattering eigenchannels (a)are identical to the eigenstates of the shortrange Hamiltonian, the same orthogonal transformation Uiareduces both K and H to diagonal form. Put another way, the matrix element Viashould be

0 FIG.2. Potential and kinetic energies of an electron outside an ion core. The total energy E corresponds to a situation where the electron is bound with respect to the three ionization limits E , , E2, and E,.

QUANTUM DEFECT THEORY

65

interpreted as the projection of a fragmentation-channel wave function Ii) onto an eigenstate la) of the collision (or of the short-range Hamiltonian). For example, in electron scattering by a core having a spin-orbit splitting, the fragmentation channels i must be characterized in j j coupling, since the asymptotic electron wave vector kiis different for the different core statesj,. On the other hand, the predominance of the exchange interaction in the short-range dynamics implies that the eigenchannels (a)are LS-coupled solutions. A factor of the frame transformation matrix U, should thus be given by the standard recoupling coefficient (jjlLS).Examples of this “finestructure frame transformation” have been most extensive in the contexts of atomic photoionization and negative ion photodetachment, as in the studies by Lee and Lu ( 1973), Lee ( 1979, Johnson et al. (1 980), and Rau and Fano (1971). Perhaps even more striking applications of the frame transformation viewpoint are the treatments of vibrational and rotational interactions in diatomic molecules (Fano, 1970; Atabek et al., 1974; Jungen and Atabek, 1977; Dill and Jungen, 1980; Jungen and Dill, 1980). Here the fragmentation channel labels (i) are just the ionic rotational and vibrational quantum numbers (u+, N + ) , since together they identify the energy of a distant electron. The crucial, nontrivial idea which permits a surprisinglysimple formulation of complex rovibrational interactions among Rydberg levels has been are (at r < ro)precisely the realization that the short-range eigenchannels (a) the wave functions obtained in the Born - Oppenheimer approximation. This is ensured by the small electron -nucleus mass ratio (Born and Oppenheimer, 1927) or by greatly different time scales in other contexts (Chase, 1956). Moreover, the short-range electronic phase shift p(R) is generally a slowly varying function of the energy, whereby the electronic time delay is negligible on the vibrational time scale. In this event the electron experiences the instantaneous field of the nuclei, which can then be regarded as frozen in this while the electron moves within the core. The eigenchannels labels (a} case are then (R,A), A representing the projection of the electronic angular momentum onto the internuclear axis I?, with R representing the internuclear separation. This deceptively simple statement should not be confused with the conventional Born - Oppenheimer approximation, which assumes that the nuclei are frozen in space for all electronic radii r, and which clearly fails to describe molecular Rydberg states properly. The eigenquantum defects in this case are Mulliken’s p,,(R) (1969), and they can be calculated from the (known) Rydberg-state Born - Oppenheimer potential curves U,(R): The reaction matrix K (e.g., a 20 X 20 matrix for the H, npA, J # 0, states if 10 vibrational channels are included) would be essentially impossible to obtain without this simplification. Besides this “practical” significance ofthe eigenchannel formulation, it also shows that the eigenchannel parame-

66

Chris H . Greene and Ch. Jungen

tersFa and U , and the state vectors have a clear physical significance- in this case coinciding with the fixed-nuclei parameters and wave functions.

111. Rovibrational Channel Interactions A. ADAPTATIONOF THE QUANTUM DEFECTFORMALISM TO MOLECULAR PROBLEMS

The photoabsorption spectra of diatomic molecules in the vicinity of numerous rovibrational thresholds can be quite complicated. In a zerothorder picture of the spectrum, one might imagine a simple Rydberg series of levels converging to each ionic threshold from below, with a smooth adjoining continuum above each threshold. While simple, this picture is not even qualitatively correct. The actual spectra are dominated by severe perturbations of level positions and intensities, and the photoionization continua display broad and narrow autoionization profiles interwoven in a complex fashion. We discuss in this section the adaptation of quantum defect theory which has accounted for these complex spectral features to a remarkable extent in H, and Na,. As indicated above in Section II,D, the fragmentation channel labels for this problem are ( i ) = (u+, N + ) ,and the complementary eigenchannel labels are (a)= (R, A). Each element of the frame transformation matrix Viaof Section I1 consequently factors into a product of two projections

uia uu+~+, R A = (V+lR)(N+)(N+IA)('J)

(33) The factor (v+IR)(") is the vibrational wave function corresponding to a of the molecular ion in its ground electronic state. (Interspecific level Eu+N+ actions among different electronic channels are discussed in Section IV below.) The rotational factor of the frame transformation in Eq. (33), denoted (N+lA)("),transforms a state from the Hund's coupling case relevant within the reaction zone at small r (typically case b) to the Hund's coupling case relevant at large r (case d). (Different Hund's coupling cases are described by Herzberg, 1950.)If the ground ionic state is a Z+ state, as is true for H2+, this factor is given by (N+IA)(lJ) = (-

[2/( 1

dAo)]"2(N+oll- A, JA)

(34) This result and its generalization to non-X+ core states are derived by Chang and Fano (1972). In Eqs. (33) and (34) lis the orbital angular momentum of the outermost electron. As it stands, Eq. (33) presumes that spin effects are negligible and that only one l has appreciable amplitude in the fragmentation 1)J+A-N'

67

QUANTUM DEFECT THEORY

zone. This is an excellent approximation in the case of the npa and npn Rydberg levels of H,, but for other symmetries and for many other molecules an expansion of the outer wave function in I is often needed. (Even in H,, for example, s-d interactions may be significant.) When this more complicated situation occurs, the full-frame transformation matrix, Eq. (33), must of course be modified to reflect the fact that the orbital momentum of the outer electron can be changed when it collides with the core. Moreover, this full-frame transformation matrix is no longer a purely geometrical quantity, but contains dynamical information which may require a separate calculation. Likewise ionization thresholds may exhibit splittings due to spin -orbit coupling, as in the rare gases, or to spin -rotation coupling (e.g., in a Z ionic core). This results in additional channel interactions not considered in this review. The eigenquantum defectspa are related to the usual Born-Oppenheimer potential energy curves of the neutral molecular Rydberg states, U,,(R), and the ground-state ionic Born - Oppenheimer potential U+(R)by U,,(R) = U+(R)- (2[n - P*(N12)-1 (35) All energies in Eq. (35) are given in atomic units. Some nA potential curves of H2are given in Figure 3a to show the range of energies and internuclear radii of interest. Despite the quite different shapes of these potential curves, they are accurately described by Eq. (35) withp,,(R) independent ofn. As Fig. 3b shows, a residual n dependence is in fact present, particularly at IargeR for the lowest Z state, but for our present discussion we will neglect this bodyframe energy dependence. Through Eqs. (33) and (35) the short-range quantum defect parameters are thus known, whereby the treatment of Sections II,C and II,D can now be ' -1 H'+ H(ls) e

n.2.

R (a

u)

-05 0

I

I

2

I

I

4

1

4

6

I

R h u )

FIG.3. Potential energy (a) and quantum defect (b) curves for the lowest ungerade singlet Rydberg states of molecular hydrogen.

68

Chris H . Greene and Ch. Jungen

used to find the energy levels and the photoabsorption spectrum of the molecule. The actual implementation of this procedure involves an additional complication not anticipated in Section 11, associated with the fact that the eigenchannel index (a)= (R, A} is partly continuous. Consequently the system, Eq. ( I7), is apparently underdetermined, as it involves an infinite number of unknowns A,, but only a finite number (N) of equations. This might be resolved in a variety of ways, but the most straightforward approach is to represent the A R A themselves as a finite superposition of rovibrational channel functions with unknown Bi = In practice any orthonormal vibrational basis (RI v) can be used in the superposition [Eq. (36)] to express the problem in terms of unknown coefficients BUN+, By a suitable choice of the vibrational basis it is possible to bring the system Eq. (17) [or Eq. (37) below] into nearly diagonal form locally, in a given energy range. This circumstance has been exploited in the applications to low Rydberg states which are described in Section III,D,l below (see Jungen and Atabek, 1977). With any such choice this modifies the system [Eq. (17)] to the form

2 Fii@)B,(E) N

=0

(37)

i'- I

with Fij.(E)=

+

sin PiCiit cos pi§ii., -sin n7Cii, cos nrSi,,,

+

i€Q

i EP

(38)

Equation (28) is also modified and takes the form

with

(39)

With Eq. (36), the matrix elements C,, of Eq. (38) take the form

and an analogous expression applies for §,i, with sin np. The reduced dipole matrix elements DLJ) used in Eq. (39) have been

69

QUANTUM DEFECT THEORY

further expressed in terms of purely body-frame matrix elements dA(R)in Eqs. ( 1 7)-( 19) of Dill (1972). [See also Eqs. (18)-(23) of Jungen and Dill ( 1980).] The resulting expression is

DL’)

= (2J

+ 1)1’2d~(R)(~,IR)(’o’(AlJO)(L’)

(41) where J, and (u,lR) are the total angular momentum and the vibrational wave function of the initial state, respectively. The superscript 1 in the matrix element (AIJo)(lfirepresents the multipolarity of the incident electric dipole photon and not the orbital angular momentum of the escaping photoelectron. B. CHANNEL INTERACTIONS INVOLVING HIGHLYEXCITED BOUNDLEVELS 1. Rotational Perturbations in H2

The observation by Herzberg, in 1969, of high Rydberg levels of H, assohas ciated with series converging to alternative rotational levels of Hz+, historically been the first documented example of rovibrational channel interactions (Herzberg, 1969; Herzberg and Jungen, 1972). Figure 4 displays E (crn-I) 124,300

I ‘

22

1124,200

21 16

20 unidentified 1.0

0.5

PO

0

np0,v.o

np2,v.O

FIG.4. Rotational perturbations between highly excited Rydberg levels in H,. The spectrogram is adapted from Herzberg and Jungen (1972) and representsabsorption to J = 1, odd-parity final-state levels. To its right are shown the (strong) np series converging to the v + = 0, N + = 0 level of H,+ and the perturbing(weak) seriesconverging to the v+ = 0, N+ = 2 level. The stick spectrum to the left of the spectrogram has been calculated using quantum defect theory. Far left: observed (circles) and calculated (full lines) quantum defects of the individual levels evaluated with respect to the v+ = 0, N + = 0 limit.

70

Chris H . Greene and Ch. Jungen

a section of the spectrum obtained by Henberg. By working with cold para-H, ,and owing to the fact that the H, ground-state rotational levels have a comparatively wide spacing, he obtained a gas sample in which virtually all molecules were in the (Jo=) J” = 0 (even-parity) level. Dipole absorption thus yields the excited system of J = 1 states with odd parity. The outer electron in H, has virtually pure 1 = 1 character and therefore only the N + = 0 and N + = 2 rotational levels of H,+ can be formed in photoionization under these circumstances. Correspondingly, the Rydberg series seen in Fig. 4 are associated with the u+ = 0, N + = 0 and 2 thresholds. It is however evident from Fig. 4 that strong perturbations occur which affect both level positions and intensities in the two series. These perturbations stand out in the level pattern drawn at the right of the figure where an attempt is made to associate each observed line with either the N + = 0 or the N + = 2 threshold. To the left of the experimental spectrum in Fig. 4 is shown a theoretical stick spectrum obtained by solving the linear system [Eq. (37)] with the matrix elements [Eq. (40)] and the analytic frame transformation [Eq. (33)]. The transition matrix elements have been evaluated using the united-atom approximation, by setting d,(R) = dn(R)= d. The quantum defect curves were initially extracted from the potential curves shown in Fig. 3a by use of the Rydberg Eq. (35) with n = 3 and 4, but a slight adjustment oftheir values (= 5%) was subsequently found necessary and reveals a slight energy dependence of the p’s. The good agreement between observed and calculated spectra underlines the correctness of the concepts outlined above in Sections II,E and III,A. Although the spectral range shown in Fig. 4 exhibits only lines associated with the interacting levels of the u+ = 0, N+ = 0 and 2 Rydberg series, the influence of vibrational interactions with channels u+ > 0 is not negligible: In order to attain the quality of the theoretical calculation shown (residual discrepancies of less than one wave number unit, see Jungen and Raoult, 1981 ), it has been necessary to include 10 vibrational channels in the calculation. 2. Lu - Fano Plols

From a more qualitative point of view we can nevertheless regard Fig. 4 as representing a two-channel, two-limit system. It is instructive to see how the level perturbations in the two Rydberg series affect the empirical quantum defect as given by Eq. (6). This is shown on the left of Fig. 4: Here the quantum defect pN+-ocalculated for each successive line with the known ionization limit E(u+ = 0, N + = 0) is plotted. (Historically the inverse procedure has been followed since the multichannel treatment of the rotational interactions was used to determine the ionization potential of H, .) The plot

71

QUANTUM DEFECT THEORY

does not distinguish between levels associated with the N + = 0 and 2 limits since strictly this distinction is meaningless. The overall effect of this procedure is that each time a level belonging in a first approximation to theN+ = 2 series occurs, the quantum defect P,+-~ rises by unity. This arises because one level “too many” has been counted in the N + = 0 Rydberg series so that the quantum defect must also increase in order to keep v = n - p constant. The interesting fact displayed in Fig. 4 is that in this way the level perturbations can be made to stand out very clearly. Since the “extra” N+ = 2 level interacts with the nearby unperturbed N + = 0 levels and pushes them away from their unperturbed positions, the increase of the quantum defect is not just a step function but exhibits a characteristic dispersion pattern. The stronger the interaction, the broader is this pattern. It is repeated each time a zeroth-order N + = 2 level occurs. Further, it can easily be appreciated that the dispersion patterns, if plotted with respect to the effective quantum number v,+-~ = [2E(v+= 0, N+ = 2) - 2E]-’12,will occur in periodic intervals as do the zeroth-order N + = 2 levels. The dispersion curves obtained in this way can finally be telescoped into a single unit square by plotting the itself. The result of these points versus v,,,+-~ (modulo 1) instead of v,+, operations is shown in Fig. 5, where it can be seen that all observed points now come to lie on (or close to) a single cuwe. This curve then represents in condensed form all of the perturbations arising from the interaction between the two channels. This type of diagram has been introduced by Lu and Fano (1 970) and has

- 0.5

0

v2(mod I )

0.5

FIG.5. Lu-Fano plot representing the perturbations shown in Fig. 4.

72

Chris H . Greene and Ch. Jungen

both practical and theoretical significance. The use of Lu-Fano plots assists the experimentalist in the spectral analysis of perturbed Rydberg series; it helps to detect channel interactions and to assess their strength on an absolute scale without any calculation effort. Numerous Rydberg levels can be represented graphically in compact form. Theoretically, the Lu - Fano curve represents the solution of the linear system, Eq. ( 17), for a bound two-channel, two-limit system, namely,

[

+

det cos 8 sin a(v, p , ) -sin8sina(v2+pl) cos 0 = U , , = U22,

+

sin 8 sin n(vl p2) cos8sin n ( v 2 + p 2 )] = o

(42)

sin 8 = UI2= - U2,

The effective quantum numbers v, and v2( v ~ + -and ~ v,+,~ in the example of Fig. 5 ) are regarded here as variables, disconnected from the channel electron energies E , and c2in terms of which they were originally defined through

E = E i + e i = E i - 1/2~: (43) Read in this way, Eq. (42) and the equivalent graphical curve depend just on the dynamical parameters pa and U,, which are nearly independent of the energy, while Eq. (43) includes the strong energy dependences. In other words, the Lu-Fano plot can be viewed as an elegant graphical way of removing the boundary conditions on the wave function at infinity inasmuch as they govern the detailed level positions, and hence to make stand out the short-range dynamics common to all the levels. The detailed properties of Lu- Fano plots have been discussed by Lu and Fano ( 1 970) and by Giusti-Suzor and Fano (1 984). Here we mention only that the channel interaction is reflected by the departure of Fig. 5 from a step function, and that, as is clear from Eq. (42), the intersection points of the curve with the subsidiary diagonal straight line v1 (modulo 1 ) = v2 (modulo 1) give directly the values of pa, a = 1, 2. At low energy (small v values), observed levels will lie near this diagonal if the core levels are sufficiently closely spaced so that their differences can be neglected in Eq. (43) and hence v, = v2.These levelsconform to the Born -0ppenheimer approximation. At higher energy Eq. (43) is represented in Fig. 5 by lines with negative slope <- 1 (not drawn in the figure for clarity) whose intersections with the Lu Fano curve determine the positions of the bound levels. When there is no channel interaction the Lu -Fano curve degenerates into two perpendicular straight lines, each of which represents an unperturbed Rydberg series. Inspection of Eq. (42) indicates that this can arise in two ways, either when the two eigenquantum defects pa coincide or when the frame transformation matrix is diagonal, i.e., the frame transformation angle 8 is zero. As an example Fig. 6 shows the Lu-Fano plot describing

QUANTUM DEFECT THEORY

73

n; (mod I )

05

00 He2 ndA

05

t

I

5P

FIG.6. Lu-Fano plot representingthe u = 0, N = 6, odd-parity triplet levels of He, arising from n d n - and ndA- states. The values n: and n: refer to the u+ = 0, N+ = 5 and 7 limits, respectively.(From Ginter and Ginter, 1983.)

excited levels with J = 6, 1 = 2, and negative parity of molecular helium observed by Ginter and Ginter (1983). (For related work see Ginter and Ginter, 1980; Ginter et al., 1984.) The two interacting Rydberg series have ll- and A- symmetry, respectively, in the Born-Oppenheimer limit and converge toward the N + = 5 and 7 limits of He,+, u+ = 0. Although the frame transformation angles 0 are nearly identical in the H, and He, examples, namely 54.7’ in H2 as compared with 5 1.7” in He,, it can be seen that the channel interaction is much weaker in He,. The reason is that the eigenquantum defects pn and p Adiffer by less than 0.1, as can be read from the plot. This is characteristic of an electron with a high orbital angular momentum 1 which is largely kept away from the core by the centrifugal potential l(l+ 1)/2r2 and thus cannot probe the noncentral core structure efficiently. Quite generally, molecular Rydberg states corresponding to a high I value appear in the spectra as compact “I complexes” (see Dieke, 1929) owing to the small values of all pu,’s, and they are characterized by weak channel interactions. They tend to conform to Hund’s case d even for small values of the principal quantum number, because, even when vI = v2 in Eq. (43), the linear system corresponding to Eq. (42) still yields coefficientsA, far from 1 or 0, again as a consequence of the smallness of the pa’s. In graphical terms, even for small differences vI - v,, the levels will remain on the horizontal or vertical parts of the Lu-Fano curve.

74

Chris H . Greene and Ch. Jungen

3. Rotational Channel Interactions in a Heavy Molecule Labastie, Bordas, Broyer, Martin, and collaborators (Martin et al., 1983; Labastie et al., 1984; Bordas et al., 1984) have recently observed highly excited Rydberg states of the Na, molecule using the method of opticaloptical double resonance, and they have applied quantum defect theory to their highly resolved data. An example of their spectra is shown in the lower part of Fig. 7 corresponding to excitation of 1 = 2 Rydberg states near the v+ = 4 threshold from the intermediate A ‘Zf, v+ = 4, J’ = 22 even-panty level. The final levels thus must have J = 2 1,22, and 23 and odd parity, and therefore involve series associated with odd N + values ranging from 19 to 25. In all, eight different channels are represented in the spectrum of Fig. 7 and involve two three-channel systems and one two-channel system. These excited Rydberg levels are subject to vibrational preionization since they lie above the u+ = 0 ionization threshold-in fact they are detected experimentally by the ionization signal which they produce. However, the resonance widths are smaller than the experimental resolution; so that in a good

24

25

26

I

39,850

27

28

29

30

31

I

32 33 3 4

39,900

35

-rE 3;,950

[Em-’]

FIG.7. Rydberg spectrum of Na, excited from the A Ci:, v = 4, J’ = 22 even-parity level (top: theory; bottom: experiment). The spectrum corresponds to excitation of I = 2 Rydberg states near the v+ = 4 limit and illustrates the fringe effect. The energies corresponding to integer values of v are marked below the experimental spectrum. (After Bordas ef al.. 1984.)

QUANTUM DEFECT THEORY

75

approximation we can disregard the ionization process for the present purposes. In the upper part of Fig. 7 is shown a theoretical spectrum based on the methods outlined in Section 11, in particular the algebraic linear system given in Eqs. ( 17) and ( 1 8). All open channels are omitted and, in fact, the R dependence of p,,(R) has been disregarded altogether, whereby Eq. (40) becomes proportional to S,,”+, .As for H,, Eq. (34) was used to evaluate the rotational 1-uncouplingframe transformation. A fit of the quantum defects p, to the experimental data yielded the valuespd, = 0 . 2 1 , = ~ -0.04, ~ ~ and pda = 0.43. The ionization limits E,+,,,+have also been fitted and can be found in the paper of Martin et al. (1983). The quite good agreement between the experimental and theoretical spectra demonstrates that theory accounts for the rotational channel interactions in Na, as it does in H, and He,. The spectra of Na, illustrate a striking new aspect of channel interactions which is not obvious in the H, example discussed above. The new feature of the spectrum in Fig. 7 is its periodicity, which is related simultaneously to several series. By contrast, the conspicuous level perturbations seen in H, (Fig. 4) arise whenever a N + = 2 level occurs; that is, they are related to the periodicity of just one series. The periodicity of the Na, spectrum manifests itself by simplified, comblike structures, or fringes, which are numbered k = 1,2, . . . . Between the fringes the intensity is distributed over numerous channel components, and no outstanding structure can be discerned. For the present discussion it is sufficient to consider a two-channel situation to which Eqs. (42) and (43) apply. Labastie et al. (1984) have shown that the fringes occur whenever

vI-v2=k, k=l,2,. .. (44) Equation (42) is indeed periodic for integer values of k; one of the coefficients A , of the corresponding linear system is then unity; i.e., a periodic return to the Born-Oppenheimer limit occurs (Hund’s case b), as had been stressed by Fano ( 1975) and Jungen and Raoult ( 1 98 1). A favorable circumstance in Na, is that the dipole intensity for excitation from the intermediate A lZ; state is dominated by the l’l- 2 transition moment since Z- Z is quite weak and A -2is dipole forbidden. As a consequence all intensity is concentrated near energies corresponding to vi = ni - pdn, and different lines coincide owing to Eq. (44). The condition [Eq. (44)] is, however, not yet sufficient to explain the appearance of the fringes. For these to become clearly visible it is also necessary that the interfering Rydberg series remain “in step” over several cycles of nv. Thus, according to Labastie et al., one must also have dv,/dv,

-

1

(45)

76

Chris H. Greene and Ch. Jungen

Since Eq. (43) implies that V , = v,( 1 - 2 ~ AE)-'I2 : (46) where AE is the splitting between the two ionization limits, it follows that, in order to fulfillthe conditions of Eqs. (44) and (45) simultaneously, one must have

2k K v, K ( 2 AE)-'I2 (AE in a.u.) (47) Clearly, then, fringes of the type shown in Fig. 7 are better visible for smaller AE. This explains why they appear in the heavy system Na,, which has a much narrower rotational structure than H, or He,. The fringe pattern becomes progressively narrower as k increases. From Eq. (47) we estimate that the last fringe, k,, , corresponds roughly to AE =r 250/k2,, (AE in cm-') (48) This shows that fringes are not to be expected in vibrational or electronic channel interactions where AE usually is much larger than 250 cm-I. Also, the fringe effect has obviously no counterpart in the continuum, unlike the level perturbations of the type seen in H, which extend into the continuum in the form of preionization profiles (and present also, of course, in Na,). C. CHANNEL INTERACTIONS INVOLVING CONTINUA

The main conceptual feature of quantum defect theory has been the recognition that Rydberg bound states lying below an ionization threshold and the continuum states just above this threshold are governed by the same physics at short distances. This concept has been used widely in other physical problems, of course, such as effective-range theory applications in nuclear and atomic physics. Indeed Sommerfeld ( I 93 1) had long ago exploited this fact in his studies of atomic Rydberg electrons. Now we turn to an explicit demonstration of this unity, whereby the same parameters used to calculate accurate H, bound levels will equally serve to characterize the photoionization cross sections of H,. 1. The Total Photoionization Cross Section

Besides the eigenquantum defects p, and frame-transformation matrix elements U,,given above, the photoabsorption spectrm depends on dipole matrix elements DL? = D E connecting the ground state of H, to the finalstate eigenchannels w,. These are given in terms of Born-Oppenheimer matrix elements d,,(R) in Eq. (4 I). In the absence of an ab inifiocalculation

QUANTUM DEFECT THEORY

77

of these matrix elements, studies to date have chosen d,(R) = dAR) to be a constant, independent of R but slowly varying as a function of energy. The equality of d, and dn is suggested by the united-atom limit (Herzberg and Jungen, 1972). The assumption of R independence should not be taken too seriously,as the H2ground state is confined to a narrow range of internuclear separations and therefore the R dependence is hardly probed by ground-state photoabsorption experiments. It is now relatively straightforward to calculate the photoionization cross section by solving the linear equations of quantum defect theory, Eqs. (37)(38), and then using the general cross-section formula, Eq. (31). At any energy E of interest there are typically both continuum (i E P)and discrete channels (i E Q). The continua provide the path to ionization, that is, to the actual escape to infinity of the electron from the molecular ion. The closed channels provide a mechanism for the formation of short-lived quasibound states which typically autoionize in to sec. In simple atoms, autoionization is often said to result from the electron -electron repulsion terms in the Hamiltonian. In this vein rovibrational autoionization in small molecules must be attributed to a qualitatively different dynamic feature; namely, the R dependence of the quantum defect function,u,,(R) is responsible for “vibrational” autoionization, while its A dependence is responsible for “rotational” autoionization. (Most rovibrational autoionization features involve both physical mechanisms, making it impossible to distinguish the two processes.) This distinction can at least be made in a mathematical sense since the rotational factor of the frame-transformation matrix [Eq. (33)] would induce no autoionization if ,u, were equal to Pn. Similarly, if p,,(R) were independent of R, then the vibrational factor of Eq. (33) would induce no autoionization. When both of these conditions are fulfilled, the matrices Sii,and C,,,of Eq. (40) are proportional to the unit matrix, in which case the absence of channel interactions is obvious. But this weak-coupling limit need not be satisfied, as evidenced for example by H, photoabsorption where channel interactions are indeed so strong as to render any perturbative scheme useless. Only quantum defect theory seems capable of dealing with these multichannel interactions which disturb the positions and intensities of an infinite number of Rydberg and continuum levels at once. Figure 8 displays a quite simple photoionization spectrum measured under high resolution by Dehmer and Chupka (1976). The spectral range shown includes the 8pa and 9pa, u = 2, J = 1 levels ofH, . This spectrum has again been obtained using a cooled sample of para-H, so that it corresponds for the most part to an excited system with J = 1, odd parity. The Rydberg channels involved in this range are represented by Fig. 9: All discrete Rydberg structures at this energy are embedded in the v+ = 0 and 1 , N + = 0 and 2 continua. Between the v = 2 levels with n = 8 and 9 lie numerous “inter-

78

-

'>

Chris H. Greene and Ch. Jungen

0.8 0.4 -

H~fm*=lJcolc. I-

0

mou. 50 1

h

0.04

nfn=2I+nflr J co /c

122300

127,200

127,100 E (ern-')

127,000

12 6,900

FIG.8. Preionization and predissociation near the u+ = I , N+ = 2 ionization ' threshold ( 1 26773.6 cm-l) in H, ( J = 1, J" = 0). Top: observed photoionization spectrum of Dehmer and Chupka (1976). Bottom: calculated partial vibrational photoionization spectra and photodissociation spectrum (see text for details).

lopers" which are associated with higher ionization limits and correspond to n < 8. Note that all the assignments given in Figs. 8 and 9 have no strict meaning owing to the strong channel interactions which take place between all the channels; they are useful just for "bookkeeping" purposes since of course the number of levels occurring in the whole range is not changed by the couplings. The quantum defect calculation (Raoult and Jungen, 1981; see also Takagi and Nakamura, 1981) is based on exactly the same parameters and d used in the calculation described in Section III,B, 1. The calculation yielded all the partial cross sections point by point on a conveniently chosen energy mesh. These theoretical partial cross sections are shown in the lower part of Fig. 8 for each v+ final state, summed over the N + = 0 and 2 rotational contributions. Since Dehmer and Chupka (1 976) did not discriminate between the different photoelectron groups, they obtained the total photoionization cross section. More precisely, their spectrum gives the photoionization efficiency, that is, the ratio of the total ionization to the transmitted light, but this is very nearly proportional to the ionization cross section except near the very strongest resonances. For the exact comparison between experiment and theory, the theoretical partial cross sections should be summed up and convoluted with the experimental resolution width of 0.016 A. The result of this procedure at the wavelengths corresponding to the peak maxima is indicated in the experimental spectrum by horizontal arrows. Quite good agreement is found on

QUANTUM DEFECT THEORY

79

Fic. 9. Schematic illustration of vibrational-rotational preionization and predissociation in H, (J = 1, odd panty). Ionization and dissociation continua are indicated by vertical and oblique hatching, respectively. For each given u+ of the H,+ion there are two continua corresponding to rotational quantum numbers N+ = 0 and 2 of the ion ( J = 1). Selected Rydberg levels are indicated below the vibrational ionization limit with which they are associated.

the whole. (The theoretical spectrum is normalized to coincide with the experimental one at the 8p0, u = 2 peak.) Similarly, it can be seen that the continuum background is quite accurately reproduced; one must remember that the vibronic perturbations which affect the Rydberg levels are quite strong. These perturbations are not so obvious in Fig. 8, but their presence has been established empirically by Herzberg and Jungen ( 1972) by the use of Lu-Fano plots, and they are also well borne out in the channel coefficients [Eq. (37)]resulting from the calculations. The vertical arrows in the top spectrum of Fig. 8 mark the peak positions measured by Herzberg and Jungen in the absorption spectrum. Their wavelength determination is more accurate than can be attained in the photoionization experiment: It can be seen that the theoretical peak energiesgenerally agree better with these values.

80

Chris H . Greene and Ch. Jungen

Turning now to the discussion of the resonance widths, we remark that, both in the experiment and in the calculation, the broadest features correspond to the lowest vibrational quantum numbers. This means that the ionization process is favored when only a single quantum of vibrational energy has to be exchanged. Quantitatively this can be understood with reference to Eq. (40):Assuming a linear R dependence of sin np,(R) and cos np,,(R), and harmonic-oscillator vibrational wave functions, one obtains coupling matrices C, ,+, and S , , ,,,+,whichobey the selection rule Av+ = k 1 familiar from optical infrared spectra. In reality, vibrational anharmonicity as well as the interactions among the closed channels break this selection rule, but its approximate validity is still clearly borne out in the simpler portions of the H2photoionization spectrum, such as that shown in Fig. 8, where the resonances are relatively well isolated. The Av+ = k 1 selection rule as applied to photoionization spectra has first been discussed by Berry ( 1966) and Bardsley ( 1967a). Berry termed it a “propensity rule” because in a first approximation the strongest channel couplings (e.g., the off-diagonal elements of a reaction matrix) indicate the preferred decay paths: Indeed, Fig. 8 shows that the v+ = 1 ionization channel is calculated to carry the bulk of the ion signal arising from the resonance features. By contrast, the flat parts of the total cross section are shared by the ionization channels according to the Franck - Condon factors for excitation from the ground state of H,; they correspond more nearly to a “direct” photoionization process. An experimental test of these predictions has been made by Dehmer and Chupka (1 977) and more recently by Ito et al. ( 198 1 ) for a few selected resonances. The agreement between experiment and theory is again satisfactory. An example is furnished by the 6pn, v = 6, J = 1 (odd-parity) resonance at 752.866 A, which is reproduced in Figs. 10 and 1 1 . The observed (Dehmer and Chupka, 1977) and calculated (Jungen and Raoult, 198 1 ) final vibrational-state distributions in photoionization are

Theoretical (%) Experimental (%)

v+ = 4

v+ = 3

82 82

16 15

v+

=2

2 3

Close inspection of Fig. 8 reveals some details which have not been accounted for in the discussion so far. For one thing, in the excitation energy range corresponding to the figure, the H2molecule is subject not only to preionization but also to predissociation. The latter process occurs within the p-type molecular Rydberg channel because the Rydberg electron might transfer some of its energy to the core vibrational motion, enabling the molecule to dissociate into atoms. This is the exact opposite of vibrational

81

QUANTUM DEFECT THEORY

preionization, where the electron picks up energy from the core. Energetically this is possible since the dissociation limit lies lower than the ionization limit (cf. Fig. 3). Figure 8 includes at the bottom the theoretical photodissociation cross section, calculated using an extension of quantum defect theory which will be discussed in Section III,D,2 below. Notice that only resonances corresponding to high vibrational quantum numbers and low principal quantum numbers are noticeably affected by predissociation; that is, these are the levels whose “vibrational energy” most closely approaches that of the vibrational continuum state leading to dissociation. Thus we see that predissociation in the present situation conforms to a rule similar to the Au+ = 1 rule for preionization: In both cases the preferred decay paths are those which correspond to exchange of the minimum amount ofenergy. Note the analogy with the “energy-gap law” familiar from the theory of nonradiative transitions in larger molecules (see, e.g., Robinson and Frosch, 1963). The correctness of the quantum defect treatment of the competition between ionization and dissociation processes is demonstrated by the 4pu, u = 6 and the overlapping 3pn, u = 8 and 6pu, u = 3 peaks: These resonances absorb very strongly but are calculated and observed to appear only weakly in photoionization. (Note the very different scales of the dissociation and ionization cross sections in Fig. 8.) Figure 10 displays a more complex section of the photoionization curve of

1

-- -5 -

I

r6 p 6n

I

I

I

I

7

6PU v . 6 1

vr

1.0-

v,

$

.-0

9pn

- v=5 : I

;P!j .’

I

#

FIG.10. Preionization near the u+ = 4, N + = 0 and 2 thresholds in H, ( J = 1, J" = 0). The observed and calculated total oscillator strengths are shown as functions of photon wavelength. The experimental points from Dehmerand Chupka (1976) have been shifted by -0.068 A so as to bring the observed and calculated 9pq u = 5 peaks into coincidence. The calculated spectrum is broadened to a resolution of 0.0 16 A to correspond to the experimental measurements. (After Jungen and Raoult, 1981.)

82

Chris H . Greene and Ch. Jungen

H, corresponding to excitation wavelengths near 754 A. At this energy the u+ = 0 to 4 channels are all open; the u+ = 4, N + = 0 and 2 limits fall into the range shown and are indicated, but they are not directly visible in the spectrum as a threshold discontinuity since the averaged below-threshold resonance structure matches smoothly onto the continuum cross section (Gailitis, 1963). However, a new type of feature appears just below the u+ = 4, N + = 2 threshold. An interloper with low n and high u (6~17,u = 6) falls among the dense manifold of the high n/low u series (np2, u+ = 4). Interaction between these closed channels leads to a transfer of intensity from the strong 6pa, u = 6 line to the very weak np2, u+ = 4 lines. The remarkable result is the formation of a “complex resonance” whose width exceeds by far the widths ofits individual components. Dehmer and Chupka( 1976) in their experiment were just able to resolve some of the fine structure ofthe complex resonance (cf. the figure), but in an earlier spectrum (Chupka and Berkowitz, 1969) the resonance is not resolved and has an apparently perfect Lorentzian shape with a width of about 15 cm-’. As a result one might mistakenly take this global width as the one governing the preionization decay rate. It is interesting to examine the calculated partial cross sections near the 6pa, u+ = 6 complex resonance. They are shown in Fig. 1 1 . Obviously this is an instance where the Av+ = 1 selection rule, as applied to such processes, is bound to break down: As a consequence of the strong perturbations affecting the Rydberg levels, no definite vibrational quantum number u+ can be ascribed to any preionized level. We see indeed that the complex resonance is calculated to appear as such in the u+ = 4, 3, and 2 ionization channels, although with diminishing intensity. It virtually disappears, however, in the u+ = 1 channel where only the narrow central peak (labeled 6pu, u = 6) subsists. Complex resonances have been found not only in H, (Jungen and Dill, 1980; Jungen and Raoult, 198 1) but also in N, (Dehmer et al., 1984) and in atoms (Connerade, 1978; Gounand et al., 1983). We think that they are probably a more typically molecular phenomenon because in molecules, owing to the vibrational and rotational degrees of freedom, dense manifolds of levels are always present. We may suspect that many unresolved resonance features observed in molecular spectra at high energy are actually of this type. Giusti-Suzor and Lefebvre-Brion ( 1984) and independently Cooke and Cromer ( 1985) have used quantum defect theory to study analytically the simplest case of a complex resonance, involving two closed and one open ionization channel. They showed how the global or “effective” width of the resonance depends on the coupling between the closed channels. The significance of complex resonances, however, seems to lie more in the role they play in the competition between alternative decay processes. Indeed, the level scheme underlying the complex resonance, corresponding to an iso-

83

QUANTUM DEFECT THEORY

0.0.4

0.00 0.08

0.04

0.00

-->

0.08

-

I

0)

5

0)

0.0.4

c aI c L In

;0.00

I

,

I

I

I

I

1

I

I

I

I

I

I

1

1

1

1

m

z .u

1

1

1

1

1

1

1

1

1

-71

+-

0.8

n p 2 v =I nPO

u)

0

0.4

I

0.0

08

04

00

[I

-

max.x

I

I 753

'

75 5

754 Wavelength (

I

)

FIG.1 1. Calculated partial vibrational photoionization oscillator distributionsfor the spectral range shown in Fig. 10. Near the u+ = 4, N + = 2 limit the data have been averaged over the dense Rydberg structure arising from the np2, u = 4 (n > 36) Rydberg series (broken line). (After Jungen and Raoult, 1981.)

84

Chris H. Greene and Ch. Jungen

lated level facing a dense set of states, both being embedded in a set of continua, has been the building block for the description of the unimolecular decay of metastable molecules (see, e.g., Mies and Krauss, 1966; Rice et al., 1968; Lahmani et al., 1974). In that theory a time-dependent language is usually adopted and attention is focused, for example, on nonexponential decay patterns. The analog in the present context of complex resonances is their departure from Lorentzian shape: Note how in the example shown in Fig. 1 I the effective width is different by orders of magnitude depending on whether the u+ = 4 or the u+ = 1 ionization channel is monitored. 2. Photoelectron Angular Distributions The angular distribution of ejected photoelectrons provides an additional experimental handle on the short-range molecular dynamics of interest. For processes excited by an electric dipole photon linearly polarized along an axis i, the fraction of photoelectrons observed in a particular channel i along an axis k is proportional to 1 /3iP2(cosO), in terms of the second Legendre k. The quantity ofinterest here is polynomial P2(x);its argument is cos 0 = i* the so-called “asymmetry parameter” p. The importance of measuring p stems largely from the fact that it depends not only on the relative amplitude but also on the relativephaseofthe alternative partial waves lcomprising the final continuum state. The dynamic information needed to calculate pi is contained in the reduced dipole matrix elements D:!-)of Eq. (30). We will refer to this element here with the notation DIN-)to indicate its dependence on the orbital angular momentum 1 of the escaping photoelectron explicitly. This amplitude connects the initial state having angular momentum Joto an incoming-wave, energy-normalized final state of angular momentum J, where J = Jo,Jok I . These amplitudes correspond to an angular momentum coupling,

+

Jo +j, = J

= Nt

+1

(49)

The angular momentum contributed to the photoionization process by the incident electric dipole photon is designated by j ? = 1; that of the final ionic fragment state i is N t . In writing Eq. (49) the spin angular momentum s = f of the photoelectron and the spin Si of the molecular ion have not been written explicitly. This amounts to the assumption that these spins remain coupled throughout the ionization process to the value of the spin Soin the initial state. [More precisely, it presumes the relation So= S, s to hold in addition to Eq. (49).] The implications and limitations of this frequent assumption are discussed by Watanabe ef al. ( 1984). In small molecules it is

+

QUANTUM DEFECT THEORY

85

an excellent approximation except in very limited energy ranges, such as the range between two spin-rotation thresholds of the molecular ion. As mentioned in Section II,D, the reduced dipole amplitudes D Y - ) determine the asymmetry parameter Pi. This relationship takes the general form

as can be extracted from Jacobs ( 1972), Lee ( 1974b),or from references cited in these works. The symbol o( ) denotes a complicated geometrical quantity, involving Wigner 3j and 6j coefficients.Notice that the expression, Eq. (50), represents a coherent summation over 1 and J, which complicates the analysis and requires considerable attention to ensure that the phases of the amplitudes DIN-) are correct. In two papers, Fano and Dill ( 1 972) and Dill and Fano ( 1972) recognized that the coherence originates in the quantum-mechanical incompatibility of the operators l2 and J 2with the observable 0. Dill and Fano use an alternative set of amplitudes S,( j,) characterized not by J 2but by j:, where j, is the angular momentum transferred between unobserved photofagments. Since j: commutes with 8, amplitudes having differentj, then contribute incoherently to P. More explicitly this defines j, by

---

j , = N t - JO = j Y - 1

(51)

The second equality follows from Eq. (49), and implies that at most three partial waves 1 =j,, 1 =j , k 1 can be present for a given angular momentum transferj,. Parity conservation also implies that only even 1or else only odd 1 can contribute coherently in any given ionization channel i. The net result of the angular momentum recoupling is that the expression for /3 in terms of the Sl(jt)is now incoherent:

Here each j, is classified as parity-favoredor parity-unfavored according to whether j , - 1 is odd (k1) or even (0), respectively. For parity-unfavored transfers j,,

86

Chris H . Greene and Ch. Jungen

while for parity-favored transfers

In Eqs. (53) the S&) are abbreviations for S,,,(j,). Whereas parity-unfavored contributions to p are fixed at the value - 1, parity-favored contribu6 2. Finally, the amplitions can lie anywhere in the full range - 1 G pfav(jt) tudes S,(j,) have been expressed in terms of reduced dipole amplitudes by Dill ( 1973),

The preceding results were utilized by Dill (1972) to study the angular distribution of photoelectrons ejected from H, (earlier work by Buckingham et al., 1970, had introducedj, as a dummy summation index, but without the physical interpretation stressed by Dill and Fano). The dominance of the 1 = 1 partial wave in the asymptotic wave function of H, photoelectrons was pointed out above in Sec. III,B,I. If all other partial waves 1 # 1 are neglected, the asymmetry parameter pi appropriate to each ionic rotational channel N t can be evaluated explicitly. In particular, when the target molecule has J, = 0, only the two values IVt = 0, 2 are permitted by angular momentum conservation, with asymmetry

p ( N t = 0 ) = 2.0;

P ( N t = 2) = 0.2

(55)

independently of both the energy and the vibrational state of the ion. Table I compares the above predictions to some ofthe more recent experiTABLE I EXPERIMENTAL ASYMMETRYPARAMETERS IN H2PHOTOIONIZATION

I(nrn) P(N+ = 0, u+ 58.4

1.918"

73.6

I .903"

= 0)

Ruf et al. (1983). Pollard et al. ( 1982).

f l N + = 2, U + = 0 )

0.81 f0.17' 0.87 f 0. 19b 0.54 f 0.16' 0.08 f 0. I 5 b

87

QUANTUM DEFECT THEORY

mental results. While p values close to 2 are observed in the N + = 0 channel as expected, a sizable discrepancy is present in the N+ = 2 channel where p = 0.2 is not observed at either photon energy. This discrepancy is of considerable interest since it has immediate implications about the amplitudes of 1 # 1 partial waves in the final-state wave function. In view ofthe excellent agreement between theoretical (1 = 1 only) and experimental integrated cross sections, the deviation ofp(N+ = 2, u+ = 0) from has been a surprise. Indeed, a wide range of incompatible p values has been published for this channel (e.g., at 73.6 nm in Table I). (See Ruf et nl., 1983, references therein for some discussion of the past and present experimental situation.) Theoretical efforts to understand the unexpected departure of p from 4 have been undertaken by Itikawa ( 1979) and by Ritchie ( 1982), which consider an admixture of 1 = 3. In essence the origin for the difficulty of determiningp for this channel, both theoretically and experimentally, stems from the small value of the N + = 2 partial cross section. The dominant ( I = 1) oi experiences a substantial cancellation which reduces it to an order of magnitude smaller than the N + = 0 cross section; this can be seen from the (purely 1 = 1) calculations of Raoult et al. ( I 980). This opens the door to a possible influence byfwaves despite a complete absence of (observed) 1 = 3 states in the known discrete spectra of H,. The very recent results of Hara (1985) and of Hara and Ogata (1985) are in essential agreement with the measurements of Ruf, Bregel, and Hotop (1983). At 73.6 nm, for instance, these authors calculate p(u+ = 0, N + = 0) = 1.891 and p(u+ = 0, N+ = 2) = 0.643 (see Table I). Finally, there have been several calculations (Raoult et al., 1980; Raoult and Jungen, 1981;Jungen and Raoult, 1981) of the asymmetry parameter in the vicinity of autoionizing resonances in H,. Just as this spectral range shows rapid variations in the total and partial cross sections, the rotationally unresolved p oscillates in a complicated fashion also. To date no experimental work has been done in this range to test the sensitive predictions of quantum defect theory. The effects of electronic preionization on the p parameter have been studied in a fully ab initio approach by Raoult et al. (1983). This pilot calculation accounts successfully for p in the X- and A-state channels of N,+. This work is reviewed in Section IV,B.

+

D. TREATMENT OF A CLASSOF NON-BORN - OPPENHEIMER PHENOMENA In valence states of diatomic molecules the nuclei and the outer electrons are confined within the same volume of space, typically within a radius of a few Bohr radii. In addition the molecule is bound, implying that the forces

88

Chris H . Greene and Ch. Jungen

on the nuclei and on the electrons are of comparable magnitude. This standard argument immediately suggests that the time scale of nuclear motion is orders of magnitude slower than the (valence) electronic time scale, whereby the Born-Oppenheimer concept of defining an electronic energy U(R) at each internuclear separation makes good physical sense. A rough estimate of the validity of the Born - Oppenheimer separation of electronic and nuclear variables is given by the ratio of vibrational level spacings to the spacing between successive potential curves of the same symmetry [see Dressler (1 983)1, = A U(R)/A&ib.

(56)

When y is large compared to unity (e.g., y = 25 for the H2ground state with v = 0) the Born-Oppenheimer approximation is adequate, and nonadiabatic effects embodied in electronic matrix elements such as <4;'l(d/dR)&') are of little significance. For Rydberg potential curves described by Eq. (35), however, AU(R) = K 3 ,so that in fact y approaches unity in Hz by n = 5 since A & , is nearly independent of n. The qualitative change in the physics renders the Born Oppenheimer description ineffective for high-lying Rydberg states. As we have seen, experimental manifestations of this non-Born -0ppenheimer behavior include strong rovibronic perturbations and conspicuous intensity modulations in the ionization continuum associated with vibrational autoionization. Yet another conspicuous nonadiabatic process is predissociation, to be discussed in Section 111,D,2. 1. Adiabatic and Nonadiabatic Correctionsto the Discrete Levels

Figure 12 presents energy levels of the Hz C Ill; state as calculated by Jungen and Atabek ( 1977)using the quantum defect procedures described in Section II,C. Shown are deviations between experimental values and theoretical values obtained in QDT and in the strict Born-Oppenheimer approximation. Note how the quantum defect results, derived from the Born Oppenheimer functions pA(R),agree with experiment far better than do the full Born - Oppenheimer energy levels. Among the non-Born Oppenheimer effects included in the quantum defect calculation are the well-known corrections for the H2+ion (Hunter et al., 1974); this ensures that the H2Rydberg levels converge to the proper thresholds. Here this gives the bulk of the correction in Fig. 12, with a comparatively small correction coming from the outer electron. The quantum defect formulation represents these effects in a much simpler fashion than conventional theory (Van Vleck, 1936), which would require separate evaluation of several perturbation integrals as functions of R followed by vibrational averaging.

QUANTUM DEFECT THEORY

89

FIG.12. Deviations of the observed J = 1 levels in the 2pn ‘n; state from those derived ab inirio for H2(circles) and D, (dots) in the Born-Oppenheimer approximation (BO) and using quantum defect theory (QDT). The difference between the two results represents the adiabatic and nonadiabatic corrections. The remaining deviations in QDT correspond to the specific isotope effect. (After Jungen and Atabek, 1977.)

These non-Born - Oppenheimer couplings also cause perturbations between discrete levels belonging to different Born - Oppenheimer potential curves. One reflection of such perturbations is the A doubling as given in Figure 13, for example, which is the splitting between the 3p7r D Ill: and Ill; states. Since T I: interacts with 5; levels while Ill; states experience no such interaction, the A doubling displays the effect of perturbations clearly. Notice how a seemingly random dependence on J and v, corresponding to the erratic occurrence of near degeneracies, again emerges “automatically” in this approach. While agreement with experimental values of Takezawa (1970) is not exact, the general trends are very well represented.

2. R-Matrix Treatment of Predissociation A more extreme failure of the Born- Oppenheimer approximation occurs when an excited molecular electronic state predissociates, thereby converting its internal energy from the electronic degrees of freedom into kinetic energy of the dissociating atoms. Experimentally the strength of this process is measured on an absolute scale by the “reduced” predissociation linewidth

90

Chris H. Greene and Ch. Jungen

0

I

3

5

J

FIG.13. A doubling in the 3pn D TI,state of H, (difference between n: and n; components). Dots: observed values from Takezawa ( 1970). Full lines: calculated values from Jungen and Atabek (1977). The irregular pattern of the splittings illustrates the breakdown of the hypothesis of “pure precession” (Van Vleck, 1929) arising because the vibrational structures of the interacting 3pn and 3pu states are intermingled and, in addition, different (cf. inset).

of rovibrational levels below the first ionization threshold, a dimensionless quantity -

r = r-(dE/du)-i = r / h w (57) Here w is the vibrational spacing and r is the full width at half-maximum.

Above the ionization threshold the strength ofdissociation relative to ionization can be measured more directly at each energy by observing the branchThe usual expectation is that this branching ratio remains ing ratio ud/oji. much less than one, since the photoabsorption tends to excite the electrons initially rather than the heavy nuclei. Experiment (Guyon ez al., 1979; Dehmer and Chupka, 1976) and recent theory (Jungen, 1982, 1984) both indicate, however, that this expectation is frequently violated, especially in light molecules and particularly close to resonances.

QUANTUM DEFECT THEORY

91

Predissociation can result from a variety of mechanisms, but it yields to a surprisingly simple treatment when a single electronic channel is present in the body frame. In this case the quantum defect approach gives an essentially exact wave function [Eqs. ( 14) and ( 16)] in terms of the reaction matrix K or the equivalent § and C matrices of Eq. (40). These contain the dynamic information required to describe the processes of vibrational autoionization or vibrational excitation. This type of predissociation can be viewed as a natural extension of the vibrational excitation process treated in Section III,C, e- H2+(u) e- H2+(u’),the only difference being that the final state u’ actually lies in the H H vibrational continuum of the lowest potential curves in the Rydberg series. Realized some time ago by Chang et al. (1973), this concept has been developed into a complete formulation only recently (Jungen, 1982, 1984) through an adaptation of the eigenchannel R-matrix method (Fano and Lee, 1973).The R-matrix method (Burke and Robb, 1975;Nesbet, 1980)obtains a variational solution to the Schrodinger equation at the total energy E of interest, but only within a finite reaction volume R of configuration space. Outside the surface I: which encloses this reaction volume, the solution can be represented by a channel expansion such as Eq. (10) or (14) with the proper form at large distances. By matching the inner variational R-matrix solutions to the outer region solutions on Z, the reaction matrix K is obtained. In this sense the R-matrix method links naturally with quantum defect theory, in that the variational problem is attacked only at small distances where the many-particle system is complicated; the more rapid energy variations associated with the motion at large distances are then represented analytically. The eigenchannel version of R-matrix theory (Fano and Lee, 1973; Lee, 1974a; Greene, 1983; Le Rouzo and Raseev, 1984) solves variationally for the eigenstates of the R matrix, i.e., for states having a constant normal logarithmic derivative (- b) everywhere on the reaction surface I:.The reaction volume R in this problem is the rectangular region r < r,, R < R, shown in Fig. 14. Here r2 is taken just large enough to contain the entire Born - Oppenheimer electronic wave function of each dissociative state. Instead R, is chosen just large enough so that all non-Born-Oppenheimer processes are contained within R < R,. The reaction surface enclosing this volume has two distinct parts: The line ( r = r2,R) connects to the ionization channels in regions 11, while the line (r, R = R,) connects to the dissociation channels in region 111. The wave function in region IV is neglected altogether along with the possibility of three-body fragmentation (dissociative ionization). The region denoted the “reaction zone” in Fig. 14 is characterized by complicated dynamics, but as we saw above the physical solution along the line r = r, is well represented by an R- and A-dependent quantum defect function p,,(R).

+

-

+

+

92

Chris H . Greene and Ch. Jungen

@tion

PRO zone

R (nuclei)

*

FIG.14. Schematic representation of the (r, R) configuration space relevant to the calculation of competing ionization and dissociation processes in HI.The appropriatechannel expansion terms are indicated for different zones. (After Jungen, 1984.)

The aim of the R-matrix calculation is thus to find a set of eigensolutions

w,, of the Hamiltonian with the following properties: wp

=

c

PU+N+(C

(u+N+)€P

R)(RIu+)1Tu+N+,,

r B r2, R < & (58) - Gd(R)sin az,], r r2, R 3 &

X [fu+N+(r) cos mp- gu+N+(r) sin nr,],

w, =

@$Tdg[Fd(R)cos KT,

(59) Here Eq. ( 5 8 ) coincides with the expression [Eq. (19)] for the electronic eigenchannels. Equation (59) is the exact analog on the dissociative part of the reaction surface, with @$ the electronic Born -0ppenheimer wave function in a dissociative channel d. At t g s stage the choice of two independent vibrational continuum solutions (Fd, G,,) in the relevant Born -0ppenheimer potential for the open dissociative channels is somewhat arbitrary. It is useful to choose this base pair through boundary conditions at

R

= Ro,

so that Eq. (59) takes a form familiar from R-matrix theory -aR +tann~,~,=O,

R=Ro, r s r 2

93

QUANTUM DEFECT THEORY

\

(Note that Fd and Gddo not coincide with the usual regular and irregular Born - Oppenheimer solutions Fand G, but are convenient linear combinations of them.) These eigenchannel boundary conditions supplement the behavior along r = rospecifiedby p,,(R), and the usual finitenesscondition at r 00 imposed by algebraic quantum defect procedures. It is convenient to regard the electronic phase parameter zPin Eq. (58) and the vibrational phase parameter zp in Eqs. (59) and (61) as independent variables, disconnected from the constraint ofequality implied by Eqs. (58) and (59),much as in the Lu - Fano plot discussed in Section 111,B,2. In practice the boundary condition [Eq. (6 l)] on the dissociativepart of the reaction surface is imposed first by picking a trial value 7 and thereby deJininga discrete, orthonormal vibrational basis of Hz+, (u+lR)cT).The associated energy levels EFjN+also now depend on 7 , although this dependence is negligible for the low v+ states which do not extend to R = R,,. Figure 15 compares the resulting finiterange spectrum (with Ro = 4) with the infinite-range spectrum in Hz+, showing that the finite-range level density is much lower for high v + as these states reach rapidly up into the vibrational continuum. For this particular (trial) choice of 7 the vibrational integrals are evaluated to give the S7)and C(T) matrices as before, and at the energy E of interest the closed ionization channels i E Q can be eliminated [see Eq. (23)] by solving

-

TB, = tan 7rtphB,

L

-

-0.50-

a

0

(62)

/ '

-

r

W

-0.60-

I

I

I

I

1

1

b

Chris H . Greene and Ch. Jungen

94

The net result of this elimination for each trial choice ofthe vibrational phase 7 is a set of No eigenphase shifts n7, of the open portion of the (ionization channel) reaction matrix, with p = 1 , . , . ,No. Since in general the vibrational surface parameter 7 will not equal any of these eigenvalues 7, (modulo l), it is necessary to search as a function of 7 until this consistency is established. As in the Lu-Fano plot, a convenient graphical representation of this procedure in Fig. 16 plots the electronic eigenphases 7, as functions of the vibrational eigenphase T . The example of Fig. 16 corresponds to two open ionization and two open dissociation channels 2 p n and 3pZ. Although only two electronic 7pare present for each value of T, these two curves 7 f ) display four intersections with the diagonal, giving No Nd = 4 eigensolutions tpas expected. In particular the two arrows show that the rapid variations of 7, lie close to the (unperturbed) dissociative H2Born - Oppenheimer logarithmic derivatives at this energy. In essence these appear as 7 varies across its full range 0 d 7 s 1, when low-lying Rydberg states associated with very high vibrational thresholds sweep through the energy E. A few additional remarks will now show how to extract the final 4 X 4 “grand” reaction matrix W, which refers to ionization and dissociation at once. With the eigenvalues of W given by tan n7,, we require their corresponding eigenvectors. The components of an eigenvector T, in the ioniza-

+

3pP

2pn

I 7P

0.50

0

FIG. 16. Predissociation in H1.Electronic eigenphases functions of the vibrational eigenphase 7 (cf. text).

7p (quantum

defects) plotted as

95

QUANTUM DEFECT THEORY

tion channels i are given by the standard result

The components in the dissociative channels are somewhat more complicated to evaluate in the general case, but in the limit of only one dissociative channel per value of A (i.e., d = nA), it is (see Lee, 1974a)

-

[Although this appears to be ill behaved in the special case T,, f ,there is in vanishes in that limit.] The solutions fact no divergence since (u+IR,)(~J Bi,,,= Bu+N+,p of the homogeneous system [Eq. (62)] are normalized according to

Finally, the grand reaction matrix refemng to ionization and dissociation is

R = T tan ntTT

(68)

In fact, the eigenphase shifts T,, correspond to a constant phase shift in the ionization channels i, but to a constant logarithmic derivative in the dissociation channels nA. A linear transformation [Eq. (3.3) ofGreene (1980)l gives the “usual” reaction matrix in an energy-normalized representation of ionization and dissociation channels:

K = (F’ - FR)(G’ - GR)-’

(69)

Here the diagonal matrices F, F’, G, C ’ take the values Fii,= - Giit= dii,, Gii,= Fji, = 0 in each ionization channel but instead ( F d , Gd)d,, are the values of the energy-normalized regular and irregular Born - Oppenheimer dissociative-state vibrational solutions evaluated at R = R, . Likewise (FL, G;)ddd, are their derivatives with respect to R at R = Ro (details concerning the specification of these vibrational comparison functions can be found in Greene et al., 1983; Mies, 1984). We now discuss the application of these procedures to the 3pn+, v = 3, J = 2 level of H,,to which Fig. 16 actually refers. In this energy range all ionization channels are closed,but to obtain agrand reaction matrix K which varies weakly with the energy the v+ = 3, N + = 1,3channels were artificially opened in this initial stage of the calculation (as in Jungen, 1984).In terms of the resulting 4 X 4 reaction matrix the predissociation can now be calculated in a straightforward manner using the general methods of Section 11. The

Chris H . Greene and Ch. Jungen

96 0.8L

\

-

f 0.6-

v)

W

I

a

z

-

0.4-

2 0.2W

t FIG.17. Predissociation in H1.Calculated vibrational eigenphasesum plotted as a function of the energy near the 3pn D Il:, u = 3, J = 2 level. physical boundary conditions at infinity must now be applied; i.e., in the present example the ionization channels v+ = 3, N + = 1 and 3 are at this point treated as closed channels while the dissociation channels 3 Z and 2 ll are open. Figure 17 presents the resulting variation of the eigenphase sum as a function of energy near the position of the 3pU, v = 3, J = 2 level. A dispersion pattern is obtained which arises from the predissociation by the 3pZ continuum. The resonance center and the predissociation width can be determined from it easily: we see that the calculated width is in almost perfect agreement with the measurement of Glass-Maujean et al. (1979). The resonance position is also very well reproduced by the calculation: Ecalc= 119318.3 cm-l as compared with EOb,= 119320.5 cm-I (Takezawa, 1970). Jungen (1984) has applied this approach to the more complicated situation of competition between predissociation and preionization occurring above the v+ = 1 ionization threshold. Some examplesare shown in Fig. 8 and have already been discussed. It is worth mentioning that, unlike most R-matrix calculations, the bulk of the computational effort in the present formulation is not spent in finding variational approximations to eigenfunctions of the Hamiltonian at the desired energy- these are already given to good accuracy in terms of the input function pA(R)through the vibrational frame transformation. Rather the only calculation required here is the determination of appropriate linear combinations of these eigensolutions which obey the consistency requirement [Eqs. (58), (59)] between the electronic phase shifts zp and the vibrational phase parameter z (and which obey the boundedness requirement at r 00). Surprisingly, the information contained in pA(R)alone, together

-

QUANTUM DEFECT THEORY

97

with the preceding discussion, suffices even to describe an inelastic atomic collision process.

IV. Electronic Interactions at Short Range The discussion of the preceding section has been limited to rovibrational channel interactions since it was assumed that the physically relevant fragmentation channels always correspondto the same electronic state of the ion core. We proceed now to the discussion of processes whereby conversion of electronic core energy plays a role in addition to the conversion of rovibrational energy already discussed. In Section IV,A we deal with the formal aspects of the problem. A. THEORETICAL DEVELOPMENTS 1. Rovibronic Channel Interactions

The most straightforwardmethod of including electronic channel interactions starts out from the rovibrational treatment of Section 111. The fragmentation channel labels for ionization are now (i) = (k, u+, N + ) , where k stands for both the electronic state of the ion core and the orbital angular momentum of the distant excited electron, neglecting all spin effects. The complementary eigenchannel labels are (a)= ( p , R,A). The electronic eigenchannel representation /3 may not have any simple physical interpretation; it must be determined from experiment or calculated ab initio for given R and A. The frame transformation matrix U, of Section III,A [Eq. (33)] is now complemented by an electronic factor, giving

u. = ukv+N+,BRA = (~~~)'"'(v'IR)'~"'(N'(A)'~~)

(70) This factorization isjustified by the fact that each factor results from integration over different variables (electronic, vibrational, rotational). Both the eigenchannel representationa of the full system and the relevant levelsof the free core are assumed here to conform to the Born - Oppenheimer approximation. [For diatomic ion cores this is usually a good approximation for the lowest few electronic states, but in polyatomic ions even the electronic ground state may depart significantly and systematically from the BornOppenheimer approximation if an electronic angular momentum is present (Jahn -Teller effect, Renner-Teller effect). The present treatment would

Chris H . Creene and Ch. Jungen

98

require some extension in order to include such situations.] The reaction matrix including electronic interactions now takes the form

Similarly the matrices C,,, and Sii,of Eq. (40) in the preceding section are obtained in generalized form by replacing the tangent function in Eq. (7 1) by cosine and sine functions, respectively. Correspondingly, in the formulation of Eqs. ( 17) and ( 18), the matrices 43, and S, are, e.g., 43. .C

kv+N+, BRA

= (klp)(RA)(U+IR)(kN+)(N+I~)(kJ) cos nppA(R) (72)

and similarly for S,. If nuclear motion is disregarded entirely, Eqs. (70)-(72) define a purely electronic multichannel problem forfixed R and A with frame-transformation elements

u,

,y(RN=

kg

(k

(73)

and a corresponding body-frame reaction matrix

These simplified expressions provide the link with atomic quantum defect theory and they involve the electronic quantities-the reaction matrix or alternatively its eigenphases and vectors- which can in principle be obtained in an electronic ab initio calculation for fixed geometry. For photoabsorption processes the required dipole matrix elements of Eq. (4 1) now have the structure d,,(R). For negative energy the solutions of the generalized eigenvalue problem defined by Eqs. (23)-(25) with U and p from Eqs. (73) and (74) yield the Born- Oppenheimer potential energy curves relevant to a given problem. Unlike in the preceding section, these curves are no longer given by a set of R-dependent Rydberg equations of the type of Eq. (35)with nearly energyor n-independent quantum defects. Rather, strong n dependences will occur owing to the electronic interchannel interactions, and in a potential curve plot of the type shown in Fig. 3a numerous avoidedcrossings between curves associated with different electronic core states may occur. An example is the gerade electronic symmetry ofthe H, molecule [cf. Fig. 1 of Wolniewicz and Dressler (1977)]. We note two further features of the rovibronic treatment just outlined.

99

QUANTUM DEFECT THEORY

First, by their definition the electronic transformation elements include the projections of several pure 1 states on a given eigenstate j?. In other words, 1 mixing, mentioned in Section III,A, is in principle taken into account here. Second, the flexibility in the choice of the vibrational basis used to represent the channel mixing coefficientsABRhcorresponding to Eq. (36) persists in the present extended treatment. One can alternatively use the vibrational wave functions of the different ion states or, if there are strong avoided crossings between potential curves of the neutral molecule, the vibrational wave functions corresponding to the adiabatic curves might be useful. [See the discussion following Eq. (36).] The treatment of rovibronic channel interactions sketched here has been outlined first by Jungen and Atabek (1977) and later by Raseev and Le Rouzo (1983). It has to date not been implemented. Instead another approach, due to Giusti ( 1980), has proved very useful in situations where the electronic channel interactions are relatively weak and lend themselves to a perturbation treatment. In addition, this approach can be extended very easily to treat dissociation processes involving an electron core rearrangement.

2. Two-step Treatment of Electronic Channel Interactions Giusti starts out by considering that the Hamiltonian ofthe system can be conveniently partitioned into two parts, H = Ho V Ho includes the longrange fields and those short-range interactions confined to a single electronic channel. It is useful to further indicate the separate short- and long-range contributions to Ho separately through Ho = T Vo= T VLR ( Vo- VLR), where V L R is the long-range potential (e.g., the Coulomb potential for photoionization of neutral systems). Residual interactions between electronic channels specified, e.g., in an independent-electron model, are described by the short-range operator V. The approach will be most useful when V is small. In the first step of the treatment only Ho is taken into account. The resulting total multichannel wave function t,do)can be written exactly as in Section 11, as

+

+

v(o)(E)=

x Y

+

+

W;O’(aAy(E)

Here the eigenchannel label y has been used instead of a to indicate that the influence of V has still been omitted. The summation over fragmentation

100

Chris H . Greene and Ch. Jungen

channels i is meant to include alternative rovibrational as well as electronic core states as in Section IV,A, 1. Giusti proceeds now by remarking that each factor ( A cos xpy - gi sin zpy) can be regarded as a new, phase-shifted basis functionf;(r, y ) in the longrange field, which may serve in the second step of the treatment. The linearly independent conjugate basis function g,(r, y), which l a g s j by 90", is cos lccLy +A@) sin w Equation (75) can thus be rewritten as 7) =

Fi(rY

W'O)(E)=

c Y

y

(76)

w$'(E24,(E)

The main effect of the residual interaction Vis to modify the wave function at large distances ras follows. Each factorj(r, y)A,(E)in Eq. (77) must now be replaced by a sum over degenerate eigenstates(I!of the complete Hamiltonian with an additional phase shift in each channel,

2 U$:)[j(r,Y) cos npLV' - a r , 7) sin npLV)1AII(E)

(78)

(I

Here the npLV)are the additional phase shifts due to V, and the matrix U$V,) transforms the eigenstates of Ho into those of the full Hamiltonian H. After insertion of Eq. (78) into Eq. (77) and using the full expressions forfand g, the following wave function w results (for r 2 ro): with

§ , =

2 Uiysin n ( p y+ pLV))u::) Y

This wave function has exactly the same form as the function, Eq. (79, of the one-step treatment except that each matrix element C,(I is no longer equal to Uiacos npa but involves the product of two successive unitary matrices, Uiy and UiV,),and the sum of eigenphases nhYand npf) arising from ( Vo - VLR) and V, respectively. The net effect of the short-range interactions contained can be equivalently expressed with the following reaction matrices

101

QUANTUM DEFECT THEORY

If these two reaction matrices are known at the outset, diagonalization yields their eigenvectors and the eigenphases so that the linear system, Eq. (17), can be set up with the matrices S, and C, evaluated with Eq. (80).2 The advantage of the two-step method in comparison with the approach outlined in Section IV,A, 1 is that, by including most of the interactions in Ho , the residual term V can be treated perturbatively. Electronic channel interactions leading to electronic preionization may serve as a first illustration.

3. Interactions between Ionization Channels This application of the two-step approach has been introduced by GiustiSuzor and Lefebvre-Brion ( 1980);we present it here with a slight change of emphasis. In the first step rovibrationalinteractions within a given molecular ionization channel are included, and the mixing of partial waves 1 by the noncentral core field is also taken into account. The fragmentationchannels are labeled ( i ) = (kv+N+)as in Section IV,A, 1 and the eigenchannels of Ho are labeled (7) = (BRA).The transformation matrix U,, has elements U,, = (klp)(RA)(u+lR)(kN+)(N+lA)(~') (82) in exact analogy to Eq. (70). The difference is that the electronic eigenstatesp here are not meant to diagonalizethe full electronic Hamiltonian but rather correspond to the motion of an electron in the field of a core frozen into a given electronic state. Quite a number of ab initio computer programs are available today which furnish numerical solutions of exactly this problem One may ask at this point how the matrices K and K(") of Eq. (81) relevant to a given problem are related to the matrix KCtOUl)describing the same problem in a one-step approach such as outlined in Section IV,A,l [Eq. (71)]. It is in fact simpler to derive a complex matrix S(Ww) = (1 iK(loul))/(l- iK(mw))rather than K('OW)itself. By forming the complex Jost-type matrices (see, e.g., Newton, 1966) with elements j g = C, isi, = U, exp(i inp,) we find that &. = U, exp(2inp,)(UT),. = j&(j-)ii!

+

*

x U

(I

in a one-step treatment. In a two-step approach the matricesj* become with Eq. (80) j&=

2 Uiyexp[fin(py+ p!,"))]Ui:) Y

so that the complex "short-range'' scattering matrix S(low)which combines both steps into one

has elements

$:PW)

=

c (xuiyexp(inp,)u:) u

w'

)

exp(2inp;~)(~)~),.exp(inp?,)(v)y,i,

It can be easily verified that this matrix is unitary and therefore that K(mw)is symmetric as it should be.

102

Chris H. Greene and Ch. Jungen

(e.g., Dill and Dehmer, 1974;Dehmer and Dill, 1979;Lucchese and McKoy, 1981; Raseev, 1980; Schneider and Collins, 1981 ; Burke ef al., 1983). In the second step the coupling Vbetween electronic channels associated with different cores is introduced. It arises then from the interelectronic repulsion l/ru. If the interaction is weak the matrix elements of the reaction matrix Kc?, Eq. (81), can be evaluated in good approximation as a matrix elemeny of V over the core region deriving from the LippmannSchwinger equation in first order (see Giusti, 1980, for a comprehensive discussion). Thus KpRAd,R,A, (V) -n(i/F)l Vli/$’) !‘(!I?) d(R - R’)dAAt (83) 88

The electronic interaction matrix V$!(R)is dimensionless, as can be verified from the dimensions of y / ( O ) and V As a quantity resulting from electronic short-range interactions, V$!(R) will generally vary slowly with excitation energy. Its eigenvaluestan npLT(R)and eigenvectors(Pl(~)(~”)are needed for each value of R and A. The labels of the eigenchannels are then (aRA),and the frame transformation for the second step has the form

(PRAlaR’A’)= ( p l ~ ~ )d(R ‘ ~ ”-) R ’ ) d A A ,

(84)

The analog to the matrices 43, of Eq. (72) therefore becomes

Q)k,+N+, nRA = (N+Il\)‘kJ’(U+IR)‘kN”

By comparing Eqs. (72) and (85) we see that the one- and two-step approaches differ in the electronic factor. In Section IV,B below we describe the application of this formulation to electronic preionization in N,. Section IV,C shows how the approach has been extended to dissociativeprocesses in NO and in H, for the case of weak coupling. B. ELECTRONIC PREIONIZATION I N MOLECULAR NITROGEN The preceding formulation has first been used by Lefebvre-Bnon and Giusti-Suzor (1983) for model calculations dealing with the interplay between electronic and vibrational preionization. These authors showed that, as in the case of rovibrational preionization (cf. Section III), a clear distinction between the two processes is often impossible so that one should really speak of “vibronic” preionization. Morin el al. ( 1982) applied the two-step approach to the photoionization of molecular oxygen. They presented an

QUANTUM DEFECT THEORY

103

empirical interpretation of a complicated section in the partial cross sections for vibrational excitation where electronic preionization dominates. Quite recently Raoult et al. (1983) have published a detailed ab initio study of electronic preionization in the Hopfield series of molecular nitrogen [seealso the subsequent paper by Le Rouzo and Raoult (1989, who have presented improved calculations]. We shall now discuss these results in some detail. Numerous Rydberg series are known in N, converging to the X, A, and B states of N2+.The most strikinglybeautiful of these are the so-called Hopfield series (Hopfield, 1930a,b) (Fig. 18) which appear above the N2+A ,nu threshold as regular broad preionization features converging towards the B 22: limit of Nz+. The regularity of the series (i.e., the absence of overlapping vibrational progressions and channel perturbations) no doubt arises because only v+ = 0 is excited in the B ,Stcore owing to the similarity of the potential energy curves of the N, Xand N,+ B states, and because vibrational preionization is not strong. Besides their regularity, the Hopfield series exhibit striking preionization shapes: There is one series which appears with absorption peaks, while another consists of window resonances. In order to account for these observations a multichannel treatment ofthe Rydberg series associated with the B state of N,+ as well as of the continua associated with the lower-lying X and A states must be carried out. In photoabsorption from the X '2: ground state of N,, 'Z: and 'nuelectronic states can be excited. Therefore the channels (PRA)of Section IV,A,3 are superpositions of the following partial waves 1 G 3:

:

A = '2

X 22icore: pa,, fau

(Worley- Jenkins series)

A 21"Iu core: dng

A = 'nu

B 'Z: core: sag,dog

(Hopfield series)

X ,2: core: pn,, fir,

(Worley- Jenkins series)

A

2nucore:

B ,2,, core:

sogB, dog,dS, dirg

(Hopfield series)

The spin variables have for the most part been excluded here: Strictly, the ionization channels should be labeled using a coupling scheme (Q, w ) analogous to ( j ,j ) coupling in atoms, and the triplet states should thus be included in the PRA representation used in the above table. The neglect of these is justified by the relative smallness of spin-orbit coupling (= 80 cm-l in the A state of N2+).Raoult et al. have disregarded molecular rotation and vibration altogether and evaluated the quantum defects pjA(R), transformation elements (kl/3)("^),and dipole amplitudes d,&) separately for each molecular and for the internuclear distance R = Re corsymmetry, A = 5 and 'nu,

:

$1

a

0 0 I-

0

I

a w

1

I-

a _I

w

a I

FIG.18 Relative photoionization cross section for N, X 'El, U" = 0 taken at a temperatureof 78 K. The spectrum shown correspondsto the range between the A 211uand B 2E: thresholds of N,+. (After Dehmer et al., 1984.)

QUANTUM DEFECT THEORY

105

responding to equilibrium in the N, ground state. The calculations were done with the ab initio code of Raseev (1980) which is based on the singlecenter frozen-core static exchange approximation. Partial waves up to 1 = 14 had to be included in order to obtain convergence in the core region; however, it was found that outside the core only values l d 3 contribute significantly, and the dimension of the matrix K in Eq. (8 1) is in fact considerably and to N = 5 for ‘Z:. reduced, to N = 6 for ‘nu This first step of the ab initiowork shows that strong s-dmixing occurs in the channels associated with both the A and B states of N2+,whilep-fmixing is negligible in the energy range of interest here. [At much higher energy the occurs (Plummer et well-known shape resonance associated with N2+X al., 1977; Hamnet et al., 1976), which involves strong mixing of the p andf partial waves.] For example, the transformation matrix (kip) of Eq. (82) for so and do electrons associated with the B ,Z states is calculated by Raoult et af. to have elements

: k

P

B2X:, su

B 2 X i , da

B *Xi,“s”a B lX:, “d”a

0.86 0.52

-0.52 0.86

Following these authors “I” is used here to denote the partial wave with largest amplitude I in the p representation. In the second step the residual electronic interaction V s ! ( R )is calculated by evaluating the required two-electron integrals with a computer program written by Le Rouzo (unpublished). None of these interactions exceeds 0.1, and the first-order approximation, Eq. (83), is thus justified a posteriori. The resulting additional phase shifts n,uL? and eigenvectors (pla)then serve to construct the matrices S, and 43, according to Eq. ( 8 5 ) and to set up the linear system of Eq. (1 7). The results thus obtained by Raoult et al. and Le Rouzo and Raoult with their fully ab initio approach include predictions of total and partial cross sections for photoionization, the photoelectron angular distributions, and the polarization of the fluorescence from the N,+ B 2Z: state formed in photoionization. As an example Fig. 19 compares a section of the calculated total photoionization cross section with the corresponding experimental curve recorded by Dehmer et al. ( 1984). Since these authors measured only the relative ionization cross section, it had to be calibrated with the absolute absorption cross section curve of Giirtler et al. ( 1977)assuming that photodissociation is negligible. Figure 19 shows that the observed and calculated curves are in quantitative agreement inasmuch as the continuum back-

Chris H . Greene and Ch. Jungen

106

M

20

t

t

ol-

1

670

L--1-

680

I 1

682

PHOTON WAVELENGTH

(%I

684

FIG.19. Electronic preionization in N,. Calculated (top, Le Rouzo and Raoult, 1985) and observed (bottom, Dehmer et al.. 1984) total ionization cross section near then = 5 membersof the Hopfield Rydberg series.

ground is concerned, and in semiquantitative agreement inasmuch as resonance positions, widths, and shapes are concerned. Thus Raoult et al. have been able to conclude that the Hopfield “absorption” series corresponds to excitation of a quasibound “d”a Rydberg electron while the “emission” series is due to a “ d ’ k and “s”o electron. These resonances appear either because the preionized state has a high absorption intensity (“d”a) or because the preionized state is strongly coupled to an intense continum (“d”n, “s”~). It must be stressed that, in order to be able to interpret the structure of the Hopfield series, a full treatment of the ionization process including electronic preionization has been necessary. C. PHOTODISSOCIATION AND DISSOCIATIVE RECOMBINATION 1. Theoretical Development

The two-step approach provides also a convenient description of dissociation processes which enter into competition with ionization (Giusti, 1980; Giusti-Suzor and Jungen, 1984). Figure 20 illustrates this with the help of a potential energy diagram. Shown are a set of Rydberg potential energy curves converging towards one ionic curve, plus a set of valence state (“nonRydberg”) curves which cross the former at numerous points. In the spirit of

107

QUANTUM DEFECT THEORY

I’

I

90

I

I

-

-n -- - - - -

1.P -

I 7

R (a u.)

FIG. 20. Diabatic potential energy curves of *lIstates in NO. The arrows indicate the avoided crossings occumng in the adiabatic curves as a result of Rydberg-valence-state interactions (cf. the inset). Three energy ranges are marked which correspond to the zones of I: spectral perturbations; 11: predissociation; 111: competition between predissociation and preionization. (After Gallusser and Dressler, 1982.)

quantum defect theory the Rydberg curves are regarded as representing the closed portion of a molecular ionization channel. By analogy each valence state can be viewed as representing a dissociation channel: The vibrational continuum above the dissociation limit represents its open portion while the bound vibrational spectrum corresponds to the closed portion. The situation depicted in Fig. 20 is different from that encountered in Section 111,D,2, since here dissociation proceeds through a valence state external to the Rydberg series, whose electronic structure diflers from that o f the Rydberg

108

Chris H . Greene and Ch. Jungen

states. In other words, dissociation in the present case involves electronic rearrangement in the core, while in the situation discussed previously it does not. Electronic couplings between the Rydberg manifold and the valence states are indicated in the inset of Fig. 20 by avoided crossings: These interactions produce spectral perturbations in the discrete part of the spectrum; they lead to electronic predissociation of Rydberg levels situated above the dissociation limit. Above the ionization potential there will be a competition between dissociation and ionization processes. In order to apply the general approach of Section IV,A,2 to this problem we must first define the fragmentation channel labels. The Rydberg manifold corresponds to channels ( i } = (u+N+)if we restrict the treatment to a single electronic core state and a single partial wave 1. The valence states are labeled (dA};the “core” wave function here is the electronic valence state wave function for all electrons, while the radial functionsfand g for relative motion of the fragments are just the regular and irregular vibrational wave functions FLA)(R)and GLA)(R)in each valence state potential. The “core” region or “reaction” zone extends in its electronic dimension to the edge of the ion core, and in its nuclear dimension to the radius& somewhat beyond the outermost curve crossing (see the figure). The first step of the treatment is analogous to the theory of Section 111,A: The rovibrational interactions within the ionization channel are taken into account while the dissociation channel wave functions are still FLA)(R).Thus we have labels (7) = (RAYdA), Next the Rydberg-valence-state interaction Vmust be introduced. The first-order Lippmann - Schwinger equation analogous to Eq. (83) is now an integral over all electron coordinates and the nuclear coordinate, extending throughout the reaction zone. However, since the Rth Rydberg vibrational eigenchannel function is essentially6(R - R ’), we are left with an electronic integral for each value of R , multiplied by the regular valence-state continuum wave function F,(R). Thus the nonzero elements of K(y) become p) 88’ E K (RA,&‘ V) = -n V y ( R ) F y ( R ) (86) Note that, unlike in electronic preionization (Section IV,A), the dimension of the electronic interaction VLA)is (energy)’l2,while Flf)(R)has dimension (energy X length)-’/*, whereby the element K E , is not dimensionless. As a quantity resulting from electronic short-range interactions, V$*)(R)is expected to vary slowly with excitation energy. By contrast, the matrix K(”wil1 vary on the scale of a vibrational energy quantum owing to the factor F,,(R), and it must therefore be evaluated on a sufficiently narrow energy mesh. A complication in practice is the partially continuous character of the basis set (7) = (RAYdA). This difficulty had already been encountered in Section II1,A and the remedy here is similar: Instead of expressing the matrix K(“) in

QUANTUM DEFECT THEORY

109

the representation (7) we must express it in the fragmentationchannel representation ( i ) . We then obtain (see Giusti-Suzor and Jungen, 1984)

K ("+N+, v) "+IN+' Using this reaction matrix one can then proceed with the general method outlined in Section 11. The remaining problem in practice is how to obtain the electronic interaction VLA)(R).One way would be to fit VLA)directly to experimental data. Another way consists of relating VbA) to the size of avoided crossings seen in Fig. 20. From the construction of the first-order expression of the matrix K(v), it can be seen that for small V$*)and low principal quantum numbers n (for which successiveRydberg curves are still well separated) the magnitude of the avoided crossings, 2 V S ) ,will be approximately equal to twice the interaction VLA)multiplied by the discrete Rydberg normalization factor for the nth level. Thus we may put V z ) ( R ) [n - pA(R)]-3'2V(dh)(R) (88) The energies V S )are the quantities entering vibronic perturbation calculations of the type often performed in molecular spectroscopy. A final remark concerns the vibrational wave functions F(R) and G(R). The regular function F is required for the evaluation of the first-order reaction matrix K(") in the second step of the calculation. Unlike the Coulomb functions which are known analytically, F must be calculated numerically. For positive energies this can be done using any standard integration method, while for negative energies the Milne approach can be used (see Greene et al., 1983), which immediately defines then G(R)as well. This approach yields also the negative energy phase parameter which is no longer given simply by A ( V - I) as for a Coulomb field.

2. Application to Competing Dissociation and Ionization Processes in NO Giusti-Suzor and Jungen (1984) have applied this approach to the calculation of predissociation and preionization in the threshold region of nitric oxide. The idea underlying their work was to use the very detailed information available on Rydberg- valence-stateinteractions in the discrete region in order to interpret the decay processes observed at higher energy. Precise data concerning level perturbations in the discrete range stem mainly from the work of Miescher and collaborators (see Miescher and Huber, 1976), who performed extensive analyses of the high-resolution absorption spectra

QUANTUM DEFECT THEORY

111

of several isotopes of NO. The potential energy curve diagram shown in Fig. 20 refers to the Rydberg and valence states of NO having 211 symmetry. Gallusser and Dressler ( 1982)have given a quantitative representation ofthe observed vibronic level positions and intensities in the states shown in the figure, in terms of a set of “deperturbed” potential energy curves and transition moments, and of electronic Rydberg - valence-state interaction energies V:). Their data furnish directly the main parameters required for the implementation of the approach of Section IV,C, I . Figure 1 compares the medium-resolution photoabsorption and photoionization spectra (Miescher et al., 1978;Ono ef al., 1980)at higher energy near the ionization threshold. The assignment of npn, u+ Rydberg peaks derives from the work of Miescher (1976) and is based on the spectra of several isotopes obtained under much higher resolution than the spectrogram shown in the figure. The present recording shows that these peaks dominate the absorption spectrum in this range. (A careful analysis indicates that thepn ionization channel cames nearly half ofthe total absorption cross section in this range.) The high-resolution spectrum shows also that the individual rotational lines are diffuse, with widths that vary strongly from one band to another, but this is not visible in Fig. 1 except for the 5pa, u = 3 peak whose width exceeds the rotational band structure. The ionization cross section is strikingly different: It is essentially flat, with steps occumng near each vibrational threshold u+. Resonances are visible only as fairly weak “wiggles” superimposed onto the staircase structure. Obviously the difference between the two spectra must correspond to photodissociation, i.e., to the cross section for formation of the atomic fragments, mainly, N(2D) O(3P).Note that this difference cannot be taken directly from the figure since the two curves shown are on relative scales which in fact turn out to be very different. The fact that all resonance features appear only weakly in ionization indicates thatpredissociafion plays a dominant role. In the case of the npn, v+ resonances the strong Rydberg state crossings with the B 211 and L 211 valence states furnish the mechanism for predissociation which first comes to mind. Figure 2 1 presents two sections of the spectra of Fig. 1 on an enlarged scale. Drawn also are theoretical curves resulting from the two-step approach

+

..

FIG.2 I . Two sections ofthe photoionization (top) and photoabsorptionspectrum (bottom) of NO (Fig. I ) in the region between the v+ = 0 and 1 (right) and v+ = 1 and 2 (left) ionization thresholds. The observed and calculated spectra are represented by full and broken lines, respectively.Note the difference between the cross-sectionscales in the top and bottom spectra. The calculated spectrum contains only pn resonances, but accounts for the background due to all partial waves II. (After Giusti-Suzor and Jungen, 1984.)

112

Chris H . Greene and Ch. Jungen

outlined in the preceding subsection, implemented with the parameters (in particular interaction energies V , and the R dependence of the quantum defect) taken from Gallusser and Dressler’s work. Molecular rotation has been disregarded in the initial multichannel calculations, its neglect being justified by the basically vibronic origin of preionization and predissociation in NO, and because below n = 10 1 uncoupling in the pn Rydberg series is weak. In order to draw the theoretical contours of Fig. 2 1 Giusti-Suzor and Jungen convoluted each calculated resonance feature with the rotational band structure reflecting the Boltzmann distribution present in the NO sample and the experimental resolution width. Even after this convolution the height of each peak npn, v+ depends sensitively on the level width as well as on the intensity associated with the peak. The ionization signal in turn depends sensitively on the characteristics of the competition between preionization and predissociation. The calculated curves shown in the figure do of course not include resonances other than pn. For these, however, the overall agreement between experiment and theory is quite striking, showing that the interplay between dissociation and ionization processes is on the whole correctly calculated. It is instructive to discuss the competition between the two processes with reference to the Au = 1 selection rule mentioned in Section II1,C. If this rule could be applied directly to the processes rather than only to the channel couplings,one would be tempted to conclude that only Av = 1 resonances (if any) can appear in photoionization, while IAvl> 1 resonances should be absent, i.e., appear only in photodissociation. An early analysis of the photoionization spectrum of NO (Ng et al., 1976), in contradiction with Miescher’s work, had indeed been based on this assumption. It can be seen, however, in Fig. 21 that lAol= 1, 2, and 3 resonances are observed and calculated to have comparable peak heights in the photoionization cross section. Following Giusti-Suzor and Jungen this must be attributed to an indirect process, whereby a bound Rydberg level becomes coupled to the ionization continuum via the valence-state dissociation channel. In other words, the strong electronic interaction induces preionization even where the direct coupling is extremely weak. One might also term this a continuum - continuum interaction which interferes with the discrete continuum interactions. According to the present analysis, the characteristic staircase pattern of the ionization curve of NO- one of the classic examples of this type-is also a consequence of the predominance of electronic predissociation over vibrational preionization. Unlike in Fig. 10, where no step was apparent at the o+ = 4 ionization threshold, the large step at the v+ = 1 threshold in Fig. la reflects the strong influence of predissociation, which affects the resonances far more than the direct continuum background.

QUANTUM DEFECT THEORY

113

3. Dissociative Recombination Giusti-Suzor et al. ( 1983)have applied the same approach to dissociative recombination in low-energy e-H2+ collisions. This is in turn an instance where quantum defect theory has been combinedwith ab initio theory which provided the dynamical parameters for the problem, namely quantum defects, valence-state potential energy curves, and Rydberg- valence-state interaction energies. Unlike photoabsorption in H2, which restricts the excitation to the manifold of singlet ungerade electronic states, dissociative recombination may involve singlet as well as triplet, gerade as well as ungerade, compound electronic states; in addition all partial waves 1 are present in the entrance channel. Giusti-Suzor et al. selected for their calculations those channels which appeared to be the most important for the process studied. Thus, high-1 states were omitted since their small amplitude in the reaction zone implies they are unlikely to provide an efficient reaction path. Ungerade electronicchannels were disregarded since they exhibit no avoided crossings, as shown by Fig. 3a, and the nonadiabatic mechanism discussed in Section III,D,2 is relatively weak. Triplet channels were ruled out on the basis of the same argument. Among the singlet gerade states it is only the manifold of 'Zlpotential curveswhich exhibitsavoided crossingsqualitativelysimilar to those shown in Fig. 20 but with considerably stronger electronic interactions. Among those the sagRydberg channel was omitted, being presumed unimportant, and s-d mixing was also neglected along with molecular rotation. The channels of '2: symmetry which were finally retained have dominant electronic configurations ionization (v+):

(laJ(~daJ

dissociation ( d ) :

(la,)Z

The dynamic parameters were taken from a separate study of Hazi et al.

( 1983).Thus the diabatic ( 1aJ2valence-state curve was taken in part directly

from the ab initio scattering calculations of Takagi and Nakamura (1980) and in part extracted from a deperturbation analysis performed on the very accurate adiabatic (noncrossing) '2: curves of Wolniewicz and Dressler ( 1977, 1979).This deperturbationcalculation yielded also the R dependence of the quantum defect, p..c'T(R).Finally, the electronic interaction Vd including its R dependence was obtained by the Stieltjes moment method. The calculationswere carried out in terms of the resulting parameters in a way exactly analogousto the NO example,except that, instead ofthe incoming wave boundary condition of Section II,D, the outgoing wave boundary

Chris H . Greene und Ch. Jungen

114

conditions discussed in Section II,B were employed (and of course the dipole transition moments were not required). Figure 22a presents an example of the results obtained by Giusti-Suzor et al. We notice, first, that the background cross section decreases nearly in proportion to E-’ as is expected since the background parts of the K and S matrices are nearly energy independent on this scale so that only the factor E-’ from the general cross-section formula remains. Second, all Rydberg resonances which are supenmposed on the background are window resonances. The reason is related to the weakness of the R dependence of the quantum defect in this example, as can be appreciated as follows: Electron capture constitutes the first step of the dissociative process and hence plays a role analogous to photoexcitation in all previous examples. Near threshold the population of the Rydberg levels n“d”a, v+ > 0 by electron capture is governed by their vibrational coupling to the entrance channel “ d ” q u+ = 0, and this turns out to be quite weak owing to a weak R dependence of p near equilibrium. The situation is then

L

-

0. 0 5

ENERGY ( s V

1

0.1

0.2

0.5

FIG. 22. Cross sections for dissociative recombination of electrons with Hz+ions (after Giusti-Suzor el a/., 1983). (a) Solid curve: calculated cross section for the u+ = 0 state of H2+. Dips are due to ndu, u Rydberg states, with the values of n and u indicated for the most prominent dips. Dashed curve: result obtained when all closed channels are omitted. (b) Cross section for e-H,+ recombination with a mixture of ions in the u+ = 0, I , and 2 states: solid curve, calculation; dashed curve, experiment (Auerbach et a/., 1977).

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analogous to that in which a closed Rydberg channel has a nearly vanishing optical transition moment and is embedded in a strong continuum: It is well known that window resonances appear under these circumstances (Fano, 196 1). In turn, the large widths of the resonances seen in Fig. 22a are due mainly to predissociation. In an experiment, H2+levels with v+ > 0 are also present, and for the comparison of theory with the observations an appropriate convolution of the calculated cross sections must be made which takes account also of the experimental resolution. Figure 22b presents such a convoluted cross-section curve, designed for the comparison with the merged-beam experiment of Auerbach et af.(1 977) (also shown). It can be seen that the two curves are quite similar, although neither the resonance positions nor the absolute cross-section values do quite coincide.

V. Discussion and Conclusions There appears to be some misconception about multichannel quantum defect theory being primarily “empirical” since it is based on parameters which supposedly cannot be obtained directly by ab initio computational schemes. Much in the previous sections has been designed to show that this is not the case. First of all, electron-ion scattering calculations (such as, e.g., those of Takagi and Nakamura, 1980, 1983) yield phase shifts and possibly transition moments for fixed molecular symmetry and geometry which are exactly of the type used in molecular quantum defect theory. The most recent ab initio study of doubly excited states in H, has been performed by Raseev (1985). It is true that no ab initio calculations appear to have been camed out as yet at negative energy which would yield directly quantum defects, although this is now routinely done in atomic calculations (Lee, 1974a; Greene, 1983; O’Mahony and Greene, 1985). Instead quantum defect curves have been extracted from potential energy curves calculated by quantum chemical methods as discussed in Sections 111 and IV,C. In electronic multichannel situations this procedure would become less straightforward (cf. Section IV,A,l), but there appears to be no conceptual problem. Obviously ab initio methods of the R matrix type, where the electron structure calculation is restricted to a finite volume encompassing the reaction zone, complement the quantum defect theory of the outer zone in the most natural way. But fixed-geometry ab initio results of any type can be combined with rovibronic multichannel calculations as the examples discussed here show. Reliable ab initio computations no doubt will play an ever-increasing role in the future in the analysis ofexperimental

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data relating to molecular decay processes. As an example, the analysis of the Hopfield series of N2reviewed in Section IV,B requires knowledge of, in all, 47 dynamical parameters all of which have been calculated ab initio: It is difficult to envisage how these could have been obtained empirically from experiment given the fact that molecular rotational structure is virtually washed out at these energies by the fast decay processes. Applications of quantum defect theory to polyatomic molecules have been scarce thus far. Dagata et al. (1981) have published a Lu-Fano plot pertaining to the sand d series in CH,I convergingtowards the two spin states of CH,I+. Their plot has been derived from a high-resolution absorption spectrum and resembles very closely the correspondingplot for the Xe atom, as anticipated by the single-channel analysis of Wang et al. ( 1977). Apparently vibrational motion does not completely alter the channel interactions in this case, as a remarkable similarity between the ionic fine-structure effects in the atom and in the polyatomic molecule emerged. Fano and Lu (1 984) have sketched the application of quantum defect theory to the Rydberg molecule H, . In a different context Golubkov and Ivanov (1984b) have applied quantum defect methods to the problem of slow collisions between an atom and a diatomic molecule. By taking into account simultaneously the ion core and the perturber atom, they obtained an expression for the potential surfaces of the quasimoleculeXf A. In a series of papers (Golubkov and Ivanov, 1981,1984a; Ivanov and Golubkov, 1984)a parallel reformulation of the essentials of quantum defect theory has been developed in terms of integral equations. Using a Lippman - Schwinger equation starting from the Coulomb Green’s function, integral equations for the transition matrix have been derived. The connection with the approach of Sections I1 and 111 has not been explicitly determined, and few applications of this approach have been published. This review has concentrated almost excusively on the use ofthe quantum defect method for studying photoabsorption by neutral molecules or electron scattering by positive ions. Closely related work has likewise been accomplished in parallel for the problem of electron scattering by neutrals and also for the photodetachment of molecular negative ions. For example, it was for electron- neutral scattering that the vibrational frame transformation was introduced by Chang and Fano ( 1972).An extensive review by Lane ( 1980)points out many such applications. Chang ( 1984)has recently considered the extension of frame transformation theory to polyatomic molecules. An important difference in the physics of negative ions is the absence of a long-range Coulomb potential between the outermost electron and the core. This fact alone need not invalidate the use of quantum defect theory as described in Section 11, but it implies that the outermost electron moves far more slowly in the vicinity of r = r, than it would in the presence of a charged core. Accordingly the fixed-nuclei approximation may be inappropriate

+

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even within r C r,. Some difficultieswith the frame-transformationmethod have in fact been noticed (Momson et al., 1984;Jerjian and Henry, 1985),in numerical studies of near-threshold vibrational excitation of H2by electron impact. Nevertheless the problem of resonant vibrational excitation (e.g., as in electron scattering by N2)seems to be well understood followingthe work of Herzenberg (1968) and of Schneider et al. (1979). Electron scattering by neutral polar molecules has received much attention in recent years (see particularly Norcross and Collins, 1982, and references therein). One conceptual problem is posed by an electron in the field of a polar molecule which was absent when the Coulomb attraction dominates. That is, the r2dipolar potential experienced by a distant electron in the body frame becomes converted into an r4polarization potential once the electron escapes so far that the molecule is left in a definite rotational level. Analytical studies close in spirit to quantum defect theory (Clark, 1977, 1979, 1984; Engelking, 1982)and also some numerical studies (Collins and Norcross, 1978)have sorted out some of the effects of this change of frames, but a comprehensive theory still seems to be lacking. Another area of electron - neutral scattering theory which has been explored in depth is dissociativeattachment (e.g., e- HC1+ H C1-). This is one of the simplest processes in which “electronic” kinetic energy can be converted into nuclear motion, and as such it has fundamental chemical significance. Here much progress has been made by Gauyacq (1983), who treats separately the short- and long-range physics in a formulation closely related to the frame-transformationmethod. Two opposite limiting cases, of energy-independent short-range scattering phase shifts d,(R) and of very strong energy dependences (such as near a resonance, see Bardsley, 1967b; Gauyacq and Herzenberg, 1982)have been treated. Recent progress using an R-matrix-type formulation has also been reported by Launay and Le Dourneuf( 1984).Nonetheless,further work seems desirableto encompass both of the limiting regimes and all intermediate situations as well within a single theoretical framework. The remarkable upshot of all this is the widespread applicability of the quantum defect approach, which has now reached far beyond the original intentions of Seaton (1966) or of Fano ( 1970).It has been proven capable of describing spectral features in problems approaching chemical complexity, and should thus serve as an appropriate interface between experiment and ab initio theory.

+

+

ACKNOWLEDGMENTS We thank Annick Giusti-Suzor for suggestionsand comments on this manuscript.One ofus (C. H.Greene) was supported in part by the National Science Foundation and in part by an Alfred P.Sloan Foundation Fellowship.

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