Electronic Transitions In Atomic and Molecular Dynamic Processes

J. Phys. Chem. 1984, 88, 4812-4823

4812

Electronic Transitions In Atomic and Molecular Dynamic Processes Hiroki Nakamura Division of Theoretical Studies, Institute f o r Molecular Science, Myodaiji, Okazaki 444, Japan (Received: January 30, 1984)

There are four basic interactions which govern electronic transitions in low-energyatomic and molecular processes. Theoretical treatments of these interactions are reviewed briefly. Particular attention is paid to the nonadiabatic transitions induced by radial and rotational coupling in atomic collisions. A unified treatment of these transitions is shown to be possible by using the recently proposed dynamical-state representation. A general many-state spectroscopy as well as collision problem can be dealt with uniformly by the semiclassical formulas in this representation. Some numerical applications are presented. Applicability of the Rosen-Zener formula to light-atom-transfer collinear reactions is briefly discussed. Discussions are also made on a possible generalization of the basic idea of the dynamical-state representation to a wider class of dynamic problems.

1. Introduction Dynamics of electronically excited molecules has been the subject of recent increasing interest.' Among them are included collisions (or reactions) involving electronically excited atoms and molecules and unimolecular decay processes of excited molecules. This world of science is hitherto rather poorly explored in spite of the importance of these molecules. The importance may be well understood by simply recalling the fact that a main part of the oscillator strength distribution lies usually in the region of energy higher than the lowest ionization potential of molecules.2 Besides, these molecules have some peculiar features such as strong coupling between electronic and nuclear motions and sensitivity to an external field. The collisions involving excited species known as the collisions of the second kind, on the other hand, generally have large cross sections compared to those of the ordinary collisions of the first kind between unexcited species and are important in many fields of appli~ation.~ In addition, the cross sections show, in some cases, peculiar energy dependences, which attract theoretical interest. The variety of channels and competitions among them are typical characteristics in this world, making the exploration of the world complicated but at the same time interesting and challenging. Understanding the mechanisms of electronic transitions in molecules presents a basic step, in general, to comprehend energy conversion and change of state (phase transition) in more complicated systems. Chemistry of excited molecules will thus surely open a new prosperous world of science. It is worthwhile to classify the basic mechanisms of electronic transitions, to analyze their nature, and to figure out the way for unifying their treatments. In this paper basic interactions are classified into four groups. These are (1) electron correlation (configuration interaction), (2) diabatic coupling among Rydberg states (frame transformation), (3) nonadiabatic radial coupling, and (4)nonadiabatic rotational (Coriolis) coupling. Spin-orbit interaction is not counted as a basic one, since this can be incorporated in the electronic Hamiltonian to define basis electronic states. For simplicity discussions in this paper are confined to these mechanisms in a diatomic system. The first one is responsible for the decay of molecular resonance states embedded in the electronic continuum, the most typical example of which is a two-electron excited state. Transitions among rovibrationally excited Rydberg states can be treated in an elegant way by the quantum defect theory in which the adiabatic quantum defect plays a role as a diabatic coupling. The nonadiabatic radial coupling represents the best known mechanism responsible for a transition between the adiabatic states of the same electronic (1) See, for instance, Adv. Chem. Phys., 28 (1975), 45 (1981). (2) J. Berkowitz, "Photoabsorption, Photoionization and Photoelectron Spectroscopy", Academic Press, New York, 1979. (3) H. Nakamura and M. Matsuzawa, Butsuri, 25,727 (1970) (in Japa-

nese).

0022-3654/84/2088-4812$01.50/0

symmetry. The Coriolis coupling presents another important nonadiabatic coupling which causes a transition between the adiabatic states of different symmetry. Theoretical treatments of these interactions are reviewed briefly in the following two sections. Section 2 is devoted to the discussions on the first two interactions. The local complex potential method is briefly explained to deal with the processes involving the states embedded in the continuum. The most important quantity to characterize these states is the decay width r ( R ) as a function of internuclear distance R . A limitation of the method is also discussed. The multichannel quantum defect theory (MQDT)4 is outlined in this section. The MQDT enables us to treat uniformly to some extent the first two couplings mentioned above. The key quantity in this formalism is the adiabatic quantum defect p(R). Information on r ( R )and p ( R ) of molecules is yet very scarce. Recent calculations on H2 are presented in this section. In section 3 nonadiabatic radial and rotational coupling problems are reviewed. Attention is paid to the analytic structures of the problems in the ordinary adiabatic-state representation, i.e. to the analytic difference of the two coupling problems. In section 4 the dynamical-state (DS) representation5is introduced, which makes possible a unified treatment of the nonadiabatic transitions. This representation transforms the analytic structure of a rotational coupling problem to the same structure as that of radial coupling and thus localizes all the transitions at avoided crossings of the dynamical states. It is shown that a general many-state spectroscopic as well as collisional problem involving both radial and rotational couplings can be dealt with analytically by using the path integral formalism in the DS representation. Some practical numerical applications are presented in section 5. Section 6 briefly discusses the applicability of the semiclassical theory for nonadiabatic transitions to collinear reactions. The Rosen-Zener formula is applied to an asymmetric hydrogen-atom-transfer collinear reaction. The basic idea of the DS representation can be, in principle, generalized to other more complicated systems. The internuclear distance R is replaced by the hyperradius of a system. Then a wider class of problems could be, in principle, treated uniformly. A brief discussion on this undeveloped future problem is made in section 7 . Atomic units are employed throughout the paper.

2. Electron Correlation and Diabatic Coupling among Rydberg States The excited states of molecules whose excitation energy is higher than the lowest ionization potential are called superexcited states. There are two kinds of superexcited states. The first kind is the state whose electronic energy is higher than the ionization potential at least in a certain range of R . The electronic state is embedded in the ionization continuum. Two-electron excited states and inner (4) Ch. Jungen and 0. Atabek, J . Chem. Phys., 66, 5584 (1977). (5) H. Nakamura, Phys. Reu. A , 26, 3125 (1982).

0 - 1984 American Chemical Societv

The Journal of Physical Chemistry, Vol. 88, No. 21, 1984 4813

Electronic Transitions in Dynamic Processes shell one-electron excited states are t h e typical examples. The second kind is the rotationally and/or vibrationally excited Rydberg states whose total (electronic and rovibrational) energy is higher than the ionization potential. The superexcited states of the first kind are not the eigenstates of the electronic Hamiltonian, and the principal mechanism responsible for the decay (preionization) of the states is electron correlation (electron-electron interaction). The coupling ro the continuum is given by where & is the electronic wave function of the superexcited state, 4, is that of the continuum normalized by the 6 function of energy, He,is the electronic Hamiltonian, and r represents the electron coordinates collectively. A typical example of dynamic processes relevant to the states of the first kind is the Penning ionization A*

+ B -,A + B+ + e, AB+ + e

where the excitation energy of A is larger than the ionization potential of B; thus, a quasi-molecule A*B formed during the collision is embedded in the electronic continuum AQ' + e. This state usually corresponds to an inner shell excited state. Since autoionization of molecule A*B can take place even if the two atoms are rested at R, the superexcited state can be well characterized by a local complex potential6 W ( R )= Ed@) - ( i / 2 ) r ( R )

(2.2)

where The imaginary part r(R)is responsible for the decay. The probability PI of the decay (ionization) during a collision with orbital angular momentum 1 is given by

P, = 1 - exp[-2L:l?(R)/oi(R)

dR]

(2.4)

where q(R)is the relative local velocity and & is the turning point. A more comprehensive quantity is the probability amplitude for the state to propagate, say from R , to R, (R2> Ri). This is given by

where

The local complex potential method has been used also in the resonant electron scattering by molecules, associative detachment, dissociative attachment, and dissociative recombination. In the former three cases the superexcited states are unstable negative molecular ions. Apart from the collision processes, a unimolecular decay of a superexcited state produced, for instance, by photoabsorption presents a similar problem. If there exist both attractive and dissociative superexcited states, there are a variety of decay channels as is shown in Figure 1. Since the dissociative superexcited state is expected to give relatively high vibrational states of molecular ions, one has to be careful about the existence of this type of state when one analyzes photoelectron ~ p e c t r a . ~As is easily conjectured, the local complex potential approximation is good only when the nuclear motion is slow compared to the velocity of an ejected electron whose energy is roughly equal to the potential energy difference between the states A*B and AB'. It should be noted, therefore, that the nonlocality of electronic coupling and the nonadiabatic coupling to the continuum would become nonnegligible at high collision energies. When the nuclear vibrations are involved as in the resonant electron scattering, the (6) H. Nakamura, J . Phys. SOC.Jpn., 26, 1473 (1969); W. H. Miller, J . Chem. Phys., 52, 3563 (1970). (7) H. Nakamura, Chem. Phys., 10, 271 (1975).

Internuclear Distancc

Figure 1. Schematic diagram of the decay of molecular superexcited states.

local complex potential approximation is generally supposed to be good if the electron ejection is completed during one period of vibration. In this sense the local complex potential approximation is sometimes called the "boomerang approximation". The approximation loses its validity when the superexcited state crosses the molecular ion state. In the vicinity of the crossing point the nonlccality of the interaction potential should be taken into account (see also the discussions below from eq 2.1 1 to the end of this section).* In the case of the Penning ionization in which the state A*B lies above the AB+ state at all R, the local complex potential approximation is expected to be quite good. The superexcited states of the second kind are the vibrationally and/or rotationally excited Rydberg states. Their electronic states are eigenstates of the electronic Hamiltonian, being different from the first kind of states, but the total (electronic rovibrational) energy is higher than the ionization potential. Although it is not impossible to treat the transitions among them by the nonadiabatic theory, that is not a clever way since there is no conspicuous avoided crossings. A more elegant way of treating the transitions with deep physical insight is the multichannel quantum defect theory (MQDT) which was first developed by Seatong for atomic problems. The theory was later extended by Fanototo a coupling in molecules between electronic and rotational motions and then has been extensively applied to more general molecular dynamic p r o b l e m ~ . ~ J *The J ~ coordinate space of a Rydberg electron is divided into inner and outer regions, in which the basic physics controlling electron motions are different. In the inner region the electron moves together with the molecular ion core, and the Born-Oppenheimer (BO) expansion is a good approximation to describe the total system. In the outer region, on the other hand, the electron moves rather independently from the core, and the close-coupling type of expansion holds well. The basis transformation between the two expansions and the boundary condition imposed in the outer region lead to a basic equation to be solved. Determination of energy levels of the perturbed Rydberg states and autoionization of vibrationally and/or rotationally excited Rydberg states are typical examples. The most important physical quantity is the adiabatic quantum defect p A ( R ) ,where h is the component of the electronic angular momentum along the molecular axis. In the two-channel (one open and one closed) approximation, for instance, this basic equation can be written ast2

+

K , , sin(q--8,,) K,, sin (nu +

1 =o

K,,s1nh--8~~) K,, sin (nu +

(2,7)

wherej = 1 (2) corresponds to an open (closed) channel, 7 is the phase shift, and u is a parameter to measure energy ( E = -1/(2u2)) from the higher ionization threshold K,J cos O,, = C,J

(2.8a)

(8) J. N. Bardsley, J . Phys. B, 1, 349 (1968); A. Giusti, J. N. Bardsley, and C. Derkits, Phys. Reu. A, 28, 682 (1983). (9) M. J. Seaton, Rep. Prog. Phys., 46, 167 (1983). (10) U. Fano, Phys. Rev. A, 2, 353 (1970); (11) M. Raoult and Ch. Jungen, J . Chemt Phys., 74, 3388 (1981); Ch. Jungen and Dan Dill, ibid.,73, 3338 (1980). (12) H. Takagi and H. Nakamura, J . Chem. Phys., 74 5808 (1981).

4814

The Journal of Physical Chemistry, Vol. 88, No. 21, 1984

Nakamura vibr.

?SSOc.

ioniz. excit

The function x(R) represents the vibrational wave function. Equation 2.7 gives a phase shift as a function of energy for the elastic scattering of electrons by the molecular ion core in the vibrational state nz. This phase shift includes a contribution from the resonance state, in which the incident electron is captured to a vibrationally excited (in n , ) Rydberg state. Analyzing the resonance structure, one can obtain an analytic expression for the resonance width which represents autoionization from the vibrationally excited Rydberg state (AB*(nl) AB+(n2) + e).12 The off-diagonal matrix elements defined by eq 2.8d and 2.8e represent basic coupling terms. The adiabatic quantum defect wA(R)plays a role of a kind of diabatic coupling potential. When these off-diagonal elements are small and wA(R)can be well approximated by a linear equation

-

I

Figure 2. A unified view of various dynamic processes involving superexcited states.

continuum

with Re the equilibrium internuclear distance, then the autoionization width I' can be simply expressed as

in

D vs. h.

W(R) and V: (R)

This expression coincides with that of the crude Born-Oppenheimer approximation first derived by Herzberg and Jungen.13 It should be noted, however, that the second condition (eq 2.9) for the crude BO approximation to hold is dot usually well satisfid. Because of this, the crude BO approximation becomes bad especially for a high vibrational state and for a large An = Inl - n21 transition.12 One of the advantages of the MQDT approach is its ability of a stepwise treatment of different mechanism^.'^ Take, for instance, the case that there exists a dissociative superexcited state which crosses the molecular ion state. An interaction between this dissociative state and the electronic continuum is treated first. Since the nonlocality of the coupling potential should be taken into account as was mentioned before, the following integral equation for the K matrix is solved:

i unified treatrent of

various processes by MQDT

Figure 3. Elastic scattering of electrons from molecular ions as a key system to investigate various dynamic processes.

shown in Figure 2. The S matrix for the process i written as14 Sfi= C(fle)e2".(eli)

-+

and the coupling matrix element

is given by

where xi(R) is the vibration wave function, Fdis the nuclear wave function for dissociation continuum, and Ve(R)is the electronic coupling matrix element defined by eq 2.1. This equation takes care of the electronic coupling between the dissociative state (d) and the electronic continuum (i) (or Rydberg states). Vibronic coupling among electronic continuum and Rydberg states is taken into account by the MQDT after obtaining the K matrix from eq 2.1 1. By constructing the total energy eigenfunctions, we can treat various dynamic processes uniformly. Essential mechanisms governing the various processes are the same, and differences lie only in the boundary conditions. This situation is schematically (13) G. Herzberg and Ch. Jungen, J. Mol. Spectrosc., 41, 425 (1972). (14) A. Giusti, J . Phys. E, 13, 3867 (1980).

f can be (2.14)

e

where le) represents the total energy eigenchannel with eigenphase shift 7,. Equation 2.14 can be applied to vibrational excitation by electron impact, dissociative recombination, and associative ionization. The oscillator strength for photodissociation and photoionization is given by" (2.15)

(2.1 1) where the channel indices (p, q, r) represent the dissociation channel (d) (electronic bound state plus nuclear dissociative continuum) or the ionization channel (i) (electronic continuum ( E ) plus nuclear vibration ni), E(E',E") is the total energy, Le. in the case of ionization channel E=E E, (2.12)

-

Dfi =

C ( fle)eire(qelrlO)

(2.16)

e

where 10) represents the ground state with rotational quantum numbers (Jo,Mo), \ke is the standing wave total energy eigenchannel function, and hu is the photon energy. Although the MQDT is very powerful as explained above, the theory is not almighty and is unsuitable, at the present stage, to treat the channels such as A B+ e in which two particles are far away at the same time. In those cases what one can do now is to solve eq 2.11 for the K matrix with this double-continuum channel included.Is As is easily understood from the discussions above in this section, the electronic coupling matrix element VJR) defined by eq 2.1 (or r(R)of eq 2.3) and the adiabatic quantum defect p A ( R )are the most important physical quantities for investigating the dynamics of superexcited states of molecules. Accurate information on these quantities, especially on V,(R) (or r(R)), is very scarce. These quantities can be obtained by calculating phase shifts for scattering of electrons by molecular ions. Here we give some results of recent elaborate Kohn-type variational calculations of the phase shifts for the electron scattering by H2+.I6 A study

+ +

(15) See, for instance, S. Kanfer and M. Shapiro, J . Phys. E, 16,L655 ( 1983).

Electronic Transitions in Dynamic Processes

The Journal of Physical Chemistry, Vol. 88, No. 21, 1984 4815

,

0.07,

%ZPU,)' -

8

0.06-

..:..0.051 '

-

1

0 0.04.

3

0.03,

o,021

0.01

I

-

1.0 2.0 3.0 4.0 5.0 6.0 7 . 0 Internuclear distance (a u )

Figure 4. Potential energy curves for some of the two-electron excited states of H2 (dashed lines) (taken from the results of ref 16).

of elastic scattering of electrons from molecular ions presents a basic problem for understanding the properties and dynamics of higher excited states of molecules. Namely, from the calculations of elastic scattering phase shifts in the fixed-nuclei approximation, one obtains not only the elastic scattedng amplitude and the continuum scattering wave function but also the adiabatic quantum defects and the properties of two-electron excited states. All these quantities are useful for the study of the various dynamic processes. Figure 3 illustrates this situation. Figure 4 shows the potential energy curves of some of the two-electron excited states of H2. The (2paJ2 '8,state is the well-known lowest two-electron excited state which plays an important role in dissociative recombinati~n.'~ Being optically allowed from the ground state, the (2pu,2sag) ' 2 , state is a good target of a study of dissociation and/or ionization of H2by photon impact. The (2pn,nsag) 'II, states correlate to two excited hydrogen atoms in the separated atom limit. Some of the results on the widths r are shown in Figure 5. For the '2, and the lowest states other calculations are also available.l6 It is seen from this figure that magnitudes of r vary very much from state to state and that in many cases the R dependence of is not monotonic. In some cases the nonmonotonicity seems to be associated with avoided crossings of the corresponding states. It is highly desirable to inquire more extensively properties of two-electron excited molecular states and to investigate what physics really determine the magnitudes of r. Reanalysis of the data obtained for H2 would be worthwhile by using, for instance, the hyperspherical coordinates which were successfully applied to two-electron excited states of atoms. Adiabatic quantum defects pa(R) can be obtained from the phase shifts by extrapolating them to zero energy limit. When resonance states are located far from the ionization threshold, it is all right to assume pa(R)to be energy independent. When this is not the case, however, an energy dependence of p,(R) seems to have to be incorporated in the framework. Figure 6 shows a strong energy dependence of the da-wave phase shift at R = 2ao. The energy dependence of Vc(R) should also be taken into account when we deal with eq 2.11.

3. Nonadiabatic Transitions Electronically inelastic collisions involving lower excited states are usually studied in the adiabatic-state representation except for the case of high-energy collisions in which the large impact parameter encounters are dominant. Transitions among adiabatic states are called nonadiabatic tran~iti0ns.l~In this sectipn theoretical treatments of them are reviewed briefly. The total

,

(b)

I

n,(2Pn,3S0g)

0.005r

I 'll"(2P

0.004-

310-

2.0

Internuclear distance (a.u.)

T"25

\,

v

Internuclear distance (a.u. ) Figure 5. Widths r of the two-electron excited states of H, as a function of R (p-wave partial widths for the Il states) (taken from the results of ref 16). 1.41 do-wave .?

(R=Za,)

1.0

/

-0.6

i

I

Figure 6. Elastic scattering phase shift for da-wave in e 2ao (taken from the results of ref 16).

Hamiltonian of a diatom system with reduced mass body-fixed frame is written as5J8

p

in the

where He]is the electronic Hamiltonian, H,,, represents the rotational motion of a molecule, and HcOIrepresents the Coriolis interaction. HcOIand HI are explicitly given by H cor = - - h2 (L+U+ + L-u-) (3.2) 2pR2 (3.3) where L* = L, i iL,

(3.4)

a i a u, = F- + - + L f cot 8 d8 sin 8 a(p

(3.5)

~

(16) H. Takagi and H. Nakamura, Phys. Rev. A , 27, 691 (1983). (17) See, for instance, M. S.Child, *Atom-Molecule Collision Theory", R. B. Bernstein, Ed., Plenum Press, New York, 1979, Chapter 13.

+ H2+ at R =

(18) W. R. Thorson, J . Chem. Phys., 34, 1744 (1961).

4816

The Journal of Physical Chemistry, Vol. 88, No. 21, 1984

Nakamura

TABLE I: Analytic Properties of Nonadiabatic Couplings coupling scheme A€ f V

Adiabatic-State Representation radial“ a ( R - R.)1/2 rotational (i) degeneracy at R = 0 a R2 (ii) crossing at finite Rb a ( R - R,) (iii) no crossing -constant

a

1

a

(RR2

Dynamical-State RepresentatiQn any transition a ( R - R*)Il2 a (R -

where Tj is a turning point in the adiabatic potential ti@) (t2(R) is assumed to be larger than q ( R ) ) A = R,(R*) (3.12)

‘ R . is complex. b R , is real. I

- e,(R)] PLZ

C#J~

-

= exP(-26)

= y In y - y - arg r ( l

(3.14)

+ iy) + a / 4

y = 6/a I

_________ T,

T?

__

~~

~~

A

-

Internuclear distance

Figure 7. Schematic two-state potential energy diagram ( A = Re@)).

L is the electronic orbital angular momentum, LE,,,rare its components in the molecule-fixed coordinate system with the [axis along the molecular axis, and 0 and cp are the ordinary angle variables to define molecular axis orientation. Let us define adiabatic electronic eigenfunctions &(A;r;R) and eigenvalues €(A:R) of He, by

Hel$e(A;r:R)= e(A:R)&(A;r:R)

(3.6)

As is well-known, the Neumann-Wigner noncrossing rule holds for the adiabatic states of the same symmetry (same lAl) and a nonadiabatic transition between these states is caused by the first term of eq 3.1, since other operatorb have no off-diagonal element in the manifold of the states of the same IAl (a contribution from HI is small and is usually neglected). This transition occurs predominantly at an avoided crossing. This is not only because the adiabatic energy difference becomes minimum there but also because the nonadiabatic coupling matrix element becomes maximum there. More precisely, the zero R* of order one-half of the adiabatic energy difference coincides with the pole of order unity of the coupling in the complex R plane (see Table I). The real part of R. is roughly equal to the avoided crossing point on the real axis. Because of this coincidence of pole and zero, the simple perturbation theory cannot be a good approximation. The above-mentioned analytic property of the radial coupling problem underlies the derivation of the famous Landau-Zener-Stuckelberg and Rosen-Zener formula^.^^^^^ The most sophisticated L Z S formula for the scattering matrix in the two-state approximation is given in a matrix multiplication form by5

where E A ,PA-, and PATA are diagonal propagation matrices and IA(0,) is the transition matrix corresponding to the incoming (outgoing) segment of the trajectory (see Figure 7). These matrices are defined by

(3.16) (3.17)

The r(x) is the r function, and & is the so-called Stokes phase. In the case of Rosen-Zener type of transitions the S matrix is again given by eq 3.7 with $s put equal to zero, and the transition probability pLzfor one passage of the avoided crossipg point is replaced by

The probabilities for the transition 1

-

(3.18) 2 are given by

for the LZS type of transitions and for the R Z type of transitions, where d

= uo

+ JTAkl(R) 1

1k,(R) dR A

dR -

(3.21)

T2

It should be noted that what are required in these expressions are only the adiabatic potential energies as analytic functions of R . These formulas can be derived by reducing the basic coupled equations to a Weber equation with use of the comparison equation methodz1 and by utilizing the Stokes phenomenon of the asymptotic expansions of the Weber function.22 The difference between the LZS and the R Z formulas comes from the different asymptotic expressions of the Weber function. In practical applications, the L Z S formula is frequently criticized not to be a good approximation. This is, however, mainly because the formulas used are not the full LZS formula given above but the simplified ones.2o The simplest one is the original L Z form in which the exponent 6 is approximated by (3.22) where VI, and AV are the coupling matrix element and the difference of the diagonal elements in the diabatic Pepresentation of Hel,R, is the crossing point, and u(R,) is the local velocity at R,. ‘Another simplification frequently employed is the neglect of thp Stokes phase 9,. This phase takes a value in between a / 4 (diabatic limit) and 0 (adiabatic limit). The approximation (3.22) obviously fails when a turning point is larger than R,, whereas the full L Z S formula can handle this case properly by analytic continuation. The importance of 4sis demonstrated below. The (21) S. C. Miller and R. H. Good, Phys. Rev., 91, 174 (1953).

(19) D. S. F. Crothers, Adv. Phys., 20, 405 (1971). (20) D. S. F. Crothers, Adu. At. Mol. Phys., 17, 55 (1981).

(3.13)

(22) J. Heading, “An Introduction to Phase-Integral Methods”, Methuen & Co., London, 1962.

The Journal of Physical Chemistry, Vol. 88, No. 21, 1984 4817

Electronic Transitions in Dynamic Processes

coupling problem. Besides, these are not suitable for generalization to a many-state problem, since these give only probabilities (not the probability amplitudes) and the transitions are not localized. Because of the peculiar properties of rotational coupling, there is a qualitative difference in collision energy dependences of radial transitions and rotationally induced transition^.^ Total cross sections for the latter transitions increase monotonically as a function of collision velocity, if the so-called electron translation factor is not taken into account. This can be conjectured if one looks at the rotational coupling term in a semiclassical approximation (lIH&) (bu/RZ)(llL$) (3.24) Collision energy (a.u.)

Figure 8. Transition probabilities as a function of collision energy (1 = 0) (model 3.23): (- -), exact numerical calculation;(-), LZS formula

(3.19);

-

(-e),

LZS formula (3.19) without @.,

model potential system is two attractive Morse potentials coupled by an exponential function V,,(R) = dle-a(R-R.i)(e-a(R-R.l) - 2)

V,,(R) = d2e-a(R-R.z)(e-~(R-R.2) - 2) Y l 2 ( R )= Ve-OR

(3.23)

where dlezaReI= d2eZaReZ,Rel = 2.5, Re2 = 2.0, d2 = 0.15,A€ = -0.1, and Y = 1 .O. The reduced mass is assumed to be 2000 au. Transition probabilities as a function of collision energy are shown in Figure 8. The LZS formula (3.19) works well for its rather simple form. The importance of & was also recognized in predisso~iation.’~~ The full L Z S and R Z formulas in a two-state approximation given above have one more big advantage. Namely, they can be easily extended so as to be applicable to a general many-state problem, since the transitions are well localized at avoided crossings and the S matrix can be expressed in the form of matrix multiplication as in eq 3.7. Each matrix has a simple physical meaning. One disadvantage of this theory is the necessity of analytic continuation of the adiabatic potential energies into the complex R plane. Let us next discuss a transition among the adiabatic states of different symmetry (different lAl). This transition is caused by H,, and is called a rotationally induced nonadiabatic transition. The selection rule is IAAl = 1 . The importance of rotational coupling was first recognized in high-energy ion-atom collisions. This coupling is now known to play a role even in low-energy processes. Since the noncrossing rule does not hold for the states with different 1AI, there can be the following three basic curve crossing schemes in the adiabatic-state representation: (i) potential curve crossing at the finite internuclear distance R,, (ii) degeneracy at the united atom limit (R = 0), and (iii) no curve crossing. Being different from radially induced transitions, rotationally induced transitions are delocalized between a turning point and a crossing point. This is simply because the coupling is proportional to R-, and has a pole at R = 0 (see eq 3.2). Because of this difference in analytic structure, the L Z S and the R Z formulas are not applicable. Even in case ii in which the pole coincides with the zero of adiabatic energy difference, the conventional LZS formula can not be applied, because the orders of the zero and the pole are not the same as those in the radial coupling problem. The analytic properties are summarized in Table I. Demkov et aLZ3 proposed a certain analytic formula for transition probability in case ii, noticing the similarity of its analytic property to that of the ordinary radial coupling problem. Namiki et aLZ4derived an analytic expression for probability applicable to cases i and iii. These formulas, however, cannot deal uniformly with the radial coupling problem and the three cases (i-iii) of the rotational (23) Yu. N. Demkov, C. V. Kunasz, and V. N. Ostrovskii, Phys. Reo. A, 18, 2097 (1978). (24) M. Namiki, H. Yagisawa, and H. Nakamura, J . Phys. B, 13, 743 (1980).

where u is the collision velocity and b is the impact parameter. Namely, the coupling term is proportional to the collision velocity.

4. Unified Treatment of Nonadiabatic Transitions As was explained in the previous section, the LZS theory has an advantage to be extendable to a many-state problem. This advantage has no use, however, if there exists even one rotational coupling. It is desirable, therefore, to develop a theory which can handle the radial coupling problem and all the cases of rotational coupling problems described in the previous section in a unified way. In order for such a theory in the two-state approximation to be general enough, a transition, either radially induced or rotationally induced, should be made to occur locally at a certain internuclear distance and an analytic formula should be given not for probability but for probability amplitude. The recently proposed dynamical-state r e p r e ~ e n t a t i o nmeets ~ ~ ~these ~ ~ ~require~ ments. (Essentially the same representation has been discussed by several authors.5 Their viewpoints are, however, different from the one here.) The dynamical states are defined as the eigenstates of (4.1) Hdyn E He, + Hrot + HI + H c o r Hd,,\kK(r,k:R) = EK(R) \kK(r,R:R) (4.2) where k = (0, cp) and K is the total angular momentum quantum number. Since Hdynis Hermitian and depends on R only parametrically, the eigenvalue E K ( R )is a real function of R. As is seen from eq 4.2, the dynamical states are the eigenstates of the rotating collision complex at a fixed internuclear distance (at a fixed size of the complex) and thus are dependent on the total angular momentum K = L LR,where L, is the angular momentum of relative motion of nuclei. In other words, the dynamical states are completely diagonal with respect to nuclear and electronic rotations. Let us next introduce the electronic rotational basis functions defined as @$(A) = (1/2llZ) X (@e(A+;r:R) Y ( K A + ; k )f 4e(A-;r:R) Y ( K A - ; k ) ) for A # 0 (4.3)

+

= +c(2;r:R)Y ( K Z ; k ) where Y ( K A ; k )is an eigenfunction of Hrot

(4.4)

HrotY(KA;k)= [K(K + 1 ) - 2 A 2 ] Y ( K A ; k ) (4.5) The whole set of states (@:(A), a K ( 2 )can ) be divided into two classes, namely, (@f(A), @(E+)) and (a!( A), @(Z)),which have no connection to each other. (In ref 5 the Z state is implicitly assumed to be the 2’ state.) The Coriolis interaction H,, couples the states in each class according to ~ @ 2 ~ ~ l ~ l ~ c = o r l @ ~ ~ ~ z ~ ~ fi2

--(A-(KJZ)a(AT,AZ 2wRZ

- 1)(4e(AT)IL-I4e(At))

+

where A*(K,A) = [ ( K + A ) ( K f A

+ l)]’’’

(4.7)

(25) H. Nakamura and M. Namiki, J . Phys. SOC.Jpn., 49, L843 (1980). (26) H. Nakamura and M. Namiki, Phys. Reo. A , 24, 2963 (1981).

4818

The Journal of Physical Chemistry, Vol. 88, No. 21, 1984

Nakamura

The dynamical-state wave functions \kK(r,k:R) can be expanded in terms of the functions (@$(A), @(E+)] or (@(A), @K(E-)) Namely, the eigenvalues EK(R)are obtained by diagonalizing a matrix of HdYnspanned by the electronic rotational basis functions. Since the dynamical states are the eigenstates of &,,, transitions among them are exclusively induced by the first term of eq 3.1 irrespective of the type of coupling in the adiabatic-state representation. In order to understand a more detailed nature of this representation, it is convenient to consider the simplest two-state approximation composed of one II state and one 2 state: \kK

= C$O"(Z)

+ Cf$$(rI)

ili.-i L.-.

Internuclear distance

(4.8)

Figure 9. Localization of rotationally induced transitions in the DS representation (collision velocity u = 0.5 au, impact parameter b = 0.6 au, A = R,(R.), and T is the turning point): (---), exact numerical calculation in the adiabatic-state representation; (-), exact numerical calculation in the DS representation; (- -), LZS approximation in the DS representation (taken from the results of ref 26).

The conventional method of diagonalization of a 2 X 2 matrix leads to

-

+

+

E$@) = '/z(eK(II:R) eK(Z:R)f [ ( A c ( R ) ) ~4G(R)]1/2) (4.9)

The step function in the figure represents the LZS approximation. The value in between the two steps at R = R,(R*)is simply equal to exp(-6), a transition probability for one passage of the avoided crossing. The asymptotic value on the way out (R > R,(R,)) equals PLDgiven by eq 3.19, an overall transition probability after collision. In the adiabatic-state representation (broken line) the transition is not localized anywhere. The theory developed in this section is applicable equally to problems other than collisions. The difference among the various processes lies only in boundary conditions. Basic mechanisms do not differ from process to process. The diagrammatic technique devised by Child,27for instance, is useful to describe uniformly a wide variety of spectroscopic as well as collision phenomena. In order to deal uniformly with all the nonadiabatic transitions involving both radial and rotational couplings, all we have to do is just replace the adiabatic-state energies by the dynamical-state energies in various expressions. As a typical example, a two-state problem is briefly discussed. Being different from the definitions of matrices of Child, the following is used here:

where a2

(4.10) At(R)

eK(II:R)- @(Z:R)

eK(A:R)= c(A:R)+ h Z [ K ( K 2pR2

(4.1 1 )

h2 + 1) - 2A2] + -(L2)* 2pR2 (4.12)

VO = (+e(E)IWe(H+))

(4.13)

A main term of the new couplings in the DS representation is given by

(

!$VK - z A e ) / ( E f ( R ) - E f ( R ) ) 2(4.14)

As is easily seen from eq 4.9 and 4.14, the adiabatic states f ( A R ) , the rotational coupling terms VK(R),and the dynamical states EK(R)play a role of diabatic states, diabatic coupling terms, and adiabatic states, respectively, in the conventional terminology. Thus, the rotationally coupled adiabatic states avoid crossing in the DS representation. Furthermore, it is easily shown that analytic properties of the dynamical states are the same as those of the ordinary radial coupling problem irrespective of cases i-iii mentioned in the previous section. The avoided crossings between the adiabatic states of the same symmetry remain avoided in the DS representation, although the potential energy curves can be deformed by rotational couplings with other states. Thus, once one obtains a dynamical-state potential curve system, one can apply uniformly the LZS or RZ formula given in section 3. This theory meets the requirements mentioned in the beginning of this section. Transitions induced by either radial or rotational coupling occur locally at new avoided crossings. The analytical formulas are given for probability amplitude and can be easily extended to many-state problems. The total scattering matrix can be expressed as a multiplication of two types of matrices as in eq 3.7. The first kind of matrix is a diagonal propagation matrix P which expresses an evolution of the system along a dynamical state from one avoided crossing to the other without any transition. The second kind is a transition matrix l o r 0 which involves only one nondiagonal2 X 2 submatrix representing a transition at an avoided crossing between two dynamical states. Localization of transitions in the DS representation is demonstrated in Figure 9.26 The following model potential system is used, simulating the two-state (IT, and 2a,) problem in the Ne+-Ne system: and

Vo = 1.42

and matrices I and 0 are defined by eq 3.10 and 3.1 1 for nonadiabatic transitions instead of R' and R" of Child, where (4.17) Matrices 4.15 and 4.16 are employed in order for the turning point contribution ( ~ / 4 to ) phase shift to be directly incorporated in these matrices. Nuclear wave functions in the channel j are represented as R

IC;. = -V$ exp[ i s , kf(R) d R ]

+ V'$ exp[ - i L R k f ( R ) d R ]

where

kj = kjK(R=m)

~

(27) M. S. Child, J . Mol. Spectrosc., 53, 280 (1974).

(4.20)

The Journal of Physical Chemistry, Vol. 88, No. 21, 1984 4819

Electronic Transitions in Dynamic Processes Tjjo)(A)= Lm[k/K(R)- kj] dR

- kjA + (K/2)a

(4.22)

The scattering matrix S is defined by

e,=&/

(4.23)

In the case that both channels are open (two-channel scattering problem) (see Figures 5 and 7 of ref 27), one obtains

tan 2p = -2 sin cos ( C T+ &)

+ $,)(p(1 sin (-+s+ uo) x + sin (a + &) cos (-dS+ uo)(2p - 111 (4.39) (CT

Equation 4.34 is the same as eq 72 of Colle,28who formulated predissociation in terms of the MQDT. By analyzing the resonance structure energy dependence of A, the analytic expression for the resonance width can be obtained. The semiclassical nuclear wave functions for each channel are given by

= D 1sin ( L > f ( R ) d R = a sin ( L > f ( R )

=

(E:)

(4.26)

where C1(C,) is a constant proportional to the amplitude of the wave function at R R TI (T2). Thus, the scattering matrix is given as

As is easily seen, this is exactly the same as eq 3.7 in the DS representation. In the case that both channels are closed, one naturally obtains the same equation as eq 9.4 of Child (see Figure 7 of ref 27). Finally, the case of one open channel with the other one closed presents a predissociation (see Figure 6 of ref 27). V’ = OA$(Tl,A;Tz,A)IAV”

g0)V”

(4.28)

V’, = [L*(A,T2)IlIC= exp[-iyz(A,T2)]C

(4.29)

V”, = [L*(A,T2)],,C= exp[iy,(A,T2)]C

(4.30)

where T2 is the right turning point in the dynamical potential Ef(R) and C is a constant proportional to the amplitude of the wave function in the channel 2 at R 5 T2. The total elastic scattering phase shift 7 can be obtained from (4.31) The potential scattering phase shift qfo)by the potential Ef(R) is given by

vio) = +j\O)(A)+ L A k f ( R ) dR + ?r/4

(4.32)

I

The total phase shift 7 can thus be expressed as 7=

vio) + A = ?(‘)(A)+

1 A

kf(R) dR

TI

+ a/4 + A

(4.33)

where A represents the effective phase shift due to the nonadiabatic coupling. After manipulating eq 4.31 with use of eq 4.28-4.30, one can obtain cos2 p sin

tiz2 sin (611 -

A)

+ sin2 p sin 621 sin (6],

q2 = D2 sin

dR

+ r/4)

+ A + a/4)

1

IT)f(R)

= D2 sin ( L T z k f ( R ) dR

dR

R

+ a/4)

+ a/4)

Tl

A