Nonadiabatic effects on the photodissociation of diatomic molecules to

Nonadiabatic effects on the photodissociation of diatomic molecules to open-shell atoms. Yehuda B. Band, Karl F. Freed, Sherwin J. Singer, and Carl J...
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J. Phys. Chem. 1987, 91, 5402-5409

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of TBN and possible mismatch between the splitting of the (CHJ2N and NO fragments of DMN might explain the difference between the N O spin-orbit population ratio following TBN and D M N photolysis. Summary

The analysis and comparison of the directional properties of the N O product resulting from each of the dissociation processes reported here provide essential information about the relevant

excited states: their geometry, symmetry, potential surfaces, lifetime, and possible exit channel interactions. Further experiments and analysis concerning the S2 excited state of the TBN molecule and the energy distribution among the various degrees of freedom of the N O fragment resulting from dissociation of both TBN and DMN are in progress and will be the subject of future publications. Registry No. ( C H 3 ) * N N 0 , 62-75-9; (CH,),CONO, 540-80-7; NO, 10102-43-9.

Nonadiabatic Effects on the Photodissociation of Diatomic Molecules to Open-Shell Atoms Yehuda B. Band,* Department of Chemistry, Ben Gurion University, Beer-Sheva, Israel

Karl F. Freed, The James Franck Institute and the Department of Chemistry, The University of Chicago, Chicago, Illinois 60637

Sherwin J. Singer, Department of Chemistry, The Ohio State University, Columbus, Ohio 43210

and Carl J. Williams The James Franck Institute and the Department of Chemistry, The University of Chicago, Chicago, Illinois 60637 (Received: January 5, 1987)

Photodissociation of diatomic molecules to open-shell atoms is shown to be very strongly affected by nonadiabatic interactions when the photon energy is just above the threshold for dissociation. The energy dependencesof photodissociationcross sections and various anisotropy parameters exhibit a wealth of structure in conformity with our theoretical predictions that nonadiabatic interactions between molecular states, approaching the same atomic term limit, lead to the emergence of resonance features in the spectra. Some of these features are associated with Feshbach and shape resonances on states that carry no oscillator strength in zeroth order. As an example, we consider the photodissociation cross sections for the production of individual Cf(ZP3,2,,,2) states from selected initial CH' levels and the fragment angular distribution, orientation, and alignment. We discuss how totally oriented fragments can be produced by near-threshold photodissociation in a magnetic field.

I. Introduction Nonadiabatic couplings strongly affect the near-threshold photodissociation of molecules to open-shell fragments.'q2 Molecular systems correlating to open-shell fragments have several nearly degenerate asymptotic electronic potential surfaces and therefore present the possibility of interesting consequences resulting from the fact that the Born-Oppenheimer approximation completely breaks down at intermediate fragment separations where the nonadiabatic interactions become larger than the spacings between the nearly degenerate electronic potential surfaces.2 The nonadiabatic interactions arise by virtue of spin-orbit, Coriolis, and hyperfine interactions. It is known that photodissociation provides a great deal of information regarding potential energy surfaces upon which the fragments fly Because of the effects of the nonadiabatic interactions, near-threshold photodissociation dynamics contains a wealth of information (1) Singer, S. J.; Freed, K. F.; Band, Y . B. Adu. Chem. Phys. 1985, 61, 1.

(2) Moseley, J. T. Ado. Chem. Phys. 1985, 60, 245. (3) Leone, S. R. Adu. Chem. Phys. 1982, 50, 225. (4) Shapiro, M.; Bersohn, R. Annu. Reu. Phys. Chem. 1982, 33, 409. (5) Band, Y. B.; Freed, K. F.; Kouri, D. J. Chem. Phys. Lerr. 1981, 79, 233. Band, Y. B.; Freed, K. F. Chem. Phys. Lett. 1981, 79, 238.

0022-3654/87/209 1-5402$0 1.50/0

concerning the molecular electronic potential surfaces, even the molecular surfaces which are forbidden to carry oscillator strength from the initial molecular state. The diatomic molecule dissociation to open-shell fragments is the simplest system where nonadiabatic couplings between nearly degenerate electronic surfaces affects the direct photodissociation process. The cylindrical symmetry and the lack of internal (vibration-rotation) degrees of freedom for the atomic fragments make the diatomic dissociation theoretically simpler to treat than polyatomic systems. Because these diatomic dissociations already present major theoretical and conceptual difficulties, we have developed a full quantum theory to treat their photodissociation dynamics.lv6-* Low-energy atom-atom scattering is also sensitive to the long-range intermolecular forces, the attractive regions of the potentials, and the nonadiabatic interactions between different electronic potential curves approaching the same atomic term limit. Thus, the inversion of this scattering data should permit, in principle, the determination of these potentials and couplings. (6) Singer, S. J.; Freed, K. F.; Band, Y . B. J . Chem. Phys. 1983, 79,6060. (7) Singer, S. J.; Freed, K. F.; Band, Y . B. J . Chem. Phys. 1984.81, 3064. ( 8 ) Singer, S. J.; Freed, K. F.; Band, Y . B. J . Chem. Phys. 1986,84, 3762.

0 1987 American Chemical Society

Photodissociation of Diatomic Molecules However, crossed beam scattering experiments between pairs of open-shell atoms are currently not feasible, and even if they were, the scattering data buries the information in averages over impact parameters and average over reactant kinetic energies. Photodissociation dynamics probes the same regions of electronic potential curves and the nonadiabatic couplings. However, photodissociation from a single initial bound state with a single-mode laser beam permits precise control of the total energy of the system and its angular momentum. Moreover, the use of polarized light in the photodissociation process introduces anisotropy into the diatomic system to provide for the possibility of extracting additional information from the polarization of the atomic fragments. Thus, photodissociation, in principle, provides more detailed (Le., less averaged) data than can be obtained in current crossed beam scattering experiments. It has been shown that near-threshold photodissociation results in the emergence of resonances in the energy dependence of the cross sections for production of fine structure states of the photofragments, in their angular distributions, and in the fluorescence from electronically excited f r a g m e n t s . ' ~ ~ ? ~Many - ~ ~ of these resonances arise due to nonadiabatic interactions. The orientation and alignment of open-shell fragments provide additional probes of the interactions occurring as the fragments separate from one a n ~ t h e r , s * ~and ~ - 'even ~ more detailed information is, in principle, available from coincidence measurements of fragments and their fluorescence.' Calculations show that the influence of nonadiabatic interactions in the production of open-shell fragments often persists to energies over 2 orders of magnitude larger than the asymptotic fragment fine structure splittings. This nonadiabaticity is particularly noticeable in predicted nonstatistical branching ratios for the production of atomic fragment fine structure states. The predicted nonadiabatic resonances are of two types. The first involves nonadiabatic shape resonances, arising from tunneling through a centrifugal barrier on an electronic surface other than the optically pumped surface. Feshbach resonances are associated with the formation of a bound state on an electronic surface which is energetically closed asymptotically. When this electronic surface correlates with an upper fine structure state, then Feshbach resonances formed on it may decay through nonadiabatic coupling to electronic surfaces which correlate to lower fine structure states. Our theory shows that the nonadiabatic couplings enable the observation of some of these resonances and thereby provide a method of studying near threshold "dark" electronic states which are not radiatively coupled in zeroth order to the initial (or intermediate) bound electronic state. We review in the following sections the full multichannel quantum theory of photodissociation of diatomics to atomic fragments with fine structure. The theory is illustrated by multichannel calculations for the model system

The theory has also been applied to the direct dissociation of NaH and Na?-l2 and to the predissociation of O H ? Both N a H and OH show that the branching ratio for production of fine structure fragments may remain nonstatistical at excess energies greater than 2 orders of magnitude larger than the atomic spin-orbit ~ p l i t t i n g s . ' ~Here J ~ we will focus upon low-energy calculations for the photodissociation of CH+ and discuss the dynamical effects leading to the immense and complicated structure in this system. (9) Singer, S. J.; Freed, K. F.; Band, Y. B. Chem. Phys. Lett. 1984, 105, 158. (10) Williams, C. J.; Freed, K. F. Chem. Phys. Lett. 1986, 127, 360. Williams, C. J.; Freed, K. F. J . Chem. Phys. 1986, 85, 2699. (11) Struve, W. S.; Singer, S. J.; Freed, K. F. Chem. Phys. Let?. 1984, 110, 588. (12) Singer, S . J.; Band, Y. B.; Freed, K. F. Chem. Phys. Let?. 1982, 91, 12. Singer, S. J.; Freed, K. F.; Band, Y. B. J . Chem. Phys. 1984, 81, 3091. (13) Fano, U.; Macek, J. M. Rev. Mod. Phys. 1973, 45, 553. (14) Greene, C. M.; Zare, R. N. Annu. Rev. Phys. Chem. 1982, 33, 89. Greene, C. M.; Zare, R. N. J . Chem. Phys. 1984, 48, 4304. (15) Lee, S.;Williams, C. J.; Freed, K. F. Chem. Phys. Le??.1986, 130, 271.

The Journal of Physical Chemistry, Vol. 91, No. 21. I987

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CH' provides an ideal model system for illustrating nonadiabatic effects in photodissociation because of the moderate spin-orbit splitting of the C+(22Plj2,3/2) fragment (64 cm-I), the relatively small reduced mass leading to moderately strong Coiolis coupling, and a maximal number of bound states since all electronic potential curves support bound states (see Figure 1). Even the primarily repulsive c3Z+potential has bound states because of the attractive long-range ion-atom interaction in this state. Calculations are reported of the anisotropy parameters for the production of photofragments, of parameters related to the alignment and orientation of the C+ fragment, of branching ratios, and of total and partial photodissociation cross sections. Additional impetus for theoretically studying the CH+ system comes from previous and continuing photodissociation experiments for this molecule using opticalI6-'* and infrared19s20excitation. A small fraction of the large number of experimentally observed resonances in the optical region have been assigned to shape resonances on the A'II resonances, lying in the region electronic c u r ~ e . ~No ~ ?Feshbach '~ between the two fine structure components of the C+(22P)states, have yet been assigned in the experimental spectra, although predictions of A'II and other resonances in this region have been made.9*10 The previously studied N a H and Na2 systems".I2 have a small Na(32Plj2,3i2)fine structure splitting (17 cm-') and hence do not support as many "quasi-bound" states in the energy region between the two fine structure components of the Na(32P,jz,,j2)atomic state. Further, the NaH BIIIuand c3Z,,+ states are purely repulsive and thus serve to simplify the photodissociation dynamics of this diatomic system. The large reduced mass and the resulting weak Coriolis coupling of Na2 combine with the existence of a barrier in the B'n, potential at large internuclear separations to make the Na2 system behave more adiabatically." Experiments on Na?' have measured the polarization of the Na(32P) fragment and the widths of the tunneling levels in the B'II, state. Several photodissociation experiments on alkali dimers have studied photodissociation in a magnetic field which lifts the degeneracy of the magnetic sublevels of the fragment fine structure states.22 Section I1 provides a sketch of the theory of diatomic photodissociation to atomic fragments with fine structure. Section I11 describes the results of our calculations on CH+. 11. Review of the Theory of Photodissociation

The primary advancements made by our full quantum theory of dissociation to open-shell atoms involve the complete inclusion of nonadiabatic couplings, the treatment of all the coupled angular momenta, and the analysis of the proper asymptotic scattering states. The theory utilizes two molecular basis sets which provide a zeroth-order description of the "half-collision" dynamics in the molecular and the asymptotic regions, respectively. In addition, we introduce a transformation TBbetween those two basis sets, which is independent of internuclear separation r. Elsewhere, the general theory of diatomic photodissociation is described along with a treatment of resonant two-photon dissociation processes and an analysis of the angular distribution and polarization of photofragments.1,6-8 Below a brief overview of this theory is presented. A . Basis Functions. The first basis set is a space-fixed (SF) basis set derived from one Hund's coupling case basis. Here a (16) Cosby, P. C.; Helm, H.; Moseley, J. T. Ap. J . 1980, 235, 52. (17) Helm, H.; Cosby, P. C.; Graff, M. M.; Moseley, J. T. Phys. Reu. A 1982, 25, 304. (18) Sarre, P. J.; Walmsley, J. M.; Whitham, C. J. J . Chem. SOC.,Faraday Trans. 2 1986, 82, 1243. Sarre, P. J.; Walmsley, J. M.; Whitham, C. J. Faraday Discuss. Chem. SOC.1986, 82. (19) Carrington, A,; Buttenshaw, J.; Kennedy, R. A,; Softley, T. P. Mol. Phys. 1982, 42, 747. (20) Carrington, A,; Softley, T. P. Chem. Phys. 1986, 106, 315. (21) Geber, G.; Moller, R. Phys. Rev. Let?. 1985, 55, 814. Rothe, E. W.; Krause, U.; Duren, R. Chem. Phys. Lett. 1980, 72, 100. Chawla, G. K.; Vedder, H. J.; Field, R. W. J . Chem. Phys., in press. (22) Kato, H. Faraday Discuss. Chem. SOC.,in press. Kato, H.; Onomichi, K. J . Chem. Phys. 1985,82,1642. Kato, H.; Baba, M.; Hanazaki, I. J . Chem. Phys. 1984, 80, 3936.

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The Journal of Physical Chemistry, Vol. 91, No. 21, 1987

-# 1

U

, u1/ ,

,

,

1

5 ID INTERNUCLEAR SEPARATION

,

,

,

li-3.6

-5

c I

-

0

(nu)

INTERNUCLEAR SEPARATION

(au)

Figure 1. (a) A B 0 electronic potential energy curves approaching the lowest atomic term limit C+(22P)+ H(12S) for CH'. The C+(22P)spin orbit splitting of 64 cm-' is exaggerated. (b) and (c) are diagonalized potentials, including electronic, Coriolis, and spin-orbit interactions, for the CH+ states which dissociate to the C'(2'P) + H(12S) atomic term limit. (b) is for J = 2 and even parity whereas (c) is for J = 2 and odd parity. The potentials are displayed at intermediate nuclear separations on the right side with a n expanded energy scale to show the existence of a small well in the c3Z+electronic state and to exhibit the correlations of the fully diagonalized potential with the two fine structure states of C+. Many of the low J resonances in CH+ arise from "quasi-bound vibrational states" on potential curves which approach the energetically closed C+(22P,,,) limit but which decay through couplings to states correlating with the energetically open C+(22P,i2)fine structure state.

case a basis IhsI:) is chosen, where S is the total spin of the system and A and I: are the projections of electronic and spin angular momenta onto the diatomic internuclear axis. The latter axis is termed the body-fixed (BF) z axis. Since all couplings are included in our calculations, it is possible to adopt a basis with any of the Hund's case coupling schemes, independent of which in zeroth order is most advantageous. In the molecular region the wave function can be written in a Hund's case a basis I M Z ) in terms of the basis functions IJMASZp) of inversion symmetry p

*

IJMASZp) = ( 2 - 6.i06,0)-1/2[(2J "*(CYPO)lASZ)

+ 1 ) / ( 8 ~ * ) ] ' / ~X

+ (1

-

6,,6,0)

x - AS - 2 , ) (2)

(-l)p+J-~+"d,_,*(CYpo)I

where D&(cupO) are the Wigner rotation matrices, J is the total angular momentum of the system, Q = .I Z is the projection of the total angular momentum along the internuclear axis, and

+

M is the projection of the total angular momentum along the SF z axis. The quantity u is zero unless the electronic state is of Esymmetry in which case it takes the value l . Thus the parity of the basis set IJMASZp) is given by (-1)P. The second basis set IJMjljJ,) is related to the much-ignored Hund's case e basis and which we refer to as the "atomic" basis set since this molecular basis set has the special property that as r the individual basis functions become a product of atomic fine structure states bama)b@b) and a spherical harmonic Z;,(pa) which describes rotation of the atomic fragments about their center of mass. The quantum number j , and j b are the electronic angular momentum quantum numbers of fragments a and b, respectively, with projection ma and m bonto the SF z axis; 1 is the angular momentum quantum number for rotation about the center of mass; is the quantum number associated with projection of I onto the SF z axis; j = j, + j, is the total fragment electronic angular momentum; and J is the total angular momentum quantum

-

Photodissociation of Diatomic Molecules

The Journal of Physical Chemistry, Vol. 91, No. 21, 1987 5405

+

number resulting from J = j I. It is important to note that at arbitrary internuclear separations the basis set IJMjlJb) cannot be written as a product of atomic fine structure states but is a linear combination of A B 0 electronic states. Hence, the “atomic” basis set is defined to have the limiting from

-

IJMjljJb) = C l i m j J b ) Y l ~ ( P ~ ) ( J M l i l m cr l ) , mfi

m

(3a)

where PmjJb) is given by limjJb) =

C Vama)libmb)CimliJbmamb)

(3b)

m,mb

The parity of IJMjljJb) is (-1)”’a’’b. The basis I J M A S Z p ) properly describes the zeroth-order dynamics in the molecular region since it diagonalizes the electronic Hamiltonian without the small relativistic and Coriolis interactions and radial derivative couplings between states of the same symmetry. The “atomic” basis IJMjljJb), on the other hand, appropriately describes the dynamics in the asymptotic region since it contains the proper boundary conditions for the production of atomic fragments in specific fragment states and it diagonalizes a. We refer to the former basis the total Hamiltonian for r as the molecular basis and the latter as the “atomic” basis. It is possible to obtain the transformation TB from the molecular basis to the atomic basis by recognizing that at infinite internuclear separation the Hund’s case a electronic states can be rewritten as

-

(AsZ) =

‘La&,

Ilaha)llbhb)(SZ)(Aahblh),r -

a

(4)

where A, and Ab are the quantum numbers for the projections of the total electronic orbital angular momenta I, + Ib of the fragments onto the internuclear axis. The spins are taken to remain coupled. Equation 3 is simplified when one fragment is produced in a S state (Le., either I, or /b = 0). Assuming l b = 0 the sum in (4)reduces to a single term because (AaAblh) = s(A,A). In most cases the coefficient (A,AblA) must be evaluated a t some large, but finite, internuclear separation r. The r-independent transformation matrix TB is then determined by substituting (4)into ( 2 ) and by examining how (3) projects onto this basis. Thus, a linear combination of the atomic basis functions IJMjoJb) is found to possess the same asymptotic form as obtained by substituting (4) into (2). (The details are described in ref 6 and 1.) Because TB is obtained from the r m form of the two basis sets, TB is independent of r and can be used at all internuclear separations to transform the Hund’s case basis to a molecular basis which has the appropriate asymptotic boundary conditions. B. Continuum Wave Function and Hamiltonian. The full Hamiltonian H ( r ) can be written as

-

where Helec(r)is the electronic Hamiltonian, Tnuc(r)is the nuclear kinetic energy operator, and Hrel(r)is the relativistic Hamiltonian which is taken for simplicity to be the spin-orbit Hamiltonian Hso(r). In general, hyperfine and spinspin interactions are added to Hre,(r),but this complicates the angular momentum algebra for the derivation of the transformation TBand introduces, in principle, additional channels in the close-coupling calculations. The nuclear kinetic energy operator is composed of radial and angular parts as

The Hund’s case a basis diagonalizes Helec(r)with energy eigenvalues that are the adiabatic Born-Oppenheimer (ABO) electronic curves. The off-diagonal elements of the orbital angular momentum operator l2 in this basis are the Coriolis coupling elements. At large r the atomic basis diagonalizes the full Hamiltonian H(r) since the electronic Hamiltonian Hels(r)tends in the atomic to zero. The operator Hso(r) is diagonal at r basis with diagonal matrix elements given by the atomic spin-orbit splittings of the fragment states; the radial part of TnUc(r) in this basis becomes a constant matrix for r a,and the operator l2

-

-

-

is diagonal since I is a good quantum number in our atomic basis for r a. The scattering wave function IkfiJbmamb(-))is a solution of the Schrodinger equation

[ E - H]IkJ&m,mb(-)) = 0

(7)

in which the full Hamiltonian H for dissociative motion includes all nonadiabatic couplings and A B 0 electronic curves which are needed to describe the dissociative dynamics. The scattering wave function IkJJbm,mb(-)) can be expanded in terms of energy normalized continuum wave functions IEJMjljJb(-))of total angular momentum J through the standard partial wave expansion IkJ~bmamb(-)) = IEJMjIjJb(-))( J M p l m u )CimliJbm,mb)i‘Yl,(k)*(8) Jjlmu

In practice, the close-coupled equation (7) is solved for a given value of J to obtain the spherical wave function l ~ J ~ j l j & j b ( - ) ) defined in (8). The spherical wave function lEJMjljJb(-))has the property that there is outgoing flux in the channels labeled by j , I , j , , and jb and incoming flux in all other channels. C. Transition Amplitudes and Differential Cross section^.^' The spherical continuum wave function IEJMjljJJ-)) is used to calculate the photodissociation or “half-collision’’ amplitude ( EJMjljJ,c-)Je.xlJoMovo)that describes the transition amplitude between the initial bound state and the dissociative continuum wave function with all nonadiabatic interactions between electronic states incorporated in the continuum wave function. The radiation field is assumed to be weak so that the transition between the ground and dissociative states can be described by first-order perturbation theory. The ground-state wave function lJoMovo) has initial angular momentum Jo with the electric dipole selection rule specifying the fin2.i angular momentum as J = Jo, Jo f 1 for the dissociative wave function. Mo is the projection of Jo onto the S F z axis; vo collects all other quantum numbers necessary to characterize the initial state and includes the vibrational quantum number vo, as well as the electronic quantum numbers A, S, and 2 . The photodissociation amplitude can be factored by using the Wigner-Eckart theorem to separate the dependence on Mo,M and the polarization indices q as

where the T(Jjlj&lJOlfO) are called reduced transition amplitudes. The sum in (9) can be removed if the incident light is in a state of pure polarization. For linearly polarized light it is convenient to choose the SF z axis along the polarization vector of the incident light, while for circularly polarized light the SF z axis is taken as parallel to the direction of propagation of the incident beam. Our general theory of diatomic photodissociation to open-shell atoms enables us to calculate the double differential cross section

where k is the direction in the SF coordinate system of the receding fragments and t is the polarization vector, relative to the SF z axis, of the emitted light from an excited photofragment or of the polarization vector of a probe laser for one of the photofragments. The uKsQs;KDQDare coefficients that are found through rather complicated angular momentum algebra1%7ss to be proportional to the bilinear sum of a reduced transition amplitude ~ ( J j l j J ~ l J ~ v , , ) and the complex conjugate of a second reduced amplitude (23) The basic notation of this section follows that of ref 1. The partial cross sections are those defined in ref 8 which are not identical with those of ref 1. The difference in partial cross sections from ref 1 is a result of the inclusion of cylindrically polarized light which cannot be written in terms of a spherical harmonics (see eq 9 and ref 8). The main result of this change is that many of the anisotropy parameters of ref 1 have slightly different algebraic forms in terms of the partial cross sections.

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Band et al.

The Journal of Physical Chemistry, Vol. 91, No. 21, 1987

I

-L5

I

-30

-15

0 cncrpyicm

'

15

30

L

L5

-45

0

'

energy/ctn

The anisotropy parameters (solid line) and pF(eM) (dashed line) along with the integral photodissociation cross sections for the X1Zt(u=0,J0=3) AIII(J) photodissociation of CH' to C'(22Pli2) using right-hand circularly polarized light. The orientation parameter pc(eM)for the Ct(22Pl/2)fragments is computed assuming that the C+(22P1/2)fragments are excited to the Ct(2D) state with right-hand circularly polarized light. Although both anisotropy parameters show structure, it is interesting to note the structure in BD which is not apparent in the integral cross section. Figure 2.

-15

-30

-

15

30

L5

-

Similar to Figui? 2 but for the X1Bc(v=0,J0=6) A'II(J) transition. Of special interest is the unresolved structure in pD near threshold and the structure at 10 cm-'.

Figure 3.

7*(J'j'l'jJbi,lJoqo).Here CKDQD(8,$)is a renormalized spherical harmonic,24 given in terms of the usual spherical harmonics yKQ(O*'#')by cKQ(flj'$)= [ 4 ~ / ( 2 k+ 1)1i'2YKQ(8,'#')

(1 1)

and '#'X;Q,(tS) is the photon polarization density matrix in spherical coordinates, as defined by O m ~ n tot ~be~ '#'K,Q~(~=

C(-1)1-4'(KsQsll 1q - q ' ) c q * c q ,

(12)

44'

The alignment and orientation of atomic fragments, along with various polarization ratios, can be expresed as simple algebraic expressions in terms of these the partial cross sections aKSQs;KDQD. When the polarization of final fragment states is not determined or when all spontaneously emitted light is collected regardless of its polarization, then the differential cross section for detection of atomic fragments reduces to d 4 ) / d f i k = 8*

C ~oo;KDQD~KDQD(k)

(13)

KDQD

Similarly, by integrating (10) over all fragment angles k we obtain the differential cross section for the polarization of atomic fragments as

- L45

-30

-15

0

15 . 30

-

energy c m

L5

du(k)/dfik =

+ PDP2(k'2)]

UO[~

(15)

where PD

=

uO020/'OO;O0

(16)

P2(i(.2) is the second Legendre polynomial, and 2aaois the integral photodissociation cross section. (24) Brink, D. M.; Satchler, G. R. Angular Momentum, 2nd ed.; Clarendon: Oxford, 1968. ( 2 5 ) Omont, A. Prog. Quantum Electron. 8 1977, 5 , 69.

75

90 1

The X1Z'(v=5,J0=11) AlII(J) transition for production of Ct(22P1/2)fragments (see caption to Figure 2). Structure is seen to exist well above threshold to the Ct(22P3i2)limit at 21.3 cm-I. Figure 4.

In a similar fashion an anisotropy parameter PS for the angular distribution of fluorescence is defined as Ps = -21~2~20;00/~00;00

(17)

The familiar polarization ratio

P = (111 - li)/(lll The total or integral cross section is within a constant factor uoo;oo and is related to a linear superposition of the 17(Jj~j,iblJoqo)]2. D. Anisotropy Parameters. The partial differential cross section for fragments (eq 13) can be written in terms of the anisotropy parameter PD as

60

'

- 1,)

(18)

where 11,and I , are the intensities of the spontaneously emitted (or probed) light polarized parallel and perpendicular to the original linearly polarized radiation, is related to os by p = 3 P d 4 + PSI

(19)

This parameter Ps is proportional to the alignment A , of the fragments as defined by"J2 A , = (j2 - 3jZ2)/jg'+ 1)

(20)

where the average is over the states of the emitting fragment. If right and left circularly polarized light are used for the dissociation and probing of fragments, it is possible to measure an additional anisotropy parameter which is related to the orientation 0, as given by 0 0

= ( j 2 ) / W + 1)11'2

Final fragment states with j =

(21)

have A . = 0 but because 0,

Photodissociation of Diatomic Molecules

The Journal of Physical Chemistry, Vol. 91, No. 21, 1987 5407

does not necessarily vanish, the orientation 0,may be useful to analyze the structure in cross sections when the fragment states have j = The circularly polarized photodissociation ratio is defined by Pc = (1,

- 11)/ (1, - 11)

(22)

where I , and II are the intensities of right and left circularly polarized light emitted from excited photofragments or used in the probing of photofragments. P, is, in general, a complicated function of both the orientation and the alignment. In a recent paper,I6 it is shown that a judicious choice of angles makes P, become proportional to the orientation only. The required angle is the “magic angle” e M with respect to the SF z axis at which the second Legendre polynomial Pz (cos (e)) vanishes. At this angle we have pc(eM)

=

alQOO/

(2

’2aOQOO)

(23) 2’0

which is simply proportional to 0,.

io

io

50

$0 energy/cm-’

7b

io

do

lob1

-

111. Calculational Results

The theory of section I1 is applied to the near-threshold photodissociation of CH+ with fragments a and b taken to be C+(zP) and H(2S), respectively, where 1, = IC+ = 1/2, j , = Jc+ = (3/z,1/z] for the two fine structure states of the C+ ion. The Hamiltonian (5) in the atomic basis can be rewritten as H(r) = TBVABO(r)TBt - 1/(2pr)d?rl

+ 1/(2pr2)12+ Hso (24)

where TB is the basis transformation from our “atomic” to the molecular Hund’s case a basis, and VABO(r)is a diagonal matrix of A B 0 electronic potential curves.26 The matrix of Hso is diagonal in the atomic basis with eigenvalues given by the asymptotic spin-orbit splitting of the C+ fragment, while l2 is diagonal in this basis with matrix elements equal to the eigenvalues of (J - j)2 at infinite separation. The approximation of replacing the Coriolis and spin-orbit interaction matrices by their r m form is common in scattering t h e ~ r y ~ and ’ - ~ is~ invoked because the required electronic matrix elements for C H + are currently not available. These approximations can be qualitatively justified in scattering dynamics by noting that the nonadiabatic couplings become important in CH+ only as the coupled potential surfaces become nearly degenerate. However, this arises in CH+ at intermediate r a n d is, hence, in the region where the coupling elements must begin to approach m values. This approximation can be insufficient for their r shape resonances at high J when the rotational barrier is at small r. The Schrodinger equation (7) is solved numerically at energy E to obtain the spherical continuum wave function IEJMjljc+jH(-)) as defined in (8). The photodissociation or reduced transition amplitude of (9) is evaluated simultaneously to give the amplitude for first-order electric dipole transitions from an initial bound state of angular momentum Jo to a final continuum state of angular momentum J . Experimental spectra implicitly contain a sum over all possible final states and an average over the initial states present. On the other hand, theoretical total cross sections may be evaluated for the hypothetical transition from a given initial state to a selected final J state, with the observable cross section obtained as a sum of these partial cross sections. Because interference effects between states of different J may contribute and be important, the calculation of angular distributions and

-

-

( 2 6 ) A description of the A B 0 potentials, transition dipole functions, and Coriolis and spin-orbit coupling elements for CH+ are given in ref 10 and references therein. (27) Gay, J. C.; Schneider, W. B. 2. Phys. A 1976,278,211and references therein. (28) Launay, J. M.; Roueff, E. J. Phys. B 1977,I O , 879. Mies, F. H. Phys. Reu. A 1973, 7, 942. Weisheit, J. C.; Lane, N. F. P ~ Y SRev. . A 1971,4, 171. (29) Van Vleck, J. H. Phys. Rev. 1929,33, 467.

Figure 5. The anisotropy parameters Po (solid line) and Ps (dashed line) for production of C+(22P3/2)fragments in the X1B+(u=5,Jo=11) AIII(J) transition. The integral cross section is also shown. At even higher angular momenta additional structure emerges in the anisotropy parameters and cross sections for production of C+(22P3,2)fragments?

isotropy parameters requires the inclusion of all final J = Jo f 1 states. Close-coupled calculations for the integral photofragment cross section uWm, the asymmetry parameter pD governing the angular distribution (16) of photofragments, and the orientation parameter Pc(eM) of (23) for production of oriented photofragments are shown in Figures 2-4 for the XIB+ AlII transition in CH’. These three observables are displayed for the two final fine structure states of the C+ fragment as a function of the excess dissociation energy where the zero of energy taken as the barycenter of the C’(zP) fine structure fragment states. This choice fixes Cc(zPljz) as having an energy of -42 cm-’ and C+(2P,Iz) with an energy of 21.3 cm-I. Figure 2 presents results of calculations for the principal perpendicular transition X’Z+(v=0,Jo=3) A’II(J) where C+(2P1/2)fragments are produced. Figures 3 and 4 show the calculationed u o ~ o oPo, , and Pc(eM) for Jo = 6 and Jo = 1 1 , respectively. These figures provide the first multichannel calculations for P c ( e M ) . All of the observables in Figures 2-4 exhibit a wealth of structure for the near-threshold dissociation of CH’. The resonances of Figure 2 have previously been assigned on the basis of a series of multichannel calculations.1° The individual electronic potentials are modified by adding or subtracting a small square well onto the bottom of these potentials, leading to shifts in certain resonance positions. The information from such calculations is combined with various calculations on identical A B 0 curves but where different transitions, such as a 3 n z c3Z(J), are excited to aid in the assignment of the resonances (see ref 7 for a more thorough discussion of the assignments). Comparison of such calculations allows the assignment of the major features in Figure 2 for the integral cross section. Similarly, comparison with the anisotropy parameter PD of Figure 2 shows that many of the features of PD can be related to resonances of a particular final J . Some resonances, however, are more pronounced in pD, and, therefore, Po provides additional dynamical information which may be useful when applied to actual experimental data to aid in untangling the different final J contributions to the integral cross section. The parameter Pc(eM) also has some structure, but not to the extent of pD in the calculations presented herein. The very low translational energy region-for Jo = 6 and Jo = l 1 is in Figures and and has behavior to that of Jo = 3. Once again, 80 exhibits additional structure which is not apparent in the integral cross section uwoo but which can often be related to features that exist in the partial cross sections computed for a momentum J’ Figure Presents the three basic Wantities for the production of C+(’p3p) fragments from Jo = 11. The anisotropy parameter Ps is displayed

-

-

-

5408

The Journal of Physical Chemistry, Vol. 91, No, 21, 1987

os

instead of the parameter PC(0,). Note that is not relevant for the production of C+(2Pl,,) fragments since. it has no meaning forj, < 1.

IV. Conclusions Many of the rules of diatomic spectroscopy can be applied to explain the structure in the predicted low-energy photodissociation spectra of CH+.l0 The increased number of resonances over those previously expected arises because of the breakdown of electronic selection rules due to asymptotic nonadiabatic couplings. The resonances may, however, be assigned effective vibrational quantum numbers and rotational constants. One problem in unraveling the total cross section is that many of the resonances, which are apparent in the hypothetical state to state cross sections, are washed out in the integral cross section where there is a coherent sum over final angular momenta J = Jo, Jo 5 1. This occurs because of different Franck-Condon factors for the P, Q, and R branch transitions, as well as because there are several resonances with different final J lying in the same energy region. Anisotropy parameters, such as OD,may be helpful in unraveling some of the “hidden” dynamics in systems like CH+ because the contributing resonances often enter into PD with different weights than in the total cross section. The structure present in an anisotropy parameter may point either to resonances which are not apparent from the integral cross section or to interference effects between resonances. In addition, measurements of fragment kinetic energy can be useful in analyzing resonances when several initial states are involved. The full quantum theory of diatomic photodissociation thus predicts many novel features in the near-threshold dissociation of diatomics. This theory is the first to include the angular recoupling algebra necessary to describe both the internal angular momentum of the parent molecule and that of the fragments produced in the half-collision. New experiments, such as the dissociation of diatomics in magnetic fields,22may provide additional information, beyond that described in this paper, on the near-threshold dissociation of diatomic molecules to atoms with fine structure. We are currently working on the theory of near-threshold photodissociation of diatomic molecules in the presence of a magnetic field and predict several new interesting features. Among these are the possibility of additional dynamical effects arising from coupling of states of total angular momentum J by the magnetic field with states of different total angular momentum. In addition, the presence of a magnetic field ensures that there are orientational effects for all open-shell fragments even when linear polarized light is used in the dissociation dynamics, and lastly, we predict that the presence of a magnetic field in diatomic photodissociation allows for the energetic selection of a single magnetic components of both atomic fragments.30 This last point can be understood by considering a diatomic molecule in the presence of a magnetic field which dissociates into atomic fragment states b,m,)pbmb) where m, is its projection along the magnetic field direction (which we take to be the space-fixed z axis). The atomic fragment state with the lowest energy is given by IcJlm,‘ = j a ‘ )Jc&’m{ = j { ) where j,l labels the lowest spin-orbit multiplet of the spectral term limit [for example for C+(22P) + H(12S) the state IcJa’m,’)Ic${m{)= 132P,’/2,-1/2)~12S,1/2,-1/2)]. Consider the photodissociation of the ground vibronic state of the diatomic molecule using a photon whose energy is such that the only channel open asymptotically is ~ ~ m , ’ ) ~ { mthen < ) , the fragments must be obtained in states of pure polarization given by m,’ and m i . This may be a viable method of producing sources of polarized atoms which cannot be easily produced by optical pumping or other means. Photodissociation of a single initial bound state with a singlemode laser beam permits precise control of the total energy of the system and its angular momentum. The use of polarized light in the photodissociation process introduces anisotropy into the (30) Band, Y. B.; Bersohn, R.; Freed, K. F.; Williams, C. J., to be published.

Band et al. diatomic system to provide the possibility for extracting additional information from the polarization of the atomic fragments. Thus, photodissociation dynamics is an ideal means of learning about the potential surfaces and interactions among these potential surfaces. It provides a means of learning about “dark states” (for example, triplet states) which are not accessible in ordinary bound-state spectroscopic methods when transitions from the ground state of the molecule to these states are forbidden optically. Although scattering experiments can also be used in principle to obtain similar information, the inherent average over many angular momentum channels participating in scattering experiments makes it more practical to extract information from photodissociation experiments. The theory of photodissociation of triatomic molecules on a single potential energy surface has been developed over the past 10 year^.^'^^^ Polyatomic molecule dissociation provides the possibility of transitions due to nonadiabatic coupling occurring at surface crossings occurring at finite fragment separations. One such example is the photodissociation of CH31to CH3 + I(2P,/2zj2) but many more examples In addition to the nonadiabatic transitions at interior surface crossings, nonadiabatic effects must also occur at large fragment separations in systems correlating to open-shell fragments. These nonadiabatic interactions arise by virtue of spin-orbit, nuclear kinetic energy, and hyperfine interactions just as in the diatomic photodissociation. What are the generalizations of the present theory which enable the description of nonadiabatic effects in triatomic molecules? This is an open question and work in this area is vital to properly treat the influence of nonadiabatic effects which operate at large internuclear separations as the adiabatic surfaces correlating to the open shell fragment states begin to coalesce. A sketch of what must be done can be made. The Hamiltonian for the system can be written as where H0,=(r)is the electronic Hamiltonian, Tnuc(r)is the nuclear kinetic energy operator, and Hso(r) is the spin-orbit Hamiltonian, but now r must include the internal nuclear coordinates of the diatomic fragment in addition to the internuclear fragment separation. The nuclear kinetic energy operator is composed of radial and angular interfragment motion as well as the internal kinetic energy of the diatomic fragment (including vibrational and rotational energy). We must determine two molecular basis sets which provide a zeroth-order description of the half-collision dynamics in the molecular and the asymptotic regions, respectively, and it is necessary to introduce a transformation T, between these two basis sets. The atomic molecular basis must contain the proper boundary conditions for the production of fragments in specific 03 and the fragment states. It must diagonalize H(r) for r individual basis functions become a product of fragment states multiplied by a wave function describing free fragment translation. The molecular basis must properly describe the zeroth-order dynamics in the molecular region and must diagonalize the electronic Hamiltonian without the nonadiabatic interactions. We

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(31) Band, Y. B.; Freed, K. F. J. Chem. Phys. 1975, 63, 3382; J . Cbem. Phys. 1975,63,4479. Morse, M. D.; Freed, K. F.; Band, Y. B. J . Cbem. Phys. 1979,70,3604. Morse, M. D.; Freed, K. F.; Band, Y. B. J . Cbem. Phys. 1979, 70, 3620. Band, Y. B.; Freed, K. F. J . Chem. Phys. 1977, 67, 1462. Band, Y. B.; Freed, K. F. J . Chem. Phys. 1978,68, 1292. Band, Y. B.; Morse, M. D.; Freed, K. F. J. Chem. Pbys. 1978, 68, 2702. Morse, M. D.; Freed, K. F. J . Chem. Phys. 1983, 78, 6045. Morse, M . D.; Band, Y. B.; Freed, K. F. J . Chem. Phys. 1983, 78, 6066. (32) Shapiro, M. Chem. Phys. Letr. 1977, 46, 442. Beswick, J. A.; Shapiro, M.; Sharon, R. J . Chem. Phys. 1977, 67, 4045. Heller, E. J. J . Chem. Phys. 1978, 68, 3891. Heller, E. J . J . Chem. Phys. 1978, 68, 2062. Heller, E. J. Acc. Chem. Res. 1981, 14, 368. Shapiro, M.; Bersohn, R. J. Chem. Pbys. 1980, 73, 381. Segev, E.; Shapiro, M. J . Cbem. Phys. 1980, 73, 2001. Segev, E.; Shapiro, M. J . Chem. Phys. 1982, 77, 5604. Heather, R. W.; Light, J. C. J . Chem. Phys. 1983, 78, 5513. Heather, R. W.; Light, J. C. J . Chem. Phys. 1983, 79, 147. Atabek, 0.;Beswick, J. A,; Delgado Barrio, G. J. Chem. Phys. 1985,83,2954. Schinke, R. J. Chem. Phys. 1986,85,5049. Schinke, R.; Engel, V. J . Chem. Phys. 1985, 83, 5068. Schinke, R.; Engel, V.; Andresen, P.; Hausler, D.; Balint-Kurti, G. G. Phys. Reu. Lerr. 1985, 55, 1180. (33) Hennig, S.; Engel, V.; Schinke, R J . Chem. Phys. 1986, 84, 5444. Schinke, R. J . Chem. Pbys. 1986, 84, 1487.

J. Phys. Chem. 1987, 91, 5409-5412 need to calculate a photodissociation amplitude similar to ( EJMj/jJ,,(-)lt.xlJ,&foso) that describes the transition amplitude between the initial bound state and the dissociative continuum wave function with all nonadiabatic interactions between electronic states incorporated in the continuum wave function. This am(see eq 9). plitude may be factored just as in the diatomic For a triatomic photodissociation one must work out the nonadiabatic couplings between the adiabatic channels in the continuum

5409

wave function expressed in the molecular basis set representation and to form the transformation between the molecular basis and the atomic basis. To do so will require a good deal of effort. Acknowledgment. This research is supported, in part, by N S F Grant C H E 83-17098 and the U.S.-Israel Binational Science Foundation. Registry No. CH’,

24361-82-8; C’,

14067-05-1; H , 12385-13-6.

The C0,DBr Precursor Geometry Limited Reaction of Deuterium with Carbon Dioxide S. Buelow: G. Radhakrishnan,t and C. Wittig* Chemistry Department, University of Southern California, Los Angeles, California 90089-0484 (Received: January 26, 1987)

We report nascent OD(X*II) rotational, vibrational, spin-orbit, and A-doublet excitations from reactions of deuterium atoms with C02. D atoms are produced by the 193-nm photolysis of DBr (hv - Do = 244 kJ mol-’) (i) within a weakly bonded COzDBrcomplex, and (ii) under 300 K singlecollision “bulk”conditions. The differencesbetween the resulting OD distributions are modest, and the present results are similar to those of the analogous H + COz system.

Introduction In previous publications,’ we described experiments in which a weakly bound binary complex, COZHBr,was used as a precursor for studying the reaction H(%)

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+ COz(XIZ+) OH(X211) + CO(X’Z+)

AHzs8 = 107 kJ mol-’ (1)

Photodissociation of HBr within the complex propels the H atom toward COz with a set of angles and impact parameters dictated by the equilibrium geometry and zero-point fluctuations of the precursor. Such precursor geometry limited ( E L ) reactions offer opportunities for studying elementary processes under unique conditions and with high precision, and it may even be possible to select processes which are thermodynamically unfavorable and would not be major product channels in homogeneous gas phase or crossed molecular beam environments (e.g., SCOHX + hv CS OH X).2 In the case of COzHBr, initial HBr electronic excitation remains sufficiently localized that the serial picture of HBr photodissociation and subsequent reactive scattering is valid, albeit with significant perturbations and exit channel interactions due to the nearby Br atom. However, in cases where electronic excitation of the complex is delocalized relative to that of the isolated species, opportunities exist for studying the photochemistry of excited aggregates and/or reactions involving excited states of a particular moiety. This has been exploited by Soep and co-workers3 in an elegant series of experiments which can test many of the basic premises of orbital stereospecificity in gas-phase reactions. In the present publication, we complement our previous work by providing data for the reaction

+

D(%)

-

+

+

-

COz(X’Z+) OD(XZII) CO(XIZ+)

+

AHzs8= 103 kJ mol-’ (2)

obtained under PGL and “bulk” (300 K, single-collision) conditions, where nascent OD(X*II) is detected by LIF. Since the present experiments and motivations are quite similar to those described previously, we will (i) forego further introductory comments, (ii) present only those experimental details which are germane to the present work, and (iii) present and discuss the +Present address: Los Alamos National Labs., Division CLS-4, Mail Stop 5-567, Los Alamos, N M 87545. ‘Present address: Jet Propulsion Laboratory, Pasadena, CA 91 109.

results within the context of our previous publications.]

Experimental Section The experimental conditions were similar to those described previously.lb C02DBr clusters were produced by expanding a mixture containing 0.5% DBr and 3.0% COz, diluted in 96.5% He, through a pulsed nozzle into a vacuum chamber. The dependences of the relative concentrations of the prepared complexes on backing pressure and mixture ratios were similar to that of COzHBr.Ib The complexes were monitored downstream with a separately pumped quadrupole mass spectrometer. Photolysis was done with a 193-nm ArF excimer laser, and nascent OD(X211) concentrations were detected by using laser-induced fluorescence (LIF). The photolysis and probe lasers were collinear, and we did not attempt to determine effects due to laser polarization^.^ HBr photolysis at 193 nm occurs via a perpendicular transition, producing approximately 85% Br(2P312)and 15% Br(2P,,2),5and we assume that DBr is similar. Nearly 98% of the available kinetic energy, 244 and 206 kJ mol-’ for Br(*P3/J and Br(2P,12),respectively, appears as translational energy of the D atoms. The resulting velocities of D atoms are -70% of the H-atom velocities.. The velocities corresponding to the lower and upper Br spin-orbit states are 1.5 X lo6 and 1.44 X lo6 cm s-’. O D spectroscopic constants and A22+ XzII fluorescence lifetimes are from ref 6. Deuteriation of the vacuum line and nozzle assembly was achieved by flushing the system with DzO, evacuating, and then seasoning with high concentrations of DBr. [D]/[H] ratios were determined to be >20.

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(1) (a) Buelow, S.;Radhakrishnan, G.; Catanzarite, J.; Wittig, C. J . Chem. Phys. 1985, 83,444. (b) Radhakrishnan, G.; Buelow, S.;Wittig, C. J . Chem. Phys. 1986, 84, 127. (2) Koplitz, B.; Xu, Z.; Baugh, D.; Buelow, S.; Hausler, D.; Rice, J.; Reisler, H.; Qian, C. X. W.; Noble, M.; Wittig, C. Faraday Discuss. Chem. SOC.1986, 82, 12. (3) (a) Jouvet, C.; Soep, B. Chem. Phys. Lett. 1983, 96, 426. (b) Jouvet, C.; Soep, B. Laser Chem. 1985, 5 , 157. (c) Jouvet, C.; Soep, B. J . Chem. Phys. 1984,80, 2229. (4) (a) Case, D. A,; McClelland, G. M.; Herschabach, D. R. Mol. Phys. 1978,35, 541. (b) MacPherson, M. T.; Simoms, J. P.; Zare, R. N. Mol. Phys. 1979, 38, 2049. (5) Magnotta, F.; Nesbitt, D. J.; Leone, S. R. Chem. Phys. Lett. 1981, 83, 21. (6) (a) Clyne, M. A. A,; Coxon, J. A.; Woon Fat, A. R. J . Mol. Spectrosc. 1973, 46, 146. (b) Dimpfl, W. L.; Kinsey, J . L. J . Quantum Spectrosc. Radial. Trans. 1979, 21, 233.

0022-3654/87/2091-5409$01.50/00 1987 American Chemical Society