Quantum-mechanical calculations of the dissociation of H-3 Rydberg states

Quantum-mechanical calculations of the dissociation of H, Rydberg states Jeffrey L. Krause, Kenneth C. Kulander, and John C. Light@ Theoretical Atomic and Molecular Ph#cs Group, Lawrence Livermore National Laboratory, Liver-more, Cal@ornia 94550

Ann E. Ore1 Department of ‘Applied Science* Univemity of California, Davis, Livermore, Caiifornia 94550

(Received 22 October 199 1; accepted 9 December 199 1)

We present t.hree-dimensional, time-dependent quantum-mechanical calculations of the dynamics of the dissociation of H, Rydberg states at total energies up to 6 eV. The method used in this work employs a Chebychev propagator in time, and computes the kinetic-energy operators in the discrete variable representation. We calculate the total dissociation cross section, as well as partial vibrational and rotational cross sections, and compare our results to previous two-dimensional calculations and to experiment. The results display clear three- dimensional effects, and indicate the importance of including both sheets of the H, ground potential-energy surf&ace in the dynamics.

INTRODUCTION

The I-I, molecule is unstable in its ground electronic state. However, Rydberg states of H, formed by attaching an electron to the Hjf core can be quite long-lived.’ Exten- sive information exists on the spectroscopy of H, , both experimental’ and theoretical.” The low-lying Rydberg states decay either by predissociation or bound-free photon emission to the repulsive ground electronic state.” Recently, several experiments have been performed to study the dynamics of the dissociation of I-I,. 5*6 Predissociation of H, produces vibrationally excited H,, and so it may be important in un- derstanding the eficiency of 11~ generation in hydrogen plasmas.a The Rydberg states also provide a way to probe regions of the H, ground potential-energy surface not readi- ly accessible in conventional atom-diatom reactive scattering experiments.

Experiments by Peterson and co-workers5 begin by mass selecting Hi’ from a hydrogen discharge. The Hjt is then neutralized in cesium vapor, producing electronically excited II, in a distribut.ion of Rydberg states, predominant- ly n -= 2 and 3, where n refers to the principle quantum number of the Rydberg electron. Most of the HT states quickly dissociate, yielding both two-body and three-body products,

HQ-+H2 (v,J) -+- H (la)

+H+H+H. (lb) Some of the H atoms generated by the reaction capture an electron from Cs, and Peterson and co-workers measure the kinet.ic energies of the resulting H -. These data are trans- formed to produce the center-of-mass kinetic-energy release spectrum. The spectrum shows two main peaks, which are assigned to the two-body and three-body channels of Eq. ( 1). By assuming a geometrical model for the three-body

a) Permanent address: The Department of Chemistry and the James Franck Institute, The University of Chicago, Chicago, IL 60637.

dissociation (i.e., collinear or triangular), the two-body to three-body branching ratio can be derived.

In another set of experiments, Cosby and Helm6 begin with a source of H, similar to that of Peterson and co- workers. However, they allow the beam to drift for about 1 ,us before probing it with a laser. Ry this time, all of the H, Rydberg states have decayed except for the rotationless (N = K = 0) levels of the 2p “A ; state (see Fig. 1) . Symme- try constrains this electronic state to predissociate via rotational coupling, so the rotationless levels, which eventually decay either radiatively or via spin-orbit coupling, are rela- tively long-lived.“*h*x29 The u = 0 level, for example, lives for about 640 ns.4 Cosby and Helm excite the 2p ‘A 2 state with a laser to the n = 3 Rydberg levels, which rapidly dissociate. They then use high-resolution translational spectroscopy to measure the vibrational and rotational final-state distributions of the Hz product.

The experiments by Peterson and co-workers and Cosby and Helm have yielded a wealth of detailed data which previous two-dimensional calculations’” were able to model qualitatively, but not quantitatively. In this paper we present results of three-dimensional quantum-mechanical calculations of the dynamics of the dissociation of H, . We describe briefly the method we have developed to study this problem, and compare our results to previous calculations and to experiments.

METHOD The H, system has long been a favorite for theoretical

study because of the light masses involved and the availability of accurate ab initio potential-energy surfaces for the ground electronic state. Fully converged three-dimensional (3D) quantum calculations of reactive scattering on the ground H + H, surface have now been performed,” and the results agree essentially perfectly with experiment.12 Several aspects of the H, predissociation problem make direct comparison with experiment considerably more challenging. For one, as shown in Fig. 1, the n = 3 Rydberg levels studied in the experiments of Cosby and Helm lie about 7-8 eV above

J. Ghem. Phys. 96 (6), 15 March 1992 0021-9606/92/064283-l O$OS.OO @ 1992 American institute of Physics 4283 Downloaded 16 Aug 2003 to 128.135.132.83. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

4284 Krause eta/.: Dissociation of H, Rydberg states

H+H2 n I I 1 I “0 1 2 3 4 5

r2 (bohr)

FIG, 1. Potential-energy surfaces for the “lower” H + H, and “upper” H + H + H sheets of the H, ground state, at 8 = 90” and a fixed H, bond length of 1.65 a.u., the equilibrium bond length in H:. Also shown in the figure are the energies of some of the H: Rydberg states relevant to the experiments of Cosby and Helm (Ref. 6).

the H( 1s) + H, (u = 0, J = 0) dissociation limit, which is considerably more energetic than the l-2 eV at which converged reactive scattering calculations have been accomplished to date. Also, all of the Rydberg states are unstable with respect to dissociation to both the H + XH, ‘8, and theHfbH, 32, (orH+H+H)sheetsoftheH, ground electronic potential-energy surface. These surfaces cross in a conical intersection (or seam) at D,, (equilateral triangle) geometries in the Franck-Condon region,lJ so a complete characterization of the reaction must include the effects of the dynamics on two coupled potential surfaces. Peterson and co-workers observe about a 30% contribution from the three-body channel,’ and Cosby and Helm see about 10% from the predissociation of the 3d *E o state.6 However, since three-body dissociation is possible on both sheets of the ground-state surface, the three-body branching ratio is not sufficient to estimate the magnitude of the effect of the second surface on the dynamics.

One computational advantage in studying the H, dissociation problem compared to H + H, reactive scattering is that the initial state is localized and has well-defined quantum numbers, so extensive averaging over impact parameters is not necessary. The dynamics also occur on a fairly rapid time scale with a well-defined zero of time. For these reasons, we have chosen to use a time-dependent wave-packet method. Such methods have no serious problems with coupled potential surfaces”4*‘5 or a three-body continuum,” both of which can be difficult to treat with a basis set. Wave- packet methods have the additional advantage that the cross section is obtained at all energies in a single calculation, in contrast to coupled-channel-type methods, which must be repeated at each energy. However, each initial state requires a separate calculation in time-dependent methods, while

time-independent methods can typically be extended to additional initial states with little extra effort. Recent advances in propagator technology l6 have increased the efficiency of time-dependent techniques to the point that 3D calculations are now feasible.

The method used in this work proceeds by direct integration of the time-dependent Schriidinger equation,

i&W,0 = fiWr,O, (2)

where, as throughout this paper, we have used atomic units. In Jacobi coordinates (see Fig. 2), for zero total angular momentum, the Hamiltonian can be written asI

1 ap 1 d2 H(r,,r*,B) = ------ %l ar: 34 ad

-&-J&-(sin@&) + V(r,,r*d% (3)

wherep, is the reduced mass of H, , ,u2 is the reduced mass of H + H2, and 1, the moment of inertia, is given by

1 1+1 -=- I Pl4 Z’

(4)

The time propagation in Eq. (2) is performed with the Chebychev method,‘” which proceeds by expanding the time evolution operator

U(r) =e lHr (51 in a set of complex Chebychev polynomials. In contrast to some early applications of time-dependent methods which

FIG. 2. Jacobi coordinate system r, , r, , and 0 used in this work.

J. Chem. Phys., Vol. 96, No. 6,15 March 1992 Downloaded 16 Aug 2003 to 128.135.132.83. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

used approximate propagators, or propagators exact only in special potentials, the Chebychev method is formally exact. By choosing the number of terms in the expansion, the propagation can be accomplished with a predetermined preci- sion. The Chebychev propagator has proven to be accurate and eficient in a number of applications.”

The kinetic-energy operators in Eq. (3 ) are calculated in the discrete variable representation (DVR) .” The DVR begins (formally) by expanding the wave function in a basis of orthogonal polynomials #i (x) and then performing a uni- tary transformation to a representation in which the coordinate operator X is diagonal. That is, if we define the coordinate operator in the polynomial basis as

x, - (4, (x9 I-# (x9 ), and the DVR transformation by

(69

T’XT = 2, (79 then 2 is a diagonal matrix whose elements, the eigenvalues of the coordinate operator, are the grid points of the calculation. Associat.ed with each grid point is a weight that is defined by the polynomial basis. In the case of orthogonal polynomials, X is a tridiagonal matrix that can be generated efficiently using the recursion relations of the polynomials, and the weights are well-known quantities. The kinetic-energy operators in the DVR are not diagonal. However, operators that are functions of a coordinate can be approximated to n-point Gaussian quadrature accuracy as the value of the operator at the grid points. Thus, for example, the trans- formed potential, TtV(r, ,r, ,B)T becomes V( r;,e,e “9, where cz labels the grid points, not the basis functions.

There arc several advantages to the DVR. One is that the entire Hamiltonian matrix is never explicitly construct- ed, which saves considerably on storage requirements. An- other is that the DVR can be formulated almost entirely in terms of matrix times matrix operations, which are very efficient on vector supercomputers, Perhaps most importantly, the grid can be tailored to the dynamics by choosing the polynomial basis that generates the optimal DVR for the problem. In this work we use a Chebychev basis for the DVR in the two radial dimensions, which yields an evenly spaced DVR with a constant weight at each grid point and boundary conditions of zero at the edges of the grid. In the angular dimension we use a Legendre DVR, which concentrates the integration weight near 90”. The Legendre DVR has continuous boundary conditions which allow the f 90” symmetry of the Jacobi coordinates to be built explicitly into the calculation. In addition, the kinematic symmetry of the H, system allows the even and odd Legendre solutions to be propagated separately.

In practice, the DVR is a very efficient computational method. We find that we can use a much larger grid spacing than that required by, for example, a finite difference representation of the kinetic-energy matrices. This leads to reduced storage requirements as well as reduced computation time, in spite of the increased overhead in the DVR. The grid spacing in the DVR is constrained by the accuracy required to converge the integrals, as well as the requirement that the maximum kinetic energy of interest be well characterized. For the Chebychetp DVR, the maximum energy that can be

Krause et&: Dissociation of H3 Rydberg states 4285

supported by the grid in dimension P is

Em,, = E-=/~/LA;,

where A, is the grid spacing. Equation (2) is an initial value problem and requires as

input the starting state of the wave packet on the grid. In the absence of sufficiently detailed information about the HT potential-energy surfaces, we use as the initial state an eigen- state of H3”. We obtain this wave function using a method described previously” by diagonalizing the H,* potential on the same DVR grid used in the dynamics calculations. We assume in this procedure that attaching a Rydberg electron to the H;+ core does not greatly distort the molecular geometry. As can be seen by examining the Hz potential compared to the NY Rydberg potentials,** this is probably a very good approximation, at least for the lower vibrational levels. For example, the equilibrium bond length in Hit is 1.656 a.u., compared to 1.637 a.u. in the 2p *A ; state of HT.’ For the potential of Meyer and co-workers,‘” on a typical grid used in this work, we obtain an energy of 4364.5 cm ~~ ’ for the zero-point energy of H3”, which compares well with previous theoretical values.‘”

For the ground electronic state of H, , we use the potential surfaces calculated by Varandas and co-workers.“’ The 2p *E’ ground state of H, is doubly degenerate with a conical intersection at D,, symmetry.r3 One component of this state (the ‘A,, in C,,, symmetry) correlates asymptotically to H2 (v,J) + H, and the other (the ‘B, ) to H -I- H $ H. The calculations in this paper were all performed on either the “upper” (H + H + H) or the “lower” (H + H, ) adiabatic potential surface. We will discuss the consequences of this approximation below.

As discussed by previous authors, ‘oS14~26P27 two methods exist in the time-dependent formalism to calculate the total dissociation cross section for a bound-free transition. One is based on the “short-time” behavior of the wave packet, and the other on the “long-time,” asymptotic projection of the wave packet onto final states. In the short-time method, the total dissociation cross section is given by the Fourier trans- form of the autocorrelation of the initial state with the time- evolved wave packet on the dissociative surface,“6

c(E) -P f

a dte’E’(IC(r,O)I~(r,t)). -co

(99

In this equation, Eis the total energy of the wave packet, and the initial state I$[ r,O) ) is a bound rovibrational wave function of the excited surface times an operator Y(r) that cou- ples the bound and dissociative surfaces, and P is a prefactor that depends on the dissociation mechanism. The time- evolved wave packet on the dissociative surface is then given by

IW,t)) = l’ dt’ U(t’>$(r,O). (10)

At a total energy equal to the energy of the initial Rydberg state, the molecule can predissociate, in which case Y is the nonadiabatic coupling operator. At total energies less than the energy of the initial state only radiative pathways are possible, and so ‘Y is the dipole operator. Since photons can


4286 Krause eta/.: Dissociation of H, Rydberg States

be emitted at any (positive) frequency, the radiative dissociation spectrum is continuous. However, for predissociation, Eq. (9) is valid only at total energies equal to the energy of a Rydberg state, and thus the predissociation “spectrum” is disc.rete.

We would like to note that Eqs. (9) and f. 11) are the time-dependent analogs of the time-independent expression for a transition (e.g., photodissociation) from a bound state I& ) to a scattering state [$I( n,E - ) > with “incoming” boundary conditions at energy Eand quantum numbers n,28

In the short-time method of Eq. (9)) the wave packet must be propagated only as long as some part of it remains in the Franck-Condon region. However, this method can determine only the total dissociation probability. To obtain the partial dissociation cross sections to specific fragment final states, we use a long-time method that depends on the asymptotic properties of the wave packet in the calculation.

The long-time method proceeds by evolving the wave packet until it reaches the asymptotic regions of the potential surface, and then projecting it onto appropriate translational, vibrational, and rotational functions’“*14*27

$(E)-limP$- t-m

2 X&b, )Ili(r,,r,,m , (11)

where k is the momentum of the hydrogen atom, +r are the translational functions, q5j are the rotational functions, r$Uj are the vibrational functions, ‘)’ labels the asymptotic arrangement, and P is once again a constant that depends on the coupling mechanism. The sum of the partial rotational and vibrational cross sections in Eq. ( 11) gives an alternate expression for the total dissociation cross section,

(~,(E)-l~?s,l~lW@ - )>I”, (14) and are valid under the same conditions. In particular, Eqs. (9) and ( 11) can be derived in the limits of first-order per- turbation theory and weak cw (continuous wave) laser fields.‘“,“” These equations prescribe a method for obtaining the total and partial cross sections that involves propagating the initial bound-state w-ave function on the dissociative surface. However, the equations do not imply that this is the mechanism for the dissociation process, except in the limit of a S-function laser pulse. In the case of cw excitation, the actual u,?ave function on the dissociative surface is highly de- localized, and has no resemblance to a localized wave packet. Regardless of its interpretation, the time-evolved wave packet contains all of the dynamical information in the problem. For example, the time required for the wave packet to leave the Franck-Condon region, which is related to the slope of the dissociative surface, gives the overall width of the total dissociation spectrum. Similarly, through the projection in Eq. ( 11)) the channel in which the wave packet evolves asymptotically corresponds to the correct, physical final states.

RESULTS a(E) = z a$(& (12)

C.j,Y which can be compared to Eq. (9). At energies below the opening of the three-body channel these two methods must give the same results, and so any difference between them gives an indication of the accuracy of the calc.ulation. If three-body dissociation is important, Eq. ( 11) can be extended to include an appropriate projection onto outgoing plane waves for the three hydrogen atoms.

The translational functions #1 for the Hamiltonian in Eq. (3) are spherical Bessel functions, which go asymptotically as eikr/r. The l/r term is canceled by an r from the Jacobian, so we use eikrz functions for the translational functions. For&j and $L,J, we use rotational and vibrational functions of H, . The vibrational functions are obtained by diagonalizing a cut through the H, potential surface with the addition of a term of the formi(i + 1) /(p, 4 ). This procedure assumes that the rotations and vibrations are decou- pled, which is reasonable in the asymptotic regions of the potential surface, but includes the effects of the distortion of the vibrational functions by the rotations. The rotational functions #i are obtained from the DVR transformation matrices T in IQ. (6)) which convert the finite basis representation (FBR) to the DVR. When operating on the wave function, they have the effect of projecting it onto the eigenfunctions of the FBR because (si (x) ) the basis functions in the FBR, are related to those of the DVR, bj (x, ), by”

In Fig. 3(a) we show contours of the lowest (vi ,vZ,vJ ) = (O,O,O), even-parity A, state of Hc superimposed on a cut through the lower H + H, potential surface near 0 = 90”. The A I states are forbidden in H: due to nu- clear-spin statistics but are allowed in HP under certain cir- cumstances. For example, the 2p ‘A y state of H, is formed by attaching a 2p Rydberg electron perpendicular to the plane of the three hydrogen atoms. The antisymmetry of the 2p orbital permits the e?ristence of totally symmetric vibrational levels in this electronic state.2’s Figures 3(b)-3 (d) show the evolution of the wave packet in Fig. 3 (a) in time, superimposed on a cut through the potential surface at 8 = 5” (near collinear geometry). The wave packets shown in Fig. 3 were obtained from a calculations using a grid of 40 by 150 by 14 in rl, r,, and 0, with a grid spacing of 0.12 a.u. in r, and 0.1067 a.u. in r, , and 0 < 13 < 90”. The wave packet was propagated for a total of 900 a.u. (22 fs) before projecting it onto tinal states.

+i Cxoi 1 = C 4, Cx) Ti,a* (13) I

Figure 3(a) shows that the wave packet begins at the H + H, saddle point and then spreads into the asymptotic product channels, two of which are clearly visible in Figs. 3(b)-3(d). Because the II, system is kinematically symmetric, the asymptotic analysis for a symmetric initial state can be performed in just a single channel. The total probabilities are then obtained by multiplying the single channel results by three. To reduce the size of the grid required for the calculation, we restrict the number of grid points in r, and remove the flux that reaches the boundary in that dimension with a mask function, or flux gobbler. After each time step

J. Chem. Phys,, Vol. 96, No. 6,15 March 1992 Downloaded 16 Aug 2003 to 128.135.132.83. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

Downloaded 16 Aug 2003 to 128.135.132.83. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

4288 Krause eta/.: Dissociation of H3 Rydberg states

the wave packet is multiplied by a function that goes smooth- ly to zero at the edge of the grid. Typically we use a sin’ mask over the last 10-15 points of the grid. As can be seen in Fig. 3(d), this procedure is quite effective. The outgoing wave packet contains 33% of the initial density and is localized in a single asymptotic arrangement. The projection onto final states is performed only in this channel.

Figure 4 shows the short-time and long-time total dissociation cross section, as calculated with Eqs. (9) and (12), respectively. No projection onto three-body states was performed, so any differences between these curves should be caused by three-body effects. The three-body channel opens at a total energy of 4.74 eV, an energy that is well in the tails of the spectra, and so the agreement between the two methods is nearly perfect. Had we chosen a vibrationally excited state of H3+ as our initial state, we would have seen a larger contribution from the three-body channel,” because an initial state with a greater spatial extent has an increased overlap with more energetic regions of the ground-state potential.

At the discrete energies of the HT Rydberg states, all of which lie above 5.5 eV, (see Fig. 1) the amplitudes of the spectra in Fig. 4 can be interpreted as the total predissociation probability. When a molecule predissociates it does not emit a photon, and so all of the available energy must be shared among the translational, vibrational, and rotational modes of the fragments. If the molecule decays radiatively, Fig. 4 can be interpreted as the probability for a Rydberg state to spontaneously emit a photon to the ground state, plotted as a function of the sum of the fragment energies, rather than the photon energy. The energy of the photon equals the energy of the Rydberg state minus the total frag-

Energy (eV)

FIG. 4. Total dissociation cross section for a transition from a vibrationless, rotationless initial state of HT to the lower sheet of the H, ground state as calculated with (solid line) the “short-time” method of Eq. (9) and (dashed line) the “long-time” method of Eq. ( 12). The x axis labels the total fragment energy, consisting of the sum of the translational, vibrational, and rotational energies of the fragments. The arrow marks the total energy at which the three-body dissociation channet opens. The two spectra have not been normalized.

ment energy. The photon emission spectrum can be under- stood qualitatively in terms of the Condon reflection principle, which predicts that the most probable photon emission occurs as a vertical transition from the bound to the dissociative surface. In this approximation, Fig. 4 is the lower sheet component of the bound-free photon emission spectrum from an H, Rydberg state to the ground state, plotted as a function of the fragment energy. The photon spectrum would have the same shape as that in Fig. 4, but would be shifted by an energy equal to the electronic excitation energy of the Rydberg state, assuming that the dipole transition moment is constant over the range of the initial vibrational state. Bound-free photon spectra have been observed from HT by several groups.3’,32

In reality, of course, the probabilities for predissociation and radiative emission are different, and the spectra in Fig. 4 cannot be used directly to predict either. These two pro- cesses would be expected to have matrix elements of different magnitudes, with different energy dependencies, because the coupling operators, the nonadiabatic transition operator in the case of predissociation, and the dipole operator in the case of photon emission, are different. Similarly, the matrix elements would be expected to vary depending on which Rydberg state emits the photon, or predissociates. However, in the absence of any knowledge about these operators, we have set them equal to a constant in this work. We will return ; to this point below.

Figure 5 shows the contribution from the H, u = 0 and u = 1 final states to the total dissociation cross section. The total cross section is composed almost entirely of these two vibrational products, with only small amounts of higher vibrational states. This result is a reflection of the vibrational adiabaticity of the H, surface, Had we begun with a vibrationally excited initial state, much more vibrational excitation of H, would have resulted, and the total dissociation spectrum would have been more highly structured.‘” The 3D calculations also allow us to calculate rotational final- state distributions for a given vibrational state. We present

0 I 2 3 4 5 Energy (eV)

FIG. 5. Total dissociation cross section compared to the u = 0 and u = 1 partialcross sections for the same initial state 3s in Fig. 4.


( j30 L’T~=-.;; , . . - W - 1

I 0 .2 5 - ! 0.20

3 s

3 0 .15

iz 0 .10 1 1 ;1 m . 0 .05 .q -q .\r

n n n $ 1 ‘1 ...-_ 0 2 4 6 8 1 0 1 2 1 4 1 6 1 8

.I

FIG. 6. H, rotat ional d ist r ibut ions at a total ene rgy of 5.6 e V for the H, u E 0 f inal state (so l id bars ) a n d the H, u = 1 f inal state (ha tched bars) . T h e ini t ial state in the ca lcu la t ion was as in Fig. 4.

such distr ibut ions in Fig. 6 for u = 0 a n d u = 1 at 5 .6 e V , the ene rgy of the Z p ‘A 2 Rydbe rg state. T h e distr ibut ion is con- s iderab ly hot ter in the u = 1 f inal state than in u = 0. This t rend is a lso s e e n exper imenta l ly , w h e r e rotat ional exci tat ion increases signif icant ly wi th increas ing v ibrat ional exci ta- t ion.O W ith the know ledge of the v ibrat ional a n d rotat ional compos i t ion of the total d issociat ion spect rum, ou r resul ts can b e c o m p a r e d wi th exper iment .

F igure 7 shows the ca lcu la ted center of mass kinet ic ene rgy re lease spec t rum for spon taneous d issociat ion v ia ra- d iat ive decay of the 2 p ‘A g state c o m p a r e d to the expe r imen- tal spect rum. T h e exper imenta l resul ts w e r e ob ta ined by Cosby a n d H e l m by measu r i ng the time- reso lved f ragment kinet ic energ ies f rom the long- l ived 2 p *A ; state, in the ab -

K r a u s e &al. : D issoc ia t ion of H 3 Rydbe rg states 4 2 8 9

sence of the p u m p 1aser .6 T h e theoret ical spec t rum was ob - t amed by shif t ing the state-to-state emiss ion spect ra by their rov ibrat ional exci tat ion energ ies a n d then s u m m i n g them. This p rocedu re y ie lds the d issociat ion cross sect ion as a funct ion of the kinet ic ene rgy of the h y d r o g e n a tom relat ive to the center of mass of the molecu le . For spon taneous p h o - ton emiss ion, the prefactor P in E q . ( 11 ) conta ins a factor of 03, w h e r e w is the pho ton energy . W e h a v e inc luded this factor in ca lcu lat ing the spect ra in Fig. 7. T h e ag reemen t be tween exper iment a n d the 3 D resul ts is fair ly good , t hough the exper imenta l spec t rum shows a la rger contr ibut ion f rom h igher -energy f ragments than the theoret ical spect rum. Par t of the d i f ference be tween theory a n d exper iment cou ld b e caused by predissociat ion, o r by a smal l a m o u n t of v ibrat ion- a l exci tat ion in the init ial state. T h e m a i n di f ference, h o w - ever , is p robab ly d u e to t ransi t ions to the u p p e r sheet of the H, g round-s ta te surface. S u c h transi t ions a re capab le of p ro - duc ing h y d r o g e n a toms with low kinet ic ene rgy by, for ex- amp le , th ree-body dissociat ion, as wel l as h igh kinet ic ene r - gy by, for example , c ross ing th rough the conica l in tersect ion to the lower sheet a n d creat ing H, in low v ibrat ional states. W o r k is in p rogress to ca lcu late d ipo le m a trix e lements f rom the 2 p *A ; state to the g r o u n d state. This wil l a l low us to de te rm ine the init ial b ranch ing rat io to the u p p e r sheet , a n d thus to est imate the magn i t ude of its effect o n the dynamics.

W e no te that pho ton emiss ion f rom the 2 p *A 2 state to the g r o u n d state is d ipo le fo rb idden at the equ i l ib r ium g e o m - etry, so it was init ial ly thought”” that the 2 p *A y state d isso- c ia ted v ia a d ipo le -a l lowed transi t ion to the 2s ‘A ; state, wh ich then rap id ly predissoc iated. However , the ca lcu la ted l i fet ime of the d ipo le t ransi t ion in this p rocess3 (88~4s ) is too l ong to b e consistent wi th the exper iments .’ Pred issoc ia t ion of the 2s ‘A ; state wou ld a lso l ead to a k inet ic -energy re lease spec t rum that peaks at h ighe r ene rgy than the obse rved spec- t rum.

F igure 8 shows total d issociat ion spect ra as ca lcu la ted in two a n d th ree d imens ions . T h e 2 D spec t rum labe led “d iaba -

2 .5 [--“‘..--.- --T‘--m _ _ _ _ ~) I

2.5 [ I I I I L

’ ---I 4

K inet ic E n e r g y Re lease ( e V )

2.0 -

3 ‘g 1.5 -

2 ;s 1.0 ;

0 .5 -

0.0 t 0 1 2 3 4 5 6

E n e r g y ( e V )

FIG. 7. Theory (so l id l ine) vs exper imen t (dots) for the k inet ic -energy re- l ease spec t rum ob ta ined f rom the spon taneous d issoc ia t ion v ia rad iat ive de - cay of the H T 2 p ‘A $ ’ state. T h e spect ra have b e e n no rma l i zed to the s a m e m a x i m u m intensity. T h e exper imenta l da ta a re f rom Ref. 6.

FIG. 8. H, total d issoc ia t ion cross sect ion as ca lcu la ted in three a n d two d imens ions . T h e curve labe led “3 -d” is f rom the present work. T h e curve labe led “2 -d Diabat ic” was pub l i shed prev ious ly (Ref. IO) .

J. Chem. Phys. , Vo l . 96, No. 6 ,15 M a r c h 1 9 9 2 Downloaded 16 Aug 2003 to 128.135.132.83. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

tic” was published previously. lo The integrated areas of the spectra are governed by a sum rule, and are equal. There are several reasons for the differences between 2D and 3D. First, the 2D calculations used a harmonic approximation for the initial state, rather than solving for a vibrational level of the H: potential. More importantly, the 2D calculations re- stricted the H, geometry to 6 = 90”. As we mentioned previously, the ground electronic potential of H, is doubly degenerate, with a true crossing at D3,,, so in 2D the “upper” and “lower” surfaces can be chosen to be either the adiabatic surfaces, in which the lower surface is always the surface that has the lower energy, or the diabatic, in which the higher- energy surface is chosen for obtuse isosceles triangle configu- rations, and the lower-energy surface is chosen otherwise. In 3D, at geometries away from D,, the crossing between the surfaces is avoided, so we performed the dynamics on the upper and lower adiabatic surfaces.

The radiative dissociation spectrum on the lower diabatic surface in 2D peaks at a higher energy than the 3D spectrum, and shows increased vibrational excitation of the H2 products. As can be seen by examining Fig. 1, the reason for the shift to higher energy is that the spread of the initial wave packet in r, and r, causes the wave packet to access higher- energy regions of the potential in the diabatic case than in the adiabatic. One additional consequence of using the diabatic surfaces is that the upper diabatic surface dissociates to H + H + H collinearly. In 3D, the three-body dissociation on the upper surface appears to proceed via triangular geometries. As we mentioned previously, the geometry of the three-body dissociation is important in interpreting the results of Peterson and co-workers.’ The differences between the 2D and 3D adiabatic results are caused by the distribution of the initial state in 0, and the subsequent spreading of the wave packet in that dimension. Figure 9 shows the evolution of the wave packet in 0 with time. The density is initially concentrated near 0 = 90”, but rapidly becomes quite flat.

0.5 , -l,-I.-..T . ..---. , .

0.1

60 30 0 0 (degrees)

4290 Krause et&: Dissociation of H, Rydberg States

The spread in 8 allows the wave packet to access lower-energy regions of the potential, and is responsible for the shift to lower energy in the 3D total dissociation spectrum.

CONCLUSIONS We have presented in this paper results of 3D quantum

calculations of the dissociation of H, Rydberg states. This work extends previous 2D results’” and 3D results obtained via a time-independent method at energies of l-2 eV.34 We showed that for a vibrationless initial state corresponding to, for example, the lowest vibrational level of the 2~ ‘A ; of H:, the short-time and long-time methods for calculating the total dissociation cross section agree quite well. This agreement gives a measure of the accuracy of the calculation, and indicates that the three-body breakup channel is not important for the dissociation of this initial state on the lower sheet of the H, ground-state potential. By converting the total dissociation spectrum to the kinetic-energy release spectrum, we compared our calculations to experiment and to previous 2D calculations. We found that the experiment and 3D calculation agree reasonably well. Comparing the 3D and 2D calculations, we see that the 2D diabatic spectrum is shifted to significantly higher energy, but that repeating the 2D calculation on the lower adiabatic surface lessens the disagreement. However, a clear indication remains of the importance of non-CZo geometries, as well as the contribution of the upper sheet of the H; ground state potential to the dynamics.

Considerably more work is required to make detailed comparisons with the experimental data, particularly the vibrational and rotational distributions measured by Cosby and Helm,’ and the two-body to three-body branching ratios determined by Peterson and co-workers.’ A comparison of our calculations to the experiments of Peterson and co- workers will require results for a distribution of electronically and vibrationally excited initial states, which will compli- cate the analysis. Cosby and Helm’s experiments begin in the 2p ‘A ; state, which has total angular momentum equal to zero. They then probe Rydberg states in the n = 3 manifold with total angular momentum equal to one. Thus, to calculate product-state distributions from the states relevant to the experiments, our method must be extended to treat total angular momentum greater than zero. In addition, the II = 3 Rydberg levels lie well into the tail of the total dissociation spectrum, meaning that the partial cross sections for vibrationally excited product states will be difficult to converge. In the. present calculations, the Q = 0 and u = 1 partial cross sections are well converged, but the u = 2 and higher partials are not. The U= 2 partial cross section, for example, is lower in magnitude by a factor of about 100 compared to u = 1, and its energy dependence is not completely stable with respect to the parameters of the calculation.

FIG. 9. Evolution of the wave packet in Has a function of time. The norm of the wave packet at t = 0 and t = 4 fs is 1.0. At t = 15 fs, the norm is 0.35. The initial state of the calculation was as in Fig. 4.

One factor that makes converging the product-state distributions difficult is that the method we have chosen for the final-state analysis is extremely sensitive to small amounts of flux reflecting from the r, grid boundary. As shown in Fig. 3, we have prevented most of the reflection, but it is nearly impossible to prevent a slight amount of density from reflect-


Krause etal.: Dissociation of H, Rydberg states 4291

1.2 I ~~~ --...._ - ..- r.. ~~~~i~ .-?“-.-.....,-

T i ing back into the asymptotic product channel and interfering with the projections. Several methods have been proposed to overcome such diBiculties,“5--37 including calculating the flus through a dividing surface in the asymptotic region,“’ or using a splitting algorithm to remove the asymptotic por- tion of the wave packet analytically.3h A promising new method has also recently been suggested for time-dependent grid-based methods that involves integrating the Schrij- dinger equation in the interaction representation to prevent the wave packet from reaching the grid boundaries.“”

Qwr--.+. , 1

We have assumed in this work that the total dissociation spectrum depends only on the vibrational quantum number of the initial state. That is, we have set the dipole operator or nonadtidbatic coupling operator equal to a constant. The ina- dequacy of this approsimation can be seen by examining the experimental data.3” For example, the H, 3s and 3d Ryd- berg states have nearly the same energy, and rather similar potentials near the potential minima. However, the product vibrational distributions from these two states are quite different. This is an indication of the importance of the nonadiabatic coupling between the Rydberg states and the ground state. Work is in progress to calculate these couplings ub iiiitio.

Time (fs)

FIG. 10. Square of the time-dependent overlap ($(r,t) IJ,(r,O)) as a function of time on the lower and upper sheets of the H,s ground-state potential- energy surface.

To calculate the two-body to three-body branching ratio, 3s well as the product-state distributions, the dynamics must be performed on the coupled ground-state potential surfaces. The feasibility of such calculations has now been demonstrated,“() and the couplings between the upper and lower adiabatic surfaces are known,2s but including a second surface considerably enlarges the magnitude of problem computationally. An additional factor complicating the H, system is that the upper adiabatic surface has a well in the Franck-Condon region. This well has a maximum depth of about 2 eV and is capable of temporarily trapping the wave packet. This causes several difficulties in the calculations. For one, as shown in Fig. 10, the square of autocorrelation function (J,(r,t) I$(r,O j) on the upper surface has a large recurrence, which means that the integration time required for t.he wave packet to leave the Franck-Condon region is much longer than on the lower surface. When dynamics calculations are run on only the upper surface, the wave packet leaks slowly through the barriers in rand t; and then quickly dissociates. In a coupled surface calculation, density will also leak through the conical interaction to the lower surface, causing the projections in Eq. ( 11) at a finite value of r, to change with time. The presence of a very fast time scale corresponding to direct dissociation in conjunction with a much slower time scale corresponding to escape from the well will make characterizing the wave packet on the grid difficult. Care must also be taken to assure the validity of Eqs. (9) and I1 1) for nondirect dissociation.

tions near threshold in the H, dissociation spectrum predicted a large isotope effect in the fragment branching ratios.“” We would like to determine whether or not this result per- sists in the dissociation of the n = 3 Rydberg levels. The previous work also predicted that the reactive Feshbach resonances on the H, ground-state surface should be visible in dissociation as an enhancement of the u = 1 partial cross sections.“4 Our resolution did not permit us to verify this effect in the present work, but an improved method for final- state analysis should allow us to examine this question more closely. If the resonances can be detected, stimulated emission pumping experiments in HT might prove to be a very effective way to study these resonances, in an inverse spectroscopy of the transition state.

ACKNOWLEDGMENTS We would like to thank Charlie Cerjan for helpful con-

servations, and for supplying the Chebychev propagator. The calculations presented in this work were performed at the San Diego Supercomputer Center. This work was performed under the auspices of the U.S. Department of Energy at Lawrence Livermore National Laboratory under Con- tract No. W-740%ENG-48. One of the authors (J. C. L.) also acknowledges partial support of this research under Grant No. DE-FG02-87ER 13679.

In conclusion, we believe that the present calculations are 3 first step toward a realistic 3D quantum treatment of the PI, dissociation problem. The availability of high-quality experimental data, and high-quality potential surfaces makes this an excellent test problem for comparing exact and approximate dynamical methods. Further work on this system will also allow us to address additional t.heoretical and experimental issues. For example, previous 3D calcula-

’ For a review, see G. Her&erg, Annu. Rev. Phys. Chem. 38, 27 ( 1987). ‘See, forexample, L. J. Lembo, H. Helm, and D. L. IIuestis, J. Chem. Phys. 90. 5299 (1989); W. Ketterle, H.-P. Messmer, and I-I. Walther, Eur- ophys. Lett. 8, 333 (1989).

‘H. F. King and K. Morokuma, J. Chem. Phys. 71,3213 (1979). ‘C. Bordas, P. C. Co&y, and H. Helm, J. Chem. Phys. 93,6303 ( 1990). ‘J. R. Peterson, P. Devynck, Ch. Hertzler, and W. G. Graham (unpub-

lished). OP. C. Cosby and H. Helm, Phys. Rev. Lett. 61,298 (1988). ‘5. R. Niskes, Comments At. Mol. Phys. 19, 59 ( 1987). ‘J. F. Garvey and A. Kuppermann, J. Chcm. Phys. 86,6746 (1987).


4292 Krause eta/: Dissociation of H3 Rydberg states

‘N. Bjerre, I. Hazell, and D. C. Lorents, Chem. Phys. Lett. 181, 301 (1991).

‘“A. E. Orel and K. C. Kulander, 3. Chem. Phys. 91,6086 (1989). Ii J. 2. H. Zhang and W. H. Miller, J. Chem. Phys. 91, 1528 (1989); N. C.

Blais, M. Zhao, D. G. Truhlar, D. W. Schwenke, and D. J. Kouri, Chem. Phys. Lett. 166, 11 (1990).

‘“D. A. V. Kliner, K.-D. Rinnen, and R. N. Zare, Chem. Phys. Lett. 166, 107 (1990).

“R. N. Porter, R. M. Stevens, and M. Karplus, J. Chem. Phys. 49, 2408 (1968).

“A. E. Ore1 and K. C. Kulander, Chem. Phys. Lett. 146,428 (1988 ). Is R. Heather, X.-P. Jiang, and H. Metiu, J. Chem. Phys. 90,2555 (1988). “‘C. Leforestier, R. H. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A.

Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H.-D. Meyer, N. Lipkin, 0. Roncero, and R. Kosloff, J. Comput. Phys. 94, 59 (1991).

“J, Tennyson and B. T. Sutcliffe, J. Chem. Phys. 71,406l (1982). “H. Tal-Ezer and R. Kosloff, J. Chem. Phys. 81,3967 (1984). “R. Kosloff, J. Phys. Chem. 92,2087 ( 1988). ‘@.T. V. Lill, G. A. Parker, and J. C. Light, Chem. Phys. Lett. 89,483 (1982);

J. C. Light, I. P. Hamilton, and J. V. Lill, J. Chem. Phys. 82,140O ( 1985); S. E. Choi and J. C. Light, ibid. 92,2129 ( 1990).

” R. M. Whitnell and J. C. Light, J. Chem. Phys. 90, 1774 ( 1989). ‘l K. C. Kulander and M. F. Guest, J. Phys. B 12, LSOI ( 1979). “W. Meyer, P. Botschwina, and P. G. Burton, J. Chem. Phys. 84, 891

(1986). z4 J. R. Henderson and J. Tennyson, Chem. Phys. Lett, 173, 133 ( 1990 j. “A. J. C. Varandas, F. B. Brown, C. A. Mead, D. G. Truhlar, and N. C.

Blais, J. Chem. Phys. 86,625s (1987). 26E. 6. Heller, J. Chem. Phys. 68, 3891 (1978). “K. C. Kulander and E. J. Heller, J. Chem. Phys. 69,2439 ( 1978). 28 M. Shapiro and R. Bersohn, Annu. Rev. Phys. Chem. 33,409 ( 1982). 29N. E. Henricksen, Comments At. Mol. Phys. 21, 153 (1988). 30M. V. Rama Krishna and R. D. Coalson, Chem. Phys. 120,327 (1988). 3’ A. B. Raksit, R. F. Porter, W. P. Garver, and J. J. Leventhal, Phys. Rev.

Lett. 55,378 ( 1985). 31 H. Figger, M. N. Dixit, R. Maier, W. Schrepp, H. Walther, J. R. Peterkin,

and J. K. G. Watson, Phys. Rev. Lett. 52,906 (1984). 33 G. I. Gellene and R. F. Porter, J. Chem. Phys. 79,5975 ( 1983). j4 K. C. Kulander and J. C. Light, J. Chem. Phys. 85, 1938 ( 1986 j. 35 G. G. Bahnt-Kurti, R. N. Dixon, and C. C. Marston, J. Chem. Sot. Fara-

day Trans. 86, 1741 (1990). 36R W. Heather and H. Metiu, J. Chem. Phys. 86, 5009 ( 1986 j. ” D. Neuhauser, M. Baer, R. S. Judson, and D. J. Kouri, J. Chem. Phys. 93,

1 (1990): 3*J. Z. H. Zhang, Chem. Phys. L&t. 160, 417 ( 1989); C. J. Williams, J.

Oian. and D. J. Tannor. J. Chem. Phvs. 95, 1721 (1991). 3p p‘. C!.’ Cosby and H. Heim (unpublished). wU. Manthe and H. Kijppel, Chem. Phys. Lett. 178,36 (1991).


Date post:	11-Feb-2022
Category:	Documents
Upload:	others
View:	1 times
Download:	0 times

Quantum-mechanical calculations of the dissociation of H-3 Rydberg states

Documents