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Semiclassical approach to the Rydberg emission spectra of H3 and its isotopomers
18 June 1999
Chemical Physics Letters 306 Ž1999. 387–394
Semiclassical approach to the Rydberg emission spectra of H 3 and its isotopomers S. Mahapatra ) , H. Koppel ¨ Theoretische Chemie, Physikalisch-Chemisches Institut UniÕersitat ¨ Heidelberg, Im Neuenheimer Feld 229, D-69120 Heidelberg, Germany Received 30 December 1998; in final form 20 April 1999
Abstract We investigate the optical emission spectra of Rydberg-excited H 3 and its isotopomers semiclassically in the framework of the reflection principle. The semiclassical spectra are also obtained with the approximation of a locally linear potential energy surface where the contribution from the breathing and the Jahn–Teller ŽJT. active modes are separable. While the JT mode brings out the energy splitting in the bimodal emission profiles, the breathing mode smears out their overall structure. Despite some severe limitations of the semiclassical results, the energy splitting is reproduced well. This enables us to establish a simple kinematic scaling of this quantity for different isotopomers. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction Triatomic hydrogen, the simplest neutral polyatomic system, belongs to the family of Jahn–Teller ŽJT. active species. Its ground electronic state is orbitally degenerate at the D 3 h configuration and forms a conical intersection w1–4x. It has been, and still is, a benchmark system in studies of the reactive scattering processes which occur on the repulsive lower sheet of its ground electronic manifold Žsee, e.g., Refs. w5–7x.. The importance of the conical intersection on the reactive scattering cross-sections w8–10x and transition state resonances w11,12x have been discussed in the literature. Mostly, these studies have considered the geometric phase change of the adiabatic electronic wavefunction w13x when encir) Corresponding author. Fax: q49 6221 545221; e-mail: [email protected]
cling the conical intersection in a closed path. While the impact of the conical intersection on the reactive scattering dynamics is still a debatable issue w14,15x at energies below that of the minimum of the seam of intersections, it has negligible influence on the transition state resonances which are dominated by the saddle point region Žat the collinear arrangements of the three nuclei w16x. of the potential energy surface ŽPES. w11,12x. On the other hand, highly nonadiabatic nuclear dynamics due to the conical intersection is conceivable near the perpendicular arrangement of the three nuclei. In a previous paper, we have theoretically explored H 3 in the near D 3 h configuration and reported its spectra and dynamics using wavepacket ŽWP. propagation methods w17x. The adiabatic sheets of the double many-body expansion ŽDMBE. PES w18x were used in our study. As regards the Žuncoupled. lower adiabatic sheet, resonances are not pronounced at the
0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 9 . 0 0 4 7 4 - 1
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S. Mahapatra, H. Koppelr Chemical Physics Letters 306 (1999) 387–394 ¨
C2 Õ geometry of H 3 and only minor effects of the nonadiabatic coupling to the upper adiabatic sheet are found Žbelow the minimum energy of the seam of conical intersections.. In the absence of the coupling to the lower adiabatic sheet, the upper one supports bound states w11,19x. This bound state structure changes to a broad bump in the spectral intensity distribution when the nonadiabatic coupling is turned on w17,20–22x. The large width of this bump is consistent with an extremely fast non-radiative decay of the vibrational levels of the upper sheet occurring within 3–6 fs only w17x. These are the fastest relaxation times known to date in an internal conversion process and demonstrate the strong impact of the nonadiabatic coupling in this system. The results of these studies agree with the Rydberg emission spectra of H 3 and its isotopomers as recorded by Bruckmeier et al. w23x and Azinovic et al. w24x. The dominating transition in this experiment is from the third principal quantum shell Ž n s 3. Žnear D 3 h configuration. to the ground electronic manifold of H 3 . The emission profile of D 3 exhibits two broad maxima at 240 and 310 nm which are separated by 9600 cmy1 Ž1.2 eV. in energy w23x, and otherwise reveals no structure. Azinovic et al. w24x have simulated these spectra using a phenomenological lifetime broadening of the upper adiabatic sheet in the range 3–8 fs which is fully consistent with our ab initio estimate w17x. The aforementioned accounts imply that on arrival to the ground electronic manifold H 3 dissociates on an ultrashort timescale due to strong nonadiabatic coupling effects Žthis finally yields the highly diffuse and structureless emission profile.. We are motivated by the nature of these profiles to pursue a semiclassical investigation in the framework of the reflection principle w25–28x in order to shed further light on our earlier quantal results w20–22x as well as the experimental findings of Bruckmeier et al. w23x. The venerable reflection principle has been found to be quite successful in direct photodissociation studies of diatomic and also polyatomic systems w25–28x. Inspired by this fact, we apply this principle to the present situation of a degenerate final electronic manifold. Because of the fast dissociation in the final electronic state the energy dependence of the emission profile can be represented as an image of the initial coordinate distribution of the Rydberg
molecule mediated by the potential of the final state. The width of the emission profile Ž D E . is proportional to the width of the initial coordinate distribution of the molecule and to the steepness of the potential at the Franck–Condon ŽFC. zone centre. We outline the reflection principle as applied to the present case of a degenerate electronic manifold in Section 2. The results obtained are presented and discussed in Section 3 and the main conclusions are summarized in Section 4.
2. Reflection principle applied to a degenerate electronic manifold In the Golden Rule formalism, the spectral intensity for an electronic transition can be written as
s Ž E . A dqC ) Ž q . d Ž Hˆ y E . C Ž q . ,
H
Ž 1.
where a matrixrvector notation of the Hamiltonian and wavefunction has been adopted. The quantity q collectively represents the coordinates, H is the Hamiltonian matrix of the respective states, here degenerate at the D 3 h conformation. E s E0 q " v , where E0 is the energy of the initial ŽRydberg. state and " v is the energy of the emitted photon. Finally,
t C Ž q . s ty c Ryd Ž q . , q
ž /
Ž 2.
denotes the transition dipole matrix element t. for the final electronic state in question times the initial Rydberg wavefunction. In the reflection approximation w25–28x, the short time limit prevails and therefore the kinetic energy term is dropped from the Hamiltonian in Eq. Ž1.. With this and Eq. Ž2., assuming coordinate-independent transition dipole matrix elements, Eq. Ž1. can be recast into the form
s Ž E. A Ý
H dq < c
Ryd
Ž q . < 2d Vi Ž q . y E ,
Ž 3.
i
where Vi Ž q . is the potential energy function of the ith final electronic state. Since the FC transition of Rydberg-excited H 3 directly probes the ground electronic manifold at its equilateral triangular geometry w23x, in the following we use the mass scaled normal mode coordinates pertinent to the D 3 h symmetry point group to evalu-
S. Mahapatra, H. Koppelr Chemical Physics Letters 306 (1999) 387–394 ¨
ate the integral in Eq. Ž3.. These coordinates are obtained by the FG-matrix method of Wilson et al. w29x. We define the symbols q0 for the symmetric stretching Žbreathing. mode and q1 and q2 for the components of the degenerate vibration. If r and f represent polar coordinates, i.e., magnitude and direction of this degenerate vibration, then q1 s r sin f and q2 s r cos f . In the harmonic approximation, the initial Rydberg wavefunction can be written as
c
Ryd
Ž q0 , r . s Ne
ya A 1Ž q 0yq 0e . 2 r "
e
ya E Ž r y r e . 2 r "
389
independent and therefore the dynamics of these two modes can be decoupled from each other Žsee, e.g., Ref. w34x.. In the time-dependent picture, the total time autocorrelation function is written as the product of the component autocorrelation functions for these two modes. Therefore, the Fourier convolution theorem can be utilized in describing the full emission spectrum. Following Eqs. Ž3. and Ž4., the spectrum in the JT mode can be described as
sJT Ž E . A Ý ,
y2 a E r 2 r "
d Ž Vi Ž q0e , r . y E . r d r ,
He
i
Ž 5.
Ž 4. where N is the normalization constant and a k s mpn k , with m being the reduced mass and n k Ž k s A1 , E . the frequencies of the symmetric Ž A1 . and the degenerate Ž E . modes. The quantities q0e and r e correspond to the equilibrium geometry of the Rydberg state and define the coordinates of the FC zone centre. Since r e s 0 at this centre Žsee below., hereafter, we will drop it from all equations. The full semiclassical emission profile is obtained by substituting Eq. Ž4. into Eq. Ž3.. We emphasize that in the above formulation the initial state probability distribution is treated in a quantal fashion, as a square of the initial vibrational wavefunction. This choice has been made because we compare below with low temperature Žin fact, T s 0. spectra for which it is well accepted in the literature w28,30,31x. For the same reason, the energy in the initial state is independent of the nuclear coordinate, corresponding to a vibrational eigenvalue, and only the final state potential energy appears in the argument of the d function in Eq. Ž3.. In the high temperature limit, a purely classical probability distribution would be appropriate also for the initial state and would lead to the appearance of the potential energy difference in Eq. Ž3. w32,33x. However, as already indicated above, the experimental and full quantal reference spectra addressed in Section 3 below rather correspond to the limit T s 0. We now describe the emission profile within the linear coupling approximation using a first-order Taylor series expansion of the final state potential. In such a situation, the two modes, viz. the breathing mode q0 and the Ž E = e . JT active mode r are
where Vi correspond to the adiabatic potential energies of the final electronic manifold. The quantity r d r is the volume element for the JT mode coordinate. The Vi are obtained by diagonalizing the potential part of the linear Ž E = e . JT model Hamiltonian w32x V1,2 Ž q0e , r . s V0 Ž q0e . . kr ,
Ž 6.
where . indicates the lower and upper adiabatic sheets, respectively, and V0 s V0 Ž q0m at h r m e . is the potential energy at the FC zone centre. The quantity k is the first-order JT coupling constant and represents the slope < dV1,2r d r < rs r e < of the adiabatic potential with respect to the JT mode at the equilibrium geometry. Below we extract the value of k using the adiabatic potential energies from the DMBE PES w18x for the relevant geometries of H 3 . Using Eq. Ž6. and utilizing the properties of the d function, the emission intensity in Eq. Ž5. is given by w25– 28,30,31x ey2 b E Ž EyV 0 .
sJT Ž E . A Ý
i
2
r"
< E y V0 < ,
Ž 7.
where bE s < a Erk 2 <. The emission profile, Eq. Ž7., exhibits a bimodal intensity distribution corresponding to the two sheets of the JT split PES w30,31x. The energy splitting D between the two maxima obtained from Eq. Ž7. is given by
(
Dsk
"
aE
.
Ž 8.
S. Mahapatra, H. Koppelr Chemical Physics Letters 306 (1999) 387–394 ¨
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Now we turn to the spectrum in the breathing mode:
s br Ž E . A Ý
He
y2 a A 1Ž q 0yq 0e . 2 r "
d Ž Vi Ž q0 . y E . d q0 .
i
Ž 9. It can be evaluated in a similar way by expanding the potential around the equilibrium V1,2 Ž q0 . f V0 y k g Ž q0 y q0e . ,
Ž 10 .
where k g s y dV1,2r d q0 < q 0sq 0e is the slope of the adiabatic potential with respect to the breathing mode at the equilibrium geometry. With these, the integral in Eq. Ž9. reads w25–28x
s br Ž E . A Ý i
ey2 b A1Ž EyV 0 . < kg <
2
r"
,
Ž 11 .
with bA 1 s < a A 1rk g2 <. Finally, the semiclassical emission profile is obtained by convoluting the JT profile sJT Ž E . with the profile for the breathing mode s br Ž E ..
3. Results and discussion Following our earlier work w20–22x, we consider here the transition of H 3 and its isotopomers from the Ž3d .3 EX component state of the n s 3 principle quantum shell to the degenerate ground electronic manifold. The equilibrium geometry of these species in this quantum shell is nearly equilateral triangular and is close to that of the corresponding cations in the electronic ground state w35x. The equilibrium bond length amounts to ; 1.642 a0 in the Ž3d .3 EX electronic state w35x. In the following, we use the above equilibrium bond distance and the vibrational frequencies of the corresponding cations in their electronic ground state w36x in order to calculate the emission profiles semiclassically. Cuts of the adiabatic sheets of the DMBE PES along with the Ž3d .3 EX Rydberg electronic state and r < c Ryd < 2 of H 3 as a function of the JT coordinate r Žwith q0 s q0e s 1.642 a 0 . are shown in Fig. 1. As can be seen from Fig. 1, the potential curves of the ground electronic manifold exhibit a crossing Žconical intersection. at the centre of the FC zone at r s 0. The magnitude of the potential energy V0 at
Fig. 1. Adiabatic potential energies as a function of the JT coordinate r for fixed value of q0 s1.642 a0 . The solid and dashed curves represent the lower and the upper adiabatic sheets, respectively. The point of intersection of the two at 3.089 eV corresponds to the center of the FC zone. The initial Rydberg electronic state and corresponding probability density for H 3 is also shown at the top of the figure to illustrate the optical emission process.
the FC zone centre is 3.089 eV w18x. The solid and the dashed curves in Fig. 1 correspond to the dissociative lower and bound upper adiabatic sheets of the DMBE PES, respectively. The direction f of the degenerate vibration can take values either 0 or p at the C2 Õ configuration. r < c Ryd < 2 has a bimodal shape due to the volume element in the JT mode. The reflection principle implies that this bimodal probability density distribution of the initial state in the coordinate space will be mediated through the final electronic manifold into the bimodal intensity distribution of Eq. Ž7.. The magnitude of the coupling constant k is calculated at the D 3 h equilibrium geometry of H 3 using Eq. Ž5. and the adiabatic potential energies from the DMBE PES. By definition k is the derivative of the adiabatic potential with respect to r at r s 0. We calculate this derivative numerically by a finite difference scheme, which leads to k eff s < V2 Ž r . y V1Ž r .
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energy separation obtained in the experimental recording w23,24x as well as in our earlier quantal studies w20–22x. The Ž E = e . spectrum convoluted with the Gaussian in Eq. Ž11., corresponding to the progression in the breathing mode q0 , also has the intensity minimum at 3.089 eV and is symmetric about the intensity minimum Žshown by short dashed lines.. Likewise, the positions of the maxima and the energy splitting between them remained unaltered. However, the convoluted one is more diffuse than the Ž E = e . spectrum. Therefore, only the breathing
Fig. 2. Ža. The effective first-order JT coupling constant k eff , as a function of the JT coordinate r of H 3 for different values of the breathing mode q0 Ž a 0 . indicated in the panel. Žb. The value of k eff averaged over r as a function of the breathing mode q0 .
FC zone at q0 s 1.642 a0 to the location of minimum energy on the seam at q0 s 1.973 a 0 w18x. The average value of k eff over r is plotted as a function of q0 in Fig. 2b. The variation of k eff with q0 implies the relevance of the higher-order coupling terms in the dynamics which are neglected here for the sake of simplicity. We use k eff s 3.56 eVra0 , obtained for q0 s 1.642 a 0 , in our subsequent calculations of the emission profiles. Fig. 3 in its uppermost panel displays the results of the semiclassical calculations on H 3 . The types of calculation from which the individual curves originate are denoted in the panel. All three curves exhibit a bimodal distribution of emission intensities with two maxima located at the same final state energies of ; 2.55 eV and ; 3.65 eV. The Ž E = e . JT spectrum Žshown by long dashed lines. obtained from Eq. Ž7. shows an intensity minimum at the energy corresponding to the FC zone centre Ž3.089 eV.. The two maxima are separated by 1.1 eV in energy. This is to be compared with ; 1.2 eV
Fig. 3. The relative intensity Žarbitrary units. of emission of H 3 in X the optical transition from the upper Ž3d .3 E Rydberg electronic state to the ground electronic manifold, plotted as a function of the energy in this final electronic manifold. The semiclassical results obtained within the reflection approximation are shown in the uppermost panel. Three different variants are compared: the emission profiles obtained with the JT active mode alone Žshown by long dashed lines., its convolution with the progression in the breathing mode Žshown by short dashed lines. and the result using the full DMBE PES Žshown by the full line.. The profile obtained from the quantal calculations using WP propagation methods and coupled final electronic manifold is reproduced from Refs. w20–22x and shown in the middle panel. The lower panel shows the analogous results for transition to the uncoupled adiabatic sheets of the ground electronic manifold.
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S. Mahapatra, H. Koppelr Chemical Physics Letters 306 (1999) 387–394 ¨
mode contributes to the smearing out of the overall structure of the emission profile. We remind here that we have relied on a linear coupling model to calculate the emission profiles discussed above. The essential inputs are the slopes of the adiabatic potential at the FC zone centre and the equilibrium value of the potential energy. The weak dependence of k eff on q0 is also ignored. Therefore, the asymmetries of the adiabatic potentials do not show up in the emission profiles. In order to get away from these constraints, we have also calculated the emission profile exploiting the DMBE PES in full three dimensions. Since all the relevant potential interactions are implicit in this ab initio PES, the emission profile resulting from it is expected to bring out the correct asymmetry as observed in the experimental spectrum. A discrete grid is used in Jacobi coordinates Ž R,r,g . consisting of 128 points in both R and r, ranging from 1.0 to 2.016 a 0 , and of the nodes of a 96-point Gauss– Legendre quadrature along g . The initial wavefunction and the potential energies are evaluated on each node of this three-dimensional grid, and the integral in Eq. Ž3. is evaluated using the quadrature rule. We have adopted a histogram procedure and discretized the final state energy E at an interval of d E s 0.05 eV in order to numerically evaluate the d function in Eq. Ž3.. If the potential energy at a node falls within a bin between E and E q d E then the probability density
22x. In the middle panel of Fig. 3, we have reproduced the emission profile of H 3 obtained from the three-dimensional time-dependent WP calculations of Refs. w20–22x. Since details have been given earlier, here we only mention that the above result is obtained by using Jacobi coordinates with total angular momentum J s 0 and a diabatic representation of the Hamiltonian. The most pronounced feature of the quantal result is the shift of the intensity maximum and minimum to higher final state energies as compared to the semiclassical results. Also notable is the energy splitting between the intensity maxima in the quantal profile which is ; 0.1 eV higher than that obtained from the semiclassical profile. In order to find out possible origins of the energy shifts in the quantal coupled states result we have re-calculated the quantal spectrum suppressing the nonadiabatic coupling between the lower and upper adiabatic sheets. Otherwise, the calculation is the same as in the coupled states treatment of the central panel of Fig. 3. The results are shown in the lowermost panel of Fig. 3, identifying the spectrum of the lower and upper adiabatic sheet by thick dashed and thin solid line, respectively. Due to its repulsive nature, we see no discrete structures in the spectrum of the lower adiabatic sheet. On the other hand, in the absence of coupling to the lower sheet, the upper sheet is bound in nature and, therefore, a series of discrete lines is observed in its spectrum. Since we have used a symmetry adapted initial wavefunction ŽEq. Ž4.., only the lines of A1 symmetry appear in the spectrum. The minimum of the upper adiabatic sheet coincides with the minimum of the seam of conical intersections at ; 2.75 eV w18x, but the first line in its spectrum emerges at ; 3.72 eV only. This amounts to a huge zero-point energy of ; 1 eV of this sheet that arises from the cusp of the JT-split potential energy surfaces. As a consequence, the quantal spectrum for the upper Žuncoupled. surface peaks at substantially higher energy than the semiclassical one. Comparing with the coupled states treatment, the two maxima occur at even higher energies. As discussed in Refs. w11,12,17,19x these energy shifts in the coupled state results arise from the effect of the geometric phase. This effect is more pronounced on the upper adiabatic sheet because the energy minimum of this sheet lies on the seam of conical intersections and, therefore, the impact of the
S. Mahapatra, H. Koppelr Chemical Physics Letters 306 (1999) 387–394 ¨
degeneracy is stronger on this sheet. Of course, the main effect of the nonadiabatic coupling is the dramatic change of the sharp line spectrum of the upper uncoupled sheet to a broad bump in the coupled state description. The latter quantal calculation w20–22x elegantly reproduces the experimental results w23,24x. An explicit comparison has already been given in Refs. w20–22x and is not repeated here. We conclude that the above characterization of the semiclassical spectra also pertains to the comparison with the experimental spectrum of H 3 . Despite the above shortcomings, the semiclassical model is fairly successful in reproducing the observed energy splitting D in the bimodal emission profile. The quantity D is the most important observable in the experimental measurements. This tempted us to search for a simple kinematic scaling relation for this quantity when studying heavier isotopomers of H 3 . Substituting k s 3.56 eVra0 , " s 1, m s 1836 and n E s 0.1828 = 10y2 auy1 w36x in Eq. Ž8. we obtain D H 3 s 1.1 eV, similar to that extracted from the full semiclassical emission profile. Again this is ; 0.1 eV lower than the experimental w23x as well as the quantal w20–22x value. Using Eq. Ž8., the energy splitting in the emission profiles of heavier isotopomers ŽX 3 . is given by
DX 3 s
DH 3 m1r4 X3
,
Ž 12 .
where m X 3 s m X rm H in the normal mode picture and m H s 1.008 u. We have calculated the emission profiles for the heavier isotopomers up to m X s 5.04 u semiclassically using the full DMBE PES. The energy splittings D extracted from these numerical results are plotted against m in Fig. 4 and shown by the filled circles. The energy splitting obtained by Eq. Ž12. is shown in Fig. 4 by the solid line. As can be seen from Fig. 4, the calculated energy splittings are well reproduced by the scaling relation of Eq. Ž12.. The energy splitting decreases in the heavier isotopomers and a similar trend is noticed in the experimental spectra w23x. With increasing mass, the zero-point energy decreases and the initial wavefunction becomes narrower. This causes a narrower intensity distribution of the heavier isotopomers with the two maxima approaching the centre of the FC
393
Fig. 4. The energy splitting D Žin eV. between the maxima in the bimodal emission profile as a function of the mass m Žin u. of the isotopomers of H 3 . The filled circles indicate the results obtained semiclassically using the full DMBE PES and the solid line is obtained using the scaling relation in Eq. Ž12..
zone and resulting in the decrease of D according to Eq. Ž12..
4. Conclusions We have calculated the Rydberg emission profiles of H 3 and its isotopomers semiclassically in the framework of the reflection principle. We obtained bimodal emission profiles as observed in the experimental measurements and also in our earlier quantal investigations. It is seen that the most important experimental observable, viz. the energy splitting D between the two maxima of the bimodal profile, is obtained by describing the dynamics only in the JT active mode and only considering a linear coupling between the component states of the final electronic manifold. The added contributions from the breathing mode smears out the profile and makes it more diffuse. The observed asymmetry in the emission profiles is missing in the linear coupling results. This is reproduced by calculating the spectrum numerically Žwithin the semiclassical approximation. employing the adiabatic potentials for the lower and upper sheets of the final electronic manifold. However, the absolute positions of the two maxima in the semiclassical profile occur at much lower final state energies when compared to the quantal results. Also, the semiclassical profile exhibits an intensity minimum at E s 3.089 eV which is close to that of the vertical transition. In the quantal profile likewise the
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S. Mahapatra, H. Koppelr Chemical Physics Letters 306 (1999) 387–394 ¨
intensity minimum shifts to ; 0.7 eV higher energy. The shift of the maxima and minimum to higher energies in the quantal profile arises from the zeropoint energy as well as from geometric phase effects. These are purely quantal effects which are absent in the semiclassical picture. The two maxima in the quantal profile of H 3 are ; 1.2 eV apart which is ; 0.1 eV higher than the semiclassical value. Apart from this relatively small quantal correction, the energy splitting D for any heavier isotopomer of H 3 can be obtained from the simple scaling relation in Eq. Ž12.. Although the semiclassical model qualitatively reproduces the features of the experimental findings, it is missing quantitative details. The results embodied in this Letter, therefore, illustrate both the success and shortcomings of the semiclassical model in this fundamental polyatomic molecule.
Acknowledgements SM gratefully acknowledges the Alexander von Humboldt foundation for a research fellowship. We are indebted to Professor L.S. Cederbaum for his interest in this work.
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Chemical Physics Letters 306 Ž1999. 387–394
Semiclassical approach to the Rydberg emission spectra of H 3 and its isotopomers S. Mahapatra ) , H. Koppel ¨ Theoretische Chemie, Physikalisch-Chemisches Institut UniÕersitat ¨ Heidelberg, Im Neuenheimer Feld 229, D-69120 Heidelberg, Germany Received 30 December 1998; in final form 20 April 1999
Abstract We investigate the optical emission spectra of Rydberg-excited H 3 and its isotopomers semiclassically in the framework of the reflection principle. The semiclassical spectra are also obtained with the approximation of a locally linear potential energy surface where the contribution from the breathing and the Jahn–Teller ŽJT. active modes are separable. While the JT mode brings out the energy splitting in the bimodal emission profiles, the breathing mode smears out their overall structure. Despite some severe limitations of the semiclassical results, the energy splitting is reproduced well. This enables us to establish a simple kinematic scaling of this quantity for different isotopomers. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction Triatomic hydrogen, the simplest neutral polyatomic system, belongs to the family of Jahn–Teller ŽJT. active species. Its ground electronic state is orbitally degenerate at the D 3 h configuration and forms a conical intersection w1–4x. It has been, and still is, a benchmark system in studies of the reactive scattering processes which occur on the repulsive lower sheet of its ground electronic manifold Žsee, e.g., Refs. w5–7x.. The importance of the conical intersection on the reactive scattering cross-sections w8–10x and transition state resonances w11,12x have been discussed in the literature. Mostly, these studies have considered the geometric phase change of the adiabatic electronic wavefunction w13x when encir) Corresponding author. Fax: q49 6221 545221; e-mail: [email protected]
cling the conical intersection in a closed path. While the impact of the conical intersection on the reactive scattering dynamics is still a debatable issue w14,15x at energies below that of the minimum of the seam of intersections, it has negligible influence on the transition state resonances which are dominated by the saddle point region Žat the collinear arrangements of the three nuclei w16x. of the potential energy surface ŽPES. w11,12x. On the other hand, highly nonadiabatic nuclear dynamics due to the conical intersection is conceivable near the perpendicular arrangement of the three nuclei. In a previous paper, we have theoretically explored H 3 in the near D 3 h configuration and reported its spectra and dynamics using wavepacket ŽWP. propagation methods w17x. The adiabatic sheets of the double many-body expansion ŽDMBE. PES w18x were used in our study. As regards the Žuncoupled. lower adiabatic sheet, resonances are not pronounced at the
0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 9 . 0 0 4 7 4 - 1
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C2 Õ geometry of H 3 and only minor effects of the nonadiabatic coupling to the upper adiabatic sheet are found Žbelow the minimum energy of the seam of conical intersections.. In the absence of the coupling to the lower adiabatic sheet, the upper one supports bound states w11,19x. This bound state structure changes to a broad bump in the spectral intensity distribution when the nonadiabatic coupling is turned on w17,20–22x. The large width of this bump is consistent with an extremely fast non-radiative decay of the vibrational levels of the upper sheet occurring within 3–6 fs only w17x. These are the fastest relaxation times known to date in an internal conversion process and demonstrate the strong impact of the nonadiabatic coupling in this system. The results of these studies agree with the Rydberg emission spectra of H 3 and its isotopomers as recorded by Bruckmeier et al. w23x and Azinovic et al. w24x. The dominating transition in this experiment is from the third principal quantum shell Ž n s 3. Žnear D 3 h configuration. to the ground electronic manifold of H 3 . The emission profile of D 3 exhibits two broad maxima at 240 and 310 nm which are separated by 9600 cmy1 Ž1.2 eV. in energy w23x, and otherwise reveals no structure. Azinovic et al. w24x have simulated these spectra using a phenomenological lifetime broadening of the upper adiabatic sheet in the range 3–8 fs which is fully consistent with our ab initio estimate w17x. The aforementioned accounts imply that on arrival to the ground electronic manifold H 3 dissociates on an ultrashort timescale due to strong nonadiabatic coupling effects Žthis finally yields the highly diffuse and structureless emission profile.. We are motivated by the nature of these profiles to pursue a semiclassical investigation in the framework of the reflection principle w25–28x in order to shed further light on our earlier quantal results w20–22x as well as the experimental findings of Bruckmeier et al. w23x. The venerable reflection principle has been found to be quite successful in direct photodissociation studies of diatomic and also polyatomic systems w25–28x. Inspired by this fact, we apply this principle to the present situation of a degenerate final electronic manifold. Because of the fast dissociation in the final electronic state the energy dependence of the emission profile can be represented as an image of the initial coordinate distribution of the Rydberg
molecule mediated by the potential of the final state. The width of the emission profile Ž D E . is proportional to the width of the initial coordinate distribution of the molecule and to the steepness of the potential at the Franck–Condon ŽFC. zone centre. We outline the reflection principle as applied to the present case of a degenerate electronic manifold in Section 2. The results obtained are presented and discussed in Section 3 and the main conclusions are summarized in Section 4.
2. Reflection principle applied to a degenerate electronic manifold In the Golden Rule formalism, the spectral intensity for an electronic transition can be written as
s Ž E . A dqC ) Ž q . d Ž Hˆ y E . C Ž q . ,
H
Ž 1.
where a matrixrvector notation of the Hamiltonian and wavefunction has been adopted. The quantity q collectively represents the coordinates, H is the Hamiltonian matrix of the respective states, here degenerate at the D 3 h conformation. E s E0 q " v , where E0 is the energy of the initial ŽRydberg. state and " v is the energy of the emitted photon. Finally,
t C Ž q . s ty c Ryd Ž q . , q
ž /
Ž 2.
denotes the transition dipole matrix element t. for the final electronic state in question times the initial Rydberg wavefunction. In the reflection approximation w25–28x, the short time limit prevails and therefore the kinetic energy term is dropped from the Hamiltonian in Eq. Ž1.. With this and Eq. Ž2., assuming coordinate-independent transition dipole matrix elements, Eq. Ž1. can be recast into the form
s Ž E. A Ý
H dq < c
Ryd
Ž q . < 2d Vi Ž q . y E ,
Ž 3.
i
where Vi Ž q . is the potential energy function of the ith final electronic state. Since the FC transition of Rydberg-excited H 3 directly probes the ground electronic manifold at its equilateral triangular geometry w23x, in the following we use the mass scaled normal mode coordinates pertinent to the D 3 h symmetry point group to evalu-
S. Mahapatra, H. Koppelr Chemical Physics Letters 306 (1999) 387–394 ¨
ate the integral in Eq. Ž3.. These coordinates are obtained by the FG-matrix method of Wilson et al. w29x. We define the symbols q0 for the symmetric stretching Žbreathing. mode and q1 and q2 for the components of the degenerate vibration. If r and f represent polar coordinates, i.e., magnitude and direction of this degenerate vibration, then q1 s r sin f and q2 s r cos f . In the harmonic approximation, the initial Rydberg wavefunction can be written as
c
Ryd
Ž q0 , r . s Ne
ya A 1Ž q 0yq 0e . 2 r "
e
ya E Ž r y r e . 2 r "
389
independent and therefore the dynamics of these two modes can be decoupled from each other Žsee, e.g., Ref. w34x.. In the time-dependent picture, the total time autocorrelation function is written as the product of the component autocorrelation functions for these two modes. Therefore, the Fourier convolution theorem can be utilized in describing the full emission spectrum. Following Eqs. Ž3. and Ž4., the spectrum in the JT mode can be described as
sJT Ž E . A Ý ,
y2 a E r 2 r "
d Ž Vi Ž q0e , r . y E . r d r ,
He
i
Ž 5.
Ž 4. where N is the normalization constant and a k s mpn k , with m being the reduced mass and n k Ž k s A1 , E . the frequencies of the symmetric Ž A1 . and the degenerate Ž E . modes. The quantities q0e and r e correspond to the equilibrium geometry of the Rydberg state and define the coordinates of the FC zone centre. Since r e s 0 at this centre Žsee below., hereafter, we will drop it from all equations. The full semiclassical emission profile is obtained by substituting Eq. Ž4. into Eq. Ž3.. We emphasize that in the above formulation the initial state probability distribution is treated in a quantal fashion, as a square of the initial vibrational wavefunction. This choice has been made because we compare below with low temperature Žin fact, T s 0. spectra for which it is well accepted in the literature w28,30,31x. For the same reason, the energy in the initial state is independent of the nuclear coordinate, corresponding to a vibrational eigenvalue, and only the final state potential energy appears in the argument of the d function in Eq. Ž3.. In the high temperature limit, a purely classical probability distribution would be appropriate also for the initial state and would lead to the appearance of the potential energy difference in Eq. Ž3. w32,33x. However, as already indicated above, the experimental and full quantal reference spectra addressed in Section 3 below rather correspond to the limit T s 0. We now describe the emission profile within the linear coupling approximation using a first-order Taylor series expansion of the final state potential. In such a situation, the two modes, viz. the breathing mode q0 and the Ž E = e . JT active mode r are
where Vi correspond to the adiabatic potential energies of the final electronic manifold. The quantity r d r is the volume element for the JT mode coordinate. The Vi are obtained by diagonalizing the potential part of the linear Ž E = e . JT model Hamiltonian w32x V1,2 Ž q0e , r . s V0 Ž q0e . . kr ,
Ž 6.
where . indicates the lower and upper adiabatic sheets, respectively, and V0 s V0 Ž q0m at h r m e . is the potential energy at the FC zone centre. The quantity k is the first-order JT coupling constant and represents the slope < dV1,2r d r < rs r e < of the adiabatic potential with respect to the JT mode at the equilibrium geometry. Below we extract the value of k using the adiabatic potential energies from the DMBE PES w18x for the relevant geometries of H 3 . Using Eq. Ž6. and utilizing the properties of the d function, the emission intensity in Eq. Ž5. is given by w25– 28,30,31x ey2 b E Ž EyV 0 .
sJT Ž E . A Ý
i
2
r"
< E y V0 < ,
Ž 7.
where bE s < a Erk 2 <. The emission profile, Eq. Ž7., exhibits a bimodal intensity distribution corresponding to the two sheets of the JT split PES w30,31x. The energy splitting D between the two maxima obtained from Eq. Ž7. is given by
(
Dsk
"
aE
.
Ž 8.
S. Mahapatra, H. Koppelr Chemical Physics Letters 306 (1999) 387–394 ¨
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Now we turn to the spectrum in the breathing mode:
s br Ž E . A Ý
He
y2 a A 1Ž q 0yq 0e . 2 r "
d Ž Vi Ž q0 . y E . d q0 .
i
Ž 9. It can be evaluated in a similar way by expanding the potential around the equilibrium V1,2 Ž q0 . f V0 y k g Ž q0 y q0e . ,
Ž 10 .
where k g s y dV1,2r d q0 < q 0sq 0e is the slope of the adiabatic potential with respect to the breathing mode at the equilibrium geometry. With these, the integral in Eq. Ž9. reads w25–28x
s br Ž E . A Ý i
ey2 b A1Ž EyV 0 . < kg <
2
r"
,
Ž 11 .
with bA 1 s < a A 1rk g2 <. Finally, the semiclassical emission profile is obtained by convoluting the JT profile sJT Ž E . with the profile for the breathing mode s br Ž E ..
3. Results and discussion Following our earlier work w20–22x, we consider here the transition of H 3 and its isotopomers from the Ž3d .3 EX component state of the n s 3 principle quantum shell to the degenerate ground electronic manifold. The equilibrium geometry of these species in this quantum shell is nearly equilateral triangular and is close to that of the corresponding cations in the electronic ground state w35x. The equilibrium bond length amounts to ; 1.642 a0 in the Ž3d .3 EX electronic state w35x. In the following, we use the above equilibrium bond distance and the vibrational frequencies of the corresponding cations in their electronic ground state w36x in order to calculate the emission profiles semiclassically. Cuts of the adiabatic sheets of the DMBE PES along with the Ž3d .3 EX Rydberg electronic state and r < c Ryd < 2 of H 3 as a function of the JT coordinate r Žwith q0 s q0e s 1.642 a 0 . are shown in Fig. 1. As can be seen from Fig. 1, the potential curves of the ground electronic manifold exhibit a crossing Žconical intersection. at the centre of the FC zone at r s 0. The magnitude of the potential energy V0 at
Fig. 1. Adiabatic potential energies as a function of the JT coordinate r for fixed value of q0 s1.642 a0 . The solid and dashed curves represent the lower and the upper adiabatic sheets, respectively. The point of intersection of the two at 3.089 eV corresponds to the center of the FC zone. The initial Rydberg electronic state and corresponding probability density for H 3 is also shown at the top of the figure to illustrate the optical emission process.
the FC zone centre is 3.089 eV w18x. The solid and the dashed curves in Fig. 1 correspond to the dissociative lower and bound upper adiabatic sheets of the DMBE PES, respectively. The direction f of the degenerate vibration can take values either 0 or p at the C2 Õ configuration. r < c Ryd < 2 has a bimodal shape due to the volume element in the JT mode. The reflection principle implies that this bimodal probability density distribution of the initial state in the coordinate space will be mediated through the final electronic manifold into the bimodal intensity distribution of Eq. Ž7.. The magnitude of the coupling constant k is calculated at the D 3 h equilibrium geometry of H 3 using Eq. Ž5. and the adiabatic potential energies from the DMBE PES. By definition k is the derivative of the adiabatic potential with respect to r at r s 0. We calculate this derivative numerically by a finite difference scheme, which leads to k eff s < V2 Ž r . y V1Ž r .
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energy separation obtained in the experimental recording w23,24x as well as in our earlier quantal studies w20–22x. The Ž E = e . spectrum convoluted with the Gaussian in Eq. Ž11., corresponding to the progression in the breathing mode q0 , also has the intensity minimum at 3.089 eV and is symmetric about the intensity minimum Žshown by short dashed lines.. Likewise, the positions of the maxima and the energy splitting between them remained unaltered. However, the convoluted one is more diffuse than the Ž E = e . spectrum. Therefore, only the breathing
Fig. 2. Ža. The effective first-order JT coupling constant k eff , as a function of the JT coordinate r of H 3 for different values of the breathing mode q0 Ž a 0 . indicated in the panel. Žb. The value of k eff averaged over r as a function of the breathing mode q0 .
FC zone at q0 s 1.642 a0 to the location of minimum energy on the seam at q0 s 1.973 a 0 w18x. The average value of k eff over r is plotted as a function of q0 in Fig. 2b. The variation of k eff with q0 implies the relevance of the higher-order coupling terms in the dynamics which are neglected here for the sake of simplicity. We use k eff s 3.56 eVra0 , obtained for q0 s 1.642 a 0 , in our subsequent calculations of the emission profiles. Fig. 3 in its uppermost panel displays the results of the semiclassical calculations on H 3 . The types of calculation from which the individual curves originate are denoted in the panel. All three curves exhibit a bimodal distribution of emission intensities with two maxima located at the same final state energies of ; 2.55 eV and ; 3.65 eV. The Ž E = e . JT spectrum Žshown by long dashed lines. obtained from Eq. Ž7. shows an intensity minimum at the energy corresponding to the FC zone centre Ž3.089 eV.. The two maxima are separated by 1.1 eV in energy. This is to be compared with ; 1.2 eV
Fig. 3. The relative intensity Žarbitrary units. of emission of H 3 in X the optical transition from the upper Ž3d .3 E Rydberg electronic state to the ground electronic manifold, plotted as a function of the energy in this final electronic manifold. The semiclassical results obtained within the reflection approximation are shown in the uppermost panel. Three different variants are compared: the emission profiles obtained with the JT active mode alone Žshown by long dashed lines., its convolution with the progression in the breathing mode Žshown by short dashed lines. and the result using the full DMBE PES Žshown by the full line.. The profile obtained from the quantal calculations using WP propagation methods and coupled final electronic manifold is reproduced from Refs. w20–22x and shown in the middle panel. The lower panel shows the analogous results for transition to the uncoupled adiabatic sheets of the ground electronic manifold.
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mode contributes to the smearing out of the overall structure of the emission profile. We remind here that we have relied on a linear coupling model to calculate the emission profiles discussed above. The essential inputs are the slopes of the adiabatic potential at the FC zone centre and the equilibrium value of the potential energy. The weak dependence of k eff on q0 is also ignored. Therefore, the asymmetries of the adiabatic potentials do not show up in the emission profiles. In order to get away from these constraints, we have also calculated the emission profile exploiting the DMBE PES in full three dimensions. Since all the relevant potential interactions are implicit in this ab initio PES, the emission profile resulting from it is expected to bring out the correct asymmetry as observed in the experimental spectrum. A discrete grid is used in Jacobi coordinates Ž R,r,g . consisting of 128 points in both R and r, ranging from 1.0 to 2.016 a 0 , and of the nodes of a 96-point Gauss– Legendre quadrature along g . The initial wavefunction and the potential energies are evaluated on each node of this three-dimensional grid, and the integral in Eq. Ž3. is evaluated using the quadrature rule. We have adopted a histogram procedure and discretized the final state energy E at an interval of d E s 0.05 eV in order to numerically evaluate the d function in Eq. Ž3.. If the potential energy at a node falls within a bin between E and E q d E then the probability density
22x. In the middle panel of Fig. 3, we have reproduced the emission profile of H 3 obtained from the three-dimensional time-dependent WP calculations of Refs. w20–22x. Since details have been given earlier, here we only mention that the above result is obtained by using Jacobi coordinates with total angular momentum J s 0 and a diabatic representation of the Hamiltonian. The most pronounced feature of the quantal result is the shift of the intensity maximum and minimum to higher final state energies as compared to the semiclassical results. Also notable is the energy splitting between the intensity maxima in the quantal profile which is ; 0.1 eV higher than that obtained from the semiclassical profile. In order to find out possible origins of the energy shifts in the quantal coupled states result we have re-calculated the quantal spectrum suppressing the nonadiabatic coupling between the lower and upper adiabatic sheets. Otherwise, the calculation is the same as in the coupled states treatment of the central panel of Fig. 3. The results are shown in the lowermost panel of Fig. 3, identifying the spectrum of the lower and upper adiabatic sheet by thick dashed and thin solid line, respectively. Due to its repulsive nature, we see no discrete structures in the spectrum of the lower adiabatic sheet. On the other hand, in the absence of coupling to the lower sheet, the upper sheet is bound in nature and, therefore, a series of discrete lines is observed in its spectrum. Since we have used a symmetry adapted initial wavefunction ŽEq. Ž4.., only the lines of A1 symmetry appear in the spectrum. The minimum of the upper adiabatic sheet coincides with the minimum of the seam of conical intersections at ; 2.75 eV w18x, but the first line in its spectrum emerges at ; 3.72 eV only. This amounts to a huge zero-point energy of ; 1 eV of this sheet that arises from the cusp of the JT-split potential energy surfaces. As a consequence, the quantal spectrum for the upper Žuncoupled. surface peaks at substantially higher energy than the semiclassical one. Comparing with the coupled states treatment, the two maxima occur at even higher energies. As discussed in Refs. w11,12,17,19x these energy shifts in the coupled state results arise from the effect of the geometric phase. This effect is more pronounced on the upper adiabatic sheet because the energy minimum of this sheet lies on the seam of conical intersections and, therefore, the impact of the
S. Mahapatra, H. Koppelr Chemical Physics Letters 306 (1999) 387–394 ¨
degeneracy is stronger on this sheet. Of course, the main effect of the nonadiabatic coupling is the dramatic change of the sharp line spectrum of the upper uncoupled sheet to a broad bump in the coupled state description. The latter quantal calculation w20–22x elegantly reproduces the experimental results w23,24x. An explicit comparison has already been given in Refs. w20–22x and is not repeated here. We conclude that the above characterization of the semiclassical spectra also pertains to the comparison with the experimental spectrum of H 3 . Despite the above shortcomings, the semiclassical model is fairly successful in reproducing the observed energy splitting D in the bimodal emission profile. The quantity D is the most important observable in the experimental measurements. This tempted us to search for a simple kinematic scaling relation for this quantity when studying heavier isotopomers of H 3 . Substituting k s 3.56 eVra0 , " s 1, m s 1836 and n E s 0.1828 = 10y2 auy1 w36x in Eq. Ž8. we obtain D H 3 s 1.1 eV, similar to that extracted from the full semiclassical emission profile. Again this is ; 0.1 eV lower than the experimental w23x as well as the quantal w20–22x value. Using Eq. Ž8., the energy splitting in the emission profiles of heavier isotopomers ŽX 3 . is given by
DX 3 s
DH 3 m1r4 X3
,
Ž 12 .
where m X 3 s m X rm H in the normal mode picture and m H s 1.008 u. We have calculated the emission profiles for the heavier isotopomers up to m X s 5.04 u semiclassically using the full DMBE PES. The energy splittings D extracted from these numerical results are plotted against m in Fig. 4 and shown by the filled circles. The energy splitting obtained by Eq. Ž12. is shown in Fig. 4 by the solid line. As can be seen from Fig. 4, the calculated energy splittings are well reproduced by the scaling relation of Eq. Ž12.. The energy splitting decreases in the heavier isotopomers and a similar trend is noticed in the experimental spectra w23x. With increasing mass, the zero-point energy decreases and the initial wavefunction becomes narrower. This causes a narrower intensity distribution of the heavier isotopomers with the two maxima approaching the centre of the FC
393
Fig. 4. The energy splitting D Žin eV. between the maxima in the bimodal emission profile as a function of the mass m Žin u. of the isotopomers of H 3 . The filled circles indicate the results obtained semiclassically using the full DMBE PES and the solid line is obtained using the scaling relation in Eq. Ž12..
zone and resulting in the decrease of D according to Eq. Ž12..
4. Conclusions We have calculated the Rydberg emission profiles of H 3 and its isotopomers semiclassically in the framework of the reflection principle. We obtained bimodal emission profiles as observed in the experimental measurements and also in our earlier quantal investigations. It is seen that the most important experimental observable, viz. the energy splitting D between the two maxima of the bimodal profile, is obtained by describing the dynamics only in the JT active mode and only considering a linear coupling between the component states of the final electronic manifold. The added contributions from the breathing mode smears out the profile and makes it more diffuse. The observed asymmetry in the emission profiles is missing in the linear coupling results. This is reproduced by calculating the spectrum numerically Žwithin the semiclassical approximation. employing the adiabatic potentials for the lower and upper sheets of the final electronic manifold. However, the absolute positions of the two maxima in the semiclassical profile occur at much lower final state energies when compared to the quantal results. Also, the semiclassical profile exhibits an intensity minimum at E s 3.089 eV which is close to that of the vertical transition. In the quantal profile likewise the
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intensity minimum shifts to ; 0.7 eV higher energy. The shift of the maxima and minimum to higher energies in the quantal profile arises from the zeropoint energy as well as from geometric phase effects. These are purely quantal effects which are absent in the semiclassical picture. The two maxima in the quantal profile of H 3 are ; 1.2 eV apart which is ; 0.1 eV higher than the semiclassical value. Apart from this relatively small quantal correction, the energy splitting D for any heavier isotopomer of H 3 can be obtained from the simple scaling relation in Eq. Ž12.. Although the semiclassical model qualitatively reproduces the features of the experimental findings, it is missing quantitative details. The results embodied in this Letter, therefore, illustrate both the success and shortcomings of the semiclassical model in this fundamental polyatomic molecule.
Acknowledgements SM gratefully acknowledges the Alexander von Humboldt foundation for a research fellowship. We are indebted to Professor L.S. Cederbaum for his interest in this work.
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