Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
THE JOURNAL OF CHEMICAL PHYSICS 125, 124302 共2006兲 Photodissociation of methane: Exploring potential energy surfaces Rob van Harrevelta兲 Instituut voor Theoretische Chemie, Radboud Universiteit Nijmegen, Toernooiveld 1, 6525 ED Nijmegen, The Netherlands 共Received 3 May 2006; accepted 13 July 2006; published online 22 September 2006兲 The potential energy surface for the first excited singlet state 共S1兲 of methane is explored using multireference singles and doubles configuration interaction calculations, employing a valence triple zeta basis set. A larger valence quadruple zeta basis is used to calculate the vertical excitation energy and dissociation energies. All stationary points found on the S1 surface are saddle points and have imaginary frequencies for symmetry-breaking vibrations. By studying several two-dimensional cuts through the potential energy surfaces, it is argued that CH4 in the S1 state will distort to planar structures. Several conical intersection seams between the ground state surface S0 and the S1 surface have been identified at planar geometries. The conical intersections provide electronically nonadiabatic pathways towards products CH3共X̃ 2A2⬙兲 + H, CH2共ã 1A1兲 + H2, or CH2共X̃ 3B1兲 + H + H. The present results thereby make it plausible that the CH3共X̃ 2A2⬙兲 + H and CH2共ã 1A1兲 + H2 channels are major dissociation channels, as has been observed experimentally. © 2006 American Institute of Physics. 关DOI: 10.1063/1.2335441兴 I. INTRODUCTION The present understanding of the photofragmentation of methane is far from complete, especially from the theoretical perspective. This is undoubtedly related to the multidimensional character of the problem. Methane has nine internal degrees of freedom, which could all play a role. Beside the fundamental importance of methane as a prototypical molecule, a detailed understanding is also important for photochemical models of atmospheres, especially of Saturn’s moon Titan.1,2 The methylene and methyl radicals produced by methane photolysis are building blocks for the production of higher hydrocarbon molecules and other organic molecules.3 The ultraviolet absorption of methane starts at about 140 nm.4 The spectrum between 140 and about 110 nm is rather diffuse, which is indicative of a short lifetime of the excited state. Table I gives the measured yields for various photofragments 共see Refs. 4–8兲. Unraveling the branching ratios for different fragmentation channels from these yields is quite complicated. A clue about the origin of the H fragments is provided by measurements of the the kinetic energy of the H products.9–11 About 63% of the H atoms have a velocity which is energetically only compatible with the CH3共X̃ 2A2⬙兲 + H channel.11 The slower H atoms can, in principle, also be formed by three-body dissociation 共CH + H2 + H and CH2 + H + H兲. The H2 fragments can be formed via two body dissociation to CH2 + H2 or three-body dissociation to CH + H2 + H. Since the CH + H2 + H channel is the only channel which produces CH, it follows from Table I that a small part of the H2 products is formed by three-body dissociation. The remaining part is presumably formed in combination with CH2共ã 1A1兲.11 a兲 Electronic mail: [email protected] 0021-9606/2006/125共12兲/124302/8/$23.00 The experimental results are in strong disagreement with the earliest theoretical calculations,12–15 which suggest that CH2共b̃ 1B1兲 + H2 is the dominant channel. Also more recent ab initio calculations of Mebel et al.16 reveal the presence of a barrierless pathway from the Franck-Condon region 共the region close to the ground state equilibrium geometry兲 towards the CH2共b̃ 1B1兲 + H2 asymptote. On the other hand, the main products channels, CH2共ã 1A1兲 + H2 and CH3共X̃ 2A2⬙兲 + H, do not correlate adiabatically with the excited state S1 but rather with the ground state S0. Thus, the photodissociation of methane is predominantly a nonadiabatic process with electronic transitions from the S1 state to the S0 state. However, conical intersections that can cause a strong coupling between the S0 and S1 states have not yet been identified. As CH3共X̃ 2A2⬙兲 + H also correlates with the lowest triplet state T1, spin-orbit couplings between the singlet and triplet states could provide an alternative mechanism which produces CH3共X̃ 2A2⬙兲.10,16 Based on the measured anisotropy parameters, Wang et al.6 and Cook et al.10 argue that part of the H atoms are produced via the T1 pathway. As a first step towards understanding the multidimensional dynamics, we examine the relevant potential energy surfaces 共PESs兲 using multireference singles and doubles configuration interaction 共MR-SDCI兲 calculations. Previous theoretical explorations of the PESs Ref. 12–16 have already studied adiabatic dissociation pathways. In this paper, the main objective is to study the nonadiabatic pathways. The paper also presents a normal mode analysis of the stationary points of the first excited state, based on MR-SDCI calculations. In contrast to previous multiconfiguration selfconsistent field calculations,16 no local minima have been found. In this work, we restrict our attention to the lowest two singlet surfaces, S0 and S1. For most regions of coordinate 125, 124302-1 © 2006 American Institute of Physics Downloaded 24 Oct 2008 to 130.208.167.39. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp 124302-2 J. Chem. Phys. 125, 124302 共2006兲 Rob van Harrevelt TABLE I. Experimental quantum yields, i.e., the average number of fragments produced after photolysis on one CH4 molecule. Fragment Quantum yield Wavelength 共nm兲 Reference共s兲 H H2 CH共X 2⌸兲 0.45± 0.10 0.58 0.059± 0.005 0.003 121.6 123.6 123.6 121.6 5 and 6 7 8 4 CH2共b̄ 1A1兲 space, the S1 state is well separated in energy from higher excited states. However, at the tetrahedral ground state equilibrium geometry, the first three excited electronic states S1, S2, and S3, which arise from 3s ← 1t2 Rydberg transitions, are degenerate. Other Rydberg states lie at much higher energies.17 Due to the Jahn-Teller effect, the energies of the S1, S2, and S3 states split when the symmetry is reduced. Of the three Jahn-Teller components, only the lowest BornOppenheimer state 共S1兲 is dissociative. The other components, S2 and S3, are bound states. The strong nonadiabatic couplings between the S1, S2, and S3 states at tetrahedral geometries result in a rapid predissociation of the S2 and S3 states via the S1 state. Thus, in order to study photodissociation pathways, we can focus on the adiabatic S0 and S1 states. II. DETAILS OF THE ELECTRONIC STRUCTURE CALCULATIONS The potential energy surfaces for ground state and the first excited state have been computed using the MR-SDCI approach, employing the COLUMBUS electronic structure package.18–28 The COLUMBUS package provides analytical first derivatives of the energy. The molecular orbitals used to construct the reference space are based on state-averaged multiconfiguration self-consistent field 共MCSCF兲 calculations. The carbon 1s orbital is doubly occupied. The remaining eight electrons are distributed over nine active orbitals. These nine orbitals represent the four C–H bonding orbitals, the four antibonding orbitals, and one carbon 3s Rydberg orbital. The active space is restricted by requiring that no more than three electrons are excited from the four lowest occupied valence orbitals. For the MR-SDCI calculations the same restricted active space is employed. The 1s orbital is treated as a core orbital: no excitations out of this orbital are included. The Davidson correction is employed to estimate the contribution to the correlation energy of configurations not included explicitly. The Davidson correction is not included in the normal mode analysis. Larger active spaces have been used to check that the restricted active space of nine orbitals gives an accurate description. Two larger active spaces have been considered: a complete active space of nine orbitals, and a restricted active space of 13 orbitals. In the latter restricted active space, no more than three electrons are excited for the four lowest occupied orbitals. Calculations with these larger active spaces have been performed at several selected geometries, including at asymptotic regions. The relative energies 共i.e., the energy relative to the energy of the ground state of meth- TABLE II. Properties of the S0 and S1 states at the ground state equilibrium geometry. The ground state CI energy does not include the Davidson correction. Bond distance 共re兲 共Å兲 CI energy 共hartree兲 Vertical excitation energy 共eV兲 Transition dipole moment 共a.u.兲 TZ− / Ry AQZ− / Ry AQZ− / Ry core corr. 1.089 −40.4181 10.55 0.668 1.089 −40.4426 10.57 0.667 1.086 −40.4707 10.60 0.657 ane兲 obtained with these larger active spaces differ by no more than 0.01 eV from the results obtained with the restricted active space of nine orbitals. In this paper we restrict our consideration to the lowest two singlet states, S0 and S1. However, the treatment of the tetrahedral equilibrium geometry requires special care in the MCSCF procedure. To avoid artificial symmetry breaking, all three degenerate excited states 共S1, S2, and S3兲 should be included in the MCSCF calculation. On the other hand, including the S2 or S3 state at other geometries can result in convergence problems when the S3 state is close in energy to higher excited states. This problem is particularly important for force constant calculations by finite difference methods, for which the smoothness of the potential is crucial. Thus, different regions of coordinate space require different stateaveraging procedures. In principle, this will result in discontinuities in the potential energy surface. However, in the present case we found that it did not cause difficulties. We therefore adopt the following weighting scheme. For nontetrahedral geometry, the state averaging is only performed over the S0 and S1 states, with equal weights. At the tetrahedral equilibrium geometry of the ground state, a state averaging over the first four singlet states is performed, with weight 0.5 for the S0 state and weight 0.1667 for the S1, S2, and S3 states. This weighting scheme is chosen because the ground state has the same weight 共0.5兲 for all geometries. However, the resulting potential energy surface is not sensitive to the precise weighting scheme. For example, an alternative weighting scheme with weight 0.25 for all states included 共S0, S1, S2, and S3兲 gives practically identical results 共the energy difference is about 1 meV兲. The atomic orbital basis set employed is based on Dunnings correlation consistent polarized valence triple zeta basis 共cc-pVTZ兲.29 The highest angular momenta basis functions 共f on C and d on H兲 are omitted from the cc-pVTZ basis to reduce the computational effort. Diffuse functions for C have been added to describe the Rydberg s orbital: two s functions 共exponents 0.023 and 0.007兲 and one p function 共exponent 0.021兲. This basis set abbreviated as “TZ− / Ry” in this paper. 关The “⫺” is added to emphasize that high angular momentum terms have been omitted兴. Other basis sets are used in Sec. III to examine the accuracy of this approach. III. VERTICAL EXCITATION ENERGIES, DISSOCIATION ENERGIES, AND SADDLE POINTS Table II lists some properties of the S0 and S1 states at the ground state equilibrium geometry. The vertical excita- Downloaded 24 Oct 2008 to 130.208.167.39. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp 124302-3 J. Chem. Phys. 125, 124302 共2006兲 Photodissociation of methane TABLE III. Dissociation energies 共in eV兲 for all spin-allowed dissociation channels energetically accessible for Lyman ␣ photodissociation. Experimental values are taken from Ref. 10. The electronic energies for the QZ basis set have been obtained after reoptimization of the geometry with the QZ/Ry basis. Harmonic zero-point vibrational energies are calculated using the TZ− basis. Asymptotes De 共TZ− / Ry兲 De 共QZ/Ry兲 De 共expt.兲 CH3共X̃ 2A⬙2兲 + H 4.35 4.42 4.48 CH3共B̃ 1A⬘1兲 + H CH2共ã 1A1兲 + H2 10.08 10.21 10.20 4.95 6.04 4.99 6.06 5.01 6.04 CH2共b̃ 1B1兲 + H2 CH2共X̃ 3B1兲 + 2H CH2共ã 1A1兲 + 2H CH共X̃ 2⌸兲 + H2 + H 8.91 9.13 9.14 9.36 8.72 9.51 8.96 9.52 9.06 tion energy calculated using the TZ− / Ry basis is 10.55 eV. Since the vertical excitation energy is difficult to estimate from experiment, it is useful to compare this value with the results of more accurate calculations. The AQZ− / Ry basis, which is the aug-cc-pVQZ basis29 without g functions on C and f functions on H, extended with Rydberg functions on C, is employed as reference calculation. Increasing the basis set results in an increase of the vertical excitation energy of 0.02 eV. So far, the 1s orbital was doubly occupied in the CI expansion. Including excitations of electrons from the 1s orbital increases the vertical excitation energy by 0.03 eV. The increase is partially due to the contraction of the equilibrium C–H bond length, and partially due to an increase of the energy difference between the S0 and S1 surfaces at fixed geometries. Assuming that the result of the AQZ− calculation including core correlation, 10.60 eV, is accurate, we can conclude that the TZ− / Ry result is about 0.05 eV too low. The only properties of the excited state potential energy surface which can easily be compared with experiment are dissociation energies. Table III presents dissociation energies for various dissociation channels, correlating with both the S0 and S1 states. Zero-point energies of CH4 and of the various fragments have been computed using harmonic normal mode analysis, employing the TZ− / Ry basis. Table III shows that the dissociation energies obtained using the TZ− / Ry basis are between 0.04 and 0.23 eV below the experimental values. The largest errors are found for the three-body dissociation channels. For comparison, dissociation energies have also been calculated with the cc-pVQZ basis29 plus Rydberg functions 共QZ/Ry兲. Note that now the g function on C and f function on H are included. The QZ/Ry results are in much better agreement with experiment. This suggests that discrepancies between the TZ− / Ry results and experiment is due to the basis set. As the error is not larger than 0.23 eV, it appears that the TZ− basis provides a reasonable compromise between accuracy and computational effort. This basis set is therefore employed to explore the potential energy surfaces in the remaining part of this paper. We have not found local minima on the S1 surface. All stationary points have at least one imaginary frequency. The geometries and energies of some important saddle points are listed in Table IV. The geometry is described by the parameters ri, the C–H bond distances, and ij, the H–C–H bond TABLE IV. Saddle points on the S1 surface. The energy is relative to the ground state energy at the equilibrium geometry. Energies in eV, frequencies in cm−1, bond lengths in Å, and bond angles in degrees. Structure C3v 共A1兲 Cs 共A⬘兲 D2d 共A1兲 D4h 共B2u兲 Energy Imaginary frequencies r1 r2 r3 r4 12 13 14 23 24 34 9.39 e 共1202i兲 1.31 1.10 1.10 1.10 93 93 93 120 120 120 8.99 a⬙ 共955i兲 1.19 1.10 1.13 1.13 136 80 80 113 113 131 9.05 e 共622i兲 1.13 1.13 1.13 1.13 141 96 96 96 96 141 6.98 b1g 共594i兲 1.18 1.18 1.18 1.18 180 90 90 90 90 180 angles. The imaginary frequencies correspond to bending modes for the C3v, Cs, and D2d structures and an antisymmetric stretch mode for the D4h saddle point. Mebel et al. have presented a detailed analysis of stationary points of the S1 surface of methane before, based on MCSCF calculations.16 They found two local minima on the excited state, one with C2v and one with C3v symmetry. Their structures are real minima, i.e, there are no imaginary frequencies. The present MCSCF approach 共which is used as a starting point for the MR-SDCI calculations兲, also gives these two structures, with almost the same geometry parameters. However, the C2v structure does not exist at the MR-SDCI level. This can be explained as follows. The C2v minimum is located on the CH4 → CH2共b̃ 1B1兲 + H2 dissociation path, and is the result of a small barrier for dissociation, presumably caused by Rydberg-valence mixing. Due to electron correlation, the Rydberg-valence interaction increases and the barrier disappears. Therefore, the MR-SDCI calculations do not give a stationary point with C2v symmetry. In the present study, the C3v structure has an imaginary frequency, both at MCSCF and MR-SDCI levels. Normal mode analysis based on the MR-SDCI approach has not yet been presented for the excited state of CH4. However, MRD-CI results are available for the CH+4 ion.30 The C3v, Cs, and D2d structures for CH+4 are very similar to the structures of methane, and have similar imaginary frequencies.30 The electronic state for the D4h structure is a pure valence state. The D4h structure of methane does therefore not have an analogous structure for CH+4 . IV. PHOTODISSOCIATION PATHWAYS In this section, more insight into the dynamics on the S1 surface is gained by studying several cuts through the surfaces for the adiabatic states S0 and S1. The cuts are constructed as follows. First, ab initio potential energies are computed on crude two-dimensional grids. Cubic spline interpolation is then employed to calculate the energy on a finer grid. In order to describe the photodissociation dynamics, it is convenient to employ the classical trajectory picture. In this picture, a classical trajectory starts in the Franck-Condon re- Downloaded 24 Oct 2008 to 130.208.167.39. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp 124302-4 Rob van Harrevelt J. Chem. Phys. 125, 124302 共2006兲 FIG. 2. Coordinates used to illustrate the C3v → C1 / Cs distortion. X, Y, and Z are orthogonal. The C, H2, H3, and H4 atoms lie in the XY plane and form an equilateral triangle. The X axis is parallel to the C–H2 bond. The range of the polar and azimuthal angles are −90° 艋 ␥ 艋 90° and 0 ° 艋 艋 180°. FIG. 1. The S1 potential energy surface for the C3v single-bond stretch pathway. The thick solid line is the intersection of the the A1 and E surfaces. The adiabatic S1 surface corresponds to the A1 and E surfaces in the regions below and above this line, respectively. Contour labels in eV. gion. It is then integrated in time using the excited state Hamiltonian. A classical trajectory behaves as a ball rolling of a hill without friction. The potential energy surface plays the role of a landscape with hills and valleys. We will show that trajectories can easily reach conical intersections of the S0 and S1 surfaces at planar geometries. At the conical intersection, the trajectory can hop from the S1 to the S0 surface and continue on the S0 surface. In this paper, the classical trajectory picture is only used as an intuitive framework which can explain many features of photodissociation dynamics without actually solving the classical equations of motion. The advantage of a classical picture compared to a quantum mechanical picture is that it is much easier to think in terms of trajectories than in terms of wave functions or time-dependent wave packets. A. Single-bond stretch distortion When a single CH bond is stretched, the Td symmetry is reduced to C3v. The threefold degenerate T2 states will split into one A1 state and a one degenerate E state. According to the Jahn-Teller theorem, the A1 and E surfaces must cross at tetrahedral geometries. There will also be crossings at other geometries. The adiabatic S1 state can therefore correspond to different irreducible representations of the C3v point group, depending on the nuclear geometry. Figure 1 presents the C3v single-bond stretch potential as function of r1 and ⬅ 12 = 13 = 14. Other CH bond lengths 共r2, r3, and r4兲 are fixed at the ground state equilibrium value re. The point 共 , r1兲 = 共109.47° , re兲 corresponds to the tetrahedral equilibrium geometry, where the adiabatic S1 state is one component of the threefold degenerate T2 state. Figure 1 shows a complex structure with several minima. When this paper refers to minima in figures, these minima should be considered as minima in the subspace of coordinate space corresponding to the specific cut of the surface. Thus, in this case the minima in Fig. 1 are minima with the constraint that the molecule has C3v point group symmetry, and that r2, r3, and r4 are equal to re. These minima do not necessarily correspond to minima or even stationary points in full nine-dimensional coordinate space. In fact, the Jahn-Teller theorem states that the points for which the electronic state is degenerate state 共here E兲 cannot be stationary points. Two of the minima in Fig. 1 are located relatively close to the Franck-Condon point: the minimum at 共r , 兲 ⬇ 共1.3 Å , 92° 兲, which is close to the C3v共A1兲 saddle point discussed in Sec. II, and a minimum at 共r , 兲 ⬇ 共1.08 Å , 125° 兲. At the latter geometry, the electronic wave function has E character. It is not a stationary point when the symmetry is reduced to Cs or C1 due to the Jahn-Teller effect. Geometry optimization in full coordinate space starting from this point results in dissociation to CH2共b̃ 1B1兲 + H2 via an asymmetric pathway 共C1 symmetry兲. Figure 1 also shows two minima at the CH3 + H asymptote, corresponding to different structures of the excited doublet state of methyl. The minimum at = 90° corresponds to the B̃ state of methyl. This is a Rydberg 3s state with a planar D3h equilibrium structure. It has a small barrier for dissociation to CH2共ã 1A1兲 + H.31 The other minimum, at ⬇ 130°, is a signature of the valence excited state of methyl.32 As discussed in the Appendix, the valence excited state is unstable due to a conical intersection of the ground and excited state surfaces. Thus, the valence excited state will probably decay radiationless to the ground state. Trajectories reaching the region where the excited state of methyl has valence character can also continue on the excited state to the CH2共ã 1A1兲 + H or CH + H2 asymptotic regions. The minimum energy path for the C3v single-bond stretch extension is close to the = 90° line. On this path, the e bending mode has an imaginary frequency. This means that the energy drops when the symmetry is reduced from C3v to Cs or C1. To illustrate this, we now consider the following cut though the potential energy surface. Let the C, H2, H3, and H4 atoms lie in the XY plane and form a equilateral triangle, with C–H bond lengths fixed at re. Freezing the positions of C, H2, H3, and H4, we vary the position of H1. The H1 position can be described by the coordinates r1, ␥, and as indicated in Fig. 2. For the present case it is convenient to employ somewhat unconventional ranges for the polar and azimuthal angles: −90° 艋 ␥ 艋 90° and Downloaded 24 Oct 2008 to 130.208.167.39. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp 124302-5 J. Chem. Phys. 125, 124302 共2006兲 Photodissociation of methane FIG. 3. The potential energy 共in eV兲 for the S1 state as a function of r1 and ␥, at = 0 共a兲 and = 30° 共b兲. Coordinates are defined in Fig. 2. r1, r2, and r3 are fixed at re. Contour labels in eV. 0 ° 艋 艋 180°. A value ␥ = 0 corresponds to the C3v point group. When = 0 mod 60°, the symmetry is Cs, and for other values of , it is C1. Consider first the potential for = 0° 关Fig. 3共a兲兴. The decrease in energy for C3v → Cs distortions is obvious. The region around the deep minimum at ␥ = −90° corresponds to the upper cone of a conical intersection of the S0 and S1 surfaces. The conical intersection is the result of a crossing between surfaces at the planar geometry. For 共␥ , 兲 = 共−90° , 0兲, the molecule has point group C2v, and the electronic states can be labeled according to the irreducible representations of this group. The two lowest states belong to the A1 and B1 representations 共the B1 state changes sign under reflection in the molecular plane兲. Figure 4 shows the energy surfaces for these two states. As shown in Table V, the 1 1A1 and 1 1B1 states correlate asymptotically with CH3共B̃ 2A1⬘兲 + H and CH3共X̃ 2A2⬙兲 + H, respectively. The crossing of the 1 1A1 and 1 1B1 surfaces results in a conical intersection of the adiabatic S0 an S1 surfaces. When the trajectory approaches the conical intersection on the S1 surface, it has a large probability to hop to the S0 surface and continue its path to the CH3共X̃ 2A2⬙兲 + H asymptote. The other minimum in Fig. 3共a兲, at ␥ = 90°, is not a coni- cal intersection. The intersection is located at smaller r1 distance of about 1.75 Å. Due to the high energy of the conical intersection, it is not likely that trajectories reach this point. For this type of geometry, where two H atoms 共H1 and H2兲 are quite close, it is more natural to consider dissociation into CH2 + H2. In order to study this process, it is necessary to also vary the C–H2 distance r2. Figure 5 presents the 1 1A1 and 1 1B1 surfaces at 共␥ , 兲 = 共90° , 0兲, as functions of r1 and r2. The 1 1A1 surface correlates asymptotically with CH2共ã 1A1兲 + H2. On the planar dissociation path the 1 1A1 and 1 1B1 surfaces cross, giving rise to a conical intersection. The intersection with the lowest energy is found at 共r1 , r2兲 ⬇ 共2.2 Å , 1.5 Å兲. Trajectories which pass planar geometries close to this point can hop to the S0 state and proceed to CH2共ã 1A1兲 + H2. The potential for = 30°, shown in Fig. 3共b兲, also has two equivalent minima at planar geometries. These minima are conical intersections between the S0 and S1 states. Nonadiabatic pathways can give CH3共X̃ 2A2⬙兲 + H. Since the H1 and H2 atoms are again quite close, nonadiabatic dissociation to CH2共ã 1A1兲 + H2 may also be possible. In summary, it is plausible that CH3共X̃ 2A2⬙兲 + H and CH2共ã 1A1兲 + H2 are major channels formed after single-bond stretch type of distortions. These two dissociation channels are indeed the main dissociation channels in the current experimental picture.6 B. C2v distortion In the previous section, dissociation pathways after initial Td → C3v distortions have been discussed. Of course, many more different initial distortions are possible. We now consider Td → C2v distortions. The T2 state splits into A1, B1, and B2 states. Figure 6 presents a cut of the C2v potential as TABLE V. Labeling of the singlet electronic wave functions at different asymptotes, using the irreducible representations of the C1 共no symmetry兲, planar C2v, planar Cs, and planar D2h point groups. The coordinate system is chosen so that the molecule lies in the YZ plane. C1 Planar C2v Planar Cs Planar D2h 1 A 共S0兲 1 1 B1 1 1 A⬙ ¯ CH2共X̃ 3B1兲 + 2H 1 1A 共S0兲 1 1A 共S0兲 1 1A 1 1 1B 1 1 1A ⬘ 1 1A ⬙ ¯ 1 1A u CH3共B̃ 2A⬘1兲 + H 2 1A 共S1兲 1 1A 1 1 1A ⬘ ¯ CH2共b̃ B1兲 + H2 CH2共ã 1A1兲 + 2H 2 A 共S1兲 1 1 B1 1 1A ⬙ ¯ 2 1A 共S1兲 1 1A 1 1 1A ⬘ 1 1A g Products CH3共X̃ 2A⬙2兲 + H CH2共ã 1A1兲 + H2 FIG. 4. Potential energy curve for CH4 → CH3 + H dissociation for 共␥ , 兲 = 共−90° , 0兲. Bond distances r2, r3, and r4 are fixed at re. States are labeled according to C2v symmetry. 1 1 1 Downloaded 24 Oct 2008 to 130.208.167.39. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp 124302-6 J. Chem. Phys. 125, 124302 共2006兲 Rob van Harrevelt FIG. 5. The crossing seam for 共␥ , 兲 = 共90° , 0兲 共see Fig. 2兲. Bond distances r3 and r4 are fixed at re; the H3 – C – H4 bond angle is fixed at 120°. Panel 共a兲: the 1 1A1 surface. Panel 共b兲: the 1 1B1 surface. Contour labels of the potential energy are in eV. a function of 12 and 34. The point 共12 = 34兲 = 共109.47° , 109.47° 兲 corresponds to Td symmetry. Figure 6 shows three regions which correspond to the three JahnTeller components A1, B1, and B2. The A1 region is the region with the D2d pathway 共12 = 34兲 towards the planar D4h structure 共12 , 34兲 = 共180° , 180° 兲. The deep minima of 9 eV in the B1 and B2 regions are part of the valleys to the CH2共b̃ 1B1兲 + H2 channel. The potential is completely repulsive along the pathway to CH2共b̃ 1B1兲 + H2. The planar D4h structure has an imaginary frequency for a b1g antisymmetric stretch vibration 共see Table IV兲. This is caused by the pseudo-Jahn-Teller coupling33 to the S2 state, which transforms as A2u at the D4h stationary point. Even though the energy gap between S1 and S2 is quite larger 共3 eV兲, the pseudo-Jahn-Teller coupling can still have a significant effect on the topology of the PES.33 The b1g vibration reduces symmetry from D4h to D2h. In D2h symmetry, the lowest electronic states are labeled as 1 1Ag and 1 1Au. The energies for these states are shown in Fig. 7, as functions of r1 = r2, keeping r3 and r4 fixed at re. The point r1 = r2 = re is the 共12 , 34兲 = 共180° , 180° 兲 minimum in Fig. 6. The 1 1Au and 1 1Ag surfaces cross close to the minimum of the 1 1Au surface; trajectories that reach the planar geometry close to this intersection can hop to the S0 surface. Dissociation on FIG. 6. The S1 potential energy surfaces for Td → C2v bond-angle distortions. All C–H bond lengths are fixed at re. The thick solid lines are the intersections of the A1, B1, and B2 surfaces. Contour labels in eV. the S0 surface can then produce CH2共X̃ 3B1兲 + H + H or other fragments which correlate with the S0 surface. Although the energy drops dramatically when the symmetry is distorted from Td to C2v, it decreases even more when the symmetry is further distorted to C1. Normal mode analysis at points in the Franck-Condon region yield imaginary frequencies for symmetry-breaking vibrations. For example, a normal mode analysis at the point 共12 , 34兲 = 共90° , 115° 兲 yields an imaginary frequency 共−1079i cm−1兲 for a b2 antisymmetric stretch vibration involving r1 and r2. For larger distortions along path to CH2共b̃ 1B1兲 + H2, the imaginary frequency gradually disappears. A similar instability of C3v pathways has been discussed in Sec. IV A. It can therefore be expected that most trajectories will follow asymmetric 共C1兲 pathways. V. CONCLUSION The potential energy surface of the S1 state of methane has been explored using accurate MR-SDCI calculations. Direct barrierless dissociation to CH2共b̃ 1B1兲 + H2 is possible as also has been found in previous calculations.13–16 The CH2共b̃ 1B1兲 + H2 products can be formed via different types FIG. 7. The conical intersection at the planar D2h geometry. r3 = r4 are fixed at re, and bond angles are 90°. States are labeled according to D2h symmetry. Downloaded 24 Oct 2008 to 130.208.167.39. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp 124302-7 J. Chem. Phys. 125, 124302 共2006兲 Photodissociation of methane of pathways. Besides the C2v pathway 共Sec. IV B兲 we also found, for example, a C1 pathway involving the C3v共E兲 structure 共see Sec. IV A兲. The present analysis shows that also several nonadiabatic pathways via conical intersections exist. Nonadiabatic dissociation produces CH3共X̃ 2A2⬙兲 + H, CH2共ã 1A1兲 + H2, and CH2共X̃ 3B1兲 + H + H. The first two dissociation channels are indeed believed to be the main dissociation channels.6 Based on the present work it can be expected that the CH2共X̃ 3B1兲 + H + H channel is responsible for a part of the measured H atom products. Here, only a few particular routes have been considered which illustrate the role that conical intersections could play. However, there must be many more pathways, in particular, because the calculations show that C3v, D2d, and C2v pathways are unstable at least in the Franck-Condon region. The existence of conical intersection pathways does not necessarily imply that some trajectories will reach the conical intersection seams. An illustrative example of a case where the conical intersection is missed by almost all trajectories is provided by the photodissociation of H2S.34 If the trajectories miss the conical intersection seams, adiabatic dissociation pathways will dominate. For methane, the adiabatic pathways yield CH2共b̃ 1B1兲 + H2, CH2共ã 1A1兲 + H + H, or CH + H2 + H. That experiments6 show that these are minor channels, strongly suggests that conical intersection pathways do play an important role. The present study by itself cannot determine the relative importance of adiabatic and nonadiabatic dynamics. Nuclear dynamics simulations are required for this purpose and will be presented in future work. The present work is the first step in the study of the nonadiabatic photodissociation of methane. The next step would be the construction of the full-dimensional representation of the potential energy surface, preferably using interpolation methods. This surface can then be used in dynamics calculations. Another usefully future step could be the more systematic mapping of portions of the conical intersection seam, for example, using the methods described in Refs. 35–37. ACKNOWLEDGMENTS The work is financially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek 共NWO兲. The author thanks Professor van der Avoird for proofreading the manuscript and useful suggestions. APPENDIX: THE VALENCE EXCITED STATES OF METHYL Mebel et al.32 report a stable structure of the valence excited state of methyl, which has Cs point group symmetry. The mirror plane is parallel to one CH bond; the other two bonds are reflected in this plane. The symmetry species of the wave function for the valence excited state structure is 2 A⬙. As the valence excited state is the lowest state of species 2 A⬙, we label this state as 1 2A⬙. Although the present calculations confirm the existence of the 1 2A⬙ structure 共see Table VI兲, we find that it is unstable, for two reasons. First, it has an imaginary frequency for an a⬙ vibration. This vibration thus reduces symmetry to C1 共no symmetry兲. Second, at the TABLE VI. Structures of the methyl radical discussed in the Appendix. Energies in eV, frequencies in cm−1, bond lengths in Å, and bond angles in degrees. Energies do not include the Davidson correction, which did not gave reasonable results at the 1 2A⬘ / 1 2A⬙ intersection. Structure Point group Cs label C1 label Energy Imaginary frequencies r1 r2 r3 12 13 23 X̃ B̃ 1 2A ⬙ 1 2A⬘ / 1 2A⬙ crossing D3h 1 2A ⬘ 1 2A 0 ¯ 1.08 1.08 1.08 120 120 120 D3h 2 2A ⬘ 2 2A 5.75 ¯ 1.11 1.11 1.11 120 120 120 Cs 1 2A ⬙ 1 2A 4.20 a⬙ 共3650i兲 1.11 1.25 1.25 95 95 44 Cs 4.23 1.10 1.26 1.26 96 96 44 1 2A⬙ minimum, the energy of the 1 2A⬙ state is lower than that of the 1 2A⬘ state. The 1 2A⬙ state is therefore labeled as 1 2A in C1 symmetry. Thus, the 1 2A⬙ saddle point is part of the ground state surface. However, at the ground state equilibrium geometry, the Cs-point group label of the electronic wave function is 1 2A⬘ instead of 1 2A⬙. This means that there must be a conical intersection between the ground and excited potential energy surfaces at Cs symmetries. The 1 2A⬙ saddle point is very close to the crossing seam 共see Table VI兲. After small Cs → C1 distortions of the 1 2A⬙ saddle point, geometry optimization in C1 symmetry gives the equilibrium geometry of the ground state X̃. The geometry optimization follows an asymmetric pathway. Despite extensive searching, no other minimum corresponding to a valence excited state of methyl has been found in this work. In full coordinate space, without symmetry, we only found one real minimum on the first doublet excited state of methyl 共2 2A兲: the minimum corresponding to the B̃ 3s Rydberg state. 1 Y. L. Yang, M. Allen, and J. Pinto, Astrophys. J., Suppl. Ser. 55, 465 共1984兲. C. Romanzin, M. C. Gazeau, Y. Bénilan, E. Hébrard, A. Jolly, F. Raulin, S. Boyé-Péronne, S. Douin, and D. Gauyacq, Adv. Space Res. 36, 258 共2005兲. 3 C. Sagan, Science 276, 5316 共1997兲. 4 L. C. Lee and C. C. Chiang, J. Chem. Phys. 78, 688 共1983兲. 5 R. A. Brownsword, M. Hillenkamp, T. Laurent, R. K. Vasta, H.-R. Volpp, and J. Wolfrum, Chem. Phys. Lett. 266, 259 共1997兲. 6 J.-H. Wang, K. Liu, Z. Min, H. Su, R. Bersohn, J. Preses, and J. Z. Larese, J. Chem. Phys. 113, 4146 共2000兲. 7 A. H. Laufer and J. R. McNesby, J. Chem. Phys. 49, 2272 共1968兲. 8 R. E. Rebbert and P. Ausloos, J. Photochem. 1, 171 共1972兲. 9 D. H. Mordaunt, I. R. Lambert, G. P. Morley, M. N. R. Ashfold, R. N. Dixon, L. Schnieder, and K. H. Welge, J. Chem. Phys. 98, 2054 共1993兲. 10 P. A. Cook, M. N. R. Ashfold, Y.-J. Lee, K.-H. Jung, S. Harich, and X. Yang, Phys. Chem. Chem. Phys. 3, 1848 共2001兲. 11 J.-H. Wang and K. Liu, J. Chem. Phys. 109, 7105 共1998兲. 12 S. Karplus and R. Bersohn, J. Chem. Phys. 51, 2040 共1969兲. 13 M. S. Gordon, Chem. Phys. Lett. 52, 161 共1977兲. 14 M. S. Gordon and J. W. Caldwell, J. Chem. Phys. 70, 5503 共1979兲. 15 H. U. Lee, Chem. Phys. 39, 271 共1979兲. 16 A. M. Mebel, S.-H. Lin, and C.-H. Chang, J. Chem. Phys. 106, 2612 共1997兲. 17 A. M. Velasco, J. Pitarch-Ruiz, A. M. J. S. de Merás, J. Sánchez-Marín, 2 Downloaded 24 Oct 2008 to 130.208.167.39. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp 124302-8 and I. Martin, J. Chem. Phys. 124, 124313 共2006兲. H. Lischka, R. Shepard, F. B. Brown, and I. Shavitt, Int. J. Quantum Chem., Quantum Chem. Symp. 15, 91 共1981兲. 19 R. Shepard, I. Shavitt, R. M. Pitzer, D. C. Comeau, M. Pepper, H. Lischka, P. G. Szalay, R. Ahlrichs, F. B. Brown, and J. Zhao, Int. J. Quantum Chem., Quantum Chem. Symp. 22, 149 共1988兲. 20 H. Lischka, R. Shepard, R. M. Pitzer, et al., Phys. Chem. Chem. Phys. 3, 664 共2001兲. 21 R. Shepard, Int. J. Quantum Chem. XXXI, 33 共1987兲. 22 R. Shepard, H. Lischka, P. G. Szalay, T. Kovar, and M. Ernzerhof, J. Chem. Phys. 96, 2085 共1992兲. 23 R. Shepard, in Modern Electronic Structure Theory, edited by D. R. Yarkony 共World Scientific, Singapore, 1995兲, p. 345. 24 H. Lischka, M. Dallos, and R. Shepard, Mol. Phys. 100, 1647 共2002兲. 25 P. Császár, J. Mol. Struct. 114, 31 共1984兲. 26 G. Fogarasi, X. Zhou, P. W. Taylor, and P. Pulay, J. Am. Chem. Soc. 114, 9191 共1992兲. 18 J. Chem. Phys. 125, 124302 共2006兲 Rob van Harrevelt 27 P. Pulay, G. Fogarasi, G. Pongor, J. E. Boggs, and A. Vargha, J. Am. Chem. Soc. 105, 7037 共1983兲. 28 H. Lisckha, R. Shepard, I. Shavitt et al., COLUMBUS, an ab initio electronic structure program, release 5.9.0, 2005. 29 T. H. Dunning, Jr., J. Chem. Phys. 90, 1007 共1989兲. 30 R. F. Frey and E. R. Davidson, J. Chem. Phys. 88, 1775 共1988兲. 31 H. T. Yu, A. Sevin, E. Kassab, and E. M. Evleth, J. Chem. Phys. 90, 2049 共1984兲. 32 A. M. Mebel and S.-H. Lin, Chem. Phys. 215, 329 共1997兲. 33 I. B. Bersuker, Chem. Rev. 共Washington, D.C.兲 101, 1067 共2001兲. 34 P. A. Cook, S. R. Langford, R. N. Dixon, and M. N. R. Ashfold, J. Chem. Phys. 114, 1672 共2001兲. 35 M. J. Paterson, M. J. Bearpark, M. A. Robb, and L. Blancafort, J. Chem. Phys. 121, 11562 共2004兲. 36 M. J. Paterson, M. J. Bearpark, M. A. Robb, L. Blancafort, and G. A. Worth, Phys. Chem. Chem. Phys. 7, 2100 共2005兲. 37 D. R. Yarkony, J. Chem. Phys. 123, 204101 共2005兲. Downloaded 24 Oct 2008 to 130.208.167.39. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp