Download Photodissociation of methane: Exploring potential energy surfaces

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
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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-
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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-
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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
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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
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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.
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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.
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