THE JOURNAL OF CHEMICAL PHYSICS 123, 204310 共2005兲
Simulation of a complex spectrum: Interplay of five electronic
states and 21 vibrational degrees of freedom in C5H4+
Andreas Markmanna兲
Theoretische Chemie, Technische Universität München, Lichtenbergstrasse 4, 85 747 Garching, Germany
Graham A. Worth
School of Chemistry, University of Birmingham, Edgbaston, Birmingham B15 2TT, United Kingdom
Susanta Mahapatra
School of Chemistry, University of Hyderabad, Hyderabad 500 046, India
Hans-Dieter Meyer, Horst Köppel, and Lorenz S. Cederbaum
Theoretische Chemie, Physikalisch-Chemisches Institut, Ruprecht-Karls-Universität Heidelberg,
Im Neuenheimer Feld 229, 69 120 Heidelberg, Germany
共Received 5 July 2005; accepted 12 September 2005; published online 23 November 2005兲
Using a five-state, all-mode vibronic coupling model Hamiltonian derived in a previous publication
关A. Markmann et al., J. Chem. Phys. 122, 144320 共2005兲兴, we have calculated the photoelectron
spectrum of the pentatetraene cation in the neighborhood of the B̃ 2E state, which can be represented
with charge-localized components. To this end, quantum nuclear dynamics calculations were
performed using the multiconfiguration time-dependent Hartree method, taking all 21 vibrational
normal modes into account. Compared to experiment, the main features are reproduced but higher
accuracy experiments are necessary to gauge the accuracy of the predictions for the vibronic
progressions at the rising flank of the spectrum. © 2005 American Institute of Physics.
关DOI: 10.1063/1.2104531兴
I. INTRODUCTION
Valence photoelectron spectroscopy can be used to study
the vibrational energy-level structure of ionized and neutral
molecule.1–6 Transitions to several states may occur which
may be coupled. The photoionization spectrum bears the signature of the vibronic coupling, more so if a conical
intersection7–16 is involved, where the coupling between
states is particularly important.17 In a particular class of
these, the Jahn-Teller 共JT兲 systems,18,19 an electronic degeneracy enforced by symmetry is lifted by suitable nuclear distortions. Systems in which interactions between a degenerate
and a nondegenerate state exist that lift the degeneracy are
labeled as pseudo-Jahn-Teller 共PJT兲 systems.17,20
The molecule pentatetraene 共C5H4, D2d symmetry at
equilibrium configuration兲 is of interest as a molecular wire.
The JT effect in the highest occupied molecular orbital
共HOMO兲-ionized X̃ 2E electronic manifolds of pentatetraene
and the related molecule allene 共C3H4兲 were first theoretically investigated in Ref. 21, where the impact of E ⫻ B JT
coupling was identified. Dynamical calculations were also
performed to predict photoelectron spectra for these states
that compared well to experimental results.22 E ⫻ B is a relatively rare form of JT coupling in which the orbital degeneracy is lifted by a nondegenerate pair of vibrations.
We label the main cationic states of the molecules as X̃,
Ã, B̃, etc., while satellite states 共which may be energetically
between main states兲 are given the labels S̃1, S̃2, etc. The
a兲
Electronic mail: [email protected]
0021-9606/2005/123共20兲/204310/9/$22.50
ground and first two excited main states of the pentatetraene
cation are of the 2E type, i.e., X̃ 2E, Ã 2E, and B̃ 2E. These
states are each orbitally degenerate at the D2d symmetry
equilibrium geometry of the neutral molecule. In this paper,
we use a five-state, all-mode vibronic coupling model Hamiltonian derived in a previous publication23 to calculate the
photoelectron spectrum of the pentatetraene cation C5H+4 in
the neighborhood of the B̃ 2E state. The model Hamiltonian
is based exclusively on ab initio data since experimental data
at the desired resolution is presently unavailable for the pentatetraene band of interest. All 21 degrees of freedom are
taken into account. An extended fit approach was taken in the
fitting procedure, contrasting the finite difference approach
taken in Refs. 24–26.
The B̃ 2E state is of special interest since its degenerate
components can be represented such that the hole charge is
localized at either end of the molecule. In a forthcoming
publication, we will present dynamical calculations in which
one component of the charge-localized states of allene and
pentatetraene is artificially depopulated. These simulations
will serve as generic model for intramolecular charge transfer along a conjugated chain. We will therefore refer to these
states as charge-transfer states in the following. Ultrafast
charge-transfer 共CT兲 and, more specifically, electron-transfer
共ET兲 processes between molecular systems play an important
role in biological and biochemical systems.27–29 They have
also been studied as candidates for future molecular
computational devices experimentally30–36 as well as
theoretically,37–40 forming an active field of current research.
The components of the doubly degenerate B̃ 2E state are
123, 204310-1
© 2005 American Institute of Physics
204310-2
J. Chem. Phys. 123, 204310 共2005兲
Markmann et al.
TABLE I. ADC共3兲 ionization potentials 共energies above the neutral groundstate equilibrium geometry兲 共in eV兲 of the states of interest for pentatetraene
as used for the E共k兲
0 values in the Hamiltonian 关Eqs. 共5兲–共12兲兴.
B̃ 2E
C̃ 2B2
D̃ 2A1
S̃ 13E
1,2
15.08
3
15.82
4
16.27
5
18.36
State
k
ADC共3兲
the C̃ 2B2 and the D̃ 2A1 state. A fit to ab initio data of a
vibronic coupling Hamiltonian involving these states plus a
higher-energy satellite state 共S 13E兲 was presented in Ref. 23.
Briefly, ab initio energies at the outer valence Green’s
function 共OVGF兲 level supplemented by the third-order algebraic diagrammatic construction 关ADC共3兲兴 calculations
were used as a template for fitting a vibronic coupling
Hamiltonian. The ADC共3兲 vertical ionization potentials 共IP兲
23
E共k兲
0 for the states of interest are shown in Table I.
The coupling between the states selected and the vibrational modes is modeled by the bilinear model vibronic coupling Hamiltonian
Ĥ = Ĥ0 + 共eij兲兩i,j=1,. . .,5 .
FIG. 1. Sketch of a cross section of the PESs along a coupling or tuning
mode with possible intramolecular charge-transfer mechanisms. Shown is a
cross section of the PESs along the coupling mode. A fictitious process is
considered where initially, the molecule is in a state where the hole sits only
on the right-hand side. The question is whether direct JT coupling between
the components of the E state 共small arrow兲 or indirect PJT coupling via the
B2 state 共loop-shaped arrows via higher states兲 dominates. Note that modes
␯ 苸 B2 also couple the B2 with the A1 state 共dark arrow兲, so that different PJT
pathways for charge transfer can communicate with each other.
coupled by Jahn-Teller coupling, and after removing an electron from one end of the molecule this can directly mediate a
charge transfer from one end of the chain to the other. This
state is, however, further coupled by pseudo-Jahn-Teller coupling to higher-lying electronic states 共C̃ 2B2 and D̃ 2A1兲 that
provide indirect pathways for CT. The competing mechanisms are sketched in Fig. 1.
In this paper, we present the first simulated photoelectron spectrum of the band involving the CT state of pentatetraene. Section II describes the ansatz used for the model
vibronic coupling Hamiltonian and briefly introduces the
multiconfigurational time-dependent Hartree 共MCTDH兲 quantum dynamics simulation method 共whose efficiency allows
us to take all 21 modes into account兲 and the method used to
extract the photoelectron spectrum from the simulated dynamics. The computational details are described in Sec. III.
Our results are presented and discussed in Sec. IV.
II. THEORETICAL BACKGROUND
A. The vibronic coupling Hamiltonian
The state components representable as localized at either
end of the molecule are those of the B̃ 2E state of pentatetraene. Neighboring states at higher ionization energies are
共1兲
Here Ĥ0 = K̂ + V0 consists of the kinetic energy K̂ plus the
harmonic oscillator plus quartic correction potential V0 modeling the neutral ground state of the molecule. The square
matrix 共eij兲兩i,j=1,. . .,5 is the coupling matrix. The 21 vibrational
normal modes of pentatetraene have the irreducible
representations
共2兲
⌫ = 4A1 + B1 + 4B2 + 6E.
Thus
K̂n =
兺
␯ 苸A ,B ,B
i
1
1
−
2
冉
冊
⳵2
␻i ⳵2
␻i ⳵2
+
−
+
,
兺
2 ⳵ Q2i ␯i苸E 2 ⳵ Q2ix ⳵ Q2iy
共3兲
V0 =
兺
␯ 苸A ,B ,B
i
+
1
1
␻i 2
␻i 2
共Qix + Q2iy兲
Qi + 兺
2
2
␯i苸E
2
兺
␯ 苸A ,B ,B
i
1
1
␰i 4
␰i 4
共Qix + Q4iy兲.
Qi + 兺
4!
4!
␯
苸E
2
i
共4兲
Q␣ denotes the mass-frequency-scaled coordinate 共i.e., the
dimensionless normal coordinate兲 of the degree of freedom
␣. ␻i are the harmonic frequencies of the normal modes and
␰i are quartic terms accounting for anharmonicities in the
ground-state potential.
The elements eij of the coupling matrix are expanded in
a Taylor series up to second order with additional fourthorder diagonal elements. Using selection rules derived from
symmetry considerations,17,20 they can be parameterized as
follows:
e11 = E共1兲
0 +
兺
␯ 苸A ,B
i
1
␬共1兲
i Qi +
2
1
1
␥共1兲
兺
兺 ⑀共1兲Q4 ,
ij QiQ j +
2 i,j
4! i i i
共5兲
204310-3
J. Chem. Phys. 123, 204310 共2005兲
Five electronic states in C5H4+
TABLE II. Overview of mode frequencies 共in eV兲 and linear coupling parameters 关in eV, where dimensionless mass-frequency scales coordinates Q␣ have
been used in Eqs. 共5兲–共12兲兴 used to model the state of interest of the pentatetraene cation. Note that higher-order terms are omitted for clarity. A complete
overview of the parameters can be found in Ref. 23 共note that in Ref. 23, the value of ␬共3兲
3 = 0.337 eV appears in the wrong place兲.
Symmetry
i
␻i
␭共12兲
i
␭共15兲
i
␭共13兲
i
␭共14兲
i
␭共34兲
i
␬共1兲
i
␬共3兲
i
␬共4兲
i
B1
A1
B2
E
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0.3921
0.2387
0.1801
0.0914
0.0876
0.0251
0.3920
0.2731
0.1873
0.1634
0.4033
0.1253
0.1016
0.0543
0.0367
0.0188
0.2921
0.2261
0.2311
0.0326
0.0808
0.2224
0.7000
−0.3780
−0.2440
−0.2440
e22 = E共2兲
0 +
−0.1411
−0.2266
−0.2656
兺
␯ 苸A ,B
i
1
−0.2907
0.3370
0.3658
␬共2兲
i Qi +
2
0.1823
0.3384
−0.0989
−0.0739
−0.0268
0.1033
0.0750
1
1
4
兺 ␥共2兲QiQ j + 4! 兺i ⑀共2兲
i Qi ,
2 i,j ij
共6兲
ekk艌3 = E共k兲
0 +
兺
␯ 苸A
i
␬共k兲
i Qi +
1
1
1
␥共k兲
兺
兺 ⑀共k兲Q4 ,
ij QiQ j +
2 i,j
4! i i i
共7兲
e12 = e21 =
兺
␯i苸B1
␭共12兲
i Qi ,
兺
e13 = e31 =
␯i苸E共A⬘兲
␭共13兲
i Qix,
兺
␯i苸E共A⬘兲
␭共14兲
i Qix,
兺
e23 = e32 =
␷i苸E共A⬘兲
e24 = e42 =
␷i
兺
␯i苸E共A⬙兲
␭共15兲
i Qix,
␭共13兲
i Qiy ,
兺
苸E共A⬘兲
␭共14兲
i Qiy ,
e25 = e52 =
兺
␷i苸E共A⬙兲
␭共15兲
i Qiy ,
共11兲
e34 = e43 =
兺
␯ 苸B
i
␭共34兲
i Qi ,
B. Simulated spectrum through wave-packet
dynamics with MCTDH
The time-dependent Schrödinger equation was solved by
the multiconfigurational time-dependent Hartree 共MCTDH兲
method.41,42 The basis of the method is to represent the vibronic wave function as a linear combination of timedependent basis functions, called single-particle functions
␬兲
共SPFs兲 ␰共k,
j␬ . The ansatz for the diabatic problem studied
here is the multiset formulation:
5
共10兲
e15 = e51 =
0.0761
0.2806
共8兲
共9兲
e14 = e41 =
0.3620
0.1887
共12兲
2
where ␬共k兲
i are the linear intrastate coupling constants in the
state k for the mode i, ␭共kl兲
are the linear JT and PJT coupling
i
constants between the states k and l for the mode i, ␥共k兲
ij are
共k兲
the intrastate bilinear coupling constants, and ⑀i are the intrastate quartic coupling constants. Further details are in
Ref. 23.
The B1 mode is regarded as a “coupling” mode and the
B2 modes as “tuning” modes.17 The normal mode frequencies and linear coupling constants are listed in Table II.
0.0681
␺共q1, . . . ,q p,t兲 = 兺
n共k兲
1
n共k兲
p
1
p
¯ 兺 A共k兲
兺
j ,. . .,j 共t兲
k=1 j =1
j =1
1
p
p
␬兲
⫻ 兿 ␰共k,
共q␬,t兲兩␩共k兲典
j
␬=1
␬
共k兲 共k兲
= 兺 兺 A共k兲
J ⌶J 兩␩ 典,
k
共13兲
共14兲
J
where 兩␩共k兲典 denotes the electronic state indexed by k as be␬兲
fore. Note that a different set of SPFs 兵␰共k,
j␬ 其 is used for each
electronic state and the multi-index J implicitly depends on
the state, as different numbers of SPFs can be used for each
state. Using the variational principle, equations of motion
can be derived for both the coefficients A共k兲
J and the SPFs
␬兲 43
␰共k,
.
j␬
In order to minimize the computational effort, the number of single-particle functions has to be balanced with the
dimensionality of the coordinates. The variational principle
guarantees that the optimal basis functions are found, so that
calculations with significantly increased efficiency can be
performed. The treatment of large systems inaccessible with
grid Fourier-transform-based methods has become feasible
with MCTDH, as shown in Refs. 25 and 44–51.
The photoelectron spectrum can be calculated by the
expression17
204310-4
J. Chem. Phys. 123, 204310 共2005兲
Markmann et al.
5
P共E兲 = 兺 兩␶共k兲兩22 Re
k=1
where
C共k兲共t兲 =
冕
⬁
e共i/ប兲共E−E0兲tC共k兲共t兲dt,
冓 冏 冉 冊 冏 冉 冊冔
␾共k兲
t
2
*
共15兲
0
␾共k兲
t
2
共16兲
,
␶共k兲 = 具␩共k兲兩T̂兩␩共0兲典,
共17兲
where 兩␩典 and 兩␾典 denote diabatic electronic and vibrational
wave functions, respectively, and the superscript is used to
denote the electronic state as in Table I. State zero denotes
the neutral 共 1A1兲 ground state. We use the Condon approximation 共vertical transition兲 ␾共k兲共t = 0兲 = ␾共0兲
0 . T̂ is the transition
operator mediating the interaction with the external field of
energy E.
C共k兲共t兲 is the time-autocorrelation function of the wave
packet starting in electronic state k. Equation 共16兲 共Refs. 42,
43, and 52兲 halves the propagation time needed to calculate
the photoelectron spectrum at a given resolution.
The matrix elements of the transition operator ␶共k兲 are
called the generalized oscillator strengths of the final states k
of the radical cation.17,24,25 They are known to vary only
weakly with the nuclear coordinates and are therefore treated
as constants. This is in accordance with the generalized Condon approximation in the diabatic electronic basis.53 Mixed
terms do not appear, as the main states are of different symmetry, and ␶共5兲 = 0, as the satellite state is not populated directly due to ionization from the neutral ground state.
The simulated spectra are broadened by convoluting the
spectral lines with a peaked curve, corresponding to a damping of the autocorrelation function in the time-dependent picture. The damping used here was chosen as
h共t兲 = e−兩t兩/␶ ,
leading to a Lorentzian broadening with a full width at half
maximum of ⌫ = ␶ / 2 due to the parameter ␶. As the present
experimental spectrum lacks sufficient detail, we have chosen ␶ = 55 fs, a setting which yields smooth spectra with a
good level of detail.
III. COMPUTATIONAL SETUP
Extrapolating the experimental spectrum shown in Fig.
2共a兲, it can be estimated that the band of interest stretches
from about 13.6 to 17.6 eV. Thus it stretches over an energy
range of ⌬E = 4.0 eV. The harmonic-oscillator model of the
ground state, i.e., the zeroth order of the vibronic coupling
Hamiltonian, can be used to estimate the number of open
states. This estimate is then the cardinal number of the set of
coefficients
再
冎
Ncount共⌬E兲 = # 共␷i兲 苸 N0M : 兺 ␷iបwi 艋 ⌬E ,
i
共18兲
where M is the number of degrees of freedom 共DOF兲. Due to
the large number of vibrational degrees of freedom involved,
a lower bound for this can be extracted by using a continuous
approximation for some low-frequency modes, which can be
extrapolated to arrive at an estimate of the exact result. The
FIG. 2. Comparison of 共a兲 experimental spectral band 共Ref. 22兲 of interest
of pentatetraene and 关共b兲 and 共c兲兴 our theoretical results. The sharp peak in
共a兲 at 15.7 eV has been assigned to N2 which was present in the experimental setup. In the theoretical result, the spectra due to initial population of the
main states taken into account are represented by the broken lines. The solid
lines are arrived at by summing over the component spectra, i.e., 2 兩 ␶E兩2
= 兩␶B2兩2 = 兩␶A1兩2 is assumed since solid evidence on the relative weights of the
cationic main states is currently available only through the spectrum, which
matches our predictions well with this setting. It can be seen that the overall
shape of the spectral band between 13.7 and 16.7 eV is reproduced well.
The experimental peak marked by a question mark is most likely due to
satellite states of E symmetry which exist below the B̃ 2E state and cross
into it, as has been discussed in Ref. 23. We suppose that their inclusion in
a future model may partly wash out the peak structure visible at the rising
flank of our current theoretical band 共c兲 due to coupling.
lower bound calculated is about 4.8⫻ 1015, while extrapolation yields an estimate of 7.4⫻ 1015. The lower bound and
the estimate are at the same order of magnitude, which gives
credibility to the estimation method used 共see the Appendix兲.
Although only a fraction of the harmonic states will be
accessible during the wave-function propagation, even a
value a few orders of magnitude smaller than the lower
bound is sufficient to make a direct diagonalization of the
Hamiltonian computationally unviable. Therefore, we have
204310-5
J. Chem. Phys. 123, 204310 共2005兲
Five electronic states in C5H4+
calculated the spectrum using the autocorrelation function
from a dynamical simulation, as indicated in Sec. II B.
As the model Hamiltonian used is based on a harmonic
model with anharmonic extensions, we have used a
harmonic-oscillator discrete variable representation 共DVR兲
to represent the single-particle functions.43 The dimensionless normal coordinates Qi are obtained from the massweighted normal coordinates by multiplication with the
square root of the frequency 冑␻i.54
The initial wave function is the vibrational ground state
of the harmonic model for the electronic ground state. Although the ground state is represented by a harmonic plus
quartic model, this initial state is a good approximation, as it
is localized at the minimum, where the impact of the fourthorder terms is small.
This approach implies that a sudden ionization is simulated by placing this vibrational state onto a cationic electronic state. With the harmonic DVR, this initial state is represented exactly. The sizes of the primitive and singleparticle function bases are chosen such that the highest
primitive basis functions and the last single-particle functions are only negligibly populated during the propagation,
so that the calculations are converged with respect to the
bases. The populations of the least populated single-particle
functions were about 0.1% and those of the highest primitive
basis functions were about 1%. The latter were harder to
converge, as an extension of the primitive basis necessarily
involves all degrees of freedom in a particle, which dramatically impairs runtime.
Within the MCTDH formalism, the 21 vibrational degrees
of freedom 共DOF兲 of pentatetraene were combined into
seven particles involving three DOF each, so that the effort
for propagating on the product primitive grid required for
each particle and the overhead introduced by managing the
particles remained balanced. The combinations used are
given in Table III. The particles with the largest product
primitive grid determine the speed of the computation per
particle. These are particles 3 and 7 which, with primitive
bases of 12, 16, and 16 functions, respectively, make for a
product primitive grid of 12⫻ 16⫻ 16= 3072 functions. The
full primitive basis consists of a total of 1024 points for each
electronic state. Several SPFs are used for each mode combination and each electronic state to capture the dynamics of
the wave function. There are 8 290 560 configurations
altogether.
Throughout the results presented in this paper, a propagation time of 120 fs was used, resulting in a spectrum corresponding to an autocorrelation function with a length of
240 fs. This setup results in very demanding computational
jobs, about 2.5 Gbyte random access memory 共RAM兲 and
five weeks of computational time on a 1.8 GHz Opteron
compute node are needed for every all-mode propagation.
This is comparable with the most demanding MCTDH calculations to date,45,49 which involved more DOF but a smaller
number of primitive basis functions in each.
The spectra are extracted with the method discussed in
Sec. II B using a software that is part of the MCTDH program
packages.55 Solid evidence on the relative weights of cationic main states is currently available only through the spec-
TABLE III. Details of MCTDH computational setup. Listed are the mode
combinations in the particles, the primitive bases of these modes, and the
number of single-particle functions of this type in the different electronic
states. The calculations were converged with respect to the grid such that the
population of the highest harmonic-oscillator functions dropped to 0.01.
Vibrational modes bracketed together form single particles. The primitive
basis is the number of harmonic-oscillator DVR functions. The particle
primitive basis is formed by the product grid of the primitive bases of the
normal modes involved, e.g., particle 3 has a primitive basis consisting of
12⫻ 16⫻ 16= 3072 basis functions. The full primitive basis consists of a
total of 1024 points for each electronic state. Dimensionless mass-frequencyscaled coordinates were used. The SPF basis is the number of single-particle
functions used, respectively, for the different electronic states. The A1 and S
states did not feature very complicated dynamics, so that two SPFs each
sufficed to simulate the dynamics accurately. There are 8 290 560 configurations altogether. The runs take five weeks and 2.5 Gbyte RAM on a
1.8 GHz Opteron compute node.
Particle
Normal modes
Primitive basis
SPF basis
共Ex , Ey , B2 , A1 , S兲
1
2
3
4
5
6
7
共␯5 , ␯15x , ␯15y兲
共␯11x , ␯13x , ␯13y兲
共 ␯ 2 , ␯ 3 , ␯ 9兲
共␯8 , ␯14x , ␯14y兲
共␯7 , ␯11y , ␯12y兲
共␯4 , ␯10x , ␯10y兲
共␯1 , ␯6 , ␯12x兲
关10,14,14兴
关14,14,14兴
关16,16,12兴
关10,16,16兴
关12,14,16兴
关20,12,12兴
关16,12,16兴
共8,8,10,2,2兲
共8,8,10,2,2兲
共8,8,10,2,2兲
共8,8,8,2,2兲
共8,8,8,2,2兲
共8,8,8,2,2兲
共8,8,8,2,2兲
trum, which matches our predictions well if 2 兩 ␶E兩2 = 兩␶B2兩2
= 兩␶A1兩2 is assumed. For the satellite state ␶S = 0, as its vanishing pole strength does not permit direct ionization into this
state from the neutral ground state.
A constant shift of −1.8 eV was applied to all bands of
the spectrum to match the energy ranges covered by the experimental spectrum, thereby correcting for a constant error
in the magnitude of the energies. The ratios between the
兩␶共k兲兩2 values and the constant shift were the only empirical
parameters used to obtain the results presented below.
IV. RESULTS
Figure 3 shows the magnitude of the autocorrelation
functions C共k兲共t兲 关Eq. 共16兲兴 for different initial states k. The
bottom panel shows the autocorrelation function for the
wave packet starting in the state B̃ 2E, i.e., for k = 1 , 2. The
largest partial recurrence is seen for this initial condition,
16% at 53 fs.
The autocorrelation functions for the wave packet initially in the C̃ 2B2 state, i.e., for k = 3, and D̃ 2A1, i.e., for k
= 4, are shown in the middle and top panels of Fig. 3. Note
that the scales are different from the bottom panel. While the
initial B2 state wave packet experiences non-negligible partial recurrences of up to 10% at times over 200 fs, the autocorrelation function of the wave packet initially placed on
the A1 state decays systematically and does not exceed 2.5%
towards the end of the propagation time.
This means that from the highest-energy A1 state dissipation is most effective. This is in accordance with expectation, as totally symmetric modes take the D̃ 2A1 wave packet
from the initial configuration to the conical intersection most
rapidly, where it crosses into the lower electronic states and
204310-6
Markmann et al.
FIG. 3. Modulus of the autocorrelation functions, 兩C共k兲共t兲兩 关Eq. 共16兲兴, for
different initial states k. Bottom panel: initial state B̃ 2E, k = 1 , 2. Middle
panel: initial state C̃ 2B2, k = 3. Top panel: initial state D̃ 2A1, k = 4. All functions start at unity 共not shown兲. The largest partial recurrence is 16% for the
initial E state, at 53 fs 共note different scale in bottom panel兲. While the
initial B2 state wave packet experiences non-negligible partial recurrences of
up to 10% at times over 200 fs, the autocorrelation function for the wave
packet initially placed on the A1 state decays systematically, i.e., from this
state dissipation is most effective. This is in accordance with the expectation, as totally symmetric modes take the D̃ 2A1 wave packet from the initial
configuration to the conical intersection most rapidly, where it crosses into
the lower electronic states and there dissipates energy vibrationally 共see
Ref. 23兲.
there dissipates energy vibrationally 共see Ref. 23兲. Wave
packets starting on the lower electronic states are subject to
not so steep potentials and hence have a higher chance of
recurring to their initial configuration. Wave-packet dynamics for pentatetraene 共and allene兲 and particularly the electronic state populations will be presented in more detail in a
forthcoming article.
The spectrum resulting from our all-mode dynamical
simulations is shown in Figs. 2共b兲 and 2共c兲. The sharp peak
at 15.7 eV in the experimental spectrum22 关Fig. 2共a兲兴 has
been assigned to N2 which was present in the experimental
setup. Between 14 and 16.8 eV 共where the experimental data
ends兲, our theoretical result matches the overall shape of the
experimental spectrum well. The total spectrum is the sum of
the spectra obtained from dynamical runs with the initial
electronic states E, B2, and A1, respectively. These components are indicated in the figure. It can be seen that the spectra corresponding to initial E and A1 states extend into the B2
spectrum, while the latter is the most extended of the three,
as the B2 state features strong coupling to both the E and the
A1 states.23 Unfortunately, the experimental N2 peak screens
the theoretically predicted peak due to the B2 spectrum, so
that our global structure prediction cannot be verified in this
energy range.
J. Chem. Phys. 123, 204310 共2005兲
The experimental peak with a question mark is most
likely due to satellite states which exist below the B̃ 2E state
and cross into it, as has been discussed in Ref. 23. It would
be desirable to take these satellite states into account in the
future. However, as the computational effort is already enormous, they were omitted for the present purposes in order to
obtain a computationally viable model.
The peak structure at the rising flank of the band 关shown
in more detail in Fig. 2共c兲兴 is suggestive of two alternating
series. We suppose it may wash out partly in a future model
due to coupling with electronic states additional to the current model 关particularly the satellite states below the marked
in Fig. 2共a兲 with a question mark兴. Due to the limited resolution of the experiment, the structure of the experimental
spectrum is difficult to recognize, rendering futile any comparison with our prediction at this level. Experimental results
at higher resolution without the disturbing N2 peak are therefore highly desirable to estimate how accurate the present
model is. Such results will drive the theoretical model towards the inclusion of more electronic states and higherorder model potentials, yielding a more complete description
of the system which will hopefully become viable with more
advanced computing technology.
Going the opposite way, we have attempted to describe
the system dynamics by a model taking only few vibrational
modes into account. Such an approach is successful if only a
few modes dominate the dynamics, while the others only
play a spectating role.
Reduced dimensionality spectra, obtained from dynamical simulations with only a few modes, are shown in Fig. 4.
For the results shown, modes were chosen such that a representative of each symmetry is added until one of each is
present in the bottom result. The following mode combinations were chosen: Upper panel: the lowest-frequency totally
symmetric mode ␯4共A1兲 with the coupling mode ␯5共B1兲.
Middle panel: ␯4 and ␯5 with the lowest-frequency tuning
mode ␯9共B2兲. Bottom panel: ␯4, ␯5, and ␯9 with the lowestfrequency PJT mode ␯15共E兲 共which has two components, x
and y, making for a simulation with five DOF兲. Details on
the character of the modes can be found in Ref. 23.
The spectra thus calculated were shifted by the zeropoint energy of the omitted normal modes in order to match
them with the spectrum obtained from the all-mode simulation to allow a comparison of these results with each other. It
can be seen in Fig. 4 that the three bands due to different
initial states 共from left to right E, B2, and A1兲 are well separated, which is not the case in the spectrum of the full simulation 共Fig. 2兲. The E band widens from top to bottom, which
means that the treatment of the coupling effects is improved
from one simulation to the next. Particularly the inclusion of
the ␯9 mode 共which is active within the E state only兲 widens
the E band considerably. The B2 and A1 bands appear to be
involved in very little interaction. This is because along the
modes featured here, crossings between the E state and the
B2 and A1 states are quite high energetically and hence are
inaccessible for the wave packet.
Another point to note is the energetic distance of the
minimum of the conical intersection seam from the FranckCondon point. In the bilinear coupling model, this is ⌬EFC
204310-7
Five electronic states in C5H4+
FIG. 4. Reduced dimensionality spectra, obtained from dynamical simulations with only a few modes. For the results shown, modes were chosen
such that a representative of each symmetry is added until one of each is
present in the bottom result. Upper panel: two-mode simulation involving
the lowest-frequency totally symmetric mode ␯4共A1兲 with the coupling mode
␯5共B1兲. Middle panel: three-mode simulation involving ␯4 and ␯5 with the
lowest-frequency tuning mode ␯9共B2兲. Bottom panel: simulation with five
DOF involving ␯4, ␯5, ␯9, and the lowest-frequency PJT mode ␯15共E兲 共which
has two components, x and y兲. Details on the character of the modes can be
found in Ref. 23. The three bands due to different initial states 共from left to
right E, B2, and A1兲 are well separated, which is not the case in the spectrum
due to the full simulation 共Fig. 2兲. The E band widens from top to bottom,
which means that the treatment of the coupling effects in this state is improved from one simulation to the next.
共1兲
2
= 兺␯i苸A1共␬共1兲
i 兲 / 2共␻i + ␥ii 兲. For the full model system, this is
0.85 eV, while the inclusion of ␯4 only brings 0.06 eV.
A better choice of modes is to select those whose linear
coupling parameters are large compared to their frequencies,
yielding maximal effects on the dynamics. It may seem surprising that the JT coupling mode ␯5 is omitted but its coupling is dominated by the PJT coupling along other modes.
Spectra from simulations with such mode selections are
shown in Fig. 5. Upper panel: simulation with five DOF,
involving ␯3共A1兲 ␯11共E兲 and ␯15共E兲. As before, note that the
E modes have two components. Middle panel: simulation
with seven DOF, involving ␯3共A1兲 ␯11共E兲, ␯15共E兲, and ␯12共E兲.
The totally symmetric mode ␯3 brings the largest contribution to ⌬EFC, at 0.32 eV. The inclusion of ␯12共E兲, which is
responsible for strong coupling to the satellite state, widens
and smears out the E band. Bottom panel: simulation with
seven DOF, involving ␯3共A1兲 ␯11共E兲, ␯12共E兲, and ␯14共E兲. Replacement of the lowest-frequency E mode ␯15 by ␯14, which
has a larger ratio ␭ / ␻, washes out the regular line structure
of the A1 band. The thin solid line is the sum spectrum. The
B2 and A1 bands now overlap. The inclusion of the ␯9 mode
would further wash out the structure in the E band but would
not affect the other two bands. A sharply peaked structure
J. Chem. Phys. 123, 204310 共2005兲
FIG. 5. Reduced dimensionality spectra, obtained from dynamical simulations involving modes with large linear coupling and low frequency. Upper
panel: simulation with five DOF, involving ␯3共A1兲 ␯11共E兲 and ␯15共E兲. Middle
panel: seven DOF, involving ␯3共A1兲 ␯11共E兲, ␯15共E兲, and ␯12共E兲. Inclusion of
␯12共E兲, which is responsible for strong coupling to the satellite state, widens
the E band of the spectrum. Bottom panel: seven DOF, involving ␯3共A1兲
␯11共E兲, ␯12共E兲, and ␯14共E兲. Replacement of the lowest-frequency E mode ␯15
by ␯14, which has a larger ratio ␭ / ␻, washes out the regular line structure of
the A band. The solid line is the sum of the spectra. The B2 and A1 bands
now overlap. However, it can be seen in all reduced dimensionality spectra
共including the previous figure兲 that the bands corresponding to different
initial electronic states overlap much less than for the full simulation. The
coupling between the electronic states is underestimated, as many coupling
channels are neglected. The band considered therefore represents a genuine
many-state, many-mode system necessitating the full simulation.
appears at the rising flank of the E band in this last simulation which is reminiscent of the corresponding structure in
the full spectrum. According to the distance between these
peaks, it can then be presumed that this structure is due to
progressions in the ␯12 and ␯14 modes, whose frequency difference is 65 meV.
Despite the fact that we already used seven DOF in the
last example 共which is hard to attain with, for example,
Fourier-transform-based propagation methods兲, the E, B2,
and A1 bands overlap much less than in the full simulation.
The coupling between the electronic states is underestimated,
as many coupling channels exist in pentatetraene and therefore many have to be neglected when selecting only a few
modes for the simulation.
This was indicated already by the analysis of the linear
coupling constants in Ref. 23, where a large number of pathways for population transfer between the states of interest
was proposed.
Hence we have simulated the diabatic coupling in a real
many-state, many-mode system. All totally symmetric modes
204310-8
J. Chem. Phys. 123, 204310 共2005兲
Markmann et al.
contribute strongly to vibrational dissipation while all other
modes contribute to the coupling between the electronic
states 共see Table II兲 either by JT or PJT coupling. The results
of the simulations with only a few modes indicate that the
PJT clearly dominates but it is still necessary to take all
modes into account to come up with a realistic spectrum as
seen in Fig. 2共b兲. This result could only be extracted from a
simulation involving many degrees of freedom through the
use of efficient propagation techniques, such as the MCTDH
code package we have employed.
兺
Ncount共⌬E兲 =
v M−1ប␻ M−1艋⌬E−␷ M ប␻ M
⯗
M
v1ប␻1艋⌬E− 兺 ␷iប␻i
i=2
where ␯i 苸 N0.
Assuming all frequencies were equal 共␻i ⬅ ␻兲 and ⑀
= ⌬E is a multiple of ប␻, this is equal to the M-dimensional
tetrahedral number
M
V. CONCLUSION
We have presented the first theoretical simulation of the
charge-transfer band in the photoelectron spectrum of the
pentatetraene cation. Using our five-state, all-vibrationalmode model based on fully ab initio potential-energy surfaces 共PESs兲,23 the overall shape of the experimental photoelectron spectrum is reproduced well. Higher-resolution
photoelectron spectra may serve to analyze the rising edge of
the band more accurately, giving further insight into the vibronic progressions involved.
Simulations involving only some of the lowestfrequency vibrational modes failed to reproduce the spectrum in its entirety. The system considered is therefore a
genuine multistate, multimode system in which the contributions from all normal modes have repercussions on the spectrum.
In a future work, a model involving satellite states below
the CT states is desirable in order to reproduce the peak
marked in Fig. 2共a兲 by a question mark. Experimental results
at higher resolution are desirable to estimate how accurate
the present model is. Such results will drive the theoretical
model towards the inclusion of more electronic states and
higher-order model potentials, yielding a more complete description of the system.
共A1兲
1,
␷ M ប␻ M 艋⌬E
1
Ncont共M, ⑀兲 =
兿
M! i=1
冉
⑀
ប␻ M−i+1
冊
共A2兲
+i ,
where ␰ denotes the floor of ␰ 共i.e., the largest integer
smaller or equal to ␰兲. Going back to the general case that the
␻i are not equal, the ordering of the terms in the sum makes
sure that the expression becomes minimal, i.e., a lower
bound is extracted. In the general case that ⑀ is not a multiple
of every ប␻i, this corresponds to the continuum
approximation—the M-dimensional tetrahedron with the side
length ⑀ is partitioned into small cuboids of volume
M
兿i=1
⑀ / ប␻i.
This approximation can be improved by using the continuum approximation 关Eq. 共A2兲兴 only for the DOF whose
frequency is smaller than a certain cutoff ␻cut and counting
explicitly 关Eq. 共A1兲兴 for the high-frequency DOF.
Let c be the index of the vibrational DOF just below the
cutoff, ␻c 艋 ␻cut ⬍ ␻c+1. Then the partial continuum estimate
Npart共c , ⌬E兲 艋 Ncount共⌬E兲 is defined by
Npart共c,⌬E兲
=
冉
兺
Ncont c,⌬E −
␷ M ប␻ M 艋⌬E
M
兺
i=c+1
冊
␷i q ␻i , 共A3兲
v M−1ប␻ M−1艋⌬E−␷ M ប␻ M
⯗
M
vc+1ប␻c+1艋⌬E− 兺 ␷iប␻i
ACKNOWLEDGMENTS
The authors would like to thank Christoph Cattarius for
his support using the MCTDH program package. This work
has been supported financially by the Volkswagen-Stiftung in
the Schwerpunktprogramm Intra- and Intermolecular Charge
Transfer.
APPENDIX: PARTIAL CONTINUUM ESTIMATE
FOR THE NUMBER OF OPEN STATES
To compare the effort of an explicit diagonalization of
the Hamiltonian with the effort of a dynamical approach, we
estimate the number of vibrational states within the energy
range of the observed spectrum ⌬E by counting the number
of harmonic-oscillator states with vibrational excitation energy within this range 关Eq. 共18兲兴. As for a larger number of
vibrational DOF and a wider energy range, explicit counting
becomes computationally very demanding, we have devised
a lower-bound estimate for this number.
Let M be the number of vibrational DOF and let ␻1
艋 ␻2 艋 ¯ 艋 ␻ M . Equation 共18兲 is then equivalent to
i=c+2
where
c
1
Ncont共c, ⑀兲 = 兿
c! i=1
冉
冊
⑀
+i .
ប␻c−i+1
共A4兲
Figure 6 shows the dependence of the partial continuum
estimate Npart共c , ⌬E兲 for harmonic-oscillator states in the
pentatetraene band of interest dependent on the energy in
Fig. 2 共note logarithmic scale兲. The energy range of the band
of interest is taken to be E = 13.5 eV+ ⌬E, where ⌬E
is as in Eq. 共A3兲. It can be seen that Npart共c , ⌬E兲 varies
smoothly with ⌬E. With rising ⌬E, the continuum
estimate was brought in according to the relation c
= max共0 , 2.6共⌬E / 1 eV兲 − 1.8兲. Note that as ⌬E increases,
the continuum approximation improves for a given small frequency ␻i, justifying this approach.
The inset shows the dependence on the number of DOF
c treated continuously in the calculations of Npart共c , ⌬E兲 for
⌬E = 4 eV 共solid line, note linear scale兲. The results for the
pentatetraene mode frequencies indicate a linear dependence
204310-9
J. Chem. Phys. 123, 204310 共2005兲
Five electronic states in C5H4+
15
FIG. 6. Partial continuum estimate Npart共c , ⌬E兲 for harmonic-oscillator
states in the band of interest dependent on the energy in Fig. 2 共note logarithmic scale兲. The energy range is E = 13.5 eV+ ⌬E 共see Fig. 2兲, where ⌬E
is as in Eq. 共A3兲. It can be seen that Npart共c , ⌬E兲 varies smoothly with ⌬E.
The inset shows the dependence on the number of DOF c treated continuously in Npart共c , ⌬E兲 for ⌬E = 4 eV 共solid line, note linear scale兲. For c = 8,
the exact value is reproduced to within a factor of about one-half.
on c, which lends itself to an extrapolation of the exact result. The exact result is always reproduced for c = 1 关see Eq.
共A3兲兴 which is why the linear extrapolation 共dash-dotted
line兲 is shown to level off at this value. An extrapolation of
7.4⫻ 1015 at c = 1 results. For ⌬E = 1 eV, an extrapolation
from the same c values as those shown yields 1.3⫻ 108
which compares favorably to the exact value of 1.4⫻ 108
共which is not so easily obtained for ⌬E = 4 eV兲. This means
that for c = 8 共the maximum value used in the main plot兲, the
exact value is already reproduced to within a factor of about
one-half, so that no extrapolation was necessary for the main,
logarithmic plot.
Strictly speaking, the numbers of vibrational states
would have to be added for every electronic state taking part
in the band of interest. However, for the energy differences
between the states 共see Table I兲, Fig. 6 demonstrates that the
resulting numbers for the higher states are more than an order of magnitude smaller than for the B̃ 2E state, so that the
quantity of interest—the order of magnitude of the sum—is
not affected.
1
D. W. Turner, C. Baker, A. D. Baker, and C. R. Brundle, Molecular
Photoelectron Spectroscopy 共Wiley, New York, 1970兲.
2
D. M. Neumark, Acc. Chem. Res. 26, 33 共1993兲.
3
J. C. Polanyi and A. H. Zewail, Acc. Chem. Res. 28, 119 共1995兲.
4
A. Held and E. W. Schlag, Acc. Chem. Res. 31, 467 共1998兲.
5
P. G. Wenthold and W. C. Lineberger, Acc. Chem. Res. 32, 597 共1999兲.
6
Faraday Discuss. 115, 1 共2000兲 共entire issue兲.
7
E. Teller, J. Phys. Chem. 41, 109 共1937兲.
8
G. Herzberg and H. C. Longuet-Higgins, Discuss. Faraday Soc. 35, 77
共1963兲.
9
T. Carrington, Acc. Chem. Res. 7, 20 共1974兲.
10
E. R. Davidson, J. Am. Chem. Soc. 99, 397 共1977兲.
11
F. Bernardi, M. Olivucci, and M. A. Robb, Chem. Soc. Rev. 25, 321
共1996兲.
12
W. Domcke and G. Stock, Adv. Chem. Phys. 100, 1 共1997兲.
13
D. R. Yarkony, Acc. Chem. Res. 31, 511 共1998兲.
14
W. Domcke and H. Koeppel, in Encyclopedia of Computational
Chemistry—Theoretical and Physical Chemistry, edited by H. F. Schaefer
III 共Wiley, New York, 1998兲.
Conical Intersections, Advanced Series in Physical Chemistry, Vol. 15,
edited by W. Domcke, D. R. Yarkony, and H. Köppel 共World Scientific,
Singapore, 2004兲.
16
G. A. Worth and L. S. Cederbaum, Annu. Rev. Phys. Chem. 55, 127
共2004兲.
17
H. Köppel, W. Domcke, and L. S. Cederbaum, Adv. Chem. Phys. 57, 59
共1984兲.
18
R. Englman, The Jahn-Teller Effect in Molecules and Crystals 共Wiley,
New York, 1972兲.
19
I. B. Bersuker and V. Z. Polinger, Vibronic Interactions in Molecules and
Crystals 共Springer, Berlin, 1989兲.
20
H. Köppel, L. S. Cederbaum, and W. Domcke, J. Chem. Phys. 89, 2023
共1988兲.
21
L. S. Cederbaum, W. Domcke, and H. Köppel, Chem. Phys. 33, 319
共1978兲.
22
G. Bieri, J. D. Dill, E. Heilbronner, J. P. Maier, and J. L. Ripoll, Helv.
Chim. Acta 60, 629 共1977兲.
23
A. Markmann, G. A. Worth, and L. S. Cederbaum, J. Chem. Phys. 122,
144320 共2005兲.
24
S. Mahapatra, L. S. Cederbaum, and H. Köppel, J. Chem. Phys. 111,
10452 共1999兲.
25
S. Mahapatra, G. A. Worth, H. D. Meyer, L. S. Cederbaum, and H.
Köppel, J. Phys. Chem. A 105, 5567 共2001兲.
26
G. A. Worth and L. S. Cederbaum, Chem. Phys. Lett. 348, 477 共2001兲.
27
A. J. Hoff and J. Deisenhofer, Phys. Rep. 287, 1 共1997兲.
28
K. Yoshihara, Adv. Chem. Phys. 107, 371 共1999兲.
29
K. Wynne and R. M. Hochstrasser, Adv. Chem. Phys. 107, 263 共1999兲.
30
V. Balzani, A. Credi, and M. Venturi, Chem.-Eur. J. 8, 5524 共2002兲.
31
V. Balzani, Photochem. Photobiol. Sci. 2, 459 共2003兲.
32
S. Isoda, Y. Hanazato, K. Akiyama, S. Nishikawa, and S. Ueyama, Thin
Solid Films 441, 277 共2003兲.
33
L. Zhang, J. Yang, L. Wang, G. Z. Yang, and Y. X. Weng, J. Phys. Chem.
B 107, 13688 共2003兲.
34
M. Hissler, J. E. McGarrah, W. B. Connick, D. K. Geiger, S. D. Cummings, and R. Eisenberg, Coord. Chem. Rev. 208, 115 共2000兲.
35
A. M. Moran, D. S. Egolf, M. Blanchard-Desce, and A. M. Kelley, J.
Chem. Phys. 116, 2542 共2002兲.
36
Y. Q. Xue and M. A. Ratner, Phys. Rev. B 68, 115406 共2003兲.
37
Y. Q. Xue, S. Datta, and M. A. Ratner, J. Chem. Phys. 115, 4292 共2001兲.
38
Y. Q. Xue, S. Datta, and M. A. Ratner, Chem. Phys. 281, 151 共2002兲.
39
R. V. Belosludov, H. Mizuseki, K. Ichinoseki, and Y. Kawazoe, Jpn. J.
Appl. Phys., Part 1 41, 2739 共2002兲.
40
Y. D. Li, H. H. Li, and C. K. Wang, Acta Phys. Sin. 51, 2349 共2002兲.
41
H.-D. Meyer, U. Manthe, and L. S. Cederbaum, Chem. Phys. Lett. 165,
73 共1990兲.
42
U. Manthe, H.-D. Meyer, and L. S. Cederbaum, J. Chem. Phys. 97, 3199
共1992兲.
43
M. H. Beck, A. Jäckle, G. A. Worth, and H.-D. Meyer, Phys. Rep. 324,
1 共2000兲.
44
G. A. Worth, H.-D. Meyer, and L. S. Cederbaum, J. Chem. Phys. 109,
3518 共1998兲.
45
A. Raab, G. A. Worth, H.-D. Meyer, and L. S. Cederbaum, J. Chem.
Phys. 110, 936 共1999兲.
46
H. Köppel, M. Döscher, I. Baldea, H.-D. Meyer, and P. G. Szalay, J.
Chem. Phys. 117, 2657 共2002兲.
47
F. Huarte-Larrañaga and U. Manthe, J. Chem. Phys. 116, 2863 共2002兲.
48
M. Nest and H.-D. Meyer, J. Chem. Phys. 117, 10499 共2002兲.
49
M. Nest and H.-D. Meyer, J. Chem. Phys. 119, 24 共2003兲.
50
E. V. Gromov, A. B. Trofimov, N. M. Vitkovskaya, H. Köppel, J.
Schirmer, H.-D. Meyer, and L. S. Cederbaum, J. Chem. Phys. 121, 4585
共2004兲.
51
M. D. Coutinho-Neto, A. Viel, and U. Manthe, J. Chem. Phys. 121, 9207
共2004兲.
52
V. Engel, Chem. Phys. Lett. 189, 76 共1992兲.
53
W. Domcke, H. Köppel, and L. S. Cederbaum, Mol. Phys. 43, 851
共1981兲.
54
E. B. Wilson, J. C. Decius, Jr., and P. C. Cross, Molecular Vibrations
共Dover, New York, 1955兲.
55
G. A. Worth, M. H. Beck, A. Jäckle, and H.-D. Meyer, MCTDH package,
version 8.2 共University of Heidelberg, Heidelberg, 2000兲; H.-D. Meyer,
MCTDH package, version 8.3 共University of Heidelberg, Heidelberg,
2000兲; see http://www.pci.uni-heidelberg.de/tc/usr/mctdh/