2 Nonadiabatic Dynamics ``on the Fly'' in the Framework of Time-Dependent Density Functional Theory (TDDFT)
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17 Nonadiabatic Dynamics in Complex Systems
303
processes are simulated by propagating ensembles of classical trajectories parallel
to the solution of the time-dependent Schrăodinger equation which determines the
quantum-mechanical electronic state populations. For this purpose, along each
classical trajectory an electronic wavefunction |Ψ (r; R(t)) is defined in terms of
the adiabatic electronic state wavefunctions according to:
|Ψ (r; R(t)) = ∑ CK (t) |ΨK (r; R(t)) ,
(17.1)
K
where |ΨK (r; R(t)) represents the wavefunction for the electronic state K while
the CK (t) are the time-dependent expansion coefficients. The time evolution of the
expansion coefficients for a given trajectory can be obtained by numerical solution
of the time-dependent Schrăodinger equation:
I (r; R(t))
i¯hC˙K (t) = EK (R(t))CK (t) − i¯h ∑ CI (t) ΨK (r; R(t))
∂t
I
(17.2)
where the bracket in the second term corresponds to the nonadiabatic coupling DKI
between the states I and K, which can be approximately calculated using the finite
difference approximation for the time derivative [73]:
DKI R t +
Δt
2
≈
1
( ΨK (r; R(t)) |ΨI (r; R(t + Δ t))
2Δ t
− ΨK (r; R(t + Δ t)) |ΨI (r; R(t)) )
(17.3)
where Δ t is the timestep used for the integration of the classical Newton’s equations
of motion.
The numerical solution of the Eq. 17.2, obtained e.g. using the fourth order
Runge-Kutta procedure, provides the time-dependent electronic state coefficients
CK (t) which can be used to define the hopping probabilities that are needed for the
electronic state switching procedure in the frame of the TSH approach. The hopping
probabilities PI→K for switching from state I to state K can be either calculated after
each nuclear dynamics time step Δ t or, alternatively, after each of the much smaller
time steps Δ used for the integration of the electronic Schrăodinger equation (17.2),
as recently introduced by us [18].
In the latter case, the hopping probability is defined as:
PI→K (τ ) = −2
Δ τ [Re(CK∗ (τ )CI (τ )DKI (τ ))]
.
CI (τ )CI∗ (τ )
(17.4)
An alternative procedure for calculating the hopping probabilities can be also
based only on the occupations of the electronic states, represented by diagonal
density matrix elements ρII = CI∗ (t)CI (t) and ρKK = CK∗ (t)CK (t) [74, 75].
The probability for hopping from state I to state K can then be defined as:
PI→K = Θ (−ρ˙ II )Θ (ρ˙ KK )
−ρ˙ II
ρ˙ KK
Δt
ρII ∑L Θ (ρ˙ LL )ρ˙ LL
(17.5)
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This probability is nonzero only if the population of the Ith state is decreasing
and the population of the Kth state is increasing, which is represented by the Θ
function. The summation in the denominator is performed over all states L whose
population is also growing. It should be pointed out that this equation requires only
the calculation of the hopping probabilities at each nuclear time step due to the fact
that populations generally vary more slowly than the coherences CK∗ (τ )CI (τ ) which
are employed in Eq. 17.4. This is particularly useful in the context of field-driven
multistate dynamics during which the laser field is varying very fast as it will be
shown in Sect. 17.6.
The necessary ingredients for performing TSH simulations are the forces (energy
gradients) in the ground and excited electronic states as well as the nonadiabatic
couplings DKI (R(t + Δ2 )). While the calculation of excited state forces in the
framework of TDDFT is already a standard procedure available in many commonly
used quantum chemical program packages, the procedure for the calculation of
nonadiabatic couplings in the framework of linear response TDDFT has been
developed only recently using plane wave basis sets by Răothlisberger et al. [2123],
as well as using localized Gaussian basis sets by us [17, 18]. In the following after
introducing the representation of the electronic wave function within the KohnSham linear response method, we briefly outline our approach for the calculation
of the nonadiabatic couplings using localized Gaussian basis sets.
In order to calculate nonadiabatic couplings in the framework of the TDDFT
method a representation of the wavefunction based on Kohn-Sham (KS) orbitals is
required. Since in the linear response TDDFT method the time-dependent electron
density contains only contributions of single excitations from the manifold of
occupied to virtual KS orbitals, a natural ansatz for the excited state electronic
wavefunction is the configuration interaction singles (CIS)-like expansion:
CSF
|ΨK (r; R(t)) = ∑ cKi,a Φi,a
(r; R(t)) .
(17.6)
i,a
CSF
where |Φi,a
(r; R(t)) represents a singlet spin adapted configuration state function
(CSF) defined as:
1
CSF
Φi,a
(r; R(t)) = √
2
Φiaαβ (r; R(t)) + Φiaβα (r; R(t))
,
(17.7)
where |Φiaαβ (r; R(t)) and |Φiaβα (r; R(t)) are Slater determinants in which one electron has been promoted from the occupied orbital φi to the virtual orbital φa with spin
α or β , respectively. This ansatz can be used to calculate the nonadiabatic coupling
as described below, but more generally, it can provide the expectation values of any
observable of interest, e.g. transition dipole moments between excited states as we
have shown in Ref. [19]. In the context of nonadiabatic dynamics the accuracy of
this representation of the wavefunction has been previously demonstrated in our
work on pyrazine [17, 19] and benzylideneaniline [18]. The expansion coefficients
17 Nonadiabatic Dynamics in Complex Systems
305
cKi,a in Eq. 17.6 are determined by requiring that the wavefunction in Eq. 17.6 gives
rise to the same density response as the one obtained by the linear response TDDFT
procedure. Their precise connection to the TDDFT eigenvectors has been shown in
Ref. [18].
The electronic structure of isolated molecular systems is most naturally described
by using Gaussian type atomic orbitals (AO’s) as basis functions in contrast to plane
waves, which represent the natural choice in extended periodic systems. Here we
present the approach for the calculation of the nonadiabatic couplings using KS
orbitals expanded in terms of localized Gaussian atomic basis sets. This formulation
is particularly convenient since it can be coupled with commonly used quantum
chemical DFT codes.
In order to calculate the nonadiabatic couplings according to the discrete
approximation given by Eq. 17.3 the overlap between two CI wavefunctions at times
t and t + Δ t along the nuclear trajectory R(t) is needed:
ΨK (r; R(t)) |ΨI (r; R(t + Δ t))
I
CSF
CSF
= ∑ ∑ c∗K
i,a ci ,a Φi,a (r; R(t)) Φi ,a (r; R(t + Δ t))
(17.8)
ia i a
The overlap of the CSF’s in Eq. 17.8 can be reduced to the overlap of singly
excited Slater determinants using Eq. 17.7, which can be further decomposed to the
overlap of spatial KS orbitals φi (t) and φi (t + Δ t) as described in Refs. [17, 18].
The spatial KS orbitals can be expressed in terms of atomic basis functions bk (R(t))
according to:
φi (t) =
n
∑ cik (t) bk (R(t))
(17.9)
k=1
with the molecular orbital (MO) coefficients cik (t). This leads to the final expression
for the overlap integral of two spatial KS orbitals at times t and t + Δ t:
φi (t) φ j (t + Δ t) =
n
n
∑ ∑ cik (t)c jm(t + Δ t)
bk (R(t)) bm (R(t + Δ t)) .
k=1 m=1
(17.10)
It should be noticed that since the two sets of basis functions bk (R(t)) and
bm (R(t + Δ t)) are centered at different positions R(t) and R(t + Δ t) they do
not form an orthonormal basis set. Therefore, in order to calculate nonadiabatic
couplings along each classical trajectory the overlap integrals between moving basis
functions are calculated at successive nuclear time steps and the KS MO coefficients
and linear response eigenvectors are utilized to transform the overlap integrals.
In order to eliminate possible random phase variations of the nonadiabatic coupling,
the phases of the CI-like wavefunction coefficients (cf. Eq. 17.6) and of the KohnSham orbital coefficients (cf. Eq. 17.9) are aligned in each nuclear timestep to the
phases of the previous step.
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R. Mitri´c et al.
17.3 Simulation of Time-Resolved Photoelectron
Spectra (TRPES)
The time-resolved photoelectron spectroscopy represents a powerful approach for
interrogation of nonadiabatic processes. The basic principle of this technique involves the creation of a coherent superposition of the ground and excited electronic
states of the studied system by an ultrashort laser pulse. This gives rise to a
wavepacket in the excited electronic states whose time evolution is subsequently
probed through the photoionization by a time-delayed ultrashort probe pulse.
The kinetic energy and angular distribution of the released photoelectrons reflect
therefore the character of the electronic state which has been ionized. Since during
the excited state dynamics this character can change, e.g. due to the passage through
a conical intersection, the above-mentioned observables offer a sensitive probe for
the nonadiabatic transitions [76].
Our method for the simulation of TRPES pump-probe signals in the frame of
the Wigner distribution approach [6, 56] is based on the propagation of an ensemble
of classical trajectories “on the fly”. For weak electric fields of Gaussian form, a
perturbation theory expression for the final quantum state populations leads to an
analytical formula for the pump-probe signal. This approach provides a general
tool for simulation of ultrafast processes and femtosecond signals in complex
systems, involving both adiabatic and nonadiabatic dynamics [56]. However, for the
simulation of TRPES a modification has to be introduced which takes into account
that a part of the probe-pulse energy E pr changes into the photoelectron kinetic
energy (PKE). Furthermore, the vibrational states of the ionized system can be also
taken into account as discussed in Ref. [77]. The photoionization process produces
photoelectrons with kinetic energies E ranging from zero up to the maximal value
of PKE max = E pr − IP(tD ), where IP is the ionization potential. The intensity of the
photoelectrons at a particular PKE is proportional to the electronic transition dipole
moments μik between the bound state i and the ionized continuum state k as well
as to the Franck-Condon (FC) factors Fik,ν between the neutral and the individual
cationic vibrational states. The TRPES signal at the time delay tD in the frame of
the Wigner distribution approach assumes then the following analytic form:
∞
STRPES(tD , E) ∼
dq0 dp0
×
0
d τ1 exp −
∞
0
× exp
dEk,ν Fik,ν exp
2
−σ pu
h¯ 2
2
−σ pr
h¯ 2
(τ1 −tD )2
2 +σ 2
σ pu
pr
∑ |μik (q(τ1 ; q0 , p0 ), E)|2
i,k
E pr −Vik (q(τ1 ; q0 , p0 ))−Ek,v − E
[E pu −Vi0 (q0 )]2 P00 (q0 , p0 )
2
(17.11)
17 Nonadiabatic Dynamics in Complex Systems
307
Since in our classical simulation the FC factors are not available, we have
assigned them a constant value for the whole PKE interval [0, PKE max ] [56, 78].
Therefore the integration over the vibrational levels Ek,ν of the ionized system can
be performed analytically. This approximate treatment is verified by comparison
with experimental TRPES signals [79]. In the above expression σ pu (σ pr ) and
E pu = h¯ ω pu (E pr = h¯ ω pr ) are the pulse durations and excitation energies for
the pump and probe step with time delay tD . Vki (q1 (τ1 ; q0 , p0 )) labels the timedependent energy gap between the electronic state i in which the dynamics takes
place and the ionized electronic state k that is used for probing. Both are obtained
from the ab initio MD “on the fly” [56]. The initial coordinates and momenta q0 and
p0 needed for the dynamics simulation can be sampled from a canonical Wigner
distribution for all normal modes at the given temperature according to:
αi
αi 2
(pi0 + ωi2 q2i0 ) ,
exp −
h
h
π
ω
¯
¯
i
i=1
N
P00 (q0 , p0 ) = ∏
(17.12)
where ωi represents the frequency of the i’th normal mode and αi = tanh(¯hωi /2kb T )
[56]. Vi0 (q0 ) are the excitation energies of the initial ensemble. The signal is
calculated by averaging over the whole initial distribution P00 (q0 , p0 ) given by
the ensemble of trajectories. Notice, that expression (17.11) is valid under the
assumption of weak electric fields due to the perturbation theory treatment [6, 56].
The simulation of the TRPES thus involves three steps: (1) The ensemble
of initial conditions is generated by sampling the Wigner distribution function
corresponding to the canonical ensemble at the given temperature. (2) The ensemble
of trajectories is propagated using nonadiabatic MD “on the fly”. (3) The TRPES is
calculated by averaging over the ensemble of trajectories employing the analytical
expression (17.11).
17.4 Application of the Nonadiabatic Dynamics “on the fly”
for the Simulation of Ultrafast Observables of Furan:
Comparison with Experiment
The simulation of ultrafast observables such as TRPES allows to make direct
comparison with experimental data and thus to reveal the dynamical processes
involved in the excited state relaxation and their time scales. Moreover, the new
methods for simulation of ultrafast processes challenge also the development of
novel experimental techniques with increasing resolution.
We wish to show that ultrafast time-resolved photoelectron imaging (TR-PEI)
together with nonadiabatic ab initio dynamics “on the fly” accounting for all degrees
of freedom allows to elucidate precisely the photophysics and photochemistry
of furan. Our theoretical simulation of photoionization is based on the methods
described in Sects. 17.2 and 17.3. The theoretical analysis is focused on the timedependent photoelectron signal intensity and PKE distribution. The complementary
308
R. Mitri´c et al.
Fig. 17.1 Comparison of the theoretical thermally broadened absorption spectrum of furan (red)
for the first S1 [1 A2 (π 3s)] ← S0 (1 A1 ) and second S2 [1 B2 (ππ ∗ )] ← S0 (1 A1 ) excited state obtained
from 240 structures sampled from the thermal ensemble at T = 300 K with the measured absorption
spectrum at room temperature (blue). The discrete absorption lines for each member of the
ensemble were convoluted with a Lorentzian function with a width of 0.1 eV and added together.
The equilibrium structure of furan in the neutral ground state as well as the dominant excitations
of the transitions to the S1 (π 3s) and S2 (ππ ∗ ) states are also shown
experimental data have been obtained by TR-PEI with an unprecedented time
resolution of 22 fs [79] using sub-20 fs pulses at 260 and 200 nm generated by the
multi-colour filamentation method [80, 81]. The combination of the experimental
findings with the theoretical simulations reveals ultrafast deactivation of excited
furan through internal conversion from S2 over S1 to the ground state [79].
The simulations have been performed in a manifold consisting of the ground
and the three lowest excited states. The energies, gradients as well as nonadiabatic
couplings needed to carry out the nonadiabatic dynamics have been calculated “on
the fly” using the hybrid PBE0 functional [82] combined with the 6-311G**++
basis set [83] containing also diffuse functions. This level of theory for electronic
structure describes accurately the stationary absorption properties and is suitable
for performing the dynamics simulations as discussed in Ref. [79]. Notice that
recently the accuracy of the TDDFT method for the description of nonadiabatic
dynamics in heterocyclic organic molecules has been validated against the highly
correlated multireference ab initio methods on the example of the pyrrole molecule
[84]. For the further computational details cf. Ref. [79] Based on the nonadiabatic
MD trajectories, the TRPES signal was calculated according to Eq. 17.11, assuming
a constant value for the transition dipole moments μik in the whole energy range.
The experimental photoabsorption spectrum of furan vapour at room temperature
as well as our TDDFT absorption spectrum simulated also at room temperature are
shown in Fig. 17.1. The good agreement between experiment and theory allows