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DOI: 10.1002/cphc.201200941
Trajectory-Based Nonadiabatic Dynamics with TimeDependent Density Functional Theory
Basile F. E. Curchod, Ursula Rothlisberger, and Ivano Tavernelli*[a]
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Understanding the fate of an electronically excited molecule
constitutes an important task for theoretical chemistry, and
practical implications range from the interpretation of attoand femtosecond spectroscopy to the development of lightdriven molecular machines, the control of photochemical reactions, and the possibility of capturing sunlight energy. However, many challenging conceptual and technical problems are
involved in the description of these phenomena such as 1) the
failure of the well-known Born–Oppenheimer approximation;
2) the need for accurate electronic properties such as potential
energy surfaces, excited nuclear forces, or nonadiabatic cou-
1. A Short Historical Overview
Ab initio molecular dynamics has led to a large number of theoretical predictions for molecules and solids.[1, 2] In the well-established mixed quantum/classical formulation,[3–5] and thanks
to highly efficient electronic structure methods such as Kohn–
Sham (KS) density functional theory (DFT),[6] simulations on
molecular systems with up to thousands of atoms are nowadays feasible.[7–11] Initially restricted to a single adiabatic state,
DFT-based ab initio molecular dynamics was recently extended
to the nonadiabatic regime,[12–14] and this provided an important tool for the study of photophysical and photochemical
processes.[15–24]
Among the most commonly used nonadiabatic molecular
dynamics schemes are Ehrenfest dynamics and Tully’s trajectory surface hopping[25, 26] (TSH, Table 1). In the first case, the nuclear dynamics is replaced by a single-point-like trajectory
evolving in the mean-field potential derived from the time
evolution of the electronic wavefunction. Differently, in TSH
the nuclear wavepacket is represented by a swarm of independent classical trajectories, whereas the nonadiabatic couplings (NACs) induce hops between different electronic states
that occur according to a stochastic algorithm. The classical approximation for the motion of the nuclei breaks down when
interferences,[27] wavepacket bifurcation,[28] (de)coherence, or
tunneling effects occur during the dynamics. A trajectorybased solution of the quantum dynamics able to describe
theses phenomena was introduced by Wyatt.[29] The so-called
quantum trajectory method (QTM) describes the time evolution of the nuclear wavefunction by means of the quantum hydrodynamics (Bohmian)[30–32] equations of motion.[33] In this approach, the nuclear wavepacket is split into fluid elements that
represent volume elements of configuration space carrying information about the amplitude and the phase of the nuclear
wavefunction. These fluids elements are then propagated
[a] B. F. E. Curchod, Prof. Dr. U. Rothlisberger, Dr. I. Tavernelli
Laboratory of Computational Chemistry and Biochemistry
Ecole Polytechnique Fdrale de Lausanne
1015 Lausanne (Switzerland)
E-mail: [email protected], corresponding author. Designed the research and wrote the bulk of this review article.
Supporting information for this article is available on the WWW under
http://dx.doi.org/10.1002/cphc.201200941.
2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
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pling terms; and 3) the necessity of describing the dynamics of
the photoexcited nuclear wavepacket. This review provides an
overview of the current methods to address points 1) and 3)
and shows how time-dependent density functional theory
(TDDFT) and its linear-response extension can be used for
point 2). First, the derivation of Ehrenfest dynamics and nonadiabatic Bohmian dynamics is discussed and linked to Tully’s
trajectory surface hopping. Second, the coupling of these trajectory-based nonadiabatic schemes with TDDFT is described
in detail with special emphasis on the derivation of the required electronic structure properties.
Table 1. Selected cornerstones for trajectory-based nonadiabatic dynamics based on LR-TDDFT.
Landmark
Year
Ref.
Born–Oppenheimer approximation
Ehrenfest theorem
First nonadiabatic model (LZS)[a]
Bohmian trajectories
Born–Huang Ansatz
DFT
KS DFT
TSH
TDDFT
Ab initio MD
Fewest switches TSH
LR-TDDFT
Nuclear gradient within TDDFT
NACs in TDDFT
Nonadiabatic QTM
DFT-based TSH
Approximate “TDDFT”-based TSH
LR-TDDFT-based TSH
NACs between excited states
1927
1927
1932
1952
1954
1964
1965
1971
1984
1985
1990
1995
1999
2000
2001
2002
2005
2007
2010
[45]
[46]
[47–49]
[30–32]
[50, 51]
[52]
[6]
[25]
[53]
[1]
[26]
[54]
[55, 56]
[57]
[58]
[12]
[13]
[14]
[59]
[a] Landau–Zener–Stueckelberg
along Bohmian trajectories that differ from their classical counterpart by the presence of an additional potential called the
quantum potential, which is responsible for all quantum effects. Other schemes have been proposed to approximate the
nonadiabatic dynamics of the nuclear wavefunction, such as,
for example, semiclassical approaches,[34, 35] extended surface
hopping,[36, 37] quantum-classical Liouville approaches,[27, 38] linearized nonadiabatic dynamics (LAND-map),[39] and ab initio
multiple spawning methods.[40]
As an alternative to trajectory-based approaches, quantum
dynamics methods use an exact treatment of both electronic
and nuclear wavefunctions (e.g. see refs. [41–44] for a discussion of the multiconfiguration time-dependent Hartree
method). However, the applicability of these methods is hampered by their high computational cost, which limits the
number of accessible nuclear degrees of freedom. In addition,
these methods usually require fitting of the relevant electronic
potential energy surfaces (PESs) prior to propagation.
This review is focused on the coupling of [linear response
(LR-)]TDDFT to trajectory-based nonadiabatic schemes such as
TSH and Ehrenfest dynamics. It is, however, important to note
that also semiempirical and wavefunction-based methods have
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been extensively used, especially in combination with
TSH.[21, 23, 60–65] After a short description of those most relevant
approaches able to describe the nonadiabatic dynamics of molecular systems in the unconstrained phase space, we will discuss their implementation within the framework of DFT/
(LR-)TDDFT, which allow for efficient “on-the-fly” calculation of
the required electronic structure properties such as electronic
energies (i.e. PESs), nuclear forces, and NACs. In an effort to
extend the applicability of nonadiabatic molecular dynamics
(MD) to larger systems of physical, chemical, and biological interest, we will also discuss the possibility of bridging different
time and length scales, while keeping a valid description of
the photoactive components. This is done through a hierarchical scheme, which combines the ultrafast dynamics of photo-
excited electrons [purely quantum mechanical (QM) tier] with
the nuclear dynamics of the excited molecule (mixed-quantum
classical tier) and with the reorganization of the environment
[fully classical, molecular mechanical (MM), level].
Basile F. E. Curchod was born in 1984
in Vevey, Switzerland. He earned his
Bachelor (2007) and Master (2009) degrees in Chemistry at the cole Polytechnique Fdrale de Lausanne, Switzerland. Since May 2009, he has been
working as a Ph.D. student in the Laboratory of Computational Chemistry
and Biochemistry of Prof. Ursula Rçthlisberger under the supervision of
Dr. MER Ivano Tavernelli. His research
interests are nonadiabatic ab initio molecular dynamics, nonadiabatic quantum trajectories, and excitedstate properties of inorganic and organometallic molecules.
Ivano Tavernelli was born in 1967 in
Mendrisio, Switzerland. He studied Biochemistry (M.Sc., 1991) and then Theoretical Physics (M.Sc., 1996) at the
Eidgençssische Technische Hochschule
Zrich (ETHZ), where he also obtained
his Ph.D. in 1999. In 2000, he received
a Marie Curie Fellowship and went to
work in the group of Prof. Michiel
Sprik in the department of Chemistry
at Cambridge University (UK), where
he stayed until 2002. He then moved
back to Switzerland to join the group of Prof. Ursula Rçthlisberger,
first at ETHZ (2002–2003) and later at the cole Polytechnique Fdrale de Lausanne (EPFL, 2003 until now). In 2010, he earned the
title of Matre d’Enseignement et de Recherche (MER) at EPFL. In
the field of quantum-classical dynamics, Ivano Tavernelli has developed novel theoretical frameworks able to combine electronic
structure techniques based on density [DFT and (LR-)TDDFT] with
the calculation of nonadiabatic quantum and classical trajectories.
His research interests comprise adiabatic and nonadiabatic molecular dynamics (Ehrenfest dynamics, trajectory surface hopping, and
Bohmian dynamics) for the study of photochemical and photophysical processes in molecules, condensed phase, and biological
systems.
Ursula Rothlisberger was born in Switzerland and obtained her diploma in
Physical Chemistry from the University
of Bern. She earned her Ph.D. degree
at the IBM Zurich Research Laboratory
in Rschlikon. From 1992–1995, she
worked as a postdoctoral fellow, first
at the University of Pennsylvania in
Philadelphia (USA) and then at the
Max-Planck-Institute for Solid State
Physics in Stuttgart, Germany. In 1996,
she moved as a Profile 2 Fellow of the
National Science Foundation to the ETH in Zurich. One year later,
she became Assistant Professor of Computer-Aided Inorganic
Chemistry at the ETH Zurich, and in 2002 she accepted a call for
a position as Associate Professor at the cole Polytechnique Fdrale de Lausanne (EPFL). Since 2009, she has been working as a full
Professor in Computational Chemistry and Biochemistry at the
EPFL. In 2001, she received the Ruzicka Prize, and in 2005, the
World Association of Theoretically Oriented Chemists (WATOC)
awarded her the Dirac Medal for “the outstanding computational
chemist in the world under the age of 40”. Ursula Rothlisberger is
an expert in the field of density functional based mixed quantum
mechanical/molecular mechanical molecular dynamics simulations
in the ground and electronically excited states. She has published
more than 200 original publications in peer-reviewed journals and
various review articles in specialized journals and as book chapters.
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2. Nonadiabatic Dynamics
The starting point of our derivation is the time-dependent
Schrçdinger equation [Eq. (1)]:
in which R = (R1,R2,…,RNn) is the collective vector of the Nn nuclear positions in R3Nn and r = (r1,r2,…,rNel) is the collective
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vector for the Nel electrons. (A full list with the nomenclature
used herein can be found in the Supporting Information.) In
^ mol is the molecular Hamiltonian [Eq. (2)]:
Equation (1), H
the nuclear positions. The propagation of this functional form
for Y(r,R,t) implies a restriction of the Hilbert space accessible
to the molecular wavefunction at later times. A more rigorous
factorization of Y(r,R,t) was proposed by Hunter[67] and then
further developed into an exact-factorization nonadiabatic MD
scheme by Gross and co-workers.[68]
The exponential part of Equation (4) is called the “phase
term” and is defined by:
and Y(r,R,t) is the total wavefunction of the nuclear (labeled
with g and z) and electronic (labeled with i and j) degrees of
freedom. In the following, atomic units will be used except for
the reduced Planck constant h, which will be kept for clarity. In
this first section, we will derive the equations of motion for the
nuclear (slow) and electronic (fast) degrees of freedom by
using what is known as a trajectory-based approach. In this
framework, the electrons are described at a quantum mechanical level, whereas the nuclear wavepacket is discretized into an
ensemble of points in phase space and then propagated along
classical (or Bohmian quantum) trajectories that, as we will see,
keep some essential quantum character including nonadiabatic
effects.
The first step in the derivation of the equations of motion
for the combined electron-nuclear dynamics is the definition
of a suitable representation of the total system wavefunction.
Depending on the particular choice of this expansion, we can
obtain different (approximated) solutions of the initial molecular Schrçdinger equation [Eq. (1)]. In the following, we will restrict ourselves to two main representations of the total molecular wavefunction that will give rise to two main trajectorybased nonadiabatic molecular dynamics solutions: mean-field
Ehrenfest dynamics and surface hopping dynamics.
Inserting the representation of the total wavefunction given
in Equation (3) into the total Schrçdinger equation [Eq. (1)] and
multiplying from the left-hand side by W*(R,t) and integrating
over R we get [Eq. (5)]:
P
P
Zg
Z Z
g;i R r
þ g<z R gRz
j j
j g ij
j g zj
In a similar way, multiplying by F*(r,t) and integrating over r
we obtain [Eq. (6)]:
^ RÞ ¼
in which Vðr;
P
1
i<j r r
i
j
Conservation of energy has also to be imposed through the
^ mol i
d hH
condition that dt 0.
Note that both the electronic and nuclear parts evolve according to an average potential generated by the other component (in square brackets). These average potentials are timedependent and are responsible for the feedback interaction
between the electronic and nuclear components. The set of
Equations (5) and (6) forms the so-called mean-field time-dependent self-consistent field (TDSCF) method.[4]
2.1.1. The Time-Dependent Nuclear Equation
In both approaches, the nuclei are described by classical trajectories (one in the mean-field case), and therefore, the methods belong to the class of mixed quantum/classical solutions.
The following derivation is partially inspired by the articles
of Tully (ref. [4]), the book of Marx and Hutter on ab initio molecular dynamics (ref. [2]), and Curchod et al. (ref. [66]).
We start from the polar representation of the nuclear wavefunction [Eq. (7)]:[69, 70]
in which the amplitude A(R,t) and the phase S(R,t)/h are real
functions (see Figure 1 for a simple representation of these
functions).
Inserting Equation (7) into Equation (6) gives Equation (8):
2.1. Ehrenfest Dynamics
In Ehrenfest dynamics, we make use of a single-configuration
Ansatz for the total wavefunction [Eq. (3)]:
in which F(r,t) describes the (time-dependent) electronic
wavefunction and W(R,t) describes the (time-dependent) nuclear wavefunction. In the factorization in Equation (3), the electronic wavefunction does not depend (even parametrically) on
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or, equivalently [Eq. (11)]:
Figure 1. Complex nuclear wavefunction W(R) (top) and its representation in
terms of amplitude
and
i
phase A(R) and S(R) (bottom):
W(R) = A(R)exp h SðRÞ .
Each classical atom g moves according to the law of classical
mechanics on a PES given by the expectation value of the
electronic Hamiltonian, H^el , which depends on the position of
all nuclei at time t. For practical reasons, the forces in Equation (11) are often computed by using the Hellmann–Feynman
theorem as the expectation
value of the gradient of the Hamil
tonian,[72, 73] Fðr; tÞrg H^el ðr; RÞFðr; tÞ . Notably, multitrajectory extensions of Ehrenfest dynamics have also been proposed in the literature (e.g. see refs. [74, 75]).
2.1.2. The Time-Dependent Electronic Equation
Collecting the terms that do not depend on h,[71] that is,
taking the classical limit (h !0), we obtain [Eq. (9)]:
Starting from Equation (5), we first take the classical limit of
the nuclear degrees of freedom by replacing the nuclear density j W(R,t) j 2 with a product of delta-functions centered at the
atomic positions R at time t [Eq. (12)]:
This has the form of the “Hamilton–Jacobi” (HJ) equation of
classical mechanics, which establishes a relation between the
partial differential equation for S(R,t) in configuration space
and the trajectories of the corresponding (quantum) mechanical systems.[70] The identification of S(R,t) with the “classical”
action defines a point-particle dynamics with a Hamiltonian,
Hcl, given by (minus) the right-hand side of Equation (9) and
with momenta P = rRS(R). The solutions of this Hamiltonian
system are represented by curves in the (R,t) space, which are
extrema of the action S(R,t) for given initial conditions R(t0)
and P(t0) = rRS(R) j R(t0) and are called the characteristics of the
differential equation in Equation (9). In this new formulation
and after some rearrangements,1 we obtain the Newton-like
equation [Eq. (10)] for the nuclear trajectories corresponding to
the HJ Equation (9):
Note that R2R3Nn is a vector in the configuration space and
Rg2R3 is the actual position of atom g. In this way, we recover
the time-dependent Schrçdinger equation for the many-electron wavefunction [Eq. (13)]:
1
Starting from Equation (9) we first take the derivative with respect to Rz, rz
Note that both the Hamiltonian H^el (r,R(t)) and the electronic
wavefunction F(r;R(t),t) depend parametrically on the position
of the classical nuclei. This ensures the coupling between the
time-dependent propagation of the electronic and nuclear degrees of freedom.
2.1.3. The TDDFT-Based Ehrenfest MD Scheme
The numerical implementation of the Ehrenfest MD scheme requires the simultaneous solution of the coupled differential
equations for the electronic and nuclear dynamics [Eqs. (14)
and (15)]:
The first term becomes
Rearranging these equations, one gets
and, finally, using rzS/Mz = uz and the definition of the Lagrangian time derivative
we obtain
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in which “;R(t)” denotes the parametric dependence of the
electronic Schrçdinger equation from the atomic positions.
This is accomplished through the use of a Runge–Kutta algorithm as described in refs. [16, 76]. All quantities required in
Equations (14) and (15) are, therefore, computed “on-the-fly” at
the instantaneous potential generated by the nuclear positions
(Rg(t)) and electronic wavefunctions F(r;R(t),t).
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Until this point, the formulation was kept as general as possible and did not depend on a particular choice of the electronic structure method used to solve the time-dependent
Schrçdinger equation for the electrons. In many cases, it is
however convenient to map the set of Equations (14) and (15)
into a DFT/TDDFT-based formulation, which allows efficient calculation of the molecular electronic structure and properties
for systems composed of several hundreds of atoms at a moderate computational cost.
By using the Runge–Gross theorem,[53] we can derive the
equation of motion for the electron density from the variational principle applied to the action functional [Eq. (16)]2 :
in which Exc is the DFT (ground-state) exchange and correlation
energy density functional. This approximation is sufficient for
most applications, but simulations beyond the AA are also
present in the literature.[79–82]
An alternative solution to the propagation of the electronic
wavefunction within DFT/TDDFT is to introduce the representation of the TDKS orbitals in a linear combination of static KS
orbitals, opt
: ðr; RðtÞÞ , obtained from the diagonalization of
the KS Hamiltonian at time t with effective potential us[1] j
1(r) 1(r,t) [Eq. (21)]:
!
in which H[1(r,t)] is the Hartree energy functional. In the DFT
and TDDFT context, r refers to a position in the Euclidean
space (r2R3) and should not be confused with the collective
electron coordinate used for the many-electron wavefunctions.
Applying the variational principle to Equation (17) subject to
P
the constraint 1(r,t) = k jk ðr; tÞj2 results in the time-dependent KS (TDKS) equations [Eq. (18)]:
Inserting this expansion into Equation (18) produces a set of
differential equations for the coefficients ckp(t) that, when
squared, can be interpreted as KS orbital occupations. Although this offers some additional information about the
nature of the propagated wavefunction, the diagonalization of
the KS Hamiltonian requires further computational costs that
can be avoided by using the straightforward propagations in
Equation (18) (see Appendix A in the Supporting Information).
The mapping of the nuclear equation [Eq. (15)] into the DFT
formalism is more straightforward and only requires the description of the forces rghH^el (r,R(t))i as a functional of the
time-dependent density 1(r,t). Replacing the expectation value
of the electronic Hamiltonian with the DFT energy evaluated
with the exchange-correlation potential uxc[1] j 1(r) 1(r,t), the gradient with respect to the nuclear coordinates can be performed analytically as in the case of the adiabatic Born–Oppenheimer and Car–Parrinello[1] MD schemes (BO-MD and CP-MD,
respectively).[2]
!
in which Ĥee is the exact electron-electron interaction and
F[1](t) is the time-dependent wavefunction associated to the
time-dependent density 1(r,t).3 In the KS formulation of DFT,
the action becomes a functional of the one-electron KS orbitals
[Eq. (17)]:
2.1.4. Limitations and Use of Ehrenfest MD
in which [Eq. (19)]:
P Z
in which uext ðr; tÞ ¼ g rRg þ duapp ðr; tÞ is the external poj gj
tential, and uH(r,t) is the Hartree potential. The effective potential us also depends on the initial wavefunction at time t0
(F0(r,t0)) or, equivalently, the corresponding density 1(r,t0). To
simplify the notation, in the following we remove the dependence from the initial value conditions.
In Ehrenfest MD, the nuclei evolve in time according to forces
computed as the gradient of the average energy, hH^el (r,R(t))i,
by using the instantaneous electronic wavefunction F(r;R(t),t).
Both quantum Hamiltonian and electronic wavefunction
depend parametrically on R(t). The mean-field character of this
dynamics is evident if we use the following expansion of the
wavefunction F(r;R(t),t) in the adiabatic (static) base obtained
from the solutions of the electronic time-independent Schrçdinger equation [Eq. (22); {F.(r;R(t))}, see Section 2.2 for more
details]:
The simplest approximation to the time-dependent exchange-correlation action functional Axc [1(r,t)] is the so-called
adiabatic approximation (AA) [Eq. (20)]:
2
This form of the action functional violates causality because a potential of
@v ðr;tÞ
3
xc
the form
has a functional derivative @1ðr
0 ;t 0 Þ, which issymmetric in time and, therefore, does not guarantee that information only
travels forward in time. Different solutions to the causality problem
exists,[77, 78] but they do not affect our derivation.
In the remainder of this section we drop the explicit dependence of wavefunctions and operators on the nuclear position.
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The square of the time-dependent coefficients ci(t) describes
the “occupation probability” of the different states that contribute to the nuclear forces. The validity of the “mean-field”
approximation is restricted to the case in which the classical
trajectories corresponding to the different states do not differ
significantly.[83] In fact, after having left a region of strong nonadiabatic coupling (mixing), the system keeps evolving on the
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average potential without collapsing to one of the adiabatic
states, and for the case in which trajectories on different states
would have different fates in the configuration space, the average Ehrenfest trajectory could lose its physical meaning (even
though it could still provide acceptable expectation values for
quantum observables).
For this reason, the use of Ehrenfest dynamics in molecular
computational physics and chemistry should be restricted to
cases in which atomic rearrangements along the reaction
channels associated to the different electronic states involved
in the mean-field dynamics are not significantly different. Examples are the calculation of absorption spectra and dielectric
functions of systems in the gas phase and in solution,[84, 85] for
which the electron dynamics of the perturbed electronic structure relaxes on a timescale that is much shorter than the one
of the nuclei (in many cases one can use frozen atomic positions[86]). Other applications include the investigation of ultrafast processes triggered by intense laser fields that produce
ionized states followed by fast electronic rearrangement of the
electronic structure such as photoionization and Coulomb explosion (when only dissociation channels are populated).[87–94]
Another severe deficiency of Ehrenfest MD is the violation of
detailed balance (microscopic reversibility), which is an essential property of the kinetics of physical and chemical processes.
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that is nonlocal in time and that depends on the path(s) of the
classical nuclear system, R(t). This means that the forces on the
nuclei at time t depend on the forward and backward propagation of the wavefunctions from time t’ and t’’, respectively.
However, this can only be determined when the path is already known, which introduces a self-consistency issue in the
calculation of the electronic and nuclear dynamics. Pechukas’
equations must, therefore, be solved iteratively, which limits
the efficiency of all algorithms based on this approach. Concerning the observed nuclear trajectory, R(t), Ehrenfest dynamics generates, for any given initial state, a single classical
mean-field path, whereas Pechukas’ mixed quantum–semiclassical approach gives no guarantees of convergence towards
a global path connecting the initial and final states, but in
many occasions it may produce multiple paths solutions. It
could be, therefore, very hard to find a rigorous derivation of
Ehrenfest dynamics from semiclassical arguments even though
attempts in this direction were already proposed.[97]
2.1.6. Application: Coulomb Explosion in Condensed Matter
The main approximations in Ehrenfest MD are the use of the
single-configuration Ansatz for the electronic wavefunction in
Equation (3) and the classical limit applied to the nuclear degrees of freedom. To judge the quality of these approximations, we can compare the results obtained for Ehrenfest MD
with the ones derived from the stationary phase approximation of the path-integral formulation of quantum dynamics.[95, 96] This approach due to Pechukas is derived from the
saddle-point approximation of the Feynman–Kac propagator
and, therefore, does not rely on any a priori Ansatz for the
nature of the many-electron wavefunction. As this semiclassical
method produces a single (or few) nuclear trajectory(ies), it
can be interesting to compare Pechukas’ equations of motion
with the one derived from the Ehrenfest approximation.
Following Pechukas’ derivation, the nuclear trajectories are
the solution to Equation (23):
“Hadron therapy” characterizes a type of radiotherapy in which
highly charged ion beams are used to target deep-seated
tumors.[98] This cancer therapy has shown promising results,
despite the fact that its microscopic mechanisms are far from
being understood and only few molecular theoretical studies
are available.[99, 100] Studying the interaction of a highly charged
and ultrafast cation (e.g. proton, C4 + or C6 + ) with biological
media such as DNA base pairs or simply water molecules constitute an important step towards a better rationalization of
the effects of hadron therapy. This interaction is expected to
lead to Coulomb explosion of the targeted molecules due to
extraction of one or more core electrons by the charged projectile. Ehrenfest dynamics constitutes, therefore, a method of
choice to simulate the electronic reorganization after ionization
and the subsequent nuclear dynamics. We highlight here
recent work[92] in which the Coulomb explosion of uracil surrounded by QM water molecules is simulated. The nonadiabatic dynamics is initiated by removing two electrons from a molecular orbital localized on the C=O group of uracyl (Figure 2,
left). Ehrenfest dynamics is then employed to observe the Coulomb explosion of solvated uracil2 + , which takes place after
30 fs with the ejection of the carbonyl oxygen atom (Figure 2,
^
in which U(t’’,t)
is the time evolution operator. In this case, the
initial and final electronic states Fj(r,R,t’) and Fi(r,R,t’’) as well
as the initial and final time of the propagation, t’ and t’’, respectively, need to be known before starting the propagation.
At difference to Ehrenfest MD, in Pechukas’ formulation the
electronic quantum system exerts a potential on the nuclei
Figure 2. TDDFT-based Ehrenfest dynamics of the Coulomb explosion of solvated uracil2 + . Left) Molecular orbital from which two electrons are removed.
Right) Snapshot obtained 30 fs after the ionization.
2.1.5. Discussion of Ehrenfest MD
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right). This atomic oxygen is then solvated by the water molecules surrounding the uracil. Interestingly, uracil rearrangement
after the oxygen loss produces a H3O + molecule in the first
solvation shell. Further results of the Coulomb explosion of
uracil simulated with Ehrenfest dynamics have been successfully compared to experimental data.[93]
2.2.1. The Born–Oppenheimer Approximation: The Adiabatic
Case
2.2. BO-MD and Its Nonadiabatic Extensions
which only induce a shift of the electronic PESs Ejel ðRÞ felt by
the nuclear wavefunctions [the second term of Eq. (30) is zero
for j = i, when the F·(r;R) are real].
The BO-MD equations can be derived starting from the Born–
Huang representation of the molecular wavefunction
[Eq. (24)]:[50, 51]
In this equation, {F.(r;R)} describes a complete set of orthonormal electronic wavefunctions, solutions of the time-independent Schrçdinger equation [Eq. (25)]:
with hFj j Fii = dij and for which “;R” denotes the parametric
dependence of the electronic Schrçdinger equation from the
position of the atoms. Note that only the nuclear wavefunction
depend explicitly on time, whereas H^el (r,R) and Fi(r;R) only
depend on t through the implicit time-dependence of R(t).
Inserting Equation (24) into the time-dependent Schrçdinger
equation [Eq. (1)], we obtain (after multiplying by Fj* ðr; RÞ
from the left-hand-side and integrating over dr) Equation (26):
In the Born–Oppenheimer approximation, only the diagonal
terms, F jj , are retained [Eq. (28)]:
In this approximation, the nuclei move in the potential of
a single electronic state, the PES Ejel ðRÞ, and the electronic
[Eq. (25)] and nuclear [Eq. (26)] Schrçdinger equations become
completely decoupled. The term F jj (R) is called Born–Oppenheimer diagonal correction[101, 102] and, depending on the nuclear mass, induces an isotope dependence[103–105] of the total
energy, Ejel ðRÞ + F jj (R). However, this term is usually small and
is neglected in the so-called Born–Oppenheimer adiabatic approximation.4
As in the derivation of the Ehrenfest equations of motion,
we introduce the polar representation of the nuclear wavefunction Wj(R,t) [Eq. (29)]:
Even though the same protocol as that in the Section 2.1.1
can be used to extract Born–Oppenheimer equations of
motion, it is interesting here to follow another route. Inserting
the polar representation into Equation (26) [with all F ji (R) = 0]
and separating the real and the imaginary parts, we obtain
Equations (30) and (31):
The quantities F ji (R) [Eq. (27)]:
are the NACs, with a contribution from the nuclear kinetic
energy operator and a second from the momentum operator.
In the most general case, the nondiagonal elements of F ji (R)
are non-zero and induce a coupling between different electronic states due to the motion of the nuclei.
In fact, the last term in Equation (26) brings (or gives) amplitude (F ji Wi(R,t)) from the “electronic state” with energy Eiel ðRÞ,
which is the solution of the electronic time-independent
Schrçdinger equation [Eq. (25)], to the actual state j, with
energy Ejel ðRÞ. This interpretation of the nuclear wavefunction
dynamics Equation (26) is at the basis of the surface hopping
description of nonadiabatic dynamics.
Taking the classical limit h !05 in Equation (30), we obtain
a HJ equation for what can now be interpreted as the action
function6 Sj(R,t) [Eq. (32)]:
4
5
6
2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
Note that Born–Oppenheimer approximation is often used for what is here
called Born–Oppenheimer adiabatic approximation (see also
refs. [44, 106, 107]).
The classical limit proposed in this derivation is sometimes called the “canonical condition” for enforcing classical behavior. It is mainly a mathematical
procedure with limited physical meaning. Alternative formulations with their
physical implications can be found in different references.[70, 108, 109] In Bohmian
for example,
the classical limit is more properly defined by
hdynamics,
i
rg2 Aj
h2 P
! 0.[108]
2 g M1
g
Aj
As already stated in Section 2.1.1, in classical mechanics the action function
S(R,t) also depends on the initial coordinates andR time and, therefore, should
_ 0 ÞÞdt 0 , along
be written as SR0,t0(R,t). It is defined as SR0,t0(R,t) = R~ LðRðt0 Þ; Rðt
the extremal path R̃(t) connecting the points R0 at time t0 to R at time t, in
which L(R(t’), Ṙ(t’)) is the Lagrangian. In this definition, SR0,t0(R,t) also corresponds to the Hamilton’s principal function.[110]
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2.2.2. Trajectory-Based Nonadiabatic Dynamics
which correspond to a classical point-particle time evolution of
the nuclei, and from Equation (31) a continuity equation7 for
the propagation of the amplitude on the adiabatic state of interest, d/dt(sdRjWj(R,t) j 2) = 0. Comparing this result with the
one obtained for the Ehrenfest dynamics [Eq. (9)], we observe
that the potential acting on the nuclei is derived, in this case,
from a static expectation value of the electronic Hamiltonian
computed for the time-independent state Fj(r;R) solution of
the electronic Schrçdinger equation Equation (25).
Using the relation, rg Sj RðtÞ ¼ Pgj ðtÞ, we obtain a Newton-like
equation of motion for the “classical” nuclei evolving in electronic state j (see footnote 1) [Eq. (33]:
The Born–Oppenheimer approximation breaks down when
electronic states approach in energy, which especially occurs
when the dynamics is initiated in one of the electronically excited states of the system. This is the usual situation encountered in a pump-probe experiment, in which an initial pulse
excites the system while a second one monitors its time-dependent relaxation towards the ground state (or a stable excited-state configuration).
The starting point is the time-dependent Schrçdinger equation for the molecular system given by Equation (26) that we
rewrite as Equation (36):
In summary, the BO-MD equations can be described by the
following system of coupled equations [Eqs. (34) and (35)]:
in which [Eq. (37)]:
are the first-order NACs (or NAC vectors), and [Eq. (38)]:
in which only the second one describes an explicit time evolution. The electronic energies and the forces acting on the
nuclei are computed statically by solving Equation (34) “onthe-fly” at each new position sampled along the trajectory R(t).
Contrary to what is obtained for Ehrenfest dynamics, in BO-MD
there is no explicit time dependence of the electronic degrees
of freedom. It is important to further stress that, due to the assumption that F ji = 0, BO-MD always evolves on a single electronic PES, even in the case in which the system approaches
regions of strong coupling between electronic and nuclear degrees of freedom. In practice, the state of interest is the
ground state for which the adiabatic separation from all other
states (excited states) holds in most (nonmetallic) cases.
The combination of BO-MD with DFT for the “on-the-fly” calculation of the electronic structure properties (energies and
forces) at each MD step is straightforward and can be found in
many textbooks (e.g. ref. [2]). By using the Hohenberg–Kohn
theorem, one first maps the electronic structure problem from
the wavefunction space into the density space and then,
within the KS formulation of DFT, the electronic ground-state
energy functional, E0el ½1ðr; RÞ, and its gradients are computed.
7
are the second-order coupling elements.
2.2.3. The Nonadiabatic Bohmian Dynamics (NABDY)
Equations
Using the polar representation for the nuclear wavefunction
Equation (29) in Equation (36) we obtain, after separating real
and imaginary parts, Equation (39):
and Equation (40):
The continuity equation can be easily obtained by multiplying both sides of
r S
Equation (31) by 2 Aj and by using the relation Mgg j ¼ ugj . We therefore get the
equation
in which 1j ¼ A2j and jgj is the nuclear probability flux for g.
2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
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in which both Sj(R,t) and Aj(R,t) are real fields and fij(R,t) =
1
h(Si(R,t)Sj(R,t)). Equations (39) and (40) correspond to the
exact Schrçdinger equation for a nuclear wavefunction evolving in the potential of the different electronic surfaces determined by the time-independent Schrçdinger equation. In comparison with the single surface case or with the diabatic formulation, time evolution of phases and amplitudes involves firstand second-order coupling elements that mix contributions
from other PESs. In particular, transfer of amplitude from one
electronic state to another becomes possible thanks to the
coupling terms, which make, however, the solution of the set
of Equations (39) and (40) more involved. In the dynamics that
emerges, the first equation is the equivalent of the classical HJ
equation (first two terms) for the action S(R,t), augmented with
two additional parts of quantum nature of order h and h2. The
third term is the quantum potential Q(R,t), which describes all
quantum effects in a single electronic state and introduces
nonlocality.[29]8 Finally, the fourth to sixth terms constitute the
nonadiabatic quantum potential iD(R,t), which describes interstate contributions. After applying the gradient with respect
to the nucleus b on both sides of Equation (39) and moving to
the Lagrangian frame,[33] we obtain a Newton-like equation of
motion (using the HJ definition of the momenta
rb Sj ðR; tÞRðtÞ ¼ Pbj ðtÞ [Eq. (41)]:
which describes the time evolution (trajectory) of the Rb components of a fluid element with collective variable R(t) (d/dtj =
@/@t + grgSj(R,t)/Mg·rg).
Even though complicated, the NABDY equations [Eqs. (39),
(40), and (41)] can be implemented in any DFT/LR-TDDFT code,
which allows the calculations of energies (for ground and excited states), nuclear forces, and couplings as a functional of
the electronic density for any given molecular geometry R(t)9.
The most severe limitation in the implementation of NABDY
is related to calculation of the derivatives in the configuration
space such as the one of the amplitudes, rgAj(R(t),t), and of
the phases rgSj(R(t),t). In fact, as the size of the system increases, it becomes very impractical to perform such derivatives on
a real grid of dimensions 3Nn6. Possibilities to move “off
grid” and compute the derivatives by using techniques such as
Voronoi space partitioning,[111, 112] dimensional reduction,[113] or
high-dimensional interpolations are under investigation.
8
9
The quantum potential is responsible for numerical instabilities associated
with the formation of nodes (when the nuclear amplitude tends to zero).[29]
Note that for any practical application of NABDY (or TSH, see below), the
P
summation over electronic states 1
i is truncated, and therefore, only
a finite number of electronic states are included, that is, coupled. This constitutes an approximation, called the “group Born–Oppenheimer approximation”.[44, 107]
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2.2.4. The TSH Dynamics
TSH can be seen as an approximation to the “in principle”
exact NABDY formulation of the nuclear equations of motion.
Starting from Equations (39) and (40) and keeping only firstorder terms in O(h) for the quantities Sj(R,t) and Aj(R,t) we get
Equation (42):
and Equation (43):
At this point, we look for the trajectories in the (R,t) space,
which are the extrema of the action function, dS(R,P,t) = 0, as
we did in the NABDY case. However, we now consider a “classical” type of dynamics in which the different trajectories corresponding to different initial conditions are assumed to be independent from each other. In this independent trajectory approximation (ITA) the nuclear wavepacket density is given by
the fraction of the total number of classical trajectories, Nt, that
are found in the volume dV d3NnR [Eq. (44)]:
in which N(R(t),dV,t) is the number of trajectories in the volume
element dV. In TSH, the equation of motion for the nuclear trajectories is given by the terms in Equation (42), which are independent from the NAC vectors, dgji ðRðtÞÞ. This dynamics (formulated in the HJ framework) for what we call the classical acCL
tions SCL
j ðRðtÞ; tÞ and amplitudes Aj ðRðtÞ; tÞ is exactly equivalent to Newton’s equation (in the Lagrangian frame) [Eq. (45)]:
in which Pj ðtÞ ¼ rSCL
j ðR; tÞ RðtÞ . As far as the NACs are negligible, the trajectories evolve, therefore (independently), along
the adiabatic PES, Ejel ðRðtÞÞ. However, when a region of strong
coupling is encountered, the nuclear trajectories experience an
additional force of quantum origin given by the gradient of
the last term in Equation (42). At the same time, quantum amplitude is transferred from one electronic state to another according to the third term in Equation (43). The transfer of classical probability density (ACL(R(t),t))2 between electronic states
is accomplished through a trajectory “hop” from one PES to
the other. This occurs through the switching algorithm introduced by Tully[26] (see below). Both nonadiabatic forces and
amplitude transfer have no classical counterpart and, therefore,
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must be treated separately from the classical evolution of the
trajectories.
On the basis of Equations (42) and (43), we introduce a quanQM
tum phase, SQM
j ðRðtÞ; tÞand a quantum amplitude, Aj ðRðtÞ; tÞ
governed by the following evolution equations [Eqs. (46) and
(47)]10 :
1
QM
QM
with QM
ij ðRðtÞ; tÞ ¼ h Si ðRðtÞ; tÞ Sj ðRðtÞ; tÞ . These two
equations represent the polar form of Tully’s TSH equations for
the state amplitudes [Eq. (48)]:
h
i
i QM
with Cj ðRðtÞ; tÞ ¼ AQM
j ðRðtÞ; tÞ exp h Sj ðRðtÞ; tÞ . Note that in
Equations (46), (47), and (48), the actions SQM
j ðRðtÞ; tÞ, ampliðRðtÞ;
tÞ,
and
coefficients
C
(R(t),t)
are, strictly, not
tudes AQM
j
j
fields of the extended configuration-time space, but they are
all restricted to the trajectory path, and therefore, they can be
QM½a
simply labeled with a trajectory index, ([a]), to give Sj
ðtÞ,
QM½a
½a
Aj
ðtÞ, and Cj ðtÞ, respectively. The dependence on R is recovered from the distribution of the trajectories
in the configu
2
ration-time space as done for ACL
ðRðtÞ;
tÞ
in
Equation
(44).
j
In TSH dynamics, Equations (45) and (48) are solved simultaneously with the assumption that after sampling a sufficient
number of trajectories [Eq. (49)]:
for each state j, which is the internal consistency criterion described in ref. [115]. This is achieved (approximately) by means
of the fewest-switches TSH algorithm designed by Tully,[26]11
which gives the probability [Eq. (50)]:
for a trajectory to “hop” from one reference electronic state, j,
to another state, i, in the time interval Dt measured at time t.
Transitions are decided by a Monte–Carlo-type procedure, in
which a random number z2[0,1] is generated and a hop from
state j to state i (j ¼
6 i) is made if, and only if [Eq. (51)]:
10
11
A more detailed attempt to link the NABDY equations of motion with the
TSH equations will be described in ref. [114].
In the following, we will mostly focus on the fewest-switches version of TSH.
For a summary of other TSH methodologies, refer to refs. [116, 117].
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In summary, within the quantum hydrodynamic formulation
of the quantum nuclear dynamics we obtain an extended picture of the TSH dynamics. Starting from the exact NABDY
equations, we reached TSH equations by using two main assumptions: 1) That in each electronic state j, the nuclear dynamics can be split into a classical and a quantum component
governed, respectively, by the action SCL
j ðRðtÞ; tÞ and the phase
QM
SQM
j ðRðtÞ; tÞ with the amplitude Aj ðRðtÞ; tÞ. 2) That through
½a
the hopping probability pi j [Eq. (50)], the square of the classiQM
cal and quantum amplitudes, ACL
j ðRðtÞ; tÞ and Aj ðRðtÞ; tÞ, coincide.
2.2.5. The TSH Algorithm: Advantages and Pitfalls
In TSH dynamics, a series of independent trajectories are computed starting from a previously equilibrated population distribution of initial configurations sampled at a given temperature, T, in a chosen electronic state, j (better would be the sampling of the corresponding Wigner distribution, which, in general, is however more difficult to compute; see refs. [65, 118]
for a discussion on initial conditions). All trajectories are classical in the sense that only classical forces need to be computed
from the gradient of the selected PES. For a given independent
trajectory [a] of the swarm, the quantum amplitudes are also
evolved in time by using Equation (48) together with the nuclear coordinates propagated with Equation (45). In a region of
coupling between two PESs j and i, this classical trajectory can
eventually hop from one surface to another according to the
½a
probability pi j. After a surface hop, the excess (deficient)
energy due to the transition is redistributed into (extracted
from) the motion along the direction of the NAC vectors, as
suggested by the extra force obtained from the gradient of
the third term on the left-hand side of Equation (42) and by
Pechukas’ semiclassical interpretation of the nonadiabatic transitions.[36, 96] In this way, energy conservation is guaranteed
along the entire trajectory. Frustrated hops occur when the
quantum particles have insufficient kinetic energy to compensate the potential energy loss in an upward transition.[119, 120]
This method has the advantage to be simple enough to be
implemented into any existing wavefunction-based or DFT/LRTDDFT code that offers the possibility to compute efficiently
the electronic energies (PESs), the classical forces, and the
NACs. These quantities can be either precomputed or evaluated “on-the-fly” along the growing trajectory. However, the first
approach is only suited in the case of relatively small systems
made up of only a few atoms for which the multidimensional
3Nn6 potential energy hypersurfaces (or a restricted portion
of them) can be computed for all relevant states. The “on-thefly” approach reduces dramatically the number of electronic
structure calculations to the number of integration steps of
the classical trajectory, and therefore, it can be used for the
simulation of the nonadiabatic dynamics of large systems
(made of up to hundreds of atoms[121]).
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Due to the classical nature of
the dynamics along the PESs,
TSH cannot describe nuclear
tunneling and nuclear dephasing
processes.[117, 122] The only nuclear
quantum effects reproducible
with TSH dynamics are those related to wavepacket splittings
induced by avoided crossings
and conical intersections (regions of strong nonadiabatic
couplings). Due to the ITA and
the local nature of the transitions (in space and time), quantum decoherence effects between states are difficult to capture in standard TSH dynamics.
However, variations of TSH algorithm to circumvent these limitations are available.[115, 120, 123–127]
Figure 3. Nonadiabatic dynamics of NaI. Top) Electronic energy as a function of the interatomic distance. In the
Compared to Ehrenfest MD, inset we report the full potential energy profiles (c) together with the nonadiabatic couplings (a). The nonthe trajectories in TSH evolve on adiabatic dynamics starts in the first excited state of NaI with a kinetic energy component pointing to the right. In
adiabatic PESs (except for the in- TSH, all trajectories lie on an adiabatic PES, and therefore, they are only distinguishable at the instant in time at
stantaneous transitions), which which a surface hop occurs (vertical lines). In contrast, Ehrenfest trajectory evolves on a mean-field PES that is a superposition of the two adiabatic states (b). Bottom) The two lower panels show the TSH trajectories distribusimplifies the physical interpreta- tion (histograms) and NABDY nuclear probability density (*) at four different times along the dynamics (State 1 is
tion of the results and compari- the ground state and State 2 is the first excited state).
son with experiments. In addition, TSH MD in its “fewestcontrast, in TSH the independent trajectories evolve on adiaswitches” implementation[26] obeys detailed balance approxibatic PESs (the two potential energy curves of Figure 3), and
mately, with deviations that tend to vanish in the limits of
population transfer among the states occurs through surface
small adiabatic splitting and the limit of large NACs.[128] The
hops described by Equation (50). By computing the population
use of the ITA together with the representation of the nuclear
densities at each instant of time according to Equation (44) we
wavepacket amplitude by the density of trajectories offers
can reconstruct approximate nuclear wavepackets on the two
a simple (even though approximate) solution to the problem
PESs (Figure 3, lower panels), which compare well with the
of the high dimensional derivatives that we discussed in the
“exact” quantum wavepackets obtained from the propagation
case of the implementation of the NABDY equations. More
of the Bohmian trajectories within the NABDY scheme. For
specific information can be found about TSH in the following
TSH, the time evolution of the energy and distance spreads of
references.[5, 65, 97, 122, 125, 129–133]
the wavepacket is shown in Figure 4, where the density of
curves reproduces the squared amplitude of the wavepacket
2.3. Comparison between the Nonadiabatic Schemes
(transitions between the two states are represented by vertical
lines). In the lower panel, we can see how the single Ehrenfest
As an illustrative example to show the main differences among
path follows the evolution of the “center” of the TSH trajectory
the nonadiabatic dynamics schemes proposed so far (Ehrenfest
bundle. However, when the kinetic energy is small enough to
dynamics, TSH, and NABDY), we discuss the case of photoexcitprevent the full depletion of the excited state, a wavepacket
ed sodium iodide (NaI). The PESs of this diatomic molecule exsplitting is observed (inset), which can only be reproduced by
hibit a characteristic avoided crossing between the ground
using a multitrajectory-based approach such as TSH and
state and the first excited state (Figure 3, inset) that, at the
NABDY.
equilibrium distance, have ionic and covalent character, respectively. When a nuclear wavepacket is prepared and launched in
the first electronic excited state, part of its amplitude is transferred back to the ground state. The resulting dynamics de2.4. DFT/TDDFT-Based TSH Approaches and Their
pends largely on the mixed quantum/classical method used
Applications
(see Figure 3). In Ehrenfest dynamics, the system evolves on
Several implementations of “on-the-fly” DFT/TDDFT-based TSH
a mean-field PES (gray dashed line in Figure 3) so that EquaMD are nowadays available in different software packages
tion (22) is a weighted average of the ground-state and first
such as CPMD,[134] Turbomole[135] (from version 6.4), and
excited-state PESs. The degree of mixing depends on the size
of the NACs in the region surrounding the avoided crossing. In
NEWTON-X[136] (which proposes an interface with several elec 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
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Figure 4. Nonadiabatic dynamics of NaI—Electronic energy and interatomic distances as a function of simulation
time for the same dynamics reported in Figure 3. Top) Time evolution of the electronic energies for a swarm of
TSH trajectories (black lines with different styles) and for the single Ehrenfest trajectory (b). Bottom) Time evolution of the intramolecular distances (same color code as in the top panel). Inset: Same plot but for a smaller initial
momentum. In this case, splitting of the wavepacket between the two states is observed. Note that although TSH
can describe the splitting of the wavepacket, the Ehrenfest dynamics proceeds along a single mean-field trajectory.
transition probabilities according
to fewest-switches criteria,[26] an
additional orthonormalization of
the electronic wavefunctions is
required.[12]
The ROKS surface hopping
methodology[12, 137] can be use to
study nonradiative decays between the first singlet excited
state (S1) and the ground state
of molecular systems (see
refs. [138–146] for additional examples). In recent work,[147] the
use of bridged azobenzene as
a potential photoswitch was
studied by ROKS nonadiabatic
dynamics. Azobenzene is a wellknown molecular photoswitch
that is used in many biological
and material applications. Its
photoexcitation leads to Z!E
isomerization, which consequently stretches the molecule.
It has been found that bridged
azobenzene (Figure 5) exhibits
tronic structure codes). They mainly differ in the way in which
the electronic structure calculations are performed.
2.4.1. TSH/Restricted Open-Shell Kohn–Sham (ROKS)
Among the DFT-based TSH MD implementations, the method
by Doltsinis and Marx[12] is based on the restricted open-shell
formulation of the first singlet DFT excited state (i.e. ROKS) and
is available in the software package CPMD.[134]
Briefly, in TSH/ROKS dynamics[12] the total electronic wavefunction
of
the
system
is
represented
by
iR
P
Y ¼ j aj Fj exp h Ej dt in which aj are time-dependent expansion coefficients and Fj are KS-based many-electron wavefunctions with corresponding eigenvalues Ej. By using the
closed-shell KS ground-state Slater determinant for F0 and the
ROKS S1 excited-state wavefunction F1, the following set of
coupled equations [Eqs. (52) and (53)] of motion for the expansion coefficients is obtained:
R Ej in which pj ¼
iE h dt , Sij = hFi j Fji with S01 = S10 = S and
D exp
@
Sii = 1, Dij ¼ Fi @t Fj , Hii = Ei, and H01 = H10 = E0S. This set of
equations can be easily integrated together with the equation
of motion for the nuclei by using, for instance, a fourth-order
Runge–Kutta scheme. For the calculation of the nonadiabatic
2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
Figure 5. Average value of the CNNC dihedral angle in the first excited state
of azobenzene: bridged (E)-azobenzene (*), (E)-azobenzene (*), (Z)-azobenzene (~) for which the 1808 CNNC dihedral angle is plotted. Inset) Optimized
structures of azobenzene (top) and bridged azobenzene (bottom) for the Z
(left) and E (right) isomers. Figure reprinted with permission from M. Bçckmann et al., Angew. Chem. 2010, 122, 3454; Angew. Chem. Int. Ed. 2010, 49,
3382. Copyright 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.
a much larger quantum yield for photoisomerization than the
standard, unbridged (E)-azobenzene. This rather counterintuitive fact has been studied theoretically, and the enhanced isomerization quantum yield ( 50 %) of the bridged azobenzene
has been precisely reproduced. In fact, this study revealed that
the mechanism of photoisomerization of bridged (E)-azobenzene is similar to that of unconstrained (Z)-azobenzene, which
involves specific prearrangement of the phenyl rings.
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2.4.2. TSH/TDKS
More recently, Prezhdo et al. developed a TSH/MD scheme
based on the time-dependent propagation of the KS orbitals,
which are used to approximate the many-electron Slater-type
wavefunctions for the calculation of the NACs.[13, 148] In short,
the time-dependent KS orbitals fk(r,t) are expanded in the
basis of the instantaneous adiabatic KS orbitals opt
: ðr; RðtÞÞ
[see Eq. (18)], which leads [when included into TDKS Equation (21)] to the following set of differential equations [Eq. (54)]
for the expansion coefficients:
D
E
@ opt
_ KS
in which dKS
. Within this
h opt
pm ðRðtÞÞ R ðtÞ ¼ i
p @t m
framework, the many-electron adiabatic states are generated
ad hoc by using excited Slater determinants of KS orbitals,
whereas the NAC terms (and the corresponding surface-hopping transition probabilities) are computed from the corresponding matrix elements involving electron-hole (virtual/occupied) KS pairs.
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same trend as that of the population decay. Interestingly, holes
with higher energy (closer to the Fermi level) show a slower relaxation decay (pluses in Figure 6 b).
2.4.3. TSH/LR-TDDFT
In 2007, Tavernelli and co-workers[14] derived a LR-TDDFT-based
TSH MD scheme in which all quantities required for the propagation of the trajectories [Eq. (45)] and amplitudes [Eq. (48)]
are rigorously derived from TDDFT in the LR formulation (see
Table 2) and expressed as a functional of the electronic density.
The details of this derivation are given in Section 3 after a brief
introduction to LR-TDDFT.
TSH/LR-TDDFT has been applied to a multitude of different
photophysical/photochemical processes of organic molecules
(see refs. [14, 159–164] for a nonexhaustive list). The nonradiative relaxation of the photoexcited oxirane molecule is taken
here as a prototypical example of TSH/LR-TDDFT dynamics.[20]
Even though it is well known that photoexcitation of oxirane
leads to its ring opening,[165] the mechanistic details of this
photochemical reaction remained elusive for a long time. In
ref. [14], we showed that the first electronic excited state S1, initially with the character 1(n,3s), is nonreactive, that is, it behaves as an energy reservoir during the first part of the dynamics ( 280 fs in a typical trajectory, Figure 7). Nevertheless,
dramatic changes in the TSH dynamics are observed whenever
This method is simple and efficient, but the description of
excited states by means of excited KS Slater determinants is
not rigorous, and therefore, the approximation done in the
representation of the PESs and
their couplings is not controllable and cannot be improved systematically. The adiabatic KS
states are indeed seen as
a zeroth-order approximation to
the
LR-TDDFT
adiabatic
states.[148]
We highlight here an application of TSH/TDKS in the study of
electron and hole relaxation dynamics in photoexcited (7,0)
zigzag carbon nanotubes[149] (for
further
examples,
see
refs. [17, 150–152]). Interestingly,
it was found that the electron
relaxation follows a simple exponential, whereas the hole relaxes
through more complicated pathways, which leads to multiple
timescales (Figure 6). The nonadiabatic dynamics indicates, in
addition, that the main relaxation pathways involve high-frequency longitudinal optical phonons. Holes seem to couple in
addition to lower frequency
radial breathing modes. FigFigure 6. Left) Electron and hole relaxation dynamics in (7,0) zigzag carbon nanotubes: Right) Population decay
ure 6 b shows the electron and
(a) and the average energies of the electrons and holes along the dynamics (b). Figure taken from ref. [149].
hole energy decay after photo- Figure reprinted with permission from B. F. Habenicht, C. F. Craig, O. V. Prezhdo, Phys. Rev. Lett. 2006, 96, 187401.
excitation, and it exhibits the Copyright (2006) by the American Physical Society.
2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
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Table 2. Summary of the main electronic properties required in TSH/LRTDDFT[14] and references to their derivation and validation (the labels
i and j refer to the electronic states).
Electronic property
Excited-state PESs
Excited-state forces on the nuclei
NAC vectors[a]
NAC vectors[b]
Transition dipole moments[c]
LR-TDDFT response density[d]
Ref.
el
j
E
Fj
d0i
dji
mji
d1j(r)
[54, 153]
[55, 154, 155]
[14, 156, 157]
[59]
[158]
[54, 155]
carbonyl ligand upon photoexcitation.[170] The so-formed free
coordination site can then be filled by solvent molecules in different ways.[171] Crespo-Otero et al. performed a theoretical
study of the release of one carbonyl ligand from a chromium
hexacarbonyl complex.[169] They used a swarm of 30 TSH trajectories starting from a MLCT (metal-to-ligand charge transfer)
excited electronic state (either S1, S2, or S3) to investigate the
possible reaction channels. Three main time windows dominate the relaxation process towards the ground state S0
(Figure 8). During the first 20 fs, the dynamics is confined in
[a] Ground to excited state, based on the auxiliary many-electron wavefunction. [b] Excited to excited state, based on the auxiliary many-electron wavefunction. [c] The same for matrix elements of any one-body operator, based on the auxiliary many-electron wavefunction (see Section 3.3). [d] For QM/MM applications, for example.
Figure 8. Potential energy lines as a function of time. Left) Typical TSH/LRTDDFT trajectory (top) and representative snapshots of the relevant structures (bottom). The dots indicate the state that drives the dynamics of the
nuclei. Right) Schematic illustration of the energy profiles. Figure reprinted
with permission from R. Crespo-Otero et al., J. Chem. Phys. 2011, 134,
164305. Copyright 2011, American Institute of Physics.
Figure 7. 1D cut of the PESs along the reaction path computed with TSH/
LR-TDDFT: Energies computed along the most probable, reactive path (c)
and low probability path that would continue on the nonreactive surface
(a); S0 (c), S1 (c), S2 (c), S3 (c); running states are indicated by
*). The geometries of the TSH trajectory are shown at time a) 243, b) 284,
and c) 300 fs. Figure reprinted with permission from E. Tapavicza et al., J.
Chem. Phys. 2008, 129, 124108. Copyright 2008, American Institute of Physics.
a mixing between S1 [1(n,3s)] and S2 [1(n,3pz)] occurs (t > 280 fs
in Figure 7; the probability to continue the dynamics on the
nonreactive 1(n,3s) state through a surface hop is small). Population of the 1(n,3pz) state leads to the dissociation of a CO
bond and, therefore, to ring opening. Interestingly, this constitutes the only photochemical rearrangement observed in the
excited state, and any further dissociation into two radical moieties is expected to occur only once the system has relaxed
into the ground state.
Applications of TSH/LR-TDDFT to metal-based molecules are
numerous (e.g. refs. [166–168]). In this paragraph, we present
the nonadiabatic relaxation of chromium hexacarbonyl
Cr(CO)6.[169] Metal hexacarbonyl (e.g. based on molybdenum,
chromium, or tungsten) are known to efficiently release one
2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
the manifold of MLCT states of the photoexcited Cr(CO)6 complex. Later, a splitting of the excited-state bands into two subbands is observed (region II in Figure 8), which is associated
with the dissociation of a carbonyl ligand. This process leads
to the formation of an excited Cr(CO)5 moiety and a free CO
molecule within approximately 40 fs of dynamics. Finally,
Cr(CO)5 reaches the ground state after an average time of
about 100 fs. The time constants extracted for the photodissociation dynamics agree nicely with experimental data obtained
by ultrafast optical spectroscopy.
3. TDDFT Quantities for Nonadiabatic
Dynamics
The nonadiabatic schemes presented in the previous section
depend, in their implementation, on a number of electronic
properties such as electronic energies (i.e. PESs), forces, NACs,
and others (for a complete summary see Table 2). The efficient
and reliable calculation of these terms is key to the development of “on-the-fly” nonadiabatic molecular dynamics
schemes. Matrix elements between electronic states are
straightforwardly obtained by means of wavefunction-based
methods such as, for example, configuration interaction singles
(CIS),[172] multireference configuration interaction (MR-CI),[173, 174]
and equation-of-motion excitation energy coupled-cluster singles and doubles (EOMEE-CCSD).[175] However, the evaluation
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of any matrix element within LR-TDDFT requires reformulation
of these quantities as a functional of the electronic density or,
equivalently, of the KS orbitals. This is necessary as DFT and
TDDFT are wavefunction-free theories that are based exclusively on the electronic density and its representations.
In this section, we will briefly present the main LR-TDDFT
equations and introduce the concept of auxiliary many-electron wavefunctions, which is used to describe all matrix elements required in the nonadiabatic dynamics. For a short summary of the main LR-TDDFT equations in the so-called Casida
formalism,[54, 176, 177] see Appendix B in the Supporting Information.
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one obtains, after matrix inversion, the LR-TDDFT equations in
matrix form (see Appendix B in the Supporting Information)
[Eq. (61)]:
P
in which d1ðr; wÞ ¼ ijs is ðrÞjs* ðrÞdPijs ðwÞ and e. are the KS
orbital energies. Note that we use the same symbol for a function and its Fourier transform. To which one we are referring
to is clear from the variable specified (t or w).
3.1.1. The LR Equation for TDDFT
3.1. The Casida Equations
The excitation energies of a molecular system (in the gas
phase as well as in the condensed phase) are given by the
poles of the Fourier-transform in frequency space of the density–density response function defined as [Eq. (55)]:[178]
in which d1(r,t) is the electron density variation, c(r,t,r’,t’) is the
response function, and duapp(r’,t’) is the applied perturbation.
Because c(r,t,r’,t’) depends only on the time difference tt’,
Equation (55) is usually formulated in the Fourier (energy)
space [Eq. (56)]:
which is the starting point for the derivation of LR-TDDFT
equations.
Key to the development of Casida’s equations for LR-TDDFT
is the formulation of the LR equation [Eq. (56)] in the basis of
KS orbitals {fi(r)} with occupations fi and spin si [Eq. (57)]:
Starting from the LR susceptibility for a system of noninteracting electrons (see Appendix C in the Supporting Information) [Eq. (58)]:
As we just showed in TDDFT, the LR equation [Eq. (61)] is complicated by the fact that the self-consistent potential us[1](r,t)
[see Eq. (19)] depends on the response of the density. Using
Equation (A-15) from Appendix B in the Supporting Information we get [Eq. (62)]:
in which, formally, c1 = ([cs]1K), and:
By ordering the KS orbital basis according to the occupation,
we can split Equation (62) into particle–hole (ph) and hole–particle (hp) sectors and rewrite Equation (62) in the following
form (in the following i,j,k,… label occupied orbitals and
a,b,c,… label virtual orbitals) [Eq. (63)]:
in which [Eqs. (64) and (65)]:
and using the matrix representation of the density–density response for the fully interacting system [Eq. (59)]:
and for the corresponding (noninteracting) KS system (Appendix B in the Supporting Information) [Eq. (60)]:
2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
The matrix on the left-hand side of Equation (63) is the inverse of the susceptibility, and therefore, the excitation energies wn can be obtained from the solution of the generalized
eigenvalue equation [Eq. (66)]:
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for which we now assume that the matrices A and B are frequency independent (AA for the TDDFT kernel, fxc). This equation has paired excitation (wn > 0) and de-excitation (wn < 0).[177]
The usual normalization is [Eq. (67)]:
3.1.2. The Tamm–Dancoff Approximation (TDA)
The TDA[179, 180] consists in setting B = 0 in Equation (66), which
leads to the simplified LR-TDDFT equation [Eq. (68)]:[181, 182]
Physically, this means that in the TDA all contributions to
the excitation energies coming from the de-excitation of the
correlated ground state are neglected. Even though it is an approximation, the TDA can improve the stability of the LRTDDFT calculations with most of the standard (approximated)
functionals. In particular, by decoupling the DFT ground-state
problem from the calculation of the LR-TDDFT excitation energies, TDA can provide better PESs, especially in the regions of
strong coupling with the ground state.[20, 177, 183, 184] This is of crucial importance for all nonadiabatic MD schemes based on LRTDDFT PESs.
3.1.3. The Sternheimer Formulation of LR-TDDFT
The Sternheimer time-dependent perturbative approach[185]
offers an alternative way to calculate the LR-TDDFT energies
and properties. Most importantly, this method does not require
calculation of the unoccupied KS orbitals, as in the case of Casida’s equations, but it is based on the calculation of one response orbital for each occupied KS orbital. This may become
computationally advantageous when looking for high-energy
excited states.
In the Sternheimer formulation of LR-TDDFT,[56, 155] the LR perturbation density is given by [Eq. (69)]:
n
o
in which the LR orbitals fg
satisfy Equation (70):[155]
n;i ðrÞ
P
Q ¼ 1 Ni el ji ihi j, H0 , is the KS Hamiltonian, Eij is a
Lagrangian multiplier ensuring the orthogonality of the
ground-state orbitals, and dun,SCF(r,w) is the change of the
self-consistent field (in the AA) upon perturbation [Eq. (71)]:
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3.1.4. Current Approximations in LR-TDDFT: The Pitfalls
Even though LR-TDDFT (in the AA, see Section 3.1.1) has been
successfully applied to a multitude of molecular systems,[82, 177]12
its rise in the realm of quantum chemistry and quantum physics is still hampered by several failures rooted in the approximations of the xc functional and its functional derivatives. A
complete account of those failures is far beyond the scope of
this article and can be found in specialized books[82, 186] or reviews.[177, 187] Nevertheless, we find important to highlight here
those deficiencies that are particularly deleterious in nonadiabatic dynamics and that add up to the ones already known for
ground-state DFT.
First, LR-TDDFT is not able to properly describe electronic
states with strong (> 50 %, see ref. [82]) double excitation character.[188–192] This problem is related to the use of the AA for
the TDDFT kernel, and unfortunately, at present only few potential solutions to this failure exist.[190, 193]
Second, the inaccurate description of derivative discontinuities, the problem of self-interaction error, and the incorrect
long-range properties of currently used xc potentials are at the
heart of the most important problem faced in the application
of LR-TDDFT in chemistry and physics, namely, the chargetransfer failure.[194–200] LR-TDDFT using standard local density
approximation (LDA) or generalized gradient approximation
(GGA) functionals is indeed inappropriate to describe excitations and charge-transfer excitations between spatially separated molecules. However, the use of long-range corrected functionals[201, 202] strongly improves the situation.
Finally, it is important to stress that LR-TDDFT may have difficulties in describing conical intersections between the ground
state and the first electronic state.[191] However, it has been
shown that, at least in some cases, the use of the TDA improves the description of these critical points.[20, 186]
3.2. The Nuclear Forces from LR-TDDFT
LR-TDDFT could become the method of choice for the calculation of excited-state PESs to be used in different nonadiabatic
MD schemes. In this case, the calculation of nuclear forces
within the LR-TDDFT formalism becomes an essential step.
Among the different approaches developed for the calculation
of analytic derivatives, the Lagrangian method[203] is of particular interest because of its numerical efficiency. However, the
derivation of LR-TDDFT is technically involved, and as it does
not bring any new physical insights, we simply refer the interested reader to the rich literature on this subject.[2, 155, 204–206]
12
Excited-states energies and densities are computed from the
solutions of Equation (70). For more details see Appendix D in
the Supporting Information and refs. [155, 156].
2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
See also the following special issues: “Time-Dependent Density-Functional
Theory”, Phys. Chem. Chem. Phys. 2009, 11, 4421–4688; “Time-Dependent
Density-Functional Theory for Molecules and Molecular Solids”, J. Mol.
Struct.: THEOCHEM 2009, 914, 1–138; “Open Problems and New Solutions
in Time-Dependent Density Functional Theory”, Chem. Phys. 2011, 391, 1–
176.
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3.3. The Quest for Matrix Elements in TDDFT
In Section 2, we showed that matrix elements such as the NAC
vectors [Eq. (37)] or the second-order NACs [Eq. (38)] are key to
the implementation of any nonadiabatic dynamics scheme
[Section 4.2 will also highlight the importance of transition
dipole moments, Eq. (114)]. However, such quantities are formally defined in terms of electronic wavefunctions, and the
main challenge consists, therefore, in the reformulation of
these quantities as a functional of the ground-state electronic
density (or equivalently, of the KS orbitals).
In the following, we will introduce the concept of auxiliary
many-electron wavefunctions. Those are linear combinations
of singly excited KS Slater determinants with the property that,
when used in the calculation of matrix elements of one-body
operators, they provide the formally exact results within LRTDDFT (formally exact in the sense that this approach would
give the exact matrix elements if the exact xc functional and
TDDFT kernel would be known). When used as approximations
to the “real” electronic states, the auxiliary many-electron
wavefunctions perform poorly [even though successfully applied in the context of the assignment of electronic transitions;[54] instructive in this context is the meaning of the
ground-state KS Slater determinant (built from the occupied
KS orbitals), which gives a poor representation of the groundstate wavefunction (less accurate than the corresponding Hartree–Fock Slater determinant)]. It is only when applied in the
context of the calculation of matrix elements between ground
state and excited states (as well as between pairs of excited
states, see Section 3.3.4) that the auxiliary many-electron wavefunctions provide the correct answer within the linear response.
The auxiliary many-electron wavefunctions have been used
to calculate several electronic properties such as, for example,
the NAC vectors (Section 3.5), the second-order NACs,[207] the
adiabatic excited-state nuclear forces,[208] the transition dipole
moments[158, 209] (Section 4.2), and the relativistic spin–orbit
couplings,[210, 211] which, with different flavors, are already
available within the LR-TDDFT formalism (e.g. refs. [212, 213]).
In the following sections we outline the main steps in the
derivation of the auxiliary many-electron wavefunctions. First,
in Section 3.3.1 we drive the sum-over-states (SOS) representation of the density–density response function within LR-TDDFT.
Then, by using the equivalent SOS formula derived in manybody perturbation theory (MBPT, Section 3.3.2) and by equating the LR-TDDFT and MBPT SOSs residue-by-residue, we finally
obtain the desired auxiliary many-electron wavefunctions (Section 3.3.3) that are applied in the calculation of the matrix elements in LR-TDDFT.
3.3.1. The LR-TDDFT SOS Formula
In this section we derive a SOS representation of the density–
density response function within LR-TDDFT. The starting point
is the matrix form of the TDDFT LR equations in frequency
space [Eq. (63)]. For a real perturbation of the type du’(w) =
1/2(du(w) + du*(w)) Equation (63) leads to Equation (72):[54]
2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
in which [Eq. (73)]:
and [Eq (74)]:
We can therefore define the susceptibility c(w) as [Eq. (75)]:
Using the spectral representation of the (W(w)w2I)1, we
can write [Eq. (76)]:
in which Zn are the eigenvectors of W(w) [Eq. (77)]:
and are related to the eigenvectors of Equation (66) according
to [Eq. (78)]:[177]
Note that in case of a frequency-dependent W(w), the eigenvectors Zn require the normalization [Eq. (79)]:[54]
to enforce the residues in the spectral representation [Eq. (76)]
to be equal to one. Finally, we obtain for the susceptibility
c the following SOS representation [Eq. (80)]:
Using Equation (80), we can now compute the response
^
dhOi(w)
of any one-body operator O^ due to the action of a generic time-dependent perturbation û(r,t) = û’(r)E(t). Taking the
usual representation of one-body operators in terms of the KS
creation and annihilation operators [Eqs. (81) and (82)]:
R
R
^
*is ðrÞjs ðrÞ and u0 ijs ¼ dr^
with oijs ¼ drOðrÞ
u0 ðrÞ*is ðrÞjs ðrÞ,
^
^
we first get for dO(w) = dhOi(w)
= sdrO(r)d1(r,w)
[Eq. (83)]:
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in which, according to d1(r,w) = sdru’(r’)E(w)c(r,r’,w) [Eq. (84)]:
Therefore, from Equation (80), the final LR-TDDFT SOS equation reads [Eq. (85)]:
dm
As an example, the polarizability defined as aE1,E2(w) = E2 2ðtÞ in
which E1 and E2 are two Cartesian directions in space measures
the change in dipole dmE1 = E1·r induced by a perturbation of
the form u’(r,w) = E2·rE(w). From Equation (84) we get (for E1 = x̂
and E2 = ẑ) [Eq. (86)]:
1
For finite-sized systems in which we can construct a discrete
set of “quasiparticle” one-electron orbitals to be used in Slatetype many-electron wavefunctions (Hartree–Fock or KS orbitals
in most cases), we can introduce the matrix representation
[Eq. (91)]:
in which [Eq. (92)]14 :
2
in which the factor 2 comes from the fact that (for symmetry
reasons) the summation over ij can be restricted to terms with
(fisfjs) > 0.
P
P
^yis a
^js , u
^yis a
^js . Using the symmetry
^0 ¼ ijs u0 ijs a
and O^ ¼ ijs oijs a
property [Eq. (93)]:
we finally obtain the SOS formula [Eq. (94)]:
3.3.2. The Matrix Elements in LR-TDDFT
The derivation of general formulae for the evaluation of matrix
elements of one-particle operators in LR theory [Eq. (87)]:
is based on a direct comparison with the same quantity derived by using MBPT. In Equation (87), the states j F0i and j Fni
describe the ground-state and nth electronic excited state
wavefunctions, respectively.
Comparing the residues of LR-TDDFT response function
Equation (85) with the residues of the MBPT response function
Equation (94) at equal energy wn, we obtain the following
identity [Eqs. (95) and (96)]15 :
We start, therefore, with a short outline of the main LR equations in MBPT.
From the definition of the retarded density–density response
function [Eq. (88)]13 :
the change of an observable O, under the influence of a perturbation ûext(r’,t’) in the LR regime is given by [Eq. (89)]:
This equation was derived by Casida[54] and then applied by
Tavernelli et al.[14, 156, 157] and Hu et al.[214, 215] for the calculation of
the NAC vectors between the ground state and an excited
state. A similar equation was also given in ref. [216].
3.3.3. The Auxiliary Many-Electron Wavefunction
(here we consider an interaction of the form ûext(r’,t’) =
û’(r’)E(t’)). If c depends only on the difference (tt’), the Fourier
transform in time gives [Eq. (90)]:
13
It may be useful at this point to investigate the possibility to
further simplify the definition and the calculation of matrix elements within LR-TDDFT by means of the definition of a set of
“auxiliary” multideterminantal many-electron wavefunctions
14
For the special case of o(r) = 1, we get the LR density:
15
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With no Hartree–Fock exchange contribution in the functional, (AB) is diagonal and becomes:[177]
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based on KS orbitals. This route was first explored by Casida[54]
to solve the assignment problem of the LR-TDDFT excitedstate transitions and then further developed by Tavernelli
et al.[14] in relation to the calculation of matrix elements in the
linear and second-order response regimes.[59, 156, 157]
In ref. [157], we showed that defining the ground-state
many-electron wavefunction as a Slater determinant of all ocel
cupied KS orbitals fi gNi¼1
[Eq. (97)]:
and the excited-state wavefunction corresponding to the excitation energy wn as [Eq. (98)]:
we obtain for any one-body operator of the form O^ =
P
^yps a
^qs (in which p and q are general indices) the corpqs opqs a
rect LR expression for the matrix element hF0 j O^ j Fni. Equation (98) is derived from Equation (96) for which now the index
i runs over all occupied KS orbitals and the Eindex a runs over
~ as denotes a singly
the unoccupied (virtual) KS orbitals and F
is
excited Slater determinant defined by the transition is!as.
This theory was then successfully extended to the case of the
calculation of matrix elements between two excited-state
wavefunctions, hFn j O^ j Fmi, as shown in the section on the
calculation of NAC vectors (Section 3.5).
It is important to further stress the fact that both auxiliary
functions introduced in Equations (97) and (98) have only
a physical meaning when used withinDLR-TDDFT
Efor the calcu~ 0 O^F
~ n and eventulation
Eelements of the type F
D of matrix
^ ~
~
ally Fn OFm (see Section 3.3.4). The use of these representations of the ground-state and excited-state KS many-electron
wavefunctions in other contexts is not justified.
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ðiÞ Fn and mðiÞ ¼ Fn mðiÞ Fm .
with mðiÞ
n ¼ F0 m
nm
The TDDFT second-order density–density response function
can also be obtained (see refs. [217, 220]) and then converted
into a SOS representation by setting up a classical system of
coupled harmonic oscillators (bosons) that share the same
linear and second-order response properties as those of
TDDFT.[217, 219, 220] It is beyond the scope of this review to describe the derivation of the second-order matrix elements following this approach in detail. It is, however, important to
mention that this solution is an approximation, and in particular, it does not account for contributions to the matrix elements produced by the de-excitation of the correlated ground
state j F0i.[59]
In ref. [59], Dit was
shown
E that second-order matrix elements
~ n O^F
~ m can be evaluated. This solution is
of the type F
more symmetric with respect to the ph and hp transitions
than the one derived in refs. [217, 219, 220], but only considers
terms first-order in Z in the expansion of the KS energy functional around the equilibrium (unperturbed) value. The TDDFT
second-order matrix elements are, therefore, evaluated according to [Eq. (100)]:
and match exactly the leading LR contributions obtained by
using the second-order coupled electronic oscillator approach
of Mukamel and Tretiak and co-workers.[217, 219, 220] In the
TDA[59, 179, 180] the two approaches coincide.
In terms of Casida’s eigenvectors (Xn,Yn), Equation (100) becomes [Eq. (101)]:
3.3.4. The Second-Order Response Matrix Elements
The same procedure used in the derivation of the matrix elements between the ground and excited states can be followed
for theDcalculation
E of the second-order response matrix ele ments Fn O^Fm , in which both states n and m are LR excited states. We will, therefore, evaluate the SOS second-order
density–density response function in both wavefunction-based
MBPT and TDDFT, and then equate the residues corresponding
to equal poles term-by-term.
Within the many-body formulation of quantum mechanics
in second quantization, the SOS second-order density–density
response function is obtained by using a perturbative approach applied to the molecular Hamiltonian and reads
[Eq. (99)]:[59, 217–219]
2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
This equation will be used to evaluate NAC vectors along
nonadiabatic trajectories produced with the TDDFT-based TSH
approach.[14] To this end, the general operator O^ is replaced by
the nuclear gradient of the electronic Hamiltonian (see Section 3.5).
3.4. The LR-TDDFT Response Density
As it will be discussed in Section 4.1, QM/MM nonadiabatic dynamics requires the evaluation of the electrostatic interaction
between the classical environment and the quantum system,
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which comprises the excited density 1n(r,t) = 10(r) + d1n(r,t).
Within Casida’s formulation, the density response d1n(r,t) can
be expanded in the auxiliary many-electron wavefunctions of
Equation (98) [Eq. (102)]:
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3.5.1. The Couplings with the Ground State
We start from an alternative definition of the NAC vectors[222]
(see also Chapter 5 of ref. [223] for a complete discussion) between the ground (0) state and the nth excited state for a molecular system characterized by nuclear coordinates R in the
configuration space (R3Nn) [Eq. (107)]:16
which in first-order becomes [Eq. (103)]:
where g is an atomic label, H^el is the (many-body) electronic
Hamiltonian, and @gH^el = @H^el /@Rg.
Applying the results of Sections 3.3.2
3.3.3
Eon the evaluD and
ation of matrix elements of the form F0 O^Fn in LR-TDDFT
to the NAC vector gives directly the desired expression
[Eq. (108)]:
in which f̃p = (1fp).
Using the LR-TDDFT equation for the transition density
1’(r) = pqfpf̃q(Xn+Yn)pqfp(r)yq(r) and the definition of the coeffin [14, 156, 177]
cients Cpq
we finally get, in agreement with
refs. [177, 221] [Eq. (104)]:
in which [Eq. (105)]:
and
x1 ðrÞ; . . . ; xNel ðrÞ; xNel þ1 ðrÞ; xNel þ2 ðrÞ; . . .
¼ 1 ðrÞ; . . . ; Nel ðrÞ; y1 ðrÞ; y2 ðrÞ; . . . .
Within the Sternheimer framework (see Section 3.1.3), the response density matrix elements in TDA are given by ref. [155]
[Eq. (106)]:
R
in which hgijs ¼ dr@g H^el is* ðrÞjs ðrÞ.
This formula for the NAC vectors within LR-TDDFT has been
derived several times in the literature by using slightly different
formalisms. The first derivation was by Chernyak and Mukamel[57] who used classical Liouville dynamics for the single-electron density matrix. Later, Tavernelli et al.[14, 156, 157] and Hu
et al.[214, 215] arrived at the same result [Eq. (108)] by using the
most widely used formulation based on Casida’s LR-TDDFT
equations.[54]
Concerning the numerical implementation of Equation (107),
several approaches have also been proposed that differ mainly
in the choice of the basis set and in the way in which the implicit dependence of the pseudopotentials on the nuclear positions is treated. Due to the technical nature of this subject, we
will not go through the numerical details but instead refer to
the literature, which is very rich on this subject.[156, 214, 224–227]
3.5.2. The Finite Difference Formulation of the NACs
For a detailed account of the implementation see the Appendix D in the Supporting Information and refs. [155, 156].
3.5. The NAC Vectors in LR-TDDFT
Traditionally, the computation of the nonadiabatic coupling
vectors is carried out by using wavefunction-based ab initio
quantum chemistry approaches, which, however, are not well
suited for condensed-phase applications and become computationally unaffordable when the dynamics of large molecular
systems is considered.
Using the auxiliary many-electron wavefunction approach
outlined in Section 3.3, we present here an approach for the
calculation of nonadiabatic vectors within LR-TDDFT.
2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
In TSH dynamics, the NAC terms appear as a scalar product of
the NAC vectors with the particle velocities, s0n = d0n·Ṙ, in the
time evolution of the amplitudes associated to the different
states. Starting from an equivalent definition of the NAC vectors [Eq. (109)]:
we, therefore, obtain [Eq. (110)]:[228]
16
Note that with explicit dependencies, Equation (107) would read:
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in which j F0i and j Fni are evaluated at subsequent time t
the auxiliary many-electron wavefunctions,
E
andEt + dt by using
F
~ 0 and F
~ n [Eqs. (97) and (98)], inqthe
ffiffiffiffiffiffiffiffiTDA approximation
ea ei
[14, 225]
obtained through the replacement
Because
wn ! 1.
s0n j t + dt/2 is the only quantity needed in the evaluation of the
surface hopping probability in the
[t,t + dt] beE time
E
interval
~ 0 and F
~ n , it is numerically
tween the electronic states F
more efficient to use Equation (110) instead of the cumbersome evaluation of d0n followed by the product d0n·Ṙ. However, at the position of the “surface hops”, the calculation of the
full NAC vectors is required for the redistribution of the potential energy difference to guarantee energy conservation.
3.5.3. NACs between Excited States
In the excited-state nonadiabatic dynamics of molecular systems we need to compute NAC vectors between pairs of excited states. These are beyond the reach of LR theory (see Section 3.3.4) and, therefore, cannot be evaluated by using
Eq. (96). A second-order response
Efor the evaluation of
D theory
matrix elements of the form Fn O^Fm within TDDFT was
first derived by Mukamel and co-workers[217, 219] by using an approximate mapping of the original electronic problem into
a boson system sharing the same response properties.
The use of the “auxiliary” electronic wavefunctions introduced in Section 3.3.4 offers a valid alternative to this approach and produces second-order matrix elements that include contributions from the de-excitation of the correlated
ground state,[59] which are neglected in the derivation given in
ref.[219] In fact, the two approaches coincide in the TDA up to
terms second-order in Zm.[59]
For the calculation
of the ENAC vectors between a pair of ex E
~ m , in the LR-TDDFT-based TSH dy~ n and F
cited states, F
namics of Section 2.4.3 we, therefore, use the expression given
in Equation (101) with O^ replaced by @gH^el . The quality of
these matrix elements was assessed in ref. [59]
4. Coupling with the Environment
In this section we will outline the implementation of a coupling
scheme that allows the simulation of nonadiabatic processes
coupled to external static and time-dependent fields. The first
case is suited for the investigation of all solvent effects that do
not involve charge transfer from the molecule of interest to
the solvent (or vice versa) or (photo)chemical reactions between solute and solvent molecules. The coupling with timedependent fields offers instead the possibility to investigate
the interaction of matter with any form of electromagnetic radiation from solar light to laser fields.
www.chemphyschem.org
molecule surrounded by solvent molecules or embedded in
a matrix could be strongly altered by electrostatic effects, spatial confinement, or photochemical reactions with the solvent
(e.g. see refs. [15, 229–237]). The most straightforward solution
for describing explicit condensed-phase effects in simulations
would be to include directly solvent molecules in the ab initio
simulation (ab initio means, in this context, any theory that describes, even approximately, the quantum nature of the electrons). Even though this would have the benefit of fully describing the possible electronic coupling effects between
solute and solvent, the computational cost of full (solute and
solvent) TSH/LR-TDDFT calculations is nowadays still prohibitive
and only few solvent molecules can be added (to the QM part)
to get a first indication of the role of solvent-induced nonradiative pathways.[238]
Instead of treating solvent molecules as part of the quantum
setup, classical force fields can be used for their description
within a MM approach. This is at the hearth of the QM/MM
method.[239–242] The molecule of interest is still modeled at
a QM level (QM part), whereas the solvent (MM part) is described at a classical level and no charge transfer between the
two subsets is possible. In this way, complete solvation of the
QM part can be achieved at very low computational cost. In
the so-called additive scheme,[242] the point charges associated
to each atom of the solvent molecules polarize the electronic
density of the QM part, and if a polarizable force field is used,
the QM part can eventually influence the MM region in return.
When the system of interest is coupled to its environment in
a QM/MM setup, the electrostatic interaction between the QM
subsystem (treated at TDDFT or LR-TDDFT level) and the MM
subsystem (classically described through an empirical Hamiltonian) is described by [Eq. (111)]:[168]
in which d1i(r,t) is the electron density perturbation induced
by the transition into the ith excited state and is evaluated according to Equation (104) together with Equation (105) or,
equivalently, Equation (106). The potential usC(jRgrj) is
a screened Coulomb potential generated by the atom at Rg
and modified at short range to avoid spurious overpolarization
effects.[241] In a QM/MM setup we, therefore, need, in addition
to the calculation of the (LR-)TDDFT energies, forces, and couplings, an explicit evaluation of the (LR-)TDDFT perturbed densities. When coupled to QM/MM, the TSH/LR-TDDFT/MM dynamics allow the photophysics and photochemistry of molecules in their natural environment to be described (e.g.
refs. [168, 238, 243, 244]).
4.1. Classical Environment and TDDFT/MM Coupling
The nonadiabatic schemes presented so far can be applied to
the study of the photochemistry and photophysics of molecules in the gas phase, a condition that is, for example, encountered in experiments performed under vacuum or in molecular jets. However, the nonradiative decay of a photoexcited
2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
4.2. External TD Fields
The coupling of nonadiabatic MD with an external TD electric
field is given by the interaction Hamiltonian (with no spin–
magnetic field contributions) [Eq. (112)]:
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www.chemphyschem.org
second-order quantities that transcend LR-TDDFT as we saw in
Section 3.3.4.
in which the classical vector potential A(ri,t) is related to the
1 @A
actual electric field by E = c @t . The summation in Equation (112) is over all electrons, whereas the interaction with the
nuclei is treated at the fully classical level and is, therefore, not
included in the following derivation. Ĥint can be directly added
to the Hamiltonian that governs the Ehrenfest dynamics
[Eqs. (14) and (15)].[84, 245] In TDDFT-based Ehrenfest dynamics,
the coupling is explicit and can be included by replacing the
square of the momentum operator p̂2 = h2r2 with the term
(p̂e/cA(r,t))2 in Equation (18).[71]
In TSH nonadiabatic MD cases,[158, 209] one needs to evaluate
the radiation field coupling matrix element [Eq. (113)]:17
in which A(t) = A(t)El and [Eq. (114)]:
is the transition dipole vector, and wji = (EjEi)/h. Once more,
the use of auxiliary many-electron wavefunctions within LRTDDFT simplifies the calculation of the transition dipoles,
which can be computed by using Equation (114) after replacing j Fii by j F̃ii and hFj j by hF̃j j . As discussed in Section 3.3.4, this equation can also be used to evaluate transition
dipole vectors between pairs of excited states, which is an approximate solution that becomes exact when the TDA is invoked.
Finally, the differential equation for the TSH coefficients
Equation (50) in the presence of an external radiation field interacting with the system[24, 158, 167, 243, 247–253] becomes [Eq. (115)]:
in which the last term is responsible for the amplitude transfer
among the different electronic states.
The coupling of TSH dynamics with explicit time-dependent
external fields within LR-TDDFT was the subject of a series of
recent publications.[158, 167, 209, 243] Even though everybody agrees
on the form of the coupling, in some cases no mention is
made on the meaning and the derivation of the LR-TDDFT
many-electron wavefunctions used in the calculation of the
matrix elements; this is particularly deceptive in the case of
the coupling between pairs of excited states, which are, strictly,
17
The derivation of these coupling matrix elements implies the Coulomb gauge
ri·A(ri,t) = 0 and several approximations such as the “weak-field limit” [ne1
glect of the term 2c2 A(ri,t)·A(ri,t) in Eq. (112)] and the dipole approximation.
See, for example, refs. [158, 246] for more details on those approximations.
2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
4.2.1. Local Control and Pulse Design
Of particular interest is the possibility to design laser pulses
able to induce specific physical or chemical modifications to
a molecular system of interest. To this end, several optimal
control schemes have been developed in both the theoretical
and experimental communities with the aim to control chemical reactions. In this section, we will briefly describe a pulseshaping algorithm, which is intimately coupled to the nonadiabatic dynamics schemes discussed in previous sections. In particular, TSH dynamics offers an interesting opportunity for the
implementation of what is known as local control theory.[254–262]
The aim of local control theory is to compute the electric field
that maximizes (or minimizes) the TSH population associated
to a selected electronic state “on-the-fly” by using Equation (115).
Starting from the time evolution of an arbitrary operator B^
[Eq. (116)]:
^ which is only true in
and assuming that Ĥmol commutes with B,
the absence of NACs, we obtain [Eq. (117)]:
(note that in the presence of NACs, Equation (117) will contain
an additional time-dependent term). To control the population
of a particular electronic state j Fii, we introduce the projector
operator P^i = j FiihFi j . The time evolution of the state population is simply given by [Eq. (118)]:
Using the TSH Ansatz for a classical trajectory a [Eq. (119)]:
Equation (118) becomes [Eq. (120)]:
It is now evident that choosing the electric field as
[Eq. (121)]:
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D
E
will ensure that P^i½a ðtÞ increases (or decreases) at all times.
In conclusion, when a targeted chemical process can be associated to a well-defined excited-state transition, local control
theory coupled to TSH dynamics offers a valuable tool for the
optimization of the external TD electric field able to efficiently
trigger the desired reaction.[209]
5. Conclusions
In this review, we presented a thorough description of several
trajectory-based nonadiabatic molecular dynamics schemes
based on TDDFT for “on-the-fly” calculation of all required electronic structure quantities (e.g. energies, nuclear forces, NACs,
and coupling with external fields). In particular, mean-field Ehrenfest MD and TSH MD emerge as best candidates for the description of nonadiabatic effects in DFT/TDDFT-based ab initio
MD. Alternative solutions based on a more rigorous description
of the nuclear quantum dynamics have also been explored,
and the NABDY approach represents, in our opinion, a very
promising alternative with the potential to include quantum
effects (such as tunneling, entanglement, and decoherence) in
an efficient and accurate way.[40, 58, 66, 263, 264]
What makes these trajectory-based nonadiabatic schemes
particularly efficient is the use of TDDFT for “on-the-fly” calculation of electronic properties. Being a density-based theory,
TDDFT avoids dealing with the complex realm of wavefunction-based electronic structure methods. However, despite its
apparent formal simplicity, the physical interpretation of
TDDFT remains intrinsically difficult to grasp. Although the
many-body solution of the many-electron problem could
appear cumbersome at first glance, its physical content is
transparent and does not require further investigation. In contrast, TDDFT is a density-based theory that relies on an “existence theorem” and that describes response quantities as contracted two-point functions [Eq. (56)] instead of the four-point
functions obtained from the solution of the Bethe–Salpeter
equation[265] when “physical” nonlocal potentials are present.
Therefore, rigorous derivation of most electronic properties
within LR-TDDFT can become more involved than when standard perturbation theory is used, as is traditionally done in
quantum chemistry and MBPT (based on Green’s functions
techniques. This is the reason why, in this review, we have devoted a large section to the derivation of all TDDFT-based electronic structure properties essential to nonadiabatic dynamics.
In particular, we introduced the concept of “auxiliary manyelectron wavefunction” in LR-TDDFT to emphasize their physical origin and to avoid the misleading parallel with real (Hartree–Fock and post-Hartree–Fock) many-electron wavefunctions.
Despite the success of LR-TDDFT in physics and chemistry,
the quality of the results still strongly depends on the quality
of the exchange and correlation functionals and of the LR
kernel used in the calculations. Fortunately, recent developments in the design of better-performing functionals have im 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
www.chemphyschem.org
proved the description of key physical and chemical processes
such as charge transfer, many-electron excitations, and Rydberg-type excitations and of the topography of the PESs that
are, in particular, close to conical intersections.
In conclusion, we believe that, thanks to the most recent
theoretical advances, TDDFT-based nonadiabatic dynamics MD
will become the method of choice for the investigation of photochemical and photophysical processes of complex molecular
systems in the gas and condensed phases. By combining the
efficiency of “on-the-fly” trajectory-based techniques with the
formal simplicity of TDDFT, these nonadiabatic MD approaches
have an enormous potential that is far from being fully exploited.
Acknowledgements
The authors thank Giovanni Ciccotti, Sara Bonella, Thomas J.
Penfold, Jrme F. Gonthier, and Felipe Franco de Carvalho for
useful discussions. Pascal Miville is acknowledged for graphical
representations. This work was supported by the COST Action
CM0702, the NCCR-MUST interdisciplinary research program, and
the Swiss National Science Foundation Grants Nos. 200020130082 and 200021-146396. Financial support for CADMOS and
the Blue Gene/P system is provided by the Canton of Geneva,
Canton of Vaud, Hans Wilsdorf Foundation, Louis-Jeantet Foundation, University of Geneva, University of Lausanne and cole
Polytechnique Fdrale de Lausanne.
Keywords: density functional calculations · electronic
structure properties · molecular dynamics · photochemistry ·
theoretical chemistry
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