Time-dependent Density Functional Theory
Miguel A. L. Marques and E. K. U. Gross
1
Introduction
Time-dependent density-functional theory (TDDFT) extends the basic ideas
of ground-state density-functional theory (DFT) to the treatment of excitations or more general time-dependent phenomena. TDDFT can be viewed
an alternative formulation of time-dependent quantum mechanics but, in
contrast to the normal approach that relies on wave-functions and on the
many-body Schrödinger equation, its basic variable is the one-body electron
density, n(r, t). The advantages are clear: The many-body wave-function, a
function in a 3N -dimensional space (where N is the number of electrons in
the system), is a very complex mathematical object, while the density is a
simple function that depends solely on 3 variables, x, y and z. The standard
way to obtain n(r, t) is with the help of a fictitious system of non-interacting
electrons, the Kohn-Sham system. The final equations are simple to tackle numerically, and are routinely solved for systems with a large number of atoms.
These electrons feel an effective potential, the time-dependent Kohn-Sham
potential. The exact form of this potential is unknown, and has therefore to
be approximated.
The scheme is perfectly general, and can be applied to essentially any
time-dependent situation. Two regimes can however be observed: If the timedependent potential is weak, it is sufficient to resort to linear-response theory
to study the system. In this way it is possible to calculate, e.g. optical absorption spectra. It turns out that, even with the simplest approximation to the
Kohn-Sham potential, spectra calculated within this framework are in very
good agreement with experimental results. On the other hand, if the timedependent potential is strong, a full solution of the Kohn-Sham equations is
required. A canonical example of this regime is the treatment of atoms or
molecules in strong laser fields. In this case, TDDFT is able to describe nonlinear phenomena like high-harmonic generation, or multi-photon ionization.
Our purpose, while writing this chapter, is to provide a pedagogical introduction to TDDFT1 . With that in mind, we present, in section 2, a
quite detailed proof of the Runge-Gross theorem[5], i.e. the time-dependent
generalization of the Hohenberg-Kohn theorem[6], and the corresponding
1
The reader interested in a more technical discussion is therefore invited to read
Ref. [1–4], where also very complete and updated lists of references can be found.
2
Miguel A. L. Marques and E. K. U. Gross
Kohn-Sham construction[7]. These constitute the mathematical foundations
of TDDFT. Several approximate exchange-correlation (xc) functionals are
then reviewed. In section 3 we are concerned with linear-response theory,
and with its main ingredient, the xc kernel. The calculation of excitation energies is treated in the following section. After giving a brief overlook of the
competing density-functional methods to calculate excitations, we present
some results obtained from the full solution of the Kohn-Sham scheme, and
from linear-response theory. Section 5 is devoted to the problem of atoms
and molecules in strong laser fields. Both high-harmonic generation and ionization are discussed. Finally, the last section is reserved to some concluding
remarks.
For simplicity, we will write all formulae for spin-saturated systems. Obviously, spin can be easily included in all expressions when necessary. Hartree
atomic units (e = h̄ = m = 1) will be used throughout this chapter.
2
2.1
Time-dependent DFT
Preliminaries
A system of N electrons with coordinates r = (r 1 · · · r N ) is known to obey
the time-dependent Schrödinger equation
i
∂
Ψ (r, t) = Ĥ(r, t)Ψ (r, t) ,
∂t
(1)
This equation expresses one of the most fundamental postulates of quantum
mechanics, and is one of the most remarkable discoveries of physics during the
2
XXth century. The absolute square of the electronic wave-function, |Ψ (r, t)| ,
is interpreted as the probability of finding the electrons in positions r. The
Hamiltonian can be written in the form
T̂ (r) + Ŵ (r) + V̂ext (r, t) .
(2)
The first term is the kinetic energy of the electrons
N
T̂ (r) = −
1X 2
∇ ,
2 i=1 i
(3)
while Ŵ accounts for the Coulomb repulsion between the electrons
N
1
1 X
.
Ŵ (r) =
2 i,j=1 |ri − r j |
(4)
i6=j
Furthermore, the electrons are under the influence of a generic, time-dependent
potential, V̂ext (r, t). The Hamiltonian (2) is completely general and describes
Time-dependent Density Functional Theory
3
a wealth of physical and chemical situations, including atoms, molecules, and
solids, in arbitrary time-dependent electric or magnetic fields, scattering experiments, etc. In most of the situations dealt with in this article we will be
concerned with the interaction between a laser and matter. In that case, we
can write the time-dependent potential as the sum of the nuclear potential
and a laser field, V̂TD = Ûen + V̂laser . The term Ûen accounts for the Coulomb
attraction between the electrons and the nuclei,
Ûen (r, t) = −
Nn X
N
X
ν=1 i=1
Zν
,
|ri − Rν (t)|
(5)
where Zν and Rν denote the charge and position of the nucleus ν, and Nn
stands for the total number of nuclei in the system. Note that by allowing the
Rν to depend on time we can treat situations where the nuclei move along a
classical path. This may be useful when studying, e.g., scattering experiments,
chemical reactions, etc. The laser field, V̂laser , reads, in the length gauge,
V̂laser (r, t) = E f (t) sin(ωt)
N
X
i=1
ri · α ,
(6)
where α, ω and E are respectively the polarization, the frequency and the
amplitude of the laser. The function f (t) is a envelope that shapes the laser
pulse during time. Note that in writing Eq. (6) we use two approximations:
i) We treat the laser field classically, i.e., we do not quantize the photon field.
This is a well justified procedure when the density of photons is large and the
individual (quantum) nature of the photons can be disregarded. In all cases
presented in this article this will be the case. ii) Expression (6) is written
within the dipole approximation. The dipole approximation holds whenever
(a) The wave-length of the light (λ = 2πc/ω, where c is the velocity of light
in vacuum) is much larger than the size of the system. This is certainly
true for all atoms and most molecules we are interested in. However, one
has to be careful when dealing with very large molecules (e.g. proteins) or
solids. (b) The path that the particle travels in one period of the laser field
is small compared to the wavelength. This implies that the average velocity
of the electrons, v, fulfills vT λ ⇒ v λ/T = c, where T stands for
the period of the laser. In these circumstances we can treat the laser field
as a purely electric field and completely neglect its magnetic component.
This approximation holds if the intensity of the laser is not strong enough
to accelerate the electrons to relativistic velocities. (c) The total duration of
the laser pulse should be short enough so that the molecule does not leave
the focus of the laser during the time the interaction lasts.
Although the many-body Schrödinger equation, Eq. (1), achieves, to our
best knowledge, a remarkably good description of nature, it poses a tantalizing problem to scientists. Its exact (in fact, numerical) solution has been
achieved so far for a disappointingly small number of particles. In fact, even
4
Miguel A. L. Marques and E. K. U. Gross
the calculation of a “simple” two electron system (the Helium atom) in a laser
field takes several months in a modern computer[8] (see also the work on the
++
H+
molecule[10]). The effort to solve Eq. (1) grows
2 [9] molecule and the H3
exponentially with the number of particles, therefore rapid developments regarding the exact solution of the Schrödinger equation are not expected.
In these circumstances, the natural approach of the theorist is to transform and approximate the basic equations to a manageable level that still
retains the qualitative and (hopefully) quantitative information about the
system. Several techniques have been developed throughout the years in the
quantum chemistry and physics world. One such technique is TDDFT. Its
goal, like always in density-functional theories, is to replace the solution of
the complicated many-body Schrödinger equation by the solution of the much
simpler one-body Kohn-Sham equations, thereby relieving the computational
burden.
The first step of any DFT is the proof of a Hohenberg-Kohn type theorem[6]. In its traditional form, this theorem demonstrates that there exists
a one-to-one correspondence between the external potential and the (onebody) density. With the external potential it is always possible (in principle) to solve the many-body Schrödinger equation to obtain the many-body
wave-function. From the wave-function we can trivially obtain the density.
The second implication, i.e. that the knowledge of the density is sufficient
to obtain the external potential, is much harder to prove. In their seminal
paper, Hohenberg and Kohn used the variational principle to obtain a proof
by reductio ad absurdum. Unfortunately, their method cannot be easily generalized to arbitrary DFTs. The Hohenberg-Kohn theorem is a very strong
statement: From the density, a simple property of the quantum mechanical system, it is possible to obtain the external potential and therefore the
many-body wave-function. The wave-function, by its turn, determines every
observable of the system. This implies that every observable can be written
as a functional of the density.
Unfortunately, it is very hard to obtain the density of an interacting system. To circumvent this problem, Kohn and Sham introduced an auxiliary
system of non-interacting particles[7]. The dynamics of these particles are
governed by a potential chosen such that the density of the Kohn-Sham
system equals the density of the interacting system. This potential is local
(multiplicative) in real space, but it has a highly non-local functional dependence on the density. In non-mathematical terms this means that the
potential at the point r can depend on the density of all other points (e.g.
through gradients, or through integral operators like the Hartree potential).
As we are now dealing with non-interacting particles, the Kohn-Sham equations are quite simple to solve numerically. However, the complexities of the
many-body system are still present in the so-called exchange-correlation (xc)
functional that needs to be approximated in any application of the theory.
Time-dependent Density Functional Theory
2.2
5
The Runge-Gross theorem
In this section, we will present a detailed proof of the Runge-Gross theorem[5],
the time-dependent extension of the ordinary Hohenberg-Kohn theorem[6].
There are several “technical” differences between a time-dependent and a
static quantum-mechanical problem that one should keep in mind while trying to prove the Runge-Gross theorem. In static quantum mechanics, the
ground-state of the system can be determined through the minimization of
the total energy functional
E[Φ] = hΦ| Ĥ |Φi .
(7)
In time-dependent systems, there is no variational principle on the basis of
the total energy for it is not a conserved quantity. There exists, however, a
quantity analogous to the energy, the quantum mechanical action
Z t1
∂
dt hΦ(t)| i − Ĥ(t) |Φ(t)i ,
A[Φ] =
(8)
∂t
t0
where Φ(t) is a Ne -body function defined in some convenient space. From
expression (8) it is easy to obtain two important properties of the action:
i) Equating the functional derivative of Eq. (8) in terms of Φ∗ (t) to zero, we
arrive at the time-dependent Schrödinger equation. We can therefore solve the
time-dependent problem by calculating the stationary point of the functional
A[Φ]. The function Ψ (t) that makes the functional stationary will be the
solution of the time-dependent many-body Schrödinger equation. Note that
there is no “minimum principle”, but only a “stationary principle”. ii) The
action is always zero at the solution point, i.e. A[Ψ ] = 0. These two properties
make the quantum-mechanical action a much less useful quantity than its
static counterpart, the total energy.
Another important point, often overlooked in the literature, is that a
time-dependent problem in quantum mechanics is mathematically defined as
an initial value problem. This stems from the fact that the time-dependent
Schrödinger equation is a first-order differential equation in the time coordinate. The wave-function (or the density) thus depends on the initial state,
which implies that the Runge-Gross theorem can only hold for a fixed initial state (and that the xc potential depends on that state). In contrast, the
static Schrödinger equation is a second order differential equation in the space
coordinates, and is the typical example of a boundary value problem.
From the above considerations the reader could conjecture that the proof
of the Runge-Gross theorem is more involved than the proof of the ordinary
Hohenberg-Kohn theorem. This is indeed the case. What we have to demonstrate is that if two potentials, v(r, t) and v 0 (r, t), differ by more than a purely
time dependent function c(t)2 , they cannot produce the same time-dependent
2
If the two potentials differ solely by a time-dependent function, they will produce
wave-functions which are equal up to a purely time-dependent phase. This phase
6
Miguel A. L. Marques and E. K. U. Gross
density, n(r, t), i.e.
v(r, t) 6= v 0 (r, t) + c(t) ⇒ ρ(r, t) 6= ρ0 (r, t) .
(9)
This statement immediately implies the one-to-one correspondence between
the potential and the density. In the following we will utilize primes to distinguish the quantities of the systems with external potentials v and v 0 . Due
to technical reasons that will become evident during the course of the proof,
we will have to restrict ourselves to external potentials that are Taylor expandable with respect to the time coordinate around the initial time t0
v(r, t) =
∞
X
k=0
ck (r)(t − t0 )k ,
(10)
with the expansion coefficients
ck (r) =
1 dk
v(r,
t)
.
k
k! dt
t=t0
(11)
We furthermore define the function
∂k
0
uk (r) = k [v(r, t) − v (r, t)]
.
∂t
t=t0
(12)
Clearly, if the two potentials are different by more than a purely timedependent function, at least one of the expansion coefficients in their Taylor
expansion around t0 will differ by more than a constant
∃k≥0 : uk (r) 6= const.
(13)
In the first step of our proof we will demonstrate that if v 6= v 0 + c(t) then
the current densities, j and j 0 , generated by v and v 0 are also different. The
current density j can be written as the expectation value of the current
density operator:
j(r, t) = hΨ (t)| ĵ(r) |Ψ (t)i ,
(14)
where the operator ĵ is written
i
h
io
1 nh
ĵ(r) = −
∇ψ̂ † (r) ψ̂(r) − ψ̂ † (r) ∇ψ̂(r) .
2i
(15)
We now use the quantum-mechanical equation of motion, which is valid for
any operator, Ô(t),
i
h
i
d
∂
hΨ (t)| Ô(t) |Ψ (t)i = hΨ (t)| i Ô(t) + Ô(t), Ĥ(t) |Ψ (t)i ,
dt
∂t
(16)
will, of course, cancel while calculating the density (or any other observable, in
fact).
Time-dependent Density Functional Theory
7
to write the equation of motion for the current density in the primed and
unprimed systems
i
h
d
j(r, t) = hΨ (t)| ĵ(r), Ĥ(t) |Ψ (t)i
dt
h
i
d
i j 0 (r, t) = hΨ 0 (t)| ĵ(r), Ĥ 0 (t) |Ψ 0 (t)i .
dt
i
(17)
(18)
As we start from a fixed initial many-body state, at t0 the wave-functions,
the densities, and the current densities have to be equal in the primed and
unprimed systems
|Ψ (t0 )i = |Ψ 0 (t0 )i ≡ |Ψ0 i
n(r, t0 ) = n0 (r, t0 ) ≡ n0 (r)
j(r, t0 ) = j 0 (r, t0 ) ≡ j 0 (r) .
(19)
(20)
(21)
If we now take the difference between the equations of motion (17) and (18)
we obtain, when t = t0 ,
i
h
i
d j(r, t) − j 0 (r, t) t=t = hΨ0 | ĵ(r), Ĥ(t0 ) − Ĥ 0 (t0 ) |Ψ0 i
0
dt
i
h
= hΨ0 | ĵ(r), v(r, t0 ) − v 0 (r, t0 ) |Ψ0 i
= in0 (r)∇ [v(r, t0 ) − v 0 (r, t0 )] .
(22)
Let us assume that Eq. (13) is fulfilled already for k = 0, i.e. that the two
potentials, v and v 0 , differ at t0 . This immediately implies that the derivative
on the left-hand side of Eq. (22) differs from zero. The two current densities
j and j 0 will consequently deviate for t > t0 . If k is greater than zero the
equation of motion is applied k + 1 times to yield
dk+1 j(r, t) − j 0 (r, t) t=t0 = n0 (r)∇uk (r) .
dtk+1
(23)
The right-hand side of Eq. (23) differs from zero, which again implies that
j(r, t) 6= j 0 (r, t) for t > t0 .
In a second step we prove that j 6= j 0 implies n 6= n0 . To achieve that
purpose we will make use of the continuity equation
∂
n(r, t) = −∇ · j(r, t) .
∂t
(24)
If we write Eq. (24) for the primed and unprimed system and take the difference, we arrive at
∂
[n(r, t) − n0 (r, t)] = −∇ · j(r, t) − j 0 (r, t) .
∂t
(25)
8
Miguel A. L. Marques and E. K. U. Gross
As before, we would like an expression involving the kth time derivative of
the external potential. We therefore take the (k + 1)st time-derivative of the
previous equation to obtain (at t = t0 )
∂ k+1 ∂ k+2
[n(r, t) − n0 (r, t)]t=t0 = −∇ · k+1 j(r, t) − j 0 (r, t) t=t0
k+2
∂t
∂t
= −∇ · [n0 (r)∇uk (r)] .
(26)
In the last step we made use of Eq. (23). By the hypothesis (13) we have
uk (r) 6= const. hence it is clear that if
∇ · [n0 (r)∇uk (r)] 6= 0 ,
(27)
then n 6= n0 , from which follows the Runge-Gross theorem. To show that
Eq. (27) is indeed fulfilled, we will use the versatile technique of demonstration by reductio ad absurdum. Let us assume that ∇ · [n0 (r)∇uk (r)] = 0 with
uk (r) 6= const., and look at the integral
Z
Z
2
3
d r n0 (r) [∇uk (r)] = − d3 r uk (r)∇ · [n0 (r)∇uk (r)]
(28)
Z
+ n0 (r)uk (r)∇uk (r) · dS .
S
This equality was obtained with the help of Green’s theorem. The first term
on the right-hand side is zero by assumption, while the second term vanishes
if the density and the function uk (r) decay in a “reasonable” manner when
r → ∞. This situation is always true for finite systems. We further notice that
2
the integrand n0 (r) [∇uk (r)] is always positive. These diverse conditions
can only be satisfied if either the density n0 or ∇uk (r) vanish identically.
The first possibility is obviously ruled out, while the second contradicts our
initial assumption that uk (r) is not a constant. This concludes the proof of
the Runge-Gross theorem.
2.3
Time-dependent Kohn-Sham equations
As mentioned in section 2.1, the Runge-Gross theorem asserts that all observables can be calculated with the knowledge of the one-body density. Nothing
is however stated on how to calculate that valuable quantity. To circumvent
the cumbersome task of solving the interacting Schrödinger equation, Kohn
and Sham had the idea of utilizing an auxiliary system of non-interacting
(Kohn-Sham) electrons, subject to an external local potential, vKS [7]. This
potential is unique, by virtue of the Runge-Gross theorem applied to the noninteracting system, and is chosen such that the density of the Kohn-Sham
electrons is the same as the density of the original interacting system. In the
time-dependent case, these Kohn-Sham electrons obey the time-dependent
Time-dependent Density Functional Theory
9
Schrödinger equation
∂
∇2
i ϕi (r, t) = −
+ vKS (r, t) ϕi (r, t) .
∂t
2
(29)
The density of the interacting system can be obtained from the time-dependent
Kohn-Sham orbitals
occ
X
2
n(r, t) =
|ϕi (r, t)| .
(30)
i
Eq. (29), having the form of a one-particle equation, is fairly easy to solve
numerically. We stress, however, that the Kohn-Sham equation is not a meanfield approximation: If we knew the exact Kohn-Sham potential, vKS , we
would obtain from Eq. (29) the exact Kohn-Sham orbitals, and from these
the correct density of the system.
The Kohn-Sham potential is conventionally separated in the following
way
vKS (r, t) = vext (r, t) + vHartree (r, t) + vxc (r, t) .
(31)
The first term is again the external potential. The Hartree potential accounts
for the classical electrostatic interaction between the electrons
Z
n(r, t)
.
(32)
vHartree (r, t) = d3 r0
|r − r0 |
The last term, the xc potential, comprises all the non-trivial many-body
effects. In ordinary DFT, vxc is normally written as a functional derivative
of the xc energy. This follows from a variational derivation of the KohnSham equations starting from the total energy. It is not straightforward to
extend this formulation to the time-dependent case due to a problem related
to causality[11,2]. The problem was solved by van Leeuwen in 1998, by using
the Keldish formalism to defined a new action functional[12], Ã. The timedependent xc potential can then be written as the functional derivative of
the xc part of Ã,
δ Ãxc vxc (r, t) =
,
(33)
δn(r, τ ) n(r,t)
where τ stands for the Keldish pseudo-time.
Inevitably, the exact expression of vxc as a functional of the density is
unknown. At this point we are obliged to perform an approximation. It is important to stress that this is the only fundamental approximation in TDDFT.
In contrast to stationary-state DFT, where very good xc functionals exist,
approximations to vxc (r, t) are still in their infancy. The first and simplest of
these is the adiabatic local density approximation (ALDA), reminiscent of the
ubiquitous LDA. More recently, several other functionals were proposed, from
which we mention the time-dependent exact-exchange (EXX) functional[13],
and the attempt by Dobson, Bünner, and Gross[14] to construct an xc functional with memory. In the following section we will introduce the above
mentioned functionals.
10
2.4
Miguel A. L. Marques and E. K. U. Gross
xc functionals
Adiabatic approximations There is a very simple procedure that allows
the use of the plethora of existing xc functionals for ground-state DFT in the
time-dependent theory. Let us assume that ṽxc [n] is an approximation to the
ground-state xc density functional. We can write an adiabatic time-dependent
xc potential as
adiabatic
vxc
(r, t) = ṽxc [n](r)|n=n(t) .
(34)
I.e. we employ the same functional form but evaluated at each time with the
density n(r, t). The functional thus constructed is obviously local in time.
This is, of course, a quite dramatic approximation. The functional ṽxc [n] is
a ground-state property, so we expect the adiabatic approximation to work
only in cases where the temporal dependence is small, i.e., when our timedependent system is locally close to equilibrium. Certainly this is not the case
if we are studying the interaction of strong laser pulses with matter.
By inserting the LDA functional in Eq. (34) we obtain the so-called adiabatic local density approximation (ALDA)
ALDA
HEG
vxc
(r, t) = vxc
(n)n=n(r,t) .
(35)
The ALDA assumes that the xc potential at the point r, and time t is equal
to the xc potential of a (static) homogeneous-electron gas (HEG) of density
n(r, t). Naturally, the ALDA retains all problems already present in the LDA.
Of these, we would like to emphasize the erroneous asymptotic behavior of the
LDA xc potential: For neutral finite systems, the exact xc potential decays
as −1/r, whereas the LDA xc potential falls off exponentially. Note that
most of the generalized-gradient approximations (GGAs), or even the newest
meta-GGAs have asymptotic behaviors similar to the LDA. This problem
gains particular relevance when calculating ionization yields (the ionization
potential calculated with the ALDA is always too small), or in situations
where the electrons are pushed to regions far away from the nuclei (e.g., by
a strong laser) and feel the incorrect tail of the potential.
Despite this problem, the ALDA yields remarkably good excitation energies (see sections 4.2 and 4.3) and is probably the most used xc functional in
TDDFT.
Time-dependent optimized effective potential Unfortunately, when
one is trying to write vxc as explicit functionals of the density, one encounters some difficulties. As an alternative, the so-called orbital-dependent xc
functionals were introduced several years ago. These functionals are written
explicitly in terms of the Kohn-Sham orbitals, albeit remaining implicit density functionals by virtue of the Runge-Gross theorem. A typical member of
this family is the exact-exchange (EXX) functional. The EXX action is obtained by expanding Axc in powers of e2 (where e is the electronic charge),
Time-dependent Density Functional Theory
11
and retaining the lowest order term, the exchange term. It is given by the
Fock integral
Z
Z
occ Z
ϕ∗ (r 0 , t)ϕk (r 0 , t)ϕj (r, t)ϕ∗k (r, t)
1 X t1
3
3 0 j
AEXX
=
−
. (36)
dt
d
r
d
r
x
2
|r − r 0 |
t0
j,k
From such an action functional, one seeks to determine the local Kohn-Sham
potential through a series of chain rules for functional derivatives. The procedure is called the optimized effective potential (OEP) or the optimized
potential method (OPM) for historical reasons[15,16]. The derivation of the
time-dependent version of the OEP equations is very similar to the groundstate case. Due to space limitations we will not present the derivation in this
article. The interested reader is advised to consult the original paper[13], one
of the more recent publications[17,18], or the chapter by E. Engel contained
in this volume. The final form of the OEP equation that determines the EXX
potential is
Z
occ Z t1
X
0
dt d3 r0 [vx (r 0 , t0 ) − ux j (r 0 , t0 )]
0=
(37)
j
−∞
×ϕj (r, t)ϕ∗j (r 0 , t0 )GR (rt, r 0 t0 ) + c.c.
The kernel, GR , is defined by
iGR (rt, r0 t0 ) =
∞
X
k=1
ϕ∗k (r, t)ϕk (r0 , t0 )θ(t − t0 ) ,
(38)
and can be identified with the retarded Green’s function of the system. Moreover, the expression for ux is essentially the functional derivative of the xc
action in relation to the Kohn-Sham wave-functions
ux j (r, t) =
δAxc [ϕj ]
1
.
ϕ∗j (r, t) δϕj (r, t)
(39)
Note that the xc potential is still a local potential, albeit being obtained
through the solution of an extremely non-local and non-linear integral equation. In reality, the solution of Eq. (37) poses a very difficult numerical problem. Fortunately, by performing an approximation first proposed by Krieger,
Li, and Iafrate (KLI) it is possible to simplify the whole procedure, and obtain an semi-analytic solution of Eq. (37)[19]. The KLI approximation turns
out to be a very good approximation to the EXX potential. Note that both
the EXX and the KLI potential have the correct −1/r asymptotic behavior
for neutral finite systems.
A functional with memory There is a very common procedure for the
construction of approximate xc functionals in ordinary DFT. It starts with
12
Miguel A. L. Marques and E. K. U. Gross
the derivation of exact properties of vxc , deemed important by physical arguments. Then an analytical expression for the functional is proposed, such
that it satisfies those rigorous constraints. We will use this recipe to generate
a time-dependent xc potential which is non-local in time, i.e. that includes
the “memory” from previous times[14].
A very important condition comes from Galilean invariance. Let us look
at a system from the point of view of a moving reference frame whose origin is
given by x(t). The density seen from this moving frame is simply the density
of the reference frame, but shifted by x(t)
n0 (r, t) = n(r − x(t), t) .
(40)
Galilean invariance then implies[20]
vxc [n0 ](r, t) = vxc [n](r − x(t), t) .
(41)
It is obvious that potentials that are both local in space and in time, like the
ALDA, trivially fulfill this requirement. However, when one tries to deduce
an xc potential which is non-local in time, one finds condition (41) quite
difficult to satisfy.
Another rigorous constraint follows from Ehrenfest’s theorem which relates the acceleration to the gradient of the external potential
d2
hri = − h∇vext (r)i .
dt2
For an interacting system, Ehrenfest’s theorem states
Z
Z
d2
3
d r r n(r, t) = − d3 r n(r, t)∇vext (r) .
dt2
(42)
(43)
In the same way we can write Ehrenfest’s theorem for the Kohn-Sham system
Z
Z
d2
3
d
r
r
n(r,
t)
=
−
d3 r n(r, t)∇vKS (r) .
(44)
dt2
By the construction of the Kohn-Sham system, the interacting density is
equal to the Kohn-Sham density. We can therefore equate the right-hand
sides of Eq. (43) and (44), and arrive at
Z
Z
3
d r n(r, t)∇vext (r) = d3 r n(r, t)∇vKS (r, t) .
(45)
If we Rnow insert the definition of the Kohn-Sham potential, Eq. 31, and note
that d3 r n(r, t)∇vHartree (r) = 0, we obtain the condition
Z
Z
d3 r n(r, t)∇vxc (r, t) = d3 r n(r, t)F xc (r, t) = 0 ,
(46)
Time-dependent Density Functional Theory
13
i.e. the total xc force of the system is zero. This condition reflects Newton’s
third law: The xc effects are only due to internal forces, the Coulomb interaction among the electrons, and should not give rise to any net force on the
system.
A functional that takes into account these exact constraints can be constructed[14]. The condition (46) is simply ensured by the expression
Z
1
∇ dt0 Πxc (n(r, t0 ), t − t0 ) .
(47)
F xc (r, t) =
n(r, t)
The function Πxc is a pressure-like scalar memory function of two variables.
In practice, Πxc is fully determined by requiring it to reproduce the scalar
linear response of the homogeneous electron gas. Expression (47) is clearly
non-local in the time-domain but still local in the spatial coordinates. From
the previous considerations it is clear that it must violate Galilean invariance.
To correct this problem we use a concept borrowed from hydrodynamics. It
is assumed that, in the electron liquid, memory resides not with each fixed
point r, but rather within each separate “fluid element”. Thus the element
which arrives at location r at time t “remembers” what happened to it at
earlier times t0 when it was at locations R(t0 |r, t), different from its present
location r. The trajectory, R, can be determined by demanding that its time
derivative equals the fluid velocity
∂
j(R, t0 )
R(t0 |r, t) =
,
0
∂t
n(R, t0 )
(48)
with the boundary condition
R(t|r, t) = r .
We then correct the Eq. (47) by evaluating n at point R
Z
1
F xc (r, t) =
∇ dt0 Πxc (n(R, t0 ), t − t0 ) .
n(r, t)
(49)
(50)
Finally, an expression for vxc can be obtained by direct integration of F xc
(see [14] for details).
2.5
Numerical considerations
As mentioned before, the solution of the time-dependent Kohn-Sham equations is an initial value problem. At t = t0 the system is in some initial state
described by the Kohn-Sham orbitals ϕi (r, t0 ). In most cases the initial state
will be the ground state of the system (i.e., ϕi (r, t0 ) will be the solution of
the ground-state Kohn-Sham equations). The main task of the computational
physicist is then to propagate this initial state until some final time, tf .
14
Miguel A. L. Marques and E. K. U. Gross
The time-dependent Kohn-Sham equations can be rewritten in the integral form
ϕi (r, tf ) = Û (tf , t0 )ϕi (r, t0 ) ,
(51)
where the time-evolution operator, Û , is defined by
#
" Z 0
t
0
dτ ĤKS (τ ) .
Û(t , t) = T̂ exp −i
(52)
t
Note that ĤKS is explicitly time-dependent due to the Hartree and xc potentials. It is therefore important to retain the time-ordering propagator, T̂ ,
in the definition of the operator Û . The exponential in expression (52) is
clearly too complex to be applied directly, and needs to be approximated
in some suitable manner. To reduce the error in the propagation from t0
to tf , this large interval is usually split into smaller sub-intervals of length
∆ t. The wave-functions are then propagated from t0 → t0 + ∆ t, then from
t0 + ∆ t → t0 + 2∆ t and so on.
The simplest approximation to (56) is a direct expansion of the exponential in a power series of ∆ t
h
il
k
−iĤ(t + ∆ t/2)∆ t
X
Û (t + ∆ t, t) ≈
+ O(∆ tk+1 ) .
(53)
l!
l=0
Unfortunately, the expression (53) does not retain one of the most important properties of the Kohn-Sham time-evolution operator: unitarity. In other
words, if we apply Eq. (53) to a normalized wave-function the result will no
longer be normalized. This leads to an inherently unstable propagation. Several propagation methods that fulfill the condition of unitarity exist in the
market. We will briefly mention two of these: a modified Krank-Nicholson
scheme, and the split-operator method.
A modified Krank-Nicholson scheme This method is derived by imposing time-reversal symmetry to an approximate time-evolution operator.
It is clear that we can obtain the state at time t + ∆ t/2 either by forward
propagating the state at t by ∆ t/2, or by backward propagating the state at
t+∆t
ϕ(t + ∆ t/2) = Û (t + ∆ t/2, t)ϕ(t)
= Û (t − ∆ t/2, t + ∆ t)ϕ(t + ∆ t) .
(54)
This equality leads to
ϕ(t + ∆ t) = Û(t + ∆ t/2, t + ∆ t)Û (t + ∆ t/2, t)ϕ(t) ,
(55)
where we used the fact that the inverse of the time-evolution operator Û −1 (t+
∆t, t) = Û (t−∆t, t). To propagate a state from t to t+∆t we follow the steps:
Time-dependent Density Functional Theory
15
i) Obtain an estimate of the Kohn-Sham wave-functions at time t + ∆ t by
propagating from time t using a “low quality” formula for Û (t + ∆ t, t). The
expression (53) expanded to third or forth order is well suited for this purpose.
ii) With these wave-functions construct an approximation to Ĥ(t + ∆ t) and
to Û (t + ∆ t/2, t + ∆ t). iii) Apply Eq. (55). This procedure leads to a very
stable propagation.
The split-operator method In a first step we neglect the time-ordering in
Eq. (52), and approximate the integral in the exponent by a trapezoidal rule
h
i
h
i
Û (t + ∆ t, t) ≈ exp −iĤKS (t)∆ t = exp −i(T̂ + V̂KS )∆ t .
(56)
We note that the operators exp −iV̂KS ∆ t and exp −iT̂ ∆ t are diagonal
respectively in real and Fourier spaces, and therefore trivial to apply in those
spaces. It is possible to decompose the exponential (56) into a form involving
only these two operators. The two lowest order decompositions are
h
i
exp −i(T̂ + V̂KS )∆ t = exp −iT̂ ∆ t exp −iV̂KS ∆ t + O(∆ t2 ) (57)
and
h
i
∆t
∆t
exp −iV̂KS ∆ t exp −iT̂
exp −i(T̂ + V̂KS )∆ t = exp −iT̂
2
2
+O(∆ t3 ) .
(58)
For example, to apply the splitting (58) to ϕ(r, t) we start by Fourier
trans
∆t
forming the wave-function to Fourier space. We then apply exp −iT̂ 2 to
ϕ(k, t) and Fourier
transform
back the result to real space. We proceed by
applying exp −iV̂ ∆ t , Fourier transforming, etc. This method can be made
very efficient by the use of fast-Fourier transforms.
As a better approximation to the propagator (52) we can use a mid-point
rule to estimate the integral in the exponential
h
i
Û (t + ∆ t, t) ≈ exp −iĤKS (t + ∆ t/2)∆ t .
(59)
It can be shown that the same procedure described above can be applied
with only a slight modification: The Kohn-Sham potential has to be updated
after applying the first kinetic operator [21].
3
3.1
Linear response theory
Basic theory
In circumstances where the external time-dependent potential is small, it
may not be necessary to solve the full time-dependent Kohn-Sham equations.
16
Miguel A. L. Marques and E. K. U. Gross
Instead perturbation theory may prove sufficient to determine the behavior
of the system. We will focus on the linear change of the density, that allows
us to calculate, e.g., the optical absorption spectrum.
Let us assume that for t < t0 the time-dependent potential vTD is zero –
i.e. the system is subject only to the nuclear potential, v (0) – and furthermore
that the system is in its ground-state with ground-state density n(0) . At t0
we turn on the perturbation, v (1) , so that the total external potential now
consists of vext = v (0) + v (1) . Clearly v (1) will induce a change in the density.
If the perturbing potential is sufficiently well-behaved (like almost always in
physics), we can expand the density in a perturbative series
n(r, t) = n(0) (r) + n(1) (r, t) + n(2) (r, t) + · · ·
(60)
where n(1) is the component of n(r, t) that depends linearly on v (1) , n(2)
depends quadratically, etc. As the perturbation is weak, we will only be concerned with the linear term, n(1) . In frequency space it reads
Z
n(1) (r, ω) = d3 r0 χ(r, r 0 , ω)v (1) (r 0 , ω) .
(61)
The quantity χ is the linear density-density response function of the system. In other branches of physics it has other names, e.g., in the context of
many-body perturbation theory it is called the reducible polarization function. Unfortunately, the evaluation of χ through perturbation theory is a
very demanding task. We can, however, make use of TDDFT to simplify this
process.
We recall that in the time-dependent Kohn-Sham framework, the density
of the interacting system of electrons is obtained from a fictitious system of
non-interacting electrons. Clearly, we can also calculate the linear change of
density using the Kohn-Sham system
Z
(1)
(1)
n (r, ω) = d3 r0 χKS (r, r0 , ω)vKS (r 0 , ω) .
(62)
Note that the response function that enters Eq. (62), χKS , is the density response function of a system of non-interacting electrons and is, consequently,
much easier to calculate than the full interacting χ. In terms of the unperturbed stationary Kohn-Sham orbitals it reads
0
χKS (r, r , ω) = lim+
η→0
∞
X
jk
(fk − fj )
ϕj (r)ϕ∗j (r 0 )ϕk (r 0 )ϕ∗k (r)
,
ω − (j − k ) + iη
(63)
where fm is the occupation number of the m orbital in the Kohn-Sham
ground-state. Note that the Kohn-Sham potential, vKS , includes all powers
of the external perturbation due to its non-linear dependence on the density.
The potential that enters Eq. (62) is however just the linear change of vKS ,
Time-dependent Density Functional Theory
17
(1)
vKS . This latter quantity can be calculated explicitly from the definition of
the Kohn-Sham potential
(1)
(1)
(1)
vKS (r, t) = v (1) (r, t) + vHartree (r, t) + vxc
(r, t) .
(64)
The variation of the external potential is simply v (1) , while the change in the
Hartree potential is
Z
n(1) (r0 , t)
(1)
vHartree (r, t) = d3 r0
.
(65)
|r − r0 |
(1)
Finally vxc (r, t) is the linear part in n(1) of the functional vxc [n],
Z
Z
δvxc (r, t) (1) 0 0
(1)
n (r , t ) .
vxc
(r, t) = dt0 d3 r0
δn(r 0 , t0 )
(66)
It is useful to introduce the exchange-correlation kernel, fxc , defined by
fxc (rt, r0 t0 ) =
δvxc (r, t)
.
δn(r0 , t0 )
(67)
The kernel is a well know quantity that appears in several branches of theoretical physics. E.g., evaluated for the electron gas, fxc is, up to a factor,
the “local-field correction”; To emphasize the correspondence to the effective
interaction of Landau’s Fermi-liquid theory, fxc plus the bare Coulomb interaction is sometimes called the “effective interaction”, while in the theory
of classical liquids the same quantity is referred to as the Ornstein-Zernicke
function.
Combining the previous results, and transforming to frequency space we
arrive at:
Z
n(1) (r, ω) = d3 r0 χKS (r, r 0 , ω)v (1) (r 0 , ω)
(68)
Z
Z
1
0
+
f
(x,
r
,
ω)
n(1) (r0 , ω) .
+ d3 x d3 r0 χKS (r, x, ω)
xc
|x − r0 |
From Eq. (61) and Eq. (68) trivially follows the relation
χ(r, r0 , ω) = χKS (r, r 0 , ω) +
Z
Z
3
3 0
d x d x χ(r, x, ω)
(69)
1
+ fxc (x, x0 , ω) χKS (x0 , r 0 , ω) .
|x − x0 |
This equation is a formally exact representation of the linear density response
in the sense that, if we possessed the exact Kohn-Sham potential (so that we
could extract fxc ), a self-consistent solution of (69) would yield the response
function, χ, of the interacting system.
18
3.2
Miguel A. L. Marques and E. K. U. Gross
The xc kernel
As we have seen in the previous section, the main ingredient in linear response
theory is the xc kernel. fxc , as expected, is a very complex quantity that
includes – or, in other words, hides – all non-trivial many-body effects. Many
approximate xc kernels have been proposed in the literature over the past
years. The most ancient, and certainly the simplest is the ALDA kernel
ALDA
HEG
fxc
(rt, r 0 t0 ) = δ(r − r0 )δ(t − t0 ) fxc
(n)n=n(r,t) ,
(70)
where
d HEG
(n)
(71)
v
dn xc
is just the derivative of the xc potential of the homogeneous electron gas.
The ALDA kernel is local both in the space and time coordinates.
Another commonly used xc kernel was derived by Petersilka et al. in 1996,
and is nowadays referred to as the PGG kernel[22]. Its derivation starts from
a simple analytic approximation to the EXX potential. This approximation,
that is called the Slater approximation in the context of Hartree-Fock theory,
only retains the leading term in the expression for EXX, which reads
HEG
fxc
(n) =
vxPGG (r, t) =
occ
2
X
|ϕk (r, t)|
k
n(r, t)
[ux k (r, t) + c.c] .
(72)
Using the definition (67) and after some algebra we arrive at the final form
of the PGG kernel
Pocc
2
| k ϕk (r)ϕ∗k (r 0 )|
1
1
fxPGG (rt, r0 t0 ) = −δ(t − t0 )
.
(73)
2 |r − r 0 |
n(r)n(r 0 )
As in the case of the ALDA, the PGG kernel is local in time .
Noticing the crudeness of the ALDA, especially the complete neglect of
any frequency dependence, one could expect it to yield very inaccurate results in most situations. Surprisingly, this is not the case as we will show in
section 4. To understand this numerical evidence, we have to take a step back
and study more thoroughly the properties of the xc kernel for the homogeneous electron gas[1].
HEG
In this simple system fxc
only depends on r − r 0 and on t − t0 , so
it is convenient to work in Fourier space. Our knowledge of the function
HEG
fxc
(q, ω) is quite limited. Several of its exact features can nevertheless
be obtained through analytical manipulations. The zero frequency and zero
momentum limit is given by
HEG
lim fxc
(q, ω = 0) =
q→0
d2 HEG nxc (n) ≡ f0 (n) ,
dn2
(74)
Time-dependent Density Functional Theory
19
where HEG
xc , the xc energy per particle of the homogeneous electron gas, is
known exactly from Monte-Carlo calculations[23]. Also the infinite frequency
limit can be written as a simple expression
HEG
1 d
(n)
xc (n)
4 2 d HEG
xc
HEG
lim fxc (q, ω = ∞) = − n 3
+ 6 n3
(75)
q→0
5 dn
dn
n2/3
n1/3
≡ f∞ (n) .
From these two expression, one can prove that the zero frequency limit is
always smaller than the infinite frequency limit, and that both these quantities are smaller than zero (according to the best approximations known for
HEG
Exc
), i.e.
f0 (n) < f∞ (n) < 0
(76)
From the fact that fxc is a real function when written in real space and in
real time one can deduce the following symmetry relations
HEG
HEG
<fxc
(q, ω) = <fxc
(q, −ω)
(77)
HEG
HEG
=fxc
(q, ω) = −=fxc
(q, −ω) .
From causality follow the Kramers-Kronig relations:
Z ∞
HEG
(q, ω)
dω 0 =fxc
HEG
HEG
(78)
<fxc (q, ω) − fxc (q, ∞) = P
ω − ω0
−∞ π
Z ∞
HEG
HEG
(q, ω) − fxc
(q, ∞)
dω 0 <fxc
HEG
,
=fxc
(q, ω)
= −P
0
ω−ω
−∞ π
where P denotes the principal value of the integral. Note that, as the infinite
frequency limit of the xc kernel is different from zero, one has to subtract
HEG
fxc
(q, ∞) in order to apply the Kramers-Kronig relations.
Furthermore, by performing a perturbative expansion of the irreducible
polarization to second order in e2 , one finds
HEG
lim =fxc
(q = 0, ω) = −
ω→∞
23π
.
15ω 3/2
(79)
The real part can be obtained with the help of the Kramers-Kronig relations
HEG
lim <fxc
(q = 0, ω) = f∞ (n) +
ω→∞
23π
.
15ω 3/2
(80)
It is possible to write a analytical form for the long-wavelength limit of the
imaginary part of fxc that incorporates all these exact limits[24]
HEG
=fxc
(q = 0, ω) ≈
α(n)ω
5
(1 + β(n)ω 2 ) 4
.
(81)
20
Miguel A. L. Marques and E. K. U. Gross
0
0
-1
-2
rs = 4
-5
Im fxc (a.u.)
Re fxc (a.u.)
rs = 2
-10
rs = 2
-3
-4
rs = 4
-5
-6
-7
-15
0
0.5
1
1.5
ω (a.u.)
2
2.5
3
0
1
0.5
1.5
ω (a.u.)
2
2.5
3
HEG
Fig. 1. Real and imaginary part of the parametrization for fxc
. Figure reproduced
from Ref. [25].
The coefficients α and β are functions of the density, and can be determined
uniquely by the zero and high frequency limits. A simple calculation yields
5
α(n) = −A [f∞ (n) − f0 (n)] 3
(82)
β(n) = B [f∞ (n) − f0 (n)] ,
(83)
4
3
where A, B > 0 and independent of n. Once again, by applying the KramersHEG
Kronig relations we can obtain the corresponding real part of fxc
√
2 2α
1
HEG
(84)
<fxc (q = 0, ω) = f∞ + √ 2 2E √
π βr
2
1+r
1−r
1−r 1
1+r 1
−
,
−
Π
,√
Π
,√
2
2
2
2
2
2
p
where r = 1 + βω 2 and E and Π are the elliptic integrals of second and
HEG
third kind. In Fig. 1 we plot the real and imaginary part of fxc
for two
different densities (rs = 2 and rs = 4, where rs is the Wigner-Seitz radius,
1/n = 4πrs3 /3). The ALDA corresponds to approximating these curves by
their zero frequency value. For very low frequencies, the ALDA is naturally a
good approximation, but at higher frequencies it completely fails to reproduce
HEG
the behavior of fxc
.
To understand how the ALDA can yield such good excitation energies,
albeit exhibiting such a mediocre frequency dependence, we will look at a specific example, the process of photo-absorption by an atom. At low excitation
frequencies, we expect the ALDA to work. As we increase the laser frequency,
we start exciting deeper levels, promoting electrons from the inner shells of
the atom to unoccupied states. The atomic density increases monotonically
as we approach the nucleus. fxc corresponding to that larger density (lower
Time-dependent Density Functional Theory
21
rs ) has a much weaker frequency dependence, and is much better approximated by the ALDA than the low density curve. In short, by noticing that
high frequencies are normally related to high densities, we realize that for
practical applications the ALDA is often a reasonably good approximation.
One should however keep in mind that these are simple heuristic arguments
that may not hold in a real physical system.
4
4.1
Excitation energies
DFT techniques to calculate excitations
In this section we will present a short overview of the several techniques to
calculate excitation energies that have appeared in the context of DFT over
the past years. Indeed quite a lot of different approaches have been tried.
Some are more or less ad hoc, others rely on a solid theoretical basis. Moreover, the degree of success varies considerably among the different techniques.
The most successful of all is certainly TDDFT that has become the de facto
standard for the calculation of excitations for finite systems. We will leave
the discussion of excitation energies in TDDFT to the following sections, and
concentrate for now on the “competitors”. The first group of methods is based
on a single determinant calculation, i.e. only one ground-state like calculation is performed, subject to the restriction that the Kohn-Sham occupation
numbers are either 0 or 1.
As a first approximation to the excitation energies, one can simply take
the differences between the ground-state Kohn-Sham eigenvalues. This procedure, although not entirely justifiable, is often used to get a rough idea of
the excitation spectrum. We stress that the Kohn-Sham eigenvalues (as well
as the Kohn-Sham wave-functions) do not have any physical interpretation.
The exception is the eigenvalue of the highest occupied state that is equal to
minus the ionization potential of the system[26].
The second scheme is based on the observation that the Hohenberg-Kohn
theorem and the Kohn-Sham scheme can be formulated for the lowest state
of each symmetry class[27]. In fact, the single modification to the standard
proofs is to restrict the variational principle to wave-functions of a specific
symmetry. The unrestricted variation will clearly yield the ground-state. The
states belonging to different symmetry classes will correspond to excited
states. The excitations can then be calculated by simple total-energy differences. This approach suffers from two serious drawbacks: i) Only the lowest
lying excitation for each symmetry class is obtainable. ii) The xc functional
that now enters the Kohn-Sham equations depends on the particular symmetry we have chosen. As specific approximations for a symmetry dependent
xc functional are not available, one is relegated to use ground-state functionals. Unfortunately the excitation energies calculated in this way are only of
moderate quality.
22
Miguel A. L. Marques and E. K. U. Gross
Another promising method was recently proposed by A. Görling[28]. The
so-called generalized adiabatic connection Kohn-Sham formalism is no longer
based on the Hohenberg-Kohn theorem but on generalized adiabatic connections associating a Kohn-Sham state with each state of the real system. This
formalism was later extended to allow for the proper treatment of symmetry of the Kohn-Sham states[29]. The quality of the results obtained so far
with this procedure varies: For alkali atoms the agreement with experimental
excitation energies is quite good[28], but for the Carbon atom and the CO
molecule the situation is considerably worse[29]. We note however that this
method is still in its infancy, so further developments can be expected in the
near future.
It is also possible to calculate excitation energies from the ground-state
energy functional. In fact, it was proved by Perdew and Levy[30] that “every
extremum density ni (r) of the ground-state energy functional Ev [n] yields the
energy Ei of a stationary state of the system.” The problem is that not every
excited-state density, ni (r), corresponds to an extremum of Ev [n], which
implies that not all excitation energies can be obtained from this procedure.
The last member of the first group of methods was proposed by Ziegler,
Rauk and Baerends in 1977[31] and is based on an idea borrowed from multiconfiguration Hartree-Fock. The procedure starts with the construction of
many-particle states with good symmetry, Ψi , by taking a finite superposition
of states
X
Ψi =
ciα Φα ,
(85)
α
where Φα are Slater determinants of Kohn-Sham orbitals, and the coefficients
ciα are determined from group theory. Through a simple matrix inversion we
can express the determinants as linear combinations of the many-body wavefunctions
X
Φβ =
aβj Ψj .
(86)
j
By taking the expectation value of the Hamiltonian in the state Φβ we arrive
at
X
hΦβ | Ĥ |Φβ i =
|aβj |2 Ej ,
(87)
j
where Ej is the energy of the many-body state Ψj . The “recipe” to calculate
excitation energies is then: a) Build Φβ from n Kohn-Sham orbitals (not
necessarily the lowest); b) Make an ordinary Kohn-Sham calculation for each
Φβ , and associate the corresponding total energy EβDFT with hΦβ | Ĥ |Φβ i;
c) Determine Ej by solving the system of linear equations, Eq. (87).
This methods works quite well in practice, and was frequently used in
quantum chemistry till the advent of TDDFT. We should nevertheless indicate two of its limitations: i) The decomposition (85) is not unique and the
system of linear equations can be under- or overdetermined. ii) The whole
procedure of the “recipe” is not rigorously founded.
Time-dependent Density Functional Theory
23
The next technique, the so-called ensemble DFT, makes use of fractional occupation numbers. Ensemble DFT, first proposed by Theophilou
in 1979[32], evolves around the concept of an ensemble. In the simplest case
it consists of a “mixture” of the ground state, Ψ1 , and the first excited state,
Ψ2 , described by the density matrix[33–35]
D̂ = (1 − ω) |Ψ1 i hΨ1 | + ω |Ψ2 i hΨ2 | ,
(88)
where the weight, ω, is between 0 and 1/2 (in this last case the ensemble
is called “equiensemble”). We can further define the ensemble energy and
density
E(ω) = (1 − ω)E1 + ω E2
nω (r) = (1 − ω)n1 (r) + ω n2 (r) .
(89)
(90)
At ω = 0 the ensemble energy clearly reduces to the ground-state energy.
Using the ensemble density it is possible to construct a DFT, i.e. to prove a
Hohenberg-Kohn theorem and construct a Kohn-Sham scheme. The main features of the Kohn-Sham scheme are: i) The one-body orbitals have fractional
occupations determined by the weight ω. ii) The xc functional depends on
the weight, Exc (ω). To calculate the excitation energies from ensemble DFT
we can follow two paths. The first involves obtaining the ground-state energy
and the ensemble energy for some fixed ω, from which the excitation energy
E1 − E2 trivially follows
E2 − E 1 =
E(ω) − E(0)
.
ω
(91)
The second path is obtained by taking the derivative of Eq. (89)
dE(ω)
= E2 − E1 .
dω
(92)
It is then possible to prove
E2 − E 1 =
+1
N
ω
−
N
ω
∂Exc (ω) +
.
∂ω n=nω
(93)
Naturally, we need approximations to the xc energy functional, Exc (ω). An
ensemble LDA was developed for the equiensemble by W. Kohn in 1986[36],
by treating the ensemble as a reminiscent of a thermal ensemble. He then related Exc (ω) to the finite temperature xc energy of the homogeneous electron
gas by equating the entropies of both systems. Unfortunately, the results obtained with this functional were not very encouraging. A promising approach,
recently proposed, is the use of orbital functionals within an ensemble OEP
method[37,38].
24
Miguel A. L. Marques and E. K. U. Gross
4.2
Full solution of the Kohn-Sham equations
One of the most important uses of TDDFT is the calculation of photoabsorption spectra. This problem can be solved in TDDFT either by propagating the time-dependent Kohn-Sham equations[39] or by using linearresponse theory. In this section we will be concerned by the former, relegating
the latter to the next section.
Let ϕ̃j (r) be the ground-state Kohn-Sham wave-functions for the system
under study. We prepare the initial state for the time-dependent propagation
by exciting the electrons with the electric field v(r, t) = −k0 xν δ(t), where
xν = x, y, z. The amplitude k0 must be small in order to keep the response
of the system linear and dipolar. Through this prescription all frequencies of
the system are excited with equal weight. At t = 0+ the initial state for the
time evolution reads
( Z +
)
0
h
i
dt ĤKS − k0 xν δ(t) ϕ̃j (r)
ϕj (r, t = 0+ ) = T̂ exp −i
0
= exp [ik0 xν ] ϕ̃j (r) .
(94)
The Kohn-Sham orbitals are then further propagated during a finite time.
The dynamical polarizability can be obtained from
Z
1
αν (ω) = −
d3 r xν δn(r, ω) .
(95)
k
In the last expression δn(r, ω) stands for the Fourier transform of n(r, t) −
ñ(r), where ñ(r) is the ground-state density of the system. The quantity that
is usually measured in experiments, the photo-absorption cross-section, is
essentially proportional to the imaginary part of the dynamical polarizability
averaged over the three spatial directions
σ(ω) =
4πω 1 X
=
αν (ω) ,
c 3
ν
(96)
where c stands for the velocity of light. Although computationally more demanding than linear-response theory, this method is very flexible, and is
easily extended to incorporate temperature effects, non-linear phenomena,
etc. Note also that this approach only requires an approximation to the xc
potential.
To illustrate the method, we present, in Fig. 2, the excitation spectrum
of benzene calculated within the LDA/ALDA3 . The agreement with experiment is quite remarkable, especially when looking at the π → π ∗ resonance at
3
We will use the notation “A/B” consistently throughout the rest of this article to
indicate that the ground-state xc potential used to calculate the initial state was
“A”, and that this state was propagated with the time-dependent xc potential
“B”. In the case of linear-response theory, “B” will denote the xc kernel.
Time-dependent Density Functional Theory
25
3
Averaged
z - direction
Experimental
Dipole Strength (1/eV)
2.5
2
1.5
1
0.5
5
10
20
15
Energy (eV)
25
30
35
Fig. 2. Optical absorption of the benzene molecule. Experimental results from
Ref. [40]. Figure reproduced from Ref. [41].
around 7 eV. The spurious peaks that appear in the calculation at higher energies are artifacts caused by an insufficient treatment of the unbound states.
We furthermore observe that such good results are routinely obtained when
applying the LDA/ALDA to several finite systems, from small molecules to
metallic clusters and biological systems.
4.3
Excitations from linear-response theory
The first self-consistent solution of the linear response equation (69) was
performed by Zangwill and Soven in 1980 using the LDA/ALDA [42]. Their
results for the photo-absorption spectrum of Helium for energies just above
the ionization threshold are shown in Fig. 3. Once more the theoretical curve
compares very well to experiments.
Unfortunately, a full solution of Eq. (69) is still quite difficult numerically.
Besides the large effort required to solve the integral equation, we need as
an input the non-interacting response function. To obtain this quantity it is
usually necessary to perform a summation over all states, both occupied and
unoccupied [cf. Eq. (63)]. Such summations are sometimes slowly convergent
and require the inclusion of many unoccupied states. There are however approximate frameworks that circumvent the solution of Eq. (69). The one we
will present in the following was proposed by Petersilka et al. [22].
26
Miguel A. L. Marques and E. K. U. Gross
35
30
σ(ω) (Mb)
25
20
15
10
5
0
5
6
7
8
hω (Ry)
9
10
11
Fig. 3. Total photo-absorption cross-section of Xenon versus photon energy in the
vicinity of the 4d threshold. The solid line represents TDDFT calculations and the
crosses are the experimental results of Ref. [43]. Figure adapted from Ref. [42].
The density response function can be written in the Lehmann representation
X h0| ρ̂(r) |mi hm| ρ̂(r 0 ) |0i h0| ρ̂(r0 ) |mi hm| ρ̂(r) |0i ,
−
χ(r, r0 , ω) = lim+
ω − (Em − E0 ) + iη
ω + (Em − E0 ) + iη
η→0
m
(97)
where |mi is a complete set of many-body states with energies Em . From this
expansion it is clear that the full response function has poles at frequencies
that correspond to the excitation energies of the interacting system
Ω = E m − E0 .
(98)
As the external potential does not have any special pole structure as a function of ω, Eq. (61) implies that also n(1) (r, ω) has poles at the excitation
energies, Ω. On the other hand, χKS has poles at the excitation energies of
the non-interacting system, i.e. at the Kohn-Sham orbital energy differences
j − k [cf. Eq. (63)].
By rearranging the terms in Eq. (68) we obtain the fairly suggestive equation
Z
Z
d3 r0 [δ(r − r 0 ) − Ξ(r, r0 , ω)] n(1) (r 0 , ω) = d3 r0 χKS (r, r 0 , ω)v (1) (r 0 , ω)
(99)
where the function Ξ is defined by
Z
Ξ(r, r 0 , ω) = d3 r00 χKS (r, r 00 , ω)
1
+ fxc (r 00 , r0 , ω)
|r00 − r 0 |
(100)
Time-dependent Density Functional Theory
27
As noted previously, in the limit ω → Ω the linear density n(1) has a pole,
while the right-hand side of Eq. (100) remains finite. For the equality (100)
to hold, it is therefore required that Ξ has zero eigenvalues at the excitation
energies Ω, i.e. λ(ω) → 1 when ω → Ω, where λ(ω) is the solution of the
eigenvalue equation
Z
d3 r 0 Ξ(r, r0 , ω)ξ(r 0 , ω) = λ(ω)ξ(r, ω) .
(101)
This is a rigorous statement, that allows the determination of the excitation
energies of the systems from the knowledge of χKS and fxc . It is possible
to transform this equation into another eigenvalue equation having the true
excitation energies of the system, Ω, as eigenvalues[44]. We start by defining
the quantity
Z
Z
1
3 00 ∗ 00
00
3 0
00
0
ζjk (ω) = d r d r ϕj (r )ϕk (r )
+ fxc (r , r , ω) ξ(r 0 , ω) .
|r 00 − r0 |
(102)
With the help of ζjk Eq. (101) can be rewritten in the form
X (fk − fj ) ϕj (r)ϕ∗ (r)
k
jk
ω − (j − k ) + iη
ζjk (ω) = λ(ω)ξ(r, ω) .
(103)
By solving this equation for ξ(r, ω) and inserting the result into Eq. (102),
we arrive at
X
j 0 k0
Mjk,j 0 k0
ζj 0 k0 (ω) = λ(ω)ζjk (ω)
ω − (j 0 − k0 ) + iη
(104)
where we have defined the matrix element
Z
Z
Mjk,j 0 k0 (ω) = (fk0 − fj 0 ) d3 r d3 r0 ϕ∗j (r)ϕk (r)ϕj (r0 )ϕ∗k (r 0 ) ×
1
0
+ fxc (r, r , ω) .
(105)
|r − r0 |
Introducing the new eigenvector
βjk =
ζjk (Ω)
,
Ω − (j 0 − k0 )
(106)
taking the η → 0 limit, and by using the condition λ(Ω) = 1, it is straightforward to recast Eq. (104) into the eigenvalue equation
X
(107)
[δjj 0 δkk0 (j 0 − k0 ) + Mjk,j 0 k0 (Ω)] βj 0 k0 = Ωβjk .
j 0 k0
28
Miguel A. L. Marques and E. K. U. Gross
It is also possible to derive an operator whose eigenvalues are the square
of the true excitation energies, thereby reducing the dimension of the matrix equation (107)[45]. The oscillator strengths can be obtained from the
eigenfunctions of the operator.
The eigenvalue equation, Eq. (107), can be solved in several different ways.
For example, it is possible to expand all quantities in a suitable basis and
solve numerically the resulting matrix-eigenvalue equation. As an alternative,
we can perform a Laurent expansion of the response function around the
excitation energy
χKS (r, r0 , ω) = lim
η→0+
ϕj0 (r)ϕ∗j0 (r 0 )ϕk0 (r 0 )ϕ∗k0 (r)
+ higher order .
ω − (j0 − k0 ) + iη
(108)
By neglecting the higher-order terms, a simple manipulation of Eq. (101)
yields the so-called single-pole approximation (SPA) to the excitation energies
Ω = ∆ + K(∆) ,
(109)
where ∆ is the difference between the Kohn-Sham eigenvalue of the unoccupied orbital j0 and the occupied orbital k0 ,
∆ = j0 − k0
and K is a correction given by
Z
Z
K(∆) = 2< d3 r d3 r0 ϕj0 (r)ϕ∗j0 (r0 )ϕk0 (r 0 )ϕ∗k0 (r) ×
1
0
+
f
(r,
r
,
∆)
.
xc
|r − r0 |
(110)
(111)
Although not as precise as the direct solution of the eigenvalue equation,
Eq. (107), this formula provides us with a simple and fast way to calculate
the excitation energies.
To assert how well this approach works in practice we list, in Tab. 1, the
1
S →1 P excitation energies for several atoms[22]. Surprisingly perhaps, the
eigenvalue differences, ∆, are already of the proper order of magnitude. For
other systems they can be even much closer (cf. Tab. 3). Adding the correction K then brings the numbers indeed very close to experiments for both xc
functionals tried. We furthermore notice that the EXX/PGG functional gives
clearly superior results than the LDA/ALDA. This is related to the different quality of the unoccupied states generated with the two ground-state xc
functionals. The unoccupied states typically probe the farthest regions from
the system, where the LDA potential exhibits severe deficiencies (as previously mentioned in section 2.4). As the EXX potential does not suffer from
this problem, it yields better unoccupied orbitals and consequently better
excitation energies.
Time-dependent Density Functional Theory
Atom
Be
Mg
Ca
Zn
Sr
Cd
∆LDA ΩLDA/ALDA
0.129
0.200
0.125
0.176
0.088
0.132
0.176
0.239
0.082
0.121
0.152
0.214
∆EXX ΩEXX/PGG
0.130
0.196
0.117
0.164
0.079
0.117
0.157
0.211
0.071
0.105
0.135
0.188
29
Ωexp
0.194
0.160
0.108
0.213
0.099
0.199
Table 1. 1 S → 1 P excitation energies for selected atoms. Ωexp denotes the experimental results from [46]. All energies are in Hartrees. Table adapted from Ref. [22].
In Tab. 2 we show the excitation energies of the CO molecule. This case is
slightly more complicated than the previous example due to the existence of
degeneracies in the eigenspectrum of the CO molecule. Although the KohnSham eigenvalue differences are equal for all transitions involving degenerate
states, the true excitation energies depend on the symmetry of the initial and
final many-body states. As is clearly seen from the table, this splitting of the
excitations is correctly described by the correction factor, K.
State
1
A Π
a 3Π
B 1Σ+
b 3Σ+
I 1Σ−
e 3Σ−
a’ 3 Σ +
D 1∆
d 3∆
c 3Π
E 1Π
SPA
full
∆LDA ΩLDA/ALDA
ΩLDA/ALDA
exp.
5σ → 2π 0.2523
5σ → 6σ 0.3332
1π → 2π 0.3626
4σ → 2π 0.4388
1π → 6σ 0.4436
0.3268
0.2238
0.3389
0.3315
0.3626
0.3626
0.3181
0.3812
0.3404
0.4204
0.4435
0.3102
0.2214
0.3380
0.3316
0.3626
0.3626
0.3149
0.3807
0.3396
0.4202
0.4435
0.3127
0.2323
0.3962
0.3822
0.3631
0.3631
0.3127
0.3759
0.3440
0.4245
0.4237
SPA
Table 2. Excitation energies for the CO molecule. ΩLDA/ALDA
are the LDA/ALDA
SPA
excitation energies obtained from Eq. (109), and ΩLDA/ALDA are obtained from
the solution of Eq. (107) neglecting continuum states. “exp” are the experimental
results from Ref. [47]. All energies are in Hartrees. Table reproduced from Ref. [48].
We remember that several approximations have been made to produce the
previous results. First a static Kohn-Sham calculation was performed with
an approximate vxc . Then the resulting eigenfunctions and eigenvalues were
used in Eq. (109) to obtain the excitation energies. In the last step, we used
an approximate form for the xc kernel, fxc , and we neglected the higher order
terms in the Laurent expansion of the response functions. To assert which of
30
Miguel A. L. Marques and E. K. U. Gross
these approximations is more important, we can look at the lowest excitation
energies of the He atom. For this simple system the exact stationary KohnSham potential is known[49], so we can eliminate the first source of error. We
can then test different approximations for fxc , both by performing the singlepole approximation or not. The results are summarized in Tab. 3. We first
note that the quality of the results is almost insensitive to the xc kernel used.
Both using the ALDA or the PGG yield the same mean error. This statement
seems to hold not only for atoms but also to molecular systems[50]. From the
table it is also clear that the SPA is an excellent approximation and that the
calculated excitation energies are in very close agreement to the exact values.
Why, and under which circumstances this is the case is discussed in detail
in Refs. [51,52]. This leads us to conclude that the crucial approximation to
obtain excitation energies in TDDFT is the choice of the static xc potential
used to calculate the Kohn-Sham eigenfunctions and eigenvalues.
State k0 → j0 ∆KS
23 S 1s → 2s 0.7460
21 S
33 S 1s → 3s 0.8392
31 S
43 S 1s → 4s 0.8688
41 S
23 P 1s → 2p 0.7772
21 P
33 P 1s → 3s 0.8476
31 P
43 P 1s → 4s 0.8722
41 P
Mean abs. dev.
Mean % error
exact/ALDA (xc)
SPA
full
0.7357 0.7351
0.7718 0.7678
0.8366 0.8368
0.8458 0.8461
0.8678 0.8679
0.8714 0.8719
0.7702 0.7698
0.7764 0.7764
0.8456 0.8457
0.8483 0.8483
0.8714 0.8715
0.8726 0.8726
0.0011 0.0010
0.15% 0.13%
exact/PGG
SPA full
0.7232 0.7207
0.7687 0.7659
0.8337 0.8343
0.8448 0.8450
0.8667 0.8671
0.8710 0.8713
0.7693 0.7688
0.7850 0.7844
0.8453 0.8453
0.8500 0.8501
0.8712 0.8713
0.8732 0.8733
0.0010 0.0010
0.13% 0.13%
exact
0.7285
0.7578
0.8350
0.8425
0.8672
0.8701
0.7706
0.7799
0.8456
0.8486
0.8714
0.8727
Table 3. Comparison of the excitation energies of neutral Helium, calculated from
the exact xc potential[49] by using approximate xc kernels. SPA stands for “single pole approximations”, while “full” means the complete solution of Eq. (107).
The exact values are from a non-relativistic variational calculation[53]. The mean
absolute deviation and mean percentage errors also include the transitions from
the 1s until the 9s and 9p states. All energies are in Hartrees. Table adapted from
Ref. [17].
4.4
When does it not work?
In the previous sections we showed the results of several TDDFT calculations, most of them agreeing quite well with experiment. Clearly no physical
Time-dependent Density Functional Theory
31
theory works for all systems and situations, and TDDFT is not an exception.
It is the purpose of this section to show some examples where the theory does
not work. However, before proceeding with our task, we should specify what
we mean by “failures of TDDFT”. TDDFT is an exact reformulation of the
time-dependent many-body Schrödinger equation – it can only fail in situations where quantum-mechanics also fails. The key approximation made in
practical applications is the approximation for the xc potential. Errors in the
calculations should therefore be imputed to the functional used. As a large
majority of TDDFT calculations use the ALDA or the adiabatic GGA, we will
be mainly interested in the errors caused by these approximate functional.
Furthermore, and as we already mentioned in the previous section, there are
usually two sources for the errors in the calculation: i) The functional used
to obtain the Kohn-Sham ground-state; ii) The approximate time-dependent
xc potential. In any discussion on the errors of TDDFT the effects of these
two sources have to be clearly separated. With these arguments in mind let
us then proceed.
Our first example is the calculation of optical properties of long conjugated molecular chains[54]. Using the local or gradient-corrected approximations can give overestimations of several orders of magnitude. The problem
is related to a non-local dependence of the xc potential: In a system with
an applied electric field, the exact xc potential develops a linear part that
counteracts the applied field[54,55]. This term is completely absent in both
the LDA and the GGA, but is present in more non-local functionals like the
EXX.
A related problem occurs in solids[56]. In fact, TDLDA does not work
properly for the calculation of excitations of non-metallic solids, especially in
systems like wide-band gap semiconductors. For infinite systems, the Coulomb
potential is (in momentum space) 4π/q 2 . It is then clear from the response
equation (69) that if fxc is to correct the non-interacting response for q → 0 it
will have to contain a term that behaves asymptotically as 1/q 2 when q → 0.
This is not the case for the local or gradient-corrected approximations. Several attempts have been done to correct this problem from which we mention
Refs. [57–60].
As other faults of the TDLDA we can mention the incorrect excitation
energies of the stretched H2 molecule[61], the large error in the calculation
of singlet-triplet separation energies[62], the underestimate of the onset of
absorption for some clusters[50], etc. Despite these failures, we would like to
emphasize that TDLDA does work very well for the calculation of excitations
in a large class of systems.
32
Miguel A. L. Marques and E. K. U. Gross
5
Atoms and molecules in strong laser fields
5.1
What is a “strong” laser?
Before discussing the behavior of atoms and molecules in strong laser fields,
we have to specify what the adjective “strong” means in this context. The
electric field that an electron feels in a Hydrogen atom, at the distance of one
Bohr from the nucleus, is
E=
1 e
= 5.1 × 109 V/m .
4π0 a20
(112)
The laser intensity that corresponds to this field is given by
I=
1
0 cE 2 = 3.51 × 1016 W/cm2 .
2
(113)
We can clearly consider a laser to be “strong” when its intensity becomes
comparable to (113). In this regime, perturbation theory is no longer applicable, and the theorist has to resort to non-perturbative methods. When
approaching these high intensities, a wealth of non-linear phenomena appear,
like multi-photon ionization, above threshold ionization (ATI), high harmonic
generation, etc.
The fact that allowed systematic investigation of these high-intensity phenomena was the remarkable evolution in laser technology during the past four
decades. Through a series of technological breakthroughs, scientists were able
to boost the peak intensity of pulsed lasers from 109 W/cm2 in the 1960s, to
more than 1021 W/cm2 of the current systems – 12 orders of magnitude! Besides this increase in laser intensity, very short pulses – sometimes of the order
of hundreds of attoseconds (1 as = 10−18 s) – became available at ultraviolet or soft X-ray frequencies [63,64]. In the present context we are concerned
mainly with intensities in the range 1013 −1016 W/cm2 . For higher intensities
many-body effects associated with the electron-electron interaction – which
are the main interest of DFT – become less and less important due to the
strongly dominant external field.
TDDFT is a tool particularly suited for the study of systems under the
influence of strong lasers. We recall that the time-dependent Kohn-Sham
equations yield the exact density of the system, including all non-linear effects. To simulate laser induced phenomena it is customary to start from the
ground-state of the system, which is then propagated under the influence of
the potential
vTD (r, t) = Ef (t)z sin(ωt) .
(114)
vTD describes a laser of frequency ω and amplitude E 4 . The function f (t),
typically a Gaussian or the square of a sinus, defines the temporal shape
4
The amplitude is related to the laser intensity by the relation I = 21 0 cE 2 .
Time-dependent Density Functional Theory
33
-5
|d(ω)|
2
10
10
-10
10
-15
10
20
30
40
Harmonic Order
50
60
Fig. 4. Harmonic spectrum for He at λ = 616 nm and I = 3.5 × 1014 W/cm2 . The
squares represent experimental data taken from Ref. [65] normalized to the value
of the 33rd harmonic of the calculated spectrum. Figure reproduced from Ref. [66].
of the laser pulse. From the time-dependent density it is then possible to
calculate the photon spectrum using the relation
2
σ(ω) ∝ |d(ω)| ,
(115)
where d(ω) is the Fourier transform of the time-dependent dipole of the system
Z
d(ω) = d3 r z n(r, t) .
(116)
Other observables, such as the total ionization yield or the ATI spectrum,
are much harder to calculate within TDDFT. Even though these observables
(as all others) are functionals of the density by virtue of the Runge-Gross
theorem, the explicit functional dependence is unknown and has to be approximated.
5.2
High-harmonic generation
If we shine a high-intensity laser onto an atom (or a molecule, or even a surface), an electron may absorb several photons and then return to its groundstate by emitting a single photon. The photon will have a frequency which
is an integer multiple of the external laser frequency. This process, known as
high-harmonic generation, has received a great deal of attention from both
theorists and experimentalists. As the outgoing high-energy photons maintain
a fairly high coherence, they can be used as a source for X-ray lasers.
A typical high-harmonic spectrum is shown in Fig. 4 for the Helium atom.
The squares represent experimental data taken from Ref. [65], and the solid
34
Miguel A. L. Marques and E. K. U. Gross
10
10
|d(ω)|
2
10
10
10
10
10
10
-1
13
-3
2
I = 2.0 10 W/cm
13
2
I = 4.0 10 W/cm
13
2
I = 8.0 10 W/cm
14
2
I = 1.5 10 W/cm
14
2
I = 2.5 10 W/cm
14
2
I = 3.2 10 W/cm
-5
-7
-9
-11
-13
-15
-17
10 0
0.5
1
2
1.5
Frequency ω (a.u.)
2.5
3
Fig. 5. Harmonic spectra of Hydrogen at a laser wavelength of λ = 1064 nm for
various laser intensities. Figure reproduced from Ref. [67].
line was obtained from a calculation using the EXX/EXX functional[66]. The
spectrum consists of a series of peaks, first decreasing in amplitude and then
reaching a plateau that extends to very high frequency. The peaks are placed
at the odd multiples of the external laser frequency (the even multiples are
dipole forbidden by symmetry). We note that any approach based on perturbation theory would yield a harmonic spectrum that decays exponentially, i.e.
such a theory could never reproduce the measured peak intensities. TDDFT,
on the other hand, gives a quite satisfactory agreement with experiment.
As mentioned before, high-harmonics can be used as a source of soft Xray lasers. For such purpose, one tries to optimize the laser parameters, the
frequency, intensity, etc., in order to increase the intensity of the emitted
harmonics, and to extend the plateau the farthest possible. By performing
“virtual experiments”, TDDFT can be once more used to tackle such important problem. As an illustration, we show, in Fig. 5, the result of irradiating
a Hydrogen atom with lasers of the same frequency but with different intensities. For clarity, we only show the position of the peaks, and the points
were connected by straight lines. As we increase the intensity of the laser,
the amplitude of the harmonics also increases, until reaching a maximum
at I = 1.5 × 1014 W/cm2 . A further increase of the intensity will, however,
decrease the produced harmonics. This reflects the two competing processes
that happen upon multiple absorption of photons: The electron can either
ionize, or fall back into the ground-state emitting a highly energetic photon.
Beyond a certain threshold intensity the ionization channel begins to predominate, thereby reducing the production of harmonics. Other laser parameters,
like the intensity, or the spectral composition of the laser, are also found to
influence the generation of high-harmonics in atoms[66,67].
Time-dependent Density Functional Theory
5.3
35
Multi-photon ionization
v(x)
v(x)
x
x
−Ip
−Ip
(a)
v(x)
x
−Ip
(b)
(c)
Fig. 6. Ionization in strong laser fields: (a) Multi-photon ionization; (b) Tunneling;
(c) Over the barrier.
To better understand the process of ionization of an atom in strong laser
fields, it is convenient to resort to a simple quasi-static picture. In Fig. 6
we have depicted a one-electron atom at a time t after the beginning of the
laser pulse. The dashed line represents the laser potential felt by the electron
and the solid line the total (i.e. the nuclear plus the laser) potential. Three
different regimes of ionizations are governed by the Keldish parameter,
γ=
ω
.
E
(117)
At low intensities (I < 1014 W/cm2 , γ 1) the electron has to absorb several
photons before leaving the atom. This is the so-called multi-photon ionization
regime. At higher intensities (I ≤ 1015 W/cm2 , γ ≈ 1) we enter the tunneling
regime. If we further increase the strength of the laser field (I > 1016 W/cm2 ,
γ 1), then the electron can simply pass over the barrier.
The measured energy spectrum of the outgoing photo-electrons is called
the above threshold ionization (ATI) spectrum[68]. As the electron can absorb
more photons than the necessary to escape the atom, an ATI spectrum will
consist of a sequence of equally spaced peaks at energies
E = (n + s)ω − Ip
(118)
where n is a natural integer, s is the minimum integer such that s ω − Ip > 0,
and Ip denotes the ionization potential of the system.
Another interesting observable is the number of outgoing charged atoms
as a function of the laser intensity. The two sets of points in Fig. 7 represent the yield of singly ionized and doubly ionized Helium. The solid curve
on the right is the result of a calculation assuming a sequential mechanism
for the double ionization of Helium, i.e., the He2+ is generated by first removing one electron from He, and then a second from He+ . Strikingly, this
36
Miguel A. L. Marques and E. K. U. Gross
Fig. 7. Measured He+ and He2+ yields as a function of the laser intensity. The
solid curve on the right is the calculated sequential He2+ yield. Figure reproduced
from Ref. [69].
naı̈ve sequential mechanism is wrong by six orders of magnitude for some
intensities.
Similar experimental results were found for a variety of molecules. Furthermore, in these more complex systems, the coupling of the nuclear and
the electronic degrees of freedom gives rise to new physical phenomena. As
an illustrative example of such phenomena, we refer the so-called ionization
induced Coulomb explosion[70].
5.4
Ionization yields from TDDFT
It is apparent from Fig. 7 that a simple sequential mechanism is insufficient
to describe the double ionization of Helium. In this section we will show how
one can try to go beyond this simple picture with the use of TDDFT[71].
To calculate the Helium yields we invoke a geometrical picture of ionization. We divide the three-dimensional space, 3 , into a (large) box, A,
containing the Helium atom, and its complement, B = 3 \A. Normalization of the (two-body) wave function of the Helium atom, Ψ (r 1 , r 2 , t), then
implies
Z Z
Z Z
2
2
1=
d3 r1 d2 r2 |Ψ (r 1 , r2 , t)| + 2
d3 r1 d2 r2 |Ψ (r 1 , r2 , t)| (119)
A A
A B
Time-dependent Density Functional Theory
+
Z Z
37
2
d3 r1 d2 r2 |Ψ (r 1 , r 2 , t)| ,
B B
where the subscript “X” has the meaning that the space integral is only over
region X. A long time after the end of the laser excitation, we expect that
all ionized electrons are in region B. This implies that the first term in the
right-hand side of Eq. (119) measures the probability that an electron remains
close to the nucleus; Similarly, the second term is equal to the probability of
finding an electron in region A and simultaneously another electron far from
the nucleus, in region B. This is interpreted as single ionization; Likewise, the
final term is interpreted as the probability for double ionization. Accordingly,
we will refer to these terms as p(0) (t), p(+1) (t), and p(+2) (t).
To this point of the derivation we have utilized the many-body wavefunction to define the ionization probabilities. Our goal is however to construct a density functional. For that purpose we introduce the pair-correlation
function
2
2 |Ψ (r1 , r2 , t)|
g[n](r 1 , r2 , t) =
(120)
n(r 1 , t)n(r 2 , t)
and rewrite
Z Z
1
d3 r1 d3 r2 n(r1 , t)n(r2 , t)g[n](r1 , r 2 , t)
p (t) =
2 A A
Z
p(+1) (t) = d3 r n(r, t) − 2p(0) (t)
(0)
(121)
A
p(+2) (t) = 1 − p(0) (t) − p(+1) (t) .
We recall that by virtue of the Runge-Gross theorem g is a functional of the
time-dependent density. Separating g into an exchange part (which is simply
1/2 for a two electron system) and a correlation part,
g[n](r 1 , r2 , t) =
1
+ gc [n](r 1 , r2 , t)
2
(122)
we can cast Eq. (121) into the form
2
p(0) (t) = [N1s (t)] + K(t)
p(+1) (t) = 2N1s (t) [1 − N1s (t)] − 2K(t)
p
(+2)
(123)
2
(t) = [1 − N1s (t)] + K(t)
with the definitions
Z
Z
1
d3 r n(r, t) = d3 r |ϕ1s (r, t)|2
2 A
A
Z Z
1
3
3
K(t) =
d r1 d r2 n(r 1 , t)n(r 2 , t)gc [n](r1 , r2 , t) .
2 A A
N1s (t) =
(124)
(125)
38
Miguel A. L. Marques and E. K. U. Gross
1
Probability
10
10
-2
-4
-6
10
10
TDHF
ALDA
TDSIC
SEQ
-8
15
10
10
2
Intensity (W/cm )
16
Fig. 8. Calculated double-ionization probabilities from the ground-state of Helium
irradiated by a 16 fs, 780 nm laser pulse for different choices of the time-dependent
xc potentials. Figure reproduced from Ref. [71].
In Fig. 8 we depict the probability for double ionization of Helium calculated from Eq. (123) by neglecting the correlation part of g. It is clear that
all functionals tested yield a significant improvement over the simple sequential model. Due to the incorrect asymptotic behavior of the ALDA potential,
the ALDA overestimates ionization: The outermost electron of Helium is not
sufficiently bound and ionizes too easily.
To compare the TDDFT results with experiment it is preferable to look
at the ratio of double- to single-ionization yields. This simple procedure eliminates the experimental error in determining the absolute yields. Clearly all
TDDFT results presented in Fig. 9 are of very low quality, sometimes wrong
by two orders of magnitude. We note that two approximations are involved
in the calculation: The time-dependent xc potential used to propagate the
Kohn-Sham equations, and the neglect of the correlation part of the paircorrelation function. By using a one-dimensional Helium model, Lappas and
van Leeuwen were able to prove that even the simplest approximation for
g was able to reproduce the knee structure[72]. As neither of the TDDFT
calculations depicted in Fig. 9 show the knee structure, it seems that the
approximation for the time-dependent xc potential is the most important in
obtaining the ionization yields.
Time-dependent Density Functional Theory
39
10
10
-1
+
/ He ratio
1
-2
He
+2
10
-3
10
10
TDHF
ALDA
TDSIC
Exp.
-4
10
-5
15
10
2)
Intensity (W/cm
10
16
Fig. 9. Comparison of the ratios of double- to single-ionization probability calculated for different choices of the time-dependent xc potential. Figure reproduced
from Ref. [71].
6
Conclusion
In this chapter we tried to give a brief, yet pedagogical overview of TDDFT,
from its mathematical foundations – the Runge-Gross theorem and the timedependent Kohn-Sham scheme – to some of its applications, both in the linear
and in the non-linear regimes. In the linear regime, TDDFT has become the
standard tool to calculate excitation energies within DFT, and is now incorporated into the major quantum-chemistry codes. In the non-linear regime,
TDDFT is able to describe extremely non-linear effects, like high-harmonic
generation, or multi-photon ionization. Unfortunately, some problems, like
the knee structure in the yield of doubly ionized Helium, are still beyond the
reach of modern time-dependent xc potentials. In our opinion, we should not
dismiss these problems as failures of TDDFT, but as a challenge to the next
generation of “density-functionalists”, in their quest for better approximations to the elusive xc potential.
Acknowledgments
We would like to thank L. Wirtz and A. Rubio for their useful suggestions
and comments, and also to A. Castro for his invaluable help in producing the
figures.
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THE JOURNAL OF CHEMICAL PHYSICS 123, 062206 共2005兲
Time-dependent density functional theory: Past, present, and future
Kieron Burke
Department of Chemistry and Chemical Biology, Rutgers University, Piscataway, New Jersey 08854
Jan Werschnik and E. K. U. Gross
Institut für Theoretische Physik, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany
共Received 18 August 2004; accepted 17 March 2005; published online 17 August 2005兲
Time-dependent density functional theory 共TDDFT兲 is presently enjoying enormous popularity in
quantum chemistry, as a useful tool for extracting electronic excited state energies. This article
discusses how TDDFT is much broader in scope, and yields predictions for many more properties.
We discuss some of the challenges involved in making accurate predictions for these properties.
© 2005 American Institute of Physics. 关DOI: 10.1063/1.1904586兴
Kohn–Sham density functional theory1–3 is the method
of choice to calculate ground-state properties of large molecules, because it replaces the interacting many-electron
problem with an effective single-particle problem that can be
solved much faster. Time-dependent density functional
theory 共TDDFT兲 applies the same philosophy to timedependent problems. We replace the complicated many-body
time-dependent Schrödinger equation by a set of timedependent single-particle equations whose orbitals yield the
same time-dependent density n共rt兲. We can do this because
the Runge–Gross theorem4 proves that, for a given initial
wave function, particle statistics, and interaction, a given
time-dependent density n共rt兲 can arise from at most one
time-dependent external potential vext共rt兲. We define timedependent Kohn–Sham 共TDKS兲 equations that describe N
noninteracting electrons that evolve in vs共rt兲, but produce the
same n共rt兲 as that of the interacting system of interest. Development and applications of TDDFT have enjoyed exponential growth in the last few years,5–8 and we hope this
merry trend will continue.
The scheme yields predictions for a huge variety of phenomena that can largely be classified into three groups: 共i兲
the nonperturbative regime, with systems in laser fields so
intense that perturbation theory fails, 共ii兲 the linear 共and
higher-order兲 regime, which yields the usual optical response
and electronic transitions, and 共iii兲 back to the ground-state,
where the fluctuation-dissipation theorem produces groundstate approximations from TDDFT treatments of excitations.
In the first, nonperturbative regime, we have systems in
intense laser fields with electric field strengths that are comparable to or even exceed the attractive Coulomb field of the
nuclei.5 The time-dependent field cannot be treated perturbatively, and even solving the time-dependent Schrödinger
equation for the evolution of two interacting electrons is
barely feasible with present-day computer technology.9 For
more electrons in a time-dependent field, wave function
methods are prohibitive, and in the regime of 共not too high兲
laser intensities, where the electron-electron interaction is
still of importance, TDDFT is essentially the only practical
scheme available. With the recent advent of attosecond laser
pulses, the electronic time scale has become accessible. The0021-9606/2005/123共6兲/062206/9/$22.50
oretical tools to analyze the dynamics of excitation processes
on the attosecond time scale will become more and more
important. An example of such a tool is the time-dependent
electron localization function 共TDELF兲 共Refs. 10 and 11兲.
This quantity allows the time-resolved observation of the formation, modulation, and breaking of chemical bonds, thus
providing a visual understanding of the dynamics of excited
electrons 共for an example see Fig. 1 and Ref. 12兲. The natural
way of calculating the TDELF is from the TDKS orbitals.
Recent applications in the beyond-perturbative regime range
from above-threshold ionization of metal clusters13 to coherent control of quantum wells14 to multiharmonic generation
in benzene.15
A much larger group of applications in chemistry is the
linear response to a spatially uniform electric field, i.e., the
optical response in the dipole approximation.16,17 Formal
analysis of this situation shows that TDDFT yields predictions for electronic excitations, both their position 共transition
frequency兲 and intensity 共oscillator strength兲. These are corrections to transitions between occupied and unoccupied levels of the ground-state KS potential, thus providing a simple
interpretation of those levels.18 In the area of calculating
electronic excitations, TDDFT is rapidly becoming a standard tool, complimentary to existing wave function
techniques.19 Just as in the ground-state case, it has the advantage in computational speed, allowing study of larger systems than with traditional methods, and the usual disadvantage 共or excitement兲 of being unsystematic and artful. A final
application is to write the ground-state exchange-correlation
(XC) energy in terms of the frequency-dependent response
function, and so linear response TDDFT yields approximate
treatments of the ground-state problem.20–23
A random walk through some of 2004’s papers using
TDDFT gives some feeling for the breadth of applications.
Most are in the linear response regime. In inorganic chemistry, the optical response of many transition metal
complexes24–29 has been calculated, and even some x-ray
absorption.30 In organic chemistry, the response of
thiouracil31 and s-tetrazine,32 and annulated porphyrins33
were investigated. In photobiology, potential energy curves
for the trans-cis photoisomerization of protonated Schiff base
123, 062206-1
© 2005 American Institute of Physics
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062206-2
J. Chem. Phys. 123, 062206 共2005兲
Burke, Werschnik, and Gross
FIG. 1. 共Color online兲 Snapshots of the time-dependent ELF for the excitation of acetylene by a 17.5 eV laser pulse 共Ref. 12兲. The pulse had a total length
of 7 fs, an intensity of 1.2⫻ 1014 W cm−2, and was polarized along the molecular axis. Ionization and the transition from the bonding ␲ state to the
antibonding ␲* state are clearly visible.
of retinal34 have been calculated. For these and other systems, there is great interest in charge-transfer
excitations,35–40 but 共as we later discuss兲 intermolecular
charge transfer is a demanding problem for TDDFT. Another
major area of application is clusters, large and small, covalent and metallic, and everything in between,41–47 including
Met-Cars.48 Several studies include solvation, for example,
the behavior of metal ions in explicit water.49 TDDFT in
linear response can also be used to calculate both electronic
and magnetic circular dichroism,50,51 and has been applied to
helical aromatics,52 and to artemisinin complexes in
solution.53 There have also been applications in materials54,55
and quantum dots56 but, as discussed below, the optical response of solids requires some nonlocal approximations.57
Beyond the linear regime, there is also growing interest in
second- and third-order response58–61 in all these fields.
A wonderful aspect of TDDFT is that a single approximation to the time-dependent XC potential implies predictions for all these quantities. This is analogous to the groundstate case, where a single approximation to EXC can be
applied to all ground-state properties of all electronic systems, such as dissociation energies, bond lengths and angles,
vibrational frequencies, etc., of atoms, molecules, clusters,
and solids. The starting point of most TDDFT approximations is the adiabatic local density approximation 共ALDA兲,
which approximates the XC potential at point r and time t by
that of a ground-state uniform electron gas of density n共rt兲.
This is clearly accurate when the density varies sufficiently
slowly in time and space, but works surprisingly well beyond
that limit for many systems and properties, just as LDA does
for most ground-state problems.
We make an important distinction here between the mature subject of ground-state DFT, and the developing one of
TDDFT. In the former, there is general consensus on which
properties are captured by which functionals, and the aim is
toward higher accuracy.62 One expects chemical bonds to
form in modern KS DFT calculations, and one hopes to use
better functionals to produce better accuracy and reliability.63
But time-dependent quantum mechanics probes a far more
diverse range of electronic phenomena, and in TDDFT, we
are still exploring even which properties are captured at all
by the presently available approximate functionals. Quantitative accuracy is less of an issue as yet. Most data on the
performance of TDDFT are for systems driven by some external field. Practically nothing is known about how TDDFT
performs in the description of relaxation processes, i.e., on
the time evolution of large systems starting from a nonequilibrium initial condition.64 A closely related question is the
description of density fluctuations within TDDFT.65
At this point, we introduce a few equations, to make the
discussion more precise. We use atomic units throughout,
and suppress spin indices. For brevity, we drop commas between arguments wherever the meaning is clear. We write the
TDKS equations as
i
冉
冊
ⵜ2
d␾ j共rt兲
= −
+ vs关n兴共rt兲 ␾ j共rt兲,
2
dt
共1兲
whose density n共rt兲 = 兺Nj=1兩␾ j共rt兲兩2 is precisely that of the real
system. We define the exchange-correlation potential via
vs共rt兲 = vext共rt兲 +
冕
d 3r ⬘
n共r⬘t兲
+ vXC共rt兲.
兩r − r⬘兩
共2兲
The exchange-correlation potential, vXC共rt兲 is in general a
functional of the entire history of the density n共rt兲, the initial
interacting wave function ⌿共0兲, and the initial Kohn–Sham
wave function ⌽共0兲. This functional is a very complex one,
much more so than the ground-state case. Knowledge of it
implies solution of all time-dependent Coulomb interacting
problems. If we always begin in a nondegenerate ground
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062206-3
J. Chem. Phys. 123, 062206 共2005兲
Time-dependent density functional theory
state,66 the initial-state dependence can be subsumed by the
Hohenberg–Kohn theorem,1 and the only unknown part of
vs共rt兲, the exchange-correlation 共XC兲 potential, is a functional of n共rt兲 alone.
In the special case of the response of the ground-state to
a weak external field, the system’s response is characterized
by the nonlocal susceptibility
␦n共rt兲 =
冕 ⬘冕
dt
d3r⬘␹关n0兴共rr⬘ ;t − t⬘兲␦vext共r⬘t⬘兲.
共3兲
␹ is a functional of the ground-state density n0共r兲. The central equation of TDDFT linear response16 is a Dyson-like
equation for the true ␹ of the system,
␹共rr⬘␻兲 = ␹s共rr⬘␻兲 +
⫻
冉
冕 冕
d 3r 1
d3r2␹s共rr1␻兲
冊
1
+ f XC共r1r2␻兲 ␹共r2r⬘␻兲.
兩r1 − r2兩
共4兲
Here ␹s is the Kohn–Sham response function, constructed
from KS energies and orbitals,
␹s共rr⬘␻兲 = 2 兺
q
⌽q共r兲⌽*q共r⬘兲
␻ − ␻q + i0+
+ c.c.共␻ → − ␻兲,
共5兲
where q is a double index, representing a transition
from occupied KS orbital i to unoccupied KS orbital a, ␻q
= ⑀a − ⑀i, and ⌽q共r兲 = ␾*i 共r兲␾a共r兲. Thus ␹s is purely a product
of the ground-state KS calculation. On the other hand, the
XC kernel is defined as
f XC关n0兴共rr⬘ ;t − t⬘兲 = 兩␦vXC共rt兲/␦n共r⬘t⬘兲兩n0 .
共6兲
This is a much simpler quantity than vXC关n兴共rt兲, since the
functional is only evaluated at the ground-state density. It is
nonlocal in both space and time. The nonlocality in time
manifests itself as a frequency dependence in the Fourier
transform, f XC共rr⬘␻兲.
Next, we introduce Casida’s equations,17 in which the
poles of ␹ are found as the solution to an eigenvalue problem,
兺 ⍀̃qq⬘共␻兲aq⬘ = ␻2aq ,
共7兲
q⬘
where
⍀̃qq⬘共␻兲 = ␻2q␦qq⬘ + 2冑␻q␻q⬘具q兩f HXC共␻兲兩q⬘典
共8兲
and 具q兩f HXC共␻兲兩q⬘典 is the matrix element of the 共Hartree兲-XC
kernel in the set of functions ⌽q共r兲. Eigenvalues yield the
square of transition frequencies, while eigenvectors yield oscillator strengths. Ignoring off-diagonal matrix elements can
yield much insight into the nature of the TDDFT corrections
to the KS transitions.18
Lastly, we mention how TDDFT produces sophisticated
approximations to the ground-state exchange-correlation
energy. The adiabatic connection fluctuation-dissipation
formula is
EXC关n0兴 = −
⫻
冕 冕 冕
冕 关兴
1
1
2
d␭
0
⬁
0
d 3r
d 3r ⬘
1
兩r − r⬘兩
d␻ ␭
兵␹ n0 共rr⬘␻兲 + n0共r兲␦共3兲共r − r⬘兲其,
␲
共9兲
where the coupling-constant ␭ is defined to multiply the
electron-electron repulsion in the Hamiltonian, but the external potential is adjusted to keep the density fixed.67,68 So any
model for f XC, even setting it to zero 共called the Random
Phase Approximation兲, yields a sophisticated approximation
to EXC, by solving Eq. 共4兲 for ␹ 共at each ␭兲 and inserting in
Eq. 共9兲.
All the above equations are formally exact. In any practical DFT calculation, approximations must be made. The
most common approximation in TDDFT is the adiabatic approximation, in which
adia
gs
关n兴共rt兲 = 兩vXC
关n0兴共r兲兩n0共r兲=n共rt兲 ,
vXC
共10兲
i.e., the XC potential at any time depends only on the density
at that time, not on its entire history. This becomes exact for
slow variations in time. Most applications, however, are not
in this slowly varying regime. Nevertheless, results obtained
within the adiabatic approximation are, in most cases, rather
accurate. Any ground-state approximation 共LDA, GGA, hybrid兲 automatically provides an adiabatic approximation
共e.g., ALDA兲 in TDDFT. Moreover, the XC kernel is
frequency-independent in the adiabatic approximation, taking its ␻ → 0 value.
As mentioned above, TDDFT is proving very useful in
predicting optical response properties of molecules. The
Casida equations have been encoded in most standard quantum chemical packages, and efficient algorithms developed
to extract the lowest-lying excitations. A small survey is
given by Furche and Ahlrichs.19 Typical chemical calculations are done with the B3LYP69 functional, and typical results are transition frequencies within 0.4 eV of experiment,
and structural properties of excited states are almost as good
as those of ground-state calculations 共bond lengths to within
1%, dipole moments to within 5%, vibrational frequencies to
within 5%兲. Most importantly, this level of accuracy appears
sufficient in most cases to qualitatively identify the nature of
the most intense transitions, often debunking cruder models
that have been used for interpretation for decades. This is
proving especially useful for the photochemistry of biological molecules.70 An alternative implementation, often favored by physicists, is to propagate the TDKS equations in
real time, having given the system an initial weak perturbation. Such calculations either use a real-space-grid71,72 or
plane waves.73
This article is not about the 共admittedly兲 gratifying successes of TDDFT calculations, which are discussed in recent
reviews5,6 and the recent literature. We begin from there, and
explore a much wider arena. To do this, in Fig. 2 we have
drawn a cartoon 共literally, a stick figure兲 to represent the
information in a typical calculation. Each line represents a
transition, with its position denoting the transition frequency
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062206-4
Burke, Werschnik, and Gross
J. Chem. Phys. 123, 062206 共2005兲
FIG. 2. Cartoon of the exact optical absorption spectrum of an atom or
molecule, with discrete transitions represented by straight lines 共see text兲.
FIG. 3. Same as Fig. 2, but showing higher frequencies, including the infinite Rydberg series of states as the ionization threshold is approached.
and its height proportional to the oscillator strength. For the
He atom, the 2s → 2p singlet transition is at 21.2 eV, while
for the N2 molecule, the 3⌸␥ transition is at 7.4 eV. Many
applications of TDDFT report only the positions of the few
lowest optically allowed transitions, while some report also
their oscillator strengths.
To help our analysis, we list qualitatively different
sources of error in the predictions from any TDDFT calculation. We refer to them as the four deadly sins.
calculates the optical response of N noninteracting electrons
in the exact KS ground-state potential, i.e., what we call the
KS response, its ionization threshold is in exactly the right
place, by virtue of the DFT version of Koopman’s theorem.74
From the very earliest calculations of transition
frequencies,16,17 it was recognized that the inaccuracy of
standard density functional approximations 共LDA, GGA, hybrids兲 for the ground-state XC potential leads to inaccurate
KS eigenvalues. Because the approximate KS potentials
have incorrect asymptotic behavior 共they decay exponentially, instead of as −1 / r兲, the KS orbital eigenvalues are
insufficiently negative, the ionization threshold is far too low,
and Rydberg states are often unbound. This is therefore a
ground-state sin.
Given this disastrous behavior, many methods have been
developed to asymptotically correct potentials.75,76 Any corrections to the ground-state potential are dissatisfying, however, as the resulting potential is not a functional derivative
of an energy functional. Even mixing one approximation for
vXC共r兲 and another for f XC has become popular. A more satisfying route is to use the optimized effective potential
共OEP兲 method77,78 and include exact exchange or other selfinteraction-free functionals. This produces a far more accurate KS potential, with the correct asymptotic behavior. The
chief error is simply the correlation contribution to the position of the HOMO, i.e., a small shift. All the main features
below and just above I are retained.
Why has the poor quality of ground-state potentials not
impeded the rapid growth of TDDFT calculations for excitations in quantum chemistry? For many molecules, the lowest
excitations are not Rydberg in character, and the orbitals do
not depend on the large-r behavior of the potential. But there
are important cases where the problem does show up. The
“fruitfly” of TDDFT benchmarks is the ␲ → ␲* transition in
benzene. This occurs at about 5 eV in a ground-state LDA
calculation, and ALDA shifts it correctly to about 7 eV.79
Unfortunately, this is in the LDA continuum, which starts at
about 6.5 eV! So how is it possible to get this right in
ALDA?
The answer is that ALDA usually yields good oscillator
strengths, even for states pushed into the continuum.80 The
reason is simple, and was suggested long ago in early photoabsorption calculations by Zangwill and Soven.81 The KS
LDA potential looks very much like the exact one 共especially
in the interior, as the occupied orbitals yield a good approxi-
• The sin of the ground-state: Errors in the underlying
ground-state DFT calculation. If the KS orbital energies
are wrong to begin with, TDDFT corrections cannot
produce accurate results.
• The sin of locality: Errors due to local 共or gradientcorrected兲 approximations to an adiabatic f XC共rr⬘兲, i.e.,
properties that require nonlocality in 兩r − r⬘兩.
• The sin of forgetfulness: Phenomena missing when the
adiabatic approximation is made, i.e., properties that require nonlocality in time, i.e., memory.
• The sin of the wave function: Even if the exact vXC共rt兲
is used, solution of the TDKS equations only yields the
TDKS noninteracting wave function. The true wave
function may differ so dramatically from the KS wave
function that observables evaluated on the latter may be
inaccurate.
The remainder of this essay is a discussion of the various
areas of TDDFT applications and development, and the challenges presently facing us. We begin in the middle, with the
linear response regime, where most of the applications presently are, then go to nonperturbative phenomena, and end
with ground-state applications.
We start with applications to the excitations of atoms and
molecules. An important point we wish to emphasize here is
the wealth of prediction made by any TDDFT approximation. The simplest real system of interacting electrons is the
He atom, and even it has a rich and complex optical absorption spectrum. Returning to Fig. 2, we note that a calculation
of the optical spectrum of the bare ground-state KS system,
often18 looks quite similar to the exact one, with TDDFT
corrections merely shifting and resizing peaks. In Fig. 3, we
zoom out a little, and see the ionization threshold at ␻ = I and
the infinite Rydberg series of excitations just to its left. If one
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062206-5
Time-dependent density functional theory
FIG. 4. Same as Fig. 3, but now including the second ionization threshold.
mation to the true density兲, shifted up by a constant, due to
the lack of derivative discontinuity.74 The shift pushes the
Rydberg states into the continuum, but retains their contribution to the optical spectrum. Likewise for the benzene transition. Hence ALDA can still be used and trusted for that
transition. Moral: Just because it is in the continuum, does
not mean it is not right.
The cartoon of Fig. 2 changes when bonds are stretched.
Of particular interest in biochemistry are charge-transfer excitations, especially between weakly bonded molecules.
Capturing these seems unnatural within TDDFT, for the
simple reason that the numerator in ␹S in Eq. 共5兲 vanishes as
the molecular separation goes to infinity. Thus, in the
density-density response, the oscillator strength for these
transitions is exponentially small. Recently,82 it has been
shown how to build an empirical approximation to an adiabatic f XC that can capture these effects, but it is one that
grows exponentially with 兩r − r⬘兩. Thus this is a sin of locality, in which local approximations to f XC miss a qualitative
feature.
A presently open question is the extraction of double
excitations.83,84 In the adiabatic approximation, these are lost
from the linear response equations. When a double excitation
lies close to a single excitation, elementary quantum mechanics shows that f XC must have a strong frequency
dependence.85 Thus this problem is due to the adiabatic approximation: a sin of forgetfulness. Recently, a postadiabatic
TDDFT methodology has been developed85,86 for including a
double excitation when it is close to an optically active
single excitation, and works well for small dienes.87 It had
been hoped that, by going beyond linear response, nontrivial
double excitations would be naturally included in, e.g.,
TDLDA, but it has recently been proven that, in the higherorder response in TDLDA, the double excitations occur simply at the sum of single excitations. Thus we do not currently
know how best to approximate these excitations. This problem is particularly severe for quantum dots, where the external potential is parabolic, leading to multiple near degeneracies between levels of excitation.
Concerns about both ionization potentials and double excitations are combined when we consider more of the optical
response of the He atom. Zooming out just a little more in
Fig. 4, we see that there is of course a second ionization
potential, when a second electron is stripped off the atom or
molecule. For reference, in a He atom, I = 24.2 eV, and I2
J. Chem. Phys. 123, 062206 共2005兲
= 54 eV. But the bare KS response contains only the first
threshold. It has no structure at all in the region of the second
ionization. Our simple density functional approximations to
f XC tend to shift and resize peak positions. It is very difficult
to imagine density or orbital functional approximations to
f XC that can build in the ␻-dependence needed to create a
second threshold.
Lastly in this section, we mention recent progress in developing a theory for electron scattering from molecules.
This was one of the original motivations for developing TDDFT. One approach would be to evolve a wave packet using
the TDKS equations, but a more direct approach has been
developed,88 in terms of the response function ␹ of the N
+ 1 electron system 共assuming it is bound兲. This uses similar
technology to the discrete transition case. Initial results for
the simplest case, electron scattering from He+, suggest a
level of accuracy comparable to bound-bound transitions, at
least for low energies 共the most difficult case for traditional
methods, due to bound-free correlation89兲.
A key question that often arises is the need for timedependent current DFT (TD-CDFT), or not. The Runge–
Gross theorem proceeds by first proving a one-to-one correspondence between currents and scalar potentials. Obviously,
the current is needed in the presence of time-dependent magnetic fields, but in their absence, is it necessary? By continuity, dn / dt = −⵱ · j共rt兲, so that the density is uniquely determined by the current 共up to its initial value兲, but not vice
versa. It would seem preferable to stay within the simpler
density functional theory where possible. A careful examination of the conditions of applicability of the Runge–Gross
theorem to finite systems90 shows that all atoms and molecules satisfy the necessary conditions of potentials vanishing sufficiently rapidly as r → ⬁. However, early work
showed that the gradient expansion 共the origin of GGA’s for
the ground state兲 fails within TDDFT, but behaves reasonably within the current theory, yielding the Vignale–Kohn
共VK兲 approximation91,92 for the response kernel, which has
frequency dependence.
These questions become relevant to the optical response
of bulk insulators. The Dyson-like Eq. 共4兲 becomes a matrix
equation with indices of the reciprocal lattice vectors G for
each perturbation of wave vector q. As q → 0, to find the
optical response, any local approximation to f XC produces a
negligible correction to the RPA response 共f XC = 0兲, as the
Hartree contribution 共correctly兲 blows up as 1 / q2. Thus, to
have a noticeable effect, the XC kernel must have a 1 / q2
component as q → 0. While this effect is sometimes referred
to as “ultra”-nonlocal, we prefer to call it simply nonlocal, as
the range of nonlocality is precisely that of the Hartree contribution. The optical response of the solid can be found
within TDDFT by perturbing the system with a longwavelength perturbation of wave vector q, and by carefully
taking q → 0. This requires extending the RG theorem to periodic Hamiltonians.93 On the other hand, a q = 0 calculation,
with just the period of the lattice, is possible within TDCDFT, and the nonlocal contribution in TDDFT appears as a
local contribution within TDCDFT, with no unusual nonlocality needed in the current density. For example, the VK
approximation produces a finite correction, whereas LDA
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062206-6
J. Chem. Phys. 123, 062206 共2005兲
Burke, Werschnik, and Gross
and GGA do not. Thus, in this and other cases, TDCDFT is
not strictly necessary, but provides a more direct description
and a route to extract quantities that are nonlocal in TDDFT.
Similar observations apply to the polarizabilities of long organic polymers. ALDA and GGA greatly overestimate these
quantities, but VK often does much better.
In this context, an important challenge is the proper description of excitonic peaks in the optical spectra of insulators. It was recently demonstrated94–96 that with complicated
orbital-dependent approximations for f XC, which were derived from the Bethe–Salpeter equation, excitonic effects can
be described perfectly. However, the presently available
schemes require a GW calculation 共where G is the Green’s
function and W the screened Coulomb interaction97兲 in the
first place. There remains the challenge to find sufficiently
simple 共possibly current-dependent兲 approximations that are
able to describe excitonic effects.
Another interesting question in the optical response of
insulators is the one of the gap. It is well-known from
ground-state DFT that the gap in the spectrum of KS eigenvalues 共the KS gap兲 differs from the true gap by a quantity
called the derivative discontinuity.3 Ignoring excitons within
the gap, should not TDDFT correct the KS gap to yield the
true gap? The answer is, yes, but again the XC kernel that
does this must be very sophisticated, just as in our double
ionization example. Since ␹s develops an imaginary part for
frequencies above the KS gap, the kernel must have a branch
cut that exactly suppresses this in order to widen the gap.
There is a close analogy to the problem of charge-transfer
excitations: Remove an electron from the donor to infinity.
This costs the ionization energy IDON. Then move the electron from infinity to the acceptor. In this way one gains the
energy −AACC. So the excitation energy is ⌬E = IDON − AACC
共ACC兲
共ACC兲
LUMO
HOMO
= ⑀ACC
− ⑀DON
+ ⌬XC
, where ⌬XC
is the discontinuity
in the ground-state vXC共r兲 of the acceptor. This formula is
reminiscent of the band gap in insulators.3 Furthermore, the
next-order correction is the Coulomb-interaction between the
electron on the acceptor and the hole on the donor which, in
solids, corresponds to the exciton binding energy.
On the boundary between extended systems and molecules is transport through single molecules connected to
bulk metal leads.98 There is enormous interest in this as a key
component in future nanotechnology. Present formulations
use ground-state density functionals to describe the stationary nonequilibrium current-carrying state.99 But several recent suggestions consider this as a time-dependent
problem,64,100–103 and use TD共C兲DFT for a full description of
the situation. Only time will tell if TDDFT is really needed
for an accurate description of these devices. In the special
case of weak bias, XC corrections to the Landauer formula
are missed by local approximations, the sin of locality.104
Next we turn our attention to beyond-perturbative regimes. Due to advances in laser technology over the past
decade, many experiments are now possible in regimes
where the laser field is stronger than the nuclear attraction.
There are a whole host of phenomena that TDDFT might be
able to predict: high harmonic generation, multiphoton ionization, above-threshold ionization, above-threshold dissociation, etc. For high harmonic generation, TDDFT calcula-
tions have been rather successful for atoms105,106 and
molecules.15,107 In the near future, this might become very
important for the generation of attosecond laser pulses.108–110
For multiphoton ionization, the relative proportion of double
to single ionization for He, while given much better in approximate TDDFT calculations than in previous calculations
assuming a sequential mechanism, still does not show the
same pronounced features 共the “knee”兲 seen in
experiments.111,112 The electron spectra from above-threshold
ionization have recently been calculated within TDDFT.13,113
Since the ionization yields and above-threshold ionization
spectra depend on probabilities extracted directly from the
wave function, these errors are suspected to be sins of the
wave function, mentioned above. An important task for the
future will be the design of more realistic expressions for
ionization probabilities or, more generally, transition probabilities as functionals of the time-dependent density or the
time-dependent KS orbitals. First steps in this direction can
be found in Ref. 111.
While the need for more accurate KS potentials was first
noticeable in calculating excitations, it is even more acute in
strong laser fields. To ensure an atom or molecule ionizes
accurately in an approximate TDKS calculation, Koopmans’
theorem 共I = −⑀HOMO兲 should be well-satisfied, and this again
requires using OEP exact exchange77,78 or other selfinteraction-free functionals.
The field of quantum control has, so far, mainly concentrated on manipulating the motion of the nuclear wave packet
on a given set of precalculated potential energy surfaces, the
ultimate goal being the femtosecond control of chemical
reactions.114 With attosecond pulses available, the control of
electronic dynamics has come within reach. A marriage of
optimal-control theory with TDDFT appears to be the ideal
theoretical tool to tackle this situation. However, it will bring
with it its own difficulties and challenges for approximate
functionals. Using the functional algorithms developed by
Rabitz and co-workers,115,116 we can find the optimal pulse
that drives a He atom from its ground state to its first excited
state, 1s2p. 共In practice, we do not reach exactly 100% occupation, due to a finite penalty factor.兲
Now repeat this experiment on noninteracting electrons
sitting in the same potential. Such a pulse cannot be found,
i.e., the noninteracting system is not controllable, whereas
the interacting system is. The two noninteracting electrons
must follow the same time evolution as they start from the
same initial 1s orbital and are exposed to the same laser field.
Hence the time-dependent wave function of the two noninteracting electrons must have the form
⌽共r1␴1r2␴2t兲 = ␸共r1t兲␸共r2t兲␹S共␴1␴2兲,
共11兲
where ␹s共␴1␴2兲 represents the 共antisymmetric兲 spin-singlet
part of the wave function. But we want to maximize the
occupation
兩具⌽共T兲兩⌽1s,2p典兩2
共12兲
of the time-propagated wave function ⌽共T兲 at the end, T, of
the laser pulse in the lowest excited state
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062206-7
⌽1s,2p =
J. Chem. Phys. 123, 062206 共2005兲
Time-dependent density functional theory
1
冑2
„␸1s共r1兲␸2p共r2兲 + ␸2p共r1兲␸1s共r2兲…␹S共␴1␴2兲.
共13兲
Expanding the final wave function ⌽共T兲 in the complete set
of single-particle orbitals representing the eigenfunctions of
the unperturbed system, one sees that the best possible occupation is 50%.
If TDDFT is used to describe the multielectron dynamics, how would one properly define the control target, i.e.,
the functional to be maximized? Choosing, as control target,
the overlap with an excited-state Kohn–Sham determinant
does not seem to be a good idea in view of the above dilemma. If, on the other hand, the time-dependent density of a
fully controlled 共1s兲2 to 共1s , 2p兲 transition of the interacting
system is given, would an exact TDKS calculation reproduce
such an optimal evolution? The answer is yes, but vXC共rt兲
must be very special to do so. To see this, take the density
evolution from the exact Schrödinger equation, and invert
the TDKS equation for the single 共doubly occupied兲 timedependent orbital. The final state KS potential is very odd,
producing the density of two orbitals of different symmetry
from a single doubly occupied orbital.117
For a subset of cases in which molecules are exposed to
strong fields, the nuclear motion can be treated classically.
The electrons then feel the Coulomb field of classically moving nuclei as well as the laser field. In this case the electronic
motion is well described by ordinary TDDFT. However,
when nuclear trajectories split, e.g., when a molecule has a
50% chance of dissociation in a given laser pulse, the classical treatment fails. A multicomponent TDDFT118–120 has
been developed for electrons and nuclei which, in principle,
handles such a situation. In practice, one needs to develop
appropriate approximations that can build in the appropriate
physics of, e.g., dissociating nuclei. Again, in this case,
orbital-dependent functionals appear crucial.118,120
Finally, and fondly, we return to the ground-state. The
last general application mentioned was the odd-sounding
process of using TDDFT to generate ground-state approximations, via Eq. 共9兲. By inserting an approximation for f XC,
we get an approximation to EXC. Most importantly for biological systems, Eq. 共9兲 provides a route to van der Waals
forces for separated pieces of matter, and so is being much
studied by developers. In particular, the coefficient in the
decay of the energy between two such pieces 共C6 in E →
−C6 / R6, where R is their separation兲 can be accurately
共within about 20%兲 evaluated using a local approximation to
the frequency-dependent polarizability.21,121–123 Recent work
shows that the response functions of TDDFT can yield extremely accurate dispersion energies of monomers.124 More
recently, the frequency integral in Eq. 共9兲 has been done
approximately, yielding an explicit nonlocal density
functional23 applicable at all separations.
One can go the other way, and try using Eq. 共9兲 for all
bond lengths.125,126 Such calculations are costly, as the functional is very high up on Jacob’s ladder of sophistication,
including both occupied and unoccupied KS orbitals.63 However, they have the merit of being entirely nonempirical and,
where successful, can be used as a starting point for new
approximations. In fact, Eq. 共9兲 provides a KS density functional that allows bond-breaking without artificial symmetry
breaking.22 In the paradigm case of the H2 molecule, the
binding energy curve has no Coulson–Fischer point, and the
dissociation occurs correctly to two isolated H atoms. Unfortunately, simple approximations, while yielding correct results near equilibrium and at infinity, produce an unphysical
repulsion at large but finite separations. This can be traced
back22 to the lack of double excitations in any adiabatic f XC.
We end with a summary. Most importantly, TDDFT has
become extremely popular as a method for calculating electronic excited-state energies in chemistry. In this arena, it has
become as robust 共or as flaky, depending on your perspective兲 as ground-state DFT, and is being used to really understand the photochemistry of many interesting problems. Furthermore, our favorite crude density functional
approximations from the ground-state serve well here. We
are very thankful for this, and it has led to tremendous interest in further methodological development.
In principle, however, TDDFT yields predictions for an
enormous variety of phenomena, and electronic excitations
are only the tip of an iceberg. We have mentioned a few.
Even limiting ourselves to linear response, there are double
excitations, second ionization thresholds, optical response of
solids, gaps in solids, transport through single molecules.
Combining with the fluctuation-dissipation theorem, TDDFT
yields a route to van der Waals forces and bond breaking
with symmetry problems. In strong fields, there are high harmonic generation, multiphoton ionization, above-threshold
ionization, quantum control, and quantum nuclear motion.
For some of these areas, simple application of density
functionals within the adiabatic approximation, works well,
but for many, such methods miss some qualitative features
共e.g., double excitations, or nonlocality in the response of
solids兲. A now standard step upward in sophistication is to
use orbital-dependent functionals 共at least, among developers兲, and these cure some of the difficulties 共e.g., the first
ionization threshold or the polarizability of long-chain polymers兲. But such functionals are unlikely to cure all the problems 共e.g., inclusion of double excitations or defining the
target in quantum control兲 for properties that are of interest
experimentally and technologically. We happily look forward
to many interesting years of development to come.
The authors thank Maxime Dion, Vazgen Shekoyan, and
Adam Wasserman for useful discussions. K.B. gratefully acknowledges support of the U.S. Department of Energy, under
Grant No. DE-FG02-01ER45928. This work was supported,
in part, by the Deutsche Forschungsgemeinschaft, the EXC!TiNG Research and Training Network of the European
Union and the NANOQUANTA Network of Excellence.
Some of this work was performed at the Centre for Research
in Adaptive Nanosystems 共CRANN兲 supported by Science
Foundation Ireland 共Award No. 5AA/G20041兲.
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