Time-dependent density-functional theory (TDDFT) is a
quantum mechanical
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
theory used in physics and chemistry to investigate the properties and
dynamics of many-body systems in the presence of time-dependent potentials, such as
electric
Electricity is the set of physical phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by ...
or
magnetic field
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
s. The effect of such fields on molecules and solids can be studied with TDDFT to extract features like
excitation energies, frequency-dependent response properties, and
photoabsorption spectra.
TDDFT is an extension of
density-functional theory
Density-functional theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure (or nuclear structure) (principally the ground state) of many-body ...
(DFT), and the conceptual and computational foundations are analogous – to show that the (time-dependent)
wave function
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
is equivalent to the (time-dependent)
electronic density
In quantum chemistry, electron density or electronic density is the measure of the probability of an electron being present at an infinitesimal element of space surrounding any given point. It is a scalar quantity depending upon three spatial va ...
, and then to derive the effective potential of a fictitious non-interacting system which returns the same density as any given interacting system. The issue of constructing such a system is more complex for TDDFT, most notably because the time-dependent effective potential at any given instant depends on the value of the density at all previous times. Consequently, the development of time-dependent approximations for the implementation of TDDFT is behind that of DFT, with applications routinely ignoring this memory requirement.
Overview
The formal foundation of TDDFT is the
Runge–Gross (RG) theorem (1984) – the time-dependent analogue of the Hohenberg–Kohn (HK) theorem (1964). The RG theorem shows that, for a given initial wavefunction, there is a unique mapping between the time-dependent external potential of a system and its time-dependent density. This implies that the many-body wavefunction, depending upon 3''N'' variables, is equivalent to the density, which depends upon only 3, and that all properties of a system can thus be determined from knowledge of the density alone. Unlike in DFT, there is no general minimization principle in time-dependent quantum mechanics. Consequently, the proof of the RG theorem is more involved than the HK theorem.
Given the RG theorem, the next step in developing a computationally useful method is to determine the fictitious non-interacting system which has the same density as the physical (interacting) system of interest. As in DFT, this is called the (time-dependent) Kohn–Sham system. This system is formally found as the
stationary point
In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of the function where the function's derivative is zero. Informally, it is a point where the function "stops" in ...
of an
action
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
functional defined in the
Keldysh formalism
In Non-equilibrium thermodynamics, non-equilibrium physics, the Keldysh formalism is a general framework for describing the quantum mechanical evolution of a system in a non-equilibrium state or systems subject to time varying external fields (e ...
.
The most popular application of TDDFT is in the calculation of the energies of excited states of isolated systems and, less commonly, solids. Such calculations are based on the fact that the linear response function – that is, how the electron density changes when the external potential changes – has poles at the exact excitation energies of a system. Such calculations require, in addition to the exchange-correlation potential, the exchange-correlation kernel – the
functional derivative
In the calculus of variations, a field of mathematical analysis, the functional derivative (or variational derivative) relates a change in a functional (a functional in this sense is a function that acts on functions) to a change in a function on ...
of the exchange-correlation potential with respect to the density.
Formalism
Runge–Gross theorem
The approach of Runge and Gross considers a single-component system in the presence of a time-dependent
scalar field
In mathematics and physics, a scalar field is a function (mathematics), function associating a single number to every point (geometry), point in a space (mathematics), space – possibly physical space. The scalar may either be a pure Scalar ( ...
for which the
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
takes the form
:
where ''T'' is the kinetic energy operator, ''W'' the electron-electron interaction, and ''V''
ext(''t'') the external potential which along with the number of electrons defines the system. Nominally, the external potential contains the electrons' interaction with the nuclei of the system. For non-trivial time-dependence, an additional explicitly time-dependent potential is present which can arise, for example, from a time-dependent electric or magnetic field. The many-body wavefunction evolves according to the
time-dependent Schrödinger equation under a single
initial condition
In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted ''t'' = 0). For ...
,
:
Employing the Schrödinger equation as its starting point, the Runge–Gross theorem shows that at any time, the density uniquely determines the external potential. This is done in two steps:
# Assuming that the external potential can be expanded in a
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
about a given time, it is shown that two external potentials differing by more than an additive constant generate different
current densities.
# Employing the
continuity equation
A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. S ...
, it is then shown that for finite systems, different current densities correspond to different electron densities.
Time-dependent Kohn–Sham system
For a given interaction potential, the RG theorem shows that the external potential uniquely determines the density. The Kohn–Sham approaches chooses a non-interacting system (that for which the interaction potential is zero) in which to form the density that is equal to the interacting system. The advantage of doing so lies in the ease in which non-interacting systems can be solved – the wave function of a non-interacting system can be represented as a
Slater determinant
In quantum mechanics, a Slater determinant is an expression that describes the wave function of a multi-fermionic system. It satisfies anti-symmetry requirements, and consequently the Pauli principle, by changing sign upon exchange of two elect ...
of single-particle
orbitals, each of which are determined by a single
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
in three variable – and that the kinetic energy of a non-interacting system can be expressed exactly in terms of those orbitals. The problem is thus to determine a potential, denoted as ''v''
s(r,''t'') or ''v''
KS(r,''t''), that determines a non-interacting Hamiltonian, ''H''
s,
:
which in turn determines a determinantal wave function
:
which is constructed in terms of a set of ''N'' orbitals which obey the equation,
:
and generate a time-dependent density
:
such that ''ρ''
s is equal to the density of the interacting system at all times:
:
Note that in the expression of density above, the summation is over ''all''
Kohn–Sham orbitals and
is the time-dependent occupation number for orbital
. If the potential ''v''
s(r,''t'') can be determined, or at the least well-approximated, then the original Schrödinger equation, a single partial differential equation in 3''N'' variables, has been replaced by ''N'' differential equations in 3 dimensions, each differing only in the initial condition.
The problem of determining approximations to the Kohn–Sham potential is challenging. Analogously to DFT, the time-dependent KS potential is decomposed to extract the external potential of the system and the time-dependent Coulomb interaction, ''v''
J. The remaining component is the exchange-correlation potential:
:
In their seminal paper, Runge and Gross approached the definition of the KS potential through an action-based argument starting from the
Dirac action
Distributed Research using Advanced Computing (DiRAC) is an integrated supercomputing facility used for research in particle physics, astronomy and cosmology in the United Kingdom. DiRAC makes use of multi-core processors and provides a variety of ...
:
Treated as a functional of the wave function, ''A''
� variations of the wave function yield the many-body Schrödinger equation as the stationary point. Given the unique mapping between densities and wave function, Runge and Gross then treated the Dirac action as a density functional,
:
and_derived_a_formal_expression_for_the_exchange-correlation_component_of_the_action,_which_determines_the_exchange-correlation_potential_by_functional_differentiation._Later_it_was_observed_that_an_approach_based_on_the_Dirac_action_yields_paradoxical_conclusions_when_considering_the_causality_of_the_response_functions_it_generates._The_density_response_function,_the_functional_derivative_of_the_density_with_respect_to_the_external_potential,_should_be_causal:_a_change_in_the_potential_at_a_given_time_can_not_affect_the_density_at_earlier_times._The_response_functions_from_the_Dirac_action_however_are_symmetric_in_time_so_lack_the_required_causal_structure._An_approach_which_does_not_suffer_from_this_issue_was_later_introduced_through_an_action_based_on_the_
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_of_complex-time_path_integration._An_alternative_resolution_of_the_causality_paradox_through_a_refinement_of_the_action_principle_''in_real_time''_has_been_recently_proposed_by_Giovanni_Vignale.html" "title="rho.html" ;"title="Psi[\rho">Psi[\rho,\,
and derived a formal expression for the exchange-correlation component of the action, which determines the exchange-correlation potential by functional differentiation. Later it was observed that an approach based on the Dirac action yields paradoxical conclusions when considering the causality of the response functions it generates. The density response function, the functional derivative of the density with respect to the external potential, should be causal: a change in the potential at a given time can not affect the density at earlier times. The response functions from the Dirac action however are symmetric in time so lack the required causal structure. An approach which does not suffer from this issue was later introduced through an action based on the
of complex-time path integration. An alternative resolution of the causality paradox through a refinement of the action principle ''in real time'' has been recently proposed by Giovanni Vignale">Vignale