5 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
1 Introduction
The founding principles of density functional theory (DFT) were initially formulated only for the ground states of fermionic many-body systems [1, 2]. It is
therefore commonly accepted that the excited states in the context of DFT can be
accessed by the use of some form of response formalism implemented, for instance,
in the time-dependent DFT (TD-DFT) methods [3, 4]. In principle, TD-DFT is a
rigorous formulation of the ground state DFT for time-dependent phenomena [5].
However the excitation energies of many-body systems are typically accessed with
the use of the linear response (LR) formalism, which assumes that the time
dependence stems from a weak (usually oscillatory) perturbing potential [3–5]. In
practice, LR-TD-DFT yields a very reasonable description of optical absorption
spectra with the use of the commonly available ground-state approximate density
functionals [3, 4]; however, some spectacular failures of the formalism are also
known. In particular, standard implementation of LR-TD-DFT relies on the adiabatic approximation (i.e., locality of the exchange-correlation (XC) kernel in the
time domain) and consequently cannot take proper account of multiple excitations
[6, 7], which become important, e.g., for excited states of conjugated molecular
systems [8]. Yet another failure of the standard LR-TD-DFT to describe the excited
states of strongly correlated systems, e.g., H 2 at stretched bondlength [9], can be
traced back to the use of the standard ground-state Kohn–Sham (KS) formalism [2]
which fails to take proper account of the non-dynamic electron correlation.
In the domain of wavefunction theory (WFT), the excited states of molecules can
be obtained from the ground-state response formalism as well as the variational
excited state formalism [10]. An appealing idea is to employ the (time-independent)
variational formalism to obtaining excitation energies in the context of DFT.
Indeed, the first attempts to calculate the excitation energies by taking the energy
differences between the variationally obtained ground state energy and the energy
of a state obtained by promoting an electron to unoccupied energy level, the socalled ΔSCF approach,
1 date back to the early 1970s [11, 12]. However, despite
some attempts to justify the ΔSCF approach for computing the energies of oneelectron transitions between the states of different spatial symmetry [13, 14], the
idea of variationally obtaining the energy of an individual excited state in the
context of DFT lacks firm theoretical background [15–17].
A rigorous way of developing time-independent formalism for obtaining excitation energies in the context of DFT is offered by ensemble DFT [18, 19] which
1 For more details on ΔSCF, see the chapter “A Constricted Variational Density Functional Theory
Approach to the Description of Excited States” by T. Ziegler, M. Krykunov, I. Seidu, and Y. C.
Park.
98
M. Filatov
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
1 Introduction
The founding principles of density functional theory (DFT) were initially formulated only for the ground states of fermionic many-body systems [1, 2]. It is
therefore commonly accepted that the excited states in the context of DFT can be
accessed by the use of some form of response formalism implemented, for instance,
in the time-dependent DFT (TD-DFT) methods [3, 4]. In principle, TD-DFT is a
rigorous formulation of the ground state DFT for time-dependent phenomena [5].
However the excitation energies of many-body systems are typically accessed with
the use of the linear response (LR) formalism, which assumes that the time
dependence stems from a weak (usually oscillatory) perturbing potential [3–5]. In
practice, LR-TD-DFT yields a very reasonable description of optical absorption
spectra with the use of the commonly available ground-state approximate density
functionals [3, 4]; however, some spectacular failures of the formalism are also
known. In particular, standard implementation of LR-TD-DFT relies on the adiabatic approximation (i.e., locality of the exchange-correlation (XC) kernel in the
time domain) and consequently cannot take proper account of multiple excitations
[6, 7], which become important, e.g., for excited states of conjugated molecular
systems [8]. Yet another failure of the standard LR-TD-DFT to describe the excited
states of strongly correlated systems, e.g., H 2 at stretched bondlength [9], can be
traced back to the use of the standard ground-state Kohn–Sham (KS) formalism [2]
which fails to take proper account of the non-dynamic electron correlation.
In the domain of wavefunction theory (WFT), the excited states of molecules can
be obtained from the ground-state response formalism as well as the variational
excited state formalism [10]. An appealing idea is to employ the (time-independent)
variational formalism to obtaining excitation energies in the context of DFT.
Indeed, the first attempts to calculate the excitation energies by taking the energy
differences between the variationally obtained ground state energy and the energy
of a state obtained by promoting an electron to unoccupied energy level, the socalled ΔSCF approach,
1 date back to the early 1970s [11, 12]. However, despite
some attempts to justify the ΔSCF approach for computing the energies of oneelectron transitions between the states of different spatial symmetry [13, 14], the
idea of variationally obtaining the energy of an individual excited state in the
context of DFT lacks firm theoretical background [15–17].
A rigorous way of developing time-independent formalism for obtaining excitation energies in the context of DFT is offered by ensemble DFT [18, 19] which
1 For more details on ΔSCF, see the chapter “A Constricted Variational Density Functional Theory
Approach to the Description of Excited States” by T. Ziegler, M. Krykunov, I. Seidu, and Y. C.
Park.
98
M. Filatov