operates with weighted sums (ensembles) of fractionally occupied (ground and
excited) states. The ensemble representation of the density and the energy for an
arbitrary many-body fermionic system was put on a firm theoretical ground by Lieb
[20] and Englisch and Englisch [21, 22], and was later extended to the domain of
excited states by Gross et al. [23–25].
A practical demonstration of the necessity to invoke the ensemble representation
for mapping the density of a strongly correlated system onto a non-interacting KS
reference was achieved by Baerends et al. [26, 27] in first-principles numeric
simulations employing the (nearly) exact molecular densities, which was later
confirmed by Morrison [28] in a series of first-principles atomic calculations.
Although the ensemble formalism enables one to obtain excitation energies in a
rigorous and computationally convenient way [5], the progress in this direction was
extremely slow [29–33], perhaps because of the perceived lack of suitable density
functionals capable to accommodate the densities with fractional occupation numbers (FONs).
A practically accessible approach to the calculation of the strongly correlated
ground and excited states of molecules which employs the ideas behind ensemble
DFT was achieved in the form of the spin-restricted ensemble-referenced KS
(REKS) method [34–41]. The method was initially developed for the ground states
of strongly correlated molecular systems [34–38] and was later extended to the
domain of excited state calculations [39–41]. Although the REKS method is
founded on a rigorous theoretical background [20, 21, 26] and was successfully
applied to study situations often intractable with the use of the conventional KS
DFT methods [37, 38, 40–61], the method has received a little attention in the
literature and has been largely overlooked by the computational chemistry community. In this chapter, an overview of the REKS methodology and its connection
to the ensemble DFT formalism is given with emphasis on the use of the method to
obtain excited states of molecular systems.
2 Ensemble DFT
The basic tenet of KS DFT is that any physical fermionic ground state density ρ(r)
can be uniquely mapped onto the ground state density ρ s (r) of a fictitious system of
non-interacting particles moving in a suitably modified external potential v s (r). If
such a v s (r), which is also known as the KS potential, can be found, the respective
KS Hamiltonian H ˆ
s is minimized by a single Slater determinant (KS determinant)
constructed from the lowest-energy one-electron functions (KS orbitals) φ s,i (r) and
the non-interacting density ρ s (r) is
ρ s r
ð Þ ¼
X
i
2
φ s, i r
ð Þ
2 ; 8ε i μ ;
ð1Þ
where ε i are the respective eigenvalues, μ is the Fermi level, and a closed electronic
Ensemble DFT Approach to Excited States of Strongly Correlated Molecular Systems
99
excited) states. The ensemble representation of the density and the energy for an
arbitrary many-body fermionic system was put on a firm theoretical ground by Lieb
[20] and Englisch and Englisch [21, 22], and was later extended to the domain of
excited states by Gross et al. [23–25].
A practical demonstration of the necessity to invoke the ensemble representation
for mapping the density of a strongly correlated system onto a non-interacting KS
reference was achieved by Baerends et al. [26, 27] in first-principles numeric
simulations employing the (nearly) exact molecular densities, which was later
confirmed by Morrison [28] in a series of first-principles atomic calculations.
Although the ensemble formalism enables one to obtain excitation energies in a
rigorous and computationally convenient way [5], the progress in this direction was
extremely slow [29–33], perhaps because of the perceived lack of suitable density
functionals capable to accommodate the densities with fractional occupation numbers (FONs).
A practically accessible approach to the calculation of the strongly correlated
ground and excited states of molecules which employs the ideas behind ensemble
DFT was achieved in the form of the spin-restricted ensemble-referenced KS
(REKS) method [34–41]. The method was initially developed for the ground states
of strongly correlated molecular systems [34–38] and was later extended to the
domain of excited state calculations [39–41]. Although the REKS method is
founded on a rigorous theoretical background [20, 21, 26] and was successfully
applied to study situations often intractable with the use of the conventional KS
DFT methods [37, 38, 40–61], the method has received a little attention in the
literature and has been largely overlooked by the computational chemistry community. In this chapter, an overview of the REKS methodology and its connection
to the ensemble DFT formalism is given with emphasis on the use of the method to
obtain excited states of molecular systems.
2 Ensemble DFT
The basic tenet of KS DFT is that any physical fermionic ground state density ρ(r)
can be uniquely mapped onto the ground state density ρ s (r) of a fictitious system of
non-interacting particles moving in a suitably modified external potential v s (r). If
such a v s (r), which is also known as the KS potential, can be found, the respective
KS Hamiltonian H ˆ
s is minimized by a single Slater determinant (KS determinant)
constructed from the lowest-energy one-electron functions (KS orbitals) φ s,i (r) and
the non-interacting density ρ s (r) is
ρ s r
ð Þ ¼
X
i
2
φ s, i r
ð Þ
2 ; 8ε i μ ;
ð1Þ
where ε i are the respective eigenvalues, μ is the Fermi level, and a closed electronic
Ensemble DFT Approach to Excited States of Strongly Correlated Molecular Systems
99