and ϕ b [34, 35, 77]. In (29), ^
F a and ^
F b are the Fock operators for the open-shell
orbitals and ε ab is the off-diagonal Lagrange multiplier
4 in the open-shell Lagrangian [73]. As the two states, Φ 0 and Φ 1 , are mutually orthogonal, the average of the
new energies E 0 and E 1 obtained from the above secular problem remains the same
as the average of the REKS(2,2) and ROKS energies. This implies that the orbitals
for the new approach, dubbed SI-SA-REKS or SSR, can still be obtained from the
SA-REKS orbital optimization, provided that ω ¼ 1/2 was employed in the latter. In
practical applications of the SI-SA-REKS method [40, 41, 59–61], it was found that
the described state-interaction scheme is important for obtaining the correct shape
of the ground and excited state PESs in the vicinity of conical intersections and near
avoided crossings. For other situations, when the energy gap between the ground
and excited states is sufficiently wide the SI-SA-REKS method yields nearly the
same excitation energies as the SA-REKS method [59].
The argument leading to the SI-SA-REKS method can be proposed based on
the adiabatic connection formalism for ensemble DFT as advocated by
Fromager et al. [32]. Setting the coupling strength α in the Hamiltonian
(15) to zero leads to the degeneracy of the states represented by (25) and
(26). Applying the quasi-degenerate perturbation theory results in a 2 Â 2
secular problem
E
α
0
H
α
01
H
α
10
E
α
1
, where the off-diagonal elements are given by
(29) for the intermediate coupling strength. Employing the coupling strength
integration and invoking the assumptions used in (22) one arrives at the
energy expressions for the ground and excited states of the SI-SA-REKS
method (see the paragraph above). It should be noted that, for a
homosymmetric molecule, such as H
2
, the off-diagonal matrix element vanishes by symmetry and the SI-SA-REKS description collapses to the SAREKS one.
To illustrate how the SI-SA-REKS method describes dissociation of a
heteropolar chemical bond, let us briefly review the ground and the lowest excited
singlet states of the LiH molecule. Near the equilibrium bondlength, the ground
state of the LiH molecule has ionic character with ca. 0.3 e ¯ shifted to the hydrogen
atom. When the Li–H bond dissociates, the ground state undergoes an avoided
crossing with the excited state, which has covalent character, and, at the dissociation limit, the ground state corresponds to a covalent configuration with two
electrically neutral atoms.
The potential energy curves of the ground x
1
Σ
+ and the excited a
1
Σ
+ states of
LiH are shown in Fig. 4. The results of the SI-SA-REKS calculations using the LC4 The matrix of Lagrange multipliers in open-shell SCF becomes Hermitian (but not diagonal)
upon convergence to the variational minimum [77].
114
M. Filatov
F a and ^
F b are the Fock operators for the open-shell
orbitals and ε ab is the off-diagonal Lagrange multiplier
4 in the open-shell Lagrangian [73]. As the two states, Φ 0 and Φ 1 , are mutually orthogonal, the average of the
new energies E 0 and E 1 obtained from the above secular problem remains the same
as the average of the REKS(2,2) and ROKS energies. This implies that the orbitals
for the new approach, dubbed SI-SA-REKS or SSR, can still be obtained from the
SA-REKS orbital optimization, provided that ω ¼ 1/2 was employed in the latter. In
practical applications of the SI-SA-REKS method [40, 41, 59–61], it was found that
the described state-interaction scheme is important for obtaining the correct shape
of the ground and excited state PESs in the vicinity of conical intersections and near
avoided crossings. For other situations, when the energy gap between the ground
and excited states is sufficiently wide the SI-SA-REKS method yields nearly the
same excitation energies as the SA-REKS method [59].
The argument leading to the SI-SA-REKS method can be proposed based on
the adiabatic connection formalism for ensemble DFT as advocated by
Fromager et al. [32]. Setting the coupling strength α in the Hamiltonian
(15) to zero leads to the degeneracy of the states represented by (25) and
(26). Applying the quasi-degenerate perturbation theory results in a 2 Â 2
secular problem
E
α
0
H
α
01
H
α
10
E
α
1
, where the off-diagonal elements are given by
(29) for the intermediate coupling strength. Employing the coupling strength
integration and invoking the assumptions used in (22) one arrives at the
energy expressions for the ground and excited states of the SI-SA-REKS
method (see the paragraph above). It should be noted that, for a
homosymmetric molecule, such as H
2
, the off-diagonal matrix element vanishes by symmetry and the SI-SA-REKS description collapses to the SAREKS one.
To illustrate how the SI-SA-REKS method describes dissociation of a
heteropolar chemical bond, let us briefly review the ground and the lowest excited
singlet states of the LiH molecule. Near the equilibrium bondlength, the ground
state of the LiH molecule has ionic character with ca. 0.3 e ¯ shifted to the hydrogen
atom. When the Li–H bond dissociates, the ground state undergoes an avoided
crossing with the excited state, which has covalent character, and, at the dissociation limit, the ground state corresponds to a covalent configuration with two
electrically neutral atoms.
The potential energy curves of the ground x
1
Σ
+ and the excited a
1
Σ
+ states of
LiH are shown in Fig. 4. The results of the SI-SA-REKS calculations using the LC4 The matrix of Lagrange multipliers in open-shell SCF becomes Hermitian (but not diagonal)
upon convergence to the variational minimum [77].
114
M. Filatov