where the frontier (weak) spinorbitals are those that belong to a set of frontier
orbitals and their occupancy number is smaller than 1/2 (antibonding orbitals).
BBC3 functional possesses a correct orbital structure of the two-electron functional
(39) if applied to hydrogen molecule in the dissociating limit and it is able to
reproduce very accurately the whole dissociation curve [27]. Moreover, based on
examples of small atoms and diatomic molecules at equilibrium geometries and in
their dissociation limits, it has been shown that BBC3 accounts for both dynamic
and static correlation yielding correct potential energy and recovering most of the
correlation energy.
A difficulty with practical usage of the BBC3 functional is a need to select
bonding and antibonding orbitals. In a computer implementation a strongly occupied orbital of the lowest occupancy is taken as bonding although a weakly
occupied orbital of the highest occupancy is taken as an antibonding with a
straightforward extension for this selection rule for molecules with more than one
bond. This, however, leads to numerical problems because in the optimization
procedure occupation numbers are varied and the antibonding character of orbitals
may change, which may cause problems with convergence or may result in
obtaining discontinuities in potential energy curves. To avoid the previously
described problems with selecting frontier orbitals and to improve the overall
accuracy of the BBC3 functional, it has been proposed to replace the elements
G pq present in the BBC functional, cf. (41), with a function G(n p ,n q ). The function
mimics the behavior of the G
BBC3
pq
elements but does it automatically, based on the
values of its arguments. The optimal function has been found by introducing two
parameters and fitting the resulting AC3 functional to accurate energies of two
molecules at equilibrium and stretched-bond geometries [40]. The AC3 functional
yields decent quality potential energy curves for ten-electron molecules, although
for some molecules a small hump in the curve is visible.
A two-electron wavefunction (33) is a special case of the more general closedshell N-electron ansatz (N assumed to be even) involving, apart from a reference
determinant Φ 0 , all determinants arising from diagonal double, diagonal quadruple,
etc., excitations; cf. (67). Taking the expectation value of the Hamiltonian with
such a wavefunction yields an energy expression involving only Coulomb,
exchange, and integrals of the hpp|qqi type [41]. In [42] the previously mentioned
ansatz for the wavefunction has been used in development of the extended L€ owdin–
Shull (ELS) functional. The functional is applicable to systems for which a set of
the natural spinorbitals can be partitioned into “inner” orbitals localized on atoms
and the occupancies close to 1 and “outer” orbitals including a bonding orbital and
all weakly occupied orbitals, i.e., orbitals localized on a single bond. For N-electron
(N being even) systems (N/2 À 1) strongly occupied orbitals (in a spin-restricted
formulation each orbital gives rise to two spinorbitals with opposite spins) would be
considered “inner” and the remaining strongly occupied orbital of the lowest
occupancy would belong to the “outer” orbital set. By analyzing a structure of the
energy expression resulting from the assumed ansatz, the following form of the ELS
functional has been proposed
136
K. Pernal and K.J.H. Giesbertz
orbitals and their occupancy number is smaller than 1/2 (antibonding orbitals).
BBC3 functional possesses a correct orbital structure of the two-electron functional
(39) if applied to hydrogen molecule in the dissociating limit and it is able to
reproduce very accurately the whole dissociation curve [27]. Moreover, based on
examples of small atoms and diatomic molecules at equilibrium geometries and in
their dissociation limits, it has been shown that BBC3 accounts for both dynamic
and static correlation yielding correct potential energy and recovering most of the
correlation energy.
A difficulty with practical usage of the BBC3 functional is a need to select
bonding and antibonding orbitals. In a computer implementation a strongly occupied orbital of the lowest occupancy is taken as bonding although a weakly
occupied orbital of the highest occupancy is taken as an antibonding with a
straightforward extension for this selection rule for molecules with more than one
bond. This, however, leads to numerical problems because in the optimization
procedure occupation numbers are varied and the antibonding character of orbitals
may change, which may cause problems with convergence or may result in
obtaining discontinuities in potential energy curves. To avoid the previously
described problems with selecting frontier orbitals and to improve the overall
accuracy of the BBC3 functional, it has been proposed to replace the elements
G pq present in the BBC functional, cf. (41), with a function G(n p ,n q ). The function
mimics the behavior of the G
BBC3
pq
elements but does it automatically, based on the
values of its arguments. The optimal function has been found by introducing two
parameters and fitting the resulting AC3 functional to accurate energies of two
molecules at equilibrium and stretched-bond geometries [40]. The AC3 functional
yields decent quality potential energy curves for ten-electron molecules, although
for some molecules a small hump in the curve is visible.
A two-electron wavefunction (33) is a special case of the more general closedshell N-electron ansatz (N assumed to be even) involving, apart from a reference
determinant Φ 0 , all determinants arising from diagonal double, diagonal quadruple,
etc., excitations; cf. (67). Taking the expectation value of the Hamiltonian with
such a wavefunction yields an energy expression involving only Coulomb,
exchange, and integrals of the hpp|qqi type [41]. In [42] the previously mentioned
ansatz for the wavefunction has been used in development of the extended L€ owdin–
Shull (ELS) functional. The functional is applicable to systems for which a set of
the natural spinorbitals can be partitioned into “inner” orbitals localized on atoms
and the occupancies close to 1 and “outer” orbitals including a bonding orbital and
all weakly occupied orbitals, i.e., orbitals localized on a single bond. For N-electron
(N being even) systems (N/2 À 1) strongly occupied orbitals (in a spin-restricted
formulation each orbital gives rise to two spinorbitals with opposite spins) would be
considered “inner” and the remaining strongly occupied orbital of the lowest
occupancy would belong to the “outer” orbital set. By analyzing a structure of the
energy expression resulting from the assumed ansatz, the following form of the ELS
functional has been proposed
136
K. Pernal and K.J.H. Giesbertz