functional does not follow the exact polarizability as closely as the DMLS, it has the
main advantage that no spurious divergences emerge, even when all natural orbital
transitions are taken into account in the response calculation [117, 121, 147].
Another disadvantage of the standard adiabatic approximation is that we effectively lose excitation energies. Because the original frequency-dependent response
matrix has M + M + m ¼ m
2 entries, the standard adiabatic approximation yields m
2
excitation energies, because the response matrices A and C are frequency independent. However, we only obtain 2M ¼ m(m À 1) sensible excitation energies and
m roots ω ¼ 0, which are physically meaningless, so, effectively, these m excitation
energies are lost in the standard adiabatic approximation [142].
Test calculations on the excitation spectrum of the hydrogen molecule as a
function of the bond length have been carried out to test the performance of both
LS functionals. Because divergences in the polarizability correspond to excitations,
the spurious divergences of the DMLS functional already indicate that the performance of the DMLS functional for the calculation of excitation energies is not very
good. Indeed, the test calculations on H 2 have shown that the DMLS functional
predicts many spurious low lying excitations which completely clutter the excitation spectrum when all natural orbital transitions are taken into account [117, 145,
148]. Reducing the number of transitions to only transitions from the two heaviest
occupied natural orbitals is very effective in cleaning up the DMLS excitation
spectrum [117]. Such an approach would not be desirable in practice, because it is
orthogonal to the idea that expanding a basis brings one closer to the desired result.
Because the PILS functional is dependent on the phase of the natural orbitals, the
occupation numbers are not necessarily stationary any more in the standard adiabatic approximation. However, it can be demonstrated that there are still only
2M ¼ m(m À 1) non-trivial roots of the response equations (113) and m zero excitations [117]. Nevertheless, the PILS functional gives a huge improvement over the
DMLS functional for excitation energies. Most notably, no spurious low lying
excitations appear when we exhaust the response basis by including more natural
orbital transitions. Furthermore, one can show that the
1
Σ
þ
u ,
1
Π g and
1
Π u excitations
become equal to the full CI result when all natural orbital transitions are taken into
account [117, 141, 142]. This is caused by the fact that these excitations do not need
any perturbation in the natural occupation numbers to be described exactly, which
is related to symmetry. This also holds for excitations in other irreducible representations (irreps) that do not couple to the completely symmetric irreducible part
of the response matrix, such as the xy component of the Δ g excitations. The x
2
À y
2
component does couple to the occupation numbers, however, so the Δ g excitations
of the H 2 molecule is symmetry broken when using the PILS functional: the xy
components are equal to the full CI result and the x
2
À y
2 components are not
[117]. Symmetry breaking does not occur for the DMLS functional, because the
occupation numbers are never involved in the standard adiabatic response.
164
K. Pernal and K.J.H. Giesbertz
main advantage that no spurious divergences emerge, even when all natural orbital
transitions are taken into account in the response calculation [117, 121, 147].
Another disadvantage of the standard adiabatic approximation is that we effectively lose excitation energies. Because the original frequency-dependent response
matrix has M + M + m ¼ m
2 entries, the standard adiabatic approximation yields m
2
excitation energies, because the response matrices A and C are frequency independent. However, we only obtain 2M ¼ m(m À 1) sensible excitation energies and
m roots ω ¼ 0, which are physically meaningless, so, effectively, these m excitation
energies are lost in the standard adiabatic approximation [142].
Test calculations on the excitation spectrum of the hydrogen molecule as a
function of the bond length have been carried out to test the performance of both
LS functionals. Because divergences in the polarizability correspond to excitations,
the spurious divergences of the DMLS functional already indicate that the performance of the DMLS functional for the calculation of excitation energies is not very
good. Indeed, the test calculations on H 2 have shown that the DMLS functional
predicts many spurious low lying excitations which completely clutter the excitation spectrum when all natural orbital transitions are taken into account [117, 145,
148]. Reducing the number of transitions to only transitions from the two heaviest
occupied natural orbitals is very effective in cleaning up the DMLS excitation
spectrum [117]. Such an approach would not be desirable in practice, because it is
orthogonal to the idea that expanding a basis brings one closer to the desired result.
Because the PILS functional is dependent on the phase of the natural orbitals, the
occupation numbers are not necessarily stationary any more in the standard adiabatic approximation. However, it can be demonstrated that there are still only
2M ¼ m(m À 1) non-trivial roots of the response equations (113) and m zero excitations [117]. Nevertheless, the PILS functional gives a huge improvement over the
DMLS functional for excitation energies. Most notably, no spurious low lying
excitations appear when we exhaust the response basis by including more natural
orbital transitions. Furthermore, one can show that the
1
Σ
þ
u ,
1
Π g and
1
Π u excitations
become equal to the full CI result when all natural orbital transitions are taken into
account [117, 141, 142]. This is caused by the fact that these excitations do not need
any perturbation in the natural occupation numbers to be described exactly, which
is related to symmetry. This also holds for excitations in other irreducible representations (irreps) that do not couple to the completely symmetric irreducible part
of the response matrix, such as the xy component of the Δ g excitations. The x
2
À y
2
component does couple to the occupation numbers, however, so the Δ g excitations
of the H 2 molecule is symmetry broken when using the PILS functional: the xy
components are equal to the full CI result and the x
2
À y
2 components are not
[117]. Symmetry breaking does not occur for the DMLS functional, because the
occupation numbers are never involved in the standard adiabatic response.
164
K. Pernal and K.J.H. Giesbertz