excitation energies decrease somewhat when correcting the incorrect ω ! 0 limit in
the adiabatic approximation and it depends on the particular system whether this is
an improvement. For H 2 the results are slightly worse in the alternative adiabatic
approximation, whereas for HeH
+ they are slightly better [117, 142].
Although the alternative adiabatic approximation is successful in correcting the
ω ! 0 limit of the standard adiabatic approximation, this adiabatic approximation is
still not very satisfactory, because the occupation numbers are still not truly
dynamic variables and we still lose m excitations. These m excitations correspond
to excitations which require a significant response of the occupation numbers.
Because the response of the occupation numbers corresponds to the response of
the diagonal of the 1-RDM (98), so they are referred to as the diagonal double
excitations. Other double excitations related to perturbations in the off-diagonal
parts of the 1-RDM are well accounted for, as the excitation energies in the
1
Σ
þ
u and
1
Π u are perfectly accounted for [117, 142]. It turns out that these diagonal double
excitations are important for the correct description of the lowest
1
Σ
þ
g excitation
energy in stretched H 2 [141, 142], so including the diagonal double excitations is
important.
5.5 Phase Including Natural Orbitals
It is unlikely that the missing m diagonal double excitations can be restored with
any decent adiabatic approximation to the TD-RDMFT equations. The problem is
that the excitation energies should come out of the response equations in pairs +ω
and Àω. This pairing of the frequencies is dictated by an important symmetry of the
response function χ ω
ð Þ ¼ χ* Àω
ð Þ, which follows directly from the Lehmann [150]
(sum-over-states) representation. We therefore need to increase the number of roots
by m in some manner, because m is not necessarily even. Increasing the number of
roots to 2(M + m) results in an even number of roots, which in turn ensures that all
excitations are present in both the positive and negative parts of the spectrum.
This partially explains why we had m zero excitations in the standard adiabatic
approximation, because ω ¼ 0 is the only number which is its own negative, so it
does not destroy the χ ω
ð Þ ¼ χ* Àω
ð Þsymmetry even if an odd number of these roots
is present. This does not explain why we could not have bm/2c excitation energies
occurring both at +ω and Àω in an adiabatic approximation. To explain this, we
observe that for a proper quantum evolution a quantity needs to be able to have a
complex phase. All the off-diagonal elements of the 1-RDM are able to obtain a
complex phase-factor, but because the diagonal is necessarily real, the occupation
numbers do not have a quantum phase [117]. This lack of a corresponding quantum
phase for the natural occupation numbers is not limited to the 1-RDM, but exists for
the diagonal of any p-RDM if the BBGKY hierarchy is truncated at the pth order [151].
The way to solve all these problems together is to include an additional set of
m complex phase factors which can act as the conjugate variables for the natural
166
K. Pernal and K.J.H. Giesbertz
the adiabatic approximation and it depends on the particular system whether this is
an improvement. For H 2 the results are slightly worse in the alternative adiabatic
approximation, whereas for HeH
+ they are slightly better [117, 142].
Although the alternative adiabatic approximation is successful in correcting the
ω ! 0 limit of the standard adiabatic approximation, this adiabatic approximation is
still not very satisfactory, because the occupation numbers are still not truly
dynamic variables and we still lose m excitations. These m excitations correspond
to excitations which require a significant response of the occupation numbers.
Because the response of the occupation numbers corresponds to the response of
the diagonal of the 1-RDM (98), so they are referred to as the diagonal double
excitations. Other double excitations related to perturbations in the off-diagonal
parts of the 1-RDM are well accounted for, as the excitation energies in the
1
Σ
þ
u and
1
Π u are perfectly accounted for [117, 142]. It turns out that these diagonal double
excitations are important for the correct description of the lowest
1
Σ
þ
g excitation
energy in stretched H 2 [141, 142], so including the diagonal double excitations is
important.
5.5 Phase Including Natural Orbitals
It is unlikely that the missing m diagonal double excitations can be restored with
any decent adiabatic approximation to the TD-RDMFT equations. The problem is
that the excitation energies should come out of the response equations in pairs +ω
and Àω. This pairing of the frequencies is dictated by an important symmetry of the
response function χ ω
ð Þ ¼ χ* Àω
ð Þ, which follows directly from the Lehmann [150]
(sum-over-states) representation. We therefore need to increase the number of roots
by m in some manner, because m is not necessarily even. Increasing the number of
roots to 2(M + m) results in an even number of roots, which in turn ensures that all
excitations are present in both the positive and negative parts of the spectrum.
This partially explains why we had m zero excitations in the standard adiabatic
approximation, because ω ¼ 0 is the only number which is its own negative, so it
does not destroy the χ ω
ð Þ ¼ χ* Àω
ð Þsymmetry even if an odd number of these roots
is present. This does not explain why we could not have bm/2c excitation energies
occurring both at +ω and Àω in an adiabatic approximation. To explain this, we
observe that for a proper quantum evolution a quantity needs to be able to have a
complex phase. All the off-diagonal elements of the 1-RDM are able to obtain a
complex phase-factor, but because the diagonal is necessarily real, the occupation
numbers do not have a quantum phase [117]. This lack of a corresponding quantum
phase for the natural occupation numbers is not limited to the 1-RDM, but exists for
the diagonal of any p-RDM if the BBGKY hierarchy is truncated at the pth order [151].
The way to solve all these problems together is to include an additional set of
m complex phase factors which can act as the conjugate variables for the natural
166
K. Pernal and K.J.H. Giesbertz