[15, 16]. Also GGA potentials decay too fast in the asymptotic region, e.g., like
À c/r
2 [9]. As a consequence, artificial Rydberg states appear in the spectrum of
finite (LCAO) basis set calculations and are unbound in the basis set limit. Asymptotic correction schemes have been proposed that recover the Rydberg series
without changing the spectrum of valence excitations, but they require an experimental or estimated values for the IE as input [16–18].
A more challenging problem for the adiabatic approximations to TDDFT are
charge-transfer (CT) excitations, in which an electron is promoted from a donor to
an acceptor orbital that is located in a different region of the system. Excitations
with partial CT character are most common in organic dyes, biological
chromophores and of particular interest for applications in dye-sensitised solar
cells. In these systems, the donor and acceptor states are coupled and their wave
functions overlap to create a large transition dipole moment. Depending on the
spatial separation of donor and acceptor states, adiabatic approximations drastically
underestimate such transitions. The problem is best understood for the case of
non-overlapping donor/acceptor orbitals that are located on two separate molecules
[19]. In this case, the product ϕ i (r)ϕ a (r) vanishes in Eq. (4.3) and the excitation
energy of the CT transition reduces to the difference of the Kohn–Sham energy
levels, e.g., ε LUMO
acceptor
À ε HOMO
donor . The exact excitation energy, for infinite distance
R between the molecules, would be the difference between the IE of the donor
and the EA of the acceptor site. If asymptotically corrected potentials are
used, À ε HOMO
donor indeedapproximates the exact IE, but ε LUMO
acceptor does not approximate
the EA of the acceptor, unlike in Hartree–Fock (HF) theory, because of the missing
XC part of the derivative discontinuity of the energy with respect to the particle
number [20]. This leads to a drastic underestimation of CT excitations by common
density functionals. At finite separation R between the molecules, the energy of the
CT excited state is lowered by the Coulomb attraction between the hole created on
the donor and the excess electron on the acceptor, and additional higher-order
polarisation terms. In time-dependent HF, or configuration interaction singles
(CIS), which is the same for CT states, the particle–hole term is included in the
response calculation due to the non-local exchange operator. In TDDFT, the exact
XC kernel diverges with vanishing overlap of the donor/acceptor orbitals [21],
which is not described by common density functionals. Therefore, the latter lack the
correct 1/R asymptotic behavior of CT transitions.
The underestimation of CT excitations by local density functionals within the
adiabatic approximation to TDDFT has two important implications. (1) When the
QM system is extended to include parts of the environment (solvent, protein, or a
surface), the spectrum of valence excitations on the chromophore can be messed up
by a multitude of artificially low CT excitations that appear in the same energy
region. Although these are expected to couple weakly with the long-range CT
excitations, they tend to mix strongly with the latter when the energy difference
is small [22, 23]. (2) In general, the amount of CT character depends on the
geometry and can vary substantially along certain reaction coordinates. Therefore,
the excited-state PES can be distorted such that reaction pathways towards structures
that enhance the CT character are energetically favoured [24].
4 Theoretical Methods
49
À c/r
2 [9]. As a consequence, artificial Rydberg states appear in the spectrum of
finite (LCAO) basis set calculations and are unbound in the basis set limit. Asymptotic correction schemes have been proposed that recover the Rydberg series
without changing the spectrum of valence excitations, but they require an experimental or estimated values for the IE as input [16–18].
A more challenging problem for the adiabatic approximations to TDDFT are
charge-transfer (CT) excitations, in which an electron is promoted from a donor to
an acceptor orbital that is located in a different region of the system. Excitations
with partial CT character are most common in organic dyes, biological
chromophores and of particular interest for applications in dye-sensitised solar
cells. In these systems, the donor and acceptor states are coupled and their wave
functions overlap to create a large transition dipole moment. Depending on the
spatial separation of donor and acceptor states, adiabatic approximations drastically
underestimate such transitions. The problem is best understood for the case of
non-overlapping donor/acceptor orbitals that are located on two separate molecules
[19]. In this case, the product ϕ i (r)ϕ a (r) vanishes in Eq. (4.3) and the excitation
energy of the CT transition reduces to the difference of the Kohn–Sham energy
levels, e.g., ε LUMO
acceptor
À ε HOMO
donor . The exact excitation energy, for infinite distance
R between the molecules, would be the difference between the IE of the donor
and the EA of the acceptor site. If asymptotically corrected potentials are
used, À ε HOMO
donor indeedapproximates the exact IE, but ε LUMO
acceptor does not approximate
the EA of the acceptor, unlike in Hartree–Fock (HF) theory, because of the missing
XC part of the derivative discontinuity of the energy with respect to the particle
number [20]. This leads to a drastic underestimation of CT excitations by common
density functionals. At finite separation R between the molecules, the energy of the
CT excited state is lowered by the Coulomb attraction between the hole created on
the donor and the excess electron on the acceptor, and additional higher-order
polarisation terms. In time-dependent HF, or configuration interaction singles
(CIS), which is the same for CT states, the particle–hole term is included in the
response calculation due to the non-local exchange operator. In TDDFT, the exact
XC kernel diverges with vanishing overlap of the donor/acceptor orbitals [21],
which is not described by common density functionals. Therefore, the latter lack the
correct 1/R asymptotic behavior of CT transitions.
The underestimation of CT excitations by local density functionals within the
adiabatic approximation to TDDFT has two important implications. (1) When the
QM system is extended to include parts of the environment (solvent, protein, or a
surface), the spectrum of valence excitations on the chromophore can be messed up
by a multitude of artificially low CT excitations that appear in the same energy
region. Although these are expected to couple weakly with the long-range CT
excitations, they tend to mix strongly with the latter when the energy difference
is small [22, 23]. (2) In general, the amount of CT character depends on the
geometry and can vary substantially along certain reaction coordinates. Therefore,
the excited-state PES can be distorted such that reaction pathways towards structures
that enhance the CT character are energetically favoured [24].
4 Theoretical Methods
49