calculated by the GGA functionals BP86, BLYP, and revBEP differ on average by
less than 0.05 eV from LDA estimates and introducing SAOP or GRAC with the
right asymptotic 1/r behavior does not lead to any significant change. Charge
transfer transitions can also be described well by the variational DFT-based spinrestricted ensemble referenced Kohn–Sham (REKS) method [57] (see footnote 1).
It is at this point important to note that the experimental excitation energies for
the anthracene systems were all obtained in solution with CH 3 Cl as the solvent. We
do not expect the solvent effect to be significant. In fact, theoretical calculations
[33] using a continuum model revealed that the excitation energies were lower by
only 0.05 eV. We have, as a consequence, decided to compare our gas-phase results
directly with the experimental solvent data. We obtain from such a comparison that
the RMSD is 0.06 for LDA followed by 0.07 for BP86, revPBE, and 0.08 for BLYP.
The two 1/r asymptotically corrected functionals afford 0.10 for SAOP and 0.07 for
CRAC. Stein et al. [33] “SKB” introduced in their DZP gas-phase study a uniform
correction of À0.32 eV to simulate solvation effects; see Table 8. The magnitude
and sign of this correction was given without much explanation [33]. After applying
their correction the authors obtained an RMSD of 0.1 which is quite similar to the
one found here for our gas-phase results without any solvent correction. It should be
noted that ATDDFT with the same functionals carries errors of up to 1 eV.
3 Concluding Remarks
We have here reviewed the theoretical foundation of constricted variational density
functional theory and illustrated its scope through applications. CV(n)-DFT encompasses adiabatic TDDFT and ΔSCF-DFT as special cases. Thus our variational
second order CV(2)-DFT is identical to adiabatic TDDFT ground state response
theory [29, 62] and ΔSCF-DFT is the same as RSCF-CV(1)-DFT in the case
where the transition is described by a single orbital replacement with γ of (18) equal
to π/2 [28]. CV(n)-DFT can be used as a natural extension of adiabatic TDDFT. The
first step in this direction is the perturbative P-CV(1)-DFT approach [64] in which
the U from CV(2)-DFT is used to calculate the all order energy in CV(1)-DFT [27,
64]. It is shown to work well for π ! π* transitions in conjugated systems. At a
higher level, U is optimized with respect to the all order energy in CV(1)-DFT
scheme leading to SCF-CV(1)-DFT [28]. Experience has shown [27, 63] that
optimization of U alone is insufficient. One also has to relax all the other occupied
orbitals which do not directly participate in the transition. This is done in SCF-CV
(1)-DFT by introducing orbital relaxation (RSCF-CV(1)-DFT) [26]. The RSCFCV(1)-DFT scheme differs from adiabatic TDDFT (CV(2)-DFT) by going to all
orders in U and by introducing orbital relaxation. The extra effort involved in
connection with RSCF-CV(1)-DFT compared to adiabatic TDDFT does not result
in improved accuracy for cases where adiabatic TDDFT fares well, such as for the
π ! π* transition [26]. However, it does not fail for charge transfer [30, 31] and
Rydberg transitions [66] in the way adiabatic TDDFT does for regular functionals.
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T. Ziegler et al.
less than 0.05 eV from LDA estimates and introducing SAOP or GRAC with the
right asymptotic 1/r behavior does not lead to any significant change. Charge
transfer transitions can also be described well by the variational DFT-based spinrestricted ensemble referenced Kohn–Sham (REKS) method [57] (see footnote 1).
It is at this point important to note that the experimental excitation energies for
the anthracene systems were all obtained in solution with CH 3 Cl as the solvent. We
do not expect the solvent effect to be significant. In fact, theoretical calculations
[33] using a continuum model revealed that the excitation energies were lower by
only 0.05 eV. We have, as a consequence, decided to compare our gas-phase results
directly with the experimental solvent data. We obtain from such a comparison that
the RMSD is 0.06 for LDA followed by 0.07 for BP86, revPBE, and 0.08 for BLYP.
The two 1/r asymptotically corrected functionals afford 0.10 for SAOP and 0.07 for
CRAC. Stein et al. [33] “SKB” introduced in their DZP gas-phase study a uniform
correction of À0.32 eV to simulate solvation effects; see Table 8. The magnitude
and sign of this correction was given without much explanation [33]. After applying
their correction the authors obtained an RMSD of 0.1 which is quite similar to the
one found here for our gas-phase results without any solvent correction. It should be
noted that ATDDFT with the same functionals carries errors of up to 1 eV.
3 Concluding Remarks
We have here reviewed the theoretical foundation of constricted variational density
functional theory and illustrated its scope through applications. CV(n)-DFT encompasses adiabatic TDDFT and ΔSCF-DFT as special cases. Thus our variational
second order CV(2)-DFT is identical to adiabatic TDDFT ground state response
theory [29, 62] and ΔSCF-DFT is the same as RSCF-CV(1)-DFT in the case
where the transition is described by a single orbital replacement with γ of (18) equal
to π/2 [28]. CV(n)-DFT can be used as a natural extension of adiabatic TDDFT. The
first step in this direction is the perturbative P-CV(1)-DFT approach [64] in which
the U from CV(2)-DFT is used to calculate the all order energy in CV(1)-DFT [27,
64]. It is shown to work well for π ! π* transitions in conjugated systems. At a
higher level, U is optimized with respect to the all order energy in CV(1)-DFT
scheme leading to SCF-CV(1)-DFT [28]. Experience has shown [27, 63] that
optimization of U alone is insufficient. One also has to relax all the other occupied
orbitals which do not directly participate in the transition. This is done in SCF-CV
(1)-DFT by introducing orbital relaxation (RSCF-CV(1)-DFT) [26]. The RSCFCV(1)-DFT scheme differs from adiabatic TDDFT (CV(2)-DFT) by going to all
orders in U and by introducing orbital relaxation. The extra effort involved in
connection with RSCF-CV(1)-DFT compared to adiabatic TDDFT does not result
in improved accuracy for cases where adiabatic TDDFT fares well, such as for the
π ! π* transition [26]. However, it does not fail for charge transfer [30, 31] and
Rydberg transitions [66] in the way adiabatic TDDFT does for regular functionals.
90
T. Ziegler et al.