situations where the homolytic breaking of the π-bond is energetically preferred,
and as twist-bond_length_alternation (tw-BLA) CIs, for cases when the two π-bond
breaking mechanisms, homolytic and heterolytic, either become nearly isoenergetic
or the latter mechanism is preferred [81]. An accurate description of this type of CI
requires a balanced description of the relative stability of diradical and ionic
electronic configurations; thus the ability of computational methods to deliver
such a description is probed by this type of CI.
Another type of CI shown in Fig. 3 can be described as n/π* CIs that originate
from the crossing between an electron configuration with a doubly occupied lone
pair orbital (n
2
π*
0 ) and a singly excited configuration (n
1
π*
1 ) [97]. These CIs occur
in ketene, ethylene (ethylidene or methylcarbene CI), and methyliminium cation
(methylimine CI) as shown in Fig. 3. This type of CI probes the accuracy of
description of lone-pair excitations by the tested theoretical methods.
The set of CIs shown in Fig. 3 misses some other types, in particular those found
in cyclic molecules, such as nucleobases; however, good-quality reference
MRCISD data for these CIs would be extremely difficult to obtain because of the
large size of these molecules. We therefore omit these CIs from comparisons and
focus instead on the ability of the DFT computational schemes to reproduce the
molecular geometry and relative energy of the MECI points and the shape of the S 0
and S 1 PESs in their vicinity. To this end, we compare the BP vectors obtained using
the DFT approaches with the respective MRCISD vectors, and we investigate the S 0
and S 1 PESs around the optimized MECI points by scanning the PESs (1) in the
direction of the BP vectors and (2) along a closed path around the MECI point
[98]. The latter scans enable us to evaluate the ability of the respective theoretical
methods to describe the correct dimensionality of the CI seam [43, 98].
As is evident from Table 1, all the DFT methods are capable of describing the
vertical excitations at the Franck–Condon (FC) point sufficiently accurately as
compared to the best estimates of these energies. The accuracy of the MRCISD
description is sometimes inferior to the DFT methods, probably because of certain
restrictions with regard to the primary active space and the basis set employed
[59]. The TD method tends to overestimate the vertical excitation energies as
compared to the other two DFT methods. The relative energies of the MECI points
obtained in the DFT calculations are in semi-quantitative agreement with the
MRCISD energies; deviations of ca. 0.5 eV or slightly more (e.g., for ketene or
PSB3) are observed in Table 1. However, the MRCISD energies in Table 1 cannot
be regarded as highly accurate benchmark reference data because of the aforementioned basis set and active space restrictions; these data illustrate what is currently
achievable at the computationally affordable MR-WFT level of calculation [59].
The accuracy of the optimized molecular geometries at the MECI points is
characterized in Fig. 3 in graphical form and in Table 2 in terms of root-meansquare-deviations (RMSDs) between the optimized Cartesian coordinates. With the
exception of the PSB3 cation, for which the SF and TD methods predict a much too
strong pyramidalization of the C 3 atom, there are no conspicuous differences
between the DFT and MRCISD geometries. This is reflected in the DFT vs
Description of Conical Intersections with Density Functional Methods
461
and as twist-bond_length_alternation (tw-BLA) CIs, for cases when the two π-bond
breaking mechanisms, homolytic and heterolytic, either become nearly isoenergetic
or the latter mechanism is preferred [81]. An accurate description of this type of CI
requires a balanced description of the relative stability of diradical and ionic
electronic configurations; thus the ability of computational methods to deliver
such a description is probed by this type of CI.
Another type of CI shown in Fig. 3 can be described as n/π* CIs that originate
from the crossing between an electron configuration with a doubly occupied lone
pair orbital (n
2
π*
0 ) and a singly excited configuration (n
1
π*
1 ) [97]. These CIs occur
in ketene, ethylene (ethylidene or methylcarbene CI), and methyliminium cation
(methylimine CI) as shown in Fig. 3. This type of CI probes the accuracy of
description of lone-pair excitations by the tested theoretical methods.
The set of CIs shown in Fig. 3 misses some other types, in particular those found
in cyclic molecules, such as nucleobases; however, good-quality reference
MRCISD data for these CIs would be extremely difficult to obtain because of the
large size of these molecules. We therefore omit these CIs from comparisons and
focus instead on the ability of the DFT computational schemes to reproduce the
molecular geometry and relative energy of the MECI points and the shape of the S 0
and S 1 PESs in their vicinity. To this end, we compare the BP vectors obtained using
the DFT approaches with the respective MRCISD vectors, and we investigate the S 0
and S 1 PESs around the optimized MECI points by scanning the PESs (1) in the
direction of the BP vectors and (2) along a closed path around the MECI point
[98]. The latter scans enable us to evaluate the ability of the respective theoretical
methods to describe the correct dimensionality of the CI seam [43, 98].
As is evident from Table 1, all the DFT methods are capable of describing the
vertical excitations at the Franck–Condon (FC) point sufficiently accurately as
compared to the best estimates of these energies. The accuracy of the MRCISD
description is sometimes inferior to the DFT methods, probably because of certain
restrictions with regard to the primary active space and the basis set employed
[59]. The TD method tends to overestimate the vertical excitation energies as
compared to the other two DFT methods. The relative energies of the MECI points
obtained in the DFT calculations are in semi-quantitative agreement with the
MRCISD energies; deviations of ca. 0.5 eV or slightly more (e.g., for ketene or
PSB3) are observed in Table 1. However, the MRCISD energies in Table 1 cannot
be regarded as highly accurate benchmark reference data because of the aforementioned basis set and active space restrictions; these data illustrate what is currently
achievable at the computationally affordable MR-WFT level of calculation [59].
The accuracy of the optimized molecular geometries at the MECI points is
characterized in Fig. 3 in graphical form and in Table 2 in terms of root-meansquare-deviations (RMSDs) between the optimized Cartesian coordinates. With the
exception of the PSB3 cation, for which the SF and TD methods predict a much too
strong pyramidalization of the C 3 atom, there are no conspicuous differences
between the DFT and MRCISD geometries. This is reflected in the DFT vs
Description of Conical Intersections with Density Functional Methods
461