MRCISD RMSDs in Table 2, which vary in the range between 0.061 Å (SSR) and
0.079 Å (TD). An even better agreement is seen between the geometries obtained
by the different DFT methods, especially for the SF and TD methods (0.030 Å). The
BP projections in Table 2 indicate that all methods produce nearly the same BP
vectors at the MECI geometries. The very good agreement between the DFT
methods and MRCISD seems to indicate that these much simpler methods can be
confidently used in lieu of MRCISD for locating CIs and analyzing the S 0 and S 1
PESs in their vicinity. However, there is a caveat: despite the seemingly good
agreement with the high-level MR-WFT approach, not all the DFT methods
produce the correct dimensionality of the CI seam. In the following, we focus on
three molecules and their respective CIs, ethylene and its tw-pyr MECI, cationic
PSB3 and its tw-BLA MECI, as well as ketene molecule and its n/π* MECI, as
representatives of the specific types of CIs discussed above.
4.1 Ethylene
On the S 0 /S 1 CI seam in ethylene there are two distinct MECI points that correspond
to a tw-pyr MECI (the major photorelaxation funnel) and to a MECI point with
partial transfer of a hydrogen atom; see the tw-pyr and ethylidene MECIs of C 2 H 4
in Fig. 3 [105–108]. The tw-pyr MECI originates from a crossing between the PESs
for homolytic and heterolytic π-bond breaking along the double bond torsion mode.
As the heterolytic π-bond breaking is strongly disfavored in ethylene, the tw-pyr
MECI features a strong pyramidalization of one of the carbon atoms to stabilize the
ionic electronic configuration.
All the DFT methods employed here predict the vertical excitation energy at the
FC point in a good agreement with the best theoretical estimate of 7.8 eV. The energy
level of the tw-pyr MECI is much lower than the FC point. The DFT methods yield
tw-pyr MECI energies varying between 4.96 eV (SSR) and 4.73 eV (TD), in good
agreement with the MRCISD value 4.79 eV; see Table 1. The BP vectors predicted
by the DFT methods are in very good agreement with the MRCISD BP vectors, as
can be seen from a visual comparison of the vectors in Fig. 4 and from the BP
projections (see Sect. 2) which lie in the range 0.99–0.97 in each case.
From these results it seems that all the DFT methods are capable of describing
the tw-pyr MECI in ethylene correctly. However, a closer look at the S 0 and S 1 PESs
near the MECI point reveals that the surface crossing predicted by adiabatic LRTD-DFT is not actually a conical intersection but is rather a linear crossing. As seen
in Fig. 5, the S 1 PES produced by adiabatic LR-TD-DFT falls below the S 0 PES
when moving along the x 1 direction.
5 Furthermore, the shape of the S 0 and S 1 PESs
5 The conventional KS DFT/LR-TD-DFT calculations experienced severe convergence problems
in the vicinity of the MECI point, which are reflected in the shape of the S 1 PES in the upper right
panel of Fig. 5.
Description of Conical Intersections with Density Functional Methods
465
0.079 Å (TD). An even better agreement is seen between the geometries obtained
by the different DFT methods, especially for the SF and TD methods (0.030 Å). The
BP projections in Table 2 indicate that all methods produce nearly the same BP
vectors at the MECI geometries. The very good agreement between the DFT
methods and MRCISD seems to indicate that these much simpler methods can be
confidently used in lieu of MRCISD for locating CIs and analyzing the S 0 and S 1
PESs in their vicinity. However, there is a caveat: despite the seemingly good
agreement with the high-level MR-WFT approach, not all the DFT methods
produce the correct dimensionality of the CI seam. In the following, we focus on
three molecules and their respective CIs, ethylene and its tw-pyr MECI, cationic
PSB3 and its tw-BLA MECI, as well as ketene molecule and its n/π* MECI, as
representatives of the specific types of CIs discussed above.
4.1 Ethylene
On the S 0 /S 1 CI seam in ethylene there are two distinct MECI points that correspond
to a tw-pyr MECI (the major photorelaxation funnel) and to a MECI point with
partial transfer of a hydrogen atom; see the tw-pyr and ethylidene MECIs of C 2 H 4
in Fig. 3 [105–108]. The tw-pyr MECI originates from a crossing between the PESs
for homolytic and heterolytic π-bond breaking along the double bond torsion mode.
As the heterolytic π-bond breaking is strongly disfavored in ethylene, the tw-pyr
MECI features a strong pyramidalization of one of the carbon atoms to stabilize the
ionic electronic configuration.
All the DFT methods employed here predict the vertical excitation energy at the
FC point in a good agreement with the best theoretical estimate of 7.8 eV. The energy
level of the tw-pyr MECI is much lower than the FC point. The DFT methods yield
tw-pyr MECI energies varying between 4.96 eV (SSR) and 4.73 eV (TD), in good
agreement with the MRCISD value 4.79 eV; see Table 1. The BP vectors predicted
by the DFT methods are in very good agreement with the MRCISD BP vectors, as
can be seen from a visual comparison of the vectors in Fig. 4 and from the BP
projections (see Sect. 2) which lie in the range 0.99–0.97 in each case.
From these results it seems that all the DFT methods are capable of describing
the tw-pyr MECI in ethylene correctly. However, a closer look at the S 0 and S 1 PESs
near the MECI point reveals that the surface crossing predicted by adiabatic LRTD-DFT is not actually a conical intersection but is rather a linear crossing. As seen
in Fig. 5, the S 1 PES produced by adiabatic LR-TD-DFT falls below the S 0 PES
when moving along the x 1 direction.
5 Furthermore, the shape of the S 0 and S 1 PESs
5 The conventional KS DFT/LR-TD-DFT calculations experienced severe convergence problems
in the vicinity of the MECI point, which are reflected in the shape of the S 1 PES in the upper right
panel of Fig. 5.
Description of Conical Intersections with Density Functional Methods
465