the same computational method but started from different initial conditions may not
coincide with each other.
2
When comparing BP vectors (x
0
1 , x
0
2 ) obtained using different computational
schemes, the arbitrariness in their orientation within the BP can be bypassed by
aligning the vector pairs such as to maximize the projection of one of the vectors,
e.g., x
0
1 , obtained using computational method A onto the matching vector (x
0
1 or x 1 )
obtained using method B [59, 76]. As proposed recently [59], such an alignment can
be carried out by a similarity transformation SRS
À1 that spans a 2D orthogonal
rotation R and a shear transformation S. The latter transformation is necessary
because the BP vectors are not, in general, orthogonal with respect to one another,
i.e., the inner products x
0
1 Á x
0
2
À
Á
and (x 1 · x 2 ) are not zero. The shear transformation
leaves the inner product invariant upon rotation within the plane. The same publication [59] also introduced a number of useful measures to compare the BPs obtained
using different computational methods. Thus, projections p B x
A
k
À Á
, k ¼ 1, 2 of the
vectors x k (or x
0
k ) obtained using method A onto the BP obtained by method B and
the projection r AB of a rectangle spanned by the vector pair (x
A
1 , x
A
2 ) onto the (x
B
1 , x
B
2 )
rectangle were introduced to quantify the similarity (or discrepancy) between the
BP vectors produced by different computational methods [59]. These measures are
used in the following, when comparing the computational methods addressed in
this chapter.
The use of the alignment procedure and the numerical measures described above
for comparing BP vectors become especially useful when addressing computational
schemes for which the BP vectors cannot be obtained using definitions (4) or (5),
such as the TD-DFT or SF-TD-DFT methods. For these methods, only the x
0
1 vector
can be determined explicitly by differentiation of the respective electronic energies.
The x
0
2 vector is obtained during optimization for a CI using the branching space
update method of Maeda et al. [77] as an orthogonal complement to x
0
1 in a plane
that iteratively converges to the BP of the CI. The (x
0
1 , x
0
2 ) vector pairs so obtained
have been shown to approximate sufficiently accurately the true BPs of the optimized CIs [77]. However, the resulting vectors are strictly orthogonal, a property
not shared by the explicitly calculated vectors. In the context of the REKS method,
the BP vectors are defined by (4), which uses diabatic states and, therefore, yields a
unique orientation of the vectors. This feature becomes especially important when
analyzing the Cls and their BPs in chemical terms.
2 As shown by Yarkony [74, 75], a rotation of the crossing states which orthogonalizes the BP
vectors brings the vectors to a unique orientation, especially when symmetry is present. Application of such a prescription, however, modifies the BP vectors, while leaving the BP unchanged.
Description of Conical Intersections with Density Functional Methods
451
coincide with each other.
2
When comparing BP vectors (x
0
1 , x
0
2 ) obtained using different computational
schemes, the arbitrariness in their orientation within the BP can be bypassed by
aligning the vector pairs such as to maximize the projection of one of the vectors,
e.g., x
0
1 , obtained using computational method A onto the matching vector (x
0
1 or x 1 )
obtained using method B [59, 76]. As proposed recently [59], such an alignment can
be carried out by a similarity transformation SRS
À1 that spans a 2D orthogonal
rotation R and a shear transformation S. The latter transformation is necessary
because the BP vectors are not, in general, orthogonal with respect to one another,
i.e., the inner products x
0
1 Á x
0
2
À
Á
and (x 1 · x 2 ) are not zero. The shear transformation
leaves the inner product invariant upon rotation within the plane. The same publication [59] also introduced a number of useful measures to compare the BPs obtained
using different computational methods. Thus, projections p B x
A
k
À Á
, k ¼ 1, 2 of the
vectors x k (or x
0
k ) obtained using method A onto the BP obtained by method B and
the projection r AB of a rectangle spanned by the vector pair (x
A
1 , x
A
2 ) onto the (x
B
1 , x
B
2 )
rectangle were introduced to quantify the similarity (or discrepancy) between the
BP vectors produced by different computational methods [59]. These measures are
used in the following, when comparing the computational methods addressed in
this chapter.
The use of the alignment procedure and the numerical measures described above
for comparing BP vectors become especially useful when addressing computational
schemes for which the BP vectors cannot be obtained using definitions (4) or (5),
such as the TD-DFT or SF-TD-DFT methods. For these methods, only the x
0
1 vector
can be determined explicitly by differentiation of the respective electronic energies.
The x
0
2 vector is obtained during optimization for a CI using the branching space
update method of Maeda et al. [77] as an orthogonal complement to x
0
1 in a plane
that iteratively converges to the BP of the CI. The (x
0
1 , x
0
2 ) vector pairs so obtained
have been shown to approximate sufficiently accurately the true BPs of the optimized CIs [77]. However, the resulting vectors are strictly orthogonal, a property
not shared by the explicitly calculated vectors. In the context of the REKS method,
the BP vectors are defined by (4), which uses diabatic states and, therefore, yields a
unique orientation of the vectors. This feature becomes especially important when
analyzing the Cls and their BPs in chemical terms.
2 As shown by Yarkony [74, 75], a rotation of the crossing states which orthogonalizes the BP
vectors brings the vectors to a unique orientation, especially when symmetry is present. Application of such a prescription, however, modifies the BP vectors, while leaving the BP unchanged.
Description of Conical Intersections with Density Functional Methods
451