approximately orthogonal orientation during the torsion, the interaction between
the fragment wavefunctions vanishes and this results in a vanishing H mn element.
Thus, the gradient of the interstate coupling element in the SI-SA-REKS method
(see Sect. 3.1) can be conveniently associated with the x 2 vector.
From this brief discussion, the advantage of the specific orientation of the BP
vectors as in Fig. 1 is that it offers a simple chemical interpretation of the CI and the
factors influencing its occurrence. In particular, the relative electronegativity of the
fragments connected by the double bond and, consequently, the relative preference
for one of the bond breaking mechanisms define the molecular geometry at a CI [58,
81]. For example, in alkenes, the homolytic π-bond breaking is energetically
preferred over the heterolytic bond breaking, and this results in a pronounced
pyramidalization at a CI which is needed to stabilize the ionic structure and
(along with the torsion) to reach the point of surface crossing. By contrast, in
organic molecules with strongly electron-withdrawing (or electron-donating) functional groups, the two bond breaking mechanisms may become nearly isoenergetic
and a CI can be reached without requiring pyramidalization; in this case, the
molecular geometry at the CI corresponds to double bond torsion combined with
stretching/compression of the other single and double bonds, hence a bond length
alternation (BLA) distortion [81]. Because the BP vectors obtained in the SI-SAREKS calculations have a unique orientation and a transparent chemical interpretation, the vectors obtained by other methods, TD-DFT, SF-TD-DFT, and
MRCISD, are aligned with them by using the recently introduced similarity
transformation [59].
3 Computational Methods
In the following, we give a brief overview of the computational methods used to
study the CIs in this chapter. A complete description of these computational
schemes can be found in other chapters of this book, and so we recapitulate only
those features most relevant for PES crossings. The interested reader is advised to
consult the other chapters in this book and the literature references cited therein for
more detail on the computational methodologies.
3.1 REKS Method
The REKS method [54–56] employs the ideas behind ensemble DFT to describe the
non-dynamic correlation in the ground and excited states of molecules and to access
the excited states through the application of the variational principle [42, 57,
58]. Perhaps the most important realization in ensemble DFT that was rigorously
proved by Lieb [53] and computationally verified by Schipper et al. [82, 83] and by
Morrison [84] is that the density and the ground-state energy of a strongly
Description of Conical Intersections with Density Functional Methods
453
the fragment wavefunctions vanishes and this results in a vanishing H mn element.
Thus, the gradient of the interstate coupling element in the SI-SA-REKS method
(see Sect. 3.1) can be conveniently associated with the x 2 vector.
From this brief discussion, the advantage of the specific orientation of the BP
vectors as in Fig. 1 is that it offers a simple chemical interpretation of the CI and the
factors influencing its occurrence. In particular, the relative electronegativity of the
fragments connected by the double bond and, consequently, the relative preference
for one of the bond breaking mechanisms define the molecular geometry at a CI [58,
81]. For example, in alkenes, the homolytic π-bond breaking is energetically
preferred over the heterolytic bond breaking, and this results in a pronounced
pyramidalization at a CI which is needed to stabilize the ionic structure and
(along with the torsion) to reach the point of surface crossing. By contrast, in
organic molecules with strongly electron-withdrawing (or electron-donating) functional groups, the two bond breaking mechanisms may become nearly isoenergetic
and a CI can be reached without requiring pyramidalization; in this case, the
molecular geometry at the CI corresponds to double bond torsion combined with
stretching/compression of the other single and double bonds, hence a bond length
alternation (BLA) distortion [81]. Because the BP vectors obtained in the SI-SAREKS calculations have a unique orientation and a transparent chemical interpretation, the vectors obtained by other methods, TD-DFT, SF-TD-DFT, and
MRCISD, are aligned with them by using the recently introduced similarity
transformation [59].
3 Computational Methods
In the following, we give a brief overview of the computational methods used to
study the CIs in this chapter. A complete description of these computational
schemes can be found in other chapters of this book, and so we recapitulate only
those features most relevant for PES crossings. The interested reader is advised to
consult the other chapters in this book and the literature references cited therein for
more detail on the computational methodologies.
3.1 REKS Method
The REKS method [54–56] employs the ideas behind ensemble DFT to describe the
non-dynamic correlation in the ground and excited states of molecules and to access
the excited states through the application of the variational principle [42, 57,
58]. Perhaps the most important realization in ensemble DFT that was rigorously
proved by Lieb [53] and computationally verified by Schipper et al. [82, 83] and by
Morrison [84] is that the density and the ground-state energy of a strongly
Description of Conical Intersections with Density Functional Methods
453