Keywords conical intersections • ensemble DFT • excited states • spin-flip
TD-DFT • TD-DFT
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446
2 Conical Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
2.1 Branching Plane Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
2.2 Double Bond Torsion and Conical Intersections . . . . .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 452
3 Computational Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453
3.1 REKS Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453
3.2 Linear-Response Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456
4 Application of DFT Methods to Conical Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460
4.1 Ethylene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465
4.2 Penta-2,4-Dieniminium Cation, PSB3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468
4.3 Ketene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470
5 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473
1 Introduction
The molecular potential energy surface (PES) is perhaps one of the most significant
concepts in molecular physics which enables one to model molecular structure in
terms of specific spatial arrangements of atoms (molecular geometries) [1]. Defining
(meta)stable molecular conformations (local minima) and transient species (saddle
points or transition states) in terms of the molecular electronic energy–molecular
geometry relationship [2], the PES concept is a basis for theories of molecular
structure and reactivity and serves as a starting point for even more approximate
models, such as force-field molecular mechanics [3]. Rooted in the Born–Oppenheimer approximation [4], the PES concept is naturally valid in situations where the
coupling between the motion of the nuclei and electrons is negligibly weak, that is,
when the separation between the electronic states is much greater than the characteristic energy of nuclear motion [5]. This assumption breaks down in the vicinity of
points where two (or more) PESs corresponding to distinct electronic states become
degenerate, the surface crossing points [5–8]. Near these points, the non-adiabatic
coupling between the electronic and nuclear degrees of freedom becomes decisive
for the dynamics of transformations occurring in the excited states [9–12] as well as
in the ground state [13, 14] of molecules.
Especially interesting and at the same time especially challenging for their
accurate theoretical description are situations of accidental (that is, not
symmetry-imposed) crossing of PESs of states of the same spatial and spin symmetry, the conical intersections (CIs) [6, 15–19]. Such crossings occur in the
subspace of MÀ2 internal molecular degrees of freedom (for a nonlinear molecule,
M ¼ 3 N À 6, N – the number of atoms). The degeneracy between the electronic
states is lifted along the two remaining directions, which span the branching plane
446
M. Huix-Rotllant et al.
TD-DFT • TD-DFT
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446
2 Conical Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
2.1 Branching Plane Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
2.2 Double Bond Torsion and Conical Intersections . . . . .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 452
3 Computational Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453
3.1 REKS Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453
3.2 Linear-Response Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456
4 Application of DFT Methods to Conical Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460
4.1 Ethylene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465
4.2 Penta-2,4-Dieniminium Cation, PSB3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468
4.3 Ketene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470
5 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473
1 Introduction
The molecular potential energy surface (PES) is perhaps one of the most significant
concepts in molecular physics which enables one to model molecular structure in
terms of specific spatial arrangements of atoms (molecular geometries) [1]. Defining
(meta)stable molecular conformations (local minima) and transient species (saddle
points or transition states) in terms of the molecular electronic energy–molecular
geometry relationship [2], the PES concept is a basis for theories of molecular
structure and reactivity and serves as a starting point for even more approximate
models, such as force-field molecular mechanics [3]. Rooted in the Born–Oppenheimer approximation [4], the PES concept is naturally valid in situations where the
coupling between the motion of the nuclei and electrons is negligibly weak, that is,
when the separation between the electronic states is much greater than the characteristic energy of nuclear motion [5]. This assumption breaks down in the vicinity of
points where two (or more) PESs corresponding to distinct electronic states become
degenerate, the surface crossing points [5–8]. Near these points, the non-adiabatic
coupling between the electronic and nuclear degrees of freedom becomes decisive
for the dynamics of transformations occurring in the excited states [9–12] as well as
in the ground state [13, 14] of molecules.
Especially interesting and at the same time especially challenging for their
accurate theoretical description are situations of accidental (that is, not
symmetry-imposed) crossing of PESs of states of the same spatial and spin symmetry, the conical intersections (CIs) [6, 15–19]. Such crossings occur in the
subspace of MÀ2 internal molecular degrees of freedom (for a nonlinear molecule,
M ¼ 3 N À 6, N – the number of atoms). The degeneracy between the electronic
states is lifted along the two remaining directions, which span the branching plane
446
M. Huix-Rotllant et al.