C.18.- PLANES DEL CEAD Y PLANES DE MEJORA
C.18.1) PLAN DE COMUNICACIÓN LINGÜÍSTICA, BIBLIOTECA Y RADIOS ESCOLARES
An intersystem crossing is a type of crossing that can occur between two PESs of different spin state. When two electronic states have differing spin-multipicities the derivative coupling term is zero by symmetry. Therefore the PES cross in a single coordinate, i.e. the gradient difference coordinate (gd - x1). Thus, when the PES are
plotted along the gd and any other coordinate they have the appearance of a one- dimensional seam (Figure 2.11).
Figure 2.11. Relaxation through an intersystem crossing.
Rather than the non-adiabatic terms coupling different electronic states, here the spin- orbit interaction, which is the coupling between electron spin S and orbital angular momentum L, governs the strength of the coupling. SOC is formally a relativistic phenomenon arising from the Dirac equation of quantum mechanics [3]. The Dirac equation differs from Schrödinger equation in the form of their Hamiltonians, which in the former is much more complicated:
HDirac = c! " p + #mc
2
(
)
+V (2.45)where α and β are 4×4 matrices, that represent large and small components of a wavefunction including α and β spin functions and V is the electric potential. The large component corresponds to a normal non-relativistic wavefunction. The small components correspond to a coupling with positronic states. When the speed of light c goes to infinity these become decoupled and the Schrödinger equation, is returned. The Schrödinger equation is not relativistically correct and for many electron systems a Dirac-Coulomb Hamiltonian may need to be constructed [3].
In the many-electron extensions of the Dirac equation each electron can be described as
c! " p + #mc2
(
)
and so additional terms arise which can include the spin-other-orbit, spin-spin, and orbit-orbit interactions. Spin-other-orbit terms describe the interaction ofISC seam
Spin B PES
! Spin A PES
an electron spin with the magnetic field generated by the movement of the other electrons. Spin-spin and orbit-orbit terms describe additional magnetic interactions [3]. The magnitude of these relativistic effects scales with nuclear charge so that they are extremely important for heavy elements and as such cannot be treated as a perturbation on the non-relativistic wavefunctions and energies. The scalar one-electron relativistic effects on the core orbitals for heavy atoms are most often accounted for using a psuedo-potential, which accounts for the shape and energetics of the core electrons, as obtained from relativistic calculations, but the orbitals and electrons are replaced by a smooth potential in the non-relativistic framework [3, 59-62]. The number of electrons that can be represented as a pseudo-potential can vary. It is common to treat valence electrons explicitly and the remaining core as a pseudo-potential, however, for better results one can also include orbitals from the next lower shell and treat them explicitly. The Stuttgart-Dresden (SDD) pseudo-potential mentioned previously in subsection 2.2.2 of this chapter is an example of such a pseudo-potential and will be used for some calculations in the following chapters. In the chapter 4, for the iodine atom that contains 53 electrons, the SDD potential represents the 46 core electrons, and the 7 remaining valence electrons are treated explicitly.
(1s)2(2s)2(2p)6(3s)2(3p)6(4s)2(3d)10(4p)6(4d)10(5s)2(5p)5
In the case of the platinum atom that contains 78 electrons, the SDD potential represents 60 core electrons, and 18 valence electrons are treated explicitly.
(1s)2(2s)2(2p)6(3s)2(3p)6(4s)2(3d)10(4p)6(4d)10(4f)14(5s)2(5p)6(6s)1(5d)9
It should be noted that a pseudo-potential deals with the large magnitude of relativistic effects for the core electrons, while the SOC type of effects important in the case of intersystem crossings deal with much, much weaker interactions coming from the valence electrons. For the first three rows in the periodic table relativistic effects such as valence SOC are usually quite weak.
For lighter elements (including the first row of the periodic table) a perturbative treatment of these relatively weak interactions is most appropriate. The wavefunction of
the state m, that can be perturbed due to the effect of SOC as it mixes with state k, can be written as: !m = "m+ ak"k k#m
$
(2.46) !where !mis pure-spin zero-order wave function for state m, !kpure-spin wavefunction of perturbing state k, and akare coefficients that represent the contribution of the
individual pure spin-perturbing states to the ground state wavefunction, treated by perturbation theory. Equation 2.47 gives the expression for these coefficients, where
ˆ
HSO is for example a Breit-Pauli spin-orbit operator, that allows for the mixing of pure spin states (Equation 2.47) [63].!
ak = !k H
!SO ! m
Em" Ek (2.47)
Em and Ek are energies of the states m and k respectively. These terms (Equation 2.46,
2.47) arise from the CI treatment of states described in subsection 2.2.3 of this chapter.
In organic chemistry a further approximation is often invoked whereby the spin-orbit coupling is based upon an empirically scaled one-electron term that for example can give a reasonable description of the mixing of singlet and triplet states of organic molecules. This empirically scaled model has not been extended to the first transition metal series and any evaluation of SOC needs to use the full perturbative model. We note here that when SOC is weak the position of the crossing seams as shown in figure 2.10 is the most important factor in determining whether or not a system changes spin state. For such weak coupling the pure spin-states are valid and the crossing region determines the geometries where the system can change from one spin state to another (i.e. when the denominator in equation 2.47 tends to zero). Thus the SOC can be expected to be reasonably constant over a small geometrical change. For the systems studied later, i.e. open-shell doublet and quartet states, whose photochemistry sometimes involves a spin-state change we will be concerned only with these ISC crossing regions between the pure spin-states. We do not evaluate any SOC as the full perturbative treatment is too expensive for these systems, and also as discussed we expect the SOC to be quite weak and relatively insensitive to geometry, therefore for the
non-radiative spin-changing transition the low energy crossing seams take on the same importance as conical intersections do for spin-conserving radiationless transitions.
One of the manifestations of intersystem crossing can be the radiative process called phosphorescence, which in this thesis will be considered as any general radiative non- spin conserving transition. Thus phosphorescence occurs following a non-radiative ISC from one spin-state to another on a relatively fast time scale, and the phosphorescence is the radiative return to the original spin state on a much larger time scale.