6. Resultados
6.4. Eventos de riesgos correspondientes a Lavado de Activos y Financiamiento al
The electronic state functionYa(r, R) depends not only on electron coordinates {r} but also on the nuclear coordinates {R}. The subscript a denotes a set of electronic quantum numbers. Because the mass of the electrons is much smaller than the mass of the nuclei, the electron motion follows the motion of the nuclei adiabatically, so it is customary to adopt the Born–Oppenheimer approximation, as a result of which the state function may be written as a product of electronic and nuclear state functions:
Caðr, RÞ ¼ a, RðrÞ a, ðRÞ, (1) where denotes a set of vibrational quantum numbers. The electronic state function depends parametrically on the positions of the nuclei, and this is indicated by the subscript R.
The electronic energy Ea(R), calculated at a series of values of the nuclear displacements {R}, is the potential energy Ua(R) for the vibrational motion. Ua(R), which depends on the electronic state a, is called the adiabatic potential. Being a property of the molecule, it is invariant under any symmetry operator of the molecular point group. With S¼ 0, and making use of the orthonormal property of the spin functions, the matrix element (10.1.1) for a vibrating molecule becomes
ð1Þ hCa0jDjCai ¼ h lkjDj jii (2)
which equals zero, unless the DP
l k j i ðx, y, zÞ 1: (3) For a fundamental vibrational transition i¼ 1, and kis one of the representations (Qk) to which the normal modes belong, so that the vibronic problem reduces to answering the question, does
ð1Þ l j ðx, y, zÞ ðQkÞ, (4)
where (Qk) is one or more of the IRs for which the normal modes form a basis? The vibronic interaction is a perturbation term Hˆenin the Hamiltonian, and so transitions that are symmetry forbidden but vibronically allowed can be expected to be weaker than the symmetry-allowed transitions. Consequently, consideration of vibronic transitions is usually limited to E1 transitions. If a transition is symmetry allowed then the vibronic
10.2 Vibronic coupling 173
interaction Hˆenbroadens the corresponding spectral line into a broad band, and this is the reason why absorption and emission spectra consist of broad bands in liquids and solids.
Example 10.2-1 Find if any of the symmetry-forbidden transitions in benzene can become vibronically allowed, given that in the benzene molecule there are normal modes of B2gand E2gsymmetry.
In D6h symmetry the dipole moment operator forms a basis for the representations A2u E1u. The ground-state electronic state function belongs to A1gand
B1u ðA2u E1uÞ ¼ B2g E2g, (5) B2u ðA2u E1uÞ ¼ B1g E2g: (6) Since there are normal modes of B2gand E2gsymmetry, both the transitions1A1g!1B1u
and1A1g!1B2u(which are forbidden by symmetry in a rigid molecule) become allowed through vibronic coupling. These transitions account for the two weaker bands in the benzene spectrum at 2000 and 2600 A˚ .
Example 10.2–2 The ground-state configuration of an nd1octahedral complex is t2g1, and the first excited configuration is eg1 so that optical transitions between these two config-urations are symmetry-forbidden by the parity selection rule. Nevertheless, TiðH2OÞþ36 shows an absorption band in solution with a maximum at about 20 000 cm1and a marked
‘‘shoulder’’ on the low-energy side of the maximum at about 17 000 cm1. Explain the source and the structure of this absorption band.
From the character Table for Oh in Appendix A3, we find that the DP T2g Eg¼ T1g T2g does not contain (x, y, z)¼ T1u, so that the transition t2g1! e1g is symmetry-forbidden (parity selection rule). Again using the character table for Oh, ð4Þ T2g Eg T1u¼ A1u A2u 2Eu 2T1u 2T2u: (7) The normal modes of ML6form bases for A1g Eg T2g 2T1u T2u. Since the DP of two g representations can give only g IRs, we may work temporarily with the group O:
T2g Eg¼ 6 2 0 0 0f g ¼ T1g T2g:
The DP does not contain (r)¼ T1u, so the transition t2g! egis symmetry-forbidden. This is an example of a parity-forbidden transition. We now form the DP T2g Eg T1u¼ 18 2 0 0 0f g: The direct sum must consist of u representations.
Still working with O,
cðT2uÞ ¼ ð1=24Þ½54 6 ¼ 2, cðT1uÞ ¼ ð1=24Þ½54 6 ¼ 2, cðEuÞ ¼ ð1=24Þ½36 þ 12 ¼ 2, cðA2uÞ ¼ ð1=24Þ½18 þ 6 ¼ 1, cðA1uÞ ¼ ð1=24Þ½18 þ 6 ¼ 1,
and so the parity-forbidden transition becomes vibronically allowed through coupling to the odd-parity vibrational modes of T1u and T2u symmetry. The vibronic transition
2T2g ! 2Eg accounts for the observed absorption band, but why does it show some structure? This indicates additional splitting of energy levels associated with a lowering of symmetry (the Jahn–Teller effect). The Jahn–Teller theorem (see, for example, Sugano, Tanabe, and Kamimura (1970)) states that any non-linear molecule in an orbitally degener-ate stdegener-ate will undergo a distortion which lowers the energy of the molecule and removes the degeneracy. Consider here a lowering in symmetry from Oh to D4h. The effect on the energy levels is shown in Figure10.2. The single d electron is now in a b2gorbital. The Jahn–Teller splitting between b2gand egis too small for a transition between these states to appear in the visible spectrum. The relatively small splitting of the main absorption band tells us that we are looking for a relatively small perturbation of the t2g! egtransition, which is forbidden in Ohsymmetry. So the observed structure of the absorption band is due to the2T2g ! 2Egtransition in Ohsymmetry being accompanied by 2B2g ! 2Blgand
2B2g!2Algin D4hsymmetry due to the dynamical Jahn–Teller effect.
Example 10.2-3 Since it is the nearest-neighbor atoms in a complex that determine the local symmetry and the vibronic interactions, trans-dichlorobis(ethylenediamine)cobalt(III) (Figure10.3(a)) may be regarded as having D4hsymmetry for the purpose of an analysis of its absorption spectrum in the visible/near-ultra-violet region (Ballhausen and Moffitt (1956)). The fundamental vibrational transitions therefore involve the 21 6 ¼ 15 normal modes of symmetry: 2Alg, Blg, B2g, Eg, 2A2u, Blu, 3Eu.
(1) In Ohsymmetry the ground-state configuration of this low-spin complex would be t62g. Determine the ground-state and excited-state spectral terms in Ohsymmetry.
(2) Now consider a lowering of symmetry from Ohto D4h. Draw an energy-level diagram in Ohsymmetry showing the degeneracy and symmetry of the orbitals. Then, using a correlation table, show the splitting of these levels when the symmetry is lowered to D4h. Determine the ground-state and excited-state terms, and show how these terms
a1g eg
b1g
b2g eg t2g
Oh D4h
Figure 10.2. Splitting of the t2gand egenergy levels due to the Jahn–Teller effect in an octahedral d1 complex. The short arrow indicates that in the ground state the b2glevel is occupied by one electron.
10.2 Vibronic coupling 175
correlate with the corresponding terms in Oh symmetry. Determine the symmetry-allowed vibronic transitions in D4hsymmetry between the ground state and the four excited states, noting the corresponding polarizations.
(3) Figure10.3(b) shows the absorption spectrum: the continuous line shows the optical absorption for light polarized (nearly) parallel to OZ (the Cl–Co–Cl axis) and the dashed line indicates the absorption for light polarized perpendicular to this axis, namely in the xy plane. Assign transitions for the observed bands. [Hints: (i) The highest-energy band in the spectrum is a composite of two unresolved bands. (ii) The oscillator strengths for parallel and perpendicular transitions are not necessarily equal. (iii) The additional D4h crystal-field splitting is less than the Oh splitting, named or 10Dq.]
(1) Co (atomic number 27) has the electron configuration 3d84s1 and Co3þ has the configuration d6. In Oh symmetry, the configuration is t2g6 when the crystal-field splitting is greater than the energy gain that would result from unpairing spins, as in the present case. The ground-state term is therefore1Alg. The first excited state has the configuration t2g5 e1g. Since all states for d6are symmetric under inversion, we may use the character table for O. As already shown in Example10.2-2, T2g Eg¼ T1gþ T2g
so the excited state terms are 1T1g, 3T1g, 1T2g, 3T2g. Though parity-forbidden, the
1A1g ! 1T1g,1T2g are vibronically allowed in Oh symmetry, it being known from calculation that the T1glevel lies below T2g.
(2) Figure10.4shows the splitting of the one-electron orbital energies and states as the symmetry is lowered from Ohto D4h. The ground state is eg4b2g2:1A1g. Since all states for d6 have even parity under inversion, we may use the character table for D4in AppendixA3. The four excited states and their symmetries are
120000 20000 30000 40000
Figure 10.3. (a) The trans-dichlorobis(ethylenediamine)cobalt(III) ion, showing only the nearest-neighbor atoms in the ligands. (b) Absorption spectra of the trans-[Co(en)2Cl2]+ion showing the dichroism observed with light polarized nearly parallel to the Cl–Co–Cl axis OZ (——), and with light polarized in the xy plane perpendicular to that axis (- - - - ). After Yamada and Tsuchida (1952) and Yamada et al. (1955).
b2ga1g B2g A1g¼ B2g, b2gb1g B2g B1g¼ A2g, eg b1g Eg B1g¼ Eg, eg a1g Eg A1g¼ Eg:
There are therefore excited singlet and triplet states Xgof A2g, B2g, and Egsymmetry. In D4hthe dipole moment operatorer forms a basis for (r) ¼ A2u¼ Eu. Since all states in the DP Xg (A2uþ Eu1A1g) are odd under inversion, we may continue to work with the D4 character table in evaluating DPs. The symmetries of the dipole moment matrix elements for the possible transitions are shown in Table10.1. All the transitions in Table 10.1are forbidden without vibronic coupling. Inspection of the given list of the symmetries of the normal modes shows that there are odd-parity normal modes of A2u, B1u, and Eu
symmetry, and consequently four allowed transitions for (x, y) polarization, namely
1A1g!1A2g,1B2gand1Eg(2), there being two excited states of Egsymmetry, one correlat-ing with T1gin Ohand the other with T2g. These transitions become allowed when there is
eg
a1g b1g
T2g B2g
Eg
T1g
Eg A2g
D4h Oh
1A1g t2g
eg
b2g D4h Oh
(b) (a)
Figure 10.4. Splitting of (a) the one-electron orbital energy levels and (b) the electronic states, as the symmetry is lowered from Ohto D4h.
Table 10.1. Symmetry of the dipole moment matrix elements in trans-dichlorobis(ethylenediamine)cobalt(III) in D4hsymmetry.
Polarization Symmetry of operator Final state
A2g B2g Eg
Z A2u A1u B1u Eu
x, y Eu Eu Eu A1u A2u B1u B2u
10.2 Vibronic coupling 177
simultaneous excitation of a normal mode of symmetry Eu, Euand A2u, or B1u, respectively.
Only three bands are actually observed because the two highest energy bands overlap.
(3) In z polarization (the continuous line in the spectrum) there should be three bands due to
1A1g!1B2g,1Eg, and1Egtransitions, but only two bands are resolved. Disappearance of band 2 with z polarization identifies this with the1A1g!1A2gtransition. Given that T1g
lies below T2gin Ohsymmetry, and since the D4hsplitting is less than , we deduce that band 1 is due to1A1g! 1Eg(T1g), which is allowed in both polarizations, but with somewhat different oscillator strengths. The highest-energy absorption consists of the unresolved bands 3 and 4 due to1A1g!1Eg(T2g) and1A1g!1B2g, which are both allowed in both polarizations. These assignments lead to the approximate energy-level diagram shown in Figure10.5, which also shows the observed transitions. In interpreting this diagram one must bear in mind that the energy differences given are optical energies, which are greater than the corresponding thermal energies because the minima in the adiabatic potential energy curves for ground and excited states do not coincide.