1. Theoretical Chemistry for Electronic Excited States
Supplementary information for Theoretical Chemistry for Electronic Excited States© The Royal Society of Chemistry 2018
Theoretical Chemistry for Electronic Excited States
Supplementary File: Chapter 3

2. Supplementary information for Theoretical Chemistry for Electronic Excited States© The Royal Society of Chemistry 2018
Figure 3.1 Partition of the orbital space into inactive orbitals, active orbitals and virtual orbitals.

3. Supplementary information for Theoretical Chemistry for Electronic Excited States© The Royal Society of Chemistry 2018
Figure 3.2 The diagrammatic representation of eqn (3.8). The rules for the diagram are given in Figures 3.3 and The three states Φ1, Φμ and Φ2 (notation from eqn (3.4)) correspond to levels 1, 2 and 3, respectively, in the diagram. The energy denominator comes from the intermediate level (level 2). The formulae for the interaction lines are given in Figure 3.3. The formula for the energy denominator is given in Figure 3.4.

4. Supplementary information for Theoretical Chemistry for Electronic Excited States© The Royal Society of Chemistry 2018
Figure 3.3 Rules for the evaluation of two and one electron interaction (dashed lines between the two dots).

5. Supplementary information for Theoretical Chemistry for Electronic Excited States© The Royal Society of Chemistry 2018
Figure 3.4 A worked example of the evaluation of a diagram. The rule for the interaction lines is given in Figure 3.3: interactions: 〈ad|be〉 〈ad for inward lines ad, be〉 for outward lines. Energy denominator: wrap free lines around a cylinder, shown as dashed lines, retaining the labels of the upward lines. The downward lines are positive orbital energies; upward lines are negative orbital energies.

6. Supplementary information for Theoretical Chemistry for Electronic Excited States© The Royal Society of Chemistry 2018
Figure 3.5 Partial cancellation of the correlation energy of the core, inactive, electrons (a), the core correlation energy that contributes to the diagonal element 〈|αβ••••pipj|Heff|αβ••••pipj〉 (b), and the core correlation
energy as a diagonal shift (c) and (d). Figure 3.5a equals the core correlation energy (b) plus the two terms (c) and (d).

7. Supplementary information for Theoretical Chemistry for Electronic Excited States© The Royal Society of Chemistry 2018
Figure 3.6 Some representative terms in the perturbation expansion of one electron terms in eqn (3.3).2

8. Supplementary information for Theoretical Chemistry for Electronic Excited States© The Royal Society of Chemistry 2018
Figure 3.7 Some representative terms in the perturbation expansion of two electron terms in eqn (3.3).2

9. Supplementary information for Theoretical Chemistry for Electronic Excited States© The Royal Society of Chemistry 2018
Figure 3.8 (a) Semi-internal correlation, (b) valence (active)-core (inactive) correlation energy.

10. Supplementary information for Theoretical Chemistry for Electronic Excited States© The Royal Society of Chemistry 2018
Figure 3.9 A representative diagram for the effective Hamiltonian in the space of all single particle ionizations
(the reference states are shown on the right-hand side of the figure).

11. Supplementary information for Theoretical Chemistry for Electronic Excited States© The Royal Society of Chemistry 2018
Figure 3.10 A typical second-order diagram in the space of particle hole excitations
; (a) is the zeroth order interaction and (b) core (inactive)-virtual excitations.

12. Supplementary information for Theoretical Chemistry for Electronic Excited States© The Royal Society of Chemistry 2018
Figure 3.11 Comparison of (a) QM-MM and (b) a hybrid method MMVB. In (a) the CH2–CH2 fragment and the C=O–O–C=O fragments are treated with molecular mechanics. The fragments with the pπ orbitals are treated with QM. In QM-MM the QM-MM fragment as a whole is treated with QM. (b) In MMVB just the atoms pπ orbitals are treated as hybrid atoms.

13. Supplementary information for Theoretical Chemistry for Electronic Excited States© The Royal Society of Chemistry 2018
Figure 3.12 An example of a diagram at third order that contributes to an effective three electron integral.

14. Supplementary information for Theoretical Chemistry for Electronic Excited States© The Royal Society of Chemistry 2018
Figure 3.13 CASSCF diagrams: core (inactive)→virtual (max nh = 1, max np = 1) diagrams: (a) zeroth order, (b) second order.

15. Supplementary information for Theoretical Chemistry for Electronic Excited States© The Royal Society of Chemistry 2018
Figure 3.14 CASSCF diagrams: active→virtual (max nh = 1, max np = 1) diagrams: (a) zeroth order, (b) second order.

16. Supplementary information for Theoretical Chemistry for Electronic Excited States© The Royal Society of Chemistry 2018
Figure 3.15 CASSCF diagrams: core (inactive)→active (max nh = 1, max np = 1) diagrams: (a) zeroth order, (b) second order.

17. Supplementary information for Theoretical Chemistry for Electronic Excited States© The Royal Society of Chemistry 2018
Figure 3.16 The orbital spaces associated with RASSCF.

18. Supplementary information for Theoretical Chemistry for Electronic Excited States© The Royal Society of Chemistry 2018
Figure 3.17 The neutral GFP molecule.

19. Supplementary information for Theoretical Chemistry for Electronic Excited States© The Royal Society of Chemistry 2018
Figure 3.18 NBO of the double bond linkage showing the occupancy in S2.

20. Supplementary information for Theoretical Chemistry for Electronic Excited States© The Royal Society of Chemistry 2018
Figure 3.19 Four of theNBOs of the phenyl group linkage showing the occupancy in S2.

21. Supplementary information for Theoretical Chemistry for Electronic Excited States© The Royal Society of Chemistry 2018
Figure 3.20 Two NOs showing the occupancy in S2.

22. Supplementary information for Theoretical Chemistry for Electronic Excited States© The Royal Society of Chemistry 2018
Figure 3.21 Four NOs showing the occupancy in fully optimized CAS-SCF (8 active orbitals) wavefunction for S2.

23. Supplementary information for Theoretical Chemistry for Electronic Excited States© The Royal Society of Chemistry 2018
Figure 3.22 RAS1: 3×2pσ orbitals; RAS2: 4×2pπ orbitals; RAS3: 4×3pπ, a3×2pσ* NBO.
Adapted from V. Santolini, J. P. Malhado, M. A. Robb, M. Garavelli and M. J. Bearpark, Mol. Phys., 2015,27 © 2015 The Author(s). Published by Taylor & Francis. Published under the terms of the CC BY 4.0 license,

24. Supplementary information for Theoretical Chemistry for Electronic Excited States© The Royal Society of Chemistry 2018
Figure 3.23 Optimized orbitals for the B2 state.
Adapted from V. Santolini, J. P. Malhado, M. A. Robb, M. Garavelli and M. J. Bearpark, Mol. Phys., 2015,27 © 2015 The Author(s). Published by Taylor & Francis. Published under the terms of the CC BY 4.0 license,

25. Supplementary information for Theoretical Chemistry for Electronic Excited States© The Royal Society of Chemistry 2018
Figure 3.24 The Kekule´ (a) and anti-Kekule´ (b) resonance structures of para-xylene. In each case there are two almost degenerate resonance structures With +- vs. -+ positions at positions one and four.
Adapted from M. Vacher, J. Meisner, D. Mendive-Tapia, M. J. Bearpark and M. A. Robb, J. Phys. Chem. A, 2015, 119, 5165– Copyright 2015 American Chemical Society; and M. Vacher, L. Steinberg, A. J. Jenkins,
M. J. Bearpark and M. A. Robb, Phys. Rev. A, 92, , 2015, Copyright (2015) by the American Physical Society.

26. Supplementary information for Theoretical Chemistry for Electronic Excited States© The Royal Society of Chemistry 2018
Figure 3.25 A schematic representation of the conical intersection in the paraxylene radical cation. The vertical red bar gives the equilibrium geometry of the neutral species and the conical intersection is slightly displaced from this geometry.
Adapted from M. Vacher, J. Meisner, D. Mendive-Tapia, M. J. Bearpark and M. A. Robb, J. Phys. Chem. A, 2015, 119, 5165– Copyright 2015 American Chemical Society; and M. Vacher,
L. Steinberg, A. J. Jenkins, M. J. Bearpark and M. A. Robb, Phys. Rev. A, 92, , 2015, Copyright (2015) by the American Physical Society.

27. Supplementary information for Theoretical Chemistry for Electronic Excited States© The Royal Society of Chemistry 2018
Figure 3.26 A schematic representation of the resonance structures and the molecular structures of para-xylene as one goes around the moat of the conical intersection shown in Figure The geometry on the X1 axis is indicated by the diamond. The initial superposition (eqn (3.44)) is indicated by the dot on the X2 axis.
Adapted from M. Vacher, J. Meisner, D. Mendive-Tapia, M. J. Bearpark and M. A. Robb, J. Phys. Chem. A, 2015, 119, 5165– Copyright 2015 American Chemical Society; and M. Vacher,
L. Steinberg, A. J. Jenkins, M. J. Bearpark and M. A. Robb, Phys. Rev. A, 92, ,
2015, Copyright (2015) by the American
Physical Society.

28. Supplementary information for Theoretical Chemistry for Electronic Excited States© The Royal Society of Chemistry 2018
Figure 3.27 The evolution of spin density of para-xylene on ring atoms 3 and 6 vs. 2 and 5. The corresponding evolution of the wavefunction is shown in the line emanating from the dot along the X2 axis in Figure The structure oscillates between the electronic resonance structures with a period of about 9 fs.