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First-order derivative couplings between excited states from adiabatic TDDFT response theory
5.Conical Intersections: Electron Structure Dynamics and Spectroscopy, edited by W. Domcke, D. R. Yarkony, and H. Köppel (World Scientific Publishing Co. Pte. Ltd., Singapore, 2004).
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30. It is worth noting that although Refs. 19–21 do not produce the correct final derivative coupling, Tavernelli et al. did invoke the correct formalism for deriving such a derivative coupling (i.e., by comparing many-body response theory with TD-DFT). Along the way, however, the authors incorrectly reduced the derivative coupling to a simple one-electron operator between singly excited auxiliary wavefunctions (via Casida’s assignment).
32.X. Zhang and J. M. Herbert, J. Chem. Phys. 142, 064109 (2015).
34. This fact follows because is Hermitian; see Eqs. (43) and (42).
36. In Ref. 33, the sign of ωα is not consistent with the definitions of and . In this work, we adopt the exact same formalism and nomenclature as in Ref. 33 but change the sign in front of ωα in Eq. (73) to be a plus sign to correct this earlier typo.
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40.Y. Shao, L. Fusti-Molnar, Y. Jung, J. Kussmann, C. Ochsenfeld, S. T. Brown, A. T. B. Gilbert, L. V. Slipchenko, S. V. Levchenko, D. P. O’Neill et al., Phys. Chem. Chem. Phys. 8, 3172 (2006).
41.E. Alguire, Q. Ou, and J. E. Subotnik, “Calculating derivative couplings between time-dependent Hartree–Fock excited states with pseudo-wavefunctions,” J. Phys. Chem. B (published online).
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We present a complete derivation of derivative couplings between excited states in the framework of adiabatic time-dependent density functional response theory. Explicit working equations are given and the resulting derivative couplings are compared with derivative couplings from a pseudo-wavefunction ansatz. For degenerate excited states, i.e., close to a conical intersection (CI), the two approaches are identical apart from an antisymmetric overlap term. However, if the difference between two excitation energies equals another excitation energy, the couplings from response theory exhibit an unphysical divergence. This spurious behavior is a result of the adiabatic or static kernel approximation of time-dependent density functional theory leading to an incorrect analytical structure of the quadratic response function. Numerical examples for couplings close to a CI and for well-separated electronic states are given.
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