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Letter
The excitation number: Characterizing multiply-excited states Giuseppe M. J. Barca, Andrew T.B. Gilbert, and Peter M.W. Gill J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.7b00963 • Publication Date (Web): 22 Dec 2017 Downloaded from http://pubs.acs.org on December 26, 2017
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The excitation number: Characterizing multiply-excited states Giuseppe M.J. Barca, Andrew T.B. Gilbert, and Peter M.W. Gill∗ Research School of Chemistry, Australian National University, ACT 2601, Australia Received December 22, 2017; E-mail:
[email protected] Abstract: How many electrons are excited in an electronic transition? In this Letter, we introduce the excitation number η to answer this question when the initial and final states are each modelled by a single-determinant wave function. We show that calculated η values lie close to positive integers, leading to unambiguous assignments of the number of excited electrons. This contrasts with previous definitions of excitation quantities which can lead to mis-assignments. We consider several examples where η provides improved excited-state characterizations. The electronic states of a molecular system are completely characterized by their wave functions Ψk which satisfy the time-independent Schr¨ odinger wave equation ˆ k = E k Ψk HΨ
(0.1)
ˆ is the Hamiltonian for the system and Ek is where H the energy corresponding to Ψk . Often, however, we are less interested in individual states than in the changes between two states, A and B, of the system. Many of these changes, such as vertical excitation energies and oscillator strengths, can be determined both experimentally and theoretically. However, the answer to one apparently benign question remains elusive: how many electrons are excited in an electronic transition? The issue is not trivial, for the absorption of even a single photon can excite more than one electron. 1,2 Moreover, a two-photon absorption can result in either the double excitation of a single electron, or the single excitations of two electrons. 3,4 Multiply-excited states play a key role in important fields such as optoelectronics, 5 but only rarely can the number of excited electrons be measured experimentally, and there is a dearth of reliable, general theoretical methods to compute an “excitation number”. In part, the reason for this theoretical deficiency can be traced to the successful modeling of excited states 6 by configuration interaction (CI) expansions Ψk ≈ c0 Φ0 +
occ X virt X i
r
cri Φri
+
occ X virt X
rs crs ij Φij + . . . (0.2)
i
