## Abstract

We report a rigorous formulation of density functional theory for excited states, providing a theoretical foundation for a multistate density functional theory. We prove the existence of a Hamiltonian matrix functional H[D] of the multistate matrix density D(r) in the subspace spanned by the lowest N eigenstates. Here, D(r) is an N-dimensional matrix of state densities and transition densities. Then, a variational principle of the multistate subspace energy is established, whose minimization yields both the energies and densities of the individual N eigenstates. Furthermore, we prove that the N-dimensional matrix density D(r) can be sufficiently represented by N2 nonorthogonal Slater determinants, based on which an interacting active space is introduced for practical calculations. This work establishes that the ground and excited states can be treated on an equal footing in density functional theory.

Original language | English (US) |
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Pages (from-to) | 7762-7769 |

Number of pages | 8 |

Journal | Journal of Physical Chemistry Letters |

Volume | 13 |

Issue number | 33 |

DOIs | |

State | Published - Aug 25 2022 |

### Bibliographical note

Funding Information:This work was supported in part by grants from Shenzhen Municipal Science and Technology Innovation Commission (KQTD2017-0330155106581), the Key-Area Research and Development Program of Guangdong Province (Grant 2020B0101350001), the National Natural Science Foundation of China (Grant 21533003), and the NIH (GM046736), the latter for developing applications to treat excited states in biological systems. We thank Professors Wenjian Liu and Weitao Yang for discussion and insightful comments.

Publisher Copyright:

© 2022 American Chemical Society.

## PubMed: MeSH publication types

- Journal Article