In this Perspective, we introduce a minimal active space (MAS) for the lowest N eigenstates of a molecular system in the framework of multistate density functional theory (MSDFT), consisting of no more than N 2 nonorthgonal Slater determinants. In comparison with some methods in wave function theory in which one seeks to expand the ever increasing size of an active space to approximate the wave functions, it is possible to have an upper bound in MSDFT because the auxiliary states in a MAS are used to represent the exact Ndimensional matrix density function D(r). Here, we partition the total Hamiltonian matrix functional [ ] D into an orbital-dependent part, including multistate kinetic energy T ms and Coulomb-exchange energy E Hx plus an external potential energy ∫ dr v(r)D(r), and a correlation matrix density functional [ ] D c . The latter accounts for the part of correlation energy not explicitly included in the minimal active space. A major difference from Kohn−Sham DFT is that state interactions are necessary to represent the N-matrix density D(r) in MSDFT, rather than a noninteracting reference state for the scalar ground-state density ρ o (r). Two computational approaches are highlighted. We first derive a set of nonorthogonal multistate self-consistent-field (NOSCF) equations for the variational optimization of [ ] D . We introduce the multistate correlation potential, as the functional derivative of [ ] D c, which includes both correlation effects within the MAS and that from the correlation matrix functional. Alternatively, we describe a nonorthogonal state interaction (NOSI) procedure, in which the determinant functions are optimized separately. Both computational methods are useful for determining the exact eigenstate energies and for constructing variational diabatic states, provided that the universal correlation matrix functional is known. It is hoped that this discussion would stimulate developments of approximate multistate density functionals both for the ground and excited states.

Cu/SSZ-13 is the current state-of-the-art catalyst for selective catalytic reduction of NO x with NH 3 (NH 3 -SCR) in diesel after-treatment systems. Recent investigations under in situ or operando conditions yielded rich molecular level understanding about dynamic transformations of Cu sites during reactions. However, in situ investigations aiming at the distinction of the two SCR-active Cu species in this catalyst, that is, Z 2 Cu II and ZCu II OH, are still scarce. Herein, we apply in situ UV−vis spectroscopic studies combined with theoretical calculations to investigate the two Cu sites under SCR-relevant conditions at low temperatures. We demonstrate that in the presence of O 2 , two isolated ZCu II OH sites readily transform into a double O-bridged dicopper species with a Cu−Cu distance of 3.37 Å, whereas Z 2 Cu II species cannot undergo such a transformation. In addition, ZCu II OH displays stronger activity than Z 2 Cu II during both reduction by NH 3 and NO oxidation to bidentate nitrates. Despite these differences, Z 2 Cu II and ZCu II OH sites exhibit similar spectroscopic features under both NH 3 oxidation and NH 3 -SCR conditions. These findings demonstrate that UV−vis spectroscopy is a powerful tool to be used in situ to provide rich information on the NH 3 -SCR mechanism and on the rational design of Cu/SSZ-13 catalysts.

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A flexible self-consistent field method, called target state optimization (TSO), is presented for exploring electronic excited configurations and localized diabatic states. The key idea is to partition molecular orbitals into different subspaces according to the excitation or localization pattern for a target state. Because of the orbital-subspace constraint, orbitals belonging to different subspaces do not mix. Furthermore, the determinant wave function for such excited or diabatic configurations can be variationally optimized as a ground state procedure, unlike conventional ΔSCF methods, without the possibility of collapsing back to the ground state or other lower-energy configurations. The TSO method can be applied both in Hartree–Fock theory and in Kohn–Sham density functional theory (DFT). The density projection procedure and the working equations for implementing the TSO method are described along with several illustrative applications. For valence excited states of organic compounds, it was found that the computed excitation energies from TSO–DFT and time-dependent density functional theory (TD-DFT) are of similar quality with average errors of 0.5 and 0.4 eV, respectively. For core excitation, doubly excited states and charge-transfer states, the performance of TSO-DFT is clearly superior to that from conventional TD-DFT calculations. It is shown that variationally optimized charge-localized diabatic states can be defined using TSO-DFT in energy decomposition analysis to gain both qualitative and quantitative insights on intermolecular interactions. Alternatively, the variational diabatic states may be used in molecular dynamics simulation of charge transfer processes. The TSO method can also be used to define basis states in multistate density functional theory for excited states through nonorthogonal state interaction calculations. The software implementing TSO-DFT can be accessed from the authors.

Atom typing is the first step for simulating molecules using a force field. Automatic atom typing for an arbitrary molecule is often realized by rule-based algorithms, which have to manually encode rules for all types defined in this force field. These are time-consuming and force field-specific. In this study, a method that is independent of specific force field based on graph representation learning is established for automatic atom typing. The topology adaptive graph convolution network (TAGCN) is found to be an optimal model. The model does not need manual enumeration of rules but can learn the rules just through training using typed molecules prepared during the development of a force field. The test on the CHARMM general force field gives a typing correctness of 91%. A systematic error of typing by TAGCN is its inability of distinguishing types in rings or acyclic chains. It originates from the fundamental structure of graph neural networks and can be fixed in a trivial way. More importantly, analysis of the rationalization processes of these models using layer-wise relation propagation reveals how TAGCN encodes rules learned during training. Our model is found to be able to type using the local chemical environments, in a way highly in accordance with chemists' intuition.