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We extend ground-state density-functional theory to excited states and provide the theoretical formulation for the widely used $\Delta SCF$ method for calculating excited-state energies and densities. As the electron density alone is insufficient to characterize excited states, we formulate excited-state theory using the defining variables of a noninteracting reference system, namely (1) the excitation quantum number $n_{s}$ and the potential $w_{s}(\mathbf{r})$ (excited-state potential-functional theory, $n$PFT), (2) the noninteracting wavefunction $\Phi$ ($\Phi$-functional theory, $\Phi$FT), or (3) the noninteracting one-electron reduced density matrix $\gamma_{s}(\mathbf{r},\mathbf{r}')$ (density-matrix-functional theory, $\gamma_{s}$FT). We show the equivalence of these three sets of variables and their corresponding energy functionals. Importantly, the ground and excited-state exchange-correlation energy use the \textit{same} universal functional, regardless of whether $\left(n_{s},w_{s}(\boldsymbol{r})\right)$, $\Phi$, or $\gamma_{s}(\mathbf{r},\mathbf{r}')$ is selected as the fundamental descriptor of the system. We derive the excited-state (generalized) Kohn-Sham equations. The minimum of all three functionals is the ground-state energy and, for ground states, they are all equivalent to the Hohenberg-Kohn-Sham method. The other stationary points of the functionals provide the excited-state energies and electron densities, establishing the foundation for the $\Delta SCF$ method.

We propose a new Perturbation Theory framework that can be used to help with the projective solution of the Schr\"odinger equation for arbitrary wavefunctions. This Flexible Ansatz for N-body Perturbation Theory (FANPT) is based on our previously proposed Flexible Ansatz for N-body Configuration Interaction (FANCI). We derive recursive FANPT expressions including arbitrary orders in the perturbation hierarchy. We show that the FANPT equations are well-behaved across a wide range of conditions, including static correlation-dominated configurations and highly non-linear wavefunctions.

Electron pairs have an illustrious history in chemistry, from powerful concepts to understanding structural stability and reactive changes, to the promise of serving as building blocks of quantitative descriptions of the electronic structure of complex molecules and materials. However, traditionally, two-electron wavefunctions (geminals), have not enjoyed the popularity and widespread use of the more standard single-particle methods. This has changed recently, with a renewed interest in the development of geminal wavefunctions as an alternative to describing strongly-correlated phenomena. Hence, there is a need to find geminal methods that are accurate, computationally-tractable, and do not demand significant input from the user (particularly, via cumbersome and often ill-behaved orbital optimization steps). Here we propose new families of geminal wavefunctions, inspired by the pair Coupled Cluster (pCCD) ansatz. We present a new hierarchy of two-electron wavefunctions that extend the one-reference orbital idea to other geminals. Moreover, we show how to incorporate single-like excitations in this framework, without leaving the quasiparticle picture. We explore the role of imposing seniority restrictions on these wavefunctions, and benchmark these new methods on model strongly-correlated systems.

We provide an inductive proof of Borchardt's theorem for calculating the permanent of a Cauchy matrix via the determinants of auxiliary matrices. This result has implications for antisymmetric products of interacting geminals (APIG), and suggests that the restriction of the APIG coefficients to Cauchy form (typically called APr2G) is special in its tractability.

Direct approaches to the quantum many-body problem suffer from the so-called "curse of dimensionality": the number of parameters needed to fully specify the exact wavefunction grows exponentially with increasing system size. This motivates the develop of accurate, but approximate, ways to parametrize the wavefunction, including methods like couple cluster theory and correlator product states (CPS). Recently, there has been interest in approaches based on machine learning both direct applications of neural network architecture and the combinations of conventional wavefunction parametrizations with various Boltzmann machines. While all these methods can be exact in principle, they are usually applied with only a polynomial number of parameters, limiting their applicability. This research's objective is to present a fresh approach to wavefunction parametrization that is size-consistent, rapidly convergent, and robust numerically. Specifically, we propose a hierarchical ansatz that converges rapidly (with respect to the number of least-squares optimization). The general utility of this approach is verified by applying it to uncorrelated, weakly-correlated, and strongly-correlated systems, including small molecules and the one-dimensional Hubbard model., Comment: 18 pages

We develop a bivariational principle for an antisymmetric product of nonorthogonal geminals. Special cases reduce to the antisymmetric product of strongly-orthogonal geminals (APSG), the generalized valence bond-perfect pairing (GVB-PP), and the antisymmetrized geminal power (AGP) wavefunctions. The presented method employs wavefunctions of the same type as Richardson-Gaudin (RG) states, but which are not eigenvectors of a model Hamiltonian which would allow for more freedom in the mean-field. The general idea is to work with the same state in a primal picture in terms of pairs, and in a dual picture in terms of pair-holes. This leads to an asymmetric energy expression which may be optimized bivariationally, and is strictly variational when the two representations are consistent. The general approach may be useful in other contexts, such as for computationally feasible variational coupled-cluster methods.