Understanding the electronic structure and excited-state dynamics of photochemical systems is a challenging problem in computational chemistry. Features of the adiabatic potential energy surfaces such as avoided crossings and conical intersections play important roles in events such as non-radiative recombination of excited fluorophores and photovoltaic cells. Nuclear geometries corresponding to points of (near) degeneracy of adiabatic potential energy surfaces represent regions where the Born-Oppenheimer approximation breaks down. A consequence of an erosion of the ability to separate the nuclear and electronic degrees of freedom in the wavefunction is the need for multireference electronic structure methods. The state-averaged complete active space self-consistent field (SA-CASSCF) and the complete active space configuration interaction (CASCI) methods are standard tools for generating multireference wavefunctions. A common component of each of these methods is a configuration interaction (CI) calculation step. The CI method is computationally demanding, however, and imposes hard limits on the dimension of the correlated region. In this work we describe our direct, graphical processing unit (GPU) vectorized full CI implementation, allowing us to calculate CI wavefunctions and energies for systems having $\mathcal{O}(10^3)$ basis functions and $\mathcal{O}(10^8)$ determinants on timescales of minutes. We apply our direct CI method to a series of molecular benchmark systems and demonstrate that our atomic orbital to molecular orbital basis integral transformation scales approximately quadratically with respect to basis set size. Analysis of the scaling behavior of the rate-limiting component of the CI iterations, the matrix-vector product $\bm{\sigma} \mathbf{=Hc}$, where $\mathbf{H}$ is the electronic Hamiltonian and $\mathbf{c}$ is the CI vector, reveals that our algorithm scales approximately linearly with respect to the number of determinants. We have developed a GPU-based implementation of the 1- and 2-particle reduced density matrices (1- and 2-RDMs), both of which are necessary for fast evaluation of analytical nuclear energy gradients and nonadiabatic coupling vectors. Calculation of certain properties such as the dipole moment and the transition dipole moment also require the 1-RDM, and SA-CASSCF energy and gradient calculations require the 1-RDM and 2-RDMs, respectively. Formulation and implementation of several spin-purification schemes that are useful in the context of open-shell determinantal CI are described next. These approaches counteract the numerical instabilities associated with high-accuracy open-shell CI calculations. A GPU accelerated direct $\mathbf{S^2c}$ algorithm, where $\mathbf{S}$ is the spin matrix, enables robust spin purification of trial vectors entering the (Krylov) subspace for purification approaches relying on projection or, conversely, modification of the $\bm{\sigma}$ vector to correspond to a spin-penalized Hamiltonian for penalty-based approaches. To demonstrate the utility of spin-purification methods we include a study of the multireference and multi-excitation character of plasmonic open-shell silver clusters. The rank-reduced CI (rrCI) method is described for extremely large configuration spaces. The rrCI method allows ground state singlet and triplet calculations having configuration spaces on the order of $\mathcal{O}(10^{16})$ determinants while achieving mH accuracy relative to full CI (FCI). Single point energies of acenes having $2-5$ aromatic rings are reported using HF-CAS-rrCI and compared with density matrix renormalization group (DMRG) calculated energies as an additional verification of the accuracy of rrCI. All methods described in this work have been implemented in the TeraChem GPU accelerated electronic structure software package. We conclude with a description of large-scale CI calculations that have been performed using methods described in this manuscript.
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