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(4)where E ls,0 and E S,0 are the nonpenalized ground-state energies of the targeted low-spin (ls) and the higher spin ( S) state and Δ⟨ Ŝ 2⟩ = S( S + 1) – S ls( S ls + 1) corresponds to the difference in their spin expectation values. (37,38) Other sparse FCI solvers (2,39−50) could also benefit from the enhanced sparsity of the Hamiltonian and wave functions that follow the above-mentioned strategy. (36) In the latter cases, a connection with the concept of alternancy symmetry can be envisioned. (34) This strategy has been applied to exchange-coupled polynuclear transition metal clusters with a large number of localized open-shell orbitals (32−35) and to one-dimensional Heisenberg and Hubbard model Hamiltonians. (34,35) These properties arise from a unique block-diagonal structure of the GUGA Hamiltonian matrix, even within the same spin-symmetry sector, that follows chemically/physically motivated molecular orbital transformations. Recently, we have discovered a strategy within GUGA that allows an unprecedented reduction of the multireference character (compression) (32−35) of ground- and excited-state wave functions and the unique possibility to perform state-specific optimizations of ground- and excited-state wave functions. The Graphical Unitary group approach (GUGA) (17−26) is one such example of a fully spin-adapted approach, which was implemented within the stochastic full-CI quantum Monte Carlo (FCIQMC) (9,27−30) and the Stochastic-CASSCF (10,31,32) frameworks.
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(1−16) In these approaches, Ŝ 2 symmetry is explicitly enforced, ensuring zero spin contamination, and enabling the targeting of any desired spin state. For these reasons, there has been much interest in recent years in developing spin-adapted approaches, which work in Hilbert spaces of configuration state functions (CSFs), rather than Slater determinants (SDs). Furthermore, when a high-spin state is the ground-state, states of the same symmetry but with lower spin are impossible to obtain with ground state projective techniques. Such systems usually exhibit a strong multireference character, with numerous dominant electronic configurations featuring similar weights in the configuration interaction (CI) expansion. In these systems, spin contamination is a major problem for an accurate description of their electronic spectrum. Strongly open-shell molecules present a number of challenges to quantum chemical methods, arising from the large number of nearly degenerate states with different total spin quantum number, S, which exist in such systems and are in general hard to resolve. It also provides spin-resolved reduced density matrices, important for the investigation of spin-dependent properties in polynuclear transition metal clusters, such as the hyperfine-coupling constants. Moreover, in contrast to the GUGA-FCIQMC technique, this strategy features both spin projection and total spin adaptation, making it appealing when solving anisotropic Hamiltonians. This strategy can be coupled to other algorithms that use the Slater determinant based FCIQMC algorithm as configuration interaction eigensolver, including the Stochastic Generalized Active Space, the similarity-transformed FCIQMC, the tailored-CC, and second-order perturbation theory approaches. In the absence of a robust spin adaptation/purification technique, both applications would be unattainable by Slater determinant based ground state methods, with any starting wave function collapsing into the higher-spin ground state during the optimization. Two crucial applications are presented to demonstrate the validity and robustness of this scheme: the 1Δ g ← 3Σ g vertical excitation in O 2 and key spin gaps in a cluster. I’ve worked as a content curator and writer on IBMblr since 2013.In this article, we demonstrate that a first-order spin penalty scheme can be efficiently applied to the Slater determinant based Full-CI Quantum Monte Carlo (FCIQMC) algorithm, as a practical route toward spin purification.
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