Riemannian Trust Region Method for Minimization of the Fourth Central Moment for Localized Molecular Orbitals.

The importance of localized molecular orbitals (MOs) in correlation treatments beyond mean-field calculation and in the illustration of chemical bonding (and antibonding) can hardly be overstated. However, the generation of orthonormal localized occupied MOs is significantly more straightforward than obtaining orthonormal localized virtual MOs. Orthonormal MOs allow facile use of highly efficient group theoretical methods (e.g., graphical unitary group approach) for calculation of Hamiltonian matrix elements in multireference configuration interaction calculations (such as MRCISD) and in quasi-degenerate perturbation treatments, such as the Generalized Van Vleck Perturbation Theory. Moreover, localized MOs can elucidate qualitative understanding of bonding in molecules, in addition to high-accuracy quantitative descriptions. We adopt the powers of the fourth moment cost function introduced by Jørgensen and coworkers. Because the fourth moment cost functions are prone to having multiple negative Hessian eigenvalues when starting from easily available canonical (or near-canonical) MOs, standard optimization algorithms can fail to obtain the orbitals of the virtual or partially occupied spaces. To overcome this drawback, we applied a trust region algorithm on an orthonormal Riemannian manifold with an approximate retraction from the tangent space built into the first and second derivatives of the cost function. Moreover, the Riemannian trust region outer iterations were coupled to truncated Conjugate Gradient inner loops, which avoided any costly solutions of simultaneous linear equations or eigenvector/eigenvalue solutions. Numerical examples are provided on model systems, including the high-connectivity H10 set in 1-, 2-, and 3-dimensional arrangements, and on a chemically realistic description of cyclobutadiene (c-C4H4) and the propargyl radical (C3H3). In addition to demonstrating the algorithm on occupied and virtual blocks of orbitals, the method is also shown to work on the active space at the MCSCF level of theory.

Design and Synthesis of Kekulè and Non-Kekulè Diradicaloids via the Radical Periannulation Strategy: The Power of Seven Clar's Sextets.

This work introduces an approach to uncoupling electrons via maximum utilization of localized aromatic units, i.e., the Clar's π-sextets. To illustrate the utility of this concept to the design of Kekulé diradicaloids, we have synthesized a tridecacyclic polyaromatic system where a gain of five Clar's sextets in the open-shell form overcomes electron pairing and leads to the emergence of a high degree of diradical character. According to unrestricted symmetry-broken UCAM-B3LYP calculations, the singlet diradical character in this core system is characterized by the y0 value of 0.98 (y0 = 0 for a closed-shell molecule, y0 = 1 for pure diradical). The efficiency of the new design strategy was evaluated by comparing the Kekulé system with an isomeric non-Kekulé diradical of identical size, i.e., a system where the radical centers cannot couple via resonance. The calculated singlet-triplet gap, i.e., the ΔEST values, in both of these systems approaches zero: -0.3 kcal/mol for the Kekulé and +0.2 kcal/mol for the non-Kekulé diradicaloids. The target isomeric Kekulé and non-Kekulé systems were assembled using a sequence of radical periannulations, cross-coupling, and C-H activation. The diradicals are kinetically stabilized by six tert-butyl substituents and (triisopropylsilyl)acetylene groups. Both molecules are NMR-inactive but electron paramagnetic resonance (EPR)-active at room temperature. Cyclic voltammetry revealed quasi-reversible oxidation and reduction processes, consistent with the presence of two nearly degenerate partially occupied molecular orbitals. The experimentally measured ΔEST value of -0.14 kcal/mol confirms that K is, indeed, a nearly perfect singlet diradical.

Theoretical Calculations of the 242 nm Absorption of Propargyl Radical.

The propargyl radical, the most stable isomer of neutral C3H3, is important in combustion reactions, and a number of spectroscopic and reaction dynamics studies have been performed over the years. However, theoretical calculations have never been able to find a state that can generate strong absorption around 242 nm as seen in experiments. In this study, we calculated the low-lying electronic energy levels of the propargyl radical using the highly accurate multireference configuration interaction singles and doubles method with triples and quadruples treated perturbatively [denoted as MRCISD(TQ)]. Calculations indicate that this absorption can be attributed to a Franck-Condon-allowed electronic transition from the ground 2B1 state to the Rydberg-like excited state 12A1. Further insight into the behavior of the multireference perturbative theory methods, GVVPT2 and GVVPT3, on a very challenging system are also obtained.

Challenges for variational reduced-density-matrix theory with three-particle N-representability conditions.

The direct variational optimization of the two-electron reduced density matrix (2RDM) can provide a reference-independent description of the electronic structure of many-electron systems that naturally capture strong or nondynamic correlation effects. Such variational 2RDM approaches can often provide a highly accurate description of strong electron correlation, provided that the 2RDMs satisfy at least partial three-particle N-representability conditions (e.g., the T2 condition). However, recent benchmark calculations on hydrogen clusters [N. H. Stair and F. A. Evangelista, J. Chem. Phys. 153, 104108 (2020)] suggest that even the T2 condition leads to unacceptably inaccurate results in the case of two- and three-dimensional clusters. We demonstrate that these failures persist under the application of full three-particle N-representability conditions (3POS). A variety of correlation metrics are explored in order to identify regimes under which 3POS calculations become unreliable, and we find that the relative squared magnitudes of the cumulant three- and two-particle reduced density matrices correlate reasonably well with the energy error in these systems. However, calculations on other molecular systems reveal that this metric is not a universal indicator for the reliability of the reduced-density-matrix theory with 3POS conditions.

Data-Driven Refinement of Electronic Energies from Two-Electron Reduced-Density-Matrix Theory.

The exponential computational cost of describing strongly correlated electrons can be mitigated by adopting a reduced-density matrix (RDM)-based description of the electronic structure. While variational two-electron RDM (v2RDM) methods can enable large-scale calculations on such systems, the quality of the solution is limited by the fact that only a subset of known necessary N-representability constraints can be applied to the 2RDM in practical calculations. Here, we demonstrate that violations of partial three-particle (T1 and T2) N-representability conditions, which can be evaluated with knowledge of only the 2RDM, can serve as physics-based features in a machine-learning (ML) protocol for improving energies from v2RDM calculations that consider only two-particle (PQG) conditions. Proof-of-principle calculations demonstrate that the model yields substantially improved energies relative to reference values from configuration-interaction-based calculations.

Challenges for Variational Reduced-Density-Matrix Theory: Total Angular Momentum Constraints.

The variational two-electron reduced density matrix (v2RDM) method is generalized for the description of total angular momentum (J) and projection of total angular momentum (MJ) states in atomic systems described by nonrelativistic Hamiltonians, and it is shown that the approach exhibits serious deficiencies. Under ensemble N-representability constraints, v2RDM theory fails to retain the appropriate degeneracies among various J states for fixed spin (S) and orbital angular momentum (L), and for fixed L, S, and J, the manifold of MJ states is not necessarily degenerate. Moreover, a substantial energy error is observed for a system for which the two-electron reduced density matrix is exactly ensemble N-representable; in this case, the error stems from violations in pure-state N-representability conditions. Unfortunately, such violations do not appear to be good indicators of the reliability of energies from v2RDM theory in general. Several states are identified for which energy errors are near zero and yet pure-state conditions are clearly violated.