Over the course of the last century, electronic structure theory (or, alternatively, computational quantum chemistry) has grown from being a fledgling field to being a "full partner with experiment" [Goddard Science 1985, 227 (4689), 917--923]. Numerous instances of theory matching experiment to very high accuracy abound, with one excellent example being the high-accuracy ab initio thermochemical data laid out in the 2004 work of Tajti and co-workers [Tajti et al. J. Chem. Phys. 2004, 121, 11599] and another being the heats of formation and molecular structures computed by Feller and co-workers in 2008 [Feller et al. J. Chem. Phys. 2008, 129, 204105]. But as the authors of both studies point out, this very high accuracy comes at a very high cost. In fact, at this point in time, electronic structure theory does not suffer from an accuracy problem (as it did in its early days) but a cost problem; or, perhaps more precisely, it suffers from an accuracy-to-cost ratio problem. We can compute electronic energies to nearly any precision we like, as long as we are willing to pay the associated cost.

And just what are these high computational costs? For the purposes of this work, we are primarily concerned with the way in which the computational cost of a given method scales with the system size; for notational purposes, we will often introduce a parameter, N, that is proportional to the system size. In the case of Hartree-Fock, a one-body wavefunction-based method, the scaling is formally N⁴, and post-Hartree-Fock methods fare even worse. The coupled cluster singles, doubles, and perturbative triples method [CCSD(T)], which is frequently referred to as the "gold standard" of quantum chemistry, has an N⁷ scaling, making it inapplicable to many systems of real-world import.

If highly accurate correlated wavefunction methods are to be applied to larger systems of interest, it is crucial that we reduce their computational scaling. One very successful means of doing this relies on the fact that electron correlation is fundamentally a local phenomenon, and the recognition of this fact has led to the development of numerous local implementations of conventional many-body methods. One such method, the DLPNO-CCSD(T) method, was successfully used to calculate the energy of the protein crambin [Riplinger, et al. J. Chem. Phys 2013, 139, 134101].

In the following work, we discuss how the local nature of electron correlation can be exploited, both in terms of the occupied orbitals and the unoccupied (or virtual) orbitals. In the case of the former, we highlight some of the historical developments in orbital localization before applying orbital localization robustly to infinite periodic crystalline systems [Clement, et al. 2021, Submitted to J. Chem. Theory Comput.]. In the case of the latter, we discuss a number of different ways in which the virtual space can be compressed before presenting our pioneering work in the area of iteratively-optimized pair natural orbitals ("iPNOs") [Clement, et al. J. Chem. Theory Comput. 2018, 14 (9), 4581--4589].

Concerning the iPNOs, we were able to recover significant accuracy with respect to traditional PNOs (which are unchanged throughout the course of a correlated calculation) at a comparable truncation level, indicating that our improved PNOs are, in fact, an improved representation of the coupled cluster doubles amplitudes. For example, when studying the percent errors in the absolute correlation energies of a representative sample of weakly bound dimers chosen from the S66 test suite [Řezác, et al. J. Chem. Theory Comput. 2011, 7 (8), 2427--2438], we found that our iPNO-CCSD scheme outperformed the standard PNO-CCSD scheme at every truncation threshold (τ_PNO) studied. Both PNO-based methods were compared to the canonical CCSD method, with the iPNO-CCSD method being, on average, 1.9 times better than the PNO-CCSD method at τ_PNO = 10⁻⁷ and more than an order of magnitude better for τ_PNO < 10⁻¹⁰ [Clement, et al. J. Chem. Theory Comput 2018, 14 (9), 4581--4589]. When our improved PNOs are combined with the PNO-incompleteness correction proposed by Neese and co-workers [Neese, et al. J. Chem. Phys. 2009, 130, 114108; Neese, et al. J. Chem. Phys. 2009, 131, 064103], the results are truly astounding. For a truncation threshold of τ_PNO = 10⁻⁶, the mean average absolute error in binding energy for all 66 dimers from the S66 test set was 3 times smaller when the incompleteness-corrected iPNO-CCSD method was used relative to the incompleteness-corrected PNO-CCSD method [Clement, et al. J. Chem. Theory Comput. 2018, 14 (9), 4581--4589].

In the latter half of this work, we present our implementation of a limited-memory Broyden-Fletcher-Goldfarb-Shanno (BFGS) based Pipek-Mezey Wannier function (PMWF) solver [Clement, et al. 2021 }, Submitted to J. Chem. Theory Comput.]. Although orbital localization in the context of the linear combination of atomic orbitals (LCAO) representation of periodic crystalline solids is not new [Marzari, et al. Rev. Mod. Phys. 2012, 84 (4), 1419--1475; Jònsson, et al. J. Chem. Theory Comput. 2017, 13} (2), 460--474], to our knowledge, this is the first implementation to be based on a BFGS solver. In addition, we are pleased to report that our novel BFGS-based solver is extremely robust in terms of the initial guess and the size of the history employed, with the final results and the time to solution, as measured in number of iterations required, being essentially independent of these initial choices. Furthermore, our BFGS-based solver converges much more quickly and consistently than either a steepest ascent (SA) or a non-linear conjugate gradient (CG) based solver, with this fact demonstrated for a number of 1-, 2-, and 3-dimensional systems. Armed with our real, localized Wannier functions, we are now in a position to pursue the application of local implementations of correlated many-body methods to the arena of periodic crystalline solids; a first step toward this goal will, most likely, be the study of PNOs, both conventional and iteratively-optimized, in this context.