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One way to reduce the computational cost of electronic structure calculations is to use auxiliary basis expansions to approximate four-center integrals in terms of two- and three-center integrals, usually by using the variationally optimum Coulomb metric to determine the expansion coefficients. However, the long-range decay behavior of the auxiliary basis expansion coefficients has not been characterized. We find that this decay can be surprisingly slow. Numerical experiments on linear alkanes and a toy model both show that the decay can be as slow as 1/r in the distance between the auxiliary function and the fitted charge distribution. The Coulomb metric fitting equations also involve divergent matrix elements for extended systems treated with periodic boundary conditions. An attenuated Coulomb metric that is short-range can eliminate these oddities without substantially degrading calculated relative energies. The sparsity of the fit coefficients is assessed on simple hydrocarbon molecules and shows quite early onset of linear growth in the number of significant coefficients with system size using the attenuated Coulomb metric. Hence it is possible to design linear scaling auxiliary basis methods without additional approximations to treat large systems.

One way to reduce the computational cost of electronic structure calculations is to use auxiliary basis expansions to approximate four-center integrals in terms of two- and three-center integrals, usually by using the variationally optimum Coulomb metric to determine the expansion coefficients. However, the long-range decay behavior of the auxiliary basis expansion coefficients has not been characterized. We find that this decay can be surprisingly slow. Numerical experiments on linear alkanes and a toy model both show that the decay can be as slow as 1/r in the distance between the auxiliary function and the fitted charge distribution. The Coulomb metric fitting equations also involve divergent matrix elements for extended systems treated with periodic boundary conditions. An attenuated Coulomb metric that is short-range can eliminate these oddities without substantially degrading calculated relative energies. The sparsity of the fit coefficients is assessed on simple hydrocarbon molecules and shows quite early onset of linear growth in the number of significant coefficients with system size using the attenuated Coulomb metric. Hence it is possible to design linear scaling auxiliary basis methods without additional approximations to treat large systems. [PUBLICATION ABSTRACT]

A short review is given of newly developed fast electronic structure methods that are designed to treat molecular systems with strong electron correlations, such as diradicaloid molecules, for which standard electronic structure methods such as density functional theory are inadequate. These new local correlation methods are based on coupled cluster theory within a perfect pairing active space, containing either a linear or quadratic number of pair correlation amplitudes, to yield the perfect pairing (PP) and imperfect pairing (IP) models. This reduces the scaling of the coupled cluster iterations to no worse than cubic, relative to the sixth power dependence of the usual (untruncated) coupled cluster doubles model. A second order perturbation correction, PP(2), to treat the neglected (weaker) correlations is formulated for the PP model. To ensure minimal prefactors, in addition to favorable size-scaling, highly efficient implementations of PP, IP and PP(2) have been completed, using auxiliary basis expansions. This yields speedups of almost an order of magnitude over the best alternatives using 4-center 2-electron integrals. A short discussion of the scope of accessible chemical applications is given.

Given an initial point \\(x_0 \\in \\mathbb{R}^d\\) and a sequence of vectors \\(v_1, v_2, \\dots\\) in \\(\\mathbb{R}^d\\), we define a greedy sequence by setting \\(x_{n} = x_{n-1} \\pm v_n\\) where the sign is chosen so as to minimize \\(\\|x_n\\|\\). We prove that if the vectors \\(v_i\\) are chosen uniformly at random from \\(\\mathbb{S}^{d-1}\\) then elements of the sequence are, on average, approximately at distance \\(\\|x_n\\| \\sim \\sqrt{\\pi d/8}\\) from the origin. We show that the sequence \\((\\|x_n\\|)_{n=1}^{\\infty}\\) has an invariant measure \\(\\pi_d\\) depending only on \\(d\\) and we determine its mean and study its decay for all \\(d\\). We also investigate a completely deterministic example in \\(d=2\\) where the \\(v_n\\) are derived from the van der Corput sequence. Several additional examples are considered.

The interferon-induced transmembrane (IFITM) proteins broadly inhibit the entry of diverse pathogenic viruses, including Influenza A virus (IAV), Zika virus, HIV-1, and SARS coronaviruses by inhibiting virus-cell membrane fusion. IFITM3 was previously shown to disrupt cholesterol trafficking, but the functional relationship between IFITM3 and cholesterol remains unclear. We previously showed that inhibition of Influenza A virus entry by IFITM3 is associated with its ability to promote cellular membrane rigidity, and an amphipathic helix (AH) encoded by IFITM3 is required for both activities. Furthermore, it has been shown that the AH of IFITM3 alters lipid membranes in vitro in a cholesterol-dependent manner. Therefore, we aimed to elucidate the relationship between IFITM3 and cholesterol in more detail. Using a fluorescence-based in vitro binding assay, we found that a peptide derived from the AH of IFITM3 directly interacted with the cholesterol analog, NBD-cholesterol. In comparison, a peptide derived from a putative cholesterol recognition motif in the transmembrane domain of IFITM3 exhibited minor cholesterol binding activity. Importantly, previously characterized mutations within the AH of IFITM3 that strongly disrupt antiviral activity (F63Q and F67Q) disrupted AH structure and inhibited cholesterol binding. Our data suggest that direct interactions with cholesterol may contribute to the inhibition of membrane fusion pore formation by IFITM3. These findings may facilitate the design of therapeutic peptides for use in broad-spectrum antiviral therapy. Competing Interest Statement The authors have declared no competing interest.