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Kohn–Sham density functional theory calculations using conventional diagonalization based methods become increasingly expensive as temperature increases due to the need to compute increasing numbers of partially occupied states. We present a density matrix based method for Kohn–Sham calculations at high temperatures that eliminates the need for diagonalization entirely, thus reducing the cost of such calculations significantly. Specifically, we develop real-space expressions for the electron density, electronic free energy, Hellmann–Feynman forces, and Hellmann–Feynman stress tensor in terms of an orthonormal auxiliary orbital basis and its density kernel transform, the density kernel being the matrix representation of the density operator in the auxiliary basis. Using Chebyshev filtering to generate the auxiliary basis, we next develop an approach akin to Clenshaw–Curtis spectral quadrature to calculate the individual columns of the density kernel based on the Fermi operator expansion in Chebyshev polynomials and employ a similar approach to evaluate band structure and entropic energy components. We implement the proposed formulation in the SPARC electronic structure code, using which we show systematic convergence of the aforementioned quantities to exact diagonalization results, and obtain significant speedups relative to conventional diagonalization based methods. Finally, we employ the new method to compute the self-diffusion coefficient and viscosity of aluminum at 116 045 K from Kohn–Sham quantum molecular dynamics, where we find agreement with previous more approximate orbital-free density functional methods. [ABSTRACT FROM AUTHOR]

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Hyperphosphorylation of the protein tau is one of the biomarkers of neurodegenerative diseases in the category of tauopathies. However, the molecular level, mechanistic, role of this common post-translational modification (PTM) in enhancing or reducing the aggregation propensity of tau is unclear, especially considering that combinatorial phosphorylation of multiple sites can have complex, non-additive, effects on tau protein aggregation. Since tau proteins stack in register and parallel to elongate into pathological fibrils, phosphoryl groups from adjacent tau strands with 4.8 Å separation must find an energetically favorable spatial arrangement. At first glance, this appears to be an unfavorable configuration due to the proximity of negative charges between phosphate groups from adjacent neighboring tau fibrils. However, this study tests a counterhypothesis that phosphoryl groups within the fibril core-forming segments favorably assemble into highly ordered, hydrogen-bonded, one-dimensionally extended wires under biologically relevant conditions. We selected two phosphorylation sites associated with neurodegeneration, serine 305 (S305 p ) and tyrosine 310 (Y310 p ), on a model tau peptide jR2R3-P301L (tau295-313) spanning the R2/R3 splice junction of tau, that readily aggregate into a fibril with characteristics of a seed-competent mini prion. Using multiple quantum spin counting (MQ-SC) by 31 P solid-state NMR of phosphorylated jR2R3-P301L tau peptide fibrils, enhanced by dynamic nuclear polarization, we find that at least six phosphorous spins must neatly arrange in 1D within fibrils or in 2D within a protofibril to yield the experimentally observed MQ-coherence orders of four. We found that S305 p stabilizes the tau fibrils and leads to more seeding-competent fibrils compared to jR2R3 P301L or Y310 p . This study introduces a new concept that phosphorylation of residues within a core forming tau segment can mechanically facilitate fibril registry and stability due a hitherto unrecognized role of phosphoryl groups to form highly ordered, extended, 1D wires that stabilize pathological tau fibrils.