Orbital-optimised opposite-spin scaled second-order perturbation theory (O2) generates a single-reference wave function composed of approximate Brueckner orbitals with fourth-order computational scaling. While O2 provides significantly improved treatment of radicals by reducing spin contamination, it has been shown to suffer from first derivative discontinuities for bond stretching near the unrestriction point. That qualitative failure is resolved in this work by the implementation of regularised O2, which includes a regularisation parameter in the denominator of its second-order term. The value of the regularisation parameter is semi-empirically chosen to qualitatively describe bond stretching energetics of hydrogen, ethane and ethene, while also considering the effect of the regularisation parameter on thermochemical errors for the well-known Gaussian-2 (G2) test set. The generality of the empirical scaling and semi-empirical regularisation parameter is studied by application to the 3dMLBE20, DBH24, RSE43 and W4-11 test sets. We demonstrate that accuracy of O2 is roughly maintained and sometimes even improved by regularisation, with root mean squares of regularised O2 between factors of 1.6 and 0.8 from corresponding root mean squares of O2.

The protein folding problem was first articulated as question of how order arose from disorder in proteins: How did the various native structures of proteins arise from interatomic driving forces encoded within their amino acid sequences, and how did they fold so fast? These matters have now been largely resolved by theory and statistical mechanics combined with experiments. There are general principles. Chain randomness is overcome by solvation-based codes. And in the needle-in-a-haystack metaphor, native states are found efficiently because protein haystacks (conformational ensembles) are funnel-shaped. Order-disorder theory has now grown to encompass a large swath of protein physical science across biology.

We perform targeted attack, a systematic computational unlinking of the network, to analyze its effects on global communication across the brain network through its giant cluster. Across diffusion magnetic resonance images from individuals in the UK Biobank, Adolescent Brain Cognitive Development Study and Developing Human Connectome Project, we find that targeted attack procedures on increasing white matter tract lengths and densities are remarkably invariant to aging and disease. Time-reversing the attack computation suggests a mechanism for how brains develop, for which we derive an analytical equation using percolation theory. Based on a close match between theory and experiment, our results demonstrate that tracts are limited to emanate from regions already in the giant cluster and tracts that appear earliest in neurodevelopment are those that become the longest and densest.

Brain aging is associated with hypometabolism and global changes in functional connectivity. Using functional MRI (fMRI), we show that network synchrony, a collective property of brain activity, decreases with age. Applying quantitative methods from statistical physics, we provide a generative (Ising) model for these changes as a function of the average communication strength between brain regions. We find that older brains are closer to a critical point of this communication strength, in which even small changes in metabolism lead to abrupt changes in network synchrony. Finally, by experimentally modulating metabolic activity in younger adults, we show how metabolism alone-independent of other changes associated with aging-can provide a plausible candidate mechanism for marked reorganization of brain network topology.