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In recent years, van der Waals (vdW) heterostructures and homostructures, which consist of stacks of two-dimensional (2D) materials, have risen to prominence due to their association with exotic quantum phenomena. Atomistic scale relaxation effects play an extremely important role in the electronic scale quantum physics of these systems. We investigate such structural relaxation effects in this work using atomistic and mesoscale models, within the context of twisted bilayer graphene -- a well-known heterostructure system that features moire patterns arising from the lattices of the two graphene layers. For small twist angles, atomic relaxation effects in this system are associated with the natural emergence of interface dislocations or strain solitons, which result from the cyclic nature of the generalized stacking fault energy (GSFE), that measures the interface energy based on the relative movement of the two layers. In this work, we first demonstrate using atomistic simulations that atomic reconstruction in bilayer graphene under a large twist also results from interface dislocations, although the Burgers vectors of such dislocations are considerably smaller than those observed in small-twist systems. To reveal the translational invariance of the heterointerface responsible for the formation of such dislocations, we derive the translational symmetry of the GSFE of a 2D heterostructure using the notions of coincident site lattices (CSLs) and displacement shift complete lattices (DSCLs). The workhorse for this exercise is a recently developed Smith normal form bicrystallography framework. Next, we construct a bicrystallography-informed and frame-invariant Frenkel-Kontorova model, which can predict the formation of strain solitons in arbitrary 2D heterostructures, and apply it to study a heterostrained, large-twist bilayer graphene system.

The ground state electron density -- obtainable using Kohn-Sham Density Functional Theory (KS-DFT) simulations -- contains a wealth of material information, making its prediction via machine learning (ML) models attractive. However, the computational expense of KS-DFT scales cubically with system size which tends to stymie training data generation, making it difficult to develop quantifiably accurate ML models that are applicable across many scales and system configurations. Here, we address this fundamental challenge by employing transfer learning to leverage the multi-scale nature of the training data, while comprehensively sampling system configurations using thermalization. Our ML models are less reliant on heuristics, and being based on Bayesian neural networks, enable uncertainty quantification. We show that our models incur significantly lower data generation costs while allowing confident -- and when verifiable, accurate -- predictions for a wide variety of bulk systems well beyond training, including systems with defects, different alloy compositions, and at unprecedented, multi-million-atom scales. Moreover, such predictions can be carried out using only modest computational resources.

We introduce carbon Kagome nanotubes (CKNTs) -- a new allotrope of carbon formed by rolling up sheets of Kagome graphene, and investigate the properties of this material using first principles calculations. Based on the direction of rolling, we identify two principal varieties of CKNTs -- armchair and zigzag, and find that the bending stiffness associated with rolling Kagome graphene into either type of CKNT is about a third of that associated with rolling conventional graphene into carbon nanotubes (CNTs). Ab initio molecular dynamics simulations indicate that both types of CKNTs are likely to exist as stable structures at room temperature. Each CKNT explored here is metallic and features dispersionless states (i.e., flat bands) throughout its Brillouin zone, along with an associated singular peak in the electronic density of states, close to the Fermi level. We calculate the mechanical and electronic response of CKNTs to torsional and axial strains and compare against conventional CNTs. We show in particular, that upon twisting, degenerate dispersionless electronic states in CKNTs split, Dirac points and partially flat bands emerge from the quadratic band crossing point at the Fermi level, and that these features can be explained using a relatively simple tight-binding model. Overall, CKNTs appear to be unique and striking examples of realistic elemental quasi-one-dimensional (1D) materials that can potentially display fascinating collective material properties arising from the presence of strongly correlated electrons. Additionally, distorted CKNTs may provide an interesting material platform where flat band physics and chirality induced anomalous transport effects may be studied together.

The Posner molecule (calcium phosphate trimer), has been hypothesized to function as a biological quantum information processor due to its supposedly long-lived entangled $^{31}$P nuclear spin states. This hypothesis was challenged by our recent finding that the molecule lacks a well-defined rotational axis of symmetry -- an essential assumption in the proposal for Posner-mediated neural processing -- and exists as an asymmetric dynamical ensemble. Following up, we investigate here the spin dynamics of the molecule's entangled $^{31}$P nuclear spins within the asymmetric ensemble. Our simulations show that entanglement between two nuclear spins prepared in a Bell state in separate Posner molecules decays on a sub-second timescale -- much faster than previously hypothesized, and not long enough for super-cellular neuronal processing. Calcium phosphate dimers however, are found to be surprisingly resilient to decoherence and are able to preserve entangled nuclear spins for hundreds of seconds, suggesting that neural processing might occur through them instead.

We formulate and implement a spectral method for solving the Schrodinger equation, as it applies to quasi-one-dimensional materials and structures. This allows for computation of the electronic structure of important technological materials such as nanotubes (of arbitrary chirality), nanowires, nanoribbons, chiral nanoassemblies, nanosprings and nanocoils, in an accurate, efficient and systematic manner. Our work is motivated by the observation that one of the most successful methods for carrying out electronic structure calculations of bulk/crystalline systems -- the plane-wave method -- is a spectral method based on eigenfunction expansion. Our scheme avoids computationally onerous approximations involving periodic supercells often employed in conventional plane-wave calculations of quasi-one-dimensional materials, and also overcomes several limitations of other discretization strategies, e.g., those based on finite differences and atomic orbitals. We describe the setup of fast transforms to carry out discretization of the governing equations using our basis set, and the use of matrix-free iterative diagonalization to obtain the electronic eigenstates. Miscellaneous computational details, including the choice of eigensolvers, use of a preconditioning scheme, evaluation of oscillatory radial integrals and the imposition of a kinetic energy cutoff are discussed. We have implemented these strategies into a computational package called HelicES (Helical Electronic Structure). We demonstrate the utility of our method in carrying out systematic electronic structure calculations of various quasi-one-dimensional materials through numerous examples involving nanotubes, nanoribbons and nanowires. We also explore the convergence, accuracy and efficiency of our method. We anticipate that our method will find numerous applications in computational nanomechanics and materials science.

We present a machine learning based model that can predict the electronic structure of quasi-one-dimensional materials while they are subjected to deformation modes such as torsion and extension/compression. The technique described here applies to important classes of materials such as nanotubes, nanoribbons, nanowires, miscellaneous chiral structures and nano-assemblies, for all of which, tuning the interplay of mechanical deformations and electronic fields is an active area of investigation in the literature. Our model incorporates global structural symmetries and atomic relaxation effects, benefits from the use of helical coordinates to specify the electronic fields, and makes use of a specialized data generation process that solves the symmetry-adapted equations of Kohn-Sham Density Functional Theory in these coordinates. Using armchair single wall carbon nanotubes as a prototypical example, we demonstrate the use of the model to predict the fields associated with the ground state electron density and the nuclear pseudocharges, when three parameters - namely, the radius of the nanotube, its axial stretch, and the twist per unit length - are specified as inputs. Other electronic properties of interest, including the ground state electronic free energy, can then be evaluated with low-overhead post-processing, typically to chemical accuracy. We also show how the nuclear coordinates can be reliably determined from the pseudocharge field using a clustering based technique. Remarkably, only about 120 data points are found to be enough to predict the three dimensional electronic fields accurately, which we ascribe to the symmetry in the problem setup, the use of low-discrepancy sequences for sampling, and presence of intrinsic low-dimensional features in the electronic fields. We comment on the interpretability of our machine learning model and discuss its possible future applications.

The Posner molecule, $\text{Ca}_9(\text{PO}_4)_6$, has long been recognized to have biochemical relevance in various physiological processes. It has found recent attention for its possible role as a biological quantum information processor, whereby the molecule purportedly maintains long-lived nuclear spin coherences among its ${^{31}\text{P}}$ nuclei (presumed to be symmetrically arranged), allowing it to function as a room temperature qubit. The structure of the molecule has been of much dispute in the literature, although the $\text{S}_6$ point group symmetry has often been assumed and exploited in calculations. Using a variety of simulation techniques (including ab initio molecular dynamics and structural relaxation), rigorous data analysis tools and by exploring thousands of individual configurations, we establish that the molecule predominantly assumes low symmetry structures ($\text{C}_\text{s}$ and $\text{C}_\text{i}$) at room temperature, as opposed to the higher symmetry configurations explored previously. Our findings have important implications on the viability of this molecule as a qubit., Comment: Posner Molecule, Symmetry, Biological Qubit, Density Functional Theory

We present a real-space formulation and implementation of Kohn-Sham Density Functional Theory suited to twisted geometries, and apply it to the study of torsional deformations of X (X = C, Si, Ge, Sn) nanotubes. Our formulation is based on higher order finite difference discretization in helical coordinates, uses ab intio pseudopotentials, and naturally incorporates rotational (cyclic) and screw operation (i.e., helical) symmetries. We discuss several aspects of the computational method, including the form of the governing equations, details of the numerical implementation, as well as its convergence, accuracy and efficiency properties. The technique presented here is particularly well suited to the first principles simulation of quasi-one-dimensional structures and their deformations, and many systems of interest can be investigated using small simulation cells containing just a few atoms. We apply the method to systematically study the properties of single-wall zigzag and armchair group-IV nanotubes, as they undergo twisting. For the range of deformations considered, the mechanical behavior of the tubes is found to be largely consistent with isotropic linear elasticity, with the torsional stiffness varying as the cube of the nanotube radius. Furthermore, for a given tube radius, this quantity is seen to be highest for carbon nanotubes and the lowest for those of tin, while nanotubes of silicon and germanium have intermediate values close to each other. We also describe different aspects of the variation in electronic properties of the nanotubes as they are twisted. In particular, we find that akin to the well known behavior of armchair carbon nanotubes, armchair nanotubes of silicon, germanium and tin also exhibit bandgaps that vary periodically with imposed rate of twist, and that the periodicity of the variation scales in an inverse quadratic manner with the tube radius.

Chiral degrees of freedom occur in matter and in electromagnetic fields and constitute an area of research that is experiencing renewed interest driven by recent observations of the chiral-induced spin selectivity (CISS) effect in chiral molecules and engineered nanomaterials. The CISS effect underpins the fact that charge transport through nanoscopic chiral structures favors a particular electronic spin orientation, resulting in large room-temperature spin polarizations. Observations of the CISS effect suggest opportunities for spin control and for the design and fabrication of room-temperature quantum devices from the bottom up, with atomic-scale precision. Any technology that relies on optimal charge transport, including quantum devices for logic, sensing, and storage, may benefit from chiral quantum properties. These properties can be theoretically and experimentally investigated from a quantum information perspective, which is presently lacking. There are uncharted implications for the quantum sciences once chiral couplings can be engineered to control the storage, transduction, and manipulation of quantum information. This forward-looking perspective provides a survey of the experimental and theoretical fundamentals of chiral-influenced quantum effects, and presents a vision for their future roles in enabling room-temperature quantum technologies.

We formulate and implement Helical DFT -- a self-consistent first principles simulation method for nanostructures with helical symmetries. Such materials are well represented in all of nanotechnology, chemistry and biology, and are expected to be associated with unprecedented material properties. We rigorously demonstrate the existence and completeness of special solutions to the single electron problem for helical nanostructures, called helical Bloch waves. We describe how the Kohn-Sham Density Functional Theory equations for a helical nanostructure can be reduced to a fundamental domain with the aid of these solutions. A key component in our mathematical treatment is the definition and use of a helical Bloch-Floquet transform to perform a block-diagonalization of the Hamiltonian in the sense of direct integrals. We develop a symmetry-adapted finite-difference strategy in helical coordinates to discretize the governing equations, and obtain a working realization of the proposed approach. We verify the accuracy and convergence properties of our numerical implementation through examples. Finally, we employ Helical DFT to study the properties of zigzag and chiral single wall black phosphorus (i.e., phosphorene) nanotubes. We use our simulations to evaluate the torsional stiffness of a zigzag nanotube ab initio. Additionally, we observe an insulator-to-metal-like transition in the electronic properties of this nanotube as it is subjected to twisting. We also find that a similar transition can be effected in chiral phosphorene nanotubes by means of axial strains. Notably, self-consistent ab initio simulations of this nature are unprecedented and well outside the scope of any other systematic first principles method in existence. We end with a discussion on various future avenues and applications.

We present a symmetry-adapted real-space formulation of Kohn-Sham density functional theory for cylindrical geometries and apply it to the study of large X (X=C, Si, Ge, Sn) nanotubes. Specifically, starting from the Kohn-Sham equations posed on all of space, we reduce the problem to the fundamental domain by incorporating cyclic and periodic symmetries present in the angular and axial directions of the cylinder, respectively. We develop a high-order finite-difference parallel implementation of this formulation, and verify its accuracy against established planewave and real-space codes. Using this implementation, we study the band structure and bending properties of X nanotubes and Xene sheets, respectively. Specifically, we first show that zigzag and armchair X nanotubes with radii in the range 1 to 5 nm are semiconducting. In particular, we find an inverse linear dependence of the bandgap with respect to the radius for all nanotubes, other than the armchair and zigzag type III carbon variants, for which we find an inverse quadratic dependence. Next, we exploit the connection between cyclic symmetry and uniform bending deformations to calculate the bending moduli of Xene sheets in both zigzag and armchair directions. We find Kirchhoff-Love type bending behavior for all sheets, with graphene and stanene possessing the largest and smallest moduli, respectively. In addition, other than graphene, the sheets demonstrate significant anisotropy, with larger bending moduli along the armchair direction. Finally, we demonstrate that the proposed approach has very good parallel scaling and is highly efficient, enabling ab initio simulations of unprecedented size for systems with a high degree of cyclic symmetry. In particular, we show that even micron-sized nanotubes can be simulated with modest computational effort., Comment: 24 pages, 8 figures, 4 tables

We describe a novel iterative strategy for Kohn-Sham density functional theory calculations aimed at large systems (> 1000 electrons), applicable to metals and insulators alike. In lieu of explicit diagonalization of the Kohn-Sham Hamiltonian on every self-consistent field (SCF) iteration, we employ a two-level Chebyshev polynomial filter based complementary subspace strategy to: 1) compute a set of vectors that span the occupied subspace of the Hamiltonian; 2) reduce subspace diagonalization to just partially occupied states; and 3) obtain those states in an efficient, scalable manner via an inner Chebyshev-filter iteration. By reducing the necessary computation to just partially occupied states, and obtaining these through an inner Chebyshev iteration, our approach reduces the cost of large metallic calculations significantly, while eliminating subspace diagonalization for insulating systems altogether. We describe the implementation of the method within the framework of the Discontinuous Galerkin (DG) electronic structure method and show that this results in a computational scheme that can effectively tackle bulk and nano systems containing tens of thousands of electrons, with chemical accuracy, within a few minutes or less of wall clock time per SCF iteration on large-scale computing platforms. We anticipate that our method will be instrumental in pushing the envelope of large-scale ab initio molecular dynamics. As a demonstration of this, we simulate a bulk silicon system containing 8,000 atoms at finite temperature, and obtain an average SCF step wall time of 51 seconds on 34,560 processors; thus allowing us to carry out 1.0 ps of ab initio molecular dynamics in approximately 28 hours (of wall time)., Comment: Resubmitted version (version 2)

The Discontinuous Galerkin (DG) electronic structure method employs an adaptive local basis (ALB) set to solve the Kohn-Sham equations of density functional theory (DFT) in a discontinuous Galerkin framework. The adaptive local basis is generated on-the-fly to capture the local material physics, and can systematically attain chemical accuracy with only a few tens of degrees of freedom per atom. A central issue for large-scale calculations, however, is the computation of the electron density (and subsequently, ground state properties) from the discretized Hamiltonian in an efficient and scalable manner. We show in this work how Chebyshev polynomial filtered subspace iteration (CheFSI) can be used to address this issue and push the envelope in large-scale materials simulations in a discontinuous Galerkin framework. We describe how the subspace filtering steps can be performed in an efficient and scalable manner using a two-dimensional parallelization scheme, thanks to the orthogonality of the DG basis set and block-sparse structure of the DG Hamiltonian matrix. The on-the-fly nature of the ALBs requires additional care in carrying out the subspace iterations. We demonstrate the parallel scalability of the DG-CheFSI approach in calculations of large-scale two-dimensional graphene sheets and bulk three-dimensional lithium-ion electrolyte systems. Employing 55,296 computational cores, the time per self-consistent field iteration for a sample of the bulk 3D electrolyte containing 8,586 atoms is 90 seconds, and the time for a graphene sheet containing 11,520 atoms is 75 seconds., Comment: Submitted to The Journal of Chemical Physics

We formulate and implement Cyclic Density Functional Theory (Cyclic DFT) -- a self-consistent first principles simulation method for nanostructures with cyclic symmetries. Using arguments based on Group Representation Theory, we rigorously demonstrate that the Kohn-Sham eigenvalue problem for such systems can be reduced to a fundamental domain (or cyclic unit cell) augmented with cyclic-Bloch boundary conditions. Analogously, the equations of electrostatics appearing in Kohn-Sham theory can be reduced to the fundamental domain augmented with cyclic boundary conditions. By making use of this symmetry cell reduction, we show that the electronic ground-state energy and the Hellmann-Feynman forces on the atoms can be calculated using quantities defined over the fundamental domain. We develop a symmetry-adapted finite-difference discretization scheme to obtain a fully functional numerical realization of the proposed approach. We verify that our formulation and implementation of Cyclic DFT is both accurate and efficient through selected examples. The connection of cyclic symmetries with uniform bending deformations provides an elegant route to the ab-initio study of bending in nanostructures using Cyclic DFT. As a demonstration of this capability, we simulate the uniform bending of a silicene nanoribbon and obtain its energy-curvature relationship from first principles. A self-consistent ab-initio simulation of this nature is unprecedented and well outside the scope of any other systematic first principles method in existence. Our simulations reveal that the bending stiffness of the silicene nanoribbon is intermediate between that of graphene and molybdenum disulphide. We describe several future avenues and applications of Cyclic DFT, including its extension to the study of non-uniform bending deformations and its possible use in the study of the nanoscale flexoelectric effect., Comment: Version 3 of the manuscript, Accepted for publication in Journal of the Mechanics and Physics of Solids, http://www.sciencedirect.com/science/article/pii/S0022509616303684

Pulay's Direct Inversion in the Iterative Subspace (DIIS) method is one of the most widely used mixing schemes for accelerating the self-consistent solution of electronic structure problems. In this work, we propose a simple generalization of DIIS in which Pulay extrapolation is performed at periodic intervals rather than on every self-consistent field iteration, and linear mixing is performed on all other iterations. We demonstrate through numerical tests on a wide variety of materials systems in the framework of density functional theory that the proposed generalization of Pulay's method significantly improves its robustness and efficiency., Comment: Version 2 (with minor edits from version 1)

Starting from the observation that one of the most successful methods for solving the Kohn-Sham equations for periodic systems -- the plane-wave method -- is a spectral method based on eigenfunction expansion, we formulate a spectral method designed towards solving the Kohn-Sham equations for clusters. This allows for efficient calculation of the electronic structure of clusters (and molecules) with high accuracy and systematic convergence properties without the need for any artificial periodicity. The basis functions in this method form a complete orthonormal set and are expressible in terms of spherical harmonics and spherical Bessel functions. Computation of the occupied eigenstates of the discretized Kohn-Sham Hamiltonian is carried out using a combination of preconditioned block eigensolvers and Chebyshev polynomial filter accelerated subspace iterations. Several algorithmic and computational aspects of the method, including computation of the electrostatics terms and parallelization are discussed. We have implemented these methods and algorithms into an efficient and reliable package called ClusterES (Cluster Electronic Structure). A variety of benchmark calculations employing local and non-local pseudopotentials are carried out using our package and the results are compared to the literature. Convergence properties of the basis set are discussed through numerical examples. Computations involving large systems that contain thousands of electrons are demonstrated to highlight the efficacy of our methodology. The use of our method to study clusters with arbitrary point group symmetries is briefly discussed., Comment: Manuscript submitted (with revisions) to Journal of Computational Physics