Solution of the Schrödinger equation for quasi-one-dimensional materials using helical waves. (English) Zbl 07807977
Summary: We formulate and implement a spectral method for solving the Schrödinger 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. The basis functions in our method – called helical waves (or twisted waves) – are eigenfunctions of the Laplacian with symmetry adapted boundary conditions, and are expressible in terms of plane waves and Bessel functions in helical coordinates.
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 properties of our method, and assess its accuracy and computational efficiency by comparison against reference finite difference, transfer matrix method and plane-wave results. We anticipate that our method will find applications in computational nanomechanics and multiscale modeling, for carrying out transport calculations of interest to the field of semiconductor devices, and for the discovery of novel chiral phases of matter that are of relevance to the burgeoning quantum hardware industry.
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 properties of our method, and assess its accuracy and computational efficiency by comparison against reference finite difference, transfer matrix method and plane-wave results. We anticipate that our method will find applications in computational nanomechanics and multiscale modeling, for carrying out transport calculations of interest to the field of semiconductor devices, and for the discovery of novel chiral phases of matter that are of relevance to the burgeoning quantum hardware industry.
MSC:
| 65Fxx | Numerical linear algebra |
| 65Dxx | Numerical approximation and computational geometry (primarily algorithms) |
| 65Nxx | Numerical methods for partial differential equations, boundary value problems |
Keywords:
helical waves; electronic structure calculations; nanomaterials; nanostructures; chiral materials; spectral methodSoftware:
BLOPEX; lobpcg.m; CheFSI; quadgk; WebPlotDigitizer; ClusterES; FastGaussQuadrature; BLAS; MatlabReferences:
| [1] | Bhushan, B.; Bhushan Baumann, B., Springer Handbook of Nanotechnology, vol. 2 (2007), Springer |
| [2] | Cao, G., Nanostructures & Nanomaterials: Synthesis, Properties & Applications (2004), Imperial College Press |
| [3] | Kang, I.; Schulz, M. J.; Kim, J. H.; Shanov, V.; Shi, D., A carbon nanotube strain sensor for structural health monitoring. Smart Mater. Struct., 3, 737 (2006) |
| [4] | Chopra, S.; McGuire, K.; Gothard, N.; Rao, A.; Pham, A., Selective gas detection using a carbon nanotube sensor. Appl. Phys. Lett., 11, 2280-2282 (2003) |
| [5] | Baughman, R. H.; Cui, C.; Zakhidov, A. A.; Iqbal, Z.; Barisci, J. N.; Spinks, G. M.; Wallace, G. G.; Mazzoldi, A.; De Rossi, D.; Rinzler, A. G., Carbon nanotube actuators. Science, 5418, 1340-1344 (1999) |
| [6] | Kong, L.; Chen, W., Carbon nanotube and graphene-based bioinspired electrochemical actuators. Adv. Mater., 7, 1025-1043 (2014) |
| [7] | Sun, H.; Zhang, Y.; Zhang, J.; Sun, X.; Peng, H., Energy harvesting and storage in 1d devices. Nat. Rev. Mater., 6, 1-12 (2017) |
| [8] | Fan, F. R.; Tang, W.; Wang, Z. L., Flexible nanogenerators for energy harvesting and self-powered electronics. Adv. Mater., 22, 4283-4305 (2016) |
| [9] | Xu, C., Chiral Nanoprobes for Biological Applications (2022), John Wiley & Sons |
| [10] | James, R. D., Objective structures. J. Mech. Phys. Solids, 11, 2354-2390 (2006) · Zbl 1120.74312 |
| [11] | Giamarchi, T., Quantum Physics in One Dimension, vol. 121 (2003), Clarendon Press |
| [12] | Bockrath, M.; Cobden, D. H.; Lu, J.; Rinzler, A. G.; Smalley, R. E.; Balents, L.; McEuen, P. L., Luttinger-liquid behaviour in carbon nanotubes. Nature, 6720, 598-601 (1999) |
| [13] | Egger, R.; Zazunov, A.; Yeyati, A. L., Helical Luttinger liquid in topological insulator nanowires. Phys. Rev. Lett., 13 (2010) |
| [14] | Zaitsev-Zotov, S.; Kumzerov, Y. A.; Firsov, Y. A.; Monceau, P., Luttinger-liquid-like transport in long InSb nanowires. J. Phys. Condens. Matter, 20, L303 (2000) |
| [15] | Arutyunov, K.; Golubev, D.; Zaikin, A., Superconductivity in one dimension. Phys. Rep., 1-2, 1-70 (2008) |
| [16] | Qin, F.; Shi, W.; Ideue, T.; Yoshida, M.; Zak, A.; Tenne, R.; Kikitsu, T.; Inoue, D.; Hashizume, D.; Iwasa, Y., Superconductivity in a chiral nanotube. Nat. Commun., 1, 1-6 (2017) |
| [17] | Krusin-Elbaum, L.; Newns, D.; Zeng, H.; Derycke, V.; Sun, J.; Sandstrom, R., Room-temperature ferromagnetic nanotubes controlled by electron or hole doping. Nature, 7009, 672-676 (2004) |
| [18] | Liu, M.; Wang, J., Giant electrocaloric effect in ferroelectric nanotubes near room temperature. Sci. Rep., 1, 1-7 (2015) |
| [19] | Zhang, D.; Hua, M.; Dumitrica, T., Stability of polycrystalline and wurtzite Si nanowires via symmetry-adapted tight-binding objective molecular dynamics. J. Chem. Phys., 8 (2008) |
| [20] | Yu, H. M.; Banerjee, A. S., Density functional theory method for twisted geometries with application to torsional deformations in group-iv nanotubes. J. Comput. Phys. (2022) · Zbl 07518101 |
| [21] | Ding, J.; Yan, X.; Cao, J., Analytical relation of band gaps to both chirality and diameter of single-wall carbon nanotubes. Phys. Rev. B, 7 (2002) |
| [22] | Tans, S. J.; Verschueren, A. R.; Dekker, C., Room-temperature transistor based on a single carbon nanotube. Nature, 6680, 49-52 (1998) |
| [23] | Li, S.; Yu, Z.; Yen, S.-F.; Tang, W.; Burke, P. J., Carbon nanotube transistor operation at 2.6 GHz. Nano Lett., 4, 753-756 (2004) |
| [24] | Aiello, C. D.; Abendroth, J. M.; Abbas, M.; Afanasev, A.; Agarwal, S.; Banerjee, A. S.; Beratan, D. N.; Belling, J. N.; Berche, B.; Botana, A.; Caram, J. R.; Celardo, G. L.; Cuniberti, G.; Garcia-Etxarri, A.; Dianat, A.; Diez-Perez, I.; Guo, Y.; Gutierrez, R.; Herrmann, C.; Hihath, J.; Kale, S.; Kurian, P.; Lai, Y.-C.; Liu, T.; Lopez, A.; Medina, E.; Mujica, V.; Naaman, R.; Noormandipour, M.; Palma, J. L.; Paltiel, Y.; Petuskey, W.; Ribeiro-Silva, J. C.; Saenz, J. J.; Santos, E. J.G.; Solyanik-Gorgone, M.; Sorger, V. J.; Stemer, D. M.; Ugalde, J. M.; Valdes-Curiel, A.; Varela, S.; Waldeck, D. H.; Wasielewski, M. R.; Weiss, P. S.; Zacharias, H.; Wang, Q. H., A chirality-based quantum leap. ACS Nano (2022) |
| [25] | Naaman, R.; Waldeck, D. H., Spintronics and chirality: spin selectivity in electron transport through chiral molecules. Annu. Rev. Phys. Chem., 1, 263-281 (2015) |
| [26] | Zhang, H.; Liu, D. E.; Wimmer, M.; Kouwenhoven, L. P., Next steps of quantum transport in majorana nanowire devices. Nat. Commun., 1, 1-7 (2019) |
| [27] | Martin, R. M., Electronic Structure: Basic Theory and Practical Methods (2004), Cambridge University Press · Zbl 1152.74303 |
| [28] | Marx, D.; Hutter, J., Ab Initio Molecular Dynamics: Basic Theory and Advanced Methods (2009), Cambridge University Press |
| [29] | Banerjee, A. S., Density functional methods for Objective Structures: Theory and simulation schemes (2013), University of Minnesota: University of Minnesota Minneapolis, Ph.D. thesis |
| [30] | Banerjee, A. S., Ab initio framework for systems with helical symmetry: theory, numerical implementation and applications to torsional deformations in nanostructures. J. Mech. Phys. Solids (2021) |
| [31] | D’Arco, P.; Noel, Y.; Demichelis, R.; Dovesi, R., Single-layered chrysotile nanotubes: a quantum mechanical ab initio simulation. J. Chem. Phys., 20 (2009) |
| [32] | Dovesi, R.; Saunders, V.; Roetti, C.; Orlando, R.; Zicovich-Wilson, C.; Pascale, F.; Civalleri, B.; Doll, K.; Harrison, N.; Bush, I., Crystal17 (2017) |
| [33] | White, C. T.; Robertson, D. H.; Mintmire, J. W., Helical and rotational symmetries of nanoscale graphitic tubules. Phys. Rev. B, 5485-5488 (1993) |
| [34] | Chang, E.; Bussi, G.; Ruini, A.; Molinari, E., First-principles approach for the calculation of optical properties of one-dimensional systems with helical symmetry: the case of carbon nanotubes. Phys. Rev. B (2005) |
| [35] | Noel, Y.; D’arco, P.; Demichelis, R.; Zicovich-Wilson, C. M.; Dovesi, R., On the use of symmetry in the ab initio quantum mechanical simulation of nanotubes and related materials. J. Comput. Chem., 4, 855-862 (2010) |
| [36] | Chang, E.; Bussi, G.; Ruini, A.; Molinari, E., Excitons in carbon nanotubes: an ab initio symmetry-based approach. Phys. Rev. Lett. (2004) |
| [37] | Balabin, R. M., Communications: intramolecular basis set superposition error as a measure of basis set incompleteness: can one reach the basis set limit without extrapolation?. J. Chem. Phys., 21 (2010) |
| [38] | Gutowski, M.; Chałasiński, G., Critical evaluation of some computational approaches to the problem of basis set superposition error. J. Chem. Phys., 7, 5540-5554 (1993) |
| [39] | Simon, S.; Duran, M.; Dannenberg, J., How does basis set superposition error change the potential surfaces for hydrogen-bonded dimers?. J. Chem. Phys., 24, 11024-11031 (1996) |
| [40] | Ghosh, S.; Banerjee, A. S.; Suryanarayana, P., Symmetry-adapted real-space density functional theory for cylindrical geometries: application to large group-IV nanotubes. Phys. Rev. B, 12 (2019) |
| [41] | Sharma, A.; Suryanarayana, P., Real-space density functional theory adapted to cyclic and helical symmetry: application to torsional deformation of carbon nanotubes. Phys. Rev. B, 3 (2021) |
| [42] | Gygi, F.; Galli, G., Real-space adaptive-coordinate electronic-structure calculations. Phys. Rev. B, 4 (1995) |
| [43] | Banerjee, A. S.; Suryanarayana, P., Cyclic density functional theory: a route to the first principles simulation of bending in nanostructures. J. Mech. Phys. Solids, 605-631 (2016) · Zbl 1482.74135 |
| [44] | Saad, Y., Numerical Methods for Large Eigenvalue Problems (2011), SIAM · Zbl 1242.65068 |
| [45] | Banerjee, A. S.; Elliott, R. S.; James, R. D., A spectral scheme for Kohn-Sham density functional theory of clusters. J. Comput. Phys., 226-253 (2015) · Zbl 1351.81014 |
| [46] | Matlab, 9.7.0.1190202 (R2019b) (2019), the MathWorks Inc.: the MathWorks Inc. Natick, Massachusetts |
| [47] | Broglia, R.; Coló, G.; Onida, G.; Roman, H., Solid State Physics of Finite Systems: Metallic Clusters, Fullerenes, Atomic Wires. Advanced Texts in Physics (2004), Springer |
| [48] | D’yachkov, P. N.; Makaev, D., Linear augmented cylindrical wave method and its applications to nanotubes electronic structure. J. Nanophotonics, 1 (2010) |
| [49] | Hussain, S. R.; Kharitonov, S.; Pavelyev, V., Calculation of the band structure of a non-chiral semiconductor and metallic carbon nanotubes. J. Phys. Conf. Ser., 012109 (2018) |
| [50] | D’Yachkov, P., Cylindrical wave method for pure and doped nanotubes, 87-100 |
| [51] | Makaev, D.; D’yachkov, P., Linearized augmented cylindrical wave method for chiral nanotubes. Dokl., Phys. Chem., 47-52 (2008) |
| [52] | Jüstel, D.; Friesecke, G.; James, R. D., Bragg-von Laue diffraction generalized to twisted X-rays. Acta Crystallogr. A, Found. Adv., 2, 190-196 (2016) · Zbl 1370.82086 |
| [53] | Friesecke, G.; James, R. D.; Jüstel, D., Twisted X-rays: incoming waveforms yielding discrete diffraction patterns for helical structures. SIAM J. Appl. Math., 3, 1191-1218 (2016) · Zbl 1342.78028 |
| [54] | Chaplain, G.; De Ponti, J.; Craster, R., Elastic orbital angular momentum. Phys. Rev. Lett., 6 (2022) |
| [55] | Chaplain, G. J.; De Ponti, J. M., The elastic spiral phase pipe. J. Sound Vib. (2022) |
| [56] | Walter, J. P.; Cohen, M. L., Wave-vector-dependent dielectric function for Si, Ge, GaAs, and ZnSe. Phys. Rev.B, 6, 1821 (1970) |
| [57] | Fong, C.; Cohen, M. L., Energy band structure of copper by the empirical pseudopotential method. Phys. Rev. Lett., 7, 306 (1970) |
| [58] | Hoffmann-Ostenhof, M.; Hoffmann-Ostenhof, T.; Ahlrichs, R.; Morgan, J., On the exponential fall off of wavefunctions and electron densities, 62-67 · Zbl 0446.35017 |
| [59] | Ahlrichs, R.; Hoffmann-Ostenhof, M.; Hoffmann-Ostenhof, T.; Morgan, J. D., Bounds on the decay of electron densities with screening. Phys. Rev. A, 5, 2106-2117 (1981) |
| [60] | Waldron, R., A helical coordinate system and its applications in electromagnetic theory. Q. J. Mech. Appl. Math., 4, 438-461 (1958) · Zbl 0082.40502 |
| [61] | Hochberg, D.; Edwards, G.; Kephart, T. W., Representing structural information of helical charge distributions in cylindrical coordinates. Phys. Rev. E, 3, 3765 (1997) |
| [62] | Saad, Y.; Chelikowsky, J. R.; Shontz, S. M., Numerical methods for electronic structure calculations of materials. SIAM Rev., 1, 3-54 (2010) · Zbl 1185.82004 |
| [63] | Orszag, S. A., Transform method for the calculation of vector-coupled sums: application to the spectral form of the vorticity equation. J. Atmos. Sci., 6, 890-895 (1970) |
| [64] | Orszag, S. A., Spectral methods for problems in complex geometries. J. Comput. Phys., 1, 70-92 (1980) · Zbl 0476.65078 |
| [65] | Shen, J.; Tang, T.; Wang, L.-L., Spectral Methods: Algorithms, Analysis and Applications. Springer Series in Computational Mathematics (2011), Springer: Springer Berlin, New York · Zbl 1227.65117 |
| [66] | Gottlieb, D.; Orszag, S., Numerical Analysis of Spectral Methods: Theory and Applications (1977), SIAM-CBMS: SIAM-CBMS Philadelphia · Zbl 0412.65058 |
| [67] | Abramowitz, M.; Stegun, I. A.; Romer, R. H., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (1988) |
| [68] | Teodorescu, P. P.; Stanescu, N.-D.; Pandrea, N., Numerical Analysis with Applications in Mechanics and Engineering (2013), John Wiley & Sons |
| [69] | Bylaska, E.; Tsemekhman, K.; Govind, N.; Valiev, M., Large-scale plane-wave-based density functional theory: formalism, parallelization, and applications, 77-116 |
| [70] | Cancès, E.; Chakir, R.; Maday, Y., Numerical analysis of the planewave discretization of some orbital-free and Kohn-Sham models. ESAIM: Modél. Math. Anal. Numér., 341-388 (2012) · Zbl 1278.82003 |
| [71] | Monkhorst, H. J.; Pack, J. D., Special points for Brillouin-zone integrations. Phys. Rev. B, 12, 5188 (1976) |
| [72] | Geru, I. I., Time-Reversal Symmetry (2018), Springer · Zbl 1412.81005 |
| [73] | Lemoine, D., The discrete Bessel transform algorithm. J. Chem. Phys., 5, 3936-3944 (1994) |
| [74] | Johansen, H.; Sørensen, K., Fast Hankel transforms. Geophys. Prospect., 4, 876-901 (1979) |
| [75] | Bisseling, R.; Kosloff, R., The fast Hankel transform as a tool in the solution of the time dependent Schrödinger equation. J. Comput. Phys., 1, 136-151 (1985) · Zbl 0578.65127 |
| [76] | Key, K., Is the fast Hankel transform faster than quadrature?. Geophysics, 3, F21-F30 (2012) |
| [77] | Blackford, L. S.; Petitet, A.; Pozo, R.; Remington, K.; Whaley, R. C.; Demmel, J.; Dongarra, J.; Duff, I.; Hammarling, S.; Henry, G., An updated set of basic linear algebra subprograms (BLAS). ACM Trans. Math. Softw., 2, 135-151 (2002) · Zbl 1070.65520 |
| [78] | Stewart, G. W., A Krylov-Schur algorithm for large eigenproblems. SIAM J. Matrix Anal. Appl., 3, 601-614 (2002) · Zbl 1003.65045 |
| [79] | Payne, M. C.; Teter, M. P.; Allan, D. C.; Arias, T. A.; Joannopoulos, J. D., Iterative minimization techniques for ab initio total-energy calculations: molecular dynamics and conjugate gradients. Rev. Mod. Phys., 1045-1097 (1992) |
| [80] | Zhou, Y.; Chelikowsky, J. R.; Gao, X.; Zhou, A., On the “preconditioning” function used in planewave dft calculations and its generalization. Commun. Comput. Phys., 1, 167-179 (2015) · Zbl 1388.65195 |
| [81] | Knyazev, A., Locally optimal block preconditioned conjugate gradient (2022) |
| [82] | Knyazev, A., Toward the optimal preconditioned eigensolver: locally optimal block preconditioned conjugate gradient method. SIAM J. Sci. Comput., 2, 517-541 (2001) · Zbl 0992.65028 |
| [83] | Knyazev, A. V.; Argentati, M. E.; Lashuk, I.; Ovtchinnikov, E., Block locally optimal preconditioned eigenvalue xolvers (BLOPEX) in HYPRE and PETSc. SIAM J. Sci. Comput., 5, 2224-2239 (2007) · Zbl 1149.65026 |
| [84] | Duersch, J. A.; Shao, M.; Yang, C.; Gu, M., A robust and efficient implementation of lobpcg. SIAM J. Sci. Comput., 5, C655-C676 (2018) · Zbl 1401.65038 |
| [85] | Teter, M. P.; Payne, M. C.; Allan, D. C., Solution of Schrödinger’s equation for large systems. Phys. Rev.B, 12255-12263 (1989) |
| [86] | Mayer, A., Band structure and transport properties of carbon nanotubes using a local pseudopotential and a transfer-matrix technique. Carbon, 10, 2057-2066 (2004) |
| [87] | Banerjee, A. S.; Lin, L.; Hu, W.; Yang, C.; Pask, J. E., Chebyshev polynomial filtered subspace iteration in the discontinuous Galerkin method for large-scale electronic structure calculations (2016), arXiv preprint |
| [88] | Banerjee, A. S.; Lin, L.; Suryanarayana, P.; Yang, C.; Pask, J. E., Two-level Chebyshev filter based complementary subspace method: pushing the envelope of large-scale electronic structure calculations. J. Chem. Theory Comput., 6, 2930-2946 (2018) |
| [89] | Zhou, Y.; Chelikowsky, J. R.; Saad, Y., Chebyshev-filtered subspace iteration method free of sparse diagonalization for solving the Kohn-Sham equation. J. Comput. Phys., 770-782 (2014) · Zbl 1351.82098 |
| [90] | Laturia, A. A.; Van de Put, M. L.; Vandenberghe, W. G., Generation of empirical pseudopotentials for transport applications and their application to group iv materials. J. Appl. Phys., 3 (2020) |
| [91] | Van de Put, M. L.; Fischetti, M. V.; Vandenberghe, W. G., Scalable atomistic simulations of quantum electron transport using empirical pseudopotentials. Comput. Phys. Commun., 156-169 (2019) · Zbl 07674839 |
| [92] | Chelikowsky, J. R., Introductory Quantum Mechanics with MATLAB: For Atoms, Molecules, Clusters, and Nanocrystals (2019), John Wiley & Sons · Zbl 1406.81002 |
| [93] | Rohatgi, A., Webplotdigitizer: version 4.5 (2021) |
| [94] | Mayer, A.; Vigneron, J.-P., Accuracy-control techniques applied to stable transfer-matrix computations. Phys. Rev. E, 4, 4659 (1999) |
| [95] | Pendry, J.; MacKinnon, A., Calculation of photon dispersion relations. Phys. Rev. Lett., 19, 2772 (1992) |
| [96] | Tamura, H.; Ando, T., Conductance fluctuations in quantum wires. Phys. Rev.B, 4, 1792 (1991) |
| [97] | Van de Put, M. L.; Laturia, A. A.; Fischetti, M. V.; Vandenberghe, W. G., Efficient modeling of electron transport with plane waves, 71-74 |
| [98] | Troullier, N.; Martins, J. L., Efficient pseudopotentials for plane-wave calculations. Phys. Rev.B, 3, 1993 (1991) |
| [99] | Kohn, W.; Sham, L. J., Self-consistent equations including exchange and correlation effects. Phys. Rev., A1133-A1138 (1965) |
| [100] | Perdew, J. P.; Wang, Y., Accurate and simple analytic representation of the electron-gas correlation energy. Phys. Rev.B, 13244-13249 (1992) |
| [101] | Zhang, D.; James, R. D.; Dumitrica, T., Electromechanical characterization of carbon nanotubes in torsion via symmetry adapted tight-binding objective molecular dynamics. Phys. Rev.B, 11 (2009) |
| [102] | Yang, L.; Anantram, M.; Han, J.; Lu, J., Band-gap change of carbon nanotubes: effect of small uniaxial and torsional strain. Phys. Rev.B, 19 (1999) |
| [103] | Yang, L.; Han, J., Electronic structure of deformed carbon nanotubes. Phys. Rev. Lett., 1, 154 (2000) |
| [104] | Agarwal, S.; Banerjee, A. S., On the Solution of the Poisson Problem for Twisted Geometries (2022), (in preparation) |
| [105] | Pathrudkar, S.; Yu, H. M.; Ghosh, S.; Banerjee, A. S., Machine learning based prediction of the electronic structure of quasi-one-dimensional materials under strain. Phys. Rev. B (2022) |
| [106] | Hakobyan, Y.; Tadmor, E.; James, R., Objective quasicontinuum approach for rod problems. Phys. Rev.B, 24 (2012) |
| [107] | (Le Bris, C., Computational Chemistry. Computational Chemistry, Handbook of Numerical Analysis, vol. X (2003), North-Holland) · Zbl 1052.81001 |
| [108] | Nikiforov, I.; Hourahine, B.; Aradi, B.; Frauenheim, T.; Dumitrică, T., Ewald summation on a helix: a route to self-consistent charge density-functional based tight-binding objective molecular dynamics. J. Chem. Phys., 9 (2013) |
| [109] | Liebman, O.; Agarwal, S.; Banerjee, A., Helical waves for self-consistent first principles calculations of chiral one-dimensional nanomaterials. Bull. Am. Phys. Soc. (2022) |
| [110] | Release 1.1.5 of 2022-03-15 |
| [111] | Laslett, L. J.; Lewish, W., Evaluation of the zeros of cross-product Bessel functions. Math. Comput., 78, 226-232 (1962) · Zbl 0107.11101 |
| [112] | Hellmann, H., Hans Hellmann: Einführung in Die Quantenchemie: Mit Biografischen Notizen von Hans Hellmann Jr. (2015), Springer-Verlag |
| [113] | Feynman, R. P., Forces in molecules. Phys. Rev., 4, 340 (1939) · Zbl 0022.42302 |
| [114] | Deaño, A.; Huybrechs, D.; Iserles, A., Computing Highly Oscillatory Integrals (2017), SIAM |
| [115] | Ma, Y.; Xu, Y., Computing highly oscillatory integrals. Math. Comput., 309, 309-345 (2018) · Zbl 1376.65027 |
| [116] | Iserles, A.; Nørsett, S. P., Efficient quadrature of highly oscillatory integrals using derivatives. Proc. R. Soc. A, Math. Phys. Eng. Sci., 2057, 1383-1399 (2005) · Zbl 1145.65309 |
| [117] | Iserles, A.; Nørsett, S. P., On quadrature methods for highly oscillatory integrals and their implementation. BIT Numer. Math., 4, 755-772 (2004) · Zbl 1076.65025 |
| [118] | Chen, R., Numerical approximations to integrals with a highly oscillatory Bessel kernel. Appl. Numer. Math., 5, 636-648 (2012) · Zbl 1241.65116 |
| [119] | Ralston, A.; Rabinowitz, P., A First Course in Numerical Analysis (2001), Courier Corporation · Zbl 0976.65001 |
| [120] | Hale, N.; Townsend, A., Fast and accurate computation of Gauss-Legendre and Gauss-Jacobi quadrature nodes and weights. SIAM J. Sci. Comput., 2, A652-A674 (2013) · Zbl 1270.65017 |
| [121] | Laurie, D., Calculation of Gauss-Kronrod quadrature rules. Math. Comput., 219, 1133-1145 (1997) · Zbl 0870.65018 |
| [122] | Shampine, L. F., Vectorized adaptive quadrature in Matlab. J. Comput. Appl. Math., 2, 131-140 (2008) · Zbl 1134.65021 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
