Classical mechanics technique for quantum linear response
E.V. Tsiper
Department of Chemistry, Princeton University, Princeton, NJ 08544
It is shown that the lowest random phase approximation (RPA)
excitation energies of a quantum many-fermion system can be obtained
by minimizing an effective classical energy functional. The idea is
based on an analogy between RPA and classical Hamiltonian equations of
motion. Generalized Lanczos recursion allows the minimum to be found
very efficiently. The technique is used to find the electronic
excitation spectrum of the C60 molecule.
The ONETEP linear-scaling density functional theory program.
The Journal of chemical physics, 152(17), p.174111, 2020.
Zhong, H.X., Chen, G.L. and Shen, W.Q.,
Shift and invert weighted Golub-Kahan-Lanczos bidiagonalization algorithm for linear response eigenproblem.
Journal of Computational Analysis & Applications, 26(1), 2019.
Teng, Z. and Zhong, H.X.,
Rayleigh-Ritz Majorization Error Bounds for the Linear Response Eigenvalue Problem.
Open Mathematics, 17(1), pp.653-667, 2019.
Zhong, H.X. and Chen, G.L.,
Thick restarting the weighted harmonic Golub-Kahan-Lanczos algorithm for the linear response eigenvalue problem.
Electronic Transactions on Numerical Analysis, 51, pp.529-546, 2019.
ZHONG, H.X.,
A weighted harmonic Golub-Kahan-Lanczos with deflated restarting method for linear response eigenvalue problems.
Researchgate preprint, 2018.
Vecharynski, E., Brabec, J., Shao, M., Govind, N. and Yang, C.,
Efficient block preconditioned eigensolvers for linear response time-dependent density functional theory.
Computer Physics Communications, 221, pp.42-52, 2017.
Teng, Z. and Zhang, L.H.,
A block Lanczos method for the linear response eigenvalue problem.
Electron. Trans. Numer. Anal, 46, pp.505-523, 2017.
Wang, W.G., Zhang, L.H. and Li, R.C.,
Error bounds for approximate deflating subspaces for linear response eigenvalue problems.
Linear Algebra and its Applications, 528, pp.273-289, 2017.
Xu, H. and Zhong, H.X.,
Weighted Golub-Kahan-Lanczos algorithms and applications.
Electronic Transactions on Numerical Analysis. 47, pp.153–178, 2017.
Teng, Z., Zhou, Y. and Li, R.C.,
A block Chebyshev-Davidson method for linear response eigenvalue problems.
Advances in Computational Mathematics, 42(5), pp.1103-1128, 2016.
Gorni, T.,
Spin-fluctuation spectra in magnetic systems: a novel approach based on TDDFT.
Doctoral dissertation, SISSA - International School for Advanced Studies, Trieste, Italy, 2016.
Zuehlsdorff, T.J., Hine, N.D.M., Payne, M.C. and Haynes, P.D.,
Linear-scaling time-dependent density-functional theory beyond the Tamm-Dancoff approximation: Obtaining efficiency and accuracy with in situ optimised local orbitals.
The Journal of Chemical Physics, 143(20), p.204107, 2015.
Zhang, L.H., Lin, W.W. and Li, R.C.,
Backward perturbation analysis and residual-based error bounds for the linear response eigenvalue problem.
BIT Numerical Mathematics, 55(3), pp.869-896, 2015.
Teng, Z., Lu, L. and Li, R.C.,
Perturbation of partitioned linear response eigenvalue problems.
Electronic Transactions on Numerical Analysis, 44, pp.624-638, 2015.
Bai, Z. and Li, R.C.,
Recent progress in linear response eigenvalue problems.
In International Workshop on Eigenvalue Problems: Algorithms, Software and Applications in Petascale Computing (pp. 287-304). Springer, Cham. 978-3-319-62426-6, 2015.
Mehl, C. and Xu, H.,
Canonical forms of structured matrices and pencils.
In Numerical Algebra, Matrix Theory, Differential-Algebraic Equations and Control Theory ed. by P. Benner et al. (pp. 131-159). Springer, Cham. ISBN: 978-3-319-15260-8, 2015.
Zuehlsdorff, T.J., Hine, N.D., Payne, M.C. and Haynes, P.D.,
Linear-scaling time-dependent density-functional theory (TDDFT) beyond the Tamm-Dancoff approximation: obtaining efficiency and accuracy with in situ optimised local orbitals.
arXiv preprint arXiv:1507.08157, 2015.
Zuehlsdorff, T.J. and Hine, N.D.M.,
Calculating excited states of extended systems in LR-TDDFT.
eCSE02-15 Technical Report PDF, 2015.
Zuehlsdorff, T.J.,
Linear-Scaling TDDFT in ONETEP.
Doctoral Thesis, Imperial College London, UK, 2015.
Challacombe, M.,
Linear scaling solution of the time-dependent self-consistent-field equations.
Computation, 2(1), pp.1-11, 2014.
Zhang, L.H., Xue, J. and Li, R.C.,
Rayleigh--Ritz Approximation For the Linear Response Eigenvalue Problem.
SIAM Journal on Matrix Analysis and Applications, 35(2), pp.765-782, 2014.
Linear-scaling time-dependent density-functional theory in the linear response formalism.
The Journal of Chemical Physics, 139(6), p.064104, 2013.
Bai, Z. and Li, R.C.,
Minimization principles for the linear response eigenvalue problem II: Computation.
SIAM Journal on Matrix Analysis and Applications, 34(2), pp.392-416, 2013.
Teng, Z. and Li, R.C.,
Convergence analysis of Lanczos-type methods for the linear response eigenvalue problem.
Journal of Computational and Applied Mathematics, 247, pp.17-33, 2013.
Timrov, I.,
Ab initio study of plasmons and electron-phonon coupling in bismuth: from free-carrier absorption towards a new method for electron energy-loss spectroscopy
Doctoral dissertation, Ecole Polytechnique X, 2013.
Bai, Z. and Li, R.C.,
Minimization principles for the linear response eigenvalue problem I: Theory.
SIAM Journal on Matrix Analysis and Applications, 33(4), pp.1075-1100, 2012.
Rocca, D., Bai, Z., Li, R.C. and Galli, G.,
A block variational procedure for the iterative diagonalization of non-Hermitian random-phase approximation matrices.
The Journal of Chemical Physics, 136(3), p.034111, 2012.
Huang, C. and Huang, Y.C.,
Unification theory of classical statistical uncertainty relation and quantum uncertainty relation and its applications.
Physics Letters A, 375(3), pp.271-275, 2011.
Bai, Z. and Li, R.C.,
Minimization principle for linear response eigenvalue problem with applications.
Technical Report 2011-06, Department of Mathematics, University of Texas at Arlington, 2011.
Tretiak, S., Isborn, C.M., Niklasson, A.M. and Challacombe, M.,
Representation independent algorithms for molecular response calculations in time-dependent self-consistent field theories.
The Journal of chemical physics, 130(5), p.054111, 2009.
Rocca, D., Gebauer, R., Saad, Y. and Baroni, S.,
Turbo charging time-dependent density-functional theory with Lanczos chains.
The Journal of Chemical Physics, 128(15), p.154105, 2008.
Walker, B.G., Hendy, S.C., Gebauer, R. and Tilley, R.D.,
Application of Lanczos-based time-dependent density-functional theory approach to semiconductor nanoparticle quantum dots.
The European Physical Journal B, 66(1), pp.7-15, 2008.
Artinyan, E., Habets, F., Noilhan, J., Ledoux, E., Dimitrov, D., Martin, E. and Le Moigne, P.,
Modelling the water budget and the riverflows of the Maritsa basin in Bulgaria.
Hydrology and Earth System Sciences Discussions, 12(1), pp.21-37, 2008.
Papakonstantinou, P.,
Reduction of the RPA eigenvalue problem and a generalized Cholesky decomposition for real-symmetric matrices.
EPL (Europhysics Letters), 78(1), p.12001, 2007.
Walker, B., Saitta, A.M., Gebauer, R. and Baroni, S.,
Efficient approach to time-dependent density-functional perturbation theory for optical spectroscopy.
Physical review letters, 96(11), p.113001, 2006.
Tsiper, E.V.,
Fractional charge revealed in computer simulations of resonant tunneling in the fractional quantum Hall regime.
Physical review letters, 97(7), p.076802, 2006.
Huang, Y.C., Ma, F.C. and Zhang, N.,
Generalization of classical statistical mechanics to quantum mechanics and stable property of condensed matter.
Modern Physics Letters B, 18(26n27), pp.1367-1377, 2004.
Mehl, C., Mehrmann, V. and Xu, H.,
On doubly structured matrices and pencils that arise in linear response theory.
Linear algebra and its applications, 380, pp.3-51, 2004.
Tretiak, S. and Chernyak, V.,
Resonant nonlinear polarizabilities in the time-dependent density functional theory.
The Journal of chemical physics, 119(17), pp.8809-8823, 2003.
Tretiak, S. and Mukamel, S.,
Density matrix analysis and simulation of electronic excitations in conjugated and aggregated molecules.
Chemical reviews, 102(9), pp.3171-3212, 2002.