Abstract
Metrics have been used to investigate the relationship between wavefunction distances and density distances for families of specific systems. We extend this research to look at random potentials for time-dependent single-electron systems, and for ground-state two-electron systems. We find that Fourier series are a good basis for generating random potentials. These random potentials also yield quasi-linear relationships between the distances of ground-state densities and wavefunctions, providing a framework in which density functional theory can be explored.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
We denote atomic units using a.u., where \(\hbar = m = e = 4 \pi \varepsilon _{0} = 1 \).
The Runge-Gross theorem does not prohibit two systems, propagated by (possibly different) time-dependent potentials from different initial many-body states from having the same density at some later instant. Such a case would provide a trajectory that touches the very bottom of the lower triangle in Fig. 2: \(D_{n}= 0\) while Dψ≠ 0. This provides further support for time-evolution in the metric space of Fig. 2. being predominantly concentrated away from the upper triangle.
References
R. Parr, W. Lang, Density-Functional Theory of Atoms and Molecules (Oxford University Press, Oxford, 1994)
K. Burke, J. Chem. Phys. 136, 150901 (2012)
P. Hohenberg, W. Kohn, Phys. Rev. 136, B864 (1964)
I. D’Amico, J.P. Coe, V.V. França, K Capelle, Phys. Rev. Lett. 106, 050401 (2011)
P. Sharp, I. D’Amico, Phys. Rev. B. 89, 115137 (2014)
P. Sharp, I. D’Amico, Phys. Rev. A. 92, 032509 (2015)
A.H. Skelt, R.W. Godby, I. D’Amico, Phys. Rev. A. 98, 012104 (2018)
W. Kohn, L.J. Sham, Phys. Rev. 140, A1133 (1965)
M. Herrera, R.M. Serra, I. D’Amico, Scientific Reports. 7, 4655 (2017)
M.J.P. Hodgson, J.D. Ramsden, J.B.J. Chapman, P. Lillystone, R.W. Godby, Phys. Rev. B. 88, 241102(R) (2013)
P.M. Sharp, I. D’Amico, Phys. Rev. A . 94, 062509 (2016)
E. Runge, E.K.U. Gross, Phys. Rev. Lett. 52, 997–1000 (1984)
M. Marques, C. Ullrich, F. Nogueira, A. Rubio, K. Burke, E. Gross. Time-Dependent Density Functional Theory (Springer, New York, 2006)
Acknowledgements
We acknowledge helpful discussions with P. Sharp, J. Wetherell and M. Hodgson, and advice on the iDEA code from J. Wetherell. AHS acknowledges support from EPSRC; IDA acknowledges support from the Conselho Nacional de Desenvolvimento Cientfico e Tecnologico (CNPq, grant: PVE Processo: 401414/2014-0) and from the Royal Society (grant no. NA140436). All data published in this research is available on request from the York Research Database http://dx.doi.org/10.15124/ba6b5031-5ec6-4e12-ab17-d8813551a04f
Author information
Authors and Affiliations
Corresponding author
Appendix A: System Parameters
Appendix A: System Parameters
Table 1 gives details of the parameters of (1) for the systems used in Figs. 2, 3 and 4. Half the system size, L, is 15 a.u., and so a to f represent number drawn from a uniform random distribution from \(-\)0.5 to 0.5, where \({\Lambda } = 0.1\) has been incorporated into the random number distribution for this table.
The single-electron random-potential systems span a range of characteristics which, in the presence of the applied field, range from ballistic motion (including reflections from the system edge) within a broad well, to field-induced tunnelling through a barrier.
Rights and permissions
About this article
Cite this article
Skelt, A.H., Godby, R.W. & D’Amico, I. Metrics for Two Electron Random Potential Systems. Braz J Phys 48, 467–471 (2018). https://doi.org/10.1007/s13538-018-0589-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13538-018-0589-1