Abstract
An efficient utilization of the numerical resources available is crucial for successful nonequilibrium Green’s function (NEGF) calculations for spatially inhomogeneous systems. Typically, the limiting factor is the computer memory. This fact has its origin in the non-Markovian structure of the Kadanoff-Baym equations which requires to remember (and hence to store) the dynamics history.
After a brief review of NEGF applications for homogeneous systems, the third chapter discusses the pros and cons of grid and basis representations of the nonequilibrium Green’s function in respect of reducing the computational effort and, in turn, of enabling larger propagation times and/or system sizes. A combination of both (grid and basis) strategies emerges as being particularly promising and advantageous. This is exemplified by using the finite element-discrete variable representation which remarkably simplifies the computation of the self-energies.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
This means, that the Green’s function (or self-energy) does not decay sufficiently fast when “looking” back into the past from the current time.
- 2.
Aside from the generic two-time structure.
- 3.
Note, that the correlation functions are in general complex. Assuming double precision, we need 16 bytes for a single complex number.
- 4.
Potentially, including an adaptive time step size.
- 5.
- 6.
- 7.
In the presence of correlations, that is beyond the HF level.
- 8.
Note that the runtime itself can be a bottleneck at large memory consumption as we have to account for matrix instead of scalar multiplications.
- 9.
Quadratic integrability is required to well-define operator matrix elements.
- 10.
Similar to methods using B-splines.
- 11.
As the roots of Legendre polynomials are symmetric about x=0, the same symmetry applies to the Gauss-Lobatto points (weights) x m (w m ).
- 12.
With the short-hand notation \(\delta _{mm'}^{ii'}=\delta_{ii'}\delta_{mm'}\).
- 13.
The same result holds for any other operator that is local in space.
- 14.
This becomes clear after a thorough analysis of Eg. (3.21).
- 15.
For a spin-polarized system, we have ξ=1.
- 16.
Restore all orbital indices and insert the full matrix for w (2).
References
M. Bonitz, Quantum Kinetic Theory (Teubner, Stuttgart, 1998)
D. Hochstuhl, M. Bonitz, Two-photon ionization of helium studied with the multiconfigurational time-dependent Hartree-Fock method. J. Chem. Phys. 134, 084106 (2011)
F. Jensen, Introduction to Computational Chemistry (Wiley, Chichester, 1999)
A. Szabo, N.S. Ostlund, Modern Quantum Chemistry (Dover Publications, New York, 1996)
H. Haug, A.-P. Jauho, Quantum Kinetics in Transport and Optics of Semiconductors (Springer, Heidelberg, 1996)
P. Danielewicz, Quantum theory of nonequilibrium processes I. Ann. Phys. 152, 239 (1984)
N.E. Dahlen, R. van Leeuwen, Self-consistent solution of the Dyson equation for atoms and molecules within a conserving approximation. J. Chem. Phys. 122, 164102 (2005)
D. Semkat, M. Bonitz, D. Kremp, Relaxation of a quantum many-body system from a correlated initial state. A general and consistent approach. Contrib. Plasma Phys. 43, 321 (2003)
D. Kremp, D. Semkat, M. Bonitz, Short-time kinetics and initial correlations in quantum kinetic theory. J. Phys. Conf. Ser. 11, 1 (2005)
D. Kremp, M. Schlanges, W.D. Kraeft, Quantum Statistics of Nonideal Plasmas (Springer, Berlin, 2005)
N.E. Dahlen, A. Stan, R. van Leeuwen, Nonequilibrium Green function theory for excitation and transport in atoms and molecules. J. Phys. Conf. Ser. 35, 324 (2006)
M. Bonitz, D. Kremp, Kinetic energy relaxation and correlation time of nonequilibrium many-particle systems. Phys. Lett. A 212, 83 (1996)
N.H. Kwong, M. Bonitz, R. Binder, H.S. Köhler, Semiconductor Kadanoff-Baym equation results for optically excited electron-hole plasmas in quantum wells. Phys. Status Solidi (b) 206, 197 (1998)
H.S. Köhler, N.H. Kwong, H.A. Yousif, A Fortran code for solving the Kadanoff-Baym equations for a homogeneous fermion system. Comput. Phys. Commun. 123, 123 (1999)
M. Hartmann, W. Schäfer, Real time approach to relaxation and dephasing processes in semiconductors. Phys. Status Solidi (c) 173, 165 (1992)
W. Schäfer, Influence of electron-electron scattering on femtosecond four-wave mixing in semiconductors. J. Opt. Soc. Am. B 13, 1291 (1996)
H.S. Köhler, Memory and correlation effects in nuclear collisions. Phys. Rev. C 51, 3232 (1995)
K. Balzer, S. Hermanns, M. Bonitz, The generalized Kadanoff-Baym ansatz. Computing nonlinear response properties of finite systems, arXiv:1211.3036 (2012)
R. Binder, H.S. Köhler, M. Bonitz, N. Kwong, Green’s function description of momentum-orientation relaxation of photoexcited electron plasmas in semiconductors. Phys. Rev. B 55, 5110 (1997)
D. Semkat, Kurzzeitkinetik und Anfangskorrelationen in nichtidealen Vielteilchensystemen. Dissertation, Universität Rostock, Germany, 2001
D. Semkat, D. Kremp, M. Bonitz, Kadanoff-Baym equations with initial correlations. Phys. Rev. E 59, 1557 (1999)
N.-H. Kwong, M. Bonitz, Real-time Kadanoff-Baym approach to plasma oscillations in a correlated electron gas. Phys. Rev. Lett. 84, 1768 (2000)
M. Bonitz, Correlation time approximation in kinetic theory. Phys. Lett. A 221, 85 (1996)
M. Bonitz, N.H. Kwong, D. Semkat, D. Kremp, Generalized Kadanoff-Baym theory for non-equilibrium many-body systems in external fields. An effective multi-band approach. Contrib. Plasma Phys. 39, 37 (1999)
D.O. Gericke, S. Kosse, M. Schlanges, M. Bonitz, T-matrix approach to equilibium and nonequilibrium carrier-carrier scattering in semiconductors. Phys. Rev. B 59, 10639 (1999)
E.K.U. Gross, J.F. Dobson, M. Petersilka, Density functional theory of time-dependent phenomena. Top. Curr. Chem. 181, 81 (1996)
G. Onida, L. Reining, A. Rubio, Electronic excitations: density-functional versus many-body Green’s-function approaches. Rev. Mod. Phys. 74, 601 (2002)
C.A. Ullrich, High Performance Computing in Science and Engineering, Garching/Munich 2009 (Oxford University Press, New York, 2012)
K. Balzer, M. Bonitz, R. van Leeuwen, A. Stan, N.E. Dahlen, Nonequilibrium Green’s function approach to strongly correlated few-electron quantum dots. Phys. Rev. B 79, 245306 (2009)
N.E. Dahlen, R. van Leeuwen, A. Stan, Propagating the Kadanoff-Baym equations for atoms and molecules. J. Phys. Conf. Ser. 35, 340 (2006)
A. Stan, N.E. Dahlen, R. van Leeuwen, Fully self-consistent GW calculations for atoms and molecules. Europhys. Lett. 76, 298 (2006)
N.E. Dahlen, R. van Leeuwen, Solving the Kadanoff-Baym equations for inhomogeneous systems: application to atoms and molecules. Phys. Rev. Lett. 98, 153004 (2007)
M.S. Lee, M. Head-Gordon, Polarized atomic orbitals for self-consistent field electronic structure calculations. J. Chem. Phys. 107, 9085 (1997)
N. Marzari, D. Vanderbilt, Maximally localized generalized Wannier functions for composite energy bands. Phys. Rev. B 56, 12847 (1997)
P.L. Silvestrelli, N. Marzari, D. Vanderbilt, M. Parrinello, Maximally-localized Wanner functions for disordered systems: application to amorphous silicon. Solid State Commun. 107, 7 (1998)
I. Schnell, G. Czycholl, R.C. Albers, Hubbard-U calculations for Cu from first-principle Wannier functions. Phys. Rev. B 65, 075103 (2002)
F. Aryasetiawan, K. Karlsson, O. Jepsen, U. Schönberger, Calculations of Hubbard U from first-principles. Phys. Rev. B 74, 125106 (2006)
A. Georges, G. Kotliar, W. Krauth, M.J. Rozenberg, Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions. Rev. Mod. Phys. 68, 13 (1996)
T.N. Rescigno, C.W. McCurdy, Numerical grid methods for quantum-mechanical scattering problems. Phys. Rev. A 62, 032706 (2000)
L.A. Collins, S. Mazevet, J.D. Kress, B.I. Schneider, D.L. Feder, Time-dependent simulations of large-scale quantum dynamics. Phys. Scr. T110, 408 (2004)
J.C. Light, I.P. Hamilton, J.V. Lill, Generalized discrete variable approximation in quantum mechanics. J. Chem. Phys. 82, 1400 (1985)
J.C. Light, T. Carrington, Discrete-variable representations and their utilization. Adv. Chem. Phys. 114, 263 (2007)
B.I. Schneider, L.A. Collins, S.X. Hu, Parallel solver for the time-dependent linear and nonlinear Schrödinger equation. Phys. Rev. E 73, 036708 (2006)
S.X. Hu, Quantum study of slow electron collisions with Rydberg atoms. Phys. Rev. A 74, 062716 (2006)
L. Tao, C.W. McCurdy, T.N. Rescigno, Grid-based methods for diatomic quantum scattering problems: a finite-element discrete-variable representation in prolate spheroidal coordinates. Phys. Rev. A 79, 012719 (2009)
L. Tao, C.W. McCurdy, T.N. Rescigno, Grid-based methods for diatomic quantum scattering problems. II. Time-dependent treatment of single- and two-photon ionization of \(\mathrm{H}_{2}^{+}\). Phys. Rev. A 80, 013402 (2009)
L. Tao, C.W. McCurdy, T.N. Rescigno, Grid-based methods for diatomic quantum scattering problems. III. Double photoionization of molecular hydrogen in prolate spheroidal coordinates. Phys. Rev. A 82, 023423 (2010)
S.X. Hu, L.A. Collins, Strong-field ionization of molecules in circularly polarized few-cycle pulses. Phys. Rev. A 73, 023405 (2006)
D.J. Haxton, K.V. Lawler, C.W. McCurdy, Multiconfiguration time-dependent Hartree-Fock treatment of electronic and nuclear dynamics in diatomic molecules. Phys. Rev. A 83, 063416 (2011)
B.I. Schneider, L.A. Collins, The discrete variable method for the solution of the time-dependent Schrödinger equation. J. Non-Cryst. Solids 351, 1551 (2005)
K. Balzer, Solving the two-time Kadanoff-Baym equations. Application to model atoms and molecules. Dissertation, Universität Kiel, Germany, 2012
D.E. Manolopoulos, R.E. Wyatt, Quantum scattering via the log derivative version of the Kohn variational principle. Chem. Phys. Lett. 152, 23 (1988)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Balzer, K., Bonitz, M. (2013). Representations of the Nonequilibrium Green’s Function. In: Nonequilibrium Green's Functions Approach to Inhomogeneous Systems. Lecture Notes in Physics, vol 867. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35082-5_3
Download citation
DOI: https://doi.org/10.1007/978-3-642-35082-5_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-35081-8
Online ISBN: 978-3-642-35082-5
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)