Variational and nonvariational principles in quantum transport calculations

Abstract

A variational principle is not generally satisfied in steady-state quantum transport as opposed to the case of ground-state problems. We show that for a short-range potential, a functional for the scattering amplitude can be introduced that is stationary for arbitrary variations about the exact scattering wave function. However, except for the special case of spherically symmetric potentials, the functional does not satisfy any minimum principle even in linear response and for single-channel scattering. The absence of a minimum principle puts severe limitations on the choice of trial wave functions in transport calculations. Examples of electronic transport in selected quantum wires will be presented to illustrate the problem. DOI: 10.1103/PhysRevB.66.041405 PACS number͑s͒: 73.40.Jn, 73.40.Cg, 73.40.Gk, 85.65.ϩh The variational method is a powerful tool to estimate the ground-state energy of a particle with Hamiltonian H. 1 The method relies on the use of a trial wave function ͑or a family of trial wave functions͒ which reproduces as closely as possible the actual ground-state wave function GS . The functionalhas then the property that ͑2͒If the trial wave function depends on certain parameters, then due to Eq. ͑2͒, the best estimate for the ground-state wave function and consequently the ground-state energy can be obtained by minimizing the functional ͓Eq. ͑1͔͒ with respect to these parameters. Total-energy calculations in condensed-matter physics, for instance, are based on such a principle. 2 The families of trial wave functions used in actual calculations vary from linear combinations of atomic orbitals to plane waves, and the choice of one family with respect to another depends mainly on numerical convenience. 2 In all choices, however, confidence in the convergence of the results with respect to the best estimate of the ground-state wave function is guaranteed by the existence of the variational principle ͓Eq. ͑2͔͒. Electronic transport in a given sample connected to external reservoirs is neither an equilibrium nor a ground-state problem. The physical observable that one needs to calculate in this case is the electric current across the sample. In general, therefore, one should not expect to have a minimum principle from which the best estimate of the scattering amplitudes can be obtained. Note that the absence of such a principle puts severe limitations on the choice of trial wave functions in transport calculations since a minimum principle is necessary to uniquely determine the best estimate from a family of trial wave functions. 3 This problem was recognized a long time ago. 4 It was demonstrated 4 that a minimum principle still applies to the scattering amplitudes for a spherically symmetric potential at zero energy for the very special case when ͑i͒ a single s-wave channel contributes to scattering at that energy and (ii) the potential does not support bound states or, if it does, the radial part of the Hamiltonian has a negative expectation value on the bound states-trial wave functions. 4 In r...

Effect of the contact distance on transport properties of an organic molecular device

“…nally, what we would like to make is that in this present work we adopt a simplified model for the two metallic electrodes, namely the differences of the various atomic configurations of metallic surface and binding sites of a molecule to the metallic surface are not taken into account [10][11][12][13][14][15][16][17][18][19][20] . As well as from the above calculations one can see that the asymmetry of I(G)-V characteristics increases with the increase of molecule-metal distance on one of the two contacts, one hence can expect that the asymmetry of I(G)-V curves will become more significant if the asymmetry of a device further increases.…”

“…Our calculations use the complete self-consistently theoretical method without any empirical adjustment. The detail of the theoretical method used in this paper has been described in previous literatures [10][11][12][13][14][15][16][17][18][19][20] . Here, we only give a brief description for this calculating approach.…”

Density-functional theory of nonequilibrium tunneling

“…The LSC DFT is based on formal collision theory 41 for the interacting many-body problem 42,43 and allows an exact discussion resting on the LS variational principle. [44][45][46] The LSC DFT is expressed through universal density functionals that characterize the variational form of the noninteracting and interacting many-body T matrices.…”

Comment on “Molecular Transport Junctions: Clearing Mists”

“…However, the difference may have little to do with the jellium model, and more with the different basis set used in the two types of calculations. [4,5] To the short list in Table 2 of ref. [1], one can add several other results, obtained using static DFT with various computational details for the same junction geometry.…”