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arXiv:1408.4346 · Physica status solidi. C, Conferences and critical reviews/Physica status solidi. C, Current topics in solid state physics

Towards the explicit computation of Bohm velocities associated to N‐electron wave‐functions with arbitrary spin‐orientations

A. Alarcón, Xavier Cartoixà, Xavier Oriols

Published Aug 16, 2010• 9 citations
arXivHTMLPDF

Abstract

Abstract The direct solution of the many‐particle Schrödinger equation is computationally inaccessible for more than very few electrons. In order to surpass this limitation, one of the authors [X. Oriols, Phys. Rev. Lett. 2007, 98 (066803)] has recently proposed a new model to study electron‐electron correlations from Bohm trajectories associated to time‐dependent wave‐packets solutions of pseudo single‐particle Schrödinger equations. In the aforementioned paper only the orbital exchange interaction is considered assuming that all electrons have the same spin orientation. Then, the many‐particle wave function is a complex Slater determinant of the single‐particle wave‐packets. In the present work the previous formalism is extended to study many‐particle wave functions where the electrons have different spin orientations. The main difficulty to treat N different electron spin orientations with time‐dependent wave‐packets is that one must study all the possible N!N! products of permutations among spin states. To overcome this computationally inaccessible problem, in this article the total wave function is treated as a separated product of two many‐particle wave functions, the first with spin up and the second with spin down. In order to numerically justify this approximation, the Bohm velocity in different antisymmetric total wave‐function scenarios is computed. The computational results confirms the accurate validity of our approximation under a large number of cases (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

DOI
10.1002/pssc.200983841

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Abstract The direct solution of the many‐particle Schrödinger equation is computationally inaccessible for more than very few electrons. In order to surpass this limitation, one of the authors [X. Oriols, Phys. Rev. Lett. 2007, 98 (066803)] has recently proposed a new model to study electron‐electron correlations from Bohm trajectories associated to time‐dependent wave‐packets solutions of pseudo single‐particle Schrödinger equations. In the aforementioned paper only the orbital exchange interaction is considered assuming that all electrons have the same spin orientation. Then, the many‐particle wave function is a complex Slater determinant of the single‐particle wave‐packets. In the present work the previous formalism is extended to study many‐particle wave functions where the electrons have different spin orientations. The main difficulty to treat N different electron spin orientations with time‐dependent wave‐packets is that one must study all the possible N!N! products of permutations among spin states. To overcome this computationally inaccessible problem, in this article the total wave function is treated as a separated product of two many‐particle wave functions, the first with spin up and the second with spin down. In order to numerically justify this approximation, the Bohm velocity in different antisymmetric total wave‐function scenarios is computed. The computational results confirms the accurate validity of our approximation under a large number of cases (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
Following Oriols (2007 Phys. Rev. Lett. 98 066803), an algorithm to deal with the exchange interaction in non-separable quantum systems is presented. The algorithm can be applied to fermions or bosons and, by construction, it exactly ensures that any observable is totally independent of the interchange of particles. It is based on the use of conditional Bohmian wave functions which are solutions of single-particle pseudo-Schrödinger equations. The exchange symmetry is directly defined by demanding symmetry properties of the quantum trajectories in the configuration space with a universal algorithm, rather than through a particular exchange-correlation functional introduced into the single-particle pseudo-Schrödinger equation. It requires the computation of N(2) conditional wave functions to deal with N identical particles. For separable Hamiltonians, the algorithm reduces to the standard Slater determinant for fermions (or permanent for bosons). A numerical test for a two-particle system, where exact solutions for non-separable Hamiltonians are computationally accessible, is presented. The numerical viability of the algorithm for quantum electron transport (in a far-from-equilibrium time-dependent open system) is demonstrated by computing the current and fluctuations in a nano-resistor, with exchange and Coulomb interactions among electrons.
Many-electron theory is treated here with much attention focused on the synthesis of the density functional and density matrix theory. This synthesis shows that the exchange–correlation potential, normally obtained as the functional derivative with respect to the electron density of an assumed approximate exchange–correlation energy, can be written exactly in terms of the “near-diagonal” behavior of the fully interacting first-order density matrix and the diagonal of the second-order density matrix, thereby bypassing the functional derivative. Also involved, since the exchange–correlation potential is an “artifact” of reducing the many-electron Schrödinger equation to one-body Slater–Kohn–Sham equations, are the one-body orbitals derived from these latter equations, not just the density thereby obtained. This situation has then motivated a discussion of what constraints are placed on the low-order density matrices of atoms and molecules, described by an assumed spin-independent Hamiltonian, by the requirement that these matrices are constructed from an N-electron wave function which is an eigenfunction of the spin operators S2 and Sz. Some briefer discussion is added on the optimally convergent determinantal expansion of such an N-electron wave function, the relation between Brueckner and Löwdin natural orbitals being commented on as a by-product. In this latter context, it is noted that the exchange-only potential of density functional theory can be represented as a sum of two line integrals, each of which is path-dependent, the sum being, of course, path-independent. But in addition to knowledge of Slater–Kohn–Sham orbitals appearing as in the Harbola–Sahni exchange-only potential (which is one of the path-dependent terms), there is a kinetic correction. This involves Löwdin natural orbitals and their occupation numbers, but only now in the weak coupling limit of the electron–electron interaction, as emphasized in the kinetic “correction” to the Harbola–Sahni formula exhibited by Levy and the writer. Some directions for further work are briefly set out. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 74: 163–176, 1999
Abstract A method for propagating multi‐dimensional wave packets in time using a product ansatz for the wave function is presented. Other than in the time‐dependent Hartree approach which is based on a variational scheme, the current method is based on the stochastic Ito calculus. The correlations between the different one‐dimensional wave functions are taken into account via stochastic processes such that on average the original Schrödinger equation is reproduced. Numerical tests of the proposed scheme have been performed in two and four dimensions. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
The de Broglie–Bohm interpretation of quantum mechanics with and without spin-dependence is used to determine electron trajectories, arrival-time distribution of electrons and the mean arrival time of spin-1/2 particles (including electrons) in the presence of a uniform field and a barrier separately. The difference for the mean arrival times, which are calculated with different guidance equations, is examined versus mass of the arriving particle and versus group velocity of the arriving electron wave packet and also versus the width of the barrier. Numerical calculations show that these differences are of the order of 10 −18 –10 −17 s. Another feature of the modified guidance law is that Bohmian trajectories cross each other, but this does not contradict the single-valuedness of the wavefunction.
A system of NNN antisymmetric particles, moving under the influence of a fixed potential and their mutual many-particle interactions, is investigated in the ordinary Hartree-Fock scheme, having the total wave function approximated by a single Slater determinant. It is shown that all the density matrices of various orders, the wave function, and the entire physical situation depends only on a fundamental invariant $\ensuremath{\rho}({x}_{1}, {x}_{2})$, which is identical with the first-order density matrix. The Hartree-Fock equations are expressed in terms of this quantity.The Hartree-Fock equations are also solved by expanding the eigenfunctions in a given complete set, and applications to the MO-LCAO theory of the electronic structure of molecules, and crystals are given. It is shown that, in this scheme, the entire physical situation depends on a charge- and bond-order matrix $R(\ensuremath{\nu}\ensuremath{\mu})$ with respect to the ordinary atomic spin-orbitals involved. The Hartree-Fock equations for this matrix are investigated.Finally, the ionized and excited states are investigated, and it is shown that the Hartree-Fock scheme has a high degree of physical visuality also in case of many-particle interactions. The excitation energy of the system is the difference (${{\ensuremath{\omega}}_{j}}^{\ensuremath{'}}\ensuremath{-}{\ensuremath{\omega}}_{i}$) between two "spin-orbital energies," being eigenvalues to the effective Hamiltonians associated with the two states under consideration.
Extending the one-particle Schrodinger equation from one to three spatial dimensions enabled us to describe electron states in the hydrogen atom and in other hydrogen-like systems. This chapter generalizes the Schrodinger equation from one to two and then to many particles bound as a single system. For all particles of half-integral spin, the total wave function, as for electrons, is always antisymmetric with respect to interchange, whereas with all particles of integral spin the total wave function is always symmetric. The chemical and spectroscopic characteristics of an element depend on the details of the quantum state of its outermost electrons. The chemical valence depends upon the extent to which the total number of electrons in an atom represents an excess or a deficit with respect to the more stable configuration of a completed shell. The Schrödinger equation for helium contains no explicit mention of the spin states of the electrons.