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International Journal of Quantum Chemistry

Phase associated with the single‐particle density of many‐electron systems

N. Sukumar, Bimalendu Deb

Published Oct 1, 1991• 6 citations
Publisher

Abstract

Abstract In excited states of atoms and molecules, as well as in time‐dependent situations, the one‐electron density no longer suffices to completely characterize the electronic state; in addition, one now requires information about the electronic phase or the current density. We show that, for a stationary electronic state, the continuity equation of quantum fluid dynamics represents a differential equation for the electronic phase, which must be solved subject to certain periodicity conditions. These periodicity conditions arise from the nodal topology of the wave function and give rise to quantized vortices of current. The consequences of writing an electronic “wave function” for a many‐electron system directly in terms of the single‐particle density and phase have been investigated. We have shown that such a procedure leads to the appearance of an “internal magnetic vector potential.” We also establish the connection between the electronic phase and the geometrical (“Berry”) phase accompanying the adiabatic transport of a quantal system around a closed loop in parameter space. This leads to a generalization of the current density concept and allows us to discuss the geometrical phase in terms of the circulation of this current in parameter space.

DOI
10.1002/qua.560400406

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Abstract In excited states of atoms and molecules, as well as in time‐dependent situations, the one‐electron density no longer suffices to completely characterize the electronic state; in addition, one now requires information about the electronic phase or the current density. We show that, for a stationary electronic state, the continuity equation of quantum fluid dynamics represents a differential equation for the electronic phase, which must be solved subject to certain periodicity conditions. These periodicity conditions arise from the nodal topology of the wave function and give rise to quantized vortices of current. The consequences of writing an electronic “wave function” for a many‐electron system directly in terms of the single‐particle density and phase have been investigated. We have shown that such a procedure leads to the appearance of an “internal magnetic vector potential.” We also establish the connection between the electronic phase and the geometrical (“Berry”) phase accompanying the adiabatic transport of a quantal system around a closed loop in parameter space. This leads to a generalization of the current density concept and allows us to discuss the geometrical phase in terms of the circulation of this current in parameter space.
Some consequences of the quantum fluid dynamics formulation are discussed for excited states of atoms and molecules and for time-dependent processes. It is shown that the conservation of electronic current density j(r) allows us to manufacture a gauge potential for each excited state of an atom, molecule or atom in a molecule. This potential gives rise to a tube of magnetic flux carried around by the many-electron system. In time-dependent situations, the evolution of the electronic density distribution can be followed with simple, site-dependent cellular automaton (CA) rules. The CA consists of a lattice of sites, each with a finite set of possible values, here representing finite localized elements of electronic charge and current density (since the charge density r no longer suffices to fully characterize a time-dependent system, it needs to be supplemented with information about the current density j).Our numerical results are presented elsewhere and further development is in progress.
The nodal structure of the Schr\"odinger wave function is used to describe an NNN-particle system in a stationary quantum state. In terms of nodal lines (quantized flux lines) together with their circulation numbers, two fields are defined: the microscopic, singular nodal flux field n (r) and the macroscopic, regular nodal flux density field N (r). Their definition arises in a natural way from a quantum-mechanical vector potential defined by the gradient of the multivalued phase function associated with the pattern of quantized flux lines. The flux of the fields n (r) and N (r) is quantized and shown to be proportional to the circulation of the velocity in the system. The unit circulation for a system of bosons with a particle spin s=0s=0s=0 and mass m0{m}_{0}m0​ is equal to hm0\frac{h}{{m}_{0}}m0​h​. In the case of charged particles, the field (cq)N(r)(\frac{c}{q})\mathbf{N}(\mathbf{r})(qc​)N(r) resembles Jehle's formulation of the magnetic field of a lepton as a superposition of quantized flux lines. In our case, however, the presence of quantized flux lines, forming closed loops follows from the mere existence of a current density and does not have to be assumed. By a simple argument, using permutation symmetry, it is shown that a flux quantum of the size hc2q\frac{\mathrm{hc}}{2q}2qhc​ is possible.
, which are relevant to properly describe phenomena such as molecular dissociation with density-functional theory, to the case in which the electronic wavefunction corresponds to a complex current-carrying state. In such a case, an extra term in the form of a vector potential appears which cannot be gauged away. Both scalar and vector potentials are introduced via the exact factorization formalism which allows us to express the given Schrödinger equation as two coupled equations, one for the marginal and one for the conditional amplitude. The electronic vector potential is directly related to the paramagnetic current density carried by the total wavefunction and to the diamagnetic current density in the equation for the marginal amplitude. An explicit example of this vector potential in a triplet state of two non-interacting electrons is showcased together with its associated circulation, giving rise to a non-vanishing geometric phase. Some connections with the exact factorization for the full molecular wavefunction beyond the Born-Oppenheimer approximation are also discussed.
A time-dependent density-functional formalism is developed for many-electron systems subjected to external electric and magnetic fields with arbitrary time dependence. The single-particle current density is shown to determine uniquely the time-dependent scalar and vector potentials characterizing the system, and hence also the many-particle wave function. A Levy-type universal functional is defined, and practical schemes for the calculation of electron density and the current density through hydrodynamical as well as a set of single-particle Kohn-Sham-like equations are proposed. The gauge invariance of the present self-consistent formalism is also proved.
Abstract In presence of external electric and magnetic fields, the Schrödinger equation for many‐electron systems is transformed into a continuity equation and an Euler‐type equation of motion in configuration space. Then, using the natural‐orbital Hamiltonian, as defined by Adams, the two fluid‐dynamical equations are derived in the three‐dimensional space. This generates a “classical” view of such quantum systems, corresponding to an MCSCF wave function: The many‐electron Schrödinger fluid consists of individual fluid components, each corresponding to a natural orbital and having its own charge density and current density. The local observables, viz., the net charge density and net current density, are obtained by merely summing over the natural orbitals, with the occupation numbers as weight factors; but, the net velocity field cannot be so obtained. Further, although each fluid component moves irrotationally in the absence of a magnetic field, the net velocity field is not irrotaional. The irrotational character of each velocity component is destroyed by rotation of the nuclear framework of the system while electron spin introduces an additional term, the spin magnetization moment, into each component current density. The physical significance of the fluid‐dynamical equations as well as their advantages and disadvantages are discussed.
For an N-electron system, a connection is explored between density-functional theory and quantum fluid dynamics, through a dynamical extension of the former. First, we prove the Hohenberg–Kohn theorem for a time-dependent harmonic perturbation under conditions which guarantee the existence of the corresponding steady (or quasiperiodic) states of the system. The corresponding one-particle time-dependent Schrödinger equation is then variationally derived starting from a fluid-dynamical Lagrangian density. The subsequent fluid-dynamical interpretation preserves the ’’particle’’ description of the system in the sense that the N-electron fluid has N components each of which is an independent-particle Schrödinger fluid characterized by a density function ρj and an irrotational velocity field uj, j = 1,⋅⋅⋅,N. However, the mean velocity u of the fluid is not irrotational, in general. The force densities and the stress tensor occurring in the Navier–Stokes equation are physically interpreted. The present work is another step towards the interpretation of physicochemical phenomena in three-dimensional space.