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Allowed permutation symmetry in atomic and molecular wavefunctions: simple examples

Francisco Marcelo Fernández

Published Jan 1, 2020• 0 citations
OpenAlex

Abstract

It is well known that the allowed wavefunctions for an N-electron system should be antisymmetric with respect to the permutation of any pair of electron labels. On the other hand, the Hamiltonian for such system is invariant under any permutation of electron labels and, consequently, its eigenfunctions are basis for the irreducible representations of the symmetric group SN.Here, we investigate which symmetry species of the SN group are compatible with the antisymmetry principle. We illustrate the conclusions by means of simple N-particle one-dimensional models with harmonic interactions.

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Allowed permutation symmetry in atomic and molecular wavefunctions: simple examples

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It is well known that the allowed wavefunctions for an N-electron system should be antisymmetric with respect to the permutation of any pair of electron labels. On the other hand, the Hamiltonian for such system is invariant under any permutation of electron labels and, consequently, its eigenfunctions are basis for the irreducible representations of the symmetric group SN.Here, we investigate which symmetry species of the SN group are compatible with the antisymmetry principle. We illustrate the conclusions by means of simple N-particle one-dimensional models with harmonic interactions.
Abstract It is well known that the allowed wavefunctions for an N -electron system should be antisymmetric with respect to the permutation of any pair of electron labels. On the other hand, the Hamiltonian for such a system is invariant under any permutation of electron labels and, consequently, its eigenfunctions are the basis for the irreducible representations of the symmetric group S N . Here, we investigate which symmetry species of the S N group are compatible with the antisymmetry principle. We illustrate the conclusions by means of simple N -particle one-dimensional models with harmonic interactions.
In this paper we analyze a recently proposed approach for the construction of antisymmetric functions for atomic and molecular systems. It is based on the assumption that the main problems with Hartree-Fock wavefunctions stem from their lack of proper permutation symmetry. This alternative building approach is based on products of a space times a spin function with opposite permutation symmetry. The main argument for devising such factors is that the eigenfunctions of the non-relativistic Hamiltonian are either symmetric or antisymmetric with respect to the transposition of the variables of a pair of electrons. However, since the eigenfunctions of the non-relativistic Hamiltonian are basis for the irreducible representations of the symmetric group they are not necessarily symmetric or antisymmetric, except in the trivial case of two electrons. We carry out a simple and straightforward general analysis of the symmetry of the eigenfunctions of the non-relativistic Hamiltonian and illustrate our conclusions by means of two exactly-solvable models of N=2N=2N=2 and N=3N=3N=3 identical interacting particles.
The symmetry properties of the N-particle multilevel system are studied. For indistinguishable particles, the total wave function must be either symmetric or antisymmetric in the exchange of any two particles. The internal states of the multilevel system are combined with suitable spatial wave functions to produce a totally symmetric or antisymmetric wave function. The problem of degeneracy and the method of combining a given permutational symmetry of the internal wave function with the same or the conjugate symmetry for the spatial part are described. It is found that Young’s diagram corresponding to the given irreducible representation of the permutation group plays an important role.
A group-theoretical description of N identical harmonic oscillators on a line is presented. It provides a scheme for labeling the energy eigenstates that are invariant under the permutation group SN and for obtaining the symmetric operators that transform these degenerate eigenfunctions among themselves. The symmetry algebra that these generators form is in general polynomial. The 2- and 3-particle cases are considered in detail. For the simple 2-body problem the invariance algebra is found to be the cubic SU(2) algebra: [J0,J±]=±J±, [J+,J−]=2J0−αJ03. In the 3-body case, the permutational invariant states are characterized with the help of the subgroup chain U(3)⊃U(2)⊃O(2). The labeling and step operators are obtained from determining an integrity basis for the S3 scalar in U(U(3)). Generating functions techniques are used to that end; an eight-dimensional basis is found whose elements span the symmetry algebra of the three identical oscillator problem. These constants of motion are seen to generate a nonlinear algebra whose representation on the symmetric states is provided.
If a system in atomic physics contains a number of particles of the same kind, e.g. a number of electrons, the particles are absolutely indistinguishable one from another. No observable change is made when two of them are interchanged … A satisfactory theory ought, of course, to count any two observationally indistinguishable states as the same state and to deny that any transition does occur when two similar particles exchange places. ( Dirac, 1958, p. 207 ) Introduction In our contribution to this volume we deal with discrete symmetries: these are symmetries based upon groups with a discrete set of elements (generally a set of elements that can be enumerated by the positive integers). In physics we find that discrete symmetries frequently arise as ‘internal’, non-spacetime symmetries. Permutation symmetry is such a discrete symmetry, arising as the mathematical basis underlying the statistical behaviour of ensembles of certain types of indistinguishable quantum particle (e.g. fermions and bosons). Roughly speaking, if such an ensemble is invariant under a permutation of its constituent particles (i.e. permutation symmetric) then one doesn't ‘count’ those permutations which merely ‘exchange’ indistinguishable particles; rather, the exchanged state is identified with the original state. This principle of invariance is generally called the ‘indistinguishability postulate’ (IP), but we prefer to use the term ‘permutation invariance’ (PI).
The symmetrization postulate (SP) states that wave functions are either completely symmetric or completely antisymmetric under permutations of identical particles. It is shown by one-dimensional counter-examples that SP is not demanded by the usual physical interpretation of the mathematical formalism of wave mechanics unless one makes use of further physical properties of real systems; the error in a standard proof of SP which ignores these properties is pointed out. It is then proved that SP is true for those systems of spinless particles which have the following properties: (a) probability densities are permutation-invariant, (b) allowable wave functions are continuous with continuous gradient, (c) the 3n3n3n-dimensional configuration space is connected, (d) the Hamiltonian is symmetric, and (e) the nodes of allowed wave functions have certain properties. The counterexamples show that SP is not a necessary property of those systems which do not have property (c). The proof is extended to particles with internal degrees of freedom (including spin) by noting that any observable commutes with every permutation and making use of a particular observable acting only on internal variables. Extraneous mathematical assumptions, such as that of the existence of a "complete" set of commuting observables, criticized by Messiah and Greenberg, are not employed. Some comments are made on the conventional nature of the concept of identity for similar particles; the equivalence between the usual formulation in which different species of similar particles are treated as distinct, and that in which they are regarded as identical particles in different internal states, is emphasized.