Antisymmetry, permutational

Electronic Wavefuntions Must be Constructed to Have Permutational Antisymmetry Because the N Electrons are Indistinguishable Eermions... 

In particular, within the orbital model of eleetronie strueture (whieh is developed more systematieally in Seetion 6), one ean not eonstruet trial waveflmetions whieh are simple spin-orbital produets (i.e., an orbital multiplied by an a or P spin funetion for eaeh eleetron) sueh as lsalsP2sa2sP2pia2poa. Sueh spin-orbital produet funetions must be made permutationally antisymmetrie if the N-eleetron trial funetion is to be properly antisymmetrie. This ean be aeeomplished for any sueh produet wavefunetion by applying the following antisymmetrizer operator ... 

Three distinct sets of linear mappings for the partial 3-positivity matrices in Eqs. (31)-(36) are important (i) the contraction mappings, which relate the lifted metric matrices to the 2-positive matrices in Eqs. (27)-(29) (ii) the linear interconversion mappings from rearranging creation and annihilation operators to interrelate the lifted metric matrices and (iii) antisymmetry (or symmetry) conditions, which enforce the permutation of the creation operators for fermions (or bosons). Note that the correct permutation of the annihilation operators is automatically enforced from the permutation of the creation operators in (iii) by the Hermiticity of the matrices. 

This is the first term on the right-hand side of Eq. (10), for the case p = 2.) The first term on the right-hand side in Eq. (39) is obviously connected, and we may deduce that the second term is unconnected because its trace equals N (h)/2. The third term, which constitutes a transvection [27, 62] of Ai with itself, is actually connected, but differs from the second term by a coordinate permutation. If the second term is removed from CSE(2), then the third term ought to be removed as well, for otherwise we destroy the antisymmetry of O2. This example illustrates the complexity of formulating an extensive version of CSE(2). It is not enough to eliminate unconnected terms one must ehminate their exchange counterparts as well. 

N electrons with the same values of quantum numbers n,7 (LS coupling) or tijljji (jj coupling) are called equivalent. The corresponding configurations will be denoted as nlN (a shell) or nljN (a subshell). A number of permitted states of a shell of equivalent electrons are restricted by the Pauli exclusion principle, which requires antisymmetry of the wave function with respect to permutation of the coordinates of the electrons. 

When constructing many-electron wave functions it is necessary to ensure their antisymmetry under permutation of any pair of coordinates. Having introduced the concepts of the CFP and unit tensors, Racah [22, 23] laid the foundations of the tensorial approach to the problem of constructing antisymmetric wave functions and finding matrix elements of operators corresponding to physical quantities. 

Each pair of unique, external hole or particle lines introduces a permutation function, P(pq) (as defined earlier in Eq. [154]), to ensure antisymmetry of the final expression. 

The permutation operators appear in order to maintain the antisymmetry of the algebraic expressions, as explained earlier. Note that the factors of lA appearing in the second and third terms result from both a pair of equivalent lines and a pair of equivalent vertices in each of the corresponding diagrams. 

Equation (121) was obtained from Eq. (120) by use of the antisymmetry of the test function h. Now let us consider the permutation of indices between topologically distinct regions ... 

To maintain full antisymmetry of an amplitude, the algebraic expression for a diagram should be preceded by a permutation operator permuting the open lines in all distinct ways, 2P(— )pP. 

Cases of three or more electrons were very difficult to treat by the above methods. For instance, for three-electron systems, it is required to have six terms in the expansion of each basis function in order to comply with the antisymmetry criterion, and each term must have factors containing ri2, ri3, r23, etc., if we want to accelerate the convergence. There is, indeed, a real problem with the size of each trial wave function. A symmetrical wavefunction requires that the trial basis set for helium contain two terms to guarantee the permutation of electrons. For an N-electron system, this number grows as N . For a ten-electron system like water, it would be required that each basis set member have more than 3 million terms, and this is in addition to the dependence on 3N variables of each of the terms. These conditions make the Schrodinger equation intractable for systems of even a few electrons. Just the bookkeeping of these terms is practically impossible. 

What is true is that, for a given type of spin function, the requirement of antisymmetry imposes a particular form of permutation symmetry on the spatial form of any acceptable wavefunction which is to be variationally optimised, and this form must be retained during the variation procedure. That is, for a given form of spin function the spatial part of the wavefunction must behave in a predetermined way when the spatial coordinates of electrons are permuted. 

The MO approximation is based on the assumption that an individual spin orbital (pioc or (piP (where cpi is the function of space coordinates and a (m = 1/2) or P(nis= — 1/2) are the functions of spin coordinates) corresponds to each electron. The full wave function of a many-electron system ip in the Hartree-Fock approximation is written as a Slater determinant whose form provides for the property of antisymmetry of ip, required by the Pauli principle, with respect to the pairwise permutation of any electron... 

In the present context, the term symmetry refers to the permutational symmetry of electrons, that is to the Pauh principle. This problem plays a central role in perturbation theories of intermolecular interactions since the antisymmetry of the perturbed wave functions has to be ensured. The symmetry-adapted PT is called also as exchange-PT , because the antisymmetry results exchange interactions between molecules A and B. Several formulations of the exchange-PT have been developed (Van der Avoird, 1967, Amos Musher 1967, Hirschfedler 1967, Murrel Shaw 1967, Salewicz Jeziorski 1979) which will not be discussed in detail. In the spirit of the present treatment, we shall focus on the application of second quantization to this problem. This formalism eo ipso guarantees the proper antisymmetry of any wave function expressed in terms of anticommuting fermion operators, thus the symmetry adaptation is done automatically and it does not require any further discussion. 

For a system of either bosons or fermions, the wavefunction must have the correct properties of symmetry and antisymmetry. Particles with half-integral spin, such as electrons, are fermions and require antisymmetric wavefunctions. Particles with integral spin, such as photons, are bosons and require symmetric wavefunctions. The complete space-spin wavefunction of a system of two or more electrons must be antisymmetric to the permutation of any two electrons. Except in the simplest cases, the wavefunction for a system of n fermions is positive and negative in different regions of the 3 -dimensional space of the fermions. The regions are separated by one or more (3 - 1 )-dimensional hypersurfaces that cannot be specified except by solution of the Schrodinger equation. 

The Hamiltonian (O Eq. 2.23) maintains full symmetry and is invariant under electronic permutations and under rotation-reflections of the electronic coordinates. Trial functions are usually constructed from atomic orbitals and from their spin-orbitals. Permutational antisymmetry is achieved by forming Slater determinants from the spin-orbitals. Rotational symmetry is usually realized by vector coupling of orbitals that form bases for representations of the rotation group SO(3). Spin-eigenfunctions too are achieved by vector coupling. ... 

In discussing the helium atom (Section 1.2) the antisymmetry requirement on the electronic wavefunction was easily satisfied for with only two electrons the function would be written as a product of space and spin factors, one of which had to be antisymmetric, the other symmetric, lliis is possible even for an exact eigenfunction of the Hamiltonian (1.2.1), as well as for an orbital product. The construction of an antisymmetric many-electron function is less easy. We have seen in Section 1.2 that for a general permutation (involving both space and spin variables) an antisymmetric function has the property... 

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