Antisymmetric/antisymmetries wavefunction

The Pauli antisymmetry principle is a requirement a many-electron wavefunction must obey. A many-electron wavefunction must be antisymmetric (i.e. changes sign) to the interchange of the spatial and spin coordinates of any pair of electrons i and/, that is ... 

The requirement of overall exchange antisymmetry of the /V-clcct.ron wavefunction [Pg.36]

As was mentioned previously, simple orbital products (electron configurations) must be converted into antisymmetrized orbital products (Slater determinants) in order to satisfy the Pauli principle. Thus, proper many-electron wavefunctions satisfy constraints of exchange antisymmetry that have no counterpart in pre-quantum theories. 

Under the Hartree-Fock (i.e., HF) approximation, the function of in variables for the solutions of the electronic Hamiltonian is reduced to n functions, which are referenced as molecular orbitals (MOs), each dependent on only three variables. Each MO describes the probability distribution of a single electron moving in the average field of all other electrons. Because of the requirements of the Pauli principle or antisymmetry with respect to the interchange of any two electrons, and indistinguishability of electrons, the HF theory is to approximate the many-electron wavefunction by an antisymmetrized product of one-electron wavefunctions and to determine these wavefunctions by a variational condition applied to the expected value of the Hamiltonian in the resulting one-electron equations,... 

These defects of the Hartree SCF method were corrected by Fock (Section 4.3.4) and by Slater2 in 1930 [8], and Slater devised a simple way to construct a total wavefunction from one-electron functions (i.e. orbitals) such that will be antisymmetric to electron switching. Hartree s iterative, average-field approach supplemented with electron spin and antisymmetry leads to the Hartree-Fock equations. 

The symmetric and antisymmetric squares have special prominence in molecular spectroscopy as they give information about some of the simplest open-shell electronic states. A closed-shell configuration has a totally symmetric space function, arising from multiplication of all occupied orbital symmetries, one per electron. The required antisymmetry of the space/spin wavefunction as a whole is satisfied by the exchange-antisymmetric spin function, which returns Fq as the term symbol. In open-shell molecules belonging to a group without... 

The antisymmetry feature of the trial wavefunction is required because the Schrodinger equation does not exclude those trial functions that do not fulfill the antisymmetry requirement. Therefore, not all solutions to eq. (1) are acceptable the search for a solution must be restricted to those trial wavefunctions that are antisymmetric with respect to the exchange of any two electrons. This implies that the basis set should have a form that allows the wavefunction to be antisymmetric to the interchange of any two electrons. As a result, the Vee energy has two parts a direct term,... 

MOs first appear in the framework of the Hartree-Fock (HF) method, which is a mean-field treatment [17,22]. The basic idea is to start from an A-particle wave-function that is appropriate for a system of non-interacting electrons. Having fixed the Ansatz for the A-particle wavefunetion in this way, the variational principle is used in order to obtain the best possible approximation for the fully interacting system. Such independent particle wavefunctions are Slater-determinants, which consist of antisymmetrized products of single-particle wavefunctions (x)J (the antisymmetry brought about by the determinantal form is essential in order to satisfy die Pauh principle). Thus, the Slater-determinant is written as... 

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 procedures described above for DQMC and GFQMC lead to the lowest energy solutions for boson systems that are nodeless ground state wave-functions. They also lead to the ground state in the case of two electrons (fermions) of opposite spin for which the wavefunction is symmetric to the exchange of the two electrons. For a system of two or more electrons of the same spin, the wavefunction must be antisymmetric to the exchange of electrons of the same spin and must contain one or nodal hypersurfaces. The treatment of systems with nodes requires that the solutions be constrained to the appropriate antisymmetry. 

In general, the spin of each electron is specified in advance of a calculation. For instance, with H2O, electrons 1-5 are spin up and electrons 6-10 are spin down. This places the system in one region of spin-coordinate space. Other regions of 5-up/5-down would give the same result. For electrons of the same spin, antisymmetry in the space coordinates is imposed and produces the nodal structure in the space of those electrons. One can devise an overall space-spin wavefunction that is antisymmetric by combining with spin functions. 

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... 

In the development of the Slater method (Section 3.1) it was noted that the Pauli principle in the form (1.2.27) could always be satisfied by constructing the electronic wavefunction from determinants (i.e. antisymmetrized products) of spin-orbitals. In an earlier section, however, it was shown that for a two-electron system the antisymmetry principle could also be satisfied by writing the wavefunction as a product of individually symmetric or antisymmetric factors—one for spatial variables and the other for spin variables. Since, in the usual first approximation the Hamiltonian does not contain spin variables, it is natural to enquire whether a corresponding exact N-electron wavefunction might be written as a space-spin product in which the spatial factor is an exact eigenfunction of the spinless Hamiltonian (1.2.1). To investigate this possibility, we need a few basic ideas from group theory (Appendix 3). 

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