Symmetric wavefunctions

Two orthogonal orbitals that interact, yAB 0, are not completely localized they correspond to Lowdin orthogonal orbitals. Two symmetrical wavefunctions mix ( lA2 + B2) and AB)) giving rise to S0 and S2 while antisymmetric wavefunction A2 -B2) remains an eigenstate ... 

In a centrosymmetric molecule the wavefunctions can be split up in a symmetric and antisymmetric set regarding the inversion operation. The symmetric wavefunctions are labeled as nAg, the antisymmetric ones as nBu. The above selection rule of no transitions within the symmetry subset applies to all directions. It is impossible to reach a higher Ag state from an Ag ground state by a simple one-photon absorption. 

The quantitative determination of the magnitude of the tensor component is based on the explicit knowledge of the spatial dependence of the electronic wavefunctions. As pointed out in Section 2 the Z , w, II) functions transform as the basis functions of the irreps of the symmetry group G and can be obtained with the modified projector technique method [23] in tight binding approximation. The so obtained symmetrized wavefunctions preserve the transformation properties as dictated by the irreps of the symmetry group G. 

Electron Exchange. The need for an anti-symmetric wavefunction for incident and bound electrons gives rise to nonlocal interactions which greatly complicate the solution of the scattering equations. These exchange interactions have the form,... 

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. 

Parlides that have antisymmetric wavefunctions are known as fermions They include protons and C nuclei, as wel as electrons. Other particles, such as photons and nuclei, have symmetric wavefunctions that do not change sign when identical particles are interchanged. These are known as bosons... 

To satisfy the Pauli principle the antisymmetric wavefunction must be combined with one of the three symmetric spin states, given by equations (7.26)-(7.28). This particular excited state can therefore exist in three different forms. These have slightly different energies because of the small magnetic interactions which occur between the spin and orbital motions of the electrons, and this causes any spectral lines involving this state to be split into three. For this reason it is known as a triplet state. The symmetric wavefunction, yr, combines with the single antisymmetric spin state, and it is said to form a singlet state. ... 

We have introduced a symmetry that was not present in the original a(l)j8(2) and jS(l)a(2) functions hence the term symmetrized wavefunctions. 

Let s pause a moment to point out two features of these symmetrized wavefunctions. First, if the electrons in a wavefunction are indistinguishable, then we can start to see how many terms must appear in the wavefunction. For example, the Is ground state of He is constructed from two distinct one-electron states Isa and lsj8. The electrons must both be in the Is, according to the electron configuration, and one must be spin up while the other is spin down, according to the Pauli exclusion principle. There are two ways to put the two electrons into those two states, and hath ways must appear in the complete wavefunction for the electrons to be... 

Write one properly symmetrized wavefunction in terms of and rg for each electron, including the spin wavefunctions a and p, for the lcr 2a- MO configuration in H2. 

Another example of a forbidden transition is that between two different s-type states of a hydrogen atom. Such states have spherically symmetric wavefunctions, but (the electric field) is antisymmetric for reflection through a plane (to within an additive constant), and so must vanish for reasons of symmetry. It is easy... 

This analysis is a little misleading. From a spherical polar viewpoint, there is very little volume of space close to the nucleus, because for all values of 6 and (f) a small value of r sweeps out a very tiny sphere. The total probability of the electron existing in such a small volume of space should be small. However, as the radius increases, the spherical volume swept out by the spherically symmetric wavefunction gets larger and larger, and one would expect an increase in probability that the electron will be located at greater distances from the nucleus. 

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 approximations used in the tight-bonding approach are only valid for nondegenerate s-electron (spherically symmetric) wavefunctions. It is introduced here only... 

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