Eigenfunctions, antisymmetric

Curve 1 represents the total energy of the hydrogen molecule-ion as calculated by the first-order perturbation theory curve 2, the naive potential function obtained on neglecting the resonance phenomenon curve 3, the potential function for the antisymmetric eigenfunction, leading to elastic collision. 

This resonance energy leads to molecule formation only if the eigenfunction is symmetric in the two nuclei. The perturbation energy for the antisymmetric eigenfunction is... 

We shall next consider whether or not the antisymmetric eigenfunction Hl for two hydrogen atoms (Equation 29b) would lead to an excited state of the hydrogen molecule. The perturbation energy is found to be... 

The observed structure of the spectra of many-electron atoms is entirely accounted for by the following postulate Only eigenfunctions which are antisymmetric in the electrons , that is, change sign when any two electrons are interchanged, correspond to existant states of the system. This is the quantum mechanics statement (26) of the Pauli exclusion principle (43). 

It is equivalent to saying that two electrons cannot occupy the same orbit. Thus there is no antisymmetric eigenfunction composed of -p (1) a (1) and p (2) a (2), and no such state exists... 

In dealing with systems containing only two electrons we have not been troubled with the exclusion principle, but have accepted both symmetric and antisymmetric positional eigenfunctions for by multiplying by a spin eigenfunction of the proper symmetry character an antisymmetric total eigenfunction can always be obtained. In the case of two hydrogen atoms there are three... 

P (2) — p (1) a (2). The last is required to make the symmetric positional eigenfunction of Equation 29a conform to Pauli s principle, and the first three for the antisymmetric 4>H2- Since the a priori probability of each eigenfunction is the same, there... 

Strictly speaking, the complete eigenfunction for the molecule should be made antisymmetric before the charge densities at the various positions are calculated. It is easily shown, however, that this further refinement in the treatment does not alter the results obtained. 

In crystals for which n0 is large, such as iodine, the lowest symmetric and the lowest antisymmetric state have practically the same energy and properties, and each corresponds to one eigenfunction only. As a result a mixture of symmetric and antisymmetric molecules at low temperatures will behave as a perfect solid solution, each molecule having just its spin quantum weight, and the entropy of the solid will be the translational entropy plus the same entropy of mixing and spin entropy as that of the gas. This has been verified for I2 by Giauque.17 Only at extremely low temperatures will these entropy quantities be lost. 

To see why this is so, let us attempt to apply the procedure of Section II.B to a bound-state wave function. This is illustrated schematically in Fig. 19. It is clear immediately that we cannot construct an unsymmetric in the double space, because each bound-state eigenfunction must be an irreducible representation of the double-space symmetry group. Thus a bound-state function in the double space is necessarily symmetric or antisymmetric under R2k, and is thus either a Fq or a Fn function. For a Fq function, we have Fn = 0 (since and Fn cannot form a degenerate pair), which implies [from Eq. (6)] that... 

The wave funetion obtained eorresponds to the Unrestricted Hartree-Fock scheme and beeomes equivalent to the RHF ease if the orbitals (t>a and (()p are the same. In this UHF form, the UHF wave funetion obeys the Pauli prineiple but is not an eigenfunction of the total spin operator and is thus a mixture of different spin multiplicities. In the present two-eleetron case, an alternative form of the wave funetion which has the same total energy, which is a pure singlet state, but whieh is no longer antisymmetric as required by thePauli principle, is ... 

The A-particle eigenfunctions I v(l, 2,. .., A) in equation (8.47) are not properly symmetrized. For bosons, the wave function (1, 2,. .., N) must be symmetric with respect to particle interchange and for fermions it must be antisymmetric. Properly symmetrized wave functions may be readily con-... 

Equation (5.15b) is the fundamental assumption underlying London s theory, which is essential both for numerical evaluation and for physical interpretation of the perturbative expressions. Whereas short-range intramolecular interactions in (5.16a) and (5.16b) must be described with properly antisymmetric eigenfunctions satisfying... 

Mention should be made here of recent attempts by Piepho, Schatz and Krausz (46) to give a general interpretation of intervalence bandshapes in terms of a Hamiltonian equivalent to that of eq 6. They use vibronic eigenfunctions (following the method of solution of Merrifield (47)) rather than adiabatic Born-Oppenheimer (ABO) functions. Thus, the aim is to interpret an observed spectrum in terms of one vibrational coupling mode, which is antisymmetric. Their analysis of the spectrum of the Creutz-Taube ion yields a value of 0 of 1.215, i.e., a rather weakly localized ground state. Using their assumed unperturbed... 

As the exchange energy, the polarization-exchange energy (.poi-txch is also nonadditive. The standard PT cannot be applied to the calculation of the poi-exch- The reason is that the antisymmetrized functions of zeroth order (Ai/>o. ..) are not eigenfunctions of the unperturbed Hamiltonian Ho as long as the operator Ho does not commute with the antisymmetrizer operator A. Many successful approaches for the symmetry adapted perturbation theory (SAPT) have been developed for a detailed discussion see chapter 3 in book, the modern achievements in the SAPT are described in reviews . 

We have so far said little about the nature ofthe space function, S. Earlier we implied that it might be an orbital product, but this was not really necessary in our general work analyzing the effects of the antisymmetrizer and the spin eigenfunction. We shall now be specific and assume that S is a product of orbitals. There are many ways that a product of orbitals could be arranged, and, indeed, there are many of these for which the application of the would produce zero. The partition corresponding to the spin eigenfunction had at most two rows, and we have seen that the appropriate ones for the spatial functions have at most two columns. Let us illustrate these considerations with a system of five electrons in a doublet state, and assume that we have five different (linearly independent) orbitals, which we label a, b,c,d, and e. We can draw two tableaux, one with the particle labels and one with the orbital labels. 

Antisymmetric eigenfunctions of the spin one obtains for the matrix system,... 

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