Energy eigenfunctions antisymmetric

When the solution is carried out, the factors i/f i (1), V 2(2), V 3 (3), and V 4(4) are harmonic oscillator energy eigenfunctions with quantum numbers vi, V2, vs, and V4. Since the oscillators are distinguishable from each other, the system wave function does not have to be symmetrized or antisymmetrized, and there are no restrictions on the values of the quantum numbers. The system energy eigenvalues corresponding to Eq. (25.1-2) are... 

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... 

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. 

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 ... 

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 . 

Recall that the minimum requirement for a many-electron wave function is that it be written as a suitably antisymmetrized sum of products of one-electron wave functions, that is, as a Slater determinant of MOs [see equation (A.68)] In Chapter 2 and Appendix A, we find that the condition that this be the best possible wave function of this form is that the MOs be eigenfunctions of a one-electron operator, the Fock operator [recall equation (A.42)], from which one can choose the appropriate number of the lowest energy. The Fock operator in restricted form, F( 1) [RHF, the UHF form was given in equation (A.41)], is given by... 

P correspond to the same energy value E. We conclude that in order to incorporate into quantum theory the indistinguishability of the two identical particles, we must restrict the allowable wave functions to those that are symmetric and antisymmetric, i.e., to those that are simultaneous eigenfunctions of H(, 2) and P. 

Since the reference system s consists of noninteracting particles, the results of Section 6.2 and the Pauli principle show that the ground-state wave function of the reference system is the antisymmetrized product (Slater determinant) of the lowest-energy Kohn-Sham spin-orbitals up of the reference system, where the spatial part 0P(r,) of each spin-orbital is an eigenfunction of the one-electron operator AP that is. 

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