Antisymmetrization associativity

It is beyond the scope of these introductory notes to treat individual problems in fine detail, but it is interesting to close the discussion by considering certain, geometric phase related, symmetry effects associated with systems of identical particles. The following account summarizes results from Mead and Truhlar [10] for three such particles. We know, for example, that the fermion statistics for H atoms require that the vibrational-rotational states on the ground electronic energy surface of NH3 must be antisymmetric with respect to binary exchange... 

Infrared The absorptions of interest m the IR spectra of amines are those associated with N—H vibrations Primary alkyl and arylammes exhibit two peaks m the range 3000-3500 cm which are due to symmetric and antisymmetric N—H stretching modes... 

In order that i/ r A always antisymmetric for IT2 the antisymmetric ij/ are associated with even J states and the symmetric with odd J states, as shown in Figure 5.18. Interchange between states with symmetric and antisymmetric is forbidden so that IT2 can be regarded as consisting of two distinct forms ... 

We now take vibronic interactions into account. In this case, we must determine vibronic states rather than the electronic and vibrational ones. For example, if X3 in a degenerate E vibration is singly excited in an E electronic state, we obtain the vibronic states evA evA 2 evE, since VE eE = evA evA 2 evE . If the same vibration is doubly excited (e.g., if v 2 = 2, with the symmetric product being [vE v E = VA VE Note that the associated antisymmetric product is M ), we get the vibronic species ( Aj VE ) eE = evA evA 2 2evE. Table XIII shows the symmetries of the lowest 25 vibrational and vibronic states. In turn, the lowest 26 levels calculated for Li3... 

These two transcendental equations define a pair of closely spaced energy levels, respectively associated with symmetric and antisymmetric wave functions as defined by the arbitrary choice of D = C. 

Another well-defined configuration of the classical three body Coulomb problem with unambiguous quantum correspondence is the collinear antisymmetric stretch configuration, where the electrons are located on opposite sides of the nucleus. In contrast to the frozen planet orbit, the antisymmetric stretch is unstable in the axial direction (G.S. Ezra et.al., 1991 P. Schlagheck et.al., 2003), with the two electrons colliding with the nucleus in a perfectly alternating way (Fig. 3 (left)). Hence, already the one dimensional treatment accounts for the dominant classical decay channel of this configuration. As for the frozen planet, there are doubly excited states of helium associated to the periodic orbit of the ASC as illustrated in Fig. 3 (left). 

Figure 3. Contour piot of the eiectronic density of a (tripiet) eigenstate strongly scarred by the antisymmetric stretch orbit (left), in 2D configuration space (spanned by the electrons distances ri and r-2 from the nucleus, in the collinear configurations considered here). This eigenstate belong to the N = 9 series. The solid lines depict the associated classical periodic orbit. Autoionization rates of antisymmetric stretch singlet states (right) of the Nth autoionizing series of the helium spectrum, in ID (squares), 2D (circles), and 3D (diamonds) configuration space.

At this point, it is necessary to say a few words about the v-representability of the electron density. An electron density is said to be v-representable if it is associated with the antisymmetric wave function of the ground state, corresponding to an external potential v(r), which may or may not be a Coulomb potential. Not all densities are v-representable. Furthermore, the necessary and sufficient conditions for the v-representability of an electron density are unknown. Fortunately, since the /V-representability (see Section 4.2) of the electron density is a weaker condition than v-representability, one needs to formulate DFT only in terms of /V-representable densities without unduly worrying about v-representability. 

In view of the preceding considerations it should be emphasized that it is incorrect to talk about the self-consistent-field molecular orbitals of a molecular system in the Hartree-Fock approximation. The correct point of view is to associate the molecular orbital wavefunction of Eq. (1) with the N-dimen-sional linear Hilbert space spanned by the orbitals t/2,... uN any set of N linearly independent functions in this space can be used as molecular orbitals for forming the antisymmetrized product. 

Typical potential energies associated with such a Hamiltonian are shown in Figure 4 as a function of the parameter 0 = x/2J. The coordinate is the antisymmetric combination. The symmetric mode clearly adds a term to the total energy independent of coupling. 

For a system of N identical fermions in a state ij/ there is associated a reduced density matrix (RDM) of order p for each integer p, 1 Hermitian operator DP, which we call a reduced density operator (RDO) acting on a space of antisymmetric functions of p particles. The case p = 2 is of particular interest for chemists and physicists who seldom consider... 

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