Electron permutation symmetry

Just as permutational symmetry was not considered in the work of Bom and his collaborators, neither is it considered in the later work. With the choice of transla-tionally invariant coordinates made above, it is a simple matter to incorporate electronic permutational symmetry and, without any diminution of mathematical generality, to require that the electronic part of the wave function includes spin and be properly antisymmetric. On this understanding, it is perfectly reasonable to assume that the potential at the minimum should not be degenerate. If it seems sufficient to treat the nuclei as distinguishable particles, then it can confidently be asserted that the Bom-Oppenheimer approach offers a perfectly satisfactory account of molecular wave functions whose energy is close to a minimum in the potential. 

The wave function [j used in the RS formulation does not fully reflect the electron permutational symmetry of the dimer the permutations among electrons of A and B are neglected. This leads to severe inconsistencies and large errors when RS PT is applied over the whole range of distances. One has to rework the perturbation theory in the search of other approaches. The simplest way would simply replace Oq where... 

Stabilizing resonances also occur in other systems. Some well-known ones are the allyl radical and square cyclobutadiene. It has been shown that in these cases, the ground-state wave function is constructed from the out-of-phase combination of the two components [24,30]. In Section HI, it is shown that this is also a necessary result of Pauli s principle and the permutational symmetry of the polyelectronic wave function When the number of electron pairs exchanged in a two-state system is even, the ground state is the out-of-phase combination [28]. Three electrons may be considered as two electron pairs, one of which is half-populated. When both electron pahs are fully populated, an antiaromatic system arises ("Section HI). 

A symmetry that holds for any system is the permutational symmetry of the polyelectronic wave function. Electrons are fermions and indistinguishable, and therefore the exchange of any two pairs must invert the phase of the wave function. This symmetry holds, of course, not only to pericyclic reactions. 

PERMUTATIONAL SYMMETRY AND THE ROLE OF NUCLEAR SPIN IN THE VIBRATIONAL SPECTRA OF MOLECULES IN DOUBLY DEGENERATE ELECTRONIC STATES ... 

IT. Total Molecular Wave Functdon TIT. Group Theoretical Considerations TV. Permutational Symmetry of Total Wave Function V. Permutational Symmetry of Nuclear Spin Function VT. Permutational Symmetry of Electronic Wave Function VIT. Permutational Symmetry of Rovibronic and Vibronic Wave Functions VIIT. Permutational Symmetry of Rotational Wave Function IX. Permutational Symmetry of Vibrational Wave Function X. Case Studies Lis and Other Systems... 

As pointed out in the previous paragraph, the total wave function of a molecule consists of an electronic and a nuclear parts. The electrons have a different intrinsic nature from nuclei, and hence can be treated separately when one considers the issue of permutational symmetry. First, let us consider the case of electrons. These are fermions with spin and hence the subsystem of electrons obeys the Fermi-Dirac statistics the total electronic wave function... 

The permutational symmetry of the rotational wave function is determined by the rotational angular momentum J, which is the resultant of the electronic spin S, elecbonic orbital L, and nuclear orbital N angular momenta. We will now examine the permutational symmetry of the rotational wave functions. Two important remarks should first be made. The first refers to the 7 = 0 rotational... 

In this chapter, we discussed the permutational symmetry properties of the total molecular wave function and its various components under the exchange of identical particles. We started by noting that most nuclear dynamics treatments carried out so far neglect the interactions between the nuclear spin and the other nuclear and electronic degrees of freedom in the system Hamiltonian. Due to... 

In Chapter VIII, Haas and Zilberg propose to follow the phase of the total electronic wave function as a function of the nuclear coordinates with the aim of locating conical intersections. For this purpose, they present the theoretical basis for this approach and apply it for conical intersections connecting the two lowest singlet states (Si and So). The analysis starts with the Pauli principle and is assisted by the permutational symmetry of the electronic wave function. In particular, this approach allows the selection of two coordinates along which the conical intersections are to be found. 

The notation < i j k 1> introduced above gives the two-electron integrals for the g(r,r ) operator in the so-called Dirac notation, in which the i and k indices label the spin-orbitals that refer to the coordinates r and the j and 1 indices label the spin-orbitals referring to coordinates r. The r and r denote r,0,( ),a and r, 0, ( ), a (with a and a being the a or P spin functions). The fact that r and r are integrated and hence represent dummy variables introduces index permutational symmetry into this list of integrals. For example,... 

Most particles of interest to physicists and chemists are found to be antisymmetric under permutation. They include electrons, protons and neutrons, as well as positrons and other antiparticles These particles, which are known as Fermions, all have spins of one-half. The relation between the permutation symmetry and the value of the spin has been established by experiment and, in the case of the electron, by application of relativistic quantum theory. 

Wigner rotation/adiabatic-to-diabatic transformation matrices, 92 Electronic structure theory, electron nuclear dynamics (END) structure and properties, 326-327 theoretical background, 324-325 time-dependent variational principle (TDVP), general nuclear dynamics, 334-337 Electronic wave function, permutational symmetry, 680-682 Electron nuclear dynamics (END) degenerate states chemistry, xii-xiii direct molecular dynamics, structure and properties, 327 molecular systems, 337-351 final-state analysis, 342-349 intramolecular electron transfer,... 

Dirac theory, 266—268 nonrelativistic states, 263- 265 Electron spin, permutational symmetry, 711-712 Electron transfer ... 

HCCS radical, Renner-Teller effect, tetraatomic molecules, II electronic states, 633-640 H2D molecule, non-adiabatic coupling, two-state molecular system, 107-109 HD2 molecule, permutational symmetry isotopomers, 713-717 potential energy surfaces, 692-694 Heaviside function ... 

Planar molecules, permutational symmetry electronic wave function, 681-682 rotational wave function, 685-687 vibrational wave function, 687-692... 

---