Permutation motion

An important difference between the BO and non-BO internal Hamiltonians is that the former describes only the motion of electrons in the stationary field of nuclei positioned in fixed points in space (represented by point charges) while the latter describes the coupled motion of both nuclei and electrons. In the conventional molecular BO calculations, one typically uses atom-centered basis functions (in most calculations one-electron atomic orbitals) to expand the electronic wave function. The fermionic nature of the electrons dictates that such a function has to be antisymmetric with respect to the permutation of the labels of the electrons. In some high-precision BO calculations the wave function is expanded in terms of basis functions that explicitly depend on the interelectronic distances (so-called explicitly correlated functions). Such... 

The algebraic equations and efficient computational sequences were derived by smith and reported by us [33] for CCSD-, CCSDT-, and CCSDTQ-R12, their excited-state analogues via the equation-of-motion (EOM) formalisms (EOM-CC-R12 up to EOM-CCSDTQ-R12), and the so-called A equations for the analytical gradients and response properties, again up to A-CCSDTQ-R12. The full CCSD-, CCSDT-, and CCSDTQ-R12 methods [34,35] were implemented by smith into efficient computer codes that took advantage of spin, spatial, and index-permutation symmetries. 

There are numerous other examples of stereochemical correspondence in propeller molecules. The propellane 19> 1e has only one energetically reasonable isomerization mechanism available a twisting motion about the C3 axis which corresponds to the three-ring flip as well as to the trigonal twist rearrangement (Fig. 5). Similarly, only one isomerization mechanism (stereochemically correspondent in the permutational sense to the zero-ring flip) is energetically reasonable for tri-o-thymotide (Id). 8,20)... 

From the general considerations presented in the previous section, one can expect that the many-body non-adiabatic wave function should fulfill the following conditions (1) All particles involved in the system should be treated equivalently (2) Correlation of the motions of all the particles in the system resulting from Coulombic interactions, as well as from the required conservation of the total linear and angular momenta, should be explicitly incorporated in the wave function (3) Particles can only be distinguishable via the permutational symmetry (4) The total wave function should possess the internal and translational symmetry properties of the system (5) For fixed positions of nuclei, the wave functions should become equivalent to what one obtains within the Born-Oppenheimer approximation and (6) the wave function should be an eigenfunction of the appropriate total spin and angular momentum operators. 

The problem is conventionally sidestepped by assuming that nuclear and electronic motions are decoupled, but despite many efforts this condition has never been shown to yield a rigid molecular shape either. The insurmountable problem is permutational invariance. In molecular-orbital calculations that decouple electronic from nuclear motion the nuclei are identified in order to support the definition of molecular structure, but then permutation of identical nuclei implies rearrangement of bonds and a new set of calculated electronic energies. There is little hope of ever overcoming these problems ... 

The theoretical investigation of these reactions requires quantum-mechanical methods, in particular, the study of chemical bonds in initial, final, and intermediate compounds, as well as the consideration of nuclear motions. Yet frequently the important information can be obtained without analysis of electronic structure of molecules and investigation of actual motion of nuclei, only resorting to the graph theory and employing the group-theoretical conceptions. Here we should define two terms permutation isomers and permutation isomerism reaction. 

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