Spatial function symmetry three-electron

In the present contribution the interpretation of the energy-level structure of quasi-one-dimensional quantum dots of two and three electrons is reviewed in detail by examining the polyad structure of the energy levels and the symmetry of the spatial part of the Cl wave functions due to the Pauli principle. The interpretation based on the polyad quantum number is applied to the four electron case and is shown to be applicable to general multi-electron cases. The qualitative differences in the energy-level structure between quasi-one-dimensional and quasi-ta>o-dimensional quantum dots are briefly discussed by referring to differences in the structure of their internal space. 

The conclusion above that optimisation of the non-linear parameters in the AO basis leads to a basis with correct spatial symmetry properties cannot be true for all intemuclear separations. At R = 0 the orbital basis must pass over into the double-zeta basis for helium i.e. two different 1 s orbital exponents. It would be astonishing if this transition were discontinuous at R = 0. While considering the variation of basis with intemuclear distance it is worth remembering that the closed-shell spin-eigenfunction MO method does not describe the molecule at all well for large values of R the spin-eigenfunction constraint of two electrons per spatial orbital is completely unrealistic at large intemuclear separation. With these facts in mind we have therefore computed the optimum orbital exponents as a function of R for three wave functions ... 

Pj, is a projection operator ensuring the proper spatial symmetry of the function. The above method is general and can be applied to any molecule. In practical application this method requires an optimisation of a huge number of nonlinear parameters. For two-electron molecule, for example, there are 5 parameters per basis function which means as many as 5000 nonlinear parameters to be optimised for 1000 term wave function. In the case of three and four-electron molecules each basis function contains 9 and 14 nonlinear parameters respectively (all possible correlation pairs considered). The process of optimisation of nonlinear parameters is very time consuming and it is a bottle neck of the method. 

To generate the proper Aj, A2, and E wavefunctions of singlet and triplet spin symmetry (thus far, it is not clear which spin can arise for each of the three above spatial symmetries however, only singlet and triplet spin functions can arise for this two-electron example), one can apply the following (un-normalized) symmetry projection operators (see Appendix E where these projectors are introduced) to all determinental wavefunctions arising from the e2 configuration ... 

In this set the functions can be classified into two types in the right column the spatial multiplier is symmetric with respect to transpositions of the spatial coordinates and the spin multiplier is antisymmetric with respect to transpositions of the spin coordinates in the left column the spatial multiplier is antisymmetric with respect to transpositions of the spatial coordinates and the spin multipliers are symmetric with respect to transpositions of the spin coordinates. Because in the second case the spatial (antisymmetric) multiplier is the same for all three spin-functions, the energy of these three states will be the same i.e. triply degenerate - a triplet. The state with the antisymmetric spin multiplier is compatible with several different spatial wave functions, which probably produces a different value of energy when averaging the Hamiltonian, thus producing several spin-singlet states. From this example one may derive two conclusions (i) the spin of the many electronic wave function is important not by itself (the Hamiltonian is spin-independent), but as an indicator of the symmetry properties of the wave function (ii) the symmetry properties of the spatial and spin multipliers are complementary - if the spatial part is symmetric with respect to permutations the spin multiplier is antisymmetric and vice versa. 

This result is a consequence of the existence of two independent functions from each of eig and e2 thus the product of them gives four independent functions. The first excited configuration of benzene gives rise to three spatial states that are accidentally degenerate to the approximation of zero electronic interactions. The formal method for obtaining the correct linear combinations of the product functions from purely symmetry arguments is straightforward. The direct product matrices Dj Jl(R) formed from the eig and 2u matrices and... 

In the cubic case the symmetry of the JT modes follows the reducible representation e + t2 (in Td and O) or Cg + t2g (in Oh) (see Eq. 13). The epikemels of these JT coordinates may be found in Tables 2 and 3. In view of the high dimensionality of the distortion spaces it is difficult to visualize the spatial distribution of these epikemels, not to mention the pictorial representation of a complicated potential surface in the corresponding distortion space. One possibility is to draw cross-sections of the most relevant subspaces [3]. Another more comprehensive alternative, wich combines all essential features in one drawing, may be realized by using a projection technique. The projection space is the space of the electronic functions. This space may be defined by three cartesian axes which represent the three components Tx>, iy> and Tz> of the electronic T state. An eigenfunction, say Ta >, of the distorted hamiltonian H (Eq. 14) can be expressed as a linear combination of these three basis functions. 

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