Spin eigenfunctions symmetry properties

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 ... 

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. 

Before proceeding to the details of the MCSCF formalism, we first impose some restrictions on the matrices A and U. We will later restrict the MCSCF wavefunction to be an eigenfunction of the spin operators S, and S. The orbital transformations that are applied to this wavefunction should not destroy these symmetry properties. This condition will be satisfied if the operator A commutes with these spin operators. If the spin components of the operator are written explicitly... 

The symmetry properties of wavefunctions stem from the symmetry properties of the Hamiltonian as defined by its commutation properties with symmetry operators. Accordingly, the wavefunctions, being eigenfunctions of the Hamiltonian, transform irreducibly under the operators in the symmetry group of the molecule. This applies to both space and spin symmetry, but this article will mainly be concerned with spatial symmetry. 

The most obvious new feature of the Dirac equation as compared with the standard nonrelativistic Schrodinger equation is the explicit appearance of spin through the term a p. Any spin operator trivially commutes with a spin-free Hamiltonian, but the introduction of spin-dependent terms may change this property, as demonstrated in the case of ji/-coupling. A further scrutiny of spin symmetry is therefore a natural first step in discussing the symmetry of the Dirac Hamiltonian. This requires a basis of spin functions on which to carry out the various operations, and a convenient choice is the familiar eigenfunctions of the operator, i, i) and j, — ), also called the a and spin functions. 

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