Molecular potential spin symmetry

The second approach to treating nondynamical correlation has an air of the ostrich about it ignore the spin symmetry of the wave function and use unrestricted Haxtree-Fock (UHF) theory as the single configuration description [7]. Since the UHF wave function comprises one spin-orbital for each electron, a molecular UHF wave function should dissociate to atomic UHF wave functions, for example. This is certainly not the case for spin-restricted Hartree-Fock (RHF) molecules and atoms in general. And there is an attractive simplicity about UHF — no active orbitals to identify, and so forth. However, where nondynamical correlation would be important in an RHF-based treatment, the UHF method will suffer from severe spin-contamination, while where nondynamical correlation is not important the RHF solution may be lower in energy than any broken-symmetry UHF solution, so potential curves and surfaces may have steps or kinks where the spin symmetry is broken in the UHF treatment. 

Our discussion of spatial inversion symmetry concludes our project of demonstrating the symmetry properties of the Dirac equation under transformations of spin and spatial coordinates. With a spherical potential, the symmetry group is SU(2) 0 0(3). For potentials of lower (e.g., molecular) symmetries the appropriate group is SU (2)0G, where G is the nonrelativistic (spatial) point group of the potential—that is, the... 

Reestablishment of the equilibrium population is associated with changes in the spin state, through the influence of the exclusion principle on the space-spin symmetry of the molecular states. As a consequence, the signals of methyl Cs may appear with different sign compared to the other resonances. As a consequence, the absolute intensity of the spectrum is not the same any more for microwave frequencies of constructive interference of quantum rotor and DNP effects. It should be mentioned that quantum rotor mechanisms constitute a separate polarization mechanism with great potential [13,15,16]. 

The single Slater determinant wavefunction (properly spin and symmetry adapted) is the starting point of the most common mean field potential. It is also the origin of the molecular orbital concept. 

Figure 1. Cartoon depicting symmetry considerations of the asymptotic wavefunction that influences which potential energy surface is accessed. Representative angular momentum quantum numbers relevant to the process are the total molecular angular momentum exclusive of nuclear spin (f) and the relative orbital angular momentum (/).

The one-center approximation allows for an extremely rapid evaluation of spin-orbit mean-field integrals if the atomic symmetry is fully exploited.64 Even more efficiency may be gained, if also the spin-independent core-valence interactions are replaced by atom-centered effective core potentials (ECPs). In this case, the inner shells do not even emerge in the molecular orbital optimization step, and the size of the atomic orbital basis set can be kept small. A prerequisite for the use of the all-electron atomic mean-field Hamiltonian in ECP calculations is to find a prescription for setting up a correspondence between the valence orbitals of the all-electron and ECP treatments.65-67... 

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