Spin wavefunctions and

The spin Hamiltonian operates only on spin wavefunctions, and all details of the electronic wavefunction are absorbed into the coupling constant a. If we treat the Fermi contact term as a perturbation on the wavefunction theR use of standard perturbation theory gives a first-order energy... 

Symmetry dictates that the representations of the direct product of the factors in the integral (3 /T Hso 1 l/s2) transform under the group operations according to the totally symmetric representation, Aj. The spin part of the Hso spin-orbit operator converts triplet spin to singlet spin wavefunctions and singlet functions to triplet wavefunctions. As such, the spin function does not have a bearing on the symmetry properties of Hso- Rather, the control is embedded in the orbital part. The components of the orbital angular momentum, (Lx, Ly, and Lz) of Hso have symmetry properties of rotations about the x, y, and z symmetry axes, Rx, Ry, and Rz. Thus, from Table 2.1, the possible symmetry... 

Many-Electron Spatial Wavefunctions 157 TOOLS OF THE TRADE Photoeiectron Spectroscopy 163 Approximate Solution to the Schrodinger Equation 164 BIOSKETCH Syivia Ceyer 179 Spin Wavefunctions and Symmetrization 179 Vector Model of the Many-Electron Atom 186 Periodicity of the Elements 190 Atomic Structure The Key to Chemistry 191... 

Because symmetric and antisymmetric properties follow the same rules of multiplication as odd/even or positive/negative multiplication (that is, sym X sym = sym, sym X antisym = antisym, antisym X antisym = sym), all we need to do is identify the symmetry properties of the spatial and spin wavefunctions and combine antisymmetric parts with symmetric parts to get an overall antisymmetric wave-function. The following summarizes the properties, which can easily be verified ... 

Note that the complete wavefunction as written in Eq. (2.47) changes sign if the labels of the electrons (1 and 2) are interchanged. W. PauU pointed out that the wavefunctions of all multielectronic systems have this property. The overall wavefunction invariably is antisymmetric for an interchange of the coordinates (both positional and spin) of any two electrons. This assertion rests on experimental measurements of atomic and molecular absorption spectra absorption bands predicted on the basis of antisymmetric electrOTiic wavefunctirais are seen experimentally, whereas bands predicted on the basis of symmetric electronic wave-functions are not observed. Its most important implication is the Pauli exclusion principle, which says that a given spatial wavefunction can hold no more than two electrons. This follows if an electron can be described completely by specifying its spatial and spin wavefunctions and electrons have only two possible spin wave-functions (a and fi). 

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