Spatial function symmetry orbital approximation

From the Table I, moreover, we can see that the contributions of Cv-3d orbital are significantly different between the MOs with ng and eg symmetry. This result indicates that the difference of interaction between the impurity and the ligand orbitals according to the direction of spatial extension of the orbitals is very important for the calculation of multiplet structure. Why the empirical method has been seemed to be successfully explained the multiplet structure, nevertheless it approximates the radial parts of wave functions to be the same one, is supposed that it "hides" and "rounds off the errors by adjusting the parameters arbitrarily to the experimental data. 

Thus this model maps density over atoms rather than spatial coordinates. If overlap is included some other definition of charge density such as Mulliken s17 may be employed. Eq. (30) and (31) are then used with this wave function to calculate the hyperfine constants as a function of the pn s. If symmetry is high enough, there will be enough hyperfine constants to determine all the p s, otherwise additional approximations may be necessary. For transition metal complexes, where spin-orbit effects are appreciable, it is necessary to include admixtures of excited-state configurations that are mixed with the ground state by the spin-orbit operator. To determine the extent of admixture, we must know the value of the spin-orbit constant X and the energy of the excited states. 

Generally speaking, it is probably impossible to give any general recommendations concerning the search for the observables, the operators of which commute with the Hamiltonian. The exceptions are the cases when the observable to be found characterizes the properties of the spatial symmetry of the ket-vector v /(t)> (in Schroedinger s spatial presentation this vector is called the wave function). It should be noted that a more or less precise pattern of the wave function /(t) is known for very few molecular system. At present the most widespread is the i /(t) presentation in the Hartree-Fock approximation as a symmetrized linear combination of atomic orbitals (LCAO) [16]. 

However, classifying the wave function of polymerization by the spatial symmetry features and taking into account the above specificity of addition polymerization, it is advisable for simplicity to introduce as a supplement to LCAO the approximation of the polymerization wave function in the form of a linear combination of molecular orbitals of fragments (LCMOF). The validity of introduction of this approximation is based on the general quantum-mechanical principle of superposition. 

By Restricted we mean that two electrons with different spins a and P do occupy the same spatial orbital. The horizontal mirror symmetry (D h) of H2 requires that the two active orbitals belong to two different irreducible representations Og and Ou. Hence, for a minimal basis set calculation, the composition of the MOs is given by symmetry and the wave function does thus have no degree of freedom at all. The situation is quite different in case of unrestricted MO calculation vide infra). As expected, the dissociation behavior is not correctly reproduced because of the restricted approximation. But it is worth to note, that the singlet-triplet separation, for which the correct behavior is to vanish when the inter-atomic distance increases. 

A computationally convenient approach, but by no means the only one, is to approximate each atomic radial function by a sum of i x Gaussian functions, ip. These functions are then multiplied by an angular function in order to create the three-dimensional symmetry of the s, p, d, etc. spatial orbitals. This collection... 

The use of restricted orbital rotations not only ensures that the approximate electronic wave function has the desired spin and spatial symmetries. As an additional benefit, the computational cost is lowered by reducing the number of free parameters. 

Within the Bom-Oppenheimer approximation, the exact stationary states form a basis for an irreducible representation of the molecular point group. We may enforce the same spatial symmetry on the approximate state by expanding the wave function in determinants constmcted from a set of symmetry-adapted orbitals. For atoms, in particular, the use of point-group... 

In the unrestricted Hartree-Fock (UHF) approximation, on the other hand, the wave function is not required to be a spin eigenfunction and different spatial orbitals are used for different spins. Occasionally, the spatial symmetry restrictions are also lifted, and the MOs are then no longer required to transform as irreducible representations of the molecular point group. It should be noted that, in particular for systems close to the equilibrium geometry, the symmetries of the exact state are sometimes present in the UHF state as well even though they have not been imposed during its optimization. In such cases, the UHF and RHF states will coincide. 

---