Permutational symmetry of the basis

It must be emphasised that these ideas are restricted to those cases where the computational problem can be reduced to the diagonalisation of an effective one-electron matrix, unlike the methods of integral calculation based on the permutational symmetry of the basis which are applicable to any method since all that these latter methods do is produce a file of integrals in a non-standard order which, if the method of processing the electron-repulsion integrals is well-designed, is immaterial. 

In Chapter VIII, Haas and Zilberg propose to follow the phase of the total electronic wave function as a function of the nuclear coordinates with the aim of locating conical intersections. For this purpose, they present the theoretical basis for this approach and apply it for conical intersections connecting the two lowest singlet states (Si and So). The analysis starts with the Pauli principle and is assisted by the permutational symmetry of the electronic wave function. In particular, this approach allows the selection of two coordinates along which the conical intersections are to be found. 

Determination of a wave function for a system that obeys the correct permutational symmetry may be ensured by projection onto the irreducible representations of the symmetry groups to which the systems in question belong. For each subset of identical particles i, we can implement the desired permutational symmetry into the basis functions by projection onto the irreducible representation of the permutation group, for total spin 5, using the appropriate projection operator T,. The total projection operator would then be a product ... 

If the TV-particle basis were a complete set of JV-electron functions, the use of the variational approach would introduce no error, because the true wave function could be expanded exactly in such a basis. However, such a basis would be of infinite dimension, creating practical difficulties. In practice, therefore, we must work with incomplete IV-particle basis sets. This is one of our major practical approximations. In addition, we have not addressed the question of how to construct the W-particle basis. There are no doubt many physically motivated possibilities, including functions that explicitly involve the interelectronic coordinates. However, any useful choice of function must allow for practical evaluation of the JV-electron integrals of Eq. 1.7 (and Eq. 1.8 if the functions are nonorthogonal). This rules out many of the physically motivated choices that are known, as well as many other possibilities. Almost universally, the iV-particle basis functions are taken as linear combinations of products of one-electron functions — orbitals. Such linear combinations are usually antisymmetrized to account for the permutational symmetry of the wave function, and may be spin- and symmetry-adapted, as discussed elsewhere ... 

The relationships between the one-electron transformations involving the orbital basis and the two-electron transformations involving the basis-products are useful for formal purposes but would certainly never be implemented as a practical way of solving the SCF equations as they stand since they involve x matrices which contain much redundant information. However, techniques of storage compression, analogous to use of the permutational symmetries of the repulsion integral labels, can be used to enable what is known as the supermatrix formulation of the SCF equations to be implemented in an economical way. 

We therefore conclude that, for a combination of model, numerical and conceptual reasons the OHAO basis is well-adapted to a theory of valence. The hybrid orbital basis (for simple molecules) has a distinctive symmetry property it carries a permutation representation of the molecular symmetry group the equivalent orbitals are always sent into each other, never into linear combinations of each other. This simple fact enables the hybrid orbital basis to be studied in a way which is physically more transparent than the conventional AO basis. 

The function F(l,2) is in fact the space part of the total wave function, since a non-relativistic two-electron wave function can always be represented by a product of the spin and space parts, both having opposite symmetries with respect to the electrons permutations. Thus, one may skip the spin function and use only the space part of the wave function. The only trace that spin leaves is the definite per-mutational symmetry and sign in Eq.(14) refers to singlet as "+" and to triplet as Xi and yi denote cartesian coordinates of the ith electron. A is commonly known angular projection quantum number and A is equal to 0, 1, and 2 for L, II and A symmetry of the electronic state respectively. The linear variational coefficients c, are found by solving the secular equations. The basis functions i(l,2) which possess 2 symmetry are expressed in elliptic coordinates as ... 

The symmetry group of NH2D (ND2H) is the C2V group of tire permutations and permutation-inversions of the elements E, (12), E, and (12). By the same arguments as described above for NH3 we find that the symmetry coordinates for NH2D form the basis of a reducible representation ... 

In the case of the electron-repukion integrals, we noted that the electron-repulsion operator (l/ri2) was sphe ic dly symmetric" and so it is only the permutation (transformation) properties of the basis functions which mattered in using molecular symmetry. The situation is similar in the case of the one-electron integrals ... 

The kinetic energy integrals involve the operator which, like the electron-repulsion operator has much higher symmetry than the molecular point group so, again, only the lower symmetry of the permutations of the basis functions is involved. 

The nuclear-attraction integrals contain the nuclear attraction terms from the electrostatic Hamiltonian, which, of course( ), has the symmetry of the nuclear framework and so is left invariant by exactly the same permutation operations which we are considering. Again, we may consider only the symmetry properties of the permutations of the basis functions. 

The simplest case of all when the basis is simply permuted by the symmetry transformations of the molecular point group (the case considered earlier in treating the integral computation problem). The matrices V are unit matrices the centres are permuted (and so the basis is permuted, but no further mixing occurs). 

The number of two-electron integrals formally grows as the fourth power of the size of the basis set. Owing to permutation symmetry (the following integrals are identical... 

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