Antisymmetric/antisymmetries matrices

However, billiard balls are a pretty bad model for electrons. First of all, as discussed above, electrons are fermions and therefore have an antisymmetric wave function. Second, they are charged particles and interact through the Coulomb repulsion they try to stay away from each other as much as possible. Both of these properties heavily influence the pair density and we will now enter an in-depth discussion of these effects. Let us begin with an exposition of the consequences of the antisymmetry of the wave function. This is most easily done if we introduce the concept of the reduced density matrix for two electrons, which we call y2. This is a simple generalization of p2(x1 x2) given above according to... 

When constructing many-electron wave functions it is necessary to ensure their antisymmetry under permutation of any pair of coordinates. Having introduced the concepts of the CFP and unit tensors, Racah [22, 23] laid the foundations of the tensorial approach to the problem of constructing antisymmetric wave functions and finding matrix elements of operators corresponding to physical quantities. 

Each of the MOs in Fig. 10-3 is symmetric or antisymmetric for some of the operations that apply to a tetrahedron. 02 is symmetric for rotations about the z axis by 27t/3, and also for reflection through the xz plane. This same reflection plane is a symmetry plane for 03 and 04, but neither of these MOs shows symmetry or antisymmetry for rotation about the z axis. Each MO contains one p AO and, perforce, has the symmetry of that AO. We shall refer to these as p-type MOs. Note that hydrogen Is AOs lying in the nodal plane of a p AO do not mix with that p AO in formation of MOs. This results from zero interaction elements in H, which, in turn, results from zero overlap elements in S. Note also that the MO 02 is the MO that we anticipated earlier on the basis of inspection of the matrix H. Because of phase agreements between the hydrogen Is AOs and the adjacent lobes of the p AOs, 0j, 02, and 03 are C-H bonding MOs. 

Although Eqs. (8) to (10) provide formal expressions to isolate the physical eigenstates from the unphysical states in which they can be embedded, and to correspondingly demonstrate equivalence with results obtained from prior antisymmetry (f ), an efficient recursive scheme using appropriately chosen antisymmetrized starting functions is sufficient to construct Specifically, a transformation of the Hamiltonian matrix which is equivalent to that of Eq. (8) is obtained from the recurrence equations 18)... 

---