Permutation operator case functions

The exchange sum can be expressed in a way which enables a single operator K to be defined using a permutation operator P(iJ) which permutes the ftpin-orbitals Xi and Xj which are functions of the same coordinate (xi in the case below) ... 

Margolus [marg84] points out that second-order reversible CA may also be constructed by using operations other than the subtraction modulok we used in our example. The actual operation could in fact be a function of the neighbor s values at time f . In the most general case, the neighborhood at time C can be used to choose a permutation on the set of allowed site values. The permutation is then applied to the site value at time t-V to obtain its next state. 

For the irreducible representation A the symmetrized combination is easily found to be 0-1+02+ <73 The application of Eq. (47) for the representation E yields - 02 - 03. As the shhplMed projectifota operator has beefi employed in this example, the second combination of species E is not given directly. However, it is sufficient in this case to coi tiiict a secOiid litiear cOfffbtiia-don that is compatible with the symmetry Et and ofttiOgohal to the first. A direct method to find the appropriate combination is to permute cycliC iiy the functions obtained above, viz. 

The point symmetry group of the molecule is denoted by 9 (Dnu or Cnv in the present case), and it is necessary to produce from the functions (35) wavefunctions which form bases for irreducible representations A of rd. We note first of all that since all the orbitals are localized on one or other of the atoms forming the molecule, the application of a spatial symmetry operation 52 of rS is equivalent to a permutation of the orbitals on the equivalent atoms amongst themselves, possibly multiplied by a rotation of the orbitals on the central atom. Hence with every operation 52 we may associate a certain permutation of the orbitals, Pr, in which the bar emphasizes that one permutes the orbitals themselves and not the electron co-ordinates. Thus,... 

The actual spin-coupled wavefunction corresponds to none of these cases, but to six identical highly localized orbitals on each of the six carbon atoms. One of these is shown in Fig. 11, and it can be seen that it is essentially a C(2p,) orbital which is deformed by the two adjacent C atoms. The orbitals are permuted by the operation Cg. All five of the spin functions participate, giving a function which corresponds to a symmetric coupling of the spins around the ring. 

When a S-f symbol contains an odd F an odd number of times the corresponding integral vanishes because of its ungerade character. So in this case one of the F s must be represented by a corresponding axial operator (and the others still by their spherical harmonic standard basis functions) in order that the 3-r symbol can be represented by an integral. The 3-F symbol defined in this way will then automatically change sign under odd permutations of its columns, so we have an odd 3-F symbol. 

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 full case when the symmetry operations of the point group induce linear transformations amongst the basis functions. The matrices V are non-trivial orthogonaJ matrices. The centres are permuted and the basis functions are mixed by the transformations. 

We will now derive a Dyson equation by expressing the inverse matrix of the extended two-particle Green s function Qr,y, u ) by a matrix representation of the extended operator H. We already mentioned that the primary set of states l rs) spans a subspace (the model spaice) of the Hilbert space Y. Since the states IVrs) are /r-orthonormal they are also linearly independent and thus form a basis of this subspace. Here and in the following the set of pairs of singleparticle indices (r, s) has to be restricted to r > s for the pp and hh cases (b) and (c) where the states are antisymmetric under permutation of r and s. No restriction applies in the ph case (a). The primary set of states Yr ) can now be extended to a complete basis Qj D Yr ) of the Hilbert space Y. We may further demand that the states Qj) are /r-orthonormal ... 

The quantum number v embodies the set of nuclear dynamic states with their labels (see below) and /c stands for the electronic quantum state. Thus, the nuclear wave function is always determined relatively to particular electronic states which, in turn, must be correlated to the (point) symmetries of the system. This stationary wave function may define, for particular cases, a class of geometric elements having an invariant center of mass. Actually, the (equivalence) class of configurations are those for which symmetry operations leave invariant this center of mass. This framework shares the discrete symmetries, such as permutation and space reflection invariances that are properties of the molecular eigenstates. There exists, then, a specific geometric framework pok- At this point, the expectation value ofH ,. taken with respect to the universal wave function is stationary to any geometric variation. 

The assumption, known as Kleinman symmetry, that the quadratic (and higher-order) response function is symmetric imder interchange of any pair of operators is only true at zero frequency, as is easily verified from O Eq. 5.31. Note that the quadratic response function is symmetric when permuting the operators B and C at co = co, which is the case for, for example. Second Harmonic Generation j jjk -2(o-,co,(o) = Pikj -2(o-,(o,co). Kleinman symmetry is often assumed in calculations of the electric dipole hyperpolarizabUity at low frequencies where it is approximately vahd. This reduces the number of independent tensor elements and thus the computational effort... 

The basis functions introduced in Problem 2.11 are said to be symmetry-adapted or to be symmetry functions . Use a similar procedure to obtain MOs and orbital energies for (i) linear H4 and (ii) square-planar H4. [Hint Orbitals that are exchanged (more generally permuted) by a symmetry operation are said to be equivalent they can be combined to give symmetry functions. In case (ii) you may use two reflections, classifying the functions as ( ] ]) (-r-f- ) etc.]... 

---