Kronecker product, transformation

The mn product functions (9.127) are thus transformed into linear combinations of one another by the symmetry operators of the group they therefore form a basis for some mn-dimensional representation Ty of the group. The representation T, is called the direct product (or Kronecker product) of the representations and TG this is symbolized by... 

The direct (Kronecker) product of the SU (2) matrices representing the q- and p-pararotations acts in this space with the notion that the q-dependent matrix eq. (3.48) acts on the states of the first particle and the p-dependent one on the states of the second particle in the product state. Then we form linear combinations of the above states, which correspond to specific values of the total spin and desired spatial symmetry. The combination which corresponds to the zero total spin of two particles transforms as a scalar i.e. (singlet) s-function. Those which correspond to the total spin equal to unity form the basis in the three-dimensional (triplet) space of />functions. The coordinate (x-, y-, and z-) functions are obtained as the following combinations of the states with the definite spin projections (the above product states) ... 

Traaisformations of the basis now induce transformations of the operator and density matrices in terms of the Kronecker product of the transformation matrix with itself ... 

A similar transformation is induced in the electron-repulsion integrals which may be written in Kronecker-product form or, more commonly, simply written out in full ... 

Once the electric or magnetic dipole nature of the transitions is known, selection rules are determined from group theory. The components of the electric and magnetic field vectors transform according to definite representations of the point group (often different ones, as E is a polar vector and B is an axial vector). A particular transition is allowed (forbidden) if the irreducible representation of the final state is (is not) contained in the Kronecker product of the initial state representation and the representation of the appropriate component of E or B (Tinkham, 1%4). 

The outer or Kronecker product of two matrices (C = A X B) is defined by Cik ii = AijBki (double-index labelling of rows and columns). By writing a set of spin-orbitals in the form rl) = Xa, where < > is a row of orbitals and o = (spin factors are separately transformed. [Hint The matrix product of two outer products is... 

The rest follows without too much trouble. The products of spherical harmonics arising from P2 coscOij), P4(costransform under the operations of SO(7) and Gj like those parts of the Kronecker squares (200) and (20) that contain an 80 (3) scalar. From tables D-4, E-2 and E-3 of Wybourne (1970), we find that the acceptable irreducible representations are given by... 

In the MO basis this is a product of Kronecker delta functions. In the case of the V intermediate needed in CCSD(F12) theory, the indices p and q are in the AO basis, and the overlap integrals are not the unity matrices anymore. Instead, they must be expressed as partially transformed overlap integrals... 

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