Product Kronecker

APPENDIX A1 THE KRONECKER PRODUCT OF MATRICES AND THE vec(o) OPERATOR The Kronecker Product... 

The mass matrix M enters the Hamiltonian for convenience of expression and is an n X n matrix with on the diagonal elements and 2 on all of the off-diagonal elements the M notation for any matrix will mean a Kronecker product with the 3 X 3 identity matrix, M = M h. 

Effectively, vector r has 3 x n x 1 components since each r, in (47) is itself a three-dimensional vector. Technically speaking, in place of Ak in (46), one should write the Kronecker product A with being the 3 x 3 identity matrix. However, to simplify notations and avoid writing routinely this obvious Kronecker product, below in this section we will be using the following convention for matrix-vector multiplications involving such vectors ... 

Other notation used diagB is the diagonal n x n matrix consisting of the diagonal elements of the square matrix B. The trace of B is denoted trB, and the determinant of B is denoted B. The Kronecker product of two matrices is denoted by symbol (g). Other notation will be introduced as needed. 

The Kronecker product with the identity ensures rotational invariance (sphericalness) elliptical Gaussians could be obtained by using a full n x n A matrix. In the former formulation of the basis function, it is difficult to ensure the square integrability of the functions, but this becomes easy in the latter formulation. In this format, all that is required is that the matrix, A, be positive definite. This may be achieved by constructing the matrix from a Cholesky decomposition A), = Later in this work we will use the notation... 

A = A h to indicate the Kronecker product with the 3x3 identity matrix. 

To maintain the parallel with 2-component theory, it is expedient to express the Dirac space as a Kronecker product of two Pauli spaces. R om... 

T TW is called the direct product (or Kronecker product) of the representations r" and r The sign does not mean multiplication, it is simply a signal that the direct product of two representations has been formed in the manner given above. 

The inner direct product (DP) (or inner Kronecker product)... 

Exercise 14.3-1 Write down the non-zero CG coefficients for the inner DPs of the point group mm2 (C2V). [Hints See Table 14.4. Recall that %3 (R) means x(r3) and that for this group T3 = T4.] Using Table 14.6 derive expressions for the non-zero CG d coefficients of the magnetic point group 4mm in terms of the cijk and evaluate these. Hence write down the CG decomposition for the Kronecker products of the IRs T of 4mm. 

The tensor (Kronecker) product of two vectors is written as follows... 

Appendix A. Composite indices. Some properties of direct (Kronecker) products. 284... 

APPENDIX A. COMPOSITE INDICES. SOME PROPERTIES OF DIRECT (KRONECKER) PRODUCTS... 

Wig51] Wigner EP 1951 On the matrices which reduce the Kronecker products of representations of simply-reducible groups (unpublished), cited in [App68]. 

Constructing an 50(4) matrix in terms of two SU(2) matrices parametrized by q and p is done as follows each of the SU(2) matrices corresponding to q and p, respectively, acts in a separate space of states of two particles with -spins [28,29]. Since the 50(4) group is a direct product of two 50(3) (or of SU(2) locally isomor-phous to 50(3)) groups the matrix representing an element of 50(4) is the direct (Kronecker) product of two SU(2) matrices. The space in which it acts is a direct product of two spaces spanned by the basis states +5), — 5) eac 1- configu-... 

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) ... 

In the above equation the operation , called the Kronecker Product, is defined for A (an mxn matrix) and B (a p x q matrix) matrixes as partitioned mp x nq matrix whose ijth partition is ay times B ... 

Here r is a 3n x 1 vector of Cartesian coordinates for the n particles, Lk is an n x n lower triangular matrix of rank n and I3 is the 3x3 identity matrix, k would range from 1 to A where N is the number of basis functions. The Kronecker product with I3 is used to insure rotational invariance of the basis functions. Also, integrals involving the functions k are well defined only if the exponent matrix is positive definite symmetric this is assured by using the Cholesky factorization LkL k. The following simplifications will help keep the notation more compact ... 

The symmetric group S(n) is of fundamental importance in quantum chemistry as well in nuclear models and symplectic models of mesoscopic systems. One wishes to discuss the properties of the symmetric group for general n and concentrate on stable results that are essentially n—independent. Here the reduced notation(6)-(9) proves to be very useful. The tensor ir-reps A of S(n) are labelled by ordered partitions(A) of integers where A I- n. In reduced notation the label Ai, A2,. .., Ap for S(n) is replaced by (A2,...,AP). Kronecker products can then be fully developed in a n-independent manner and readily programmed. Thus one finds, for example, the terms arising in the reduced Kronecker product (21) (22) are... 

Looking at the above list one is immediately struck by the observation that the list is self-associated. That is every partition (A) in the list either has a conjugate partner (A) where the rows and columns of the Young frame of the partition (A) have been interchanged or the partition (A) is self-conjugate. Some Kronecker products are self-associated while others are not. Is there... 

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