Matrices direct product

Alternative and less tedious methods by use of matrix direct products can also be used when the basis functions are also the eigenfunctions, as shown in Appendix D, where a number of spin matrices are given.) By following the procedure of Eq. 11.39, we obtain the density matrix at the end of the pulse ... 

Molecular point-group symmetry can often be used to determine whether a particular transition s dipole matrix element will vanish and, as a result, the electronic transition will be "forbidden" and thus predicted to have zero intensity. If the direct product of the symmetries of the initial and final electronic states /ei and /ef do not match the symmetry of the electric dipole operator (which has the symmetry of its x, y, and z components these symmetries can be read off the right most column of the character tables given in Appendix E), the matrix element will vanish. 

The reason for this is simple. If the reaction chemistry is not "clean" (meaning selective), then the desired species must be separated from the matrix of products that are formed and that is costly. In fact the major cost in most chemical operations is the cost of separating the raw product mixture in a way that provides the desired product at requisite purity. The cost of this step scales with the complexity of the "un-mixing" process and the amount of energy that must be added to make this happen. For example, the heating and cooling costs that go with distillation are high and are to be minimized wherever possible. The complexity of the separation is a function of the number and type of species in the product stream, which is a direct result of what happened within the reactor. Thus the separations are costly and they depend upon the reaction chemistry and how it proceeds in the reactor. All of the complexity is summarized in the kinetics. 

If the system contains symmetry, there are additional Cl matrix elements which become zero. The symmetry of a determinant is given as the direct product of the symmetries of the MOs. The Hamilton operator always belongs to the totally symmetric representation, thus if two determinants belong to different irreducible representations, the Cl matrix element is zero. This is again fairly obvious if the interest is in a state of a specific symmetry, only those determinants which have the correct symmetry can contribute. 

The permutations of 0[H] have a special effect on the rows of the matrix (1.37) if a permutation moves an element of one row into another row, then the permutation moves all the elements of the one row into the other row. The rows of (1.37) are imprimitive domains of 0[H]. The permutations of 0[H] which leave the r imprimitive domains invariant (the gross permutation of which is the identity) form a subgroup it has order it is the direct product H xHxHx...xH with r factors and is a normal subgroup of C [H], with factor group. 

We choose to use the primitive coupled basis, rather than the 712 coupled basis that is often employed, primarily because the primitive coupled basis has a direct product form. This form makes it easy to accomplish the transformation from ji, j2, fci to 0i, 02, - The potential matrix, V, may be written as a numerical quadrature over the angular grid points in the following matrix form ... 

While the matrix multiplication defined by Eq. (28) is the more usual one in matrix algebra, there is another way of taking the product of two matrices. It is known as the direct product and is written here as A <8> 1 . If A is a square matrix of order n and B is a square matrix of order m, then A<8>B is a square matrix of order tun. Its elements consist of all possible pairs of elements, one each from A and B, viz. 

The arrangement of die elements in the direct-product matrix follows certain conventions. They are illustrated in the following chapter, where the direct product of matrices is employed in the theory of groups. 

Suppose now that A) and B) belong to an electronic representation I ,. Since H is totally symmetric, Eq. (6) implies that the matrix elements (A II TB) belong to the representation of symmetrized or anti-symmetrized products of the bras (A with the kets 7 A). However, the set TA) is, however, simply a reordering of the set ( A). Hence, the symmetry of the matrix elements in the even- and odd-electron cases is given, respectively, by the symmetrized [Ye x Te] and antisymmetrized Ff x I parts of the direct product of I , with itself. A final consideration is that coordinates belonging to the totally symmetric representation, To, cannot break any symmetry determined degeneracy. The symmetries of the Jahn-Teller active modes are therefore given by... 

If Ai, A2, Bi, and B2 are any matrices whose dimensions are such that the ordinary matrix products AiA2 and BiB2 are defined, then the direct product has the property... 

The operation of direct product of matrices is both associative and also distributive with respect to matrix addition, and hence finally... 

The matrix element < [Equation (7.9)] is zero unless the direct product... 

These matrix elements are nonzero by spatial symmetry only if the direct products r, (8)rj and share a common irreducible representation [58]. 

The diagonal elements in a direct-product matrix Ffg are those giving the coefficient of a particular term figj in the hnear combination expressing A figj). This element of Ffg A) is equal to... 

INTRODUCTION TO SYMMETRY AND GROUP THEORY EOR CHEMISTS The trace of the direct product matrix is ... 

In order to apply the direct product representation to the derivation of selection rules, recognize that a matrix element of the form ipi, O lpj) will be equal to zero for symmetry reasons if there is even one symmetry operation that takes the integrand into its negative. The argument follows exactly the course of that of section 10.2. Thus the matrix element will vanish unless the direct product representation is totally symmetric (Ai), or contains A upon reduction. 

Simplification of secular equations. Because the Hamiltonian is totally symmetric - that is, for a molecule of C2v symmetry such as H2O, of symmetry species Ai - the matrix elements Hij = ipi, Ti. ipj) as well as the overlap integrals Sij = (tpi, ipj) will be equal to zero unless the direct product representation r. contains Ai. This is the basis for the assertion that states of different symmetry do not mix. ... 

The matrix Rij,kl = Rik Rjl represents the effect of R on the orbital products in the same way Rjk represents the effect of R on the orbitals. One says that the orbital products also form a basis for a representation of the point group. The character (i.e., the trace) of the representation matrix Rij,kl appropriate to the orbital product basis is seen to equal the product of the characters of the matrix Rjk appropriate to the orbital basis %e2(R) = Xe(R)%e(R)i which is, of course, why the term "direct product" is used to describe this relationship. 

The problem with these equations is that they correspond to infinite different Hamiltonians so that the solutions for different electronic quantum numbers are incommensurate. To do away with these objections, use instead the complete set of functions rendering the kinetic energy operator Kn diagonal. The set, within normalization factors, is fk(Q) exp(ik Q) k is a vector in nuclear reciprocal space. Including the system in a box of volume V, the reciprocal vectors are discrete, ki, and the functions f (Q) = (1/Vv) exp(iki Q) form an orthonormal set with the completeness relation 8(Q-Q ) = Si fi(Q) fi(Q )- The direct product set ( )j(q)fki(Q) is complete. The matrix elements of eq. (8) reads ... 

The direct product of two matrices is quite different from the ordinary matrix product. First, let us consider how the indices of the various matrix elements are related, By comparing eqns (8-3.2) and (8-3.4) we have f, = ... 

The direct product of two matrices is quite different from the ordinary matrix product. 

As an example, consider the direct product of a 2x2 matrix A and a 3x3 matrix B. Table (9.135) gives for the indices... 

This direct-product matrix A B can be partitioned into four submatrices as follows ... 

Thus the set of functions Xyk, called the direct product of A, and Yk, also forms a basis for a representation of the group. Tlie zjUk are the elements of a matrix X of order (mn) x (mn). 

It should be clear from the associative property of matrix multiplication that what has been said regarding direct products of two representations can be extended to direct products of any number of representations. 

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