Identifying Nonzero Matrix Elements

Integrals of the general type 5.2-4 occur frequently in quantum mechanical problems. They are often termed matrix elements, since they occur as such in the secular equations which commonly provide the best way of formulating the problem (see Chapters 7, 8, and 10 for examples of secular equations). In order to give the results just presented in Section 5.2 some concrete meaning, we shall discuss here the two commonest examples of the type of matrix element represented in 5.2-4. 

We thus have an explicit expression for an energy, which can be thought of as the energy of interaction between two states described by the wave functions Wi and y/r If the integral that occurs in the numerator of the left-hand side of this equation is in fact required to have a value identically equal to zero, it will be helpful to know this at the earliest possible stage of a calculation so that no computational effort will be wasted on it. This information may be obtained very simply from a knowledge of the irreducible representations to which the wave functions y/, and y/j belong. 

An energy integral f y/iHy/j dr may be nonzero only if y/, and y/t belong to the same irreducible representation of the molecular point group. 

Perhaps the second commonest case in which the simple question of whether or not a matrix element is required by symmetry considerations to vanish occurs in connection with selection rules for various types of transition from one stationary state of a system to another with the gain or loss of a quantum of energy. If the energy difference between the states is represented by , - , then radiation of frequency v will be either absorbed or emitted by the transition, if it is allowed, with r being required to satisfy the equation 

In general the intensity, /, of a transition from a state described by y/x to another described by y/, is given by an equation of the type 

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To evaluate the first contribution, the addressing rules of Section 11.8.2 are employed to identify the nonzero matrix elements Kp E Jp). For each nonzero element, the elements of with a fixed index Jp constitute a vector, the elements of which are addressed by K . This vector is then scaled by the nonzero element Kp E Jp) and added to the corresponding vector obtained by fixing the indices rs and Kp m Drs K Kfi- The same method is used for the second contribution in (11.8.16). This procedure, which uses elementary vector operations, is well suited to modem computers. The number of operations required to constmct the matrix D is equal to the number of nonzero matrix elements K Kp Ers JciJp), which is given by L of (11.7.11). 

Actually, only one matrix need be stored if the adjacency matrix is stored initially and thereafter multiplied by itself. Matrix elements are replaced by the resulting product elements as they are computed. The product matrix obtained in this manner for the fcth power may contain some nonzero elements which correspond to paths longer than k steps instead of strictly k step paths, but this will not affect the final matrix obtained corresponding to the nth power, since these paths would eventually be identified in any case. All of the modifications to the methods of Section II mentioned above simplify the calculations needed to obtain the reachability matrix. The procedure for identifying the maximal loops given in Section II remains the same. 

A new matrix M(1) is formed from M(0) in the same manner that M(0) was formed from the occurrence matrix. The column of M<0) containing the most nonzero elements is identified, and M(1> is made up of rows identical to each row of M(0), which contains a zero in column k and one final row, which is the Boolean union of the remaining rows in M(0). Figure 13b illustrates M(1). A record is kept of which rows of the original occurrence matrix have been combined to form each row of M(0), M(1), and so on. 

C REAOING. THE MATRIX OUTPUT SET IS ON THE MAIN DIAGONAL. AND THE ROWS C ARE IN THE FINAL PRECEDENCE ORDER. THE ELEMENTS IN A MATRIX ROW ARE CM THOSF VARIABLES ASSOCIATED WITH THE EQUATION GIVEN UNDER THE C COLUMN HEADED EONS. THE COLUMN HEADED VARS. IDENTIFIES THE OUTPUT c VARIABLE FOR THAT EQUATION. BY TRANSPOSING THE VECTOR OF NUMBERS C UNDER THE COLUMN VARS. AND PLACING IT ACROSS THE TQP OF THE MATRIX(X.X) C M YOU CAN IDENTIFY EACH OF THE VARIABLES BY NUMBER. EACH COLUMN WITH C M NONZERO ELEMENTS ABOVE THE MAIN DIAGONAL REPRESENTS AN ITERATED C m VARIABLE. 

Identification requires that the rank of each matrix be M-l =3. The second is obviously not identified. In (1), none of the three columns can be written as a linear combination of the other two, so it has rank 3. (Although the second and last columns have nonzero elements in the same positions, for the matrix to have short rank, we would require that the third column be a multiple of the second, since the first cannot appear in the linear combination which is to replicate the second column.) By the same logic, (3) and (4) are identified. 

These index assignments also help to identify the nonzero elements of the rather sparse macromolecular density matrix, based on the also rather sparse fragment density matrices. By looping over only the nonzero elements of each fragment density matrix P ((p(ATt)), the macromolecular density matrix P(J is assembled iteratively ... 

To simplify the representation, one needs to identify the smallest square submatrix pattern that occurs in the same place for all matrix representations of the symmetry operations. The numbers will be on the main diagonal of these smaller submatrices. The blocks may have nonzero off-diagonal elements or sometimes zeros for diagonal elements that doesn t matter. In the example above, a 3 X 3 section in the middle of each matrix, in the same position in each matrix, will serve. In the case of E, the block is outlined as follows ... 

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