This problem deals with the matrix solution of the linear-variation method when the basis functions are nonorthogonal. (a) If {(i} in ( = (ni=1 ci(i is not an orthonormal set, we take linear combinations of the functions {(i} to get a new set of functions {gm} that are orthonormal. We have gm = (k akm fk, m = 1, 2,((((, n, where the coefficients akm are constants and where (gj | gm ) = (jm. [One procedure to choose the akm coefficients is the Schmidt method (Section 7.2); another method is discussed in Prob. 8.58.] (a) In (gj ( gm) = (jm, substitute the summation expression for each g and show that the resulting equation is equivalent to the matrix equation A†SA = I, where I is a unit matrix, S is the overlap matrix with elements Sjk, and A is the matrix of coefficients (km. (b) Verify that the set of equations (8.53) can be written as Hc = WSc, where c is the column vector of coefficients c1, c2, ( ( ( ( cn. As we did in going from (8.79c) to (8.95), we introduce the index i to label the various eigenvalues and eigenvectors and we write Hc = WSc as Hc(i) = WiSc(i) for i = 1, 2, ( ( ( ( n. Verify that HC = SCW, where C and W are the matrices in (8.86). (c) Since AA-1 = I, we can write HC = SCW as HAA-1C = SAA-1CW. Multiply this last equation by A† on the left. Use the result of (a) to show that we get H(C( = C(W, where C( K A-1C and H( K A†HA. Comparison with (8.87) shows that H(C( = C(W is the eigenvalue equation for the H( matrix. The matrix procedure to solve the linear variation problem HC = SCW with nonorthogonal basis functions is then as follows: (1) Compute the matrix elements of H and S using the nonorthogonal basis. (2) Use the overlap integrals and a procedure such as the Schmidt method to find a matrix A whose elements akm transform the nonorthogonal functions {(i} to orthonormal functions {gi}. (3) Calculate H( using H( ( A†HA. (4) Find the eigenvalues Wi and the eigenvectors c((i) of the H( matrix. (5) Use C = AC( to compute the coefficient matrix C. The eigenvalues Wi found in step 4 and the coefficients found in step 5 are the desired energy estimates and variation-function coefficients.