(a) Use Mathcad to create an animation showing how contours and three-dimensional plots of the H+2 LCAO MOs ϕ1 and ϕ2 [Eqs. (13.57) and (13.58)] for a plane containing the nuclei change as R changes from 3.8 to 0.1 bohr. Proceed as follows. Define the function U1R, k2 by adding the inter-nuclear repulsion to (13.63). Include a parameter b in U, such that for b = 1 we get U for ϕ1 and b = -1 we get U for ϕ2. Specify the b value before defining U. Set R equal to 3.8 - FRAME/10, where the animation variable FRAME will later be defined to go from 0 to 37. To find the optimum k value at each R, use the Mathcad function root (f (k), k), which finds the k value that makes ϕ1k2 = 0 provided we enter an initial guess for k. Take ϕ(k) as the derivative of U1R, k2 with respect to k. Do not find this derivative yourself but use the d/dx facility in Mathcad to have Mathcad take the derivative. Define an initial value for k before setting k equal to the root function. [In some versions of Mathcad, the root function will fail to find the solution for certain initial values of k. Use trial-and-error to find k values that work, and if a single initial k value does not work at all R values, use the if function (or nested if functions) to specify various k values at various R values. Different k values may be needed for the bonding and anti-bonding MOs.] To make the plot, define xi and zi to vary from -2.5 to 2.5 bohrs with increments of 1/6 bohr. Then use (13.57) and (13.58) to define phi(x, z) as the MO's value in the xz plane. Define the array (matrix) M by defining Mi j as phi(xi, zi). Create a contour plot, entering -2.5 and 2.5 as the limits for the axes. Also create a three-dimensional-surface plot taking appropriate values for the limits of y. Then create the animation.
(b) By adding statements to the worksheet of part (a), use the solve block facility to find the predicted Re for ϕ1 and the optimum k at Re. The conditions to be satisfied are that the derivative of U with respect to R must be zero and the derivative of U with respect to k must be zero. Also include conditions that R and k must be positive.