I. The Schrodinger equation (in atomic units) of an electron moving in one-dimension under the influence of the potential —6(x) is
( 1 d2 2 dx2— 6(X)) = EW. Here 6(x) is the Dirac delta function. Use the variational principle with the trial function qtr = Ne-ax' to show that —tr-1 is an upper bound to the exact ground state energy (which is -0.5). You will need the following integral
j.co f(x)8(x)dx = f(0)
j co (2m)!X1/2 xThle -RX2dx — 22mm! am* 1/2
- Suppose the spin orbitals xi and x, are eigenfunctions of a one-electron operator It with eigenvalues c, and el. One can construct the Hartree product wave functions ,II2(xl, x2) = Xi(x1)Xi(x2) and Lir (x2, x2) = Xax2)Xi(x2) . The Slater-determinant wave function that satisfies the antisymmetry principle is
"(Xi, X2) = = Xi(X2)Xj(xi)). V2 v 2 (1) Show that W(xl, x2) is normalized. Note that the spin orbitals xi and x, are orthonormal, (x)x i(x)dx = du (2) Show that 4f2P(xt. x2). WAP (xt, x2) and IP(xi, x2) are all eigenfunctions of the independent-particle Hamiltonian H = h(x1) + h(x2) and they have the same eigenvalue et + ej.