9780030839931, Chapter 17, Problem 2P

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Chapter 17, Problem 2P Problem Derivation of the Hartree-Fock Equations from the Variational Principle (a) Show that the expectation value of the Hamiltonian (17.2) in a state of the form (17.13) is given by (17.14). (b) Show that when applied to Eq. (17.14) the procedure described in Problem 1(b) now leads to the Hartree-Fock equations (17.15). Step-by-step solution Step of 4 (a) The one-electron Schrodinger equation involving the potential U(r) is 2m Here, mass of the electron is Planck's constant is h and h h For N electrons in the metal, the corresponding wave function of the above equation is For equation HY = (1) Here, Step of 4 The wave function can be found as (2) Here, is a set of N orthonormal one-electron wave functions. The above wave function violated the Pauli's principle. It requires the sign of when any two of its arguments are interchanged. Step of 4 This is a linear combination of product equation All other products are obtained from it by permutation of among themselves. If added together with weights +1 or it follows = The above anti-symmetrized product can be written as the determinant of an Nx N matrix: (3) It can be shown that if energy of the equation (1) is evaluated in a state of form equation (3), with orthonormal single electron wave function then (4) The last term of the above is negative and involves the product (r) Step of 4 (b) If equation (4) is minimized with respect to it leads to a generalization of the Hartree equations. It is given as 2m This equation is known as the Hartree-Fock equations.