Purely electronic Hamiltonian

Hel is the purely electronic Hamiltonian VNN is the nuclear repulsion. Since is constant for a given nuclear configuration, it can be omitted from (1.274) this gives the same wave functions pcl and simply reduces the energy values by the constant VNN. Thus we have... 

For the hydrogen molecule, with the two protons at a fixed distance = R, the purely electronic Hamiltonian is given by (13.2) with the first, second, and fifth terms... 

Figure 13.3 shows H. The nuclei are at a and b R is the intemuclear distance and Tf, are the distances from the electron to nuclei a and b. Since the nuclei are fixed, we have a one-particle problem whose purely electronic Hamiltonian is [Eq. (13.5)]... 

The hydrogen molecule is the simplest molecule containing an electron-pair bond. The purely electronic Hamiltonian (13.5) for H2 is... 

The proof of the Hohenberg-Kohn theorem is as follows. The ground-state electronic wave function ipQ of an n-electron molecule is an eigenfunction of the purely electronic Hamiltonian of Eq. (13.5), which, in atomic units, is... 

The first term is the electronic kinetic-energy operator the second and third terms are the attractions between the electron and the nuclei. In atomic units the purely electronic Hamiltonian for is... 

Without consulting the text, write down the complete nonrelativistic Hamiltonian operator for the H2 molecule. Then write down the purely electronic Hamiltonian operator for H2. 

The expression for the Hartree-Fock molecular electronic energy hf is given by the variation theorem as = D H i + FawI-D), where D is the Slater-determinant Hartree-Fock wave function, and the purely electronic Hamiltonian and the inter-nuclear repulsion are given by (13.5) and (13.6). Since doesn t involve electronic coordinates and D is normalized, we have D Vnn D) = Vnn D D) = The operator is the sum of one-electron operators /, and two electron operators gj/, we have gij, where (in atomic units)... 

Now we will evaluate the average energies of these two molecular orbitals for H2. Using the first wavefunction and assuming the purely electronic Hamiltonian where the nuclei are separated at some distance R ... 

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