Task 3: Expectation values

There are two main quantities that we wish to compute for the ground state in this project. They are the full many-body ground state energy \(E\) from the restricted Hartree-Fock method, and the particle density (also known as the one-body density or the electron density) \(\rho(x)\).

The restricted Hartree-Fock energy

We wish to compute the ground state energy from the Hartree-Fock ansatz, \(| \Phi \rangle = |\phi_1\dots\phi_N\rangle\). This is done by

(14)\[\begin{align} E &= \langle \Phi | \hat{H} | \Phi \rangle = \sum_{i = 1}^{n} \langle \phi_i | \hat{h} | \phi_i \rangle + \frac{1}{2} \sum_{i, j = 1}^{n} \langle \phi_i \phi_j | \hat{u} | \phi_i \phi_j \rangle_{AS}, \end{align}\]

where the molecular orbitals are the optimized Hartree-Fock orbitals. Inserting the basis expansion the energy can be found as a function of the coefficient matrices and the atomic orbital matrix elements.

The particle density

The particle density can be computed from the single-particle functions evaluated on a grid, and the one-body density matrix. We have already found the one-body density matrix when constructing the Fock matrix, it is given by \(D_{\nu \mu} = \sum_{i = 1}^{n} C^{*}_{\mu i} C_{\nu i}\). We can then compute the particle density from

(15)\[\begin{align} \rho(x) = \sum_{\mu, \nu = 1}^{l} \psi^{*}_{\mu}(x) D_{\nu \mu} \psi_{\nu}(x) \end{align}\]