Task 5: Time-dependent expectation values#

Time-dependent overlap#

We compute the time-dependent overlap between two Slater determinants (with orthonormal atomic orbital basis functions) by

(21)#\[\begin{align} P(t, t_0) &= \bigl|\langle \Phi(t) | \Phi(t_0) \rangle \bigr|^2 = \bigl|\det\bigl(C^{\dagger}_o(t) C_o(t_0)\bigr)\bigr|^2, \end{align}\]

where \(C_o(t)\) is the coefficient matrix of the time-evolved state with \(o\) denoting the occupied columns.

One-body expectation values#

Given a one-body operator \(\hat{A}\) we wish to compute the time-dependent expectation values.

Note

A one-body operator is an operator that acts on a single particle at a time. For example, the position operator \(\hat{x}\) is a one-body operator, whereas the Coulomb interaction \(\hat{u}\) is a two-body operator as it applies to the interaction between two particles at a time.

Having already found the matrix elements of \(\hat{A}\) in a static (time-independent) atomic basis \(\{\psi_{\mu}\}_{\mu = 1}^{l}\), we can compute the expectation value of this operator by

(22)#\[\begin{align} \langle A(t) \rangle &= \langle \psi_{\mu} | \hat{A} | \psi_{\nu} \rangle D_{\nu \mu}(t), \end{align}\]

where \(D_{\nu \mu}(t) = \sum_{i = 1}^{n} C^{*}_{\mu i}(t) C_{\nu i}(t)\) is the time-evolved Hartree-Fock density matrix. Alternatively, by transforming the one-body operator to the time-evolved molecular orbital basis \(\{\psi_p(t)\}_{p = 1}^{k}\) we can compute the expectation value from

(23)#\[\begin{align} \langle A(t) \rangle &= \langle \phi_{p}(t) | \hat{A} | \phi_q(t) \rangle \rho_{q p}, \end{align}\]

where \(\rho_{q p}\) is the one-body density matrix in the time-evolved Hartree-Fock basis. This matrix is given by

(24)#\[\begin{align} \boldsymbol{\rho} &= \begin{pmatrix} \boldsymbol{I}_{n \times n} & \boldsymbol{0}_{n \times (k - n)} \\ \boldsymbol{0}_{(k - n) \times n} & \boldsymbol{0}_{(k - n) \times (k - n)} \end{pmatrix}, \end{align}\]

that is, the matrix is the identity in the occupied block and zero elsehwere. Both ways of computing the expectation value give the same value.