Task 4: Time-dependent restricted Hartree-Fock solver

Having constructed and solved the restricted Hartree-Fock ground state problem, we now set out to shine a monochromatic dipole laser on the system. This means appending a time-dependent operator to the Hamiltonian. The operator describes a semi-classical electric field in the dipole approximation in the length gauge. For a more thorough discussion of this operator check out chapters 2 - 2.4 in [JKP09]. The operator will in the one-dimensional case be described by

(16)\[\begin{align} \hat{h}_I(t) = -f(t) \hat{d}, \end{align}\]

where \(\hat{d} \equiv q\hat{x}\) is the dipole moment operator with \(q = -1\) the electron charge, and \(f(t)\) the time-dependent laser field. From [ZKBS04] we have \(f(t) = \mathcal{E}_0 \sin(\omega t)\), (note that \(\omega\) is not necessarily the same as the harmonic oscillator well frequency) which describes a monochromatic laser field that is always active.

Time-dependent Hartree-Fock

In the time-dependent Hartree-Fock method we use an ansatz for the many-body wave function as

(17)\[\begin{align} | \Psi(t) \rangle = | \Phi(t) \rangle = | \phi_1(t) \phi_2(t) \dots \phi_n(t) \rangle, \end{align}\]

where the time-dependence is kept in the molecular orbitals.

Note

Add derivation of the time-dependent Hartree-Fock equations.

The time-evolution of the molecular orbitals is described by the time-dependent Hartree-Fock equation

(18)\[\begin{align} i \frac{\text{d}}{\text{d} t} | \phi_p(t) \rangle = \hat{f}(t) | \phi_p(t) \rangle, \end{align}\]

where \(\hat{f}(t)\) is the time-dependent Fock operator and we have set \(\hbar = 1\). We have chosen to expand the time-independent molecular orbitals in a known basis of atomic orbitals (the harmonic oscillator eigenfunctions). Here as well we choose to expand our time-dependent molecular orbitals in a time-independent basis of atomic orbitals, and let the time-evolution occur in the coefficients. That is,

(19)\[\begin{align} | \phi_p(t) \rangle = \sum_{\mu = 1}^{l} C_{\mu p}(t) | \psi_{\mu} \rangle, \end{align}\]

where \(\{\psi_{\mu}\}_{\mu = 1}^{l}\) is a time-independent atomic orbital basis which we assume to be orthonormal. Inserting this expansion into the time-dependent Hartree-Fock equations and left-projecting with \(\langle \psi_{\mu}|\) we get

(20)\[\begin{gather} i \frac{\text{d}}{\text{d} t} \sum_{\nu = 1}^{l} C_{\nu p}(t) | \psi_{\nu} \rangle = \hat{f}(t) \sum_{\nu = 1}^{l} C_{\nu p}(t) | \psi_{\nu} \rangle \\ \implies i \sum_{\nu = 1}^{l} \dot{C}_{\nu p}(t) \langle \psi_{\mu} | \psi_{\nu} \rangle = i \dot{C}_{\mu p}(t) = \langle \psi_{\mu} | \hat{f}(t) | \psi_{\nu} \rangle C_{\nu p}(t) = f_{\mu \nu}(t) C_{\nu p}(t), \end{gather}\]

where we now need to find the matrix elements of the time-dependent Fock operator in the atomic orbital basis. As the laser field interaction operator \(\hat{h}_I(t)\) is a one-body operator, the time-dependent Fock operator will need to add this term.