Many-body theories

For an accurate description of multi-electron systems such as atoms, molecules, and solids, it is essential to be able to describe the many-body processes in the system. The most prominent many-body process is the electron-electron correlation effect which is mediated by the Coulomb repulsion between the electrons. Many-body theories have been developed since the early days of quantum mechanics in the 1930s to cope with the exponentially increasing demand when going to larger systems with more electrons [1]. Today, many sophisticated theories exists. The main three directions to solve the electronic structure of multi-electron systems are wavefunction-based theories, density-functional theory, and quantum Monte-Carlo calculations. Within the last 20 years, most of these theories have been extended from time-independent/static scenarios to time-dependent scenarios. Often, however, only properties that are accessible via linear response are calculated. Treating non-perturbative light-matter interactions is especially difficult where an external light field triggers complex electronic dynamics and an explicit time propagation of the \(N\)-body system is requiring. An top, non-perturbative light-matter interactions results in electronic dynamics that occupied a large phase space, meaning the spatial extent and the kinetic energy spectrum of the dynamics is exceptionally large. All of these aspects (many-body, time-dependence, and non-perturbative light-matter interaction) are difficult for themselves. But, especially, the combination of them poses a tremendous challenge for theory.

Hartree-Fock Method

The main idea about Hartree-Fock is to find an optimal set of orbitals that describe an (N)-electron state in terms of a Slater determinant (written in second quantization)
\begin{aligned}
\left| \Phi_0 \right> &= \prod_{i=1}^N c^\dagger_i \left|\textrm{vacuum} \right> = \left| \varphi_1,\ldots, \varphi_N \right>
\end{aligned}
such that the energy \(E_\textrm{HF}\))of this state \(\left|\Phi_0\right>\) is as close as possible to the exact ground state energy (lowest energy state in the system).
Consequently, \(\left|\Phi_0\right>\) and \(E_\textrm{HF}\) are called Hartree-Fock ground state and Hartree-Fock ground state energy.

Here, I focus on the wavefunction-based theories that generally start with the Hartree-Fock (HF) ground state as a reference states. Therefore, these theories are also called post-Hartree-Fock methods. First, I briefly explain the main idea behind Hartree-Fock before I discuss post-Hartree-Fock methods and, in particular, the method I am using to tackle many-body problems in the strong-field regime (namely time-dependent configuration interaction singles or short TDCIS). With the variational method

\begin{aligned}
\delta \left< \Phi_0 | \hat H | \Phi_0 \right> = \left< \delta \Phi_0 | \hat H | \Phi_0 \right> + c.c. \overset{!}{=} 0,
\end{aligned}

the optimal state \(\left| \Phi_0 \right>\) and therefore the optimal shape of the one-particle orbitals \(\left|\varphi_i\right>=c^\dagger_i \left|\textrm{vacuum}\right>\) is found that minimizes the energy of \(\left| \Phi_0 \right>\) within the trial configuration space. In the case of the Hartree-Fock method, this trial configuration space is the one spanned by the single \(N\)-body Slater determinates.

More to come …

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References:

[1] V. Fock, Z. Phys. 61, 126 (1930); J. C. Slater, Phys. Rev. 81, 385 (1951).