In this chapter we introduce the main ideas of the many-body theory of molecular systems. This theory is used to compute geometries and the electronic structure, including spectra, of molecules. We will firstly introduce the part of many-body theory that is relevant for molecules, the theory of many-electron systems, one of the most important approximations in the theory and one of the first approaches to solving the many-electron problem. Thereafter we will discuss three important, more modern, approaches for solving parts of the problem. The first is Density Functional Theory (DFT) a method that is good in predicting ground state properties of materials. The second is known as $GW$-BSE which deals well with excited states and the third method is time-dependent DFT (TDDFT), which is also used for excited state calculations.

Many-Electron Systems

From a chemistry perspective all materials are surprisingly similar on the nanoscale, they consist of only two different types of particles (bodies), the atomic nuclei $l$ of mass $M_l$, charge $Z_l$ and at position $\mathbf{R}_l$ and electrons $i$ with mass $m$, charge $-e$, spin $\sigma_i$ and position $\mathbf{r}_i$. Furthermore there is predominantly only one interaction between the particles namely the electrostatic or Coulomb interaction. The potential corresponding to this interaction for a particle with unit charge at the origin is

where we assumed atomic units, i.e. $m_e = e = \hbar = 4 \pi \epsilon_0 = 1$. The state of the system is described by a wavefunction $\Psi(x,t)$, where $x$ contains all time-independent variables (i.e. $x = \{\{\mathbf{r}_i, \sigma_i\},\ \{Z_l,R_l\}\}$) and $t$ represents time. The dynamics of the system are governed by the system’s Hamiltonian $\hat{H}_\text{sys}$ via the Schrödinger equation

with $\hat{H}_\text{sys}$ given by

where we introduced the momentum operators $P_l = -i\nabla_{R_l}$ and $p_i = -i\nabla_{r_i}$. In equation (3) we have from left to right, the kinetic energy of the nuclei, the kinetic energy of the electrons, the nucleus-nucleus interactions, the nucleus-electron interactions and finally the electron-electron interactions. Note that the Hamiltonian (3) does not depend on time. We therefore split the wavefunction in a product of spatial and temporal terms, i.e. $\Psi(x,t) = \Psi(x)f(t)$. This allows us to apply separation of variables

and since the left side only depends on $t$ and the right only on $x$, both sides must be equal to a constant. This constant is the energy $E$ of the system, solving both parts separately gives the time-independent Schrödinger equation

and the time evolution of the system

Since the Hamiltonian does not depend on the spin of the electrons, we will ignore the spin of the electrons for now and reintroduce it when it is necessary (i.e. $\Psi = \Psi(\{\mathbf{r}_i\},\ \{Z_l,R_l\})$).

In principle all properties (geometric structure, spectra etc.) of the system can be computed from equation (5). In practice however we run into problems quickly. Due to all the interacting terms, equation (5) forms a system of coupled differential equations that cannot be split into smaller systems and the number of variables in the system is large. For example, a single benzene molecule, see Figure [benzene]{reference-type=“ref” reference=“benzene”}, with 12 nuclei and 42 electrons already leads to a problem of 162 spatial variables. Therefore solving equation (5) for any reasonable molecule is impossible no matter what numerical methods are used.