Most of our current understanding of the structure and dynamics of matter on the atomic scale is based on quantum mechanics. However, a straightforward solution of the interacting Schrödinger equation for complex quantum systems, such as biomolecules, is not possible in practice, since our current computers are not able to handle the enormous amount of data and the exponential scaling of such an exact solution. Many-body theory tries to circumvent this problem by using a closed set of equations for reduced quantities, which do not involve the explicit solution of the many-body Schrödinger equation and in which the many-body correlations can be approximated efficiently. Pursuits in this direction have led to various approaches such as density-functional theory (DFT), reduced density-matrix-functional theory (RDMFT) and and many-body perturbation theory (MBPT). These approaches differ in the complexity of the reduced quantity, i.e. the functional variable, which is used to calculate the various observables of interest. In the theory department we investigate and advance the foundations of these three closely connected many-body approaches.

### Time-Dependent Density-Functional Theory

In ground-state DFT and its time-dependent formulation TDDFT only the electronic charge density of the quantum system has to be determined, which can be computed very efficiently. This makes calculations for complex quantum systems feasible. Hence (TD)DFT has become one of the major approaches to determine the properties and dynamics of realistic many-body systems. The main drawback of this exact reformulation of (time-dependent) quantum mechanics is that it usually employs an effective potential, the so-called Kohn–Sham potential, that is not known explicitly, and in practice often uncontrolled approximations have to be used. Therefore (TD)DFT simulations currently cannot be used in a black-box manner, but need the practitioner to have an already good understanding of the physical situation.

Besides the construction of more reliable approximations to the time-dependent Kohn–Sham potential, we are developing a hitherto missing, precise mathematical framework for TDDFT based on the local-force equation of many-body quantum mechanics.

### Conditional (TD)DFT and Non-Adiabatic Electron-Ion Dynamics

The correlated motion of electrons and ions is a challenging problem yet one that is increasingly topical due to the advent of experimental techniques that allow to visualize the “molecular movie”. Theoretical methods mostly rely on the Born-Huang expansion of the molecular wavefunction. Hence, the concepts of Born-Oppenheimer potential-energy surfaces (BOPES) and nonadiabatic couplings (NAC) arise naturally, suggesting the picture of a nuclear wavepacket evolving on many static BOPESs. This approach is, however, very expensive due to the computational costs associated to the calculation (and parametrization) of all BOPESs and NACs involved in the dynamics.

We work on an alternative approach to molecular dynamics based on the use of conditional wavefunctions. The exact electron-nuclear dynamics is described by means of an ensemble of single, time-dependent, potential energy surfaces (C-TDPESs) that drive each component of the total wavefunction. While keeping the theory at the full configuration level, this approach allows for the use of trajectory-based techniques to circumvent the calculation of the BOPESs and NACs, and allows to draw clear connections between different exact frameworks. We have investigated features of the C-TDPESs in the presence of strong nonadiabatic couplings and proposed a universal mechanism for the explanation of quantum nonadiabatic effects. The paradigm shift associated with the transition from the many static BOPESs to the single time-dependent potentials can open new avenues in the understanding of quantum dynamics. Furthermore, the combination of the CD approach with the inherent scalability of other techniques such as TDDFT could lead to a breakthrough in the efficiency of ab-initio molecular dynamics methods.

### Reduced Density-Matrix-Functional Theory

One of the main challenges for ground-state DFT is the description of so-called static correlations. These static correlations are most pronounced in system in which the ground-state wave function has more than one dominant Slater determinant, e.g., for stretched molecules. Reduced Density-Matrix-Functional Theory (RDMFT), which employs the one-particle-reduced density matrix as fundamental variable, is a very promising approach to efficiently describe strong static correlations.

While RDMFT can describe total energies accurately, quite surprisingly it does not directly provide an effective single-particle picture of the many-body state, such as the Kohn-Sham electrons in DFT or the quasi-particle concept of MBPT. Also the minimization of the ground-state energy functional is done directly and not via a self-consistent solution of an effective single-particle problem. Therefore our research investigates how a quasi-particle concept can be introduced in RDMFT. This will not only allow for a more intuitive approach to construct approximations to the exchange-correlation energy functional, but may also also lead to more efficient numerical schemes to obtain the ground-state energy.

### Many-Body Perturbation Theory

In MBPT one deals with approximations to the many-body Green’s functions; these objects describe how the system reacts when particles are removed or added. The advantage of this approach is that more accurate approximations can easily be constructed by including higher-order Feynman diagrams. However, the numerical costs quickly become prohibitive, and hence often a combination with density-functional methods is employed. Indeed, MBPT can be used to find more accurate approximations to the effective potentials of DFT and TDDFT, and Kohn-Sham calculations are often used as input for MBPT calculations. We therefore investigate the connection between density-functional and perturbation-theory methods, especially in the time-dependent case.

#### Further research lines pursued in the group

- Properties of Kohn-Sham potentials: memory and initial-state dependence
- Lattice density-functional theories
- Study of model systems on lattices or in reduced dimensions
- Current-density-functional theories

#### Software development

- The group is one of the main developers of the real-space real-time TDDFT code OCTOPUS.

#### Books and special journal issues on Foundations of TDDFT and many-body theory

- Review J. Phys. Condens. Matter on foundations of TDDFT : Vol. 27 (2015)
- Book Fundamentals of Time-Dependent Density Functional Theory; Springer Lecture Notes in Physics, Vol. 837 (2012)
- Book Time-Dependent Density Functional Theory; Springer Lecture Notes in Physics, Vol. 706 (2006)
- Special issue of EPJ B on GW : Vol. 10051 (2012)