Quantum Chemistry

There are currently two very popular approaches to the electronic structure (many-body) problem:

Density Functional Theory (DFT) maps the exact problem to an independent particle model. Although formally exact, DFT aims at an economical, approximate solution of the electronic structure problem in terms of a density matrix and an unknown correlation-exchange functional. DFT is applicable to ground states, only, possibly subject to additional constraints. Time-dependent (TD) DFT connects excited state properties to the ground state by a response to some perturbation. DFT based methods have certain systematic short-comings such as dispersion interaction and unreliable energy barriers; in part semi-empirical corrections, addition of Hartree-Fock exchange and more target-specific functionals may rectify these problems. Particular difficult benchmark cases are electronic structure problems with two or more electronic states of widely different character (e.g. ionic versus bi-radical) are present in a narrow energy range and their energetic order flips depending on small changes in the geometrical structure. So far, there is no systematic way to reduce or to predict the inherent error of a DFT calculation.

With respect to wave-function based methods, there is the distinction between single-reference and multi-reference methods, discriminating between a wavefunction qualitatively well described by a single n-particle function (e.g. Slater determinant) and those not. Single-reference methods such as Coupled Cluster offer the great advantage that they can be used as „black box“ schemes – the calculation contains few parameters that can be choosen easily.

Wavefunction-based multi-reference electronic structure methods, on the other hand rely on a partitioning of the electron correlation somewhat vaguely into a static and a dynamic contribution, which makes them more difficult to use. The static term accounts for (large) near-degeneracy effects while the dynamic term accounts for a vast number of mainly small contributions. There is a variety of methods starting from variational MRCISD over selected MRCI+PT2 involving different selection and correction schems, arbitrary order MRCI and MRCC, DMRG and ultimatively various Quantum MC schemes each having distinct advantages and disadvantages. Multi-reference methods come into play if an accurate potential energy landscape for multiple electronic states is required or the state of interest is of multi-reference nature. The multi-reference character can become even more important when considering electron correlation and relativistic effects simultaneously. Although scalar relativistic effects are not only relevant for the tightly bound core electrons but have side-effects on the valence electron distribution as well. In addition the spin-orbit coupling term is of crucial importance e.g. in single-atom magnets where both electron correlation and relativistic effects combine to a delicately balanced system.