Background and problem

In quantum mechanics, any system can be described by the time-independent Schrödinger equation. However, this eigenvalue problem is only exactly solvable for a few very specific systems like the hydrogen atom or the quantum harmonic oscillator. This is due to the exponential scaling of the Hilbert space with the number of particles. Consequently, one has to rely on approximate methods.

A great variety of approximations has been developed each with their own strengths and weaknesses. Most methods start from a particular wave function ansatz and determine the parameters through variation or pertubation. A completely different approach would be to start from the 2-electron reduced density matrix (2-RDM) instead of the wave function. It was found that the expectation value of any one- or two-particle operator can be calculated from the 2-RDM. Moreover, the energy of a quantumchemical system can be expressed as a linear function of the 2-RDM. Compared to the wave function the 2-RDM is much more compact but it still contains most of the relevant information. Thus the ground state energy can be determined by variational optimization of the 2-RDM, or in short the v2DM method. Although one gets rid of the exponential scaling of the wave function, now we will have to deal with the so-called N-representability problem. The optimization of the 2-RDM has to be constrained to the class of 2RDMs that can be derived from a physical (ensemble of) N-electron wave function(s). The necessary and sufficient conditions for N-representability are know but they are of no practical use. Consequently, one has to use a set of necessary (but in general not sufficient) conditions to enforce that the resulting 2-RDM approximates a wave function derivalble 2-RDM as best as possible.

The resulting constrained optimization problem is known as a semidefinite programming problem. This is a well-known class of convex optimization problems for which many general purpose solvers already exist. Specific solvers for v2DM using a variety of necessary conditions have also been written. Unfortunately, on the whole, the v2DM technique is not competitive, both in speed and accuracy, with other standard methods.

Goal

In our research group, an alternative algorithm was recently proposed to solve a problem which closely resembles the variational optimization of the energy in v2DM theory: find the closest N-representable 2-RDM to a given 2-RDM. N-representable here means that it fulfills a certain set of sufficient N-representability conditions. This is an iterative procedure which is based on the positive semidefinite property of the matrices involved in the N-representability conditions.
The goal of this thesis is to implement this alternative algorithm in the framework of variational 2-RDM theory and compare the speed and accuracy with the existing semidefinite programming algorithms. This thesis combines computational work with theory development.