In Quantum mechanics, we want to correctly describe the behaviour of an interacting N-particle system by means of solving the N-particle Schrödinger equation. Unfortunately, the dimension of the (Hilbert)space of the problem grows exponentially with the system size making exact solutions impossible.

One can avoid the exponential scaling of the many-body Hilbert space by replacing the full wave function with the low-dimensional two-particle density matrix (2DM). For any pair-wise interacting Hamiltonian the energy can be expressed exactly as a function of the 2DM. This means the 2DM can be directly determined by minimizing the energy. This naive approach, however, yields terrible results because one needs to take into account that the 2DM has to be derivable from a physical N-particle wave function. This is known as the N-representability problem. Over the years people have come up with a host of necessary N-representability conditions known as two-index, and the computationally more intensive three-index constraints, which are expressible as a linear matrix map of the 2DM which has to be positive semidefinite. The variational determination of the 2DM under these constraints can be formulated as a standard numerical optimization technique called semidefinite programming. Several groups have developed codes for the determination of the 2DM (called v2DM), and applied them to atoms, molecules and lattice system with some success. A bottleneck for the larger application of this method, however, is the computational cost associated with the semidefinite optimization program.

In recent years there has been an increased interest in solving the electronic structure problem in the doubly-occupied space (DOCI). The reason for this is the fact that a large part of the static correlation present in a molecule can be captured by this approximation. The remaining dynamical correlation can be recovered with multi-reference perturbation theory. When the theory of v2DM is formulated for doubly-occupied states it turns out that the 2DM and the N-representability constraints simplify significantly. As with the standard v2DM, DOCI-v2DM can be solved through a semidefinite programming algorithm. The computational complexity, however, decreases from O(M⁶)to O(M³)with Mthe size of the single-particle Hilbert space. We have recently derived new constraints on the three-particle density matrix (3DM), specific to DOCI states, and found that the semidefinite program emerging from it only scales as O(M⁴). This makes it possible for us to compare results from v3DM with v2DM including three-index conditions for realistic systems, which until now has been hindered by the huge computational complexity of v3DM. The 3DM which emerges from this approach is interesting in itself, since it is used in many multi-reference perturbation schemes in quantum chemistry. It would be interesting to see if the simplifications in the structure of the 3DM carry through on this level.

A major complication in DOCI calculations is the orbital dependency: as the orbital can only be double occupied, the choice of orbitals is crucial. This means that every DOCI solver needs an orbital optimizer. Finding the best orbitals is a very hard problem: it means finding the global minima in a very rough landscape. Many techniques exists, we currently use a local minimizer using Jacobi Rotations.

Goal This subject/challenge, should you chose to accept it, can lead to many different destinations. One can try new orbital optimizers for which several new ideas are currently investigated. The conditions on the 3DM can be implemented and applied to a variety of molecular and condensed matter systems. A third option is to derive the multi-reference perturbation for the DOCI density matrix.

Depending on your personal preferences for numerical or theoretical based work, we will find you a worthy challenge!