Objective 1:
As a preliminary study, the so-called extended coupled-cluster method (ECC) was analyzed. The ECC method was the original context of the BIVP in Arponen's work, and is a computational method for atoms and molecules. Proving that the ECC method works was a step forwards in understanding how the BIVP must be approached from a mathematical angle. The work was published in SIAM Journal of Numerical Analysis in 2018.
As a next step, we extracted from the ECC analysis a mathematical theorem on the BIVP. This theorem is sufficiently powerful to prove that many computational methods in use today actually work. A journal article is in preparation, detailing the mathematical analysis of the BIVP.
The next major step in Objective 1 is to find a mathematical theorem for the multireference situation, which is lacking in the previous theorem. In order to achieve this, the so-called tailored coupled-cluster method has been studied. The tailored coupled-cluster method is able to deal with some aspects of the multireference problem, and the study was published in SIAM Journal of Numerical Analysis in 2019.
Objective 2:
The real power of the BIVP lies in the potential for addressing the multireference problem in quantum chemistry. We have devised a computational method based on the BIVP that may deal with this situation, the bivar-MRCC method. This method has been thoroughly tested, and we believe it may become a useful tool for quantum chemists once an efficient computer implementation has been done. The method was published in the Journal of Chemical Physics in 2020.
Objective 3:
The orbital-adaptive coupled-cluster method is a variation on the aforementioned "gold standard of quantum chemistry", improving its flexibility at moderate cost. It is hoped that the OACC method can be a viable computational method that also can deal with some multireference cases. So far, we have derived equations that must be programmed on a computer, and have begun the process of writing such a program.
As part of the mathematical study of OACC, a simplified version of the method called the non-orthogonal orbital-optimized coupled-cluster method has been thoroughly analyzed. It was found that, indeed, the simplified method can approach the exact answer if enough computing resources are available. This study was published in The Journal of Chemical Physics in 2018.
Spin-off research:
A feature of the BIVP is that when applied to complex Hamiltonians, such as those arising from interactions with magnetic fields, one obtains complex valued wavefunctions and physical predictions for methods such as the coupled-cluster method. These predictions are on one hand fundamentally unphysical, and thus their interpretation is of utmost importance. The study of magnetic and other complex quantum mechanical systems led to two spin-offs: The study of time-dependent bivariational principles for dynamical theories, where every quantity is complex valued, and to density-functional theory for magnetic systems, whose fundamental formulation yields great insight into the structure of complex-valued wavefunctions, if density-functional theory is not formulated with the bivariational principle.
Several important publications were made in both spin-off topics. For example, in one publication in the Journal of Chemical physics in 2020, we studied the interpretation of time-dependent coupled-cluster theory, and in one publication in the Journal of Chemical Physics in 2018, we studied uniform magnetic fields in density-functional theory.
