# Simen Kvaal: Rewriting the rules of quantum chemistry using the bivariational principle: The ERC Starting Grant project BIVAQUM [Departmental Colloquium]

Researcher Simen Kvaal from our own department is giving this talk.

Simen Kvaal

Starting time as usual at 15:00, with Coffee and Cake from 14:30.

## Abstract

### Rewriting the rules of quantum chemistry using the bivariational principle: The ERC Starting Grant project BIVAQUM

*Simen Kvaal - Department of Chemistry, UiO*

I was recently awarded an ERC Starting Grant for the proposal BIVAQUM — “Bivariational Approximations in Quantum Mechanics and Applications to Quantum Chemistry”. This talk is a presentation of my project: the background, what I hope to achieve, how, and why.

Already in 1743, the famous mathematician Leonhard Euler postulated that “all natural laws can be formulated as maximum or minimum principles”. From Fermat’s principle for light rays to the standard model of elementary particles, the laws of physics have beautiful and elegant formulations using such “variational principles.” Indeed, the Schrödinger equation for a molecule can be derived by such a principle which we quantum scientists simply call “the variational principle” (VP): the ground-state of the molecule minimizes the energy.

The VP is thus the cornerstone of quantum mechanics. Virtually all approximate schemes are derived from the VP in some way. From a computational scientist’s point of view, the power of the VP comes from the ability to make well-behaved approximations that can be systematically refined to obtain arbitrary precision, at least in principle.

The VP has one fundamental shortcoming: “the curse of dimensionality,” or exponential scaling of the computational procedure with respect to the number of constituent atoms and electrons. This is one of the major obstacles for large-scale computations.

There is a little known generalization of the VP usually called “the bivariational principle” (BIVP). This principle is able to overcome the curse of dimensions — indeed the very popular coupled-cluster (CC) method has a neat formulation using the BIVP. This formulation is considered unconventional, and has found little use. Its mathematical foundation is not even studied in detail. However, recently I developed a novel computational method using the BIVP, demonstrating a proof of concept: the derivation of CC using the BIVP is not a fluke.

Thus, by using the BIVP as the cornerstone rather than the standard VP, one may be able to create more efficient and accurate ab initio algorithms, expanding the horizons of quantum chemistry. This is the topic of the ERC project.