Bivariational Approximations in Quantum Mechanics and Applications to Quantum Chemistry

Project Information

BIVAQUM

Grant agreement ID: 639508

DOI

10.3030/639508

Project closed

EC signature date 19 February 2015

Start date 1 April 2015

End date 31 May 2021

Funded under

• EXCELLENT SCIENCE - European Research Council (ERC)

Total cost

€ 1 499 572,00

EU contribution

€ 1 499 572,00

Coordinated by

UNIVERSITETET I OSLO
Norway

Periodic Reporting for period 4 - BIVAQUM (Bivariational Approximations in Quantum Mechanics and Applications to Quantum Chemistry)

Reporting period: 2019-10-01 to 2021-05-31

Summary of the context and overall objectives of the project

The world around us is built from atoms and molecules. In the field of quantum chemistry, the properties of these are studied by solving the Schrödinger equation – the fundamental law of quantum mechanics. Quantum chemistry can guide, and in some cases completely replace, costly and dangerous experiments. Additionally, it increases our understanding of how matter is structured and interacts on a fundamental level. This is the setting of the BIVAQUM project, a interdisciplinary project in the junction of mathematics and quantum chemistry.

In quantum mechanics, matter has both particle and wave properties, and it is the wave aspect which is responsible for most chemical phenomena. The fundamental problem quantum chemists are faced with is the extremely complicated nature of the so-called wavefunction. Since it describes chemical phenomena, it is of utmost importance. On the other hand, it is so complicated that one needs to resort to approximate calculations on computers. Quantum chemists strive to develop the fastest and most accurate computational techniques in order to be able to study more complicated molecules and phenomena.

The ERC Starting Grant project BIVAQUM explores an unconventional way of expressing the Schrödinger equation using the so-called bivariational principle (BIVP), introduced independently around 1983 by the Finnish physicist Jouko Arponen and the Swedish quantum chemist Per-Olov Löwdin. The formulation attracted little attention in the following decades. The bivariational principle is a reformulation of quantum mechanics as an energy optimization problem: The wavefunction of the electrons is the one that optimizes a certain energy function. (This is similar to soap bubbles taking the shape that minimizes its surface area.) However, the principle has hitherto lacked a mathematical sound footing. The latter is important when one constructs approximate computational methods to be programmed on a computer, and is one of the aims of BIVAQUM.

The research question in the project is therefore: Can we understand the BIVP mathematically? Can the BIVP be used to devise other useful computational schemes for atoms and molecules?

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

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.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

In Objective 1, the mathematical theorem obtained represents a step forward from the state of the art: the theorem on the BIVP allows for a unified view of a wide range of methods, providing both conditions for when these will provide accurate results and a measure for the accuracy. For methods like the explicitly correlated coupled-cluster method, such an analysis has not been available up to this point. This is a substantial improvement of the state of the art in mathematical analysis of quantum chemical methods.

In Objective 2, the analysis of the tailored coupled-cluster method and the development of the bivar-MRCC method both represent significant improvements of the state of the art. In fact, the analysis of the tailored coupled-cluster method is the first mathematical error analysis of any multireference coupled-cluster method in the literature, and the bivar-MRCC method is unique in that it is as accurate in practical situations as established methods, while avoiding some problems that these methods have.

The study of time-dependent coupled-cluster theory has led to a systematic way to interpret complex bivariational wavefunctions, which is a first in the literature. We expect that this will make time-dependent coupled-cluster calculations an increasingly important tool in the future.

The development of density-functional theory for systems with magnetic fields is of great importance to many subfields of chemistry and physics. The theoretical foundation provided by our team is expected to have great impact on the community.

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