## Periodic Reporting for period 2 - BIVAQUM (Bivariational Approximations in Quantum Mechanics and Applications to Quantum Chemistry)

**Reporting period:**2016-10-01

**to**2018-03-31

## 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 is important since it 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 wave-function. 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, 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. In short, 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.

Why is the bivariational principle so important? Most computational methods today are based on the original variational formulation of the Schrödinger equation that has been around since Erwin Schrödinger formulated his equation in 1926. However, these methods tend to have serious shortcomings. For example, it is difficult to formulate a method that gives properties that scale correctly with the size of the molecule, and this introduces systematic errors. Having an alternative, mathematically sound way of generating computational schemes would be very attractive. Additionally, we have the curious fact that the so-called ""gold standard of quantum chemistry"", the coupled-cluster method, cannot be obtained by the conventional way, but it fits neatly into the unconventional picture using the bivariational principle. Importantly, the coupled-cluster method has fewer of the shortcomings of other methods. We remark that it is not conventional to treat the coupled-cluster method with the bivariational principle.

The research question in the project is therefore: Can the bivariational principle be used to devise other useful computational schemes for atoms and molecules?

Apart from finding a mathematical characterization of how the bivariational principle works, the BIVAQUM project also aims to develop novel computational schemes. There are in principle unlimited ways to do this, since the principle is very general. One method that is in development in the project is a so-called multireference method.

In quantum chemistry, the phenomenon of multireference systems is a great challenge, hitherto without a good universal solution. Most molecules, even huge ones, are well described by a simple starting point called the Hartree-Fock wavefunction. However, some molecules, even some very simple ones like the nitrogen gas molecule, refuse to be treated in this way. The concept of the bivariational principle has never been applied to such problems, a"

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 wave-function. 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, 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. In short, 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.

Why is the bivariational principle so important? Most computational methods today are based on the original variational formulation of the Schrödinger equation that has been around since Erwin Schrödinger formulated his equation in 1926. However, these methods tend to have serious shortcomings. For example, it is difficult to formulate a method that gives properties that scale correctly with the size of the molecule, and this introduces systematic errors. Having an alternative, mathematically sound way of generating computational schemes would be very attractive. Additionally, we have the curious fact that the so-called ""gold standard of quantum chemistry"", the coupled-cluster method, cannot be obtained by the conventional way, but it fits neatly into the unconventional picture using the bivariational principle. Importantly, the coupled-cluster method has fewer of the shortcomings of other methods. We remark that it is not conventional to treat the coupled-cluster method with the bivariational principle.

The research question in the project is therefore: Can the bivariational principle be used to devise other useful computational schemes for atoms and molecules?

Apart from finding a mathematical characterization of how the bivariational principle works, the BIVAQUM project also aims to develop novel computational schemes. There are in principle unlimited ways to do this, since the principle is very general. One method that is in development in the project is a so-called multireference method.

In quantum chemistry, the phenomenon of multireference systems is a great challenge, hitherto without a good universal solution. Most molecules, even huge ones, are well described by a simple starting point called the Hartree-Fock wavefunction. However, some molecules, even some very simple ones like the nitrogen gas molecule, refuse to be treated in this way. The concept of the bivariational principle has never been applied to such problems, a"

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

"There are three main objectives of the BIVAQUM project: Objective 1 deals with the mathematical characterization of the bivariational principle (BIVP). Objective 2 deals with the multireference problem in the context of the BIVP. Finally, Objective 3 implements and applies a novel computational scheme called the orbital-adaptive coupled-cluster method (OACC), based on the BIVP.

Objective 1:

As a preliminary study, the so-called extended coupled-cluster method (ECC) has been 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 is submitted for publication in the journal SIAM Journal of Numerical Analysis, and is currently in peer review.

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 works. A manuscript is in preparation, and will be submitted early 2018.

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 is studied. The tailored coupled-cluster method is able to deal with some aspects of the multireference problem. This study will probably yield sufficient knowledge to formulate the theorem. A manuscript is currently in preparation, summarizing our findings about the tailored coupled-cluster method.

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. We have formulated mathematical equations that must be programmed on a computer. However, some work remains before the results can be disseminated to the community: a computer program must be written and tested to measure the usefulness of the method, and ideally we would like to have a mathematical theorem that guarantees the latter.

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. A manuscript has been submitted to The Journal of Chemical Physics, and is currently in peer review.

"

Objective 1:

As a preliminary study, the so-called extended coupled-cluster method (ECC) has been 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 is submitted for publication in the journal SIAM Journal of Numerical Analysis, and is currently in peer review.

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 works. A manuscript is in preparation, and will be submitted early 2018.

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 is studied. The tailored coupled-cluster method is able to deal with some aspects of the multireference problem. This study will probably yield sufficient knowledge to formulate the theorem. A manuscript is currently in preparation, summarizing our findings about the tailored coupled-cluster method.

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. We have formulated mathematical equations that must be programmed on a computer. However, some work remains before the results can be disseminated to the community: a computer program must be written and tested to measure the usefulness of the method, and ideally we would like to have a mathematical theorem that guarantees the latter.

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. A manuscript has been submitted to The Journal of Chemical Physics, and is currently in peer review.

"

## 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. From the preliminary data obtained, it is expected that Objective 1 will be fully solved during the project lifetime, obtaining a complete mathematical characterization of the BIVP, applicable to both the complicated multireference situation and to the more common and simpler situation. This is a substantial improvement of the state of the art in mathematical analysis of quantum chemical methods.

In Objective 2, the proposed multireference-resolving method may represent a significant improvement over the state of the art. If successful, it represents a relatively simple method based on the BIVP with proven mathematical error bounds. In contrast, all multireference methods of coupled-cluster type up to this point has serious shortcomings and complications. There exists very little work in the way of mathematical justification for these, and no method has utilized the BIVP. It is expected that there will be significant progress on Objective 2 in the remainder of the project lifetime, pushing the state-of-the-art in multireference calculations.

In Objective 3, it is expected that in the project lifetime we will have a finished computer program for the OACC method. It is expected that we can perform computations on medium-sized molecules that rival those of more common techniques. It is also expected that we will have studied the method's performance, in particular the degree of multireference resolution the OACC method can provide.

The progress on the BIVAQUM project so far indicates that the BIVP will indeed obtain a solid mathematical footing, and significantly move the horizon of what types of calculations can be done in quantum chemistry. This is true, even if some of the work packages will meet roadblocks or only partially succeed.

In Objective 2, the proposed multireference-resolving method may represent a significant improvement over the state of the art. If successful, it represents a relatively simple method based on the BIVP with proven mathematical error bounds. In contrast, all multireference methods of coupled-cluster type up to this point has serious shortcomings and complications. There exists very little work in the way of mathematical justification for these, and no method has utilized the BIVP. It is expected that there will be significant progress on Objective 2 in the remainder of the project lifetime, pushing the state-of-the-art in multireference calculations.

In Objective 3, it is expected that in the project lifetime we will have a finished computer program for the OACC method. It is expected that we can perform computations on medium-sized molecules that rival those of more common techniques. It is also expected that we will have studied the method's performance, in particular the degree of multireference resolution the OACC method can provide.

The progress on the BIVAQUM project so far indicates that the BIVP will indeed obtain a solid mathematical footing, and significantly move the horizon of what types of calculations can be done in quantum chemistry. This is true, even if some of the work packages will meet roadblocks or only partially succeed.