## Final Report Summary - VASPT2 (VASPT2: a method for targeted quantum dynamics of hydrogen transfer reactions)

The VASPT2 project was motivated by the importance of hydrogen/proton transfer reactions and by the fact that the reliable descriptions of them are still challenging. Hydrogen/proton transfer reactions are one of the most fundamental reactions in chemistry. They often appears in organic and inorganic chemical reactions, which includes enzymatic reactions, and play crucial roles. Indeed the rate determining steps of some enzymatic reactions are considered to be hydrogen/proton transfer reactions. In spite of their importance, it is difficult to observe hydrogen transfer reactions directly by experiments since they are ultra fast phenomena and the systems such as enzymes are too large to obtain fine spectra. Computational simulations, on the other hand, can provide an alternative way to study such reactions; they allow us to analyse chemical reactions in atomistic resolutions and femtosecond time scales. However, it is also not easy to perform a reliable computational simulation for a hydrogen/proton transfer reaction, because of proton's quantum character. Although there have been enormous efforts to develop methods to describe proton’s quantum dynamics so far, a method that is reliable yet applicable for a large system is still missing. The goal of this project was to develop such a method and to investigate the role of quantum effects in enzymes.

At first, we devised a second order multi-reference perturbation theory (VASPT2) for molecular dynamics using analogies with electronic structure theory. In general, methods to solve Schrödinger equations may be classified into two groups. One is a perturbational method and the other is a variational method. The former is applicable for weakly correlated large systems, whereas the latter can treat strongly correlated small systems where more than one degree of freedom strongly coupled to each other. Both of the two approaches are often not appropriate for hydrogen transfer reaction in a large system, as hydrogen transfer reactions can be considered as a strong correlation problem. In other words, a method that is scalable and can treat strong correlation is highly desirable. We developed such a new scheme based on the assumption that the number of strongly correlated degrees of freedoms is not increased with respect to the size of a system. The new scheme divides a system into active and bath regions, and an expensive variational method is only used for small-size active regions. The rest of the system and the coupling between two regions are treated by a mean-field approach and a successive second-order perturbation theory. The resulting VASPT2 method corresponds to CASPT2 (to be more precise MRMP2) in electronic structure theory. VASPT2 was trialled on formic acid, as a prototype of a system with strong and weak correlation and the estimated fundamental bands were matched with experimental values within the errors of electronic structure calculations (RMSD 9 wavenumbers). Work has also been undertaken to combine the VASPT2 approach with the reaction path Hamiltonian, and results from preliminary calculations on malonadehyde as an important prototype for tunnelling in intermolecular hydrogen transfer are expected in the near future.

We also developed a new procedure to generate semi-global potential energy surfaces to describe hydrogen transfer reactions. The reliability of the description of quantum dynamics depends not only on the model of a wavefunction but also on a model and accuracy of potential energy surface. Thanks to the recent advancement of electronic structure theory, electronic energy at a single point geometry can be computed with a sufficiently small error even for a sizable molecule. Nonetheless it is time demanding and impractical to compute potential energies every time directly from electronic structure calculations. In most cases it is, therefore, essential to construct a model potential or force field to do a molecular dynamics simulation on computers. Our method employed a new linear regression approach to fit model surface using the least absolute shrinkage and selection operator (LASSO) method, which suppresses overfitting problems and has no unphysical holes. For formic acid, the potential generated via this procedure showed only less than 0.25 % cumulative RMS errors from electronic structure calculations. In order to check the quality of this scheme we constructed a full dimensional potential for malonaldehyde and are now computing the zero point tunneling splitting using fixed-node diffusion Monte Carlo. As a preliminary result, we obtained 21.0 wavenumbers for the splittings, which agreed excellently with the experimental value of 21.6 wavenumbers. These results will be submitted for publication soon.

Through the project, we proposed new frameworks for vibrational wavefunctions and potential energy surface that opened the way to computationally describe hydrogen transfer reactions in a reliable fashion. They were implemented into a new program suite for quantum dynamics ‘DYNAMOL.' We have been developing DYNAMOL during the project, and have implemented lots of functionalities from finding a reaction path to computing vibrational wavefunctions with several Hamiltonians. Although the development is still in the alpha stage, it will be beneficial tools for chemists in order to do research on hydrogen/proton transfer reactions. As to deepen our knowledge on hydrogen transfer reactions is vital to design more efficient and useful catalysis, the potential impact of this project is high. Additionally, the products of this project would contribute to providing a conclusive answer to an open question in (bio-) chemistry "How quantum effects are important for enzymatic reactions?" For example, toward this end, we are currently using DYNAMOL to study quantum effects on the hydrogen transfer reaction of Glutamate Mutase with other research groups in the Centre for Computational Chemistry at Bristol.

At first, we devised a second order multi-reference perturbation theory (VASPT2) for molecular dynamics using analogies with electronic structure theory. In general, methods to solve Schrödinger equations may be classified into two groups. One is a perturbational method and the other is a variational method. The former is applicable for weakly correlated large systems, whereas the latter can treat strongly correlated small systems where more than one degree of freedom strongly coupled to each other. Both of the two approaches are often not appropriate for hydrogen transfer reaction in a large system, as hydrogen transfer reactions can be considered as a strong correlation problem. In other words, a method that is scalable and can treat strong correlation is highly desirable. We developed such a new scheme based on the assumption that the number of strongly correlated degrees of freedoms is not increased with respect to the size of a system. The new scheme divides a system into active and bath regions, and an expensive variational method is only used for small-size active regions. The rest of the system and the coupling between two regions are treated by a mean-field approach and a successive second-order perturbation theory. The resulting VASPT2 method corresponds to CASPT2 (to be more precise MRMP2) in electronic structure theory. VASPT2 was trialled on formic acid, as a prototype of a system with strong and weak correlation and the estimated fundamental bands were matched with experimental values within the errors of electronic structure calculations (RMSD 9 wavenumbers). Work has also been undertaken to combine the VASPT2 approach with the reaction path Hamiltonian, and results from preliminary calculations on malonadehyde as an important prototype for tunnelling in intermolecular hydrogen transfer are expected in the near future.

We also developed a new procedure to generate semi-global potential energy surfaces to describe hydrogen transfer reactions. The reliability of the description of quantum dynamics depends not only on the model of a wavefunction but also on a model and accuracy of potential energy surface. Thanks to the recent advancement of electronic structure theory, electronic energy at a single point geometry can be computed with a sufficiently small error even for a sizable molecule. Nonetheless it is time demanding and impractical to compute potential energies every time directly from electronic structure calculations. In most cases it is, therefore, essential to construct a model potential or force field to do a molecular dynamics simulation on computers. Our method employed a new linear regression approach to fit model surface using the least absolute shrinkage and selection operator (LASSO) method, which suppresses overfitting problems and has no unphysical holes. For formic acid, the potential generated via this procedure showed only less than 0.25 % cumulative RMS errors from electronic structure calculations. In order to check the quality of this scheme we constructed a full dimensional potential for malonaldehyde and are now computing the zero point tunneling splitting using fixed-node diffusion Monte Carlo. As a preliminary result, we obtained 21.0 wavenumbers for the splittings, which agreed excellently with the experimental value of 21.6 wavenumbers. These results will be submitted for publication soon.

Through the project, we proposed new frameworks for vibrational wavefunctions and potential energy surface that opened the way to computationally describe hydrogen transfer reactions in a reliable fashion. They were implemented into a new program suite for quantum dynamics ‘DYNAMOL.' We have been developing DYNAMOL during the project, and have implemented lots of functionalities from finding a reaction path to computing vibrational wavefunctions with several Hamiltonians. Although the development is still in the alpha stage, it will be beneficial tools for chemists in order to do research on hydrogen/proton transfer reactions. As to deepen our knowledge on hydrogen transfer reactions is vital to design more efficient and useful catalysis, the potential impact of this project is high. Additionally, the products of this project would contribute to providing a conclusive answer to an open question in (bio-) chemistry "How quantum effects are important for enzymatic reactions?" For example, toward this end, we are currently using DYNAMOL to study quantum effects on the hydrogen transfer reaction of Glutamate Mutase with other research groups in the Centre for Computational Chemistry at Bristol.