Systematic QM Region Construction in QM/MM Calculations Based on Uncertainty Quantification

While QM/MM studies of enzymatic reactions are widely used in computational chemistry, the results of such studies are subject to numerous sources of uncertainty, and the effect of different choices by the simulation scientist that are required when setting up QM/MM calculations is often unclear. In particular, the selection of the QM region is crucial for obtaining accurate and reliable results. Simply including amino acids by their distance to the active site is mostly not sufficient as necessary residues are missing or unimportant residues are included without evidence. Here, we take a first step towards quantifying uncertainties in QM/MM calculations by assessing the sensitivity of QM/MM reaction energies with respect to variations of the MM point charges. We show that such a point charge variation analysis (PCVA) can be employed to judge the accuracy of QM/MM reaction energies obtained with a selected QM region, and devise a protocol to systematically construct QM regions that minimize this uncertainty. We apply such a PCVA to the example of catechol \textit{O}-methyltransferase, and demonstrate that it provides a simple and reliable approach for the construction of the QM region. Our PCVA-based scheme is computationally efficient and requires only calculations for a system with a minimal QM region. Our work highlights the promise of applying methods of uncertainty quantification in computational chemistry.

Students’ understanding of the inverse square law in electrostatics

One problem of learning Electrostatics is that students often learn from their commonsense beliefs about electric force and electric field. This study investigated students’ conceptual understanding of finding electric force, electric field, and electric potential of point charge after learning an introductory physics course. We administered the Electrostatics Conceptual Evaluation Test to four lecture-based classes in high school. The first question was a comparison of the electric force from two-point charges at two different positions and the second question was a comparison of the electric field from a point charge at two different positions. The use of the inverse square law is required to find the electric force and the electric field at various positions. It was found that many students answered incorrectly. They described that the electric force and the electric field decrease whereas the distance increases by neglecting the inverse square law. This finding can be particularly used to suggest high school teachers to develop their effective strategy to support student learning.

New relationships between spherical and spheroidal harmonics and applications

<p>This thesis deals with solutions to Laplace's equation in 3D, finding new relationships between solutions, manipulating these to find new approaches to physical problems, and proposing a new class of solutions. We mainly consider spherical and prolate spheroidal geometry and their corresponding solutions - spherical and spheroidal solid harmonics. We first present new relationships between these, expressing for example spherical harmonics as a series of spheroidal harmonics. Similar relationships are known but we work with the spherical and spheroidal coordinate systems being offset from each other. We also propose a new class of solutions which we call logopoles which have many links with spherical and spheroidal harmonics, and are related to the potential created by simple finite line charge distributions. Through the logopoles we find another relationship between the spheroidal harmonics and the often discarded alternate spherical harmonics. Then we apply one of the new spherical-spheroidal harmonic relationships to problems involving a point charge/dipole outside a dielectric sphere. We find new solutions where the potential is expanded as a series of spheroidal harmonics instead of the standard spherical ones, and we show that the convergence is much faster. We also solve these problems with logopoles and the solutions converge even faster, although they are more complicated as they involve a combination of logopoles and spherical harmonics.</p>

New relationships between spherical and spheroidal harmonics and applications

<p>This thesis deals with solutions to Laplace's equation in 3D, finding new relationships between solutions, manipulating these to find new approaches to physical problems, and proposing a new class of solutions. We mainly consider spherical and prolate spheroidal geometry and their corresponding solutions - spherical and spheroidal solid harmonics. We first present new relationships between these, expressing for example spherical harmonics as a series of spheroidal harmonics. Similar relationships are known but we work with the spherical and spheroidal coordinate systems being offset from each other. We also propose a new class of solutions which we call logopoles which have many links with spherical and spheroidal harmonics, and are related to the potential created by simple finite line charge distributions. Through the logopoles we find another relationship between the spheroidal harmonics and the often discarded alternate spherical harmonics. Then we apply one of the new spherical-spheroidal harmonic relationships to problems involving a point charge/dipole outside a dielectric sphere. We find new solutions where the potential is expanded as a series of spheroidal harmonics instead of the standard spherical ones, and we show that the convergence is much faster. We also solve these problems with logopoles and the solutions converge even faster, although they are more complicated as they involve a combination of logopoles and spherical harmonics.</p>

Systematic QM Region Construction in QM/MM Calculations Based on Uncertainty Quantification

While QM/MM studies of enzymatic reactions are widely used in computational chemistry, the results of such studies are subject to numerous sources of uncertainty, and the effect of different choices by the simulation scientist that are required when setting up QM/MM calculations is often unclear. In particular, the selection of the QM region is crucial for obtaining accurate and reliable results. Simply including amino acids by their distance to the active site is mostly not sufficient as necessary residues are missing or unimportant residues are included without evidence. Here, we take a first step towards quantifying uncertainties in QM/MM calculations by assessing the sensitivity of QM/MM reaction energies with respect to variations of the MM point charges. We show that such a point charge variation analysis (PCVA) can be employed to judge the accuracy of QM/MM reaction energies obtained with a selected QM region, and devise a protocol to systematically construct QM regions that minimize this uncertainty. We apply such a PCVA to the example of catechol \textit{O}-methyltransferase, and demonstrate that it provides a simple and reliable approach for the construction of the QM region. Our PCVA-based scheme is computationally efficient and requires only calculations for a system with a minimal QM region. Our work highlights the promise of applying methods of uncertainty quantification in computational chemistry.

Dirac Spatial Search with Electric Fields

Electric Dirac quantum walks, which are a discretisation of the Dirac equation for a spinor coupled to an electric field, are revisited in order to perform spatial searches. The Coulomb electric field of a point charge is used as a non local oracle to perform a spatial search on a 2D grid of N points. As other quantum walks proposed for spatial search, these walks localise partially on the charge after a finite period of time. However, contrary to other walks, this localisation time scales as N for small values of N and tends asymptotically to a constant for larger Ns, thus offering a speed-up over conventional methods.

Models of Extended Electron with a Point Charge

The discovery of the electron insisted and inspired a number of theoretical predictions which are formulated in terms of models. A lot of models in wide variety were proposed for more than a century time. An attempt of a mini-review of them is taken here. But as a shorter version here only extended models with point charge are discussed instead of all.