Sheng Der Chao

In previous studies we have established a new basis‐set expansion method, called the kinetic energy partition (KEP) method, for solving general quantum eigenvalue problems, and applied it to several one‐body or two‐body systems, such as the dihydrogen cation and helium‐like atoms. Extending this method to larger atomic and molecular systems, although straightforward, confronts additional technical problems. Some of the difficulties are associated with the Coulomb interactions, so here we study several model atoms, such as the harmonium (Hookium), the Dirackium, and the Moshinsky atoms, in which a partial or full substitution of Coulomb interactions among charges with model potentials yields a simplifying solution scheme. In this way we can clearly identify the main technical issues associated with the KEP method, with the hope to point toward a streamlined solution procedure. Our results show that with a variety of complexity of interaction patterns, the calculation procedure is still well under control and the resulting errors are all within the required accuracy level. The purpose of this paper is, therefore, to advocate the utility of the KEP method in front‐line researches in quantum chemistry problems.

Based on the idea of adiabatic symmetry, we present a novel basis set expansion method—the kinetic energy partition (KEP) method—for solving quantum eigenvalue problems. Broken symmetry is responsible for quantum entanglement in many-body systems via parametric non-adiabatic corrections. Starting from simple one-particle-in-one-dimension problems, we gradually increase the complexity in the number of particles and the interaction patterns. Our goal in the mini-review is to advocate for the utility of the KEP method in front-line research, in particular for research beginners in quantum many-body problems.

Accurate energy data from noncovalent interactions are essential for constructing force fields for molecular dynamics simulations of bio-macromolecular systems. There are two important practical issues in the construction of a reliable force field with the hope of balancing the desired chemical accuracy and working efficiency. One is to determine a suitable quantum chemistry level of theory for calculating interaction energies. The other is to use a suitable continuous energy function to model the quantum chemical energy data. For the first issue, we have recently calculated the intermolecular interaction energies using the SAPT0 level of theory, and we have systematically organized these energies into the ab initio SOFG-31 (homodimer) and SOFG-31-heterodimer datasets. In this work, we re-calculate these interaction energies by using the more advanced SAPT2 level of theory with a wider series of basis sets. Our purpose is to determine the SAPT level of theory proper for interaction energies with respect to the CCSD(T)/CBS benchmark chemical accuracy. Next, to utilize these energy datasets, we employ one of the well-developed machine learning techniques, called the CLIFF scheme, to construct a general-purpose force field for biomolecular dynamics simulations. Here we use the SOFG-31 dataset and the SOFG-31-heterodimer dataset as the training and test sets, respectively. Our results demonstrate that using the CLIFF scheme can reproduce a diverse range of dimeric interaction energy patterns with only a small training set. The overall errors for each SAPT energy component, as well as the SAPT total energy, are all well below the desired chemical accuracy of ~1 kcal/mol.

Accurate determination of intermolecular non-covalent-bonded or non-bonded interactions is the key to potentially useful molecular dynamics simulations of polymer systems. However, it is challenging to balance both the accuracy and computational cost in force field modelling. One of the main difficulties is properly representing the calculated energy data as a continuous force function. In this paper, we employ well-developed machine learning techniques to construct a general purpose intermolecular non-bonded interaction force field for organic polymers. The original ab initio dataset SOFG-31 was calculated by us and has been well documented, and here we use it as our training set. The CLIFF kernel type machine learning scheme is used for predicting the interaction energies of heterodimers selected from the SOFG-31 dataset. Our test results show that the overall errors are well below the chemical accuracy of about 1 kcal/mol, thus demonstrating the promising feasibility of machine learning techniques in force field modelling.

We propose an extension of the Wentzel-Kramers-Brillouin (WKB) approximation for solving the Schr\"odinger equation. A set of coupled differential equations has been obtained by considering an ansatz of wave function with two auxiliary conditions on gauging the first and the second derivatives of the wave function, respectivey. It is shown that an alternating perturbation method can be used to decouple this set of equations, yielding the well known Bremmer series. We derive an analytical scheme to approximate the wave function which consists of the usual WKB approximation as the zeroth order solution with extra correction terms up to the first order perturbation. Therefore, we call this alternative methodology the alternating WKB (a-WKB) approximation. We apply the a-WKB method to solve the eigenvalue problem of a harmonic oscillator and the scattering problem of a repulsive inverse-square (centrifugal) potential and demonstrate its supremacy over the usual WKB approximation.

We extend our previous quantum chemistrycalculations of interaction energies for 31 homodimers of smallorganic functional groups (the SOFG-31 data set) by including239 heterodimers with monomers selected within the SOFG-31data set, thus resulting in the SOFG-31+239 data set. The minimum-level theoretical scheme contains (1) the basis set superposition errorcorrected supermolecule (BSSE-SM) approach for intermolecularinteractions; (2) the second-order Moller-Plesset perturbationtheory (MP2) with the Dunning's aug-cc-pVXZ (X = D, T, Q)basis sets for the geometry optimization and correlation energycalculations; and (3) the single-point energy calculations with thecoupled cluster with single, double, and perturbative triple excitationsmethod at the complete basis set limit [CCSD(T)/CBS] using thewell-tested extrapolation methods for the MP2 energy calibrations.In addition, we have performed a parallel series of energy decomposition calculations based on the symmetry adapted perturbationtheory (SAPT) in order to gain chemical insights. That the above procedure cannot be further reduced has been proven to be verycrucial for constructing reliable data sets of interaction energies. The calculated CCSD(T)/CBS interaction energy data can serve asa benchmark for testing or training less accurate but more efficient calculation methods, such as the electronic density functionaltheory. As an application, we employ a segmental SAPT model previously developed for the SOFG-31 data set to predict bindingenergies of large heterodimer complexes. These model energy"quanta"can be used in coarse-grained molecular dynamicssimulations by avoiding large-scale calculations

We have developed a general method, dubbed the split beam method, to solve Euler–Bernoulli equations for cantilever beams under multiple loading conditions. This kind of problem is, in general, a difficult inhomogeneous eigenvalue problem. The new idea is to split the original beam into two (or more) effective beams, each of which corresponds to one specific load and bears its own Young’s modulus. The mode shape of the original beam can be obtained by linearly superposing those of the effective beams. We apply the split beam method to simulating mechanical responses of an atomic force microscope probe in the “dynamical” operation mode, under which there are a stabilizing force at the positioner and a point-contact force at the tip. Compared with traditional analytical or numerical methods, the split beam method uses only a few number of basis functions from each effective beam, so a very fast convergence rate is observed in solving both the resonance frequencies and the mode shapes at the same time. Moreover, by examining the superposition coefficients, the split beam method provides a physical insight into the relative contribution of an individual load on the beam.