Symmetry-adapted Basis Research Articles

How symmetry helps to improve the estimation of the hyperfine splitting of torsional levels due to tunneling. The case of the HSSSH molecule

In this study, we analyze a series of molecules belonging to the C2V(M) molecular symmetry group which are characterized by several conformers. The use of molecular symmetry at each stage of calculating the energy of stationary torsional states is demonstrated. In particular, the importance is shown of preliminary symmetrization of physical characteristics of the molecules obtained by quantum chemical calculations. For the first time, symmetry-adapted basis functions for the diagonal kinetic coefficients are presented, which for the analyzed molecules do not satisfy all symmetry operations of the C2V(M) group. Using the HSSSH molecule as an example, it is shown how the full or partial accounting for molecular symmetry influences the calculated values of ultra-small tunneling splittings of the ground torsional states of the trans- and cis-conformers. It has also been established that the Hamiltonian matrix is characterized by symmetry which, when taken into account, makes it possible to halve the time of calculation of its elements.

Automated Search Strategy for Novel Ordered Structures of Block Copolymers.

Block copolymers with different architectures can possibly generate innumerable stable or metastable structures and thus provide an irreplaceable platform for theoretically exploring novel structures. Self-consistent field theory (SCFT) is a powerful tool to predict the ordered structures of block copolymers; however, it is sensitively dependent on its initial condition. Here we propose to use multiple symmetry-adapted basis functions to generate the initial conditions of SCFT and then apply Bayesian optimization to search for ordered structures by navigating the coefficient space of these basis functions. Without any prior knowledge, our scheme can automatically recover hundreds of ordered structures for two simple block copolymers, including most of the common structures and complex Frank-Kasper structures, together with many novel structures. By applying the automated scheme to various block copolymers, a huge number of novel structures can be obtained to expand the structural library, which may create new opportunities for the scientific community.

Using nested tensor train contracted basis functions with group theoretical techniques to compute (ro)-vibrational spectra of molecules with non-Abelian groups.

In this paper, we use nested tensor-train contractions to compute vibrational and ro-vibrational energy levels of molecules with five and six atoms. At each step, we fully exploit symmetry by using symmetry adapted basis functions obtained from an irreducible tensor method. Contracted basis functions are determined by diagonalizing reduced dimensional Hamiltonian matrices. The size of matrices of eigenvectors, used to account for coupling between groups of coordinates, is reduced by discarding rows and columns. The size of the matrices that must be diagonalized is thus substantially reduced, making it possible to use direct eigensolvers, even for molecules with five and six atoms. The symmetry-adapted contracted vibrational basis functions have been used to compute J = 0 energy levels of the CH3CN (C3v) and J > 0 levels of CH4.

Fast evaluation of spherical harmonics with sphericart.

Spherical harmonics provide a smooth, orthogonal, and symmetry-adapted basis to expand functions on a sphere, and they are used routinely in physical and theoretical chemistry as well as in different fields of science and technology, from geology and atmospheric sciences to signal processing and computer graphics. More recently, they have become a key component of rotationally equivariant models in geometric machine learning, including applications to atomic-scale modeling of molecules and materials. We present an elegant and efficient algorithm for the evaluation of the real-valued spherical harmonics. Our construction features many of the desirable properties of existing schemes and allows us to compute Cartesian derivatives in a numerically stable and computationally efficient manner. To facilitate usage, we implement this algorithm in sphericart, a fast C++ library that also provides C bindings, a Python API, and a PyTorch implementation that includes a GPU kernel.

Symmetry-adapted modeling for molecules and crystals

We have developed a symmetry-adapted modeling procedure for molecules and crystals. By using the completeness of multipoles to express spatial and time-reversal parity-specific anisotropic distributions, we can generate systematically the complete symmetry-adapted multipole basis set to describe any of electronic degrees of freedom in isolated cluster systems and periodic crystals. The symmetry-adapted modeling is then achieved by expressing the Hamiltonian in terms of the linear combination of these bases belonging to the identity irreducible representation, and the model parameters (linear coefficients) in the Hamiltonian can be determined so as to reproduce the electronic structures given by the density-functional computation. We demonstrate our method for the modeling of graphene, and emphasize usefulness of the symmetry-adapted basis to analyze and predict physical phenomena and spontaneous symmetry breaking in a phase transition. The present method is complementary to de-facto standard Wannier tight-binding modeling, and it provides us with a fundamental basis to develop a symmetry-based analysis for materials science.

Ab initio symmetry-adapted emulator for studying emergent collectivity and clustering in nuclei

We discuss emulators from the ab initio symmetry-adapted no-core shell-model framework for studying the formation of alpha clustering and collective properties without effective charges. We present a new type of an emulator, one that utilizes the eigenvector continuation technique but is based on the use of symplectic symmetry considerations. This is achieved by using physically relevant degrees of freedom, namely, the symmetry-adapted basis, which exploits the almost perfect symplectic symmetry in nuclei. Specifically, we study excitation energies, point-proton root-mean-square radii, along with electric quadrupole moments and transitions for 6Li and 12C. We show that the set of parameterizations of the chiral potential used to train the emulators has no significant effect on predictions of dominant nuclear features, such as shape and the associated symplectic symmetry, along with cluster formation, but slightly varies details that affect collective quadrupole moments, asymptotic normalization coefficients, and alpha partial widths up to a factor of two. This makes these types of emulators important for further constraining the nuclear force for high-precision nuclear structure and reaction observables.

Time-reversal symmetry adaptation in relativistic density matrix renormalization group algorithm.

In the nonrelativistic Schrödinger equation, the total spin S and spin projection M are good quantum numbers. In contrast, spin symmetry is lost in the presence of spin-dependent interactions, such as spin-orbit couplings in relativistic Hamiltonians. Therefore, the relativistic density matrix renormalization group algorithm (R-DMRG) only employing particle number symmetry is much more expensive than nonrelativistic DMRG. In addition, artificial breaking of Kramers degeneracy can happen in the treatment of systems with an odd number of electrons. To overcome these issues, we propose time-reversal symmetry adaptation for R-DMRG. Since the time-reversal operator is antiunitary, this cannot be simply achieved in the usual way. We introduce a time-reversal symmetry-adapted renormalized basis and present strategies to maintain the structure of basis functions during the sweep optimization. With time-reversal symmetry adaptation, only half of the renormalized operators are needed, and the computational costs of Hamiltonian-wavefunction multiplication and renormalization are reduced by half. The present construction of the time-reversal symmetry-adapted basis also directly applies to other tensor network states without loops.

Generalization of microscopic multipoles and cross-correlated phenomena by their orderings

The generalization of the atomic-scale multipoles is discussed. By introducing the augmented multipoles defined in the hybrid orbitals or in the site/bond-cluster, any of electronic degrees of freedom can be expressed in accordance with the crystallographic point group. These multipoles are useful to describe the cross-correlated phenomena, band-structure deformation, and generation of effective spin–orbit coupling due to antiferromagnetic ordering in a systematic and comprehensive manner. Such a symmetry-adapted multipole basis set could be a promising descriptor for materials design and informatics.

Molecular frames for a symmetry-adapted rotational basis set

In molecular ro-vibrational calculations, it is desirable to build a symmetry-adapted basis set where both the rotational and vibrational functions are each separately symmetrised. Enabling rotational symmetrisation requires the use of body-fixed frames whose rotations under molecular symmetry operations are constant. The familiar Eckart and Sayvetz frames are examples of such frames, as are bisector frames for certain molecules. In this work, we introduce alternative frames for CH3Cl-type molecules and C2H6 that can be used for rotational symmetrisation. These are defined by the vibrational coordinates without the need for a reference geometry. We also provide one method to find such frames. Finally, we state a straightforward transformation of the s-vectors when changing frames which facilitates the evaluation of kinetic energy operators in these non-standard frames.

Rotation, Electric-Field Responses, and Absolute Enantioselection in Chiral Crystals.

The microscopic origin of chirality, possible electric-field induced static rotational lattice deformation, and rotation-field induced electric polarization are investigated. By building up a realistic tight-binding model for the elemental Te crystal in terms of a symmetry-adapted basis, we identify the microscopic origin of the chirality and essential couplings among polar and axial vectors with the same time-reversal properties. Based on this microscopic model, we elucidate quantitatively that an interband process, driven by nearest-neighbor spin-dependent imaginary hopping, is the key factor in the electric-field induced rotation and its inverse response. From the symmetry point of view, these couplings and responses are characteristic and common to any chiral material, leading to a possible experimental approach to achieve absolute enantioselection by simultaneously applied electric and rotation fields, or a magnetic field and electric current, and so on, as a conjugate field of the chirality.

Efficacy of the symmetry-adapted basis for ab initio nucleon-nucleus interactions for light- and intermediate-mass nuclei

We study the efficacy of a new ab initio framework that combines the symmetry-adapted (SA) no-core shell-model approach with the resonating group method (RGM) for unified descriptions of nuclear structure and reactions. We obtain ab initio neutron-nucleus interactions for 4He, 16O, and 20Ne targets, starting with realistic nucleon-nucleon potentials. We discuss the effect of increasing model space sizes and symmetry-based selections on the SA-RGM norm and direct potential kernels, as well as on phase shifts, which are the input to calculations of cross sections. We demonstrate the efficacy of the SA basis and its scalability with particle numbers and model space dimensions, with a view toward ab initio descriptions of nucleon scattering and capture reactions up through the medium-mass region.

Quantum spin solver near saturation: QS3

We develop a program package named QS3▪ based on the (thick-restart) Lanczos method for analyzing spin-1/2 XXZ-type quantum spin models on spatially uniform/non-uniform lattices near fully polarized states, which can be mapped to dilute hardcore Bose systems. All calculations in QS3, including eigenvalue problems, expectation values for one/two-point spin operators, and static/dynamical spin structure factors, are performed in the symmetry-adapted bases specified by the number N↓ of down spins and the wave number k associated with the translational symmetry without using the bit representation for specifying spin configurations. Because of these treatments, QS3 can support large-scale quantum systems containing more than 1000 sites with dilute N↓. We show the benchmark results of QS3 for the low-energy excitation dispersion of the isotropic Heisenberg model on the 10×10×10 cubic lattice, the static and dynamical spin structure factors of the isotropic Heisenberg model on the 10×10 square lattice, and the open-MP parallelization efficiency on the supercomputer (Ohtaka) based on AMD Epyc 7702 installed at the Institute for the Solid State Physics (ISSP). Theoretical backgrounds and the user interface of QS3 are also described. Program summaryProgram Title: QS3CPC Library link to program files:https://doi.org/10.17632/yc6jytvf8v.1Developer's repository link:https://github.com/QS-Cube/EDLicensing provisions: MITProgramming language: Fortran90External routines/libraries: BLAS, LAPACKNature of problem: Physical properties (such as total energy, magnetic moment, two-point spin correlation function, and dynamical structure factor)Solution method: Application software based on the full diagonalization method and the exact diagonalization method using the Lanczos and thick-restart Lanczos techniques for quantum spin S=1/2 models such as the XXZ model.Restrictions: Spin S=1/2 systems with U(1) symmetry.Unusual features: Massively large quantum spin systems with U(1) symmetry near the saturation field can be solved with or without considering translational symmetry, which is difficult to treat using standard exact diagonalization libraries with the bit representation for specifying spin configurations.

Lattice-symmetries: A package for working with quantum many-body bases

Exact diagonalization (ED) is one of the most reliable and established numerical methods of quantum many-body theory. The main limiting factor of the method is the exponential scaling of Hilbert space dimension with system size. Fortunately, by symmetry considerations the effective dimension can be reduced by multiple orders of magnitude. Here, we present lattice-symmetries, a package for working with such symmetry-adapted quantum many-body bases and operators. It supports bases for spin-1/2 particles with arbitrary user-defined symmetries and generic 1-, 2-, 3-, and 4-point operators. As an example application we discuss SpinED program which allows to easily diagonalize clusters of at least 42 sites on a single node thus making large-scale ED easily accessible to people with no background in numerical methods and computational physics.

Using nondirect product Wigner D basis functions and the symmetry-adapted Lanczos algorithm to compute the ro-vibrational spectrum of CH4-H2O.

By doing calculations on the methane-water van der Waals complex, we demonstrate that highly converged energy levels and wavefunctions can be obtained using Wigner D basis functions and the Symmetry-Adapted Lanczos (SAL) method. The Wigner D basis is a nondirect product basis and, therefore, efficient when the kinetic energy operator has accessible singularities. The SAL method makes it possible to exploit symmetry to label energy levels and reduce the cost of the calculation, without explicitly using symmetry-adapted basis functions. Line strengths are computed, and new bands are identified. In particular, we find unusually strong transitions between states associated with the isomers of the global minimum and the secondary minimum.

Symmetry-adapted decomposition of tensor operators and the visualization of coupled spin systems

We study the representation and visualization of finite-dimensional, coupled quantum systems. To establish a generalized Wigner representation, multi-spin operators are decomposed into a symmetry-adapted tensor basis and are mapped to multiple spherical plots that are each assembled from linear combinations of spherical harmonics. We explicitly determine the corresponding symmetry-adapted tensor basis for up to six coupled spins 1/2 (qubits) using a first step that relies on a Clebsch–Gordan decomposition and a second step which is implemented with two different approaches based on explicit projection operators and coefficients of fractional parentage. The approach based on explicit projection operators is currently only applicable for up to four spins 1/2. The resulting generalized Wigner representation is illustrated with various examples for the cases of four to six coupled spins 1/2. We also treat the case of two coupled spins with arbitrary spin numbers (qudits) not necessarily equal to 1/2 and highlight a quantum system of a spin 1/2 coupled to a spin 1 (qutrit). Our work offers a much more detailed understanding of the symmetries appearing in coupled quantum systems.

Ab initio construction of symmetry-adapted k·p Hamiltonians for the electronic structure of semiconductors

While $\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ Hamiltonians are frequently used for the description of electronic states in quantum nanostructures, a method is lacking to obtain them in their symmetrized form directly from ab initio band structure calculations of bulk material. We developed a method for obtaining the parameters and the symmetry-adapted form of the $\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ Hamiltonian from the output of an ab initio band structure calculation. The method consists of (i) evaluation of momentum matrix elements between the wave functions obtained from band structure calculation; (ii) identification of the unitary transformation that transforms these wave functions to the symmetry-adapted basis; (iii) transformation of the $\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ Hamiltonian to the symmetry-adapted basis. We illustrate the methodology by obtaining $\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ Hamiltonians that describe the band structure of zinc-blende CdSe and then we use the Hamiltonians obtained to calculate the electronic states in CdSe quantum wells. Excellent agreement between density functional theory and $\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ is obtained for the electronic structure, even for quite thin wells.

Benchmark calculations of electromagnetic sum rules with a symmetry-adapted basis and hyperspherical harmonics

We demonstrate the ability to calculate electromagnetic sum rules with the \textit{ab initio} symmetry-adapted no-core shell model. By implementing the Lanczos algorithm, we compute non-energy weighted, energy weighted, and inverse energy weighted sum rules for electric monopole, dipole, and quadrupole transitions in $^4$He using realistic interactions. We benchmark the results with the hyperspherical harmonics method and show agreement within $2\sigma$, where the uncertainties are estimated from the use of the many-body technique. We investigate the dependence of the results on three different interactions, including chiral potentials, and we report on the $^4$He electric dipole polarizability calculated in the SA-NCSM that reproduces the experimental data and earlier theoretical outcomes. We also detail a novel use of the Lawson procedure to remove the spurious center-of-mass contribution to the sum rules that arises from using laboratory-frame coordinates. We further show that this same technique can be applied in the Lorentz integral transform method, with a view toward studies of electromagnetic reactions for light through medium-mass nuclei.

Modal Tracking Based on Group Theory

Issues in modal tracking in the presence of crossings and crossing avoidances between eigenvalue traces are solved via the theory of point groups. The von Neumann-Wigner theorem is used as a key factor in predictively determining mode behavior over arbitrary frequency ranges. The implementation and capabilities of the proposed procedure are demonstrated using characteristic mode decomposition as a motivating example. The procedure is, nevertheless, general and can be applied to an arbitrarily parametrized eigenvalue problems. A treatment of modal degeneracies is included and several examples are presented to illustrate modal tracking improvements and the immediate consequences of improper modal tracking. An approach leveraging a symmetry-adapted basis to accelerate computation is also discussed. A relationship between geometrical and physical symmetries is demonstrated on a practical example.

A quantum solution for efficient use of symmetries in the simulation of many-body systems

A many-body Hamiltonian can be block-diagonalized by expressing it in terms of symmetry-adapted basis states. Finding the group orbit representatives of these basis states and their corresponding symmetries is currently a memory/computational bottleneck on classical computers during exact diagonalization. We apply Grover’s search in the form of a minimization procedure to solve this problem. Our quantum solution provides an exponential reduction in memory, and a quadratic speedup in time over classical methods. We discuss explicitly the full circuit implementation of Grover minimization as applied to this problem, finding that the oracle only scales as polylog in the size of the group, which acts as the search space. Further, we design an error mitigation scheme that, with no additional qubits, reduces the impact of bit-flip errors on the computation, with the magnitude of mitigation directly correlated with the error rate, improving the utility of the algorithm in the Noisy Intermediate Scale Quantum era.

Multipole expansion for magnetic structures: A generation scheme for a symmetry-adapted orthonormal basis set in the crystallographic point group

We propose a systematic method to generate a complete orthonormal basis set of multipole expansion for magnetic structures in arbitrary crystal structure. The key idea is the introduction of a virtual atomic cluster of a target crystal, on which we can clearly define the magnetic configurations corresponding to symmetry-adapted multipole moments. The magnetic configurations are then mapped onto the crystal so as to preserve the magnetic point group of the multipole moments, leading to the magnetic structures classified according to the irreducible representations of crystallographic point group. We apply the present scheme to pyrhochlore and hexagonal ABO3 crystal structures, and demonstrate that the multipole expansion is useful to investigate the macroscopic responses of antiferromagnets.