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Thomas D. Kühne

Publications and source records attributed to Thomas D. Kühne.

At least 19 recordsLinked to original sources
arXiv2026

From Wavefunction Regularity to Eigenvector Conditioning: Accuracy--Cost Trade-offs of Quantum Algorithms for Non-Hermitian Transcorrelated Hamiltonians

The electron--electron Coulomb singularity forces cusp structure in electronic wavefunctions. This cusp structure limits wavefunction regularity and slows the convergence of expansions in smooth bases, requiring large basis sets to reach a target accuracy. The transcorrelated (TC) method incorporates this short-range behavior into the Hamiltonian, thereby improving the regularity of the transformed wavefunction and accelerating convergence toward the complete-basis-set limit. The resulting Hamiltonian is non-Hermitian and non-normal, which limits the direct applicability of standard quantum algorithms. For such non-normal operators, the complexity of several recent quantum algorithms depends explicitly on quantities related to eigenvector conditioning and eigenvalue sensitivity. Understanding these quantities is essential for assessing both algorithmic complexity and quantum resource requirements. In this work, we address these issues in two stages. We first study a solvable model, which provides a controlled setting for isolating how transcorrelation affects basis convergence and eigenvector conditioning. Within this setting, we investigate two distinct one-parameter Jastrow correlators. We derive the asymptotic energy-error scaling: $N^{-1/2}$ for the bare projection and $N^{-3/2}$ for both TC correlators at generic fixed parameters, with $N$denoting the basis size. Furthermore, we show that this faster convergence can yield a polynomial improvement in the asymptotic query-complexity upper bound. We then extend the analysis to second-quantized electronic-structure Hamiltonians and use particle-number-sector diagnostics to distinguish target-sector from auxiliary-sector conditioning. Our work provides a framework for understanding the central accuracy--cost trade-off in TC Hamiltonians.

quant-ph↗
arXiv2026

From Bellissard's gap labeling to atomistic quasiperiodic bilayers

Bellissard's framework uses covariant observables to describe aperiodic solids. We examine how butterfly-like spectra in twisted bilayers can be related to bulk gap labels and topological response. A bulk gap permits a spectral projector and an integrated density of states. A topological response requires an additional pairing of that projector. In an illustrative finite honeycomb bilayer, an exact second-moment identity isolates the interlayer contribution to spectral variance. Angle-resolved spectra, finite state counts, an uncoupled reference, and size and broadening checks show how coupling redistributes spectral weight without assigning a bulk gap label. We then examine the requirements for atomistic electronic-structure calculations, including nonorthogonal orbitals, projected densities of states, and basis functions that move during relative sliding. The resulting framework specifies the steps needed to connect atomistic spectra with bulk state counting and topological response.

physics.comp-ph↗
arXiv2026

Molecular Implementation of the Machine-Learned Skala Exchange-Correlation Functional in CP2K through GauXC

Machine-learned exchange--correlation (XC) functionals offer a route to improve Kohn--Sham density-functional theory without incurring the cost of explicitly correlated electronic-structure methods. Their use in production simulation codes, however, requires a well-defined mapping between the learned model and the host-code density representation. We formulate and implement a Skala-1.1 interface in CP2K through the external GauXC library. CP2K supplies the geometry, Gaussian basis, spin-resolved atomic-orbital density matrix, and communicator, while GauXC evaluates the XC energy, atomic-orbital potential matrix, and available nuclear derivatives. The interface accepts both all-electron and valence-only density matrices. The latter may arise from separable dual-space pseudopotentials or molecular effective-core potentials. Implementation errors are isolated from functional differences by comparing the Perdew--Burke--Ernzerhof (PBE) functional evaluated through GauXC with native CP2K PBE. The resulting interface gives consistent energies, forces validated against finite-difference total-energy checks, and force-based molecular-virial diagnostics for representative molecular cases. The dietGMTKN55 benchmark suite is evaluated with an all-electron Gaussian augmented plane-wave treatment for elements up to bromine and def2 effective-core potentials for the heavier elements. The resulting aggregate mean absolute deviation of 1.255 kcal/mol is within 0.012 kcal/mol of the corresponding Skala reference value of 1.243 kcal/mol. This work establishes a validated molecular implementation of Skala in CP2K through GauXC.

physics.chem-ph↗
arXiv2026

Native implementation of the machine-learned Skala exchange-correlation functional in CP2K: Unified one-centre reconstruction for molecular and condensed-phase calculations

We implement the machine-learned Skala exchange-correlation (XC) functional natively in CP2K using its Gaussian and plane-wave (GPW) and Gaussian and augmented-plane-wave (GAPW) methods. A joint one-centre reconstruction of density, density gradients, and kinetic-energy density before functional evaluation preserves mixed gradient terms and nonlocal couplings between smooth and atom-local contributions in all-electron and pseudopotential calculations. An XC-specific GAPW representation resolves rapidly varying local contributions on atom-centred grids, reducing plane-wave requirements for pseudopotential calculations. The implementation includes Brillouin-zone sampling and point-group symmetry reduction. With sufficiently flexible orbital bases, it retains the molecular benchmark accuracy of our earlier GauXC formulation. All-electron Skala-D3(BJ) calculations yield a mean absolute error (MAE) of 1.54 kJ mol$^{-1}$ against diffusion Monte Carlo for crystalline CO$_2$, NH$_3$, and urea, spanning dispersion, quadrupolar electrostatics, and hydrogen bonding. For all thirteen DMC-ICE13 phases, the MAEs are 1.16 kJ mol$^{-1}$ for absolute lattice energies and 0.82 kJ mol$^{-1}$ for the twelve relative energies to ice Ih. At fixed experimental geometries, all-electron/mixed-core calculations give a band-gap MAE of 0.43 eV on a 15-material set, substantially below commonly used semilocal functionals and comparable to widely used hybrid functionals. The LC10 benchmark nevertheless reveals systematically underestimated equilibrium lattice constants, indicating structural overbinding. This framework connects molecular Skala to condensed-phase electronic structure and enables future development using periodic many-body reference data.

cond-mat.mtrl-sci↗
arXiv2026

On-Water Surface Catalysis: From Hydrogen Bonding to Charge-Transfer Activation

On-water catalysis accelerates reactions between poorly soluble organic substrates in aqueous suspensions, but its molecular origin remains debated. This Account argues that hydrogen bonding and proton transfer can both enhance charge-transfer stabilization between the organic reactants. Hydrogen bonds from surface water polarize the reacting complex, whereas protonation can perturb the same donor-acceptor interaction more strongly without requiring identical reaction pathways. We connect simulations of the water/vapor interface, finite-temperature cycloaddition studies, and energy decomposition analysis based on absolutely localized molecular orbitals. Dangling OH groups establish reactive contacts, but an increased transition-state hydrogen-bond count alone does not explain activation. Electronic analysis instead identified hydration-enhanced charge-transfer stabilization between organic partners by approximately 30%. Comparisons of related dienophiles, a protonated limiting model, and oxygen/sulfur substitution link substrate-dependent electronic response to hydration and bond formation. A hydration-induced charge-transfer response metric provides a testable molecular hypothesis, not a standalone measure of catalytic acceleration. Activation depends on differential stabilization of reactant and transition-state ensembles. Increasing the accessible reactive interface can amplify this local mechanism without requiring exceptionally strong additional electric fields, connecting conventional on-water chemistry with selected microdroplet reactions while distinguishing adsorption, concentration, and intrinsic molecular activation.

physics.chem-ph↗
arXiv2026

Capturing Nuclear Quantum Effects in Hydrogen Diffusion through MoS2 via Machine-Learning-Enhanced Path-Integral Simulations

Hydrogen transport through layered two-dimensional (2D) materials is central to technologies such as hydrogen storage, fuel cells, and isotope separation. Among these materials, MoS2 exhibits tunable interlayer diffusion properties, whose accurate theoretical description requires accounting for nuclear quantum effects (NQEs), including zero-point motion and tunneling. Here, we present a machine-learning-enhanced atomistic study of hydrogen and deuterium diffusion in layered MoS2 based on interatomic potentials trained on r2SCAN+rVV10 density-functional-theory data. Combining well-tempered metadynamics with path-integral molecular dynamics, we investigate diffusion across multiple MoS2 polytypes and twisted bilayer structures while explicitly incorporating NQEs. Our simulations show that NQEs substantially lower free-energy barriers for hydrogen diffusion at 300 K, significantly increasing the hydrogen self-diffusion coefficient compared to classical nuclei simulations. We further identify a pronounced kinetic isotope effect, with a 35 meV difference between hydrogen and deuterium quantum free-energy barriers. In twisted bilayer MoS2, hydrogen transport exhibits strong spatial variations governed by the local stacking environments within the moiré superlattices. These results highlight the critical role of NQEs in hydrogen transport through layered materials and provide atomistic insight intoisotope-selective diffusion in structurally complex 2D systems.

cond-mat.mtrl-sci↗
arXiv2026

CP2K: An electronic structure and molecular dynamics software package - Dynamics, Transport, and Spectroscopic Response

One of the distinguishing aspects of CP2K is its seamless integration of diverse structural and transition-state optimization techniques with advanced sampling approaches including Monte Carlo, molecular dynamics, and metadynamics, enabling the efficient exploration of complex potential- and free-energy landscapes, including rare events. These capabilities are combined with a broad hierarchy of energy and force evaluation methods, ranging from classical and machine-learned interaction potentials and mixed quantum-classical multiscale and semiempirical schemes, to highly accurate quantum-mechanical electronic-structure approaches. At the heart of the latter lies the Gaussian and plane-wave framework, along with its augmented all-electron generalization, which have been described in detail in our previous code review [T. D. Kühne et al., J. Chem. Phys. 152, 194103 (2020)]. Building on this foundation, the present work revisits the methods within CP2K that turn electronic structure into dynamics, transport, and spectroscopic response. Particular emphasis is placed on the coupling between static response calculations and nuclear motion: spectra may be evaluated at optimized structures, averaged over thermally sampled configurations, obtained from time-correlation functions along ab-initio or path integral molecular trajectories, or followed in real time together with electronic and nuclear dynamics. The same modular structure also enables equilibrium and biased transport simulations, from Kubo-type linear response to open-boundary approaches under external potentials, highlighting CP2K's unique capability to unify quantum chemistry with quantum and statistical mechanics within a versatile, holistic simulation environment.

physics.chem-ph↗
arXiv2026

MANDALA: An E(3)-Equivariant Graph Neural Network Framework for Learning Electronic-Structure Operators with Observable Guidance

Electronic-structure calculations based on Kohn-Sham density functional theory remain indispensable in computational materials science and chemistry. Their computational cost, however, limits accessible system sizes and simulation times. At the same time, conventional machine-learning interatomic potentials (MLIPs), which are becoming the workhorse of large-scale materials modeling, usually target only energies and forces. They therefore leave out the quantum-operator-level information required to reconstruct band structures, densities of states, spatial charge distributions, and other electronic observables. \texttt{Mandala} fills this methodological gap. It is a modular software framework for learning block-sparse electronic-structure matrices with E(3)-equivariant graph neural networks. The framework is built around a unified representation of atom-resolved Hamiltonian, overlap, and density matrices, together with reusable abstractions for basis conversion, sparse block handling, irreducible representation mapping, graph construction, model definition, and training. This design allows \texttt{Mandala} to support heterogeneous chemical compositions, a wide range of neural architecture variants within one workflow, and multiple electronic-structure backends. \texttt{Mandala} evaluates selected observables directly from the predicted operators, including band energy, electron count, density of states, and band structure. This connects electronic-structure learning and observable-guided modeling while retaining a representation tied to quantum-mechanical operators rather than only scalar or vector targets as in MLIPs. In this form, \texttt{Mandala} is intended to complement atomistic interatomic potential workflows by resolving electronic structure and operator-derived observables within one scalable implementation.

cond-mat.mtrl-sci↗
arXiv2026

GPU-accelerated finite-temperature Lanczos method for Heisenberg spin systems with non-Abelian permutation symmetries

We extend a recent GPU implementation of the finite-temperature Lanczos method (FTLM) for Heisenberg spin Hamiltonians, which uses only total-magnetization symmetry, to the full permutation symmetry, including non-Abelian point and space groups. For highly symmetric clusters such as icosahedral polyhedra, the multidimensional irreducible representations (irreps) of the permutation group reduce the block dimensions substantially further than any Abelian subgroup. Here, we provide a matrix-free formalism (avoiding explicit storage of the projected Hamiltonian) for using such symmetries. The implementation applies equally to Abelian groups, retaining the advantages of GPU acceleration. Several cluster topologies are built in; beyond these, symmetry groups can be supplied as site-permutation generators, with irrep matrices computed automatically, so that user-defined clusters run without any code changes. We demonstrate the approach in production runs on a single NVIDIA B200 accelerator for the s=3/2 Heisenberg antiferromagnet on the dodecahedron (Hilbert-space dimension 4^20 ~ 1.1x10^12) and for the s=1/2 square lattice of 5x8 sites (M=0 dimension 1.4x10^11). With R=24 random vectors per symmetry block and 60 Lanczos steps, the dodecahedron campaign (with the largest iterated symmetry block having a dimension of 3.6x10^9) requires about 50 GPU-hours. The code is openly available under the Apache-2.0 license at https://github.com/ghasdeke/ftlm-pg-gpu, archived at DOI: 10.5281/zenodo.21445872.

cond-mat.str-el↗
arXiv2026

The UZH protocol: Separating errors and constructing improved CP2K basis sets and pseudopotentials

Reliable density-functional simulations require numerical settings whose residual errors are smaller than the chemical and materials trends being interpreted. In CP2K/Quickstep, this requirement is complicated by the joint use of atom-centered Gaussian basis sets and norm-conserving pseudopotentials: a code-to-code discrepancy usually contains both contributions. We present the UZH protocol, a closed-loop CP2K workflow that calibrates molecularly optimized Gaussian basis sets on small molecules, validates the resulting settings in unary-crystal equation-of-state benchmarks, identifies whether the limiting approximation is the Gaussian basis or the pseudopotential. The diagnosis is then used to revise the parameter files. The central diagnostic is a three-way comparison between production CP2K-GTH-UZH calculations, SIRIUS calculations using the same Goedecker--Teter--Hutter pseudopotential in a systematic plane-wave representation, and all-electron full-potential linearized augmented-plane-wave SIRIUS references. This construction decomposes the practical CP2K error into a Gaussian-basis component and a pseudopotential component. The protocol distinguishes basis-limited noble-gas and heavy-element cases from pseudopotential-limited transition-metal cases, guides targeted revisions with the CP2K basis and pseudopotential optimizers, and produces improved MOLOPT basis sets and GTH pseudopotentials as explicit outputs of the workflow. The UZH protocol is therefore constructive: it does not merely measure or reduce errors a posteriori, but allows turning verification outliers into validated CP2K parameter files for simulations across molecules and condensed phases.

physics.chem-ph↗
arXiv2026

Finite-temperature Fe K-edge X-ray absorption simulations reveal local structural dynamics of an iron(II) photosensitizer in solution and the crystalline phase

Interpreting metal K-edge spectra of flexible photosensitizers requires a structural model that separates electronic signatures from thermal motion, solvent disorder, and crystal-packing effects. We combine Fe K-edge X-ray absorption measurements with second-generation Car--Parrinello ab initio molecular dynamics and all-electron Gaussian and augmented-plane-wave simulations for an iron(II) N-heterocyclic carbene photosensitizer in acetonitrile solution and in the crystalline phase. Ensemble-averaged spectra reproduce the main near-edge features in both environments and preserve the experimentally observed similarity of the first Fe coordination shell upon dissolution. Comparison with radial distributions extracted from extended fine-structure measurements validates the Fe--N and Fe--C coordination shells sampled by the trajectories, while element-resolved pair distributions explain why higher-shell experimental contrast is rapidly lost. The same dynamical ensembles reveal a broad out-of-plane distribution of the terpyridine nitrogen atom and a nearly octahedral distribution of the Fe-centered coordination planes. The results show that finite-temperature X-ray absorption simulations can provide a compact structural-dynamics picture of molecular transition metal photosensitizers by linking local spectra, solvent-phase ligand motion, and medium-range structural disorder within one trajectory-based description.

cond-mat.mtrl-sci↗
arXiv2026

Best practices for second-generation Car-Parrinello ab initio molecular dynamics with CP2K/Quickstep

Second-generation Car--Parrinello \textit{ab initio} molecular dynamics (CP2G AIMD) combines a Born--Oppenheimer-like nuclear equation of motion with a predictor-corrector propagation of the one-particle density matrix and a modified Langevin equation to ensure an accurate sampling of the Boltzmann distribution. In the CP2K implementation, this approach can substantially reduce the number of self-consistent-field iterations required per molecular-dynamics step, but only if the density-matrix propagation, the orbital-transformation corrector, and the modified Langevin friction parameters are tuned consistently. This tutorial gives a practical workflow for setting up CP2G AIMD calculations in CP2K, starting from Born--Oppenheimer pre-equilibration and proceeding through corrector-step-size selection, always-stable predictor-corrector order optimization, final relaxation, and the calibration of the dissipative and stochastic terms used for canonical sampling. The protocol is demonstrated for a 32-molecule liquid-water benchmark and is written for users who already know the basics of CP2K input files but want a reliable route from a conventional AIMD setup to a production-ready CP2G simulation.

physics.comp-ph↗
arXiv2026

Terahertz-induced local-field dynamics and transient birefringence in aqueous electrolytes

Terahertz Kerr-effect (TKE) spectroscopy provides a time-domain optical probe of the intermolecular structural dynamics of liquids, but the measured birefringence can only be interpreted microscopically if transient molecular structure, local electric fields, and nonlinear optical response are treated on the same footing. We combine terahertz-driven non-equilibrium molecular dynamics with a hyperpolarizability-enhanced dipole-induced-dipole (DID) response model to connect the measured TKE signal of water and aqueous electrolytes to molecular-scale hydration dynamics. Starting from a multipolar expansion of the field-dependent molecular energy, the optical response is written as a differential polarizability in which first and second molecular hyperpolarizabilities are coupled to the instantaneous local field generated by the evolving liquid structure. For neat water, these local-field terms convert an insufficiently structured intramolecular response into the experimentally observed bipolar TKE line shape. For aqueous MgCl2, polarizable force-field trajectories combined with the extended optical response reproduce the concentration-dependent enhancement of the negative birefringence, whereas non-polarizable trajectories yield the wrong trend. Species-resolved analysis assigns the high-concentration response to strongly field-polarized anions and, more importantly, to water molecules bridging cations and anions. The work identifies TKE birefringence as a sensitive readout of ion-hydration local-field dynamics and shows that induced polarization in both the trajectory and the optical response is required to recover the structural dynamics encoded in electrolyte TKE measurements.

physics.chem-ph↗
arXiv2026

GPU-accelerated finite-temperature Lanczos method for spin Hamiltonians

We present a GPU implementation of the finite-temperature Lanczos method (FTLM) for Heisenberg spin Hamiltonians that targets workstation hardware rather than distributed-memory clusters. The Hamiltonian action is evaluated matrix-free in a row-wise gather formulation. We introduce and compare two state-to-index strategies: a compressed lookup table (CLT), which reduces lookup memory by a factor of 16 relative to a full table while retaining a fixed, branch-light access pattern, and a GPU-adapted combinatorial-ranking scheme that removes the lookup table altogether. Numerical tests against FP64 CPU references show that FP32 GPU arithmetic changes heat capacities and magnetic susceptibilities by amounts several orders of magnitude below the stochastic uncertainty of the FTLM trace estimator at typical sample sizes. Benchmarks show speedups of up to about one order of magnitude over optimized multicore CPU calculations and enable Hilbert-space sectors of dimension ~10^8 on a single workstation GPU. The MATLAB/CUDA implementation, including example input files and benchmark scripts, is openly available at https://github.com/ghasdeke/ftlm-gpu (archived at DOI: 10.5281/zenodo.20378647) under the Apache-2.0 license.

cond-mat.str-el↗
arXiv2026

Separable Force Matching of PBE0 Hybrid-Functional Reference Forces for Path-Integral Simulations of Liquid Water

Force-matched water models provide a practical route from first-principles reference data to long classical and path-integral molecular simulations. Previous flexible four-site potentials in the spirit of q-TIP4P/F showed that fitting analytic models to density-functional-theory forces can reproduce key structural features of liquid water while retaining the efficiency required for quantum-nuclear sampling. Here we introduce two refinements aimed at making this strategy more accurate and more reproducible for molecular simulation. First, the production fit is based on PBE0 hybrid-functional reference forces and therefore includes the Hartree--Fock exact-exchange contribution in the electronic-structure target. Second, the parametrization is formulated as a separable nonlinear least-squares problem in which all linear force-field amplitudes are eliminated analytically for every trial set of nonlinear shape parameters. The resulting force-matching protocol lowers the dimension of the nonlinear search, reduces compensation between heterogeneous parameters, and enables a controlled comparison of Lennard-Jones and Buckingham oxygen--oxygen repulsion terms. Applied to CP2K reference forces for liquid water, the projected optimization reproduces target force distributions and yields radial distribution functions close to first-principles and neutron-scattering benchmarks. The Buckingham representation gives a more flexible short-range repulsive wall than a purely Lennard-Jones form, and the final flexible model remains stable in path-integral simulations. The method provides a transparent workflow for deriving simulation-ready water potentials from accurate hybrid density-functional reference forces.

physics.chem-ph↗
arXiv2026

Accelerated "on-the-fly" coupled-cluster path-integral molecular dynamics: Impact of nuclear quantum effects on an asymmetric proton

We present an accelerated ``on-the-fly'' coupled-cluster path-integral molecular dynamics (PIMD) method for finite-temperature simulations in which electron correlation and nuclear quantum effects are treated simultaneously. The approach is based on our quantum ring-polymer contraction (qRPC) technique, in which the inexpensive Hartree-Fock potential is evaluated on the full ring-polymer, while the expensive coupled-cluster correction is evaluated on the centroid only. This qRPC decomposition is combined with a second-generation Car-Parrinello-like dynamics of the Hartree-Fock reference and a basis-consistent extrapolation of the coupled-cluster and de-excitation amplitudes. The combination of all three acceleration layers is essential for making correlated PIMD calculations feasible. We apply the method to a proton shared by water and formaldehyde. Relative to classical nuclei, nuclear quantum effects broaden covalent X--H bond-length distributions, reduce the bias of the shared proton toward formaldehyde, and shift the mean proton-transfer coordinate from 0.206 to 0.135A. The probability of finding the proton closer to formaldehyde decreases from 81.7$\%$ to 61.1$\%$. The corresponding nuclear magnetic shielding tensors show that electron correlation and nuclear quantum effects are of comparable magnitude and can act in opposite directions. These results demonstrate that predictive simulations of asymmetric hydrogen bonds require a simultaneous treatment of correlated electronic structure and nuclear quantum fluctuations.

physics.chem-ph↗
arXiv2026

Importance of nuclear quantum effects on the structure of supercooled water around its liquid--liquid critical point

Supercooled water is expected to exhibit a liquid--liquid phase transition between low- and high-density liquid states, possibly terminating in a liquid--liquid critical point in the experimentally difficult no man's land. Because the hydrogen atoms are light, nuclear quantum effects (NQE) may alter the structural signatures used to identify this transition. Here, we compare classical molecular dynamics and path-integral molecular dynamics simulations of a flexible q-TIP4P/F-like water model in the deeply supercooled regime. The classical simulations show a pronounced density change at 180 K between 180 and 220 MPa, whereas the path-integral simulations exhibit a smoother pressure dependence. Radial distribution functions and bond-order parameters show that NQE broaden pair correlations, reduce the tetrahedral order of the first hydration shell, and slightly increase the Steinhardt $Q_6$ parameter. These results demonstrate that NQE modify both low- and high-density liquid structures and therefore need to be included when interpreting structural signatures of the liquid--liquid transition in supercooled water.

physics.chem-ph↗
arXiv2026

Basis-free neural-network geminal and Jastrow factors for variational Monte Carlo

Neural-network quantum states offer a flexible route to compact many-electron wave functions, but their practical accuracy depends strongly on how fermionic antisymmetry, electron correlation, and optimization noise are treated. Here we combine an antisymmetrized geminal power (AGP) determinant with feed-forward neural networks that replace conventional basis-set expansions in the geminal and in two Jastrow-factor constructions. The resulting basis-free Jastrow--AGP ansatz is optimized by variational Monte Carlo and is designed to separate two tasks: the AGP part defines the nodal surface, while the neural-network Jastrow factor recovers dynamical correlation at fixed nodes. This separation makes it possible to distinguish errors associated with dynamical correlation from those caused by static, multireference correlation. Applications to the hydrogen molecule and the rectangular hydrogen tetramer show sub-millihartree accuracy when the AGP nodes are adequate, and expose the residual nodal limitation near the large-radius square geometry of the hydrogen tetramer. These results clarify where neural-network building blocks can improve a compact geminal ansatz and where additional nodal flexibility is required.

physics.comp-ph↗