Single Diagonalization Research Articles

Westerfit: A new program for spin–torsion–rotation spectra

A new program, westerfit, has been developed to treat Cs molecules with internal rotation and spin angular momentum. It implements a single diagonalization Rho Axis Method approach for the torsion–rotation alongside a complete treatment of nuclear quadrupole interaction and spin–rotation coupling. Unlike other programs designed for internal rotation with spin effects, westerfit includes matrix elements off-diagonal in the rotational angular momentum quantum number, N, rather than the perturbative treatment of the spin–rotation and quadrupole interactions. This full combined approach allows fitting of all symmetrically allowed terms in both the spin–rotation and the quadrupole tensors as well as inclusion of higher order terms coupling the large amplitude motion to the spin angular momentum. The program was benchmarked against other published programs to test molecular cases of torsion–rotation, spin–rotation, and spin–torsion-rotation. All three tests produced a lower RMS. westerfit paves a way forward for complete treatment of spin–torsion–rotation problems regardless of barrier height or quadrupole moment.

DFTB Simulation of Charged Clusters Using Machine Learning Charge Inference.

We present a modification to self-consistent charge density functional-based tight binding (SCC-DFTB), which allows computation based on approximate atomic charges. We obtain these charges by means of a machine learning (ML) process that combines a Coulomb model with a neural network. This allows us to avoid the SCC cycles in the SCC-DFTB calculation while maintaining its accuracy. The main input of the model is the atomic positions characterized by a set of atom-centered symmetry functions. The charge inference from our ML algorithm is as close as 10-2 units of charge from the exact SCC solution. Our ML-DFTB approach provides a good approximation of the density matrix and of the energy and forces with only a single diagonalization. This is a significant computational saving with respect to the complete SCC algorithm, which allows us to investigate a bigger ensemble of atoms. We show the quality of our approach in the case of charged silicon carbide (SiC) clusters. The ML-DFTB potential energy surface (PES) mimics the SCC-DFTB PES rather well, despite its simplicity. This allows us to obtain the same geometric structure ordering with respect to energy for small clusters. The dissociation barriers for ion emission are well-reproduced, which opens the way to investigating ion field emission and charged cluster stability. The ML-DFTB approach is obviously not limited to charged clusters or SiC materials. It opens a new route to investigate larger clusters than those investigated by standard SCC-DFTB, as well as surface and solid-state chemistry at the atomic level.

Finding Excited-State Minimum Energy Crossing Points on a Budget: Non-Self-Consistent Tight-Binding Methods.

The automated exploration and identification of minimum energy conical intersections (MECIs) is a valuable computational strategy for the study of photochemical processes. Due to the immense computational effort involved in calculating non-adiabatic derivative coupling vectors, simplifications have been introduced focusing instead on minimum energy crossing points (MECPs), where promising attempts were made with semiempirical quantum mechanical methods. A simplified treatment for describing crossing points between almost arbitrary diabatic states based on a non-self-consistent extended tight-binding method, GFN0-xTB, is presented. Involving only a single diagonalization of the Hamiltonian, the method can provide energies and gradients for multiple electronic states, which can be used in a derivative coupling-vector-free scheme to calculate MECPs. By comparison with high-lying MECIs of benchmark systems, it is demonstrated that the identified geometries are good starting points for further MECI refinement with ab initio methods.

The single-point Berry phase in condensed-matter physics

The Berry phase plays an outstanding role in condensed-matter physics, where it provides the main entry for defining and computing a key observable: macroscopic electrical polarization. This phase obtains as a loop integral in reciprocal space, which has the topology of a torus. I show here that in some circumstances the loop may be contracted to a point, and polarization can be computed in practice from a single Hamiltonian diagonalization.

Electronic-Structure Properties from Atom-Centered Predictions of the Electron Density.

The electron density of a molecule or material has recently received major attention as a target quantity of machine-learning models. A natural choice to construct a model that yields transferable and linear-scaling predictions is to represent the scalar field using a multicentered atomic basis analogous to that routinely used in density fitting approximations. However, the nonorthogonality of the basis poses challenges for the learning exercise, as it requires accounting for all the atomic density components at once. We devise a gradient-based approach to directly minimize the loss function of the regression problem in an optimized and highly sparse feature space. In so doing, we overcome the limitations associated with adopting an atom-centered model to learn the electron density over arbitrarily complex data sets, obtaining very accurate predictions using a comparatively small training set. The enhanced framework is tested on 32-molecule periodic cells of liquid water, presenting enough complexity to require an optimal balance between accuracy and computational efficiency. We show that starting from the predicted density a single Kohn-Sham diagonalization step can be performed to access total energy components that carry an error of just 0.1 meV/atom with respect to the reference density functional calculations. Finally, we test our method on the highly heterogeneous QM9 benchmark data set, showing that a small fraction of the training data is enough to derive ground-state total energies within chemical accuracy.

In-medium T matrix in case of pairing instability

The diagonalization technique for two-body propagator in nuclear matter is generalized by inclusion of the two-hole continuum. In the approach, the total effective two-body Hamiltonian is introduced which allows us to find the complete propagator as well as the reaction $T$ matrix at many energies and relative momenta simultaneously from a single diagonalization procedure for the Hamiltonian matrix. Explicit treatment of bound and continuum states allows us to determine forward- and backward-propagating parts of the $T$ matrix in a closed form which simplifies strongly the evaluation of single-particle and single-hole self-energies. Special attention is paid to the bound-state contribution to the reaction matrix in case of pairing instability when complex conjugated eigenvalues of the effective Hamiltonian appear. A generalization of the approach to calculation of pairing gaps in case of nonzero center-of-mass momentum of the pair is also presented.

Prediction of many-electron wavefunctions using atomic potentials: Refinements and extensions to transition metals and large systems.

For a given many-electron molecule, it is possible to define a corresponding one-electron Schrödinger equation, using potentials derived from simple atomic densities, whose solution predicts fairly accurate molecular orbitals for single-determinant and multideterminant wavefunctions for the molecule. The energy is not predicted and must be evaluated by calculating Coulomb and exchange interactions over the predicted orbitals. Transferable potentials for first-row atoms and transition metal oxides that can be used without modification in different molecules are reported. For improved accuracy, molecular wavefunctions can be refined by slightly scaling nuclear charges and by introducing potentials optimized for functional groups. The accuracy is further improved by a single diagonalization of the Fock matrix constructed from the predicted orbitals. For a test set of 20 molecules representing different bonding environments, the transferable potentials with scaling give wavefunctions with energies that deviate from exact self-consistent field or configuration interaction energies by less than 0.05 eV and 0.02 eV per bond or valence electron pair, respectively. On diagonalization of the Fock matrix, the corresponding errors are reduced by a factor of three to less than 0.016 eV and 0.006 eV, respectively. Applications to the ground and excited states of a Ti18O36 nanoparticle and chlorophyll-a are reported.

Prediction of many-electron wavefunctions using atomic potentials: extended basis sets and molecular dissociation.

A one-electron Schrödinger equation based on special one-electron potentials for atoms is shown to exist that produces orbitals for an arbitrary molecule that are sufficiently accurate to be used without modification to construct single- and multi-determinant wavefunctions. The exact Hamiltonian is used to calculate the energy variationally and to generate configuration interaction expansions. Earlier work on equilibrium geometries is extended to larger basis sets and molecular dissociation. For a test set of molecules representing different bonding environments, a single set of invariant atomic potentials gives wavefunctions with energies that deviate from configuration interaction energies based on SCF orbitals by less than 0.04 eV per bond or valence electron pair. On a single diagonalization of the Fock matrix, the corresponding errors are reduced 0.01 eV. Atomization energies are also in good agreement with CI values based on canonical SCF orbitals. Configuration interaction applications to single bond dissociations of water and glycine, and multiple bond dissociations of ethylene and oxygen produce dissociation energy curves in close agreement with CI calculations based on canonical SCF orbitals for the entire range of internuclear distances.

Sturmian bases for two-electron systems in hyperspherical coordinates

We give a detailed account of an ab initio spectral approach for the calculation of energy spectra of two active electron atoms in a system of hyperspherical coordinates. In this system of coordinates, the Hamiltonian has the same structure as the one of atomic hydrogen with the Coulomb potential expressed in terms of a hyperradius and the nuclear charge replaced by an angle dependent effective charge. The simplest spectral approach consists in expanding the hyperangular wave function in a basis of hyperspherical harmonics. This expansion however, is known to be very slowly converging. Instead, we introduce new hyperangular Sturmian functions. These functions do not have an analytical expression but they treat the first term of the multipole expansion of the electron–electron interaction potential, namely the radial electron correlation, exactly. The properties of these new functions are discussed in detail. For the basis functions of the hyperradius, several choices are possible. In the present case, we use Coulomb–Sturmian functions of half integer angular momentum. We show that, in the case of H−, the accuracy of the energy and the width of the resonance states obtained through a single diagonalization of the Hamiltonian, is comparable to the values given by state-of-the-art methods while using a much smaller basis set. In addition, we show that precise values of the electric-dipole oscillator strengths for transitions in helium are obtained thereby confirming the accuracy of the bound state wave functions generated with the present method.

Extended Lagrangian Density Functional Tight-Binding Molecular Dynamics for Molecules and Solids

A computationally fast quantum mechanical molecular dynamics scheme using an extended Lagrangian density functional tight-binding formulation has been developed and implemented in the DFTB+ electronic structure program package for simulations of solids and molecular systems. The scheme combines the computational speed of self-consistent density functional tight-binding theory with the efficiency and long-term accuracy of extended Lagrangian Born-Oppenheimer molecular dynamics. For systems without self-consistent charge instabilities, only a single diagonalization or construction of the single-particle density matrix is required in each time step. The molecular dynamics simulation scheme can be applied to a broad range of problems in materials science, chemistry, and biology.

Thermostating extended Lagrangian Born-Oppenheimer molecular dynamics.

Extended Lagrangian Born-Oppenheimer molecular dynamics is developed and analyzed for applications in canonical (NVT) simulations. Three different approaches are considered: the Nosé and Andersen thermostats and Langevin dynamics. We have tested the temperature distribution under different conditions of self-consistent field (SCF) convergence and time step and compared the results to analytical predictions. We find that the simulations based on the extended Lagrangian Born-Oppenheimer framework provide accurate canonical distributions even under approximate SCF convergence, often requiring only a single diagonalization per time step, whereas regular Born-Oppenheimer formulations exhibit unphysical fluctuations unless a sufficiently high degree of convergence is reached at each time step. The thermostated extended Lagrangian framework thus offers an accurate approach to sample processes in the canonical ensemble at a fraction of the computational cost of regular Born-Oppenheimer molecular dynamics simulations.

Solution to the multichannel scattering problem via the discrete spectral shift formalism

A brief description of a new approach to solving scattering problems on the basis of continuum discretization and the formalism of the spectral shift function is given. A discrete representation for this function admits a direct generalization to the multichannel case, where the total multichannel S matrix at many energy values can be found from a single diagonalization of the total Hamiltonian of the system. The model problem of the elastic scattering of a composite particle on a nucleus is considered as an illustration with inclusion of the intermediate breakup of this particle.

Extended Lagrangian Born-Oppenheimer molecular dynamics in the limit of vanishing self-consistent field optimization

We present an efficient general approach to first principles molecular dynamics simulations based on extended Lagrangian Born-Oppenheimer molecular dynamics [A. M. N. Niklasson, Phys. Rev. Lett. 100, 123004 (2008)] in the limit of vanishing self-consistent field optimization. The reduction of the optimization requirement reduces the computational cost to a minimum, but without causing any significant loss of accuracy or long-term energy drift. The optimization-free first principles molecular dynamics requires only one single diagonalization per time step, but is still able to provide trajectories at the same level of accuracy as "exact," fully converged, Born-Oppenheimer molecular dynamics simulations. The optimization-free limit of extended Lagrangian Born-Oppenheimer molecular dynamics therefore represents an ideal starting point for robust and efficient first principles quantum mechanical molecular dynamics simulations.

Low-energy electron collisions with O2: Test of the molecularR-matrix method without diagonalization

Electron collisions with O${}_{2}$ at scattering energies below 1 eV are studied in the fixed-nuclei approximation for a range of internuclear separations using the ab initio molecular R-matrix method. The ${}^{2}{\ensuremath{\Pi}}_{\mathrm{g}}$ scattering eigenphases and quantum defects are calculated. The parameters of the resonance and the energy of the bound negative ion are then extracted. Different models of the target that employ molecular orbitals calculated for the neutral target are compared with models based on anionic orbitals. A model using a basis of anionic molecular orbitals yields physically correct results in good agreement with experiment. An alternative method of calculation of the R matrix is tested, where instead of performing a single complete diagonalization of the Hamiltonian matrix in the inner region, the system of linear equations is solved individually for every scattering energy. This approach is designed to handle problems where diagonalization of an extremely large Hamiltonian is numerically too demanding.

Characterizing disordered fermion systems using the momentum-space entanglement spectrum.

The use of quantum entanglement to study condensed matter systems has been flourishing in critical systems and topological phases. Additionally, using real-space entanglement one can characterize localized and delocalized phases of disordered fermion systems. Here we instead propose the momentum-space entanglement spectrum as a means of characterizing disordered models. We show that localization in one dimension can be characterized by the momentum space entanglement between left and right movers and illustrate our methods using explicit models with spatially correlated disorder that exhibit phases which avoid complete Anderson localization. The momentum space entanglement spectrum clearly reveals the location of delocalized states in the energy spectrum, can be used as a signature of the phase transition between a delocalized and localized phase, and only requires a single numerical diagonalization to yield clear results.

Internal rotation: Single diagonalization approach versus standard approaches, application to the methyl peroxy radical $\widetilde{\hspace*{-4pt}A}\leftarrow \widetilde{\hspace*{-2pt}X}$Ã←X̃ transition

A new approach avoiding double (two-step) diagonalization is proposed to deal with internal rotation. The development of this method was stimulated by the jet-cooled high resolution spectrum of the vibrationless Ã←X̃ transition of the deuterated species of the methyl peroxy radical. This spectrum, originally analyzed with a rigid rotor Hamiltonian including spin-rotation but neglecting internal rotation, has been revisited in the previous paper (P. Dupré, J. Chem. Phys. 134, 244308 (2011)) and a determinable yaw of the molecular principal axes of inertia about the c-axis (axis-switching) during the electronic transition was established. The spectral resolution of the jet-cooled data of the vibrationless transition (∼7355 - 7390 cm(-1)) does not allow the observation of splitting due to internal rotation of the methyl top, but when these data are combined with the low resolution room temperature data (∼7200 - 8000 cm(-1)) accurate fits or simulations of the two sets of data are possible. A recent study of the room temperature data has been reported in this journal (G. M. P. Just et al., J. Chem. Phys. 127, 044310 (2007)) showing evidence of the internal rotation coupling by analyzing the intensity of the torsional mode energy progression. That investigation combined ab initio quantum chemistry calculations and the rho-axis-method (RAM) to model the internal rotation. Here, a comparison of full spectral intensity analyses based on both the usual RAM and on the new approach requiring a single-diagonalization principal-axis-method is presented. The comparison favors the single-diagonalization approach. Axis-switching and spin-rotation coupling are incorporated in the analysis, in which the use of the principal axes of inertia is maintained. Symmetries, energy levels, and advantages are carefully discussed for all methods.

Correlated n 1,3 S states for coulomb three-body systems

n1,3S (n = 1 − 4) states for atomic three-body systems are studied with the Angular Correlated Configuration Interaction method. A recently proposed angularly correlated basis set is used to construct, simultaneously and with a single diagonalization, ground and excited states wave functions which: (i) satisfy exactly Kato cusp conditions at the two-body coalescence points; (ii) involve only linear parameters; (iii) show a fast convergency rate for the energy; and (iv) form an orthogonal set. The efficiency of the method is illustrated by the study a variety of three-body atomic systems [mmm] with two negatively charged light particles, with diverse masses m and m, and a heavy positively charged nucleus m. The calculated ground 11S and excited n1,3S (n = 2 − 4) state energies are compared with those given in the literature, when available. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2011

Intramolecular energy transfer dynamics in 24-mode pyrazine by partitioning technique: A time-dependent approach

We investigate the intramolecular energy transfer dynamics of the S(2) excited electronic state of pyrazine due to radiationless transitions to energetically lower-lying singlet electronic states using a new time-dependent method. The femtosecond decay of S(2) to the S(1) excited state and the picosecond decay of S(2) to the ground electronic state S(0) are studied within an efficient methodology for computing the intramolecular dynamics in multidimensional configurational spaces. Our method is based on partitioning the full configuration space into the (small) subspace of interest Q and the rest, the subspace P. The exact equations of motion for the states in Q, under the influence of P, are derived in the time domain in form of a system of integrodifferential equations. Their numerical solution is readily obtained when the Q space consists of just a few states. Otherwise, the integrodifferential equations for the states in Q are transformed into a (larger) system of ordinary differential equations, which can be solved by a single diagonalization of a general complex matrix. The former approach is applied to study the pyrazine picosecond S(2)→S(0) dynamics and the latter is applied to the study of the ultrafast pyrazine S(2)→S(1) decay dynamics.

New approach toward a direct evaluation of the multichannel multienergySmatrix without solving the scattering equations

A novel method which allows us to find observables at many energies simultaneously in a multichannel scattering problem using just a single diagonalization of the total Hamiltonian matrix in some ${L}_{2}$ basis is developed. The approach is based on the multichannel generalization of ideas due to Lifshits and Birman-Krein on the tight relationship between a spectral shift function and a single-channel $S$ matrix in the combination with ${L}_{2}$ discretization of the continuum. As a clear illustration for the efficiency of the suggested method, the differential cross sections of neutron scattering off heavy nuclei in the case of a complex nonlocal interaction potential as well as energy dependence of partial $\mathit{NN}$ phase shifts and mixing parameters for the two-nucleon scattering with tensor force are calculated.

Ground and excited states for exotic three-body atomic systems

An Angular Correlated Configuration Interaction method is extended and applied to exotic threebody atomic systems with general masses. A recently proposed angularly correlated basis set is used to construct, simultaneously and with a single diagonalization, ground and excited states wave functions which: (i) satisfy exactly Kato cusp conditions at the two-body coalescence points; (ii) have only linear parameters; (iii) show a fast convergency rate for the energy; (iv) form an orthogonal set. The efficiency of the construction is illustrated by the study a variety of three-body atomic systems [m1− m2− m3z3+ ] with two negatively charged light particles, with 123 diverse masses m1− and m2−, and a heavy positively charged nucleus m3z3+. The calculated ground 11S and several excited n1,3S state energies are compared with those given in the literature, when available. We also present a short discussion on the critical charge necessary to get a stable three-body system supporting two electrons, an electron and a muon, or two muons.