Regular Hamiltonian Research Articles

Bridging two quantum quench problems — local joining quantum quench and Möbius quench — and their holographic dual descriptions

We establish an equivalence between two different quantum quench problems, the joining local quantum quench and the Möbius quench, in the context of (1 + 1)-dimensional conformal field theory (CFT). Here, in the former, two initially decoupled systems (CFTs) on finite intervals are joined at t = 0. In the latter, we consider the system that is initially prepared in the ground state of the regular homogeneous Hamiltonian on a finite interval and, after t = 0, let it time-evolve by the so-called Möbius Hamiltonian that is spatially inhomogeneous. The equivalence allows us to relate the time-dependent physical observables in one of these problems to those in the other. As an application of the equivalence, we construct a holographic dual of the Möbius quench from that of the local quantum quench. The holographic geometry involves an end-of-the-world brane whose profile exhibits non-trivial dynamics.

Coordination of Mercury(II) in Water Promoted over Hydrolysis in Solvated Clusters [Hg(H2O)1-6](aq)2+: Insights from Relativistic Effects and Free Energy Analysis.

Understanding the nature of the interaction between mercury(II) ions, Hg2+, and water molecules is crucial to describe the stability and chemical behavior of structures formed during solvation, as well as the conditions that favor the Hg2+ coordination or inducing water hydrolysis. In our study, we explored exhaustively the potential energy surface of Hg2+ with up to six water molecules. We analyzed electronic and Gibbs free energies, binding, and nuclear magnetic resonance parameters. We used the zeroth-order regular approximation Hamiltonian, including scalar and spin-orbit relativistic corrections for free energy calculations and geometry optimizations to explore the interplay between electron correlation and relativistic effects. We analyzed intermolecular interactions with energy decomposition analysis, quantum theory of atoms in molecules, and natural bond orbital. Additionally, we used the four-component Dirac Hamiltonian to compute solvent effect on the magnetic shielding and J-coupling constants. Our results revealed that the water hydrolysis by Hg2+ requires a minimum of three water molecules. We found that the interaction between Hg2+ and water molecules is an orbital interaction due to relativistic effects and the most stable structures are opened-shape clusters, reducing the number of oxygen-mercury contacts and maximizing the formation of hydrogen bonds among water molecules. In these types of clusters, Hg2+ promotes the water hydrolysis over coordination with oxygen atoms. However, when we considered the change associated with the transfer of a cluster from the ideal gas to a solvated system, our solvation free energy analysis revealed that closed-shape clusters are more favorable, maximizing the number of oxygen-mercury contacts and reducing the formation of hydrogen bonds among water molecules. This finding suggests that, under room conditions, the coordination of Hg2+ is more favorable than hydrolysis. Our results have significant implications for understanding Hg2+ behavior in water, helping to develop targeted strategies for mercury remediation and management, and contributing to advancements in the broader field of environmental chemistry.

Symmetries and conservation laws of Hamiltonian systems

In this paper we study the infinitesimal symmetries, Newtonoid vector fields, infinitesimal Noether symmetries and conservation laws of Hamiltonian systems. Using the dynamical covariant derivative and Jacobi endomorphism on the cotangent bundle we find the invariant equations of infinitesimal symmetries and Newtonoid vector fields and prove that the canonical nonlinear connection induced by a regular Hamiltonian can be determined by these symmetries. Finally, an example from optimal control theory is given.

Classical and quantum analysis of 3D electromagnetic pp-wave spacetime

The general classical solution of the 3D electromagnetic pp-wave spacetime has been obtained. The relevant line element contains an arbitrary essential function providing an infinite number of in-equivalent geometries as solutions. A classification is presented based on the symmetry group. To the best of our knowledge, the solution corresponding to only one of the Classes is known. The dynamics of some of the Classes was also derived from a minisuperspace Lagrangian which has been constructed. This Lagrangian contains a degree of freedom (the lapse) which can be considered either as dynamical or non-dynamical (indicating a singular or a regular Lagrangian correspondingly). Surprisingly enough, on the space of classical solutions, an equivalence of these two points of view can be established. The canonical quantization is then used in order to quantize the system for both the singular and regular Hamiltonian. A subsequent interpretation of quantum states is based on a Bohm-like analysis. The semi-classical trajectories deviate from the classical only for the regular Hamiltonian and in particular for a superposition of eigenstates (a Gaussian initial state has been used). Thus, the above mentioned equivalence is broken at the quantum level. It is noteworthy that the semi-classical trajectories tend to the classical ones in the limit where the initial wavepacket is widely spread. Hence, even with this simple superposition state, the classical solutions are acquired as a limit of the semi-classical.

Quantitative validation of Ames IR intensity and new line lists for 32/33/34S16O2, 32S18O2 and 16O32S18O

The quality of Ames-296K SO2 Infrared (IR) line list intensities is first validated by quantitative exploration of several dipole moment surfaces (DMSs) and partition sum convergence. The DMSs are computed with several of Dunning's correlation-consistent basis sets and their vibrational dependence are compared to the empirical model derived from Stark effect experiments reported by D. Patel, D. Margolese, and T.R. Dyke [J. Chem. Phys.70, 2740 (1979)]. The effective dipole deviations from the DMS adopted in the Ames IR lists is 0.2–0.4% for vibrational states up to 3ν3. The vibrational dependence of the dipole moment is also in good agreement, except for nν1. Partition sum convergence at 296 K is confirmed by new calculations with rotational quantum number J up to 150 and upper state E’ up to 8000 cm−1. The isotopologue consistency of the Ames IR line lists is superior relative to the regular Effective Hamiltonian (EH) models and Effective Dipole Moment (EDM) models. The ν1 + ν2 and ν2 + ν3 intensity consistency check reveals the recently reported experimental intensities need significant improvement or re-analysis. After the accuracy, convergence, and isotopologue consistency have been confirmed, the theoretical Ames-296K intensities are combined with the experimental line positions or EH models that experimental spectroscopists published after 2009. Three high-resolution IR line sets are reported for the 32/33/34S16O2, 32S18O2 and 16O32S18O isotopologues: (1) the “New Lines Sets” include experimentally measured line positions; (2) the “Expanded Line Sets” include possible transitions among new rovibrational levels assigned in experiments and ground state (GS) levels predicted by reliable EH models; (3) the “Ames + MARVEL Sets” include possible transitions among all those levels reported in a recent MARVEL analysis. [Tóbiás et al, JQSRT 208, 152 (2018)]. Compared to the limited data in High-resolution TRANsmission molecular absorption database (HITRAN), these line sets have significantly improved the data coverage up to 4000 cm−1. Some missing bands can be traced to the unpublished experimental data. The isotopologue consistency of these line sets will help identify the uncertainties and defects in the experimental EH and EDM models. These line sets are good candidates for the next HITRAN update, if line shape parameters are available. The line sets can be downloaded from supplementary files or from the Ames Molecular Spectroscopic Database at http://huang.seti.org.

Some developments of reductiontheory for controlled Hamiltonian system with symmetry

A regular controlled Hamiltonian (RCH) system is aHamiltonian system with external force and control. In general, an RCHsystem under the action of external force and control is notHamiltonian, however, it is a dynamical system closely related to aHamiltonian system, and it can be explored and studied by extendingthe methods for external force and control in the study of Hamiltonian systems.In this paper, we introducebriefly some recent developments in the study of reductiontheory for the RCH systems with symmetries and applications,in terms of the completeness ofMarsden-Weinstein reduction and the change of geometrical structures.

On Nonuniqueness of Solutions of Hamilton\u2013Jacobi\u2013Bellman Equations

An example of a nonunique solution of the Cauchy problem of Hamilton–Jacobi–Bellman (HJB) equation with surprisingly regular Hamiltonian is presented. The Hamiltonian H(t, x, p) is locally Lipschitz continuous with respect to all variables, convex in p and with linear growth with respect to p and x. The HJB equation possesses two distinct lower semicontinuous solutions with the same final conditions; moreover, one of them is the value function of the corresponding Bolza problem. The definition of lower semicontinuous solution was proposed by Frankowska (SIAM J. Control Optim. 31:257–272, 1993) and Barron and Jensen (Commun. Partial Differ. Equ. 15(12):1713–1742, 1990). Using the example an analysis and comparison of assumptions in some uniqueness results in HJB equations is provided.

Geometrical structures on the cotangent bundle

In this paper, we study the geometrical structures on the cotangent bundle using the notions of adapted tangent structure and regular vector fields. We prove that the dynamical covariant derivative on [Formula: see text] fix a nonlinear connection for a given [Formula: see text]-regular vector field. Using the Legendre transformation induced by a regular Hamiltonian, we show that a semi-Hamiltonian vector field on [Formula: see text] corresponds to a semispray on [Formula: see text] if and only if the nonlinear connection on [Formula: see text] is just the canonical nonlinear connection induced by the regular Lagrangian.

Spin-orbit ZORA and four-component Dirac-Coulomb estimation of relativistic corrections to isotropic nuclear shieldings and chemical shifts of noble gas dimers.

Hartree-Fock and density functional theory with the hybrid B3LYP and general gradient KT2 exchange-correlation functionals were used for nonrelativistic and relativistic nuclear magnetic shielding calculations of helium, neon, argon, krypton, and xenon dimers and free atoms. Relativistic corrections were calculated with the scalar and spin-orbit zeroth-order regular approximation Hamiltonian in combination with the large Slater-type basis set QZ4P as well as with the four-component Dirac-Coulomb Hamiltonian using Dyall's acv4z basis sets. The relativistic corrections to the nuclear magnetic shieldings and chemical shifts are combined with nonrelativistic coupled cluster singles and doubles with noniterative triple excitations [CCSD(T)] calculations using the very large polarization-consistent basis sets aug-pcSseg-4 for He, Ne and Ar, aug-pcSseg-3 for Kr, and the AQZP basis set for Xe. For the dimers also, zero-point vibrational (ZPV) corrections are obtained at the CCSD(T) level with the same basis sets were added. Best estimates of the dimer chemical shifts are generated from these nuclear magnetic shieldings and the relative importance of electron correlation, ZPV, and relativistic corrections for the shieldings and chemical shifts is analyzed.

What factors influence the metal-proton spin-spin coupling constants in mercury- and cadmium-substutited rubredoxin?

The indirect metal-proton spin-spin coupling constants between protons in cysteine groups and the mercury or cadmium nucleus have been calculated for a small model of Me-rubredoxin complex (Me = Cd, Hg) by means of density functional theory with zeroth-order regular approximation Hamiltonian (DFT-ZORA). The calculated spin-spin coupling constants, in spite of the moderate size of the model system, are in good agreement with the values measured in NMR experiment, which are in the 0.29-0.56 Hz range for the Cd complex and in the 0.57-2.20 Hz range for the Hg complex. The robustness of the chosen method has been verified by calculations with a number of different exchange-correlation functionals and basis sets. Additionally, it has been shown that the short- and long-distance metal-proton coupling constants are affected mainly by the values of the metal-proton distance and the H-N-C-C dihedral angle.

Regular reduction of controlled Hamiltonian system with symplectic structure and symmetry

In this paper, our goal is to study the regular reduction theory of regular controlled Hamiltonian (RCH) systems with symplectic structure and symmetry, and this reduction is an extension of regular symplectic reduction theory of Hamiltonian systems under regular controlled Hamiltonian equivalence conditions. Thus, in order to describe uniformly RCH systems defined on a cotangent bundle and on the regular reduced spaces, we first define a kind of RCH systems on a symplectic fiber bundle. Then we introduce regular point and regular orbit reducible RCH systems with symmetry by using momentum map and the associated reduced symplectic forms. Moreover, we give regular point and regular orbit reduction theorems for RCH systems to explain the relationships between RpCH-equivalence, RoCH-equivalence for reducible RCH systems with symmetry and RCH-equivalence for associated reduced RCH systems. Finally, as an application we regard rigid body and heavy top as well as them with internal rotors as the regular point reducible RCH systems on the rotation group SO(3) and on the Euclidean group SE(3), as well as on their generalizations, respectively, and discuss their RCH-equivalence. We also describe the RCH system and RCH-equivalence from the viewpoint of port Hamiltonian system with a symplectic structure.

Alternative separation of exchange and correlation energies in multi-configuration range-separated density-functional theory

The alternative separation of exchange and correlation energies proposed by Toulouse et al. [Theor. Chem. Acc. 114, 305 (2005)] is explored in the context of multi-configuration range-separated density-functional theory. The new decomposition of the short-range exchange-correlation energy relies on the auxiliary long-range interacting wavefunction rather than the Kohn-Sham (KS) determinant. The advantage, relative to the traditional KS decomposition, is that the wavefunction part of the energy is now computed with the regular (fully interacting) Hamiltonian. One potential drawback is that, because of double counting, the wavefunction used to compute the energy cannot be obtained by minimizing the energy expression with respect to the wavefunction parameters. The problem is overcome by using short-range optimized effective potentials (OEPs). The resulting combination of OEP techniques with wavefunction theory has been investigated in this work, at the Hartree-Fock (HF) and multi-configuration self-consistent-field (MCSCF) levels. In the HF case, an analytical expression for the energy gradient has been derived and implemented. Calculations have been performed within the short-range local density approximation on H2, N2, Li2, and H2O. Significant improvements in binding energies are obtained with the new decomposition of the short-range energy. The importance of optimizing the short-range OEP at the MCSCF level when static correlation becomes significant has also been demonstrated for H2, using a finite-difference gradient. The implementation of the analytical gradient for MCSCF wavefunctions is currently in progress.

Dual Structures on the Prolongations of a Lie Algebroid

Abstract In the present paper we study the properties of dual structures on the prolongations of a Lie algebroid. We introduce the dynamical covariant derivative on Lie algebroids and prove that the nonlinear connection induced by a regular Lagrangian is compatible with the metric and symplectic structures. The notions of mechanical structure and semi-Hamiltonian section are introduced on the prolongation of the Lie algebroid to its dual bundle and their properties are investigated. Finally, we prove the equivalence between the metric nonlinear connection and semi-Hamiltonian section, using the Legendre transformation induced by a regular Hamiltonian.

Spin–Spin Artificial DNA Intercalated with Silver Cations: Theoretical Prediction

The indirect nuclear spin-spin coupling constants of Ag(+) cation intercalated between imidazole rings in DNA chains are calculated by means of DFT with relativistic effects taken into account by the use of the zeroth-order regular approximation Hamiltonian (DFT-ZORA). The calculations model how the (1)J((15)N,(109)Ag) coupling constant is affected by different types of geometry deformations and by the presence of water, which is simulated by means of the polarizable continuum model and explicitly present water molecules. Calculations for systems containing two and three imidazole pairs are also carried out to model the influence of stacking interactions. The computed (1)J((15)N,(109)Ag) spin-spin coupling constant is in the range of 85-105 Hz (depending on the computational model) and is in good agreement with the experimental value (ca. 92 Hz). This coupling constant is very little affected by the presence of solvent, stacking interactions, and geometry deformations. Such behavior is explained by visualization of the coupling path by means of coupling energy density (CED). Bigger models allow the coupling constant between two adjacent silver ions to be computed, and give a value of approximately 1 Hz, which is probably too small to be of practical interest. The (2)J((15)N,(15)N) value is calculated to be about 2.5 Hz, and is therefore of measurable magnitude.

Full configuration-interaction calculations with the simplest iterative complement method: Merit of the inverse Hamiltonian

The simplest iterative complement (SIC) calculations starting from Hartree-Fock and giving full configuration interaction (CI) at convergence were performed using regular and inverse Hamiltonians. Each iteration step is variational and involves only one variable. The convergence was slow when we used the regular Hamiltonian, but became very fast when we used the inverse Hamiltonian. This difference is due to the Coulomb singularity problem inherent in the regular Hamiltonian; the inverse Hamiltonian does not have such a problem. For this reason, the merit of the inverse Hamiltonian over the regular one becomes even more dramatic when we use a better-quality basis set. This was seen by comparing the calculations due to the minimal and double-\ensuremath{\zeta} basis sets. Similar problematic situations exist in the Krylov sequence and in the Lanczos and Arnoldi methods.

Spin-orbit relativistic long-range corrected time-dependent density functional theory for investigating spin-forbidden transitions in photochemical reactions

A long-range corrected (LC) time-dependent density functional theory (TDDFT) incorporating relativistic effects with spin-orbit couplings is presented. The relativistic effects are based on the two-component zeroth-order regular approximation Hamiltonian. Before calculating the electronic excitations, we calculated the ionization potentials (IPs) of alkaline metal, alkaline-earth metal, group 12 transition metal, and rare gas atoms as the minus orbital (spinor) energies on the basis of Koopmans' theorem. We found that both long-range exchange and spin-orbit coupling effects are required to obtain Koopmans' IPs, i.e., the orbital (spinor) energies, quantitatively in DFT calculations even for first-row transition metals and systems containing large short-range exchange effects. We then calculated the valence excitations of group 12 transition metal atoms and the Rydberg excitations of rare gas atoms using spin-orbit relativistic LC-TDDFT. We found that the long-range exchange and spin-orbit coupling effects significantly contribute to the electronic spectra of even light atoms if the atoms have low-lying excitations between orbital spinors of quite different electron distributions.

Reordering Hydrogen Bonds In The Protein Backbone In Hamiltonian Exchange MD Enhances Sampling Of Conformational Changes

Molecular simulation enables detailed insight into the mechanisms underlying protein conformational changes and is complementary to experiments. Studying a protein folding reaction at atomistic resolution with conventional Molecular Dynamics (MD) is unpractical due to the long time scales involved. These long time scales originate from the presence of local free energy minima from which it is not trivial to escape.The replica exchange MD (REMD) method allows for the escape of such local minima by letting multiple replicas diffuse through temperature space while maintaining the canonical distribution at the temperature of interest. Instead of switching temperatures, the replicas can also exchange (part of) their Hamiltonians. A smooth interpolation between the regular Hamiltonian and one that enables swift conformational changes of the protein (e.g. by lowering barriers) enhances sampling of the conformational space.As hydrogen bonds play an important role in conformational changes of proteins, including folding, it seems natural to bias these bonds specifically. We have developed biasing potentials that allow the breaking and formation of bonds between arbitrary hydrogen bond donors and acceptors in the protein backbone. Employing these potentials in a Hamiltonian REMD scheme we aim to reorder hydrogen bonds in the protein backbone, and thus speed up the conformational sampling.For a simple beta-heptapeptide test system, we show that our hydrogen bond switching scheme is four times more efficient in sampling the conformational space of the peptide as conventional temperature REMD, while it also has the advantage of not having to know the stable states in advance.Finally, we discuss the slow convergence for larger systems, which is often observed in REMD or Hamiltonian exchange. This is caused by replicas having a preference for specific biasing potentials.

Gradients for two-component quasirelativistic methods. Application to dihalogenides of element 116

The authors report the implementation of geometry gradients for quasirelativistic two-component Hartree-Fock and density functional methods using either the zero-order regular approximation Hamiltonian or spin-dependent effective core potentials. The computational effort of the resulting program is comparable to that of corresponding nonrelativistic calculations, as it is dominated by the evaluation of derivative two-electron integrals, which is the same for both types of calculations. Besides the implementation of derivatives of matrix elements of the one-particle Hamiltonian with respect to nuclear displacements, the calculation of the derivative exchange-correlation energy for the open shell case involves complicated expressions because of the noncollinear approach chosen to define the spin density. A pilot application to dihalogenides of element 116 shows how spin-orbit coupling strongly affects the chemistry of the superheavy p-block elements. While these molecules are bent at a scalar-relativistic level, spin-orbit coupling is so strong that only the 7p3/2 atomic orbitals of element 116 are involved in bonding, which favors linear molecular geometries for dihalogenides with heavy terminal halogen atoms.

Analytic energy derivatives for regular approximations of relativistic effects applicable to methods with and without correlation corrections

Analytic expressions are derived for the evaluation of derivatives of the total molecular energy with respect to external parameters (nuclear coordinates, external electric fields, etc.) within the relativistic regular approximation. The presented formalism employs the spectral resolution of the identity avoiding, however, the explicit use of an auxiliary basis set in the calculation of the matrix elements of the regular relativistic Hamiltonian. The final formulas for the total energy and energy derivatives are presented in matrix form suitable for implementation into standard quantum chemical packages. Results of benchmark calculations for gold containing diatomic molecules and for xenone hexafluoride performed at the Hartree–Fock and various correlation corrected levels of theory are presented and discussed.

Solvent effects on heavy atom nuclear spin-spin coupling constants: a theoretical study of Hg-C and Pt-P couplings.

The computation of indirect nuclear spin-spin coupling constants, based on the relativistic two-component zeroth order regular approximate Hamiltonian, has been recently implemented by us into the Amsterdam Density Functional program. Applications of the code for the calculation of one-bond metal-ligand couplings of coordinatively unsaturated compounds containing (195)Pt and (199)Hg, including spin-orbit coupling or coordination effects by solvent molecules, show that relativistic density functional calculations are able to reproduce the experimental findings with good accuracy for the systems under investigation. Spin-orbit effects are rather small for these cases, while coordination of the heavy atoms by solvent molecules has a great impact on the calculated couplings. Experimental trends for different solvents are reproduced. An orbital-based analysis of the solvent effect is presented. The scalar relativistic increase of the coupling constants is of the same order of magnitude as the nonrelativistically obtained values, making a relativistic treatment essential for obtaining quantitatively correct results. Solvent effects can be of similar importance.