Nonrelativistic Hamiltonian Research Articles

DAMA/LIBRA-phase2 in WIMP effective models

The DAMA/LIBRA collaboration has recently released updated results from their search for the annual modulation signal expected from Dark Matter (DM) scattering in their NaI detectors. We have fitted the updated DAMA result for the modulation amplitudes in terms of a Weakly Interacting Massive Particle (WIMP) signal, parameterizing the interaction with nuclei in terms of the most general effective Lagrangian for a WIMP particle spin up to 1/2, systematically assuming dominance of one of the 14 possible interaction terms, and assuming for the WIMP velocity distribution a standard Maxwellian. We find that most of the couplings of the non-relativistic effective Hamiltonian can provide a better fit compared to the standard Spin Independent interaction case, and with a reduced fine-tuning of the three parameters (WIMP mass, WIMP-nucleon effective cross-section and ratio between the WIMP-neutron and the WIMP-proton couplings). Moreover, effective models for which the cross section depends explicitly on the WIMP incoming velocity can provide a better fit of the DAMA data at large values of mχ compared to the standard velocity-independent cross-section due to a different phase of the modulation amplitudes. All the best fit solutions are in tension with exclusion plots of both XENON1T and PICO60.

Testing Lorentz and CPT invariance with ultracold neutrons

In this paper we investigate, within the standard model extension framework, the influence of Lorentz- and CPT-violating terms on gravitational quantum states of ultracold neutrons. Using a semiclassical wave packet, we derive the effective nonrelativistic Hamiltonian which describes the neutrons vertical motion by averaging the contributions from the perpendicular coordinates to the free falling axis. We compute the physical implications of the Lorentz- and CPT-violating terms on the spectra. The comparison of our results with those obtained in the GRANIT experiment leads to an upper bound for the symmetries-violation $c_{\mu\nu} ^{n}$ coefficients. We find that ultracold neutrons are sensitive to the $a _{i} ^{n}$ and $e _{i} ^{n}$ coefficients, which thus far are unbounded by experiments in the neutron sector. We propose two additional problems involving ultracold neutrons which could be relevant for improving our current bounds; namely, gravity-resonance-spectroscopy and neutron whispering gallery wave.

Accurate ab initio calculations of spectroscopic constants and properties of BeLi+

We have calculated the spectroscopic constants and the molecular properties, for the ground-state (X1Σ+) of BeLi+, such as dipole moment, quadrupole moment and dipole polarizability at different levels of correlation: many-body perturbation theory (MP2), coupled cluster method with single and double excitations (CCSD) and CCSD with perturbative triples (CCSD(T)). The correlation consistent polarized valence cc-pVXZ (X = D, T, Q) basis sets and also their augmented counterparts are used together with the non-relativistic and relativistic Hamiltonians. The results are extrapolated to the complete basis set limit (CBS) using exponential-Gaussian functions. Thus, accurate and reliable results for BeLi+ with the maximum possible error bars on them are reported. The effects of diagonal Born-Oppenheimer corrections (DBOC) and of electron correlations beyond the CCSD(T) level of approximation are also estimated. In addition, by solving the vibrational Schrödinger equation using the potential energy curve obtained at CCSD(T)/QZ level, we have calculated the vibrational parameters such as energies, rotational constants, permanent dipole moments, spontaneous and black-body radiation (BBR) induced transition rates between different vibrational levels of the electronic ground state. The lifetime of the lowest rovibrational state (v=0 and J=0) at room temperature, is found to be 4.67s. Further, the electronic and the vibrational parameters of a few low-lying electronic excited states: 13Σ+, 13Π, 2–41Σ+, and 1–21Π are calculated using the EOM-CCSD/QZ method. The dipole moments for the electronic transitions from the ground- to the excited singlet states are also reported.

Spin\u2013spin coupling constants in hbox {HC}{equiv }hbox {CXH}_3 molecules; hbox {X}{=}hbox {C}, Si, Ge, Sn and Pb

The nuclear spin–spin coupling constants in a series of acetylene derivatives hbox {HC}{equiv }hbox {CXH}_3, where X is C, Si, Ge, Sn, Pb, have been calculated employing both coupled-cluster theory (CC) and density functional theory (DFT), the latter in nonrelativistic and relativistic four-component approach with different exchange–correlation functionals and basis sets. In addition, property derivatives with respect to molecular geometry parameters have been computed with nonrelativistic and relativistic Hamiltonians in order to evaluate the usefulness of the calculated nonrelativistic vibrational corrections. Generally, the CC method reproduces the experimental values somewhat better than DFT. In the case of the latter, the performance of B3LYP functional was the most satisfactory. The relativistic effects become noticeable for the couplings including heavy atom in tin- and lead-containing molecules. The calculations of the derivatives of the coupling constants indicate that these derivatives are even more sensitive to the relativistic effects than the corresponding coupling constants.

Relativistic pseudo-Jahn-Teller effect in the square-planar molecules with a central atom

ABSTRACTWe consider the relativistic multi-mode pseudo-Jahn-Teller effect (2Eu + 2A2u) × (big + b2g + eu) for square-planar molecular complexes with a heavy central atom and the odd number of electrons.All 32 elements of the double spin group D4h are determined in the form of space-matrix operators. The 6 × 6 vibronic matrix is derived in the quadratic (with respect to the normal coordinates) approximation for the contributions of electrostatic (non-relativistic) Hamiltonian and in linear approximation for contribution of spin-orbital coupling. Vibronic matrix is represented on the basis of double-value irreducible representations of the symmetry group D4h

Inclusion of orbital relaxation and correlation through the unitary group adapted open shell coupled cluster theory using non-relativistic and scalar relativistic Hamiltonians to study the core ionization potential of molecules containing light to medium-heavy elements.

The orbital relaxation attendant on ionization is particularly important for the core electron ionization potential (core IP) of molecules. The Unitary Group Adapted State Universal Coupled Cluster (UGA-SUMRCC) theory, recently formulated and implemented by Sen et al. [J. Chem. Phys. 137, 074104 (2012)], is very effective in capturing orbital relaxation accompanying ionization or excitation of both the core and the valence electrons [S. Sen et al., Mol. Phys. 111, 2625 (2013); A. Shee et al., J. Chem. Theory Comput. 9, 2573 (2013)] while preserving the spin-symmetry of the target states and using the neutral closed-shell spatial orbitals of the ground state. Our Ansatz invokes a normal-ordered exponential representation of spin-free cluster-operators. The orbital relaxation induced by a specific set of cluster operators in our Ansatz is good enough to eliminate the need for different sets of orbitals for the ground and the core-ionized states. We call the single configuration state function (CSF) limit of this theory the Unitary Group Adapted Open-Shell Coupled Cluster (UGA-OSCC) theory. The aim of this paper is to comprehensively explore the efficacy of our Ansatz to describe orbital relaxation, using both theoretical analysis and numerical performance. Whenever warranted, we also make appropriate comparisons with other coupled-cluster theories. A physically motivated truncation of the chains of spin-free T-operators is also made possible by the normal-ordering, and the operational resemblance to single reference coupled-cluster theory allows easy implementation. Our test case is the prediction of the 1s core IP of molecules containing a single light- to medium-heavy nucleus and thus, in addition to demonstrating the orbital relaxation, we have addressed the scalar relativistic effects on the accuracy of the IPs by using a hierarchy of spin-free Hamiltonians in conjunction with our theory. Additionally, the contribution of the spin-free component of the two-electron Gaunt term, not usually taken into consideration, has been estimated at the Self-Consistent Field (ΔSCF) level and is found to become increasingly important and eventually quite prominent for molecules with third period atoms and below. The accuracies of the IPs computed using UGA-OSCC are found to be of the same order as the Coupled Cluster Singles Doubles (ΔCCSD) values while being free from spin contamination. Since the UGA-OSCC uses a common set of orbitals for the ground state and the ion, it obviates the need of two N5 AO to MO transformation in contrast to the ΔCCSD method.

On the exact rotational and internal Hamiltonian for a non-relativistic closed many-body system

Without applying Born-Oppenheimer approximation, the non-relativistic Hamiltonian can be separated into Hamiltonians for the translation of the center of mass and for the rotational and internal motions of the closed many-body system. This exact rotational and internal Hamiltonian can be expressed in terms of three Euler angles for three independent rotations of the system and the rotated Jacobi coordinates for the internal motions.

Short note: Hamiltonian for a particle with position-dependent mass

An approach for obtaining a Schrodinger equation for a spinless nonrelativistic particle with a position-dependent mass is proposed. Rather than starting with the nonrelativistic hamiltonian for a free particle, we begin with its relativistic completion in the form of a Klein–Gordon equation and then reduce it to obtain the nonrelativistic limit. This type of procedure avoids the usual ordering ambiguities that commonly arise in obtaining a Schrodinger equation for a particle with position-dependent mass.

Relativistic Dynamics of a Quantum System

In this paper, we examine quantum systems with relativistic dynamics. We show that for a successful description of these systems, the application of Galilei invariant nonrelativistic Hamiltonian is necessary. To modify this Hamiltonian to relativistic dynamics, we require precise relativistic kinetic energy operators instead of nonrelativistic ones for every internal (Jacobi) coordinate. Finally, we introduce and investigate the Schr&oumldinger equation with relativistic dynamics for two-particle systems with harmonic oscillator and Coulomb potentials.

All-electron Gaussian basis sets of double zeta quality for the actinides.

For the actinides, two segmented all-electron basis sets of valence double zeta quality plus polarization functions (DZP) are developed. One of them must be used along with the non-relativistic Hamiltonian, whereas the other with the Douglas-Kroll-Hess (DKH) one. Adding diffuse functions of s, p, d, f, and g symmetries to the non-relativistic and relativistic sets, augmented basis sets are developed. These functions are essential to describe correctly electrons far away from the nuclei. For some compounds, geometric parameters, atomic charges and valence orbital populations of the actinides, and bond dissociation energies are calculated using the Becke 3-parameter (exchange) and the Lee, Yang, and Parr (correlation) functional in conjunction with the DZP-DKH basis set. For Am and No, the static electric mean dipole polarizabilities are also reported. Comparison with benchmark theoretical and experimental values found in the literature is carried out. It is verified that the performances of the relativistic compact size basis sets generated in this work are regular, efficient, and reliable. They will be extremely helpful in molecular property calculations that need explicitly to consider the core electrons.

The Relativistic Effects on the Carbon-Carbon Coupling Constants Mediated by a Heavy Atom.

The (2)JCC, (3)JCC, and (4)JCC spin-spin coupling constants in the systems with a heavy atom (Cd, In, Sn, Sb, Te, Hg, Tl, Pb, Bi, and Po) in the coupling path have been calculated by means of density functional theory. The main goal was to estimate the relativistic effects on spin-spin coupling constants and to explore the factors which may influence them, including the nature of the heavy atom and carbon hybridization. The methods applied range, in order of reduced complexity, from the Dirac-Kohn-Sham (DKS) method (density functional theory with four-component Dirac-Coulomb Hamiltonian), through DFT with two- and one-component zeroth-order regular approximation (ZORA) Hamiltonians, to scalar effective core potentials (ECPs) with the nonrelativistic Hamiltonian. The use of DKS and ZORA methods leads to very similar results, and small-core ECPs of the MDF and MWB variety reproduce correctly the scalar relativistic effects. Scalar relativistic effects usually are larger than the spin-orbit coupling effects. The latter tend to influence the most the coupling constants of the sp(3)-hybridized carbon atoms and in compounds of the p-block heavy atoms. Large spin-orbit coupling contributions for the Po compounds are probably connected with the inverse of the lowest triplet excitation energy.

Tune-out wavelength around 413 nm for the helium23S1state including relativistic and finite-nuclear-mass corrections

The tune-out wavelength at 413 nm for the $2{\phantom{\rule{0.28em}{0ex}}}^{3}{S}_{1}$ state of helium is expected to be sensitive to finite nuclear mass, relativistic, and quantum electrodynamic (QED) corrections, which provides a scheme for testing atomic structure theory [J. Mitroy and L.-Y. Tang, Phys. Rev. A 88, 052515 (2013)]. In the present work, a large-scale full-configuration-interaction calculation based on both the Dirac-Coulomb-Breit Hamiltonian and the nonrelativistic Hamiltonian is performed for the dynamic dipole polarizabilities of helium in the $2{\phantom{\rule{0.28em}{0ex}}}^{3}{S}_{1}$ state. The tune-out wavelengths for the magnetic sublevels ${M}_{J}=0$ and ${M}_{J}=\ifmmode\pm\else\textpm\fi{}1$ are determined to be 413.0801(4) nm and 413.0859(4) nm, respectively, at sub-ppm accuracy, including finite nuclear mass and relativistic corrections. Our value for the ${M}_{J}=1$ sublevel agrees with the measured value of 413.0938(20)(9) nm [B. M. Henson et al., Phys. Rev. Lett. 115, 043004 (2015)] at the level of 19 ppm. The discrepancy between these two values is mainly due to the uncalculated QED contribution. Our current value confirms quantitatively the prediction of Mitroy and Tang. Also, for the state of $2{\phantom{\rule{0.28em}{0ex}}}^{3}{S}_{1}$ we find that the corrections due to finite nuclear mass and relativistic effects to the static dipole polarizability of 315.7227(4)${a}_{0}^{3}$ are about 600 ppm and 310 ppm, respectively, which are about 1.4 and 5.4 times larger than those for the ground state. A measurement at the level of 10 ppm for the static dipole polarizability of helium in $2{\phantom{\rule{0.28em}{0ex}}}^{3}{S}_{1}$ can be used to determine the transition matrix element between $2{\phantom{\rule{0.28em}{0ex}}}^{3}S$ and $2{\phantom{\rule{0.28em}{0ex}}}^{3}P$ at the level of ${10}^{\ensuremath{-}5}$.

Heavy Element Effects in the Diagonal Born-Oppenheimer Correction within a Relativistic Spin-Free Hamiltonian.

Methodologies beyond the Born-Oppenheimer (BO) approximation are nowadays important to explain high precision spectroscopic measurements. Most previous evaluations of the BO correction are, however, focused on light-element molecules and based on a nonrelativistic Hamiltonian, so no information about the BO approximation (BOA) breakdown in heavy-element molecules is available. The present work is the first to investigate the BOA breakdown for the entire periodic table, by considering scalar relativistic effects in the Diagonal BO correction (DBOC). In closed shell atoms, the relativistic EDBOC scales as Z(1.25) and the nonrelativistic EDBOC scales as Z(1.17), where Z is the atomic number. Hence, we found that EDBOC becomes larger in heavy element atoms and molecules, and the relativistic EDBOC increases faster than nonrelativistic EDBOC. We have further investigated the DBOC effects on properties such as potential energy curves, spectroscopic parameters, and various energetic properties. The DBOC effects for these properties are mostly affected by the lightest atom in the molecule. Hence, in X2 or XAt molecule (X = H, Li, Na, K, Rb, and Cs) the effect of DBOC systematically decreases when X becomes heavier but in HX molecules, the effect of DBOC seems relatively similar among all the molecules.

ConstrainingCPT-even and Lorentz-violating nonminimal couplings with the electron magnetic and electric dipole moments

We analyze some dimension-five $CPT$-even and Lorentz-violating nonminimal couplings between fermionic and gauge fields in the context of the Dirac equation. After evaluating the nonrelativistic Hamiltonian, we discuss the behavior of the terms under discrete symmetries and analyze the implied effects. We then use the anomalous magnetic dipole moment and electron electric dipole moment measurements to reach upper bounds of 1 part in ${10}^{20}$ and ${10}^{24}\text{ }\text{ }(\mathrm{eV}{)}^{\ensuremath{-}1}$, improving the level of restriction on such couplings by at least 8 orders of magnitude. These upper bounds are also transferred to the Sun-centered frame by considering the Earth's rotational motion.

Stability of the complex generalized Hartree-Fock equations

For molecules with complex and competing magnetic interactions, it is often the case that the lowest energy Hartree-Fock solution may only be obtained by removing the spin and time-reversal symmetry constraints of the exact non-relativistic Hamiltonian. To do so results in the complex generalized Hartree-Fock (GHF) method. However, with the loss of variational constraints comes the greater possibility of converging to higher energy minima. Here, we report the implementation of stability test of the complex GHF equations, along with an orbital update scheme should an instability be found. We apply the methodology to finding the local minima of several spin-frustrated hydrogen rings, as well as the non-collinear molecular magnet Cr3, illustrating the utility of the broken symmetry GHF method and some of its lesser-known nuances.

Bottomonium spectroscopy and radiative transitions involving theχbJ(1P,2P)states atBaBar

We use $(121\pm1)$ million $\Upsilon(3S)$ and $(98\pm1)$ million $\Upsilon(2S)$ mesons recorded by the BABAR detector at the PEP-II $e^+e^-$ collider at SLAC to perform a study of radiative transitions involving the $\chi_{b\mathrm{J}}(1P,2P)$ states in exclusive decays with $\mu^+\mu^-\gamma\gamma$ final states. We reconstruct twelve channels in four cascades using two complementary methods. In the first we identify both signal photon candidates in the Electromagnetic Calorimeter (EMC), employ a calorimeter timing-based technique to reduce backgrounds, and determine branching-ratio products and fine mass splittings. These results include the best observational significance yet for the $\chi_{b0}(2P)\rightarrow\gamma\Upsilon(2S)$ and $\chi_{b0}(1P)\rightarrow\gamma\Upsilon(1S)$ transitions. In the second method, we identify one photon candidate in the EMC and one which has converted into an $e^+e^-$ pair due to interaction with detector material, and we measure absolute product branching fractions. This method is particularly useful for measuring $\Upsilon(3S)\rightarrow\gamma\chi_{b1,2}(1P)$ decays. Additionally, we provide the most up-to-date derived branching fractions, matrix elements and mass splittings for $\chi_b$ transitions in the bottomonium system. Using a new technique, we also measure the two lowest-order spin-dependent coefficients in the nonrelativistic QCD Hamiltonian.

On planar quantum dynamics of a magnetic dipole moment in the presence of electric and magnetic fields

The planar quantum dynamics of a neutral particle with a magnetic dipole moment in the presence of electric and magnetic fields is considered. The criteria to establish the planar dynamics reveal that the resulting nonrelativistic Hamiltonian has a simplified expression without making approximations, and some terms have crucial importance for the system dynamics.

Dirac equation in very special relativity for hydrogen atom

In this work, we study the modified Dirac equation in the framework of very special relativity (VSR). The low-energy regime is accessed and the nonrelativistic Hamiltonian is obtained. It turns out that this Hamiltonian is similar to that achieved from the Standard Model Extension (SME) via coupling of the spinor field to a Lorentz-violating term, but new features arise inherited from the non-local character of the VSR. In addition, the implications of the VSR-modified Lorentz symmetry on the spectrum of a hydrogen atom are determined by calculating the first-order energy corrections in the context of standard quantum mechanics. Among the results, we highlight that the modified Hamiltonian provides non-vanishing corrections which lift the degeneracy of the energy levels and allow us to find an upper bound upon the VSR-parameter.

Microscopic origin of the relativistic splitting of surface states

Spin-orbit splitting of surface states is analyzed within and beyond the Rashba model using as examples the (111) surfaces of noble metals, Ag2Bi surface alloy, and topological insulator Bi2Se3. The ab initio analysis of relativistic velocity proves the Rashba model to be fundamentally inapplicable to real crystals. The splitting is found to be primarily due to a spin-orbit induced in-plane modification of the wave function, namely to its effect on the nonrelativistic Hamiltonian. The usual Rashba splitting -- given by charge distribution asymmetry -- is an order of magnitude smaller.

Complex Absorbing Potential Method for the Perturbed Dirac Operator

The Complex Absorbing Potential (CAP) method is widely used to compute resonances in quantum chemistry, both for nonrelativistic and relativistic Hamiltonians. In the semiclassical limit ℏ → 0 we consider resonances near the real axis and we establish the CAP method rigorously for the perturbed Dirac operator by proving that individual resonances are perturbed eigenvalues of the nonselfadjoint CAP Hamiltonian, and vice versa. The proofs are based on pseudodifferential operator theory and microlocal analysis.