Nonrelativistic Wave Functions Research Articles

A four-component quadratic vibronic coupling approach to the Renner–Teller effect in linear triatomic molecules. The , manifold of BrCN+ and ClCN+

In this work, we present the four-component quadratic vibronic coupling model for the description of the Renner–Teller effect (RTE) in the presence of the spin–orbit coupling. The interaction of the two potential energy surfaces emerging from the cationic 2Π states of singly ionized linear triatomic molecules is described by the quadratic coupling constant c for the genuine RT repulsion and the second parameter, d, for a nonconstant spin–orbit coupling varying with the bond angle of the triatomic. The emergence of a linear RT constant in the presence of the spin–orbit operator was originally shown by Poluyanov and Domcke (2004 Chem. Phys. 301 111–27) and is based on the application of the Breit–Pauli Hamiltonian in combination with nonrelativistic wavefunctions. In contrast to this methodology, we generate the diabatic RT Hamiltonian in a 4-spinor basis where the symmetry transformation properties of the electronic and vibrational wavefunctions completely determine the RT matrix structure. Explicit access to highly correlated wavefunctions is not required in our approach. In addition, the four-component vibronic coupling model takes into account the full spatial orbital relaxation upon the inclusion of the spin–orbit coupling and is therefore well suited for heavy systems. The third parameter, p, accounting for a possible pseudo-Jahn–Teller interaction is not considered here, but it does not introduce a principal difficulty. As the initial systems for this study, we considered the BrCN+ and ClCN+ cations and determined the c and d parameters by a numerical fit to accurate adiabatic potential energy surfaces obtained by the relativistic Fock-space coupled-cluster method. New values for the computed linear RT parameter d amount to 14.7 ± 0.5 cm−1 for ClCN+ and 73.2 ± 0.7 cm−1 for BrCN+.

Beyond relativistic mean-field studies of low-lying states in neutron-deficient krypton isotopes

Neutron-deficient krypton isotopes are of particular interest due to the coexistence of oblate and prolate shapes in low-lying states and the transition of ground-state from one dominate shape to another as a function of neutron number. A detailed interpretation of these phenomena in neutron-deficient Kr isotopes requires the use of a method going beyond a mean-field approach that permits to determine spectra and transition probabilities. The aim of this work is to provide a systematic calculation of low-lying state in the even-even 68-86Kr isotopes and to understand the shape coexistence phenomenon and the onset of large collectivity around N=40 from beyond relativistic mean-field studies. The starting point of our method is a set of relativistic mean-field+BCS wave functions generated with a constraint on triaxial deformations (beta, gamma). The excitation energies and electric multipole transition strengths of low-lying states are calculated by solving a five-dimensional collective Hamiltonian (5DCH) with parameters determined by the mean-field wave functions. To examine the role of triaxiality, a configuration mixing of both particle number (PN) and angular momentum (AM) projected axially deformed states is also carried out within the exact generator coordinate method (GCM) based on the same energy density functional. The energy surfaces, the excitation energies of 0^+_2, 2^+_1, 2^+_2 states, as well as the E0 and E2 transition strengths are compared with the results of similar 5DCH calculations but with parameters determined by the non-relativistic mean-field wave functions, as well as with the available data...

Relativistic Two-Dimensional Harmonic Oscillator Plus Cornell Potentials in External Magnetic and AB Fields

The Klein-Gordon (KG) equation for the two-dimensional scalar-vector harmonic oscillator plus Cornell potentials in the presence of external magnetic and Aharonov-Bohm (AB) flux fields is solved using the wave function ansatz method. The exact energy eigenvalues and the wave functions are obtained in terms of potential parameters, magnetic field strength, AB flux field, and magnetic quantum number. The results obtained by using different Larmor frequencies are compared with the results in the absence of both magnetic field (ωL= 0) and AB flux field (ξ=0) cases. Effect of external fields on the nonrelativistic energy eigenvalues and wave function solutions is also precisely presented. Some special cases like harmonic oscillator and Coulombic fields are also studied.

Relativistic quantum mechanics and the quark-pair creation model

In the context of a ${}^{3}\phantom{\rule{-0.16em}{0ex}}{P}_{0}$ quark-pair creation model for the process $\ensuremath{\rho}\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}$, a method is developed for taking relativity into account in the calculation of hadron decays. Following a brief review of relativistic quantum mechanics, an expression is derived for the general relation between a momentum-space, two-particle, instant form wave function in an arbitrary frame and the wave function associated with the c.m. frame. This relation is used to develop relativistic wave functions for the $\ensuremath{\pi}$ and $\ensuremath{\rho}$ mesons. Second quantized state vectors for $\ensuremath{\pi}\ensuremath{\pi}$ and $\ensuremath{\rho}$ states are constructed with the help of these relativistic wave functions. The $\ensuremath{\rho}\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}$ transition amplitude is obtained by using these state vectors to calculate matrix elements of a second quantized ${}^{3}\phantom{\rule{-0.16em}{0ex}}{P}_{0}$ quark-pair creation operator derived from a scalar Lagrangian density. The amplitude differs from the one obtained using nonrelativistic wave functions in the appearance of Wigner rotations. In spite of the complications arising from these rotations the calculation of the relativistic amplitude is reduced to carrying out a two-dimensional integral. The amplitude is of the same form as one derived from an effective $\ensuremath{\rho}\ensuremath{\pi}\ensuremath{\pi}$ Lagrangian except for the presence of a form factor that depends on the magnitude of the three-momentum of a final-state pion. The shape of the form factor is determined by the relativistic $\ensuremath{\pi}$ and $\ensuremath{\rho}$ wave functions. Using the $\ensuremath{\rho}\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}$ transition amplitude as a vertex interaction in a relativistic model of $\ensuremath{\pi}\ensuremath{\pi}$ scattering, the $p$-wave, $\ensuremath{\pi}\ensuremath{\pi}$ scattering amplitude is calculated and fit to data by adjusting the interaction strength and the $\ensuremath{\rho}$ bare mass. This leads to a mass shift and decay width for the $\ensuremath{\rho}$ meson. Using nonrelativistic wave functions to calculate the form factor leads to a negligible mass shift, whereas using the relativistic wave functions leads to a bare $\ensuremath{\rho}$ mass of 855.7 MeV, corresponding to a physical $\ensuremath{\rho}$ mass of 775.5 MeV. The quark-pair creation operator strength parameter for the relativistic case is roughly a factor of 2 larger than that for the nonrelativistic case.

Molecular structure calculations: A unified quantum mechanical description of electrons and nuclei using explicitly correlated Gaussian functions and the global vector representation

We elaborate on the theory for the variational solution of the Schrödinger equation of small atomic and molecular systems without relying on the Born-Oppenheimer paradigm. The all-particle Schrödinger equation is solved in a numerical procedure using the variational principle, Cartesian coordinates, parameterized explicitly correlated Gaussian functions with polynomial prefactors, and the global vector representation. As a result, non-relativistic energy levels and wave functions of few-particle systems can be obtained for various angular momentum, parity, and spin quantum numbers. A stochastic variational optimization of the basis function parameters facilitates the calculation of accurate energies and wave functions for the ground and some excited rotational-(vibrational-)electronic states of H(2) (+) and H(2), three bound states of the positronium molecule, Ps(2), and the ground and two excited states of the (7)Li atom.

Electron-induced proton knockout from neutron rich nuclei

We study the evolution of the (e,e'p) cross section on nuclei with increasing asymmetry between the number of neutrons and protons. The calculations are done within the framework of the nonrelativistic and relativistic distorted-wave impulse approximation. In the nonrelativistic model phenomenological Woods-Saxon and Hartree-Fock wave functions are used for the proton bound-state wave functions, in the relativistic model the wave functions are solutions of Dirac-Hartree equations. The models are first tested against experimental data on 40 Ca and 48Ca nuclei, and then they are applied to a set of spherical calcium isotopes.

Nonrelativistic limit of quantum field theory in inertial and noninertial frames and the principle of equivalence

We discuss the non-relativistic limit of quantum field theory in an inertial frame, in the Rindler frame and in the presence of a weak gravitational field, highlighting and clarifying several subtleties. We study the following topics: (a) While the action for a relativistic free particle is invariant under the Lorentz transformation, the corresponding action for a non-relativistic free particle is not invariant under the Galilean transformation, but picks up extra contributions at the end points. This leads to an extra phase in the non-relativistic wave function under a Galilean transformation, which can be related to the rest energy of the particle even in the non-relativistic limit. (b) We show how the solution to the generally covariant Klein-Gordon equation in a non-inertial frame, which has a time-dependent acceleration, reduces to the quantum mechanical wave function in the presence of an appropriate (time-dependent) gravitational field, in the non-relativistic limit. The extra phase acquired by the non-relativistic wave function in an accelerated frame actually arises from the gravitational time dilation and survives in the non-relativistic limit. (c) We provide a detailed description of the non-relativistic limit of the Feynman propagator in a weak gravitational field, and discuss related issues. [Abridged Abstract]

Quasifree (e,e′p) reactions on nuclei with neutron excess

We study the evolution of the $(e,{e}^{\phantom{\rule{0.16em}{0ex}}\ensuremath{'}}p)$ cross section on nuclei with increasing asymmetry between the number of neutrons and protons. The calculations are done within the framework of the distorted-wave impulse approximation, by adopting nonrelativistic and relativistic models. We compare the results obtained with three different approaches based on the mean-field description for the proton bound-state wave function. In the nonrelativistic model phenomenological Woods-Saxon and Hartree-Fock wave functions are used; in the relativistic model the wave functions are solutions of Dirac-Hartree equations. The models are first tested against experimental data on ${}^{16}$O, ${}^{40}$Ca, and ${}^{48}$Ca nuclei, and then they are applied to calculate $(e,{e}^{\phantom{\rule{0.16em}{0ex}}\ensuremath{'}}p)$ cross sections for a set of spherical calcium and oxygen isotopes. From the comparison of the results obtained for the various isotopes we can infer information about the dependence of the various ingredients of the models on the neutron to proton asymmetry.

Subshell Fitting of Relativistic Atomic Core Electron Densities for Use in QTAIM Analyses of ECP-Based Wave Functions

Scalar-relativistic, all-electron density functional theory (DFT) calculations were done for free, neutral atoms of all elements of the periodic table using the universal Gaussian basis set. Each core, closed-subshell contribution to a total atomic electron density distribution was separately fitted to a spherical electron density function: a linear combination of s-type Gaussian functions. The resulting core subshell electron densities are useful for systematically and compactly approximating total core electron densities of atoms in molecules, for any atomic core defined in terms of closed subshells. When used to augment the electron density from a wave function based on a calculation using effective core potentials (ECPs) in the Hamiltonian, the atomic core electron densities are sufficient to restore the otherwise-absent electron density maxima at the nuclear positions and eliminate spurious critical points in the neighborhood of the atom, thus enabling quantum theory of atoms in molecules (QTAIM) analyses to be done in the neighborhoods of atoms for which ECPs were used. Comparison of results from QTAIM analyses with all-electron, relativistic and nonrelativistic molecular wave functions validates the use of the atomic core electron densities for augmenting electron densities from ECP-based wave functions. For an atom in a molecule for which a small-core or medium-core ECPs is used, simply representing the core using a simplistic, tightly localized electron density function is actually sufficient to obtain a correct electron density topology and perform QTAIM analyses to obtain at least semiquantitatively meaningful results, but this is often not true when a large-core ECP is used. Comparison of QTAIM results from augmenting ECP-based molecular wave functions with the realistic atomic core electron densities presented here versus augmenting with the limiting case of tight core densities may be useful for diagnosing the reliability of large-core ECP models in particular cases. For molecules containing atoms of any elements of the periodic table, the production of extended wave function files that include the appropriate atomic core densities for ECP-based calculations, and the use of these wave functions for QTAIM analyses, has been automated.

Jahn–Teller and spin–orbit coupling effects in transition-metal trifluorides

The effects of linear and higher-order Jahn–Teller couplings as well as spin–orbit coupling in orbitally degenerate electronic states of the transition-metal trifluorides MnF 3 and CoF 3 have been systematically explored with multi-configuration ab initio methods. The adiabatic potential-energy surfaces of low-lying 5 E′ and 5 E″ states have been calculated with the CASSCF method. The spin–orbit coupling is described by matrix elements of the Breit–Pauli operator with nonrelativistic CASSCF wave functions. The Jahn–Teller coupling parameters of the Jahn–Teller active in-plane bending and stretching modes have been determined up to sixth order and fourth order, respectively, in the normal-mode expansion. Vibronic spectra have been computed, employing a Jahn–Teller Hamiltonian up to sixth order in the degenerate bending mode and fourth order in the degenerate stretching mode. These results represent the first ab initio study of the dynamical Jahn–Teller effect in transition-metal trifluorides with inclusion of spin–orbit coupling.

Complete pure vibrational spectrum of HD calculated without the Born-Oppenheimer approximation and including relativistic corrections

All 18 bound pure vibrational levels of the HD molecule have been calculated within the framework that does not assume the Born-Oppenheimer (BO) approximation. The nonrelativistic energies of the states have been corrected for the relativistic effects of the order of ${\ensuremath{\alpha}}^{2}$ (where $\ensuremath{\alpha}$ is the fine structure constant), calculated using the perturbation theory with the nonrelativistic non-BO wave functions being the zero-order approximation. The calculations were performed by expanding the non-BO wave functions in terms of one-center explicitly correlated Gaussian functions multiplied by even powers of the internuclear distance and by performing extensive optimization of the Gaussian nonlinear parameters. Up to 10 000 basis functions were used for each state.

Calculation of electric multipole transition radial matrix elements, oscillator strengths and Einstein coefficients over nonrelativistic radial wave function using Slater type orbitals

In this study, a new method is proposed for evaluating electric multipole transition (radial) matrix elements of the generalized type H nl , n ′ l ′ k in hydrogenic atom and ions using the Slater type orbitals (STOs). The formula obtained allows the determination of all multipole transition matrix elements between two different nonrelativistic radial wave functions R nl and R n ′ l ′ . A comparative study carried out between the results of analytical computations and other numerical simulations shows that the methods agree well and emphasizing thus the effectiveness and accuracy of the proposed analytical expressions. The simple equation thus obtained has been found to be remarkable accurate and has shown a wide range of applicability.

Heavy Quark Spin Symmetry and Heavy Baryons: Electroweak Decays

Heavy quark spin symmetry is discussed in the context of single and doubly heavy baryons. A special attention is paid to the constraints/simplifications that this symmetry imposes on the non-relativistic constituent quark model wave functions and on the b->c semileptonic decays of these hadrons.

The Bondons: The Quantum Particles of the Chemical Bond

By employing the combined Bohmian quantum formalism with the U(1) and SU(2) gauge transformations of the non-relativistic wave-function and the relativistic spinor, within the Schrödinger and Dirac quantum pictures of electron motions, the existence of the chemical field is revealed along the associate bondon particle B̶ characterized by its mass (mB̶), velocity (vB̶), charge (eB̶), and life-time (tB̶). This is quantized either in ground or excited states of the chemical bond in terms of reduced Planck constant ħ, the bond energy Ebond and length Xbond, respectively. The mass-velocity-charge-time quaternion properties of bondons’ particles were used in discussing various paradigmatic types of chemical bond towards assessing their covalent, multiple bonding, metallic and ionic features. The bondonic picture was completed by discussing the relativistic charge and life-time (the actual zitterbewegung) problem, i.e., showing that the bondon equals the benchmark electronic charge through moving with almost light velocity. It carries negligible, although non-zero, mass in special bonding conditions and towards observable femtosecond life-time as the bonding length increases in the nanosystems and bonding energy decreases according with the bonding length-energy relationship , providing this way the predictive framework in which the B̶ particle may be observed. Finally, its role in establishing the virtual states in Raman scattering was also established.

Isotope shifts of the1s22s2(1S0)→1s22p2(1S0) transition in the doubly ionized carbon ionC2+

Highly accurate quantum mechanical calculations are performed for the $1{s}^{2}2{s}^{2}$ (${}^{1}{S}_{0}$) $\ensuremath{\rightarrow}$ $1{s}^{2}2{p}^{2}$ (${}^{1}{S}_{0}$) transition energy in the isotopomers of ${\mathrm{C}}^{2+}$ ion to determine the isotope shifts. Explicitly correlated Gaussian functions and a variational approach that explicitly includes the nuclear motion are employed in the calculations. The leading relativistic and quantum electrodynamics corrections to the transition energy are also calculated using the perturbation theory with the nonrelativistic wave function as the zero-order approximation. It is determined that the ${}^{12}{\mathrm{C}}^{2+}$ transitions energy, which is obtained from the calculations to be 182 519.031 cm${}^{\ensuremath{-}1}$ (vs the experimental value of 182 519.88 cm${}^{\ensuremath{-}1}$, an excellent sub-wave-number agreement) up-shifts by 1.755 cm${}^{\ensuremath{-}1}$ for ${}^{13}{\mathrm{C}}^{2+}$ and by additional 1.498 cm${}^{\ensuremath{-}1}$ for ${}^{14}{\mathrm{C}}^{2+}$. Those shifts are sufficiently large to be measured experimentally.

Isotope shifts of the three lowest S1 states of the B+ ion calculated with a finite-nuclear-mass approach and with relativistic and quantum electrodynamics corrections

We present very accurate quantum mechanical calculations of the three lowest S-states [1s(2)2s(2)((1)S(0)), 1s(2)2p(2)((1)S(0)), and 1s(2)2s3s((1)S(0))] of the two stable isotopes of the boron ion, (10)B(+) and (11)B(+). At the nonrelativistic level the calculations have been performed with the Hamiltonian that explicitly includes the finite mass of the nucleus as it was obtained by a rigorous separation of the center-of-mass motion from the laboratory frame Hamiltonian. The spatial part of the nonrelativistic wave function for each state was expanded in terms of 10,000 all-electron explicitly correlated Gaussian functions. The nonlinear parameters of the Gaussians were variationally optimized using a procedure involving the analytical energy gradient determined with respect to the nonlinear parameters. The nonrelativistic wave functions of the three states were subsequently used to calculate the leading alpha(2) relativistic corrections (alpha is the fine structure constant; alpha=1/c, where c is the speed of light) and the alpha(3) quantum electrodynamics (QED) correction. We also estimated the alpha(4) QED correction by calculating its dominant component. A comparison of the experimental transition frequencies with the frequencies obtained based on the energies calculated in this work shows an excellent agreement. The discrepancy is smaller than 0.4 cm(-1).

DIRAC: A new version of computer algebra tools for studying the properties and behavior of hydrogen-like ions

During recent years, the Dirac package has proved to be an efficient tool for studying the structural properties and dynamic behavior of hydrogen-like ions. Originally designed as a set of Maple procedures, this package provides interactive access to the wave and Green's functions in the non-relativistic and relativistic frameworks and supports analytical evaluation of a large number of radial integrals that are required for the construction of transition amplitudes and interaction cross sections. We provide here a new version of the Dirac program which is developed within the framework of Mathematica (version 6.0). This new version aims to cater to a wider community of researchers that use the Mathematica platform and to take advantage of the generally faster processing times therein. Moreover, the addition of new procedures, a more convenient and detailed help system, as well as source code revisions to overcome identified shortcomings should ensure expanded use of the new Dirac program over its predecessor.

Five lowestS1states of the Be atom calculated with a finite-nuclear-mass approach and with relativistic and QED corrections

We have performed very accurate quantum mechanical calculations of the five lowest $S$ states of the beryllium atom. In the nonrelativistic part of the calculations we used the variational method and we explicitly included the nuclear motion in the Schr\odinger equation. The nonrelativistic wave functions of the five states were expanded in terms of explicitly correlated Gaussian functions. These wave functions were used to calculate the leading ${\ensuremath{\alpha}}^{2}$ relativistic correction ($\ensuremath{\alpha}$ is the fine structure constant) and the ${\ensuremath{\alpha}}^{3}$ quantum electrodynamics (QED) correction. We also estimated the ${\ensuremath{\alpha}}^{4}$ QED correction by calculating its dominant component. A comparison of the experimental transition frequencies with the frequencies obtained based on the energies calculated in this work shows an excellent agreement.

A comparison of the hydrogenlike dipole radial matrix elements with overlap integrals and a step toward explicit expressions of the multipole matrix elements

For nonrelativistic radial hydrogenlike wavefunctions Rnl(r), the ratio ⟨Rνλ|r|Rnl⟩/⟨Rνλ|Rnl⟩ of the dipole to the overlap integral is proportional to (l ⋅ ΔE)−1 for l = λ + 1. ΔE is the energy difference between the shells ν and n. This property holds for discrete or continuous states, and can be derived as a particular case of the Feneuille recurrence formula (Feneuille 1970 C. R. Acad. Sci., Paris B 271 992). This relation and Feinberg's comparable one for ⟨Rνλ|r−2|Rnl⟩ suggest the existence of explicit expressions for the matrix elements ⟨Rνλ|rt|Rnl⟩. For ν ≠ n we are not yet able to write down a general formula valid for any l and λ, but we show with examples (λ = 0, l = 1 and λ = 1, l = 0) how explicit formulae can be obtained for fixed values of l and λ.

Validity of the mass-velocity term in the Breit-Pauli hamiltonian

The mass-velocity term, Hmv = –p4/8m3ec2, of the Breit-Pauli Hamiltonian is studied for its range and capability in providing relativistic corrections to the nonrelativistic kinetic energy, Hke = p2/2me. For this purpose, after a brief critical review of the characteristics of Hmv, the expectation values (Hkr), (Hmv), and (H0) are presented for a series of ground-state neutral atoms (Z = 1–92), where H0 is the relativistic kinetic energy, mec2[(1 + p2/m2ec2)1/2 – 1]. All expectation values are with respect to nonrelativistic wavefunctions. As expected, (Hke) has a larger value than (H0) for all atoms considered, whereas the mass velocity–corrected kinetic energy, (Hke) + (Hmv), is always (i.e., for all values of Z) smaller than (H0), even for the hydrogen atom. Our results also indicate that the efficiency of Hmv in bringing about relativistic corrections to (Hke), decreases linearly as a function of the atomic number, Z. This efficiency starts at 98.41% for Z = 1 (hydrogen) and decreases to 0.639% for Z = 60 (neodymium), beyond which the efficiency is negative and the inclusion of Hmv in the relativistic Hamiltonian produces more error than correction.