Infinite Nuclear Mass Research Articles

Ground-state energy of H−: a critical test of triple basis sets

We report an improved variational upper bound for the ground state energy of H− using Hylleraas-like wave functions in the form of a triple basis set having three distinct distance scales. The extended precision DQFUN of Bailey, allowing for 70 decimal digit arithmetic, is implemented to retain sufficient precision. Our result exceeds the previous record [A. M. Frolov, Euro. J. Phys. D 69, 132 (2015)], indicating that the Hylleraas triple basis set exhibits comparable convergence to the widely used pseudorandom all-exponential basis sets, but the numerical stability against roundoff error is much better. It is argued that the three distance scales have a clear physical interpretation. The new variational bound for infinite nuclear mass is –0.527 751 016 544 377 196 590 814 478 a.u. New variational bounds are also presented for the finite mass cases of the hydrogen, deuterium, and tritium negative ions H−, D−, and T−.

Influence of hydrogen-like nucleus mass on electronic state

In an isolated two-body composite system, the discussion of how the change of one body affects the state of the other will help to know the relation of a single particle's mixed and pure states. Given 5 one-dimensional hydrogen-like atoms models, each Coulomb interaction potential keeps invariant, while the masses of the nuclei are different. These two-body composite systems stay in their respective entangled state, each electron stays in a mixed state. If we suppose a one-dimensional hydrogen atom model having infinite nuclear mass, the electron stays in a pure state. We select position representation. The wave function of the ground state of the atom has the form of the square root of a <i>δ</i> function. To avoid calculation trouble of <i>δ</i> function, we select the first excited state and the superposed state of the first and the second excited states. We compare the two pure states, the first excited state and the superposed state, with those corresponding mixed states by fidelity and <i>l</i><sub>1</sub> norm coherence, and calculate the purities of the mixed states. The summations become integrations due to the position variable having a continuous eigenvalue spectrum. We investigate the changes in these quantities with the increase of the nuclear mass. The results show that the purities of the mixed states and the fidelities increase with the nuclear mass increasing. However, the coherences of the mixed states decrease with the nuclear mass increasing. This can be explained as that a mixed state with non-zero coherence may approach to a pure eigenstate, while the latter has zero coherence in the eigenspace. These mean that the greater a nuclear mass is, the closer the mixed state approaches to the corresponding pure state. Therefore, the two pure states are the approximations of the corresponding mixed states. The entangled state of the electron and the nucleus is related with the nuclear mass and the Coulomb interaction potential. So, each electron mixed state and its coherence are related with the nucleus and their Coulomb interaction potential. If the nuclear mass is great, the nucleus is approximately motionless or its state is approximately unchanged, and the Coulomb interaction potential approximates to the external Coulomb potential of the electron. The electron approximately stays in a pure state. The state and its coherence are related with the nucleus and the Coulomb interaction. From other point of view, the entangled state of the nucleus and the electron approximates to the product state of the unchanged nucleus state and the electron state. In this case, an electron mixed state approximates to its corresponding pure state, and then these states and their coherences are all related with the nucleus and the Coulomb interaction.

On highly accurate, bound state calculations of the molecular diatomic ions

Results of highly accurate calculations obtained recently for a number of Coulomb two-centre ions with the unit electrical charges and are presented. In particular, we have determined (to very high accuracy) the total energies and other bound state properties of the ground (bound) states in the two-centre adiabatic (or molecular) H, D, HD, HT, T and DT ions. In these computations we used the new masses of hydrogen isotopes, which have been measured in recent high-energy experiments. Theory of soft adiabatic three-body systems with the unit electrical charges is also discussed. An accurate mass-interpolation formula is derived and tested for the total energies of model two-centre ions with very heavy masses of the point particles ( 100,000 ). This formula allows one to evaluate (to high accuracy) the total energies and other bound state properties of arbitrary two-centre ions, including very heavy systems. We also determine the upper mass limit for applicability of our universal expansion in the relative and/or perimetric coordinates. Highlights An effective procedure is developed for highly accurate numerical computations of bound states in the two-center molecular ions. Properties of many actual two-center ions, including the both and ions, have been determined to very high accuracy. Theory of the soft adiabatic (two-center) three-body systems is developed. Properties of these ions are now known to high accuracy. Accurate numerical method which allows one to predict the total energies and other properties of truly adiabatic systems with very large (or infinite) nuclear masses has been proposed, applied and studied in detail.

Upper and lower bounds to atomic radial position moments

A procedure is developed to generate rigorous rough upper bounds to radial moments [r2n] with integer n ≥ − 2 for the ground and excited states of atoms and molecules. These rough upper bounds to [r2n] enable the calculation of accurate upper and lower bounds to the lesser moment [rn] through existing and new formulas that utilize a lower bound to the square overlap of a trial function and true wave function (in this case through an energy lower bound via the Eckart formula). Error bars to [r−1], [r], [r2], and [r4] are calculated for the ground state of the non-relativistic infinite nuclear mass helium atom to yield expectation values within 0.037%, 0.018%, 0.039%, and 0.24% respectively, of the true values. As a byproduct of this investigation, a new formula for the error bar to observables is derived which is a slight improvement upon similar error bars. Also Lieb’s limit on the number of electrons that an atom can bind is reproduced.

Magnetic orbit-orbit interaction involving electrons and the nucleus orbiting around the center of mass in 1S and 1P Rydberg states of helium. Finite-nuclear-mass calculations with explicitly correlated Gaussian functions.

An finite-nuclear-mass (FNM) approach is employed to describe the magnetic orbit-orbit interaction of the electrons and the nucleus orbiting around the center of mass of the helium atom in 1S and 1P Rydberg states. The calculations are performed using explicitly correlated Gaussian functions to expand the states’ wave functions. The non-linear parameters of the Gaussians are optimized using a variational approach employing the analytical energy gradient. An FNM and INM (infinite-nuclear-mass) approaches are used in the optimization and compared.

Properties of the finite-mass helium ground state

ABSTRACTWe describe a simple method of incorporating the finite mass of the nucleus directly into atomic variational Monte Carlo calculations. To test this algorithm we computed the energy and 20 other properties of 4He. We then compared these values with those obtained from our earlier infinite nuclear mass algorithm. All of our expectation values are in excellent agreement with previous results on this system.

Electron–positron pair production in ion collisions at low velocity beyond Born approximation

We derive the spectrum and the total cross section of electromagnetic $e^{+}e^{-}$ pair production in the collisions of two nuclei at low relative velocity $\beta$. Both free-free and bound-free $e^{+}e^{-}$ pair production is considered. The parameters $\eta_{A,B}=Z_{A,B}\alpha$ are assumed to be small compared to unity but arbitrary compared to $\beta$ ($Z_{A,B}$ are the charge numbers of the nuclei and $\alpha$ is the fine structure constant). Due to a suppression of the Born term by high power of $\beta$, the first Coulomb correction to the amplitude appears to be important at $\eta_{A,B}\gtrsim \beta$. The effect of a finite nuclear mass is discussed. In contrast to the result obtained in the infinite nuclear mass limit, the terms $\propto M^{-2}$ are not suppressed by the high power of $\beta$ and may easily dominate at sufficiently small velocities.

The adiabatic limit of the exact factorization of the electron-nuclear wave function.

We propose a procedure to analyze the relation between the exact factorization of the electron-nuclear wave function and the Born-Oppenheimer approximation. We define the adiabatic limit as the limit of infinite nuclear mass. To this end, we introduce a unit system that singles out the dependence on the electron-nuclear mass ratio of each term appearing in the equations of the exact factorization. We observe how non-adiabatic effects induced by the coupling to the nuclear motion affect electronic properties and we analyze the leading term, connecting it to the classical nuclear momentum. Its dependence on the mass ratio is tested numerically on a model of proton-coupled electron transfer in different non-adiabatic regimes.

Analysis of a two dimensional molecular Berry phase system

Discontinuities in the Born–Oppenheimer (infinite nuclear masses) limit of an artificial molecular system, absent for finite masses, have led Min et al. (2014) to a questioning of Berry phases in some molecular systems. We show here by a rigorous following of the analysis of Longuet-Higgins et al. (1958), that (a) the same phenomena of discontinuities occurs in typical degenerate molecular systems with regular Berry-phase effects, (b) they arise from the non-unique definition of the key quantity, electronic ‘vector field’, and (c) they can be removed by a different choice of normalization.

Effect of nuclear motion on the critical nuclear charge for two-electron atoms

A variational method for calculating the critical nuclear charge ${Z}_{C}$ required for the binding of a nucleus to two electrons is reported. The method is very effective and performs well compared to the traditional variational principle for calculating energy. The critical nuclear charge, which corresponds to the minimum charge required for the atomic system to have at least one bound state, has been calculated for heliumlike systems with both infinite and finite nuclear masses. The value of ${Z}_{C}=0.911\phantom{\rule{0.16em}{0ex}}028\phantom{\rule{0.16em}{0ex}}2(3)$ is in very good agreement with recent values in the literature for two-electron atoms with an infinite nuclear mass. When nuclear motion is considered, the value for ${Z}_{C}$ varies from 0.911 030 3(2) for that with a nuclear mass of Ne (the largest heliogenic system considered) to $0.921\phantom{\rule{0.16em}{0ex}}802\phantom{\rule{0.16em}{0ex}}4(4)$ for a system with the nuclear mass of a positron. In all cases the energy varies smoothly as $Z\ensuremath{\rightarrow}0$. It is found that for the finite nuclear mass case, in agreement with previous work for the fixed nucleus mass system, the outer electron remains localized near the nucleus at $Z={Z}_{C}$. Additionally, the electron probability distribution is calculated to determine the behavior of the electrons at low $Z$.

Is the Molecular Berry Phase an Artifact of the Born-Oppenheimer Approximation?

We demonstrate that the molecular Berry phase and the corresponding nonanalyticity in the electronic Born-Oppenheimer wave function is, in general, not a true topological feature of the exact solution of the full electron-nuclear Schrödinger equation. For a numerically exactly solvable model we show that a nonanalyticity, and the associated geometric phase, only appear in the limit of infinite nuclear mass, while a perfectly smooth behavior is found for any finite nuclear mass.

The two-component quantum theory of atoms in molecules (TC-QTAIM): tensor formulation and its implications

In this contribution, the tensor formulation of the two-component quantum theory of atoms in molecules is addressed in detail. The subsystem hypervirial and the regional atomic theorems are presented in their local forms enabling one to introduce the tensor formulation easily. Accordingly, the two-component force, virial, torque, power, current and continuity theorems are introduced. Then, by proposing a model wavefunction for quantum nuclei, the nuclear stress tensor density is analytically evaluated. Based on the derived analytical stress tensor density, the regional nuclear properties are also computed analytically and their asymptotic trend at the infinite nuclear mass limit is inspected. It is demonstrated that in the infinite nuclear mass limit, the nuclear contribution in basin properties is null, while electronic contributions reduce to that computed within context of the orthodox quantum theory of atoms in molecules. Subsequently, the regional energies as well as the regional components of the potential energy are derived for two-component systems. The computed analytical equation for the nuclear contribution in basin energies is of particular importance. Finally, the electric dipole moment of a typical two-component system is decomposed into electronic/nuclear charge transfer and polarization contributions revealing the fine structure of the charge density distribution for both electrons and quantum nuclei.

Variational upper bounds for low-lying states of lithium

We present improved calculations of variational energy eigenvalues for the $1{s}^{2}2s {}^{2}S$, $1{s}^{2}3s {}^{2}S$, and $1{s}^{2}2p {}^{2}P$ states of lithium using basis sets with up to 30 224 terms in Hylleraas coordinates. The nonrelativistic energies for infinite nuclear mass are $\ensuremath{-}7.478 060 323 910 143 7(45)$ a.u. for $1{s}^{2}2s {}^{2}S$, $\ensuremath{-}7.354 098 421 444 316 4(32)$ a.u. for $1{s}^{2}3s {}^{2}S$, and $\ensuremath{-}7.410 156 532 651 6(5)$ a.u. for $1{s}^{2}2p {}^{2}P$, which represent the most accurate variational upper bounds to date. An important advantage of the basis sets with multiple distance scales is their exceptional numerical stability.

Relativistic corrections to He I transition ratesThis paper was presented at the International Conference on Precision Physics of Simple Atomic Systems, held at École de Physique, les Houches, France, 30 May – 4 June, 2010.

The relativistic corrections to the theoretical oscillator strengths of light elements such as helium are typically less than 0.1% and usually are ignored. However, they can be important for comparisons with the most accurate experiments, and they rapidly increase in magnitude with increasing nuclear charge. We have begun with the spin-forbidden electric-dipole transitions of neutral helium, using calculations consisting of (1) extremely accurate wave functions without relativistic corrections for both infinite and finite nuclear mass, (2) spin-changing matrix elements through the perturbations of the wave functions by the spin-orbit and spin-other-orbit Breit operators, (3) the use of pseudostates in the sums over all the intermediate states including the continuum, and (4) the inclusion as perturbers of the 1S0 and 3S1 states the pseudostates corresponding to the doubly excited npn′p 3P 0e and npn′p 1P 1e terms, respectively. As examples of these calculations, we present oscillator strengths for the transitions 1 1S0–2 3P1, 2 1S0–2 3P1, 2 3S1–2 1P1, 2 1P1–3 3D1,2, and 2 3P1,2–3 1D2.

Finite- to zero-range relativistic mean-field interactions

We study the relation between the finite-range (meson-exchange) and zero-range (point-coupling) representations of effective nuclear interactions in the relativistic mean-field framework. Starting from the phenomenological interaction DD-ME2 with density-dependent meson-nucleon couplings, we construct a family of point-coupling effective interactions for different values of the strength parameter of the isoscalar-scalar derivative term. In the meson-exchange picture this corresponds to different values of the σ-meson mass. The parameters of the isoscalar-scalar and isovector-vector channels of the point-coupling interactions are adjusted to nuclear matter and ground-state properties of finite nuclei. By comparing results for infinite and semi-infinite nuclear matter, ground-state masses, charge radii, and collective excitations, we discuss constraints on the parameters of phenomenological point-coupling relativistic effective interaction.

Which masses are vibrating or rotating in a molecule?

The traditional formulation of a molecular wave function describing the motion of the electrons and the nuclei in terms of the Born hierarchy does not allow us to account for the participation of the electrons in vibrational or rotational motion, except by resorting to a fully non-adiabatic treatment, abandoning completely the concept of a single potential energy surface for one electronic state of the molecule. An alternative approach is presented that is based on an exact separation in all possible dissociation limits. One obtains a wave function of a generalized valence bond form in terms of neutral fragments with the adiabatic ansatz in the limit of infinite nuclear masses. An energy expression is obtained correct to O(M −2), where M is a representative nuclear mass in units of the electron mass. This expression involves a single potential energy surface, but a geometry-dependent reduced mass for vibration and rotation, in which the participation of the electrons is taken care of. The vibrational mass differs from the rotational mass. For and H2, approximations to these effective masses are obtained for the first time in a simple way in terms of a LCAO or a Heitler–London formalism, respectively. In the electron participates to roughly 60% in vibration, but only about 30% in rotation. Similar values are found for H2. Such a fractional participation appears typical for valence electrons, whereas inner-shell electrons are likely to participate fully both in vibration and rotation.

Adiabatic Jacobi corrections for H2+-like systems

The Coulomb three-body problem in Jacobi coordinates was solved by treating the distance of the particles having equal charge as a parameter. This method allows computation of electronic energies with finite nuclear masses while maintaining the notion of a potential energy curve. The rotationless ground-state electronic and the so-called adiabatic Jacobi correction (AJC) energies are presented for H2+, D2+, and HD+ at fixed internuclear separations. The AJCs are defined as the difference between the results obtained from calculations using proper finite and infinite nuclear masses. Except at the united atom limit, the AJCs are smaller than the traditional first-order diagonal Born-Oppenheimer corrections. Expectation values of proton-electron, p-e, and deuteron-electron, d-e, distances for HD+ have been computed as a function of internuclear separation. Similarly to the fully nonadiabatic approach, the present method is able to follow the symmetry breaking in HD+. Exact and approximate analytical and numerical results are given for counterfactual systems as well. In these cases changes are allowed for the values of the electron rest mass or the elementary charge, as well as for the mass or charge of the unique particle (electron).

Extracular densities of the non-Born–Oppenheimer Hookean H 2 molecule

The exact nuclear and electronic extracular non-Born–Oppenheimer density functions and their associated probability functions have been obtained for the spherical ground and excited states of the Hookean hydrogen molecule. The salient features of these complex nodal functions are discussed aided by their mean expectation values and fluctuations. The infinite nuclear mass limit is shown to appear naturally from the derived equations.

The diamagnetic Coulomb problem: an eigenvalue problem with two singularities

By using the angular oblate spheroidal functions as basis functions a bounded wavefunction is constructed in the singularities of the Schrodinger equation of the diamagnetic Coulomb problem with infinite nuclear mass. The expansion in terms of these functions is a model to resolve singularities in an eigenvalue problem of non-separable partial differential equations of non-relativistic quantum mechanics. A comprehensive asymptotic analysis reveals the complete set of asymptotic solutions, makes possible a uniform numerical treatment of the bound, autoionizing continuum and continuum levels, and indicates how to find hitherto unknown low-lying stationary levels. An example, the splitting of the ground level, has been found numerically by an iterative shooting method.

A program for accurate solutions of two-electron atoms

We present a comprehensible computer program capable of treating non-relativistic ground and excited states for a two-electron atom having infinite nuclear mass. An iterative approach based on the implicitly restarted Arnoldi method (IRAM) is employed. The Hamiltonian matrix is never explicitly computed. Instead the action of the Hamiltonian operator on discrete pair functions is implemented. The finite difference method is applied and subsequent extrapolations gives the continuous grid result. The program is written in C and is highly optimized. All computations are made in double precision. Despite this relatively low degree of floating point precision (48 digits are not uncommon), the accuracy in the results can reach about 10 significant figures. Both serial and parallel versions are provided. The parallel program is particularly suitable for shared memory machines such as the Sun Starcat series. The serial version is simple to compile and should run on any platform. Program summary Title of program: corr2el Catalogue identifier: ADUX Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADUX Program obtainable from:CPC Program Library, Queen's University of Belfast, N. Ireland Distribution format: tar.gz Computer for which the program is designed and others on which it has been tested: Computers: Sun Fire 15K StarCat, Sun Ultra SPARC III, PC Operating systems or monitors under which the program has been tested: Sun Solaris 9, Linux Programming language used: ANSI C Memory required to execute with typical data: 3 Mwords or more No. bits in a word: 32 No. processors used: arbitrary Has the code been vectorized or parallelized: parallelized Number of lines in distributed program, including test data, etc.:5885 Number of bytes in distributed program, including test data, etc.: 26 199 Nature of physical problem: The Schrödinger equation for two-electron atoms is solved using finite differences. Method of solution: An iterative eigenvalue-solver that requires only the action of the Hamiltonian on a trial function is applied. The two-electron wave function is expanded in a sum of partial waves. The finite difference method is then applied to approximate the derivatives of the pair functions. The total action of the Hamiltonian on the partial waves, including correlation effects, is computed using highly optimized routines. Restriction on the complexity of the problem: The Hamiltonian employed here does not take relativistic or finite nuclear mass effects into account. The amount of computing time may become unreasonable for excited states far above the ground state. The use of double precision puts a limit on the accuracy obtainable. Typical running time: This ranges from half a minute (to obtain 10 significant figures for the s-limit of the Helium ground state) to perhaps a day for advanced examples depending on the level of parallelization. Unusual features of the program: The implicitly restarted Arnoldi method used to obtain the eigenvalues is implemented by using the ARPACK/PARPACK program library [ http://www.netlib.org/arpack]. This package also depends on the standard numerical libraries BLAS and LAPACK [ http://www.netlib.org/lapack]. Good performance is obtained by using Sun's optimized performance libraries [ http://www.sun.com]. By using a 64-bit environment (Ultra SPARC III and Solaris 9), memory limitations are non-problematic. Shared memory is used in the parallel version. Fast communication between the nodes is made over shared memory using Sun's implementation of MPI.