Three-body Coulomb Research Articles

Use of double-bound three-body Coulomb distorted-wave-like basis sets for two-electron wave functions

The double-bound analog of the double-continuum three-body Coulomb (C3) wave function, recently introduced by the authors, allows us to propose two different basis sets of correlated functions which can be used for describing double-bound states. The components of the two basis diagonalize the Coulomb interactions but differ in the definition of the distortion factor which depends only on the interelectronic coordinate. In the first set, this factor includes a nonlinear parameter, while the second set contains parameter-free functions. Through an angular correlated configuration interaction method it is possible to generate highly correlated wave functions for $S$ bound states of heliumlike systems. By construction, these states form an orthogonal set, satisfy all two-body cusp conditions, and retain the C3 double-continuum character. Even when a limited number of configurations is included, rather good energy values for the ground and excited states are obtained. A systematic improvement by inclusion of more configurations can be easily achieved, in particular with the computationally efficient, parameter-free basis.

The interaction of HD+2，H+3 and D+3 with solid foils and the three-body wake effect

The mechanism to form HD+2，H+3 and D+3 from RF ion source is analyzed by experimental study of the interaction of these ions with solid foils. The structural pattern of HD+2 and the mean internuclear separations are determined using the Coulomb explosion technique. In our experiment，there is no line-shaped structure for HD+2. A theoretical model to deal with the wake effect of three-body Coulomb explosion is proposed and applied to the interaction of HD+2，H+3 and D+3 with solid foils. The model is proved to be successful and the structure of HD+2 is confirmed.

Influence of the three-body coulomb effects on the values of asymptotic normalization coefficients (nuclear vertex constants) of astrophysical interest extracted from the modified DWBA approach for the peripheral proton-transfer reactions

From the exact three-body distorted wave amplitude obtained by other authors for the one-charged-particle-transfer reaction A(x, y)B within the strict three-body (x = y + a and B = A + a, where a is a transferred particle) model, its part is separated in which the three-body Coulomb dynamics of the transfer mechanism is taken into account correctly. The contribution of the three-body Coulomb dynamics of the transfer mechanism to the partial wave amplitudes at l ≫ 1 for the peripheral proton-transfer reactions 13C(3He, d)14N, 13C(14N, 13C)14N, and 9Be(10B, 9Be)10B is estimated within the three-body approach combining the dispersion method and the DWBA approach. For these reactions, the Coulomb renormalization factors, arising owing to correctly taking into account the three-body Coulomb dynamics of the proton-transfer mechanism in the DWBA cross sections, are calculated. A new estimate is obtained for the values of the asymptotic normalization factors for p + 13C → 14N, which also have astrophysical interest.

Dalitz plot analysis of Coulomb exploding O3 in ultrashort intense laser fields

The three-body Coulomb explosion of O3, O3(3+)-->O++O++O+, in ultrashort intense laser fields (2x10(15) W/cm2) is studied with two different pulse durations (9 and 40 fs) by the coincidence momentum imaging method. In addition to a decrease in the total kinetic energy release, a broadening in the Dalitz plot distribution [Philos. Mag. 44, 1068 (1953)] is observed when the pulse duration is increased from 9 to 40 fs. The analysis based on a simple Coulomb explosion model shows that the geometrical structure of O3 remains almost unchanged during the interaction with the few-cycle intense laser fields, while a significant structural deformation along all the three vibrational coordinates, including the antisymmetric stretching coordinate, is identified in the 40 fs intense laser fields. The observed nuclear dynamics are discussed in terms of the population transfer to the excited states of O3.

Charged exciton resonances in two and three dimensions

Resonances of three-body Coulomb systems are investigated in two and three dimensions. The complex scaling method combined with the stochastic variational approach has been used to calculate the position (energy and width) of the resonance states. The dependence of the resonance states on the mass ratio of the constituents and the dimensionality of the space is studied. It is found that the width of the resonance states behaves very differently in 2D and 3D for molecule-like cation systems. The calculated lifetimes of the resonances are in the nano- and picosecond region, and these states might be experimentally observable in excitonic trions in semiconductor quantum dots.

Faddeev-Noble Versus Sasakawa-Sawada Equations for the Nuclear Three-Body Coulomb Problem

We compare the efficiency of two Faddeev-type approaches for the nuclear three-body Coulomb problem. The first one is a modification of Noble’s approach, the second one is due to Sasakawa and Sawada. In an integral-equation formulation both of these methods rely on the same Green’s operators and driving terms. The differences lie in the treatment of the long-range Coulomb potentials. Numerical examples show that the modified Faddeev-Noble approach provides for faster convergence.

Probing ultrafast electron correlation with double attosecond pulses

Double ionization of He by two intense attosecond pulses has been investigated by using the time-dependent close-coupling method. By adjusting the time delay between the two attosecond pulses so as to coordinate the sequential two-photon absorption in He, we can examine electron correlation effects in this three-body Coulomb system. The double-ionization cross-sections from sequential two-photon absorption indicate that the correlation time between the two electrons [partially/fully ionized], on the order of ∼1.5 fs in the specific case of photon energy eV, can be measured. This demonstrates the feasibility of directly probing ultrafast electron correlation in the time domain.

Description of the continuous spectrum of a three-body Coulomb system within the J-matrix approach

The contribution of the continuous spectrum of two-body Coulomb sybsystems to reaction matrix elements is studied. The respective Faddeev-Merkuriev differential equations are solved within the J-matrix approach. The differential cross sections for the double ionization of a helium atom by a direct electron impact is calculated by way of example. The results agree well with experimental data both in shape of the cross sections and in magnitude.

Double-bound equivalent of the three-body Coulomb double-continuum wave function

In cross-section calculations of electron or photon double ionization processes of two-electron atoms, it is desirable to have a symmetric description of initial and final states. In this contribution, we search the doubly bound analog, for $S$ states, of the well-known and widely used three-body Coulomb (C3) double-continuum wave function. This is performed with two alternative approaches: through an analytic continuation of the C3 continuum and through a ``C3 approach'' of the Hylleraas equation for $S$ bound states. The double-bound analog consists of the product of two Coulomb bound states (one for each electron-nucleus interaction) multiplied by a Coulomb distortion factor which describes the electron-electron correlation. Our result differs from Pluvinage's wave function which is commonly misbelieved to be the bound counterpart of the C3 double continuum. With a rigorous treatment of the analytic continuation of the distortion factor, the double-bound equivalent of the C3 model is also found. Though the purpose is not to obtain good two-electron bound states, the found bound wave functions are tested, in the case of helium, through various local and mean quantities which probe different regions of the configuration space, in particular those close to the two-particle coalescence points.

Kinetic energy operator approach to the quantum three-body problem with Coulomb interactions

We present a non-variational approach to the solution of the quantum three-body problem, based on the decomposition of the three-body Laplacian operator through the use of its intrinsic symmetries. With the judicious choice of angular momentum eigenfunctions, a clean separation of spatial rotation from kinematic rotation is achieved, leading to a finite set of coupled PDEs in terms of the canonical variables. Numerical implementation of this approach to the three-body Coulomb problem is shown to yield accurate ground state eigenvalues and wavefunctions, together with those of low-lying excited states. We present results on some typical three-body systems. In particular, the eigenvalues and wavefunctions of the even-parity P e 3 state of the negative hydrogen ion are detailed for the first time. The issue of computational efficiency is also briefly discussed.

Observation of multiple ionization pathways for OCS in an intense laser field resolved by three-dimensional covariance mapping and visualized by hierarchical ionization topology

The two- and three-body Coulomb explosion of carbonyl sulfide (OCS) by 790 nm, 50 fs laser pulses focused to {approx_equal}10{sup 16} W cm{sup -2} has been investigated by the three-dimensional covariance mapping technique. In a triatomic molecule, a single charge state, in this case the trication, has been observed to dissociate into two distinct energy channels. With the aid of a three-dimensional visualization technique to reveal the ionization hierarchy, evidence is presented for the existence of two sets of ionization pathways resulting from these two initial states. While one group of ions can be modeled using a classical enhanced ionization model, the second group, consisting of mainly asymmetric channels, cannot. The results provide clear evidence that an enhanced ionization approach must also be accompanied by an appreciation of the effects of excited ionic states and multielectronic processes.

Theoretical study of the CS2+ dication

The potential energy and spin-orbit coupling functions of 11 lowest electronic states of CS(2+) dication have been calculated using internally contracted multireference configuration method. Using these functions, the positions and widths of the corresponding vibronic levels have been evaluated by means of the stabilization and log-phase-amplitude methods. The states governing the second step in the sequential pathway CS2(3+)-->S++CS2+-->S++C++S+ of the overall three-body Coulomb explosion of CS(2)3+ have been determined.

Two- and three-charged-particle nuclear scattering in momentum space: A two-potential theory and a boundary condition model

The two- and three-charged-particle nuclear scattering problems are investigated in momentum space. The three-body equations with a short-range nuclear potential, a three-body force potential, and the long-range Coulomb potential are presented in a mathematically rigorous way within a generalized two-potential theory. To remove the serious singularity in the two- and three-body Coulomb problems in momentum space, we have proposed a novel boundary condition to the phase shift that arises from a potential difference, the so-called auxiliary potential (AP), between the Coulomb potential and the screened Coulomb one: ${V}^{\ensuremath{\phi}}={V}^{C}\ensuremath{-}{V}^{R}$. Furthermore, we point out the importance of the off-shell amplitude for the AP, by which one can uniquely obtain the two-body on-shell and off-shell Coulomb amplitude in momentum space. Therefore, this formulation is also useful for atomic systems, which is another benefit. It is recalled that the traditional phase-shift renormalization theory, in which the screened Coulomb amplitude is sandwiched by renormalized phase factors ${e}^{i\ensuremath{\phi}}$, is not consistent with two-potential theory. Some ambiguities or misunderstandings of the traditional methods for handling the Coulomb problem are clarified. Finally, the three-body unitarity relation is proved for the amplitude generated from these three kinds of potential. Moreover, a generalized two-potential theory is presented. It is pointed out that the asymptotic three-body nuclear wave function with the Coulomb potential could not be written by a product of two individual wave functions ${\stackrel{~}{\ensuremath{\psi}}}_{x}(x){\stackrel{~}{\ensuremath{\psi}}}_{y}(y)$ defined in terms of Jacobi coordinates $x$ and $y$, respectively.

On the validity range of the Born-Oppenheimer approximation: A semiclassical study for all-particle quantization of three-body Coulomb systems

The validity range of the Born-Oppenheimer (BO) approximation is studied with respect to the variation of the mass (m) of negatively charged particle by substituting an electron (e) with muon (mu) and antiproton (p) in hydrogen molecule cation. With the use of semiclassical quantization applied to these (ppe), (ppmu), and (ppp) under a constrained geometry, we estimate the energy difference of the non-BO vibronic ground state from the BO counterpart. It is found that the error in the BO approximation scales to the power of 3/2 to the mass of negative particles, that is, m(1.5). The origin of this clear-cut relation is analyzed based on the original perturbation theory due to Born and Oppenheimer, with which we show that the fifth order term proportional to m(5/4) is zero and thereby the first correction to the BO approximation should arise from the sixth order term that is proportional to m(6/4). Therefore, the validity range of the Born-Oppenheimer approximation is wider than that often mistakenly claimed to be proportional to m(1/4).

Three-body sequential Coulomb explosions of CH 3OD 3+ induced by intense laser fields

Three-body Coulomb explosion of triply charged methanol- d (CH 3OD 3+), CH 3 OD 3 + → H + + H 2 + + COD + induced by a laser field of (≈2 × 10 14 W/cm 2), is investigated by the coincidence momentum imaging (CMI) method. It is securely identified that the explosion proceeds in a stepwise manner and that there are two types of sequential pathways: one is a pathway in which H + is ejected first, and the other is a pathway in which H 2 + is ejected first. Dynamical information on the two types of explosion processes is extracted from the analysis of CMI correlation maps.

Leading asymptotic terms of the three-body Coulomb scattering wave function

The asymptotic wave function derived by Alt and Mukhamedzhanov [Phys. Rev. A 47, 2004 (1993)] and Mukhamedzhanov and Lieber [Phys. Rev. A 54, 3078 (1996)] has been refined in the region where the pair $ (\beta,\gamma)$ remains close to each other while the third particle $\alpha$ is far away from them $(\rho_{\alpha} \to \infty$, $r_{\alpha}/\rho_{\alpha}\to 0)$. The improved wave function satisfies the Schr\"odinger equation up to the terms of order $O(1/{\rho_{\alpha}}^{3})$, provides the leading asymptotic terms of the three-body scattering wave function with Coulomb interactions and gives further insight into the continuum behavior of the three-charged-particle wave function and helps to obtain $3\to 3$ scattered wave. This opens up further ways of solving and analysing the three-body Schr\"odinger equation by numerical means.

Channel-coupling, target-structure, and second-order effects in electron-impact ionization ofAr(3p)andAr(3s)

The importance of second-order effects in the projectile-target interaction, channel coupling in the ejected-electron--residual-ion interaction, and multiconfiguration effects in the target description is investigated for electron-impact ionization of argon. For incident projectile energies of 200 and $113.5\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$, a scattering angle of 15\ifmmode^\circ\else\textdegree\fi{}, and ejected-electron energies between 2 and $20\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$, we find a strong sensitivity of the theoretical predictions to the target structure, channel coupling, and the proper treatment of nonlocal exchange. In some cases, accounting for these effects seems more important than extending a local-exchange, single-configuration distorted-wave theory by including the accurate three-body Coulomb asymptotic boundary condition. Despite encouraging improvement in agreement between our predictions and recent experimental results in some cases, severe discrepancies remain in others. More experimental data, preferably absolute but at least cross-normalized to each other for a variety of kinematic conditions, seem necessary to further guide ongoing theoretical efforts.

Fast electron correlation methods for molecular clusters in the ground and excited states*

An efficient and accurate electronic structure method for clusters of weakly interacting molecules has been proposed, on the basis of the pair-interaction method of Kitaura et al., and combined with density functional, many-body perturbation, coupled-cluster, equation-of-motion coupled-cluster, configuration-interaction singles, and time-dependent density functional theories. The method retains the one- and two-body (and, if necessary, also three-body) Coulomb, exchange, and correlation energies exactly and higher-order Coulomb energies in the leading order of multipole expansion (hence the dipole polarization or induction effects). The latter makes the combination of this method with existing implementations of any electronic structure theory extremely easy. It typically recovers the total energies within 0.001%, binding energies within a few kilocalories per mole, and excitation energies within a few hundredths of an electron volt of the conventional implementations. The size dependence of the computational cost of the method is asymptotically linear for total energies and constant for excitation energies. The method has been applied to the total energies of water clusters, to the total energies of zwitterionic and neutral glycine–water clusters, and to the excitation energies of formaldehyde–water clusters. The largest calculation was performed at an equation-of-motion coupled-cluster singles and doubles level for a formaldehyde–(H2O)81 cluster containing 247 atoms that predicted the solvatochromic shift of 1360 cm−1 in the lowest transition energy of formaldehyde in water.

Accumulation of Three-Body Resonances above Two-Body Thresholds

We calculate resonances in the three-body system with attractive Coulomb potential by solving homogeneous Faddeev-Merkuriev integral equations for complex energies. The equations are solved using the Coulomb-Sturmian separable expansion method. This approach allows us to study the exact behavior of the three-body Coulomb systems near the threshold. A negatively charged positronium ion is used as a test case. In addition to locating all previously known S-wave resonances of the positronium ion, we also find a large number of new resonant states that accumulate just slightly above the two-body thresholds. The pattern of accumulation of resonant states above the two-body thresholds suggests that probably they are infinite in number. We conjecture that this may be a general property of the three-body system with an attractive Coulomb potential.

Laser spectroscopy of antiprotonic helium atoms and ions

Laser spectroscopy of an antiprotonic helium ([Formula: see text]He+) atom, a neutral three-body Coulomb system consisting of an antiproton, a helium nucleus, and an electron has so far contributed to the determination of antiproton mass and charge to a precision of 10-8. Recently, we have succeeded in producing long-lived (τ [Formula: see text] 100 ns) antiprotonic helium ions (two-body system: [Formula: see text]He++). This was done by stopping ∼50 keV antiprotons decelerated by using an "inverse linac" (RFQD) in a very low-density helium gas target and selectively populating the ionic level by using laser tagging. It may be possible to use this two-body ion for future high-precision work.PACS Nos.: 36.10.–k, 34.90.+q, 25.43.+t