Nuclear Potential Energy Surface Research Articles

Diffractive Optics Based Four-Wave, Six-Wave, ..., ν-Wave Nonlinear Spectroscopy

A detailed understanding of chemical processes requires information about both structure and dynamics. By definition, a reaction involves nonstationary states and is a dynamic process. Structure describes the atomic positions at global minima in the nuclear potential energy surface. Dynamics are related to the anharmonicities in this potential that couple different minima and lead to changes in atomic positions (reactions) and correlations. Studies of molecular dynamics can be configured to directly access information on the anharmonic interactions that lead to chemical reactions and are as central to chemistry as structural information. In this regard, nonlinear spectroscopies have distinct advantages over more conventional linear spectroscopies. Because of this potential, nonlinear spectroscopies could eventually attain a comparable level of importance for studying dynamics on the relevant time scales to barrier crossings and reactive processes as NMR has for determining structure. Despite this potential, nonlinear spectroscopy has not attained the same degree of utility as linear spectroscopy largely because nonlinear studies are more technically challenging. For example, unlike the linear spectrometers that exist in almost all chemistry departments, there are no "black box" four-wave mixing spectrometers. This Account describes recent advances in the application of diffractive optics (DOs) to nonlinear spectroscopy, which reduces the complexity level of this technology to be closer to that of linear spectroscopy. The combination of recent advances in femtosecond laser technology and this single optic approach could bring this form of spectroscopy out of the exclusive realm of specialists and into the general user community. However, the real driving force for this research is the pursuit of higher sensitivity limits, which would enable new forms of nonlinear spectroscopy. This Account chronicles the research that has now extended nonlinear spectroscopy to six-wave processes and to a completely generalized "nu-wave" mixing form to fully control state preparation and coherences. For example, direct observation of global protein motions and energetics has led to the collective mode coupling model to understand structure-function correlations in biological systems. Direct studies of the hydrogen bond network of liquid H(2)O have recently shown that both intramolecular and intermolecular degrees of freedom are strongly coupled so that the primary excitations of water have an excitonic-like character. This fundamentally different view of liquid water has now resolved a 100-year-old problem of homogeneous versus inhomogeneous broadening of the vibrational line shapes. By adding programmable pulse shaping, we can access new information about the many-body interactions directly relevant to chemical reaction dynamics. We can also steer the course of the reaction along multidimensional surfaces to provide information about fluctuations far from the equilibrium, which are most relevant to chemical reactivity.

Symmetry-adapted polynomial basis for global potential energy surfaces-applications to XY4 molecules

The nuclear potential energy surfaces (PES) used in quantum chemistry inherit the symmetry of the whole molecular system, and are therefore invariant under the action of the nuclear permutation-translation-rotation-inversion group. One can take advantage of this property, both theoretically and numerically. The present article is the prolongation of the works of Schmelzer and Murrell and of Collins and Parsons on the subject. It presents a simplified technique to obtain symmetry-adapted polynomial basis for global PES, together with algorithmic recipes that make the problem computationally tractable. The method is illustrated in detail on XY 4 type of molecules for which a full description of the algebra of invariant polynomials under the full symmetry group of the molecule is given.

Laser-induced charge transfer and photodesorption of Cs at Cu(111): quantum dynamical model simulations

We present electronic and nuclear quantum model simulations for the direct, laser-induced charge transfer from a Cu(111) surface state to an unoccupied resonance state of an adsorbed Cs atom, and the resulting nuclear dynamics. Based on a modified one-electron model potential adopted from Chulkov et al., we determine energies and lifetimes of electronic states of Cs/Cu(111) at low coverage. In addition, semiempirical nuclear potential energy surfaces of the electronic ground and the antibonding excited states along the Cs-Cu distance are designed and used for nuclear dynamics. For both models, electronic and nuclear, we perform open-system quantum dynamics with the goals (i) to estimate the excitation (charge transfer) and Cs desorption probabilities and (ii) to optimize the charge-transfer process by pulse shaping and the desorption of Cs from the Cu surface by vibrationally exciting the Cs-Cu bond.

β-delayed proton decays and spin assignments for 133Sm and 149Yb

The proton-rich isotope 133Sm was produced via the fusion evaporation reaction 40Ca + 96Ru. Its β-delayed proton decay was studied by p-γ coincidence in combination with a He-jet tape transport system, and half-lives, proton energy spectra, γ-transitions following the proton emission, as well as β-delayed proton branching ratios to the low-lying states in the grand-daughter nucleus were determined. Comparing the observed β-delayed proton branching ratios with statistical model calculations, the best agreement is found assuming that only one level with the spin of 3/2 in 133Sm decays or two levels with the spins of 1/2 and 5/2 decay with similar half-lives. The configuration-constrained nuclear potential energy surfaces of 133Sm were calculated using the Woods-Saxon-Strutinsky method, which suggests a 1/2- ground state and a 5/2+ isomer with an excitation energy of 120keV. Therefore, the simple (EC+β+) decay scheme of 133Sm in Eur. Phys. J. A 11, 277 (2001) has been revised. In addition, our previous experimental data on the β-delayed proton decay of 149Yb reported in Eur. Phys. J. A 12, 1 (2001) was also analyzed using the same method. The spin-parity of 149Yb is suggested to be 1/2-.

β-delayed proton decays and spin assignments for 140Tb, 141Dy and 143Dy

The proton-rich isotopes 140Tb and 141Dy were produced via the fusion evaporation reaction 40Ca + 106Cd. Their β-delayed proton decays were studied by p-γ coincidence in combination with a He-jet tape transport system, and half-lives, proton energy spectra, γ-transitions following the proton emission, as well as β-delayed proton branching ratios to the low-lying states in the grand-daughter nuclei were determined. Comparing the experimental data with statistical model calculations, the ground-state spins of 140Tb and 141Dy were found to be consistent with 7 and 9/2, respectively. The configuration-constrained nuclear potential energy surfaces (NPES) of 140Tb and 141Dy were calculated using the Woods-Saxon-Strutinsky method, which suggest the ground-state spins and parities of 140Tb and 141Dy to be 7+ and 9/2-, respectively. In addition, the configuration-constrained NPES of 143Dy were calculated, which predict a 1/2+ ground state and a 11/2- isomer with excitation energy of 198keV. These findings are consistent with our previous experimental data on 143Dy reported in Eur. Phys. J. A 16, 347 (2003).

β-delayed proton decays near the proton drip line

We briefly reviewed and summarized the experimental study on $\ensuremath{\beta}$-delayed proton decays published by our group over the last 8 years, namely the experimental observation of $\ensuremath{\beta}$-delayed proton decays of nine new nuclides in the rare-earth region near the proton drip line and five nuclides in the mass 90 region with $N~Z$ by utilizing the $p\text{\ensuremath{-}}\ensuremath{\gamma}$ coincidence technique in combination with a He-jet tape transport system. In addition, important technical details of the experiments were provided. The experimental results were compared to the theoretical predictions of some nuclear models, resulting in the following conclusions. (1) The experimental half-lives for $^{85}\mathrm{Mo},\phantom{\rule{0.3em}{0ex}}^{92}\mathrm{Rh}$, as well as the predicted ``waiting point'' nuclei $^{89}\mathrm{Ru}$ and $^{93}\mathrm{Pd}$ were 5--10 times longer than the macroscopic-microscopic model predictions of M\"oller et al. [At. Data Nucl. Data Tables $66,131(1997)$]. These data considerably influenced the predictions of the mass abundances of the nuclides produced in the rp process. (2) The experimental assignments of spin and parity for the drip-line nuclei $^{142}\mathrm{Ho}$ and $^{128}\mathrm{Pm}$ could not be well predicted by any of the nuclear models. Nevertheless, the configuration-constrained nuclear potential-energy surfaces calculated by means of a Woods-Saxon-Strutinsky method could reproduce the assignments. (3) The ALICE code overestimated by one or two orders of magnitude the production-reaction cross sections of the nine studied rare-earth nuclei, while the HIVAP code overestimated them by approximately one order of magnitude.

β-delayed proton decays near the proton drip line

We briefly reviewed the experimental study on β-delayed proton decays near the proton drip line published by our group during the period of 1996–2004, namely the first observation of the β-delayed proton decays of 9 new nuclides in the rare-earth region and the new measurements of β-delayed proton decays of 5 nuclides in the mass ∼90 region near theN=Z line with the aid of the “p-y” coincidence in combination with a He-jet tape transport system. In the meantime some important experimental technique details were supplemented. The experimental results, including the half-lives, spins, parities, deformations and production reaction cross sections for the 14 nuclei were summarized and compared with the current nuclear-model predictions, and then the following points were represented. (1) The experimental half-lives for85Mo and92Rh as well as the predicted “waiting point” nuclei89Ru and93Pd are 5–10 times longer than the theoretical predictions given by Moller et al. using a macroscopic-microscopic model. It considerably influences the predictions of the abundances of the nuclides produced in the rp-process. (2) The current-model predictions are not consistent with the experimental assignments of the spins and parities for the proton drip-line nuclei142Ho and128Pm. However, the nuclear potential energy surface (PES) calculated by using a Woods-Saxon-Strutinsky method reproduced the experimental results. (3) The Alice code overestimated the production reaction cross sections of the studied 9 rare-earth nuclei by one order of magnitude or two, while HIVAP code overestimated them by one order of magnitude approximately.

First decay study of the new isotope 129Pm near the proton drip line

The very neutron-deficient nuclide 129Pm was produced via the 92Mo(40Ca, p2n) reaction and identified for the first time by using the X- $\gamma $ coincidence in combination with a He jet tape transport system. According to the decay curve of a 99 keV $\gamma $ -ray which corresponds to the known $5/2^{-} \rightarrow 1/2^{-}$ transition in the daughter nucleus 129Nd of the 129Pm decay, the half-life of 129Pm was determined to be 2.4(9) s. Based on the nuclear potential energy surface (PES) calculations, the ground-state spin-parity of 129Pm was predicted as 5/2- which is favorable to feed a 5/2- low-lying state in the daughter nucleus 129Nd via the (EC $ + \beta^{ + }$ ) decay.

Highly deformed band inNd138

High-spin states in $^{138}\mathrm{Nd}$ have been investigated with the reaction $^{94}\mathrm{Zr}(^{48}\mathrm{Ca,4}n)$, using the GASP array. A new rotational band has been identified, which has close similarities with the well known highly deformed bands of the lighter Nd isotopes, and is therefore interpreted as built on the second minimum of the nuclear potential energy surface. From comparison with calculations a configuration is suggested which involves two neutrons in the ${i}_{13∕2}$ deformation driving orbital.

Efficient linear scaling geometry optimization and transition-state search for direct wavefunction optimization schemes in density functional theory using a plane-wave basis

Two linear scaling schemes for the search of stationary points on the nuclear potential energy surface have been developed and implemented for density functional theory programs using plane waves: a geometry optimizer based on the limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) method and a linear scaling method for transition-state search based on the microiterative scheme using the partitioned rational function optimizer (P-RFO) and L-BFGS. These optimizers are written with parallelized execution in mind. It is shown that the electronic wavefunction does not need to be fully optimized in the earlier stages of geometry optimization. The reasons for the robustness and good performance of the proposed schemes are identified. Test calculations are presented that use our implementation in the CPMD code.

Efficient characterization of stationary points on potential energy surfaces

Traditional methods for characterizing an optimized molecular structure as a minimum or as a saddle point on the nuclear potential energy surface require the full Hessian. However, if f denotes the number of nuclear degrees of freedom, a full Hessian calculation is more expensive than a single point geometry optimization step by the order of magnitude of f. Here we present a method which allows to determine the lowest vibrational frequencies of a molecule at significantly lower cost. Our approach takes advantage of the fact that only a few perturbed first-order wave functions need to be computed in an iterative diagonalization scheme instead of f ones in a full Hessian calculation. We outline an implementation for Hartree–Fock and density functional methods. Applications indicate a scaling similar to that of a single point energy or gradient calculation, but with a larger prefactor. Depending on the number of soft vibrational modes, the iterative method becomes effective for systems with more than 30–50 atoms.

Spin assignments for the β-delayed proton precursors142Hoand128Pm

The previous experimental data on \ensuremath{\beta}-delayed proton decays of ${}^{142}\mathrm{Ho}$ and ${}^{128}\mathrm{Pm}$ were compared with a statistical-model calculation. As a result, the spins of both ${}^{142}\mathrm{Ho}$ and ${}^{128}\mathrm{Pm}$ were assigned as 5, 6, or 7. Nuclear potential-energy surface calculations were performed using the Woods-Saxon-Strutinsky method. The calculated results support the assignments of ${7}^{\ensuremath{-}}$ to ${}^{142}\mathrm{Ho}$ and ${6}^{+}$ to ${}^{128}\mathrm{Pm}.$

Magnetic dipole bands in [formula omitted]: first evidence of excitations across the Z=82 sub-shell in Hg nuclei

An experiment aimed at studying high-spin states in 190 Hg was performed with the Eurogam II array. The data have revealed the presence of cascades of magnetic dipole transitions with some unexpected properties. Unlike the M1 bands previously observed in the heavier Hg isotopes, these structures have extremely large B( M1)/ B( E2) ratios. The observation of a third dipole band with much lower B( M1)/ B( E2) values in the same spin/excitation energy regime suggests that the bands may represent configurations occurring in different minima in the potential energy surface. Configuration-dependent Cranked Nilsson–Strutinsky calculations predict the presence of a minimum in the nuclear potential energy surface at a deformation of ε≈0.2, γ≈−90°, occurring when a proton is excited across the Z=82 shell-gap into an h 9/2 orbital. It is suggested that the bands exhibiting anomalously large B( M1)/ B( E2) ratios may be associated with this minimum.

Calculation of the line shapes of electronic transitions at defects using the frozen Gaussian technique

Calculations of the optical properties of defects in semiconductors or multiphonon transition rates presently make far more severe approximations than standard electronic structure calculations. One major challenge is how one can handle realistically the lattice vibrations, including quantum nuclear dynamics when those are necessary. The semi-classical frozen Gaussian technique allows us to calculate the line shapes of electronic transitions at defects using molecular dynamic simulation data from the initial and final states. Approximate nuclear wave functions are constructed from classical trajectories on nuclear potential energy surfaces for the two states. The method expands the system initial and final states (functions of position and momentum), as a sum of Gaussians, whose centres evolve classically on the relevant potential surface. An expression for the transition probability is then derived from the time-dependent overlap of the two wave functions. The frozen Gaussian method has been tested on the core exciton in diamond. The potential energy surfaces are calculated using an approximate but self-consistent molecular orbital technique, while simultaneously performing molecular dynamics. The results of our calculations show a Stokes shift of 3.64eV, similar to the value of “up to 5eV” obtained experimentally. We predict a fine structure with much narrower lines than those observable in the (low-resolution) experimental spectra.

Search for a phase transition in the nuclear shape at finite temperature and rapid rotation

At high spin the strength function for γ-decay of the Giant Dipole Resonance in highly excited 45Sc compound nuclei exhibits a shoulder at E γ ≈ 25 MeV, in addition to the main peak at E γ ≈ 16 MeV, suggesting the presence of large deformations β > 0.45. Calculated spectral shapes including thermal shape fluctuations provide a good description of the data when the nuclear potential energy surface includes the oblate → triaxial phase transition predicted by the rotating liquid drop model at high spin.

Vibration-rotation coordinates and kinetic energy operators for polyatomic molecules

In molecular vibration-rotation calculations it is frequently desirable to work in a nuclear coordinate system tailored to the nuclear potential energy surface. It is now possible to derive the kinetic energy operator for virtually any coordinate system by using computer algebra programs. We discuss how to choose coordinates for use in practical calculations, in particular variational energy level calculations, with special consideration of the vibration-rotation separation. As an example we present a kinetic energy operator suitable for calculating the rovibrational spectrum of sequentially bonded four-atomic molecules, using valence coordinates. We carry out a rigorous symmetry classification using the MS group, and also consider how the problem of singularities affects the choice of basis set.

Rotational band identification: A model‐based detection and estimation approach

AbstractNuclear theory predicts that secondary minima develop in the nuclear potential energy surface of some nuclei as they are stretched or elongated. Some minima are deep enough to support quasistationary states. The states are metastable because the decay is inhibited by the required shape change. Detection of these metastable states and their associated rotational bands allows us to investigate nuclei at large deformations and test existing theoretical predictions. The identification of these difficult‐to‐locate structures will permit the detailed study of nuclei at extraordinarily high deformation, providing new insights into nuclear structure. We discuss the application of model‐based image processing techniques to detect and enhance the nuclear rotational bands in noisy experimental measurements. The aim of the experiment is to identify the shape isomer by detecting the presence of the associated rotational band pattern. Unfortunately, the band signal is only ≈︁1% or less of the amplitude of similar disturbance signals due to the nuclear states of normal deformation; thus the signal‐to‐noise ratio is quite poor. Thus it is necessary to investigate image processing techniques to detect and extract the predicted band pattern. Using a model‐based approach, that is, characterizing a particular band by a simple mathematical model and incorporating it into an image processing scheme, we show that we are able to solve the band pattern detection and enhancement problems from both synthesized and measured experimental data.

Beta-delayed fission and neutron emission calculations for the actinide cosmochronometers.

The Gamow-Teller beta-strength distributions for 19 neutron-rich nuclei, including ten of interest for the production of the actinide cosmochronometers, are computed microscopically with a code that treats nuclear deformation explicitly. The strength distributions are then used to calculate the beta-delayed fission, neutron emission, and gamma deexcitation probabilities for these nuclei. Fission is treated both in the complete damping and WKB approximations for penetrabilities through the nuclear potential-energy surface. The resulting fission probabilities differ by factors of 2 to 3 or more from the results of previous calculations using microscopically computed beta-strength distributions around the region of greatest interest for production of the cosmochronometers. The indications are that a consistent treatment of nuclear deformation, fission barriers, and beta-strength functions is important in the calculation of delayed fission probabilities and the production of the actinide cosmochronometers. Since we show that the results are very sensitive to relatively small changes in model assumptions, large chronometric ages for the Galaxy based upon high beta-delayed fission probabilities derived from an inconsistent set of nuclear data calculations must be considered quite uncertain.

A b i n i t i o calculation of reaction energies. III. Basis set dependence of relative energies on the FH2 and H2CO potential energy surfaces

The relative energies of the stationary points on the FH2 and H2CO nuclear potential energy surfaces relevant to the hydrogen atom abstraction, H2 elimination and 1,2-hydrogen shift reactions have been examined using fourth-order Mo/ller–Plesset perturbation theory and a variety of basis sets. The theoretical absolute zero activation energy for the F+H2→FH+H reaction is in better agreement with experiment than previous theoretical studies, and part of the disagreement between earlier theoretical calculations and experiment is found to result from the use of assumed rather than calculated zero-point vibrational energies. The fourth-order reaction energy for the elimination of hydrogen from formaldehyde is within 2 kcal mol−1 of the experimental value using the largest basis set considered. The qualitative features of the H2CO surface are unchanged by expansion of the basis set beyond the polarized triple-zeta level, but diffuse functions and several sets of polarization functions are found to be necessary for quantitative accuracy in predicted reaction and activation energies. Basis sets and levels of perturbation theory which represent good compromises between computational efficiency and accuracy are recommended.

Nuclear Shapes and Shape Transitions

We study nuclear potential-energy surfaces, ground-state masses and shapes calculated by use of the Yukawa-plus-exponential macroscopic model and a folded-Yukawa single-particle potential for 4023 nuclei ranging from 16O to 279112. We present an overview of the results in the form of four colour contour diagrams vs. proton number Z and neutron number N. The four diagrams show calculated values of |ϵ2| and ϵ4 at the ground state, ground-state microscopic shell-plus-pairing corrections and the deviations between experimental and calculated masses. The diagrams vividly display the regions of magic and deformed nuclei. In particular, the plot of |ϵ2| vs. Z and N clearly shows the well-known deformed actinide and rare-earth regions and the two new deformed regions around A = 80 and A = 100. The plots indicate differences between the various deformed regions. For instance, there are differences in the magnitude of the deformation and in the character of the transition from spherical to deformed shapes. We discuss extensively the transition from spherical to deformed shapes and study the relation between shape changes and the mass corresponding to the ground-state minimum, and the significance of additional minima in the nuclear potential-energy surface. For a few illustrative cases we discuss the effect of angular momentum on the nuclear shape. The calculated values for the ground-state mass and shape show good agreement with experimental data throughout the periodic system, but some discrepancies remain that deserve further study.