Atomic bound state and scattering properties of effective momentum-dependent potentials

Effective classical dynamics provide a potentially powerful avenue for modeling large-scale dynamical quantum systems. We have examined the accuracy of a Hamiltonian-based approach that employs effective momentum-dependent potentials (MDPs) within a molecular-dynamics framework through studies of atomic ground states, excited states, ionization energies, and scattering properties of continuum states. Working exclusively with the Kirschbaum-Wilets (KW) formulation with empirical MDPs [C. L. Kirschbaum and L. Wilets, Phys. Rev. A 21, 834 (1980)], optimization leads to very accurate ground-state energies for several elements (e.g., N, F, Ne, Al, S, Ar, and Ca) relative to Hartree-Fock values. The KW MDP parameters obtained are found to be correlated, thereby revealing some degree of transferability in the empirically determined parameters. We have studied excited-state orbits of electron-ion pair to analyze the consequences of the MDP on the classical Coulomb catastrophe. From the optimized ground-state energies, we find that the experimental first- and second-ionization energies are fairly well predicted. Finally, electron-ion scattering was examined by comparing the predicted momentum transfer cross section to a semiclassical phase-shift calculation; optimizing the MDP parameters for the scattering process yielded rather poor results, suggesting a limitation of the use of the KW MDPs for plasmas.

Figure 1

Plots of $r^2\left[V(r,p)=\frac{{\varepsilon }^2}{4\alpha r^2}{e}^{\alpha \left[1-{(rp/\varepsilon )}^4\right]}\right]$ (in atomic units) where $\alpha =1$ and $\varepsilon =1$ (a) and $\alpha =1$ and $\varepsilon =2$ (b) show that for $rp\ll \varepsilon$, the potential becomes very repulsive. Since $V^H$ and $V^P$ have similar functional forms, common notations are used to denote the potentials and their corresponding variables: $V(r,p)$ denotes $V^H$ or $V^P,\alpha$ denotes ${\alpha }_H$ or ${\alpha }_P,\varepsilon$ denotes ${\varepsilon }_H$ or ${\varepsilon }_P,r$ denotes $r_i$ or $r_{ij}$, and $p$ denotes $p_i$ or $p_{ij}$. Therefore, the highly repulsive behavior of $V(r,p)$ as $rp\ll \varepsilon$ enforces the Heisenberg and Pauli principles within the classical framework.

Figure 2

Projected surface of ${\mathrm{\Delta }}_{HF}(\%)=100|(E-E_{HF})/E_{HF}|$ in (${\varepsilon }_H,{\varepsilon }_P$) plane for Al (a) and Ca (b) showing the region of minima. (${\varepsilon }_H,{\varepsilon }_P$) points (white dots) corresponding to ${\mathrm{\Delta }}_{HF}\lesssim 4\%$ follow a curve indicating that there is a correlation between ${\varepsilon }_H$ and ${\varepsilon }_P$ values that result in ground-state energies which are in very good agreement with HF values. The correlation between the parameters reveal some degree of transferability to ground-state energies of other elements that are not tested.

Figure 3

(a) Relative error ${\mathrm{\Delta }}_{HF}$ (%) of MDP prediction of ground-state energies with respect to HF values (green circles connected by dashed green line) for N, Ne, Al, Ar, and Ca show that the KW formulation is an excellent model for ground-state energies (lines are to guide the eye). (b) Correlated (${\varepsilon }_H,{\varepsilon }_P$) points (circles in shades of red from light to dark corresponding to increasing atomic number) extracted from the ${\mathrm{\Delta }}_{HF}$ surfaces. Also shown are curve fits (solid curves in shades of red increasing from light to dark corresponding to increasing atomic number) using ${\varepsilon }_P=A{\varepsilon }_H^2+B{\varepsilon }_H+C$ for N, Ne, Al, Ar, and Ca. For all the elements considered, the curve fits match well with the points extracted from the corresponding ${\mathrm{\Delta }}_{HF}$ surfaces, suggesting a possible transferability with respect to the atomic number.

Figure 4

(a) Coefficients A, B, and C of the fit ${\varepsilon }_P=A{\varepsilon }_H^2+B{\varepsilon }_H+C$ vary as a function of atomic number (Z). The pattern in the data points of A (green circles), B (black star), and C (red squares) corresponding to N, Ne, Al, Ar, and Ca are captured well by fourth degree polynomial fits for A (solid green curve), B (dotted black curve), and C (dashed red curve). (b) Correlated (${\varepsilon }_H,{\varepsilon }_P$) curves for F (blue dashed) and S (blue solid) computed with A, B, and C interpolated using their corresponding fourth degree polynomial fits. Ground state energies for F and S from their correlated (${\varepsilon }_H,{\varepsilon }_P$) are in good agreement with HF values (with a maximum ${\mathrm{\Delta }}_{HF}$ of $\sim 10\%$ and $\sim 5\%$ for F and S, respectively). This confirms the transferability of the trained (${\varepsilon }_H,{\varepsilon }_P$) to ground-state energies of elements that were not included in the training.

Figure 5

(a) Electron trajectory around ion (black dot) corresponding to different initial conditions: Kepler-like motion from $t=0$ to $t=50$ a.u. (blue curve), zig-zag motion from $t=0$ to $t=50$ a.u. (dark green curve), zig-zag motion from $t=50$ to $t=231$ a.u. (light green curve), and reciprocating motion from $t=0$ to $t=50$ a.u. (red curve). Reciprocating motion corresponds to zero initial angular momentum, which would result in an unstable trajectory in the absence of Heisenberg MDP. (b) Magnitude of position ($r$) and magnitude of momentum ($p$) corresponding to the trajectories in (a) from $t=0$ to $t=50$ a.u. are superimposed on the Hamiltonian surface. (c) Region 1 of (b) magnified to show the bandlike structure in ($r,p$) dynamics. (d) Region 2 of (b) magnified to show a similar bandlike structure in ($r,p$) dynamics.

Figure 6

For N, F, Ne, Al, S, Ar, and Ca, first and second ionization energies were computed using MDPs with the correlated (${\varepsilon }_H,{\varepsilon }_P$) optimized to give accurate neutral ground-state energies. MDP prediction of first ionization energies (red curve) is in good agreement with NIST data (dashed blue curve); second ionization energies using MDPs (green curve) yield mixed results compared to NIST data (dashed cyan curve). Therefore, the parameters trained on neutral ground-state energies transfer fairly well to the prediction of first and second ionization energies with some outliers.

Figure 7

Comparison of the classical and quantum MTCS for $Z=1,{\kappa }_B=1$, and $E=10$ to 100 a.u. The classical MTCS using semianalytic method (blue dashed) and the trajectory method (cyan dashed) are in very good agreement. They also match with the asymptotic limit of the classical MTCS (red dashed) given by Eq. (27). The quantum MTCS (black dashed) differ significantly from the classical result. Also shown is the asymptotic limit of the quantum MTCS (pink dashed) given by Eq. (28) with the difference from the numerical quantum MTCS decreasing as energy increases.

Figure 8

Scattering angle $\theta$ vs impact parameter $b$ corresponding to $Z=1,{\kappa }_B=1$, and $E=10$ a.u. for the purely classical case (black dashed curve) and with Heisenberg MDP (blue curve: ${\alpha }_H=1,{\varepsilon }_H=0.125$; green curve: ${\alpha }_H=1,{\varepsilon }_H=0.25$; red curve: ${\alpha }_H=1,{\varepsilon }_H=0.5$). Scattering at low impact parameters is highly influenced by the MDP, resulting in a structure in the scattering angles with the angle decreasing to zero and then increasing as impact parameter increases. Scattering angles of all the curves overlap with each other for large impact parameters.

Figure 9

(a) Scattering angle vs impact parameter obtained using MDP with ${\alpha }_H=1,{\epsilon }_H=0.125$ for $Z=1,{\kappa }_B=1$, and $E=10$ a.u. There is a structure in the scattering angles with the angles becoming nearly zero for an impact parameter of $b$ = 0.028 a.u. (b) Electron trajectory for $b$ = 0.028 a.u. (red curve) reveals that though there is a strong interaction between the electron and the screened ion (black dot), the interaction is such that the scattering angle is nearly zero in the asymptotic limit of the trajectory. Electron trajectories for $b$ = 0.026 a.u. (blue curve) and $b$ = 0.03 a.u. (green curve) indicated by the vertical lines in (a) show that though the corresponding scattering angles are similar in magnitude, the nature of the trajectories are different with the electron scattered upward for b = 0.026 a.u. while the electron is scattered downward for b = 0.03 a.u.

Figure 10

(a) For $Z=1,{\kappa }_B=1$, and $E=10$ to 100 a.u. the classical MTCS (blue curve) and the quantum MTCS (green curve) are compared with the MTCS using MDP (red dots) optimized with respect to its free parameters ${\alpha }_H$ and ${\varepsilon }_H$ to minimize the squared difference between the quantum MTCS and the MTCS using MDP. (b) Filled contour of MTCS using MDP for a range of ${\alpha }_H$ and ${\varepsilon }_H$ for $Z=1,{\kappa }_B=1$, and $E=25$ a.u. The lowest value on the MTCS contour is 0.016 a.u., which is still large compared to the corresponding quantum MTCS value of 0.0122 a.u. Though the MDP significantly influences the electron-ion scattering for small impact parameters, its MTCS predictions are close to the classical MTCS suggesting the MDP's limitation for modeling electron-ion scattering in dense plasmas.