Nonlinear response theories and effective pair potentials

We present a general method based on nonlinear response theory to obtain effective interactions between ions in an interacting electron gas, which can also be applied to other systems where an adiabatic separation of time scales is possible. Nonlinear contributions to the interatomic potential are expressed in terms of physically meaningful quantities, giving insight into the physical properties of the system. The method is applied to various test cases and is found to improve the standard linear and quadratic response approaches. It also reduces the discrepancies previously observed between perturbation theory and density functional theory results for proton-proton pair potentials in metallic environments.

Figure 1

(Color online) Pair potentials for $\delta$-function potentials in one-dimensional noninteracting electron gas: exact, linear response, and the SPM and SSPM results. The Fermi wave vector is set to $k_f=0.785∕a_0$. The residuals (difference between estimated and exact pair potentials) are shown in the inset. Note the difference in behavior of the SSPM near the origin between attractive (first three figures) and repulsive (last figure) $\delta$-function potentials.

Figure 2

(Color online) Upper figure: dependence of the effective pair potential between two repulsive colloidal particles on the interaction strength parameter $x=\beta V_0$: exact result versus the SPM and SSPM, and perturbations at various orders. Lower figure: relative errors $\mid f_n\left(x\right)-f\left(x\right)\mid ∕f\left(x\right)$.

Figure 3

(Color online) Comparison of the SSPM results with and without contributions from Eq. (13) to quadratic response and ab initio results [from Bonev and Ashcroft (Ref. 18)] for proton-proton pair potentials at $r_s=2$.

Figure 4

(Color online) Contribution of the diagram from Fig. 5a to the proton-proton pair potential for various values of $r_s$.

Figure 5

(a) Lowest-order diagram contributing to the interaction (13). (b) Nonreducible diagram equal to diagram (a) in the limit $R_{ab}\to 0$, for identical ions, here protons (diagrammatic conventions are explained in Appendix and Fig. 7).

Figure 6

(Color online) Contributions of the Coulombic and exchange parts to the effective electron-electron potential $\tilde{v},$ for $r_s=2$.

Figure 7

Some typical diagrams from the free energy expansion contributing to the pair potential. Loops with $n+1$ legs represent response function of order $n$. The correlation function $c_0^{\left(n\right)}$ is represented by a solid circle surrounded by $n+1$ wavy lines and the external potential by solid lines. Diagram (a) is a quadratic response contribution. Diagrams (b) and (c) represent some third-order contributions taken into account by Eqs. (12, 13), respectively. Diagrams (d), (e), and (f) are examples of cubic-response contributions which are not taken into account in the approach presented here.

Figure 8

Diagrammatic expansion for the dressed ${\tilde{c}}_0^{\left(1\right)}$ in terms of the bare $c_0^{\left(1\right)}$ and linear response functions.