Simple charge-transfer model to explain the electrical response of hydrogen chains

Linear ${\mathrm{H}}_{2n}$ chains of ${\mathrm{H}}_2$ units are experimentally unrealizable but simple and widely studied exemplars of the response coefficients (polarizabilities and hyperpolarizabilities) of polymer chains in uniform longitudinal electric fields. They show two surprising features: (1) Their response coefficients, unlike those of bulk solids, show a strong nonlinear dependence upon length or $n$. (2) Their response coefficients are seriously too large when computed with standard local or semilocal density functionals, but are much more correct when computed with self-interaction-free approaches (including Hartree-Fock). We propose a simple charge-transfer model which explains both of these effects in analytic terms. In this model, charge is transferred between ${\mathrm{H}}_2$ units paired up at equal distances from but on opposite sides of the chain center. All symmetric pairs of ${\mathrm{H}}_2$ units, not just the one for the chain ends, are included. This transfer is driven by the external electric field, and opposed by the chemical hardness of each ${\mathrm{H}}_2$ unit. Unlike the situation in a bulk solid, this charge transfer is not suppressed (or even much affected) by electrostatic interactions among the transferred charges for $n⩽7$. Self-interaction-free approaches increase the chemical hardness of an ${\mathrm{H}}_2$ unit in comparison with semilocal density functionals, and so reduce the charge transfer. The physical picture behind the model is validated and its limitations are revealed by an analysis of the charge density from self-consistent electronic structure calculations. An appendix presents an accurate method to extract the hyperpolarizability from self-consistent calculations, and its results.

Figure 1

Energy change of Eq. (1) vs electric field $F$ for an ${\mathrm{H}}_{14}$ chain. The upper panel is the polarizability term, and the lower panel is the hyperpolarizability term. $\alpha =165.7\mathrm{a.u.}$ and $\chi =202900\mathrm{a.u.}$ from the self-consistent HF calculations of Appendix and Ref. 16.

Figure 2

Energy change $\Delta$ vs transferred electron number $\delta$ for an ${\mathrm{H}}_4$ chain, using model parameters from Table . The upper panel is without electric field, and the lower panel is for electric field $F=0.008\mathrm{a.u.}$

Figure 3

Electron number $\delta$ transferred to the $j\text{th}$ chain unit in ${\mathrm{H}}_{14}$ vs ${\mathrm{H}}_2$ unit index $j$, for ${\mathrm{H}}_{14}$ in electric field $F=0.008\mathrm{a.u.}$, using model parameters from Table . The electric field points to the right-hand-side and drives electrons to the left-hand-side.

Figure 4

Polarizability $\alpha$ vs number $n$ of chain units in ${\mathrm{H}}_{2n}$. The dashed and solid curves are based on the model with parameters from Table , fitted to self-consistent LSD and HF values of Appendix (shown as circles and diamonds, respectively).

Figure 5

Longitudinal second hyperpolarizability $\chi$ (in a.u.) vs number $n$ of chain units in ${\mathrm{H}}_{2n}$. Here $\chi ∕{10}^3$ is displayed. The dashed and solid curves are based on the model with parameters from Table , fitted to self-consistent LSD and HF values of Appendix (shown as circles and diamonds, respectively). The off-scale $n=7$ LSD values are self-consistent 2270, model 2325.

Figure 6

Dependence of $F_{\max }$ on $n$ for the LSD second hyperpolarizability $\chi$, for ${\mathrm{H}}_{2n}$ chains, from Eq. (A3).

Figure 7

LSDA second hyperpolarizability $\chi$ (a.u.): Dependence on the electric field strength for the ${\mathrm{H}}_{14}$ chain.