The Electron Bubble and the $He_{60}$ Fullerene: A First-Principles Approach

Abstract

Helium has a light atomic mass and, as a closed shell element, shows minimal interaction with other particles of the environment. Such properties favor the capture of an electron by liquid helium, leading to the formation of an electron bubble. The helium bubbles are of theoretical importance since different levels of quantum mechanical models can be tested for the correct prediction of a single quantum particle trapped in a cage. In this work, we propose a first-principles model of the electron bubble that takes, for the first time, the electronic structure of the cage into consideration. The model consists of a fullerene-type cage made of He atoms with an additional electron. The solution of the many-body Schroedinger equation is then performed using density functional theory, with a small and an extra-large atomic basis set. Several major improvements over the model of a particle in a rigid or soft spherical potential are assessed in this way, such as the localization and delocalization of the electron in the helium bubble, the transition of the electron to the continuum, the polarization of the He atoms building the wall, the ionic state of the electron bubble, besides the determination of relations of the volume-pressure type.

Appendix

The purpose of this section is to evaluate the dispersion effects on the He bubble energy, and also estimate the accuracy of energy differences by using different DFT energy functionals. To do this, we depart from the fact that an energy functional of DFT is an approximation, whose energies are usually adapted to reproduce the exchange-correlation and dispersion energies measured in the experiments. There is not an exchange-correlation energy functional or dispersion energy functional exclusively designed for helium. In spite of that, there are functionals specially designed to provide the correct long-range attractive and repulsive interactions. One of these is the $APF-D$ functional, which includes dispersion effects as an empirical correction to the APF functional [38]. The dispersion expression contained in the $APF-D$ functional is shown below.

$$\begin{aligned} E_\mathrm{disp}= \sum _{A>B} V (R_{AB}); \quad V (R_{AB})= \left\{ \begin{array} {l} 0; \quad R_{AB} \le R_{d,AB}\\ \\ \frac{C_{6,AB}}{[R_{AB}^2 -R_{s,AB}^2]^3}\, f(R_{AB})\, g(R_{AB}); \quad R_{AB} > R_{d,AB} \end{array}\right. \nonumber \\ \end{aligned}$$

(9)

The dispersion term contains the damping function f and switching function g for the attenuation of the Coulomb interactions, and with the purpose of producing continuous derivatives. There are nine parameters $P_i$ adjusted to reproduce the ionization potentials ($\varepsilon _i$), and effective atomic polarizabilities ($\alpha _i$) of 15 noble gas dimers (several of them containing He), plus some hydrocarbon dimers.

$$\begin{aligned} C_{6,AB}= & {} \frac{3}{2}\, P_1 \left( \frac{\varepsilon _{H,A}\, \varepsilon _{H,B}}{\varepsilon _{H,A} + \varepsilon _{H,B}} \right) \alpha _A\, \alpha _B\nonumber \\ R_{s,AB}= & {} \frac{P_i}{\sqrt{-(\varepsilon _{H,A} + \varepsilon _{H,B})/2}} \quad ; \quad R_{d,AB}= \frac{P_j}{\sqrt{-(\varepsilon _{H,A} + \varepsilon _{H,B})/2}}- P_6\nonumber \\ P_1= & {} 1.18604 \quad ; \quad P_6= 0.234859 \end{aligned}$$

(10)

The $P_i$ factor is one of the $P_2$, $P_3$, $P_4$, $P_5$ parameters, and the $P_j$ factor is one of the $P_7$, $P_8$, $P_9$ parameters (presented in Table 3 of Ref. [38]), decided according to the dimer type AB. The accuracy of the APF-family of functionals is comparable to that of more refined levels of theory, such as CCSD(T)/aug-cc-pVTZ, and the energy values are found in excellent agreement with experimental measurements.

In the present work, we have been mainly concerned with energy differences (such as the ionization potential, energy gap, pressure, and more). Therefore, we consider the difference between the electron bubble, $E (He_{60}^-)$, and the energy of the empty cage, $E (He_{60})$, for the evaluation of the dispersion effects. In Table 4, we present a list of energies using the energy functional $APF-D$, it includes dispersion effects in the energy, and the energy functional APF, which is the counterpart functional that includes no dispersion effects in the energy [38]. The basis set is $aug-cc-pV5Z$, it was used in the main text. Table 4 also includes the level of theory used in the main text ($DFT-KS/B88LYP/aug-cc-pV5Z$) with the purpose of comparing energies. The computations are performed for a helium bubble with radius 8.227 Å.

Table 4 Energies of the He cage with and without electron

The results of Table 4 indicate that the energy difference $E(He_{60}^-) -E(He_{60})$ using the functional $APF-D$, which includes dispersion effects, is 0.0159 Hartree. It is essentially identical to the energy obtained from the functional APF, which includes no dispersion effects. Thereby, there are no noticeable energy changes (by the inclusion of the dispersion effects) up to the fourth significant figure on the bubble energy.

On the other side, we find a value of 0.0191 Hartree of the energy difference $E(He_{60}^-) -E(He_{60})$ using the level of theory reported in the body text. There is an absolute difference of 0.0031 Hartree between the energies given by the functionals APF and B88LYP, both without including dispersion effects. In principle, the value 0.0031 Hartree may be considered an upper limit in the error when one uses different energy functionals. Such a number is an order of magnitude smaller than the energy differences presented in the main text. The reason to use the B88LYP functional, instead of the APF functional, in the main text is the relatively rapidity of the energy convergence in the self-consistent process.