Communication: Superatom molecular orbitals: New types of long-lived electronic states Free

We present ab initio calculations of the quasiparticle decay times in a Buckminsterfullerene based on the many-body perturbation theory. A particularly lucid representation arises when the broadening of the quasiparticle states is plotted in the angular momentum (ℓ) and energy (ɛ) coordinates. In this representation the main spectroscopic features of the fullerene consist of two occupied nearly parabolic bands, and delocalized plane-wave-like unoccupied states with a few long-lived electronic states (the superatom molecular orbitals, SAMOs) embedded in the continuum of Fermi-liquid states. SAMOs have been recently uncovered experimentally by Feng et al. [Science 320, 359 (2008) https://doi.org/10.1126/science.1155866] using scanning tunneling spectroscopy. The present calculations offer an explanation of their unusual stability and unveil their long-lived nature making them good candidates for applications in the molecular electronics. From the fundamental point of view these states illustrate a concept of the Fock-space localization [B. L. Altshuler, Y. Gefen, A. Kamenev, and L. S. Levitov, Phys. Rev. Lett. 78, 2803 (1997) https://doi.org/10.1103/PhysRevLett.78.2803] with properties drastically different from the Fermi-liquid excitations.

Superatom molecular orbitals (SAMOs) were recently discovered¹ as universal characteristics of C₆₀ molecules and their aggregates. Further experiments on endohedral systems² and calculations for a series of quasi-spherical molecules³ showed that SAMOs are associated with the whole system (rather than with a particular atom), have a well defined spherical symmetry (Fig. 1, angular momentum ℓ = 0, 1, 2, principal quantum number n = 3), and are capable of forming chemical bonds. Being markedly different from other unoccupied delocalized states they hold a promise for unique applications in molecular electronics. Hitherto theory was not able to answer why these states are so resilient to the chemical environment and why their character is not washed out by the hybridization. The clarification of these issues is of a critical importance for the SAMO-mediated charge transport.⁴ 

To resolve these issues we report here results of quantum chemical calculations for a prototypical C₆₀ using a recently developed self-energy formalism.⁵ When a transport electron is injected into the unoccupied SAMO (i.e., the single-particle (1p) state ϕ_α), the molecule undergoes a transition from the ground state

$ΨI(N)$ into the mixed quantum state

$\Psi _F^{(N+1)}$

$ΨF(N+1)$ that decays in time. By expanding the final-state wave-function in terms of the so called Dyson orbitals

one can view this as a quantum evolution with the initial condition

At later instances the decay is exponential

$exp(−γαt)$ with

${\bm {\gamma }}_\alpha$

$γα$ given by the imaginary part of the on-shell self-energy

${\bm {\gamma }}_\alpha =\Im {\bm {\Sigma }}_{\alpha \alpha }(\epsilon )$

$γα=ℑΣαα(ε)$ (Ref. 6). However, it is the initial stage of the decay that is manifested in ultrafast nonequilibrium processes of technological relevance. We have shown⁵ that the spectral function for a large class of relevant electronic systems can be represented in the form

where the set-in time of the exponential decay is given by

$τα=2γα/σαα2$⁠. The spectral function

${\bm {A}}_\alpha (t)$

$Aα(t)$ has the following short and long time-limits:

meaning that for short times (t ≪ τ_α) we have the quadratic decay

$σαα2$ is the central quantity for our theory. It can be determined as follows. A simple substitution of the asymptotic expansions into the Dyson equation leads to the exact relations between the spectral moments of the self-energy

${\bm {\rm \Sigma }}(\omega )$

$Σ(ω)$ and of the spectral function

${\bm {\rm A}}(\omega )$

$A(ω)$⁠:⁷ 

where

$Σ∞$ is the frequency independent real part of the self-energy.⁸ 

${\bm {\rm \varepsilon }}$

$ɛ$ is a diagonal matrix with the elements given by the zeroth-order state energies. By defining the matrix of the spectral functions in terms of the imaginary part of the single-particle Green function (⁠

${\bm {\rm A}}(\omega )=\frac{1}{\pi }|{\rm Im}{\bm {\rm G}}(\omega )|$

$A(ω)=1π| Im G(ω)|$⁠) and likewise for the spectral function of the self-energy (⁠

${\bm {\rm S}}(\omega )=\frac{1}{\pi }|{\rm Im}{\bm {\rm \Sigma }}(\omega )|$

$S(ω)=1π| Im Σ(ω)|$⁠), and by the use of the superconvergence theorem,⁹ one can redefine the matrices in terms of the frequency integrals:

Equation (8) expressed in the basis of the Hartree-Fock states is central to our discussion. In order to be feasible it should be written in a finite basis and coupled with an appropriate approximation for the self-energy operator. It is well known that the self-energy can be recast in terms of the 2p1h (2h1p) six-point correlation function.^8,10 Our approach corresponds to its factorization into the product of p-h and p(h) correlation functions (Fig. 2) linking it to the established GW approximation.^11,12 It is remarkable that after all complicated Lehmann-type expressions for the correlation functions are substituted into the

$Σ=iGW$ and the frequency integral in Eq. (10) is performed, we obtain an expression for the energy-uncertainty of the particularly simple form:

where

$ρij=⟨ΨI(N)|ci†cj|ΨI(N)⟩$ is the single-particle density matrix and the Coulomb matrix elements are defined as

$\langle \alpha \beta |\gamma \delta \rangle =\int d({\bm {\rm r}}_1{\bm {\rm r}}_2)\phi _\alpha ({\bm {\rm r}}_1) \phi _\beta ({\bm {\rm r}}_1)\phi _\gamma ({\bm {\rm r}}_2)\phi _\delta ({\bm {\rm r}}_2)/|{\bm {\rm r}}_1\break -{\bm {\rm r}}_2|$

$⟨αβ|γδ⟩=∫d(r1r2)φα(r1)φβ(r1)φγ(r2)φδ(r2)/|r1−r2|$⁠. The prefactor 2 arises from the spin degeneracy. Apart from the factorization of the six-point correlation function Eq. (11) is exact. It is sufficient to determine only the density matrix of the correlated ground state. For small molecules or clusters this can be done at any desired level of accuracy by the configuration interaction (CI) approach which, however, currently is not feasible for fullerenes. Therefore, we consider a further approximation widely known as GW⁰ (Ref. 13) and treat the screened Coulomb interaction on the random phase approximation level. This is equivalent to the configuration interaction singles treatment of the excited states. In view of the Brillouin's theorem¹⁴ the ground state density matrix is ρ_ij = n_iδ_ij leading to the energy-uncertainty expressible in terms of the Coulomb matrix elements only:

We note here important distinctions from the commonly used GW approach. If a fully self-consistent calculation is performed (i.e., the Green's function used to construct the self-energy is the same as the one resulting from the solution of the Dyson equation), the initial guess for the Green's function is, in principle, not relevant. In contrast, for partially self-consistent schemes (their accuracy was recently evaluated by Stan et al.¹⁵) the choice of the basis is important. Typically, Kohn-Sham states are used to construct the initial approximation for the Green's function.¹² In our method we require the Hartree-Fock states as a basis for the expansion of all operators. This is dictated by some important simplifications (e.g., possibility to use the Brillouin's theorem) or by the possibility to perform more accurate CI calculations of the p-h correlation function.¹⁶ 

The main computational burden in our approach is the transformation of the Coulomb matrix elements from the atomic to molecular orbital basis which scales as

$O(N5)$ in the brute force implementation. To obtain convergent results we used very large basis sets (up to 6-311++G(3d3f,3p3d)) resulting in N = 2340 functions for C₆₀. Integral transformations for this number of basis functions is not feasible with any standard quantum chemistry package and required a parallelized implementation fully accounting for the symmetry of the system.

Our calculations clearly indicate a peculiarity of SAMOs: a strong localization in the energy domain (cf.,

$σ HOMO =16.0$ eV and

${\bm {\rm \sigma }}_{\rm SAMO_d}=6.1$

$σ SAMO d=6.1$ eV) (Fig. 1) or their extended lifetime even in comparison with the lifetime of HOMO or LUMO states. This finding endorses the potential of SAMOs as transport channels in molecular electronic devices, since the energy is hardly dissipated during the short transport time. Our results are also unexpected from the Landau's theory of Fermi liquids¹⁷ and illustrate that finite systems possess electronic excitations that differ drastically from quasiparticles in an extended matter. Namely, a particle excitation may decay into two particles and one hole. This process may recur for many generations or it may stop after a few. For the former case a large number of many-particle peaks form a Lorentzian envelope, the so-called quasiparticle. If the decay stops after a finite number of branchings, only Dirac peaks appear in the single particle spectrum (denoting a localization in the Fock space). A decisive discriminating factor was shown to be the particle's energy (ε) (Ref. 18). Our ab initio approach suggests that it is actually the kinetic energy ε_K that governs the decay. We find that even though the energies of electrons with different radial character may be very similar, their kinetic energy is substantially different (as follows from the specific form of the Kohn-Sham potential for this system, Ref. 19), which imposes strong restrictions on the Coulomb matrix elements and leads to the localization of SAMOs in the Fock space. This hints on the prolonged lifetimes and stipulates the observed high stability of these states.

We thank the Deutsche Forschungsgemeinschaft (DFG) for financial support through SFB 762 and H. Petek for fruitful discussions. J.B. acknowledges financial support by the Stanford Institute for Materials & Energy Science and the Stanford Pulse Institute for ultrafast science where parts of this work were completed.