Electron–vibration coupling induced renormalization in the photoemission spectrum of diamondoids

The development of theories and methods devoted to the accurate calculation of the electronic quasi-particle states and levels of molecules, clusters and solids is of prime importance to interpret the experimental data. These quantum systems are often modelled by using the Born–Oppenheimer approximation where the coupling between the electrons and vibrational modes is not fully taken into account, and the electrons are treated as pure quasi-particles. Here, we show that in small diamond cages, called diamondoids, the electron–vibration coupling leads to the breakdown of the electron quasi-particle picture. More importantly, we demonstrate that the strong electron–vibration coupling is essential to properly describe the overall lineshape of the experimental photoemission spectrum. This cannot be obtained by methods within Born–Oppenheimer approximation. Moreover, we deduce a link between the vibronic states found by our many-body perturbation theory approach and the well-known Jahn–Teller effect.


Supplementary Tables
Supplementary Table 1. Adamantane related vibration modes and their coupling to the given electronic state. HOMO (eV) SF 26-28, t2 № of peak coupled state mode symmetry frequency (meV) residual  Table 3), respectively. The tables can be interpreted as follows. The first column labels the electronic level such as the highest occupied molecular orbital (HOMO), the level below it (HOMO− 1), etc. In the spectral function the breakdown of the electron quasi-particle (QP) picture may result in several peaks for a given electronic level. For instance, there are two peaks for HOMO in adamantane (see Supplementary Table 1). The peaks are labeled by № symbol of 1, 2, etc., and the energetic position of these broadened peaks (in eV) are also shown beside the index of the peak in parenthesis. In the top row of the second column we provide the index of the given electronic state ('SF') together with its symmetry. In the case of degenerate state, all the degenerate states' indexes are given (e.g., SF 26-28 in adamantane means that the HOMO is triple degenerate). In the "coupled state" column we give the indexes of states that couple to the given electronic states by vibrations. If the vibration couples to the same states (diagonal term in the electron-vibration Hamiltonian) then the text is written by a normal black ink otherwise it is indicated by red color.
We note that more than one state may couple to the given peak in the spectral function. These multiple states are separated by a semicolon in the row. In the column "mode" we provide the indexes of the vibration modes that contribute the most to the electron-vibration coupling for the given peak of the given electronic state. A larger index indicates a larger vibration frequency.
More-than-one vibration mode is separated by semicolon in the row. The corresponding symmetry labels and their frequency in meV unit are given in the columns "symmetry" and "frequency", respectively. Finally, we give the contribution of the given pairs of states coupled by the given vibration in the corresponding peak of spectral function (column "residual"). We provide here only the most significant contributions.

Supplementary Methods
Ab initio calculation of the broadening of the quasi-particle levels According to Born-Oppenheimer approximation, the electrons follow the motion of nuclei instantly, giving the possibility to define the potential energy surface (PES). If this surface can be approximated with a multidimensional parabola then the wavefunction of nuclei will be quantum harmonic oscillators. We assume that the Born-Oppenheimer approximation holds both in the neutral and ionized states, and their potential energy surfaces can be calculated by density functional theory (DFT). The potential energy surface of the neutral and ionized states differs. As a consequence, vibration assisted broadening of the quasi-particle levels may appear in the PES where its amplitude for a given quasi-particle state may be calculated as where Φ N and Φ I wavefunctions are the many-body harmonic oscillators of the neutral (N ) and the ionized (I) systems, respectively. We apply Huang-Rhys (HR) approximation to calculate the integral in Supplementary Eq. (1) which assumes displaced multidimensional parabolas between the neutral and ionized systems. In this case the (3M − 6) 2 dimensional problem reduces to (3M − 6) one-dimensional problems where M is the number of atoms in the nanodiamonds with (3M − 6) independent harmonic oscillators. For a given harmonic oscillator λ the overlap can be calculated where the H n (x) (λ) is the nth Hermite polynomial, and φ m (x) (λ) is the wavefunction of the dimensionless quantum harmonic oscillator. The displacement d λ of along mode λ is related directly to the partial HR factors S λ where the corresponding q λ is defined in the Method section of the main text. With the properties of the Hermite polynomials, the overlap (O λ ) for the mode λ in Supplementary Eq. 2 can be transformed into a finite series, where m and n label the vibration quantum numbers of the neutral and ionized systems, respectively. The summation of k goes up to either n or m, whichever is larger. Here we can further simplify this equation by recognizing the fact that the lowest energy of the vibration modes are larger in all the considered diamondoids than the measurement temperature (room temperature).
This means that m = 0 for each λ in practice, that is, the room temperature spectrum should be almost equal to the zero kelvin spectrum. As a consequence, the summation in Supplementary Eq. 4 reduces, and the amplitude of the vibration assisted sidebands, 0 → n can be calculated as where n = 0 gives the amplitude of the pure quasi-particle state and the rest are the vibration assisted bands that broaden quasi-particle peak. It is clear from Supplementary Eq. 5 that the amplitude of the vibration sidebands decays exponentially as a function of the quantum number of n for a given λ. The amplitude becomes approximately zero at n = 10 for any vibration mode λ for the considered diamondoids. The partial HR spectral function can be defined as The total vibration sideband can be calculated as a convolution ( * ) of (3M − 6) partial HR spectral functions, Finally, by defining the total HR factor as S = λ S λ and an average vibration mode as ω = λ ω λ S λ /S one can approximate a continuous envelope function of A ( ω) by where n can be chosen to be continuous by taking n! = Γ(n + 1), and evaluate the Γ function between n = [−1..∞]. We applied this simplified function to broaden the quasi-particle levels and the corresponding many-body perturbation theory spectral functions.
For the HOMO levels of adamantane, diamantane and urotropine we show the calculated spectrum in detail in Supplementary Figure 1. In the detailed spectrum some peaks can be identified in the photo-emission spectrum (black curve) of adamantane and diamantane as the pure quasiparticle peak and the vibration assisted peaks. The red curve plots the calculated A ( ω) in Supplementary Eq. 6 which shows a non-symmetrical function with fine structure that originates from vibration modes coupled strongly to the ionization process. In the case of adamantane and diamantane the calculated fine structure is visible for n = 4 of their corresponding average vibration modes. Whereas some peaks in PES may be identified as these resonances, nevertheless, we rather focus on the broadening of the spectrum as defined in Supplementary Eq. 7 (blue curve).
The overall width of the broadening of the HOMO is ≈1.2 eV in the considered diamondoids.