Ab Initio Random Matrix Theory of Molecular Electronic Structure

Abstract

We use ab initio electronic-structure methods to investigate random-matrix theory (RMT) universality in molecular electronic structure. Using single-reference electronic structure methods, including Hartree-Fock, configuration-interaction singles (CIS), density functional theory, and linear-response time-dependent density-functional theory, we compute single-particle orbital energies and many-electron excitations of several representative molecules (benzene, alanine, 1-phenylethylamine, methyloxirane, and helicene chains). For generic low-symmetry geometries, the unfolded spectra of these ab initio Hamiltonians exhibit Wigner-Dyson level statistics of the Gaussian orthogonal ensemble (GOE). For extended helicene chains we explicitly restrict to bound valence excitations below the ionization threshold and still observe GOE statistics, indicating that the RMT universality is present for physical states of direct relevance to real molecules. We further explore the electric and magnetic field dependence of the molecular electronic spectra. The variance of electric polarizability (level curvature K) is predicted to be non-analytic in the magnetic field which serves as an infrared cutoff, <K^2> proportional to log(1/|B|). We observe a transition to the Gaussian unitary ensemble (GUE) by increasing the magnetic fields, although it occurs only at magnetic fields far beyond experimentally accessible scales. Our results indicate that random matrix universality provides a general framework for organizing ab initio predictions of interacting electron spectra in complex systems.

Figures (8)

• Figure 1: Nearest-neighbor level spacing histograms of the unfolded single-particle energies for (a) methyloxirane, (b) alanine, (c) 1‑phenylethylamine, (d) $C_{1}$ benzene, and (e) $D_{6h}$ benzene. The molecular structures are shown from left to right in the same order and share the same Cartesian system (x,y,z) with identical axis orientations. The magenta arrows represent the atomic displacement in benzene, which reduces its group symmetry from $D_{6h}$ to $C_{1}$. Carbon atoms are shown in dark gray, hydrogen in light gray, nitrogen in blue, and oxygen in red.
• Figure 2: Three representative eigenfunctions of the perturbed benzene ($n=697$ - top panel, $n=704$ - middle panel, and $n=861$ - lower panel). The left panel displays the isosurface plots (isovalue = 0.01 a.u.) of molecular virtual orbital wavefunction, with red and blue colors representing opposite signs. The middle panel displays the position space $\int|\psi(\bm r)|^2 dr_z$ and the right panel is the momentum space projection onto the xy-plane $\int|\tilde{\psi}(\bm p)|^2dp_z$. The red circle plotted in the right panel represents equimomentum contour defined by the average kinetic energy of the orbital $p^2_x+p^2_y = 2m_e(\epsilon-\langle\hat{V}_{eff}\rangle)$.
• Figure 3: Nearest-neighbor level spacing statistics for (a)-(e) interacting and (f)-(j) non-interacting many particle excitations of (a)(f) methyloxirane, (b)(g)alanine, (c)(h) 1-phenylethylamine, (d)(i) $C_{1}$ benzene , and (e)(j) $D_{6h}$ benzene. The three chiral molecules and the perturbed benzene molecule have no spatial symmetries and the nearest-neighbor level spacings show GOE distributions. The symmetric benzene molecule is known to have $D_{6h}$ group symmetry and mixing excited states across symmetry groups disrupt the GOE distribution.
• Figure 4: Spectral form factors calculated with unfolded energy levels for alanine (blue), $C_{1}$ benzene (orange), $D_{6h}$ benzene (green), methyloxirane (red), 1-phenylethylamine (purple). The molecules that exhibit WD nearest-level statistics (all except $D_{6h}$ benzene, shown in green) also display a ramp following the initial dip, consistent with a hallmark signature of quantum chaos.
• Figure 5: Nearest-neighbor level spacing statistics for interacting CIS excitations of (a)-(b) alanine, (c)-(d) perturbed benzene in an external magnetic field along the z direction with $B_z = 0.001$ a.u. in (a)(c) and $B_z = 0.01$ a.u. in (b)(d). The x, y, and z axes are shown in the molecular frames in orange, green, and blue. The transition from GOE to GUE becomes more apparent with $B_{z}=10^{-2}$ a.u..