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Register a new userCubic-scaling iterative solution of the Bethe-Salpeter equation for finite systems
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Mathias Per Ljungberg
Publication date
2015
fields
Physics
and research's language is
English
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The Bethe-Salpeter equation (BSE) is currently the state of the art in the description of neutral electron excitations in both solids and large finite systems. It is capable of accurately treating charge-transfer excitations that present difficulties for simpler approaches. We present a local basis set formulation of the BSE for molecules where the optical spectrum is computed with the iterative Haydock recursion scheme, leading to a low computational complexity and memory footprint. Using a variant of the algorithm we can go beyond the Tamm-Dancoff approximation (TDA). We rederive the recursion relations for general matrix elements of a resolvent, show how they translate into continued fractions, and study the convergence of the method with the number of recursion coefficients and the role of different terminators. Due to the locality of the basis functions the computational cost of each iteration scales asymptotically as $O(N^3)$ with the number of atoms, while the number of iterations is typically much lower than the size of the underlying electron-hole basis. In practice we see that , even for systems with thousands of orbitals, the runtime will be dominated by the $O(N^2)$ operation of applying the Coulomb kernel in the atomic orbital representation
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We present a method to compute optical spectra and exciton binding energies of molecules and solids based on the solution of the Bethe-Salpeter equation (BSE) and the calculation of the screened Coulomb interaction in finite field. The method does not require the explicit evaluation of dielectric matrices nor of virtual electronic states, and can be easily applied without resorting to the random phase approximation. In addition it utilizes localized orbitals obtained from Bloch states using bisection techniques, thus greatly reducing the complexity of the calculation and enabling the efficient use of hybrid functionals to obtain single particle wavefunctions. We report exciton binding energies of several molecules and absorption spectra of condensed systems of unprecedented size, including water and ice samples with hundreds of atoms.
We present a hybrid approach for GW/Bethe-Salpeter Equation (BSE) calculations of core excitation spectra, including x-ray absorption (XAS), electron energy loss spectra (EELS), and non-resonant inelastic x-ray scattering (NRIXS). The method is based on {it ab initio} wavefunctions from the plane-wave pseudopotential code ABINIT; atomic core-level states and projector augmented wave (PAW) transition matrix elements; the NIST core-level BSE solver; and a many-pole GW self-energy model to account for final-state broadening and self-energy shifts. Multiplet effects are also accounted for. The approach is implemented using an interface dubbed OCEAN (Obtaining Core Excitations using ABINIT and NBSE). To demonstrate the utility of the code we present results for the K-edges in LiF as probed by XAS and NRIXS, the K-edges of KCl as probed by XAS, the Ti L_2,3-edge in SrTiO_3 as probed by XAS, and the Mg L_2,3-edge in MgO as probed by XAS. We compare the results to experiments and results obtained using other theoretical approaches.
We present first-principles many-body perturbation theory calculations of the quasiparticle electronic structure and of the optical response of HfO$_2$ polymorphs. We use the $GW$ approximation including core electrons by the projector augmented wave (PAW) method and performing a quasiparticle self-consistency also on wavefunctions (QS$GW$). In addition, we solve the Bethe-Salpeter equation on top of $GW$ to calculate optical properties including excitonic effects. For monoclinic HfO$_2$ we find a fundamental band gap of $E_g = 6.33$ eV (with the direct band gap at $E_g^d = 6.41$ eV), and an exciton binding energy of 0.57 eV, which situates the optical gap at $E^o_g = 5.85$ eV. The latter is in the range of spectroscopic ellipsometry (SE) experimental estimates (5.5-6 eV), whereas our electronic band gap is well beyond experimental photoemission (PE) estimates ($< 6$ eV) and previous $GW$ works. Our calculated density of states and optical absorption spectra compare well to raw PE and SE spectra. This suggests that our predictions of both optical and electronic gaps are close to, or at least lower bounds of, the real values.
It is well known that the ambient environment can dramatically renormalize the quasiparticle gap and exciton binding energies in low-dimensional materials, but the effect of the environment on the energy splitting of the spin-singlet and spin-triplet exciton states is less understood. A prominent effect is the renormalization of the exciton binding energy and optical strength (and hence the optical spectrum) through additional screening of the direct Coulomb term describing the attractive electron-hole interaction in the kernel of the Bethe-Salpeter equation (BSE). The repulsive exchange interaction responsible for the singlet-triplet slitting, on the other hand, is unscreened within formal many-body perturbation theory. However, Loren Benedict argued that in practical calculations restricted to a subspace of the full Hilbert space, the exchange interaction should be appropriately screened by states outside of the subspace, the so-called $S$ approximation cite{Benedict2002}. Here, we systematically explore the accuracy of the $S$ approximation for different confined systems, including a molecule and heterostructures of semiconducting and metallic layered materials. We show that the $S$ approximation is actually exact in the limit of small exciton binding energies (i.e., small direct term) and can be used to significantly accelerate convergence of the exciton energies with respect to the number of empty states, provided that a particular effective screening consistent with the conventional Tamm-Dancoff approximation is employed. We further find that the singlet-triplet splitting in the energy of the excitons is largely unaffected by the external dielectric environment for most quasi-two-dimensional materials.
We present a systematic investigation of the role and importance of excitonic effects on the optical properties of transitions metal oxide perovskites. A representative set of fourteen compounds has been selected, including 3$d$ (SrTiO$_3$, LaScO$_3$, LaTiO$_3$, LaVO$_3$, LaCrO$_3$, LaMnO$_3$, LaFeO$_3$ and SrMnO$_3$), 4$d$ (SrZrO$_3$, SrTcO$_3$ and Ca$_2$RuO$_4$) and 5$d$ (SrHfO$_3$, KTaO$_3$ and NaOsO$_3$) perovskites, covering a band gap ranging from 0.1 eV to 6.1 eV and exhibiting different electronic, structural and magnetic properties. Optical conductivities and optical transitions including electron-hole interactions are calculated through the solution of the Bethe-Salpeter equation (BSE) with quasi-particle energies evaluated by single-shot $G_0W_0$ approximation. The exciton binding energies are computed by means of a model-BSE (mBSE), carefully benchmarked against the full BSE method, in order to obtain well-converged results in terms of k-point sampling. The predicted results are compared with available measured data, with an overall satisfactory agreement between theory and experiment.
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