Feynman diagram for the GW self-energy expressed in our local LCAO and dominant products basis.
(a) Density of states of benzene computed from different Green's functions using as an input the results of a DFT-LDA calculation performed with the SIESTA package. A DZP basis set, with orbital radii determined using a value of the energy shift parameter of 3 meV, has been used. The results shown in this figure are obtained with a single energy window. GW x refers to the results obtained with only the instantaneous part of the self-energy (only exchange), while GW xc labels the results obtained with the whole self-energy (incorporating additional correlation effects). (b) Ball and stick model of benzene produced with the XCRYSDEN package (see Ref. 39).
The density of states of benzene computed with a uniformly discretized spectral function and using the second window technique. The peak positions are very weakly perturbed by using the two windows technique. The parameters of the calculation are identical with those of Fig. 2. The two windows technique allowed to reduce the number of frequency points from N ω = 1024 to N ω = 192.
(a) Comparison of the screened interaction calculated for benzene using our original dominant product basis and the screened interaction projected to a compressed product basis (see Eq. (38)). We plot the sum of all the matrix elements of the imaginary part of the screened interaction. (b) A plot of the difference of the functions represented in panel (a). The change in spectral weight of the screened interaction due to compressing the space of dominant products is seen to be small. Please notice the different scales of the y-axis in both panels.
(a) Density of states for naphthalene. The results have been obtained with our most extended basis orbitals (corresponding to an energy shift of 20 meV (see Ref. 35)). We can appreciate the accuracy of the second window technique. (b) Ball and stick model of naphthalene produced with the XCRYSDEN package (see Ref. 39).
(a) Density of states for anthracene. The results have been obtained using the extended basis orbitals corresponding to an energy shift of 20 meV (see Ref. 35). Here, we compare calculations using the instantaneous (exchange-only) self-energy and the full self-energy (including correlation effects). The correlation component of the self-energy is crucial to reproduce the experimental observation that anthracene is an acceptor. In contrast, the exchange-only calculation locates the LUMO level above the vacuum level. (b) Ball and stick model of anthracene produced with the XCRYSDEN package (see Ref. 39).
Ionization potentials and electron affinities for benzene versus the extension of the basis functions. The extension of the atomic orbitals is determined using the energy shift parameter of the SIESTA method (see Ref. 35). Note that rather extended orbitals are necessary to achieve converged results. Differences associated with the use of the second window technique introduced in Sec. V B are of the order of 0.1 eV (see also Table II). The experimental ionization potential is taken from the NIST server (see Ref. 40). The electron affinity of benzene is taken from Ref. 41.
The ionization potentials and electron affinities computed with different first frequency windows in the second window technique. The second energy window extends over a range of ±80 eV. Calculations were performed using atomic orbitals whose extension is defined by a 3 meV energy shift.
Electron affinities for benzene versus the compression parameters and eigenvalues cutoff κ min in Eq. (39). In brackets the dimension of compressed subspace is given, . The dimension is governed by and was 39, 297, and 765 for eV, eV, and eV, accordingly. The number of atomic orbitals is , while the number of dominant products is .
Convergence of the calculated ionization potential and affinity level as a function of the number N orb of numerical atomic orbitals (see Ref. 10) in the basis set describing a benzene molecule. The convergence of the total energy with respect to the TZDP result ΔE total = E total, TZDP − E total, Basis and that of the HOMO and LUMO levels in the DFT-LDA calculations is also displayed. This allows to check the improvement of the description of the ground state of the molecule as we increase the size of the basis. The calculations are performed for a fixed geometry obtained using the TZDP basis. DZP is the standard basis used in ground-state SIESTA (see Ref. 10) calculations. See the text for the definition of the acronyms describing the different basis. N prod is the number of dominant products, i.e., before performing a non-local compression, for different basis sizes.
Ionization potentials and electron affinities for naphthalene and anthracene and their dependence on the extension of the atomic orbitals. For naphthalene we compared the results obtained with spectral functions discretized in one or two windows. The experimental data has been taken from the NIST server (see Ref. 40). For naphthalene and anthracene, vertical ionization potentials are not available at the NIST database. Therefore, we give experimental ionization energies including effects of geometry relaxation.
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