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WIREs Comput Mol Sci
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Periodic and fragment models based on the local correlation approach

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A rigorous treatment of dynamical electron correlation in crystalline solids is one of the main challenges in today's materials quantum chemistry and theoretical solid state physics. In this study, we address this problem by using the local correlation approach and exploring a variety of methods, ranging from the full periodic treatment through embedded fragments to finite clusters. Apart from the computational advantages, the direct‐space local representation for the occupied space allows one to partition the system into fragments and thus forms a natural basis for a hierarchy of embedding models. Furthermore, a subset of localized orbitals in a cluster or a fragment can be chosen to mimic the unit cell of the reference periodic system. Introduction of such subsets allows one to define a formal quantity “the correlation energy per unit cell”, which is directly related to the correlation energy per unit cell in the crystal. The orbital pairs, where neither of the two localized orbital indices belongs to the “unit cell” do not explicitly contribute to the “energy per cell”: Their role is to provide correlated embedding via the couplings in the amplitude equations. The periodic, fragment and finite‐cluster approaches can be combined in a form of high precision computational protocols, where progressively higher‐level corrections are evaluated using lower‐level embedding models. We apply these techniques to investigate the importance of Coulomb screening in dispersively interacting systems on the examples of the phosphorene bilayer and the adsorption of water on 2D silica.

This article is categorized under:

  • Structure and Mechanism > Computational Materials Science
  • Electronic Structure Theory > Ab Initio Electronic Structure Methods
  • Software > Quantum Chemistry
Comparison of (a) periodic and (b) fragment classification of interlayer pairs, in the case of a model bilayer system with two WFs per cell. The WFs of the lower and upper layers are displayed by smaller blue and larger red circles, respectively, and are denoted by combined intracell (1, 2) and cell‐counting 2¯1¯2 indices. The latter are written in italic font and decorated with an overbar if the cell corresponds to a “negative” translation. Panel (a) presents the full periodic system with a restricted pair list (including pairs involving up to second nearest‐neighbor WFs), and panel (b) a fragment cut out from it. The pairs are schematically represented by arrows: squares denote the first orbital index in a pair and arrowheads the second. In panel (b) the black solid arrows refer to pairs with both orbitals inside the reference cell, the blue dashed arrows to pairs with just one WF inside it, and the green dotted arrows to pairs with both WFs outside the reference cell. (Reprinted with permission from Masur et al. (). Copyright 2016 American Chemical Society)
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Comparison of the three embedding models, presented in the sections 2.2, 2.3, and 2.4, respectively
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Correlation interaction energies ΔEcorr. (a) and HL–LL corrections δEHL‐LL (b) for the water monolayer as functions of fragment's or cluster's size. ΔEcorr. was computed via Equation . The corrections δEHL‐LL are defined by Equation with LL = LMP2, SCS‐LMP2 and LdrCCD, and HL = LCCSD(T0)|LCCD[S]‐R−6. The periodic LMP2 and SCS‐LMP2 results are shown by black and red dotted lines, respectively. The fragment results are given with dashed lines and filled symbols, while the finite‐cluster ones with solid lines and open symbols: LMP2, squares; SCS‐LMP2, circles; LdrCCD, asterisks; LCCSD(T0)|LCCD[S]‐R−6, diamonds. The fragment LMP2 and SCS‐LMP2 values were computed in the periodic‐fragment model (cf. section 2.2), while LdrCCD in the embedded‐fragment model (cf. section 2.3)
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Correlation interaction energies ΔEcorr. (a) and HL–LL corrections δEHL‐LL (b) for the water monolayer adsorbed on 2D silica as functions of fragment's or cluster's size. ΔEcorr. was computed via Equation . The corrections δEHL‐LL are defined by Equation with LL = LMP2, SCS‐LMP2 and LdrCCD, and HL = LCCSD(T0)|LCCD[S]‐R−6. The periodic LMP2 and SCS‐LMP2 results are shown by black and red dotted lines, respectively. The fragment results are given with dashed lines and filled symbols, while the finite‐cluster ones with solid lines and open symbols: LMP2 – squares, SCS‐LMP2 – circles, LdrCCD – asterisks, LCCSD(T0)|LCCD[S]‐R−6 – diamonds. The fragment LMP2 and SCS‐LMP2 values were computed in the periodic‐fragment model (cf. section 2.2), while LdrCCD in the embedded‐fragment model (cf. section 2.3). The “number of correlated orbitals” directly reflects the size of the fragment or cluster
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Correlation interaction energies ΔEcorr. (a), the increments ΔEincr. thereof by the fragment expansion (logarithmic scale) (b), and HL–LL corrections δEHL‐LL (c) for the phosphorene bilayer as functions of fragment's or cluster's size. ΔEcorr. was computed via Equation . ΔEincr. are the correlation interaction energy increments, obtained by stepwise extensions of the fragments by increasing Rcutintra. The corrections δEHL‐LL were defined by Equation with LL = LMP2, SCS‐LMP2 and LdrCCD, and HL = LCCSD(T0)|LCCD[S]‐R−6. The periodic LMP2 and SCS‐LMP2 results are shown by black and red dotted lines, respectively. The fragment results are given with dashed lines and filled symbols, while the finite‐cluster ones with solid lines and open symbols: LMP2 – squares, SCS‐LMP2 – circles, LdrCCD – asterisks, LCCSD(T0)|LCCD[S]‐R−6 – diamonds. The fragment LMP2 and SCS‐LMP2 were computed in the periodic‐fragment model (cf. section 2.2), while LdrCCD in the embedded‐fragment model (cf. section 2.3). The embedded‐fragment LMP2 results are also given in the panel (a) by the gray dashed line and crosses. (Panel (a): Reprinted with permission from Schütz et al. (). Copyright 2017 American Chemical Society)
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The cluster models used for the 2D silica–water system (a), and water monolayer (b). The silica atoms are shown in blue, oxygen in red and hydrogen in pink
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The cluster models used for the phosphorene bilayer. The gray, blue, and red atoms are phosphorus, and yellowish – hydrogens. The red and blue atoms denote the unit cell atoms of type A and B, respectively (cf. section 2.5). (Reprinted with permission from Schütz, Maschio, Karttunen, and Usvyat (). Copyright 2017 American Chemical Society)
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The systems studied in this work: phosphorene bilayer (side view, panel a), and a water monolayer adsorbed on 2D silica (top view, panel b). The phosphorus atoms are given in yellow color, the silicon atoms in blue, the water's oxygen in red, the silica's oxygens in pink, the hydrogens in gray. In the 2D silica–water system, half of the water molecules are located over the silicon atoms, while the other half over the centers of the silica's six‐membered rings. The green lines indicate the unit cells
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