Random phase approximation with exchange for an accurate description of crystalline polymorphism

We determine the correlation energy of BN, ${\mathrm{SiO}}_2$, and ice polymorphs employing a recently developed random phase approximation with exchange (RPAx) approach. The RPAx provides larger and more accurate polarizabilities as compared to the random phase approximation (RPA), and captures the effects of anisotropy. In turn, the correlation energy, defined as an integral over the density-density response function, gives improved binding energies without the need for error cancellation. Here, we demonstrate that these features are crucial for predicting the relative energies between low- and high-pressure polymorphs of different coordination number as, e.g., between $\alpha$-quartz and stishovite in ${\mathrm{SiO}}_2$, and layered and cubic BN. Furthermore, a reliable ${\left({\mathrm{H}}_2\mathrm{O}\right)}_2$ potential energy surface is obtained, necessary for describing the various phases of ice. The RPAx gives results comparable to other high-level methods such as coupled cluster and quantum Monte Carlo, also in cases where the RPA breaks down. Although a higher computational cost than RPA, we observe a faster convergence with respect to the number of eigenvalues in the response function.

Figure 1

Diagrammatic representation of the two terms in Eq. (4). Full lines correspond to KS Green's functions.

Figure 2

Cohesive energy (eV) as a function of nearest-neighbor distance (Å) in solid (a) Ar and (b) Kr. The correlation energy (Ec-RPAx) is also shown, together with a fitted $y=k/x^6$ curve. Green and blue squares correspond to CCSD(T) results [42].

Figure 3

Upper row: Wurtzite ($w$-BN), cubic ($c$-BN), and layered rhombohedral ($r$-BN) boron nitride. Lower row: Cristobalite and $\alpha$-quartz with fourfold coordinated silicon, and stishovite with sixfold coordinated silicon. Silicon in gold and oxygen in red.

Figure 4

Interlayer binding energy (eV) of $r$-BN as a function of interlayer separation (Å).

Figure 5

(a) Correlation energy difference of $r$-BN and $c$-BN as a function of eigenvalues per electron within RPA and RPAx. (b) Relative energies (meV) of $w$-BN and $r$-BN with respect to $c$-BN within LDA, PBE0, RPA, SOSEX, and RPAx. CC results are from Ref. [38]. $c$-BN is the reference value at zero.

Figure 6

(a) Correlation energy difference of stishovite and $\alpha$-quartz as a function of eigenvalues per electron within RPA and RPAx. (b) Relative energies (meV) between cristobalite and $\alpha$-quartz (red) and between stishovite and $\alpha$-quartz (yellow) within LDA, PBE0, RPA, SOSEX, and RPAx. QMC results are from Refs. [5, 63].

Figure 7

Error in the binding energy (meV) with respect to CCSD(T) for the ten Smith stationary points.

Figure 8

Binding energy (meV) as a function of O-O distance (Å) of water dimers in the (a) SP1 and (b) SP6 configuration. Red square corresponds to the CCSD(T) result [72].

Figure 9

Structures of the four ice polymorphs studied in this work (XI, IX, II, VIII). Blue-dashed boxes indicate water dimers in non-hydrogen bonded configurations, but with comparable O-O distance.

Figure 10

The $\alpha$-parameter of the hybrid functional was chosen so as to minimize the total energy of the water molecule. The minimum occurs at 20% in RPA and at 35% in RPAx.

Figure 11

Convergence of the correlation energy difference with respect to the number of eigenvalues per electron in the response function. (a) Correlation energy difference between solid Ar and the isolated Ar atom, (b) between layered $r$-BN and an isolated BN layer, (c) between $\alpha$-quartz and cristobalite, and (d) between the water dimer and an isolated water molecule.

Figure 12

Convergence of the correlation energy of (a) Ar, (b) $c$-BN, (c) stishovite, and (d) Ice-XI with respect to the number of $k$-points. Curves have been shifted vertically by subtracting the value at the last point.

Figure 13

Illustration of the ten Smith stationary points of the (${\mathrm{H}}_2{\mathrm{O})}_2$ potential energy surface.