# Solid-State Testing of a Van-Der-Waals-Corrected Exchange-Correlation Functional Based on the Semiclassical Atom Theory

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## Abstract

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## 1. Introduction

## 2. Method and Computational Details

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## Abbreviations

KS | Kohn–Sham |

DFT | Density Functional Theory |

XC | Exchange-Correlation |

GGA | Generalized Gradient Approximation |

LDA | Local Density Approximation |

PBE | Perdew-Burke-Ernzerhof |

ME | Mean Error |

MAE | Mean Absolute Error |

MARE | Mean Absolute Relative Error |

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**Figure 1.**Mean absolute errors, obtained from SG4 + rVV10m as functions of the value of the parameter b in the rVV10m non-local functional (keeping $C={10}^{-4}$), for the lattice constants of the NOB4 and ALK5 test sets.

**Figure 2.**Relative variation of different properties of Xe as functions of the parameter b in the rVV10m correction. The inset reports the energy vs. volume curves for two values of b.

**Figure 3.**Mean absolute errors of different functionals for the lattice constants (

**upper**panel, Å), bulk moduli (

**middle**panel; GPa), and cohesive energies (

**bottom**panel, eV or eV/bohr${}^{2}$) of various test sets.

**Figure 4.**LDA, PBE, PBEsol, and SG4 xc enhancement factors ${F}_{xc}(s,{r}_{s},\zeta =0)$ versus the reduced gradient s, for the spin-unpolarized case ($\zeta =0$) and the bulk parameter ${r}_{s}=10$.

**Figure 5.**Plot of the non-covalent interaction(NCI) indicator [93] (density vs. reduced gradient) for MoTe${}_{2}$ and WS${}_{2}$.

**Table 1.**Lattice constants, bulk moduli, and cohesive energies of the NOB4 test set as obtained with different functionals. The reference values are taken from Reference [63]. For each property, we also report the mean error (ME), the mean absolute error (MAE), and the mean absolute relative error (MARE).

SG4 | SG4-rVV10m | PBE + rVV10L | rVV10 | Reference | ||
---|---|---|---|---|---|---|

$\mathit{b}=\mathbf{8.5}$ | $\mathit{b}=\mathbf{12.0}$ | |||||

Lattice constants (Å) | ||||||

Ne | 5.226 | 4.432 | 4.668 | 4.346 | 4.164 | 4.464 |

Ar | 6.854 | 5.555 | 5.832 | 5.508 | 5.181 | 5.300 |

Kr | 7.459 | 5.856 | 6.219 | 5.892 | 5.549 | 5.646 |

Xe | 8.299 | 6.195 | 6.535 | 6.375 | 6.037 | 6.132 |

ME | 1.574 | 0.124 | 0.428 | 0.145 | −0.153 | |

MAE | 1.574 | 0.140 | 0.428 | 0.204 | 0.153 | |

MARE | 28.46% | 2.57% | 7.83% | 3.72% | 3.06% | |

Bulk moduli (GPa) | ||||||

Ne | 0.40 | 2.10 | 1.40 | 2.50 | 4.00 | 1.10 |

Ar | 0.20 | 1.40 | 1.00 | 2.30 | 4.10 | 2.70 |

Kr | 0.20 | 1.50 | 0.70 | 2.20 | 4.50 | 3.60 |

Xe | 0.10 | 1.90 | 0.80 | 2.30 | 5.10 | 3.64 |

ME | −2.54 | −1.04 | −1.79 | −0.44 | 1.67 | |

MAE | 2.54 | 1.54 | 1.94 | 1.14 | 1.67 | |

MARE | 86.98% | 61.30% | 62.20% | 54.45% | 95.15% | |

Cohesive energies (eV) | ||||||

Ne | 0.01 | 0.10 | 0.06 | 0.05 | 0.04 | 0.02 |

Ar | 0.01 | 0.13 | 0.08 | 0.08 | 0.11 | 0.08 |

Kr | 0.01 | 0.15 | 0.09 | 0.09 | 0.15 | 0.12 |

Xe | 0.01 | 0.19 | 0.11 | 0.13 | 0.22 | 0.16 |

ME | −0.09 | 0.05 | −0.01 | −0.01 | 0.04 | |

MAE | 0.09 | 0.05 | 0.03 | 0.02 | 0.04 | |

MARE | 84.18% | 121.01% | 65.08% | 44.71% | 56.25% |

**Table 2.**Lattice constants, bulk moduli, and cohesive energies of the ALK5 test set as obtained with different functionals. The reference values are taken from References [64,65] and are corrected for thermal and zero-point vibrational effects. For each property, we also report the mean error (ME), the mean absolute error (MAE), and the mean absolute relative error (MARE).

SG4 | SG4-rVV10m | PBE + rVV10L | rVV10 | Reference | ||
---|---|---|---|---|---|---|

$\mathit{b}=\mathbf{8.5}$ | $\mathit{b}=\mathbf{12.0}$ | |||||

Lattice constants (Å) | ||||||

Li | 3.452 | 3.438 | 3.443 | 3.429 | 3.407 | 3.449 |

Na | 4.212 | 4.163 | 4.180 | 4.180 | 4.132 | 4.210 |

K | 5.291 | 5.196 | 5.229 | 5.231 | 5.136 | 5.212 |

Rb | 5.665 | 5.543 | 5.585 | 5.600 | 5.477 | 5.576 |

Cs | 6.131 | 5.927 | 6.001 | 6.029 | 5.846 | 6.039 |

ME | 0.053 | −0.044 | −0.010 | −0.003 | −0.098 | |

MAE | 0.053 | 0.044 | 0.020 | 0.020 | 0.098 | |

MARE | 0.95% | 0.83% | 0.40% | 0.45% | 1.90% | |

Bulk moduli (GPa) | ||||||

Li | 13.00 | 13.30 | 13.20 | 13.80 | 14.10 | 13.30 |

Na | 7.30 | 7.80 | 7.60 | 7.80 | 8.20 | 7.50 |

K | 3.40 | 3.70 | 3.60 | 3.70 | 4.10 | 3.70 |

Rb | 2.60 | 3.00 | 2.80 | 2.90 | 3.30 | 2.90 |

Cs | 1.80 | 2.20 | 2.00 | 2.10 | 2.50 | 2.10 |

ME | −0.28 | 0.10 | −0.06 | 0.16 | 0.54 | |

MAE | 0.28 | 0.10 | 0.10 | 0.16 | 0.54 | |

MARE | 7.53% | 2.44% | 2.60% | 1.55% | 11.80% | |

Cohesive energies (eV) | ||||||

Li | 1.54 | 1.63 | 1.60 | 1.61 | 1.58 | 1.67 |

Na | 1.01 | 1.15 | 1.10 | 1.11 | 1.05 | 1.13 |

K | 0.83 | 0.99 | 0.94 | 0.95 | 0.94 | 0.94 |

Rb | 0.74 | 0.91 | 0.85 | 0.87 | 0.87 | 0.85 |

Cs | 0.68 | 0.89 | 0.82 | 0.83 | 0.86 | 0.80 |

ME | −0.12 | 0.04 | −0.02 | −0.01 | −0.02 | |

MAE | 0.12 | 0.05 | 0.02 | 0.03 | 0.05 | |

MARE | 11.93% | 5.36% | 1.89% | 2.53% | 4.22% |

**Table 3.**Lattice constants, bulk moduli, and cohesive energies of several layered materials as obtained with different functionals. The reference lattice constants are experimental values [21] (corrected for zero-point vibrational effects), reference bulk moduli are VV10 results [20], and reference cohesive energies are Random Phase Approximation (RPA) values [20,21]. For each property, we also report the mean error (ME), the mean absolute error (MAE), and the mean absolute relative error (MARE).

SG4 | SG4-rVV10m | PBE + rVV10L | rVV10 | Reference | ||
---|---|---|---|---|---|---|

$\mathit{b}=\mathbf{8.5}$ | $\mathit{b}=\mathbf{12.0}$ | |||||

Interlayer lattice constants (Å) | ||||||

Graphite | 10.751 | 6.582 | 6.798 | 6.900 | 6.721 | 6.635 |

H-BN | 10.808 | 6.449 | 6.654 | 6.850 | 6.620 | 6.635 |

MoS${}_{2}$ | 13.177 | 12.083 | 12.200 | 12.570 | 12.448 | 12.283 |

MoTe${}_{2}$ | 13.848 | 13.572 | 13.641 | 14.130 | 14.261 | 13.961 |

VS${}_{2}$ | 5.866 | 5.535 | 5.624 | 5.920 | 5.854 | 5.748 |

VSe${}_{2}$ | 6.196 | 5.870 | 5.940 | 6.290 | 6.228 | 6.101 |

PdTe${}_{2}$ | 5.002 | 4.962 | 4.975 | 5.130 | 5.282 | 5.110 |

WS${}_{2}$ | 17.927 | 12.132 | 12.283 | 12.680 | 12.487 | 12.308 |

NbTe${}_{2}$ | 6.591 | 6.492 | 6.519 | 6.840 | 6.929 | 6.606 |

NbSe${}_{2}$ | 12.316 | 12.013 | 12.093 | 12.650 | 12.711 | 12.534 |

ME | 1.456 | −0.223 | −0.119 | 0.204 | 0.162 | |

MAE | 1.547 | 0.223 | 0.156 | 0.204 | 0.165 | |

MARE | 18.64% | 2.57% | 1.82% | 2.48% | 2.01% | |

Intralayer lattice constants (Å) | ||||||

Graphite | 2.452 | 2.451 | 2.451 | 2.470 | 2.466 | 2.456 |

H-BN | 2.497 | 2.498 | 2.498 | 2.510 | 2.514 | 2.510 |

MoS${}_{2}$ | 3.133 | 3.119 | 3.124 | 3.170 | 3.218 | 3.162 |

MoTe${}_{2}$ | 3.487 | 3.464 | 3.473 | 3.530 | 3.617 | 3.518 |

VS${}_{2}$ | 3.115 | 3.108 | 3.111 | 3.170 | 3.229 | 3.221 |

VSe${}_{2}$ | 3.263 | 3.254 | 3.259 | 3.320 | 3.385 | 3.358 |

PdTe${}_{2}$ | 4.053 | 4.033 | 4.040 | 4.080 | 4.159 | 4.024 |

WS${}_{2}$ | 3.139 | 3.124 | 3.130 | 3.170 | 3.217 | 3.153 |

NbTe${}_{2}$ | 3.626 | 3.606 | 3.614 | 3.670 | 3.760 | 3.680 |

NbSe${}_{2}$ | 3.405 | 3.383 | 3.391 | 3.460 | 3.509 | 3.442 |

ME | −0.035 | −0.048 | −0.043 | 0.003 | 0.055 | |

MAE | 0.041 | 0.050 | 0.047 | 0.022 | 0.055 | |

MARE | 1.23% | 1.51% | 1.39% | 0.66% | 1.57% | |

Bulk moduli (GPa) | ||||||

Graphite | 0.50 | 30.00 | 20.40 | - | 42.00 | 34.00 |

H-BN | 0.50 | 27 | 18.90 | - | 37.00 | 37.00 |

MoS${}_{2}$ | 1.20 | 49.00 | 39.50 | - | 52 | 53.00 |

MoTe${}_{2}$ | 42.40 | 75.00 | 65.70 | - | 47.70 | 50.90 |

VS${}_{2}$ | 20.50 | 43.40 | 34.40 | - | 49.40 | 55.60 |

VSe${}_{2}$ | 23.00 | 52.20 | 40.40 | - | 45.10 | 49.00 |

PdTe${}_{2}$ | 92.70 | 103.90 | 100.30 | - | 64.70 | 72.30 |

WS${}_{2}$ | 0.40 | 52.20 | 38.80 | - | 55.90 | 62.10 |

NbTe${}_{2}$ | 48.50 | 77.00 | 68.60 | - | 39.80 | 57.10 |

NbSe${}_{2}$ | 43.50 | 82.20 | 70.90 | - | 51.80 | 55.00 |

ME | −25.28 | 6.59 | −2.81 | −4.06 | ||

MAE | 29.36 | 14.61 | 16.85 | 5.66 | ||

MARE | 59.13% | 26.61% | 32.44% | 10.74% | ||

Cohesive energies (meV/Å${}^{2}$) | ||||||

Graphite | 0.14 | 25.18 | 16.80 | 16.04 | 26.58 | 18.32 |

H-BN | 0.14 | 23.51 | 15.80 | 14.43 | 24.66 | 14.49 |

MoS${}_{2}$ | 0.10 | 28.86 | 20.90 | 19.24 | 28.65 | 20.53 |

MoTe${}_{2}$ | 6.32 | 35.22 | 27.60 | 21.40 | 28.65 | 20.80 |

VS${}_{2}$ | 3.22 | 35.83 | 26.80 | 20.20 | 31.08 | 25.61 |

VSe${}_{2}$ | 4.27 | 37.19 | 28.24 | 20.02 | 31.08 | 22.26 |

PdTe${}_{2}$ | 39.40 | 70.09 | 61.69 | 41.71 | 49.14 | 40.1 |

WS${}_{2}$ | 0.08 | 28.87 | 20.90 | 19.59 | 28.74 | 20.20 |

NbTe${}_{2}$ | 10.48 | 39.23 | 31.60 | 23.51 | 30.95 | 23.03 |

NbSe${}_{2}$ | 6.70 | 40.21 | 31.10 | 21.96 | 31.67 | 19.57 |

ME | −15.41 | 13.93 | 5.65 | −0.68 | 8.63 | |

MAE | 15.41 | 13.93 | 5.96 | 1.70 | 8.63 | |

MARE | 75.72% | 61.01% | 23.68% | 7.45% | 41.46% |

**Table 4.**Lattice constants and cohesive energies of several molecular crystals as obtained with different functionals. The reference values are taken from References [67,68,69,70,71]. For each property, we also report the mean error (ME), the mean absolute error (MAE), and the mean absolute relative error (MARE).

SG4 | SG4-rVV10m | PBE+rVV10L | rVV10 | Reference | ||
---|---|---|---|---|---|---|

$\mathit{b}=\mathbf{8.5}$ | $\mathit{b}=\mathbf{12.0}$ | |||||

Lattice constants (Å) | ||||||

CO${}_{2}$ | 6.810 | 5.616 | 5.801 | 5.722 | 5.561 | 5.624 |

NH${}_{3}$ | 5.066 | 4.880 | 4.916 | 5.033 | 4.962 | 5.048 |

Urea (a) | 5.665 | 5.448 | 5.526 | 5.600 | 5.493 | 5.565 |

Urea (c) | 4.634 | 4.628 | 4.629 | 4.676 | 4.676 | 4.684 |

Naphthalene (a) | 11.426 | 8.019 | 8.273 | 8.406 | 7.972 | 8.108 |

Naphthalene (b) | 6.750 | 5.863 | 5.938 | 5.951 | 5.858 | 5.940 |

Naphthalene (c) | 9.536 | 8.606 | 8.688 | 8.686 | 8.596 | 8.647 |

Pyrazine (a) | 9.039 | 9.167 | 9.223 | 9.345 | 9.241 | 9.325 |

Pyrazine (b) | 5.664 | 5.701 | 5.757 | 5.759 | 5.684 | 5.850 |

Pyrazine (c) | 5.405 | 3.663 | 3.855 | 3.847 | 3.630 | 3.733 |

ME | 0.976 | −0.090 | 0.030 | 0.061 | −0.095 | |

MAE | 1.022 | 0.083 | 0.096 | 0.088 | 0.095 | |

MARE | 17.01% | 1.61% | 1.73% | 1.45% | 1.67% | |

Interaction energies energies (KJ/mol) | ||||||

CO${}_{2}$ | −5.00 | −33.40 | −22.40 | −22.60 | −35.11 | −27.20 |

NH${}_{3}$ | −25.50 | −40.90 | −39.90 | −38.40 | −46.00 | −39.20 |

Urea | −72.60 | −98.70 | −108.40 | −101.50 | −118.70 | −103.60 |

Naphthalene | −5.90 | −95.20 | −62.40 | −58.30 | −91.60 | −70.40 |

Pyrazine | −16.02 | −76.70 | −54.70 | −52.30 | −74.60 | −61.50 |

ME | 35.38 | −8.60 | 2.82 | 5.76 | −12.82 | |

MAE | 35.38 | 10.56 | 5.02 | 5.76 | 12.82 | |

MARE | 62.41% | 18.36% | 9.30% | 10.63% | 22.48% |

**Table 5.**Mean absolute errors on lattice constants, bulk moduli, and cohesive energies of different classes of iono-covalent bulk solids as obtained with different functionals.

SG4 | SG4-rVV10m | PBE+rVV10L | rVV10 | ||
---|---|---|---|---|---|

$\mathit{b}=\mathbf{8.5}$ | $\mathit{b}=\mathbf{12.0}$ | ||||

Lattice constants (Å) | |||||

simple metals | 0.038 | 0.065 | 0.052 | 0.035 | 0.091 |

transition metals | 0.013 | 0.024 | 0.019 | 0.051 | 0.110 |

semiconductors | 0.012 | 0.022 | 0.015 | 0.052 | 0.133 |

ionic crystals | 0.062 | 0.030 | 0.040 | 0.080 | 0.074 |

insulators | 0.012 | 0.012 | 0.012 | 0.024 | 0.037 |

overall | 0.028 | 0.034 | 0.030 | 0.049 | 0.093 |

Bulk moduli (GPa) | |||||

simple metals | 0.86 | 0.33 | 0.51 | 1.36 | 1.53 |

transition metals | 9.96 | 16.14 | 13.20 | 19.42 | 37.30 |

semiconductors | 12.35 | 8.78 | 10.12 | 19.80 | 23.87 |

ionic crystals | 9.20 | 4.88 | 6.54 | 7.24 | 8.60 |

insulators | 8.88 | 7.13 | 7.78 | 19.55 | 31.70 |

overall | 7.83 | 6.99 | 7.19 | 12.59 | 18.90 |

Cohesive energies (eV) | |||||

simple metals | 0.22 | 0.35 | 0.29 | 0.12 | 0.06 |

transition metals | 0.33 | 0.91 | 0.69 | 0.17 | 0.12 |

semiconductors | 0.13 | 0.44 | 0.33 | 0.09 | 0.27 |

ionic crystals | 0.26 | 0.11 | 0.15 | 0.12 | 0.09 |

insulators | 0.38 | 0.59 | 0.50 | 0.31 | 0.10 |

overall | 0.25 | 0.46 | 0.38 | 0.15 | 0.13 |

LDA | SG4 | PBE | PBEsol | SG4-VV10m | PBE-VV10 | ||
---|---|---|---|---|---|---|---|

$\mathit{b}=\mathbf{8.5}$ | $\mathit{b}=\mathbf{12}$ | ||||||

Hydrogen-bond complexes | |||||||

NH${}_{3}$-NH${}_{3}$ | 1.9 | −0.7 | −0.4 | 0.2 | 0.0 | −0.3 | −0.1 |

H${}_{2}$O-H${}_{2}$O | 2.8 | −0.2 | −0.1 | 0.7 | 0.4 | 0.1 | 0.2 |

Formic acid dimer | 7.8 | 0.2 | −0.9 | 2.3 | 2.3 | 1.4 | 0.2 |

Formamide dimer | 5.6 | −1.1 | −1.5 | 0.8 | 0.9 | 0.1 | −0.5 |

Uracil dimer | 5.3 | −1.9 | −2.4 | 0.2 | 1.0 | −0.1 | −0.9 |

2-Pyroxine-Aminopyridine | 5.7 | −1.3 | −1.8 | 0.8 | 1.9 | 0.7 | −0.2 |

Adenine-thymine | 5.1 | −2.2 | −2.5 | 0.0 | 1.4 | 0.1 | −0.7 |

MAE | 4.9 | 1.1 | 1.4 | 0.7 | 1.1 | 0.4 | 0.4 |

Dispersion complexes | |||||||

CH${}_{4}$-CH${}_{4}$ | 0.3 | −0.8 | −0.4 | −0.4 | −0.2 | −0.5 | −0.2 |

C${}_{2}$H${}_{4}$-C${}_{2}$H${}_{4}$ | 0.9 | −1.8 | −1.2 | −0.8 | −0.5 | −1.1 | −0.6 |

C${}_{6}$H${}_{6}$-CH${}_{4}$ | 0.5 | −1.9 | −1.5 | −1.0 | −0.2 | −0.9 | −0.7 |

PD benzene dimer | 0.0 | −5.2 | −4.5 | −3.3 | −0.7 | −2.4 | −2.1 |

Pyrazine dimer | 0.2 | −5.5 | −4.9 | −3.5 | −0.9 | −2.7 | −2.5 |

Stacked uracil dimer | 0.4 | −7.9 | −7.1 | −5.1 | −1.0 | −3.6 | −3.4 |

Stacked indole-benzene | −0.3 | −7.5 | −6.8 | −4.9 | −1.2 | −3.6 | −3.4 |

Stacked adenine-thymine | 0.2 | −11.0 | −10.3 | −7.1 | −1.6 | −5.2 | −5.1 |

MAE | 0.4 | 5.2 | 4.6 | 3.3 | 0.8 | 2.5 | 2.2 |

Mixed character complexes | |||||||

Ethene-ethyne | 0.8 | −0.6 | −0.3 | −0.1 | 0.1 | −0.2 | 0.0 |

C${}_{6}$H${}_{6}$-H${}_{2}$O | 1.1 | −1.6 | −1.3 | −0.7 | −0.1 | −0.7 | −0.6 |

C${}_{6}$H${}_{6}$-NH${}_{3}$ | 0.7 | −1.8 | −1.4 | −0.9 | −0.2 | −0.9 | −0.7 |

C${}_{6}$H${}_{6}$-HCN | 1.2 | −1.9 | −1.8 | −0.9 | −0.1 | −0.8 | −0.9 |

T-shaped benzene dimer | 0.3 | −3.0 | −2.6 | −1.9 | −0.3 | −1.3 | −1.2 |

T-shaped indole-benzene | 0.6 | −3.9 | −3.6 | −2.4 | −0.2 | −1.5 | −1.7 |

phenol dimer | 1.8 | −3.6 | −3.3 | −1.9 | −0.3 | −1.5 | −1.6 |

MAE | 0.9 | 2.4 | 2.1 | 1.2 | 0.2 | 1.0 | 1.0 |

Overall statistics | |||||||

MAE | 2.0 | 3.0 | 2.8 | 1.8 | 0.7 | 1.3 | 1.2 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Terentjev, A.V.; Cortona, P.; Constantin, L.A.; Pitarke, J.M.; Della Sala, F.; Fabiano, E.
Solid-State Testing of a Van-Der-Waals-Corrected Exchange-Correlation Functional Based on the Semiclassical Atom Theory. *Computation* **2018**, *6*, 7.
https://doi.org/10.3390/computation6010007

**AMA Style**

Terentjev AV, Cortona P, Constantin LA, Pitarke JM, Della Sala F, Fabiano E.
Solid-State Testing of a Van-Der-Waals-Corrected Exchange-Correlation Functional Based on the Semiclassical Atom Theory. *Computation*. 2018; 6(1):7.
https://doi.org/10.3390/computation6010007

**Chicago/Turabian Style**

Terentjev, Aleksandr V., Pietro Cortona, Lucian A. Constantin, José M. Pitarke, Fabio Della Sala, and Eduardo Fabiano.
2018. "Solid-State Testing of a Van-Der-Waals-Corrected Exchange-Correlation Functional Based on the Semiclassical Atom Theory" *Computation* 6, no. 1: 7.
https://doi.org/10.3390/computation6010007