# Double-Hybrid DFT Functionals for the Condensed Phase: Gaussian and Plane Waves Implementation and Evaluation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background

## 3. Computational Details

#### 3.1. Gaussian and Plane Waves Method (GPW) and Integral Evaluation

#### 3.2. Test Systems

_{3}and HCN crystals additionally contain hydrogen bonds which are crucial for the discription of proteins. In contrast to that, there are only dispersion interactions within the rare-gas crystals. This results in low cohesive energies and the need for well-balanced functionals.

#### 3.3. Parameters of the Calculations

_{3}and HCN, and 10,000 Ry for the rare-gas crystals Ar and Ne (see Section 4.1 for more details) and a relative cutoff of 50 Ry. For the rare-gas crystals, we set the parameters $EPS\_DEFAULT$, $EPS\_PGF\_ORB$, $EPS\_SCF$, and $EPS\_SCHWARZ$ in the HF section to 10

^{−30}, 10

^{−50}, 10

^{−5}, and 10

^{−10}, respectively, for the molecular crystals, we were using for the same parameters 10

^{−20}, 10

^{−40}, 10

^{−5}, and 10

^{−9}, respectively (see the CP2K manual for the meaning of these parameters). HF calculations for the bulk systems were using a truncated Coulomb potential with a cutoff radius of roughly half the super cell size. All densities have been smoothed using the $NN10$ method.

#### 3.4. Choice of Functionals and Implementation

#### 3.5. Basis Sets and Pseudopotentials

#### 3.6. Cohesive Energies and Basis Set Superposition Error

## 4. Results

#### 4.1. General Remarks

#### 4.2. Convergence with Respect to Super Cell Size

^{−1}. For weakly-interacting systems such as rare-gas crystals with cohesive energy of less than chemical accuracy, the order of magnitude is set by the cohesive energy itself. As the error of a method should be not larger than chemical accuracy, the allowed error of the supercell method must be at least one order of magnitude smaller then the methodological error, i.e., not larger than 0.4 kJ·mol

^{−1}. We find that a $3\times 3\times 3$ super cell provides sufficient accuracy for all functionals and test systems. This behaviour is in agreement with the literature [80]. Sometimes, the total energy per formula unit of the $4\times 4\times 4$ super cell has a higher magnitude than this of the $3\times 3\times 3$ supercell, which may be due to numerical issues. For PBE, a cubic fit does not seem to be appropriate, and an exponential fit should be used instead.

#### 4.3. Convergence of the BSSE

#### 4.4. Convergence with Respect to Basis Set Size

_{3}and HCN. They are bound together by covalent bonds, dipole-dipole interactions, and dispersion interactions. For both systems, the results with the RPA and MP2 methods significantly improve the results over GGA DFT functionals, MP2 even achieving chemical accuracy. The $\omega $B97M-V functional also provides very accurate numbers. The PW6B95 functional, as PBE, systematically underestimates the cohesive energies with errors compatible to PBE. The PWRB95 functional significantly improves upon the results of its relative PW6B95, bringing them within 1 kJ·mol

^{−1}from the experiment. The same holds for the $\omega $B97X-2 functional compared with the $\omega $B97M-V functional, although the DHDF is based on the non-meta-GGA HDF $\omega $B97X [86]. One of the worst performing functionals is SOS-PBE0-2.

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DFT | Density Functional Theory |

DHDF | Double-Hybrid Density Functional |

DZ | Double Zeta |

GTO | Gaussian Type Orbital |

HDF | Hybrid Density Functional |

HF | Hartree-Fock |

KS | Kohn-Sham |

MP2 | second order Moller-Plesset Perturbation theory |

PW | Plane Wave |

RI | Resolution of the Identity |

RPA | Random-Phase Approximation |

SOS-MP2 | Scaled Opposite-Spin MP2 |

TZ | Triple Zeta |

WFC | Wave-Function Correlation |

WFT | Wave-Function Theory |

XC | Exchange-Correlation |

## Appendix A. Pseudopotentials

#### Appendix A.1. PBE Pseudopotential

H | 1 | 1 | |

0.20059317301776 | −4.17806832477260 | 0.72440924243368 | |

C | 6 | 4 | |

0.33855479630051 | −8.80455195420776 | 1.33837678314185 | |

0.30260967537284 | 9.62286249628669 | ||

N | 7 | 5 | |

0.28382600053810 | −12.41517350030142 | 1.86813618209744 | |

0.25541754972811 | 13.63124869974610 | ||

O | 8 | 6 | |

0.24446328480160 | −16.67548222363837 | 2.48908598241780 | |

0.22097110943471 | 18.33446866406285 | ||

Ne | 10 | 8 | |

0.19013599957922 | −27.11394809602904 | 4.36380616652624 | |

0.17606810618356 | 28.17757050106910 | 0.83365740579601 | −1.07616503335814 |

0.19546571702831 | −0.23610573844687 | ||

Ar | 18 | 8 | |

0.40003082668805 | −7.08796199095755 | ||

0.31882882990174 | 17.25258480082010 | −5.58549698978039 | 7.21031819907047 |

0.35335630753415 | 4.97482100660240 |

#### Appendix A.2. PBE0 Pseudopotential

H | 1 | 1 | |

0.20049539759096 | −4.17780338804233 | 0.72403926676805 | |

C | 6 | 4 | |

0.34015230644208 | −8.75626046428525 | 1.33212403341974 | |

0.30255799930084 | 9.58980355283555 | ||

N | 7 | 5 | |

0.28405138134082 | −12.39652421586226 | 1.86372383417056 | |

0.25538070446385 | 13.63073438324169 | ||

O | 8 | 6 | |

0.24671011902360 | −16.65533253748591 | 2.50854752353111 | |

0.22100154713718 | 18.34370453997725 | ||

Ne | 10 | 8 | |

0.19050423878827 | −27.40404160755363 | 4.42644219542327 | 0.00312528175949 |

0.17609378094694 | 28.18364816845336 | 0.83365182689679 | −1.06378213860369 |

0.19427680906964 | −0.23683812086750 | ||

Ar | 18 | 8 | |

0.39979462541098 | −7.23417721420866 | 0.00616780402846 | |

0.31880599007091 | 17.21513221796928 | −5.58548607072547 | 7.21072495408957 |

0.35343634803616 | 4.97384170460103 |

#### Appendix A.3. HF Pseudopotential

H | 1 | 1 | |

0.20049539759096 | −4.17780338804233 | 0.72403926676805 | |

C | 6 | 4 | |

0.34816792458406 | −8.54312820557867 | 1.33276540541946 | |

0.30230247000627 | 9.59710582360109 | ||

N | 7 | 5 | |

0.28300476743411 | −12.39840200798251 | 1.86939057420079 | |

0.25539202567537 | 13.64483766978610 | ||

O | 8 | 6 | |

0.24676969870316 | −16.66528269564613 | 2.52030687064467 | |

0.22121058101998 | 18.39425181647437 | ||

Ne | 10 | 8 | |

0.19050265092574 | −27.39590696172339 | 4.41958869715540 | 0.01834396326683 |

0.17637388496062 | 28.18533818441574 | 0.83365996989179 | −1.04842942962620 |

0.19585379054851 | −0.27609661906079 | ||

Ar | 18 | 8 | |

0.39771927261258 | −7.21348927487361 | 0.01323122557817 | |

0.31872450490949 | 17.20921819285275 | −5.58549109340678 | 7.19978913165534 |

0.35357441343299 | 4.98951929408379 |

## Appendix B. Primary Basis Sets

**Table A4.**Parameters of the cc-DZVP basis set of Hydrogen. Parameters taken from [80].

Shell Type | Exponents | Contraction Coefficients |
---|---|---|

s | 8.3744350009 | −0.0283380461 |

1.8058681460 | −0.1333810052 | |

0.4852528328 | −0.3995676063 | |

s | 0.1658236932 | 1.0000000000 |

p | 0.7270000000 | 1.0000000000 |

**Table A5.**Parameters of the cc-TZVP basis set of Hydrogen. Parameters taken from [80].

Shell Type | Exponents | Contraction Coefficients |
---|---|---|

s | 10.8827241585 | −0.0167058885 |

3.0968750876 | −0.0627538300 | |

0.9874518162 | −0.1917521975 | |

s | 0.3450687533 | 1.0000000000 |

s | 0.1492693554 | 1.0000000000 |

p | 1.4070000000 | 1.0000000000 |

p | 0.3880000000 | 1.0000000000 |

d | 1.0570000000 | 1.0000000000 |

**Table A6.**Parameters of the cc-DZVP basis set of Carbon. Parameters taken from [80].

Shell Type | Exponents | Contraction Coefficients | |
---|---|---|---|

sp | 4.3362376436 | 0.1490797872 | −0.0878123619 |

1.2881838513 | −0.0292640031 | −0.2775560300 | |

0.4037767149 | −0.6882040510 | −0.4712295093 | |

sp | 0.1187877657 | 1.0000000000 | 1.0000000000 |

d | 0.5500000000 | 1.0000000000 |

**Table A7.**Parameters of the cc-TZVP basis set of Carbon. Parameters taken from [80].

Shell Type | Exponents | Contraction Coefficients | |
---|---|---|---|

sp | 5.3685662937 | 0.0974901974 | −0.0510969367 |

1.9830691554 | 0.1041996677 | −0.1693035193 | |

0.6978346167 | −0.3645093878 | −0.3579933930 | |

sp | 0.2430968816 | 1.0000000000 | 1.0000000000 |

sp | 0.0812865018 | 1.0000000000 | 1.0000000000 |

d | 1.0970000000 | 1.0000000000 | |

d | 0.3180000000 | 1.0000000000 | |

f | 0.7610000000 | 1.0000000000 |

**Table A8.**Parameters of the cc-DZVP basis set of Nitrogen. Parameters taken from [80].

Shell Type | Exponents | Contraction Coefficients | |
---|---|---|---|

sp | 6.1526903413 | 0.1506300537 | −0.0950603476 |

1.8236332280 | −0.0360100734 | −0.2918864295 | |

0.5676628870 | −0.6942023212 | −0.4739050050 | |

sp | 0.1628222852 | 1.0000000000 | 1.0000000000 |

d | 0.8170000000 | 1.0000000000 |

**Table A9.**Parameters of the cc-TZVP basis set of Nitrogen. Parameters taken from [80].

Shell Type | Exponents | Contraction Coefficients | |
---|---|---|---|

sp | 7.6227447102 | 0.0983924689 | −0.0561654555 |

2.7970605447 | 0.1045217098 | −0.1798165209 | |

0.9909765447 | −0.3742661352 | −0.3653986185 | |

sp | 0.3417314862 | 1.0000000000 | 1.0000000000 |

sp | 0.1116822743 | 1.0000000000 | 1.0000000000 |

d | 1.6540000000 | 1.0000000000 | |

d | 0.4690000000 | 1.0000000000 | |

f | 1.0930000000 | 1.0000000000 |

**Table A10.**Parameters of the cc-DZVP basis set of Oxygen. Parameters taken from [80].

Shell Type | Exponents | Contraction Coefficients | |
---|---|---|---|

sp | 8.3043855492 | 0.1510165999 | −0.0995679273 |

2.4579484191 | −0.0393195364 | −0.3011422449 | |

0.7597373434 | −0.6971724029 | −0.4750857083 | |

sp | 0.2136388632 | 1.0000000000 | 1.0000000000 |

d | 1.1850000000 | 1.0000000000 |

**Table A11.**Parameters of the cc-TZVP basis set of Oxygen. Parameters taken from [80].

Shell Type | Exponents | Contraction Coefficients | |
---|---|---|---|

sp | 10.2674419938 | 0.0989598460 | −0.0595856940 |

3.7480495696 | 0.1041178339 | −0.1875649045 | |

1.3308337704 | −0.3808255700 | −0.3700707718 | |

sp | 0.4556802254 | 1.0000000000 | 1.0000000000 |

sp | 0.1462920596 | 1.0000000000 | 1.0000000000 |

d | 2.3140000000 | 1.0000000000 | |

d | 0.6450000000 | 1.0000000000 | |

f | 1.4280000000 | 1.0000000000 |

Shell Type | Exponents | Contraction Coefficients | |
---|---|---|---|

sp | 13.8523672900 | 0.1501498200 | 0.10214300 |

4.0685498000 | −0.0314908700 | 0.3058092400 | |

1.2730584300 | −0.7070497300 | 0.4766050400 | |

sp | 0.3565013600 | 1.0000000000 | 1.0000000000 |

d | 2.2020000000 | 1.0000000000 |

Shell Type | Exponents | Contraction Coefficients | |
---|---|---|---|

sp | 17.4276488400 | 0.073686700 | 0.0702714400 |

6.3439264100 | 0.0969132500 | 0.2167849700 | |

2.2823205800 | −0.3010470300 | 0.4317763300 | |

sp | 0.7945993700 | 1.0000000000 | 1.0000000000 |

sp | 0.2560537300 | 1.0000000000 | 1.0000000000 |

d | 4.0140000000 | 1.0000000000 | |

d | 1.0960000000 | 1.0000000000 | |

f | 2.5440000000 | 1.0000000000 |

Shell Type | Exponents | Contraction Coefficients | |
---|---|---|---|

sp | 2.6724631600 | 0.1547491900 | 0.2663267700 |

1.5750569800 | −0.1300613000 | −1.0821938600 | |

0.5528926600 | −0.1247859600 | 0.1177549000 | |

sp | 0.1720724500 | 1.0000000000 | 1.0000000000 |

d | 0.7380000000 | 1.0000000000 |

Shell Type | Exponents | Contraction Coefficients | |
---|---|---|---|

sp | 3.5650652500 | −0.03560400 | −0.0341601300 |

2.8711385000 | 0.107453000 | 0.0274003200 | |

0.928908200 | −0.070935800 | 0.1084604500 | |

sp | 0.3762992800 | 1.0000000000 | 1.0000000000 |

sp | 0.1388133000 | 1.0000000000 | 1.0000000000 |

d | 1.2540000000 | 1.0000000000 | |

d | 0.4100000000 | 1.0000000000 | |

f | 0.8900000000 | 1.0000000000 |

## Appendix C. Auxiliary Basis Sets

**Table A16.**Parameters of the cc-DZVP auxiliary basis set of Hydrogen. Parameters taken from [89].

Shell Type | Exponents | Contraction Coefficients |
---|---|---|

s | 5.1153315245 | 1.0000000000 |

s | 1.1472440266 | 1.0000000000 |

s | 0.3203181150 | 1.0000000000 |

p | 1.9149400132 | 1.0000000000 |

p | 0.9859513111 | 1.0000000000 |

d | 1.1714848284 | 1.0000000000 |

**Table A17.**Parameters of the cc-TZVP auxiliary basis set of Hydrogen. Parameters taken from [89].

Shell Type | Exponents | Contraction Coefficients |
---|---|---|

s | 8.5115919487 | 1.0000000000 |

s | 1.8744684087 | 1.0000000000 |

s | 0.5632515602 | 1.0000000000 |

s | 0.3698299759 | 1.0000000000 |

p | 2.3711712242 | 1.0000000000 |

p | 1.1794161391 | 1.0000000000 |

p | 0.6050431621 | 1.0000000000 |

d | 1.8092525711 | 1.0000000000 |

d | 1.1433220615 | 1.0000000000 |

f | 1.8065804513 | 1.0000000000 |

**Table A18.**Parameters of the cc-DZVP auxiliary basis set of Carbon. Parameters taken from [89].

Shell Type | Exponents | Contraction Coefficients |
---|---|---|

s | 13.8045000000 | 1.0000000000 |

s | 4.7727700000 | 1.0000000000 |

s | 1.5133300005 | 1.0000000000 |

s | 0.7826969986 | 1.0000000000 |

s | 0.4090720022 | 1.0000000000 |

s | 0.2067960415 | 1.0000000000 |

p | 6.0052300018 | 1.0000000000 |

p | 1.7206000311 | 1.0000000000 |

p | 0.7544648237 | 1.0000000000 |

p | 0.3216662007 | 1.0000000000 |

d | 2.6784400611 | 1.0000000000 |

d | 0.9225147829 | 1.0000000000 |

d | 0.3408412315 | 1.0000000000 |

f | 2.7429299969 | 1.0000000000 |

f | 0.8957560323 | 1.0000000000 |

**Table A19.**Parameters of the cc-TZVP auxiliary basis set of Carbon. Parameters taken from [89].

Shell Type | Exponents | Contraction Coefficients |
---|---|---|

s | 22.2608165950 | 1.0000000000 |

s | 7.1315246807 | 1.0000000000 |

s | 3.5380450775 | 1.0000000000 |

s | 1.2333453175 | 1.0000000000 |

s | 0.2821517353 | 1.0000000000 |

s | 0.3468258230 | 1.0000000000 |

p | 7.2975063903 | 1.0000000000 |

p | 3.7896065213 | 1.0000000000 |

p | 1.0633834831 | 1.0000000000 |

p | 0.2356430320 | 1.0000000000 |

p | 0.5078423493 | 1.0000000000 |

d | 9.9000557486 | 1.0000000000 |

d | 2.3408375066 | 1.0000000000 |

d | 1.5195338451 | 1.0000000000 |

d | 0.5788522388 | 1.0000000000 |

d | 0.3721345858 | 1.0000000000 |

f | 1.9332589728 | 1.0000000000 |

f | 1.1560553410 | 1.0000000000 |

f | 0.4987261239 | 1.0000000000 |

g | 1.2175667359 | 1.0000000000 |

**Table A20.**Parameters of the cc-DZVP auxiliary basis set of Nitrogen. Parameters taken from [89].

Shell Type | Exponents | Contraction Coefficients |
---|---|---|

s | 20.4678978643 | 1.0000000000 |

s | 7.6243888531 | 1.0000000000 |

s | 2.3446722210 | 1.0000000000 |

s | 1.1234062160 | 1.0000000000 |

s | 0.7258555682 | 1.0000000000 |

s | 0.3516451521 | 1.0000000000 |

p | 8.5530798511 | 1.0000000000 |

p | 2.5349440268 | 1.0000000000 |

p | 1.0857134625 | 1.0000000000 |

p | 0.4193736786 | 1.0000000000 |

d | 3.4384121802 | 1.0000000000 |

d | 1.2761051199 | 1.0000000000 |

d | 0.3971760294 | 1.0000000000 |

f | 3.4010871209 | 1.0000000000 |

f | 1.3350506486 | 1.0000000000 |

**Table A21.**Parameters of the cc-TZVP auxiliary basis set of Nitrogen. Parameters taken from [89].

Shell Type | Exponents | Contraction Coefficients |
---|---|---|

s | 21.6812818892 | 1.0000000000 |

s | 7.9027065688 | 1.0000000000 |

s | 2.4447520737 | 1.0000000000 |

s | 1.2617706294 | 1.0000000000 |

s | 0.8067419821 | 1.0000000000 |

s | 0.2885279906 | 1.0000000000 |

p | 10.3296673020 | 1.0000000000 |

p | 2.9182107455 | 1.0000000000 |

p | 1.3834177164 | 1.0000000000 |

p | 0.7162830530 | 1.0000000000 |

p | 0.3296257918 | 1.0000000000 |

d | 13.9094333585 | 1.0000000000 |

d | 4.5822351942 | 1.0000000000 |

d | 2.1943496520 | 1.0000000000 |

d | 0.8349245145 | 1.0000000000 |

d | 0.4510857395 | 1.0000000000 |

f | 3.4744937308 | 1.0000000000 |

f | 1.5532348673 | 1.0000000000 |

f | 0.8522508678 | 1.0000000000 |

g | 1.7674440596 | 1.0000000000 |

**Table A22.**Parameters of the cc-DZVP auxiliary basis set of Oxygen. Parameters taken from [89].

Shell Type | Exponents | Contraction Coefficients |
---|---|---|

s | 25.5779913844 | 1.0000000000 |

s | 9.5515670675 | 1.0000000000 |

s | 2.9409752222 | 1.0000000000 |

s | 1.3964896911 | 1.0000000000 |

s | 0.9105756313 | 1.0000000000 |

s | 0.4821009543 | 1.0000000000 |

p | 10.8823093197 | 1.0000000000 |

p | 3.2132775587 | 1.0000000000 |

p | 1.3802086101 | 1.0000000000 |

p | 0.4601246170 | 1.0000000000 |

d | 4.5934895346 | 1.0000000000 |

d | 1.7871052175 | 1.0000000000 |

d | 0.4206288858 | 1.0000000000 |

f | 4.2218855419 | 1.0000000000 |

f | 1.7894864633 | 1.0000000000 |

**Table A23.**Parameters of the cc-TZVP auxiliary basis set of Oxygen. Parameters taken from [89].

Shell Type | Exponents | Contraction Coefficients |
---|---|---|

s | 24.5595006061 | 1.0000000000 |

s | 8.3254503805 | 1.0000000000 |

s | 2.8895585562 | 1.0000000000 |

s | 1.3383587201 | 1.0000000000 |

s | 0.8797495165 | 1.0000000000 |

s | 0.2902204697 | 1.0000000000 |

p | 15.0341204959 | 1.0000000000 |

p | 3.9838033442 | 1.0000000000 |

p | 2.2151496463 | 1.0000000000 |

p | 0.8979637674 | 1.0000000000 |

p | 0.4128471304 | 1.0000000000 |

d | 15.8683289847 | 1.0000000000 |

d | 5.3913486662 | 1.0000000000 |

d | 2.5385447175 | 1.0000000000 |

d | 1.0911199995 | 1.0000000000 |

d | 0.3766843343 | 1.0000000000 |

f | 4.6812603411 | 1.0000000000 |

f | 2.1656106741 | 1.0000000000 |

f | 1.0331835741 | 1.0000000000 |

g | 2.3079719899 | 1.0000000000 |

Shell Type | Exponents | Contraction Coefficients |
---|---|---|

s | 0.4283083846 | 1.0000000000 |

s | 0.9127866030 | 1.0000000000 |

s | 1.4054659820 | 1.0000000000 |

s | 2.7308004917 | 1.0000000000 |

s | 10.8821914790 | 1.0000000000 |

s | 27.4563627600 | 1.0000000000 |

p | 0.7929884909 | 1.0000000000 |

p | 2.5008596627 | 1.0000000000 |

p | 5.9151957867 | 1.0000000000 |

p | 15.3673109753 | 1.0000000000 |

d | 0.8283650682 | 1.0000000000 |

d | 3.3870444721 | 1.0000000000 |

d | 10.9778901482 | 1.0000000000 |

f | 3.1170471491 | 1.0000000000 |

f | 6.8656878672 | 1.0000000000 |

Shell Type | Exponents | Contraction Coefficients |
---|---|---|

s | 0.4218054070 | 1.0000000000 |

s | 0.9220306716 | 1.0000000000 |

s | 1.8281445142 | 1.0000000000 |

s | 4.0122843245 | 1.0000000000 |

s | 9.3557953735 | 1.0000000000 |

s | 21.2448698799 | 1.0000000000 |

p | 0.7819113369 | 1.0000000000 |

p | 1.2774995450 | 1.0000000000 |

p | 4.0929391223 | 1.0000000000 |

p | 6.2303237413 | 1.0000000000 |

p | 16.9114662817 | 1.0000000000 |

d | 0.5593906817 | 1.0000000000 |

d | 1.1997449519 | 1.0000000000 |

d | 2.6389336208 | 1.0000000000 |

d | 4.9434479195 | 1.0000000000 |

d | 14.1003353302 | 1.0000000000 |

f | 1.5593705359 | 1.0000000000 |

f | 3.3654768720 | 1.0000000000 |

f | 8.0058977735 | 1.0000000000 |

g | 3.9789077704 | 1.0000000000 |

Shell Type | Exponents | Contraction Coefficients |
---|---|---|

s | 0.2021934524 | 1.0000000000 |

s | 0.5951644570 | 1.0000000000 |

s | 0.9713367515 | 1.0000000000 |

s | 2.1348414404 | 1.0000000000 |

s | 8.3055209987 | 1.0000000000 |

s | 24.4838599910 | 1.0000000000 |

p | 0.3806171008 | 1.0000000000 |

p | 1.0795771281 | 1.0000000000 |

p | 1.5490430664 | 1.0000000000 |

p | 4.5409363372 | 1.0000000000 |

d | 0.4407961817 | 1.0000000000 |

d | 1.2550515264 | 1.0000000000 |

d | 5.8159089208 | 1.0000000000 |

f | 1.2170432674 | 1.0000000000 |

f | 9.1164484253 | 1.0000000000 |

Shell Type | Exponents | Contraction Coefficients |
---|---|---|

s | 0.2374198920 | 1.0000000000 |

s | 0.4593140129 | 1.0000000000 |

s | 0.8759707445 | 1.0000000000 |

s | 1.6509598569 | 1.0000000000 |

s | 3.0758784521 | 1.0000000000 |

s | 5.6934758267 | 1.0000000000 |

p | 0.2795695538 | 1.0000000000 |

p | 0.5599111995 | 1.0000000000 |

p | 1.1757695117 | 1.0000000000 |

p | 2.6036947304 | 1.0000000000 |

p | 5.6658077981 | 1.0000000000 |

d | 0.2898716779 | 1.0000000000 |

d | 0.6972610882 | 1.0000000000 |

d | 1.2166500341 | 1.0000000000 |

d | 2.4487532956 | 1.0000000000 |

d | 5.4801182100 | 1.0000000000 |

f | 0.6819926505 | 1.0000000000 |

f | 1.4331910682 | 1.0000000000 |

f | 3.3654529504 | 1.0000000000 |

g | 1.4464903837 | 1.0000000000 |

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**Figure 1.**Total energies per formula unit relative to extrapolated total energy in kJ·mol

^{−1}against inverse number of unit cells in supercell with basis sets of DZ and TZ quality for the systems Ne, Ar, NH

_{3}, HCN.

**Figure 2.**Negative Basis set superposition errors in kJ·mol

^{−1}with basis sets of DZ and TZ quality for the systems Ne, Ar, NH

_{3}, HCN.

**Figure 3.**Errors in cohesive energies in kJ·mol

^{−1}with respect to the experimental values with basis sets of DZ and TZ quality for the systems Ne, Ar, NH

_{3}, HCN.

**Table 1.**Structural information about the bulk structures used in this study. ${n}_{fu}$ is the number of formula units per unit cell. References for the geometrical information of the respective system are provided in the last column. Please note that there was a mistake in the cell parameters of CO

_{2}provided in reference [63].

System | a;b;c (Å) | ${\mathit{n}}_{\mathit{fu}}$ | References |
---|---|---|---|

NH_{3} | 5.048 | 4 | [65] |

HCN | 4.13; 4.85; 4.34 | 2 | [66] |

Ne | 4.464 | 4 | [67,68,69] |

Ar | 5.300 | 4 | [70,71] |

**Table 2.**Cohesion energies ${E}_{coh}$ and absolute relative error with respect to experimental results for all considered methods and systems in kJ/mol exploiting basis sets of DZ quality. The statistical indicators are the mean absolute error (MAE) and the mean absolute relative error (MARE).

^{a}This work.

^{b}Values by Sansonse et al. [63].

^{c}Experimental values [68,71,84], corrected for zero-point energy (ZPE) and thermal effects at 298 K [63,85].

Functional | ${\mathit{E}}_{\mathit{coh}}(\mathit{Ne})$ | % | ${\mathit{E}}_{\mathit{coh}}(\mathit{Ar})$ | % | ${\mathit{E}}_{\mathit{coh}}({\mathit{NH}}_{3})$ | % | ${\mathit{E}}_{\mathit{coh}}(\mathit{HCN})$ | % | MAE | MARE |
---|---|---|---|---|---|---|---|---|---|---|

PBE ^{a} | −2.92 | 48 | 2.47 | 131 | −39.9 | 9 | −29.0 | 32 | 7.09 | 0.55 |

PBE ^{b} | −0.27 | 86 | 2.36 | 130 | −26.9 | 25 | −28.0 | 34 | 8.95 | 0.69 |

RPA ^{a} | 0.82 | 141 | 4.61 | 159 | −19.0 | 47 | −28.1 | 33 | 11.7 | 0.96 |

MP2 ^{a} | 0.83 | 141 | 2.95 | 138 | −27.6 | 23 | −35.2 | 17 | 7.40 | 0.80 |

MP2 ^{b} | 0.22 | 111 | 3.13 | 140 | −24.2 | 33 | −31.7 | 25 | 9.01 | 0.78 |

$\omega $B97M-V ^{a} | −2.65 | 34 | −8.30 | 7 | −41.2 | 13 | −58.0 | 36 | 5.37 | 0.23 |

PW6B95 ^{a} | −1.99 | 1 | −0.03 | 99 | −27.1 | 25 | −33.8 | 20 | 6.43 | 0.37 |

PWRB95 ^{a} | −2.29 | 16 | −4.15 | 46 | −34.8 | 4 | −40.3 | 5 | 1.91 | 0.18 |

$\omega $B97X-2 ^{a} | −0.43 | 78 | −3.60 | 53 | −35.2 | 3 | −42.3 | 0 | 1.77 | 0.34 |

SOS-PBE0-2 ^{a} | 0.57 | 128 | 2.76 | 135 | −25.0 | 31 | −31.0 | 27 | 8.98 | 0.81 |

Expt. ^{c} | −1.97 | −7.73 | −36.3 | −42.6 |

**Table 3.**Same as Table 2, but with basis sets of TZ quality.

^{b}exploits basis sets of augmented DZ quality.

Functional | ${\mathit{E}}_{\mathit{coh}}(\mathit{Ne})$ | % | ${\mathit{E}}_{\mathit{coh}}(\mathit{Ar})$ | % | ${\mathit{E}}_{\mathit{coh}}({\mathit{NH}}_{3})$ | % | ${\mathit{E}}_{\mathit{coh}}(\mathit{HCN})$ | % | MAE | MARE |
---|---|---|---|---|---|---|---|---|---|---|

PBE ^{a} | −1.22 | 37 | 0.49 | 106 | −28.5 | 21 | −29.5 | 30 | 7.47 | 0.49 |

PBE ^{b} | −0.40 | 79 | 0.42 | 105 | −26.2 | 27 | −29.7 | 30 | 8.18 | 0.61 |

RPA ^{a} | 0.32 | 116 | 0.44 | 105 | −24.1 | 33 | −34.9 | 17 | 7.58 | 0.68 |

MP2 ^{a} | 0.05 | 102 | −4.48 | 42 | −34.4 | 5 | −43.9 | 3 | 2.13 | 0.38 |

MP2 ^{b} | −1.10 | 44 | −6.45 | 16 | −31.8 | 12 | −41.4 | 2 | 1.96 | 0.19 |

$\omega $B97M-V ^{a} | −3.51 | 78 | −9.28 | 20 | −39.5 | 8 | −50.9 | 19 | 3.65 | 0.32 |

PW6B95 ^{a} | −2.31 | 17 | −1.95 | 74 | −25.6 | 29 | −32.6 | 23 | 6.72 | 0.36 |

PWRB95 ^{a} | −2.67 | 35 | −3.10 | 59 | −36.1 | 0 | −44.0 | 3 | 1.73 | 0.25 |

$\omega $B97X-2 ^{a} | −2.59 | 31 | −8.15 | 5 | −38.6 | 6 | −43.0 | 0 | 0.93 | 0.11 |

SOS-PBE0-2 ^{a} | 0.10 | 105 | −0.12 | 98 | −26.4 | 27 | −33.0 | 22 | 7.28 | 0.63 |

Expt. ^{c} | −1.97 | −7.73 | −36.3 | −42.6 |

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Stein, F.; Hutter, J.; Rybkin, V.V. Double-Hybrid DFT Functionals for the Condensed Phase: Gaussian and Plane Waves Implementation and Evaluation. *Molecules* **2020**, *25*, 5174.
https://doi.org/10.3390/molecules25215174

**AMA Style**

Stein F, Hutter J, Rybkin VV. Double-Hybrid DFT Functionals for the Condensed Phase: Gaussian and Plane Waves Implementation and Evaluation. *Molecules*. 2020; 25(21):5174.
https://doi.org/10.3390/molecules25215174

**Chicago/Turabian Style**

Stein, Frederick, Jürg Hutter, and Vladimir V. Rybkin. 2020. "Double-Hybrid DFT Functionals for the Condensed Phase: Gaussian and Plane Waves Implementation and Evaluation" *Molecules* 25, no. 21: 5174.
https://doi.org/10.3390/molecules25215174