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## Contents |

The exchange-hole dipole moment (XDM) dispersion model was developed by Axel Becke and myself in a series of papers from 2005-2007 [1, 2, 3, 4, 5, 6]. For the final versions of the XDM equations, refer to these two papers: [6, 7] since the earlier XDM references propagated an algebraic error inherited from prior works.

In general, XDM uses the interaction of induced dipoles (and higher-order multipoles) to model dispersion. The source of the instantaneous dipole moments is taken to be the dipole moment of the exchange hole.

The XDM damping function coefficients (*a*_{1} and *a*_{2}) control the shape of the Becke-Johnson damping function:

with:

and:

They are obtained by minimizing the mean average percent error of the calculated binding energies of a training set of non-covalently bound dimers for which very accurate reference data is available. The *a*_{1} and *a*_{2} parameters depend on the functional and, to a lesser degree, on the basis set used. This is because they are correcting for both varying contributions to non-covalent binding by the functional and for basis-set superposition error effects.

The damping parameters have to be entered in the codes calculating the XDM dispersion energy. In the next sections, we provide damping parameters for:

- Pseudopotential/plane-waves (PS/PW) calculations: for Quantum ESPRESSO.
- Gaussian basis sets: for Gaussian/postg, nwchem.

We used the training set proposed by Kannemann and Becke or subsets of it. The particular subset is indicated in the "nset" column: 65 (the original KB set with the noble-gas dimers), 49 (the original set minus all dimers involving noble gases), and 12 (only dimers involving stack-stack interactions). In the PS/PW, supercells have been used to model the gas-phase dimers.

For periodic solids, XDM has been incoroporated into the core distribution of Quantum Espresso as of version 5.0.2.

To add the XDM dispersion energy in Quantum ESPRESSO (QE), use:

vdw_corr="xdm"

in the `&system`

namelist of your pw.x input. You will also need the damping function parameters from this list (assuming that you are using PAW):

functional | a1 | a2 (Å) | MAPD | nset | notes |
---|---|---|---|---|---|

pw86pbe | 0.6836 | 1.5045 | 11.7 | 49 | |

b86bpbe | 0.6512 | 1.4633 | 11.8 | 49 | |

pbe | 0.3275 | 2.7673 | 14.4 | 49 | |

blyp | 0.4502 | 1.6210 | 14.8 | 49 | |

revpbe | 0.3454 | 1.9225 | 14.9 | 49 | old |

pbesol | 0.0000 | 4.1503 | 18.9 | 49 | fix,old |

which have to be passed to QE using the `xdm_a1`

and `xdm_a2`

variables in `&system`

. For instance, for a B86bPBE functional, use:

vdw_corr="xdm", xdm_a1=0.6512, xdm_a2=1.4633,

In atomic and cell relaxations, the dispersion coefficients are calculated once at the beginning of the run, then frozen during the entire procedure. This is necessary because the XDM forces are coded assuming that the dispersion coefficients are independent of the geometry, and the optimization procedure is sensitive to the mismatch between these forces and the actual energies. Provided the initial geometry is not too far from equilibrium, this should not represent a problem.

Previously, we implemented XDM in an in-house version of QE (version 4.3.2), which can be downloaded here:

The damping function parameters listed above correspond to the newer version, but the changes were not that important, so using these parameters with the 4.3.2 version shouldn't be a problem.

Please email Erin Johnson (erin.johnson@dal.ca) or Alberto Otero de la Roza (aoterodelaroza@gmail.com) if you find problems with this software. The compilation and installation follow the same procedure as the corresponding version of Quantum ESPRESSO.

When using XDM in QE, please cite:

- A. Otero de la Roza and E. R. Johnson. J. Chem. Phys.
**136**, 174109 (2012) link

We have also parameterized XDM for use with hybrid functionals and norm-conserving pseudopotentials using QE 6.3. XDM parameters for GGAs with NC pseudopotentials are also given for completeness.

functional | %HF | a1 | a2 (Å) | MAPD | nset |
---|---|---|---|---|---|

pbe | 0 | 0.4283 | 2.4690 | 15.2 | 49 |

pw86pbe | 0 | 0.7825 | 1.2077 | 12.6 | 49 |

b86bpbe | 0 | 0.7767 | 1.0937 | 12.4 | 49 |

blyp | 0 | 0.6349 | 1.0486 | 11.2 | 49 |

hse | 25/0 | 0.4206 | 2.4989 | 11.9 | 49 |

pbe0 | 25 | 0.4590 | 2.3581 | 11.1 | 49 |

b3lyp | 20 | 0.6070 | 1.3862 | 7.4 | 49 |

bh&hlyp | 50 | 0.2292 | 2.9698 | 10.4 | 49 |

b86bpbe-10x | 10 | 0.7272 | 1.2674 | 10.8 | 49 |

b86bpbe-20x | 20 | 0.6898 | 1.4072 | 9.6 | 49 |

b86bpbe-25x | 25 | 0.6754 | 1.4651 | 9.2 | 49 |

b86bpbe-30x | 30 | 0.6627 | 1.5181 | 9.0 | 49 |

b86bpbe-40x | 40 | 0.6465 | 1.5981 | 8.9 | 49 |

b86bpbe-50x | 50 | 0.6434 | 1.6405 | 9.1 | 49 |

When using XDM in with hybrid functionals in QE, please cite:

- A. Otero-de-la-Roza, L. M. LeBlanc, and E. R. Johnson. J. Chem. Theory Comput. 15, 4933-4944 (2019) link

For gas-phase molecules, we use a combination of Gaussian and the postg code for XDM calculations. See Tale 1 in the Anthology of interest for an example. The parameters in the following table correspond to various basis sets built-into Gaussian:

functional | a1 | a2 (Å) | MAPD | nset | notes |
---|---|---|---|---|---|

pw86pbe | 0.7564 | 1.4545 | 11.1 | 65 | In g09: iop(3/74=809) |

b3lyp | 0.6356 | 1.5119 | 6.4 | 49 | |

b3pw91 | 0.6002 | 1.4043 | 13.1 | 49 | |

b3p86 | 1.0400 | 0.3741 | 12.4 | 49 | |

pbe0 | 0.4186 | 2.6791 | 10.4 | 49 | |

camb3lyp | 0.3248 | 2.8607 | 9.3 | 65 | |

b97-1 | 0.1998 | 3.5367 | 12.3 | 49 | |

bhandhlyp | 0.5610 | 1.9894 | 9.8 | 65 | |

blyp | 0.7647 | 0.8457 | 9.4 | 49 | |

pbe | 0.4492 | 2.5517 | 14.2 | 49 | |

lcwpbe | 1.0149 | 0.6755 | 7.3 | 49 | |

tpss | 0.6612 | 1.5111 | 10.8 | 49 | |

b86bpbe | 0.7443 | 1.4072 | 13.1 | 49 | Psi4 |

functional | aX | a1 | a2 (Å) | MAPD |
---|---|---|---|---|

blyp | 0.0 | 0.7557 | 0.8734 | 9.22 |

blyp | 0.1 | 0.7004 | 1.1398 | 6.99 |

blyp | 0.2 | 0.6205 | 1.4885 | 5.70 |

blyp | 0.3 | 0.5108 | 1.9379 | 5.48 |

blyp | 0.4 | 0.3825 | 2.4548 | 6.13 |

blyp | 0.5 | 0.2460 | 3.0104 | 7.52 |

blyp | 0.6 | 0.1011 | 3.6060 | 9.45 |

blyp | 0.7 | -0.0709 | 4.3069 | 11.89 |

blyp | 0.8 | -0.2939 | 5.1940 | 14.60 |

blyp | 0.9 | -0.6016 | 6.3837 | 17.46 |

blyp | 1.0 | -1.0242 | 7.9897 | 20.54 |

pw86 | 0.0 | 0.8674 | 1.1425 | 11.67 |

pw86 | 0.1 | 0.8399 | 1.2449 | 10.06 |

pw86 | 0.2 | 0.8269 | 1.3121 | 9.42 |

pw86 | 0.3 | 0.8251 | 1.3485 | 9.10 |

pw86 | 0.4 | 0.8334 | 1.3574 | 9.01 |

pw86 | 0.5 | 0.8511 | 1.3427 | 9.62 |

pw86 | 0.6 | 0.8742 | 1.3140 | 10.54 |

pw86 | 0.7 | 0.9035 | 1.2711 | 11.66 |

pw86 | 0.8 | 0.9372 | 1.2193 | 12.98 |

pw86 | 0.9 | 0.9744 | 1.1610 | 14.46 |

pw86 | 1.0 | 1.0147 | 1.0987 | 16.00 |

pbe | 0.0 | 0.4468 | 2.5618 | 14.11 |

pbe | 0.1 | 0.4172 | 2.6640 | 12.08 |

pbe | 0.2 | 0.4044 | 2.7189 | 11.03 |

pbe | 0.3 | 0.4114 | 2.7162 | 10.33 |

pbe | 0.4 | 0.4420 | 2.6438 | 10.11 |

pbe | 0.5 | 0.4963 | 2.5028 | 10.30 |

pbe | 0.6 | 0.5732 | 2.2949 | 10.85 |

pbe | 0.7 | 0.6703 | 2.0297 | 11.77 |

pbe | 0.8 | 0.7810 | 1.7273 | 12.94 |

pbe | 0.9 | 0.8969 | 1.4131 | 14.44 |

pbe | 1.0 | 1.0148 | 1.0985 | 16.00 |

functional | a1 | a2 (Å) | MAPD | nset | notes |
---|---|---|---|---|---|

pw86pbe | 0.6736 | 1.9327 | 16.8 | 49 | In g09: iop(3/74=809) |

b3lyp | 0.6224 | 1.7068 | 10.4 | 49 | |

pbe0 | 0.1389 | 3.8310 | 14.2 | 49 | |

camb3lyp | 0.1849 | 3.5140 | 13.6 | 49 | |

b97-1 | 0.0000 | 4.4443 | 17.4 | 49 | fix |

bhandhlyp | 0.1247 | 3.5725 | 10.1 | 49 | |

blyp | 0.9742 | 0.3427 | 11.0 | 49 | |

pbe | 0.2061 | 3.5486 | 19.3 | 49 | |

lcwpbe | 1.1800 | 0.4179 | 8.3 | 49 |

functional | a1 | a2 (Å) | MAPD | nset | notes |
---|---|---|---|---|---|

lcwpbe | 0.5922 | 1.9441 | 6.5 | 49 | |

b3lyp | 0.5166 | 1.8829 | 7.9 | 49 | |

blyp | 0.7065 | 1.0273 | 11.7 | 49 | |

pbe | 0.2280 | 3.2444 | 16.8 | 49 | |

pbe0 | 0.1980 | 3.3552 | 12.2 | 49 |

functional | a1 | a2 (Å) | MAPD | nset | notes |
---|---|---|---|---|---|

lcwpbe | 0.5313 | 2.2665 | 8.1 | 49 | |

b3lyp | 0.4376 | 2.1607 | 8.1 | 49 | |

blyp | 0.6988 | 1.0776 | 10.8 | 49 | |

bhandhlyp | 0.0112 | 3.7782 | 10.6 | 49 |

functional | a1 | a2 (Å) | MAPD | nset | notes |
---|---|---|---|---|---|

pw86pbe | 0.6336 | 1.9148 | 16.9 | 49 | In g09: iop(3/74=809) |

b3lyp | 0.4515 | 2.1357 | 11.4 | 49 | |

pbe0 | 0.0845 | 3.7940 | 14.5 | 49 | |

camb3lyp | 0.2315 | 3.2123 | 13.3 | 49 | |

b97-1 | 0.0118 | 4.1784 | 16.3 | 49 | |

bhandhlyp | 0.1483 | 3.3435 | 12.2 | 49 | |

blyp | 0.5942 | 1.4555 | 15.3 | 49 | |

pbe | 0.2445 | 3.2596 | 18.6 | 49 | |

lcwpbe | 0.8134 | 1.3736 | 10.0 | 49 |

functional | a1 | a2 (Å) | MAPD | nset | notes |
---|---|---|---|---|---|

pw86pbe | 0.6935 | 1.7519 | 16.6 | 49 | In g09: iop(3/74=809) |

b3lyp | 0.4306 | 2.2076 | 11.2 | 49 | |

pbe0 | 0.1163 | 3.7191 | 14.3 | 49 | |

camb3lyp | 0.2365 | 3.2081 | 12.8 | 49 | |

b97-1 | 0.0429 | 4.1090 | 15.8 | 49 | |

bhandhlyp | 0.1432 | 3.3705 | 11.8 | 49 | |

blyp | 0.5653 | 1.5460 | 14.9 | 49 | |

pbe | 0.2746 | 3.1857 | 17.8 | 49 | |

lcwpbe | 0.8934 | 1.1466 | 10.1 | 49 |

6-31G* on H,B,C,Si and 6-31+G* on all other atoms, targeted for geometry optimizations on large organic systems, where full 6-31+G* optimizations are impractical.

functional | a1 | a2 (Å) | MAPD | nset | notes |
---|---|---|---|---|---|

blyp | 0.1753 | 2.9480 | 22.7 | 49 | |

b3lyp | 0.0000 | 3.7737 | 19.0 | 49 | fix |

bhandhlyp | 0.0000 | 4.0821 | 18.4 | 49 | fix |

lcwpbe | 0.6889 | 1.9452 | 14.1 | 49 |

functional | a1 | a2 (Å) | MAPD | nset | notes |
---|---|---|---|---|---|

pw86pbe | 0.0255 | 3.8471 | 8.7 | 12 | In g09: iop(3/74=809) |

b3lyp | 0.3795 | 2.4516 | 5.3 | 12 | |

pbe0 | 0.2817 | 3.1852 | 4.2 | 12 | |

camb3lyp | 0.5549 | 2.3185 | 5.0 | 12 | |

b97-1 | 0.6134 | 2.2096 | 7.0 | 12 | |

bhandhlyp | 1.2730 | -0.1701 | 8.8 | 12 | |

blyp | 0.0090 | 3.4136 | 11.6 | 12 | |

pbe | 0.0073 | 3.9745 | 9.0 | 12 | |

lcwpbe | 0.6611 | 1.9747 | 3.3 | 12 |

**A note about using XDM in nwchem**: An implementation of XDM is available in
nwchem, since version 6.5. To use it, put
a `xdm`

keyword in the `dft`

block. The syntax
is:

xdm a1 <a1 parameter> a2 <a2 parameter>

Although the damping function parameters given in the following sections can be used in nwchem without catastrophic results, the implementation is slightly different from postg. Hence, we recommend using parameters specifically fitted to nwchem to minimize errors. The parameters in the following table correspond to the complete-basis-set limit (aug-cc-pVTZ):

functional | a1 | a2 (Å) | MAPD | nset | notes |
---|---|---|---|---|---|

b3lyp | 0.8957 | 0.7796 | 10.8 | 49 | |

bhandhlyp | 0.3229 | 3.2920 | 10.7 | 49 | |

blyp | 0.8068 | 0.8359 | 16.8 | 49 | |

lcwpbe | 1.3483 | -0.4244 | 9.4 | 49 | |

pbe | 0.2939 | 3.1839 | 13.3 | 49 | |

pbe0 | 0.5815 | 2.2580 | 10.5 | 49 | |

pw86pbe | 0.8829 | 1.1669 | 12.7 | 49 | |

revpbe | 0.7326 | 1.0557 | 18.9 | 49 |

If you need parameters for a smaller basis set, or if your functional is not in the table above, please contact me.

A word of warning: we have found that for some of the dimers and
monomers in the
Kannemann-Becke set, the
SCF converges to a spurious minimum. The output does not show any
sign of trouble, but the energies are wrong. Using the
`convergence rabuck`

option in the `dft`

block
seems to solve this problem.

[fix] Fixed a1 or a2 parameter in the optimization.

[old] Fitted with a previous version of the code or with old benchmark data. Can be re-parametrized on demand (send me an e-mail).

- A. D. Becke, E. R. Johnson,
*Exchange-Hole Dipole Moment and the Dispersion Interaction*, J. Chem. Phys.**122**, 154104 (2005) link

- E. R. Johnson, A. D. Becke,
*A Post-Hartree-Fock Model of Intermolecular Interaction*, J. Chem. Phys.**123**, 024101 (2005) link

- A. D. Becke, E. R. Johnson,
*A Density-Functional Model of the Dispersion Interaction*, J. Chem. Phys.**123**, 154101 (2005) link

- A. D. Becke, E. R. Johnson,
*Exchange-hole Dipole Moment and the Dispersion Interaction: High-Order Dispersion Coefficients*, J. Chem. Phys.**124**, 014104 (2006) link

- E. R. Johnson, A. D. Becke,
*A Post-Hartree-Fock Model of Intermolecular Interactions: Inclusion of Higher-Order Corrections*, J. Chem. Phys.**124**, 174104 (2006) link

- A. D. Becke, E. R. Johnson,
*Exchange-Hole Dipole Moment and the Dispersion Interaction Revisited*, J. Chem. Phys.**127**, 154108 (2007) link

- A. D. Becke, E. R. Johnson,
*A Unified Density-Functional Treatment of Dynamical, Nondynamical and Dispersion Correlations*, J. Chem. Phys.**127**, 124108 (2007) link