# XDM

## The exchange-hole dipole moment (XDM) model

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 (a1 and a2) control the shape of the Becke-Johnson damping function:

$f_n(R_{ij}) = \frac{R_{ij}^{n}}{R_{ij}^{n} + R_{vdw,ij}^{n}}$

with:

$R_{vdw,ij} = a_1 \times R_{c,ij} + a_2$

and:

$R_{c,ij} = \frac{1}{3} \left[\left(\frac{C_{8,ij}}{C_{6,ij}}\right)^{1/2} + \left(\frac{C_{10,ij}}{C_{6,ij}}\right)^{1/4} + \left(\frac{C_{10,ij}}{C_{8,ij}}\right)^{1/2} \right]$

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 a1 and a2 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:

1. Pseudopotential/plane-waves (PS/PW) calculations: for Quantum ESPRESSO.
2. 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.

## XDM in periodic solids (PAW)

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

### Hybrid functionals (NC)

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

## XDM in gas-phase molecules

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:

### aug-cc-pvTZ (complete basis set)

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

### aug-cc-pvTZ (hybrids series)

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

### aug-cc-pvDZ

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

### pc-2-spd

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

### 6-311+G(2d,2p)

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

### 6-31+G*

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

### 6-31+G**

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

### Mixed Basis

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

### 6-31G* (for stacks only!)

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

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).

1. A. D. Becke, E. R. Johnson, Exchange-Hole Dipole Moment and the Dispersion Interaction, J. Chem. Phys. 122, 154104 (2005) link
2. E. R. Johnson, A. D. Becke, A Post-Hartree-Fock Model of Intermolecular Interaction, J. Chem. Phys. 123, 024101 (2005) link
3. A. D. Becke, E. R. Johnson, A Density-Functional Model of the Dispersion Interaction, J. Chem. Phys. 123, 154101 (2005) link
4. 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
5. 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
6. A. D. Becke, E. R. Johnson, Exchange-Hole Dipole Moment and the Dispersion Interaction Revisited, J. Chem. Phys. 127, 154108 (2007) link
7. A. D. Becke, E. R. Johnson, A Unified Density-Functional Treatment of Dynamical, Nondynamical and Dispersion Correlations, J. Chem. Phys. 127, 124108 (2007) link