3.22 Exchange-hole dipole moment (XDM) dispersion method

The exchange-hole dipole moment (XDM) model [26] is a method to calculate the dispersion energy in molecules and solids. The dispersion energy in XDM is calculated as a damped asymptotic expression:

E_{\text{disp}}=-\sum_{i>j}\frac{C_{6,ij}f_{6}(R_{ij})}{R_{ij}^{6}}+\frac{C_{8% ,ij}f_{8}(R_{ij})}{R_{ij}^{8}}+\frac{C_{10,ij}f_{10}(R_{ij})}{R_{ij}^{10}}

(3.67)

where $i$ and $j$ run over atoms, and $R_{ij}$ is the interatomic distance. The XDM contribution is used to correct the energy from the base density functional approximation for missing dispersion:

E_{\text{total}}=E_{\text{base}}+E_{\text{DFT}}

(3.68)

The dispersion coefficients are calculated from the self-consistent density and kinetic energy density. For instance, the leading dispersion coefficients are:

C_{6,ij}=\frac{\alpha_{i}\alpha_{j}\langle M_{1}^{2}\rangle_{i}\langle M_{1}^{% 2}\rangle_{j}}{\langle M_{1}^{2}\rangle_{i}\alpha_{j}+\langle M_{1}^{2}\rangle% _{j}\alpha_{i}}

(3.69)

where $\langle M_{1}^{2}\rangle_{i}$ are the multipole moments of the electron plus exchange-hole dipole distribution and $\alpha_{i}$ are the in-molecule atomic polarizabilities. The Becke-Johnson damping functions, $f_{n}$ ( $n=6,8,10$ ), contain two adjustable parameters $a_{1}$ and $a_{2}$ (in Å), which are determined for each basis-functional combination with which XDM is coupled by fitting to a small set of molecular gas-phase dimers, KB49. More details can be found in the original reference [26], the periodic XDM implementation [237], and recent reviews [160].

For additional information, example usage, and step-by-step instructions on how to fit XDM’s damping parameters, see the FHIaims XDM Tutorial.

Tags for general section of control.in:

Tag: xdm(control.in)

Usage: xdm [basis]
Usage: or
Usage: xdm [a1 a2]
Purpose: Calculate the XDM dispersion energy and its derivatives.
Required Parameters:

•

[basis]: If using a default basis from species_defaults/, specify it.
Usage: XDM auto-detects your exchange-correlation functional, xc, and your exact-exchange mixing parameter hybrid_xc_coeff. By also specifying your basis, XDM can use these to set the optimal damping parameters for this combination automatically. Supported basis defaults are listed below.
•

[a1 a2]: XDM’s Becke-Johnson damping function parameters ( $a_{2}$ in Å).
Usage: These optional parameters, entered as two floating-point numbers, can be used to manually set the damping coefficients. This may be useful if XDM does not support your basis or functional, or if you want to override XDM’s default damping parameters.

Default: Previously, if neither a [basis] nor [a1 a2] are specified, XDM proceeded with a warning, assuming you were using either the defaults_2020/lightdense or defaults_2020/tight basis defaults. To prevent accidental misusage of XDM, it now halts the calculation with an error.

Supported Basis Sets:

You may specify your basis defaults, corresponding to species_defaults/, by setting the [basis] option as follows:

  xdm defaults_2020_light
  xdm defaults_2020_intermediate
  xdm defaults_2020_tight
  xdm defaults_2020_lightdense
  xdm defaults_next_lightdenser
  xdm non-standard_tier2_aug2

Additionally, the following aliases are also supported:

  light        -> defaults_2020_light
  intermediate -> defaults_2020_intermediate
  tight        -> defaults_2020_tight
  lightdense   -> defaults_2020_lightdense
  lightdenser  -> defaults_next_lightdenser
  aug2         -> non-standard_tier2_aug2

Supported Functionals:

XDM supports the following functionals in combination with the above basis defaults:

GGAs
- –
  
  b86bpbe
- –
  
  pbe
- –
  
  revpbe
Global Hybrids
- –
  
  b3lyp (For the default hybrid_xc_coeff = 0.20)
- –
  
  bhlyp (For the default hybrid_xc_coeff = 0.50)
- –
  
  b86bpbe0 (For any value of hybrid_xc_coeff)
- –
  
  b86bpbe-25 (For the default hybrid_xc_coeff = 0.25)
- –
  
  b86bpbe-50 (For the default hybrid_xc_coeff = 0.50)
- –
  
  pbe0 (For any value of hybrid_xc_coeff)
- –
  
  revpbe0 (For the default hybrid_xc_coeff = 0.25, and for 0.50)
Range-Separated Hybrids
- –
  
  hse06 0.11 ( $\omega=0.11~{}\text{b}^{-1}$ and default hybrid_xc_coeff = 0.25)
- –
  
  lc_wpbeh 0.2 0.0 ( $\omega=0.20~{}\text{b}^{-1}$ and hybrid_xc_coeff = 0.00)
- –
  
  lc_wpbeh 0.2 0.2 ( $\omega=0.20~{}\text{b}^{-1}$ and hybrid_xc_coeff = 0.20)
- –
  
  lc_wpbeh 0.4 0.0 ( $\omega=0.40~{}\text{b}^{-1}$ and hybrid_xc_coeff = 0.00)
- –
  
  lc_wpbeh 0.4 0.2 ( $\omega=0.40~{}\text{b}^{-1}$ and hybrid_xc_coeff = 0.20)

HSE and LC- $\omega$ PBEh use hse_unit b .

Notes:

•

The [basis] aliases will be updated over time to point to the most recent basis defaults for each basis type. Thus, these aliases may not be backwards compatible between versions. If you choose to use these, always check the XDM section of your version’s FHI-aims manual for the up-to-date alias list.
•

If your system’s basis is not one of the above defaults, or is a mixed basis, you should re-fit XDM’s damping parameters and manually specify [a1 a2] in your control.in. Re-fitting instructions, as well as inputs and fit scripts, are available in the FHIaims XDM Tutorial. The $a_{1}$ and $a_{2}$ damping parameters converge with basis size and are already well-converged by tight. If your basis is larger than tight (e.g. really_tight), it should be safe to use the tight defaults.
•

For the non-standard_tier2_aug2 basis, only B86bPBE and PBE, functionals are supported.
•

The recently developed screening potential used for PBE0prime, described in Ref. [174], is also, in principle, supported for the above listed global hybrid functionals. To enable this, include exx_lr_approximation 0.11 and hse_unit b in your control file. In testing, we determined that the difference between the screened and unscreened damping parameters was negligible, thus XDM currently assigns the damping parameters for the unscreened global hybrid. Note that only values of $\omega=0.11~{}\text{b}^{-1}$ have been tested in combination with XDM, and higher values could result in loss of accuracy due to divergence of the optimal screened and unscreened damping parameters.

Example Usage:

  xdm defaults_2020_light
  xdm lightdenser
  xdm 0.45447758 2.43090048