Interface to DFTD3 by S. Grimme¶

Code author: Lori A. Burns

Section author: Lori A. Burns

Module: Samples

Installation¶

Binary

DFTD3 is available as a conda package for Linux and macOS (and Windows, through the Ubuntu shell).
If using the Psi4conda installer, DFTD3 has already been installed alongside.
If using the PSI4 conda package, the dftd3 conda package can be obtained through conda install dftd3 -c psi4 or conda install psi4-rt -c psi4.
If using PSI4 built from source, and anaconda or miniconda has already been installed (instructions at Quick Installation), the dftd3 executable can be obtained through conda install dftd3 -c psi4.
To remove a conda installation, conda remove dftd3.

Source

If using PSI4 built from source and you want to build DFTD3 from from source also, follow the instructions provided with the source (essentially, download the freely available tarball, unpack the source, edit the Makefile to select a Fortran compiler, and run make). From version 3.1.0 onwards, DFTD3 can be used as-is; for earlier versions, patches are available: psi4/psi4/share/psi4/scripts/patch_grimme_dftd3.3.0.2.

To be used by PSI4, the program binary (dftd3) must be found in your PSIPATH or PATH (in that order). If PSI4 is unable to execute the binary, an error will be reported. To preferentially use a particular dftd3 compilation, simply adjust its position in the path environment variables.

Theory¶

The local or semilocal character of conventional density functionals necessarily leads to neglect of the long-range correlation interactions which capture attractive van der Waals forces. Initially proposed by Yang [Wu:2002:515] and assiduously developed by Grimme, [Grimme:2004:1463] [Grimme:2006:1787] [Grimme:2010:154104] the DFT+Dispersion method appends to the base functional a scaled, damped, and fitted leading term to the well-known dispersion energy series, $E_{d i s p} = - C_{6} / R^{6} - C_{8} / R^{8} - C_{10} / R^{10} - \dots$ . The DFT-D2 [Grimme:2006:1787] variant takes the explicit form below. Here, dispersion coefficients, $C_{6}^{i j}$ , obtained from the geometric mean of tabulated elemental values, are summed over interatomic distances, $R_{i j}$ , modulated by a damping function, $f_{d a m p} (R_{i j})$ , that gradually activates the dispersion correction (at a rate characterized by $α_{6}$ ) over a distance characterized by the sum of the two atomic vdW radii, $R_{v d W}$ , while an overall scaling term, $s_{6}$ , is optimized to be unique to each $E_{x c}$ functional. ( $α_{6}$ is sometimes allowed to vary as well.)

(1)¶

E_{d i s p}^{D2} = - s_{6} \sum_{i, j > i}^{N_{a t}} \frac{C_{6}^{i j}}{(R_{i j})^{6}} f_{d a m p} (R_{i j})

f_{d a m p} (R_{i j}) = \frac{1}{1 + e^{- α_{6} (R_{i j} / R_{v d W} - 1)}}

Grimme recently presented a refined method, DFT-D3, [Grimme:2010:154104] which incorporates an additional $R^{- 8}$ term in the dispersion series and adjusts the $C_{6}^{i j}$ combination formula and damping function. The individual atomic $C_{6}^{i}$ are interpolated from several reference values based upon coordination numbers extracted from the molecular structure, rather than assigned solely by atomic identity as in DFT-D2, and thereby incorporate some awareness of the chemical environment into an otherwise largely heuristic correction. The -D3 dispersion has the following form, where $s_{r, 6}$ and $s_{8}$ are the customary nonunity parameters fitted for individual functionals.

(2)¶

E_{d i s p}^{D3ZERO} = - \sum_{n = 6, 8} s_{n} \sum_{i, j > i}^{N_{a t}} \frac{C_{n}^{i j}}{(R_{i j})^{n}} f_{d a m p} (R_{i j})

f_{d a m p} (R_{i j}) = \frac{1}{1 + 6 (R_{i j} / (s_{r, n} R_{0}^{i j}))^{- α_{n}}}

A modified damping scheme for DFT-D3 using the rational damping form of Becke and Johnson was introduced in [Grimme:2011:1456]. The parameters fit for individual functionals are now $s_{6}$ , $s_{8}$ , $a_{1}$ , and $a_{2}$ .

E_{d i s p}^{D3BJ} = - \sum_{n = 6, 8} s_{n} \sum_{i, j > i}^{N_{a t}} \frac{C_{n}^{i j}}{(R_{i j})^{n} + (f_{d a m p})^{n}}

f_{d a m p} = a_{1} \sqrt{\frac{C_{8}^{i j}}{C_{6}^{i j}}} + a_{2}

All parameters characterizing the dispersion correction are taken from Grimme’s website or else from the literature.

Running DFTD3 or DFTD4¶

A number of a posteriori dispersion corrections are available in PSI4. While some are computed within PSI4‘s codebase (-D1, -D2, -CHG, -DAS2009, -DAS2010), the -D3 or -D4 corrections and their variants are available only through the DFTD3 or DFTD4 programs. Once installed, the dftd3/PSI4 and dftd4/PSI4 interfaces are transparent, and all corrections are interfaced exactly alike.

Dispersion corrections are built into DFT functionals, so appending an a posteriori correction to a computation is as simple as energy('b2plyp-d') vs. energy('b2plyp'). For example, the following input file computes (with much redundant work) for water a B3LYP, a B3LYP-D2, a B3LYP-D3 (zero-damping), and a B3LYP-D4 (Becke-Johnson damping) energy.

molecule h2o {
     O
     H 1 1.0
     H 1 1.0 2 104.5
 }
 set {
     basis cc-pVDZ
 }
 energy('b3lyp')
 energy('b3lyp-d')
 energy('b3lyp-d3')
 energy('b3lyp-d4')

Consult the table -D Functionals to see for each functional what corrections are available and what default parameters define them. The dispersion correction is available after a calculation in the PSI variable DISPERSION CORRECTION ENERGY. By default, the output from the dftd3 program is suppressed; to see it in the output file, set print > 2. No text output is available from the dftd4 program.

Variants of dispersion corrections¶
Extension 1	Variant	Computing Program (engine)	DFT_DISPERSION_PARAMETERS 11
-D	alias to -D2
-D1	-D1 2	PSI4‘s libdisp	[ $s_{6}$ ]
-D2	-D2 3	PSI4‘s libdisp OR `dftd3`	[ $s_{6}$ , $α_{6}$ , $s_{r, 6}$ ]
-D3	alias to -D3ZERO
-D3ZERO	-D3 4 w/ original zero-damping	`dftd3`	[ $s_{6}$ , $s_{8}$ , $s_{r, 6}$ , $α_{6}$ , $s_{r, 8}$ ]
-D3BJ	-D3 5 w/ newer Becke-Johnson rational damping	`dftd3`	[ $s_{6}$ , $s_{8}$ , $a_{1}$ , $a_{2}$ ]
-D3(BJ)	alias to -D3BJ
-D3M	alias to -D3MZERO
-D3MZERO	-D3 6 w/ reparameterized and more flexible original zero-damping	`dftd3`	[ $s_{6}$ , $s_{8}$ , $s_{r, 6}$ , $β$ ]
-D3MBJ	-D3 6 w/ reparameterized newer Becke-Johnson rational damping	`dftd3`	[ $s_{6}$ , $s_{8}$ , $a_{1}$ , $a_{2}$ ]
-D3M(BJ)	alias to -D3MBJ
-NL	Grimme’s -NL (DFT plus VV10 correlation) 7	PSI4‘s nl	[ $b$ , $c$ ] via NL_DISPERSION_PARAMETERS
-CHG	Chai & Head-Gordon dispersion formula 8	PSI4‘s libdisp	[ $s_{6}$ ]
-DAS2009	Podeszwa & Szalewicz dispersion formula 9	PSI4‘s libdisp	[ $s_{6}$ ]
-DAS2010	Podeszwa & Szalewicz dispersion formula 10	PSI4‘s libdisp	[ $s_{6}$ ]
-D4	alias to -D4BJEEQATM
-D4BJ	alias to -D4BJEEQATM
-D4BJEEQATM	-D4 12	`dftd4`	[ $a_{1}$ , $a_{2}$ , $a l p$ , $s_{6}$ , $s_{8}$ , $s_{9}$ ]

Three-Body Dispersion Corrections¶

In addition to the previously discussed two-body dispersion corrections, the dftd3/PSI4 interface enables computations of three-body dispersion corrections. In DFT-D3, three-body dispersion is approximated with the Axilrod-Teller-Muto model:

E_{d i s p}^{(3)} = - \frac{1}{6} \sum_{A \neqB \neqC} \frac{C_{9}^{A B C} (3 \cos θ_{a} \cos θ_{b} \cos θ_{c} + 1)}{(r_{A B} r_{B C} r_{A C})^{3}} f_{d a m p} ({\bar{r}}_{A B C})

where $θ_{a}$ is the angle at atom A corresponding to the triangle formed by atoms A, B, and C, and ${\bar{r}}_{A B C}$ is the geometric mean of the corresponding atomic-pair distances. The dispersion coefficients are defined as

C_{9}^{A B C} = \sqrt{C_{6}^{A B} C_{6}^{B C} C_{6}^{A C}}

See the DFT-D3 documentation for more details.

For now, the three-body correction can be called by using the run_dftd3() function with d3-atmgr as the passed functional string. For example, the three-body ATM dispersion correction for a neon trimer could be computed with:

molecule ne3 {
Ne 0.0 0.0 0.0
Ne 0.0 0.0 1.0
Ne 0.0 1.0 1.0
}
ne.update_geometry()
energy = m.run_dftd3('d3-atmgr', dertype=0)
print(energy)

Footnotes

1: Note that there are functionals with these extensions (e.g., wB97X-D) that, not being Grimme corrections, won’t follow this table exactly.
2: [Grimme:2004:1463]
3: [Grimme:2006:1787]
4: [Grimme:2010:154104]
5: [Grimme:2011:1456]
6(1,2): [Smith:2016:2197]
7: [Hujo:2011:3866]
8: [Chai:2010:6615]
9: [Pernal:2009:263201]
10: [Podeszwa:2010:550]
11: Keyword not used for user-defined functionals where the dft_dict["dispersion"]["params"] is easily editable for this purpose. See Advanced Functional Use and Manipulation
12: [Caldeweyher:2019:154122]

A few practical examples:

DFT-D2 single point with default parameters (dftd3 not called)
```
energy('bp86-d')
```
DFT-D3BJ optimization with default parameters
```
optimize('pbe-d3bj')
```

DFT-D2 optimization with custom s6 parameter

set dft_dispersion_parameters [1.20]
optimize('b3lyp-d2')

DFT-D3ZERO single point (b3lyp) with custom s8 parameter (reset all four values)

set dft_dispersion_parameters [1.0, 2.0, 1.261, 14.0]
energy('b3lyp-d3')

DFT-D2 single point with dftd3 instead of PSI4‘s libdisp
```
energy('pbe-d2', engine='dftd3')
```

If only dispersion corrections (rather than total energies) are of interest, the dispersion programs can be run independently of the scf through the python function run_dftd3() or run_dftd4(). (These functions call QCEngine, which is the same PSI4 + dftd3/dftd4 interface that is called during an scf job.) This “D-only” route is much faster than running a DFT-D energy.

Note that in a DFT+D energy or gradient calculation, user-specified dispersion parameters override any information provided about the functional. The same holds true for a dftd3 “D-only” calculation. But in a dftd4 “D-only” calculation, functional information overrides any user-specified dispersion parameters.

Some set-up:

molecule nene {
Ne
Ne 1 2.0
}

nene.update_geometry()

The same four dispersion corrections/gradients as the section above:

>>> print nene.run_dftd3('bp86', 'd', dertype=0)
-7.735e-05

>>> E, G = nene.run_dftd3('pbe', 'd3bj')
>>> print G
[[0.0, 0.0, -1.1809087569358e-05], [0.0, 0.0, 1.1809087569358e-05]]

>>> E, G = nene.run_dftd3('b3lyp', 'd2', {'s6': 1.20})
>>> print E
-8.84e-05

>>> E, G = nene.run_dftd3(dashlvl='d3', dashparam={'s8': 2.0, 'alpha6': 14.0, 'sr6': 1.261, 's6': 1.0})
>>> print E
-0.00024762

qcdb.Molecule.run_dftd3(self, func=None, dashlvl=None, dashparam=None, dertype=None, verbose=1)¶

Compute dispersion correction via Grimme’s DFTD3 program.

Parameters

func (Optional[str]) – Name of functional (func only, func & disp, or disp only) for which to compute dispersion (e.g., blyp, BLYP-D2, blyp-d3bj, blyp-d3(bj), hf+d). Any or all parameters initialized from dashcoeff[dashlvl][func] can be overwritten via dashparam.
dashlvl (Optional[str]) – Name of dispersion correction to be applied (e.g., d, D2, d3(bj), das2010). Must be key in dashcoeff or “alias” or “formal” to run.
dashparam (Optional[Dict]) – Values for the same keys as dashcoeff[dashlvl][‘default’] used to override any or all values initialized by func. Extra parameters will error.
dertype (Union[int, str, None]) – Maximum derivative level at which to run DFTD3. For large molecules, energy-only calculations can be significantly more efficient. Influences return values, see below.
verbose (int) – Amount of printing.

Returns

energy (float) – When dertype=0, energy [Eh].
gradient (ndarray) – When dertype=1, (nat, 3) gradient [Eh/a0].
(energy, gradient) (tuple of float and ndarray) – When dertype=None, both energy [Eh] and (nat, 3) gradient [Eh/a0].

class psi4.driver.procrouting.empirical_dispersion.EmpiricalDispersion(*, name_hint=None, level_hint=None, param_tweaks=None, engine=None, save_pairwise_disp=False)[source]¶

Lightweight unification of empirical dispersion calculation modes.

dashlevel¶

{‘d1’, ‘d2’, ‘d3zero’, ‘d3bj’, ‘d3mzero’, ‘d3mbj’, ‘chg’, ‘das2009’, ‘das2010’, ‘nl’, ‘dmp2’, “d4bjeeqatm”} Name of dispersion correction to be applied. Resolved from name_hint and/or level_hint into a key of empirical_dispersion_resources.dashcoeff.

Type: str

dashparams¶

Complete set of parameter values defining the flexible parts of dashlevel. Number and parameter names vary by dashlevel. Resolved into a complete set (keys of dashcoeff[dashlevel][‘default’]) from name_hint and/or dashcoeff_supplement and/or user param_tweaks.

Type: dict

fctldash¶

If dashparams for dashlevel corresponds to a defined, named, untweaked “functional-dashlevel” set, then that functional. Otherwise, empty string.

Type: str

description¶

Tagline for dispersion dashlevel.

Type: str

dashlevel_citation¶

Literature reference for dispersion dashlevel in general, not necessarily for dashparams.

Type: str

dashparams_citation¶

Literature reference for dispersion parameters, if dashparams corresponds to a defined, named, untweaked “functional-dashlevel” set with a citation. Otherwise, empty string.

Type: str

dashcoeff_supplement¶

See description in qcengine.programs.empirical_dispersion_resources.from_arrays. Used here to “bless” the dispersion definitions attached to the procedures/dft/<rung>_functionals-defined dictionaries as legit, non-custom, and of equal validity to qcengine.programs.empirical_dispersion_resources.dashcoeff itself for purposes of validating fctldash.

Type: dict

engine¶

{‘libdisp’, ‘dftd3’, ‘nl’, ‘mp2d’, “dftd4”} Compute engine for dispersion. One of Psi4’s internal libdisp library, external Grimme or Beran projects, or nl.

Type: str

disp¶

Only present for engine =libdisp. Psi4 class instance prepared to compute dispersion.

Type: Dispersion

ordered_params¶

Fixed-order list of relevant parameters for dashlevel. Matches DISPERSION CORRECTION ENERGY ordering. Used for printing.

Type: list

Parameters

name_hint (Optional[str]) – Name of functional (func only, func & disp, or disp only) for which to compute dispersion (e.g., blyp, BLYP-D2, blyp-d3bj, blyp-d3(bj), hf+d). Any or all parameters initialized from dashcoeff[dashlevel][functional-without-dashlevel] or dashcoeff_supplement[dashlevel][functional-with-dashlevel] can be overwritten via param_tweaks.
level_hint (Optional[str]) – Name of dispersion correction to be applied (e.g., d, D2, d3(bj), das2010). Must be key in dashcoeff or “alias” or “formal” to one.
param_tweaks (Union[Dict, List, None]) – Values for the same keys as dashcoeff[dashlevel][‘default’] (and same order if list) used to override any or all values initialized by name_hint. Extra parameters will error.
engine (Optional[str]) – Override which code computes dispersion. See above for allowed values. Really only relevant for -D2, which can be computed by libdisp or dftd3.