3.5. Dispersion Corrections¶

A major weakness of conventional density functional theory (and Hartee-Fock) is the lacking description of long-range correlation effects including London dispersion. London dispersion represents the attractive component of the van-der-Waals interaction and even though it is a rather weak interaction, its correct description can be crucial for many cases. This is specifically the case for non-covalent interactions and extended molecules. However, there are several well-established dispersion-corrections available to largely correct for this shortcoming. Within ORCA various options are available that include Grimme’s D3 [270, 271] and D4 [159, 272] corrections as well as the non-local variant of VV10 (NL, [273, 274] ) are available.

Some notes on dispersion corrections

Dispersion corrections do not only affect non-covalent complexes, but also conformational energies and structures which are heavily influenced by intramolecular dispersion. Therefore, including the dispersion correction is almost always recommended or even required (except for a handful of cases where it cannot, should not or need not be used, see below).
DFT calculations with small basis sets (such as double-\(\zeta\) basis sets) often yield attractive potential energy surfaces even without the dispersion correction. However, this is due to basis set superposition error (BSSE), and the interaction energy brought about by the BSSE frequently does not match the true interaction energy due to dispersion (because they have completely different origins). Therefore, although a DFT double zeta calculation without the dispersion correction may appear to give qualitatively correct results, or occasionally even better results than a double zeta calculation with dispersion corrections (because in the latter case one typically overestimates the total attraction), it is still highly recommended to “get the right answer for the right reason” by reducing the BSSE and turning on the dispersion correction. The BSSE can be corrected by a variety of means, for example (1) by using a larger basis set; (2) by using the counterpoise correction (Counterpoise Corrections); or (3) by using the geometrical counterpoise correction (section Geometrical Counterpoise Correction (gCP)). Of these, (3) is available at almost no cost (including analytic gradient contributions), and is especially suitable for geometry optimization of large molecules. Otherwise (1) (or its combination with (2)) may be more appropriate due to its higher accuracy.
Functionals that contain VV10-type non-local dispersion (in general, these are the functionals whose names end with “-V”) do not need (and cannot be used together with) dispersion corrections. The same holds for post-HF and multireference methods, like MP2, CCSD(T), CASSCF and NEVPT2. However, one can add a dispersion correction on top of HF.
Certain functionals, especially the Minnesota family of functionals (e.g. M06-2X), describe medium-range dispersion but miss long-range dispersion. They give reasonable dispersion energies for small to medium systems but may slightly underestimate the dispersion energies for large systems. As the uncorrected functional already accounts for the bulk of the dispersion in this case, the dispersion correction is much less important than e.g. the case of B3LYP, and should in general be considered as beneficial but not mandatory.

3.5.1. Grimme’s DFT-D3 and DFT-D4¶

DFT-D3 and DFT-D4 are atom-pairwise (atom-triplewise) dispersion corrections which can be added to the KS-DFT energies and gradient [270]. In ORCA, D3 is available for all elements up to Pu (Z = 94) and D4 up to Lr (Z = 103).

(3.9)¶\[E_{\text{DFT-D}} = E_{\text{KS-DFT}} + E_{\text{disp}} \]

\(E_{\text{disp}}\) is then the sum of the two- and three-body contributions to the dispersion energy, \(E_{\text{disp} } =E^{(2) }+E^{(3) }\). Most important is the two-body term, which is given at long range by:

(3.10)¶\[E_{\text{disp} } = - \sum_{A < B} \sum_{n=6,8} s_{n} \frac{C_{n}^{AB} }{r_{AB}^{n} } \]

where \(C_{n}^{AB}\) denotes the averaged (isotropic) \(n^{\text{th}}\)-order dispersion coefficient for atom pair AB and \(r_{AB}\) is their internuclear distance. \(s_{n}\) is a functional-dependent scaling factor (see below). In the general case, an adequate damping function must be employed. The more recent D4 model basically uses a similar theory but further introduces an atomic partial charge dependence of the atomic reference polarizabilities.

3.5.1.1. Damping Functions¶

In order to avoid near-singularities for small \(r_{AB}\), the dispersion contribution needs to be damped at short distances. One possible way is to use rational damping as proposed by Becke and Johnson [275, 276, 277]:

(3.11)¶\[E^{(2) }= - \sum_{A < B} \sum_{n=6,8} s_{n} \frac{C_{n}^{AB} }{r_{AB}^{n} +f(R_{0}^{AB} )^{n} } \]

with [277]

(3.12)¶\[R_{0}^{AB} =\sqrt{ \frac{C_{8}^{AB} }{C_{6}^{AB} } } \]

and

(3.13)¶\[f(R_{0}^{AB} )=a_{1} R_{0}^{AB} +a_{2} . \]

Damping the dispersion contribution to zero for short ranges (as in Ref. [270]) is also possible:

(3.14)¶\[E^{(2) }= - \sum_{A < B} \sum_{n=6,8} s_{n} \frac{C_{n}^{AB} }{r_{AB}^{n} }f_{d,n} (r_{AB} ) \]

with

(3.15)¶\[f_{d,n} =\frac{1}{1+6(\frac{r_{AB} }{s_{r,n} R_{0}^{AB} })^{-\alpha_{n} } } \]

Note that the \(R_{0}^{AB}\) used with this damping are from Ref. [270]. For more information on the supported damping functions, see Ref [271].

3.5.1.2. Three-body term¶

It is possible to calculate the three-body dispersion contributions according to

(3.16)¶\[E^{(3) }= - \sum_{A < B < C} \frac{C_{9}^{ABC} (3\cos{\theta_{a} } \cos{\theta_{b} } \cos{\theta_{c} } +1) }{(r_{AB} r_{BC} r_{CA} )^{3} } f_{d,(3) }(\overline{r}_{ABC}) \]

where \(\theta_{a}\), \(\theta_{b}\) and \(\theta_{c}\) are the internal angles of the triangle formed by \(r_{AB}\), \(r_{BC}\) and \(r_{CA}\). The \(C_{9}\) coefficient is approximated by

(3.17)¶\[C_{9}^{ABC} \approx -\sqrt{ C_{6}^{AB} C_{6}^{AC} C_{6}^{BC} } \]

The three-body contribution has a small effect on medium-sized molecules and is damped according to equation (3.16). The damping function \(f_{d,(3) }(\overline{r}_{ABC})\) is similar to the one shown in equation (3.15) with \(\overline{r}_{ABC}\) being the geometric mean of \(r_{AB}\), \(r_{BC}\) and \(r_{CA}\). It can be used with both variants of the \(E^{(2) }\) term, although the three-body term itself will always be calculated using the zero damping scheme. Adding the three-body correction has proven to give quite accurate results for the thermochemistry of supramolecular systems[278]. Note that the three-body term is included in D4 by default.

3.5.1.3. Basic Usage¶

The D3 correction can be invoked by the !D3 keyword that will automatically make use of the default Becke-Johnson damping and is thus equivalent to !D3BJ. The zero damping scheme can be used with !D3ZERO. Note that the Becke-Johnson scheme is generally recommended.

!PBE D3

The three-body term can be invoked via the keyword !ABC.

!PBE D3 ABC

Note

Note that not all methods are parameterized for all damping options. E.g. Hartree-Fock (HF) is only parametrized with BJ-damping, while M06 is only parameterized with zero damping.

The more recent and generally recommended D4 correction can be used with the !D4 keyword. Note that D4 uses Becke-Johnson damping and the three-body term by default.

!PBE D4

The D4 correction can also be defined in the %method block of the input:

! PBE def2-TZVP

%method
  DFTDOpt     5     # 5 = D4; 4 = D3(BJ)
end

3.5.1.4. Custom parameters¶

If there is a functional for which parameters are known but not yet implemented into ORCA, they can be sepcified manually via the %method block.

Warning

Usage of custom parameters is at own risk and is generally not recommended. A list of some available parameters for D3 can be found at the website of the Grimme group.

Important

GGA and hybrid functionals should only be used with \(s_{6} =1.0\) to ensure asymptotically correct behavior. Double-hybrid functionals already account for parts of the dispersion interaction and should therefore be used with \(s_{6} < 1.0\).

For D3 with Becke-Johnson damping, the \(s_6\), \(a_1\), \(s_8\), and \(a_2\) parameters can be adjusted:

!B2PLYP D3

%method
  D3S6 0.6400
  D3S8 0.9147
  D3A1 0.3065
  D3A2 5.0570
end

For D4, the \(s_9\) parameter of the three-body term can be adjusted as well.

!B2PLYP D4

%method
  D4S6 0.6400
  D4S8 1.1689
  D4A1 0.4415
  D4A2 4.7311
  D4S9 1.0000
end

For the D3 variant with zero damping the parameters \(s_6\), \(rs_6\), \(s_8\), and \(\alpha_{6}\) can be adjusted.

!BLYP D3ZERO

%method
  D3S6     1.000
  D3RS6    1.094
  D3S8     1.682
  D3alpha6 14
end

For some functionals, there are also special parameters, which were optimized for use with the def2-TZVPP triple-zeta basis set. This option is only available with zero damping and can be invoked by the keywords !D3ZERO D3TZ. Preliminary results in the SI of Ref. [270] indicate that results are only slightly worse than with quadruple-zeta basis sets using the default parameters. This option should be carefully tested for future use in large computations.

3.5.1.5. Example¶

In this first example, the DFT-D4 dispersion correction is used.

!PBE def2-TZVP D4

* xyz 0 1
  C     0.000000     0.000000     0.000000
  O     0.000000     0.000000     1.400000
  O     0.000000     0.000000    -1.400000
*

The output for the D4 dispersion correction in the ORCA output will look like this:

-------------------------------------------------------------------------------
                          DFT DISPERSION CORRECTION                            
                                                                               
                                DFTD4 V3.4.0                                   
-------------------------------------------------------------------------------
The PBE functional is recognized
Active option DFTDOPT                   ...         5   

-------------------------   ----------------
Dispersion correction           -0.000936618
-------------------------   ----------------


-------------------------   --------------------
FINAL SINGLE POINT ENERGY      -188.360452990031
-------------------------   --------------------

The total dispersion energy \(E_{\text{disp} }\) is given as the Dispersion correction and is automatically included in the final single point energy. As discussed above, the individual parameters may be adjusted manually.

!PBE def2-TZVP D4

%method
  D4S6 1.0
  D4A1 0.4289
  D4S8 0.7875
  D4A2 4.4407
  D4S9 0.9000
end

* xyz 0 1
  C     0.000000     0.000000     0.000000
  O     0.000000     0.000000     1.400000
  O     0.000000     0.000000    -1.400000
*

This results in the same output as above, but with additional messages that user inputs were found for the parameters:

A user input s6-coefficient scaling factor has been recognized
A user input a1-coefficient scaling factor has been recognized
A user input s8-coefficient scaling factor has been recognized
A user input a2-coefficient scaling factor has been recognized
A user input s9-coefficient scaling factor has been recognized

3.5.1.6. Keywords¶

Table 3.13 Simple input keywords for the DFT-D corrections.¶
Keyword	Description
`D2`	Activates D2 correction
`D3`	Activates D3 correction with BJ damping (D3(BJ))
`D3BJ`	Activates D3 correction with BJ damping (D3(BJ))
`D3ZERO`	Activates D3 correction with zero damping (D3(0))
`D30`	Activates D3 correction with zero damping (D3(0)), equivalent to `D3ZERO`
`D3TZ`	Activates special triple-\(\zeta\) D3 parameterization (only with `D3ZERO`)
`D4`	Activates D4 correction (BJ and ATM included)
`ABC`	Activates three-body term
`ATM`	Activates three-body term, equivalent to `ABC`
`NOVDW`	Deactivates DFT-D dispersion corrections

Table 3.14 `%method` block input keywords for the DFT-D corrections.¶
Keyword	Options	Description
`DFTDOPT`	`1`	Activate the original D correction[279] (deprecated)
	`2`	Activate the D2 correction (deprecated)
	`3`	Activate the D3ZERO correction
	`4`	Activate the D3BJ correction
	`5`	Activate the D4 correction
`D3S6`	`<real>`	Controls \(s_6\) parameter for D3
`D3S8`	`<real>`	Controls \(s_8\) parameter for D3
`D3A1`	`<real>`	Controls \(a_1\) parameter for D3
`D3A2`	`<real>`	Controls \(a_2\) parameter for D3
`D3RS6`	`<real>`	Controls \(rs_6\) parameter for zero damping
`D3alpha6`	`<real>`	Controls \(\alpha_{6}\) parameter for zero damping
`D4S6`	`<real>`	Controls \(s_6\) parameter for D4
`D4S8`	`<real>`	Controls \(s_8\) parameter for D4
`D4A1`	`<real>`	Controls \(a_1\) parameter for D4
`D4A2`	`<real>`	Controls \(a_2\) parameter for D4
`D4S9`	`<real>`	Controls \(s_9\) parameter of ABC term for D4
`DFTDScaleC6`	`<real>`	Defines the C6 scaling coefficient

3.5.1.7. Available Parameters¶

Table 3.15 DFT-D4 (`!D4`) parameters used for various functionals.¶
Functional	\(s_{6}\)	\(s_{8}\)	\(a_{1}\)	\(a_{2}\)	\(s_{9}\)
HF	1.0000	1.61679827	0.44959224	3.35743605	1.0000
BLYP	1.0000	2.34076671	0.44488865	4.09330090	1.0000
BPBE	1.0000	3.64405246	0.52905620	4.11311891	1.0000
BP86	1.0000	3.35497927	0.43645861	4.92406854	1.0000
BPW91	1.0000	3.24571506	0.50050454	4.12346483	1.0000
GLYP	1.0000	4.23798924	0.38426465	4.38412863	1.0000
LB94	1.0000	2.59538499	0.42088944	3.28193223	1.0000
mPWLYP	1.0000	1.25842942	0.25773894	5.02319542	1.0000
mPWPW	1.0000	1.82596836	0.34526745	4.84620734	1.0000
OLYP	1.0000	2.74836820	0.60184498	2.53292167	1.0000
OPBE	1.0000	3.06917417	0.68267534	2.22849018	1.0000
PBE	1.0000	0.95948085	0.38574991	4.80688534	1.0000
RPBE	1.0000	1.31183787	0.46169493	3.15711757	1.0000
REVPBE	1.0000	1.74676530	0.53634900	3.07261485	1.0000
PW86PBE	1.0000	1.21362856	0.40510366	4.66737724	1.0000
RPW86PBE	1.0000	1.12624034	0.38151218	4.75480472	1.0000
PW91	1.0000	0.77283111	0.39581542	4.93405761	1.0000
PW91P86	1.0000	0.32801227	0.35874687	6.05861168	1.0000
XLYP	1.0000	1.62972054	0.11268673	5.40786417	1.0000
B97BECKE	1.0000	0.87854260	0.29319126	4.51647719	1.0000
TPSS	1.0000	1.76596355	0.42822303	4.54257102	1.0000
revTPSS	1.0000	1.53089454	0.44880597	4.64042317	1.0000
SCAN	1.0000	1.46126056	0.62930855	6.31284039	1.0000
B1LYP	1.0000	1.98553711	0.39309040	4.55465145	1.0000
B3LYP	1.0000	2.02929367	0.40868035	4.53807137	1.0000
BHANDHLYP	1.0000	1.65281646	0.27263660	5.48634586	1.0000
B1P	1.0000	3.36115015	0.48665293	5.05219572	1.0000
B3P86	1.0000	3.08822155	0.47324238	4.98682134	1.0000
B1PW91	1.0000	3.02227550	0.47396846	4.49845309	1.0000
B3PW91	1.0000	2.88364295	0.46990860	4.51641422	1.0000
O3LYP	1.0000	1.75762508	0.10348980	6.16233282	1.0000
REVPBE0	1.0000	1.57185414	0.38705966	4.11028876	1.0000
REVPBE38	1.0000	1.66597472	0.39476833	4.39026628	1.0000
PBE0	1.0000	1.20065498	0.40085597	5.02928789	1.0000
PWP1	1.0000	0.60492565	0.46855837	5.76921413	1.0000
PW1PW	1.0000	0.96850170	0.42427511	5.02060636	1.0000
mPW1PW	1.0000	1.80841716	0.42961819	4.68892341	1.0000
mPW1LYP	1.0000	1.15591153	0.25603493	5.32083895	1.0000
PW6B95	1.0000	-0.31926054	0.04142919	5.84655608	1.0000
TPSSh	1.0000	1.85897750	0.44286966	4.60230534	1.0000
TPSS0	1.0000	1.62438102	0.40329022	4.80537871	1.0000
X3LYP	1.0000	1.54701429	0.20318443	5.61852648	1.0000
M06L	1.0000	0.59493760	0.71422359	6.35314182	1.0000
M06	1.0000	0.16366729	0.53456413	6.06192174	1.0000
wB97	1.0000	6.55792598	0.76666802	8.36027334	1.0000
wB97X	1.0000	-0.07519516	0.45094893	6.78425255	1.0000
B97M-D4	1.0000	0.66330000	0.42880000	3.99350000	1.0000
CAM-B3LYP	1.0000	1.66041301	0.40267156	5.17432195	1.0000
LC-BLYP	1.0000	1.60344180	0.45769839	7.86924893	1.0000
B2PLYP	0.6400	1.16888646	0.44154604	4.73114642	1.0000
B2GP-PLYP	0.5600	0.94633372	0.42907301	5.18802602	1.0000
mPW2PLYP	0.7500	0.45788846	0.42997704	5.07650682	1.0000
PWPB95	0.8200	-0.34639127	0.41080636	3.83878274	1.0000
B97-D	1.0000	1.69460052	0.28904684	4.13407323	1.0000
RSCAN	1.0000	0.87728975	0.49116966	5.75859346	1.0000
r2SCAN	1.0000	0.60187490	0.51559235	5.77342911	1.0000
r2SCANh	1.0000	0.83240000	0.49440000	5.90190000	1.0000
r2SCAN0	1.0000	0.89920000	0.47780000	5.87790000	1.0000
r2SCAN50	1.0000	1.04710000	0.45740000	5.89690000	1.0000
wB97X-D4rev	1.0000	0.44850000	0.33060000	4.27900000	1.0000
wB97M-D4rev	1.0000	0.84200000	0.35900000	4.66800000	1.0000
wR2SCAN	1.0000	1.00000000	0.38340000	5.78890000	1.0000
r2SCAN0-DH	0.9424	0.38560000	0.42710000	5.85650000	1.0000
r2SCAN-CIDH	0.8666	0.53360000	0.41710000	5.91250000	1.0000
r2SCAN-QIDH	0.7867	0.29550000	0.40010000	5.83000000	1.0000
r2SCAN0-2	0.7386	0.00000000	0.40300000	5.51420000	1.0000
Pr2SCAN50	0.7964	0.34210000	0.46630000	5.79160000	1.0000
kPr2SCAN50	0.8402	0.12120000	0.43820000	5.82320000	1.0000
wPr2SCAN50	0.8143	0.38420000	0.41350000	5.87730000	1.0000
Pr2SCAN69	0.7167	0.00000000	0.46440000	5.25630000	1.0000
revDSD-PBEP86/2021	0.5917	0.00000000	0.37100000	4.20140000	1.0000
revDOD-PBEP86/2021	0.6158	0.00000000	0.34400000	4.24270000	1.0000
LRC-PBE	1.0000	1.17000000	0.37800000	4.81600000	1.0000

Table 3.16 DFT-D3(BJ) (`!D3BJ`) parameters used for various functionals.¶
Functional	\(s_{6}\)	\(s_{8}\)	\(a_{1}\)	\(a_{2}\)	\(s_{9}\)
HF	1.0000	0.91710000	0.33850000	2.88300000	1.0000
BP86	1.0000	3.28220000	0.39460000	4.85160000	1.0000
BLYP	1.0000	2.69960000	0.42980000	4.23590000	1.0000
REVPBE	1.0000	2.35500000	0.52380000	3.50160000	1.0000
B97-D	1.0000	2.26090000	0.55450000	3.22970000	1.0000
PBE	1.0000	0.78750000	0.42890000	4.44070000	1.0000
RPBE	1.0000	0.83180000	0.18200000	4.00940000	1.0000
RPW86PBE	1.0000	1.38450000	0.46130000	4.50620000	1.0000
B3LYP	1.0000	1.98890000	0.39810000	4.42110000	1.0000
BHANDHLYP	1.0000	1.03540000	0.27930000	4.96150000	1.0000
TPSS	1.0000	1.94350000	0.45350000	4.47520000	1.0000
TPSS0	1.0000	1.25760000	0.37680000	4.58650000	1.0000
PBE0	1.0000	1.21770000	0.41450000	4.85930000	1.0000
REVPBE38	1.0000	1.47600000	0.43090000	3.94460000	1.0000
PW6B95	1.0000	0.72570000	0.20760000	6.37500000	1.0000
B2PLYP	0.6400	0.91470000	0.30650000	5.05700000	1.0000
mPWLYP	1.0000	2.00770000	0.48310000	4.53230000	1.0000
OLYP	1.0000	2.62050000	0.52990000	2.80650000	1.0000
BPBE	1.0000	4.07280000	0.45670000	4.39080000	1.0000
OPBE	1.0000	3.38160000	0.55120000	2.94440000	1.0000
B3PW91	1.0000	2.85240000	0.43120000	4.46930000	1.0000
REVPBE0	1.0000	1.75880000	0.46790000	3.76190000	1.0000
TPSSh	1.0000	2.23820000	0.45290000	4.65500000	1.0000
CAM-B3LYP	1.0000	2.06740000	0.37080000	5.47430000	1.0000
B2GP-PLYP	0.5600	0.25970000	0.00000000	6.33320000	1.0000
PWPB95	0.8200	0.29040000	0.00000000	7.31410000	1.0000
SCAN	1.0000	0.00000000	0.53800000	5.42000000	1.0000
RSCAN	1.0000	1.08860000	0.47020000	5.73410000	1.0000
R2SCAN	1.0000	0.78980000	0.49480000	5.73080000	1.0000
R2SCANh	1.0000	1.12360000	0.47090000	5.91570000	1.0000
R2SCAN0	1.0000	1.18460000	0.45340000	5.89720000	1.0000
R2SCAN50	1.0000	1.32940000	0.43110000	5.92400000	1.0000
WR2SCAN	1.0000	1.00000000	0.38340000	5.78890000	1.0000
r2SCAN0-DH	0.9424	0.38560000	0.42710000	5.85650000	1.0000
r2SCAN-CIDH	0.8666	0.53360000	0.41710000	5.91250000	1.0000
r2SCAN-QIDH	0.7867	0.29550000	0.40010000	5.83000000	1.0000
r2SCAN0-2	0.7386	0.00000000	0.40300000	5.51420000	1.0000
Pr2SCAN50	0.7964	0.34210000	0.46630000	5.79160000	1.0000
kPr2SCAN50	0.8402	0.12120000	0.43820000	5.82320000	1.0000
wPr2SCAN50	0.8143	0.38420000	0.41350000	5.87730000	1.0000
Pr2SCAN69	0.7167	0.00000000	0.46440000	5.25630000	1.0000
revDSD-PBEP86/2021	0.5917	0.00000000	0.37100000	4.20140000	1.0000
revDOD-PBEP86/2021	0.6158	0.00000000	0.34400000	4.24270000	1.0000
DSD-BLYP	0.5000	0.21300000	0.00000000	6.05200000	1.0000
DSD-BLYP/2013	0.5700	0.00000000	0.00000000	5.40000000	1.0000
DSD-PBEB95	0.6100	0.00000000	0.00000000	6.20000000	1.0000
DSD-PBEP86	0.4180	0.00000000	0.00000000	5.65000000	1.0000
DSD-PBEP86/2013	0.4800	0.00000000	0.00000000	5.60000000	1.0000
B97M-D3BJ	1.0000	0.13840000	-0.07800000	5.59460000	1.0000
wB97X-D3BJ	1.0000	0.26410000	0.00000000	5.49590000	1.0000
wB97M-D3BJ	1.0000	0.39080000	0.56600000	3.12800000	1.0000
wB97X-2	0.5470	0.00000000	3.52000000	7.79500000	1.0000
PBE0DH	0.8800	1.08900000	0.00000000	6.38500000	1.0000
PBE02	0.5400	0.51500000	0.00000000	8.34500000	1.0000
PBE-QIDH	0.6100	0.56600000	0.11400000	7.53800000	1.0000

Table 3.17 DFT-D3(0) (`!D3ZERO`) parameters used for various functionals.¶
Functional	\(s_{6}\)	\(rs_{6}\)	\(s_{8}\)	\(\alpha_{6}\)
HF	1.0000	1.1580	1.7460	14
BLYP	1.0000	1.0940	1.6820	14
BP86	1.0000	1.1390	1.6830	14
B97-D	1.0000	0.8920	0.9090	14
REVPBE	1.0000	0.9230	1.0100	14
PBE	1.0000	1.2170	0.7220	14
RPBE	1.0000	0.8720	0.5140	14
TPSS	1.0000	1.1660	1.1050	14
B3LYP	1.0000	1.2610	1.7030	14
PBE0	1.0000	1.2870	0.9280	14
PW6B95	1.0000	1.5230	0.8620	14
TPSS0	1.0000	1.2520	1.2420	14
B2PLYP	0.6400	1.4270	1.0220	14
B2GP-PLYP	0.5600	1.5860	0.7600	14
PWPB95	0.8200	1.5570	0.7050	14
mPWLYP	1.0000	1.2390	1.0980	14
BPBE	1.0000	1.0870	2.0330	14
BHANDHLYP	1.0000	1.3700	1.4420	14
TPSSh	1.0000	1.2230	1.2190	14
REVPBE0	1.0000	0.9490	0.7920	14
REVPBE38	1.0000	1.0210	0.8620	14
RPW86PBE	1.0000	1.2240	0.9010	14
B3PW91	1.0000	1.1760	1.7750	14
M06L	1.0000	1.5810	0.0000	14
M06	1.0000	1.3250	0.0000	14
M062X	1.0000	1.6190	0.0000	14
wB97X-D3	1.0000	1.2810	1.0000	14
CAM-B3LYP	1.0000	1.3780	1.2170	14
SCAN	1.0000	1.3240	0.0000	14
wB97X-2	0.5470	4.7330	-0.1090	14
PBE0DH	0.8800	1.1280	0.0000	14
PBE02	0.5400	1.2430	-0.2280	14
PBE-QIDH	0.4000	1.1140	0.0540	14

3.5.2. Non-Local Dispersion Correction (VV10): DFT-NL¶

A different route to correct for London dispersion is followed by the van der Waals Density Functional (vdW-DF) as pioneered by Langreth and Lundquist [280]. These methods use only the electron density to include such dispersion/correlation effects and can be used in a self-consistent way also correction the molecular orbitals and not only energies. The vdW correlation functional VV10 of Vydrov and Van Voorhis [273] currently seems to be the most promising candidate for a general and accurate electronic structure method.

We use the term DFT-NL for any density functional in combination with the non-local part of the VV10 functional with an optimized parameter \(b\), which will be defined below. The performance of these methods has been evaluated in Ref. [274] using the GMTKN30 [248, 281, 282] database and the S66 set [283]. The performance of weak hydrogen bonds were evaluated in Ref. [284].

DFT-NL and DFT-D3/D4 perform very similarly, but NL is to be preferred for metallic systems or when the basic electronic structure changes significantly (e.g. oxidations or ionizations). As a recent example, Janes and Iron showed that for functionals such as wB97X-V, including VV10 correlation results in very high quality reaction barriers when metals are involved [285].

The total exchange-correlation (XC) energy of VV10-type functionals is defined in eq. (3.18). It is composed of standard exchange (X) and correlation (C) parts and the non-local (NL) term, which covers (mainly) long-range dispersive energy:

(3.18)¶\[E_{\text{XC} }^{\text{DFT-NL} } = E_{\text{X} }^{\text{(hybrid)GGA} } +E_{\text{C} }^{\text{GGA} } +E_{\text{C-NL} }^{\text{VV10} } \]

The NL term is given by the following double integral:

(3.19)¶\[E_{\text{C-NL} }^{\text{VV10} } = \int dr\rho (r)\left[{ \beta +\frac{1}{2}\mathop \int\nolimits dr'\rho ( { r'} ) \varphi (r,r') } \right]\]

where \(\rho\) is the total electron density, and the definitions of the kernel \(\varphi (r,r')\) and \(\beta\) are as follows (in a.u.):

(3.20)¶\[\varphi \left({ r,r'} \right) = -\frac{3}{2gg'\left({ g+g'} \right)} \]

(3.21)¶\[\beta = \frac{1}{32}\left[{ \frac{3}{b^{2} }} \right]^{3/4} \]

with

(3.22)¶\[\begin{split}\begin{aligned} g\left( r \right) &= \omega_{0} \left( r \right)R^{2}+\kappa (r) \\ R &= \left|{ r-r'} \right| \\ \omega_{0} \left( r \right) &= \sqrt{ C\left|{ \frac{\nabla \rho (r) }{\rho (r) }} \right|^{4}+\frac{4\pi }{3}\rho (r) } \\ \kappa \left( r \right) &= b\frac{3\pi }{2}\left[{ \frac{\rho (r) }{9\pi } } \right]^{1/6} \end{aligned} \end{split}\]

In the original definition, the short-range attenuation parameter \(b\) appearing in \(\kappa\) and \(\beta\) was fitted to the S22 set [286] of non-covalent interactions (\(b = 5.9\) for the rPW86PBE GGA). The other parameter \(C = 0.0093\), appearing in \(\omega_{0}\), determines the long-range behavior, and was set to its original value. Other DFT-NL functionals are constructed analogously. For a detailed discussion of the derivation of the formulas and their physical meaning and basis, see the references given above.

The defined energy of the non-local DFT-NL exchange-correlation functional can be computed in two ways: as a post-SCF addition based on a converged SCF density or in a self-consistent treatment. In our implementation of the non-self-consistent DFT-NL functional, a self-consistent DFT computation is performed as the first step. In the second step, the optimized electron density from the DFT computation is taken as input for the energy calculation of the non-local part. In the self-consistent treatment, the orbitals and density are optimized in the presence of the full DFT + VV10 exchange-correlation potential. According to many test calculations, an self-consistent treatment is rarely necessary for normal energy evaluations. The computation of the double integral given in eq. (3.19) requires using an integration grid, just like for normal exchange-correlation functionals. This grid is built similarly to the regular grids available in the ORCA, see Sec. Numerical Integration.

3.5.2.1. Basic usage¶

The non-self-consistent variant of NL can be invoked by the !NL keyword.

!B3LYP NL

The self-consistent variant by !SCNL.

!B3LYP SCNL

In the current version of ORCA, NL parameters are implemented for various functionals and HF:

Table 3.18 Methods with available NL parameters. If no values are given, the default parameters are used.¶
Method	\(b\)	\(C\)	\(NLScal\)
Hartree-Fock
`HF`	3.9
(meta-)GGA
`BLYP`	4.0
`BP86`	4.4
`PBE`	6.4
`REVPBE`	3.7
`RPBE`	4.0
`RPW86PBE`	5.9
`SCANfunc`	6.4
`RSCAN`	10.8
`R2SCAN`	12.3
`TPSS`	5.0
`B97M-V`	6.0	0.01
Hybrid
`B3LYP`	4.8
`B3LYP/G`	4.8
`B3P86`	5.3
`B3PW91`	4.5
`mPW1PW`	5.3
`PBE0`	6.9
`PW1PW`	7.7
`PW6B95`	9.0
`REVPBE0`	4.3
`REVPBE38`	4.7
`TPSSh`	5.2
`TPSS0`	5.5
`R2SCANh` (see [236])	11.9
`R2SCAN0` (see [236])	11.4
`R2SCAN50` (see [236])	10.8
Range-separated hybrid
`LRC-PBE`	6.4
`WR2SCAN` (see [246])	9.2612
`WB97X-V`	6.0	0.01
`WB97M-V`	6.0	0.01
Double-hybrid
`B2PLYP` (see [287])	7.8
`DSD-BLYP` (see [288])	12.0
`DSD-PBEP86` (see [288])	14.2
`DSD-BLYP/2013`	12.0
`DSD-PBEP86/2013`	14.2
`PWPB95` (see [288])	11.1
`PR2SCAN50` (see [246])	10.9207		0.75
`KPR2SCAN50` (see [246])	10.6723		0.70
`PR2SCAN69` (see [246])	9.0691		0.5556
Range-separated double-hybrid
`WPR2SCAN50` (see [246])	9.4149		0.65
`WB97M(2)`	10.0	0.01	0.65904

Important

Head-Gordon’s \(\omega\)B97X-V[242], B97M-V [211], and \(\omega\)B97M-V[245] functionals are specially fitted variants that use the non-local VV10 kernel to capture London-dispersion effects. They employ the non-self-consistent NL correction by default and no additional NL keywords have to be used! A self-consistent treatment can be invoked by the !SCNL keyword.

Some notes on the NL corrections in ORCA

Self-consistent calculations are not possible with the Hartree-Fock method.
Analytic gradients are available, thus geometry optimizations with numerical frequencies can be computed.
TD-DFT calculations are not yet available.
Any calculations that require second derivatives of the NL functional are not yet possible. These are needed for real type perturbations in the CP-SCF solutions, e.g. for analytic Hessians, dipole polarizabilities, and double-hybrid gradients.
Strictly imaginary perturbations such as NMR shielding and EPR g-tensors (both also with GIAOs), and hyperfine couplings are available.

3.5.2.2. Custom parameters¶

For NL, the parameters \(b\), \(C\), and a global scaling parameter of the NL correlation energy term can be adjusted via the %method block. The global scaling parameter NLScal should be used to scale the NL correction for methods that already partly cover long-range correlatione like double-hybrids.

!B3LYP NL

%method
  NLb    5.0
  NLC    0.0083
  NLScal 1.0
end

Note

In order to improve the scaling for larger systems, a distance cutoff was also introduced, controlled by the vdWdistTCUT flag in the %method block. The default value is 10 Å, so two grid points more than 10 Å away from each other do not correlate via the VV10 potential. This is already very conservative and has practically zero effect on the final energy, but can be altered if needed.

3.5.2.3. Example¶

In the following, we compute the energy of the argon dimer at a distance of 3.76 Å with the def2-TZVP basis set and using the B3LYP hybrid functional as an example. Here, we choose the non-self-consistent variant of the DFT-NL dispersion correction.

!B3LYP NL def2-TZVP

*xyz 0 1
  Ar  0.0 0.0 0.0
  Ar  0.0 0.0 3.76
*

The DFT-NL output for this example is shown below:

-------------------------------------------------------------------------------
                    post-SCF DFT-NL dispersion correction                      
-------------------------------------------------------------------------------

SCF Energy:    -1054.960511496 
NL    Energy:      0.209449910 
SC+NL Energy:  -1054.751061586 
NL done in  :              0.7 sec
-------------------------------------------------------------------------------

[...]

----------------
TOTAL SCF ENERGY
----------------

[...]

DFT components:
N(Alpha)           :       17.999996328440 electrons
N(Beta)            :       17.999996328440 electrons
N(Total)           :       35.999992656880 electrons
E(X)               :      -47.880920358714 Eh       
E(C)               :       -1.761923687962 Eh       
NL Energy, E(C,NL) :        0.209449910227 Eh       
E(XC)              :      -49.433394136450 Eh

Here, we find the B3LYP total energy (SCF Energy), the non-local contribution (NL Energy), and their sum (SC+NL Energy), which is the final total energy. In the DFT components section, the non-local contribution is listed separately (NL Energy, E(C,NL)) in order to be consistent with the !SCNL output.

3.5.2.4. Keywords¶

Table 3.19 Simple input keywords for the NL correction.¶
Keyword	Description
`NL`	Activates non-self-consistent NL correction
`SCNL`	Activates self-consistent NL correction

Table 3.20 `%method` block input keywords for the NL correction.¶
Keyword	Options	Description
`NLb`	`<real>`	Controls \(b\) parameter for NL
`NLC`	`<real>`	Controls \(C\) parameter for NL
`NLScal`	`<real>`	Controls the global scaling parameter for NL
`vdWdistTCUT`	`<real>`	Controls the correlation distance cutoff for NL