3.7. Composite Methods (3c methods)

Composite HF and DFT methods utilize relatively small tailored basis sets and special corrections to achieve high accuracies at a fraction of the computational cost of a calculation approaching the basis set limit. The most prominent composite methods are the so-called “3c” methods by Grimme and co-workers.

3.7.1. HF-3c

HF-3c is a fast Hartree-Fock based method developed for computation of structures, vibrational frequencies and non-covalent interaction energies in huge molecular systems [323]. The starting point for evaluating the electronic energy is a standard HF calculation with a small Gaussian AO basis set. The used so-called MINIX basis set consists of different sets of basis functions for different groups of atoms as shown in table Table 3.30. In ORCA, HF-3c is available for all elements up to Pu (Z = 94).

Table 3.30 Composition of the MINIX basis set.

element

basis

H-He, B-Ne

MINIS

Li-Be

MINIS+1(p)

Na-Mg

MINIS+1(p)

Al-Ar

MINIS+1(d)

K-Zn

SV

Ga-Kr

SVP

Rb-Rn

def2-SVP with def-ECPs

Fr-Lr

def-SVP with def-ECPs

Three terms are added to correct the HF energy \(E_{\text{tot} }^{\text{HF/MINIX} }\) in order to include London dispersion interactions, to account for the BSSE and to correct for basis set deficiencies, i.e. overestimated bond lengths. The corrected total energy is therefore calculated as

(3.23)\[ E_{\text{tot} }^{\text{HF-3c} } = E_{\text{tot} }^{\text{HF/MINIX} } + E_{\text{disp} }^{\text{D3(BJ) }} + E_{\text{BSSE} }^{\text{gCP} } + E_{\text{SRB} }. \]

The first correction term \(E_{\text{disp} }^{\text{D3(BJ) }}\) is the atom-pair wise London dispersion energy from the D3 dispersion correction scheme[270] applying Becke-Johnson (BJ) damping [275, 276, 277] (see section Grimme’s DFT-D3 and DFT-D4). The second term \(E_{\text{BSSE} }^{\text{gCP} }\) denotes the geometrical counterpoise (gCP) correction [124] to treat the BSSE (see section Geometrical Counterpoise Correction (gCP)). For the HF-3c method, the three usual D3 parameters \(s_8\), \(a_1\) and \(a_2\) were re-fitted using reference interaction energies of the complexes of the S66 test set [283]. This results in \(s_8=0.8777\), \(a_1=0.4171\) and \(a_2=2.9149\). The parameter \(s_6\) was set to unity as usual to enforce the correct asymptotic limit and the gCP correction was already applied in this fitting step.

The last term \(E_{\text{SRB} }\) is a short-ranged correction to deal with basis set deficiencies which occur when using small or minimal basis sets. It corrects for systematically overestimated covalent bond lengths for electronegative elements and is calculated as a sum over all atom pairs:

\[E_{\text{SRB} } = -s \sum_{A}^{\text{Atoms} } \sum_{B \neq A}^{\text{Atoms} }(Z_A Z_B)^{3/2}\exp(-\gamma (R_{AB}^{0,\text{D3} })^{3/4} R_{AB})\]

Here, \(R_{AB}^{0,\text{D3} }\) are the default cut-off radii as determined ab initio for the D3 scheme [270] and \(Z_A\), \(Z_B\) are the nuclear charges. This correction is applied for all elements up to argon. The empirical fitting parameters \(s=0.03\) and \(\gamma= 0.7\) were determined to produce vanishing HF-3c total atomic forces for B3LYP-D3(BJ)/def2-TZVPP equilibrium structures of 107 small organic molecules. More details can be found in the original publication [323].

The HF-3c method can only be invoked with a simple keyword:

! HF-3c

! HF-3c is a compound keyword and equals ! HF MINIX D3BJ GCP(HF/MINIX) PATOM, hence no basis set etc. has to be specified. The PATOM guess is selected since the grid construction for the default guess can take more time than an actual SCF step. The guess can only be overwritten manually in the %method section.

The default mode for the integral handling is set to Conventional. The storing of the two-electron integrals on disk or in memory if possible leads to large computational savings. In case one want to use the Direct mode, this has to be specified in the %scf input section:

%scf
  SCFmode Direct
end

The output gives the used parameters and the correction itself for D3 and gCP separately. As the SRB correction is also calculated with the otool_gcp, the results are given in the gCP output section. The Total correction to HF/MINIX is the sum of all three corrections (D3, gCP and SRB) and FINAL SINGLE POINT ENERGY is the total HF-3c energy as given in equation (3.23).

-------------------------------------------------------------------------------
                          DFT DISPERSION CORRECTION

                                DFTD3 V2.1  Rev 6
                          USING Becke-Johnson damping
-------------------------------------------------------------------------------
The default Hartree-Fock  is recognized
Active option DFTDOPT                   ...         4

molecular C6(AA) [au] = 1689.256597


            DFT-D V3
 parameters
 using HF/MINIX parameters
 s6 scaling factor         :     1.0000
 a1 scaling factor         :     0.4171
 s8 scaling factor         :     0.8777
 a2 scaling factor         :     2.9149
 ad hoc parameters k1-k3   :    16.0000     1.3333    -4.0000

 Edisp/kcal,au: -32.163184627631  -0.051255291794
 E6   /kcal   : -18.007221978
 E8   /kcal   : -14.155962649
 % E8         :  44.012938437

-------------------------   ----------------
Dispersion correction            -0.051255292
-------------------------   ----------------


------------------------------------------------------------------------------

                  g C P  - geometrical counterpoise correction

------------------------------------------------------------------------------
Method:  hf/minix

Parameters:   sigma        eta         alpha        beta
              0.1290      1.1526      1.1549       1.1763
Egcp:       0.0723150636 a.u.
Ebas:      -0.0636976872 a.u.
------------------   -----------------
gCP+bas correction         0.008617376
------------------   -----------------
----------------------------   ----------------
Total correction to HF/MINIX       -0.042637915
----------------------------   ----------------


-------------------------   --------------------
FINAL SINGLE POINT ENERGY      -163.002895262171
-------------------------   --------------------

For the elements up to Xe, the default initial guess is a Hückel guess. Beyond Xe, the guess mode is changed to HCORE since no Hückel parameters for the respective ECP bases are available and other models are not implemented at the moment. For calculations with only lighter elements and therefore no ECPs, the ECP printouts in the output file can be ignored.

3.7.2. B97-3c

B97-3c[324] is another composite DFT method designed for thermochemistry, structures, and noncovalent interactions specifically also for transition metal chemistry and other stronger correlated systems. It is based on the B97 GGA including the D3(BJ) dispersion correction with three-body contribution, a short range bond length correction, and a modified, stripped-down triple-\(\zeta\) basis termed def2-mTZVP, the computational cost of this method is between that of HF-3c and PBEh-3c (for large systems roughly two times more expensive than HF-3c). In ORCA, B97-3c is available for all elements up to Pu (Z = 94).

B97-3c can be invoked via simple input keyword:

! B97-3c

3.7.3. \(r^2\)SCAN-3c

The \(r^2\)SCAN-3c composite method[221] is available as robust “Swiss army knife” electronic structure method for thermochemistry, geometries and non-covalent interactions and has shown in preliminary tests consistent performance for both open and closed shell transition metal complexes. It is based on the \(r^2\)SCAN[216] meta-GGA combined with the D4 dispersion correction[325] and the geometrical counter poise-correction[124]. The modified triple-\(\zeta\) basis set, def2-mTZVPP, is larger and more consistent for the light main-group elements and almost as computationally efficient as the def2-mTZVP basis set of B97-3c. The computational cost of \(r^2\)SCAN-3c is slightly larger than B97-3c. In ORCA, \(r^2\)SCAN-3c is available for all elements up to Lr (Z = 103). It is invoked with the simple keyword

! r2SCAN-3c

3.7.4. PBEh-3c

PBEh-3c is a highly efficient electronic structure approach performing particularly well in the optimization of geometries and for interaction energies of non-covalent complexes.[237] Here, a global hybrid variant of the Perdew-Burke-Ernzerhof (PBE) functional with a relatively large amount of non-local Fock-exchange (42%) is employed with a valence-double-zeta Gaussian AO basis set (def2-mSVP). Basis set superposition errors (BSSE) and London dispersion effects are accounted for by the gCP and D3 schemes, respectively (see above). The basis set is constructed such that:

Table 3.31 Composition of the def2-mSVP basis set.

element

basis

H

def2-SV(P) (\(\alpha\) scaled by 1.2)

He

def2-SVP(-p)

Li-Be,Na-Kr

def2-SV(P)

B,Ne

Ahlrichs’ DZ + Polarization from def2-SVP

C-F

Ahlrichs’ DZ + Polarization from 6-31G*

Rb-Rn

def2-SVP with def2 ECPs

Fr-Lr

def-SVP with def-ECPs

For inter- and intramolecular BSSE the gCP expression from Eq. (2.25) is used but with a damping function (similar to the zero-damping in Eq. (3.14)). This damping improves the thermochemistry of the method significantly compared with the non-damped version. London dispersion effects are accounted for by the DFT-D3 (BJ-damping) scheme including the three-body term. Compared to the related HF-3c approach, the PBEh-3c is somewhat more costly, however, it yields much better geometries. These are roughly of MP2-quality (or even better for non-covalent structures) but may be computed at much lower cost. Due to the moderate amount of non-local Fock exchange, the method is less prone to self-interaction errors (as in GGAs) but still applicable in cases when Hartree-Fock fails (strongly correlated systems). In ORCA, PBEh-3c is available for all elements up to Pu (Z = 94).

The PBEh-3c method may be invoked with the simple keyword:

! PBEh-3c

Identical to HF-3c, the default initial guess for all elements up to Xe is a Hückel guess. Beyond Xe, the guess mode is changed to HCORE. For calculations with only lighter elements and therefore no ECPs, the ECP printouts in the output file can be ignored.

3.7.5. B3LYP-3c

B3LYP-3c is a method combination introduced by Grimme and co-workers to efficiently calculate gas-phase infrared spectra.[238] It combines the standard B3LYP functional with the D3(BJ)-ATM dispersion correction, a def2-mSVP basis set, and a geometrical counterpoise correction. In ORCA, B3LYP-3c is available for all elements up to Pu (Z = 94). B3LYP-3c can be invoked via simple input keyword:

! B3LYP-3c

or it can be constructed manually from its components via:

!B3LYP D3BJ GCP(DFT/SV(P)) def2-mSVP ABC

3.7.6. \(\omega\)B97X-3c

The \(\omega\)B97X-3c composite method[49] is based on the \(\omega\)B97X-V functional and combines a tailored and molecule-optimized polarized valence double-\(\zeta\) (vDZP) basis set and a specifically adapted D4 dispersion correction. The vDZP basis set employs large-core ECPs and shows only very small basis set superposition and incompleteness errors compared to conventional double-\(\zeta\) basis sets. In thorough tests on standard benchmarks sets, the \(\omega\)B97X-3c method was shown to be on par with well-performing hybrid DFT methods in a quadruple-\(\zeta\) basis set at a fraction of their computational cost. \(\omega\)B97X-3c is consistently available for all elements up to Rn (Z = 1–86).

It is invoked with the simple keyword:

! wB97X-3c

The vDZP basis set alone is utilized as follows (note that the corresponding large-core ECPs are called automatically):

! vDZP

3.7.7. Keywords

Table 3.32 Simple input keywords for the 3c composite methods.

Keyword

Description

HF-3C

Invokes the HF-3c method

B97-3C

Invokes the B97-3c method

R2SCAN-3C

Invokes the \(r^2\)SCAN-3c method

PBEH-3C

Invokes the PBEh-3c method

B3LYP-3C

Invokes the B3LYP-3c method

WB97X-3C

Invokes the \(\omega\)B97X-3c method