1. Introduction
Electronic structure calculations for realistic condensed-phase systems are generally more involved than those for molecules. The former include more atoms and are performed under periodic boundary conditions (PBC), implying interactions between periodic images. Therefore, condensed-phase electronic structure modelling often relies on simple approximations. Tight-binding approaches—semiempirical methods, density functional based tight-binding (DFTB)—used to be the work horse in the field. With increased computational power Kohn-Sham density functional theory (KS DFT) [
1] became a standard approach. Recently, implementations of wave function theories (WFT) became available, although their application is far from routine.
In DFT, energy is given as unique functional of electron density alone (Hohenberg-Kohn theorem) [
2]. Although the exact functional is unknown, several approximation levels are available, often classified as rungs of the Jacob’s ladder of accuracy [
3]. The most simple approximation includes only local information on the density (local density approximation, LDA) [
4,
5,
6,
7,
8]. More eleborate theories take more properties of the density into account. Including the density gradient yields generalized-gradient (GGA) approximations (LYP [
9], PBE functionals [
10]), whereas including the kinetic energy density gives meta-GGA functionals [
11,
12,
13] (e.g., TPSS functional [
14]).
Incorporating a portion of exact exchange (non-local) leads to hybrid functionals (e.g., PBE0 [
15], B3LYP [
16]). Exact exchange energy is not density-dependent, but is rather a non-local quantity (dependent on the density matrix) borrowed from WFT,
viz. from Hartree-Fock (HF) theory [
17,
18]. Hence, the term “hybrid” functional means including quantities from WFT, i.e., the Hartree-Fock exchange energy, into DFT functionals. Further examples of this approach are range-separated methods (HSE [
19], WB97X [
20]) and double-hybrid functionals, the latter can also involve range separation. Whereas hybrid functionals depend on the occupied KS orbitals, double-hybrid functionals include additionally virtual orbitals. They account for electron correlation in both DFT fashion via exchange-correlation functional and WFT fashion via excited determinants. We will refer to WFT methods which include correlation energy as Wave-function correlation (WFC) method.
Double-hybrid functionals [
21,
22,
23,
24,
25,
26] can potentially take “the best of the two worlds”. GGA-, meta-GGA- and hybrid DFT functionals are relatively fast and accurate for covalently and ionically bound systems. However, they intrinsically fail to describe long-range dispersion interactions (which is often coped with by explicit dispersion corrections [
27,
28,
29,
30,
31,
32] and non-local functionals [
33,
34]) and strong correlations. WFC methods, on the other hand, inherently include the correct asymptotic
behaviour. Their significant disadvantage is the high computational cost:
scaling and higher in the canonical formulations. Reduced-cost methods allow decreasing the scaling, although with high prefactors. Consequently, the cost of a double-hybrid DFT calculation is defined by the cost of its WFT part. The question may then arise: why not use pure WFT instead of double-hybrid functionals? The answer is that the
–
scaling of WFT methods used for double-hybrid functionals (second-order Møller-Plesset perturbation theory, MP2 [
35]; random phase approximation, RPA [
36]) are relatively crude approximations, and despite capturing long-range interactions they can be outperformed by DFT functionals. Thus, inclusion of electron correlation in WFT and DFT fashion may lead to the improved accuracy of both at moderate price as compared to highly precise WFT approaches, such as coupled-cluster methods [
37] scaling as
and higher.
Most condensed-phase implementations of electronic structure methods are based either on the use of plane waves (PW) or Gaussian basis sets. Plane waves constitute a basis in a strict mathematical sense: they are orthogonal and complete. In PW basis DFT and correlation energies converge systematically with basis size [
38]. However, due to the fact that PW do not reflect the character of chemical bonding, a larger number of basis functions is needed for accurate calculations, which is detrimental for calculations with WFC methods as virtual space becomes huge. Since atom-centered Gaussian functions reasonably approximate atomic orbitals, good accuracy can be achieved with compact basis sets, i.e., at a lower computational price, especially for DFT. WFC energies are more sensitive to basis set size and exhibit slow convergence with basis set size [
39], especially for long-range dispersion interactions [
40]. WFC methods and DHDFs are available for PW basis sets in VASP [
38,
41], for Gaussian basis sets in CP2K [
42], CRYSTAL [
43] and GAMESS (US) [
44] and for Slater type basis sets in ADF [
45].
2. Theoretical Background
In the following,
are virtual orbital indices,
occupied orbital indices,
general orbital indices, and
auxiliary function indices. In DFT, the total energy is given as a functional of the total ground-state density
:
where
is the total energy functional,
is the kinetic energy of a reference system of non-interacting electrons,
is the nuclei-electron interaction energy,
is the Hartree energy describing the classical electron-electron interaction energy, and
is the exchange-correlation energy describing the quantum mechanical contributions of the electron-electron interaction. The ground-state density is expressed in terms of orbital functions
where
i runs over all occupied orbitals. The orbital functions fulfill the orthonormality constraint:
with the Kronecker delta
. The orbitals are solutions of the Kohn-Sham (KS) equation:
with the potential arising from the nuclei
, the Hartree potential
, the exchange- correlation (XC) potential
, and the orbital energy
of orbital
i. In this article, we will consider Gaussian functions centered at the atoms only.
Because the total energy functional is not known explicitly in terms of the ground-state density, we rely on approximations of the XC functional. These approximate energy functionals are given as integrals of a function explicitly depending on the ground-state density, its gradient and its Laplacian. For convenience, the XC functional is split into an exchange functional and a correlation functional .
The more complex hybrid density functionals (HDFs) [
16] include explicit information of the occupied orbitals. They modify the exchange functional by including a certain amount
of Hartree-Fock (HF) exchange
providing the exchange functional
We introduce the amount
of DFT exchange
to reflect that the DFT exchange functional is an already known GGA or meta-GGA functional (compare [
15,
16]). Using the Mulliken notation (chemists’ notation) for electron repulsion integrals
the HF exchange energy can be written as
Non-HDFs suffer from self-interaction errors [
46]. These are reduced in HDFs but usually not fully cancelled since
in general case. This self-interaction error results in erroneous description of charge-separation processes and transition states. But even hybrid methods and HF lack a reasonable description of dispersion interactions decaying like
with
R being a measure of charge separation.
For increased flexibility, we can further split the exchange functional in a long-range and a short-range functional and describe both with a given mixture of HF theory and DFT resulting in range-separated HDFs [
19].
The highest flexibility is achieved by including virtual orbitals
. Double-hybrid density functionals (DHDFs) are HDFs in which the correlation functional is composed of a mixture of a DFT correlation functional
with ratio
and correlation energy
of a WFC method with ratio
providing a functional
Because WFC methods are computationally more demanding than HDFs or standard DFT functionals, most DHDFs exploit the MP2 theory, the SOS-MP2 theory, or the RPA method.
The correlation energy within the MP2 theory for closed-shell systems is
The computationally most expensive step of the MP2 method is given by the transformation of the electron interaction integrals from atom orbital basis to molecular orbital basis leading to a
scaling with
N being a measure of system size. The prefactor can be reduced by the resolution-of-the-identity (RI) approach introducing an auxiliary basis in which densities are expanded giving the equation
with
This method is called RI-MP2 [
47,
48].
A simplified version of the RI-MP2 method is the Scaled-Opposite-Spin(SOS)-MP2 method [
49] given by
with
The integration is carried out numerically using a Minimax quadrature. The RI-SOS-MP2 method scales like .
Another correlation method with increasing popularity is the Random Phase Approximation (RPA) method [
50,
51] within the RI approximation
with
RI-RPA scales like
. As with the RI-SOS-MP2 method, the integration is carried out numerically using a Clenshaw-Curtis grid [
52] or a Minimax grid [
53,
54].
All WFC methods and all DHDFs correctly reproduce the
energy behaviour of long-range interactions. Comparable to range-separated HDFs, there are DHDFs with range-separated exchange functionals like the
B97X-2 functional [
55]. Further, there are DHDFs with range-separated correlation functionals [
56]. In this article, we will not focus on DHDFs with range-separated correlation functionals and refer to the literature [
40,
57,
58,
59,
60] for more details.
4. Results
4.1. General Remarks
We found the convergence of total energies of meta-HDFs PW6B95 and PWRB95 requires very tight energy cutoffs for the auxiliary PW basis of at least 4000 Ry. In contrast to that, calculations with the other meta-HDFs in our benchmark study, B97M-V, provided reasonable results with a cutoff of only 1200 Ry. Because the basis functions for the elements argon and neon are more localized than those for hydrogen, carbon and nitrogen, higher cutoffs for the noble gases were needed for an adequate representation of the basis functions of these elements on the grid.
It is well-known that GGA functionals and especially meta-GGA functionals require very tight integration grids for convergence and thus accurate results. Such cutoffs reflect numerical issues and the need for very fine integration grids when using the PW6B95 and PWRB95 functionals. Such grids are not necessary for the
B97M-V functional which has been optimized with coarser integration grids in mind [
72]. Thus, energy differences converged faster with
B97M-V and PBE. Nevertheless, the total energies were not converged. To remove any possible problems due to incomplete convergence with respect to cutoffs, we utilized unusually high cutoffs for all density functionals.
Furthermore, we have found convergence problems with the PW6B95 and PWRB95 functionals, which can be resolved with density smoothing. Unfortunately, in some cases an increase of the energy cutoff for the density resulted in SCF convergence issues which could not be resolved with tighter filter thresholds. Nevertheless, we were able to achieve convergence by restarting the calculations with a higher cutoff starting from the converged SCF results with a lower cutoff. This was not possible for argon, where we exploited a cutoff of 4000 Ry for the PW6B95 and PWRB95 functional. Thus, some numbers for the PW6B95 and PWRB95 functionals are not fully converged with respect to the density cutoff.
Due to the higher computational costs, we have not carried out calculations of the supercells on the TZ level.
4.2. Convergence with Respect to Super Cell Size
In
Figure 1, we compiled the differences in total energies per formula unit relative to the extrapolated total energies. In general, we expect the total energies to decrease with increasing supercell size and the extrapolated value is a lower bound for the total energies of the super cells. Our results show exactly this behaviour.
An important question is for which supercell size the error becomes negligible. A useful magnitude is given by the chemical accuracy of 4 kJ·mol
−1. For weakly-interacting systems such as rare-gas crystals with cohesive energy of less than chemical accuracy, the order of magnitude is set by the cohesive energy itself. As the error of a method should be not larger than chemical accuracy, the allowed error of the supercell method must be at least one order of magnitude smaller then the methodological error, i.e., not larger than 0.4 kJ·mol
−1. We find that a
super cell provides sufficient accuracy for all functionals and test systems. This behaviour is in agreement with the literature [
80]. Sometimes, the total energy per formula unit of the
super cell has a higher magnitude than this of the
supercell, which may be due to numerical issues. For PBE, a cubic fit does not seem to be appropriate, and an exponential fit should be used instead.
4.3. Convergence of the BSSE
The BSSEs for the different test systems are compiled in
Figure 2. First, we would like to point out that the BSSE is significantly larger for the molecular crystals than for the rare-gas crystals. This might be related to the larger number of atoms per molecule and to the spread of the basis functions. Since the effective core charge of rare-gas atoms is larger than for carbon or nitrogen, the basis functions are more localized which results in weaker overlap with neighbouring atoms. This is supported by the smaller reduction in BSSE for Ar and Ne when we exploit larger basis sets. Thus, augmentation of basis sets must significantly reduce BSSEs of Ar and Ne. Indeed, diffuse basis functions actually improve cohesive energies as shown by Sansone et al. [
63].
Molecular crystals are thus more suitable objects to study BSSE than rare-gas crystals. For both molecular crystals in the test set, the non-DHDFs PBE, PW6B95 and B97M-V, provide the smallest BSSEs whereas the two WFC methods MP2 and RPA have the largest BSSEs, as expected. The DHDFs have a BSSE between both classes of methods because they employ a mixture of DFT and WFC methods.
Furthermore, we note that the WFC methods in CP2K are implemented within the RI approximation employing an auxiliary basis set. This leads to an additional source of BSSE for RPA, MP2 and all the DHDFs because the addition of the auxiliary functions of the ghost atoms increases the overall accuracy.
4.4. Convergence with Respect to Basis Set Size
In numerous studies, it was shown that total energies from DFT calculations converge exponentially with respect to basis set size. In contrast to that, total energies from WFC methods converge cubically with respect to basis set size when employing correlation-consistent basis sets. Thus, most DHDFs are expected to have a cubic convergence with respect to basis set size but with a smaller prefactor. DHDFs employing a long-ranged Coulomb operator only and describing short-ranged interactions with a density functional, converge exponentially [
40]. This behaviour is confirmed with our data compiled in
Figure 3.
Since larger basis sets systematically reduce total energies, cohesive energies increase. We observe this behaviour for the WFC methods and almost all DHDFs. The slight difference for PWRB95 in case of Ar may be due to not full convergence with respect to super cell size. For the other functionals—PBE, B97M-V and PW6B95—the cohesive energy from the TZ basis set is sometimes higher, i.e., the system is weaker bound. One problem might be that the super cells are not yet fully converged or the extrapolation scheme using a linear fit of the total energies versus the inverse of the volume is not appropriate and an exponential fit might be more suitable.
Next, we would like to discuss the results obtained for the molecular crystals NH
3 and HCN. They are bound together by covalent bonds, dipole-dipole interactions, and dispersion interactions. For both systems, the results with the RPA and MP2 methods significantly improve the results over GGA DFT functionals, MP2 even achieving chemical accuracy. The
B97M-V functional also provides very accurate numbers. The PW6B95 functional, as PBE, systematically underestimates the cohesive energies with errors compatible to PBE. The PWRB95 functional significantly improves upon the results of its relative PW6B95, bringing them within 1 kJ·mol
−1 from the experiment. The same holds for the
B97X-2 functional compared with the
B97M-V functional, although the DHDF is based on the non-meta-GGA HDF
B97X [
86]. One of the worst performing functionals is SOS-PBE0-2.
For the rare-gas crystals, the picture is more complicated because the absolute values of the cohesive energies are of the order of the chemical accuracy. As pointed out by Sansone et al. [
63], augmented basis sets are required for these systems. Our cohesive energies from MP2 with a TZ basis set are only slightly lower than their result with a DZ basis set but still much worse than those with an augmented basis set for both Ne and Ar. Consequently, our results for Ne do not allow for an evaluation of the performance of these functionals and further studies employing either quadruple or augmented basis sets (which are to be constructed) are needed. Nevertheless, our results for Ar show that the
B97X-2 functional provides a good description. The same holds for MP2, PW6B95 and PWRB95, although one needs further investigations with augmented basis sets.
This issue does not apply to the molecular crystals. Indeed, our cohesive energies with a TZ basis set are even lower than those with an augmented basis set. Thus, the use of augmented basis sets is not necessary for the molecular crystals. This result is important for reducing computational costs of HF calculations and low-scaling WFC methods.
5. Discussion
Because DHDFs can be considered to be a mixture of DFT and WFC methods, the flexibility of DHDF parametrizations can yield approaches more accurate than the parent DFT and WFC functionals. At the same time, they inherit the shortcomings of both classes. Due to the dependence on the grid parameters, the functionals PW6B95 and PWRB95 are more difficult to use than others: care must be taken to check whether the results are converged with respect to the grid parameters, in CP2K, the density cutoff.
As expected, PBE can only provide the order of magnitude for weakly interacting systems, although it converges fast with respect to basis set size and has a low BSSE. MP2 and RPA are more sensitive to the basis set size and exhibit large BSSEs. These methods provide a moderate accuracy for different systems with small basis sets.
Non-DHDFs benefit from lower BSSEs. The PW6B95 functional has high demands on integration grids. Both considered functionals also provide a moderate accuracy and should be favourable over MP2 and RPA with their higher computational costs.
The double-hybrid functionals PWRB95 and B97X-2 show excellent performance with moderate BSSEs and lower basis set incompleteness errors. Both have computational costs compatible to full MP2 or RPA calculations and inherit the need of fine integration grids for accurate results, especially for PWRB95.
The non-empirical SOS-MP2 based DHDF, SOS-PBE0-2, does not provide any advantage as compared to the original methods. It was pointed out by different authors [
87,
88] that non-empirical DHDFs usually perform worse than empirical DHDFs.