Semilocal density functional obeying a strongly tightened bound for exchange

Significance

Efficient calculation of the properties of atoms, molecules, and solids on the computer requires a semilocal approximation to the density functional for the exchange-correlation energy, which becomes thereby a single integral over three-dimensional space. A recent, strongly tightened lower bound on the exchange energy has been built into the approximation “meta-generalized gradient approximations made very simple,” or MGGA-MVS, with accurate results for heats of formation, energy barriers, and weak interactions of molecules, and for lattice constants of solids. This would not have been possible without the use of a third ingredient (the local kinetic energy density) in addition to the standard two (the local electron density and its gradient). This third ingredient permits accurate energies even with the drastically tightened bound.

Abstract

Because of its useful accuracy and efficiency, density functional theory (DFT) is one of the most widely used electronic structure theories in physics, materials science, and chemistry. Only the exchange-correlation energy is unknown, and needs to be approximated in practice. Exact constraints provide useful information about this functional. The local spin-density approximation (LSDA) was the first constraint-based density functional. The Lieb–Oxford lower bound on the exchange-correlation energy for any density is another constraint that plays an important role in the development of generalized gradient approximations (GGAs) and meta-GGAs. Recently, a strongly and optimally tightened lower bound on the exchange energy was proved for one- and two-electron densities, and conjectured for all densities. In this article, we present a realistic “meta-GGA made very simple” (MGGA-MVS) for exchange that respects this optimal bound, which no previous beyond-LSDA approximation satisfies. This constraint might have been expected to worsen predicted thermochemical properties, but in fact they are improved over those of the Perdew–Burke–Ernzerhof GGA, which has nearly the same correlation part. MVS exchange is however radically different from that of other GGAs and meta-GGAs. Its exchange enhancement factor has a very strong dependence upon the orbital kinetic energy density, which permits accurate energies even with the drastically tightened bound. When this nonempirical MVS meta-GGA is hybridized with 25% of exact exchange, the resulting global hybrid gives excellent predictions for atomization energies, reaction barriers, and weak interactions of molecules.

Density functional theory (DFT) is a powerful tool for predicting structural, energetic, electronic, and magnetic properties of molecules, nanoparticles, surfaces, and solids in quantum chemistry, condensed matter physics, and materials science (1–3). It delivers in principle the exact ground-state energy and electron density for a system of electrons under an external potential, via self-consistent solution of auxiliary one-electron Schrödinger equations, the so-called Kohn–Sham equations. In practice, the density functional for the exchange-correlation energy, a typically small fraction of the total energy but a large fraction of the binding energy between atoms, has to be approximated. Because of their inexpensive computational cost and reasonable accuracy, semilocal approximations, which approximate the exact double integral for this energy with a single integral over 3D space, are widely used for a variety of properties, especially for large systems. They are also the bases for nonlocal corrections (4–6) that account for the nonlocality missing in the single integral. Nonlocal corrections are important for some systems and properties, e.g., strong correlation (4) and the long-range van der Waals interaction (6).

Exact constraints on the exchange-correlation functional play essential roles in the development of semilocal functionals. The local spin-density approximation (LSDA) (1) was derived from the uniform electron gas limit (4, 7, 8). It predicts reasonable lattice constants and many other properties for solids, but yields unacceptably high atomization energies for molecules and thus is not used much for chemistry. To remedy the overestimation of atomization energy, the electron density gradient is introduced in the generalized gradient approximation (GGA) to account for the inhomogeneity of electron densities, and most GGAs respect the second-order gradient expansion for slowly varying densities (1, 9, 10) to some extent. The widely used Perdew–Burke–Ernzerhof (PBE) GGA (11) regularizes the second-order gradient expansion with the Lieb–Oxford bound (12) and other constraints. PBE yields reasonably accurate atomization energies for molecules, while providing generally at least comparable (and better in most cases) predictions of properties for solids in comparison with LSDA. However, GGAs are still not flexible enough to satisfy simultaneously some different constraints relevant for different properties. For example, to have good energetic predictions, the PBE GGA has a coefficient for the second-order gradient expansion of the exchange energy for slowly varying densities that is about twice the first-principles one (10). This choice of the coefficient was initially designed to reproduce the good LSDA linear response of the uniform electron gas (11), and it provides a decent fit to the results of the real-space cutoff construction of the Perdew–Wang 1991 GGA for exchange (13, 14). It is also justified by the large-Z asymptotic expansion of the exchange energy of neutral atoms with atomic number Z (15–17). However, PBE usually generates too-long lattice constants for solids, with about equal but opposite error compared with LSDA. By restoring the first-principles coefficient, the PBEsol GGA (18) significantly improves the lattice constants over PBE, but unfortunately it strongly degrades the atomization energies of molecules and dramatically violates the asymptotic expansion of the exchange energy of neutral atoms.

Perdew (19) used the Lieb–Oxford lower bound on the indirect Coulomb energy (12) to derive looser lower bounds on the exchange-correlation (20–22) and exchange energies; the exchange-energy bound was later used in the construction of PBE (11) and other density functionals. Recently, a strongly and optimally tightened bound on the exchange energy was proved for one- and two-electron densities, and conjectured for all densities (23). Whereas LSDA automatically satisfies it, all GGAs strongly violate this constraint and have to violate it for good energetic predictions (23). However, we present here a realistic meta-GGA that, to our knowledge, respects this constraint for the first time, and simultaneously satisfies many other constraints, including the second-order gradient expansion with the first-principles coefficient and the large-Z asymptotic expansion of the exchange energy for neutral atoms.

Meta-GGAs are computationally semilocal and thus efficient, but they build the integrand of the single integral for the exchange-correlation energy from the kinetic energy density $\tau$ of the Kohn–Sham orbitals, a fully nonlocal functional of the electron density, and not just from the electron density $n$ and its gradient $\nabla n$.

Some useful functionals have been developed with only minimal satisfaction of exact constraints, e.g., the successful Minnesota functionals (24, 25). They usually gain predictive power by being trained on a large set of experimental and/or high-level theoretical data. Without relying much on exact constraints, these functionals can be exploratory. For example, the M06L meta-GGA is, to our knowledge, the first semilocal functional that captures the intermediate-range van der Waals interaction (24, 25). However, to describe different properties contained in the data set, heavy parameterizations of the functionals are usually inevitable, causing problems like numerical stability and lack of transferability to systems outside of the training data set, as also seen in M06L (24).

The ability of M06L to describe the intermediate-range van der Waals interaction is due to the inclusion of the kinetic energy density, which helps identify weak noncovalent bonds (26). Our recent work (27, 28) further showed that a meta-GGA can recognize different chemical bonds (covalent single, metallic, and weak) by choosing the right dimensionless ingredient

$$\alpha =\frac{\tau -{\tau }^{\text{W}}}{{\tau }^{\text{unif}}},$$ [1]

the dimensionless deviation from a single orbital shape (23). Here,

$$\tau =2{\displaystyle \sum _{\text{i}}^{\text{occup}}\frac{1}{2}{\left|\nabla {\phi }_{\text{i}}\right|}^2}$$ [2]

is the positive kinetic energy density of the occupied Kohn–Sham orbitals ${\phi }_{\text{i}}$, which reduces to the Weizsäcker kinetic energy density ${\tau }^{\text{W}}={\left|\nabla n\right|}^2/8n$ for a two-electron singlet density ($\alpha =0$) and to ${\tau }^{\text{unif}}=\left(3/10\right){\left(3{\pi }^2\right)}^{2/3}{n}^{5/3}$ for a uniform electron gas (UEG) ($\alpha =1$), and $n=2{\displaystyle {\sum }_{\text{i}}^{\text{occup}}{\left|{\phi }_{\text{i}}\right|}^2}$ is the electron density.

The semilocal exchange energy of a spin-unpolarized density with the right dimensionless ingredients can be written as

$$E_{\text{x}}\left[n\right]={\displaystyle \int n{\epsilon }_{\text{x}}^{\text{unif}}\left(n\right)F_{\text{x}}\left(s,\alpha \right)d^{3}r}.$$ [3]

For a spin-polarized density, we can use the exact spin-scaling relation (29). Here,

$$s=\frac{\left|\nabla n\right|}{2{\left(3{\pi }^2\right)}^{1/3}{n}^{4/3}}$$ [4]

is the reduced density gradient and ${\epsilon }_{\text{x}}^{\text{unif}}\left(n\right)=-\left(3/4\pi \right){\left(9\pi /4\right)}^{1/3}/r_{\text{s}}$ is the exchange energy per particle of a UEG with the Seitz radius $r_{\text{s}}={\left(4\pi n/3\right)}^{-\left(1/3\right)}$. $F_{\text{x}}(s,\alpha )$ is the exchange enhancement factor, which reduces to 1 for a UEG. For GGAs, $F_{\text{x}}$ is only dependent on $s$, a measure of the inhomogeneity of the density. As shown in Fig. 1, the GGA enhancement factors have to recover the second-order gradient expansion, i.e., $F_{\text{x}}=1+\mu s^2$, for small $s$. The first-principles coefficient derived by Antoniewicz and Kleinman (10) for densities that vary slowly over space is ${\mu }_{\text{GE}}=10/81=0.12346$. However, ${\mu }_{\text{PBE}}=0.21951$ was used in PBE to recover the good LSDA linear response of the UEG (11). It was shown recently that the coefficient ${\mu }_{\text{AE}}$ has to be about twice ${\mu }_{\text{GE}}$(${\mu }_{\text{AE}}=2.109{\mu }_{\text{GE}}$) if a GGA is required to satisfy the large-Z asymptotic expansion of the exchange energy of neutral atoms (14, 15). The B88 GGA (30), which is popular in chemistry, has ${\mu }_{\text{B}88}=2.208{\mu }_{\text{GE}}$, very close to ${\mu }_{\text{AE}}$. However, it is argued that ${\mu }_{\text{GE}}$ is more relevant and has to be restored for solids, as in the PBEsol GGA (18). Fig. 1 also shows that the enhancement factors of GGAs monotonically increase with $s$ to values much larger than 1 (e.g., 1.804 for PBE) for $s\gg$ 1, to better describe atoms and molecules which are far from having slowly varying densities (11, 27). With their choices of the gradient coefficient, PBE and B88 have much stronger dependences on $s$ than PBEsol does. Table 1 shows that PBE yields better exchange energies of rare-gas atoms than PBEsol, whereas B88 is the best among them for that property.

Fig. 1.

Exchange enhancement factor $F_{\text{x}}\hspace{0.17em}\text{vs}.\hspace{0.17em}s$ of LDA and GGAs.

Table 1.

Errors ($E_{\text{h}}$) in exchange energies of rare-gas atoms with different functionals

Atom	Exact	LDA	PBEsol	PBE	B88	MVS
He	−1.026	0.142	0.058	0.012	0.001	−0.005
Ne	−12.108	1.074	0.443	0.041	−0.03	−0.013
Ar	−30.188	2.325	1.045	0.192	0.04	0.061
Kr	−93.890	5.266	2.444	0.465	0.02	−0.002
Xe	−179.200	8.638	4.158	0.960	0.2	0.028
MAE		3.489	1.630	0.334	0.058	0.022

MAE is the mean absolute error. The considered functionals are LDA (7), PBEsol GGA (18), PBE GGA (11), B88 GGA (30), and MVS meta-GGA (this work).

The other variable of $F_{\text{x}}\left(s,\alpha \right)$ in meta-GGAs is $\alpha$. For two-electron systems, where $\alpha =0$, it has been proved (23) that an optimal lower bound on the exchange energy will be satisfied for all possible densities if and only if

$$F_{\text{x}}(s,\alpha =0)\le 1.174,$$ [5]

and it is conjectured that

$$F_{\text{x}}(s,\alpha )\le F_{\text{x}}(s,\alpha =0)\le \mathrm{1.174.}$$ [6]

Eq. 6 is much tighter than the older bound $F_{\text{x}}\le 1.804$ (11, 19). At the GGA level, the tight bound of Eq. 6 is not compatible with any second-order gradient expansion [$F_{\text{x}}\left(\text{s}\right)=1+\mu s^2+\cdots$], because the enhancement factor would be too small in comparison with those of PBE and B88 for a large range of $s$ as shown in Fig. 1. A GGA respecting the tight bound would not be useful for atoms and molecules, even less so than PBEsol (18). However, at the meta-GGA level, the restriction on the s dependence can be compensated by the freedom brought by the $\alpha$-dependence, as already demonstrated by the meta-GGA made simple (27) (MGGA_MS), which has $F_{\text{x}}^{\text{MGGA}\_\text{MS}}\le 1.29$, a value rather close to 1.174.

Meta-Generalized Gradient Approximation Made Very Simple

Here we propose a meta-GGA enhancement factor that respects the tight bound and other constraints, including the second-order gradient expansion for the slowly varying density with the first-principles coefficient and the asymptotic expansion of the exchange energy of neutral atoms. It can be written as

$$F_{\text{x}}\left(s,\alpha \right)=\frac{h_{\text{x}}^1+f_{\text{x}}\left(\alpha \right)[h_{\text{x}}^0-h_{\text{x}}^1]}{{(1+b{s}^4)}^{1/8}},$$ [7]

where

$$h_{\text{x}}^0=1+k_0,$$ [8]

with $k_0=0.174$ for the tight lower bound ($\alpha =0$) and

$$h_{\text{x}}^1=1$$ [9]

for the UEG limit ($\alpha =1$). The function

$$f_{\text{x}}\left(\alpha \right)=\frac{\left(1-\alpha \right)}{{[{\left(1+e_1{\alpha }^2\right)}^2+c_1{\alpha }^4]}^{1/4}}$$ [10]

interpolates between them and extrapolates to a constant as $\alpha \to \infty$. The denominator of $F_{\text{x}}\left(s,\alpha \right)$ in Eq. 7 is used to satisfy the constraint

$$\underset{s\to \infty }{\lim }{F}_{\text{x}}\left(s,\alpha \right)\propto s^{-\left(1/2\right)},$$ [11]

which is required to yield a finite nonzero exchange energy per electron for the nonuniform scaling limit of a density (23, 31); see also refs. 13, 32–36. For single-orbital systems, $F_{\text{x}}\left(s,\alpha =0\right)=(1+k_0)/{(1+b{s}^4)}^{1/8}$, where $b$ = 0.0233 is fitted to the exchange energy of the hydrogen atom.

Remarkably, we can recover the second-order gradient expansion for the exchange energy of a slowly varying density, using only $\alpha$ and not $s$. For slowly varying densities, with $s\approx 0$ and $\alpha \approx 1$, $\alpha$ can be represented by its second-order gradient expansion (37, 38). Then, under integration by parts,

$${\displaystyle \int d^{3}r{\epsilon }_{\mathrm{x}}^{\mathrm{unif}}\left(n\right)(1-\alpha )}={\displaystyle \int d^{3}r{\epsilon }_{\mathrm{x}}^{\mathrm{unif}}\left(n\right)\left(\frac{20}{27}{s}^2\right)}.$$ [12]

To recover the second-order gradient expansion, we expand Eq. 3 (with Eqs. 7–10) to first order in $1-\alpha$ about $\alpha =1$, integrate by parts, equate the coefficient of $s^2$ to ${\mu }_{\text{GE}}$, and find $c_1={\left(20k_0/27{\mu }_{\text{GE}}\right)}^4-{\left(1+e_1\right)}^2$.

Finally, $e_1=-1.6665$ is fixed by the large-Z asymptotic expansion of the exchange energy for neutral atoms following the procedure of ref. 16. For each value of $e_1$, we compute the exchange energies for the rare-gas series Z = 10, 18, 36, 54, then find the coefficient of the term of order Z in equation 13 of ref. 16. We select the value of $e_1$ that yields the correct coefficient. Then $c_1=0.7438$ and $F_{\text{x}}\left(s=0,\alpha \to \infty \right)=0.873$. We call this form “meta-GGA made very simple (MVS),” because it is simpler than previous meta-GGAs.

Results and Discussion

Except for the exchange energies of rare gas atoms in Table 1, where the density functional results were based on the Hartree-Fock densities (39), all other density functional results in this paper are self-consistent. For meta-GGAs, we achieve self-consistency by the procedure of Neumann et al. (40).

Table 1 shows that MVS delivers the best exchange energies of rare-gas atoms among the considered functionals, even better than B88. This is impossible for a GGA that respects the tight lower bound and any second-order gradient expansion [$F_{\text{x}}\left(\text{s}\right)=1+\mu s^2+\cdots$] for slowly varying densities, demonstrating that the tight lower bound is not too tight at the meta-GGA level. Fig. 2A shows that $F_{\text{x}}$ of MVS is smaller than 1.174, varies little with $s$ for the range of $0<s<2$, and monotonically decreases with $s$. It is qualitatively different from that of PBE and other GGAs. Atoms have important regions with $\alpha \approx 0$ and $s\le 1$, where MVS shows more exchange enhancement than any GGA, and this fact explains how MVS achieves accurate exchange energies for atoms. On the other hand, Fig. 2B shows that the $F_{\text{x}}$ of MVS monotonically and strongly decreases with $\alpha$, consistent with the finding of MGGA-MS (27).

Fig. 2.

Exchange enhancement factor $F_{\text{x}}$ of PBE and MVS. (A) $F_{\text{x}}\hspace{0.17em}\text{vs}.\hspace{0.17em}s$; (B) $F_{\text{x}}\hspace{0.17em}\text{vs}.\hspace{0.17em}\alpha$.

It is interesting to note that MVS is weakly dependent on $s$ and thus almost local for single-orbital systems. This corroborates the idea of a “generalized local density approximation” for real (confined) two-electron densities, proposed by Loos et al. when they studied two electrons in a finite 3D curved space (41, 42). However, our work on meta-GGAs (27), including the current work, suggests that a generalized local density approximation, i.e., a meta-GGA without a dependence upon $\nabla n$, cannot be highly accurate for most real molecules and solids.

Now let us turn to molecules and solids. We combine the MVS exchange with the modified PBE GGA correlation, which is also used as the correlation part of the MGGA-MS meta-GGA (27). Table 2 compares the resulting meta-GGA (also denoted as MVS) against the PBEsol and PBE GGAs for heats of formation of the G3 set (43), barrier heights of the BH76 set (44, 45), weak interactions of the S22 set (46), bond lengths of the T-96R set (47), and lattice constants of the LC20 set (48), where the first four are molecular test sets and the last is a solid-state test set. For computational details, see refs. 47–49. It can be seen that MVS matches PBE for the heats of formation and matches PBEsol for the bond lengths and lattice constants, as expected from consideration of the satisfied constraints. Note that MVS is much better than PBE for the barrier heights and the weak interactions, with a remarkable mean absolute error (MAE) of 0.8 kcal/mol for the S22 set. The performance of MVS for the G3 heats of formation is not especially impressive compared with other meta-GGAs (24, 27, 35, 37, 49) and even some GGAs [e.g., the PBE-large s (LS) GGA (36) shown in Table 2], but paradoxically its PBE-like overbinding of atoms in molecules opens the door to a successful MVS hybrid functional to be discussed later in this section.

Table 2.

Errors of different density functionals on heats of formation (G3) (43), barrier heights (BH76) (44, 45), weak interactions (S22) (46), and bond lengths of molecules (T-96R) (47), and on lattice constants of solids (LC20) (48)

Functional	G3 (kcal/mol)		BH76 (kcal/mol)		S22 (kcal/mol)		T-96R ($\mathrm{Å}$)		LC20 ($\mathrm{Å}$)
	ME	MAE	ME	MAE	ME	MAE	ME	MAE	ME	MAE
PBEsol	−58.7	58.8	−11.5	11.5	−1.3	1.8	0.010	0.013	−0.012	0.036
PBE	−21.6	22.2	−9.1	9.2	−2.8	2.8	0.015	0.016	0.051	0.059
PBE-LS	1.6	9.5	−7.5	7.6	−4.0	4.0	0.019	0.019	0.096	0.096
TPSS	−6.0	6.5	−8.6	8.7	−3.7	3.7	0.014	0.014	0.035	0.043
MGGA_MS	−2.4	8.5	−6.0	6.3	−1.7	1.7	0.005	0.008	0.016	0.023
MVS	−19.2	20.0	−4.4	4.6	−0.6	0.8	−0.002	0.013	−0.018	0.038
B3LYP	3.3	4.8	−4.6	4.7	−4.0	4.0	0.005	0.010	—	—
M06	−3.7	4.5	−2.0	2.3	−1.0	1.0	−0.003	0.011	—	—
PBEh	−4.9	6.8	−4.0	4.2	−2.5	2.5	−0.001	0.010	—	—
MVSh	−2.7	7.1	−0.7	1.9	−0.8	1.0	−0.011	0.020	—	—

Note that the errors of the heats of formation are by construction essentially minus those of the atomization energies. ME: mean error; MAE: mean absolute error. The considered functionals are PBEsol GGA (18), PBE GGA (11), PBE-LS GGA (36), TPSS meta-GGA (37), MGGA_MS meta-GGA (27), MVS meta-GGA (this work), B3LYP hybrid GGA (30, 51, 52), M06 hybrid meta-GGA (24), PBEh hybrid GGA (53–55), and MVSh hybrid meta-GGA (this work). For computational details, see refs. 47–49. Note that only MVS and MVSh satisfy the tight lower bound on the exchange energy for all possible electron densities (23), including all possible two-electron densities for which the tight bound is rigorously derived.

A local exchange approximation with $F_{\text{x}}=1$ overbinds atoms in molecules and weakly bound complexes, and one with $F_{\text{x}}=1.174$ overbinds even more. We will now explain how this overbinding can be corrected either by an $F_{\text{x}}$ that starts from 1 at $s=0$ and increases with increasing $s$, as in the PBE GGA, or by an $F_{\text{x}}$ that starts from around 1.174 at $\alpha =0$ and decreases with increasing $\alpha$, as in the MVS meta-GGA. The key fact is that free atoms tend to have regions in which $s$ is larger and $\alpha$ is smaller than in the corresponding molecule or complex. So, passing from $F_{\text{x}}=1$ to the PBE GGA lowers the energies of the free atoms relative to the corresponding molecule or complex, and passing from $F_{\text{x}}=1.174$ to the MVS meta-GGA raises the energy of the molecule relative to its constituent atoms. This confirms that the $s$- and $\alpha$-dependences of $F_{\text{x}}$ can complement each other: the $s$ dependence can be reduced if the $\alpha$-dependence is enhanced (27).

Table 2 illustrates the energy–geometry dilemma of GGAs (50): As the GGA atomization energies or G3 heats of formation improve from PBEsol to PBE to PBE-LS, the bond lengths and lattice constants lengthen and worsen.

One way of adding nonlocal corrections is to hybridize a semilocal approximation with the exact exchange functional. The Becke-3–Lee–Yang–Parr (B3LYP) hybrid GGA (5, 30, 51, 52), which minimally fits to experimental data, is the most popular functional in chemistry, whereas the PBEh hybrid GGA (53–55) is a nonempirical functional that has 25% of the exact exchange functional. The M06 hybrid meta-GGA (25) with a heavy parameterization is also widely used in chemistry.

Inspired by the success of the PBEh hybrid GGA, we hybridize MVS with 25% of the exact exchange. The mixing fraction 25% is not an optimized parameter, but is motivated by the similar performance of MVS and PBE on the G3 set. No exact constraint is lost when MVS exchange is replaced by 75% of MVS exchange plus 25% of exact exchange. As shown for the molecule sets in Table 2, the resulting hybrid meta-GGA (denoted as MVSh) predicts reasonably accurate bond lengths for the T-96R set, whereas its performance for the energetic properties is remarkable, considering that the mean energies of the G3, BH76, and S22 sets are ∼500 kcal/mol, ∼50 kcal/mol, and ∼10 kcal/mol, respectively. MVSh is comparable to the other hybrid functionals for the heats of formation, but much better than B3LYP and PBEh (and slightly better than M06) for the barrier heights and weak interactions. Note that MVSh achieves this accuracy without fitting to any molecular data sets. It is, to our knowledge, the first hybrid nonlocal functional based on nonempirical meta-GGAs that allows so much (25%) exact exchange mixing (56). The nonempirical Tao–Perdew–Staroverov–Scuseria (TPSS) (37, 38) and revised TPSS can only allow about 10% exact exchange mixing.

In this article, we have constructed a realistic meta-GGA exchange functional (MVS) that satisfies the tight lower bound of Eq. 6 for the first time, to our knowledge, beyond LSDA. It also respects other constraints, including the second-order gradient expansion for slowly varying densities with the first-principles coefficient and the large-Z asymptotic expansion of the neutral-atom energy, which no realistic GGA can simultaneously satisfy along with the tight bound. MVS exchange is radically different from that of other GGAs and meta-GGAs: The MVS exchange enhancement factor decreases monotonically with both increasing $s$ and increasing $\alpha$. When MGGA-MVS exchange is combined with modified PBE correlation, the resulting meta-GGA exchange-correlation functional is reasonably accurate for molecules and solids. After further hybridizing the meta-GGA exchange-correlation functional with 25% exact exchange, the resulting hybrid meta-GGA MVSh has an excellent accuracy for molecules, especially for the barrier heights (within 2 kcal/mol) and the weak interactions (within 1 kcal/mol). It will be interesting to perform solid-state tests of MVSh, and subsequently its range-separated version, although this will require further code development. The proof-of-principle in the present article encourages us to try to develop a meta-GGA for exchange and correlation that respects all possible known constraints and is more accurate than MVS at the stand-alone meta-GGA level.