Accuracy and Scaling Factors of Non‐Empirical Double‐Hybrid Density Functionals for Harmonic and Fundamental Frequencies (And ZPVE)

ABSTRACT

We obtain here scaling factors for harmonic and fundamental vibrational frequencies, as well as for Zero‐Point Vibrational Energies (ZPVE), using the HFREQ2014 dataset of reference values. We consider a set of (non‐empirical) double‐hybrid density functionals, also including spin‐scaled and range‐separated versions, namely PBE‐QIDH, RSX‐PBE‐QIDH, SOS1‐PBE‐QIDH, and SOS1‐RSX‐PBE‐QIDH. We also analyze the dependence of the results with respect to the size of def‐nVP(PD) basis sets of increasing angular momentum, with the def2‐TZVPD basis set providing the best trade‐off between accuracy and computational cost for all the functionals considered. Actually, (SOS1‐)PBE‐QIDH double‐hybrid functionals are able to provide average errors as low as $20 {\text{cm}}^{-1}$ for both types of frequencies, and lower than 0.1 kcal/mol for ZPVE, considerably better than the corresponding range‐separated (spin‐scaled or not) versions (SOS1‐)RSX‐PBE‐QIDH. Therefore, accurate thermochemical calculations or vibronically‐resolved spectra will expectedly benefit from the scaling factors derived here and the assessment of double‐hybrid density functionals done along this work.

Scaled factors for non‐empirical double‐hybrid density functionals (i.e., the QIDH family of models) provide average errors of $20 {\text{cm}}^{-1}$ for ahrmonic and fundamental frequencies and 0.1 kcal/mol for ZPVE.

1. Introduction

The search of the best possible accuracy for quantum‐chemical calculations, taken into account the bottleneck imposed by the scaling of highly correlated methods with the system size, has always inspired the development of composite or extrapolated methods [1], such as W4 [2], HEAT [3], FPA [4], FPD [5], Gn theories [6], ccCA [7], to name just a few of them, which are able to deliver sub‐kcal ${\text{mol}}^{-1}$ or even sub‐kJ ${\text{mol}}^{-1}$ accuracy for example, thermochemical calculations. A common aspect of all the existing procedures is the use of high‐level wavefunction‐based methods, together with several corrections, namely core‐valence, relativistic, spin‐orbit, non‐Born‐Oppenheimer, and Zero‐Point Vibrational Energy (ZPVE), as well as extrapolation/acceleration of the convergence of the results with respect to the basis set size. This field has also substantially benefited from the increasing capacity of modern electronic structure calculation methods to reach such an accuracy for larger and larger systems, thanks also to recent developments employing localized orbitals, such as the Domain Local Pair Natural Orbitals (DLPNO) technique [8], or more in general PNO approximations [9], which can also be applied to gradients [10] and all kinds of thermochemical and kinetics calculations [11].

Generally speaking, the accuracy of ZPVE is a critical factor affecting the final quality of the results, among all the corrections listed before, and its importance can not be underestimated. Furthermore, ZPVE errors increase with the system size (i.e., it can reach 100–200 kcal ${\text{mol}}^{-1}$ for medium‐sized organic compounds [12, 13]) and a small error (i.e., just a 1% error, or in other words 1–2 kcal/mol) in its determination might thus compromise the whole strategy in the pursuit of that sub‐kcal ${\text{mol}}^{-1}$ or even sub‐kJ ${\text{mol}}^{-1}$ accuracy for systems composed up to hundreds of atoms [14, 15] or with heavy main group elements [16]. Therefore, obtaining ZPVE values with that degree of sub‐kcal ${\text{mol}}^{-1}$ or sub‐kJ ${\text{mol}}^{-1}$ accuracy requires the consideration of vibrational anharmonicity effects [17]. The importance of this factor can be better visualized if the ZPVE is expanded following the expression casted by Barone [18] or Schaefer et al. [19]:

$$\begin{array}{c}\begin{array}{ll}\text{ZPVE}\hfill & =\frac{1}{2}\sum _i{\tilde{\omega }}_i-\frac{1}{32}\sum _{ijk}\frac{{\lambda }_{iik}{\lambda }_{jjk}}{{\omega }_k}-\frac{1}{48}\sum _{ijk}\frac{{\lambda }_{ijk}^2}{{\omega }_i+{\omega }_j+{\omega }_k}\\ \hfill & +\frac{1}{32}\sum _{ij}{\lambda }_{iijj}+\text{⋯},\end{array}\hfill \end{array}$$ (1)

which depends not only on the harmonic frequencies ${\tilde{\omega }}_i$, but also on the cubic (i.e., ${\lambda }_{ijk}$) or semidiagonalized quartic (i.e., ${\lambda }_{iijj}$) force constants. However, cubic and quartic terms are too costly for molecules containing more than just a few non‐H atoms [19], which prompts for some alternative in practical calculations.

A straightforward approach to obtain more accurate frequencies and ZPVE involves the use of scaling factors, which implicitly take into account not only vibrational anharmonicity effects but also the bias of the particular theoretical model employed for the calculations, be it the harmonic frequencies or for the ZPVE correction. This strategy has been widely and very successfully employed before at all levels of theory, for wavefunction‐based and Density Functional Theory (DFT) methods [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]. Note also the interest of having as accurate as possible frequencies calculated by cost‐effective methods, for example, themochemistry calculations assessed by a tailored basis set (DH‐SVPD) [33], vibrationally resolved absorption and fluorescence spectra of dyes with industrial or technological interest [34, 35, 36], 0‐0 energies of chromophores [37], systems violating Hund's rule for which a small error can alter the energy order of the singlet and triplet electronics pairs and thus the sign of the gap [38], equilibrium isotopic fractionation in large organic molecules [39], advanced spectroscopies of biomolecular building blocks [40], or absorption and resonance Raman spectra, to name just a few examples. Additionally, some known pathological cases [41, 42] for high‐level wavefunction‐based methods can benefit from more accurate thermochemical estimates employing double‐hybrid functionals.

Hence, we will obtain in the following a set of scaling factors for the harmonic frequencies (${\tilde{\omega }}_{\text{corrected}}=$ ${\tilde{\omega }}_{\text{calculated}}·f_{\text{harmonic}}$), fundamental (or true anharmonic) frequencies (${\tilde{\nu }}_{\text{corrected}}\approx {\tilde{\nu }}_{\text{calculated}}·f_{\text{fundamental}}$), and ZPVE (${\text{ZPVE}}_{\text{corrected}}={\text{ZPVE}}_{\text{calculated}}·f_{\text{ZPVE}}$) for a set of non‐empirical double‐hybrid functionals (i.e., the Quadratic Integrand Double‐Hybrid, QIDH, family of non‐empirical models [43], vide infra) competing in accuracy with high‐level wavefunction‐based methods for thermochemical [44] (i.e., the GMTKN55 database for main group thermochemistry) and kinetics [45] (i.e., the BH9 dataset composed of a large number of forward and reverse barrier heights) applications. Previously to the consideration here of those non‐empirical QIDH‐based double‐hybrid functionals, some authors [25, 27, 29, 30] have also derived the optimal scaling factors for other examples of double‐hybrid functionals such as B2‐PLYP [46], B2GP‐PLYP [47], or the DSD‐based family of expressions [48, 49], with $n$‐$\xi$ basis sets of diverse type (cc‐pVnZ, cc‐pV(n+d)Z, aug‐cc‐pVnZ, def2‐nVP(PD), and so forth, with $n$ referring to the double‐ξ, triple‐ξ or quadruple‐ξ size). However, up to the best of our knowledge, the assessment of non‐empirical double‐hybrid functionals for harmonic and fundamental frequencies is still missed, which will conveniently be tackled here, seen also the recent implementation of these QIDH‐based functional in common codes for both ground‐ and excited‐state calculations [50, 51, 52, 53] and their thermochemical and kinetics applications.

2. Theoretical and Computational Details

2.1. The PBE‐QIDH Functional

We have considered here the set of modern density functionals based on the Quadratic Integrand Double‐Hybrid (QIDH) approach, both the original and some recently developed variants, in which the exchange‐correlation functional form is derived in a completely non‐empirical fashion and includes the following terms common to all double‐hybrid density functionals [46, 54, 55, 56]:

$$\begin{array}{c}\begin{array}{ll}{E}_{xc}^{\text{QIDH}}[\rho ]\hfill & =a_x{E}_x^{\text{EXX}}[\left\{{\psi }_i\right\}]+(1-a_x)E_x[\rho ]\\ \hfill & +a_c{E}_c^{\mathrm{PT}\mathrm{2}}[\left\{{\psi }_i,{\psi }_a\right\}]+(1-a_c)E_c[\rho ],\end{array}\hfill \end{array}$$ (2)

with $a_x$ ($a_c$) being the weight given to the EXact‐eXchange (EXX) and to the 2nd. order Perturbation Theory (PT2) terms, respectively, complementing the usual exchange and correlation density functionals, $E_x[\rho ]$ and $E_c[\rho ]$, respectively, chosen here to be those given by the PBE [57] expressions, which corresponds to the model termed PBE‐QIDH. The wavefunction‐based terms are given by the standard (for simplicity, only the closed‐shell case is shown) expressions:

$$\begin{array}{ll}\hfill & E_x^{\text{EXX}}=-\frac{1}{2}\sum _{ij}^N\int \int {\psi }_i^{\star }(\mathbf{\text{x}}){\psi }_j^{\star }({\mathbf{\text{x}}}^{\prime })\frac{1}{|\mathbf{\text{r}}-{\mathbf{\text{r}}}^{\prime }|}{\psi }_j(\mathbf{\text{x}}){\psi }_i({\mathbf{\text{x}}}^{\prime })d\mathbf{\text{x}}d{\mathbf{\text{x}}}^{\prime },\end{array}$$ (3)

$$\begin{array}{ll}\hfill & E_c^{\text{PT}2}=-\frac{1}{4}\sum _{i,j}^N\sum _{a,b}^{N\ge N+1}\frac{|⟨{\psi }_i{\psi }_j||{\psi }_a{\psi }_b⟩{|}^2}{\left({ϵ}_a+{ϵ}_b\right)-\left({ϵ}_i+{ϵ}_j\right)},\end{array}$$ (4)

with $\int d\mathbf{\text{x}}={\sum }_{\sigma }\int d\mathbf{\text{r}}$, and depending on the set of occupied ($\left\{{\psi }_i\right\}$) and virtual ($\left\{{\psi }_a\right\}$) spinorbitals self‐consistently built along the calculation. The weights $a_x=3^{-1/3}$ and $a_c=1/3$ were originally [58] obtained without any fitting to reference or external data yet based on the adiabatic connection model and the need to reduce the self‐interaction error as much as possible.

2.2. Spin‐Scaled and Range‐Separated Variants

If the $E_c^{\mathrm{PT}\mathrm{2}}$ term is further decomposed [59] into same‐spin and opposite‐spin energy contributions, $E_c^{ss}$ and $E_c^{os}$, respectively, one can also consider to give additional weights to these components too, in the hope of reducing the error due to the truncation of the perturbation operator:

$$E_c^{\mathrm{PT}\mathrm{2}}=c_{ss}{E}_c^{ss-\mathrm{PT}\mathrm{2}}+c_{os}{E}_c^{os-\mathrm{PT}\mathrm{2}},$$ (5)

allowing to consequently define a spin‐scaled double‐hybrid density functional. Our previous work has allowed to also non‐empiricallly determine the optimal values ($c_{ss}=0$, $c_{os}=4/3$) as those leading to a performance similar to that of the pristine PBE‐QIDH model, giving rise to the corresponding Spin‐Opposite‐Scaled (SOS1‐)PBE‐QIDH expression [60]. Furthermore, the exchange energy term can also be decomposed into a short‐ and a long‐range contribution after splitting the two‐electron operator [61] as:

$$\frac{1}{r_{ij}}=\frac{1-\left[\alpha +\beta f\left(\omega r_{ij}\right)\right]}{r_{ij}}+\frac{\alpha +\beta f\left(\omega r_{ij}\right)}{r_{ij}},$$ (6)

with $f$ being the error function, $\omega$ (often denoted also as $\mu$ in the literature) the range separation coefficient, which controls the spatial extension of both short‐ and long‐range regime [62], and the $\alpha$ and $\beta$ coefficients identified here as $\alpha =a_x$ and $\beta =1-\alpha$. Again, the value of $\omega$ associated to the PBE‐QIDH form, thus defining the corresponding Range‐Separated eXchange (RSX‐PBE‐QIDH) double‐hybrid functional, was also determined non‐empirically with much success [63], and thus not fitted to minimize errors on specific datasets [64]. Finally, one can also combine both strategies, SOS1‐ and RSX‐, into the SOS1‐RSX‐PBE‐QIDH functional. Table 1 summarizes the values of the $a_x$, $a_c$, $c_{ss}$, $c_{os}$, and $\omega$ for every of the QIDH‐based models considered here.

TABLE 1.

Notation and non‐empirical $a_x$, $a_c$, $c_{ss}$, $c_{os}$, and $\omega$ values (${\text{bohr}}^{-1}$) for QIDH‐based double‐hybrid density functionals.

Functional	$a_x$	$a_c$	$c_{ss}$	$c_{os}$	$\omega$
PBE‐QIDH	$3^{-1/3}$	$1/3$	1	1	—
RSX‐PBE‐QIDH	$3^{-1/3}$	$1/3$	1	1	0.27
SOS1‐PBE‐QIDH	$3^{-1/3}$	$1/3$	0	$4/3$	—
SOS1‐RSX‐PBE‐QIDH	$3^{-1/3}$	$1/3$	0	$4/3$	0.27

2.3. Computational Details

All the calculations are done with the Gaussian'16 software [65] and tight numerical thresholds, using the PBE‐QIDH keyword and non‐standard routes (see the Supporting Information) for the SOS1‐PBE‐QIDH, RSX‐PBE‐QIDH, and SOS1‐RSX‐PBE‐QIDH calculations. We choose to work with the Weigend‐Ahlrichs family of basis sets [66], given their accuracy and massive use today for ground‐ and excited‐state DFT calculations, including the def2‐SVP, def2‐SVPD, def2‐TZVP, def2‐TZVPD, def2‐TZVPP, def2‐TZVPPD, def2‐QZVP, def2‐QZVPD, def2‐QZVPP, and def2‐QZVPPD members of the family, thus covering all the steps of the hierarchy and reaching the nearly‐exact limit with the largest one (def2‐QZVPPD); we also included the historically used 6‐31G(d) and 6‐31G(2df,p) basis sets for completeness and for comparison with previous results in the literature. We have considered the dataset and reference values (obtained at the CCSD(T*)(F₁₂*)/cc‐pVQZ‐F₁₂ level) previously used for the benchmarking of other related computational methods [29], which is a slight modification of the HFREQ2014 dataset previously employed by the same authors [67], and includes the diatomics Cl₂, ClF, CO, CS, S₂, HCl, HF, N₂, SiO, and SO, and polyatomics C₂H₂, C₂H₄, CH₃OH, H₂CS, CH₄, ClCN, CO₂, H₂CO, H₂O, H₂S, HCN, HOCl, N₂O, NH₃, OCS, PH₃, SO₂, CS₂, BH₃, CCl₂.

The scaling factors were obtained by regression through the origin, that is, taken the case of the corrected harmonic frequencies:

$${\tilde{\omega }}_{i,\text{corrected}}={\tilde{\omega }}_{i,\text{calculated}}·f_{harmonic}+y_i,$$ (7)

being $f_{harmonic}$ the scaling factor, ${\tilde{\omega }}_{i,\text{calculated}}$ the calculated harmonic frequencies by the particular model chemistry (i.e., combination of basis set and exchange‐correlation functional) used, and the residual origin $y_i$. The interceptless or zero intercept value in that equation is evaluated with the ${\sigma }^2$ function as defined here:

$${\sigma }^2=\frac{{\sum }_i^n{y}_i^2}{n-2}.$$ (8)

This fit has been implemented with the LINEST function of Microsoft Excel. The correlation coefficient $r^2$ was found to be $0.9996-0.9998$ for all the cases.

3. Results and Discussion

3.1. Harmonic Frequencies and Their Scaling Factors

Table 2 lists the ‘Root Mean‐Squared Deviation’ (RMSD) values obtained for the harmonic frequencies calculated with the PBE‐QIDH, RSX‐PBE‐QIDH, SOS1‐PBE‐QIDH, and SOS1‐RSX‐PBE‐QIDH models, together with all the considered basis sets, with respect to the reference values for the HFREQ2014 dataset. First of all, we note some dependence of the results with respect to the basis set size, spanning errors from $45-90$ ${\text{cm}}^{-1}$, thus confirming the previous results of Reference [29] where it was already stated that hybrid functionals showed much lower sensitivity to the basis set than some wavefunction‐based methods such as MP2 or its spin‐scaled SCS‐MP2 version. However, the presence of the $E_c^{\mathrm{PT}\mathrm{2}}$ term in double‐hybrid functionals (here with an $a_c=1/3$ weight) situates these models at an intermediate position and explains the dependence found. Overall, the computational effort required as the size of the basis set increases compensates for the decrease in the RMSD value. As a matter of example, the additional computational cost going from the def2‐SVP to the def2‐TZVP reduces considerably the RMSD values, while going to the next step in the hierarchy from the def2‐TZVP to the def2‐QZVP does not imply the same degree of improvement. Actually, the lowest deviation is found for the def2‐TZVPD basis set for all the functionals analysed, which might be thus considered as the basis set of choice for generalized applications involving the calculation of (harmonic) frequencies.

TABLE 2.

RMSD (${\text{cm}}^{-1}$) for unscaled harmonic frequencies.

Functional	6‐31G(d)	6‐31G(2df,p)	def2‐SVP	def2‐SVPD	def2‐TZVP	def2‐TZVPD	def2‐TZVPP	def2‐TZVPPD	def2‐QZVP	def2‐QZVPD	def2‐QZVPP	def2‐QZVPPD
PBE‐QIDH	68.08	64.05	66.97	56.67	48.67	45.87	51.81	50.04	50.99	50.46	51.04	50.46
RSX‐PBE‐QIDH	88.93	86.01	90.48	79.52	72.84	70.33	75.80	74.14	75.06	74.58	75.06	74.58
SOS1‐PBE‐QIDH	65.16	62.60	65.01	55.36	47.62	45.16	51.28	49.64	50.87	50.36	50.87	50.36
SOS1‐RSX‐PBE‐QIDH	86.05	84.38	88.20	77.97	71.37	69.12	74.82	73.28	74.39	73.96	74.39	73.96

Looking at the dependence of the results with respect to the functional choice, we found deviations between 45 and 90 ${\text{cm}}^{-1}$ as said before, roughly speaking, although these results can be gathered into two groups: (SOS1‐)PBE‐QIDH one hand and RSX‐(SOS1‐)PBE‐QIDH on the other hand, with smaller deviations for the former functionals independently of the basis set fixed for the comparison between the two groups. The decomposition of the exchange energy term into short‐ and long‐range contributions (i.e., RSX‐PBE‐QIDH) disfavours the calculation of harmonic frequencies, which also translates to the RSX‐SOS1‐PBE‐QIDH case. The corresponding optimal scaling factors are presented in Table 3. The scaling factors, always smaller than unity, show a generalized overestimation of the frequencies. For the QIDH‐based double‐hybrid density functionals, using the Weigend‐Ahlrichs family of basis sets, these scaling factors are between 0.96 and 0.98, slightly lower than those found for B2(GP)‐PLYP or DSD‐PBEP86 comparing the same basis sets [29]. After the corresponding scaling, the dependence with respect to the basis set is, not unsurprisingly, reduced for all the QIDH‐based functionals. RMSD around 20 ${\text{cm}}^{-1}$ are found for (SOS1‐)PBE‐QIDH and slightly above 30 ${\text{cm}}^{-1}$ for (SOS1‐)RSX‐PBE‐QIDH, see Table 4, in line with the general trend found before. These deviations are situated between those previously found [29] for example, B2(GP)‐PLYP ($10-15$ ${\text{cm}}^{-1}$) and those obtained for popular hybrid functionals such as PBE0 or M06‐2X, with RMSD values between 32 and 37 ${\text{cm}}^{-1}$ after scaling.

TABLE 3.

Optimal scaling factors ($f_{\text{harmonic}}$) for harmonic frequencies.

Functional	6‐31G(d)	6‐31G(2df,p)	def2‐SVP	def2‐SVPD	def2‐TZVP	def2‐TZVPD	def2‐TZVPP	def2‐TZVPPD	def2‐QZVP	def2‐QZVPD	def2‐QZVPP	def2‐QZVPPD
PBE‐QIDH	0.97025	0.97106	0.97107	0.97639	0.97847	0.97993	0.97697	0.97779	0.97746	0.97773	0.97745	0.97773
RSX‐PBE‐QIDH	0.96053	0.96206	0.96159	0.96692	0.96907	0.97042	0.96761	0.96836	0.96809	0.96834	0.96809	0.96834
SOS1‐PBE‐QIDH	0.96053	0.97190	0.97229	0.97710	0.97937	0.98070	0.97749	0.97821	0.97780	0.97805	0.97780	0.97780
SOS1‐RSX‐PBE‐QIDH	0.99978	0.96305	0.96295	0.96786	0.97016	0.97141	0.97749	0.96902	0.96867	0.96890	0.96867	0.96890

TABLE 4.

RMSD (${\text{cm}}^{-1}$) for harmonic frequencies after scaling.

Functional	6‐31G(d)	6‐31G(2df,p)	def2‐SVP	def2‐SVPD	def2‐TZVP	def2‐TZVPD	def2‐TZVPP	def2‐TZVPPD	def2‐QZVP	def2‐QZVPD	def2‐QZVPP	def2‐QZVPPD
PBE‐QIDH	27.28	20.66	28.47	28.13	19.05	19.15	19.60	19.17	19.97	19.99	20.04	19.99
RSX‐PBE‐QIDH	30.30	30.97	39.76	38.48	32.97	33.18	33.29	32.84	33.71	33.70	33.71	33.70
SOS1‐PBE‐QIDH	28.11	21.39	29.39	28.13	20.76	20.87	20.83	20.24	21.28	21.22	21.28	21.22
SOS1‐RSX‐PBE‐QIDH	32.29	32.03	40.64	38.93	34.32	34.53	34.25	33.74	34.71	34.70	34.71	34.70

3.2. Anharmonic Frequencies and Their Scaling Factors

Table 5 gathers now the RMSD values obtained for the fundamental frequencies calculated with the PBE‐QIDH, RSX‐PBE‐QIDH, SOS1‐PBE‐QIDH, and SOS1‐RSX‐PBE‐QIDH models, together with all the considered basis sets, and again with respect to the reference values for the HFREQ2014 dataset. The trend disclosed above for harmonic frequencies is also found here, with the (SOS1‐)PBE‐QIDH functionals leading to lower errors than the (SOS1‐)RSX‐PBE‐QIDH variants, and again with the def2‐TZVPD basis set arising as an optimal compromise between accuracy and computational cost. The scaling factors are included in Table 6, again smaller than unity and not showing a marked dependence with respect to the basis set size. Looking now at the RMSD values upon scaling, see Table 7, low values around 20 ${\text{cm}}^{-1}$ are found for (SOS1‐)PBE‐QIDH once a triple‐$\xi$ basis set is imposed, but slightly larger for the (SOS1‐)RSX‐PBE‐QIDH variants in agreement with the results found for harmonic frequencies too. One can compare the best available results for PBE‐QIDH (with the def2‐TZVPD basis set) of 22 ${\text{cm}}^{-1}$ with values of 24–26 ${\text{cm}}^{-1}$ for B2(GP)‐PLYP using the same basis set [29], thus showing a similar degree of accuracy.

TABLE 5.

RMSD (${\text{cm}}^{-1}$) for unscaled fundamental frequencies.

Functional	6‐31G(d)	6‐31G(2df,p)	def2‐SVP	def2‐SVPD	def2‐TZVP	def2‐TZVPD	def2‐TZVPP	def2‐TZVPPD	def2‐QZVP	def2‐QZVPD	def2‐QZVPP	def2‐QZVPPD
PBE‐QIDH	67.61	66.16	68.87	60.41	52.65	49.09	55.00	52.84	54.08	53.46	53.97	53.43
RSX‐PBE‐QIDH	91.67	89.99	94.17	76.35	77.63	66.48	80.01	69.48	79.13	70.52	79.50	70.41
SOS1‐PBE‐QIDH	56.97	64.72	66.84	57.71	55.68	48.74	54.19	52.22	53.79	53.26	53.95	53.24
SOS1‐RSX‐PBE‐QIDH	88.56	88.23	91.73	77.33	76.19	65.72	78.96	69.61	78.47	70.45	79.39	70.54

TABLE 6.

Optimal scaling factors ($f_{\text{fundamental}}$) for fundamental frequencies.

Functional	6‐31G(d)	6‐31G(2df,p)	def2‐SVP	def2‐SVPD	def2‐TZVP	def2‐TZVPD	def2‐TZVPP	def2‐TZVPPD	def2‐QZVP	def2‐QZVPD	def2‐QZVPP	def2‐QZVPPD
PBE‐QIDH	0.97008	0.96910	0.96920	0.97398	0.97615	0.97796	0.97496	0.97599	0.97552	0.97582	0.97558	0.97585
RSX‐PBE‐QIDH	0.95813	0.95872	0.95834	0.96837	0.96554	0.97200	0.96434	0.97015	0.96479	0.96980	0.96469	0.96983
SOS1‐PBE‐QIDH	0.97447	0.96996	0.97059	0.97553	0.97813	0.97906	0.97569	0.97665	0.97596	0.97626	0.97589	0.97628
SOS1‐RSX‐PBE‐QIDH	0.96010	0.95974	0.95973	0.96803	0.96662	0.97292	0.96507	0.97049	0.96541	0.97026	0.96508	0.97024

TABLE 7.

RMSD (${\text{cm}}^{-1}$) for fundamental frequencies after scaling.

Functional	6‐31G(d)	6‐31G(2df,p)	def2‐SVP	def2‐SVPD	def2‐TZVP	def2‐TZVPD	def2‐TZVPP	def2‐TZVPPD	def2‐QZVP	def2‐QZVPD	def2‐QZVPP	def2‐QZVPPD
PBE‐QIDH	30.48	22.01	29.62	30.29	22.05	21.58	22.34	21.80	22.69	22.50	22.69	22.56
RSX‐PBE‐QIDH	33.04	31.63	40.43	41.78	33.92	35.25	34.15	34.69	34.09	35.50	34.54	35.40
SOS1‐PBE‐QIDH	24.85	22.72	30.86	30.35	34.43	24.99	23.67	23.18	23.88	23.93	23.93	23.96
SOS1‐RSX‐PBE‐QIDH	35.07	32.39	40.91	42.48	35.21	36.80	34.93	36.14	35.28	36.99	35.93	37.07

Finally, we graphically represent the deviations $\mathrm{\Delta }\tilde{\omega }$ (harmonic frequencies) and $\mathrm{\Delta }\tilde{\nu }$ (fundamental frequencies) between the QIDH‐calculated and the reference values, in Figures 1 and 2, respectively. First of all, the scatter for harmonic frequencies seems to be more moderate independently of the functional considered, which can also be corroborated by the sum of maximum positive and negative errors (see the Supporting Information). The PBE‐QIDH arises as the most robust functional among all the variants considered, with that sum of maximum positive and negative errors being less than 100 ${\text{cm}}^{-1}$ once a triple‐$\xi$ basis set is considered, followed by SOS1‐PBE‐QIDH and the RSX‐based versions with considerable larger deviations. A previous investigation [27] of scaling factors for a large variety of functionals, including B2GP‐PLYP among them, in conjunction with the correlation consistent cc‐pVnZ basis sets, imposed separate scaling factors for the high (above 1000 ${\text{cm}}^{-1}$) and low (below 1000 ${\text{cm}}^{-1}$) harmonic frequencies. However, we do not see any reason to follow that strategy here, seen the somehow homogeneous scattering of harmonic frequencies peaking between $\pm$ 75 ${\text{cm}}^{-1}$ independently of their values, or at least below 3000 ${\text{cm}}^{-1}$. The sum of maximum positive and negative errors for fundamental frequencies (see the Supporting Information) gives higher values for any of the functionals considered, but with the original PBE‐QIDH arising again as the most robust with the sum of maximum positive and negative errors around 130 ${\text{cm}}^{-1}$ once a triple‐$\xi$ basis set is at least considered.

FIGURE 1.

Scatter of harmonic frequencies (scaled by the respective optimal scaling factor) with respect to reference results for all the QIDH‐based models, with the def2‐QZVPPD basis set.

FIGURE 2.

Scatter of fundamental frequencies (scaled by the respective optimal scaling factor) with respect to reference results for all the QIDH‐based models, with the def2‐QZVPPD basis set.

3.3. Extension to Harmonic and Anharmonic ZPVE

The optimal scaling factors for harmonic ZPVE (between 0.95 and 0.97 roughly speaking) for the PBE‐QIDH, RSX‐PBE‐QIDH, SOS1‐PBE‐QIDH, and SOS1‐RSX‐PBE‐QIDH models, together with all the considered basis sets, are included in Table 8. Once scaled, RMSD values for ZPVE as low as 0.06–0.14 kcal/mol are obtained independently of the functional and basis set used, see Table 9, with weak deviations in this case between pristine or spin‐scaled (range‐separated or not) versions. These small errors are comparable to those obtained before for B2(GP)‐PLYP or DSD‐PBEP86 in Reference [29], using the same family of def2‐nVP(PD) basis sets, and considerably smaller than those obtained with hybrid functionals [27]. Actually, B2GP‐PLYP with the sequence of (aug‐)cc‐pVnZ basis set led to deviations [27] between 0.10 and 0.12 kcal/mol, thus showing again the minor dependence with the basis set size once an optimal scaling factor is used. The extension to anharmonic ZPVE leads, see Table 10, to slightly larger scaling factors (between 0.96 and 0.98 roughly speaking) but very similar RMSD (and low) values for ZPVE (see Table 11).

TABLE 8.

Optimal scaling factors for harmonic ZPVE.

Functional	6‐31G(d)	6‐31G(2df,p)	def2‐SVP	def2‐SVPD	def2‐TZVP	def2‐TZVPD	def2‐TZVPP	def2‐TZVPPD	def2‐QZVP	def2‐QZVPD	def2‐QZVPP	def2‐QZVPPD
PBE‐QIDH	0.95854	0.95370	0.96011	0.96611	0.96550	0.96691	0.96437	0.96512	0.96493	0.96519	0.96493	0.96519
RSX‐PBE‐QIDH	0.95101	0.94556	0.95222	0.95816	0.95761	0.95899	0.95653	0.95725	0.95711	0.95734	0.95711	0.95734
SOS1‐PBE‐QIDH	0.95930	0.95560	0.96116	0.96682	0.96623	0.96759	0.96475	0.96545	0.96518	0.96541	0.96518	0.96541
SOS1‐RSX‐PBE‐QIDH	0.95185	0.94745	0.95334	0.95899	0.95845	0.95979	0.95704	0.95772	0.95750	0.95771	0.95750	0.95771

TABLE 9.

RMSD (kcal/mol) for ZPVE after scaling.

Functional	6‐31G(d)	6‐31G(2df,p)	def2‐SVP	def2‐SVPD	def2‐TZVP	def2‐TZVPD	def2‐TZVPP	def2‐TZVPPD	def2‐QZVP	def2‐QZVPD	def2‐QZVPP	def2‐QZVPPD
PBE‐QIDH	0.07	0.10	0.09	0.09	0.06	0.07	0.07	0.07	0.07	0.07	0.07	0.07
RSX‐PBE‐QIDH	0.12	0.10	0.14	0.12	0.12	0.12	0.12	0.12	0.12	0.12	0.12	0.12
SOS1‐PBE‐QIDH	0.07	0.09	0.10	0.09	0.07	0.07	0.08	0.07	0.08	0.08	0.08	0.12
SOS1‐RSX‐PBE‐QIDH	0.12	0.10	0.14	0.13	0.12	0.13	0.13	0.12	0.13	0.12	0.13	0.13

TABLE 10.

Optimal scaling factors for anharmonic ZPVE.

Functional	6‐31G(d)	6‐31G(2df,p)	def2‐SVP	def2‐SVPD	def2‐TZVP	def2‐TZVPD	def2‐TZVPP	def2‐TZVPPD	def2‐QZVP	def2‐QZVPD	def2‐QZVPP	def2‐QZVPPD
PBE‐QIDH	0.97123	0.96577	0.97338	0.97918	0.97843	0.98011	0.97757	0.97838	0.97815	0.97836	0.97816	0.9784
RSX‐PBE‐QIDH	0.9636	0.9575	0.9652	0.9765	0.9704	0.9775	0.9695	0.9755	0.9701	0.9754	0.9700	0.97544
SOS1‐PBE‐QIDH	0.97209	0.96792	0.97455	0.97994	0.97923	0.98080	0.97802	0.97875	0.97846	0.97876	0.97845	0.97879
SOS1‐RSX‐PBE‐QIDH	0.96451	0.95964	0.96817	0.97734	0.97131	0.97827	0.97007	0.97618	0.97051	0.97613	0.97050	0.97612

TABLE 11.

RMSD (kcal/mol) for anharmonic ZPVE.

Functional	6‐31G(d)	6‐31G(2df,p)	def2‐SVP	def2‐SVPD	def2‐TZVP	def2‐TZVPD	def2‐TZVPP	def2‐TZVPPD	def2‐QZVP	def2‐QZVPD	def2‐QZVPP	def2‐QZVPPD
PBE‐QIDH	1.08	0.08	0.10	0.09	0.08	0.07	0.08	0.07	0.08	0.07	0.07	0.07
RSX‐PBE‐QIDH	0.13	0.10	0.15	0.12	0.13	0.11	0.13	0.11	0.13	0.11	0.13	0.11
SOS1‐PBE‐QIDH	0.08	0.08	0.11	0.10	0.09	0.09	0.08	0.08	0.08	0.08	0.08	0.08
SOS1‐RSX‐PBE‐QIDH	0.14	0.11	0.17	0.12	0.14	0.12	0.14	0.11	0.14	0.12	0.14	0.12

4. Conclusions

We have determined optimal scaling factors for harmonic and fundamental vibrational frequencies, as well as ZPVE, of a family of non‐empirical double‐hybrid functionals, namely PBE‐QIDH, RSX‐PBE‐QIDH, SOS1‐PBE‐QIDH, and SOS1‐RSX‐PBE‐QIDH, including pristine (PBE‐QIDH), range‐separated (RSX‐PBE‐QIDH), spin‐scaled (SOS1‐PBE‐QIDH) and the combination of range‐separation and spin‐scaling (SOS1‐RSX‐PBE‐QIDH) to assess all the implemented possibilities of modern codes. The HFREQ2014 dataset of nearly‐exact results was selected for reference values, including diatomic and polyatomic molecules of moderate size. The dependence with respect to the basis set size and type has been incorporated through the def2‐nVP(PD) family of basis sets: def2‐SVP, def2‐SVPD, def2‐TZVP, def2‐TZVPD, def2‐TZVPP, def2‐TZVPPD, def2‐QZVP, def2‐QZVPD, def2‐QZVPP, and def2‐QZVP(PD). Among the various basis sets considered, the def2‐TZVPD offers the best compromise between accuracy and computational cost for all the QIDH‐based models. On the other hand, the historically used Pople's style basis sets can be discarded as they always provide higher RMSD values in all cases, in agreement with the recent trend not employing those basis sets for calculations with double‐hybrid density functionals. Overall, RMSD values as low as $20 {\text{cm}}^{-1}$ for harmonic or fundamental frequencies, and less than 0.1 kcal/mol for ZPVE, can be safely obtained at for example, PBE‐QIDH/def2‐TZVPD level. However, we also note that the values taken here as reference are also affected [67] by a RMSD of 7 ${\text{cm}}^{-1}$, which can be thus considered as the lower limit for any accurate purposes, which thus situates this PBE‐QIDH/def2‐TZVPD scheme as an affordable, widely implemented, model chemistry for highly accurate thermochemical, kinetics, and spectroscopic calculations.

Supporting information

Data S1. Supporting Information.

Miguel B., Brémond E., Pérez‐Jiménez A. J., Adamo C., and Sancho‐García J. C., “Accuracy and Scaling Factors of Non‐Empirical Double‐Hybrid Density Functionals for Harmonic and Fundamental Frequencies (And ZPVE),” Journal of Computational Chemistry 47, no. 1 (2026): e70300, 10.1002/jcc.70300.

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.