Ground-State Gas-Phase Structures of Inorganic Molecules Predicted by Density Functional Theory Methods

Abstract

We tested a battery of density functional theory (DFT) methods ranging from generalized gradient approximation (GGA) via meta-GGA to hybrid meta-GGA schemes as well as Møller–Plesset perturbation theory of the second order and a single and double excitation coupled-cluster (CCSD) theory for their ability to reproduce accurate gas-phase structures of di- and triatomic molecules derived from microwave spectroscopy. We obtained the most accurate molecular structures using the hybrid and hybrid meta-GGA approximations with B3PW91, APF, TPSSh, mPW1PW91, PBE0, mPW1PBE, B972, and B98 functionals, resulting in lowest errors. We recommend using these methods to predict accurate three-dimensional structures of inorganic molecules when intramolecular dispersion interactions play an insignificant role. The structures that the CCSD method predicts are of similar quality although at considerably larger computational cost. The structures that GGA and meta-GGA schemes predict are less accurate with the largest absolute errors detected with BLYP and M11-L, suggesting that these methods should not be used if accurate three-dimensional molecular structures are required. Because of numerical problems related to the integration of the exchange–correlation part of the functional and large scattering of errors, most of the Minnesota models tested, particularly MN12-L, M11, M06-L, SOGGA11, and VSXC, are also not recommended for geometry optimization. When maintaining a low computational budget is essential, the nonseparable gradient functional N12 might work within an acceptable range of error. As expected, the DFT-D3 dispersion correction had a negligible effect on the internuclear distances when combined with the functionals tested on nonweakly bonded di- and triatomic inorganic molecules. By contrast, the dispersion correction for the APF-D functional has been found to shorten the bonds significantly, up to 0.064 Å (AgI), in Ag halides, BaO, BaS, BaF, BaCl, Cu halides, and Li and Na halides and hydrides. These results do not agree well with very accurate structures derived from microwave spectroscopy; we therefore believe that the dispersion correction in the APF-D method should be reconsidered. Finally, we found that inaccurate structures can easily lead to errors of few kcal/mol in single-point energies.

1. Introduction

The relative energies and other molecular properties of inorganic molecules are known to converge very slowly, often requiring expensive but rigorous and robust wave function theory (WFT) calculations in the complete basis set limit.^1,2 By contrast, the molecular structures of these molecules are expected to converge more rapidly,^1,2 and they can usually be obtained from relatively cheap density functional theory (DFT) methods with moderate basis sets of triple-ζ quality with polarization functions. To speed up routine quantum chemical calculations, Pople and co-workers³⁻⁷ proposed the so-called composite schemes in which sophisticated WFT methods are utilized only in a single-point (SP) fashion on the molecular structures optimized with much cheaper protocols, often based on DFT methods. Although the accuracy of SP energy evaluation methods is routinely tested to reproduce the thermochemistry of individual substances⁸⁻²² as well as chemical reactions,^{16,20,23−37} the systematic study of the quality of the molecular structures predicted by DFT is far less common. Selecting the particular method for geometry optimization and estimating the potential inaccuracies arising from such a choice remain difficult.

An analysis of studies^27,37−56 testing the accuracy of DFT molecular structures suggests that, first, hybrid-, double-hybrid-, meta-, and hybrid-meta-generalized gradient approximation (GGA) functionals generally provide better structures than classical GGA schemes; and, second, dispersion-corrected DFT functionals usually outperform their standard counterparts if significant noncovalent intra- or intermolecular interactions can be found in the system. It is impossible to identify the method that results in the best molecular structures because the performance of individual functionals is not homogeneous. Often, a functional performing well for one class of compounds provides moderate or even quite poor accuracy for another group of compounds. Even if the above-mentioned studies can provide a rough idea for the most suitable protocol for a given chemical system, these studies often have at least one of the two following shortcomings. First, in many studies benchmarking the quality of three-dimensional molecular structures, the most recent DFT functionals are not included, which is not surprising, as many functionals appeared in the last decade and became available through the latest releases of quantum chemistry computer codes. Second, the referenced experimental structures quite often are from diffraction methods and cannot, strictly speaking, be directly compared with the calculated structures in a rigorous way. The molecular structures obtained from DFT or WFT methods correspond to the potential energy minimum. Often such structures are abbreviated as r_e (equilibrium three-dimensional structure). Unfortunately, there are no experimental methods that provide direct access to the equilibrium structure.⁵⁷ Thus, the result of a gas-phase electron diffraction (GED) experiment is the three-dimensional molecular structure averaged over all molecular vibrations at a certain temperature, r_a. Another popular diffraction method in chemistry, X-ray analysis, results in structures of molecules that, in addition to being averaged over vibrations and librations, are also influenced by effects arising from crystal-packing forces. Clearly both kinds of structures cannot be directly compared to equilibrium structures, r_e, obtained from quantum chemistry methods. Several approximations exist to modify the experimental structures from diffraction methods to make them more comparable to their r_e counterparts, for example, through inclusions of the effects from vibrations (e.g., r_h1, r_α, etc. in GED).⁵⁸⁻⁶² However, the accuracy of such approximations remains open to debate.

Here, we seek to contribute to existing studies that benchmark three-dimensional molecular structures in two ways. First, along with the standard well-tested functionals, including PBE^63,64 and B3LYP,⁶⁵⁻⁶⁸ we also include the most recent functionals, in particular, the latest methods from the Minnesota group on which many fewer molecular geometry validation studies are available in the literature. As reference structures, we use molecular structures from gas-phase microwave spectroscopy.

2. Computational Details

2.1. Benchmark Set

We extracted the reference molecular structures from microwave spectroscopy measurements from the National Institute of Standards and Technology (NIST) Diatomic⁶⁹ and Triatomic⁷⁰ Spectral Databases. We selected molecules for which only the most reliable measurements were available. They are listed in Tables 1 and 2. In particular, we include only diatomic molecules for which r_e molecular structures are provided in the dataset. Similarly, we include only triatomic molecules for which r_e or r_s (distances obtained through isotopic substitution experiments)⁵⁷ structures are available in the dataset. We give preference to r_e if both r_e and r_s (isotopic substitution) structures are available.

Table 1. Experimental Internuclear Distances of Selected Diatomic Molecules Derived from Microwave Spectroscopy Measurements.

moleculea	re (Å)b	refs
AgBr (1Σ+)	2.393109(2) (107Ag79Br)	(71)
AgCl (1Σ+)	2.2807916(6) (107Ag35Cl)	(72)
AgF (1Σ+)	1.983179(1) (107Ag19F)	(73)
AgI (1Σ+)	2.5446165(12) (107Ag127I)	(71,74)
AlBr (1Σ+)	2.294859(3) (27Al79Br)	(75)
AlCl (1Σ+)	2.1301663(4) (27Al35Cl)	(75)
AlF (1Σ+)	1.6543688(4) (27Al19F)	(76−78)
AlI (1Σ+)	2.5371027(3) (27Al127I)	(75)
BCl (1Σ+)	1.7194925(5) (11B35Cl)	(79)
BF (1Σ+)	1.2625c	(80)
BaO (1Σ+)	1.9396299(34)d (138Ba16O)	(81,82)
BaS (1Σ+)	2.5073221(8) (138Ba32S)	(83,84)
BrCl (1Σ+)	2.1360399(8) (79Br35Cl)d	(85)
BrF (1Σ+)	1.7589200(18) (79Br19F)	(85)
CO (1Σ+)	1.128229(1) (12C16O)d	(86)
CS (1Σ+)	1.5348192(12) (12C32S)c	(87)
CSe (1Σ+)	1.67618(9) (12C80Se)c	(88)
CaO (1Σ+)	1.8222034(3) (40Ca16O)	(89)
ClF (1Σ+)	1.628 2718(5) (35Cl19F)d	(85)
CuBr (1Σ+)	2.173441(8) (63Cu79Br)d	(90)
CuCl (1Σ+)	2.051192(1) (63Cu35Cl)d	(91)
CuF (1Σ+)	1.744930(2) (63Cu19F)	(73)
CuI (1Σ+)	2.3383236(15) (63Cu127I)d	(92)
GaBr (1Σ+)	2.3524633(14) (69Ga79Br)	(93)
GaCl (1Σ+)	2.201690(3) (69Ga35Cl)	(94)
GaF (1Σ+)	1.7743691(9) (69Ga19F)	(78)
GaI (1Σ+)	2.5746370(6) (69Ga127I)	(95)
GeO (1Σ+)	1.624647(2) (73Ge16O)	(96,97)
GeS (1Σ+)	2.0120863(1) (74Ge32S)	(98)
GeSe (1Σ+)	2.1346292(1) (74Ge80Se)	(98)
GeTe (1Σ+)	2.340165(12) (74Ge130Te)	(99)
HBr (1Σ+)	1.414435(3) (1H81Br)	(100)
HCl (1Σ+)	1.274552(6) (1H35Cl)	(101,102)
HF (1Σ+)	0.916809(27) (1H19F)	(103−105)
HI (1Σ+)	1.6090231(3) (1H127I)d	(102,106)
IBr (1Σ+)	2.4689735(6) (127I79Br)d	(85)
ICl (1Σ+)	2.320877(7) (127I35Cl)	(107)
IF (1Σ+)	1.909759(5) (127I19F)	(108)
InBr (1Σ+)	2.5431804(55) (115In79Br)	(109)
InCl (1Σ+)	2.401168(2) (115In35Cl)	(110−112)
InF (1Σ+)	1.9853964(9) (115In19F)	(78)
InI (1Σ+)	2.753639(54) (115In127I)	(113)
LiBr (1Σ+)	2.170428(4) (6Li79Br)	(114,115)
LiCl (1Σ+)	2.0206676(4) (6Li35Cl)	(116,117)
LiF (1Σ+)	1.5638601(3) (6Li19F)	(116,118)
LiH (1Σ+)	1.5949131(8) (6Li1H)d	(119)
LiI (1Σ+)	2.391907(14) (6Li127I)	(120)
MgO (1Σ+)	1.7483805(1) (24Mg16O)	(121)
NaBr (1Σ+)	2.5020379(12) (23Na79Br)	(117,122)
NaCl (1Σ+)	2.360795(2) (23Na35Cl)	(123,124)
NaF (1Σ+)	1.9259465(3) (23Na19F)	(125,126)
NaH (1Σ+)	1.88652(9) (23Na1H)d	(127,128)
NaI (1Σ+)	2.711452(2) (23Na127I)	(117,122)
PN (1Σ+)	1.4908663(5) (31P14N)	(129−131)
PbO (1Σ+)	1.9218134(17) (207Pb16O)	(132,133)
PbS (1Σ+)	2.2868628(12) (207Pb32S)	(133,134)
PbSe (1Σ+)	2.4022353(22) (208Pb80Se)	(99,135)
PbTe (1Σ+)	2.594975(4) (208Pb130Te)	(99,136)
SiO (1Σ+)	1.5097375(2) (28Si16O)	(97,137)
SiS (1Σ+)	1.9293212(2) (28Si32S)	(138,139)
SiSe (1Σ+)	2.058327(4) (28Si80Se)	(140,141)
SnO (1Σ+)	1.8325054(20) (120Sn16O)	(133,142)
SnS (1Σ+)	2.2090267(3) (120Sn32S)	(133,143)
SnSe (1Σ+)	2.325601(3) (120Sn80Se)	(144,145)
SnTe (1Σ+)	2.522814(1) (120Sn130Te)	(145,146)
SrO (1Σ+)	1.919 849(1) (88Sr16O)c	(82)
TlBr (1Σ+)	2.618184(11) (205Tl79Br)	(147,148)
TlCl (1Σ+)	2.4848260(4) (205Tl35Cl)	(149,150)
TlF (1Σ+)	2.0844377(1) (205Tl19F)	(151,152)
TlI (1Σ+)	2.813676(10) (205Tl127I)	(147,148)
BaCl (2Σ+)	2.6827325(5) (138Ba35Cl)	(153)
BaF (2Σ+)	2.1592708(6) (138Ba19F)	(154)
CN (2Σ+)	1.171807(20) (12C14N)c	(155)
CaBr (2Σ+)	2.5935838(4) (40Ca79Br)	(156)
CaCl (2Σ+)	2.4367580(4) (40Ca35Cl)	(156)
SrCl (2Σ+)	2.575854(1) (88Sr35Cl)	(157)
SrF (2Σ+)	2.075366(1) (88Sr19F)	(157)
BiBr (3Σ+)	2.6095031(24) (209Bi79Br)	(158)
BiCl (3Σ+)	2.4715223(8) (209Bi35Cl)	(159)
BiF (3Σ+)	2.0515431(3) (209Bi19F)	(160)
BiI (3Σ+)	2.8005010(3) (209Bi127I)	(161)
SO (3Σ+)	1.4809899(13) (32S16O)d	(162)
SeO (3Σ+)	1.6394792(18) (78Se16O)d	(163)

^aGround-state molecular term.

^bCalculated as r_e = [505 379.006(51)/μ_rY₀₁]^1/2 as in the NIST database unless otherwise noted.

^cTaken from the original work.

^dCalculated as r_e = [505379.006(51)/μ_rB_e]^1/2 as in the NIST database.

Table 2. Experimental Molecular Structures of Selected Triatomic Molecules Derived from Microwave Spectroscopy Measurements.

molecule (ABC)a	type	rAB (Å)	rBC (Å)	∠ABCb (deg)	refs
BrNO (1A′)	rs	2.140(2)	1.146(10)	114.50(50)	(164)
ClCN (1Σ+)	re	1.629(6)	1.160(7)	180	(165)
ClNO (1A′)	rs	1.975(5)	1.139(12)	113.3(6)	(166)
FCN (1Σ+)	rs	1.262(2)	1.159(2)	180	(167)
FNO (1A′)	rs	1.512(5)	1.136(3)	110.1(2)	(168)
GeF2 (1A1)	re	1.7321	1.7321	97.148	(169)
H2O (1A1)	re	0.9587(1)	0.9587(1)	103.89(0.06)	(170)
H2S (1A1)	re	1.3356	1.3356	92.11	(171,172)
H2Se (1A1)	re	1.4605(30)	1.4605(30)	90.92(12)	(173)
HBS (1Σ+)	rs	1.1692(4)	1.5995(4)	180	(174)
HCN (1Σ+)	re	1.0655(5)	1.15321(10)	180	(175)
HCP (1Σ+)	re	1.0692(8)	1.5398(2)	180	(176)
HNC (1Σ+)	rs	0.98607(9)	1.17168(22)	180	(177,178)
NNO (1Σ+)	rs	1.1286(3)	1.1876(3)	180	(179)
NSCl (1A′)	rs	1.450	2.161	117.7	(180)
O3 (1A1)	re	1.2717(2)	1.2717(2)	116.78	(181)
OCS (1Σ+)	re	1.1543(10)	1.5628(10)	180	(182)
OCSe (1Σ+)	re	1.1535(1)	1.7098(1)	180	(183)
OF2 (1A1)	re	1.4053(4)	1.4053(4)	103.07(06)	(184)
SO2 (1A1)	re	1.43076(13)	1.43076(13)	119.33(1)	(185)
SeO2 (1A1)	re	1.6076(6)	1.6076(6)	113.83	(186)
SiF2 (1A1)	re	1.5901(1)	1.5901(1)	100.77(2)	(187)
ClO2 (2B1)	rs	1.471	1.471	117.5	(188−190)
NO2 (2A1)	re	1.1945(5)	1.1945(5)	133.85(10)	(191)

^aGround-state molecular term.

^bFor comparisons with r_e structures from DFT and WFT methods, we modify the structural description with the two distances and a bend angle in favor of three distances.

2.2. Geometry Optimizations

We performed all geometry optimizations with the most popular DFT functionals listed in Table 3 as implemented in the Gaussian 09¹⁹² suite of programs. We adopted the default values for the self-consistent-field and geometry optimization convergence criteria. We performed numerical integration of the exchange–correlation (XC) terms using the tighter-than-default “ultrafine” option (pruned, 99 radial shells and 590 angular points per shell) to eliminate the potential numerical noise in the energy derivatives. We applied Grimme’s D3(BJ)¹⁹³ dispersion correction to PBE, TPSS, BP86, B3PW91, B3LYP, BLYP, BPBE, BMK, CAM-B3LYP, LC-ωPBE, and PBE0 to arrive at the corresponding DFT-D3(BJ) methods and to account for the possible influence of the dispersion interactions not (or only partly) covered by the standard functionals. However, because in practically all cases we observed only insignificant differences in internuclear distances of only ca. 0.001 Å (or less) upon inclusion of the dispersion correction, we discuss only the dispersion-uncorrected results for the above-mentioned functionals. In addition to DFT functionals, we also optimized all structures with Møller–Plesset perturbation theory of the second order (MP2) and single and double excitation coupled-cluster (CCSD) methods for which analytical electronic energy first derivatives (gradients) are available. In both WFT methods, we included all electrons in the correlation treatment as the core–valence and core–core correlation effects have been shown to be important for some inorganic species.^{8,26,194−200}

Table 3. Summary of the Methods Used in the Present Work to Optimize the Molecular Structures.

method	typea	refs
PBE	GGA	(63,64)
BP86	GGA	(65,212)
BLYP	GGA	(65,66)
BPBE	GGA	(63,65)
HCTH/407	GGA	(213−215)
B97-D	GGA+D	(216)
B97-D3	GGA+D	(216,217)
SOGGA11	GGA	(218)
N12	NGA	(219)
OLYP	GGA	(66,220−222)
TPSS	mGGA	(223)
VSXC	mGGA	(224)
M06-L	mGGA	(225)
M11-L	mGGA	(226)
MN12-L	mNGA	(227)
B3PW91	GH-GGA	(65,68,228−232)
B3LYP	GH-GGA	(65,68)
mPW1PW91	GH-GGA	(233)
SOGGA11-X	GH-GGA	(234)
PBE0	GH-GGA	(63,64)
mPW1LYP	GH-GGA	(66,222,233)
mPW1PBE	GH-GGA	(63,64,233)
X3LYP	GH-GGA	(66,222,235)
B971	GH-GGA	(213)
B972	GH-GGA	(236)
B98	GH-GGA	(237,238)
B1LYP	GH-GGA	(239)
APF	GH-GGA	(240)
APF-D	GH-GGA+D	(240)
LC-ωPBE	RSH-GGA	(241)
CAM-B3LYP	RSH-GGA	(242)
N12-SX	RSH-GGA	(243)
ωB97X-D	RSH-GGA+D	(244)
BMK	GH-mGGA	(245)
TPSSh	GH-mGGA	(223,246)
M06	GH-mGGA	(247)
M11	RSH-mGGA	(248)
MN12-SX	RSH-mGGA	(243)
MP2	WFT	(249)
CCSD	WFT	(250−253)

^a+D = functional corrected for dispersion interactions, NGA = nonseparable gradient approximation, mGGA = meta-GGA, mNGA = meta-NGA, GH = global hybrid, and RSH = range-separated hybrid.

We used the following triple-ζ correlation-consistent basis sets. We used Dunning’s cc-pVTZ basis sets to describe hydrogen.²⁰¹ We described lithium, sodium, and magnesium with Wilson et al.’s all-electron (AE) correlation-consistent core–valence cc-pwCVTZ basis sets.²⁰² We used Dunning and Peterson’s AE correlation-consistent core–valence cc-pwCVTZ basis sets to describe boron, carbon, nitrogen, oxygen, fluorine, aluminum, silicon, phosphorus, sulfur, chlorine, and argon.²⁰³ We used Li et al.’s pseudopotential-based correlation-consistent core–valence cc-pwCVTZ-PP basis sets to describe calcium, strontium, and barium.²⁰⁴ The 10 core electrons of Ca, the 28 core electrons of Sr, and the 46 core electrons of Ba were described with Stuttgart-type fully relativistic effective core potentials of Schwerdtfeger and co-workers.²⁰⁵ Copper and silver were described with the pseudopotential-based correlation-consistent core–valence cc-pwCVTZ-PP basis sets of Peterson and Puzzarini.²⁰⁶ The 10 core electrons of Cu and the 28 core electrons of Ag were described with Stuttgart-type fully relativistic effective core potentials of Stoll and co-workers.²⁰⁷ Gallium, germanium, selenium, bromine indium, tin, tellurium, iodine, thallium, lead, and bismuth were described with the pseudopotential-based correlation-consistent core–valence cc-pwCVTZ-PP basis set of Peterson and Yousaf.²⁰⁸ The 10 core electrons of Ga, Ge, Se, and Br, the 28 core electrons of In, Sn, Te, and I, and the 60 core electrons of Tl, Pb, and Bi were described with Stuttgart-type fully relativistic effective core potentials of Dolg and co-workers.^209−211 Even when the core–valence basis sets were used for AE MP2 and CCSD calculations to describe the subvalence correlation, we applied exactly the same basis sets for the DFT methods for an unbiased comparison. Structures were characterized as true energy minima by the eigenvalues of the analytically calculated Hessian matrix.

2.3. Calculation of Errors

To quantitatively measure the deviation of the theoretical equilibrium internuclear distances from their experimental counterparts, we used a few common protocols. First, for each method, we calculated the mean unsigned error (MUE) and the mean signed error (MSE) defined as follows

1

2

where M is the number of molecules in the analyzed dataset, N is the number of atoms in the molecule (N = 2 for diatomic and N = 3 for triatomic molecules), R_ij(Exp., m) is the experimental internuclear distance between nuclei i and j in the molecule, m, and R_ij(Theor., m) is its theoretical counterpart.

In addition to the two above-mentioned criteria, for the errors in internuclear distances obtained from each method, we also calculated the standard deviation (σ) and the difference between the largest positive and negative deviation (δ = Max – Min). These two values are necessary to understand the dispersion of the errors.

3. Results and Discussion

We organize the results and discussion section as follows. First, we comment on the overall performance obtained for the combined dataset involving the internuclear distances of all di- and triatomic molecules covered in the present study. Then, we assess an ability of methods in reproducing specific internuclear distances as those involving the s-metals, d-metals, and light and heavy elements. Afterward, we estimate the error in the absolute electronic energy arising from the use of different methods for geometric optimization and the SP energy evaluation, which corresponds to the maximum error associated with composite schemes.

3.1. Performance in Predicting All Internuclear Distances

The MSE and MUE for the 164 internuclear distances of the 83 diatomic and 27 triatomic DFT- and WFT-optimized molecules listed in Tables 1 and 2 relative to the corresponding values derived from microwave spectroscopy measurements are presented in Figure 1. Among the local GGA DFT methods, the smallest MUE of 0.012 Å was detected for the recently developed nonseparable gradient N12 functional from the Minnesota group followed closely by the HCTH/407 functional with an MUE of 0.017 Å. While N12, with an MSE of −0.007 Å, exhibits a small tendency toward underestimation of the distances, HCTH/407, with an MSE of 0.014 Å, results in distances that are, in general, too long. The largest errors in molecular structures were obtained by the BLYP functional. With the MUE and MSE of 0.035 Å, BLYP was found to overestimate the distances, which agrees with previous reports.^{47,48,254−257} All other tested local GGA DFT methods, including SOGGA11, PBE, OLYP, BPBE, BP86, B97-D3, and B97-D, resulted in similar MUEs in the range of 0.020–0.027 Å. With all seven methods, positive MSEs that were practically equal in magnitude to their corresponding MUEs were obtained, indicating the pronounced tendency toward overestimation of internuclear distances. Interestingly, the B97-D3 functional with an MUE of 0.024 Å turned out to be superior to the B97-D functional with an MUE of 0.027 Å, indicating an improvement of the dispersion parameterization of D3 over D2. Despite the good accuracy obtained by the SOGGA11 method, this functional turned out to be particularly sensitive to an integration grid. The harmonic frequency calculation with the SOGGA11 functional on the three-dimensional structure optimized with the same method and the tighter-than-G09-default “ultrafine” integration grid used in this work (pruned, 99 radial shells and 590 angular points per shell) resulted in the two imaginary frequencies of i1102.9 cm^–1, both of which correspond to bending vibrations distorting the molecule from its linear structure. As this molecule was experimentally found to be linear with confirmation from all but the SOGGA11 functional, we concluded that the SOGGA11 method predicts incorrect harmonic frequencies. The origin of this issue is the use of reduced-quality integration grids by the G09 code during calculation of the second derivatives of the electronic energy. Even if the “ultrafine” integration grid was explicitly specified, a grid of lower quality was actually generated to calculate the second derivatives. Indeed, the harmonic frequency calculation that explicitly specified “CPH(Grid = ultrafine)” was the only option that resulted in real frequencies.

Figure 1.

MSEs and MUEs for the 164 internuclear distances of the 83 diatomic and 27 triatomic DFT- and WFT-optimized molecules relative to the corresponding microwave spectroscopy data.

From the local meta-GGA functionals, the lowest MUE of 0.012 Å was found for the M06-L functional. The MSE of 0.007 Å for M06-L indicates that the functional has a small tendency to overestimate internuclear distances. Perhaps surprisingly, we found that the M11-L functional from the Minnesota group had the largest MUE of 0.031 Å. This functional’s negative MSE of −0.020 Å suggests that it has a tendency to shorten the internuclear distances. All other meta-GGA functionals, including TPSS, MN12-L, and VSXC, performed rather uniformly with MUEs in the range of 0.018–0.019 Å. While the TPSS and VSXC methods with MSEs of 0.017 and 0.019 Å provided distances that were too long, the MN12-L functional with an MSE of −0.007 Å predicted a distance that was slightly short.

Among the hybrid and hybrid meta-GGA functionals, the lowest MUE of 0.010 Å and practically vanishing MSEs of 0.001 and −0.002 Å were, respectively, found with the B3PW91 and APF functionals. An MUE of only 0.001 Å higher was found with the mPW1PW91, PBE0, mPW1PBE, B972, B98, and TPSSh functionals. The first three of these functionals, namely mPW1PW91, PBE0, and mPW1PBE, with the MSE of −0.004 Å exhibited a small tendency toward overestimation of the internuclear distances. The B972 method predicted a practically vanishing MSE. The B98 and TPSSh functionals predicted positive MSEs of 0.006 and 0.008 Å, which were in general too long. Slightly higher MUEs of 0.012–0.014 Å were detected for the B971 (MUE/MSE = 0.012/0.007 Å), SOGGA11-X (MUE/MSE = 0.013/–0.006 Å), X3LYP (MUE/MSE = 0.013/0.009 Å), APF-D (MUE/MSE = 0.013/–0.006 Å), CAM-B3LYP (MUE/MSE = 0.013/–0.005 Å), N12-SX (MUE/MSE = 0.013/–0.009 Å), BMK (MUE/MSE = 0.013/–0.002 Å), B3LYP (MUE/MSE = 0.014/0.011 Å), mPW1LYP (MUE/MSE = 0.014/0.008 Å), B1LYP (MUE/MSE = 0.014/0.009 Å), ωB97XD (MUE/MSE = 0.014/–0.003 Å), and M06 (MUE/MSE = 0.014/–0.001 Å) functionals. A larger MUE of 0.015 Å was detected for the MN12-SX functional, which suggests a small tendency to underestimation as indicated by MSE of −0.007 Å. On the other side of the scale, the largest MUE of 0.018 Å was predicted by the M11 and CAM-ωPBE functionals for which the MSEs turned out to be −0.003 and −0.016 Å, respectively. Interestingly, practically no influence of the Grimme dispersion correction was found for the PBE0, B3LYP, PBE, and other functionals. However, a significant difference in the predicted internuclear distances was obtained between APF and its dispersion-corrected counterpart, APF-D, as illustrated by a difference in the MUE of 0.003 Å. A detailed analysis revealed that the difference between the APF and APF-D functionals was evident only for a limited number of molecules, including Ag halides, BaO, BaS, BaF, BaCl, Cu halides, and Li and Na halides and hydrides. For these molecules, the nonzero “Petersson–Frisch” dispersion energy²⁴⁰ was found whereas for all other species, the absolute zero dispersion energy was calculated. The observation that nonzero dispersion energy was obtained for all studied Li, Na, and Ba complexes whereas absolute zero dispersion energy was found for Mg, Ca, and Sr complexes is remarkable. Because the APF-D functional provides a larger MUE than APF does, we believe that, perhaps, additional adjustments of the dispersion parameters for this functional might be needed. Finally, for the two WFT methods for which AE were correlated, MP2 (AE) and CCSD (AE), MUEs of 0.013 and 0.012 Å were found, which is comparable to that of the hybrid and hybrid-meta-GGA DFT methods (Av. MUE of 0.013 Å) and significantly better than that of the meta-GGA methods (Av. MUE of 0.020 Å) and local GGA methods (Av. MUE of 0.022 Å). This indicates that higher correlated WFT methods with larger basis sets have to be used to benchmark the performance of DFT methods in reproducing inorganic three-dimensional molecular structures.

Apart from the MUEs and MSEs, we also analyzed the dispersion of errors obtained for every method tested. For this purpose, in Figure 2, we show the standard deviation (σ) and the absolute difference (δ) between the most positive and negative deviations. As expected, we found the largest dispersion of errors for the heavily empirically parameterized Minnesota functionals: M11-L resulted in σ/δ of 0.033/0.198 Å, followed by MN12-L with σ/δ of 0.022/0.152 Å and SOGGA11 with σ/δ of 0.024/0.135 Å. Interestingly, a relatively large dispersion of errors (σ/δ of 0.019/0.135 Å) was detected for the MP2 WFT method. The smallest dispersion of errors was found for the TPSSh, APF, mPW1PBE, PBE0, mPW1PW91, and B3PW91 functionals.

Figure 2.

Standard deviation (σ) and absolute difference between the largest positive and negative deviations (δ) for the 164 internuclear distances of the 83 diatomic and 27 triatomic DFT- and WFT-optimized molecules relative to the corresponding microwave spectroscopy data.

3.2. Analysis of Specific Internuclear Distances

We also analyzed the difference in performance of the tested methods in reproducing the structures of the molecules consisting of the lighter elements with 1–3 periods and the heavier elements with 4–6 periods. The MSEs obtained for the molecules comprising the elements from the 1–3 groups and the elements from the 4–6 groups are presented in Figure S1; the corresponding MUEs are given in Figure S2; and the difference between the MUEs of the molecules comprising the lighter (periods 1–3) and heavier (periods 4–6) elements is depicted in Figure S3. In general, all the methods resulted in larger MUEs for the structures of the molecules comprising heavier elements from the 4–6 groups. The average difference between the MUEs in reproducing the molecular structures comprising the heavier (4–6 groups) and lighter (1–3 groups) elements turned out to be 0.007 Å. The smallest difference in the MUEs was obtained for N12-SX (ΔMUE = 0.001 Å), mPW1PW91 (ΔMUE = 0.002 Å), SOGGA11-X (ΔMUE = 0.002 Å), PBE0 (ΔMUE = 0.002 Å), and mPW1PBE (ΔMUE = 0.002 Å), indicating the robust performance of these methods across the different groups in the periodic table (see Figure S3). On the other hand, the largest difference in MUEs obtained for the structures comprising the elements of the 1–3 and 4–6 groups was obtained for SOGGA11 (ΔMUE = 0.019 Å), BLYP (ΔMUE = 0.015 Å), VSXC (ΔMUE = 0.014 Å), and B97-D (ΔMUE = 0.014 Å), indicating significantly better performance of these methods for the structures of the molecules comprising lighter elements. The larger MUEs obtained for the molecules comprising the heavier elements belonging to the 4–6 groups can be explained by a variety of factors. For the correlated WFT methods, such as MP2 and CCSD, the core–valence correlation effects become more important for the molecules comprising elements from group 4 and higher, requiring larger and more flexible basis sets, especially in the subvalence region.^194,258 For the semiempirical DFT methods, the absence of the heavy elements from the fourth period and above in the fitting set, as in the B3LYP method, might result in the larger errors documented for the molecules comprising these elements. Indeed, practically no difference in performance for molecules involving the lighter (1–3 groups) and heavier (4–6 groups) elements was found with the PBE and PBE0 methods for which no empirical parameters were used.

Finally, for all the methods, relativistic effects become more important and start influencing the molecular structures significantly. In the current work scalar relativistic effects were accounted for through using relativistic effective core potentials; although popular in the density functional theory research community, it might lead to some discrepancies with the AE scalar relativistic calculations.²⁵⁸ Moreover, the spin–orbit effects become nonnegligible even for the closed-shell species when the molecules comprise heavy elements, especially 5p/6p elements.^41,259−267 These effects may cancel themselves out when treated partially or neglected, leading to small errors. A detailed investigation of all these effects and their interplay is beyond the scope of the present study. Here, we restrict ourselves to the analysis of the errors in the structures of the molecules comprising the 5p and 6p elements. The MUEs and MSEs for the internuclear distances of the DFT- and WFT-optimized molecules comprising the elements of the 5p and 6p block relative to the corresponding microwave spectroscopy data are given in Figure S4. Because the structures of these molecules are influenced by spin–orbit coupling, which we neglected in the present study, the best performing methods in our work most likely experienced some fortuitous cancellation of errors. The best-performing methods for these complexes turned out to be N12 (MSE/MUE = 0.000/0.014 Å), mPW1PW91 (MSE/MUE = 0.001/0.014 Å), PBE0 (MSE/MUE = 0.000/0.014 Å), mPW1PBE (MSE/MUE = −0.001/0.014 Å), and APF (MSE/MUE = 0.003/0.014 Å) followed closely by CCSD (MSE/MUE = −0.004/0.015 Å). Because the calculations involving spin–orbit coupling are significantly more computationally expensive, the best-performing methods can be recommended to perform geometry optimizations while the spin–orbit contributions can be introduced later in a SP fashion. Similarly to what was found for the molecules comprising the elements of the 4–6 groups, quite large errors have been documented for SOGGA11, which might be explained by the fact that molecules containing the 5p/6p elements were not a part of the training set used to parameterize this method.

Further, we analyzed the errors obtained for distances involving the 1 and 2 group metals as well as d metals. In Figure S5, the MSEs and MUEs for the internuclear distances involving the s-metals (1/2 group elements) of the DFT- and WFT-optimized molecules relative to the corresponding microwave spectroscopy data are shown. The MSE/MUE averaged over all the tested methods turns out to be 0.008/0.016 Å, which is comparable with the average errors of 0.005/0.016 Å obtained for a dataset containing all molecules (see section 3.1). Vanishing MSEs and small MUEs were obtained for PBE0 (MUE = 0.008 Å), followed closely by MPW1PW91 (MUE = 0.009 Å), SOGGA11-X (MUE = 0.009 Å), mPW1PBE (MUE = 0.009 Å), APF (MUE = 0.009 Å), and N12-SX (MUE = 0.009 Å). The CCSD and MP2 methods slightly overestimated the distances of MSE/MUE of 0.013/0.013 and 0.012/0.014 Å, respectively. Finally, the largest errors were found for the B97D (MSE/MUE = 0.038/0.040 Å) functional. Remarkably, with the MSE/MUE of −0.018/0.019 Å, the APF-D functional was found to provide the distances that are in general too short; meanwhile, its dispersion-free counterpart, APF, resulted in a significantly better MUE of 0.009 Å and practically vanishing MSE. This indicates that the dispersion term in the APF-D functional should probably be re-evaluated. In Figure S6, the MSEs and MUEs for the internuclear distances involving the d-metals (Cu and Ag) of the DFT- and WFT-optimized molecules relative to the corresponding microwave spectroscopy data are shown. The MSE/MUE averaged over all the tested methods tuned out to be 0.007/0.018 Å, which is comparable to that obtained for the distances involving the s-metals. The best performing methods with an MUE of only 0.004 Å and vanishing MSE turned out to be TPSS and its hybrid version, TPSSh, and the mPW1PBE method. Particularly, small MUEs of only 0.005 Å and vanishing MSEs have also been obtained for the PBE, BP86, BPBE, mPW1PW91, PBE0, and APF methods. The worst performing method with an MSE/MUE of −0.052/0.052 Å turned out to be the MN12-L functional followed closely by SOGGA11 with an MSE/MUE of 0.050/0.050 Å. Remarkably, similarly to that found for the s-metals, the dispersion-corrected APD-F functional with an MSE/MUE of −0.035/0.037 Å was found to perform significantly worse compared with its dispersion-free counterpart, strongly indicating a need for re-evaluation of the dispersion term in the APF-D method. When it comes to the WFT methods, the CCSD resulted in acceptable MSE/MUE both equal to 0.011 Å. Meanwhile, the MP2 methods with an MSE/MUE of −0.037/0.037 Å predicted distances that were rather too short. The poor performance of the MP2 method for transition metal complexes that we found here is in agreement with earlier reports.^257,268

3.3. Errors in Absolute Electronic Energies Arising from Geometry Optimization and SP Energy Evaluation with Various Methods

We performed a series of additional calculations to estimate the potential deviations in the absolute electronic energy because of the different methods used for geometry optimization and SP energy evaluation. The deviation in the absolute energy of a single compound corresponds to the maximum error in the relative energies obtained with the composite schemes because no cancellation between the reactants and products is assumed. First, we calculated the CCSD SP energy of three-dimensional molecular structures obtained with one of the best performing DFT functionals, PBE0. Then, we calculated the CCSD SP energy on the structures from one of the worst performers for molecular structures in the current study, M11-L. Finally, we calculated the difference between the CCSD SP energies from the two protocols described above, that is, CCSD/aug-cc-pwCVTZ//PBE0/aug-cc-pwCVTZ and CCSD/aug-cc-pwCVTZ//M11-L/aug-cc-pwCVTZ, and the CCSD SP energy on a consistently optimized geometry. That is, CCSD/aug-cc-pwCVTZ//CCSD/aug-cc-pwCVTZ. The results are summarized in Figure 3 and Table S1. The average difference in electronic energy obtained with the CCSD/aug-cc-pwCVTZ//PBE0/aug-cc-pwCVTZ protocol compared with the CCSD energies on consistently optimized geometries turned out to be 0.0 kcal/mol, with a maximum deviation of 0.5 kcal/mol obtained for the CaO molecule. By contrast, the difference in energy obtained with the CCSD/aug-cc-pwCVTZ//M11-L/aug-cc-pwCVTZ protocol compared with CCSD energies on consistently optimized geometries was significantly larger, 0.3 kcal/mol, with a maximum deviation of 2.3 kcal/mol obtained for the GeF₂ molecule. These results suggest that an error of a few kcal/mol in SP energies can easily be obtained even for relatively small systems if inaccurate three-dimensional structures are used.

Figure 3.

Difference in CCSD/aug-cc-pwCVTZ(-PP) SP energies on PBE0/aug-cc-pwCVTZ(-PP) and M11-L/aug-cc-pwCVTZ(-PP) compared with CCSD/aug-cc-pwCVTZ(-PP) SP energies on consistently optimized geometries.

4. Conclusions

We used very accurate gas-phase three-dimensional molecular structures of di- and triatomic molecules from microwave spectroscopy measurements to assess the accuracy of a battery of DFT and WFT methods with aug-cc-pwCVTZ basis sets. When we consider the performance of the methods in reproducing all internuclear distances in all the molecules, we find that the most accurate molecular structures were obtained by hybrid and hybrid meta-GGA functionals, with the B3PW91, APF, TPSSh, mPW1PW91, PBE0, mPW1PBE, B972, and B98 functionals providing the lowest errors. Among these functionals, we recommend APF, mPW1PW91, mPW1PBE, and PBE0 because they resulted in the smallest errors for the subsets of internuclear distances that we studied as well as for the distances involving s-metals, d-metals, and 5p/6p-containing molecules. Also, we found a negligible difference in performance for molecular structures containing lighter (1–3 periods) and heavier (4–6 elements) for these methods. The CCSD method predicted molecular geometries of similar quality although at considerably higher computational cost. Quite low errors were also found for the nonseparable gradient local-GGA functional N12, which we recommend for projects with low CPU budgets. We detected the largest errors with local GGA and meta-GGA functionals, namely BLYP and M11-L, suggesting that these methods not be used if accurate three-dimensional molecular structures are the goal. Because we encountered some numerical problems with integrating the exchange–correlation part^269,270 and obtained large dispersion of errors with some Minnesota functionals, particularly MN12-L, M11, M06-L, SOGGA11, and VSXC, we also recommend that these methods not be used for geometry optimizations. The large and nonsystemic nature of the errors in internuclear distances is especially concerning in quantitative structure–activity relationship studies that correlate such parameters with chemical activity, for example, the correlation of the length of the Ru=C bond in Grubbs complexes with their catalytic activity in olefin metathesis.²⁷¹ Apart from these methods, we found that B97D performs poorly for distances involving s-metals and should probably be avoided as well. The DFT-D3 dispersion correction was found to have a negligible effect on internuclear distances. By contrast, the APF-D dispersion correction led to significant shortening up to 0.064 Å (AgI) of the bonds in Ag halides, BaO, BaS, BaF, BaCl, Cu halides, and Li and Na halides and hydrides. Given that these results do not agree well with very accurate structures derived from microwave spectroscopy, we believe that the dispersion correction in the APF-D method should be reconsidered.

When it comes to relatively large and nonrigid molecules for which intramolecular interactions are critical, significantly influencing the structure, particular recommendations on the geometric optimization method remain difficult to give. In our opinion, because of the large dispersion of errors that we found and problems related to noise in the integration of the exchange–correlation part of the functionals, the Minnesota models should either be avoided or applied with extreme caution. The DFT-D3 methods are perhaps a better choice than the Minnesota models. Experimental structures in the size range discussed here are available only from diffraction methods, which are significantly contaminated by the effects of molecular vibrations (GED) or librations and crystal-packing forces (X-ray), which may be comparable or even larger than the errors in internuclear distances for particular functional, making validation of DFT results a difficult prospect. Moreover, application of accurate WFT methods to obtain accurate geometries of relatively large molecules of 50–100 atoms to validate DFT methods is also impractical as robust results can be obtained only with the CCSD(T) method, relative large basis sets, and correlated subvalence electrons. Creation of a database of accurate molecular geometries for benchmarking of methods that require less-expensive computational chemistry methods would be a good subject for future work. Finally, we performed a series of additional calculations to estimate the potential errors in the absolute electronic energy from different methods used for geometry optimization and SP energy evaluation. Our calculations showed that an error of few kcal/mol in SP energies can easily be obtained even in relatively small systems if inaccurate geometries are used.

Supporting Information Available

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsomega.7b01203.

• Figures S1–S6, Table S1 (mentioned in the main text), tabulated errors forming the basis of Figures 1–3 and S1–S6, and Cartesian coordinates (Å) of PBE, BP86, BLYP, BPBE, HCTH/407, B97-D, B97-D3, SOGGA11, N12, OLYP, TPSS, VSXC, M06-L, M11-L, MN12-L, B3PW91, B3LYP, mPW1PW91, SOGGA11-X, PBE0, mPW1LYP, mPW1PBE, X3LYP, B971, B972, B98, B1LYP, APF, APF-D, LC-ωPBE, CAM-B3LYP, N12-SX, ωB97X-D, BMK, TPSSh, M06, M11, MN12-SX, MP2, and CCSD optimized geometries (PDF)

The authors declare no competing financial interest.