A Molecular “Thermometer” for Measuring Effective Non‐Local Exchange

ABSTRACT

Non‐local exchange (NLX) is a key ingredient for accurate density functional calculations, but its effective strength is often hard to quantify beyond simple global hybrids. To this end, we introduce a molecular probe based on the isomerization of hexaethynylbenzene to carbo‐benzene, a reaction with exceptional sensitivity to exchange effects. Using the isomerization energy, we define a simple relative measure, $t_{\mathrm{X}}$, that gauges the effective NLX of a method on a scale ranging from 0 for the local density approximation to 100 for Hartree‐Fock. Across a broad set of density functional approximations, $t_{\mathrm{X}}$ closely tracks the formal Hartree‐Fock exchange content of global hybrids, while revealing that most common GGAs, meta‐GGAs, and global hybrids still provide too little effective exchange. Several range‐separated hybrid, double hybrid, and some local hybrid functionals perform much better and closely approach the near‐basis‐set‐limit coupled‐cluster reference value of $t_{\mathrm{X}}=60$. The $t_{\mathrm{X}}$ values of various semiempirical quantum mechanical and machine‐learned interatomic potential methods indicate that they empirically account for NLX effects to a varying degree. Thus, the proposed reaction serves as a compact thermochemical benchmark for assessing and developing methods that aim to reliably describe NLX.

The isomerization of hexaethynylbenzene to carbo‐benzene acts as a sensitive “thermometer” for the effective treatment of non‐local exchange effects in density functional, semiempirical, and machine‐learning methods.

1. Introduction

Density functional theory (DFT), in its efficiently applicable Kohn‐Sham formulation, is the most widely used method in computational chemistry. The key ingredient in modern density functional approximations (DFAs) [1, 2] is the description of the exchange‐correlation (XC) energy. According to J. Perdew's metaphoric picture of Jacob's ladder (see Table 1) [3], DFAs are classified into rungs according to increasing accuracy starting from a purely local density approximation (LDA) [4, 5], to a generalized‐gradient approximation (GGA) on rung two [6], meta (m)‐GGA on rung three [7, 8], and hybrid [9] or double hybrid (DH) [10] on rungs four and five, respectively. Especially, the admixture of non‐local Fock‐exchange from Hartree‐Fock (HF) theory [11, 12] introduced by A. Becke in 1993 (see Equation 1) marked a milestone in the development of chemically accurate DFAs [13, 14]. So‐called hybrid functionals are among the most widely used DFAs, including functionals such as B3LYP or PBE0 [15, 16].

TABLE 1.

Contribution of non‐local Fock exchange ($a_X$) in different types of DFAs according to “Jacob's ladder.” Here, $r$ denotes a real‐space coordinate, $r_{12}$ is the inter‐electronic distance, and $f$ is DFA‐specific function.

Rung on “Jacob's ladder”	$a_X$
Local (LDA)	0
Semi‐local GGA	0
Semi‐local meta‐GGA	0
Hybrid	> 0
Range‐separated hybrid	$f\left(r_{12}\right)$
Local hybrid	$f\left(r\right)$

Since there is a wide variety of approximations for the XC functional, comprehensive benchmark studies [17] and best practice guidelines [18] are indispensable. However, for an initial assessment of practical applicability, classification based on the admixture of non‐local Fock‐exchange is often already helpful. Overwhelming empirical evidence suggests that the treatment of non‐local exchange (NLX) effects is crucial for accurately describing thermochemistry and reaction barriers [17, 19, 20, 21], spectroscopic parameters [22, 23, 24, 25, 26, 27], and even equilibrium structures (e.g., bond‐length alternation [28, 29]).

Over the past few decades, many DFAs have been proposed that go beyond a simple linear mixing of local and NLX components in so‐called global hybrids (GHs).

$$E_{\mathrm{X}}=\left(1-a_X\right)\left(E_{\mathrm{X}}^{\mathrm{LDA}/\left(\mathrm{m}\right)\mathrm{GGA}}\right)+a_X{E}_{\mathrm{X}}^{\text{Fock}}.$$ (1)

By parameterizing the mixing based on the inter‐electronic distance $r_{12}$, range‐separated hybrids (RSHs) [30] combine the correct long‐range exchange behavior of HF with beneficial semi‐local DFA exchange at short range. Local hybrids (LHs) [31] further increase flexibility by replacing the parameter $a_X$ in Equation (1) with a function of the real‐space electronic coordinate, tailoring the exchange admixture to the local electronic structure. In recent years, machine learning (ML) has also entered the field of DFA development [32, 33], frequently by targeting an optimal admixture of NLX in GHs and LHs [34, 35, 36, 37, 38]. The recent Skala mGGA [39] completely avoids admixture of Fock exchange in favor of a non‐local ML‐based XC functional, which may in the future warrant the addition of further intermediate rungs in Jacob's ladder. Nevertheless, it is often unclear how such functionals should be categorized and compared, given the lack of a single numerical value $a_x$ quantifying the extent of NLX effects.

In the course of current tight‐binding electronic structure developments [40], we stumbled over a seemingly simple hydrocarbon isomerization reaction of hexaethynylbenzene to so‐called carbo‐benzene [41, 42] that shows extreme sensitivity to NLX effects. Notably, the number of C‐C and C‐H bonds, as well as the number of sp (12) and sp² (6) hybridized carbon atoms, remain unchanged during the reaction. We therefore refer to it as an (almost) homodesmotic bond rearrangement isomerization (BRI) [43, 44]. Strictly, the reaction is classified as isogyric due to the allene substructure in the resonance structures of carbo‐benzene (see Scheme 1). However, we prefer the nomenclature as almost homodesmotic based on the looser definition of equal numbers of spⁿ‐hybridized carbons without explicit reference to the bond order. The computed reaction energy $\mathrm{\Delta }E=E\left(\mathbf{2}\right)-E\left(\mathbf{1}\right)$ for the isomerization of hexaethynylbenzene (1) to carbo‐benzene (2), as shown in Scheme 1, varies from about 66 kcal mol⁻¹ at the HF level to −14 kcal mol⁻¹ for LDA. The AO‐basis‐set‐converged coupled‐cluster (CC) reference value of 34.5 kcal mol⁻¹ lies roughly in the middle of this range (vide infra). Normally, such huge effects are only observed in species with fractional charges and/or spin, for example, the prototypical H₂ ⁺ dissociation [50, 51], making their occurrence in a reaction involving only neutral closed‐shell molecules particularly striking.

SCHEME 1.

Bond rearrangement isomerization (BRI) of hexaethynylbenzene to carbo‐benzene. Top: Equilibrium geometries of both compounds on the RI‐MP2 [45, 46]/def2‐TZVPP [47, 48, 49] level of theory. Bottom: Corresponding Lewis structures.

Qualitatively, this observation can be rationalized by more pronounced $\pi$‐electron delocalization in (2) compared to (1), as indicated by the anisotropy of the current‐induced density (ACID) [52, 53, 54] and magnetically‐induced current density (MICD) [55, 56, 57] shown together in Figure 1. The integrated magnetically‐induced ring current increases more than fourfold for the BRI (from 8.1 nA T⁻¹ in (1) to 34.1 nA T⁻¹ in (2), see Supporting Information for details), which is in line with extensive previous studies on the aromaticity of (2) and various related compounds [59, 60, 61, 62, 63, 64]. Local or semi‐local DFAs generally overstabilize states with such delocalized electron densities, leading to the so‐called delocalization error [65, 66]. Analysis of the magnetically‐induced ring current and density‐corrected DFT calculations show that the resulting energetic error of semi‐local DFAs for the BRI is largely functional‐driven, that is, it arises from the limitations of the approximate energy functional rather than an excessively delocalized density (see Section S1 of the Supporting Information for details) [67].

FIGURE 1.

Combined scalar ACID (yellow, isovalue: 0.002) and vectorial MICD plot (red, selectively at 1 Bohr above the molecular plane vectors with a magnitude between 0.001 and 0.01 a.u., and scaled by a factor 5 for visibility) calculated at the PBE0 [16]/GIAO‐def2‐TZVPP [47, 48, 58] level of theory for hexaethynylbenzene (a) and carbo‐benzene (b).

Here, we propose the above reaction as a quantitative measure of NLX effects in thermochemistry. The idea is to provide a convenient test of the treatment of NLX effects in existing or newly proposed DFAs, even if $a_X$ is zero or given by a potentially complicated ML procedure. This approach also applies to commonly used RSHs and more recent LHs, for which the effective amount of exact exchange is often unclear, or even to semiempirical quantum mechanical (SQM) or machine‐learned interatomic potential (MLIP) methods without explicit (or only very approximate) account for NLX effects. Orbital energy gaps are often regarded as an alternative metric that focuses more on the XC potential than on the energy, and thus complements the thermochemical approach pursued here. While none of these approaches can replace comprehensive thermochemical benchmarking, they can provide valuable guidance in selecting a problem‐specific method or DFA.

2. Computational Details

The geometries of (1) and (2) were optimized at the RI‐MP2 [45, 46]/def2‐TZVPP [47, 48, 49] level of theory using the ORCA program package in version 6.0.1 [68]. All DFT calculations (except for the composite “‐3c” methods) consistently employ the def2‐TZVPP basis set and were mostly performed with ORCA using the libXC library version 7.0.0 [69] for the XC functionals. As an exception, LH calculations were performed with the Turbomole program version 8.0 [70], while selected DHs ($\omega$B97M(2), PTPSS‐D3, and XYG3) were calculated with Q‐Chem version 6.3.1 [71]. The Skala and DM21 ML DFAs were used in PySCF version 2.11.0 [72, 73].

The xtb program version 6.7.1 [74] in combination with the tblite 0.4.0 library was used for GFN1‐xTB [75] and GFN2‐xTB [76] calculations, while g‐xTB calculations employ the standalone development program [40]. PM6‐D3H4X calculations were performed with MOPAC 2016 [77] via a wrapper in xtb 6.7.1. MLIPs were accessed via ASE [78] interface to the AIMNet2 [79] (version parametrized with $\omega$B97M‐D3 reference values) and UMA‐sm [80] (version 1.0, with the task parameter set to omol) models.

Reference reaction energy values were computed with PNO‐LCCSD(T)‐F12b [81, 82, 83]/aug‐cc‐pV5Z [84, 85] (cc‐pV5Z for H; employing corresponding aug‐cc‐pV5Z+/JKFIT [86, 87] and aug‐cc‐pV5Z/MP2FIT [88] auxiliary basis sets) [83, 84, 85] with VTIGHT domain settings [89] and CABS singles correction using Molpro 2025.1 [90, 91]. The residual error for this highly accurate relative energy is estimated to be below 0.25 kcal mol⁻¹ (see Supporting Information for details).

To investigate possible multi‐reference character of the electronic wavefunction, fractional occupation number weighted electron density (FOD) analysis [92, 93, 94] was performed at the PBE0‐D4/def2‐TZVPPD [95] level of theory with ORCA 6.0.1, employing an electronic temperature of $T_{\mathrm{el}}$ = 8830.5 K. Additionally, the first triplet excited states were calculated at the $\omega$B97M‐V/def2‐TZVPPD UKS level of theory using the Turbomole program.

For the analysis of the delocalization and aromaticity in both systems, ACID [52, 53, 54] and MICD [55, 56, 57] were calculated at the PBE0/def2‐TZVPP level of theory using gauge‐including atomic orbitals (GIAOs) [58, 96] via the TURBOMOLE interface [97] to the GIMIC program version 2.2.1 [98]. Here, the magnetic field was applied perpendicular to the molecular plane, and the resulting MICD was integrated through one of the six half‐planes intersecting the carbon–carbon bonds of the original hexagonal benzene structure (see Supporting Information for details). ACID and MICD were plotted using ParaView version 5.13.2 [99].

3. Results and Discussion

The systems involved in the BRI are chemically simple, have high symmetry ($D_{6h}$) that enables fast calculations, and are insensitive to changes in the basis set and the optimized geometry used (see Supporting Information for details). Possible multi‐reference character of the electronic wavefunctions, especially for (2), has been investigated but found to be relatively mild: an analysis based on the FOD revealed only moderate amounts of “hot” electrons for (2) (slightly less than in, e.g., tetracene) and even less in (1) (see Supporting Information for details). This is further supported by relatively large first triplet excitation energies of 66.7 and 36.6 kcal mol⁻¹ for (1) and (2), respectively. The multi‐reference diagnostics derived from the CCSD(T) calculations, such as T1, D1, and %TAE[(T)] [100], also show no signs for particularly difficult electronic structures. The corresponding values are well below the usual thresholds (T1 < 0.05, D1 < 0.150, %TAE[(T)] < 5%) [101] for both 1 (T1 = 0.013, D1 = 0.030, %TAE[(T)] = 2.84%) and 2 (T1 = 0.013, D1 = 0.032, %TAE[(T)] = 3.24%) and can therefore be clearly classified as single‐reference systems. We note, however, that with higher amounts of Fock exchange, the $D_{6h}$ symmetry reduces to $D_{3h}$ as the triple and double bonds localize and the mesomerization depicted in Scheme 1 breaks down. The effect of long‐range (London) dispersion interactions is relatively small (e.g., 10.1 kcal mol⁻¹ with B3LYP‐D4 or 16.2 kcal mol⁻¹ with HF‐D4). Thus, dispersion corrections are included for consistency in the following, but not further discussed (see Supporting Information for the actual dispersion contributions for the tested DFA).

The proposed relative (percentage) NLX measure $t_{\mathrm{X}}$ is defined as

$$t_{\mathrm{X}}=100\cdot \frac{\mathrm{\Delta }E-\mathrm{\Delta }{E}_{\mathrm{LDA}}}{\mathrm{\Delta }{E}_{\mathrm{HF}}-\mathrm{\Delta }{E}_{\mathrm{LDA}}},$$ (2)

where $\mathrm{\Delta }E$ is the computed reaction energy ($\mathrm{\Delta }E=E\left(\mathbf{2}\right)-E\left(\mathbf{1}\right)$), such that $t_{\mathrm{X}}=100$ for HF and $t_{\mathrm{X}}=0$ for LDA. $t_{\mathrm{X}}$ assumes a linear relation of the BRI reaction energy and NLX effects, motivated by previous observations that energetic quantities can show approximately linear sensitivities to the fraction of Fock exchange in hybrid density functionals [19, 102, 103]. However, we emphasize that $t_{\mathrm{X}}$ represents an effective measure, which is conveniently mapped onto the extreme scale between LDA and HF and not a literal percentage of NLX. It should offer a helpful scale to compare a DFA's ability to capture NLX effects and inform the DFAs choice, given a problem‐specific NLX sensitivity. Since the CC reference reaction energy amounts to 34.5 kcal mol⁻¹, the correct (CC‐based) effective NLX for this particular reaction is $t_{\mathrm{X}}=60.0$. We note that defining $t_{\mathrm{X}}$ via the reaction energy is not the only possible way to map the strong NLX dependence of the BRI into a relative measure. Quantities such as the gap between the highest‐occupied and lowest‐unoccupied molecular orbitals in 1 and 2 show a clear correlation with the $t_{\mathrm{X}}$ value of a given method (see Section S7 of the Supporting Information for details), but remain restricted to molecular orbital‐based mean‐field methods. Table 2 collects both thermochemical data and the resulting $t_{\mathrm{X}}$‐measure for a series of common DFAs from all rungs of Jacob's ladder (see Supporting Information for further examples).

TABLE 2.

Formal amount of Fock exchange $a_{\mathrm{X}}$ in %, reaction energies $\mathrm{\Delta }E$ in kcal mol⁻¹, and the NLX measure $t_{\mathrm{X}}$ defined in Equation (2) for a selection DFAs and HF (see Supporting Information for further examples). The methods are ordered based on the $t_{\mathrm{X}}$ measure. All DFT calculations employ the def2‐TZVPP basis set and a functional‐specific dispersion correction.

DFA	$a_{\mathrm{X}}$	$\mathrm{\Delta }E$	$t_{\mathrm{X}}$
LDA [4, 104]	0	−13.6	0.0
PBE‐D4 [9]	0	−10.7	3.6
M06L‐D4 [105]	0	−9.2	5.4
TPSS‐D4 [106]	0	−7.8	7.2
r2SCAN‐D4 [107]	0	−6.5	8.8
BLYP‐D4 [108, 109]	0	−6.2	9.2
B97M‐V [110]	0	−4.7	11.1
revPBE‐D4 [111]	0	−4.4	11.4
SCAN‐D4 [112]	0	−4.4	11.5
RPBE‐D4 [113]	0	−3.8	12.3
MN15‐L‐D3(0) [114]	0	−3.7	12.4
M11‐L‐D3(0) [115]	0	−3.3	12.9
TPSSh‐D4 [116]	10	−1.9	14.6
r2SCANh‐D4 [117]	10	−1.1	15.6
HSE06‐D4 [118]	25–0	2.7	20.3
B3LYP‐D4 [15]	20	4.5	22.6
PBE0‐D4 [16]	25	5.2	23.4
PW6B95‐D4 [119]	28	7.1	25.9
r2SCAN0‐D4 [117]	25	7.4	26.1
M06‐D4 [105]	27	10.1	29.5
PBE0‐DH [120]	50	12.6	32.7
TMHF‐D4 [121]	0–100	13.0	33.2
B2PLYP‐D4 [10]	53	13.2	33.5
PBE38‐D4 [17]	37.5	13.5	33.8
LH20t‐D4 [122]	0–100	14.3	34.8
SCS‐B2PLYP‐D4 [123] a	53	17.1	38.3
MN15‐D3(BJ) [124]	44	18.0	39.4
Pr2SCAN50‐D4 [125]	50	18.9	40.5
PWPB95‐D4 [126]	50	18.9	40.6
SOS‐B2PLYP‐D4 [123] a	53	19.1	40.8
CF22D [35]	46.3	19.2	41.0
PBE‐QIDH [127]	69	19.8	41.7
LH25nP‐D4 [38]	0–100	20.2	42.2
Skala‐D3 [39]	0	20.9	43.1
LH24n‐D4 [37]	0–100	21.1	43.3
r2SCAN50‐D4 [117]	50	21.8	44.2
CAM‐B3LYP‐D4 [128]	19–65	22.7	45.3
M06‐2X‐D3(0) [105]	54	23.4	46.1
BHHLYP‐D4 [13]	50	23.4	46.2
$\kappa$Pr2SCAN50‐D4 [125]	50	24.9	48.0
DM21‐D3(BJ) [34]	0–100	26.6	50.2
revDSD‐PBEP86‐D4 [129]	69	26.6	50.2
$\omega$B97XD [130]	19.6–100	28.3	52.3
$\omega$LH23tdE‐D4 [131]	0–100	31.8	56.7
$\omega$r2SCAN‐D4 [125]	0–100	31.9	56.9
$\omega$Pr2SCAN50‐D4 [125]	50–100	32.6	57.7
LC‐BLYP‐D4 [132]	0–100	32.7	57.8
$\omega$LH22t‐D4 [133]	0–100	33.4	58.6
$\omega$LH25tdE‐D4 [134]	0–100	34.0	59.4
CCSD(T) reference	—	34.5	60.0
RSX‐QIDH [135]	69–100	35.1	60.8
M11‐D3(BJ) [136]	42.8–100	35.6	61.4
$\omega$B97M(2) [137]	62.2	36.0	61.9
CAM‐QTP‐01‐D4 [138]	23–100	36.3	62.3
$\omega$B97M‐V [139]	15–100	36.7	62.8
$\omega$B97X‐V [140]	16.7–100	37.4	63.7
RSX‐0DH [141]	50–100	39.5	66.3
$\omega$B97X‐D4 [142]	16.7–100	41.2	68.5
SCS‐$\omega$B2PLYP‐D4 [123] a	53–100	41.5	68.8
LC‐PBE‐D4 [143]	0–100	49.8	79.2
HF‐D4 [11, 12]	100	66.5	100.0

^a DFT‐D4 parameters adapted from the canonical DH.

Figure 2 visualizes the proposed NLX measure, $t_{\mathrm{X}}$, as a function of the formally admixed exact Fock exchange fraction, $a_{\mathrm{X}}$. We focus initially on the subset of GHs (blue), where the non‐locality of the exchange effects is unambiguously quantified by $a_{\mathrm{X}}$. Most notably, $t_{\mathrm{X}}$ correlates closely with the $a_{\mathrm{X}}$ fraction for the investigated GHs with a high Pearson correlation coefficient (PCC) of 0.961. Data points above the regression line indicate that the DFA intrinsically accounts for NLX effects through its semi‐local components. Linear regression maps the range from (semi‐)local DFAs ($a_{\mathrm{X}}=0\%$) to HF ($a_{\mathrm{X}}=100\%$) onto a slightly compressed $t_{\mathrm{X}}$ range of roughly 7–83. This compression likely arises from the predominant use of semi‐local DFAs in the construction of GHs, which already slightly reduce the delocalization error of the purely local LDA used to define $t_{\mathrm{X}}$ (vide infra). When comparing GHs based on different DFAs with the same $a_{\mathrm{X}}$ fraction, we observe a slight scatter in the $t_{\mathrm{X}}$ values (e.g., between 23.4 (PBE0‐D4) and 26.2 (r²SCAN0‐D4) at $a_{\mathrm{X}}=25\%$). Such differences arise from NLX effects already present in the semi‐local components of the GH, which seem surprisingly small at only a few percent. Thus, even for modern third‐rung mGGA components, exact exchange admixture still plays the predominant role in counteracting delocalization error. However, none of the commonly employed GHs considered here admixes enough exact exchange to reach the reference value of $t_{\mathrm{X}}=60.0$ (BHHLYP‐D4 with $a_{\mathrm{X}}=50\%$ achieves the highest $t_{\mathrm{X}}$ value for a GH at 46.2). Extrapolation of the linear $t_{\mathrm{X}}$ versus $a_{\mathrm{X}}$ trend for the tested global hybrids suggests that a conventional GH would require approximately 76.5% exact exchange to do so.

FIGURE 2.

Proposed relative NLX measure $t_{\mathrm{X}}$ (see Equation 2) versus the formal amount of exact Fock exchange ($a_{\mathrm{X}}$) for a series of GGA (orange), mGGA (red), and GH (blue) DFAs (corresponding $t_{\mathrm{X}}$ values in Table 2 and the Supporting Information). The dashed yellow line corresponds to a linear regression of $t_{\mathrm{X}}$ with $a_{\mathrm{X}}$ for the hybrid DFAs with a confidence interval of 95% (translucent band). The dashed horizontal green line corresponds to the expected $t_x$ for the reference reaction energy.

After validating the $t_{\mathrm{X}}$ measure for GHs, we can now compare DFAs where the degree of non‐locality in the exchange treatment is less apparent. Figure 3 plots $t_{\mathrm{X}}$ for the tested DFAs grouped by their respective functional classes. As in the case of GHs, all semi‐local (m)GGAs (orange and red) underestimate NLX effects, yielding mostly $t_{\mathrm{X}}$ values below 20. While third‐rung mGGAs have, on average, a slightly higher $t_{\mathrm{X}}$ than rung‐two GGAs, the differences are negligible compared to the remaining error. The ML‐based Skala functional represents the only striking exception: although $a_{\mathrm{X}}$ is formally zero, its effective $t_{\mathrm{X}}$ value of 43.1 is comparable to GHs with large fractions of Fock exchange. This suggests that the non‐local construction of the Skala XC functional effectively emulates the role of exact exchange in hybrids, even though it is still not quite sufficient for the present example.

FIGURE 3.

Proposed relative NLX measure $t_{\mathrm{X}}$ (see Equation 2) for various functionals grouped as GGA (orange), mGGA (red), GH (blue), RSH (turquoise), LH (pale green), DH (green) DFAs. The dashed vertical green line corresponds to the expected $t_x$ needed to exactly match the CC reference reaction energy.

According to the introduced “thermometer” for NLX, DFAs require $t_{\mathrm{X}}$ values between 55 and 65 to reproduce the CC reference reaction energy with reasonable accuracy (10% or ±3.5 kcal mol⁻¹). Among all tested DFAs, most RSHs and several DHs fall into this region, whereas only a few LHs do so. CAM‐B3LYP‐D4 ($t_{\mathrm{X}}=45.2$) and LC‐PBE‐D4 ($t_{\mathrm{X}}=79.2$) are notable exceptions among the RSHs, illustrating two distinct issues. The former admixes Fock exchange only up to 65% at large inter‐electronic distances, rather than the full 100% expected from Hartree‐Fock theory [128], whereas the latter includes too much Fock exchange already at short range due to its unusually large range‐separation parameter of 0.47 Bohr⁻¹ [143]. In particular, the long‐range admixture of 100% Fock exchange appears to be crucial, as is also evident from the tested LHs and DHs. While all tested range‐separated LHs fall within the narrow $t_{\mathrm{X}}$ range between 55 and 60, a purely local modification of the Fock exchange admixture appears insufficient (e.g., LH20t‐D4 with $t_{\mathrm{X}}=34.8$). Even ML‐based local mixing functions, such as those used in LH25nP‐D4 ($t_{\mathrm{X}}=42.2$) and DM21‐D3(BJ) ($t_{\mathrm{X}}=50.2$), cannot fully eliminate the delocalization error. For DHs, a similar picture emerges, with all range‐separated DHs having $t_{\mathrm{X}}$ values between 55 and 69, while global DHs yield lower values (e.g., B2PLYP with $t_{\mathrm{X}}=33.5$ and revDSD‐PBEP86‐D4 with $t_{\mathrm{X}}=50.2$). Taken together, these results suggest that the $t_{\mathrm{X}}$ measure is predominantly determined by long‐range NLX effects and is much less sensitive to other details of a functional, such as the treatment of correlation.

To assess whether the $t_{\mathrm{X}}$ measure extends meaningfully beyond systems characterized by strong $\pi$‐electron delocalization, we compare $t_{\mathrm{X}}$ with the mean deviations (MDs) for the BH76 barrier‐height benchmark set [17, 144]. This benchmark comprises transition states of small‐molecule reactions, including radicals and charged species. For selected functionals, the MDs for BH76 and $t_{\mathrm{x}}$ show moderate correlation, with $R^2=0.573$ (see Section S6 of the Supporting Information for detailed results). This correlation is mainly weakened by a small number of well‐performing (m)GGAs functionals, most notably BLYP and the Minnesota functionals MN15‐L, M11‐L, and M06‐L. Excluding these functionals increases the correlation to $R^2=0.858$, suggesting that $t_{\mathrm{X}}$ can serve as a useful qualitative indicator for missing NLX contributions in the description of transition states.

Finally, we extend the use of the NLX measure beyond the comparison of DFAs within a fixed basis set. The composite DFT (“‐3c”), SQM, and MLIP methods in Table 3 span a remarkably wide range of effective NLX performance, revealing both the promise and limitations of computationally efficient approaches. Among the composite DFT methods, the results follow the expected trend with the rungs of Jacob's ladder. B97‐3c, lacking any Fock exchange, yields a strongly negative reaction energy and a near‐zero $t_{\mathrm{X}}$ of 1.4, essentially behaving like a standard GGA. r²SCAN‐3c improves only modestly to $t_{\mathrm{X}}$ = 10.2, consistent with its mGGA character. PBEh‐3c, with 42% formal Fock exchange, reaches $t_{\mathrm{X}}$ = 33.4, placing it within the range of GHs but still well below the target of 60.0. Among the composite methods, the range‐separated $\omega$B97X‐3c performs best with $t_{\mathrm{X}}$ = 64.8, thereby slightly overshooting the CC reference. The pathological behavior of HF‐3c ($t_{\mathrm{X}}$ = 138.5) reflects the well‐known overcorrection of pure HF for delocalized systems, combined with limitations due to the minimal basis set used.

TABLE 3.

Formal amount of Fock exchange $a_{\mathrm{X}}$ in %, reaction energies $\mathrm{\Delta }E$ in kcal mol⁻¹, and the NLX measure $t_{\mathrm{X}}$ for a set of composite DFT, SQM, and MLIP methods.

DFA/SQM/MLIP	$a_{\mathrm{X}}$	$\mathrm{\Delta }E$	$t_{\mathrm{X}}$
B97‐3c [145]	0	−12.5	1.4
r2SCAN‐3c [146]	0	−5.4	10.2
PBEh‐3c [147]	42	13.2	33.4
$\omega$B97X‐3c [148]	16.7–100	38.3	64.8
HF‐3c [149]	100	97.3	138.5
GFN1‐xTB [75]	0	14.0	34.5
GFN2‐xTB [76]	0	9.8	29.2
g‐xTB [40]	15–100	36.7	62.7
PM6‐D3H4X [150, 151]	100	44.0	72.0
AIMNet2 [79]	—	6.7	25.3
UMA‐sm [80]	—	37.4	63.7

The tested tight‐binding (TB) methods show a clear progression with increasing sophistication. GFN1‐xTB and GFN2‐xTB, both formally without Fock exchange, yield $t_{\mathrm{X}}$ values of 34.5 and 29.2, respectively, already substantially above the GGA baseline. This suggests that their semiempirical parameterization implicitly captures some NLX‐like effects, as expected from the use of higher‐level reference data during their fitting. Most notably, g‐xTB achieves a $t_{\mathrm{X}}$ value of 62.7, approaching the CC target region impressively for an SQM method. Given that g‐xTB employs a range‐dependent approximate exchange treatment, this result is encouraging and suggests that the method captures a meaningful fraction of the physical NLX effects, even if the delocalization error is not fully corrected. PM6‐D3H4X, with formally 100% HF exchange in its semiempirical framework, overshoots with $t_{\mathrm{X}}$ = 72.0, although not to the same extent as HF itself.

The MLIP results are striking and illustrative of the current state of the field. UMA‐sm achieves a $t_{\mathrm{X}}$ value of 63.7, essentially matching the CC reference, which is remarkable for a method with no explicit quantum mechanical treatment of exchange. This implies that the training data and architecture, which is aimed at universality, have led to an implicit encoding of NLX‐like effects. In contrast, AIMNet2 yields a clearly incorrect reaction energy, suggesting that the electronic reorganization in this BRI is not captured accurately. Since both UMA‐sm and AIMNet2 aim to reproduce RSH‐DFT reference data, it appears reasonable to assume that such significant changes in electron delocalization are better represented in the larger OMol25 data set [152] used for UMA‐sm.

4. Conclusion

We presented a simple molecular “thermometer” that quantifies effective non‐local exchange (NLX) via the isomerization energy of hexaethynylbenzene to carbo‐benzene. The newly introduced relative $t_{\mathrm{X}}$ measure quantifies NLX effects on a scale ranging from 0 for local density approximation to 100 for Hartree‐Fock and requires only two comparably inexpensive energy evaluations. This thermochemistry‐based approach provides a practical complement to orbital‐gap or potential‐based diagnostics. Moreover, it is based on electronically relatively simple hydrocarbons, thereby avoiding interference with charge transfer or static correlation effects.

To test the $t_{\mathrm{X}}$ measure, we computed a high‐level coupled‐cluster reference value for the “thermometer” reaction and compared results across a comprehensive set of DFAs, SQM methods, and MLIPs. $t_{\mathrm{X}}$ closely correlates with the formal Fock exchange admixture in global hybrids (GHs), and we showed that the effective NLX in commonly used GGAs, mGGAs, and GHs is insufficient to describe the investigated reaction accurately. Meanwhile, range‐separated hybrids, as well as their local hybrid and double hybrid variants, closely approach the reference $t_{\mathrm{X}}$ value of 60. In particular, the full treatment of long‐range NLX effects via the admixture of 100% long‐range Fock exchange is crucial for removing the delocalization error dominant in this isomerization. These results imply that the introduced molecular “thermometer” may serve as a practical initial probe for choosing and developing DFAs, SQM methods, and MLIPs that accurately capture NLX in chemically relevant problems.

Conflicts of Interest

The authors declare no conflicts of interest.

Supporting information

Data S1: jcc70436‐sup‐0001‐Supinfo.pdf.

Data S2: jcc70436‐sup‐0002‐Supinfo.zip

Grimme S., Müller M., Froitzheim T., and Hansen A., “A Molecular “Thermometer” for Measuring Effective Non‐Local Exchange,” Journal of Computational Chemistry 47, no. 17 (2026): e70436, 10.1002/jcc.70436.

Data Availability Statement

The data that supports the findings of this study are available in the Supporting Information of this article.