Three decades of quantum science: how quantum chemistry transformed thermochemical database generation for benchmarking DFT and machine learning

Amir Karton

doi:10.1071/CH24130

RESEARCH ARTICLE (Open Access)

Previous Contents Vol 78(3)

Three decades of quantum science: how quantum chemistry transformed thermochemical database generation for benchmarking DFT and machine learning

Amir Karton

^A ^*

+ Author Affiliations

- Author Affiliations

^A School of Science and Technology, University of New England, Armidale, NSW 2351, Australia.

^* Correspondence to: amir.karton@une.edu.au

Handling Editor: Curt Wentrup

Australian Journal of Chemistry 78, CH24130 https://doi.org/10.1071/CH24130

Submitted: 12 September 2024 Accepted: 10 February 2025 Published online: 17 March 2025

© 2025 The Author(s) (or their employer(s)). Published by CSIRO Publishing. This is an open access article distributed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND)

Abstract

In celebration of the United Nations’ declaration of 2025 as the International Year of Quantum Science and Technology, marking 100 years since the development of quantum mechanics, this review highlights how accurate quantum mechanical calculations have transformed gas-phase thermochemistry. In particular, the developments of high-level composite ab initio methods over the past 30 years enable the calculations of thermochemical properties with confident chemical accuracy (i.e. with 95% confidence intervals ≤1 kcal mol⁻¹) for molecules with up to 12 non-hydrogen atoms. Lower-level composite ab initio methods can be applied to molecules containing up to ~50 non-hydrogen atoms; however, they cannot achieve confident chemical accuracy in terms of 95% confidence intervals. Over the past three decades, hundreds of composite ab initio methods have been developed, covering different theoretical frameworks, levels of accuracy and computational costs. To guide users in selecting an appropriate composite ab initio method for a given system size and level of accuracy, we present a general approach for categorising the accuracy of these methods. This approach places composite ab initio methods on four rungs of Jacob’s Ladder. Lower rungs offer less accuracy but are applicable to larger systems, and higher rungs offer greater accuracy but are applicable to smaller systems. Each consecutive rung of this ladder represents an improvement in the treatment of the one-particle space, n-particle space, or both, leading toward the exact solution of the relativistic Schrödinger equation. The Jacob’s Ladder of composite ab initio methods can be considered as an extension to the Jacob’s Ladder of density functional theory (DFT), which leads from ‘Hartree Hell’ to the ‘Heaven’ of double-hybrid DFT methods.

Keywords: CCSD(T), CCSDTQ, chemical databases, composite ab initio methods, density functional theory, machine-learning, quantum chemistry.

Experimental thermochemical databases

Experimental thermochemistry is a foundational branch of chemistry that is critical to understanding the heat and energy changes involved in chemical transformations. This field generates valuable thermochemical properties through various experimental techniques such as calorimetric, spectrophotometric and mass-spectrometric measurements. These measurements assist in determining reaction mechanisms and predicting reaction outcomes and have broad applications in material science, biochemistry, environmental science and pharmaceuticals. Experimental thermochemistry has been a highly active field of chemical research throughout the 20th Century. Towards the last quarter of the century, the critical mass of thermochemical determinations was compiled into several experimental thermochemical databases. These databases included thermochemical properties, such as heats of formation, bond dissociation energies, electron affinities and ionisation potentials. Popular experimental databases include the National Institute of Standards and Technology (NIST),¹ NIST-JANAF Thermochemical Tables,²^,³^,^A Gas-Phase Ion and Neutral Thermochemistry (GIANT),⁴ Thermodynamics Research Center (TRC),⁵ CODATA,⁶ Gurvich et al. ⁷^–⁹ and Pedley et al.¹⁰ There data underpin the prediction of reaction feasibility and elucidation of reaction mechanisms across diverse chemical disciplines. However, with the advancement of correlated wavefunction methods and computer hardware in the 1980s and 1990s, the availability of reliable thermochemical data served another critical role – namely, as reference data in the development of quantum chemical methods capable of accurate thermochemical predictions – the so-called composite ab initio methods (also known as composite wavefunction methods). The development of the Active Thermochemical Tables (ATcT) by Ruscic and co-workers in 2004 represented a significant breakthrough in experimental thermochemistry.¹¹^,¹² In contrast to traditional thermochemical tabulations, which compile thermochemical experimental determinations, ATcT combines thermochemical data from many experiments and high-level theoretical calculations forming a network where multiple pathways lead to the heat of formation of the same molecule in the network. Since ATcT considers a vast quantity of thermochemical data in one interconnected network, it can identify inconsistencies and potential errors in the data and find the most consistent set of enthalpies for the molecules involved in the network. Overall, this approach leads to thermochemical determinations that are more robust and reliable than the traditional thermochemical databases. Nevertheless, although highly accurate, this approach is still limited to the molecules in the thermochemical network. High-accuracy computational thermochemistry offers an attractive alternative since it is applicable to a much wider range of systems, offering a broader perspective on the world of thermochemistry. Although they do not replace experiments, computational tools have become indispensable, allowing chemists to explore and understand thermodynamic phenomena with unprecedented speed and efficiency. Composite ab initio methods are also starting to play a critical role in obtaining reliable thermochemical properties for machine learning databases with hundreds of thousands of thermochemical determinations.¹³

Bird’s eye view of composite ab initio methods

In the broadest sense, composite ab initio methods are multistep theoretical procedures designed to obtain energetic or spectroscopic chemical properties directly comparable to experimental data. The quest for the development of these methods began in the late 1980s with the pioneering work of Pople and colleagues, who introduced the so-called Gaussian-1 (G1) theory.¹⁴^–¹⁷ G1 theory established a framework for combining a series of wavefunction electron correlation methods with secondary energetic corrections to obtain thermochemical properties for small molecules. G1 theory was soon followed by Gausian-2 (G2) theory,¹⁸ which provided a precise recipe for obtaining thermochemical data. In short, G2 theory calculates the electronic energy using quadratic configuration interaction theory (QCISD(T)) with the 6-311G(d,p) basis set, whereas basis set corrections to account for the effect of diffuse and higher polarisation functions are calculated using fourth-order Møller–Plesset (MP4) perturbation theory. The core-valence correction is calculated using second-order Møller–Plesset (MP2) perturbation theory and the zero-point vibrational energy at the Hartree–Fock level. G2 theory also includes an empirical higher-level correction (HLC) term to account for some of the remaining deficiencies in the theoretical model. The empirical parameter involved in the HLC term was parameterised against the 55 experimental total atomisation energies (TAEs) in the G2 test set. For the entire G2 test set, which includes 55 TAEs, 38 ionisation energies, 25 electron affinities and 7 proton affinities, G2 theory attains a mean absolute deviation (MAD) of 1.24 kcal mol⁻¹. This pioneering work demonstrated the critical role that reliable experimental thermochemical data play in the development and evaluation of composite ab initio methods. Incidentally, 1 kcal mol⁻¹ has become the yardstick for ‘chemical accuracy’ in the computational determination of thermochemical properties (Karton¹⁹ provides a detailed overview of the term ‘chemical accuracy’). G2 theory was followed by G3 theory²⁰ in 1998 – the year John Pople was awarded the Nobel Prize in Chemistry for the ‘development of computational methods in quantum chemistry’. Undoubtedly, the Nobel Prize was also awarded for the development of the Gaussian-n composite ab initio methods, which were noted in Pople’s Nobel Lecture (8 December 1998). Here, we must note the highly influential computational and theoretical chemists who had major contributions to these remarkable developments, namely Larry Curtiss, Paul Redfern and Krishnan Raghavachari.²¹ For a comprehensive review of the Gaussian-n methods by these authors, see Curtiss et al.²² Overall, dozens of Gaussian-n type methods have been developed over the years, including the reduced order perturbation theory Gn(MP2) methods,²³^–²⁶ which are computationally more economical and are applicable to systems as large as buckminsterfullerene (C₆₀) and its isomers.²⁷^,²⁸ The development of Gn-type methods by other internationally renowned groups, such as the group of Leo Radom, should also be noted here.²⁹^–³⁵

The development of the early variants of the Gaussian-n methods has sparked extensive theoretical developments in the area of computational thermochemistry, leading to the development of many more types and variants of composite ab initio methods. These include the complete basis set (CBS) model chemistries,³⁶^–⁴² focal-point analysis (FPA),⁴³^–⁴⁷ Weizmann-n (Wn),⁴⁸^–⁵³ WnX,⁵⁴^–⁵⁶ multi-coefficient correlation methods (MCCMs),⁵⁷^–⁶² high-accuracy extrapolated ab initio thermochemistry (HEAT),⁶³^–⁶⁷ correlation consistent composite approach (ccCA),⁶⁸^–⁷⁷ Feller–Peterson–Dixon (FPD),⁷⁸^–⁸⁴ ab initio thermochemistry using optimal-balance models with isodesmic corrections (ATOMIC),⁸⁵^–⁸⁷ interference-corrected explicitly correlated second-order perturbation theory (INT-MP2-F12)⁸⁸ and the so-called cheap composite scheme (ChS)⁸⁹^,⁹⁰ procedures. For in-depth reviews of composite ab initio methods, see Curtiss et al.,¹⁸ Raghavachari,²¹ Karton,⁹¹^,⁹² Patel et al.,⁹³ Chan,⁹⁴ C. Peterson et al.,⁹⁵ Feller et al. ⁷⁹^,⁹⁶, Dixon et al.,⁷⁸ K. A. Peterson et al.,⁹⁷ Jiang and Wilson,⁹⁸ Klopper et al.,⁹⁹ DeYonker et al.,¹⁰⁰ Helgaker et al.,¹⁰¹^,¹⁰² and Martin.¹⁰³^,¹⁰⁴ Here, we will give a bird’s eye view of the different types of composite ab initio methods and their expected accuracies. Broadly speaking, composite ab initio methods can be classified based on the highest level of coupled-cluster excitation considered (e.g. CCSD(T), CCSDT(Q), CCSDTQ, CCSDTQ5 and CCSDTQ56) and the level of basis set completeness it approximates. With numerous composite ab initio methods, spanning dozens of variants across multiple families (as outlined above), it is essential to establish clear guidelines for assessing their expected accuracy. Here, we propose a Jacob’s Ladder framework, where the methods are placed on rungs based on their expected accuracy. Each successive rung of Jacob’s Ladder represents a more rigorous treatment of the one-particle space, n-particle space, or both. Fig. 1 depicts the Jacob’s Ladder of composite ab initio methods (for further details, see also Karton¹⁹).

Fig. 1.

Jacob’s Ladder of composite ab initio methods illustrates the four categories of these methods. Each successive rung represents a more rigorous treatment of the one-particle space, the n-particle space, or both. Methods on the first rung (e.g. G4(MP2) theory) are suitable for large systems such as C₆₀ isomers, whereas those on the fourth rung (e.g. W4-F12 theory) are limited to much smaller molecules like pentane (C₅H₁₂).

The hierarchical structure of composite ab initio methods

The first rung of Jacob’s Ladder represents methods that approximate the CCSD(T)/CBS energy using MP2-based basis-set additivity schemes. This rung includes computationally economical composite ab initio methods such as Gn, CBS and ccCA. These methods are commonly denoted as CCSD(T)/CBS(MP2)¹⁰⁵^–¹⁰⁷ and use the following simple basis set additivity scheme:

(1)

CCSD (T) / Large = CCSD (T) / Small + MP2 / Large - MP2 / Small + [additional corrections]

In Eqn 1, Small and Large denote different basis set sizes, typically corresponding to at least double-ζ and triple-ζ quality, respectively. For example, in one of the most computationally efficient CCSD(T)/CBS(MP2) methods (i.e. G4(MP2)),¹⁰⁸ ‘Small’ refers to the 6-31G(d) basis set and ‘Large’ refers to the G3MP2LargeXP basis set (i.e. a modified version of the 6-311+G(3df,2p) basis set). The computationally most demanding step in the G4(MP2) method is typically the CCSD(T)/6-31G(d) calculation. Thus, the G4(MP2) method can be routinely applied to large systems with over 40 carbon atoms, e.g. the entire set of C₄₀ isomers¹⁰⁹^,¹¹⁰ and even to a subset of C₆₀ isomers.²⁶^,²⁷ It is also noteworthy that the G4(MP2) method has been applied to the 133,000 molecules with up to nine non-hydrogen first-row atoms,¹³^,¹¹¹ thus providing a valuable database for benchmarking density functional theory (DFT) methods.¹¹¹ For comparison, the ccCA-PS3 method is an example of a computationally more demanding CCSD(T)/CBS(MP2) method.⁷¹ In this method, ‘Small’ refers to the cc-pV(T+d)Z basis set and ‘Large’ refers to a basis set extrapolation from basis sets of up to aug-cc-pV(Q+d)Z quality.¹¹²^–¹¹⁴ Thus, the ccCA-PS3 method has been applied to smaller systems than G4(MP2). Large systems to which the ccCA-PS3 method was applied to include up to ~20 non-hydrogen atoms, e.g. highly energetic molecules such as HMX (C₄H₈N₈O₈) and PETN (C₅H₈N₄O₁₂),¹¹⁵ melatonin conformers (C₁₃H₁₆N₂O₂)¹¹⁶ and ditetrazinetetroxide (C₂N₈O₄).¹¹⁷ Depending on the specific CCSD(T)/CBS(MP2) composite ab initio at hand, additional corrections to the electronic energy in Eqn 1 may include basis set correction terms for the effects of diffuse and higher polarisation functions, a complete basis set extrapolation correction for the Hartree–Fock energy and a core-valence correction.

The second rung of Jacob’s Ladder involves CCSD(T)/CBS composite ab initio methods that do not involve Moller–Plesset perturbation theory (MPn) based corrections terms. That is, methods that extrapolate the components of the CCSD(T) energy to the complete basis set limit by the following general expression:

(2)

CCSD (T) / CBS = HF / CBS + {CCSD}^{corr} / CBS + (T) / CBS + [additional corrections]

Here CCSD^corr and (T) are the CCSD and (T) correlation energies respectively. Methods on rung two of Jacob’s Ladder involve a more rigorous treatment of the one-particle space when compared to methods from the first rung. Examples of such methods include the lower members of the Weizmann-n family of composite ab initio methods (e.g. W1, W2, W1-F12 and W2-F12).⁴⁸^,⁵²^,¹⁰⁴^,¹¹⁸ W1-F12 theory is an example of a computationally economical method of this category. In this method, the HF energy and CCSD-F12 correlation energy are extrapolated separately to the CBS limit from the cc-pVDZ-F12 and cc-pVTZ-F12 basis sets.¹¹⁹ The (T) correlation energy is extrapolated from the regular jul-cc-pV{D,T}Z basis set pair.¹²⁰ The computationally most demanding steps in W1-F12 theory are the CCSD-F12/cc-pVTZ-F12 and CCSD(T)/jul-cc-pV(T+d)Z calculations. Therefore, methods such as W1 and W1-F12 theories are still applicable to reasonably large systems. For example, molecules with up to 21 non-hydrogen atoms such as sumanene (C₂₁H₁₂),²⁷^,¹²¹ dodecahedrane ((CH)₂₀),¹²² corannulene (C₂₀H₁₀),²⁷^,¹²¹ terphenyl (C₁₈H₁₄)¹²³ and chrysene (C₁₈H₁₂).¹²³ W2-F12 theory is an example of a higher-end CCSD(T)/CBS method, which attains results closer to the true basis set limit compared to W1-F12 theory. The computationally most demanding steps in W2-F12 theory are the CCSD-F12/cc-pVQZ-F12 and CCSD(T)/cc-pVTZ-F12 calculations. The largest systems W2-F12 theory was applied to include molecules with ~10 non-hydrogen atoms such as methionine (C₅H₁₁NO₂S),¹²⁴ adenine (C₅H₅N₅)⁵² and cubane ((CH)₈).¹²²

Rung three of Jacob’s Ladder additionally incorporates post-CCSD(T) contributions up to the CCSDT(Q) level into Eqn 2:

(3)

CCSDT (Q) / CBS = HF / CBS + {CCSD}^{corr} / CBS + (T) / CBS + T - (T) / CBS + (Q) / CBS + [additional corrections]

Thus, moving from rung two to rung three represents a more rigorous treatment of the n-particle space. Methods that belong to this rung typically build on the top-end methods of rung two by adding higher-order triples excitations CCSDT–CCSD(T) (T–(T)) and quasiperturbative quadruple excitations ((Q)). For example, W3-F12 theory, which approximates the CCSDT(Q)/CBS energy, uses W2-F12 theory as a baseline and adds a T–(T) contribution extrapolated from the cc-pV{D,T}Z basis set pair and a (Q) contribution calculated with the cc-pVDZ basis set. It is important to mention here that successively higher coupled cluster expansion terms (∆CCSD, ∆(T), ∆T–(T), ∆(Q), ∆Q–(Q), ∆(5), etc.) converge increasingly faster with the basis set size.¹⁹^,⁵⁰^,⁵¹ Indeed, this is the reason that composite ab initio methods from rungs three and four of Jacob’s Ladder can be performed at a realistic computational cost. The computationally most demanding step in W3-F12 theory is typically the CCSDT/cc-pVTZ calculation.¹²⁵ Therefore, rung three methods are not generally applicable to systems with over 10 non-hydrogen atoms. The largest systems W3-F12 (or W3) theory have been applied to include the benzene dimer (C₁₂H₁₂),¹⁰⁷ bullvalene ((CH)₁₀),¹²⁶ octasulfur ring (S₈),¹²⁷ phosphorus sulfide cages (P₄S₃ and P₄S₄)¹²⁸ and hexahaloethanes (C₂X₆).¹²⁹^,¹³⁰

Composite ab initio methods on the fourth rung of Jacob’s Ladder represent the most accurate methods in contemporary quantum chemistry that are capable of confident sub-benchmark accuracy even for pathologically multireference systems such as O₃ and $C_{2} (\sum_{g}^{+})$ .⁵⁰ Moving from rung three to rung four typically involves a more rigorous treatment of both the one-particle and n-particle spaces in order to achieve confident sub-benchmark accuracy. To illustrate this, it is instructive to compare the CCSD(T) and post-CCSD(T) components in W3 and W4 theories. In W3 theory, which sits on rung three of Jacob’s Ladder, the CCSD(T) energy is obtained by CCSD(T)/CBS = HF-AV{Q,5}Z+CCSD/AV{Q,5}Z+(T)/AV{T,Q}Z. Whereas in W4 theory (rung four), all the basis sets are upgraded by one angular momentum, namely CCSD(T)/CBS = HF/AV{5,6}Z+CCSD/AV{5,6}Z+(T)/AV{Q,5}Z. Likewise, the basis set used for calculating the parenthetical connected quadruples ((Q)) is cc-pVDZ in W3 and cc-pVTZ in W4. Moving to the n-particle space, W3 does not include correlation contributions beyond CCSDT(Q), whereas W4 theory calculates the CCSDTQ–CCSDT(Q) component with a cc-pVDZ basis set and the CCSDTQ5–CCSDTQ component with a truncated version of the cc-pVDZ basis set.¹⁹^,⁵⁰^,⁹² The significantly improved treatment of the one-particle and n-particle spaces in rung four composite ab initio methods results in a prohibitive computational cost. For example, the largest systems W4 theory has been applied to include up to five non-hydrogen atoms (e.g. tetrafluorosilane, tetrachloromethane, acetic acid, tetrahedrane and n-butane).⁵³

So far, we have categorised the composite ab initio methods on the rungs of Jacob’s Ladder based on the levels of accuracy used for calculating the non-relativistic electronic energy. In order to obtain thermochemical and kinetic data that are directly comparable to experimental measurements, composite ab initio methods include secondary energetic corrections such as the core-valence, scalar relativistic, spin-orbit, Born–Oppenheimer and zero-point vibrational energy (ZPVE) corrections. Excluding the ZPVE correction, these calculations are typically computationally less demanding than those required for the non-relativistic electronic energy. For a comprehensive overview of secondary energetic corrections in composite ab initio methods, the reader is referred to several recent reviews.¹⁹^,⁹¹^,⁹²

The performance of composite ab initio methods has been extensively benchmarked relative to accurate and reliable experimental and high-level theoretical data. It is well established that the performance of electronic structure methods depends on both the chemical properties being considered (e.g. heats of formation, reaction barrier heights, conformational energies or non-covalent interactions) and the specific composition of the evaluation dataset (e.g. elemental composition and multireference character of the species involved). The atomisation energy is the energy required to break a molecule into its constituent atoms in the gas phase. As such, TAEs are one of the most challenging thermochemical properties and, therefore, are commonly used for the evaluation of composite ab initio methods.¹³¹ In this context, it should be pointed out that TAEs can be converted to molecular heats of formation at 0 K from the atomic heats of formation at 0 K (which can be taken from experiments) and the zero-point vibrational energy (which can be obtained from CCSD(T) harmonic frequencies and DFT quartic force fields).⁵⁰^,⁹²^,¹³²^,¹³³

The performance of composite ab initio methods from rungs one and two of Jacob’s can be evaluated relative to a wide and diverse set of TAEs obtained from methods on the fourth rung of Jacob’s Ladder. The W4-17 database of 200 TAEs represents such a dataset.¹²⁹ It comprises first- and second-row molecules with up to eight non-hydrogen atoms, which cover a broad spectrum of bonding situations, electronic states and multireference characters. Since methods from rungs one and two approximate the CCSD(T) energy, which is not suitable for highly multireference systems, pathologically multireference species (e.g. halogen oxides and atomic clusters)¹⁹^,⁵³^,⁹¹^,⁹²^,¹²⁹^,¹³⁴^–¹³⁷ are excluded from the evaluation dataset. Two recent reviews¹⁹^,⁹¹ give a comprehensive overview of the performance of popular composite ab initio methods for TAEs. As customary in experimental (and high-level computational) thermochemistry, we will use 95% confidence intervals (CIs) rather than root-mean-square deviations (RMSDs) or MADs for robust uncertainty quantification of composite ab initio methods.¹³⁸^,¹³⁹ We note that the 95% CI is approximately equal to twice the RMSD for a normal distribution. As previously noted,¹⁹ methods from rung one attain 95% CIs ranging between 2 and 5 kcal mol⁻¹ and methods from the second rung attain 95% CIs ranging between 1 and 2 kcal mol⁻¹. Thus, in terms of 95% CIs, methods from rung one are not capable of chemical accuracy for TAEs (arbitrarily defined as deviations of ~1.0 kcal mol⁻¹), and methods from rung two are capable of near chemical accuracy for TAEs. Methods from the third and fourth rungs can only be evaluated against highly accurate and reliable experimental TAEs, which are typically taken from the ATcT network of Ruscic and co-workers.¹¹^,¹² Relative to a subset of ATcT TAEs, methods from rung three of Jacob’s Ladder attain 95% CI of near or sub-benchmark accuracy for TAEs (arbitrarily defined as deviations of ~1.0 kJ mol^–1), and methods from rung four are capable of confident sub-benchmark accuracy.¹⁹ These results show a progressive improvement in performance as we ascend the rungs of Jacob’s Ladder, which validates the more rigorous theoretical framework. It should be emphasised that the above 95% CIs are obtained for one of the most challenging thermochemical properties and that the performance of the composite ab initio methods can improve significantly for less challenging thermochemical properties, in particular, thermochemical properties associated with reactions that conserve large molecular fragments on both sides of the reaction.¹²²^,¹³⁰^,¹⁴⁰^–¹⁴⁸ Finally, it is important to note that composite ab initio methods are extensively used not just for obtaining accurate energy-based thermochemical properties, but also spectroscopic properties (e.g. equilibrium geometries, vibrational frequencies and rotational constants)⁸⁹^,⁹⁰^,⁹⁶^,¹⁴⁹^–¹⁵⁸ and electrical properties (e.g. dipole moments, polarisabilities and hyperpolarisabilities).⁸⁴^,¹⁵⁴^,¹⁵⁹

Theorretical thermochemical databases

As mentioned in the previous section, experimental thermochemical databases such as G2,¹⁸ G2/97,¹⁶⁰ G3/99,¹⁶¹ Database/3⁵⁹ and G3/05,¹⁶² played a foundational role in the development of composite ab initio methods. Since the development of the ATcT network in 2004, ATcT has become the dominant thermochemical database because it provides highly accurate, reliable and internally consistent thermochemical values. In addition, ATcT is a dynamic database that is regularly updated, for example, the current version of ATcT includes ~3000 heats of formation determined both at 0 and 298 K (ATcT Network, Argonne National Laboratory, see https://atct.anl.gov/). For comparison, the largest of the test sets listed above,G3/05, includes 454 thermochemical determinations, namely 270 enthalpies of formation, 105 ionisation energies, 63 electron affinities, 10 proton affinities and 6 hydrogen-bonded complexes.

When it comes to benchmarking electronic structure methods, one limitation of the above experimental databases is that they focus on a small number of thermochemical properties (e.g. ∆H_f, TAEs, IPs, EAs, PAs), most of which are obtained for small species with 1–9 non-hydrogen atoms. Theoretical databases generated using composite ab initio methods overcome these limitations since they are accessible to a much wider range of thermochemical and kinetic properties, and composite ab initio methods from rungs one and two of Jacob’s Ladder are applicable to much larger systems with dozens of non-hydrogen atoms. Contemporary composite ab initio methods span from methods capable of near chemical accuracy and applicable to systems with ~20 non-hydrogen atoms, to methods capable of sub-benchmark accuracy and applicable to systems with 6 non-hydrogen atoms.¹⁹^,⁹¹^,⁹² These theoretical developments, along with advances in high-performance computer technology, have led to a dramatic increase in the scale and diversity of theoretical thermochemical databases since the mid-2000s.¹⁶³ The databases generated by composite ab initio methods are chemically more diverse and cover a wider range of chemical properties than the experimental ones. For example, they paved the way for the generation of databases focusing on chemical properties that are typically not included in experimental databases such as isomerisation energies,¹³⁰^,¹⁶⁴^,¹⁶⁵ conformational energies,¹⁶⁶^–¹⁷⁰ various reaction energies,¹³⁰^,¹⁷¹^,¹⁷² (including hypothetical species),¹⁷³ reaction barrier heights¹⁷⁴^–¹⁸¹ and other chemical properties (e.g. self-interaction errors¹⁸² and radical stabilisation energies).¹⁸³^–¹⁸⁵

There are many dozens of examples of high-quality theoretical thermochemical databases. For an overview, the reader is referred to Goerigk and Grimme,¹⁸²^,¹⁸⁶ Goerigk et al.,¹⁸⁷ Mardirossian and Head-Gordon,¹⁸⁸ and Goerigk.¹⁸⁹ Here, we will only describe two extreme cases the W4-11 database,¹³⁰ which includes close to 1000 reaction energies calculated by W4 theory from rung four of Jacob’s Ladder, and the GDB-9 database, which includes over 133,000 TAEs calculated by G4(MP2) theory from rung one of Jacob’s Ladder.¹³

Let us begin with the W4-11 database, which includes the following sets of reaction energies: 140 TAEs, 99 bond dissociation energies, 707 heavy-atom transfer reaction energies, 20 isomerisation energies and 13 nucleophilic substitution reaction energies, totalling 979 reaction energies. All the reaction energies are all-electron, relativistic, ZPVE-inclusive and DBOC-inclusive CCSDTQ5/CBS energies. The largest systems represented in the W4-11 database are molecules such as CF₄, FOOF, acetic acid, SiF₄, SO₃, P₄ and S₄. In the original work, this extensive set of 797 highly accurate reaction energies was used to benchmark the performance of composite ab initio methods from rungs one and two of Jacob’s Ladder as well as a range of contemporary DFT and DHDFT methods.¹³⁰ In 2017, this set of 140 TAEs was extended to include 27 additional TAEs from W4 theory (rung four) and 33 TAEs from W4lite (rung three of Jacob’s Ladder) in the W4-17 database.¹²⁹ Although the W4-11 and W4-17 databases achieve the most accurate thermochemical properties by composite ab initio methods, the high computational cost associated with rung four methods, and in particular the CCSDTQ/cc-pVDZ and CCSDTQ5/DZ calculations, limits their applicability to relatively small systems with up to 5–6 non-hydrogen atoms. It should be pointed out that the atomisation energies in the W4-11 and W4-17 databases are not more accurate than the most accurate TAEs in the ATcT database. However, unlike experiments, W4 theory is applicable to any arbitrary system with up to 5–6 first- and second-row atoms, whether it is poisonous, explosive, short-lived or hypothetical.

On the other extreme, methods such as G4(MP2) are applicable to systems with dozens of non-hydrogen atoms. Therefore they have been applied for the calculation of the isomerisation energies of 40 C₄₀ isomers¹⁰⁹^,¹¹⁰ and 8 C₆₀ isomers.²⁶^,²⁷ A particularly impressive application of the G4(MP2) method has been the calculation of the atomisation energies of over 133,000 molecules with 9 non-hydrogen atoms in the GDB-9 database.¹³^,¹⁹⁰ This makes this database of atomisation energies highly diverse, albeit at the price of reduced accuracy relative to the W4-11 and W4-17 databases.¹²⁹^,¹³⁰ (For a comprehensive discussion of the composition of the molecules in the GDB-9 database, see Narayanan et al.,¹³ Ramakrishnan et al.,¹⁹⁰ and Huang and von Lilienfeld¹⁹¹.) The largest systems W1 and W1-F12 theories have been applied to include medium-sized hydrocarbons such as corannulene (C₂₀H₁₀),¹²¹ sumanene (C₂₁H₁₂),¹²¹ dodecahedrane ((CH)₂₀)¹²² and the C₂₀ and C₂₄ carbon clusters.¹⁹²

Finally, it should be emphasised that computational acceleration techniques such as the resolution-of-the-identity (RI)¹⁹³^–¹⁹⁶ and explicitly correlated approximations⁵²^,¹⁹⁷^–²⁰¹ and localised-orbital approaches²⁰²^–²⁰⁸ have enabled the application of composite ab initio methods to large molecular systems by reducing their computational cost. Examples of composite ab initio methods that use these acceleration techniques include Wn-F12,⁵²^,⁵³ WnX,⁵⁴^–⁵⁶ L-W1X,²⁰⁹ ccCA-F12,⁷⁶ G4(MP2)-XK,²⁶ G4(MP2)-XK-D³⁵ and cc-G4-n.²¹⁰

A significant advantage of using theoretical rather than experimental data for benchmarking electronic structure methods is that the theoretical benchmark data are directly comparable with the more approximate data obtained from DFT and low-level electronic structure calculations. Namely, electronic structure methods calculate gas-phase, non-relativistic electronic energies at the bottom of the well. By contrast, thermochemical properties obtained from experiments include additional components such as relativistic, zero-point vibrational, thermo-statistical, entropic and solvation corrections. In order to compare the electronic structure and experimental data, these components have to be added to the theoretical calculations or backtracked from the experimental values.²¹¹ Alternatively, reference values that are calculated by non-relativistic high-level wavefunction methods (e.g. CCSD(T) or CCSDT(Q)) are directly comparable to those obtained from DFT (e.g. B3LYP, PBE, M06-2X, ωB97X-D) or low-level ab initio (e.g. HF, MP2, SCS-MP2) calculations. Therefore, data obtained from high-level wavefunction methods can be readily used for the evaluation and development of lower-level electronic structure methods.

It should be emphasised that backtracking the above secondary energetic contributions from experimental determinations (or adding them to non-relativistic, bottom-of-the-well theoretical determinations) is not a trivial task. This process is likely to increase the uncertainty of the experimental values when the secondary energetic contributions are not calculated with sufficient accuracy (e.g. when they are obtained from DFT calculations). Indeed, much of the research into high-level composite ab initio methods has been dedicated to obtaining secondary energetic contributions (such as scalar relativistic, spin-orbit, zero-point vibrational and Born–Oppenheimer corrections) with sufficient levels of accuracy (see Karton¹⁹^,⁹¹^,⁹² for an overview). As an illustrative example, let us consider the zero-point vibrational energy (ZPVE) component of small-to-medium-sized molecules. The ZPVE for simple organic molecules ethane, pentane, arginine and dodecahedrane is 46.39, 98.88, 138.18 and 222.97 kcal mol⁻¹ respectively.¹²²^,¹²⁴^,¹⁶⁴^,²¹² Thus, even a 1% error in the calculated ZPVE due to the neglect of explicit anharmonicity can translate to errors on the order of 1–2 kcal mol⁻¹ in the ZPVE for small molecules.

Electronic structure calculations are generally easier to perform than experiments and can be applied to a wide range of systems that may be difficult or impossible to study experimentally, including poisonous, explosive, short-lived or hypothetical species. Electronic structure calculations can be applied to a broad spectrum of thermochemical and kinetic properties, including those that are difficult to measure accurately, such as some reaction barrier heights and arbitrary bond dissociation energies. This allows the construction of large, diverse and systematic datasets. However, a significant limitation of low-level electronic structure methods, such as DFT, is their inability to consistently and confidently achieve high accuracy across a wide range of chemical properties and systems. This underscores the need to regularly benchmark DFT methods to ensure reliability and precision across diverse applications.¹⁸⁶^–¹⁸⁹ With the exception of multireference systems, composite ab initio methods from rungs one and two of Jacob’s Ladder largely overcome this limitation. Whereas composite ab initio methods from rungs three and four of Jacob’s Ladder can be safely applied to moderately and highly multireference systems respectively.¹⁹^,⁹¹^,⁹²

An advantage of theoretical databases over experimental ones is that high-level thermochemical data can be obtained reasonably easily at a reasonable computational cost if the rung of Jacob’s Ladder is chosen according to the system size. For instance, rung four methods are applicable for systems with up to ~5 non-hydrogen atoms, rung three for systems with up to ~10 non-hydrogen atoms, rung two for systems with up to ~16 non-hydrogen atoms and rung one for systems with up to ~32 non-hydrogen atoms. This ease of data generation offers significant flexibility in designing diverse and targeted databases for specific chemical systems and properties. This allows the generation of systematic databases targeted at certain elements, compounds and substituents, as well as tackling specific chemical properties.

Synergy of accurate computational thermochemistry databases and machine learning

Machine learning (ML) is transforming the field of chemistry by enabling novel approaches to predict and analyse the chemical properties of molecules, proteins and materials. A recent testament to the impact of ML in chemistry was the 2024 Nobel Prize in Chemistry awarded to David Baker, a computational biochemist from the University of Washington, and to Demis Hassabis and John Jumper from Google, for their groundbreaking work in computational protein design and protein structure prediction. ML applications in chemistry span from designing functional molecules, proteins and materials with tailored properties to optimising reaction conditions and synthetic routes. A key approach to achieving these advances is through the prediction of critical properties, such as binding energies, chemical stabilities and catalytic activities. However, for reliable and accurate predictions, ML models require vast and diverse datasets of high-quality chemical data.¹⁹¹^,²¹³^–²²⁴

Over the past two decades, composite ab initio methods have been extensively used for generating highly accurate thermochemical, kinetic and non-covalent interaction databases for (i) training empirical DFT methods and (ii) benchmarking the performance of empirical and non-empirical DFT methods.⁹¹^,¹⁶³^,¹⁸⁹ These databases are typically calculated using composite ab initio methods from rungs one to four of Jacob’s Ladder of composite ab initio methods (Fig. 1). Over the past decade, there have been several compilations of many small databases into large general-purpose ones that cover a wide range of chemical systems and properties. Examples include the GMTKN55,¹⁸⁸ MGCDB84¹⁸⁹ and 2015B²²⁵ databases (for a broader overview, see Karton and de Oliveira¹⁶³ and Goerigk¹⁸⁹). Together, these general-purpose databases include well over 5000 accurate thermochemical, kinetic and non-covalent interaction determinations. These compilations have proved to be highly valuable for training and benchmarking DFT methods over the past decade.¹⁶³^,¹⁸⁹ Nevertheless, in terms of their size, these compilations are still not comparable to databases used for training ML models, which typically require tens to hundreds of thousands of data points to capture complex patterns and dependencies across diverse chemical systems.

DFT has become the workhorse of quantum chemistry due to its attractive accuracy-to-computational cost ratio. DFT methods enable the efficient generation of extensive databases encompassing hundreds of thousands (or more) of molecular systems that are invaluable for training and testing ML models. Consequently, databases for training ML algorithms have been largely generated using DFT methods.¹⁹⁰^,¹⁹¹^,²²⁶^–²³² However, this scalability comes with a trade-off in accuracy compared to composite ab initio methods.¹¹¹^,¹⁸⁷^,¹⁸⁸ Needless to say, this limits the accuracy of ML models trained on these data. The accuracy of any ML algorithm is constrained by the quality of the data it is trained on. Therefore, care must be taken with the accuracy of the reference data in databases used for training ML models for applications demanding high precision. A clear example is the prediction of chemical kinetics, where even an ~1 kcal mol⁻¹ variation in the reaction barrier height leads to an order-of-magnitude change in reaction rate at room temperature. This sensitivity to the accuracy of the reaction barrier height becomes even more critical for reactions at lower temperatures. Another example is the development of ML-based DFT methods designed to outperform conventional DFT methods. Achieving such enhanced performance is a primary driving force behind the development of ML-DFT methods, aiming to surpass the limitations of conventional DFT in both accuracy and computational efficiency.

Currently, the availability of thermochemical, kinetic and non-covalent interaction datasets with hundreds of thousands of chemical properties calculated by composite ab initio methods is an underdeveloped area. To the best of our knowledge, the main such database is the recalculation of the 133,000 total atomisation energies in the GDB-9 database, originally calculated at the B3LYP/6-31G(2df,p) level of theory,¹⁹⁰ with the rung one composite ab initio method G4(MP2).¹³ This important work by Curtiss and co-workers, demonstrates that with contemporary supercomputer hardware, it is possible to generate large thermochemical databases containing hundreds of thousands of systems with up to nine non-hydrogen atoms, with composite ab initio methods from rung one of Jacob’s Ladder of composite ab initio methods (Fig. 1). Excluding multireference systems, these methods provide reliable thermochemical data, superior to that generated by DFT and DHDFT methods.¹¹¹ Continuation of the development of such databases will improve the accuracy of machine learning-based classical and quantum mechanical methods, such as ML-based force fields²³³^–²³⁶ and ML-DFT methods.²³⁷^–²⁴⁰ Nevertheless, an inherent limitation of highly accurate databases obtained by composite ab initio methods is that they will always be biased towards smaller (or fewer) systems compared to databases obtained from DFT calculations. Thus, it might be beneficial to augment such databases with thermochemical data obtained from DHDFT calculations that are applicable to larger systems and capable of achieving an intermediate accuracy between composite ab initio and conventional DFT methods.

When dealing with large and diverse chemical databases with tens or even hundreds of thousands of molecules generated by either composite ab initio or DFT methods, manual analysis of the structural and energetic data becomes impractical. Machine learning algorithms can efficiently analyse this vast data, for example, by identifying structure-function relationships and uncovering hidden patterns within the thermochemical data. This synergy between machine learning and large-scale thermochemical databases will ultimately lead to advances across various fields of chemistry, such as the design and optimisation of more selective and efficient catalysts and drugs with enhanced efficacy and fewer side effects. Nevertheless, we stress that the accuracy of the thermochemical data is critical to making reliable predictions and designing effective molecules and materials.

Conclusions

This review examines the synergy between experimental and computational thermochemistry. It highlights the historical role of experimental thermochemistry in developing highly accurate multilevel wavefunction theories (also known as composite ab initio methods) capable of near-kilocalories per mole to sub-kilojoules per mole thermochemical predictions. It continues to introduce the framework of Jacob’s Ladder of composite ab initio methods, which categorises the large number of composite ab initio methods developed over the past three decades based on the treatment of the one-particle and n-particle spaces. Each rung of the ladder represents progressively more accurate and computationally demanding methods:

Rung 1 includes CCSD(T)/CBS(MP2) methods, which involve CCSD(T) calculations with a reasonably small basis set and MP2-based basis-set corrections calculated with larger basis sets. This rung includes methods such as G4(MP2), which are applicable to systems as large as C₆₀.
Rung 2 includes CCSD(T)/CBS methods that offer a more rigorous treatment of the one-particle space and extrapolate the HF, CCSD and (T) components individually to the complete basis set limit using reasonably large basis sets. This rung includes methods such as W1-F12, which are applicable to systems as large as dodecahedra (CH)₂₀.
Rung 3 offers a more rigorous treatment of the n-particle space by introducing higher-order triples and quasiperturbative quadruples excitations (i.e. methods that approximate the CCSDT(Q)/CBS energy). This rung includes methods such as W3-F12, which are applicable to systems as large as bullvalene (CH)₁₀.
Rung 4 represents the most accurate composite ab initio methods that approximate the FCI/CBS energy and are capable of confident sub-benchmark accuracy. These methods (e.g. W4-F12 theory) treat the one-particle and n-particle spaces with exceptional rigor but are limited to small molecules such as tetrahedrane (CH)₄.

This Jacob’s Ladder framework facilitates better understanding and usage of the wide range of composite ab initio methods, which will assist researchers in selecting appropriate methods for different system sizes and levels of accuracy. We continue to explore how the landscape of thermochemical predictions has evolved due to the development of the above composite ab initio methods, which enable the creation of large theoretical thermochemical databases critical for DFT development and benchmarking. Finally, the growing synergy between quantum chemistry and machine learning is explored, with large chemical databases generated by composite ab initio methods representing the pinnacle of this approach. Machine learning models trained on large databases of highly accurate thermochemical data could revolutionise predictions for larger or more complex systems that are otherwise computationally prohibitive. These developments are set to accelerate advancements across various fields of chemistry, such as the design of novel materials, catalysts and bioactive molecules.

Data availability

This review is based on previously published research.

Conflicts of interest

Amir Karton is an Associate Editor of the Australian Journal of Chemistry but did not at any stage have editor-level access to this manuscript while in peer review, as is the standard practice when handling manuscripts submitted by an editor to this journal. Australian Journal of Chemistry encourages its editors to publish in the journal and they are kept totally separate from the decision-making processes for their manuscripts. The author has no further conflicts of interest to declare.

Declaration of funding

This research did not receive any specific funding.

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Footnotes

A Data are also accessible online at http://webbook.nist.gov/chemistry and http://srdata.nist.gov/cccbdb.