## Abstract

We have developed the W*n*X series of quantum chemistry composite protocols for the computation of highly-accurate thermochemical quantities with advanced efficiency and applicability. The W1X-type methods have a general accuracy of ~3–4 kJ mol^{−1} and they can currently be applied to systems with ~20–30 atoms. Higher-level methods include W2X, W3X and W3X-L, with the most accurate of these being W3X-L. It can be applied to molecules with ~10–20 atoms and is generally accurate to ~1.5 kJ mol^{−1}. The W*n*X procedures have opened up new possibilities for computational chemists in pursue of accurate thermochemical values in a highly-productive manner.

## Introduction

During the early days of computational quantum chemistry, it was a dream for theoretical chemists to calculate thermochemical quantities (e.g. heats of formation, reaction energies, activation barriers) with an accuracy that could rival or surpass precise experimental measurements. Owing to continuous development of computer technology and quantum chemistry methodologies, substantial progresses have been made toward this goal [1]. With the first-principle nature of many contemporary quantum chemistry methods, they have enabled highly-accurate prediction of thermochemical quantities in an unbiased manner. Thus, computational quantum chemistry has now become an indispensable tool for many chemists, by contributing to chemical knowledge in some situations where conducting practical experiments are difficult or undesirable.

Modern quantum chemistry computations [2] typically involve protocols that combine a theoretical method and a basis set. The many widely-available theoretical methods belong to one of the two general classes, namely (1) wavefunction and (2) density functional theory (DFT) methods. Wavefunction methods typically employ a “Hartree–Fock” (HF) reference wavefunction, on which more advanced treatment for “electron correlation” can be applied. The use of a more sophisticated correlation method, in conjunction with a larger basis set, generally leads to a systematic improvement over a protocol that employs a less advanced method and/or with a smaller basis set. The gain in accuracy through this systematic improvement, however, comes at the expense of dramatically increased demand for computer resources.

A major breakthrough in addressing this challenge is the introduction of a class of methods generally called “composite procedures” [3]. These methods combine the electronic energies obtained with a number of computationally economical protocols to approximate the energy of a higher-level method with a larger basis set. This usually involves a base energy obtained with a low-level method in conjunction with a large basis set, with higher-order electron correlation effects included using increasingly higher-level methods in combination with progressively smaller basis sets. Additional corrections (e.g. for core-valence correlation and scalar-relativistic effects) are sometimes also incorporated using values obtained at relatively low-levels of theory.

The performance of such an approach depends to a considerable extent on the validity of the assumption that the various effects that contribute to the total energy of a molecule can be treated in an additive manner. In addition, it is also important that the individual components can be calculated with good accuracy using relatively modest computational procedures. In general, the demonstrated accuracy of many composite methods supports the validity of these types of approximations. For instance, the first general-purpose composite method, namely the G1 procedure [4] of Pople and co-workers, has an accuracy of approximately 10 kJ mol^{−1}; its mean absolute deviation (MAD) for the original G2 set [5] is 6.4 kJ mol^{−1}. Recent variants of the G*n* protocols (e.g. G4 [6], G4(MP2) [7], ROG4(MP2) [8] and G4(MP2)-6X [9]) have improved general accuracies of about 5 kJ mol^{−1} (e.g. the MAD for the E2 set [9], [10] is 3.6 kJ mol^{−1} for G4(MP2)-6X). Some of the most accurate composite protocols (e.g. W4 [11] and HEAT [12]) have been demonstrated [13] to have sub-kJ mol^{−1} accuracy for the highly-accurate “Active Thermochemical Tables” (ATcT) database [14]. In general, most of the widely-used contemporary composite methods such as the G*n* [15], W*n* [3], ccCA [16] and CBS [17] series of procedures are associated with accuracies in the range of about 1–10 kJ mol^{−1}.

Another key feature of the aforementioned composite methods is that each one is well-defined in terms of the contributing components, rather than providing a general strategy but leaving the exact choice of quantum chemistry methods, basis sets and other factors to the user. Such a characteristic enables straightforward implementation of these protocols into computational chemistry software packages. In some cases, they are implemented as a “general keyword” in an input file for a software package (e.g. for some of the G*n* and W*n* methods) [18]. In some other cases, input file templates (e.g. for W*n*-F12 [3] and W*n*X [19], [20], [21]) that are conveniently adaptable are provided [22]. Thus, these composite methods often can be used with ease by a wide range of computational chemists including novices. We note that the concept of composite protocol is utilized in the FPA [23] and FPD [24] strategies in a more flexible but less strictly-defined manner, and these strategies have also been shown to be useful, perhaps in an arguably more limited fashion.

The variation in the accuracy of the different composite methods is associated with a variation in the computational requirement, which in turn leads to a difference in the range of systems that can be calculated with different procedures. It is now feasible to perform routine calculations on systems up to about 40 atoms with relatively low-cost composite methods such as G4(MP2) and G4(MP2)-6X using a modest high-performance-computer cluster. Thus, these methods are applicable to the study of thermochemistry for many systems using realistic chemical models, i.e. without substantial truncation of the system. However, the original W*n* procedures are limited to systems with approximately 10–15 atoms when their computations are carried out on comparable hardware. Despite this, their improved accuracy (MAD~1–4 kJ mol^{−1} for various test sets such as E2 and W4-11) and robustness (i.e. consistent accuracy) over G4(MP2)-type procedures (MAD~5–7 kJ mol^{−1}) are quite desirable. This is because such a seemingly small difference can in fact leads to, for example, almost an order of magnitude difference in the calculated product distribution and/or reaction rate [25].

With these two series of composite methods, we are still facing the dilemma of trading one accuracy for another accuracy. On the one hand, we can use a very good but not outstanding method [G4(MP2)-type protocols] on a realistic model. On the other hand, we can apply an outstanding procedure (W*n*) to a truncated system, with the gain in accuracy in the quantum chemistry calculations quite possibly offset by the error introduced with the truncation. Such a situation has in part motivated the continuous development and refinement of composite protocols. To this end, the W*n* series of methods have undergone some key alterations in recent years, and transformed into new methodologies with similar accuracies and, at the same time, widened applicabilities. Among the new W*n*-type methods are the W*n*-F12 [3] and W*n*X [19], [20], [21] series of procedures. A major feature for both sets of composite protocols is the use of the so-called “explicitly-correlated” methods [26] to enable the use of smaller basis sets. An excellent review on the W*n*-F12 methods has been published elsewhere [3]. In this account, we focus on W*n*X that we have developed in the past few years.

## The original W*n* methods and the challenge of applying them economically

### The W1 protocol as an approximation to CCSD(T)/CBS

The W*n* series of protocols employ, exclusively, high-level (conventional) “coupled-cluster” (CC) methods together with fairly large basis sets in their formulations. This strategy provides a solid foundation for their high accuracy and outstanding robustness, but also leads to the high computational requirements for the W*n* methods. The W1 procedure [27] contains four components, which, when combined together, yields an excellent approximation to the so-called “gold standard” of computational chemistry, namely CCSD(T) [28] at the “complete-basis-set” (CBS) limit:

The “frozen-core” approximation is used in the calculation of both Δ*E*[CCSD] and Δ*E*[(T)], i.e. correlation effects are included only for the valence electrons. The CBS limits are obtained using a “two-point” extrapolation technique [29]. For *E*[HF] and Δ*E*[CCSD], a triple-ζ (TZ) and a (larger) quadruple-ζ (QZ) basis sets are used. The calculation of the computationally more demanding Δ*E*[(T)] term employs smaller double-ζ (DZ) and TZ basis sets. The last term in the *E*[W1] formula, i.e. Δ*E*[C+R], accounts for core-valence correlation plus scalar-relativistic effects:

In this equation, “Full,DKH” indicates that all electrons are correlated, and the “Douglas–Kroll–Hess” [30] treatment for scalar-relativistic effect is applied, in a single CCSD(T) calculation. The “FC,NR” signifies the use of frozen-core approximation in a non-relativistic computation. The MTsmall basis set [27] is a “decontracted” TZ basis set. In comparison with typical basis sets that are “contracted”, calculations with decontracted basis sets are significantly more demanding on computational resources.

Among all the calculations that are required in order to obtain all the components in W1, two individual ones are notably more demanding on computational resources than the others. These are the CCSD/QZ and CCSD(T)(Full,DKH)/MTsmall calculations. The former calculation employs the largest (QZ) basis set within the W1 protocol, and the computation of this component is often associated with the largest requirements on computer memory and disk space. The all-electron scalar-relativistic calculation is often the most time consuming. For instance, it amounts to about 80% of the computer time required for the entire W1 calculation of benzene [31]. Quite obviously, these two bottlenecks represent logical targets for further development of more cost-effective W1-type protocols.

### Higher-level W*n* (*n*=2, 3 and 4) protocols

We now continue our brief introduction to the W*n*-type methods. The W2 method [27] is similar to W1. It employs larger basis sets (up to quintuple-ζ, i.e. 5Z) to provide an improved approximation to the CCSD(T)(Full,DKH)/CBS energy. The W3 protocol [32] builds on the foundation of W2, and incorporates additional correlation effects using higher-level CC procedures up to “CCSDTQ” [33] with progressively smaller basis sets for the progressively more expensive higher-order CC calculations:

The Δ*E*[T] term represents the CCSDT – CCSD(T) energy difference, and it is extrapolated to the CBS limit using a DZ and a TZ basis sets. The Δ*E*[Q] term is obtained by taking the difference between the CCSDTQ and CCSDT energies using a DZ basis set, and multiply this relative energy by a fitted coefficient. The W4 procedure [11] improves upon W3 in terms of accuracy and robustness by using even larger basis sets (up to sextuple-ζ) in its formulation, together with the inclusion of even higher-order correlation effects (up to CCSDTQ5 [33]). At this end of the scale, W4 has been demonstrated to yield sub-kJ mol^{−1} accuracy for a set of small molecules that can be viably computed.

## The W*n*X-type procedures as our answer to the challenge

### Some earlier attempts in reducing the computational cost of W*n*

We will shortly provide a brief overview of the formulation of the W*n*X series of composite methods. Before we proceed to describing W*n*X, it is noteworthy that there have been other attempts in reducing the computational requirements of W*n* in order to broaden their applicability. For instance, in recognition of the major bottleneck associated with the all-electron CCSD(T) calculation in W1, the W1c procedure [34] has been proposed. It involves a dramatic simplification by replacing the CCSD(T)-based core-valence correlation component with empirical terms, thus bypassing quantum chemistry calculation for this constituent altogether. The W*n*-F12 procedures developed in recent years employ explicitly-correlated CCSD(T)-F12-type methods [35] in the valence coupled-cluster calculations. They are computed in conjunction with smaller basis sets when compared with the original W*n* protocols. Thus, W1-F12 [36] uses only DZ and TZ basis sets, whereas for W1, as mentioned earlier, contains a QZ basis set in its formulation.

There are also more economical options for the post-CCSD(T) W3 and W4 protocols. These alternative methods typically employ smaller basis sets and/or lower-level procedures to reduce the computational demand for the expensive post-CCSD(T) calculations. For example, the W3.2 protocol [11] replaces CCSDTQ used in W3 with the lower-cost “CCSDT(Q)” method [33], while W3.2lite [37] makes further simplification by using smaller basis sets in an additive manner. The W3-F12 method [36] combines the post-CCSD(T) component of W3.2 with the W2-F12 procedure [36], which employs CCSD(T)-F12-type methods to provide an economical variant of W2. Very recently, the W4-F12 protocol [38] has been proposed as an alternative to W4. It also employs CCSD(T)-F12-type methodologies for the calculation of the CCSD(T)/CBS component.

With the perspectives given by the above background information, in the next section, we will detail the approach that we use in the various W*n*X-type methodologies in order to tackle the challenges. Before we begin, for the convenience of our readers, we provide a brief summary of the major computations that are employed to obtain the components of these methods (Table 1). In addition, as W*n*X shares certain design philosophies with the analogous W*n*-F12-type methods, we also show the components in W1-F12 and W4-F12 procedures for comparison purpose. These two methods represent at the one end the most cost-effective (W1-F12) and at the other end the most accurate (W4-F12) protocols within the W*n*-F12 series of composite methods.

Component | W1X-2 | W1X-1 | W2X | W3X | W3X-L | W1-F12 | W4-F12 |
---|---|---|---|---|---|---|---|

Geometry | B3-LYP/PVTZ+d | B3-LYP/PCTZ+d | B3-LYP/PVTZ+d | B3-LYP/PVTZ+d | B3-LYP/PVTZ+d | B3-LYP/PVTZ+d | CCSD(T)/PVTZ+d |

E[HF] and ΔE[CCSD] | CCSD(T)^{b}/A′PV[D,T]Z | CCSD^{b}/PV[D,T]Z^{c} | CCSD^{b}/A′PVQZ | CCSD^{b}/PV[D,T]Z^{c} | CCSD^{b}/A′PVQZ | CCSD^{b}/PV[D,T]Z^{c} | CCSD^{b}/PV[Q,5]Z^{c} |

ΔE[(T)] | CCSD(T)/A′PV[D,T]Z+d | CCSD(T)^{b}/A′PVTZ | CCSD(T)/A′PV[D,T]Z+d | CCSD(T)^{b}/A′PVTZ | CCSD(T)/A′PV[D,T]Z+d | CCSD(T)/A′PV[Q,5]Z+d | |

ΔE[C+R] | MP2/PCVTZ | MP2/PCVTZ | MP2/PCVTZ(3d)+ CCSD(T)/PCVDZ(2d) | MP2/PCVTZ | MP2/PCVTZ(3d)+ CCSD(T)/PCVDZ(2d) | CCSD/PWCVTZ+ CCSD(T)/PWCVTZ(– f)+ CCSD(T)/A′PVDZ-DK | CCSD(T)/A′WCPV[T,Q]Z |

ΔE[(T)+] | CCSDT/VTZ+ CCSDT(Q)/VDZ | CCSDT/PVTZ(d)+ CCSDT(Q)/PVDZ(d) | CCSDT/PV[D,T]Z +CCSDT(Q)/PVTZ +CCSDTQ/PVDZ +CCSDTQ5/VDZ |

### The W1X-type protocols

We now proceed to the description of our W*n*X protocols. First of all, we note that all of them employ geometries and associated vibrational frequencies obtained using the DFT method B3-LYP [39] with the cc-pVTZ+d basis set [40]. This methodology is economical and fairly accurate, and is used in other W1-type procedures [27], [36]. We have thus employed such a strategy without modification. In our early W*n*X papers [19], [20], we also adopted the scale factor of 0.985 for the B3-LYP vibrational frequencies for the calculation of zero-point vibrational energies (ZPVEs), vibrational contributions to 298 K enthalpies [Δ*H*_{298}(vib)] and vibrational entropies [*S*_{298}(vib)]. In our more recent investigations [21], [41], we have re-determined the scale factors [0.9886, 0.9926 and 0.9970, respectively, for ZPVE, Δ*H*_{298}(vib) and *S*_{298}(vib)], owing to evidences suggesting the original value being not fully adequate [42], [43]. We will come back to the issue of geometry and associated vibrational thermochemical quantities later.

Let us now turn our attention to the aspect of electronic energy in W*n*X. They combine the strategies of the economical alternatives of W*n* mentioned in the previous section; each of them individually addresses some but not all of the computational bottlenecks in the various W*n* methods. Our endeavor began with the development of two W1X procedures, namely W1X-1 and W1X-2 [19]. The W1X-1 protocol makes use of the approach used for the valence CCSD(T)/CBS quantum chemistry calculations of W1-F12:

Here, *E*[HF(CABS)] is the HF energy with the “complementary auxiliary basis set” (CABS) correction [35], which is obtained in the CCSD-F12b [35] calculation used for the Δ*E*[CCSD-F12b] term. The use of the CABS approach enables more rapid basis set convergence. Thus, the DZ-type cc-pVDZ-F12 and the TZ-type cc-pVTZ-F12 basis sets [44] are employed in the calculation of *E*[HF(CABS)]/CBS (and Δ*E*[CCSD-F12b]/CBS). The standard two-point extrapolation formula: *E _{L}*=

*E*

_{CBS}+

*A L*

^{−α}[29] is used to obtain the CBS limit, where

*L*=2 for DZ and 3 for TZ and so on, and α has a value of five, which is typically used for the extrapolation of HF energies. Note that

*A*is canceled when one rearranges the formula, such that only

*E*and

_{L}*E*

_{L}_{+1}are required to calculate

*E*

_{CBS}.

For the calculation of Δ*E*[CCSD-F12b]/CBS, the same extrapolation formula is used but with a different α value. The value for this term in W1X-1 is 3.67 obtained by fitting to reliable experimental data, namely the G2/97 set [45], [46] of thermochemical quantities. The Δ*E*[(T)]/CBS component is calculated using conventional CCSD(T) and CCSD energies without applying the F12 correction. For this term, the aug′-cc-pV(D+d)Z and aug′-cc-pV(T+d)Z basis sets [47] are used in the CBS extrapolation using the same formula, but with α=2.04, which was determined simultaneously together with the α value for Δ*E*[CCSD-F12b]/CBS.

As mentioned earlier, in terms of the consumption of computer time, the major bottleneck in W1 is the CCSD(T)(Full,DKH)/MTsmall calculation used to obtain the Δ*E*[C+R] term. The W1c method tackles the challenge by empirical replacement of this component, while this issue is somewhat addressed in W1-F12 in a less dramatic manner with the use of more manageable basis sets. The W1c approach is somewhat restrictive because it requires subjective bonding information, which is ill defined in many situations. The W1-F12 approach is fully objective (i.e. no adjustable parameter) and highly accurate, but might not have gone far enough to substantially alleviate the high computational demand for this component. In W1X-1, we attempted to strike a better compromise by obtaining Δ*E*[C+R] at the “MP2” level [48] with the more manageable cc-pCVTZ basis set [49]. This calculation is significantly less demanding on computational resources than the other ones in W1X-1. Typically, the computer time required for the Δ*E*[C+R] calculations is less than 5% of the total time for W1X-1. We note that the ccCA methodology [16] and related procedures such as ccCA-F12 [50] also employ MP2 for their core-valence correlation calculations.

This protocol for calculating the Δ*E*[C+R] term is also used in the W1X-2 procedure [19], which is a simplified alternative to W1X-1. It employs two calculations at the CCSD(T)-F12b level using the aug′-cc-pVDZ and aug′-cc-pVTZ basis sets [49] to obtain the three terms, namely *E*[HF(CABS)]/CBS, Δ*E*[CCSD-F12b]/CBS and Δ*E*[(T)-F12b]/CBS in the approximation for the CBS limit of valence CCSD(T). The values for α are 5, 4.74 and 2.09 for *E*[HF(CABS)], Δ*E*[CCSD-F12b] and Δ*E*[(T)-F12b], respectively. It is noteworthy that, while the significant saving in cost for W1X-1 and W1X-2 is advantageous, one should be aware of some potential challenges associated with the use of more economical components such as MP2 for Δ*E*[C+R] and aug′-cc-pV*n*Z (as opposed to cc-pV*n*Z-F12) for basis-set extrapolation [27], [38].

### The two directions in enhancing W1X: W2X and W3X

In subsequent studies, we have built on the foundation of W1X and devised W*n*X protocols with improved accuracy and/or robustness, and at the same time with the intention of keeping the increased demands on computational resources to minimum. Two directions have been taken in this regard. In W2X [21], we have employed a more accurate approximation for CCSD(T)/CBS. Thus, the valence component is obtained with energies computed at the CCSD-F12b and CCSD(T)-F12b levels with the larger aug′-cc-pVTZ and/or aug′-cc-pVQZ basis sets. The *E*[HF(CABS)] component is provided by *E*[HF(CABS)]/aug′-cc-pVQZ, the Δ*E*[CCSD-F12b] term is obtained by extrapolating the energies with the aug′-cc-pVTZ and aug′-cc-pVQZ basis sets with an α value of 5.88, and Δ*E*[(T)-F12b] is calculated with the aug′-cc-pVTZ basis set with the resulting energy difference scaled by 1.06. In W2X, the Δ*E*[C+R] component itself is obtained in a “composite” manner:

The Δ*E*[C+R][MP2] term is the base energy obtained at the MP2/cc-pCVTZ(3*d*) level. It is very similar to the Δ*E*[C+R] component in W1X-1 and W1X-2, but calculated with a somewhat larger basis set. The “3*d*” in this basis set indicates that, for non-hydrogen atoms, the *d* functions in cc-pCVTZ are replaced by those from cc-pCVQZ. The Δ*E*[C+R][CCSD(T)] component is a correction for higher-order correlation effects obtained as Δ*E*[C+R][CCSD(T)] – Δ*E*[C+R][MP2] using the more manageable cc-pCVDZ(2*d*) basis set, with “2*d*” signifies the replacement of the *d* functions in cc-pCVDZ by those in cc-pCVTZ.

The W3X method [20] adds post-CCSD(T) effects to W1X-1. The formulation for the post-CCSD(T) term is similar to that for W3.2 and W3.2lite, in that the highest-level procedure used is CCSDT(Q). In order to keep the computational requirement for the post-CCSD(T) calculations to minimum, highly truncated basis sets are used:

In this formula, the cc-V*n*Z basis sets are truncated cc-pV*n*Z basis sets with the “polarization functions” (*d*,*p* for cc-pVDZ and 2*df*,2*pd* for cc-pVTZ) completely removed; hence the letter “p”, signifying “polarized”, is dropped in the nomenclature.

### W3X-L: Where the two paths (W2X and W3X) converge

When both types of enhancements, i.e. the use of larger basis sets and the addition of post-CCSD(T) effects, are applied to W1X, we arrive at our (currently) highest-level W*n*X-type protocol, namely W3X-L [21]. It employs the W2X energy as its CCSD(T)/CBS component. To this, a post-CCSD(T) component is added. It is obtained in a more rigorous manner than the one in W3X, while keeping the associated increase in computational cost to minimum:

The CBS limit for the Δ*E*[T] component is obtained using the cc-VDZ (i.e. cc-pVDZ but without polarization functions) and cc-pVTZ(*d*) basis sets. The cc-pVTZ(*d*) basis set is a modified cc-pVTZ basis set. It contains no polarization function for hydrogen; for other atoms, the 2*df* functions are replaced by the *d* function of cc-pVDZ. An α value of 2.61 is used for the basis set extrapolation. For the calculation of the Δ*E*[(Q)] component, we used the cc-pVDZ(*d*) basis set, which is cc-pVDZ with the omission of the *p* polarization functions on hydrogen.

## Performance of W*n*X: have we met our challenge?

### How accurate are the W*n*X methods in general?

Having defined the W*n*X protocols, we now turn our attention to their performance. Specifically, we ask two questions: are they able to preserve the accuracy of the analogous W*n* procedures, and if so, is this achieved together with a significant gain in computational efficiency? We will first look at the former issue, i.e. their accuracy in comparison with other W*n*-type methods.

The W1X-1 and W1X-2 protocols have been assessed with the E2 set [9], [10] of over 500 accurate thermochemical data consist of a wide range of properties, plus the BDE261 set [51] of bond dissociation energies, and the CEPX set [52] of complexation energies and proton exchange barriers. We provide the full set of mean absolute deviations (MADs, kJ mol^{−1}) for the various W1-type methods in Table 2. While many interesting comparisons can be drawn with this set of statistical data, herein we simply note that the various W1-type procedures, including W1X-1 and W1X-2, show comparable accuracies. The MAD values for all methods are ~2–3 kJ mol^{−1} for the E2 set and most of its subsets. Nonetheless, it has been noted that [19] W1X-2 may perform slightly less well than other W1-type procedures for systems with a large number of highly-electronegative atoms such as fluorine.

Test set | Property | W1X-2 | W1X-1 | W1-F12 | W1w |
---|---|---|---|---|---|

E2 | 2.7 | 2.4 | 2.9 | ||

Atomization energies | |||||

W4/08 | Atomization energy | 3.8 | 2.7 | 4.1 | 3.7 |

G2′ ∆_{f}H | Heat of formation | 3.0 | 2.7 | 3.2 | 4.4 |

G3′ ∆_{f}H | Heat of formation | 3.7 | 3.5 | 4.9 | |

Other fundamental properties | |||||

G2 IE | Ionization energy | 2.7 | 2.8 | 2.5 | 2.7 |

G2 EA | Electron affinity | 1.7 | 1.9 | 2.1 | 1.7 |

G2 PA | Proton affinity | 2.2 | 2.2 | 2.1 | 1.8 |

Reaction energies | |||||

ADD | Radical addition | 1.4 | 1.0 | 1.3 | |

ABS | Radical abstraction | 0.9 | 0.8 | 1.2 | |

Barrier heights | |||||

DBH24 | Atom transfer | 1.3 | 1.3 | 0.9 | 0.8 |

PR8 | Pericyclic reaction | 1.4 | 1.6 | 2.1 | |

Non-covalent interactions | |||||

HB16 | Hydrogen bond | 1.0 | 0.6 | 0.6 | 0.4 |

WI9/04 | Weak interaction | 0.7 | 0.7 | 0.7 | 0.7 |

Test sets in addition to E2 | |||||

BDE261 | Bond energy | 1.0 | 0.9 | 1.0 | |

CEPX | Proton exchange | 4.6 | 1.3 | 1.0 | 0.7 |

The W2X procedure has been evaluated using the W4-11 set [13] of highly-accurate atomization energies (with sub-kJ mol^{−1} accuracy) [21]. The values in this set were computed at the W4 level rather than taken from experiment. Thus, it enables comparison with, not only full W4 energies, but also the component CCSD(T)/CBS energies. The W2X protocol, along with other W*n*-type procedures, has been assessed in both regards (Table 3). We can see that, when compared with benchmark CCSD(T)/CBS values, the W2-type methods all have similar MADs of ~0.5 kJ mol^{−1}, which is considerably better than those for W1-type protocols (~2 kJ mol^{−1}). Importantly, W2X shows an accuracy that is similar to the other two W2-type methods (W2w [32] and W2-F12) in this assessment. When full W4 atomization energies are used as the benchmark, the W2-type methods have larger MADs of ~1–2 kJ mol^{−1}, which can largely be attributed to the lack of post-CCSD(T) effects in these methods. Interestingly, the MADs for the W1-type methods remain approximately 2 kJ mol^{−1}.

Method | Benchmark energies | |
---|---|---|

CCSD(T)/CBS | W4 | |

W1w | 1.8 | 2.0 |

W1-F12 | 1.6 | 2.1 |

W1X-1 | 2.0 | 2.1 |

W1X-2 | 2.2 | 2.2 |

W2w | 0.5 | 1.1 |

W2-F12 | 0.7 | 1.3 |

W2X | 0.6 | 1.8 |

W3.2 | 0.6 | |

W3-F12 | 0.8 | |

W3X | 1.9 | |

W3X-L | 0.8 |

Among the two post-CCSD(T) W*n*X methods, the lower-cost W3X method is based on a design philosophy of improving the robustness over W1X-1 for difficult cases for which CCSD(T) is inadequate, while maintaining the general accuracy for systems for which CCSD(T) performs well. In this regard, we note that the systems in the W4-11 set by-and-large fall under the latter category, with difficult systems contribute to about 10% of the total [13]. We can see that W3X has achieved its intended goal for “non-problematic” systems (i.e. those without severe multi-reference characters), with an MAD of 1.9 kJ mol^{−1} that is similar to those for W1-type methods (Table 3). In comparison, we have devised the W3X-L procedure with an objective for it to be all-round more accurate than other W*n*X methods. For the W4-11 set, its MAD of 0.8 kJ mol^{−1} is notably smaller than those for other W*n*X protocols. Importantly, it is also quite competitive with the W3.2 and W3-F12 methods, which have MADs of 0.6 and 0.8 kJ mol^{−1}, respectively.

### How well do W*n*X methods work for more challenging systems?

Having established the degree of accuracy of W*n*X for typical systems, let us now discuss their performance on some of the more challenging ones. Specifically, the W*n*X methods have been examined [20], [21] for systems that are known to represent problems for CCSD(T). These tests have been undertaken in order to distinguish the accuracy and robustness between the CCSD(T)-based protocols (W1X-1, W1X-2 and W2X) and the post-CCSD(T) ones (W3X and W3X-L) under more challenging circumstances. The difficult cases that have been tested in the original W*n*X publications include the homolytic dissociation of F_{2}, the reaction barrier for the automerization of cyclobutadiene, and post-CCSD(T) contributions to the atomization energies of chromium oxides.

For F_{2} dissociation (Fig. 1), the CCSD(T)-based W1X-1 and W2.2 [32] procedures were examined, with both of them showing large deviations of ~70–100 kJ mol^{−1} at an F–F distance of 3 Å when compared with the proper dissociation behavior. Such large deviations are substantially reduced with the incorporation of post-CCSD(T) effects in W3-type protocols. Among the W3-type methods, the deviations for W3X and W3.2lite are not very large but still quite noticeable. In comparison, those for W3X-L and W3.2 are virtually negligible. The second system in our test, namely the automerization of cyclobutadiene, has been a subject of numerous theoretical studies [53 and refs therein]. The best estimates point to a value of ~45 kJ mol^{−1} for the barrier when basis set effects are taken into account (Table 4). For this quantity, W1X-1 yields a value of 77.7 kJ mol^{−1}, while a value of 50.4 kJ mol^{−1} has been obtained with W3X.

cc-pVDZ | cc-pVTZ | |
---|---|---|

RMRCCSD(T) | 30.1 | 39.7 |

MkCCSD(T) | 32.6 | 37.2 |

CAS-BCCC4 | 31.8 | 36.4 |

MR-AQCC | 30.5 | 35.1 |

EOM-CCSD[+2] | 34.7 | 39.7 |

W1X-1 | 77.7 | |

W3X | 50.4 |

The third system that we have examined, i.e. chromium oxide (CrO, CrO_{2} and CrO_{3}) atomizations, represents the most challenging case among the three tests. It has been the subject of a previous computational chemistry study [54] using composite strategies with high-level methods up to CCSDTQ5. In that study, it has been emphasized that CrO_{3} is particularly problematic due to its highly-multi-reference character. For instance, post-CCSD(T) contributions to atomization energies are significantly larger for CrO_{3} than for the other two, with values of 2.4, 8.9 and 24.9 kJ mol^{−1} for CrO, CrO_{2} and CrO_{3}, respectively. In addition, an examination of the basis set effects indicates a slower convergence for CrO_{3} than for CrO and CrO_{2}. Table 5 shows the post-CCSD(T) contributions obtained with the various W3-type protocols in comparison with those obtained previously with higher-level procedures. It can be seen that the W3-type methods are reasonably adequate for the computation of CrO, but their performances for CrO_{2} and (especially) CrO_{3} are rather poor.

W3X | W3.2lite | W3.2 | Best estimate | |
---|---|---|---|---|

CrO | 8.9 | 6.4 | 9.7 | 2.4 |

CrO_{2} | 13.3 | 21.5 | 24.8 | 8.9 |

CrO_{3} | 14.9 | −1.3 | 8.4 | 24.9 |

### Do W*n*X methods have improved computational efficiency?

Let us recall that our objective for developing the W*n*X protocols is to enable quantum chemistry calculations with W*n*-level accuracies at considerably lower computational costs than those for alternative W*n*-type methods. In the two sections above, we have shown that W*n*X have similar accuracies when compared with other W*n*-type procedures. The W*n*X protocols are suitable for the accurate calculation of most systems, except those with very significant multi-reference characters, such as first-row transition metal species.

We now turn our attention to the aspect of computational efficiency of W*n*X. Table 6 shows the memory and disk usable for the calculation of benzene using various W*n*-type protocols [21]. These values should be fairly independent of the hardware platform, so long as the same software packages (Molpro [22] and MRCC [33]) are employed. The actual computer time required for the calculations would have stronger dependency on the hardware, and we therefore provide a relative timing comparison in Table 6, with that for W1X-1 assigned as unity.

Memory (GB) | Disk (GB) | Relative time | |
---|---|---|---|

W1 and W2-type procedures | |||

W1X-2 | 0.6 | 6.5 | 0.7 |

W1X-1 | 0.7 | 10.0 | 1.0 |

W1w | 2.2 | 59.2 | 4.6 |

W2X | 2.2 | 59.2 | 3.0 |

W2w | 7.3 | 413.1 | 38.3 |

Δ(T)+ component for W3-type procedures | |||

W3X | 0.7 | 106.0 | 27.9 |

W3X-L | 1.9 | 284.8 | 75.5 |

CCSDT/cc-pVDZ | 1.3 | 196.5 | 35.0 |

For benzene, which is a molecule of fairly moderate size, the memory requirement is quite small (a few GB). Nonetheless, we can already see that W1X-1 and W1X-2 consume just a fraction of memory when compared with that for W1w. The same level of difference can be seen between W2X and W2w. In terms of scratch disk usages, there is a considerably larger variation from less than 10 GB for the two W1X methods to over 400 GB for W2w. For this quantity, the saving of using W*n*X over W*n* is more substantial. Computer time is another aspect in which significant gain in efficiency is achieved with W*n*X. For instance, the calculation with W2X in fact took less time than that for W1w, while W2w consumed more than 10 times of the computer time used for W2X.

The post-CCSD(T) calculations in the W3-type methods have significant time requirements when compared with those for the CCSD(T)/CBS calculations, even in the case of benzene. One important result that we will highlight is that, with our computational resources at the time of undertaking the calculations, we were not able to complete the tasks of CCSDT/cc-pVTZ and CCSDT(Q)/cc-pVDZ computations employed in the post-CCSD(T) component of W3.2. This illustrates the importance of using adequate methodologies that are as economical as possible especially when post-CCSD(T) computations are required.

## Illustrative examples of possibilities opened up with W*n*X

The improved computational efficiency of the W*n*X protocols enables, in a straightforward manner, highly-accurate quantum chemistry calculations on systems that may be too large for alternative W*n* methods. For instance, W1, W1-F12, W1X-1 and W1X-2 have been applied to the calculation of barriers for a set of pericyclic reactions [19], among which the two largest ones were only feasible with W1X-1 and W1X-2 (Table 7). For this set, we also see that the calculated values with the four methods are generally in good agreement with one another.

W1X-2 | W1X-1 | W1-F12 | W1 | |
---|---|---|---|---|

Cyclobutene→cis-butadiene | 139.2 | 138.9 | 137.5 | 140.9 |

cis-1,3,5-hexatriene→1,3-cyclohexadiene | 122.5 | 122.5 | 123.0 | 126.4 |

o-xylylene→benzocyclobutene | 110.7 | 110.8 | 110.9 | 113.3 |

cis-1,3-pentadiene sigmatropic shift | 151.8 | 151.3 | 153.7 | 152.9 |

cyclopentadiene sigmatropic shift | 109.9 | 110.3 | 111.1 | 108.5 |

cis-1,5-hexadiene sigmatropic shift | 144.9 | 140.7 | 140.8 | 143.3 |

Ethane+cis-butadiene→cyclohexene | 94.0 | 94.0 | 93.6 | 96.7 |

Ethane+cyclopentadiene Diels–Alder reaction | 78.2 | 79.0 | 78.0 | 80.8 |

cyclopentadiene dimerization | 62.3 | 63.6 | ||

cis-triscyclopropanocyclohexane→1,4,7-cyclononatriene | 106.0 | 100.5 |

The W1X-type methods have subsequently been applied to some of our other studies [55], [56], [57], [58], [59], [60]. These include, for example, the investigations into the chemistry of amino acid derivatives [58] and polycyclic aromatic hydrocarbons (PAHs) [56], [57], [59] that are reasonably large in size. For one of the PAHs, namely C_{13}H_{10}^{•+}, the computer time consumed for its W1X-2 calculation was 512 h. This is not dramatically longer than that for the W1w calculation for the (approximately half-sized) C_{6}H_{5}Me^{•+} molecule (221 h) if we consider the “standard” *N*^{7} scaling for CCSD(T) (*N* is proportional to the system size) could lead to a time of ~15 000 h for the W1w calculation of C_{13}H_{10}^{•+}.

The W3X-type procedures, with their inclusion of post-CCSD(T) components, are substantially more demanding than the CCSD(T)-based methods. Thus, W3X and W3X-L are still only applicable to relatively small systems such as benzene and toluene. Nonetheless, for systems that calculations with W3X-type protocols are feasible, the computations can often be completed in a reasonably rapid manner. The strength of these procedures therefore lies in their capacity for conducting highly-accurate calculations on a large number of moderately-sized molecules in a timely fashion.

For instance, W3X has been applied to the MB08 set [61] of 165 artificially-generated molecules, with some of them possessing notable post-CCSD(T) effects [20]. The W3X-L procedure has been employed to obtain more accurate values for the G2/97 set of thermochemical quantities with over 300 data points [21], with an estimated general uncertainty of ~1.5 kJ mol^{−1} for this set of systems. The utility of the computational efficiency of W3X-L has also been demonstrated in its application to over 120 species [62] for which highly-accuracy ATcT [14] values are also available. The high-level of accuracy of W3X-L has enabled its use to cross-validate the ATcT values in that study. In another study [63], the two W3X-type methods have been used in a comprehensive investigation of the reaction mechanism and kinetics of the Criegee intermediate, which illustrates the utility of W3X and W3X-L for atmospheric chemistry.

## Additional considerations and recommendations

### On the appropriate W*n*X protocol for a particular problem

Among the existing W*n*X methods, the W1X-type procedures would obviously give rise to the highest productivity in research output. In many cases, they may be the only feasible W*n*X methodologies to use. In other cases, there may be sufficient resources to perform higher-level calculations. Nonetheless, under such circumstances, one may wish to evaluate the potential benefits offered by the more expensive W2X and W3X-type procedures. To this end, we have examined diagnostics that may shed some light on this issue [20].

There are three factors that may be involved in cases where W1X-type methods are not sufficiently accurate: (1) the failure of CCSD(T) itself for multi-reference systems, (2) an inadequate valence CCSD(T)/CBS approximation, and (3) inaccurate core-valence correlation effects associated with the use of MP2. It has been demonstrated that, for a given species, the percentage contribution of Δ*E*[(T)] to the total atomization energy (TAE), denoted “%(T)”, provides a reasonable assessment for the importance of the first factor. Thus, a value larger than 5% suggests that caution should be taken [11]. Our analysis also supports the adoption of this recommendation in this aspect. For the two remaining factors, we have devised a diagnostic that we call “#CCSD(T)” to evaluate their combined importance:

The #VCBS term is an estimate for the quality of the valence CCSD(T)/CBS term. It is defined as the difference between the two Δ*E*[(T)]_{TAE} terms, obtained with a TZ basis set and at the CBS limit, where Δ*E*[(T)]_{TAE} is the contribution of Δ*E*[(T)] to the TAE in kJ mol^{−1}. The #CR term is defined as the MP2 Δ*E*[C+R] contribution to the TAE, also in kJ mol^{−1}. Note that we employ absolute values for both #VCBS and #CR. The linear combination and their coefficients were determined by fitting to the difference from benchmark values in order to give the most reliable diagnostic for using W1X-1 as an approximation to CCSD(T)(Full,DKH)/CBS. We suggest a value larger than 3 kJ mol^{−1} to be an indicator for potential problem in this regard. Our recommended W*n*X protocols are given in Table 8 according to the classification of a molecule into one of the four groups using the #CCSD(T) and %(T) diagnostics.

#CCSD(T) | %(T) | Suggested method |
---|---|---|

≤3 | ≤5 | W1X-1 |

>3 | ≤5 | W2X |

≤3 | >5 | W3X |

>3 | >5 | W3X-L |

### On the accuracy of geometry and vibrational frequencies used in W*n*X

We have mentioned at the beginning of our description of the W*n*X protocols that B3-LYP/cc-pVTZ+d is the prescribed method for obtaining geometries and vibrational frequencies in all W*n*X procedures. This choice is based mostly on its general accuracy for calculating these quantities with a very good computational efficiency. For highly-accurate composite protocols, however, the uncertainty associated with a low-level geometry and related quantities may well dominate the total uncertainty. Indeed, in our analysis of the uncertainties in our calculated W3X-L values for the G2/97 set, we find that approximately half of the total uncertainty is associated with the use of B3-LYP geometries and scaled vibrational frequencies [21]. We have attempted to address this challenge in a recent study [41]. Our findings show that, among the major sources of errors associated with geometries and scaled vibrational frequencies [electronic energies (*E*_{elec}) on low-level geometries, ZPVE, Δ*H*_{298}(vib) and *S*_{298}(vib)], the largest ones are those for *E*_{elec} and ZPVE.

We have assessed a series of “double-hybrid” (DH) DFT procedures [64] in conjunction with a variety of basis sets as alternatives to B3-LYP/cc-pVTZ+d for geometry optimization and vibrational frequency calculations within W*n*X. In general, DH-DFT methods are more accurate than conventional DFT owing to the additional higher-level MP2 component in DH-DFT. They are computationally more demanding than DFT but not excessively so within the context of high-level composite protocols. Our findings indicate that the DSD-PBE-P86/aug′-cc-pVTZ+d procedure [65] yields somewhat better agreements with benchmark values. More importantly, it reduces some of the largest deviations by ~2 kJ mol^{−1}, which is fairly significant given that, when one employs the highest-level W3X-L protocol, the desired accuracy would often be 2 kJ mol^{−1} or better. For the lower-level members of W*n*X, we feel that B3-LYP may still represent the sweet spot between accuracy and efficiency for geometry optimizations.

## Concluding remarks

Computational quantum chemistry has come a long way to become an important and widely applicable tool for obtaining highly-accurate thermochemical data in an objective manner. In part, the continuous development of the so-called “composite methods” has made a significant contribution to this achievement. In this regard, the W*n*X protocols that we have developed in recent years have further pushed the boundary of the versatility of highly-accurate composite procedures. The currently most economical methods within the W*n*X family, namely W1X-1 [19] and W1X-2 [19], are capable of treating molecules with ~20–30 atoms in a reasonably timely manner, while keeping a general accuracy of ~3–4 kJ mol^{−1}. Higher-level W*n*X protocols include W2X [21], W3X [20] and W3X-L [21]. Among these, W3X-L represents the highest-level procedure, with an accuracy of ~1–2 kJ mol^{−1} and an applicability to systems with ~10–20 atoms. Its strength lies in its reasonably rapid turnover rate, especially when compared with other composite methods of similar accuracy. The W*n*X series of composite procedures have opened up new possibilities for computational chemists in pursue of accurate thermochemical quantities in a highly-productive manner.

## Article note:

A collection of invited papers based on presentations at the 23^{rd} IUPAC Conference on Physical Organic Chemistry (ICPOC-23), Sydney, Australia, 3–8 July 2016.

## Acknowledgements

We thank the Australian Research Council for financial support, and National Computational Infrastructure and Intersect Australia Limited for their generous provision of computational resources throughout the course of these studies.

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**Published Online:**2017-02-18

**Published in Print:**2017-06-27

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