On the hydrogen activation by frustrated Lewis pairs in the solid state: benchmark studies and theoretical insights

Abstract

Recently, the concept of small molecule activation by frustrated Lewis pairs (FLPs) has been expanded to the solid state showing a variety of interesting reactivities. Therefore, there is a need to establish a computational protocol to investigate such systems theoretically. In the present study, we selected several FLPs and applied multiple levels of theory, ranging from a semi-empirical tight-binding Hamiltonian to dispersion corrected hybrid density functionals. Their performance is benchmarked for the computation of crystal geometries, thermostatistical contributions, and reaction energies. We show that the computationally efficient HF-3c method gives accurate crystal structures and is numerically stable and sufficiently fast for routine applications. This method also gives reliable values for the thermostatistical contributions to Gibbs free energies. The meta-generalized gradient approximated TPSS-D3 evaluated in a projector augmented plane wave basis set is able to produce sufficiently accurate reaction electronic energies. The established protocol is intended to support experimental studies and to predict new reactions in the emerging field of solid-state FLPs.

This article is part of the themed issue ‘Frustrated Lewis pair chemistry’.

1. Introduction

Typically, when Lewis acids and bases are brought together in solution, they rapidly react with each other and form classic adducts. In 2006, Stephan and co-workers [1] found that such neutralization reaction can be changed by the presence of bulky organic substituents on the Lewis acid and base centres. The Lewis acid and base components form a complex mainly because of the secondary interactions between the organic substituents (inter-molecular FLPs), or by bridging atoms between the active centres (intra-molecular FLPs). In both cases, the reactivities of the Lewis acids and bases remain and the pair was called a ‘frustrated Lewis pair’ (FLP) [1]. Interestingly, the FLP has been shown to be able to activate molecular hydrogen, and serves as a non-metal catalyst for H₂ activation. Later on, the FLPs were extended to include other Lewis systems, mainly consisting of P/B or N/B combinations. Moreover, FLPs have been shown to activate other small molecules, such as CO₂, SO₂ and NO [2–7]. The mechanism of H₂ activation by FLP in solution has been subsequently analysed via density functional theory (DFT) calculations [8–13] and molecular dynamics simulations [14,15]. Up to now, there have been two typical reaction mechanisms proposed in the literature: (i) the electron transfer model in which the H₂ molecule is polarized by the electron transfer from the lone pair of tBu₃P to the σ*(H₂) orbital and from the σ(H₂) to the empty orbital of B(C₆F₅)₃; [8] and (ii) the electric field model in which the H₂ molecule is polarized by the electric field created by the FLPs [11]. Additionally, the solvation contribution to the Gibbs free energy has been shown to be crucial for stabilizing the charge separated products [16]. However, it is important to point out that one decade after their first discovery, all reported FLP reactivity was observed in solution, which limits the possible industrial application of such reactions.

Recently, O'Hare and co-workers expanded the concept of frustrated Lewis pairs to the solid state [17]. In their work, a silica-supported Lewis acid (SiOB(C₆F₅)₂ was combined with a Lewis base (tBu₃P) forming a solid-phase FLP and such FLPs showed H₂ activation at mild conditions (i.e. 65°C and 72 h). While quantum chemical methods are routinely used to compute and analyse the contributions to the reaction Gibbs free energies for molecules (i.e. reactivities of FLPs in solution), the corresponding solid-state treatments are underdeveloped [18]. We think that more in depth benchmark studies of computational methods to theoretically investigate in particular the reactivities of FLPs in the solid state are mandatory and in the present study, we intend to close this gap. We first formulate some of seemingly contradictory requirements (wish list) for a reasonable protocol.

1. The final reaction Gibbs free energies need a sufficient accuracy of about 2–3 kcal mol⁻¹ enabling a meaningful analysis of the results.

2. In order to screen several possibilities in terms of structures, conformations, chemical modifications, etc., the computational effort has to be limited. In practice we want to be able to perform all calculations in a time frame well below one month on a single computer node with about eight processor units.

3. The methods should be consistently applicable to both the solid and the solution state and require only minimal adjustments for new target systems.

Apparently, we need to find a good compromise between computational efficiency and overall accuracy. Possible methods applicable to the routine electronic structure computation of organic crystals were recently reviewed [19]. We analysed several established and new quantum chemical methods. These are with increasing computational effort the density functional tight-binding Hamiltonian DFTB3-D3 [20], the minimal basis set Hartree–Fock HF-3c method [21], the generalized gradient approximated (GGA) functional (PBE) evaluated in small orbital sets [22,23], the screened exchange hybrid functional in small orbital expansion HSE-3c [24,25], the meta-GGA TPSS-D3 [26], and the hybrid functional PBE0-D3 [27] both evaluated in a converged orbital expansion. The task of computing the reaction Gibbs free energy can be roughly separated into three main steps that can be treated with different methods:

1. generation of accurate crystal geometries

2. computation of the phonon spectrum for the thermostatistical contributions

3. calculation of the electronic reaction energy.

The final reaction Gibbs free energy can be expressed as follows:

1.1

with G = E_elec + ZPE + E_therm + PV − T × S and where E_elec is the electronic energy, ZPE is the zero point vibrational energy, E_therm is the thermal energy, and P, V, T and S denote pressure, volume, temperature and entropy, respectively.

The different computational tasks produce different computational workloads, growing in the order electronic energy (1), geometry (2) and phonon spectrum (3). However, their impact on the final Gibbs reaction energy is not equally strong. Thus, we discuss each task independently and analyse the accuracy of feasible methods while keeping the computational costs in mind. To this end we selected four experimentally studied FLPs (scheme 1) and present a benchmark study in order to establish a computational protocol for the above mentioned purpose. The systems have been chosen carefully to represent typical inter- and intra-molecular FLP systems.

Scheme 1.

Four studied frustrated Lewis pairs (FLPs). Mes,1,3,5-trimethylbenzene; Cy, cyclohexyl.

2. Computational details

We list the quantum mechanical methods according to their relative computational cost starting with the most affordable. DFTB3-D3: The density functional tight binding Hamiltonian is used in its third-order variant with self-consistent charge distribution, and D3 London dispersion correction [20]. We use the 3OB Slater--Koster files provided by Elstner and co-workers and damp all hydrogen containing pair-potentials with an exponent of 4.2 [28–30]. HF-3c: The empirically modified minimal basis set Hartree–Fock method accounts for London dispersion and basis set errors in a consistent and cost-effective way (only nine global empirical parameters) [21]. While it has been introduced for molecular complexes, its extension and accuracy for various organic crystals has been demonstrated recently [31,32]. PBE-D3/def2-SVP: As cross check we apply the dispersion corrected GGA PBE functional [22] in a common split-valence orbital set without further corrections [23]. The method suffers from basis set incompleteness and self-interaction errors (see below for a discussion). HSE-3c: In the spirit of the ‘3c’ methods, the latest variant is a screened exchange hybrid functional evaluated in a modified double-zeta type orbital set, which is ideally suited to describe organic complexes and crystals. The screened exchange reduces the self-interaction error, while keeping the method numerically robust even for small gap materials [24,25]. TPSS-D3/‘CBS’: The well-established dispersion-corrected meta-GGA TPSS [26] is applied in a well-converged projector augmented plane wave (PAW) basis sets with high energy cut-off of 800 eV [33,34]. PBE0-D3/‘CBS’: For reaction energies, exact, non-local Fock (one-determinant) exchange can be important and we use the dispersion corrected hybrid functional PBE0 [27] as the most reliable method. It is evaluated in the same tightly converged PAW basis set expansion. For molecular calculations, we employ Gaussian type orbitals of triple zeta quality def2-TZVP [23]. All semi-local electronic structure methods cannot describe the London dispersion interaction. Thus, the D3 correction is used in the Becke–Johnson damping scheme including the triple-dipole three-body term [35–37]. A comprehensive overview of various types of dispersion corrections for mean field methods is given in a recent review article [37]. Fragment based approached might be applicable too, but will not be considered here [38].

The Brillouin zone is sampled with Γ-centred k-meshes with density of about 0.03 A⁻¹, which corresponds to the Γ-Point only due to large unit cell dimensions. Harmonic vibrational frequency calculations are used to characterize the nature of the stationary points along the reaction coordinates and no imaginary frequencies were found for the local minima. The thermostatistical contributions to the reaction Gibbs free energy in the gas phase were obtained from a harmonic oscillator approximation at a temperature of 298.15 K and a pressure of 1 atm. HF-3c frequencies are scaled by 0.86 as recommended in the original publication [21], while all otherwise computed frequencies are used unscaled.

Different program codes are used for the various methods: DFTB3 energy and gradient contributions are computed with the dftb+ code [39–41]. For HF-3c, PBE-D3/def2-SVP, and HSE-3 a developer version of CRYSTAL14 [42] is employed. This program is ideally suited for cost-effective DFT calculations in small basis sets as it can exploit full point and space group symmetry [43] as well as massive parallelism, reducing the computation (wall clock) time dramatically [44]. TPSS and PBE0 in the PAW basis set are calculated with the VASP 5.4 program suite [45,46]. All molecular calculations are conducted with TURBOMOLE 7.0 [47] together with the density-fitting RI-J [48,49] approach for the Coulomb integrals.

3. Results and discussions

(a). Crystal structures

In order to study the reactivities of FLPs in the solid state, i.e. to calculate the reaction Gibbs free energies, we first need to find out a method which could represent the experimental crystal structures with acceptable errors. It is important to point out that the inclusion of dispersion corrections generally improves the performance of density functionals for organic crystal structures and is thus always included in this study [50–52]. Unfortunately, only for the FLPs 2 and 3 in scheme 1 are experimental crystal structures available [53]. Hence, we first adapted the experimental crystal structures of FLPs 2 and 3 to validate four methods, namely HSE-3c, HF-3c, PBE-D3/def-SVP and DFTB3-D3. The data are provided in table 1. Note that we compare theoretical 0 K, zero-point exclusive structures with finite temperature measurements. The thermal expansion can typically increase the unit cell volumes by about 2–6% [54–57]. Our results indicate that both HSE-3c and HF-3c show a good agreement with experimental data in terms of unit cell parameters and crystal mass densities. The average errors for the length of the unit cell are less than 8% and the errors of the density are less than 6% which are in the acceptable range for our target accuracy. In contrast, the errors of PBE-D3/def2-SVP are substantially larger (i.e. producing mass density values that are too high). For instance, the errors of the mass density can be up to 18% in the case of FLP 3. A previous study reveals that the dispersion corrected PBE (PBE-D3) method is able to produce reasonable crystal structures, but with a well-converged PAW basis set. This indicates that the bad performance of the PBE-D3/def2-SVP method is attributed to basis set errors from the employed single particle basis set of only double zeta quality [58,59]. HSE-3c and HF-3c yield good crystal structure as they explicitly include corrections for basis set superposition error. Note that the good agreement between HF-3c and X-ray mass densities can be attributed to fortuitous error compensation, i.e. the slightly too large unit cells from HF-3c are compensated by the neglected thermal expansion effects. This might explain some of the differences between HF-3c and HSE-3c. The HSE-3c structures are expected to be of higher quality, which agrees with the observation that they are systematically denser compared to the finite temperature measurement. The crystal structures of FLPs 1 and 4 are experimentally not available. Therefore, we have predicted reasonable structures for these two FLPs based on the known crystals and by employing the HF-3c method (for details, see electronic supplementary material) (figure 1).

Table 1.

Experimental and calculated unit cell parameters and crystal mass densities for the reactants FLPs 1 to 4 of which molecular formulae are provided in scheme 1.

					density
		a (Å)	b (Å)	c (Å)	(g cm−3)	error (%)
1	X-ray	—	—	—	—	—
	HSE-3c	12.1	10.4	23.4	1.49	—
	HF-3c	12.1	10.6	25.4	1.38	—
	PBE-D3/def-SVP	12.0	10.1	23.2	1.58	—
	DFTB3-D3	12.1	10.0	22.3	1.64	—
2	X-ray	12.1	10.4	24.1	1.48
	HSE-3c	12.1	10.4	23.7	1.52	2.7
	HF-3c	12.0	10.3	25.9	1.41	4.7
	PBE-D3/def-SVP	11.9	10.1	23.6	1.59	7.4
	DFTB3-D3	11.9	10.0	22.8	1.66	12.2
3	X-ray	8.6	20.2	19.2	1.34
	HSE-3c	8.4	20.0	18.7	1.43	6.0
	HF-3c	8.6	20.6	18.7	1.34	0.3
	PBE-D3/def-SVP	8.8	18.7	17.1	1.58	17.9
	DFTB3-D3	8.4	19.1	18.1	1.54	14.9
4	X-ray	—	—	—	—	—
	HSE-3c	10.6	9.6	21.3	1.36	—
	HF-3c	10.9	10.4	21.6	1.18	—
	PBE-D3/def-SVP	10.0	9.0	20.8	1.46	—
	DFTB3-D3	10.4	9.3	20.9	1.48	—

Figure 1.

Optimized crystal structures (HF-3c level) of the studied FLPs in scheme 1. Colour legend: P yellow, B pink, F green, C black and H white. (Online version in colour.)

Typically for the FLPs in scheme 1, full relaxation at the HF-3c level including all atomic positions and unit cell parameters requires several days of CPU time. Faster computations can be conducted at the DFTB3-D3 level of theory. However, DFTB3-D3 gives quite large errors compared to experimental data or other methods. For example, the error of the mass density can be up to 15% compared to the experimental data in the case of FLP 3. The error is even larger compared to the HF-3c result which reaches 25% in the case of FLPs 1. This is most likely due to the neglect of three- and four-centre integrals in the DFTB method. A similar behaviour has been also observed in DFTB3-D3 benchmarks for the structures of several ice polymorphs [50]. In short, the HF-3c method turns out to be the best choice to provide accurate geometries for the study of reactivity of FLPs in the solid state. The required computational times are also acceptable, which normally are in a range of days.

(b). Thermostatistical contributions

After the geometry optimization, one has to compute harmonic vibrational frequencies in order to characterize optimized structures to be local minima. Additionally, they are needed to compute the thermostatistical contributions to the reaction Gibbs free energy, i.e. the zero-point vibrational energy, thermal corrections to enthalpy and entropy (equation (1.1)). In the previous section we concluded that the HF-3c method yields quite reasonable geometries for the crystal structures. In order to test the accuracy of the thermostatistical corrections at the HF-3c level, we first compare it to more reliable TPSS-D3 molecular (gas phase) data for four FLPs listed in scheme 1. The TPSS-D3 approach has been successfully applied in a number of previous studies of FLP systems in solution and has been demonstrated to yield reasonable values for the thermostatistical contributions [53,60,61]. We computed the thermostatistical contributions to the reaction Gibbs free energies of H₂ activation by FLPs in solution at both HF-3c and TPSS-D3/def2-TZVP level (figure 2).

Figure 2.

Optimized molecular structures (HF-3c level) of the studied FLPs in scheme 1. Colour legend: P yellow, B pink, F green, C black and H white. Hydrogen atoms are omitted for clarity. (Online version in colour.)

As shown in table 2, the HF-3c method indeed gives values close to the TPSS-D3/def2-TZVP level of theory. The mean error is below 1 kcal mol⁻¹, indicating the HF-3c thermostatistical contributions can be used without introducing significant errors. The solid-state frequency calculations at the HF-3c level are, however, quite time consuming with a CPU demand of several weeks. Thus, in addition we tested the performance of DFTB3-D3 for frequency calculations. The gas phase thermostatistical contributions from DFTB3-D3 differ only by 2–3 kcal mol⁻¹ from the TPSS-D3 ones. In contrast the deviations for the solids are more pronounced when using the fully relaxed structures with a maximum deviation to the HF-3c result of 6 kcal mol⁻¹ (FLP 1). This is probably related to the too dense crystal structures as shown in table 1. To circumvent this problem, we additionally computed the DFTB3-D3 frequencies on structures with unit cells constrained to the HF-3c optimizations. The resulting thermostatistical contributions show a much better agreement with the ones calculated at HF-3c level and the mean error is only about 1 kcal mol⁻¹. Specifically, the difference in thermostatistical contributions between DFTB3-D3 (full relaxation) and HF-3c for the H₂ activation by FLP 1 in the solid state is about 6 kcal mol⁻¹ and this difference is reduced to 1 kcal mol⁻¹ if we constrain the unit cell. In short, we found that the HF-3c method yields reasonable values for the thermostatistical contributions compared to those obtained at the TPSS-D3 level with a requirement of an average CPU time of several weeks. In principle, the frequency calculations are trivially parallelizable on many computer nodes, i.e. these computations are still feasible with sufficient resources. However, similar quality results can be obtained much faster at the DFTB3-D3 level with only 1 kcal mol⁻¹ average deviation. The typical CPU time to compute the thermostatistical contributions at the DFTB3-D3 is only about few days.

Table 2.

The thermostatistical contributions for the H₂ activation by four FLPs at HF-3c, DFTB3-D3 and TPSS-D3/def2-TZVP level of theory. All values are given in kcal mol⁻¹.

		thermostatistical contributions
		molecule	crystal
1	HF-3c	11.5	9.6
	DFTB3-D3a	13.9	15.4
	DFTB3-D3b	—	10.6
	TPSS-D3/def2-TZVP	11.8	—
2	HF-3c	9.3	8.2
	DFTB3-D3a	12.1	12.0
	DFTB3-D3b	—	7.6
	TPSS-D3/def2-TZVP	10.2	—
3	HF-3c	8.1	7.6
	DFTB3-D3a	11.3	13.3
	DFTB3-D3b	—	7.9
	TPSS-D3/def2-TZVP	8.9	—
4	HF-3c	7.4	7.1
	DFTB3-D3a	11.2	11.4
	DFTB3-D3b	—	10.3
	TPSS-D3/def2-TZVP	8.1	—

^aFull relaxation, including atomic positions and cell parameters.

^bConstrain relaxation, i.e. adopting crystal structures from HF-3c level, and only allowing atoms to relax while the unit cell is fixed.

(c). Electronic energies

The most dominant contribution to the reaction Gibbs free energy is the reaction electronic energy, ΔE. We compute the reaction energies for eight reactions (H₂ activation by FLP 1−4 in gas phase and in solid state), and the results are summarized in table 3. Currently there are no experimental data reported for reaction energies of the studied FLPs. However, it has been demonstrated that the hybrid PBE0-D3 functional with a large PAW basis set gives very accurate lattice energies with a mean absolute deviation of about 1 kcal mol⁻¹ compared to back-corrected experimental values [62]. A recent study analysed frustrated Lewis pairs and classical Lewis adducts at the ‘gold standard’ coupled cluster level (DLPNO-CCSD(T)) and confirmed the good accuracy of PBE0-D3 [63]. Therefore, we assume that in our context PBE0-D3 gives accurate single point reaction electronic energies, and use it as reference in the following.

Table 3.

Calculated reaction electronic energies (ΔE) for H₂ activation by four FLPs at different levels of theory. All geometries were optimized at HF-3c level and all values are given in kcal mol⁻¹.

		reaction electronic energies
		molecule	crystal
1	PBE0-D3a	−7.1	−9.4
	TPSS-D3a	−6.2	−8.6
	HSE-3c	−5.2	−3.8
	HF-3c	−8.7	−6.3
2	PBE0-D3a	2.5	1.9
	TPSS-D3a	3.2	1.5
	HSE-3c	6.2	4.2
	HF-3c	13.4	12.8
3	PBE0-D3a	4.2	−0.7
	TPSS-D3a	5.2	−1.3
	HSE-3c	7.3	4.5
	HF-3c	15.3	16.0
4	PBE0-D3a	−14.1	−25.9
	TPSS-D3a	−11.8	−25.1
	HSE-3c	−14.6	−25.2
	HF-3c	0.3	−6.2

^aGaussian basis set of def2-TZVP quality is used in the molecular case and a PAW basis set with 800 eV cut-off is used in the crystal case.

In general, the results given in table 3 show that HF-3c gives quite large errors which can be up to about 10 kcal mol⁻¹. This can be attributed to the complete lack of semi-local electron correlation, which is crucial to describe chemical reactions. The meta-GGA TPSS-D3 is significantly more reliable with 1 kcal mol⁻¹ mean absolute deviations to PBE0-D3. The new HSE-3c method performs reasonable as well. Note the good transferability of the method performances from the gas phase to the solid state. This seems to be a general trend and highlights that gas phase benchmarks can indeed be used to judge the performance of molecular crystal thermochemistry. We have additionally examined the effects of numerical settings on the accuracy of the reaction electronic energies (electronic supplementary material, table S2). By increasing the number of k points, we have not obtained any significant effect on the single point energies. For example, the energy differences are below 0.1 kcal mol⁻¹ when the k point grid is increased from 1 × 1 × 1 to 2 × 2 × 1. In contrast, we obtained an energy difference of 1 kcal mol⁻¹ when increasing the PAW basis set energy cut-off from 800 eV to 900 eV at TPSS-D3/PAW level of theory, and this difference becomes negligible at PBE0-D3/PAW level of theory (ca. 0.03 kcal mol⁻¹). In short, a setting with Γ-centred 1 × 1 × 1 k points and 800 eV energy cut-off provides reasonably accurate single point reaction electronic energies. In addition, we found that the HF-3c method yields too large reaction energy errors due to the neglect of electron correlation effects. The meta-GGA functional (TPSS-D3) seems to be sufficient to yield comparable single point energy to that of the hybrid functional (PBE0-D3), i.e. self-interaction errors seem to be small in the studied systems. It is important to point out that the CPU time requirements for TPSS-D3 are less demanding compared to PBE0-D3, in particular when evaluated in large plane wave basis sets (half a day versus several days). We thus recommend TPSS-D3/PAW single point calculations on top of HF-3c structures as a routine theoretical level which can be improved with higher computational demands (or if high accuracy is required) by PBE0-D3/PAW calculations. The HSE-3c method can be employed for double-checking purposes as well.

(d). Combined error estimates and final Gibbs free energies

In order to provide an overview about the performance of the different proposed multi-level strategies, we summarize the error estimates and computational requirements for the three calculation steps (geometry optimization, harmonic frequencies and electronic energy) and the final results for the reaction Gibbs free energies for H₂ activation by FLP 1−4 in tables 4 and 5. In general, the reaction Gibbs free energies based on HF-3c geometries and frequencies and PBE0-D3 electronic energies seem to be reliable with estimated errors below 3 kcal mol⁻¹. However, the computational demands are substantial with about two weeks. Computing the frequencies at the more efficient DFTB3-D3 level and replacing the electronic energy by the TPSS-D3 energies increases the overall error only slightly while reducing the computational costs significantly to about 1.5 days. Finally, HSE-3c is also applicable with reasonable accuracy and might be especially useful in cross-checking the TPSS-D3 values in systems where the self-interaction error can be large.

Table 4.

Error estimates and CPU time in computing reaction Gibbs free energies of H₂ activation by FLP crystals arising from different computational contributions. Errors are assumed to be uncorrelated. All values are given in kcal mol⁻¹, respectively.

				overall
	geometry	harmonic frequency	electronic energy	error	CPU timea
combination A
level	HF-3c	HF-3c	PBE0-D3d
error	1.5b	0.7c	2.1e	2.7f	287.7 h
CPU time	22.7 h	186.0 h	79.0 h
combination B
level	HF-3c	HF-3c	TPSS-D3d
error	1.5b	0.7c	0.7g	2.8f	214.5 h
CPU time	22.7 h	186.0 h	5.8 h
combination C
level	HF-3c	DFTB3-D3	TPSS-D3d
error	1.5b	1.3h	0.7g	3.0f	33.8 h
CPU time	22.7 h	5.3 h	5.8 h
combination D
level	HF-3c	DFTB3-D3	HSE-3cd
error	1.5b	1.3h	2.8g	4.1f	29.1 h
CPU time	22.7 h	5.3 h	1.1 h

^aTaken the product of H₂ activation by FLP 1 as an example. Hardware: One shared memory node with 16 Intel(R) Xeon(R) E5-2650 cores (20 MB Cache, 64 GB Ram, 2 GHz base frequency, 2.8 GHz max frequency).

^bEstimated as HSE-3c energy differences between fixed HF-3c and fully relaxed structure.

^cMean absolute deviation (MAD) to gas phase thermostatistical contributions at TPSS-D3 level.

^dSingle-point energy on HF-3c geometry.

^eDeviation to DLPNO-CCSD(T) reaction energies on 14 FLPs and classical Lewis adducts (ref. [63]).

^fVia Gauss error propagation.

^gMAD to PBE0-D3 solid state reaction energies.

^hMAD to solid-state thermostatistical contributions at HF-3c level.

Table 5.

The computed reaction Gibbs free energies for H₂ activation by FLP 1−4 in the solid state. The corresponding error estimates and CPU time are provided in table 4. All values are given in kcal mol⁻¹.

	combinations	A	B	C	D
	geometry	HF-3c	HF-3c	HF-3c	HF-3c
	harmonic frequency	HF-3c	HF-3c	DFTB3-D3	DFTB3-D3
	electronic energy	PBE0-D3	TPSS-D3	TPSS-D3	HSE-3c
1	ΔGtherm	9.6	9.6	10.6	10.6
	ΔE	−9.4	−8.6	−8.6	−3.8
	ΔGtotal	0.2	1.0	2.0	6.8
2	ΔGtherm	8.2	8.2	7.6	7.6
	ΔE	1.9	1.5	1.5	4.2
	ΔGtotal	10.1	9.7	9.1	11.8
3	ΔGtherm	7.6	7.6	7.9	7.9
	ΔE	−0.7	−1.3	−1.3	4.5
	ΔGtotal	6.9	6.3	6.6	12.4
4	ΔGtherm	7.1	7.1	10.3	10.3
	ΔE	−25.9	−25.1	−25.1	−25.2
	ΔGtotal	−18.8	−18.0	−14.8	−14.9

(e). Crystal field contributions

Although the reaction electronic energies (ΔE) for H₂ activation vary at different levels of theory, the H₂ activation is always more favoured in solid state than in solution (table 3 and figure 3). Specifically, the reaction electronic energies for H₂ activation by FLP 1 is −7.1 kcal mol⁻¹ in solution and it decreases to −9.4 kcal mol⁻¹ in the solid state. As a result, we gain about 2 kcal mol⁻¹ in the reaction electronic energy when transferring the same reaction from solution to solid state. The same trend has been also found for H₂ activation by FLP 2 and 3, where the solution--solid state difference is 0.6 and 4.9 kcal mol⁻¹, respectively. Moreover, we found that the crystal field contributions are larger for inter-molecular compared to intra-molecular FLPs. This is most likely to be due to the larger charge separation in the case of inter-molecular FLPs. For example, the computed reaction electronic energies for H₂ activation by the inter-molecular FLP 4 in solution and in solid state are −14.1 and −25.9 kcal mol⁻¹, respectively. Thus, its crystal field contribution is about 12 kcal mol⁻¹. However, among our studied systems, the crystal field contributions from intra-molecular FLPs are often less than 5 kcal mol⁻¹. In short, when transferring the reactivity of FLPs in solution to solid state, we lose on the one hand the stabilization contributions from the solvent, which is typical about 10 kcal mol⁻¹ [16]. On the other hand, we benefit from the crystal field and gain stabilizations for the zwitterionic product. Overall, the H₂ activation reaction can still proceed in solid state.

Figure 3.

Calculated reaction electronic energies (ΔE) for H₂ activation by four FLPs at the PBE0-D3/PAW//HF-3c level of theory. The values calculated at different levels of theory are provided in table 3. (Online version in colour.)

4. Conclusion

In this study, we have established a computational protocol for the theoretical study of reactivities of FLPs in the molecular crystal (solid) state. We selected several experimentally investigated FLPs and conducted calculations at various levels of theory to benchmark their performance for the crystal structures, thermostatistical contributions, as well as single point electronic reaction energies. By comparing the computational results with either experimental data or data from a high level of theory, we conclude that:

• (1) HF-3c is a reasonable method for crystal structure optimizations in terms of computational time as well as accuracy. The mean error of this method compared to experimental data is less than 8% in the unit cell dimensions and less than 5% for the mass densities.

• (2) Thermostatistical contributions can be reasonably computed by either HF-3c or by DFTB3-D3, when using the constrained HF-3c unit cell parameters. Their effect on the accuracy of the final Gibbs free reaction energies is intermediate, i.e. more computation time might first be invested in the geometry optimization or the electronic energy steps before this part is further improved.

• (3) The single point energies from TPSS-D3 level are quite close to that calculated from the PBE0-D3 level of theory, and HSE-3c might be useful for fast cross-checks.

Overall, we have presented a guideline how to study the reactivity of FLP in the solid state. The H₂ activation reaction is taken as an example in the present work but the conclusions are very likely be transferable to other small molecule activations. Moreover, it has been demonstrated that the crystal field replaces the role of the solvent in enhancing the FLP reactivity. Our computational efficient method combinations with their corresponding error estimates may be helpful in future theoretical investigations and screening of more systems, which is already in progress in our laboratory.

Data accessibility

Supporting information on computational details including generation of plausible crystal structures and test calculations on numerical settings is available.

Authors' contributions

L.L. and J.G.B. contributed equally to the work by carrying out all computations and performing the data analysis. S.G. conceived of and designed the study. All authors drafted the manuscript and approved the final form of the manuscript.

Competing interests

The authors declare that they have no competing interests.

Funding

S.G. acknowledges support of DFG in the framework of the ‘Gottfried-Wilhelm-Leibniz’ prize. J.G.B. acknowledges support of the Humboldt foundation within the Feodor-Lynen program.