Theoretical Prediction of Activation Free Energies of Various Hydride Self-Exchange Reactions in Acetonitrile at 298 K.

Hydride transfer reactions are very important chemical reactions in organic chemistry. It has been a chemist's dream to predict the rate constants of hydride transfer reactions by only using the physical parameters of the reactants. To realize this dream, we have developed a kinetic equation (Zhu equation) in our previous papers to predict the activation free energies of various chemical reactions using the activation free energies of the corresponding self-exchange reactions and the related bond dissociation energies or redox potentials of the reactants. Because the activation free energy of the hydride self-exchange reaction is difficult to measure using the experimental method, in this study, the activation free energies of 159 hydride self-exchange reactions in acetonitrile at 298 K were systematically computed using an accurately benchmarked density functional theory method with a precision of 1.1 kcal mol-1. The results show that the range of the activation free energies of the 159 hydride self-exchange reactions is from 16.1 to 46.6 kcal mol-1. The activation free energies of 25 122 hydride transfer reactions in acetonitrile at 298 K can be estimated using the activation free energies of the 159 hydride self-exchange reactions and the corresponding heterolytic bond dissociation free energies of the reactants. The effects of the heteroatom, substituent, and aromaticity on the activation free energies of hydride self-exchange reactions were examined. The results show that heteroatoms, substituents at the reaction center, and the aromaticity of reactants, all have remarkable effects on the activation free energy of hydride self-exchange reactions. All kinetic information provided in this work on the hydride self-exchange reactions in acetonitrile at 298 K should be very useful in chemical labs and chemical industry.

Introduction

Hydride transfer reactions (eq ) are very important organic chemical reactions and have been widely used for the mutual conversion between organic unsaturated and saturated compounds[1−3] as well as the preparation and conversion of hydrogen gas as green energy.[4−8] It is well-known that if a hydride transfer reaction can be used, there are two most fundamental scientific problems that need to be solved first: one is the thermodynamic problem, that is, how much is the thermodynamic driving force of the hydride transfer reaction, ΔGo(XH/Y+)? The other is the kinetic problem, that is, how much is the activation free energy of the hydride transfer reaction, ΔG⧧(XH/Y+)? As to the thermodynamic problem of hydride transfer reactions, as long as the molar free energy change of the hydride donor (XH) to release hydride anions, ΔGo(XH), and the molar free energy change of the hydride acceptor (Y) to accept hydride anions, ΔGo(Y+), are available, the thermodynamic driving force of the hydride transfer reaction, ΔGo(XH/Y+), can be obtained by using eq . Because the ΔGo(XH) values of many important organic hydride donors[9−13] and the ΔGo(Y+) values of many important organic hydride acceptors, such as olefins,[14,15] imines,[16] and carbonyl[17] compounds have been determined in acetonitrile at room temperature in our lab, the thermodynamic problems of many important organic hydride transfer reactions, in fact, have been well-solved in our previous work. However, unlike the thermodynamic driving force of hydride transfer reactions, ΔGo(XH/Y+), the activation free energy of hydride transfer reactions, ΔG⧧(XH/Y+), cannot be predicted using the related physical parameters of reactants up to now. The main reason is that the factors affecting the kinetics of reactions are much more complicated and diversified than those affecting the thermodynamics of reactions. In fact, it has been a chemist’s dream to predict the activation free energy of hydride transfer reactions only by using the physical parameters of reactants.

In our previous work,[18] we reported a new kinetic equation (eq and named as Zhu equation) to quantitatively estimate the activation free energy of various chemical reactions using the activation free energies of the corresponding self-exchange reactions and the related bond dissociation energies or redox potentials of the reactants. The validity of the Zhu equation has been verified by predicting the kinetic isotope effect values of 4556 hydride transfer reactions[18] and the activation free energies of 5886 hydrogen atom transfer reactions[19] in acetonitrile at 298 K.

For hydride transfer reactions, in eq , ΔG⧧(XH/Y+) and ΔGo(XH/Y+) are the activation free energy and thermodynamic driving force of the hydride transfer reactions, respectively. ΔG⧧(XH/X+) and ΔG⧧(YH/Y+) are the activation free energies of the corresponding hydride self-exchange reactions for reactant XH and reactant Y, respectively. Because the thermodynamic driving forces of many hydride transfer reactions, ΔGo(XH/Y+), have been measured in our lab, evidently, as long as the activation free energy of the corresponding hydride self-exchange reactions is available, the activation free energies of the hydride transfer reactions can be obtained by using eq . Because the activation free energy value of hydride self-exchange reactions, ΔG⧧(XH/X+), like the molar free energy change of the hydride donor (XH) to release hydride anions, ΔGo(XH), is also the characteristic parameter of the hydride donor (XH) or the hydride acceptor (X), it is clear that as long as the ΔG⧧(XH/X+) values of the reactants (XH and X) are available, the chemist’s dream that the activation free energy of hydride transfer reactions can be predicted only by using the characteristic physical parameters of the reactants can be realized. However, since the activation free energy values of hydride self-exchange reactions in solution have so far been scarce,[20] there is an urgent need in chemical labs and chemical industries to determine these values in the same solvent and at the same temperature.

Zhu equation

It is well-known that for electron-,[21−24] proton-,[25−28] and hydrogen atom-[29−31] self-exchange reactions, the activation free energy values can be directly measured using dynamic electron paramagnetic resonance (EPR) or dynamic nuclear magnetic resonance (NMR) techniques because the rates of these self-exchange reactions are generally fast enough to meet the requirements of dynamic EPR or dynamic NMR techniques. However, for hydride self-exchange reactions, the reaction rates are generally too slow to be measured using dynamic NMR techniques (requiring k2 to be within 102–106 M–1 s–1).[20] To the best of our knowledge, except some indirect experimental methods,[20] no direct experimental methods were reported in the literature to accurately determine the activation free energy of hydride self-exchange reactions in solution.

Owing to the development of density functional theory (DFT) in recent years and the large enhancement of computational capability, the activation free energies of many organic reactions in solution can be computed by employing suitable density functionals, which means that the activation free energy values of hydride self-exchange reactions in solution could also be obtained by the DFT method. In this work, we try to use a suitable DFT method to estimate the activation free energies of hydride self-exchange reactions in acetonitrile at room temperature. Because dihydropyridine, dihydroquinoline, dihydroarcridine, and dihydroimidazole as well as their various derivatives are all well-known good hydride donors and many of them have been extensively applied as efficient reducing agents in reductions of various organic unsaturated compounds,[32−35] 159 hydride self-exchange reactions involving dihydropyridine, dihydroquinoline, dihydroarcridine, and dihydroimidazole as well as their various derivatives and analogues as hydride donors are examined in this work. The parent structures and marks of the 159 hydride donors are shown in Scheme .

Scheme 1

Parent Structures and Marks of Hydride Donors (XH) Examined in This Work

Results

Because it is the first time to systematically compute the activation free energies of various hydride self-exchange reactions in acetonitrile at room temperature by using the DFT method, it is necessary to examine the reliability of the computational method by comparing the computational results with the corresponding experimental results. In this work, six hydride cross-transfer reactions (eqs –9) were selected because they are all well-known hydride transfer reactions, and the activation free energies values in acetonitrile at 298 K can be directly measured using the conventional experimental methods[18,20,36] (see Table ).

Table 1

Experimental and Computational Activation Free Energies (ΔG⧧) of Six Hydride Transfer Reactions in Acetonitrile at 298 K (kcal mol–1)

		ΔGcomp.⧧
XH/Y+	ΔGexp⧧.a	method 1b	method 2b	method 3b	method 4b	method 5b	method 6b
75H/79+	23.12	23.2	23.6	22.8	24.3	17.9	16.4
35H(Me)/72+	19.90	21.5	21.6	19.8	23.7	17.8	14.8
71H/35+(Me)	17.91	18.5	18.8	17.2	20.3	16.3	13.8
69H/35+(Me)	14.88	15.6	16.0	14.4	17.6	13.2	10.7
67H/35+(Me)	14.41	12.1	12.7	10.4	14.8	11.0	4.7
35H(Me)/39+	11.20	12.2	12.5	10.4	14.2	9.6	4.6
MADc		1.1	1.2	1.1	2.3	2.6	6.1
RMSDc		1.3	1.3	1.7	2.5	2.9	6.4

Experimental activation free energies directly derived from refs (18)(20), and (36) using Eyring equation.[39]

Computational activation free energies by using different methods to compute the gas-phase electronic energy (method 1: M06-2X-D3/def2-TZVPP; method 2: M06-2X-D3/def2-QZVP; method 3: M06-2X-D3/MG3S; method 4: M06-2X/def2-TZVPP; method 5: B3LYP-D3(BJ)/def2-TZVPP; and method 6: BMK-D3(BJ)/def2-TZVPP).

MAD, mean absolute deviation; RMSD, root-mean-square deviation.

Because optimization by using M06-2X/6-31+G** can give reliable geometries for organic compounds[37] and also it has been proved that the solvation energy computation protocol M05-2X/6-31G*/SMD can achieve the smallest error for general computation,[38] we only need to evaluate the reliability of different methods to compute the gas-phase electronic energy for the six hydride cross-transfer reactions (eqs –9). Three different density functionals (M06-2X, B3LYP, and BMK) were compared in the evaluation with or without Grimme’s D3 dispersion correction in combination with the def2-TZVPP basis set, and the optimal density functional was further evaluated in combination with MG3S and def2-QZVP to test the basis set effect on computational performance. The detailed results of different methods are listed in Table (and Table S1 in the Supporting Information).

Without D3 correction, the performances of B3LYP and BMK in combination with def2-TZVPP were extremely poor with the MAD value up to 16.3 and 10.3 kcal mol–1 (Table S1 in the Supporting Information), whereas the MAD of B3LYP-D3(BJ) and BMK-D3(BJ) were 2.6 and 6.1 kcal mol–1 respectively. M06-2X gave a comforting result (MAD 2.3 kcal mol–1) because of its parameterization against barrier heights and noncovalent interactions, but the deviation exceeds 2 kcal mol–1 in four reactions. After D3 correction, M06-2X-D3/def2-TZVPP reduced the MAD to 1.1 kcal mol–1, and its estimation error exceeded 2.0 kcal mol–1 in one reaction, indicating the importance of including dispersion description in these reactions with rings. As for the basis set, in combination with the optimal M06-2X-D3 density functional, Ahlrichs basis set (def2-TZVPP) outperformed the Pople type basis set (MG3S) a little, and there is no improvement from def2-TZVPP to def2-QZVP in these reactions. Therefore, the method of M06-2X-D3/def2-TZVPP was chosen in this work to compute the gas-phase electronic energies for the 159 hydride self-exchange reactions (XH/X). We chose the same method as the optimal method for the computation of reaction Gibbs free energy change. To be concise, the assessment of different computational methods on the computation of reaction Gibbs free energy change are provided in the Supporting Information (Table S2). The final computed activation energies of the 159 hydride self-exchange reactions in acetonitrile at 298 K are listed in Table . To examine the contributions of activation enthalpy and activation entropy to the activation free energy and the effect of the heterolytic bond dissociation free energy of the hydride donors on the activation free energy, activation enthalpies and activation entropies of the 159 hydride self-exchange reactions as well as the heterolytic bond dissociation free energy of 159 hydride donors (XH) in acetonitrile at 298 K were also computed and are provided in Table .

Table 2

Activation Free Energies, Activation Enthalpies, and the Contribution of Activation Entropy to Gibbs Free Energy for 159 Hydride Self-Exchange Reactions Together with the Heterolytic Bond Dissociation Free Energies of 159 Hydride Donors in Acetonitrile at 298 K (kcal mol–1)

no.	XH	ΔG⧧	ΔH⧧	–TΔS⧧	ΔGo(XH)	no.	XH	ΔG⧧	ΔH⧧	–TΔS⧧	ΔGo(XH)
1H(G)						77	22H	24.5	10.9	13.6	80.5
1	H	24.2	12.9	11.4	52.5	78	23H	18.5	5.6	12.9	82.9
2	Me	24.3	12.5	11.8	50.1	79	24H	23.7	9.6	14.2	81.3
3	tBu	27.1	14.3	12.8	49.1	80	25H	17.8	4.8	13.0	93.5
4	Ph	23.8	10.0	13.8	56.8	81	26H	22.7	8.7	14.0	90.0
5	NH2	25.1	12.7	12.4	51.7	82	27H	23.6	11.0	12.6	58.9
6	CHO	25.4	12.7	12.7	71.7	83	28H	23.5	11.1	12.4	82.4
7	CN	25.0	12.7	12.2	78.2	84	29H	22.1	9.3	12.8	84.5
8	NO	27.8	15.2	12.6	78.0	85	30H	17.3	4.4	12.9	95.6
9	NO2	27.2	14.3	12.9	80.3	86	31H	23.7	11.5	12.2	60.9
2H(G)						87	32H	24.5	12.4	12.1	84.9
10	Me	23.4	11.3	12.2	50.7	88	33H	22.5	9.7	12.8	86.7
11	tBu	23.8	10.7	13.1	50.7	89	34H	17.7	5.4	12.3	98.8
12	Ph	24.3	10.8	13.6	52.9	35H(R)
13	NH2	26.9	14.4	12.6	44.2	90	H	21.1	7.8	13.3	74.1
14	CHO	23.6	10.8	12.8	60.7	91	Me	20.7	7.0	13.7	76.2
15	CN	24.1	11.8	12.2	64.3	92	tBu	30.8	16.1	14.7	84.5
16	NO	24.2	11.5	12.6	66.9	93	Ph	21.9	7.5	14.4	75.5
17	NO2	24.5	11.7	12.8	67.4	36H(R)
3H(G)						94	Me	23.6	9.3	14.2	72.6
18	Me	23.6	11.5	12.1	51.7	95	tBu	46.6	31.5	15.1	90.7
19	tBu	24.1	11.4	12.6	51.2	96	Ph	27.9	13.4	14.5	74.4
20	Ph	23.6	10.0	13.7	55.3	97	37H	18.2	4.9	13.3	90.0
21	NH2	24.6	12.7	11.9	48.3	98	38H	24.9	10.4	14.6	88.9
22	CHO	22.4	10.0	12.4	65.9	99	39H	19.5	6.0	13.4	92.2
23	CN	24.3	12.1	12.2	67.5	100	40H	29.6	14.7	14.9	92.6
24	NO	23.2	10.7	12.4	75.0	101	41H	16.1	2.5	13.6	97.0
25	NO2	23.5	11.1	12.4	72.6	102	42H	24.4	9.6	14.8	96.2
4H(G)						103	43H	22.3	9.1	13.2	66.4
26	Me	26.2	14.6	11.6	48.3	104	44H	28.8	14.5	14.3	65.9
27	tBu	33.4	20.6	12.8	51.4	105	45H	22.8	9.5	13.3	89.6
28	Ph	26.8	13.7	13.1	50.1	106	46H	28.1	13.9	14.2	86.9
29	NH2	35.1	23.5	11.6	36.6	107	47H	22.1	8.4	13.7	93.5
30	CHO	21.9	9.8	12.1	56.3	108	48H	29.7	15.2	14.5	93.1
31	CN	27.7	15.9	11.8	62.1	109	49H	16.4	2.7	13.7	103.7
32	NO	25.0	13.2	11.8	64.0	110	50H	25.4	10.3	15.1	100.8
33	NO2	31.9	19.8	12.1	72.3	111	51H	21.8	8.9	12.9	67.0
34	5H	21.8	10.4	11.4	76.5	112	52H	23.2	9.9	13.3	57.2
35	6H	24.6	11.6	13.0	71.7	113	53H	23.0	9.8	13.2	61.8
36	7H	19.5	7.9	11.7	75.9	114	54H	24.2	11.0	13.2	58.3
37	8H	22.5	9.3	13.3	71.5	115	55H	24.3	11.2	13.1	67.5
38	9H	19.5	7.7	11.8	90.5	116	56H	29.8	18.6	11.2	50.6
39	10H	21.8	8.5	13.3	83.3	117	57H	32.1	19.4	12.7	48.7
40	11H	27.3	16.4	10.9	59.3	118	58H	26.2	14.6	11.7	70.8
41	12H	25.0	14.1	10.9	49.0	119	59H	30.4	17.3	13.1	66.3
42	13H	24.2	13.4	10.8	74.4	120	60H	24.0	11.9	12.1	70.8
43	14H	23.1	11.9	11.2	77.2	121	61H	25.9	12.6	13.3	67.4
44	15H	18.1	6.4	11.7	89.6	122	62H	21.8	9.4	12.3	76.6
16H(R)						123	63H	23.3	9.9	13.4	70.8
45	H	22.4	9.7	12.7	62.2	124	64H	23.2	10.5	12.8	50.8
46	Me	22.7	9.7	13.1	60.5	125	65H	23.0	9.6	13.4	59.5
47	tBu	29.2	14.5	14.7	61.4	126	66H	19.0	3.7	15.3	65.2
48	Ph	24.3	10.0	14.3	64.4	127	67H	19.7	3.0	16.7	64.6
17H(G)						128	68H	22.2	5.5	16.6	63.6
49	Me	22.6	9.6	13.0	59.9	129	69H	24.7	10.3	14.4	60.1
50	tBu	23.5	10.0	13.5	60.3	130	70H	23.4	9.4	14.0	64.8
51	Ph	21.2	7.4	13.8	62.1	131	71H	25.2	11.3	13.9	66.3
52	NH2	27.0	14.1	12.8	50.8	132	72H	21.3	7.2	14.2	75.4
53	CHO	19.9	6.5	13.4	71.0	133	73H	20.4	6.0	14.4	73.5
54	CN	20.9	7.8	13.1	74.0	134	74H	19.1	4.1	15.1	82.1
55	NO	21.8	8.2	13.6	77.1	135	75H	26.5	15.3	11.2	79.3
56	NO2	19.4	5.8	13.5	77.0	136	76H	30.4	19.2	11.2	74.1
18H(G)						137	77H	35.4	22.4	13.0	81.6
57	Me	22.7	9.4	13.2	61.5	138	78H	25.5	13.8	11.7	94.4
58	tBu	24.9	11.5	13.4	61.8	139	79H	31.6	17.8	13.9	91.0
59	Ph	21.1	7.0	14.1	65.3	140	80H	22.0	8.9	13.1	67.8
60	NH2	24.6	11.6	12.9	59.0	141	81H	21.1	8.4	12.7	87.1
61	CHO	20.2	6.9	13.3	74.8	142	82H	27.1	13.3	13.9	89.6
62	CN	22.3	9.0	13.3	75.6	143	83H	21.4	9.6	11.9	86.5
63	NO	21.9	8.7	13.2	83.3	144	84H	26.4	14.5	11.9	78.3
64	NO2	20.7	7.0	13.7	80.4	145	85H	24.5	11.6	12.9	83.6
19H(R)						146	86H	19.8	6.9	12.8	88.5
65	Me	24.7	11.7	12.9	57.7	147	87H	22.2	10.0	12.2	81.2
66	tBu	39.1	25.1	14.0	65.5	148	88H	22.4	8.7	13.7	83.9
67	Ph	26.7	12.8	13.9	61.5	149	89H	26.3	15.3	11.1	64.9
20H(G)						150	90H	26.7	14.1	12.5	68.4
68	Me	22.7	9.9	12.9	60.6	151	91H	31.0	17.1	13.9	73.0
69	tBu	22.4	8.7	13.7	60.5	152	92H	23.4	9.7	13.7	78.7
70	Ph	23.5	9.5	14.1	62.4	153	93H	32.0	17.7	14.4	75.7
71	NH2	25.1	11.4	13.8	55.9	154	94H	31.5	17.0	14.4	73.1
72	CHO	23.1	9.8	13.3	65.1	155	95H	40.2	25.4	14.8	73.6
73	CN	23.3	9.9	13.4	67.4	156	96H	29.7	16.0	13.6	61.5
74	NO	23.2	9.8	13.4	66.6	157	97H	28.5	14.8	13.8	63.8
75	NO2	23.5	9.7	13.8	68.4	158	98H	39.2	25.6	13.6	66.5
76	21H	20.7	8.0	12.6	83.4	159	99H	26.6	12.7	13.9	88.2

Scale of ΔG⧧(XH/X+)

From Table , it is clear that although the thermodynamic driving forces of the 159 hydride self-exchange reactions in acetonitrile at 298 K, ΔGo(XH/X+), are all equal to zero, the activation free energy values of the 159 hydride self-exchange reactions in acetonitrile at 298 K, ΔG⧧(XH/X+), are all quite different and range from 16.1 kcal mol–1 for 41H/41 to 46.6 kcal mol–1 for 36H(/36(. Such a large range of ΔG⧧(XH/X+) not only indicates the mistake of linear free energy relationship between ΔG⧧ and ΔGo for various chemical reactions but also shows that the activation free energy of hydride self-exchange reactions is strongly dependent on the structure and composition of the reactants. To elucidate the main factors in structure and composition of the reactants that affect ΔG⧧(XH/X+), a plot of ΔG⧧(XH/X+) values with 53 typical hydride donors (XH) are presented in Figure . From Figure , it is clear that the magnitude of ΔG⧧(XH/X+) depends not only on the structure but also on the composition of reactants, and no main structural factors can be easily found, which means that the factors affecting ΔG⧧(XH/X+) are quite complex. There is no doubt that the intrinsic logic relations between ΔG⧧(XH/X+) and the structure and composition of the reactants (XH and X+) should be one of the most important issues in physical chemistry for further investigations. However, several main factors that affect ΔG⧧(XH/X+) for some special reactants can still be examined here.

Figure 1

Visual comparison of ΔG⧧(XH/X+) for the 54 representative hydride donors in Table .

Main Factors Affecting ΔG⧧(XH/X+)

Heteroatom Effect on the Activation Free Energy

It is well-known that if heteroatoms, such as nitrogen, oxygen, sulfur, and so forth, are used to replace the carbon atom in a chemical compound, the heteroatom can not only change the physical property of the compound but also change the chemical property of the compound, which means that the heteroatom in the reactants should have great influence on the activation free energy of the hydride self-exchange reactions. From Table , it is clear that when the carbon atom at the para position to the reaction center of the reactant (9H) is replaced by nitrogen (1H), oxygen (5H), and sulfur (7H), the activation free energy values of the corresponding hydride self-exchange reactions (9H/9, 1H/1, 5H/5, and 7H/7) are altered, as shown in Figure . From Figure a, it is clear that the heteroatoms, especially the nitrogen atom (1H) and the oxygen atom (5H), can significantly increase the activation free energy of the corresponding hydride self-exchange reactions, and the increase order is that N atom (24.2 kcal mol–1 for 1H/1) > O atom (21.8 kcal mol–1 for 5H/5) > S atom (19.5 kcal mol–1 for 7H/7) ≥ C atom (19.5 kcal mol–1 for 9H/9). In the same way, if the carbon atom at the ortho position of the reactant (15H) is replaced by nitrogen (12H), oxygen (13H), and sulfur (14H), it is found that the heteroatoms at the ortho position to the reaction center can also significantly increase the activation free energy of the hydride self-reactions, and the increase order is N (25.0 kcal mol–1 for 12H/12) > O (24.2 kcal mol–1 for 13H/13) > S (23.1 kcal mol–1 for 14H/14) > C (18.1 kcal mol–1 for 15H/15) [Figure b]. Because the activation free energy changes of 14H/14 (5.0 kcal mol–1), 13H/13 (6.1 kcal mol–1), and 12H/12 (6.9 kcal mol–1) from that of 15H/15 are all larger than the corresponding activation free energy changes of 7H/7 (0.01 kcal mol–1), 5H/5 (2.3 kcal mol–1), and 1H/1 (4.7 kcal mol–1) from that of 9H/9, respectively, it is clear that the effect of heteroatoms at the 2-position on the activation free energy is much larger than that of the heteroatoms at the 4-position. The main reason is that the position of the heteroatom at the 2-position is closer to the reaction center of the reactant than that of the heteroatom at the 4-position.

Figure 2

Heteroatom effect on the activation parameters of 1,4-dihydrobenzene and 1,2-dihydro-isomer (energy unit: kcal mol–1).

If the activation enthalpy and activation entropy for the hydride self-exchange reactions with 1,4-dihydro-compounds as hydride donors (9H/9, 7H/7, 5H/5, and 1H/1) as well as 1,2-dihydro-isomers as hydride donors (15H/15, 14H/14, 13H/13, and 12H/12) are examined together, it is found that for the hydride self-exchange reactions with 1,4-dihydro-compounds as reactants, the activation entropies have no significant change, but the activation enthalpies have a significant change: 12.9 kcal mol–1 for 1H/1, 10.4 kcal mol–1 for 5H/5, 7.9 kcal mol–1 for 7H/7, and 7.7 kcal mol–1 for 9H/9, which indicates that the effect of the heteroatoms at the 4-position on the activation free energy is mainly due to the activation enthalpy change. Little or no effect of the heteroatom at the 4-position on the activation entropy is found; the main reason could be that the position of the heteroatom at the 4-position in the molecule is further from the reaction center in the reactants than that at the 2-position.

Because the activation entropy change of reactions mainly depends on the conformation change of the reactant system from the initial state to the transition state (TS) and the activation enthalpy of reactions mainly depends on the bond energy change of the reactant system from the initial state to the TS, it is conceived that for the 1,4-dihydro-compounds as hydride donors, the conformation change of the reactant system from the initial state to the TS should be similar for each other, but for the 1,2-dihydro-isomer as hydride donors, the conformation change of the reactant system from the initial state to the TS could be a little different from each other.

In addition, from Figure , it is interesting to find that ΔGo(XH) values of the hydride donors decrease in the following order: 9H (90.5 kcal mol–1) > 5H (76.5 kcal mol–1) > 7H (75.9 kcal mol–1) > 1H (52.5 kcal mol–1) for the 1,4-dihydro hydride donors and 15H (89.6 kcal mol–1) > 14H (77.2 kcal mol–1) > 13H (74.4 kcal mol–1) > 12H (49.0 kcal mol–1) for the 1,2-dihydro hydride donors, but the corresponding activation free energy changes decrease in the opposite direction. The smaller the heterolytic bond dissociation free energy of the reactants, the greater the activation energy of the corresponding hydride self-exchange reactions; what is the reason for this anomaly?

In our previous paper,[19] we reported a new version of Zhu equation (eq ) from eq , the purpose of which is to realize chemists’ dream that the activation free energy of a chemical reaction can be directly estimated only using the corresponding characteristic physical parameter of the reactants. In eq , ΔG⧧o(XH) and ΔG⧧o(X+) are called the thermo-kinetic parameters of reactants XH and X+, respectively, because both the physical parameters are made of the thermodynamic parameter of the reactants, ΔGo(XH), and the kinetic parameter of the reactants, ΔG⧧(XH/X+), (see eqs and 12, respectively). The physical meanings of ΔG⧧o(XH) and ΔG⧧o(X+) are shown in Scheme and Figure . Because ΔG⧧o(XH) is the free energy absorbed by XH from the initial state of the reaction to the TS of the reaction and ΔG⧧o(X+) is the free energy released by X+ from the initial state of the reaction to the TS of the reaction, obviously, ΔG⧧o(XH) is the real resistance of the reaction, and ΔG⧧o(X+) is the real power of the reaction. Because the magnitudes of ΔG⧧o(XH) and ΔG⧧o(X+) not only depend on the heterolytic bond dissociation free energy of the reactants, ΔGo(XH), but also depend on the activation free energy of the hydride self-exchange reaction, ΔG⧧(XH/X+), we should not be amazed at the anomaly that the greater the heterolytic bond dissociation free energy of the reactant, the smaller the activation free energy of the hydride self-exchange reaction. To elucidate the reason for the anomaly, the thermo-kinetic parameters of 159 XH to release the hydride ion and the thermo-kinetic parameters of 159 X+ to capture the hydride ion in acetonitrile at 298 K were calculated according to eqs and 12, respectively, and the results are listed in Table .

Scheme 2

Physical Meanings of ΔG⧧o(XH) and ΔG⧧o(X+)

Figure 3

Kinetic model of hydride transfer from XH to X+ to form X+ and XH, described by two reverse Morse-type free energy curves: the left one (red) refers to the chemical process of XH to release the hydride ion; the right one (black) refers to the chemical process of X+ to capture the hydride ion; the intersecting point refers to the TS.

Table 3

Thermo-kinetic Parameter Values of 159 Hydride Donors and 159 Hydride Acceptors in Acetonitrile at 298 K (kcal mol–1)

no.	XH	ΔG⧧o(XH)	ΔG⧧o(X+)	no.	XH	ΔG⧧o(XH)	ΔG⧧o(X+)
1H(G)				77	22H	52.5	–28.0
1	H	38.4	–14.1	78	23H	50.7	–32.2
2	Me	37.2	–12.9	79	24H	52.5	–28.8
3	tBu	38.1	–11.0	80	25H	55.6	–37.8
4	Ph	40.3	–16.5	81	26H	56.4	–33.6
5	NH2	38.4	–13.3	82	27H	41.3	–17.7
6	CHO	48.6	–23.2	83	28H	53.0	–29.4
7	CN	51.6	–26.6	84	29H	53.3	–31.2
8	NO	52.9	–25.1	85	30H	56.4	–39.1
9	NO2	53.7	–26.5	86	31H	42.3	–18.6
2H(G)				87	32H	54.7	–30.2
10	Me	37.1	–13.6	88	33H	54.6	–32.1
11	tBu	37.2	–13.4	89	34H	58.2	–40.5
12	Ph	38.6	–14.3	35H(R)
13	NH2	35.6	–8.6	90	H	47.6	–26.5
14	CHO	42.2	–18.6	91	Me	48.5	–27.7
15	CN	44.2	–20.1	92	tBu	57.7	–26.8
16	NO	45.5	–21.4	93	Ph	48.7	–26.8
17	NO2	45.9	–21.5	36H(R)
3H(G)				94	Me	48.1	–24.5
18	Me	37.6	–14.1	95	tBu	68.7	–22.1
19	tBu	37.6	–13.6	96	Ph	51.2	–23.3
20	Ph	39.5	–15.8	97	37H	54.1	–35.9
21	NH2	36.5	–11.9	98	38H	56.9	–32.0
22	CHO	44.2	–21.7	99	39H	55.8	–36.4
23	CN	45.9	–21.6	100	40H	61.1	–31.5
24	NO	49.1	–25.9	101	41H	56.6	–40.4
25	NO2	48.0	–24.6	102	42H	60.3	–35.9
4H(G)				103	43H	44.4	–22.1
26	Me	37.3	–11.0	104	44H	47.3	–18.5
27	tBu	42.4	–9.0	105	45H	56.2	–33.4
28	Ph	38.4	–11.7	106	46H	57.5	–29.4
29	NH2	35.8	–0.7	107	47H	57.8	–35.7
30	CHO	39.1	–17.2	108	48H	61.4	–31.7
31	CN	44.9	–17.2	109	49H	60.0	–43.7
32	NO	44.5	–19.5	110	50H	63.1	–37.7
33	NO2	52.1	–20.2	111	51H	44.4	–22.6
34	5H	49.2	–27.4	112	52H	40.2	–17.0
35	6H	48.1	–23.6	113	53H	42.4	–19.4
36	7H	47.7	–28.2	114	54H	41.3	–17.0
37	8H	47.0	–24.5	115	55H	45.9	–21.6
38	9H	55.0	–35.5	116	56H	40.2	–10.4
39	10H	52.5	–30.8	117	57H	40.4	–8.3
40	11H	43.3	–16.0	118	58H	48.5	–22.3
41	12H	37.0	–12.0	119	59H	48.3	–18.0
42	13H	49.3	–25.1	120	60H	47.4	–23.4
43	14H	50.1	–27.1	121	61H	46.6	–20.8
44	15H	53.8	–35.8	122	62H	49.2	–27.4
16H(R)				123	63H	47.1	–23.8
45	H	42.3	–19.9	124	64H	37.0	–13.8
46	Me	41.6	–18.9	125	65H	41.3	–18.2
47	tBu	45.3	–16.1	126	66H	42.1	–23.1
48	Ph	44.4	–20.1	127	67H	42.1	–22.5
17H(G)				128	68H	42.9	–20.7
49	Me	41.2	–18.6	129	69H	42.4	–17.7
50	tBu	41.9	–18.4	130	70H	44.1	–20.7
51	Ph	41.7	–20.4	131	71H	45.8	–20.6
52	NH2	38.9	–11.9	132	72H	48.3	–27.0
53	CHO	45.5	–25.5	133	73H	46.9	–26.6
54	CN	47.4	–26.5	134	74H	50.6	–31.5
55	NO	49.4	–27.7	135	75H	52.9	–26.4
56	NO2	48.2	–28.8	136	76H	52.2	–21.9
18H(G)				137	77H	58.5	–23.1
57	Me	42.1	–19.4	138	78H	60.0	–34.4
58	tBu	43.4	–18.4	139	79H	61.3	–29.7
59	Ph	43.2	–22.1	140	80H	44.9	–22.9
60	NH2	41.8	–17.2	141	81H	54.1	–33.0
61	CHO	47.5	–27.3	142	82H	58.4	–31.2
62	CN	49.0	–26.7	143	83H	54.0	–32.5
63	NO	52.6	–30.7	144	84H	52.3	–26.0
64	NO2	50.6	–29.8	145	85H	54.1	–29.6
19H(R)				146	86H	54.2	–34.4
65	Me	41.2	–16.5	147	87H	51.7	–29.5
66	tBu	52.3	–13.2	148	88H	53.2	–30.8
67	Ph	44.1	–17.4	149	89H	45.6	–19.3
20H(G)				150	90H	47.5	–20.9
68	Me	41.7	–19.0	151	91H	52.0	–21.0
69	tBu	41.5	–19.1	152	92H	51.1	–27.7
70	Ph	42.9	–19.4	153	93H	53.9	–21.9
71	NH2	40.5	–15.4	154	94H	52.3	–20.8
72	CHO	44.1	–21.0	155	95H	56.9	–16.7
73	CN	45.4	–22.0	156	96H	45.6	–15.9
74	NO	44.9	–21.7	157	97H	46.2	–17.7
75	NO2	45.9	–22.4	158	98H	52.8	–13.7
76	21H	52.1	–31.4	159	99H	57.4	–30.8

Definition

From Table , we can find that the thermo-kinetic parameters of XH in acetonitrile at 298 K are 55.0, 47.7, 49.2, and 38.4 kcal mol–1 for 9H, 7H, 5H, and 1H, respectively; and the thermo-kinetic parameters of X in acetonitrile at 298 K are −35.5, −28.2, −27.4, and −14.1 kcal mol–1 for 9, 7, 5, and 1, respectively. Although the thermo-kinetic parameters of 9H, 5H, 7H, and 1H decrease successively as the ΔGo(XH) values of 9H (90.5 kcal mol–1), 5H (76.5 kcal mol–1), 7H (75.9 kcal mol–1), and 1H (52.5 kcal mol–1) decrease, the thermo-kinetic parameters of 9, 7, 5, and 1 increase successively as ΔGo(XH) decreases, and the values of the latter ones increase relative to that of 9 (7.3 kcal mol–1 for 7, 8.1 kcal mol–1 for 5, and 21.4 kcal mol–1 for 1, respectively) are all larger than the values of the corresponding former ones that decrease relative to that of 9H (−7.3 kcal mol–1 for 7H, −5.8 kcal mol–1 for 5H, and −16.6 kcal mol–1 for 1H, respectively). It is this reason that leads to the anomaly that the greater the heterolytic bond dissociation free energy of reactants, the smaller the activation free energy of the hydride self-exchange reactions.

Substituent Effect on the Activation Free Energy

It is well-known that the substituent effect is the thermodynamic and/or kinetic parameter change of a reaction when one atom or atom group in the reactants is substituted by another atom or atom group. For a bimolecular reaction, when one reactant is modified by a substituent, the substituent effect (electronic effect and steric effect) on the reaction rate can be used to rationalize the reaction mechanism. However, for a hydride self-exchange reaction, as both the reactants carry the same substituent, the substituent effect on the reaction rate could become more complex. To elucidate the effect of the substituent nature, position, and size on the rate of hydride self-exchange reactions, the activation free energies of hydride self-exchange reactions, with 2H(Me), 3H(Me), 1H(H), 1H(Me), and 4H(Me) as hydride donors [Figure a], and the activation free energies of hydride self-exchange reactions, with 2H(CN), 3H(CN), 1H(H), 1H(CN), and 4H(CN) as hydride donors [Figure b], as well as the activation free energies of hydride self-exchange reactions with 1H(H), 4H(Me), and 4H( as hydride donors [Figure c] are compared together.

Figure 4

Effects of nature, position, and size of the substituent groups on the activation free energy (energy unit: kcal mol–1).

From Figure a, it is clear that the effects of the substituent at 1-, 2-, and 3-positions are all quite small, but the effect of the substituent at the 4-position is prominent; the reason is that the substituent at the 4-position is closest to the reaction center. However, when the activation enthalpy and activation entropy of the hydride self-exchange reactions for the 4-substituted reactants are compared, an unexpected finding is that the main change is enthalpy rather than entropy.

When Figure a and 4b are compared, it is found that although CN is a strong electron-withdrawing group, its effect does not make the hydride self-exchange reaction much slower. The reason is that although CN reduces the hydride-donating ability of the hydride donor, at the same time, CN can also enhance the hydride-accepting ability of the hydride acceptor. The cancelation leads to the fact that the electronic effects of the substituent is not a major factor to affect hydride self-exchange reactions (eq ).

From Figure c, it is clear that when the hydrogen atom at the 4-position in 1,4-dihydropyridine is replaced by different bulky substituents (Me and tBu), the activation free energy increases significantly as the size of the substituent increases. If the activation enthalpy change and the activation entropy change are compared, it is surprising to find that the main cause of the bulky effect of the substituent is the activation enthalpy change rather than the activation entropy change.

Aromatization Effect of the Reactant on the Activation Free Energy

From the structure of the hydride donors examined in this work (Scheme ), it is clear that the 159 organic compounds are mostly heterocyclic compounds. Because the aromaticity of the heterocyclic compounds is an important component of the thermodynamic driving force for the heterocyclic compounds to release hydride ions, obviously, the aromatization of heterocyclic compounds should be an important factor to affect the activation free energy values of the corresponding hydride self-exchange reactions. To elucidate the dependence of the activation free energy of hydride self-exchange reactions on the aromatization of heterocyclic compounds, some related physical parameters of 1H, 16H, and 35H for their corresponding hydride self-exchange reactions are compared together in Figure . From Figure , it is interesting to find that the greater the aromaticity of the hydride acceptor, the smaller the activation free energy of the corresponding hydride self-exchange reaction.

Figure 5

Aromatization effect of the reactants on the hydride self-exchange reaction activation free energy.

Application of ΔG⧧(XH/X+)

Prediction of the Activation Free Energy of Various Hydride Transfer Reactions

Because the thermokinetic parameters of the 159 hydride donors [ΔG⧧o(XH)] and 159 hydride acceptors [ΔG⧧o(X+)] in acetonitrile at 298 K can be obtained from the activation free energy of the corresponding 159 hydride self-exchange reactions and the molar free energy change of the 159 hydride donors to release hydride ions in acetonitrile at 298 K (Table ), the activation free energy values of 25 122 (A1592 = 25 122) hydride transfer reactions in acetonitrile at 298 K can be estimated from these thermo-kinetic parameters of hydride donors (XH) and acceptors (X+), according to the new version of Zhu equation (eq ). Because of the limitation of the paper space, the detailed results of the 25 122 activation free energy values are not provided in this paper. However, to verify the 25 122 predictions, the activation free energies of 41 representative cross-reactions among them were examined by DFT computation, and the results are provided in Table . As we can see, the activation free energies derived from Zhu equation (eq ) are all quite close to those directly computed by the benchmarked DFT method.

Table 4

Activation Free Energies of 41 Hydride Transfer Reactions Predicted by Zhu Equation (eq ) and the Benchmarked DFT Method Together with the Thermo-kinetic Parameters of Hydride Donors and Acceptors (kcal mol–1)

entry	XH	Y+	ΔGDFT⧧	ΔGZhu eq⧧	d	ΔG⧧ο(XH)	ΔG⧧ο(Y+)
1	1H	2+(Me)	24.9	24.7	–0.1	38.4	–13.6
2	1H	2+(tBu)	24.9	24.9	0.1	38.4	–13.4
3	1H	2+(NH2)	30.0	29.7	–0.2	38.4	–8.6
4	1H	2+(CHO)	19.7	19.8	0.1	38.4	–18.6
5	1H	2+(CN)	18.0	18.3	0.3	38.4	–20.1
6	1H	2+(NO)	16.3	17.0	0.7	38.4	–21.4
7	1H	2+(NO2)	16.5	16.9	0.4	38.4	–21.5
8	1H	3+(Me)	24.4	24.3	0.0	38.4	–14.1
9	1H	3+(tBu)	24.9	24.8	–0.1	38.4	–13.6
10	1H	3+(NH2)	26.4	26.5	0.1	38.4	–11.9
11	1H	3+(CHO)	16.7	16.6	0.0	38.4	–21.7
12	1H	3+(CN)	16.8	16.8	–0.1	38.4	–21.6
13	1H	3+(NO)	12.8	12.5	–0.3	38.4	–25.9
14	1H	3+(NO2)	13.8	13.8	0.0	38.4	–24.6
15	16H	18+(Me)	22.8	22.9	0.1	42.3	–19.4
16	16H	18+(tBu)	23.8	23.9	0.1	42.3	–18.4
17	16H	18+(NH2)	25.2	25.1	–0.1	42.3	–17.2
18	16H	18+(CHO)	15.0	15.0	0.0	42.3	–27.3
19	16H	18+(CN)	15.5	15.6	0.1	42.3	–26.7
20	16H	18+(NO)	11.9	11.6	–0.3	42.3	–30.7
21	16H	18+(NO2)	12.6	12.5	–0.1	42.3	–29.8
22	1H	16+	18.4	18.5	0.1	38.4	–19.9
23	1H	35+	12.3	11.8	–0.5	38.4	–26.5
24	16H	35+	15.9	15.8	–0.2	42.3	–26.5
25	1H	5+	11.7	11.0	–0.7	38.4	–27.4
26	1H	7+	10.6	10.2	–0.4	38.4	–28.2
27	16H	21+	11.6	10.9	–0.7	42.3	–31.4
28	16H	23+	10.8	10.1	–0.6	42.3	–32.2
29	35H	37+	12.2	11.7	–0.5	47.6	–35.9
30	35H	39+	11.6	11.2	–0.4	47.6	–36.4
31	37H	39+	18.2	17.7	–0.5	54.1	–36.4
32	75H	78+	18.6	18.4	–0.2	52.9	–34.4
33	75H	79+	23.2	23.2	0.0	52.9	–29.7
34	66H	99+	11.0	11.3	0.3	42.1	–30.8
35	66H	90+	21.4	21.2	–0.2	42.1	–20.9
36	66H	91+	21.8	21.1	–0.7	42.1	–21.0
37	66H	93+	19.6	20.3	0.7	42.1	–21.9
38	66H	94+	20.7	21.3	0.7	42.1	–20.8
39	43H	91+	24.2	23.4	–0.9	44.4	–21.0
40	43H	94+	24.1	23.6	–0.6	44.4	–20.8
41	61H	91+	26.5	25.6	–0.9	46.6	–21.0
MAD				0.4

Because compounds XH or corresponding compounds X+ in Scheme can all be extensively used as hydride donors or acceptors, it is clear that the activation free energy values of the 25 122 hydride transfer reactions in acetonitrile at 298 K should be very useful in chemical labs and in chemical industry.

Quest for the Oxidation–Reduction Center Structure of Nicotinamide Coenzyme

It is well-known that nicotinamide-adenine dinucleotide coenzyme (NADH and NAD+) plays a vital role in bioreductions in living bodies by cycle transfer of an apparent hydride ion (eq ). In the past decade, studies on the oxidation–reduction center structure of NADH have been an interesting issue. In the early days, the oxidation–reduction center structure of NADH was generally believed to be 1,2-dihydropyridine.[40,41] Until the late 1950s, the center structure of NADH was unambiguously identified to be 1,4-dihydropyridine rather than 1,2-dihydropyridine.[41,42] This naturally introduces an interesting question: why NADH coenzyme chooses 1,4-dihydropyridine rather than the corresponding 1,2-dihydroisomer as its redox active center structure in the hydride transfer cycle? The reason is obviously related to the thermodynamic and kinetic difference between 1,4-dihydropyridine and 1,2-dihydropyridine in their hydride transfer cycles. To elucidate this question, in this work, we used 1,4-dihydropyridine (1H) and 1,2-dihydropyridine (12H) as the models of NADH and its 1,2-dihydroisomer and chose 80 as the hydride acceptor for 1H and 12H. At the same time, we chose 2H(NH) as the hydride donor for the regeneration of 1H and 12H from 1 and 12, respectively. The two hydride transfer cycles for 1H and 12H are constructed in Scheme .

Scheme 3

Hydride Transfer Cycles between 1,4-Dihydropyridine and Its 1,2-Dihydroisomer (Energy Unit: kcal mol–1)

From Scheme , it is clear that during the hydride transfer from 1H or 12H to 80, the activation free energy of the hydride transfers from 1H to 80 (15.5 kcal mol–1) is greater than that of the hydride transfer from 12H to 80 (14.1 kcal mol–1), which means that the hydride transfer from 1H to 80 is more unfavorable than that from 12H to 80. But during the regeneration of 1H or 12H by hydride transfer from 2H(NH) to pyridinium salt, the activation free energy of the hydride attacking the 4-position on the pyridinium ring is smaller than that of the hydride attacking the 2-position on the pyridinium ring by 2.1 kcal mol–1, which indicates that the pyridinium cation formed from 1,2-dihydropyridines cannot return to the original reduced form, 1,2-dihydropyridine, by hydride transfer cycle, that is, 1,2-dihydropyridines cannot regenerate from the corresponding cation by the hydride transfer cycle. Obviously, the much more negative thermo-kinetic parameter value of 1 to yield 1H (−14.1 kcal mol–1) than that of 12 to yield 12H (−12.0 kcal mol–1) should be a key reason that makes NADH coenzyme choose 1,4-dihydropyridine rather than 1,2-dihydropyridine as its redox active center structure in vivo.

Conclusions

In this work, the activation free energies, activation enthalpies, and activation entropies of 159 hydride self-exchange reactions as well as the heterolytic bond dissociation free energies of the corresponding 159 hydride donors in acetonitrile at 298 K were systematically computed by the DFT method: M06-2X-D3/def2-TZVPP//M06-2X-D3/6-31+G**/SMD with solvation energy computed at M05-2X/6-31G*/SMD level of theory. After analysis of the activation free energies scale of the 159 hydride self-exchange reactions in acetonitrile at 298 K and examination of the effects of the heteroatom, substituent, and aromatic property of reactants on the activation free energies of the hydride self-exchange reactions, the following conclusions can be made: (i) The activation free energy values of the 159 hydride self-exchange reactions range from 16.1 to 46.6 kcal mol–1, which strongly indicates the incorrectness of the linear free energy relationship between the activation free energies (ΔG⧧) and the corresponding thermodynamic driving forces of reactions (ΔGo) for chemical reactions. (ii) For the 159 hydride self-exchange reactions, the activation free energy differences among them are mainly due to the differences in the activation enthalpies. (iii) Heteroatoms (N, O, and S) in the reactants can effectively change the activation free energies of hydride self-exchange reactions. In general, the order of decrease of the activation energy as heteroatoms change is N > O > S > C. (iv) The effects of substituents on the activation free energies of hydride self-exchange reactions are generally quite small (smaller than 1 kcal mol–1), except the substituent at the position of the reaction center where both electron-withdrawing and electron-donating substituents significantly increase the activation free energies of hydride self-exchange reactions. The steric substituent at the reaction center can also effectively increase the activation free energies of hydride self-exchange reactions. In general, the larger the substituent size, the larger the activation free energy. (v) The substituent effects on the activation enthalpy are much larger than those on the activation entropy. (vi) The aromatization of reactants also has an evident effect on the activation free energies of hydride self-exchange reactions. In general, the larger the aromaticity change from the hydride donor (XH) to the hydride acceptor (X+), the smaller the activation energy of the corresponding hydride self-exchange reaction. It is evident that these important and hard-to-get activation free energies of various hydride self-exchange reactions in acetonitrile and the conclusions on the effects of heteroatom, substituent, and aromatic property of reactants on the activation free energies of the hydride self-exchange reactions could provide very important clues to choose a suitable reducing agent from Scheme and predict the reaction rate of various hydride transfer reactions.

Computational Method

All structures were optimized by using M06-2X[43]-D3[44]/6-31+G**[45,46] in acetonitrile with the SMD solvation model[38] accounting for solvent effects. Each optimized structure was confirmed to be a real minimum for a reactant or a first order saddle point with only one imaginary frequency corresponding to the hydride transfer process for the TS by frequency analysis at the same level of theory as optimization. Electronic energies were computed in the gas phase at M06-2X/def2-TZVPP,[47] M06-2X-D3/def2-TZVPP, M06-2X-D3/MG3S,[48] M06-2X-D3/def2-QZVP,[47] B3LYP[49]/def2-TZVPP, B3LYP-D3(BJ)[50]/def2-TZVPP, BMK[51]/def2-TZVPP, and BMK-D3(BJ)[50]/def2-TZVPP levels. Solvation energies were computed on the optimized structures in acetonitrile at M05-2X[52]/6-31G* level with the SMD solvation model. The thermal corrections to the Gibbs free energy were obtained at 298.15 K with a vibrational frequency scale factor of 0.967,[53] whereas entropic contributions to the free energy were calculated using Truhlar’s quasi-harmonic correction[54] by setting all vibrational frequencies less than 100–100 cm–1 to reduce errors arising from treating low frequency vibrations as harmonic oscillators. The reference state for gas-phase computation is 1 atm, whereas for solution phase, it is 1 M. The ultrafine integral grid was used throughout the study. All computations were performed by using Gaussian 09 program package.[55]

The activation free energy of a hydride self-exchange reaction, ΔG⧧(XH/X+), is defined as the Gibbs free energy difference between TS and isolated reactants (XH and X) by eq . The Gibbs free energy of each species (XH, X, and TS) in solution is calculated by eq , where Egas is the gas phase electronic energy, ΔGsol is the solvation energy, ΔGcorr is the thermal correction to Gibbs free energy, and the last term 1.9 kcal mol–1 accounts for the change of reference state from 1 atm to 1 M.

The heterolytic bond dissociation free energy of hydride compound (XH), ΔGo(XH), is defined in this work as the molar Gibbs free energy change of XH releasing the hydride ion (H) in acetonitrile at 298 K. The isodesmic reaction (eq ) is constructed to compute ΔGo(XH) to avoid the inaccuracy in directly computing the energy of a hydride ion in acetonitrile. A reliable experimental ΔGo(XH) value (76.2 kcal mol–1) of 10-methyl-9,10-dihydroacridine (AcrH2) in acetonitrile[20] was used in the computation. ΔGo(XH) of XH in acetonitrile is computed by eq .