Theoretical Study of Multidimensional Proton Tunnelling in Benzoic Acid Dimer

Abstract

Ab initio B3LYP/6-311++G** calculations have been carried out for the benzoic acid dimer for the stable and saddle point structures. The energy barrier for the proton tunneling amounts to 6.5 kcal/mol. The normal mode frequencies have been computed including modes coupled to the proton tunneling mode. Two-dimensional model potentials, formed from symmetric mode coupling potential and squeezed double well potential, have been fitted to the calculated energy barrier, geometries and frequencies, and used to analyze proton dynamics. The calculated proton tunneling energy splitting in the vibrationally ground states of the low-frequency modes is ~230 cm^-1. The two-dimensional model PES predict monotonic increase of the tunneling splitting with the excitation of the planar modes. Depending of the sign of the coupling parameter out-of-plane modes can either suppress or promote the splittings.

Introduction

The importance of proton tunneling in chemical and biological systems is well known, e.g. for the DNA base pairing, as discussed by Löwdin [1]. The phenomenon of potential barrier penetration plays an important role in many branches of physics: quantum field theory, fission of atomic nuclei, scanning tunneling microscopy and solid state physics [2]. Theoretical studies of proton tunneling require the knowledge of multi-dimensional potential energy surfaces (PES’s) which are difficult to obtain from ab initio calculations, especially for electronically excited states. From the theoretical point of view proton transfer in the ground electronic state is more easily tractable. Extensive experimental and theoretical studies have been reported for multidimensional proton tunneling in tropolone [3,4,5,6,7,8,9,10,11,12,13]. Vener et al. [10] studied theoretically multidimensional proton tunneling in tropolone by using adiabatic separation of variables. Smedarchina et al. [11] used the instanton approach to account for tunnelling splittings. Takada and Nakamura [12] on the base of high accuracy ab initio calculations proposed a model potential energy surface (PES) for the electronically ground $\tilde{X}$ state and employed it to analyze the proton tunnelling dynamics. Wójcik et al. [13] reported the results of the high accuracy ab initio MO calculations of the potential energy surfaces in the excited à state of tropolone, and by fitting the two- and three-dimensional analytical model potentials to these surfaces and solving the multidimensional vibrational problems, interpreted the existing experimental data. Other systems for which tunnelling have been studied include malonaldehyde [14,15,16,17], formic acid [18], hydrogen-oxalate anion [16], substituted tropolone [19,20] and 5-methyl-9-hydroxyphenalenone (OH and OD) [21], methanol tetramer [22] and hydrogen carbonate dimer ion [23]. Recently a mixed quantum-classical approach has been used to study dynamics of hydrogen-bonded systems [24,25].

System which draws our attention in the present paper is benzoic acid dimer. It is present in the structure of the crystal [26,27,28]. Its proton transfer has been recently studied by inelastic neutron scattering by Plazanet et al. [29] and Fillaux et al. [30]. Vibrational spectra of benzoic acid have been reported in Refs. [31,32]. In this article we present the results of high accuracy ab initio MO and DFT calculations of the potential energy surfaces for the ground state of benzoic acid dimer in the stable and saddle point structures, and by fitting the two-dimensional analytical model potentials to these surfaces and solving the multidimensional vibrational problems, we predict the effects of excitations of the low-frequency in- and out-of-plane modes on the proton tunnelling splittings.

This paper is organized as follows. The results of our quantum chemical calculations for the ground state of benzoic acid dimer are presented in Sec. II. Model studies of the tunnelling are discussed in Sec. III. Concluding remarks are given in Sec. IV.

Quantum Chemical Calculations

We performed ab initio B3LYP/6-311++G** calculations for the ground state of benzoic acid dimer using the Gaussian 98 program package [33]. The calculations have been done in the Computer Center of Hokkaido University in Sapporo, Japan and the Department of Materials Chemistry, Uppsala University in Sweden. The calculations have been performed for the stable and the saddle point structures. The geometry of benzoic acid dimer in the stable and the saddle point structures, calculated by the B3LYP/6-311++G** method is shown in Fig. 1. Optimized geometry is summarized in Table 1. The complex is planar in the stable and the saddle point structures. The calculated values reproduce the experimental bond lengths and angles reasonably well. There are small discrepancies for C₁-O₂ and some C-C distances. Experimental C-H distances taken from Ref. [26] are only estimates. The angles are well reproduced except O₂C₁C₂. The influence of the saddle point structure on geometry in hydrogen-bonded cyclic structure is considerable.

Figure 1. Benzoic acid in the stable and saddle point structures.

Figure 1. Benzoic acid in the stable and saddle point structures.

The calculated frequencies and symmetries of the normal modes at the stable and the saddle point structure are summarized in Table 2. The modes calculated by the B3LYP/6-311++G** method and used in the model calculations are pictured in Fig. 2.. The frequencies have not been scaled. The agreement between experimental and calculated frequencies is generally good, especially in the low and medium-frequency regions. The calculated frequencies in the region of O-H and C-H stretchings are overestimated by about 100-150 cm^-1 and the frequency of the O-H symmetric stretching mode ν₇₃ is overestimated by ~500 cm^-1. This is partly due to harmonic approximation and effects of intermolecular interactions on this mode in the crystal.

Table 1. Optimized geometries of the stable and saddle point structures of benzoic acid by the B3LYP/6-311++G** (DFT) method.

	Stable structure			Saddle point structure
	DFT	EXP.		DFT
Bond lengths (Å)
O1··· O2’	2,663	2,64a; 2,629b; 2,633c		2,410
O1 – H1	1,663	1,64b		1,205
O2 – H2	1,000	0,988b		1,205
C1 – O1	1,230	1,24a; 1,268b; 1,263 c		1,272
C1 – O2	1,323	1,29a; 1,275b; 1,275 c		1,271
C2 – C1	1,486	1,48a; 1,484 c		1,486
C2 – C3	1,400	1,39a; 1,390 c		1,400
C3 – C4	1,391	1,42a; 1,387 c		1,391
C4 – C5	1,395	1,36a; 1,379 c		1,395
C5 – C6	1,395	1,37a; 1,384 c		1,395
C6 – C7	1,390	1,41a; 1,401 c		1,391
C7 – C2	1,400	1,39a; 1,392 c		1,400
C3 – H3	1,082	0,79a		1,082
C4 – H4	1,084	0,96a		1,084
C5 – H5	1,084	0,91a		1,084
C6 – H6	1,084	0,96a		1,084
C7 – H7	1,083	0,79a		1,082
Bond angles (o)
O1H1O2’	177,14			179,63
C1O1H1	126,84			116,66
C1O2H2	110,27			116,71
O1C1O2	123,26	122 a; 123,2c		123,78
O2C1C2	114,50	118 a; 119,9 c		118,12
C2C1O1	122,24	122 a; 120,2 c		118,11
C1C2C3	121,40	122 a; 118,0 c		120,04
C3C2C7	119,90	119 a; 119,9 c		119,92
C7C2C1	118,70	119 a; 118,8 c		120,04
C2C3C4	119,86	118 a; 120,1 c		119,94
C3C4C5	120,02	123 a; 119,9 c		120,03
C4C5C6	120,15	118 a; 120,3 c		120,16
C5C6C7	119,98	122 a; 119,7 c		120,03
C6C7C2	120,02	120 a; 119,8 c		119,94
C2C3H3	119,48			119,11
H3C3C4	120,66			120,95
C3C4H4	119,85			119,89
H4C4C5	120,07			120,08
C4C5H5	119,91			119,92
H5C5C6	119,94			119,92
C5C6H6	120,09			120,08
H6C6C7	119,93			119,89
C6C7H7	121,12			120,95
H7C7C2	118,86			119,12

^a Ref. 26, ^b Ref. 27, ^c Ref. 28.

Table 1. Optimized geometries of the stable and saddle point structures of benzoic acid by the B3LYP/6-311++G** (DFT) method.

	Stable structure			Saddle point structure
	DFT	EXP.		DFT
Bond lengths (Å)
O1··· O2’	2,663	2,64a; 2,629b; 2,633c		2,410
O1 – H1	1,663	1,64b		1,205
O2 – H2	1,000	0,988b		1,205
C1 – O1	1,230	1,24a; 1,268b; 1,263 c		1,272
C1 – O2	1,323	1,29a; 1,275b; 1,275 c		1,271
C2 – C1	1,486	1,48a; 1,484 c		1,486
C2 – C3	1,400	1,39a; 1,390 c		1,400
C3 – C4	1,391	1,42a; 1,387 c		1,391
C4 – C5	1,395	1,36a; 1,379 c		1,395
C5 – C6	1,395	1,37a; 1,384 c		1,395
C6 – C7	1,390	1,41a; 1,401 c		1,391
C7 – C2	1,400	1,39a; 1,392 c		1,400
C3 – H3	1,082	0,79a		1,082
C4 – H4	1,084	0,96a		1,084
C5 – H5	1,084	0,91a		1,084
C6 – H6	1,084	0,96a		1,084
C7 – H7	1,083	0,79a		1,082
Bond angles (o)
O1H1O2’	177,14			179,63
C1O1H1	126,84			116,66
C1O2H2	110,27			116,71
O1C1O2	123,26	122 a; 123,2c		123,78
O2C1C2	114,50	118 a; 119,9 c		118,12
C2C1O1	122,24	122 a; 120,2 c		118,11
C1C2C3	121,40	122 a; 118,0 c		120,04
C3C2C7	119,90	119 a; 119,9 c		119,92
C7C2C1	118,70	119 a; 118,8 c		120,04
C2C3C4	119,86	118 a; 120,1 c		119,94
C3C4C5	120,02	123 a; 119,9 c		120,03
C4C5C6	120,15	118 a; 120,3 c		120,16
C5C6C7	119,98	122 a; 119,7 c		120,03
C6C7C2	120,02	120 a; 119,8 c		119,94
C2C3H3	119,48			119,11
H3C3C4	120,66			120,95
C3C4H4	119,85			119,89
H4C4C5	120,07			120,08
C4C5H5	119,91			119,92
H5C5C6	119,94			119,92
C5C6H6	120,09			120,08
H6C6C7	119,93			119,89
C6C7H7	121,12			120,95
H7C7C2	118,86			119,12

^a Ref. 26, ^b Ref. 27, ^c Ref. 28.

Table 2. Calculated normal mode frequencies for the stable and saddle point structures of benzoic acid by the B3LYP/6-311++G** (DFT) method.

No.	Symmetry	Frequency (cm-1)
		Stable structure	Saddle point structure	Literature Infrared
1	AU	22	22	25b
2	AU	33	35	35b
3	BG	41	17	41b
4	BU	60	75	71b
5	BG	64	62	79b
6	AU	87	79	94a
7	AG	106	120	110b
8	AG	114	201	127b
9	AU	172	174	146a
10	BG	179	177
11	AG	259	294
12	BU	283	363
13	BU	389	385
14	BG	414	413
15	AU	415	415
16	AG	423	588
17	AU	446	448	421a
18	BG	448	451
19	AG	512	529
20	BU	546	723	491a
21	AG	632	632
22	BU	632	631	615a
23	AG	669	729
24	BU	679	698	669a
25	BG	694	699
26	AU	696	697	687a
27	BG	707	680
28	AU	724	729	711a
29	BG	804	788
30	AG	804	824
31	BU	816	866	767a
32	AU	831	838	813a
33	AU	866	865
34	BG	866	865	856a
35	BG	886	1203
36	BG	961	961
37	AU	963	965	937a
38	AU	982	1327	960a
39	BG	1000	999
40	AU	1000	999	974a
41	BG	1010	1009	998a
42	AU	1011	1009
43	BU	1018	1015	1002a
44	AG	1018	1018
45	BU	1046	1044	1027a
46	AG	1046	1047
47	AG	1101	1101
48	BU	1101	1101	1066a
49	BU	1146	1121	1027a
50	AG	1150	1173
51	BU	1184	1184	1164a
52	AG	1184	1184
53	BU	1198	1185	1185a
54	AG	1199	1202
55	AG	1309	1436	1316a
56	BU	1317	1563	1322a
57	AG	1344	1337
58	BU	1346	1338	1297a
59	AG	1350	1350
60	BU	1350	1350	1380a
61	BU	1453	1449	1430a
62	AG	1474	1477
63	BU	1480	1478	1456a
64	AG	1486	1697
65	AG	1524	1523
66	BU	1524	1526	1496a
67	BU	1620	1616	1590a
68	AG	1621	1614
69	BU	1643	1642	1606a
70	AG	1643	1641
71	AG	1686	1696	1699a
72	BU	1731	1690	1738a
73	AG	3102	1194 i	2605a
74	BU	3166	3167	3012a
75	AG	3166	3167
76	BU	3179	3179	3041a
77	AG	3179	3179
78	BU	3187	3189	3068a
79	AG	3188	3189
80	BU	3197	1216	3312a
81	AG	3203	3206
82	BU	3203	3206	3079a
83	BU	3210	3207	3098a
84	AG	3210	3207

^a Ref. 30, ^b Ref. 31.

Table 2. Calculated normal mode frequencies for the stable and saddle point structures of benzoic acid by the B3LYP/6-311++G** (DFT) method.

No.	Symmetry	Frequency (cm-1)
		Stable structure	Saddle point structure	Literature Infrared
1	AU	22	22	25b
2	AU	33	35	35b
3	BG	41	17	41b
4	BU	60	75	71b
5	BG	64	62	79b
6	AU	87	79	94a
7	AG	106	120	110b
8	AG	114	201	127b
9	AU	172	174	146a
10	BG	179	177
11	AG	259	294
12	BU	283	363
13	BU	389	385
14	BG	414	413
15	AU	415	415
16	AG	423	588
17	AU	446	448	421a
18	BG	448	451
19	AG	512	529
20	BU	546	723	491a
21	AG	632	632
22	BU	632	631	615a
23	AG	669	729
24	BU	679	698	669a
25	BG	694	699
26	AU	696	697	687a
27	BG	707	680
28	AU	724	729	711a
29	BG	804	788
30	AG	804	824
31	BU	816	866	767a
32	AU	831	838	813a
33	AU	866	865
34	BG	866	865	856a
35	BG	886	1203
36	BG	961	961
37	AU	963	965	937a
38	AU	982	1327	960a
39	BG	1000	999
40	AU	1000	999	974a
41	BG	1010	1009	998a
42	AU	1011	1009
43	BU	1018	1015	1002a
44	AG	1018	1018
45	BU	1046	1044	1027a
46	AG	1046	1047
47	AG	1101	1101
48	BU	1101	1101	1066a
49	BU	1146	1121	1027a
50	AG	1150	1173
51	BU	1184	1184	1164a
52	AG	1184	1184
53	BU	1198	1185	1185a
54	AG	1199	1202
55	AG	1309	1436	1316a
56	BU	1317	1563	1322a
57	AG	1344	1337
58	BU	1346	1338	1297a
59	AG	1350	1350
60	BU	1350	1350	1380a
61	BU	1453	1449	1430a
62	AG	1474	1477
63	BU	1480	1478	1456a
64	AG	1486	1697
65	AG	1524	1523
66	BU	1524	1526	1496a
67	BU	1620	1616	1590a
68	AG	1621	1614
69	BU	1643	1642	1606a
70	AG	1643	1641
71	AG	1686	1696	1699a
72	BU	1731	1690	1738a
73	AG	3102	1194 i	2605a
74	BU	3166	3167	3012a
75	AG	3166	3167
76	BU	3179	3179	3041a
77	AG	3179	3179
78	BU	3187	3189	3068a
79	AG	3188	3189
80	BU	3197	1216	3312a
81	AG	3203	3206
82	BU	3203	3206	3079a
83	BU	3210	3207	3098a
84	AG	3210	3207

^a Ref. 30, ^b Ref. 31.

In Table 3 we present the calculated energies and potential barriers. The energy barrier for the proton transfer amounts to 6.497 kcal/mol, no ZPE and only 1.867 kcal/mol, with ZPE.

Figure 2. Normal modes of benzoic acid calculated by the B3LYP/6-311++G** method.

Figure 2. Normal modes of benzoic acid calculated by the B3LYP/6-311++G** method.

Table 3. Calculated energies and barrier for benzoic acid by the B3LYP/6-311++G** (DFT) method.

	DFT
Energy, stable structure	No ZPE, a.u.	-841,923
	With ZPE, a.u.	-841,691
Energy, saddle point structure	No ZPE, a.u.	-841,912
	With ZPE, a.u.	-841,688
Energy barrier	No ZPE, kcal/mol	6,497
	No ZPE, cm-1	2272
	With ZPE, kcal/mol	1,867
	With ZPE, cm-1	653

Table 3. Calculated energies and barrier for benzoic acid by the B3LYP/6-311++G** (DFT) method.

	DFT
Energy, stable structure	No ZPE, a.u.	-841,923
	With ZPE, a.u.	-841,691
Energy, saddle point structure	No ZPE, a.u.	-841,912
	With ZPE, a.u.	-841,688
Energy barrier	No ZPE, kcal/mol	6,497
	No ZPE, cm-1	2272
	With ZPE, kcal/mol	1,867
	With ZPE, cm-1	653

Model Calculations

On the basis of the ab initio calculations we constructed two-dimensional model PES for the proton tunnelling ν₇₃ coupled to chosen low-frequency modes of benzoic acid dimer which largely affect the tunnelling. These are the low-frequency planar hydrogen-bond stretching and bending modes ν₄, ν₇, ν₈, ν₁₁, ν₁₃ and ν₁₆ and the lowest-frequency out-of-plane modes ν₁, ν₂, ν₃, ν₅ and ν₆. They are shown in Fig. 2 and their calculated vibrational frequencies are listed in Table 2. We assumed the symmetric (synchronous) double proton transfer in benzoic acid dimer in accordance with the results of ref. [18].

The two-dimensional model potentials used to simulate the couplings are [12,13]:

(a) the symmetric mode coupling potential (SMC) describing couplings of the proton tunneling mode ν₁ with the planar modes ν₈, ν₉ and ν₂₁:

$$V_{SMC}=\frac{1}{8x_0^2}{\left(x-x_0\right)}^2{\left(x+x_0\right)}^2+\frac{1}{2}\frac{{\omega }_y}{{\omega }_x}{\left[y+\alpha \left(x^2-x_0^2\right)\right]}^2$$

(1)

and

(b) the squeezed double well potential (SQZ) describing couplings of the proton tunneling mode ν₁ with the out-of-plane modes ν₁₂, ν₁₃ and ν₂₄:

$$V_{SQZ}=\frac{1}{8x_0^2}{\left(x-x_0\right)}^2{\left(x+x_0\right)}^2+\frac{1}{2x_0^2}\left[\frac{{\omega }_z}{{\omega }_x}{x}_0^2-\gamma \left(x^2-x_0^2\right)/\left(\frac{{\omega }_z}{{\omega }_x}\right)\right]z^2,$$

(2)

where x and y and z, denote the coordinates of the proton tunneling and the low-frequency modes, respectively, ω_x, ω_y and ω_z are the angular frequencies, 2x_o the distance between the two minima, α and γ the coupling strengths. In the formulas (1) and (2) the potentials are expressed in the units of the quantum ћω_x and the coordinates x, y and z are dimensionless:

$$\begin{array}{l}x=\tilde{x}\sqrt{\frac{m_x{\omega }_x}{ћ}},\\ y=\tilde{y}\sqrt{\frac{m_y{\omega }_y}{ћ}},\\ z=\tilde{z}\sqrt{\frac{m_z{\omega }_z}{ћ}}.\end{array}$$

(3)

$\tilde{x},\tilde{y}$ and $\tilde{z}$ denote the dimensional coordinates and m_x, m_y and m_z are the effective masses.

The parameters x_o, α and γ of the potentials (1) and (2) were estimated from the formulas:

$$\begin{array}{l}\text{Δ}E=\frac{x_0^2}{8}ћ{\omega }_x,\\ y_s=\alpha x_0^2,\\ \gamma =\frac{\left({\omega }_z^s-{\omega }_z\right){\omega }_z}{{\omega }_x^2},\end{array}$$

(4)

where ΔE, y_s and ${\omega }_z^s$ denote the energy barrier, the value of the normal coordinate y of the coupled mode of the SMC potential at the saddle point structure and the angular frequency of the mode z at the saddle point structure, respectively. We used no ZPE energy barriers. The parameters used in subsequent calculations of the energy splittings have been calculated from the results of the B3LYP/6-311++G** data and are listed in Table 4. The parameter α describing coupling between the O-H stretching mode ν₇₃ and the mode ν₄, ν₇, ν₈, ν₁₁, ν₁₃ or ν₁₆ represent an analogue of linear distortion parameters b used for theoretical reproduction of the X-H infrared band shapes of tropolone [34], salicylaldehyde [35] or aspirin [36]. Both different spectroscopic facts thus have the same origin, the anharmonic coupling in the potential energy between the two X-H and X...Y vibrations.

Tunnelling energy splittings have been calculated variationally. The convergence has been confirmed by reproducing exactly the energy splittings reported by Takada and Nakamura [12] and calculated by the DVR method [37,38]. The results are presented in Table 5 for the two-dimensional model potentials (1) and (2). The proton tunnelling energy splitting in the vibrationally ground states of the low-frequency modes is ca. 230 cm^-1. This compares reasonably well with the experimental value 172 cm^-1 obtained from the inelastic neutron scattering measurements for benzoic acid - D₅H and recently reported by Fillaux et al. [30]. The two-dimensional model PES qualitatively explain increase of the tunnelling splitting with the excitation of the planar ν₄, ν₇, ν₈,ν₁₁, ν₁₃ and ν₁₆ modes. For the out-of-plane modes the monotonic change depend on a sign of the coupling parameter γ. When the sign is positive (for modes ν₁ and ν₂) one obtains monotonic decrease of the splitting with the excitation, and when it is negative (for the ν₃, ν₅ and ν₆ mode) the excitations cause increase of the splittings. This remains in agreement with our previous theoretical results for the excited state of tropolone [13] and hydrogen carbonate dimer [23]. Experimentally observed in tropolone promotion of the tunnelling by the excitation of the planar modes and its suppression by the excitation of the out-of-plane modes [7,8] are qualitatively confirmed in the present two-dimensional model calculations. Our previous results on the excited state of tropolone [13], hydrogen carbonate dimer ion [23] and the present ones do not require any adjustments and present pure quantum-mechanical approach to the problem of tunnelling splittings.

Table 4. Parameters of the two-dimensional models.

Mode		xo	α	γ
Stable	Saddle point
ν4		2,42	0,178
ν7		2,42	0,096
ν8		2,42	0,122
ν11		2,42	0,040
ν13		2,42	0,056
ν16		2,42	0,138
	ν1	2,42		0,00000017
	ν2	2,42		0,00000597
	ν3	2,42		-0,00010000
	ν5	2,42		-0,00000807
	ν6	2,42		-0,00007099

Table 4. Parameters of the two-dimensional models.

Mode		xo	α	γ
Stable	Saddle point
ν4		2,42	0,178
ν7		2,42	0,096
ν8		2,42	0,122
ν11		2,42	0,040
ν13		2,42	0,056
ν16		2,42	0,138
	ν1	2,42		0,00000017
	ν2	2,42		0,00000597
	ν3	2,42		-0,00010000
	ν5	2,42		-0,00000807
	ν6	2,42		-0,00007099

Table 5. Energy splittings (cm^-1) calculated for two-dimensional model potentials.

Quantum number	SMC							SQZ
	ν4	ν7	ν8	ν11	ν13	ν16		ν1	ν2	ν3	ν5	ν6
0	229,24	229,50	229,36	229,67	229,65	229,54		229,71	229,63	230,99	229,78	230,10
1	229,43	229,68	229,69	229,86	230,52	235,50		229,71	229,47	233,54	229,89	230,87
2	229,62	229,86	230,02	230,05	231,38	241,32		229,70	229,30	236,12	230,01	231,64
3	229,82	230,04	230,35	230,25	232,23	247,01		229,69	229,14	238,74	230,13	232,41
4	230,01	230,22	230,68	230,44	233,09	252,57		229,69	228,97	241,39	230,25	233,18
5	230,20	230,40	231,01	230,63	233,94	258,02		229,68	228,81	244,08	230,37	233,96
6	230,39	230,57	231,34	230,82	234,79	263,36		229,67	228,64	246,80	230,49	234,74
7	230,58	230,75	231,67	231,02	235,63	268,59		229,66	228,48	249,55	230,60	235,52
8	230,77	230,93	231,99	231,21	236,48	273,72		229,66	228,31	252,34	230,72	236,30
9	230,96	231,11	232,32	231,40	237,31	278,76		229,65	228,15	255,16	230,84	237,09
10	231,15	231,29	232,65	231,59	238,15	283,70		229,64	227,98	258,01	230,96	237,88

Table 5. Energy splittings (cm^-1) calculated for two-dimensional model potentials.

Quantum number	SMC							SQZ
	ν4	ν7	ν8	ν11	ν13	ν16		ν1	ν2	ν3	ν5	ν6
0	229,24	229,50	229,36	229,67	229,65	229,54		229,71	229,63	230,99	229,78	230,10
1	229,43	229,68	229,69	229,86	230,52	235,50		229,71	229,47	233,54	229,89	230,87
2	229,62	229,86	230,02	230,05	231,38	241,32		229,70	229,30	236,12	230,01	231,64
3	229,82	230,04	230,35	230,25	232,23	247,01		229,69	229,14	238,74	230,13	232,41
4	230,01	230,22	230,68	230,44	233,09	252,57		229,69	228,97	241,39	230,25	233,18
5	230,20	230,40	231,01	230,63	233,94	258,02		229,68	228,81	244,08	230,37	233,96
6	230,39	230,57	231,34	230,82	234,79	263,36		229,67	228,64	246,80	230,49	234,74
7	230,58	230,75	231,67	231,02	235,63	268,59		229,66	228,48	249,55	230,60	235,52
8	230,77	230,93	231,99	231,21	236,48	273,72		229,66	228,31	252,34	230,72	236,30
9	230,96	231,11	232,32	231,40	237,31	278,76		229,65	228,15	255,16	230,84	237,09
10	231,15	231,29	232,65	231,59	238,15	283,70		229,64	227,98	258,01	230,96	237,88

Conclusions

The proton tunnelling dynamics in benzoic acid dimer in the ground electronic state has been studied by performing quantum mechanical calculations of the potential energy surface and fitting it by two-dimensional model potentials. The tunnelling energy splittings for different vibrationally excited states have been calculated. The calculated proton tunnelling energy splitting in the vibrationally ground states of the low-frequency modes (~230 cm^-1) compares reasonably well with the experimental value 172 cm^-1 obtained from the inelastic neutron scattering measurements for benzoic acid - D₅H [29]. Our model calculations predict the promotion of the proton tunnelling by the excitation of the planar modes. Depending of the sign of the coupling parameter γ, out-of-plane modes can either supress or promote the tunnelling. This results are confirmed by existing experimental data [7,8]. Our model presents a pure quantum-mechanical approach to the problem of the proton tunnelling splittings.