Journal of Chemical Software, Vol.5, No.2 (1999)

Evaluation of Enthalpies of Benzene-Monosubstituted Benzene interactions by Semi-Empirical MO Calculations. Their Relationship to the Experimental Enthalpies Determined by GLPC

Tadashi ENDO, Takahiro IIDA, Nobuhide FURUYA, Yumiko YAMADA and Masato M. ITO

1 Introduction

Precise molecular recognition is essential to living systems. For example, the following biological discrimination is well known: enzyme-substrate [1], antigen-antibody [2], nucleic acid-protein [3], drug-receptor [4], and hormone-receptor recognition [5]. In contrast, a very large number of studies have recently been made on chemical recognition mainly using various model compounds of host-guest type such as crown ethers [6], criptands [7], calixarenes [8], and convergent cleft molecules [9]. It is now accepted that molecular recognition, whether biological or chemical [10], is due to specific weak (noncovalent) interactions between interacting groups in the respective molecules.
Concerning weak interactions between groups, comparable in energies (or enthalpies) to van der Waals interactions, experimental studies have extensively been made. These include dipole-induced dipole interaction between the 1,3-dioxane and phenyl rings [11], an alkyl-phenyl interaction [12], an alkyl-alkyl interaction [13] including that between two t-butyl groups [14], and so on. In addition, recent theoretical studies on aromatic-aromatic interactions suggested an energetically favorable orientation of two aromatic rings as well as their interaction energies [15 - 17].
We have been studying factors controlling such precise molecular recognition as occurs in living systems, by use of a pair of acylurea derivatives [open chain analogues of pyrimidine bases (e.g., uracil and thymine)] which can associate strongly with each other as well as with themselves [18]. As a result, it has been clarified that (i) intermolecular association is the first requirement for molecular recognition [19], (ii) three-dimensional shape similarity between interacting groups in reacting molecules is responsible for more specific and precise molecular recognition than would otherwise be achieved [20].
These findings led us to investigate weak interactions between interacting groups. Using GLPC, it has been demonstrated that (i) enthalpies of weak interactions of a phenyl group (DDH^t)with substituted benzenes [21], alkanes [22], alkenes [23], ethers [23], and carbonyl compounds [21] range from -2.7 to +0.1 kcal mol^-1 (1 cal = 4.184 J) and (ii) methylation or methoxylation of a benzene ring weakens or strengthens, respectively, the phenyl-phenyl interaction [21]. Further examinations have revealed that degree of the recognition between molecules each having a nonpolar group can be controlled by the strength of shape-specific weak interactions between nonpolar groups in the respective molecules [24].
The above-mentioned experimental enthalpies of the interactions [21] can only be determined with a phenyl group as one component, because of limitation of stationary liquids available in GLPC. Thus, some theoretical approaches to evaluate weak interactions between relatively nonpolar molecules will help to elucidate an entire picture of the weak interactions including their shape-specificity. This approach need not be very accurate and quantitative; it should, however, be (i) precise enough to bring about good correlation with experimental results and (ii) convenient enough to make it possible to calculate the interaction energies between the phenyl and a wide variety of groups within a realistic time.
Ab initio calculations seem not to be suitable for this purpose; they require enormous time and resources of the computer because of complicated calculations of configuration interactions (CI) using a large number of basis sets, the calculation methods being still a subject of theoretical chemists even for the intermolecular interaction of a methane dimer [25]. There appear to have been no calculation methods available for electron correlation which enable calculations of interactions between benzene and substituted benzenes to be precise enough within a realistic time. On the other hand, for semi-empirical methods such as PM3 in MOPAC where (i) CI calculations are not implemented explicitly and (ii) the effect corresponding to electron correlation is supposed to be incorporated implicitly in the set of adjustable parameters, the use of which is characteristic of semi-empirical calculations. This is evident from the fact that heats of formation for various organic compounds can be evaluated with considerable accuracy by means of MOPAC [26].
With this situation in mind, MOPAC was used to evaluate enthalpies of intermolecular interactions, and the effectiveness and limitations of the semi-empirical method were examined. In this paper, we wish to report that the calculated enthalpies of weak interactions between benzene and various substituted benzenes have close correlation with the corresponding experimental enthalpies determined by GLPC.

2 Calculation methods

All calculations were carried out with the MOPAC93 program [27] for Power Macintosh (CS MOPAC, Fujitsu Ltd.) on Apple Power Macintosh 7200/120.
As monosubstituted benzenes (PhX), anisole (PhOMe), benzene (PhH), chlorobenzene (PhCl), N,N-dimethylaniline (PhNMe₂) , ethylbenzene (PhEt), fluorobenzene (PhF), nitrobenzene (PhNO₂) , and toluene (PhMe) were used.
As initial arrangements of a pair of benzene and a monosubstituted benzene (PhH-PhX system), one parallel (P) and four vertical arrangements (V_r , V_p , V_m , and V_b) were chosen (Figure 1a). In P arrangement, two benzene rings of PhH and PhX molecules are stacked parallel to each other. In V_r arrangement, the lowest H atom of PhH points to the center of the benzene ring of PhX. In V_p (Figure 1b) or V_m arrangement, the lowest H atom at the para or meta position, respectively, to substituent X is located above the centroid of PhH. In V_b arrangement, substituent X lying at the lowest position ("bottom") of PhX is located above the centroid of PhH.

Figure 1. (a) Schematic drawings of initial arrangements (P, V_r , V_p , V_m , and V_b) of benzene (PhH)-monosubstituted benzene (PhX) systems; (b) a drawing of initial arrangement V_p. The symbol r_I represents an initial intermolecular distance; for definition, see the text. The symbol X shows a substituent of benzene. In the case of V_b arrangement, the r_I for X = Me, Et, MeO, NMe₂, or NO₂ is the distance between the H atom in these groups (except for a NO₂ group [the O atom instead of the H atom]) and the "closest" C atom in PhH; for X = NMe₂ or NO₂, the N atom in these two groups is located above the centroid of PhH. In view of van der Waals radii of Cl (1.80 A), F (1.35 A), and O (1.40A), the r_I values set up for these are 3.50, 3.05, and 3.1 A, respectively. For the other type of initial arrangement V_r (V_r^R) where a PhH molecule located above a PhX molecule is rotated by 90° relative to PhX, the DDH_f values are also evaluated.

In the case of V arrangements, an intermolecular distance is defined as the distance between the centroid of a benzene ring lying on a plane and the lowest atom (H or X) of the other PhH or PhX molecule located above the centroid of the former benzene ring (Figure 1). With P arrangement, an intermolecular distance is defined as the distance between the facing "above" and "below" carbon atoms of two benzene rings of parallel orientation.
Setting up of an initial intermolecular distance (r_I) was performed using Chem3D (CambridgeSoft Corp.) ver. 3.5 by geometrical calculation so that the r_I may become the desired value (e.g., 2.60, 2.75, 3.05, 3.20, or 3.40 A). In the case of V_b arrangement, setting up of the r_I is shown in the caption of Figure 1.
Energy minimization (geometry optimization) was carried out in the following way both for a single molecule (PhH or PhX) and for a pair of molecules (PhH-PhX system), thereby giving their heats of formation (DH_f) : (1) in the case of a single molecule, the structure preliminarily optimized by Chem3D was subject to geometry optimization with MOPAC calculation; (2) with a PhH-PhX system, (i) its initial arrangement (Figure 1) was set up, using the optimized structures of a PhH molecule and a single PhX one mentioned above, (ii) the r_I was then set up, and (iii) energy minimization was performed with MOPAC calculation.
In general, enthalpy of interaction (DDH_f) between A and B molecules can be defined as Eq. 1, where DH_f (A-B) is heat of formation of coexisting A and B molecules, DH_f (A) is that of A
DDH_f = DH_f (A - B) - {DH_f (A) + DH_f (B)} (1)
molecule, and DH_f (B) is that of B molecule.

3 Results and discussion

PhH-PhH Interactions
Evaluation of interaction enthalpies (DDH_f) of benzene-benzene interaction for initial arrangement V was performed with three methods (MNDO, AM1, and PM3) and the results were compared with one another (Table 1). In view of the fact that (i) a half of thickness of a benzene ring is 1.7 A and (ii) van der Waals radius of an H atom is 1.2 A, heats of formation (DH_f) were calculated, with structures optimized for various initial intermolecular distances (r_I) of 2.60, 2.75, 3.05, and 3.20 A. The intermolecular distance ( r_o ) optimized with PM3 method, for instance, was ca. 2.5 A, irrespective of the r_I. The corresponding DH_fs for initial structure V were 46.53 (r_I = 3.20 A), 46.48 (3.05 A), 46.35 (2.75 A), and 46.29 kcal mol^-1 (2.60 A), whereas those for the optimized structures were all 46.26 kcal mol^-1, regardless of the r_I. It has been clarified that (i) MNDO and AM1 methods tend to overestimate the intermolecular interactions and (ii) PM3 method is most suitable since it gives the DDH_f values close to the experimental interaction enthalpy (DDH^t) of -1.32 kcal mol^-1 (X = H) [28].

Table 1 Effect of calculation methods on the DDH_f for initial arrangements V and P of PhH-PhH system^a

r_I /A DDH_f / kcal mol^-1
MNDO
(V) AM1
(V) PM3
(P) PM3
(V)
2.60 -4.29 -2.96 0.06 -0.52
2.75 -4.30 -2.94 0.06 -0.52
3.05 -4.29 -2.95 0.06 -0.52
3.20 -4.29 -2.96 0.06 -0.52
^a The following key words were chosen for PM3 method. (1) For a PhH or PhX molecule: PM3 EF XYZ PRECISE GNORM = 0.01; (2) for a PhH-PhX system: PM3 EF XYZ PRECISE GNORM = 0.05 LET DDMIN = 0.0.

r_I /A	DDH_f / kcal mol^-1
MNDO (V)	AM1 (V)	PM3 (P)	PM3 (V)
2.60	-4.29	-2.96	0.06	-0.52
2.75	-4.30	-2.94	0.06	-0.52
3.05	-4.29	-2.95	0.06	-0.52
3.20	-4.29	-2.96	0.06	-0.52

It is also evident from the data in Table 1 that the DDH_f values by PM3 method for V and P arrangements (X = H) are -0.52 and 0.06 kcal mol^-1, respectively, regardless of the r_I. Since the experimental interaction enthalpy (DDH^t) for X = H is -1.32 kcal mol^-1, V arrangement appears to be more suitable as an initial arrangement of the system than P arrangement.
An optimized intermolecular distance for the benzene dimer was shown to be 5.0 A between their centroids (d_cb-cb) by ab initio calculations [16, 29], molecular dynamics simulations [30], and Monte Carlo simulations [31]. In the present work, the d_cb-cb becomes 4.978 A [32], the distance practically equal to that obtained by the previous work mentioned above.
Ab initio calculations of the benzene dimer indicated that interplanar angles (dihedral angles) between interacting benzene molecules approaching 90°, the "T-stacked" arrangement, are enthalpically favorable [16]. Moreover, when they adopt edge-to-face stacked orientation [17], electrostatic interaction between two benzene rings is reported to be attractive (-1.4 kcal mol^-1). These findings agree with our result that the optimized geometry, which leads to close relationship between the DDH_f and DDH^t , is similar to the initial arrangement of vertical type (V). The T-shaped orientation of two benzene molecules is well recognized in four kinds of studies: (i) crystal structures of benzene [33], peptides (and proteins) [15], and organic compounds [34], (ii) molecular beam electric deflection studies [35], (iii) NMR studies [36], and (iv) theoretical calculations [30, 31, 37].
As to interaction energy, ab initio calculations of the benzene dimer for various orientations [16] revealed that the interaction energy minimum was ca. -2.4 kcal mol^-1; a similar result was obtained by more recent calculations that the interaction energy for the T-shaped structure was -2.64 kcal mol^-1 [29, 38, 39]. Moreover, NMR studies showed the enthalpy (DH ) of the benzene-benzene interaction to be ca. -2.0 kcal mol^-1 [40], the DH value corresponding to theDDH^t of -1.32 kcal mol^-1 for X = H; transfer of a benzene molecule from an environment of saturated hydrocarbon molecules into an environment of benzene molecules is energetically favorable to an extent ranging to ca. -l kcal mol^-1 [41]. These theoretical and experimental results suggest the enthalpy of PhH-PhH interaction to be ca. -l to -2 kcal mol^-1. Accordingly, the calculated enthalpy (DDH_f) of the PhH-PhH interaction by PM3 method might be more positive by ca. 0.5 kcal mol^-1 or more.
PhH-PhX Interactions
As mentioned above, the calculated enthalpy (DDH_f) of the PhH-PhH interaction was satisfactory in comparison with interaction energies (or enthalpy) obtained with ab initio calculations and with the experiment. Thus, the DDH_f values were further calculated for various PhH-PhX systems in the case of the five initial arrangements. As the r_I, 2.75 A was chosen, because the optimized intermolecular distance ( r_o ) for X = H has been shown to be all 2.5 A except for P arrangement r_o =5.41A), regardless of the r_I [42], and because the sum -- (a half of thickness of a benzene ring) + (van der Waals radius of a H atom) -- is 2.9 A.
Figure 2 depicts a plot of the DDH_f against the DDH^t for initial arrangement P ( r_I = 2.75 A). Arrangements after geometry optimization (optimized geometries) have been found to resemble the initial arrangement P. For most pairs, the PhH-PhX interactions are slightly repulsive. Optimized intermolecular distances ( r_o ) change from 4.7 to 5.5 A. There exists no appreciable relationship between the DDH_f and DDH^t.
In the case of initial arrangement V_r ( r_I = 2.75 A), the PhH-PhX interactions are all attractive, ranging in enthalpies markedly from -0.5 to -5.5 kcal mol^-1. Optimized geometries resemble initial arrangement V_r except for (i) an unreasonable vertical arrangement[VP arrangement](X = Me) where two H atoms facing each other (an H atom of the methyl group in PhMe and an H atom in PhH) are penetrated to each other within the van der Waals radius; (ii) an unreasonable arrangement where the two benzene rings are aligned on the same plane ("coplane") in an edge-to-edge manner, with the two closest H atoms in the respective ring being penetrated to each other within the van der Waals radius[CP arrangement](X = Et); (iii) twisted CP -- another CP-like abnormal arrangement where the two benzene rings are twisted perpendicularly to each other[CP* arrangement](X = F). Furthermore, the r_o values alter from 1.7 to 2.5 A. There is no appreciable relationship between theDDH_f and DDH^t , even if the points for the above-mentioned unreasonable arrangements are excluded.
For initial arrangement V_b , the PhH-PhX interactions are repulsive for X = Me, F, and NO₂ , while attractive for the other Xs. Optimized geometries are similar to initial arrangement V_b except for (i) X = Et ( CP* arrangement) and (ii) X = OMe and NMe₂ (arrangements somewhat different from V_b arrangement). Moreover, the r_o values alter greatly from 1.7 to 5.6 A. The DDH_f bears no appreciable relationship to the DDH^t .
The DDH_f values for initial arrangement V_p (r_I = 2.75 A) are listed in Table 2, together with the DDH^t values for reference. It has been clarified that (i) optimized geometries are similar to initial arrangement V_p regardless of both the r_I and X and (ii) weak interactions between PhH and PhX are all attractive; the r_o values for each system remain almost unaltered with the r_I (ca. 2.4 to 2.5 A). A close relationship has been found to exist between the DDH_f and DDH^t (Figure 3), the correlation coefficient (r) being as large as 0.94.

Table 2 Initial arrangements V^a, optimized intermolecular distances ( r_o ), The DDH_f (r_I = 2.75 A), and the DDH^t [21]

PhX V_p V_m
r_o /A DDH_f r_o /A DDH_f DDH^t
kcal mol^-1 kcal mol^-1 kcal mol^-1
PhEt 2.51 -0.49 2.51 -0.50 -1.09
PhMe 2.51 -0.50 2.51 -0.50 -1.18
PhF 2.48 -0.64 2.48 -0.67 -1.26
PhCl 2.49 -0.61 2.48 -0.63 -1.29
PhH 2.50 -0.52 2.50 -0.52 -1.32
PhOMe 2.45 -0.80 2.50 -0.53 -1.51
PhNMe₂ 2.43 -0.80 2.51 -0.48 -1.58
PhNO₂ 2.45 -0.91 2.54 -1.24 -1.84
^a In the case of V_p arrangement, differences in the r_o between the r_I values of 2.75 and 3.05 A are all within 0.01 A, and differences in the DDH_f between the r_I values of 2.75 and 3.05 A are all within 0.01 kcal mol^-1.

PhX	V_p	V_m
r_o /A	DDH_f	r_o /A	DDH_f	DDH^t
kcal mol^-1	kcal mol^-1	kcal mol^-1
PhEt	2.51	-0.49	2.51	-0.50	-1.09
PhMe	2.51	-0.50	2.51	-0.50	-1.18
PhF	2.48	-0.64	2.48	-0.67	-1.26
PhCl	2.49	-0.61	2.48	-0.63	-1.29
PhH	2.50	-0.52	2.50	-0.52	-1.32
PhOMe	2.45	-0.80	2.50	-0.53	-1.51
PhNMe₂	2.43	-0.80	2.51	-0.48	-1.58
PhNO₂	2.45	-0.91	2.54	-1.24	-1.84

In the case of initial arrangement V_m (r_I = 2.75A), PhH-PhX interactions are also all attractive (Table 2). Optimized geometries are all similar to initial arrangement V_m . Further, the r_o values remain almost unaltered with the systems (ca. 2.5A). However, the DDH_f appears to have no appreciable relationship to the DDH^t (Figure 4).
The DDH_f for the electron-donating group (OMe or NMe₂) is more negative with V_p than with V_m , whereas the DDH_f for the NO₂ group is more negative with V_m than with V_p (Table 2). Further, the V_p and V_m arrangements are interconvertible simply by rotating the upper benzene ring in PhX (Figure 1b) around its C₆ axis by 60 degrees. Thus each system would be able to adopt either V_p or V_m arrangement, for which the DDH_f value is more negative. On this basis, of the DDH_f values for the two arrangements, the more negative value was selected for each system and plotted against the DDH^t values (Figure 5). A close relationship has also proved to exist between the DDH_f and DDH^t , with the r being 0.94. In addition, the differences between the experimental DDH^t values and the "DDH^t" values calculated from the equation shown in Figure 5 have turned out to be ca. 0.1 kcal mol^-1 or smaller.

Figure 2. A plot of the DDH_f optimized from initial arrangement P (r_I = 2.75 A) against the DDH^t. The numbers show substituents X of PhX: 1, ethyl; 2, methyl; 3, fluoro; 4, chloro; 5, H; 6, methoxy; 7, dimethylamino; 8, nitro group. All optimized geometries are similar to P arrangement (). In the case of r_I = 3.40 A, (1) the DDH_f values are equal to those for r_I = 2.75 A except for (i) X = NMe₂ (more positive by 0.07 kcal mol^-1) and (ii) X = Et (more negative by 2.07 kcal mol^-1) and (2) optimized geometries are similar to P arrangement except for X = Et (an arrangement similar to V_r).

Figure 3. A plot of the DDH_f optimized from initial arrangement V_p (r_I = 2.75 A) against the DDH^t. PhX molecules used are as in Figure 2. Optimized geometries are similar to V_p arrangement () except for X = MeOand NMe₂ (arrangements somewhat different from V_p ).

Figure 4. A plot of the DDH_f optimized from initial arrangement V_m (r_I = 2.75 A) against the DDH^t. PhX molecules used are as in Figure 2. All optimized geometries are similar to V_m arrangement.

Figure 5. A plot of the DDH_f optimized from initial arrangement V_m or V_p (r_I = 2.75 A) against the DDH^t. PhX molecules used are as in Figure 2. The DDH_f values are taken from that for either V_m or V_p arrangement: , from V_p; , from V_m.

The results described here demonstrate that it is the initial arrangement V_p or V_m that brings about close relationship between the theoretical interaction enthalpies (DDH_f) and the experimental ones (DDH^t).

4 Conclusion

The present work has clarified that enthalpies of weak interactions between benzene (PhH) and monosubstituted benzenes (PhX) can be evaluated by MOPAC93 semi-empirical MO method, the calculated enthalpies having close relationship to the experimental ones, when (i) PM3 method is used as a calculation method and (ii) the initial arrangement of the PhH-PhX system is of the vertical type, in which the lowest H atom at the para or meta position to X in PhX is located above the centroid of PhH. Further, for X = H , the optimized intermolecular distance and optimized geometry [43] are consistent with those obtained with the ab initio calculations.
Evaluation of intermolecular interactions by means of ab initio MO method requires incorporation of electron correlation, for which calculations of considerably large configuration interactions (CI) are necessary. On the other hand, in the case of MOPAC methods such as PM3, it would not be unreasonable to consider that the DDH_f (a measure of weak intermolecular interactions), essentially the difference between the DHs (heats of formation), is expected to be evaluated nearly as precisely as the DH themselves.
This is the first example in which theoretical enthalpies of the weak interactions between various pairs of aromatic molecules bear close correlation with the experimental enthalpies. The results presented here encourage us to evaluate enthalpies of weak interactions between benzene and a variety of molecules such as other substituted benzenes and alkenes by semi-empirical MO methods.

We thank Mr. Mitsuru Koike for valuable assistance.

References

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