Analysis of the Problems regarding the Role of p Electrons in Benzene, and a Solution Based on Mechanics

 

                                               Kojin Harayama*

                          Yatsu Institute of Molecular Information

(This article was published in Res. Devel. Mol. Structure, 1, 93-115 (2002).)

 

ABSTRACT:     The controversial problems concerning the role of p electrons in benzene were analyzed. It was found that the problems stemmed from four erroneous approaches: (1) To obtain the cyclohexatriene-like electronic structure, the geometry was distorted where the effect of the electrostatic potentials toward the energy of p electrons was not taken into account. (2) The energy of p electrons was defined without examining whether the defined energy actually expresses ‘p energy’. (3) To obtain the wave functions for the Kekulé electronic structure, unstationary wave functions were used. (4) The vertical and adiabatic processes in conjugation were confusedly considered.

To solve those problems, we first ascertained that the so-called ‘p energy’ is the kinetic energy of p electrons, and then we adopted mechanical approaches: (1) to know the preference of p electrons without changing the geometry, the partial derivatives of every energy component with respect to the b2u coordinate and p bond order were examined. (2) The energy components, including various combinations that may express ‘p energy’, were obtained as functions of the p bond order to know the optimized distribution of p electrons for them in benzene and distorted benzene. In the above approaches, (3) the constrained Hartree-Fock method was used to determine the stationary wave function of the targeted p electronic structure.

We applied these methods to interpret the preferable distribution of p electrons in both normal (D6h) and distorted benzene (D3h). We found no evidence that the uniform distribution of p electrons in benzene is unstable. Further, it was confirmed that p electrons in distorted benzene have a preference towards more uniform delocalization. Thus, we concluded that the p electrons of benzene are the most stable at the equivalent delocalization on all C-C bonds and that the p electrons behave as the corrective source of distorted benzene towards regular hexagonal benzene. Consequently, the traditional concept of p electrons in benzene was proved to be valid.

 

Key words: benzene; stability of p electrons; definition of p energy; kinetic energy of p electrons; role of p electrons; constrained Hartree-Fock equation;

 

Introduction

Very fundamental is that any scientific discussion must be based on explicitly defied common technical terms and on the results achieved by the correct methodology. In chemistry, phenomena have often been discussed without regard to this requirement to result in considerable confusion and vain, time-consuming efforts. The recent dispute on the role of p electrons in benzene is a typical example. Since the subject concerns conjugation/aromaticity that is a very basic concept in organic chemistry, confusion arises not only among researchers but also in the undergraduate education media. Thus the situation has been severe. This article will review the problem and point out what has been wrong. At the same time, we propose a method to solve subtle problems in organic chemistry. The general idea of our approach is that chemical phenomena are interpreted in terms of mechanics.

The p electrons in benzene have long been believed, by both organic and theoretical chemists [1, 2], to play a decisive role in determining the characteristics of benzene: particular stability with uniform delocalization of the cyclic 6p electron system has been a firmly established idea and is used to explain the properties of various aromatic compounds. However, spectroscopical and computational investigations concluded that the delocalized p system of benzene is not stable and is not responsible for the D6h geometry. This conclusion has, of course, been refuted. To make a long story short, the following background is reported.

Benzene has two fundamental modes of vibration that keep benzene's planarity: a1g and b2u modes. The former is a vibration that keeps the D6h symmetry and the latter is that between the D6h and D3h geometries. More than four decades ago, Hornig first pointed out the importance of the stable Kekulé form (D3h) to explain the abnormally low force constant concerning the b2u vibrations. Namely, the potential energy toward the Kekulé structure changes slowly, giving rise to a low force constant [3]. Here, it should be pointed out that the D3h-geometry-distorted benzene is taken for granted as the Kekulé form: there already was confusion between electronic and geometrical structures.

In 1961, Berry questioned the stability of uniformly distributed p electrons. He claimed that the p electrons are not the major source of stability for the regular hexagon, and the p electrons might well have a lower energy if the ring approaches a cyclohexatriene-like structure [4]. In his article, he referred to the results by Snyder who performed Hückel model calculations of p and s contributions to the energy of the b2u distortion of benzene [5].

More than twenty years after Berry’s claim, Hiberty and Shaik and their co-workers computationally examined the same question, where they focused on the changes of ‘p and s energies’ based on ab initio molecular orbital theories (the word in single quotation marks indicates the key words that are used without explicitly explaining the definition)when the D6h benzene is deformed to D3h [6-8]. They repeatedly asserted that electronic delocalization of p electrons is a byproduct of the s-imposed geometric symmetry and not a ‘driving force’ by itself, but they did not show the values of such driving forces.

Their results have been supported by or are coincident with those by a number of other groups [9-14]. However, Baird showed a case in which the conclusion was reversed [15], which was immediately rejected [16]. Meanwhile, Aihara confirmed that the equalization of the CC bond lengths in benzene was caused primarily by a high degree of aromaticity, and benzene still is highly aromatic even if it is deformed artificially [17].

Back in 1979, Kollmar proposed a method for direct calculation of resonance energy by substituting p MOs that are localized in another system. According to the method, a considerable amount of resonance energy is figured out in the D6h benzene [18].

Glendenig et al. also studied this problem [19]. They concluded that benzene favors a bond alternating geometry when its canonical p MOs are replaced by three localized ethylenic orbitals, revealing that delocalization is largely responsible for the equilibrium symmetric structure. In contrast, a s-p energy partitioning analysis suggests that the s framework of benzene is responsible for the symmetric structure, the p system preferring a distorted geometry. However, it is also shown that the ‘p-energy component’ contains a sizable and strongly geometry-dependent contribution from the localized wave function. Hiberty et al. have tried to remove geometry-dependent contribution to the ‘p energy component’ [8, 16, 20]. However, as will be pointed out, their methods are irrelevant to such removal.

Ichikawa and Kagawa showed that there are countless numbers of reaction coordinates for distortion that either lower or increase the energy of p electrons. The minimum-energy path (MEP) was found for the distortion from the D6h to D3h benzene and was not the one previously investigated [21]. Along such an MEP, the energy of p electrons was traced. It was found that the energy was the most stable at the D6h symmetry. Considering the molecular virial theorem [22], however, they emphasized that to distort the molecular geometry is not right way in order to perturb the p-electronic structure.

In the above discussion, four problems have been overlooked. To understand these points, it may be effective to show the energy components that build up the total energy (E). They are most simply illustrated by using the H2 system, where all kinds of energy components are involved, as shown in Fig. 1. <T>, <VeN>, <Vee>, and VN are the expectation values of the kinetic energy of electrons, one-electron potential energy (attractive interaction between electrons and nuclei), two-electron potential energy (repulsive interaction between electrons), and the classical internuclear repulsion energy. The angle brackets indicate that they are given as expectation values (i. e. quantum-mechanical quantities).

 

 

FIGURE 1. Mechanical components of molecular total energy. Where <T>, <VeN>, and <Vee> are the kinetic energy of electrons, the one-electron potential energy, and the two-electron potential energy, which are given as expectation values while VN is given as a classical quantity.

 

 

 

 

The four erroneous approaches are:

1.          The manner of the distribution of p electrons gives the conjugation/aromaticity. Thus if one wants to investigate the role of p electrons in benzene, only the vertical process must be examined. Namely, the geometry must not be changed because change in nuclear configuration affects the energy of p electrons. However, to make an alternating p electronic structure, the geometry has been distorted [6-8, 10, 12-16, 19, 20] (hereafter we call this the geometry-distortion method). Considerable influence caused by the nuclear displacement is exerted on the p electrons through the potential fields. Removal of the effect of nuclear displacement has been tried by choosing nuclear configurations that keep the internuclear repulsion energy constant between benzene and distorted benzene [8,16] and by a new method, the quasiclassical (QC) method without anti-symmetrized wave function[20]. It should be pointed outthat keeping internuclear repulsion energy (VN in Fig. 1) constant is irrelevant to keeping the electrostatic fields towards p electrons constant. As will be demonstrated, change in nuclear configuration surely varies the electrostatic fields that p electrons receive (not only through <VeN>p but also through <Vee>p,and <Vee>ps in Eq. 2). Another problem of the geometry-distortion method is that benzene still is highly aromatic even if it is distorted artificially [17]: the b2u distorted benzene does not correspond to the Kekulé electronic structure.

2.          The energy of p electrons that is the role-playing term has been defined without examining whether or not it is qualified to express ‘p energy’ [6-10, 12-16, 18-20]. This is important because simple partitioning due to the concept of ‘p energy’ was shown not to express actual ‘p energy’ [23, 24]. Further, force may have a clear mechanical meaning but is used without quantification [8,13,20].

3.          The wave functions for a Kekulé electronic structure, were obtained by another system are used [18,19]. The problem of this method is that such wave functions are not stationary concerning the Kekulé electronic structure. The reason for this requirement will be seen in the later section.

4.          The geometry of benzene apparently ranges from benzene to distorted benzene since benzene vibrates, and should be regarded as a completely different system from the quantum mechanical viewpoint based on the Born-Oppenheimer approximation [25]. However, most chemists accept distorted benzene as benzene. This chemical idea often drives one to make a mishmash discussion concerning the role of p electrons in the vertical and adiabatic processes in conjugation. These problems will be analyzed in detail and their solutions are given in the following sections.

 

Determination of ‘p energy’ in unsaturated hydrocarbons

Let us consider the case of conjugated unsaturated hydrocarbons in which all atoms are placed on one molecular plane such as that of benzene. The role of p electrons has been discussed in connection with their energy. Although the concept of a p electron need not be defined, its energy is conceptually vague. The reason is that, as will be discussed, the environment does not affect the image of a p electron but its energy. First, let us examine what ‘p energy’ is, based on the literature knowledge.

‘p Energy’ appeared when the Hückel molecular orbital theory was proposed [26] and has been regarded as the energy obtained by the Hückel, Paiser-Parr-Pople (PPP) [27] or its related molecular orbital theories [28]. The vagueness of ‘p energy’ stems from the fact that the Hamiltonian of the Hückel theory is not mechanically or explicitly defined while those of PPP and its related-theories contain abbreviations and parameters to coincide the calculated results with the observed values. Namely, their Hamiltonians consist of vague operators. Despite this fact, ‘p energy’ seems to be generally accepted as the energy when p electrons move in the potentials from p and s electrons and nuclei of the whole system. Here, we regard ‘p energy’ as the energy by the Hückel theory that may represent those by PPP and its related theories in that they share a common feature that the energies in linear polyenes are linearly additive with respect to the ethylenic unit: this linearity is the most important characteristic of ‘p energy’ and “aromaticity” has been defined based on this additivity in polyenes [28,29].

It should be noted that the linear additivity of ‘p energy’ indicates that ‘p energy’ includes information of internuclear repulsion energy, in spite of the fact that such Hamiltonians are made up by considering the circumstance of p electrons only, where the contribution of nuclear repulsion energy to ‘p energy’ has not been taken into account. *

 

Partitioning of the total energy into mechanical components.  The preference of p electrons may be determined by considering its energetic advantage. Thus, correct determination of ‘p energy’ in terms of the ab initio MO theory is a must. In the Hartree-Fock-Roothaan-Hall self-consistent field (SCF) molecular orbital theory [30-33], every mechanical component can be obtained as an expectation value of its quantum-mechanical operator.

The molecular total energy (E) under the Born-Oppenheimer approximation [25] is the sum of the electronic energy (Eel) and the internuclear repulsion energy (VN) (Eq. 1), where Eel can be expressed in terms of four energy components that have clear mechanical meanings as Fig. 1 shows.

 

           　                         (1)

 

Eel is broken down as small as,

          (2)

This is possible because the SCF electronic wave function (Y) is a Slater determinant of molecular orbitals (MOs: yi) expanded in terms of a linear combination of AOs (cr) and since, in a planar unsaturated hydrocarbon, the total bond order (Prs) is the sum of p- (Prsp) and s-bond orders (Prss). Concretely, the partitioned energies of the p electrons for a closed-shell system are obtained as expectation values by a general expression,

 

                          (3)


where Crpi is the coefficient of cr at the ith p MO. X is the operator of the corresponding energy (<T>, <VeN>, or <Vee>), while Xrs is its element in its matrix form.

 

Are the partitioned energy components usable?     In 1962 Ruedenberg et al. carried out the partitioning of the total energy into its components when they studied the covalent bond formation in the H+ + H system [34]. The approach to chemical phenomena on this line extends to Hund’s rule [35], Jahn-Teller effect [36], reaction mechanisms [37], p-bond formation [38] steric effect [39], and already to conjugation [38, 39, 41-43]and aromaticity [23, 24, 44]. However, special attention must be paid when using this method because the energy components are expectation values. The expectation values sensitively fluctuate depending on the molecular geometry, the adopted basis set, the scale factor of the wave function, and anything that affects the size of the considered space. This is because any MO theory is based on a truncated Hilbert space. (The geometry dependence will be discussed in the next section.) If a geometry is optimized with respect to any nuclear coordinate, the virial ratio, <V>/<T>, must be exactly -2. Usual basis sets do not satisfy this requirement, while rescaling of the wave function using a universal scale factor z does it. This is possible because the degree of the kinetic-energy operator is -2 with respect to the coordinate while that of the potential energy is -1.

Pedersen and Morokuma have expressed the changes in E, <T>, and <V> in a power series of scale error, g(= z -1) as [45],

                  (4)

which show that the deviation of the total energy is in the second-order of g, whereas those of <T> and <V> are in the first-order. Since rescaling of the wave function means changing the space that expresses the considered system, the energy components might include a large amount of deviation in the unscaled wave function and also in the wave function of the incompletely optimized geometry because of the reason given in the next section. The actual degree of deviation was reported to be up to 103 times more than that of the total energy [45-47]. This may preclude putting any meaning to the calculated expectation values.

However, Ichikawa et al., after examining the effect of the scale factor on the calculated expectation values, found that the calculated energy components can be reliable only if the differences are considered in both vertical [48] and adiabatic processes [21]. Some other related papers show that the above idea was acceptable [49, 50]. Therefore, if conditions of the SCF convergence and geometry optimization are carefully set* and only if the differences of energy are considered, such energy-components are usable in the analysis of chemical phenomena.

 

Conceptual partitioning of the total energy into p and s energies.     To study the role of p electrons in benzene, the energy of p electrons must be defined. One method is to partition E into the p-energy part (Ep) and the skeletal part (Es') where Ep depends on the concept of ‘p energy’.

                    (5)

If one adopts the traditional concept of ‘p energy’ as the energy of p electrons that move in the fields of other p electrons, nuclei, and the s electrons that are fixed around the nuclei, Ep and E’s may be expressed, as Shaik and Hiberty and their coworkers did [7, 8, 16], as,

        (6)

     (7)

where, <Vp-s>(= <Vs-p>) ­is the interaction energy between p and s electrons.

However, s electrons actually shift to avoid the average repulsion from p electrons; the s energy part must take this into account. Thus, the following equations are obtained.

                        (8)

              (9)

The term Epel of Eq. 8 can be called “p-electronic energy” according to the literal meaning of “electronic energy”. In ab initio electronic theories, Epel is the only faithful expression of the idea of ‘p energy’, the energy of p electrons that move in the potentials from p and s electrons and nuclei of the whole system. However, it has been shown that Epel fails to satisfy the linear additivity requirement as ‘p energy’ [23, 24].

 

Comparison of ‘p energy’ with partitioned energies.  A notable character of ‘p energy’ is its linear additivity in linear polyenes concerning the ethylenic unit. This fact has been used in the modern definition of aromaticity [28, 29]. “Being linear” indicates that ‘p energy’ includes the internuclear repulsion energy that only the distance between any two nuclei gives.* There is a clear contradiction between the concept of ‘p energy’ and its observed character.

Jug et al. also have been aware of the necessity that ‘p energy’ must include some part of the nuclear repulsion energy and proposed a partitioning scheme as [13],

        (10)

where, ZA, RAB, nAs, and nAp are the number of nuclear charge of atom A, the distance between atoms A and B, the number of s electrons of atom A, and that of p electrons.

They stated “each negative charge of an electron has a corresponding positive charge in a molecule”. Intuitively, such an idea may be understandable but is apparently erroneous if one is aware of the fact that electrons are indistinguishable. Fundamentally, VN is not given by the electrostatic energy of electrons but of nuclei. Ridiculous results are easily seen if all electrons are taken out: according to Eq. 10, the internuclear repulsion energy disappears even if nuclei exist. Thus, the idea of incorporation of the nuclear repulsion energy into ‘p energy’ must be abandoned.

In 1985, Ichikawa and Ebisawa compared Hückel energy with those partitioned energies and found that the kinetic energies (<T>s  as well as <T>p) have an almost exact linear correlation with Hückel energies [23]. Any other term or combination with “physical meaning” including Epel fails to satisfy this linearity. They also showed this relationship holds in a variety of conjugated hydrocarbons [24]. Thus, they insisted that the so-called ‘p energy’ or Hückel energy must be the kinetic energy of p electrons. Their recent work showed that the total energies of polyenes have the character of linear additivity and the kinetic energy of p electrons reflects the total energy through the viral theorem as will be shown in the next section [51]. This conclusion is very significant because the kinetic energy of p electrons can be a measure of aromaticity and because the kinetic energy of p electrons is not a compound quantity but a basic term that has a clear mechanical meaning.

 

‘p Energy’ as the kinetic energy of p electrons.  Why ‘p energy’ includes information on the nuclear configuration of the system has been an unanswered question. Recently, Ichikawa et al. examined this [50-52]. They showed that (1) ‘p energy’ or Hückel energy is a quantitative expression of the kinetic energy of free-moving electrons in a one-dimensional box given by CH=CH units. (2) Such kinetic energy is linear-additive by nature with respect to the unit. (3) The ratio of the kinetic energy of p electrons to the total kinetic energy in linear polyenes is constant. (4) Thus, the kinetic energy of p electrons includes information on the nuclear configuration through the virial theorem. Namely, the relationship EHückel ∝ <T>p∝ <T> = -E = <V>/2 holds.

The key to understanding such a relationship is to understand the constancy of the ratio <T>p/<T> or the linear property of the kinetic energy of free-moving electrons in a linear one-dimensional box. The findings by Ichikawa et al. are that the Hückel energy as well as the kinetic energy of particles (electrons) that move in a one-dimensional box has the property of linear additivity with respect to the unit length. Although such linearity is not perfect, deviation from perfect linearity was found to be limited within a small number of units. This quasi-linear property makes the constancy of <T>p/<T> understandable, since <T> (= -E) has a linear additivity with respect to the ethylene unit; inclusion of the internuclear repulsion energy in EHückel is rationalized [51]. Thus, we may fairly say that the so-called Hückel energy or ‘p energy’ is a quantitative quantity of the kinetic energy of free-moving electrons as well as that of p electrons in polyene.

 

The linearity of the kinetic energy of p electrons in vertical and adiabatic conjugations.     Since measurement of aromaticity is carried out in an adiabatic process, the linear additivity of the kinetic energy of p electrons in the adiabatic process is also required to hold the property as a measure of aromaticity. The linear additivities of the kinetic energy of p electrons have been shown in both vertical and adiabatic processes where such relationships were obtained by varying n in the system CH2=CH-


 

FIGURE 2.　Linearity of the kinetic energy of p electrons concerning number of conjugation (n) in the C10H12 system.

 

(CH=CH)n-CH=CH2 [23, 24, 51]. However, the influence of the s skeleton on the kinetic energy of p electrons must be removed. This leads us to use the unsaturated hydrocarbon system with the same number of carbon atoms, where only the number of conjugations between double bonds is varied.

Thus, we seek the relationship between n, the number of conjugations, and the kinetic energy of p electrons in the linear C10H12 system. Figure 2 shows the energy lowering of the kinetic energy of p electrons in both vertical and adiabatic processes. The basis set is STO-6G [53,54].

The average lowering by a single ethylene unit is given 82.0 (vertical) and 68.7 kJ/mol (adiabatic). Although the universal scaling is not applied, deviations are as small as 2.0% (vertical) and 1.7% (adiabatic). Thus, the kinetic energy of p electrons can be a measure of aromaticity in both vertical and adiabatic conjugations.

 

Geometry dependence of partitioned energies

p Electrons in benzene have their clear image. Such an image is not varied even if the geometry of benzene is changed. This often leads to a misunderstanding that the energy of p electrons may be changed not by a change in the nuclear configuration but by that in the electronic structure. The reality is, however, that the energy of p electrons is greatly varied by a small change in the geometry. This is simply because the potential fields are varied: the energy of, say, a stone on the seashore is different from that when it is brought to the top of a mountain, the latter having a large (potential) energy.

The total energy is the sum of all energy components. Generally, the change in the total energy by geometry distortion is small. This also gives rise to an illusion on the energy components. First, mathematically we will discuss the difference in geometry dependence between the total energy (eigenvalue) and partitioned energies (expectation values).

The total energy (E) is given as a functional,

                                  (11)

where C and R represent the LCAO coefficients and a set of nuclear coordinates. Under the variational conditions, W/C=0, the direct differentiation of Eq. 11 gives

                         (12)

Thus, it is understood that the derivative of the total energy with respect to a nuclear coordinate (Ra) is simply the force along the coordinate, where there is no contribution from C. If the geometry is optimized, W/Ra is null as well. These facts indicate the change in the total energy induced by the geometry change around the optimized geometry may be small.

Let us consider the case where the total energy is partitioned into s (E’s) and p parts (Ep = Epelor EpShaik). Such energy is also a functional concerning the nuclear coordinate. The partial derivative of Ep with respect to Ra is given similarly as,

                           (13)

where (Ep /Ra)0 is the derivative under the condition that the coefficients are regarded as constants. Since Y is not an eigenfunction of the p-electron operator, all terms in Eq. 13 are not null. On change in the geometry (Ra), Ep includes the effects from the coefficients C of both s and p electrons, in addition to direct change in Ep (Ep /R).

To substantiate the above discussion, let us take the case when bond lengths between adjacent carbon atoms are changed by 0.01Å along the b2u coordinate. Table I shows changes of all energy components by STO-6G: The total energy (E) varies by only 0.6 kJ/mol. Look at the change of <VeN>p. This term, which is given by change of the nuclear configuration, gives rise to 4.1 kJ/mol, as much as 6.8 times more than that in the total energy. As a result, the changes of Epel and EpShaik are 5.0 and 3.0 times more than that in E. The changes in <Vee>p and <T>p are rather small, because they are mainly due to the change in the p electronic structure.

 

 

 

 

 

 

 

TABLE I

 Changes in Energy Components When D6h Benzene is Distorted to D3h Benzene along the b2u Coordinatea

term

benzeneb

distortedc

dif.d

E

-230.131181

-230.130948

0.6

Eel

-434.895341

-434.899241

-10.2

VN

204.764160

204.768293

10.9

<T>

229.990483

229.992533

5.4

<VeN>

-946.860745

-946.869539

-23.1

<Vee>

281.974922

281.977766

7.5

Epel e

-40.067251

-40.068398

-3.0

EpShaik f

-6.405036

-6.405731

-1.8

<T>p

7.471762

7.471929

0.4

<VeN>p

-85.535949

-85.537529

-4.1

<Vee>p

4.334722

4.334535

-0.5

Eselg

-394.828090

-394.830843

-7.2

<T>s

222.518721

222.520603

4.9

<VeN>s

-861.324796

-861.332010

-18.9

<Vee>s

210.315770

210.317897

5.6

2X<Vee>ps

67.324430

67.325334

2.4

E'sh

-190.063930

-190.062550

3.6

E'sShaik i)

-223.726145

-223.725217

2.4

aSTO-6G. bAu. Geometry-optimized benzene (D6h). The bond lengths of C-C and C-H are 1.38585 and 1.07867 Å. cAu. Displacement by 0.01Å along the b2u coordinate (D3h). dDifference in kJ/mol. ep Electronic energy (Eq. 8). f‘p Energy’ defined by eq 6. gs Electronic energy. hSkeletal energy (Eq. 9). iSkeletal energy by Eq. 7.

TABLE II

Changes in Energy Components for the Adiabatic Process with Constant Internuclear Repulsion Energya

term

benzeneb

distortedc

diff.d

E

-230.129758

-230.121619

21.4

Eel

-433.129000

-433.120864

21.4

VN

202.999242

202.999245

0.0

<T>

229.853050

229.914492

161.3

<VeN>

-943.318178

-943.333486

-40.2

<Vee>

280.336128

280.298130

-99.8

Epel e

-39.810960

-39.828767

-46.8

EpShaikf)

-6.3404505

-6.3595885

-50.3

<T>p

7.483757

7.489013

13.8

<VeN>p

-85.071045