Matta, Boyd. The quantum theory of atoms in molecules

372 13 Interactions Involving Metals: From ‘‘Chemical Categories’’ to QTAIM, and Backwards

References

1	G. N. Lewis, J. Am. Chem. Soc. 1916,		Cremer, E. Kraka, Angew. Chem.,
	38, 762–785.		Int. Engl. Ed., 1984, 23, 627–628.
2	R. F. W. Bader, Atoms in Molecules: A	15	V. G. Tsirelson, Acta Crystallogr. 2002,
	Quantum Theory, 1990, Oxford		B58, 632–639.
	University Press, Oxford.	16	Y. A. Abramov, Acta Crystallogr., 1997,
3	N. A. Richardson, Y. Xie, R. B. King,		A53, 264–272.
	H. F. Schaefer J. Phys. Chem. A, 2001,	17	P. Macchi, A. Sironi, Coord. Chem.
	105, 11134–11143.		Rev., 2003, 238/239, 383–412.
4	(a) A. D. Becke, J. Chem. Phys., 1993,	18	B. N. Figgis, Introduction to Ligand
	98, 5648–5652. (b) C. Lee, W. Yang,		Fields, 1964, J. Wiley and Sons, New
	R. G. Parr, Phys. Rev. B, 1988, 37,		York.
	785–789.	19	G. Frenking, N. Fro¨hlich, Chem. Rev.,
5	(a) A. D. Becke, Phys. Rev. A 1988,		2000, 100, 717–774.
	38, 3098–3100. (b) J. P. Perdew,	20	I. B. Bersuker, Electronic structure
	Phys. Rev. B, 1986, 33, 8822–		and properties of transition metal
	8824.		compounds, 1996, J. Wiley and Sons,
6	(a) R. F. W. Bader, H. Essen, J. Chem.		New York.
	Phys., 1984, 80, 1943–1960; (b) R. F. W.	21	J. Volker, G. Frenking, M. T. Reetz,
	Bader, P. J. MacDougall, C. D. H.		J. Am. Chem. Soc., 1994, 116,
	Lau, J. Am. Chem. Soc., 1984, 106,		8741–8753.
	1594–1605.	22	K. Morokuma, Acc. Chem. Res., 1977,
7	E. Clementi, C. Roetti, At. Data Nucl.		10, 294–300.
	Data Tables 1974, 14, 177–478.	23	(a) J. S. Dewar, Bull. Soc. Chim. Fr.
8	(a) H. Schmider, R. P. Sagar, V. H.		1951, 18, C71. (b) J. Chatt., L. A.
	Smith, J. Chem. Phys., 1991, 94,		Duncanson, J. Chem. Soc. 1953,
	8627–8629; (b) K. D. Sen, M. Slamet,		2939–2947.
	V. Sahni, Chem. Phys. Lett., 1993, 205,	24	G. Frenking, K. Wichmann, N.
	313–316.		Fro¨hlich, C. Loschen, M. Lein,
9	(a) R. P. Sagar, A. C. T., Ku, V. H.		J. Frunzke, V. M. Rayo´n, Coord.
	Smith, A. M. Simas, J. Chem. Phys.,		Chem. Rev. 2003, 238–239, 55–82.
	1988, 88, 4367–4374; (b) Z. Shi, R. J.	25	L. J. Farrugia, C. Evans, J. Phys.
	Boyd, J. Chem. Phys., 1988, 88,		Chem. A, 2005, 109, 8834–8848.
	4375–4377.	26	T. S. Koritsanszky, P. Coppens, Chem
10	(a) P. J. MacDougall, PhD Thesis,		Rev., 2001, 101, 1583–1627.
	McMaster University, 1990; (b) P. J.	27	F. Corte´s-Guzma´n, R. F. W. Bader,
	MacDougall, M. B. Hall, Trans. Am.		Coord. Chem. Rev., 2005, 249,
	Crystallogr. Ass., 1990, 26, 105–123;		633–662.
	(c) C. Bo, J. P. Sarasa, J. M. Poblet,	28	P. Macchi, D. M. Proserpio, A. Sironi,
	J Phys. Chem., 1993, 97, 6362–6366.		J. Am. Chem. Soc., 1998, 120,
11	(a) R. J. Gillespie, I. Hargittai, Allyn		13429–13435.
	and Bacon, Boston, MA, 1991;	29	H. B. Davis, F. W. B. Einstein, P. J.
	(b) R. W. F. Bader, G. L. Heard,		Glavina, T. Jones, R. K. Pomeroy,
	J. Chem. Phys.; 1999, 111, 8789–		P. Rushman, Organometallics. 1989, 8,
	8797.		1030–1039.
12	R. F. W. Bader, M. E. Stephens, J. Am.	30	D. A. Brown, W. J. Chambers, N. J.
	Chem. Soc., 1975, 97, 7391–7399.		Fitzpatrick, R. M. Rawilson, J. Chem.
13	X. Fradera, M. A. Austen, R. F. W.		Soc. A, 1971, 720–725.
	Bader, J. Phys. Chem. A, 1999, 103,	31	M. Martin, B. Rees, A. Mitschler, Acta
	304–314.		Crystallogr. 1982, B38, 6–15.
14	(a) D. Cremer, E. Kraka, Croat. Chem.	32	P. C. Leung, P. Coppens, Acta
	Acta, 1984, 57, 1259–1281; (b) D.		Crystallogr. 1983, B39, 535–542.

			References	373
33	(a) A. A. Low, K. L. Kunze, P. J.		(c) R. Stegmann, A. Neuhaus,
	MacDougall, M. B. Hall, Inorg. Chem.,		G. Frenking, J. Am. Chem. Soc., 1993,
	1991, 30, 1079–1086; (b) M. B. Hall,		115, 11930–11938.
	Electron Distribution and the Chemical	48	P. Macchi, D. M. Proserpio, A. Sironi,
	Bond, P. Coppens, M. B. Hall Editors,		J. Am. Chem. Soc., 1998, 120,
	Plenum Press, New York, 1982,		1447–1455.
	p. 205.	49	W. Scherer, G. Eickerling,
34	(a) R. Bianchi, G. Gervasio, D.		D. Shorokhov, E. Gullo, G. S.
	Marabello, Helv. Chim. Acta, 2001, 84,		McGrady, P. Sirsch, New J. Chem.
	722–734; (b) R. Bianchi, G. Gervasio,		2006, 30, 309–312.
	D. Marabello, Acta Crystallogr. 2001,	50	T. A. Albright, R. Ho mann, J. C.
	B57, 638–645; (c) L. J. Farrugia,		Thibeault, D. L. Thorn, J. Am. Chem.
	Chem. Phys. Lett., 2005, 414, 122–126.		Soc., 1979, 101, 3801–3812.
35	(a) R. Bianchi, G. Gervasio,	51	R. H. Crabtree, Angew. Chem. Int. Ed.
	D. Marabello, Chem. Commun. 1998,		1993, 32, 789–805.
	1535–1536; (b) R. Bianchi,	52	G. J. Kubas, Metal Dihydrogen and
	G. Gervasio, D. Marabello, Inorg.		s-bond Complexes, 2001, Kluwer
	Chem. 2000, 39, 2360–2366.		Academic, New York.
36	L. J. Farrugia, P. R. Mallinson,	53	F. Maseras, A. Lledo´s, M. Costas, J. M.
	B. Stewart, Acta Crystallogr., 2003,		Poblet, Organometallics 1996, 15,
	B59, 234–247.		2947–2953.
37	W. Uhl, S. Melle, G. Frenking,	54	S. Dapprich, G. Frenking, Angew.
	M. Hartmann, Inorg. Chem., 2001, 40,		Chem., 1995, 34, 354–357.
	750–755.	55	(a) J. Toma`s, A. Lledo´s, Y. Jean,
38	G. Gervasio, R. Bianchi, D. Marabello,		Organometallics 1998, 17, 190–195;
	Chem. Phys. Lett., 2004, 387, 481–484.		(b) J. Toma`s, A. Lledo´s, Y. Jean,
39	E. Espinosa, I. Alkorta, J. Elguero,		Organometallics 1998, 17, 4932–
	E. Molins, J. Chem. Phys., 2002, 117,		4939.
	5529–5542.	56	P. L. A. Popelier, J. Logothetis,
40	P. Mori-Sanchez, A. M. Penda´s,		J. Organomet. Chem., 1998, 555,
	V. Luan˜a, J. Am. Chem. Soc. 2002,		101–111.
	124, 14721–14723.	57	W. Scherer, G. S. McGrady, Angew.
41	R. F. W. Bader, C. F. Matta, Inorg.		Chem. Int. Ed., 2004, 43, 1782–1806
	Chem., 2001, 40, 5603–5611.		and references therein.
42	However, with the recent experi-	58	W. Scherer, W. Hieringer, M. Spiegler,
	mentally constrained wave function		P. Sirsch, G. S. McGrady, A. J.
	approach (D. Jayatilaka, D. J. Grim-		Downs, A. Haalandd, B. Pedersen,
	wood, Acta Crystallogr. 2001, A57,		Chem. Commun., 1998, 2471–2472.
	76–86) the pair density is accessible.		The original electron density model
43	R. Ponec, G. Yuzhakov, M. R.		characterized by a weak bcp was
	Sundberg, J. Comput. Chem. 2005, 26,		afterwards revised and a TiaHb bond
	447–454.		path could not be anymore located.
44	R. F. W. Bader, C. F. Matta, F. Corte´s-		Analogously, on changing the level of
	Guzma´n, Organometallics, 2004, 23,		theory two di erent molecular graphs
	6253–6263.		are obtained.
45	M. Bo¨hme, T. Wagener, G. Frenking,	59	D. L. Lichtenberger, Organometallics
	J. Organomet. Chem. 1996, 520,		2003, 22, 1599–1602.
	31–43.	60	P. Macchi, D. Donghi, A. Sironi,
46	D. Cremer, E. Kraka, J. Am. Chem.		J. Am. Chem. Soc., 2005, 127,
	Soc., 1985, 107, 3800–3810.		16494–16504.
47	(a) U. Pidun, G. Frenking, Organo-	61	R. Ho mann, Angew. Chem., Int. Ed.
	metallics 1995, 14, 5325–5336; (b) G.		Engl., 1982, 21, 711–724.
	Frenking, U. Pidun, J. Chem. Soc.,	62	C. W. Bauschlicher, J. Chem. Phys.,
	Dalton Trans., 1997, 1653–1662.		1986, 84, 872–875.

374 13 Interactions Involving Metals: From ‘‘Chemical Categories’’ to QTAIM, and Backwards

63	L. Pauling, Z. S. Herman, J. Chem.	66	P. Macchi, L. Garlaschelli, S.
	Educ., 1984, 61, 582–587.		Martinengo, A. Sironi, J. Am. Chem.
64	C. Mealli, D. M. Proserpio,		Soc., 1999, 121, 10428–10429.
	J. Organomet. Chem., 1990, 386,	67	P. Macchi, L. Garlaschelli, A. Sironi,
	203–208.		J. Am. Chem. Soc., 2002, 124,
65	R. H. Summerville, R. Ho mann,		14173–14184.
	J. Am. Chem. Soc., 1979, 101,	68	O. Kluge, M. Finger, J. Reinhold,
	3821–3831.		Inorg. Chem., 2005, 44, 6494–6496.

375

14

Applications of the Quantum Theory of Atoms in Molecules in Organic Chemistry –

Charge Distribution, Conformational Analysis and Molecular Interactions

Jesu´s Herna´ndez-Trujillo, Fernando Corte´s-Guzma´n,

and Gabriel Cuevas

14.1 Introduction

This chapter deals with description of the structure and reactivity of several organic molecules by use of the quantum theory of atoms in molecules (QTAIM) [1]. For some topics, emphasis is put on the information that can be obtained from the bond critical points (BCPs) and the delocalization of the Fermi hole density, as accounted for by the delocalization index between two atoms. Relationships between these two types of descriptor are discussed to illustrate how the QTAIM can account for their known experimental chemical behavior. The rest of the chapter analyzes the 3JHH coupling constants of some aliphatic and aromatic molecules (Section 14.2), conformational processes including rotational barriers of XCH2CH2X molecules and the anomeric e ect of heterocyclohexanes (Section 14.3), and the electronic structure of aromatic molecules (Section 14.4). Finally, closing remarks from a global perspective are presented in Section 14.5.

14.2

Electron Delocalization

14.2.1

The Pair-density

Information about the electronic structure of a molecule in a given state is provided by the wavefunction Cðx1; x2 . . . ; xNÞ of the N-electron system, were x ¼ ðr; sÞ, represents the space and spin coordinates of an electron. According to the postulates of quantum mechanics, Cðx1; x2 . . . ; xNÞ, contains all the information that can be known about the system [2, 3]. As an alternative, characterization can also be performed in terms of the properties of the electron-density functions derived from the wavefunction. One of these provides the probability density for

376 14 Applications of the Quantum Theory of Atoms in Molecules in Organic Chemistry

finding any of the N electrons in a volume element that includes both its space and spin coordinates, dx1, irrespective of the position and spin of the remaining electrons:

ð

gðx1Þ ¼ N C ðx1; x2; . . . ; xN ÞCðx1; x2; . . . ; xN Þ dx2 . . . dxN ð1Þ

By means of an integration of this function over the spin coordinate s1 one obtains the one-density:

ð

rðr1Þ ¼ gðx1Þ ds1 ð2Þ

i.e. the probability density of finding an electron in a spatial volume element dr1. Integration of rðr1Þ over all space yields the number of electrons in the system, N. This scalar field is the electron density obtained experimentally from crystallographic experiments. Because the underlying theory of the topological properties of the electron density has been discussed in detail elsewhere [1], the rest of this subsection will be devoted to a brief description of the pair density, because its use for analysis of electron delocalization from the viewpoint of the QTAIM is spread throughout several papers. The pair density is defined by:

pðx1; x2Þ ¼ NðN 1Þ ð C ðx1; x2; . . . ; xN ÞCðx1; x2; . . . ; xN Þ dx3 . . . dxN ð3Þ 2

This function determines the probability of finding any two electrons in space– spin volume elements dx1 and dx2, irrespective of the spin and position of the remaining electrons. The corresponding spinless function is obtained when integration over the spin coordinates s1 and s2 is performed. Thus:

ð

Pðr1; r2Þ ¼ pðx1; x2Þ ds1 ds2 ð4Þ

represents the probability of any two electrons being one at r1 and the other at r2 simultaneously, irrespective of the position of the remaining electrons and of their spin. Because the molecular Hamiltonian involves only one-electron and two-electron operators, no higher than pair density functions are necessary to describe the interactions. Double integration of Pðr1; r2Þ yields the number of electron pairs of the N-electron system. From these equations, rðr1Þ can be obtained from Pðr1; r2Þ by integration as follows:

rðr1Þ ¼		2		ð Pðr1; r2Þ dr2	ð5Þ
	N		1

14.2 Electron Delocalization 377

The one-density function can be expressed in terms of spin contributions:

rðr1Þ ¼ raðr1Þ þ rb ðr1Þ

ð6Þ

in which the spin state of the electrons is given by the spin functions a and b. The spin density can also be defined as the di erence raðr1Þ rb ðr2Þ, a scalar field that accounts for the excess of spin up ðaÞ over spin down ðbÞ contributions to the one-density.

Pðr1; r2Þ can also be resolved into its spin contributions by integration of pðx1; x2Þ over all spin possibilities:

Pðr1; r2Þ ¼ Paaðr1; r2Þ þ Pab ðr1; r2Þ þ Pbaðr1; r2Þ þ Pbb ðr1; r2Þ

ð7Þ

For example, Pab ðr1; r2Þ is the probability density of having an electron at dr1 with spin a and another at dr2 with spin b. In addition, the conditional probability of finding an electron at r1 if another electron is at r2, irrespective of the position of the remaining electrons, is obtained from the pair density:

P2ðr1; r2Þ

¼

r

r

Þ½

1

þ

h

r

; r

Þ&

ð

8

Þ

rðr1Þ

ð

2

ð

1

2

The term hðr1; r2Þ is a distribution function determined by inclusion of Coulomb correlation and, to a greater extent, by the spin distribution of the electrons, as required by the Pauli exclusion principle. In the absence of Coulomb correlation, exchange correlation is the only one present and hx ðr1; r2Þ is called the Fermi hole density in which the subindex x emphasizes its exchange correlation dependence. This density vanishes for the contributions of a–b interactions, because the motion of two electrons with di erent spin is uncorrelated and is di erent from zero for two electrons with the same spin. At the restricted Hartree– Fock level (RHF), the Fermi hole density for a electrons is [4]:

a		Xi; j	fi ðr1Þfj ðr2Þfjðr1Þfiðr2Þ
hx	ðr1; r2Þ ¼			:	ð9Þ
			raðr1Þ

Double integration of the product raðr1Þhx aðr1; r2Þ over the basin of a given atom A defined by the QTAIM provides a measure of the number of electrons located on that atom, whereas double integration over the basins of two di erent atoms A and B in the molecule, not necessarily sharing an interatomic surface, accounts for the number of electrons shared between them. In this manner, localization and delocalization indices lðAÞ and dðA; BÞ, respectively, are defined by [5]:

lðAÞ ¼ Xhi1=2hj1=2SijðAÞ2; dðA; BÞ ¼ 2 Xhi1=2hj1=2SijðAÞSijðBÞ ð10Þ

i; j i; j

378 14 Applications of the Quantum Theory of Atoms in Molecules in Organic Chemistry

In these definitions, hi and hj denote the occupation numbers of natural orbitals fi and fj and SijðAÞ is the overlap integral of fi and fj over the basin of atom A. For RHF, the summations run over the occupied orbitals, hi ¼ hj ¼ 2, and reduce to [6]:

X X

lðAÞ ¼ 2 SijðAÞ2; dðA; BÞ ¼ 4 SijðAÞSijðBÞ ð11Þ

i; j i; j

These latter expressions have also been used in the Kohn–Sham approximation (KS) and it has been found that the corresponding numerical values are similar to those found with RHF [7], although its use can be criticized because of the N-representability problem of KS orbitals.

The localization and delocalization indices add up to the atomic population,

P

NðAÞ ¼ lðAÞ þ 12 B0A dðA; BÞ, and the total number of electrons of atom A that are delocalized is DðAÞ ¼ NðAÞ lðAÞ. The delocalization of the Fermi hole has been used to explain several chemical e ects. Of these, two important examples are:

1.their relationship with the Lewis model of electron pairs [6]; and

2.their ability to provide a physical foundation for the VSERP model of molecular geometry [8].

In addition, the delocalization indices and the properties of the one-density, which in what follows will be referred to as the electron density rðrÞ, enable appropriate characterization of the chemical bond, as is illustrated in the examples discussed in this chapter.

14.2.2

3JHH Coupling Constants and Electron Delocalization

One direct application of the delocalization index is in the study of coupling constants between vicinal H atoms of organic molecules. Beginning from the Hamiltonian for the electron interactions in the field of nuclei with magnetic moments, four main contributions are identified for the nuclear–nuclear spin-coupling constants – the Fermi contact, the paramagnetic spin–orbit, the spin–dipolar, and the diamagnetic spin–orbit interactions [9]. Of these, the Fermi contact contribution has frequently been found to be dominant, for example for protons not directly bonded to each other. For this example, by use of molecular orbital theory [10], it has been found that the coupling constant, J, for nuclei with coordinates

Rn and Rn0 is proportional to										i; j fi ðRnÞfj ðRn 0 ÞfjðRnÞfiðRn 0 Þ and, from Eq. (9),
to the product ra			ð	R		nÞ	ð	n	n 0 Þ		. J has, moreover, also been written in terms of
						ha	R		; RP
Pab Rn; Rn 0	Þ	Paa			ð	Rn; Rn0		Þ	, the excess number of b over a electrons at Rn 0 given
ð

that there is an a electron at Rn; this di erence can be related to orbital-based definitions of bond order [10]. Consequently, the Fermi contact contribution to 3JHH results mainly from the coupling of nuclear spins mediated by the electronic

14.2 Electron Delocalization 379

Fig. 14.1 Relationship between d(H, H0) and the HaCaCaH0 dihedral angle for ethane. QCISD/6-311G(d,p) wavefunctions used.

spins and, to a great extent, from the values of the s-type orbitals at the nuclear positions. From this discussion, it is reasonable to assume proportionality between 3JHH 0 and d(H, H0) [11], an assumption that is supported by the empirical correlations described below.

An important feature of the Fermi contact contribution is its sensitivity to geometric changes. This is illustrated by the well-known Karplus-type behavior [12]

of 3JHH0 of ethane as a function of the HaCaCaH dihedral angle, f – the 3JHH 0 values obtained from valence bond calculations were successfully subject to non-

linear fitting of type cos2 f. Interestingly, the corresponding values of d(H, H0) shown in Fig. 14.1 have the same trend as a function of f [4]. Accordingly, nonlinear fit of d(H, H0) against f of the type dðH; H 0Þ ¼ A cos2 f þ B can be achieved. Following Karplus [12], the fitting yields the data given in Table 14.1 for f less than or greater than 90 degrees. From this it can be concluded that electron delocalization is responsible for the conformational behavior of 3JHH0 of ethane.

This conclusion can be extended beyond ethane, as can be observed from the good empirical correlations reported [11] between 3JHH and d(H, H0), with correlation coe cients, r2, of 0.990 and 0.975, for several polybenzenoid and aliphatic hydrocarbons, respectively, using RHF/6-31G(d,p) wavefunctions. These results

Table 14.1	Nonlinear fit of the form dðH; H0Þ ¼ A cos2 f þ B for
dðH; H0Þ for the dihedral angle, f, of ethane.[a]
f	A	B	Correlation coe cient
0–90	0.0067	0.0016	0.9997
90–180	0.0079	0.0018	0.9988

a QCISD/6-311þþG(2d,2p) wavefunctions used.

380 14 Applications of the Quantum Theory of Atoms in Molecules in Organic Chemistry

support the conclusion that proton–proton vicinal coupling constants are a consequence of electron delocalization and exemplify how the Fermi exchange density contains information related to nonbonded atoms.

It is interesting to note that satisfactory empirical correlations have also been reported for FaF coupling constants [13], even though higher angular momentum functions included on the F atoms involve larger contributions from dipolar terms. This probably works because of error cancellation between other than Fermi contact terms, with the latter remaining as the most important contribution, or because of the very tight electron density in the neighborhood of the F nuclei. Further study is necessary to clarify these points.

14.3

Conformational Equilibria

14.3.1

Rotational barriers

14.3.1.1 Rotational Barrier of Ethane

Internal rotation about single bonds is discussed in many textbooks as a relevant feature of molecular conformation and its dynamics, and ethane is often considered in theoretical work as a prototype for analysis of this phenomenon. After Kemp and Pitzer [14] proposed the existence of a rotational barrier in ethane in 1937, several explanations have been invoked [15]; but the discussion of its origin is far from complete.

Of several experimental measurements reported [16, 17], a value of 2.93 kcal mol 1 [18] can be regarded as the most accepted value for the barrier and 2.79 kcal mol 1 seems to be a reasonable best estimate from CCSD(T)/6- 311G(3df,2p) calculations [19]. The barrier is usually attributed to a steric e ect, because of repulsion between CaH bonds or between vicinal H atoms in the eclipsed conformation [20]. A popular explanation of the barrier has been given with the use of natural bond orbital analysis in which hyperconjugation is regarded as the source of the conformational preference of the molecule [21–23] (by means of sCaH ! s CaH vicinal interactions) although such a model has been criticized [24].

Bader et al. [25] provided an alternative explanation based on the QTAIM using the energy partitioning included in the molecular Hamiltonian without resort to any hypothetical reference state. In this partition, the contributions to the potential energy of the molecule are the attractive (nuclear–electron) and repulsive (nuclear–nuclear plus electron–electron) terms. Accordingly, the barrier results from the decrease of magnitude of the attractive interaction in the eclipsed conformation, despite the accompanying reduction in the repulsion contribution.

The origin of the rotational barrier in ethane can be traced back to the behavior of the atomic energies along the barrier, one of the main results being that the energy of a carbon atom has the same trend as the barrier (Fig. 14.2). The carbon

14.3 Conformational Equilibria 381

Fig. 14.2 (a) Molecular and atomic energies of carbon and hydrogen atoms along the rotational barrier of ethane. (b) Localization and delocalization indices for ethane. QCISD/6-311þþG(2d,2p) wavefunctions used. Values relative to the staggered conformer.

atoms are more stable for the staggered conformer by 5.1 kcal mol 1 and H atoms are more stable in the eclipsed conformation with a total contribution to the barrier of 2.28 kcal mol 1. The change of the potential energy of the carbon atom is mainly because of the nuclear-electron contribution of the carbon atom, þ62.06 kcal mol 1, which indicates a decrease in the magnitude of attraction, compared with a reduction of 57.09 kcal mol 1 for the atomic repulsive contribution. These values are in accordance with both the molecular results reported by Bader et al. [25] and with the Ehrenfest force – the force exerted on the electron distribution by the other electrons and the nuclei [26]. It was found there is a decrease in the attractive Ehrenfest force acting on the C atoms when rotation from the staggered to the eclipsed conformation occurs. Because in these two conformations the Feynman forces acting on the nuclei vanish, the Ehrenfest force explains the barrier. This evidence makes it unnecessary to use any repulsive model to explain the rotational barrier (for example one based on Pauli repulsions, for which no force operator can be defined) [27].

Small but decisive charge redistribution accompanies all these energy changes. The electron population of the carbon atom increases by 0.007 e (Table 14.2), as does the CaC bond distance (by 0.0141 A˚ ) during rotation from the staggered to the eclipsed conformer. As a consequence, rbðrÞ at the CaC BCP decreases by 0.007 au. The electronic energy density, HðrÞ ¼ GðrÞ þ VðrÞ, the sum of the positive definite kinetic energy density plus the potential energy density, is useful for characterization of chemical bonding [28]. HbðrÞ at the CaC BCP of ethane becomes less negative by 0.011 au, indicative of lower CaC bond stability in the eclipsed conformation. Additional evidence of the charge polarization occurring