A Model for the Chemical Bond - DePa

Research: Science and Education

A Model for the Chemical Bond1 Valerio Magnasco Dipartimento di Chimica e Chimica Industriale dell’Università, Via Dodecaneso 31, 16146 Genova, Italy; [email protected]

The Orbital Model The forces keeping electrons and nuclei together in atoms and molecules are essentially electrostatic in nature and, at the microscopic level, satisfy the principles of quantum mechanics. Experimental evidence (1) suggests formulating a planetary model of the atom, composed of a point-like nucleus (with a diameter of 1012–1013 cm) making up the mass and the positive charge +Ze. The nucleus is surrounded by electrons, each having a negative elementary charge e and a mass about 2000 times smaller than that of the proton. The electrons make up the negative charge Ne of the atom (N = Z for neutral atoms), distributed as a charge-cloud in an atomic volume with a diameter of the order of 108 cm = 1 Å. The distribution of the electrons is apparent from the density contours obtained from X-ray diffraction spectra in polycyclic hydrocarbons (2). The electron density in atoms can be described in terms of atomic orbitals (AOs), which are one-electron functions ψ(r) depending on a single center (the nucleus of the atom). The electron density in molecules can be described in terms of molecular orbitals (MOs), which are many-center, oneelectron functions depending on the different nuclei of the molecule. The physical meaning of ψ(r) is such that |ψ(r)|2dr is the probability of finding the electron in state ψ(r) in dr, provided we satisfy the normalization condition ∫dr|ψ(r)|2 = 1 where integration is extended over all space. This implies some physical restrictions on the form of the mathematical functions ψ(r) (single valued, continuity of the function with its first derivative, quadratic integrability) that are obtained as permissible solutions of a variety of eigenvalue equations, which can be traced to some space form of Schrödinger-type differential equations. If (r, θ, ϕ) are the spherical coordinates specifying electron position in space, the AOs are functions having the form:

magnetic quantum number. These numbers fully specify the state of the electron in atoms and the form of the corresponding AO. Of the utmost importance is the fact that, while ns functions (l = 0) are spherical, np functions (l = 1) have a directional character. In real form, px, py, pz are directed along three Cartesian coordinate axes, while orbitals obtained from the mixing of s and p AOs onto the same center (hybrid orbitals) have a still more marked directional character, as shown schematically in Figure 1. This fact is of great relevance in the geometric structure of polyatomic molecules and in stereochemistry, in general.3 An Elementary Molecular Orbital Model The many-center, one-electron space functions φ(r) describing electron distribution in molecules are called molecular orbitals (MOs). In the independent-electron approximation convenient MOs are constructed by linear combination of atomic orbitals, by the so-called LCAO method where coefficients are determined by the Ritz method (6).

ψ (r , θ, ϕ ) = R (r )Y (θ, ϕ )

where, Rn (r ) ∝ r n − 1e −c r or Rn (r ) ∝ r n −1e −c r

(STO)

2

(GTO)

is the radial part, which can have an exponential (Slater-type orbital, STO) or Gaussian (Gaussian-type orbital, GTO) decay with the distance r of the electron from the nucleus, c > 0 an orbital exponent, and Yl,m(θ, ϕ), the angular part of the orbital, known as spherical harmonic, which in real form can be expressed in terms of trigonometric functions of angles θ and ϕ. The radial and angular parts of the orbitals are specified by the three integers (n, l, m) called quantum numbers (4, 5): n = 1, 2, 3,…, principal quantum number; l = 0, 1, 2, 3,…, (n − 1), orbital quantum number; m = 0, ±1, ±2,…, ±l, www.JCE.DivCHED.org

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Figure 1. Schematic drawing of s, p, and sp (ti) hybrid AOs.

Vol. 81 No. 3 March 2004

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Journal of Chemical Education

427


If we combine N AOs we shall obtain N MOs. We shall limit ourselves for the present to consider the formation of two two-center MOs obtained from two normalized real fixed AOs, χ1(r) on atom A and χ2(r) on atom B,

φ1 = χ1C11 + χ 2C 21 = C11 ( χ1 + λ1χ 2 ) where,

λ1 =

C 21 C11

is the polarity parameter of MO φ1 and C11 a normalization factor, and

φ 2 = χ 2C 22 + χ1C12 = C 22 ( χ 2 + λ 2 χ1 ) with,

λ2 =

C12 C 22

the polarity parameter of MO φ2. Optimization of the linear coefficients in a simple Hückel scheme including overlap (7) yields the 2 × 2 secular equation, α1 − ε β − ε S = 0 β − εS α 2 − ε where α1 < 0, α2 < 0 are atomic integrals4 (8) specifying the energy levels of AOs χ1 and χ2; β < 0, a bond integral5 describing formation of a bond between χ1 and χ2; S =

dr χ1(r ) χ 2(r )

the overlap integral giving the superposition between the normalized AOs χ1 and χ2 that depends in an exponentially decreasing way on the internuclear distance R between atoms A and B, S ∝ exp(aR ) with a > 0. It is important to note that β depends on S and that no bond can be formed between AOs for which S = 0 by symmetry as shown in Figure 2. Expanding the second-order determinant we obtain,

(1 − S ) ε 2

2

− (α1 + α 2 − 2 β S ) ε

(

)=

α1 + α 2 − 2 βS ± ∆

(

)

where,

α + β β − αS = α + 1+ S 1 + S β − αS < 0 1+ S

φ1 =

χ1 + χ 2 , 2 + 2S

ε2 =

α − β β − αS = α − 1− S 1− S

λ1 = 1

∆ε 2 = ε 2 − α = − 2 ∆ =  (α 2 − α1 )  1 2 + 4 (β − α1S ) (β − α 2 S ) > 0 

428

ε1 =

0

an algebraic equation quadratic in ε having the real roots, 2 1 − S2

The roots εi of the secular equation are called molecular orbital energies, while the differences ∆εi = εi − αi are assumed to give the contribution of the ith MO to the bond energy. The chemical bond will, in general, depend on β, α1, α2, and S. The solutions become particularly simple in the following two cases, which are shown schematically in Figure 3. If α1 = α2 = α (Figure 3A), we have degeneracy of the atomic levels, and readily obtain for orbital energies and MOs:

∆ε1 = ε1 − α =

+ α1α 2 − β 2

ε =

Figure 2. Elementary overlap S between AOs. The phases of the directed orbitals are chosen so that the resulting overlap (dashed area) is positive. No bonding is possible in the last two cases since physical overlap is zero by symmetry.


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φ2 =

χ 2 − χ1 , 2 − 2S


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orbital energy of the bonding MO attractive bonding orbital interaction normalized bonding MO orbital energy of the antibonding MO

repulsive antiβ − αS > 0 bonding orbital 1− S interaction

λ 2 = −1

normalized antibonding MO

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For the lower orbital energy we obtain,

ε1 ≈

α1 + α 2 − 2βS − (α 2 − α1 )

(

2 1 − S2

(β

−

=

= α1 −

= α1 −

− α1S ) (β − α 2 S )

(1 − S ) (α 2

α1 − βS 1 − S

)

(β

−

2

2

− α1 )

− α1S ) (β − α 2 S )

(1 − S ) (α 2

2

− α1 )

β − α1S  β − α 2 S  S + 2  α 2 − α1   1 − S 

(β

− α1S )

(1 − S ) (α 2

2

2

− α1 )

≈ α1 −

2

(β

− α1S ) α 2 − α1

for S small. In this case, the orbital interaction is second order in β (small interaction):

ε1 ≈ α1 −

Figure 3. Bonding (ε1) and antibonding (ε2) molecular splittings for (A) first-order and (B) second-order interactions.

1

∆ = (α 2

 4 (β − α1S ) (β − α 2 S )  − α1 )  1 +  (α 2 − α1 )2    

 2 (β − α1S ) (β − α 2 S )  ≈ (α 2 − α1 )  1 +    (α 2 − α 1 ) 2   = (α 2 − α1 ) +

2 (β − α1S ) (β − α 2 S ) α 2 − α1

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2

− α1S ) α 2 − α1

∆ε1 = ε1 − α1 = −

φ1 =

We notice that the resulting MOs are orthogonal, that φ2 has a nodal plane at the midpoint of the internuclear axis whereas φ1 has no nodes, and that for S ≠ 0 the molecular levels are nonsymmetric with respect to the atomic levels (the bonding level being less bonding, the antibonding level more antibonding than the case S = 0 where splitting is symmetric). Finally the orbital interaction energy is first order in β (strong interaction). This is still true for the case of near degeneracy, with (α2 − α1) 0 small α 2 − α1

The (normalized) bonding MO φ1 is little different from χ1. ε 2 ≈ α2 +

(β

− α2S ) α 2 − α1

∆ε2 ≈ ε 2 − α 2 ≈

φ2 =

2

(β

− α2S ) α 2 − α1

χ 2 + λ 2 χ1 1 + λ2

2

orbital energy of the antibonding MO

+ 2λ 2 S

2

, λ2 ≈

> 0

repulsive antibonding orbital interaction

β − α2S < 0 small α 2 − α1

The normalized antibonding MO φ2 is not very different from χ2. The greater ∆α = α2 − α1 the smaller the orbital interaction: this explains why the chemical bond always occurs at the valence level, where energy differences between AOs are smaller. The shapes of different MOs are given schematically in Figure 4, where bonding and antibonding MOs are shown for H2, and σ and π bonding MOs for CO. Details about the diagrams for CO are given in ref 9.


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Bond Energies and Pauli Repulsions in Diatomics When two atoms interact two potential energy curves are obtained as a function of the interatomic distance R as shown in Figure 5.6 Only for the lower curve can a bond be formed. According to our simple model, the energy of a chemical bond (in short, the bond energy) in homonuclear diatomics can be obtained by adding, at the valence level, the contributions from the different MOs that are occupied by ni electrons, taking into account the Aufbau principle7 and the fact that degenerate levels must be filled according to Hund’s rule (4): occ

∆E =

∑ ni ∆εi i

The results for σ- and π-electron systems of a few homonuclear diatomics of the first row are depicted in Figure 6. We can easily see that the bond energies of the whole series are expressible in terms of the single electron bond energy parameter,

β − αS 1+ S

< 0

different for σ- and π-systems, which represents the one-electron part of the exchange–overlap component of the interaction (8, 10). In Figure 6A (σ-systems), we see that the first three interaction energies are attractive (negative), so that we can properly speak of one-electron σ-bond (H2+), two-electron σ-bond (H 2), and three-electron σ-bond (He 2+). Bond strength has a maximum for H2, since the bonding level is full and the antibonding level empty. The addition of a further electron (He2+) requires occupation of the antibonding level, so the bond strength reduces to a value8 similar to that observed for the one-electron case. No four-electron σ-bond is possible, however, since both bonding and antibonding MO levels would be full. The antibonding is more repulsive, so for He2 we observe a Pauli repulsion between σ closed shells. This repulsion will generally be observed whenever the number of electrons in antibonding MOs equals the number of electrons in bonding MOs. A similar trend is expected for the series of 2σ systems Li2+, Li2, Be2+, and [Be2]2σ that involve superposition of 2s AOs instead of 1s AOs and a weaker [(β − αS)兾(1 + S)]2σ bond energy parameter. Li2 has a weak two-electron σ-bond of 24.3 kcal mol1 at the rather long bond length Re = 2.67 Å (11). Experimental data are lacking for Li2+ and Be2+, for which still weaker one-electron and three-electron σ-bonds are expected, whereas Be2 is a van der Waals molecule. The B atom (one 2p electron in its valence shell) would give B2+, B2, B2−, where only one-electron, two-electron, and threeelectron single π-bonds are formed, with no underlying σbonds. B2 has a triplet ground state (g.s.) with De = 84.5 kcal mol1 at Re = 1.59 Å (11). In Figure 6B (π-systems), we observe that molecules and molecular ions all have an underlying σ-skeleton of 10 electrons, comprising four inner-shell electrons, a two-electron σ-bond, and four nonbonding electrons (two lone pairs). The

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Figure 4. Schematic drawing of (top) bonding and antibonding homopolar σ-MOs for H2, and (bottom) heteropolar σ- and π-bonding MOs for ground state CO.

Figure 5. The two potential energy curves for a diatomic molecule. Lower curve: a chemical bond of strength |De| is formed at Re. Upper curve: Pauli repulsion (a scattering state, no chemical bond can be formed).

series is then characterized by the filling of π MO levels that can contain up to a maximum of eight electrons. So we can speak of one-electron π-bond (C2+), two-electron π-bond (C2), where electrons with same spin occupy the doubly degenerate πu bonding level, up to the four-electron π-bond (N2), where maximum strength is obtained because of the full occupancy of the π bonding levels. In view of the un-


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Research: Science and Education Table 1. Comparison of the Exchange Part (S = 0) of Model Bond Energies with Experimental Results for Some Homonuclear Diatomics Molecule

|De|/ kcal mol1

N

Re/Å

H2+ a

1

1.06

64.4

60

H2 a,c,e

2

0.74

109.5

120

3

1.08

62.2

60

4

3.00

C2+ b

1

1.30

-----

C2 b

2

1.24

113.0

110

N2+ a

3

1.12

149.3

165

Model

σ-systems

He2+ a [He2]

f

0.02

-----

π-systems

N2

a,c–e

4

1.09

225.1

220

O2+ a

5

1.12

152.1

165

O2 a,e

6

1.21

117.2

110

F2+ b

7

1.30

65.0

55

F2 c–e

8

1.42

39.0

a

Ref 11.

Figure 6. Electron configurations of σ- and π-systems of simple firstrow homonuclear diatomics. Bonding is possible as long as the number of electrons in bonding MOs is greater than that of electrons in antibonding MOs.

derlying σ-bond, we can appropriately speak of a triple bond in N2 (one σ and two π two-electron bonds, as in acetylene C2H2). The same is seen to occur for the heteropolar molecule of carbon monoxide CO, isoelectronic with N2, whose atoms must however be considered bonded by what might be considered as an ionic triple bond C−⬅O+, in view of the fact that seven electrons must be attributed either to C (C−) or to O (O+). This is confirmed by the small value observed for the electric dipole moment of CO (≈ 0.11 D),9 which is directed from C to O contrary to the strong C+O− polarity expected for the bonding MOs on electronegativity grounds (12). Adding a further electron (O2+) decreases the bond strength to the same level as in N2+, with a similar result for the isoelectronic NO molecule, whose 2Π g.s. has a small dipole moment, 0.25 D (13, 14), suggesting that the unpaired electron is mostly localized on the N atom, giving the ionic bond N−⬅O+ (eight electrons on N, N−, seven electrons on O, O +). Next case is the ground state O 2 molecule, with its characteristic triplet state due to the two unpaired electrons in the antibonding πg MO. Bonding in g.s. O2 can hence be viewed as resulting from a two-electron σ-bond and two three-electron π-bonds, in full agreement with an early valence bond (VB) suggestion by Wheland (15). While F2+ is just like C2+, [F2]π exemplifies the repulsive

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55

b

Ref 12.

c

Ref 16.

d

Ref 17.

e

Ref 18.

----f

Ref 19.

nature (Pauli repulsion) of π closed-shell interactions (eight electrons). While [He2]1σ can form a weak bond in the van der Waals region because of London attraction, a weak chemical bond is nonetheless observed for g.s. F2 because the energy of its underlying σ-bond overtakes Pauli repulsion of its π electrons. [Ne2] cannot exist as a chemically bounded molecule because its valence g.s. configuration σg2πu4πg4σu2 shows both σ and π Pauli repulsions—rather it must be considered a weakly bonded van der Waals molecule because of London attraction. Bond energies and bond lengths in both series of molecules shown in Figure 6 strictly follow the filling of electrons in their MOs, showing the same periodicity that can be observed for the periodic table of the elements when filling electrons into AO levels. The results collected in Table 1 are striking in this sense and show that the model bond energies, assuming S = 0, |βσ| = 60 kcal mol1 and |βπ| = 55 kcal mol1, reproduce experimental atomization energy results (11, 12, 16–19) for our homonuclear diatomics with an average error not exceeding 10%. Multiple Bonds We have seen that bonds can be σ, when the bond is directed along the internuclear axis, or π, when the bond is directed perpendicularly to the internuclear axis. When both bonds exist we can speak of multiple bonds. As an example, ethylene C2H4 has a conventional σπ double bond between carbons that severely hinders rotation of one part of the molecule with respect to the other,10 giving rise to the possibility of cis–trans isomerism in substituted derivatives. N2 and acetylene C2H2 have a σπxπy homopolar conventional triple bond, while we have already noted that the smallness of the electric dipole moments observed for the heteropolar molecules CO and NO suggests the presence of ionic triple bonds.


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Bond energies for some conventional multiple bonds (20) are given in Table 2. We observe that multiple bonds have larger bond energies and consequently the bond lengths are shorter. For example, shorter bond lengths are found with increasing bond multiplicity in the CC bond: CC, Re = 1.54 Å; CC, Re = 1.34 Å; C⬅C, Re = 1.20 Å. Of course, the concept of conventional single, double, and triple bonds as formed by σ2, σ2π2, and σ2π4 electrons refers to the most typical case of the strongest two-electron bonds, but we have seen that we can have weaker one-electron (H2+) and threeelectron (He2+) σ bonds, and a variety of π bonds ranging from one-electron (C2+) to seven-electrons (F2+) in our firstrow series of homonuclear diatomics. This shows that the concept of multiple bond is rather more extended than is usually assumed.

Table 2. Bond Energies for Some Conventional Multiple Bonds Type of Bond

Bond Energies/(kcal mol1) Single

Double

Triple

CC

83

146

201

NN

40

100

225

OO

34

118

---

CN

73

147

212

CO

79

158

---

Bond Stereochemistry in Polyatomic Molecules and the Bonding in Methane In polyatomic molecules bonds are directed in space, making interbond angles that can be measured experimentally. In these molecules, bonds between atoms tend to be straight because bent bonds are weaker. Consider, the formation of a bond between an AO on B, sB, and an AO on A, pA, making an angle θ with the interbond axis, usually chosen as z axis as shown in Figure 7A. Because of its directional character, the p-orbital can be resolved into a σ-orbital directed along the bond and a π-orbital directed perpendicularly to it:

pA = σ A cos θ + π A sin θ The bond integral is βAB = βσscos θ, since the π component vanishes by symmetry (see Figure 2). βσs is now a quantity characteristic of the bond and does not depend on the angle. For the sake of simplicity, we shall neglect explicit overlap S in the secular equation.11 For the two-electron bond we then have:

(

)

E (θ ) = 2 ε(θ ) = α p + α s − ∆ (θ )

∆ (θ ) =  α p − α s  

(

)

2

1

+ 4βσs 2 cos 2 θ   

2

> 0

Taking the first and second derivative of E(θ) versus θ:

2βσs2 d∆ dE dε = 2 = − = sin(2θ ) dθ ∆ dθ dθ

d 2E d θ2

= −

d 2∆ dθ2

=

 β σs 2 2 4 βσs 2  sin (2θ )   cos(2θ ) + 2 ∆  ∆ 

we see that the first derivative vanishes for θ = 0, where the second derivative is positive, so that the energy reaches its minimum for the straight bond (bond strength is a maximum). Since overlap has a maximum for θ = 0 (Figure 7A), the principle of largest bond energy is often referred to as principle of maximum overlap.12

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Figure 7. (A) The overlap between pA and sB depends on the orientation θ of the directed p-orbital. (B) The CH4 molecule is inscribed in a cube having the C atom at its center and the four H atoms at noncontiguous vertices of the cube.

This principle is of the utmost importance in the stereochemistry of bonds in polyatomic hydrides, as we can see in the case of the CH4 molecule that has the tetrahedral symmetry shown in Figure 7B. Let us consider the formation of four CH bonds in CH4. Using a short self-explanatory notation for the eight valence AOs, we call s, x, y, z the 2s and 2p orbitals on C, and h1, h2, h3, h4 the 1s orbitals on the H atoms at the vertices of the tetrahedron. Molecular symmetry suggests to use combinations of H orbitals transform-


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ing as (s, x, y, z), which we can write by inspection: hs =

1

hx =

1

(h1

+ h 2 + h3 + h4 )

The lowest roots of the secular equations will give the Hückel energy of the eight valence electrons as a function of angle θ,

2 ( h1

+ h 2 − h3 − h4 )

E (θ ) = 2∑ ε i = (α s + α h ) + 3 α p + αh

2

occ

(

i

1

hy =

2

(h1

− h2 + h3 − h4 )

)

2

−  α p − α h  

)

2

+ 16 βxh 2   

)

2

+ 16 βyh2   

)

2

+ 16 β zh 2   

(

hz = 1 2 (h1 − h2 − h3 + h4 ) The four bonding (unnormalized) MOs are then appropriately written as,

(

a1 = s + λ hs t2 x = x + µ hx

−  α p − α h  

(

t2 y = y + µ hy t2 z = z + µ hz

−  α p − α h  

(

where the coefficients (polarity parameters) are found by solving four 2 × 2 Hückel secular equations: Hss − ε Hshs = 0 Hshs Hhshs − ε

Symmetry A1

Hzz − ε H zhz = 0 H zhz Hhz hz − ε

Symmetry T2(x, y, z)

H shs = 2 βsh

Hhshs = α h + 3β hh ≈ α h

H zz = α p

H zhz = 2βzh

Hhz hz = αh − βhh ≈ α h

with similar expressions for the remaining x, y components, and where βhh has been neglected in the spirit of Hückel theory (nonadjacent atoms). The β integrals involving (x, y, z) AOs on carbon can be expressed in terms of the more convenient set (σ, πx, πy), where σ is a p orbital directed along the CH1 bond, and π’s are orbitals perpendicular to the bond, x =

1 σ sin θ + 2

1 π x cos θ − 2

1 πy 2

y =

1 σ sin θ + 2

1 π x cos θ + 2

1 πy 2

z = σ cos θ − π x sinθ

(

−  α p − αh  

(

1 sin θ 2

2

1

1

+ 16 βs h2    2

2

2

) 2

1

+ 8 β ph2 sin 2 θ   

1

+ 16 β ph2 cos 2 θ   

cos 2 θ =

1 , sin θ = 3

cos θ =

2

2

1

2 sin

2

θ

2 ⇒ 2θ = 109.5° 3

that is, the tetrahedral angle. The resulting MOs are symmetry MOs delocalized over the entire molecule. A description more adherent to the chemical picture of four localized CH bonds can however be obtained in terms of the unitary transformation13 connecting occupied MOs, the first relation being,

( a1

B1 =

1

=

1

2  s

=

1

2  s

2

+ t2x + t2 y + t2z

 + 

where βph is a quantity characteristic of the bond.


)

λ + 3µ  + (x + y + z ) + h1 2  λ − µ + (h2 + h3 + h4 )  2 

+

•

2

which has a minimum dE兾dθ = 0, d2E兾dθ2 > 0 for,

β zh = βph cos θ

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)

)

2

 

1

2

− 2  α p − α h  

where 2θ is the interbond (valence) angle. Then, β xh = βyh = βph

)

2 

1

(

(

Hss = α s

+ 16 βsh

= α s + 3α p + 4 α h

−  α s − α h  

Matrix elements are,

1

−  α s − α h  

)

•

3 p1 +

λ + 3µ h1 2

λ − µ (h2 + h3 + h4 )  2 


433


and so forth for B2, B3, B4. If λ ≈ µ, we obtain:

B1 ≈

s +

3 p1 2

ing coefficients of the various elementary densities gives, qA =

+ λ h1 = t1 + λ h1

2 2

1 + λ

+ 2 λS 2

a (unnormalized) localized CH1 bond orbital, where p1 = (1兾√3)(x + y + z) is a 2p orbital on C pointing along the (111) diagonal of the cube, and,

qB = q AB + q BA =

t1 =

s +

3 p1 2

a tetrahedral sp3 hybrid on C (25% s, 75% p) pointing towards H1. So the usual chemical picture of CH4 in terms of four equivalent CH bonds having tetrahedral symmetry is simply recovered from the requirement of lowest energy. These considerations can be extended to the remaining hydrides of the first row, where a little hybridization on the heavy atoms allows the interbond angle to open beyond 90 (4), maintaining straight XH bonds satisfying the principle of maximum bond energy. The Distribution of the Electronic Charge in Molecules The distribution of the electronic charge in molecules can be analyzed in terms of its AOs and the coefficients obtained by solving the secular equation. We shall illustrate the main features in the case of just two electrons in a heteropolar σ bonding MO. If λ is the polarity parameter and S the overlap between two normalized real AOs, a(r) on A and b(r) on B, the normalized two-center MO will be:

1 + λ2 + 2 λ S

1 + λ2 + 2λS

Q A = qA + q AB =

Q B = qB + qBA =

δA = 1 − Q A =

qA + qB + qAB + qBA = 2

the total number of electrons in the bond orbital. Identify-


•

1 + λ2 + 2 λ S

λ2 − 1 1 + λ 2 + 2λ S λ > 1

δA = δ > 0

The distribution of the formal charges in AB, A+δ B−δ, gives rise to an electric dipole (21). For homonuclear diatomics λ equals 1 and

qAB = q BA =

1 + λ2 + 2 λ S

and can be analyzed in terms of normalized atomic densities a2(r), b2(r), and overlap densities [a(r)b(r)]兾S, where qA, qB are atomic charges on A and B and qAB + qBA the overlap charge in AB, with,

2λ2 + 2 λSS

1 + λ2 + 2 λ S

δ B = 1 − QB = −

a 2 (r ) + λ2 b 2(r ) + 2 λ a (r ) b(r ) a (r ) b (r ) S

1 + λ2 + 2λ S

λ2 − 1

qA = qB =

= qA a 2 (r ) + qB b 2 (r ) + (q AB + q BA )

2 + 2λS

and the formal charges, δ, on A and B are

P ( r ) = P α (r ) + P β ( r )

434

4 λS

δ B = − δA = − δ

The electron density, P(r), contributed by the two electrons occupying the MO with opposite spin is a function of the space point r,

= 2σ 2 ( r ) = 2

1 + λ2 + 2λS

where qA is the fraction of electronic charge on A distributed with density a2(r), qB the fraction of charge on B distributed with density b2(r), qAB + qBA the fraction of electronic charge distributed in the interbond region AB with the normalized overlap density [a(r)b(r)]兾S. Gross charges, Q, on atoms A and B are

a (r ) + λ b ( r )

σ (r ) =

2λ

1 < 1 1+ S S > 0 1 + S

For S > 0, electronic charge is pushed from the atomic into the bond region, where the increased electron density will act so as to reduce internuclear repulsion during the formation of the bond.14 Since the molecule has now a center of symmetry, just at the midpoint of the bond, the distribution of formal charges will represent a molecular quadrupole, A+δ −2δ B+δ. These elementary considerations can be extended to the Mulliken population analysis (22), and to consideration of P α(r) ≠ Pβ(r), allowing to define the spin density Q(r) = P α(r) − P β(r), describing the excess of α (up) over β (down) spin in nonsinglet states of atoms and molecules (23).


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Concluding Remarks The fundamentals of chemical bonding in molecules are established, within an elementary orbital model, in terms of the quantum mechanical principle of superposition of the valence AOs of the interacting atoms, yielding, for the case of two AOs, a 2 × 2 Hückel-type secular equation including overlap. The two roots specify the bonding and antibonding nature of the resulting molecular levels that can be occupied by electrons with appropriate spin following the Aufbau principle. The resulting bond energies strictly follow the filling up of such levels by electrons, Pauli repulsion occurring when both levels are full. Most of the bond properties of simple first-row homonuclear diatomics are qualitatively recovered by the model, which shows the same periodicity of that observed for the experimental atomization energies of these molecules. Bond stereochemistry in polyatomic hydrides is explained in terms of the principle of bond energies maximization, which yields XH straight bonds and suggests the formation of appropriate sp hybrids on the central atom, as in CH4. An elementary introduction to the electron charge distribution in molecules is given in terms of a two-center MO doubly occupied by electrons, and atomic, overlap, gross and formal charges are introduced to explain the origin of molecular electric moments. Acknowledgment Financial support by the Italian National Research Council (CNR) and by University funds is gratefully acknowledged. Notes 1. Based on UNI.T.E. lectures delivered at the University of Genoa during the academic year 2001–2002. 2. Introduced by Boys (3) mostly for computational reasons and largely used today. 3. Whenever necessary, we shall complete our description by assuming for each electron two possible spin states, spin up (α) and spin down (β). 4. As explained in ref 8, in our model the Coulomb and resonance integrals of the original theory are reinterpreted as atomic and bond integrals, respectively. The α’s are different for neutral atoms or ions. 5. As a characteristic parameter of a given bond, β is essentially negative, apart from its possible dependence on angular factors. Anyway, we define β < 0 for S > 0. 6. This R-dependence is followed by the bond energy parameter (see later) in the bond region. The bond energy De is calculated from the bottom of the potential well, the bond length Re being the value of the interatomic distance R corresponding to this minimum. The bond strength is usually referred to as |De| > 0. 7. According to this principle, orbitals in atoms and molecules (AOs and MOs) are filled by electrons with opposite spin starting from those of lowest energy so as to satisfy Pauli’s exclusion principle. As far as possible, degenerate levels are occupied by electrons with same spin in different sublevels (Hund’s rule).

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8. Actually a bit less, because of the nonsymmetry of MO levels with respect to the atomic levels for S ≠ 0. 9. The Debye is the unit for the electric dipole moment: 1D = 1018esu cm. 10. We must break the π carbon–carbon bond, which would require about 55 kcal mol1, compared with the 3 kcal mol1 of the torsional barrier around the CC single bond in ethane C2H6. 11. This is not in contradiction with what we have previously assumed, since orthogonal does not mean not interacting. In other words, S = 0 does not automatically imply β = 0. 12. More properly, we should speak of principle of maximum exchange–overlap interaction (8, 10). 13. This leaves the whole physical description invariant. 14. The previous definition of formal charge does no longer apply in this case, where δA = (N兾2) − qA = qAB.

Literature Cited 1. Karplus, M.; Porter, R. N. Atoms and Molecules; Benjamin: London, 1970; pp 18–20. 2. Bacon, G. E. Endeavour 1969, 25, 129. 3. Boys, S. F. Proc. Roy. Soc.(London) 1950, A200, 542. 4. Coulson, C. A. Valence, 2nd ed.; Oxford University: Oxford, 1961. 5. McWeeny, R. Coulson’s Valence, 3rd ed.; Oxford University: Oxford, 1979. 6. McWeeny, R. Methods of Molecular Quantum Mechanics, 2nd ed.; Academic: London, 1989. 7. Hoffmann, R. J. Chem. Phys. 1963, 39, 1397. 8. Magnasco, V. Chem. Phys. Lett. 2002, 363, 544. 9. Huo, W. M. J. Chem. Phys. 1965, 43, 624. 10. Magnasco, V.; McWeeny, R. Weak Interactions between Molecules and Their Physical Interpretation. In Theoretical Models of Chemical Bonding, Part 4; Maksic, Z. B., Ed.; Springer: Berlin, 1991; p 133. 11. Herzberg, G. Molecular Spectra and Molecular Structure: Spectra of Diatomic Molecules, 2nd ed.; Van Nostrand: Princeton, NJ, 1957. 12. Mulliken, R. S.; Ermler, W. C. Diatomic Molecules; Academic: New York, 1977. 13. Neumann, R. S. Astrophys. J. 1970, 161, 779. 14. Green, S. Chem. Phys. Lett. 1973, 23, 115. 15. Wheland, G. W. Trans. Faraday Soc. 1937, 33, 1499. 16. Bak, K. L.; Jørgensen, P.; Olsen, J.; Helgaker, T.; Klopper, W. J. Chem. Phys. 2000, 112, 9229. 17. Noga, J.; Valiron, P.; Klopper, W. J. Chem. Phys. 2001, 115, 2022. 18. Feller, D.; Dixon, D. A. J. Chem. Phys. 2001, 115, 3484. 19. Feltgen, R.; Kirst, H.; Köhler, K. A.; Pauly, H.; Torello, F. J. Chem. Phys. 1982, 76, 2360. 20. Darwent, B. de B. Bond Dissociation Energies in Simple Molecules; NSRDS-NBS 31; U.S. National Bureau of Standards, U.S. Government Printing Office: Washington, DC, 1970. 21. Coulson, C. A. Electricity, 5th ed.; Oliver and Boyd: Edinburgh, U.K., 1958. 22. Mulliken, R. S. J. Chem. Phys. 1955, 23, 1833. 23. McWeeny, R. Rev. Mod. Phys. 1960, 32, 335.


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A Model for the Chemical Bond - DePa

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