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4. Valence Bond (VB) Theory READING: Chapter 2, Sections 2.4 – 2.6 The first quantum mechanical theory of bonding (Pauling, Heitler, London, etc. 1930s) Only considers the bonding between two atoms at a time – “localized bonds”. Essential features of VB theory: • To a first approximation, we are considering only electrons in valence orbitals. • An electron pair between atoms forms a (covalent) bond. Other electron pairs are lone

pairs. (cf. Lewis dot structures) • All bonds must consist of an underlying σ bond, and may also have additional π, δ, or φ bonds.

• π resonance structures allow for alternate designations of electron pairs (as lone pairs

or as π bond pairs). • All bonds are constructed by the overlap of atomic orbitals of the same symmetry.

(Atomic orbital of atom A overlaps with atomic orbital of atom B.) • The underlying σ bond is directional. Lone pairs are also directional. • The Schrödinger atomic orbital functions (s, p, d, f) are centrosymmetric. To arrive at

directional σ bonding / lone pairs, we must develop an alternate set of “directional” atomic orbitals → hybridized atomic orbitals (sp, sp2, sp3, sp3d, sp3d2).

• Repulsive forces between electron pairs can be used as a guide to basic structure. • Molecular structures are characterized according to symmetry (point groups). • From the relationship between atomic orbital symmetry and the molecular point

group, we can define specific hybrid atomic orbitals to describe the bonding. • Each electron pair is described by a wavefunction formed of the product (or sum of

products) of the atomic orbital wave functions. e.g. for the H2 molecule (acknowledging indistinguishable electrons):

(cf. CHEM 206 for some further considerations on H2, H2O, NH3)

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4.1 Hybridization CONCEPT:

n atomic orbitals (orthogonal)

centrosymmetric

→ n hybrid orbitals

(orthogonal) directional

Consider BeH2 as an example (cf. CHEM 206). VSEPR predicts: H : Be : H a linear molecule

Be: 1s22s22p0 • To rationalize the bonding, we need to construct two alternate atomic orbitals from

the centrosymmetric valence orbitals (2s and 2p) that should be directed equally towards the two hydrogen atoms (1s valence orbitals).

The total electron density in BeH2 looks something like this:

… and consists of an overlap of the electron density of the two Be-H bonds, which look something like this:

(Source: Purcell + Kotz, Inorganic Chemistry, 1977) • We can develop alternate Be atomic orbitals that will describe such localized electron

pairs between each H atom and the central Be atom using symmetry arguments! BeH2 is point group D∞h

Some explanations about this point group: - Infinite number of C2 axis ⊥ to the bond axis, which is a C∞ axis. - Mirror plane ⊥ to C∞: σh - Infinite number of mirror planes || to the bond axis C∞

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Strategy: 1) Define two bond vectors h1 and h2 that represent the two Be-H bond directions:

(Source: Purcell + Kotz, Inorganic Chemistry, 1977)

2) Apply symmetry operations of D∞h to the two vectors if R (hn) = hn → character = 1 if R (hn) = hm≠n → character = 0 where R = symmetry operations of the point group. h1 = 1 1 1 0 0 0 0 h2 = 1 1 1 0 0 0 0 2) Sum up to yield reducible representation

(h1, h2) = 2 2 2 0 0 0 0 3) Deconvolute into irreducible representations (by inspection or formula)

Σ+g = 1 1 1 1 1 1 1

Σ+u = 1 1 1 -1 -1 -1 -1

4) Identify the transformation basis

Σ+g transformation bases x2 + y2; z2 → s orbital

Σ+u transformation basis z → pz orbital

5) These bases identify the appropriate atomic orbitals: s + pz → 2 sp hybrid orbtials

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NOTE: Treating the special point groups D∞h and C∞v: • The linear infinite point groups D∞h and C∞v are difficult to work with and somewhat

confusing, therefore: • To make is easier, represent D∞h by D2h and C∞v by C2v. The overall results in the

context of VB and/or MO theory (see later) will be the same! The previous example worked out again in point group D2h: D2h E C2(z) C2(y) C2(x) i σ(xy) σ(xz) σ(yz) Ag 1 1 1 1 1 1 1 1 x2, y2, z2 B1g 1 1 -1 -1 1 1 -1 -1 Rz xy B2g 1 -1 1 -1 1 -1 1 -1 Ry xy B3g 1 -1 -1 1 1 -1 -1 1 Rx yz Au 1 1 1 1 -1 -1 -1 -1 B1u 1 1 -1 -1 -1 -1 1 1 z B2u 1 -1 1 -1 -1 1 -1 1 y B3u 1 -1 -1 1 -1 1 1 -1 x

Operating with D2h on h1 and h2: D2h E C2(z) C2(y) C2(x) i σ(xy) σ(xz) σ(yz)

h1 1 1 0 0 0 0 1 1 h2 1 1 0 0 0 0 1 1 Γred 2 2 0 0 0 0 2 2 By inspection: Γred = Ag + B1u Result: Use 2s and 2pz orbital to construct hybrid orbitals h1 and h2. (Same result as using D∞h.) HOMEWORK: Draw a Lewis Dot Structure for H2O. What is the symmetry of H2O? Using the above method, figure out the hybridization of the valence atomic orbitals of the oxygen atom. (HINT: both single bonds and lone pairs are directional!!)

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Okay, now let’s work from one s and one pz orbital to form h1 and h2 • From two orthogonal centrosymmetric atomic orbitals (φs and φp), we must form two

new directional, mutually orthogonal (hybridized) atomic orbitals (h1 and h2). • Ad-mix some portion a of φs with some portion b of φp to get:

h1 = aφs + bφp

…and the orthogonal combination: h2 = bφs - aφp

(a2 + b2 = 1 :each hybrid orbital holds exactly one electron)

Proof:

• We can rationalize these two new orbitals by their symmetry properties:

i operating on h1 must give h2 (see figure above table)

→ i (aφs + bφp) must be equal to bφs - aφp = h2 i (aφs + bφp) = aφs - bφp must be equal to bφs - aφp

aφs - bφpz = bφs - aφp

→ This can only be if a = b

→ a2 + b2 = 2a2 = 1 → a = b = √1/2

• Therefore: h1 = √1/2 (φ s + φp) and h2 = √1/2 (φ s - φp)

… and they look like this : … the sp hybrid orbitals: (Source: Purcell + Kotz, Inorganic Chemistry, 1977) Note: h1

2 + h22 = φ s

2 + φz2

i.e. there is no real physical distinction between the original AO and the resulting hybrid orbitals.

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Some general comments on VB theory: • The hybrid orbitals hn are only all equal if all X in EXn are equal. • The molecular structure is determined by the σ-bonds, the π-bonds modify it • VB gets complicated for non-EXn or EXnYoZp structures, where E is a central atom. The general VB solution for the series of EXn molecules and examples are give in the following tables:

Source for all tables: Purcell + Kotz, Inorganic Chemistry, 1977)

Huheey, Keiter, Keiter, Inorg. Chem. 4th Ed., 1993.

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