Unit 3: Quantum Mechanics of Chemical Bonding

1. The Variation Theorem

The variation theorem is a fundamental quantum mechanical principle used to approximate the ground state energy of systems for which the Schrödinger equation cannot be solved exactly. Since most chemical systems (except the hydrogen atom) involve multiple electrons and cannot be solved analytically, approximation methods like the variation method are essential.

The Variation Principle: If an arbitrary wave function Ψ (called a trial wave function) is used to calculate the expectation value of the energy E for a system described by a Hamiltonian operator H, the calculated energy E_trial will always be greater than or equal to the true ground state energy E₀ of the system.

Mathematically, this is expressed as:

E_trial = ( ∫ Ψ^* H Ψ dτ ) / ( ∫ Ψ^* Ψ dτ ) ≥ E₀

If the trial wave function is normalized, the denominator is equal to 1, and the expression simplifies to:

E_trial = ∫ Ψ^* H Ψ dτ ≥ E₀

Key Steps in Applying the Variation Theorem

1. Set up a trial wave function Ψ containing one or more adjustable parameters (variational parameters), such as α, β, etc.
2. Calculate the energy expectation value E(α, β, ...) as a function of these parameters.
3. Minimize the energy with respect to each parameter by setting the partial derivatives to zero: ∂E/∂α = 0, ∂E/∂β = 0, etc.
4. Substitute the optimized parameters back into the energy equation to find the lowest possible energy, which represents the best approximation to the true ground state energy.

2. Valence Bond (VB) and Molecular Orbital (MO) Approaches

To describe chemical bonding in molecules, two primary quantum mechanical approaches are used: the Valence Bond (VB) theory and the Molecular Orbital (MO) theory.

3. LCAO-MO Treatment of H₂⁺ and H₂

The Linear Combination of Atomic Orbitals (LCAO) is an approximate method for representing molecular orbitals as a sum of atomic orbitals.

3.1. The Hydrogen Molecule Ion (H₂⁺)

The H₂⁺ ion is the simplest molecular system, consisting of two nuclei (A and B) and a single electron. Let φ_A and φ_B be the 1s atomic orbitals of hydrogen atoms A and B, respectively.

By LCAO-MO, the trial molecular wave functions Ψ are written as linear combinations of these atomic orbitals:

Ψ = c_Aφ_A ± c_Bφ_B

Due to the homonuclear symmetry of the system, c_A² must equal c_B², meaning c_A = ±c_B. This gives rise to two normalized wave functions:

1. Bonding Molecular Orbital (Ψ_g):

Ψ_g = [2(1 + S)]^-1/2 (φ_A + φ_B)

Here, S is the overlap integral, defined as S = ∫ φ_A φ_B dτ. The subscript "g" stands for gerade (even under inversion symmetry).

2. Antibonding Molecular Orbital (Ψ_u):

Ψ_u = [2(1 - S)]^-1/2 (φ_A - φ_B)

The subscript "u" stands for ungerade (odd under inversion symmetry).

3.2. Bonding vs. Antibonding Orbitals

The physical distinction between bonding and antibonding states is evident from their electron density distributions:

• Bonding Orbital (Ψ_g): In this state, electron density accumulates in the region between the two nuclei, acting as a shield that reduces nuclear-nuclear repulsion and holds the nuclei together. It has lower energy than the individual atomic orbitals.
• Antibonding Orbital (Ψ_u): In this state, there is a nodal plane between the nuclei where the electron density is exactly zero. The electrostatic repulsion between the bare nuclei is unshielded, leading to a higher energy state than the constituent atomic orbitals.

3.3. LCAO-MO Treatment of the H₂ Molecule

The H₂ molecule contains two electrons (1 and 2) and two nuclei (A and B). In the simple LCAO-MO approach, both electrons are placed into the lowest-energy bonding molecular orbital, Ψ_g:

Ψ_g(1) = N (φ_A(1) + φ_B(1))
Ψ_g(2) = N (φ_A(2) + φ_B(2))

The overall spatial wave function for the two-electron system is the product of these individual one-electron molecular orbitals:

Ψ_MO(1,2) = Ψ_g(1) Ψ_g(2) = N² [φ_A(1) + φ_B(1)] [φ_A(2) + φ_B(2)]

Expanding this product yields:

Ψ_MO(1,2) = N² [ φ_A(1)φ_B(2) + φ_B(1)φ_A(2) + φ_A(1)φ_A(2) + φ_B(1)φ_B(2) ]

The terms can be classified physically as:

• Covalent terms: φ_A(1)φ_B(2) + φ_B(1)φ_A(2) (where electron 1 is on atom A and electron 2 is on atom B, or vice versa).
• Ionic terms: φ_A(1)φ_A(2) + φ_B(1)φ_B(2) (where both electrons are on atom A, representing H_A^-H_B⁺, or both are on atom B, representing H_A⁺H_B^-).

4. Comparison of LCAO-MO and VB Treatments of H₂ and Their Limitations

The primary difference between the two treatments lies in their description of the ground-state wave functions.

4.1. Wave Functions

Valence Bond (VB) Wave Function (Heitler-London model):

The VB approach starts with the premise that electrons remain localized on their respective atoms. The spatial wave function is:

Ψ_VB = N_VB [ φ_A(1)φ_B(2) + φ_B(1)φ_A(2) ]

Note that this function contains only covalent terms.

Molecular Orbital (MO) Wave Function:

As derived in section 3.3, the expanded MO wave function contains both covalent and ionic components:

Ψ_MO = N_MO [ {φ_A(1)φ_B(2) + φ_B(1)φ_A(2)} + {φ_A(1)φ_A(2) + φ_B(1)φ_B(2)} ]
Ψ_MO = Ψ_covalent + Ψ_ionic

4.2. Comparison and Limitations

• Weight of Ionic Terms: The MO wave function gives equal weight (50% covalent, 50% ionic) to the covalent and ionic states. In reality, ionic states are much higher in energy and contribute far less to the actual ground state of H₂ (experimentally, only about 5-10%). Thus, simple MO theory overestimates the ionic character.
• Behavior at Dissociation: As the distance between the two hydrogen atoms approaches infinity (R → ∞):
  • The VB wave function correctly predicts that the molecule dissociates into two neutral hydrogen atoms (covalent limit).
  • The MO wave function incorrectly predicts that the molecule dissociates into a 50-50 mixture of neutral atoms (H + H) and ions (H⁺ + H^-), which is physically incorrect because of the immense energy required to form gas-phase ions from neutral H₂.
• Corrections:
  • To correct the VB wave function, ionic terms must be manually added to account for ionic-covalent resonance.
  • To correct the MO wave function, Configuration Interaction (CI) must be introduced, which mixes the ground state MO configuration with excited state configurations to reduce the ionic contribution.

5. Spin State of Two-Electron Systems: Singlet and Triplet States

The total wave function of a multi-electron system must satisfy the Pauli Exclusion Principle, which states that the total wave function (spatial × spin) must be antisymmetric (change sign) with respect to the exchange of any two electrons.

Ψ_total(1,2) = Ψ_spatial(1,2) × χ_spin(1,2)

For a two-electron system, the individual spin states of the electrons can be either α (spin up, m_s = +1/2) or β (spin down, m_s = -1/2). This leads to four possible spin combinations, which are grouped into symmetric and antisymmetric spin states:

5.1. Singlet State (Antisymmetric Spin State, S = 0)

There is only one antisymmetric spin wave function:

χ_singlet(1,2) = 2^-1/2 [ α(1)β(2) - β(1)α(2) ]

Since the spin part is antisymmetric, the corresponding spatial wave function must be symmetric under electron exchange to make the total wave function antisymmetric:

Ψ_{total, singlet} = Ψ_symmetric(1,2) × χ_singlet(1,2)

This corresponds to paired spins (total spin quantum number S = 0, multiplicity 2S + 1 = 1).

5.2. Triplet State (Symmetric Spin States, S = 1)

There are three symmetric spin wave functions (multiplicity 2S + 1 = 3):

χ_{triplet, +1}(1,2) = α(1)α(2)
χ_{triplet, 0}(1,2) = 2^-1/2 [ α(1)β(2) + β(1)α(2) ]
χ_{triplet, -1}(1,2) = β(1)β(2)

Since the spin part is symmetric, the corresponding spatial wave function must be antisymmetric under electron exchange to make the total wave function antisymmetric:

Ψ_{total, triplet} = Ψ_{antisymmetric}(1,2) × χ_triplet(1,2)

This corresponds to unpaired spins with parallel alignments.

6. Schrödinger Equation for Many-Electron Atoms (He, Li)

Setting up the Schrödinger equation for multi-electron atoms requires considering the kinetic energy of the electrons, and the electrostatic interactions between all charged particles. Under the Born-Oppenheimer approximation, the nucleus is assumed to be stationary, so only the electronic Hamiltonian is constructed.

6.1. Helium Atom (He, Z = 2)

The helium atom has a nucleus of charge +2e at the origin and two electrons (labeled 1 and 2) at distances r₁ and r₂ from the nucleus. The distance between the two electrons is r₁₂.

The Hamiltonian operator H for the helium atom (in atomic units) is:

H = - (1/2)∇₁² - (1/2)∇₂² - 2/r₁ - 2/r₂ + 1/r₁₂

Where:

• - (1/2)∇₁² and - (1/2)∇₂² are the kinetic energy operators of electron 1 and electron 2.
• - 2/r₁ and - 2/r₂ are the attractive potential energies between the nucleus (+2) and the electrons.
• + 1/r₁₂ is the repulsive potential energy between the two electrons (interelectronic repulsion term).

The Schrödinger equation is:

H Ψ(1,2) = E Ψ(1,2)

Note: The presence of the interelectronic repulsion term (1/r₁₂) makes it impossible to separate the variables in the Schrödinger equation, meaning it cannot be solved analytically.

6.2. Lithium Atom (Li, Z = 3)

The lithium atom has a nucleus of charge +3e and three electrons (1, 2, and 3) at distances r₁, r₂, and r₃ from the nucleus, with interelectronic distances r₁₂, r₁₃, and r₂₃.

The Hamiltonian operator H for the lithium atom (in atomic units) is:

H = ∑ [ - (1/2)∇_i² - 3/r_i ] + 1/r₁₂ + 1/r₁₃ + 1/r₂₃

The Schrödinger equation is:

H Ψ(1,2,3) = E Ψ(1,2,3)

7. Qualitative Description of LCAO-MO Treatment of Diatomic Molecules

The LCAO-MO approach can be applied qualitatively to predict the electronic structures, magnetic properties, and bond orders of homonuclear and heteronuclear diatomic molecules.

7.1. Homonuclear Diatomic Molecules

In homonuclear diatomic molecules (e.g., N₂, O₂, F₂), the two bonding atoms are identical, meaning their corresponding atomic orbitals have the same energy. When they combine, they form symmetric bonding and antisymmetric antibonding molecular orbitals.

• s-p Mixing: For lighter homonuclear diatomic molecules (such as B₂, C₂, N₂), the energy gap between the 2s and 2p atomic orbitals is small, leading to significant s-p mixing. This shifts the σ_2pz orbital to a higher energy level than the π_2px and π_2py orbitals.
• For heavier molecules (like O₂ and F₂), the 2s-2p energy gap is larger, so there is negligible s-p mixing, and the σ_2pz orbital remains lower in energy than the π_2p orbitals.

7.2. Heteronuclear Diatomic Molecules

In heteronuclear diatomic molecules, the two atoms have different electronegativities, meaning their atomic orbitals have different initial energy levels. The more electronegative atom has lower-energy atomic orbitals.

1. Lithium Hydride (LiH)

• The active atomic orbitals are the Li 2s orbital and the H 1s orbital.
• Since hydrogen is more electronegative than lithium, the H 1s orbital is lower in energy than the Li 2s orbital.
• The bonding molecular orbital (σ) is closer in energy to the H 1s orbital, which means it has more H 1s character (giving it polar covalent character: Li^δ+H^δ-).
• The antibonding molecular orbital (σ^*) is closer in energy to the Li 2s orbital, meaning it has more Li 2s character.

2. Hydrogen Fluoride (HF)

• Fluorine is highly electronegative, so its atomic orbitals are significantly lower in energy than the H 1s orbital.
• The H 1s orbital overlaps primarily with the F 2p_z orbital (assuming the molecular axis is along z) because they are closer in energy. This forms a bonding σ orbital and an antibonding σ^* orbital.
• The F 2p_x and F 2p_y orbitals have no compatible symmetry to overlap with the H 1s orbital along the z-axis, so they remain as non-bonding orbitals (lone pairs on fluorine).
• The bonding σ orbital is heavily weighted toward the lower-energy F 2p_z orbital, making the bond highly polar (H^δ+F^δ-).

8. Calculation of Bond Order

Bond order indicates the stability and strength of a covalent bond.

Formula: Bond Order = (N_b - N_a) / 2

Where N_b is the total number of electrons in bonding molecular orbitals, and N_a is the total number of electrons in antibonding molecular orbitals.