Graduate
  1. Physical Chemistry  1.1. Thermodynamics  1.1.1. Laws of Thermodynamics  1.1.2. Enthalpy and heat capacity  1.1.3. Entropy and free energy  1.1.4. Phase Equilibrium  1.1.5. Statistical thermodynamics  1.1.6. Thermodynamic cycle  1.1.7. Fugacity and activity  1.2. Quantum Chemistry  1.2.1. Principles of quantum mechanics  1.2.2. Schrödinger Equation  1.2.3. Particle in a box  1.2.4. Operators and eigenvalues  1.2.5. Atomic orbitals  1.2.6. Molecular orbital theory  1.2.7. Perturbation theory  1.2.8. Valence bond theory  1.2.9. Hybridization and chemical bonding  1.3. Chemical kinetics  1.3.1. Rate laws and reaction mechanisms  1.3.2. Collision theory  1.3.3. Transition state theory  1.3.4. Enzyme Kinetics  1.3.5. Chain Reactions and Polymerization  1.3.6. Photochemical reactions  1.4. Statistical mechanics  1.4.1. Partitioning functions  1.4.2. Molecular distribution functions  1.4.3. Boltzmann Distribution  1.4.4. Bose–Einstein and Fermi–Dirac statistics  1.5. Spectroscopy  1.5.1. Rotational Spectroscopy  1.5.2. Vibrational Spectroscopy  1.5.3. Electronic Spectroscopy  1.5.4. Nuclear magnetic resonance spectroscopy  1.5.5. Mass Spectrometry  1.5.6. Raman Spectroscopy  1.5.7. Electron paramagnetic resonance spectroscopy  1.6. Surface and colloidal chemistry  1.6.1. Absorption isotherm  1.6.2. Surface tension and wetting  1.6.3. Colloidal Stability  1.6.4. Catalysis  1.6.5. Emulsions and micelles  1.7. Electrochemistry  1.7.1. Nernst equation  1.7.2. Electrochemical cells  1.7.3. Conductivity and mobility  1.7.4. War  1.7.5. Fuel cells  1.7.6. Electrolysis
  2. Organic chemistry  2.1. Reaction mechanism  2.1.1. Nucleophilic substitution reactions  2.1.2. Electrophilic addition reactions  2.1.3. Elimination reactions  2.1.4. Rearrangement reactions  2.1.5. Radical reactions  2.1.6. Pericyclic reactions  2.2. Spectroscopy and structural determination  2.2.1. UV-Vis Spectroscopy  2.2.2. IR Spectroscopy  2.2.3. NMR Spectroscopy  2.2.4. Mass Spectrometry  2.2.5. X-ray crystallography  2.3. Stereoscopic  2.3.1. Chirality and optical activity  2.3.2. Structural analysis  2.3.3. Geometrical isomerism  2.3.4. Dynamic Stereochemistry  2.4. Organometallic Chemistry  2.4.1. Organolithium and organomagnesium reagents  2.4.2. Palladium-catalyzed cross-coupling reactions  2.4.3. Transition Metal Complexes  2.4.4. Metal–carbon bond  2.5. Polymer chemistry  2.5.1. Polymerization mechanism  2.5.2. Polymer Characteristics  2.5.3. Biodegradable Polymer  2.5.4. Conducting polymer  2.5.5. Supramolecular polymers  2.6. Medicinal chemistry  2.6.1. Drug design and development  2.6.2. Pharmacokinetics and pharmacodynamics  2.6.3. Structure-activity relationships  2.6.4. Molecular docking and drug screening
  3. Inorganic chemistry  3.1. Coordination chemistry  3.1.1. Crystal field theory  3.1.2. Ligand field theory  3.1.3. Spectrochemical Series  3.1.4. Chelation and stability  3.2. Organometallic Chemistry  3.2.1. Metal Carbonyls  3.2.2. Catalysis by organometallic complexes  3.2.3. Metallocenes  3.3. Bio-inorganic chemistry  3.3.1. Metalloproteins and enzymes  3.3.2. The role of metals in biological systems  3.3.3. Metal ion transport and storage  3.4. Solid state chemistry  3.4.1. Crystal Structures  3.4.2. Band theory of solids  3.4.3. Superconductors  3.4.4. Defects in the crystal  3.5. Lanthanides and Actinides  3.5.1. Electronic configuration in lanthanides and actinides  3.5.2. Coordination chemistry of the lanthanides  3.5.3. Magnetic Properties of Lanthanides
  4. Analytical chemistry  4.1. Chromatography  4.1.1. Gas Chromatography  4.1.2. High Performance Liquid Chromatography  4.1.3. Thin Layer Chromatography  4.2. Spectroscopic Techniques  4.2.1. Atomic absorption spectroscopy  4.2.2. X-ray diffraction  4.2.3. Inductively coupled plasma spectroscopy  4.3. Electroanalytical methods  4.3.1. Potentiometry  4.3.2. Voltammetry  4.3.3. Colometry  4.4. Mass Spectrometry  4.4.1. Ionization Technique  4.4.2. Fragmentation Pattern  4.5. Chemometrics  4.5.1. Multidisciplinary analysis  4.5.2. Machine Learning in Chemistry
  5. Theoretical and Computational Chemistry  5.1. Molecular dynamics simulation  5.1.1. Force fields and energy minimization  5.1.2. Monte Carlo simulation in molecular dynamics simulations  5.2. Quantum chemical methods  5.2.1. Hartree–Fock theory  5.2.2. Density functional theory  5.2.3. Semi-empirical methods  5.3. Computational drug design  5.3.1. Molecular Docking  5.3.2. QSAR Modeling  5.3.3. Virtual Screening
  6. Biochemistry  6.1. Enzyme Kinetics  6.1.1. Michaelis-Menten kinetics  6.1.2. Inhibition mechanism  6.2. Metabolism and Bioenergetics  6.2.1. Glycolysis  6.2.2. Citric acid cycle  6.2.3. Electron transport chain  6.3. molecular Biology  6.3.1. DNA replication and repair  6.3.2. Protein synthesis  6.3.3. Gene Regulation  6.4. Structural Biochemistry  6.4.1. Protein Folding  6.4.2. Membrane Biophysics
  7. Environmental Chemistry  7.1. Atmospheric Chemistry  7.1.1. Greenhouse gases  7.1.2. Ozone depletion  7.1.3. Air pollutants and smoke  7.2. Water Chemistry  7.2.1. pH and water hardness  7.2.2. Wastewater treatment  7.2.3. Aquatic Chemistry  7.3. Soil Chemistry  7.3.1. Heavy metal contamination  7.3.2. Soil pH and Buffering  7.3.3. Nutrient cycling  7.4. Toxicology and Chemical Safety  7.4.1. Toxicokinetics  7.4.2. Risk Assessment in Toxicology and Chemical Safety in Environmental Chemistry  7.4.3. Effects of pollutants on the environment

Graduate → Physical Chemistry → Quantum Chemistry ↓


Molecular orbital theory


Molecular orbital theory (MOT) is a method in quantum chemistry that helps us understand the behavior of electrons in molecules. Unlike other models, MOT describes electrons as being distributed across multiple atoms rather than being localized between two atoms. This widespread distribution of electrons leads to molecular orbitals, which can extend throughout the molecule. Let us understand this concept in depth with simple language and illustrative examples.

The concept of atomic orbitals

Before exploring molecular orbitals, it is necessary to understand atomic orbitals. In an atom, electrons reside in regions of space around the nucleus called orbitals. These orbitals are solutions to the Schrödinger equation for electrons in atoms. They have different shapes and energies, and are represented as s, p, d, and f orbitals.

s orbitals: spherical shape
p orbitals: dumbbell shape with three orientations (px, py, pz)
d orbitals: more complex shapes with five orientations (dxy, dyz, dxz, dx2-y2, dz2)

According to the Pauli exclusion principle, each atomic orbital can hold a maximum of two electrons with opposite spins.

Molecular orbitals

When atoms combine to form molecules, their atomic orbitals overlap to form molecular orbitals. The key aspect of MOT is that these molecular orbitals are formed from linear combinations of atomic orbitals (LCAO). Molecular orbitals can be classified into bonding and antibonding orbitals.

Bonding molecular orbitals: When atomic orbitals combine in phase, they reinforce each other, leading to low-energy, constructive interference. This results in a bonding molecular orbital, which increases the probability of the electron between the nuclei and holds the atoms together.

Antibonding molecular orbitals: In contrast, if atomic orbitals combine out of phase, they interfere destructively, resulting in higher energy levels, known as antibonding molecular orbitals. These molecular orbitals have nodes where the electron density is minimal or zero, reducing the bond strength.

The overlapping of two atomic orbitals can be represented as:

Ψ_molecular = c1Ψ_A + c2Ψ_B

Here Ψ_molecular denotes the molecular orbital, Ψ_A and Ψ_B are atomic orbitals, and c1 and c2 are coefficients indicating the contribution of each atomic orbital to the molecular orbital.

Visualization of molecular orbitals for diatomic molecules

Let's visualize the formation of molecular orbitals using diatomic hydrogen (H 2) as a simple example.

For H2, each hydrogen atom has a 1s atomic orbital. When they combine, they form two molecular orbitals:

Bonding (σ 1s) and antibonding (σ* 1s) molecular orbitals:

               1s 1s
                ,
                 ,
              1s 1s 1s 1s

The bonding orbital (σ 1s) has less energy than the original atomic orbital, while the antibonding orbital (σ* 1s) has more energy. In H 2, both electrons occupy the bonding molecular orbital, forming a stable molecule.

Litmus test: bond order and stability

Molecular orbital theory introduces the concept of bond order to assess molecular stability:

Bond order = (Electrons in bonding orbitals - Electrons in blocking orbitals) / 2

For H2, the bonding molecular orbital is filled with two electrons and the antibonding orbital is empty. Therefore, the bond order is:

Bond order = (2 - 0) / 2 = 1

Positive bond order indicates the stability of the molecule. If the bond order is zero or negative, the molecule is unlikely to exist under normal conditions.

Heteroatomic diatomic molecules

Molecular orbital theory is not limited to similar atoms. Let us consider the case of hydrogen fluoride (HF). The process is similar but there are differences due to the electronegativities disparity between hydrogen and fluorine.

The subjective view of overlapping is depicted as follows:

                       F H
                      2p 1c
                     ,
                    ,
                σ(2p-1s) π(2p)

Due to the high electronegativities of fluorine, its atomic orbitals are lower in energy than the 1s orbitals of hydrogen. Therefore, the molecular orbitals in HF are tilted towards fluorine, indicating the significant influence of fluorine atomic orbitals in the molecule.

Note: Only orbitals with similar symmetry and comparable energy combine significantly. Therefore, HF forms a bonding molecular orbital that is primarily dominated by the 2p orbital of fluorine and the 1s orbital of hydrogen.

Application of molecular orbital theory to polyatomic molecules

Extending the MOT to polyatomic molecules involves more complex interactions, but it follows the same principles. In molecules such as water (H 2 O), the 1s orbitals of the hydrogen atoms interact with the 2p orbitals of the oxygen, forming new molecular orbitals spanning all three atoms.

The formation can be generalized as follows:

O(2s,2p) + H(1s) + H(1s) → Molecular orbitals

Constructive combination results in bonding orbitals that hold the atoms together, while destructive combination results in restricting orbitals.

Conclusion

Molecular orbital theory connects quantum chemistry and real-world chemical bonding, providing profound insight into the nature of molecules. Extending MOT to complex molecules increases the complexity of calculations, but it provides an incredibly detailed understanding of electron distribution and bond strength. Despite its complexity, the central idea is that molecular orbitals formed by overlapping atomic orbitals drive molecule stability and structure.

Hopefully through this exploration the world of molecular orbitals will seem more tangible and intuitive.


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Molecular orbital theory

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