PHD
  1. Inorganic chemistry  1.1. Coordination chemistry  1.1.1. Ligand field theory  1.1.2. Crystal field theory  1.1.3. Molecular Orbital Theory for Coordination Compounds  1.1.4. Stability of Coordination Compounds  1.1.5. Electronic spectra of coordination compounds  1.1.6. Isomerism in Coordination Compounds  1.1.7. Kinetics and mechanism of coordination reactions  1.2. Organometallic Chemistry  1.2.1. Metal Carbonyls  1.2.2. Metal alkyl and aryl  1.2.3. Fischer and Schrock carbonaceans  1.2.4. Oxidative Addition and Reductive Elimination  1.2.5. Metal Hydrides  1.2.6. Catalysis by organometallic compounds  1.2.7. Metallocenes and Sandwich Complexes  1.2.8. CH activation  1.3. Solid state chemistry  1.3.1. Crystal structures and lattices  1.3.2. Band theory and electrical properties  1.3.3. Defects in the crystal  1.3.4. Synthesis of solid state materials  1.3.5. Magnetic and optical properties of solids  1.3.6. Superconductivity  1.3.7. Solid state ionics  1.4. Bio-inorganic chemistry  1.4.1. Metalloproteins and enzymes  1.4.2. Metal ion transport and storage  1.4.3. The role of metals in medicine  1.4.4. Bioinspired catalysis  1.4.5. Metal–DNA interactions  1.4.6. Metal Complexes in Medical Science  1.5. Lanthanides and Actinides  1.5.1. Electronic configuration and oxidation states in lanthanides and actinides  1.5.2. Spectral and Magnetic Properties in Lanthanides and Actinides  1.5.3. Separation and Extraction of Lanthanides and Actinides  1.5.4. Coordination chemistry of f-block elements  1.6. Main Group Chemistry  1.6.1. Chemistry of boron compounds  1.6.2. Chemistry of Silicon and Germanium Compounds  1.6.3. Chemistry of Phosphorus and Sulfur Compounds  1.6.4. Organosilicon chemistry  1.6.5. Noble Gas Chemistry
  2. Organic chemistry  2.1. Reaction mechanism  2.1.1. Nucleophilic substitution reactions  2.1.2. Electrophilic addition and substitution reactions  2.1.3. Elimination reactions  2.1.4. Rearrangement reactions  2.1.5. Radical reactions  2.1.6. Pericyclic Reactions  2.1.7. Photochemical reactions  2.2. Stereoscopic  2.2.1. Chirality and optical activity  2.2.2. Structural analysis  2.2.3. Stereoelectronic effects  2.2.4. Asymmetric synthesis  2.2.5. Dynamic Stereochemistry  2.3. Organometallic Chemistry in Organic Synthesis  2.3.1. Organolithium and organomagnesium reagents  2.3.2. Transition metal catalyzed coupling reactions  2.3.3. Palladium-catalyzed reactions  2.3.4. Olefin Metathesis  2.3.5. Gold and silver catalysis  2.4. Natural products chemistry  2.4.1. Alkaloids and terpenoids  2.4.2. Steroids and Flavonoids  2.4.3. Total synthesis of natural products  2.4.4. Biosynthesis of natural products  2.5. Supramolecular Chemistry  2.5.1. Host-Guest Chemistry  2.5.2. Self-assembly and molecular recognition  2.5.3. Supramolecular Catalysis  2.5.4. Molecular Machines
  3. Physical Chemistry  3.1. Thermodynamics  3.1.1. Laws of Thermodynamics  3.1.2. Chemical Potential and Equilibrium  3.1.3. Statistical thermodynamics  3.1.4. Non-equilibrium thermodynamics  3.2. Quantum Chemistry  3.2.1. Schrödinger equation and its applications  3.2.2. Molecular orbital theory  3.2.3. Density functional theory  3.2.4. Post-Hartree–Fock methods  3.3. Chemical kinetics  3.3.1. Reaction rate theory  3.3.2. Mechanism and rate laws  3.3.3. Catalysis and enzyme kinetics  3.3.4. Reaction dynamics  3.4. Spectroscopy and molecular structure  3.4.1. Infrared Spectroscopy  3.4.2. UV-visible spectroscopy  3.4.3. Nuclear Magnetic Resonance Spectroscopy  3.4.4. Mass Spectrometry  3.4.5. Raman Spectroscopy  3.5. Surface and Colloid Chemistry  3.5.1. Absorption and Catalysis  3.5.2. Surfactants and micelles  3.5.3. Colloidal Stability  3.5.4. Nanomaterials and Interfaces
  4. Analytical chemistry  4.1. Chromatography  4.1.1. Gas Chromatography  4.1.2. High Performance Liquid Chromatography  4.1.3. Thin Layer Chromatography  4.1.4. Ion Chromatography  4.2. Electroanalytical techniques  4.2.1. Voltammetry and Polarography  4.2.2. Conductometry and potentiometry  4.2.3. Colometry  4.3. Spectroscopic Methods  4.3.1. Atomic absorption spectroscopy  4.3.2. Fluorescence Spectroscopy  4.3.3. X-ray diffraction  4.3.4. Electron paramagnetic resonance spectroscopy  4.4. Chemometrics  4.4.1. Data processing and statistical methods  4.4.2. Multidisciplinary analysis  4.4.3. Machine Learning in Analytical Chemistry
  5. Theoretical and Computational Chemistry  5.1. Molecular dynamics simulation  5.2. Quantum chemistry methods  5.3. Computational drug design  5.4. Machine Learning in Chemistry  5.5. Ab initio molecular dynamics
  6. Biophysics and Medicinal Chemistry  6.1. Drug discovery and design  6.2. Enzyme kinetics and inhibition  6.3. Chemical biology approach  6.4. Protein-ligand interactions  6.5. Bioorthogonal chemistry
  7. Materials chemistry  7.1. Polymer chemistry  7.1.1. Polymerization mechanism  7.1.2. Functional Polymers  7.1.3. Biodegradable Polymer  7.1.4. Conducting and semiconducting polymers  7.2. Nano Chemistry  7.2.1. Nanoparticles and nanostructures  7.2.2. Carbon-based nanomaterials  7.2.3. Quantum dots  7.3. Energy and Environmental Chemistry  7.3.1. Battery and Fuel Cell Chemistry  7.3.2. Green chemistry and sustainable materials  7.3.3. Photocatalysis

PHD → Physical Chemistry → Quantum Chemistry ↓


Schrödinger equation and its applications


The development of quantum mechanics has revolutionized our understanding of the microscopic world, providing a framework for describing the behavior of atoms and molecules that classical physics could not explain. Central to quantum mechanics is the Schrödinger equation, a fundamental partial differential equation that provides a way to calculate the wave function of a quantum system and thus provide all possible information about the system.

Introduction to the Schrödinger equation

The Schrödinger equation was formulated by Austrian physicist Erwin Schrödinger in 1925. It serves as the foundation of non-relativistic quantum mechanics. This equation plays a role similar to Newton's laws and energy conservation in classical mechanics.

The time-dependent Schrödinger equation is given as:

iħ ∂ψ(x, t) / ∂t = Ĥψ(x, t)

Where:

  • i is the imaginary unit.
  • ħ (h-bar) is the reduced Planck constant.
  • ψ(x, t) is the wave function of the quantum system.
  • Ĥ is the Hamiltonian operator representing the total energy of the system.

The time-independent version, which is more commonly used in chemistry, especially for steady states, is:

Ĥψ(x) = Eψ(x)

where E is the energy eigenvalue associated with the wave function ψ(x).

Basic concepts of quantum mechanics

Before taking a deeper look at the applications of the Schrödinger equation, let's review the basic concepts of quantum mechanics:

  • Wave-particle duality: Particles like electrons exhibit both particle-like and wave-like behaviour.
  • Quantum state: It is represented by the wave function ψ, which contains all the information about the system.
  • Probability density: The square of the magnitude of the wave function |ψ|^2 gives the probability density of finding a particle at a given position.

Applications in quantum chemistry

Quantum chemistry applies quantum mechanics and the Schrödinger equation to chemical systems to understand and predict chemical behavior. Here, we explore important applications in quantum chemistry.

Particle in a box

The particle in the box is a classic example used to illustrate quantum mechanics principles. It depicts a particle free to move around within a rigid, one-dimensional box with impenetrable walls. The solution of the Schrödinger equation for this model gives insight into quantization.

Consider a one-dimensional box of length L, then the wave function solution is:

ψ_n(x) = √(2/L) sin(nπx/L)

Where n is a quantum number, which can be any positive integer, and n = 1, 2, 3...

The corresponding energy levels are quantized as follows:

E_n = n²h²/8mL²
Particle

This model helps explain electron behavior in quantum dots, which are used in modern electronic and photonic devices.

Quantum harmonic oscillator

The harmonic oscillator model describes a particle subject to a restoring force proportional to its displacement, similar to a mass on a spring. The Schrödinger equation provides solutions for the vibrational motion of molecules.

The energy levels for a harmonic oscillator are:

E_n = (n + 1/2)ħω

Where ω is the angular frequency of oscillation and n = 0, 1, 2...

Vibrations

It is important in predicting the vibrational spectrum of molecules, which provides information about molecular structures.

Hydrogen atom

The Schrödinger equation for the hydrogen atom is a cornerstone of quantum chemistry because it can be solved exactly. The solutions describe the electron orbitals, and the derived models are used to explain the arrangement of the periodic table.

The energy levels are given as:

E_n = - (me⁴)/(8ε_0²h²n²)

where m is the electron mass and ε₀ is the permittivity of free space.

Nucleus Electron

Analysis of the hydrogen atom provides information about chemical bonds and molecular interactions.

Conclusion

The Schrödinger equation is an indispensable tool in quantum chemistry, helping scientists understand atomic and molecular structures and interactions. Its diverse applications, from particles in a box, harmonic oscillators and hydrogen atoms, establish its centrality in explaining chemical phenomena and predicting chemical reactivity.

Today, solutions to the Schrödinger equation are inspiring innovations in materials science, medicine, and nanotechnology, demonstrating the equation's power to unlock the mysteries of the microscopic world.


PHD → 3.2.1


U
username
0%
completed in PHD

← Prev (3.2)
Quantum Chemistry
Next (3.2.2) →
Molecular orbital theory

Comments 



Schrödinger equation and its applications

https://www.chemistryelite.com/phd/3.2.1?ceal=en