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 ↓


Density functional theory


Density functional theory (DFT) is a quantum mechanical framework widely used in physical chemistry, materials science, and condensed matter physics to investigate the electronic structure of many-body systems, particularly atoms, molecules, and solids. Unlike traditional approaches that attempt to solve the many-electron Schrödinger equation directly, DFT focuses on the electron density as the primary quantity, making it computationally more feasible for large systems.

Historical background

DFT has its roots in the early 20th century. However, its modern development began in the 1960s with the Hohenberg–Kohn theorems, which laid the foundation by proving that the ground-state properties of a many-electron system are uniquely determined by its electron density.

Major theorems

The two major Hohenberg–Cohn theorems can be summarized as follows:

  1. Existence theorem: The ground state properties of a system are uniquely determined by its electron density, which means that all information contained in the wave function is also contained in the electron density.
  2. Variational principle: A universal functional of the density exists, such that the energy obtained by using the exact electron density is minimized.

Fundamentals of DFT

At the core of DFT is the concept of the "energy functional", which is a function of the electron density. The exact form of the functional is unknown, which requires the use of approximations. The essential idea is to transform the many-electron problem into the problem of a single electron moving in an effective potential field.

Kohn–Sham equation

An important development in DFT was the introduction of the Kohn-Sham (KS) equations, which propose to express the total electron density in terms of a set of single-particle orbitals. These equations are given as:

    - (1/2) ∇² ψ_i(r) + v_eff(r) ψ_i(r) = ε_i ψ_i(r)

Where:

  • ψ_i(r) are Kohn–Sham orbitals
  • v_eff(r) is the effective potential, which includes the extrinsic and exchange-correlation potentials
  • ε_i are the orbital energies

These KS equations are solved self-consistently to determine the ground state electron density.

Exchange-correlation functional

The exact exchange-correlation functional is unknown and forms the "holy grail" of DFT. Various approximations have been developed over the years to model this functional:

  • Local density approximation (LDA): This assumes that the exchange-correlation energy at a point depends only on the density at that point, which is accurate for systems with slowly changing densities.
  • Generalized gradient approximation (GGA): Extends LDA by incorporating the gradient of the density, which applies to a wider range of systems.
  • Hybrid functionals: Combine a portion of the exact exchange energy with the DFT exchange, improving accuracy for molecular systems.

Applications of DFT

DFT is a versatile tool used for a wide range of applications ranging from fundamental research to industrial purposes. Some of the major areas include:

  • Molecular chemistry: Calculations of molecular structures, energies, and reaction pathways.
  • Condensed Matter Physics: Study of band structure, lattice dynamics, and properties of solids.
  • Materials Science: Exploration of new materials, surfaces, interfaces, and defects.
  • Biochemistry: Interaction of small molecules with biocomposites, drug design, and enzyme catalysis.

Example: water molecule

Consider the calculation of the geometry of a water molecule using DFT. The task is to find the optimal geometry that minimizes the energy of the system within the chosen approximation to the exchange-correlation functional.

    H
     ,
      Oh

The calculations will involve using a hybrid functional such as B3LYP with a suitable basis set. The result will provide bond lengths, angles, and partial charges, which will provide insight into molecular interactions.

Challenges and limitations

Despite its success, DFT is not devoid of limitations. Its applicability is often hindered by the choice of the exchange-correlation functional, which may not accurately capture certain features, such as van der Waals forces or strongly correlated electrons.

Semi-local and non-local functionals

Recent developments have focused on extending DFT to include more accurate representations of non-local interactions and dispersion forces, which are important for accurately describing chemical systems.

Future directions

The future of DFT lies in the continued refinement of exchange-correlation functionals, the incorporation of machine learning techniques, and the development of multi-scale methods that can seamlessly integrate DFT with other methods.

Integration with Machine Learning

Machine learning is being integrated with DFT calculations to predict properties based on existing data, optimize computational resources, and propose new functional forms.

Conclusion

Density functional theory remains a cornerstone of quantum chemistry and theoretical materials science due to its balance of accuracy and computational efficiency. As our understanding of the exchange-correlation functional improves and new computational techniques emerge, DFT is set to become even more powerful, revealing more mysteries of the quantum world.


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Density functional theory

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