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 → Theoretical and Computational Chemistry → Quantum chemical methods ↓


Density functional theory


Density functional theory (DFT) is a powerful computational quantum mechanical modeling method used in physics, chemistry, and materials science to investigate the electronic structure of many-body systems, especially atoms, molecules, and condensed phases. Its importance lies in its ability to explain all kinds of chemical properties using electron densities rather than wave functions.

Basic concepts of DFT

The basic idea behind DFT is contained in its name itself: the density functional approach. In DFT, the properties of a many-electron system are uniquely determined by its electron density, which is a much simpler quantity than the many-body wave function. The electron density, ρ(r), describes the probability of finding an electron at a point, r, in space.

ρ(r) = ∑ i |ψ i (r)| 2

This equation defines the electron density ρ(r) in terms of the molecular orbitals ψ i (r).

Historical development

DFT was developed from the Thomas–Fermi model, which was based on a semiclassical approximation of electron gases. A leap forward came in 1964 with two important theorems by Hohenberg and Kohn, which laid the mathematical foundation for modern DFT.

Hohenberg–Cohn theorem

1. Existence theorem: It states that the ground state properties of a many-electron system are uniquely determined by the electron density ρ(r).

2. The variational principle: for any density ρ(r) that can be physically realized, the energy functional, E[ρ], attains its minimum value at the true ground state density. In other words, if you can guess the true density, you can calculate the ground state energy.

E[ρ] = T[ρ] + V[ρ] + V ne [ρ] + E xc [ρ]

Where, E[ρ] is the energy as a functional of the density ρ(r), T[ρ] denotes the kinetic energy part, V[ρ] denotes the electron-electron interaction, V ne [ρ] is the electron-nuclear interaction, and E xc [ρ] is the exchange-correlation functional.

Kohn–Sham approach

Practical implementations of DFT are primarily based on the Kohn–Sham (KS) formalism, introduced in 1965. Kohn and Sham provided a way to treat interacting systems as non-interacting systems of particles with the same density.

Kohn–Sham DFT reduces the problem of interacting electrons to solving a set of self-consistent field equations for non-interacting electrons:

[-frac {1}

in which V eff (r) is the effective potential, ε i are the orbital energies, and ψ i (r) are the Kohn–Sham orbitals. The electron density is expressed in terms of these orbitals.

Exchange-correlation functional

An important element of the DFT is the exchange-correlation energy, E xc [ρ]. This is where the tricky part of the DFT lies: finding an exact expression for the exchange-correlation functional. All terms except E xc [ρ] can be derived directly.

T[ρ] V[ρ] v has [ρ] e x c [ρ]

Several approximations to the exchange–correlation functional exist, including:

  • Local density approximation (LDA): This assumes that the exchange-correlation at each point in space depends only on the electron density at that point.
  • Generalized gradient approximation (GGA): This includes not only the density but also its gradient, providing greater flexibility and accuracy.
  • Hybrid functionals: combine a part of the exact exchange from Hartree–Fock theory, aiming to further improve accuracy.

The choice of functional significantly affects the reliability of DFT calculations, and choosing the right functional often depends on the trade-off between accuracy and computational cost.

Applications of DFT

DFT is used extensively in various fields of chemistry and materials science due to its computational efficiency and reliability. It helps in predicting molecular structures, vibrational spectra, thermochemical properties, reaction mechanisms, and much more.

Atoms Molecules solids
  • Atoms and Molecules: DFT is widely used to calculate electron distribution and energy states in atoms and molecules. It plays an important role in quantum chemistry to understand chemical bonding and reactivity.
  • Solid State Physics: In solids, DFT helps calculate band structures, lattice dynamics, and mechanical properties, which proves essential in materials science for new materials design.
  • Catalysis: DFT enables the study of catalytic surfaces and reaction mechanisms at the atomic level, helping in the design of more efficient catalysts.

Limitations of DFT

Despite its wide application and popularity, DFT has its limitations, mainly due to the approximations in the exchange-correlation functional.

  • Exchange-correlation: choosing inappropriate functionals can lead to erroneous results, especially for systems with significant electron correlation.
  • Van der Waals interactions: DFT usually underestimates these weak, long-range interactions unless specifically corrected for.
  • Open systems: Modeling open systems with non-conserved particle number can be challenging within the DFT framework.

Researchers are continually developing new functionalities and approaches to overcome these limitations and further extend the applicability of DFT.

Main advantages of DFT

The main strength of the DFT lies in its computational efficiency and accuracy for a wide range of systems compared to traditional wave function methods.

  • Efficiency: The computational cost of DFT is typically lower than that of wave function-based methods such as Hartree–Fock or post-Hartree–Fock methods, making it possible to study larger systems.
  • Scalability: DFT scales more favorably with the size of the system, making it suitable for systems containing hundreds of atoms.
  • Versatility: It is applicable to a wide variety of systems - from isolated atoms to large molecular complexes and solids.

Conclusion

Density functional theory is a cornerstone of computational chemistry and physics. With its balance of computational efficiency and accuracy, DFT has been widely adopted to tackle complex electronic structure problems in chemistry and physics. Despite inherent challenges and limitations, advances in the methodology continue to advance its capabilities, strengthening its role in future discoveries and technological innovations.


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Density functional theory

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