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 ↓


Hartree–Fock theory


In the field of quantum chemistry, Hartree-Fock (HF) theory stands as a cornerstone method for studying the electronic structures of atoms and molecules. This foundational technique provides a means of approximating the wave function and energy of a quantum many-body system in a computationally efficient manner. Since its development, Hartree-Fock theory has been used to gain insights into chemical bonding and reactions, providing a starting point for more sophisticated approaches.

Background and basics

The journey to understanding molecules begins with solving the Schrödinger equation for systems of interacting electrons and nuclei. For systems with N electrons, this involves taking into account electron-electron interactions, which are complex and cannot be solved analytically for systems beyond the hydrogen atom. Hartree-Fock theory simplifies this problem by using a mean-field approach.

Schrödinger equation

The time-independent Schrödinger equation for a system is given as:

ĤΨ = EΨ

where Ĥ is the Hamiltonian operator, Ψ is the wave function of the system, and E is the energy of the system.

Wave function and Slater determinant

In HF theory, the wave function of several electrons is approximated as a single Slater determinant, which is an antisymmetrized product of a one-electron wave function or orbitals. The Slater determinant ensures that the wave function satisfies the antisymmetry requirement due to the Pauli exclusion principle.

The Slater determinant for N electrons is expressed as:

Ψ(1,2,...,N) = (1/√N!) * | ψ₁(1) ψ₂(1) ... ψ_N(1) | | ψ₁(2) ψ₂(2) ... ψ_N(2) | | . . . | | . . . | | ψ₁(N) ψ₂(N) ... ψ_N(N) |

Each ψ_i refers to a molecular orbital, which is itself a linear combination of atomic orbitals.

Hartree–Fock method

Mean-field approximation

In the HF framework, each electron effectively moves in the average field created by all the other electrons, thereby simplifying the many-electron problem into a set of one-electron problems. This approach introduces the Fock operator ( F ), which replaces the Hamiltonian operator in the Schrödinger equation, turning it into the Fock equations:

Fψ_i = ε_iψ_i

where F includes the kinetic energy of the electrons, their interactions with nuclei, and the average interactions with other electrons.

Self-consistent field (SCF) method

The Fock operator contains terms dependent on its solution, making the equations non-linear. As a result, HF uses an iterative procedure known as the self-consistent field method. The steps are as follows:

  1. Begin by estimating the molecular orbitals.
  2. Construct the Fock matrix using stream orbitals.
  3. Solve the Fock equations to obtain the new orbitals.
  4. Check for convergence; if not convergence update the orbitals and repeat.

Visual representation of the iterative SCF process

Estimate the orbitals Construction of the Fock Matrix Solve the Fock equation Check convergence Converged? Done

Boundaries and extent

Hartree-Fock theory, although invaluable, has its limitations. Electron correlation, the interaction between simultaneous electron motions, is excluded as a result of the main assumption of the mean-field approach. As a result, HF theory often underestimates the total energy. More accurate methods such as configuration interaction (CI), Møller-Plesset perturbation theory (MP), and coupled cluster (CC) theory have been developed to address this by incorporating electron correlation effects.

Mathematical formalism

The Fock operator F has several components:

F = Ĥ_core + J - K
  • Ĥ_core: the core Hamiltonian, including kinetic energy and nuclear attraction.
  • J: Coulomb operator, representing the electrostatic repulsion between electrons.
  • K: exchange operator, accounting for antisymmetry and exchange effects.

The Coulomb and exchange terms are central to HF calculations:

J_j(ψ_i) = ∫|ψ_j(r')|²|rr'|⁻¹dr'

For Exchange Operator:

K_j(ψ_i) = ∫(ψ_j(r')*ψ_i(r'))|rr'|⁻¹ψ_j(r)dr'

These integrals are complex, and often require computational resources to evaluate accurately.

Illustration of orbital interaction

Nucleus Coulomb ψ₁ exchange ψ₂

Application

Hartree–Fock theory serves as an important tool in computational chemistry, helping to predict a variety of chemical properties and behaviors, including bond lengths, dipole moments, and electronic spectra. It serves as a stepping stone to more complex ab initio methods and has wide applications in fields ranging from drug design to materials science. The accuracy of results from HF is often improved by using larger basis sets, which expand the space of possible wave functions by using more atomic orbitals.

Conclusion

Hartree-Fock theory has enduring relevance in the field of quantum chemistry. Although it makes a number of approximations that limit its accuracy, particularly with regard to electron correlation, it remains an important starting point for understanding and simulating molecular systems. For those entering the theoretical and computational aspects of chemistry, mastering HF theory is an essential step toward exploring more sophisticated methods that capture the complexity of electron interactions in greater detail.


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Hartree–Fock theory

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