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 ↓


Post-Hartree–Fock methods


Post-Hartree–Fock methods are a series of computational approaches in quantum chemistry designed to improve the Hartree–Fock (HF) method. The HF method, although revolutionary, approximates the many-electron wave function as a single Slater determinant, which inherently limits its accuracy. Post-Hartree–Fock methods attempt to go beyond this limitation by considering electron correlation, which is a key factor in obtaining more accurate results.

Understanding the limitations of the Hartree-Fock method

The Hartree-Fock method is the starting point for many quantum chemistry simulations. It provides an approximation to the solution of the Schrödinger equation by considering the interactions between electrons in an averaged way. The wave function in HF is expressed as a single Slater determinant, which constructs a many-electron wave function from a single-particle wave function (orbitals). The main limitation comes from its account of electron correlation.

Ψ_HF = |ψ_1ψ_2...ψ_n|

The above formula shows the Slater determinant for n electrons.

Electron correlation refers to how the motion of electrons is correlated due to their mutual repulsion. The HF method takes into account correlation (average field) to some extent, but it ignores dynamic correlation, which requires considering not just a single determinant but a combination of several determinants.

Post-Hartree–Fock methods

Several post-Hartree-Fock methods have been designed to overcome these shortcomings, introducing corrections for electron correlation. Some of the most common methods include configuration interaction (CI), Møller-Plesset perturbation theory (MP2), coupled cluster (CC), and quantum Monte Carlo (QMC). Each of these approaches has its own methodology to incorporate correlation effects and different levels of computational complexity.

Configuration interaction (CI)

Configuration interactions involve considering a wave function expressed as a linear combination of several Slater determinants. The determinants include the ground state and various excited states. This approach systematically improves HF by taking virtual excitations into account.

Ψ_CI = c_0 Ψ_HF + c_1 Ψ_2^* + c_2 Ψ_3^* + ...

where c_0, c_1, c_2... are the coefficients to be determined, and Ψ_2^*, Ψ_3^* are excited determinants.

The main drawback of CI is its scaling; as the number of electrons increases, the number of determinants also grows factorially. The full configuration interaction (FCI) represents an exact solution within a given basis set, but its calculation becomes impractical for large systems.

Møller–Plesset perturbation theory (MP2)

This method applies perturbation theory to the Hartree–Fock energy, leading to improvements. MP2, the second-order approximation, is often used because it is computationally less demanding than CI and provides a good balance between accuracy and cost.

E_MP2 = E_HF + E^(2)

where E_MP2 is the corrected energy and E^(2) is the second-order correction term.

MP2 works well for systems where electron correlation is not very obvious. However, it can be inaccurate or diverge in cases where the HF itself provides a poor starting point, such as near-degenerate systems.

Coupled cluster (CC) methods

Coupled cluster methods are among the most accurate post-HF approaches, representing a completely different way to consider correlation. The exponential form of the wave function involves excitations as clusters and is effectively characterized by its size expansion, i.e., its energy grows linearly with the number of electrons in a system.

Ψ_CC = exp(T) Ψ_HF

where T is the cluster operator with single (T_1), double (T_2), or higher excitations.

One of the most popular forms is the CCSD(T) method, which includes single and double excitations, and a perturbative treatment for triple excitations, which is often referred to as the "gold standard" for quantum chemistry due to its balance of accuracy and computational cost.

Quantum Monte Carlo (QMC)

Quantum Monte Carlo represents a fundamentally different approach to increasing the accuracy of quantum chemical calculations using statistical methods. It is particularly useful for large systems and can provide highly accurate results.

Consider a function that uses random sampling to find a solution to a many-electron problem. Accuracy and efficiency depend largely on the quality of the random sampling.

Although QMC methods are very powerful, they are complex to implement and interpret compared to more deterministic methods such as CI or CC.

Visual representation

To further understand the difference in the correlation of electrons when using these advanced methods compared to HF, consider the following diagram.

Electron 1 Electron 2 Repulsion is ignored in HF

This diagram shows a simple representation of the electron interaction in Hartree–Fock.

Electron 1 Electron 2 CI/CC Correlation in effect

This visualization shows how post-HF methods create correlations, and mediate electron-electron interactions.

Strengths and applications

Each of these methods has its own unique strengths and best-suited applications. CI, while accurate, often requires extensive computational resources and is better suited for small systems. MP2 provides a good compromise between computational cost and accuracy for medium-sized systems. CCSD(T) is highly reliable and is often preferred for accurate calculations at a reasonable computational cost. QMC is used for systems where electron correlation effects are strong and complex conjugate gradient solutions are required.

Challenges of post-Hartree–Fock methods

Although these methods offer significant improvements compared to the Hartree–Fock baseline, several challenges remain:

  • Computational cost: As the size of the system grows, the computational resources required for methods such as CC and CI grow exponentially. Strategies to simplify or approximate the methods can reduce, but not eliminate, these bottlenecks.
  • Convergence issues: Some methods, particularly perturbative methods such as MP2, can struggle with convergence for systems where the HF is not a good baseline solution.
  • Implementation complexity: These methods require significant expertise to implement and the selection of appropriate basis sets and computational parameters is critical.

Conclusion

Post-Hartree-Fock methods stand out as essential tools for anyone seeking a deeper understanding of molecular systems and their properties in quantum chemistry. They build on the foundation laid by the Hartree-Fock method by introducing corrections for electron correlation – a more reliable representation of real molecular systems. Through methods such as CI, MP2, CC and QMC, chemists and physicists can make more accurate predictions of chemical phenomena, leading to applications in fields ranging from materials science to pharmaceuticals.

The choice of method depends largely on the specific requirements of the problem, including the desired accuracy and available computational resources. Continuing advances in computational algorithms and computing power promise to make these powerful techniques even more accessible in the future.


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