Introduction to Hartree–Fock and Post–Hartree–Fock Methods

One of the ultimate goals of modern quantum chemistry is to solve the Schrödinger equation, which describes the electronic structure and properties of atoms and molecules based on fundamental physical principles. This ab initio approach provides the ability to make theoretical predictions without relying on experimental data. Typically, the Born–Oppenheimer approximation is adopted, assuming that nuclei move much more slowly than electrons, thereby reducing the problem to solving the electronic structure for a fixed nuclear configuration. However, even after this simplification, an analytical solution to the electronic Schrödinger equation for many-electron systems is not available. This limitation has led to the development of a hierarchy of methods in electronic structure theory that approach reality with varying degrees of accuracy. At the foundation of this hierarchy lies the Hartree–Fock (HF) theory, which also forms the conceptual backbone of molecular orbital (MO) theory.

Theoretical Foundations of the Hartree–Fock (HF) Approach

The Hartree–Fock method simplifies the complex wave function of an N-electron system by representing it with a single Slater determinant. The Slater determinant is an antisymmetric combination of single-electron wave functions known as spin–orbitals, and it naturally ensures compliance with the Pauli exclusion principle, which requires that the wave function change sign under the exchange of any two electrons (1).

Figure 1. Pauli Exclusion Principle (2).

The foundation of this method lies in the variational principle. According to this principle, the energy calculated using any trial wave function is always greater than or equal to the true ground-state energy of the system. The goal of the Hartree–Fock procedure is to find the best set of spin–orbitals that minimize the energy within the space of wave functions represented by a single Slater determinant. This minimization is carried out through an iterative process known as the Self-Consistent Field (SCF) procedure. The SCF process involves solving the Hartree–Fock–Roothaan equations, formulated by C. C. J. Roothaan for molecular systems, which are a set of one-electron equations (3). These equations are expressed as a pseudo–eigenvalue problem of the Fock operator, which includes the kinetic energy of each electron, its attraction to the nuclear potential, and its interaction with the average potential field created by all other electrons. The average electron–electron repulsion in the Fock operator includes both the classical Coulomb repulsion (Coulomb operator) and a quantum mechanical term known as the exchange interaction (exchange operator), which arises from the Pauli exclusion principle and acts only between electrons with the same spin. The iterations terminate when the field generated by these orbitals (the Fock operator) becomes consistent with the orbitals obtained from that field.

Limitations of Hartree–Fock Theory: The Electron Correlation Phenomenon

Although the HF method provides a qualitatively valuable picture, the mean-field nature of the approach inherently neglects electron correlation. This term refers to the situation in which the instantaneous positions and motions of electrons are interdependent. In the HF framework, this instantaneous repulsion is neglected because each electron is assumed to experience only the average repulsive potential of all other electrons. Consequently, the motion of electrons which, in reality, tend to avoid each other is not accurately described, leading to incomplete energy estimations. According to the definition by P.-O. Löwdin, the difference between the exact nonrelativistic energy of a system and its Hartree–Fock limit energy is called the correlation energy (4). Electron correlation can be divided into two main categories.

Figure 2. Correlation Energy (Image generated using artificial intelligence)

Dynamic Correlation: This refers to the instantaneous effect describing the tendency of electrons to avoid each other due to their momentary Coulomb repulsion. In the mean-field of HF, each electron responds only to the average charge distribution of the others, which increases the likelihood of electrons being closer than they actually are in reality and, consequently, raises the total energy of the system.

Static (or Non-Dynamic) Correlation: This situation arises when a single Slater determinant (a single configuration) is fundamentally inadequate to describe the ground electronic state of a system. In cases such as bond dissociation or molecules with degenerate or near-degenerate orbitals (e.g., diradicals or certain excited states), a mixture of several low-energy electronic configurations is required. HF can completely fail to describe such systems.

The post–Hartree–Fock methods discussed here primarily aim to correct dynamic correlation. Systems dominated by static correlation generally require the use of multi-reference methods.

Post-Hartree-Fock Methods and Correlation Corrections

To account for electron correlation and achieve more accurate results, a series of systematic methods have been developed that use the HF solution as a starting point.

Configuration Interaction (CI): CI expands the wave function as a linear combination of determinants representing electronically excited configurations, in addition to the HF ground state. Because it is a variational method, the computed energy is always an upper bound to the true energy. However, besides its computational cost, truncated CI methods (such as CISD) have a significant theoretical weakness: they are not size-consistent. This means that the total energy obtained for two non-interacting subsystems is not equal to the sum of the energies obtained from separate calculations on each subsystem.

Møller-Plesset Perturbation Theory (MPn): This theory treats the Hartree–Fock Hamiltonian as the zeroth-order reference and considers the correlation component of electron–electron interaction as a perturbation. The approach was first proposed by C. Møller and M. S. Plesset in 1934 (5). The energy is corrected as a series expansion in the order of this perturbation. MP2 (second-order correction) is widely used because it captures a significant portion of the dynamic correlation energy at a relatively low computational cost.

Coupled–Cluster (CC) Theory: One of the most powerful and reliable methods in modern quantum chemistry for achieving highly accurate results. Adapted to quantum chemistry by J. Čížek (6), this theory constructs the wave function by applying an exponential excitation operator (eᴛ̂) to the reference HF determinant. This exponential form naturally ensures that CC theory is size-consistent, which is a major advantage over truncated CI. The most basic version, CCSD, includes single (T₁) and double (T₂) excitations. The CCSD(T) method—considered the “gold standard” for achieving chemical accuracy (1 kcal/mol)—adds the effect of triple excitations perturbatively to the CCSD calculation and was developed by Raghavachari and co-workers (7).

Figure 3. Schematic Representation of the Approach from the HF Limit to the Full CI Solution (8).

References

1.Fock, V. (1930). Näherungsmethode zur Lösung des quantenmechanischen Mehrkörperproblems. Zeitschrift für Physik. 61 (1–2), 126–148. https://doi.org/10.1007/BF01340294

2.The Pauli Exclusion Principle. (t.y.). In Physics. Lumen Learning. Erişim 29 Eylül 2025, adres:https://courses.lumenlearning.com/suny-physics/chapter/30-9-the-pauli-exclusion-principle/

3.Roothaan, C. C. J. (1951). New Developments in Molecular Orbital Theory. Reviews of Modern Physics. 23 (2), 69–89. https://doi.org/10.1103/RevModPhys.23.69

4.Löwdin, P.-O. (1955). Quantum Theory of Many-Particle Systems. III. Extension of the Hartree-Fock Scheme to Include Correlation Effects. Physical Review. 97 (6), 1509–1520. https://doi.org/10.1103/PhysRev.97.1509

5.Møller, C.; Plesset, M. S. (1934). Note on an Approximation Treatment for Many-Electron Systems. Physical Review. 46 (7), 618–622. https://doi.org/10.1103/PhysRev.46.618

6.Čížek, J. (1966). On the Correlation Problem in Atomic and Molecular Systems. Calculation of Wavefunction Components in Ursell-Type Expansion Using Quantum-Field Theoretical Methods. The Journal of Chemical Physics. 45 (11), 4256–4266. https://doi.org/10.1063/1.1727484

7.Raghavachari, K.; Trucks, G. W.; Pople, J. A. (1989). Head-Gordon, M. A fifth-order perturbation comparison of electron correlation theories. Chemical Physics Letters. 157 (6), 479–483.

8.Shikano, Y., Watanabe, H.C., Nakanishi, K.M. et al. (2021). Post-Hartree–Fock method in quantum chemistry for quantum computer. Eur. Phys. J. Spec. Top. 230, 1037–1051. https://doi.org/10.1016/S0009-2614(89)87395-6