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A certain amount of electron correlation is already considered within the HF approximation, found in the electron exchange term describing the correlation between electrons with parallel spin. This basic correlation prevents two parallel-spin electrons from being found at the same point in space and is often called Pauli correlation.
An alternative notation that is commonly used for the Pauli matrices is to write the vector index k in the superscript, and the matrix indices as subscripts, so that the element in row α and column β of the k-th Pauli matrix is σ k αβ.
For an electron in an electron gas, the exchange symmetry creates an "exchange hole" in its vicinity, which other electrons with the same spin tend to avoid due to the Pauli exclusion principle. This decreases the energy associated with the Coulomb interactions between the electrons with same spin. [3]
An example is the neutral helium atom (He), which has two bound electrons, both of which can occupy the lowest-energy states by acquiring opposite spin; as spin is part of the quantum state of the electron, the two electrons are in different quantum states and do not violate the Pauli principle.
Thomas' result convinced Pauli that electron spin was the correct interpretation of his two-valued degree of freedom, while he continued to insist that the classical rotating charge model is invalid. [34] [6] In 1927, Pauli formalized the theory of spin using the theory of quantum mechanics invented by Erwin Schrödinger and Werner Heisenberg.
The Slater determinant arises from the consideration of a wave function for a collection of electrons, each with a wave function known as the spin-orbital (), where denotes the position and spin of a single electron. A Slater determinant containing two electrons with the same spin orbital would correspond to a wave function that is zero everywhere.
While spin was known about ever since the publication of Stern and Gerlach's results in 1922, with the Pauli exclusion principle being formulated in 1925, the importance of 'spin correlation' for understanding when and why electrons form pairs in molecules was not understood until the work of Lennard-Jones in the 1950s. [5]
The appropriate two-band effective Hamiltonian is + = + where is the 2 × 2 identity matrix, , the Pauli matrices and the electron effective mass. The spin–orbit part of the Hamiltonian, H R {\displaystyle H_{\text{R}}} is parametrized by α {\displaystyle \alpha } , sometimes called the Rashba parameter (its definition somewhat varies ...