Many-Electron Atoms — formula sheet

The three new ideas of Part G. (i) Electrons carry an intrinsic angular momentum of magnitude √3ℏ/2 — spin — that has no classical origin and does not come out of the Schrödinger equation. (ii) Electrons are strictly indistinguishable, so the wavefunction can only depend on which states are occupied, not on which electron is where. (iii) For electrons, exchanging any two of them must reverse the sign of the total wavefunction. From (iii) alone the Pauli exclusion principle, the exchange energy and Hund's first rule all follow.

Spin: an intrinsic angular momentum carried by the electron itself, of fixed quantum number s = ½, independent of its motion. It is not the electron spinning; it is an internal degree of freedom with the mathematics of an angular momentum. It contributes to the magnetic moment of the atom and to the angular-momentum bookkeeping, and it is not derivable from the Schrödinger equation.

Spin-orbital: the product of a spatial orbital and a spin function — 1sα, 1sβ, 2p_zβ, and so on. A spin-orbital is what an electron actually occupies. Each spatial orbital supports exactly two spin-orbitals, which is why an orbital ‘holds two electrons’. The complete label of an electron in an atom is therefore (n, l, m_l, m_s) — four numbers, not three.

Indistinguishability: the statement that no observable quantity can depend on which of two identical particles is which. Formally, all measurable properties must be unchanged when the labels of any two identical particles are interchanged.

The antisymmetry principle (the Pauli principle, properly stated): the total wavefunction of a system of electrons must be antisymmetric with respect to the interchange of the complete set of coordinates — space and spin — of any two electrons. This is a postulate of quantum mechanics of the same standing as the Schrödinger equation itself.

The Pauli exclusion principle: no two electrons in an atom may have the same set of all four quantum numbers (n, l, m_l, m_s); equivalently, no spin-orbital may be occupied more than once. It is a consequence of the antisymmetry principle, valid only for the special case of a wavefunction written as a single product of orbitals, and it is the weaker of the two statements.

Slater determinant: the antisymmetric N-electron wavefunction built from N spin-orbitals χ_1, …, χ_N as the determinant of the N × N matrix whose i-th row is spin-orbital χ_i and whose j-th column is electron j, normalised by 1/√(N!). It is the simplest wavefunction consistent with antisymmetry, and it is what the orbital picture of chemistry actually means.

Coulomb integral J: J = ∫∫ |a(1)|² (1/r_12) |b(2)|² dτ_1dτ_2. The ordinary electrostatic repulsion between the charge cloud of orbital a and the charge cloud of orbital b. It is always positive and it has a perfectly classical meaning.

Exchange integral K: K = ∫∫ a*(1)b(1) (1/r_12) b*(2)a(2) dτ_1dτ_2. Built not from either charge density but from the overlap density a(r)b(r). It has no classical analogue whatsoever, it is positive for real orbitals, and it vanishes if the two orbitals do not overlap.

Hund's first rule: among the terms arising from a given configuration, the one with the highest spin multiplicity lies lowest in energy. Equivalently, within a set of degenerate orbitals, electrons occupy them singly with parallel spins before any orbital is doubly occupied.

The orbital approximation (the independent-particle model): the assumption that the many-electron wavefunction can be written as a product (properly, an antisymmetrised product — a Slater determinant) of one-electron functions. It is false for any atom with more than one electron. It is nevertheless the foundation of every orbital picture in chemistry, and G.5 shows how to make it as accurate as it can possibly be.

Screening (shielding): the reduction in the nuclear charge experienced by one electron because of the electron density of the others between it and the nucleus. Quantified by the effective nuclear charge Z_eff = Z − σ, where σ is the screening constant. It is not a fudge factor: for helium it is derivable exactly, and it is 5/16.

Self-consistent field (SCF): the iterative procedure of guessing a set of orbitals, using them to build the average field, solving the one-electron equations in that field to obtain new orbitals, and repeating until the orbitals stop changing. When the orbitals that come out are the same as those that went in, the field is self-consistent and the calculation has converged.

The Fock operator: F̂ = ĥ + Σ_j(2Ĵ_j − K̂_j) for a closed shell, where ĥ is the one-electron (kinetic + nuclear) operator, Ĵ_j is the Coulomb operator representing the average repulsion from the charge cloud of orbital j, and K̂_j is the exchange operator, which has no classical analogue and acts by swapping the orbitals of the two electrons.

The Hartree–Fock limit: the energy that would be obtained with a complete (infinite) basis set at the Hartree–Fock level. It is the best a single Slater determinant can do, and it is not the exact energy. The gap between the two is the correlation energy, defined precisely in G.6.

Koopmans' theorem: within the frozen-orbital approximation, the energy required to remove an electron from occupied orbital i is −ε_i, and the energy released on adding an electron to virtual orbital a is −ε_a. It gives ionisation energies directly from a single calculation on the neutral species.

Correlation energy: E_corr = E_exact, non-relativistic − E_Hartree–Fock limit. It is negative by construction (the exact energy is lower), it is defined only relative to a specified Hamiltonian and basis limit, and it is not a physical observable — it is the error of a particular approximation.

Configuration: a statement of which orbitals are occupied and by how many electrons. Term: a group of states of the same energy, arising from one configuration, characterised by definite total orbital angular momentum L and total spin S. Level: a subdivision of a term by the total angular momentum J. State: a single quantum state, labelled additionally by M_J. One configuration → several terms → several levels → many states.

Russell–Saunders (LS) coupling: the scheme in which the individual orbital angular momenta are first coupled together to give a total L, the individual spins are coupled together to give a total S, and only then are L and S coupled to give J. It is appropriate when the electrostatic repulsion between electrons is much larger than the spin–orbit interaction, which is true for light atoms.

Hund's rule 1 (maximum multiplicity): the term with the largest S lies lowest. Reason (G.3): high S forces a symmetric spin function, hence an antisymmetric spatial function, hence a Fermi hole, hence a smaller ⟨1/r_12⟩. It is electrostatics enforced by symmetry, not magnetism.

Hund's rule 2 (maximum L): among terms of equal S, the one with the largest L lies lowest. Reason: a large L means the electrons are circulating in the same sense and so tend to stay on opposite sides of the nucleus, which again reduces the average repulsion. This rule is on weaker ground than the first and has genuine exceptions.

Hund's rule 3 (J): for a subshell that is less than half full, the level with the smallest J lies lowest (a regular multiplet); for one more than half full, the largest J lies lowest (an inverted multiplet). Reason: the sign of the spin–orbit coupling constant reverses when electrons are replaced by holes. At exactly half filling L = 0, only one J exists, and the rule is silent.

j–j coupling: the scheme used when spin–orbit coupling is stronger than the electrostatic repulsion. Each electron's own l and s are coupled first to give an individual j_i, and the j_i are then coupled to give J. L and S cease to be meaningful labels; only J survives.