8: Multielectron Atoms
Electrons with more than one atom, such as Helium (He), and Nitrogen (N), are referred to as multi-electron atoms. Hydrogen is the only atom in the periodic table that has one electron in the orbitals under ground state. We will learn how additional electrons behave and affect a certain atom.
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- 8.1: Atomic and Molecular Calculations are Expressed in Atomic Units
- This page discusses the benefits of using atomic units (au) in atomic physics, emphasizing their role in simplifying calculations compared to SI units. It details how atomic units standardize mass, charge, and Planck's constant, allowing for a clearer focus on important physical factors. The article compares the Hamiltonian of a helium atom in both unit systems, illustrating the clarity atomic units provide.
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- 8.2: Perturbation Theory and the Variational Method for Helium
- This page explores methods for solving the helium atom's electron structure using perturbation theory and variational methods. It highlights the refinement of energy estimates by incorporating electron-electron interactions, notably showing that using complex trial wavefunctions leads to results within 0.08% of experimental values. The Chandrasakar wavefunction notably reduces electron-electron repulsion and achieves an accuracy of 0.07%.
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- 8.3: Hartree-Fock Equations are Solved by the Self-Consistent Field Method
- This page explores the Hartree approximation's role in determining wavefunctions and energies for multi-electron atoms by treating electrons as independent entities interacting through an average potential. It outlines the Self-Consistent Field (SCF) method and its iterative approach to solving the Schrödinger equation. The Hartree-Fock method enhances accuracy with antisymmetrized wavefunctions and addresses the challenges of electron-electron repulsion.
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- 8.4: An Electron has an Intrinsic Spin Angular Momentum
- This page explores electron spin, an intrinsic angular momentum linked to magnetic properties and quantum mechanics. It highlights two spin states (α and β) with different energies influenced by magnetic fields, demonstrated by the Zeeman effect and the Stern-Gerlach experiment. The g-factor, approximately 2.0023, relates to the spin gyromagnetic ratio, indicating that electron spin is more complex than simple orbital motion.
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- 8.5: Wavefunctions must be Antisymmetric to Interchange of any Two Electrons
- This page explores quantum mechanics principles for multi-electron atoms, highlighting indistinguishable particles and the Pauli Exclusion Principle. It explains the unchanged probability density when electrons swap positions and introduces the Schrödinger equation for helium. Additionally, it discusses exchange symmetry in particles, distinguishing between bosons and fermions through the exchange operator, \(\hat{P}_{12}\).
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- 8.6: Antisymmetric Wavefunctions can be Represented by Slater Determinants
- This page covers the Pauli Exclusion Principle and its application in constructing antisymmetric wavefunctions for multi-electron atoms like helium and carbon. It details the importance of Slater determinants for ensuring antisymmetry in wavefunctions due to electron indistinguishability, illustrating various configurations and the complexity of calculations.
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- 8.7: Hartree-Fock Calculations Give Good Agreement with Experimental Data
- This page explains the distinction between the Hartree and Hartree-Fock methods in quantum mechanics, highlighting their treatment of indistinguishable electrons. The Hartree method is flawed for neglecting antisymmetry, while Hartree-Fock improves accuracy by using Slater determinants to account for the Pauli exclusion principle and electron interactions. It discusses the calculation of orbital energies and their relation to ionization energy through Koopmans' theorem.
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- 8.8: Term Symbols Gives a Detailed Description of an Electron Configuration
- This page explains the relationship between electron configurations and angular momentum, focusing on quantum numbers such as total orbital angular momentum (L), total magnetic quantum number (M_l), total spin magnetic quantum number (M_s), and total intrinsic spin quantum number (S).
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- 8.9: The Allowed Values of J - the Total Angular Momentum Quantum Number
- This page discusses L-S coupling in multi-electron atoms, particularly for lighter elements, where orbital and spin angular momenta combine to define total angular momentum and corresponding term symbols formatted as \(^{2S+1}L_J\). It illustrates this with the hydrogen atom's ground state term symbol \(^2S_{1/2}\), reflecting its spin states and angular momentum. Additionally, the text provides exercises on writing term symbols for various quantum numbers.
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- 8.10: Hund's Rules Determine the Term Symbols of the Ground Electronic States
- This page explains Hund's three rules for electron configurations in multi-electron systems. It emphasizes maximizing spin multiplicity, orbital angular momentum, and stability based on orbital fill levels. The first rule states electrons occupy empty orbitals to reduce repulsion. The second maximizes total angular momentum, and the third differentiates between less than and greater than half-filled configurations.
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- 8.11: Using Atomic Term Symbols to Interpret Atomic Spectra
- This page explores spin-orbit coupling in atomic spectroscopy, detailing its role in fine structure, which explains closely spaced spectral lines in hydrogen and sodium atoms. It discusses selection rules for electronic transitions, particularly in heavier atoms, and provides specific examples, such as the sodium D-line (589.0 nm and 589.6 nm) and the hydrogen \(H_{\alpha}\) line (656.279 nm), both influenced by spin-orbit interactions.
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- 8.E: Multielectron Atoms (Exercises)
- This page discusses solutions to various physics problems in atomic structures and quantum mechanics, focusing on the speed of electrons in the Bohr model, angular dependencies in Hartree-Fock approximations, and term symbols for elements like carbon and halogens. It details calculations for term symbols of electron configurations, including \(2P\) and \(2S\), with specific results for calcium and vanadium.
Thumbnail: Neon Atom. (CC BY 3.0 Unported; BruceBlaus via Wikipedia )