5: Many Electron Atoms
- Page ID
- 455290
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- 5.1: The Helium Atom Cannot Be Solved Exactly
- The second element in the periodic table provides our first example of a quantum-mechanical problem which cannot be solved exactly. Nevertheless, as we will show, approximation methods applied to helium can give accurate solutions in perfect agreement with experimental results. In this sense, it can be concluded that quantum mechanics is correct for atoms more complicated than hydrogen. By contrast, the Bohr theory failed miserably in attempts to apply it beyond the hydrogen atom.
- 5.2: Atomic and Molecular Calculations are Expressed in Atomic Units
- Atomic units form a system of natural units which is especially convenient for atomic physics calculations. There are two different kinds of atomic units, Hartree atomic units and Rydberg atomic units, which differ in the choice of the unit of mass and charge. This article deals with Hartree atomic units, where the numerical values of the following four fundamental physical constants are all unity by definition.
- 5.3: The Variational Method Approximation
- In this section we introduce the powerful and versatile variational method and use it to improve the approximate solutions we found for the helium atom using the independent electron approximation.
- 5.4: Single-electron Wavefunctions and Basis Functions
- Finding the most useful single-electron wavefunctions to serve as building blocks for a multi-electron wavefunction is one of the main challenges in finding approximate solutions to the multi-electron Schrödinger Equation. The functions must be different for different atoms because the nuclear charge and number of electrons are different. The attraction of an electron for the nucleus depends on the nuclear charge, and the electron-electron interaction depends upon the number of electrons.
- 5.5: An Electron has an Intrinsic Spin Angular Momentum
- Spin is one of two types of angular momentum in quantum mechanics, the other being orbital angular momentum. The orbital angular momentum operator is the quantum-mechanical counterpart to the classical angular momentum of orbital revolution. The existence of spin angular momentum is inferred from the Stern–Gerlach experiment, in which particles are observed to possess angular momentum that cannot be accounted for by orbital angular momentum alone.
- 5.6: Wavefunctions must be Antisymmetric to Interchange of any Two Electrons
- The probability |Ψ(r1, r2)|² should be identical to the probability |Ψ(r2, r1)|² because the electrons have no label and they cannot be told apart because of Heisenberg principle. You can naively think that Ψ(r1, r2)=±Ψ(r2, r1) but it turns out that the sign must always be minus for the electrons. This is an additional postulate of quantum mechanics.
- 5.7: Antisymmetric Wavefunctions can be Represented by Slater Determinants
- John Slater introduced an idea of a Slater determinant that is a relatively simple scheme for constructing antisymmetric wavefunctions of multi-electron systems from a product of one-electron functions in the form of a determinant.
- 5.8: Electron Configurations, The Pauli Exclusion Principle, The Aufbau Principle, and Slater Determinants
- The assignment of electrons to orbitals is called the electron configuration of the atom. We extend that idea to constructing multi-electron wavefunctions that obeys the Pauli Exclusion Principle, which requires that each electron in an atom or molecule must be described by a different spin-orbital. The mathematical analog of this process is the construction of the approximate multi-electron wavefunction as a product of the single-electron atomic orbitals.
- 5.9: The Self-Consistent Field Approximation (Hartree-Fock Method)
- In this section we consider a method for finding the best possible one-electron wavefunctions that was published by Hartree in 1948 and improved two years later by Fock.
- 5.10: Hartree-Fock Calculations Give Good Agreement with Experimental Data
- The Hartree–Fock method is a method of approximation for the determination of the wave function and the energy of quantum many-body systems. The Hartree–Fock method often assumes that the exact, N-body wave function of the system can be approximated by a single Slater determinant of N spin-orbitals. By invoking the variational method, one can derive a set of N-coupled equations for the N spin orbitals. A solution of these equations yields the Hartree–Fock wave function and energy of the system.
- 5.11: Term Symbols Gives a Detailed Description of an Electron Configuration
- The term symbol is an abbreviated description of the (total) angular momentum quantum numbers in a multi-electron atom (however, even a single electron can be described by a term symbol). Each energy level of an atom with a given electron configuration is described by not only the electron configuration but also its own term symbol, as the energy level also depends on the total angular momentum including spin. The usual atomic term symbols assume LS coupling (Russell-Saunders coupling).
- 5.12: The Allowed Values of J - the Total Angular Momentum Quantum Number
- The total angular momentum quantum number parameterizes the total angular momentum of a given particle, by combining its orbital angular momentum and its intrinsic angular momentum (i.e., its spin).
- 5.13: Hund's Rules Determine the Term Symbols of the Ground Electronic States
- The allocation electrons among degenerate orbitals can be formalized by Hund’s rule: For an atom in its ground state, the term with the highest multiplicity has the lowest energy. Hund's first rule states that the lowest energy atomic state is the one that maximizes the total spin quantum number for the electrons in the open subshell. The orbitals of the subshell are each occupied singly with electrons of parallel spin before double occupation occurs.