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Orbital Angular Momentum, R = 0, 1, 2,

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Orbital Angular Momentum, R = 0, 1, 2, Atomic Spectroscopy of Polyelectronic Atoms Atoms, Energy Levels and Spectroscopy Energy Levels in Atoms (13.2, 13.3, 13.4, 13.5) Coupling of Angular Momenta (13.7, 13.8) Term Symbols and Selection Rules (13.9) Russell-Saunders Coupling (13.9) Examples Hydrogen and Alkali Metal Atoms Helium Selection rules and other polyelectronic atoms Supplementary Notes Coupling of Angular Momenta Hyperfine Splitting Zeeman Interaction Stark Effect Atomic Spectroscopy Electronic spectroscopy is the study of absorption and emission transitions between electronic states in an atom or a molecule. Atomic spectroscopy is concerned with the electronic transitions in atoms, and is quite simple compared to electronic molecular spectroscopy. This is because atoms only have translational and electronic degrees of freedom, and sometime influenced by nuclear spin (molecules also have vibrations and rotations). Recall (Lecture 5) that for hydrogen and hydrogen-like ions (i.e., He+, Li2+, Be3+, etc.) with a single electron and a nucleus with charge +Ze, the hamiltonian is: £ 2 Ze 2 ,'& L2 & 2µ 4B,0r µ = (memp)/(me + mp) reduced mass e = 1.6021 × 10-19 C charge on an electron r = distance between +e and -e -12 2 2 -1 -3 ,0 = 8.854 × 10 C s kg m permitivity of free space For a polyelectronic atom, the hamiltonian becomes: 2 2 2 £ 2 Ze e ,'& j Li & j % j 2me i i 4B,0ri i<j 4B,0rij summed over i electrons. The new term describes coulombic repulsions between pairs of electrons which are distance rij apart (second term is coulombic attraction between nucleus and electrons) SCF Method The electron-electron repulsion term creates a many-body problem, in which the hamiltonian cannot be broken down into a sum of contributions from each electron - the result is that the Schroedinger equation cannot be solved exactly. Methods of approximation have been devised, including that of Hartree, who replaces the electron repulsions with a sum of potential energy functions from separate electrons. This is known as the self-consistent field (SCF) method, under which the Schroedinger equation can be solved. 2 2 £ 2 Ze , • & j Li & j % j V(ri) 2me i i 4B,0ri i The electron repulsions remove degeneracy of orbitals with different orbital angular momenta (like 3s, 3p, 3d, which are degenerate in the H atom) - see next page The atomic orbital (AO) energies vary with the principal quantum number n = 1, 2, 3, ... and the orbital angular momentum, R = 0, 1, 2, ... for s, p, d, ... orbitals. Plots of the SCF Hartree-Fock radial distribution functions are shown for sodium - electron density is grouped into shells, as anticipated by early chemists! Electron Configurations The value of the orbital energy increases with the nuclear charge of the atom (the energy to remove an electron from the 1s orbital, the ionization energy, is 13.6 eV for He and 870.4 eV for Ne). A comparison of the AO energy levels in a hydrogen-like atom/ion and a polyelectronic atom/ion are shown below: The aufbau or building-up principle: electrons are “fed” into the orbitals in order of increasing energy until the electrons are used up - this gives the ground configuration of the atom. The Pauli-exclusion principle: no two electrons may have the same set of quantum numbers: n, R, mR, ms. Since mR can have 2R + 1 possible values and ms = ±½, each orbital characterized by n and R can have 2(2R + 1) electrons. Configurations & States Orbital: particular values of n and R Shell: all orbitals with same value of n For n = 1, 2, 3, 4, ... shells are labelled K, L, M, N, ... A configuration describes the manner in which the electrons are distributed among the orbitals, but a given configuration may give rise to more than one state - the nature of the states depend on how the electrons couple with one another, resulting in states of different energies. Trends in the periodic table: alkali metals: outer ns1 configuration, monovalent alkaline earth metals: outer ns2 configuration, divalent noble gases: outer np6 configuration, filled orbital sub-shell results in their characteristic chemical inertness first transition series: characterized by filling of the 3d orbital (3d and 4s are similar in energy, but their separation changes along the series). Examples: Cu has a KLM4s1 or KL3d104s1 ground configuration, the completely filled 3d10 orbital has an “innate stability” Cr has a KL3s23p63d54s1 ground configuration, in some cases half-filled d-orbitals confer stability onto a particular configuration lanthanides: filling of the 4f orbital is characteristic of the lanthanides, but since 4f and 5d are of similar energy, in some cases one electron will go into the 5d orbital Angular Momenta & Magnetic Moments An electron in an atom has two possible sources of angular momenta: orbital angular momentum and spin angular momentum. The orbital angular momentum vector for a single electron is given by: [R(R % 1)]1/2£ ' R(£ where R = 0, 1, 2, ... (the quantity [A(A+1)]1/2 happens so much in these discussions that we denote it as A*) The spin angular momentum vector for a single electron is: [s(s % 1)]1/2£ ' s (£ where s = ½. For an electron with both types of angular momentum, there is a quantum number j, assigned to the total angular momentum, which is a vector quantity: [j(j % 1)]1/2£ ' j (£ j ' R % s, R % s & 1, ..., |R & s| For one-electron atoms, where s = ½, j is not very useful (unless considering fine structure), but the quantum number J (for polyelectronic atoms) is very important. If the nucleus has a non-zero spin quantum number I, there may be a coupling from nuclear spin angular momentum: [I(I % 1)]1/2£ ' I (£ Since the nucleus is so large compared to the electron, this angular momentum is considerably smaller, so can normally be neglected - however, if looking at fine structure, this additional angular momentum results in hyperfine splitting in atomic spectra. Angular Momenta & Magnetic Moments, 2 A charge -e circulating in an orbit is equivalent to current flowing through a circular wire. The latter causes a magnetic field perpendicular to the plane of the loop: while the former results in a magnetic moment. R R s µs µ s s µR µR The magnetic moment µR from orbital angular momentum is opposite direction to the orbital angular momentum vector, R. The classical picture of the electron spinning about its own axis gives a magnetic moment µs in the opposite direction to the spin angular momentum vector s. µR and µs may be parallel or antiparallel, and can be regarded as tiny classical bar magnets that interact Singlet & Triplet States He: 1s12s1 excited configuration (ground: 1s2), so electrons do not have to be paired since they are in different orbitals. According to Hund’s rules, the state of the atom with spins parallel lies lower in energy than the state in which they are paired (though both states are allowed). Parallel and antiparallel spins differ in overall angular momentum. When they are paired, zero net spin, called the singlet state: &1/2 F&(1,2) ' 2 ["(1)$(2) & "(2)$(1)] The spin angular momenta of two parallel spins add together for non-zero total spin, this is the triplet state: "(1)"(2) $(1)$(2) &1/2 F%(1,2) ' 2 ["(1)$(2) % "(2)$(1)] In the triplet state, the parallel pair of spins are not precisely parallel; rather, the angle between the spin angular momentum vectors is always In the singlet state, the constant, so all three anti-parallel pair of arrangements have spins are precisely anti- the same total spin parallel, and the angular momentum. resultant vector is zero. Coupling of Angular Momenta The interaction of µR and µs (like small classical bar magnets) is referred to as coupling of angular momenta. The larger the magnetic moments involved, the greater the size or strength of the coupling. The coupling of two vectors a and b produces a resultant vector c. If these vectors represent angular momenta, then a and b undergo precession around c. The rate of precession (or precession frequency) increases as the strength of the coupling increases. Normally, c precesses about some arbitrary direction in space, unless in the presence of an electric or magnetic field (Stark and Zeeman effects, respectively), in which case space quantization may be observed. The spin of one electron can interact with: (a) spins of other electrons (b) its own orbital motion (ls or jj- coupling, often important only for few states of heavy atoms) (c) orbital motion of other electrons (very small) Russell-Saunders Coupling If we make the approximation that (i) the coupling between the spin of an electron and its orbital momentum can be neglected (no jj-coupling), but that (ii) coupling between orbital momenta is significant and (iii) coupling between spin momenta is weak but appreciable, then this is the opposite extreme to jj-coupling known as the Russell- Saunders coupling approximation, and provides a useful way of describing the states of most atoms. Non-equivalent electrons are those which have different values of n or R: e.g., 3p13d1 or 3p14p1 configurations have non-equivalent electrons, but 2p2 electrons are equivalent. Consider the coupling of orbital angular momenta of two non-equivalent electrons, which is known as RR coupling. For example, the He atom with excited configuration 2p13d1, where the 2p and 3d electrons are labelled 1 and 2 so that R1 = 1 and R2 = 2, which have vectors with magnitudes of 21/2£ and 61/2£, respectively. These vectors couple to give a resultant L of magnitude: [L(L % 1)]1/2£ ' L (£ The values of the total orbital angular momentum number L are limited, since the relative orientations of R1 and R2 are limited to the Clebsch-Gordan series: L ' R1 % R2, R1 % R2 & 1, ..., |R1 & R2| In this example, L = 3, 2 or 1 and magnitude of L is 121/2£, 61/2£ or 21/2£ (see next page).
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