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Multi-electron Atom Pauli Exclusion Principle Atomic Structure and the Periodic Table Total Angular Momentum Anomalous Zeeman Effect Beyond Hydrogen Atom CHAPTER 9 Beyond Hydrogen Atom What distinguished Mendeleev was not only genius, but a passion for the elements. They became his personal friends; he knew every quirk and detail of their behavior. - J. Bronowski Dimitri Mendeleev

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Independent Particle Approximation The potential of an electron depends on its distance from the nucleus as well as the its distance from other electron. The independent Particle approximation simply assumes that the potential for each particle is the same as that of a single electron orbiting the nucleus like that in the Hydrogen atom.

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Multi-electron atoms When more than one electron is involved, the potential and the wave function are functions of more than one position: Solving the Schrodinger Equation in this case can be very hard. But we can approximate the solution as the product of single- particle wave functions: And it turns out that, for electrons (and other spin ½ particles), all the i ’s must be different. This is the Pauli Exclusion Principle.

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Pauli Exclusion Principle To understand atomic spectroscopic data, Pauli proposed his exclusion principle: No two electrons in an atom may have the same set of quantum numbers (n, ℓ, m ℓ, m s ). It applies to all particles of half-integer spin, which are called fermions, and particles in the nucleus are fermions. The periodic table can be understood by two rules: The electrons in an atom tend to occupy the lowest energy levels available to them. The Pauli exclusion principle.

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Symmetry of the Wave function Two electron Schrodinger equation The probability density does not change if we interchange the particles The total wave function has two possibilities symmetric Antisymmetric

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Symmetry of the Wave function If a and b are the four quantum numbers of the two electrons then we can construct linear combinations Symmetric Antisymmetric For two electrons with symmetric wave function the two particles can exist simultaneously with a=b, however the anitsymmetric wave function is zero if a=b, the electrons cannot be in the same state. So according to quantum mechanics the anitsymmetric wave function is consistent with the Pauli's exclusion principle.

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Example 9.1 An atom with four electrons is in an excited state. One electron is in energy level 1.04 eV, one electron is in energy level 4.16 eV and two are in energy level 9.36 eV. When the atom returns to its ground state what would the possible energies of the photons.

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Atomic Structure Hydrogen: (n, ℓ, m ℓ, m s ) = (1, 0, 0, ±½) in ground state. In the absence of a magnetic field, the state m s = ½ is degenerate with the m s = −½ state. Helium: (1, 0, 0, ½) for the first electron. (1, 0, 0, −½) for the second electron. Electrons have anti-aligned (m s = +½ and m s = −½) spins. The principle quantum number also has letter codes. n = Letter = KLMN… n = shells (eg: K shell, L shell, etc.) nℓ = subshells (e.g.: 1s, 2p, 3d) Total number of electrons in any shell is Electrons for H and He atoms are in the K shell. H: 1s 2 He: 1s 1 or 1s

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Atomic Structure How many electrons may be in each subshell? Recall: ℓ = … letter = s p d f g h … ℓ = 0, (s state) can have two electrons. ℓ = 1, (p state) can have six electrons, and so on. The lower ℓ values have more elliptical orbits than the higher ℓ values. Electrons with higher ℓ values are more shielded from the nuclear charge. Electrons lie higher in energy than those with lower ℓ values. 4s fills before 3d. Total For each m ℓ : two values of m s 2 For each ℓ: (2ℓ + 1) values of m ℓ 2(2ℓ + 1)

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Examples 9.2 write down all possible sets of quantum numbers for an electron in a 4f and 2p subshels. 9.3 Suppose the outer electron in a potassium atom is in a state with l=2. Compute the magnitude of L. What are the possible values of j and the possible magnitudes of J? 9.4 write down the electron configuration of Carbon. 9.5 what element has this ground state electron configuration 1s 2 2s 2 2p 6 3s 2 2p which of the following atoms would you expect to have its ground state split by the spin orbit interaction: Li, B, Na

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The Periodic Table

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Groups and Periods Groups: Vertical columns. Same number of electrons in an ℓ orbit. Can form similar chemical bonds. Periods: Horizontal rows. Correspond to filling of the subshells.

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The Periodic Table Inert Gases: Last group of the periodic table Closed p subshell except helium Zero net spin and large ionization energy Their atoms interact weakly with each other Fig a) Closed shell at 2, 10, 18, 36 and 54 Fig b) Increase in atomic radius

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The Periodic Table Halogens: Need one more electron to fill outermost subshell Form strong ionic bonds with the alkalis More stable configurations occur as the p subshell is filled Transition Metals: Three rows of elements in which the 3d, 4d, and 5d are being filled Properties primarily determined by the s electrons, rather than by the d subshell being filled Have d-shell electrons with unpaired spins As the d subshell is filled, the magnetic moments, and the tendency for neighboring atoms to align spins are reduced

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The Periodic Table Lanthanides (rare earths): Have the outside 6s 2 subshell completed As occurs in the 3d subshell, the electrons in the 4f subshell have unpaired electrons that align themselves The large orbital angular momentum contributes to the large ferromagnetic effects Actinides: Inner subshells are being filled while the 7s 2 subshell is complete Difficult to obtain chemical data because they are all radioactive Have longer half-lives

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Coupling of Angular Momenta Corrections to Energy Levels Orbital – Orbital Coupling, spin-spin coupling and spin orbit coupling, jj coupling Orbital-Orbital Coupling Important because coulomb repulsion between electrons means the electron probability density is dependent on L. Atoms with low Z have the largest effect on energies For 2 electron system: States with larger value of l have lower energy

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Coupling of Angular Momenta Spin-Spin Coupling Magnetic moment between electrons is weak, but for atoms with low Z values, the Pauli exclusion principle is important. For 2 electron system If spins are parallel the energy levels are further apart then when spins are antiparallel. Total wave function is antisymmetric so if the sping part is symmetric, the space part must be antisymmetric and thus Pauli principle requires the energy level to be further apart. One of the energy level is lower than normal

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Coupling of Angular Momenta Spin-Orbit Coupling Much smaller effect on energy levels if Z is not very large. Often called LS coupling or Russell-Saunders coupling.

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Coupling of Angular Momenta J-J Coupling For large Z values the spin-orbit coupling is much stronger than the spin-spin and orbital-orbital coupling. In this case we first find the spin-orbit coupling correction and do the other two next.

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Many-Electron Atoms Hund’s rules: The total spin angular momentum S should be maximized to the extent possible without violating the Pauli exclusion principle. Insofar as rule 1 is not violated, L should also be maximized. For atoms having subshells less than half full, J should be minimized. For a two-electron atom There are LS coupling and jj coupling to combine four angular momenta J.

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Examples 9.7 Consider a system of two electron each with l=1 and s=1/2. a) what are possible values of the quantum number L for the total angular momentum? b) what are the possible values of the quantum number S for the total spin? c) find the possible quantum numbers j for the combination J=L+S. d) what are the possible quantum number j 1 and j 2 for the total angular momentum of each particle? e) find the possible values of j from the combination j 1 and j 2

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