Lecture 11 continue Building Atoms and Molecules +e r +e yeven yodd d Plane of hydrogen atoms Content Molecular Wavefunctions Example H + H H2 Atomic Configurations Building atoms with the Pau[.]
Lecture 11_continue: Building Atoms and Molecules +e +e r d yeven yodd Plane of hydrogen atoms Content Atomic Configurations Building atoms with the Pauli exclusion principle Selection rules Molecular Wavefunctions Example: H + H H2 Pauli Exclusion Principle We now want to start building more complicated atoms to study the Periodic Table For atoms with many electrons (e.g., carbon: 6, iron: 26, etc.) - what energies they have? From spectra of complex atoms, Wolfgang Pauli (1925) deduced a new rule: “Pauli Exclusion Principle” “In a given atom, no two electrons* can be in the same quantum state, i.e they cannot have the same set of quantum numbers n, l, ml , ms” I.e., every “atomic orbital with n,l,ml” can hold electrons: () Therefore, electrons not pile up in the lowest energy state, i.e, the (1,0,0) orbital They are distributed among the higher energy levels according to the Pauli Principle Particles that obey the Pauli Principle are called “fermions” *Note: More generally, no two identical fermions (any particle with spin of ħ/2, 3ħ/2, etc.) can be in the same quantum state Filling the atomic orbitals according to the Pauli Principle n s p d f g l=0 4 Example: Na Energy En 13.6 eV Z n2 is valid only for one electron in the Coulomb potential of Z protons The energy levels shift as more electrons are added, due to electron-electron interactions Nevertheless, this hydrogenic diagram helps us keep track of where the added electrons go Z = 11 l label 1s22s22p63s1 s 1 p d f Z = atomic number = number of protons #orbitals (2l+1) exercise 1: Pauli Exclusion Principle Which of the following states (n,l,ml,ms) is/are NOT allowed? (a) (b) (c) (d) (e) (2, 1, 1, -1/2) (4, 0, 0, 1/2) (3, 2, 3, -1/2) (5, 2, 2, 1/2) (4, 4, 2, -1/2) Which of the following atomic electron configurations violates the Pauli Exclusion Principle? (a) (b) (c) (d) (e) 1s2, 2s2, 2p6, 3d10 1s2, 2s2, 2p6, 3d4 1s2, 2s2, 2p8, 3d8 1s1, 2s2, 2p6, 3d5 1s2, 2s2, 2p3, 3d11 exercise 1: Pauli Exclusion Principle Which of the following states (n,l,ml,ms) is/are NOT allowed? (a) (b) (c) (d) (e) (2, 1, 1, -1/2) (4, 0, 0, 1/2) (3, 2, 3, -1/2) (5, 2, 2, 1/2) (4, 4, 2, -1/2) ml = -l, -(l -1), … (l-1), l n>l Which of the following atomic electron configurations violates the Pauli Exclusion Principle? (a) (b) (c) (d) (e) 1s2, 2s2, 2p6, 3d10 1s2, 2s2, 2p6, 3d4 1s2, 2s2, 2p8, 3d8 1s1, 2s2, 2p6, 3d5 1s2, 2s2, 2p3, 3d11 2(2l +1) = allowed electrons 2(2l +1) = 10 allowed electrons Filling Procedure for Atomic Orbitals example: Bromine Due to electron-electron interactions, the hydrogen levels fail to give us the correct filling order as we go higher in the periodic table The actual filling order is given in the table below Electrons are added by proceeding along the arrows shown Bromine is an element with Z = 35 Find its electronic configuration (e.g 1s2 2s2 2p6 …) As you learned in chemistry, the various behaviors of all the elements (and all the molecules made up from them) is all due to the way the electrons organize themselves, according to quantum mechanics Optical Transitions between Atomic Levels Consider the hydrogenic picture: r U(r) DE f n = n = photon DE c h hc 1240 eV nm DE DE In the field of a photon, the electron may be considered as being in a superposition of two stationary states The time-dependent solution of the SEQ shows the wave function oscillating between the two eigenstates Not all transitions are possible must conserve angular momentum (and photon has ħ!) Superpositions : Stationary States: Forbidden No electric1s ± 2s dipole moment transition Dl = 1s 2s 2p www.falstad.com/qmatomrad 1s ± 2p Oscillating electric-dipole couples to photons Allowed transition Dl = ±1 Each photon carries 1ħ of angular momentum Allowed Transitions for H (You observed some of these transitions in Lab 4.) n s p d f g l=0 Energy (eV) 0.00 -0.85 -1.51 -3.40 En Selection Rule for electric-dipole transitions: Dl 1 (A few representative transitions are shown.) 13.6 eV n2 Selection Rule on m: Dm 0, 1 -13.6 eV photon is linearly polarized photon is circularly polarized ... Building atoms with the Pauli exclusion principle Selection rules Molecular Wavefunctions Example: H + H H2 Pauli Exclusion Principle We now want to start building more complicated atoms. .. complicated atoms to study the Periodic Table For atoms with many electrons (e.g., carbon: 6, iron: 26, etc.) - what energies they have? From spectra of complex atoms, Wolfgang Pauli (1925) deduced a new... (e.g 1s2 2s2 2p6 …) As you learned in chemistry, the various behaviors of all the elements (and all the molecules made up from them) is all due to the way the electrons organize themselves, according