Understanding Multi-Electron Atoms and Electron Spin in Quantum Mechanics

1. MULTIELECTRON ATOMS • ELECTRON SPIN: • Electron has intrinsic angular momentum - “spin”; it behaves as if it were “spinning”, i.e. rotating around its axis. • rotating charge  magnetic field - the electron is a magnetic dipole. S = sħ with s =  • remember: ħ = h/2 is the “unit of angular momentum” • (In units of angular momentum,) the electron has “spin”; it is a member of the family of particles with “half-integer spin”, called “Fermions”. • Fermions obey Pauli's exclusion principle. • direction of spin is “quantized”, i.e. only certain directions are allowed. For spin 1/2, only two directions are allowed - called “up” and “down”. • PAULI'S EXCLUSION PRINCIPLE • No quantum mechanical state can be occupied by more than one fermion of the same kind (e.g. more than one electron). • MULTIELECTRON ATOMS • The hydrogen atom, having only one electron, is a very simple system; this is why Bohr's simple model worked for it. • Complications in atoms with many electrons: • in addition to the force between electron and nucleus, there are also forces between the electrons; • “shielding”: electrons in outer orbits are shielded from the force of the nucleus by electrons in the inner orbits. •  need to solve Schrödinger equation to describe multi-electron atoms.

2. QUANTUM NUMBERS • Schrödinger equation applied to atom : • electron's energy, magnitude and direction of angular momentum are quantized (i.e. only certain values allowed); • no well-defined orbit, only probability of finding electron at given position “orbital”; • the state of an electron is described by a set of four quantum numbers: • n, the principal quantum number • l , the orbital quantum number • ml= lz, the orbital magnetic quantum number • ms= sz, the spin quantum number • Meaning of quantum numbers: • the energy level of a state is determined by n andl • most probable value of distance grows with n • n = 1,2,3,…. • l= 0,1,2,3,…, n -1 (i.e. n different values); measures magnitude of angular momentum in units of ħ ; value oflinfluences the energy and the shape of the orbital: • l= 0 : spherical • l= 1 : dumbbell shaped,.... • ml= 0, 1,2, l, i.e. (2l+ 1) different values; specifies direction of angular momentum (gives component of angular momentum vector in specified direction)  determines orientation of orbital • ms = denotes direction of spin: ms = + “spin up” ms =  “spin down”

3. orbitals • Orbital shapes

4. Electron “shells” • Some definitions: • collection of orbitals with same n: “electron shell”; shells named K,L,M,N,.. • one or more orbitals with same n and l : “subshell”; • spectroscopic notation for orbitals: orbitals denoted by value of n and a letter code for the value of l : • l = 0: s • l = 1: p • l = 2: d • l = 3: f • l = 4: g • and alphabetic after that; • e.g.: “2s” refers to the subshell n=2, =0; • number of electrons in the subshell is added as a superscript; • e.g.: “2p6 means a configuration where 6 electrons are in the subshell 2p, i.e the subshell with n = 2 and l = 1. • PAULI EXCLUSION PRINCIPLE  : • No two electrons in the same atom can have the same set of four quantum numbers •  the number of electrons in a given subshell is limited to 2x(2l+ 1) (factor of 2 is due to orientation of spin), • number of electrons in a given shell is limited to 2n2 . •  in multi-electron atom, electrons cannot all sit in the lowest energy levels.

5. Electron configurations of the elements • Periods 1,2, 3, 4 • period 5

6. Electron configuration, cont’d • Period 6

7. Electron configuration, cont’d • Period 7