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PHY.K02UF Molecular and Solid State Physics

## Review of atomic physics (Many-electron atoms)

Many-electron atoms
The Hamiltonian for an atom with many electrons is,

$$$\label{eq:many} H_{\text{total}}= - \sum \limits_i^N \frac{\hbar^2}{2m}\nabla_i^2 - \sum \limits_i^N \frac{Ze^2}{4\pi\epsilon_0 |\vec{r}_i|} + \sum \limits_{i\lt j}\frac{e^2}{4\pi\epsilon_0 |\vec{r}_i-\vec{r}_j| }.$$$

The first sum are the kinetic energies of the electrons, the second sum are the attractive Coulomb interactions between the electrons and the nucleus, the third sum are the repulsive electron-electron interactions. A reasonable first guess for the many-electron wave function for an atom is to construct a Slater determinant from the spin orbitals that make up that wave function. To take electron screening into account, the effective nuclear charge for the atomic orbitals should determined by Slater's rules. This approximate wavefunction can be evaluated with the total Hamiltonian to determine the energy. The electron configuration with the lowest energy for each atom is listed below.

 1 H hydrogen 1s1 2 He helium 1s2 3 Li lithium 1s2 2s1 4 Be beryllium 1s2 2s2 5 B boron 1s2 2s2 2p1 6 C carbon 1s2 2s2 2p2 7 N nitrogen 1s2 2s2 2p3 8 O oxygen 1s2 2s2 2p4 9 F fluorine 1s2 2s2 2p5 10 Ne neon 1s2 2s2 2p6 11 Na sodium 1s2 2s2 2p6 3s1 =  [Ne] 3s1 12 Mg magnesium 1s2 2s2 2p6 3s2 =  [Ne] 3s2 13 Al aluminium 1s2 2s2 2p6 3s2 3p1 =  [Ne] 3s2 3p1 14 Si silicon 1s2 2s2 2p6 3s2 3p2 =  [Ne] 3s2 3p2 15 P phosphorus 1s2 2s2 2p6 3s2 3p3 =  [Ne] 3s2 3p3 16 S sulfur 1s2 2s2 2p6 3s2 3p4 =  [Ne] 3s2 3p4 17 Cl chlorine 1s2 2s2 2p6 3s2 3p5 =  [Ne] 3s2 3p5 18 Ar argon 1s2 2s2 2p6 3s2 3p6 =  [Ne] 3s2 3p6 19 K potassium 1s2 2s2 2p6 3s2 3p6 4s1 =  [Ar] 4s1 20 Ca calcium 1s2 2s2 2p6 3s2 3p6 4s2 =  [Ar] 4s2 21 Sc scandium 1s2 2s2 2p6 3s2 3p6 3d1 4s2 =  [Ar] 3d1 4s2 22 Ti titanium 1s2 2s2 2p6 3s2 3p6 3d2 4s2 =  [Ar] 3d2 4s2 23 V vanadium 1s2 2s2 2p6 3s2 3p6 3d3 4s2 =  [Ar] 3d3 4s2 24 Cr chromium 1s2 2s2 2p6 3s2 3p6 3d5 4s1 =  [Ar] 3d5 4s1 25 Mn manganese 1s2 2s2 2p6 3s2 3p6 3d5 4s2 =  [Ar] 3d5 4s2 26 Fe iron 1s2 2s2 2p6 3s2 3p6 3d6 4s2 =  [Ar] 3d6 4s2 27 Co cobalt 1s2 2s2 2p6 3s2 3p6 3d7 4s2 =  [Ar] 3d7 4s2 28 Ni nickel 1s2 2s2 2p6 3s2 3p6 3d8 4s2 =  [Ar] 3d8 4s2 29 Cu copper 1s2 2s2 2p6 3s2 3p6 3d10 4s1 =  [Ar] 3d10 4s1 30 Zn zinc 1s2 2s2 2p6 3s2 3p6 3d10 4s2 =  [Ar] 3d10 4s2 31 Ga gallium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p1 =  [Ar] 3d10 4s2 4p1 32 Ge germanium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p2 =  [Ar] 3d10 4s2 4p2 33 As arsenic 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p3 =  [Ar] 3d10 4s2 4p3 34 Se selenium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p4 =  [Ar] 3d10 4s2 4p4 35 Br bromine 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p5 =  [Ar] 3d10 4s2 4p5 36 Kr krypton 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 =  [Ar] 3d10 4s2 4p6 37 Rb rubidium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 5s1 =  [Kr] 5s1 38 Sr strontium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 5s2 =  [Kr] 5s2 39 Y yttrium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d1 5s2 =  [Kr] 4d1 5s2 40 Zr zirconium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d2 5s2 =  [Kr] 4d2 5s2 41 Nb niobium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d4 5s1 =  [Kr] 4d4 5s1 42 Mo molybdenum 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d5 5s1 =  [Kr] 4d5 5s1 43 Tc technetium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d5 5s2 =  [Kr] 4d5 5s2 44 Ru ruthenium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d7 5s1 =  [Kr] 4d7 5s1 45 Rh rhodium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d8 5s1 =  [Kr] 4d8 5s1 46 Pd palladium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 =  [Kr] 4d10 47 Ag silver 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s1 =  [Kr] 4d10 5s1 48 Cd cadmium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 =  [Kr] 4d10 5s2 49 In indium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p1 =  [Kr] 4d10 5s2 5p1 50 Sn tin 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p2 =  [Kr] 4d10 5s2 5p2 51 Sb antimony 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p3 =  [Kr] 4d10 5s2 5p3 52 Te tellurium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p4 =  [Kr] 4d10 5s2 5p4 53 I iodine 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p5 =  [Kr] 4d10 5s2 5p5 54 Xe xenon 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 =  [Kr] 4d10 5s2 5p6 55 Cs caesium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 6s1 =  [Xe] 6s1 56 Ba barium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 6s2 =  [Xe] 6s2 57 La lanthanum 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 5d1 6s2 =  [Xe] 5d1 6s2 58 Ce cerium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f1 5d1 6s2 =  [Xe] 4f1 5d1 6s2 59 Pr praseodymium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f3 6s2 =  [Xe] 4f3 6s2 60 Nd neodymium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f4 6s2 =  [Xe] 4f4 6s2 61 Pm promethium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f5 6s2 =  [Xe] 4f5 6s2 62 Sm samarium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f6 6s2 =  [Xe] 4f6 6s2 63 Eu europium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f7 6s2 =  [Xe] 4f7 6s2 64 Gd gadolinium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f7 5d1 6s2 =  [Xe] 4f7 5d1 6s2 65 Tb terbium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f9 6s2 =  [Xe] 4f9 6s2 66 Dy dysprosium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f10 6s2 =  [Xe] 4f10 6s2 67 Ho holmium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f11 6s2 =  [Xe] 4f11 6s2 68 Er erbium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f12 6s2 =  [Xe] 4f12 6s2 69 Tm thulium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f13 6s2 =  [Xe] 4f13 6s2 70 Yb ytterbium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f14 6s2 =  [Xe] 4f14 6s2 71 Lu lutetium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f14 5d1 6s2 =  [Xe] 4f14 5d1 6s2 72 Hf hafnium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f14 5d2 6s2 =  [Xe] 4f14 5d2 6s2 73 Ta tantalum 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f14 5d3 6s2 =  [Xe] 4f14 5d3 6s2 74 W tungsten 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f14 5d4 6s2 =  [Xe] 4f14 5d4 6s2 75 Re rhenium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f14 5d5 6s2 =  [Xe] 4f14 5d5 6s2 76 Os osmium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f14 5d6 6s2 =  [Xe] 4f14 5d6 6s2 77 Ir iridium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f14 5d7 6s2 =  [Xe] 4f14 5d7 6s2 78 Pt platinum 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f14 5d9 6s1 =  [Xe] 4f14 5d9 6s1 79 Au gold 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f14 5d10 6s1 =  [Xe] 4f14 5d10 6s1 80 Hg mercury 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f14 5d10 6s2 =  [Xe] 4f14 5d10 6s2 81 Tl thallium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f14 5d10 6s2 6p1 =  [Xe] 4f14 5d10 6s2 6p1 82 Pb lead 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f14 5d10 6s2 6p2 =  [Xe] 4f14 5d10 6s2 6p2 83 Bi bismuth 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f14 5d10 6s2 6p3 =  [Xe] 4f14 5d10 6s2 6p3 84 Po polonium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f14 5d10 6s2 6p4 =  [Xe] 4f14 5d10 6s2 6p4 85 At astatine 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f14 5d10 6s2 6p5 =  [Xe] 4f14 5d10 6s2 6p5 86 Rn radon 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f14 5d10 6s2 6p6 =  [Xe] 4f14 5d10 6s2 6p6 87 Fr francium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f14 5d10 6s2 6p6 7s1 =  [Rn] 7s1 88 Ra radium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f14 5d10 6s2 6p6 7s2 =  [Rn] 7s2 89 Ac actinium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f14 5d10 6s2 6p6 6d1 7s2 =  [Rn] 6d1 7s2 90 Th thorium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f14 5d10 6s2 6p6 6d2 7s2 =  [Rn] 6d2 7s2 91 Pa protactinium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f14 5d10 6s2 6p6 5f2 6d1 7s2 =  [Rn] 5f2 6d1 7s2 92 U uranium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f14 5d10 6s2 6p6 5f3 6d1 7s2 =  [Rn] 5f3 6d1 7s2 93 Np neptunium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f14 5d10 6s2 6p6 5f4 6d1 7s2 =  [Rn] 5f4 6d1 7s2 94 Pu plutonium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f14 5d10 6s2 6p6 5f6 7s2 =  [Rn] 5f6 7s2 95 Am americium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f14 5d10 6s2 6p6 5f7 7s2 =  [Rn] 5f7 7s2 96 Cm curium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f14 5d10 6s2 6p6 5f7 6d1 7s2 =  [Rn] 5f7 6d1 7s2 97 Bk berkelium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f14 5d10 6s2 6p6 5f9 7s2 =  [Rn] 5f9 7s2 98 Cf californium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f14 5d10 6s2 6p6 5f10 7s2 =  [Rn] 5f10 7s2 99 Es einsteinium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f14 5d10 6s2 6p6 5f11 7s2 =  [Rn] 5f11 7s2 100 Fm fermium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f14 5d10 6s2 6p6 5f12 7s2 =  [Rn] 5f12 7s2 101 Md mendelevium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f14 5d10 6s2 6p6 5f13 7s2 =  [Rn] 5f13 7s2 102 No nobelium 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f14 5d10 6s2 6p6 5f14 7s2 =  [Rn] 5f14 7s2

There are some anomolies in the table of electron configurations. For instance, nickel has 28 electrons and an electron configuration of 1s2 2s2 2p6 3s2 3p6 3d8 4s2 while copper has 29 electrons and an electron configuration of 1s2 2s2 2p6 3s2 3p6 3d10 4s1. The electrons fill the 4s shell in nickel but in copper one of these electrons is removed from the 4s shell. The mysteries of how the electron shells fill can be explained by the Schrödinger equation. To find the ground state electron configuration of copper, guess a some candidate configurations with 29 electrons such as 1s2 2s2 2p6 3s2 3p6 3d10 4s1 and 1s2 2s2 2p6 3s2 3p6 3d9 4s2, construct the corresponding many-electron wave functions and evaluate their energies using the total Hamiltonian. The electron configuration with the lowest energy is the one that will be observed.

It becomes increasing difficult to evaluate the energies of many-electron wavefunctions as the number of electrons increases. This is because the determinant of an $N\times N$ matrix has $N!$ terms. For gold with 79 electrons, there are 79! = 8.94 × 10116 237-dimensional integrals that need to be evaluated. Fortunately it is usually only necessary to include the valence electrons in the calculation.

A description of the hydrogen atom and many-electron atoms can be found in The Physics of Atoms and Quanta by H. Haken and H. C. Wolf.