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PHY.K02UF Molecular and Solid State Physics | ||||
Many-electron atoms
The Hamiltonian for an atom with many electrons is,
In first sum are the kinetic energies of the electrons, in the second sum are the attractive Coulomb interactions between the electrons and the nucleus, and in 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 anomalies 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 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 increasingly 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.