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

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

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.