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,

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

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.