Lecture physics a2 building atoms and molecules huynh quang linh

Lecture 11 continue Building Atoms and Molecules +e r +e yeven yodd d Plane of hydrogen atoms Content  Molecular Wavefunctions  Example H + H  H2 Atomic Configurations Building atoms with the Pau[.]

Lecture 11_continue: Building

Atoms and Molecules

+e

r

+e

yeven

yodd

d

Plane of hydrogen atoms

 Molecular Wavefunctions

 Example: H + H  H 2

Atomic Configurations

Building atoms with the Pauli exclusion principle

Selection rules

 Therefore, electrons do not pile up in the lowest energy state, i.e,

the (1,0,0) orbital

 They are distributed among the higher energy levels according to the Pauli Principle

 Particles that obey the Pauli Principle are called “fermions”

Pauli Exclusion Principle

From spectra of complex atoms, Wolfgang Pauli (1925) deduced a new rule:

“In a given atom, no two electrons* can be in the same quantum state, i.e they cannot have the same set of quantum numbers n, l, ml , ms”

I.e., every “atomic orbital with n,l,m l” can hold 2 electrons: ()

“Pauli Exclusion Principle”

We now want to start building more complicated atoms to study the Periodic Table For atoms with many electrons (e.g., carbon: 6, iron: 26, etc.) - what energies do they have?

*Note: More generally, no two identical fermions (any particle with

spin of ħ/2, 3ħ/2, etc.) can be in the same quantum state

Filling the atomic orbitals according to the

Pauli Principle

2 2

n

eV 6 13



is valid only for one electron in the Coulomb potential of Z

protons The energy levels shift

as more electrons are added, due to electron-electron

interactions Nevertheless, this hydrogenic diagram helps us

keep track of where the added electrons go.

Energy

n



4

3

2

1

l = 0 1 2 3 4

s p d f g Example: Na Z = 11 1s 2 2s 2 2p 6 3s 1 Z = atomic number = number of protons l label #orbitals (2l+1) 0 s 1

1 p 3

2 d 5

3 f 7

exercise 1: Pauli Exclusion Principle

1 Which of the following states (n,l,ml,ms) is/are NOT allowed?

2 Which of the following atomic electron configurations violates the Pauli Exclusion Principle?

(a) (2, 1, 1, -1/2) (b) (4, 0, 0, 1/2) (c) (3, 2, 3, -1/2) (d) (5, 2, 2, 1/2) (e) (4, 4, 2, -1/2)

(a) 1s2, 2s2, 2p6, 3d10

(b) 1s2, 2s2, 2p6, 3d4

(c) 1s2, 2s2, 2p8, 3d8

(d) 1s1, 2s2, 2p6, 3d5 (e) 1s2, 2s2, 2p3, 3d11

2(2l +1) = 6 allowed electrons 2(2l +1) = 10 allowed electrons

exercise 1: Pauli Exclusion Principle

1 Which of the following states (n,l,ml,ms) is/are NOT

allowed?

2 Which of the following atomic electron configurations

violates the Pauli Exclusion Principle?

(a) (2, 1, 1, -1/2) (b) (4, 0, 0, 1/2) (c) (3, 2, 3, -1/2) (d) (5, 2, 2, 1/2) (e) (4, 4, 2, -1/2)

ml = -l, -(l -1), … (l-1), l

n > l

(a) 1s2, 2s2, 2p6, 3d10

(b) 1s2, 2s2, 2p6, 3d4

(c) 1s2, 2s2, 2p8, 3d8

(d) 1s1, 2s2, 2p6, 3d5 (e) 1s2, 2s2, 2p3, 3d11

Filling Procedure for Atomic Orbitals

example: Bromine

Due to electron-electron interactions, the hydrogen levels fail to give

us the correct filling order as we go higher in the periodic table

The actual filling order is given in the table below Electrons are

added by proceeding along the arrows shown

Bromine is an element with Z = 35 Find its electronic configuration (e.g 1s2 2s2 2p6 …)

As you learned in chemistry, the various behaviors of all the elements (and all the molecules made up from them) is all due to the way the electrons organize themselves, according to quantum mechanics

Optical Transitions between Atomic Levels

Consider the

hydrogenic picture:

In the field of a photon, the electron may be considered as being in a

superposition of two stationary states The time-dependent solution of the SEQ shows the wave function oscillating between the two eigenstates Not all

transitions are possible  must conserve angular momentum (and photon has ħ!)

1s 2s 2p

Stationary States: Superpositions :

1s ± 2s

couples to photons

No electric-dipole moment

Forbidden transition

Dl = 0

Allowed transition

Dl = ±1

E

nm eV

1240 E

hc

c h

E f

D D





D











n = 2

n = 1

DE

photon

r

U(r)

www.falstad.com/qmatomrad

Allowed Transitions for H

Selection Rule for electric-dipole transitions:

1





Dl

(A few representative transitions are shown.)

Energy (eV)

0.00 -0.85 -1.51

-3.40

-13.6 eV

n



4

3

2

1

l = 0 1 2 3 4

2

6 13

n

eV

E n  

s p d f g

(You observed some of these transitions in Lab 4.)

Selection Rule on m:

0, 1

m

D  

photon is linearly polarized

photon is circularly polarized