Multi-Electron Atoms

Complete Description of a Ground State Wavefunction ψ

A total of three quantum numbers appear from the solution of

n = principal quantum number

l = angular momentum quantum number

ml = magnetic quantum number

nlm(r,θ,Φ)

What will a ground state wave-function be called??

Correlation of Wavefunctions to Orbitals

Using the terminology of chemists:100(r,,) is instead called the 1s orbital.n designates the shell (1, 2, 3, 4,…..)l designates the sub-shell (s, p, d, f….)ml completes the description of the

orbitall = 0 (s orbital) l = 1 (p orbital)

l = 2 (d orbital) l = 3 (f orbital)

When l =1 and ml = 0 then orbital is pz When l=1 and ml= ±1 then orbital is px or py

Hydrogen Atom Wavefunctions

n = 2l = 1m = 0

n = 2l = 0m = 0

n = 2l = 1m = +1

n = 1l = 0m = 0

n = 2l = 1m = -1

100

200

211

210

21-1

100 1s -RH / 12-2.18 x 10-18J

200

211

210

21-1

2s

2px (or 2py)

2pz

2py (or 2px)

-RH / 22

-RH / 22

-RH / 22

-RH / 22

-5.45 x 10-19J

- 5.45 x 10-19J

- 5.45 x 10-19J

- 5.45 x 10-19J

What is the corresponding orbital for a 4,1,0 state?

1. 1s2. 2s3. 4s4. 5s5. 4px

6. 4py

7. 4pz

8. 4dz

Arrangement of Shells/Subshells/Orbitals and Corresponding Quantum Numbers

For a H-atom, orbitals with same valueof n have equal energy.

En = -RH

n2

For any shell n there are n2 degenerate orbitals.

Energy Levels For a Hydrogen Atom:

1s

2s

3s

2px 2pz 2py

3px 3py3pz 3dxy 3dyz 3dz2 3dxz 3dx

2-y

2

Concept Check!

How many orbitals in a single atom can have the following two quantum numbers: n = 4, ml = -2

1. one2. two3. three4. four5. five6. six7. seven8. eight9. zero

Degeneracy of states

States having the same energy are called degenerate.

For every value of n there are n2 degenerate states.

Physical Interpretation of Ψ

Quantum world is very different from the macroscopic world that we are used to seeing.

Therefore unfortunately a physical interpretation of Ψ does not exist. However a physical interpretation for Ψ2 does exist!

| nlm(r,,)|2 = Probability Density

Max Born

Probability / Volume

Figure from MIT Open CourseWare

Electron cloudsAlthough we cannot know how the electron travels around

the nucleus we can know where it spends the majority of its time (thus, we can know position but not trajectory).

The “probability” of finding an electron around a nucleus can be calculated.

Relative probability is indicated by a series of dots, indicating the “electron cloud”.

• 90% electron probability or cloud for 1s orbital (notice higher probability toward the centre)

Solution to the Wave function for a H-atomAny wave function Ψ can be divided into two components

radial Ψ angular Ψ

Rnl(r) Ylm(,)

Rnl(r)Ylm(,)

Where a0 = Bohr radius (constant) = 52.9 pmFor all s orbitals (1s, 2s, 3s, etc,) Y is a constant.

Shape of an s-orbital

The shape of an s-orbital is spherically symmetrical, independent of and .

Figure 1.23 Chem Principles

Probability Density Plots of s-orbitals

Figures from MIT OCW

NODE a value of r, and for which both and 2 = 0

Radial Nodes = n - 1 - l

For 1s radial nodes = 1 – 1 – 0 = 0

For 2s radial nodes = 2 – 1 – 0 = 1

For 3s radial nodes = 3 – 1 – 0 = 2

2a0

RADIAL NODE: a value of r for which both and 2 = 0

How many radial nodes does a hydrogen atom 3d orbital have?

1. One2. Two 3. Three4. Four5. Five6. Six 7. Seven 8. Eight9. Zero

Radial Probability Distribution

The probability of finding an electron in a shell of thickness dr at a distance r from the nucleus.

For s-orbitals RPD = 4πr22 dr

rmp = Bohr radius = 0.529Å

http://www.emu.edu.tr/mugp101/PHYSLETS/physletprob/ch10_modern/radial.html

RPD for a 2s and a 3s orbital for H-atom:

volume

rmp = 6a0

rmp = 11.5a0

nodenode node

as n increases rmp also increases.

Concept Check

Identify the correct RPD plot (and radial node number) for a 4s orbital

1.

2.

3.

4.

Radial Probability Distributions for other orbitals:

<

3d 3p 3s

Quantum Tunneling

Scanning Tunneling Microscope

p-orbitals (l = 1):

For any sub-shell l = 1 there are three p orbitals; m = +1 or -1 (px or py) and m = 0 (pz)

Difference from the s-orbitals lies in the fact that p-orbitals wave-functions depend on and .

p-orbitals are not spherically symmetrical!

p and d-orbitals

p-orbitals look like a dumbell with 3 orientations: px, py, pz (“p sub z”).

p-orbitals consist of two lobes seperated by a nodal plane.

There is zero probability of finding a p-electron at the nucleus.

Only electrons in the s orbitals have a substantial probability of being very close to the nucleus

Electrons in the s orbitals are LEAST shielded.

d-orbitalsFour of the d orbitals resemble two dumbells in a clover shape.

The last d orbital resembles a p-orbital with a donut wrapped around the middle!

Multi-Electron Atoms and the spin quantum number ms

When the Schrodinger wave equation is solved for multielectron atoms, a fourth quantum number ms the spin quantum number also appears.

ms = +1/2 (spin up) or ms = -1/2 (spin down)

ms completes the description of an electron and is NOT dependant on the orbital.

Atoms with Many Electrons and the Periodic Table

The underlying physical laws necessary for …the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble! Paul Dirac (1929)

Paul Dirac at a Super-Collider workshop in the early 1930s.

Discovery of Electron Spin:

1s2 2s2 2px2 2py

2 2pz2

Ne

http://www.ilorentz.org/history/spin/goudsmit.html

Uhlenbeck and Goudsmit

Wolfgang Pauli

Pauli Exclusion Principle

No two electrons in the same atom can have the same four quantum numbers.

The Pauli exclusion principle limits us to two electrons per orbital.

1s2 2s2 2px2 2py

2 2pz2

Ne

Concept Check!

How many electrons in a single atom can be in a 2p state?

1. one2. two3. three4. four5. five6. six7. seven8. eight9. zero

How many electrons in a single atom can have the following two quantum numbers: n = 4, ml = -2

1. One2. Two 3. Three4. Four5. Five6. Six7. Seven8. Eight9. Zero

Shrodinger Equation for Multielectron Atoms

(r111r222) (r111r2 22)

(r111r222r333) (r111r2 22r333)

NEED AN APPROXIMATION!

Hartree Orbitals

One electron orbital approximation:

e- # 1 e- # 2

1s (1) 1s (2)

1s(2) 2s(1)

100+1/2 100-1/2 200+1/2

Electronic Configurations

Electronic configurations are basically short hand notations for different wavefunctions, using the “1 electron orbital approximation”.

1s22s2 1s22s22p1

Multi-electron vs. Hydrogen Atom Wave Functions

e.g. Ar 1s22s22p63s23p6

Similarities to H-atom Wave functions:

Differences to H-atom Wave functions:

• Similar in shape• Identical nodal structure

• Each multi-electron orbital is smaller than the corresponding hydrogen atom orbital.• In multi-electron atoms, orbital energies depend not only on n (shell) they also depend on l (sub-shell).

Multi-electron vs. Hydrogen Atom Energy Levels

more negative

Zeff ≠ Z

Zeff differs from Z because of shielding.

Shielding and Zeff Case A

•Electron #2 cancels part of the charge experienced by electron #1. •Electron #1 experiences a force on average of Zeff = ___ , not Zeff = +2e. •The energy of electron #1 is that of an electron in a H (1-electron) atom. (2.18 x 10-18 J)

+1

total Shielding

# 2

2

# 1

Shielding and Zeff Case B

•Electron #2 does not cancel the charge experienced by electron #1.•Electron #1 experiences a force on average of Zeff = ___ •The energy of electron #1 is that of an electron in a He+1(1-electron) ion. (8.72 x 10-18 J)

+2

No Shielding

# 1

2

# 2

Extreme case A: Zeff = 1, IEHe= 2.18 x 10–18 J total shielding Extreme case B: Zeff = 2, IEHe= 8.72 x 10–18 J no shielding

Experimental IEHe = 3.94 x 10–18 J

So the reality is somewhere between total shielding and no shielding.

We can calculate the Zeff from the experimentally determined IE:

Our calculated Zeff should be a reasonable value, it should fall between total shielding and no shielding.

Which value(s) below is a possible Zeff for the 2s electron in a Li (Z = 3) atom?

1. Zeff = 0.39

2. Zeff = 0.87

3. Zeff = 1.42

4. Zeff = 3.19

5. Option 1 and 2

6. Option 1, 2, and 3

7. Option 2 and 4

Why is E2s<E2p and E3s<E3p<E3d?

Energy differences of s and p-orbitals:

can get

Also for a given n state, electrons in the s-orbitals are less shielded from the nucleusas compared to the p-electrons and hence experience a greater Zeff.

single multi

Radial Probability Distributions for other orbitals:

<

3d 3p 3s

Consider why the electronic configuration for Li is 1s2 2s1 and not 1s2 2p1.

The s-orbital is less shielded.

Averaging over the RPD yields Zeff2p < Zeff

2s

E2s < E2p

Aufbau (building up) principle

Fill energy states that depend on (n & l) one electron at a time, starting at the lowest energy state.

parallel

O (Z = 8)

1s22s22p4

Identify the correct electron configuration for the carbon (Z = 6) atom.

1s

2s

3s

1. 1s22s23s2

2. 1s22s22px2

3. 1s22s22py2

4. 1s22s22pz2

5. 1s22s22px12pz

1

6. 1s22s22px12pz

12py1

2px 2pz 2py

E

Periods in the Periodic Table Period in a periodic table refers to the value of the principal quantum

number “n”.

corevalence

3d103d5

Writing Down the Electronic Configurations:

Electron Configurations for Ions

3d24s2

3d2

Concept Check!

Refer to the periodic table and determine which element has the following electron configuration: [Ar]4s13d10

1. Cu

2. Zn

3. Ga

4. Ag

5. Cd

6. In

Concept Check!

Select the correct electronic configuration for V1+ (Z = 23)

1. [Ar] 4s23d3

2. [Ar] 3d2

3. [Ar] 4s13d3

4. [Ar] 4s23d2

5. [Ar]4s14d3

6. [Ar]3d3

Photo-Electron Spectroscopy (PES)

K.E.1s22s22p6 1s22s22p5 +

K.E.1s22s22p6 1s22s12p6 +

1s22s22p6 1s12s22p6 + K.E.

(Recall for the photoelectric effect: Ei = φ + KE)

IE = _____ -_____ Ei KE

22

48

870

Which electron ejection requires the least amount of energy?

1. Si ([Ne]3s23p2) Si+ ([Ne]3s13p2) + e-

2. Si ([Ne]3s23p2) Si+ ([Ne]3s23p1) + e-

3. Si+ ([Ne]3s23p1) Si+2 ([Ne]3s2) + e-

If a certain element being studies by x-ray PES displays an emission spectrum with five distinct kinetic energies. What are all of the possible elements that could produce this spectrum?