Atoms to molecules
Today:
The Periodic Table.
Molecular orbitals
HWK 13 due at 5PM. Please fill out the online
participation survey. Worth 10points on HWK 13.
EXTRA CREDIT HWK 14 study questions for Final Exam
on the website.
Final Exam is Monday, Dec. 15 10:30A-1P HERE
Duane G1B20
Can Schrodinger make sense of the periodic table?
Schrodinger’s solution for multi-electron atoms
Need to account for all the interactions among the electrons
Must solve for all electrons at once! (use matrices)
V (for q1) = kqnucleus*q1/rn-1 + kq2q1/r2-1 + kq3q1/r3-1 + ….
Schrodinger’s solution for multi-electron atoms
What’s different for these cases?
Potential energy (V) changes!
(Now more protons AND other electrons)
V (for q1) = kqnucleusq1/rn-1 + kq2q1/r2-1 + kq3q1/r3-1 + ….
Need to account for all the interactions among the electrons
Must solve for all electrons at once! (use matrices)
Gets very difficult to solve … huge computer programs!
Solutions change:
- wave functions change
higher Z  more protons  electrons in 1s more strongly
bound  radial distribution quite different
general shape (p-orbital, s-orbital) similar but not same
- energy of wave functions affected by Z (# of protons)
higher Z  more protons  electrons in 1s more strongly
bound (more negative total energy)
For a given atom, Schrodinger predicts allowed wave functions
and energies of these wave functions. SIMILAR STRUCTURE:
l=0
l=1
4p
Energy
2s
n=2
1s
n=1
Principal
quantum
number.
Angular momentum
quantum numbers
3d
m=-2,-1,0,1,2
4s
3s
l=2
Li (3 e’s)
3p
Na (11 e’s)
2p
m=-1,0,1
Why would behavior of Li be similar to Na?
a. because shape of outer most electron is similar.
b. because energy of outer most electron is similar.
c. both a and b
d. some other reason
Wave functions for Li vs Na
Li (3 e’s)
3s Na (11 e’s)
2p
1s
2s
In case of Na, what will energy of outermost electron be and WHY?
a. much more negative than for the outermost electron in Li
b. similar to the energy of the outermost electron in Li
c. much less negative than for the outermost electron in Li
Wave functions for sodium
What affects total energy of outermost electron?
3s 1. The effective charge (force) it feels towards center
2p
of atom.
1s 2s 2. It’s distance from the nucleus.
What effective charge does 3s electron feel
pulling it towards the nucleus?
Close to 1 proton… 10 electrons closer in
shield (cancel) a lot of the nuclear charge.
What about distance?
In H, 3s level is on average 9x further than 1s, so 9*Bohr radius.
In Na, 11 protons pull 1s, 2s, 2p closer to nucleus
distance of 3s not as far out.
Electron in 3s is a bit further than 1s in H, but ~same as 2s in Li.
Proximity of electrons in 1s, 2s, 2p is what makes 3s a bit bigger.
In case of Na, what will energy of outermost electron be and WHY?
b. very similar to the energy of the outermost electron in Li
AND somewhat (within a factor of 3) of the ground state of H
Schrodinger predicts wave functions and energies of these
wave functions.
l=1
l=0
4p
Energy
4s
3s
2s
1s
3p
l=2
3d
m=-2,-1,0,1,2
Li
Na
2p
m=-1,0,1
Why would behavior of Li be similar to Na?
a. because shape of outer most electron is similar.
b. because energy of outer most electron is similar.
c. both a and b
d. some other reason
Why does
ionization
energy
increase and
size
decrease as
add electrons
in p orbitals?
Ionization energy
Size (distance of outermost e)
2p
2s
1s
As go from Li to N,
end up with 3 electrons in 2p (one
in each orbital),
Why is ionization energy larger
and size smaller than in Li?
(Develop reasoning)
P orbitals each have direction…
electrons in px do not effectively shield
electrons in py from the nucleus.
So electrons in p orbitals:
1. feel larger effective positive charge
2. are held closer to nucleus.
All atoms in this row have common filling of outer
most shell (valence electrons), common shapes,
similar energies … so similar behavior
l=0 (s-orbitals)
l=1 (p-orbitals)
Valence (n)
l=2 (d-orbitals)
l=2 (f-orbitals)
Boron (5p, 5e’s)
NOT TO SCALE!
Hydrogen (1p, 1e)
n=3
n=2
l=0
(s)
l=1
(p)
l=2
(d)
3s
3p
3d
2s
4p
2p
3d
4s
3p
2p
1s2
2s2
3s
2p m=-1,0,1
n=1
1s
l=0,m=0
Energy only
depends on n
ENERGY
2s
Splitting of s and p
energy levels (shielding)
Energy depends
on n and l
1s
Energy
In multi-electron atoms, energy of electron level depends on
n and l quantum numbers:
l=1
l=0
l=2
m=-1,0,1
m=-2,-1,0,1,2
4p
3d
4s
3s
2s
1s
3p
What is electron configuration for
atom with 20 electrons?
Write it out (1s2 etc… !
a. 1s2, 2s2, 2p6, 3s2, 3p4
b. 1s2, 2s2, 2p6, 3s2, 3p6, 3d2
2p
c. 1s2, 2s2, 2p6, 3s2, 3p6, 4s2, 3d6
d. 1s2, 2s2, 2p6, 3s2, 3p6, 4s2
e. none of the above
Answer is d! Calcium: Fills lowest energy levels first
Which orbitals are occupied effects:
chemical behavior (bonding, reactivity, etc.)
In multi-electron atoms, energy of electron level depends on
n and l quantum numbers:
l=0
l=1
l=2
m=-1,0,1
m=-2,-1,0,1,2
4p
3d
Energy
4s
3p
3s
Calcium has 3 complete shells.
4th Shell
Incomplete shell:
Chemical behavior & bonding
determined by electrons in outer
most shell (furthest from the
rd
3 Shell
nucleus).
4
2p
2s
1s
2st Shell
1st Shell
2
1
3
Electronic structure of atom determines its form
(metal, semi-metal, non-metal):
- related to electrons in outermost shell
- how these atoms bond to each other
Semiconductors
Lecture stopped here.
Bonding
- Main ideas:
1. involves outermost electrons and their wave functions
2. interference of wave functions
(one wave function from each atom) that produces situation where atoms want to stick
together.
3. degree of sharing of an electron across 2 or more atoms determines the type of
bond
Degree of sharing of electron
Ionic
Covalent
electron completely
electron equally shared
transferred from one atom to between two adjacent atoms
the other
Li+ F-
H2
Metallic
electron shared between
all atoms
in solid
Solid Lead
Ionic Bond (NaCl)
Na (outer shell 3s1)
Cl (outer shell 3s23p5)
Has one weakly bound electron
Low ionization energy
Needs one electron to fill shell
Strong electron affinity
Na+ ClV(r)
Attracted by coulomb attraction
Separation
of ions
Energy
Na+ Cl-
Repulsion of electrons
Cl-
Na+
Coulomb attraction
Covalent Bond
Sharing of an electron… look at example H2+
(2 protons (H nuclei), 1 electron)
Protons far apart …
1
Wave function if electron bound to
proton 1
Proton 1
Potential energy curve
Proton 2
Covalent Bond
Sharing of an electron… look at example H2+
(2 protons (H nuclei), 1 electron)
Protons far apart …
1
Wave function if electron bound to
proton 1
Proton 1
Proton 2
2
Wave function if electron bound to
proton 2
Proton 1
Proton 2
Covalent Bond
Sharing of an electron… look at example H2+
(2 protons (H nuclei), 1 electron)
If 1 and 2 are both valid solutions,
then any combination is also valid solution.
+ = 1 + 2
1
(molecular orbitals)
2
Add solutions
(symmetric):
+ = 1 + 2 and
- = 1-2
-2
Subtract solutions
(antisymmetric):
- = 1-2
Look at what happens to these wave functions as bring protons closer…
Visualize how electron cloud is distributed… for which wave function would
this cloud distribution tend to keep protons together? (bind atoms?) … what is
your reasoning?
a. S or +
b. A or -
Look at what happens to these wave functions as bring protons closer…
+ puts electron density between protons ..  … no electron density between protons
glues together protons.
… protons repel (less / not stable)
Bonding Orbital
Antibonding Orbital
+ = 1 + 2
1
2
(molecular orbitals)
- = 1-2
-2
Energy (molecule)
V(r)
Energy of - as distance decreases
Separation of protons
Energy of + as distance decreases
(more of electron cloud between them)
Quantum Bound State Sim
Now FIX the protons: what does the electron energy look like
What would you expect
for two square wells?
For two atoms?
Same idea with p-orbital bonding … need constructive interference of wave
functions between 2 nuclei.
Sign of wave function matters!
Determines how wave functions interfere.
Why doesn’t He-He bond?
Not exact same molecular orbitals as H2+, but similar.
With He2, have 4 electrons …
fill both bonding and anti-bonding orbitals. Not stable.
So doesn’t form.
Demo
• Which is more reactive?
• He
• H2
Big Picture.
Now almost infinite power!
Know how to predict everything about behavior of atoms and electrons or
anything made out of them:
1. Write down all contributions to potential energy,
includes e-e, nuc.-nuc., nuc.-e for all electrons and nuclei.
q1q2/r1-2 + q2q3/r1-3 + qnuc1qnuc2/rqnuc1-qnuc2 +q1qnuc1/r1-nuc1 +
one spin up and one down electron per state req....
(plus little terms involving spin, magnetism, applied voltage)
2. Plug potential energy into Schrod. eq., add boundary. cond.
3. Solve for wave function elec1,(r1, r2, rnuc1, ...)
elec2,
nuc1,
nuc2, ...
get energy levels
for system
calculate/predict everything there is to know!!
almost
why "almost"...one little problem...
Limitations of Schrodinger
• With three objects (1 nuclei + 2 electrons) solving eq. very hard.
• Gets much harder with each increment in number of electrons and nuclei !!
Give up on solving S. E. exactly-Use various models and approximations.
Not perfect but very useful, tell a lot.
(lots of room for cleverness, creativity, intuition)
Quantum Mechanics to understand (predict, control, etc.)
flow of electricity through materials.
The foundation of modern technology
insulators, conductors,
QM  control current flow in semiconductors
results: transistors, cell phones, iPods,…
Where to start in understanding flow of electrons in object
at QM level?
V
V
What is important for flow of current from QM perspective?
a. electrons move through material as classical particles, so QM effects are only a
minor effect.
b. spacing of electron energy levels is important because big spacing between levels
means electrons can move easily.
c. spacing of electron energy levels is important because small spacing between
levels means electrons can move easily.
d. QM is important because the shape of the wave function determines the direction
in which electron can move.
small
to what?
e. some other
QMcompared
effect
from class 20 months ago : )
Nanotechnology: how small does a wire have to be
before movement of electrons starts to depend on size
and shape due to quantum effects?
How to start?
Need to look at
Energy level spacing compared to thermal energy, kT.
Almost always focus on energies in QM.
Electrons, atoms, etc. hopping around with random energy kT.
Larger than spacing, spacing irrelevant. Smaller, spacing big deal.
So need to calculate energy levels.
pit depth compared
to kT?
How does atom-atom interaction lead to band structure?
1. Energy levels and spacings in atoms  molecules  solids
2. How energy levels determine how electrons move.
Insulators, conductors, semiconductors.
3. Using this physics for nifty stuff like copying machines, diodes and transistors (all
electronics), light-emitting diodes.
Spacing of gap to the next higher, open energy level for electron is the critical
feature.
Small, large, in middle compared to kT (~1/40 eV)?
What happens to energy levels as put bunch of atoms together?
Look at what happens to these wave functions as bring protons closer…
+ puts electron density between protons ..  … no electron density between protons
glues together protons.
… protons repel (not stable)
Bonding Orbital
Antibonding Orbital
+ = 1 + 2
1
2
(molecular orbitals)
- = 1-2
-2
V(r)
Energy
Energy of - as distance decreases
Separation of protons
Energy of + as distance decreases
(more of electron cloud between them)
V = -ke2/r
Potential energy of electron due to single proton:
(r) ~ e-r
Ground state wave function of
electron in this potential:
Eatom
Potential energy of electron due to two protons:
+
=
Ground state wave function of electron (symmetric/bonding):
+
=
1st excited state wave function (antisymmetric/antibonding):
+
=
For every energy level for 1 proton, 2 energy levels for 2 protons.
If protons far away, symmetric and antisymmetric state both have same energy as
ground state of electron bound to single proton:
Eatom
As protons get closer together, symmetric and antisymmetric state become more
distinct and energy levels split:
Eatom + 
Eatom – 
As separation decreases, energy
splitting  increases
QM of electrical conduction
multielectron atoms
energy levels of atoms  molecules  solids
Energy
inner electrons stick close
to nuclei. Outer e’s get
shared.
at 1
at1-at2 molec
at 2
Quantum Bound State Sim
Now FIX the protons: what does the electron energy look like
What would you expect
for two square wells?
For two atoms?
QM of electrical conduction
energy levels of atoms  molecules  solids
Energy
top energy wave functions spread waaaay out
at 2
at 1
many levels!
at 3
at 4
Bound State Sim.. Many Wells
In solid, `1022 atoms/cm3, many!! electrons, and levels
countless levels smeared together, individual levels
indistinguishable. "bands" of levels. Each level filled with 2
electrons until run out.
empty
empty
“conduction
band”
Energy
“band gap” ~ few eV
3
filled with electrons
2
filled with electrons
1
bands
atom level
more atoms
“valence
band”
Which band structure goes with which material?
(be ready to give reasoning)
1. Diamond
2. copper
empty
full
3. germanium (poor conductor)
a. 1=w, 2=x, 3=y b. 1=z, 2=w, 3=y c. 1=z, 2=y, 3=x
d. 1=y, 2= w, 3=y. e. 1=w, 2=x, 3=y
Energy
25 eV
element w
x
0
only top 2 filled and lowest 2 empty bands shown
y
z