Embed Size (px)
Citation preview
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?
V (for q1) = kqnucleus*q1/rn-1 + kq2q1/r2-1 + kq3q1/r3-1 + ….
Schrodinger’s solution for multi-electron atomsNeed to account for all the interactions among the electrons
Must solve for all electrons at once! (use matrices)
What’s different for these cases?
Potential energy (V) changes!
(Now more protons AND other electrons)
Need to account for all the interactions among the electrons
Must solve for all electrons at once! (use matrices)
Schrodinger’s solution for multi-electron atoms
V (for q1) = kqnucleusq1/rn-1 + kq2q1/r2-1 + kq3q1/r3-1 + ….
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)
n=1
n=2
For a given atom, Schrodinger predicts allowed wave functions
and energies of these wave functions. SIMILAR STRUCTURE:
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
1s
2s
3s
l=0 l=1 l=2
4s
2p
3p
4p3d
Energ
y
Li (3 e’s)
Na (11 e’s)
m=-1,0,1
m=-2,-1,0,1,2
Principal
quantum
number.
Angular momentum
quantum numbers
2s
2p
1s
3s
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
Li (3 e’s)
Na (11 e’s)
Wave functions for Li vs Na
2s2p
1s
3s
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
Wave functions for sodium
What affects total energy of outermost electron?
1. The effective charge (force) it feels towards center
of atom.
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.
Schrodinger predicts wave functions and energies of these
wave functions.
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
1s
2s
3s
l=0 l=1 l=2
4s
2p
3p
4p3d
Energ
y
m=-1,0,1
Li
Na
m=-2,-1,0,1,2
Why does
ionization
energy
increase and
size
decrease as
add electrons
in p orbitals?
Ionization energy
Size (distance of outermost e)
2s
2pAs 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)1s
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.
l=0 (s-orbitals) l=1 (p-orbitals)
l=2 (d-orbitals)
l=2 (f-orbitals)
Vale
nce (
n)
All atoms in this row have common filling of outer
most shell (valence electrons), common shapes,
similar energies … so similar behavior
n=1
n=2
n=3
l=0
(s)
l=1
(p)
l=2
(d)
l=0,m=01s
2s 2p
3s 3p 3d
Hydrogen (1p, 1e) Boron (5p, 5e’s)
NOT TO SCALE!
1s
2s
3s
2p
3p
m=-1,0,1
4s
4p 3d
EN
ER
GY Splitting of s and p
energy levels (shielding)
2s2
2p
1s2
Energy only
depends on nEnergy depends
on n and l
1s
2s
3s
l=0 l=1
m=-1,0,1l=2
m=-2,-1,0,1,2
In multi-electron atoms, energy of electron level depends on
n and l quantum numbers:
4s
2p
3p
4p3d
Ene
rgy
What is electron configuration for
atom with 20 electrons?
Write it out (1s2 etc… !
c. 1s2, 2s2, 2p6, 3s2, 3p6, 4s2, 3d6
b. 1s2, 2s2, 2p6, 3s2, 3p6, 3d2a. 1s2, 2s2, 2p6, 3s2, 3p4
d. 1s2, 2s2, 2p6, 3s2, 3p6, 4s2
e. none of the above
Which orbitals are occupied effects:
chemical behavior (bonding, reactivity, etc.)
Answer is d! Calcium: Fills lowest energy levels first
3123
In multi-electron atoms, energy of electron level depends on
n and l quantum numbers:
1s
2s
3s
l=0 l=1
m=-1,0,1 l=2
m=-2,-1,0,1,2
4s
2p
3p
4p3d
Energ
y
1st Shell
2st Shell
3rd Shell
4th Shell
Calcium has 3 complete shells.
Incomplete shell:
Chemical behavior & bonding
determined by electrons in outer
most shell (furthest from the
nucleus). 4
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
Ionic MetallicCovalent
electron completely
transferred from one atom to
the other
electron equally shared
between two adjacent atoms
electron shared between
all atoms
in solid
Degree of sharing of electron
Li+ F-H2 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+ Cl-
Attracted by coulomb attraction
Coulomb attraction
Energ
y
Separation
of ions
V(r)
Na+Cl-
Repulsion of electrons
Na+ Cl-
Covalent BondSharing of an electron… look at example H2
+
(2 protons (H nuclei), 1 electron)
Proton 1 Proton 2
1
Wave function if electron bound to
proton 1
Protons far apart …
Potential energy curve
Covalent BondSharing of an electron… look at example H2
+
(2 protons (H nuclei), 1 electron)
Proton 1 Proton 2
Proton 1 Proton 2
1
2
Wave function if electron bound to
proton 1
Protons far apart …
Wave function if electron bound to
proton 2
Covalent BondSharing of an electron… look at example H2
+
(2 protons (H nuclei), 1 electron)
1 2
If 1 and 2 are both valid solutions,
then any combination is also valid solution.
Subtract solutions
(antisymmetric):
- = 1-2
(molecular orbitals)
Add solutions
(symmetric):
+ = 1 + 2 and
-2
+ = 1 + 2
- = 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 ..
glues together protons.- … no electron density between protons
… protons repel (less / not stable)
Bonding Orbital Antibonding Orbital
Energ
y (
mo
lecu
le)
Separation of protons
V(r)
Energy of + as distance decreases
(more of electron cloud between them)
Energy of - as distance decreases
1 2(molecular orbitals)
-2
+ = 1 + 2
- = 1-2
Quantum Bound State Sim
What would you expect
for two square wells?For two atoms?
Now FIX the protons: what does the electron energy look like
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
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!!
why "almost"...one little problem...
Big Picture.
almost
• 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)
Limitations of Schrodinger
Quantum Mechanics to understand (predict, control, etc.)
flow of electricity through materials.
insulators, conductors,
QM control current flow in semiconductors
results: transistors, cell phones, iPods,…
The foundation of modern technology
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.
e. some other QM effectsmall compared to what?
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?
from class 20 months ago : )
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?
How does atom-atom interaction lead to band structure?
Look at what happens to these wave functions as bring protons closer…
+ puts electron density between protons ..
glues together protons.- … no electron density between protons
… protons repel (not stable)
Bonding Orbital Antibonding Orbital
Energ
y Separation of protons
V(r)
Energy of + as distance decreases
(more of electron cloud between them)
Energy of - as distance decreases
1 2(molecular orbitals)
-2
+ = 1 + 2
- = 1-2
Potential energy of electron due to two protons:
Potential energy of electron due to single proton: V = -ke2/r
Ground state wave function of
electron in this potential:
(r) ~ e-r
Eatom
+ =
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
energy levels of atoms molecules solids
at1-at2 molec
inner electrons stick close
to nuclei. Outer e’s get
shared.
at 1
Energ
y
at 2
multielectron atoms
Quantum Bound State Sim
What would you expect
for two square wells?For two atoms?
Now FIX the protons: what does the electron energy look like
QM of electrical conductionenergy levels of atoms molecules solids
at 1
Energ
y
at 2at 3
at 4
many levels!
top energy wave functions spread waaaay out
Bound State Sim.. Many Wells
countless levels smeared together, individual levels
indistinguishable. "bands" of levels. Each level filled with 2
electrons until run out.
1
2
3
In solid, `1022 atoms/cm3, many!! electrons, and levels
En
erg
y
atom level bands
more atoms
“band gap” ~ few eV
“valence
band”
“conduction
band”empty
filled with electrons
filled with electrons
empty
Which band structure goes with which material?
(be ready to give reasoning)
empty
full
1. Diamond 2. copper 3. germanium (poor conductor)
Energ
y
only top 2 filled and lowest 2 empty bands shown
x y zelement w
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
25 eV
0
