Download do with electron orbitals?

What does probability have to
do with electron orbitals?
Today:
The physics behind chemistry. How wavefunctions
describe the rules of chemical bonding.
Fill out the Faculty Course Questionnaire
HWK 13 up later today on the website.
Final Exam is Monday, Dec. 15 10:30A-1P HERE
Duane G1B20
Review Models of the Atom
–
–
• Thomson – Plum Pudding
– Why? Known that negative charges can be removed from atom.
– Problem: Rutherford showed nucleus is hard core.
–
–
–
• Rutherford – Solar System
– Why? Scattering showed hard core.
– Problem: electrons should spiral into nucleus in ~10-11 sec.
–
+
• Bohr – fixed energy levels
– Why? Explains spectral lines, gives stable atom.
– Problem: No reason for fixed energy levels
• deBroglie – electron standing waves
– Why? Explains fixed energy levels
– Problem: still only works for Hydrogen.
• Schrodinger – quantum wave functions
– Why? Explains everything!
– Problem: None (except that it’s hard to calculate)
+
+
Schrödinger’s Solutions for Hydrogen
How is it same or different than Bohr, deBroglie?
(energy levels, angular momentum, interpretation)
What do wave functions look like? What does that mean?
Extend to multi-electron atoms,
Extend to molecules and bonding,
Extend to solids (metals, insulators, etc.)
Extend to neutron stars…
2




How does
  x, t   V  x, t   x, t   i   x, t 
2
2m x
t
Relate to atoms?
Next:
Apply Schrödinger Equation to atoms
and make sense of chemistry!
(Reactivity/bonding of atoms and Spectroscopy)
Schrodinger predicts: discrete energies
and wave functions for electrons in atoms
How atoms bond, react, form solids?
Depends on:
the shapes of the electrons wave functions
the energies of the electrons in these wave functions, and
how these wave functions interact as atoms come together.
Next:
Apply Schrödinger Equation to atoms
and make sense of chemistry!
(Reactivity/bonding of atoms and Spectroscopy)
Schrodinger predicts: discrete energies
and wave functions for electrons in atoms
Which colors emitted by atoms/molecules and the probability of
transitions (the brightness of the line)?
Depends on:
the energies of the electrons in these wave functions, and
how the shapes of these wave functions relate
A brief review of chemistry
First, what is you chemistry background?
a. no chem
b. regular HS chem
c. AP or IB HS chem
d. college chem
What is Schrodinger Model of
Hydrogen Atom?
Electron is cloud of probability whose
wave function (x,t) is the solution to the
Schrodinger equation:
2  2 2 2 
 2  2  2  ( x, y, z , t )

2m  x y z 

 V ( x , y , z )  ( x , y , z , t )  i  ( x , y , z , t )
t
where:
V
Zke2
Zke2
V ( x, y , z )  
 2
r
( x  y 2  z 2 )1/ 2
r
Can get rid of time dependence and simplify:
Equation in 3D, looking for (x,y,z,t):
2  2 2 2 
 2  2  2  ( x, y, z , t )

2m  x y z 

 V ( x , y , z )  ( x , y , z , t )  i  ( x , y , z , t )
t
Since V(x,y,z) not function of time:
 ( x, y , z , t )   ( x, y , z ) e
 iEt / 
E ( x, y, z )e
 iEt / 
Time independent Schrödinger Equation:
2  2 2 2 
 2  2  2  ( x, y, z )  V ( x, y, z ) ( x, y, z )  E ( x, y, z )

2m  x y z 
Since potential spherically
symmetric “easier” to solve w/
spherical crds:
2
V (r )  Zke / r 
Schrödinger’s Equation in
Spherical Coordinates & no time:
z

r

y
x
 1   2  
(x,y,z) =

r


2
(rsincos, rsinsin, rcos
2m r r  r 
2
2


1  
 
1 

V (r )  E
 sin 
 2
2 
2 
2mr  sin   
  sin   
2
Technique for solving = “Separation of Variables”
 (r, ,  )  R(r ) f ( ) g ( )
 iEt / 
(r , ,  , t )  R(r ) f ( ) g ( ) e
Have you seen this technique for solving different equations?
A. yes B. no
z
In 1D (electron in a wire):
we got quantization from applying
boundary conditions in terms of x.
In 3D, now have 3 degrees of freedom:
Boundary conditions in terms of r,,

r
x

y
What are the boundary conditions on the wavefunction () in r ?
a.  must go to 0 at r=0
b.  must go to 0 at r=infinity
c.  at infinity must equal  at 0
d. A and B
e. A, B, and C
 must be normalizable, so needs to go to zero …
Also physically makes sense … not probable to find electron there
z
In 1D (electron in a wire):
Have 1 quantum number (n).
Need to specify value of n to know
what state electron is in.
(x, t)   n ( x) n (t) 

n


r
2
nx  iE n t / 
sin(
)e
L
L
In 3D, now have 3 degrees of freedom:
Boundary conditions in terms of r,,
x

y
How many quantum numbers are there in 3D?
In other words, how many numbers do you
need to specify unique wave function? And
why? (We’ll ask you to explain your reasoning!)
a. 1
r: n
Answer: 3 – Need one
b. 2
quantum number for
: l
c. 3
each dimension:
d. 4
: m
e. 5
(If you said 4 because you were thinking about
spin, that’s OK too. We’ll get to that later.)
z
In 1D (electron in a wire):
Have 1 quantum number (n)
In 3D, now have 3 degrees of freedom:
Boundary conditions in terms of r,,
Have 3 quantum numbers (n, l, m)

r
x

y
 nlm (r , ,  )  Rnl (r ) f lm ( ) g m ( )
Shape of  depends on n, l ,m. Each (nlm) gives unique 
2p
n=1, 2, 3 … = Principle Quantum Number
l=0, 1, 2, 3 …= Angular Momentum Quantum Number
=s, p, d, f
(restricted to 0, 1, 2 … n-1)
n=2
m = ... -1, 0, 1.. = z-component of Angular Momentum
l=1
(restricted to –l to l)
m=-1,0,1
Comparing H atom & Infinite Square Well:
Infinite Square Well: (1D)
• V(x) = 0 if 0<x<L
∞ otherwise
∞
r
∞
0
L
x
• Energy eigenstates:
n 2 2  2
En 
2mL2
2
L
sin(
nx
L
• Energy eigenstates:
mZ 2 k 2 e 4
En  
2 2 n 2
• Wave functions:
 n ( x) 
H Atom: (3D)
• V(r) = -Zke2/r
)
 n ( x, t )  n( x)e  iE nt / 
• Wave functions:
 nlm (r , ,  )  Rnl (r ) f lm ( ) g m ( )
 nlm (r ,  ,  , t )   nlm (r ,  ,  )e  iE nt / 
What do the wave functions look like?
 nlm (r , ,  )  Rnl (r ) f lm ( ) g m ( )
n = 1, 2, 3, …
l (restricted to 0, 1, 2 … n-1)
m (restricted to –l to l)
Much harder to draw in 3D
than 1D. Indicate amplitude
of  with brightness.
1s
Increasing n
2s
Increases distance from nucleus, www.orbital.com
Increases # of radial nodes
See pictures:
m=-3
3s
4s (l=0) 4p (l=1) 4d (l=2)
Increasing l
Increases angular nodes
Decreases radial nodes
Changes
4f (l=3, m=0) angular
distribution
m=3
Shapes of hydrogen wave functions:
 nlm (r , ,  )  Rnl (r ) f lm ( ) g m ( )
Look at s-orbitals (l=0): no angular dependence
n=1
n=2
Higher n  average r bigger
 more spherical shells stacked within each other
 more nodes as function of r
n=1
l=0
Probability finding
electron as function of r
n=2
l=0
n=3
l=0
0.05nm
Radius (units of Bohr radius, a0)
Shapes of hydrogen wave functions:
 nlm (r , ,  )  Rnl (r ) f lm ( ) g m ( )
l=1, called p-orbitals: angular dependence (n=2)
l=1, m=0: pz = dumbbell shaped.
l=1, m=-1: bagel shaped around z-axis (traveling wave)
l=1, m=+1

r  r / 2 a0 
3


n  2, l  1, m  0 
 211 
e
cos



4
2 6a03 a 0


1

r  r / 2 a0 
3
i

n  2, l  1, m  1 
 211 
e
sin e 

3 a
8
2 6a0 0


1
Superposition applies:
px=superposition (addition of m=-1 and m=+1)
py=superposition (subtraction of m=-1 and m=+1)
w/time dependence
eim+it/h
Dumbbells
(chemistry)
Physics vs Chemistry view of orbits:
2p wave functions
(Physics view)
(n=2, l=1)
Dumbbell Orbits
(chemistry)
px
m=1
m=-1
m=0
pz
py
px=superposition
(addition of m=-1 and m=+1)
py=superposition
(subtraction of m=-1 and m=+1)
Chemistry: Shells – set of orbitals with similar energy
1s2
2s2, 2p6 (px2, py2, pz2)
3s2, 3p6
3d10
These are the wave functions (orbitals) we just found:
n=1, 2, 3 … = Principle Quantum Number
En   E1 / n
2
(for Hydrogen, same as Bohr)
l=s, p, d, f … = Angular Momentum Quantum Number
=0, 1, 2, 3
(restricted to 0, 1, 2 … n-1)
| L | l (l  1) 
m = ... -1, 0, 1.. = z-component of Angular Momentum
(restricted to –l to l)
Lz  m
n=1, 2, 3 … = Principle Quantum Number
En   E1 / n
2
(for Hydrogen, same as Bohr)
l=s, p, d, f … = Angular Momentum Quantum Number
=0, 1, 2, 3
(restricted to 0, 1, 2 … n-1)
| L | l (l  1) 
m = ... -1, 0, 1.. = z-component of Angular Momentum
(restricted to -l to l)
Lz  m
An electron in hydrogen is excited to Energy = -13.6/9 eV. How
many different wave functions nlm in H have this energy?
[graded indep. but use groups]
a. 1 b. 3 c. 6 d. 9 e. 10
An electron in hydrogen is excited to Energy = -13.6/9 eV. How
many different wave functions in H have this energy?
a. 1 b. 3 c. 6 d. 9 e. 10
n= Principle Quantum Number:
l=(restricted to 0, 1, 2 … n-1)
m=(restricted to -l to l)
n
3
3
3
3
3
3
3
3
3
l
0
1
1
1
2
2
2
2
2
En   E1 / n 2
n=3
l=0,1,2
Answer is d:
m
0 3s states 9 states all with the same energy
-1
0 3p states (l=1)
Isn’t this cool…
1
Chemists had already
-2
figured out rules for how
-1
many electrons could be in
0 3d states (l=2)
each shell. Didn’t know
1
why. Solving Schrödinger
2
equation explains WHY!
Energy Diagram for Hydrogen
l=0
(s)
n=3
n=2
3s
2s
l=1
(p)
3p
l=2
(d)
3d
2p
In HYDROGEN, energy only
depends on n, not l and m.
(NOT true for multi-electron atoms!)
n=1
1s
l=0,m=0
n=1, 2, 3 … = Principle Quantum Number
En   E1 / n
2
(for Hydrogen, same as Bohr)
l=s, p, d, f … = Angular Momentum Quantum Number
=0, 1, 2, 3
(restricted to 0, 1, 2 … n-1)
| L | l (l  1) 
m = ... -1, 0, 1.. = z-component of Angular Momentum
(restricted to -l to l)
Lz  m
What is the magnitude of the angular momentum of the ground
state of Hydrogen?
a. 0 b. h c. sqrt(2)h
d. not enough information
Answer is a.
n=1 so l=0 and m=0 ... Angular momentum is 0 …
Schrodinger finds quantization of energy and angular momentum:
n=1, 2, 3 …
l=0, 1, 2, 3 (restricted to 0, 1, 2 … n-1)
En   E1 / n
2
| L | l (l  1) 
How does Schrodinger compare to what Bohr thought?
same
I. The energy of the ground state solution is ________
II. The angular momentum of the ground state solution is different
_______
different
III. The location of the electron is _______
a. same, same, same
b. same, same, different
c. same, different, different
d. different, same, different
e. different, different, different
Bohr got energy right,
but he said angular
momentum L=nh, and
thought the electron was
a point particle orbiting
around nucleus.
How does Schrödinger model of
atom compare with other models?
Why is it better?
Student ideas…
•
•
•
•
•
•
Solved S’s equation for hydrogen:
wave functions, energies, angular momentum
In atom with multiple electrons, what do you expect to change
in the way you set up the problem?
and in the solutions?
Student Ideas:
A.
B.
C.
D.
E.
F.
G.