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Chem 104A, UC, Berkeleys orbital (l=0, m=0)
H 1s orbital:
)2
1)(2(100
re
r
100 1
),()( ,,,, ll mllnmln YrR
MT: pp22
Chem 104A, UC, BerkeleyProbability Density
rerP
22
100)(
r
2100
1
This probability function gives the probability offinding the electron at any point in space.
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Chem 104A, UC, Berkeley
allspace
dV 12
At what radius is it most probable to find the electron?
r
2100
1
Is it at r=0?
rerP
22
100)(
Chem 104A, UC, Berkeley
Most probable radius: r +dr
Volume of the thin shell with thickness dr
drr 24
22
22
4)(/
4)(Pr
rrRdrP
drrrRobability
P/dr, radial distribution function (RDF)compares the probability of finding electron at different r
Also called radial probability function
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Chem 104A, UC, Berkeley
H 1s orbital:
rr ererRDF 2222 16)2(4 Most probable radius:
0)(
r
RDF
unit: a0
Chem 104A, UC, Berkeley
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Chem 104A, UC, BerkeleyAmplitude of the wavefunction: R(r)Whenever the function changes sign, there is a Radial Node:Radius at which the probability of finding the electronis zero.
No. of radial nodes=n-l-1
0
2/
2/3
0s2
/
)2(2
R
aZr
ea
Z
Chem 104A, UC, Berkeley
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Chem 104A, UC, Berkeley
Chem 104A, UC, Berkeley
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Chem 104A, UC, Berkeley
No. of radial nodes=n-l-1
Chem 104A, UC, Berkeley
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Chem 104A, UC, Berkeley
Angular component: YNo. of angular nodes =l
),()( ,,,, ll mllnmln YrR
l=0 s orbital:
2
100 Y
No angular dependenceGerade (g): even with respect to inversion
AllowedOnly one spatial orientation for a sphere
0 angular node
0lm
Chem 104A, UC, Berkeley
l=1 p orbital
cos2
310 Y
+
-
Ungerade (u) : odd with respect to inversion
Allowed
Three spatial orientations.
1 angular node
1,0 lm r
z
3
2
1
r
x
3
2
1
r
y
3
2
1
In cartesian coordinates
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Chem 104A, UC, Berkeley
l=2 d orbital
Gerade (g)
Allowed
Five spatial orientations.
2 angular nodes
cossincos22
3020 Y
2,1,0 lm
2
22225
4
1
r
yxz
2
15
4
1
r
xz
Chem 104A, UC, Berkeley
Contour diagram
•Orbital depiction is based on •With the sign of indicted (very important for bonding considerations)•List all node planes
2
3pz
r
z
3
2
1
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Chem 104A, UC, Berkeley
2
22225
4
1
r
yxz
Chem 104A, UC, Berkeley
Orbital Radial Function
R(r)
Angular
Function
Y(x,y,z)
Angular
Function
Y(,)
zp2 2/
62
1 rre
2
)/(3 rx
2
cos3
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Chem 104A, UC, Berkeley
Orbital Radial Function
R(r)
Angular
Function
Y(x,y,z)
Angular
Function
Y(,)
2
)/(3 rx
2
cos3zp3 3/2 )6(
681
4 rerr
For hydrogen, r with unit of a0
For multiple electron systems,Replace r with
oaZr /
Chem 104A, UC, Berkeley
Orbital Radial Function
R(r)
Angular
Function
Y(x,y,z)
Angular
Function
Y(,)
23z
d 3/2
3081
4 rer
4
/)3(5 222 rrz
4
)1cos3(5 2
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Chem 104A, UC, Berkeley
All atomic orbitals (on same atom) are mutually orthogonal.
021 dV
Chem 104A, UC, Berkeley
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Chem 104A, UC, Berkeley
1 electron system: Hydrogen
How about many-electron system?
Z+
1 electron with nuclear charge Z
Increasing electrostatic interactionOrbital contractOrbital energy drops
r
ZerV
2
)(
2
2 6.13
n
eVZEn
Example: For He+, Z=2, E1=-54.4 eV
Chem 104A, UC, Berkeley
For multi-electron atoms, Schrodinger equation can be set up, but can not be solved exactly.Use approximation.Build many-electron system:1. Pauli exclusion principle:
No two electrons in an atom can have the same set of four quantum numbers (n, l, ml, ms)
ms: spin quantum numberQuantized, +1/2 or –1/2
Each atomic orbital can contain at most two electrons.
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Chem 104A, UC, Berkeley
Aufbau principle: electron fill the atomic orbitals from the lowest energy up.
Chem 104A, UC, Berkeley
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Chem 104A, UC, Berkeley
1s 2s 2p 3s 3p 4s 3d 4p 5s 4d…..
Example:
He: 1,0,0, ½ & 1,0,0, -1/2
Li: 1,0,0, ½; 1,0,0,-1/2; 2,0,0, ½
Al: 13 electrons
21s12 21 ss
1212622 33][33221 psNepspss
Coreelectron
Valenceelectron
Chem 104A, UC, Berkeley
Recall in Hydrogen: 222
422 2
2 n
k
hn
em
r
eE e
nn
Why are they different in multi electron system?
E(2s)=E(2p)
Electron-electron repulsion
2+
He = He + 1 e+
First electron:
2
2 6.13
n
eVZEn
Second electron??
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Chem 104A, UC, Berkeley
Ionization Energy (IE): energy required to remove an electron from the gaseous atom.
IE(He) =24.6 eV
eVHeE 4.54)(1
Each electron feels +2 nucleus and 1 electron.
In a many-electron atom, each electron is simultaneously: •attracted to the protons in the nucleus •repelled by other electrons (like-charge repulsion)
Chem 104A, UC, BerkeleyThe net positive charge attracting the electron is called
the effective nuclear charge
For He: 34.1effZ
eVn
ZE eff 6.13
2
2
= -24.4 eV
IE(He) =24.6 eV
Now, Li?
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Chem 104A, UC, Berkeley12 21 ss
2s orbital penetrates the inner 1s electron shell better than 2p.
2s electron feel a greater effective nuclear charge lower E for 2s
)2()2( 11 pZsZ effeff E(2s) <E(2p)
Ground State: 12 21 ss
Chem 104A, UC, Berkeley
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Chem 104A, UC, Berkeley
For Z >1:
E(ns) < E(np) < E(nd) < E (nf) for given n
Zeff:
Penetrating ability:
low
poor
ZZeff
Shielding constant
Chem 104A, UC, BerkeleySlater's Rules:
1) Write the electron configuration for the atom using the following design;
(1s)(2s,2p)(3s,3p) (3d) (4s,4p) (4d) (4f) (5s,5p)
2) Any electrons to the right of the electron of interest contributes no shielding.
3) All other electrons in the same group as the electron of interest shield to an extent of 0.35 nuclear charge units
4) If the electron of interest is an s or p electron: All electrons with one less value of the principal quantum number (n-1 shell) shield to an extent of 0.85 units of
nuclear charge. All electrons with two less values of the principal quantum number (n-2 shell) shield to an extent of 1.00 units.
5) If the electron of interest is an d or f electron: All electrons to the left shield to an extent of 1.00 units of nuclear charge.
6) Sum the shielding amounts from steps 2 through 5 and subtract from the nuclear charge value to obtain the effective nuclear charge.
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Chem 104A, UC, Berkeley
Example:
O: 422 221 pss
45.355.48
55.4)35.0(5)85.0(22
ZZeff
p
