radial functions - indian institute of technology bombaynaresh/courses/ch107-l5.pdf · radial...
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Radial functions
100
1s N e
200 22 2s N e
0
r
a
210 22 cos
zp N e
For s-Orbitals the maximum probability denisty of finding the electron is on the nucleus
For s-Orbitals the probability of finding the electron on the nucleus zero
Surface plots
Surface plot of the 2s ; 2s wavefunction (orbital) of the hydrogen atom. The height of any point on the surface above the xy plane (the nuclear plane) represents the magnitude of the 2s function at the at point (x,y) in the nuclear plane. Note that there is a negative region (depression) about the nucleus; the negative region begins at r=2a0 an goes asymptotically to zero at r=.
Surface plot of the |2s|2; the probability density associated with
the 1s wavefunction of the hydrogen atom. Note that the negative region of the 2s plot on the left now appears as positive region.
Surface plot of the 1s wavefunction (orbital) of the hydrogen atom. The height of any point on the surface above the xy plane (the nuclear plane) represents the magnitude of the 1s function at the at point (x,y) in the nuclear plane. The nucleus is located in the xy place immediately below the ‘peak’
Surface plot of the |1s|2; the probability density associated with
the 1s wavefunction of the hydrogen atom.
1s
2s
(1s)2
(2s)2
Surface plots
R(2pz)
(2pz)2
Surface plot of radial portion of a 2p wavefunction of the hydrogen atom. The gird lines have been left transparent so that the inner ‘hollow’ portion is visible.
Profile of the radial portion of a 2p wavefunction of the hydrogen atom.
Profile of the 2pz orbital along the z-azis. Surface plot of the 2pz wavefunction (orbital) in the xz (or yz) plane for the hydrogen atom. The ‘pit’ represents the negative lobe and the ‘hill’ the positive lobe of a 2p orbital.
Surface plot of the (2pz)2; the probability density
associated with the 2pz wavefunction of the hydrogen atom. Each of the hills represents and area in the xz (or yz) plane where the probability density is the highest, The probability density along the x (or y) axis passing through the nucleus (0,0) is everywhere zero.
2pz
2pz
R(2pz)
Surface plots
Surface plot of the 3dz2 wavefunction (orbital) in the xz (or yz) plane for the hydrogen atom. The large hills correspond to the positive lobes and the small pits correspond to the negative lobes.
Surface plot of the (3dz2 )2 the probability density associated with the 3dz2
orbital of the hydrogen atom. This figure is rotated with respect to the figure on the left so that the small hill will be clearly visible. Another smaller hill is hidden behind the large hill.
Surface plot of the 3dxy wavefunction (orbital) in the xz plane for the hydrogen atom. The hills and the pits have same amplitude. Surface plot of the (3dxy )
2 the probability density associated with the 3dxy orbital of the hydrogen atom. Pits in the figure to the left appear has hills.
Radial and Radial Distribution Functions
2 2
2
2 2 2
Probability of finding the electron
anywhere in a shell of thickness
at radius is 4 ( ) (for )
increasing function
4 ( ) 0 as 4 0
nl
nl
dr r r R r dr s
r
r R r dr r dr
Radial Distribution Functions
2 24 ( )nlr R r
3s: n=3, l=0 Nodes=2
3p: n=3, l=1 Nodes=1
3d: n=3, l=2 Nodes=2
ns nsr r
Number of radial nodes = n-l-1
Shapes and Symmetries of the Orbitals
s-Orbitals
3 2 3 2
2
1 2
0
1 1 1 1 2
4 2o o
r ra a
s s
o o
re e
a a a
Function of only r; No angular dependence Spherical symmetric
n-l-1=0 l=0
n-l=0
radial nodes angular nodes
Total nodes
n-l-1=1 l=0
n-l=1
Shapes and Symmetries of the Orbitals
p-Orbitals
Function of only r , (and ) Not Spherical symmetric 2pz Orbital: No dependence Symmetric around z-axis
radial nodes angular nodes Total nodes
n-l-1=0 l=1
n-l=1
3 2
2
210 2
0
1 1cos
4 2o
z
ra
p
o
re
a a
xy nodal plane Zero amplitude at nucleus
Angular Distribution Functions
p-Orbitals
3 2
2
210 2
0
1 1cos 0 case
4 2o
z
ra
p
o
re m
a a
+
–
cos
0 1.000
30 0.866
60 0.500
90 0.000
120 -0.500
150 -0.866
180 -1.000
210 -0.866
240 -0.500
270 0.000
300 0.500
330 0.866
360 1.000
2210 2 cos
zp N e
Angular part: Polar plot of 2pz --- cos
x
z
p-Orbitals
2210 2 cos
zp N e
22
22
22
cos
sin cos
sin sin
z
x
x
p
p
p
N e
N e
N e
Color/shading are related to sign of the wavefunction
d-Orbitals
2
2 2
2 2 313
2 33 2
2 33 3
2 2 33 4
2 2 33 5
(3cos 1)
(sin cos cos )
(sin cos sin )
(sin cos2 )
(sin sin2 )
z
xz
yz
x y
xy
d
d
d
d
d
N e
N e
N e
N e
N e
Angular part
Blue: -ve Yellow: +ve
Angular + Radial
n=3; l=2; m=0,±1, ±2
f-Orbitals
n=4; l=3; m=0,±1, ±2, ±3
Green: -ve Red: +ve
Cross-sections of Orbitals
Hydrogen Wavefunctions
Orbitals: External Structure
Orbitals: Internal Stucture
Hydrogen atom & Orbitals
Hydrogen atom has only one electron, so why bother about all these orbitals? 1. Excited states 2. Spectra 3. Many electron atoms
Many Electron Atoms
Helium is the simplest many electron atom
+
-
-
r1
r2
r12= r1- r2
2 22 2 2 22 2 2
1 2
0 1 2 12
1
2 2 2 4N N
N
N e e
Z e Z e eH
m m m r r r
KE of Nucleus
KE of Electron1
KE of Electron2
Attraction between nucleus and Electron1
Attraction between nucleus and Electron1
Repulsion between Electron1 and Electron2
Helium Atom
2 22 2 2 22 2 2
1 2
0 1 2 12
2 22 2 2 22 2 2
1 2
1 2 12 0
2 22 2 2 22 2 2
1 2
1 2 12
1
2 2 2 4
1;
2 2 2 4
2 2 2
N NN
N e e
N NN
N e e
N NN eN
N e e
N eN n N
Z e Z e eH
m m m r r r
QZ e QZ e QeH Q
m m r m r r
QZ e QZ e QeH H
m m r m r r
H E H
e e eE
Helium Atom
2 22 2 22 2
1 2
1 2 12
2
1 2
12
2 22 22 2
1 21 2
1 2
2 2
and 2 2
N Ne
e e
e
N N
e e
QZ e QZ e QeH
m r m r r
QeH H H
r
QZ e QZ eH H
m r m r
The Hamiltonians Ĥ1 and Ĥ1 are one electron Hamiltonians similar to that of hydrogen atom
1 21 1 1 2 2 2 1 1 1 2 2 2 1 1 1 2 2 2
2
1 1 1 2 2 2
12
( , , , , , ) ( , , , , , ) ( , , , , , )
( , , , , , )
e e e e
e
H r r H r r H r r
Qer r
r
Orbital Approximation
1 1 1 2 2 2 1 1 1 1 2 2 2 2( , , , , , ) ( , , ) ( , , )e e er r r r
(1,2,3,... ) (1) (2) (3) ( )e n n
Orbital is a one electron wavefunction The total electronic wavefunction of n number of electrons can be written as a product of n one electron wavefunctions
1 21 1 1 2 2 2 1 1 1 2 2 2 1 1 1 2 2 2
2
1 1 1 2 2 2
12
( , , , , , ) ( , , , , , ) ( , , , , , )
( , , , , , )
e e e e
e
H r r H r r H r r
Qer r
r
1 1 1 2 2 2 1 1 1 1 2 2 2 2( , , , , , ) ( , , ) ( , , )e e er r r r
1 21 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2
2
1 1 1 1 2 2 2 2
12
( , , ) ( , , ) ( , , ) ( , , )
( , , ) ( , , )
e e e e e e
e e
H H r r H r r
Qer r
r
Helium Atom: Orbital Approximation
1 21 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2
2
1 1 1 1 2 2 2 2
12
( , , ) ( , , ) ( , , ) ( , , )
( , , ) ( , , )
e e e e e e
e e
H H r r H r r
Qer r
r
Helium Atom: Orbital Approximation
1 1 1 1 1 2 2 2 2 2 1 1 1 1 2 2 2 2
2
1 1 1 1 2 2 2 2
12
( , , ) ( , , ) ( , , ) ( , , )
( , , ) ( , , )
e e e e e e
e e
H r r r r
Qer r
r
2
1 2 1 1 1 1 2 2 2 2
12
( , , ) ( , , )e e e e
QeH r r
r
2 4 2
1 2 2 2 2 2
0
13.6
8
Z e ZeV
h n n
Helium Atom: Orbital Approximation
2
1 2 1 1 1 1 2 2 2 2
12
( , , ) ( , , )e e e e
QeH r r
r
If we ignore the term
2
12
Qe
r
1 2 1 1 1 1 2 2 2 2( , , ) ( , , )e e e eH r r
He 1 2E = 108.8eV
3 2 3 2
1 1
1 1(1) (2)o o
Zr Zra a
e s s
o o
Z Ze e
a a
Helium Atom: Orbital Approximation
He 1 2
He
E = (54.4 54.4) 108.8
E (24.59 54.4) 78.99 (Experimental)
eV eV
eV eV
Ignoring is not justified! Need better approximation
2
12
Qe
r