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Lecture 18: Orbitals of the Hydrogen
AtomThe material in this lecture covers the following in Atkins.
The structure and Spectra of Hydrogenic Atoms
13.2 Atomic orbitals and their energies(c) Shells and subshells(b) s-orbitals(c) Radial distribution function(d) p-orbitals(e) d-orbitals
Hydrogenic atomic orbitals
Lecture on-lineHydrogenic atomic orbitals (PDF Format)
Hydrogenic atomic orbitals (PowerPoint)
Handout for this lecture
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Audio-visuals on-line key hydrogen orbitals (PowerPoint)(From the Wilson Group,***)key hydrogen orbitals (PDF)(From the Wilson Group,***)
Vizualization of atomic hydrogen orbitals (PowerPoint)
(From the Wilson Group,***)Vizualization of atomic hydrogen orbitals (PDF)(From the Wilson Group,***)
Slides from the text book (From the CD included in Atkins ,**) Interactive Hydrogen Orbital Plots (For Mac users only)( Visualizes all the
angular and radial wavefunctions of the hydrogen atom, *****)
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Orbitals of Hydrogenic Atom
Thenlm
orbitals for the hydrogenic atom are given by(r, , R(r) Y ( , ; n = 1,2,3 ...l < n - 1; m = - l, - l+ 1, ....l -1, l
nl l,m ) )=
Wherel ml m
im
Y , ) =2l+14
P
are eigenfunctions to L and L
l,m l|m|
2 z
(( | !|( | !|)
(cos ) exp[ ] +
With the radial part given as
R r Nn
L r enl nll
n ln( ) ( ),
/ = 2
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Hydrogen Levels
nn l m=1,
(, ,l = 0
( , ) = R (r)Y ( , )nl l,m
R r e1 0
3 22
,
//
( ) =
2
Za o
No rnodes. R everywhere positive1,0 ( )
For l = 0 we have m = 0;
Yo,o =1
4
Value of Y is uniform oversphere
oo1s orbital
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Orbitals of Hydrogenic Atom...prop.dens.
The probability density(r, , ) = (r, , (r, ,nlm nlm nlm PP ) )*
A constant-volume electron-sensitivedetector (the small cube) gives its greatest
reading at the nucleus, and a smallerreading elsewhere. The same reading isobtained anywhere on a circle of given
radius: the s orbital is sphericallysymmetrical.
PP 100
3 22
3 22
3,
/ / *
/ /
( , . )r e Y
e Y e
oo
oo
=
=
2Z
a
2Z
a1 Z
a
o
o o
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Representations of the1 s and 2 s hydrogenicatomic orbitals in termsof their electron densities
(as represented by thedensity of shading).
The probability density
(r, , ) = r, , r, ,nlm nlm nlmPP
) )
PP
PP
1 00
3
200
32 22
12
,
/
( )
( ) ( )
r e
r e
=
=
1 Z
a
132
Za
o
o
Orbitals of Hydrogenic Atom...prop.dens.
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Orbitals of Hydrogenic Atom
constant-volume electron-sensitiveetector (the small cube) gives its greatesteading at the nucleus, and a smallereading elsewhere. The same reading is
btained anywhere on a circle of givenadius: the s orbital is sphericallyymmetrical.
PP 1 00
3 22
3 22
3
,
//
//
( , . )r e Y
e Y e
oo
oo
=
=
2Z
a
2 Za
1 Za
o
o o
Radial probability density
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Orbitals of Hydrogenic Atom
he radial distribution function P gives the probability that the electron
ill be found anywherein a shell of radius r. For a 1 s
lectron in hydrogen, P
is aaximum when r is equal tohe Bohr radius a 0. The valuef P is equivalent to the reading
hat a detector shaped like apherical shell would give asts radius was varied.
Radial probability density
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Orbitals of Hydrogenic Atom
The probability of finding the electronbetween r and r + dr is :
| R (
| R (
nlm
nlm
nl
nl
P r dr r
r r d d r dr
r Y d d r dr
r r dr
Thus
P
nl oo
oo
lmoo
( ) ( , , )
( , , )
( ) | ( , ) |
( ) |
= =
=
=
PP
PP
dv
sin
sin
1
the radial probability density is
2
22
2 2 22
2 2
nlnl r r r( ) ( ) |=| R (nl2 2
Radial probability density
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Orbitals of Hydrogenic Atom
The radial probability density is
most probable radius corrsponds
to the maxima for1s
r max
P r r r
The
P rFor
aZ
nl
nl
o
( ) ( ) |
( ).
=
=
| R (nl2 2
Radial probability density
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Orbitals of Hydrogenic Atom
Orbitals with increasing l are called
p - orbitals m = - 1, 0,1n1m =R r Yn m1 1( ) ( , )
n2m d - orbitals m = - 2, -1, 0,1, 2=R r Yn m2 2( ) ( , )
n3m f - orbitals m = - 3, -2, -1, 0,1, 2, 3=R r Yn m3 3( ) ( , )
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Orbitals of Hydrogenic Atom
R
R (r) R (r)R (r) R (r)
nl
nl nlnl nl
is a solution to
+ + + + =h h2 2
2
2
222
41
2
{ ) {
( )}
r r rZe
rl lmr
Eo
Veff
As l increases the centrigugal term
becomes more repulsive near nucleus at small r.Hence R tend to zero as r o with increasingspead as l becomes larger
nl
h 2
21
2
l l
mrr
( )
( )
+
Orbital near nucleus
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Orbitals of Hydrogenic Atom
Close to the nucleus, p orbitals are proportional to r ,d orbitals are proportional to r 2,and f orbitals are proportional to r 3.
Electrons are progressively excludedfrom the neighbourhood of the nucleusas l increases. An s orbital has afinite, nonzero value at the nucleus.
Orbital near nucleus
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Orbitals of Hydrogenic Atom
he variation of the mean radius of aydrogenic atom with the principal andrbital angular momentum quantum
umbers. Note that the mean radius lies inhe order d < p < s for a given value of n.
Mean radius
The mean radius is given as theespectation value
< >= r r dvnlm nlm
For the hydrogenic atom
< r > nl = + +
nl l
naZo2
2112
11
(( )
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Orbitals of Hydrogenic Atom
The boundary surface of an s orbital,within which there is a 90 per cent
probability of finding the electron.
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Orbitals of Hydrogenic Atom
nlm (r, , R(r) Y ( ,n = 2 , l = 1
nl l,m) )=
211
1 238
(r, , R(r) Y ( , = -R(r)21 1,1 21) ) sin/
= e
i
m = 1 0 1, ,
2101 23
4(r, , R(r) Y ( , = R(r)21 1,0 21) ) cos
/
=
21 1
1 238
= (r, , R(r) Y ( , = R(r)21 1,-1 21) ) sin/
e i
Let us now look at the angular part of the p - functions
Angular part
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Orbitals of Hydrogenic Atom
2101 23
4(r, , = R(r) 21) cos
/ l = 1 , m = 0
xy
z
L2 =h 2l(l +1) Lz = hm l
m=0
The electron has an angular momentum L
such that L L =z - component of L is L
2
z
r
hs +
=l l
The o( )1
Angular part
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211
1 23
8(r, , -R(r) 21) sin
/
= ei
Orbitals of Hydrogenic Atom
x
y
z
L2 = h 2l(l +1)
Lz= h
m l
x
yz
r v
vL =
vr
vv
l = 1 , m = 1
The electron has an angular momentum L
such that L L =z - component of L is L
2
z
r
hs
h
+=
l lThe
( )1
Angular part
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Orbitals of Hydrogenic Atom
21 1
1 23
8 (r, , = R(r) 21) sin
/e i l = 1 , m = - 1
x
yz
L2 =h 2l(l +1)
Lz = hm lm=-1
The electron has an angular momentum L
such that L L =z - component of L is L
2
z
r
hs
h +
= l l
The( )1
Angular part
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Plotting
p z
angular part of
2101 23
42(r, , = R(r) 21) cos
/
=
Orbitals of Hydrogenic Atom
R =k cos +
-
Draw vector R of lengthkcos through each point
( , ) on sphere
r
Draw surface through allvectors R
Real orbitals
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2111 23
8(r, , -R(r) 21) sin [cos sin ]
/
= +i
Orbitals of Hydrogenic Atom
21 11 23
8
(r, , = R(r) 21) sin [cos sin ]/
i
The remaining two orbitals are complex
However by linear combinations we can get the real orbitals
22
38211 21 1
1 2p
iy =
{ + } = R(r) 21 /
sin sin
212
38211 21 1
1 2p x =
{ - } = R(r)21 /
sin cos
Angular part
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22
3
8211 21 1
1 2p
iy =
{ + } = R(r) 21
/
sin sin
2 12 38211 21 11 2
p x = { - } = R(r)21 /
sin cos
Orbitals of Hydrogenic Atom
The orbitals 2p and 2p have the same energies and eigenvalues to
L as andx y
2211 21-1 . However only the latter are eigenfunctions of L z
+
-
x
z
+-
y
2p y 2p x
Angular part
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Orbitals of Hydrogenic Atom
The boundary surfacesof p orbitals. A nodalplane passes throughthe nucleus and
separates the two lobesof each orbital.The dark and light areas denoteregions of opposite signof the wavefunction.
Angular part
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Orbitals of Hydrogenic Atom
Grotrian diagram that summarizes theppearance and analysis of the spectrum of tomic hydrogen. The thicker the line, theore intense the transition.
In transitions betweendifferent energy levels of
the hydrogenic atom
the following selectionrules apply
l = 1m = o, 1
Transitions
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What you need to learn fromthis lecture about the hydrogen atom
for the quizz and the final exam
The
R r
definition of the radialdistribution functionP(r) = r and how itis used to calculate the
expaction values < r
2
n
( )2
>
The rbehaviour of R nearthe nucleus (Fig 13.16)
nl ( )
Understand how the
balance between kineticenergy and potentialenergy shapes R
3.10nl
Fig.1
Selection rules for electronictransitions in thehydrogen atom
l = 1m = o, 1
Memorize the relation betweenreal p - orbitals (p and
imaginary p - orbitals (pUnderstand the physicaldifference. Understandplots of (p
xo,
x
, , )
, ).
, , ).
p p
p p
p p
y z
y z
1 1
Same for d and f great,but not required
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Appendix : Orbitals of Hydrogenic Atom..why nodes
R r e20
3 242
1
2( ) ( )
//=
1
2 2
Z
a o
n n on l m
= ,(, ,
l = 0
( , ) = R (r)Y ( , )nl l,m
R r e30
3 22 66 2
19
( ) ( )/
/=
+
19 3
Za
o
One node at
= 2Zr/anode : 2a
o
o
( )
/
2 12
0 =with
Z
two nodes at
= 2Zr/anodes : 1.9a
7.1Zr/a
o
o
o
( )
/
6 21
902 + =
withZ
and
For l = 0 we have m = 0;
Yo,o = 14
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n n on o oo
= ,(, ,
l = 0( , ) = R , (r)Y ( , )n o o,oo
Thus Y Y
R Y R Y dv
oo oo
n o oo n o oo
any two s - orbitals R and Rmust be orthorgonal
n'o no
' '000
2
0
=
Why has R n - 1 nodes ?nl ( )r
We have seen previously that
any two independent solutionsto the Schrdinger equation mustbe orthogonal :
i =j ijdv
Appendix : Orbitals of Hydrogenic Atom..why nodes
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R Y R Y dv
Y Y d d R r R r r dr
n o oo n o oo
oooo
oo n oo
r
no
' '
'sin ( ) ( )
000
2
22
0
0
=
=
1
normalized
R r R r r dr
Y
n oo
r
no
oo
' ( ) ( ) =2 0
The
R r e
R function is positiveevery where
2Z
a
1o
o1 0
3 22
,
//( ) =
Appendix : Orbitals of Hydrogenic Atom..why nodes
f
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Thus for R to be orthogonal
to R
must have positive andnegative regions
2o
1o
R r R r r dr
R
oo
r
o1 22
20
0 =( ) ( )
R r e20
3 242
12
( ) ( )/
/=
12 2
Za o
For R to be orthogonal toR and R two nodes arerequired etc...
3o
1o 20
Appendix : Orbitals of Hydrogenic Atom..why nodes
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e1
e 2
e3e 4
e 5
Appendix : Drawing Orbitals of Hydrogenic Atom An asideHow do we plot Y ?lm
Consider a unit spherewithradius 1. Draw a large
number ofunit vectors fromthe origin ofthe sphere indifferent directions(e.i. different , ) e e ee e
1 2 3
4 n
r r r
r r
, , ,...
Calculateat
where R eDraw
n
n n
the value of Ythe position of each e
(e.i for each Constructthe vectors : R R R R
YR
lm
n
n
1 2 3 n
lm
n
( , )
, )., , , ..
: ( , )
r
rr=
R2
R 3
1R
R4
R 5
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An aside
R2
R 3
1R
R4
R 5
Draw
This
a surface through the
endpoints of all Rsurface represents Y
nlm
r
.( , )
Appendix : Drawing Orbitals of Hydrogenic Atom
A idd b l f d
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An aside
For Y we have
R
oo
n
( , )
r r= 1
2for all ne n
We
n n
have that Y is sphericale.u. the same for all
oo ( , ),
THus Y representing the angularpart of a ns function is representedby a sphere
oo
ns - orbital
Appendix : Drawing Orbitals of Hydrogenic Atom