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THE HYDROGEN ATOM
Juan I. Rodríguez Hernández
Escuela Superior de Física y Matemáticas
Instituto Politécnico Nacional
Mexico City
August 2010 Part of the Course: Molecular Modelling
THE HYDROGEN ATOM
Natural
abundance
1H 99.985 %
2H 0.015 %
3H unstable
~75% of universe’s
mass is 1H !!
~90% of universe’s
atoms are 1H !!
Experimental ionization energy: IE= 1312 kJ/mol = 13.5eV 2
The hydrogen atom The Schrodinger equation
EH
VTH ˆˆˆ
must be a “well behaved” function:
T is the kinetic energy operator
V is the potential energy operator
● single-valued
● continuous function and continuous n-th’s partial derivatives
● must be finite everywhere (in its dominium)
● normalized:
12
* dd
3
The Hamiltonian function (in SI):
epe
e
p
p
hydrogenrr
e
m
p
m
pH
2
0
22
4
1
2
)(
2
)(
),,(
),,(
eee
ee
ppp
pp
zyxiip
zyxiip
The Hamiltonian operator:
ep
e
e
p
p
hydrogenrr
e
mmH
2
0
22
22
4
1
22ˆ
Transformation of H to
Quantum Operator
pr
er
4
The hydrogen atom The Schrodinger equation
EH
),(),(}4
1
22{
2
0
22
22
epep
ep
e
e
p
p
rrErrrr
e
mm
pr
er
5
The hydrogen atom The Schrodinger equation
),(),(}4
1
22{
2
0
22
22
epep
ep
e
e
p
p
rrErrrr
e
mm
Step 1: Solving SE
Step 2:Computing Properties
6
Solving SE:
Transforming to CMS
),(),(}4
1
22{
2
0
22
22
epep
ep
e
e
p
p
rrErrrr
e
mm
ep
ep
eepp
rrr
mm
rmrmR
rmm
mRr
rmm
mRr
ep
p
e
ep
ep
? 7
Transforming to CMS
),,ˆ,ˆ(ˆepep rrppH
Chain rule
),,ˆ,ˆ('ˆ rRppH rR
),,,( epep rrppH
),,,(' rRppH rR
),,ˆ,ˆ('ˆ rRppH rR
8
It is better doing the transformation “before” (transformation of the
Hamiltonian FUNCTION:
Better transforming
Hamiltonian function H
rmm
mRr
rmm
mRr
ep
p
e
ep
ep
r
ep
p
Re
r
ep
eRp
vmm
mvv
vmm
mvv
222222
2
1
2
1
2
1
2
1
2
)(
2
)(rReepp
e
e
p
pvMvvmvm
m
p
m
pT
Total mass: ep mmM
Reduced mass: ep
ep
mm
mm
9
The hydrogen atom SE
in the CMS system
),(),()}(22
{ 22
22
epepepe
e
p
p
rrErrrrVmm
),(),()}(22
{ 22
22
rRErRrVM
rR
10
Separation of Schrodinger Equation
),(),()}(22
{ 22
22
rRErRrVM
rR
)()(2
22
RERM
RR
)()()}(
2{ 2
2
rErrV rr
rR EEE
rRrR
)()(),(
11
Separating the SE in the CMS
),(),()}(22
{ 22
22
rRErRrVM
rR
)()(2
22
RERM
RR
)()()}(
2{ 2
2
rErrV rr
12
The Schrodinger Equation for the
reduced mass particle
)()()}(
2{ 2
2
rErrV rr
Fictitious particle with mass μ,
e
ep
epm
mm
mm9994557.0
Note: Equation very similar to Born Oppenheimer approximation equation
Subjected to Coulomb potential
r
erV
1
4)(
0
2
kgm
kgm
e
p
31
27
10109.9
10673.1
ep mm 1837
13
14
Remember that this transformation can be
applied for any CENTRAL POTENTIAL
)()()}(2
{ 22
rErrV rr
Transforming to spherical coordinates
)()()}(
2{ 2
2
rErrV rr
)()()}(]sin
1cot
112[
2{
2
2
2222
2
22
22
rErrVrrrrrr
r
)()()}(ˆ2
1]
2[
2{ 2
22
22
rErrVLrrrr
r
}sin
1cot{ˆ
2
2
22
222
L
15
Separating the SE in the CMS
),(),(ˆ2 ffL
)()()}(ˆ2
1]
2[
2{ 2
22
22
rErrVLrrrr
r
)()()})((2]2
[{ 2
2
222 rRrRErVr
rrrr r
),()(),,( frRr
16
The angular equation
)()( 2
2
2
m
d
d
)()(),( f
),(),(ˆ2 ffL
),(),(}sin
1cot{
2
2
22
22
ff
)()(}sin
cot{22
2
2
2
m
d
d
d
d
17
Solution of the angular equation:
Spherical Harmonics
imm
l
m
l ePml
mllYf )(cos
)!(4
)!)(12(),(),(
)()1()( 22
lm
mm
m
l Pd
dP
])1[(!2
1)( 2 l
l
l
lld
d
lP
Associated Lengendre Polynomials
,...2,1,0l llllm ,1,...,2,1,0,...,1,
Lengendre Polynomials
18
Solution of the angular equation:
Spherical Harmonics
,...2,1,0l llllm ,1,...,2,1,0,...,1,
),()1(),(ˆ 22 m
l
m
l YllYL
Conditions coming from the well behaved requirement on ψ
''
'
'
0
2
0
sin),(),(* mmll
m
l
m
l ddYY
19
The “radial” equation
)()1()()})((2]
2[{ 22
2
222 rRllrRErVr
rrrr r
1
0
21
12
1212
!)!12()!1(
})!{()1()()(
ln
k
kk
rl
ll
lnkklkln
lnL
d
dL
,...3,2,1n 1,...,2,1,0 nl
)2(})!{(2
)!1()
2()
2()( 12
3
3 naZrLrelnn
ln
na
Z
na
ZrR l
ln
lnaZrl
nl
2
2
04
ea
''
0
2
'' )()( llnnlnnl drrrRrR
20
The associated Laguerre polynomials:
Laguerre polynomials:
)()(
ed
deL r
r
r
r
Hydrogen Wave Functions
and Energies
,...2,1,0l
llllm ,1,...,2,1,0,...,1,
,...3,2,1n
),()(),,( m
lnlnlm YrRr
222
0
42
222
0
42 1
8
1
)4(2 nh
eZ
n
eZEE rn
Principal quantum number
Azimuthal quantum number
magnetic quantum number 21
,...2,1,0l llllm ,1,...,2,1,0,...,1, ,...3,2,1n
Hydrogen eigenfunctions
and eigenvalues
),,(),,(ˆ rErH nlmnnlm
r
eH r
1
42ˆ
0
22
2
),,( rnlm eigenfunctions
nE Eigen values
22
,...2,1,0l llllm ,1,...,2,1,0,...,1, ,...3,2,1n
Eigen -functions and -values
degeneracy
),,(),,(ˆ rErH nlmnnlm
),,( rnlm
nEThe degree of DEGENRACY is
equal to n2
23
Once the problem (ES) is solved,
what else?
24
Hydrogen Properties!
Energy!
222
0
42
222
0
42 1
8
1
)4(2 nh
eZ
n
eZEE rn
eV
J
JsmNC
CkgZ
h
eZEn
598.13
1017868.2
)1062607.6)(/108541878.8(8
)6021765.1)(1010938.99994557.0(
8
18
2342212
4312
22
0
42
1
25
eVE 598.131 ???
Energy??
26
3
0
2
4 r
reF
Ley de Coulomb:
pr
er
r
r
dr
rdVVF
)(
Sabemos que el potencial que produce esta fuerza es central:
2
0
2 1
4
)(
r
e
dr
rdV
Energy??
27
2
0
2 1
4
)(
r
e
dr
rdV
Cr
drerV 2
0
2
4)(
0)( rVr
erV
1
4)(
0
2
Energy??
28
0)( rV
r
0V
Hydrogen Properties!
Energy!
222
0
42
222
0
42 1
8
1
)4(2 nh
eZ
n
eZEE rn
eV
J
JsmNC
CkgZ
h
eZEn
598.13
1017868.2
)1062607.6)(/108541878.8(8
)6021765.1)(1010938.99994557.0(
8
18
2342212
4312
22
0
42
1
eVEn 598.131 eVIE 573.13exp !!
Theory Experiment 29
eVEn 598.131 eVIE 573.13exp !!
Theory Experiment
30
eVEn 598.131 eVIE 598.13exp !!
Theory Experiment
31
eVEn 598.131 eVIE 598.13exp !!
Theory Experiment
¿¿ Sera ésto simulación??
32
eVEn 598.131 eVIE 598.13exp !!
Theory Experiment
epe
e
p
p
hydrogenrr
e
m
p
m
pH
2
0
22
4
1
2
)(
2
)(
pr
er
33
Simulación: Acción de simular
Simular: Representar algo,
fingiendo o imitando
lo que no es
34
eVEn 598.131 eVIE 598.13exp !!
Theory Experiment
¿¿ Sera ésto simulación??
Hydrogen energies:
quantum states
36
photon
Hydrogen energies:
quantum states
37
photon
Series n 1
38
)1
1(8 222
0
42
11nh
eZEEE nn
)1
1(8
11232
0
42
1nch
eZE
chn
1710520973731568.1 mRtheory
32
0
42
8 ch
eZR
Lyman series
39
)1
1(1
2nR
17
exp 1009737315.1 mR eriment
1710520973731568.1 mRtheory
40
)1
1(1
2nR
17
exp 1009737315.1 mR eriment
1710520973731568.1 mRtheory
¿¿ Sera ésto simulación??
Hydrogen energies:
quantum states
41
Spectral series n m
42
(ultraviolet)
(visible)
(infrared)
1,...,2,1,0 nl llm ,0,...,,...3,2,1n
Hydrogen eigenfunctions
),()(),,( m
lnlnlm YrRr
43
pr
er
222
0
42
222
0
42 1
8
1
)4(2 nh
eZ
n
eZEE rn
),,(),,(ˆ rErH nlmnnlm
1,...,2,1,0 nl llm ,0,...,,...3,2,1n
Hydrogen-like eigenfunctions
),()(),,( m
lnlnlm YrRr
44
pr
er
222
0
42 1
8 nh
eZEn
EH ˆ
N
),,(),,(ˆ rErH nlmnnlm
1Z ZZ
ep
ep
mm
mm
eN
eN
mm
mm
? He+
Li2+
Be3+
B4+
Hydrogen-like eigenfunctions
),()(),,( m
lnlnlm YrRr
45
)2(})!{(2
)!1()
2()
2()( 12
3
3 naZrLrelnn
ln
na
Z
na
ZrR l
ln
lnaZrl
nl
2
2
04
ea
222
0
42 1
8 nh
eZEn
Dependence on Z and :
Hydrogen-like eigenfunctions
),()(),,( m
lnlnlm YrRr
46
imm
l
m
l ePml
mllY )(cos
)!(4
)!)(12(),(
Complex
Real Spherical Harmonics
47
imm
l
m
l ePml
mllY )(cos
)!(4
)!)(12(),(
0
0)(cos2))1((2
1
cos)(cos2))1((2
1
0
),(
0
mif
mifsenmPNYYi
mPNYY
mifY
Y
m
llm
m
l
mm
l
m
llm
m
l
mm
l
l
m
l
Notation:
,...,,,,,
,...6,5,4,3,2,1
hgfdps
l
Real hydrogen-like functions
48
A linear combination of eigenfunctions of the same
degenerate eigenvalor is eigenfunction.
),()(),,( m
lnlnlm YrRr
Are they eigen funtions of the hydrogen-like Hamiltonian?
Real hydrogen-like functions
49
n l m Symbol for
orbital
1 0 0 1s
2 0 0 2s
2 1 1 2p+1 +
2 1 0 2p0 px, py, pz
2 1 -1 2p-1 +
3 0 0 3s
3 1 1 3p+1 +
3 1 0 3p0 px, py, pz
3 1 -1 3p-1 +
3 2 2 3d+2 ++
3 2 1 3d+1 |
3 2 0 3d0 dz2, dxz , dyz , dxy , dx
2-y2
3 2 -1 3d-1 +´+
3 2 -2 3d-2 +
Real hydrogen-like functions
50
),()(),,( m
lnlnlm YrRr
na
Z
Notation
Real hydrogen-like functions
51
),()(),,( m
lnlnlm YrRr
na
Z
Real hydrogen-like functions
52
),()(),,( m
lnlnlm YrRr
Hydrogen eigenfunctions
),()(),,( m
lnlnlm YrRr
53
Fourth Postulate:
rdrnlm
2),,(
Probability of finding the electron in a
infinitesimal volumen around at ),,(),,( zyxrr
v
What information????
Hydrogen eigenfunctions
),()(),,( m
lnlnlm YrRr
54
Fourth Postulate:
rdrnlm
2),,(
Probability of finding the electron in a
infinitesimal volumen around at ),,(),,( zyxrr
v
What information????
Radial Distribution Function
55
r
ddrdrYYrRrD m
l
m
lnlnl sin),(),()()(
2
0 0
2*
22)]([)( rrRrD nlnl
1sin),(),(
2
0 0
*
ddYY m
l
m
l
Radial Distribution Function
56
Amme
a 529.010529.04 10
2
2
00
a0 is the Bohr radious:
Radial Distribution Function
57
Radial distribution functions for the 2s and 3s density distributions
drrRrdrrrDr nlnl
0
3
0
)()(
)})1(
1(2
11{
2
0
2
n
ll
Z
anr
Electron Density
58
)(1)( electronsofnumberNrdr
gives the probability of finding an electron at position )(r r
2)()()( rerer
Charge density:
0
2 )()(
rrV
2)()( rr
Experimental quantity !!
59
2
1 )()( rr s
Electron density
contour maps: 1s case
60
Electron density contour maps: 2s & 2p cases
Orbital Density
61
Electron density contour maps: 3d and 4f cases
Orbital (3dxz) Density
Orbital (4fxz2)
Average values:
Properties
62
rdrPrPP nlmnlm
),,(ˆ),,(ˆ *
a the property operator P
Atomic Units (a.u.)
63
1 1em 02 e=
)()(}1
42{
0
22
2
rErr
er
)()(}1
2
1{ 2 rEr
rr
Atomic Units (a.u.)
64
5 1 15 31 27.2 2.20 10 6.58 10 2.63 10 / .)Hartree eV cm Hz kJ mol
)()(}1
2
1{ 2 rEr
rr
HartreesuaE 5.0..5.0
mABohr 111029.5529.01
Energy:
Lenth:
Mass: kgme
31101095.9
Charge:
Ce 19106022.1
Many Electron Atoms
65
),(),(}1
2
1
2{
11
2
1
222
ii
N
i
N
ij ji
N
i j
N
i
rR
n
rRErRrrrR
e
M i
EH
For gold N=79, so we have 3*79=237 independent variables !!!
