The hydrogen atom - w3.esfm.ipn.mxw3.esfm.ipn.mx/.../juan_The_hydrogen_atom2011OK.pdf · The...
Transcript of The hydrogen atom - w3.esfm.ipn.mxw3.esfm.ipn.mx/.../juan_The_hydrogen_atom2011OK.pdf · The...
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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
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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
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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
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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
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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
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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
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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
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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:
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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
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The hydrogen atom SE
in the CMS system
),(),()}(22
{ 22
22
epepepe
e
p
p
rrErrrrVmm
),(),()}(22
{ 22
22
rRErRrVM
rR
10
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Separation of Schrodinger Equation
),(),()}(22
{ 22
22
rRErRrVM
rR
)()(2
22
RERM
RR
)()()}(
2{ 2
2
rErrV rr
rR EEE
rRrR
)()(),(
11
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Separating the SE in the CMS
),(),()}(22
{ 22
22
rRErRrVM
rR
)()(2
22
RERM
RR
)()()}(
2{ 2
2
rErrV rr
12
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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
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14
Remember that this transformation can be
applied for any CENTRAL POTENTIAL
)()()}(2
{ 22
rErrV rr
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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
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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
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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
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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
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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
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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
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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
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,...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
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,...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
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Once the problem (ES) is solved,
what else?
24
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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 ???
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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
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Energy??
27
2
0
2 1
4
)(
r
e
dr
rdV
Cr
drerV 2
0
2
4)(
0)( rVr
erV
1
4)(
0
2
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Energy??
28
0)( rV
r
0V
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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
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eVEn 598.131 eVIE 573.13exp !!
Theory Experiment
30
eVEn 598.131 eVIE 598.13exp !!
Theory Experiment
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31
eVEn 598.131 eVIE 598.13exp !!
Theory Experiment
¿¿ Sera ésto simulación??
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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
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33
Simulación: Acción de simular
Simular: Representar algo,
fingiendo o imitando
lo que no es
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34
eVEn 598.131 eVIE 598.13exp !!
Theory Experiment
¿¿ Sera ésto simulación??
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Hydrogen energies:
quantum states
36
photon
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Hydrogen energies:
quantum states
37
photon
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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
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Lyman series
39
)1
1(1
2nR
17
exp 1009737315.1 mR eriment
1710520973731568.1 mRtheory
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40
)1
1(1
2nR
17
exp 1009737315.1 mR eriment
1710520973731568.1 mRtheory
¿¿ Sera ésto simulación??
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Hydrogen energies:
quantum states
41
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Spectral series n m
42
(ultraviolet)
(visible)
(infrared)
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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
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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+
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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 :
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Hydrogen-like eigenfunctions
),()(),,( m
lnlnlm YrRr
46
imm
l
m
l ePml
mllY )(cos
)!(4
)!)(12(),(
Complex
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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
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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?
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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 +
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Real hydrogen-like functions
50
),()(),,( m
lnlnlm YrRr
na
Z
Notation
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Real hydrogen-like functions
51
),()(),,( m
lnlnlm YrRr
na
Z
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Real hydrogen-like functions
52
),()(),,( m
lnlnlm YrRr
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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????
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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????
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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
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Radial Distribution Function
56
Amme
a 529.010529.04 10
2
2
00
a0 is the Bohr radious:
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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
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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 !!
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59
2
1 )()( rr s
Electron density
contour maps: 1s case
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60
Electron density contour maps: 2s & 2p cases
Orbital Density
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61
Electron density contour maps: 3d and 4f cases
Orbital (3dxz) Density
Orbital (4fxz2)
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Average values:
Properties
62
rdrPrPP nlmnlm
),,(ˆ),,(ˆ *
a the property operator P
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Atomic Units (a.u.)
63
1 1em 02 e=
)()(}1
42{
0
22
2
rErr
er
)()(}1
2
1{ 2 rEr
rr
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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
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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 !!!