Post on 06-Jul-2020
National Aeronautics and Space Administration
www.nasa.gov
To ∞ and beyond: diatomic molecules above the dissociation limit"
David W. Schwenke"NASA Ames Research Center"
Applied Modeling & Simulation Seminar Series July 29, 2014
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Re-entry is Hot!!!"
To infinity and beyond 2
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Free
Stre
am
N2,
O2 T
=300
K
Pay
load
Car
bon
base
d H
eat S
hiel
d
Bou
ndar
y La
yer
N2,
O2,
NO
, CN
, C2
T~10
00K
Equ
ilibr
ium
Reg
ion
N, N
+ , O
, O+ ,
N2,
N2+ ,
NO
, e-
T~60
00K
Sho
ck
N, N
+ , O
, O+ ,
e-
T=20
000-
8000
0K
hν
v=~11 km/s
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HyperRad Development team"Alan Wray"Yen Liu"Duane Carbon"Winifred Huo"Galina Chaban"Richard Jaffe"David Schwenke"
To infinity and beyond 4
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National Aeronautics and Space Administration! To infinity and beyond 6
+
=
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Solar spectrum"
To infinity and beyond 7
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Spectrum for each compound is unique ! remote detection""Helium: solar in 1868, terrestrial in 1895"What causes the spectrum?"Newton didn’t know""
To infinity and beyond 8
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Simplest spectrum: Hydrogen Atom"
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Visible Spectrum
Ultraviolet Spectrum 1880s:
1/λ=R[(1/n”)2-(1/n’)2]
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More complicated: Na"
To infinity and beyond 10
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Quantum Mechanics"!! "
To infinity and beyond 11
ih ∂∂tΨα x , t( ) = − 1
2 ∇x2 +V x( )%& '(Ψα x , t( ) ≡ H x( )Ψα x , t( )
Ψα x , t( ) = e−iEαt/hψα x( )H x( )ψα x( ) = Eαψα x( )c = λνhνβα = Eβ − Eα
ψβ*∫ x( )M x( )ψα x( ) dx
2
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Quantum numbers"n: principle quantum number!!1,2,3,4,…!l: electronic angular momentum quantum number!!0 (S for sharp), 1 (P for principle), 2 (D for diffuse), 3 (F), 4 (G), …!Λ: electronic angular momentum along diatomic axis! 0 (Σ±), ±1 (Π), ±2 (Δ), ±3 (Φ), …!S: electron spin angular momentum quantum number!!0,1/2, 1, 3/2, …!Spin multiplicity: 2S+1!!1 (S=0, singlets), 2 (S=1/2, doublets), 3 (S=1, triplets), …!Interchange symmetry: g,u!!!!
To infinity and beyond 12
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Divide and Conquer"1: Freeze nuclei, solve for electron wave functions!! !high dimensionality, low density of states!2: Allow nuclei to move under forces of electron wave functions!! !low dimensionality, high density of states!Born and Oppenheimer 1927!H=Helec+Hnuc+Hcross!
To infinity and beyond 13
National Aeronautics and Space Administration!
Divide and Conquer"1: Freeze nuclei, solve for electron wave functions!! !high dimensionality, low density of states!2: Allow nuclei to move under forces of electron wave functions!! !low dimensionality, high density of states!Born and Oppenheimer 1927!H=Helec+Hnuc+Hcross!
!!
To infinity and beyond 14
National Aeronautics and Space Administration!
Divide and Conquer"1: Freeze nuclei, solve for electron wave functions!! !high dimensionality, low density of states!2: Allow nuclei to move under forces of electron wave functions!! !low dimensionality, high density of states!Born and Oppenheimer 1927!H=Helec+Hnuc+Hcross!
! !justify by low density of states, mN/me >>1!
To infinity and beyond 15
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Solving for electron motion"Integral-differential equation with homogenous boundary conditions at infinity!èexpand in terms of analytic basis functions!• Assign electrons to basis functions in all possible ways & diagonalize:!
!full configuration interaction (FCI)!!For 10 e-, 92 basis functions, !
• Or!• Optimize molecular orbitals with small FCI: 536 variables!
• Allow 0, 1, or 2 e- outside of small FCI space: 1,729,808 functions!
• Contract pair excitations: 47,375!
To infinity and beyond 16
925
!
"#
$
%&
2
≈ 2.5 ×1015
National Aeronautics and Space Administration! To infinity and beyond 17
-92.6
-92.55
-92.5
-92.45
-92.4
0 1 2 3 4 5 6
Tota
l Ene
rgy
(a.u
.)
Bond Length R (a.u.)
CN Potential Curves
1000/cm=10 micron
10,000/cm=1 micron
blue=4750 A
2Sig+2Pi
4Sig+
National Aeronautics and Space Administration!
-92.6
-92.5
-92.4
-92.3
-92.2
-92.1
0 1 2 3 4 5 6
Total
Energ
y (a.u
.)
Bond Length R (a.u.)
CN Potential Curves
1000/cm=10 micron
10,000/cm=1 micron
100,000/cm=1000 A
blue=4750 A
2Sig+2Pi
4Sig+4Del
4Pi4Sig-2Sig-2Del2Phi
6Sig+4Sig-
6Pi4Del4Phi
18
National Aeronautics and Space Administration!
Advances in potential energy curve calculation"A) Dynamic weighting for valence MO determination!
A) Adjust weighting by value of energy!
B) Rydberg Orbital Calculation!A) First compute valence MOs: inward!
B) Freeze valence MOs, compute Rydberg MOs: outward!
C) Diabatic States!A) Hcross remains small!
B) Phase factors!!
C) Splitting!
D) Easier to interpolate!
E) Easier to adjust!
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-150.2
-150.15
-150.1
-150.05
-150
-149.95
-149.9
-149.85
-149.8
1 2 3 4 5 6
Ener
gy in
Har
tree
Bond Length in Bohr
O2 1Sig+g
National Aeronautics and Space Administration!
Advances in potential energy curve calculation"A) Dynamic weighting for valence MO determination!
A) Adjust weighting by value of energy!
B) Rydberg Orbital Calculation!A) First compute valence MOs: inward!
B) Freeze valence MOs, compute Rydberg MOs: outward!
C) Diabatic States!A) Hcross remains small!
B) Phase factors!!
C) Splitting!
D) Easier to interpolate!
E) Easier to adjust!
To infinity and beyond 20
-129.8
-129.7
-129.6
-129.5
-129.4
-129.3
1.5 2 2.5 3 3.5 4 4.5
Ener
gy in
Har
tree
Bond Length in Bohr
NO 2Pi
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Adiabatic vs. Diabatic states"Thermodynamics: without gain or loss of heat"Adiabatic from the Greek “incapable of being crossed” "" "Nuclei are stationary, solve for electron motion: "Diabatic – not adiabatic"""""" ""
To infinity and beyond 21
-129.8
-129.75
-129.7
-129.65
-129.6
-129.55
-129.5
1.6 1.8 2 2.2 2.4 2.6 2.8 3
Ener
gy (a
.u.)
Bond Length (a.u.)
NO 2Pi diabatic states
E1 0 H11 H12
= UT U
0 E2 H21 H22
H r( )ψαa = Eα r( )ψα
a
ψαd does not "change" with r
National Aeronautics and Space Administration!
Advances in potential energy curve calculation"A) Dynamic weighting for valence MO determination!
A) Adjust weighting by value of energy!
B) Rydberg Orbital Calculation!A) First compute valence MOs: inward!
B) Freeze valence MOs, compute Rydberg MOs: outward!
C) Diabatic States!A) Hcross remains small!
B) Phase factors!!
C) Easier to interpolate!
D) Easier to adjust!
E) Splitting!
To infinity and beyond 22
-129.8
-129.75
-129.7
-129.65
-129.6
-129.55
-129.5
1.6 1.8 2 2.2 2.4 2.6 2.8 3
Energ
y (a.u.)
Bond Length (a.u.)
NO 2Pi diabatic states
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-129.8
-129.7
-129.6
-129.5
-129.4
-129.3
-129.2
1 2 3 4 5 6
Ener
gy in
Har
tree
Bond length in Bohr
NO 2Pi adiabatic/molecular diabatic states
To infinity and beyond 23
National Aeronautics and Space Administration!
-129.8
-129.7
-129.6
-129.5
-129.4
-129.3
-129.2
-129.1
-129
1 2 3 4 5 6
Ener
gy in
Har
tree
Bond length in Bohr
Splitting NO 2Pi
To infinity and beyond 24
National Aeronautics and Space Administration!
-129.8
-129.7
-129.6
-129.5
-129.4
-129.3
-129.2
-129.1
-129
1 2 3 4 5 6
Ener
gy in
Har
tree
Bound length in Bohr
Split NO 2Pi
To infinity and beyond 25
National Aeronautics and Space Administration! To infinity and beyond 26
-75.4
-75.35
-75.3
-75.25
-75.2
-75.15
-75.1
-75.05
-75
1 2 3 4 5 6 7 8 9 10
Ene
rgy
(a.u
.)
Bond length (a.u.)
C2+
4sig-g2piu4pig
2delg2sig-g2sig+g4sig-u
2pig2sig+u4delu
4sig+u2sig-u2delu
6piu4piu
4delg4sig+g2phig
National Aeronautics and Space Administration!
-75.4
-75.35
-75.3
-75.25
-75.2
-75.15
-75.1
-75.05
-75
1 2 3 4 5 6 7 8 9 10
Ener
gy (a
.u.)
Bond length (a.u.)
C2+
3P
1D
1S
5S
4sig-g2piu4pig
2delg2sig-g2sig+g4sig-u
2pig2sig+u4delu
4sig+u2sig-u2delu
6piu4piu
4delg4sig+g2phig
To infinity and beyond 27
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Solving for nuclear motion"All states are coupled! (except ug)"Rotation strongly coupled to vibration"Finite Difference Boundary Value method""Bound states: diagonalize a banded matrix""Free states: solve banded linear equations""""
To infinity and beyond 28
0
0.1
0.2
0.3
0.4
0.5
1 2 3 4 5 6 7
Ener
gy (a
.u.)
Bond Length (a.u.)
N2 effective potential
200
100
50
National Aeronautics and Space Administration! To infinity and beyond 29
How to find energies"""""Bound state:"""Free state:"""
Hψα = Eαψα, ψα = ηη
∑ Fηα r( )
H = −h2
2µη
η
∑ ∂ 2
∂r2η + η
η !η
∑ Vη !η r( ) !η
limr→0Fηα r( ) = 0 and limr→∞Fη
α r( ) = 0b.c. determine energy
limr→0F "ηηE r( ) = 0
limr→∞F "ηηE r( ) = k "η
− 12 exp −ik "η r( )δη "η − exp ik "η r( ) Sη "η E( )%& '(
kη2 = 2µ E − eη( )
energy determines b.c.
incoming outgoing
complex
National Aeronautics and Space Administration! To infinity and beyond 30
How to find energies"""""Bound state:"""Free state:"""
Hψα = Eαψα, ψα = ηη
∑ Fηα r( )
H = −h2
2µη
η
∑ ∂ 2
∂r2η + η
η !η
∑ Vη !η r( ) !η
limr→0Fηα r( ) = 0 and limr→∞Fη
α r( ) = 0b.c. determine energy
limr→0F "ηηE r( ) = 0
limr→∞F "ηηE r( ) = k "η
− 12 exp −ik "η r( )δη "η − exp ik "η r( ) Sη "η E( )%& '(
kη2 = 2µ E − eη( )
energy determines b.c.
incoming outgoing
complex
acoustics: echoes, whisper rooms, …
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0
0.05
0.1
0.15
0.2
0.25
0.3
1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
Ener
gy in
Har
tree
Bond Lenth in Bohr
O2 Schumann-Runge
To infinity and beyond 31
3Σ-g
3Σ-u
1Πu
National Aeronautics and Space Administration! To infinity and beyond 32
1e-20
1e-15
1e-10
1e-05
1
100000
1e+10
48000 50000 52000 54000 56000 58000 60000
|S|^
2 in
arb
itrar
y un
its
Frequency omega in 1/cm
Γα / 2E − Eα( )2 + Γα
2 / 4
Odd Parity"
National Aeronautics and Space Administration!
Even Parity"
1e-20
1e-15
1e-10
1e-05
1
100000
1e+10
48000 50000 52000 54000 56000 58000 60000
|S|^
2 in
arb
itrar
y un
its
Frequency omega in 1/cm
To infinity and beyond 33
National Aeronautics and Space Administration!
0
0.05
0.1
0.15
0.2
0.25
0.3
1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
Ener
gy in
Har
tree
Bond Lenth in Bohr
O2 Schumann-Runge
To infinity and beyond 34
3Σ-g
3Σ-u
1Πu
National Aeronautics and Space Administration!
0
0.05
0.1
0.15
0.2
0.25
0.3
1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
Ener
gy in
Har
tree
Bond Lenth in Bohr
O2 Schumann-Runge
To infinity and beyond 35
3Σ-g
3Σ-u
1Πu
National Aeronautics and Space Administration!
-150.2
-150.15
-150.1
-150.05
-150
-149.95
-149.9
1 2 3 4 5 6 7
Ener
gy in
Har
tree
Bond Length in Bohr
03P+ 03P+
01D+ 01D+
01D+ 03P+
01S+ 03P+
1sig-u3delu
3sig+u3piu1piu5piu
3sig-u
To infinity and beyond 36
National Aeronautics and Space Administration!
Line shapes…"Bound-bound"" "Voigt profile is Gold-Standard"""Bound-free"
To infinity and beyond 37
L ν( ) = A ν0( ) exp0
∞
∫ −ν − $ν( )2
2σ 2
%
&''
(
)**
γ / 2$ν − ν0( )2 + γ 2 / 4
d $ν
L ν( ) = A ν0( ) exp0
∞
∫ −ν − $ν( )2
2σ 2
%
&''
(
)**
$γ / 2$ν − ν0( )2 + $γ 2 / 4
d $ν ,
$γ = γ + Γ
?
National Aeronautics and Space Administration!
Line shapes…"Bound-bound"" "Voigt profile is Gold-Standard"""Bound-free"
To infinity and beyond 38
L ν( ) = A ν0( ) exp0
∞
∫ −ν − $ν( )2
2σ 2
%
&''
(
)**
γ / 2$ν − ν0( )2 + γ 2 / 4
d $ν
L ν( ) = A ν0( ) exp0
∞
∫ −ν − $ν( )2
2σ 2
%
&''
(
)**
$γ / 2$ν − ν0( )2 + $γ 2 / 4
d $ν ,
$γ = γ + Γ
? σ =!νc
kTm
A ν0( ) = 4ν03
3gc3hµ3
National Aeronautics and Space Administration!
Review"• Hypersonic Re-entry!
• Spectroscopy!
• Quantum Mechanics!
• Born-Oppenheimer Approximation!
• Electronic Structure Calculations!
• Diabatic States!
• Bound vs. Free states!
• Line shapes!
To infinity and beyond 39
-129.8
-129.75
-129.7
-129.65
-129.6
-129.55
-129.5
1.6 1.8 2 2.2 2.4 2.6 2.8 3
Ener
gy (a
.u.)
Bond Length (a.u.)
NO 2Pi diabatic states
1e-20
1e-15
1e-10
1e-05
1
100000
1e+10
48000 50000 52000 54000 56000 58000 60000
|S|^
2 in
arb
itrar
y un
its
Frequency omega in 1/cm