To infinity and beyong: diatomic molecules above the ... › assets › pdf › ams › 2014 › AMS... · 7/29/2014 · To ∞ and beyond: diatomic molecules above the dissociation

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

National Aeronautics and Space Administration!

Re-entry is Hot!!!"

To infinity and beyond 2

National Aeronautics and Space Administration! To infinity and beyond 3

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


HyperRad Development team"Alan Wray"Yen Liu"Duane Carbon"Winifred Huo"Galina Chaban"Richard Jaffe"David Schwenke"




+

=


Solar spectrum"



Spectrum for each compound is unique ! remote detection""Helium: solar in 1868, terrestrial in 1895"What causes the spectrum?"Newton didn’t know""



Simplest spectrum: Hydrogen Atom"


Visible Spectrum

Ultraviolet Spectrum 1880s:

1/λ=R[(1/n”)2-(1/n’)2]


More complicated: Na"



Quantum Mechanics"!! "


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


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!!!!



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!



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!


925

!

"#

$

%&

2

≈ 2.5 ×1015


-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+


-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


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!


-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








C)  Splitting!

D)  Easier to interpolate!

E)  Easier to adjust!


-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


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"""""" ""


-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








C)  Easier to interpolate!

D)  Easier to adjust!

E)  Splitting!


-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.)



-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



-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



-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



-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


-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



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""""


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


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


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, …


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


3Σ-g

3Σ-u

1Πu


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"


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




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


3Σ-g

3Σ-u

1Πu


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


3Σ-g

3Σ-u

1Πu


-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



Line shapes…"Bound-bound"" "Voigt profile is Gold-Standard"""Bound-free"


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

?


Line shapes…"Bound-bound"" "Voigt profile is Gold-Standard"""Bound-free"


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


Review"•  Hypersonic Re-entry!

•  Spectroscopy!

•  Quantum Mechanics!

•  Born-Oppenheimer Approximation!

•  Electronic Structure Calculations!

•  Diabatic States!

•  Bound vs. Free states!

•  Line shapes!


-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.)


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


Documents

To infinity and beyong: diatomic molecules above the ... › assets › pdf › ams › 2014 › AMS... · 7/29/2014 · To ∞ and beyond: diatomic molecules above the dissociation