hamiltonian is the kinetic and potential energy of the...
TRANSCRIPT
![Page 1: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/1.jpg)
Vibrational Energy Structure
Hamiltonian is the kinetic and potential energy of the atoms vibrating in the potential set up by the positions of the nuclei.
HΨ=EΨ
![Page 2: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/2.jpg)
Diatomic molecules
What is the possible vibrational motion?
![Page 3: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/3.jpg)
Classically the frequency of oscillation for 2 atoms of mass m1 and m2 connected by a spring with constant k is
where µ is the reduced mass of the system
ν = 12π
kµ
µ= m1m2
m1+m2
![Page 4: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/4.jpg)
Quantum Solutionsolution is frequency ν0 of a harmonic oscillator. This maps out a potential
V
![Page 5: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/5.jpg)
Energy
where v is an integer (quantum number) v = 0, 1, 2, 3, …
Ei = vi + 12( ) hν i
the energy of the harmonic oscillator i is
![Page 6: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/6.jpg)
![Page 7: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/7.jpg)
Vibrational Energy
see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II. Infrared and Raman Spectra of Polyatomic Molecules," Van Nostrand, New York, 1960.) for details
E ν1,ν2,ν3( )= v1 + 12( )ω1+ v2 + 12( )ω 2 + v3 + 12( )ω 3 iii
![Page 8: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/8.jpg)
Linear triatomic
What motions are possible?
![Page 9: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/9.jpg)
Basis Set
To map all motions is complicated. Is there a set of basis vectors that will allow us to describe any motion?
![Page 10: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/10.jpg)
normal modes
![Page 11: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/11.jpg)
CO2
![Page 12: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/12.jpg)
Vibrational Energy
bend mode: ν2 = 667.37998 cm-1
!
symmetric stretch mode: ν1 = 1388.18435 cm-1
!
asymmetric stretch mode: ν3 = 2349.14295 cm-1
![Page 13: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/13.jpg)
Non-linear triatomic
H2O as an example
![Page 14: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/14.jpg)
Vibrational basis vectorssymmetric stretch mode ν1 = 3657.0532 cm-1
![Page 15: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/15.jpg)
Vibrational basis vectors
asymmetric stretch mode ν3 = 3755.9287 cm-1
![Page 16: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/16.jpg)
Vibrational basis vectors
bend mode ν2 = 1594.7498 cm-1
![Page 17: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/17.jpg)
![Page 18: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/18.jpg)
Molecules are not harmonic oscillators
![Page 19: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/19.jpg)
Anharmonic Corrections
E ν1,ν2,ν3( )= v1 + 12( )ω1+ v2 + 12( )ω 2 + v3 + 12( )ω 3 +
x11 v1 + 12( )2 + x22 v2 + 12( )2 + x33 v3 + 12( )2 +x12 v1 + 12( ) v2 + 12( )+ x13 v1 + 12( ) v3 + 12( )+x23 v2 + 12( ) v3 + 12( )+ iii
![Page 20: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/20.jpg)
Large molecules - HNO3
![Page 21: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/21.jpg)
Large molecules - HNO3
normal modes in cm-1 ω1 = 3550. ω6 = 647. ω2 = 1710. ω7 = 579. ω3 = 1326. ω8 = 762. ω4 = 1303. ω9 = 458. ω5 = 879.
![Page 22: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/22.jpg)
Intensities
units are cm-1/(molecule cm-2)
Sf←i T( ) = 8π3
3hcdi e
−Ei kT
Qν f←i 1−e
−hν f←ikT⎛
⎝⎜⎞⎠⎟1di
Rfξ←iξ '
2
ξ , ξ '∑
S is temperature dependent
![Page 23: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/23.jpg)
Line Shape
The lines we see in the spectra have a particular shape. The features can be understood in terms of line shape theory.
![Page 24: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/24.jpg)
γ
δ
ν0
![Page 25: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/25.jpg)
A. A. Michelson in the 1890s
![Page 26: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/26.jpg)
26
Interruption of radiation
� cm-1
![Page 27: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/27.jpg)
27
a single radiating molecule surrounded by a gas of perturbers
The System
![Page 28: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/28.jpg)
28
Connecting states
i i�
f
f�
optical transitions
collisionally induced transitions
J2
J2� ΔEi!i� ΔEJ2!J2
�
Absorbing Molecule Perturbing Molecule V
![Page 29: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/29.jpg)
29
γ − iδ( ) f← i =n22πc
v × 1 − e− iI S1 f ,i,J2v,b( )+ I S2 f ,i,J2v,b( ){ }e
RS2⎡⎣⎢
⎤⎦⎥ v,b,J2
Semi-classical Robert-Bonamy formalism
![Page 30: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/30.jpg)
γ and δ are temperature dependent
By making calculations or measurements at different temperatures we can model the temperate dependence of the line shape parameters
![Page 31: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/31.jpg)
Temperature Dependence of γ
■ Power law form !!!!
■ In practice plot (fit) !
!!
( ) ( )N
TTTT !"
#$%
&= 00γγ
( )( ) !
"#
$%&=
!"#
$%&
TTN
TT 00
ln lnγγ
![Page 32: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/32.jpg)
Temperature dependence of γ(CO2-N2)
![Page 33: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/33.jpg)
33
Validation of the CRB Calculations CO2-CO2, 296K 30012-00001
![Page 34: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/34.jpg)
uncertainty
How well can we simulate a spectra?
How well can we do a retreival?
![Page 35: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/35.jpg)
uncertainty
line positions - 4th place after decimal or better
S generally between 1 and 5 percent for most molecules.
γ and δ least well known.
![Page 36: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/36.jpg)
![Page 37: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/37.jpg)
![Page 38: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/38.jpg)
38
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, 43, 1102-1108, 2005
![Page 39: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/39.jpg)
39
![Page 40: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/40.jpg)
ν, S, γ, δ, E”
These are the spectral parameters needed to simulate a spectra. Given these parameters how do we simulate a spectra? What happens when light passes through a gaseous medium?
![Page 41: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/41.jpg)
Absorption of Radiation
![Page 42: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/42.jpg)
Observation
The intensity of the light is reduced after passing through a medium. The reduction is a function of frequency (wavelength, wavenumber) Proportional to amount of molecules in the path, length of the path, and temperature.
![Page 43: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/43.jpg)
Beer–Lambert–Bouguer law
the change in intensity along a path (of gas) ds is proportional to the amount of matter (gas) along the path.
dI ν( )=− kv ν( ) I ν( )ds
I ν( )= I0 ν( )e−kv ν( )ds
where kv(ν) is the volume absorption coefficient
![Page 44: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/44.jpg)
The absorption coefficient depends on the density of the gas
• the molecular absorption coefficient, kn(ν) = kv(ν)/n, where n is the number density of the absorbing molecules
• the mass absorption coefficient, km(ν) = kv(ν)/ρa where ρa is the density of the absorbing gas
• the absorption coefficient at s.t.p, ks(ν) = kv(ν)n/ns, where ns is Loschmidt's number (nL = NA/Vm = 2.686 763 x 1025 molecules/m3 at STP).
![Page 45: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/45.jpg)
What is important is the product kn(ν) ds must be unit less: So for each type of coefficient there is a corresponding different measure of path length.
![Page 46: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/46.jpg)
we will rewrite
I ν( )= I0 ν( )e−kv ν( )ds
to emphasize the path
Iν s2( )= Iν s1( )e−kv ν( )ds
s1 s2Iν Iν + dIνds
gas
![Page 47: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/47.jpg)
This equation is often written as
Iν s2( )= Iν s1( )Tν s1, s2( )where Ts(ν) is the monochromatic transmission function
Tν s1, s2( ) =e− kv ν( )dss1
s2
∫
![Page 48: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/48.jpg)
Assuming we know the radiation incident at some point along a path (i.e., at s1) and we make a measurement of the radiation flowing from the atmosphere at some other level along this path (i.e., at s2), this provides enough information to obtain the transmission.
![Page 49: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/49.jpg)
At this point, it is convenient to introduce the quantity
which we refer to as the optical path. This quantity is basic to our mathematical description of how radiation interacts with matter.
τ s1, s2( )= kv ν( )dss1
s2
∫
![Page 50: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/50.jpg)
To simplify the discussion we will consider an ideal atmosphere taken as a horizontally stratified medium.
Z2
Z=0
dzZ1
θ
Ps
P1
P2
ds= 1ηdz
ds
![Page 51: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/51.jpg)
The transmission along a path tilted from the vertical by an angle θ, the zenith angle, is simply related to the transmission along the vertical path according to
where τν(z1, z2) is now measured along the vertical and is referred to as the optical depth.
Tν s1, s2( ) =Tν z1, z2,η = cosθ( )=e−τν z1,z2( )
η
![Page 52: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/52.jpg)
It is common to use the mass absorption coefficient in the definition of optical depth in describing the transmission along a path through an absorbing gas. From the previous Eqs. the slant path transmission function is
Tν s1, s2( ) = e− 1η kv ν( )ρa dz
z1
z2
∫⎛
⎝⎜⎜
⎞
⎠⎟⎟
![Page 53: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/53.jpg)
The optical mass is defined as
which is often quoted in units of grams per square centimeter.
u z1, z2( )= ρaz1
z2
∫ dz
![Page 54: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/54.jpg)
Using the hydrostatic assumption, dp/dz=-mg and the mixing ratio r = ρa/ρ, where ρ is the density of air gives
where g is the acceleration of gravity, and where p1 and p2 are the pressures associated with the altitudes z1 and z2, respectively
u p1, p2( )= 1g
rp2
p1
∫ dp
![Page 55: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/55.jpg)
Take a path from a satellite to the ground
when you apply the limits
u p1, p2( )= rg 0
ps
∫ dp
u p1, p2( )= r psg
![Page 56: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/56.jpg)
The absorption path for a uniformly mixed gas is directly proportional to the atmospheric surface pressure ps. !
This relationship that has been proposed as a basis for the remote sensing of surface pressure.
![Page 57: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/57.jpg)
For remote sensing, it is important to distinguish between monochromatic transmission functions and band transmission functions.
The former represents the transmission of radiation at one selected wavelength, whereas the latter is the transmission averaged over a range of wavelengths as specified, for example, by the spectral response of a particular instrument.
![Page 58: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/58.jpg)
suppose that the radiation received at a detector is of the form
where g(ν) is the spectral response function of the instrument over its spectral band pass Δν. In terms of the transmission function, the intensity measured at s2 is
IΔν = g(ν ) Iν dν∫
![Page 59: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/59.jpg)
If the spectral band Δν is sufficiently narrow that the incident intensity is constant across the band, then the band transmission function becomes
IΔν s2( )= g ν( ) Iν s1( )Tν s1, s2( )dνΔν∫
IΔν s2( )IΔν s1( )
⎛
⎝⎜⎞
⎠⎟=TΔν s1, s2( ) = g ν( )Tν s1, s2( )dν
Δν∫
![Page 60: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/60.jpg)
Therefore instrument properties [in this case the response function g(ν)] directly influence the transmission derived from measurements and must be accounted for in retrieval schemes.
![Page 61: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/61.jpg)
![Page 62: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/62.jpg)
![Page 63: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/63.jpg)
![Page 64: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/64.jpg)
![Page 65: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/65.jpg)
![Page 66: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/66.jpg)
![Page 67: Hamiltonian is the kinetic and potential energy of the ...faculty.uml.edu/robert_gamache/85.503/QM_2.pdf · see Herzberg (Herzberg, G, "Molecular Spectra and Molecular Structure II](https://reader033.vdocument.in/reader033/viewer/2022060210/5f0480237e708231d40e477e/html5/thumbnails/67.jpg)