Molecule Types Linear Molecules | <Prev Next> |

The key parameters required for a
simulation of a linear molecule spectrum are summarized in Making a Linear Molecule Data File. The
steps below show how to do this for part of the Schumann-Runge
Bands of O_{2}, which are responsible for atmospheric
absorption around 190nm.

The molecular parameters are
required for any simulation; an excellent compilation of constants
for diatomic molecules can be found in the NIST Chemistry web book
at http://webbook.nist.gov/chemistry/.
This database includes an electronic version of information in the
book K.P. Huber and G. Herzberg, "Constants of Diatomic
Molecules". A formula search on O_{2} with "Constants of
diatomic molecules" selected yields many states; the
Schumann-Runge Bands arise from the B^{3}Σ_{u}^{-}
- X^{3}Σ_{g}^{-}. (The ν_{00}
value can be a helpful guide here as to which band to use.) An
abbreviated version of the table is presented below, with just the
B and X state entries.

The footnotes contain the source of the information, and also some
additional constants that are often essential to obtaining a
reasonable simulation including the the spin orbit coupling
constant, A (required if
the electron spin and orbital angular momentum are both non zero)
and and the spin-spin coupling constant, λ (required if the
electron spin is >= 1; LambdaSS
in PGOPHER). The
more important ones are referenced in the in the table above:

a) ω_{e}y_{e}
= -0.139, γ_{e} = -0.00055_{6} from a low order
fit to v ≤ 4._{
}b)_{ }The spin splitting
constants at low v are λ = 1.5, -γ ~ 0.04 cm^{-1}.

c) β =0.22E-6 for low v.

d) ω_{e}z_{e}
= -0.00127_{3
}e) Spin splitting constants λ_{0}
= +1.9847511, γ_{0} = -0.00842536;

f) B_{1}
= 1.42192

## Vibrational Dependence of The Constants

Much of this information is concerned with the variation of
constants with vibrational number, v. The numbers in the main
table are not suitable for direct entry into PGOPHER, as they are
equilibrium values. For each vibronic transition, with a
particular v' and v" a separate set of constants is required. To
choose a specific case we will simulate the B^{3}Σ_{u}^{-}
v' = 2 - X^{3}Σ_{g}^{-} v" = 0 transition.

### The band origin

The first step is to calculate the band origin. For this purpose the
following expression must be evaluated for each vibrational state:State | T_{e} |
ω_{e} |
ω_{e}x_{e} |
ω_{e}y_{e} |
B_{e} |
α_{e} |
γ_{e} |
D_{e} |
β_{e} |
r_{e} |
Trans. | ν_{00} |
---|---|---|---|---|---|---|---|---|---|---|---|---|

B ^{3}Σ_{u}^{-} |
49793.28 | 709.31 a Z | 10.65 a | -0.139 | 0.8190_{2} a b |
0.01206 a | -5.5_{6}E-4 |
4.55E-6 c | 1.6042_{6} |
B ↔ X R | 49358.15 Z | |

X ^{3}Σ_{g}^{-} |
0 | 1580.19_{3} Z |
11.98_{1} d |
.0474_{7} |
[1.4376766] e | 0.0159_{3} f |
[4.839E-6] e | 1.20752 |

a) ω

c) β =0.22E-6 for low v.

d) ω

f) B

E(v)
=
T_{e} + ω_{e}(v + ½) − ω_{e}x_{e}(v + ½)^{2} + ω_{e}y_{e} (v + ½)^{3} + ω_{e}z_{e} (ν + ½)^{4}

For BOrigin = 49793.28 + 709.31(2 +
½) − 10.65(2 + ½)^{2} + -0.139 (2 +
½)^{3} = 51497.82 cm^{-1}.

Note that ωFor X

Origin = 1580.19_{3}(0 +
½) − 11.98_{1}(v + ½)^{2} +
0.0474_{7}(0 + ½)^{3} + -0.001273(0 +
½)^{4} = 787.11 cm^{-1}. Here ω_{e}z_{e} is given.

The band origin for the transition is then 51497.82 - 787.11 =
50710.71 cmB_{v} = B_{e} − α_{e}(v +
½) + γ_{e}(v +
½)^{2} + ...

For BB_{2} = 0.8190_{2}
− 0.01206(v + ½) + -5.5_{6}E-4(v +
½)^{2} = 0.785395 cm^{-1}.

For XThe centrifugal distortion constants follow a similar equation:

D_{v}
= D_{e} − β_{e}(v + ½) + ...

though this can be omitted for a rough simulation.For B^{3}Σ_{u}^{-}
v' = 2, λ = 1.5

For X^{3}Σ_{g}^{-} v" = 0, λ = +1.9847511

The spin-rotation constant, γ, is also given for these two states,
though this is normally less important.For X

- Click on File, New, Linear Molecule
- Select View, Constants
- Click on LinearMolecule. This sets the parameters common to
all states, and the ones that need to be changed from the
default values are:

- Symmetric = "True" (as the molecule has a centre of symmetry)
- AsymWt = 0 (This is
the statistical weight of antisymmetric states, and the zero
reflects the fact that half the levels of O
_{2}are missing. - Click on "v=0" and enter the information for X
^{3}Σ_{g}^{-}v = 0 - Lambda = Sigma-
- S = 1 (the electron spin)
- gerade = True (the default, indicating g symmetry)
- B = 1.4376766
- LambdaSS (λ) = 1.9847511
- Click on "v=1" and enter the information for B
^{3}Σ_{u}^{-}v = 2 - Lambda = Sigma-
- S = 1 (the electron spin)
- gerade = False (to indicate u symmetry)
- Origin = 50710.71
- B = 0.785395
- LambdaSS (λ) = 1.5
- To avoid confusion the "v=1" name should be changed; right
click on "v=1" and select rename to do this. The name does not
affect the calculation, but "v=2" would be the logical choice!

This should be enough to for a
basic simulation; press the simulate button () and then the all button () and you should see a simulation.
With a small adjustment of the plot range it should look like
this:

## Checking and refining the simulation

It is important to check the simulation, not
only because it is easy to make mistakes in the steps above, but
also because there can be differences in the definitions of the
constants used. Checking against tabulated line positions or
published spectra is an effective check; note the facility for overlaying pictures
(from a Journal for example) onto a simulation.

In this case a numerical spectra is available from the Harvard-Smithsonian Center for Astrophysics website (http://cfa-www.harvard.edu/), specifically the CfA Molecular Data. For oxygen, measured cross sections are available; follow the links Measured cross sections: Schumann-Runge bands of all isotopes, 179 nm - 203 nm, (2,0) band, 50050-50720 cm^{-1} . This gives a table
of numbers which can be directly copied into PGOPHER; in your browser
use "select all", "copy" and then in PGOPHER use "Overlays, Paste". The observed
spectrum will appear above the simulation.

Following the instructions above gives a spectrum a reasonable simulation, and a good visual match can be obtained with a Lorentzian linewidth of 0.65 cm^{-1} (The
"Lor" box on the main window toolbar). The width does not take
effect in the simulation if the display is too compressed; zooming
in to the portion around the band origin gives the following plot:

Here the red line is the Harvard data and the black is the
simulation. The most obvious discrepancy is on the right, and is
most likely a transition from another band. The line positions are
not absolutely perfect, but this could be improved by adding D and γ to the simulation and
checking the literature for more accurate constants. The overall
experimental spectrum also shows additional lines at the other
end, probably from the (4-1) band. (An obvious exercise for the
reader is to check this using by following the procedure above for
this band.)

In this case a numerical spectra is available from the Harvard-Smithsonian Center for Astrophysics website (http://cfa-www.harvard.edu/), specifically the CfA Molecular Data. For oxygen, measured cross sections are available; follow the links Measured cross sections: Schumann-Runge bands of all isotopes, 179 nm - 203 nm, (2,0) band, 50050-50720 cm

Following the instructions above gives a spectrum a reasonable simulation, and a good visual match can be obtained with a Lorentzian linewidth of 0.65 cm