MHz to THz: Rotational Line Catalogue Frequencies

Placing a catalogue line on a terahertz axis

Rotational spectroscopy has a split personality about units. Pull a transition out of the JPL microwave catalogue or the Cologne Database and it arrives in megahertz, printed to four decimal places — CO J = 1–0 at 115 271.2018 MHz. Plot a survey of the same source and the horizontal axis almost always runs in terahertz, or in wavenumbers derived from it. Feeding a line frequency in below turns the catalogue value into the figure your plot, your receiver tuning table or your far-infrared colleague is expecting.

Conversion factor: THz = MHz ÷ 1 000 000. That CO ground-state line at 115 271.2018 MHz becomes 0.1152712 THz; the water line at 556 936.002 MHz becomes 0.556936 THz.

Two catalogues, one ladder of lines

Where the numbers come from

Catalogue entries are fitted from laboratory microwave and submillimetre measurements, then extended up the rotational ladder with a molecular Hamiltonian. Their native unit has always been megahertz, because that is what the lab spectrometers read out.

Rungs of a ladder, not a scatter

For a linear molecule the rotational lines are near-multiples of one another: CO climbs 115 271, 230 538 and 345 796 MHz, which is 0.115 271, 0.230 538 and 0.345 796 THz. Spot that spacing and the carrier is often identifiable from the pattern alone.

The third unit in the room

Far-infrared work adds wavenumbers, where 1 THz equals 33.356 cm⁻¹. Going from megahertz to terahertz first makes that last step a single multiplication instead of a chain of them.

From a line list to a plotted spectrum

Paste the catalogue frequency unchanged

Line lists print values with digit-group spacing and sometimes a comma decimal marker. Both are handled: spaces are stripped and a comma is read as a decimal point, so 345 795.9899 goes in exactly as it appears in the file.

Take the THz figure for the axis

Up to eight decimals are shown, enough to keep a transition separable from its neighbour at submillimetre frequencies. The output tracks every keystroke, so working down a list means editing one number rather than reloading the page.

Copy it straight into your plotting script

Each field has a copy button that hands over the bare number with no unit and no grouping spaces, which is what a marker position or an axis limit in a notebook wants. Ctrl+C inside a field behaves the same way.

Reverse it to query the catalogue

Catalogue searches expect megahertz, so use the swap button (↔) — or type into the right-hand field — to turn a 0.4922 THz feature back into 492 200 MHz before setting a search range. Either dropdown searches the full frequency list if you need a different unit.

Rotational lines as the catalogues list them

These are standard reference transitions, quoted at catalogue precision in megahertz and converted to terahertz. The wavelength column is a reminder of why the millimetre and submillimetre bands carry the names they do.

Transition	Catalogue frequency (MHz)	Same line (THz)	Vacuum wavelength
NH₃ (1,1) inversion	23 694.4955	0.02369450 THz	12.65 mm
HCN J = 1–0	88 631.602	0.08863160 THz	3.38 mm
HCO⁺ J = 1–0	89 188.525	0.08918853 THz	3.36 mm
CS J = 2–1	97 980.953	0.09798095 THz	3.06 mm
CO J = 1–0	115 271.2018	0.11527120 THz	2.60 mm
CO J = 2–1	230 538.000	0.23053800 THz	1.30 mm
CO J = 3–2	345 795.9899	0.34579599 THz	0.867 mm
Atomic carbon ³P₁–³P₀	492 160.651	0.49216065 THz	0.609 mm
H₂O 1(1,0)–1(0,1)	556 936.002	0.55693600 THz	0.538 mm
CO J = 7–6	806 651.806	0.80665181 THz	0.372 mm

Catalogue decimals survive the trip

A value such as 345 795.9899 MHz keeps enough digits on the terahertz side to stay distinguishable from a neighbouring transition, instead of collapsing into a rounded 0.3458.

A whole ladder in one sitting

Because the field re-converts as you edit, stepping J = 1–0 up to J = 7–6 takes a few keystrokes and gives every rung of the series in the notation your survey plot uses.

Down to the receiver’s own units

The searchable dropdowns reach kHz and Hz as well, which helps when a linewidth or a spectrometer channel width is quoted far below the line frequency itself.

Questions that come up with a line list

Why are line catalogues tabulated in megahertz at all?

Because megahertz is where the measurements were made and where the uncertainties live. A well-determined rotational transition is known to a few kilohertz, sometimes better, and in megahertz that reads as four tidy decimal places. Written in terahertz the same entry would need nine or ten decimals before the meaningful digits appeared, and a printed table of leading zeros is nearly unreadable. The convention dates back to the earliest microwave spectroscopy, and both the JPL and Cologne catalogues keep to it.

How wide is a line once Doppler broadening is included?

The fractional width equals the velocity spread divided by the speed of light, so the broadening scales with the line frequency. A cold cloud with a 1 km/s spread smears CO J = 1–0 by about 0.385 MHz, and J = 3–2 at three times the frequency by about 1.15 MHz. A 20 km/s outflow widens J = 1–0 to roughly 7.7 MHz. That figure is what you compare against a spectrometer channel width — resolution far finer than the astrophysics delivers buys nothing.

How precise are catalogue frequencies, and does it matter?

It matters a great deal, because a frequency error turns straight into a velocity error. Near 345 796 MHz an uncertainty of 50 kHz corresponds to about 43 m/s, and 10 kHz to roughly 8.7 m/s. Lab-measured lines are usually good to that level or better, while lines predicted by extrapolating a fit to high rotational states can be an order of magnitude worse — which is why every catalogue carries an explicit uncertainty column. Anyone deriving infall or rotation velocities has to read that column, not just the frequency.

Where do rotational transitions give way to vibrational ones?

Pure rotation owns the millimetre and submillimetre region; for a light molecule such as CO the ladder is still climbing past 0.8 THz and continues into the low terahertz. Ordinary stretching vibrations sit far higher — a fundamental at 300 cm⁻¹ already corresponds to about 9 THz, and most lie well above 10 THz, deep in the mid-infrared. The stretch in between is not empty: large-amplitude bends, internal torsions, hydrogen-bond modes and lattice vibrations in solids all live there, which is why the far-infrared is worked by both communities at once.

How do I turn a line frequency into a wavelength?

Divide the speed of light by the frequency, and converting to terahertz first keeps the arithmetic tidy. At 0.1152712 THz the answer is 2.60 mm, which is why CO J = 1–0 is spoken of as a 3 mm-band line; at 0.80665181 THz it is 0.372 mm, firmly submillimetre. For wavenumbers, multiply the terahertz figure by 33.356 — that same CO line comes out at 3.845 cm⁻¹. These are vacuum values, which is the convention every catalogue uses.