Glycolaldehyde                        MethylFormate                         Acetic Acid                           Trans isomer glycolaldehyde

 The location and abundance of glycolaldehyde in the the ISM is determined by the temperature of the atmosphere. Horn (2004) discovered that the bulk of the glycolaldehyde was distributed about the colder areas of the LMH of the Sg BR2 (N)2.   Calculations that permit a prediction of the number of molecules in a given temperature range are derived using the Boltzman Distribution. Calculations of this sort are facilitated by programs such as WebMO as they provide quick access to the rotational constants for a given molecules. Once known, the rotational constant can be substituted in the equation for rotational energy levels to derive the energy levels that correspond to rotational transitions within the molecule. The equation for this calculation is
  where h is Planck's constant and I is the moment of inertia.  Once known, the difference in energy levels can be used in the Boltzman equation to determine the  number of molecules that populate the various rotational energy states. Since temperature is a variable within the boltzman equation, it can be used to predict the number of molecules in a given energy state.  
 The goal of this webmo calculation is to use the rotational constants derived from WebMo and thence use these constants to determine the energy levels of the first eight transitions of  glycolaldehyde and methyl formate. Once known, the difference in energy levels can be used in the boltzman distribution equation to determine which energy levels are populated at a given temperature. This is of interest as the literature suggests that transitions between the first four energy levels are the source of the radiation through which the spatial scale of compound was determined (1). Transitions within the J=8 level were the initial emissions through which the molecule was detected (2).  Additionally the literature suggested that ratio of the methyl formate:acetic acid: glycolaldehyde dynamic is temperature dependant (3).  My goal in these caluclations is to verify which  energy levels are preferentially populated as a function of temperature using the WebMo values and the Boltzman equation.

My calculations were based on a hypothetical population of 1 E 9 molecules.  Using the rotational constants from the WebMo I calculated the energy levels for the first eight rotational transitions. I then used the Boltzman equation with a hypothetical population of 1 E 9 molecules.  I performed the calculations at temperature intervals of 10K, 20K 50K and 100K.  My initial calculations using the first rotational constant suggested that I had made an error as the number of molecules in the first three states totaled more than 1 E9 molecules (see WebMoExcel)   Upon further study, I discovered that I needed to include all three rotational constants in the calculation: I  had only used one.  The difficulty begins the low symmetry (C1) of the glycolaldehyde molecule.    This result of this low symmetry is  a molecule with three distinct moments of inertia:  Ia>Ib>Ic . This type of  molecule ( know as ( oblate asymmetrical rotor) has three rotational constants where: A> B>C.   There is at present no method for calculations using the three rotational constants at once (4), thus calculations with but one constant would necessarily yield anomalous results.  A further complication exists with  the WebMo program as it only measured rotational constants for J level transitions (4).  Astrochemical spectroscopy tends to use transitions on the k ladder structure.  These transitions arise as a result of the shift in symmetry as the molecule rotates .  These small variations in asymmetry produce a splitting of the J levels. Thus transitions between k levels are reported as subscripts to the J energy level (i.e.. J 8₀₇ - 7₁₆).  These are the results reported in the literature (see rotational transitions of glycolaldehyde (Hollis 2000)  & Table 1 ( Hollis 2003).      The k value
(known as Ray's asymmetry parameter) is calculated using the three rotational constants: k = (2B-A-C)/ (A-C) and varies between k =+1 to k = -1. If the k value is near +1 the molecule is near oblate, if near to -1 it is near prolate.  The k value for glycoladehyde is -0.75133: near the limit of prolate tops (4) .