Sample problem 2: nprCH2

Fig. QS3-1. CH₂ torsion of n-propyl radical.

bx1fitPlls input:  nprCH2.fit

1. The first line beginning with '#' is a title line:

   # n-propyl radical -.CH2 intramolecular rotation

2. Next non-blank line is an output option control;

   outputOption FourierSeries

   which tells bx1fitPlls to generate input file for bx1HRsol named 'nprCH2.inp'.   The program bx1HRsol requires potential function in Fourier series.
3. The first block;

   optFuncs{
     const
     cos   2
     cos   4
     cos   6
     cos   8
     cos  10
     cos  12
     cos  14
     cos  16
     cos  18
   }

   tells bx1fitPlls that the regression function is:   y = a₀ + a₁ cos(2x) + a₂ cos(4x) + a₃ cos(6x) + ... + a₉ cos(18x).
4. Next block is a data table:

   xyTable{
            0.           57.1562
     deriv2 0.                         283.094
            0.414428431  69.7012
     deriv1 0.414428431           0.
     deriv2 0.414428431               -454.686
            1.095630438   0.0000
     deriv1 1.095630438           0.
     deriv2 1.095630438                714.797
            1.570796327  28.1917
     deriv2 1.570796327               -580.236
   }

   Each line between block header 'xyTable{' and terminator '}' defines a datum point.   The two-column format datum line is a regular datum point containing x (coordinate) and y (potential energy), as explained in the previous step.   The three-columned line is a derivative datum.   deriv1, deriv2, etc. stand for first, second, etc. derivative. The second column is x and third is y', y", or etc., depending on the first column.
   Figure QS3-2 shows the potential energies calculated at four stationary points along the CH₂ torsion coordinate.
   
   Fig. QS3-2. Input potential energy points.
5. The last block is a 'continue' block:

   continue{
     maxAbsQN  50        ! maximum absolute J
     rotConst  10.575    ! cm-1
     nuSpinSt  2  1 3    ! period = 2, alternation is 1,3,1,3,...
   ! symNumbr  2
   !
     output all
     tempRange  200.  500.  50.
     tempRange  600. 1200. 100.
     tempRange 1400. 2000. 200.
     tempRange 2500. 3000. 500.
     tempList   298.
   }

   Any line in the block is copied to the bx1HRsol input, 'nprCH2.inp'.

bx1fitPlls console output

Function:
 y = a0 + a1 * cos(2*x) + a2 * cos(4*x) + a3 * cos(6*x) + a4 * cos(8*x) + a5 * cos(10*x) + a6 * cos(12*x) + a7 * cos(14*x) + a8 * cos(16*x) + a9 * cos(18*x)
 y' = a1 * -2*sin(2*x) + a2 * -4*sin(4*x) + a3 * -6*sin(6*x) + a4 * -8*sin(8*x) + a5 * -10*sin(10*x) + a6 * -12*sin(12*x) + a7 * -14*sin(14*x) + a8 * -16*sin(16*x) + a9 * -18*sin(18*x)
 y" = a1 * -4*cos(2*x) + a2 * -16*cos(4*x) + a3 * -36*cos(6*x) + a4 * -64*cos(8*x) + a5 * -100*cos(10*x) + a6 * -144*cos(12*x) + a7 * -196*cos(14*x) + a8 * -256*cos(16*x) + a9 * -324*cos(18*x)

bx1HRsol input:  nprCH2.inp

1. The first line is a title line copied from nprCH2.fit:

   # n-propyl radical -.CH2 intramolecular rotation

2. Also, following non-blank lines are copy of continue block contents:

   maxAbsQN  50        ! maximum absolute J
   rotConst  10.575    ! cm-1
   nuSpinSt  2  1 3    ! period = 2, alternation is 1,3,1,3,...
   ! symNumbr  2
   !
   output all
   tempRange  200.  500.  50.
   tempRange  600. 1200. 100.
   tempRange 1400. 2000. 200.
   tempRange 2500. 3000. 500.
   tempList   298.

   i) Here, 'maxAbsQN', the maximum absolute quantum number, is set to 50. That is, the free rotor basis functions of J = −50 to J = 50 (101 functions) are used as expansion basis.
   ii) The next key 'rotConst' sets the rotational constant ( = ² / 2I ).
   - Note that the exclamation mark '!' can be used to insert comments.
   iii) Third line beginning with 'nuSpinSt' is nuclear spin statistic information input.   Here, for two indistinguishable hydrogen atoms of CH₂ rotor, the statistical weights for even and odd J's are 1 and 3, respectively.   Alternatively, as shown in the next comment line, a key 'symNumber' can be used to set the rotational symmetry number.
   iv) The key output controls the output, as has been described in the previous step.   Here, all possible outputs are generated by bex1HRsol.
   v) The last five lines controls the temperatures at which the partition functions are calculated.   The example sets the temperatures, 200, 250, 298, 300, 350, 400, 450, 500, 600, 700, 800, 900, 1000, 1100, 1200, 1400, 1600, 1800, 2000, 2500, and 3000 K
3. The last part is potential function input:

   potPars{           ! potential parameters
     0   37.021012
     2   31.151841
     4   4.5381994
     6   -17.268128
     8   1.2278534
     10   0.69347043
     12   -0.23513126
     14   -0.20930961
     16   0.12201598
     18   0.11437619
   }

   Each line in the block specifies the order n_i and coefficient a_i for a term of Fourier.   Zero or positive order, n_i ≥ 0, indicates a cosine term, a_i cos(n_i x), and negative order, n_i < 0, indicates a sine term, a_i sin(|n_i| x).

bx1HRsol output

  The first few eigen values are shown in Fig. QS3-3.   Because of low barriers for internal rotation, the eigenvalues are well approximated by the free rotor eigenvalues except for the first few eigenstates.

Fig. QS3-3. Calculated eigen values (horizontal dotted lines)

Fig. QS3-4. Calculated eigen functions of first few eigenstates.