Vibrational Correction

For the calculation of the thermodynamics of radicals within the JTHERGAS system, the loss of vibrational states due to the loss of a hydrogen atom to form the radical is computed. The vibrational states lost are translated to corrections to the temperature dependent heat capacities.

The estimation of vibrational states is also based on a table of structures. A vibrational mode is identified by a given structure. Associated with the structure is a vibrational frequency (For example, see Table A.13 in Benson's book[18]). This vibrational frequency is then translated to contributions to the entropies and the heat capacities as a function of temperature (see Table A.15 and A.17) in Benson's book.

Also associated with each vibrational structure is a numerical factor. This factor serves two purposes. The first is to take care of symmetries in the vibrational structure. For example, the bend structure, H-C-H, due to its symmetry, matches twice as many times. To ensure the counting, the number of matches has to be divided by two. The other purpose of the numerical factor is to determine whether the structure should be added (positive factor) or subtracted (negative factor). In counting the vibrational structures both matches in the parent molecule and the radical are made. The number of times a certain vibrational mode appears in the parent mode and the number of times a vibrational mode appears in the radical are subtracted. A positive numerical factor keeps this relationship and a negative numerical factor reverses this relationship. This was incorporated to give more flexibility in the definition of which modes should be counted.

Within JTHERGAS, the implementation goes beyond what is outlined in Table A.13 in Benson's book[18] for carbon radicals. The loss of tetrahedral modes is calculated for loss of a hydrogen atom in primary, secondary and tertiary carbon configurations. This replaces the simple CCH and HCH bond bends listed in Benson's tables. In addition, corrections due to resonance are also included. For example, for the calculation of a carbon next to a double bond, i.e. a resonant structure, the structure ^.CH₂=C is matched. The associated frequencies are: 1300 (resonant stretch), two 1150 (twist and rock primary bends) and 1450 (primary bend). These frequencies are then translated to corrections in entropies and temperature dependent heat capacities.

For the calculation of the thermodynamics of radicals within the JTHERGAS system, the loss of vibrational states due to the loss of a hydrogen atom to form the radical is computed. The vibrational states lost are translated to corrections to the temperature dependent heat capacities.

The estimation of vibrational states is also based on a table of structures. A vibrational mode is identified by a given structure. Associated with the structure is a vibrational frequency (For example, see Table A.13 in Benson's book). This vibrational frequency is then translated to contributions to the entropies and the heat capacities as a function of temperature (see Table A.15 and A.17) in Benson's book.

Also associated with each vibrational structure is a numerical factor. This factor serves two purposes. The first is to take care of symmetries in the vibrational structure. For example, the bend structure, H-C-H, due to its symmetry, matches twice as many times. To ensure the counting, the number of matches has to be divided by two. The other purpose of the numerical factor is to determine whether the structure should be added (positive factor) or subtracted (negative factor). In counting the vibrational structures both matches in the parent molecule and the radical are made. The number of times a certain vibrational mode appears in the parent mode and the number of times a vibrational mode appears in the radical are subtracted. A positive numerical factor keeps this relationship and a negative numerical factor reverses this relationship. This was incorporated to give more flexibility in the definition of which modes should be counted.

Within JTHERGAS, the implementation goes beyond what is outlined in Table A.13 in Benson's book for carbon radicals. The loss of tetrahedral modes is calculated for loss of a hydrogen atom in primary, secondary and tertiary carbon configurations. This replaces the simple CCH and HCH bond bends listed in Benson's tables. In addition, corrections due to resonance are also included. For example, for the calculation of a carbon next to a double bond, i.e. a resonant structure, the structure ^.CH₂=C is matched. The associated frequencies are: 1300 (resonant stretch), two 1150 (twist and rock primary bends) and 1450 (primary bend). These frequencies are then translated to corrections in entropies and temperature dependent heat capacities.

Implementation Database Notes

The vibrational tables uses the mySQL table: VibrationalStructures. and if an extra structure is needed, StructureTypes.

CH Stretch example: The need for two structures to represent the stretch loss

A type ASCII input for vibrational information is:

CH-Stretch CH-Bond 3100.0 1

where:

CH-Stretch: The name of the vibrational correction
CH-Bond: The corresponding structure in CMLStructures that is matched in the molecule and radical
3100.0: The frequency of this particular vibration.
1: The symmetry associated with the structure (this compensates for not only the structure symmetry, but also to get the proper correction).

The represents the CH stretch in the Benson's book Table A.13 with frequency of 3100.0.

This vibrational structure is used when the structure, in this case, CH-Bond, matches in the molecule and the radical:

select CMLStructure from CMLStructures where ElementName="CH-Bond";

<?xml version="1.0" encoding="ISO-8859-1"?>

</atomArray>

</bondArray>

</molecule>

1 row in set (0.00 sec)

Whenever this structure matches within the molecule or radical structure, then the frequency correction is applied.

In the calculation of thermodynamics within JThermodynamics, the vibrational correction is only used in the calculation of the radical by the difference method. That is to say, only the 'lost' frequency modes in going from the molecule where the radical is replaced by a hydrogen to the radical are needed. Thus the structure for the frequency, in this case CH-Bond, has to match in both the molecule and the radical.

For example, for the radical, ch3/ch2(.), meaning ethyl radical, the corresponding molecule is ch3/ch3, or ethane. The 'difference' is the loss of one CH vibrational mode, in going from the ethyl radical to the ethane molecule. If we apply our rule, then matching the C-HBond structure we see:

ch3/ch3: Matches 6 time, one for each hydrogen in both the methyl groups.
ch3/ch2(.): Match 3 times, one for each hydrogen on the non-radical methyl group.

The difference is 6-3=3, meaning the loss of three hydrogens, which is incorrect. The two hydrogens on the radical were not accounted for with just the structure C-HBond. Therefore, in order to correctly account for the another structure is needed to match the CH on the radical, namely C.H-Bond:

select CMLStructure from CMLStructures where ElementName="C.H-Bond";

<?xml version="1.0" encoding="ISO-8859-1"?>

</atomArray>

</bondArray>

</molecule>

The ASCII input in the input accounting for this extra structure is:

CH-Stretch-RadicalCompC.H C.H-Bond 3100.0 1

Matching C.H-Bond in the ethyl radical and the molecule:

ch3/ch3: Matches 0 times, because there are no radical carbons
ch3/ch2(.): Match 2 times, one for each hydrogen on the radical carbon methyl group.

so, in total we have 6+0 matches in the ethane, 6 for CH-Bond and 0 for C.H-Bond, and 3+2=5, 6 for CH-Bond and 0 for C.H-Bond, matches in ethyl radical. The difference is what it should be, namely 1.

In summary, to represent the CH stretch two entries are needed, here corresponding to the ASCII input:

CH-Stretch CH-Bond 3100.0 1

CH-Stretch-RadicalCompC.H C.H-Bond 3100.0 1

This correction show up in the calculation with the following lines in the output:

BensonThermodynamicBase: Reference: 'Correction: CH-Stretch-RadicalCompC.H: 3100.0 Count=2.0'

H298: 0.00 S298: -0.00 [ 300.0, -0.00] [ 400.0, -0.01] [ 500.0, -0.04] [ 600.0, -0.13] [ 800.0, -0.47] [ 1000.0, -0.93] [ 1500.0, -1.99]

BensonThermodynamicBase: Reference: 'Correction: CH-Stretch: 3100.0 Count=-3.0'

H298: 0.00 S298: 0.00 [ 300.0, 0.00] [ 400.0, 0.01] [ 500.0, 0.06] [ 600.0, 0.19] [ 800.0, 0.71] [ 1000.0, 1.40] [ 1500.0, 2.99]

The calculation finds a difference of 2 C.HBond structures between the ethane (0 matches) and the ethyl radical (2 matches) and a difference of 6-3=3 CHBond structures between the ethane (6 matches) and ethyl (3 matches). Thus the total correction is due to a single CH stretch with frequency 3100.

The principle of having to account for matching for the molecule and for the radical applies to most corrections. For example:

No Contribution elements

Unfortunately, due to this type of matching there can be many non-radical and radical pairs matches resulting in no contribution. For example, for the case of propane, there are many matches giving no contribution:

BensonThermodynamicBase: Reference: 'Correction: CCHSecondary(op)-Bend: 700.0 Count=1.0'

H298: 0.00 S298: 0.21 [ 300.0, 0.21] [ 400.0, 0.30] [ 500.0, 0.36] [ 600.0, 0.39] [ 800.0, 0.44] [ 1000.0, 0.46] [ 1500.0, 0.48]

BensonThermodynamicBase: Reference: 'Correction: CCHSecondary(op)-BendRadical.CCH: 700.0 Count=-1.0'

H298: 0.00 S298: -0.41 [ 300.0, -0.42] [ 400.0, -0.60] [ 500.0, -0.72] [ 600.0, -0.79] [ 800.0, -0.87] [ 1000.0, -0.91] [ 1500.0, -0.96]

BensonThermodynamicBase: Reference: 'Correction: CCHSecondary(rk)-Bend: 700.0 Count=-1.0'

H298: 0.00 S298: 0.21 [ 300.0, 0.21] [ 400.0, 0.30] [ 500.0, 0.36] [ 600.0, 0.39] [ 800.0, 0.44] [ 1000.0, 0.46] [ 1500.0, 0.48]

BensonThermodynamicBase: Reference: 'Correction: CCHSecondary(rk)-BendRadical.CCH: 700.0 Count=1.0'

H298: 0.00 S298: -0.41 [ 300.0, -0.42] [ 400.0, -0.60] [ 500.0, -0.72] [ 600.0, -0.79] [ 800.0, -0.87] [ 1000.0, -0.91] [ 1500.0, -0.96]

BensonThermodynamicBase: Reference: 'Correction: CCHSecondary(tw)-Bend: 1150.0 Count=-2.0'

H298: 0.00 S298: 0.24 [ 300.0, 0.24] [ 400.0, 0.56] [ 500.0, 0.86] [ 600.0, 1.09] [ 800.0, 1.41] [ 1000.0, 1.59] [ 1500.0, 1.80]

BensonThermodynamicBase: Reference: 'Correction: CCHSecondary(tw)-BendRadical.CCH: 1150.0 Count=2.0'

H298: 0.00 S298: -0.48 [ 300.0, -0.49] [ 400.0, -1.12] [ 500.0, -1.71] [ 600.0, -2.18] [ 800.0, -2.81] [ 1000.0, -3.18] [ 1500.0, -3.59]

BensonThermodynamicBase: Reference: 'Correction: CCHSecondary-Bend: 700.0 Count=1.0'

H298: 0.00 S298: 0.21 [ 300.0, 0.21] [ 400.0, 0.30] [ 500.0, 0.36] [ 600.0, 0.39] [ 800.0, 0.44] [ 1000.0, 0.46] [ 1500.0, 0.48]

BensonThermodynamicBase: Reference: 'Correction: CCHSecondary-BendRadical.CCH: 700.0 Count=-1.0'

H298: 0.00 S298: -0.41 [ 300.0, -0.42] [ 400.0, -0.60] [ 500.0, -0.72] [ 600.0, -0.79] [ 800.0, -0.87] [ 1000.0, -0.91] [ 1500.0, -0.96]

And only these actually give contributions:

BensonThermodynamicBase: Reference: 'Correction: CCHPrimary(rk)-Bend: 1100.0 Count=-2.0'

H298: 0.00 S298: 0.19 [ 300.0, 0.19] [ 400.0, 0.41] [ 500.0, 0.61] [ 600.0, 0.76] [ 800.0, 0.96] [ 1000.0, 1.08] [ 1500.0, 1.21]

BensonThermodynamicBase: Reference: 'Correction: CCHPrimary(tw)-Bend: 1150.0 Count=2.0'

H298: 0.00 S298: 0.16 [ 300.0, 0.16] [ 400.0, 0.37] [ 500.0, 0.57] [ 600.0, 0.73] [ 800.0, 0.94] [ 1000.0, 1.06] [ 1500.0, 1.20]

BensonThermodynamicBase: Reference: 'Correction: CH-Stretch-RadicalCompC.H: 3100.0 Count=2.0'

H298: 0.00 S298: -0.00 [ 300.0, -0.00] [ 400.0, -0.01] [ 500.0, -0.04] [ 600.0, -0.13] [ 800.0, -0.47] [ 1000.0, -0.93] [ 1500.0, -1.99]

BensonThermodynamicBase: Reference: 'Correction: CH-Stretch: 3100.0 Count=-3.0'

H298: 0.00 S298: 0.00 [ 300.0, 0.00] [ 400.0, 0.01] [ 500.0, 0.06] [ 600.0, 0.19] [ 800.0, 0.71] [ 1000.0, 1.40] [ 1500.0, 2.99]

BensonThermodynamicBase: Reference: 'Correction: CHH-Bend-RadicalCompC.HH: 1450.0 Count=1.0'

H298: 0.00 S298: -0.04 [ 300.0, -0.05] [ 400.0, -0.15] [ 500.0, -0.27] [ 600.0, -0.39] [ 800.0, -0.58] [ 1000.0, -0.70] [ 1500.0, -0.85]

BensonThermodynamicBase: Reference: 'Correction: CHH-Bend: 1450.0 Count=-3.0'

H298: 0.00 S298: 0.13 [ 300.0, 0.14] [ 400.0, 0.44] [ 500.0, 0.82] [ 600.0, 1.18] [ 800.0, 1.74] [ 1000.0, 2.10] [ 1500.0, 2.54]

meaning contributions from

CCH(rk) 1100 with -2
CCH(tk) 1150 with +2
CH stretch with 3100 with 2-3=-1
CHH Bend with 1450 with 1-3=-2

CHH (tw) Bend: Correction for Symmetry

The symmetry listed in the vibrational mode definition depends on three factors:

The structure used to identify the mode and the combinatorics of matching.
How many times the particular mode (frequeny) should be counted
Whether the mode should be added or subtracted, whether the symmetry is greater than 0 or less than zero.

For example, associated with the vibrational modes C-C-H bend(tw) and C-C-H bend(rk) is the structure GeneralPrimaryCH3C. The importance of matching of this substructure within the molecule is how many times it matches. Where it matches is not of importance here. GeneralPrimaryCH3C has three identical hydrogens and thus, due to the combinatorics of macthing , there are 6 ways to match GeneralPrimaryCH3C with a primary methyl in a hydrocarbon. Thus to count the number of places this structure can be found in the molecule, number of matches should be divided by 6.

For both of these these vibrational modes, they should be counted twice. This means that the factor of 6 should be divided by 2 giving 3. In other words, dividing the number of matches by 3, will compensate for the combinatorics of matching and for the factor of 2 of counting.

For example, the vibrational modes of C-C-H bend(tw) and C-C-H bend(rk) have a frequencies of 1150 and 1100, respectively. Their influence on the entropy are opposite, namely the C-C-H bend(tw) is subtracted and the C-C-H bend(rk) is added. This is reflected in the sign of the symmetry factor.

The final definition with the symmetry factor is:

CCHPrimary(tw)-Bend GeneralPrimaryCH3C 1150.0 -3

CCHPrimary(tw)-BendRadical.CCH GeneralPrimaryCH3C. 1150.0 -3

CCHPrimary(rk)-Bend GeneralPrimaryCH3C 1100.0 3

CCHPrimary(rk)-BendRadical.CCH GeneralPrimaryCH3C. 1100.0 3

Applying these four representations to, for example ethane, gives:

BensonThermodynamicBase: Reference: 'Correction: CCHPrimary(rk)-Bend: 1100.0 Count=-4.0'

H298: 0.00 S298: 0.37 [ 300.0, 0.38] [ 400.0, 0.82] [ 500.0, 1.22] [ 600.0, 1.53] [ 800.0, 1.93] [ 1000.0, 2.16] [ 1500.0, 2.41]

BensonThermodynamicBase: Reference: 'Correction: CCHPrimary(rk)-BendRadical.CCH: 1100.0 Count=2.0'

H298: 0.00 S298: -0.19 [ 300.0, -0.19] [ 400.0, -0.41] [ 500.0, -0.61] [ 600.0, -0.76] [ 800.0, -0.96] [ 1000.0, -1.08] [ 1500.0, -1.21]

BensonThermodynamicBase: Reference: 'Correction: CCHPrimary(tw)-Bend: 1150.0 Count=4.0'

H298: 0.00 S298: 0.32 [ 300.0, 0.33] [ 400.0, 0.75] [ 500.0, 1.14] [ 600.0, 1.45] [ 800.0, 1.88] [ 1000.0, 2.12] [ 1500.0, 2.39]

BensonThermodynamicBase: Reference: 'Correction: CCHPrimary(tw)-BendRadical.CCH: 1150.0 Count=-2.0'

H298: 0.00 S298: -0.16 [ 300.0, -0.16] [ 400.0, -0.37] [ 500.0, -0.57] [ 600.0, -0.73] [ 800.0, -0.94] [ 1000.0, -1.06] [ 1500.0, -1.20]

The difference, using the previous section as a reference, of CCHPrimary(rk)-Bend and CCHPrimary(rk)-BendRadical.CCH is -4+2=2, meaning that there is a contribution of the entropy of 0.37 due to the frequency of 1100. For CCHPrimary(rk)-Bend the structure GeneralPrimaryCH3C was matched 12 times for ethane and 0 times for the radical (The carbons in C-C-H cannot not match with the radical in ethyl). Twelve is then divided by -3 to give -4. For the CCHPrimary(rk)-BendRadical.CCH mode, the structure GeneralPrimaryCH3C. is used. This is matched 0 times in the ethane and 6 times in the radical. Six is divided by 3, giving 2.

The difference, again using the previous section as a reference, of CCHPrimary(tw)-Bend and CCHPrimary(tw)-BendRadical.CCH is also -4+2=2, meaning that there is a contribution of the entropy of 0.32 due to the frequency of 1150.