The moments of inertia are calculated using I_{A} = Σ_{i}m_{i}(R_{Ai})^{2}, where i runs over all atoms in the system, m_{i }is the mass of the atom in amu, and R_{Ai} is the distance from the axis of rotation, A, to atom i in Ångstroms.
The axes of rotation are calculated as follows:
First, a 3 by 3 matrix, t, is constructed, with the elements of t being:
t_{1,1} = Y^{2} + Z^{2 }=^{
}Σ_{i}m_{i}(y_{i}^{2}
+ z_{i}^{2})
t_{1,2} = X.Y^{ }=^{ }Σ_{i}m_{i}x_{i}y_{i
}t_{2,2} = X^{2} + Z^{2 }=^{
}Σ_{i}m_{i}(x_{i}^{2}
+ z_{i}^{2})
t_{1,3} = X.Z^{ }=^{ }Σ_{i}m_{i}x_{i}z_{i
}t_{2,3} = Y.Z^{ }=^{ }Σ_{i}m_{i}y_{i}z_{i
}t_{3,3} = X^{2} + Y^{2 }=^{
}Σ_{i}m_{i}(x_{i}^{2}
+ y_{i}^{2})
where m_{i }is the mass of the atom in amu, and x_{i}, y_{i}, and z_{i}, are the Cartesian coordinates of the atoms, in Ångstroms. Then t is diagonalized. The resulting eigenvalues, (amu Ångstrom^{2}), are divided by N.A^{2}, where N=Avogardo's number and A = number of Ångstroms in a centimeter, to give the moments of inertia in g.cm^{2}. Because a useful unit is 10^{40}.g.cm^{2}, the moments of inertia are multiplied by 10^{40} before being printed. The eigenvectors associated with the eigenvalues are the axes of rotation, A, B, and C.
Useful conversion factors
1 g cm^{2 } =
1.660540x10^{40} (amu Ångstrom^{2})
Rotational constants in cm^{1}:
A = hN10^{16}/(8π^{2}c)/(amu Ångstrom^{2})
A (in MHz) =
5.053791x10^{5}/(amu Ångstrom^{2})
A (in cm^{1}) = 5.053791x10^{5}/c(amu Ångstrom^{2}) =
16.85763/(amu Ångstrom^{2})
Ab initio MO methods provide total energies, E_{eq}, as the sum of electronic and nuclearnuclear repulsion energies for molecules, isolated in vacuum, without vibration at 0 K.
From the 0 K potential surface and using the harmonic oscillator approximation, we can calculate the vibrational frequencies, ν_{i}, of the normal modes of vibration. Using these, we can calculate vibrational, rotational and translational contributions to the thermodynamic quantities such as the partition function and heat capacity which arise from heating the system from 0 to T K.
Q: partition function, E: energy, S: entropy,
and C: Heat capacity at constant pressure = C_{p}. In ab initio
calculations, the heat capacity calculated is C_{v}.
The relationship between C_{p} and C_{v} (in cal.degree^{1}.mol^{1}) is:
The vibrational contribution to the internal energy arises from population of the vibrational energy levels. The vibrational partition coefficient, Q_{vib,} is given by:
E_{vib}, for a molecule at the temperature T as:
where h is Planck's constant, ν_{i }the ith normal vibration frequency, and k the Boltzmann constant. For 1 mole of molecules, E_{vib }should be multiplied by the Avogadro number N_{a} = R/k. Thus:
Equation 1 
Note that the first term in the above equation is the zeropoint vibration energy, E_{zpe}. Hence, the second term is the additional vibrational contribution due to the temperature increase from 0 K to T K. Namely,
E_{vib}  =  E_{zpe}+_{ }E_{vib }(T)  
E_{zpe}  =  
E_{vib}(T)  = 
Equation 2 
The value of
E_{vib }from GAUSSIAN 82 and 86
includes
E_{zpe }as defined by Equation 1
and Equation 2.
S_{vib} 
= 

C_{vib}  = 
At temperature T>0 K, a molecule rotates about the x, y, and zaxes and translates in x, y, and zdirections. By assuming the equipartition of energy, energies for rotation and translation, E_{rot }and E_{tr}, are calculated.
Σ is the symmetry number (Examples of
symmetry numbers are shown in the Table). I is moment of inertia. I_{A},
I_{B}, and I_{C} are moments of
inertia about A, B, and C axes.
C_{1}  C_{I}  C_{S}:  1  D_{2}  D_{2d}  D_{2h}:  4  C:  1  
C_{2}  C_{2v}  C_{2h}:  2  D_{3}  D_{3d}  D_{3h}:  6  D :  2  
C_{3}  C_{3v}  C_{3h}:  3  D_{4}  D_{4d}  D_{4h}:  8  T, T_{h} T_{d}:  12  
C_{4}  C_{4v}  C_{4h}:  4  D_{5}  D_{5d}  D_{5h}:  10  O, O_{h}:  24  
C_{5}  C_{5v}  C_{5h}:  5  D_{6}  D_{6d}  D_{6h}:  12  I, I_{h}:  60  
C_{6}  C_{6v}  C_{6h}:  6  D_{7}  D_{7d}  D_{7h}:  14  S_{4}:  2  
C_{7}  C_{7v}  C_{7h}:  7  D_{8}  D_{8d}  D_{8h}:  16  S_{6}:  3  
C_{8}  C_{8v}  C_{8h}:  8  S_{8}:  4 
Values for Q_{rot}, E_{rot}, and S_{rot} for a linear molecule are defined below:
Q_{ro}  =  
E_{ro}  =  2/2)RT  
S_{rot}  =  
= 
where .
Values for Q_{rot}, E_{rot}, and S_{rot}
for a nonlinear molecule are defined below:
Q_{rot}  =  
=  
E_{rot}  =  (3/2)RT 
S_{rot}  =  
= 
Here, 5.386 3921 is calculated as:
Given that M is the molecular weight, then the values for Q_{tra},
E_{tra}, S_{tra}, and C_{tra
}for a molecule are as defined below:
or H_{tra }= (5/2)RT due to the PV term (cf. H = U + PV). The internal energy U at T is:
U = E_{eq} + [E_{vib }+ E_{rot} + E_{tra}]
or, in terms of the zero point energy, E_{zpe}, and real vibrations, E_{vib }(T),
U = E_{eq} + [(E_{zpe }+_{ }E_{vib }(T))+ E_{rot} + E_{tra}].
Enthalpy H for one mole of gas is defined as
Assumption of an ideal gas (i.e., PV = RT) leads to
H = U + PV = U + RT
Thus, Gibbs free energy G can be calculated as:
G = H  T S
It should be noted that M.O. parameters for MNDO, AM1, etc., are optimized so as to reproduce the experimental heat of formation (i.e., standard enthalpy of formation or the enthalpy change to form a mole of compound at 25^{o}C from its elements in their standard state) as well as observed geometries (mostly at 25^{o}C), and not to reproduce the E_{eq} and equilibrium geometry at 0 K.
In this sense, E_{SCF} (defined as Heat of formation, ΔH_{f}), force constants, normal vibration frequencies, etc. are all related to the values at 25^{o}C, not to 0 K. Therefore, the E_{0 }calculated in FORCE is not the true E_{0}. Its use as E_{0 }should be made at your own risk, bearing in mind the situation discussed above.
Since E_{SCF} is standard enthalpy of formation (at 25C):
E_{SCF} = E_{eq} + E_{zpe} + E_{vib}+ E_{rot} + E_{tra} + PV + Σ[E_{elec}(atom) + ΔH_{f}(atom)].
E_{eq}: Electronic plus nuclear energy for the equilibrium geometry at 0 K; E_{zpe}: Zeropoint energy; E_{vib}: Vibrational energy at 298.15 K; E_{rot}: Rotational energy at 298.15 K; E_{tra}: Translational energy at 298.15 K.
To avoid the complication arising from the definition of E_{SCF}, within the thermodynamics calculation the Standard Enthalpy of Formation, ΔH, is calculated by
ΔH = E_{SCF} + (H_{T}  H_{298}).
Here, E_{SCF} is the heat of formation (at 25^{o}C) given in the output list, and H_{T} and H_{298} are the enthalpy contributions for the increase of the temperature from 0 K to T and 298.15, respectively. In other words, the enthalpy of formation is corrected for the difference in temperature from 298.15 K to T.
There is a problem in that H_{T} is the heat of formation at T relative to the heat of formation of the elements in their standard state at 298K. This involves mixing standard and not standard terms. There is no easy way to get the correct value for H_{T}, but for rough work H_{T} is useful. For more correct work, calculate ΔH for the elements in their standard state at T, and use these ΔH's to get the ΔH for the compound you're working with (or use tables from the literature).
This problem is, however, not normally important, because the most common use of H_{T} is for calculating the thermodynamics of reactions at temperatures other than 298K. For all reactions, the types and number of atoms must be the same in reactants and products, therefore the fact that the H_{T} are relative to the elements in their standard state at 298K is irrelevant. Consider the simple DielsAlder reaction:
C_{2}H_{4} + C_{4}H_{6} = C_{6}H_{10}
The heat of this reaction at 298K is H_{298}( C_{6}H_{10})  H_{298}(C_{2}H_{4})  H_{298}( C_{4}H_{6}). At any other temperature, the heat of reaction would be:
H_{R }=_{ }H_{T}( C_{6}H_{10})  H_{T}(C_{2}H_{4})  H_{T}( C_{4}H_{6}).
Care must, however, be taken to account for changes in volume  if any of the reactants or products are gaseous, then appropriate corrections must be made to H_{R} . Complications arise only if absolute heats of formation are needed. Thus, if the heat of formation of benzoic acid (C_{7}H_{6}O_{2}) at 398K (100C) is needed, the H_{398}(C_{7}H_{6}O_{2}) generated by MOPAC would be for the reaction:
7H_{298}(graphite) + 3H_{298}(H_{2}) + H_{298}(O_{2}) => H_{398}(C_{7}H_{6}O_{2})
0 + 0 + 0 =>H_{398}(C_{7}H_{6}O_{2})
Note that on the left side, the temperatures are 298K. For H_{2} and O_{2}, the heats of formation at 398K can readily be calculated, but for graphite the calculation is more complicated. The easiest way to generate a balanced equation would be to use tables of heats of formation of the elements at nonstandard temperatures.
Finally, as mentioned above, changes in volume must also be taken into account: if the reaction volume changes, then RΔN(T298) must be added or subtracted, where R is the gas constant (~ 2cals/degree/mol), and ΔN is the change in volume. Thus for the formation of methane from graphite and hydrogen, 2 volumes of reactant (H_{2} + graphite) yield 1 volume of methane, therefore ΔN = 1.
The method of calculation for T and H_{298} will be given below.
In MOPAC, the variables defined below are used:
The wavenumber, , in cm^{1}:
The rotational constants A, B, and C in cm^{1}:
Energy and Enthalpy in cal/mol, and Entropy in cal/mol/K. Thus, the earlier Equations can be written as follows:
= 

E_{zpe} 
= 


= 

=  

= 


= 


= 


= 
Rotation:
= 

= 
(2/2)RT  
= 

= 
(2/2)R 

= 


= 
(3/2)RT 

= 


= 


= 


= 
(3/2)R 

= 


= 
(3/2)RT 

= 


= 


= 
In MOPAC:
(Note: is not included in ; is not derived from forceconstants at 0 K) and for T:
H_{T} = [H_{vib} + H_{rot} + H_{tra}]
while for T=298.15 K:
H_{298} = [H_{vib} + H_{rot} + H_{tra}]
Note that H_{T} (and H_{298}) is equivalent to:
(E_{vib}E_{zpe}) + E_{rot }+ (E_{tra}+ pV)
except that the normal frequencies are those obtained from force constants at 25C, or at least not at 0 K.
Thus, Standard Enthalpy of Formation, ΔH, can be calculated using U and H and Escf, thus:
ΔH = E_{SCF} + (H_{T}  H_{298})
Note that is already counted in E_{scf}.
By using
H = U + pV = U + RT 
Standard Internal Energy of Formation, , can be calculated as:

Standard Gibbs FreeEnergy of Formation, , can be calculated by taking the difference from that for the isomer or that at different temperature:
Taking the difference is necessary to cancel the unknown values of standard entropy of formation for the constituent elements.