Introduction: Unlike a singlegeometry calculation or even a geometry optimization, verification of a DRC trajectory is not a simple task. In this section a rigorous proof of the DRC trajectory is presented; it can be used both as a test of the DRC algorithm and as a teaching exercise. Users of the DRC are asked to follow through this proof in order to convince themselves that the DRC works as it should.
For the nitrogen molecule (using MNDO) the equilibrium distance is 1.103816Å, the heat of formation is 8.25741 kcal/mol and the vibrational frequency is 2738.8 cm^{1}. For small displacements, the energy curve versus distance is parabolic and the gradient curve is approximately linear, as is shown in the Table. A nitrogen molecule is thus a good approximation to a harmonic oscillator.
Table:
Stretching Curve for Nitrogen Molecule
NN DIST 

GRADIENT 
(Ångstroms) 
(kcal/mol) 
(kcal/mol/Ångstrom) 
1.11800 
8.69441 
60.84599 
1.11700 
8.63563 
56.70706 
1.11600 
8.58100 
52.54555 
1.11500 
8.53054 
48.36138 
1.11400 
8.48428 
44.15447 
1.11300 
8.44224 
39.92475 
1.11200 
8.40444 
35.67214 
1.11100 
8.37091 
31.39656 
1.11000 
8.34166 
27.09794 
1.10900 
8.31672 
22.77620 
1.10800 
8.29611 
18.43125 
1.10700 
8.27986 
14.06303 
1.10600 
8.26799 
9.67146 
1.10500 
8.26053 
5.25645 
1.10400 
8.25749 
0.81794 
1.10300 
8.25890 
3.64427 
1.10200 
8.26479 
8.12993 
1.10100 
8.27517 
12.63945 
1.10000 
8.29007 
17.17278 
1.09900 
8.30952 
21.73002 
1.09800 
8.33354 
26.31123 
1.09700 
8.36215 
30.91650 
1.09600 
8.39538 
35.54591 
1.09500 
8.43325 
40.19953 
1.09400 
8.47579 
44.87745 
1.09300 
8.52301 
49.57974 
1.09200 
8.57496 
54.30648 
1.09100 
8.63164 
59.05775 
1.09000 
8.69308 
63.83363 
The period of vibration (time taken for the oscillator to undertake one complete vibration, returning to its original position and velocity) can be calculated in three ways. Most direct is the calculation from the energy curve; using the gradient constitutes a faster, albeit less direct, method, while calculating it from the vibrational frequency is very fast but assumes that the vibrational spectrum has already been calculated.
where k is the force
constant. The reduced mass, μ,
(in amu) of a nitrogen molecule is
14.0067/2 = 7.00335,
and the forceconstant, k, can be calculated from:
Given
R_{o} = 1.1038, R = 1.092,
c = 8.25741 and
E = 8.57496 kcal/mol
then:
k = 4561.2 kcal/mol/A^{2} (per mole)
k = 1.9084x10^{30} ergs/cm^{2} (per mole)
k = 31.69x10^{5} dynes/cm (per molecule)
(Experimentally, for N_{2}, k = 23x10^{5} dynes/cm )
Therefore:
If the frequency is calculated using the other half of the curve ( R=1.118, E=8.69441), then k=12.333 fs, or k, average, = 12.185 fs.
Since we are using discrete points, the force constant is best
obtained from finite differences:
For x_{2} = 1.1100, G_{2} = 27.098 and for x_{1} = 1.0980, G_{1} = 26.311, giving rise to k = 4450.75 kcal/mol/Å^{2} and a period of 12.185 fs.
or as 12.179 fs.
Summarizing, by three different methods the period of oscillation of N_{2} is calculated to be 12.1851, 12.185 and 12.179 fs, average 12.183 fs.
A useful check on the dynamics of N_{2} is to calculate the initial acceleration of the two nitrogen atoms after releasing them from a starting interatomic separation of 1.094 Å.
At R(NN) = 1.094 Å, G = 44.877 kcal/mol/Å or erg/cm. Therefore acceleration, cm/sec/sec, or cm/s^{2}, which is Earth surface gravity.
Distance from equilibrium = 0.00980 Å. After 0.1 fs, velocity is cm/sec or 1340.5 cm/s.
In the DRC the timeinterval between points calculated is a complicated function of the curvature of the local surface. By default, the first timeinterval is 0.105fs, so the calculated velocity at this time should be cm/s, in the DRC calculation the predicted velocity is 1407.6 cm/s.
The option is provided to allow sampling of the system at constant timeintervals, the default being 0.1 fs. For the first few points the calculated velocities are given in Table 1.
As the calculated velocity is a fourthorder polynomial of the acceleration, and the acceleration, its first, second and third derivatives, are all changing, the predicted velocity rapidly becomes a poor guide to future velocities.
For simple harmonic motion the velocity at any time is given by:
By fitting the computed velocities to simple harmonic motion, a much better fit is obtained (Table 2).
Calculated  Simple Harmonic  Diff.  
Time  Velocity  25325.Sin(0.5296t)  
0.000  0.0  0.0  0.0 
0.100  1340.6  1340.6  0.0 
0.200  2678.0  2677.4  +0.6 
0.300  4007.0  4006.7  +0.3 
0.400  5325.3  5324.8  +0.5 
0.500  6628.4  6628.0  +0.4 
0.600  7912.7  7912.5  0.0 
The repeattime required for this motion is 11.86 fs, in good agreement
with the three values calculated using static models. The repeat time
should not be calculated from the time required to go from a minimum to a
maximum and then back to a minimumonly half a cycle. For all real systems
the potential energy is a skewed parabola, so that the potential energy
slopes are different for both sides; a compression (as in this case) normally
leads to a higher forceconstant, and shorter apparent repeat time (as in
this case). Only the addition of the two halfcycles is meaningful.