Knowledge Bank

We have calculated equilibrium force constants $fe=-(dF/dr)e=(d2U/dr2)e[F=force,r=$ internuclear separation, U = potential energy, e = equilibrium] from the Rydberg-Klein-Rees potential energy curves for 11 diatomic molecules. The results are in good agreement with the values previously calculated from $fe=4\pi 2C2\omega e2\mu$, which was derived from the harmonic potential energy function [c = speed of light in vacuum, $c\omega e=$ classical vibrational frequency in cycles/sec for infinitesimal amplitudes, $\mu =$ reduced mass]. This is as it should be since the force per unit displacement at $re$ should be the same for any valid potential energy curve. The values of $fe$ for isotopes of these molecules were nearly the same as for the ordinary molecules. For HCI, the $fe$ values were: $\mathrm{<mrow\; data-mjx-texclass="ORD">1</mrow>}H-Cl35,51.674\pm 0.002;2H-Cl35,51.634\pm 0.002;3H-Cl35,51.604\pm 0.003;3H-Cl37,51.60\pm 0.01$ microdynes/picometer. If the molecule were a harmonic oscillator, the force derivative function $f(r)=-dF/dr=d2U/dr2$ would be the same for all values of r, but for the actual molecule the values of this function vary with r. At $re,f(r)=fe$; at $Ri,f(r)=0[Ri$ is the value of r at the inflection point i on the potential energy curve]. The average value $fv$ of $f(r)$ for the vibrational state v can be taken as the effective force constant in that vibrational state. We have calculated values of $f¯v$ from the R-K-R curve of $H2$ for several values of v. As v increases from 0 to 9, $f¯v$ decreases from $57.28\pm 0.01$ to $53.9\pm 0.2\mu dyn/pm$. Effective force constants $f¯e$ and $f¯1$ for $v=0$ and $v=1$ were calculated for the 11 molecules. In all cases, $f¯e$ and $f¯2$ were larger than $f\sigma =4\pi 2c2\sigma 2\mu$ and smaller than $fe=4\pi 2c2\omega e2\mu [\sigma =$ observed wave number in cycles/cm]. For $H2$, the values are: $fe=57.3967\pm 0.0003,f0=57.28\pm 0.01,f¯1=56.990\pm 0.008,f\sigma =51.3971\pm 0.0003\mu dyn/pm$.