## FORCE CONSTANTS FROM RYDBERG-KLEIN-REES POTENTIAL ENERGY CURVES.

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Ohio State University##### Abstract:

We have calculated equilibrium force constants $f_{e} = -(dF/dr)_{e} = (d^{2}U/dr^{2})_{e} [F = force, r =$ 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 $f_{e} = 4\pi^{2}C^{2}\omega_{e}^{2}\mu$, 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 $r_{e}$ should be the same for any valid potential energy curve. The values of $f_{e}$ for isotopes of these molecules were nearly the same as for the ordinary molecules. For HCI, the $f_{e}$ values were: $^{1}H-Cl^{35}, 51.674 \pm 0.002; ^{2}H-Cl^{35}, 51.634 \pm 0.002; ^{3}H-Cl^{35}, 51.604 \pm 0.003; ^{3}H-Cl^{37}, 51.60 \pm 0.01$ microdynes/picometer. If the molecule were a harmonic oscillator, the force derivative function $f(r) = -dF/dr = d^{2}U/dr^{2}$ would be the same for all values of r, but for the actual molecule the values of this function vary with r. At $r_{e}, f(r) = f_{e}$; at $R_{i}, f(r)= 0 [R_{i}$ is the value of r at the inflection point i on the potential energy curve]. The average value $f_{v}$ 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 $\bar{f}_{v}$ from the R-K-R curve of $H_{2}$ for several values of v. As v increases from 0 to 9, $\bar{f}_{v}$ decreases from $57.28 \pm 0.01$ to $53.9 \pm 0.2 \mu dyn/pm$. Effective force constants $\bar{f}_{e}$ and $\bar{f}_{1}$ for $v = 0$ and $v = 1$ were calculated for the 11 molecules. In all cases, $\bar{f}_{e}$ and $\bar{f}_{2}$ were larger than $f_{\sigma} = 4 \pi^{2}c^{2}\sigma^{2}\mu$ and smaller than $f_{e} = 4\pi^{2}c^{2}\omega e^{2}\mu [\sigma =$ observed wave number in cycles/cm]. For $H_{2}$, the values are: $f_{e} = 57.3967 \pm 0.0003, f_{0} = 57.28 \pm 0.01, \bar{f}_{1} = 56.990 \pm 0.008, f_{\sigma} = 51.3971 \pm 0.0003 \mu dyn/pm$.

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Author Institution: Physics Department, Illinois Institute of Technology

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