Toward Understanding Vibrations of Polyatomic Molecules

For a polyatomic molecule in the Born–Oppenheimer approximation, the quantum‐mechanical virial theorem takes the form ${\text{V\hspace{0.17em}=\hspace{0.17em}2W\hspace{0.17em}+\hspace{0.17em}Σ}}_{\mathrm{\alpha \hspace{0.17em}<\hspace{0.17em}\beta }}{R}_{\mathrm{\alpha \beta }}{\mathrm{(\partial W\hspace{0.17em}/\hspace{0.17em}\partial R}}_{\mathrm{\alpha \beta }})$ or ${\mathrm{T\hspace{0.17em}=\hspace{0.17em}-\hspace{0.17em}W\hspace{0.17em}-\hspace{0.17em}\Sigma }}_{\mathrm{\alpha \hspace{0.17em}<\hspace{0.17em}\beta }}{\mathrm{(\partial W\hspace{0.17em}/\hspace{0.17em}\partial R}}_{\mathrm{\alpha \beta }})$ , where $W$ is the vibrational potential‐energy function, expressed as a function of the internuclear distances $R_{\mathrm{\alpha \beta }}$ , and $V$ and $T$ are the electronic potential energy and electronic kinetic energy, as functions of $R_{\mathrm{\alpha \beta }}$ . If $W$ for a particular case has the form $$\text{W\hspace{0.17em}=\hspace{0.17em}W(0)\hspace{0.17em}+\hspace{0.17em}W(−\hspace{0.17em}1)\hspace{0.17em}+\hspace{0.17em}W(−\hspace{0.17em}2)\hspace{0.17em}+\hspace{0.17em}···,}$$ where $\mathrm{W(P)}$ is a function of the $R_{\mathrm{\alpha \beta }}$ homogeneous of degree $P$ , then $$\text{V\hspace{0.17em}=\hspace{0.17em}2W(0)\hspace{0.17em}+\hspace{0.17em}W(−\hspace{0.17em}1)\hspace{0.17em}+\hspace{0.17em}···}$$ and $$\text{T\hspace{0.17em}=\hspace{0.17em}−\hspace{0.17em}W(0)\hspace{0.17em}+\hspace{0.17em}W(−\hspace{0.17em}2)\hspace{0.17em}+\hspace{0.17em}···.}$$ Also, (B) would imply (A) and (C), and (C) would imply (A) and (B). These facts suggest use of empirical vibrational potential functions of the form (A), for then the terms $\text{W (−\hspace{0.17em}1)}$ will represent the Coulomb‐like conformation‐dependent part of the electronic potential energy, and the terms $\text{W (−\hspace{0.17em}2)}$ will be the particle‐in‐a‐box‐like conformation‐dependent part of the electronic kinetic energy. To illustrate the method, the vibrations of the CO₂ molecule are treated. With $R_1$ and $R_2$ the two CO distances, $R_3$ the OO distance, and $\gamma$ the OCO angle, the vibrational potential is written as $${\mathrm{W\hspace{0.17em}=\hspace{0.17em}W}}_0{\mathrm{\hspace{0.17em}+\hspace{0.17em}W}}_1{\mathrm{(R}}_1^{\text{−1}}{\mathrm{\hspace{0.17em}+\hspace{0.17em}R}}_2^{\text{−1}}{\mathrm{)\hspace{0.17em}+\hspace{0.17em}W}}_{11}{\mathrm{(R}}_1^{\text{−2}}{\mathrm{\hspace{0.17em}+\hspace{0.17em}R}}_2^{\text{−2}}{\mathrm{)\hspace{0.17em}+\hspace{0.17em}W}}_{111}{\mathrm{(R}}_1^{\text{−3}}{\mathrm{\hspace{0.17em}+\hspace{0.17em}R}}_2^{\text{−3}}{\mathrm{)\hspace{0.17em}+\hspace{0.17em}W}}_3{\mathrm{(R}}_3^{\text{−1}}{\mathrm{)\hspace{0.17em}+\hspace{0.17em}W}}_{12}{\mathrm{(R}}_1{\mathrm{\hspace{0.17em}+\hspace{0.17em}R}}_2{)}^{\text{−2}}{\mathrm{\hspace{0.17em}+\hspace{0.17em}W}}_{\gamma }{\mathrm{(R}}_1{R}_2{)}^{\text{−1}}{\tan }^2[\frac{1}{2}\mathrm{(\pi \hspace{0.17em}-\hspace{0.17em}\gamma )]}$$ . Values of the six (five independent) parameters $W_1$ to $W_{\gamma }$ are found which approximate 12 constants in the Overend–Suzuki valence‐force potential function. Values of force constants determined from (D) are as follows (empirical values in parentheses): quadratic constants, 10.3 (10.3), 1.7 (1.7), 0.4 (0.4); cubic constants, − 29.6 (− 29.6), − 2.9 ( − 2.9), − 1.0 (− 0.7); quartic constants, 57.2 (47.3), 4.6 (4.3), 3.1 (6.0), 1.6 (1.1), 0.4 (3.5), 0.1 (0.0).