Vibraciones moleculares#
(327)#\[\left(\hat{T}_n + E_i(\{\vec{R}\})\right)\Psi_n^{\alpha}=E_{i,\alpha}\Psi_n^{\alpha}\]
(328)#\[ K=\frac{d^2 E}{d R^2}\]
(329)#\[\left(-\frac{\hbar^2}{2m} \frac{d^2}{dx^2} +\frac{1}{2} K x^2\right) \chi_\nu= E_\alpha \chi_\nu\]
(330)#\[\omega = \sqrt{\frac{K}{m}}\]
(331)#\[\omega = \sqrt{\frac{K}{m}}
E_\nu = \left(\nu+\frac{1}{2}\right) \hbar \omega, \nu=0, 1, 2, \cdots\]
(332)#\[H_\nu \rightarrow \chi_\nu = C_\nu H_{\nu}(y)e^{-y^2/2}, y=\sqrt{\frac{\omega}{\hbar}}\]
(333)#\[\hat{T}_n = -\sum_{\alpha^\prime} \frac{\hbar^2}{2M_\alpha^\prime} \frac{d^2}{dR^2_{\alpha^\prime}} \Rightarrow \hat{T}_n = -\sum_{\alpha^\prime} \frac{\hbar^2}{2} \frac{d^2}{d\tilde{R}^2_{\alpha^\prime}}\]
(334)#\[E_{i}(\{\tilde{R}\})\approx E_{i}(\tilde{R}_0)+\sum_{\alpha^\prime} 0 \tilde{R}_\alpha^\prime + \frac{1}{2}\sum_{\alpha^\prime,\beta^\prime} \left.\frac{d^2 E_i}{d\tilde{R}_{\alpha^\prime} d\tilde{R}_{\beta^\prime}} \right\vert_{R=R_0}\tilde{R}_{\alpha^\prime} \tilde{R}_{\beta^\prime} + \cdots\]
(335)#\[\mathcal{H}=\frac{1}{2} \left. \frac{d^2 E_i}{d\tilde{R}_{\alpha^\prime} d\tilde{R}_{\beta^\prime}} \right\vert_{R=R_0} \rightarrow \mathcal{H}Q_\alpha = K_\alpha Q_\alpha \rightarrow Q_\alpha = \sum_{\alpha^\prime} c_{\alpha \alpha^\prime} \tilde{R}_{\alpha^\prime}\]
(336)#\[ \left(\hat{T}_n + E_i(\{\vec{R}\})\right)\Psi_n^{\gamma}\approx \sum_{\alpha} \left(\frac{\hbar^2}{2} \frac{d^2}{dQ_\alpha^2}+\frac{1}{2}K_\alpha Q_\alpha^2\right)\Psi_n^{\gamma} = E_{i,\gamma}\Psi_n^{\gamma}\]
(337)#\[ \left(\hat{T}_n + E_i(\{\vec{R}\})\right)\Psi_n^{\gamma}\approx \sum_{\alpha} \left(\frac{\hbar^2}{2} \frac{d^2}{dQ_\alpha^2}+\frac{1}{2}K_\alpha Q_\alpha^2\right)\Psi_n^{\gamma} = E_{i,\gamma}\Psi_n^{\gamma}\]
(338)#\[\Psi_n^{\gamma}=\chi_\text{tras}(x,y,z) \chi_\text{rot} (\alpha,\beta,\gamma)\prod_{\alpha}^{3N-6} \chi_\alpha (Q_\alpha)\]
(339)#\[\begin{split}\mathcal{H}=\begin{pmatrix}
K_{CO}+K_{OO} & -K_{CO} & -K_{OO} \\
-K_{CO} & 2K_{CO} & -K_{CO} \\
-K_{OO} & -K_{CO} & K_{CO} + K_{OO} \end{pmatrix}\end{split}\]
(340)#\[\begin{split}\vec{\tilde{Z}}^T=M^{1/2}\vec{Z}^T \rightarrow \begin{pmatrix} \tilde{Z}_{O_1} \\ \tilde{Z}_C\\ \tilde{Z}_{O_2}\end{pmatrix} = \begin{pmatrix}
\sqrt{M_O} & 0 & 0 \\
0 & \sqrt{M_C} & 0 \\
0 & 0 & \sqrt{M_O} \end{pmatrix}\begin{pmatrix} Z_{O_1} \\ Z_C\\ Z_{O_2}\end{pmatrix}\end{split}\]
(341)#\[\hat{H}=-\frac{\hbar^2}{2}\frac{\partial^2 }{\partial \tilde{Z}_{O_1}^2}-\frac{\hbar^2}{2}\frac{\partial^2 }{\partial \tilde{Z}_{O_2}^2}-\frac{\hbar^2}{2}\frac{\partial^2 }{\partial \tilde{Z}_{C}^2}+\frac{1}{2}\vec{\tilde{Z}} \tilde{\mathcal{H}}\vec{\tilde{Z}}^T\]
(342)#\[\hat{H}=-\frac{\hbar^2}{2}\frac{\partial^2 }{\partial \tilde{Q}_{u1}^2}-\frac{\hbar^2}{2}\frac{\partial^2 }{\partial \tilde{Q}_{u2}^2}-\frac{\hbar^2}{2}\frac{\partial^2 }{\partial \tilde{Q}_{g}^2}+\frac{1}{2}\omega_g^2 Q_g^2+\frac{1}{2}\omega_{u1}^2 Q_{u1}^2+\frac{1}{2}\omega_g^2 Q_{u2}^2\]
(343)#\[\begin{split} \omega_g^2 = \sqrt{2} \frac{K_{CO}+K_{OO}}{M_O} \\
\omega_{u1}^2 = 0 \\
\omega_{u2}^2 = \left(\frac{1}{M_O} + 2\frac{1}{M_C}\right)K_{CO}\end{split}\]
(344)#\[E=D\left[1-e^{-a(R-R_0)}\right]\]
(345)#\[E_\nu=-\frac{a^2\hbar^2}{2m}\left(\lambda-\nu-\frac{1}{2}\right)^2, \lambda=\frac{\sqrt{2mD}}{a\hbar}\]
Definición#
The divergence is an operator that is applied on a vector field.
(346)#\[\vec{\nabla}\cdot \vec{f}=\frac{\partial f_x}{\partial x}+\frac{\partial f_y}{\partial y}+\frac{\partial f_z}{\partial z}\]
As can be seen, the result of the applying the divergence is a scalar field.
Interpretation#
The divergence measures the difference between the number of field lines that enter and leave from/to a particular point.
Ejemplo
Fig. 204 The divergence measures the number of lines entering/exiting a point.#
A positive divergence is associated with a majority of lines coming out (leaving) from a point. This point is a source of field lines. On the other hand, a point with a negative divergence is seen to come from a majority of lines entering into a point. This point is called a sink of field line.