26.1: The Molecular Hamiltonian
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For a molecule, we can decompose the Hamiltonian operator as:
\[ \hat{H} = \hat{K}_N +\hat{K}_{e} + \hat{V}_{NN} + \hat{V}_{eN} + \hat{K}_{ee} \label{27.1.1} \]
where we have decomposed the kinetic energy operator into nuclear and electronic terms, \(\hat{K}_N\) and \(\hat{K}_e\), as well as the potential energy operator into terms representing the interactions between nuclei, \(\hat{V}_{NN}\), between electrons, \(\hat{V}_{ee}\), and between electrons and nuclei, \(\hat{V}_{eN}\). Each term can then be calculated using:
\[\begin{equation} \begin{aligned}
\hat{K}_N &=-\sum_i^{\text {nuclei }} \dfrac{\hbar^2}{2 M_i} \nabla_{\mathbf{R}_i}^2 \\
\hat{K}_e &=-\sum_i^{\text {electrons }} \dfrac{\hbar^2}{2 m_e} \nabla_{\mathbf{r}_i}^2 \\
\hat{V}_{N N} &=\sum_i \sum_{j>i} \dfrac{Z_i Z_j e^2}{4 \pi \varepsilon_0\left|\mathbf{R}_i-\mathbf{R}_j\right|} \\
\hat{V}_{e N} &=-\sum_i \sum_j \dfrac{Z_i e^2}{4 \pi \varepsilon_0\left|\mathbf{R}_i-\mathbf{r}_j\right|} \\
\hat{V}_{e e} &=\sum_i \sum_{i<j} \dfrac{e^2}{4 \pi \varepsilon_0\left|\mathbf{r}_i-\mathbf{r}_j\right|}
\end{aligned}\end{equation} \label{27.1.2} \]
where \(M_i\), \(Z_i\), and \(\mathbf{R}_i\) are the mass, atomic number, and coordinates of nucleus \(i\), respectively, and all other symbols are the same as those used in Equation 26.1 for the many-electron atom Hamiltonian.
Small terms in the molecular Hamiltonian
The operator in Equation \ref{27.1.1} is known as the “exact” nonrelativistic Hamiltonian in field-free space. However, it is important to remember that it neglects at least two effects. Firstly, although the speed of an electron in a hydrogen atom is less than 1% of the speed of light, relativistic mass corrections can become appreciable for the inner electrons of heavier atoms. Secondly, we have neglected the spin-orbit effects, which is explained as follows. From the point of view of an electron, it is being orbited by a nucleus which produces a magnetic field (proportional to \({\bf L}\)); this field interacts with the electron’s magnetic moment (proportional to \({\bf S}\)), giving rise to a spin-orbit interaction (proportional to \({\bf L} \cdot {\bf S}\) for a diatomic.) Although spin-orbit effects can be important, they are generally neglected in quantum chemical calculations, and we will neglect them in the remainder of this textbook as well.