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| The interaction energy of a cluster of $N$ interacting //rigid// molecules is defined to be $\begin{align} E_{\rm int} & = E_{ABC\dots} - E_A - E_B - E_C - \cdots. \label{eq:Eint-cluster} \end{align}$ | The interaction energy of a cluster of $N$ interacting //rigid// molecules is defined to be $\begin{align} E_{\rm int} & = E_{ABC\dots} - E_A - E_B - E_C - \cdots. \label{eq:Eint-cluster} \end{align}$ |
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| Non-rigid molecules undergo a deformation in the cluster, with an associated //deformation energy// cost: $\delta E_X=E_X(x_X^\ast)-E_X(x_X^0)$, where $x_X^0$ is the geometry of monomer $X$ in isolation and $x_{X}^\ast$ is the geometry in the cluster. This deformation energy cost should be included as part of the interaction energy defined above, but since this chapter is mainly concerned with the interactions of rigid molecules, we will assume that the deformation energies $\delta E_X$ are obtained in a suitable manner, and concern ourselves only with the interaction energy defined by the above equation. |
Non-rigid molecules undergo a deformation in the cluster, with an associated //deformation energy// cost: $\delta E_X=E_X(x_X^\ast)-E_X(x_X^0)$, where $x_X^0$ is the geometry of monomer $X$ in isolation and $x_{X}^\ast$ is the geometry in the cluster. This deformation energy cost should be included as part of the interaction energy defined above, but since this chapter is mainly concerned with the interactions of rigid molecules, we will assume that the deformation energies $\delta E_X$ are obtained in a suitable manner, and concern ourselves only with the interaction energy defined by the above equation. |
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| $\begin{equation} E_{\rm int}(ABC\cdots) = \sum_{X \lt Y} E_{\rm int}(XY) + \sum_{X \lt Y \lt Z} \Delta E_{\rm int}(XYZ) + \cdots, \label{eq:many-body} \end{equation}$ where $\Delta E_{\rm int}(XYZ)$ is the three-body correction, defined as \begin{align} \Delta E_{\rm int}(XYZ) & = E_{\rm int}(XYZ) - E_{\rm int}(XY) - E_{\rm int}(XZ) - E_{\rm int}(YZ). \label{eq:3body} \end{align} In the same way, we can define four-body corrections, five-body corrections, and so on. For a cluster of $N$ molecules, this expansion terminates at the $N$-body correction. |
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| The many-body expansion would not be of much use if we had to evaluate all terms given above. But experience has shown that the expansion converges quickly and terms beyond those involving three bodies are not so important. This is fortunate as the 2-body interactions are well understood and can be evaluated for moderate-sized molecules using a variety of methods, while good approximations are available for the terms involving three and more bodies, which usually arise from the effects of polarization in the cluster. |
$\begin{equation} E_{\rm int}(ABC\cdots) = \sum_{X \lt Y} E_{\rm int}(XY) + \sum_{X \lt Y \lt Z} \Delta E_{\rm int}(XYZ) + \cdots, \label{eq:many-body} \end{equation}$ where $\Delta E_{\rm int}(XYZ)$ is the three-body correction, defined as $\begin{align} \Delta E_{\rm int}(XYZ) & = E_{\rm int}(XYZ)-E_{\rm int}(XY) - E_{\rm int}(XZ) - E_{\rm int}(YZ). \label{eq:3body} \end{align}$. In the same way, we can define four-body corrections, five-body corrections, and so on. For a cluster of $N$ molecules, this expansion terminates at the $N$-body correction. The many-body expansion would not be of much use if we had to evaluate all terms given above. But experience has shown that the expansion converges quickly and terms beyond those involving three bodies are not so important. This is fortunate as the 2-body interactions are well understood and can be evaluated for moderate-sized molecules using a variety of methods, while good approximations are available for the terms involving three and more bodies, which usually arise from the effects of polarization in the cluster. |
Index:
Definition of the intermolecular energy
The interaction energy of a cluster of $N$ interacting //rigid// molecules is defined to be $\begin{align} E_{\rm int} & = E_{ABC\dots} - E_A - E_B - E_C - \cdots. \label{eq:Eint-cluster} \end{align}$
Here $E_{ABC\dots}$ is the energy of the cluster and $E_{X}$, $X=A$, $B$, $C$, etc., is the energy of molecule $X$.
Non-rigid molecules undergo a deformation in the cluster, with an associated //deformation energy// cost: $\delta E_X=E_X(x_X^\ast)-E_X(x_X^0)$, where
$x_X^0$ is the geometry of monomer $X$ in isolation and $x_{X}^\ast$
is the geometry in the cluster. This deformation energy cost should be included as part of the interaction energy defined above, but since this chapter is mainly concerned with the interactions of rigid molecules, we will assume that the deformation energies
$\delta E_X$ are obtained in a suitable manner, and concern ourselves only with the interaction energy defined by the above equation.
$E_{\rm int}$ can be evaluated directly, but for computational efficiency as well as physical interpretation it is worthwhile to partition the $N$-body interaction energy into contributions from dimers, trimers and so on.
This leads to an exact re-formulation of $E_{\rm int}$ that is known as the many-body expansion:
$\begin{equation} E_{\rm int}(ABC\cdots) = \sum_{X \lt Y} E_{\rm int}(XY) + \sum_{X \lt Y \lt Z} \Delta E_{\rm int}(XYZ) + \cdots, \label{eq:many-body} \end{equation}$
where $\Delta E_{\rm int}(XYZ)$ is the three-body correction, defined as
$\begin{align} \Delta E_{\rm int}(XYZ) & = E_{\rm int}(XYZ)-E_{\rm int}(XY) - E_{\rm int}(XZ) - E_{\rm int}(YZ). \label{eq:3body} \end{align}$.
In the same way, we can define four-body corrections, five-body corrections, and so on. For a cluster of $N$ molecules, this expansion terminates at the $N$-body correction.
The many-body expansion would not be of much use if we had to evaluate all terms given above. But experience has shown that the expansion converges quickly and terms beyond those involving three bodies are not so important. This is fortunate as the 2-body interactions are well understood and can be evaluated for moderate-sized molecules using a variety of methods, while good approximations are available for the terms involving three and more bodies, which usually arise from the effects of polarization in the cluster.