|
Size: 2586
Comment:
|
Size: 2409
Comment:
|
| Deletions are marked like this. | Additions are marked like this. |
| Line 1: | Line 1: |
| ## page was renamed from AJMGrpOnly/teaching/intermolecular-interactions/two-body-energy ## page was renamed from ajm/teaching/intermolecular-interactions/two-body-energy #acl apw185:read,write,delete Known:read All: |
#acl +All:read apw185:read,write,delete Known:read All: |
| Line 8: | Line 6: |
| * [[ajm/teaching/intermolecular-interactions|Intermolecular Interactions]] | * [[AJMPublic/teaching/intermolecular-interactions|Intermolecular Interactions]] |
| Line 12: | Line 10: |
| \begin{align} E_{\rm int} & = E_{AB} - E_{A} - E_{B}. \label{eq:Eint2body} \end{align} If we calculate all three energies using a suitable electronic structure method---some of which will be described later---this definition provides a simple method for calculating the interaction energy of a pair of interacting molecules. This approach, known as the //supermolecular// method, provides us with a single number: the interaction energy $E_{\rm int}$. We gain considerable physical insight into $E_{\rm int}$ by evaluating it not as the difference in energies suggested by the above definition, but through //perturbation theory//, which enables us to partition $E_{\rm int}$ into physical components: the electrostatic, induction (or polarization), dispersion and exchange-repulsion energies. This breakup proves invaluable not only as an aid to interpretation, but also forms the basis for all analytic atom--atom potentials (See //Intermolecular Forces// by Anthony Stone, OUP (2012)). |
\begin{align} E_{\rm int} & = E_{AB} - E_{A} - E_{B}. \label{eq:Eint2body} \end{align}$ If we calculate all three energies using a suitable electronic structure method---some of which will be described later---this definition provides a simple method for calculating the interaction energy of a pair of interacting molecules. This approach, known as the //supermolecular// method, provides us with a single number: the interaction energy $E_{\rm int}$. We gain considerable physical insight into $E_{\rm int}$ by evaluating it not as the difference in energies suggested by the above definition, but through //perturbation theory//, which enables us to partition $E_{\rm int}$ into physical components: the electrostatic, induction (or polarization), dispersion and exchange-repulsion energies. This breakup proves invaluable not only as an aid to interpretation, but also forms the basis for all analytic atom--atom potentials (See //Intermolecular Forces// by Anthony Stone, OUP (2012)). |
| Line 30: | Line 16: |
| \begin{equation} H(\lambda) = H_0 + \lambda V, \label{eq:Hamiltonian} \end{equation} where $H_0 = H_A + H_B$ is the sum of the Hamiltonians of the unperturbed monomers $A$ and $B$, and $V$ is the intermonomer interaction operator that consists of electron-electron, electron-nuclear, and nuclear-nuclear interactions between the monomers. We may now carry out a symmetrized Rayleigh-Schrodinger perturbation theory ((Jeziorski, Moszynski & Szalewicz, Chem. Rev. 1994)) using as our zeroth-order wave function the unsymmetrized product $\Phi_0 = \Phi^{\rm A}_{0} \Phi^{\rm B}_{0}$ and applying a suitable projection to retain only the fully antisymmetric states. The zeroth-order energy is then $E_0 = E^{\rm A}_{0} + E^{\rm B}_{0}$. The interaction energy appears at first and higher orders in perturbation theory (setting $\lambda=1$): \begin{align} E_{\rm int} &= E_{\rm elst}^{(1)} + E_{\rm exch}^{(1)} + E_{\rm ind}^{(2)} + E_{\rm disp}^{(2)} + \cdots \end{align} |
$\begin{equation} H(\lambda) = H_0 + \lambda V, \label{eq:Hamiltonian} \end{equation}$ where $H_0 = H_A + H_B$ is the sum of the Hamiltonians of the unperturbed monomers $A$ and $B$, and $V$ is the intermonomer interaction operator that consists of electron-electron, electron-nuclear, and nuclear-nuclear interactions between the monomers. We may now carry out a symmetrized Rayleigh-Schrodinger perturbation theory ((Jeziorski, Moszynski & Szalewicz, Chem. Rev. 1994)) using as our zeroth-order wave function the unsymmetrized product $\Phi_0 = \Phi^{\rm A}_{0} \Phi^{\rm B}_{0}$ and applying a suitable projection to retain only the fully antisymmetric states. The zeroth-order energy is then $E_0 = E^{\rm A}_{0} + E^{\rm B}_{0}$. The interaction energy appears at first and higher orders in perturbation theory (setting $\lambda=1$): $\begin{align} E_{\rm int} &= E_{\rm elst}^{(1)} + E_{\rm exch}^{(1)} + E_{\rm ind}^{(2)} + E_{\rm disp}^{(2)} + \cdots \end{align}$ |
Contents
Index
The two-body interaction energy
The interaction energy of a pair of interacting molecules is defined as
\begin{align} E_{\rm int} & = E_{AB} - E_{A} - E_{B}. \label{eq:Eint2body} \end{align}$
If we calculate all three energies using a suitable electronic structure method---some of which will be described later---this definition provides a simple method for calculating the interaction energy of a pair of interacting molecules. This approach, known as the //supermolecular// method, provides us with a single number: the interaction energy $E_{\rm int}$. We gain considerable physical insight into $E_{\rm int}$ by evaluating it not as the difference in energies suggested by the above definition, but through //perturbation theory//, which enables us to partition $E_{\rm int}$ into physical components: the electrostatic, induction (or polarization), dispersion and exchange-repulsion energies. This breakup proves invaluable not only as an aid to interpretation, but also forms the basis for all analytic atom--atom potentials (See //Intermolecular Forces// by Anthony Stone, OUP (2012)).
Consider the dimer Hamiltonian partitioned as
$\begin{equation} H(\lambda) = H_0 + \lambda V, \label{eq:Hamiltonian} \end{equation}$
where $H_0 = H_A + H_B$ is the sum of the Hamiltonians of the unperturbed monomers $A$ and $B$, and $V$ is the intermonomer interaction operator that consists of electron-electron, electron-nuclear, and nuclear-nuclear interactions between the monomers. We may now carry out a symmetrized Rayleigh-Schrodinger perturbation theory ((Jeziorski, Moszynski & Szalewicz, Chem. Rev. 1994)) using as our zeroth-order wave function the unsymmetrized product $\Phi_0 = \Phi^{\rm A}_{0} \Phi^{\rm B}_{0}$ and applying a suitable projection to retain only the fully antisymmetric states.
The zeroth-order energy is then $E_0 = E^{\rm A}_{0} + E^{\rm B}_{0}$. The interaction energy appears at first and higher orders in perturbation theory (setting $\lambda=1$):
$\begin{align} E_{\rm int} &= E_{\rm elst}^{(1)} + E_{\rm exch}^{(1)} + E_{\rm ind}^{(2)} + E_{\rm disp}^{(2)} + \cdots \end{align}$
As indicated, at first and second order the terms take on well-defined physical meanings.