Direct Exchange Interaction in Heitler-London Model

The idea of direct exchange interaction was originally derived in Heitler-London model which explained the binding energy of the hydrogen molecule. This model inspired Heisenberg's theory of ferromagnetism which based on an exchange interaction between the spins of two neighboring atoms.

Consider two hydrogen atoms \(a\) and \(b\) with their isolated atomic orbitals represented as \(\phi_{a}(\vec{r})\) and \(\phi_{b}(\vec{r})\), respectively. The hamiltonian of an isolated hydrogen atom is given as: \[ H = - \frac{\hbar^{2}}{2m}\nabla^{2} - \frac{e^{2}}{r}\] The hamiltonian of the hydrogen molecule is given by the sum of those for the isolated atoms: \[ H = - \frac{\hbar^{2}}{2m}\nabla_{1}^{2} - \frac{\hbar^{2}}{2m}\nabla_{2}^{2} - \frac{e^{2}}{r_{a1}} - \frac{e^{2}}{r_{b2}} + \left( \frac{e^{2}}{r_{12}} + \frac{e^{2}}{r_{ab}} - \frac{e^{2}}{r_{b1}} - \frac{e^{2}}{r_{a2}} \right)\] where \(r_{1}\) and \(r_{2}\) represent coordinates of the two electrons in the molecule; \(- e^{2}/r_{a2}\) represent electrostatic attraction between the nucleus of atom \(a\) and the electron of atom \(b\); \(- e^{2}/r_{b1}\) represent electrostatic attraction between the nucleus of atom \(b\) and the electron of atom \(a\). The first four terms on the right side of the equation represent simple addition of hamiltonians for isolated atoms, whereas, the additional terms in parentheses represent the interaction between the two constituent atoms. We can rewrite the above expression for hamiltonian as: \[ H = H_{0} + \Delta H\] where \begin{eqnarray*} H_{0} & = & - \frac{\hbar^{2}}{2m}\nabla_{1}^{2} - \frac{\hbar^{2}}{2m}\nabla_{2}^{2} - \frac{e^{2}}{r_{a1}} - \frac{e^{2}}{r_{b2}} \\ \Delta H & = &\frac{e^{2}}{r_{12}} + \frac{e^{2}}{r_{ab}} - \frac{e^{2}}{r_{b1}} - \frac{e^{2}}{r_{a2}} \end{eqnarray*}

The wavefunctions of electrons in a hydrogen molecule can be represented as superposition of bilinear and higher order non-linear combinations of the wavefunctions of isolated atoms in accordance with perturbation theory. The Heitler-London model takes the zeroth order approximation of wavefunction of hydrogen molecule as: \[ \psi_{s} (\vec{r}_{1},\vec{r}_{2}) \, = \, \frac{1}{\sqrt{2(1 + |\left\langle a | b\right\rangle|^{2})}} ( \phi_{a}(\vec{r}_{1})\phi_{b}(\vec{r}_{2}) + \phi_{a}(\vec{r}_{2})\phi_{b}(\vec{r}_{1}))\] \[ \psi_{a} (\vec{r}_{1},\vec{r}_{2}) \, = \, \frac{1}{\sqrt{2(1 - |\left\langle a | b\right\rangle|^{2})}} ( \phi_{a}(\vec{r}_{1})\phi_{b}(\vec{r}_{2}) - \phi_{a}(\vec{r}_{2})\phi_{b}(\vec{r}_{1}))\] where \(s\) and \(a\) represent the symmetric and anti-symmetric combination of bilinear wavefunction \(\phi_{a}(\vec{r}_{1})\phi_{b}(\vec{r}_{2})\), with respect to interchange of \(r_{1}\) and \(r_{2}\) electron co-ordinates. Note that above equations consider only the orbital component of the electron wavefunctions, and do not account for spin. In accordance with Pauli exclusion principle, the molecular state with parallel-spin (symmetric spin combination) electrons \(must\) correspond to anti-parallel combination of orbital wavefunctions. (In order for Pauli exclusion principle to hold for a system of fermions, the total wavefunction of the system \(must\) be anti-symmetric. This means for a symmetric combination of spins, the orbital functions must combine in anti-symmetric fashion, and vice-versa, so that the resultant wavefunction (spin \(\times\) orbital) is anti-symmetric.). The state with parallel-spins (\(\psi_{a} (\vec{r}_{1},\vec{r}_{2})\)) is known as the triplet state, and the one anti-parallel spins \(\psi_{s} (\vec{r}_{1},\vec{r}_{2})\) the singlet state. The coefficient on right-side of the equation is the normalization factor; \(\left\langle a | b\right\rangle\) is the overlap integral between \(\phi_{a}\) and \(\phi_{b}\). \[ \left\langle a | b \right\rangle = \int \phi_{a}^{*}(\vec{r})\phi_{b}(\vec{r}) dr \] The energies of the spin-singlet and spin-triplet states is given by expectation values of the hamiltonian with respect to the symmetric and anti-symmetric wavefunctions, respectively, as: \begin{eqnarray} \left\langle H \right\rangle_{s} & = &\frac{\left\langle ab|H|ab \right\rangle + \left\langle ab|H|ba \right\rangle}{1 + |\left\langle a | b\right\rangle|^{2}} \\ & = & 2\epsilon_{1s} + \frac{\left\langle ab|\Delta H|ab \right\rangle + \left\langle ab|\Delta H|ba \right\rangle}{1 + |\left\langle a | b\right\rangle|^{2}} \end{eqnarray} \begin{eqnarray} \left\langle H \right\rangle_{a} & = &\frac{\left\langle ab|H|ab \right\rangle - \left\langle ab|H|ba \right\rangle}{1 - |\left\langle a | b\right\rangle|^{2}} \\ & = & 2\epsilon_{1s} + \frac{\left\langle ab|\Delta H|ab \right\rangle - \left\langle ab|\Delta H|ba \right\rangle}{1 - |\left\langle a | b\right\rangle|^{2}} \end{eqnarray} Here \(\epsilon_{1s}\) is the energy of the \(1s\) ground state for the hydrogen atom. The direct exchange integral is given by half the energy difference between the symmetric (singlet) and anti-symmetric (triplet) states i.e. \((\left\langle H \right\rangle_{s}-\left\langle H \right\rangle_{a})/2\). In the original Heitler-London model, this difference (between singlet nd triplet energies) was negative up to a moderately large values of interatomic distances but changes to positive for distances greater than 50\(a_{H}\) (\(a_{H}\) is Bohr radius of hydrogen atom.). However, according to the theory of Sturm-Liouville functions*. The eigenfunction corresponding to the lowest eigenvalue of an operator of Sturm-Liouville type has no node; therefore, \(\left\langle H \right\rangle_{s}\) must always be less than \(\left\langle H \right\rangle_{a}\). Thus, the Heitler-London model for hydrogen molecule does not predict the correct behavior of direct exchange integral even though it introduces and takes into account the exchange interaction between the constituting atoms.

The reason for this discrepancy is the term \(\left\langle ab|e^{2}/r_{12}|ba \right\rangle\) (exchange integral of \(e^{2}/r_{12}\) - exchange interaction between electrons, where \(r_{12}\) is the distance between the two electrons). At large interatomic distances, electron-electron exchange dominates other terms in \(\Delta H\). This matrix element is an approximation of \[\left\langle \phi|e^{2}/r_{12}|P_{12} \phi \right\rangle\] where \(\phi\) is the sum of \textbf{exact} functions for singlet and triplet states \(\phi_{s}\) and \(\phi_{a}\), respectively, and \(P_{12}\) is the permutation operator which interchanges the coordinates of electrons 1 and 2. When the exact functions \(\phi\) are considered, the electrons the two electrons keep away from each other to reduce the contribution of \(e^{2}/r_{12}\). \(\phi_{a} \phi_{b}\) in Heitler-London model ignores the electron correlation effect and gives unreasonable asymptotic behavior of the exchange integral.

* Sturm-Liouville theory is the theory of real second-order linear differential equation of the form \(\frac{d}{dx}\left[p(x) \frac{dy}{dx}\right] + q(x) = -\lambda w(x)y \) where \(y\) is a function of \(x\) and \(p(x)\), \(q(x)\), and \(w(x)>0\) are given. Bessel equation and Legendre equations are examples of this type of differential equations.

ADDENDUM \[\boxed{\mbox{Derivation of} \,\,\, \left\langle H \right\rangle_{s} = 2\epsilon_{1s} + \frac{\left\langle ab|\Delta H|ab \right\rangle + \left\langle ab|\Delta H|ba \right\rangle}{1 + |\left\langle a | b\right\rangle|^{2}}}\]

\begin{eqnarray} \left\langle H \right\rangle_{s} & = &\frac{\left\langle ab|H|ab \right\rangle + \left\langle ab|H|ba \right\rangle}{1 + |\left\langle a | b\right\rangle|^{2}} \\ & = & \frac{\left\langle ab|H_{0} + \Delta H|ab \right\rangle + \left\langle ab|H_{0} + \Delta H|ba \right\rangle}{1 + |\left\langle a | b\right\rangle|^{2}} \\ & = & \frac{\left\langle ab|H_{0}|ab \right\rangle + \left\langle ab|\Delta H|ab \right\rangle + \left\langle ab|H_{0}|ba \right\rangle + \left\langle ab| \Delta H|ba \right\rangle}{1 + |\left\langle a | b\right\rangle|^{2}} \hspace{2cm} \textbf{(A)} \end{eqnarray} \begin{eqnarray*} \left\langle ab|H_{0}|ba \right\rangle & = & \int \phi^{*}_{a}(\vec{r}_{1})\phi^{*}_{b}(\vec{r}_{2}) H_{0} \phi_{b}(\vec{r}_{1})\phi_{a}(\vec{r}_{2}) d^{3}\vec{r}_{1} d^{3}\vec{r}_{2} \\ & = & \int \phi^{*}_{a}(\vec{r}_{1})\phi^{*}_{b}(\vec{r}_{2}) \left[\left(-\frac{\hbar^{2}}{2m}\nabla_{1}^{2} - \frac{e^{2}}{r_{a1}}\right) +\left(-\frac{\hbar^{2}}{2m}\nabla_{2}^{2}-\frac{e^{2}}{r_{b2}} \right)\right] \phi_{b}(\vec{r}_{1})\phi_{a}(\vec{r}_{2}) d^{3}\vec{r}_{1} d^{3}\vec{r}_{2} \\ & = & \int \phi^{*}_{a}(\vec{r}_{2})\phi^{*}_{b}(\vec{r}_{2}) d^{3}\vec{r}_{2} \int \phi^{*}_{a}(\vec{r}_{1}) \left(-\frac{\hbar^{2}}{2m}\nabla_{1}^{2} - \frac{e^{2}}{r_{a1}}\right) \phi_{b}(\vec{r}_{1}) d^{3}\vec{r}_{1} \\ & + & \int \phi^{*}_{a}(\vec{r}_{1})\phi^{*}_{b}(\vec{r}_{1}) d^{3}\vec{r}_{1} \int \phi^{*}_{b}(\vec{r}_{2}) \left(-\frac{\hbar^{2}}{2m}\nabla_{2}^{2}-\frac{e^{2}}{r_{b2}} \right) \phi_{a}(\vec{r}_{2}) d^{3}\vec{r}_{2} \end{eqnarray*} \(\int \phi^{*}_{a}(\vec{r}_{1})\phi^{*}_{b}(\vec{r}_{1}) d^{3}\vec{r}_{1} = \int \phi^{*}_{a}(\vec{r}_{2})\phi^{*}_{b}(\vec{r}_{2}) d^{3}\vec{r}_{2} = \left\langle a|b \right\rangle << 1\) when interatomic distances are large so that electron-electron exchange term dominate. Thus, \(\left\langle ab|H_{0}|ba \right\rangle \approx \) 0. Therefore, the third term in the numerator can be safely ignored. \begin{eqnarray*} \left\langle ab|H_{0}|ab \right\rangle & = & \int \phi^{*}_{a}(\vec{r}_{1})\phi^{*}_{b}(\vec{r}_{2}) H_{0} \phi_{a}(\vec{r}_{1})\phi_{b}(\vec{r}_{2}) d^{3}\vec{r}_{1} d^{3}\vec{r}_{2} \\ & = & \int \phi^{*}_{a}(\vec{r}_{1})\phi^{*}_{b}(\vec{r}_{2}) \left[\left(-\frac{\hbar^{2}}{2m}\nabla_{1}^{2} - \frac{e^{2}}{r_{a1}}\right) +\left(-\frac{\hbar^{2}}{2m}\nabla_{2}^{2}-\frac{e^{2}}{r_{b2}} \right)\right] \phi_{a}(\vec{r}_{1})\phi_{b}(\vec{r}_{2}) d^{3}\vec{r}_{1} d^{3}\vec{r}_{2} \\ & = & \int \phi^{*}_{b}(\vec{r}_{2})\phi^{*}_{b}(\vec{r}_{2}) d^{3}\vec{r}_{2} \int \phi^{*}_{a}(\vec{r}_{1}) \left(-\frac{\hbar^{2}}{2m}\nabla_{1}^{2} - \frac{e^{2}}{r_{a1}}\right) \phi_{a}(\vec{r}_{1}) d^{3}\vec{r}_{1} \\ & + & \int \phi^{*}_{a}(\vec{r}_{1})\phi^{*}_{a}(\vec{r}_{1}) d^{3}\vec{r}_{1} \int \phi^{*}_{b}(\vec{r}_{2}) \left(-\frac{\hbar^{2}}{2m}\nabla_{2}^{2}-\frac{e^{2}}{r_{b2}} \right) \phi_{b}(\vec{r}_{2}) d^{3}\vec{r}_{2} \\ & = & \left\langle b|b \right\rangle \epsilon_{1s} + \left\langle a|a \right\rangle \epsilon_{1s} \\ & = & 2\epsilon_{1s} \end{eqnarray*} Substituting these results in (A), we obtain \[\left\langle H \right\rangle_{s} = \frac{2\epsilon_{1s}}{1 + |\left\langle a | b\right\rangle|^{2}} + \frac{\left\langle ab|\Delta H|ab \right\rangle + \left\langle ab|\Delta H|ba \right\rangle}{1 + |\left\langle a | b\right\rangle|^{2}} \] Since \(\left\langle a|b \right\rangle << 1\) and \(2\epsilon_{1s}\) is large, we can omit \(\left\langle a|b \right\rangle\) from the denominator of the first term. Therefore, \[\left\langle H \right\rangle_{s} = 2\epsilon_{1s} + \frac{\left\langle ab|\Delta H|ab \right\rangle + \left\langle ab|\Delta H|ba \right\rangle}{1 + |\left\langle a | b\right\rangle|^{2}} \]