Exchange force

Exchange force is the interaction between particles of a system, as a result of the symmetry of the wavefunction describing the system. This interaction results in a change in the expectation values of inter-particle distances, and hence, a change in energy eigenvalues.

Consider a system of two electrons. The exchange force between electrons is also known as spin correlation. Since the electrons are indistinguishable, the normalised spatial wavefunction can be expressed generally as:

$\psi_{\pm}(r_1,r_2)=\frac{1}{\sqrt{2}}[\phi_a(r_1)\phi_b(r_2)\pm\phi_b(r_1)\phi_a(r_2)]$

where $\psi_+$ and $\psi_-$ are symmetric spatial wavefunction and anti-symmetric spatial wavefunction respectively of the system of a pair of identical electrons, $\phi_a$ and $\phi_b$ are orthonormal one-electron wavefunctions, and the notation of $\phi_a(r_1)\phi_b(r_2)$ denotes electron 1 in the state $\phi_a$ and electron 2 in the state $\phi_b$ .

To evaluate the effect of wavefunction symmetry on the expectation values of inter-particle distances, we analyse, for convenience, the average value of the square of the inter-particle distance:

$\langle(\boldsymbol{\mathit{r}}_1-\boldsymbol{\mathit{r}}_2)^{2}\rangle=\frac{1}{2}\int \psi_{\pm}^{\;*}(r_1^{\;2}+r_2^{\;2}-2\boldsymbol{\mathit{r}}_1\cdot\boldsymbol{\mathit{r}}_2)\psi_{\pm} dr_1dr_2$

Substitute $\psi_{\pm}$ in the above equation and expanding,

$\langle(\boldsymbol{\mathit{r}}_1-\boldsymbol{\mathit{r}}_2)^{2}\rangle=\frac{1}{2}\left ( \langle r_1^{\;2}\rangle_a\pm\langle r_2^{\;2}\rangle_a+\langle r_2^{\;2}\rangle_b\pm\langle r_1^{\;2}\rangle_b\right )-\langle\boldsymbol{\mathit{r}}_1\rangle_a\langle\boldsymbol{\mathit{r}}_2\rangle_b \mp\langle\boldsymbol{\mathit{r}}_1\rangle_b\langle\boldsymbol{\mathit{r}}_2\rangle_a\mp\int \phi_a^{\;*}(r_1)\boldsymbol{\mathit{r}}_1\phi_b(r_1)dr_1\int \phi_b^{\;*}(r_2)\boldsymbol{\mathit{r}}_2\phi_a(r_2)dr_2\mp\int \phi_b^{\;*}(r_1)\boldsymbol{\mathit{r}}_1\phi_a(r_1)dr_1\int \phi_a^{\;*}(r_2)\boldsymbol{\mathit{r}}_2\phi_b(r_2)dr_2$

where $\langle r_n^{\;2}\rangle_k=\int \phi_k^{\;*}(r_n) r_n^{\;2}\phi_k(r_n)dr_n$ and $\langle \boldsymbol{\mathit{r}}_n \rangle_k=\int \phi_k^{\;*}(r_n)\boldsymbol{\mathit{r}}_n\phi_k(r_n)dr_n$ , with $n=1,2$ and $k=a,b$ .

Applying the Hermitian property of $\boldsymbol{\mathit{r}}_1$ and $\boldsymbol{\mathit{r}}_2$ for the last two terms of the above equation,

$\langle(\boldsymbol{\mathit{r}}_1-\boldsymbol{\mathit{r}}_2)^{2}\rangle=\frac{1}{2}(\langle r_1^{\;2}\rangle_a\pm\langle r_2^{\;2}\rangle_a+\langle r_2^{\;2}\rangle_b\pm\langle r_1^{\;2}\rangle_b)-\langle\boldsymbol{\mathit{r}}_1\rangle_a\langle\boldsymbol{\mathit{r}}_2\rangle_b\mp\langle\boldsymbol{\mathit{r}}_1\rangle_b\langle\boldsymbol{\mathit{r}}_2\rangle_a\mp\langle\boldsymbol{\mathit{r}}_1\rangle_{ab}\langle\boldsymbol{\mathit{r}}_2\rangle_{ab}^{\;*}\mp\langle\boldsymbol{\mathit{r}}_1\rangle_{ab}^{\;*}\langle\boldsymbol{\mathit{r}}_2\rangle_{ab}$

where $\langle\boldsymbol{\mathit{r}}_n\rangle_{ab}=\int \phi_a^{\;*}(r_n)\boldsymbol{\mathit{r}}_n\phi_b(r_n)dr_n$ .

Since the two particles are indistinguishable, $\langle r_1^{\;2}\rangle_k=\langle r_2^{\;2}\rangle_k=\langle r^{2}\rangle_k$ . Similarly, $\langle \boldsymbol{\mathit{r}}_1\rangle_k=\langle \boldsymbol{\mathit{r}}_2\rangle_k=\langle \boldsymbol{\mathit{r}}\rangle_k$ and $\langle \boldsymbol{\mathit{r}}_1\rangle_{ab}=\langle \boldsymbol{\mathit{r}}_2\rangle_{ab}=\langle \boldsymbol{\mathit{r}}\rangle_{ab}$ . So,

$\langle(\boldsymbol{\mathit{r}}_1-\boldsymbol{\mathit{r}}_2)^{2}\rangle=\langle r^{2}\rangle_a+\langle r^{2}\rangle_b-2\langle\boldsymbol{\mathit{r}}\rangle_a\langle\boldsymbol{\mathit{r}}\rangle_b\mp2\vert\langle\boldsymbol{\mathit{r}}\rangle_{ab}\vert^{2}\; \; \; \; \; \; \; \; 246$

where we have used the logic that $\frac{1}{2}\langle r^{2}\rangle_k\pm\frac{1}{2}\langle r^{2}\rangle_k=\langle r^{2}\rangle_k$ because the difference of the terms is zero.

The consequence of eq246 is that $\langle(\boldsymbol{\mathit{r}}_1-\boldsymbol{\mathit{r}}_2)^{2}\rangle_{\psi_+}<\langle(\boldsymbol{\mathit{r}}_1-\boldsymbol{\mathit{r}}_2)^{2}\rangle_{\psi_-}$ , which implies that

$\frac{1}{\; \; \vert(\boldsymbol{\mathit{r}}_1-\boldsymbol{\mathit{r}}_2)\vert_{\psi_+}}>\frac{1}{\; \; \vert(\boldsymbol{\mathit{r}}_1-\boldsymbol{\mathit{r}}_2)\vert_{\psi_-}}\; \; \; \; \; \; \; \; 247$

The eigenvalues of the Hamiltonian of eq239 are therefore expected to be different for a singlet state and a triplet state because the total wavefunction of the system, according to Pauli’s exclusion principle, must be anti-symmetric with respect to particle exchange.

Next article: Many-electron systems

Previous article: Non-relativistic multi-electron Hamiltonian

Content page of quantum mechanics

Content page of advanced chemistry

Main content page

Leave a Reply Cancel reply