Consider two Slater determinants \(\Phi_{1}\) and \(\Phi_{2}\) that differ by only one spinorbital; that is, $$\begin{array}{l} \Phi_{1}=\left\|\cdots \varphi_{m} \varphi_{i} \cdots\right\| \\ \Phi_{2}=\left\|\cdots \varphi_{p} \varphi_{i} \cdots\right\| \end{array}$$ Derive the following Condon-Slater rule: $$\begin{aligned} \left\langle\Phi_{1}|H| \Phi_{2}\right\rangle=&\left\langle\varphi_{m}(1)\left|h_{1}\right| \varphi_{p}(1)\right\rangle+\sum_{i}\left\\{\left[\varphi_{m} \varphi_{i} | \varphi_{p} \varphi_{i}\right]\right.\\\ &\left.-\left[\varphi_{m} \varphi_{i} | \varphi, \varphi_{p}\right]\right\\} \end{aligned}$$ where we have used the notation $$\left[\varphi_{4} \varphi_{b} | \varphi_{c} \varphi_{d}\right]=j_{0} \int \varphi_{a}^{*}(1) \varphi_{b}^{*}(2)\left(\frac{1}{r_{12}}\right) \varphi_{c}(1) \varphi_{d}(2) \mathrm{d} r_{1} \mathrm{d} r_{2}$$

Step by step solution

01

Expression for \(\Phi_{1}\) and \(\Phi_{2}\)

The exercise has provided two Slater determinants, \(\Phi_{1}\) and \(\Phi_{2}\), which differ by only one spinorbital. \(\Phi_{1}\) has \(\varphi_{m}\) and \(\Phi_{2}\) has \(\varphi_{p}\). Write down these expressions.

02

Expression for Hamiltonian Operator

The Hamiltonian operator, H, in the bra-ket notation, is acting on \(\Phi_{2}\) and the result is an inner product with \(\Phi_{1}\). The operator H applies to the functions within the brackets and adds them or subtracts depending on the function.

03

Definition of Two-Electron Integral

The exercise uses a two-electron integral in Dirac notation to represent electron-electron interaction. Understand the form of notation that represents the integral \(\int \varphi_{a}^{*}(1) \varphi_{b}^{*}(2)\left(\frac{1}{r_{12}}\right) \varphi_{c}(1) \varphi_{d}(2) \mathrm{d} r_{1}\ \mathrm{d} r_{2}\).

04

Expand the Hamiltonian Operator

Substitute the definition of the two-electron integral into the expression obtained in Step 2. Use the antisymmetry property of the Slater determinants to expand the Hamiltonian operator.

05

Simplify the Expression

Now, simplify the expression by cancelling terms and rearranging to obtain the Condon-Slater rule. This gives the transition probabilities between different quantum states.

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