In the second quantized notation, a two body operator $\hat{O}$ can be written as

$$\hat{O} = \sum\limits_{x_1,x_2,x_3,x_4} O_{x_1,x_2,x_3,x_4} a^\dagger_{x_1}a^\dagger_{x_2}a_{x_4}a_{x_3} ,$$ where $$ O_{x_1,x_2,x_3,x_4} = \langle x_1x_2 \rvert\hat{O}\rvert x_3x_4\rangle .$$

However, if we write $\langle ij \rvert\hat{O}\rvert kl\rangle$ as

$$\langle \Omega \rvert a_{i}a_{j} \hat{O}a^\dagger_{k}a^\dagger_{l} \rvert \Omega\rangle = \sum\limits_{x_1,x_2,x_3,x_4} O_{x_1,x_2,x_3,x_4} \langle \Omega \rvert a_{i}a_{j}a^\dagger_{x_1}a^\dagger_{x_2} a_{x_4}a_{x_3} a^\dagger_{k}a^\dagger_{l} \rvert \Omega\rangle $$

The vacuum expectation can be calculated using Wick's theorem to be

$$\langle \Omega \rvert a_{i}a_{j}a^\dagger_{x_1}a^\dagger_{x_2} a_{x_4}a_{x_3} a^\dagger_{k}a^\dagger_{l} \rvert \Omega\rangle = \delta_{i x_1}\delta_{j x_2}\delta_{k x_4}\delta_{l x_3}-\delta_{i x_1}\delta_{j x_2}\delta_{k x_3}\delta_{l x_4}-\delta_{i x_2}\delta_{j x_1}\delta_{k x_4}\delta_{l x_3}+\delta_{i x_2}\delta_{j x_1}\delta_{k x_3}\delta_{l x_4}$$

Then $$\langle ij \rvert\hat{O}\rvert kl\rangle = O_{ijlk}- O_{ijkl}-O_{jilk}+O_{jikl} ,$$ not just $O_{ijkl}$!

Where am I committing a mistake that results in this apparent contradiction?


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