I started to learn some basics of second quantisation and specifically its use in quantum chemistry. Currently I'm reading this book by Péter R. Surján, and here is small excerpt from it.
If one has two electrons numbered as 1 and 2 which occupy the orbitals $\phi_{i}$ and $\phi_{k}$ our "first quantized" wave function is a Slater determinant \begin{equation*} \Phi(1,2) = \frac{1}{\sqrt{2}} \bigg| \begin{matrix} \phi_{i}(1) & \phi_{k}(1) \\ \phi_{i}(2) & \phi_{k}(2) \end{matrix} \bigg| \, . \end{equation*} The same entry is denoted in the second quantized notation as: $$ \lvert ik \rangle = a_{i}^{+} a_{k}^{+} \lvert \text{vac} \rangle \, . $$ In words, an electron is created on the vacuum in state $\lvert k \rangle$, then another one in state $\lvert i \rangle$. One has the correspondence: $$ \Phi(1,2) \leftrightarrow a_{i}^{+} a_{k}^{+} \lvert \text{vac} \rangle \, . $$
As far as I understand the order of creation operators here is just a convention: since we are dealing with antisymmetric fermionic states here, we have to to agree on a specific order of the creation operators and the author's choice was that the order of creation operators in the second quantized expression for an electronic state is the same as the order of spin-orbitals in the corresponding first quantized Slater determinant.
Obviously, once a convention is chosen we have to stick with it, but I fell like the author failed to do so, since later in the text he writes:
Now. creating an $N$-electron wave function $\Psi$ in which spinorbitals $\psi_{1}, \psi_{2}, \dotsc, \psi_{N}$ are occupied, one writes: $$ \Psi(1, 2, \dotsc, N) \leftrightarrow a_{n}^{+} \dotsc a_{2}^{+} a_{1}^{+} \lvert \text{vac} \rangle \, . $$
And here the order of creation operators is the reverse of the order spin-orbitals in Slater determinant (although Slater determinant is not written explicitly, I think we can safely assume that it is $| \begin{matrix} \psi_{1} & \psi_{2} & \cdots & \psi_{n} \end{matrix} |$, i.e. that orbitals follow in natural order). But with this convention $\Phi(1,2)$ mentioned above should be equivalent to $a_{k}^{+} a_{i}^{+} \lvert \text{vac} \rangle$ rather than $a_{i}^{+} a_{k}^{+} \lvert \text{vac} \rangle$.
Am I right or totally confused?