## Secular equation ### MO by LCAO Since we use $\Phi = \sum_{i=1}^{N}a_{i}\phi_{i}$ as our molecular orbitals, the energy $\frac{\bra{\Phi}\hat{H}\ket{\Phi}}{\braket{ \Phi | \Phi } }$ becomes $E=\frac{\sum_{i=1}^{N}\sum_{j=1}^{N}a_{i}a_{j}\bra{\phi_{i}}\hat{H}\ket{\phi_{j}}}{\sum_{i=1}^{N}\sum_{j=1}^{N}a_{i}a_{j}\braket{ \phi_{i} | \phi_{j} } } =\frac{\sum_{i,j}a_{i}a_{j}H_{ij}}{\sum_{i,j}a_{i}a_{j}S_{ij}}$ compute $\frac{\partial E}{\partial a_{i}}=0$, we get $\sum_{i=1}^{N}a_{i}(H_{j,i}-ES_{{j,i}})=0$ for any $j$. This lead to Secular equation, with its determinant having the form $\begin{vmatrix} H_{11}-ES_{11} &H_{12}-ES_{12} & \cdots & H_{1N}-ES_{1N}\\ H_{21}-ES_{21} &H_{22}-ES_{22} & \cdots & H_{2N}-ES_{2N}\\ \vdots&\vdots & \ddots &\vdots\\ H_{N1}-ES_{N1} &H_{N2}-ES_{N2} & \cdots & H_{NN}-ES_{NN} \\ \end{vmatrix}=0$ There will be $N$ roots $E_{j}$, which permit the Secular equations to be true (or meaningful, to give physically reasonable results), each with a different set of coefficient $a_{ij}$, or eigenvectors, which define an optimal wave function within the given basis set $\Phi_{j} = \sum_{i=1}^{N}a_{ij}\psi_{i}$ and this is LCAO, if we take basis as atomic orbitals and consider $\Phi$ as WF of molecules/solid. >[!Notice] >Secular eq is just a general eq solving $E$, it does not have to be related to LCAO. It does not only give the $E_{0}$, but all energies from the selected basis. > >Remember, this stationary picture is also based on [Born-Oppenheimer approximation](https://en.wikipedia.org/wiki/Born%E2%80%93Oppenheimer_approximation), i.e., only consider motions of electrons, and assume nuclei are stationary. >[!Note] > Above $\sum_{i=1}^{N}a_{i}(H_{j,i}-ES_{{j,i}})=0$ is exactly the secular equation, and we solve the determinant. So the process is: - for $N$ basis set functions, - we have $N\times N$ secular determinant, i.e., $N^{2}$, with $H$, $S$, - by solving it, we have $N$ energies and $N$ (linear combined) wave functions. Some principles are better being remarked here, though will be discussed in other chapters in a more detailed way. - Aufbau principle: filling from the lowest to the highest. - Hund's rules: max spin multiplicity, consists with the lowest energy. - Pauli principle: no two electron has identical quantum number. And a typical illustration for LCAO forming MO is ![[Drawing 2023-10-02 23.04.23.excalidraw.svg]] We linear combined AOs to form the same number of MOs. ### Bond order Bond order is defined as $B_{0} =\frac{1}{2}(N_{el}^{bonding}-N_{el}^{antibonding})$ This $N_{el}$ is the number of electrons. In LCAO related community, people typically write $\beta=S_{ij}$, the overlap, and $\alpha = H_{ij}$, the interaction. This gives the band as $\alpha + 2\beta \cos(ka)$. ### Hückel theory Consider $\rm C_{3}H_{5}$, with $\rm C-C-C$ structure, we only consider $\pi$ bonds, then we can solve $\begin{vmatrix} \alpha-E & \beta & 0 \\ \beta & \alpha -E & \beta \\ 0 & \beta & \alpha -E \end{vmatrix}=0$ and have $E=\begin{cases} \alpha+\sqrt{2}\beta \\ \alpha \\ \alpha - \sqrt{2}\beta \end{cases}$ and $\begin{cases} a_{11}=\frac{1}{2} \\ a_{21}= \frac{\sqrt{2}}{2} \\ a_{31}=\frac{1}{2} \end{cases}$ This gives $\phi_{1}=\frac{1}{2}p_{1}+\frac{\sqrt{2}}{2}p_{2}+\frac{1}{2}p_{3}$, similarly we can get other sets of coefficients for other energies. >[!Notice] >The assumptions we have here are: >- only consider $\pi$ bond >- all AO has the same energy (i.e., all $\pi$ bond energy) >- only neighboring atoms have interactions >- no motion of nuclei (Born-Oppenheimer) >- ignore correlation - In the extended Hückel theory, we ignore the core electrons. Their changes as a function of environment was considered of no chemical result. - Each remaining valence orbital is represented by a so-called "Slater-type" orbital, having the same angular dependence as the solution of the Schrödinger equation. This leads to the [[Basis sets|STO basis set]]. >[!Info] >See more on [Linear combination of atomic orbitals](https://en.wikipedia.org/wiki/Linear_combination_of_atomic_orbitals), [Hückel method](https://en.wikipedia.org/wiki/H%C3%BCckel_method), [Molecular orbital theory](https://en.wikipedia.org/wiki/Molecular_orbital_theory.