'molecules are complete now'

ch08/note02.md

@@ -31,11 +31,7 @@ constructed, one with + sign and the other with $-$ sign. Normalization of the w
 
 $${\int{\psi_\pm^*\psi_\pm d\tau} = 1}$$
 
-:::{figure-md} markdown-fig
-<img src="./images/b_vs_u.png" alt="electro" class="bg-primary mb-1" width="400px">
 
-Bonding vs anti-bonding wavefunctions (Molecular Orbitals). Show are wavefunctions and probability densities (squares of wavefunctions)
-:::
 
 ### Overlap integral
 
@@ -65,10 +61,15 @@ Note that the antibonding orbital has \underline{zero} electron density between
 
 ### Bonding vs antibonding orbitals
 
-- Recall that the square of the wavefunction gives the electron density. In the left hand side figure (the + wavefunction), the electron density is amplified between the nuclei whereas in the $-$ wavefunction the opposite happens. 
+:::{figure-md} markdown-fig
+<img src="./images/b_vs_u.png" alt="electro" class="bg-primary mb-1" width="400px">
+
+Bonding vs anti-bonding wavefunctions (Molecular Orbitals). Show are wavefunctions and probability densities (squares of wavefunctions)
+:::
 
 - The main feature of a chemical bond is the increased electron  density between the nuclei. This identifies the + wavefunction as a  **bonding orbital** and $-$ as an **antibonding orbital**.
 
+
 - When a molecule has a center of symmetry (here at the half-way between the 
 nuclei), the wavefunction may or may not change sign when it is inverted
 through the center of symmetry. If the origin is placed at the center of 
@@ -78,7 +79,7 @@ $g$ (even parity) and for $\psi(x, y, z) = -\psi(-x, -y, -z)$ we have $u$ label
 (odd parity). 
 -  According to this notation the $g$ symmetry orbital is the 
 bonding orbital and the $u$ symmetry corresponds to the antibonding orbital.
-Later we will see that this is \underline{reversed} for $\pi$-orbitals!
+Later we will see that this is {reversed} for $\pi$-orbitals!
 
 - The overlap integral $S(R)$ can be evaluated analytically (derivation not shown):
 

ch08/note03.md

@@ -17,7 +17,7 @@ used to denote the integrals occurring in variational treatment of the problem.
 
 ### Variational solution 
 
-- To minimize the energy expectation value in Eq. (\ref{eq11.14}) with respect to $c_1$ and $c_2$, we have to calculate the partial derivatives of energy with respect to these parameters:
+- To minimize the energy expectation value  with respect to $c_1$ and $c_2$, we have to calculate the partial derivatives of energy with respect to these parameters:
 
 
 $${E\times (c_1^2 + 2c_1c_2S + c_2^2) = c_1^2H_{AA} + 2c_1c_2H_{AB} + c_2^2H_{BB}}$$
@@ -70,6 +70,11 @@ $${E_g(R) = E_{1s} + \frac{J(R) + K(R)}{1 + S(R)}}$$
 
 $${E_u(R) = E_{1s} + \frac{J(R) - K(R)}{1 - S(R)}}$$
 
+:::{figure-md} markdown-fig
+<img src="./images/Energies.png" alt="electro" class="bg-primary mb-1" width="400px">
+
+Energies of bonding and antibonding orbitals
+:::
 
 - Since energy is a relative quantity, it can be expressed relative to separated
 nuclei:
@@ -96,11 +101,28 @@ $2p_y$, $2p_z$ etc. orbitals.
 - It is a common practice to represent the molecular orbitals by molecular 
 orbital (MO) diagrams. The formation of bonding and antibonding orbitals can be visualized as follows:
 
+:::{figure-md} markdown-fig
+<img src="./images/h2plus_diag.png" alt="electro" class="bg-primary mb-1" width="400px">
+
+Molecular Orbital diagram
+:::
+
 - $\sigma$ orbitals When two $s$ or $p_z$ orbitals interact,
 a $\sigma$ molecular orbital is formed. The notation $\sigma$ specifies
 the amount of angular momentum about the molecular axis (for $\sigma$, $\lambda = 0$ with $L_z = \pm\lambda\hbar$). In many-electron systems, both bonding and antibonding $\sigma$  orbitals can each hold a maximum of two electrons. Antibonding orbitals are often  denoted by *.
 
+:::{figure-md} markdown-fig
+<img src="./images/MO_variety.png" alt="electro" class="bg-primary mb-1" width="400px">
+
+MOs for homonuclear molecules have distinct symmetry
+:::
+
 - $\pi$ orbitals. When two $p_{x,y}$ orbitals interact, a $\pi$ molecular orbital forms. $\pi$-orbitals are doubly degenerate: $\pi_{+1}$ and $\pi_{-1}$ (or alternatively $\pi_x$ and $\pi_y$), where the $+1/-1$ refer to the eigenvalue of the $L_z$ operator ($\lambda = \pm1$). In many-electron systems a bonding $\pi$-orbital can therefore hold a maximum of 4 electrons (i.e. both $\pi_{+1}$ and $\pi_{-1}$ each can hold two electrons). The same holds for the antibonding $\pi$ orbitals. Note that only the atomic orbitals of the same symmetry mix to form molecular orbitals (for example, $p_z - p_z$, $p_x -  p_x$ and $p_y - p_y$). When atomic $d$ orbitals mix to form molecular orbitals, $\sigma (\lambda = 0)$, $\pi (\lambda = \pm 1)$ and $\delta (\lambda = \pm 2)$ MOs form. 
 
 - Excited state energies of $H_2^+$ resulting from a calculation employing an extended basis set (e.g. more terms in the LCAO) are shown on the left below. The MO energy diagram, which includes the higher energy molecular orbitals, is shown on the right hand side. Note that the energy order of the MOs depends on the molecule.
 
+:::{figure-md} markdown-fig
+<img src="./images/MO_summary.png" alt="electro" class="bg-primary mb-1" width="400px">
+
+Mo diagram of homonuclear molecules follows similiar pattern with alternating bonding and anti-bonding MOs. 
+:::