Search
- Filter Results
- Location
- Classification
- Include attachments
- https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Chemical_Group_Theory_(Miller)/01%3A_Translational_Symmetry/1.03%3A_Periodic_Boundary_ConditionsNow, the electron density ρ(r) of a crystal has the full periodicity of the lattice, ρ(r+Tn1n2)=ρ(r) for all integers n 1 a...Now, the electron density ρ(r) of a crystal has the full periodicity of the lattice, ρ(r+Tn1n2)=ρ(r) for all integers n 1 and n 2 . On the other hand, the electronic wavefunctions (crystal orbitals) do not, but they must obey the periodic boundary conditions, i.e.,
- https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Chemical_Group_Theory_(Miller)/00%3A_IntroductionGroup theory provides the mathematical framework for applying the symmetry of a chemical structure to characterize its various physical states and properties. Therefore, this section of the course is ...Group theory provides the mathematical framework for applying the symmetry of a chemical structure to characterize its various physical states and properties. Therefore, this section of the course is divided into two subsections:
- https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Chemical_Group_Theory_(Miller)/00%3A_Front_Matter
- https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Chemical_Group_Theory_(Miller)/05%3A_Blochs_Theorem/5.01%3A_Introduction_to_Blochs_Theorem\[\left( 1 \middle| \boldsymbol{T} \right)\psi_{n\boldsymbol{k}}\left( \boldsymbol{r} \right) = \psi_{n\boldsymbol{k}}\left( \left( 1 \middle| \boldsymbol{T} \right)^{- 1}\boldsymbol{r} \right) = \psi...(1|T)ψnk(r)=ψnk((1|T)−1r)=ψnk((1|−T)r)=ψnk(r−T)
- https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Chemical_Group_Theory_(Miller)/05%3A_Blochs_Theorem/5.05%3A_Vibrational_States_and_Phonon_Dispersion_CurvesVn(ma)=12Kvib[(un(ma)−un((m+1)a))2+(un(ma)−un((m−1)a))2] In the...Vn(ma)=12Kvib[(un(ma)−un((m+1)a))2+(un(ma)−un((m−1)a))2] In the figures illustrating each phonon mode, the arrows signify the direction and “size” of the atomic displacements, which are, in fact, oscillations about the equilibrium positions of the atoms.
- https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Chemical_Group_Theory_(Miller)/05%3A_Blochs_Theorem/5.02%3A_The_First_Brillouin_ZoneNevertheless, since ψn(−k)(r) and ψ∗nk(r) are basis functions for the same IR \(\Gamma^{( - \bold...Nevertheless, since ψn(−k)(r) and ψ∗nk(r) are basis functions for the same IR Γ(−k), ψn(−k)(r) and ψnk(r) are degenerate eigenfunctions of the Hermitian Hamiltonian operator, so that En(−k)=En(k).
- https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Chemical_Group_Theory_(Miller)/05%3A_Blochs_Theorem
- https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Chemical_Group_Theory_(Miller)/00%3A_Front_Matter/01%3A_TitlePageChemical Group Theory Modules
- https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Chemical_Group_Theory_(Miller)/03%3A_Space_Groups/3.03%3A_Glide_Reflections\(\left( R \middle| \boldsymbol{\tau} \right) = \left( 1 \middle| \frac{\boldsymbol{a} + \boldsymbol{b} + \boldsymbol{c}}{4} \right)\left( {\overline{4}}_{\boldsymbol{c}} \middle| \boldsymbol{0} \righ...\(\left( R \middle| \boldsymbol{\tau} \right) = \left( 1 \middle| \frac{\boldsymbol{a} + \boldsymbol{b} + \boldsymbol{c}}{4} \right)\left( {\overline{4}}_{\boldsymbol{c}} \middle| \boldsymbol{0} \right)\left( 1 \middle| \frac{- \boldsymbol{a} - \boldsymbol{b} - \boldsymbol{c}}{4} \right) = \left( 1 \middle| \frac{\boldsymbol{a} + \boldsymbol{b} + \boldsymbol{c}}{4} \right)\left( {\overline{4}}_{\boldsymbol{c}} \middle| \frac{\boldsymbol{b} - \boldsymbol{a} + \boldsymbol{c}}{4} \right) = \left( …
- https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Chemical_Group_Theory_(Miller)/03%3A_Space_Groups/3.07%3A_Volume_A_of_the_International_Tables_of_Crystallography\(\left( \left( m_{010} \middle| \alpha_{2}\ 0\ \gamma_{2} \right)\left( 2_{010} \middle| 0\ \beta_{1}\ 0 \right) \right)^{2} = \left( \overline{1} \middle| \alpha_{2}\ \left( - \beta_{1} \right)\ \ga...((m010|α2 0 γ2)(2010|0 β1 0))2=(¯1|α2 (−β1) γ2)2=(1|0 0 0): no other constraints.
- https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Chemical_Group_Theory_(Miller)/03%3A_Space_GroupsSpace groups identify the possible ways to describe the rotational and translational symmetry of crystalline structures in real space. As we have seen, these aspects of 3-d crystalline symmetry are se...Space groups identify the possible ways to describe the rotational and translational symmetry of crystalline structures in real space. As we have seen, these aspects of 3-d crystalline symmetry are separately described by 32 crystallographic point groups and 14 Bravais lattices. For any space group, these two types of symmetry must be compatible with each other.
