By Gaston Bachelard
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Additional info for Le rationalisme appliqué, 3e édition
On the other hand, in crystalline solids, there are ∼1023 atoms in a 3D lattice structure. LCAO theory based on linear algebra is no longer useful. 6 Schematic illustration of energy level in a multiatom molecule with chain structure, showing that electrons are mostly occupying the energy level below ␣ and that a gap exists between HOMO and LUMO. ) 36 Fundamentals and Applications of Nanomaterials imagine trying to use it to solve such a big and sparse matrix, not to mention that the lattice is now 3D so we might need a tensor instead of a matrix.
8-eV semiconductor, it can only absorb and emit the red light spectrum. However, when we have quantum dots of nanocrystals ranging from 30 to 200 nm, the entire visible light spectrum can be covered. 8 eV on bulk materials to almost 4 eV at the 30-nm size of quantum dots, which leads to a wide range of light-emitting and -absorbing spectra. 11. 11 Energy bandgap increase with decrease of quantum well thickness, indicating that the quantum effect is significant at small feature size. 2 Bottom-Up Method for Carbon-Based Nanocrystals The bottom-up approach uses the same LCAO theory that we used when we studied molecules, but extends it to a much larger number of atoms.
7, when Li atoms are isolated, the energy levels for the 1s and 2s orbitals are E 1s and E 2s , respectively. When we bring N atoms together to form a solid, the 2s level will split to N finite energy levels. From discussions based on LCAO theory in previous sections, all of those N energy levels now locate between ␣ + 2 ␤ and ␣ − 2 ␤ . The maximum separation is ∼4 ␤ on the order of 5 to 10 eV. On the other hand, N is such a large number (on the order of 1023 ) that the discrete energy levels in the range of ␣ + 2 ␤ and ␣ − 2 ␤ can be considered as a continuous energy band.