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Quantum Confinement and Electronic Structure of Quantum Dots
Published in Vinod Kumar Khanna, Introductory Nanoelectronics, 2020
Hitherto, our attention has been focused on the top-down view of the band structure of crystals. Now we reverse our thinking. In the bottom-up approach, the nanocrystal is built up starting from individual atoms. Molecular orbital theory proposes that individual atomic orbitals combine to form molecular orbitals. The unification of atomic orbitals is accomplished by linearly conjoining the atomic orbitals. This approach is referred to as linear combination of atomic orbitals (LCAO). Suppose the atomic orbital of an atom A is described by the wave function ΨA while that of atom B is assigned the wave function ΨB. Let the electron clouds of the atoms A and B overlap. Then two molecular orbitals are formed: One molecular orbital is formed by the addition of wave functions of atoms A and B.Another molecular orbital is obtained by subtraction of these wave functions.
Quantum Theory, Energy Bands of Solids, Effective Mass, and Holes
Published in Jyoti Prasad Banerjee, Suranjana Banerjee, Physics of Semiconductors and Nanostructures, 2019
Jyoti Prasad Banerjee, Suranjana Banerjee
In the TB model, the crystal orbital, i.e., the wave function of electrons in the periodic field of a crystalline solid, is expressed as a linear superposition of atomic orbitals or the wave functions of electrons in free isolated atoms. This method is known as Linear Combination of Atomic Orbitals (LCAO).
Generation of reflected second harmonic: Basic theoretical concepts
Published in O.A. Aktsipetrov, I.M. Baranova, K.N. Evtyukhov, Second Order Non-linear Optics of Silicon and Silicon Nanostructures, 2018
O.A. Aktsipetrov, I.M. Baranova, K.N. Evtyukhov
As the basis functions we can consider the atomic wave functions for the valence electrons in the s-, p- and other states of the atom. For silicon, the basis can be represented by the wave functions of sp3-hybridized orbitals. In these cases, we can talk about the implementation in solid-state physics of the well-known (in chemistry) method of linear combinations of atomic orbitals (LCAO). Sometimes the basis functions are represented by the wave functions of binding and antibinding orbitals that are formed during merger of atoms in the molecule. The use of the method of molecular orbitals (MO) should also be considered.
Distributed Gaussian orbitals for molecular calculations: application to simple systems
Published in Molecular Physics, 2020
Stefano Battaglia, David Bouet, Alexis Lecoq, Stefano Evangelisti, Noelia Faginas-Lago, Thierry Leininger, Andrea Lombardi
In practice, therefore, a single-centre Gaussian expansion does not appear to be the most suitable choice to perform actual calculations on multi-atomic systems. In particular, the nuclear cusp of an s-type orbital of a given atom can hardly be described unless the expansion contains Gaussians with very large exponents that are centred on the nucleus of the atom itself. For this reason, a very common computational strategy is to expand the molecular orbitals (MOs) of the system on a set of multi-centred atomic orbitals (AOs) of Gaussian type, centred on each one of the atoms that belong to the molecule. This is the very well known Linear Combination of Atomic Orbitals (LCAO) strategy, first used by Linus Pauling to describe the H system, and in a more systematic way by Lennard-Jones, to describe the bonds of atoms belonging to the first main row of the periodic table. The reason why this choice is so effective is that the intra-molecular interactions are relatively weak, and therefore the different atoms maintain their individuality in a molecular system.
Double exponential transformation for computing three-center nuclear attraction integrals
Published in Molecular Physics, 2020
A common approach for calculating molecular orbitals (MOs) is to build each MO as a linear combination of atomic orbitals (LCAO), where atomic orbitals (AOs) are the solutions to the Schrodinger equation for the hydrogen atom or a hydrogen-like ion. This technique is often referred to as the LCAO-MO method. The complexity of the three-center nuclear attraction integrals, as well as that of other multi-center integrals, can be mitigated by strategically choosing the basis of atomic orbitals with which to perform the calculations. It has been shown that a suitable atomic orbital basis should satisfy two conditions for analytical solutions to the Schrödinger equation: exponential decay at infinity [2] and a cusp at the origin [3].