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The characteristic roots (roots of the characteristic equation) also provide qualitative information about the behavior of the variable whose evolution is described by the dynamic equation. For a differential equation parameterized on time, the variable's evolution is stable if and only if the real part of each root is negative.
Stereochemistry, a subdiscipline of chemistry, studies the spatial arrangement of atoms that form the structure of molecules and their manipulation. [1] The study of stereochemistry focuses on the relationships between stereoisomers, which are defined as having the same molecular formula and sequence of bonded atoms (constitution) but differing in the geometric positioning of the atoms in space.
In stereochemistry, a chiral auxiliary is a stereogenic group or unit that is temporarily incorporated into an organic compound in order to control the stereochemical outcome of the synthesis. [1] [2] The chirality present in the auxiliary can bias the stereoselectivity of one or more subsequent reactions. The auxiliary can then be typically ...
There are typically three mathematical forms for the radial functions R(r) which can be chosen as a starting point for the calculation of the properties of atoms and molecules with many electrons: The hydrogen-like orbitals are derived from the exact solutions of the Schrödinger equation for one electron and a nucleus, for a hydrogen-like atom.
In crystallography, crystal structure is a description of ordered arrangement of atoms, ions, or molecules in a crystalline material. [1] Ordered structures occur from intrinsic nature of constituent particles to form symmetric patterns that repeat along the principal directions of three-dimensional space in matter.
In theoretical and computational chemistry, a basis set is a set of functions (called basis functions) that is used to represent the electronic wave function in the Hartree–Fock method or density-functional theory in order to turn the partial differential equations of the model into algebraic equations suitable for efficient implementation on a computer.
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It was later extended to conjugated molecules such as pyridine, pyrrole and furan that contain atoms other than carbon and hydrogen (heteroatoms). [4] A more dramatic extension of the method to include σ-electrons, known as the extended Hückel method (EHM), was developed by Roald Hoffmann .