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In order to solve the equation of an electron in a spherical potential, Hartree first introduced atomic units to eliminate physical constants. Then he converted the Laplacian from Cartesian to spherical coordinates to show that the solution was a product of a radial function () / and a spherical harmonic with an angular quantum number , namely = (/) (,).
The presence of the sign in the Appleton–Hartree equation gives two separate solutions for the refractive index. [6] For propagation perpendicular to the magnetic field, i.e., , the '+' sign represents the "ordinary mode," and the '−' sign represents the "extraordinary mode."
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.
The origin of the Hartree–Fock method dates back to the end of the 1920s, soon after the discovery of the Schrödinger equation in 1926. Douglas Hartree's methods were guided by some earlier, semi-empirical methods of the early 1920s (by E. Fues, R. B. Lindsay, and himself) set in the old quantum theory of Bohr.
The Legendre ordinary differential equation is frequently encountered in physics and other technical fields. In particular, it occurs when solving Laplace's equation (and related partial differential equations) in spherical coordinates. Associated Legendre polynomials play a vital role in the definition of spherical harmonics.
Within Hartree–Fock theory, the wave function is approximated as a Slater determinant of spin-orbitals. For an open-shell system, the mean-field approach of Hartree–Fock theory gives rise to different equations for the α and β orbitals.
In quantum chemistry, Brillouin's theorem, proposed by the French physicist Léon Brillouin in 1934, relates to Hartree–Fock wavefunctions. Hartree–Fock, or the self-consistent field method, is a non-relativistic method of generating approximate wavefunctions for a many-bodied quantum system, based on the assumption that each electron is exposed to an average of the positions of all other ...
The Hartree–Fock method is also used in atomic physics and condensed matter physics as Density Functional Theory, DFT. The process of solving the Hartree–Fock equations can only be iterative, since these are in fact a Schrödinger equation in which the potential depends on the density, that is, precisely on the wavefunctions to be