Algebraic solutions electronic orbital

Under the first assumption, each electron moves as an independent particle and is described by a one-electron orbital similar to those of the hydrogen atom. The wave function for the atom then becomes a product of these one-electron orbitals, which we denote P (r,). For example, the wave function for lithium (Li) has the form i/ atom = Pa ri) Pp r2) Py r3). This product form is called the orbital approximation for atoms. The second and third assumptions in effect convert the exact Schrodinger equation for the atom into a set of simultaneous equations for the unknown effective field and the unknown one-electron orbitals. These equations must be solved by iteration until a self-consistent solution is obtained. (In spirit, this approach is identical to the solution of complicated algebraic equations by the method of iteration described in Appendix C.) Like any other method for solving the Schrodinger equation, Hartree s method produces two principal results energy levels and orbitals. 

The algebraic approximation results in the domain of the operator being restricted to a finite-dimensional subspace S of Hilbert space. For an AT-electron system, the algebraic approximation may be implemented by defining a suitable orthonormal basis set of M > N) one-electron spin orbitals (most often solutions of the matrix Hartree-Fock equations) and then constructing all unique IV-electron determinants. The number of unique determinants that can be formed is given by... 

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