Wave function multielectron atom

The relationship between alternative separable solutions of the Coulomb problem in momentum space is exploited in order to obtain hydrogenic orbitals which are of interest for Sturmian expansions of use in atomic and molecular structure calculations and for the description of atoms in fields. In view of their usefulness in problems where a direction in space is privileged, as when atoms are in an electric or magnetic field, we refer to these sets as to the Stark and Zeeman bases, as an alternative to the usual spherical basis, set. Fock s projection onto the surface of a sphere in the four dimensional hyperspace allows us to establish the connections of the momentum space wave functions with hyperspherical harmonics. Its generalization to higher spaces permits to build up multielectronic and multicenter orbitals. 

Now the parity of a multielectron, ionic, or atomic wave function is given by TT ( — l)/fc, where lk (the angular-momentum quantum number)... 

In atoms with more than one electron, wave functions should include the coordinates of each particle, and a new term representing the electrostatic interactions between electrons. Even for the case of only two electrons, such a wave equation is so complex that it has never been solved exactly. To analyse multielectron atoms some approximations have to be made. The most practical one is to assume that the electron considered moves in an electrical potential that is a combination of all other electrons and the nucleus, and that this potential has spherical symmetry. This approximation has proven very useful, as it allows a description of energy states in a similar manner to that employed for the H atom by using a comparable set of four quantum numbers. An important, additional condition appears no two electrons can have the same set of quantum numbers in other words, no more than one electron can occupy the same energy state. This is Pauli s exclusion principle. 

Spin orbitals are products of spatial and spin wave-functions, but correct antisymmetric forms of wavefunctions for multielectron atoms are sums and differences of spatial wavefunctions. Explain why acceptable antisymmetric wave-functions are sums and differences (that is, combinations) instead of products of spatial wavefunctions. 

Note that the complete wavefunction as written in Eq. (2.47) changes sign if the labels of the electrons (1 and 2) are interchanged. W. PauU pointed out that the wavefunctions of all multielectronic systems have this property. The overall wavefunction invariably is antisymmetric for an interchange of the coordinates (both positional and spin) of any two electrons. This assertion rests on experimental measurements of atomic and molecular absorption spectra absorption bands predicted on the basis of antisymmetric electrOTiic wavefunctirais are seen experimentally, whereas bands predicted on the basis of symmetric electronic wave-functions are not observed. Its most important implication is the Pauli exclusion principle, which says that a given spatial wavefunction can hold no more than two electrons. This follows if an electron can be described completely by specifying its spatial and spin wavefunctions and electrons have only two possible spin wave-functions (a and fi). 

In addition to the conditions for the electronic structures of multielectron atoms established by the monoelectronic wave functions and their relative energies mentioned above, other restrictions should also be considered. One of them is the Pauli principle stating that no two electrons can have the same quantum numbers. Thus one orbital can contain a maximum of two electrons provided they have different spin quantum numbers. Other practical rules or restrictions refer to the influence of interelectronic interactions on the electronic structures established by Hund s rules. The electrons with the same n and / values will occupy first orbitals with different nti and the same rris (paired spins). 

The electronic wave function for a multielectron atom must he antisymmetric. That is, the wave function must change sign if the coordinates of two electrons are exchanged. 

There are several commonly used approximation schemes that can be applied to the electronic states of multielectron atoms. The first approximation scheme was the variation method, in which a variation trial function is chosen to minimize the approximate ground-state energy calculated with it. A simple orbital variation trial function was found to correspond to a reduced nuclear charge in the helium atom. This result was interpreted to mean that each electron in a helium atom shields the other electron from the full charge of the nucleus. A better variation trial function includes electron correlation, a dependence of the wave function on the electron lectrcm distance. ... 

If orbital wave functions are used to calculate transition dipole moments, the following selection rules are obtained for multielectron atoms ... 

The quantum mechanical wave function for a multielectron atom can be approximated as a superposition of orbitals, each bearing some resemblance to those describing the quantum states of the hydrogen atom. Each orbital in a multielectron atom describes how a single electron behaves in the field of a nucleus under the average influence of all the other electrons. 

Multielectron Atoms— The wave function of a multielectron atom can be approximated as a sujjerposition of orbitals, each of which is qualitatively similar to a hydro-gen-like orbital. In multielectron atoms, orbitals with different values of are not degenerate (Fig. 8-34). The loss of degeneracy within a principal shell is explained in terms of the different effective nuclear charge, Z, experienced by electrons in different subshells. 

Realistic values of can be obtained from an analysis of the wave functions of multielectron atoms, as described in Are You Wondering 9-1. Values of Zgg for the valence electrons for the first 36 elements are shown in Figure 9-7. The following points can be established by careful examination of these values. 

These estimates come from an analysis of the wave functions of multielectron atoms. An exact solution of the Schrodinger equation can be obtained for the H atom, but for multielectron atoms, only approximate solutions are possible. The principle of the calculation is to assume each electron in the atom occupies an orbiM much like those of the hydrogen atom. However, the functional form of the orbital is based on another assumption that the electron moves in an effective or average field dictated by all the other electrons. With this assumption, the complicated multielectron Schrodinger equation is converted into a set of simultaneous equations—one for each electron. Each equation contains the unknown effective field and the unknown functional form of the orbital for the electron. The approach to solving such a set of equations is to guess at the functional forms of the orbitals, calculate an average... 

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