Wave Functions for Atoms

To describe atoms with several electrons, one has to consider the interaction between the electrons, adding to the Hamiltonian a term of the form Ei . Despite this complication it is common to use an approximate wave function which is a product of hydrogen-like atomic orbitals. This is done by taking the orbitals in order of increasing energy and assigning no more than two electrons per orbital. 

The wave function, constructed from the atomic orbitals must be antisymmetric with respect to interchange of electrons in order to satisfy the Pauli exclusion principle, having different spin quantum numbers (a and J3) for two electrons which are in the same orbital. 

For example, the helium atom electron wave function can be written as 

This type of wave function, which is clearly antisymmetric with respect to exchange of electron 1 and 2, can be also written in a determinant form 

Such a wave function is known as a Slater determinant. In general, when we deal with antisymmetrized wave functions, we use a compact notation for the Slater determinant  

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Note that the lability principle is formulated first of all for transferable electrons and atoms. An increase in their lability leads as a rule to an increase in the overlapping of the wave functions. For atoms the latter means a decrease in the Franck-Condon barrier. 

This wave function describes a bonding orbital of the a type that arises from the combination of two Is wave functions for atoms 1 and 2. To make that point clear, the wave function could be written as... 

J. Gerratt, Adv. At. Mol. Phys. 7, 141 (1971). General Theory of Spin-Coupled Wave Functions for Atoms and Molecules. 

F. O. Ellison, and P. G. Lykos Generalized antisymmetrized wave functions for atoms and molecules. J. Chem. Phys. 24, 1106 (1956). 

The problem of calculating accurate ground-state wave functions for atoms, and to a lesser degree for molecules, is tractable with modern computers for relatively small numbers of electrons N. 

Note the language subtleties At least within the field of ab initio quantum chemistry, the term wave function is almost exclusively used for a many-electron wave function T. A one-electron wave function for atoms and molecules, however, goes by the name (atomic or molecular) orbital ip, not wave function. The exception proves the rule, though. 

Another motivation for studying H2+ in a magnetic field is related to fundamental questions How is the chemical bond affected by the external magnetic field What is the appropriate wave function for atoms and molecules in a magnetic field The simplest molecule, H2", is a good starting point for investigating these issues. 

The valence bond theory suggests that the bonding in metals occurs due to the overlap of localized electrons only (see Chapter 6 for a more in-depth description of valence bond theory). Because the electrons are considered to be localized, they are essentially pinned down in space, making it somewhat easier to calculate the electron wave function. This theory was developed to explain the waveform, or more precisely, the wave function of atoms. Wherever the wave functions for atoms overlap, this represents bond formation. When it was applied using hydrogen as a model, two new concepts regarding chemical bonding were developed — resonance and orbital hybridization. 

Like the sine function, the more complicated wave functions for atomic orbitals can also have phases. Consider, for example, the representations of the Is orbital in T FIGURE 9.38. Note that here we... 

M. H. Kalos, Stochastic wave function for atomic helium, /. Comp. Phys. 1,251-211 (1967). 

Like the sine function, the more complicated wave functions for atomic orbitals can also have phases. Consider, for example, the representations of the Is orbital in T Figure 9.38. Note that here we plot this orbital a bit differently from what is shown in Section 6.6. The origin is the point where the nucleus resides, and the wave function for the Is orbital extends from the origin out into space. The plot shows... 

LCAO (linear combination of atomic orbitals. Section 1.11) A mathematical method for arriving at wave functions for molecular obitals that involves adding or subtracting wave functions for atomic orbitals. 

Hartree set forth to calculate wave functions for atoms with many electrons soon after the discovery of the SE of the hydrogen atom. His idea was to write down the repulsive average potential for the electrons, which is easily done using electrostatic... 

Bunge, C. F. (1970). Electronic wave functions for atom. II. Some aspects of convergence of configuration interaction expansion for ground states of He isoelectronic series. Theoretica Chimica Acta, 16,126. 

Equation (2.109) gives the Einstein coefficient for a one-electron system if r = (x, y, z) is the vector from the nucleus to the electron and the u (x, y, z) are the one-electron wave functions. For atoms with N electrons all of which may contribute to the dipole moment, the matrix element has to be generalized to /N... 

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