Primer for Point and Space Groups pp 136-159 | Cite as

# Atoms in Crystals and Correlation Diagrams

Chapter

## Abstract

Consider an atom with which has the eigenstates

*Z*protons in its nucleus and with*Z*outer electrons (the*atomic number*of the atom is*Z*). In the central-field approximation, atomic electrons are assumed to be independent of one another. The Hamiltonian of the ith atomic electron is given by$${H_{i}}({r_{i}}) = \frac{{p_{i}^{2}}}{{2m}} - \frac{{Z{e^{2}}}}{{{r_{i}}}}$$

(6.1a)

$${\Psi _{nlm}}({r_i}) = {R_{nl}}({r_i}){Y_{lm}}({\theta _i},{\phi _i}){\xi _i}$$

(6.1b)

## Keywords

Point Group Orbital Angular Momentum Young Diagram Principal Quantum Number Symmetry Operation
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## References

- 1.See, for example, R.L. Liboff,
*Introductory Quantum Mechanics*, 4th ed. Addison-Wesley, San Francisco, CA (2002), Table 10.3.Google Scholar - 2.See, for instance, D.M. Bishop,
*Group Theory and Chemistry*(see Bibliography).Google Scholar - 3.For further discussion see, A.F. Cotton,
*Chemical Applications of Group Theory*, 3rd ed.,*ipid*(see Bibliography); D.M. Bishop,*Group Theory and Chemistry*(see Bibliography); B.N. Figgis,*Introduction to Ligand Fields*(see Bibliography).Google Scholar - 4.For further discussion, see J.D. Jackson,
*Classical Electrodynamics*, 3rd ed., Wiley, New York (1999).MATHGoogle Scholar - 5.A similar situation occurs in quantum mechanics. The coupled spin states of three electrons are combinations of the tensor forms
*F*_{ijk}*= α*_{i}*β*_{j}*γ*_{k}where*α*_{i}is, say, the spin state of the ‘*α*’ electron and (*i*,*j, k*)*=*1,2. So*F*_{ijk}is a tensor with*r =*3 and*n =*2. It is known that antisymmetric coupled spin states of three or more electrons do not exist. [See, R.L. Liboff,*Am. J. Physics***52**, 561 (1984).]ADSCrossRefGoogle Scholar

## Copyright information

© Springer-Verlag New York, Inc. 2004