Ferromagnetism and the Band Theory

Two rival methods of handling molecular problems, the Heitler-London and the molecular orbital methods, have been used for many years, and they have their counterparts in magnetism in the Heisenberg and the energy-band methods. The first part of this paper shows that the Heitler-London and Heisenberg methods, as usually applied, are not appropriate for problems containing more than a few electrons. The reason is the lack of orthogonality of the one-electron orbitals of atomic type. This involves the presence of overlap integrals which, if included in the calculation, make the method essentially divergent and unsuitable for numerical calculation if the product of the number of electrons in the system, times an overlap integral between nearest neighbors, is large compared to unity, which in practice confines the application to problems of a few electrons. Most applications of the method, including all applications of the Heisenberg method to ferromagnetism and antiferromagnetism, disregard many integrals which are not as a matter of fact negligible, and for this reason cannot be regarded as valid.

Van Vleck many years ago tried to overcome this difficulty, but his results are only partially satisfactory on account of lack of generality. The straightforward way to overcome it is to use really orthogonal one-electron orbitals, either molecular orbitals or Wannier functions, and the remainder of the paper considers how the familiar facts of ferromagnetism are to be explained on this basis. If we use a single determinantal wave function composed of energy-band orbitals, we have the familiar energy-band or collective electron theory of ferromagnetism. This simple method may well be fairly accurate at small internuclear distances; it predicts correctly that ferromagnetism must be impossible when the atoms are close enough together to broaden the energy bands greatly. For large internuclear distances, however, it leads to quite wrong limiting behavior, since it gives a wave function involving a considerable contribution of ionic states. To eliminate these ionic states, at large internuclear distances, we must make linear combinations of different determinantal functions, corresponding to different assignments of electrons to orbitals. The results of such a calculation are sketched.

To treat magnetic problems properly, we must make such calculations for different total spins, calculate the energy of the states of different magnetizations as a function of internuclear distance, and see which states lie lower, the magnetic or nonmagnetic. At infinite separation, the energy will be independent of total magnetization, since in this limit the spins of different atoms can be oriented in any arbitrary way without affecting the energy. At very small distances, the elementary band theory shows that the nonmagnetic state must lie below the magnetized state, so that ferromagnetism is impossible. At intermediate distances, the elementary theory can show only that ferromagnetism is possible, not that it is necessary; to find whether it actually exists, we should really have to carry out a calculation of the type described. If we find ferromagnetism, we shall necessarily find that the energy difference between nonferromagnetic and ferromagnetic states was zero at infinite internuclear distance, increased as the internuclear distance decreased, went through a maximum, and then went to zero and changed sign. This behavior is often postulated for the Heisenberg exchange integral, although we do not feel that this integral has the direct theoretical meaning often ascribed to it.

These problems are closely related to the correlation energy, which is discussed. We do not feel that the familiar Wigner-Seitz calculation of correlation energy is very accurate; the correlation energy is in fact a function of magnetization, decreasing numerically with decreasing internuclear distance, and its interpretation cannot be given properly except by considering its intimate connection with magnetization. The net result of the present discussion is that to give a proper account of ferromagnetism, we are forced to use orthogonal orbitals and, therefore, must make close connection with the band theory. We must, however, carry our theory further than has usually been done, though the prediction of the elementary band theory, i.e., that ferromagnetism is impossible for broad bands and can exist only for narrow bands and electronic wave functions which overlap only slightly, is verified by the present more accurate approach.

As an illustration of the general method of calculating magnetic energies described in this paper, we may mention the recent calculation of Meckler, not yet published, on the oxygen molecule. This has been carried out by setting up many determinantal wave functions formed from orthogonal atomic orbitals made out of the original nonorthogonal orbitals. The secular equations involving these determinantal functions were solved by use of a digital computer. The resulting energy levels and wave functions reduce to the proper type to represent the ${}_{}{}^3{}_{}{}^{}P$ ground state of the oxygen atom at infinite internuclear distance. Meckler's energy for the triplet ground state is very accurate; that for the next singlet excited state is quite good. The energy separation between singlet and triplet shows the behavior expected for the energy difference between a ferromagnetic and nonferromagnetic state, being zero at infinite distance, then increasing, going to a maximum, then decreasing again. As far as the writer is aware, this is the first time when such a behavior has been found as a result of straightforward calculation from quantum mechanics.