exchange. Comparing the predictions of theory with the results of experiment
shows [ 7 ] that electronic wavefunctions are actually antisymmetric with respect
to exchange (such particles are called fermions, after the physicist Enrico Fermi;
particles like photons whose wavefunctions are exchange-symmetric are called
bosons, after the physicist S. Bose). Any rigorous attempt to approximate the wave-
functioncshould use an antisymmetric function of the coordinates of the electrons
1, 2,...n, but the Hartree product is symmetric rather than antisymmetric; for
example, if we approximate a helium atom wavefunction as the product of two
hydrogen atom 1sorbitals, then ifca¼ 1 s(x 1 ,y 1 ,z 1 )1s(x 2 ,y 2 ,z 2 ) andcb¼ 1 s(x 2 ,y 2 ,z 2 )
1s(x 1 ,y 1 ,z 1 ), thenca¼cb.
These defects of the Hartree SCF method were corrected by Fock (Section 4.3.4)
and by Slater^2 in 1930 [ 8 ], and Slater devised a simple way to construct a total
wavefunctioncfrom one-electron functions (i.e. orbitals) such thatcwill be
antisymmetric to electron switching. Hartree’s iterative, average-field approach
supplemented with electron spin and antisymmetry leads to the Hartree–Fock
equations.
5.2.3 The Hartree–Fock Equations.....................................
5.2.3.1 Slater Determinants
The Hartree wavefunction (above) is a product of one-electron functions called
orbitals, or, more precisely,spatialorbitals: these are functions of the usual space
coordinatesx,y,z. The Slater wavefunction is composed, not just of spatial orbitals,
but ofspin orbitals. A spin orbitalc(spin) is the product of a spatial orbital and a
spin function,aorb: The spin orbitals corresponding to a given spatial orbital are
cðspinaÞ¼cðspatialÞa¼cðx;y;zÞa
and cðspinbÞ¼cðspatialÞb¼cðx;y;zÞbAs the functionc(spatial) has as its variables the coordinatesx,y,z, so the spin
functionsaandbhave astheirvariables a spin coordinate, sometimes denotedx
(Greek letterkziorzi) oro(Greekomega). We know that a wavefunctioncfits in
with an operator and eigenvalues, say the energy operator and energy eigenvalues,
according to the equationH^c¼Ec. Analogously, the spin functionsaandbare
(^2) John Slater, born Oak Park, Illinois, 1900. Ph.D. Harvard, 1923. Professor of physics, Harvard,
1924–1930; MIT 1930–1966; University of Florida at Gainesville, 1966–1976. Author of 14
textbooks, contributed to solid-state physics and quantum chemistry, developed X-alpha method
(early density functional theory method). Died Sanibel Island, Florida, 1976.
5.2 The Basic Principles of the ab initio Method 181