Schr€odinger equation for helium into two 1-electron equations which, like the
hydrogen atom equation, could be solved exactly (for treatments of the hydrogen
and helium atoms see the appropriate sections of references 1 ). This problem arises
in any system with three or more interacting moving objects, whether subatomic
particles or planets. In fact themany-body problemis an old one even in classical
mechanics, going back to eighteenth century studies in celestial mechanics [ 5 ]. The
three-particle hydrogen molecule ion, HH(+, with two heavy particles and one light
one, can be solved “exactly” – but only within the Born–Oppenheimer approxima-
tion [ 6 ]. The impossibility of an analytic solution to polyelectronic systems
prompted Hartree’s approach to calculating wavefunctions and energy levels for
atoms.
Hartree’s method was to write a plausible approximate polyelectronic wavefunc-
tion (a “guess”) for an atom as the product of one-electron wavefunctions:
C 0 ¼c 0 ð 1 Þc 0 ð 2 Þc 0 ð 3 Þ...c 0 ðnÞð 5 : 7 ÞThis function is called a Hartree product. HereC 0 is a function of the coordi-
nates of all the electrons in the atom,c 0 (1) is a function of the coordinates of
electron 1,c 0 (2) is a function of the coordinates of electron 2, etc.; the one-electron
functionsc 0 (1),c 0 (2), etc. are called atomic orbitals (molecular orbitals if we were
dealing with a molecule). The initial guess,c 0 , is our zeroth approximation to the
true total wavefunctionc, zeroth because we have not yet started to refine it with
the Hartree process; it is based on the zeroth approximationsc 0 (1),c 0 (2), etc. To
apply the Hartree process we first solve for electron 1 aone-electronSchr€odinger
equation in which the electron–electron repulsion comes from electron 1 and an
average, smeared-out electrostatic field calculated fromc 0 (2),c 0 (3),...,c 0 (n),
due to all theotherelectrons. The only moving particle in this equation is electron 1.
–1 / 2∇ 22–1 / 2∇ 12kinetic energy potential energy from attraction, stabilizing
potential energy from repulsion, destabilizing++electron 1electron 21/r 12Z = 2- Z/r 1
- Z/r 2
Fig. 5.1 The terms in the helium atom Hamiltonian,H^¼#^12 r^21 #^12 r^22 #Zr 1 #Zr 2 þr^112
5.2 The Basic Principles of the ab initio Method 179