F^c 1 ð 1 Þ¼e 1 c 1 ð 1 Þ
F^c 2 ð 1 Þ¼e 2 c 2 ð 1 Þ
F^c 3 ð 1 Þ¼e 3 c 3 ð 1 Þ
...F^cnð 1 Þ¼encnð 1 Þ$ð 5 : 47 ÞThese equations (5.47) are the Hartree–Fock equations; the matrix form is
Eq.5.44or Eq.5.46. By analogy with the Schr€odinger equationH^c¼Ec, we
see that they show that the Fock operator acting on a one-electron wavefunction (an
atomic or molecular orbital) generates an energy value times the wavefunction.
Thus the Lagrangian multipliersliiturned out to be (with the#1/2 factor) the
energy values associated with the orbitalsci. Unlike the Schr€odinger equation the
Hartree–Fock equations are not quite eigenvalue equations (although they are
closer to this ideal than is Eq.5.35), because inF^ci¼kcithe Fock operatorF^is
itself dependent onci; in a true eigenvalue equation the operator can be written
down without reference to the function on which it acts. The significance of the
Hartree–Fock equations is discussed in the next section.
5.2.3.5 The Meaning of the Hartree–Fock Equations
The Hartree–Fock equations (5.47) (in matrix form Eqs.5.44and5.46) arepseu-
doeigenvalueequations asserting that the Fock operatorF^acts on a wavefunctionci
to generate an energy valueei, timesci.Pseudoeigenvaluebecause, as stated above,
in a true eigenvalue equation the operator is not dependent on the function on which
it acts; in the Hartree–Fock equationsF^depends oncbecause (Eq.5.36) the
operator containsJ^andK^, which in turn depend (Eqs.5.29and5.30) onc. Each
of the equations in the set (5.47) is for a single electron (“electron 1” is indicated,
but any ordinal number could be used), so the Hartree–Fock operatorF^is a one-
electron operator, and each spatial molecular orbitalcis a one-electron function (of
the coordinates of the electron). Two electrons can be placed in a spatial orbital
because thefulldescription of each of these electrons requires aspin functionaorb
(Section 5.2.3.1) and each electron “moves in” a different spin orbital. The result is
that the two electrons in the spatial orbitalcdo not have all four quantum numbers
the same (for an atomic 1sorbital, for example, one electron has quantum numbers
n¼1,l¼0,m¼0 ands¼1/2, while the other hasn¼1,l¼0,m¼0 ands¼#1/2),
and so the Pauli exclusion principle is not violated.
Thefunctionscare the spatial molecular (or atomic) orbitals or wavefunctions
that (along with the spin functions) make up the overall or total molecular (or
atomic) wavefunctionc, which can be written as a Slater determinant (Eq.5.12).
Concerning theenergiesei, from the fact that
ei¼Z
ciF^cidv ð 5 : 48 Þ194 5 Ab initio Calculations