(this follows simply from multiplying both sides of a Hartree–Fock equation by
ciand integrating, noting thatciis normalized) and the definition ofF^(Eq.5.36)
we get
ei¼Z
cið 1 ÞH^core
ð 1 Þcið 1 ÞdvþXnj¼ 1ð 2 Jijð 1 Þ#Kijð 1 ÞÞð 5 : 49 Þi.e.
ei¼HiicoreþXnj¼ 1ð 2 Jijð 1 Þ#Kijð 1 ÞÞð 5 : 50 Þ(theoperatorsJ^andK^in Eq.5.36have been transformed by integration into
theintegrals JandKin Eq.5.49). Equation5.50shows thateiis the energy of an
electron incisubject to interaction with all the other electrons in the molecule:
Hcoreii (p. 13) is the energy of the electron due only to its motion (kinetic energy) and
to the attraction of the nuclear core (electron–nucleus potential energy), while the
sum of 2J#Kterms represents the exchange-corrected (viaK) coulombic repulsion
(throughJ) energy resulting from the interaction of the electron with all the other
electrons in the molecule or atom [ 19 ].
In principle the equations5.47allow us to calculate the molecular orbitals
(MO’s)cand the energy levelse. We could start with “guesses” (possibly obtained
by intuition or analogy) of the MO’s (the zeroth approximation to the MOs) and use
these to construct the operatorF^(Eq.5.36), then allowF^to operate on the guesses
to yield energy levels (the first approximation to theei) and new, improved func-
tions (the first calculated approximations to theci). Using the improved functions in
F^and operating on these gives the second approximations to theciandei, and the
process is continued untilciandeino longer change (within preset limits), which
occurs when the smeared-out electrostatic field represented in Eq.5.17by∑∑(2J#K)
(cf. Fig.5.3) ceases to change appreciably – is consistent from one iteration cycle to
the next, i.e. is self-consistent. How do we know that iterationsimprovepsi and
epsilon? This is usually, but not invariably, the case [ 20 ]; in practice “initial guess”
solutions to the Hartree–Fock equations usually converge fairly smoothly to give
the best wavefunction and orbital energies (and thus total energy) that can be
obtained by the HF method from the particular kind of guess wavefunction (e.g.
basis set; Section 5.2.3.6.5).
To expand a bit on Dewar’s cautiousendorsementoftheSCFprocedure[ 20 ]
(“SCF calculations are by no means foolproof; ...Usually one finds a reasonably
rapid convergence to the required solution”): occasionally a wavefunction is
obtained that is not the best one available from the chosen basis set. This
phenomenon is calledwavefunction instability.Toseehowthiscouldhappen
note that the SCF method is an optimization procedure somewhat analogous to
geometry optimization (Section 2.4). Ingeometry optimizationweseekarelative
minimum or a transition state on a hypersurface in a mathematical energy versus
nuclear coordinates space defined byE¼f(nuclear coordinates); in wavefunction
5.2 The Basic Principles of the ab initio Method 195