The transformed Fock matrixF^0 satisfiesF^0 ¼C^0 eC^0 #^1 ð 5 : 69 Þ(cf.Eq. 4.104). The overlap matrixSis readily calculated, so ifFcan be
calculated it can be transformed toF^0 , which can be diagonalized to giveC^0
ande, which latter yields the MO energy levelsei.- Transformation ofC^0 toC(Eq. 4.100) gives the coefficientscsiin the expansion
of the MO’scin terms of basis functionsf:
C¼S#^1 =^2 C^0 ð 5 : 70 ÞEquations5.63–5.66show that to calculateF, i.e. each of the matrix elementsF,
we need the wavefunctionsci, because J^and K^, the coulomb and exchange
operators (Eq. (5.65) and Eq. (5.66)), are defined in terms of thec’s. It looks like
we are faced with a dilemma: the point of calculatingFis to get (besides thee’s) the
c’s (thec’s with the chosen basis set {f} make up thec’s), but to getFwe need the
c’s. The way out of this is to start with a set of approximatec’s, e.g. from an
extended H€uckel calculation, which needs noc’s to begin with because the
extended H€uckel “Fock” matrix elements are calculated from experimental ioniza-
tion potentials (Section 4.4.1). Thesec’s, theinitial guess, are used with the basis
functionsfto in effect (Section 5.2.3.6.3) calculate initial MO wavefunctionsc,
which are used to calculate theFelementsFrs. Transformation ofFtoF^0 and
diagonalization gives a “first- cycle” set ofe’s and (after transformation ofC^0 toC)
a first-cycle set ofc’s. Thesec’s are used to calculate newFrs, i.e. a newF, and this
gives a second-cycle set ofe’s andc’s. The process is continued until things–thee’s,
thec’s (as the density matrix – Section 5.2.3.6.3), the energy, or, more usually, some
combination of these – stop changing within certain pre-defined limits, i.e. until the
cycles have essentially converged on the limitinge’s andc’s. Typically, about ten
cycles are needed to achieve convergence. It is because the operatorF^depends on
the functionsfon which it acts, making an iterative approach necessary, that the
Roothaan–Hall equations, like the Hartree–Fock equations, are called pseudoei-
genvalue (see end ofSection 5.2.3.4and start ofSection 5.2.3.5).
Now, in the Hartree–Fock method (the Roothaan–Hall equations represent one
implementation of the Hartree–Fock method) each electron moves in anaverage
fielddue to all the other electrons (see the discussion in connection with Fig.5.3,
Section 5.2.3.2). As thec’s are refined the MO wavefunctions improve and so
this average field that each electron feels improves (sinceJandK, although not
explicitly calculated (Section 5.2.3.6.3) improve with thec’s ). When thec’s no
longer change the field represented by this last set ofc’s is (practically) the same
as that of the previous cycle, i.e. the two fields are “consistent” with one another,
i.e. “self-consistent”. This Roothaan–Hall–Hartree–Fock iterative process (initial
guess, firstF, first-cyclec’s, secondF, second-cyclec’s, thirdF, etc.) is therefore a
self-consistent-fieldprocedure orSCFprocedure, like the Hartree–Fock procedure
5.2 The Basic Principles of the ab initio Method 205