coordinates), depending on the program. In practice a virtual molecule would
likely be created with an interactive model-building program (usually by click-
ing together groups and atoms) which would then supply the EHM program with
either Cartesian or internal coordinates.- The EHM program calculates the overlap integralsSand assembles the overlap
matrixS. - The program calculates the Fock matrix elementsHij¼ fijH^jfj
DE
(Eqs.4.91
and4.92) using stored values of ionization energiesI, the overlap integralsS,
and the proportionality constantKof that particular program. The matrix ele-
ments are assembled into the Fock matrixH.- The overlap matrix is diagonalized to giveP,DandP"^1 (Eq.4.105) andD"1/2is
then calculated by finding the inverse square roots of the diagonal elements ofD.
The orthogonalizing matrixS"1/2is then calculated fromP,D"1/2andP"^1
(Eq.4.107). - The Fock matrixHin the atom-centered nonorthogonal basis {f} is transformed
into the matrixH^0 in the delocalized, linear combination orthogonal basis {f^0 }
by pre- and postmultiplyingHby the orthogonalizing matrixS"1/2(Eq.4.102).
6.H^0 is diagonalized to giveC^0 ,«andC^0 "^1 (Eq.4.104). We now have the energy
levelse(the diagonal elements of the«matrix).
7.C^0 must be transformed to give the coefficientscof the original, atom-centered
set of basis functions {f} in the MOs (i.e. to convert the elementsc^0 toc). To get
the c’s in the MOs cj¼c 1 jf 1 þc 2 jf 2 þ..., we transformC^0 toC by
premultiplying byS"1/2(Eq.4.100).
4.4.1.4 Molecular Energy and Geometry Optimization in the Extended
H€uckel Method
Steps 1–7 take an input geometry and calculate its energy levels (the elements of«)
and their MOs or wavefunctions (thec’s; from thec’s, the elements ofC, and the
basis functionsf). Now, clearly any method in which the energy of a molecule
depends on its geometry can in principle be used to find minima and transition states
(seeChapter 2). This brings us to the matter of how the EHM calculates the energy
of a molecule. The energy of a molecule, that is, the energy of a particular nuclear
configuration on the potential energy surface, is the sum of the electronic energies
and the internuclear repulsions (EelectronicþVNN).
In comparing the energies of isomers, or of two geometries of the same mole-
cule, one should, strictly, compareEtotal¼EelectronicþVNN. The electronic energy
is the sum of kinetic energy and potential energy (electron–electron repulsion and
electron–nucleus attraction) terms. The internuclear repulsion, due to all pairs of
interacting nuclei and trivial to calculate, is usually represented byV, a symbol for
potential energy. The EHM ignoresVNN. Furthermore, the method calculates elec-
tronic energy simply as the sum of one-electron energies (Section 4.4.4.2, Weak-
nesses), ignoring electron–electron repulsion. Hoffmann’s tentative justification
4.4 The Extended H€uckel Method 159