C 2 n¼
1
ffiffiffiffiffiffiffiffiffiffi
ð 2 nÞ!p'
c 1 ð 1 Það 1 Þ c 1 ð 1 Þbð 1 Þ c 2 ð 1 Það 1 Þ c 2 ð 1 Þbð 1 Þ(((cnð 1 Þbð 1 Þ
c 1 ð 2 Það 2 Þ c 1 ð 2 Þbð 2 Þ c 2 ð 2 Það 2 Þ c 2 ð 2 Þbð 2 Þ(((cnð 2 Þbð 2 Þ
... ... ... ...
c 1 ð 2 nÞað 2 nÞc 1 ð 2 nÞbð 2 nÞc 2 ð 2 nÞað 2 nÞc 2 ð 2 nÞbð 2 nÞ(((cnð 2 nÞbð 2 nÞ(^)
(^)
ð 5 : 12 Þ
The Slater determinant for the total wavefunctionCof a 2n-electron atom or
molecule is a 2n'2ndeterminant with 2nrows due to the 2nelectrons and2n
columns due to the 2nspin orbitals (you can interchange the row/column format);
since these are closed-shell species, the number of spatial orbitalscis half the
number of electrons. We use the lowestnoccupied spatial orbitals (the lowest 2n
spin orbitals) to make the determinant.
The determinant (¼total molecular wavefunctionC) just described will lead to
(remainder ofSection 5.2)noccupied, and a number of unoccupied, component
spatial molecular orbitalsc. These orbitalscfrom the straightforward Slater
determinant are calledcanonical(in mathematics the word means “in simplest or
standard form”) molecular orbitals. Since each occupied spatialccan be thought of
as a region of space which accommodates a pair of electrons, we might expect that
when the shapes of these orbitals are displayed (“visualized”;Section 5.5.6) each
one would look like a bond or a lone pair. However, this is often not the case; for
example, we do not find that one of the canonical MOs of water connects the O with
one H, and another canonical MO connects the O with another H. Instead most of
these MOs are spread over much of a molecule, i.e. delocalized (lone pairs, unlike
conventional bonds, do tend to stand out). However, it is possible to combine the
canonical MOs to get localized MOs which look like our conventional bonds and
lone pairs. This is done by using the columns (or rows) of the SlaterCto create aC
with modified columns (or rows): if a column/row of a determinant is multiplied by
kand added to another column/row, the determinant remainskD(Section 4.3.3).
We see that if this is applied to the Slater determinant withk¼1, we will get a
“new” determinant corresponding to exactly the same total wavefunction, i.e. to the
same molecule, but built up from different component occupied MOsc. The newC
and the newc’s are no less or more correct than the previous ones, but by
appropriate manipulation of the columns/rows thec’s can be made to correspond
to our ideas of bonds and lone pairs. These localized MOs are sometimes useful.
5.2.3.2 Calculating the Atomic or Molecular Energy
The next step in deriving the Hartree–Fock equations is to express the energy of the
molecule or atom in terms of the total wavefunctionC; the energy will then be
minimized with respect to each of the component molecular (or atomic; an atom is a
184 5 Ab initio Calculations