Computational Chemistry

5.2.3.3 The Variation Theorem (Variation Principle)

The energy calculated from Eq.5.14is theexpectation valueof the energy operator
H^, i.e. the expectation value of the Hamiltonian operator. In quantum mechanics an
integral of a wavefunction “over” an operator, like CjH^jC
in Eq.5.14, is the
expectation valueof that operator. The expectation value is the value (strictly,
the quantum-mechanical average value) of the physical quantity represented by the
operator. Every “observable”, i.e. every measurable property of a system, is thought
to have a quantum mechanical operator from which the property could be calcu-
lated, at least in principle, by integrating the wavefunction over the operator. The
expectation value of the energy operatorH^(for which a better symbol might have
beenEˆ) is the energyEof the molecule or atom. Of course this energy will be the
exact, true energy of the molecule only if the wavefunctionCand the Hamiltonian
H^are exact. The variation theorem states thatthe energy calculated from Eq.5.14
must be greater than or equal to the true ground-state energy of the molecule.The
theorem [ 17 ] (it can be stated more rigorously, specifying thatH^must be time-
independent andCmust be normalized and well-behaved) assures us that any
ground state (we examine electronic ground states much more frequently than we
do excited states) energy we calculate “variationally”, i.e. using Eq.5.14, must be
greater than or equal to the true energy of the molecule. This is useful because it
tells us that a test for the quality of a wavefunction is the value of the energy
calculated from it variationally: the lower the better. We can try to improve our
wavefunction, checking the variational energy against that from previous functions.
In practice, any molecular wavefunction we insert into Eq.5.14is always only an
approximation to the true wavefunction and so the variationally calculated molecu-
lar energy will always be greater than the true energy. The Hartree–Fock energy
is variational, but as we will see, not all quantum chemical energies are. The
Hartree–Fock energy levels off at a value above the true energy as the Hartree–Fock
wavefunction, based on a Slater determinant, is improved; this is discussed in
Section 5.5, in connection with post-Hartree–Fock methods.

5.2.3.4 Minimizing the Energy; the Hartree–Fock Equations

The Hartree–Fock equations are obtained from Eq.5.17by minimizing the energy
with respect to the atomic or molecular orbitalsc. The minimization is carried out
with the constraint that the orbitals remain orthonormal, for orthonormality was
imposed in deriving Eq.5.17. Minimizing a function subject to a constraint can be
done using the method of undetermined Lagrangian multipliers [ 18 ]. For ortho-
normality the overlap integralsSmust be constants (¼dij, i.e. 0 or 1) and at the
minimum the energy is constant (¼Emin). Thus atEminany linear combination ofE
andSijis constant:

Eþ

Xn

i¼ 1

Xn

j¼ 1

lijSij¼constant ð 5 : 23 Þ

5.2 The Basic Principles of the ab initio Method 189