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For a given Hamiltonian
we minimise the expectation value of the energy over a sub-set of states
that are linear combinations of
given states
,
The
are assumed to be normalised but not necessarily mutually orthogonal, i.e., one can have
.
The energy
is therefore minimized with respect to the
coefficients
,
. It can be written as
where one has introduced the matrices
and
with matrix elements
|
|
|
(3.4) |
We find the minimum of
|
|
|
(3.5) |
by setting the gradient to zero. We treat
and its complex conjugate
as independent variables and calculate
Correspondingly,
Thus,
since
at the minimum! A necessary condition for a minimum therefore is the equation
, which has a solution for
only if
|
|
|
(3.9) |
Exercise: Check which equations one obtains when taking the derivative
instead of
!
We summarise:
The minimization problem thus led us to an eigenvalue problem.
Next: Bonding and Antibonding
Up: The Hydrogen Molecule Ion
Previous: Hamiltonian for
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Tobias Brandes
2005-04-26