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Chem 3502/5502
Physical Chemistry II (Quantum Mechanics)
3 Credits
Fall Semester 2009
Laura Gagliardi
Lecture 25, November 11, 2009
(Some material in this lecture has been adapted from
Cramer, C. J.
Essentials of
Computational Chemistry
, Wiley, Chichester:
2002; pp. 96109.)
Recapitulation of the Variational Principle and the Secular Equation
Recall that for any system where we cannot determine exact wave functions by
analytical solution of the Schrödinger equation (because the differential equation is
simply too difficult to solve), we can make a guess at the wave function, which we will
designate
Φ
, and the variational principle tells us that the expectation value of the
Hamiltonian for
Φ
is governed by the equation
!
"
*
H
!
d
r
!
"
*
!
d
r
#
E
0
(251)
where
E
0
is the
correct
groundstate energy.
Not only does this lowerlimit condition provide us with a convenient way of
evaluating the quality of different guesses (lower is better), but it also permits us to use
the tools of variational calculus to identify minimizing values for any parameters that
appear in the definition of
Φ
.
In the LCAO (linear combination of atomic orbitals) approach, the parameters are
coefficients that describe how molecular orbitals (remember, orbital is another word for a
one
electron wave function contributing to a
many
electron wave function) are built up as
linear combinations of atomic orbitals. In particular, manyelectron wave functions
Φ
can
be written as antisymmetrized Hartree products—i.e., Slater determinants—of such one
electron orbitals
φ
, where the oneelectron orbitals are defined as
!
=
a
i
"
i
i
=
1
N
#
(252)
where the set of
N
atomicorbital basis functions
ϕ
i
is called the “basis set” and each has
associated with it some coefficient
a
i
, where we will use the variational principle to find
the optimal coefficients.
To be specific, for a given oneelectron orbital we evaluate
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E
=
a
i
*
!
i
*
i
"
#
$
%
%
&
’
(
(
)
H
a
j
!
j
j
"
#
$
%
%
&
’
(
(
d
r
a
i
*
!
i
*
i
"
#
$
%
%
&
’
(
(
)
a
j
!
j
j
"
#
$
%
%
&
’
(
(
d
r
=
a
i
*
a
j
ij
"
!
i
*
)
H
!
j
d
r
a
i
*
a
j
ij
"
!
i
*
)
!
j
d
r
=
a
i
*
a
j
ij
"
ij
H
a
i
*
a
j
ij
"
ij
S
.
(253)
where the shorthand notation
H
ij
and
S
ij
is used for the resonance and overlap integrals in
the numerator and denominator, respectively.
If we impose the minimization condition
!
E
!
a
k
=
0
"
k
(254)
we get
N
equations which must be satisfied in order for equation 254 to hold true,
namely
a
i
i
=
1
N
!
H
ki
–
ES
ki
( ) =
0
"
k
.
(255)
these equations can be solved for the variables
a
i
if and only if
H
11
–
ES
11
H
12
–
ES
12
!
H
1
N
–
ES
1
N
H
21
–
ES
21
H
22
–
ES
22
!
H
2
N
–
ES
2
N
"
"
#
"
H
N
1
–
ES
N
1
H
N
2
–
ES
N
2
!
H
NN
–
ES
NN
=
0
(256)
where equation 256 is called the secular equation. There are
N
roots (i.e.,
N
different
values of
E
) which permit the secular equation to be true. For each such value
E
j
there
will be a different set of coefficients,
a
ij
, which can be found by solving the set of linear
equations 255 using that specific
E
j
, and those coefficients will define an optimal
associated wave function
φ
j
within the given basis set.
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This note was uploaded on 12/14/2010 for the course CHEM 3502 taught by Professor Staff during the Spring '08 term at Minnesota.
 Spring '08
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