Biomedical Engineering Reference
In-Depth Information
I have of course assumed that all the wavefunctions are real rather than complex; it would
not have made any difference to my argument.
This latter quantity Equation (15.12) times dτ
1
gives the probability of finding electron
1indτ
1
with either spin, and the other electrons anywhere and again with either spin. Since
there are two indistinguishable electrons in dihydrogen, the total electron density must be
twice my result:
ρ (
r
)
=−
2
e
{
ψ
+
(
r
)
}
2
The charges and bond orders matrix is therefore just twice what we found for the hydrogen
molecular ion:
⎛
⎝
⎞
⎠
1
1
1
+
S
1
+
S
P
LCAO
=
(15.13)
1
1
1
+
S
1
+
S
A corresponding analysis for the VB function gives
⎛
⎞
1
S
⎝
⎠
+
+
1
S
2
1
S
2
P
VB
=
(15.14)
S
1
1
+
S
2
1
+
S
2
and the gross Mulliken population for each hydrogen nucleus is once again 1/2, just as it
should be.
There is a nice way to remember the result, provided you are happy with matrices. If we
write an
overlap matrix
S
for the two atomic orbitals 1s
A
and 1s
B
as
⎛
⎝
⎞
⎠
1s
A
1s
A
dτ
1s
A
1s
B
dτ
S
=
1s
B
1s
A
dτ
1s
B
1s
B
dτ
then a little analysis shows that
P
ij
S
ij
=
number of electrons
This can be restated in terms of the trace of the matrix product as
Tr (
PS
)
=
number of electrons
(15.15)
References
Bates, D.R., Ledsham, K. and Stewart, A.L. (1953)
Phil. Trans. R. Soc. A
,
246
, 215.
Born, M. and Oppenheimer, J.R. (1927)
Ann. Phys.
,
84
, 457.
Burrau, O. (1927)
Kgl Danske Videnskab Selskab
,
7
,1.
Eyring, H., Walter, J. and Kimball, G.E. (1944)
Quantum Chemistry
, John Wiley & Sons, Inc.,
New York.
Heitler, W. and London, F. (1927)
Z. Physik
,
44
, 455.
