Chemistry Reference
In-Depth Information
ForZ
¼
2 (He atom), Z
eff
¼
Z
ð
5
=
16
Þ¼
27
=
16
¼
1
:
6875, and (4.32)
gives
«
w
(best)
¼
2.847 656E
h
, which is more than 98% of the accurate
value of
2
903 724E
h
given by Pekeris (1958). Eckart (1930) gave as two
nonlinear parameter approximations to the ground-state energy of the
He atom the following improved 'split-shell' result:
:
« ¼
2
:
875 661E
h
c
1
¼
2
:
183 171
c
2
¼
1
:
188 531
ð
4
:
33
Þ
which is within 99% of Pekeris's result. We end this section by noting:
(i) that Equation 4.32 describes in part the 'screening effect' of the second
electron on the nuclear charge Z,
5
showing that the variation theorem
accounts as far as possible for real physical effects; (ii) that the calculated
energy is not an observable quantity, so that comparison with experi-
mental results is possible only through the values of the ionization
potential, defined as
I
¼ «ð
He
þ
Þ«ð
He
Þ
ð
4
:
34
Þ
Since the ionization potential I is smaller than the absolute energies of
either the atom or the ion, its approximate values are affected by larger
errors.
4.3 LINEAR PARAMETERS AND THE
RITZ METHOD
This famous method of linear combinations is due to the young Swiss
mathematicianRitz (1909) and, therefore, is usually referred to as the Ritz
method. From the variational point of view, flexibility in the trial function
is now introduced through the coefficients of the linear combination of
a given set of regular functions. Usually, the basis functions are fixed,
but they can be successively optimized even with respect to the nonlinear
parameters present in their functional form. We shall see that the Ritz
5
In the He atom, an electron near to the nucleus sees the whole nuclear charge Z; far from it, the
nuclear charge is
ð
Z
1
Þ
, as if it were fully screened by the other electron. The variational result
averages between these two extreme cases, privileging the regions near to the nucleus (hence,
Z
eff
1
:
7, closer to 2 rather than 1).
