Introduction
Nearly 150 years ago an atomistic explanation of the periodic table had
been given by Mandeleev [1]. Very recently a quantum aspect of the
periodic table has been questioned [2]. As is already pointed out in
that article, 5 scientists separately tried to explain the Madelung’s
rule with group theoretical approach [3, 4, 5, 6, 7, 8]. Among these
studies Fet’s and Barut’s approach were better in many aspect. In other
words the dynamical Group theoretical explanation of the periodic system
was mainly from [4, 5]. We will not going to discuss the group
theoretical studies in this study but in future work we would explain
some observer anomalies from the periodic table by dynamical Group
theoretical method. In this study we mainly focus on some numeric
calculation together with some other work with the literature. Using the
calculation we would try to explain some discrepancies from the
expectations. Although we will study the group theoretical aspect of the
problem in some future work, we should remember that Barut like Fet were
forced to treat the element, not as composite particle, but as state of
superparticle [2], where group structure was investigated on the
basis of the group SO(4,2) which is also discussed by Odabasi [9].
These superparticles (supermultiplets) later named as Baruton [10].
By starting from the fact that a special representation of the dynamical
group SO(4,2) provides the quantum numbers n, l, m for the hydrogen atom
[11] and by taking each atom or ion in its normal configuration as a
state, Barut demonstrated that hydrogenic order can be explained by the
chain
SO(4,2)-SO(4,1)-SO(4)-SO(3)
While Madelung rule is given by the reductions,
SO(4,2)-SO(3,2)-SO(3)XSO(2)
Barut also showed that [4, 5] increasing order of element does not
always corresponds to the increasing order of energy in the hydrogen
atom. In the Hydrogen atom the energy is increased with increasing n and
when relativistic spin-orbit is included it increases with l, and this
rule is generally denoted (n+l,n) rule[4, 5]. From the Barut’s work
it is important to figure out that for ions, the electron shells are
also filled in different order from that for atoms, the hydrogenic order
being the limit for highly ionized case. In future work we would try to
apply the Barut’s method for some specific unexpected filling. In this
study we mainly concentrate on some unusual filling especially filling
up the Cr atom by calculating the self-consistent field. As is pointed
out earlier, the filling up principle was discovered empirically by
Madelung [12] for atoms as:
1-When consecutive atoms are considered, the electron shells are filled
in the order of the sum of two quantum numbers n and l, that is, (n+l).
2-Shelss with equal n+l numbers are filled in the order of quantum
number n.
This rule is very successful for how the energy levels are filled for
the neutral atoms. However experiments show that there are some
deviations from this rule.
Madelung’s second rule predicts that the 4s shell will be filled before
3d shell and that 4f shell and 5f shell would be occupied first at Z=57
and Z=89 respectively. From the atomic spectroscopic data it is quite
evident that there are same expectations for this rule. For example for
Cr atom the expected occupation is
4s23d4 but the correct occupation
was 4s13d5. In this study we mainly
concentrate on this anomaly and try to find out why the second
configuration lies lowest. The energies of the 3d and 4s states are so
close together that small effects can shift the balance either way [4,
5]. It is assumed that by putting four electrons into the 3d state
their repulsion raises 4s level such that one electron shifts over. The
requirement of parallel spins for electrons that do occupy different
orbitals is a consequence of a quantum mechanical effect called spin
correlation effect with the tendency for two electrons with parallel
spins to stay apart from one another and hence to repel each other less.
One consequence of this effect is that half-filled shells of electrons
with parallel spins are particularly stable. For example, the ground
state of the chromium atom is 4s13d5rather than 4s23d4. An additional
feature, another consequence of spin correlation, is that in some cases
a lower total energy may be obtained by forming a half-filled or filled
d subshell, even though that may mean moving an s electron into the d
subshell. Therefore, as a half-filled d shell is approached the
ground-state configuration is likely to be
d5s1 and not
d4s2 (as for Cr).