Four-dimensional pyramid of chemical elements and multidimensional symmetries of filling atomic shells as a function of nuclear charge

3. To the Problem
of Four-Dimensional Symmetries
of the Properties of Chemical Elements
and the Interpretation of this Phenomenon
F.Bogomolov, Y.Magarshak, On commuting operators related to asymptotic
symmetries in the atomic theory; Scientific Israel-Technological
Advantages, vol 8, issues 1-2, pp. 161-165 (2006)
• Y.Magarshak, "Four-Dimensional Pyramidal Structure of the Periodic
Properties of Atoms and Chemical Elements", Scientific Israel -
Technological Advantages vol. 7, No.1,2 , pp. 134-150 (2006)
• Yu. Magarshak, J.Malinsky,"A Three-Dimensional Periodic Table ", Nature,
vol.360: 113-114 (1992).

4. Орбитальное Рис. 1
Квантовое Порядок заполнения оболочек
число
Порядок
заполнения
термов (подоболочек)
14F
10D
6P
2S
1 2 3 4 5 6 7 8 Главное
квантовое
число
),()( ϕθlmYrR=Ψ
[ ] 0)(
2
2
=Ψ−+∆Ψ rUE
m


9. EMPIRIC HUND RULE: Electronic configuration, which has
the the largest (for that electronic configuration) spin value
and the greatest possible at this value of the spin orbinal momentum
has the lowest energy.
***
Experimentally proven order of atomic orbitals energy encrease:
1s<2s<2p<3s<3p<4s<3d<4p<5s<4d<5p<6s<4f <5d <6p <7s <5f <6d <7p < 8s

10. 5f < 6d < 7p < 8s …
4f < 5d < 6p < 7s <
4d < 5p < 6s <
3d < 4p < 5s <
3p < 4s <
2p < 3s <
2s <
1s <

11. 5g 6F 7D 8P 8s
5f 6d 7p 8s
Восьмой уровень
Четвертый
суперцикл
4f (14 элементов) 5d (10 элементов) 6p (6 элементов) 7s (2 элемента)
Седьмой уровень
4d (10 элементов) 5p (6 элементов) 6s (2 элемента)
Шестой уровень
Третий
суперцикл 3d (10 элементов) 4p(6 элементов) 5s (2 элемента)
Пятый уровень
3p (6 элементов) 4s (2 элемента)
Четвертый уровень
Второй
суперцикл
2p (6 элементов) 3s (2 элемента)
Третий уровень
2s (2 элемента)
Второй уровень
Первый
суперцикл
1s (2 элемента)
Первый уровень
Рис. 2
Copiright Yuri Magarshak
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12. 8s
7S
6 S 7P
5S 6P
4 S 5P 6D
3 S 4P 5D
2S 3P 4D 5F
1S 2P 3D 4F
1 2 3 4 5 6 7 8 Level

13. 8S
7S
6S 7P
5S 6P
4S 5P 6D
3S 4P 5D
2S 3P 4D 5F
1S 2P 3D 4F
1 2 3 4 5 6 7 8 Level

14. 8s
7s
6s 7p
5s 6p
4s 5p 6d
3s 4p 5d
2s 3p 4d 5f
1s 2p 3d 4f
1 2 3 4 5 6 7 8 Level

15. Fig. 7б
Copiright Yuri Magarshak
All rights reserved
Fr Rb Na H
Ti Ga B
Lu Sc
La
Uut
Lr
Ac
Y
Al
Cs
In
K Li

16. Fig. 7б
Copiright Yuri Magarshak
All rights reserved
Fr Rb Na H
Ti Ga B
Lu Sc
La
Uut
Lr
Ac
Y
Al
Cs
In
K Li

17. Copiright Yuri Magarshak
All rights reserved
Fr Rb Na H
Ti Ga B
Lu Sc
La
Uut
Lr
Ac
Y
Al
Cs
In
K Li

18. (1s)2
<
(2s)2
<
(2p)6
< (3s)2
<
(3p)6
< (4s)2
<
(3d)10
< (4p)6
< (5s)2
<
(4d)10
< (5p)6
< (6s)2
<
(4f)14
< (5d)10
< (6p)6
< (7s)2
<
(5f)14
< (6d)10
< (7p)6
< (8s)2

21. Порядок
заполнения
термов
элементами
Порядок Порядок
заполнения заполнения
периодов 3-7 периодов 2 и 3
термами термами
Copiright Yuri Magarshak
All rights reserved
Li HNaKCs RbFr
Al BGaIn
Y ScLuLr
TiUut
LaAc

25. The Basic Features
of the 4- Dimensional Pyramid
of Periodically Repeating Properties of Chemical
Elements
All chemical elements can be subdivided by
• Eight cycles having 2,2; 8,8;18,18, 32, 32
elements each
• Four Supercycles having 4, 16, 36 и 64
elements each
• Two clusters: M-cluster and F-cluster
• Paired clusters, cycles, subshells and
elements

26. 8S
7S
6S 7P
5S 6P
4S 5P 6F
3S 4P 5D
2S 3P 4D 5F
1S 2P 3D 4F
1 2 3 4 5 6 7 8 Level

27. 8s
7s
6s 7p
5s 6p
4s 5p 6d
3s 4p 5d
2s 3p 4d 5f
1s 2p 3d 4f
1 2 3 4 5 6 7 8 Level

28. n (principal quantum number)
8S
7S
6S 7P
5S 6P
4S 5P 6F
3S 4P 5D
2S 3P 4D 5F CYCLE
1S 2P 3D 4F CYCLE
1 2 3 4 5 6 7 8 Level
l (orbital quantum number)

32. Рис. 9a
Copiright Yuri Magarshak
All rights reserved
Порядок
заполнения
термов
элементами
Порядок Порядок
заполнения заполнения
периодов 3-7 периодов 2 и 3
термам термами
Li HNaKCs RbFr
Al BGaIn
Y ScLuLr
TiUut
LaAc

33. Copiright Yuri Magarshak
All rights reserved
Порядок
заполнения
термов
элементами
Порядок
заполнения Циклы
термов
каждого цикла
Li HNaKCs RbFr
Al BGaIn
Y ScLuLr
TiUut
LaAc

34. RaSr BaMg CaHe Be
Uub
HgCdZn
UuoRuXeCrAr
Ne
Yb No
O

35. СHEMISTRY NUMBERS. Each chemical element can be
characterized by the set of four chemical numbers.
CLUSTER NUMBER , having one of two values: either M or F
SUPERCYCLE NUMBER (or, in short, super-number) , which
determines to which pyramid super-level element belongs and on
which super-cycle is disposed ( = 1, 2, 3, 4).
LAYER NUMBER  which determines the layer number of the
cycle in which subshell  is disposed.  has integer values from -1
(bottom layer, which in the cycle is filled first) to 0 (surface layer,
щелочные и щелочноземельные металлы, подоболочки которых
заполняются в циклах последними).
NODE NUMBER , which enumerates 2(2  + 1) elements of the
subslell.
CERTAINITY PRICIPLE: TWO CHEMICAL
ELEMENTS CAN NOT HAVE FOUR
IDENTICAL CHEMICAL NUMBERS

36. ),()( ϕθlmYrR=Ψ
[ ] 0)(
2
2
=Ψ−+∆Ψ rUE
m

lm
l
x
mml
ml
xP
mP
ml
mll
mP
ml
mll
m
m
m
l
m
j
m
j
,...,2,1,0
;,...,2,1,0
)1(
!)!(
)!(
2
1
)(
cos)(cos
)!(2
)!(12
sin)(cos
)!(2
)!(12
2/2
=
=
−
−
+
=
+
−+
+
−+
ϕϑ
π
ϕϑ
π

37. 0
2)1(2
222
2
=





++
+
−+ R
r
E
m
R
r
ll
dr
dR
rdr
Rd α

1
2 22
2
−−=
−=
lnn
n
m
E
r

α
where n – radial quantum number, equal to the number of
nonzero nodes of the radial part of the wave function at
finite non-zero r.

39. Identity of the structure of the M-cluster (as well as that of
the F-clusterl) with the dependence of n(l) for hydrogen
atom, наводят на мысль о том, that D symmetry of the
periodic elements properties on the charge of the nuclei all
elements can be considered as the sates of the single
particles.
However the necessity
a) Getting two grids, which are rotated relative to each
other by /4;
б) filling the nodes on the slide 1 (hydrogen atom) and
slide 2 (for the set of all chemical elements)in opposite
directions, and
c) The presence of paired cycles, supercycles and clusters
in periodically repeating elements properties

48. Let the structures and symmetries that are described above and shown
in Fig. 1 be generated by a pair of Hamiltonian-type operators, which
naturally appear in the presence of two commutative Lie symmetry
algebras in the quantum system. From the existence of simple
commutative Lie symmetry algebras in a physical problem, it follows
the existence of a single commuting pair of energy-type operators for
localized quantum states.
Let us consider a finite-dimensional complex representation of the
product of semisimple Lie algebras. Any such representation V of
g1
 g2
decomposes into a direct sum of tensor products Vk
 Wj
where Vk
are irreducible representations of g1
and Wj
are representations of g2
.
Theorem: Assume that g1
,g2
are simple Lie algebras. Then there is a
canonically defined pair of commuting operators 1
, 2
such that both
i
, i=1,2 commute also with the action of g1
,g2
and have integer
nonnegative eigenvalues on V.

49. Note that the above decomposition may exist in the compexification of
the symmetry algebra and not in the algebra itself. Since finite-
dimensional representations of the complexified Lie algebra are the same
as of it's real form and the above decomposition holds anyway. In our
application the complexification of both real algebras are presumably
isomoprhic to so(3,C)= sl(2,C) after a complexification and hence have
rank 1. Thus there are two natural options:
1) both algebras are coming from the independent rotation invariance of
both the nucleus and the electron envelope.
2) the local symmetry group is the Lorentz group SO(3,1) and
though it's Lie algebra so(3,1) has no such a decompostion but the
complexification so(3,1) x C = sl(2,C) + sl(2,C).
The actual Hamiltonian operator of the problem splits into a sum of
two commuting energy time operators.
In our opinion second case is physically more plausible

50. How could such a symmetry appear in reality? Currently the standard
answer to such question is symmetry break. Namely the existence of
symmetry within an ensemble of different particles is usually explained
by considering them as different states of an "ideal object" with the
above symmetry so that the particles are degenerate states of the above
object where the symmetry is broken due to some process of natural
degeneration.
If we try to apply similar explanation we are brought to the idea to view
the atom as one of the possible states of an ideal particle which we will
be denoting as I-particle. This particle has Lorentz symmetry algebra
within a bigger algebra of local symmetries and supersymmetries
corresponding to other potential fields. The I-particle exists in this
ideal state only 0-time which in practice means a very short time
depending on the total energy of the I-particle according to the
"uncertainty principle". While degenerating the I-particle creates an
avalanche of intermediate particles which retain only pieces of the
initial symmetry but previous symmetry is manifested in the set of
possible degenerate states.