Quantum Matter-Wave Explanation of the Periodic Table: The Freefall Observer’s View of Chemical “Basic Substances” M. J. Lindeman August 6, 2018 Abstract The Madelung Rule is Bohr’s two-part rule that describes the overall electron configurations of the elements in the periodic table. It also has been used to predict the filling sequence as individual electrons are added to configurations. However, the Rule is based on empirical observations rather than first principles. This paper describes how the combination of (a) four symmetrical wavefactors of Einstein’s 1905 electromagnetic-interval equation and (b) Galileo’s acceleration law with a gravitational coefficient of “1” yields a principled explanation of the Madelung Rule. Einstein’s equation defines the rows and Galileo’s law defines the columns of an expanded version of Janet’s left-step form of the periodic table. Janet’s table models Solutions of Einstein’s equation and the expanded version models the individual waves (handedness) of the electron orbital pairs within each Solution. This doubles the number of rows in the table. This new approach perfectly maps the four quantum number (n, l, m, and m s ) to the arrangement of the periodic table. It also provides a visualization of patterns of filling anomalies. It groups the ∼20 anomalies into loops that break and then restore the Madelung Rule’s filling sequence. This explanation of the Periodic Table, which is based on the matter waves of Schr¨odinger’s quantum-wave mechanics, opens the door to new ways of modeling physical reality. Keywords: Periodic table, Madelung Rule, quantum numbers, interval, Galileo’s acceleration law, freefall, Mazurs, Janet, left-step, matter waves

1

Identifying L¨owdin’s Challenge

the arrangement of the periodic table. Mazurs (1974, pp. 96-105, esp. Tables 3, 4, & 6) discusses in detail the transition from modeling the

Bohr proposed a two-part (n + l, n) empirical heuristic rule to explain the periodic system and its filling sequence: 1. The energy of a state is the sum of the first quantum number n and second quantum number l (the n + l part of the rule). 2. When two states have the same value for n + l, the larger energy state has the larger n (the n part of the rule). Bohr’s rule has various names, include the Aufbau rule or the Madelung rule. It has two uses: (1) identifying the overall electron configuration of an atom of each chemical element, and (2) identifying the sequence of elements created as electrons are added to configurations. Thus it is used to describe and empirically explain

Shells>Subshells>Orbitals arrangement of the periodic table to the Periods>Subperiods>Orbitals arrangement. The result is Janet’s left-step form of the periodic table that is based on electron configurations. A configuration is a grouped set of orbital pairs of electrons. The energy levels of subperiod groups (e.g., 2p, 3s, 3p, 4s, 3d) are ordered by the two parts of the Madelung Rule. Subperiod names are the first part of the Madelung Rule without the plus sign, the first (principal) quantum number n and a letter representing the 1

2

QUANTUM MODELING OF THE PERIODIC TABLE

second quantum number l (s = 0, p = 1, d = 2 and f = 3). For example, the two subperiods 3s = 3 + 0 and 2p = 2 + 1 both have the value n + l = 3. Because 2 < 3, the subperiod 2p has a smaller energy and is filled before 3s. For 4s = 4 + 0 and 3p = 3 + 1, the value is n + l = 4. Thus 3p and then the 4s are filled. Because 3d = 3 + 2 = 5 and 4 < 5, 4s fills before 3d and this has been experimentally observed. In his discussion of Bohr’s rule, L¨owdin challenged his readers to study (derive) the rule from first principles.(1969, p. 334) He suggested starting with the manyelectron Schr¨odinger’s equation, but attempts that start with quantum mechanics have not been successful. Scerri et al. describe four types of attempts to explain Bohr’s rule: • approximate (heuristic) solutions to the original equations, • new (heuristic) equations, • symmetry groups, and • solving the corresponding equations of quantum mechanics. However, none of these approaches successfully explain Bohr’s (Madelung) Rule. “In short: in spite of all known attempts, the problem of explaining the periodic system is still far from being solved”.(1998) Scerri also writes, “Although the above Madelung rule succeeds in giving the overall configuration of the transition metals beginning with scandium, or element 21, the order of filling is not provided by this rule.”(2016, p. 135) This paper presents a new principled approach using Einstein’s 1905 electromagnetic-interval equation and Galileo’s law of acceleration. Combining these two equations provides a principled explanation of Bohr’s (Madelung) Rule and the Janet left-step form of the periodic table. It also perfectly maps the quantum numbers into the periodic table. In addition, it identifies patterns of observed anomalies in the expected filling sequence as 10 anomaly loops (from sequence deviation to restoration) that involve multiple elements. The loops range in size from the minimum of two elements to loops of six elements.

2

2 Quantum Modeling of the Periodic Table

When Bohr modeled the atom (1913), he modeled the orbits of non-radiating electrons. He wrote that the electrons’ motions will describe stationary elliptical orbits. The frequency of revolution ω and the majoraxis of the orbit 2a will depend on the amount of energy W which must be transferred to the system in order to remove the electron to an infinitely great distance apart from the nucleus. However, Bohr’s original theory “places no restriction on the eccentricity of the orbit, but only determines the length of the major axis.” (Bohr 1922, p. 17) Thus it was inadequate to describe the electron configurations of atoms completely. He wrote: (1922, p. 25) For a simply periodic motion such as we meet in the pure harmonic oscillator, and at least to a first approximation, in the motion of an electron about a positive nucleus, the manifold of stationary states can be simply co-ordinated to a series of whole numbers. For motions of the more general class mentioned above, the so-called multiply periodic motions, however, the stationary states compose a more complex manifold, in which, according to these formal methods, each state is characterized by several whole numbers, the so-called ”quantum numbers”. Sommerfeld used three quantum numbers to describe the electron configurations of atoms.(1923, pp. 234, 392) Physicists have updated Sommerfeld’s work and the interpretations of his quantum numbers are now based on Schr¨odinger’s equations. • The first (principal) quantum number n originally described the major axis of the ellipses modeling electrons’ orbits. It has been interpreted as describing an orbital’s size, which also represents the electron’s energy level. Its integer values are ”1” or larger. • The second (azimuthal or angular or orbital) quantum number l originally described the minor axis of an orbit’s ellipse. It is now interpreted as described an orbital’s shape. It can take integer values from zero to n − 1. Zero values represent a

2

QUANTUM MODELING OF THE PERIODIC TABLE sphere, and larger integers represent more complex three-dimensional shapes. • The third (magnetic) quantum number m or mi quantized the elliptical orbit into a set of spatial regions located at different positions in the orbit. It is interpreted as an orbital’s orientation in space. It can take integer values from −l to +l.

In 1925 Pauli introduced the fourth (spin) quantum number σ or m s with the binary values ± 12 for electrons. The sign indicates the direction of ‘spin’. Each electron must have a unique set of four quantum numbers, so the spin number distinguishes the electrons with left- or right-handed spin in an orbital pair. (The assignment of direction is arbitrary.) Bohr’s work was based on the unique set of atomicspectral lines emitted by each chemical element. He used this information to create a principled explanation of the periodic table (system). (1922, p. 31) The ideas of the origin of spectra outlined in the preceding have furnished the basis for a theory of the structure of the atoms of the elements which has shown itself suitable for a general interpretation of the main features of the properties of the elements, as exhibited in the natural system. Although his theory is outdated, it contains the fundamental principle of symmetrical harmonics explored further in this paper. Bohr wrote: (1950) Let us consider an electrodynamic system and inquire into the nature of the radiation which would result from the motion of the system on the basis of the ordinary conceptions. We imagine the motion to be decomposed into purely harmonic oscillations, and the radiation is assumed to consist of the simultaneous emission of series of electromagnetic waves possessing the same frequency as these harmonic components and intensities which depend upon the amplitudes of the components. An investigation of the formal basis of the quantum theory shows us now that it is possible to trace the question of the origin of the radiation processes which accompany the various transitions back to an investigation of the various harmonic com-

3 ponents, which appear in the motion of the atom. . . . This principle of correspondence at the same time throws light upon a question mentioned several times previously, . . . the relation between the number of quantum numbers, which must be used to describe the stationary states of an atom, and the types to which the orbits of the electrons belong. The classification of these types can be based very simply on a decomposition of the motion into its harmonic components. [italics added]

Quantum theory, whether in the form originated by Bohr, Sommerfeld, Heisenberg or Schr¨odinger, is based on quantum waves and their harmonic relationships. That is why Born had to introduce probabilities to explain particle collisions.(Rosenfeld 1983) The quantum waves are de Broglie’s “phase waves”. (de Broglie 1962, p. 109) They are also called “matter waves” because they comprise the mass of material objects. Thus the principled explanation of the arrangement of the periodic table is based on matter waves. Heisenberg wrote: (1974, p. 14) Physicists had long since been acquainted with such fields of force as the gravitational field and electromagnetic forces. To these had been added in the present [20th ] century the matter waves, which can also be described as force fields of the chemical bond. He also wrote, “[T]he electron may also be considered to be a plane de Broglie wave”.(1949, p. 23) Matter waves are waves of potential in Maxwell’s A field, the field described by Schr¨odinger’s quantumwave equation. As Feynman wrote, there is no way to remove the A field from Schr¨odinger’s equation even though many physicists have attempted to remove it. (1964, p. 15-12) Other quotes from Nobel Laureates in Physics emphasize the importance and use of de Broglie’s matter waves to describe physical objects. • “As early as 1928 it was shown by Jordan, Klein and Wigner that the mathematical scheme can be interpreted not only as a quantization of particle motion but also as a quantization of three-

3

MATTER-WAVE FACTORS OF EINSTEIN’S EQUATION dimensional matter waves; therefore, there is no reason to consider these matter waves as less real than the particles.”(Heisenberg 2007, pp. 107-108) • “[O]nly the waves in configuration space, that is the transformation matrices, are probability waves in the usual interpretation, while the three-dimensional material waves are not. The latter, according to Bohr and to Klein, Jordan and Wigner, have just as much (and just as little) ‘objective reality’ as particles; they have no direct connection with probability waves, but have a continuous density of energy and of momentum, like a Maxwell field.”(Heisenberg 1955, p. 24) • “The electron can no longer be conceived as a single, small granule of electricity; it must be associated with a wave and this wave is no myth; its wavelength can be measured and its interferences predicted.”(de Broglie 1929, p. 256) • “If our concern is with macroscopic masses (billiard balls or stars), we are operating with very short de Broglie waves, which are determinative for the behavior of the center of gravity of such masses.”(Einstein 1970, p. 682) • In 1950 Schr¨odinger wrote to Einstein, “I have really believed for a long time that the ψ-waves are to be identified with waves representing disturbances of the gravitational potential; not, of course, with those you [Einstein] studied first, but rather with ones that transport real mass, i.e., a non-vanishing T ik .”(Einstein 1967, p. 36)

In summary, matter waves are the A-field quantum waves of Schr¨odinger’s equation. According to Heisenberg, an electron is a plane de Broglie matter wave, and according to Einstein matter waves determine the behavior of macroscopic masses. Those statements indicate that correctly modeling matter waves is the key to principled modeling of the Periodic Table of chemical elements. How matter waves determine the Madelung Rule for the overall energy of electron configurations of atoms is described in the remainder of this paper. 3

Matter-Wave Factors of Einstein’s Equation

One of the papers Einstein published in 1905 (Einstein 1998) defined the same type of electromagnetic volumes of space and time as those used by human sensory perception. However, his model of finite volumes was not

4

explored by physicists. In 1908 the mathematician Hermann Minkowski changed Einstein’s equation to model imaginary four-dimensional (4D) spacetime.* † Einstein’s 1905 electromagnetic volumes are real intervals in which the diagonal of the observed spatial volume equals the observed time t multiplied by lightspeed c. His Equation 1 (Page 135 and endnote 3, p. 160 in Einstein 1998) defines three-dimensional volumes (length x width x height) as shown in Figure 1. x2 + y2 + z2 = (ct)2

lS

P1

spacex

(1)

P2

na ct go ce a i n d a l a ist ati td spacez h sp g li o s l &a spacey

Fig. 1: Einstein’s electromagnetic “ct” vector that rep-

resents separation in time decomposes into the three orthogonal directions of space Taking the square root of both sides of Equation 1 defines the volume’s diagonal distance (the separation between opposite corners) in space and in time. In other words, ‘space’ is the vectorial decomposition of the ct distance vector for ‘time’. Equation 1 can be rewritten and rearranged so that it is the factorable polynomial equation in Equation 2c. The summary form of the rearranged equation factors into four components—two entangled pairs of symmetrically opposing circularly polarized matter waves (Equations 2d and 2e). Einstein: s2x + s2y + s2z − (lightspeed c · time)2 = 0 (2a) * Minkowski classified matter and radiation as one “substance” and required that the interval be timelike (> 1). He also introduced an imaginary number ict for measurements of time that, when squared, converted Einstein’s minus sign (subtraction) to a plus sign (addition).(Minkowski 1952) † Minkowski’s change converted Einstein’s original finite volumes in three dimensions to lightlike cones in four dimensions. Misner, Thorne and Wheeler’s Box 2.1 (Farewell to ‘ict’) discusses this and other changes Minkowski made to the equation. They note that for Einstein’s electromagnetic (zero) intervals, the two separated points may be in different galaxies.(Misner et al. 1973, pp. 51, 437).

4

GALILEO’S LAW DEFINES THE SUBPERIODS

5

s2x + s2y + s2z = S 2D & substitute S 2D into equation. (2b) Factor (S 2D − ctime2 ) into symmetrical solutions. (2c) + S D solution : (+S D + ctime)(+S D − ctime) (2d) − S D solution : (−S D + ctime)(−S D − ctime) (2e) The four waves are ±S ± ct, and the two entangled pairs (interval solutions) are +S ±ct and −S ±ct. The ±S term indicates the starting location and the two solutions differ in phase by one-half cycle (Figure 2). The sign of the ct term indicates the direction of rotation. All assignment of direction is arbitrary because there is no conventional consensus. Fig. 3: The four wave-factors of Einstein’s 1905 special-

relativity interval equation.

Fig. 2: Initial (phase 0) points for the wavepair Solutions

are at P1 for +S and P2 for −S . The direction of rotation is +ct for counter-clockwise and −ct for clockwise. The positive spin is counter-clockwise when viewed from the position of the receiver (“sink”). The same spin appears clockwise when viewed from the position of the sender (“source”). This uses a handedness rule for determining direction—when the fingers are curled in the direction of spin, then the thumb points in the direction of travel (propagation). Figure 2 shows the waves organized by initial-phase starting point. Figure 3 shows the waves organized by matching + or − signs. The top row represents the −S solution, and the bottom row the +S solution. The left column shows the up and down clockwise (-ct) waves, and the right column the up and down counterclockwise (+ct) waves. When a wave is coming towards the observer, then the handedness shown is valid. The arbitrary labels “up” and “down” define the direction from the +S or -S zerophase point.

The factoring of Einstein’s equation identifies a threelevel hierarchy of Interval > Solution > Wave. Each Interval has a +S and a −S solution, and each Solution has a +ct and a −ct spin wave. Thus each Interval contains a set of four matter waves entangled as two solutions. The first four intervals, each containing four waves, define a hierarchical arrangement of 16 rows for the Madelung Rule electron configurations of the periodic table (Table 1 on Page 6). It has the same rows as the Janet left-step table but with separate rows for the individual waves. Thus the Janet table has eight rows for the eight solutions (the first four intervals), and the Interval table has 16 rows. 4

Galileo’s Law Defines the Subperiods

However, Einstein’s interval equation does not define the length of an interval (i.e., the number of electrons that will fit within it). According to the Pauli exclusion principle, each electron has to have a unique address as a unique set of the four quantum numbers. That issue is resolved by considering the chemical elements as “basic substances” (Scerri 2007, p. 286) that are in freefall with the Earth around the Sun. In this general-relativity approach, Einstein’s definition of freefall acceleration allows ignoring local forces and their effects. Einstein’s visualization for the core idea of general relativity was a man and his tools falling from the roof of a house.(2002, p. 136)

4

GALILEO’S LAW DEFINES THE SUBPERIODS

6

Table 1: The circularly polarized wavefactors of the first

squares of the time-intervals employed in traversing these distances.”(1954, p. 174) In words, Galileo’s law of freefall acceleration is

four intervals (Mazurs’ “cycles”) arrange the periodic table as four symmetrical sets of four rows. Interval Cycle

Solution Period −S or 1

I or 1 +S or 2 −S or 3 II or 2 +S or 4 −S or 5 III or 3 +S or 6 −S or 7 IV or 4 +S or 8

Wave Spin +ct −ct +ct −ct +ct −ct +ct −ct +ct −ct +ct −ct +ct −ct +ct −ct

Columns by Galileo Subperiods

Because for an observer in free-fall from the roof of a house there is during the fall— at least in his immediate vicinity—no gravitational field. Namely, if the observer lets go of any bodies, they remain relative to him, in a state of rest or uniform motion, independent of their special chemical or physical nature. The observer, therefore, is justified in interpreting his state [and the state of the objects with him] as being “at rest”. [Einstein’s italics] Astronauts experience this rest state as weightlessness when their vehicle is in the freefall motion of a gravitational orbit. Similarly, the Earth is in a freefall orbit around the Sun. When local forces are ignored, everything falling with the Earth is also in a freefall orbit. This includes all of the chemical elements comprising the Earth and everything with it. In his book Dialogues Concerning Two New Sciences, Galileo describes the motions of freely falling bodies. His Theorem II, Proposition II states, “The spaces described by a body falling from rest with a uniformly accelerated motion are to each other as the

Total distance in space = (constant acceleration) (time spent in freefall)2 (3) For the purposes of this paper, the constant-acceleration value is “1”. The Observer’s unit of time is called a Duration, and the unit of space is called a Slot. The “time spent in freefall” is the number of durations required for an object to freefall the observed number of slots. A combination of Durations (time) and Slots (space) is called a Configuration. Figure 4 illustrates Galileo’s acceleration law for the first five configurations. A configuration starts at zero and continues to the end of a duration. The use of spatial slots quantizes time and space into integer values.

Fig. 4: Galileo’s law of acceleration applied to define

configurations of electrons. The size of a Configuration (its name) is the number of Durations it contains. Slots: A slot is a region of space that can contain one and only one electron (i.e., one unit matter wave). Durations: A Duration is the number of slots in a unit time of accelerated motion. Corollary 1 of Galileo’s Theorem states the number of slots (spatial units) increases by two in each subsequent duration. Because Duration 1 contains only one slot, Duration 2 has three slots available for electrons to fill. The pattern continues with Duration 3 having five available

5

THE INTERVAL VIEW OF THE PERIODIC TABLE

slots, Duration 4 with seven available slots, etc. Durations determine the size of subshells (subperiods) in the electron configurations of elements. Configurations: A Configuration represents one interval wave and the Durations that define it. Configuration 1 contains only Duration 1 and therefore has only one (1+0=1) available slot. The first four elements, atomic numbers 1-4, are all Configuration 1 elements with each additional electron filling the single slot of one of the four interval waves. In the Janet left-step table, they are divided into wave pairs as the Interval solutions. Thus the two smallest rows of the Janet table each contain two elements. Configuration 2 adds Duration 2 and therefore has four (1+3=4) slots available for filling. Configuration 3 adds Duration 3 and therefore has nine (1+3+5=9) slots. Configuration 4 adds Duration 4 and therefore has sixteen (1+3+5+7=16) slots. The number of available slots in a Configuration is the square of its Interval’s number. Thus the first four intervals have 120 available slots that can be filled sequentially by electrons. Interval Number Interval I Interval II Interval III Interval IV

Calculation 4 waves · 12 slot 4 waves · 22 slots 4 waves · 32 slots 4 waves · 42 slots

Slots 4 16 36 64

An electron’s energy is the energy of a unit matter wave. Figure 5 shows the 120 waves comprising Element 120. Notice how the matter waves must be accelerated (spatially compressed) to fit into the equal freefall durations. This is related to the Madelung Rule though Mazurs’ work with the Janet left-step periodic table.(1974, pp. 96105) His “Cycles” are another name for Intervals, and his “Periods” are another name for Solutions. The doubling of Periods is because Einstein’s electromagneticinterval equation factors into two solutions. For example, in Mazurs’ Table 4 (p. 100), Cycle 4 contains Periods (Solutions) 7 and 8. Each Solution can contain up to 32 elements as the sum of its two 16-slot Configurations. Thus Interval IV (Mazurs’ Cycle 4) can contain up to 64 elements. Mazurs (1974) calls the first part of the Madelung Rule the “Periodic Law”. The Einstein-Galileo definition of Intervals and Solutions is mathematically related

7 120 spatial slots for 120 electrons +S Gravitational Interval (Counterclockwise) 7 slots 5 slots 3 slots 1 slot

l

= 3 “f”

l

= 2 “d”

l

= 1 “p”

l

= 0 “s”

−S Gravitational Interval (Clockwise) 1 slot 3 slots 5 slots 7 slots

Subshell

l

= 0 “s”

l

= 1 “p”

l

= 2 “d”

l

= 3 “f”

Fig. 5: Element 120 contains 120 waves, which is equiv-

alent to 120 electrons. Gravitation attempts to maintain a spherical shape because it takes the least energy to enclose the largest volume. to the Madelung Rule by Mazurs’ definition of a Period. He writes, “The above mentioned mathematical expression of the Periodic Law is as follows:” t =n+l

(4)

His “ordinal number of the period” is t, and n + l is the sum of the first and second quantum numbers. In Interval terminology, n + l is a Solution number plus a Duration number. 5

The Interval View of the Periodic Table

Table 2 shows the condensed periodic-table layout created by using Einstein’s electromagnetic-interval equation for the rows and Galileo’s acceleration equation for the columns. The numbers in the cells are the number of spatial slots in Durations 1, 2, 3, and 4. The last column is the number of available slots in each configuration of that interval. When the condensed layout is expanded to show the individual slots in each duration, the result is the periodic table in Figure 6. (The full-page image is provided on page 13). Each cube in the model of an element represents an orbital pair filled with two electrons. A partial cube represents either the left- or right-handed electron of a pair. Solutions (±S ) grouped under Interval type (I-IV) create the layout of the traditional “Janet left-step” periodic table. It focuses on atomic number rather than the physical and chemical properties of the material. Scerri identifies it as the extreme “Platonic” view of the periodic table.(2007, pp. 282-286)

5

THE INTERVAL VIEW OF THE PERIODIC TABLE

+S or 6 −S or 7 IV (4) +S or 8

Subshell identifiers (e.g., 3d(10)) show principal quantum number n with subshell name and number of electrons. Column headers show quantum numbers “l ” and “m”. The sign of ct for the row is the sign of the spin quantum number. For example, Radium (88) is 7, 0, 0, -½. In the Radium diagram below, each individual block represents one orbital pair of electrons, so its picture includes 44 blocks (orbital pairs).

−S −ct

−S −ct

‡

#8

+S −ct

21 Sc Scandium

G8

l =3

m = –1

m=0

+S −ct

m = +1

m = +2

m = +3

‡

G3

G3

G3

G3

G3

13 Al Aluminum

G16

7 N Nitrogen

G14

10

G17

G5

22 Ti Titanium

G9

‡

G6

23 V Vanadium

24 Cr Chromium

G10

G11

‡

G7

Ne

31 Ga Gallium

G16

G12

Sodium

Na

15 P Phosphorus

17 Cl Chlorine

+‒

Wave

‒+

Wave

12 Mg Magnesium

4s(2) G1 19 K Potassium

G18

G2

18 Ar Argon

20 Ca Calcium

‒‒

Wave

++ Wave

G13 16 S Sulfur

G13

25 Mn Manganese

11

G2 Neon

G15

14 Si Silicon

Wave

Be Beryllium

G1

G18

9 F Fluorine

G13

m = +2

+‒

5s(2) G14

G15

32 Ge Germanium

33 As Arsenic

G17

G18

G1

Wave 37 Rb Rubidium

‒+

G2

Wave 26

Iron

Fe

27 Co Cobalt

28 Ni Nickel

29 Cu Copper

30

Zinc

Zn

G4

39 Y Yttrium

G8

‡

34 Se Selenium

35 Br Bromine

36 Kr Krypton

38 Sr Strontium

5p(6) G5

40 Zr Zirconium

G9

‡

‡

41 Nb Niobium

G10

‡

‡

G6

42 Mo Molybdenum

G11

‡

G7

G13

43 Tc Technetium

49 In Indium

G16

G12

‒‒

6s(2) G15

G14

50

Tin

Sn

G17

51 Sb Antimony

G18

G1

Wave

55 Cs Cesium

++

G2

Wave 44 Ru Ruthenium

45 Rh Rhodium

46 Pd Palladium

47

Ag Silver

48 Cd Cadmium

5d(10) G3

G17

8 O Oxygen

4

‒‒

Wave

++

G3

52 Te Tellurium

53 I Iodine

54 Xe Xenon

6p(6) G4

G5

G6

G7

G13

56

Barium

Ba

+‒

7s(2) G15

G14

G1

Wave

‒+

57 La Lanthanum

G3

‡

64 Gd Gadolinium

G3

9

m = +1

G15

6 C Carbon

5 B Boron

G16

3p(6)

G2

3s(2)

4p(6) G4

G3

59 Pr 60 Nd 58 Ce Praseodymium Neodymium Cerium

G3

G3

65 Tb Terbium

G3

66 Dy Dysprosium

61 Pm 62 Sm Promethium Samarium

G3

67 Ho Holmium

G3

68 Er Erbium

69 Tm Thulium

63 Eu Europium

G3

70 Yb Ytterbium

5f(14)

+S +ct

G3

+S +ct

4f(14) G3

−S +ct

−S +ct

#6

n=5

m = –2

l =2 m=0

4d(10)

n=6

m = –3

m = –1

3d(10)

#5

n=7

n=4 n=3 n=2 n=1

m = –2

G14

Li Lithium

‡

71 Lu Lutetium

G8

76 Os Osmium

72 Hf Hafnium

G9

77 Ir Iridium

73 Ta Tantalum

74 W Tungsten

‡

‡

G10

78 Pt Platinum

G11

79

Gold

Au

75 Re Rhenium

G12

G16

80 Hg Mercury

6d(10) G3

‡

G3

‡ G3

‡

G3

‡

G3

G3

G3

81 Tl Thallium

84 Po Polonium

82

Lead

Pb

G17

85 At Astatine

83 Bi Bismuth

G18

86 Rn Radon

7p(6) G4

G5

G6

G7

G13

87 Fr Francium

G2

Wave

‒‒

88 Ra Radium

8s(2) G14

G15

Wave

++ 89 Ac Actinium

G3

‡

96 Cm Curium

m = –3

90 Th Thorium

G3

91 Pa Protactinium

92 U Uranium

93 Np Neptunium

94 Pu Plutonium

95 Am 103 Lr Americium Lawrencium

G3

G3

G3

G3

97 Bk Berkelium

98 Cf Californium

99 Es Einsteinium

100 Fm Fermium

101 Md 102 No Mendelevium Nobelium

m = –2

m = –1

m=0 l =3

m = +1

m = +2

G3

m = +3

104 Rf 105 Db Rutherfordium Dubnium

G8

G9

108 Hs Hassium

109 Mt Meitnerium

m = –2

m = –1

106 Sg Seaborgium

G3

107 Bh Bohrium

Predictions of configurations vary.

G10

‡

G11

‡

110 Ds 111 Rg Darmstadtium Roentgenium

m=0 l =2

m = +1

113 Nh Nihonium

114 Fl Flerovium

G12

G16

112 Cp Copernicium

116 Lv Livermorium

G17

117 Ts Tennessine

m = +2

m = –1

m=0 l =1

115 Mc Moscovium

Element 119

G18

118 Og Oganesson

m = +1

Wave

+‒

Solution 8

III (3)

This value indicates spin direction predicted by the ideal pattern rather than actually measured.

G13

m = +1

2s(2)

Interval IV

−S or 5

Orbital Pair

3(f )

+S +ct +S −ct

#4

=

2(d)

#7

#3

2p(6)

He

G1 3

‒+

Wave

Helium

Solution 7

+S or 4

−S +ct −S −ct

+

l =1 m=0

m = –1

All assignments of signs and directions are arbitrary.

ms = -½ ms = +½

+S +ct +S −ct

#2

Electron-configuration illustrations were drawn from 2005 standard text list at http://pearl1.lanl.gov/ periodic/downloads/periodictable.pdf and IUPAC.

Spin Number “ms”: The identifier for each of two electrons in an orbital pair.

0 (s) 1(p)

4

Symbol

2

Wave

Solution 6

II (2)

Each cube is an electron pair.

1 H Hydrogen

G18

Interval III

−S or 3

1

Azimuthal Number “l ” Old letters s, p, d and f for “subshell” & now quantum numbers 0, 1, 2, & 3 are the numbered units of time.

−S −ct

Solution 5

+S or 2

d2

nucleus

0

−S +ct

#1

Exception to ideal filling sequence.

12 Mg Magnesium

Name

+1 +2 -2 -3 -1 +1 +3 0 +2 -2 -1 +1 0

f = 3 d= 2 p=1 s= 0

G2

Atomic Number

0 -1

‡

Solution 4

I (1)

Duration d 1 2 3 4 1 1 1 1 1 3 1 3 1 3 1 3 1 3 5 1 3 5 1 3 5 1 3 5 1 3 5 7 1 3 5 7 1 3 5 7 1 3 5 7

n=7 n=6 n=5 n=4 n=3 n=2 n=1

IUPAC Group Membership

G1

Interval II

−S or 1

Wave Spin +ct −ct +ct −ct +ct −ct +ct −ct +ct −ct +ct −ct +ct −ct +ct −ct

Magnetic Number “m” Identifies each orbital electron pair within a subshell (time unit).

l =0 m=0

1s(2)

Solution 3

Solution Period

Principal Number “n” The “shell” number shown by distance from nucleus.

Element Key

Solution 2

Interval Cycle x

e− =

Periodic Table Organized by SR Intervals and Solutions Relating the Four Quantum Numbers: n, l , m and ms

Interval I

Galileo’s acceleration law defines the number of electrons in each freefall Configuration. Four intervals, each with four waves, defines 16 rows with (Cycle x)2 electrons in each row, (1x4)+(4x4)+(9x4)+(16x4)=4+16+36+64=120 total.

Solution 1

Table 2:

8

Element 120

m=0 l =0

Fig. 6: The Interval version of the left-step (Janet) tradi-

16

Modeling Solutions within Intervals is the most logical layout because it groups the orbital pairs (right- and left-handed waves) under their solutions. It defines the Madelung filling sequence shown in Figure 7. Two example elements, Radium 88 and Element 120, are shown in Figure 8. Each cube is one orbital pair and there are no unpaired electrons in either element. The middle image shows the locations of the 16 orbital pairs that must be added to Radium (on the left) to obtain Element 120 (on the right). The 16 orbital pairs are added in the subperiod sequence 5f, 6d, 7p, and 8s. They create a stair-step arrangement (the top arrow shown in Figure 7). To make visualization easier, the element depictions map all four waves into a single quadrant of a sphere. If the mapping showed one wave per quadrant, the overall shape would be spherical. The four quantum numbers map perfectly to this arrangement of elements in the periodic table. The first quantum number has “1” as the lowest energy level and increments by “1” for each higher level. These are the

tional form of the periodic table. A full-page image is provided after Section 7.

Fig. 7: The filling diagram for the empirical Madelung

energy ordering of the orbitals. It shows how the filling process attempts to maintain the most efficient structure in spacetime, a sphere.

5

THE INTERVAL VIEW OF THE PERIODIC TABLE 8s 7s 6p

5d 4f 3d 2p 1s Radium 88

+

+

7p 6d 5f

16 Orbital Pairs

=

=

Element 120

9 The third quantum number identifies the orbital pair (cube) within a subperiod. The central cube is designated as zero (no angular momentum), and the plus and minus values are to each side. The right pane illustrates the representation of the fourth quantum number, which identifies the handedness of the two members of the orbital pairs. The layout of the interval table represents a pair’s two electrons in the same column in the contiguous rows of a solution.

Fig. 8: The element on the left is Radium with 88 elec-

numbers written to the left of Radium and to the left of the added 16 orbital pairs in Figure 8. They are also the energy levels written on the vertical axis in the left pane of Figure 9. The second quantum number is designated by the older subperiod letters to keep the symbols unambiguous. For Radium’s seven energy (shell) levels in Figure 8, the subperiod sequence of the visible cubes (orbital pairs) is s, p, d, f, d, p, s. Figure 9 is the legend explanation that illustrates how the four quantum numbers define a specific electron location (chemical elements) in the periodic table. The left pane shows a sideways view of the first and second quantum numbers. The first quantum number identifies the energy level (Solution) in the stack of eight levels on the vertical axis. The second quantum number would be represented as a plane perpendicular to the axis point for each energy level. However, for ease of visualization, all four interval waves (±S ± ct) were mapped into a single quadrant rather than one wave per quadrant. Imagine the subperiods as a set of concentric rings with the s subperiod at the central axis of rotation. The second quantum number identifies the ring containing the electron. Each ring larger than s is represented as a set of connected cubes. The middle pane shows the top view of the quadrant with the second and third quantum numbers. The single cube labeled s is at the center. The sets of connected cubes marked p, d and f are the 3, 5, and 7 orbital pairs in those subperiods. The numbers written in parentheses by the letters are the second quantum number.

1st & 2nd numbers (Side View) 1st Principal Number “n”, the primary energy level; the shell label

7654321

trons. Adding 16 orbital pairs in the stair-step configuration of four n levels forms Element 120. The spherical shape is not shown because all four waves are being modeled in a single quadrant.

0 1 2 3

Azimuthal 2nd Number “l ”, orbital angular momentum; the subshell label

2nd & 3rd numbers (Top View) s(0) 0 p(1) 2nd -1 +1 d(2)

4th quantum number (Block View)

Magnetic Number “m”, the block location relative to the 0 baseline for its angular momentum

ms value indicates spin direction relative to the orbital angular momentum of its block

Each HALF-BLOCK is one ELECTRON.

-2 0 +2 f(3) + -3 -1 +1 +3 4th -2 0 +2 ms = -½ ms = +½ -1 +1 3rd 0 Spin Number “ms”, the

Fig. 9: Each electron has an address determined by its

four quantum numbers. The left pane shows the first and second quantum numbers as energy level (vertical axis) and orbital angular momentum (horizontal axis). The third quantum number is defined in the middle pane. Its sign and value identifies the location of an orbital pair (a block) from the central zero location for its subperiod. The right pane splits an orbital pair to show how the left- and right-handed electrons fit together into a block. This represents the fourth quantum number. The rows of the Janet left-step periodic table in Figure 12 on page 13 are perfectly organized by the electromagnetic intervals, solutions within intervals, and waves within solutions. The interval-wave information is in the rightmost columns of the table. The solution and wave types are also shown to the left of the element display. The columns are organized from right to left by the size of the configuration for each interval. The stair-step form of the layout occurs because of the increasing configuration size (1, 4, 9 and 16 slots). The group header for a Duration provides the second quantum number. The individual column headers in the periodic table provide the third (magnetic) quantum number for each column. The fourth (spin) quantum number is pictorially represented in the models of the elements with a

6

EXPLAINING THE FILLING ANOMALIES

10

half-cube representing one electron and the whole cube representing the completed orbital pair. The Element Key in Figure 10 identifies additional information available for each element. It includes the element’s name, symbol, atomic number, IUPAC group membership, and whether it is an exception to the filling sequence predicted by the Madelung Rule.

Element Key IUPAC Group Membership Atomic Number Name

G2

‡

Exception to ideal filling sequence.

12 Mg Magnesium

Each cube is an electron pair. Symbol

Fig. 10: The Element Key identifies the additional infor-

mation about each element. The anomalies in the positioning of electrons during the filling sequence occur when the real configuration differs from the ideal positioning. In an enlarged view, these are visually obvious from the positions of the electrons. Anomalies are identified by the symbol O ‡. 6

Fig. 11: The exceptions to the ideal filling sequence.

The circled elements have an electron that goes early to a higher energy level. The arrows indicate the slot relationships that are filled out of sequence.

Explaining the Filling Anomalies

Changing the hierarchy of variables provides new arrangements of the periodic table for visual exploration. There are three organizing variables: Intervals (I-IV), Solutions (±S ), and Handedness (±ct). The unusual layout in Figure 11 shows the repeated patterns created by anomalies in the filling sequence. This layout stacks the increasing energy levels (Solutions 1-8) from bottom to top, like the stem of a plant. The ‘leaves’ on the left are the +ct waves, and those on the right are the −ct waves. Text describing each anomaly is in Table 3 on page 12. The four different patterns of parallel arrows in Figure 11 indicate different anomaly loops that repeat once or several times at higher energy levels. The anomalous filling sequences for Chromium (24) and Copper (29) have been used as examples of the failure of the quantum numbers to predict, or even justify, the experimentally observed sequence. Scerri wrote: (2004, p. 103) Only if shells filled sequentially, which they do not, would the theoretical relationship between the quantum numbers provide a

purely deductive explanation of the periodic system. The fact that the 4s orbital fills in preference to the 3d orbitals is not predicted in general for the transition metals but only rationalized on a case by case basis as we have seen. In some cases the correct configuration cannot even be rationalized, as in the cases of chromium and copper, at least at this level of approximation. In other words, the Madelung Rule “cannot be strictly derived from first principles, although approximations used in quantum chemistry can account, after the facts, for all spectroscopically observed configurations.”(Scerri 2000) However, the layout in Figure 11 is organized by Solutions, not Shells. The obviously repeated patterns of anomalies provide logical reasoning rather than caseby-case empirical facts. In addition, Table 3 on page 12 identifies the 10 patterned loops of anomalies, and all of them involve either starting or completing a d subperiod early.

7

EXPLORING WAVE-BASED MODELS OF REALITY

This unusual layout groups together the waves with the same direction of rotation. These wave pairs create the quantum unit of angular momentum (action). Given that the quantum of action ~ is also the atomic unit of angular momentum,(Cardarelli 1997) this needs to be mathematically explored with Schr¨odinger’s equation and perhaps with other methods. Also, as described in Section 1, the Madelung Rule based on period (Solutions) does predict in general that the “4s orbital fills in preference to the 3d orbitals”. The 1946 “Periodic Law”, which is Ordinal number f or period = 1 st quantum number + 2nd quantum number, (5) is now justified by acknowledging the periods as pairs of Interval solutions. The values (1,2,...8) of the first quantum number n is the Solution number incremented from “1” for each solution in the first four intervals. Thus the Madelung Rule (n + l, n) predicts the following “Aufbau” filling sequence: 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4 f < 5d < 6p < 7s < 5 f < 6d < 7p < 8s Similarly, Mazurs’ rule for the number of elements in each cycle (Interval) is Lcycle = 4x2 . Four is the number of waves, and x is interval number. The square is the result of Galileo’s acceleration law. Thus the interval sizes as number of elements held are I:4, II:16, III:36, and IV:64, for a total of 120 elements. Behind all of this is the ruling principle of symmetry. The perfect symmetry of electromagnetic intervals (light waves) governs reality. Any result that indicates broken symmetry indicates relevant data are being omitted. It is then useful to reframe the problem and identify the symmetry(-ies). 7

Exploring Wave-Based Models of Reality

The factoring of Einstein’s electromagnetic-interval equation provides a principled reason for requiring photons to be orbital pair intersections (interval solutions) rather than continually existing particles. Einstein described particles as denser regions of a field.(Einstein 2002/1920, p. 180) Also, the focus is on symmetrical pairs of waves rather than individual waves. That is, each solution of an Interval contains two waves—a single wave with its

11

canonical conjugate. This is normal in quantum-wave mechanics, but here it is a principled definition of an interval solution rather than just an aid to calculation. Schr¨odinger wrote: (1982, p. 123) Meantime, there is no doubt a certain crudeness in the use of a complex wave function. If it were unavoidable in principle, and not merely a facilitation of the calculation, this would mean that there are in principle two wave functions, which must be used together in order to obtain information on the state of the system. Matter waves are a perfectly correlated representation of material objects because the product of a wavephase speed and its associated particle’s speed is the constant lightspeed c2 .(de Broglie 1929) This wave-based view of physical reality is an unusual perspective. Two more quotes from Schr¨odinger are particularly relevant to modeling a wave-based reality: It is hardly necessary to emphasize how much more congenial it would be to imagine that at a quantum transition the energy changes over from one form of vibration to another, than to think of a jumping electron. (1982, pp. 10-11) [L]ight of a definite frequency is always capable of producing the same physical effects as electrons of a definite velocity. From the fact of this equivalence the opposite conclusion can, however, be drawn with the same inevitability, or lack of inevitability: the electron moving with a definite velocity must be a wave phenomenon whose frequency is that of the light which is experimentally equivalent to it with regard to the excitation of resonance.(1967, p. 67) Physicists have chosen the particle-based probabilistic interpretation of Schr¨odinger’s “wave mechanics”. (Schr¨odinger 1982) However, as Schr¨odinger, Heisenberg, Einstein and Feynman strongly stated in the quotes in this paper, matter waves are physically real. There is much more to be explored with Maxwell’s A-field matter waves defined as the building blocks of both physics and chemistry.

7

EXPLORING WAVE-BASED MODELS OF REALITY

12

Table 3: Anomaly loops that break and then restore the Madelung filling sequence for the 19 anomalies identified

by Meek and Allen. There are only two types of anomalies: (a) a d shell pulls an electron from the s shell, or (b) a d subperiod receives what should go into the f subperiod. The first element in each loop identifies the anomaly and the other row(s) identify how it is resolved. When there is a shift in location for an existing electron, it occurs with the addition of the new electron. Electron-configuration models were drawn using information from the 2005 standard text list at http://pearl1.lanl.gov/periodic/downloads/periodictable.pdf and IUPAC. (Link now broken.) # Loop 1 24 25 Loop 2 29 30 Loop 3 41 42 43 Loop 4 44 45 46 47 48 Loop 5 57 58 59 Loop 6 64 65 Loop 7 78 79 80 Loop 8 89 90 91 92 93 94 Loop 9 96 97 Loop 10 110 111 112

Name Anomaly 24 Chromium Manganese Anomaly 29 Copper Zinc Anomaly 41 Niobium Molybdenum Technetium Anomaly 44 Ruthenium Rhodium Palladium Silver Cadmium Anomaly 57 Lanthanum Cerim Praseodymium Anomaly 64 Gadolinium Terbium Anomaly 78 Platinum Gold Mercury Anomaly 89 Actinium Thorium Protactinium Uranium Neptunium Plutonium Anomaly 96 Curium Berkelium Anomaly 110 Darmstadtim Roentgenium Copernicium

Abb.

Normal

Shift

To

Explanation

Cr Mn

d4

s2

d5 s2

Fills 3d 1e− orbitals early Restores s2 in 4s

Cu Zn

d9

s2

d10 s2

Completes 3d orbitals early Restores s2 in 4s

Nb Mo Tc

d3 d5

s2

d4 s2

Will fill 4d 1e− orbitals early Fills 4d 1e− orbitals early Restores s2 in 5s

Ru Rh Pd Ag Cd

d6

s2

d9

s1

d7 d8 d10 s1 s2

Will complete 4d early Completes next orbital Completes 4d orbitals early Restores s1 Restores s2 Start filling 5d before starting 4f Start filling 4f Restore normal filling 4f

La Ce Pr

f3

d1

d1 f1 f2

Gd Tb

f2

d1

d1 f1

Start completing 5d before completing 4f Restore normal completing 4f

d8

s2

d9 d10 s2

Will complete 5d early Completes 5d early Restores s2 Start filling 6d before starting 5f Continue filling 6d Start filling 5f Continue filling 5f Continue filling 5f Restores normal filling

Pt Au Hg Ac Th Pa U Np Pu

f6

d1

d1 d2 f1 & f2 f3 f4 f5

Cm Bk

f2

d1

d1 f1

Start filling 6d before completing 5f Restores normal filling

d8 d10

s2

d9

Will complete 6d early Completes 6d early Restores s2

Ds Rg Cp

d2

s2

7

EXPLORING WAVE-BASED MODELS OF REALITY

13

Fig. 12: The Interval version of the left-step (Janet) traditional form of the periodic table. The intervals I-IV are

identified in the rightmost column, and each interval’s two solutions (periods) are grouped inside it. Then each set of two waves (an orbital pair) are grouped inside each solution. The columns of elements are the spatial slots within each duration (subperiod/2). The mapping of columns and rows to quantum numbers is explained in the figure legend and in the text.

REFERENCES References Bohr, Niels (1913). “On the Constitution of Atoms and Molecules, Part 1.” In: Philosophical Magazine 26.151, pp. 1–24. doi: doi:10.1080/14786441308634955 (cit. on p. 2). — (1922). ”Niels Bohr- Nobel Lecture: The Structure of the Atom: Nobelprize.org. Nobel Media AB 2014. url: http://www.nobelprize.org/nobel_ prizes/physics/laureates/1965/feynmanlecture.html 19 Mar. 2017 (cit. on pp. 2, 3). — (1950). “The Structure of the Atom and the Physical and Chemical Properties of the Elements.” In: Autobiography of Science. Ed. by Moulton, Forest Ray et al. Originally published as Fysisk Tidsskrift 19, 153 (1921) translated by A. D. Udden, ”The Theory of Spectra and Atomic Constitution–Three Essays” (Cambridge, 1922). Doubleday. url: https: //web.lemoyne.edu/giunta/bohr2.html 30 July 2018 (cit. on p. 3). Cardarelli, Franc¸ois (1997). Scientific unit conversion. Springer (cit. on p. 11). de Broglie, Louis (1929). ”Louis de Broglie - Nobel Lecture: The Wave Nature of the Electron”. Nobelprize.org. Nobel Media AB 2014. url: http://www.nobelprize.org/nobel_ prizes/physics/laureates/1929/broglielecture.html 17 June 2017 (cit. on pp. 4, 11). — (1962). New Perspectives in Physics: Where Does Physical Theory Stand Today? Trans. by Pomerans, A. J. New York: Basic Books (cit. on p. 3). Einstein, Albert (2002/1920). “Ether and the Theory of Relativity.” In: The Collected Papers of Albert Einstein, Volume 7: The Berlin Years Writings, 1918-1921. Trans. by Engel, Alfred. Lecture quotes from 1922 book, Sidelights on Relativity. Princeton, NJ: Princeton University Press, Doc. 38, 160–182 (cit. on p. 11). — (1967). Letters on Wave Mechanics: Correspondence with H. A. Lorentz, Max Planck, and Erwin Schr¨odinger. Ed. by Przibram, K. 2011 edition distributed by

14 Open Road Integrated Media. New York: Open Road (cit. on pp. 4, 11). — (1970). “Remarks Concerning the Essays Brought Together in This Co-operative Volume.” In: Albert Einstein: Philosopher-Scientist. Ed. by Schilpp, P. 3rd. Open Court, pp. 665–688 (cit. on p. 4). — (1998). “On the Electrodynamics of Moving Bodies.” In: Einstein’s Miraculous Year: Five Papers That Changed the Face of Physics. Princeton, NJ: Princeton Univ., pp. 123–160 (cit. on p. 4). — (2002). “Fundamental Ideas and Method of the Theory of Relativity, Presented in Their Development.” In: The Collected Papers of Albert Einstein, Volume 7: The BerlinYears Writings, 1918-1921. Trans. by Engel, Alfred. Princeton University Press, Doc., 31, 113–150 (cit. on p. 5). Feynman, Richard et al. (1964). The Feynman Lectures on Physics: Mainly Electromagnetism and Matter. Vol. 2. Reading, MA: Addison-Wesley (cit. on p. 3). Galilei, Galileo (1954). Dialogues Concerning Two New Sciences. Trans. by Crew, Henry et al. New York, NY: Dover (cit. on p. 6). Heisenberg, Werner (1949). The Physical Principles of the Quantum Theory. Trans. by Eckart, Carl et al. Dover (cit. on p. 3). — (1955). “The Development of the Interpretation of the Quantum Theory.” In: Niels Bohr and the Development of Physics. Ed. by Pauli, Wolfgang. New York, NY: McGraw-Hill, pp. 12–29 (cit. on p. 4). — (1974). Across the Frontiers. Ed. by Anshen, Ruth Nanda. Trans. by Heath, Peter. Harper (cit. on p. 3). — (2007). Physics and Philosophy: The revolution in modern science. New York, NY: HarperPerennial (cit. on p. 4). L¨owdin, Per-Olov (1969). “Some Comments on the PePeriod System of the Elements.” In: International Journal of Quantum Chemistry III S, pp. 331–334 (cit. on p. 2).

REFERENCES Mazurs, Edward (1974). Graphic representations of the periodic system during one hundred years. Revised 2nd. University AL: University of Alabama Press (cit. on pp. 1, 7). Minkowski, H. (1952). “Space and Time.” In: The Principle of Relativity. Ed. by Lorentz, H. A. Trans. by Perrett, W. et al. New York: Dover. Chap. V, pp. 75–91 (cit. on p. 4). Misner, Charles W. et al. (1973). Gravitation. W. H. Freeman (cit. on p. 4). Rosenfeld, L. (1983). “Born’s Probabilistic Interpretation.” In: Quantum Theory and Measurement. Ed. by Wheeler, John Archibald et al. Princeton University Press, pp. 50–51 (cit. on p. 3). Scerri, Eric R. (2000). “The Failure of Reduction and How to Resist Disunity of the Sciences in the Context of Chemical Education.” In: Science & Education 9, pp. 405–425. doi: https://doi.org/10.1023/A: 1008719726538 (cit. on p. 10). — (2004). “Just how ab initio is ab initio quantum Chemistry?” In: Foundations of Chemistry 6. A previous version of this article appeared as ‘L¨owdin’s Remarks on the Aufbau Principle and a Philosopher’s View of Ab Initio Quantum Chemistry’ in E.J. Br¨andas, E.S. Kryachko (Eds.) Fundamental World of Quantum Chemistry, Vol. II, 675–694, Kluwer, Dordrecht, 2003., pp. 93–116 (cit. on p. 10). — (2007). The Periodic Table: Its Story and Its Significance. Oxford: Oxford University Press 1936. url: http://amzn.to/wYizkM (cit. on pp. 5, 7). — (2016). “The changing vview of a philosopher of chemistry on the question of reduction.” In: Essays in the Philosophy of Chemistry. Ed. by Scerri, E. R. et al. Oxford University Press (cit. on p. 2). Scerri, Eric R. et al. (1998). “Ordinal Explanation of the Periodic System of Elements.” In: International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 6.4, pp. 387–399. doi: https:

15 //doi.org/10.1142/S0218488598000318 (cit. on p. 2). Schr¨odinger, Erwin (1982). Collected papers on wave mechanics: together with his four lectures on wave mechanics. 3rd Augmented. Originally published in Annalen der Physik (4), vol. 79, 1926. Providence RI: AMS Chelsea (cit. on p. 11). Sommerfeld, Arnold (1923). Atomic Structure and Spectral Lines. Trans. by Brose, Henry L. London: Methuen and Co. url: https://archive.org/stream/ AtomicStructureAndSpectralLines/ (cit. on p. 2).

1

Identifying L¨owdin’s Challenge

the arrangement of the periodic table. Mazurs (1974, pp. 96-105, esp. Tables 3, 4, & 6) discusses in detail the transition from modeling the

Bohr proposed a two-part (n + l, n) empirical heuristic rule to explain the periodic system and its filling sequence: 1. The energy of a state is the sum of the first quantum number n and second quantum number l (the n + l part of the rule). 2. When two states have the same value for n + l, the larger energy state has the larger n (the n part of the rule). Bohr’s rule has various names, include the Aufbau rule or the Madelung rule. It has two uses: (1) identifying the overall electron configuration of an atom of each chemical element, and (2) identifying the sequence of elements created as electrons are added to configurations. Thus it is used to describe and empirically explain

Shells>Subshells>Orbitals arrangement of the periodic table to the Periods>Subperiods>Orbitals arrangement. The result is Janet’s left-step form of the periodic table that is based on electron configurations. A configuration is a grouped set of orbital pairs of electrons. The energy levels of subperiod groups (e.g., 2p, 3s, 3p, 4s, 3d) are ordered by the two parts of the Madelung Rule. Subperiod names are the first part of the Madelung Rule without the plus sign, the first (principal) quantum number n and a letter representing the 1

2

QUANTUM MODELING OF THE PERIODIC TABLE

second quantum number l (s = 0, p = 1, d = 2 and f = 3). For example, the two subperiods 3s = 3 + 0 and 2p = 2 + 1 both have the value n + l = 3. Because 2 < 3, the subperiod 2p has a smaller energy and is filled before 3s. For 4s = 4 + 0 and 3p = 3 + 1, the value is n + l = 4. Thus 3p and then the 4s are filled. Because 3d = 3 + 2 = 5 and 4 < 5, 4s fills before 3d and this has been experimentally observed. In his discussion of Bohr’s rule, L¨owdin challenged his readers to study (derive) the rule from first principles.(1969, p. 334) He suggested starting with the manyelectron Schr¨odinger’s equation, but attempts that start with quantum mechanics have not been successful. Scerri et al. describe four types of attempts to explain Bohr’s rule: • approximate (heuristic) solutions to the original equations, • new (heuristic) equations, • symmetry groups, and • solving the corresponding equations of quantum mechanics. However, none of these approaches successfully explain Bohr’s (Madelung) Rule. “In short: in spite of all known attempts, the problem of explaining the periodic system is still far from being solved”.(1998) Scerri also writes, “Although the above Madelung rule succeeds in giving the overall configuration of the transition metals beginning with scandium, or element 21, the order of filling is not provided by this rule.”(2016, p. 135) This paper presents a new principled approach using Einstein’s 1905 electromagnetic-interval equation and Galileo’s law of acceleration. Combining these two equations provides a principled explanation of Bohr’s (Madelung) Rule and the Janet left-step form of the periodic table. It also perfectly maps the quantum numbers into the periodic table. In addition, it identifies patterns of observed anomalies in the expected filling sequence as 10 anomaly loops (from sequence deviation to restoration) that involve multiple elements. The loops range in size from the minimum of two elements to loops of six elements.

2

2 Quantum Modeling of the Periodic Table

When Bohr modeled the atom (1913), he modeled the orbits of non-radiating electrons. He wrote that the electrons’ motions will describe stationary elliptical orbits. The frequency of revolution ω and the majoraxis of the orbit 2a will depend on the amount of energy W which must be transferred to the system in order to remove the electron to an infinitely great distance apart from the nucleus. However, Bohr’s original theory “places no restriction on the eccentricity of the orbit, but only determines the length of the major axis.” (Bohr 1922, p. 17) Thus it was inadequate to describe the electron configurations of atoms completely. He wrote: (1922, p. 25) For a simply periodic motion such as we meet in the pure harmonic oscillator, and at least to a first approximation, in the motion of an electron about a positive nucleus, the manifold of stationary states can be simply co-ordinated to a series of whole numbers. For motions of the more general class mentioned above, the so-called multiply periodic motions, however, the stationary states compose a more complex manifold, in which, according to these formal methods, each state is characterized by several whole numbers, the so-called ”quantum numbers”. Sommerfeld used three quantum numbers to describe the electron configurations of atoms.(1923, pp. 234, 392) Physicists have updated Sommerfeld’s work and the interpretations of his quantum numbers are now based on Schr¨odinger’s equations. • The first (principal) quantum number n originally described the major axis of the ellipses modeling electrons’ orbits. It has been interpreted as describing an orbital’s size, which also represents the electron’s energy level. Its integer values are ”1” or larger. • The second (azimuthal or angular or orbital) quantum number l originally described the minor axis of an orbit’s ellipse. It is now interpreted as described an orbital’s shape. It can take integer values from zero to n − 1. Zero values represent a

2

QUANTUM MODELING OF THE PERIODIC TABLE sphere, and larger integers represent more complex three-dimensional shapes. • The third (magnetic) quantum number m or mi quantized the elliptical orbit into a set of spatial regions located at different positions in the orbit. It is interpreted as an orbital’s orientation in space. It can take integer values from −l to +l.

In 1925 Pauli introduced the fourth (spin) quantum number σ or m s with the binary values ± 12 for electrons. The sign indicates the direction of ‘spin’. Each electron must have a unique set of four quantum numbers, so the spin number distinguishes the electrons with left- or right-handed spin in an orbital pair. (The assignment of direction is arbitrary.) Bohr’s work was based on the unique set of atomicspectral lines emitted by each chemical element. He used this information to create a principled explanation of the periodic table (system). (1922, p. 31) The ideas of the origin of spectra outlined in the preceding have furnished the basis for a theory of the structure of the atoms of the elements which has shown itself suitable for a general interpretation of the main features of the properties of the elements, as exhibited in the natural system. Although his theory is outdated, it contains the fundamental principle of symmetrical harmonics explored further in this paper. Bohr wrote: (1950) Let us consider an electrodynamic system and inquire into the nature of the radiation which would result from the motion of the system on the basis of the ordinary conceptions. We imagine the motion to be decomposed into purely harmonic oscillations, and the radiation is assumed to consist of the simultaneous emission of series of electromagnetic waves possessing the same frequency as these harmonic components and intensities which depend upon the amplitudes of the components. An investigation of the formal basis of the quantum theory shows us now that it is possible to trace the question of the origin of the radiation processes which accompany the various transitions back to an investigation of the various harmonic com-

3 ponents, which appear in the motion of the atom. . . . This principle of correspondence at the same time throws light upon a question mentioned several times previously, . . . the relation between the number of quantum numbers, which must be used to describe the stationary states of an atom, and the types to which the orbits of the electrons belong. The classification of these types can be based very simply on a decomposition of the motion into its harmonic components. [italics added]

Quantum theory, whether in the form originated by Bohr, Sommerfeld, Heisenberg or Schr¨odinger, is based on quantum waves and their harmonic relationships. That is why Born had to introduce probabilities to explain particle collisions.(Rosenfeld 1983) The quantum waves are de Broglie’s “phase waves”. (de Broglie 1962, p. 109) They are also called “matter waves” because they comprise the mass of material objects. Thus the principled explanation of the arrangement of the periodic table is based on matter waves. Heisenberg wrote: (1974, p. 14) Physicists had long since been acquainted with such fields of force as the gravitational field and electromagnetic forces. To these had been added in the present [20th ] century the matter waves, which can also be described as force fields of the chemical bond. He also wrote, “[T]he electron may also be considered to be a plane de Broglie wave”.(1949, p. 23) Matter waves are waves of potential in Maxwell’s A field, the field described by Schr¨odinger’s quantumwave equation. As Feynman wrote, there is no way to remove the A field from Schr¨odinger’s equation even though many physicists have attempted to remove it. (1964, p. 15-12) Other quotes from Nobel Laureates in Physics emphasize the importance and use of de Broglie’s matter waves to describe physical objects. • “As early as 1928 it was shown by Jordan, Klein and Wigner that the mathematical scheme can be interpreted not only as a quantization of particle motion but also as a quantization of three-

3

MATTER-WAVE FACTORS OF EINSTEIN’S EQUATION dimensional matter waves; therefore, there is no reason to consider these matter waves as less real than the particles.”(Heisenberg 2007, pp. 107-108) • “[O]nly the waves in configuration space, that is the transformation matrices, are probability waves in the usual interpretation, while the three-dimensional material waves are not. The latter, according to Bohr and to Klein, Jordan and Wigner, have just as much (and just as little) ‘objective reality’ as particles; they have no direct connection with probability waves, but have a continuous density of energy and of momentum, like a Maxwell field.”(Heisenberg 1955, p. 24) • “The electron can no longer be conceived as a single, small granule of electricity; it must be associated with a wave and this wave is no myth; its wavelength can be measured and its interferences predicted.”(de Broglie 1929, p. 256) • “If our concern is with macroscopic masses (billiard balls or stars), we are operating with very short de Broglie waves, which are determinative for the behavior of the center of gravity of such masses.”(Einstein 1970, p. 682) • In 1950 Schr¨odinger wrote to Einstein, “I have really believed for a long time that the ψ-waves are to be identified with waves representing disturbances of the gravitational potential; not, of course, with those you [Einstein] studied first, but rather with ones that transport real mass, i.e., a non-vanishing T ik .”(Einstein 1967, p. 36)

In summary, matter waves are the A-field quantum waves of Schr¨odinger’s equation. According to Heisenberg, an electron is a plane de Broglie matter wave, and according to Einstein matter waves determine the behavior of macroscopic masses. Those statements indicate that correctly modeling matter waves is the key to principled modeling of the Periodic Table of chemical elements. How matter waves determine the Madelung Rule for the overall energy of electron configurations of atoms is described in the remainder of this paper. 3

Matter-Wave Factors of Einstein’s Equation

One of the papers Einstein published in 1905 (Einstein 1998) defined the same type of electromagnetic volumes of space and time as those used by human sensory perception. However, his model of finite volumes was not

4

explored by physicists. In 1908 the mathematician Hermann Minkowski changed Einstein’s equation to model imaginary four-dimensional (4D) spacetime.* † Einstein’s 1905 electromagnetic volumes are real intervals in which the diagonal of the observed spatial volume equals the observed time t multiplied by lightspeed c. His Equation 1 (Page 135 and endnote 3, p. 160 in Einstein 1998) defines three-dimensional volumes (length x width x height) as shown in Figure 1. x2 + y2 + z2 = (ct)2

lS

P1

spacex

(1)

P2

na ct go ce a i n d a l a ist ati td spacez h sp g li o s l &a spacey

Fig. 1: Einstein’s electromagnetic “ct” vector that rep-

resents separation in time decomposes into the three orthogonal directions of space Taking the square root of both sides of Equation 1 defines the volume’s diagonal distance (the separation between opposite corners) in space and in time. In other words, ‘space’ is the vectorial decomposition of the ct distance vector for ‘time’. Equation 1 can be rewritten and rearranged so that it is the factorable polynomial equation in Equation 2c. The summary form of the rearranged equation factors into four components—two entangled pairs of symmetrically opposing circularly polarized matter waves (Equations 2d and 2e). Einstein: s2x + s2y + s2z − (lightspeed c · time)2 = 0 (2a) * Minkowski classified matter and radiation as one “substance” and required that the interval be timelike (> 1). He also introduced an imaginary number ict for measurements of time that, when squared, converted Einstein’s minus sign (subtraction) to a plus sign (addition).(Minkowski 1952) † Minkowski’s change converted Einstein’s original finite volumes in three dimensions to lightlike cones in four dimensions. Misner, Thorne and Wheeler’s Box 2.1 (Farewell to ‘ict’) discusses this and other changes Minkowski made to the equation. They note that for Einstein’s electromagnetic (zero) intervals, the two separated points may be in different galaxies.(Misner et al. 1973, pp. 51, 437).

4

GALILEO’S LAW DEFINES THE SUBPERIODS

5

s2x + s2y + s2z = S 2D & substitute S 2D into equation. (2b) Factor (S 2D − ctime2 ) into symmetrical solutions. (2c) + S D solution : (+S D + ctime)(+S D − ctime) (2d) − S D solution : (−S D + ctime)(−S D − ctime) (2e) The four waves are ±S ± ct, and the two entangled pairs (interval solutions) are +S ±ct and −S ±ct. The ±S term indicates the starting location and the two solutions differ in phase by one-half cycle (Figure 2). The sign of the ct term indicates the direction of rotation. All assignment of direction is arbitrary because there is no conventional consensus. Fig. 3: The four wave-factors of Einstein’s 1905 special-

relativity interval equation.

Fig. 2: Initial (phase 0) points for the wavepair Solutions

are at P1 for +S and P2 for −S . The direction of rotation is +ct for counter-clockwise and −ct for clockwise. The positive spin is counter-clockwise when viewed from the position of the receiver (“sink”). The same spin appears clockwise when viewed from the position of the sender (“source”). This uses a handedness rule for determining direction—when the fingers are curled in the direction of spin, then the thumb points in the direction of travel (propagation). Figure 2 shows the waves organized by initial-phase starting point. Figure 3 shows the waves organized by matching + or − signs. The top row represents the −S solution, and the bottom row the +S solution. The left column shows the up and down clockwise (-ct) waves, and the right column the up and down counterclockwise (+ct) waves. When a wave is coming towards the observer, then the handedness shown is valid. The arbitrary labels “up” and “down” define the direction from the +S or -S zerophase point.

The factoring of Einstein’s equation identifies a threelevel hierarchy of Interval > Solution > Wave. Each Interval has a +S and a −S solution, and each Solution has a +ct and a −ct spin wave. Thus each Interval contains a set of four matter waves entangled as two solutions. The first four intervals, each containing four waves, define a hierarchical arrangement of 16 rows for the Madelung Rule electron configurations of the periodic table (Table 1 on Page 6). It has the same rows as the Janet left-step table but with separate rows for the individual waves. Thus the Janet table has eight rows for the eight solutions (the first four intervals), and the Interval table has 16 rows. 4

Galileo’s Law Defines the Subperiods

However, Einstein’s interval equation does not define the length of an interval (i.e., the number of electrons that will fit within it). According to the Pauli exclusion principle, each electron has to have a unique address as a unique set of the four quantum numbers. That issue is resolved by considering the chemical elements as “basic substances” (Scerri 2007, p. 286) that are in freefall with the Earth around the Sun. In this general-relativity approach, Einstein’s definition of freefall acceleration allows ignoring local forces and their effects. Einstein’s visualization for the core idea of general relativity was a man and his tools falling from the roof of a house.(2002, p. 136)

4

GALILEO’S LAW DEFINES THE SUBPERIODS

6

Table 1: The circularly polarized wavefactors of the first

squares of the time-intervals employed in traversing these distances.”(1954, p. 174) In words, Galileo’s law of freefall acceleration is

four intervals (Mazurs’ “cycles”) arrange the periodic table as four symmetrical sets of four rows. Interval Cycle

Solution Period −S or 1

I or 1 +S or 2 −S or 3 II or 2 +S or 4 −S or 5 III or 3 +S or 6 −S or 7 IV or 4 +S or 8

Wave Spin +ct −ct +ct −ct +ct −ct +ct −ct +ct −ct +ct −ct +ct −ct +ct −ct

Columns by Galileo Subperiods

Because for an observer in free-fall from the roof of a house there is during the fall— at least in his immediate vicinity—no gravitational field. Namely, if the observer lets go of any bodies, they remain relative to him, in a state of rest or uniform motion, independent of their special chemical or physical nature. The observer, therefore, is justified in interpreting his state [and the state of the objects with him] as being “at rest”. [Einstein’s italics] Astronauts experience this rest state as weightlessness when their vehicle is in the freefall motion of a gravitational orbit. Similarly, the Earth is in a freefall orbit around the Sun. When local forces are ignored, everything falling with the Earth is also in a freefall orbit. This includes all of the chemical elements comprising the Earth and everything with it. In his book Dialogues Concerning Two New Sciences, Galileo describes the motions of freely falling bodies. His Theorem II, Proposition II states, “The spaces described by a body falling from rest with a uniformly accelerated motion are to each other as the

Total distance in space = (constant acceleration) (time spent in freefall)2 (3) For the purposes of this paper, the constant-acceleration value is “1”. The Observer’s unit of time is called a Duration, and the unit of space is called a Slot. The “time spent in freefall” is the number of durations required for an object to freefall the observed number of slots. A combination of Durations (time) and Slots (space) is called a Configuration. Figure 4 illustrates Galileo’s acceleration law for the first five configurations. A configuration starts at zero and continues to the end of a duration. The use of spatial slots quantizes time and space into integer values.

Fig. 4: Galileo’s law of acceleration applied to define

configurations of electrons. The size of a Configuration (its name) is the number of Durations it contains. Slots: A slot is a region of space that can contain one and only one electron (i.e., one unit matter wave). Durations: A Duration is the number of slots in a unit time of accelerated motion. Corollary 1 of Galileo’s Theorem states the number of slots (spatial units) increases by two in each subsequent duration. Because Duration 1 contains only one slot, Duration 2 has three slots available for electrons to fill. The pattern continues with Duration 3 having five available

5

THE INTERVAL VIEW OF THE PERIODIC TABLE

slots, Duration 4 with seven available slots, etc. Durations determine the size of subshells (subperiods) in the electron configurations of elements. Configurations: A Configuration represents one interval wave and the Durations that define it. Configuration 1 contains only Duration 1 and therefore has only one (1+0=1) available slot. The first four elements, atomic numbers 1-4, are all Configuration 1 elements with each additional electron filling the single slot of one of the four interval waves. In the Janet left-step table, they are divided into wave pairs as the Interval solutions. Thus the two smallest rows of the Janet table each contain two elements. Configuration 2 adds Duration 2 and therefore has four (1+3=4) slots available for filling. Configuration 3 adds Duration 3 and therefore has nine (1+3+5=9) slots. Configuration 4 adds Duration 4 and therefore has sixteen (1+3+5+7=16) slots. The number of available slots in a Configuration is the square of its Interval’s number. Thus the first four intervals have 120 available slots that can be filled sequentially by electrons. Interval Number Interval I Interval II Interval III Interval IV

Calculation 4 waves · 12 slot 4 waves · 22 slots 4 waves · 32 slots 4 waves · 42 slots

Slots 4 16 36 64

An electron’s energy is the energy of a unit matter wave. Figure 5 shows the 120 waves comprising Element 120. Notice how the matter waves must be accelerated (spatially compressed) to fit into the equal freefall durations. This is related to the Madelung Rule though Mazurs’ work with the Janet left-step periodic table.(1974, pp. 96105) His “Cycles” are another name for Intervals, and his “Periods” are another name for Solutions. The doubling of Periods is because Einstein’s electromagneticinterval equation factors into two solutions. For example, in Mazurs’ Table 4 (p. 100), Cycle 4 contains Periods (Solutions) 7 and 8. Each Solution can contain up to 32 elements as the sum of its two 16-slot Configurations. Thus Interval IV (Mazurs’ Cycle 4) can contain up to 64 elements. Mazurs (1974) calls the first part of the Madelung Rule the “Periodic Law”. The Einstein-Galileo definition of Intervals and Solutions is mathematically related

7 120 spatial slots for 120 electrons +S Gravitational Interval (Counterclockwise) 7 slots 5 slots 3 slots 1 slot

l

= 3 “f”

l

= 2 “d”

l

= 1 “p”

l

= 0 “s”

−S Gravitational Interval (Clockwise) 1 slot 3 slots 5 slots 7 slots

Subshell

l

= 0 “s”

l

= 1 “p”

l

= 2 “d”

l

= 3 “f”

Fig. 5: Element 120 contains 120 waves, which is equiv-

alent to 120 electrons. Gravitation attempts to maintain a spherical shape because it takes the least energy to enclose the largest volume. to the Madelung Rule by Mazurs’ definition of a Period. He writes, “The above mentioned mathematical expression of the Periodic Law is as follows:” t =n+l

(4)

His “ordinal number of the period” is t, and n + l is the sum of the first and second quantum numbers. In Interval terminology, n + l is a Solution number plus a Duration number. 5

The Interval View of the Periodic Table

Table 2 shows the condensed periodic-table layout created by using Einstein’s electromagnetic-interval equation for the rows and Galileo’s acceleration equation for the columns. The numbers in the cells are the number of spatial slots in Durations 1, 2, 3, and 4. The last column is the number of available slots in each configuration of that interval. When the condensed layout is expanded to show the individual slots in each duration, the result is the periodic table in Figure 6. (The full-page image is provided on page 13). Each cube in the model of an element represents an orbital pair filled with two electrons. A partial cube represents either the left- or right-handed electron of a pair. Solutions (±S ) grouped under Interval type (I-IV) create the layout of the traditional “Janet left-step” periodic table. It focuses on atomic number rather than the physical and chemical properties of the material. Scerri identifies it as the extreme “Platonic” view of the periodic table.(2007, pp. 282-286)

5

THE INTERVAL VIEW OF THE PERIODIC TABLE

+S or 6 −S or 7 IV (4) +S or 8

Subshell identifiers (e.g., 3d(10)) show principal quantum number n with subshell name and number of electrons. Column headers show quantum numbers “l ” and “m”. The sign of ct for the row is the sign of the spin quantum number. For example, Radium (88) is 7, 0, 0, -½. In the Radium diagram below, each individual block represents one orbital pair of electrons, so its picture includes 44 blocks (orbital pairs).

−S −ct

−S −ct

‡

#8

+S −ct

21 Sc Scandium

G8

l =3

m = –1

m=0

+S −ct

m = +1

m = +2

m = +3

‡

G3

G3

G3

G3

G3

13 Al Aluminum

G16

7 N Nitrogen

G14

10

G17

G5

22 Ti Titanium

G9

‡

G6

23 V Vanadium

24 Cr Chromium

G10

G11

‡

G7

Ne

31 Ga Gallium

G16

G12

Sodium

Na

15 P Phosphorus

17 Cl Chlorine

+‒

Wave

‒+

Wave

12 Mg Magnesium

4s(2) G1 19 K Potassium

G18

G2

18 Ar Argon

20 Ca Calcium

‒‒

Wave

++ Wave

G13 16 S Sulfur

G13

25 Mn Manganese

11

G2 Neon

G15

14 Si Silicon

Wave

Be Beryllium

G1

G18

9 F Fluorine

G13

m = +2

+‒

5s(2) G14

G15

32 Ge Germanium

33 As Arsenic

G17

G18

G1

Wave 37 Rb Rubidium

‒+

G2

Wave 26

Iron

Fe

27 Co Cobalt

28 Ni Nickel

29 Cu Copper

30

Zinc

Zn

G4

39 Y Yttrium

G8

‡

34 Se Selenium

35 Br Bromine

36 Kr Krypton

38 Sr Strontium

5p(6) G5

40 Zr Zirconium

G9

‡

‡

41 Nb Niobium

G10

‡

‡

G6

42 Mo Molybdenum

G11

‡

G7

G13

43 Tc Technetium

49 In Indium

G16

G12

‒‒

6s(2) G15

G14

50

Tin

Sn

G17

51 Sb Antimony

G18

G1

Wave

55 Cs Cesium

++

G2

Wave 44 Ru Ruthenium

45 Rh Rhodium

46 Pd Palladium

47

Ag Silver

48 Cd Cadmium

5d(10) G3

G17

8 O Oxygen

4

‒‒

Wave

++

G3

52 Te Tellurium

53 I Iodine

54 Xe Xenon

6p(6) G4

G5

G6

G7

G13

56

Barium

Ba

+‒

7s(2) G15

G14

G1

Wave

‒+

57 La Lanthanum

G3

‡

64 Gd Gadolinium

G3

9

m = +1

G15

6 C Carbon

5 B Boron

G16

3p(6)

G2

3s(2)

4p(6) G4

G3

59 Pr 60 Nd 58 Ce Praseodymium Neodymium Cerium

G3

G3

65 Tb Terbium

G3

66 Dy Dysprosium

61 Pm 62 Sm Promethium Samarium

G3

67 Ho Holmium

G3

68 Er Erbium

69 Tm Thulium

63 Eu Europium

G3

70 Yb Ytterbium

5f(14)

+S +ct

G3

+S +ct

4f(14) G3

−S +ct

−S +ct

#6

n=5

m = –2

l =2 m=0

4d(10)

n=6

m = –3

m = –1

3d(10)

#5

n=7

n=4 n=3 n=2 n=1

m = –2

G14

Li Lithium

‡

71 Lu Lutetium

G8

76 Os Osmium

72 Hf Hafnium

G9

77 Ir Iridium

73 Ta Tantalum

74 W Tungsten

‡

‡

G10

78 Pt Platinum

G11

79

Gold

Au

75 Re Rhenium

G12

G16

80 Hg Mercury

6d(10) G3

‡

G3

‡ G3

‡

G3

‡

G3

G3

G3

81 Tl Thallium

84 Po Polonium

82

Lead

Pb

G17

85 At Astatine

83 Bi Bismuth

G18

86 Rn Radon

7p(6) G4

G5

G6

G7

G13

87 Fr Francium

G2

Wave

‒‒

88 Ra Radium

8s(2) G14

G15

Wave

++ 89 Ac Actinium

G3

‡

96 Cm Curium

m = –3

90 Th Thorium

G3

91 Pa Protactinium

92 U Uranium

93 Np Neptunium

94 Pu Plutonium

95 Am 103 Lr Americium Lawrencium

G3

G3

G3

G3

97 Bk Berkelium

98 Cf Californium

99 Es Einsteinium

100 Fm Fermium

101 Md 102 No Mendelevium Nobelium

m = –2

m = –1

m=0 l =3

m = +1

m = +2

G3

m = +3

104 Rf 105 Db Rutherfordium Dubnium

G8

G9

108 Hs Hassium

109 Mt Meitnerium

m = –2

m = –1

106 Sg Seaborgium

G3

107 Bh Bohrium

Predictions of configurations vary.

G10

‡

G11

‡

110 Ds 111 Rg Darmstadtium Roentgenium

m=0 l =2

m = +1

113 Nh Nihonium

114 Fl Flerovium

G12

G16

112 Cp Copernicium

116 Lv Livermorium

G17

117 Ts Tennessine

m = +2

m = –1

m=0 l =1

115 Mc Moscovium

Element 119

G18

118 Og Oganesson

m = +1

Wave

+‒

Solution 8

III (3)

This value indicates spin direction predicted by the ideal pattern rather than actually measured.

G13

m = +1

2s(2)

Interval IV

−S or 5

Orbital Pair

3(f )

+S +ct +S −ct

#4

=

2(d)

#7

#3

2p(6)

He

G1 3

‒+

Wave

Helium

Solution 7

+S or 4

−S +ct −S −ct

+

l =1 m=0

m = –1

All assignments of signs and directions are arbitrary.

ms = -½ ms = +½

+S +ct +S −ct

#2

Electron-configuration illustrations were drawn from 2005 standard text list at http://pearl1.lanl.gov/ periodic/downloads/periodictable.pdf and IUPAC.

Spin Number “ms”: The identifier for each of two electrons in an orbital pair.

0 (s) 1(p)

4

Symbol

2

Wave

Solution 6

II (2)

Each cube is an electron pair.

1 H Hydrogen

G18

Interval III

−S or 3

1

Azimuthal Number “l ” Old letters s, p, d and f for “subshell” & now quantum numbers 0, 1, 2, & 3 are the numbered units of time.

−S −ct

Solution 5

+S or 2

d2

nucleus

0

−S +ct

#1

Exception to ideal filling sequence.

12 Mg Magnesium

Name

+1 +2 -2 -3 -1 +1 +3 0 +2 -2 -1 +1 0

f = 3 d= 2 p=1 s= 0

G2

Atomic Number

0 -1

‡

Solution 4

I (1)

Duration d 1 2 3 4 1 1 1 1 1 3 1 3 1 3 1 3 1 3 5 1 3 5 1 3 5 1 3 5 1 3 5 7 1 3 5 7 1 3 5 7 1 3 5 7

n=7 n=6 n=5 n=4 n=3 n=2 n=1

IUPAC Group Membership

G1

Interval II

−S or 1

Wave Spin +ct −ct +ct −ct +ct −ct +ct −ct +ct −ct +ct −ct +ct −ct +ct −ct

Magnetic Number “m” Identifies each orbital electron pair within a subshell (time unit).

l =0 m=0

1s(2)

Solution 3

Solution Period

Principal Number “n” The “shell” number shown by distance from nucleus.

Element Key

Solution 2

Interval Cycle x

e− =

Periodic Table Organized by SR Intervals and Solutions Relating the Four Quantum Numbers: n, l , m and ms

Interval I

Galileo’s acceleration law defines the number of electrons in each freefall Configuration. Four intervals, each with four waves, defines 16 rows with (Cycle x)2 electrons in each row, (1x4)+(4x4)+(9x4)+(16x4)=4+16+36+64=120 total.

Solution 1

Table 2:

8

Element 120

m=0 l =0

Fig. 6: The Interval version of the left-step (Janet) tradi-

16

Modeling Solutions within Intervals is the most logical layout because it groups the orbital pairs (right- and left-handed waves) under their solutions. It defines the Madelung filling sequence shown in Figure 7. Two example elements, Radium 88 and Element 120, are shown in Figure 8. Each cube is one orbital pair and there are no unpaired electrons in either element. The middle image shows the locations of the 16 orbital pairs that must be added to Radium (on the left) to obtain Element 120 (on the right). The 16 orbital pairs are added in the subperiod sequence 5f, 6d, 7p, and 8s. They create a stair-step arrangement (the top arrow shown in Figure 7). To make visualization easier, the element depictions map all four waves into a single quadrant of a sphere. If the mapping showed one wave per quadrant, the overall shape would be spherical. The four quantum numbers map perfectly to this arrangement of elements in the periodic table. The first quantum number has “1” as the lowest energy level and increments by “1” for each higher level. These are the

tional form of the periodic table. A full-page image is provided after Section 7.

Fig. 7: The filling diagram for the empirical Madelung

energy ordering of the orbitals. It shows how the filling process attempts to maintain the most efficient structure in spacetime, a sphere.

5

THE INTERVAL VIEW OF THE PERIODIC TABLE 8s 7s 6p

5d 4f 3d 2p 1s Radium 88

+

+

7p 6d 5f

16 Orbital Pairs

=

=

Element 120

9 The third quantum number identifies the orbital pair (cube) within a subperiod. The central cube is designated as zero (no angular momentum), and the plus and minus values are to each side. The right pane illustrates the representation of the fourth quantum number, which identifies the handedness of the two members of the orbital pairs. The layout of the interval table represents a pair’s two electrons in the same column in the contiguous rows of a solution.

Fig. 8: The element on the left is Radium with 88 elec-

numbers written to the left of Radium and to the left of the added 16 orbital pairs in Figure 8. They are also the energy levels written on the vertical axis in the left pane of Figure 9. The second quantum number is designated by the older subperiod letters to keep the symbols unambiguous. For Radium’s seven energy (shell) levels in Figure 8, the subperiod sequence of the visible cubes (orbital pairs) is s, p, d, f, d, p, s. Figure 9 is the legend explanation that illustrates how the four quantum numbers define a specific electron location (chemical elements) in the periodic table. The left pane shows a sideways view of the first and second quantum numbers. The first quantum number identifies the energy level (Solution) in the stack of eight levels on the vertical axis. The second quantum number would be represented as a plane perpendicular to the axis point for each energy level. However, for ease of visualization, all four interval waves (±S ± ct) were mapped into a single quadrant rather than one wave per quadrant. Imagine the subperiods as a set of concentric rings with the s subperiod at the central axis of rotation. The second quantum number identifies the ring containing the electron. Each ring larger than s is represented as a set of connected cubes. The middle pane shows the top view of the quadrant with the second and third quantum numbers. The single cube labeled s is at the center. The sets of connected cubes marked p, d and f are the 3, 5, and 7 orbital pairs in those subperiods. The numbers written in parentheses by the letters are the second quantum number.

1st & 2nd numbers (Side View) 1st Principal Number “n”, the primary energy level; the shell label

7654321

trons. Adding 16 orbital pairs in the stair-step configuration of four n levels forms Element 120. The spherical shape is not shown because all four waves are being modeled in a single quadrant.

0 1 2 3

Azimuthal 2nd Number “l ”, orbital angular momentum; the subshell label

2nd & 3rd numbers (Top View) s(0) 0 p(1) 2nd -1 +1 d(2)

4th quantum number (Block View)

Magnetic Number “m”, the block location relative to the 0 baseline for its angular momentum

ms value indicates spin direction relative to the orbital angular momentum of its block

Each HALF-BLOCK is one ELECTRON.

-2 0 +2 f(3) + -3 -1 +1 +3 4th -2 0 +2 ms = -½ ms = +½ -1 +1 3rd 0 Spin Number “ms”, the

Fig. 9: Each electron has an address determined by its

four quantum numbers. The left pane shows the first and second quantum numbers as energy level (vertical axis) and orbital angular momentum (horizontal axis). The third quantum number is defined in the middle pane. Its sign and value identifies the location of an orbital pair (a block) from the central zero location for its subperiod. The right pane splits an orbital pair to show how the left- and right-handed electrons fit together into a block. This represents the fourth quantum number. The rows of the Janet left-step periodic table in Figure 12 on page 13 are perfectly organized by the electromagnetic intervals, solutions within intervals, and waves within solutions. The interval-wave information is in the rightmost columns of the table. The solution and wave types are also shown to the left of the element display. The columns are organized from right to left by the size of the configuration for each interval. The stair-step form of the layout occurs because of the increasing configuration size (1, 4, 9 and 16 slots). The group header for a Duration provides the second quantum number. The individual column headers in the periodic table provide the third (magnetic) quantum number for each column. The fourth (spin) quantum number is pictorially represented in the models of the elements with a

6

EXPLAINING THE FILLING ANOMALIES

10

half-cube representing one electron and the whole cube representing the completed orbital pair. The Element Key in Figure 10 identifies additional information available for each element. It includes the element’s name, symbol, atomic number, IUPAC group membership, and whether it is an exception to the filling sequence predicted by the Madelung Rule.

Element Key IUPAC Group Membership Atomic Number Name

G2

‡

Exception to ideal filling sequence.

12 Mg Magnesium

Each cube is an electron pair. Symbol

Fig. 10: The Element Key identifies the additional infor-

mation about each element. The anomalies in the positioning of electrons during the filling sequence occur when the real configuration differs from the ideal positioning. In an enlarged view, these are visually obvious from the positions of the electrons. Anomalies are identified by the symbol O ‡. 6

Fig. 11: The exceptions to the ideal filling sequence.

The circled elements have an electron that goes early to a higher energy level. The arrows indicate the slot relationships that are filled out of sequence.

Explaining the Filling Anomalies

Changing the hierarchy of variables provides new arrangements of the periodic table for visual exploration. There are three organizing variables: Intervals (I-IV), Solutions (±S ), and Handedness (±ct). The unusual layout in Figure 11 shows the repeated patterns created by anomalies in the filling sequence. This layout stacks the increasing energy levels (Solutions 1-8) from bottom to top, like the stem of a plant. The ‘leaves’ on the left are the +ct waves, and those on the right are the −ct waves. Text describing each anomaly is in Table 3 on page 12. The four different patterns of parallel arrows in Figure 11 indicate different anomaly loops that repeat once or several times at higher energy levels. The anomalous filling sequences for Chromium (24) and Copper (29) have been used as examples of the failure of the quantum numbers to predict, or even justify, the experimentally observed sequence. Scerri wrote: (2004, p. 103) Only if shells filled sequentially, which they do not, would the theoretical relationship between the quantum numbers provide a

purely deductive explanation of the periodic system. The fact that the 4s orbital fills in preference to the 3d orbitals is not predicted in general for the transition metals but only rationalized on a case by case basis as we have seen. In some cases the correct configuration cannot even be rationalized, as in the cases of chromium and copper, at least at this level of approximation. In other words, the Madelung Rule “cannot be strictly derived from first principles, although approximations used in quantum chemistry can account, after the facts, for all spectroscopically observed configurations.”(Scerri 2000) However, the layout in Figure 11 is organized by Solutions, not Shells. The obviously repeated patterns of anomalies provide logical reasoning rather than caseby-case empirical facts. In addition, Table 3 on page 12 identifies the 10 patterned loops of anomalies, and all of them involve either starting or completing a d subperiod early.

7

EXPLORING WAVE-BASED MODELS OF REALITY

This unusual layout groups together the waves with the same direction of rotation. These wave pairs create the quantum unit of angular momentum (action). Given that the quantum of action ~ is also the atomic unit of angular momentum,(Cardarelli 1997) this needs to be mathematically explored with Schr¨odinger’s equation and perhaps with other methods. Also, as described in Section 1, the Madelung Rule based on period (Solutions) does predict in general that the “4s orbital fills in preference to the 3d orbitals”. The 1946 “Periodic Law”, which is Ordinal number f or period = 1 st quantum number + 2nd quantum number, (5) is now justified by acknowledging the periods as pairs of Interval solutions. The values (1,2,...8) of the first quantum number n is the Solution number incremented from “1” for each solution in the first four intervals. Thus the Madelung Rule (n + l, n) predicts the following “Aufbau” filling sequence: 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4 f < 5d < 6p < 7s < 5 f < 6d < 7p < 8s Similarly, Mazurs’ rule for the number of elements in each cycle (Interval) is Lcycle = 4x2 . Four is the number of waves, and x is interval number. The square is the result of Galileo’s acceleration law. Thus the interval sizes as number of elements held are I:4, II:16, III:36, and IV:64, for a total of 120 elements. Behind all of this is the ruling principle of symmetry. The perfect symmetry of electromagnetic intervals (light waves) governs reality. Any result that indicates broken symmetry indicates relevant data are being omitted. It is then useful to reframe the problem and identify the symmetry(-ies). 7

Exploring Wave-Based Models of Reality

The factoring of Einstein’s electromagnetic-interval equation provides a principled reason for requiring photons to be orbital pair intersections (interval solutions) rather than continually existing particles. Einstein described particles as denser regions of a field.(Einstein 2002/1920, p. 180) Also, the focus is on symmetrical pairs of waves rather than individual waves. That is, each solution of an Interval contains two waves—a single wave with its

11

canonical conjugate. This is normal in quantum-wave mechanics, but here it is a principled definition of an interval solution rather than just an aid to calculation. Schr¨odinger wrote: (1982, p. 123) Meantime, there is no doubt a certain crudeness in the use of a complex wave function. If it were unavoidable in principle, and not merely a facilitation of the calculation, this would mean that there are in principle two wave functions, which must be used together in order to obtain information on the state of the system. Matter waves are a perfectly correlated representation of material objects because the product of a wavephase speed and its associated particle’s speed is the constant lightspeed c2 .(de Broglie 1929) This wave-based view of physical reality is an unusual perspective. Two more quotes from Schr¨odinger are particularly relevant to modeling a wave-based reality: It is hardly necessary to emphasize how much more congenial it would be to imagine that at a quantum transition the energy changes over from one form of vibration to another, than to think of a jumping electron. (1982, pp. 10-11) [L]ight of a definite frequency is always capable of producing the same physical effects as electrons of a definite velocity. From the fact of this equivalence the opposite conclusion can, however, be drawn with the same inevitability, or lack of inevitability: the electron moving with a definite velocity must be a wave phenomenon whose frequency is that of the light which is experimentally equivalent to it with regard to the excitation of resonance.(1967, p. 67) Physicists have chosen the particle-based probabilistic interpretation of Schr¨odinger’s “wave mechanics”. (Schr¨odinger 1982) However, as Schr¨odinger, Heisenberg, Einstein and Feynman strongly stated in the quotes in this paper, matter waves are physically real. There is much more to be explored with Maxwell’s A-field matter waves defined as the building blocks of both physics and chemistry.

7

EXPLORING WAVE-BASED MODELS OF REALITY

12

Table 3: Anomaly loops that break and then restore the Madelung filling sequence for the 19 anomalies identified

by Meek and Allen. There are only two types of anomalies: (a) a d shell pulls an electron from the s shell, or (b) a d subperiod receives what should go into the f subperiod. The first element in each loop identifies the anomaly and the other row(s) identify how it is resolved. When there is a shift in location for an existing electron, it occurs with the addition of the new electron. Electron-configuration models were drawn using information from the 2005 standard text list at http://pearl1.lanl.gov/periodic/downloads/periodictable.pdf and IUPAC. (Link now broken.) # Loop 1 24 25 Loop 2 29 30 Loop 3 41 42 43 Loop 4 44 45 46 47 48 Loop 5 57 58 59 Loop 6 64 65 Loop 7 78 79 80 Loop 8 89 90 91 92 93 94 Loop 9 96 97 Loop 10 110 111 112

Name Anomaly 24 Chromium Manganese Anomaly 29 Copper Zinc Anomaly 41 Niobium Molybdenum Technetium Anomaly 44 Ruthenium Rhodium Palladium Silver Cadmium Anomaly 57 Lanthanum Cerim Praseodymium Anomaly 64 Gadolinium Terbium Anomaly 78 Platinum Gold Mercury Anomaly 89 Actinium Thorium Protactinium Uranium Neptunium Plutonium Anomaly 96 Curium Berkelium Anomaly 110 Darmstadtim Roentgenium Copernicium

Abb.

Normal

Shift

To

Explanation

Cr Mn

d4

s2

d5 s2

Fills 3d 1e− orbitals early Restores s2 in 4s

Cu Zn

d9

s2

d10 s2

Completes 3d orbitals early Restores s2 in 4s

Nb Mo Tc

d3 d5

s2

d4 s2

Will fill 4d 1e− orbitals early Fills 4d 1e− orbitals early Restores s2 in 5s

Ru Rh Pd Ag Cd

d6

s2

d9

s1

d7 d8 d10 s1 s2

Will complete 4d early Completes next orbital Completes 4d orbitals early Restores s1 Restores s2 Start filling 5d before starting 4f Start filling 4f Restore normal filling 4f

La Ce Pr

f3

d1

d1 f1 f2

Gd Tb

f2

d1

d1 f1

Start completing 5d before completing 4f Restore normal completing 4f

d8

s2

d9 d10 s2

Will complete 5d early Completes 5d early Restores s2 Start filling 6d before starting 5f Continue filling 6d Start filling 5f Continue filling 5f Continue filling 5f Restores normal filling

Pt Au Hg Ac Th Pa U Np Pu

f6

d1

d1 d2 f1 & f2 f3 f4 f5

Cm Bk

f2

d1

d1 f1

Start filling 6d before completing 5f Restores normal filling

d8 d10

s2

d9

Will complete 6d early Completes 6d early Restores s2

Ds Rg Cp

d2

s2

7

EXPLORING WAVE-BASED MODELS OF REALITY

13

Fig. 12: The Interval version of the left-step (Janet) traditional form of the periodic table. The intervals I-IV are

identified in the rightmost column, and each interval’s two solutions (periods) are grouped inside it. Then each set of two waves (an orbital pair) are grouped inside each solution. The columns of elements are the spatial slots within each duration (subperiod/2). The mapping of columns and rows to quantum numbers is explained in the figure legend and in the text.

REFERENCES References Bohr, Niels (1913). “On the Constitution of Atoms and Molecules, Part 1.” In: Philosophical Magazine 26.151, pp. 1–24. doi: doi:10.1080/14786441308634955 (cit. on p. 2). — (1922). ”Niels Bohr- Nobel Lecture: The Structure of the Atom: Nobelprize.org. Nobel Media AB 2014. url: http://www.nobelprize.org/nobel_ prizes/physics/laureates/1965/feynmanlecture.html 19 Mar. 2017 (cit. on pp. 2, 3). — (1950). “The Structure of the Atom and the Physical and Chemical Properties of the Elements.” In: Autobiography of Science. Ed. by Moulton, Forest Ray et al. Originally published as Fysisk Tidsskrift 19, 153 (1921) translated by A. D. Udden, ”The Theory of Spectra and Atomic Constitution–Three Essays” (Cambridge, 1922). Doubleday. url: https: //web.lemoyne.edu/giunta/bohr2.html 30 July 2018 (cit. on p. 3). Cardarelli, Franc¸ois (1997). Scientific unit conversion. Springer (cit. on p. 11). de Broglie, Louis (1929). ”Louis de Broglie - Nobel Lecture: The Wave Nature of the Electron”. Nobelprize.org. Nobel Media AB 2014. url: http://www.nobelprize.org/nobel_ prizes/physics/laureates/1929/broglielecture.html 17 June 2017 (cit. on pp. 4, 11). — (1962). New Perspectives in Physics: Where Does Physical Theory Stand Today? Trans. by Pomerans, A. J. New York: Basic Books (cit. on p. 3). Einstein, Albert (2002/1920). “Ether and the Theory of Relativity.” In: The Collected Papers of Albert Einstein, Volume 7: The Berlin Years Writings, 1918-1921. Trans. by Engel, Alfred. Lecture quotes from 1922 book, Sidelights on Relativity. Princeton, NJ: Princeton University Press, Doc. 38, 160–182 (cit. on p. 11). — (1967). Letters on Wave Mechanics: Correspondence with H. A. Lorentz, Max Planck, and Erwin Schr¨odinger. Ed. by Przibram, K. 2011 edition distributed by

14 Open Road Integrated Media. New York: Open Road (cit. on pp. 4, 11). — (1970). “Remarks Concerning the Essays Brought Together in This Co-operative Volume.” In: Albert Einstein: Philosopher-Scientist. Ed. by Schilpp, P. 3rd. Open Court, pp. 665–688 (cit. on p. 4). — (1998). “On the Electrodynamics of Moving Bodies.” In: Einstein’s Miraculous Year: Five Papers That Changed the Face of Physics. Princeton, NJ: Princeton Univ., pp. 123–160 (cit. on p. 4). — (2002). “Fundamental Ideas and Method of the Theory of Relativity, Presented in Their Development.” In: The Collected Papers of Albert Einstein, Volume 7: The BerlinYears Writings, 1918-1921. Trans. by Engel, Alfred. Princeton University Press, Doc., 31, 113–150 (cit. on p. 5). Feynman, Richard et al. (1964). The Feynman Lectures on Physics: Mainly Electromagnetism and Matter. Vol. 2. Reading, MA: Addison-Wesley (cit. on p. 3). Galilei, Galileo (1954). Dialogues Concerning Two New Sciences. Trans. by Crew, Henry et al. New York, NY: Dover (cit. on p. 6). Heisenberg, Werner (1949). The Physical Principles of the Quantum Theory. Trans. by Eckart, Carl et al. Dover (cit. on p. 3). — (1955). “The Development of the Interpretation of the Quantum Theory.” In: Niels Bohr and the Development of Physics. Ed. by Pauli, Wolfgang. New York, NY: McGraw-Hill, pp. 12–29 (cit. on p. 4). — (1974). Across the Frontiers. Ed. by Anshen, Ruth Nanda. Trans. by Heath, Peter. Harper (cit. on p. 3). — (2007). Physics and Philosophy: The revolution in modern science. New York, NY: HarperPerennial (cit. on p. 4). L¨owdin, Per-Olov (1969). “Some Comments on the PePeriod System of the Elements.” In: International Journal of Quantum Chemistry III S, pp. 331–334 (cit. on p. 2).

REFERENCES Mazurs, Edward (1974). Graphic representations of the periodic system during one hundred years. Revised 2nd. University AL: University of Alabama Press (cit. on pp. 1, 7). Minkowski, H. (1952). “Space and Time.” In: The Principle of Relativity. Ed. by Lorentz, H. A. Trans. by Perrett, W. et al. New York: Dover. Chap. V, pp. 75–91 (cit. on p. 4). Misner, Charles W. et al. (1973). Gravitation. W. H. Freeman (cit. on p. 4). Rosenfeld, L. (1983). “Born’s Probabilistic Interpretation.” In: Quantum Theory and Measurement. Ed. by Wheeler, John Archibald et al. Princeton University Press, pp. 50–51 (cit. on p. 3). Scerri, Eric R. (2000). “The Failure of Reduction and How to Resist Disunity of the Sciences in the Context of Chemical Education.” In: Science & Education 9, pp. 405–425. doi: https://doi.org/10.1023/A: 1008719726538 (cit. on p. 10). — (2004). “Just how ab initio is ab initio quantum Chemistry?” In: Foundations of Chemistry 6. A previous version of this article appeared as ‘L¨owdin’s Remarks on the Aufbau Principle and a Philosopher’s View of Ab Initio Quantum Chemistry’ in E.J. Br¨andas, E.S. Kryachko (Eds.) Fundamental World of Quantum Chemistry, Vol. II, 675–694, Kluwer, Dordrecht, 2003., pp. 93–116 (cit. on p. 10). — (2007). The Periodic Table: Its Story and Its Significance. Oxford: Oxford University Press 1936. url: http://amzn.to/wYizkM (cit. on pp. 5, 7). — (2016). “The changing vview of a philosopher of chemistry on the question of reduction.” In: Essays in the Philosophy of Chemistry. Ed. by Scerri, E. R. et al. Oxford University Press (cit. on p. 2). Scerri, Eric R. et al. (1998). “Ordinal Explanation of the Periodic System of Elements.” In: International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 6.4, pp. 387–399. doi: https:

15 //doi.org/10.1142/S0218488598000318 (cit. on p. 2). Schr¨odinger, Erwin (1982). Collected papers on wave mechanics: together with his four lectures on wave mechanics. 3rd Augmented. Originally published in Annalen der Physik (4), vol. 79, 1926. Providence RI: AMS Chelsea (cit. on p. 11). Sommerfeld, Arnold (1923). Atomic Structure and Spectral Lines. Trans. by Brose, Henry L. London: Methuen and Co. url: https://archive.org/stream/ AtomicStructureAndSpectralLines/ (cit. on p. 2).