From Quantum Numbers to The Periodic Table

Quantum Numbers to Periodic Tables:
The Electronic Structure of Atoms

The electronic structure of atoms can be understood in terms of the Schrödinger wave equation.

Introduction

Massive atomic nuclei are born naked, but their net positive charge, Z, quickly attracts Z comparatively mass-less electrons to produce neutral atoms.

By way of example: The nucleus of a neon atom – atomic number 10, and so the Ne¹⁰⁺ ion – attracts 10 negatively charged electrons to give an atom of neon, Ne.

The negatively charged electrons do not nuclear react with the positively charged protons of the nucleus, as they do inside neutron stars, instead they associate as three dimensional resonant standing waves.

The electrons move about the point positive charge
in a beautiful and subtle quantum mechanical dance

The modes of resonance for single electron systems such as the hydrogen atom are described by the Schrödinger wave equation.
Schrödinger knew of de Broglie's proposal that a moving particle has wavelength, l, proportional to Plank's constant, h, and momentum p so that l = h/p, a property we now known as wave-particle duality.
Schrödinger constructed a differential equation for a wavelike electron resonating in three dimensions about a point positive charge (Wikipedia), the time-independent Schrödinger wave eqn.:
Solutions to the Schrödinger wave equation correspond to modes of electron resonance and are formally called wavefunctions.
Wavefunctions assume discrete, or quantised, energies and have energies which correspond to the spectral lines of one electron atoms and ions of the type: H^•, He⁺, Li²⁺, Be³⁺, etc.
Although not exactly the same, chemists tend to call wavefunctions "orbitals".

1-Dimensional resonant standing waves, the vibrating string:

2-Dimensional resonant standing wave, a vibrating drum skin (see QuickTime movies here and here):

3-Dimensional resonant standing waves, atom wavefunctions from the Schrödinger wave equation:

Quantum Numbers to Orbitals

Chemists recognise s, p, d and f-orbitals. The topologies of these orbitals: the shape, phase & electron occupancy are described by four quantum numbers:

n	The principal quantum number
l	The subsidiary or azimuthal or angular momentum or orbital shape quantum number
ml	The magnetic quantum number
ms	The electron spin quantum number

From Wikipedia:

These quantum numbers conspire to give spherical s-orbitals, dumbbell shaped p-orbitals that come in sets of three, double dumbbell d-orbitals that come in sets of five, etc.:

Electrons enter and fill orbitals according to four rules:

Pauli Exclusion Principal	Orbitals can contain a maximum of two electrons which must be of opposite spin.
Aufbau or Build-up Principle	Electrons enter and fill lower energy orbitals before higher energy orbitals.
Hund's Rule	When there there are degenerate (equal energy) orbitals available, electrons will enter the orbitals one-at-a-time to maximise degeneracy, and only when all the orbitals are half filled will pairing-up occur. This is the rule of maximum multiplicity.
Madelung's Rule	Orbitals fill with electrons as n + l, where n is the principal quantum number and l is the subsidiary quantum number. This rule 'explains' why the 4s orbital has a lower energy than the 3d orbital, and it gives the periodic table its characteristic appearance.

Certain 'magic' numbers of electrons of electrons exhibit energetic stability: 2, 10, 18, 36, 54, 86 and, one assumes, 118, are associated with the Group 18 inert or noble gases: He, Ne, Ar, Kr, Xe, Rn & Uuo.

The 'magic' numbers inevitably arise from the underlying quantum mechanics, but as Richard Feynman told us (here): "I think I can safely say that nobody understands quantum mechanics."

We can predict quantum mechanical patterns, but we don't know why we can predict the patterns. We do not understand QM in terms of a deeper theory.

Now available on the web are Richard Feynman's "Messenger Lectures" where he looks at the nature of physical theory and its relationship with mathematics. Highly recommended!

Quantum Patterns

The pattern of orbital structure is very rich and can be mapped onto the two dimensions of paper in many different ways.

Some mappings emphasize how the orbitals are ordered and filled with electrons, others stress how the chemical elements and their orbitals are ordered with respect to atomic number Z. Each tells us something different about atomic orbital structure and/or elemental periodicity.

Orbital Filling

Madelung's Rule says the orbitals fill in the order n + l (lowest first). This gives the sequence:

(n = 1) + (l = 0) = 1    1s

(n = 2) + (l = 0) = 2    2s

(n = 2) + (l = 1) = 3    2p

(n = 3) + (l = 0) = 3    3s

(n = 3) + (l = 1) = 4    3p

(n = 4) + (l = 0) = 4    4s

(n = 3) + (l = 1) = 4    3d

(n = 4) + (l = 1) = 5    4p

(n = 5) + (l = 0) = 5    5s

and so on...

A nice electron shell representation from the Encyclopedia Britannica:

Thus, the orbital filling sequence is, from the bottom of this diagram, upwards because the lowest energies fill first:

Electron Shells

As electrons are added, the quantum numbers build up the orbitals. Read this diagram, from the top downwards:

Elements by Orbital, And Some Subtleties...

The electronic structure can be illustrated adding electrons to boxes (to represent orbitals). This representation shows the Pauli exclusion principle, the aufbau principle and Hund's rule in action.

There are some subtle effects with the d block elements chromium, Cr, and copper, Cu. Hund's rule of maximum multiplicity lowers the energy of the 3d orbital below that of the the 4s orbital, due to the stabilisation achieved with a complete and spherically symmetric set of five 3d orbitals containing five or ten electrons. Thus,

Chromium has the formulation: [Ar] 3d⁵ 4s¹ and not: [Ar] 3d⁴ 4s²
Copper has the formulation: [Ar] 3d¹⁰ 4s¹ and not: [Ar] 3d⁹ 4s²

The Periodic Table

A periodic table – of sorts – can be constructed by listing the elements by n and l quantum number:

The problem with this mapping is that the generated sequence is not contiguous with respect to atomic number atomic number, Z, and so is NOT a periodic table. Try counting the numbers past the red lines | on this representation:

Named after a French chemist who first published in the formulation in 1929, the Janet or Left-Step Periodic Table uses a slightly different mapping that is contiguous with atomic number:

While the Janet periodic table is very logical and clear, it does not separate metals from non-metals as well as the Mendeleev version, and helium is problem chemically.

However, it is a simple mapping to go from the Janet or Left-Step periodic table to a modern formulation of Mendeleev's periodic table (the movement of He to group 18 needs is not shown in this formulation):

The Devil In The Detail

Matters are considerably more involved than implied above.

Wavefunctions are called orbitals by chemists, but this ignores the existence of real and imaginary portions of the complex wave function – where the term 'complex' is being used in the mathematical sense.

Orbitals should not be considered as being physically real, and in principle no experimental technique can directly observe orbitals.

The wavefunction is Ψ, and the normalised wavefunction is |Ψ|.
Electron density is |Ψ|², and this is a measurable quantity.
However, the orbital is the square root of the electron density, and on taking the root all phase information is lost, for the same reason the square root of minus 1 cannot be known.

Solutions for the isolated hydrogen atom must be physically spherically symmetric (because the isolated hydrogen atom force field is spherically symmetric) and this gives rise to the 1s, 2s, 3s.. 'shells'. But mathematically one also gets 2p, 3p... solutions (lobes) that are not spherically symmetric because theta and phi are defined. These angles presuppose x, y, z axes that exist mathematically and allow non-symmetrical solutions. Physically these lobes don't exist for any isolated atom, but chemists talk about them when atoms are closely interacting, i.e. they exist in non-uniform fields, where axes and lobes can make physical sense.

The Schrödinger wave equation can only be solved analytically for one electron systems. For multi-electron systems it is necessary to use approximations, and the more electrons are involved, the more severe are the approximations. That said, mathematically manipulated orbitals are the foundation of much computational quantum chemistry software.

We chemists can fool ourselves into thinking that we understand s and p-orbitals, because they can be superimposed upon (mapped to) the easily understandable three dimensional x, y, z Cartesian coordinate system. But d-orbitals project into 5-dimensional space, and f-orbitals into 7-dimensional space. d-Orbitals can be mapped to the 3-dimensional Cartesian coordinate system, but a mathematical artifact is the 'strange looking' dz² orbital.

Real, physical chemical structure and reactivity systems also exist in three dimensional Cartesian coordinate space. Mother Nature must also solve the 5-dimensional to Cartesian dimensional reduction when forming transition metal complexes, such as [Cu(H₂O)₆]²⁺, that use d-orbitals. Understanding comes from crystal field theory that is able to predict what happens when a Cartesian array of charges interacts with d-orbitals that are degenerate in 5-dimensional space. The geometry causes the 5 d-orbitals to split in energy to give three lower energy orbitals and two higher energy orbitals.

Thanks to DD, an ex-physicist/philosopher of science, for his input into this section.

Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles (2nd Ed.) by Robert Eisberg & Robert Resnick, 1985 (John Wiley & Sons, NY. pp. 252)

Segré Chart

The Periodic Table: What Is It Showing?

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