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4…The Structure of the Atom

4…The Structure of the Atom

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P. Douglas et al.

this very small volume by the strong nuclear force. This is a very strong, but short

range (ca. fm range), attractive force between proton–proton, proton–neutron, and

neutron–neutron, which is independent of electric charge. As the atomic number

increases, charge repulsion between protons increases also, and the number of

neutrons required for a stable nucleus, in which the strong force between all

particles is greater than the electrostatic repulsion between protons, increases.

Thus, the neutron to proton ratio for the elements increases rapidly with Z. But this

nucleus stabilisation by neutrons is limited in effect, and above an atomic number

of 83 (Bi) all nuclides are unstable to radioactive decay (209Bi, Z = 83, is the

heaviest stable nucleus).

The way the spins of the protons and neutrons in the nucleus combine can lead to

nuclei with spin quantum numbers (I), varying from 0 to 6, and isotopes of the same

element can have, and usually do have, different spin quantum numbers. Isotopic

nuclear spin is important in NMR. For example, 12C has I = 0 and 13C has I = ‘,

so that NMR spectroscopy is not possible with 12C nuclei but is possible with 13C.

Fortunately for chemists the natural carbon isotopes include *1 % 13C, and 13C

NMR is a very important tool in chemical structure elucidation [14]. Nuclear spin

can also have subtle effects in other spectroscopies [15], and nuclear mass is

important in molecular vibrational and rotational spectroscopies.

There is an electrostatic force of attraction between electrons and the nucleus

and for atoms with more than one electron, there is also electrostatic repulsion

between electrons. Since the electron mass is much smaller than that of the

nucleus, the centre of mass of an electron-nucleus system lies much more closely

to the nucleus than the electron. The force of attraction between a nucleus and

electron acts on both bodies equally, and since F = ma, the electron is accelerated

much more by this attraction than the nucleus. This gives us a simple atomic

model: a very small, yet relatively massive, slow moving positively-charged

nucleus around which there are relatively light, rapidly moving, negativelycharged electrons, all held together by electrostatic forces. These electrostatic

forces, nucleus–electron and electron–electron, control the momentum, and also,

through the de Broglie relationship, the wavelength of an electron as it moves

around the nucleus. So between them, the nature of the electrostatic force and the

mass of the electron determine the shapes of the stable electron waves in atoms.

1.4.2 Electron Waves in Atoms: Atomic Orbitals Electron Waves

The physical boundaries of a musical instrument, the length of a string or column

of air, or the area of a drum, restrict the possible shapes and frequencies, and hence

energies, of the sound waves the instrument can produce. In a similar way the

‘boundaries’ of a molecule created by the electrostatic forces between electrons

1 Foundations of Photochemistry


and nuclei restrict the shapes and energies of the electron waves within the molecule. In a musical instrument the different wave shapes and energies give rise to

the fixed frequencies of the fundamental and overtones generated by the instrument. In chemical structures, the restriction of possible wave shapes and energies

controls the behaviour of electrons, and hence chemical properties. The mathematics for an electron in a chemical structure is more complex than a wave on a

string or in a column of air, but the principle is the same; the combination of

constraints and boundary conditions, naturally limits the properties of the electron

waves in atoms and molecules to discrete, fixed, ‘quantised’, values. The shapes of

the waves are limited, and the important properties of: electron wave energy,

absolute electron wave orbital angular momentum, and orientation of the electron

wave orbital angular momentum, are also naturally quantised.

The answer to the question of what stable electron waves are allowed in any

chemical structure is given by the time-independent Schrödinger wave equation

for that structure. (The time-independent wave equation is used to obtain the stable

electron waves around atoms and other chemical structures, and the time-dependent wave equation is used for calculations of electron waves as they undergo

‘transitions’ from one wave into another). The Schrödinger equation is not

derivable directly from any previous equations; it combines ideas of wave and

particle behaviour that were previously considered mutually exclusive. This

combination of particle and wave properties can be illustrated by discussion of the

equation for the hydrogen atom. The Hydrogen Atom

The particles. The hydrogen atom is made up of only two particles, the proton and

electron. The mathematics of the relative movement of two particles is such that

we can reduce the problem to only one moving particle of reduced mass moving

around the centre of gravity of the pair. Because of the difference in masses of the

proton and electron, it is convenient to identify the electron as the moving particle

(with the reduced mass, le, given by le = (memn)/(me ? mn), where me and mn are

the mass of an electron and the nucleus, respectively). The difference in mass and

reduced mass of the electron is less than 0.1 %, and the centre of gravity of the

atom lies very close to the centre of the proton, so that, for all but detailed

calculations, we can keep the simple model of an electron moving around a proton.

The energy of the negatively charged electron moving around the positivelycharged proton is made up of two parts: (1) the potential energy, which is the

electrostatic energy of the electron associated with its position with regard to the

nucleus, PE; and (2) the kinetic energy, KE, due to its velocity, The sum of the two

is the total energy of the system, i.e. Etotal = PE ? ‘mV2. As the electron moves

about the nucleus there is a constant interchange between these two types of

energy: the potential energy is greatest when the electron is furthest from the

nucleus and is moving slowly, and the kinetic energy is greatest when the electron


P. Douglas et al.

is moving rapidly, close to the nucleus. Conservation of energy requires their sum,

the total energy, to be constant.

The waves. The wave equations for two charged particles moving with respect

to one another can be obtained exactly. In Schrödinger’s equation the wavelength

of the electron at any point and time is given by its momentum (p = mv) using the

de Broglie equation k = h/p. The electron is constantly changing velocity as it

moves around the proton so the wavelength of the electron wave varies as it

moves. Only some electron waves will ‘fit’ and be stable to interference as they

oscillate. The energy of the electron wave, i.e. the sum of the potential and kinetic

energy, is equated with the frequency of the oscillation of the wave by E = hm.

The mathematics of the Schrödinger wave equation determines which electron

energies give waves that are stable to destructive interference. It also gives the

shapes and angular momenta of the waves. As for all waves that are constrained in

space, only certain waves are stable, only certain electron waves can fit within the

atom and these waves have fixed energies. The electron can only have one of these

fixed energies and no other; the electron energy is quantised, The electron energy

is not continuously variable unless the electron escapes from the constraints of the

atom itself, i.e. it is ionised and becomes a free moving electron.

Just as there are a number of waves which will fit on the string of a musical

instrument—the fundamental and the overtones, there are a number of solutions to

the Schrödinger wave equation for an atom, each one a different electron wave

shape with a different energy. Each solution is called a wavefunction, given the

symbol W. The wavefunction gives the amplitude of the electron wave as a

function of the position of the electron around the nucleus when in that electron

wave. (The wavefunction for a wave on a string gives the amplitude of the displacement of the string, as a function of position on the string as in Fig. 1.5). The

amplitude can be positive or negative, i.e. the sign of the wavefunction can vary in

different regions of the wave. This property of sign or phase is critically important

when electron waves overlap in space. If both waves have the same sign then the

waves interfere constructively in that region of space, when the waves are of

opposite sign they interfere destructively. It is this wave interference that is

responsible for the formation of chemical bonds between atoms.

In atoms, these three dimensional electron waves are referred to as atomic

orbitals (AO’s); in molecules, they are molecular orbitals (MO’s). In hydrogen,

the solutions of the Schrödinger equation give a series of orbitals, which are

characterised by three properties, each with an associated quantum number:

1. Energy (quantum number n).

2. Orbital angular momentum (quantum number l). Like orbital energy this is

quantised and can only take fixed values. The orbital angular momentum

controls the shape and symmetry of the wave.

3. Orientation of the orbital angular momentum (quantum number ml). The orientation of the waves is limited to those in which the angular momentum along

any chosen axis is quantised in integral units.

1 Foundations of Photochemistry


n is the principal quantum number and can only take integral values 1, 2, 3, …

etc. For a single electron atom, such as hydrogen, it defines the energy of the wave;

as n increases, the energy of the wave increases and the wave moves further away

from the nucleus. The value of n determines the total number of nodal surfaces

(surfaces where the amplitude of the wave function is zero, and across which the

wave function changes sign) in the wave; the number of such surfaces is given by

n-1 (see also Fig. 1.5 where the number of nodes is also given by n-1). The value

of n also controls the possible values that the total orbital angular momentum

quantum number, l, can take. All orbitals of the same n make up the n shell.

The orbital angular momentum quantum number, l, defines the total angular

momentum of the orbital. l defines the shape (symmetry) of the wave. As n increases

the orbitals can have more angular momentum, i.e. the greater the value of n, the

more shapes are available (Fig. 1.6). l can take values of l = 1, 2, 3…(n-1). Atomic

Fig. 1.6 Parity and the effect of inversion on the sign of the wavefunctions for some s, p, d and

f orbitals


P. Douglas et al.

orbitals are usually described by a letter associated with each l value. When l = 0 the

orbital is an s orbital, when l = 1 it is a p orbital, when l = 2 is it a d orbital, and when

l = 3 it is an f orbital (these letters arise from spectroscopic studies of emission lines

which were classified as: sharp (s), principal (p), diffuse (d), and fundamental (f)).

Orbitals with l [ 3 (g orbitals etc.) are not important in general chemistry or photochemistry, but are very important in atoms excited close to their dissociation limit,

so-called Rydberg atoms.

The angular momentum of each orbital is given by:

orbital angular momentum ẳ ẵll ỵ 1ފ1=2 Áh=2p:


l determines the number of nodal planes which go through the nucleus. If l = 0

or is even (2, 4…), then the orbitals are symmetric with respect to inversion

through the nucleus, i.e. the sign of the wavefunction at any point is the same at the

corresponding point across the nucleus. For l = odd number (1, 3…), the orbitals

are anti-symmetric with respect to inversion, i.e. the sign of the wavefunction

changes across the nucleus. This is of particular relevance when the molecules

have a centre of symmetry and these properties are sometimes described as parity;

the orbitals which are symmetric by inversion have even (symmetric) parity, while

those which are asymmetric have odd (anti-symmetric) parity. The German terms

gerade, g, (even) and ungerade, u, (odd) are also commonly used. Figure 1.6

shows the parity of some s, p, d and f orbitals in a centrosymmetric system.

The orbital magnetic quantum number, ml, specifies the orientation of the orbital

in space and the magnitude of the orbital angular momentum component along any

specified axis. Orbitals with the same n and l, but different ml, are all equal in energy

in the absence of a magnetic field, (orbitals of the same energy are described as

degenerate), but, because their magnetic moments point in different directions, they

align with slightly different energies in the presence of an applied magnetic field.

The values of ml are limited by the value of l to: ml = l, (l-1), …, -l.

The orbital description of the electron can then be given as either a series of the

three quantum numbers, n, l, ml, or, more usually, the principal quantum number,

n, followed by the letter corresponding to the angular momentum quantum number, l, i.e. s, p, d or f. Grouped by principal quantum number, the atomic orbitals

for the first four n shells are:

1sn ẳ 1; l ẳ 0ị;

2sn ẳ 2; l ẳ 0ị; 2pn ẳ 2; l ẳ 1ị;

3sn ẳ 3; l ¼ 0Þ; 3pðn ¼ 3; l ¼ 1Þ; 3dðn ¼ 3; l ẳ 2ị;

4sn ẳ 4; l ẳ 0ị; 4pn ¼ 4; l ¼ 1Þ; 4dðn ¼ 4; l ¼ 2ị; 4f n ẳ 4; l ẳ 3ị:

While the electron wavefunction can be used to obtain the energy and other

properties of the electron, the question arises, in quantum mechanics generally, as

to what the wavefunction itself ‘‘means’’. This has been, and still is, the subject of

much debate and there is currently intense research activity into using attosecond

spectroscopy to probe atomic wavefunctions [16]. The most useful interpretation

of the wavefunction for chemistry is that due to Born, who, by analogy to a light

wave, where the intensity is proportional to the square of the amplitude, suggested

1 Foundations of Photochemistry


Fig. 1.7 Three-dimensional shapes of the s, p and d atomic orbitals for the hydrogen atom. The

‘‘+’’ and ‘‘-’’ signs represent the sign of the electron wavefunction in that region of space

that the square of the amplitude of the wavefunction in a small volume of space

gives the probability of finding the electron there. Thus, the square of the wavefunction can be used to calculate electron density maps and surfaces.

Figure 1.7 summarises the atomic orbitals for the hydrogen atom, and shows

their three dimensional shape using the usual chemist’s representation where the

outer surface is all at the same electron density, i.e. they are isoelectronic density

plots, such that the probability of finding the electron in the volume drawn is some

fixed value—usually 95 or 90 %. The orbitals are often marked with the sign of the

wavefunction in that region of space (not to be confused with ionic charges), or

regions of different sign are shown in two different colours.

The lines in the emission and absorption spectra of the atoms correspond to the

electronic transitions as electrons move from one orbital to another. The energy of

the photon emitted, or absorbed, corresponds to the difference in energies of the

electron in the two orbitals involved. The yellow/orange glow of sodium lamps or

street lights corresponds to the photons emitted as excited sodium atoms with the

outermost electron in the 4p orbital fall down to the ground atomic state in which it

is in the 3s orbital.

The Schrödinger equation (more precisely the refined version incorporating both

relativity and quantum electrodynamics), and those obtained from it, describe the

physical and chemical features of the hydrogen atom with an accuracy limited only

by the precision to which the fundamental constants required are known. Unfortunately, the hydrogen atom is the only chemical structure for which the Schrödinger

equation can be solved exactly; everything else requires approximation. For small

atoms and very small molecules the approximations can be very good, but for any


P. Douglas et al.

structure of much complexity, and that means most interesting structures, they are

not good enough to allow the calculation of many important chemical properties to a

useful level of accuracy. Because of this, chemistry and photochemistry remain

essentially experimental subjects. Theory is, however, invaluable in our understanding of the underlying phenomena. Multi-Electron Atoms

For multi-electron atoms the Schrödinger equation cannot be solved analytically,

so methods involving approximations, of varying degrees of sophistication, are

used instead. The first approximation is that the wave equation for the whole

system is made up of individual wave equations for each electron, and each

electron is in an orbital of a form modelled on those of the hydrogen atom. For

purposes of calculation and modelling, sophisticated approximations such as Slater

orbitals or Gaussian orbitals are used. However, because orbital symmetry is often

of paramount importance when combining orbitals to give molecular structures,

the simple orbital shapes and symmetries given in Fig. 1.7 provide an excellent

basis for the understanding and visualisation of chemical bonding and chemical

structures. Also, usually we are only concerned with the electrons in the highest

energy, outer, or valence, orbitals of an atom. Electrons in lower energy orbitals,

the core orbitals, are held closely to the nucleus and are not generally involved in

chemical or photochemical transformations.

The first question for multi-electron atoms is: ‘‘how are the electrons placed in

the orbitals of different energies?’’ In the absence of any other information, we

might expect that in the state of lowest energy all of the electrons in the atom

would be in the lowest orbital, although we would also be aware that putting

electrons together in the same region of space will result in some increase in

overall energy due to electrostatic repulsion.

However, we have not yet incorporated electron spin into our atomic model.

The alignment of the electron spin angular momentum along any axis can only be

such to give a resultant spin angular momentum along the axis of either = ? ‘"h

or -‘"

h. We can describe which of these two spin states an electron adopts using

the additional quantum number, ms, which can take values of +‘ or -‘. Thus, the

complete description of the electron in an atom requires four quantum numbers: n,

l, ml, and ms. It is a general property of electrons (and all fermions), expressed in

the Pauli Exclusion Principle, that no two can have the same four quantum

numbers, Thus, it is possible to place two, but only two, electrons in the same n, l,

ml state and when they are so placed they must have electron spins arranged in

opposing directions, i.e. one with ms = ? ‘ and one with ms = -‘. If we use an

arrow to indicate the electron spin direction, up or down, when two electrons are in

the same orbital the electron spins are arranged thus, :;.

So the first two electrons can go into the 1s orbital, but with two electrons the

orbital is full and the next electron must go into the next highest orbital, the

1 Foundations of Photochemistry


2s. Putting electrons together in the same orbital raises the energy of both electrons

due to electron–electron repulsion, the total increase in energy this causes is called

the electron pairing energy. Therefore, when filling orbitals of equal energy, i.e.

degenerate orbitals, such as the three p orbitals, the electrons go into the individual

orbitals singly at first (Hund’s rule). Also, for two different singly occupied orbitals

the arrangement in which the two electron spins are parallel, ::, is lower in energy

than that in which they are opposed, :;, because in the parallel spin arrangement the

electrons tend to spend more time apart in space. So when electrons go singly into

different orbitals the arrangement with spins parallel is lower in energy than that with

spins opposed. The magnitude of the separation between these spin opposed and

spin-paired energy states (sometimes referred to as the exchange energy integral) is

determined by the spatial overlap of the two orbitals, i.e. how likely the two electrons

are to occupy the same region of space. If the orbitals have little overlap in space this

energy difference is small. We will see later how the same ideas apply to molecules

and are important in their excited state behaviour.

The arrangement of electrons is thus governed by four factors: the energy of the

orbitals, an orbital occupancy of at most two electrons spin paired per orbital, the

magnitude of the electron–electron repulsion spin pairing energy, and the direction

of the spin angular momenta of the electrons. Using these ideas, and a few

refinements in the relative energy of s, p, d and f orbitals in many electron-atoms,

the electronic structure of all atoms can be understood [17]. Size of Atoms and Molecules Compared to Wavelength of Light

Atoms are not hard incompressible spheres and so the definition of their size is

somewhat arbitrary. A variety of atomic sizes can be measured from intermolecular distances, or liquid or solid volumes, and can also be calculated from the

Schrödinger equation. In the lowest electronic state, atomic diameters are of the

order of one hundred to a few hundred pm. The Schrödinger equation shows, that

for the H atom, the orbital radii increase in proportion to n2, so if an electron is

promoted into a higher energy orbital the size of the electronically excited atom

will be significantly larger than the ground state. (In terms of our analogy of waves

in boxes, the atomic energy levels corresponding to different principal quantum

number n, are not so much the different energy levels arising from waves in a box

of fixed size, which would increase in energy separation as n increases as shown in

Fig. 1.5, but rather the different energy levels corresponding to waves in boxes of

different size which decrease in energy separation as the box size increases.) The

importance of atomic/molecular size for the interaction between UV/Vis photons

and chemical species is that, since the wavelength of UV/Vis radiation is one

thousand times larger than an atomic diameter then, even for a large or excitedstate atom, or molecule, the wavelength of a UV/Vis photon is big enough that the

electromagnetic field of the photon acts on all electrons and nuclei in the atom/

molecule with, to a very good approximation, the same phase. (Although the fact


P. Douglas et al.

that the electromagnetic phase is not exactly the same across an atom or molecule

is one reason why nominally forbidden transitions (see Sect. 1.12.1) are allowed to

some degree.)

1.4.3 Description of Atomic Electronic States: Term

Symbols, Spin Multiplicity, Angular Momenta, Spin

Orbit Coupling and the Heavy Atom Effect

Depending upon the precise electronic arrangement, the various combinations of

electron spin and orbital angular moment can lead to atomic energy levels of

different spin, orbital and total angular momenta. Term symbols are a useful

shorthand representation of the angular momenta of these individual atomic states.

The values which need to be represented are: the total electron spin angular

momentum, described by the quantum number S; the total orbital angular

momentum, described by the quantum number L; and the total angular momentum

resulting from the combination of L and S, described by the quantum number J.

The term symbol gives these three angular momenta in the following form:


corresponding letter for Li:e: S; P; D; FịJvalue;or values


using upright font for the letter to help distinguish it from the italic quantum


The first superscript does not give S directly but rather the spin multiplicity, the

number of possible values of electron spin momentum along a specific axis, which

is 2S ? 1. Thus if S = 0, (all electrons spin paired), there is only one arrangement;

the total spin is zero and therefore also zero along the axis, and the state is called a

singlet. For S = ‘ (one unpaired electron), there are two arrangements, the total

spin is [‘(‘ ? 1)]1/2"

h and this can point up or down the axis to give values +‘"h

and -‘"h along the axis, and the state is a doublet. For S = 1 (two unpaired

electrons), there are three arrangements, the total spin is [1(1 ? 1)]1/2"h and this

can point up, to give spin +1"

h along the axis, point down to give -1"h, or point at


h along the axis (i.e. at 908 to the axis), and the state is a triplet. A schematic

diagram showing the formation of singlet and triplet states from the spin momenta

of two electrons is shown in Fig. 1.8.

As a simple example, an atom with only one unpaired electron, consider a

ground state sodium atom (11 electrons). This has a full n = 1 shell, a full n = 2

shell and the single outermost electron in a n = 3 s-orbital, i.e. l = 0. The full

shells have zero spin and angular momenta and therefore do not contribute to the

term symbol, which, since S = 1/2 and L = 0, is 2S1/2 (read as: doublet S one half).

If the electron is excited into an n = 3 p-orbital, then since S = 1/2, L = 1 the spin

and angular momenta can reinforce or oppose each other giving J = 3/2 or 1/2 to

give two term symbols 2P3/2 and 2P1/2 (read as: doublet P three halves, doublet P

one half); see Fig. 1.8 for a diagram of this case). These two levels with J = 3/2

1 Foundations of Photochemistry


Fig. 1.8 Combining spin–spin, and spin-orbital, angular momenta. a The resultant total electron

spin angular momentum and spin orientations arising from the combination of electron spin from

two electrons, each s = 1/2, which gives a singlet (S = 0, i.e. no resultant spin) and a triplet state

(S = 1). The triplet state is triply degenerate because the spin angular momentum vector can

point so as to have three values along the z-axis. b The resultant total angular momentum and

orientations from the combination of the spin of one electron, s = 1/2, and orbital angular

momentum from a p orbital (l = 1) which gives rise to J = 1/2 and J = 3/2


P. Douglas et al.

and J = 1/2 differ slightly in energy, and close examination of the yellow/orange

line at 589 nm produced by sodium atoms in a flame, or a sodium street lamp,

shows that it is in fact made up of two lines very close together, one at 589.76 nm

and one at 589.16 nm, corresponding to the transition from the two 2P3/2,1/2 levels

to the single 2S1/2 ground state.

Consider now a magnesium atom (12 electrons). In the ground state it has a full

n = 1 shell, a full n = 2 shell and the two ‘outermost electrons’ are in an n = 3 sorbital, i.e. l = 0. By the Pauli exclusion principle, the two electrons in the n = 3 sorbital must be spin paired and so S = 0, L = 0 and the term symbol is 1S0 (read:

singlet S nought). The atom can be excited by promoting one of the n = 3 selectrons to an n = 3 p-orbital, for which l = 1. There are now two electrons in

different orbitals and these are not restricted by the Pauli exclusion principle. If they

are spin paired S = 0, L = 1 and the term symbol is 1P1, (singlet P one) if they are

not spin paired but spin parallel, then S = 1, L = 1, and the term symbols are 3P2

(triplet P two), 3P1 (triplet P one), and 3P0 (triplet P nought). The triplet states are of

lower energy than the singlet due to the spin-pairing energy of the singlet.

Most molecules have a ground state with electrons paired, i.e. a singlet ground

state, and just like the Mg atom, promotion of an electron to a higher orbital leads

to two possible spin states, i.e. singlet and triplet states. For virtually all organic

systems and many (but far from all) inorganic ones, the singlet and triplet spin

states are the only ones which are relevant for photochemical processes. Note that

the three levels of a triplet state are degenerate (i.e. they have the same energy)

under normal conditions, but have different energies in a magnetic field. We can

obtain information on the properties of triplet states by studying them in a magnetic field (magnetic field effects).

In general, spin- and orbital- angular momenta for atoms can be combined in

one of two ways, depending upon the strength of coupling between the spin and

orbital angular momenta for the electrons in the atom. Where the coupling between

spin and orbital momenta is weak (the most important case) Russell–Saunders

coupling is used. Due to weak coupling, spin and orbital angular momenta are first

of all treated separately, i.e. all the electron spin momenta are combined together

to give S, all the orbital angular momenta are combined to give L and these are

then combined to give J. Strong coupling between spin and orbital momenta, gives

jj coupling, in which the individual electron spin and orbital angular momenta

combine first, to give an electron overall angular momentum, j, for each electron,

and these j momenta combine to give the total overall angular momentum of the

atom, J. The strength of spin–orbit coupling increases rapidly with atomic number,

depending as it does on Zn, (where n C 4); thus Russell–Saunders coupling is

found for light and moderate Z atoms, while jj coupling is found for the heavier

atoms of the Periodic Table. A strong spin-orbit coupling is evident in the relatively easy interchange of spin angular momenta and orbital angular momenta,

known as the heavy atom effect. The heavy atom effect mixes electron spin and

electron orbital angular momenta. For atoms, and molecules composed of atoms,

which have weak spin orbit coupling, i.e. those of low or medium atomic number,

electron spin angular momenta and orbital momenta can be considered distinct and

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