Mitin V.V., Sementsov D.I., Vagidov N.Z. Quantum mechanics for nanostructures

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6.4 Many-electron atoms 173

lines due to spin–orbit coupling is called the ﬁne structure of spectral lines, and

the splitting caused by the interaction between the intrinsic magnetic moments

of the electron and the nucleus is called the hyperﬁne structure of spectral lines.

6.4 Many-electron atoms

6.4.1 The Pauli exclusion principle and quantum states

Let us assume that in many-electron atoms the electron–electron interaction

can be neglected. This means that each electron moves independently in the

spherically-symmetric ﬁeld of the nucleus:

U (r) =−

, (6.62)

where Z is the atomic number. In this approximation the energy of electron

stationary states in a many-electron atom should have the same form as the

energy (6.7) of the hydrogen atom:

= E

=−13.6

eV. (6.63)

If all electrons were in the state with n = 1, then the atom’s ionization energy

should be deﬁned by the expression

= 13.6Z

eV.

Thus, for a mercury atom, Hg, with Z = 80 the ionization energy should be

equal to E

= 8.7 × 10

eV, whereas the correct value of the ionization energy

is E

= 10 eV. Therefore, such a big difference between the calculated and

experimental ionization energies shows that the approximation of non-interacting

electrons is not valid. The drawbacks of such an approximation are determined

by the fact that we do not take into account two important properties of the

behavior of a system of electrons in a many-electron atom.

First of all, when the atomic number, Z, increases, and therefore the number

of electrons increases, the ﬁlling of energy states by electrons occurs as follows.

As a new electron is added it does not occupy the ground state but instead

occupies the lowest of the available unoccupied energy states. The formation of

many-electron atoms happens according to one of the main principles of quantum

physics – the Pauli exclusion principle, according to which a particular state,

which is characterized by a given set of quantum numbers (in the case of an atom

the quantum numbers are n, l, m, m

), may be occupied only by one electron.

Because of this principle, the outer electrons, for example, in a mercury (Hg)

atom can occupy only states with n = 6.

Second, it is necessary to take into account that the energy levels in this

approximation are strongly degenerate, because their position depends only on

the principal quantum number, n, and does not depend on the other quantum

numbers. Electron repulsion lifts this degeneracy, and as a result the energy

174 Quantum states in atoms and molecules

levels depend not only on the principal quantum number, n, but also on the orbital

quantum number, l. Electrons with the same quantum number n may be grouped

in shells, and electrons with a given n, which have the same quantum number l,

may be grouped in subshells. Thus, the ﬁlling of energy levels by electrons in a

many-electron atom is reduced to their distribution over the shells and subshells.

Therefore, for n = 1 the values of the other quantum numbers are l = 0, m = 0,

and m

=±1/2. For the n = 1 shell, which has only a single subshell with l = 0,

we have that only two electrons with opposite spin projections may occupy it.

At n = 2, two subshells can exist, with l = 0 and l = 1. The quantum numbers

m = 0 and m

=±1/2 correspond to the ﬁrst subshell, and two electrons may

occupy it. The second subshell, with l = 1, corresponds to the quantum numbers

m =−1, 0, 1 and each value of m corresponds to m

=±1/2. As a result this

subshell may contain six electrons. Therefore, the total number of electrons in

the shell with n = 2 is equal to 8. In the general case the capacity of the subshell

withagivenl is equal to 2(2l + 1) and for l = 0, 1, 2, 3, and 4 the capacity of

subshells is equal to 2, 6, 10, 14, and 18 electrons, respectively. Correspondingly,

the capacity of the shell is equal to

n−1



l=0

2(2l + 1) = 2[1 +3 +5 +···+(2n − 1)] = 2n

1 + (2n − 1)

= 2n

. (6.64)

For the principal quantum numbers n = 1, 2, 3, 4, and 5 the capacity of the shells

is equal to 2, 8, 18, 32, and 50, respectively. Thus, in accordance with the Pauli

exclusion principle, in many-electron atoms, after ﬁlling shells with lower n,

the electrons are distributed over the shells and subshells with increasing n and

l. The properties of atoms with closed shells drastically differ from those of

atoms with half-ﬁlled shells. It is difﬁcult to remove an electron from atoms

with completely ﬁlled shells (for example, helium (He) or neon (Ne) with Z = 2

and 10, respectively). Atoms with one extra electron (for example, lithium (Li)

and sodium (Na) with Z = 3 and 11, respectively) easily lose their outermost

electron. Atoms lacking one electron for completion of their outer shell (for

example, hydrogen (H) and ﬂuorine (F) with Z = 1 and 9, respectively) easily

acquire one more electron. In the next section we will consider the classiﬁcation

of the quantum states of many-electron atoms.

6.4.2 The total angular momentum of an atom

For many-electron atoms the angular momenta and spin angular momenta are

deﬁned by the expressions

= h



L(L + 1), (6.65)

= h



S(S + 1), (6.66)

where L and S are quantum numbers for the atomic angular momenta and spin

angular momenta. The quantum number L is always either an integer or equal

6.4 Many-electron atoms 175

Table 6.4. Spectroscopic classiﬁcation of quantum states

L 012 34 5

SymbolSPDFGH

to zero. The quantum number S for an even number of electrons is an integer,

whereas for an odd number of electrons it is a half-integer.

The total angular momentum of an atom, M

, is a result of quantum-

mechanical summation of momenta, which depends substantially on the type

of interaction between the momenta. For the most common normal coupling the

angular momenta are added to give the resulting angular momentum, M

, and

the spin angular momenta to give the resulting spin angular momentum, M

.The

interaction of these momenta deﬁnes the total angular momentum of an atom,

, whose magnitude is equal to

= h



J (J + 1). (6.67)

Here, the quantum number J takes on one of the following values:

J = L + S , L + S − 1 , ..., |L − S|. (6.68)

This number takes on integer values if S is an integer, i.e., for an even number

of electrons in an atom, and it takes on half-integer values if S is a half-integer,

i.e., for an odd number of electrons.

The projection of the total angular momentum of an atom onto the chosen

direction is deﬁned by the formula of space quantization:

= m

, (6.69)

where the quantum number m

takes on 2J + 1 values:

=−J, −J + 1, ..., J − 1, J. (6.70)

For the designation of different quantum states of many-electron atoms the

spectroscopic classiﬁcation of quantum states is used. This consists of the follow-

ing: each value of the total orbital quantum number, L, corresponds to a certain

capital letter of the Latin alphabet (see Table 6.4). As a superscript to the left of

the Latin letter the multiplicity of states, which is equal to 2S + 1 and deﬁnes the

number of sublevels to which the level can be split, is written. The multiplicity

is related to the spin quantum number S (do not confuse this with the symbol of

state with L = 0). As a subscript to the right of the symbol the quantum number

J , which deﬁnes the total angular momentum of the atom, is written. Thus, the

ground state of the helium atom, He, is designated by the symbol

,forwhich

S = 0, L = 0, and J = 0. In an atom with two electrons, two types of states

with S = 0 (the electron spins are anti-parallel) and with S = 1 (electron spins

are parallel) may be possible. In the ﬁrst case J = L = 0 and the multiplicity is

equal to 2S + 1 = 1. In the second case the multiplicity is equal to 2S + 1 = 3

176 Quantum states in atoms and molecules

Table 6.5. Symbolic representation of quantum states of

many-electron atoms

S 000 1 1 1

L 012 0 1 2

J 0 1 2 1,0 2,1,0 3,2,1

Symbol

Table 6.6. The total number of states in shells

Symbol of the shell K L M N O

Quantum number n 12 3 4 5

Total number of states 2n

2 8 18 32 50

and three values of J are possible: L + 1, L, and |L − 1|. The designations of

the above-mentioned states are listed in Table 6.5.

According to the Pauli exclusion principle and Eq. (6.64), the quantum state

with a given principal quantum number, n, corresponds to 2n

states with different

values of the quantum numbers l, m, and m

. Those electrons in an atom with

the same quantum number n form the so-called shell. In correspondence to the

values of n, the shells are designated by capital Latin letters. The ﬁrst ﬁve atomic

shells are shown in Table 6.6. The shells are divided into subshells, which differ

only by orbital quantum number, l. The number of different states in a subshell

with different quantum numbers m and m

is equal to 2(2l + 1). The subshells

with l = 0, 1, 2, 3,...include the quantum states 2, 6, 10, 14, ..., respectively,

and may be designated as

1s, 2s 2p, 3s 3p 3d, 4s 4p 4d 4f,..., (6.71)

where the number designates the quantum number n, i.e., the afﬁliation with the

corresponding shell. Completely ﬁlled shells and subshells have L = 0, S = 0,

and J = 0. Each subsequent atom is built up from the previous one by changing

the charge of the nucleus by adding a positive elementary charge, e, and by

adding one electron, which according to the Pauli exclusion principle ﬁlls the

next vacant state with the minimal energy. Thus, the next element after hydrogen

(H), namely helium (He), ﬁlls the K shell by adding a second 1s-electron with

spin opposite to the spin of the ﬁrst electron. The third element, lithium (Li), has

in addition to a ﬁlled K shell one more electron in the 2s subshell. The fourth

element, beryllium (Be), has a ﬁlled K shell and 2s subshell.

The distribution of electrons over the states is called the electron conﬁguration.

Thus, the electron conﬁguration of the ground state of silicon (Si), which contains

6.4 Many-electron atoms 177

14 electrons, has the form

This means that the K and L shells and 3s subshell are completely ﬁlled, and

there are two electrons in the 3p subshell.

6.4.3 Hund’s rules

The ﬁfth element of the Periodic Table of the elements, boron (B), has in its

2p subshell one electron with L = 1 and S = 1/2. These values correspond to

values of the quantum number J = 3/2 and 1/2, i.e., to the two quantum states

of an atom

3/2

and

1/2

. Which of these states is the ground state is decided

by Hund’s rules, which deﬁne the algorithm for the ﬁlling of subshells.

1. A given electron conﬁguration has the smallest energy in the state with the largest

possible S and L, which corresponds to the value of spin, S.

2. The quantum number J is equal to J =|L − S| if the subshell is less than half-ﬁlled,

and to J = L + S if the subshell is more than half-ﬁlled.

Taking into account the second of Hund’s rules, it is clear that for the boron

atom (B) the ground state should be

1/2

since with one electron in the 2p

subshell the largest possible m

and m

are m

= 1/2 and m

= 1. Therefore,

S = 1/2, L = 1, and J =|L − S|=1/2. In Table 6.7 the result of the algorithm

for ﬁlling shells and subshells with electrons is shown for the ﬁrst 36 elements.

6.4.4 Optical spectra

Atoms of the alkali elements, i.e., metals of the ﬁrst group (Li, Na, and K) of

the Periodic Table of the elements (see Fig. 6.11), have one electron in their

outer orbits, which is called the valence electron. These atoms have the simplest

emission spectra. Their spectra, which are similar to the hydrogen-atom spectrum,

consist of a great number of spectral lines. The systematization of these lines

allowed one to group them in series, each of which is related to the transition of

the excited atom to a certain energy level.

An atom of an alkali element has Z electrons, of which Z − 1formafairly

strong core with the nucleus. The outer (valence) electron moves in the electric

ﬁeld of this core and is weakly connected to it. Therefore, atoms of the alkali

elements can be considered as hydrogen-like atoms. However, in contrast to the

hydrogen atom, where the electron moves in a spherically-symmetric potential, in

the alkali elements the ﬁeld of the core is not completely spherically symmetric.

The inner electrons deform the electron core and break the spherical symmetry

of its ﬁeld. The violation of the potential’s symmetry leads to bound states whose

energy depends not only on the principal quantum number, n, as in the case of

the hydrogen atom, but also on the orbital quantum number, l. The solution of

178 Quantum states in atoms and molecules

Table 6.7. The ﬁlling of shells and subshells according to Hund’s rules

Number (Z ) Element K (1s) L (2s 2p) M (3s 3p 3d) N (4s 4p) Ground state

1H1

1/2

2He2

3Li21

1/2

4Be22

5B221

1/2

6C222

7N223

3/2

8O224

9F225

3/2

10 Ne 2 2 6

11 Na 2 2 6 1

1/2

12 Mg 2 2 6 2

13 Al 2 2 6 2 1

1/2

14 Si 2 2 6 2 2

15 P 2 2 6 2 3

3/2

16 S 2 2 6 2 4

17 Cl 2 2 6 2 5

3/2

18 Ar 2 2 6 2 6

19 K 2 2 6 2 6 1

1/2

20 Ca 2 2 6 2 6 2

21 Sc 2 26 261 2

3/2

22 Ti 2 26 262 2

23 V 2 26 263 2

3/2

24 Cr 2 26 264 1

25 Mn 2 26 265 2

5/2

26 Fe 2 26 266 2

27 Co 2 26 267 2

9/2

28 Ni 2 26 268 2

29 Cu 2 26 2610 1

1/2

30 Zn 2 26 2610 2

31 Ga 2 26 2610 21

1/2

32 Ge 2 26 2610 22

33 As 2 26 2610 23

3/2

34 Se 2 26 2610 24

35 Br 2 26 2610 25

3/2

36 Kr 2 26 2610 26

6.4 Many-electron atoms 179

Figure 6.11 The Periodic

Table of the elements.

, eV

l =2

l =1

l =0

Figure 6.12 A schematic

representation of energy

levels and transitions for a

lithium atom, Li: the

principal series is shown

by solid lines, the sharp

series by dashed lines,

and the diffuse series by

dash–dotted lines.

the Schr¨odinger equation gives the following expression for this energy:

=−

(n + b

)

(n + b

)

, (6.72)

where b

, which depends on l, is the quantum correction to the formula for the

energy levels in a hydrogen atom (see Eq. (6.7)).

The dependence of the electron energy in the alkali elements on the orbital

quantum number, l, is the principal difference of the energy spectrum of the alkali

elements from the spectrum of the hydrogen atom. This dependence means that

for the alkali elements the degeneracy of the orbital quantum number, l, is lifted.

Figure 6.12 shows schematically the energy levels and the corresponding tran-

sitions for lithium (Li). The ground state for lithium is the state 2s, for which n = 2

180 Quantum states in atoms and molecules

because the state with n = 1 is occupied by two electrons, which form the lithium

core. Three series of spectral lines of the lithium atom are shown in Fig. 6.12:

principal, sharp, and diffuse. The lines of the principal series correspond to the

transitions from p-levels to the 2s-level. The frequencies of the spectral lines of

the principal series, taking into account Eq. (6.72), can be found as follows:

ω = 2π cR

∞



(2 + b

)

−

(n + b

)



, n = 2, 3, 4,..., (6.73)

where R

∞

is the Rydberg constant. The lines of the sharp series correspond to

transitions from the s-levels to the 2p-level, and the frequencies of the spectral

lines of this series are given by the expression

ω = 2π cR

∞



(2 + b

)

−

(n + b

)



, n = 3, 4, 5,... (6.74)

The lines of the diffuse series correspond to the transitions from the d-levels to

the 2p-level. The frequencies of the spectral lines of this series are given by the

expression

ω = 2π cR

∞



(2 + b

)

−

(n + b

)



, n = 3, 4, 5,... (6.75)

We note that the quantum corrections b for each series are practically the same,

but they may change slightly from series to series.

As was the case for the hydrogen atom, the spectral lines of the lithium atom

have ﬁne doublet structure, which is related to the splitting of its energy levels

due to the existence of the electron spin. The splitting of energy levels is caused

by the spin–orbit coupling, i.e., by interaction of magnetic moments µ

and µ

Since the core of the lithium atom has a completely ﬁlled 1s-level, its angular

momentum is equal to zero. Therefore, the angular momentum of the atom in

general is equal to the angular momentum of its external electron, which is

deﬁned by the orbital quantum number l. The total momentum of this electron

is equal to the sum of the orbital magnetic and spin magnetic moments and is

deﬁned by the quantum number j = l ± s, where l and s = 1/2 are the orbital

and spin quantum numbers. Thus, each energy level of p-states (l = 1) splits into

two sublevels with j = 1/2 and 3/2.

Each energy level of d-states (l = 2) splits into sublevels with j = 3/2 and

5/2. The levels of s-states (l = 0) do not split since they correspond to the single

value of j = 1/2.

Example 6.4. Find the maximum possible total angular momentum and corre-

sponding spectral symbol for the atomic state with the electron conﬁguration

2p 3d.

Reasoning. The total maximal angular momentum will be formed from the

maximal orbital magnetic and spin magnetic moments. Since for s-electrons

6.5 The wavefunction of a system of identical particles 181

l = 0, for p-electrons l = 1, and for d-electrons l = 2,

max

= 1 + 2 = 3.

For the ﬁlled 1s

shell spin s = 0, and therefore

max

= 1.

As a result, for the maximum quantum number which deﬁnes the total angular

momentum, we get

max

= L

max

+ S

max

= 4. (6.76)

The total maximal angular momentum of the atom is equal to

max

= h



max

+ 1) =

√

20h

. (6.77)

Since the multiplicity of the state is 2S

max

+ 1 = 3, the symbol of this state

Example 6.5. Give the spectral symbol of the atomic state whose degeneracy

with respect to J is equal to seven, with multiplicity equal to ﬁve, and for which

the value of the orbital quantum number is maximal.

Reasoning. From the multiplicity of the state,

2S + 1 = 5,

we ﬁnd spin quantum number S = 2.

From the order of degeneracy,

2J + 1 = 7,

we ﬁnd J = 3. The quantum number J can take all the integer values from

L + S to |L − S|. Therefore, the values of J and S are J = 3 and S = 2. The

requirement for L to be maximal corresponds to

J = L − S.

Therefore, L = J + S = 5 and the spectral symbol of this state is

6.5 The wavefunction of a system of identical particles

For particles that obey the laws of classical physics, it is not important whether

they are identical or not. If we assign numbers to these particles we can track

their motion along each particle’s trajectory. In quantum physics the situation

radically changes since microscopic particles do not follow trajectories and thus it

is impossible to track their motion. Because of the uncertainty principle, identical

particles in the same region become indistinguishable and their numbering does

not make sense. Therefore, the description of the behavior of a system of identical

particles is subject to an additional requirement. Besides normalization and

182 Quantum states in atoms and molecules

orthogonality, for the wavefunction of a system of particles we must take into

account the fact that they are indistinguishable.

To illustrate this, let us consider a system of two identical particles. For a

complete description we need the set of variables that deﬁne the position of each

particle in space, r

, and the projection of spin on the chosen axis, σ

.Thewave-

function of the system, ψ (r

,σ

; r

,σ

), must depend on these variables. Since

the particles are indistinguishable, any permutation of the particles would have

no effect on the physical results. This means that permutation of the arguments

,σ

) and (r

,σ

) can change only the phase factor of the wavefunction, i.e.,

ψ(r

,σ

; r

,σ

) = e

iθ

ψ(r

,σ

; r

,σ

). (6.78)

As a result of the second permutation of arguments on the right-hand side of this

equation, the wavefunction takes the initial form but with a new additional phase

factor, i.e.,

ψ(r

,σ

; r

,σ

) = e

2iθ

ψ(r

,σ

; r

,σ

). (6.79)

Since the second permutation of particles (or arguments) returns the system to

its initial physical state, the following equalities have to be satisﬁed:

2iθ

= 1 ⇒ e

iθ

=±1. (6.80)

Therefore, the permutation of the wavefunction’s arguments requires the satis-

faction of the following condition:

ψ(r

,σ

; r

,σ

) =±ψ (r

,σ

; r

,σ

). (6.81)

It follows from Eq. (6.81) that the wavefunction of a system of two identical

particles either does not change after permutation of its particles or changes its

sign. In the ﬁrst case the wavefunction is called symmetric and in the second,

antisymmetric. Thus, because the particles are indistinguishable, the wavefunc-

tion of the system of particles must have some kind of symmetry with respect to

permutations.

Relativistic quantum theory shows that the symmetry of the wavefunction

is unambiguously deﬁned by the spin of the particles. A system of particles

with integer spins (bosons) must be described by a symmetric wavefunction,

whereas a system of particles with half-integer spins (fermions) must be described

by an antisymmetric wavefunction. Since electrons have half-integer spin, the

wavefunction of a system of electrons must be antisymmetric.

Let us consider the algorithm for ﬁnding the wavefunction which satisﬁes

the above-mentioned condition, using the example of two free non-interacting

electrons. The wavefunction of a free electron without taking into account its spin

has the form (3.18). Taking into account the spin variables, the wavefunction of

the system of two electrons with momenta p

and p

must be represented (without

taking into account its dependence on time) as

ψ(r

,σ

; r

,σ

) = Ae

)

(σ

)ϕ

(σ

), (6.82)