01 10

±= ; β= . (2.48)

0 ’1

10

Hermitian (±† = ±) because the eigenvalues must be real, anticommuting because ±β + β± =

3

0, unitary because ±2 = ±† ± = 1.

28 Quantum mechanics: principles and relativistic e¬ects

Inserting this choice into (2.44) yields the following matrix di¬erential equa-

tion:

‚ ΨL mc pˆ ΨL

i =c . (2.49)

p ’mc

dt ΨS ˆ ΨS

We see that in a coordinate frame moving with the particle (p = 0) there

are two solutions: ΨL corresponding to particles (electrons) with positive

energy E = mc2 ; and ΨS corresponding to antiparticles (positrons) with

negative energy E = ’mc2 . With non-relativistic velocities p mc, the

wave function ΨS mixes slightly in with the particle wave function ΨL (hence

the subscripts L for “large” and S for “small” when we consider particles).

The eigenfunctions of the Hamiltonian matrix are

ˆ

H = ±c(m2 c2 + p2 )1/2 ,

ˆ (2.50)

which gives, after expanding the square root to ¬rst order in powers of p/mc,

the particle solution

p2

‚Ψ ˆ

≈ (mc +

2

i )Ψ, (2.51)

‚t 2m

in which we recognize the Schr¨dinger equation for a free particle, with an

o

extra constant, and irrelevant, zero-energy term mc2 .

In the case of three spatial dimensions, there are three ±-matrices for

each of the spatial components; i.e., they form a vector ± of three matrices

±x , ±y , ±z . The simplest solution now requires four dimensions, and Ψ

becomes a four-dimensional vector. The Dirac equation now reads

‚Ψ ˆ

= c(± · p + βmc)Ψ = HΨ,

ˆ

i (2.52)

‚t

where ±x , ±y , ±z and β are mutually anti-commuting 4 — 4 matrices with

their squares equal to the unit matrix. One choice of solutions is:

0σ

±= , (2.53)

σ0

0 ’i

01 10

σx = , σy = , σz = , (2.54)

0 ’1

10 i0

while β is a diagonal matrix {1, 1, ’1, ’1} that separates two solutions

around +mc2 from two solutions around ’mc2 . The wave function now also

has four components, which refer to the two sets of solutions (electrons and

positrons) each with two spin states. Thus spin is automatically introduced;

it gives rise to an angular momentum S and an extra quantum number

S = 1/2. By properly incorporating electromagnetic interactions, the small

2.3 Relativistic energy relations for a free particle 29

spin-orbit interaction, arising from magnetic coupling between the electron

spin S and angular orbital momentum L , is included in the solution of the

Dirac equation. This term makes it impossible to exactly separate the spin

and orbital momenta; in fact there is one quantum number for the total

angular momentum.

Let us now look at the relativistic e¬ects viewed as a perturbation of the

non-relativistic Schr¨dinger equation. We may ¬rst remark that spin can

o

be separately and ad hoc introduced into the non-relativistic case as a new

degree of freedom with two states. Each electron spin has associated with it

an angular momentum S and a magnetic moment μ = ’γe S , where γe is

the electron™s gyromagnetic ratio. The spin-orbit interaction term can then

be computed from the classical interaction of the electron magnetic moment

with the magnetic ¬eld that arises at the electron due to its orbital motion

around a charged nucleus.

The relativistic e¬ects arising from the high velocity of the electron can

be estimated from a Taylor expansion of the positive solution of (2.42):

E

m2 c2 + p2 ,

= (2.55)

c

p2 p4

’ + ··· ,

2

E = mc 1 + (2.56)

2m2 c2 8m4 c4

p2 p4

’ + ···

2

= mc + (2.57)

2m 8m3 c2

The ¬rst term is an irrelevant zero-point energy, the second term gives us the

non-relativistic Schr¨dinger equation, and the third term gives a relativistic

o

correction. Let us estimate its magnitude by a classical argument.

Assume that the electron is in a circular orbital at a distance r to a nucleus

with charge Ze. From the balance between the nuclear attraction and the

centrifugal force we conclude that

p2 = ’2mE, (2.58)

where E is the (negative) total energy of the electron, not including the

term mc2 (this also follows from the virial

equation valid for a central Coulombic ¬eld: Epot = ’2Ekin , or E =

’Ekin ). For the expectation value of the ¬rst relativistic correction we ¬nd

a lower bound

p4 p2 2 E2

≥ = . (2.59)

8m3 c2 8m3 c2 2mc2

The correction is most important for 1s-electrons near highly charged nuclei;

since ’E is proportional to Z 2 , the correction is proportional to Z 4 . For

30 Quantum mechanics: principles and relativistic e¬ects

the hydrogen atom E = ’13.6 eV while mc2 = 511 keV and hence the

correction is 0.18 meV or 17 J/mol; for germanium (charge 32) the e¬ect is

expected to be a million times larger and be in the tens of MJ/mol. Thus

the e¬ect is not at all negligible and a relativistic treatment for the inner

shells of heavy atoms is mandatory. For molecules with ¬rst-row atoms

the relativistic correction to the total energy is still large (-146 kJ/mol for

H2 O), but the e¬ects on binding energies and on equilibrium geometry are

small (dissociation energy of H2 O into atoms: -1.6 kJ/mol, equilibrium OH

distance: -0.003 pm, equilibrium angle: -0.08 deg).4

In addition to the spin and energetic e¬ects, the 1s-wave functions contract

and become “smaller”; higher s-wave functions also contract because they

remain orthogonal to the 1s-functions. Because the contracted s-electrons

o¬er a better shielding of the nuclear charge, orbitals with higher angular

momentum tend to expand. The e¬ect on outer shell behavior is a secondary

e¬ect of the perturbation of inner shells: therefore, for quantum treatments

that represent inner shells by e¬ective core potentials, as in most practical

applications of density functional theory, the relativistic corrections can be

well accounted for in the core potentials without the need for relativistic

treatment of the outer shell electrons.

Relativistic e¬ects show up most clearly in the properties of heavy atoms,

such as gold (atom number 79) and mercury (80). The fact that gold has

its typical color, in contrast to silver (47) which has a comparable electron

con¬guration, arises from the relatively high energy of the highest occupied

d-orbital (due to the expansion of 5d3/2 -orbital in combination with a high

spin-orbit coupling) and the relatively low energy of the s-electrons in the

conduction band (due to contraction of the 6s-orbitals), thus allowing light

absorption in the visible region of the spectrum. The fact that mercury is a

liquid (in contrast to cadmium (48), which has a comparable electron con-

¬guration) arises from the contraction of the 6s-orbitals, which are doubly

occupied and so localized and “buried” in the electronic structure that they

contribute little to the conduction band. Mercury atoms therefore resemble

noble gas atoms with weak interatomic interactions. Because the enthalpy

of fusion (2.3 kJ/mol) is low, the melting point (234 K) is low. For cad-