relativistic quantum chemistry · relativistic effects in structural chemistry ekka pyykko...

Relativistic Quantum Chemistry

Valérie Vallet ([email protected])

CNRS, Lille University, France

mailto:[email protected]

Relativistic Quantum Chemistry | Valérie Vallet @ CNRS Uni. Lille 2

Outline

▸ Why do relativistic effects become significant when the atomic number Z increases?

▸ How to account for relativistic effects in quantum chemistry?

▸ Which role do relativistic effects to play in the chemistry across the periodic table ?

✴ Bond lengths, frequencies

✴ Binding energies, reaction barriers

✴ Spectroscopy

✴ Ionization potentials, electron affinities

✴ NMR properties

Relevance and nature of relativistic effects


Can you spot the difference?

Property Ag AuElectronic configuration [Kr] 4d105s1 [Xe] 4f14 5d10 6s1

Crystal structure FCC FCCColor silver golden

Why does gold look so… golden?


Have a look at relativistic effects in chemistry

Chem. Rev. 1988. 88. 563-594 583

Relativistic Effects in Structural Chemistry

EKKA PYYKKO

Department of chemistry. Uinh'ersity of Helsinki, Et. Hesperiankatu 4, 00100 Helsinki, FWand'

Received March 17, 1987 (Revised Manuscript Received July 7, 1987)

Contents I. Introduction

11. Theory and Methods A. Fundamental Questions B. Properties of Individual Atoms C. Available Quantum Chemical Methods for

Molecules 0. Effects on Bond Lengths

111. Molecular Geometries A. Bond Lengths B. Bond Angles C. Catalysis and Reaction Pathways D. Solids E. The Inert-Pair Effect

A. Force Constants B. Dlssoclatlon Energies C. Finestructure Splittings D. Ionization Potentials, Electron Affinities,

IV. Other Properties

and Photoelectron Spectra

563 565 565 565 567

571 573 573 577 577 578 578 580 580 580 582 582

E. Colors 583 F. Charge Distributions and Molecular 563

Moments G. Magnetic Resonance Parameters 583

V. Further Possible Examples and Open Problems 585 VI. Summary: Relativistic Effects in the Periodic 587

VII. References 587 Table

I . Introducflon

The two basic theories of modem physics are the theory of relativity and quantum mechanics. While the importance of the latter in chemistry was instantly recognized, i t was not until the 19708 that the full relevance of relativistic effects in heavy-element chemistry was discovered.

For very precise calculations, relativistic energy contributions are already needed for H2+ or H,. They increase, for valence shells, roughly like .??. Depending on the accuracy achieved in the calculation, they become relevant again around Cu, or perhaps Ag. For the sixth row (around W to Bi), relativistic effects are comparable to the usual shell-structure effects and provide an explanation for much of the basic freshman chemistry of these elements. For the existing actinoids relativistic effects are essential.

The relativistic effects can be defined as anything arising from the finite speed of light, c = 137.035 989 5 (61) au4I5, as compared to c = m. The basic theory is discussed in section 11.

It has become a tradition to introduce the reader to the qualitative effects of relativity as follows: Due to

0009-2665/88/078&0563$06.50/0

pekka pwkko was bom in 1941 in Hinnerjoki. Finland. and recehed his FiLKand.. FiLLic.. and FIl.dr from the Unhrersity of Turku. After working at We U n W i s of Aafhus, Gatebwg. Helsinki. Jyviisky!A, Paris XI, and Oulu (1968-1974). he became Associate Professor of Quantum Chemistry at Abo Akademi in Turku in 1974. Since 1984 he has been Professor of Chemistry at the University of Hekinki. Finland. where he holds We "Swedish Chair of Chemistry". founded in 1908 as a parallel one to Johan Gadolin's former chair, established in 1761. His main interests have been NMR meorY and relativistic quantum chemistry

the relativistic mass increase m = mo/(l - (U/C))1'* (la)

mo being the rest mass and u the speed of the electron, the effective Bohr radius,

a. = (4rc0)(h2/me2) (W will decrease for inner electrons with large average speeds. For a 1s shell a t the nonrelativistic limit, this average speed is 2 au. Thus the 1s electron of Hg has a u/c of 80/137 = 0.58, implying a radial shrinkage by 23%. Because the higher s shells have to be orthogonal against the lower ones, they will suffer a similar contraction. Due to interacting relativistic and shell- structure effects, their contraction can in fact he even larger; for gold, the 6s shell has larger (percental) relativistic effects than the Is shell.

For readers not convinced by a qualitative argument, we can consider an exactly solvable problem: the hydrogen-like atom with 2 = 80 in Figure 1. As seen, the contractions are comparable for the three firsts shells.

Alternatively, to understand the valence electron effects in a many-electron atom, one can compare (see ref 35) the relativistic (Dirac) and nonrelativistic (Schrodinger) dynamics for the valence electron in a given atomic potential, to study the importance of the direct relativistic effect. The (originally surprising) result" was that this is the main effect for the 6s of Au or the 6p* (=6p,/,) of TI. The relativistic change of the atomic potential mattered less than the direct, dy-

0 1988 American Chemical Society

Relativistic Quantum Chemistry | Valérie Vallet @ CNRS Uni. Lille

m = �m0 =m0r1� Z2

c2

6

Relativistic effects in chemistry

▸ Relativistic effects arise from the finite speed of light (c ≈ 137 a.u.)

▸ v(1s) ∼ Z a.u.

▸ The relativistic mass increase for electrons with rest mass m0 and speed v is:

▸ The Bohr radius is inversely proportional to electron mass:

▸ Relativity will contract orbitals of one-electron atoms, e.g. Au78+ (Z/c = 57%), 18% relativistic contraction of the 1s orbital

Chem. Rev. 1988. 88. 563-594 583

Relativistic Effects in Structural Chemistry

EKKA PYYKKO

Department of chemistry. Uinh'ersity of Helsinki, Et. Hesperiankatu 4, 00100 Helsinki, FWand'

Received March 17, 1987 (Revised Manuscript Received July 7, 1987)

Contents I. Introduction

11. Theory and Methods A. Fundamental Questions B. Properties of Individual Atoms C. Available Quantum Chemical Methods for

Molecules 0. Effects on Bond Lengths

111. Molecular Geometries A. Bond Lengths B. Bond Angles C. Catalysis and Reaction Pathways D. Solids E. The Inert-Pair Effect

A. Force Constants B. Dlssoclatlon Energies C. Finestructure Splittings D. Ionization Potentials, Electron Affinities,

IV. Other Properties

and Photoelectron Spectra

563 565 565 565 567

571 573 573 577 577 578 578 580 580 580 582 582

E. Colors 583 F. Charge Distributions and Molecular 563

Moments G. Magnetic Resonance Parameters 583

V. Further Possible Examples and Open Problems 585 VI. Summary: Relativistic Effects in the Periodic 587

VII. References 587 Table

I . Introducflon

The two basic theories of modem physics are the theory of relativity and quantum mechanics. While the importance of the latter in chemistry was instantly recognized, i t was not until the 19708 that the full relevance of relativistic effects in heavy-element chemistry was discovered.

For very precise calculations, relativistic energy contributions are already needed for H2+ or H,. They increase, for valence shells, roughly like .??. Depending on the accuracy achieved in the calculation, they become relevant again around Cu, or perhaps Ag. For the sixth row (around W to Bi), relativistic effects are comparable to the usual shell-structure effects and provide an explanation for much of the basic freshman chemistry of these elements. For the existing actinoids relativistic effects are essential.

The relativistic effects can be defined as anything arising from the finite speed of light, c = 137.035 989 5 (61) au4I5, as compared to c = m. The basic theory is discussed in section 11.

It has become a tradition to introduce the reader to the qualitative effects of relativity as follows: Due to

0009-2665/88/078&0563$06.50/0

pekka pwkko was bom in 1941 in Hinnerjoki. Finland. and recehed his FiLKand.. FiLLic.. and FIl.dr from the Unhrersity of Turku. After working at We U n W i s of Aafhus, Gatebwg. Helsinki. Jyviisky!A, Paris XI, and Oulu (1968-1974). he became Associate Professor of Quantum Chemistry at Abo Akademi in Turku in 1974. Since 1984 he has been Professor of Chemistry at the University of Hekinki. Finland. where he holds We "Swedish Chair of Chemistry". founded in 1908 as a parallel one to Johan Gadolin's former chair, established in 1761. His main interests have been NMR meorY and relativistic quantum chemistry

the relativistic mass increase m = mo/(l - (U/C))1'* (la)

mo being the rest mass and u the speed of the electron, the effective Bohr radius,

a. = (4rc0)(h2/me2) (W will decrease for inner electrons with large average speeds. For a 1s shell a t the nonrelativistic limit, this average speed is 2 au. Thus the 1s electron of Hg has a u/c of 80/137 = 0.58, implying a radial shrinkage by 23%. Because the higher s shells have to be orthogonal against the lower ones, they will suffer a similar contraction. Due to interacting relativistic and shell- structure effects, their contraction can in fact he even larger; for gold, the 6s shell has larger (percental) relativistic effects than the Is shell.

For readers not convinced by a qualitative argument, we can consider an exactly solvable problem: the hydrogen-like atom with 2 = 80 in Figure 1. As seen, the contractions are comparable for the three firsts shells.

Alternatively, to understand the valence electron effects in a many-electron atom, one can compare (see ref 35) the relativistic (Dirac) and nonrelativistic (Schrodinger) dynamics for the valence electron in a given atomic potential, to study the importance of the direct relativistic effect. The (originally surprising) result" was that this is the main effect for the 6s of Au or the 6p* (=6p,/,) of TI. The relativistic change of the atomic potential mattered less than the direct, dy-

0 1988 American Chemical Society

P. Pyykkö, Chem. Rev. 1988, 88, 563

a0 =4⇡"0~2

m


Relativistic effects on orbitals

▸ The semi-quantitative calculation on the previous slide showed that 1s orbital of Au would be contracted by ~18% due to relativistic mass increase

▸ The higher s shells are orthogonal to the 1s shell and must contract, too

✴ The higher s shells, up to the valence shell, contract roughly as much as 1s because their electron speeds near the nucleus are comparable and the contraction of the inner part of the wave function affects the outer part, too

▸ p-orbitals are also contracted due to relativity (and split into p1/2 and p3/2 due to spin-orbit coupling)

▸ d and f electrons never come close to the nucleus and they will be screened more strongly by the contracted s and p orbitals

▸ Conclusions

✴ s and p orbitals are contracted and stabilized due to relativity

✴ d and f orbitals are expanded and destabilized due to relativity

▸ The relativistic effects for the valence orbitals increase as Z2


Relativity in the periodic table8A1A

2A

3B 4B 5B 6B 7B 8B 11B 12B

3A 4A 5A 6A 7A

element names in blue are liquids at room temperatureelement names in red are gases at room temperatureelement names in black are solids at room temperature

Periodic Table of the Elements

Los Alamos National Laboratory Chemistry Division

39

*

**

As39.10

85.47

132.9

(223)

9.012

24.31

40.08

87.62

137.3

(226)

44.96

88.91

47.88

91.22

178.5

(267) (268) (269) (270) (277) (278) (281) (282) (285) (289) (289)(286) (293) (294) (294)

50.94

92.91

180.9

52.00

95.96

183.9

54.94

(98)

186.2

55.85

101.1

190.2 192.2

102.9

58.93 58.69

106.4

195.1 197.0

107.9

63.55 65.39

112.4

200.5

10.81

26.98

12.01

28.09

14.01

69.72 72.64

114.8 118.7

204.4 207.2

30.97

74.92

121.8

209.0 (209) (210) (222)

16.00 19.00 20.18

4.003

32.06 35.45 39.95

78.96 79.90 83.79

127.6 126.9 131.3

140.1 140.9 144.2 (145) 150.4 152.0 157.2 158.9 162.5 164.9 167.3 168.9 173.0 175.0

232

138.9

(227) 231 238 (237) (244) (243) (247) (247) (251) (252) (257) (258) (259) (262)

Tc43

11

1

3 4

12

19 20 21 22 23 24 25 26 27 28 29 30

37 38 40 41 42 44 45 46 47 48

55 56

58 59 60

72 73 74 75 76 77 78 79 80

87 88

90

57

89 91 92 93 94 95 96

104 105 106 107 108 109 110 111 112

61 62 63 64 65 66 67

97 98 99

68 69 70 71

100 101 102 103

31

13 14 15 16 17 18

32 33 34 35 36

49 50 51 52 53 54

81 82 83 84 85 86

5 6 7 8 9 10

2

114 115113 116 117 118

3/13/17

Th

Li

Na

K

Rb

Cs

Fr

Be

Mg

Ca

Sr

Ba

Ra

Sc Ti V Cr Mn Fe Co Ni Cu Zn

Y Zr Nb Mo Ru Rh Pd Ag Cd

Hf Ta W Re Os Ir Pt Au Hg

HeH

Ne

Ar

Kr

Xe

F

Cl

ON

Br

Rf Db Sg Bh Hs Mt Ds Rg Cn

B C

Al Si P S

Ga Ge Se

In Sn Sb Te I

Tl Pb Bi Po At Rn

Fl LvMcNh Ts Og

Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb LuLa

Ac Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr

hydrogen

barium

francium radium

strontium

sodium

vanadium

berylliumlithium

magnesium

potassium calcium

rubidium

cesium

helium

boron carbon nitrogen oxygen fluorine neon

aluminum silicon phosphorus sulfur chlorine argon

scandium titanium chromium manganese iron cobalt nickel copper zinc gallium germanium arsenic selenium bromine krypton

yttrium zirconium niobium molybdenum technetium ruthenium rhodium palladium silver cadmium indium tin antimony tellurium iodine xenon

hafnium

cerium praseodymium neodymium promethium samarium europium gadolinium terbium dysprosium holmium erbium thulium ytterbium lutetium

tantalum tungsten rhenium osmium iridium platinum gold mercury thallium lead bismuth polonium astatine radon

thorium

lanthanum

actinium protactinium uranium neptunium plutonium americium curium berkelium californium einsteinium fermium mendelevium nobelium lawrencium

rutherfordium dubnium seaborgium bohrium hassium meitnerium darmstadtium roentgenium copernicium flerovium moscovium livermorium tennessine oganessonnihonium

1.008

6.94

22.99

Lanthanide Series*

Actinide Series**

1s1

[Ar]4s23d104p3[Ar]4s23d3[Ar]4s13d10

[Ne]3s23p6[Ne]3s23p4

[Ar]4s1[Ar]4s23d10

1s2

[He]2s1 [He]2s2

[Ar]4s23d7

[Ne]3s23p5

[He]2s22p1 [He]2s22p2 [He]2s22p3

[Ar]4s23d5

[He]2s22p4 [He]2s22p5 [He]2s22p6

[Ar]4s23d104p5

[Ne]3s1 [Ne]3s23p1 [Ne]3s23p3[Ne]3s23p2

[Rn]7s25f146d2

[Ne]3s2

[Ar]4s2 [Ar]4s23d1 [Ar]4s23d2 [Ar]4s13d5 [Ar]4s23d6 [Ar]4s23d8 [Ar]4s23d104p1 [Ar]4s23d104p2 [Ar]4s23d104p4 [Ar]4s23d104p6

[Kr]5s1 [Kr]5s2 [Kr]5s24d1 [Kr]5s24d2 [Kr]5s14d4 [Kr]5s14d5 [Kr]5s24d5 [Kr]5s14d7 [Kr]5s14d8 [Kr]4d10 [Kr]5s14d10 [Kr]5s24d10 [Kr]5s24d105p1 [Kr]5s24d105p2 [Kr]5s24d105p3 [Kr]5s24d105p4 [Kr]5s24d105p5 [Kr]5s24d105p6

[Xe]6s1 [Xe]6s2

[Xe]6s24f15d1 [Xe]6s24f3 [Xe]6s24f4 [Xe]6s24f5 [Xe]6s24f6 [Xe]6s24f7 [Xe]6s24f75d1 [Xe]6s24f9 [Xe]6s24f10 [Xe]6s24f11 [Xe]6s24f12 [Xe]6s24f13 [Xe]6s24f14 [Xe]6s24f145d1

[Xe]6s24f145d2 [Xe]6s24f145d3 [Xe]6s24f145d4 [Xe]6s24f145d5 [Xe]6s24f145d6 [Xe]6s24f145d7 [Xe]6s14f145d9 [Xe]6s14f145d10 [Xe]6s24f145d10 [Xe]6s24f145d106p1 [Xe]6s24f145d106p2 [Xe]6s24f145d106p3 [Xe]6s24f145d106p4 [Xe]6s24f145d106p5 [Xe]6s24f145d106p6

[Rn]7s1 [Rn]7s2

[Rn]7s26d2

Xe]6s25d1

[Rn]7s26d1 [Rn]7s25f26d1 [Rn]7s25f36d1 [Rn]7s25f46d1 [Rn]7s25f6 [Rn]7s25f7 [Rn]7s25f76d1 [Rn]7s25f9 [Rn]7s25f10 [Rn]7s25f11 [Rn]7s25f12 [Rn]7s25f13 [Rn]7s25f14 [Rn]7s25f146d1

[Rn]7s25f146d3 [Rn]7s25f146d4 [Rn]7s25f146d5 [Rn]7s25f146d6 [Rn]7s25f146d7 [Rn]7s15f146d9 [Rn]7s15f146d9 [Rn]7s15f146d9 [Rn]7s27p15f14

6d10 (predicted)[1][Rn]7s27p25f14

6d10(predicted)[2][Rn]7s27p35f14


6d10(predicted)[1][Rn]7s27p55f146d10(predicted)[4] [Rn]7s27p65f14

6d10(predicted)[1][2]

Clearly relevant

Starts to be relevant


Gold maximum of relativistic effects

P. Pyykkö and J. P. Desclaux, Acc. Chem. Res. 12, 276 (1979)

⟨r⟩rel/⟨r⟩non-rel for the 6s orbital (Z=55-100)

⟨r⟩non-rel is greater than ⟨r⟩rel


Why is gold yellow?

Au absorbs blue andreflects yellow/red

R = relativisticNR = non-relativistic


Spin-orbit interaction

▸ The spin-orbit interaction is not the interaction between spin and angular momentum of an electron. An electron moving alone in space is subject to no spin-orbit interaction!

▸ The basic mechanism of the spin-orbit interaction is magnetic induction:

✴ An electron which moves in a molecular field will feel a magnetic field in its rest frame, in addition to an electric field. The spin-orbit term describes the interaction of the spin of the electron with this magnetic field due to the relative motion of the charges.

▸ This operator couples the degrees of freedom associated with spin and space and therefore makes it impossible to treat spin and spatial symmetry separately.

hSO =1

2m2c2s · [rV ⇥ p]

with V = �Z

r

hSO =Z

2m2c2r3s · l


Spin-orbit coupling in atoms

▸ In the absence of spin-orbit coupling atomic electronic states are characterized by total orbital angular momentum L and total spin S and denoted as 2S+1L.

▸ With spin-orbit interaction only the total angular momentumJ = |L − S|; · · · ;|L + S| is conserved.

▸ The ground state configuration of oxygen is 1s22s22p4 which in a non-relativistic framework (LS-coupling) gives rise to three states.

Term L S J Level (cm-1)3P

1 12 0.0001 158.2650 226.977

1D 2 0 2 15867.8621S 0 0 0 33792.583


Spin-orbit coupling in molecules

▸ Open shells of radicals, transition metal compounds, lanthanide and actinide compounds, and excited states undergo spin-orbit splitting.

▸ There are often a multitude of low-lying electronic states for some or all conformations, especially along bond-breaking parts of reaction paths. Intrasystem interactions make them all relevant to reactivity.

▸ Example: Oxidative addition/recombination mechanism of dioxygen insertion into the NHCAuH bond

C. A. Gaggioli, L. Belpassi, F. Tarantelli, J. N. Harvey, P. Belanzoni, “Spin-Forbidden Reactions: Adiabatic Transition States Using Spin-Orbit Coupled Density Functional Theory” Chem. Eur. J. 23 (2017), p. 1. DOI: 10.1002/chem.201704608

http://dx.doi.org/10.1002/chem.201704608


Atomic oxygen emissions in atmospheric aurora

Transition Wavelength (Å) Type Lifetime (s)

Green line 1S0 → 1D2 5577 E2 0.75

Red line 1D2 → 3P2 6300 M1 110

Formalism for relativistic quantum chemistry


Three cornerstones of Quantum Chemistry

▸ Assume molecular systems follow the Born-Oppenheimer Approximation allowing the use of the PES concept

▸ Assume nuclear charge can be described by a finite-size model (e.g. use of gaussian type basis functions)

▸ Assume electrons move slow enough to be described by a non-relativistic theory

Relativistic Quantum Chemistry | Valérie Vallet @ CNRS Uni. Lille

▸ Transforms coordinates between two different reference frames

▸ Velocities are additive and time is constant.

▸ No “speed limit”

17

Galileo transformation

x = x0 + vt

y = y0

z = z0

t = t0

Motion in the x-direction


Lorentz transformationIn the x-direction

x = �(x0 + vt0)

y = y0

z = z0

t = �(t0 +vx0

c2)

r = r0 + v

✓v · r0(� � 1)

v2+ �t0

◆

t = �

✓t0 +

(v · r0)c2

◆

Generalized to 3 dimensions

� =

✓1� v2

c2

◆� 12


Recall in non-relativistic quantization

▸ Spin-Free Non-relativistic Hamiltonian:

▸ Non-relativistic Hamiltonian including spin:

H = T + V =⇡2

2m+ q�(r)

⇡ = p� qA

H =(� · ⇡)2

2m+ q�(r)


Including spin in the NR Hamiltonian

▸ Introduction of spin may appear ad hoc

▸ No spin-orbit coupling

▸ Not Lorentz invariant

▸ Linear in φ but quadratic in A

H =(� · ⇡)2

2m+ q�(r)

=⇡2

2m� q~

2m� ·B+ q�

H = i~ @

@t

Non-relativistic Pauli Hamiltonian


Relativistic quantization (1)

▸ Classical relativistic energy expression including electromagnetic-fields.

▸ Quantization results in the Klein-Gordon (KG) Equation

▸ Lorentz invariant but doesn’t include spin

▸ KG-equation is not suitable for electrons

(E � q�)2 = m2c4 + c2⇡2

⇡ = p� qA

(i~ @

@t� q�)2 = (m2c4 + c2⇡2)


Relativistic quantization (2)

▸ Generalize the concept of the square root in E − qφ equation

▸ Require cross-terms drop out by introducing parameters

▸ Quantization yields the Dirac equation which can be used to describe electrons!

E � q� = �mc2 + c↵ · ⇡

�2 = 1

[↵i,↵j ]+ = 2�ij

[↵i�]+ = 0


A way to describe electrons relativistically

▸ Lorentz invariant

▸ First derivative with respect to time AND position

▸ Linear in scalar AND vector potentials

(�mc2 + c↵ · ⇡ + q�) (r, t) = i~ @

@t (r, t)

The Dirac equation


What are the parameters?

Parameterization includes the Pauli matrices

↵x =

0 �x

�x 0

�

↵y =

0 �y

�y 0

�

↵z =

0 �z

�z 0

�

� =

I2 00 �I2

�


Requires a four-component wavefunction

▸ Spin doubles the components

▸ 2 positive energy solution: E > mc2: electrons

▸ 2 negative energy solutions: E < −mc2: positrons

H = �mc2 + c↵ · ⇡ + q�

The one-electron Dirac Hamiltonian

H =

2

664

mc2 + q� 0 c⇡z c(⇡x � i⇡y)0 mc

2 + q� c(⇡x + i⇡y) �c⇡z

c⇡z c(⇡x � i⇡y) �mc2 + q� 0

c(⇡x + i⇡y) �c⇡z 0 �mc2 + q�

3

775


Positive and negative eigenvalues of Dirac HF

D. Cremer and W. Zou and M. Filatov WIREs Comput. Mol. Sci. 4 436 (2014)


Can we treat many electron systems?

▸ Exact relativistic many-electron Hamiltonian is unknown

▸ Born-Oppenheimer Approximation

✴ Theory remains relativistic

✴ Decouples nuclear and electronic motion

✴ Simplifies relationship between time and space coordinates

▸ Add two-electron terms via the Coulomb potential?

▸ Which relativistic effects should the external potential include?

✴ Electron feels the motion of other electrons after some time (i.e. a retarded, velocity-dependent potential)

✴ Electron generates B as it moves and can interact with other electron spins (i.e. spin-orbit coupling, SOC)


Dirac-Coulomb Breit for many-el. systems

▸ Correction to the Coulomb operator obtained as a leading term in the Quantum Electrodynamics (QED) expansion.

▸ Coulomb term O(c0) (including spin-”same” orbit (SSO))

▸ Breit term O(c-2)

▸ Gaunt term (including spin-”other” orbit (SOO))

▸ Gauge-dependent term:

gCoulij =

I4 · I4rij

; charge-charge interaction

gBreitij = gGaunt

ij + ggaugeij

gGauntij = �c↵i · c↵j

c2rij; current-current (magnetic) interaction

ggaugeij = � (c↵i ·ri)(c↵j ·rj)rij2c2

; retardation term


Rewriting the Dirac Eq. in a 2-component form

▸ Shift the diagonal of the Dirac Hamiltonian by −mc2

▸ This aligns the relativistic and non-relativistic energy scales

▸ In a two-component form:

▸ ΨL(r) (large components) and ΨS(r) (small components) are the exact eigenfunctions of the Dirac equation

▸ For simplification, A = 0 and π = p

V c� · ⇡

c� · ⇡ �2mc2 + V

� L(r) S(r)

�= E

L(r) S(r)

�

Objective: eliminate the positronic states (which are coupled to the electronic states!)


Nature of the small components

▸ Exploit the highly local and atomic nature of the small components

▸ Use the no-pair approximation (neglect electron-positron pair creation and annihilation)

▸ Atomic character of the contributions of the S components⇒ neglect the multi-center integral block in the S components

▸ Transforming the 4-component equation to a 2-component one.

▸ Beware: Do not ignore the non-negligible contributions of the S-components since they are coupled!

Contributions from the large and small component densities

⇢4c(r) = ⇢L(r)⇢S(r)


Unnormalized Elimination of the S-Components

▸ Find a relationship between the two components

▸ Eliminate ψS(r) to obtain a 2-component problem

▸ Exact but not an eigenvalue equation

▸ Starting point for approximations

▸ Non-relativistic Pauli Hamiltonian is the simplest approximation, K(E,r) = 1

S(r) = � L(r)

� = K(E, r)� · p2mc

⇢1

2m(� · p)K(E, r)(� · p) + V

� L(r) = E L(r)

K(E, r) =

✓1� V � E

2mc2

◆�1


Breit-Pauli Hamiltonian

▸ Good to first order, but variational collapse at higher order

▸ Used for light elements (Z<20) where first order is sufficient

▸ Expansion invalid near the nucleus: (|V-E|/2mc2 > 1)

K(E, r) =

✓1� V � E

2mc2

◆�1Expansion in (E-V)/2mc2

HBP =H

Pauli +HDarwin +H

MV +HSO

HDarwin =

1

8m2c2(r2

V )

HMV =� p4

8m3c2

HSO =

1

4m2c2� · ((rV )⇥ p)


The regular approximations

▸ ZORA Hamiltonian contains no mass velocity term, only parts of the Darwin term, but all spin-orbit interactions arising from nuclei

Expansion in E/(2mc2-V)

K(E, r) =

✓1� V � E

2mc2

◆�1

=

✓1� V

2mc2

◆�1 ✓1 +

E

2mc2 � V

◆�1

Zeroth-Order Regular Approximation (ZORA)

HZORA = V +

1

2m(� · p) 2mc

2

2mc2 � V(� · p)

= V + T +1

4m2c2(� · p)V (� · p) + . . .


Douglas-Kroll Hess

▸ Matrix decoupling using the exact free-particle decoupling

▸ Provides regularized and variationally stable Hamiltonians, but not exact decoupling

▸ Properties of DKH

✴ Variationally stable

✴ Scalar DKH2 is used most often

✴ Good results in practice

✴ Matrix elements cannot be computed analytically due to the complicated operators

✴ If including an external field, need to transform the operators

� = (2mc2 + E)�1c� · p


eXact 2-Component (X2C) Hamiltonian

1. Solve the Dirac equation in matrix form

2. Extract the coupling χ from the solutions

3. Construct the transformation matrix U, next hX2C

▸ Advantages of X2C

✴ reproduces exactly the positive-energy spectrum of the Dirac Hamiltonian

✴ all matrix manipulations; no new operators to program

✴ explicit representation of transformation matrix;

✴ any property operator can be transformed on the fly, no picture change error


4c and 2c methods

▸ Dirac Hartree Fock (DHF) or 2c-HF methods

▸ Post HF methods: MP2, Coupled-Cluster methods (CC), CISD, Full CI

▸ Multi-configuration Self Consistent field

▸ MRCI methods (GAS-CI), IH(FSCC)

▸ Density Functional Theory

4c QC programs

2c QC programs

✴ DIRAC http://wiki.chem.vu.nl/dirac/✴ UTChem, http://utchem.qcl.t.u-tokyo.ac.jp/✴ BERTHA, L. Belpassi et al.

✴ DIRAC http://wiki.chem.vu.nl/dirac/✴ UTChem, http://utchem.qcl.t.u-tokyo.ac.jp/✴ ADF http://www.scm.com/✴ TURBOMOLE http://www.cosmologic.de/✴ ReSpect http://rel-qchem.sav.sk/✴ NWCHEM http://www.emsl.pnl.gov/capabilities/computing/nwchem/


1-component methods

▸ Split scalar relativistic effects and spin-orbit coupling

▸ In two-component or pseudopotentials (cf. slide 40) we can separate scalar relativistic effects from spin-orbit coupling terms

▸ First run a scalar relativistic (DFT, HF, post-HF, multiconfigurational, etc.)

▸ Treat spin-orbit coupling a posteriori


1-component scalar correlated + SO coupling

Scalar relativistic HF, MCSCF

All-electron basis set

electron correlation DFT or WFT correlated method

1) Couple the correlated spin-free states 2) Small SO-CI matrix with diagonal elements

from DFT/WFT correlated results

Pseudopotential (PP) valence basis set

✴ Since SO converges faster (small CI space)✴ MOLCAS (RASSI module) http://www.teokem.lu.se/molcas/✴ MOLPRO (MRCI module) http://www.teokem.lu.se/molcas/✴ EPCISO (interface with MOLCAS)[email protected]

SO-PPAll-electron SO integrals

(AMFI)


Need for frozen-core approximation

▸ In the context of chemical applications

✴ Core orbitals are not involved

✴ In heavy elements (second, third transition series, lanthanides, actinides, superheavy elements...) large number of core orbitals

▸ Core orbitals are essentially atomic like in a molecule or material

▸ Core orbitals supply a non-local static potential that can be evaluated once in the calculation

▸ Relativistic effects are to a large extent localized in the core region

✴ Include relativistic effects in the core potential

✴ Treat valence orbitals non-relativistically (but in the field of a relativistic core)


Frozen-core models

▸ To speed up a molecular or periodic calculation, inner electrons of atoms can be frozen.

▸ This is not a pseudopotential: all electrons are still present, the cores are just frozen at their optimized atomic configuration.

▸ Valence-only model

▸ Replace the core-electrons by an effective potential (pseudopotential)

Frozen-core Approximation (e.g. with ADF software)

Pseudopotentials

Hv = �1

2

nvX

i

�i +nvX

i<j

1

rij+

nvX

i

NX

I

VIcv(i) +

NX

I<J

QIQJ

RIJ+ VCPP


Pseudopotentials

▸ Use Stuttgart-Köln-Dresden (SDD in Gaussian) Pseudopotentialshttp://www.tc.uni-koeln.de/PP/index.en.htmlcoherent parameterization for the whole periodic table

▸ Parameters AIljk and aIljk adjusted to reproduce atomic relativistic calculations (Multi-configuration Wood-Boring, Multi-configurational Dirac HF) for several atomic (neutral and near neutral) configurations

V Icv(i) = �QI

riI+

lmaxX

l=0

j=l+1/2X

j=|l�1/2|

X

k

AIljk exp (�aIljkr

2iI)P

Ilj(i) with P I

lj(i) =jX

m=�j

|ljm, Iihljm, I|

http://www.tc.uni-koeln.de/PP/index.en.html


Features of pseudopotentials

▸ Radially nodeless orbitals

▸ Database: http://www.tc.uni-koeln.de/PP/index.en.html

✴ Scalar-relativistic PP, SO-PP, (Core-Polarization-Potentials) for various core sizes

✴ Must use the matching valence basis sets:

✴ default basis sets

✴ cc-pVnZ basis sets under development (K. Peterson)

▸ PP available GAUSSIAN, GAMESS, MOLCAS, MOLPRO, TURBOMOLE, NWCHEM, DIRAC,…

M. Dolg, X Cao, “Relativistic Pseudopotentials: Their Development and Scope of Applications”, In: Chem. Rev. 112 (2012), p. 403–480: DOI: 10.1021/cr2001383

Uranium valence orbitalsNeed for specific valence basis

sets

http://www.tc.uni-koeln.de/PP/index.en.html

http://dx.doi.org/10.1021/cr2001383

Illustrations of relativistic effects in chemistry & physics:

more common than you thought


Relativistic effects in chemistry

▸ Direct effects

▸ Contraction of the core-penetrating orbitals

▸ S orbitals, p1/2, p3/2 orbitals in core

▸ Energetic stabilization: higher ionization energy, higher electron affinity, smaller polarizability

▸ Indirect effects

▸ Indirect effects on d, f, orbitals and valence p orbitals

▸ Nuclear charge is shielded to a larger extent because of direct effect on core-penetrating orbitals (in particular of the semi-core)

▸ Relativistic expansion of core non-penetrating orbitals

▸ Energetic destabilization

▸ Smaller ionization energy, larger polarizability in turn, stabilization of core-penetrating orbitals in next shell

▸ Gold maximum

Direct effects

Indirect effects


Relativistic effects on atomic shell structure

▸ Direct effects dominate for Au (6s) and Tl (6p1/2)

▸ Compensation of direct and indirect effects for Tl (6p3/2)

▸ Indirect effects dominate for Lu

Rel NRAu (6s) -7.94 -6.01

Tl (6p1/2) -5.81 -5.24Tl (6p3/2) -4.79 -5.24Lu (5d3/2) -5.25 -6.63Lu (5d5/2) -5.01 -6.63

Direct and indirect effects on orbital energies (eV)

S. J. Rose, I. P. Grant, and N. C. Pyper. “The direct and indirect effects in the relativistic modification of atomic valence orbitals”. In: J. Phys. B: At. Mol. Phys. 11.7 (1978), p. 1171. DOI: 10.1088/0022-3700/11/7/016

http://dx.doi.org/10.1088/0022-3700/11/7/016


Relativistic effects on molecular structures

C. L. Collins, K. G. Dyall, and H. F. Schaefer III. “Relativistic and correlation effects in CuH, AgH, and AuH: Comparison of various relativistic methods”. In: J. Chem. Phys. 102 (1995), p. 2024. DOI: 10.1063/1.468724

Moleculee

Method re(HF) re(MP2) De(HF) De(MP2) ωe(HF) ωe(MP2)CuH NR 1.569 1.454 1.416 2.585 1642 2024

DKH 1.542 1.429 1.476 2.708 1498 2100RECP 1.543 1.429 1.465 2.696 1690 2095

DC 1.541 1.468 1.477 2.711 1699 2101Exp 2.85 1941

AuH NR 1.831 1.711 1.084 1.901 1464 1169DKH 1.576 1.498 1.727 3.042 2045 2495RECP 1.571 1.495 1.751 3.075 2076 2512

DC 1.570 1.497 1.778 3.114 2067 2496Exp 1.524 3.36 2305

✴ Relativistic effects larger in Au than in Cu✴ Huge effect on re, De, and ωe✴ Douglas-Kroll Hess, RECP and DC in good agreement

8A1A

2A

3B 4B 5B 6B 7B 8B 11B 12B

3A 4A 5A 6A 7A

element names in blue are liquids at room temperatureelement names in red are gases at room temperatureelement names in black are solids at room temperature

Periodic Table of the Elements

Los Alamos National Laboratory Chemistry Division

39

*

**

As39.10

85.47

132.9

(223)

9.012

24.31

40.08

87.62

137.3

(226)

44.96

88.91

47.88

91.22

178.5

(267) (268) (269) (270) (277) (278) (281) (282) (285) (289) (289)(286) (293) (294) (294)

50.94

92.91

180.9

52.00

95.96

183.9

54.94

(98)

186.2

55.85

101.1

190.2 192.2

102.9

58.93 58.69

106.4

195.1 197.0

107.9

63.55 65.39

112.4

200.5

10.81

26.98

12.01

28.09

14.01

69.72 72.64

114.8 118.7

204.4 207.2

30.97

74.92

121.8

209.0 (209) (210) (222)

16.00 19.00 20.18

4.003

32.06 35.45 39.95

78.96 79.90 83.79

127.6 126.9 131.3

140.1 140.9 144.2 (145) 150.4 152.0 157.2 158.9 162.5 164.9 167.3 168.9 173.0 175.0

232

138.9

(227) 231 238 (237) (244) (243) (247) (247) (251) (252) (257) (258) (259) (262)

Tc43

11

1

3 4

12

19 20 21 22 23 24 25 26 27 28 29 30

37 38 40 41 42 44 45 46 47 48

55 56

58 59 60

72 73 74 75 76 77 78 79 80

87 88

90

57

89 91 92 93 94 95 96

104 105 106 107 108 109 110 111 112

61 62 63 64 65 66 67

97 98 99

68 69 70 71

100 101 102 103

31

13 14 15 16 17 18

32 33 34 35 36

49 50 51 52 53 54

81 82 83 84 85 86

5 6 7 8 9 10

2

114 115113 116 117 118

3/13/17

Th

Li

Na

K

Rb

Cs

Fr

Be

Mg

Ca

Sr

Ba

Ra

Sc Ti V Cr Mn Fe Co Ni Cu Zn

Y Zr Nb Mo Ru Rh Pd Ag Cd

Hf Ta W Re Os Ir Pt Au Hg

HeH

Ne

Ar

Kr

Xe

F

Cl

ON

Br

Rf Db Sg Bh Hs Mt Ds Rg Cn

B C

Al Si P S

Ga Ge Se

In Sn Sb Te I

Tl Pb Bi Po At Rn

Fl LvMcNh Ts Og

Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb LuLa

Ac Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr

hydrogen

barium

francium radium

strontium

sodium

vanadium

berylliumlithium

magnesium

potassium calcium

rubidium

cesium

helium

boron carbon nitrogen oxygen fluorine neon

aluminum silicon phosphorus sulfur chlorine argon

scandium titanium chromium manganese iron cobalt nickel copper zinc gallium germanium arsenic selenium bromine krypton

yttrium zirconium niobium molybdenum technetium ruthenium rhodium palladium silver cadmium indium tin antimony tellurium iodine xenon

hafnium

cerium praseodymium neodymium promethium samarium europium gadolinium terbium dysprosium holmium erbium thulium ytterbium lutetium

tantalum tungsten rhenium osmium iridium platinum gold mercury thallium lead bismuth polonium astatine radon

thorium

lanthanum

actinium protactinium uranium neptunium plutonium americium curium berkelium californium einsteinium fermium mendelevium nobelium lawrencium

rutherfordium dubnium seaborgium bohrium hassium meitnerium darmstadtium roentgenium copernicium flerovium moscovium livermorium tennessine oganessonnihonium

1.008

6.94

22.99

Lanthanide Series*

Actinide Series**

1s1

[Ar]4s23d104p3[Ar]4s23d3[Ar]4s13d10

[Ne]3s23p6[Ne]3s23p4

[Ar]4s1[Ar]4s23d10

1s2

[He]2s1 [He]2s2

[Ar]4s23d7

[Ne]3s23p5

[He]2s22p1 [He]2s22p2 [He]2s22p3

[Ar]4s23d5

[He]2s22p4 [He]2s22p5 [He]2s22p6

[Ar]4s23d104p5

[Ne]3s1 [Ne]3s23p1 [Ne]3s23p3[Ne]3s23p2

[Rn]7s25f146d2

[Ne]3s2

[Ar]4s2 [Ar]4s23d1 [Ar]4s23d2 [Ar]4s13d5 [Ar]4s23d6 [Ar]4s23d8 [Ar]4s23d104p1 [Ar]4s23d104p2 [Ar]4s23d104p4 [Ar]4s23d104p6

[Kr]5s1 [Kr]5s2 [Kr]5s24d1 [Kr]5s24d2 [Kr]5s14d4 [Kr]5s14d5 [Kr]5s24d5 [Kr]5s14d7 [Kr]5s14d8 [Kr]4d10 [Kr]5s14d10 [Kr]5s24d10 [Kr]5s24d105p1 [Kr]5s24d105p2 [Kr]5s24d105p3 [Kr]5s24d105p4 [Kr]5s24d105p5 [Kr]5s24d105p6

[Xe]6s1 [Xe]6s2

[Xe]6s24f15d1 [Xe]6s24f3 [Xe]6s24f4 [Xe]6s24f5 [Xe]6s24f6 [Xe]6s24f7 [Xe]6s24f75d1 [Xe]6s24f9 [Xe]6s24f10 [Xe]6s24f11 [Xe]6s24f12 [Xe]6s24f13 [Xe]6s24f14 [Xe]6s24f145d1

[Xe]6s24f145d2 [Xe]6s24f145d3 [Xe]6s24f145d4 [Xe]6s24f145d5 [Xe]6s24f145d6 [Xe]6s24f145d7 [Xe]6s14f145d9 [Xe]6s14f145d10 [Xe]6s24f145d10 [Xe]6s24f145d106p1 [Xe]6s24f145d106p2 [Xe]6s24f145d106p3 [Xe]6s24f145d106p4 [Xe]6s24f145d106p5 [Xe]6s24f145d106p6

[Rn]7s1 [Rn]7s2

[Rn]7s26d2

Xe]6s25d1

[Rn]7s26d1 [Rn]7s25f26d1 [Rn]7s25f36d1 [Rn]7s25f46d1 [Rn]7s25f6 [Rn]7s25f7 [Rn]7s25f76d1 [Rn]7s25f9 [Rn]7s25f10 [Rn]7s25f11 [Rn]7s25f12 [Rn]7s25f13 [Rn]7s25f14 [Rn]7s25f146d1

[Rn]7s25f146d3 [Rn]7s25f146d4 [Rn]7s25f146d5 [Rn]7s25f146d6 [Rn]7s25f146d7 [Rn]7s15f146d9 [Rn]7s15f146d9 [Rn]7s15f146d9 [Rn]7s27p15f14

6d10 (predicted)[1][Rn]7s27p25f14



6d10(predicted)[1][Rn]7s27p55f146d10(predicted)[4] [Rn]7s27p65f14

6d10(predicted)[1][2]

http://dx.doi.org/10.1063/1.468724


High oxidation states and electron affinities

▸ 5d metals do have higher oxidation states than their 4d analogs because of the relativistic destabilization of their d shell

▸ A striking example is the predicted HgF4, which is the first Hg(IV) compound

▸ Another example is Ir(VIII)O4. The predicted IrO4+ would have the first oxidation state +IX.

▸ Review on high oxidation states:

▸ The 5d metal hexafluorides WF6 through AuF6 have high electron affinities and are extraordinary oxidizers and Lewis acids; the SO increased the EA.

X.F. Wang, L. Andrews, S. Riedel, M. Kaupp “Mercury is a transition metal: the first experimental evidence for HgF4.” In Angew. Chem. Int. Ed. Engl. 46 (2007), p. 8371. DOI: 10.1002/anie.200703710.

S. Riedel, M. Kaupp. “The highest oxidation states of the transition metal elements.” Coord. Chem. Rev. 253 (2009), p. 606

Y. Gong, M. F. Zhou, M. Kaupp, S. Riedel. “Formation and characterization of the iridium tetroxide molecule with iridium in the oxidation state +VIII.” Angew. Chem. Int. Ed. Engl. 48 (2009), p.7879. D. Himmel, C. Knapp, M. Patzschke, S. Riedel, “How far can we go? Quantum-chemical investigations of oxidation state +IX.” ChemPhysChem 11 (2010), p. 865.

http://dx.doi.org/10.1002/anie.200703710


Relativistic effect on chemical reactions

▸ Large relativistic effects in Pb

▸ Relativistic of the 6s and destabilization of 6p decreases sp3 hybridization in PbH4, thus making the reaction exothermic

▸ In some systems, spin-orbit coupling can enable crossing between states of different multiplicities (inter-system crossings)

K. G. Dyall. “All-electron molecular Dirac-Hartree-Fock calculations: Properties of the XH4 and XH2 molecules and the reaction energy XH4 → XH2 + H2, X=Si, Ge, Sn, Pb”. In: J. Chem. Phys. 96 (1992), p. 1210. DOI: 10.1063/1.462208

SCF reaction energies in kJ mol-1 for the reaction XH4 → XH2 + H2

Method Si Ge Sn Pb

NR 263 190 129 89

RECP 195 102 -31

DC 261 177 97 -26

http://dx.doi.org/10.1063/1.462208


Relativistic effects on NMR shieldings

Peter Hrobarik et al. “Relativistic Four-Component DFT Calculations of 1H NMR Chemical Shifts in Transition-Metal Hydride Complexes: Unusual High-Field Shifts Beyond the Buckingham–Stephens Model”. In: J. Phys. Chem. A 115 (2011), pp. 5654–5659. DOI: 10.1021/jp202327z

http://dx.doi.org/Peter%20Hrobarik%20et%20al.%20%C2%B4%20%E2%80%9CRelativistic%20Four-Component%20DFT%20Calculations%20of%201H%20NMR%20Chemical%20Shifts%20in%20Transition-Metal%20Hydride%20Complexes:%20Unusual%0D%0AHigh-Field%20Shifts%20Beyond%20the%20Buckingham%E2%80%93Stephens%20Model%E2%80%9D.%20In:%20J.%20Phys.%20Chem.%20A%20115.22%20(2011),%20pp.%205654%E2%80%935659.%20DOI:%2010.1021/jp202327z


Relativity and the lead-acid batteryPb(s) + PbO2(s) + 2H2SO4(aq) → 2PbSO4(s) + 2H2O(l), ΔcellG0,

✴ NR: +0.35 V✴ Average fully relativistic DFT value: + 2.13 V✴ experimental: 2.107 V✴ Hence cars start because of relativity.

Rajeev Ahuja et al. “Relativity and the Lead-Acid Battery”. In: Phys. Rev. Lett. 106 (2011), p. 018301. DOI: 10.1103/PhysRevLett.106.018301

http://dx.doi.org/10.1103/PhysRevLett.106.018301


Spin-orbit effect in structural chemistry

Ulrich Wedig et al. “Homoatomic Stella Quadrangula [Tl8]6– in Cs18Tl8O6, Interplay of Spin-Orbit Coupling, and Jahn-Teller Distortion”. In: J. Am. Chem. Soc. 132.35 (2010), pp. 12458–12463. DOI: 10.1021/ja1051022

✴ SO coupling makes Ptn clusters flat (n = 2–5)✴ In Cs18Tl8O6 the system exhibit an open-shell degenerate HOMO within a scalar

relativistic approximation.✴ With SO coupling a closed-shell electronic system is obtained in accordance

with the diamagnetic behavior of this crystal

Td D3d

Level Td D3d

SR triplet @ 23.8 kJ mol-1 Singlet @ 0.0 kJ mol-1SO Sz = 0 @ 0.0 kJ mol-1 Sz = 0 @ 12.7 kJ mol-1

http://dx.doi.org/10.1021/ja1051022


Spin-orbit contribution to properties of solids

▸ SO influences the phonon spectra of PbX (X = S, Se, Te) and the specific heat of Bi

A. H. Romero, M. Cardona, Kremer RK, Lauck R, Siegle G, et al. "Lattice properties of Pb X (X = S, Se, Te): experimental studies and ab initio calculations including spin-orbit effects.” In Phys. Rev. B 78 (2008), p. 224302. DOI: 10.1103/PhysRevB.78.224302 L. E. Díaz-Sanchez, A. H. Romero, M. Cardona, R. K. Kremer, X. Gonze. “Effect of the spin-orbit inter action on the thermodynamic properties of crystals: specific heat of bismuth”. In Phys. Rev. Lett. 99 (2007), 165504. DOI: 10.1103/PhysRevLett.99.165504

✴ circle : Inelastic Neutron Scattering✴ ◼optical spectroscopy

PbTe Bi

✴ Black solid line with SO✴ Blue dashed line without SO

http://dx.doi.org/10.1103/PhysRevB.78.224302

http://dx.doi.org/10.1103/PhysRevLett.99.165504

Concluding remarks


Check-list before starting relativistic QC calcs

▸ How to choose the relativistic description?

✴ Best method depends on the system studied

✴ Which property are you interested in?

✴ What is the accuracy you are looking for? It depends on whether you look at chemical reactions, spectroscopy, or molecular properties

✴ Which computational capacities do you have access to?

▸ For closed shell systems, one-component methods work well

▸ Don’t use non-relativistically contracted basis sets for relativistic calculations

▸ As usual electron correlation is important (large number of electrons, close lying electronic states)

▸ There are a lot of pseudopotentials on the market.

✴ Choose the right number of core electrons

✴ If you are not sure, compare to some all-electron method, perhaps even four-component calculations


Key summary points

▸ Beware of the importance of relativistic effects

▸ The classical examples of relativistic effects in chemistry remain and have been included in most chemistry textbooks.

▸ One of the oldest examples, which deserves more attention, is the SO-induced NMR heavy-atom shift.

▸ Investigators continue to discover new examples, e.g. as the heavy-element batteries.

▸ Catalysis is one of the most important applications of relativistic quantum chemistry.

▸ The SO effects in structural chemistry have been identified only recently after technical progress (possibility of 4c or 2c geometry optimizations)

▸ Include other effects if relevant:

✴ environment (crystal, surface, solvent)

✴ dynamics


Further reading

▸ Relativistic Quantum Mechanics

✴ R. E. Moss. Advanced Molecular Quantum Mechanics - An Introduction to Relativistic Quantum Mechanics and the Quantum Theory of Radiation. Springer Netherlands, 1973. ISBN: 978-94-009-5690-2. DOI: 10.1007/978-94-009-5688-9

✴ P. Strange. Relativistic Quantum Mechanics with Applications in Condensed Matter and Atomic Physics. Cambridge Univ. Press, 1998, p. 594. ISBN: 9780521565837

▸ Relativistic Quantum Chemical methods

✴ P. Schwerdtfeger. Relativistic Electronic Structure Theory: Part 1, Fundamentals. Ed. by P. Schwerdtfeger. Amsterdam: Elsevier, 2002. ISBN: 9780444512499

✴ K. G. Dyall and K. Fægri. Introduction to relativistic quantum chemistry. New York: Oxford University Press, 2007

✴ M. Reiher and A. Wolf. Relativistic Quantum Chemistry: The Fundamental Theory of Molecular Science, 2nd Edition. WILEY-VCH Verlag, 2014. ISBN: 978-3-527-33415-5

▸ Applications

✴ P. Schwerdtfeger. Relativistic Electronic Structure Theory: Part 2, Applications. Ed. by P. Schwerdtfeger. Amsterdam: Elsevier, 2004. ISBN: 978-0-444-51299-4

✴ Pekka Pyykkö. ¨ “Relativistic Effects in Chemistry: More Common Than You Thought”. In: Ann. Rev. Phys. Chem. 63.1 (2012), pp. 45–64. DOI: 10.1146/annurev-physchem-032511-143755

http://dx.doi.org/10.1007/978-94-009-5688-9

http://dx.doi.org/10.1146/annurev-physchem-032511-143755

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