RELATIVISTIC QUANTUM CHEMISTRY: FROM THE YELLOW COLOUR OF GOLD TO THE LEAD BATTERY

Pekka PYYKKÖ (Department of Chemistry, University of

Helsinki, Finland)

CSC, 9 March 2012 (1 h)

INDEX

1. Various forms of the PT . 2. The recent lead-battery case . 3. Relativity in chemistry . 4. ’Fine structure’ in the PT. 5. Magic numbers: 8, 18 and 32. 6. Predicting new species. 7. New sets of covalent radii.

THE PERIODIC SYSTEM 2010

THE PERIODIC SYSTEM, Z = 1 - 172

SOME RECENT SOURCES

E. R. Scerri, The Periodic Table, Oxford U. P. (2007), 346 p.

History aspects well told. Perhaps overemphasises the ”window

aspect” , the ”correct form of the PT” and the Madelung n+l rule. S-G. Wang and W. H. E. Schwarz: Icon of chemistry: The Periodic

System of chemical elements in the new century, Angew. Chem.

Int. Ed. 48 (2009) 3404-3415. P. Pyykkö: A suggested Periodic Table up to Z ≤ 172, based on

Dirac-Fock calculations on atoms and ions, Phys. Chem. Chem.

Phys. 13 (2011) 161-168.

My 15 minutes of celebrity

The Economist, 15 January 2011.

Al The T

The EconoThemist

The Economist

Original paper: Ahuja, Blomqvist, Larsson, Pyykkö, Zaleski-Ejgierd,

Physical Review Letters 106 (2011) 018301.

The Economist, 15 January 2011.

Also C & E News, Fyysika.ee, ing.dk, Nature, New Scientist, Physical

Review Focus, PhysicsToday, Science@ORF.at, Science et

Vie,Tekniikka ja Talous, Times of India, ...WissensLogs.de, ...

Albert Einstein’s special relativity coupled to Dmitrii Mendeleyev’s Periodic System !

:

5th-Row versus 6th-Row Compounds

9

Accurate structures for molecular MCN, M=Cu,Ag,Au

Microwave molecular structures exist for Cu [1], Ag, Au [2]. Carry out large-basis relativistic pseudotential CCSD(T) calculations,

correlating the 5s5p semicore and adding BSSE and spin-orbit

corrections. cc-pVQZ basis. 19-VE Figgen pseudopotential. Final M-C bond-lengths agree with experiment within 0.7 pm.

1. D.B. Grotjahn, M.A. Brewster, L.M. Ziurys, JACS 124 (2001) 5895.

2. T. Okabayashi, E. Y. Okabayashi, F. Koto, T. Ishida, M. Tanimoto, JACS 131 (2009) 11712.

3. P. Zaleski-Ejgierd, M. Patzschke, P. Pyykkö, J. Chem. Phys.128 (2008) 224303.

CuCN AgCN AuCN

Exp 182.962(4) 203.1197(23) 191.22519(84)

Calc. 182.36 202.42 191.05

10

11

RELATIVISTIC APPROACHES

Full Dirac equation. Can be done with a numerical basis

or with a discrete basis (Gaussian functions). (’Dirac’, ’Bertha’).

Many solid-state codes.

Transformed Hamiltonians. ZORA (’ADF’), DKH. X2C.

Pseudopotentials. With or without spin-orbit. Available for most

ab initio programs.

For a point nucleus, the 1s solution disappears at Z = 137.036, E= -mc2. For a finite nucleus, that happens around Z = 172.

12

5th-Period versus 6th-Period Compounds

13

RELATIVISTIC EFFECTS

”Relativistic effects”: Anything depending on the speed of light. Alternatively: The difference between using a Dirac or a Schrödinger

one-electron equation. Alternatively: Keep Dirac equation but let c increase from 137.036

atomic units to a very big value. Explain many chemical differences between 5th-Row and 6-th Row

elements. Ag/Au. Current textbook explanation, together with the

lanthanide contraction. New: Deeper physics (QED effects) will only change the previous

conclusions by -1% for heavy elements. The QED was the last train

from physics to chemistry. Dirac-Coulomb ’101% right’.

P. Pyykkö: ’The Physics behind Chemistry, and the Periodic Table’,

Chem. Rev. 112 (2012) 371-384. (QED aspects etc.)

P. Pyykkö, Ann. Rev. Phys. Chem. 63 (2012). (Some chemistry).

14

WHY RELATIVITY?

The innermost electrons move fast in heavy elements. The average

radial 1s velocity in atomic units (c = 137.036 au), <vr>1s = Z = 80 for Hg. (1)

This leads to a mass increase, m = m0 /[1 – (v/c)2 ] 1/2. (2)

The increased mass gives a smaller Bohr radius, a0 = ћ2 / m e2 . (3) →

a relativistic contraction and stabilization of all s and p

orbitals. Exact solution of the Dirac equation: The higher s and p states are

also strongly ’relativistic’. Due to stronger screening of the nuclear attraction by s and p shells,

the d and f shells will have a relativistic expansion and

destabilization. For valence shells, effects increase as Z 2 .

15

HYDROGEN-LIKE ATOM Hg79+

V. M. Burke, I. P. Grant, Proc. Phys. Soc. (London) 90 (1967) 297.

16

THE ”GOLD MAXIMUM” OF RELATIVISTIC EFFECTS

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

17

Data from J. P. Desclaux, P. Pyykkö, Chem. Phys. Lett. 39 (1976) 300.

18

Relativity and the Periodic System

P. Pyykkö, Chem. Rev. 88 (1988) 563-594.19

CHEMISTRY TEXTBOOKS : A PARADIGM CHANGE

G. Wulfsberg (1987, 1991). F. A. Cotton, G. Wilkinson (1988, 1999). K. M. Mackay, R. A. Mackay (1989, 1996). R. H. Petrucci (1989) + W. S. Harwood (1993). A. G. Massey (1990). W. L. Jolly (1991). A. G. Sharpe (1992). J. E. Huheey, E. A. Keiter, R. L. Keiter (1993). J. B. Umland (1993) (+ J. M. Bellama !996)). T. M. Klapötke, I.C. Tornieporth-Oetting (1994). N. C. Norman (1994, 1997). School text. ’Hollemann-Wiberg’, 101. Auflage (1995) , 102. (2007) S. S. Zumdahl, (1995, 1998).

20

CHEMISTRY TEXTBOOKS (continued)

N. N. Greenwood, A. Earnshaw, 2nd Ed. (1997). D.M.P. Mingos (1998). N. Kaltsoyannis, P. Scott (1999). G. Rayner-Canham, 2nd Ed. (1999). C. E. Housecroft, A.G. Sharpe (2001). J. Barrett (2002).

Three fronts: Chemistry, Physics, Mathematics.

21

CATALYSIS ?

Gas-phase activation of methane by Pt+

CH4 + Pt+ → Pt=CH2+ + H2 .

Relativistic stabilization as much as 50 out of 112 kcal/mol,

C. Heinemann, H. Schwarz, W. Koch, K.G. Dyall, JCP 104 (1996) 4642.

The homogeneous catalysis by Au(I, III) species in liquids:

Review: D. Gorin & F. D.Toste, Nature 446 (2007) 395-403.

Some driving forces behind the reactions: Strong Lewis acidity of both Au(I) and Au(III). The occasional aurophilic attraction between two or more Au(I):s. The strengthening of Au-L bonds. The tendency of Au(I) to two-coordination (eliminates easily further

ligands). The already mentioned stability of the carbenoids. All can be related to the relativistic mechanisms.

. 22

SEVEN RULES THAT EXPLAIN THE PERIODIC SYSTEM

1. Main vertical rule. First shell with every l (1s, 2p, 3d, 4f, 5g) is

anomalously small. <r> increases with n for others. 2. Main horisontal trend: <r> decreases with Z. 3. Main periodicity: Filled shells stable. NR half-filled ones also. 4. Partial screening effects. Lanthanide contraction due to filling the

4f shell on 6s and 6p shells. Analogous 3d, 2p and 1s effects. 5. Relativistic contraction and stabilization. (s, p). 6. Relativistic expansion and destabilization. (d, f). 7. Spin-orbit splitting. (p, d, f shells).

23

RELATIVISTIC BOND-LENGTH CONTRACTION

P. Pyykkö, J. P. Desclaux, Chem. Phys. Lett. 42 (1976) 545. Contraction increases as Z 2 . First found for PbH4

(1974).

24

BOND-LENGTH CONTRACTION NOT DUE TO ORBITAL CONTRACTION

Consider as example the isoelectronic CsH or BaH+ molecules.

One valence σ MO:

|σ> = c1 |6s> + c2 |5d> + c3 |1sH > + c4 |core>. (1)

ΔE(1) = < σ | h(BP) | σ > (2)

= ΔE(1) (core-core) + ΔE(1) (core-val) + ΔE(1) (val-val) .

The core-core term (<0) becomes larger with decreasing bond length, R.

It provides a driving force for the contraction, already with the NR, uncontracted

orbitals.

P. Pyykkö, J. G. Snijders, E. J. Baerends, CPL 83 (1981) 432.

25

26

Au(I) versus Au(III)

AuX4 - → AuX2

– +2X

P. Schwerdtfeger, J. Am. Chem. Soc. 111 (1989) 7261.

X ΔU/kJ mol-1

NR R

F -81 117

Cl -100 80

Br -152 -13

I -152 -39

27

TIN, LEAD AND RELATIVITY

P. Pyykkö, Chem. Rev. 88 (1988) 563.

28

RELATIVITY AND THE LEAD BATTERY

Rajeev Ahuja, Andreas Blomqvist, Peter Larsson. Pekka Pyykkö and

Patryk Zaleski-Ejgierd. Uppsala and Helsinki. Physical Review Letters

106 (2011) 018301. Key conclusion: About 1.7-1.8 V of the 2.1 V standard cell voltage come

from relativity. Cars, indeed, start due to relativity. Compared to tin, relativistic effects once again scale as Z2 . Largest contribution comes from PbO2

. Next one from PbSO4 .

29

RELATIVITY AND THE LEAD BATTERY / 2

A comproportionation (Gadolin 1788) reaction of the type

Pb(0) + Pb(IV) → 2 Pb(II). (1)

More specifically, Pb(s) + PbO2(s) + 2 H2SO4(l) → 2 PbSO4(s) + 2 H2O(l). (2)

Trick: Avoid liquid-state simulations by using experimental data for the

light-element reaction H2O(l) + SO3(g) → H2SO4(l) (3)

Combining (2) + 2 x (3), we get the working equations Pb(s) + PbO2(s) + 2 SO3(g) → 2 PbSO4(s). (4)

Obtain the voltage from

∆E(2) = ∆E(4) – 2 ∆E(3). (5) Use several different theoretical methods to treat the solids:

ADFBAND (ZORA relativity; VWN or PBEsol-D DFT) or

FPLO (SR or Full Dirac relativity; PW92 w/wout volume relaxation.)

30

Pb(s) + PbO2(s) + 2H2SO4(aq, 6M H2SO4) --> 2PbSO4(s) + 2H2O(l, 6M H2SO4) (1)

31

Relativistic energy changes for the reactants: DFT/PBEsol-D level. The non-relativistic energy, NR = Ef(NR), is chosen as reference; ΔSR = Ef(SR) - Ef(NR), and ΔFR = Ef(FR) - Ef(NR). Note the smallness of the effect on SO3(g).

32

Densities of States for β-PbO2 : Total and partial DOS of Pb in β-PbO2 calculated at DFT/VWN level (Fermi level Ef is set to 0). Note the pronounced relativistic shift in both occupied and unoccupied Pb 6s-states. Non-relativistic (NR), scalar-relativistic (SR), fully-relativistic (FR). Orange: Pb 6s, green: Pb 6p. Actually the metallicity of β-PbO2 is an impurity effect.

33

The Z2 argumentIt is known that valence-shell relativistic effects of the Periodic Table roughly scale

as Z2. For tin and lead it corresponds to:

while the relativistic voltage changes, ΔFR, calculated for the ‘toy models’ M + MO2 → 2MO yield: Δ(Sn)/Δ(Pb) = +0.34 V/+0.86 V = 0.395.

Newest results: Also the mercury battery has substantial

relativistic effects. At DFT level, about 30% of the +1.35 V is

’relativistic’. (Zaleski-Ejgierd & Pyykkö, PCCP 13 (2011)

16510)

HgO + Zn → Hg + ZnO, +1.35 V.

34

THE RELATIVISTIC COLOURS

BiPh 5 , violet: LUMO shift down.

PbCl 6 2- , yellow: LUMO shift down.

Metallic gold: 5d band shifts up, 6s Fermi level shifts down. Pb(NO 2 )2 , yellow. Singlet-triplet mixing of the nitrite, due to spin-orbit

coupling of the heavy metal.

35

Experiment: Gold (1) yellow but silver (5) white

K. E. Saeger, J. Rodies, Gold Bull. 10 (1) (1977) 10;

H. Fukutani, O. Sueoka, in Optical Properties and Electronic Structure of

Metals and Alloys, ed. F. Abelès, North-Holland, A’dam (1966) 565-573.

↑

Ag

↑

Au

36

Relativity indeed explains the yellow colour of gold

P. Romaniello and P. L. de Boeij, J. Chem. Phys. 122 (2005) 164303;

127 (2007) 174111. Bulk and surface plasmons similar (priv. comm.).

K. Glantschnig, C Ambrosch-Draxl, New J. Phys. 12 (2010) 103048.

↑

NR

↑

R

↑

R

37

TRENDS AMONG ALKALI METALS

1. B. Fricke, J. T. Waber, J. Chem. Phys. 56 (1972) 3246.38

’RIPPLES ON PERIODICITY’: FINE STRUCTURE

Secondary periodicity (Biron 1915). Lanthanide contraction (Goldschmidt 1925). Spin-orbit subshells and Bi(I). Alkali metals, the beginning. ’Honorary d-metals’: Cs, Ca-Ba.

Pre-s, pre-p, pre-d, and pre-f elements.

Examples: E118, Be and Zn, Ba and Th, respectively. Au as ’halogen’, Pt as ’oxygen’, Ir as ’nitrogen’.

39

SECONDARY PERIODICITY

40

UNDERSTANDING SECONDARY PERIODICITY

P. Pyykkö, J. Chem.Res. (S) (1979) 380. 41

THE LANTHANIDE CONTRACTION

Skrifter Norske Vid. Ak., I. Mat. Naturvid. Klasse, No. 7 (1925). 42

THE LANTHANIDE CONTRACTION

Skrifter Norske Vid. Ak., I. Mat. Naturvid. Klasse, No. 7 (1925). 43

MAGIC NUMBERS: 8, 18, 32 !

44

The 18-e principle

Why are the molecules or complexes with effectively 18

valence electrons around a transition metal atom

particularly stable? Examples: Mo(CO)6, Fe(CO)5, Ni(CO)4 [1], Co(CO)3(NO)

[2], Fe3(CO)12, OsCl64-, OsCl5(NO)2-, Fe(CN)64- [3].

Proposed explanation: Effectively s2p6d10 el. conf.

Sometimes little or no p !

New explanation: Ligand nodal structure suffices [4] !

1. I. Langmuir, Science 54 (1921) 59-67.

2. E. Reiff, Z. Anorg. Allg. Chem. 202 (1931) 375-381.

3. N. V. Sidgwick, R. W. Bailey, Proc. Roy. Soc. (London)

A144 (1934) 521-537.

4. P. Pyykkö, J. Organomet. Chem. 691 (2006) 4336-40.

45

46

Pu@Pb12: The first 32-e species?

1. J. P. Dognon, C. Clavaguéra, P. Pyykkö, Angew. Chem. Int.

Ed. 46 (2007) 1427.47

Pu@Pb12

1. J. P. Dognon, C. Clavaguera,

P. Pyykkö, Angew. Chem.

Int. Ed. 46 (2007) 1427.

Pb122- is experimentally known

as a stiff shell structure. Could an endohedral actinide

atom, like Pu2+ bring in six more electrons and a t2u

orbital? A: Yes! This theoretical proposal is the

first 32-electron species.

48

U@C282+ : An old new 32e system

1. J. P. Dognon, C. Clavaguéra, P. Pyykkö, JACS 131 (2009) 238.

2. K. Zhao, R.S. Pitzer, JPC 100 (1996) 4798. (Earlier analysis).

3. T. Guo, M.D. Diener, Y. Chai, M.J. Alford, R.E. Haufler, S.M.

McClure, T. Ohno, J.H. Weaver, G.E. Scuseria, R.E. Smalley,

Science 257 (1992) 1661. (Exp. discovery).49

PREDICTING NEW CHEMICAL SPECIES:

Review (No305) by P. Pyykkö in PCCP,(Advance Article, 15 February 2012).

50

WAu12

Predicted in 2002 [1]. Stabilized

by relativity, the 18-electron

principle and the ’metallophilic’

attraction between the Au 5d

shells. Prepared in 2002 [2]. Metal

evaporation to helium carrier gas, MS, PES. Also MoAu12; VAu12

-

NbAu12-, TaAu12

-.

1. P. Pyykkö, N. Runeberg, Angew. Chem. Int. Ed. 41 (2002) 2174.

2. X. Li et al., Angew. Chem. Int. Ed. 41 (2002) 4786.

51

METALLOACTINYLS: PLATINUM AS ’OXYGEN’

1. L. Gagliardi, P. Pyykkö, Angew. Chem. Int. Ed. 43 (2004) 1573.

2. M. Santos, J. Marçalo, A. Pires de Matos, J.K. Gibson, R.G.

Haire, Eur. J. Inorg. Chem. (2006) 3346. Make OUIr+.

OUIr+ prepared [2] !

52

SIMPLEST PREDICTIONS FOR CHEMICAL BONDING:Molecular, self-consistent covalent radii

R(AB) = rA + rB.

53

MEAN-SQUARE DEVIATION ONLY 3 pmfor both single-, double- and triple-bond radii.For tetrahedral radii for crystals only 0.67 pm,P. Pyykkö, Phys. Rev. B 84 (2012) 024115.

54

END OF ’RQC’ TALK .

55

The Kumpula Campus, University of Helsinki, Finland

Faculty of Science.

Government labs:

- Meteorology

- Marine Research

Including students, about 9000

people.

Entire U of H: 38000 students.

8 national CoE:s, including ’Finnish

Centre of Excellence of

Computational Molecular Science’

(2006-2011). (CMS)

CMS groups: Pyykkö-Sundholm,

Halonen, Räsänen, Vaara,

Nordlund. About 60 people.

Nordic ’umbrella’ of CoE:s.56