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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
Original paper: Ahuja, Blomqvist, Larsson, Pyykkö, Zaleski-Ejgierd,
Physical Review Letters 106 (2011) 018301.
The Economist,
Economist, 15
15 January
January
The EconoThemist
2011.
The
2011.
The Economist
Al C The
Also
& E News,
Fyysika.ee, ing.dk, Nature, New Scientist,
Physical
T
Review Focus, PhysicsToday, [email protected], 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
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.
11
 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 e 2 .
(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)
X
ΔU/kJ mol-1
NR
R
F
-81
117
Cl
-100
80
Br
-152
-13
I
-152
-39
AuX4 - → AuX2 – +2X
P. Schwerdtfeger, J. Am. Chem. Soc. 111 (1989) 7261.
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 nonrelativistic 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 βPbO2 is an impurity effect.
33
The Z2 argument
It 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
↑
↑
Au
Ag
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.
36
Relativity indeed explains the yellow colour of gold
↑
R
↑
R
↑
NR
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.
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
P. Pyykkö, J. Organomet. Chem. 691 (2006) 4336-4340.
46
[email protected]: The first 32-e species?
1. J. P. Dognon, C. Clavaguéra, P. Pyykkö, Angew. Chem. Int. Ed.
46 (2007) 1427.
47
[email protected]
 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.
1. J. P. Dognon, C. Clavaguera,
P. Pyykkö, Angew. Chem. Int.
Ed. 46 (2007) 1427.
48
[email protected]+ : 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; VAu12NbAu12-, 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’
OUIr+ prepared [2] !
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+.
52
SIMPLEST PREDICTIONS FOR CHEMICAL BONDING:
Molecular, self-consistent covalent radii
R(AB) = rA + rB.
53
MEAN-SQUARE DEVIATION ONLY 3 pm
for 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
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