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From the Potential-Well Analogy to the “Boron Connection”
Or Physical and Chemical Analogies and
Similarities
Aristides D. Zdetsis, department of Physics University of Patras, Gr.
June 2011 WavePro Soukoulis 60
From the Potential-Well Analogy to the “Boron Connection”
or The Consequences o Murphy’s Law:
I broke my leg I broke my laptop I lost the final edition of my presentation But I fortunately(?) am still here.
No matter. Costas you are worth it. Happy Birthday!
Mathematical, Physical, and Chemical similarities, which are very simple, transparent and pedagogic, have been proven at the same time to be very efficient, successful and constructive in frontier scientific research. The potential- well analogy is a classical example of such similarity. The present work focuses on Chemical similarities based on isoelectronic (isovalent) analogies. The framework of the particular chemical similarities presented here is simple, transparent and powerful. These analogies and similarities could be considered as conceptual extensions of the periodical table of the elements, assuming that two atoms or molecules having the same number of valence electrons would be expected to have similar or homologous properties. In addition, such similar moieties should be able, in principle, to replace each other in more complex structures and nanocomposites. This is only partly true and only occurs under certain conditions, which are investigated and reviewed here. Applications include the design of silicon, carbon, and silicon- carbon based rings, nanowheels, nanorodes, nanocages and multidecker
sandwiches, as well as silicon planar rings and “fullerenes”
Abstract
[1] E. N. Economou and C. M. Soukoulis, Phys. Rev. B 28, 1093 (1983); E. N. Economou, Green’s Functions in Quantum Physics, 2nd ed. (Springer, Heidelberg, 1983) [2] E. N. Economou, C. M. Soukoulis, A. D. Zdetsis, Phys. Rev. B 30, 1686 (1984) [3] A. D. Zdetsis, J. Chem. Phys., 127, 014314 (2007) [4]A. D. Zdetsis, J. Chem. Phys., 127, 214306 (2007) [4] A. D. Zdetsis, J. Chem. Phys., 127, 244308 (2007) [5] A. D. Zdetsis , Phys. Rev. B 76, 075402 (2007) [6] A. D. Zdetsis, J. Chem. Phys. 128, 184305 (2008) [7] A. D. Zdetsis, J. Phys. Chem. A 112, 5712 (2008) [8] A. D. Zdetsis, Inorg. Chem. 47, 8823 (2008) [9] A. D. Zdetsis, J. Chem. Phys. , 130, 064303 (2009) [10] E. N. Koukaras and A. D. Zdetsis, Organometallics, 28, 4308 (2009) [11] A. D. Zdetsis, J. Chem. Phys. , 131, 224310 (2009) [12] A. D. Zdetsis, Phys. Rev. B 80 195417 (2009) [13] A. D. Zdetsis, J. Phys. Chem. C 114, 10775 (2010) [14] A. D. Zdetsis Chem. Phys. Lett. 493 45 (2010) [15] A. D. Zdetsis, J. Chem. Phys. 134, 094312 (2011) [16] A. D. Zdetsis Chem. Phys. Lett. 508, 252 (2011) [17] A. D. Zdetsis, Nanoscale Research Letters 6:362 (2011)
REFERENCES/PUBLICATIONS
Structure of the talk
1. Historical Introduction: The potential-well analogy 2. Introduction to Chemical Similarities & analogies 3. Global and Local Chemical Analogies:/Rules of
correspondence 4. The “boron connection” analogy 5. Other isolobal analogies: Silicon “Fullerenes” & Aromatic
Silicon Planar Rings: The SiLi (silly) rule of thumb 6. Conclusions, Further Applications and Implications (A Bacterium that can grow by using Arsenic instead of
Phosphorous)
Economou and Soukoulis1 have pointed out that the equation for the localization length λ has exactly the same structure as the equation which determines the decay length of an eigenstate bound to a potential well. Hence, they deduced the equivalence between localized states in disordered d-dimensional systems and bound states in local potential wells in d-dimensional space. This equivalence was subsequently tested by Economou, Soukoulis and Zdetsis2 against numerical data and the agreement was found to be satisfactory.
[1] E. N. Economou and C. M. Soukoulis, Phys. Rev. B 28, 1093 (1983); E. N. Economou, Green’s Functions in Quantum Physics, 2nd ed. (Springer, Heidelberg, 1983) [2] E. N. Economou, C. M. Soukoulis, A. D. Zdetsis, Phys. Rev. B 30, 1686 (1984)
Historical Introduction:Almost 30 years ago
The importance of this observation is that it allows one to bypass the rather complicated formalism on which localization theory is based and to reduce the localization problem to the elementary problem of a bound state in a potential well.
For example, based on the potential-well analogy we can deduce that: Since a weak potential well always binds a quantum particle in d dimensions, where d ≤ 2, it follows that all eigenstates are localized in disordered d-dimensional systems for d ≤ 2. The d =2 system is a borderline case producing an exponentially long localization length. For d > 2 a critical strength must be exceeded (corresponding to a critical disorder for each given energy) in order to produce a bound (i.e., a localized) state.
Historical Introduction: Then and now
From the knowledge of a simple and/or well-known physical (or chemical) system one can deduce or predict the properties of more complicated and less-known (or as yet unknown) systems. This is the main advantage of physical and chemical analogies and similarities
Historical Conclusion (s)
Introduction to Similarities and Regularities in Chemistry
• Global and local relationships • The periodical table • Within the periodic table:
(Diagonal Relationship)
1 H
2 He
3 Li
4 Be
5 B
6 C
7 N
8 O
9 F
10 Ne
11 Na
12 Mg
13 Al
14 Si
15 P
16 S
17 Cl
18 Ar
19 K
20 Ca
31 Ga
32 Ge
33 As
34 Se
35 Br
36 Kr
37 Rb
38 Sr
49 In
50 Sn
51 Sb
52 Te
53 I
54 Xe
55 Cs
56 Ba
81 Tl
82 Pb
83 Bi
84 Po
85 At
86 Rn
1 H
2 He
3 Li
4 Be
5 B
6 C
7 N
8 O
9 F
10 Ne
11 Na
12 Mg
13 Al
14 Si
15 P
16 S
17 Cl
18 Ar
19 K
20 Ca
31 Ga
32 Ge
33 As
34 Se
35 Br
36 Kr
37 Rb
38 Sr
49 In
50 Sn
51 Sb
52 Te
53 I
54 Xe
55 Cs
56 Ba
81 Tl
82 Pb
83 Bi
84 Po
85 At
86 Rn
Diagonally adjacent elements (mainly of the second and third periods) have similar
properties (in particular those
related with Size and electronegativity).
Crossing and descending the periodic table have opposite effects on these properties canceling out for diagonal pairs of elements such as B and Si
1) Simple and transparent pedagogically attractive. 2) They are, most of the times, very efficient and
effective. 3) They are often good or very good approximations 4) Even when they fail, they can serve (often) as good
starting points and/or as zeroth order approximations, which can usually improved in a systematic way.
5)They are not empirical (consider for example the periodical table) but they are usualy based in basic principles and symmetries.
1 H
2 He
3 Li
4 Be
5 B
6 C
7 N
8 O
9 F
10 Ne
11 Na
12 Mg
13 Al
14 Si
15 P
16 S
17 Cl
18 Ar
19 K
20 Ca
31 Ga
32 Ge
33 As
34 Se
35 Br
36 Kr
37 Rb
38 Sr
49 In
50 Sn
51 Sb
52 Te
53 I
54 Xe
55 Cs
56 Ba
81 Tl
82 Pb
83 Bi
84 Po
85 At
86 Rn
(BH)→Si isovalent relation involving 4 valence electrons . Motivation: 1)The structure of Si6 cluster Fluxional3-4 , similar to boranes 2) The geometrical (Oh cubic symmetry) and electronic structure of [Si6 ] 2- dianion. Similar to [BH]6
2- borane
In fact [(BH)n ]2-→[Si]n2-
Motivation: 1) Boron Chemistry is very well known and understood. 2) There are very powerful structural lows for boranes 3) There are well-established and well-tested stability rules 4) There are many powerful concepts motivated and developed for boranes and carboranes , such as spherical aromaticity (used also later for carbon fullerenes and many more) .
1)[ Lipscomb WN: Framework rearrangement in boranes and carboranes. Science 1966, 153: 373. 2)Wade K: The structural significance of the number of skeletal bonding electron-pairs in carboranes, the higher boranes and borane anions, and various transition-metal carbonyl cluster compounds. J Chem Soc Chem Commun 1971, 22: 792. 3) Fox MA, Wade K: Evolving patterns in boron cluster chemistry. Pure Appl Chem 2003, 75: 1315. 4) Williams RE: Polyborone, carborane, carbocation continuum: architectural patterns. Chem Rev 1992, 92: 177. 5)Stephan M, Müller P, Zenneck U, Pritzkow H, Siebert W, Grimes RN: Organotransition-matall metallacarbones: Paramagnetic iron-cobalt dicobalt triple-decker sandwich complexes. Inorg Chem 1995, 34: 2058. 6)Bluhm M, Pritzkow H, Siebert W, Grimes RN: Benzene-centred tri- and tetramettala carborane sandwich complexes organotransition-metal metalla carboranes, Part 56. Angew Chem Int Ed 2000, 39: 4562-4564. 7) Brook MA: Silicon in Organic, Organometallic and Polymer Chemistry. New York: John Wiley & Sons; 2000. 8)Yang X, Jiang W, Knobler CB, Hawthorne MF: Rigid-rod molecules: carborods. Synthesis of tetrameric p-carboranes and the crystal structure of bis(tr-n-butylsilyl) tetra-p-carborane. J Am Chem Soc 1992, 114: 9719.
Some Important borane/carborane references
1 H
2 He
3 Li
4 Be
5 B
6 C
7 N
8 O
9 F
10 Ne
11 Na
12 Mg
13 Al
14 Si
15 P
16 S
17 Cl
18 Ar
19 K
20 Ca
31 Ga
32 Ge
33 As
34 Se
35 Br
36 Kr
37 Rb
38 Sr
49 In
50 Sn
51 Sb
52 Te
53 I
54 Xe
55 Cs
56 Ba
81 Tl
82 Pb
83 Bi
84 Po
85 At
86 Rn
(BH)→C ? No
(or C→Si ?)No (CH4 → SiH4)
CH→Si1- Yes BH1- → CH Yes
(BH)→Si OK
(BH)→Si and 2BH1-→2CH substitutions
1 H
2 He
3 Li
4 Be
5 B
6 C
7 N
8 O
9 F
10 Ne
11 Na
12 Mg
13 Al
14 Si
15 P
16 S
17 Cl
18 Ar
19 K
20 Ca
31 Ga
32 Ge
33 As
34 Se
35 Br
36 Kr
37 Rb
38 Sr
49 In
50 Sn
51 Sb
52 Te
53 I
54 Xe
55 Cs
56 Ba
81 Tl
82 Pb
83 Bi
84 Po
85 At
86 Rn
More Questions:
Si →Ge→Sn→Pb
What about
(BH) → Ge ?
(BH) → Sn ?
(BH) → Pb ? Inert pair effect: Lower valency on descending down the period (s pair of electrons)
More Similarities
1 H
2 He
3 Li
4 Be
5 B
6 C
7 N
8 O
9 F
10 Ne
11 Na
12 Mg
13 Al
14 Si
15 P
16 S
17 Cl
18 Ar
19 K
20 Ca
31 Ga
32 Ge
33 As
34 Se
35 Br
36 Kr
37 Rb
38 Sr
49 In
50 Sn
51 Sb
52 Te
53 I
54 Xe
55 Cs
56 Ba
81 Tl
82 Pb
83 Bi
84 Po
85 At
86 Rn
2BH1- →2As 2BH1- →2P
(BH)n2- : 2BH1-→2CH→2P… & (n-2)BH→(n-2)Si
2BH1- →2Sb
2BH1- →2Bi
2Si1- →2P
2BH1- →2CH
1 H
2 He
3 Li
4 Be
5 B
6 C
7 N
8 O
9 F
10 Ne
11 Na
12 Mg
13 Al
14 Si
15 P
16 S
17 Cl
18 Ar
19 K
20 Ca
31 Ga
32 Ge
33 As
34 Se
35 Br
36 Kr
37 Rb
38 Sr
49 In
50 Sn
51 Sb
52 Te
53 I
54 Xe
55 Cs
56 Ba
81 Tl
82 Pb
83 Bi
84 Po
85 At
86 Rn
The isolobal analogy: Operational Definition
Isolobal analogy: Formal Definition
Isolobal analogy: Electron equivalent groups
Applications: Comparison of Si62- with
(B6H6)2-
Both Oh structure, “similar” orbitals
“Isolobal”
The Boron connection part 2: Carboranes
The Boron connection part 2 (b)
The Boron connection part 2 (c)
n = 3
Isovalent ,
isostructural,
and isolobal
Multidecker sandwiches
The geometric (top left) and electronic (right) structure of C14B3Fe2H17, C14Si3Fe2H14 tripledecker sandwiches, and the geometry of C14B3Fe3H22, C14Si3Fe3H16 tetradecker sandwiches (bottom left)
Other applications Si/Bi core/shell nanoparticles –Bi nanolines in Si surfaces15-17
Extension to Si-Bi clusters of the form Bi2Sin-2 (and Bi2Gen-2?). For n=7 we have:
Sn122- Functionalization:
Bi2Sn10 and Sb2Sn10 “alloy clusters” part 1
Nanowheels and Nanorods
Nanorods functionalized nanostructures
Sn122- Functionalization:
Bi2Sn10 and Sb2Sn10 “alloy clusters” part 3
Conclusions 3: The plan of the “boron connection” 1 H
2 He
3 Li
4 Be
5 B
6 C
7 N
8 O
9 F
10 Ne
11 Na
12 Mg
13 Al
14 Si
15 P
16 S
17 Cl
18 Ar
19 K
20 Ca
31 Ga
32 Ge
33 As
34 Se
35 Br
36 Kr
37 Rb
38 Sr
49 In
50 Sn
51 Sb
52 Te
53 I
54 Xe
55 Cs
56 Ba
81 Tl
82 Pb
83 Bi
84 Po
85 At
86 Rn
The Si-Li (silly) rule of thumb
Comparison
of HOMO
π - orbitals
1b3g 1e1g
1b2g 1e1g
1b1u 1a2u
Si6Li6 C6H6 Si66- C6H6
Or
Si1- CH
Conclusions: • The fluxionality of Si6 in full analogy to isovalent boranes, lead to the so called (by the present author) “boron connection”, which can help the molecular engineering of new stable (hopefully) silicon based molecules and materials
• More analogies and similarities could be invoked.
• Further Applications and Implications? … A Bacterium that can grow by using Arsenic instead of Phosphorous [Wolfe-Simon et al. http://www.scienceexpress.org 2010]
The Physical and Chemical similarities presented here are very efficient and effective for the design of Si-based nanomaterials
Even when they fail, they can serve as good first order approximations which can usually improved in a systematic way, and be verified by experiment.
Nevertheless, to paraphrase a well known motto of public relations theory, If experiment does not verify these rules, so much the worse for experiment . Happy birthday Costas
