Graph Theoretic Approaches to Atom ic Vibrations in Fullerenes - PowerPoint PPT Presentation

Graph Theoretic Approaches to Atom ic Vibrations in Fullerenes ERNESTO ESTRADA Department of Mathematics & Statistics, Department of Physics University of Strathclyde, Glasgow w w w .estradalab.org

“A fullerene , by definition , is a closed convex cage molecule containing only hexagonal and pentagonal faces.” R. E. Smalley

β θ m = 1 + Lx = M   x 0 (1) θ    ( ) = (2) T L V x x x 2

β 4 θ 1 3 2 m = 1 Edge weight: βθ − −   2 1 0 1   − − − 1 3 1 1 = − =   L D A − − 0 1 2 1   − − −   1 1 1 3

  ( ) ∫ ∆ ≡ = 2 2 (3) x x x P x d x i i i  )  ∫ = x x x x P ( x d x (4) i j i j βθ −     ∫ =   T M Z d y exp y y   2 (5) βθ −  n + ∞ ∏∫ = µ   2 dy exp y . α α α − ∞   2 α = 1 = µ < µ ≤ ≤ µ  0 1 2 n

µ 1 = 0 − βθ  n ~ + ∞ ∏∫ =  µ  2 Z dy exp y α α α − ∞   2 α = 2 (6) π n 2 ∏ = . βθµ α = α 2 − βθ  − βθ  n ( ) n ~ + ∞ + ∞ ∑∫ ∏∫ ≡ µ × µ 2  2   2  I dy U y exp y dy exp y ν ν ν ν ν α α α i i − ∞   − ∞   2 2 ν = α = 2 2 α ≠ ν π π 2 n n U 8 2 ∑ ∏ = ν × i ( ) βθµ βθµ 3 2 ν = α = ν ν 2 2 α ≠ ν 2 n U ~ ∑ = × ν i (7) Z . βθµ ν = ν 2

~ 2 n I U ∑ ∆ ≡ = = ν 2 i i (8) x x , ~ βθµ i i Z ν = ν 2 ( ) ii ( ) 1 + ∆ = 2 x L (9) βθ i + L Estrada, E., Hatano, N. Chem. Phys. Lett . 486 2 0 1 0 , 166.

U U n = ∑ ν ν i j x x βθλ i j ν = ν 2 (10) ( ) 1 = + L βθ ij Estrada, E., Hatano, N. Chem. Phys. Lett . 486 2 0 1 0 , 166.

Estrada, E., Hatano, N. Chem. Phys. Lett . 486 2 0 1 0 , 166.

Estrada, E., Hatano, N. Chem. Phys. Lett . 486 2 0 1 0 , 166.

4 ( ) ( ) ( ) 2 + + + Ω = + − L L 2 L 1 3 1 , 2 1 , 1 2 , 2 1 , Ω 1 , 2 2 ( ) 1 ∑∑ ∑ = Ω = Kf G R i , j i 2 i j i ∑ = Ω R i i , j ( ) ∈ j V G Klein, D.J. , Randić , M. J. Math. Chem . 1 2 (1993) 81.

( )   ( ) 1 Kf + ∆ = = − 2   L x R (11) i ii i   n n ( ) ( )   1 1 2 Kf + = = − Ω − + +   x x L R R (12) i j ij ij i j  2  2 n n ( )  ( ) n 1 1 ∑ ∑ = − + − Ω V x k R R R n (13) i i i j ij 2 n 2 n = ∈ i 1 i , j E Estrada, E., Hatano, N. Physica A . 389 2 0 1 0 , 3648.

[ ] ( ) ( ) ( ) Ω = ∆ + ∆ − = − 2 2 2 (14) x x 2 x x x x ij i j i j i j [ ] n ( ) ( ) ( ) ( ) ∑ = ∆ + ∆ = ∆ + ∆ 2 2 2 2 R n x x n x x (15) i i i i = i 1 = ∑ ( ) ( ) n ∆ = ∆ 2 2 2 Kf n x n x (16) i = i i Estrada, E., Hatano, N. Physica A . 389 2 0 1 0 , 3648.

S { } ( ) ∂ = ⊂ ∈ ⇒ ∈ ∈ S U E u , v U u S , v S S Set of edges connecting S to its complement A graph G is said to have edge expansion ( K , φ ) if ( ) ∂ ≥ φ ∀ ⊆ ≤ S S , S V with S K ∂ S φ ≡ : min ≤ S S S n / 2

µ = λ − λ j 1 j λ ≥ λ ≥ ≥ λ  1 2 n βθ ≡ ∆ = λ − λ 1 1 2 ( ) 2 2 = φ 2 i φ j ( i ) 2     n ( ) ∑ ∆ x i + (17) . ∆ λ 1 − λ j j = 3 Estrada, E.; Hatano, N.; Matamala, A. R. I n the book : Mathematics and Topology of Fullerenes, A. Graovac; O. Ori; F . Cataldo, Eds.; Springer, 2010 to appear.

 ( )    2 φ 2 i φ j ( i ) 2   2 = 1   n n n ( ) ∑ ∑ = 1 1 ∑ 1 ∆ x i + ∆ +     . ∆ λ 1 − λ j λ 1 − λ j     n n i = 1 j = 3 j = 3 Among all graphs with n nodes, those having good expansion display the smallest topological displacements for their nodes. Estrada, E.; Hatano, N.; Matamala, A. R. I n the book : Mathematics and Topology of Fullerenes, A. Graovac; O. Ori; F . Cataldo, Eds.; Springer, 2010 to appear.

ω ω ˆ ( ) ( ) 2 p ∑ ∑ ∑ = + − − − ˆ 2 s H x x A k 1 x (18) r s rs s s 2 m 2 2 s r , s s  d ≡ k ˆ p s s i dx     1 i 1 i   −   = ϖ + + = ϖ − ˆ ˆ ˆ ˆ a x m p a x m p     s s s s s s ϖ ϖ       2 m 2 m ω ϖ ≡ m

 ω  ˆ p ∑ ∑ ˆ =  +  − ω 2 s H  x  x A x s r rs s   2 m 2 s r , s (19) ( ) ( ) ϖ    ∑ 1 ∑ + − + − + − = ϖ + − + +    ˆ ˆ ˆ ˆ ˆ ˆ a a a a A a a s s r r rs s s   2 2 s r , s ∑ = ˆ ˆ H H (20) j j

( ) ϖ    1 2 + − + − ˆ ≡ ϖ ˆ ˆ + − λ ˆ + ˆ    H b b b b (21) j j j j j j   2 2 ( ) ϖ    1 + − + 2 − 2 + − = ϖ ˆ ˆ + − λ ˆ + ˆ + ˆ ˆ +    b b b b 2 b b 1 j j j j j j j   2 2 − ∑ ( ) − ( ) + ∑ ˆ ˆ + = ϕ = ϕ ˆ ˆ b s a b s a j j j j j j s s [ ] [ ] ( ) ( ) ∑ ˆ − ˆ + = ϕ − ϕ + ˆ ˆ b , b r a , s a j j j r l s Eigenvectors of H r , s ( ) ( ) ∑ = ϕ ϕ δ r s j l rs r , s ( ) ( ) ∑ = ϕ ϕ = δ s s , j l jl s

1 − β ≡ ˆ H G x x 0 x e x 0 (22) rs r s r s Z − β ˆ ≡ H Z 0 e 0 . (23) ( ) ( ) ( ) ( )   1 ∑ ∏  − β ˆ  = ϕ ϕ ˆ + + ˆ − ˆ + − ˆ − H G r s 0 b b e b b 0  i  rs j l j j l l   Z j , l i ( ) ( ) 1 ∑ ∏ − β ˆ − β = ϕ ϕ δ ˆ − ˆ + × ˆ H H (24) r s 0 b e j b 0 0 e 0 i j l jl j j Z ( ) ≠ j , l i j β ϖ ( ) β ϖ     ( ) ( )   ( )  1 3 ∑ ∏ = ϕ ϕ − + λ × − + λ r s exp 1 exp 1    j j j i     Z 2 2 ( ) ≠ j i j

ϖ ≡ = −  1 H A ( ) ( ) ( ) ∑ βλ β = = ϕ ϕ A j G C e C r s e . (25) rs rs j j j ( ) ∑ ( ) βλ β A ≡ = = j Z EE G tr e e (26) j [ ] ( ) ∑ βλ ∆ = ϕ 2 x r e j (27) r j j Estrada, E., Rodriguez-Velazquez, A. Phys. Rev . E 71 2 0 0 5 , 056103. Estrada, E., Hatano, N. Phys. Rev . E 77 2 0 0 8 , 036111. Estrada, E., Hatano, N. Chem. Phys. Lett 439 2 0 0 7 , 247.

B C A A C B

P Q Q P

A classical m echanics approach to vibrations in m olecules can be form ulated on the basis of the generalised Moore-Penrose inverse of the graph Laplacian. A quantum m echanics approach to vibrations in m olecules based on coupled harm onic oscillators can be form ulated on the basis of the exponential adjacency m atrix of the graph representing the m olecular system . Both approaches give im portant inform ation about the energetic and stability of fullerene isom ers, as w ell as provide a theoretical fram ew ork for em pirical observations such as the ‘ isolated pentagon rule ’.

NAOMI CHI HATANO Institute of Industrial Sciences University of Tokyo Japan ADELI O R. MATAMALA Department of Chemistry University of Concepcion Chile PATRI CK W . FOW LER Department of Chemistry University of Sheffield U.K.