1 1.1,,,.. (, ),..,. (Fig. 1.1). Macro theory (e.g. Continuum mechanics) Consideration under the simple concept (e.g. ionic radius, bond valence) Stru

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1 MD MD 1

2 1 1.1,,,.. (, ),..,. (Fig. 1.1). Macro theory (e.g. Continuum mechanics) Consideration under the simple concept (e.g. ionic radius, bond valence) Structural relaxation by means of model potential Detail investigation by nonempirical calculation Figure 1.1:,.,. ( )

3 ,,... Fig. 1., Table I). (a) Experiment or Obsevations (b) Non-empirical Calculations (c) Non-empirical Calculations Interatomic Potentials Interatomic Potentials Simulation of Structures and Physical Properties Simulation of Structures and Physical Properties Simulation of Structures and Physical Properties Figure 1.: Table I: method MD reliability Empirical (Classical) >1 4 low Non-empirical (Quantum, PWPP) 1 high Non-empirical (Quantum, All-electron) 1 1 very high Quantum-Classical >1 4 moderately high,.,.,.,.,, ( )..., 3

4 1... 1,..,,, LJ BMH., ( ) () ,..,..,, 1,., R α (α ), M α (α ), r i (i ), m e ( )., H = α h α M α i h m e i + V (r 1,, r i, ; R 1,, R α, ). (1.1) V, V (r 1,, r i, ; R 1,, R α, )= α,i Z α e r i R α + i>j e r i r j + α>β Z α Z β e R α R β (1.). Z. HΦ(r 1,, r i, ; R 1,, R α, )=ɛφ(r 1,, r i, ; R 1,, R α, ) (1.3). Φ(r 1, ; R 1, )=Ψ(r 1, ; R 1, )φ(r 1, ) (1.4). Ψ (1.1), H e = i h m e i α,i Z α e r i R α + i>j e r i r j (1.5) 4

5 , H e Ψ(r 1, ; R 1, )=E(R 1, )Ψ(r 1, ; R 1, ) (1.6). E. (1.4) (1.1) (1.3), HΨ =Φ h α + E(R 1, )+ Z α Z β e M α α R α R β φ h ( α Ψ α φ + φ α Ψ) M α>β α α (1.7). (1.1) Ψ, (1.7), φ, h α + E(R 1, )+ Z α Z β e M α α R α R β φ(r 1, )=ɛφ(r 1, ) (1.8) α>β. (1.8),. (Born-Oppenheimer ). (1.8), 3,. R.,, E R., R., ( ). W (R 1, ), i f i = W R i. (1.9) i j, W Ψ αβ (i, j) = (1.1) u α (i) u β (j). α, β =(x, y, z), u = R R. ( ). ( ) (LD).,... -,,, Grüneisen ( ).,,,,. 5

6 :, W = 1 c ijkl ɛ ij ɛkl (1.11) ijkl. 3 3, 4 c ijkl. ( ) 1( ). x 3 ε 33 ε 3 ε 13 ε ε 31 1 ε 1 ε ε 11 ε 3 x x 1 Figure 1.3: R ψ(r 1, )., ψ ij (R i, R j ). dψ ij R i R j f ij = d R i R j R i R j. (1.1) i, f i = dψ ij R i R 1 j d R i R j R i R j = i j i ψ ij (1.13) R i j i. f i = U T R i (1.14). (1.9) U T W C,,. W = U T + C. (1.15), ψ ij (p 1,p, ; R i, R j ), (1.15). 6

7 1.3,.,.,... ( ),.,. LCAO HF (DFT) - (LDA), (GGA) + (PWPP) (DFT) - (LDA), (GGA) (FLAPW, FPLMTO, KKR) (DFT) - (LDA), (GGA) ( ),. LCAO,. HF, DFT. SCF., PWPP, (Table I).,. (Troullier-Martins(TM) ).,.,., 1 (Table I). ( smooth Hunkel.), PWPP., LDA GGA( Perdew, Burke, Ernzerhof 1996, PBE). LDA GGA. LDA GGA, 7

8 GGA.,. LDA,,, ( ) GGA (CI), LDA GGA 1 ( ),., (1.15),., w = X {W (X) U T (p 1,p, ; X)} (1.16) (p 1,p, )., X, X,., 1. ( ) X.. ɛ ( )..., (Fig. 1.4, Table II). 3., ( ).,. ( ), ( ) 8

9 1: SiO, +BMH + +Morse. ψ ij = q ( ) iq j Ai + A j r ij +f(b i +B j )exp C ic j r ij B i + B j rij 6 +D ij [exp { β(r ij rij)} exp{ β(r ij rij)}] (1.17) E (kj/mol) -9.9 ε=(δ,δ,δ,,,) E (kj/mol) -9.9 ε=(δ,,,,,) E (kj/mol) ε=(,,δ,,,) E (kj/mol) ε=(δ,,δ,,,) E (kj/mol) -9.9 ε=(δ,,-δ,,,) LDA Model empirical Model fit Figure 1.4: SiO. ɛ =(ɛ 11,ɛ,ɛ 33,ɛ 3,ɛ 31,ɛ 1 ) Table II: 3K MD Model a (Å) c (Å) c/a B (GPa) α (1 5 K 1) Exp Empirical PWPP-LDA fit PWPP-LDA Fig (, ), α-. Empirical a. Fig. 1.4 δ. LDAfit PWPP-LDA. PWPP-LDA c. Empirical, LDA fit (?). 9

10 : (Mg(OH) ) H O,,..,.. d O-H Figure 1.5: brucite ( ) PWPP-GGA brucite - ( ) Table III: brucite - Interaction type of d O H length (Å) hydrogen bond W-W 1.89 B-B Non 1.93 PWPP-GGA.13 GGA fit.56 (1.16),..,..,, ( ). ( ). 1

11 MD Ridid Ion Model(RIM)... ( ),, L = T part + T param V part,param V param (.18) MD. T, V, part,param.. (EEM) MD MD. Embeded Atom Model ( ), Breathing Shell Model..1, ( ) ( ) ( ) F E S µ = = θ (.19) N N N, ( ) E µ = N ( ) F =lim θ N (.). F, E.. E(q) =E + aq + bq + O(q 3 ), (.1). 3 ( )., E(+1) E() = a + b E() E( 1) = a b (.), I A. µ I + A (.3) 11

12 . Mulliken, χ = I + A (.4), µ χ..,,. n, χ 1 = χ = = χ n (.5). (Electronegativity Equalization Method)., E ({q i }) = E ({q i}) = = E ({q i}) (.6) q 1 q q n... n, (.6) n 1 q i = n q (.7) i n, ( n q = ). (Charge Equilibration) (Mortier et al. 1986, Rappé and Goddard 1991, etc.)., (.1) (, J ii ) (J ij ), E({q i })=J ii (q i )+ J ij (r ij ; q i,q j ) (.8) i j>i. J ij (i j) ρ i, ρ j J ij (r ij ; q i,q j )= ρ i (r 1,q i ) 1 ρ j (r,q j )dr 1 dr (.9) r 1.. R i δ, ρ i (r,q i ) = Z i eδ( r R i )+(q i Z i )ef i ( r R i ) (.3). Z, f. f i (r) = [ ζ 3 i π exp( ζ ir) ] (.31) 1

13 . (.3)(.31), (.9), MD. (.8) i = j., (.1). (.1) q 1 (.) a = I + A b = I A χ (.3).,, J ii (q) =χ q + η q (.33). (Fig..6). η E A (ev) Na E A (ev) K E A (ev) 6 4 H q (e) Mg q (e) Ca Cl q (e) E A (ev) 1 E A (ev) 1 E A (ev) Al q (e) 1 q (e) 1 8 Si O q (e) E A (ev) 3 E A (ev) 6 4 E A (ev) q (e) q (e) q (e) Figure.6:. 13

14 , n J 1 q η1 η J1 J 3 q 1 q J13 J q 1 n q J1n q 1 J 31 q 3 η1 J 3 q 3 J1 q 1 η3 J13 J q 1 3n q 3 J1n q q 1 q q 3.. = n q χ 1 χ χ 1 χ 3. J n1 q n η 1 J n q n J1 q 1 J n3 q n J13 q 1 η n J1n q 1 q n χ 1 χ n (.34) 1, Ewald. J ij / q i, G,i,j 3. DFCMD.. (Fig..7). 4 Coulomb Atomic Total E (kj/mol) q cation (e) Figure.7:.3 (DFC)MD.,. n n, MD., MD (Dynamical Fluctuating Charge)MD (Rick et al. 1994).. N 1 N L = m iṙi 1 N + M q q i U(r, q) λ q i. (.35) i i i 14

15 m i, M q, λ i,,. 1,. 3 MD, (.9) U(r, q) m i r i = (.36) r i U(r, q) M i q i = λ = χ i λ (.37) q i N q = (.38) i λ = 1 N N χ i (.39) λ., i M i q i = 1 N N (χ i χ j ) (.4) j. DFC,.,.. 1:MgO(51 ) I. ( ): =, =fix, Mg( ), O( )1 4 3 q (e) 1 1 Step 15

16 II. ( ): =±1.7, fix q (e) Step III. + : = 1.8 q (e) Step IV. (Cauchy relation) Method c 11 (GPa) c 1 (GPa) c 44 (GPa) c /c 44 Exp Empirical RIM LDA fit RIM LDA fit DFC FPLMTO-LDA

17 :MgO( ) I. (q i >q bulk :, q i <q bulk : ) Mg O II. (q i >q bulk :, q i <q bulk :, :Mg, :O) V O " V Mg - 17

18 3:DFC-MD I. (N =51 ) 1 8 Time consuming Number of atoms II. Ewald (N (G) = 15 ) 3 Time consuming Number of G 18

n (1.6) i j=1 1 n a ij x j = b i (1.7) (1.7) (1.4) (1.5) (1.4) (1.7) u, v, w ε x, ε y, ε x, γ yz, γ zx, γ xy (1.8) ε x = u x ε y = v y ε z = w z γ yz

n (1.6) i j=1 1 n a ij x j = b i (1.7) (1.7) (1.4) (1.5) (1.4) (1.7) u, v, w ε x, ε y, ε x, γ yz, γ zx, γ xy (1.8) ε x = u x ε y = v y ε z = w z γ yz 1 2 (a 1, a 2, a n ) (b 1, b 2, b n ) A (1.1) A = a 1 b 1 + a 2 b 2 + + a n b n (1.1) n A = a i b i (1.2) i=1 n i 1 n i=1 a i b i n i=1 A = a i b i (1.3) (1.3) (1.3) (1.1) (ummation convention) a 11 x