(2) N elec = D p,q p,q χ q χ p dr = p,q D p,q S q,p Mulliken PA D Mull p = p = group A D p,p 1 + D p,q S q,p p q p [ r A D Mull p ] group χ p G Mull A

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1 7 - (Electron-Donor Acceptor) : Charge-Transfer ( CT) ( (Charge-Transfer) - (electron donor-electron acceptor) [1][2][3][4] Van der Waals CT [5] Population Analysis population analysis ( ), observable physical property population analysis - (electron donor-electron acceptor) population analysis 7.1 Population Analysis Population analysis(pa) Population analysis Mulliken PA[6] Löwdin PA[7] Weinhold Natural PA[8][9] PA 1 ( ) ρ(r) i = i n i ϕ i ϕ i (1) n i c p,i χ p χ q c q,i D p,q χ p χ q p,q (2) p,q D CΩ C, [Ω] i,j = n i δ ij ϕ i n i χ p D p,q D ( { χ p }) ρ(r) ρ(r)dr =N elec 138

2 (2) N elec = D p,q p,q χ q χ p dr = p,q D p,q S q,p Mulliken PA D Mull p = p = group A D p,p 1 + D p,q S q,p p q p [ r A D Mull p ] group χ p G Mull A A D Mull p (3) G Mull A (4) ( ( p A ( ) S q,p D Mull p D Mull p = χ p ρ(r) χ p PA χ p Löwdin PA Löwdin PA χ L = χs 1 2 (5) D Lw r = χ L r ρ(r) χ L r = D p,q χ L r χ p χ q χ L r p,q = p,q ] D p,q [S 1 2 s,t = D p,q [S 1 2 p,q [ ] = S DS 2 r,r ] r,p r,s S s,p S q,t [S 1 2 ] t,r [ ] S 1 2 = [ ] [ ] S 1 2 D p,q S 1 2 q,r r,p q,r p,q (6) (7) (8) S 1 2 DS Natural Population Analysis Weinhold Natural PA Natural Atomic Orbital Gaussian Gamess Weinhold know-how [10] ρ(r) χ NPA r [10] 1 139

3 { χ p } { χ A p } D A 1 pre-nao(natural Atomic Orbital) 1-1: (p x, y, z d x 2, y 2, z 2, xy, xz, yz) (l, m), χ sph = χt sph (9) C sph = T sph C (10) D sph = T sph DT sph = T sph CΩ CT sph = C sph Ω C sph (11) S sph = T sph ST sph (12) D sph (S sph ) A, (l, m) D (A,l,m) (S (A,l,m) ) 1-2: A m A (l = 0, 1, 2, ) M (A,l) [ D (A,l)] p,q 1 2l + 1 [ S (A,l)] p,q 1 2l + 1 l m= l l m= l [ D (A,l,m)] p,q [ S (A,l,m)] p,q D (A,l), S (A,l) M (A,l) (13) (14) D (A,l) U (A,l) = S (A,l) U (A,l) Ω (A,l) (15) M (A,l) pre-nao η (A,l,m) = χ sph(a,l,m) U (A,l) (16) Ω (A,l) pre-nao [ 1-3: ANO Ω (A,l)] k = 1,, M (A,l) pre-ano k,k pre-ano natural minimal basis (NMB) Li Be s (l = 0) 2 B Ne s (l = 0) 2 (1s 2s ) p (l = 1) 1 (2p ) M (A,l) NMB pre-ano natural Rydberg Basis (NRB) Weinhold NMB NRB 2 pre-ano Population analysis AO (weighted 140

4 interatomic orthogonalization) NMB NRB NRB NMB Schmidt µ = {µ i ; i = 1, M} ν = {ν i ; i = 1, M} ν= µv (17) νν= : NMB A l,m M (A,l) k [ M ν i µ i 2 = minimum (18) i M w i ν i µ i 2 = minimum (19) i Ω (A,l)] η (A,l,m) k k,k O M W NMB η (A,l,m) 2 = minimum (20) k χ ONMB = χt sph U (A,l) O M W (21) 2-2: NRB NMB Schmidt O S NRB NRB [ Ω (A,l)] NMB k,k 1-2,1-3 U (A,l) O S NRB T sph U (A,l) O S U (A,l) χ NRB = χt sph U (A,l) O S U (A,l) (22) χ ONRB = χ NRB O R W (23) 141

5 3 ( χ ONMB, χ ONRB) Natural AO 1-2,1-3 N restore ANO χ ANO χ ANO (6) χ NRB χ ONRB N restore (24) D ANO r = χ ANO ρ(r) χ ANO r r (25) Population analysis 7.2 Mulliken [4] Mulliken (Molecular Orbital Theory) Mulliken school 1966 Mulliken MO VB MO ( D A D + A DA (26) DA DA Mulliken DA Ψ N = aφ 0 (D A) + bφ 1 (D + A ) (27) (no-bond) (dative) resonance (2 2) Φ 0 Ĥ Φ 0 Φ 0 Ĥ Φ 1 Φ 1 Ĥ Φ 0 Φ 1 Ĥ Φ 1 a b = E 1 Φ 0 Φ 1 Φ 1 Φ 0 1 a b (28) W 0 Φ 0 H Φ 0, W 1 Φ 1 H Φ 1, W 01 Φ 1 H Φ 0 = Φ 0 H Φ 1, S Φ 1 Φ 0 = Φ 0 Φ 1 142

6 W 0 E W 01 SE W 01 SE W 1 E E (W 01 SW 0 ) SE (W 01 SW 0 ) SE E = 0 (29) = 0, E E W 0 (30) E ± = 1 W 0 + W 1 (1 S 2 SW 01 ± ) 2 ( ) 2 + β 0 β 1 2 W 1 W 0 β 0 W 01 W 0 S, β 1 W 01 W 1 S β 0, β W 0 < W 1 Ψ N Φ 0 (D A) Ψ V Φ 1 (D + A ) Charge-transfer transition 3 Φ 1 (D + A ) Ĥ Φ 1 (D + A ) Φ 0 (D A) Ĥ Φ 0(D A) [ Φ(D + ) Ĥ D Φ(D + ) Φ(D) Φ(D) ] ĤD [ Φ(A + ) Ĥ A Φ(A ) ] Φ(A) ĤA Φ(A) + [ V (D + A ) V (D A) ] (31) H D (H A ) D(A) V (D + A ), V (D A) D + A D A D I E (D) I E (D) = Φ(D + ) Ĥ D Φ(D + ) Φ(D) ĤD Φ(D) 143

7 A A E (A) A E (A) = Φ(A) ĤA Φ(A) Φ(A ) Ĥ A Φ(A ) C(D + A ) [ V (D + A ) V (D A) ] (32) 1 R(D A) (33) I E (D) A E (A) C(D + A ) β 0 (β 1 ) (4 29) v QP P S Φ 1 Φ 0 S ( ) 2 β 0 β 1 2 ( ) 2 ( ) [ ] + β 0 β β 0β ( ) 2 2 ( ) W1 W 0 = + β 0β 1 2 W 1 W 0 β 0 β 1 = β 0 (W 01 W 1 S) = β 0 {W 01 W 0 S + (W 0 W 1 ) S} = β 2 0 β 0 (W 1 W 0 ) S = β β 1 (W 1 W 0 ) S E N E W 0 β2 0 E V E + W 1 + β2 1 (34) (35) hν CT = E V E N = W 1 W 0 + β2 1 + β 2 0 = I E (D) A E (A) C(D + A ) + β β 2 0 I E (D) A E (A) C(D + A ) (36) A D I E 1 I E A E 144

8 a, b (28) b N a N = W 0 E N W 01 SE N (37) a V b V = W 1 E V W 01 SE V (38) 34 (35) b N a N = β2 0 β 0 (1 + S β2 0 ) = (a 2 N + b2 N + 2a N b N S = 1) β 0 + Sβ 0 β 0 > 0 (39) a V β 1 = β 1 b V Sβ 1 < 0 (40) a N = 1 1 β, b 0 N = β 0 a V = β β1 1 1 β 0 b V = β1 (x ) (41) (42) x Ψ N x Ψ N = a 2 N Φ 0 x Φ 0 + b 2 N Φ 1 x Φ 1 + 2a N b N Φ 0 x Φ 1 D A x D A x ( β 0 x b 2 N Φ1 (D + A ) x Φ 1 (D + A ) ( ) 2 ( ) 2 β 0 1 β0 1 β R 0 D A R D A (43) ) 2 1 R 1 β 0 D A Ψ V x Ψ N b N b V Φ1 (D + A ) x Φ 1 (D + A ) ( β0 ( ) β0 ) R D A (44) 7.3 EDA(CT) DA 145

9 [11] 7-2 tetracyano-benzene 1:N,N,N N -tetramethyl-p-phenylene-diamine(tmpd), 2 N,N-dimethyl-aniline, 3:hexamethyl-benzene tetracyano-benzene(tcnb) 7-3 [11] TCNB trinitro-benzene(tnb) TCNB ( 36 1 I E ( ) 36 A E 146

10 7-3 trinitro-benzene(tnb) tetracyano-benzene(tcnb) ( I 2 C=O N=O 7-4 I 2 N (C 5 H 5 NO) CCl 4 1 I 2 2,3, C 5 H 5 NO 490nm( mµ (isobestic point) (σ σ ) ( ) 380nm 7-4 I 2 C 5 H 5 NO CCl 4 1 I 2 2,3, C 5 H 5 NO 147

11 [12] 7-5 Cl 2 Cl-Cl I E 10.13eV HF/6-31 () [12] N Cl-Cl 7-6 C 2 H 4 C-C [13] 148

12 7-6 C=C [13] 43 Q s Q s 2 x Q s 2β 0R D A 2 β 0 Q s (45) Q s C=C D-A 7.4 EDA(CT) Horváth [14] Lewis 149

13 7-1 BH 3 NH 3 [14] 7-8 N-B B-H N-H 7-7 BH 3 NH3 N-H [14] [14] 7-2 LP MO 3rdSPT + dispersion [15] [16] NH 3 -SO 2 ) E 3SPT+Disp CCSD(T)/CBS 1kcal mol E 3SPT E 3SPT ( 150

14 7-2 Hobza (ref.31 [16]) LP MO [15] kj mol 1. π tetracyanoethylene(tcne) acenaphtylene [17] 7-9 TCNE-Acenaphtylene[17] DFT [18] π T π 151

15 7.5 2 π Pulay [19] aug-pvqz CCSD(T), QCISD(T) coupled cluster T Parallel displaced (PD) 2 Note µ = {µ i ; i = 1, M} µ = χu µµ =ŨSU T (46) ν={ν i ; i = 1, M} = µv ν i = µ i v i 152

16 νν =ṼŨSUV =ṼTV = 1 (19) I = i,.j [δ ij w i (ṽ i µ i )(v i µ i ) + λ ji ṽ i v j ] ṽ i I = w i (v i µ i ) + ṽ i.j λ ji v j = 0 i = 1, M µ i w i =.j v j (δ ji w i + λ ji ) = 0 µ i w i =.j v j w j λ ji (47) λ ji λ ji δ jiw i + λ ji 21 µω = µvωλ µ TΩ = TVΩΛ (48) Λ ṼTV = 1 ṼTΩ = ṼTVΩΛ = ΩΛ 22 TΩ = TVṼTΩ V = Ω(ΩTΩ) 1/ [1][2][3] 4. ( β 0, β 1 W 0, W 1 (W 0, W 1 β 0, β 1 β i = Φ 1 Ĥ Φ 0 S Φ i Ĥ Φ i 153

17 Ω ) ) β i = Φ 1 (Ĥ Ω Φ 0 S Φ i (Ĥ Ω Φ i ( ) ( ) = Φ 1 Ĥ Φ 0 ΩS S Φ i Ĥ Φ i SΩ Ω W i = Φ i Ĥ Φ i SW i S β i Φ i H Ĥ Φ i = W i Φ i + Υ i i Υ i i β 0 = Φ 1 Ĥ Φ 0 S Φ 0 Ĥ Φ 0 = Φ 1 Υ 0 0 β 1 = Φ 0 Υ 1 1 [1] R. S. Mulliken, J. Am. Chem. Soc., 72, 600 (1950). [2] R. S. Mulliken, J. Am. Chem. Soc., 74, 811 (1952). [3] R. S. Mulliken, J. Phys. Chem., 56, 801 (1952). [4] R. S. Mulliken, Willis B. Person, Molecular Complexes, Wiley Interscience, (1969). [5] Noboru Mataga, Tanekazu Kubota, Molecular Interactions and Electronic Spectra, Marcel Dekker Inc., (1970). [6] R. S. Mulliken, J. Chem. Phys., 23, 1833 (1952). [7] P. O. Lowdin, Phys. Rev., 97, 1474 (1955). [8] A. E. Reed, L.. A. Curtis, F. Weinhold, Chem. Rev., 88, 899 (1988). 154

18 [9] F. Weinhold, Encyclopedia of Computational Chemistry Edited by von Schleyer, 3, 1792 (1998). [10] A. E. Reed, R. B. Weinstock, F. Weinhold, J. Chem, Phys., 83, 735 (1985). [11] S. Iwata, J.Tanaka, S. Nagakura, J. Am. Chem. Soc., 88, 894 (1966). [12] H. Matsuzawa, S. Iwata, Chem. Phys., 163, 297 (1992). [13] H. Matsuzawa, H. Yamashita, M. Ito, S. Iwata, Chem. Phys., 147, 77 (1990). [14] V. Horvth, A. Kovcs, I. Hargittai, J. Phy. Chem., 107, 1197 (2003). [15] S. Iwata, Phys. Chem. Phys. Chem., 14, 7787 (2012). [16] S. Karthikeyan, R. Sedlak, P. Hobza, J. Phys. Chem. A, 115, 9422 (2011). [17] H.-B. Yi, X.-Y. Li, S.-Y. Yang, X.-H. Duan, Int. J. Quant. Chem, 94, 23 (2003). [18] M.-S. Liao, Y. Lu, S. Scheiner, J. Comp. Chem., 24, 624 (2003). [19] T. Janowski, P. Pulay, Chem. Phys. Letters, 447, 27 (2007). 155

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