Charge-Charge, Charge-Dipole, Dipole-Charge, Dipole-Dipole Interaction

charge-charge, charge-dipole, dipole-charge, dipole-dipole interaction

Masahiro Yamamoto Department of Energy and Hydrocarbon Chemistry, Graduate School of Engineering, Kyoto University, Kyoto 606-8501, Japan (Dated: December 1, 2008)

In the bulk and at the interface of polar solvents and/or ionic liquids, molecules feel the strong electric field and this strong field distort the electron cloud of molecule and induces dipole around atoms. To consider this polarization effect we should consider charge-charge, charge-dipole, dipole-charge, dipole-dipole interaction.

I. CHARGE-CHARGE (C-C) INTERACTION

The coulomb potential φ from the point charge zie at ri is given by

0 . 2

0 . 4

0 . 6

0 . 8

1 0

FIG. 1: Coulomb potential of negative charge

zie 1 φC(|r − ri|) = (1) 4πϵ0 |r − ri| The electric ﬁeld is given by the gradient of the potential − i −∇ | − | zie r ri EC(r) = φC( r ri ) = 3 (2) 4πϵ0 |r − ri|

If we put the other point charge zje at at rj the electrostatic energy Vcc is given by

2 zizje 1 Vcc = zjeφC(|rj − ri|) = (3) 4πϵ0 |ri − rj|

II. CHARGE-DIPOLE (C-D) INTERACTION

In this interaction we consider two cases, i.e. (I) the coulomb potential interact with dipole ⃗µj = ezjpj and (II) dipole ﬁeld interact with point charge. In the case of (I), the C-D interaction becomes

"#$%&'

*+,-.'()'&"

! i *+,-.'

! j

"#$%&'()'&"

FIG. 2: charge-dipole(C-D) and dipole-charge(D-C) interaction

" # 2 · p2 ≅ − zizje − 1 2rji pj 1 j − 3 · 2 2 2 2 1 1 + 2 + 2 (2rji pj/rji + pj /rji) (6) 4πϵ0rji 2 rji 2 rji 8

if we assume rji >> pj 2 · 2 · · −zizje rji pj zizje rij pj zie rij ⃗µj ≡ Vcd = 3 = 3 = 3 , ⃗µj zjepj (7) 4πϵ0 rji 4πϵ0 rij 4πϵ0 rij

In the ordinary method the potential is given by −⃗µ · EC − i −∇ | − | zie r ri EC(r) = φC( r ri ) = 3 (8) 4πϵ0 |r − ri| · · − · i − zie rji ⃗µj zie rij ⃗µj Vcd = ⃗µj EC(rj) = 3 = 3 (9) 4πϵ0 rji 4πϵ0 rij which gives the same results. In the case of (II), the D-C interaction becomes

2 2 (−zi)zje 1 zizje 1 Vdc = + (10) 4πϵ0 |rj − ri| 4πϵ0 |rj − ri − pi|   " # z z e2 1 1 z z e2 1 = − i j  − q  = − i j 1 − − · 2 2 2 1/2 4πϵ0 rji 2 − · 2 4πϵ0rji (1 2rji pi/rji + pi /rji) rji 2rji pi + pi " # 2 − · 2 ≅ − zizje − 1 2rji pi 1 pi − 3 − · 2 2 2 2 1 1 + 2 + 2 ( 2rji pi/rji + pi /rji) . if we assume rji >> pj 4πϵ0rji 2 rji 2 rji 8 2 · 2 · · zizje rji pi −zizje pi rij − 1 ⃗µi rijzje ≡ = 3 = 3 = 3 , ⃗µi ziepi (11) 4πϵ0 rji 4πϵ0 rij 4πϵ0 rij

Please note that C-D and D-C interaction the sign is diﬀerent. Since the interaction energy is given by zjeφ(|rj − ri|), the dipoler ﬁeld is given by · − − 1 ⃗µi (r ri) φD(r ri) = 3 (12) 4πϵ0 |r − ri|

0 . 4

0 . 2

0 . 4

1 0

FIG. 3: dipole ﬁeld

!"#$%&'(&%!

'!"#$%&'

j i

FIG. 4: dipole-dipole interaction

III. DIPOLE-DIPOLE(D-D) INTERACTION

The dipole-dipole(D-D) interaction can be obtained in the same way. Again we assume rij >> pi, pj.

(−z )(−z )e2 1 (−z )z e2 1 z (−z )e2 1 z z e2 1 V = i j + i j + i j + i j (13) dd 4πϵ |r − r | 4πϵ |r + p − r | 4πϵ |r − r − p | 4πϵ |r + p − r − p | 0 j i 0 j j i 0 j i i 0 j j i i  z z e2 1 1 1 1 = i j  − q − q + q  4πϵ0 rji 2 · 2 2 − · 2 2 − · · 2 2 − · rji + 2rji pj + pj rji 2rji pi + pi rji 2rji pi + 2rji pj + pi + pj 2pi pj " z z e2 1 1 = i j 1 − − 2 2 2 1/2 2 2 2 1/2 4πϵ0rji (1 + 2rji · pj/r + p /r ) (1 − 2rji · pi/r + p /r ) ji j ji ji i #ji 1 + 2 2 2 2 2 2 2 1/2 (1 − 2rji · pi/r + 2rji · pj/r + p /r + p /r − 2pi · pj/r ) " ji ji i ij j ji ji 2 · p2 zizje − 1 2rji pj 1 j − 3 · 2 2 2 2 = 1 1 + 2 + 2 (2rji pj/rji + pj /rji) 4πϵ0rji 2 rji 2 rji 8 1 −2r · p 1 p2 3 −1 + ji i + i − (−2r · p /r2 + p2/r2 )2 2 r2 2 r2 8 ji i ji i ji ji ji   1 −2r · p 1 2r · p 1 p2 1 p2 1 2p · p 3 −2r · p 2r · p p2 p2 2p · p  +1 − ji i − ji j − i − j + i j + ( ji i + ji j + i + j − i j )2 2 r2 2 r2 2 r2 2 r2 2 r2 8 r2 r2 r2 r2 r2  ji ji ji ji | {zji } | ji {z ji } ji ji ji " # survive cross term survive 2 · · · zizje pi pj − (rji pi)(rji pj) = 2 3 4 4πϵ0rji r r " ji ji # · · 1 · − (⃗µi rij)(rij ⃗µj) Vdd = 3 ⃗µi ⃗µj 3 2 (14) 4πϵ0rij rij 4

i If electric ﬁeld ED from i-th dipole can be calculated from φD given above, and the interaction energy can be obtained − · i by ⃗µj ED(rj). µ ¶ ∂ ∂ ∂ Ei = −∇φ (r − r ) = − i + j + k φ (r − r ) D D i ∂x ∂y ∂z D i 1 µ (x − x ) + µ (y − y ) + µ (z − z ) φ (r − r ) = ix i iy i iz i D i 2 2 2 3/2 4πϵ0 [(x − xi) + (y − yi) + (z − zi) ] 1 1 µ i −∇φ (r − r )| = − ix D i x 3 2 2 2 3/2 4πϵ0 |r − ri| [(x − xi) + (y − yi) + (z − zi) ] 1 ⃗µ · (r − r ) −3 − i i 2(x − x )i 4πϵ − 2 − 2 − 2 5/2 2 i 0 [(x xi)· + (y yi) + (z zi) ] ¸ 1 ⃗µ · (r − r )(r − r ) Ei (r) = − ⃗µ − 3 i i i (15) D 4πϵ |r − r |3 i |r − r |2 0 i " i # · − − · − · i 1 · − ⃗µi (ri rj)(ri rj) ⃗µj Vdd = ⃗µj Ed(rj) = 3 ⃗µi ⃗µj 3 2 (16) 4πϵ0rij rij

This equation is the same as the equation given above.

IV. UNIFIED DEFINITION OF C-C, C-D, D-C, AND D-D INTERACTIONS

The total electrostatic potential is given by [check sign→OK]     X  X X X  4πϵ V = q T q −q T αµ + µ T αq − µ T αβµ  (17) 0 tot |i {zij }j i ij j,α i,α ij j i,α ij j,β i>j  α α α,β  C−C | {z } | {z } | {z } C−D D−C D−D

qi = ezi, ⃗µi = (µix, µiy, µiz) = ezipi = ezi(pix, piy, piz) (18) Here the interaction tensors are given by [check this→OK] 1 Tij = (19) rij α ∇ − −3 Tij = αTij = rij,αrij (20) αβ ∇ ∇ − 2 −5 Tij = α βTij = (3rij,αrij,β rijδαβ)rij (21) αβγ ∇ ∇ ∇ − − 2 −7 Tij = α β γ Tij = [15rij,αrij,βrij,γ 3rij(rij,αδβγ + rij,βδγα + rij,γ δαβ)]rij (22)

∂ αβγ Here ∇α means for α = x, y, z. T will be used in the force formulation. ∂rij,α ij

V. POLARIZATION

When a strong electric ﬁeld is applied to an atom i, the electrons around atom i starts to deform and a dipole moment may be induced. j The dipole moment of atom i in the α direction may be written as the superposition of the electric ﬁeld EC generated j by the charge of atom j and that ED by the dipole of atom j stat ind ≅ ind µi,α = µi,α + µi,α µi,α (23)

static Usually we take µi,α = 0. The electric ﬁeld at atom i can be written as X j j E(ri) = [EC(ri) + ED(ri)] (24) j(≠ i)

+ +

d i p o l e

i n d u c e d

FIG. 5: What’s polarization?

( " #) X ind 1 r − r 1 ⃗µ · (ri − rj)(ri − rj) = z e i j − ⃗µind − 3 j (25) 4πϵ j |r − r |3 |r − r |3 j |r − r |2 0 ̸ i j i j i j j(=i)   X X 1 − α αβ ind Eα(ri) = Tij qj + Tij µj,β (26) 4πϵ0 j(≠ i) β

ind The induced dipole moment µi,α is given by

ind i µi,α = αα,βEβ(ri) (27)

i Here αα,β polarizability tensor of atom i. If we assume   αi 0 0 [αi] =  0 αi 0  (28) 0 0 αi then, we have

⃗µind = αiE(r ), µind = αiE (r ) (29) i i  i,α α i  i X X ind α − α αβ ind  µi,α (out) = Tij qj + Tij µj,β (input) (30) 4πϵ0 ̸ β j(=i)   i X X ind α − x xβ ind  µi,x (out) = Tijqj + Tij µj,β (input) (31) 4πϵ0 ̸ β j(=i)   i X X α xij 3xijrij,β − δxβ ind  = 3 qj + [ 5 3 ]µj,β (input) (32) 4πϵ0 r r r ̸ ij β ij ij j(=i)   i X X µind(input) α xij 3xij ind − j,x  = 3 qj + 5 rij,βµj,β (input) 3 (33) 4πϵ0 r r r j(≠ i) ij ij β ij

The last equations should be solved self-consistently. In the ﬁxed charge model the interaction of C-C is calculated by Ewald method. Then when we consider the induced dipoles at the atom sites we should C-D, D-C, and D-D interaction. The total C-C, C-D, D-C, D-D interaction is given by [check this]   X X X X  − α ind ind α − ind αβ ind 4πϵ0Vtot = qiTijqj qi Tij µj,α + µi,α Tij qj µi,α Tij µj,β (34) i>j α α α,β 6

From Böttcher’s book (Theory off Electric Polarization vol 1 Dielectrics in static field 2nd ed. p110), the energy U of the induced dipole system

− ind · U = ⃗µi E(ri) + Upol (35) where Upol is the work of polarization. At equilibrium, the energy will be minimal with an inﬁnitesimal change of induced moment

ind dU = 0, for all d⃗µi (36) Then we have ⃗µind 1 dU = −d[−⃗µind · E(r )] = E(r ) · d⃗µind = i · d⃗µind = d[(µind)2] (37) pol i i i i αi i 2αi i The induced dipole is formed in a reversible process

Z Z ind µi 1 ind 2 1 ind 2 Upol = dUpol = i d[(µi ) ] = i (µi ) (38) 2α 0 2α If we count all the cotribution X 1 U = ⃗µind · ⃗µind (39) pol 2αi i i i The MD code with this polarization scheme is available, e.g. Amber or Lucretius. 7

FIG. 6: long-range electrostatic interaction 8

FIG. 7: Ewald sum convergence