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

Masahiro YamamotoDepartment 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 eld andthis strong eld distort the electron cloud of molecule and induces dipole around atoms. To consider this polarizationeect we should consider charge-charge, charge-dipole, dipole-charge, dipole-dipole interaction.

I. CHARGE-CHARGE (C-C) INTERACTION

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

1 0

5

0

5

1 0

1 0

5

0

5

1 0

1

0 . 8

0 . 6

0 . 4

0 . 2

0

FIG. 1: Coulomb potential of negative charge

C(jr rij) = zie401

jr rij (1)

The electric eld is given by the gradient of the potential

EiC(r) = rC(jr rij) =zie

40r rijr rij3 (2)

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

Vcc = zjeC(jrj rij) = zizje2

401

jri rj j (3)

II. CHARGE-DIPOLE (C-D) INTERACTION

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

Vcd =zi(zj)e2

401

jrj rij +zizje

2

401

jrj + pj rij (4)

rj ri rji; jrjij = rji; (1 + x)1=2 ' 1 x=2 + 3x2=8::: (5)

Vcd = zizje2

40

0@ 1rji

1qr2ji + 2rji pj + p2j

1A = zizje240rji

"1 1

(1 + 2rji pj=r2ji + p2j=r2ji)1=2#

2

+

+

dipole eld

i

charge

j

charge eld

i

dipole

j

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

' zizje2

40rji

"1 1 + 1

22rji pj

r2ji+

12p2jr2ji

38(2rji pj=r2ji + p2j=r2ji)2

#(6)

if we assume rji >> pj

Vcd = zizje2

40rji pjr3ji

=zizje

2

40rij pjr3ij

=zie

40rij ~jr3ij

; ~j zjepj (7)

In the ordinary method the potential is given by ~ EC

EiC(r) = rC(jr rij) =zie

40r rijr rij3 (8)

Vcd = ~j EiC(rj) = zie

40rji ~jr3ji

=zie

40rij ~jr3ij

(9)

which gives the same results.In the case of (II), the D-C interaction becomes

Vdc =(zi)zje2

401

jrj rij +zizje

2

401

jrj ri pij (10)

= zizje2

40

0@ 1rji

1qr2ji 2rji pi + p2i

1A = zizje240rji

"1 1

(1 2rji pi=r2ji + p2i =r2ji)1=2#

' zizje2

40rji

"1 1 + 1

22rji pi

r2ji+

12p2ir2ji

38(2rji pi=r2ji + p2i =r2ji)2

#: if we assume rji >> pj

=zizje

2

40rji pir3ji

= zizje2

40pi rijr3ij

= 140

~i rijzjer3ij

; ~i ziepi (11)

Please note that C-D and D-C interaction the sign is dierent. Since the interaction energy is given by zje(jrjrij),the dipoler eld is given by

D(r ri) = 140~i (r ri)jr rij3 (12)

3

1 0

5

0

5

1 0

1 0

5

0

5

1 0

0 . 4

0 . 2

0

0 . 2

0 . 4

FIG. 3: dipole eld

dipole eld

dipole

i

j

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 .

Vdd =(zi)(zj)e2

401

jrj rij +(zi)zje2

401

jrj + pj rij +zi(zj)e2

401

jrj ri pij +zizje

2

401

jrj + pj ri pij (13)

=zizje

2

40

0@ 1rji

1qr2ji + 2rji pj + p2j

1qr2ji 2rji pi + p2i

+1q

r2ji 2rji pi + 2rji pj + p2i + p2j 2pi pj

1A=

zizje2

40rji

"1 1

(1 + 2rji pj=r2ji + p2j=r2ji)1=2 1

(1 2rji pi=r2ji + p2i =r2ji)1=2

+1

(1 2rji pi=r2ji + 2rji pj=r2ji + p2i =r2ij + p2j=r2ji 2pi pj=r2ji)1=2#

=zizje

2

40rji

"1 1 + 1

22rji pj

r2ji+

12p2jr2ji

38(2rji pj=r2ji + p2j=r2ji)2

1 + 122rji pi

r2ji+

12p2ir2ji

38(2rji pi=r2ji + p2i =r2ji)2

+1 122rji pi

r2ji 1

22rji pj

r2ji 1

2p2ir2ji

12p2jr2ji

+122pi pj

r2ji| {z }survive

+38(2rji pi

r2ji+

2rji pjr2ji| {z }

cross term survive

+p2ir2ji

+p2jr2ji

2pi pjr2ji

)2

37775=

zizje2

40rji

"pi pjr2ji

3(rji pi)(rji pj)r4ji

#

Vdd =1

40r3ij

"~i ~j 3(~i rij)(rij ~j)

r2ij

#(14)

4If electric eld EiD from i-th dipole can be calculated from D given above, and the interaction energy can be obtainedby ~j EiD(rj).

EiD = rD(r ri) = i@

@x+ j

@

@y+ k

@

@z

D(r ri)

D(r ri) = 140ix(x xi) + iy(y yi) + iz(z zi)[(x xi)2 + (y yi)2 + (z zi)2]3=2

rD(r ri)jx = 1401

jr rij3ixi

[(x xi)2 + (y yi)2 + (z zi)2]3=2

140

~i (r ri)[(x xi)2 + (y yi)2 + (z zi)2]5=2

322(x xi)i

EiD(r) = 1

40jr rij3~i 3~i (r ri)(r ri)jr rij2

(15)

Vdd = ~j Eid(rj) =1

40r3ij

"~i ~j 3~i (ri rj)(ri rj) ~j

r2ij

#(16)

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]

40Vtot =Xi>j

0BBBB@qiTijqj| {z }CC

qiX

Tijj;| {z }CD

+X

i;Tijqj| {z }

DC

X;

i;Tij j;| {z }

DD

1CCCCA (17)qi = ezi; ~i = (ix; iy; iz) = ezipi = ezi(pix; piy; piz) (18)

Here the interaction tensors are given by [check this!OK]

Tij =1rij

(19)

Tij = rTij = rij;r3ij (20)Tij = rrTij = (3rij;rij; r2ij)r5ij (21)Tij = rrrTij = [15rij;rij;rij; 3r2ij(rij; + rij; + rij;)]r7ij (22)

Here r means @@rij; for = x; y; z. Tij will be used in the force formulation.

V. POLARIZATION

When a strong electric eld is applied to an atom i, the electrons around atom i starts to deform and a dipolemoment may be induced.The dipole moment of atom i in the direction may be written as the superposition of the electric eld EjC generated

by the charge of atom j and that EjD by the dipole of atom j

i; = stati; + indi; ' indi; (23)

Usually we take statici; = 0. The electric eld at atom i can be written as

E(ri) =Xj(6=i)

[EjC(ri) +EjD(ri)] (24)

5

+

+

+

+

+

+

d i p o l e

i n d u c e d

+ +

FIG. 5: What's polarization?

=1

40

Xj(6=i)

(zje

ri rjjri rj j3

1jri rj j3

"~indj 3

~indj (ri rj)(ri rj)jri rj j2

#)(25)

E(ri) =1

40

Xj(6=i)

0@Tijqj +X

Tij indj;

1A (26)The induced dipole moment indi; is given by

indi; = i;E(ri) (27)

Here i; polarizability tensor of atom i. If we assume

[i] =

0@ i 0 00 i 00 0 i

1A (28)then, we have

~indi = iE(ri); indi; =

iE(ri) (29)

indi; (out) =i

40

Xj( 6=i)

0@Tijqj +X

Tij indj; (input)

1A (30)indi;x (out) =

i

40

Xj( 6=i)

0@T xijqj +X

T xij indj; (input)

1A (31)=

i

40

Xj( 6=i)

0@xijr3ij

qj +X

[3xijrij;

r5ij x

r3ij]indj; (input)

1A (32)=

i

40

Xj( 6=i)

0@xijr3ij

qj +3xijr5ij

X

rij;indj; (input)

indj;x (input)r3ij

1A (33)The last equations should be solved self-consistently.In the xed 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 isgiven by [check this]

40Vtot =Xi>j

0@qiTijqj qiX

Tijindj; +

X

indi;Tijqj

X;

indi;Tij

indj;

1A (34)

6From Bottcher's book (Theory o Electric Polarization vol 1 Dielectrics in static eld 2nd ed. p110), the energy Uof the induced dipole system

U = ~indi E(ri) + Upol (35)where Upol is the work of polarization. At equilibrium, the energy will be minimal with an innitesimal change ofinduced moment

dU = 0; for all d~indi (36)

Then we have

dUpol = d[~indi E(ri)] = E(ri) d~indi =~indii

d~indi =12i

d[(indi )2] (37)

The induced dipole is formed in a reversible process

Upol =Z

dUpol =12i

Z iind0

d[(indi )2] =

12i

(indi )2 (38)

If we count all the cotribution

Upol =Xi

12i

~indi ~indi (39)

The MD code with this polarization scheme is available, e.g. Amber or Lucretius.

7FIG. 6: long-range electrostatic interaction

8FIG. 7: Ewald sum convergence