· Karplus dependence of spin-spin coupling constants revisited theoretically. Part 1: Second-order double perturbation theory Irina L. Rusakova and Leonid B. Krivdin * Received

Page 1

Karplus dependence of spin-spin coupling constants

revisited theoretically. Part 1: Second-order double

perturbation theory

Irina L. Rusakova and Leonid B. Krivdin*

Received (in XXX, XXX) Xth XXXXXXXXX 20XX, Accepted Xth XXXXXXXXX 20XX

DOI: 10.1039/b000000x

Supplementary Information

Derivation of Eq. (64).

The general resulting equation for the FC contribution to vicinal spin-spin coupling constant in

MO LCAO approximation reads as follows:

⋅−

= ∑ ∑ ∑∑CDAB

mlnmln

mlnmln

Dmlin

Cmlan

Bmlan

Amlin

ia iaI

FCHH

CCCCHJ

222

111

444

333

44433322211121

1

3

16][

20

εεπβ

)()()()(2444233312221111

DH

D

mlnCH

C

mlnBH

B

mlnAH

A

mlnrrrr ϕϕϕϕ⋅ (1)

Here, A, B, C, and D run over all atoms =Q {1H ,

2H ,

1C ,

2C }. Indexes ...,,, cba and ...,,, kji are

assigned to unoccupied and occupied molecular orbitals, respectively; )(2,1

QH

Q

mlniii

rϕ is the

spatial atomic orbital localized on atom Q characterized with a set of three quantum numbers

{iiimln ,, }: principal, azimuthal and magnetic;

iaεε − appearing in Eq. (1) denotes the energy

differences between Hartree-Fock unoccupied and occupied molecular orbitals; 2,1

QHr is a short

notation of the vector difference: QHQHrrr −=

2,12,1

. Based on Eq. (1), it is easy to evaluate the

number of atomic sums to be equal to 4

4 , which thus can be separated into 16 general types:

∑ ∑∑−

=

222

111

444

33321

1

3

16][

20

mln

mln

mln

mlnia iaI

FC

HHHJ

εεπβ

+)()()()( 2

444

2

333

1

222

1

111

2

444

2

333

1

222

1

111

0000H

mln

H

mln

H

mln

H

mln

H

mlin

H

mlan

H

mlan

H

mlinCCCC ϕϕϕϕ

∑≠

++

22444

2

333

1

222

1

111444

2

333

1

222

1

111

)()()()(HD

DH

D

mln

H

mln

H

mln

H

mln

D

mlin

H

mlan

H

mlan

H

mlinCCCC r000 ϕϕϕϕ

Electronic Supplementary Material (ESI) for Physical Chemistry Chemical PhysicsThis journal is © The Owner Societies 2013

Page 2

∑≠

++

2

2

4442333

1

222

1

111

2

444333

1

222

1

111

)()()()(HC

H

mlnCH

C

mln

H

mln

H

mln

H

mlin

C

mlan

H

mlan

H

mlinCCCC 0r00 ϕϕϕϕ

∑≠≠

++

2

224442333

1

222

1

111444333

1

222

1

111

)()()()(

HD

HCDH

D

mlnCH

C

mln

H

mln

H

mln

D

mlin

C

mlan

H

mlan

H

mlinCCCC rr00 ϕϕϕϕ

∑≠

++

1

2

444

2

3331222

1

111

2

444

2

333222

1

111

)()()()(HB

H

mln

H

mlnBH

B

mln

H

mln

H

mlin

H

mlan

B

mlan

H

mlinCCCC 00r0 ϕϕϕϕ

∑≠≠

++

2

12444

2

3331222

1

111444

2

333222

1

111

)()()()(

HD

HBDH

D

mln

H

mlnBH

B

mln

H

mln

D

mlin

H

mlan

B

mlan

H

mlinCCCC r0r0 ϕϕϕϕ

∑≠≠

++

2

1

2

44423331222

1

111

2

444333222

1

111

)()()()(

HC

HB

H

mlnCH

C

mlnBH

B

mln

H

mln

H

mlin

C

mlan

B

mlan

H

mlinCCCC 0rr0 ϕϕϕϕ

∑

≠≠≠

++

2

2

1244423331222

1

111444333222

1

111

)()()()(

HD

HC

HBDH

D

mlnCH

C

mlnBH

B

mln

H

mln

D

mlin

C

mlan

B

mlan

H

mlinCCCC rrr0 ϕϕϕϕ

∑≠

++

1

2

444

2

333

1

2221111

2

444

2

333

1

222111

)()()()(HA

H

mln

H

mln

H

mlnAH

A

mln

H

mlin

H

mlan

H

mlan

A

mlinCCCC 000r ϕϕϕϕ

∑≠≠

++

2

12444

2

333

1

2221111444

2

333

1

222111

)()()()(

HD

HADH

D

mln

H

mln

H

mlnAH

A

mln

D

mlin

H

mlan

H

mlan

A

mlinCCCC r00r ϕϕϕϕ

∑≠≠

++

2

1

2

4442333

1

2221111

2

444333

1

222111

)()()()(

HC

HA

H

mlnCH

C

mln

H

mlnAH

A

mln

H

mlin

C

mlan

H

mlan

A

mlinCCCC 0r0r ϕϕϕϕ

∑

≠≠≠

++

2

2

124442333

1

2221111444333

1

222111

)()()()(

HD

HC

HADH

D

mlnCH

C

mln

H

mlnAH

A

mln

D

mlin

C

mlan

H

mlan

A

mlinCCCC rr0r ϕϕϕϕ

∑≠≠

++

1

1

2

444

2

33312221111

2

444

2

333222111

)()()()(

HB

HA

H

mln

H

mlnBH

B

mlnAH

A

mln

H

mlin

H

mlan

B

mlan

A

mlinCCCC 00rr ϕϕϕϕ

∑

≠≠≠

++

2

1

12444

2

33312221111444

2

333222111

)()()()(

HD

HB

HADH

D

mln

H

mlnBH

B

mlnAH

A

mln

D

mlin

H

mlan

B

mlan

A

mlinCCCC r0rr ϕϕϕϕ

∑

≠≠≠

++

2

1

1

2

444233312221111

2

444333222111

)()()()(

HC

HB

HA

H

mlnCH

C

mlnBH

B

mlnAH

A

mln

H

mlin

C

mlan

B

mlan

A

mlinCCCC 0rrr ϕϕϕϕ

+ ∑ ∑≠≠

≠≠

)()()()(2444233312221111444333222111

1

1

2

2

DH

D

mlnCH

C

mlnBH

B

mlnAH

A

mln

D

mlin

C

mlan

B

mlan

A

mlin

HB

HA

HD

HC

CCCC rrrr ϕϕϕϕ

(2)

Electronic Supplementary Material (ESI) for Physical Chemistry Chemical PhysicsThis journal is © The Owner Societies 2013

Page 3

Taking into account only the s-type functions on hydrogen atoms, one can eliminate the

summations over the quantum numbers {iiimln ,, } in the terms containing )(2,1 0

H

mlniii

ϕ and replace

)(2,1 0H

mlniii

ϕ by

00

100

1)(2,1

aa

H

πϕ =0 . In that way, all sums in Eq. (2) can be combined into nine

groups:

+⋅−

= ∑ 2211

211001001001006

0

2

20 11

3

16][

H

i

H

a

H

a

H

iia ia

I

FC

HHCCCC

aHJ

πεεπβ

∑∑≠

+

↔++

11

122 )(1

100100100

0

4

0HA

AH

A

nlm

A

anlm

H

i

H

i

H

anlm

aiCCCCaa

rϕππ

∑∑≠

+

↔++

22

211 )(1

100100100

0

4

0HA

AH

A

nlm

A

anlm

H

i

H

a

H

inlm

aiCCCCaa

rϕππ

∑∑≠≠

++

2

222222111222111

11

222

111

)()(1

1001003

0HB

HABH

B

mlnAH

A

mln

B

mlin

A

mlan

H

a

H

i

mln

mln

CCCCa

rr ϕϕπ

∑∑≠≠

++

1

112221111222111

22

222

111

)()(1

1001003

0HB

HABH

B

mlnAH

A

mln

B

mlan

A

mlin

H

i

H

a

mln

mln

CCCCa

rr ϕϕπ

)()(1

22221111

2

1222

2

111

1

222

111

1001003

0

BH

B

mlnAH

A

mln

HB

HA

B

mlin

H

a

A

mlan

H

i

mln

mln

aiCCaiCCa

rr ϕϕπ

∑∑≠≠

↔+⋅

↔++

∑∑

≠≠≠

+

↔++

2

2

1233322221111111

1

333222

333

222

111

)()()(1

100

00

HC

HB

HACH

C

mlnBH

B

mlnAH

A

mln

A

mlan

H

i

C

mlin

B

mlan

mln

mln

mln

aiCCCCaa

rrr ϕϕϕπ

∑∑

≠≠≠

+

↔++

2

1

1233312221111333

2

222111

333

222

111

)()()(1

100

00

HC

HB

HACH

C

mlnBH

B

mlnAH

A

mln

C

mlin

H

a

B

mlan

A

mlin

mln

mln

mln

aiCCCCaa

rrr ϕϕϕπ

∑ ∑∑≠≠

≠≠

+

1

12444233312221111444333222111

2

2

444

333

222

111

)()()()(

HB

HADH

D

mlnCH

C

mlnBH

B

mlnAH

A

mln

D

mlin

C

mlan

B

mlan

A

mlin

HD

HC

mln

mln

mln

mln

CCCC rrrr ϕϕϕϕ

(3)

Herewith, the Cartesian left-handed coordinate system is chosen, so that C1C2 bond lies on Z

axis and C1H1 bond lies on Y axis. Provided that bond angles ∠ HCC equal to 2/π , the C2H2

Electronic Supplementary Material (ESI) for Physical Chemistry Chemical PhysicsThis journal is © The Owner Societies 2013

Page 4

bond lies in the plane parallel to XY plane and revolves around Z axis with the azimuth φ which

in that case equals to the dihedral angle ∠ H1C1C2H2.

Further, the short notation )(CH

C

nlmrϕ denoting the part of )(

CH

C

nlmrϕ which is not depend onφ ,

namely,

⋅=φφ

ϕϕcos

sin)()(

CH

C

nlmCH

C

nlmrr , is used throughout.

A general form of )(2,1

AH

A

nlmrr −ϕ with a factorized φ -dependence is as follows:

22

021

0

21

2,1

)cos1(21

00

1

00

100

11|)(|

CCCHHHLL

aa

HH

He

aae

aa

+−−−−

⋅=⋅=−φ

ππϕ

rr

rr (4)

)()()()(22

2

22

2

11

1

11

1

200200200200 HC

C

HC

C

HC

C

HC

Crrrr ϕϕϕϕ === (5)

)()(11

1

11

1

121121 HC

C

HC

Crr

−−=ϕϕ (6)

)()()()(12

2

12

2

21

1

21

1

200200200200 HC

C

HC

C

HC

C

HC

Crrrr ϕϕϕϕ === (7)

)()()()(12

2

12

2

21

1

21

1

210210210210 HC

C

HC

C

HC

C

HC

Crrrr ϕϕϕϕ −=−== (8)

⋅=±± φ

φϕϕ

cos

sin)()(

21

1

21

1

121121 HC

C

HC

Crr (9)

)()(12

2

12

2

121121 HC

C

HC

Crr

−−=ϕϕ (10)

⋅=±± φ

φϕϕ

cos

sin)()(

22

2

22

2

121121 HC

C

HC

Crr (11)

0)()()()(22

2

12

2

11

1

11

1210121121210 ==== ++ HCC

HCC

HCC

HCC

rrrr ϕϕϕϕ (12)

Further we consider each group in (3) using a set of Eqs. (4)-(12). If )()(CH

C

nlmCH

C

nlmrr ϕϕ = ,

then notation )(CH

C

nlmrϕ is used for )(

CH

C

nlmrϕ .

1st sum

21

2211

21,

2

3

0

1001001001006

0

2

20)1(

3

81

3

16][

HHSS

ia ia

H

i

H

a

H

a

H

i

I

FC

HH a

CCCC

aAHJ π

βεεπ

πβ

−=

−

== ∑ (13)

Here µν

π is the mutual polarizability of atomic orbitals:

Electronic Supplementary Material (ESI) for Physical Chemistry Chemical PhysicsThis journal is © The Owner Societies 2013

Page 5

∑−

−=ia ia

aaiiCCCC

εεπ νµνµ

µν4 (14)

2nd sum

∑∑∑≠

=

↔+⋅

−

=

11

122 )(1

3

161][

100100100

2

0

4

0

0)2(

'HA

AH

A

nlm

A

anlm

H

i

H

i

H

anlmia ia

I

FC

NNaiCCCC

aaHJ rϕ

εεπβ

ππ

∑∑ +

↔+

−=

nlmHH

H

nlm

H

anlm

H

i

H

i

H

aia ia

aiCCCCaa

)(1

9

256

12

22122

100100100

0

4

0

2

rϕεεπ

πβ

=

↔++

↔++ )()(

12

22122

11

11122

100100100100100100 HC

C

nlm

C

anlm

H

i

H

i

H

aHC

C

nlm

C

anlm

H

i

H

i

H

aaiCCCCaiCCCC rr ϕϕ

+

↔+

−=

+−−

∑22

0

12

22122

)cos1(21

100100100100100

0

4

0

2

)(1

9

256 CCCHLL

a

HH

HH

a

H

i

H

i

H

aia ia

eaiCCCCaa

φ

ϕεεπ

πβr

+

↔++

↔++

−−)()(

11

11122

11

11122

121121100100100200200100100100 HC

CC

a

H

i

H

i

H

aHC

CC

a

H

i

H

i

H

aaiCCCCaiCCCC rr ϕϕ

+

↔++

↔++ )()(

12

22122

12

22122

210210100100100200200100100100 HC

CC

a

H

i

H

i

H

aHC

CC

a

H

i

H

i

H

aaiCCCCaiCCCC rr ϕϕ

↔++

−−)(

12

22122

121121100100100 HC

CC

a

H

i

H

i

H

aaiCCCC rϕ (15)

Eq. (15) takes a general form:

22

0

21

)cos1(21

0)2(][

CCCHLL

a

I

FC

HHeBAHJ

+−−

+=φ

(16)

where coefficients A and B are as follows:

+

⋅

↔+

−

=

−

−∑)(

)(

,1

3

16

11

1

11

1

11122

121

200

121200100100100

2

2

00 HC

C

HC

C

CC

a

H

i

H

i

H

aia ia

aiCCCCCaa

Ar

r

ϕ

ϕ

εεβπ

⋅

↔+

+

−

−

)(

)(

)(

,,

12

2

12

2

12

2

222122

121

210

200

121210200100100100

HC

C

HC

C

HC

C

C

a

C

a

C

a

H

i

H

i

H

aaiCCCCCC

r

r

r

ϕ

ϕ

ϕ

(17)

∑−

↔+

=

ia ia

H

a

H

i

H

i

H

aaiCCCC

aB

εεβ

2122

100100100100

2

3

03

16 (18)

Electronic Supplementary Material (ESI) for Physical Chemistry Chemical PhysicsThis journal is © The Owner Societies 2013

Page 6

3rd sum

∑∑∑≠

=

↔+⋅

−

=

22

211

21

)(1

3

161][

100100100

2

0

4

0

0)3(

HAAH

A

nlm

A

anlm

H

i

H

a

H

inlmia ia

I

FC

HHaiCCCC

aaHJ rϕ

εεπβ

ππ

∑∑ +

↔+⋅

−=

nlmHH

H

nlm

H

anlm

H

i

H

a

H

iia ia

aiCCCCaa

)(1

9

256

21

11211

100100100

0

4

0

2

rϕεεπ

πβ

=

↔++

↔++ )()(

22

22211

21

11211

100100100100100100 HC

C

nlm

C

anlm

H

i

H

a

H

iHC

C

nlm

C

anlm

H

i

H

a

H

iaiCCCCaiCCCC rr ϕϕ

+

↔+⋅

−=

+−−

∑22

0

21

11211

)cos1(21

100100100100100

0

4

0

2

)(1

9

256 CCCHLL

a

HH

HH

a

H

i

H

a

H

iia ia

aiCCCCaa

φ

ϕεεπ

πβr

+

↔++

↔++ )()(

21

11211

21

11211

210210100100100200200100100100 HC

CC

a

H

i

H

a

H

iHC

CC

a

H

i

H

a

H

iaiCCCCaiCCCC rr ϕϕ

+

↔++

++φϕ sin)(

21

11211

121121100100100 HC

CC

a

H

i

H

a

H

iaiCCCC r

+

↔++

−−φϕ cos)(

21

11211

121121100100100 HC

CC

a

H

i

H

a

H

iaiCCCC r

+

↔++ )(

22

22211

200200100100100 HC

CC

a

H

i

H

a

H

iaiCCCC rϕ

+

↔++

++φϕ sin)(

22

22211

121121100100100 HC

CC

a

H

i

H

a

H

iaiCCCC r

↔++

−−φϕ cos)(

22

22211

121121100100100 HC

CC

a

H

i

H

a

H

iaiCCCC r (19)

Eq. (19) takes a general form:

22

0

21

)cos1(21

0)3(cossin][

CCCHLL

a

I

FC

HHeDCBAHJ

+−−

+++=φ

φφ (20)

The coefficients A, B, C and D are as follows:

+

⋅

↔+

−

= ∑

)(

)(

,1

3

16

21

1

21

1

11211

210

200

210200100100100

2

2

00 HC

C

HC

C

C

a

C

a

H

i

H

a

H

iia ia

aiCCCCCaa

Ar

r

ϕ

ϕ

εεβπ

↔++ )(

22

222

200200100 HC

CC

a

H

iaiCC rϕ (21)

+

↔+

−

=

±±∑ )(

1

3

16

21

11211

121121100100100

2

2

00HC

CC

a

H

i

H

a

H

iia ia

aiCCCCaaC

Brϕ

εεβπ

↔++

±±)(

22

22211

121121100100100 HC

CC

a

H

i

H

a

H

iaiCCCC rϕ (22)

Electronic Supplementary Material (ESI) for Physical Chemistry Chemical PhysicsThis journal is © The Owner Societies 2013

Page 7

∑−

↔+

=

ia ia

H

a

H

i

H

a

H

iaiCCCC

aD

εεβ

1211

100100100100

2

3

03

16 (23)

4th sum

∑∑∑≠≠

=⋅−

=

2

222222111222111

11

222

11121

)()(1

3

161][

100100

2

3

0

0)4(

HB

HABH

B

mlnAH

A

mln

B

mlin

A

mlan

H

a

H

i

mln

mlnia iaI

FC

HHCCCC

aHJ rr ϕϕ

εεπβ

π

+

⋅

−

= ∑∑ )()(

1

3

16

21

1

22221

1

111

1

222

1

111

11

222

111

100100

2

3

0

HH

H

mlnHH

H

mln

H

mlin

H

mlan

H

a

H

i

mln

mlnia ia

CCCCa

rr ϕϕεε

βπ

++ ∑ )()(21

1

22221

1

111

1

222

1

111

11

222

111

100100 HC

C

mlnHH

H

mln

C

mlin

H

mlan

H

a

H

i

mln

mln

CCCC rr ϕϕ

++ ∑ )()(22

2

22221

1

111

2

222

1

111

11

222

111

100100 HC

C

mlnHH

H

mln

C

mlin

H

mlan

H

a

H

i

mln

mln

CCCC rr ϕϕ

++ ∑ )()(21

1

22221

1

111

1

222

1

111

11

222

111

100100 HH

H

mlnHC

C

mln

H

mlin

C

mlan

H

a

H

i

mln

mln

CCCC rr ϕϕ

++ ∑ )()(21

1

22221

1

111

1

222

1

111

11

222

111

100100 HC

C

mlnHC

C

mln

C

mlin

C

mlan

H

a

H

i

mln

mln

CCCC rr ϕϕ

++ ∑ )()(22

2

22221

1

111

2

222

1

111

11

222

111

100100 HC

C

mlnHC

C

mln

C

mlin

C

mlan

H

a

H

i

mln

mln

CCCC rr ϕϕ

++ ∑ )()(21

1

22222

2

111

1

222

2

111

11

222

111

100100 HH

H

mlnHC

C

mln

H

mlin

C

mlan

H

a

H

i

mln

mln

CCCC rr ϕϕ

++ ∑ )()(21

1

22222

2

111

1

222

2

111

11

222

111

100100 HC

C

mlnHC

C

mln

C

mlin

C

mlan

H

a

H

i

mln

mln

CCCC rr ϕϕ

+ ∑ )()(

22

2

22222

2

111

2

222

2

111

11

222

111

100100 HC

C

mlnHC

C

mln

C

mlin

C

mlan

H

a

H

i

mln

mln

CCCC rr ϕϕ (24)

+

⋅

−

=

+−−

∑22

01111

21

)cos1(22

1001001001003

0

2

3

0

0)4( 11

3

16][

CCCHLL

aH

i

H

a

H

a

H

iia ia

I

FC

HHeCCCC

aaHJ

φ

πεεβπ

+ +

++−−

)()(1

21

11

21

11111

22

0

210210200200100100100

)cos1(21

00

HC

CC

iHC

CC

i

H

a

H

a

H

i

LLa

CCCCCeaa

CCCH

rr ϕϕπ

φ

Electronic Supplementary Material (ESI) for Physical Chemistry Chemical PhysicsThis journal is © The Owner Societies 2013

Page 8

+↔+

++

−−++aiCC

HC

CC

iHC

CC

iφϕφϕ cos)(sin)(

21

11

21

11

121121121121rr

+

++−−

)(1

22

22111

22

0

200200100100100

)cos1(21

00

HC

CC

i

H

a

H

a

H

i

LLa

CCCCeaa

CCCH

rϕπ

φ

+↔+

++

−−++aiCC

HC

CC

iHC

CC

iφϕφϕ cos)(sin)(

22

22

22

22

121121121121rr

+

⊗

⋅

⊗

+

)(

)(

)(

)(

,,

21

1

21

1

21

1

21

1

111111

210

200

210

200

210200210200100100

HC

C

HC

C

HC

C

HC

C

C

i

C

i

C

a

C

a

H

a

H

iCCCCCC

r

r

r

r

ϕ

ϕ

ϕ

ϕ

+⋅

⋅

↔+

+

++φϕ

ϕ

ϕsin)(

)(

)(

,21

1

21

1

21

1

11111

121

210

200

121210200100100 HC

C

HC

C

HC

C

C

i

C

a

C

a

H

a

H

iaiCCCCC r

r

r

+⋅

⋅

↔+

+

−−φϕ

ϕ

ϕcos)(

)(

)(

,21

1

21

1

21

1

11111

121

210

200

121210200100100 HC

C

HC

C

HC

C

C

i

C

a

C

a

H

a

H

iaiCCCCC r

r

r

+

⋅

↔++

±+−φϕ 2sin)(

2

12

12112112110010021

11111

HC

CC

i

C

a

H

a

H

iaiCCCC r

( )+−

⋅+

+++φϕ 2cos1)(

2

12

12112112110010021

11111

HC

CC

i

C

a

H

a

H

iCCCC r

( )++

⋅+

−−−φϕ 2cos1)(

2

12

12112112110010021

11111

HC

CC

i

C

a

H

a

H

iCCCC r

+⋅

⋅

↔+

+ )(

)(

)(

,22

2

21

1

21

1

21111

200

210

200

200210200100100 HC

C

HC

C

HC

C

C

i

C

a

C

a

H

a

H

iaiCCCCC r

r

r

ϕϕ

ϕ

+⋅

⋅

↔+

+

++φϕ

ϕ

ϕsin)(

)(

)(

,22

2

21

1

21

1

21111

121

210

200

121210200100100 HC

C

HC

C

HC

C

C

i

C

a

C

a

H

a

H

iaiCCCCC r

r

r

+⋅⋅

↔++

++φϕϕ sin)()(

21

1

22

22111

121200200121100100 HC

C

HC

CC

i

C

a

H

a

H

iaiCCCC rr

+⋅

⋅

↔+

+

−−φϕ

ϕ

ϕcos)(

)(

)(

,22

2

21

1

21

1

21111

121

210

200

121210200100100 HC

C

HC

C

HC

C

C

i

C

a

C

a

H

a

H

iaiCCCCC r

r

r

+⋅⋅

↔++

−−φϕϕ cos)()(

21

1

22

22111

121200200121100100 HC

C

HX

CC

i

C

a

H

a

H

iaiCCCC rr

+⋅

↔+++

±±−++−φϕϕ 2sin)()(

2

1

22

2

21

1212111

121121121121121121100100 HC

C

HC

CC

i

C

a

C

i

C

a

H

a

H

iaiCCCCCC rr

+−⋅

↔++

++++)2cos1)(()(

2

1

22

2

21

12111

121121121121100100φϕϕ

HC

C

HC

CC

i

C

a

H

a

H

iaiCCCC rr

Electronic Supplementary Material (ESI) for Physical Chemistry Chemical PhysicsThis journal is © The Owner Societies 2013

Page 9

++⋅

↔++

−−−−)2cos1)(()(

2

1

22

2

21

12111

121121121121100100φϕϕ

HC

C

HC

CC

i

C

a

H

a

H

iaiCCCC rr

++ )()(22

2

22

22211

200200200200100100 HC

C

HC

CC

i

C

a

H

a

H

iCCCC rr ϕϕ

+

↔++

++φϕϕ sin)()(

22

2

22

22211

121200121200100100 HC

C

HC

CC

i

C

a

H

a

H

iaiCCCC rr

+

↔++

−−φϕϕ cos)()(

22

2

22

22211

121200121200100100 HC

C

HC

CC

i

C

a

H

a

H

iaiCCCC rr

+

↔++

−+−+φϕϕ 2sin)()(

2

1

22

2

22

22211

121121121121100100 HC

C

HC

CC

i

C

a

H

a

H

iaiCCCC rr

+−+++++

)2cos1)(()(2

1

22

2

22

22211

121121121121100100φϕϕ

HC

C

HC

CC

i

C

a

H

a

H

iCCCC rr

++

−−−−)2cos1)(()(

2

1

22

2

22

22211

121121121121100100φϕϕ

HC

C

HC

CC

i

C

a

H

a

H

iCCCC rr (25)

The ba⊗ arising in Eq. (25) denotes the Cartesian product of the two sets N×1a and

M×1b , defined

as: ( )MNNNMMMNbababababababababa ...,,,...;;...,,,;...,,,

21222121211111=⊗

××ba for the row vector

and T

M

T

N

T

MN

⊗=⊗

×××× 1111baba for the column vector.

Thus Eq. (25) takes a general form:

++++++=+−− 22

0

21

)cos1(21

0)4()cos'sin''(cossin][

CCCHLL

a

I

FC

HHeCBACBAHJ

φ

φφφφ

22

0

)cos1(22

CCCHLL

aEe

+−−

+φ

(26)

where the first coefficient reads as follows:

⋅

⊗

⋅

−

= ∑ 1111

11

210200210200

1001002

3

0

,,3

16 C

i

C

i

C

a

C

aia ia

H

a

H

i CCCCCC

aA

εεβπ

⋅+

⋅+

⊗

⋅−−+++

11

21

111

21

1

21

1

21

1

21

1

121121

2

121121121

210

200

210

200

2

1)(

2

1

)(

)(

)(

)(C

i

C

aHC

CC

i

C

a

HC

C

HC

C

HC

C

HC

C

CCCC rr

r

r

r

ϕϕ

ϕ

ϕ

ϕ

+⋅

⋅

↔+

+

⋅

−)(

)(

)(

,)(22

2

21

1

21

1

211

21

1

200

210

200

200210200

2

121 HC

C

HC

C

HC

C

C

i

C

a

C

aHC

CaiCCC r

r

r

r ϕϕ

ϕϕ

+⋅

↔++

++++)()(

2

1

22

2

21

121

121121121121 HC

C

HC

CC

i

C

aaiCC rr ϕϕ

+⋅

↔++

−−−−)()(

2

1

22

2

21

121

121121121121 HC

C

HC

CC

i

C

aaiCC rr ϕϕ

++ )()(22

2

22

222

200200200200 HC

C

HC

CC

i

C

aCC rr ϕϕ

Electronic Supplementary Material (ESI) for Physical Chemistry Chemical PhysicsThis journal is © The Owner Societies 2013

Page 10

++

−−−−++++)()(

2

1)()(

2

1

22

2

22

222

22

2

22

222

121121121121121121121121 HC

C

HC

CC

i

C

aHC

C

HC

CC

i

C

aCCCC rrrr ϕϕϕϕ (27)

Taking into account that )()()(21

1

21

1

21

1

121121121 HC

C

HC

C

HC

Crrr

±−+== ϕϕϕ ,

+

⊗

⋅

⊗

⋅

−

= ∑

)(

)(

)(

)(

,,3

16

21

1

21

1

21

1

21

1

1111

11

210

200

210

200

210200210200

1001002

3

0 HC

C

HC

C

HC

C

HC

C

C

i

C

i

C

a

C

aia ia

H

a

H

i CCCCCC

aA

r

r

r

r

ϕ

ϕ

ϕ

ϕ

εεβπ

+

⋅

++

±−−++

2

121121121121121)(

2

1

21

11111

HX

CC

i

C

a

C

i

C

aCCCC rϕ

+⋅

⋅

↔+

+ )(

)(

)(

,22

2

21

1

21

1

211

200

210

200

200210200 HC

C

HC

C

HC

C

C

i

C

a

C

aaiCCC r

r

r

ϕϕ

ϕ

+⋅

↔+++

±±−−++)()(

2

1

22

2

21

12121

121121121121121121 HC

C

HC

CC

i

C

a

C

i

C

aaiCCCC rr ϕϕ

++ )()(22

2

22

222

200200200200 HC

C

HC

CC

i

C

aCC rr ϕϕ

++

±±−−++)()(

2

1

22

2

22

22222

121121121121121121 HC

C

HC

CC

i

C

a

C

i

C

aCCCC rr ϕϕ (28)

⋅

⋅

↔+

⋅

−

=

+∑)(

)(

,1

3

16

21

1

21

1

11111

210

200

121210200100100

2

3

0 HC

C

HC

C

C

i

C

a

C

a

H

a

H

iia ia

aiCCCCCa

Br

r

ϕ

ϕ

εεβπ

+⋅

⋅

↔+

+⋅

+++)(

)(

)(

,)(22

2

21

1

21

1

21111

21

1

121

210

200

121210200100100121 HC

C

HC

C

HC

C

C

i

C

a

C

a

H

a

H

iHC

CaiCCCCC r

r

r

r ϕϕ

ϕϕ

+⋅⋅

↔++

++)()(

21

1

22

22111

121200200121100100 HC

C

HC

CC

i

C

a

H

a

H

iaiCCCC rr ϕϕ

↔++

++)()(

22

2

22

22211

121200121200100100 HC

C

HC

CC

i

C

a

H

a

H

iaiCCCC rr ϕϕ (29)

⋅

⋅

↔+

⋅

−

=

−∑)(

)(

,1

3

16

21

1

21

1

11111

210

200

121210200100100

2

3

0 HC

C

HC

C

C

i

C

a

C

a

H

a

H

iia ia

aiCCCCCa

Cr

r

ϕ

ϕ

εεβπ

+⋅

⋅

↔+

+⋅

−−−)(

)(

)(

,)(22

2

21

1

21

1

21111

21

1

121

210

200

121210200100100121 HC

C

HC

C

HC

C

C

i

C

a

C

a

H

a

H

iHC

CaiCCCCC r

r

r

r ϕϕ

ϕϕ

+⋅⋅

↔++

−−)()(

21

1

22

22111

121200200121100100 HC

C

HC

CC

i

C

a

H

a

H

iaiCCCC rr ϕϕ

Electronic Supplementary Material (ESI) for Physical Chemistry Chemical PhysicsThis journal is © The Owner Societies 2013

Page 11

↔++

−−)()(

22

2

22

22211

121200121200100100 HC

C

HC

CC

i

C

a

H

a

H

iaiCCCC rr ϕϕ (30)

+

⋅

−

= ∑

)(

)(

,1

3

16'

21

1

21

1

11111

210

200

210200100100100

2

2

00 HC

C

HC

C

C

i

C

i

H

a

H

a

H

iia ia

CCCCCaa

Ar

r

ϕ

ϕ

εεβπ

↔+

+ aiC

HC

CC

i)(

22

22

200200rϕ (31)

+ +

⋅

−

=

±±±±∑ )()(

1

3

16

'

'

22

22

21

11111

121121121121100100100

2

2

00HC

CC

iHC

CC

i

H

a

H

a

H

iia ia

CCCCCaaC

Brr ϕϕ

εεβπ

↔+ ai (32)

11

1111

,

2

3

0

100100100100

2

3

03

8

3

16

HHSS

ia ia

H

i

H

a

H

a

H

i

a

CCCC

aE π

βεε

β

−=

−

= ∑ (33)

5th sum

∑∑∑≠≠

=−

=

1

112221111222111

22

222

11121

)()(1

3

16][

100100

2

3

0

0)5(

HB

HABH

B

mlnAH

A

mln

B

mlan

A

mlin

H

i

H

a

mln

mlnia iaI

FC

HHCCCC

aHJ rr ϕϕ

εεβπ

+

−

= ∑∑ )()(

1

3

16

12

2

22212

2

111

2

222

2

111

22

222

111

100100

2

3

0

HH

H

mlnHH

H

mln

H

mlan

H

mlin

H

i

H

a

mln

mlnia ia

CCCCa

rr ϕϕεε

βπ

++ )()(11

1

22212

2

111

1

222

2

111

22

100100 HC

C

mlnHH

H

mln

C

mlan

H

mlin

H

i

H

aCCCC rr ϕϕ

++ )()(12

2

22212

2

111

2

222

2

111

22

100100 HC

C

mlnHH

H

mln

C

mlan

H

mlin

H

i

H

aCCCC rr ϕϕ

++ )()(12

2

22211

1

111

2

222

1

111

22

100100 HH

H

mlnHC

C

mln

H

mlan

C

mlin

H

i

H

aCCCC rr ϕϕ

++ )()(11

1

22211

1

111

1

222

1

111

22

100100 HC

C

mlnHC

C

mln

C

mlan

C

mlin

H

i

H

aCCCC rr ϕϕ

++ )()(12

2

22211

1

111

2

222

1

111

22

100100 HC

C

mlnHC

C

mln

C

mlan

C

mlin

H

i

H

aCCCC rr ϕϕ

++ )()(12

2

22212

2

111

2

222

2

111

22

100100 HH

H

mlnHC

C

mln

H

mlan

C

mlin

H

i

H

aCCCC rr ϕϕ

++ )()(11

1

22212

2

111

1

222

2

111

22

100100 HC

C

mlnHC

C

mln

C

mlan

C

mlin

H

i

H

aCCCC rr ϕϕ

+ )()(

12

2

22212

2

111

2

222

2

111

22

100100 HC

C

mlnHC

C

mln

C

mlan

C

mlin

H

i

H

aCCCC rr ϕϕ (34)

Electronic Supplementary Material (ESI) for Physical Chemistry Chemical PhysicsThis journal is © The Owner Societies 2013

Page 12

+

−

=

+−−

∑22

022

22

21

)cos1(22

1001003

0

1001002

3

0

0)5( 1

3

16][

CCCHLL

aH

a

H

iia ia

H

i

H

a

I

FC

HHeCC

a

CC

aHJ

φ

πεεβπ

+

↔++

+−−

∑22

0

11

112

)cos1(21

100

00

)(1 CCCH

LLa

HC

C

nlm

C

anlm

H

inlm

eaiCCaa

φ

ϕπ

r

+

↔++

+−−

∑22

0

12

222

)cos1(21

100

00

)(1 CCCH

LLa

HC

C

nlm

C

anlm

H

inlm

eaiCCaa

φ

ϕπ

r

++ ∑ )()(11

1

22211

1

111

1

222

1

111

222

111

HC

C

mlnHC

C

mln

C

mlan

C

mlin

mln

mln

CC rr ϕϕ

+

↔++ ∑ )()(

12

2

22211

1

111

2

222

1

111

222

111

HC

C

mlnHC

C

mln

C

mlan

C

mlin

mln

mln

aiCC rr ϕϕ

+ ∑ )()(

12

2

22212

2

111

2

222

2

111

222

111

HC

C

mlnHC

C

mln

C

mlan

C

mlin

mln

mln

CC rr ϕϕ (35)

Summing up the terms in Eq. (35) over the quantum numbers, we obtain

+

−

=

+−−

∑22

022

22

21

)cos1(22

1001003

0

1001002

3

0

0)5( 1

3

16][

CCCHLL

aH

a

H

iia ia

H

i

H

a

I

FC

HHeCC

a

CC

aHJ

φ

πεεβπ

+

↔+

+

+−−

−

−

22

0

11

1

11

1

112

)cos1(21

121

200

121200100

00 )(

)(

,1 CCCH

LLa

HC

C

HC

C

C

a

C

a

H

ieaiCCC

aa

φ

ϕ

ϕ

π r

r

+

↔+

+

+−−

−

−

22

0

12

2

12

2

12

2

2222

)cos1(21

121

210

200

121210200100

00

)(

)(

)(

,,1 CCCH

LLa

HC

C

HC

C

HC

C

C

a

C

a

C

a

H

ieaiCCCC

aa

φ

ϕ

ϕ

ϕ

πr

r

r

+

⊗

⋅

⊗

+

−−

−−)(

)(

)(

)(

,,

11

1

11

1

11

1

11

1

1111

121

200

121

200

121200121200

HC

C

HC

C

HC

C

HC

C

C

a

C

a

C

i

C

iCCCC

r

r

r

r

ϕ

ϕ

ϕ

ϕ

+

⊗

⋅

↔+

⊗

+

−

−

−−

)(

)(

)(

)(

)(

,,,

12

2

12

2

12

2

11

1

11

1

22211

121

210

200

121

200

121210200121200

HC

C

HC

C

HC

C

HC

C

HC

C

C

a

C

a

C

a

C

i

C

iaiCCCCC

r

r

r

r

r

ϕ

ϕ

ϕ

ϕ

ϕ

Electronic Supplementary Material (ESI) for Physical Chemistry Chemical PhysicsThis journal is © The Owner Societies 2013

Page 13

⊗

⋅

⊗

+

−−

−−

)(

)(

)(

)(

)(

)(

,,,,

12

2

12

2

12

2

12

2

12

2

12

2

222222

121

210

200

121

210

200

121210200121210200

HC

C

HC

C

HC

C

HC

C

HC

C

HC

C

C

a

C

a

C

a

C

i

C

i

C

iCCCCCC

r

r

r

r

r

r

ϕ

ϕ

ϕ

ϕ

ϕ

ϕ

(36)

As follows from Eq. (36), the fifth contribution takes a general form:

22

0

22

0

21

)cos1(22

)cos1(21

0)5(][

CCCHCCCHLL

aLL

a

I

FC

HHCeBeAHJ

+−−+−−

++=φφ

(37)

where coefficients A, B and C are as follows:

⋅

⊗

−

=

−−∑ 1111

22

121200121200

1001002

3

0

,,3

16 C

a

C

a

C

i

C

iia ia

H

i

H

a CCCCCC

aA

εεβπ

⋅

↔+

⊗

+

⊗

⋅−−

−−

aiCCCCCC

a

C

a

C

a

C

i

C

i

HC

C

HC

C

HC

C

HC

C

22211

11

1

11

1

11

1

11

1

121210200121200

121

200

121

200,,,

)(

)(

)(

)(

r

r

r

r

ϕ

ϕ

ϕ

ϕ

⋅

⊗

+

⊗

⋅−−

−

−

222222

12

2

12

2

12

2

11

1

11

1

121210200121210200

121

210

200

121

200,,,,

)(

)(

)(

)(

)(C

a

C

a

C

a

C

i

C

i

C

i

HC

C

HC

C

HC

C

HC

C

HC

C

CCCCCC

r

r

r

r

r

ϕ

ϕ

ϕ

ϕ

ϕ

⊗

⋅

−−)(

)(

)(

)(

)(

)(

12

2

12

2

12

2

12

2

12

2

12

2

121

210

200

121

210

200

HC

C

HC

C

HC

C

HC

C

HC

C

HC

C

r

r

r

r

r

r

ϕ

ϕ

ϕ

ϕ

ϕ

ϕ

(38)

+

↔+

−

=

−

−∑)(

)(

,3

16

11

1

11

1

112

22

121

200

121200100

100100

2

2

00 HC

C

HC

C

ia

C

a

C

a

H

iia

H

i

H

a aiCCCCC

aaB

r

r

ϕ

ϕ

εεβπ

↔+

+

−

−

)(

)(

)(

,,

12

2

12

2

12

2

2222

121

210

200

121210200100

HC

C

HC

C

HC

C

C

a

C

a

C

a

H

iaiCCCC

r

r

r

ϕ

ϕ

ϕ

(39)

Electronic Supplementary Material (ESI) for Physical Chemistry Chemical PhysicsThis journal is © The Owner Societies 2013

Page 14

22

2222

,

2

3

0

100100100100

2

3

03

8

3

16

HHSS

ia ia

H

a

H

i

H

i

H

a

a

CCCC

aC π

βεε

β

−=

−

= ∑ (40)

6th sum

⋅−

↔+⋅

↔+

= ∑ ∑ ∑

≠≠ia ia

B

mlin

H

a

A

mlan

H

i

mln

mln

HB

HAI

FC

HH

aiCCaiCC

aHJ

εεπβ

π222

2

111

1

222

111

2

121

1001002

3

0

0)6(

3

161][

)()(22221111

BH

B

mlnAH

A

mlnrr ϕϕ⋅ (41)

which gives

⋅

↔+⋅

↔+⋅

−

= ∑∑ aiCCaiCC

aHJ

H

mlin

H

a

H

mlan

H

i

mln

mlnia iaI

FC

HH1

222

22

111

1

222

11121

100100

2

3

0

0)6( 1

3

16][

εεβπ

⋅

↔+⋅

↔++⋅ aiCCaiCC

C

mlin

H

a

H

mlan

H

iHH

H

mlnHH

H

mln1

222

22

111

1

21

1

22212

2

111100100

)()( rr ϕϕ

⋅

↔+⋅

↔++⋅ aiCCaiCC

C

mlin

H

a

H

mlan

H

iHC

C

mlnHH

H

mln2

222

22

111

1

21

1

22212

2

111100100

)()( rr ϕϕ

⋅

↔+⋅

↔++⋅ aiCCaiCC

H

mlin

H

a

C

mlan

H

iHC

C

mlnHH

H

mln1

222

21

111

1

22

2

22212

2

111100100

)()( rr ϕϕ

⋅

↔+⋅

↔++⋅ aiCCaiCC

C

mlin

H

a

C

mlan

H

iHH

H

mlnHC

C

mln1

222

21

111

1

21

1

22211

1

111100100

)()( rr ϕϕ

⋅

↔+⋅

↔++⋅ aiCCaiCC

C

mlin

H

a

C

mlan

H

iHC

C

mlnHC

C

mln2

222

21

111

1

21

1

22211

1

111100100

)()( rr ϕϕ

⋅

↔+⋅

↔++⋅ aiCCaiCC

H

mlin

H

a

C

mlan

H

iHC

C

mlnHC

C

mln1

222

22

111

1

22

2

22211

1

111100100

)()( rr ϕϕ

⋅

↔+⋅

↔++⋅ aiCCaiCC

C

mlin

H

a

C

mlan

H

iHH

H

mlnHC

C

mln1

222

22

111

1

21

1

22212

2

111100100

)()( rr ϕϕ

⋅

↔+⋅

↔++⋅ aiCCaiCC

C

mlin

H

a

C

mlan

H

iHC

C

mlnHC

C

mln2

222

22

111

1

21

1

22212

2

111100100

)()( rr ϕϕ

⋅ )()(

22

2

22212

2

111HC

C

mlnHC

C

mlnrr ϕϕ (42)

Taking off the abundant summations over the quantum numbers in some terms of Eq. (42), we

have

⋅

↔+⋅

↔+⋅

−

= ∑ aiCCaiCC

aHJ

H

i

H

a

H

a

H

iia ia

I

FC

HH1221

21100100100100

2

3

0

0)6( 1

3

16][

εεβπ

⋅

↔+⋅

↔++⋅ ∑ aiCCaiCC

C

inlm

H

a

H

a

H

inlm

HH

H

HH

H1221

21

1

12

2

100100100100100)()( rr ϕϕ

Electronic Supplementary Material (ESI) for Physical Chemistry Chemical PhysicsThis journal is © The Owner Societies 2013

Page 15

⋅

↔+⋅

↔++⋅ ∑ aiCCaiCC

C

inlm

H

a

H

a

H

inlm

HC

C

nlmHH

H2221

21

1

12

2

100100100100)()( rr ϕϕ

⋅

↔+⋅

↔++⋅ ∑ aiCCaiCC

H

i

H

a

C

anlm

H

inlm

HC

C

nlmHH

H1211

22

2

12

2

100100100100)()( rr ϕϕ

⋅

↔+⋅

↔++⋅ ∑ aiCCaiCC

C

mlin

H

a

C

mlan

H

i

mln

mlnHH

H

HC

C

nlm1

222

21

111

1

222

11121

1

11

1

100100100)()( rr ϕϕ

⋅

↔+⋅

↔++⋅ ∑ aiCCaiCC

C

mlin

H

a

C

mlan

H

i

mln

mlnHC

C

mlnHC

C

mln2

222

21

111

1

222

11121

1

22211

1

111100100

)()( rr ϕϕ

⋅

↔+⋅

↔++⋅ ∑ aiCCaiCC

H

i

H

a

C

anlm

H

inlm

HC

C

mlnHC

C

mln1221

22

2

22211

1

111100100100

)()( rr ϕϕ

⋅

↔+⋅

↔++⋅ ∑ aiCCaiCC

C

mlin

H

a

C

mlan

H

i

mln

mlnHH

H

HC

C

nlm1

222

22

111

1

222

11121

1

12

2

100100100)()( rr ϕϕ

⋅

↔+⋅

↔++⋅ ∑ aiCCaiCC

C

mlin

H

a

C

mlan

H

i

mln

mlnHC

C

mlnHC

C

mln2

222

22

111

1

222

11121

1

22212

2

111100100

)()( rr ϕϕ

⋅ )()(

22

2

22212

2

111HC

C

mlnHC

C

mlnrr ϕϕ (43)

Substitution of )(21

2,1

100 HH

Hrϕ into Eq. (43) gives

⋅

↔+⋅

↔+⋅

−

= ∑ aiCCaiCC

aaHJ

H

i

H

a

H

a

H

iia ia

I

FC

HH1221

211001001001003

0

2

3

0

0)6( 11

3

16][

πεεβπ

⋅⋅

↔+⋅

↔++⋅ ∑

+−−

)(1

21

11221

22

0

100100100

00

)cos1(22

HC

C

nlm

C

inlm

H

a

H

a

H

inlm

LLa

aiCCaiCCaa

eCCCH

rϕπ

φ

⋅⋅

↔+⋅

↔++⋅ ∑

+−−

)(1

22

22221

22

0

100100100

00

)cos1(21

HC

C

nlm

C

inlm

H

a

H

a

H

inlm

LLa

aiCCaiCCaa

eCCCH

rϕπ

φ

⋅⋅

↔+⋅

↔++⋅ ∑

+−−

)(1

11

11211

22

0

100100100

00

)cos1(21

HC

C

nlm

H

i

H

a

C

anlm

H

inlm

LLa

aiCCaiCCaa

eCCCH

rϕπ

φ

⋅⋅

↔+⋅

↔++⋅ ∑

+−−

)(11

1

111

1

222

21

111

1

222

111

22

0

100100

)cos1(21

HC

C

mln

C

mlin

H

a

C

mlan

H

i

mln

mln

LLa

aiCCaiCCeCCCH

rϕφ

⋅⋅

↔+⋅

↔++⋅ ∑ )()(

11

1

111

2

222

21

111

1

222

11121

1

222100100 HC

C

mln

C

mlin

H

a

C

mlan

H

i

mln

mlnHC

C

mlnaiCCaiCC rr ϕϕ

)(1

)(12

21221

22

2

222100100100

00

HC

C

nlm

H

i

H

a

C

anlm

H

inlm

HC

C

mlnaiCCaiCC

aarr ϕ

πϕ ⋅

↔+⋅

↔++⋅ ∑

Electronic Supplementary Material (ESI) for Physical Chemistry Chemical PhysicsThis journal is © The Owner Societies 2013

Page 16

⋅

↔+⋅

↔++⋅ ∑

+−−

aiCCaiCCeC

mlin

H

a

C

mlan

H

i

mln

mln

LLa CCCH

1

222

22

111

1

222

111

22

0

100100

)cos1(21

φ

⋅

↔+⋅

↔++⋅ ∑ aiCCaiCC

C

mlin

H

a

C

mlan

H

i

mln

mlnHC

C

mlnHC

C

mln2

222

22

111

1

222

11121

1

22212

2

111100100

)()( rr ϕϕ

⋅ )()(

22

2

22212

2

111HC

C

mlnHC

C

mlnrr ϕϕ (44)

Taking the sum over the quantum numbers in Eq. (44) gives

⋅

↔+⋅

↔+⋅

−

= ∑ aiCCaiCC

aaHJ

H

i

H

a

H

a

H

iia ia

I

FC

HH1221

211001001001003

0

2

3

0

0)6( 11

3

16][

πεεβπ

⋅

↔++⋅

+−−+−−

aiCCeaa

eH

a

H

i

LLa

LLa CCCHCCCH

21

22

0

22

0

100100

)cos1(21

00

)cos1(22

1φφ

π

⋅

↔+⋅

↔++

⋅

↔+

+aiCCaiCCaiCCC

C

i

H

a

H

a

H

i

HC

C

HC

C

C

i

C

i

H

a1221

21

1

21

1

112

121100100100

210

200

210200100)(

)(

,r

r

ϕ

ϕ

+

⋅

↔+⋅

↔++⋅

−−+φϕφϕ cos)(sin)(

21

11221

21

1

121121100100100121 HX

CC

i

H

a

H

a

H

iHC

CaiCCaiCC rr

⋅

↔+⋅

↔+

++−−

aiCCaiCCeaa

C

i

H

a

H

a

H

i

LLa CCCH

2221

22

0

200100100100

)cos1(21

00

1φ

π

+

↔+⋅

↔++⋅

++φϕϕ sin)()(

22

22221

22

2

121121100100100200 HC

CC

i

H

a

H

a

H

iHC

CaiCCaiCC rr

+

↔+⋅

↔++

−−φϕ cos)(

22

22221

121121100100100 HC

CC

i

H

a

H

a

H

iaiCCaiCC r

⋅

↔+⋅

↔+

+

−

+−−

aiCCaiCCCeaa

H

i

H

a

C

a

C

a

H

i

LLa CCCH

12111

22

0

100100121200100

)cos1(21

00

,1

φ

π

⋅

↔+

⊗

↔+

+

⋅−

−

aiCCCaiCCCC

i

C

i

H

a

C

a

C

a

H

i

HC

C

HC

C

112111

11

1

11

1

210200100121200100

121

200,,

)(

)(

r

r

ϕ

ϕ

⋅

↔+⋅

↔+

+

⊗

⋅+−

−

aiCCaiCCCC

i

H

a

C

a

C

a

H

i

HC

C

HC

C

HC

C

HC

C

12111

21

1

21

1

11

1

11

1

121100121200100

210

200

121

200,

)(

)(

)(

)(

r

r

r

r

ϕ

ϕ

ϕ

ϕ

Electronic Supplementary Material (ESI) for Physical Chemistry Chemical PhysicsThis journal is © The Owner Societies 2013

Page 17

⋅

↔+⋅

↔+

+⋅⋅

⋅−−+

−

aiCCaiCCCC

i

H

a

C

a

C

a

H

iHC

C

HC

C

HC

C

12111

21

1

11

1

11

1

121100121200100121

121

200,sin)(

)(

)(

φϕϕ

ϕr

r

r

+⋅⋅

⋅−

−

φϕϕ

ϕcos)(

)(

)(

21

1

11

1

11

1

121

121

200

HC

C

HC

C

HC

C

rr

r

+⋅

⋅

↔+⋅

↔+

+

−

−)(

)(

)(

,22

2

11

1

11

1

22111

200

121

200

200100121200100 HC

C

HC

C

HC

C

C

i

H

a

C

a

C

a

H

iaiCCaiCCC r

r

r

ϕϕ

ϕ

+⋅

⋅

↔+⋅

↔+

+

+

−

+−φϕ

ϕ

ϕsin)(

)(

)(

,22

2

11

1

11

1

22111

121

121

200

121100121200100 HC

C

HC

C

HC

C

C

i

H

a

C

a

C

a

H

iaiCCaiCCC r

r

r

+⋅

⋅

↔+⋅

↔+

+

−

−

−−φϕ

ϕ

ϕcos)(

)(

)(

,22

2

11

1

11

1

22111

121

121

200

121100121200100 HC

C

HC

C

HC

C

C

i

H

a

C

a

C

a

H

iaiCCaiCCC r

r

r

⋅

↔+⋅

↔+

⋅+

−

+−−

aiCCaiCCCCeaa

H

i

H

a

C

a

C

a

C

a

H

i

LLa CCCH

122221

22

0

100100121210200100

)cos1(21

00

,,1

φ

π

⋅

↔+

⋅⊗

↔+

+

⋅−

−

aiCCCaiCCCCC

i

C

i

H

a

C

a

C

a

C

a

H

i

HC

C

HC

C

HC

C

1122221

12

2

12

2

12

2

210200100121210200100

121

210

200

,,,

)(

)(

)(

r

r

r

ϕ

ϕ

ϕ

⋅

↔+⋅

↔+

+

⊗

⋅+−

−

aiCCaiCCCCC

i

H

a

C

a

C

a

C

a

H

i

HC

C

HC

C

HC

C

HC

C

HC

C

122221

21

1

21

1

12

2

12

2

12

2

121100121210200100

210

200

121

210

200

,,)(

)(

)(

)(

)(

r

r

r

r

r

ϕ

ϕ

ϕ

ϕ

ϕ

⋅

↔+⋅

↔+

+

⋅−−+

−

aiCCaiCCCCC

i

H

a

C

a

C

a

C

a

H

iHC

C

HC

C

HC

C

HC

C

122221

21

1

12

2

12

2

12

2

121100121210200100121

121

210

200

,,sin)(

)(

)(

)(

φϕ

ϕ

ϕ

ϕ

r

r

r

r

⋅

↔+⋅

↔+

+

⋅−−

−

aiCCaiCCCCC

i

H

a

C

a

C

a

C

a

H

iHC

C

HC

C

HC

C

HC

C

222221

21

1

12

2

12

2

12

2

200100121210200100121

121

210

200

,,cos)(

)(

)(

)(

φϕ

ϕ

ϕ

ϕ

r

r

r

r

Electronic Supplementary Material (ESI) for Physical Chemistry Chemical PhysicsThis journal is © The Owner Societies 2013

Page 18

⋅

↔+⋅

↔+

+

⋅+−

−

aiCCaiCCCCC

i

H

a

C

a

C

a

C

a

H

iHX

C

HC

C

HC

C

HC

C

222221

22

1

12

2

12

2

12

2

121100121210200100200

121

210

200

,,)(

)(

)(

)(

r

r

r

r

ϕ

ϕ

ϕ

ϕ

⋅

↔+⋅

↔+

+⋅

⋅−−+

−

aiCCaiCCCCC

i

H

a

C

a

C

a

C

a

H

iHC

C

HC

C

HC

C

HC

C

222221

22

1

12

2

12

2

12

2

121100121210200100121

121

210

200

,,sin)(

)(

)(

)(

φϕ

ϕ

ϕ

ϕ

r

r

r

r

⋅

⋅−

−

φϕ

ϕ

ϕ

ϕ

cos)(

)(

)(

)(

22

1

12

2

12

2

12

2

121

121

210

200

HC

C

HC

C

HC

C

HC

C

r

r

r

r

(45)

Eq. (45) can be generalized into the following expression:

( )+++⋅+⋅=+−−+−−

φφφφ

cossin][

22

0

22

0

21

)cos1(21

)cos1(22

0)6(DCBeeAHJ

CCCHCCCHLL

aLL

a

I

FC

HH

φφ cos'sin'' DCB +++ (46)

Coefficient A in Eq. (46) takes the form:

+++⋅

⋅−

=

−

+⋅

+

=

=−

↔+⋅

↔+

=

∑∑

∑

1221122112211221

12122121

1221

100100100100100100100100100100100100100100100100

2

3

0

100100100100100100100100

2

3

0

100100100100

2

3

0

1

3

16

3

16

3

16

H

a

H

i

H

i

H

a

H

i

H

a

H

i

H

a

H

a

H

i

H

a

H

i

H

i

H

a

H

a

H

i

ia iaia ia

H

a

H

i

H

i

H

a

H

i

H

a

H

a

H

i

ia ia

H

i

H

a

H

a

H

i

CCCCCCCCCCCCCCCC

a

CCCCCCCC

a

aiCCaiCC

aA

εεβ

εεβ

εεβ

(47)

Using the definition of the mutual polarizability given by Eq. (14), one can reduce Eq. (47) into a

compact form:

−

−

↔+⋅

= ∑

iaSS

ia

H

i

H

a

H

a

H

i

HH

aiCCCC

aA

21

1221

,

100100100100

2

3

02

1

3

16π

εεβ

(48)

Electronic Supplementary Material (ESI) for Physical Chemistry Chemical PhysicsThis journal is © The Owner Societies 2013

Page 19

The coefficient B in Eq. (46) takes the form:

⋅

↔+

⋅

↔+⋅

−

= ∑ aiCCCaiCC

aaB

C

i

C

i

H

a

H

a

H

iia ia

11221

210200100100100

2

2

00

,1

3

16

εεβπ

+⋅

↔+⋅

↔++

⋅ )()(

)(

22

22221

21

1

21

1

200200100100100

210

200

HC

CC

i

H

a

H

a

H

i

HC

C

HC

C

aiCCaiCC rr

r

ϕϕ

ϕ

+

⋅

↔+⋅

↔+

+

−

−)(

)(

,

11

1

11

1

12111

121

200

100100121200100

HC

C

HC

C

H

i

H

a

C

a

C

a

H

iaiCCaiCCC

r

r

ϕ

ϕ

⋅

↔+⋅

↔+

+

−

−

)(

)(

)(

,,

12

2

12

2

12

2

122221

121

210

200

100100121210200100

HC

C

HC

C

HC

C

H

i

H

a

C

a

C

a

C

a

H

iaiCCaiCCCC

r

r

r

ϕ

ϕ

ϕ

(49)

and coefficients C and D in Eq. (46) are

+⋅

↔+

⋅

−

↔+

=

±±∑ )(

3

16

21

112

21

121121100

100100

2

2

00HC

CC

i

H

aia ia

H

a

H

iaiCC

aiCC

aaD

Crϕ

εεβπ

↔++

±±)(

22

222

121121100 HC

CC

i

H

aaiCC rϕ (50)

Coefficients 'B 'C and 'D in Eq. (46) take the form:

⋅

↔+

⊗

↔+

⋅

−

=

−∑ aiCCCaiCCCa

BC

i

C

i

H

a

C

a

C

a

H

iia ia

112111

210200100121200100

2

3

0

,,1

3

16'

εεβπ

⋅

↔+⋅

↔+

+

⊗

⋅−

−

aiCCaiCCCC

i

H

a

C

a

C

a

H

i

HC

C

HC

C

HC

C

HC

C

22111

21

1

21

1

11

1

11

1

200100121200100

210

200

121

200,

)(

)(

)(

)(

r

r

r

r

ϕ

ϕ

ϕ

ϕ

+⋅

⋅

−

)()(

)(

22

2

11

1

11

1

200

121

200

HC

C

HC

C

HC

C

rr

r

ϕϕ

ϕ

⋅

↔+

⋅⊗

↔+

+

−aiCCCaiCCCC

C

i

C

i

H

a

C

a

C

a

C

a

H

i1122221

210200100121210200100,,,

+

⊗

⋅

−

)(

)(

)(

)(

)(

21

1

21

1

12

2

12

2

12

2

210

200

121

210

200

HC

C

HC

C

HC

C

HC

C

HC

C

r

r

r

r

r

ϕ

ϕ

ϕ

ϕ

ϕ

Electronic Supplementary Material (ESI) for Physical Chemistry Chemical PhysicsThis journal is © The Owner Societies 2013

Page 20

⋅

↔+⋅

↔+

+

−

−)(

)(

)(

)(

,,22

1

12

2

12

2

12

2

222221

200

121

210

200

200100121210200100 HC

C

HC

C

HC

C

HC

C

C

i

H

a

C

a

C

a

C

a

H

iaiCCaiCCCC r

r

r

r

ϕ

ϕ

ϕ

ϕ

(51)

⋅

↔+⋅

↔+

⋅

−

=

±−∑ aiCCaiCCC

aD

C C

i

H

a

C

a

C

a

H

iia ia

12111

121100121200100

2

3

0

,1

3

16

'

'

εεβπ

⋅

↔+⋅

↔+

+⋅

⋅±−±

−

aiCCaiCCCC

i

H

a

C

a

C

a

H

iHC

C

HC

C

HC

C

22111

21

1

11

1

11

1

121100121200100121

121

200,)(

)(

)(

rr

r

ϕϕ

ϕ

⋅

↔+⋅⋅

↔+

+⋅

⋅±−±

−

aiCCaiCCCCC

i

H

a

C

a

C

a

C

a

H

iHC

C

HC

C

HC

C

122221

22

2

11

1

11

1

121100121210200100121

121

200,,)(

)(

)(

rr

r

ϕϕ

ϕ

⋅

↔+⋅

↔+

+

⋅±−±

−

aiCCaiCCCCC

i

H

a

C

a

C

a

C

a

H

iHC

C

HC

C

HC

C

HC

C

222221

21

1

12

2

12

2

12

2

121100121210200100121

121

210

200

,,)(

)(

)(

)(

r

r

r

r

ϕ

ϕ

ϕ

ϕ

⋅

⋅±

−

)(

)(

)(

)(

22

1

12

2

12

2

12

2

121

121

210

200

HC

C

HC

C

HC

C

HC

C

r

r

r

r

ϕ

ϕ

ϕ

ϕ

(52)

The latter two coefficients can be simplified to a form:

⋅

↔+

+⋅

−

=

±±±±∑ aiCCC

aD

C

HC

CC

iHC

CC

i

H

aia ia

)()(1

3

16

'

'

22

12

21

112

121121121121100

2

3

0

rr ϕϕεε

βπ

↔+

+

⋅

−

−

−

−aiCCCCCC

HC

C

HC

C

HC

C

C

a

C

a

C

a

HC

C

HC

C

C

a

C

a

H

i

)(

)(

)(

,,)(

)(

,

12

2

12

2

12

2

222

11

1

11

1

111

121

210

200

121210200

121

200

121200100

r

r

r

r

r

ϕ

ϕ

ϕ

ϕ

ϕ(53)

7th sum:

Electronic Supplementary Material (ESI) for Physical Chemistry Chemical PhysicsThis journal is © The Owner Societies 2013

Page 21

∑∑∑

≠≠≠

⋅

↔+⋅

−

=

2

2

1111

1

333222

333

222

11121

100

2

00

0)7( 1

3

161][

HC

HB

HA

A

mlan

H

i

C

mlin

B

mlan

mln

mln

mlnia iaI

FC

HHaiCCCC

aaHJ

εεπβ

π

)()()(233322221111

CH

C

mlnBH

B

mlnAH

A

mlnrrr ϕϕϕ⋅ (54)

⋅

↔++⋅

↔+⋅

−

= ∑∑

aiCCCC

aiCCCCaa

HJ

C

mlan

H

i

H

mlin

H

mlanHH

H

mlnHH

H

mlnHH

H

mln

H

mlan

H

i

H

mlin

H

mlan

mln

mln

mlnia iaI

FC

HH

1

111

11

333

1

22221

1

33321

1

22212

2

111

2

111

11

333

1

222

333

222

11121

100

100

2

00

0)7(

)()()(

1

3

161][

rrr ϕϕϕ

εεπβ

π

⋅

↔++⋅ aiCCCC

C

mlan

H

i

H

mlin

H

mlanHH

H

mlnHH

H

mlnHC

C

mln2

111

11

333

1

22221

1

33321

1

22211

1

111100

)()()( rrr ϕϕϕ

⋅

↔++⋅ aiCCCC

H

mlan

H

i

C

mlin

H

mlanHH

H

mlnHH

H

mlnHC

C

mln2

111

11

333

1

22221

1

33321

1

22212

2

111100

)()()( rrr ϕϕϕ

⋅

↔++⋅ aiCCCC

C

mlan

H

i

C

mlin

H

mlanHC

C

mlnHH

H

mlnHH

H

mln1

111

11

333

1

22221

1

33321

1

22212

2

111100

)()()( rrr ϕϕϕ

⋅

↔++⋅ aiCCCC

C

mlan

H

i

C

mlin

H

mlanHC

C

mlnHH

H

mlnHC

C

mln2

111

11

333

1

22221

1

33321

1

22211

1

111100

)()()( rrr ϕϕϕ

⋅

↔++⋅ aiCCCC

H

mlan

H

i

C

mlin

H

mlanHC

C

mlnHH

H

mlnHC

C

mln2

111

12

333

1

22221

1

33321

1

22212

2

111100

)()()( rrr ϕϕϕ

⋅

↔++⋅ aiCCCC

C

mlan

H

i

C

mlin

H

mlanHC

C

mlnHH

H

mlnHH

H

mln1

111

12

333

1

22222

2

33321

1

22212

2

111100

)()()( rrr ϕϕϕ

⋅

↔++⋅ aiCCCC

C

mlan

H

i

C

mlin

H

mlanHC

C

mlnHH

H

mlnHC

C

mln2

111

12

333

1

22222

2

33321

1

22211

1

111100

)()()( rrr ϕϕϕ

⋅

↔++⋅ aiCCCC

H

mlan

H

i

H

mlin

C

mlanHC

C

mlnHH

H

mlnHC

C

mln2

111

11

333

1

22222

2

33321

1

22212

2

111100

)()()( rrr ϕϕϕ

⋅

↔++⋅ aiCCCC

C

mlan

H

i

H

mlin

C

mlanHH

H

mlnHC

C

mlnHH

H

mln1

111

11

333

1

22221

1

33321

1

22212

2

111100

)()()( rrr ϕϕϕ

⋅

↔++⋅ aiCCCC

C

mlan

H

i

H

mlin

C

mlanHH

H

mlnHC

C

mlnHC

C

mln2

111

11

333

1

22221

1

33321

1

22211

1

111100

)()()( rrr ϕϕϕ

⋅

↔++⋅ aiCCCC

H

mlan

H

i

C

mlin

C

mlanHH

H

mlnHC

C

mlnHC

C

mln2

111

11

333

1

22221

1

33321

1

22212

2

111100

)()()( rrr ϕϕϕ

⋅

↔++⋅ aiCCCC

C

mlan

H

i

C

mlin

C

mlanHC

C

mlnHC

C

mlnHH

H

mln1

111

11

333

1

22221

1

33321

1

22212

2

111100

)()()( rrr ϕϕϕ

⋅

↔++⋅ aiCCCC

C

mlan

H

i

C

mlin

C

mlanHC

C

mlnHC

C

mlnHC

C

mln2

111

11

333

1

22221

1

33321

1

22211

1

111100

)()()( rrr ϕϕϕ

⋅

↔++⋅ aiCCCC

H

mlan

H

i

C

mlin

C

mlanHC

C

mlnHC

C

mlnHC

C

mln2

111

12

333

1

22221

1

33321

1

22212

2

111100

)()()( rrr ϕϕϕ

⋅

↔++⋅ aiCCCC

C

mlan

H

i

C

mlin

C

mlanHC

C

mlnHC

C

mlnHH

H

mln1

111

12

333

1

22222

2

33321

1

22212

2

111100

)()()( rrr ϕϕϕ

⋅

↔++⋅ aiCCCC

C

mlan

H

i

C

mlin

C

mlanHC

C

mlnHC

C

mlnHC

C

mln2

111

12

333

1

22222

2

33321

1

22211

1

111100

)()()( rrr ϕϕϕ

Electronic Supplementary Material (ESI) for Physical Chemistry Chemical PhysicsThis journal is © The Owner Societies 2013