Mc simulation on acetone-carbon disulphide (1:124) with a quantum mechanical intermolecular potencial

Journal of Molecular Liquids, 28 (1984) 73--85

73

Elsevier Science Publishers B.V., A m s t e r d a m -- Printed in The Netherlands

MC

SII.IULATION ON A C E T O N E - C A R B O N

AECHANICAL

L.M.

INTE~MOLECULAR

DISULPHIDE

(i:124)

WIT}{ A Q U A N T U M

POTENTIAL.

SESff

Dpto.

Qulmica-Fisica,

Facultad

de C i e n c i a s ,

U.N.E.D.,

Madrid

(Spain).

;i. F E R N A N D E Z

Dpto.

Estructura

At6mico-Molecular,

U. C o m p l u t e n s e , (Received

Facultad

de C i e n c i a s

Qulmicas,

Madrid.

6 July

1983)

ABSTRACT

This using

paper


tions.

shows

~fe p r o p o s e

Theory,

simulations.

The

of

this

potential

which

obvious

calculations

ture

applications

of

coupled

of

into

Furthermore

we g i v e

structure

this

of A c e t o n e

interac-

potential,

based

with Monte-Carlo

potential

account the M C

several

studies

intermolecular

intermolecular

advantage

for a s o l u t i o n

Finally,

liquid

about

is e a s i l y

is to take

the m o l e c u l e s .

summarized.

to p e r f o r m

information

an a n a l y t i c a l

Perturbation

out MC

a way

Chemical

for c a r r y i n g

the e l e c t r o n i c

results

obtained

in C a r b o n

improvements

on (HC)

strucusing

Disulphide for

are

further

the m o d e l .

INTRODUCTION

In some for

recent

studying

environmental cular

effects.

orbitals

was

applied

[2]

the

introducing

interest

model

to c a r b o n

theory

gurations

works

[i]

the v a r i a t i o n s

was

This

and

was

developed

properties

based

Mechanics.

over

state

by

the

moletheory

liquid.

framework nuclear

system.

At p r e s e n t

of

© 1984 Elsevier Science Publishers B.V.

ideas

explained

In

by

accesible

of the

the v e r i f i c a t i o n

In [i] of Fure

rigorous

all

a theory

induced

on a p e r t u r b e d

as an e x a m p l e in a m o r e

averages

the e q u i l i b r i u m on

we have

Statistical

reformulated

is c e n t e r e d

0167-7322/84/$03.00

theory

disulphide

statistical

for

, [2],

in m o l e c u l a r

confiour

in [i]

74 and [2]

for

The

solutions.

first

to f o r m u l a t e statistical well

for

studies which

calculations,

are

out

is to s h o w t h a t ble

to o b t a i n

tials

fittings,

leads

than

INTERMOLECULAR

ble

step

for

los et al. dures

of

into

with

because

in

ensemble

in [7]

the

of

the

in M C

paper

is p o s s ! poten-

assumptions.

simulations

currently

As

of

the

used.

account

three

, and w h i c h

energy

In this

cited

[4]

[3]

body

effects

is a g e n e r a l i z a t i o n the e n e r g y

of

simulation

to c a r r y

short distances,

of

the

out

this

total

po-

by Ko-

of p r o c e -

in c e r t a i n

is n o t u s e f u l

approach,

In o r d e r

fitting

approach

computational

of

out

kind

that,

be

obtai-

intermolecular

supermolecular

in the

can

two r e p r e -

form was

, this

molecules

the energy

for a g r e a t

of

to p r o h i b i t i v e

implied

for

b y an a d e q u a t e

A supermolecular

or m o r e

of a r e l i a -

additivity)

for e a c h p a i r

et al.

many

, is the c a l c u l a t i o n

t h a t at no

pair-wise

as has b e e n p o i n t e d

importance.

simulation,

to a c c o u n t

interaction

function.

~olos

selection

try

followed

However,

by M C

in t h e c o n f i g u r a t i o n a l

the previously

to the

method.

is the

positions

sample,

it l e a d s

of m o l e c u l e s

perturbation suppose

[6]

this

numerical

solutions

If we w a n t

(assuming

a n d by C l e m e n t i ,

An alternative proposed

way

water.

m a y be of g r e a t

practice,

For

information,

a i m of

than

, for

intermolecu

intermolecular

potentials

the m o l e c u l e s

relative

et al.

liquid

on a calculation

sed

of

calculation

do not have

cases,

times

[3]

interactions)

and C N D O - l i k e

in s t u d y i n g

to a n a n a l y t i c a l

ned by C l e m e n t i tential

systems

Chemical

satisfactory

colnputation

potential.

molecules

of the r e s u l t s

theory

for

as

simulation

potentials.

. The

(weak

in the c a l c u l a t i o n s

of d i f f e r e n t

sentative

[4]

cases

of e m p i r i c a l

a possible

a supermolecular number

et al.

is

the

is a p p l i c a b l e

the

a realistic

of Q u a n t u m

expression

interested

structure

evaluation,

out

out

POTENTIAL

intermolecular

electronic

to s i m p l e

is m o r e

to

those

W h e n we are an important

limited

to s o l u t i o n s

to c a r r y

itself

So far,

intermolecular

by C l e m e n t i

this

theory

suitable

theory

to w r i t e

on perturbation

and

same o r d e r

the

in w e l l - b e h a v e d

show later

our

liquids.

to be the u s e

an a n a l y t i c a l

founded

we w i l l

been

the o n l y w a y

seems

carried

since

some e m p i r i c a l

systems,

lar p o t e n t i a l

have

to a p p l y potential

as for p u r e

solutions

there

as w a s

in o r d e r

intermolecular

solutions of

complex

problem

an

based in

times.

that was a method

first propo-

the e n t i r e

by m e a n s energy

of

the

we w i l l

electronic

wavefunc-

75 tion

~T of a s u p e r m o l e c u l a r

fixed,

system

is b u i l t up as a p r o d u c t

in a n u c l e a r

of R H F - S l a t e r ' s

the i s o l a t e d m o l e c u l e s

plus a c o r r e c t i o n

tion b e t w e e n

pairs

different

the N m o l e c u l e s exchange ~T:

N ~[ a:l

i

~ a
molecule

(that is,

spln-orbital

biexcitations ce,

Z X c iea jeb ijab

because

N

the w a v e f u n c t i o n

and o t h e r

energetic

o

obtained

(i) excitation

are of v e r y

and we do not

of the

by r e p l a c i n g

of the same spin).

poliexcitations

reasons,

...,

the e l e c t r o n i c

(c/a,b1[~c )

the i-th s i n g l e - e l e c t r o n

by a v i r t u a l - o n e

a, b, c,

we can put:

.

Ca ~

for

to the p e r t u r b a -

Labelling

and n e g l e c t i n g

molecules,

°

~a +~

i Ca r e p r e s e n t s

where

pied

different

determinants

term due

of m o l e c u l e s .

of the sample,

between

configuration

small

introduce

an o c c u -

Probably, importan

them

in our

calculations. Assuming cules will

CT n o r m a l i z e d ,

the

o

EI=

interaction

energy

for the N m o l e

be: o

o

< @T I az Ha+ a - < ~ T I aZ H a° I ~T > o

where

H a are

the i s o l a t e d m o l e c u l e

intermolecular actual

interaction

electronic

term,

hamiltonian

( 2 )

hamiltonians

and a{b Wab the

g i u e n by the d i f f e r e n c e

for the N m o l e c u l e s

between

and the sum

o

aE H a . For the c a l c u l a t i o n of Wab m a t r i x M u l l i k e n a p p r o x i m a t i o n [8] :

( ~ 1 ~°)

1

~ ~

[

2

2

(

<~lv>

(v21x 2) (~2102) (v21o2)



Assulning for e a c h m o l e c u l e methods),

and e m p l o y i n g

+ + ]

orthonormal

for the b i e l e c t r o n i c

<×U (i) ×t (2)

Ii ~ ; r12

I'-i [ ×v

~IB

basis

intermolecular

atraction

(i) > ~ R -I 6 AB uv

functions

(as in CNDO

approximation

Xc ( 2 ) × (i) > =(pvIlo)

and for the n u c l e a r - e l e c t r o n (i)

(3)

the a d d i t i o n a l

we can w r i t e

<~

we can use the

I~ ) <~1o>+

<~l ~>



elements,

=

R -I AB 6 ~

integrals: 6 0

5)

~,v~Aca ; B~b ---

6)

integrals:

76 ApL~lying raction

the

former

ener]y

assumptions,

formula

+2

j~ -i ~ 2 Z TB~ARAB, Bcb

( E Aea

E

j#0

3

neglecting

terms,

the

inte

is:

a -I EI= Z Z y QAQBRAB + 2 Y a
b

and

T i @ R-I. 2 A~B AB )

( >~ ~; B~b A~a

+ o

•

E -E l a a

a +2

E°_E ° b j

X T i j -i 2 Bcb ATBRAB)

( E AEa

b

E

E

i,0

j~0

o

i

o

]

.

(7)

E a -Ea+Eb-E ~

where QA:

Z u~A

E n c 2 -Z p P ~p A

are

the

net

charges

i p-~t Z TA = TA = p~A are

the

sums

E°-Ei: a a

p

out

-

MC

itt to a t o m

6

of

A of

from

the

application

would

computation

time

dimensional

array

be

energy

exorbitant.

pair

V -a g

~0

E°-E±aa d i f f e r e n c e s

can

be

extension: (10)

formula

the

(7)

computation

In d e s p i t e reduced

of

is n o t time

this

useful for

result,

by d e f i n i n g

to

its the

a four

of m o l e c u l e s :

T A TA'

:

orbital,

- K~ I ~t(1) >

because

each

The

theorem

be d r a s t i c a l l y

for

elements.

molecular .

orbital.

interaction

can

matrix

occupied p

a Koopmans

simulations,

transition

the ~

+ < ~t(1) I J~

t

(9)

(i:p÷t)

arises

by means a

and

c ~JP

~ip t h e ~t v i r t u a l

However, carry

c

c

c t from

estimated

on A atoms,

related

Coefficients and

(8)

~

+

B

B'|

j/o

a

/

+

i T i )]

a

b

i~0

j~0

(ll)

which

is

electronic the

fixed

for

each

information

interaction

energy

pair about (7)

of

rigid

them.

in t h e

molecules

This form:

trick

and enables

includes us

the

to w r i t e

77

EI

=

~:

Eab=

Z

[

v

~:

1

IQ AQB +

,.

a*b

a
A£a

Beb

RAB

Z

- -1

Z

A'¢a

B'~b

Pab(A,A',B,B' ))]

RA,B, (12)

where

and

Pab

will

Of c o u r s e ,

QA,QB

the

interaction

valid

for

rent

the o v e r a l l

molecules.

interactions

grows

calculation for

small

hard

of

approximation molecules.

furthermore,

0C(3) S,

OHC, we

tl

3 R A B ~ 0AB

because

(12)

implies

This

intermolecular

requirements tempered

APPLICATION

with

for e v e r y

TO A C E T O N E

has

been

a sample

, and

IN C A R B O N

°C(1)S' at this

°OC'

°OS'

level

intermolecular

of

poten-

(13)

for

choice

interactions.

In the

for

the

OAB p a r a m e t e r s .

at least,

the

two n e c e s s a r y

potential:

a)

non-collapsing,

carried

was A.

fixed

The

is s h o w n

out

in the C a n o n i c a l

were

was

of

K and

these

of

positions the

and

2. M o l e c u l a r

Ensemble 124 m o l e c u l e s

simulation

selection density

orientations

CS 2 c r y s t a l

in the c e n t r e

and

the

N,V

to an a p p r o x i m a t e

placed

in F i g u r e

of a c e t o n e

at 2 8 2 . 1 6

goodness

corresponds

molecule

DISULPHIDE

of one m o l e c u l e

those

acetone as

in CS 2 we

acetone

three

b:

intermolecular

124 CS 2 m o l e c u l e s

[12]:

to d i f f e r e n t

of

i) . Thus,

holds

initial

cube,

of

of

<}C(1)C'

additivity

a possible

10 -2 m o l e c u l e s / A 3. The

one

distances

[9].

23.231199

in [i0]

pairwise

potential

of CS 2. T e m p e r a t u r e

ted

that,

the v a l u e s

following a and

suppose

rAta, VB,b

we e x p l a i n

Simulation

was

(See F i g u r e

the the

otherwise

section

edge

a solution

for

by an a d e q u a t e

belonging

need

parameters:

of m o l e c u l e s

~ext

(N,V,T)

For

we

be

small,

is not v a l i d

the m i n i m a l

we

not

for d i f f e -

became

be r e p l a c e d

of a t o m s

liquid

set up the

pair

if

be

pair

the

0HS

can

for e a c h

ab

b)

each

will

will

distances

Consequently can

simulation.

(12)

distances method

energy.

potential

parameters

between

approximation,

Iab=

the

0CC , OCS , OSS.

need,

tial

internuclear these

In the CS 2 pure

parameters:

GC(3) c,

of of

perturbation

the

formula

range

interaction

whose

through

energy

anyone

and

distances

core,

will

When

be c o n s t a n t s

of

lattice the

geometries

cube

was

tes-

number

of

of

the

[ii]

and

simulation were [ii]

,

78

Z

/ / / /

4H/

........

/

O"--C

X

/ y

Figure

i. [4ost s t a b l e

1(C-S)

= 1.55

conformation

of

the

o

= 1.085

assuming gas

;

1(C-O)

A

;

sy~netry energy

(~II) t r u n c a t i o n

been

was

the

discussion

assumed

for

the

a fundamental

of

~ a
l(C-C)

= 1.507

;

HCH

to D h an

acetone

[13] the

the

previous

C2v

groups

molecule

In t h e

for

= 108°46 '

potential

calculation energy

(the

is l e f t the

in Mi-

evaluation

reason

for

core"

in o u r

calculation

exact

values

for

(provided

they

the form

t h e OAB v a l u e s intermolecular

section

energy

of

the

pair-wise

additi-

molecular

ensemble:

Iab

hard

We choose which the

the

potential

tem, tance

belonging

Thus,

configuration

U = Eabc...=

Since

;

used.

From vity

A

= 117012 '

rigid,

groups).

its m i n i m a l

has

molecule

/k

CCC

nimal

Image

= 1.222

/~

the m o l e c u l e s

phase

Acetone

o

A o

1(C-H)

isolated

(14)

the is

introduction

to a v o i d

OAB p a r a m e t e r s a reasonable

the are

set

of

a "modified

collapse not

of

of

the

capital

of v a l u e s ,

of

sys-

impor-

course).

proportionals to the d i s t a n c e s , R~B, for interaction between the two molecules

79

-X ] I

/

MET

~ ~ . . . . . . . . . . . . . . .

,y /

I

Figure

2. S i t u a t i o n

(a,b),

evaluated

appreciably

by the s u p e r m o l e c u l a r

the e l e c t r o n i c

we have c o n s i d e r e d when

of the A c e t o n e m o l e c u l e

populations

electronic

they w e r e changed,

in the s i m u l a t i o n

approach,

b e g a n to a f f e c t

of the atoms.

populations

cube.

In this p a p e r

"appreciably

affected"

for any a t o m of the c o n s i d e r e d

molecules ,

in 10 -3

a.u. or more.

by c o m p a r i n g CS 2 pure from

We find the p r o p o r c i o n a l i t y

an a v e r a g e

liquid,

of e x p e r i m e n t a l l y

say ~ S S = 3 . 4 0

supermolecular

CNDO/2-RHF

admited

factor

~=OAB/RAB

values

of

OSS in

A, w i t h the R s s ( C S 2 - C S 2) o b t a i n e d calculations

RSS = 3.85 A Generally

the v a l u e s

tive o r i e n t a t i o n

for RAB have a s l i g h t d e p e n d e n c e

of m o l e c u l e s ,

mun v a l u e

of the r e s u l t s

this way,

we have o b t a i n e d

-Carbon

Disulphide

OCC = 3.23 A

J

OCS = 3.36 A

I

OSS = 3.40 A

on the r e l a -

and we h a v e c h o s e n for RAB the m i n i

f r o m a serie of r e l a t i v e the f o l l o w i n g

system:

CS 2 ..... CS 2

orientations.

parameters

interactions

In

for the A c e t o n e

80 ~C (i) -C = 3.23

A

OC(1)_S

= 3.23

A

aO (2)-C

= 3.35

A

OO(2)_S

= 4.24

A

0C(3)_C

: 4.15 A

OC(3)_S

=

3.36

A

OH_ C

= 3.01 A

OH_ S

= 2.62

A

The really

physical

important

meaning region

using an e x p e r i m e n t a l The

using

During

of

matrices

CNDO-RHF

the

periodical

were

simulations((N,V,T)

number The

of p a r t i c l e s

positions

tion

parameters

6@=6@ than

of

the u s u a l

tential.

6r= 0.075

is due of

the f l u c t u a t i o n acceptation lations

a.u.

are

to the h i g h

the r i g i d

of

classical

for

20%

Due

of

instead

of

with

imposed.

with

simula

of m a s s e s

and

parameters,

po-

in the

a reduction

of r e a s o n s

the u s u a l

smaller

of our

functions

implies

kind a gre~

conditions

the Eab

kind

its own

in this

by the n a t u r e

, that

to this

potentials

hence

generated

these

imposed

[14]

and

the c e n t e r s of

been

the molecules.

samples

boundary

randomly

slope

core

probability.

rate was

with

were

for

have

for

assumption,

is v a l i d

The v a l u e s

MC p a r a m e t e r s ,

This

neighborhoods

molecule,

This

to the p e r i o d i c

=2 ° for o r i e n t a t i o n s .

PCS2_(CH3)2CO

wavefunctions

fixed.

of CS 2 m o l e c u l e s

the

molecules,

parameter.

the a c e t o n e

left

is to d e l i m i t

between

P C S 2 _ C S 2 and

ensemble),

due

approximation

electronic

simulation

images,

this

the a t t r a c t i o n

well-known

interaction

obtained

of

for

~CS 2 ...... (CH3)2CO Interactions

value

our

of

best

for MC

simu-

(50%).

Results

Markov's lattice number 3) and since

chain

fussion

and

with

significative

the

structure

liquid

invariant. sulphide

molecules,

In o r d e r

leads

potential

After

occurs

by a d d i t i o n a l

fuctions

between

to c r y s t a l

cycles.

fluctuations

is no c h a n g e d distribution

differences

to e v a l u a t e

according {H4,H 3} the

energy

the a v e r a g e

{C3,C 7}

To c a l c u l a t e

potential

in 1 . 7 8 x i 0 6

(RDF)

acetone energy

and

this (Figure

cycles, remain

carbon

di-

by m o l e c u l e ,

kcal/(mole.mol).

classified ;

radial

Neglecting

is t h e n - 2 8

our

thermalization

of cycles,

the a t o m - a t o m

/N,

builted

the a t o m - a t o m

to their

chemical

; {I15,H6,H9,HI0}

RDFs,

the m a i n

RDFS,

solute

equivalence: (See F i g u r e

expression

was:

atoms {C I}

were

; {02}

;

1 for n u m e r a t i o n ~

81 3 AnAD

gAD(r)=

(15) 4~p D [ (r+Ar) 3 -

where

D can be an C or S atom,

PD is the d e n s i t y atoms tred

located in A.

number

between

For

us,

Of

course,

structure

of

liquid

Elau

,

T

T

21

26

35

I

42

A is a n y o n e

of D atoms and

two

only

interest.

(r-Ar) 3]

spheres

RDFs

AnAD

should

the

acetone

(r+Ar)

gAC

atoms,

is the n u m b e r

of r a d i i

solute-solvent

other

of

and

and gAS

be n e c e s s a r y

of D

(r-At)

RDFs

cen-

are of

if c o m p l e t e

was d e s i r e d

T

T

49

56

6

3

T

70

77

84

91

98

T

105 112

~

119 126 133 140 1'47 1,54 161 168 175 162 169 196 CYCLEE

, 104"

30 -3 5

4.01

Figure

3. C r y s t a l

The

lattice

structure

sampling

med

in the e q u i l i b r i u m

and

for

RDFs.

the

All

Ar w i d t h

the

the d i f f e r e n t Figure gAD

shapes

RDFs

on c a n o n i c a l

state, has

been

classes

1.35xi06

taken

0.2646

averaging

{CI~ ..... {H5,

the g C I _ C

show very

ensemble

using

are o b t a i n e d

4 shows do not

fussion.

RDF.

H6,

been

perfor-

cycles,

A for all A a t o m s the g A D

H9,

Excepting

remarkable

has

simulation

functions

and over

HI0}. for

this

regularities

function,

(copies

can

the be

82

obtained ted,

from

the authors).

since acetone

alterations surrounds

This b e h a v i o u r

is a ramified

in the structure

molecule

would which

of CS 2 liquid.

the acetone m o l e c u l e

according

have

been expec-

introduces

to M a r k o v ' s

chain

(builted

by m e a n s

of the quantum m e c h a n i c a l

potential)

periodic

boundary

conditions

sample.

shows an aspect position effects

that would

of the C 1 atom. produced

and bottoms produced

influence)

of the steric effects

causes

origines

effects

and

These

spheres

the atractive

between

due

and to the

The gc -C to t h e l c e n t r a l

displays

irregular

atoms

shapes

with radii r ~ r In short,

, used

potential

form of each gAD

function,

24 ?

0 2

r (A)

atom-atom

radial

distribution

and

of the sample.

i!

4. g C ( 1 ) - C

in

the c o m b i n a t i o n

long range

the m o l e c u l e s

(tops are

(or its r e p u l s i v e

;

Figure

the

of the acetone m o l e c u l e

functions.

the c h a r a c t e r i s t i c

the interlocking

"classic",

of solute

each pair of

of the gAD(r)

to the

gAD functions

distributed).

interposition

between

the c a l c u l a t i o n

The other

by the r a m i f i c a t i o n

irregularly

by the

imposed

be called

great

The CS 2 m o l e c u l e s

function.

88

As g e n e r a l

features

i° gAS ~is

we can

functions

remark:

became

L~)lies ~ t

non

zero

before

the gAC

in the neig~iorhoods of ~ e

functions.

acetone,~lecule,

sulphur atc[ns are preferently oriented toward acetone atoms. 2 ° Roughly, there good 3 ° All

for e a c h

test gAD(r)

verifying test The

of

of

nAD(r)

These

the

loss

nAD(r)

are

functions

sphere

tend

of

of r a d i u s

of

defined

of

to a gAC m a x i m u n No doubt,

for g r e a t

order.

the gAD

This

functions

values

by: (16)

the n u m b e r

r. As e x a m p l e ,

of D a t o m s Figure

inside

5 shows

65

50

0

t-

25

0

:~

~" r(A)

Figure

5. n C ( 1 ) _ C

integrated

of r,

evaluation.

gAD (r) r 2 dr

give

is a

is a n o t h e r

function.

2

this

the c a l c u l a t i o n s .

to u n i t y

long-range

the r e l i a b i l i t y

= 4~PD [ r

corresponding

and v i c e v e r s a .

reliability

functions the

nAD (r) f u n c t i o n s

centred

radius

is a gAS m i n i m u n

atom-atom

number

function.

the

the A nCI_C

84 PERSPECTIVES

out,

In p r e s e n t

paper

in c e r t a i n

cases,

quantum

potential

employing

;

structure

is b u i l t

~EA~a .

of a b - i n i t i o as

Internal

account

easily.

lecular

integrals,

computational

studies

of c a r r y i n g

from

from C N D O / 2

analytical

wavefunctions

On the o t h e r improved

tions)

and

trying

the use

of

of

the use

molecule

without

range

can

used

by

bimo-

too m u c h

the

representation

unestabilizing

spin

into

expansions.

single-excitations

be r e p l a c e d

to e v a l u a t e

potential

(to a l l o w m o r e

expanding basis

be t a k e n

increasing

of m u l t i p o l e

short

too,

of o r t h o n o r m a l

approximations

simply

can

is p o s s i b l e

solute

improved

the

very much

the t h e o r y

lcB~b

combination

the

be

hand,

;

~Beb

wavefunctions

rotation

may

;

Aea

linear

Moreover

time

can be

of

here

-i ~ RAB

orbitals

functions.

ment

used

--i

The use

possibility

potentials.

112 ) ~-Z h RAB

molecular

the

the a p p r o x i m a t i o n s :

(~2ik2) (A

shown

liquid

intermolecular

The and

we have

fluctua-

in the d e v e l o p

-

eigenfunctions.

ACKNOWLEDGEMENTS

The

authors

G. A l v a r e z

for

are g r a t e f u l l y

helpfulness

indebted

discussions

to Dr.

and

A.

Ba~6n

for r e a d i n g

and

to

the m a n u s -

cript.

REFERENCES

1

L.M.Ses6,

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231

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L. M.

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