Investigation of the interaction between molecules at medium distances

Chemical Physics 8 (1975) 192-200 Q North-Holland Publish@ Company

lNVESTIGATlON 1. SCF LCAO

OF THE INTERACTlON

h10 superuwlecule,

for two inters&q

BETWEEN MOLECULES AT MEDIUhf DkXN’JCES

perturbational

and mutually consistent chh~~fls

HF and CHzO molecules

I. Introduction T)I~ calcu)~tjon

of jntrractjorl

energies

brtrseen

molecuks

is ofgcner;ll

intcrcst.

NOst

calculations

performed distances 1 I 1. Thw jnvest@tlans have shown that at large intermolecular distances inttxsction energy csprcssions based on pert\& bational cspansions are sufficient to obtain 3 reasonable qproxtmation. whereas at small distance betwscn tk interacting molecules (I/< 2.5 AI the two molecules have to be treated a~ one common supermolecule [ 11. CInd\e other hand problems of real interest for the orsanic chemist or molecular biologist are interacttons of organic: mola cults at medium (3-4 A) distances. For tlw interaction oforynic molecules at medium distances until now only perturbsfional calculations have been performed using 3s unperturbed one-electron functions the MOs of (he single molecllks obtajncd fronl difkrcnl semiempiricd calculations [ 2) _(mw prrpers cited in II] give also t)~e usscl P.Yprrssions of Ihr different perrurharic& terms [4] *.) On the other hand. Lischka 131 performjng ab initio ~+alcu. lations with a large basis XI and taking Into sczoun~ a considaablc pxt of the correlation hs foufjd for t\,~ interacting He .*. HF. He ... H-&I H-, ... HF and Hz ... GO systems at intermolecular distances of 3-4 9, that the pcrtutbation inrerxtion ener$x calculared wvith the Ad of the wrrect perturbational expressicms including second order yield only about SO per cent of the inrermion energies provided by the supermolecule approach. 7% iw ‘WnlS to be not WfOUn&d that by interactions of organic molecules the error may became even larger at till’ s3rw disranccs. jn a I~igl~er

approsimaIion

refer,

howtwr.

to small mokcuks

at largc.or small intclrmohxu~ar

TO investigate this rathtzr inrricarr! problem WCstarted an investigation of the interactions between mokculc5 3t medium distances. AS CI first sfep we iwe taken two interacting I+F and Cl-l,0 molecules, respectively. Tfw inrzraction energies have be?n computed in three different uays: f 1) ive ~NC: takn

the two

interacting molecules as common superrnolecule

and an &

initio

SCF LCAI) ?I0

f! Ozro. J. LadiklJrzreracrrorz

calculation (2) The

bcrwwrz nzolccrrles az rrzcdrrznzdzsrazzccr

for the supermolecule and the constituent

has been performed

193

molecules.

inrernction energies hirve been compured takirlg Into xcounl

perturbational

the clectrosta~ic (in LIIC mOnopOle approximation), the polarization. the exchange and the charge transl’er wms [-I] siarting from (11~ah inilio SCF LCAO hlOs of the unperturbed single molecules. (3) A further model (see below) has been clpplred which takes into account auwmafica11y the rnulual pokwztion of the interacting molecules and so provides better starting wave funckns for the cddxiorr of the rctruining terms in the perturbational scheme. In the present starting stage of Ihe invcsligation WCh~vc noI included the dispersion rcrm in ~ltc inlerrrction energies because [he ab initio supcrmoksuk calculations have not [aken into account the corrclarion. The proposed model could. however, be modified IO include correlation and rhen if has 10 bc f&n into account also in Ihr two other methods. This will be discusssd in a subsequent paper. The ;I(

aim

of these

investigations

is CO work

results 2nd ~‘3” be mow easily hnndlcd

IO llle supcrrnolcculc the heavy have ab

been the

For

atoms

wd

contracted

orbilnl

For rhc crestmcnc

7/3

to 4,

I,

+ 4 and 7/j

2 s functions

1. I s funcrions

and conlracrlon

and 2,

brt\rccn

approtich,

Iargcr

but gives

mol~.ul~

nrorc KCW,~~C

cxprrssmns.

f 7 gausslan

for the hydrogen

I p functions

cocfiicicnrs

oiinrcrxtions

~IKIII tlw supermolcculo

thorn the pcrcurbarional

cslculstions 3, respectively

esponcnts

inirio calcuhions

out ;I model

distance whicll is Its: evpwsive

medwm mrermolecular

lobe

basis (7s

;lloms)

for the lwauy

rhc vrrlues given

and 3 pX. pY 3~d

p_ lunctions

havs been used. The primirive atoms

by tiuL[naga

and to 3. 151 have

IIWL’ been CSL’CU!C~oft 311 IBM 360/91 L’OII)~UIP~using rhc program

written

for

funclions

1 6 functions

for J-1.

been upplicd. by Meyer

TIIC and

Pulay [61. For

3 HF

molecule

have been takn

the bond

d P u= I. 13 ,a, o = I 16.5’) &&les energy

was taken

oTO.92

S 113s been

= $CH,O

the ald of Ihe 7/J

_ +H20 f 4 basis.

in li_c.

D = 0.48 A.

I. The goornctry of a single CH20 molcculc (dc-0 of the supcrrnoleculc,

i.e., niaxirniLing

11ilZ

inwraction

,

The two CH,O molecules

Fig. 1. ‘The relzGve position of rhc tRo nOer~ctrnc HF molecules; R = 3.704,

uSed (71.

lvhcreas the IWO intsracfing HF molecules = I .?I & from Takagi and Oka (81, The relative geomelry of IIVO interacting CH,O _ shun

WIS obtained by ntrntmrzing the 1~31 crwrg of the two CHzO molrculcs, dctkd as &.A’.

with

distance

in [he rclarivt’ position

(1) wrc’

rakw

in parallel

phnh

in 311 ~rr~ngcf~iei~l

35

FIN. 2. TIC ~CI~IIV~powion ai IIICtno mrcrdmg CH,O molcculcs. ,\losr %!ble conformalion: K = 3.?OA. D = -0.30A,13=0’.

shown in fig. 2. Varping the interplane distance R, the distance D idched we

Ca.11 Set

rrom

fig.

j,

jar

the

srsblc configurntion

most

in fig. 2) and the angle 13one gets, 3s .A. D = -0.30 3 ;~nclfl= 0’.

of 111~IH’O molecules R = 3.X

(The latter angle is defined 3s the roution

angle of one of the molecules around the axis perpendicular

planes of t[lc nlo\ecules ;rnd going

one

till:

in(crac(ion

pcrlbrm

th?

energ

I7

iS -3.

calculations

with

~11subsequent calculstions

thruu&

kCai/n~ole.

all

the

rhrce

of

7Iux@

the

carbon,

IO ckCk

rcspectivd)’ ti7~’ proposed

OXv!$n

mod4

methods for the whole inreraclion

atoms.)

it would It~c

curve. 3s first

resprlctivrly

IO 1112

COIlfi$mltiOn

been better 10

step wr: hsve cx~ured

by this peometq’.

To investigate the dependence of the total energy on the basis set besides the 713 SCTL(9s

thij

In

13 s and 5 respectiwly

the hydrogen atoms) was applied

ior

\l\e

7 pA, pY and p_ functions single CH$I

also

;I

9/5 and ;1 IX/7 basis

for the heavy 31oms and 1 s functions

molecule. The primitive

functions

ior

have been contrscrc~

IO

I. I, 1 respecrivzly 3, J, 3, 2 s functions nod 4, I respectively 4.3 p functions for rhe hcnv atoms and IO 3, 1 s functions for H. The corresponding computer times were 0.5, \ .5 Jnd 3.8 minutes, rcspecriwly, If as SCF

6.

criteria I&+‘)

- pr,rl (‘0 G 10-lowae

used for all &mentSp,, t”) (o btained in the jtrh itcrnrion

The supermolccule calculation of the two CHZO molecules usuq the 7/3 basis set 1135 t&en IBhf 36Op

the pzrturbation

berwcm

rheory wilh overlap of Murrcll

Slater A05 belotlging IO difierrnt

rncrgy 3s 3 power series expsnsion of i?S2

13 minutes

on the

I computer.

Following intqyds

step) simultaneously.

ct al. [4] (which

can be applied until

the owrlap

molecules are smaller than 0.1) one gets the intrracrion

in the intermolecular

potential

0 and the overlap integral S. Up lo th? order

the interaction cnrr&g terms 3re

&P.T.

=E

CLSi. * E pot

+ 4~

Hew the ternIs EeLLI,, KpoL 3rd Ed,, laP wrurbation tributionsoforder

hvY.

+ &ch.rr

+ E dq.

are of zzroth

The contribution

+ kwh.poL

.

order in overlap and havs the same form, as in the zero owr-

of order US2 leads to the e.u&nge

U-7Sz may bt! subdivided into the charge transfer

energy, EeYcll. and the cnsrgy an-

energy Ech,_. and into the exchange’polariz)

195

tion and dispersion terms due to the exchange. Since the latter have neglected

this term

7Te electrostnfic

term is much smaller

thsn tllc o~ltcr ants

WC

[4j.

in our calculations.

term EccLa is in the monopole

approximation

(3) where 2:

is the nuclear

molecule

A and B. respectively.

of the

charge

92

ath nucleus

is the total

in molecule

clcctronic

A. IV,

and hfB stand far the number of nuclei in

charge of the ath 3tom

in molecule

A. whrch

we have

hfulliken’s population analysis 191 of lhc ab inilio SCF results and finally rwl is the distance Ir, bctwcen the 0th ato111in mokcule 11 and the /31l1 0112 in mokcule U. Ofcoursc. in more sophisticated future

taken

from

calculations hutions

use inslcad

unc could

of the two ntoleculcs

the first-order

density

The polarization

of the

applyirrg

monopolr:

either

appro~imaricm

multipole

expansions,

(3) a more refined

analysis

or use 3 three-dimensional

-

r,,l

of the clrsrgc dtstri. mesh to dcscribc

rn~~tnces.

term E,,o,. has the exprcssiun

I’_c)

(5)

II:

and rrk xc

ener@s

the numba

of molecule

for the Coulomb In zschnnge

of filled

hlOs in molcculc

A and B, respectively.

and cxch~n~~ term Ec,ch

A73

inrcgrals.

L$. $‘.

A and B. rcsprctively. I$

and $

E,‘, E,!‘, E! and E: arc one-electron

zrc the corresponding

hlOs and .I,‘,. h’p,. etc.. stand

rrspectwly.

IUS tllc Form 1-I)

411’ terms

TheEclLu,= Ecll.t1.+ hrr.

can be drrivcd

from

lhc

cxprchons

given ifl 141 as

196

0)

(In eqs.(6)

and (7) I~IC convention

Iqk~i *) =(~,(1,~,(7)lr;~‘l~~(1)~~(7))

C~,~,l’ij

was used.)

one

compares perturbational interaction energlcs Finally the dispersion term Em, should not be included if wrirh those obtained in the supcrlnolecule approach in the Hxtree-Fock level. and thereiore we do not write down here its esplicit Expressing

espression

(given

terms of the AOs and LCAO

coefficients.

which have been used in the actual

Instead

or solving

p’“p;

of the free single molecules of the partner

To we

it is easy to derive

we do not write

down

the corresponding hew

these rather

cspressions lengthy

in

iormulae

equations

Z&B I I

one can modify

molecule.

form

spxz

calculations.

the Hartrec-Fock

P$=+$.

trntial

in ref. [IO] ).

all AlOs itt (1). (6) and (7) in al LCAO

(9) their

Fock operxors

This means if we introduce

F’ and FB making them dcpcndent on tlrl: ~3. again LCAO hlOs WL’ have to solve the matrix equa-

tions F”zP

I

=$&r\

I

I ’

FBzB

I

= &jBeR

I

where the elements of the matrices FA and FE. respectively. F$=r;;:, Hew

+(&VBI$).

(IO)

I ’

p;s = F$+@IrAI~;).

are now defined as Ill)

P. Otto, 1. Ladik/hrtcraction is

the

usual

Fock-matrix

element

with

electron

term crTs has to be calculated

solution

of the modified

4

pc = -3 c 1I.U

1.1,

Since

wlrh

n~olecrrles at nwdlum dnronccs

change that the charge-bond

order

157

matrix

the aid of the eigenvector-components

element

T,:, which

pz”,

of the two-

one obiains

from

the

problem,

&c

1=l

the only

bctwem

(C

I.”

A, B).

=

(13)

the potentials

e=(r) depend which

=Jfi

(C = A, B)

df’

on the charge distributions requires

mutually

YB, respectively,

pC(r’)

consistcnr

the simple

of the parrrrtp III tk

solutions.

monopolc

( I I) dcfinc a problem

cqs. (IO) wit11 clcmcnts

molecule.

present

we have used for the potentials

calculations

I’*

and

approsimarion

(15)

where HF

the total

problem

present

electronic

with

only

niuhipolr

the aid ofa

with

and +F

can ayin

population

ltle nrsr ~pprosnnarlons

with the aid ot’a pC(r’)

charges $$

the aid of hlullikcn’s

IO rhe potcntlals

espansion

large number

oi

be obtained

tinalysis.

from

It should

(14) which

rl~c charge distribtirions

the vector

be emphasized, be obtuincd

could

or SIIII bctrcr

of point chargss in a Ihrcc-dimensional

Z, * and Z,B of the modified

however,

rhnt eqs. (I 5) re-

in a belter

approxtmation

by approsimating

rhc functions

mesh and performing

the mtcgrs-

[ion (I 4) nunicrically. Returning IWO

to the calculations

molecules

relative position

consistent

(as it was the exe

solutions,

with

Ihe aid of rhe simple

in our calculations,

if we solve the SCF problem

iteration

step besides the newly

ab initio

program,

for

the HF

both

performed

A and B arc of the same type and they are from

calculated

&fv

basis set and SCF crireris moleculss

and the CHzO

for one of thz molecules

also rhc newly

as described

molecules,

(I 5), we can obscrvc

computed

?:I.

(The

into

For this procedure ?.I),

above (see section

rcspcctlvcly.

substituting

solution

15-20

iterations

of the problem

that if the

in a svmmetric

of view of interactions

the mutually

1 and 2), we can obtain immediately

see figs.

(IO)

cspressions

the point

F”

in each

using the same were needed of the free singe

molecules required rhc same numbers of irerations.) Having rlie total

obtained

the murually

consistent

solutions

of eqs. (I 0) with (I I),

(I 2) and ( 15) w hv~ IO compute

energies

(C = A, B) Taking pzV

the difference

of the original

of the total unmodified

energies EC and EC (calculated

problem)

and adding

with

(16)

the charge-bond-order

to it (3). we c3n write

for the imenction

matrix

clcmcnrs

enera

(17)

$ Of course this advnntngc CM be used also III the ca>cii WC apply for 111spotentials SKBIIS than (15).

1“(C

= A. 8) ntorc complicated chprcr

bcrwccrr nroledes

I? Otto. 1. Ladikjllrtcractron

198

electronic

where the Fjt, etc., are again the total Eq. (17) gives in addition

charges obtained

at medim

from

dimma

the mutudlY

Consistent

Solutions~S,

polarization contribution also- Thus we do not have to calculate the complicated expression (6) for this Intter quantity, which the proposed new method gives in a simpler (and more accurate) (kxch.~

&?CF

= ‘%I, is defined

been taken

into

+ ‘&h. by cq.

account.

JJSO the murua]

The

term the

in this way wz also get better

Further

in eq.(z).

where E,, 10 fleai

way.

IT*.)

-%h.tr:

to the electrostatic

IIUS

nl:e can write

+ ‘%h.tr. + %‘hirp. +

‘%,ch.poL

(17). In our calculations,

corr&tion

wave functiOnS

for

remaining scheme

the

energy in the proposed

terms

(19)

’

described

here. in (19) OdY Eint., Eekch. and &h.lr. have of the idea of the mutually consistent field

2nd the ehtznsion

of .I?+,,

~dcuh!ion

starting

for the interaction

of the electrons

in interactirlg

molecules

will be dtscussed in 3 subsequent

paper.

3. Results In table

I we give the tot31 energies of CH,O

(in kcal/moles)

+ -1 his

set (and in rhr

case of HF also wirh 3 7/3 + 2 basis set). Finally

lated with

a 7/j

individual

terms of LL!T~.~. and LU?~~~’ .o gel 3 better

Table I Tile lolal enerpc, ofCHIO

obminsd

energies basis sets. Table 7 gives the interaction in the relative position given above, calcll. resp cctivcly,

using different

of the two HF and twu CHZO molecules,

with dtilcrcnr

bzsls scls

insighht into

in table 3 WC give 111,~

the problem.

T’abk 2 TIIC lntcrxtwn

cncrgics (in kcd/molc)

oi t\\o Hl’~nd

~\to

CH,O molcculcs

(in alomic unirs)

_-----I___ Basis set

Compufcr

Totd

cncrg

Sglem

Llasis set

time (s) 7/3 + 4 3) 915 + 4 13/7+4 -_____-

64. ____

-30 90

-I 13.68889 -I 13.76855

228

- I 13.82730

--

d 7 E and 3 pSu.p! and pz iuncuons lor the hcdvy amnlr. and 4 s functions ior tltc hydrog,x aloms

&.hl. (1)) ___--

tip.‘.

&ICF

bi

_-____

ZHF

713 f 2

-2.44

-1.93

‘HI:

713 + 4

-2.57

7/3 + 4 _____.

-3.17

- 1.49 - 4.02

2x,0 __

a)

- I.91 -1.47 --3.86

3) The liujt four terms of rq. (3) with the .q3prok~m~tc e\. pression (3) for Ecl.st.b) Tltc lirst three terms of cq. (19) wh

tltc appro~m3rti

ehprcssion (I 7).

T1 In (I 7) I\< could UW. of course. ho

prorimst?d

better (SW cq. (14)j.

a barer approlimstion

than ths monopole one if the potrntuls

In tfut casscWC would obtain instcJd of (I 7)

I rC (C = A. BJ !\ew .ip-

199

Table 3 The different --

calculated constituents

of &‘*T.

and ,&‘CF

(in kcal/mole) _.

GP.T. System

Basis

;\EhKF

E cl.w3)

E pol. b,

931

Ec,ch.C)

%~t,.~)

@A

- EA)

%.s,.

+ Epot.“)

~e,ctI.‘~

‘%,.rr.@

-1.96 -1.46

0.00 0.00

0.00 -0.0’

-3.56

0.09

-0.68 __-___

_____-

ZHF ?HF ‘CH,O

713 + 2 713 + 4

-1.89 -1.43

0.0-l -0.04

0.00 0.00

7/3 + 4

-3.17

0.00 -0.02

-0.25

010

-0.70

3) Cabulatcd b) Calculated c, Calculated d) Calculacd

from rq. (3). from cq. (4). from cq. (6).

g) CalculJtcd

from eq. (7) with 11x mutually

to.05 +0.03 +0.19 -_____

irom q(7). c) The fifth term (the double sum) of cq. (I 7). 11 Calculakd from q. (6) uith IIIC mutually conslrtcnt solutions. conslstent solutions.

4. Discussion

hokill:: 31 txble 1 we c3n 2 3.S eV which is nbout t&k

same basis set has bwn

thorough

the atomization larger [ion

calculation

Judgrmcnt energy

well. Looking between

interaction

the intcrtiction

encrg

bctwccn

only

111~’quality

rnthcr

bxis

Further,

Fo&

linlt[

was to corn--Ï

calculations.

with

of 3 CH,O

molecuk

litc

one has 10

investigation

cspensivc

(SW

c~;actly

To

obtain

the aid of each one

the exact value. This we have not done, because we wanted

with

the perturbetional

and

tllc

MCF

smaller values. than the supermolecule one should

Ire not without

the intramolecular

IO

the basis set is much

with

observe

error.

enera

schemes give for the iotcrar-

results which

thet in IIIC case of the smaller

the somewhat

Sarnely

sybsystcm

cncrgy.

31 least for two interacting

each

7/3 + 3 basis SCI the

larger 713 + 4 basis showing thet using a limited

the basis functions

of the other

agree with

and the charge trans-

tllatin this case both the exchange

than with

gets too large an interaction

the ealculotions

mo]cculcs

if in a]] calcul~[ions

of IIIC basis sets used one had to calculate

into table 3 WC cm SW further

SCI one usually

to repeat

two CH20

expect)

au

encrgics.

HF molccul~s

calculations

is (3s one would

] j/7 basis set is -0.1-l

and

close 10 [llc Har[rsc-

the ahovc mentioned

11121tlte chsngc of the to131 cncrgy

small.

found

schemes can bz compared

the three schemes is better

partly used to improve future

about

encrgics

molecule

the 7/j

energics are basis set dependent. II should be pointed out. however.

the supermolecule limited

forntaldehydr:

with LI large basis set. Since IIKZ aim of the present

interaction

of the polar

fer terms are vnnishingly agreement

of a single

rhc inreractmn

cncr~

to tables ? and 3 we can SW that troth

energies

other very

than

and to compare

than the c~lculatcd

Turning

lnryer

interxtion

sclwmes. we have not pcrformcd

in table 1 only.

demonstrate

enorgy

the total

used and to obtan

3. supermolccule

pare the different ;I more

20 times

different

2). Ikrefwe

perform

SCClh3t

OII Ihe b3sts. The difference in total energy between

rather dcpendrnt

that also the basis SC~ SO

on one of the siibsystems

are

and vice versa. In this tray ustng a

To ovrtrcomr this difficulty

wc intend in tltc

HF molecules with a basis set large ettouglt to provide

total energies near the Hartree-Fock limit. In the ease of two interacting CH,O nml~culcs hlCF

model give larger

worthwhile

to point

out

interaction that

in both

cases the crude

WC Kind that both tllc perturbational sclicmc and the proposed than the supermolecul c v~~luctlw hlCF result bring closer to it. It I$

the charg c transficr

and in the MCF schemes. From aI] these we c3n conelude using

energies

that

tttonop&

term

is rarhzr

large (-

-0.7

kcnl/mola)

tlte proposed model works sontewl~at bcttcr tktn apptosilxttion.

In subscqucnt

publications

both

tk

in rhc perturbational

perturbational

one

WCshall report the resultsofa

200

P. Urro. 1. Ladik/lrilrracrrotl

berwen

n~olenrles

ar medrum

disrances

more sophisticated version of the method using a three-dimensional mesh of Point ChWFS instead of the monopolc approximation, and Slater’s local exchange potential of the partner molecules built into the one-electron part of the Fock operator of the molecule under consideration. If in this way the exchange term can be approximated well enough and the charge rransfer cuntribution is small. one can expect that in this way one can get rather near to the Supern&c&

result Ms.“‘.

ob:ained with a renscnably large basis set to avoid Ihe “superposition

The model would have the advantage [o avoid the espensivc c&ulation

occur both in the supermolecule and perturbational bational treatment,

3pproxh)

especially if the number of interacting

and

of the illtL?rrTlOkCUhr

could be still more mUrate, is larger than two-

integrrrh

error’..

(w!lich

than the pertur-

moloculcs

Acknowlegement We should like to express our gratitude to Professor G.L. Hofacker for this continuous interest and support whicll made It possible to perform tllis investigation. WC are further indebted to Professor W. hleycr for his help in putting his programs at our disposal and for useful discussions.

The financial support Finally.

of the DeuMle

Forschungsgemeinschafr

we should like lo thank the Rechenzcntrum

should gratefully be acknowledged. fur Plasmaphysik for giving us

des hlas-Planck-lnstituts

compurer time on their IDhI 360/91 computer.

References

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