Journal of Molecular Structure (Theochem), 149 (1987) 193-200 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

POTENTIAL ENERGY SURFACES FOR THE X+* l * COz (X = Na, K) SYSTEMS*

FERNANDO MOTA and JUAN J. NOVOA

Dept. Q&mica Fisica, Fat. Quimica, Univ. de Barcelona, Au. Diagonal 647, 08028- Barcelona (Spain)

JUAN J. PEREZ

Dept. de Q&mica and Centro de1 Medio Ambiente, ETS d%nginyers Industrials, Au. Diagonal 647, 08028 Barcelona (Spain)

(Received 13 January 1986; in final form 10 July 1986)

ABSTRACT

In the present work we report the computed potential energy surfaces for the Na+... CO, and K+* * * CO, systems, obtained at the Hartree-Fock level, using the recently proposed MINI-l basis set. The interaction energy of both systems is computed at the same level and also, with the inclusion of the electronic correlation along with second order MCller-Plesset, using the MIDI-l* and further extended basis sets. These results support the accuracy of those obtained with the MINI and MIDI-l* basis sets when the counterpoise correction is carried out.

INTRODUCTION

Ions present in the atmosphere are believed to play an important role as centres for nucleation in the processes of aerosol formation [l, 23. The mechanism of such processes, although not yet well understood, may proceed via a reaction of attachment which, in the presence of a third body, yields either AB+ or AB- species, followed by consecutive clustering reac- tions giving rise to either the AB,’ or AB; systems. A knowledge of the importance of the interaction energy as well as the kinetic data for these reactions would be helpful in trying to understand the formation and evolu- tion of both the pollutant and natural particulate matter which constitute atmospheric aerosols.

In the last few years a tremendous effort has been made in order to sum- marize the existing thermodynamic and kinetic data for the ion-molecule reactions which occur in the troposphere [3]. Nevertheless, there are serious difficulties in attempting to acquire these data experimentally. These are due to the sophisticated equipment required for these types of measurement and their cost. For these reasons, in this report we will use standard quantum

*A contribution from the “Grup de Quimica Quantica de Catalunya”.

0166-1280/87/$03.50 0 1987 Elsevier Science Publishers B.V.

194

chemical methods as an alternative method for data acquisition in our at- tempt to improve understanding of the process on hand. In what follows we will show that the interactions involved in the description of the K’s . l CO* and Na+.** CO, system can be properly described within the Hartree-Fock framework when MINI-l and MIDI-l* basis sets are employed. Both systems are known to be potential participants in the gas-to-particle formation mechanism in marine aerosols.

COMPUTATIONAL METHOD

This study was initiated with a search, at the Hartree-Fock level, of the potential energy surface for the interaction between the Na’ and K+ ions and the CO, molecule, as the most important contribution to the ion-molecule interaction energy comes from the Coulomb term between the cation and the molecular multipole moments; they are correctly described at this level of theory. The dispersion energy accounts for only a tiny percentage of the total, as is shown, for example, in the Na’* . * Nz and K’* * . Nz systems when the correlation is computed. In any case, we will test in this work its impor- tance for the systems studied using a second order M$ller-Plesset method.

The basis sets selected for the surface search are the MINI-l and MIDI-l* [4-lo]. As has been shown recently by Hobza et al. [ll] , such basis sets seems to ‘give reasonable results for this type of system. In terms of the stan- dard notation commonly employed, the MINI-l base is of the type (6,3) on the carbon and oxygen atoms, (9,3) on sodium, and (12,6) on potassium, all of which are contracted to a minimum base, while the MIDI-l* uses the same set of primitives in order to obtain a contracted split valence set and then adds a set of uncontracted polarization functions.

The quality of the basis sets employed in this study has been tested using pilot calculations on the systems Nz l * l Na’ and Nz l - l K+, in order to com- pare our results with those recently reported by Pullman et al. [12] using extended bases. The calculations have been performed at one of the near minimum geometries quoted in ref. 12, that is, taking the distance N-N as 2.074 ao* and the distance X’ -N as 4.689 a, for X.= Na and 5.726 a0 for X = K. In both cases a C!,, symmetry was employed. These and all the other calculations reported here were carried out using the GAUSSIAN-80 program [13]. Table 1 shows the calculated interaction energy values for the Nz - * * X’ systems, along with the different basis sets that were used. The basis set superposition error was corrected using the counterpoise method [ 141.

As can be seen from Table 1, for any of the basis sets, the superposition

*The atomic system of units is used: a, = 52.918 pm, E, = 4.3598 x lo-” J, 1 a.u. of dipole polarizability = 1.6488 x lo-” C m J-l, 1 a.u. of quadrupole moment = 4.0319 X

10eS9 Cm*.

TA

BL

E

1

Tot

al

ener

gy (

in a

tom

ic

un

its)

an

d in

tera

ctio

n

ener

gy

(in

k

cal

mol

-I)

for

the

Na+

. . .

N,

and

K+

. . *

N,,

com

pute

d w

ith

th

e R

artr

ee-

Foc

k

and

seco

nd

ord

er

Mdl

ler-

Ple

sset

(i

n

pare

nth

eses

) m

eth

ods.

E

(X+

),

E(C

O,)

, an

d E

(X+ .

. .C

O,)

ar

e th

e to

tal

ener

gies

, w

hil

st

E(S

CF

) is

th

e in

tera

ctio

n

ener

gy

com

pute

d fr

om

the

prev

iou

s va

lues

, an

d E

(CP

) is

th

e in

tera

ctio

n

ener

gy

corr

ecte

d

by

the

basi

s se

t su

perp

osit

ion

er

ror

usi

ng

the

cou

nte

rpoi

se

met

hod

.

Sys

tem

Na+

. . *

N,

Bas

is s

et

MIN

I-l

MID

I-1

MID

I-l

*

Ext

. B

asis

Exp

.

E(X

+)

E(N

,)

E(X+ *e-N,)

-160

.697

419

-108

.090

275

-268

.804

143

-160

.697

985

-108

.273

315

-268

.989

117

-160

.708

902

-108

.371

749

-269

.096

267

(-16

0.70

9340

) (-

108.

6843

61)

(-26

9.41

152)

-1

61.6

7043

5 -1

08.9

7713

6 -2

70.6

5873

3

E( SC

F)

-10.

3 -1

1.2

-9.8

(-

11.0

-7

.o

(-

7.5)

E(C

P)

Ref

.

-9.2

T

his

wor

k

-9.3

T

his

wor

k

-7.8

T

his

wor

k

(-8.

1)

Th

is w

ork

-6

.8

WI

(-7.

4)

1121

-8

.0

f 0.

5 [2

3]

-9.5

12

41

K+

. .

. N

M

INI-

l -5

96.2

5835

2 -1

08.0

9027

5 -7

04.3

5824

8 -6

.0

-5.2

T

his

wor

k

1 M

IDI-

l -5

96.3

3835

7 -1

08.2

7721

5 -7

04.6

2134

7 -6

.1

-5.2

T

his

wor

k

MID

I-l

* -5

96.3

4548

3 -1

08.3

7174

9 -7

04.7

2625

6 -5

.7

-3.9

T

his

wor

k

(-59

6.37

6306

) (-

108.

6843

61)

(-70

5.07

0751

) (-

6.3)

(-

4.0)

E

xt.

Bas

is

-599

.008

870

-108

.977

136

-707

.992

323

-4.0

-3

.6

WI

(-4.

4)

(-4.

0)

[l21

196

corrections are not particularly large for these systems when compared with results obtained using other basis sets of the same level of quality (STO-3G, for example); this is in accordance with previous results obtained by Hobza et al. [ll] . Furthermore, the results for the MINI-l and MIDI-l* show an artificial stabilization of about 1 kcal mol-’ in comparison with the available experimental data. This may be due to the poor description of the molecular polarizability of the molecule as this property contributes to the leading factors of the molecular interaction (the Hartree-Fock MINI-l values of the electric properties for Nz are: O,, = 1.03 e&j, O,, = -2.07 ea& a1 = 2.8 ai,

ali = 7.8 a; whereas the corresponding experimental values are O,, = 1.04 ea& @Z, = -2.08 eq& crl = 9.79 a& crll = 16.06 a& respectively [15], where the polarizability is computed using the finite field method [ 161 ). Nevertheless, by comparison of the yield/effort ratio for the different basis sets shown in Table 1, it can be concluded that MINI-1 is a realistic compromise when a large number of computations on such systems are required.

RESULTS AND DISCUSSION

Using the MINI-l basis set, we have computed the interaction energy for a grid of points on the X+-CO2 potential surface, which were selected in order to cover the whole surface. In all the calculations, the COZ geometry was fixed according to the experimental values (a bond length of 2.192 ao, D -h symmetry), and the cation position was defined by means of the C... X’ distance and the angle 0 formed by the C-X’ and C-O bonds. The results of this surface scanning are shown in Table 2 for X = Na, and Table 3 for X = K. All the values in both tables were obtained at the Hartree-Fock level, and have been corrected for the superposition error by the counter- poise method.

TABLE 2

Potential energy surface grid of the CO, . *. Na’ system in kcal mol-‘. The total energy values for the CO, and Na+ are -186.19158 En and -160.69742 Eh respectively. R is the C-Na+ distance, and 0 is as defined in the text.

R/a, 0 (degrees)

90.0 78.5 66.1 53.1 42.1 31.8 18.4 0.0

3.0 132.9 144.6 199.5 387.0 803.4 1541.8 4199.6 -

4.0 27.3 27.8 31.9 54.6 119.5 265.3 636.3 1056.6 5.0 9.3 8.5 6.0 3.0 3.8 13.0 41.9 74.8 6.0 5.1 4.5 2.3 -1.6 -5.6 -9.1 -11.9 -12.7 7.0 3.3 2.9 1.4 -1.2 -4.0 -6.9 -10.1 -12.1 8.0 2.3 2.0 1.0 -0.8 -2.6 -4.4 -6.5 -7.8 9.0 1.6 1.4 0.7 -0.5 -1.7 -2.9 -4.3 -5.1

11.0 0.9 0.8 0.4 -0.2 -0.9 -1.5 -2.2 -2.6 13.0 0.6 0.5 0.3 -0.1 -0.5 -0.9 -1.3 -1.5

197

TABLE 3

Potential energy surface grid of the CO, . .. K+ system in kcal mol-‘. The total energy values for the CO, and K’ are -186.19158 Eh and -596.25835 Eh respectively. R is the C-K+ distance, and B is as defined in the text.

R/a, 0 (degrees)

90.0 78.5 66.1 53.1 42.1 31.8 18.4 0.0

3.0 336.8 374.5 540.4 1019.9 1564.4 2813.7 7597.9 -

4.0 62.3 68.8 98.2 190.5 382.5 714.2 1334.1 2018.3 5.0 14.3 14.5 16.3 25.6 49.7 97.5 200.3 301.0 6.0 5.7 5.2 3.7 1.7 1.2 3.3 10.8 19.1 7.0 3.4 2.9 1.6 -0.8 -3.2 -5.3 -7.3 -8.2 8.0 2.3 2.0 1.0 -0.7 -2.5 -4.2 -6.2 -7.4 9.0 1.6 1.4 0.7 -0.5 -1.7 -2.9 -4.3 -5.1

11.0 0.9 0.8 0.4 -0.2 -0.9 -1.5 -2.2 -2.6 13.0 0.6 0.5 0.3 -0.1 -0.5 -0.9 -1.3 -1.5

As can be seen from Tables ‘2 and 3 both potential energy surfaces show a minimum only when the cation is placed along the line of the internuclear axis of the COZ molecule. The optimization of the C-X’ distance, upon freezing the COZ geometry, gives a value of 6.240 a0 when X = Na, and 7.153 a0 for X = K. The interaction energy values computed at these points are -14.3 kcal mol-’ for Na’, and -8.5 kcal mol-’ for K’.

Taking as geometry for the complexes those that were computed by op- timizing the C-X+ distance using the MINI-l basis set, we completed the study of the X’ - - - CO2 systems with some computations using basis sets of better quality and including the correlation energy by means of a second order M$llerPlesset method. The results of these computations are shown in Table 4.

The best basis set included in Table 4 is that labelled as SV + P. This con- sists of a “split valence” plus a set of five-membered polarization functions on all the atoms. It was constructed from Veillard’s (12s, 9p) basis set [ 173 for the sodium cation with an exponent value of 0.6 for the a! functions, whilst for the carbon and oxygen atoms the basis set is Van Duijneveldt’s (lls, 7~) [ 181 taking as exponent values for the d functions a value of 0.88 for the oxygen and 1.16 for the carbon. The general scheme of contraction has been used [19]. The total energy obtained with this base is, in all the cases, near to the Hartree-Fock limit. For the COZ molecule, the reported value differs only by 0.0458 E, from the Yoshimine and McLean [20] value, whose value is believed to be within 0.003 E, of the Hartree-Fock limit. For the sodium cation, the total energy value is more accurate than that reported by Pullman et al. [ 121 who employed a basis set constructed from the Veillard’s (12s, 6~) basis set contracted into [6,2,2,1/4,2,1] and supplemented by two d functions. Their value is within 0.0064 E, of the Hartree-Fock limit [21], ours being more accurate by 0.000352 Eh.

TA

BL

E

4 I-

’ s

Tot

al

ener

gy

(in

ato

mic

u

nit

s) a

nd

inte

ract

ion

en

ergy

(i

n

kca

i m

ol-‘)

fo

r th

e N

a’ .

* ’ C

O,

and

K+

- . .

CO

, as

soci

atio

ns,

co

mpu

ted

wit

h

the

Har

tree

- F

ock

an

d se

con

d or

der

M

olle

rPle

sset

(i

n p

aren

thes

es)

met

hod

s.

E(X

+),

E

(CO

,),

and

E(X

+ .

* . C

O,)

are

the

tota

l en

ergi

es,

wh

ilst

E(S

CF

) is

th

e in

tera

ctio

n

ener

gy

com

pute

d fr

om

the

prev

iou

s va

lues

, an

d E

(CP

) is

th

e in

tera

ctio

n

ener

gy c

orre

cted

by

th

e ba

sis

set

supe

rpos

itio

n

erro

r u

sin

g th

e co

un

terp

oise

m

eth

od.

Sys

tem

Na+

...C

O,

Bas

is s

et

MIN

I-l

MID

I-1

*

6-31

G*

sv

+ P

E(X

+)

E(C

C,)

-160

.697

419

-186

.191

577

-160

.708

902

-186

.630

306

(-16

0.70

9340

) (-

187.

1166

65)

-161

.659

454

-187

.632

878

(-16

1.65

9812

) (-

188.

1061

35)

-161

.670

787

-187

.677

000

(-16

1.73

2630

) (-

188.

1057

78)

E(

X’

. + . C

O,)

-346

.916

303

-347

.366

841

(-34

7.85

3038

) -3

49.3

1129

1 (-

349.

7855

05)

-349

.364

897

(-34

9.85

5530

)

E(S

CF)

W

P)

Ref

.

-17.

1 -1

7.3

(-17

.0)

-11.

9 (-

12.3

) -1

0.7

(-10

.7)

-14.

3 -1

3.0

(-11

.4)

-11.

2 (-

10.8

) -1

0.2

(-9.

6)

Th

is w

ork

K+

...C

O

1

Exp

t.

MIN

I-1

MID

I-l

*

Exp

t.

-596

.258

352

-186

.191

577

-596

.345

483

-186

.630

306

(-59

6.37

6306

) (-

187.

1166

65)

-782

.466

477

-10.

4 -7

82.9

9083

8 -9

.4

(-78

3.51

0910

) (-

11.3

)

-13.

7 f

1.0

v31

-8.5

T

his

wor

k

-7.4

(-

8.4)

-8.5

11

1

199

Using the SV + P basis set, the value of the interaction energy for the Na+ . . . CO2 system, computed with the second order M@ller-Plesset (MP/2), is, once corrected by the basis set superposition error, -9.6 kcal mol-‘. The corresponding value computed using the Hartree-Fock method is -10.2 kcal mol-‘, that is, it is only 0.6 kcal mol-’ more stable than the result ob- tained using MP/2. Therefore, the inclusion of correlation energy does not greatly affect the interaction energy in this system. This is in full agreement with the results already shown in Table 1 for the Na’ - - * Nz and K’ * * - N2 systems using the extended basis set.

Another remarkable point is the relatively high quality of the other results quoted in Table 4 for the Na’ - - - CO2 system when compared to the SV + P ones. One can see that the MP/2 corrected interaction energies computed with the standard 6-31G* and the MIDI-l* basis sets differ by 1.2 and 1.6 kcal mole*, respectively, from the SV + P value. Such values are small when one realises that the total energies computed using the 6-31G* basis set are lower than the MIDI-l* values by almost 1 Eh in all the cases. Further- more, the superposition error in the MIDI-l* basis set is also higher (about 4-5 kcal mol-*) than the error in the 6-31G* basis set (between 0.7 and 1.5 kcal mol-‘), but the MIDI-l* basis set is less expensive computationally. So, it seems that the MIDI-l* is a reliable basis set for computation of the interaction energy of these systems. The same fact is also deduced from results in Table 1 for the systems included there.

Based on the previous conclusions, we finished the computations using the MIDI-l* basis set on the K’ - - * CO2 system, with and without the inclu- sion of correlation energy at the MP/2 level. The results, shown in Table 4, agree with the experimental value when the basis set superposition error is taken into account at the MP/2 level. At the Hartree-Fock level, the MINI-l basis set also gives a surprisingly good value. In general, this basis set gives, at the Hartree-Fock level, values of the interaction energy corrected by the superposition error which are not far from the experimental value, as is also shown in Table 1.

A comparison of the computed interaction energies with the experimental values, shows that the agreement is fairly good. Discrepancies can be assoc- iated with the fact that the experimental values correspond to enthalpies at 300 K. Estimations of ‘the translational, rotational and vibrational contribu- tions to the enthalpy are sufficient to stabilize the system by about 1 kcal mol-*. When such comparisons are undertaken, the agreements between the experimental and computed values of the interaction energies are quite good, with a maximum error of about 3 kcal moli, in the worst case. This could be due to the fact that the X’ - * . CO2 systems are not fully optimized, or could arise from the need for more flexible bases in order to obtain a proper description of the second order properties, such as the polarizability. For the COZ molecule the computed value of the polarizability at the Hartree-Fock level is alI = 14.8 a: and (Ye = 2.9 ai obtained using theMINI- basis set, the experimental values being alI = 27.3 a$ and (Ye = 13.2 ai [22].

200

Finally, it is worthwhile to comment upon two facts arising from the computations presented here: the size of the Na’. * * CO2 interaction, which is about twice that of K’ - - - COz, and the lack of a barrier for formation of any of the two associations studied here. Therefore, the most important fac- tor in the kinetics of formation will be the concentration of the C02, Na”, and K’ in the aerosol. It has been estimated that the concentration of car- bon dioxide in the unpolluted atmosphere is about 5.49 X lo5 pg rne3. On the other hand, the concentration of the sodium cation is estimated to be about 1.5 Mg mm3 with that for the potasium ion being fifteen times less. Therefore, the association Na’ * - * CO2 can be argued to play an important role in the process of particle formation, being even more important than that played by the association between Na’ and N2 given the natural occur- rence of N2 and CO2 in the atmosphere.

ACKNOWLEDGEMENTS

Financial support given by the CAICYT under grant No. 0657/81 is fully acknowledged.

REFERENCES

1 A. W. Castleman, J. Aerosol Sci., 13 (1982) 73. 2 H. S. W. Massey, in E. W. McDaniel and B. Bederson, (Eds.), Atmospheric Physics and

Chemistry, Vol. 1. Applied Atomic Collision Physics, Academic Press, New York, 1982, p. 105.

3 J. T. Herron, R. E. Huie and J. A. Hodgeson (Eds.), Chemical Kinetic Data Needs for Modeling the Lower Troposphere, NBS Special Publications No. 557, Washington DC, 1979.

4 H. Tatewaki and S. Huzinaga, J. Chem. Phys., 71 (1979) 4339. 5 H. Tatewaki and S. Huzinaga, J. Chem. Phys., 72 (1980) 399. 6 H. Tatewaki and S. Huzinaga, J. Comput. Chem., 1 (1980) 205. 7 H. Tatewaki and S. Huzinaga, J. Comput. Chem., 2 (1981) 96. 8 Y. Sakai, H. Tatewaki and S. Huzinaga, J. Comput. Chem., 2 (1981) 100. 9 Y. Sakai, H. Tatewaki and S. Huzinaga, J. Comput. Chem., 2 (1981) 108.

10 Y. Sakai, H. Tatewaki and S. Huzinaga, J. Comput. Chem., 3 (1982) 6. 11 P. Hobza and J. Sauer, Theor. Chim. Acta, 65 (1984) 279. 12 A. Pullman, H. Sklenar and S. Ranganathan, Chem. Phys. Lett., 110 (1984) 346. 13 J. S. Binkley, R. A. Whiteside, R. Krishnan, R. Seeger, D. J. DeFrees, H. B. Schlegel,

S. Topiol, L. R. Kahn and J. A. Pople, Dept. of Chemistry, Carnegie-Mellon Univer- sity, Pittsburgh, PA 15212 (U.S.A.)

14 S. F. Boys and F. Bernardi, Mol. Phys., 19 (1970) 553. 15 A. D. Buckingham, R. L. Disch and D. A. Dunmur, J. Am. Chem. Sot., 90 (1968) 3104. 16 H. D. Cohen and C. C. J. Roothaan, J. Chem. Phys., 43 (1965) 534. 17 A. Veillard, Theor. Chim. Acta, 12 (1968) 405. 18 F. van Duijneveldt, IBM Res. Report, RJ945 (1971). 19 R. C. Raffenetti, J. Chem. Phys., 58 (1973) 4452. 20 M. Yoshimine and A. D. McLean, Int. J. Quantum Chem., Symp., 1 (1967) 313. 21 H. Kistenmacher, H. Popkie and E. Clementi, J. Chem. Phys., 58 (1973) 1689. 22 N. J. Bridge and A. D. Buckingham, Proc. R. Sot. Ser. A, 295 (1966) 334. 23 R. A. Perry, B. R. Rowe, A. A. Viggiano, D. L. Albritton, E. E. Ferguson and

F. C. Fehsenfeld, Geophys. Res. Lett., 7 (1980) 693. 24 K. G. Spears, J. Chem. Phys., 57 (1972) 1850.