Dispersion energy in rare gas dimers from an ab initio perturbative procedure: Ne + Ne, Ar + Ar

Voluine 3 1, number 2

CHEMICAL k-iY+CS LETTEkS ..

‘.

1 March 1975

.:

,.

.”

D~S~ERS~O~E~~R~Y ~~~GAS~~RS FROMM AB.INITI[O PERTURBATIVE

.:

‘. PROCEDURE:

Ne + Ne, Ar -Fir

Eke FbXANSKI Equ.$pe de Recherche No. 139 du ‘CN.R.S,, Instituf.Le Bel, Urliversite”Lorrk Pasteur, 67008 S~asbourg;Fronce Received 11 November 19j4

The disper$ion energy between two neon and two argon atoms is computed from ;~nab initio pertuzbative procedure not based on the multipote expansion A comparison with the multipofe expansion provides C6 = 5.36, for Ne and 76.6 for AI. -it is seen that one d polarization function provides the main part of the C’&? contribution, the exponent of this function probably being related to the polarizability of the molecule. The multipole expansion seems acceptable up to the van der Waak miqimum but quite invtid for smaller distances, and,doubtful at the van de1 W&s minimum its&T.

direction afong the ~terato~c axis [1,2]. Other studies have shown that, using a double zeta basis set for the s functions, a p polarkrtion function with exponent 0.2 is necessary to describe the dispersion energy between two hydrogen mote&es [S], while in the case of Hz f Li* [4] where Li’ is not very polarizable, we need a p function with exponent 0.775 for the Iit&m. In the present paper, this study is extended to the case of two argon atoms. We then have occupied

1. lnfroducticn

In a previous note [l] we have given a description of the dispersion energy between two neon atoms from an ab initio perturbative procedure. Though the C, coefkient-obtained is slightly smaIIer than most of the other detestations proposed in the Literature, it is nevertheless within the range of possible values. Besides the quantirative accuracy, one of the interesting fearures

of the method

in terms of molecutcrr

is to provide

orbit&,

about

information,

that part

3p orbitals.

correlation energy which.corresponds to the. dispersiori energy of the system. One striking result is that, using a double zeta basis set for the s and p functions, ~one d polarization function with exponent 0.4-F is .. reLsponsible for the main part of the dispersion energy, ., More sipecificaliy, we showedthat each occu$ed p orbital cs.n be.q.ssociated with d orbitals which correlate it ii two.ways, for inst&rce a pZ orbital with a ‘d, (or c&) corresponding to a description of angular conel&ion, or with a d orbital pointing alcng the z ‘tis (dZz_Y2or daz_+r2), these d orbitalS providing only : a small contribution to the atomic correlation’enegy.: ‘Each pair of such orbit& in one molecuIe interacts :..’with similar pairs in the other molecule, the interac.. tion_,being siionger’ if the orbit& have some of their ,,:.

.’

..

,.

.,’

‘.

: ‘.

‘.:

.:,:_

Stiecizl

is paid to the prJssibi&y

expansion

is a fti

approximation

up to the van der

Wagsminimum [5]. Our calculations seem to show that this approximation is still acceptable near the van der Waals minimum, but doubtful at the van der Waals minimum i&lfand quite i&&d for smaller distances.

2. Details of the calculations - the role 6f the $oku&afion function ‘Using a ‘dcubfe ~~~~~~~i~* scheme, the so-called “H&tree-Fock dispersion term” is given by the expreqion [1;3] ”

.’

‘,

‘.

:.,, ,,. ‘.,

attention

of a seriesexpansionof the fobrmCeR-” f C;SR-” * C$? 7”. It is often contidered that the multipole

of tfie

.

._’ -. : ..,

,

: ”

.,

301

Dispersion energy between two neon atoms a? d = 5 au

Basisa) .

corresponding to the use of an Epstein-Nesbet partition. Q b,-,are the occupied orbitals of the molecules A and B respectively, ai, bj the corresponding virtual .orbitals, with energies eaD,ebo, ea., eb _.These orbitals, built fro,rn linear combinations of ga&ian functions, are-obtained from an SCF calcul.ation performed on the isolated molecules or atoms, As shown in ref. [I] for the case of Ne, the effect of one d. function is particularly striking: We can see from table 1 that bases Bl, B2’and B3 which have no d functions provide only a very small part of the dis-..persion energy. Bases B1 and B2 are built from the uncontracted and contracted functions respectively taken from the work of Huzinaga [6]. In basis B3, one diffuse s and one diffuse p function are added to basis-B2. Bases B4 and B5 show the importance of one d fin-ction, with exponents 1 and 0.45 respectiveIy. In the rest’of the calculations, we have used basis B6 where one d function with exponent 0.45 is added to basis B3. The-results obtained with bases 555 and B6 are close-to the first term of the multipole expansion using the (=6 coefficient recommended by Dal: garno [7]. Foliowing these results for two neon atoms, we have treated similarly two aigon atotis using a double zeta basis set for the’s and p.functions [S], extended with one d function. We have found that the best exponent for the d function is 0.28 while OS.5 jrovides only half of the dispersion energy’. For both dirners, ‘Lheoptimization of the-basis set was first performed, for one interatomic distance and then checked for other points, before and around the van der Waalsminimum as.well as for large distances. The same exponent is the best for every point. This leads us to think that this polarization function is related to an atomic characteristic involved in this interatomic interaction. As mentioned above, we also need one p polarization function for H2 and Li’, with exponents 0.2 and 6.775, respectively. We need a d function with exponent ,0.45 for Ne and one with 0.28 for Ar: it seems that the more polarizabie the molecule, fhi more~,diffi+e the polarization function: Though ,,

.302

. .

,EiF (10e4 au)

ES(y (au)

Bl

-0.201

-128.52673

B2

-0.129

I33

-0.275

- 123.51821 -125.51925

B4

-1.905

-3.926

-128.51863 -128.51’875

B6

4.256

-128.51928

ref. [7]

-t.o32

t B.5

a) The notation (95) [42] means that we have 9 s and 5 p functions contracted respectively into 4 s and 2 p. In ;his contraction, the more diffuse p function is kept alone. Bl = (95) [95];B2 = (95) [42]; B3 = B2 + one s and one p function with exponent 0.2; B4 = B2 + one d function with exponent 1; B5 = B2 + one d function with exponent 0.45; B6 = B3 -:‘one d function with exponent 0.45.

these polarization functions are not very important for the SCF energy of each atom *, they are essential to build the first order wavefunction. The annlysis lecular

:, -,

.-.-

of the contribution

shows

in terms

that we can neglect

of mo-

tlie inner

shells: or& the valence shell provides dispersion energy. A!so, v/e have the same preferential coupling between certain pairs of orbitals for Ar, as we found for Ne.

3. Comparison with the multipole expansion - range of validity of this expansion

Table 2 allows a comparison between our calculations

;;;-iJ ,!;ome other

determinations

based

on the

multjpoie expansion for two neon atoms. As mentioned in ref. ]I] many values have been proposed for C, mainly in the course of the past few years. Most of the values he between 5 and 6. The interest in this problem has notdecreased since besides the values. 6.352 [llj, 6.35 [12], 6.48 [9], 6.55 [9], 5.44 [i3], 6.925 [14:1,6.882 [15], 7.727 [5], 5.308‘[5];6.3 [7], 7.03 [16];7.75 [16], 6.68 [17] given recently (other values obttied from older works can also be found * These d functions contribute to the SCF energies tlirough the comb+ation x2 + y* + a* whic‘nis not. eliminated in ttLisealcu~ation~ ‘. ‘.:.

..

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ortitals

Volume 3 1, number 2

Table 2 Dispersion ensr,y

1

CHEMlCAL. PHYSICS LETTERS

bztwean two neon atoms

(in IO4

au) 3)

From ref. [5]d)

From refs [9,1O]c)

d (au)

work

From ref. [ 7]b) (6)

5 5.5 6 6.5 .7 10 20

-4.256 -2.337 -1.348

4.032 2.276 1.350

4.192 2366 1.404

-0.814

0.835

0.868

-0.511 -0.0544 -0.846(-3)

0.535 0.0630 0.984(-3)

0.557 0.0655 1.023(-3)

Present’.

e,

March 1975

(6!

(6,8)

(6,8,10)

(6)

(6.8)

(6, a, 10)

5.656 3.049 1.744 1.048 0.656 0.0712 1.046(-3)

6.371 .3.325 1.860 1.100 -0.681 0.0719 l-053(-33

3.392 1.915 1.136 0.703 0.450 0.0530 O&28(-3)

4.958 2.645

5.988 3.042 1.667 0.969 0.592 0.0601 Q.862(-3)

a) The notation (6) corresponds to the tern C& d; (6,8) corresponds sum C,R -G + C8R-8 + C,oR-‘o. b) C6 = 6.3. c, C6 = 6.55; C8 = 57.2; Cl3 = 698; CB/C6 = 8.73; c,,/C8 = 12.20. d) C6 = 5.3; C8 = 61.18; C 10 = 1006; C&6 = 11.52; C&8 = 16.44. e, -0.848(-3) =-0.848 X 10-3.

to the sum

CsRd

+ C8Rb;

1.500

0.895 0.557 0.0591 0.8.52(-3)

(6,8.10)

corresponds

to the

in ref. [9,17]), we can now add: 6.11 [18], 6.27 [19], 7.24 [20]. Having obtained respectively 5.38,5.368 and 5.365 ford = 40,60 and 80 au, we consider that 5.36 is a good extrapolation’ofC6 from our proce-

be compared with 8.73 [lo], 8.714 [19], 15.6 [2I], 16.8 [21], 8.943 [Xl, 11.53 [S], 10.73 [lS]. As with C6, we should check whether our small values are an

dure.

mean

As we mentioned

in ref. [l]

,this

-efficient

is

rather smaller than most of the other values proposed. In the case of two hydrogen molecules [3], or Li” + H2 [4], the dispersion energies at large distances obtained with a double zeta s set plus one polarization function are very close to the values found with larger basis sets. We have no guarantee that it is the same for two neon atoms..The program is now being modified to reduce the computation time in view to checking if a very large basis set would increase this value of C6. In table 2 we cornTare our results with the values obtained from the coefficients proposed by Dalgarrio [7] and by Starkshall and Gordon 19, lo], which are the most used in the literature. We have also included the results of Broussard and Kestner [5] where c6 is very close to our own value and therefore allows us to see more easily the behaviour of ‘the higher order terms of the multipole expansion. We can see that the discrepancy between the first and the 7th column is more important at small distances than at large distames. There are two reasons for this: - The first one is that oyr C, is smaller than the other values generally proposed. Taking C, = 5.36 and considering that CloR-lo is negligible at d = 20 au gives C, = 28.243 and C,/Ce = 5.270 (a:). This can

effect

of the basis set. If this were

the case it would

that this b&s set gives a better description of the induced dipole moment than of the higher order terms. - The second point seems to be independent of the basis set. If we determine Cs in the same way for other interatomic distances, we see that it reaches a maximum at d = 5.5 au and decreases sharply at shorter distances. Repeating the same procedure with ce = 5.36 and Ca = 28.248 to determine CL,, we have then a sim.iIar behaviour of Cl0 as can be seen in table 3.

Table 3 Determination

d (auj

of Cl0 (in au) Neon

Argon

c, = 5.36

c, = 76.6 C, = 393.68 C&/C, = S-L39

C, = 28.248 C&Z6 = 5.270 -5 5.5 6 6.5 7 7.5

100.1 160,8 187.1 196.4 193.5

142.1 6532-L 7215.4 7312.6 719 :.4 6457.4

a

‘164.1

10 ‘.

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303

Vohlmc31;

number 2.

1 Much 1975

CHEMICAL PHYSICS LEtiRS

:

It is striking that the decrease begins around the position of the van der Waals minimum, the change beco,&g steep-for smaJler distances (see d = 5.5 au). This leads usto think that between large distances .’tid the van de; Waals ‘minimum, the increase is due .to the neglect of the higher order terms of the multipole expansion, while the decrease at smaller distarices comes from the overlap between the two ,atoims. Such a sudden change at the position of the van der Waals.minimum,‘due fo the overlap, has also been seen for the dispersion and the first order 1 ,tetis [23] iu the study of the effect of the nonorthogonahty of the molecular orbit& of two interacting hydrogen mole&es. In the present case, ‘we would conclude that the multipole expylsion is acceptable frornlarge distances to near the van der. Waals m&mum, is.zriously doubtful at ‘Lhevan der Raals niinirimm itself, and quite inadequate at ,shorter distances. A similar expansion has.been made for two argon a‘t oms. Table’4 gives our calculated values together with the C6Rm6 tern of Palgamo [7], and the C6Rw6 + C&f A Cl$?‘O expansion of Starkshall and Gordon [?, lo]. From.our procedure, we have obtained respectively 76.820,.76.688 and.76647 ford = 40,60 and 80 au, so that 76.6 should be a good extrapolation of C6 at very large distances. This coeffi-. ‘cient is rather larger than the &her values proposed: 55 [7],67.68 [9],64.5 [24], 69.3 [16],70.2 [20], 62.5 [18],.%.8 [25], 62.2 [26], 66.89 [27]. At d = 20 au, we obtain Cg = 393.68 which makes C&b =

5.1.39. This ratio can be cornpared with 23 [21], 24 [21], 16.65 [lo], 7.75 [24], 16.713 [25], 17.59 1261,. 16.02 [22:), i6.75.1261. As for Ne, we must keep in mmd that the large C’, and the small ratio C& could be an effect of the basis-set, even if our results : are in rather good agreement with the total expansion .of Starkshall and Gordon at distances larger than the van der WklS rrkimum position (table 4). The disagreement begins to be important at 7.5 au and table 3 shows that this distance corresponds to. the decrease of C,, which is probably due to the overlap as for Ne. This effect would be still more important if the non-orthogonality of the atomic orbitals were taken

into

account

since it has been shown

overLp decreases the d&r&n

that

energy

Table4 Dispersion energy between two argon atoms (in 10q au) Present work

Ref. [7]a)

From refs. [9,lO]b)

6)

(6)

6,8)

6 8.1’3

5.5 7

32.825 11.877 7.449

23.482 8.619 5.524

24.450 8.974 5.753

31.934 11.517 7.711

47.645 14.344 8.582 5.367

d (au) is 7.5

4.827

3.652

3.803

‘J-930

8

3.224,

2#419

‘3,255

3.484.

8.5

2.211 0.812 0.0121

2,582 1.794 0.677 0.0105

3.209

2.334

0.790 O.OllO

0.814 0.0110

10

:.

,20

91 C6 = es.‘W C, = 67;68; Cs = 1129; &= ;. ‘.

3oi4

:,,

‘,

‘-

1.723 0.650 0.0102

24600; C, /C, = 16.68; CIO/C; = 21.78.

‘-

this

[23,28]. This neglect of the overlap is not the oriiy source of error: Thakkar and Smith [22] using the expectation values calculated by Waber conclude that the relativistic effects are not negligible. Also the neglect of the intraatomic correlation energy can affect our values of Ch and C,. In conclusion, besides the advantage that this procedure is E.direct ab initio.deteimination of the dispersion energy, we think that, our description c’dnbe of use boti for semi-empirical determinations and for other ab initio methods. For the former, always based on the multipole expansion, we think that we provide complementary information about the physica phenomena (effect of the overlap, nature of the interactions). We hope ti,at we shall be able in the near future to give more accurate results, from larger basis

_’

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Volunie 31, number 2

CHEM;CAL PHYSICS LETTERS

sets and including the effect of the overlap and of tithe interatomic correlation. For other ab initio methods, which generally start from a supermolecule treatment .evkn if they afterwards separate .the different contributions, it could be of some help to know the role of the polarization functions and the coupling of pairs of orbit&. This is clear in our procedure because we treat the dispersion energy independently. But, as we also found in calculations where the overlap is included [23], a general treatment does not allow a so simple description. : Acknowledgement

L6l S. Huzinaga, J. Chem. Phys 42 (1955) 1297. 171 A Dalgarno, Advan Chern. Phyr 12 (1967) 143. [RIB. ROOS and P. Siegbahn, Theo;& Chim. Actr 17 (1970) 209. PI G. Starkshall and R.G. Gordon, J. Chem. Fhyr 54 (1971) 663. 1101G. Starkshall and R.G. Gordon, J. Chem. Phys 56 (1972) 2801. [Ill R.P. Saxon, J. Chem. Phyr 59.(1973) 1.539. [I21 R.J. Bell and A.E. Kingston, Proc. Phys Sac (London) 88 (1966) 901. i131 S. Kaneko, J. Chem. Phys 56 (1972) 3417. 1141N.C. Dutta, CM. Dutta and J.P. Dss, fntcrn I. Quanturn Chem. 4S (1971) 299. 1151M.B. Doran, J. Phys. B 5 (1972) L 151. 1151 P.W. Langhoff and M. Ku-plus, I. Chem. Phyr 53 (1970) 233.

The calculationshave been performedon the Umvat 1108 of the Centre de Calcul de StrasbourgCronenbourg (Centre de Recherches Nucle’aires du C.N.R.S.). The fmancial support ofthe C.N.R.S. is gratefully acknowledged.

References [l] E. Kochanski, [2] E. Kochansb, be published[3] E. Kochanski, [4] E Kochanski, [S] J.T. Broussard (1973) 3593.

1 March 197.5

Chem. Phys. Letters 2.5 (1974) 381. Intern. J. Quantum Chem. 8S (1974), to J. Chem. Phys. 58 (1973) 5823. Cl-rem. Phys. Letters 28 (1974) 471. and N.R. Keshxr, J. Chem. Phys 58

.

tii/

P,W, Laughoff, Chem. Phys. Letkrs 12 (1971) 217. R.

Ahlberg

iUld

0.

GorinsXi,

1191 A.J. ‘Ibakkar ar.d V.H. Smith.

J. Phyr

B 7 C1974)

hloL whys

1194-

27 (1974)

593.

WI S.F. Abdulnur, J. Chem. Phys. 56 (1973) LLO9. 1211N. Alvarez-Rizatti and E.A. hfason, J. Chem. Phys

59 (1973) 518. 1221A.J. Thakkar and V.H. Smith, J. Phyc B 7 (1974) L 321. [231 E. Kochanski and J.F. Gouyet, Mol. Phys, to be published 1241 J.M. Parson, P.E. Siska and UT. Lee, J. Chcm. Phys 56 (1972) 1511. WI A.J. Thakkar and V.H. Smith, MaL Phys 27 (1974) 191. 1261R.E. Caligaris, O.H. SceIise, J.P. Grigera, V.H. Smith and A.J. Thakkar, Can I. Cl-rem. 52 (1974) 2444. (271 MB. Doran, J. Phyr B 7 (1974) 558. [28; M.P. Briggs, J.N. Murrell and. D.R. Sdahub, hfoL Phys 22 (1971) 907.