Perturbative Formulation of DispersionContributions to Interaction Energy ofvan der Waals Systems of‘‘Closed-Shell]Open-Shell’’ Type

´ ˇ 1 1 ˇ 2VLADIMIR LUKES, VILIAM LAURINC, STANISLAV BISKUPICDepartments of 1Chemical Physics and 2Physical Chemistry, Faculty of Chemical Technology, SlovakUniversity of Technology, 812 37 Bratislava, Slovakia

Received 23 November 1997; accepted 11 January 1999

ABSTRACT: Restricted Hartree]Fock molecular orbitals have been used forthe evaluation of dispersion contributions to the van der Waals interactionbetween closed-shell and open-shell systems represented by a simple Slaterdeterminant. The dispersion energy has been computed for the ground state ofHe—H and He—O . The dispersion energy contributions obtained in larger2basis sets are comparable with multipole expansion data and with ab initiocalculations reported in the literature. Q 1999 John Wiley & Sons, Inc.J Comput Chem 20: 857]866, 1999

Keywords: dispersion energy; weak van der Waals complexes; He—H; He—O2

Introduction

ong-range forces between atoms andL molecules play an important role in a varietyof physical phenomena, such as the equilibriumstructure of molecular and rare-gas crystals, trans-port properties of gases and liquids, and low-en-

Correspondence to: V. Lukes; e-mail: lukes@theochem.ˇchtf.stuba.sk

Contractrgrant sponsor: Slovak Grant Agency for Science;contractrgrant numbers: VEGA 1r4205r97, VEGA 1r4199r97

ergy scattering experiments. From a purely theo-retical point of view, the long-range forces havethe advantage of being a well-defined problem.Contributions to the interaction energy can be clas-

Ž .sified in two categories: 1 those of a classicalŽ .origin and 2 those of a quantum-mechanical

origin. Four of the most important terms are theelectrostatic, induction, dispersion, and exchange-repulsion energies.1 ] 5 The origin of the first threecontributions can be traced to monomer propertiesŽpermanent multipole moments and polarizabili-

.ties . The electrostatic induction energies are classi-cal long-range contributions, first considered by

( )Journal of Computational Chemistry, Vol. 20, No. 8, 857]866 1999Q 1999 John Wiley & Sons, Inc. CCC 0192-8651 / 99 / 080857-10

ˇ ˇLUKES, LAURINC, AND BISKUPIC

Debye6 and Keesom.7 The exchange interactionenergy is repulsive and it is a result of the Pauliprinciple, which forbids the electrons of onemonomer to penetrate into the occupied space ofthe partner. It may be conceptually related to theelectron charge densities of interacting monomers,which avoid each other. The dispersion energywas introduced by London8,9 and has a long-rangecharacter. It originates from a mutual polarizationof the electronic charge distributions of interacting

Žmonomers interactions of instantaneous multi-poles that are related to dynamic multipole polar-

.izabilities . An additional type of interaction en-ergy called the ‘‘charge transfer’’ energy is some-times postulated.3,5 So far, the term has eludedrigorous definition and is strongly dependent onthe theoretical formalism and the basis set effects.

For the monomers without permanent dipolemoments the dispersion forces dominate in theintermolecular attraction. They depend on theirmutual orientations in nonsymmetric interactingsystems. If the internuclear distance between twospherical symmetric atoms A and B is large, then

Ž .the nonretarded potential energy, V R , betweenABthem may be expressed as a power series of theinverse of distance R:

Ž . y6 y8 y10 Ž .V R s yC R y C R y C R y ??? 1AB 6 8 10

where coefficient C describes the dipole]dipole6interaction, C the dipole]quadrupole interaction,8and C and the quadrupole]quadrupole and10dipole]octupole interactions. Because the dipole]dipole interaction is dominant, it has attracted thegreatest attention. Reliable numerical results for C6are now available for a variety of van der Waalssystems.3,10,11 However, the accuracy of the C and8C coefficients is much more difficult to deter-10mine because there are no direct experimentalmeasurements of quadrupole and octupole polar-izabilities.

The ab initio methods used for calculations ofinteraction energies can be classified as super-molecular, perturbational, and hybrid ones. In asupermolecular approach, the interaction energy isobtained as:

Ž .E s E y E y E 2int AB A B

Ž .where E E and E are approximations to theA B ABŽ .exact ground-state energies of the A B monomer

and of the AB complex, respectively. It is clearthat, in the supermolecular approach, the interac-tion energy is obtained as a single number. Its

decomposition into terms with clear physicalmeaning is not straightforward, and additionalcalculations are necessary for a better physicalunderstanding of the interaction.5

The perturbational methods compute the inter-action energy directly as the sum of distinct physi-cal corrections:

Ž1. Ž1. Ž2. Ž2. Ž2. Ž .E s E q E q E q E q E q ??? 3int elst exch ind disp exch

where EŽ1. is the classical electrostatic interactionelstenergy, EŽ2. is the classical induction energy, andindEŽ2. is the quantum-mechanical dispersion en-disp

Žergy. The exchange-overlap corrections subscript.‘‘exch’’ may be approximated by different

ways.12 ] 17

The intermolecular perturbation methods usedto calculate the interaction energy contributionscan be developed in the framework of the re-

Ž . Ž .stricted RHF or unrestricted UHF Hartree]Fockapproaches. The application of these approachesbrings certain problems and advantages. The RHFsolutions for open-shell monomers would providethe matrix element of the Hamiltonian which, ingeneral, is not invariant with respect to orthogo-nalization procedures.18,19 The use of the UHFwave function, in describing monomers and thedimer, brings forth the difficulties resulting fromtheir different spin contamination.20 Another com-mon difficulty is that not only that the UHF proce-dure may fail to converge but it may converge to asymmetry-breaking solution.21 The efficiency of theinteraction energy components up to the secondorder of perturbation expansion using UHF treat-ment has been established recently for severalopen-shell cases.20 ] 22

The aim of this work is to use single determi-nant restricted Møller]Plesset perturbation theoryfor the evaluation of the dominant dispersion con-tributions to the interaction energy of van derWaals systems of the ‘‘closed-shell]open-shell’’type. The presented formulas are tested for theground state of He—H and He—O systems,2which are bounded by the dispersion effect. Theresults are compared with the literature data.

Theory

Let us consider two systems A and B formingtogether a composite quantum-mechanical system.The total Hamiltonian in the Møller]Plesset parti-tioning23 for intermolecular problems24,25 has the

VOL. 20, NO. 8858

PERTURBATIVE FORMULATION

form:

Ž .H s F q F q W q W q V 4AB A B A B AB

where F is the Fock operator and W the correla-X XŽ .tion or fluctuation potential for the system X s

A, B. The mutual interaction between systems Aand B is given by the perturbation term V . TheABdispersion energy can be expressed via the expan-sion26 ] 28:

Žn. Žnij. Ž .E s E 5Ýdisp disp

Ž .where n is the order of perturbation V and i jABindicates the order of the Møller]Plesset fluctua-

Ž .tion potential for the A B system. A short de-scription and the explicit formulas for the disper-sion contributions in terms of molecular orbitalsare presented in what follows. Single occupied

Ž .orbitals are denoted by S, whereas D V denoteŽ .spaces over doubly occupied virtual orbitals. We

consider that A corresponds to the closed-shellsystem and B to the open-shell system.

The separation of the EŽ2. terms can be per-dispformed easily by invoking the diagrammatic repre-sentation of perturbation theory corrections. In thissection, we present the derivation and diagram-matic description of the EŽ200., EŽ210, EŽ201., anddisp disp dispEŽ300. contributions to the dispersion energy. Wedisp

Žuse the Goldstone-type diagrams the second- and. Ž .third-order type , and the orbital lines of the B A

Ž .system are drawn conventionally above belowthe horizontal bar separating the monomers. The

Ž .occupied virtual orbitals are labeled i, j, k, lŽ . p qp, q, r, s indices, and n denotes the two-par-i j

² < :ticle molecular integrals in the form pq ij . Thep q p q Žsymbol v corresponds to n r « q « y « yi j i j i j p

.« , where « are the orbital energies.q x

SECOND-ORDER HARTREE]FOCKDISPERSION ENERGY

If the intramonomer correlation is completelyneglected, then the dispersion energy may be ap-

Ž200. Žproximated by E second-order HF dispersiondisp.energy . The corresponding diagram for this term

is presented in Figure 1, and the final orbitalexpression is:

un occŽ200. p q p q Ž .E s D n v 6Ý Ýdisp 1 i j i j

p , q i , j

where the numerical factor D obtained from the1spin summation is given in Table I. The above

FIGURE 1. Diagrammatic representation for E (200).disp

expression is very simple and the HF dispersionenergy can also be calculated easily for largerpolyatomic molecules.

THIRD-ORDER HARTREE]FOCK DISPERSIONENERGY

The value of this correction indicates the rate ofconvergence for perturbation expansion in many-electron systems. This term is visualized by fourGoldstone diagrams, which contain the pure

Ž .Coulombic terms only see Fig. 2a]d . The finalorbital expression for the sum of particle]particle,particle]hole, hole]hole, and hole]particle interac-tion is:

un occŽ300. pr qs qsE s D v n vÝ Ýdisp 1 i j p r i j

p , q , r , s i , j

un occq jpr qry D v n vÝ Ý 2 i j pk i k

p , q , r i , j , k

un occp q i k p qq D v n vÝ Ý 3 i k jl jl

p , q i , j , k , l

un occprp q i r Ž .y D v n v 7Ý Ý 4 i k jq jk

p , q , r i , j , k

where the coefficients D ]D originating from the1 4spin summations are given in Table II.

TABLE I.Multiplication Factors for E(200).disp

System A System B

D p i q j1

4 V D V D2 V D V S2 V D S D

JOURNAL OF COMPUTATIONAL CHEMISTRY 859

ˇ ˇLUKES, LAURINC, AND BISKUPIC

FIRST-ORDER INTRAMONOMERCORRELATION CORRECTIONS

The leading intramonomer correlation correc-tion to EŽ2 i j. corresponds to the first-order pertur-dispbation in W and is represented by the sum ofXEŽ210. and EŽ201.. These terms describe the couplingdisp dispof the intramonomer correlation effect with theintermolecular dispersion interaction. The dia-grams that are assigned to the EŽ210. correction todispthe HF dispersion energy are shown in Figure3a]f. We note that, by evaluating the diagramsŽ .see Fig. 3a]f , using the symbols for border linesin parentheses, we obtain the algebraic representa-tion for the term EŽ201.. The expansions of EŽ210.

disp disp

FIGURE 2. Diagrammatic representation for E (300).disp

FIGURE 3. Diagrammatic representation for E (210)disp

( (201))E .disp

and EŽ201. in terms of molecular integrals can bedispperformed using the standard diagrammatic tech-nique and the final form of these terms is:

un occqr pr p q p jv v D n y D nŽ .Ý Ý jk i k 1 i j 2 q i

p , q , r i , j , k

un occqrpr p q q pŽ . Ž .q 2v n D v y D v 8Ý Ý i k jk 1 i j 2 i j

p , q , r i , j , k

The multiplication factors D and D for EŽ210.1 2 disp

and EŽ201. are given in Table III. It should bedispstressed that the Figure 3c and d is algebraicallyidentical to Figure 3e and f.

TABLE II.Multiplication Factors for E(300).disp

System A System B System A System B

D p q i r s j D p q i r j k1 2

4 V V D V V D 4 V V D V D D4 V V D V S D 4 V V D V D S2 V V D S S D 2 V V D V S S2 V V D V V S 2 V V D S D D

System A System B System A System B

D p i j q k l D p i j r q k3 4

4 V D D V D D 4 V D D V V D4 V D D V D S 4 V D D V S D2 V D D S D D 2 V D D V V S2 V D D V S S 2 V D D S S D

VOL. 20, NO. 8860

PERTURBATIVE FORMULATION

TABLE III.Multiplication Factors for E(210) and for E(201).disp disp

(210)E System A System Bdisp

D D p q i j r k1 2

8 4 V V D D V D4 2 V V D D V S4 2 V V D D S D

(201)E System B System Adisp

D D p q i j r k1 2

8 4 V V D D V D8 4 V S D D V D8 4 V V D S V D2 2 V V S S V D2 2 S S D D V D4 0 V S S D V D

Numerical Results

HE—H SYSTEM

The heteronuclear closed-shell]open-shell sys-tem He—H represents the simplest and an ex-

Ž .tremely weak van der Waals vdW complex. Theknowledge of the interaction contributions andtheir properties for the He—H pair is also ofinterest in astrophysics. The absorption of infraredradiation by He—H is likely to be an importantsource of opacity of cooler stellar atmospheres.29,30

A reliable empirical potential of He—H for thewell region has been obtained by Jochemsen et al.Ž w xD s 22.03 microhartrees mH at R s 6.784e e

. 31bohr . It is based on the low-temperature diffu-sion coefficients measured by Hardy et al.32 Aclarification from the theoretical point of view isvery desirable, because the analysis of the experi-mental data has uncertainties. The diffusion cross-sections are measured with an accuracy of 5%,which leaves an uncertainty of about 10% for thepotential.31 The well energy of the Jochemsen em-pirical model31 is about 20% deeper than the full

ŽCI ab initio calculation of Knowles et al. D s 18.28e.33mH at R s 6.88 bohr using a relatively largee

basis set. These authors felt confident enough toreject the empirical value of the well depth31 onthe basis of their computational results, along witha similarly deep well suggested by Scoles’s so-

Ž .called HFD-B HF plus dispersion model poten-tial.34 Two decades ago, Das et al.35 performed

calculations at the MCSCF level. These calculationshave not been corrected for the basis set superposi-

Ž .tion error BSSE . It is known now that the MCSCFprocedure employed slightly overestimates thebonding contributions in the well region.36 Therecently published ab initio calculations localizedthe vdW minimum at about 6.7 bohr, which is 22.6mH deep.30,36 These results were obtained usinglarge basis sets and the BSSE was eliminated.

The selection of the basis set for the descriptionof vdW complexes is a very complicated task. Theoptimal basis set must contain polarization func-tions to describe the dispersion attractions and theBSSE must be made very small. In our work, fourdifferent basis sets are used. The B1]B3 sets repre-sent augmented correlation consistent basis setsŽ .aug-cc-PVDZ, aug-cc-PVTZ, aug-cc-PVQZ . Thedetails of these sets have been given by Dunningand coworkers.37 ] 39 In addition, basis set B4, con-

w xsisting of 6s5p4d3f functions on helium andw x Ž6s3p2d1f functions on hydrogen g functions were

30 .omitted from the original basis sets of Meyerwas used. This basis set was chosen for a betterdescription of the dispersion interaction. All calcu-lations were performed by the program codes de-veloped in our laboratory and are interfaced withthe G92 program system.40 The Boys]Bernardicounterpoise correction was used41 to eliminateBSSE from the supermolecular calculations. Allcalculations of the interaction contributions weredone using dimer-centered basis sets.

The purpose of our study was not to obtainvery accurate interaction energies, but rather toestimate the role of the developed dispersion en-ergy contributions and the importance of termEŽ200. as an attractive part of the restricted second-disp

Ž RMP2 .order interaction correlation energy D E .However, it should be noted that the predictedposition and depth of the minimum significantlydepend on the basis set used. Their values con-verge monotonously with the increasing of basisset quality to the aforementioned literature data.At the MP242,43 level, we obtained D s 8.39 mHeat R s 7.4 bohr for B1, and D s 15.44 mH ate eR s 6.86 bohr for B4.e

The calculations have been performed for inter-Ž .system distances R ranging from 5.5 to 10 bohr.

In Table IV, the supermolecular interaction energyis separated into repulsive restricted HF interac-

Ž RHF .tion D E and attractive second-order correla-Ž RMP2 .tion interaction D E energies. In particular,

the D ERHF is saturated with the present basis setsand the results are almost identical to the resultsof Meyer et al.30 The so-called HF electrostatic

JOURNAL OF COMPUTATIONAL CHEMISTRY 861

ˇ ˇLUKES, LAURINC, AND BISKUPIC

energy 44 ] 46 can be evaluated easily for closed-shelland open-shell systems. The value of the HF elec-

Ž Ž100..trostatic energy E is almost insensitive to theelsŽ .basis set and, even for medium basis sets B2 ,

Ž Ž100.tends to the limit value e.g., E s y4.43 mH atels. Ž100. RHFR s 6.75 bohr and the ratio E rD E is aboutelst

y0.229 at the same distance. For comparison witha typical weak closed-shell van der Waals system,

Ž HF .the limit HF interaction energy D E for the He2dimer at 5.6 bohr is 29.196 mH,47 and EŽ100.rD EHF

elstis approximately y0.169.48

The applicability of the evaluated contributionsw Ž . Ž .xto the dispersion energy see eqs. 6 ] 8 can be

estimated from the data in Table IV. EŽ200. repre-dispsents the leading term of the dispersion energyand the main attractive part of D ERMP2. The limitratio, D ERMP2rEŽ200., is about 0.94 at R s 6.75 bohr.dispFor He—He at R s 5.6 bohr, the limit value forEŽ200. is found to be y54.095 mH,49 and the ratiodispD ERMP2rEŽ200. is approximately 0.93.47,48,50 In ourdisp

Ž .study, the total dispersion energy E is approx-dispimated as the sum EŽ200. q EŽ210. q EŽ201. q EŽ300..disp disp disp dispThe obtained results are compared with the data

wobtained from the multipole expansion see eq.Ž .x1 . The first three long-range coefficients C , C ,6 8

Žand C 2.823, 41.83, and 871.3 in atomic units,10.respectively were taken from work of Koide et

51 Ž .al. The coefficient C 25512.3 a.u. was obtained12from the empirical recursion formula.30,52

As seen in Table IV, our dispersion energiesŽ .calculated with the B3 and B4 basis sets are inagreement with the data from the multipole ex-pansion and from MR]CI supermolecular calcula-

30 Žtions. These results are slightly lower about.7]9% than the data obtained from MCSCF calcu-

35 Ž .lations. The values in parentheses Table IV rep-Ž300. Ž Ž300. Ž200..resent E . The ratio E rE is small anddisp disp disp

indicates a relatively good convergence of the per-turbation expansion of the dispersion energy.

HE—O SYSTEM2

The triatomic vdW complexes containing thenoble gas atom have been studied intensively inmodern spectroscopic and scattering experiments.The stable ground state He—O complex has2emerged as a convenient model for developing anab initio strategy for interaction between closed-shell and open-shell species.53 ] 55

The best empirical estimates of the global mini-mum parameters R and D for the T-shapede egeometry are 6.18 bohr and 119.8 mH 53 and 6.29bohr and 107.0 mH,54 respectively. No empirical

potentials reveals any local minimum for the linearorientation and the interaction energy variesmonotonously between the linear and the T struc-ture. However, the ab initio studies indicate theexistence of secondary minima in the linear orien-tation. For example, Keil et al.56 predict 86.36 mHat 7.12 a.u., Faubel et al.54 64.6 mH at 7.3 a.u., andCybulski et al.21 116.4 mH at 7.0 bohr, respectively.This discrepancy between experiment and theorymight be explained by the fact that the experimen-tal characterization of the secondary minima ismuch more difficult, because the scattering experi-ments efficiently probe only the primary minimumregion.21

To compare the performance of the restrictedversus unrestricted Møller]Plesset perturbation

Ž .theory MPPT for the selected interaction energycontributions, we use the same geometries and5s3p2d1f basis set for the O atom and 4s1p1d forthe He atom as in the recent work of Cybulski etal.21 The definition of the geometrical parametersof the He—O complex is the same as in their2work21: R denotes the distance between the centerof mass of the oxygen molecule and He atom, andQ corresponds to the angle between R vector andthe O bond axis. The details of the basis sets used2Ž .denoted LD are given in the Appendix of ref. 18.

According to Cybulski et al.,21 the LD basis setat the UMP2 calculation level provides two min-ima: D s 81.3 mH at R s 6.0 bohr for the T-e eshaped geometry and D s 79.5 mH at R s 7.4e e

bohr for the linear geometry. The interaction en-ergy calculated using the RPT2 level at R s 6.0bohr differs form the UHF spin-contaminated cal-

Žculations. The minimum is slightly shifted D se.71.1 mH at R s 6.4 bohr . The projected UHFe

Žresults are between the RHF and UHF data see T.structure in Table V .

Similar to the He—H vdW system, the interac-Žtion energy in our case calculated at the MP2

.level is determined by the delicate balance be-tween the repulsive HF and the attractive second-order correlation interaction energies. The selectedinteraction energy components obtained at the re-stricted and unrestricted MP2 level of the theory

Ž .for angles varying from 08 to 1808 at R s 6 bohrare shown in Figure 4. The second-order interac-

Ž RMP2 UMP2 .tion correlation energies D E or D E re-veal minima at Q s 08 and 1808. The UHF calcula-tions offer a very flat maximum in the middle ofthe oxygen bond. Similar to the closed-shell case,the difference between the second-order super-molecular interaction correlation and the HF dis-

VOL. 20, NO. 8862

PERTURBATIVE FORMULATION

TAB

LEIV

.R

HF

RM

P2

()

DE

,D

E,a

ndD

isp

ersi

on

Co

ntri

but

ions

for

He

—H

syst

emE

nerg

ies

inm

H.

aR

HF

6c

Bas

isS

et1

Bas

isS

et2

Bas

isS

et3

Bas

isS

et4

MC

SC

FE

/MR

]C

IM

E(2

00

)(2

00

)(2

00

)(2

00

)R

yE

yE

yE

yE

yE

yE

yE

yE

yC

dis

pd

isp

dis

pd

isp

dis

pd

isp

dis

pd

isp

(6,8

,10

)R

HF

RM

P2

(30

0)

RH

FR

MP

2(3

00

)R

HF

RM

P2

(30

0)

RH

FR

MP

2(3

00

)R

HF

RH

F(

)[

](

)(

)(

)[

][

](

)[

][

bohr

DE

yD

EE

DE

yD

EE

DE

yD

EE

DE

yD

EE

DE

yE

DE

yE

yC

dis

pd

isp

dis

pd

isp

dis

pd

isp

(6,8

,10

,12

)

5.50

199.

3996

.32

126.

8719

8.43

127.

3816

3.24

198.

4413

9.41

176.

6019

8.12

143.

8618

2.17

198.

0916

2.94

186.

34(

)(

)(

)(

)(

)(

)(

)(

)(

)76

.82

y0.

0710

5.73

2.40

116.

364.

1612

0.34

4.95

219.

646.

0079

.60

56.2

474

.48

78.7

471

.88

93.1

078

.82

78.0

199

.97

78.7

381

.79

105.

0878

.481

78.7

194

.64

99.8

2(

)(

)(

)(

)(

)(

)(

)(

)(

)48

.07

0.00

62.9

21.

0968

.33

1.98

72.0

22.

0811

1.54

6.50

31.4

933

.88

45.0

730

.91

42.0

855

.02

30.9

545

.22

58.6

330

.94

47.8

861

.82

30.7

5130

.93

56.3

857

.03

()

()

()

()

()

()

()

()

()

30.4

50.

0238

.39

0.49

41.4

80.

8843

.86

1.19

61.5

26.

7519

.74

26.6

135

.47

19.3

132

.66

42.8

719

.32

34.9

245

.54

19.3

337

.07

48.1

719

.32

43.9

643

.99

()

()

()

()

()

()

()

()

()

24.4

70.

0130

.31

0.33

32.4

70.

5634

.53

0.82

46.8

47.

0012

.34

21.0

828

.14

12.0

425

.56

33.6

912

.04

27.2

335

.66

12.0

628

.93

37.7

611

.832

12.0

434

.55

34.3

4(

)(

)(

)(

)(

)(

)(

)(

)(

)19

.62

0.01

24.0

70.

2225

.76

0.39

27.2

70.

5736

.18

7.50

4.79

13.5

518

.13

4.66

16.1

421

.39

4.65

17.0

422

.49

4.66

18.0

723

.77

4.6

214.

6521

.81

21.5

9(

)(

)(

)(

)(

)(

)(

)(

)(

)13

.11

0.01

15.5

50.

1016

.46

0.17

17.4

30.

2722

.39

8.00

1.83

9.00

12.0

71.

7910

.54

14.0

51.

7911

.05

14.6

71.

7911

.67

15.4

61.

815

1.79

14.1

914

.07

()

()

()

()

()

()

()

()

()

8.74

0.00

10.3

20.

0210

.76

0.08

11.4

00.

1214

.44

10.0

0.04

2.28

3.06

0.04

2.53

3.39

0.04

2.60

3.49

0.04

2.69

3.58

0.04

3.31

3.33

()

()

()

()

()

()

()

()

()

2.27

0.00

2.47

0.00

2.59

0.00

2.65

0.04

3.35

a3

5b

RH

F(

)3

0c

MC

SC

Fse

para

tion.

DE

and

MR

]C

Iint

ra-a

tom

corr

elat

ion

cont

ribut

ions

disp

ersi

onte

rms

.M

ultip

ole

expa

nsio

n.P

aram

eter

sC

,C,a

ndC

are

take

nfro

mre

f.51

,6

810

Cob

tain

edfro

mth

eem

piric

alre

curs

ion

form

ula.

30

,52

12

JOURNAL OF COMPUTATIONAL CHEMISTRY 863

ˇ ˇLUKES, LAURINC, AND BISKUPIC

TABLE V.( ) 21Energy Characteristics mH of Two Minima

aof He—O Complex.2

( )R bohr 6.0 7.0( ) ( )Q 8 90 0 180

RHFDE 104.76 70.05RMP2DE y157.46 y157.89

(2)E y52.70 y87.84in t(10 0)E y24.82 y16.79els(20 0)E y212.86 y178.91disp(210) (201)E q E y17.61 y2.05disp disp(3 0 0)E 1.82 1.45disp

E y228.65 y179.51dispUHF ( ) ( )DE 130.42 141.56 73.14 74.81UMP2 ( ) ( )DE y211.69 y207.85 y152.62 y152.73

(2) ( ) ( ) ( )E UHF y81.27 y66.29 y79.48 y77.92in t(10 0) ( )E UHF y32.03 y17.40els(20 0) ( )E UHF y210.89 y170.00disp

aUHF data obtained from the work Cybulski et al.,21 and thevalues in parentheses represent the projected UHF results.

w Ž200. Ž200. Ž .xpersion E or E UHF energies containsdisp dispmainly the second-order electrostatic correlationenergy, the second-order HF exchange-dispersion,and the deformation correlation contributions.These remaining terms are small near the T config-uration and are negligible especially in the UHFapproximation. The dispersion effect in the per-pendicular orientation is small but it increases byincreasing the deviation from 908. However, the

Ž200. Ž .differences between E and E see Fig. 4disp dispreveal that contributions of the higher orders of

FIGURE 4. The Q dependence of RMP2 / UMP2interaction energy and its selected components for theground state He—O system at R s 6.0 bohr. UHF data2taken from the work Cybulski et al.20 The symbols for

( ) RHF ( ) UHF ( ) (2) ( ) (2)left y-axis: B DE ; I DE ; ' E ; ^ Eint int( ) ( ) (100) ( )UHF . The symbols for right y-axis: % E ; \els

(100) ( ) ( ) (200) ( ) (200) ( ) ( ) RMP2E UHF ; l E ; e E UHF ; v DE ;els disp disp( ) UMP2 ( ) (200) (210) (201)` DE ; = E s E q E q E qdisp disp disp dispE (300).disp

RHF dispersion energies decrease with increasingEŽ200.. The magnitude of the sum EŽ201. q EŽ210.

disp disp dispŽ300. Ž .qE is y0.76 mH at Q s 08 1808 and y15.79disp

mH at Q s 908. The data in Table V indicate thesame trend.

w Ž100. Ž100.The HF electrostatic terms E and Eels elsŽ .xUHF show an angular dependence with a maxi-mum at 908 and minima at 08 and 1808. They are ofa pure charge-overlap nature in the present caseand show distinct flattening in the linear configu-ration. The shape of both curves is in agreementwith the contour maps analysis of Bader and Es-sen.57

Summary and Conclusions

In this study, we have presented explicit formu-las for select contributions to the dispersion energyin terms of orbitals generated by the RHF]SCFprocedure for nondegenerate closed-shell andopen-shell van der Waals systems. The basis setdependence of the presented dispersion energyterms was tested for the He—H system. The nu-merical results indicate that the ‘‘best’’ values ofEŽ200. near the vdW minimum represent roughlydi s p79% of the value obtained from the multipoleexpansion. The EŽ210. and EŽ201. terms were muchdi s p di s pmore difficult to calculate than EŽ200., because theydi s prequire additional calculations of the molecular

wintegrals for separated A and B systems see eq.Ž .x8 . However, the importance of these terms andof the exchange-dispersion contributions is quite

Ž .significant about 25% . To obtain more preciseresults for the total dispersion energy, the selectionof a better basis set, as well as inclusion of higherterms to the dispersion energy and exchange-dis-persion energy is necessary. These terms can beevaluated by the presented diagrammatic schemeor by a suitable version of the coupled clustermethod.58,59

The angular anisotropy and the performance ofthe restricted versus unrestricted MP2 interactionenergy contributions was discussed for the tri-atomic vdW complex He—O at R s 6.0 bohr.2The obtained RHF and UHF results were of com-parable quality for the HF interaction, HF electro-static, and second-order HF dispersion energies.However, significant differences were observed forsecond-order interaction correlation energies in theT-shaped structure. D ERMP2 and D EUMP2 may dif-fer mainly in the exchange- and electrostatic-corre-lation energies. It is necessary to emphasize that

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the UHF values were spin contaminated and thusthe physical reliability of the interaction energycontributions may be questionable and spin projec-tion or annihilation techniques may be desirable.

Finally, we conclude that a qualitatively correctŽdescription of the dispersion energy near the vdW

.minimum for interacting systems of the closed-shell]open-shell type, together with relativelysmall computational costs, suggest that the MBPTformalism presented offers a possible alternativeto unrestricted MPPT treatment.

Acknowledgments

The authors thank Professor V. Kvasnicka forˇstimulating discussions and the referees for criticaland inspiring remarks.

References

1. Claverie, P. From diatomic to biopolymers. In: Pullman, B.,ed., Intermolecular Interactions; Wiley: New York, 1978.

2. Jeziorski, B.; Kolos, W. In: Ratajczak, H.; Orville-Thomas,W. J., eds., Molecular Interactions, Vol. 3; Wiley: New York,1982.

3. Hobza, P.; Zahradnık, R. Chem Rev 1988, 88, 871.´4. Chalasinski, G.; Gutovski, M. Chem Rev 1988, 88, 943.

5. Chalasinski, G.; Szczesniak, M. M. Chem Rev. 1994, 94,1723.

6. Debye, P. Phys Z 1920, 21, 178.

7. Keesom, W. H. Phys Z 1921, 22, 129.

8. London, F. Z Phys Chem B 1930, 11, 222.

9. London, F. Trans Faraday Soc 1937, 33, 8.

10. Tang, K. T.; Norbeck, J. M.; Certain, P. R. J Chem Phys 1976,64, 3063.

11. Buckingham, A. D.; Fowler, P. W.; Hutson, J. M. Chem Rev1988, 88, 963.

12. Rybak, S.; Jeziorski, B.; Szalewicz, K. J Chem Phys 1991, 95,6576.

13. Moszynski, R.; Jeziorski, B.; Szalewicz, K. J Chem Phys1994, 100, 1312.

14. Basilevsky, M. V.; Berenfeld, M. M. Int J Quantum Chem1972, 6, 23.

15. Daudey, J. P.; Claverie, P.; Malrieu, J. P. Int J QuantumChem 1974, 8, 1.

16. Mayer, I. Int J Quantum Chem 1983, 23, 341.

17. Laurinc, V.; Lukes, V.; Biskupic, S.; Theor Chem Acc 1998,ˇ ˇ99, 53.

18. van Lenthe, J. H.; van Duijneveldt, F. B. J Chem Phys 1984,81, 3168.

19. Bussery, B.; Wormer, P. E. S. J Chem Phys 1993, 99, 1230.

20. Cybulski, S. M.; Burcl, R.; Chalasinski, G.; Szczesniak, M.M. J Chem Phys 1995, 103, 10116.

21. Cybulski, S. M.; Burcl, R.; Szczesniak, M. M.; Chalasinski,G. J Chem Phys 1996, 104, 7997.

22. Kendall, R. A.; Chalasinski, G.; Klos, J.; Bukowski, R.; Sev-erson, M. W.; Szczesniak, M. M., Cybulski, S. M. J ChemPhys 1998, 108, 3235.

23. Møller, C.; Plesset, M. S.; Phys Rev 1934, 46, 618.

24. Amos, A. T.; Grispin, R. J. Mol Phys 1976, 31, 159.

25. Jeziorski, B.; van Hammert, M. Mol Phys 1976, 31, 713.

26. Musher, J. J.; Amos, A. T. Phys Rev 1967, 31, 164.

27. Broussard, J. T.; Kestner, N. R. J Chem Phys 1970, 53, 1507.

28. Broussard, J. T.; Kestner, N. R. J Chem Phys 1973, 58, 3593.

29. Uhlrich, B. T.; Ford, L.; Browne, J. C. J Chem Phys 1972, 57,2906.

30. Meyer, W.; Frommhold, L. Theor Chim Acta 1994, 88, 201.

31. Jochemsen, R.; Berlinsky, A. J.; Hardy, W. N. Can J Phys1984, 62, 751.

32. Hardy, W. N.; Morrow, M.; Jochemsen, R.; Statt, B. W.;Kubik, P. R.; Marsolais, R. M.; Berlinsky, A. J.; Landesman,A. Phys Rev Lett 1980, 45, 453.

33. Knowles, D. B.; Murrell, J. N.; Braga, J, P. Chem Phys Lett1984, 110, 40.

34. G. Scoles, see Appendix of ref. 31.

35. Das, G.; Wagner, A. F.; Wahl, A. C. J Chem Phys 1978, 68,4917.

36. Partridge, H.; Schwenke, D. W.; Bauschlicher, C. W. J ChemPhys 1993, 99, 9776.

37. Dunning, T. H., Jr. J Chem Phys 1989, 90, 1007.

38. Kendall, R. A.; Dunning, T. H., Jr.; Harrison, R. J. J ChemPhys 1992, 96, 6796.

39. Woon, D. E.; Dunning, T. H., Jr. J Chem Phys 1994, 100,2975.

40. Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Gill, P. M. W.;Johnson, B. G.; Wong, M. W.; Foresman, J. B.; Robb, M. A.;Head-Gordon, M.; Replogle, E. S.; Gomperts, R.; Andres, J.L.; Raghavachari, K.; Binkley, J. S.; Gonzalez, C.; Martin, R.L.; Fox, D. J.; Defrees, D. J.; Baker, J.; Stewart, J. J. P.; Pople,J. A. GAUSSIAN-92rDFT, Rev. G.3, Gaussian: Pittsburgh, PA,1993.

41. Boys, S. F.; Bernardi, F. Mol. Phys 1970, 19, 553.

42. Kvasnicka, V.; Laurinc, V.; Biskupic, S. Mol Phys 1981, 42,ˇ ˇ1345.ˇ43. Carsky, P.; Zahradnık, R.; Hubac, I.; Urban, M.; Kello, V.´ ´ ˇ ¨Theor Chim Acta 1980, 55, 315.

44. Hirschfelder, J. O.; Meath, W. J. J Adv Chem Phys 1976, 12,3.

45. Jeziorski, B.; Bulski, M.; Piela, L. Int J Quantum Chem 1976,10, 281.

46. Claverie, P. In: Pullman, B., ed. Intermolecular Interactions:From Diatomics to Biopolymers; Wiley: New York, 1978; p69.

47. Klopper, W.; Noga, J. J Chem Phys 1995, 103, 6127.

48. Szalewiczs, K.; Jeziorski, B. Mol Phys 1979, 38, 191.

49. van Mourik, T.; van Lenthe, J. H. J Chem Phys 1995, 102,7479.

50. Williams, H. L.; Korona, T.; Bukowski, R.; Jeziorski, B.;Szalewicz, K. Chem Phys Lett 1996, 262, 431.

JOURNAL OF COMPUTATIONAL CHEMISTRY 865

ˇ ˇLUKES, LAURINC, AND BISKUPIC

51. Koide, A.; Meath, W. J.; Allnatt, A. R. J Phys Chem 1982, 86,1222.

52. Tang, K. T.; Toennies, J. P. J Chem Phys 1978, 68, 786.53. Beventi, L.; Casavecchia, P.; Volpi, G. G. J Chem Phys 1986,

85, 7011.54. Faubel, M.; Kohl, K. H.; Toennies, J. P.; Gianturco, F. A. J

Chem Phys 1982, 78, 5629.55. Heaven, M. C. J Phys Chem 1993, 97, 8567.

56. Keil, M.; Slankas, J. T.; Kuppermann, A. J Chem Phys 1979,70, 541.

57. Bader, R. F. W.; Essen, H. J Chem Phys 1984, 80, 1943.

58. Jeziorski, B.; Moszynski, R.; Rybak, S.; Szalewicz, K. In:Kaldor, U., ed. Many-Body Methods in Quantum Chem-istry; Lectures Notes in Chemistry, vol. 52; Springer: NewYork, 1989; p 65.

59. Williams, H. L.; Szalewicz, K. J Chem Phys 1995, 103, 4586.

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