Jonathan Tennyson- ATOMDIAT2 and GENPOT: Adaptations of ATOMDIAT for the Ro-Vibrational Levels of...

8/3/2019 Jonathan Tennyson- ATOMDIAT2 and GENPOT: Adaptations of ATOMDIAT for the Ro-Vibrational Levels of any Floppy

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Computer Physics Communications 32 (1984) 109114 109

North-Holland, Amsterdam

ATOMDIAT2 ANDGENPOT: ADAPTATIONS OF ATOMDIAT FOR THE RO-VIBRATIONALLEVELS OF ANY FLOPPY TRIATOMIC USING A GENERAL POTENTIAL FUNCTION

Jonathan TENNYSON

SERC, Daresbury Laboratory, Daresbury, Warrington, Cheshire WA4 4AD, UK

Received 8 March 1983; in revised form 25 March 1983

ADAPTION SUMMARY ADAPTION SUMMARY

Titleofadaptation: ATOMDIAT2 Title ofadaptation: GENPOT addendum to ATOMDIAT(2)

Adaptation number: ACEN 0001 Adaptation number: ACEN 0002

Programs obtainablefrom: CPC Program Library, Queens Uni- Programs obtainablefrom: CPC Program Library, Queens Uni-

versity of Belfast, N. Ireland (see application in this issue) versity ofBelfast, N. Ireland (see application in this issue).

Reference tooriginalprogram: CPC Library File ACEN [1] Referenceto originalprogram: CPC Library files ACEN [1] and

ACEN 0001 [2]Author oforiginalprogram: J. Tennyson [1]

Author oforiginalprogram: J. Tennyson [1,2]

Computer: NAS 7000; Installation: Daresbury Laboratory Computer: NAS 7000; Installation: Daresbury Laboratory

Other machines on which the adaption has been tested: CRAY-i Other machines on which the adaption has been tested: CRAY-i

and 1BM4341/2No. ofbits in a byte: 8

No. ofbits in a byte:8

No. oflines required to effect adaptation: 190

No. oflines required to effect adaptation: 215Additional keywords: spherical oscillator functions

Additional keywords: GaussLegendre quadrature, isotopicsubstitution

NatureofphysicalproblemATOMDIAT2 solves the ro-vibrational problem for an atom Nature ofphysicalproblem

(1)diatom system (23) for which the linear 213 structure is GENPOT generalises ATOMDIAT [1] or ATOMDIAT2 [2] so

significant. that thero-vibrational states may be obtained for any triatomic

potential function, not just one fitted in Legendre polynomials.Method ofsolution It can also be used to perform shifts of a potential fitted as aSpherical oscillator-like functions are used for the R-coordinate Legendre expansion caused by isotopic substitution [3].

basis [2]. An option allowing the problem to be embeddedalong ris also included. Methodofsolution

GENPOT uses GaussLegendre quadrature to obtain a

References Legendre expansion for each R and r[4].

[1] J. Tennyson, Comput. Phys. Commun. 29 (1983) 307.

[2] J. Tennyson and B.T. Sutcliffe, J. Mol. Spectr. 101 (1983) References

71 [1] J. Tennyson, Comput Phys. Commun. 29 (1983) 307.[2] J. Tennyson, this article, first adaptation.[3] W.-K. Liu, J,E. Grabenstetter, R.J. LeRoy and F.R. Mc-

Court, J. Chem. Phys. 68 (1978) 5028.[4] J. Tennyson and B.T. Sutcliffe, J. Mol. Spectr. 101 (1983)

71.

OO1O-4655/84/$03.OO Elsevier Science Publishers B.V.(North-Holland Physics Publishing Division)


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110 J. Tennyson /A TOMDIA T2 and GENPOT, adaptations toA TOMDIAT

LONG WRITE-UP necessary to re-expand (1) in a new series. Anapproximate transformation for this based on a

1. Infroduction truncated Taylor series was proposed by Kreek

and LeRoy [8] and employed in several calcula-Recently, I published a program, ATOMDIAT tions on diatomic-rare gas Van der Waals mole-

[11, which performed variational ro-vibrational cules [8il]. GENPOT performs this transforma-

calculations on floppy triatomics using the tion exactly, limited only by the number of termsmethod of Tennyson and Sutcliffe [2,3]. Two con- taken in the Legendre expansion. This means that

straints on this program prevent it from providing the adapted ATOMDIAT canconveniently be useda completely general method for obtaining the for calculations on several isotopic species with the

ro-vibrational levels of any triatomic. same potential. Indeed, GENPOT itselfprovides aFirstly, the Morse oscillator-like functions used suitable subroutine suite for the transformation of

by ATOMDIAT for the basis of the atom (1)di- Liu et al. [12].

atom (23) stretching coordinate R are only valid The adaptions thus provide a suite ofprograms

if the vibrational wavefunctions have effectively (ATOMDIAT and ATOMDIAT2) which can be

vanished at R = 0. This problem was first noted by used to find the bound ro-vibrational levels of any

Ter Haar [41and is associated with the use of the triatomic system. It is anticipated, however, thatincorrect integration range for the Morse problem. the programs will find widest application for sys-

For most triatomics this condition does not pre- tems, such as Van der Waals complexes, which

sent a problem; however, for systems which have have one or more large amplitude vibrational mode

significant amplitude for the 2 1 3 linear struc- so called floppy molecules. ATOMDIAT2 hasture the Morse-like functions are no longer ap- been successfully applied to CH~[5] and gavepropriate. Examples of such systems are X

3, X~ better results than those obtained using more con-

(X= rare gas), CH~,CO 2 and highly excited H20. ventional (Eckart) methods [13,14].

Recently, Tennyson and Sutcliffe [5] have shown The choice between ATOMDIAT and

how the use of spherical oscillator-like functions ATOMDIAT2 depends on the physics of the prob-

[6] can overcome this problem and ATOMDIAT2 lem: ATOMDIAT being for molecules for which a

is an adaptation of ATOMDIAT which uses these 2 1 3 linear structure is not energetically accessi-

basis functions. ble. An indication of whether ATOMDIAT isSecondly, a requirement of ATOMDIAT is a suitable for a problem is given by the stability of

subroutine (47) POT containing the (analytic) the numerical integrals. In particular, if the in-potential energy surface in the form of a Legendre tegrals over R 2 are not stable to increasing the

expansion number of GaussLaguerre integration points,

then use of ATOMDIAT2 is recommended. Con-V

1R O~ V R ~ versely, if large values for a2 and I~2(see below)

k r, ~ r1 A~COS ~ ~ are required by ATOMDIAT2 then ATOMDIAT

might well prove more efficient.

This requirement limits the utility ofATOMDIAT Finally, ATOMDIAT contains an option which

as many surfaces do not give obvious analytic allows the off-diagonal Coriolis terms to be ne-

expressions for the V~(R, r)of such an expansion. glected, a simplifying approximation which hasGENPOT provides the subroutines which gener-

oftenbeen found useful [2,11,15,16]. These Corio-

alise ATOMDIAT or ATOMDIAT2 allowing the lis interactions depend on the form of the coordi-

triatomic potential energy surface to be presented nate embedding. ATOMDIAT2 contains an op-

in any convenient (analytic) form, such as pair tion which allows the user to choose betweenpotentials or a SorbieMurrell potential [7]. embedding the axis along R (as in ATOMDIAT)

The Legendre expansion of eq. (1) is based on or r (in the fashion of Istomin et al. [17]). The

the diatomic centre of mass. Ifresults are required latter embedding is more appropriate for the nearfor different isotopes of a molecule, then it is often linear systems that ATOMDIAT2 is designed for


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J. Tennyson /ATOMDIA T2 and GENPOT, adaptations to ATOMDJA T 111

and was found to give errors an order of magni- and those over the potential are performed using

tude smaller when the off-diagonal Coriolis inter- GaussLaguerre integrations [2,19] based onL~21/2actions in CH~were neglected [5].The results of a . M

2 is input and the points and weightsfull calculation are, of course, independent of the are generated automatically.embedding used. In ATOMDIAT, the z axis is embedded along

R [113].When the off-diagonal Coriolis terms areneglected, k, the projection of the total angular

2. Method momentumJ along the R axis, is a good quantum

number. If the z axis is embedded along r [5,171,ATOM DIAT2 follows the method of Q, the projection ofJ along r, is a good quantum

ATOMDIAT [1] with the exception of the func- number in this approximation. Which approxima-tions used for the R (atomdiatom stretch) coordi- tion is appropriate will depend on the system in

nate basis functions. These are now based on question and in particular on the relative magni-

spherical oscillator functions and are defined as tude of the (2~t1r2 ) ~ and (2~t

2R2 ) terms.

ATOMDIAT2 contains an option which allows

H,, = 2~2$3~4N,,,,e~2/2y~2/2L~2+/2(y

2), (2) either embedding to be employed when the off-di-

Y2 = $2R2, (3) agonal Coriolis interactions are neglected. For afull calculation the choice of embedding is im-

/32 = (~

2w2)l.z~

2, (4) material and ATOMDIAT2 uses the same embed-ding as ATOMDIAT.

where N,,,,L~~2is a normalised Laguerre poly- GENPOT allows the surface to be fitted to anomial [18] and w

2 is the frequency of the R Legendre expansion for each R and r at executionstretching fundamental. In practice, the parame- time, rather than requiring a pre-fitted surface in

ters a2 and w 2 are optimised variationally to give the form of eq. (1). This is done using A point

the best basis for each problem. GaussLegendre integration [19]. When the re-

It should be noted that all functions with a2 = 0 quest for a potential in the form of eq. (1) is issuedhave amplitude at R = 0, whilst all functions with by CALL POT(V0, Vi, Ri, R2), the amended POT

a2> 0 are zero at R = 0. Symmetry considerations performs a A point (A > Xmax) integration with

can thus determine an optimum value of a2 and (r, R) fixed. This gives the best Legendre expan-this value will not necessarily be appropriate for sion for each set of radial separations, and provid-

all quantum numbers of a particular problem [51. ing ~max is chosen large enough, entails no loss ofKinetic energy integrals over these functions are accuracy over using the exact potential.

performed analytically [5,6] If a transformed Legendre expansion is re-

$2h2 quired to cope with isotopic substitution, the

(n~Ik2In2)= ~ ~,~2_1(n2(n2 +a2 +fl)I/

2 potential given by the original expansion can sim-

ply be given in POTV and the correctly trans-+ a

2 +~) formed potential is automatically generated. Careneed only be taken with the definition of R2.

+~n~2, (5)

3. Organisation~

3.1. ATOMDIAT2/3h21n! F(n~+a2+3/2)~2

= 2 1 z 2 [ny F(n2 + a2 +3/2) Although several of the ATOMDIAT sub-

routines are adapted in ATOMDIAT2, none~ F(a+a2+1/2) n~.

+ n2>n2, (6) change their basic function. One new subroutine,,,~ F(n2 + a2 + 3/2) ~! belonging to overlay 2, is required by


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112 J. Tennyson /ATOMDIA T2 and GENPOT, adaptations to ATOMDIAT

L ~L~UERJ~L6ROOTI~LGREGRI =F gives z embedded along rCCMAINh~L AGP T ~iiroi~fi1 ~ (ignored unless JROT < 0).

K E I N T S *~tL::~~I~ L E G P T I Card 9 RE2, ALF2, WE2

E~KE1NT2 I Ls~~LPI LE G E N D I RE2 is ignored, ALF2 = a2, WE2 = ~2 see eqs.O V E R L A Y 2 (2)-(4).

Fig. 1. Adapted program structure. The dotted call is made by

ATOMDIAT2 and the dashed calls by the addendum 4.2. Card inputfor GENPOTGENPOT.

Card 5 NPNT2, NMAX2, JROT, NEVAL,

LMAX, LPOT, IDIA, KMIN, NPNT1 NMAXIATOMDIAT2. LPOT has a slightly altered meaning. It is now the


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J. Tennyson /A TOMDJA T2 and GENPOT, adaptations to ATOMDIAT 113

References [11] S.L. Holmgren, M. Waldman and W. KJemperer, J. Chem.

Phys. 69 (1978) 1661.

~1] J. Tennyson, Comput. Phys. Commun. 29 (1983) 307. [121 W.-K. Liu, J.E. Grabenstetter, Ri. LeRoy and FR. Mc-[2] J. Tennyson and B.T. Sutcliffe, J. Chem. Phys. 77 (1982) Court, J. Chem. Phys. 68 (1978) 5028.

4061. [13] R. Bartholomae, D. Martin and B.T. Sutcliffe, J. Mol.

[3] J. Tennyson and B.T. Sutcliffe, J. Chem. Phys. 79 (i983) Spectr. 87 (1981) 367.

43. [14] S. Carter and N.C. Handy, J. Mol. Spectr. 95 (1982) 9.

[4] D. ter Haar, Phys. Rev. 70 (1946) 222. [15] J. Tennyson and A. van der Avoird, J. Chem. Phys. 77

[5] J. Tennyson and B.T. Sutcliffe, J. Mol. Spectr. 101 (1983) (1982) 5664.

71. [i6] G. Brocks and J. Tennyson, J. Mol. Spectr. 99 (1983) 263.

[6] S. Bell, J. Phys. B 2 (1969) iooi. [17] V.A. Istomin, N.F. Stepanov and B.I. Zhilinskii, J. Mol.[7] KS. Sorbie and J.N. Murrell, Mol. Phys. 29 (i975) i387. Spectr. 67 (~977)265.

[8] H. Kreek and R.J. LeRoy, J. Chem. Phys. 63 (1975) 338. [18] IS. Gradshteyn and 1 . 1 1 . Ryzik, Tables of Integrals,Series[9] R.J. LeRoy, iS. Charley and J.E. Grabenstetter, Faraday and Products (Academic Press, New York, 1980).

Disc. Chem. Soc. 62 (1977) 169. [19] A.H. Stroud and D. Secrest, Gaussian Quadrature For-

[10] AM. Dunker and R.G. Gordon, J. Chem. Phys. 64 (1976) mulae (Prentice-Hall, London, 1965).

354.


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114 J. Tennyson /A TOMDIA T2 and GENPOT, adaptations toA TOMDIA T

TEST RUN OUTPUT

INTEGRATION P OINTS WEIGHTS CORRESPONDING P.

0.147399184616311 u. 114136764489014 0.10358

0.590901811243187 u.29463184O271D14 0.20739

1.334487511614576 0.318113482748014 0.31166

2.385011552004653 u.200459576450014 0.41665

3.752567873814167 u . 812054431487015 0.52263

2.451062939568396 u.218775201292D15 u.62990,.499085532907371 u.394842091O59D16 0.73881

~.92l2191 36072429 u.473236189494D17 0.84979

12.750055460117065 0.368556402828018 0.9633516.029386360375124 0.179985370322019 1.0b016

19.81951287/102021 u.522559600232O21 1.20109

24. 206680643468303 U. 833308727828023 1.32738

29. 321456103352332 U.646096973584025 1.4609))

35.379550767175560 U.198952300171D27 1.60474

42. 793255970754640 0. 16561 569586903)) 1.7648952.618366255753244 o.141730912O91034 1.95703

COMPUTED SU M OF POINTS 0. 2640000000000000+03 & WEIGHTS 0. 1U348889935258201 3

EXACT SU M OF POINTS U. 2640000000000000+03 & ))EIG)ITS 0.11)3488899352659013

22 POINT GAUSSLEGENDRE INTEGRATION

INTEGRATION POINTS WEIGHTS

u.994294585462399 U. 146279952982001

0.97006049/835429 U.331/49015844)301

0.926956772167174 U. 522933351525001

0.86581257 7720301) 0. 697964684245001

0.787816805979208 u.859416O62171D01

0.694487263186683 U . 1004141444430+00

0.587640403506912 0.1129322960810+00

0.46935583798675/ 0.1232523168110+00

0.341935820892084 0. 131173504187D+0U

0. 207860426688221 0.1365414983460+0))

0.069739273319722 0.1392518726560+00

0.994294585482399 0.146279952982001

0.970060497835429 0 . 337749Ot5844D01

0.926956772187174 0.522933351525001

0.865812577720300 0.6979646842450010. 787816805979208 1 ) . 8594160621 71001

0.694487263186683 0.1004141444430+))))

0.587640403506912 U. 1129322960)310+00

0.46935583798675/ 0.1232523768110+00

0.34193582U8Y2084 0.13117350478/0+0))0.21)7860426688221 0. 136541498346D+OU

0.069739213319122 U. 1392518/28560+00

COMPUTED SU M OF WEIGHTS U. 1999999999998580+01

EXACT S L I M OF WEIGHTS U. 20000000U0000000+U 1

MOMENT OF INERTIA IAIRIX CALCULATED NUMERICALLY

1 2 3 4 2

0.0000356

2 0. 0000039 0.0000367

3 1 ) . 0 00 00 01 0. 0000056 0.0000378

4 0. 0000000 0.0000002 0.0000069 0.01)00389

5 0. 0000000 0.000000)) 0.0000(303 0.1)001)082 0.0)1(8)401

LUWUST F EIGENVALIJLS IN HARTR)-.LS

0.321)42131)9830+00 0. 3169)339838750+ 11.3123/919U9221)+)l(J 0. 3))6/86931)3430+D0 3). 3),52691684)3D+OU

0. 3033229636)(5D+(J()

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