Jonathan Tennyson- ATOMDIAT: A Program for Calculating Variationally Exact Ro-Vibrational Levels of Levels of "Floppy" Triatomics

8/3/2019 Jonathan Tennyson- ATOMDIAT: A Program for Calculating Variationally Exact Ro-Vibrational Levels of Levels of "Floppy" Triatomics

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Computer Physics Communications 29 (1983) 307319 307

North-Holland Publishing Company

ATOMDIAT - A PROGRAM FOR CALCULATINGVAlUATIONALLY EXACTRO-VIBRATIONAL LEVELS OF FLOPPY TRIATOMICS

Jonathan TENNYSON *

Instituut voor Theoretische Chemie, Katholieke Universiteit, ToernooiveId~6525 EDN~megen,The Netherlands

Received 20 September 1982; in revised form 1 5 November 1982

PROGRAM SUMMARY

Title ofprogram. ATOMDIAT Method ofsolution

A basis set is constructed as the product ofMorse oscillator

Catalogue numbers: ACEN functions (associated Laguerre polynomials) for the radial coor-dinate(s) and associated Legendre polynomials for the bending

Program obtainable from: CPC Program Library, Queens Uni- coordinate, with rotation matrices representing the rotations.

versity ofBelfast, N. Ireland(see application form in this issue) The method is variational and the parameters used in con-structing the radial basis set(s) can be optimized [I ,2]. Asecular

Computer: IBM 4341/2: Installation: Universitair Re- matrix is constructed by use ofGaussLaguerre integration forkencentrum Nijmegen the radial coordinate(s) and analytic integration for the angular

coordinates. This matrix is diagonalised to give thesolutions. A

Operatingsystems: MVS release 3.7 choice of GivensHouseholder and orthogonalised Lanczosprocedures are supplied with the program [2].

Programming language used: FORTRAN IVRestrictions on the complexity of theproblem

High speed storage required: case dependent The size ofmatrix that can practically be diagonalised. The

program allocates arrays dynamically at execution time and in

No. ofbits in a byte: 8 the present version the total space available is a single parame-ter which can be set as required.

Overlay structure: yesTypical running time

Peripherals used: card reader, line printer, up to 4 optional disk Running times are case dependentbut dominated by the time

files required for diagonalising the secular matrix. A problem with

350 basis functions (which can be held in 512 K) takes 7 m mNo. ofcards in program and test desk: 3706 on the IBM 434 1/2.

Cardpunching code: EBCDIC Unusualfeatures of the program

A user supplied subroutine containing the potential energy asKeywords: ro-vibrational, body-fixed, associated Laguerre an (analytic) Legendre polynomial expansion is a program

polynomials, associated Legendre polynomials,GaussLaguerre requirement.

quadrature, orthogonalised Lanczos, LC-RAMP, variational,close-coupled equations References

[1] J. Tennyson and B.T. Sutcliffe, J. Chem. Phys. 77 (1982)

Nature ofphysical problem 4061.

ATOMDIAT calculates the bound ro-vibrational levels of a [2] J. Tennyson and B.T. Sutcliffe, J. Chem. Phys. (1983) in

triatomic system using body-fixed coordinates. The embedded press.coordinates are appropriate to atomdiatom systems and the

diatom bondlength may be frozen [11.

* Present address: SERC Daresbury Laboratory, Daresbury, Warrington, Cheshire WA4 4AD, UK.

001 0-4655/83/00000000/$03.00 ~ 1983 North-Holland


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308 J. Tennyson / Variationally exact ro -vibrational levels

LONG WRITE-UP

1. Introduction

The calculation of the bound ro-vibrational levels of molecules with one or more large amplitudeinternal modes has been a centre of much recent research [19].In particular, the variational treatment ofthe problem by setting up and diagonalising a secular matrix in the style of Le Roy and Van Kranendonk[1] has been a particular area of interest. The acronym LC-RAMP (linear combination of radial andangular momentum function products) has been coined [9] to distinguish the basis functions used in thesemethods.

Recently, Tennyson and Sutcliffe [7,8] have developed a general LC-RAMP method for atomdiatom

systems. this method is characterised by its use of a body-fixed coordinate and polynomial basis sets for theradial coordinate(s). Use of the polynomial basis allows the full three-dimensions to be tackled, making the

method appropriate not only for atomdiatom collision complexes such as Van der Waals molecules, but

also any triatomic system which performs large amplitude motions. Unlike methods based on the BOARSapproximation [3,6], the method is exact within the usual limitations of the BornOppenheimer approxima-tion and the variational principle, for both bent and linear molecules. For rotationally excited states, thebody-fixing of the coordinate system causes the coupling between basis functions differing in k (thecomponent of the total angular momentum on the body-fixed z axis) to be small and neglect of these

Coriolis interactions has been found to be an useful approximation [4,7,9,10] as it leads to a considerablesimplification of the computational problem involved.

ATOMDIAT is a general program which uses the method of Tennyson and Sutcliffe to solvethe ro-vibrational problem for triatomic systems. It can perform calculations on full triatomic systems,

atom-rigid diatom complexes and simple diatomics (this is useful for optimizing basis sets); options allow

the Coriolis interactions to be included or neglected as desired. For given quantum numbers, the programautomatically generates both angular and radial basis sets, the appropriate quadrature scheme for theradial integration and hence the secular matrix. A choice of diagonalisers [11,12] is provided depending onwhether the secular matrix can or cannot be retained in high speed storage.

ATOMDIAT has been applied to molecules as diverse as KCN [7], H2Ne [7], HF [8] and HFHe [8]. Ageneralisation of the program has been used for rigid-diatom rigid-diatom calculations on the (N2)2 Van

der Waals dimer [9]. A similar program in space-fixed coordinates was used for calculations on thespinorbit coupling and weak field Zeeman interactions in 02 Ar [13], ATOMDIAT generates vibrationalwavefunctions suitable for the calculation of transition intensities [13,1 4 ] and vibrationally averagedproperties. The wavefunctions have also been used as input to plot routines which given physical insight

into the nature of the large amplitude vibrations [9,15].

2. Method

Fig. 1 . illustrates the body-fixed coordinate system used by Tennyson and Sutcliffe [7,8]. R, whichconnects the diatom centre-of-mass to the atom, lies along the body-fixed z axis and the whole molecule in

the x z plane. This embedding defines three Euler angles a, $ and y. A symmetrized wavefunctionappropriate to this system can be written

~ ~flfl(r,R)[9,k(O)DMk(a,$,y)

k=p mn,!

+( l)~O,k(O)D (a, $, y)](i + ~k.o)/22~2, (1)


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J. Tennyson/Variationally exactro -vibrational levels 309

where D~kis a rotation matrix element as defined by Brink and Satchler [16] and9/k and associated

Legendre polynomial [17]. p equals 0 or 1 for ( l)JPparity under inversion. J is the total angular

momentum and kis its projection on the body-fixed z axis. 1 is the angular momentum of the diatomic.

2

Fig. I. Body-fixed coordinate system ofATOMDIAT.

Following the close-coupling approach of Arthurs and Dalgarno [18], integration over the angular basis

functions yields an effective radial Hamiltonian [7,8]:

_____ a 2 ~ \ h2 ~ 2 ~ \ h2 1 1 J(J+i)H=~kk4, 2~t

2R2-~(~R~) 2~.t

1r2-a-kr ~) +-~-i(l+ ~

1.t2R2 +2 + 2~t

2R2

+8kk,(81~k~~l~!k)

2~2R

28,!,[6kk,2k2 + ~k~,k+Ic~C7~ (1 + 8k0) + ~k~.k_ IcJkClk (1 +~kO)]~ (2)

where ~i~ ~j1 +~ ~ = m~+ (m2 +m3) with m, as the mass of atom I and

c~4~= [J(J+ l)k(k1)]h/~2. (3)

Ifthe potential is given in form of a Legendre expansion,

V(R,r,O)=~V~(R,r)Px(cosO), (4)A

the angular integral becomes analytic

K O ! ~ k I J 1 O / k )=~Jg~(l,1 , k)V,,,(R, r), (5a)

g~(l,1 , k) = (- 1)k[(21~+ 1)]1/2(~ A ~

~),(Sb)

where the 3 jsymbols in the Gaunt coefficient are standard [16]. Ifthe diatom is homonuclear only termswith even A appear in (4) and the problem block factorises according to whether 1 is odd or even.

Tennyson and Sutcliffe expanded solutions of the two-dimensional coupled differential equations given

by H acting on Pm,,, (R, r) in Morse oscillator like functions,

~ R) = rH,,(r)RH,,(R), (7)

H,, = $,I~~2j~,e~2y~~2L(y

1) i = 1 2 (8)

where N,,,,L,,~is a normalised associated Laguerre polynomial [19] and


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310 J. Tennyson / Variationally exactro -vibrational levels

y1=A1exp($1(ri~)), (9a)

(9b)

The constants (A1, $~,,~)and (A2, $2, Re) are those associated with the solutions of the Morse potentialfor diatomic molecules [20]. IfD. is the dissociation energy and w , the fundamental frequency ofdiatomic i,

then

A. =4D1/w,, $, =co~(~s1/4D,)2. (10)

Choosing a, as the nearest integer toA. gives an orthonormal basis complete for all n. As the method isvariational, the constants (re, D

1, w1) and (Re, D2, 2) can be optimized to give the best basis set for aparticular problem. Experience [79]has shown the choice of R~(and re) to be critical and optimal values

to generally be larger than the equilibrium separation.With these radial functions, the matrix elements

=~ {8,,~,,(2nj(aj+ n~+ 1) +a~+ 1) 8,, ~~2((a1 +n1)(a~+n~+ i)n~(n~ l))~2j,

~ n, 1=1,2 (Ii)

are analytic where k, is the appropriate second derivative operator from (2) [7]. Other radial matrixelements, those over r 2 R 2 and VA(r, R), are obtained by Gaussian quadrature. For this a modificationof the program given by Stroud and Secrest [21] has been used, this allows problems with a> 20 to betackled and involves rewriting the quadrature formula for normalised Laguerre polynomials. ForM, point

GaussLaguerre integration the formula for the weight of the pth point becomes

= [(dL~(x~)/dx)L~(x~)] . (12)

With a in the quadrature scheme equal to a. in the basis set, the numerical integration performs well ifM>~rnax + 3 [7].

Diagonalising the secular matrix forms the computational bottleneck of the problem. It dominates the

time taken and if a procedure is used which requires the whole matrix to be retained in fast store, it alsogives a limit to the size of problem that can be tackled. In core diagonalisation is the simplest for smallproblems and comparison of several out of core diagonalisation methods [8] showed the orthogonalised

Lanczos procedure [22] to be particularly well suited to LC-RAMP secular matrices. A choice of

diagonalisers has thus been included in ATOMDIAT, although it is anticipated that the user might wish touse other diagonalising packages particularly if vectorisation is desired.

3. Program organisation

ATOMDIAT is divided into five segments which form the basis of the overlayed structure, see fig. 2.

The subroutines have been numbered sequentially as they appear in the listing and these numbers are usedin the following subsection where each section of the overlay is discussed.

ATOMDIAT uses the convention that variable names beginning with the letters AH and OY are for

double precision reals, IN are for integers and variables whose names begin with Z are logical. Where

possible constants have been placed in data statements to ease conversion to machines with different wordlengths.


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J. Tennyson/ Variationally exact ro -vibrational levels 311

IN S IZ E N S E CM A IN F T X F V V

C O R E

6 T M A IN D Y N A M

S E T C O N

S E T F A C O V E R L A Y 1

L A IP T ~ L G R O D T L G R E C R

k E IN T S O V E R L A Y 2C C M A IN ~

sssoui O V E R L A Y 3

O T 1 C ~ E S I V T W AtV E C O V E R L A Y 1 +

S A lT E R O IA L A N et at

G E IR O W O V E R L A Y 5

Fig. 2. Program structure. Service routines SYMOUT, TIMER, OUTROW and DATETI, which are in the main segment, have not be

depicted.

3.1. Routines always retained in core

These comprise the main program(s) and several service routines. Subroutine CORE Calculates storage requirements and sets up array pointers. Entry DYNAM calls CC-MAIN if sufficient array space is available.

CCMAIN Driver routine which calls the overlayed branches. Calls ERRSET an IBM suppliedroutine which suppresses underfiow errors.

SYMOUT Prints a symmetric matrix stored in lower triangular form. TIMER Calls CPUSED and prints CPU time used. MAIN.

Integer data which characterizes the size of the problem is read in (3) INSIZE, prior to array allocation.Real data is read in (7> SETCON. The other subroutines in this segment are concerned with initialization.


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MATRG if nodiskfile is allocated for the secular matrix.

3.3. Overlay 2 : radial basisfunctions and integration

This segment sets up matrices VO , VI, RM21, RM22, HBL1 and HBL2 which contain all the matrix

elements of the radial basis functions needed in setting up the Hamiltonian. Note that performing theradial integration before the angular integration is the reverse of the order in which the formalism is

generally presented, eq. (2) and refs. [1,5,710,15,18].

(11) KEINTS forms the analytic matrix elements of eq. (Il), stored inHBL1 and HBL2. LAGPT sets up the radial basis set(s) at the numerical integration points and forms matrix

elements over r2 (RM21), R2 (RM22), J~(R,r) (VO) and VX(R, r) (VI). Checks the

sum of integration points and weights against analytic formulae.(l4)LAGUER adaptation of the GaussLaguerre integration points and weights routine of Stroud and

Secret [211for large a. Note that the initial guess formulae are arbitrary and may fail e.g.if a> 1000. The ith call to entry XPOINT returns the ith point and weight; i must be inascending order.


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J. Tennyson/ Variationally exact ro - vibrational levels 313

(22> DGCORE routine which controls in calls to the GivensHouseholder diagonaliser and prints theresults.

GIVTWO GivensHouseholder diagonalising routine due to Prosser [11]. Entry EVEC(I) returnsthe Ith eigenvector.

3.6. Overlay 5: iterative diagonalisation

This segment controls the iterative diagonalisation of the secular matrix for which 3 diskfiles arenecessary, see section 4.2. The method used is orthogonalised Lanczos [8,22]and routines are adapted from

a Paige style Lanczos program [12].

(23> DGITER provides a starting vector, calls the orthogonalised Lanczos routines and prints theanswers.

GETROW Fast unformatted read. DIALAN (43> SET! Suite to perform orthogonalised Lanczos diagonalisation.

4. Program use

ATOMDIAT requires not only card input, but a subroutine giving the potential.

4.1. Thepotential

The potential must be specified as a Legendre expansion, eq. (4). It should be provided for each system

studied as

SUBROUTINE POT (VO, Vl, Rl, R2)

yielding for Rl = r and R2 =R, VO =V(R, r) and Vl = VX(R, r). IfIDIA =2, only even V~are required.

IfNCOORD = 1 , Rl and Vl are dummies. IfNCOORD = 2, Rl is the rigid diatom bondlength, ,~,. IfNCOORD> 1 , the dimension of VI is LPOT. Note that this version of ATOMDIAT works in atomicunits, so VO and VI must be in Hartrees for Rl and R2 in Bohr. To change the units it is necessary to altersubroutines (7) SETCON, (22> DGCORE, DGITER and (47) POT only.

4.2. Card input

The program requires 9 lines of card input for all runs although if NCOORD 3, some of the data isnot used; Table 1 gives the structure of the data input. Defaults given in parentheses are assigned to all

namelist input in (2> BLOCK DATA.

Card 1 NAMELIST/PRT/ZHAMIL [F]=T requests printing of the Hamiltonian matrix.ZRAD [F]=T request printing of the radial matrix elements.

Card2 Namelist/ UNT/IHAM [2] stream to which the Hamiltonian is written if ZLANC =T.ISCRI [1] Lanczos scratchfile used if ZLANC =T.

ISCR2 [3] Lanczos scratchfile used if ZLANC =T, these scratchfiles must be retained if the Lanczosprocess is to be restarted.


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Table 1Format ofdata input

c ACENCC6O

C MAIN PROGRAM #001 ACENCC61c ACENCO66

COMMON /OUTP/ ZHAM1L,ZRAD ACENCO69NAI4ELIST /PRT/ZHAMIL,ZRAD ACENGO7O

COMMON /UNI TS / I HAN,I VEC,I SCR1, I SCR2,ZVEC,ZL ANC ACENCC7S

NAMELIST /UNT/ IUAM,IVEC,ISCRS,ISCR2,ZVEC,ZLANC ACENCO72

COMMON /LANC/ EPSV,MAXCYC,LPRINT, IER,NSTRT,INCR ACENCO73

NAt1ELIST / LNC/EPSV,MAXCYC,LPR1NT,IER,NSTRT,INCR ACENCO74C READ IN NAMELIST INPUT DATA (DEFAULTS IN BLOCK DATA) ACENCc8I

READ(5,PRT) ACENC~E2

READ(5,UNT) ACENCt~3READ(5,LNC) ACENCC84

C ACENc2c9

C SUBROUTINE INSIZE #OO3ACENG21C

C ACENc2IIREAD(5,5) NCOORD ACENC248

5 FORMAT(1C15) ACENC249

READ(5,5) NPNT2,NMAX2,JROT,NEVAL,LMAX,LPOT,IDIA,KMIN,NPNT1,NMAXI ACENW54

READ(5,500) TITLE ACENC295

5CC FORMAT(18A4) ACENI296

C ACENC645

C SUBROUTINE SETCON #007ACEN0646C ACENC647

READ(5,5) XMASS ACENC66I

5 FORMAT(3F2 0. C) ACENC662READ(5,5) RE1,DISS1,WE1 ACENC666

READ(5,5) RE2,DISS2,WE2 ACENC673

IVEC [4] stream to which the eigenvector and eigenvalues are written if ZVEC =T.

ZVEC [F] =T request writing to stream IVEC.

ZLANC [F] false for in core diagonalisation, true for Lanczos diagonalisation.

Card3 NAMELIST/LNC/Ignored if ZLANC =F, except that IER must not be


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J. Tennyson/ Variationally exact ro -vibrational levels 315

JROT IJROTI is the total rotational quantum number of the system. If JROT> 0 the off diagonalConolis terms are included, see eq. (2). IfJROT < 0 they are neglected and k(KMIN) is treated

as a good quantum number. IfJROT =0 or NCOORD= 1 there are no Coriolis terms.NEVAL the number of eigenvalues and vectors required.

IfNCOORD= 1 the rest of the card is ignored.

LMAX order of the highest associated Legendre polynominal in the basis.LPOT highest value of A in the Legendre expansion, eq. (4).IDIA 1 for a heteronuclear diatomic and 2 for a homonuclear diatomic. For homonuclears, the parity

of I is taken from the parity of LMAX.KMIN For JROT 0, KMIN= 1 p, see eq. (1).

IfNCOORD =2, the rest of the card is ignored.

NPNT1 order of Gaussian quadrature in the r coordinate (must be greater than NMAX1).NMAX 1 order of the largest radial basis function Hm(r) giving a diatomic basis of NMAXI + I

functions.

Card6 TITLEA 72 character title.

Card 7 (XMASS(I), I= 1, 3)XMASS(I) contains the mass of atom I (numbering as in fig. 1) in atomic mass units. IfNCOORD= 1,

XMASS (3) is set to zero, thus treating the diatom as atoms 1 and 2, not 2 and 3.

CARD 8 RE], DISSJ, WE]IfNCOORD= I, this card is read but ignored.

IfNCOORD =2, RE1 is the fixed diatomic bondlengths, DISSI and WE1 ignored.

IfNCOORD =3, RE1 =.~,DISSI =D1 and WE1 = c o 1 , see eqs. (8)(10).

Card9 RE2, DISS2, WE2

RE2=Re, DISS2=D2 and WE2= ~~2see eqs. (8)(l0).

4.3. Test cases

Three test cases have been included corresponding to runs with NCOORD = 1,2 and 3. The testsubroutine POT contains a full surface for the HeHF Van der Waals molecule [4]. POT has been writtenfor all 3 values of NCOORD, although it is anticipated that this will not usually be the case. Some pagesfrom these runs which have the print parameters set for maximal printing are reproduced.

Acknowledgements

I wish to thankDr. Brian Sutcliffe, Dr. Paul Wormer and Prof. A. van der Avoird forhelpful discussionsand progranmiing advice. I thankWalter Ravenek for his reading of the manuscript.


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References

[1] R.J. LeRoy and J. van Kranendonk, J. Chem. Phys. 61(1974) 4750.[2] R.G. Gordon and A.M. Dunker, J. Chem. Phys. 64 (1976) 4984.

[3] S.L. Holmgren, M. Waldman and W. Klemperer, J. Chem. Phys. 67 (1977) 4414.[4] S.L. Holmgren, M. Waldman and W. Kiemperer, J. Chem. Phys. 69 (1978) 1661.[5] R.J. LeRoy and J.S. Carley, Advan. Chem. Phys. 42 (1980) 353.

[6] J.M. Hutson and B.J. Howard, Mol. Phys. 41(1980) 1123.[7] J. Tennyson and B.T. Sutcliffe, J. Chem. Phys. 77 (1982) 4061.

[8] J. Tennyson and B.T. Sutcliffe, J. Chem. Phys. (1983) in press.

[91J. Tennyson and A. van der Avoird, J. Chem. Phys. 77 (1982) 5664.[10] J. Tennyson, in: Properties and Symmetry ofNon-rigid Molecules, ed. J. Maruam (Elsevier, Amsterdam, 1983).

[11] GIVTWO, written by F. Prosser, Indiana University (1967).

[12] J.M. van Kats and H.A. van der Vorst, Technical report 1 8 (1976) and (1977), Computer Centre, University of Utrecht, The

Netherlands.

[13] J. Tennyson and J. Mettes, Chem. Phys. (1983) in press.

[14] G.H.L.A. Geerts and J. Tennyson, J. Mol. Spectry. (1983) in press.

[15] J. Tennyson and A. van der Avoird, J. Chem. Phys. 76 (1982) 5710.

[16] D.M. Brink and G.R. Satchler, Angular Momentum, 2nd ed. (Clarendon Press, Oxford, 1968).[17] EU. Condon and G.H. Shortley, The Theory ofAtomic Spectra (Cambridge Univ. Press, Cambridge, 1935).

[18] A.M. Arthurs and A. Dalgarno, Proc. Roy. Soc. (London) A256 (1960) 540.

[19] IS. Gradshteyn and OH. Ryzik, Tables ofIntegrals, Series and Products (Academic Press, New York, 1980).

[20] P.M. Morse, Phys. Rev. 34 (1929) 57.

See also D. ter Haar, Phys. Rev. 70 (1946) 222.[21] A.H. Stroud and D. Secrest, Gaussian Quadrature Formulas (Prentice-Hall, London, 1966) chap. 2.[22] B.N. Parlett, The Symmetric Eigenvalue Problem (Prentice-Hall, Englewood Cliffs, NJ, 1980) chap. 13.


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J. Tennyson /Variationally exactro-vibrational levels 317

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Documents

Jonathan Tennyson- ATOMDIAT: A Program for Calculating Variationally Exact Ro-Vibrational Levels of Levels of "Floppy" Triatomics