James R. Henderson and Jonathan Tennyson- DVR1D: programs for mixed pointwise/ basis set calculation...

8/3/2019 James R. Henderson and Jonathan Tennyson- DVR1D: programs for mixed pointwise/ basis set calculation of ro-vibrational spectra

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Computer Physics Communications 75 (1993) 365378 Computer PhysicsNorth-Holland Communicat ions

DVR1D: programs for mixed pointwise/ basis set calculationofro-vibrational spectra

James R. Henderson and Jonathan TennysonDepartment ofPhysics andAstronomy, University College London, Gower St., London WCJE 6BT, UK

Received 18 September 1992

DVR1D calculates vibrational energy levels and wavefunctions for triatomic molecules in either scattering (Jacobi) or

Radau coordinates for a given potential. The program uses a discrete variable representation (DVR) for the angular

coordinate and a choice ofbasis functions for the radial coordinates. The program is particularly appropriate for high lyingvibrational states. The accompanying program ROTLEV2 is driven by DVR1D and calculates rotationally excited states of

AB2 molecules such as water using Radau coordinates and a bisector axis embedding which properly reflects the

symmetry ofthe system. DVR1D can alsodrive ROTLEVD, DIPOLE and hence SPECTRA from the TRIATOM program

suite (Tennyson et al., Comput. Phys. Commun., previous article).

PROGRAM SUMMARY

Titleofprogram: DVR1D No. oflines in distributed program, including test data, etc.:

3709

Catalogue number: ACNCKeywords: ro-vibrational, body-fixed, discrete variable repre-

Program obtainable from: CPC Program Library, Queens sentation, finite elements, vectorised

University ofBelfast, N. Ireland (see application form this

issue)

Nature ofphysical problemLicensing provisions: none DVR1D calculates the bound vibrational levels ofa triatomic

system using body-fixed coordinates (either Jacobi or Radau)

Computer: Convex C3840 running BSD Unix; Installation: [1]

UniversityofLondon Computer Centre

Method ofsolutionOther machines on which program has been tested: Cray- The angular coordinate is treated using a discrete variableXMP/48, Cray-YMP8i, IBM RS6000 representation (DVR) based on (associated) Legendre poly-

nomials and the radial coordinates are represented by a basis

Programming language used: FORTRAN 77 constructed as a product of either Morse oscillator-like or

spherical oscillator functions. Intermediate diagonalisation

Memory required to execute with typical data: case dependent and truncation is used to construct the final secular problem.

For rotationally excited states DVR1D provides data neces-

Peripherals used: card reader, line printer, disk files sary to drive ROTLEV2 [21or ROTLEVD [3].

Restrictions on the complexityofthe problem

Correspondence to: J. Tennyson, Department ofPhysics and The size of matrix that can practically be diagonalised.Astronomy, University College London, Gower St., London DVRID dimensions arrays dynamically at execution time and

WC1E 6BT, UK.

0010-4655/93/$06.00 1993 Elsevier Science Publishers By. All rights reserved


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366 J.R. Henderson, J. Tennyson /Mixedpointwise/basis set calculation ofro-iihrational spectra

in the present version the total space available is a single an analytic function (optionally a Legendre polynomial expan-

parameter which can be reset as required. sion) is a program requirement.

Typical running time Reterences

Case depenent hut dominated by matrix diagonalisation. The [I] J. Tennyson and JR. 1-lenderson, J. (Them. Phys. 91(1989)sample data takes 23 s for J 0 and 47 s for J = I (Coriolis 3815.

decoupled) on the Convex C3840. [2] JR. Henderson and J. Tennyson, this article, second

program (ROTLEV2).

Unusual features ofthe program [3] J. Tennyson, S. Miller and C.R. Le Sueur. Comput. Phys.A user supplied subroutine containing the potential energy as Commun. 75 (1993) 339, this issue.

PRO G RAM S U M M A R Y

Title ofprogram: ROTLEV2 Nature of physical problem

ROTLEV2 performs the second step in a two-step variational

Catalogue numbers: ACND calculation for the bound ru-vibrational levels of an AB~

molecule using Radau coordinates [1].

Program obtainable from: CPC Program Library, Queens

University ofBelfast, N. Ireland (see application form thisMethod of solution

issue) .A basis is constructed from the energy selected solutions of

the Coriolis decoupledproblem. The resulting sparse matrix isLicensing proilsions: none .

then diagonalised to give the solutions.

Computer: Convex C3840 running BSD unix: Installation:

University ofLondon Computer Centre Restrictions on the complexity ofthe problem

The time taken to transform the required matrix elements.Other machines on which program has been tested: Cray-

XMP48, Cray-YMP8i

Typical running timeProgramming language used: FORTRAN 77 Highly case dependent. The sample data takes 873 s on the

Convex C3840.

Memory required to execute with typical data: case dependent

Unusualfeatures oftheprogramPeripherals used: card reader, line printer, at least two disk Most data is read directly from DVRID [2].files

No. lines in distributedprogram, including test data, etc.: 4401 References

[I] J. Tennyson and B.T. Sutcliffe, lnt. J. Quantum Chem. 42

Keywords: rotationally excited state, Coriolis coupling, see- (1992) 941.

ondary variational method, sparse matrix, vectorised, discrete [2] JR. Henderson and J. Tennyson, first program, this

variable representation article.

LONG WRITE-UP

1 . Introduction

The use of the discrete variable representation (DVR) by Light, Bai and co-workers [1] has givennew impetus to the theoretical study of highly excited vibrational states of small molecules. This isdevelopment has proved timely because of the great increase in experimental activity in this area.

In practice most DVR calculations (see e.g. refs. [2111)on the ro-vibrational levels of triatomic

systems have been hybrids. One coordinate, usually theangular coordinate,

has been treated using a


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JR. Henderson, J. Tennyson /Mixedpointwise/basis set calculation ofro-vibrational spectra 367

DVR and the two remaining, radial, coordinates have been expanded within a finite basis representation(FBR). This method has proved highly successful and the only significant computational difference

between the works cited above has been between the form of the functions used to carry the radialmotion. These calculations show that the DVR

1FBR2 [12] method can routinely deliver an accurate

representation of several hundred vibrational states of most triatomics. This is significantly more thancan easily be obtained using comparable pure FBR (FBR3) methods [8].

In this work we present our DVR1FBR2 program, DVR1D, which has been successfully used tostudy the vibrational states ofHI~[6], LiCN [71,Na

3 [8], H20 [10], KCN [11] and H2Se [13]. Rotationalexcitation ofHj~[61,H25 [14], H20 [15] and H2Se [13] have been studied using either ROTLEVD [161or ROTLEV2 (this article). The computationally efficient [15] option of using ROTLEVD involvesback-transforming the DVR to an FBR. By this means it also possible to calculate transition intensities

using DIPOLE [161either for individual ro-vibrational transitions [17] or for complete vibrational bands

[18].

2. Method

2.1. The vibrational problem: DVR1D

Following Sutcliffe and Tennyson [14,19] one write can an effective radial Hamiltonian operator

14(r

1, r2) =KV+KVR+~kk(jkI V(r1, r2, 0)11k)9. (1)

This Hamiltonian is obtained by letting the full body-fixed Hamiltonian of the problem act on theangular functions, multiplying from the left by the complex conjugate ofthese functions and integratingover all angular variables. Appropriate angular functions are given by

If, k)=~Jk(0)lJ, .M , k), (2)

where @Jk(0) is a normalised associated Legendre polynomial with the Condon and Shortley [20] phaseconventions. J, M, k) is an angular momentum eigenfunction [21] given in terms of the Euler angles(a, /3, y) which are defined by the embedding. In eq. (2), J is the total angular momentum which is agood quantum number of the system; M is the projection ofJon the space-fixed z-axis and will not beconsidered further; k is the projection of Jonto the body-fixed z-axis.

For orthogonal coordinates, such as scattering (Jacobi) or Radau coordinates, the effective vibrationalkinetic energy operator of eq. (1), which is independent of the axis embedding, is given by

ha2 ~ 1 1

K~=~jj6kk 2 + ~hJ(.l + 1) i +i , (3)~ 8r

1 2~2ar2 p~1r1 ji2r2

where the reduced masses are given in terms ofthe atomic masses, rn:

~j1 =g~m~+m~+ (1 g

2)2m~, ~1 =m~+g~m~+ (1 g

1)2m~. (4)

The parameters (g1, g2) are defined by the coordinate system. For scattering (Jacobi) coordinates:

m2g1= , g~=0; (5)

m2 +m3


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368 JR. Henderson, J. Tennyson /Mixedpointwise/basisset calculation ofro-cibrational spectra

and for Radau coordinates [191:

a a 1/2 rn~g1=1 , g2=l , a= - , /3= . (6)

a+B a/3 1 /3+a/3 m1+m2+m5 m1+m2

For systems with two like atoms (AB2 molecules), the identical atoms are 2 and 3 in scattering

coordinates when g1 = ~. In Radau coordinates the like atoms are 1 and 2; in this case g1 =g-,.Transformation ofthe Hamiltonian of eq. (1) to a DVR in 0 is achieved by

HD=TTHT. (7)

The feature which distinguishes the DVR from other finite element approaches is that grid points aredefined by an appropriate Gaussian quadrature. The transformation matrix for the angular coordinate is

thus

(8)

where kand ~0k are the points and weights of N-point Gauss-associated Legendre quadrature points

for associated Legendre polynomials of order k .An important feature of the DVR is the quadrature approximation [221.This is used to evaluate the

contribution due to the potential which is diagonal in the DVR:

N+k I

~ 7~~(jkIV(r

1,r2,0)Ijk)97 ~V(r1,r2,

0ka) (9)

This means that all angular off-diagonal contributions arise from the kinetic energy operator via the Lmatrix which is given by

Nk1

L~,a= ~ T~j(j + 1)1-;-~. (10)1=4

The resulting J=k =0 effective radial Hamiltonian is

h a 2 h 2 a 2 iH,~=l~a ~ ~ + V ( r

1 , r 2 , 0 4 , ) + -~h2i + ~ L~. (11)

2~~1a r 1 2~2a r 2 ,a1r1 /L7r~

The solution strategy in the DVR is to solve the effective radial Hamiltonian for each a on the DVRgrid. The lowest solutions, selected either by number or energy cut-off, are then used to construct a full3D Hamiltonian matrix which is diagonalised to yield the eigenenergies and values for the wavefunctionsat the DVR grid points. In this approach there is choice as to whether the diagonal parts of the angularkinetic energy term (the W-matrix of ref. [6]) are included in the 2D or 3D problems [23]. Our

experience has shown that the latter gives better convergence.The 2D radial Hamiltonians are diagonalised using radial basis functions similar to those employed in

our FBR3 program [16]. These are either Morse oscillator-like functions or spherical oscillators. The

Morse oscillator-like functions are defined as [24]

n)=Hfl(r)=I312Nfl,~+l/2exp(+y)y~~/

2L~(y),y=Aexp[13(rre)], (12)

where

A =4De/13, /3e(~/

21)e)Y2, a=integer(A). (13)


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JR. H e n d e r s o n , ITennyson /Mixedpointwise/basis set calculation ofro-vibrational spectra 369

The parameters p,, re, ~0e and Dc can be associated with the reduced mass, equilibrium separation,

fundamental frequency and dissociation energy of the relevant coordinate respectively. In practice(re, ~0e De) are treated as variational parameters and optimised accordingly. NnaL~ is a normalisedassociated Laguerre polynomial [25].

The spherical oscillator functions are particularly useful for systems which have significant amplitudefor r

2 = 0. These functions are defined by [26]

In)=H~(r)=2~2/314Nflal/

2exp(~y)y(a+l)/2Lal/2(y) y=f3r2, (14)

where

(15)

and (a, 0e) are treated as variational parameters.

For either set of radial functions, integration over the potential is performed using GaussLaguerrequadrature [27]. Details ofhow the integrals over the radial kinetic energy operators are computed canbe found in the previous article [161.

For an AB2 system in scattering coordinates the permutation symmetry ofthe like atoms is carried by

the angular coordinate. In the DVR this leads to a modified, symmetrised L matrix [6]:

N/2+k 1

~ = 2 ~ T2kiqa~(2j+ q)(2j+q + 1)T~j+q,a, q =0, 1. (16)fk

In a symmetrised DVR only the term which differs between even (q =0) and odd (q = 1) calculations isprovided by the L matrix which means that the same solutions of the 2D problems can be used for bothq = 0 and q = 1 calculations if the diagonal portion of L is only included in the 3D Hamiltonian.

In Radau coordinates symmetry is carried by the radial functions provided the same functions areused for the r 1 and r2 coordinates (denoted I m) and I n), respectively). Symmetrised radial functionscan be written as

rn, n, q) =21/2(1 +8mn)~1~2(Im)In)+(_1)~~n)~m)),mn +q, q=0,1. (17)

2 . 2 . R o t a t i o n a l e x c i t a t i o n

2 . 2 . 1 . D r i v i n g ROTLEVD

Ifthe z-axis is taken to lie parallel to either r1 or r2, then the vibration-rotation term in Hamiltonian

(1) is given by [19]

KVR

6fJ8k~k_h2_

2(J(J+ 1) 2k2) ~jf6k,k1 h2_2(1 +6kO+~kO)~2C~CJ~, (18)

where= [J(J+ 1) k(k1)J1~~2. (19)

The first, diagonal in k, term in (18) can be included as part of the effective vibrational problemdiscussed in the previous section. The second, off-diagonal in k, Coriolis coupling term is then the only

remaining operator that needs to be considered. This is the problem solved by ROTLEVD [16].The ithsolution, c1~of the effective vibrational operator in the DVR can be transformed to an FBR, using [6]

= ~ (20)


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370 JR. Henderson, J. Tennyson /Mixedpointwise/basis set calculation ofro-iibrational spectra

The FBR eigenvector coefficients, d~,and corresponding eigenenergies, ~ form the necessary input toROTLEVD. When J= k =0 these coefficients can also be used to drive DIPOLE [16] directly.

2 . 2 . 2 .B i s e c t o r e r n b e d d i n g : ROTLEV2If, in Radau coordinates, the z-axis is placed along a direction ~0 from either r

1 or r2, the

vibrationrotation operator in (1) is given by [14]

KVRkk~(J(J+ 1) _3k2)(~ +

+64~4~(J(J+l)_k2)~11(__~ +

~kki +(k~)iJ~14)

+~kk2 ~CJ~lCJk +~)(2~I,~,J,k 1fYk.J,k) (21)

where C~is defined above and the other angular factors are

(j+k+l)(j+k+2) 1/2alk= ( 2 j + 1 ) ( 2 j +3) (22)

(jk)(jk1) 1/2blk = (4J2 1) - (23)

The integrals in (21) are

=Kjk~(1- cos 0) 1k), (24)

1 + cos 01k) (25)

and

1~~~,1.4=Kikiik>. (26)

The integrals I~and j(2) are in fact singular for k=k= 0, although inspection of(21) shows this only

to be serious in the case of jo), However, for k= k one can apply the quadrature approximation in theDVR:

N+kI 1

T~ j(I) T < ~ 27f,j=k jk,J,k ~ ~ (1 cos Oka)

In all cases where k* k, the integrals can be evaluated quickly and accurately using an appropriate

Gauss-associated Legendre quadrature scheme and then transformed into the DVR.Ofcourse evaluating in the DVR does not actually remove the singularity, it is simply avoided. For a

general molecule in Radau coordinates the singularity occurs when 0 = 00 or 180.However, for an AB2


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JR. Henderson, ITennyson /Mixedpointwise/basis set calculation ofro-vibrational spectra 371

system using the symmetrised basis functions (17), the singularity is confined to 0 =0:see Tennyson andSutcliffe [14] for a detailed discussion of how to symmetrise (21). This corresponds to a linear ABBgeometry which is energetically unfavourable for many molecules. If this is so the effects of the

singularity can be suppressed by forcing the wavefunction to be have zero amplitude near 0 =0.This isthe procedure used by ROTLEV2 which is thus only appropriate for AB

2 molecules in Radau

coordinates.

3. Program structure

The key program in the suite is DVR1D which has to be run in all cases. Ifrotationally excited statesare being considered then eigenvalues, eigenvectors and some matrix elements generated by DVR1D are

passed to either ROTLEVD [16] or ROTLEV2. The input wavefunctions for DIPOLE [16] come from

either DVR1D or ROTLEVD.Card input is needed for both DVR1D and ROTLEV2. The amount required is kept to a minimum by

passing as much information as possible from previous modules and by the use of default settings for

many parameters. Both programs follow the convention that names beginning with letters AH and OY

are for 8-byte real variables, IN are for integers and variables whose name begins with Z are logicals.The calling sequences of DVR1D and ROTLEV2 are given in figs. 1 and 2. The role ofthe individual

subroutines is described in comment cards included in the source programs.

DVR1D and ROTLEV2 used dynamical assignment of array space in which one big vector issub-divided in routine CORE. In the current versions, this array is a single fixed length array ARRAY of

dimension NAVAIL (set to 500 000 in the versions supplied) in subroutine GTMAIN. For efficientstorage management a call to a local GETMAIN, MEMORY or HPALLOC command should beimplemented.

The CPU time requirement of DVR1D is dominated by matrix diagonalisation. The required

diagonaliser has to give all eigenvalues and eigenvectors of a real symmetric matrix. The presentimplementation uses subroutine EGVQR [28] to mimic NAG routine FO2ABF [29]. We would strongly

recommend that EGVQR is replaced either by the local NAG implementation or by some diagonaliser

appropriate to the architecture of the machine.ROTLEV2 uses NAG routine FO2FJF [29] which is a sparse matrix diagonaliser based on the

algorithm of Nikolai [30]. The supplied source for this should be discarded if a local implementation ofNAG is available. In ROTLEV2 the major consumer of CPU time is routine MXMB which performsvectormatrix and matrixmatrix multiplies for both the transformation and diagonalisation step.Replacing MXMB by a version customized for any particular machine is strongly recommended.

4. Program use

4.1. The potential subroutine

DVR1D requires a user supplied potential energy subroutine. There are two ways of supplying the

potential. If it is specified as a Legendre expansion,

V(r1, r2, 0) = ~VA(rl, r2) PA(O), (28)A

which corresponds to option ZLPOT = .TRUE., then the expansion must be supplied by


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372 J.R. Henderson, J. Tennyson /Mixedpointwise/basis set calculation ofro-tibrational spectra

Dvrld }_*um4insize I

CoreGtmain h~[DYnam

~ S e t c o n iCcmain

~ S e t f a c i ~1Norms

~ LagptH

~I Laguer] ~iLgroot*P-~

Lgrecr

Keints

~ Keint2

~ Basgen

p U r l Jacobi ~ Root Recur

,~ A.llpts~

~IAslegPin Llnatrxl

Blocks Potv

I Bickr ~ Pivl~ P ij Le~

Pi~Dgblk ________

_______ Pot

Cutoff

I Choose I

~IMkehain

~rIDgore ]

_____ Trans

Fig. 1 . Structure ofprogram DVR1D. Service routines TIMER, GETROW and OUTROW have been omitted.


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JR. Henderson, IT e n n y s o n /Mixedpointwise/basis setcalculation ofro-vibrational spectra 373

jRot].ev2] ~1insize1

ftcore]

I Gtmain.~ ~[D~amlJ~1 Select II ppyarn

2I

RadintH~~1Mkrad . 1Vrmain ________ ________ ________ ________

________ An ml Gasleg FPi~1 Root I P i 4 Recur

Soirt ____________

~ Angin2 I I Asleg II Solofd~

~l Wrthaml I vacvecf Pu Sdot I

_________ 1 __________

Dgrot } ~ 1 FO2fjf I ~IM a t v e c l

_______ pin Dstorel

F i g . 2 . S t r u c t u r e ofprogram ROTLEV2. Service routines TIMER, GETROW, OUTROW an d MXMB have been omitted.

SUBROUTINE POT(V0, VL, R i , R 2 )

w h i c h r e t u r n s VO = V0(r1, r2) a n d V L ( A ) = V 5 ( r i , r 2 ) i n H a r t r e e f o r Ri = r1 and R2= r2 i n B o h r . I f

I IDIA = 2 , o n l y e v e n V~a r e n e e d e d . I f NCOORD = 2 , Ri c on t a in s t h e rigid diatom bondlength re. I f

NCOORD> 1 t he n ~A has dimensions LPOT.I f a general potential function, ZLPOT = .FALSE., i s t o b e u s e d t he n

SUBROUTINE POTV(V, R i , R 2 , XCOS)

must be supplied. POTV returns the potential V i n H a r t r e e f o r a n a r b i t r a r y p o i n t g i v e n b y Ri =r1,R2 =r2 (b ot h i n B o h r ) a n d XCOS =cos 0.

DVR1D i n c l u de s COMMON/ MASS/XMASS(3), G i , G2 w h e r e XMASS c on t a in s t h e atomic

m a s s e s i n a t o m i c u n i t s (not a m u ) , Gi =g 1 a n d G2 = g 2 . T h i s e n a b l e s u s e r s t o wr i t e f l e x i b l e p o t e n t i a l

s u b r ou t i n e s w h i c h a l l o w f o r c h a n g e s i n c oo r di n a t e s or i s o t o p i c substitution. See, for example, the version

o f POTV s u p p l i e d .

4 . 2 . I n p u t f o r DVR1D

DVR1D requires 9 lines of card input for all r u n s . C a r d s giving data not required or fo r which t h edefaults [given below in parenthesis] are sufficient should be left blank.

Card 1 : NAMELIST/PRT/

ZPHAM[F] = T, r e q u e s t s p r i n t i n g o f t h e H a m i l t o n i a n m a t r i x .Z P R A D [ F ] = T , r e q u e s t s p r i n t i n g o f t h e r a d i a l m a t r i x e l e m e n t s .

ZP2D [F] =T, requests printing of the solutions ofthe 2D radial problems.


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374 IR. Henderson, J. Tennyson /Mixedpointwise/basis set calculation ofro-Librational spectra

Z P MIN[ F ] = T , r e q u e s t s o n l y m i n i m a l p r i n t i n g .

ZPVEC[F] = T, requests printing of the eigenvectors.ZLMAT[F] = T, requests printing of the L-matrix.Z C U T [ F ] = T , f i n a l b a s i s s e l e c t e d u s in g a n e n e r g y c u t - o f f gi v e n b y EMAX.

= F, LOWEIG (LOWEIG2) final basis functions selected.

ZDIAG[T] =T, requests diagonalisation of the final Hamiltonian matrix.ZROT[F] = T, DVR1D to perform first step in a two-step variational calculation.ZLADD[F] only used if ZROT =T:

= T, number ofDVR points (NDVR) fixed with (J, k);=F, number decreases with (J, k).

ZEMBED[T] Used only in conjunction with ROTLEVD= T, z-axis embedded along r

2

= F, z-axis embedded along r1 .ZMORSE[T] =T, use Morse oscillator-like functions for r2 coordinate;

= F, use spherical oscillator functions.

ZLPOT[F] = T, potential supplied in POT;= F, potential supplied in POTV.

ZWBLK[F] =T, include W-matrix in the 2D Hamiltonians.ZTEST[F] Used only in conjunction with IDIA = 2 and JROT> 0.

=T, force suppression of functions at last DVR point.

ZTWOD[F] =T, perform 2D calculation only at specified grid point.Z V EC[ F ] =T, data for ROTLEVD or ROTLEV2 to be written to stream IVEC(IVEC2).

I V E C [ 4 ] s t r e a m f o r ROTLEVD o r ROTLEV2 d a t a .

IVEC2[7] second stream for ROTLEVD data i f IPAR = 2 .Z P F U N [ F ] =T, eigenvalues concatenated on stream ILEV.

W a r n i n g , t h e f i r s t e i g e n v a l u e s o n t h i s f i l e m u s t b e f o r J= 0 , q = 0 .

I L E V [ 1 4 ] s t r e a m f o r e i g e n v a l u e s d a t a .

IHAM[25] stream for intermediate vectors.IHAM2[i] stream for partial Hamiltonian matrix if IPAR =2.

C a r d 2 : NCOORD ( 1 5 )

NCOORD[3] t h e number of vibrational coordinates of the problem:= i f o r a d i a t om i c ( u s e f u l f o r b a s i s s e t o p t i m i s a t i o n ) ,

= 2 f o r an atom rigid diatom system (not valid for IDIA = 2 ) ,

= 3 for a f u l l t r i a t o m i c .

C a r d 3 : NPNT2, NMAX2, JROT, NEVAL, NDVR, LOWEIG, IDIA, KMIN, NPNTI, NMAXI, IPAR,LOWEIG2, LPOT ( 1 4 1 5 )

N P N T 2 [ 2 * NMAX2 + 1 ] o r d e r of Gaussian quadrature in the r2 coordinate.NMAX2 order of t h e l a r g e s t r a d i a l b a s i s fu n c t io n H,,(r2), giving an r7 b a s i s o f NMAX2 + 1

functions.JROT[0] total angular momentum quantum number of the system.NEVAL[10} number ofeigenvalues and eigenvectors required.

IfNCOORD = i the rest ofthe card is ignored.

NDVR number of DVR points in 0 f r om G a u s s ( a s s o c i a t e d ) L e g e n d r e q u a d r a t u r e .

LOWEIG Maximum dimension oflargest final Hamiltonian.


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JR. Henderson, J. Tennyson /Mixedpointwise/basisset calculation ofro-vibrational spectra 375

IfZCUT = F, it is the actual dimension,

if ZCUT =T, LOWEIG must be than the number offunctions selected.IDIA = 1 for scattering coordinates with a hetronuclear diatomic,

= 2 for scattering coordinates with a homonuclear diatomic,

= 1 for Radau coordinates with an ABC molecule, or Radau coordinates for an AB2

system with J> 0 driving ROTLEVD,

= 2 for Radau coordinates with an AB2 molecule.KMIN[0] =k for JROT> 0 and ZROT = F,

=(i p) f o r JROT> 0 and ZROT = T.Note: if KMIN=2 i n ROTLEV2 or ROTLEVD, KMIN must be i in DVR1D.

NPNT1[2*NMAX1 + 1] order of Gaussian quadrature in the r1 c o o r d i n a t e .NMAX1 o r d e r o f the largest radial basis function Hm(ri), giving an r 1 basis of NMAX1 + 1

f u n c t i o n s .IPAR[0] parity of basis if diatomic homonuclear (I IDIA I = 2).

IPAR = 0 for even parity and = i for odd.IPAR= 2: do both even and odd in same calculation (IDIA = 2 only).

LOWEG2[LOWEIG] Maximum dimension of smaller final Hamiltonian. Only used if ZROT =T and

IDIA= 2.

LPOT IfZLPOT =T, highest value of A in Legendre expansion of the potential must beconsistent with subroutine POT.

I f ZLPOT = F, ignored.

Card 4: TITLE (9A8)

A 72 character title.

Card 5: FIXCOS (F20.0)IfZTWOD = T, FIXCOS i s th e fi x e d value of cos 0 for the run.

IfZTWOD = F, t h i s card is read but ignored.

Card 6: (XMASS(I), I=

i, 3) (3F20.0)XMASS(I) contains the mass ofatom I in atomic mass units.IfNCOORD = 1, XMASS(3) is set to zero, the diatom comprising atoms i and 2.

Card 7: EMAX (F20.0)IfZCUT =T, EMAX is the cut-off energy in cm with the same energy zero as the potential.IfZCUT = F, this card is read but ignored.

Card 8: RE1, DISS1, WE1 (3F20.0)

IfNCOORD = 1, this card is read but ignored.IfNCOORD =2, RE1 is the fixed diatomic bondlength, DISS1 and WEi ignored.

IfNCOORD = 3, RE1 =r~,DISS1=Dc and WE1 =We are Morse parameters for the r1 coordinate ina.u.

Card 9: RE2, DISS2, WE2 (3F20.0)IfIDIA = 2, this card is read but ignored.IfZMORSE = T, RE2= re, DISS2 =De and WE2 =We are Morse parameters for the r2 coordinate.IfZMORSE = F, RE2 is ignored; DISS2 =a and WE2 =We are spherical oscillator parameters for the

r2 coordinate in a.u.


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4.3. Card inputforROTLEV2

Most of the data for ROTLEV2, which must have been prepared previously by DVR1D, is read from

streams IVEC (and IVEC2). Note that when J> I a copy of steam IVEC must be supplied on IVEC2.Three lines of data are read from cards.

Card 1 : NAMELIST/PRT/

TOLER[0.OdO] tolerance for convergence of the eigenvalues, zero gives machine accuracy.1.OD-4 is usually sufficient for most applications.

Z P V E C [ F ] =T, requests printing of the eigenvectors.THRESH[0.1d0] threshold for printing eigenvector coefficients, zero requests the full vector (only used if

ZPVEC =T).ZPHAM[F] =T, requests printing of the Hamiltonian matrix.ZPRAD[F] =T, requests printing of the radial matrix elements.IVEC[4] stream for input data from TRIATOM.IVEC2[7] second stream for input data from TRIATOM.

Note that the files on units IVEC and IVEC2 should be identical.ZVEC[F] =T, eigenvalue and eigenvector data to be written to disk fle.

(= T , fo r c e d i f ZTRAN = T).

JVEC[31 stream for first set ofeigenvalue and eigenvector output.

JVEC2[2] stream for second set of eigenvalue and eigenvector output. (KMIN = 2 only.)I S C R [ i ] s t r e a m f o r s c r a t c h f i l e s t o r i n g a r r a y OFFDG.

I R E S [ 0 ] r e s t a r t flag:

= 1, full restart.=2, restart second diagonalisation only (for KMIN = 2 only).= 1, perform vector transformation only (stream JVEC must be supplied).

Z P F U N [ F ] = T , e i g e n v a l u e s c o n c a t e n a t e d o n s t r e a m I L E V . The f i r s t e i g e n v a l u e s

on this file must (with J= 0, q = 0) be already present.ILEV[14] stream for eigenvalue data.

Card 2: NVIB, NEVAL, KMIN, 1BASS, NEVAL2, NPNT (615)NVIB number ofvibrational levels from DVR1D for each k to be read, and perhaps selected

from, in the second variational step.

NEVAL[iO] the number ofeigenvalues required for the first set.KMIN[0] = 0, f or p = I parity calculation.

= 1, e or p =0 parity calculation.= 2, do both e and fparity calculation.

IBASS[0] = 0 or > NVIB*(JROT +KMIN), use all the vibrational levels.

Otherwise, select IBASS levels with the lowest energy.NEVAL2[NEVAL] the number of eigenvalues required for the second set.NPNT[NDVR from DVR1D] number ofGauss associated Legendre quadrature points used for integral

evaluation.

Card 3: TITLE (9A8)A 72 character title.


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JR. Henderson, I. Tennyson /Mixedpointwise/basis set calculation ofro-vibrational spectra 377

4 . 4 . T e s t o u t p u t

Two test decks have been prepared. Both use Radau coordinates and the H2S potential of

Senekowitch et a!. [31]. The first run use DVR1D to drive programs ROTLEVD, DIPOLE andSPECTRA [16].B e c a u s e o f c o n s t r a i n t s o n ROTLEVD, t h i s r u n d o es n ot u s e t h e p e r m u t a t i o n s y m m e t r y

o f t h e H2S m o l e c u l e .

The s e c o n d t e s t r u n , w h i c h i s f o r e v e n ( q =0) s y m m e t r y , u s e s DVR1D t o d r i v e ROTLEV2. B o t h t he s e

r u n s m i m i c benchmark c a l c u l a t i o n s o n H2S transisitions performed by Carter et al. [32],a l t h o u g h t h e

size of the calculations have been reduced in the test data.The LjCN (CN frozen) scattering coordinate surface of Essers et al. [33] i s s u pp l ie d i n s u b r o u t i n e

POT.

Acknowledgements

JRH thanks SERC for a Fellowship. We thank NAG for permission to publish their routines. Thisworkwas supported by SERC grants GR/G34339 and GR/H41744.

References

[1] Z . Ba~i~and J.C. Light, Ann. Rev. Phys. Chem. 40 (1989) 469.

[2] Z . Ba~i~and iC. Light, J. Chem. Phys. 85 (1986) 4594.[3] Z . Ba~iand J.C. Light, J. Chem. Phys. 86 (1987) 3065.

[4] Z . Ba~i~,D. Watt and iC. Light, J. Chem. Phys. 89 (1988) 947.

[5] SE. Choi and J.C. Light, J. Chem. Phys. 92 (1990) 3065.[61 J. Tennyson and JR. Henderson, J. Chem. Phys. 91(1989) 3815.

[7] JR. Henderson and J. Tennyson, Mol. Phys. 69 (1990) 639.

[8] JR. Henderson, S. Miller and J. Tennyson, J. Chem. Soc. Faraday Trans. 86 (1990) 1963.[9] M. Mladenovic and Z. Ba~i,J. Chem. Phys. 93 (1990) 3039.

[101 J.A. Fernley, S. Miller and J. Tennyson, J. Mol. Spectrosc. 150 (1991) 597.

[11] JR. Henderson, H.A. Lam and J. Tennyson, J. Chem. Soc. Faraday Trans. 88 (1992) 3187.

[12] J.C. Light, R.M. Whitnell, T.J. Pack and S.E. Choi, in : Supercomputer Algorithms for Reactivity, Dynamics and Kinetics of

Small Molecules, ed. A. Lagan, NATO ASI series C, Vol. 277, (Kluwer, Dordrecht, 1989) pp. 187213.[13] J. Tennyson and O.L. Polyansky, unpublished.

[14] J. Tennyson and B.T. Sutcliffe, Intern. J. Quantum Chem. 42 (1992) 941.

[15] JR. Henderson, J. Tennyson and B.T. Sutcliffe, Phil. Mag., in press.[16] J. Tennyson, S. Miller and C.R. Le Sueur, Comput. Phys. Commun. 75 (1993) 339, this issue.

[17] S. Miller, J. Tennyson and B.T. Sutcliffe, Mol. Phys. 66 (1989) 429.

[18] CR. Le Sueur, S. Miller, J. Tennyson and B.T. Sutcliffe, Mol. Phys. 76 (1992) 1147.

[19] B.T. Sutcliffe and J. Tennyson, Intern. J. Quantum Chem. 29 (1991) 183.[20] EU. Condon and G.H. Shortley, The theory ofAtomic Spectra (Cambridge Univ. Press, Cambridge, 1935).

[21] D.M. Brink and G.R. Satchier, Angular Momentum, 2nd ed. (Clarendon Press, Oxford, 1968).

[22] A.S. Dickinson andP.R. Certain,

J. Chem. Phys. 49 (1968) 4204.

[23] JR. Henderson, Ph.D. thesis, UniversityofLondon (1990).

[24] J. Tennyson and B.T. Sutcliffe, J. Chem. Phys. 77 (1982) 4061.

[25] IS. Gradshteyn and I.H. Ryzhik, Tables ofIntegrals, Series and Products (Academic, New York, 1980).

[26] J. Tennyson and B.T. Sutcliffe, J. Mol. Spectrosc. 101 (1983) 71.[27] A.H. Stroud and D. Secrest, Gaussian Quadrature Formulas (Prentice-Hall, London, 1966).

[28] Subroutine EGVQR, originally due to T.J. Dekker, Amsterdam (1968).

[29] NAG Fortran Library Manual, Mark 11, Vol. 4 (1983).

[30] P.J. Nikolai, ACM Trans. Math. Software 5 (1979) 403.

[31] J. Senekowitsch, S. Carter, A. Zilch, H.-J. Werner, N.C. Handy and P. Rosmus, J. Chem. Phys. 90 (1989) 783.[32] S. Carter, P. Rosmus, N.C. Handy, S. Miller, J. Tennyson and B.T. Sutcliffe, Comput. Phys. Commun. 55 (1989) 71.

[33] R. Essers, J. Tennyson and P.E.S. Wormer, Chem. Phys. Lett. 89 (1982) 223.


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378 JR. Henderson, J. Tennyson /Mixedpointwise/basis set calculation ofro-cibrational spectra

TEST RUN OUTPUT

Program DVRID (verijon of 16 sept 1992)TITLE: H2S: J 0 even USING Radau COORDINATES

FULL TRIATOMIC VIBRATIONAL PR O BL EM W IT H

13 POINT NUMERICAL INTEGRATION FOR6 T N O R D ER 91 RADIAL BASIS FUNCTIONS

13 POINT NUMERICAL INTEGRATION FOR

6 T N O R D E R R 2 RADIAL BASIS FUNCTIONS24 ANGULAR DV R POINTS USED, W IT H

5 LOWEST EIGENVECTORS CH O S EN FROM

U P T O 250 DIMENSION SECULAR PROBLEM

250 EIGENVALUES SELECTED FROM 0.1234570024001 TO 0.11240497460+00

LOWEST 5 EIGENVALUES IN WAVENUMBERS:

0.3297458063750+04 0.4487898790460+04 0.566945382446D+04 0.5917813652530+04 0.684111928092D+04

TITLE: H25: J 1 even USING Radau COORDINATES

Solution, with J 1 k 0

LOWEST 120 EIGENVALUES TN NAVENUMBERS:

0.3311159190720+04 0.450177106869D+04 0.5683511483380+04 0.5931320043040+04 0.6855378099840+04

0.7105697815540+04 0.801405916722D+04 0.8271377203180+04 0.8465002577570+04 0.8562015910640+04

Solution, with S 1 k 1

LOWEST 120 BIGENVALUES IN WAVENUMBERS:

0.5943241767510+04 0.7112406111960+04 0.0272950038400+04 0.8467494364690+04 0.9423344775320+04

0.9620108566330+04 0.105644696838D+05 0.1076427028500+05 0.1090121291290+05 0.1110103992940+05

PROGRAM ROTLEV2 (VERSION OF 15 Sept 1992):

ROTATIONAL P A R T O F ROT-VIB CALCULATION W IT H :120 LOWEST VIBRATIONAL EIGENVECTORS SUPPLIED FROM

672 DIMENSION VIBRATION SECULAR PROBLEM W IT H24 A NG ULA R D V R POINTS.

120 LOWEST VIBRATIONAL EIGENVECTORS A C T UA LLY USE D24 POINT GAUSS-ASSOCIATED LEGENDRE INTEGRATION10 LOWEST R OT A T I ONA L EIGENVECTORS R E QUI R E D FOR

200 DIMENSION ROTATION SECULAR PROBLEMW IT H BASIS SELECTED BY ENERGY OR D E R I NG

TITLE: fl2S .1 1 even. RadaU CO-ORDINATES

200 FUNCTIONS SELECTED FROM K 0.1508675185001 TO 0.9494182951001

J 1 R OT A T I ONA L STATE. 200 BASIS FUNCTIONSE PARITY. SYMME T R I C (SE> + S-K> FUNCTIONS IN BASISE V E N PARITY RADIAL FUNCTIONS IN BASIS SET

LOWEST 10 EIGENVALUZS IN WAVENUMBERS

0.3311148666240+04 0.4501723410480+04 0.5683426414010+04 0.5931307850020+04 0.594324844233D+04

0.6855255572840+04 0.7105644627150+04 0.7112300358920+04 0.8013898713570+04 0.S27125402553D+04

James R. Henderson and Jonathan Tennyson- DVR1D: programs for mixed pointwise/ basis set calculation...

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