Jonathan Tennyson- TRIATOM, SELECT and ROTLEV: For the Calculation of the Ro-Vibrational Levels of...

8/3/2019 Jonathan Tennyson- TRIATOM, SELECT and ROTLEV: For the Calculation of the Ro-Vibrational Levels of Triatomic Molecules

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Computer Physics Communications 42 (1986) 257270 257

North-Holland, Amsterdam

TRIATOM, SELECT ANDROTLEV - FOR THE CALCULATION OF THE RO-VIBRATIONAL

LEVELS OF TRIATOMIC MOLECULES

Jonathan TENNYSON

DepartmentofPhysics andAstronomy, University College London, Gower Street, London WCIE6BT, UK

Received 18 April 1986

PROGRAM SUMMARY

Titleofprogram: TRIATOM MethodofsolutionA basis is constructed as a product of radial (either Morse

Catalogue number: AALO oscillator-like or spherical oscillator) functions and associated

Legendre polynomials for the bending coordinate, with rota-Program obtainablefrom: CPC Program Library, Queens Uni- tion matrices carrying the rotational motion. A secular matrix

versity of Belfast, N. Ireland (see application form in this is constructed using Gaussian quadrature and diagonalised to

issue) give the solutions. The method is variational allowing basis set

parameters to be optimised. Input can either be direct or fromComputer: CRAY-I; Installation: University of London Corn- SELECT [2]. TRIATOM gives the data necessary to drive

puter Centre ROTLEV [3].

Other machines on which program tested: NAS7000 at Dares-

bury Laboratory Restrictions on thecomplexityoftheproblemThe size of matrix that can practically be diagonalised. TRI-

Programming language used: FORTRAN 77 ATOM allocates arrays dynamically at execution time and inthe present version the total space available is a single parame-

High speedstorage required: case dependent ter which can be reset as required.

Overlaystructure: optional Typical rimning time

Case dependent but dominated by matrix diagonalisation. APeripherals used: card reader, line pnnter, optional disk files . .

problem with 533 basis functions (requinng 350000 wordsstorage) takes 8 s on the CRAY-I.

No. oflines inprogram and test deck: 8062 ofwhich 2685 form

TRIATOM

Unusualfeaturesofthe programKeywords: ro-vibrational, body-fixed, associated Laguerre A user supplied subroutine containing the potential energy as

polynomials, associated Legendre polynomials, Gaussian an analytic function (optionally a Legendre polynomial expan-

quadrature, variational, close-coupled equations, vectorised sion) is a program requirement.

Natureofphysical problem References

TRIATOM calculates the bound ro-vibrational levels of a [11 B.T. Sutcliffe and J. Tennyson, MoI. Phys. 58 (1986) 1053.triatomic system using the generalised body-fixed coordinates (2] J. Tennyson, this article, second program (SELECT).

developed by Sutcliffe and Tennyson [1]. [3] J. Tennyson, this article, third program (ROTLEV).

OO1O-4655/86/$03.50 Elsevier Science Publishers B.V.(North-Holland Physics Pub]ishing Division)


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258 J. Tennyson /Ro-vibrational levelsoftriatomic molecules

PROGRAM SUMMARY

Title ofprogram: SELECT Nature ofphysical problemSELECT selects basis sets for TRIATOM [I].

Catalogue number: AALP

Methodofsolution

Programobtainablefrom: CPC Program Library, Queens Uni- A basis function is selected either according its quantum

versity of Belfast, N. Ireland (see application form in this numbers and/or the value ofits diagonal elements.

issue)Restrictions on thecomplexityoftheproblem

Computer: CRAY-i; Installation: University ofLondon Corn- The size of matrix that can be handled by TRIATOM.puter Centre

Typical running time

Other machines on which program tested: NAS7000 at Dares- Case dependent but much less than TRIATOM.

bury Laboratory

Unusualfeaturesofthe program

Programming languageused: FORTRAN 77 A user supplied subroutine containing the potential energy asan analytic function (optionally a Legendre polynomial expan-

High speedstorage required: case dependent sion) may be needed. SELECT produces a file which can beread directly into TRIATOM.

Peripherals used: card reader, line printer and one disk file

ReferenceN o. oflines in program: 1915 [1] J. Tennyson, this article, first program (TRIATOM).

Keywords:basis set selection, first-order perturbation theory

PROGRAM SUMMARY

Titleofprogram: ROTLEV calculation for the bound ro-vibrational levels of a triatomic

system, especially those with large total angular momentum,

Catalogue number: AALQ using the generalised body-fixed coordinates developed by

Sutcliffe and Tennyson [1].

Programobtainablefrom: CPC Program Library, Queens Uni-

versity of Belfast, N. Ireland (see application form in this Methodofsolutionissue) A basis is constructed from Coriolis decoupledsolutions ofthe

problem [2]. The resulting sparse secular matrix is then di-

Computer: CRAY-I; Installation: University ofLondon Corn- agonalised to give the solutions.puter Centre

Restrictions on the complexityoftheproblem

Other machines on which program tested: NAS7000 at Dares- The size of matrix that can practically be diagonalised.bury Laboratory ROTLEV allocates arrays dynamically at execution time and

in the present version the total space available is a single

Programming languageused: FORTRAN 77 parameter which can be preset as required.

High speed storage required: case dependent Typical running timeCase dependent. A problem with 902 basis functions takes 20 s

Peripherals used:card reader, line printer, one disk file on theCRAY-i [2].

No. oflines inprogram: 3424 Unusualfeaturesoftheprogram

Most ofthedata input is read in directly from TRIATOM [3].

Keywords: rotationally excited state, Coriolis coupling, sec-

ondary variational method, sparse matrix, vectorised References

[1] B.T. Sutcliffe and J. Tennyson, Mol. Phys. 58 (1986) 1053.Natureofphysicalproblem [2] J. Tennyson and B.T. Sutcliffe, Mol. Phys. 58 (1986) 1067.

ROTLEV performs the second step in a two-step variational [3] J. Tennyson, this article, first program (TRIATOM).


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J. Tennyson /Ro-vibrational levelsoftriatomic molecules 259

LONG WRITE-UP

1. Introduction scattering coordinates and bond lengthbond an-

gle coordinates are special cases of this gener-

The calculation of the bound ro-vibrational alised coordinate system.

levels of small, especially t~iatomic,molecules has Another development has been the use of a

been an area of much recent research activity, two-step variational procedure for the calculation

Much of this work has focused on the use of of rotational excited states [14]. This has greatly

variational techniques which allow one to obtain extended the range ofrotational states that can be

the eigen energies and wavefunctions of a given practicably considered. Progress has also been

potential to high accuracy. Typical of this ap- made in the use of basis set selection to give moreproach is Whitehead and Handys [1] use of Wat- compact basis sets for the representation ofvibra-

sons [2] form of the Eckart Hamiltonian [3]. How- tionally excited states [12,15]. Finally, improve-ever, the realisation that this Hamiltonian is un- ments have been made in the algorithms used for

satisfactory for systems with large amplitude construction of the secular matrices in ATOMDI-

vibrational modes, which includes most highly AT and ATOMDIAT2, a step which could be-

excited vibrational states, has led to the develop- come rate limiting if the codes were used with ament of techniques which do not rely on the fast, vectorised diagonaliser [12].

concept of a special (equilibrium) geometry for In this work a suite of programs are presented

defining the internal coordinates of the system. which allow ro-vibrational calculations to be per-An example of this approach for triatomics is the formed on any user supplied triatomic potential

one used by Carter and Handy who express their using the generalised (r1, r2, 0) coordinates. The

Hamiltonian in terms of two bond lengths and the core of this suite is program TRIATOM which

angle between them [46]. can be used on its own for ro-vibrational calcula-

An alternative representation that has proved tions; it replaces both ATOMDIAT and ATOM-

popular is the use of scattering coordinates. In DIAT2, also including GENPOT [11] as an op-

these, a triatomic is representated as a diatomic tion. TRIATOM can be used for either fully-cou-

bond length, the distance of the third atom to the pled ro-vibrational calculations, calculations which

diatomic centre of mass and the angle between neglect off-diagonal Coriolis interactions or tothese two coordinates. This coordinate system was, loop over such calculations thus providing the

the basis of the secular equation method devel- data necessary to drive ROTLEV.

oped by Tennyson and Sutcliffe [79],and imple- ROTLEV performs the second variational step

mented in programs ATOMDIAT [10] and AT- in a two-step variational procedure [14]. Program

OMDIAT2 [11]. The method has recently been SELECT allows the basis set used by TRIATOMreviewed [12].

Since programs ATOMDIAT and ATOMDI- AT2 were published there has been a several sig- finificant developments in the calculation of ro-

vibrational levels using this methodology. The most x

fundamental of these is the derivation of a para- Z 2 2

meterised Hamiltonian by Sutcliffe and Tennyson

[13] which allows suitable coordinates to be

selected from a continuum of coordinate sets.These coordinates, given in fig. 1, consist of all B

those in which a triatomic molecule is represented

by a diatomic bondlength, r1, the distance of the 2 r~ 3third atom to any fixed point on this bond, r2, Fig. 1. Body-fixed coordinate system: axes (x,, z,) refer to

and the angle between r 1 and t~.Clearly, both embedding i. 0 u r1.


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260 J. Tennyson/Ro-vibrational levelsoftriatomic molecules

to be preselected. Criteria used are based on the ~~(2) h2 jj+16k k( 8 (j+1) \

total number of quanta of excitation or the value v ~ r1 )

!~12of the diagonal matrix element for a particular

product function. A mixture ofcriteria can also be a (j + ~ ~xlemployed. SELECT can be used to generate all ~~2 r2 / P~12the input required by program TRIATOM, which a ~( ~ +I ~, (4)is designed flexibly to allow users to implement X( +

~8r1 r1j~8r2 r21alternative basis set selection procedures if de-

sired. _____~(1)8kk6jj h2

2(J(J+1)2k2)

~ yR2j~~r

1

2. Method h2~kk 1~jj 2 (1 +8k0+~kO )1/2Using the generalised coordinates of fig. 1, a 2L

1r1body-fixed Hamiltonian can be written < c~~, (5)

VR

j~(2)=+V(r1, r2, 0), (1) VR k~kl~j~j1(1+ko+k,o)

2~s12where V represents the potential. Suitable sym- > < ~ ~ I(~+ 1) 8 )metrised angular basis functions for the Hamilto- r2 8r2nian are

+

6k~k1~j~i_1

1 , k)=(1+8ko)/22h/~2x (1 + 8k0 + ~kO)

x [~,k(0)D~k(a, /3 , y)

+(1)~k(0)D~k(a, /3 , 7)], (2) +~--~. (6)r1 ~r2 8r2)

where D~kis a rotation matrix element as defined

by Brink and Satchler [16] and an associated In the above, the auxiliary quantities are definedLegendre polynomial [17]. The total parity under as follows *

inversion is given by (_1).+P, for p = 0 or 1. Jis

the total angular momentum and k is its projec- ci~= [J(J+ 1) k(k1)11/2, (7)tion on the body-fixed z axis. jis the angular dfk= [(j k+1)(j + k+ 1)momentum of the diatomic represented by atoms

2 and 3. The Euler angles (a, / 3 , y) are those /(2j+1)(2j+3)]h/2, (8)required to place the z axis along r1 and r2 in the afk = [(j +k+1)(j + k+ 2)

positive x z plane embedding 1. Similar func-

tions may be written down for embedding 2 which /(2j +1)(2j + 3)] 1/2 (9)places z parallel to r2 and r1 in the positive x z

plane. ~/k_[(jk)(jk1)/(4j1)1, (10)

Following the close-coupling approach ofwhich are special cases of ClebschGordon coeffi-

Arthurs and Dalgarno [18] yields an effective ra-

dial Hamiltonian [13]: dents. The reduced masses are defined in terms ofthe atomic masses (m,) and g, the parameter

~ h2 ~2 h2 82 which determines the internal coordinates

~ 1=m~1+m~1 (11)

/ 1 1+ ~j(j + i)( __~~ + _~)], (3) * Note that eq. (8) is given incorrectly in ref. [13].\,.tlrl ~tL2l2


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J. Tennyson/Ro-vibrational levelsoftriatomic molecules 2 61

p.12 =g(m~+m~1) ~ (12) integration can be used to express a general poten-

tial function in the form of eq. (16) [9,11].=mj + g2m~1+

(1g)2m~~,

(13)Integration

over angularcoordinates yields

ag = (r

2 u )/r2, (14) set of two-dimensional coupled differential equa-tions, solutions of which can be expanded in terms

where u is the distance from particle 2 to the of one dimensional basis functions

intersection of r1 and r2 , see fig. 1. For scattering

coordinates Im,n(Ti, r2)=rj1Hm(r

1)r~1H,,(r

2). (18)

g =m2/(m2 + m3) (15) These radial basis functions can be expressed interms of known analytic functions. The best suited

which means that r1 cuts r2 at the diatomic centre of these to such problems are the Morse oscilla-of mass. In these coordinates l~~l2=~ =K~= 0 tor-like functions [7,10]

and the Hamiltonian reduces to the one used in

program ATOMDIAT. If g =0 or 1, one obtains I =H,,(r)

the bond lengthbond angle Hamiltonian used by =$~2N exp(y/2)y~~~~2L(y),(19)

Carter and Handy. Other values of g between 0

and 1 yield different coordinate systems which can j=A exp[/ 3 ( rbe used as appropriate. For example g = ~has wherebeen recommended for Van der Waals complexes 41~e ( p . 1/2

a =integer(A)whose symmetry has been reduced by isotopic A = , =We ~, ~75~)substitution [13].

The form of the kinetic energy operators given (20)above in (5) and (6) is appropriate for embedding and the parameters f~~e C O e and D c can be

1. Embedding 2 is obtained simply by making the associated with the reduced mass, equilibrium sep-

exchanges r1 ~-s r2 and jx 1 P2 aration, fundamental frequency and dissociation

So far only the kinetic energy operators have energy of the coordinate, respectively. In practicebeen considered. It is of course possible to write (i~,W~,Dc) are usually treated as variational

the potential solely in terms of the internal coordi- parameters and optimised accordingly. N,,aL is a

nates (r1, r2 , 0). In these coordinates one can normalised Laguerre polynomial [22]. With thesemake the Legendre expansion functions the matrix elements of the differential

V(r1, r2 , 0) =~ V~(r1,r2)Px(cos 0). (16) terms in the kinetic energy operators (3)(6) canbe calculated analytically [7,13]

For the angular basis functions, matrix elements / 82

over such a potential expansion can be computed I n Ianalytically \


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26 2 J. Tennyson /Ro-vibrational levelsoftriatomic molecules

There is no simple closed form for the matrix Having computed the matrix elements, TRI-

elements of r~,r2 or V~(r

1,r2), but these in- ATOM constructs a secular matrix which is thentegrals are evaluated using GaussLaguerre in- diagonalised to give the solutions of the problem.

tegration [20] using the normalised basis functions SELECT can optionally use the diagonal elements

to ensure numerical stability [10]. of the secular matrix to select the LBASS lowestHowever, because Morse functions do not obey basis functions. First-order perturbation theory

the correct boundary conditions at r=0, these suggests that these functions should be the most

functions are not satisfactory for the special case important. Alternatively, the quantum numbers ofthat the wavefunction has amplitude at r=0. This the basis functions can be inspected according to

is unlikely to occur for the r 1 coordinate because the formula

of the strongly repulsive nature of diatomic poten-tials at small separations, but is possible for the r2 Nmax> + + -~- (27)

coordinate when scattering coordinates are used d 3 d rn d~[9,11]. Spherical oscillator functions do not suffer

from this problem and are offered in TRIATOM where j, m, n are the quantum numbers associ-as an alternative for this special case. These func- ated with the basis functions in the in 0, r1 and r2

tions are defined by coordinates, respectively, which are weighted byd., d and d to allow more functions to be

n\=Hiri j m/ ? / selected for low energy modes.=r2

1~2$3~4Nnaexp(_y/2)y0/2L~/2(y), ROTLEV solves the second step in a two-step

j~2 ~23~ variational procedure [14]. The first variational / step is performed by TRIATOM using the Ham-

where iltonian

/3 = (p.to~)~2, (24) ~J,k = + +

p. , We and NnaL are as defined for the Morse-like + V( r1, r2 , 0) (28)

functions. Again a and w can be treated as

variational parameters. For scattering coordinates, for which both Jand k (the projection of Jonall the kinetic energy matrix elements over the the body-fixed axis) are good quantum numbers.

spherical oscillator functions can be computed Solutions of this Coriolis decoupled problem are

analytically [9 11] obtained for all the appropnate values of k.ROTLEV then solves the full Hamiltonian of eq.

/ 2 (1) using these solutions as the basis functions.I n -~--~ n =/3(~~~~(2n+ a + 4 ) Matrix elements for this step are given by

3r ~ ~i, kIHIi, k) 1/2

2 / \ =~i 1~

8kk~i +klk,(1 +~kO+~ko)+(n Ir In) nn, ~25)1 2 x C J.k.i

2 n! F(n + a + 3/2) / J.k Cj,m.n j.rn.n=/3 J,nnjmn

n! F(n+a+3/2)

+ ~ __________ x o .~ .C.~/m 1 ma! T(n + a + 3/2)) ~ V \ 2 p .

1 r12

nn. (26)aJk/ , 1

Radial integrals over the potential are again +6J+1J~2 m -~- mevaluated using GaussLaguerre quadrature. 12


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J. Tennyson /Ro-vibrational levels oftrialomic molecules 26 3

x

[1 \ d \ 1 M A IN IN S IZ E N S E Cr~ dr2 TMA I N b~AMC O R E(j+1) n1n1+ n1n

~,kK

i1~ \ S E T C O N~ ~ m1~m~

- . 2p.12 r1 ,,, S E T F A C ~ ~ -

x[~1 \ d I 1 ~ L AG P T LAGU ER LGR O OTn~_1n1_~fl~~_~fl\)~ (29) C C MA I N O IN r LG R EC Rr2 dr 2 ~ P O T L E G P T

where the ith solution of Hik has eigen energy KEINTS POTV LEGEND

cJ.k and eigenvector c~,and the radial matrix KEINT2 _ S E ~ M E N U2 -

elements are the same as those used in solving thefully coupled problem. In constructing the secular BASOUT BASGEN

matrix I kI runs from p toJand i runs from I to M A T R G

N for each k. IfN is chosen as the number of R T H A V T O T G A U NT

functions of the first variational step then the SL~MENI~

two-step procedure yields results identical to di- CORE EIGSFMrectly solving the fully-coupled problem, see ref. NO R M S SEGMENT 4

[14] for a numerical example.The two-step procedure has the advantage that Fig. 2. Structure of program ATOMDIAT. Service routines

OUTROW, TIMER, SECOND and SYMOUT have beennot all the solutions of the first variational step omitted.

are required to obtain good convergence in thesecond step and that the resulting secular matrix be the main program if there was no dynamic

has a tridiagonal blocked structure. Further de- array allocation (


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3 .2 . Overlay 1: data input and intialisation


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J. Tennyson /Ro-vibrational levels oftriatomic molecules 2 65


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266 J. Tennyson/Ro-vibrationallevels oftriatomic molecules

4. Program use ZPVEC[FJ =T requests printing of the eigenvec-

tors.

TRIATOM requires both card input (which can ZROT[F] =T TRIATOM to perform the first stepgenerated by SELECT) and a subroutine giving in a two-step variational calculation.the potential. ZLADD[F] =T maximum j in angular basis

(LMAX) incremented with (J, k); =F maxi-

4.1. Thepotential mum jfixed (only used if ZROT =T).ZEMBED[TJ =T z axis embedded along r

2 =F z

For both TRIATOM and SELECT there are axis embedded along r~.

two ways ofsupplying the potential. If the poten- ZMORSE[T] =T use Morse oscillator-like func-

tial is specified as a Legendre expansion, eq. (16), tions for r2 coordinate; =F use spherical oscil-option ZLPOT = .TRUE., then the expansion lator functions (only allowed if IDIA > 0).

must be supplied by ZLPOT[F] =T potential supplied in POT; =FSUBROUTINE POT(V0,VL,RI ~ potential supplied in POTV.

ZVEC[F] =T data for ROTLEV to be wntten to

which returns VO = V0(r1, r2) and VL(X) = stream IVEC.

Vx(r1, r2) in Hartree for Ri =r1 and R2 = r2 in IVEC[4] stream for ROTLEV data.

Bohr. If I IDIA I =2, only even V ,., are required. IfNCOORD = 1, Ri and Vi are dummies. If Card2 : NCOORD (15)

NCOORD = 2, Ri contains the rigid diatom bond NCOORD [3] is the number ofvibrational coordi-length, ,~. If NCOORD >1, VL has dimension nates of the problem: 1 for a diatomic (this

LPOT. option is useful for basis set optimisation), 2If a general potential function, ZLPOT = for an atom rigid diatom system (not valid for

.FALSE., is to be used I ISYM I =2), 3 for a full triatomic.

SUBROUTINE POTV(V,R1,R2,XCOS) Card 3 : NPNT2,NMAX2,JROT,NEVAL,LMAX,

must be supplied, which returns the potential V in LPOT,IDIA,KMIN,NPNTI,NMAXJ,ISYM,

Hartree for an arbitrary point given by Ri = r1, NBASS(1215)

R2 = r2 (both in Bohr) and XCOS =cos 0. NPNT2[2 *NMAX2 + 1] order of GaussianCOMMON/MASS/ XMASS(3), G is included quadrature in the r2 coordinate.

in 04.2 . Card inputforTRIA TOM the off-diagonal Coriolis terms are included. If

JROT


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J. Tennyson /Ro-vibrational levelsoftriatomic molecules 2 67

Legendre expansion, eq. (16). If ZLPOT =F, IfNCOORD = 3, RE1 =re, DISS1 =D c and WE1

LPOT + i + MOD (LPOT,2) point GaussLe- = W e are Morse parameters for the r1

coordi-gendre integration is used for the 0 coordinate. nate, see eqs. (19) and (20).IDIA = 1 for generalised coordinates, = 1 for

scattering coordinates with a hetronuclear di- Card 8: RE2,DISS2, WE2 (3F20.O)

atomic, = 2 for scattering coordinates with a If ISYM = 2, this card is read but ignored.homonuclear diatomic, = 2 for midpoint co- IfZMORSE =T, RE2 =re, DISS2 =D c and WE2

ordinates with a symmetric potential (e.g. where = are Morse parameters for the r2 coordi-the symmetry has been broken by isotopic sub- nate, see eqs. (19) and (20).

stitution). If ZMORSE=F, RE2 is ignored; DISS2=a and

KMIN[0] =kfor JROT 0 (including ZROT =T). see eqs. (23) and (24).

NPNT1[2 *NMAX1 + 1] order of Gaussian quad-

rature in the r1 coordinate. Card 9 onwards: (IK(I),IL(I),IM(I),IN(I),1 =1,

NMAX1 order of the largest radial basis function NBASS) (3612)

Hm(ri), giving an r1 basis ofNMAX1 + I func- If NBASS = 0, not read.

tions. If NBASS >0, basis set labels as generated by

ISYM[0] # 0 for bond lengthbond angle coor- SELECT: IK(I) =k IL(I) =j, IM(I) =m + 1dinates (g =0 or 1): = 1 for hetronuclear and IN(I)= n + 1 for the Ith basis function.

case, = 2 for symmetric AB2 case, = 2 foranti-symmetric AB2 case. ISYM I =2 cannot 4.3 . Data inputfor SELECTbe used with JROT> 0 or ZROT =T [13].

NBASS[0] number of basis function in the secular If the basis is to be selected using the diagonal

problem: =0 determined internally, >0 basis elements of the secular matrix, then a potential

preselected and to be read in (see card 9). subroutine (either POT or POTV) must be sup-plied, see section 4.1.

Card4: TITLE (9A8) Card 1: LBASS,NQMAX,NQJ,NQM,NQN,

A 72 character title. IFLA G,IOUT (715)

LBASS[0] select the LBASS lowest basis functions

CardS: (XMASS(I),1 = 1,3) (3F20.O) ordered by their diagonal elements.XMASS(I) contains the mass of atom I (number- NQMAX[0] =N

m~,see eq. (27).

ing as in fig. 1) in atomic mass units. If NQJ[i] =d~,see eq. (27).

NCOORD =i, XMASS(3) is set to zero, the NQM[i] = dm, see eq. (27).

diatom comprising atoms 1 and 2. NQN[i] =d~,see eq. (27).

IFLAG[0] * 0: select basis for different (J, k) or

Card6: G (F20. 0) symmetery than the full calculation.Parameter g determines the coordinate system, see IOUT[7] output stream for TRIATOM data file.

eq. (i4), needed if IDIA = 1 and ISYM =0.Otherwise this card is ignored and: Cards 29

These are the same as cards 1 8 of the TRI-

if IDIA >0 G =m2/(m2+ m3), ATOM input, with the exceptions:

ifIDIA = 2 G= If IFLAG # 0, then TRIATOM card 3 is re-ifISYM # 0 G =i. peated: first, to charactense the basis for the

selection run and second to characterise the

Card7: REJ,DISSJ,WE] (3F20.0) TRIATOM run.

If NCOORD = 1, this card is read but ignored. If NQMAX>0, then NMAX1 and NMAX2 de-If NCOORD = 2, RE1 is the fixed diatomic fault to NQM*NQMAX and NQN*NQMAX,

bondlength, DISSi and WE1 ignored. respectively.


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2 68 J. Tennyson / Ro-vibrational levelsoftriatomic molecules

4.4. Data inputforROTLEV References

Most of the data for ROTLEV, which must [1] R.J. Whitehead and N.C. Handy. J. Mol. Spectr. 55 (1975)have been prepared previously by TRIATOM, is 356, 59 (1976) 459.

read from stream IVEC. Three lines of data are [21C. Eckart, Phys. Rev. 47 (1935) 552.[3] J.K.G. Watson, Mol. Phys. 15 (1968) 476.

read from cards. [4] 5. Carter and NC. Handy, Mol. Phys. 47 (1982) 1445, 57(1986) 175.

Card1: NAMELIST/PRT/ [5] S. Carter, NC. Handy and B.T. Sutcliffe, Mol. Phys. 49TOLER[0.OdO] tolerance for convergence of the (1983) 745.

eigenvalues, zero gives machine accuracy [22]. [6] S. Carter, Report CCP1/84/2, Daresbury Laboratory,UK (1984).

ZPHAM[F] =T requests printing of the Hamilto- [71J. Tennyson and B.T. Sutcliffe, J. Chem. Phys. 77 (1982)

man matrix. 4061.ZPVEC[F] =T requests printing of the eigenvec- [81J. Tennyson and B.T. Sutcliffe, J. Chem. Phys. 79 (1983)

43.tors.IVEC[4] stream for data from TRIATOM. [9] J. Tennyson and B.T. Sutcliffe, J. Mo!. Spectr. 101 (1983)

71.

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

Card2 : NVIB,NEVAL,KMIN [11] J. Tennyson, Comput. Phys. Commun. 32 (1983) 109.NVIB number ofvibrational levels (N) from TRI- [12] J. Tennyson, Comput. Phys. Rep. 4 (1986) 1 .

ATOM for each k to be used in the second [13] B.T. Sutcliffe and J. Tennyson, Mo!. Phys. 58 (1986) 1053.[14] J. Tennyson and B.T. Sutciffe, Mol. Phys. 58 (1986) 1066.

variational step.[15] J. Tennyson

andS.C. Farantos, work in progress.

NEVAL[iO] the number of eigenvalues required. [16] D.M. Brink and G.R. Satchler, Angular Momentum, 2nd

KMIN[0] =(1 p), see eq. (2). ed. (Clarendon Press, Oxford, 1968).[17] EU. Condon and G.H. Shortley, The Theory ofAtomic

Card3 : TITLE (9A8) Spectra (Cambridge Univ. Press, Cambridge, 1935).[18] AM. Arthurs and A. Dalgarno, Proc. Roy. Soc. (London)

A 72 character title.A256 (1960) 540.

[19] IS. Gradshteyn and !.H. Ryzhik, Tables of Integrals,

4.5. Test output Series and Products (Academic Press, New York, 1980).[20] A.H. Stroud and D. Secrest, Gaussian Quadrature For-

A test run of SELECT, TRIATOM and mulas (Prentice-Hall, London, 1966) chap. 2.[21] B.S. Garbow, J.M. Boyle, J.J. Dongarra and C.B. Moler,ROTLEV for the D

2H~molecule has been pre- Matrix Eigensystem Routines - EISPACK Guide Exten-

pared. The potential used is the BVDH potential sion, Lecture Notes in Computer Science, vol. 51

of Martire and Burton [24]. (Springer-Verlag, New York, 1977).[22] NAG Fortran Library Manual, Mark 11, vol. 4 (1983).

[23] P.J. Nikolai, ACM Trans. Math. Software 5 (1979) 4 0 3 .[24] B . Martire and PG. Burton, Chem. Phys. Lett. 121 (1985)

Acknowledgements 479.

I would like to thank Dr. Brian Sutcliffe for

helpful discussions during the course of this work

and NAG for permission to use their routines.


13/14

J. Tennyson /Ro-vibrational levelsoftriatomic molecules 2 69

TEST RUN OUTPUT

BASIS SET SELECTION PROGRAM:9 LINES OF INPUT DATA TR AN SFER ED TO STREAM 7

SELECTION CRITERIA:LOWEST 100 BASIS FUNCTIONS CHOSENFUNCTIONS WITH U P T O 5 QUANTA CHOSEN USING

3 ANGULAR FUNCTIONS PER QUANTA

I RI FUNCTIONS PER QUANTA

1 R2 FuNCTIONS PER QUANTA

PRESELECTION PERFO~4EDFOR J = 0 CASE

FULL TRIATOMIC VIBRATIONAL PROBLEM WITH

1 1 POINT NUMERICAL INTEGRATION FOR

5 TH ORDER Ri RADIAL BASIS FUNCTIONS


. 5 TR ORDER R2 RADIAL BASIS FUNCTIONS

1 4 TH ORDER ANGULAR BASIS FUNCTIONS

28 T E l ~ 1 S IN TH E POTENTIAL EXPANSION

288 CANDIDATE BASIS FUNCTIONS

LOWEST 10 0 FUNCTIONS SELECTED FROM -~O.2662624634E+OOHARTREE TO O.1672416929E+0O HARTREE

13 FUNCTIONS SELECTED WITH LESS THAN 5 QUANTA

NBASS = 11 3 FUNCTIONS CHOSEN WITH TH E REVISED PARAMETERS

F U L l . TRIATOMIC VIBRATIONAL PROBLEM WITH


5 TH ORDER Ri RADIAL BASIS FUNCTIONS


5 TH ORDER R2 RADIAL BASIS FUNCTIONS

1 4 TH ORDER ANGULAR BASIS FUNCTIONS

28 TEUMS IN TH E POTENTIAL EXPANSION

40 LOWEST EIGENVECTORS REQUIRED FOR

11 3 DIMENSION SECULAR PROBLEM

*** VIBRATIONAL PART OF ROT--VIB CALCULATION ***J= 1 KS 0

*** OPTION TO NEGLECT CORIOLIS INTERACTIONS ***

LOWEST 40 EIGENVALUES IN WAVENUMBERS

O.714068722794E+05 O.694353775883E-f 05 -O.686699830589F.+O5 --O.675546976196E+05 O.673727788630E405

O.667371987610E-4-05 - - 0 . 66OO27939589ui~O5 O.656524442688E+05 - - 0 . 653812343100E+05 0 . 64927451 1 9 4 7 F - f 05

0.648063468882E+05 0,640898430523E-f05 O.6379019283O8E+O5 -~O.635199I44447E+O5 --O.633582357493E+05

O.632857100691E+05 O.62942258O436F.-1-05 O.6267R6554230E+O5 0.624231139330F05 O.619670654139E-i-O5

O.618224216590E+05 0.614O88206123F.-f05 0.612594598808E+O5 -O.611221346112E-4-O5 O.6O7263121757EO5

O.605715288293E-i-05 0.600599870895E+05 0.5987Q87tL4578E~05 O.59499766R185E+O5 O.592R92560815F+05

O.5924O7515111E+05 O.590072038907E+05 --O.589297440620E+05 O.586953483929E~O5 --O.585282529019E+O5

0.582417480770E+05 O.578329290682F-i-O5 --O.576358738867F+05 O.572559026414E-i-05 O.567924926990E+05


14/14


*** VIBRATIONAl, P A R T O F ROTVIE CAlCULATION ***

J= I K= I

***OPTION TO NEGLECT CORIOLIS INTERACTIONS ***

LOWEST 40 FIGEN V AL UES IN W A VEN LIM BER S

0.693434778845E+O5 O.673520908937E+05 O.667367529737E05 O.65568O535567E405 O.653695235544E+05

-0.6475812O7137E-F05 O.64i58465O351E-i-05 O.6365O6785O96E+O5 O.633898691954E+O5 O.63I2iO4969O8E+O5

--O.62735O7O5008E+O5 0. 62O451O74873E-~O5 o.619321:1004785+os 0 . 6i5l3187273IE+05 O.6l44OiO83279E+05

O.611i3789197OF-~05 -O.6O9561219989E+05 --0.6O5059997466E+O5 O.602762609382E+05 O.59696961Ft421E+05

O.594707724476E+0-5 -0.591486300333E-fOS O.5856805O2052E+05 --O.582846334O58F.+O5 O.579434392730E+O5

--O.577146293484E+O5 -O.570798691057E-+O5 -O.569522O30011E-fO5 -O.567372233855E-4-O5 0.562657059258EO5

O.557006622625E+05 --O.549539344639E+O5 O.5475i6O93382E+05 O.543753631137E+05 O.540765764232E+05

O.53O367647568E+05 O.524147046677E+05 O,52O570286981E-i-O5 O.516503198823E+05 0.51i88O33OO27E~05

R O TATIO NAL P AR T OF ROTVIB CALCULATION WITH:40 LOWEST VIBRATIONAL EIGENVECTORS SUPPLIED FROM

11 3 DIMENSION VIBRATION SECULAR PROBLEM

40 LOWEST VIBRATIONAL EIGENVECTORS ACTUALLY USED

6 LOWEST ROTATIONAL EIGENVECTORS REQUIRED FO R

80 DIMENSION ROTATION SECULAR PROBLEM

TITLE: D2H+ : E PARITY

LOWEST 6 EIGENVALUES IN WAVENUMBERS

- 0 . 7 i4 27 70 79 81 4E+O 5 O. 69 46 2 73 8 08 5 1E+O 5 O. 69 36 07 8 60 94 4E+0 5 O. 68 6 89 32 1 42 8 5E+O 5 O. 67 57 8 05 56 79 0E+0 5

-0. 6741O1182911E+05

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