James R. Henderson, C. Ruth Le Sueur and Jonathan Tennyson- DVR3D: programs for fully pointwise...

8/3/2019 James R. Henderson, C. Ruth Le Sueur and Jonathan Tennyson- DVR3D: programs for fully pointwise calculation of vibrational spectra

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Computer Physics Communications 75 (1993) 379395 Computer PhysicsNorth-Holland Communicat ions

DVR3D: programs for fully pointwise calculationofvibrational spectra

James R. Henderson, C. Ruth Le Sueur and Jonathan Tennyson

DepartmentofPhysics andAstronomy, University College London, GowerSt., London WCJE 6BT, UK

Received 18 September 1992

DVR3D calculates rotationless (J=0) vibrational energy levels and wavefunctions for triatomic systems using scattering

(Jacobi) coordinates, or optionally unsymmetrised Radau coordinates, for a given potential energy surface. The program

uses a discrete variable representation (DVR) based on GaussLegendre and GaussLaguerre quadrature for all 3 internal

coordinates and thus yields a fully pointwise representation ofthe wavefunctions. Successive diagonalisation and truncationis implemented for 4 ofthe possible 6 possible coordinate orderings. DVR3D is best used for problems for which many

(several hundred) vibrational states are required. Given appropriate dipole surfaces, the accompanying program DIPJODVR

computes vibrational band intensities for wavefunctions generated by DVR3D.

PROGRAM SUMMARY

Title ofprogram: DVR3D No. oflines in distributed program including test data, etc.: 3848

Catalogue number: ACNE Keywords: vibrations, body-fixed, discrete variable representa-tion, Coriolis decoupled, finite elements, Gaussian quadra-

Program obtainable from: CPC Program Library, Queens ture, vectorisedUniversity of Belfast, N. Ireland (see application form this

issue) Nature ofphysical problem

DVR3D calculates the bound vibrational or Coriolis decou-

Licensing provisions: none pled rovibrational states of a triatomic system in body-fixed

Jacobi (scattering) or Radau coordinates coordinates [1].Computer: Convex C3840 running BSD unix; Installation:

University ofLondon Computer Centre MethodofsolutionAll co-ordinates are treated in a discrete variable representa-

Othermachines on which program has been tested: Cray-YMP8i, tion (DVR). The angular coordinate uses a DVR based onIBM RS6000 (associated) Legendre polynomials and the radial coordinates

utilise a DVR based on either Morse oscillator-like or

Programming language used: FORTRAN 77 spherical oscillator functions. Intermediate diagonalisationand truncation is performed on the hierarchical expression of

Memory required to execute with typical data: case dependent the Hamiltonian operator to yield the final secular problem.DVR3D provides the data necessary for DIPJODVR [2] to

Peripherals used: card reader, line printer, disk files, calculate vibrational band intensities.

Restrictions on the complexity oftheproblemCorrespondence to: J. Tennyson, Department of Physics and . . . .(1) The size of the final Hamiltonian matrix that canAstronomy, University College London, Gower St., London, . .

WC1E 6BT UK practically be diagonalised. DVR3D allocates arrays dynami-cally at execution time and in the present version the total

0O10-4655/93/$06.0O 1993 Elsevier Science Publishers By. All rights reserved


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380 J.R. Henderson etal. / Fullypointwise calculation ofjibrational spectra

space available is a single parameter which can be reset as Unusual features oftheprogram

required. (2) The order ofintegration in the radial co-ordinates A user supplied subroutine containing the potential energy as

that can be dealt within the machine exponent range. Some an analytic function (optionally a Legendrc polynomial expan-

adjustment in the code may be necessary when large order sion) is a program requirement.

Gauss-Laguerre quadrature is used.

References

Typical running time [I] J.R. Henderson, PhD Thesis, University ofLondon (1990).

Case dependent but dominated by the final (3D) matrix [2] J.R. Henderson, C.R. Le Sueur and J. Tennyson, thisdiagonalisation. The test data takes 229 s on a Convex C3840. article, second program (DIPJODVR).

PROGRAM SUMMARY

Title ofprogram: DIPJODVR Methodofsolution

The amplitude ofthe eigenstates at the DVR points arc read

Catalogue number: ACNF in . The value ofthe dipole components are then calculated at

these points and transformed into components along the

Program obtainable from: CPC Program Library, Queens Eckart axes [1]. The vibrational band intensity is the sum of

University of Belfast, N. Ireland (see application form this the products ofthe dipole and the eigenstate amplitudes at allissue) these points.

Computer: Convex C3840 running BSD unix; Installation: .Restrictions on the complexity of the problem

University ofLondon Computer Centre The number of transitions to he calculated. DIPJODVRallocates arrays dynamically at execution time and in the

Othermac/ones on which program has been tested: Cray-YMP8i. .present version the total space available is a single parameterwhich can be reset as required.

Programming language used: FORTRAN 77

Memory required to run with typical data: case dependent Typical running time

7 seconds to run test data on Convex C3840.

Peripherals used: card reader, line printer, at least two disk

files Unusual features ofthe program

Most data is read directly from DVR3D [2]. A user suppliedNo. oflines in distributedprogram, including test data, etc.: . .. . .

1999 subroutine providing the dipole at any point in configurationspace is a program requirement.

Keywords: vibrational band intensities, veetorised, discrete

variable representation Refrrences

[I] C.R. Le Sueur, J. Tennyson, S. Miller and B.T. Sutcliffe.Nature ofphysical problem Mol. Phys. 76 (1992) 1147.

DIPJODVR calculates the vibrational band intensities be- [2] J.R. Henderson, C.R. Le Sueur and J. Tennyson, this

tween eigenstates calculated using the DVR method, article, first program (DVR3D).

LONG WRITE-UP

1. Introduction

The discrete variable representation, DVR, along with other pointwise methods, is becoming more

and more popular and successful as a technique for determining highly excited bound rotationvibration


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JR. Henderson et al. /Fullypointwise calculation ofvibrational spectra 381

states of triatomic molecules [1]. The experimental community is also enjoying new found enthusiasm

and success in this area of spectroscopy and molecular dynamics, due largely to advanced lasertechnology and techniques such as stimulated emission pumping.

The DVR was introduced by Harris et al. [2] and further developed by Light and co-workers [1,3,4]who showed the power of the DVR for obtaining many vibrational bound states oftriatomic molecules.

Henderson, Tennyson and co-workers subsequently exploited this [591using mixed DVR and finitebasis representation (FBR) studies, which also included rotationally excited states in a DVR (DVRt_

FBR2) [5,111. There are now a number of triatomic systems for which estimates of more than 500

vibrational band origins have been made using a DVR. This is a goal more traditional FBR based

methods struggle hard to meet. More recently Whitnell and Light [101 and us [1214]have studied

triatomic species using a full 3-dimensional (3D) DVR, ie. treating all three internal co-ordinates in a

pointwise manner. This method has proved remarkably successful in determining all the vibrational

(J=

0) bound states ofH~[141and even more powerful than a comparable DVRFBR2 study [5].We present here our DVR3 program, DVR3D, which utilises a DVR in each of the scattering (orRadau) coordinates r

1, r2 and 9. The code calculates vibrational (J= 0) eigenenergies and wavefunctionvalues, and has been used successfully on H~[14], HCN/HNC [13] and H2S. Optionally the code will

also calculate Coriolis decoupled rotationally excited states of the molecule, although there is as yet no

means ofcoupling these states to give a full rotational calculation.

Using the wavefunctions provided by DVR3D, we have developed a further program to calculate

vibrational band intensities using the theory of Le Sueur et al. [151. This program, DIPJODVR, is

included as part of the suite.

2. Method

2.1. The vibrational problem:DVR3D

2.1.1. Formulating and solving the 3D DVR Hamiltonian

We use a multidimensional DVR in scattering (Jacobi) or Radau co-ordinates. In scattering coordi-

nates r1 represents the diatom distance between atom 1 and atom 2, and r2 the separation ofthe thirdatom from the diatom centre of mass. The angle between r1 and r2 is 0. A formal definition of (r1, r2, 0)

in Radau coordinates can be found elsewhere [161.

Using a finite basis representation (FBR), the zero rotational angular momentum (J= 0) Hamiltonian

matrix can be written as [17]

(m, n, j I H m, n,f) =Km I m)8~~ + K n ~(2)

+(Km ,~(1)m)6~~~+ K n I F2)(5m,m)J(J +1)6~~

+ 0 and the projection of Jalong the body-fixed z-axis is designated k, then an extra, diagonal in k

term is added to (1):

m, n, j, J, k J~Im, n, j, J, k)= Km, n, j I~Im, n, j)

+ K t I ~(i) t~8~1~85~(J(J+ 1) 2k2). (2)

In (2), if the body-fixed z-axis is taken along r1 then It) = m), s = n and i = 1; conversely if z is along

r2, It>= In), s=m and i=2.


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382 1 . R. Henderson et al. /Fully pointwise calculation ofvibrational spectra

A DVR is essentially a unitary transformation of an FBR defined for some quadrature scheme

associated with the FBR polynomials. When using a multidimensional DVR (i.e. a totally pointwiserepresentation), one needs always to start (at least notionally anyway) by defining the FBR. Further, it is

comforting to know that one can, at any time, transform between the two representations uniquely.In this work the angular basis functions I) are Legendre polynomials, or associated Legendre

functions if k* 0. The radial basis functions are similar to those employed in our FBR3 and DVRFBR2

programs [6,19]. These are either Morse oscillator-like functions or spherical oscillators. The Morse

oscillator-like functions are defined as [181

n) =H~(r) ~i/2N exp(~y)y( V2L~(y). y =A exp[/3(r~-re)1, (3)

where

A=4De/13, I~=We(/.L/2De)~2, a = integer(A). (4)The parameters ,a, re, We and D~can be associated with the reduced mass, equilibrium separation,

fundamental frequency and dissociation energy of the relevant coordinate, respectively. In practice

(re, w, Dc) are treated as variational parameters and optimised accordingly. N,IaL~~is a normalised

associated Legendre polynomial [20].The spherical oscillator functions are particularly useful for systems which have significant amplitude

for r2 = 0. These functions are defined by [21]

In)=H,,(r)=2~2~4N,,(,/

7exp(~y)y(~+i)/2J]~l/2(y)

where

(6)

and (a, w~)are treated as vibrational parameters.

It should be noted that our usual practice is to optimise the parameters for both Morse-like and

spherical oscillator functions using an FBR isomorphic to the DVR in which the final calculation is to be

performed. To this end optimisation is generally performed using the FBR3 code TRIATOM [19]or thetwo-dimensional (ZTWOD = T) option in the DVRFBR2 code DVRID [11].

In (1) V is the potential, and the radial kinetic energy integrals are given by

h2 82KtIh~It)= 2 ~ ~ (7)

2~x1r~8 r ,

/ h2

(t ~ It) = (t 2 ~ (8)

\ 2j . t . , r 1

where It) = m) for i = 1 and It) = In) for i = 2. ~x, are the appropriate reduced masses given by [16]

~ =g~m~+m~+ (1 g2)2m~, ~i =m~+g~m~+ (1 gi)2m~. (9)

where for scattering coordinates

m 2g

1= g20 ( 1 0 )

m2 +

m ~


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J.R. Henderson etal. /Fully pointwise calculation ofvibrational spectra 383

and for Radau coordinates [16]:

a a m 1/2 mg~=1 , a= ___________ 2 (11)

a+/3af3 113+a/3 m1+m2+m3 m1+m2

In principle it is possible to find other combinations of (g1, g2) which give other orthogonal coordinate

systems [161,but this possibility has not been investigated.

A 1D DVR transformation for either of r1, r2 or U is defined in terms ofpoints, 7J, and weights, w ,7, ofthe N-point Gaussian quadrature associated with the orthogonal polynomials used for the FBR in that

coordinate [6]:

T~=(w~)U2It(~)), (12)

where It)=

m),In), Ii) for s~

=

a, f3 , y, r e s p e c t i v e l y . DVR3D automatically generates Gaussianquadrature schemes using routines adapted [211from Stroud and Secrest [22].The required composite transformation is written as a product of 1D transformations:

T = T~~= TaT/3TY ~13m,n,j m n

A three-dimensional DVR is obtained by applying the transformation TTHT. Thus t h e 3D J = 0

Hamiltonian is

(3D)Ha,a~,i

3,i3~,.y,.y~= ~ T,~JKm,n, IIH m, n, j)T,/~,~. (14)

m,n,j m,n,j

The transformed Hamiltonian can be written at the DVR grid points as [12]

(3D) (1) ~ (2) ~ ,q ,~ (1) .s _~.. (2) s~

a,a,/3,/3,y,y r,a /3/3 y,y /3/3 a,a y,y a,a,-y,-y /3/3 /3,/3,y,y a,a

+ (J(J+ 1) k2)M~T,/3/3&,i~+V(r

1~,r2 /3 , Oy)6aa~p&yy, ( 1 5 )

where i = 1 for z embedded along r1 and i = 2 for z along r2.

In (15), the potential energy operator is diagonal because ofthe quadrature approximation

[251~ T,~,~,JKm,n, IIV(r1, r2, 0) m, n, j)T,~,~,~V(ri~,r2~,Oy)~aa~pp&yy,(16)

m,n,j m,n,j

where (r1,~,r2p, O~)i s t h e v a l u e o f (r1, r2, 0) at (a, f3 , y). Note a major attraction here, in that no

integration at all is required over the potential; it is diagonal in every co-ordinate.The kinetic energy terms in (15) are represented by

K~,=ET~~KtI~flt)I~, (17)

~ =J~~~EI~Kt I tY17? (18)

J h2

2r,~~~ (19)

again applying the quadrature approximation, and where

J,= ~TYj(j+1)P. (20)


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384 JR. Henderson etat/Fullypointwise calculation ofvibrational spectra

F o r J> 0, the extra term is given by

M~= ~ T~(m I m)T,~ , ( 2 1 )

mum

i f i = 1 , when it is diagonal in /3, a n d

h2

M~2)3=E ~ 2 (22)

2~x2r2~

if i = 2, when it is diagonal in a.Th e c alc ula ti o n i s g e n e r a l l y s e t u p a s a s e r i e s o f di ag o n al i sat i o n s a n d tr un c ati o n s [12,23].Assume f o r

t h e moment t h a t t h e c o o r d i n a t e o r d e r i n g 0 t h e n r~t h e n r7 i s u s e d , i . e . d i ag o n al i s e o n y f i r s t a n d /3 l a s t

(0 r~ r2). With t h i s o rder i n g, t h e 1D p r o b l e m s t h a t h a v e t o b e s o lv ed f o r ea ch a a n d /3 a r e g i v e n b y

(lD)Ha~ =L,5~ +~ +V(r1,,, r2~,~ (23)

A m p l i t u d e s f o r t h e k t h l e v e l , wi t h e i g e n e n e r g y ~ a r e g i v e n a t e a c h g r i d p o i n t b y C ~ f .

The solutions with ~ E~a r e t h e n u s e d t o s o l v e 2D p r o b l e m s f o r e a c h v a l u e o f / 3 . T h i s g i v e s= ~ + ~ ~ (24)

So lu ti o n s f o r t h e / t h l e v e l , w i t h e i g e n e n e r g y e j~ a r e g i v e n b y C,~lk.The solutions with ~f3

E~a r e t h e n u sed t o s o l v e t h e f u l l 3D p r o b l e m o f d i m e n s i o n N:

(3D)[J,, = E1~0~~// + ~ C~/kC~IkE~ (25)

y

So lu ti o n s o f t h i s d i ag o n al i s at i o n a r e t h e r e qu i r e d e i g e n e n e r g i e s , ,, a n d w a v e f u n c t i o n c o e f f i c i e n t s C,,~,1.

2.1.2. Wavefunctions

Once t h e e i g e n v e c t o r s o f t h e 3D Hamiltonian have been obtained one would usually like to use them

t o e x p r e s s a more p h y s i c a l l y m e a n i n g f u l q u a n t i t y . T h i s c an be a c h i e v e d by t r a n s f o r mi n g t h e m t o y i e l dv a l u e s f o r t h e a m p l i t u d e o f t h e w a v e f u n c t i o n a t t h e DVR g r i d p o i n t s . T h e s e w a v e fu n c t i o n s c an t h e n , i n

p r i n c i p l e , b e p u t t o many quantum m e c h a n i c a l a n d s p e c t r o s c o p i c u s e s . S u c h u s e s i n c l u d e t h e c alc ula ti o n

o f v i b r a t i o n a l b a n d i n t e n s i t i e s , a s e x p l a i n e d i n t h e n ex t s e c t i o n , a n d c o n t o u r p l o t s o f t h e w a v e fu n c t i o n s

( o r l i n e a r c o m b i n a t i o n s o f t h e w a v e fu n c t i o n s [24])w h i c h we h a v e f o u n d v e r y i n f o r m a t i v e .

Th e w a v e f u n c t i o n a m p l i t u d e f o r t h e i t h ei gen s ta te a t t h e DVR g r i d p o i n t s i s s i mp l y

~1~py =

l.k

Computationally this process is more efficiently written as

It~~.=~ (26)

t k

Transformations to FBR are also fairly straightforward to write down, but may be computationally

expensive.

2. 7.3. Aproblem with the quadrature approximation

Th e q u a d r at u r e a p p r o x im a t io n c a n b e u sed when e v a l u a t i n g t h e DVR t r a n s fo r ma t i o n o f FBR m a t r i x

elemen ts o f a n o p e r a t o r t h a t l e a v e s t h e k e t u n c h a n g e d . I t s v a l i d i t y i s d u e t o t h e u n i t a r i t y o f t h e DVR


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JR. Henderson etal. /Fully pointwise calculation ofvibrational spectra 385

transformation matrix. We have experienced one particular failure ofthe quadrature approximation [14],

which occurred when evaluating the r~2integrals.

In scattering co-ordinates, it is possible for the r2 co-ordinate to be equal, or very close, to zero if this

linear geometry is energetically accessible or favourable. As stated earlier, it is desirable to use thespherical oscillator functions in this case. Under these circumstances it was found [14] that the

quadrature approximation had to be abandoned for the r~2integral because of its non-polynomial

behaviour as r2 f 0.

After extensive tests on the HI~molecular ion [14], we found a suitable alternative was to continue to

use the quadrature approximation in constructing (lD)H and then construct (3D)H using the full-matrix

transformation ofthe r~2integrals, correcting for the fact that the quadrature approximation was used

in ~ So the only change in the formulation is that now

(3D)j~,

11, (

3D)H~/3,11 + ~ C~tkC~lk(M,~J3_M,~) ~ ~ (27)

a,k,k y,y

where

%~i~i~=LKnI~2~In)I~, (28)

and M~ is given by eq. (22).Note that the FBR matrix elements above can be evaluated analytically, and are given by [21]

h 2 / 3 n! F(n+a+~) 1/2Kn I ~2) In) = ( 2 a + 1)~2 F(n+a + ~)) , nn. (29)

We have implemented a user input variable (ZQUAD2) in the program DVR3D so that either of eqs.

(25) or (27) can be used.

2.1.4. Order ofsolution

We have shown above how the 3D DVR Hamiltonian matrix ofeq. (14) can be solved by successive

diagonalisation and trucation. Above the angle 0 w a s dealt with first, and r2 last. It also is possible for

the problem to be solved in any of the 5 other orders. We have recently suggested how one may

determine in advance the most efficient order of solution for a particular molecule [131.It is thought that

the co-ordinate accomodating the highest density of states should be treated last and the co-ordinate

holding the lowest density ofstates coming first, although this was found to be less important in practice.

Four of the possible co-ordinate orders have been implemented in DVR3D. The two orders where 0

is considered second have been omitted; it is unlikely that such orders will offer significant savings over

the case in which the order of 0 and the first coordinate are swapped.

2.1.5. SymmetryScattering co-ordinates are capable of exploiting the C2~symmetry of a molecule. Thus they can split

the full permutation symmetry of an A2B system and can therefore be ofsome help in A3 molecules. Wehave shown how to do this in a DVR-in-0 approach [5,111;it involves only a property of the so-called L

matrix. In our multidimensional DVR we have exactly the same matrix in the J matrix of eq. (20). Thenthe symmetrised J matrix becomes

N/2 1

~ T~.Yj+q(2j+q)(2j+q+1)T~jq, q=0,1. (30)

j= = o


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386 J.R. Henderson eta! . /Fullypointwise calculation ofvibrational spectra

In a symmetrised DVR3D calculation even (q = 0) and odd (q = 1) calculations are done separately

unlike DVR1D [11]. If0 is t h e f i r s t c o -o r d i n a t e t o b e t r e a t e d then no saving can be made from doing

both odd and even runs at once. There could be some saving for other orders, although this has not yet

b e e n i m p l e m e n t e d i n p r o g r a m DVR3D.

I t s h o u l d b e n o t e d t h a t t h e c u r r e n t version of DVR3D does not employ the full symmetry of an AB5

system in Radau coordinates. To do this involves symmetrising points taken from different steps in the

d i a g o n a l i s a t i o n t r u n c a t i o n p r o c e d u r e a n d w i l l t h e r e f o r e r e q u i r e m a jo r c h a ng e s to the algorithm em-

ployed here. A code to solve this problem is currently under development [26].

2.2. Vibrational band intensities

Th e e q u a t i o n to be evaluated for vibrational band intensities may be written as

S~-i(~fH~j)~, (31)

where ~s is expressed in terms of components along the Eckart axes [15]. This expression is easy to

evaluate in the DVR, where the integral is reduced via the quadrature approximation to a sum:

E~1(a/3y)~(a/3y) ~1(a/3y)~. (32)

af3y

DIPJODVR u s e s w a v e fu n c t i o n s p r o d u c e d by DVR3D to evaluate this expression and hence to obtain

vibrational band intensities.

C o m p l i c a t i o n s c a n a r i s e i f t h e b r a a n d k e t w a v e fu n c t i o n s w e r e n o t c a l c u l a t e d a t t h e s a m e p o i n t s . I n

t h i s c a s e t h e t r a n s f o r ma t i o n m a t r i x m u s t b e u sed t o o bt ai n t h e f u n c t i o n e x p a n s i o n o f t h e w a v e fu n c t i o n s .

DIPJODVR c an c o p e w i t h t h i s ad di ti o n al p r o b l e m f o r a l i m i t e d number o f c a s e s [27].

3. Program structure

C a r d i n p u t i s n e e d e d f o r b o t h DVR3D a n d DIPJODVR. Both programs follow the convention thatnames b e g i n n i n g w i t h l e t t e r s AH a n d OY a r e f o r 8 -byt e r e a l v a r i a b l e s , IN a r e f o r i n t e g e r s , a n d

v a r i a b l e s w h o s e name b e g i n s wi t h Z a r e l o g i c a l s .Th e c a l l i n g s e q u e n c e s o f DVR3D and DIPJODVR a r e g i v e n i n f i g s . 1 and 2 . Th e r o l e o f t h e i n d i v i d u a l

s u b r o u t i n e s i s d e s c r i be d i n comment c ar d s i n c l u d e d i n t h e s o u r c e p r o g r a m s .

3.1. DVR3D

DVR3D u s e s d y n a m i c a l a s s i g n m e n t o f a r r a y s p a c e i n w h i c h o n e b i g ve c to r i s s u bd i v i d e d i n ro ut i n e

CORE. I n t he c u r r e n t v e r s i o n s , t h i s a r r a y i s a s i n g l e f i x e d l e n g t h a r r a y ARRAY ofdimension NAVAIL( s e t t o 5 0 0 0 0 0 i n t h e ve rs i o n s up p li ed ) i n s u b r o u t i n e GTMAIN. F o r e f f i c i e n t s t o r a g e management a c a l l

t o a l o c a l GETMAIN, MEMORY o r HPALLOC command s h o u l d b e i m p l e m e n t e d .

Th e CPU t i me r e q u i r e m e n t o f DVR3D i s d o m i n a t e d b y t h e f i n a l di ag o n al i sat i o n o f t h e 3D

Hamiltonian matrix HAM3. The required diagonaliser has to give all eigenvalues and eigenvectors of a

r e a l s y m m e t r i c m a t r i x . Th e p r e s e n t i m p l e m e n t a t i o n u s e s s u b r o u t i n e EGVQR [28] to mimic NAG routineFO2ABF [30].All of the intermediate diagonalisations oft h e va ri o us HAM1 a n d HAM2 ma t r i c e s us e t h i sr o u t i n e . We w o u l d s t ro n gl y recommend that EGVQR is replaced either by the local NAG i m p l e m e n t a -

tion or by some diagonaliser appropriate to the architecture of the machine.


9/17


Dvr3d j_..4lnsize

Gitain

~IS e t c o n ~Ccmain

P 1 S e t f a c l ~INorms

~ Lagpt ~ Laguer~ PI.(Lgroot~ PiI1~Lgrecr~

~ Keints

I I Keint2l

P 1 Rtwts

K1k2

~I Jacobil ~l Root I P 1 1 Recur~IA11pts~~ Asleg

~IL m a t r x ~ Potv

~I~ham1Iu~ Diag I Potvig~ P u r l Legv

P u r l CutlD I P u r l PotMkham2

~ Diag

Cut2D

I Choosel

~ ~ I

P 1 Diag3D~p p . 4 Trans

Fig. 1 . Structure ofprogram DVR3D. Service routines TIMER, GETROWandOUTROW have been omitted.


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388 J.R. Henderson etal. /Fullypointwise calculation ofvibrational specira

3.2. DJPJODVR

DIPJODVR a l s o u s e s d y n a m i c a l a s s i g n m e n t o f a r r a y s p a c e i n w h i c h o n e b i g ve c to r i s s u b d i v i d e d . B u t

i n t h i s p r o g r a m t h e f i x e d l e n g t h ar r ay ARRAY, o f l e n g t h NAVAIL (set to 500 000 in this version) is

subdivided in routine GTMAIN. For technical reasons, part of this array is also used in the main

Rddata H ConverDipjOdvr ________

_________ Messge

Gtmain

. 0 .4 Rdphi

~ Oldphi l-~4Getphi I

I Conver IGetznu Dipcal _________

_________ Dipd

Glagpt

Gettra Basis Lgrecr

Inmain 0. Diffmu Glagpt ________ ________

I ~ Lgroot l . . 4Lgrecr

~ Getbas

Basis__}..j_Sl4aaf

4

I ConverDomult Dipcal _________

_______

I_________ Dipd

Gett

o.~j Difft

Writet I

Fig. 2. Structure ofprogram DIPJODVR. Service routines TIMER, GETROW and OUTROW have been omitted.


11/17


program to read in the DVR points. Each array of DVR points is assumed to be shorter than MAXQ

( s e t t o 1 00 i n t h i s v e r s i o n ) .

For a typical problem, DIPJODVR uses a negligible quantity of CPU time. For example in the test

run, 7.5 s is used, of which 7.3 s is time used to calculate the dipole at the various DVR points. If,

however, the bra and ket use different DVR points, then a substantial proportion of time is spent

calculating the transition intensities (see below).

The dipole is provided via a user-supplied subroutine DIPD in the internal coordinate system of theDVR calculation. It is subsequently rotated by DIPJODVR into the Eckart system [15]. In order to do

t h i s , th e e qu i l i br i u m values of r1, r2 and cos 0 are required. The values of the transition moments can be

qui te s e n s i t i v e t o t h e s e v a l u e s , a n d s o i t i s i m p o r t a n t t h a t t h e y a r e c o r r e c t . A c c o r d i n g l y , t h e p r o g r a m

prints out the cartesian coordinates of the atoms corresponding to the given equilibrium values. It is

recommended that these are checked before placing any credence on the rest of the output!

The greatest part of the program is taken up with a type ofcalculation that has not been generalisedy e t , w h i c h w a s u sed t o calculate transition moments for H~[271. Because the particular boundary

conditions for this ion require that the basis functions for r2 are different for even and odd s y m m e t r y

runs, there are parts of the program (governed by ZSAME = F) which can cope with the bra and ket

functions having different points for r2. This option has only been implemented for spherical oscillator

functions in the r2 coordinate as in our experience the same Morse-like functions are always used foreven and odd symmetries. In this case additional input is required and the program is much slower.

When calculating the transition moments, the program is written to use quadrature points specified by

both the even and odd r2 functions, and to take the average. However, if as in the original work done on

H~[14], the even and odd functions differ by integer powers of r2, then it is important to use onlyquadrature points resulting from the ones specifying lower powers of r2 in order to avoid inaccuracies of

quadrature [27].

Calculations run with ZSAME = F require F functions. Subroutine S14AAF is a NAG routine [29]which calculates F functions. The present implementation uses routine GAMMLN from Numerical

Recipes [30] to mimic the operation of S14AAF.

4. Program use

4.1. The potential subroutine

DVR3D requires a user supplied potential energy subroutine. There are two ways of supplying the

potential. Ifit is specified as a Legendre expansion,

V(r1, r2, 0) = ~V5(r1, r2) PA(O), (33)A

which corresponds to option ZLPOT = .TRUE., then the expansion m u s t b e s up p l i e d b ySUBROUTINE POT(V0, VL, Ri, R2 )

which returns VO = V0(r1, r2) and VL(A) = I~(r1,r2) in Hartree for Ri = r1 and R2 = r2 in Bohr. If

NCOORD = 2, Ri contains the rigid diatom bondlength re.

Ifa general potential function, ZLPOT = .FALSE., is to be used then

SUBROUTINE POTV(V, Ri, R2, XCOS)

must be supplied. POTV returns the potential V in Hartree for an arbitrary point given by RI = r1 ,

R2 = r2 (both in Bohr) and XCOS = cos 0.


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390 JR. Henderson etal. /Fullypointwise calculation ofvibrational spectra

DVR3D includes COMMON/MASS/XMASS(3), Gi, G2 where XMASS contains the atomic masses

in atomic units (not amu), Gi =g1 and G2=g2. This enables users to write flexible potential

subroutines which allow for changes in coordinates or isotopic substitution. See, for example, the version

of POTV supplied.

4.2. Input forDVR3D

DVR3D r e q u i r e s 9 l i n e s o f c a r d i n p u t f o r a l l r u n s . C a r d s g i v i n g d a t a n o t r e q u i r e d o r f o r w h i c h t h e

defaults [given below in parenthesis] are sufficient should be left blank.

Card i: NAMELIST/PRT/ZPHAM[F] = T , r e q u e s t p ri 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 .

ZPRAD[F] = T , r e qu e s t s p ri 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 me n t s .ZP i D[ F] = T, requests printing ofthe results o f 1D c a l c u l a t i o n s .

ZP2D[F] = T, requests printing ofthe results of 2D calculations.

Z P M I N [ F ] = T , r e qu e s t s o n l y minimal printing.

ZPVEC[F] = T , r e qu e s t s p ri n t i n g o f t h e ei gen vec to rs .

ZLMAT[F] = T, requests printing o f t h e L - m a t r i x .

Z C U T [ F ] = T, final d i m e n s i o n s e l e c t e d u s i n g a n e n e r g y c u t - o f f g i ve n b y EMAX2.

= F, final dimension determined by NHAM3.

ZEMBED[T] Used o n l y i f J> 0.

= T, z-axis embedded along r2= F, z - a x i s embedded a l o n g r1 .

ZMORS1[T] = T , u s e More o s c i l l a t o r - l i k e fu n c ti o n s f o r r1 c o o rdi n ate ;

= F, use s p her i c al o s c i l l a t o r f u n c t i o n s .

ZMORS2[T] = T, use Morse oscillator-like functions for r2 c o o r d i n a t e ;

= F, u s e s p her i c al o s c i l l a t o r f u n c t i o n s .

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

= F, potential supplied in POTV.ZVEC[F] = T, store the eigenvector from all the parts of the calculation (ID, 2D and 3D) on

stream IOUT2. Further information relating to this (arrays IV1 and 1V2) is stored on

stream IOUT1.

ZALL[F] = T, requests no truncation of the intermediate solution.

ZTHETA[T] = T, let 0 be first in the order of solution;= F, let 0 be last in the order of solution.

ZR2Ri[T] = T, let r2 come before r1 in the order of solution;= F, l e t r1 come before r2 in the order of solution.

ZTRAN[F] = T, perform the transformation of the solution coefficients to the expression for the

wavefunction amplitudes at the grid point, eq. ( 2 6 ) . S t o r e t h e d a t a o n s t r e a m IWAE.ZTRAN = T automatically sets ZVEC = T.

ZQUAD2[T] = T, use the DVR quadrature approximation for the integrals of the r~2matrix, and

h e n c e make i t s DVR t r a n s fo r m a t i o n d i ago n al.

= F , e v a l u a t e t h e r~2integrals fully and perform the DVR transformation on t h e m .

N o t e t h a t ZQUAD2 = F is only implemented for ZMORS2 = F and for ZTHETA = T.

ZDIAG[T] = F , d o n o t d o f i n a l di ago n al i sa ti o n , i n st ea d t h e f i n a l H a m i l t o n i a n m a t r i x i s w r i t t e n o nunits IDIAG [20] and IDIAG [21]. For further details see the source code.

IEIGS[7] stream for eigenvalues of the 1D solutions.IVECSi[31 stream for eigenvectors of the iD solutions.


13/17

JR. Henderson etal. /Fullypointwise calculation ofvibrational spectra 391

I E I G S2 [2 ] s t r e a m f o r e i g e n v a l u e s o f t h e 2D s o l u t i o n s .

I VE CS 2 [4 ] s t r e a m f o r e i g e n v e c t o r s of the 2D solutions.

I VI N T[i 4] s c r a t c h f i l e u sed f o r s t o r i n g i n t e r m e d i a t e vectors in building the final Hamiltonian.I B A N D [ 1 5 ] s c r a t c h f i l e u sed for storing bands of the final Hamiltonian.

I N T V E C [ i 6 ] s c r a t c h f i l e f o r i n t e r m e d i a t e s t o r a g e o f t h e 2D vectors.

I OU T 1 [ 2 4 ] s t r e a m f o r a r r a y s I V 1 a n d 1 V2 , w h i c h r e c o r d t h e s i z e s o f t h e truncated vectors.

U s e d when ZVEC = T.

IOUT2[25] stream for the 1D, 2D and 3D vectors for use when ZVEC = T.

IWAVE[26] stores the wavefunction amplitudes at the grid points when ZTRAN = T.

Card 2: NCOORD (15)NCOORD[3] the number ofvibrational coordinates ofthe problem:

= 2 for an atom rigid diatom system,= 3 for a full triatomic.

C a r d 3: N PN T 2, JROT , NEVAL, NALF, MAX2D, MAX3D, IDIA, KMIN, NPNT1, IPAR (1015)

NPNT2 number o f DVR p o i n t s i n r2 Gauss(associated) Laguerre quadrature.

JROT[0] total angular momentum quantum number ofthe system, J.

NEVAL[i0] number of eigenvalues and eigenvectors required.

NALF number of DVR points in 0 from Gauss(associated) Legendre quadrature.

MAX2D maximum dimension ofthe largest intermediate 2D Hamiltonian.MAX3D maximum dimension ofthe final Hamiltonian.IfZCUT = F, it is the actual dimension,

if ZCUT = T, MAX3D is than the number of function selected.IDIA = 0 f o r Radau c o o r d i n a t e ,

= 1 f o r s c a t t e r i n g coordinates with a heteronuclear diatomic,= 2 f o r scattering coordinates with a homonuclear diatomic.

KM I N [ 0 ] = k f o r JROT> 0 .

NPNTi number o f DVR p o i n t s i n r1 from G a u s s ( a s s o c i a t e d ) L a g u e r r e q u a d r a t u r e .

I P AR [0 ] p a r i t y o f b a s i s f o r t h e h o m o n u c l e a r d i a t o mi c ( I D I A = 2) case:IPAR = 0 f o r e v e n p a r i t y a n d = i f o r o d d .

C a r d 4 : TITLE ( 9 A 8 )

A 72 character title.

C a r d 5 : ( XM . A S S ( I ) , I = i, 3) ( 3F 2 0 . 0 )

XMASS(I) c o n t ai n s t h e m a s s o f a t o m I i n a t o m i c m a s s u n i t s .

C a r d 6 : EMAX1, EMAX2 ( 2 F2 0 . 0 )

EMAX1 i s t h e f i r s t c u t - o f f e n e r g y i n cmt w i t h t h e s a m e e n e r g y ze r o a s t h e potential. Thisd e t e r m i n e s th e t r un c at i o n o f t h e 1D s o l u t i o n s .

EMAX2 i s t h e s e c o n d c u t - o f f e n e r g y i n cm1 w i t h t h e s a m e e n e r g y z e r o a s t h e p o t e n t i a l .

This controls the truncation of the 2D solutions (i.e. the size of the final basis). If

ZCUT = F it is ignored and the size ofthe final Hamiltonian is MAX3D.

C a r d 7: RE1, DISS1, WEi (3F20.0)IfNCOORD = 2, RE1 is the fixed diatomic bondlength, DISS1 and WE1 ignored


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392 JR. Henderson C t al. /Fully pointwise calculation ofvibrational spectra

IfNCOORD = 3, RE1 = re, DISS1 = Dc and WE1 = We are Morse parameters for the r1 coordinate

when ZMORS1 = T, and are spherical oscillator parameters when ZMORSI = F.

C a r d 8 : R E 2 , D I S S 2 , WE2 ( 3F 2 0 . 0 )IfZMORS2 = T, RE2 = re, DISS2 =D~and WE2 = are Morse parameters for the r2 coordi -

n a t e .

IfZMORS2 = F, RE2 is ignored; DISS2 = a and WE2 = We are spherical oscillator parameters for

the r2 coordinate.

Card 9: GSTATE (F20.0)

The ground state of the system in cm relative to the energy zero.

This must be supplied when IDIA = 2, IPAR = 1 and JROT = 0, and can be blank otherwise.

4.3. The dipole subroutine

DIPJODVR processes batches of transitions between states defined by the input file. This file is

produced by DVR3D stream IWAVE. Usually there will only be one input file, but if the Hamiltonian

has two symmetry blocks which were calculated separately, then two may be given. In that case, theinputs for the two DVR runs must agree absolutely except for IPAR which will obviously be different,

an d , i f ZSAME = F, the value of a used to define spherical oscillator functions in the r- , coordinate.

The stream IWAVE from the DVR run contains most of the data necessary to characterise the run.The rest must be supplied in the input for DIPJODVR. The program uses streams IBRAO, IKETO,IBRAI, IKET1, IWAVEO and IWAVEI, and will stop if any two of them are set of the same channel.

All arrays are internally allocated within an area of memory, the size of which is specified by a parameter

NAVAIL. Th e p r o g r a m w i l l s t o p i f NAVAIL i s n o t l a r g e e n o u g h .

DIPJODVR requires a subroutine DIPD(DIPC,R1,R2,XCOS,NU) which must supply the dipoles in

the x and z direction of the DVR coordinate embedding. The specifications of DIPD are as follows:

DIPC (DOUBLE PRECISION): value of the x dipole in atomic units if NU = 1 ;v a l u e o f t h e z dipole i n a t o m i c u n i t s i f NU = 0.

Ri (DOUBLE PRECISION): value of r1 in Bohr.R2 (DOUBLE PRECISION): value of r2 in Bohr.

XCOS (DOUBLE PRECISION): value of cos 0 .NU (INTEGER): 0 or 1 .

DIPJODVR requires 5 lines of card input for most runs. One extra card is required if the original

calculation is done in two symmetry blocks (even and odd) and these use different points for r2. Cards

giving data not required or for which the defaults [given below in square brackets] are sufficient should

b e l e f t b l a n k .

4.4. Card inputforDIPJODVR

C a r d 1 : NAMELIST/PRT/I P TO T[ 0 ] o n l y r e l e v a n t if the Hamiltonian has two symmetry blocks which w e r e c a l c u l a t e d sepa-

rately in the DVR calculation.

= 0 results from just one block are to be calculated.= 2 results from both blocks are to be calculated.

IBRA[50] the stream on which the wavefunctions for the bra are stored.

IKET[5i] the stream on which the wavefunctions for the ket are stored.


15/17


I BR A [ 5 2 ] the stream on which the wavefunctions from the second (odd) DVR calculation for theb r a a r e s t o r e d . Th i s i s only relevant if the Hamiltonian has two s y m m e t r y b lo c k s w h i c h

w e r e c a l c u l a t e d s e p a r a t e l y i n t h e DVR calculation.

IKET1[53] the stream on which the wavefunctions from the second (odd) DVR calculation for theket are stored. This is only relevant i f t h e H a m i l t o n i a n h a s two symmetry blocks which

were calculated separately in the DVR calculation.

IWAVEO[26] the stream on which the results written to IWAVE (default 26 ) in the DVR calculationare stored.

IWAVE1[55] the stream on which the results written to IWAVE (default 26) in the second (odd) DVRcalculation are stored. This is only relevant if the Hamiltonian has two symmetry blocks

which were calculated separately in the DVR calculation.

ZSAME[T] = F if the Hamiltonian has two symmetry blocks which were calculated separately and

which used different points for r2.

C a r d 2 : T i t l e ( A8 0 )

An e i g h t y c h a r a c t e r t i t l e .

C a r d 3 : LBRAO, LKETO, LBRA1, LKET1 ( 4 1 5 )

LBRAO number o f lo we s t s t a t e t o b e used in the bra.

LKETO number o f lo we s t s t a t e t o b e u sed i n t h e k e t .

LBRAI number of lowest state to be used from the second symmetry block in the bra (only

n e e d e d i f t h e H a m i l t o n i a n h a s two s y m m e t r y bl o c k s w h i c h were calculated separately int h e DVR c a l c u l a t i o n ) .

LKET1 number o f lo we s t s t a t e t o b e u sed f r o m t h e s e c o n d s y m m e t r y bl o c k i n t h e k e t (only

needed if the Hamiltonian has two symmetry blocks which were calculated separately in

the DVR calculation).

Card 4: NBRAO, NKETO, NBRA1, NKET1 (415)NBRAO number o f h i g h e s t s t a t e t o be used in the bra. ( 1 means u s e t h e v a l u e o f NEVAL f r o m

IWAVE).

NKETO number o f h i g h e s t s t a t e t o b e used in the ket.

NBRAi number o f h i g h e s t s t a t e t o b e used from the second symmetry block in the bra (onlyneeded i f t h e H a m i l t o n i a n h as t w o s y m m e t r y b lo c k s w h i c h were calculated separately in

t h e DVR c a l c u l a t i o n ) .

NKET1 number o f h i g h e s t s t a t e t o b e used from the second symmetry block in the ket (only

n e e d e d i f t h e H a m i l t o n i a n h as t w o s y m m e t r y b lo c k s w h i c h were calculated separately in

t h e DVR c a l c u l a t i o n .

C a r d 5 : R 1 E , R 2 E , XCOSE ( 3F 2 0 . 0 )RiE i s t h e equilibrium value of r1 in a t o m i c u n i t s .

R2E i s t h e e qu i l i br i u m va lu e o f r2 in atomic u n i t s .

XCOSE i s t h e e q u i l i br i u m v a l u e o f c o s 0 .

C a r d 6 : NQE, NQO ( 2 1 5 )

NQE i s t h e number o f q u a d r a t u r e p o i n t s t o b e generated using the parameters from the evensymmetry calculation.

NQO i s t h e number o f q u a d r a t u r e p o i n t s t o be generated using the parameters from the odd

symmetry calculation.


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394 JR. Henderson eta! . /Fullypointwise calculation ofvibrational spectra

4.5. Test output

A test deckwhich runs both DVR3D and DIPJODVR has been prepared. This uses the H2S potential

energy and dipole surfaces of Senekowitch et al. [31]. The run mimics the benchmark calculations onH2S performed by Carter et al. [32] and the vibrational band intensity calculations of Le Sueur et al. [15],

although the size of the calculations have been reduced in the test data.

The LiCN (CN frozen) scattering coordinate surface of Essers et al. is supplied in subroutine POT.

Acknowledgements

We would like to thank Dr. Brian Sutcliffe for helpful discussions during the course of this work. JRHt h a n k s SERC f o r a Fe l l o ws h i p . This workwas supported by SERC grants GR/G34339 and GR/H4i744.

References

[1] Z. Bai and J.C. Light, Ann. Rev. Phys. Cheni. 40 (1989) 469.

1 2 1 DO. Harris, GO. Engerholm and W. Gwinn, J. Chem. Phys. 43 (1965) 1515.[3] Z. Ba~iand iC. Light, J. Chem. Phys. 85 (1986) 4594.

[4] iC. Light, I.P. Hamilton and J.V. Lill, J. Chem. Phys. 92 (1985) 1400.

[5] J. Tennyson and JR. Henderson, J. Chem. Phys. 91(1989) 3815.[6] JR. Henderson and J. Tennyson, Mol. Phys. 69 (1990) 639.

[71J.R. Henderson, S. Miller and J. Tennyson, J. Chem. Soc. Faraday Trans. 86 (1990) 1963.18] JR. Henderson, HA. Lam and J. Tennyson, J. Chem. Soc. Faraday Trans. 88 (1992) 3187.[9] J.A. Fernley, S. Miller and J. Tennyson, J. Mol. Spectrosc. 150 (1991) 597.

[101 R.M. Whitnell and J.C. Light, J. Chem. Phys. 90 (1989) 1774.

[11] JR. Henderson and J. Tennyson, Comput. Phys. Commun. 75 (1993) 365, this issue.

[12] JR. Henderson, PhD Thesis, University ofLondon (1990).

[13] JR. Henderson, CR. Le Sueur, S.G. Pavett and J. Tennyson, Comput. Phys. Commun. 74 (1993) 93.

114] JR. Henderson, J. Tennyson and B.T. Sutcliffe, J. Chem. Phys., in press.[15] CR. Le Sueur, S. Miller, J. Tennyson and B.T. Sutcliffe, Mol. Phys. 76 (1992) 1147.

[16] B.T. Sutcliffe and J. Tennyson, mt. J. Quantum Chem. 29 (1991) 183.[17] J. Tennyson, Comput. Phys. Rep. 4 (1986) 1.

[18] J. Tennyson and B.T. Sutcliffe, J. Chem. Phys. 77 (1982) 406!.[19] J. Tennyson, S. Miller and CR. Le Sueur, Comput. Phys. Commun. 75 (1993) 339. this issue.

[20] IS. Gradshteyn and I.H. Ryzhik, Tables ofIntegrals, Series and Products (Academic, New York, 1980).[21] J. Tennyson and B.T. Sutcliffe, J. Mol. Spectrosc. 101 (1983) 71.

[22] A.H. Stroud and D. Secrest, Gaussian Quadrature Formulas (Prentice-Hall, London, 1966).

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

Small Molecules, ed. A. Lagan, NATO ASI series C, Vol. 277 (Kluwer, Dordrecht, 1989) p. 187.[24] D. Sadavoski, N.G. Fulton, JR. Henderson, J. Tennyson and B.l. Zhilinski. J. Chem. Phys., submitted.[25] AS. Dickinson and P.R. Certain, J. Chem. Phys. 49 (1968) 4204.

[26] N.G. Fulton, JR. Henderson and J. Tennyson, work in progress.[27] CR. Le Sueur, JR. Henderson and J. Tennyson, Chem. Phys. Lett., submitted.

[28] Subroutine EGVQR, originally due to Ti. Dekker, Amsterdam (1968).[29] NAG Fortran Library Manual, Mark 14 (1990).

[30] W.H. Press, B.P. Flannery, S.A. Teukolsky and W.T. Vetterling, Numerical Recipes (Cambridge Univ. Press, Cambridge.

1986).[31] J. Senekowitsch, S. Carter, A. Zilch, H.-J. Werner, NC. Handy and P. Rosmus, J. Chem. Phys. 90 (t989) 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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TEST RUN OUTPUT

PROGRAM DVR3D (vERSION OF 15 Sept 1992)

FULL TRIATOMIC VIBRATIONAL PROBLEM WITH

18 RADIAL Ri DVR POINTS USED.18 RADIAL R2 DVR POINTS USED.40 ANGULAR DV R POINTS USED. WITH

10 LOWEST EIGENVECTORS CHOSEN FROMUP TO 600 DIMENSION SECULAR PROBLEM

TITLE: H2S J0 scattering coordinates 3D DVR

PROBLEM SOLVED IN THE ORDER: THETA -> Ri -> R2

600 EIGENVALUES SELECTED FROM 0.1106398948001 TO I.1246613142D+00

LOWEST 10 EIGENVALUES IN CM-i:

0.3297458149300+04 0.448789782008D +04 0.5669474484780+04 0.591781492896D +04 0.6840932152320+04

0.709201630442D+04 0.800114410352D +04 0.8257559623590+04 0.8451661215520+04 0.854865836085D+04

600 EIGENVALUES SELECTED FROM 0.1106398948D01 TO 0.1246613142D+00

0 MINS 0.00 SECS CPU TIME USED

0 MINS 0.00 SECS CPU TIME USED

LOWEST 10 EIGENVALUES IN HARTREES:

0.150243253147D01 0.20448367735lD 01 0.258320273272001 0.269631496857001 0.311695813741D01

0.323136049862DOl 0.364558961656D01 0.376242112782001 0.385086032339D01 0.389505547614001

LOWEST 10 EIGENVALUES IN CMi:

0.329745814930D+04 0.448789782008D +04 0.5669474484780+04 0.5917814928960+04 0.6840932152320+04

0.7092016304420+04 0.8001144103520+04 0.8257559623590+04 0.8451661215520+04 0.8548658360850+04

Program DIPJODVR - version 1.1 (September 1992)

test data for H2S even eigenstates only

The equilibrium values of rl. r2 . and cos(theta)correspond to the following atom coordinates:

x SAtom 1 (mass 31.972071 amu): 0.000000 0.105563 Atom 2 (mass 1.007825 amu): 1.870000 1.674437Atom 3 (mass 1.007825 amu): 1.870000 1.674437

PRINT OUT OF DIPOLE TRANSITION MOMENTS AND S(Fi)

FREQUENCIES IN WAVENUMBERS

TRANSITION MOMENTS IN DEBYE (2.54174A.U.)S(F-I( IN DEBYE**2EINSTEIN A-COEFFICIENT IN SECi

IE1 IE2 RET ENERGY BRA ENERGY FREQUENCY I TRANSITION X TRANSITION DIPOLE S(FI) A-COEFFICIENT

1 1 0.000 0.000 0.000 0.982740E+00 0.000000E +OI 0.982740E +00 0,965779E +O0 0.000000E +001 2 0.000 1190.440 1190.440 0.147225E 01 0.000000E +00 0.147225E01 0.216751E03 0.11.4679E+001 3 0.000 2372.016 2372.016 0.367882E02 0.000000E+00 0.367982502 0.135337504 S.566464E01

1 4 0.000 2620.357 2620.357 I.721566E 02 0.000000E+00 0.721566E02 0.520658E04 0.293789E+II1 5 0.000 3543.474 3543.474 0.106987E 02 0.000000E+00 0.106987E02 0.114463EI5 0.159718E011 6 0.000 3 79 4. 55 8 3 79 4. 55 8 0.676743E 02 0.000000E +00 0.676743E 02 0.457980E 04 0.784752E +001 7 0.000 4703.686 4703.686 0.225805E03 0.000000E+00 0.225805E03 0,509878507 0.1664110021 8 0.000 4 96 0. 10 1 4 96 0. 10 1 0 .3 54 32 1E 0 3 0.000000E+00 0.354321003 0.125543E06 0.480470E021 9 0.000 5154.203 5154.203 0.257982E02 0.000000E+00 0.257982E02 0.665546005 0.2858025+00

1 10 0.000 5 25 1. 20 0 5 25 1. 20 0 0.862820E03 0.000000E+00 0.862820E03 0.744459E06 0.338079EOi

James R. Henderson, C. Ruth Le Sueur and Jonathan Tennyson- DVR3D: programs for fully pointwise...

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