James R. Henderson et al- Coordinate ordering in the discrete variable representation

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Computer Physics Communications 74(1993)193—198 Computer PhysicsNorth-Holland Communicat ions

Coordinate ordering in the discrete variable representation

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

Department of Physics and Astronomy, University College London, Gower St., London WC1E 6BT, UK

Received 10 July 1992

The problem of what order to treat the coordinates in the scheme of successive diagonalisations and truncations

commonly used in multidimensional discrete variable representation (DVR) calculations is analysed. Test calculations in 4

different coordinate orderings are performed for the vibrational band origins of the HCN and H~molecules. These testsshow that calculations which place the coordinate with the densest DVR grid last require considerably less computer time

and converge significantly faster than the other options.

1. Introduction [9],which is particularly efficient at treating casesof strong mode coupling.

The discrete variable representation (DVR) So far most DVR calculations on multidimen-

has proved very powerful for studying highly ex- sional problems have actually been FBR—DVR

cited vibrational states of small molecules. The hybrids with a DVR used only for one coordi-

application of the DVR to molecular vibrations nate. However, a number of studies onH~[10—was pioneered by Light and coworkers [1] follow- 1 2 1 have used a DVR in all three vibrationaling original developments in the sixties [2,31.More coordinates. The most recent of these studies

recently DVR based methods have also been have yielded spectacular results with estimates of

used to study heavy particle scattering [4,5]. all the bound vibrational states of this system.The advantage of the DVR over the conven- This represents a fivefold increase in the number

tional finite basis representation (FBR) results of converged states compared to an early studyfrom writing the problem as a heirachy of Hamil- which used an FBR for two coordinates and a

tonians. The Hamiltonian matrices are diago- DVR in the third [131.

nalised in turn and the lowest energy solutions When using a DVR in more than one dimen-

used as a basis for the next Hamiltonian in the sion, there is a choice over how one orders theheirachy. This diagonalisation and truncation coordinates for the successive diagonalisation and

technique leads to final Hamiltonians which are truncation procedure. So far the question of

heavily truncated compared to the initial set of whether calculations are sensitive to how coordi-

grid points [4,6,7]. It should be noted that al- nates are ordered has received little attention. It

though prediagonalisation and truncation (“con- has been stated that the best method should be to

traction”) has been used in FBR techniques [8], treat the coordinates in order of decreasing den-the DVR gives an optimal method of doing this sity of associated vibrational states [11,141. Con-

versely it has also been suggested that the order-ing of the coordinates in the calculations is of

little significance [10].Correspondence to: J.R. Henderson Department of Physicsand Astronomy, University College London, Gower st., ion- In this paper we present a number of testdon WC1E 6BT, UK. calculations on the molecules HCN and H ~ de-

0010-4655/93/$06.00 © 1993 — Elsevier Science Publishers B.V. All rights reserved



194 J.R. Henderson et al. / Coordinate ordering in the discrete variable representation

signed to illustrate the behaviour of multidimen- A three-dimensional DVR is obtained by ap-

sional DVR calculations as a function of coordi- plying the transformation

nate order. We show that the efficiency of the

calculation is strongly influenced by the ordering Haa~i3j~~,y,y~

of the coordinates and we suggest that the opti- — ~

mal procedure is given by placing the coordinate — ~ (Tm ,n,j)m,n,j m’,n’,j’with the largest number of DVR points (andhence the densest associated vibrational spec- X Km, n, f I H I m’, n’, j’)T,~’,~,’;X’. (4)

trum) last.The 3D transformation can be written as a prod-

uct of 1D transformations

2. Theory T”~’~=T~T ’3 TY (5)m,n,j m n j

The 1D transformations are defined in terms of Consider a triatomic vibrational problem with

total angular momentum J = 0 and coordinate points, i~,and weights,w~,of the N-point Gauss-

system (r 1, r2, 0). As the DVR is very much more ian quadrature associated with the orthogonal

polynomial used for the FBR in that coordinate:efficient in orthogonal coordinates, these coordi-nates could be Radau (see ref. [15]) or scattering ~ = t(i~)), (6)

(Jacobi) coordinates [1]. For these coordinates anFBR Hamiltonian matrix element can be written where I t)= m), I n), I j) for 1 1 =a, / 3 , y, re-

[16], spectively.

The transformed Hamiltonian is written at the

(m, n, jI H m’, n’, ~l’) DVR grid points as

=Km I A~(1)~~ +Kn ‘il’)6 6 (

3D)Haa~pI3fyy~

I m,m’ j,j’

+(Km I m’)ô~,~~ =K~~6I3I3’&~ y’ +K~2~,I5 6

I3~3 a,a’ y,y’

+Kn I fl’)6m,m’)J(J +1)6..~ +L~1~ ~6 +~

jj a,a’,-y,-y j3,f3’

+(m, n, j V(r 1, r2, 0)1 m’, n’, j’), (1) +V(ria, r213 , 07)6aa~

6l3,pf6yy~, (7)

where it has been assumed that the angular basis where the potential energy operator is diagonal

functions If) are Legendre polynomials and it because of the quadrature approximation [3] andwill be assumed that the radial basis functions (na, r

2~,O~)is the value of (r1, r2, 0) at grid

can also be expressed as weighted orthogonal point (a, /3 , y). The kinetic energy terms arepolynomials. In eq. (1), V is the potential and the represented by

kinetic energy integrals are given by ~ = ~(7’1)T(tI~(1)It’)7~’, (8)~1,?J’ —h2 ~2

KtI~°lt’)=( t i I t ’ ) , (2)2p~r~a r

1

2 and, again applying the quadrature approxima-tion,

wherelt)=Im)fori=land!t)=In)fori=2,

and _____

L~’~, = ~ 2p~~r~(TY)T ( +1)TT’6,~,~~.(9)‘l,’l’ 77

I I t ’ ) =<t 2~r~ I t ’ ) , (3) To illustrate the diagonalisation and trunction

schemes, consider the case where the coordinateswhere p~,is the appropriate reduced mass [16]. are treated in the order r 1 then r2 then 0, i.e. a



J.R. Henderson et aL / Coordinate ordering in thediscrete variable representation 195

then /3 then ‘y . The 1D problems, which need to there is no problem in principle with having 0 asbe solved for each value of f3 and ‘y , are given by the second coordinate, we have not included this

option as it is would require considerable repro-(1D)H13,y =K~’~+V(Tia, r

2p, o~)8aa~. (10) gramming. As we show below, the major consid-a,a a,a’

eration is the final coordinate and the aboveAmplitudes for the kth level, with eigenenergy options allow for all three possibilities.

7

are given at the grid points by C~,’k. Our calculations focussed on the triatomicSolutions with eigenenergies above a certain molecules HCN and H ~. All calculations were

cutoff value, E~,are discarded prior to the next performed in scattering coordinates. In general

step. The N 2D solutions with ~‘~‘ EI~ are we did not aim to produce fully converged calcu-

then used to solve 2D problems for each value of lations as these would have been prohibitivelyy. This gives expensive for some coordinate orderings. Instead

(

2D)HI37I3,kk, =E~’~6I3I3~6kkf + ~ ~ we chose to compare the results between order-

a I 3 ’ I 3 a,k ings, both in terms of computational resources(11) used and convergence, for standardized calcula-

tions using the same grid points and cutoff pa-

Solutions for the lth level, with eigenenergy 7’, rameters. In fact cutoffs can be expressed both in

are given by C~lk. terms of energies (E1’~E2D) and numbers of max’ max

The solutions with 7 E~are then used to functions selected (N2D, N3D). For ease of corn-

solve the full 3D problem of dimension N3’~: parison with the vibrational band origins quotedbelow, all the energy cutoffs are given relative to

(3D)H

77,11, the vibrational ground state of the relevant sys-

tem.

= ~ + ~ {CI3YI,~CI3Y:I,,~~

f3,k,k’ [ 3.1. HCN

x (L~I377~+~ [c~’~Lf’) ~c~7)] (12) HCN was chosen for initial studies as scatter-a,k a,a,y,7 a,k

a ing coordinates give a reasonable approximation

to the vibrational motions of the molecule. ThusDiagonalisation of this yields final energies and 0 approximates the bending mode for both HCN

wavefunction coefficients, and HNC isomers, and the reaction coordinateClearly it is possible to formulate similar diag- between the two minima. As the bending modes

onalisation and truncation schemes for the five are of much lower frequency than the stretching

other possible orderings of (r1, n2 , 0). Note that modes one would expect the 0 coordinate toit is possible to include the diagonal (in ~‘)contri- represent the densest set of vibrational states.

butions earlier on this solution scheme, for exam- The calculations used the HCN potential of

ple L~7,7~in the a step and L~77~in the / 3 Murrell et al. [17] used in previous FBR [18] and

step. However, previous experience [14] shows DVR [9,19,20]studies on the high-lying vibra-that this procedure produces intermediate vectors tional states of HCN. The calculations used Morse

which are less well adapted and have worse con- oscillator-like functions [16] in both coordinates.

vergence properties. All calculations here used N1 = 30 points in the

n1 coordinate, N2= 40 points in r2 and N0 = 74in 0. This grid is almost certainly denser than is

3. Calculations necessary to obtain useful results for HCN — for

example Ba~i~and Light used N0 = 45 in theirThe 3D DVR program developed by us has calculations — but this serves our purpose for

four possible coordinate orderings: (r1, r2, 0), illustrating the effect of coordinate ordering. Note(n2, n1 , 0), (0, n1 , r2) and (0, r2, n1). Although that the number of grid points in each coordinate



196 J.R. Henderson et aL / Coordinate ordering in thediscrete variable representation

Table 1 Table 2

Band origins in cm’ for HCN as a function of coordinate Band origins, in cm1, for HCN as a function of coordinate

order. All calculations used the same grid and energy cutoffs, order. All calculations used the same grid and value of E,2~

________________________________________________________ but E~n the 0 first calculations was lowered to give Nm~

Order: r

1 —~r2 —~9 r2 —~r1 —~8 0 —, r~ —~r2 0 —* r2 —~ri similar to those of the 0 last calculations.

N,~: 94 89 301 381Order: r1—*r2—*0 r2—’r1—’0 0—*r1—*r2 0—,r2—*r1

N~: 4231 3730 4455 N

21~’ 94 89 82 115max

N30: 589 597 734 642 N20’ 4239 3730 738 932

(s): 289 325 690 580 tot

N30: 589 597 580 538Band Band origins(cmt) t (s): 270 312 450 476

1 1418.5 1418.4 1420.5 1418.8Band Band origins (cm1)

21 6554.6 6532.8 6570.5 6570.5 _____________________________________________________41 8305.6 8300.1 8370.2 8348.4 1 1418.5 1418.4 1460.3 1445.2

61 9673.9 9644.3 9795.0 9742.4 21 6 5 5 4 . 5 6532.8 7628.1 7274.7

81 10654.5 10638.7 10863.0 10791.8 41 8305.6 8300.1 1110 7.2 10 05 8.9101 11590 .5 115 21.1 1 197 2. 8 1 17 89 .4 61 9673.1 9644.3 13236.3 12292.5

121 12391.4 12337.3 12834.8 12737.4 81 10 65 4.4 10 638.7 16247.9 14475.7

141 13089.0 13059.5 13876.3 13598.8 101 11589.8 11 521 .1 1 90 73.1 16730.0

121 12391.2 12337.3 2 131 9. 4 18 57 8.8

141 13088.9 13059.5 24011.5 20891.0

is an approximate measure of the density of the

vibrational spectrum in that coordinate. with those of Ba~iéand co-workers [19,20] whoTable 1 compares the four coordinate order-

ings for an unconverged calculation on HCN us- only considered band origins up to 12540 cm

ing the cutoff E~= 16144 cm~and E~= Again the 0 last calculations give much better

13 132 cm ~. For comparison the total number of convergence and require less CPU time, althoughthe difference in the size of the final Hamiltoni-1D solutions selected, ‘V~the maximum size of ans, N3D, is less marked. Indeed it is interesting

a 2D problem, N 1~ and the size of the final

Hamiltonian, N

3D, are given. The CPU time

taken, t, in Convex C240 single processor sec- Table 3

onds, is also given. It is clear from table 1 that the Band origins, in cm ~, for HCN as a function of coordinate

runs which place 0 last are superior and use less order. All calculations used the same grid and energy cutoffs.

CPU time than the calculations with 0 first. The energy cutoffs were chosen so that the (r

2, r1, 9) resultsconverged.

A feature of table 1 is the comparatively large _____________________________________________

values of Nm~used in the 0 first runs. Table 2 Order: r1 — r2 — *9 r2 —* r1 -~00 — r~ —* r2 0 —, r2 —* r1

gives an alternative comparison, again with con- N

2D. 208 154 597 815mar

stant E,2~= 13132 cm~, in which an attempt N20 11882 8282 11078 11078

has been made to make the size of N 1~compa- tot’

N

30: 2180 2200 2958 2095rable between the runs. The result of this is also

(5): 4338 4420 11055 6145to bring the values of N3D close together but to

further degrade the convergence of the 0 first Band Band origins(cmt)

calculations. Indeed these calculations no longer 1 1418.4 1418.4 1418.4 1418.4

give satisfactory results for the HCN bending 41 8295.4 8295.3 8296.8 8295.981 1 061 9. 9 10 61 9.4 10623.3 10620.3

fundamental (band origin 1). 121 12208.9 12206.9 1 22 22 .6 1 22 12. 0

Table 3 presents calculations on HCN using 161 13425.4 13 414 .2 1 344 8.5 13425.0

much larger intermediate energy cuttoffs (E~,’~= 201 14325.8 14323.1 14366.2 14345.9

26124 cm1, E,2~= 22104 cm~)which aimed 241 15344.0 15338.2 15 381.0 15 35 3.0

for convergence. At this level the band origins for 281 1 622 6. 3 16 21 8.7 16271.3 16242.7

the (r 2, r1, 0) calculations agree to within 1 cm~ 321 17002.3 16996.3 17074.0 17031.5



JR. Henderson et al. / Coordinate ordering in the discrete variable representation 19 7

Table 4Band origins in cm 1, forH~as a function of coordinate order. All calculations used the same grid and energy cutoffs. The

converged results of Henderson et al. [12] are shown for comparison.

Order: ri—*r 2—*0 r2—r1—*0 0—*ri—*r2 0—*r2—r1

N,~: 793 862 412 467 446

N~: 11731 12993 11650 11650 14287

N

30: 8 7 6 8 7 6 6 9 3 7 9 9 5 5 0 0

(s): 3 0 9 7 3 8 8 8 1 0 1 3 1 2 9 6 8 9 1 3 6

Band Band origins (cm1)

1 2521.4 2521.4 2521.3 2521.3 2521.3

41 14223.5 142215 14214.4 14225.7 14210.981 1 8 2 3 8 . 9 1 8 2 3 8 . 9 1 8 2 2 4 . 7 1 8 2 3 0 . 8 1 8 2 0 8 . 6

121 20814.1 20814.1 20806.6 20816.1 20759.4

161 22791.9 22792.0 22794.1 22819.9 22687.9

201 24572.8 24572.8 24584.4 24643.9 24356.4241 26418.2 26418.3 26340.4 26379.8 25834.1

to note that although the (0, n2, ri) calculation Henderson and Tennyson [11]. These calculations

had the smallest final Hamiltonian, this calcula- have since been criticised for displaying non-van-tion required nearly 50% more CPU time than ational behaviour [22]. In this work we only con-

the 0 last calculations. This can be attributed to sider even symmetry calculations which do be-

the significantly larger value of N1~in this calcu- have variationally [22,23]. A full discussion of the

lation. problems with variational behaviour and the 3D

Tables 1 — 3 all show that the 0 last calculations DVR can be found elsewhere [12].converge better and are quicker than the 0 first Table 4 presents even symmetry band origins

calculations for HCN. They also indicate that the for H~performed with a common set of cutoff

(r2, r1, 0) ordering converges better than (ri, r2, parameters, E~= 55936 cm~ and E1~=

0). 26091 cm* For comparison highly convertedresults, based on the study by Henderson et al.

3.2. H3~ [12],are also shown. This calculation used E~

=65936cm’~and E~x=51052cm_i.

As mentioned in the introduction H ~ has been The results of table 4 again show variations inthe focus of several 3D DVR calculations. Our convergence and CPU usage. In general theprevious calculations [11,12,14] have all used a (0, n~,r2) ordering gives the lower results, al-

(0, r1, r2) coordinate ordering * based on some though the (ni, r2, 0) and (r2, Ti, 0) calculationspreliminary tests [14]. are lower in the intermediate energy region. It

For this work, calculations were performed should be noted, however, that these 0 last calcu-using the ab initio potential due to Meyer et al. lations require more than 3 times the CPU time(MBB) [21]. The grid of points used is given by of the (0, n~,r2) calculation.

N1 = 36, N2= 40 and N9 = 32. Because of the

symmetry of the system it is only necessary to

consider half the 0 grid points [13]. 4. ConclusionsThe calculations closely followed those of

These test calculations show that 3D discretevariable calculations (DVR) show a strong sensi-

* Note that the coordinate ordering used by Henderson and tivity to the order in which the coordinates areTennyson was (0, r~,r2), not (r2, rt, 0)asstated in ref. [11]. tackled in the calculation. This sensitivity mani-



198 J.R. Henderson et a! . / Coordinate ordering in the discrete variable representation

fests itself in two ways: through convergence of [2] DO. Harris, GO. Engerhoim and W. Gwinn, J. Chem.

the eigenenergies and through the CPU time P h y s . 4 3 ( 1 9 6 5 ) 1 5 1 5 .

[3] AS. Dickinson and P.R. Certain, J. Chem. Phys. 49required to perform calculations with equivalent (1968) 4204.

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the same direction; the ordering of coordinates Supercomputer Algorithms for Reactivity, Dynamics and

which displays the best convergence properties is Kinetics of Small Molecules, ed. A. Laganà, NATO ASI

also generally the cheapest computationally. series C, Vol. 277 (Kiuwer, Dordrecht, 1989)pp. 187—213.[5] D.T. Colbert and W.H. Miller, J. Chem. Phys. 96 (1992)

Analysing simply the CPU time characteristics 1 9 8 2 .

of the results presented above, it is easy to see [6] Z. Ba~ié,R.M. Whitnell, D. Brown and J.C. Light, Com-

that the best calculations are those which place put. Phys. Commun. 51(1988) 35.

the coordinate with the largest number of grid [7] JR. Henderson, S. Miller and J. Tennyson, J. Chem.

points last. For the systems studied this coordi- Soc., Faraday Trans. 86 (1990) 1963.[8] S. Carter and N.C. Handy, Comput. Phys. Rep. 5 (1986)

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From the CPU timings the optimal coordinate 1 7 7 4 .

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in the order sparsest DVR grid to densest DVR (1990) 133.[12] J.R. Henderson, J. Tennyson and B.T. Sutcliffe, J. Chem.

grid. However, this ordering does not give the Phys., submitted.

fastest convergencç for HCN andit is clearly

[13] J. Tennyson and J.R. Henderson, J. Chem. Phys. 91

dangerous to be too prescriptive on this, espe- (1989) 3815.

cially as the increase in CPU time on swapping [14] J.R. Henderson, PhD thesis, University of London (1990).[15] Z. Ba~i6,D. Watt and J.C. Light, J. Chem. Phys. 89

the order of the first two coordinates is small. ( 1 9 8 8 ) 9 4 7 .

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

4061.

Acknowledgements [17] J.N. Murrell, S. Carter and L. Halonen, J. Mol. Spec-

trosc. 93 (1982) 307.JRH thanks SERC for a Fellowship and Prof. [18] M. Founargiotakis, S.C. Farantos and J. Tennyson, J.

John Light and David Brown for helpful discus- Chem. PHys. 88 (1988) 1598.

sions. This work was supported by SERC grants [19] Z. Baëié and J.C. Light, J. Chem. Phys. 86 (1987) 3085.GR/G34339 and GR/F14550. [20] M. Mladenovic and Z. Baëi~,J. Chem. Phys. 93 (1990)

3 0 3 9 .

[21]W. Mey er, P. Botschwina and PG. Burton, J. Chem.

References P h y s . 8 4 ( 1 9 8 6 ) 8 9 1 .

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[1] Z. Ba~iéandJ.C. Light, Ann. Rev. Phys. Chem. 40 (1989) [23] J.R. Henderson, J. Tennyson and B.T. Sutcliffe, J. Chem.

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James R. Henderson et al- Coordinate ordering in the discrete variable representation