jonathan tennyson et al- the generation of continuum orbitals for molecular r-matrix calculations...

8/3/2019 Jonathan Tennyson et al- The Generation of Continuum Orbitals for Molecular R-Matrix Calculations Using Lagrange Orthogonolisation

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Computer Physics Communications 47 (1987) 207212 207

North-Holland, Amsterdam

THE GENERATION OF CONTINUUM ORBITALS FOR MOLECULAR R-MATRIXCALCULATIONS USING LAGRANGE ORTHOGONALISATION

Jonathan TENNYSON

DepartmentofPhysics and Astronomy, University College London, GowerStreet, London WCIE 6BT, England

P.G. BURKE an d K.A. BERRINGTONDepartmentofAppliedMathematics andTheoreticalPhysics, The Queens University ofBelfast, Belfast BT7 iNN,Northern Ireland

Received 11 June 1987

A method is presented whereby a given set ofsingle-channel continuum functions can be used to generate a set of

multi-channel orbitals which are orthogonal to a specified number oftarget molecular orbitals with the same symmetry. The

procedure uses the single-channel functions as a basis set expansion which is Lagrange orthogonalised to the target orbitals by

searching for the zeros ofadeterminant. Sample calculations on a variety of diatomic targets, such asH~and CO. show themethod is reliable and does not suffer from the linear dependence problems frequently encountered with other orthogonalisa-

tion procedures. A straightforward generalisation ofthe method allows continuum orbitals to be generated which are adaptedto anisotropic terms in the potential.

I. Introduction higher energies difficulties were encountered dueto linearly dependent orbitals. Linear dependence

The R-matrix method provides a natural for- is often encountered in highly accurate electronic

malism for the representation of electronmole- structure calculations [2], but is a more severecule collisions. It allows use to be made of the problem in scattering calculations because of theexpertise developed by quantum chemists in han- need to represent a large range of continuumdung the complicated interactions when the energies.

scattered electron is close to the molecular target In response to this problem, Burke et al. [3]and the experience acquired by atomic physicists suggested the use of numerical functions to repre-in solving for the asymptotic motion of the elec- sent the continuum. These functions were ex-tron. An important addition that needs to be pressed as a partial wave expansion about themade to any electronic structure code when adapt- molecular centre of mass

ing it for scattering calculations is that functions F(r)= ~r_lum

(r)Y (?)a. (1)must be developed to represent the continuum. In 1 ,1 l,m z the R-matrix method these functions represent the where Y, is a spherical harmonic. For a diatomicportion of the discretised continuum that lies in target mbut not 1 is conserved. The radial basisthe internal region of the calculation typically a functions u7. were generated as solutions of thesphere of radius about 10a

0. model problem:The first molecular R-matrix calculations of

this type [1] represented the continuum using 1(1 + 1) V0(r) +k2 u7.(r)

functions, Slater Type Orbitals, often used to rep- dr2 resent bound electronic states. These calculations

were successful for low scattering energies but at = ~ Vi~(r)u2~~(r)+ ~X~qPP,q(r) (2)

OO1O-4655/87/$03.50 Elsevier Science Publishers B.V.

(North-Holland Physics Publishing Division)


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208 J. Tennyson etal. /Continuum orbitalsformolecularR-matrix calculations

subject to the boundary conditions, for an R-ma- large number of single channel basis functions.

trix ofradius a, This is because each solution of (2) must neces- sarily, because of the orthogonality constraint, be

U,,,~. ~ a linear combination of different u11 . The large

~ =b for all /and i. (3) number of basis functions implied by this proce-u7~ dr ~ dure means that it was found to be intractable for

more complicated problems, such as coupled elec-

The use of these fixed boundary conditions re- tronic state calculations or scattering from hetero-

quires aButtle correction [4] to be applied to the nuclear targets.final R-matrices. The orthogonality constraints in As a response to this, subsequent molecular(2) are equivalent to requiring all the solutions, u,~, R-matrix calculations [614]dropped the strictto be orthogonal to a set of orbitals: orthogonality implied by eq. (4) and used single-

a channel continuum functions which are not or-~fu7,(r)P1~(r)dr= 0 for all i and q, (4) thogonal to the occupied target MOs. These func-1 ~ tions were then orthogonalised, usually using

where in (2) and (4) P/~are the components of Schmidth orthogonalisation, to the entire (oc-the single-centred expansion of the occupied target cupied and virtual) target MO set. This simplifiedmolecular orbital (MO) ~ procedure does not suffer from either of the prob-

lems outlined above. The success of the quotedml

\1 ml

\v l~\ / \

4 q ~r1 r ,q~r)Ilm~r). . ) calculations for both electron [612]and positron/ [13]molecule collisions, an d photoionisation [14]

In (2) X7q is thus a Lagrangian multiplier which is a testimony to its usefulness. However for cer-

ensures orthogonality to the p occupied target ta m targets (see for example refs. [11,12]), theMOs. As some virtual target MOs are also usually orbital sets generatedin this fashion showed severe

retained in the orbital set, to allow for short range linear dependence problems. These problems wereeffects due to higher partial waves (often called found to be worst for lightheavy systems such ascorrelation) and polarisation, it is still necessary to HF andCH~for which no successful calculationsSchmidt orthogonalise the continuum orbitals to at all could be performed.

the complete set of target MOs. In this work we propose a new method ofIn (2), Vf,~represents the coupling due to an- obtaining continuum orbitals which are, to a good

isotropic terms in the potential; it can be ex- approximation, Lagrange orthogonalised to apressed as sums over the expansion of the molecu- specified set of target MOs without encounteringlar target potential V~[5]: any of the problems of the original procedure of

Burke et al. [3]. We demonstrate the utility oftheV/7~(r)=~g~(I, 1, m)Vx(r), (6) procedure by giving sample results for electron

scattering from H~and CO.where g~(/,1, m) is a Gaunt coefficient. In prac-

tice, all calculations have only included the iso-tropic part of this potential V~which means that 2. Theorythere is no coupling between the single channel

basis functions due to the model target potential. What we require is a set of continuum func-The calculations of Burke et al. [3] gave a good tions, u7

1,(r), which are orthogonal to the targetrepresentation of electronN

2 scattering over a MOs but can be expressed in terms of one setconsiderable energy range. However the method single channel functions for all i and a given

employed had two disadvantages. Firstly, solu- (1, m). Such a set can be writtentions of (2) satisfying the Lagrange orthogonality

constraint proved difficult to find. Secondly, and u7,(r)=

~ar11v71(r), (7)more seriously, the method gives rise to a very i


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210 J. Tennyson etal. /ContinuumorbitalsformolecularR-matrixcalculations

potential can be neglected, i.e. ~ =0, then found to give satisfactory results. Conversely it

the diagonalisation of B is no longer required. In was found that relatively few grid points werethis case eqs. (14) and (16) reduce to: needed between the neighbouring poles as once a

c xm sign change in C is found a Newton iteration /,J,JL i,q (17) procedure can be used to speedily find the poles

q~k~ k,21 to high accuracy a tolerance ofabout 0.01being

typical.and Of course, if the single channel continuum

Cqq~= ~ ~ functions are already exactly orthogonal to the(18) target MOs, the k~equals ~ and the procedure,~k~ ~ will suffer numerical difficulties because of the

It is this simplified procedure which is considered indeterminacy of C. In practice it was found forin the next section. that these problems were not great, but for the

case of a homonuclear diatomic target, separateorthogonalisation steps were implemented for

4. Computational implementation orbitals with g and u symmetry. This was donedespite the fact that the overlap integrals [16],

The theory given above allows single channel from which the C~jq were constructed (see (11)),

continuum functions to be adapted for anisotropy were read into the code only inc~symmetry.

in the potential as well as Lagrange orthogona- Table I shows the pole positions for a calcula-lised to the target MOs. We will only consider tion of electron scattering from CO in its equi-examples requiring orthogonalisation. This is be- librium geometry. The numerical functions of col-

cause experience has shown that the solutions we umn (a) correspond to the low-lying continuumobtain for scattering problems are not highly sen-sitive to the choice of model potential, providingthere are no problems with linear dependence. Forexamplein positronH2 calculations [13] very sim-ilar results were obtained when the numerical Table 1

functions were generated using a model potential Continuum function pole positions below 1 Ry for electronCOappropriate for electronH2, V0(r), and scattering calculation with ~ (m =0) symmetry. (a) k,

2~for anisotropic model CO potential [6]. (b) k~given by LagrangepositronH

2 scattering, V0(r). orthogonalisation to the 5 occupied a CO molecular orbitals;

In implementing the procedure derived above the largest expansion coefficient, a7,,, is given for each polethere are a number of numerical considerations

(a) (b)that need to be addressed. The basis of the proce-

I j k7~(Ry) i k~(Ry) / j a7 11dure is the search for zeroes in the determinant of __________________________________________________

C. However C(k2) has a pole every time k2 0 0 2.48611 1 0.05423 0 1 0.94

1 0 0.33793 2 0.09336 1 1 0.97equals k,2

1 (or k~).Thus any scan of C for the 0 1 0.02865 3 0.15139 2 0 0.99

sign change which imply a zero must avoid these 1 1 0.08152 4 0.24726 3 0 1.00poles. As the pole positions are known before the 2 0 0.14914 5 0.36743 4 0 1.00

search starts this is not a difficult problem, the 3 0 0.24724 6 0.42208 0 2 0.75search is divided into regions defined by k~+ e 0 2 0.26249 7 0.50982 5 0 1.00

4 0 0.36744 8 0.53023 1 3 0.76k2 . In this scheme s runs over all the 1 2 0.43531 9 0.60806 2 1 0.67

poles in energy order. As some continuum func- ~ 0 0.50982 10 0.67417 6 0 1.00tions, particularly those with high /and/orj, are 2 1 0.53542 11 0.75676 3 1 1.00very nearly orthogonal to the target MOs, k~may 6 0 0.67417 12 0.99215 4 1 1.00lie very close to k,2~(ork~).This means that e has 0 3 0.73982

3 1 0.75718to be chosen small or solutions will be missed, ~ 1 0.99239

values in the region of lOhl to 10_12 Ry were


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J. Tennysoneta!. /ContinuumorbitalsformolecularR-matrix calculations 211

functions used by Salvim et al. [6], who then electronic state calculation on electronHi at the

Schmidt orthogonalised them to the target MOs. H2 equilibrium separation of 1.4a0. The calcula-The functions whose pole positions are shown in tions closely followed those of Tennyson et al. [7]

column (b) were obtained by Lagrange or- and differ only in the orthogonalisation procedurethogonalisation a set of 58 functions (all those used. It can be seen that the eigenphase sums arewith k?~< 9 Ry) to the 5 occupied a orbitals of closely parallel and that the resonance parametersthe COtarget. The resulting set has 5 fewer orbitals are in good agreement. In this case the Schmidtand it can be seen from table 1 that 3 of the orthogonalisation procedure gave orbitals with

functions removed come from the low energy re- large coefficients, several greater than 100. Coeffi-gion below 1 Ry. Table 1 also shows that even in cients of. this size are associated with linear depen-

this low energy region, several of the functions are dence. For this reason, Tennyson et al. used aalmost unchanged by the Lagrange orthogonalisa- precursor of the current Lagrange orthogonalisa-tion procedure. tion procedure [15], which was found only to give

Scattering calculations performed with (a) the satisfactory results for hydrogenic systems. In thecomplete set of 58 continuum orbitals and (b) the Lagrange orthogonalised orbital set (b) there are53 continuum orbitals that result from the no coefficients greater than 4.

Lagrange orthogonalisation gave similar results.

As the expansion (7) is only exact in the limit of

an infinite set of functions V rk, one would expect 5. Concluding remarks

procedure (a), when it does not suffer from lineardependence problems, to give higher eigenphase We have developed a procedure for generatingsums. This is indeed observed. However the dif- tractable continuum orbital sets which are

ference is not great, for example the2E shape Lagrange orthogonalised to the molecular orbitals

resonance found at 1.4896 Ry in the SEP model of a given target. This procedure has been tested

[6] using procedure (a) is raised by only 3 Xi0~ for electron scattering from diatomic targets in-Ry using procedure (b). cluding H ~, H

2, N2, CO , C H +and HF and foundTable 2 compares eigenphase sums for a two satisfactory. In cases where there are no problems

with linear dependence, it can be argued on varia-

tional grounds that use of the procedure will al-ways result in a lowering of the computed eigen-

Table 2 phase sums. However, cases where this loss isEigenphase sums as a function of energy for ~ two significant are exactly those for which use of thiselectronic state electronHr calculation. (a) Schmidt ortho-

gonalisation. (b) Lagrange plus Schmidt orthogonalisation. The procedure is unnecessary. Conversely, there areenergy (E) and width (I) of the lowest resonance are also systems for which linear dependence problemsgiven have in the past proved a major obstacle to perfor-

k

2 ~j(radians) ming molecular R-matrix scattering calculations.(Ry) b These difficulties are removed by the current pro-

(a) cedure and calculations are being performed on

0.04 two such systems, HF [17] and CH~[18]. The

0.16 0:057 0:063 results of these calculations will be reported0.25 0.052 0.058 elsewhere.0.36 0.044 0.049

0.49 0.028 0.030

0.010 Acknowledgement1.00 2.702 2.628

We are grateful Dr. C.J. Noble and Dr. L.A.

Morgan for advice and use of their codes during_________________________________________ the course of this work.


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212 J. Tennyson eta!. /Continuum orb/ta/sformolecularR-matrix calculations

References [9] K.L. Baluja, C.J. Noble and J. Tennyson, J. Phys. B 18

(1985) L851.[1] C.J. Noble, P.G. Burke and S. Salvim, J. Phys. B 15 (1982) [10] L.A. Morgan, J. Phys. B 19 (1986) L439.

3779. [11] C.J. Noble and PG. Burke, J. Phys. B 19 (1986) L35.[2] S. Wilson, Advan. Chem. Phys. 67 (1987) 439. [12] J. Tennyson and C.J. Noble, J. Phys. B 19 (1986) 4025.

[3] PG. Burke, C.J. Noble and S. Salvini, J. Phys. B 16 (1983) [13] J. Tennyson, J. Phys. B 19 (1986) 4255.

L113. [14] J. Tennyson, CJ.Noble and PG. Burke, Intern. J. Quan-

[4] P.J.A. Buttle, Phys. Rev. 160 (1967) 691. turn Chem. 19 (1986) 1033.

[5] P.G. Burke andAL. Sinfailam, J. Phys. B 3 (1970) 641. J. Tennyson, J. Phys. B 20 (1987) L375.

[6] S. Salvini, PG. Burke andCJ. Noble, J. Phys. B 17 (1984) [15] S. Salvini, Ph.D. Thesis, The Queens UniversityofBelfast

2549. (1983).

[7] J. Tennyson, C.J. Noble and S. Salvini, J. Phys. B 17 [16] C.J. Noble, Daresbury Laboratory Technical Memoran-

(1984) 905. dum DL/SCI/TMT33T (1982).J. Tennyson and C.J. Noble, J. Phys. B 18 (1985) 155. [17] L.A. Morgan, to be published.

[8] L.A. Morgan and C.J. Noble, J. Phys. B 17 (1984) L369. [18] J. Tennyson, to be published.

jonathan tennyson et al- the generation of continuum orbitals for molecular r-matrix calculations...

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