A pointwise representation of the smatrix Kohn variational principle for quantum scattering
Abstract: A method is proposed for reducing the complexity of scattering calculations carried out using the Kohn variational principle. The technique is based upon the use of a pointwise representation for the L 2 part of the basis set and eliminates the need to explicitly evaluate any integrals involving such functions. Application to potential and inelastic scattering test cases show the method to be of good accuracy.
Topics: Pointwise (62%), Variational principle (61%), Scattering theory (57%), Smatrix (55%), Inelastic scattering (54%)
Summary (2 min read)
Jump to: [Introduction] – [3, Application to 1D electron scattering] – [4. Combining the pointwise description with local basis functions] – [5. Multidimensional problems] and [6. Conclusions]
Introduction
 The technique is based upon the use of a pointwise representation for the L2 part of the basis set and eliminates the need to explicitly evaluate any integrals involving such functions.
 The final expression is found to have a similar structure to the original one except that the matrices involving the L* basis now occur in point space.
 In section 3 more general basis sets and quadrature schemes are considered, and the authors also address the problem of hermiticity of the Hamiltonian matrix when calculated by an approximate quadrature.
 Application to a heavy particle potential scattering problem shows that the technique provides an accurate description when compared with the method in which all integrals are evaluated exactly.
3, Application to 1D electron scattering
 In order to test the accuracy of this pointwise representation of the Kohn expression the authors first consider the electron scattering problem used by Staszewska and Truhlar [ 191.
 The potential is attractive and has the form V(u)=exp( r) .
 This scheme evaluates the kinetic energy matrix exactly since the second derivative of eq. (10) is a linear combination of the basis functions and the Gaussian quadrature evaluates all overlap integrals exactly.
 Table 1 shows how the error in the tangent of the phase shift varies with size of the real basis for the pointwise method and compares the results with those obtained by evaluating all the integrals exactly, the variational method.
 Also, many problems are not conveniently described by a set of functions for which a Gaussian quadrature is available.
4. Combining the pointwise description with local basis functions
 Perhaps the most obvious place to put the points is at the centres of the Gaussians.
 The use of such a simple quadrature scheme dictates that the kinetic energy matrix will no longer be evaluated exactly over the basis set.
 An alternative method, favoured by Light [ 121, is to calculate the kinetic energy matrix exactly in the basis set representation and then transform to the pointwise scheme using the matrix R. This Table 2 Fractional errors in tan 6 for HeH, elastic scattering.
 This is as expected on the grounds of increasing the accuracy of the trapezoidal rule quadrature, From table 2 it is seen that the pointwise method requires a basis set about 20% greater than the variational method to attain results with a 1% error.
5. Multidimensional problems
 The extension of the pointwise representation to systems with more than one degree of freedom is straightforward and so the authors simply summarise the basic formulae.
 Such structure allows the kinetic energy matrix to be constructed from smaller matrices which have been evaluated in a space of lower dimension.
 These properties are best illustrated by a simple example.
 The parameters used are the same as for the 1D case of table 2 and five H2 vibrational functions are used.
 With these values, all matrix elements of the type ( uo&,IHEI~O#n,) could be neglected without changing the third figure of the inelastic transition probability.
6. Conclusions
 A method has been proposed for calculating the Smatrix via a pointwise representation of the Kohn variational expression.
 The method was implemented and found to be of comparable accuracy to the full variational form for degrees of freedom which are well described by functions for which a Gaussian quadrature scheme exists.
 The application to more general situations has also been considered.
 It thus appears that the method should work well for any inelastic scattering process.
 The pointwise scheme should be readily applicable to such systems and, if found to be of good accuracy, will be a powerful technique for calculating reaction rates.
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Volume 149, number 3
CHEMICAL PHYSICS LETTERS 19 August 1988
A POINTWISE REPRESENTATION
OF THE SMATRIX KOHN VARIATIONAL PRINCIPLE FOR QUANTUM SCATTERING
Andrew C. PEET and William H. MILLER
Department of Chemistry, University of California,
and Materials and Chemical Sciences Division, Lawrence Berkeley Laboratory, Berkeley, CA 94 720, USA
Received 2 June 1988
A method is proposed for reducing the complexity of scattering calculations carried out using the Kohn variational principle.
The technique is based upon the use of a pointwise representation for the L2 part of the basis set and eliminates the need to
explicitly evaluate any integrals involving such functions. Application to potential and inelastic scattering test cases show the
method to be of good accuracy.
The smatrix version of the Kohn variational principle
[
1,2] has recently enjoyed much success in describing
scattering processes of both electrons [ 31 and heavy particles
[
2,461. In particular, the method has been shown
to provide a powerful description of the full 3D reactive scattering event, with studies of H + H2
[ 5 ]
and F+ Hz
[
61 having been reported. These calculations and other related works
[ 71
represent a significant advance in
our understanding of elementary chemical reactions.
Calculation of the smatrix in the Kohn formalism may be considered in two parts. First, the evaluation of
matrix elements ofthe Hamiltonian over a basis set of scattering and
L*
functions and, second, the algebraic
manipulation of these matrices to give the Smatrix. Formally, the setting up of the matrices is a lowerorder
computational process than the matrix manipulations and so the latter will dominate the effort required for
a sufficiently large basis. However, for the problems presently being studied, the size of basis set and the nature
of the integrals dictate that the calculation of the matrices constitutes a considerable fraction of the effort
involved.
Here we present a method which greatly simplifies the application of the Kohn principle by avoiding the
need to explicitly evaluate any integrals over the
L2
basis set. The technique is based, initially, upon an Npoint
quadrature approximation to the integrals, where Nis the number of
L*
functions. After expressing the integrals
as products of square matrices, simple matrix manipulations then allow the Kahn expression to be reduced to
a form in which no quadratures over the
L*
basis functions are present. The final expression is found to have
a similar structure to the original one except that the matrices involving the
L*
basis now occur in point space.
This study constitutes a continuation of the recent interest in the use of pointwise methods to reduce the
computational complexity of large quantum mechanical calculations. These techniques are broadly based upon
the premise that a minimum of N potential evaluations is required to uniquely determine a wavefunction ex
panded in terms of N basis functions. Although such methods have existed for many years
[
8,9], it is only
recently that they have met with much popularity and been shown to provide an accurate description of real
quantum mechanical systems. Most relevantly, Light and coworkers have developed a version of the technique
which they call the discrete variable representation and successfully applied it to both scattering
[
10,111 and
bound state regions
[
12 141. Furthermore, Friesner
[
15 17
]
has managed to make great savings in the com
putation of electronic energies by a scheme which he terms the pseudospectral representation.
We begin the presentation of the technique as applied to the Kohn principle by introducing the formalism
to be used and obtaining an expression for the case of swave potential scattering. The method is illustrated
0 0092614/88/$ 03.50 0 Elsevier Science Publishers B.V.
(NorthHolland
Physics Publishing Division )
257
Volume 149, number
3
CHEMICAL PHYSICS LETTERS
19 August I988
by applying it to a model electron scattering problem which is well described by functions for which a Gaussian
quadrature scheme exists, In section 3 more general basis sets and quadrature schemes are considered, and we
also address the problem of hermiticity of the Hamiltonian matrix when calculated by an approximate quad
rature. Application to a heavy particle potential scattering problem shows that the technique provides an ac
curate description when compared with the method in which all integrals are evaluated exactly. The extension
to multidimensional systems is discussed in section 4 and the model HeH, vibrational energy transfer problem
of Secrest and Johnson
[
18
]
is investigated, with correspondingly successful results.
2. Basic formalism
We first consider swave potential scattering for which all the essential features of the pointwise method may
be illustrated. The Hamiltonian has the standard form
zI=
$g
+V(r),
with V(r) 40 as rrco. The Kohn variational expression for the Smatrix is then
[
21
(2)
where the complex quantities B and C are given by
~=(uolfWo),,t, (u~IHEluo)((u,lHEIur))‘(u~IHEIu,),
,’
(3a)
C=(GlHduo) ,/i, ~~,I~~I~~~*~~~,l~~I~~~~‘~~~I~~l~o~.
(3b)
9 ’
It should be noted that throughout this paper functions inside a bra symbol are not complex conjugated as is
usually the case; any complex conjugation (as, e.g., in eq. (3b) ) is indicated explicitly. The function uo( r) is
an incoming radial wave multiplied by a cutoff functionf( r) to regularise it at small r. The real basis functions
u,(r), I= 1, .*.,
N are square integrable and form the major part of the basis set.
From eq. (3) we see that there are two main demands on the computational effort. First, the setting up of
the matrices which involve the real basis functions and second, the inversion of the large square matrix con
structed from these functions. In this paper we address the former of these two problems. The strategy is to
reduce the integrals over the real basis functions to minimal Npoint quadratures. Choosing the points {ri} and
corresponding weights {w,} we define the matrices
("O)i=JGuO(ri) >
tDO)i=
5
JK”o”(ri)
3
(4)
where the prime denotes differentiation with respect to r. These quantities may then be used to write
B
and
C in the (approximate) forms
B=(uoIHEIuo)(R=Mo)=(R=MR)‘(R=Mo),
(5a)
C=(u~lHE~uo)(RTM;)T(RTMR)~l(R%Io),
(5b)
where the superscript T denotes the matrix transpose and
258
Volume 149. number 3
CHEMICAL PHYSICS LETTERS
19August 1988
Mo=Do+(VEl)&,
(6a)
M=T+VEl.
(6b)
The kinetic energy matrix T is given by
T=DR’ _
(7)
The two freefree matrix elements in eq.
( 5 ), ( u. (HEl uo)
and ( UZ
1 HE ( u,,
> , are assumed to be evaluated
exactly.
Since all the matrices involved are square, it is not hard to show that the matrix I? completely cancels out
of eq.
(
5) leaving the simple expressions
B=(uo(HE(uo)M$M.‘Mo,
(gal
Due to the absence of R, the final expressions for B and C no longer contain any quadratures over the real
basis functions. Furthermore, since R is a transformation matrix from the basis set to the pointwise repre
sentation, the
Lz
space now appears only in a pointwise fashion. However, it should be noted that since eqs.
(5 ) and (8) are equivalent, the pointwise method will give identical results to the basis set approach with the
integrals evaluated by the Npoint quadrature. The accuracy of the pointwise representation compared with
the full variational method is thus determined by the errors introduced from the nonexact quadrature. If these
errors are not significantly greater than those due to the truncation of the basis set, then the pointwise and
variational approaches will be of comparable accuracy.
From eqs. (6b) and (7) we see that the
L2
basis functions now appear only in the expression for the point
wise kinetic energy matrix T. The potential matrix is diagonal in point space with its elements being merely
the potential evaluated at the quadrature points. The problem of constructing the potential matrix in the basis
set representation has thus largely been replaced by one of evaluating the kinetic energy matrix in the pointwise
representation. However, as we shall see in section 5, this is often a much simpler task.
3,
Application to 1D electron scattering
In order to test the accuracy of this pointwise representation of the Kohn expression we first consider the
electron scattering problem used by Staszewska and Truhlar
[
191. The potential is attractive and has the form
V(u)=exp( r)
.
(9)
The scattering function
uo( r)
is taken to have the form of section 2 with the cutoff functionf( r) =
1 
exp
( at).
The L*
basis set is the same as used previously
[
2
1,
U,(r)=N,tl‘exp(err), I=1 ,..., N.
(10)
These real functions are linear combinations of GaussLaguerre polynomials multiplied by the conventional
weighting functions and so a Gaussian quadrature scheme exists for choosing the weights and points. This scheme
evaluates the kinetic energy matrix exactly since the second derivative of eq. (10) is a linear combination of
the basis functions and the Gaussian quadrature evaluates all overlap integrals exactly.
Table 1 shows how the error in the tangent of the phase shift varies with size of the real basis for the pointwise
method and compares the results with those obtained by evaluating all the integrals exactly, the variational
method. It is seen that the pointwise results converge with an accuracy comparable to those of the variational
calculations. This clearly demonstrates the potential of the method to reduce the complexity of the calculation
without leading to a significant increase in the basis set size. Actually, the basis set required to obtain good
259
Volume 149, number 3 CHEMICAL PHYSICS LETTERS
19 August 1988
Table 1
Fractional errors
[
41 in tan 6 for ID electron scattering test case. The scattering energy is given by
k=0.55
au [2] and LX= 2.5. The exact
result is tan 6=2.2003827
N
Variational Pointwise N Variational
Pointwise
2 0.0001 +0.1111
5
0.0003 0.0010
3 0.0019 0.0013 6 0.0001  0.0005
4 0.0011 0.0013
results in this test problem is so small that it is unwise to extrapolate such conclusions to larger systems. Also,
many problems are not conveniently described by a set of functions for which a Gaussian quadrature is avail
able. It is, therefore, important to consider the extension to a more adaptable basis and nonGaussian
quadratures.
4.
Combining the pointwise description with local basis functions
Perhaps the most adaptable of the basis sets presently available for describing onedimensional motion is
the distributed Gaussian basis proposed by Hamiltonian and Light
[
201,
q(r)= ?
( 1
l/4
exp[ A,(rr,)*]
, I=
1, .,.,N.
(11)
Since a single Gaussian quadrature scheme does not exist for this basis set, other prescriptions for choosing
the points and weights must be sought. Perhaps the most obvious place to put the points is at the centres of
the Gaussians. The weights are then most conveniently taken to be those of a trapezoidal rule quadrature, where
it is undoubtedly best to assume that the end points of the quadrature scheme lie outside the Npoint range.
The use of such a simple quadrature scheme dictates that the kinetic energy matrix will no longer be eval
uated exactly over the basis set. More importantly, the matrix will in general not be symmetric, leading to an
unsymmetric form of the pointwise matrix T. Here we avoid this problem by rewriting the kinetic energy matrix
in an inherently symmetric form. Integrating once by parts we obtain the identity

!!f
2P
t.+(r) u:(r)
dr=
!f
2P
s
u;(r,) u;,(r) dr ,
(12)
where the prime denotes differentiation with respect to r. The integral on the righthand side of this expression
may be integrated by any conventional quadrature scheme to yield a symmetric matrix. The expression for the
matrix
T
in the pointwise space then becomes
where the matrix
d
is defined by
d,,=fiu;(r,)
.
(13b)
In practice, a reasonable quadrature scheme will evaluate the kinetic energy matrix quite accurately and so
it could be used without further adaptation. However, writing
T
in a truly symmetric form has the compu
tational advantage of reduced storage and matrix inversion time as well as the conceptual nicety of preserving
the numerical hermiticity of the Hamiltonian over the L* basis. This technique for ensuring hermiticity is not,
of course, unique. An alternative method, favoured by Light
[
121, is to calculate the kinetic energy matrix
exactly in the basis set representation and then transform to the pointwise scheme using the matrix R. This
260
Volume 149, number 3
CHEMICAL PHYSICS LETTERS
19 August 1988
Table 2
Fractional errors in tan 6 for HeH, elastic scattering. The parameters used were E=6.0, (~,=0.5, RB = 10.0, rmi,= 1.0, r,,,=20.0 and
p= 0,667, all in the reduced units of Secrest and Johnson
[
181. The exact value
for tan 6 is 0.750188
N
Variational
16 0.0717
18 0.0071
20  0.0037
Pointwise N
0.5581 22
0.2758 24
0.0647 26
Variational
0.0015
 0.0004
0.0001
Pointwise
0.0100
0.000s
+0.0006
technique has also been implemented and was found to yield comparable accuracy for the examples presented
below.
The procedure outlined above was tested out on the elastic part of the HeH, scattering problem of Secrest
and Johnson
[
181, The intermolecular potential for this system has the form
Vr)=Voexp(yr),
(14)
with the parameters Vo= 12.0 au and y=O.315 au. The cutoff functionf(r) used to regularise the scattering
basis function u. was
f(r)=t{l+tanh[cu,(rr,)]},
(15)
which has been used in previous calculations
[
41. The Gaussians were distributed evenly between the values
1.0 and 20.0 a0 in r while the exponents of the Gaussians were chosen by the prescription
(16)
where Ar is the distance between the centres of the Gaussians and c is a parameter which may be varied.
The results are given in table 2, where they are also compared with the full variational calculations. The
parameter c was varied and found to be approximately optimum at 0.75 and 0.5 for the variational and point
wise schemes, respectively. It should be noted that broader Gaussians are used in the pointwise method. This
is as expected on the grounds of increasing the accuracy of the trapezoidal rule quadrature,
From table 2 it is seen that the pointwise method requires a basis set about 20% greater than the variational
method to attain results with a 1% error. This result is very encouraging especially when we consider the simple
minded nature in which the points and weights were chosen. Furthermore, if greater accuracy is desired then
fewer extra functions are required by the pointwise technique. Indeed, the size of basis which gives results cor
rect to 0.1% is the same for both methods.
5.
Multidimensional problems
The extension of the pointwise representation to systems with more than one degree of freedom is straight
forward and so we simply summarise the basic formulae. The Kohn expression now takes the form
[
21
S=;
(BC=B*‘C) ,
(17)
where B is a complex matrix which is the generalisation of eq. (3a),
B,,. = < w$,
I HEl w4z, >
(18)
261
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