Discrete variable representations of complicated ...

Discrete

variable

representations

of complicated

kinetic energy operators

Hua Weia) and Tucker Carrington, Jr. Department of Chemistry, Universitk de Monkal, C.P. 6128, succursaleA, Montr6a1, Qukbec H3C 3J7, Canada (Received 26 January 1994; accepted 14 March 1994) Probably the most important advantage of the discrete variable representation (DVR) is its simplicity. The DVR potential energy matrix is constructed directly from the potential function without evaluating integrals. For simple kinetic energy operators the DVR kinetic energy matrix is determined from transformation matrices and exact matrix representations of one-dimensional kinetic energy operators in the original delocalized polynomial basis set. For complicated kinetic energy operators, for which matrix elements of terms or factors with derivatives must be calculated numerically, defining a DVR is harder. A DVR may be defined from a finite basis representation (FBR) where matrix elements of terms or factors in the kinetic energy operator are computed by quadrature but implicating quadrature undermines the simplicity and convenience of the DVR. One may bypass quadrature by replacing the matrix representation of each kinetic energy operator term with a product of matrix representations. This product approximation may spoil the Hermiticity of the Hamiltonian matrix. In this paper we discuss the use of the product approximation to obtain DVRs of complicated, general kinetic energy operators and devise a product scheme which always yields an Hermitian DVR matrix. We test our ideas on several one-dimensional model Hamiltonians and apply them to the Pekeris coordinate Hamiltonian to compute vibrational energy levels of Hl. The Pekeris coordinate Hamiltonian seems to be efficient for H:. We use Jacobi polynomial basis sets and derive exact matrix elements for (dldx) G(x)(dldx) , r(x)(dldx), r(x), and (1 -x)‘eeJr with G(x) and r(x) rational functions. We discuss the utility of several Jacobi DVRs and introduce an improved FBR for general kinetic energy operators with more quadrature points than basis functions. We also calculate Euclidean norms of matrices to evaluate the accuracy of DVRs and FBRs.

I. INTRODUCTION Vibrational and rovibrational spectra of polyatomic molecules are often analysed using a perturbative-normal mode approach’ but to calculate energy levels accurately one must compute eigenvalues of a matrix representation of the Hamiltonian. This variational approach has been used to study many three-atom and a few four-atom molecules.2-6 To perform a variational calculation one must choose coordinates to describe the vibrations and rotations of the molecule, set up the Hamiltonian in terms of the chosen coordinates, choose functions of the coordinates as basis functions, and calculate matrix elements and eigenvalues of the Hamiltonian matrix. Of the 3N coordinates required to specify the configuration of the N nuclei of an N-atom molecule 3N- 6 coordinates describe vibration. Normal coordinates are adequate for small amplitude vibrations but for large amplitude vibrations other coordinates are better. Vibrational coordinates should be selected to facilitate the choice of good basis functions. Basis functions are good if off-diagonal matrix elements of terms in the Hamiltonian which couple two or more coordinates are small. It is important to choose coordinates which yield good basis functions because if the basis is good the “On

leave of absence from: Institute

of Modem

Physics,

sity, Xi’an, 7 10069,People’sRepublicof China.

Northwest

Univer-

dimension of the Hamiltonian matrix required to converge energy levels will be smaller. One may also reduce the size of the matrix which must be diagonalized to calculate energy levels by exploiting symmetry.’ Coordinates should, therefore, be chosen to allow one to take advantage of symmetry. As demonstrated convincingly by Bazid, Light, and coworkers it is often advantageous to use basis functions which are finite approximations to Dirac delta functions.5’8 They define the finite basis representation (FBR) as the representation obtained by evaluating potential matrix elements by Gauss quadrature and taking exact, analytic kinetic energy matrix elements. One transforms from the FBR to the discrete variable representation (DVR)5*8.9 using a transformation matrix which may be obtained by diagonalizing a matrix representation of a coordinate function.” In the DVR multiplicative operators are diagonal (and no quadratures are required). Determining the DVR of a simple kinetic energy operator is straightforward. Obtaining an Hermitian FBR (or DVR) of a complicated kinetic energy operator, whose terms involve both (noncommuting) derivatives and functions of coordinates [see for example, Eqs. (7) and (21) for onedimensional (1D) and f-dimensional (fD) examples], is somewhat tricky because analytic kinetic energy matrix elements are generally not available. We denote a kinetic energy operator as “complicated” if either (i) it is factorizable [see Eq. (40)] and matrix elements of a factor which involves

0 1994 American Institute of Physics J. Chem. Phys. 101 (2), 15 July 1994 0021-9606/94/l 01(2)/l 343/18/$6.00 1343 Downloaded 14 Mar 2002 to 129.132.216.123. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

1344

H. Wei and T. Carrington, Jr.: DVRs for complicated kinetic energy operators

derivatives must be calculated numerically, or (ii) it is not factorizable and matrix elements of a term must be evaluated numerically, or (iii) matrix representations of derivatives are not anti-Hermitian. One way to obtain an Hermitian FBR for a complicated kinetic energy operator is to use Gauss quadrature to calculate kinetic energy matrix elements.6 One might also define the FBR of a complicated kinetic energy operator by assuming that matrix representations of products of derivatives and functions of coordinates are equal to products of matrix representations of the derivatives and the functions. We refer to replacing the matrix representation of a product of operators with a product of representations as a product approximation. Once an FBR is defined the corresponding DVR is obtained by unitary transformation. As originally defined, for simple kinetic energy operators, one can compute matrix elements of the DVR Hamiltonian matrix without calculating integrals by quadrature. DVR matrices for coordinate functions are obtained directly from values of the functions at the DVR points and not by first computing an FBR matrix which is then transformed to the DVR. Not having to compute integrals makes the DVR simple and convenient. It would be advantageous not to have to evaluate quadratures even for complicated kinetic energy operators. Quadrature may be avoided only by using some sort of product approximation. Unfortunately, the product approximation Hamiltonian matrix may be non-Hermit& and it would appear that if the kinetic energy operator is complicated it is not possible to avoid quadrature and to maintain the simplicity and convenience of the original DVR. Kinetic energy operators for molecules with more than three atoms are necessarily complicated.276 Even when it is possible to evaluate kinetic energy matrix elements exactly it may be difficult to do so (see Sec. III and the Appendix). It would be much simpler to invoke the product approximation to define an FBR and a DVR. In this paper we show how the product approximation can be used to calculate an Hermitian DVR of a general kinetic energy operator without evaluating quadratures. We apply the new method to several test onedimensional problems and to three-dimensional Hl. Bond, Jacobi, hyperspherical, and Radau coordinates have all been used to calculate energy levels of triatomic molecules.5~11~12 Each of these coordinate systems has advantages and disadvantages; for a given molecule one chooses the best coordinate system.t3 Variational energy levels have also been computed using as coordinates the three internuclear distances, R t2, R23 , and R3 i , of a triatomic molecule. Early in the history of vibrational variational calculations for Hl Spirko et al. used an approximate internuclear distance Hamiltonian to calculate rovibrational energy levelsI An exact kinetic energy operator in terms of three internuclear distances was derived by Diehl et a1.l5 Recently Watson has used an internuclear distance Hamiltonian to calculate lowlying states16of H: . Potential energy surfaces are frequently obtained as functions of the three internuclear distances.‘7-‘9 These coordinates are particularly attractive for molecules with three identical atoms because they reflect the molecule’s high symmetry. Energy level calculations with either an approximate or an exact internuclear-distance Hamiltonian are plagued by problems associated with the interdependent

ranges of the coordinates. The physically allowed ranges of the coordinates are related by triangle relationships such as 31S1R12+R231. Various schemes can be de[RI,-hivised to calculate matrix elements numerically despite the interdependent ranges but the problem remains debilitating. The interdependent range problem may be avoided while maintaining the natural symmetry of the internuclear coordinates by using Pekeris coordinates20T2’ which are linear combinations of internuclear coordinates,

r3=#h+R31-W.

0)

Starting from either the Hamiltonian of Diehl et all5 or the bond coordinate Hamiltonian of Carter and Handy2 and using the chain rule we find that the vibrational Hamiltonian (in atomic units) of a triatomic molecule in terms of {rl ,r2,r3} is Hkkeris

=

-i,

$,

(2)

Gjk&+V,+V, I

where Gii=ri

1

ri+?+j+rk rk ‘i + TTIiRijRik m$kiRkj f T?ZjRjkRji ’

Gij= G,,= i,j,k=

1,2,3

rirj mkRkiRkj ’

and cyclic permutations,

(3)

1

1 va=8(R,2R23R3,)Z

’

(4)

R12=rl+r2, R23=r2+r3, R3,=r,+r3, V, is aquantum mechanical term analogous to the Watson term in the normal coordinate Hamiltonian, and V is the potential energy. The above Hamiltonian is consistent with the volume element drldr2dr3 (i.e., with a unit weight factor) so that the eigenfunctions are normalized as /omdr,lo~drJ~dr31*12=

1.

(5)

The wave functions of our Pekeris coordinate Hamiltonian are 9 = ylDiehlJS =ylDieh’J2(rl+r2)(r2+r3)(r3+rl),

(6)

where lIrDiehl is the wave function of the Hamiltonian of Diehl et al. obtained from Cartesian coordinates by the chain rule without redefining the wave function by absorbing part of the weight factor. Because the Cartesian coordinate wave

J. Chem. Phys., Vol. 101, No. 2, 15 July 1994

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TABLE

I. The

DVR

matrix

of the kinetic

b2= 1.3,

X(~+b,)/(x+b,)~], 16.8 38.0 0.0 2.5 2.2 -1.8 1.5 -1.4 1.4 2.1

-48.6 51.9 -31.3 0.7 3.2 -3.0 2.7 -2.5 2.8 4.3

bg= 1.45, 113.0 -5.9 50.6 26.8 0.5 2.3 -2.7 2.8 -3.4 -5.8

KD,p=-~~~~~DsvGP(,y)D,

energy N=

115.8 33.0 17.9 45.3 24.0 -1.6 -1.5 2.5 -3.6 -7.1

II. DVRs AND FBRs OF COMPLICATED ENERGY OPERATORS a one-dimensional

(1D)

basis,

where

D,B=(d/d~)~~R,GP(~)

=

2[( 1+x)

lO,andtheJacobibasisparametersarea=5,b=O.ThematrixisfarfrombeingHermitian(symmetric). 106.5 35.7 -11.4 20.4 41.1 -22.5 2.1 1.4 -3.5 -8.3

-95.0 -33.6 14.9 3.7 -20.7 39.3 -22.3 2.1 2.8 9.7

function (with a unit weight factor) is nonzero at linearity qDirh’ and our wave function, ?, are nonzero at linearity (r, ,r2, or r3=O). Employing Pekeris coordinates rather than simple internuclear distances enables one to avoid the interdependent range problem but another obstacle to easy and accurate calculation of energy levels remains. Derivatives with respect to Pekeris coordinates are not anti-Hermitian because vibrational wave functions (and basis functions) are in general nonzero at linearity. For H3f vibrational wave functions may have significant amplitude at linear configurations. In this article we consider FBRs and DVRs of Hamiltonians with complicated kinetic energy operators and non-anti-Hermitian representations of derivative operators.

Consider atomic units)

in a Jacobi

1345

KINETIC Hamiltonian

84.5 31.0 - 15.3 -7.6 -0.8 -21.2 40.1 -23.3 0.4 - 10.9

-76.2 -29.0 15.4 9.2 5.2 0.0 -22.3 43.7 -24.3 9.9

71.2 28.5 -16.6 -11.3 -8.3 5.9 -1.5 -23.4 48.9 16.3

73.2 32.0 -21.1 - 16.8 - 15.0 14.4 - 14.2 12.0 15.2 34.6

The original FBR was defined as the representation obtained by evaluating matrix elements of the potential with Gauss quadrature and using exact kinetic energy matrix elements.* If one uses N orthonormal basis functions of the form e,(z) =hk “2[w(z)]“2fk(z) where z is a function of x, fk(z) is a classical orthogonal polynomial, w(z) is the corresponding weight function, and hk is a normalization factor and chooses as many quadrature points as basis functions the Gauss quadrature approximation for matrix elements of zn is equivalent to assuming that the matrix representation of zn is equal to a product of n matrix representations of z.~ The product approximation Hamiltonian matrix is obtained by inserting multiple copies of the resolution of the identity between pairs of operators. The term FBR could also be used to describe the representation obtained by replacing representations of products with products of representations not only in the potential but also in the kinetic energy. In this sense the FBR of the 1D kinetic energy in Eq. (7) is

(in K zfR=-fy

D,j(jlGlk)Dk,,

00)

&j,k=O

H=K+V,

K= -;

G(x)&,

x E (x1 7x2)

(7)

with a unit weight factor so that orthogonal basis functions are normalized as Tfx I XI

e,(x)*e,(x)=

s,, ,

(8)

D,,= I and (j[Glk)

(11)

is evaluated by quadrature,

(jIGlk)=GpR=

2 f?,(x,)*G(x,)t3,(x,)WLN)lw(x,) a=0 = ( TGDVRTt)jk

(12)

with (9)

if 0,(x,) and (dldx) e,(Xj) are nonzero at boundary point Xj so that the O,(x) matrix representation of the Hamiltonian is Hermitian. If the “surface term” 6&(x)* 0,(x)1::, obtained by integrating by parts, does not vanish the matrix representing the derivative with respect to x is not antiHermitian. Despite the fact that the derivative matrix is not anti-Hermitian the kinetic energy matrix is Hermitian, if ma-

trix elementsare evaluatedexactly.

e,(x)*g e,(x)

N-l

where G and V are real and G is positive and bounded. This kinetic energy operator is general because for a unit weight factor it is always possible to write the part of the kinetic energy operator which involves derivatives in symmetric form.****” G satisfies the condition G(Xj)=O

“dx x1

Tj~=\j~ej(X,)*,

G~SBVR=G(X,)S,~

(13)

and WLN’a quadrature weight [see Eq. (8) of Ref. 91. The superscript o indicates that KoEBR is based on an ordinary product approximation. The FBR-DVR transformation matrix T diagonalizes the matrix of x.~ If GFBR and D are both banded matrices the error introduced by the product approximation is restricted to the bottom right-hand comer of

KOFBR .

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1346

where

In this article we shall use Hamiltonians written for a unit weight factor and basis functions e,(x) which are orthonormal as in Eq. (8). Using a unit weight factor the derivative matrix D is anti-Hermitian if the basis functions 19,(x) are zero at xi and x2, the largest and smallest allowed values of the coordinate. If D is anti-Hermit& the product FBR Hamiltonian matrix K°FBR is Hermitian. However, if D is not anti-Hermitian because the basis functions e,,(x) are not zero at xi or x2 the product FBR Hamiltonian matrix is not Hermitian. In Table I we show that if the basis functions do not vanish at the boundaries the DVR matrix, T+K°FBRT, is far from being Hermitian. Although an exact variational basis representation (VBR) matrix (with no product and no quadrature approximation) is Hermitian, H°FBR is not and may have complex eigenvalues. To obtain an Hermitian FBR kinetic energy matrix by invoking the product approximation one must write the kinetic energy operator in the explicitly Hermitian form

DDVR= T+DT.

(19)

The superscript b indicates that Kb is for a balanced (explicitly Hermitian) operator. We determine the T matrix by diagonalizing the coordinate matrix.’ Greek letters denote DVR labels (for either grid points or DVR basis functions) and Latin letters denote delocalized polynomial basis functions. To calculate the DVR of i it is not necessary to first calculate KFBR. Kb is computed without calculating GFBR, without quadratures, from the values of G(x) at the DVR points and DDVR. Note that although no quadratures are evaluated to obtain the DVR the quadrature approximation is implicit in our assumption that GDVR is diagonal. Similarly the DVR of K [of Bq. (7)] is obtained by transforming the ordinary product approximation FBR [Eq. (lo)] K”= - ( ~/~)DD~@“RDDVR.

(14) with

~~,=cm,~,*)=~~dxe~(x)~( g)+G(x)gB,(x),

Replacing the matrix representation of the 1D Hermitian operator K, written for a unit weight factor, with a product of matrix representations yields a non-Hermitian representation if the matrix representation of the derivative is not antiHermitian. In many dimensions the same problem may arise. It may be avoided, in general, by replacing

(15)

c

where (dldx)+ = (dldx) and the arrow denotes differentiation to the left. If integrals are evaluated exactly then

Km= km

+ WXl ,...,Xf),

1 * Z?=kf+ U=y,x j.k= + U(x*

= ( 1/2)T+D+TGDVRT+DT

c

j=l

yj=O

D(j)+Gjk(Xl ,sas,Xf)Dck) 1

(22)

,...,x*).

Note that for a unit weight factor the part of the kinetic energy operator which involves derivatives may always be written in the symmetric formZ,23 of Eq. (21). k is explicitly Hermitian and its product approximation FBR will also be Hermitian. The DVR of any term in i is obtained, without evaluating integrals by quadrature, by multiplying a transformed exact matrix representation of the adjoint of a derivative operator, a diagonal Gjk matrix, and a transformed exact representation of a derivative operator. The DVR of H is

Kb= T+KFBRT= ( 1/2)T+D+GFBRDT

i

(21)

with

where GFBR is the matrix of G with elements evaluated by quadrature. Both the exact VBR matrix K and the FBR matrix IPR obtained via the product approximation are Hermitian. The DVR matrix is obtained from K by premultiplying by Tt and postmultiplying by T

Nj-

D(j)=& I

(17)

=( 1/2)(DDVR)+GDVRDDVR,

D”‘Gjk(xI,...,xf)D’k’

J,k=l

(16)

KFBR= ( 1/2)D+GFBRD,

1 f

H=K+U=-Z,~

because at x1 and x2 either G(xj)=O or e,(Xj)=O or (dldx) 0,(x,) = 0 and the surface term obtained when one integrates by parts is zero. It is clear, therefore, that one may replace K with K. We define the FBR of K as

Hb a ,.,, af,p ,.., Bf=;

(20)

(18)

1

D(j)*G@) yjaj I,

.(j-l) a, ‘***’ aj-,

,,(A ,x(i+l) rj

aj+l

l I-Lx D$)*,Gjk(X(,‘ -, l) I ‘-Qj Jai+, 2j+k la3 ,eae,Xa

(j-l)

(j)

(j+l)

f aalp,.

(23)



1347


The DVR points for Xj are ~2; , aj= 0, 1, . . ., Nj - 1. Different Xj may have different sets of DVR points xzj distinguished by the superscript (j). It is important to note that to construct this general Hermitian DVR Hamiltonian matrix one needs only to transform exact derivative matrices (to calculate the DVR matrices D(j)); it is not necessary to calculate FBR matrices and to transform them to the DVR. We have shown that one may obtain an Hermitian DVR Hamiltonian matrix from a symmetric kinetic energy operator of the form of Eq. (21) for a unit weight factor by replacing -Do) on the left by DO’)+. What happens if the weight factor is not unity and the kinetic energy operator does not have the symmetric form of K in Eqs. (7) and (21)? Bending kinetic energy operators are frequently written with nonunit volume elements.2’24 For a nonunit weight factor, J, basis functions are normalized as

I

XZdx J(x) 1+,12= 1. XI

(24)

To avoid quadrature one might attempt to obtain an FBR matrix for a nonunit weight factor kinetic energy operator by replacing representations of products with products of representations. The FBR matrix obtained by doing so is not Hermitian. To define an Hermitian DVR without quadratures as we have described, the kinetic energy operator must be written in symmetric form with a unit weight factor. Matrix elements are unaffected by the choice of weight factor because

*‘dxJWMx)*K~~,(~) I XI = x2dx e,(x)*(~(x))1~2K~(~(x))-*~2e,(x), I XI

(25)

K is a term in the J weight where en(x) =J(x)“~+,(x), is the correfactor Hamiltonian and J(x)” $ K~/(x)-“~ sponding term in the unit weight factor Hamiltonian. Our FBR for kinetic energy operators entails two approximations: The kinetic energy matrix is written as a product of matrices and elements of the matrix which represents the coordinate function are evaluated by quadrature. The same approximations are implicit in the DVR, obtained from the FBR by unitary transformation. In some bases DFBRis a full matrix and the, error introduced by writing the matrix representation of K as a product of three matrices is not limited to the lower right-hand corner. To reduce the error introduced by the product approximation it is advantageous to write K in terms of an operator

B=dx)&, whose matrix representation is tridiagonal. For our calculations we choose to use Jacobi polynomial basis functions25-28 and q(x) = ( 1 -x2) [see Eq. (A28) in the Appendix]. For other polynomial bases similar tridiagonal operators exist (see p. 783 of Ref. 27). For the tridiagonal Morse basis” q(x) =x and for the harmonic oscillator basis

q(x) = 1, where x in each case is the argument of the polynomial in the basis function. We define the operator i’ (t is for tridiagonal) as 2=(

1/2)B+C?B,

6=q(x)-2G.

(27)

The DVR of if is K’= ( l/qBDVRt~DVRBDVR,

(28)

where @ ““&

“jJ+B~RT.

(29)

It is useful to write K’ in terms of B because the (N- 1) X (N- 1) upper left submatrix of an NX N product FEARj? matrix is unaffected by the product approximation due to the tridiagonality of B. We shall denote a representation obtained from a product of VBR matrices whose matrix elements are all exact as a pure product approximation (ppa) representation KPPG (1/2)D~Rt@‘BRD’-R.

(30)

All FBRs and their associated DVRs are based on choosing the same number of basis functions and quadrature points. The accuracy of the quadrature (for an FBR) or the extent to which functions of coordinates are diagonal (for a DVR) is therefore linked to the size of the basis. Our FBR for complicated kinetic energy operators implicates both the product approximation and quadrature. It is easy to see that the accuracy of the FBR (and the associated DVR) is increased by taking products of larger matrices and using more quadrature points. The accuracy of an FBR Hamiltonian may be improved, without increasing the dimension of the matrix, by taking the upper left hand N X N comer of a larger N4 X N4 FBR matrix. For this improved N X N FBR matrix the quadrature error is diminished because all matrix elements are computed with an N4 point quadrature. Likewise, the product error is diminished because the improved FBR matrix is obtained by truncating a product of N, X N, FBR matrices. To construct this improved FBR it is necessary to evaluate (N4 point) quadratures. A unitarily equivalent improved FBR, for which one does not have to evaluate quadratures, may be defined (from a DVR) for an operator A as Nq-1

A “,n”= 2 L,Ja”,,DvR T$, a.p=o m,n=O,N-

1;

NN (B is tridiagonal), the improved FBR eigenvalues are not the same as VBR eigenvalues because of the quadrature error. The quadrature error,


1346

H. Wei and T. Carrington, Jr.: DVRs for complicated

however,- may be systematically decreased by increasing N,. If G and the potential are polynomials of degree k or less, taking N,= N-t- N, DVR points with N,= (k - 1)/2 if k is odd and N,= k/2 if k is even will remove the residual error. III. EXACT KINETIC ENERGY MATRIX ELEMENTS IN THE JACOBI BASIS As discussed in the last section, invoking the product approximation for a symmetric kinetic energy operator [of the form of the kinetic energy of Eq. (7) or (21)] yields a non-Hermitian FBR Hamiltonian matrix if matrix representations of derivatives are not anti-Hermitian. Clearly one way to avoid the product approximation, if possible, is to use formally exact VBR expressions for kinetic energy matrix elements. In this section we derive VBR kinetic energy matrix elements for Jacobi basis functions. Deriving exact matrix element formulas is arduous but it is useful to have exact results to assess the accuracy of the DVR and improved FBR. Jacobi basis functions, Jab(x) = hiii2mPib(x) with w(x)= (1 -x)O( 1 +x)~, P:‘(x) a Jacobi polynomial and h,-db/’a normalization factor (see the Appendix) depend on two parameters; they are flexible and useful for different types of coordinates. Consider now Jacobi basis matrix elements for various values of a and b. For a>O, b*O We choose b = 0 and a > 0 if the wave function is nonzero at x = - 1 and zero at x = 1. If the wave function were nonzero at x= 1 and zero at x= - 1 one would take a =0 and b>O and matrix element formulae would be determined from equations of this section and of the Appendix by exchanging a and b. Consider the kinetic energy operator of Eq. (14). Matrix elements of this explicitly Hermitian form of the kinetic energy operator K [Eq. (7)] may be calculated exactly by splitting it into products of factors. First, consider G a Kth degree polynomial. We use the tridiagonal matrix of B in Eq. (26) for a Jacobi basis


( 1 +,x) - ’ factor is not present. Matrix elements of the terms of G (dldx) may be found in Sec. 2 of the Appendix. Second, consider a rational G which is positive and bounded. It can always be written as a sum of a polynomial part GP”’ and other terms G(x)=G~“(x)+C

k=max(O,m-

1)

d (ml(c+x)-k-ln), dx

For a=O, b=O Note that the Jacobi = (n + i) 1’2Pn(x), where P,(x)

=2(1-x)+2(

W-1)

Dmr= ~J~l~lJ3 J(2m+1)(2n+l)

I 0,

1 +x)

if

n-m=

1,3,5 ,...,

otherwise. (36)

It follows that for any F(x) and G(x) VBR

[ 1

K-3

-+gOO+(l+x)c

basis function J:‘(x) is a Legendre polynomial.

For bending coordinates, 8, Legendre basis functions with x = cos 0 are eigenfunctions of the bending kinetic energy operator and are therefore ideal basis functions. Matrix elements of terms which are diagonal in the Legendre basis are trivial but for many terms in a general Hamiltonian the following formulae should be useful. If G does not include the factor 1 -x2 then the method of Eqs. (32) and (33) cannot be used because P,( + 1) # 0 and l/( 1 tx) is not integrable. Approximate and exact calculation of Legendre matrix elements is, however, facilitated by noting that the derivative matrix in the Legendre basis is rigorously upper triangular Isee Eq. (A3O)l,

WE

(21)

(35)

where I c I > 1 because G is bounded on ( - 1, 1) . These matrix elements are given in Eq. (A43) of the Appendix.

d

G

II G=m

(34)

Matrix elements of the kinetic energy operator are obtained by writing it with one B and then expressing G(x) X( 1 -x2) - * in terms of partial fractions. In addition to the matrix elements required for a polynomial G one requires matrix elements of the form

=

=-: 5’ Bb(kl~~ln)

bj(cj+x)-kj.

= FmRD

(37)

a& KVBR= $)+G”BRD or gk(l-X)k.

k=O

K;$$ (33) Because a > 0 the basis functions are zero at x = 1 so matrix elements of (1 -x)-l are finite. Similarly if b >O the basis functions are zero at x= - 1 so matrix elements of ( 1 +x)- ’ are finite. If b = 0 the basis functions are nonzero at x = - 1 but in this case G( - 1) = 0 (which ensures the Hermiticity of the Hamiltonian) and the term with the

5’ Dj*?&=D,, j=o I=0 =i%’ D$(jlG$jn). J=o

(38)

The elements of the last row and column of GVBR do not affect KVBR because of Eq. (36). If G3 is a polynomial of




degree less than or equal to 3 only matrix elements in the last row and column of G$” computed by quadrature are in error so that the corresponding KyBR and KyR= ( 1/2)D+GyRD

G/C(XI

rX2,X3)=

(40)

2 g~~‘fr(nl)gi(x2)h,(x3) 1,i.n

matrix elements can be calculated exactly if every factor is a polynomial or rational function. The matrix elements are products of 1D matrix elements calculated as above. Most useful vibrational Hamiltonians are factorable.2v’3

r(r+k-)

dr m(r+r,)Z =--

(39)

are identical. Equations (37)-(39) may be useful for general Hamiltonians. For multidimensional kinetic energy operators with factorizable G matrix elements (e.g., for f= 3)

d

H I’, kk&S -

1349

d

-+v dr

d (I+x)(b,+x) dx ge (b2+x)2

d dx+”

(43)

where r=(x+l)r,,/2 maps r E [O,r,&j toxE[-l,l], b2= 1+(2r,lr,,), b3 = 1 + (3r,/r,,), and g, = 4/(m&,) . Equation (43) is obtained from Eqs. (2) by fixing all but one of the ri at re . The potential is VJb+ V, , we have chosen b2= 1.3, b3= 1.45, g,=39.285 274 93 cm-’ and we write,

G=Qe

(1 +x)(b,+x) (b2fX)2 =%,(l -X2)& x+bq

b3-b2 =b,+l

1 b,+l 1 -+(x+b2)2+(b2+ 1)z( x+b,

1 (45)

IV. CALCULATIONS As explained in Sec. II, if basis functions used to calculate a matrix representation of a unit weight factor kinetic energy operator do not vanish at the boundaries, an ordinary product FBR, K°FBR, and the corresponding DVR, KO, [see Eq. (20)] of the ki netic energy operator are not Hermitian. An Hermitian DVR Hamiltonian is obtained by transforming a representation which is constructed either by using an explicitly Hermitian operator and the product approximation, or, if possible, by deriving formally exact expressions for the matrix elements. In this section we calculate energy levels and compare the results obtained with different forms of onedimensional Hamiltonians. As a first example we choose the one-dimensional bending Hamiltonian H= -&

G(x)&+

V,

(41)

where V= Vlb+ V,, x= cosB,B is the triatomic bending angle, VJb is Jensen’s bending potential for water,29 VJb=ZZ~=.Jok(x-x,)k, fo2= 18 975.6 cm-‘, fo3= 1728 cm-‘, fo4 = 5 154 cm- * , x,= cos 104.439 76”, to which we have added 1 (l-x)2 va=fa [ (l-x)2+(1-xc)4

2 -(l-x,)2

4(x-x,)2 - (1-x,)4

I* (42)

The additional term V, makes the potential more repulsive when x is close to 1 but changes the potential in the vicinity of the well very little because VJx,)= VL(x,) = Van(xc)=O. G=2g,(l -x2), g,=39.285 274 93 cm-’ (determined from the masses of the hydrogen and oxygen atoms, the O-H bond length, and the H-O-H bond angle) andf,=300 cm-‘. The results are presented in Table II. We also calculate energy levels for the 1D Pekeris coordinate20*2’Hamiltonian

Exact VBR matrix elements of the 1D Pekeris kinetic energy may be computed using Eqs. (32), (45) and the formulae in the Appendix. The DVR and VBR results are presented in Table III. In Tables II and III we compare the energy levels of H” Hb,H’,Hpp”,Hab,H”‘, and HVBR computed with a basis of Ni 30 Jacobi basis functions with energy levels obtained from 99 Legendre basis functions and 114 quadrature points. The Legendre basis eigenvalues are assumed to be accurate because increasing the size of the basis from 60 to 99 and the number of quadrature points from 75 to 114 does not change the reported results. H” is obtained from K” of Eq. (20), it is not Hermitian. Eigenvalues of Ho are calculated by setting the upper triangle equal to the lower triangle. Hb is an explicitly Hermitian Hamiltonian matrix; its kinetic energy matrix is given in Eq. (18). H’ is also explicitly Hermitian, its kinetic energy matrix is in Eq. (28). Hppa is obtained from Eq. (30). Hab and Ha’ are improved FBR versions of Hb and H’ where we have increased the number of quadrature points; see Eq. (31). From the tables it is clear that energy levels computed with Ho (column 3) are very poor. Obviously it is important to ensure that the Hamiltonian matrix be Hermitian. With an explicitly Hermitian operator Hb or H’ one obtains many accurate energy levels by invoking the product appoximation and assuming that coordinate functions are diagonal (columns 4 and 5). The higher H’ energy levels are better than the corresponding Hb energy levels for the G = 2g,( 1 -x2) bond angle kinetic energy operator. For the 1D Pekeris kinetic energy operator energy levels which are well converged are slightly better with H’ than with Hb. Using B reduces the product approximation error but will increase the quadrature error if matrix elements of G are not evaluated as accurately as matrix elements of G. Calculating matrix elements of G exactly (see the Hpra results in column 6) improves very little the energy levels which are reasonably well converged. In columns 7, 8, 9, and 10 we give results obtained by taking N, extra quadrature points. Differences between the H”’ re-



1350

TABLE II. Comparison of accurate energy levels and eigenvalues of the Hamiltonian matrices Ho, Hb, H’, IIppa, Hob, and H”’ [see Eqs. (20), (18), (28), (30), is the basis size, and the Jacobi parameters are a= 15, b=O. G=2g,( 1 -x2). VBR and (3 l)]. N, + N quadrature points are used for H” b and H’ ‘, N=30 results are calculated using the formulas of the Appendix. Values of Eeigcnvalue-Eaccu are reported. All numbers are in cm-‘. DVR n

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

E,CUI,, 811.26 2 458.74 4 076.64 5 656.71 7 187.69 8 651.63 10 014.17 11 238.93 12 437.71 13 793.81 15 311.28 16 949.23 18 684.76 20 504.86 22 401.15 24 367.92 26 401.50 28 499.80 30 662.09 32 888.75 35 181.03 37 540.83 39 970.35 42 471.93 45 047.83 47 700.13 50 430.64 53 240.88 56 132.15 59 105.46

H” -30.58 -35.20 -40.75 -47.70 -57.08 -72.06 - 105.55 - 174.36 -212.04 -204.30 - 196.79 - 195.39 - 198.38 -204.92 -214.91 -228.67 -247.19 -2173.54 -2433.05 -2528.85 -2636.04 -2760.93 -2908.20 -3112.50 -3285.83 -3788.93 -3725.06 -5334.49 -3349.94 2700.85

Hb

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 O.Cil 0.00 0.00 -0.02 0.03 -2.31 -0.56 - 1041.83 -2325.75 -2551.51 -2216.03 -2499.41 - lOlQ.26 -1472.14 3707.96

Hub H’

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 -0.01 -0.47 0.25 -27.17 - 156.75 -2457.05 -2275.07 -2545.64 - 1865.48 -1471.60 -2189.92

H”’ N,=2

N,=l

HP”

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 -0.01

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.N' 0.00 0.00 0.00 0.00 0.00 0.00

0.03 -0.59 0.73 - 19.52 17.12 -932.97 - 1078.46 -2459.87 -568.26 - 1360.53 6048.3 1

0.02 0.04 0.68 1.30 14.87 22.65 203.85 224.16 1382.69 1349.12 6927.88

0.02 0.04 0.69 1.43 14.58 22.60 203.97 238.69 1335.69 1369.25 6228.21

sults and the VBR results are due only to quadrature errors in 6 and V. The Ha’ results are better than the H’ results for the highest energy levels, for which the product approximation has the largest effect. Errors due to the product approximation, to the finite size of the basis set, and to the quadrature approximation for matrix elements of G cancel each other to some extent. Energy levels of Hb,Ht,Hab, and H”’ are very close to energy levels of HVBR. If one does not modify Jensen’s water bending potential by adding V, the potential is finite at x= 1(0=0) where the two hydrogen atoms coalesce. We use a Jacobi basis with a > 0 and b = 0. Using these basis functions, which are zero at x = 1 and finite at x = - 1 ( 0= T) it is difficult to obtain an accurate description of wave functions which are nonzero at x=1. With a basis of 30 Jacobi functions (a=9.5, b=O, N,=2) we are able to converge 21 levels using Jensen’s potential and G= 2g,( 1 -x2) to within 1 cm-’ . The same size Legendre basis (finite at x= + 1) converges only 15 energy levels more poorly. However, if the basis size is increased to 79, the Legendre basis converges 72 and the Jacobi basis converges only 23 energy levels. Augmenting the Jacobi basis size from 79 to 99 changes the energy levels very little. This failure of the Jacobi basis to converge more energy levels as its size is increased is due the finiteness of the Jensen potential at x= 1. If one adds V, to

N,=

1

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 -0.01 0.62 0.35 12.31 -2.34 169.24 -42.32 865.79 192.14 -594.64

N,=2

VBR

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.02 0.02 0.66 1.08 13.52 13.58 192.69 163.77 1203.71 1134.69 2235.11

0.02 0.03 0.69 1.42 14.38 20.40 205.04 235.30 1306.93 1295.3 1 6575.29

Jensen’s potential to obtain a bending potential which is truly infinite when the hydrogen atoms coalesce, large Jacobi and Legendre bases are about equally good. We have also used the Jacobi basis to calculate energy levels of the potential (I’~~=D~[

:r;:z;;:;;]2,

X=COS@E[- 1,1-j,

(46)

which may be considered as a truncated, scaled ManningRosen potential.30 The Jacobi basis converges better than the Legendre basis for all bound states. This potential might be useful for describing bending vibrations.

V. ACCURACY OF FBRs ASSESSED BY EUCLIDEAN MATRIX NORMS In the preceding section we have compared energy levels calculated with various forms of ID Hamiltonians. It is also interesting to compare matrix representations of the kinetic energy operators and coordinate functions discussed above. To assess the deviation of the DVR or FBR matrices from the exact VBR matrices we use the Euclidean or Frobenius norm (E-NoI-I-II)~~~~~



H. Wei and T. Carrington,

Jr.: DVRs

for complicaied

kinetic

energy

operators

1351

TABLE III. Comparison of accurate energy levels and eigenvalues of the Hamiltonian matrices HO, Hh, H’, HPPa, Hab, and Ha’ [see E?qs. (20), (18), (28), is the basis size, and the Jacobi parameters are a= 15, b= 0. G= 2g, (30). and (31)]. N,+N quadrature points are used for H’ b and Ha ‘, N=30 [( 1 +x)(x+b,)/(x+L~,)~], b2= 1.3, b3= 1.45. VBR results are calculated using the formulas of the Appendix. Values of EeiscnVBI”e-EBccUIateare reported. All numbers are in cm-‘. DVR n

0 1 2 3 4 5 6 I 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 21 28 29

E MXUIXC 777.91 2 347.05 3 922.52 5 491.28 7 059.89 8 589.07 10 052.39 11 463.99 12 953.14 14 607.37 16413.02 18 343.82 20 384.71 22 527.91 24 769.11 27 109.00 29 546.62 32 084.84 34 726.94 37 476.13 40 338.14 43 314.87 46 410.23 49 626.97 52 967.3 1 56 432.99 60 025.36 63 745.43 67 593.95 71 571.49

H*

Hb

- 19.27 -26.25 -36.89 -54.73 -92.61 -216.93 -558.30 -743.02 -678.37 -608.14 -569.53 -550.53 -543.17 -544.34 -552.31 -569.60 -594.22 -988.41 -2137.79 -3317.22 -3161.00 -3762.54 -2199.03 -2883.30 -1291.51 -2981.15 3159.33 7047.39 8238.15 4827.15

H“ b H’

Hppa

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

-0.01 -0.01 0.23 0.48 -2.00 -3.08 42.88 80.54 -274.84 100.99 1006.60 1 541.59 -438.88 6 143.92 11 363.85 11 489.67 13 824.16 38 349.82

-0.01 -0.01 0.31 0.29 0.09 3.10 13.32 100.93 59.23 459.79 2 008.61 1 227.97 119.94 6 505.89 9 931.77 11 660.72 16 230.13 40 406.57

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

-0.01 0.27 0.43 2.30 -1.37 61.45 68.33 188.87 129.40 1664.92 1 761.23 1 258.86 5 024.40 12 861.61 12 430.49 13 242.35 32 566.95

0.01 0.27 0.48 2.98 6.54 62.83 99.24 302.48 553.83 1 797.06 2 494.57 3 295.40 6 556.88 13 517.32 16 464.97 14 919.24 35 342.21

0.01 0.27 0.46 3.88 6.46 64.96 92.78 365.76 540.41 852.76 388.58 176.39 381.00 659.85 076.24 190.49 121.48

(47)

Euclidean norms are the same for the DVR and the FBR because a unitary transformation does not affect the trace. We will calculate the E-Norm in the FBR. The E-norm of the difference of two matrices gives a reasonable measure of their similarity. If the E-norm of the difference of two matrix representations of a Hamiltonian is small it is likely that not only the eigenvalues but also the eigenvectors of the two matrices will be similar. It is important to have a measure of the quality of an approximate representation which reflects not only the accuracy of the eigenvalues but also the accuracy of the eigenvectors. The accuracy of eigenvectors determines how well intensities are calculated. It is easier and more compact to calculate the E-norm of the difference of two matrices than to compare the individual matrix elements or to calculate and compare their eigenvectors. In Tables IV and V we give E-norms of differences between VBR matrices and various approximate representations for several operators. For operator A we give representations A”,Ab,A’, Appa, Aab, Aa’, and Aao. A” is the representation obtained by invoking the product approximaJ. Chem.

Phys.,

N,=2

1

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

112

=[trace(AtA)]1’2.

N,=

Ha’

1 2 4 6 13 16 19 34

N,=

1

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.29 0.44 2.01 5.21 68.10 102.83 233.42 509.44 1 910.61 2 469.11 1 637.79 6 251.85 14 014.79 11 365.55 13 894.09 35 266.15

N,=2

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.28 0.45 3.02 5.61 65.66 95.53 310.37 512.04 1 869.12 2 419.67 3 137.84 6 200.95 13 867.20 15 579.12 13 760.86 34 041.73

VBR

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.02 0.27 0.44 4.5 1 7.19 66.15 90.77 408.04 566.42 1884.13 2 355.92 4 849.88 6 533.28 13 724.94 15 894.94 24 804.36 34 456.40

tion, using an N point quadrature approximation for matrices of coordinate functions and not using an adjoint derivitive operator. Ab is the representation obtained by invoking the product approximation, using an N point quadrature approximation for matrices of coordinate functions and when applicable using an explicitly Hermitian operator. APPais the representation obtained by invoking the product approximation but using exact matrices for each of the factors. A’ is the representation obtained by using the B matrix instead of a matrix representation of the derivative operator D, invoking the product approximation, using an N point quadrature approximation for matrices of coordinate functions and when applicable using an operator which is explicitly Hermitian. Aab, Aat, and Aao, are improved FBRs defined in Eq. (31). For operators without singular factors (those without an s in the table) the A” converge quickly to the exact (VBR) values as the number of additional quadrature points is increased. For operators with singular factors (those with an s in the table) the Aa appear to converge to the exact values but they do so more slowly (all matrix elements are finite). Clearly the higher the order of the singularity the more quadrature points are required to obtain convergence. Recall that (1 -~)-~(dldx) and (1 ---x)-~ have the same order singularity because a derivative introduces ( 1/2)a ( 1 -x) - ’

Vol. 101, No. 2, 15 July 1994



1352

TABLE IV. Euclidean norms of differences between approximate and exact (VBR) matrix representations in the Jacobi basis for some operators derivatives. The constant c = 1.3 and the basis size N = 10. The improved FBRs, A” b ,A” “,A” ’ are evaluated with N + N, quadrature points. IIA -

A

ljAVBRll

A”

AP”

Ab

A’

involving

AVBRll

N,=N

N,=(1/2)N

Jacobi basis parameters a = 5, b = 0

A”b d (1+x)(x+1.45) dx (x+ 1.3)2 d d - $1 - x?z

--

(c + x)-3z (c + xpdn (c + “‘-‘;i;

d ;i;

A”’

A”b

A“f

S

287.31

355.45

71.15

57.03

62.59

3.61

13.58

0.49

2.47

S

192.67

215.63

36.93

28.51

20.83

1.45

0.78

0.15

0.07

A”o

A0’

A”O

Aa’

d d d

1557.11

137.78

24.79

140.32

0.58

0.41

0.00

0.00

492.98

28.16

8.13

29.59

0.08

0.06

0.00

0.00

158.36

3.76

2.51

5.19

0.01

0.07

0.00

0.01

(1 -

x)-3;

6651.32

6837.91

6597.75

7049.86

4747.54

5007.69

(1 -

Q-z;

382.74

393.86

362.17

444.02

155.29

182.56

(1 -

x)-I&

51.26

27.35

23.44

40.07

4.01

6.51

0.68

1.12

21.97

2.50

2.50

8.33

0.00

0.31

0.00

0.03

(1 + x)$

2565.33 49.5 1

2719.21 58.59

(1 + X)(1

-

x)$

15.38

2.67

1.36

0.00

0.00

0.00

0.00

0.00

(1 + X)(1

-

x)$

18.09

4.39

1.63

2.65

0.00

0.00

0.00

0.00

(1 + X)(1

-

.&

26.30

7.35

2.12

5.77

0.00

0.00

0.00

0.00

AOb

Aa’

Aab

A”’

Jacobi basis parameters

d (1+x)(x+1.45) -dx (x+1.3)* -

21

-

x2$

u =

d z

18, b = 0

520.39

752.29

38.55

44.03

50.88

0.03

0.03

0.00

0.00

333.66

401.23

38.24

24.90

23.68

0.00

0.00

0.00

0.00

A””

A”’

A”0

A” ’

(c + p$

3117.19

187.32

21.03

189.66

0.33

0.32

0.00

0.00

(c + q2;

969.52

37.76

6.72

39.01

0.03

0.03

0.00

0.00

304.38

5.01

1.94

5.91

0.00

0.00

0.00

0.00

26.54

15.81

10.90

18.23

0.15

0.17

0.00

0.00

30.60

6.88

4.68

8.70

0.02

0.03

0.00

0.00

51.86

2.43

1.69

3.89

0.00

0.00

0.00

0.00

15.86

1.39

1.39

2.63

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

(c + x)-l& (1 -

x)-Sdx

(I

x)-z;

-

(1 -

x)-‘z

d

d

(1 + & (1 + X)(1

-

X)&

18.85

1.18

0.66

0.00

(1 + x)(1

-

&

26.01

3.80

0.87

2.7

(1 + x)(1

-

X)3&

39.75

7.91

1.42

6.75

and (l/Z)b( 1 +x)-l. Matrix elements (ml< 1 +x) x( 1 -~)~(dldx)In), k 2 I can be evaluated exactly using B instead of dldx; the matrix is a (2k+ 1)-band matrix (i.e.,

1

a banded matrix with k subdiagonals) ; its matrix elements are exact if calculated with a N,= N, + N point quadrature where N,= (k- 1)/2 if k is odd and k/2 if k is even. Matrix





1353

TABLE V. Euclidean norms of differences between approximate and exact (VBR) matrix representations in the Jacobi basis for some functions of the coordinate. The constant c = 1.3 and the basis size N= IO. The FBRs, AFBR are evaluated with N quadrature points and the improved FBRs, An are evaluated with N + N, quadrature points. Jacobi basis parameters

a = 5, b = 0

Jacobi basis parameters

I/A-A~~~//

a= 18, b=O

IIA-AVBRII A”

A

IIAVBRll

AmR

x)? x)3

344.89 39.52 6.71 4.24 1.23 13.09

278.03 22.58 1.67 0.00 0.23 0.74

85.11 2.86 0.06 0.00 0.00 0.00

(C + x)-3 (c + xl-? (c + x)-l

38.63 13.27 5.16

6.79 1.72 0.34

19.35 8.75 8.39 17.20 92.46 218.15 517.82 1235.12

(I

- x)-3

(1 (1 1 -x (1 (1 -

xj-? I)-’

S S s

(1 (1

-

x)-%-4=

(1 (1 (1 (1

-

x)-‘e-” x)-G-” X)-2@ x)-?e’X

s

(I (I

-

x)-‘e3’ x)-2e4*

S

x)-‘ee4’ x)-‘em3* x)-‘em2* x)-‘e-* x)-‘ex x)-‘e” x)-‘e-3’ x)-‘e4*

S

(1 (1 (1 (1 (1 (1 (1 (1 -

X)-2e-3x

S S S S S S

S S S S S S S

e-4’

e-3* e-2*

e-* ex ezx e3r e4’

llAVBRll

ABR

25.40 0.47 0.01 0.00 0.00 0.00

4.61 3.12 2.66 4.86 8.40 15.29

1.43 0.54 0.15 OMI 0.15 0.56

0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00

0.03 0.00 0.00

0.00 0.00 0.00

45.69 15.60 5.94

6.39 1.59 0.30

0.01 0.00 0.00

0.00 0.00 0.00

1.96 1.59 3.54 8.92 56.93 142.86 357.31 89 1.59

0.06 0.15 0.40 1.07 7.60 20.20 53.66 142.47

0.01 0.02 0.07 0.18 1.27 3.40 9.13 24.54

22.58 9.40 4.43 2.96 4.12 5.88 8.70 13.17

1.54 0.44 0.18 0.25 1.07 2.02 3.66 6.48

0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.02

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

36.71 14.90 6.56 4.37 14.03 31.11 70.50 161.78

3.86 1.19 0.43 0.63 4.49 11.82 30.50 77.69

0.00 0.00 0.01 0.02 0.17 0.46 1.26 3.41

0.00 0.00 0.00 0.00

3.43 1.00 0.24 0.08 0.37 0.79 1.54 2.84

0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00

0.11 0.29

43.09 17.41 7.43 3.70 2.86 3.72 5.23 7.63

70.56 28.10 11.58 5.20 3.67 6.49 13.08 27.80

8.30 2.64 0.74 0.13 0.12 0.62 2.05 5.88

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

82.93 32.95 13.52 5.96 2.43 2.63 3.39 4.69

7.48 2.31 0.6 1 0.10 0.06 0.24 0.58 1.17

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

N,=(l12)N

elements of operators without singular factors are better computed as A’ than A6 because of the tridiagonal form of B. Replacing dldx with (1 -x2)-‘B increases the order of the singularity and the quadrature error (particularly in the bottom right-hand comer of the FBR matrix). In Tables IV and V we present results for two different sets of Jacobi parameters to show that by varying the parameters one may improve the accuracy of the Jacobi DVRs and FBRs. Increasing the value of the parameter a reduces the effect of the singularities of some operators in Table IV because of the factor ( 1 -x)” in the Jacobi weight function. VI. APPLICATION

A”

TO 6

Pekeris coordinates have been discussed by several

authors.20’Pekeris 2’ coordinatesenableone to take full ad-

N,,=N

0.01 0.04

N,=(

112)N

N,=N

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

vantage of the symmetry of a triatomic molecule of three identical atoms. Hyperspherical coordinates33-37and intemuclear coordinates’4-16 also permit exploiting the D3h symmetry and have been used to calculate vibrational energy levels of H: . 14,‘6734-37 Calculations with internuclear coordinates are hampered by the interdependent range problem. The hyperspherical kinetic energy operator is singular at the equilateral triangle (equilibrium) configuration of H: and one must therefore be careful to choose basis functions so that kinetic energy matrix elements are always finite. Although “hyperspherical harmonics” (eigenfunctions of the angular part of the hyperspherical kinetic energy operator) would be optimal basis functions the singularity may be dealt with (albeit not as efficiently) without coping with nondirect product bases.37738 Using three sets of Jacobi coordinates it is also possibleto take advantageof the D3A symmetry.39’Many 40



1354

TABLE VI. Comparison of the levels of Hl calculated in Ref. 40 and in this work with the Pekeris Hamiltonian Eq. (2). The DVR matrix is set up as in Eq. (23) but using B(j) rather than Do). The Jacobi basis parameters are a=25, b=O, rmax= 3.3 bohr. The 1D basis size is 23. DVR points for which the potential is more than 90 000 cm-’ are excluded from the 3D basis so that the 3D basis size. is N. I&,,, values are taken from Ref. 38. Values of E-I&,, and E,,,, are reported in cm -I. Note that HTS use N*=8550 and 3850 basis functions for A, and even E states or A, and odd E states.

la

E ACcUrak A, block

17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38

13 14 14 14 14 15 15 15 15 16 16 16 17 17 17 17 17 17 18 18 18 18

Ref. 40 N*=8550

705.34 185.45 662.61 885.57 938.62 061.13 157.50 867.41 908.66 194.54 442.96 693.85 060.37 272.17 427.83 585.09 679.92 744.25 225.46 358.33 453.70 582.31

0.1 0.0 0.0 0.2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.1 0.0 0.1 0.0 0.0

A, block 5 7 9 10 11 12 13 14 15 16 17

13 15 15 16 17 17 17 17 18 18 19

0.0 0.0 0.0 0.0

N*

E block 21 29 35 39 42 44 45 46 47 48 50

13 15 16 16 17 17 17 17 18 18 18

680.88 324.39 545.30 909.03 439.61 690.63 847.80 955.82 208.49 344.92 566.81

7.1 8.5 15.9 11.1 2.3 5.6 7.1 8.7 11.0 11.8 27.0 11.0 22.4 16.7 10.9 9.3 8.7 29.0 12.2 15.7 21.4 29.7

N*=8550

746.67 178.09 950.78 577.91 076.74 670.36 805.30 845.14 315.97 865.54 173.75

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

calculations of H: energy levels employ a single set of Jacobi coordinates, which does not allow exploitation of the full D3h symmetry. 38V41-44 For H: the Pekeris Hamiltonian is appealing because of its explicit D3h symmetry and because its kinetic energy is singular only when nuclei coalesce (i.e.,

even

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.0 0.0 0.0

0.1 0.1 0.1 0.5 0.2 0.2 0.3 0.4 0.4 0.2 0.2 0.1 0.3 0.2 0.7 0.4 1.0 0.2 0.6 0.9 0.4 0.5

N*=3850 0.3 1.0 0.9 3.5 1.6 2.5 1.9 1.4 2.0 3.9 1.6 N* =

0.1 0.0 0.0 0.1 0.0 0.1 0.0 = 8550

odd

This work N=1584

N*=3850

odd 0.1 1.3 3.4 2.0 2.2 0.9 5.9 2.2 0.6 1.1 2.4

N= 1234 0.1 0.1 0.2 0.2 0.2 0.2 0.3 0.3 0.4 0.4 0.5

3850 even 8.2 22.1 13.3 25.3 16.7 26.7 31.2 31.4 25.9 12.0 28.7

N=2803 0.1 0.2 0.2 0.2 0.2 0.7 0.2 0.3 0.5 1.1 0.5

when two of r, , r2, and r3 are zero): the potential is sure to be very high close to the singularities. If in addition products of functions of the three Pekeris coordinates were good basis functions the Pekeris Hamiltonian would be excellent for Hl . At very high energies it is likely fruitless to look for a




coordinate system which decouples motions of a floppy molecule like Hl . At low energies, however, it is possible to expect to find coordinates whose associated basis functions facilitate calculation of vibrational energy levels. We use basis functions which are products of functions of a single variable. Because vibrational wave functions of Hl do not vanish at rj=O (at linearity) convergence is greatly accelerated by using basis functions which are not zero at rj=O. Morse, tridiagonal Morse, or harmonic basis functions are inappropriate. We have experimented with spherical oscillator4 and Jacobi functions25-28 and find Jacobi basis functions to be better. To devise a DVR of the Pekeris Hamiltonian we use an explicitly Hermitian form. Doing so enables us to obtain, despite the complexity of the kinetic energy operator, a DVR Hamiltonian matrix without quadratures. As basis functions we use linear combinations of DVR functions which transform like irreducible representations of the D3h point group.7*‘4 We used the same method to construct symmetry basis functions for H20.‘1’45 We transform from the unsymmetrized to the symmetrized basis as do Spirko et ~1.‘~ but we take combinations of DVR functions for each rj rather than combinations of delocalized basis functions for each bond length Rj . To construct symmetry adapted basis functions we fix the number of 1D basis functions and obtain about twice as many E basis functions as Al or A2 basis functions. This is reasonable as there are many more E states than A or A2 states.38’42Changing the parameters of our Jacobi basis functions enables us to choose basis functions with amplitude in physically important regions. In Table VI we present energy levels of H3f. We use an H’ DVR Hamiltonian. It is set up as Hb of Eq. (23) but using B(i) and & rather than D(j) and Gjk, where ~jL = ( 1 -x,‘) - ’Gjk( 1 -xi) - ’. We find that the H’ eigenvalues are more accurate than the Hb eigenvalues because of cancellation of the finite basis size error and the quadrature (DVR) error. Our results are compared with the best available calculation”8 on the same MBB potential energy surface.41’42 We use the same potential and the same mass conversion factor as Henderson, Tennyson, and Sutcliffe (HTS) so that we may compare with their results. To assess the quality of our basis we also compare our results with those obtained with similar size matrices by HTS (they also used much bigger matrices to better converge their results).42 Although our basis is somewhat smaller than the I-ITS calculation with which we compare, our energy levels are better converged. For the purpose of reducing the size of the Hamiltonian matrix and the eigenvectors the Pekeris Hamiltonian is clearly useful. By increasing the dimension of the matrices the accuracy of the Pekeris method could be made to rival that of a Jacobi coordinate calculation.38*41-44The Pekeris Hamiltonian has the important advantage that one obtains rigorous symmetry labels with the energy levels. VII. CONCLUSION The original FBR was defined as the representation obtained by taking exact analytic kinetic energy matrix ele-


1355

ments and computing potential matrix elements with Gauss quadrature. For a polynomial basis the Gauss quadrature approximation is equivalent to assuming that the matrix representation of a product of coordinate operators is equal to a product of matrix representations of the coordinate operators. To define the FBR of a Hamiltonian whose kinetic energy operator is complicated enough that its matrix elements cannot be calculated analytically one has two choices: (i) evaluate matrix elements of products of derivatives and coordinate functions by quadrature,6 or (ii) replace matrix representations of products of derivatives and coordinate functions by products of matrix representations, matrix elements of coordinate functions being evaluated with Gauss quadrature. In either case a DVR is related to the FBR by a unitary transformation. Choice (ii) has the important advantage that the DVR may be constructed directly from the transformation matrix T, matrix representations of derivatives and values of coordinate functions at DVR points without evaluating quadratures, because matrix representations of coordinate functions are diagonal in the DVR. If matrix representations of derivatives are anti-Hermitian the choice (ii) procedure is straightforward and the error introduced in the lower energy levels by making the product approximations disappears as the basis size is increased. Primitive basis functions are usually chosen as products of functions of a single coordinate. If, for a unit weightfactor Hamiltonian, single coordinate functions are not zero at the boundaries the product approximation of choice (ii) spoils the Hermiticity of H because matrix representations of derivatives are not anti-Hermitian. In this paper we have shown that even in this case if the kinetic energy operator is written in an explicitly Hermitian form one may use product approximations to define an Hermitian FBR matrix for arbitrary kinetic energy operators so that the corresponding Hermitian DVR matrix may be obtained without quadratures. Our numerical results for 1D test problems and for H: confirm that DVR matrices for complicated kinetic energy operators set up according to our prescription are accurate and useful. It is easy to set up a DVR Hamiltonian matrix using the explicitly Hermitian form of a unit weight factor Hamiltonian: It is only necessary to transform first derivative matrices to the DVR. Avoiding quadrature greatly simplifies the calculation of DVR Hamiltonian matrices.

ACKNOWLEDGMENTS We thank Matthew J. Bramley for drawing our attention to Ref. 21 in which Pekeris coordinates are discussed, for providing Hl potential subroutines, and for suggesting changes which improved the manuscript. H. W. also thanks Mark Casida, Pierre-Nicholas Roy, Matthew J. Bramley, and Andre McNichols for help with computers. This work has been supported by the Natural Science and Engineering Research Council (Canada) and the Network of Centres of Excellence in Molecular and Inter-facial Dynamics, one of the fifteen Networks of Centres of Excellence funded by the Government of Canada.


1356


APPENDIX: MATRIX ELEMENT FORMULAE IN THE JACOBI BASIS

then using the relation Jib( -x) = ( - 1 )“Ji’(x) tain

The Jacobi basis is composed of normalized weighted Jacobi polynomials (xln)=J;b(x)=h;J;2( xe[-1,

W)

I]

(Al)

y+b+l

E nab a+b+2n+

1

r(a+n+l)r(b+n+l) n!r(a+b+n+

1)

’

Therefore we may always obtain matrix elements with the factor ( 1 +x) from matrix elements with the factor ( 1 -x) and we derive only the latter. For derivatives

642) (J;bI-+lJ:b)=(-

which satisfy (m/n)=&,,

a,b>-I.

643)

In this paper we have used Jacobi bases. Jacobi basis functions have also been used in other variational calculations.25~26The two parameters a and b make the Jacobi basis flexible.27P28For a = b=O Jacobi basis functions become Legendre basis functions, for a = b = l/2 they become Chebyshev basis functions of the second kind, for a = b = cr- ( l/2) they become Gegenbauer basis functions.” Jacobi basis functions may be chosen to be zero at one boundary and nonzero at the other or to be zero at both boundaries. The parameters a and b may be varied so that they may be used to calculate wave functions which are zero or nonzero at a boundary. The formulae in this Appendix are derived from the basic properties discussed in Refs. 27 and 28. They may be used for a general domain [Ri ,Rf] by shifting and scaling: R=Ri+(1/2)(R,-R,)(X+l) where x E [-1, 11. All the formulae are written so that they are easy to program. The FORTRAN subroutines may be obtained by request. Throughout this Appendix k is a natural number, j is an integer and X and p are real numbers. We define e = 1 for a=0 but 0 for a#O, g= I- t$, (z),=r(z+A)/I’(z) , z(z-i- 1). . .(z+n-(z),E

lF i

l),

1p+n+l ( J;“i&iJ:o)

1. Matrix elements for functions

of the coordinate

In the absence of a statement to the contrary we take man and a+s,b>-1 for the matrix of (l-~)~ in this section. We have derived the following matrix elements:

(AlO)

(mlxln)=d,S,,+q,6,+,,,+q,S,,,+1, 0

if a=b b*-a2

d,=

tU+b+2n+

i

1)2-

ofiemise;

1

n==O,

(All)

4n

( n(4n2-

if a=b=O

1)-l’*

n(a+n)(b+n)(a+b+n) (a+b+2n)*1 nZ=l;

n>O

1 “*

otherwise; (AM

n=” 1

ll[(z+n)~~~(z-2)(~-1)1=~~+~)_,’

C mnab=J(a+bi-2m+

n,-, (a+b+n+

l),-,

(b+n+ (a+n+

l),-, l),-,

6413)

(m+l)(a+b+m) a-1

- n(a+b+n+l) a+1

1 I’*

I’

(A14)

’ m2n’ bw

ma- nbwe write crb then n>m, c,b-c,,-a VtO1)(2n+ 1)]“2, cri C?lO - cm00 =[(2m+ mO_ = [(n+ l),-,l(a+n+ 1),-J . We CmnOb and Cm -Cmmo also use the binomial coefficient (,“) and other symbols m d,, q,, , B,,, and D,,, etc. defined as they appear. c,,b, If one knows

Note that if +b+%n+ 1,

lrnl,l

’ -x)

kb)

C ;:

=zkb+ X

dj dk (mldxif(x);i;aln)‘F(m,n,a,b),

,

645)

and

mb,

649)

so that one may obtain the matrix elements for a = 0, b # 0 from the matrix elements for a # 0, b = 0.

644)

C na

l)m+“+i+kF(m,n,b,a).

1 -x)=‘2( 1 +X)b’*P;b(X),

with h

(ml-$f(-x)-$ln)=(-

one can ob-

ey”’ 1),-n

(_

l )‘i

:‘:i

I=o

(a+b+n+l)l(k-~),(a+b+m+2-k+l)k-I-l i!(a-k+l+ l),,+k-l

L47)

, (AW




(ml( 1 -~)~ln)=2~n!c~~ x( a+btn+l)(

,zm$.,

(- l)y

r(a+A+l+ ‘I’(a+b+A+l+m+2)

2)

=2Acmb(-i)m na m!

l,k)

6416)

@I( 1 -xPIR) (-1)l-m(

R!r-ga++bn+/;“;:

(A17)

rta+ 1 +‘I T(a+l)

I’(a+b+m+ Xr(a+b+m+2+A)

1)

+l,a+b+nCl;

Xi-l-m,a

nA+ l) r(h+l-m)

4Fd-n¶X+

‘,‘+’

L418)

where 4F3 ( a 1, LY*,CY~,a4 ; p t , /I2 ,p3 ;x) is the generalized hypergeometric function. When A = t k Eq. (A17) reduces to Equations (A16) and (A15). For matrix elements of (c+x)-k we find

mi”(k-“n) (a+b+n+ l)l(k-l),pa+lb+l n-l* 2’1!( 1 +c)k-l+m

2 ’ klR)=c:d,l=O

(-c)~F,

=c&~~~“‘~“” l=O

(a:il,~,._+::~~~~+~z)m~~~~,b+~(-~)2~~(

-cx)

1)

+l,a+b+m+2+X;l),

1’;)

l=O

(“I,,

1357


i

k-l+m,a+m+

l;a+b+2m+2;&

k-l+m,b+m+

l;a+b+2m+2;$--),

a,b>-l;lcj>l,

(A19)

where *F, (a,P; y;z) is the Gaussian hypergeometric function and m

c,,b=

(a+n+ l),-,(b+n+ [ (n+ I),-,(a+b+n+l),-,

( - 2)mcrnnab

l),-,

“* I

1 (a+b+m+l),+l

W-9

*

We find that the matrix representation of ( 1 -x)~ is a 2 k + 1 -band matrix, each matrix element being a sum of at most k + 1 terms and that the matrices of ( 1 -x)-~ and (c + X) -k are full, matrix elements being a sum of at most k terms. Because of the fast convergence of stretching potentials in Morse variables,14V34many potential functions have been fitted using them.17-‘9 The following expressions are useful for these potentials: (mI(

1 -X)he-Xf

'('+j+'+

n I

r(A+j+Z+

,=(

f’(a+A+j+l+l) XT(a+b+A+j+Z+2+m)

'1

1 -m)

w’ I!

L421)

with 6422) For an integer A =j we have E’,,(t)=(ml(

1 -x)‘,-X’ln)=~nb~j~

qnjJ(j+I,m,a,b,t),

(~23)

l=O where Cm&( - 1 )m2i, -’ “~=(a+b+m+l)j+t(a+j+l),-j (a q~j~'P*12'(a+b+m+~+j)l

+

1 +j)l '

(b+n+ l),-, (n+l),-,(a+n+l),-,(a+b+n+l),-,

112

1

’

(~24) 6425)


1358


c E(J,m,a,b,t)=g

f(l,J,m,a,b,t),

JaO

f(l,J,m,a,b,t)=

6426)

I=0

JO, D is antisymmetric, D,,=

(A30)

mO, b=O, D is not antisymmetric, D,,=

-i(a+2n+

1) # 0.

We find $&,,Ob(ml( 1 +x)-&n)=

11,

m=n;

pmn= - $#c~~,

m>n

$ &mnOb- Pnm 7 (m/(1 +x)(1 -x)*%In)=

a,baO (A31)

m 1.

(~32)

j=max(O,n-I)

We see that Eq. (A32) is a (2k+ 1 )-band matrix. We find

(4 &&In)=

(A33)

where man.

(A34)

We find 1

(“I(,-x)‘dx

PLV

m>n,a>k,bZ-0

-+n)= ~(-l)m-n’12k~1cm~~O-(m~(l-f)P+i~~)-P~m~

mi”W1’n-‘) _ c m n I=0

Pr”=&n+2k+y:9,,

(- l)‘(k-Z),(a+b+n+ I! (n-l-l)!

(A35)

m

man,

(A36)

1 tin=

-G

lrnl(

1 -X)k+

dn)+

8$-dmlj&+$,

(fIi)lh).

(A37)





1359

Another simpler expression for Eq. (A35) with more terms in the sum is a>k,baO,

(‘438)

where ~n-l (T;-I=h,b12h1,2a+b+n+1 Ul lab 2

X[(b+l+

1),-1+(-

1 (a+b+2n+ l)(a+b+21+ 1)(1+ I),-[ =? (a+/+ l),-l(b+l+l),-l(a+b+I+l),-l

l)n-l-‘(a+I+

a+b+21+ 1 b;=(a+b+l+ l),-l+2Nb+l+

1

1’2

1),-J,

6439)

+(-l)n-‘(a+l+l)n+l-J,

1),+,-l

6440)

&y is the expansion coefficient in p:+‘-b+l(-&

qp;JyX)

6441)

l=O and n-1 12,b’“(

1 -Xy2(

1 ““‘(c+x)~

1 +X&

P$(X),

dwv d, )= dx ’

c Cpll). l=O

(~42)

man (-443)

g&c+

I)+-

‘$(-

l)“-“(c-l)-kl

c,..,+(mi,,+~,,,ll~)-

Icl>l;

17:,,= &,+c;ab

min(k-*‘n-l) (a+b+n+l)l+l(k-Z), T c

d&u

mO, S,=-q,lq,+l for n> 1 we obtain the matrix element recurrence relation Ejm+,,n=SX,Eimfnl+S,Ei,,+S,Ej,_,,,

WO)

and a similar formula for E’, n+ L . We recur from Ercn+j ’ Eiii and then to EL,.

to

Method II. m+n c pynEjO;‘(t),

Ejm,(r)=C;\

(A=)

l=O

where C&= h,lL2h,-b,/2hOab and pi”” is the coefficient in P$(x) Pib(X) = x.im=+gnp~n( 1 -x)‘. Method III. Ez(r)=(ml(

1 -x)*(

1 +x)Pe-x’ln)

m+n

= C;=&2

q~“E&+“-l+A>

‘+p(r),

l=O

X+a,,x+b>-

1,

6452)

in is the coefficient where 4;ln P$(x)P;b(x) =Cy$q;““( 1 -X)m+n-l( 1 fx)‘. The accuracy of Methods I, II, and III is ruined by round-off error for large m and n. We have detected (using 64 bits) errors which are as large as 7% or 2% of the largest matrix element for methods I and II respectively even if m,n ~9. Method III works for m,nG 19, much better than Methods I or II, and Eq. (A23) gives very accurate results up to m,n=G49. The round-off error is caused by near cancellation of many positive and negative terms in the summations. In Method I the recurrent multiplier SX,=4 and the initial value OfE;jbfnfj (,2m+n+j) is very large so that after m + n iterations a term may increase to about grn+” but the sum of terms cancels to a much smaller number. The coefficients of the ( 1 -x)’ Jacobi polynomial expansion are much larger than the corresponding coefficients of the ( 1 -x)“-‘( 1 +x)’ expansion and amplify the round-off error much more. It is possible to estimate the round-off error of a quantity such as a = Zlal where the sign of al alternates by examining (A53)

v(a)=M=(bdVbl. In our case, taking t= - 1, j=2 ~(E~,)~O.lOX

77(Ei,)G0.21

we find

lo3 for Eq. (A23),

0.56X 1015 for

Method III,

X lo4

m,n