Cubic harmonic expansions using gauss integration formulas

Solid State Communications, Vol. 19, pp. 83—86, 1976.

Pergamon Press.

Printed in Great Britain

CUBIC HARMONIC EXPANSIONS USING GAUSS INTEGRATION FORMULAS* W.R. Fehiner, S.B. Nickersont and S.H. Vosko Department of Physics and Erindale College, University of Toronto, Toronto, Canada (Received 20 January 1976 by A.A. Maradudin) We present several sets of special directions for integration over solid angle of functions with full cubic symmetry which represent a major advance in precision and thus can be applied to problems which are beyond the scope of present techniques. For example, it is possible to determine the first eight moments of a Fermi surface accurately from measurements along only ten special directions. THERE ARE MANY PROBLEMS in solid state physics which are simplified by expressing a physical variable as an expansion in the appropriate symmetry adapted basis functions. For example, the inversion of de Haas—van Alphen data to obtain the Fermi surface13 and the solution of integral equations4 arising in a study of

weights A to minimize the error functional for as wide a class of functions as possible for a given value ofN. In one dimension, the Gauss procedure for solving this problem is well known.6 The Legendre polynomials, Pm(X), form a complete set of orthogonal basis functions over the interval [— 1, 1] and x 1 A are fixed by requiring RN [Pm] = 0 for as many of the basis functions as possible. This condition generates a set of coupled, non-linear equations in x, A, ,

superconductivity on such expansions. Recently, 5 has used andepend expansion of the radial moment of Bansil the spectral density to determine the electronic spectra of disordered alloys. He has pointed out that the spectrum depends only on the leading term of the expansion, and this leading term can be evaluated very efficiently by the use of special directions. However, he has obtained these special directions under certain restrictions which severely limit the accuracy of the method. By using a more general procedure for determining the special directions, we have significantly improved the accuracy of the method, so that now one can obtain accurately many coefficients in an expansion. The special directions we determine are applicable to a calculation of electronic spectra but, moreover, it is possible to apply the special directions presented here to a wider scope of.problems. After discussing the general procedure for obtaining the special directions, we will illustrate their usefulness with some applications to Fermi surface problems. One can replace an integral over a function of D

dxf(x)

N

=

~ A1f(x,) + RN

f

AjPm(Xj)

~

=

0 ~ m ~M

8m.o

—

1

(2)

1=1

where Mis the precision of the formula. We now outline the Gauss solution of equation (2) in a form which is readily generalized to many dimensions. Firstly, one determines the N points x, as the N roots of the Legendre polynomial FN. Once the x, are fixed, the first N non-linear equations become linear equations for the unknown coefficients, A,. The orthogonality of PN to FL for L
=

~

b

l1N-LJ

independent variables (represented by the vector x) by a summation of the form

$

,

N

1 (LN)P1(x)

(3)

where L = 1,2,. ,N— 1 and b1(LN) are constants, some of which may vanish. When summed .over x, with the weight factors A,, equation (3) provides N — 1 linear relations among the left-hand sides of equation(2), . .

(1)

where the error RNE f J is a functional of the integrand. The numerical problem is to choose the points x, and

sufficient to prove that the formula also integrates F1, N + 1 ~ 1 ~ 2N — 1. Finally, it is important to note that this construction fails for ~2N because

Supported in part by the National Research Council of t Canada. Based in part on work submitted by S.B. Nickerson in his Ph.D thesis.

~ N

*

0 83

=

$

2.

1

A1P~(X~)r -1 dx[PN(x)]

(4)

84

EXPANSIONS USING GAUSS INTEGRATION FORMULAS

In more than one dimension, the non-linear equations take the form N

~

A1U1(x1)

=

6~,

1

~j

~M

(5)

1=1

Vol. 19, No. I

Table 1. N-direction Gauss formulas for integrands with full cubic symmetry. M is the precision of the formula [see equation (2)] and ‘m~~cis the largest value of 1 for which all cubic harmonics are integrated exactly. Each special dfrection intersects the x’ = 1 planeaty’, z’; A is the weight

where U are the orthogonal basis functions over the region of interest. To obtain a formula for the twodimensional problem of integrations over solid angle of integrands with full cubic symmetry, Bansil attempted to solve the set of non-linear equations, equation (5) directly. However, Bansil the has weights, only given restricted problem where A solutions for a 1, are arbitrarily set equal to unity. This reduces the number of free parameters, and consequently the maximum precision to M~2N, which is less than the full power of the Gauss method. It should be pointed out that even after fixing the weights, it is a complicated nonlinear problem to find the directions, which becomes increasingly difficult for large N. Bansil presents formulas up toN= 5 withM~2N. He also obtains a 13-direction formula withM=N. The symmetry adapted basis functions in this situation are the fully symmetric cubic harmonics harmonics. as 1 express the cubic Muellercombinations and Priestly of spherical harmonics with the linear appropriate symmetry. This has the advantage of emphasizing that cubic harmonics are eigenfunctions of total angular momentum. In the remainder of this communication, we shall denote cubic harmonics by K 1, r where I is the angular momentum quantum number and r is a letter to distinguish cubic harmonics with the same total angular momentum. We order the cubic harmonics by their I-value with an arbitrary order over r. We have chosen to solve equation (4) by an extension of the Gauss solution in one dimension. ForD dimensions, in order to find isolated points x, we must determineNcommon 7roots of at points outleastD that itindependent is a nontrivial basis functions. matter to chooseStroud the best set of basis functions from which to obtain these common roots. For example, he shows that for D> 1 there is no guarantee that the common roots will all lie inside the domain of integra-

A

the coefficients, A

tion. Furthermore, the choice of the set determines how many of the Clebsch —Gordan series generate useful linear relations needed to extend the precision. Once N common roots have been found, the unknown coefficients A are determined by N linear equations as in one dimension. In Table I, we give the coordinates and weights for the 2-, 3-, 4-, 6- and 10-direction Gauss formulas for integrations over solid angle of integrands with full cubic symmetry. For convenience, we have renormalized

1, so that their sum is unity. Consequently, one must include a factor of 4ir on the righthand side of equation (1) when applying the formulas. Special directions in only a single irreducible segment of the unit sphere are listed. The coordinates are presented in the same notation used by Bansil; y’ and z’ are the intersection of the special direction with the (1 00) plane (i.e. x’ = 1). In Table 1, we also give the precision, M, of the new formulas and the maximum value of I for which a formula is precise. For1max compari= 20, son, the 13-direction formula of Bansil has

Z

N= 2,M= 5, l~,, 10 —

.345952933492 .0325470 10427 .414359813955 .780869482319 1m~~ .378238200018 = 16

.585640186045

N = 4, M = 11 .215760202206 .469573515349 .798449695976 .789950789387 N = 6, M = 15, ~

.016146179320 .221547893197 .1373853 14201 .55 1854815889 = 20

.172118837046 .350170232970 .209 108806480 .268602123505

.872176906981 .754424146006 .745849728168 .460479219778 .771725648588 .155154412714 .402980772552 .384449729798 .419045571256 .130432989168 .122522845843 .135317862711 N = 3, M = 8, ‘max 14

.072241661502 .2 16972935067 .236655092077 .115386393643 .227574607724 .131169309987

.199807337241 .198019617799 .283794324156 .709189408210 .584904897309 .250368066248 .666382648568 .19039 1670235 .465837609596 N

=

10, M = 17, ~

.843 179401584 .8 19729565563 .540340425599 .820843088776 .8206272533 19 .534443169361 .534496438814 .3058 17055924 .303308021945 .099641853625

22

.765089399536 521629445205 .529048436243 .298540198030 .0972866867 10 .303322616520 .098469866171 .300949727575 .0985 15476092 .097309393106

.063758429642 .110336568355 .062095962949 .112470423401 .112438481626 .127716882693 .129673198567 .068174692157 .140415649736 .072919710874

EXPANSIONS USING GAUSS INTEGRATION FORMULAS

Vol. 19, No. 1

comparable to the six-direction Gauss formula in Table 1. Results given below show that the errors generated by the ten-direction formula are very small for ‘m~~ To illustrate new applications, we consider the cornparison of de Haas—van Alphen data with computed band structures.13 We restrict our attention to the measurement of extremal cross-sections for closed Fermi surface sheets with full cubic symmetry about the origin and with a single valued radius vector. Under these conditions, one first calculates the coefficients of a finite cubic harmonic expansion of the radius vector squared 1>

k2(&l)

=

~

7irKi,r(fl)

l,r

eight-term cubic harmonic series (i.e., terminating with = 14), but not by a series truncated after = 10. The Fermi energy is the solution of the equation I

1

EF

J dfZ J

k2dko

~ Jd~2Ki,r(1l)k2(~2)

1

2(&2)

k

=

~

(10)

k~

Once EF is determined, we obtain the coefficients in a cubic harmonic expansion of the radius vector

13i,rKi,r(&2)

~

=

(11)

birKir(~2),

(7)

since this requires only solving the eigenvalue problem along a set of special directions if one replaces equation (7) by the appropriate Gauss formula. The use of projection integrals has the added advantage that the value of 71, r for small is independent of the upper limit in equation (6) if a Gauss formula with sufficient precision is used. This is not the case for a standard least-squares fit to a finite arbitrary sample. On the other hand, we know from the theory of orthogonal basis sets that using projection integrals to determine ‘y r gives the unit least averaged over1 the entire square error in sphere. Once the y’s are determined, one can determine the coefficients in an expansion of the cross-sectional area A(&2)

=

EF.

kF(~2)

=

— E(k)]

[EF

where E(k) is the energy dispersion determined by the pseudopotential and 0 [x] is the usual step function. Equation (10) is simply the statement that the volume enclosed by the Fermi surface must equal the volume enclosed by the free electron sphere. The integral over the magnitude of k is done analytically; the integral over (6) solid angle is replaced by an N-direction Gauss formula and equation (10) is solved iteratively for

where ~1is the solid angle. The most efficient method to determine the ‘y’s is by means of projection integrals over the surface of the sphere 7i,r

85

(8)

l,r

by using projection integrals, once again evaluating the integrals by an N-direction formula. The exact coefficients through = 14 are given in Table 2. We also list in Table 2 the fractional deviation in percent of the calculated coefficient from the exact result for N = 6, 10. 1

= l00(bL’~7— where is the coefficient obtained with an Ndirection Gauss formula. In addition, we list ~$‘,?~ bi,r)Ibi.r

(12)

b~

obtained with Bansil’s 13-direction formula. Table 2. Expansion coefficients of kF(fl). ~(N) is the % fractional deviation of calculated coefficient from exact result [see equation (12)]. N = 6, 10 are Gauss formulas. N = 13 is Bansil ‘s formula Ir ~(6) ~(1o) ~(13) bir

_____________________________________________________________

0 4 6 8

with the help of an inversion relation derived by Mueller.8 For cubic symmetry this relation takes the form

— —

2.2656 0.0558 0.0397 0.0132

0.0 —0.1 1.4 5.7

0.0 0.1 0.3 0.9

0.0 —0.3 1.5 —20.0

0.0032

—1.4

~6.0

—33.3

0.0035 0.0036 0.0021

59.8 —92.1 102.1

—

!~l.r= 7 1,r1T1’i(O)

(9)

where the F, are Legendre polynomials. The efficiency of this approach is demonstrated by a model calculation based on a pseudopotential for a bc.c. lattice with energy- and k-independent matrix elements. The matrix elements have the values Vr~~® = 0.68 Ryd and V110 = 0.20 Ryd, with all the other matrix elements vanishing. The radius of the free electron sphere, k0 =.0.639719 in atomic units which corresponds to r8 = 3.0. The resulting Fermi surface is a distorted sphere with full cubic symmetry about the origin. The Fermi surface is well represented by an —

—

10 12A 12B 14

— —

—

—

9.3 —4.1 12.6

—

1379.4 179.4 3152.1

It is clear from Table 2 that all three formulas reproduce the leading coefficient. In fact for this Fermi surface, we find that even the two-direction Gauss formula is accurate for the leading term. It is also clear that the largest i-value for which a given formula is reliable involves more than the precision of the formula. The six-direction Gauss formula has the same precision as

86

EXPANSIONS USING GAUSS INTEGRATION FORMULAS

Bansil’s 13-direction formula, but only the former gives acceptable results for b10. Moreover, errors increase more slowly for larger i-values than with the 13-direction formula. The 10-direction Gauss formula also performs well beyond its precision point. It gives good results for b14 and lower coefficients. For practical calculations, the 10-direction formula is nearly precise through l = 28. In conclusion, the integration formulas presented here are a major advance in precision and thus can be applied to problems that are beyond the scope of present techniques. As Bansil has pointed out, although traditionally directions of high symmetry have been emphasized in the physics of solids, calculations as well as measurements along special directions are more useful, because these directions represent more accurately the average properties of the solids. Our results go even further and show that information

Vol. 19, No. 1

gathered along a set of special directions presented in this paper can determine sufficient moments of a physical quantity to reproduce its anisotropic behaviour to any desired degree of accuracy. Thus measurements along only a few special directions (e.g. by positron annihilation or of Kohn anomalies) would determine the entire Fermi surface to a high degree of precision. The results presented in this paper are only a first application of these new formulas. For physical quantities where the expansion cannot be truncated at I = 14, our method of generating special directions (details to be published elsewhere) can be used to find even higher precision formulas. The general method illustrated here for systems with full cubic symmetry can easily be extended to generate special directions for other symmetries.

REFERENCES 1.

MUELLER F.M. & PRIESTLEY, M. G.,Phys. Rev. 148,638(1966).

2. 3.

WINDMILLER, L.R., KETTERSON J.B. & HORNFELDT, S.,Phys. Rev. B3, 4213 (1971). HORNFELDT, S.P., WINDMILLER, L.R. & KETTERSON, J.B.,Phys. Rev. B7, 4349 (1973).

4.

BENNETT, A.J.,Phys. Rev. 140A, 1902 (1966).

5.

BANSIL, A., Solid State Commun. 16,885 (1975).

6.

STROUD, A.H. & SECREST, D., Gaussian Quadrature Formulas. Prentice-Hall, Englewood Cliffs, NJ (1966).

7.

STROUD, A.H., Approximate Calculation ofMultiple Integrals. Prentice-Hall, Englewood Cliffs, NJ (1971).

8.

MUELLER, F.M.,Phys. Rev. 148,636(1966).