On the accuracy of fourier transformations in crystal orbitat approaches

.~ We giyqa

: ap&ox&ate Foutia transfkmation a&d pkettt

.. desaiptioi of vat&us techniques

for

am&iifi&vadon df the

I?Iott method_ A co~mparison of the differentpmadurcs and an analysisof the error&ics a& giIt is shown that ilk ,i ~_ opiim~& chaioeof a procedure dependscritically OXIthe properties of the functionto be Four&r transfonmd_ -.

~.

13n~~0~

:

The crystal orbital (CO) formalism as derived in the tight-binding approximation [l] has b&come ani&portant computational tool to &cidate the dectronic structures 6f one-dimensional (1D) periodic systems which are subject to cyclic boundary c&ditions [2,3]. The numerical a&&racy of timputati&d CO data is determined by three different convergence properties that have been &ntioned by se&-al authors [3-q: (i) com~igentiwith re$xct to t&size of the atomic basis set employed in the Co appio&h (relevant only in ab initio CC?variants); (iii &verge+x Me.y&pect. to the number of -k-pointsado~t&i in numeric@ k-space integrations; and (iii)_convergence with r&pect 6 the &mbei~of nei&hbo&g’tits &at are L&n into account in the lattice-Sum ev&mion of quasi iD: ~solids_It wilI be shown that‘ ‘the ._ . convergence problems det en&n+ by (ii) -&nd @ii c+n&t be separ&e&&om &tch bth&_ _ :One im~rtani key &p-in CO schcincs defined wi& the se&xmsistent&eld (SCF’) ap$roxi&tion i&

the determination of the bond&der matrix P,(j) .: -. ._p,(i)cr~~=dkP,(k)exp(ijk) -. --. -,/ .: -_ _-

-_

1;.

‘. ~cI _. ..- ‘- ._’

. . .. -.

: -~:--

: _--- (1) ..-

-.

betwp atogc orbital (AO) b&is fur&o& p and -P that’ &e i &t c&s a&rt; the > = 0 -term&cc&e&nd to ikracell A0 pairs.. Pp(k)-in _qi (1) & given in (2) by means of -the crys*M orbital c@ficititi F;(F)

whicharetheeigenvectorsof,the.k~ependentSCF~uations_. .;_

.:..

-,: -X-~

__-

-.

‘..:-_.-:.-'-

;_<'

-~

:.--I.

._

:

:

I

1

.-

~~~:P,(k)=2~c~(kj

iii(kj;.

:_-

-C.

_._I

:_~-

“Cm

_._

--.;

:-

f

-1

-..

_i-.

.‘IsI..;;‘;;.#.,

(2j

he

i

_‘> : ._ ; -. ~il~~~*s._~~~~~~p:~~~~~_~~_~~~~

density L&o;-&

the-? .~

operator only:h the_fxm of &X&I&

M;h’ ?wselectron_~~~~_that~d-~=_~~.~~;~~~~e~~~~_ ~Z

0301_0104/*~~330~ElsevierScia;ce.PublishasB_V.~~-~-Ho~dPaysics~b~~Division)

:.

._-

.. ..~ _

._ sotu;ions; %;entefi .~e &&__

-. -.I~_ -.- :;-. -__~

:.-, .-

-.,_; _ _ -,::

-_: e*&.-;

‘.

:.

of-: : 1’.

.. -:I -: . -:I: _+_:;t;-__:

90

P,(j) have to be det ermined by numerical integgtion- The subsequent Fourier transformation procedures have been empIoyed in prexiousIy reported ab initio or semi-empiricaI CO caIcuIationsr discrete summatioa Simpson and Gauss-Legendrc quadrature as wcII as the FiIon method_ The two prev&.ilingIyadopted “standards schemes are either based on the discrete summation or on Simpson’s rule A comparison of the capabiity of some of these integration techniquei aS a function of the lattice-sum ‘dimension(index i) has been given by DeihaUe [5l as well as by Schneider and’Ladik [7]_ The subjti of ref_ [53 is &I opposition of the widely adopted Simpson quadrature and the orig,inal Filon method where parabolas are employed as interpolating functions between EXO adjacent k-points_ An extension of the FiIon integration to higher-order polynomials is one of the topics of the fcIIowing tiysis_ Schneider and Lad&, on the other hand, investigated the accuracy of Chebyshev series as weII as sphne representations in integration procedures of osciIIatoxy integlaIs f7& It is a common attribute of the abovementioned methods that they are based on approximations for the whole integrand (exception: FiIon method and related schemes) but not for the functions P,(k) which are only known at a presekcted mesh of k-points_ Previous experieuce has shown that this restriction causes numerical errors in some Fourier transformations; the in2c-curacies are enlarged with increasing j-dimensions relative to a fixed number of k-points [5,8]_ These numerical fmdings led to the formuIation of qualitative “ruk of thumb- how to fur the lattice-sum dimension to the grid of k-points_ The numericaI instabilities encountered in the available integration procedures of the bond-order severe restriction in those 1D tight-bindin, 0 _studies where interactions beyond the nearest matricesarea neighbors have a pronounced intluence on the physical properties of the system [9,10]_ This has to be expected either in polymers with heIicaI axes allowin, 0 for s&x-t contacts between stacking units that are non-nearest neighbors or in 1D materiaIs with cxtremeiy smaII lattice spacings_ The l/j decay of the twtiectron integg in the mean-field ha&to&n has the consequence that errors of the Fourier transformation in eq_ (1) are partiahy suppressed in SCF HF CO calculationr Such an “intrinsic” attenuation, e-s is not operative in the relation (3) which defines the transformation between the spatially delocaked BIoch orb&& @f(r) and the Iocaked Wannier functions wj(r-j) ill]:

wi(r-j)

crL’=dk exp(ijk)

e:(r)_

Various procedures to det ermine the wi(r-j) functions have been reported in the Iiterature [12J3]_ The necessity for reIiabIe integration techniques is here even enIarged in comparison to the cIifIicuIties encountered in the determination of the bond-order matrix_ NumericaI integration techniques of high accuracy are a prerequisite for a reliable analysis of the asymptotic behavior of the electronic exchauge energy in one-dimensional SCF HF CO caIcuIations_ The decay properties of the exchange ekments are essentialIy determined by the faILoff of the bondorder matricez~ This has been studied by severaI authors for simpIe model poIymers with filled and half-ftied bands [M-16]_ If integgtion procedures are employed in such an investigatiori that violate the conver&nce properties of Fourier series (see heIow) an unphysical asymptotic behavior may be pretendexI that is not a manifestation of an electronic structure effect but has its origin in faihues of the adopted integgtion techuiqL= The role of the non-IocaI HF txchanSe in narrow-band mate&Is with smaII kinetic energy integrak (hoppinS elements) has been dkxssed in a prwious contribution [17]_ Computational pathways avoiding the calculation of the P,(j) ekments have been suggested by DeIhaIIe et aI. [lS] that are closely related to Fourier transform representations deveIoped by Harris and Monkhorst (19). However, it should be mentioned that these approaches are far from any routine adaption in numericaI band structure iIl\-eSti~tiOnS_ The subsequent analysis is restricted to one-dikensional materials- In section 2 it will be shown that the extremelysimplediscrete summation is an efficient integration technique in solids with finite baud gaps In

a subsequent contriiution th% resuIt wiII be used to generate’optimized k-grid3 (i-e_ selection of special .

__.:-.., _..:__._,._ : LL;

-I

1

_,

.-

:

:

;-.-z;--__

:

-F_-Pfuzdr.M_CBiihm/Forcr~rr~.~~~~~~~=

.

-. _.-

.. .

1: ‘.- ‘-_m-“]

~ointsj of two- and three&nensionaI in comparison to other integration

iekction or speiial points due to therein)_

Bravais lattices~[ZO]_ ‘I!heintegration.proeedure is here much &I&& techniques used -in the &id-state -area (Le., tetrahedron. method,: .mathematically sophisticated mi2hods, etc.;. sfqeref.-m and rqferen!azsI

.;::. _,_a_’ ._I

It is the purpose of- the present contribution to anaIyze the. accuracy of some Fonrier transformation schemes_ It wiIl be shown that the chke of an optimum method depends .on the band gap (instiIatingversus metahic materials) of the 1D system_ A modified FiIon scheme [21] has been derived.which allows for remarkable.extensions of the j-array without enhancing the absohue errors of the numerical integration. Additionally we wiU discuss different variants of attenuation factors for the evaluation of the -Fourier coefficients [22]: ?Eizzaccuracy of the numerical integration skhemes wiII be investigated by .means of simple numerical examples where: the integrals can be calculated ~analyticaIly_ ~’

.

-

..

_.

2 Description of the methods The quantities. that have to-be calculated are the Fourier coefficients c(j)

= (l/Zc)~‘hk

exp(ijk)

f(k),

j integer,

(4)

where f(k) is a periodic function, i.e. f(k + 25;) =j(k), only known at certain points kl = 2=Z/N with 1=0, l,... N_ The usual integration methods such as the Simpson rule are based on the foliowing idea_ The whole integrand I (in eq_ (4) I= exp(ijk) f(k)) is replaced by a function 1 that interpolates I exactly at the points k, and can be ktegrated analytically_ When I is a linear function wi& each interval [k,, k[,,], one gets the trapezoidal ruler cl(i)=(l/N)[l(k,)/2+I(k,)~I(k,)~___+I(k,_,)tI(k,)/2]_

(5)

Here and in the folIowing equations the index at the Fourier coefficient indicates the degree of the interpohxting polynomials. In the case I( k,) = I(k,) eq_ (5) is reduced to the discrete summation N-l co(i)

= (l/W

c

(6)

I(b)-

I-O

When

j is a parabola within-each interval [k,,, cz(j)=(1/3N)[l(ko)f41(k,)+2Z(k,)+41(k,)+...

The condition I(kN) c,(i)

= O/3+!!;

= I(k,)

k z1+2], the result is the Simpson formula +2r(kN--2)+4I(kN--1)+I(kN)]-1

(7)

aliows once again for a simplification of eq_ (7):

[3 -d--1)‘]

I(k,)-

pj

Let us now consider the asymptotic behavior of the exact Fourier coefficients for j i co. It depends on the. properties of f(k) in the following way: if f(k) is r times continuously differentiable and if the (k+ l)m derivative is still ahnost everywhere continuous in the integration intekxI, the Fourier &efficients decay at i-e_ 1c(j) I-2 M/I jr+= [,whereM-isawns~tindependent-ofj. ._ --- . . k?st with l/j”’ If f(k j shows a discontinuity (e.g., density matrices in sobaS witb@xompleteIy fiUe&bands), the k(j) fall off as 1f’_ The Fourier coefficients c(j) approach zero faster than any finite inverse power of j (e-g-. exponentiaby) in the case of a function -f(k) that is inflniteiy many times differem&bIe_ The reIations (6) -_ ..

-_

92

F_ ~

M.C. B ~ u n / Fourier wanrform,,,:,~vin cr).,.~al o r b i m l a p p r o a d ~

" "~

"

a n d (8) obviously s h o w t h a t co(j + N ) = co(j) and ca(j + N ) = ca(j)- T h e r e f o r e S i m p s o n ' s rule a s well as t h e discrete sa~mmntion c ~ n n o t give the correct a s y m p t o t i c b e h a v i o r o f the F o u r i e r coefficients. W h e n o n e is interested in large values o f j (Le. large l a t t i c e - s - m dimensions), o n e hag to increase the ntmlbc~ N Of k - p o i n t s whege f ( k ) is calculated. . . . . . A n o t h e r possibRity f o r the evaluation o f the c ( j ) is to replace f ( k ) b y an interpolating f u n c t i 0 n : f ( k ) instead o f a p p r o x i m a t i n g the w h o l e integrand, f ( k ) s h o u l d b e defined in a w a y t h a t the integral in e q . (9), E(j) = (1/'2=) [2=dk exp(ijk) f(k), ~ (9) Jo c a n b e caIculated ~-~lytically. O f course, the identity f ( k z ) = f ( k t ) ha~ t o b e f, dfilled b y the interpolntin~ function. T h i s is also the b a s i c idea. o f t h e F'don m e t h o d [21] a n d h a s t h e a d v a n t a g e t h a t the full i n f o r m a t i o n a b o u t the function e x p 0 j k ) is conserved ( e x p ( i j k ) is r a p i d l y oscillating f o r large j-values), while o n l y a c o m p a r a t i v e l y s m o o t h function has to b e interpolated. I n the original F i l o n m e t h o d f ( k ) coincides with a p a r a b o l a within e a c h interval [k21, k2/+2 ]. I t is thus a c o n t i n u o u s function, which, however, is .in general n o t everywhere differentiable. I n thiq case the a s y m p t o t i c b e h a v i o r o f the ~ ( j ) should fulfill I ~ ( j ) I < M / J 2 p r o v i d e d t h a t f(ke,. ) =f(ko)_ F o r j = 0 the F'don m e t h o d is identical to the S i m p s o n rule. F u r t h e r details o n this s c h e m e c a n b e f o u n d in ref. [5]. I t is possible to m o d i f y the I - d o n technique b y e m p l o y i n g p o l y n o m i a l s o f higher d e g r ~ as interpolating functions. F o r the interpola:-;on via fourth-degree p o l y n o m i a l s w e h a v e derived the f o r m u l a s o f the a p p r o x i m a t e Fourier t r a n s f o r m a t i o n ; they a r e s , m m a r i z e d in a p p e n d i x A_ I n the limit j = 0 this modified Filon m e t h o d coincides ~ i t h Milne's rule o f integratio:-~ [23] applied 1 o each interval [k4/, k41+4] (h = 2~/N): 0j~dkf(k)

= (4h/90) [7/(0) + 32/(h) + 12/(2h) + 32f(3h) + 7/(4h)].

(10)

B o t h the original F'don m e t h o d a n d its m o d i f i e d version c a n b e applied t o functions f ( k ) which are discontinuous at the e n d p o i n t s o f the integration interval, Le_ f ( k ~ . ) ~ jr(ko). T h i s is n o longer valid f o r the subsequent integration schemes which a r e also b a s e d o n a n i n t e r p o l a t i o n f ( k ) f o r the function f ( k ) ; these m e t h o d s m a k e use o f the gimple technique o f a t t e n u a t i o n factors_ L e t us consider t h e interpolation b y t h e pez-iodic c u b i c spline function [23] which is twice differentiable_ T h i s function f ( k ) is a p o l y n o m i a l o f third deo~ree within e a c h interval [kt, k t + ] ] with coefficients guaranteeing that f ( k z ) = f ( k t ) a n d f ( k ) itself as well as its first a n d s e c o n d derivative a r c c o n t i n u o u s at t h e points k~_ I t is well k n o w n t h a t the c u b i c spline function is i n a cerfain sense the s m o o t h e s t o f all possible twice diffexentiable interpolating functions; it minimi76,~ the integral f02= d k [ f"(k) i 2_ T h e r e f o r e t h e spline is particularly q . a l i f i e d f o r replacing s m o o t h functions. F u r t h e r m o r e it preserves all the s y m m e t r y p r o p e r t i e s o f the f ( k t ) set_ In principle it is possa-ble to calculate first t h e c o e f f i c i e n t s o f the c u b i c poly.n,o m i a l s a n d then to evaluate the integral eq_ (9)_ H o w e v e r , it turns o u t t h a t thi_~ n e e d n o t b e d o n e explicitly. Accordingly to a t h e o r e m o f Gau~ a n d R+;n~.h [22] the + ( j ) c~n b e d e t e r m i n e d in a very ~m_ple w a y directly u n d e r the premi_~e that the interpolation s c h e m e obeys the following t w o conditions: (i) f ( k ) m u s t d e p e n d linearly o n f ( k ) . (ii) T h e interpolation s c h e m e ha~ to b e Lr:an~lationally invarianL C o n d i t i o n Ca-) me~n~ t h a t the function f , ( k ) , which is the interpolation o f f , ( k ) = f [ k - - ( 2 = n ) / N ] , is s i m p l y given b y f ~ ( k ) ~ f [ k - - ( 2 - a n ) / N ] . Both conditions a r e full'died b y the periodic cubic spline interpolation_ T h e n it is possa-ble to o b t a i n the ~-(j) via ~-(j) = c o ( j ) . r ( j ) ,

.

.

.

.

.

~-

. . . . . . . . . . . .

(11)

w i t h c o ( j ) deemed by eq. (6). ~r(j) is a factor thatdepcnds only on the employed interpolation scheme. The ~-(j) are called attenuation factors: F o r the cubic spline function % ( j ) is defined by [23]: ~r3(j)=

[(sin

z)/z]4[3/(I

+ 2 cosZ z)],

.

z.= (j/N)~.

-: "

~

"

"

;

;

.(12)

-.

(15):.-

7=(j) has the character of a step function. This, however, corresponds to an interpolation via a trigonometric polynomial f(k)= j-

NE -Iv/z

exp(-ijk)

with ~3.~~ = 8--Np_ (N

4,

(16)

even) or

<*~-U/2 f(k)

= i_ _CG_l,n

exF4--ijk)

for odd IV_pi is defined by i$=

cdA%o(A-

0.5

j/N

&

w

WC~BCinz -f Fm-errmmf

F_P’

._-

--onsin

~xmrarbirarapp~

:

Fiq_ _(l5) is re&ed to ~ihe discretekunmation or the trapezoidal r&l& eq; (6), for ljl .&V/2_ In the following we want to.make a further remark on U&discrete summation (22). If this method is applied io-an integg of the form /t dkf(k) (e-g, p = 2~) where f(k +p) =j(k) isa periodic function that is~sev&al times differentiable, H is much better than its simple form might suggest- In order to verify this behavior we make use of the identity ,:.-.

.~ kpdkf(k)

=c;_pAvdkf(k).

09)

which follows from the periodicity of f(k)_ The application of Simpson’s rule ?o b&h integralsof (19) Ieads to two different numbers, the mean of khich is just the result of the discrete su~mmatiori Walogously it can be shown that S gives always some mean v&e corresponding to the different possible applications of an integgtion formu’~ if the latter is of the form

/0

Pdk-f( k) = E a,

f (&),

IL-, = pi/N (equidistant),

(20)

I-I

with the a, independent of the function f(k)_ This is did, e.g., for all formulas of the Newton-Cotes type [23]_ Therefore the error limits of all these integration schemes can also be applied to 2 if f(k) is as many times diffirentiable as required For theses procedures (includin, 0 the points k = 0 and k = p, respectively). 3_ Investigation of the accuracy

In order to investigate and compare the accuracy of the different integration procedures we have defined functions h(k) (1~ f < 7) with the Following properties: (i) f;(k +-2=)=x(k): (ii) f;(k) coincides with a polynomial of degree i; (iii) within the interval ]0,2r] f;-(k) is (i-2) times continuousIy differentiable; (iv) /z=dkL(k)=O. (v) (l/2+,‘= dk If,(k) 1’ = l_ (21) These functions are listed in appendi.. B for 1~ i < 7_ The numerical evaluation of the corresponding integrations cq_ (4) Sows for the formulation of the following rules: (13 The accuracy of the Filon method and its present extension does not depend on the continuity or difFcrentiability of I,(k) at the end points 0 and 2-s_ (ii) In table 1 we have sununaked the number OF subintervak N neccssaq to caIc&te c(i) (i SG6) for the functions x(k) (1~ i zz 7) with an error less than A_ For arbitrary Functions f(k) one can prcive the Following error limits of the Filon method in evahtating eq_ (4):

with Al, = e
I*

denotes the ith derivative of f(k)_

A = (19/45)(z/N)%f..

(24) For the modified Ron

method these Iimits are:

(W

(iii) The error of the Filon methods does in general not increase with increasing i, rather. the opposite is true_ This is in~contrast to the other integration schemes_ (iv) Both Simpson quadrature and the discrete. summation depend sensitively on the properties of f(k) at the points 0 and 29~ When ‘f’(k) ot even f(k) itself is discontinuous at these points, Simpson’s rule’ generally will be better than H and both of them are inferior to the two Filon variants With increasing differentiability of f(k) the accuracy of these techniques is increased and the discrete summation su_~asses the capability of Simpson’s rule. (v) In the case of f,(k) the discrete sununation needs just the values of N given in table 1 for the mmodified Filon method and Simpson’s quadrature slightly less than the values for the original Filon method. For f-,,(k) an accuracy better than A = 10s6 is reached when N >, 8 +i (E) or N > 12 f 2j ‘.. (Simpson). (vi) The integration schemes employing attenuation factors r(j) based on interpolations by spline functions are intermediate between Simpson’s quadrature and the discrete summation_

4. condusions It has been the purpose of the present work to give an analysis of some approximate Fourier transformation schemes necessary in connection with crystal orbital calculations and to discuss .their accuracy_ We have shown that the optimum choice of the method should be guided by the electronic properties of the solid-state system (analytical properties of e-(k)), For insulators with non-crossing bands the discrete summation (with the cut-off criterion of cq. (15)) should be the most a&rate methodFor semiconductors with a very small gap as well as for met& we propose a modified Filon iritegtationL In the case one cannot make an a priori d_ecision, it might. be worthwhiIe to -investigate the convergence properties of different integration t&miqu& by replaciu g N by a small& number, eg; ~-M/2_Of-co& care shouid be taken that one need not r&&ulate f(k) at .addition& ~poiuts The ~com$&onal : expenditure for such a test is negligible_ This procedure allows for the estimation of the numerical error of the Fourier transformation and is closely relate& t0 &trapolation- techniques ((l/k) --, 0). that -are, e.g., employed in the Rombetg integration scheme [23] which might .be interesting in connect&n with the original FiIon method_ FiiyT we do not want to omit the Gauss-Legendre quadrature [23] which might be more powerful than the FiIon schemes for sm+ j-va.hxCThe .dis+antage of this’method &. the: ~. -. : requirement of non~uidist&t k-points. -. Wee bbpe’tbat our qua+tative investigation of iurmeric+ errors of Guie~ GausFomxatid~ tecbuiqucs removes some un certainti_es~<;oncerningthe accuracy of these procedures in connection with crystal. orbital approaches where’the reliable determination of the bond-order m&ix P,(k) is one key. step encountered

:

=

~_

.. ..

~.

.~

.z __ .-

.

_-

E~M_C--/F~-~-~~q)aal~lq..

96

:

in SCF stud&s whjle the knowledge of the Wannier fupctionsis beyond the m&n-field approxi~tio~ [24]_

..

a prerequisite~in perturba&onal studies

A&xowied&ment

,--. -

-

. We are gtateful to Professor Pet& Fulde for criticaIIy reading the manuscript has been supported by the Stiftung Volkswagenwe&_

‘_

One of t&e authoi-s (MCB)

.

Appendix AZ The modified Filon metbud An approximation of the integral c(j)

= jt dkf(k)

exp(ijk)

is given by

(Al) Here y =jh_ g+=

fr = (6 - a)/.N,

(

I,-___ 25 ~ 12x2

f, = f (a f lir),and

35i 12p3

-+i5 2y=

go= “-s+(--&-5) 6-9

g_= gZ exp[ij(b

-

y5 I(

+

n = N/4

: I. The g are given by::

1 -4Y2 + ----12y’ lli

cos4y-E(s-3)

2y” 3

i 1 exp(llig), y5

sin4y,

a)] -

(A21

For small vahus of y (y = 0-l) these expressions cause numerical difficulties and therefore have io be replaced by their expansions which we Iist up toeleventh ordeE

101~5808 _ 65536 +25540515’Y9-~4465125Y

1o -1@$8576;,,+_ -2128m625’Y

--~ - -‘;

-. ..

-1 :

.-

-.;

..-Y _’ ;

:

‘I

:.

:-

.

:_ I

_: : --

.

___:

._T _-

..

=_ ,_:::; .:. :__

:

.“

~.. _.

. _-

. ._.

How

far &e

calculated

has to go- i

these

expansio& depends

oti -t!k smallest v&e

-of p foi which (k)

with sufficient accuracy.

Appendix Br The k&&&s

f;:(k)

an be : __ . . . . ._

.

Here x = k/n. f,(k)

= 6(x

- 1).

f,(k)=&/%(x4-

fr(k)=$fi(x2-2x+-$), 4x3+4x’-&),

f3(k)=$m(x3-3x+2x), 5-x4 + 9x3 - ax),

f,(k)=$#(x’-

j&c)=~~(x6-6xs+10x4-~x2t~),

.-

f,(k)=&@(x’-7x6t14xs+x3+~x)__ (B-i)

References [l] [2]

[3]

(41 [5]

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