Projection of plane-wave calculations into atomic orbitals

Solid State Communications,

Vol. 95, No. 10, pp. 685-690, 1995 Elsevier Science Ltd Printed in Great Britain

0038-1098195 $9.5Ot.O0

0038-1098(95)00341-x PROJECTION

OF PLANEWAVE

CALCULATIONS

Daniel Sanchez-Portal, Instituto

Nicolb

Cabrera

Emilio Artacho,

and Departamento

Universidad

Autonoma

INTO ATOMIC and Jose M. Soler

de Ffsica de la Materia

de Madrid,

ORBITALS

28049 Madrid,

Condensada,

C-III

Spain.

(Received 24 April 1995; accepted 9 May 1995 by F. Yndurain) The projection

of the eigenfunctions

electronic-structure

calculations

practical

link between

orbitals.

Given a candidate

the plane-wave associated analysis

plane-wave

basis sets is proposed

(ii) it is optimized

hamiltonian

procedures

first-principle as a forma1 and

based on plane waves and the ones based on atomic by maximizing

by its projection

that projection,

and energy bands are obtained,

in a natural

tight-binding

in standard

atomic basis, (i) its quality is evaluated

eigenfunctions,

tight-binding

trial-and-error obtain

the methods

is performed

obtained

into atomic-orbital

way. The proposed

of finding appropriate

hamiltonians.

method

into

(iii) the

and (iv) population

replaces

the traditional

atomic bases and the fitting of bands to

Test calculations

of some zincblende

semiconductors

are presented. Keywords:

D. electronic

Plane waves and atomic orbitals used methods of them paper

for electronic

showing

proposes

method

band structure,

A. semiconductors

provide two widely

much better

structure

advantages

calculations,

and disadvantages.

a way to use the advantages

to complement

each

wavefunctions.

This

experienced

of each

the other.

scales linearly

with system

scheme’

etc., through

large number of systems tron densities,

and energy bands.

properties

giving accurate

total energies,

is the flexibility,

state

results for elec-

and accuracy

them.

and this greatly

facilitates

the calculation

how-

ever, makes the PW basis inefficient,

requiring

a rel-

large basis size for a given accuracy.

tion, plane waves require periodic

boundary

When

(molecules,

defects)

the system

is not periodic

such a periodicity

by the use of supercells,

what further

hand,

methods

based

are

matter

is based on mentioned

are affected by the

of the completeness and This task is much worse

cutoff, and usually ends up in a very cumbersome non systematic

surfaces,

trial-and-error

and

procedure.

Thus, due to the difficulties and ambiguities of the LCAO basis construction and their less well controlled

artificially the inef-

completeness, they are frequently considered more approximate than plane waves, and they have been tradi-

on the linear

tionally used in less accurate tight-binding type of calculations. However, once a well optimized LCAO basis has been obtained, there is no reason why highly ac-

ficiency of the basis. On the other

These quantities

defined than in the case of plane waves since there are many more parameters than just a plane-wave energy

In addi-

increases

like atomic

They provide a deep physical insight.

basis: since the results of calculations

conditions.

must be imposed

analysis.’

election of the basis, a testing quality of the basis is critical.

of

everywhere,

atively

magnitudes

bond charge, charge transfer,

above is the greater difficulty in the choice of the LCAO

the energy of atomic

atomic forces. Being equally complete

effort

Local bases also of-

The price to be paid for the advantages

of the plane-

and universal com-

pleteness controlled by a single parameter, cut-off. It is a floating basis, independent

size.

system since most of the chemical language

The origin of its wide applicability simplicity,

population

because of their suitabil-

essential for the chemical analysis of a condensed

atomic forces, geometries,

wave basis set: it has a homogeneous

positions,

of a

or Bloch

in which computational

way of quantifying

charge, orbital population,

the molecular

LCAO bases have recently

a renewed interest methods4

fer a natural

the ground

In addition,

ity for order-N

On one hand, plane waves (PW), in addition to ab initio pseudopotentials’ provide a very successful to calculate

suited to represent

combination of atomic orbitals (LCAO) are far more efficient in terms of basis size, since atomic orbitals are 685

686

PROJKTION

curate

ab initio

calculations

it6 This rises the question

OF PLANE-WAVE

cannot

be performed

of whether

plane-wave

lations

can be used to find an optimal

LCAO

basis,

situations

orbitals

in molecules

constitute

and

bases

are

more

onal

or near-optimal

ones,

they

solids.

computational

non-orthogonal Even

this

kind

than

orthog-

have

to be more

sets

methods.‘!’

are

schemes

transferable

used

energies

in atoms,

(ii) the optimization experiments trast,

this

methods calculation basis. tool

and

calculations3~”

the eigenstates

LCAO

and (iii) analyze electronic

of population

within

for electron

pseudopotentials proximations

exchange

the expansion

not essential

energy

ject of a future

calculations

cutoff.

using

These ap-

wave funcin this paper

to be converged

The explicit

for the system of interest,

for the projection,

eigenstates

S measures states

the quality

tified by its ability

of an atomic

to represent

those

falls outside

basis.

For that

basis is quan-

eigenstates,

how much of the subspace

eigenstates

I&(k))

i.e.,

of the Hamil-

the subspace

spanned

we define the spilling’5

ca

k

the projection. reproduce that

I&,(k))

Nk and N, the Brillouin

are the PW are the number

calculated

eigenstates,

of calculated

zone and the number

k points

precisely,

The spilling that

the PW eigenfunctions

0 and 1.

wave-functions

exactly.

S = 1 means eigen-

states. above,

LCAO

the quality

set of Hamiltonian

basis will reproduce

the different

eigenstates

of eigenstates

relevant

the particular chemical

the occupied

to be considered.

requires

to the

basis

sets,

electronic

a larger atomic

way. Given

Slater-type

orbitals,

parameters, Taking

evaluations

a type of atomic

computational

tual PW calculation,

in in a

functions that

the prob-

minimizing

Note that

S as a

the minimiza-

of S for different

effort

selection

orbitals)

(say exponents)

that the evaluation

the

basis.

optimization

but for a single first-principles

into account

should

presented

Gaussian-type

of those parameters.

or opti-

and possible

this work allows for their systematic straightforward

density,

are the ones states

methodology

on

properties

As shown later,

analysis the

choice

of S depends

excitations

in the projection.

case typically

accuracy

The

eigenstates

For quasiparticle

In addition

(say

An

some of the lowest empty

be also included

of atomic

eigenstates.

For ground-state

(total energy, geometry,

analysis)

but

of a Hamiltonian. to the definition

ba-

system,

with very different

application.

of the system

of an LCAO

not only for a particular

the basis becomes

in

for

to the Hamiltonian

lem of optimizing

of bands considered,

of

considered

projected

by

and

eigen-

the average

S varies between

pa-

(1)

(4)

into the atomic-basis

it gives

the LCAO

the basis is orthogonal

a minimal where

= 6,“.

parameters

o=l

that satisfy

1I(1 - P) I& (k)) II2 over the eigenstates

function

y+-$&bn(k)l(l - P(k))lti,(k))

basis,

between the plane-wave

and their projection More

tion requires

S=

basis, and the vectors

depend on certain

S

(3)

= (d,(k)lY(k))

the difference

P(k)/&(k)).

and

to be considered

(4,0%44)

of the atomic

(Y‘(k)l&(k))

latter

study.

given the set of Hamiltonian

w

are the dual of the atomic

cal properties,

consideration

of the PW basis will be the sub-

Given the PW results

rameter

I&(k))

also for a particular

to the method.

of the one-particle

the PW reference

the atomic

calculations

and correlation,13

I~,(k))S;,‘(k)(~y(k)I,

= c

=

is the overlap matrix

sis is evaluated

approximation

The theory presented

of the uncompleteness

tonian

by means

by others at will, the only key fac-

tions in plane waves.

by measuring

density

that replace core electrons.14

are, however,

in plane-wave

plane-wave

the local

They can be replaced

considers

S,,(k)

As mentioned

In this work the reference

tor being

and Hamil-

structures

the

and defined as usual for non-

l&(k))(@‘(k)l

it

S = 0 means

analytical

and optimize

bands

into

k, gener-

(2)

analysis.”

are performed (LDA)

by the atomic

and practical

to (i) characterize

(ii) obtain

In con-

and LCAO

of a plane-wave

space spanned

an efficient

is applied

basis sets,

= c

local

with

into the Hilbert

which

tonians,

molecules,

work links the plane-wave

This provides

atomic

LCAO

or solids,”

by projecting

P(k)

(i) the minimization

of energy bands as compared

or with plane-wave

the present

basis,

operator

of wave vector

basis,’

for non-

throughout

to optimize

bases are based on two procedures: of total

is the projector functions

where

Non-orthogonalized

therefore

proposed

orthogonal

Vol. 95, No. 10

of

work.’ Most

P(k)

of Bloch

and to have a shorter

being thus more adequate

LCAO

basis

basis sets

though

range of interactions, parametrized

respectively. subspace

ORBITALS

effort.

to handle

shown

INTO ATOMIC

ated by the atomic

cumbersome

and less environment-dependent,

atomic

with calcu-

which can then be used in more complex

while keeping a feasible

Atomic

CALCULATIONS

values of the calculation.

of S represents

compared

with the ac-

it is obvious that this optimization

procedure

is much more convenient

and-error

procedures

than the direct trial-

using self-consitent

LCAO

codes.

PROJECTION

Vol. 95, No. 10

Using this procedure atic study comparing

we have performed

different

in different

solids.

elsewhere.16

In that analysis

properties

(spilling,

OF PLANE-WAVE

This

CALCULATIONS

stressed that the proposed

a system-

kinds of atomic

will be presented

tonian matrix elements

orbitals in detail

rameters

it is found that for many

charge density and populations,

INTO ATOMIC ORBITALS

en-

matrix

in Bloch space. By construction, tion of the matrix

take the eigenfunctions

of the

up to infinite

free atoms using the same basic approximations

as for

ture of our method

and scale them

elements

neighbors.

between

terms of its associated

band structure:

separates

atomic

is, d/(r)

from the effect of neglecting

ing Ni(Xi) the normalization factors are determined the orbitals orbitals

= Nl(&)$ptO”‘(Xlr),

factor.

The optimum

by minimization

compress

bescale

of S. Typically

which are empty in the atomic calculation

compress

substantially.

These optimized

orbitals

some scope of neighbors.

that

suitability tonians energy

utility

of the projection

for obtaining

and analyzing

technique

mixed together

LCAO Hamil-

(tight-binding parameters) and their associated bands. l1 The matrix elements of the projected

ments

space Hamiltonian,

(5)

using fast Fourier transform

and other techniques

algorit,hms

to a sufficient

G)(k + GIHPwI&(k))

lattice

calculation,

vectors

is an

number

the number

of the atomic basis.

of k points,

of points

depends

(7) on the over-

taking into account on the real space

I7 For silicon and the STOare important

up to

and a few tens of k points have

proven to be enough. In Figure 1 (a) and (b) we present the energy bands of silicon for an sp minimal

basis and for an spd basis,

a

respectively,

with interactions

It has to be

as obtained

from the projection.

to guarantee

ele-

an inverse Bloch trans-

4G basis used below, interactions

of the one used in the PW

that must be large enough

reliable representation

and E,,,

matrix

from the reciprocal-

factors which depend

range of the interactions.

(6)

cut-off independent

sep-

for clarity. The sum has to be extended

third nearest neighbours

where G are reciprocal

Hamiltonian

= ~~;,CAO(k)e’k(R+)

where normalization

that

energy

With

k

laps are omitted

of the PW method:

HLCAo(k) = ~(&Jk)lk+ I1” 1k+G12<.%..

ways of

can be analyzed

parameters) we perform

w

can be obtained

in the traditional

the real-space

(tight-binding

ffLcAo(~J = ($,(k)IHPWl&(k))

of error

formation

Hamiltonian H;:“‘(k)

beyond

two sources

each approximation

To obtain is its

These

of the basis

elements

arately.

ones used in the following, except stated otherwise. A further

it conveniently

choosing atomic orbitals from their LCAO bands. our procedure

are the

matrix

fea-

of a basis in

the effect of the uncompleteness

are usually

by a small amount except for the

directly

localized functions

for the characterization

XI (one scale factor per different

that

is obtained

it includes the informa-

This is a very interesting

down by scale factors orbital),

the Hamilno free pa-

being fitted.

results

the solid (LDA, same pseudopotential’4)

obtains

from first-principles,

The LCAO Hamiltonian

ergy bands) the type of atomic orbitals that gives better is the following:

procedure

687

FIG. 1. Electronic band st,ructure of silicon calculated projecting the plane-wave LDA Hamiltonian into an sp atomic basis (a), and into an spd atomic basis (b), both shown in solid lines. Dotted lines show the plane-wave LDA band structure. The basis functions are obtained as discussed in the text.

up to infinite

neighbors,

They are compared

688

PROJECTION

OF PLANE-WAVE

with the PW energy bands, that were calculated well converged

plane-wave

was generated orbital

basis set.

and optimized

being obtained

the silicon atom.

in Eq.

with a

configuration

of the

1) is S = 0.0076 and

S = 0.0007 for the sp and spd basis, respectively. the sp basis reproduces

figure shows that band

much better

giving a rather

than

tended.

orbitals

substantially

both

the latter.

a reduction

FIG. 2. Effect of different contractions of an sp atomic basis in the silicon band structure. Solid lines are for optimum scale factors and dotted lines for a further contraction of the basis functions of a 1.2 factor. The basis functions are obtained as discussed in the text.

ex-

of their extension

Figure 2 shows the effects of a contraction The energy

of the basis. a tendency

to establish

to the expected increase

The

improves

used above are relatively

For some purposes

is desirable.

bands,

of the band gap.

the valence band and the band gap, especially The atomic

The

the valence

the lowest conduction

poor description

of the d orbitals

inclusion

of the X point rises, showing a direct band gap, in addition

narrowing

of the bands and the overall

of the gap. The shape and the ordering

conduction

bands is very sensitive

the basis.

See for example

the s and p conduction scale factor.

the change

states

S = 0.0292 for the optimum

the result of infinite neighbors.

of the

to the contraction in ordering

of

the atomic basis obtained

of

of the atom usually tions.

at r with the change in

The charge spillings

and the contracted

the bands

basis,

ted lines in Fig. the energy bands

an overlocalization be interesting

of the atomic orbitals,

if matrix

a range are wanted

elements

basis”

with interactions

ond (b), and third

(c) nearest

from the diagonalization trix obtained

the plane-wave

this can still

Notice that

beyond

correctly

The effect of this

in Eq.

for an STO-4G

5, and they are compared

the range

for the ST0

results,

shown as dotted

the optimized

basis used for Figure

with standard

minimal

basis used in Fig.

ma-

culations

with

chosen compromising

larger than for the

1 (a) (S = 0.0076).

the extension

In LCAO cal-

of the atomic orbitals between

the quality

has to be

required

g0 w. -5 -10

X

l-L

1 (a)

sp basis sets like

5

r

with

band gap, which is not usu-

10

L

basis,

lines in Fig. 1.

15

-15

of

basis (dot-

3) is worse than for the other

4G basis is S = 0.0513, considerably

They result

of the LCAO Hamiltonian

reduces

However, the quality

the one used in Fig. 3. The charge spilling of the STO-

up to first (a), sec-

neighbors.

a longer range of interacorbitals

neighbors)

gives an indirect

ally obtained

is shown in Figure 3, where the

bands for silicon are presented

sp minimal

that

showed with solid lines in Fig. 1 (a), as compared

worse with

for neighbors

to be negligible.

kind of approximation energy

become

require

interactions.

(infinite

We have observed

from the scaled eigenfunctions

The use of STO-4G

of non-negligible

are S = 0.0076 and

respectively. Even though

Vol. 95, No. 10

above, the d

as explained

from an excited

states

INTO ATOMIC ORBITALS

The atomic basis

The spilling of charge (considering

sum over occupied

CALCULATIONS

l-

X

rL

r

FIG. 3. Band structure of silicon as a function of the range of non-neglected interactions: interactions up to first (a), second (b), and third (c) nearest neighbors. In dotted lines the result for infinite neighbors is shown as reference (in (c) they are hardly distinguishible from the solid lines). The atomic basis functions are standard STO-4G taken from Ref. 10.

X

r

for

Vol. 95, No. 10

PROJBCMON

the band-structure

and valence properties,

sibility

of neglecting

matrix

OF PLANE-WAVE

elements

The choice of basis depends

CALCULATIONS

and the pos-

beyond

a range.

on the particular

applica-

tion. Finally, we show the utility of the projection nique for the analysis

by means of LCAO population proposed

by Mulliken5

occupied

eigenstates

basis.

tech-

of the results of PW calculations analysis.

We use the one

The analysis is performed

projected

of the

It has to be done with care since the projected

eigenstates

[x0(k)) = P(k)l$,(k))

(the deviation charge

is related

spilled in the projection15).

tion of their associated is necessary to ensure ulation

analisys.

density

operator

where

are not orthonormal

from orthonormality

Proper

S

QC

&A

69

BN

3,~

0.0022

2.19 (2.14)

5.81 (5.86)

0.81 (0.86)

BP

S,P

0.0038

3.51 (3.34)

4.49 (4.66)

-.51 (-.34)

considera-

AlP

s,p

0.0035

2.15 (2.20)

5.85 (5.80)

0.85 (0.80)

SIC

s,p

0.0071

2.30 (2.19)

5.70 (5.81)

1.70 (1.81)

by defining

the

GaAs

s,p

0.0041

2.58

5.42

0.42

s,p,d

0.0010

2.78 (2.88)

5.22 (5.12)

0.22 (0.12)

as

Ix”(k)) = CpRii(k)lxp(k))

represent

tors of the dual set of the projected LCAO density

Basis

with the

overlap &p(k) = (xo(k)lxP(k)) charge conservation in the pop-

This is accomplished

matrix

the vec-

the charge associated

jection of the eigenstates

in terms

of the

into the space spanned

(Vlild”) 7

to an orbital

(9)

(iii) to obtain tight-binding

(10)

zincblende

to perform equivalent

selfconsistent agreement.

are straight-

than the previous methods

tasks. The main idea of this paper

could also be used for the improvement

for different

They are compared

LCAO calculations The basis functions

lation analysis

and (iv) to per-

The procedures

This possibility

of first-principles

is presently

being explored.

are shown in Table I, where

are calculated

materials.

from PW calculations

parameters,

analysis.

forward and more systematic

algorithms. charge transfers

obtained

by an LCAO basis can be use-

ful for multiple purposes: (i) to evaluate and optimize LCAO basis sets, (ii) to analyze LCAO band structures, form population

p then being

Q, = c P,Ar. Y Results of this procedure

it has been shown how the simple pro-

The

dual LCAO basis: P,” =

In summary,

eigenstates.

is then written

689

TABLE I. Calculated charge transfer for some zincblende semiconductors. Qo and QA stand for the valence charge on the cation and the anion, respectively, aQ for the charge transfer with respect to neutral atoms, and S for the spilling. Numbers in parenthesis were obtained from Hartree-Fock LCAO calculations,‘s except for GaAs, for which LDA LCAO was used.”

on the

into the subspace

INTO ATOMIC ORBITALS

heteropolar Acknowledgments

with results of

- We acknowledge

useful discus-

showing a remarkable

sions with F. Ynduriin.

used for these

sions with S. G. Louie. This work has been supported

are the scaled atomic orbitals

popu-

discussed

above, which showed to be best suited for the purpose.

the Direction

E. A. also acknowledges

General de Investigation

nologica of Spain (DGICYT)

Cientifica

discusby

y Tec-

under grant PB92-0169.

REFERENCES 1. D. R. Hamann, M. Schliiter, and C. Chiang, Phys. Rev. Lett. 43, 1494 (1979). 2. See, for instance,

Cohen,

J. Phys.

J. Ihm, A. Zunger, C: Solid St.

and M. L.

Phys.

12,

first-principles

quantum

See Ref. 10 for references.

chemistry.

7. P. Ordejon ‘Optimized

LCAO

Verlag (1987), and references

method’,

refereces therein.

8. E. Artacho

and F. Yndurain,

Phys.

Rev.

B 43,

and F. Yndurain,

Phys.

Rev.

B 44,

6169 (1991). 9. E. Artacho

Phys.

in

Spriger-

therein.

D. A. Drabold, R. M. Martin, and M. P. Grumbach, Phys. Rev. B 51, 1456 (1995), and

J. Chem.

calculations

4552 (1991).

4. P. Ordejon,

5. R. S. Mulliken,

used type of basis

in the very accurate

4409

(1979). 3. H. Eschrig,

6. LCAO is the most frequently

23, 1833 (1955).

and L. Milans de1 Bosch, Phys.

A 43, 5770 (1991).

Rev.

690

PROJHCI’ION

10. R. Poirier,

OF PLANB-WAVE

R. Kari, and I. G. Csizmadia,

‘Hund-

book ofgawsian basis sets’, Elsevier Science (1985), and references

therein.

11. D. J. Chadi, Phys.

tin orbitals

(LMTO)

N. E. Christensen,

instead

for linear muffin-

of plane waves.

S. Satpathy,

See

and 2. Pawlowska,

Phys. Rev. B 36, 1032 (1987). 13. P. Hohenberg

and W. Kohn,

14. N. Troullier and J. L. Martins, 1993 (1991).

15. If projecting

eigenstates,

the defined

spilling parameter gives the proportion charge Iost (spilled) when projecting.

of the total

Phys.

Rev.

Phys.

the occupied

17. H. J. Monkhorst

and J. M. Soler, to

and J. D. Pack, Phys.

Rev. 13,

5188 (1976). R. Dovesi, C. Roetti, and V. R. Saun-

: Condens.

136,

ders, J. Phys.

Rev.

19. J. R. Chelikovsky

Rev. B 43,

E. Artacho,

be published.

18. R. Orlando,

B864 (1964); W. Kohn and L. J. Sham, Phys. 140, All33 (1965).

Vol. 95, No. 10

INTO ATOMIC 0RBITAI.S

16. D. Sanchez-Portal,

Rev. B 16, 3572 (1977).

12. Similar tools have been developed

CALCULATIONS

Lett.

Matt.

2, 7769 (1990).

and J. K. Burdett,

56, 961 (1986).

Phys.

Rev.