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Geometry optimization of periodic systems using internal coordinatesTomáš Bučko, Jürgen Hafner, and János G. Ángyán Citation: The Journal of Chemical Physics 122, 124508 (2005); doi: 10.1063/1.1864932 View online: http://dx.doi.org/10.1063/1.1864932 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/122/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Ab initio studies of 1,3,5,7-tetranitro-1,3,5,7-tetrazocine/1,3-dimethyl-2-imidazolidinone cocrystal under highpressure using dispersion corrected density functional theory J. Appl. Phys. 115, 143509 (2014); 10.1063/1.4871398 Role of negatively charged defects in the lattice contraction of Al–Si–N Appl. Phys. Lett. 96, 071908 (2010); 10.1063/1.3323093 Geometry optimization of molecular clusters and complexes using scaled internal coordinates J. Chem. Phys. 122, 014104 (2005); 10.1063/1.1829043 The katoite hydrogarnet Si-free Ca 3 Al 2 ([ OH ] 4 ) 3 : A periodic Hartree–Fock and B3-LYP study J. Chem. Phys. 121, 1005 (2004); 10.1063/1.1760075 Phase diagram, chemical bonds, and gap bowing of cubic In x Al 1−x N alloys: Abinitio calculations J. Appl. Phys. 92, 7109 (2002); 10.1063/1.1518136
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Geometry optimization of periodic systems using internal coordinatesTomáš Bučkoa! and Jürgen HafnerComputational Materials Science, Institut für Materialphysik, Universität Wien, Sensengasse 8/12,A-1090 Wien, Austria
János G. Ángyánb!
Laboratoire de Cristallographie et de Modélisation des Matériaux Minéraux et Biologiques, UMR 7036,CNRS-Université Henri Poincaré, Boite Postale 239, F-54506 Vandœuvre-lès-Nancy, France
sReceived 16 November 2004; accepted 11 January 2005; published online 30 March 2005d
An algorithm is proposed for the structural optimization of periodic systems in internalschemicaldcoordinates. Internal coordinates may include in addition to the usual bond lengths, bond angles,out-of-plane and dihedral angles, various “lattice internal coordinates” such as cell edge lengths, cellangles, cell volume, etc. The coordinate transformations between Cartesiansor fractionald andinternal coordinates are performed by a generalized Wilson B-matrix, which in contrast to theprevious formulation by Kudinet al. fJ. Chem. Phys.114, 2919 s2001dg includes the explicitdependence of the lattice parameters on the positions of all unit cell atoms. The performance of themethod, including constrained optimizations, is demonstrated on several examples, such as layeredand microporous materialssgibbsite and chabazited as well as the urea molecular crystal. Thecalculations used energies and forces from theab initio density functional theory plane wavemethod in the projector-augmented wave formalism. ©2005 American Institute of Physics.fDOI: 10.1063/1.1864932g
I. INTRODUCTION
Recently, there has been an increasing interest in elec-tronic structure calculations of three-dimensional periodicsystems. Several high-performance density functional theorysRefs. 1–5d and Hartree–Fock5,6 codes are available, permit-ting the study of structural features by gradient optimization.Until very recently, geometry relaxations in solids were doneexclusively in terms of Cartesian and/or fractional coordi-nates, and lattice vector components.1,4,7–9 Although theCartesian/fractional coordinates are simple and universallyapplicable, the use of internalsatomic and latticed coordi-nates as control parameters in geometry optimizations offerseveral advantages. As it has been extensively demonstratedfor molecular examples,10–12 in internal coordinatessid it iseasy to have a good initial Hessian matrix guess,sii d thecoupling of different modes is reduced in comparison withthe Cartesian coordinates, andsiii d the handling of con-straints is simple and straightforward.
Inspired by the experience in the domain of finite mo-lecular systems, several groups have attempted to improvethe convergence of geometry optimization of solids byadopting alternative coordinate systems. Approximate nor-mal mode coordinates derived from a simple model of thedynamical matrix were proposed by Fernández-Serraet al.13
A scaling of these coordinates by estimated force constantsconsiderably improves the condition numbersratio of highestand lowest eigenvalued of the Hessian and accelerates con-
vergence. Another approach is based on the construction of aredundant set of primitive internal coordinates, such as bondlengths, valence angles, torsional angles, etc., that are trans-lationally unique. Solid optimizations can be either per-formed in these redundant coordinates directly14 or in a non-redundant linear combination of them, in delocalized internalcoordinates.15 To our knowledge, the problem of lattice vec-tor optimizations in the context of internal coordinates hasbeen discussed only in Ref. 14.
In the present work we describe an approach that usesdelocalized internal coordinates for the optimization ofatomic positions, similar to Ref. 15. The lattice parameteroptimization follows the principles of the Kudin’s method,14
with one significant improvement: while in their procedurecell parameter variations are described by a small subset ofprimitive internal coordinates, our method treats all atoms ofthe unit cell on an equal footing letting them contribute tocell parameter variations.
In Sec. II, after a short reminder of the Wilson B-matrixformalism16 for geometry optimizations in internal coordi-nates, the necessary extensions for the periodic case are dis-cussed along with the handling of constraints. Some resultsare presented in Sec. III, demonstrating that our procedureallows one to optimize efficiently atomic coordinates andcell parameters simultaneously. The gain in the number ofiterations for structures relaxed to a comparable degree ofprecision with respect to Cartesian relaxations with a unitHessian matrix guess ranges from a factor of 2 to 10. Inaddition to the better convergence properties, our approachoffers a significantly increased versatility in performing con-strained lattice optimizations.
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II. METHOD
A. Cartesian and fractional coordinates
The structure of a periodic system is characterized bythree lattice vectors, arranged in the matrixh=fa1,a2,a3gand by the 3N fractional coordinates,s=hsa
aj. The Cartesianpositions of atoma in the L =sl1, l2, l3d unit cell of the solidis given by the linear transformation
raa,L a = o
b=1
3
habssba + lb
ad. s1d
The three lattice vectors form, in the general case, an unnor-malized and nonorthogonal basis. The fractional coordinatescan be obtained from the Cartesian ones by the inverse trans-formation in terms of the reciprocal lattice vectors,shtd−1
=fa1* ,a2
* ,a3*g, defined byai
* ·aj =di j as
saa = modSo
b
hab−1rb
a,L a,laaD . s2d
The structural deformations in periodic systems arise fromthe change of atomic positions at constant lattice vectors onthe one hand, and from the deformation of the lattice vectorsat constant fractional coordinates. The variation of latticevectors is conveniently described by the strain tensor«ab as
hab8 = og
sdag + «agdhgb. s3d
Any change inhab implies a variation of atomic Cartesiancoordinates by Eq.s1d which obey an analogous relationship
ra8 = ob
sdab + «abdrb. s4d
Out of the total number of 3N+9 deformation variabless3Natomic coordinates and 9 strain tensor elementsd the energyis invariant with respect to six degrees of freedom: the posi-tion of the origin and the orientation of the lattice vectors.Although the origin of the lattice is arbitrary, it is usuallychosen on the basis of symmetry considerations. Overall ro-tations of the lattice can be avoided by taking the symmetricpart of the strain tensor,17 or the metric matrix formed fromthe six unique scalar products of lattice vectors,18 as inde-pendent lattice variables.
Most of the geometry optimization algorithms are basedon a local harmonic expansion of the total energy around aninitial structure. Using exact first and approximate secondderivatives, a convenient step is predicted that takes the sys-tem closer to the desired critical pointsminimum or saddlepointd with vanishing gradients. In periodic systems the sim-plest coordinate system is constituted from the 3N fractionalatomic positions and nine lattice vector component varia-tions, dx=hds,dhj leading to the following expansion:
Esx + dxd − Esxd = − f tdx + 12dxtFdx + ¯ + . s5d
The elements of the column vectorf are either the forces,faa =−dE/dsa
a, or the lattice vector derivatives of the totalenergy, fa
b=−dE/dhab. The latter are related to the stresstensor elements, i.e., the negative volume-normalized strainderivatives of the energy:
sab = −1
V
dE
d«ab
s6d
by the relationshipscf. the Appendixd
dE
dhab
= − Vog
sagshtdgb−1. s7d
Electronic structure codes usually provide Cartesian forcecomponents, which can be transformed to fractional coordi-nate forces as
dE
dsaa = o
b
]E
]rba
]rba
]saa = o
bog
]E
]rba
]
]saa hbgssg
a + lgad
= ob
]E
]rba hba. s8d
It should be noted that in addition to the explicit lattice vec-tor dependence of the total energy, atomic forces contributealso to the strain-derivative tensor, according to the follow-ing expression:
dE
dhab
=]E
]hab
+ oa
]E
]raa
]raa
]hab
=]E
]hab
+ oa
]E
]raa sb
a . s9d
As we shall see later, in Sec. II D, the same relationships canbe obtained directly from the B-matrix formalism, to be de-veloped below.
The extended Hessian matrixF in Eq. s5d is built upfrom several blocks: that of the second derivatives of theenergy with respect to atomic positionssrelated to the dy-namical matrixd, that of the stress-strain derivatives, as wellas cross terms.
B. Curvilinear internal coordinates for periodicsystems
In contrast to the set of external coordinates,dx=hds,dhj, including overall rotations and translations of thesystem, one can define a set of internal coordinates,dj=hdq ,dq̂j, which involve only internal degrees of freedom.Internal atomic coordinates, such as bond lengths, bondangles, torsion angles, etc., or their linear combinations are,in general, nonlinear functions of Cartesian coordinates,qi
= fshraa,L ajd, and by the virtue of the linear relationship Eq.
s1d, also of the fractional coordinates and the lattice vectors.Similarly, internal lattice coordinates, such as cell edgelength, cell angle, volume, etc., are determined by the threelattice vectors,q̂i = fshhabjd.
Internal coordinatedeformationsare related to externalcoordinate deformations by a nonlinearscurvilineard trans-formation, usually approximated by a truncated Taylor ex-pansion
dji = os
]ji
]xsdxs +
1
2ost
]2ji
]xs]rtdrsdrt + ¯
= sBdxdi + 12dxtCidx + ¯ . s10d
Inserting expansions10d into the harmonic expansion of thetotal energy with respect to internal coordinates
124508-2 Bučko, Hafner, and Ángyán J. Chem. Phys. 122, 124508 ~2005!
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Esj + djd − Esjd = − wtdj + 12djtHdj + ¯ s11d
and equating terms of the same order, we obtain a relation-ship between external and internal force/stress components
Btw = f , s12d
while the external coordinate deformations can be obtainedfrom a set of internal distortions by the first-order relation-ship
Bdx = dj. s13d
The matrixB is a generalization of Wilson’s B-matrix16 forperiodic systems. As in the molecular case, the numberM ofinternal coordinates that can be constructed for anN-atomperiodic structure is usually much more than the 3N+3 geo-metrical degrees of freedom. Sincedx, f PR3N+9 and dj, wPRM, BPRM33N+9, the B-matrixsunlessM =3N+3 and theset of internal coordinates is nonredundantd and the solutionsof the above equations are given by the Moore–Penrosepseudoinverses as
w = sBtd+f , s14d
dx = B+dj. s15d
The same B-matrix and its pseudoinverse appear in the trans-formation of the fractionalsFd and internalsHd second de-rivative matrices
F < BtHB , s16d
H < sBtd+FB+, s17d
where the correction term involving the second derivative ofthe internal coordinates with respect to the fractional oneshas been neglected.12
In the case of really large systemss.1000 atomsd, thecoordinate and force transformation steps may become thebottleneck of the optimization procedure. Various methodshave been proposed to solve efficiently the above equationsfor extended systems.19–23 In the present implementation de-localized internal coordinates15,21,24,25 are used, which areparticularly appropriate for medium-sized problems.
C. Delocalized internal coordinates
Even for small systems, the number of generated internalcoordinates is usually much larger than the number of ionicdegrees of freedomsi.e., 3N−6 for molecules and 3N+3 forsystems with three-dimensional periodicityd. The handling ofthis set of redundant coordinates may lead to a significantincrease of the computational time and may cause conver-gence problems in the coordinate back-transformation step.To avoid these problems, Bakeret al. proposed the use ofnonredundant linear combinations of primitive internalcoordinates.24 Taking the singular value decomposition ofthe B-matrix, asB=UtBDV, whereU andV are unitary ma-trices andBD is a diagonal matrix formed from the eigenvec-tors associated with the nonzero eigenvalues ofsBBtd. Aftermultiplication of the B-matrix equation from left byU, oneobtains
BDVx = Uq, s18d
which can be considered as the defining relationship of thedelocalized internal coordinate transformation:
B̃x = q̃ s19d
with B̃=BDV andq̃=Uq. Note thatB̃+, the pseudoinverse of
B̃, is simply VtBD−1. The transformation relations Eqs.
s14d–s17d, for the primitive internal coordinates remain validfor the delocalized internal coordinates after switching from
B to B̃.
D. B-matrix for periodic systems
The construction of the B-matrix for periodic systems,defined by atomic positionand lattice vector components,needs some special consideration as compared to the mo-lecular case. Kudinet al. remarked14 that primitive internalcoordinates involving atoms that belong to different unitcells are explicitly dependent on the lattice vector compo-nentshab. They have written expressions1d in the followingform:
raa,L = ra
a,0 + ob
hablb, s20d
and generated the corresponding B-matrix elements by theapplication of the chain rule. In their formulation only inter-cell coordinatessL Þ0d are allowed to contribute to the lat-tice B-matrix elements. The optimization of lattice param-eters is treated in an indirect manner, through intercelldistances between atoms and their periodic images as well asthrough angles between three replicates of the same atomlying in different unit cells. This procedure seems to be welladapted to simple molecular crystals, but it is much less con-venient for ionic systems, oxides or atomic lattices, wherethe distinction between intracell and intercell coordinates isless obvious. In the following we propose a “democratic”approach taking into consideration thatany unique internalcoordinate deformation may influence the lattice parameters.
Let us consider the augmented B-matrix equation
Sdq
dq̂D = SBqs Bqh
Bq̂s Bq̂hDSds
dhD , s21d
where the individual blocksBqs and Bq̂s describe the lineartransformation of the atomic positionssa
a, while the blocksBqh and Bq̂h describe transformations involving the latticevector componentshab to atomic qi and lattice internalq̂j
coordinates. The blocks of the augmented B-matrix can becalculated from the relationshipss1d and s2d using the chainrule.
An internal coordinate may depend on the Cartesian co-ordinates of atoms in different unit cells. Therefore theB-matrix elements between fractional atomic distortions andunique internal coordinates should be calculated as
124508-3 Geometry optimization of periodic systems J. Chem. Phys. 122, 124508 ~2005!
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Bi,aa
qs =]qisra
a,L a,rbb,L b, . . . d
]saa
= oL a
ob
]qi
]rba,L a
]rba,L a
]saa = o
L a
ob
]qi
]rba,L a
hba. s22d
Disregarding the transformation from Cartesian to fractionalcoordinates, the B-matrix elements for internal coordinatesthat “remain” entirely in the unit cell are identical to those ofa nonperiodical system. If several translated copies of thesame atom participate in a given internal coordinate, the con-tributions from different cells should be summed up. Thisleads to somewhat unexpected consequences. For instance,in the case of a monoatomic lattice, a zero B-matrix elementis obtained with respect to the atomic position, reflecting thefact that in this case the independent variable is the cell pa-rameter itself.
The proportionality between an internal coordinate de-formationdqi and a lattice vector distortiondhab is describedby the following B-matrix elements:
Bi,abqh =
]qisraa,L a,rb
b,L b, . . . d]hab
= oa,L a
og
]qi
]rga,L a
]rga,L a
]hab
= oa,L a
og
]qi
]raa,L a
dagssba + lb
ad
= oa,L a
]qi
]raa,L a
ssba + lb
ad, s23d
where the superscriptqh refers to the type of B-matrix ele-ment.
The “lattice internal” coordinatesq̂, such as cell edgelengths,a/b ratio, cell angles, volume, etc., do not dependexplicitly on the fractional atomic positions, therefore theblock of the extended B-matrix is zero,Bq̂s=0. The transfor-mation between the lattice vector changes and various “in-ternal” lattice parameters is described by the matrix elements
Bi,abq̂h =
]q̂i
]hab
. s24d
For instance, the following relationship holds forq̂i =V, theunit cell volume:
BV,abq̂h = Vshtdab
−1 . s25d
Other specific lattice B-matrix elements can also be derivedfrom the definition of lattice vector lengths and latticeangles.
In order to appreciate the role of the extended B-matrixin lattice optimizations, let us consider the special case ofCartesian coordinates as internals, i.e.,dj=hdr ,dhj. Accord-ing to the coordinate transformation relationship, Eq.s13d,
dr = Brsds+ Brhdh s26d
and using the matrix elements given by Eqs.s22d and s23d,
draa,L a = o
g
hagdsga + o
ab
ssba + lb
addhab, s27d
we retrieve the expected result that the variation of a Carte-sian coordinate can be decomposed into a variation of thefractional coordinate,dsa
a, and a variation of the lattice vec-tor components,dhab.
The transformation equations for the forces and stresses,Eq. s12d, lead to the following relationshipsfcf. Eqs.s8d ands9dg between the energy derivatives with respect to fractionalcoordinates, cell parameters, and Cartesian coordinates:
dE
dsaa = o
b
shabdt ]E
]rba , s28d
dE
dhab
= oa,L a
ssba + lb
ad]E
]raa +
]E
]hab
. s29d
The first of these equations is a simple linear transformationof the force components from a Cartesian to a lattice vectorreference system. The second equation tells that the totalderivatives with respect to the lattice parameters have a con-tribution from the partial derivative of the energy with re-spect to the lattice parameterssexplicit dependenced and a“virial contribution” proportional to the atomic forcecomponents.26 The latter term, which obviously vanishes ifthe atomic forces are zero, is analogous to the expression ofthe virial pressure discussed by several authors in a some-what different context.27–29
E. Constraints
Constrained geometry optimizations are helpful andeven necessary in most of the applications ofab initio cal-culations to chemical reactions, phase transitions, etc. Thegeneral strategy is to make vanish, exactly or at least ap-proximately, forces along the constrained coordinates toavoid deformations involving the variation of the constrainedcoordinates. The principal advantage of using internal coor-dinates is that one can imposeexactinternal coordinate con-straints during the optimization. Various algorithms, such asthe use of projection operator,30 orthogonalization,24 andLagrange multiplier31 techniques have been proposed in thepast. Our implementation follows essentially the orthogonal-ization algorithm of Ref. 24.
In the first step, the B-matrix is modified in such a waythat first derivatives of the active coordinatessthose coordi-nates which are allowed to be relaxedd are orthogonalizedwith respect to each constrained coordinateqc. The rowsB j
are modified according to the formula
B̄ j = B j − oc
sB j ·BcduBcu
Bc
uBcu, s30d
where the summation is over the constrained coordinates. Ifone of the rows of the original B-matrix is identical to aconstrained coordinate, it is exactly annihilated in this step.
Delocalized internal coordinates and corresponding gra-
dients are generated from the modified matrixB̄, as de-scribed in previous sections. The delocalized coordinates are
124508-4 Bučko, Hafner, and Ángyán J. Chem. Phys. 122, 124508 ~2005!
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now linear combinations of either constrained, or active co-ordinates, but not of both. Finally, the gradients for delocal-ized coordinates corresponding to constrained coordinatesare set to zero. The optimization then proceeds as in the caseof an unconstrained relaxation. Not only single primitive in-ternal coordinates but also the ratios and sums of primitiveinternal coordinates, the norms of vectors whose componentsare primitive internal coordinatesse.g., vibrational modesd,lattice parameters, the volume of the unit cell and many othercoordinate types can be constrained. Note that Cartesian co-ordinates can be taken as specialssingle-bodyd primitive in-ternal coordinates.
Special attention should be paid to the constrained opti-mizations where the lattice parameters are allowed tochange, but some of the internal degrees of freedom are fro-zen. The components of the Cartesian forces that correspondto the fixed internal coordinates should be subtracted fromthis expression of the stress tensor, in order to avoid their“contamination.” Our B-matrix technique, if used consis-tently, takes care of these effects.
F. Optimization strategy
The minimization algorithm used in this study is basedon the geometrical direct inversion in the iterative subspacesDIISd method by Császár and Pulay32 improved recently byFarkas and Schlegel.33 In this method the information on thepotential energy surface collected in the precedingM relax-ation steps is used to minimize the norm of the error vectordefined as a linear combination of gradientsw̃ correspondingto delocalized internal coordinatesq̃,
r k = oi=k−M
k
ciw̃i . s31d
The coefficientsci are obtained by minimizingur u2 underthe constraintoi=k−M
k ci =1 leading to a set of equations
1w̃k−M
t w̃k−M . . . w̃k−Mt w̃k 1
. . . . . .
. . . . . .
. . . . . .
w̃kt w̃k−M . . . w̃k
t w̃k 1
1 . . . 1 0
21ck−M
.
.
.
ck
l
2 =10
.
.
.
0
1
2 .
s32d
A new set of delocalized internal coordinates is calculatedusing
q̃k+1 = oi=k−M
M
ciq̃i + H̃−1 oi=k−M
M
ciw̃i . s33d
The dimensionM of the iterative subspace used in DIIShas substantial impact on efficiency of the relaxation. Whenthe structure is close to the minimumsi.e., in the harmonicregiond M should be relatively highsten or mored. Whenstarting from a poor guess and the landscape of the potentialenergy is not well described by a harmonic approximation, abetter performance can be achieved by using only a limitednumber of previous relaxation steps.
Farkas and Schleger33 suggested an improved DIIS algo-rithm, which allows one to adjust automatically the dimen-sion of the iterative subspace. The idea is that the ionic stepproduced by DIISDq̃= q̃k+1− q̃k is compared to a simple
quasi-Newton stepDq̃QN=H̃−1w̃ and should meet the follow-ing criteria.
sid The direction of the DIIS stepDq̃ deviates fromDq̃QN
by an anglef. The step is accepted if cossfd is larger than0.97, 0.84, 0.71, 0.67, 0.62, 0.56, 0.49, and 0.41 for two tonine recent relaxation steps used in DIIS. For a dimension often or higher this criterion is not taken into account.
sii d The norm of DIIS step is limited to be not more thanten times larger than that of the reference step.
siii d The sum of all positive coefficientsci should notexceed the value of 15.
sivd The magnitude ofc/ ur u2 should not exceed 108.
If one of these criteria is not fulfilled, the step is notaccepted and the most remote vector is removed from theiterative subspace. This procedure is repeated until all crite-ria are fulfilled.
One of the major advantages of the use of internal in-stead of Cartesian coordinates is that a reasonable guess forHessian matrix can be constructed. Even a very simplemodel Hessian which is just a diagonal matrix with elements0.5, 0.2, and 0.1 a.u. for bonds, angles, and torsions, respec-tively, usually works very well. Lindhet al.34 proposed amodel Hessian, constructed from force constants that aresimple functions of nuclear positions
kij = krri j , s34d
kijk = kfri jr jk, s35d
kijkl = ktri jr jkrkl, s36d
with
ri j = expfai jsr0,i j2 − r ij
2dg. s37d
For the first three rows of the periodic table one needs alto-gether 15 independent parameters for the quantitieskr, kf, kt,ai j , andr0,i j that are collected in Table I. The model Hessiancan be easily transformed to delocalized internal coordinatesusing the formula
H̃ = UtHU . s38d
In the course of the relaxation, the Hessian matrix isupdated using the Broyden–Fletcher–Goldfarb–Shanno algo-rithm:
H̃k = H̃k−1 − S Dw̃kDw̃kt
Dw̃k−1t Dq̃k
+H̃k−1Dq̃k−1Dq̃k−1
t H̃k−1
Dq̃k−1t H̃k−1Dq̃k−1
D , s39d
where Dw̃k=w̃k−w̃k−1 is the change of gradients associatedwith the the relaxation stepDq̃k= q̃k− q̃k−1.
The performance of our optimization engineGADGET ischecked against the “native” Viennaab initio simulationpackagesVASPd optimizer using the conjugate gradient al-gorithm and performed in cartesian coordinates. The conju-
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gate gradient algorithm tries to improve a simple steepestdescent stepsi.e., in the direction of gradientd by conjugatingthe search direction from the most recent step. It starts withthe steepest descent direction, but in all the following stepsthe search directionssk+1d is given by
g =fk + fk−1 · fk
fk−1 · fk−1, s40d
sk+1 = fk + gsk. s41d
The conjugate gradient algorithm requires the line mini-mization along search direction.
III. APPLICATIONS
The electronic structure calculations were done with theVASP,1 using the density functional theory in the generalizedgradient approximation. The PW91 exchange-correlationfunctional was used, using the projector-augmented waveformalism.35,36 The calculations were done with high preci-sion, i.e., the wave function was developed on a plane wavebasis with a cutoff of 800 eV, and the support grid for therepresentation of the charge density was sufficiently preciseto avoid any wraparound errors. The electronic wave func-tions were converged in each step to 2.7310−8 hartree. It isexpected that these computational parameters allow us toobtain quite reliable forces.
The geometry optimizations were done with the externaloptimizerGADGET, written in Python.GADGET reads the ge-ometry, energy, and gradients from the VASP output, sets upinternal coordinates, estimates an optimal move, calculatesthe new set of lattice parameters and Cartesian coordinates,and starts a new VASP calculation, until convergence. As weshall see, the additional overhead related to the repeated re-starts of VASP is largely compensated by the gain in thenumber of iterations. Our optimizer can be easily interfacedwith other packages that calculate total energies and forces/stresses, and such an interface is already operational with theGAUSSIAN 03package.5 Optimizations were carried out usingthe geometrical DIIS method with an iterative subspace ofdimension 5. Convergence criteria involve the simultaneousfulfillment of an energy change less than 1310−6 a.u., amaximal gradient less than 2310−4 a.u., and a volumechangesin lattice relaxationsd that is smaller than 0.05%.
In lattice parameter optimizations, involving volumechanges, we are faced with the problem of “Pulay stress,” i.e.
the plane wave basis-set incompleteness error.37 The appliedcutoff energy of 800 eV is pretty high and it is usually as-sumed that in this case the Pulay stress is negligible. Some-what unexpectedly, we have found relatively small but nev-ertheless significant residual stresses at the equilibriumvolume obtained by fitting a Murnaghan equation of state toa series of fixed-volume relaxed structures. This residualstresssPulay stressd can be added as a constant in the auto-matic relaxation procedure, in order to correct the finite-basiserror in the direct lattice optimizations.
Different kinds of optimizations were performed, allstarting from the same initial geometry.sid Only the atomicpositions are optimized at constant lattice parameters, usingdelocalized internal coordinatessGADGETd. sii d Atomic posi-tion relaxation with the native VASP optimizer.siii d Simul-taneous atomic parameter and lattice parameter optimizationin delocalized internal coordinates withGADGET. sivd Fullatomic position and lattice parameter relaxation with VASP.svd Multistep geometry relaxation withGADGET using a se-ries of fixed-volume relaxations to determine a Murnaghan-type equation of statesEOSd and reoptimizing the structureat the volume of the interpolated minimum.svid Full delo-calized internal coordinate relaxation using the Pulay stressdeduced from the EOS optimization.
A. Microporous material: SiO 2 chabazite
ChabazitesFig. 1d is microporous aluminosilicate min-eral szeolited in which every SisAl d atom is coordinated byfour O atoms. The space groups of the purely siliceous cha-
TABLE I. Parameters of the model Hessian proposed by Lindhet al. sRef. 34d. Indicesi and j designate therows of the periodic table of elements to which the atoms correspond. The universal stretch, bend, and torsioncoefficients arekr =0.45,kf=0.15, andkt=0.005. All quantities are given in atomic units.
First period Second period Third period
ai j First period 1.0000 0.3949 0.3949Second period 0.3949 0.2800 0.2800Third period 0.3949 0.2800 0.2800
r0,i j First period 1.35 2.10 2.53Second period 2.10 2.87 3.40Third period 2.53 3.40 3.40
FIG. 1. Rhombohedral unit cell of purely siliceous chabazite, Si12O24.
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bazite isR3̄m sNo. 166d, however, we have chosen a slightlyperturbed initial configuration, corresponding to a highly sili-ceous protonated framework.38 The primitive unit cell con-tains 12 SisAl d atoms and 24 O atoms giving a total of 288primitive internal coordinatessbonds, angles, and torsionsd.The results obtained after various types of relaxations aresummarized in Table II. In every case our algorithm outper-formed the native VASP optimizer by a factor of 2–3. Theminimum of the full relaxation is checked against a fit to theMurnaghan equation. In spite of the high plane wave cutoff,the effect of the Pulay stress is not negligible: the residualstress was found to be around −0.6 kB. The cell volume forthe resulting configuration is by,1 Å3 too low compared tothe true minimum.
The underestimation of the equilibrium cell volume dueto the Pulay stress is a quite general phenomenon observedin all examples given here. The full relaxation with compen-sation for a Pulay stress leads to the structure which is veryclose to a minimum of the Murnaghan equation. The sym-metry of the unit cell is not completely recoveredsaÞbÞcd in full relaxations, due to the fact that the lattice param-eters are usually “softer” than the atomic degrees of freedomand thus more difficult to relax. In order to relax to a truesymmetry, a more stringent relaxation criterion for the stresstensor would be required.
B. Layered oxide: Gibbsite
Unlike chabazite, gibbsitefAl sOHd3g ssee Fig. 2d is alayered mineral.39 The individual layers are charge balancedwhich results into very weak interlayer interactions. Theprimitive cell of gibbsite is monoclinic with space groupsymmetryP121/n1 sNo. 14d. There has been several recentab initio structural studies on gibbsite and other AlsOHd3
polymorphs40,41 and we are currently working on the theo-
retical vibrational spectrum of gibbsite and bayerite in lightof some new experimental results.42
In this system the conventional primitive internal coor-dinatessbonds, angles, and torsionsd are insufficient to de-scribe all atomic degrees of freedom. In order to describecorrectly the mutual position of subsequent layers we haveused inverse power distance coordinatess5/r coordinatesdproposed by Baker and Pulay.43 The 5/r coordinates weregenerated only for pairs of atoms lying in different structuralfragmentsslayersd and satisfying the condition that the cor-responding interatomic distance is smaller than 1.6 times thesum of the covalent radii. In this way we have generated 664conventional primitive and 179 inverse-power distance coor-dinates. Data for the initial configuration and for the relaxedstructures are collected in Table III. The performance of ourmethod is by a factor of 10 better than the conjugate gradient
TABLE II. Optimization of the chabazite structure: the initial configurationsInitiald; relaxation of the ionic degrees of freedom using delocalized internalcoordinatesshqjd and using the VASP native optimizershsjd; relaxation of the lattice parameters and ionic positions using delocalized internal coordinatesshq ,q̂jd and with the VASP native optimizershs,hjd; minimum of the fitted Murnaghan equation of statesEOSd; and the full relaxation using delocalizedinternal coordinates with stress corrected for a Pulay stressshq ,q̂ ;sjd. Listed are cell parametersfa, b, c sÅd; a, b, g sdeg.dg, energyshartreed, gradientsa.u.d,stressskBd, cell volumesÅ3d, and the number of relaxation steps.
Parameter Initial hqj hsj hq ,q̂j hs,hj EOS hq ,q̂ ;sj
a 9.301s9d 9.349s7d 9.354s1d 9.349s6d 9.351s6db 9.307s2d 9.344s5d 9.380s0d 9.349s6d 9.347s6dc 9.303s9d 9.347s7d 9.347s6d 9.349s9d 9.350s1da 94.122s4d 94.263s5d 94.113s4d 94.179s4d 94.158s3db 94.058s9d 94.286s7d 94.228s1d 94.186s0d 94.179s9dg 94.104s5d 94.280s5d 94.166s0d 94.187s8d 94.176s0dE −10.378 55 −10.535 66 −10.535 66 −10.537 09 −10.537 06 −10.537 11 −10.537 10gmax 0.132 18 0.000 15 0.000 15 0.000 11 0.000 16 0.000 05 0.000 14sxx 36.30 9.74 9.49 0.00 0.10 −0.59 −0.59syy 35.74 9.12 8.92 0.01 0.07 −0.59 −0.58szz 17.15 9.37 9.19 0.00 0.07 −0.62 −0.62sxy 2.17 -0.31 -0.60 0.00 0.01 −0.09 −0.09syz 24.34 −0.33 −0.59 −0.01 −0.02 −0.12 −0.13szx 4.17 −0.41 −0.49 0.00 −0.03 −0.10 −0.10V 799.00 799.00 799.00 809.50 809.69 810.44 810.51Nion 22 70 25 62 25
FIG. 2. Crystal structure of gibbsite, AlsOHd3.
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algorithm implemented in VASP. The full relaxation with thecompensation for a Pulay stress ofh−0.64,−1.81,−0.50jleads to the configuration close to the minimum obtainedfrom the Murnaghan equation of state.
C. Molecular crystal: Urea
Urea fsNH2d2COg sFig. 3d is an example of a molecular
crystal sspace groupP4̄21m, No. 113d, with strong electro-static and hydrogen bond interactions. In the crystal, the mol-ecules occupy a special position compatible with the fullmolecular symmetrysmm2d. High-resolution x-ray44 andneutron45,46 diffraction studies show small differences in thecell parameters, but some significant changes in the atomicpositions.
The conventional primitive internal coordinates are gen-erated only for intramolecular degrees of freedom, the mu-tual positions of individual urea molecules are described us-ing the 5/r coordinates generated as described in thepreceding section. Total number of 152 conventional primi-
tive and 160 inverse-power distance coordinates have beenused to generate the delocalized internal coordinates. Thedata concerning this example are collected in Table IV. Theinitial configuration has already the correct space group sym-
metry sP4̄21md. As pointed out by Bakeret al.,25 delocalizedinternal coordinates are automatically symmetry adapted.Therefore the symmetry is preserved if the relaxation isstarted with correct symmetry. As in the previous examples,Pulay stress causes the underestimation of the equilibriumcell volume. The structure relaxed with stress compensatedfor this effect is very close to the minimum of the Mur-naghan fit. The performance of our algorithm is by a factorof 5 better than the native optimizer in VASP.
D. Constrained relaxation: Urea
To illustrate the ability of our algorithm to impose virtu-ally any geometrical constraint on the structure, we havechosen example of urea with fixed intramolecular angles andtorsions. The relaxations have been started from the sameconfiguration as in the previous example. The relaxation cri-teria are the same as in the unconstrained relaxations. Note,however, that the contributions to the cartesian gradients cor-responding to the constrained coordinates had to be projectedout. Results of relaxations of atomic positions, atomic posi-tions, and lattice parameters, data for the minimum of theMurnaghan equation of state and for the relaxation with thecorrection for the Pulay stresssestimated in the uncon-strained relaxationsd are shown in Table V.
IV. CONCLUSIONS
We have shown that the geometry optimization of peri-odic systems in internal coordinates offers a highly advanta-geous alternative to perform structural relaxations of solids
TABLE III. Optimization of the gibbsite structure: the initial configurationsInitiald; relaxation of the ionic degrees of freedom using delocalized internalcoordinatesshqjd and using the VASP native optimizershsjd; relaxation of the lattice parameters and ionic positions using delocalized internal coordinatesshq ,q̂jd and with the VASP native optimizershs,hjd; minimum of the fitted Murnaghan equation of statesEOSd; and the full relaxation using delocalizedinternal coordinates with stress corrected for a Pulay stressshq ,q̂ ;sjd. Listed are cell parametersfa, b, c sÅd; a, b, g sdeg.dg, energyshartreed, gradientsa.u.d,stressskBd, cell volumesÅ3d, and the number of relaxation steps.
Parameter Initial hqj hsj hq ,q̂j hs,hj EOS hq ,q̂ ;sj
a 8.497s8d 8.779s2d 8.769s6d 8.774s9d 8.776s6db 4.977s5d 5.103s5d 5.102s8d 5.113s3d 5.111s2dc 9.378s5d 9.711s8d 9.699s6d 9.714s8d 9.716s3da 90.230s4d 90.024s7d 90.016s7d 89.982s6d 90.032s2db 91.955s9d 92.157s8d 92.153s5d 92.130s0d 92.142s3dg 89.995s0d 89.991s8d 89.995s3d 89.997s5d 89.987s2dE −12.037 47 −12.038 57 −12.038 52 −12.065 39 −12.065 33 −12.065 42 −12.065 41gmax 0.047 95 0.000 10 0.000 18 0.000 12 0.000 30 0.000 07 0.000 12sxx 62.48 64.80 64.64 0.01 1.09 −0.64 −0.59syy 62.08 63.06 63.07 0.15 1.02 −1.81 −1.69szz 59.54 64.03 64.06 −0.07 1.19 −0.50 −0.48sxy -0.10 0.07 0.00 0.05 0.00 0.01 −0.07syz 0.27 0.43 0.49 0.01 0.04 0.04 0.05szx −1.50 −2.08 −2.30 0.02 0.01 −0.02 0.05V 396.46 396.46 396.46 434.83 433.74 435.59 435.56Nion 13 87 15 155 15
FIG. 3. Tetragonal unit cell of urea,sNH2d2CO.
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and surfaces. It has been demonstrated that the presentimplementation is able to handle not only atomic position butalso lattice parameter relaxations. The final geometries ob-tained by the native optimizer of VASP and the internal co-ordinate optimizerGADGET are the same within the numeri-cal precision of the electronic code. For the selected modelsystemsGADGET outperforms the native optimizer by a fac-tor of 2–10. One has to be aware of the fact that a consider-able portion of this better performance can be ascribed to thequality of the initial Hessian guess and to the use of moreefficient optimization algorithms.
The most significant advantage of the internal coordinateoptimization lies undeniably in the handling of constraints,as demonstrated by the simple examples of rigid moleculeoptimization of the urea molecular crystal. In any case,the possibility of constraining virtually any physically/chemically motivated combination of internal coordinates
opens a broad field of application of periodic electronicstructure codes to complicated problems, such as phase tran-sitions, solid state chemical reactions, etc.
The internal coordinate optimizerGADGET is an indepen-dent software, easy to interface with various molecular andsolid state total energy codes. It allows the user to optimizenot only periodic but also finitesmoleculard systems in delo-calized internal coordinates by ignoring the lattice parameterblock of the extended B-matrix, i.e., by the standard methodwidely documented in the literaturessee, e.g., Ref. 12d. Inaddition to the innovative treatment of periodic systems, theGADGET program includes a large selection of algorithmspermitting the efficient handling of Hessian update, transi-tion state optimization, etc. An exhaustive overview of thesefeatures will be the subject of a forthcoming paper that de-scribes the technical aspects of our implementation of thesealgorithms inGADGET.
TABLE IV. Optimization of the urea structure: the initial configurationsInitiald; relaxation of the ionic degrees of freedom using delocalized internalcoordinatesshqjd and using the VASP native optimizershsjd; relaxation of the lattice parameters and ionic positions using delocalized internal coordinatesshq ,q̂jd and with the VASP native optimizershs,hjd; minimum of the fitted Murnaghan equation of statesEOSd; and the full relaxation using delocalizedinternal coordinates with stress corrected for a Pulay stressshq ,q̂ ;sjd. Listed are the cell parametersfa, b, c sÅd; a, b, g sdeg.dg, energyshartreed, gradientsa.u.d, stressskBd, cell volumesÅ3d, and the number of relaxation steps.
Parameter Initial hqj hsj hq ,q̂j hs,hj EOS hq ,q̂ ;sj
a 5.565s0d 5.758s2d 5.744s0d 5.790s6d 5.788s6db 5.565s0d 5.758s2d 5.744s0d 5.79s6d 5.788s6dc 4.684s0d 4.696s9d 4.695s5d 4.705s9d 4.703s6dE −3.567 49 −3.579 44 −3.579 42 −3.580 78 −3.580 75 −3.580 81 −3.580 81gmax 0.075 04 0.000 03 0.000 18 0.000 04 0.000 13 0.000 04 0.000 17sxx 79.14 9.38 9.84 −0.02 0.83 −1.17 −1.18syy 79.14 9.38 9.84 −0.02 0.83 −1.17 −1.18szz 145.60 8.18 8.63 0.04 −0.02 −1.75 −1.42sxy 0.00 0.00 0.00 0.00 0.00 0.00 0.00syz 0.00 0.00 0.00 0.00 0.00 0.00 0.00szx 0.00 0.00 0.00 0.00 0.00 0.00 0.00Volume 145.06 145.06 145.06 155.73 154.92 157.80 157.61Nion 6 27 9 49 9
TABLE V. Relaxation of urea with constrained intramolecular angles and torsions: the initial configurationsInitiald; relaxation of the ionic degrees of freedomusing delocalized internal coordinatesshqjd; relaxation of the lattice parameters and ionic positions using delocalized internal coordinatesshq ,q̂jd; minimumof the fitted Murnaghan equation of statesEOSd; and the relaxation of the lattice parameters and ionic positions using delocalized internal coordinates withstress corrected for a Pulay stressshq ,q̂ ;sjd. Listed are cell parametersfa, b, c sÅd; a, b, g sdeg.dg, energyshartreed, gradientsa.u.d, stressskBd, cell volumesÅ3d, and the number of relaxation steps.
Parameter Initial hqj hq ,q̂j EOS hq ,q̂ ;sj
a 5.565s0d 5.768s0d 5.802s4d 5.802s7db 5.565s0d 5.768s0d 5.802s4d 5.802s7dc 4.684s0d 4.710s7d 4.717s3d 4.716s9dE −3.567 49 −3.578 96 −3.580 54 −3.580 57 −3.580 57gmax 0.075 04 0.000 03 0.000 08 0.000 08 0.000 05sxx 79.14 15.17 3.53 2.18 2.12syy 79.14 15.17 3.53 2.18 2.12szz 145.60 1.20 −7.23 −8.23 −8.43sxy 0.00 0.00 0.00 0.00 0.00syz 0.00 0.00 0.00 0.00 0.00szx 0.00 0.00 0.00 0.00 0.00Volume 145.06 145.06 156.72 158.82 158.82Nion 6 19 17
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ACKNOWLEDGMENT
One of the authorssJ.G.A.d is indebted to the Computa-tional Materials Science College for having supported hisstay in Vienna.
APPENDIX: THE STRESS TENSOR
During a deformation of the lattice, all atomic positionsand virtual grid points, including the lattice vector endpoints,change according to the relationship
va8 = ob
sdab + «abdvb, sA1d
where«ab are the elements of the strain tensor. The stresstensor elementssab
are usually defined as the volume-normalized negative strain derivatives of the energy:
sab = −1
V
]E
]«ab
. sA2d
The Cartesian to fractional transformation matrixhab isformed from the three column vectorsab of the lattice ashab=sabda. Using Eq.sA1d it follows that the variation of thelattice vectors can be expressed as
hab8 = og
sdag + «agdhgb sA3d
and the strain derivative of the lattice vectors is
]hab
]«mn
= dmahnb. sA4d
The strain derivative of the energy can be written in terms ofthe lattice vector component derivatives, using Eq.sA4d, as
]E
]«ab
= omn
]E
]hmn
]hmn
]«ab
= on
]E
]han
hbn sA5d
leading to the alternative form of the stress tensor
sab = −1
Vog
]E
]hag
hgbt . sA6d
The lattice-vector derivatives of the energy can be obtainedfrom the stress tensor, since
− Vob
sabshtdbe−1 = o
gb
]E
]hag
hgbt shtdbe
−1, sA7d
so
]E
]hab
= − Vog
sagshtdgb−1. sA8d
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