Energy gradients with respect to atomic positions and cell parameters for the Kohn-Sham density-functional theory at the Γ point

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Energy gradients with respect to atomic positions and cell parameters for the Kohn-Sham density-functional theory at the ⌫ point

Valéry Weber

Department of Chemistry, University of Fribourg, 1700 Fribourg, Switzerland

and Los Alamos National Laboratory, Theoretical Division, Los Alamos, New Mexico 87545 Christopher J. Tymczak^a兲and Matt Challacombe

Los Alamos National Laboratory, Theoretical Division, Los Alamos, New Mexico 87545

The application of theoretical methods based on density-functional theory is known to provide atomic and cell parameters in very good agreement with experimental values. Recently, construction of the exact Hartree-Fock exchange gradients with respect to atomic positions and cell parameters within the⌫-point approximation has been introduced关V. Weberet al., J. Chem. Phys. 124, 214105 共2006兲兴. In this article, the formalism is extended to the evaluation of analytical ⌫-point density-functional atomic and cell gradients. The infinite Coulomb summation is solved with an effective periodic summation of multipole tensors关M. Challacombeet al., J. Chem. Phys. 107, 9708 共1997兲兴. While the evaluation of Coulomb and exchange-correlation gradients with respect to atomic positions are similar to those in the gas phase limit, the gradients with respect to cell parameters needs to be treated with some care. The derivative of the periodic multipole interaction tensor needs to be carefully handled in both direct and reciprocal space and the exchange-correlation energy derivative leads to a surface term that has its origin in derivatives of the integration limits that depend on the cell. As an illustration, the analytical gradients have been used in conjunction with the

QUICCAalgorithm关K. Németh and M. Challacombe, J. Chem. Phys. 121, 2877共2004兲兴to optimize one-dimensional and three-dimensional periodic systems at the density-functional theory and hybrid Hartree-Fock/density-functional theory levels. We also report the full relaxation of forsterite supercells at the B3LYP level of theory.

I. INTRODUCTION

The Kohn-Sham approach to density-functional theory has been proven to be a highly competitive method for a wide range of applications in solid-state physics and chem- istry. The hybrid Hartree-Fock/density-functional theory 共hybrid-HF/DFT兲 model chemistries are an important next step in accuracy beyond the generalized gradient approximation.^1–4

In preceding papers, we have developed linear scaling quantum chemical methods for construction of the periodic Coulomb, exchange-correlation,⁵and the exact Hartree-Fock exchange⁶matrices within the⌫-point approximation. In this paper, the implementation of the Coulomb and exchange- correlation energy gradients with respect to atomic positions and cell parameters at the⌫point is presented.

While the ⌫-point approximation uses only the k= 0 point to sample the Brillouin zone, it does, however, con- verge to thek-space integration limit, in the worst case with the inverse unit cell volume共see for example Refs.7and8兲. Convergence of the ⌫-point approximation to the corre- spondingk-space limit was recently demonstrated by us for DFT,⁵ HF, and hybrid-HF/DFT共Ref.6兲theories, as well as for the Hartree-Fock atomic and cell gradients.⁹ For one-

dimensional fluoric acid chains 共HF兲n, convergence of the Hartree-Fock minimum image convention共HF-MIC兲⌫-point energy, atomic, and cell forces to the converged large cell

⌫-point approximation with respect to cell length have been explicitly shown to be exponential in the cell size.⁹ A fast convergence of the total energy and geometrical parameters have also been observed for three-dimensional共3D兲systems such as MgO and urea.^5,6,9In general, the⌫-point approach enables study of very large complex and disordered systems such as liquids, low concentration defects, adsorption of large molecule on surfaces, etc., where conventional methods of sampling the Brillouin zone may become computationally too demanding, and where the⌫-point approximation is well justified.

Finding crystal structures of condensed systems can be formulated as a minimization of the total energy with respect to atomic coordinates and cell vectors. The problem is then minimization of the total energy with Ldegrees of freedom, whereL= 3N_atm+ 3;N_atmis the number of atoms, 3N_atm− 3 is the number of independent coordinates after the elimination of translation, and the number of independent vector ele- ments after the elimination of cell rotations is 6. This mini- mization can be achieved with the help of an efficient optimizer^10–12and knowledge of the gradients with respect to atomic positions and cell parameters.

The analytical cell gradient method of density-functional

a兲Electronic mail: [email protected]

Published in "The Journal of Chemical Physics 124: 224107, 2006"

which should be cited to refer to this work.

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theory using Gaussian-type atomic orbital共GTAO兲 for one- dimensional extended systems was implemented by Hirata and Iwata.¹³ The 3D case has been implemented by Kudin and Scuseria.^14,15Their approach for the Coulomb problem is based on the direct space fast multipole method. Recently Doll et al.¹⁶ presented an implementation of Hartree-Fock cell gradients¹⁷ in the CRYSTAL03共Ref. 18兲 package for 3D systems. Their code is based on GTAO and the summation of the Coulomb energy is performed with the Ewald method,¹⁹ which is a combination of direct and reciprocal cell summa- tions. The strategy to compute the analytic Hartree-Fock gra- dients for periodic systems, in the frame of the CRYSTAL03

package has been presented by Dollet al.^20,21 Their imple- mentation is based on the Hermite-Gaussian-type functions in the context of the McMurchie-Davidson algorithm.²²

This paper describes the complete implementation of the atomic and lattice gradients within the Kohn-Sham approach to density-functional theory at the⌫point. The infinite Cou- lomb summation is solved with an effective periodic summa- tion of multipole tensors.^5,23We show that, like for the exact HF exchange, the evaluation of the atomic gradients within the ⌫-point approximation does not lead to special difficul- ties. The implementation requires mainly the evaluation of the derivative of the basis function with respect to atomic positions. However, difficulties arise when the derivative of the total energy is taken with respect to lattice parameters.

For the periodic far-field Coulomb term, the derivative of the multipole interaction tensor needs to be carefully handled in both direct and reciprocal space. The exchange-correlation energy derivative leads to a surface term, which has its origin in derivatives of the integration limits that depend on the cell. We finally demonstrate the convergence of geometrical parameters and total energy for a few periodic systems within the DFT and hybrid-HF/DFT ⌫-point supercell ap- proach. The full relaxation of the geometrical parameters were done with theQUICCAalgorithm.^10,12

The remainder of this paper is organized as follows: In Sec. II, we introduce the formalism and discuss the imple- mentation of the Coulomb and exchange-correlation gradi- ents with respect to atomic positions and cell parameters at the⌫-point approximation. Then full optimization of several 3D periodic systems are given in Sec. III as an illustration of the formalism. Finally in Sec. IV we summarize our results.

II. FORMALISM

The primitive cell can be represented by the three vec- tors a, b, and c. Let us define M as the 3⫻3 matrix com- posed of the primitive cell vectors,

M=共a,b,c兲.

The position of any replicated cell is R共n兲=Mn, with n

=共na,nb,nc兲, a vector of integers. The position of atomA in the cell R共n兲 is A=M共fA+n兲, with fA=共fAa,fAb,fAc兲, the fractional coordinates of atomAin the central cell.

An unnormalized Cartesian-Gaussian-type function 共CGTF兲centered on atomAis

␾a共r兲=共x−Ax兲^ax共y−Ay兲^ay共z−Az兲^aze^{−␨a共r−A兲2},

where the triada=共a_x,ay,az兲sets the angular symmetry and the exponent ␨a is chosen to describe a particular length scale. Gaussian basis functions are often contracted to ap- proximate atomic eigenfunctions.

The total energy within the⌫-point approximation^5,6can be expressed as

E共fA,fB, . . . ,M兲=

兺

_ab ^Pab

^共

^{Tab+12共Jab+␣1Kab兲}

^兲

+␣2E^xc+h_nuc, 共1兲 wherePis the density matrix,Tis the kinetic-energy matrix, Jis the electron-electron and electron-nuclear Coulomb ma- trix, E^xc is the exchange-correlation energy, K is the exact Hartree-Fock exchange matrix, and h_nuc is the nuclear- nuclear and nuclear-electron energy. The factors ␣1 and␣2

are mixing coefficients of the exact Hartree-Fock exchange and exchange-correlation energies, respectively.

The derivative of the total energy, Eq.共1兲, with respect to a general external perturbation␭is

⳵^E

⳵␭ =

兺

_ab ^Pab

冏

^⳵⳵^T␭^ab+⳵^hnuc

⳵␭

冏

P

+1

2

兺

_ab ^Pab

冉 冏

^⳵⳵^J␭^ab

冏

P

+

冏

^␣1⳵⳵^K␭^ab

冏

P

冊

⁺

冏

^␣2⳵⳵^E␭^xc

冏

P

−

兺

_ab ^Wab⳵_⳵^S_␭^ab,

where 兩兩P means that P is held constant, W=PFP is the energy-weighted density matrix, andFthe Fockian or Kohn- Sham Hamiltonian. The gradients with respect to fractional coordinates and cell parameters are simply given by ␭=fGj

and␭=Mij, respectively. The Hartree-Fock exchange gradi- ents with respect to atomic and cell parameters, i.e.,兩⳵^Kab/⳵␭兩P, have been previously derived.⁹

The energy gradient with respect to the fractional coor- dinate fGj can be obtained through the linear transform

⳵^E

⳵^fGj

=

兺

i=x,y,z

M_ij⳵^E

⳵^Gi

,

where⳵^E/⳵^Giis the standard gradient with respect to atomic position. In the following, we describe the implementation of the Coulomb and exchange-correlation gradients with re- spect to atomic and cell parameters.

A. Coulomb integrals

The periodic quantum Coulomb sum involves the three contributions共for more detail about the formalism see Refs.

5 and23兲,

Jab=J_ab^In+J_ab^PFF+J_ab^TF,

whereJ_ab^In is the inner cell set sum evaluated by the quantum chemical tree code 共QCTC兲,²⁴ J_ab^PFF is the periodic far-field term andJ_ab^TFis a surface term which corrects the boundary at infinity.

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1. Inner sum

The inner cell set contribution to the Coulomb matrix is J_ab^In=_R

兺

苸In

冕冕

^{␳ab共r兲兩r−r⬘+R兩−1␳tot共r⬘兲dVdV⬘, 共2兲}

where

␳ab共r兲=

兺

R⬘␾a共r兲␾b共r+R⬘兲,

␳tot共r兲=

兺

_ab ^{Pab␳ab共r兲−}

兺

_A ^{ZA␦共r−A兲,}

the indexAruns over the atoms in the cell,ZA is the charge of atomA, and ␦共r兲is the Dirac delta function. The deriva- tives of Eq. 共2兲with respect to atomic position Giand cell parameterMijare

冏

^⳵⳵^JGabIni

冏

P

= 2_R

兺

_苸In

冕冕

^⳵␳⳵^abG共i^r兲

兩r−r⬘ +R兩⁻¹␳tot共r⬘兲dVdV⬘^, and

冏

⳵^⳵MJabInij

冏

P

=_R

兺

_苸In

冕冕 冉

^2⳵␳⳵^abM共ij^r兲

兩r−r⬘^+R兩⁻¹

+␳ab共r兲⳵兩r−r⬘^+R兩−1

⳵^Mij

冊

^{␳tot共r⬘兲dVdV⬘,}

where the factors of two arise from the bra/ket symmetry; the derivatives ⳵␳ab/⳵^Gi, ⳵兩r−r⬘^+R兩−1/⳵^Mij and ⳵␳ab/⳵^Mij are straightforward and do not need to be addressed in more detail. Note that similar terms arise forh_nucand are treated in an equivalent way asJ_ab^In.

2. Periodic far field

We use a spherical multipole method in order to compute the periodic far field correction to the Coulomb matrix,^5,23 which is

J_ab^PFF=

兺

lml⬘^m⬘

共− 1兲^l␳ab

lmM_l+l^m+m_⬘^⬘␳tot

l⬘^m⬘, 共3兲

where Ml

m is an effective multipole interaction tensor that accounts for the direct infinite lattice summation. The spheri- cal densities ␳ab

lm and ␳tot

lm are obtained by projecting ␳ab共r兲 and␳tot共r兲, respectively, onto the regular spherical harmonics O_l^m共r兲as

␳ab

lm=

冕

^{␳ab共r兲Olm共r兲dV and ␳totlm=␳tot共r兲Olm共r兲dV.}

The derivatives of Eq.共3兲with respect to atomic positionGi

and the cell parameter Mijare

冏

^⳵⳵^JGabPFFi

冏

P

= 2

兺

lml⬘^m⬘

共− 1兲^l⳵␳ab lm

⳵^Gi

M_l+l^m+m_⬘^⬘␳tot l⬘^m⬘, and

冏

^⳵⳵^JMabPFFij

冏

P

=

兺

lml⬘^m⬘

共− 1兲^l

冉

²⳵^⳵␳Mablmij

M_l+l^m+m_⬘^⬘

+␳ab

lm⳵M_l+l^m+m_⬘^⬘

⳵^Mij

冊

^{␳totl⬘m⬘,}

where⳵␳ab

lm/⳵^Giand⳵␳ab

lm/⳵^Mijare simple and do not need to be shown here. The calculation of⳵Ml

m/⳵^Mijis shown in the Appendix.

3. Tin foil

The tin-foil correction to the Coulomb matrix is J_ab^TF= 2␲

3V_uc共QS_ab− 2d_ab·D兲, 共4兲 whereV_ucis the unit cell volume,Qis the trace of the system quadrupole, Sab is an element of the overlap matrix, D the dipole moment of the system, anddabthe dipole moment of the distribution␳ab. The derivatives of Eq.共4兲with respect to atomic positionGiand the cell parameter Mijare

冏

^⳵⳵^JGabTFi

冏

P

= 2␲

3V_uc

冉 冏

⳵^⳵GQi

冏

P

Sab+Q⳵^Sab

⳵Gi

− 2⳵^dab

⳵^Gi

·D− 2dab·

冏

⳵^⳵GDi

冏

P

冊

^,

and

冏

⳵^⳵JMabTFij

冏

P

= 2␲

3V_uc

冉 冏

⳵^⳵MQij

冏

P

S_ab+Q⳵^Sab

⳵^Mij

− 2⳵^dab

⳵^Mij

·D− 2dab·

冏

⳵^⳵MDij

冏

P

− 1 V_uc

⳵^Vuc

⳵^Mij

共QSab− 2dab·D兲

冊

^,

where the derivatives 兩⳵^Q/⳵^Gi兩P, ⳵^dab/⳵^Gi, 兩⳵^D/⳵^Gi兩P,

⳵^Sab/⳵^Gi, 兩⳵^Q/⳵^Mij兩P, ⳵^dab/⳵^Mij, and 兩⳵^D/⳵^Mij兩P are straightforward; the terms ⳵^Sab/⳵^Mij and ⳵^Vuc/⳵^Mij are given, e.g., in Ref.16.

B. Exchange-correlation integrals

In Ref. 5, Tymczak and Challacombe showed that the exchange-correlation energy, for a periodic system, can be computed over a cubic regionV_䊐, and is completely equiva- lent to the integration over the unit cellV_uc. The exchange- correlation energy is thus

E^xc关␳兴=

冕

V_uc

f共r兲dV⬅

冕

V_䊐

f共r兲dV, 共5兲

where␳共r兲=兺abRPab␳ab共r+R兲is the periodic electronic den- sity. In the following, the subscript䊐will always refer to the cuboid cell. This simple integration volume should be con- trasted with more conventional methods for computing the exchange-correlation matrix, involving the “Becke weights,”²⁵ which requires integration over all space, V_⬁. While Eq.共5兲is written in an abstract form for simplicity, in

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practice the code relies on the approach of Popleet al.²⁶and Challacombe.²⁷

The derivatives of Eq.共5兲with respect to atomic position Giand cell parameter Mijare given by

冏

^⳵⳵^EGxci

冏

P

=

冕

V_䊐

冏

^⳵␳⳵^G共r兲i

冏

P

v_xc关␳共r兲兴dV,

and

冏

⳵^⳵MExcij

冏

P

=

冕

V_䊐

冏

^⳵␳⳵^M共r兲ij

冏

P

vxc关␳共r兲兴dV+␦ij

冕

S_j䊐

f共r兲dS, 共6兲 where S_j䊐 is the surface area of the cuboid cell with its normal vector along the jth Cartesian direction. The deriva- tive 兩⳵␳共r兲/⳵^Mij兩P is simple and does not need to be ad- dressed in more detail.

The surface integral关rightmost term in Eq.共6兲兴, which is not present in previous derivations,^15,28 has its origin in the derivative of the limits of the integral in Eq.共5兲. The surface integral is approximated by

冕

S_j䊐

f共r兲dS= 1 2h

冕

V_j䊐

f共r兲dV+O共h²兲,

where h is a small number 共typically 10⁻⁴a.u.兲 and V_j䊐

=S_j䊐⫻关−h,h兴 is a thin volume where the integration over the interval 关−h,h兴 is carried out along the normal to the surface Si䊐 共i.e., along the jth Cartesian direction兲. This simple domain, i.e., Vj䊐, can be efficiently integrated with the help of theHICUalgorithm²⁷ to any desired accuracy.

III. NUMERICAL EXAMPLES

All developments were implemented in the MONDOSCF

共Ref. 29兲 suite of linear scaling quantum chemistry pro- grams. The code was compiled using the HPFORTRANcom- pilerF95 V5.5A共Ref.30兲and the −04option and the Compaq

CcompilerCC V6.5共Ref.31兲and the −01flag. All calculations were carried out on a cluster of 256 4-CPU HP/Compaq Alphaserver ES45s with the Quadrics QsNet High Speed In- terconnect.

The TIGHT level of numerical accuracy has been used throughout this work. Thresholds that define the TIGHT

accuracy level include a matrix threshold␶= 10⁻⁶, as well as other numerical thresholds detailed in Ref.5, which deliver at least eight digits of relative accuracy in the total energy and four digits of absolute accuracy in the forces.

In order to demonstrate the capabilities of our implemen- tation of the Kohn-Sham density-functional analytic atomic and cell gradients, we present in this section full optimiza- tion studies of one-dimensional 共1D兲 and 3D periodic sys- tems without any cell or atomic positions symmetry con- straints. The optimizations were carried out with theQUICCA

algorithm.^10,12

Our first benchmark case is the optimization of 1D poly- tetrafluoroethylene 共CF₂兲2n at the Perdew-Wang³² level of theory. The PW91 functional has been obtained from the Density-Functional Repository.³³ The second benchmark is cubic MgO and was optimized at the Perdew- Burke-Ernzerhof³⁴ 共PBE兲 and Becke 3-parameter Lee-Yang-Parr³⁵ 共B3LYP兲 level of theory. Finally, ortho- rhombic magnesium orthosilicate 共Mg₂SiO₄, forsterite兲 was optimized at the B3LYP共Refs.35and36兲level of theory.

TableIshows the progression of the cell parametersa₀, bond lengths共C–C and C–F兲, bond angle共F–C–F兲, and total energies E computed for polytetrafluoroethylene 共CF2兲2n at the ⌫-point PW91 with the split-valence 6-31G^**basis set.

Comparisons are made to cell parameter, bond length, and bond angle values obtained with the CRYSTAL03 共Ref. 18兲 package and a 12k-points net at the PW91/ 6-31G^**level of theory. While the atomic and cell parameters for polytet- rafluoroethylene agree perfectly between thek-space integra- tion and the⌫-point approximation, the energies do not. The reason for this disagreement can be found in the different basis set used. CRYSTAL03 uses spherical harmonicsd shell consisting of five atomic orbitals whileMONDOSCFemploys pure Cartesian basis functions, i.e., six atomic orbitals per Cartesiandshells. This leads to a slightly lower energy in the

MONDOSCFcalculations.

Table II shows the progression of the cell parameters, total energies, and fractional coordinates of oxygen com- puted for various MgO supercells 共cubic symmetry兲 at the

⌫-point PBE共Ref.34兲and B3LYP共Ref.35兲level of theories using the 8-61G共Mg兲/8-51G共O兲 basis sets. The basis sets

TABLE I. Progression of the cell parametera₀, bond lengths共C–C and C–F兲, bond angle共F–C–F兲, and total energyEfor polytetrafluoroethylene共CF₂兲2nusing the periodic⌫-point PW91/ 6-31G^**level of theory and the

TIGHTthresholds. Lengths, angles, and energies are in angstroms, degrees, and atomic units, respectively.

n a₀ CC CF FCF E/n

MONDOSCFa 1 3.400 1.823 1.339 116.9 −475.261 618

2 2.701 1.588 1.360 109.9 −475.418 440

3 2.640 1.571 1.361 109.5 −475.433 042

4 2.634 1.570 1.361 109.5 −475.435 044

5 2.633 1.571 1.361 109.5 −475.435 451

6 2.632 1.570 1.361 109.5 −475.435 536

CRYSTAL03b 1 2.628 1.569 1.360 109.5 −475.430 945

a⌫point.

b12kpoints.

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were specially optimized at the HF level for MgO by Causà et al.³⁷and were obtained from Ref.38. The primitive cubic cell coordinates used for this system are given in Ref.39. For comparison, we report the optimized cell parameter of cubic MgO obtained with the CRYSTAL03 code and an 8⫻8⫻8 k-points integration grid as well as the experimental cell parameter.⁴⁰ For both PBE and B3LYP, the smallest system 共MgO兲4 shows a large discrepancy of the cell parameter, energy, and fractional coordinate of the oxygen with respect to its k-space integration counterpart. The larger systems give cell parameter and fractional coordinate of the oxygen in very good agreement with the CRYSTAL03results, and the energies systematical converge while the system grows.

In Table III, we present the optimization of forsterite 共orthorhombic symmetry兲 at the B3LYP/ 8-61G^*共Mg兲/ 88-31G^*共Si兲/ 8-51G^*共O兲 ⌫-point approximation. The basis sets were obtained from Ref.38. For comparison, we report the optimized cell parameters of forsterite obtained by Jo- chym et al.⁴¹ at the PW91 level of theory and a 3⫻3⫻3 k-points integration grid with the VASP 共Refs. 42 and 43兲 package. We also report the experimental cell parameters

measured by Yoder and Sahama.⁴⁴As for the MgO case, the smallest system 共Mg2SiO₄兲4 shows a large discrepancy for the cell parameters with respect to theVASPand experimental values. The larger systems, i.e., the 2⫻1⫻2 and 3⫻1⫻2 supercells, give cell parameters in very good agreement with the experimental data, and the energies systematically con- verge while the system grows. The cell parameters obtained from the ⌫-point B3LYP/ 8-61G^*共Mg兲/ 88-31G^*共Si兲/ 8-51G^*共O兲level of theory deviate less than 1% with respect to the experimental values.

IV. CONCLUSIONS

In a previous paper, construction of the analytical exact Hartree-Fock exchange gradients with respect to atomic and cell parameters within the ⌫-point approximation has been introduced. In this article, the formalism for evaluation of the density-functional analytic gradients in ⌫-point approxima- tion for Cartesian-Gaussian-type basis functions was pre- sented and implemented in theMONDOSCFpackage. We show that, like for the exact HF exchange, the evaluation of the atomic gradients within the ⌫-point approximation does not lead to difficulties. The implementation requires mainly the evaluation of the derivative of the basis function with respect to atomic positions. However, complications arise when de- rivatives of the total energy are taken with respect to lattice parameters. For the periodic far-field Coulomb term, the de- rivative of the multipole interaction tensor needs to be care- fully handled in both direct and reciprocal space. The exchange-correlation energy derivative leads to a surface term, which has its origin in the derivative of the limits of the integration over the cell.

The analytical atomic and cell gradients have been used in conjunction with theQUICCAalgorithm to optimize a few 1D and 3D periodic systems at the DFT and hybrid-HF/DFT level of theory. Convergence of bond lengths, bond angles, and cell parameters within the DFT and hybrid-HF/DFT

⌫-point supercell approach and under full relaxation with no symmetry has been demonstrated for 1D and 3D systems to better than three digits. Thus, we could show that a relative accuracy better than three digits can be already achieved with cubic cells of about 600 ų.

ACKNOWLEDGMENTS

This work has been supported by the Swiss National Science Foundation, the Swiss Office for Education and Sci- ence through the European COST Action D14, and the US Department of Energy under Contract No. W-7405-ENG-36 and the ASCI project. The advanced Computing Laboratory of Los Alamos National Laboratory is acknowledged.

TABLE II. Progression of the cell parametera₀, total energyE, and frac- tional coordinate of the oxygen f_Oin the primitive cell for cubic共MgO兲n

using the periodic ⌫-point PBE/8-61G共Mg兲/8-51G共O兲 and B3LYP/8- 61G共Mg兲/8-51G共O兲level of theories and theTIGHTthresholds. Cell param- eters and energies are in angstroms and atomic units, respectively.

n a₀ E/n f_O

MONDOSCFa PBE 4 4.331 −275.243 165 0.4998

32 4.213 −275.284 428 0.5000 108 4.212 −275.284 641 0.5000

CRYSTAL03b PBE 1 4.212 −275.284 731 1 / 2

MONDOSCFa B3LYP 4 4.328 −275.389 093 0.4972

32 4.204 −275.431 167 0.5000 108 4.204 −275.431 343 0.5000

CRYSTAL03b B3LYP 1 4.204 −275.431 235 1 / 2

Expt.^c 1 4.20

a⌫point.

b8⫻8⫻8kpoints.

cExperimental value共Ref.40兲.

TABLE III. Progression of the cell parametersa₀,b₀,c₀, and total energy E for orthorhombic 共Mg₂SiO₄兲n using the periodic ⌫-point B3LYP/ 8 -31G^*共Mg兲/ 88-31G^*共Si兲/ 8-51G^*共O兲level of theory and theTIGHTthresh- olds. The numbern= 4 , 8 , 16, 24 correspond to the共super兲cells 1⫻1⫻1, 2⫻1⫻1, 2⫻1⫻2, and 3⫻1⫻2, respectively. Cell parameters and ener- gies are in angstroms and atomic units, respectively.

n a₀ b₀ c₀ E/n

MONDOSCFa 4 4.726 10.181 6.119 −991.264 858

8 4.762 10.184 6.119 −991.260 767

16 4.787 10.258 5.996 −991.271 696

24 4.787 10.257 5.995 −991.271 841

VASPb 4 4.800 10.306 6.041

Expt.^c 4 4.756 10.195 5.981

a⌫point.

b3⫻3⫻3kpoints and PW91共Ref.41兲.

cExperimental values共Ref.40兲.

http://doc.rero.ch

APPENDIX: COMPUTATION OF THE DERIVATIVE OF THEMTENSOR WITH RESPECT TO

CELL PARAMETERS

Following the work of Challacombeet al.²³ and Tymc- zak and Challacombe,⁵ the Ml

m tensor can be written into direct and reciprocal space terms as

Ml

m=_R苸PFF

兺

^{P˜lm共cos␪R兲eim␾RGl共␤,R兲}

−_R

兺

苸In

P˜

l

m共cos␪R兲e^im␾RFl共␤,R兲

+ 4␲^3/2共i/2兲^l

V_uc⌫共l+ 1/2兲_G

兺

_{⫽兵0”其}^{Gl−2e−␲2G2/␤2}

⫻P˜

l

m共cos␪G兲e^im␾G,

whereGis a reciprocal lattice vector,G=兩G兩,R=兩R兩,˜P

l mis a scaled associated Legendre polynomial,␤⁼

冑

_␲/V_uc^1/3for con- venience,

Gl共␤^,R兲=␥共l+ 1/2,␤^2R2兲

⌫共l+ 1/2兲R^l+1 , and

Fl共␤,R兲=⌫共l+ 1/2,␤²R²兲

⌫共l+ 1/2兲R^l+1 ,

where ⌫共a,x兲=兰x⬁t^a−1exp共−t兲dt is the gamma function,

⌫共a兲=⌫共a, 0兲, and␥共a,x兲=兰₀^xt^a−1exp共−t兲dtis the incomplete gamma function.⁴⁵The derivative ofMl

mwith respect to cell parametersMijis

⳵Ml m

⳵^Mij

=_R

兺

苸PFF

e^im␾R

冉

^P˜lm

冏

⳵^⳵MGlij

冏

␤+

冉

⳵^⳵MP˜lmij

+imP˜

l m⳵␾R

⳵^Mij

冊

^Gl

冊

⁻R

^兺

苸In

e^im␾R

冉

^˜Plm

冏

⳵^⳵MFij^l

冏

␤+

冉

⳵^⳵M˜Plmij

+imP˜

l m⳵␾R

⳵^Mij

冊

^Fl

冊

+ 4␲^3/2共i/2兲^l

V_uc⌫共l+ 1/2兲_G

兺

_{⫽兵0”其}^{Gl−3e−␲2G2/␤2eim␾G⫻}

冉

^{− 1V}uc

⳵^Vuc

⳵^Mij

GP˜

l

m+

冉

^{共l− 2兲− 2}

冉

^␲␤^G

冊

²

冊

⳵^⳵MGij

P˜

l m

+G

冉

⳵^⳵MP˜lmij

+imP˜

l m⳵␾G

⳵^Mij

冊冊

^.

Note that because Ml m⫽Ml

m共␤兲, it follows that ⳵Ml m/⳵␤

= 0. The derivative of the scaled associated Legendre poly- nomialP˜

l

m共cos␪兲is

⳵^˜Pl m

⳵^Mij

共cos␪兲=lcos␪^P˜l

m−共l+m兲P˜

l−1 m

冑

1 − cos²␪

⳵␪

⳵^Mij

,

which holds for both␪=␪Rand␪=␪G. Derivation ofGl共␤,R兲 andFl共␤,R兲are given by

冏

⳵^⳵MGlij

冏

_␤⁼

冉

⌫共l^2共␤+ 1/2兲R^{R兲2le−␤2l+1R2} −Gl共␤,R兲共l+ 1兲 R

冊

⳵^⳵MRij

, and

冏

⳵^⳵MFij^l

冏

_␤^{= −}

冉

⌫共^2共␤l+ 1/2^{R兲2le−␤}兲R^2l+1R2 +Fl共␤^,R兲共l+ 1兲

R

冊

⳵^⳵MRij

. The derivatives⳵^Vuc/⳵^Mijand⳵^G/⳵^Mijcan be found in Ref.

16 and the remaining derivatives, i.e., ⳵^R/⳵^Mij, ⳵␪R/⳵^Mij,

⳵␪G/⳵^Mij,⳵␾R/⳵^Mij, and⳵␾G/⳵^Mijare easily computed.

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