Influence of the Basis Sets and Numerical Accuracy Control on Bulk MgO Energetic Properties. Hartree-Fock Calculations

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BULGARIAN CHEMICAL COMMUNICATIONS

Volume 34, Number 3/4 (pp. 362 − 383), 2002

INFLUENCE OF THE BASIS SETS AND NUMERICAL ACCURACYCONTROL ON BULK MgO ENERGETIC PROPERTIES.

HARTREE-FOCK CALCULATIONS

Viorel Chihaia1, 3*, Valentin D. Alexiev2, Nicolai M. Neshev2,Gabriel Munteanu1, Costinel I. Lepadatu1,

Woong-Ki Min3, Soong-Hyuck Suh3

1Institute of Physical Chemistry „I.G. Murgulescu”, Romanian Academy,Spl. Independentei 202, 77208, Bucharest, Romania

E-mails: [email protected] Institute of Catalysis, Bulgarian Academy of Sciences,

Acad. G. Bonchev St., Block 11, 1113 Sofia3Department of Chemical Engineering, Keimyung University, Taegu, 704-701, Korea

Dedicated to the memory of Professor Dimitar Shopov on the occasionof his 80th anniversary

Received September 11th, 2002

The ab initio periodic Hartree-Fock program CRYSTAL98 was appliedto study magnesium oxide. The influence of different basis sets and tolerationparameters on the properties of MgO are carefully investigated. Themagnesium oxide lattice constant is reproduced with a high precision underthe conditions of obtaining good virial coefficients. All the electron basis setshave the tendency to underestimate the MgO lattice constant in contrast to thevalence electron basis sets that overestimate it.

Key words: Magnesium oxide, ab initio CO-SCF-LCAO calculations, hartree-fock theory, energetic properties.

INTRODUCTION

The magnesium oxide plays an important role in a variety of technologicalapplications ranging from chemistry, catalysis and electronic industries to pharma-ceutical and food industries. It belongs to the magneto-würstite compound family(Mg, Fe)O that is an important constituent of the lower earth mantle (~10%). Itsproperties and those of the associated minerals (brucite, dolomite, hydromagnesite)are difficult to analyse experimentally under the conditions of high pressure and

2002 Bulgarian Academy of Sciences, Union of Chemists in Bulgaria

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temperature present deep inside the earth (500–1000 Km). Therefore, the numericalsimulations become necessary to complete the picture of the dynamics and theevolu-tion of the Earth's crust produced by the experiment. The simulation on largespace and time scales requires empirical potentials, while the ab initio methods areexpen-sive from computational point of view. The magnesium oxide is one of thesimplest high ionic materials, which become a testing system for the computationalmethods. The reduced number of electrons of the magnesium atom and the stronglocalization of the electrons on the ions allow the developing of accuratecoulombian 2-body and many-body simple analytical formulae [1]. The parametersof the empirical poten-tials are obtained through the fitting either of theexperimentally determined parame-ters or of the ab initio determined total energy.The transferability to different che-mical environments should be checked by abinitio calculations with various geo-metrical configurations. The choice of a highaccuracy method is an important point and the applicability of the method shouldbe based on the nature of the analysed chemical system and on the parameters andthe mechanism of interest. The ab initio methods employ different approximationsin regard to the description of the single-electron wave functions in a linear com-bination (LC) of basis sets of known elec-tronic functions plane waves (PW), ato-mic orbitals (AO), the interaction hamil-tonian hartree-fock theory (HFT), densityfunctional theory (DFT) and the numerical treatment of different integrals. Thecore electrons are not so significant as the valence electrons for the description ofthe bonding, structure and reactivity of the atoms in a chemical compound.Generally, the crystalline field does not deform the atomic core, therefore the coreAOs are taken, without modifications, from the ato-mic/molecular basis setlibraries or they are not explicitly treated in calculations. In the latter case, the coreelectrons effect on the valence electrons is accurately emu-lated using a corepotential approach, called effective core potential in quantum chemistry (ECP) orpseudo-potential in solid-state physics (PP). If there is a signi-ficant overlapbetween the core and valence electron densities, the core potential approximationwill lead to systematic errors in the total energy and reduced transfer-ability of thePP or ECP. To reach the quality of the all electron (AE) methods, the corepotentials should be consistent with the exchange and correlation potential used forthe valence elec-trons. The neglect of the electronic correlation in HFT affects alsothe quality of the results especially in regard to the parameters related with theenergy, while the absence of knowledge about the analytical form of the exchangeand correlation potentials in DFT causes some uncertainly in the final results.

The use of a plane wave (PW) basis set offers a number of advantages,including the simplicity of the basis functions , the absence of basis setsuperposition error and the ability to calculate efficiently the forces acting on theatoms. The PW method is not efficient in case of the ionic crystals because of thehigh number of plane waves required for a good description of the wave functionsnear the ionic core. In the LCAO approach the parameters can be decomposed in to

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atomic orbital contributions that can be used to interpret the interaction in thesystem. The plane waves methods imply translation periodicity of the studied sys-tems and the investigation of defects, molecules and the large supercells createsproblems for the convergence of the integrals in the reciprocal space. The LCAOmethods can be applied at the same level of accuracy to investigate molecules inquantum chemistry and also systems of low periodicity. We will adopt for thepresent study the descrip-tion of the single electron orbitals called crystallineorbitals (CO) for periodical systems within the LCAO framework. The adjustmentof the standard basis sets to the crystalline structures represents one of the mostimport-ant problems raised by the SCF-LCAO-CO methods. As the basis functionsare Bloch type of orbitals and the interatomic distances in crystals are larger than inthe molecular structures, the valence shells need reorganization the number of ganstype of orbitals (GTO) asso-ciated with each electronic shell and reoptimization ac-cording to the crystal type. The AO basis sets for molecular and covalent crystalsdo not need any substantial modification, but the valence AO must be redefined formetallic and ionic systems. In the case of ionic systems we need to adjust the basissets to the electronic configuration of the chemical elements in the res-pectivesystems by choosing a number of AOs corresponding to the occupation of theelectron shells and by optimizing the AO (contraction exponents and coefficients)in accordance with the ionic state. We will use here a standard notation for thebasis sets: the basis is designed by a sequence of figures that signify the number ofGTO corresponding to each shell; the core is separated from the valence shell by adash [2]. When the polarization AOs are included in the basis set, an asterisk isadded to the symbol, describing the set, as a superscript. The notation Ba/Bb isused throughout this paper to denote the succession of the basis set for MgO.

Dovesi modified the STO-6G [3] basis set of the oxygen atom in the ionicoxide Li2O [4]. In order to improve the virial coefficient he introduced two moreGTO functions in the contraction scheme of the 1s electron shell of oxygen. Thevalence shell was split following the scheme 411G, by removing the GTOfunctions marked with asterisk in Table 1 from the 2sp shell (columns B representsthe set for oxygen). Thus, two external layers – 3sp and 4sp are formed. Then heoptimized the exponents and coefficients of the new external layers (see Table 2).According to the above approach, this basis is denoted by 8-411G. CAUSÁ et al. [5]built basis sets for three different crystalline phases of MgO. The oxygen basis isrebuilt based on the 85-1 scheme by reintroducing the 3sp and 2sp layer withunchanged exponents and with the coefficients d(O2s) = 0.42248 and d(O2p) =0.27774 and by optimizing the new external layer at 0.210 bohr−2 for the NaClphase and 0.180 bohr−2 for the CsCl phase. For Mg Causá et al. [5] analysedseveral splitting schemes 8−6x (x = 0, 1), reaching the conclusion that the 86-1scheme is satis-factory. They showed that the population of the 3sp layer increaseswith the value of the exponent, but still the high ionic character of MgO ispreserved and it has been established that the exponents of the external layer 3sp

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must have values of about 0.400 bohr−2. The basis set is designed as 86-1G/8-51G(denoted by S1 in our work).

CATTI et al. [6] split the basis set 86-1G of the magnesium in 85-11G forsellaite MgF2. They maintained the parameters of the internal layers as determinedpreviously for the MgO (see column B in Table 1 corresponding to Mg) andoptimized the exponents of the two external layers (0.680 and 0.280 bohr−2 for 3spand 4sp, respectively). Starting with the Mg 85-11G basis set built up by Catti [6]and from the O 8-411G basis set built up by Dovesi [4], Dovesi et al. [7] totallyoptimized the 1s and 2sp core shells (columns C in Table 1) and the exponents ofthe external layers of oxygen (0.680 and 0.479 bohr−2 for 3sp and 0.280 and 0.195bohr−2 for 4sp, for magnesium and oxygen, respectively – see Table 2).

These two basis sets 86-1G/8-51G and 85-11G/8-411G and the sets, improvedwith 3d polarization shells, were used, eventually to study MgO and other similarionic systems. Thus, for investigation of the electronic and structural properties ofMgCO3, Catti et al. [8] used the basis sets 85-11G/8-411G (Mg basis set optimizedfor MgF2 – Ref. 6 and O basis set optimized for Li2O Ref. 9). In addition, 3dpolari-zation shells with exponents of 0.800 bohr−2 for both elements were added.They modified the exponents of the 3sp and 4sp shells for oxygen minimizing thetotal energy of MgCO3 (0.479 and 0.230 bohr−2 for 3sp and 4sp, respectively – seeTable 2). The MgCO3 contains two ions (Mg2+ and CO3

2-) that interact ionically,while the CO bonds are covalent. The relatively small modifications of the basis setof the oxygen show that it can describe correctly ionic systems comprising groupswith covalent bonds. These groups of MgCO3 carbonate are also present in layereddouble hydroxides systems (LDH). CATTI et al. [10] analysed by periodic Hf,besides MgCO3, also the similar CaCO3 system. The 4sp exponent of oxygen issmaller in this system than in MgCO3 (0.185 bohr−2). By including the 3d polari-zation shells the energy was lowered and some of the properties of MgCO3 andCaCO3 were affected. Using the slightly modified basis sets 85-11G/8-411G,D’ARCO et al. [12] studied brucite. This system is obtained by the interaction ofMgO with H2O. Modifying the exponents of the valence shells in the 85-11G basisset of the magnesium and adding a 3d polarization shell, BARAILLE et al. [13]studied MgH2. The results of these works show that these basis sets are adequate tostudy even more complex systems such as LDHs.

McCarthy et al. [14] extended the study on the basis set developed by Dovesiet al. [7], by including the polarization shells (for both elements the 3d exponentsare 0.650 bohr−2) and the electronic correlation effects in a post-theory correction[14]. Catti et al. [15] investigated the formation of MgAl2O4 solid solution (spinel)in the reaction between magnesia and alumina oxide, using for the magnesium andoxygen the basis set determined by Dovesi [7]. For the aluminium an optimizedbasis set 85-11G* of the isolated atom was used. The exponents of the 4sp and 3dshells were optimized for MgO, α-Al2O3 and MgAl2O4, obtaining smallimprovements (see Table 2). The introduction of the 3d layer decreases the total

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energy and influences the results obtained by the equation of state. The basis setfrom ref. 7 is considered in the study of the thin film MgO/NiO [16]. The abovedescribed basis sets were also successfully used in the study of other systems,defects in MgO [17], Mg2Si and MgSiO3 [18], Mg(OH)2 [19], Mg3Si2O5(OH)4 [20],with insignificant modifications of the exponents of the 3sp and 4sp valence shellsor of the 3d polarization layer.

Different truncation techniques, interpolation and evaluation methods are usedto reduce the computational effort in the ab initio methods. An investigation of theinfluence of the calculation procedures, of the basis set used in LCAO and of theelectronic correlation on the accuracy of the results is needed. In the present studywe analyse the influence of these factors on the bulk phase MgO properties, usingthe ab initio LCAO-SCF-CO CRYSTAL98 code. The authors of this programperformed systematic studies of the effects of the above mentioned factors on thecalculation accuracy. Here we try to establish the conditions of the calculation inorder to obtain accurate results with a reduced computational effort. In the secondsection, the MgO structure and calculation methods are represented. The results ofthe parabolic fit of the elementary cell HFT total energy versus lattice constantwith different combinations of basis sets, tolerance thresholds parameters andintegration schemes are analysed in the third section. We propose a comparativestudy of the results obtained with all electrons (AE) basis sets (in the schemes 86-1G/8-51G and 85-411G) and with the valence electrons (VE) basis sets in whichthe cores shells (1s2sp for magnesium and 1s for oxygen) are re-placed by ECP ofDurand-Barthelat type. A conclusions section closes the article.

CALCULATION METHODS

The magnesia is a highly ionic compound. It has the NaCl structure (spatialgroup Fm3m), that consists of two interpenetrated face centered cube (FCC)lattices of the Mg2+ and O2- ions. There are four oxygen and four magnesium atomsin a primitive cell. The volume of the primitive cell is four times smaller than thatof the elementary cell. The energy depends only on the lattice constant a. Theexperimentally determined value of the constant is a = 4.210 Å [21].

All calculations have been performed using the ab initio CRYSTAL98program, in which the crystalline orbitals (CO) that describe the wave function ofthe system are computed in a self-consistent way, using a linear combination ofatomic orbitals (LCAO) [22]. It allows the treatment of systems with a translationsymmetry 1d-3d, but also molecules and clusters (0d). The hamiltonians may bebuilt up within framework HF or DF theory.

The coulombian and exchange series and integrals can be exactly calculated.However, the effort is not justified, as the exponential decrease of the differentialcoverages with the internuclear distance allows the reduction of the number of thecalculated integrals. It is useful to remove the small value integrals from the calcu-lations or approximate them by multipolar expansion. Grouping the electronic

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terms for the charge distributions on electron shells may reduce the number ofterms in the infinite summations over the elementary cell indexes. The chargedistribution may be decomposed into a sum of charge distributions on electroniclayers, which is useful in selecting the bielectronic integrals and in evaluating thelarge-distance interactions. The crystal division in zones of approximation of thecoulombian and exchange integrals and of the series formed with them is con-trolled by means of the tolerance parameters TOLINTEG. In the large distancezone the electron distance ρλ of the shell λ is developed in multipoles with maxi-mal order POLEORDR, while the series are evaluated analytically by a recursiveEwald summation.

The recalculation of the charge density matrix for each SCF iteration impliesintegration in the reciprocal space. The quite regular behaviour of the integrand inthe case of oxides allows the reduction of the integrals in the reciprocal space tosuitable sums over a set of special points

k

, distributed uniformly in the first Bril-louine zone (the MONKHORST mesh [23]). Three parameters control the accuracy ofintegration in the reciprocal space: IS – the folding factor for Monkhorst mesh,ISHF – the number of plane waves used in the expansions of the single electronenergy level

)k(i∈

and ISP – the Gilat folding factor for the calculation of theFermi level and of the electron density (the GILAT mesh [24]). In the case ofinsulating crystals the first parameter is more important; the other two areimportant for the metals.

At the first step of the study we chose parameters to ensure a high accuracy ofthe results. Thus, we established the TOLINTEG parameters to be 6, 8, 6, 7, 14,under the condition of a multipolar development order POLEORDR = 4. The inte-gration parameters in the reciprocal space (IS, ISHF and ISP) are 8 6 8. Therefore,for the Monkhorst folding factor IS = 8, the summation in the reciprocal space iscarried out for 29 special

k

points. The influence of these parameters on thequality of the results will be analysed after the description of the basis sets. Toreduce the number of SCF iterations, we used a technique of mixing the elementsof the Fock matrix, calculated in the current iteration with 30% of the value of theelements, determined in the previous iteration. The relation-ship between the usedbasis sets is represented in Fig. 1, while the structures and parameters of the basissets, used in the present study, are showed in Table 1 (the shells 1s and 2sp) andTable 2 (the shells 3sp, 4sp and 3d). For some of the basis sets the exponents of thevalence electron layers have been simultaneously optimized, using a parabolicinterpolation procedure.

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Table 1. The GTO exponents and coefficients corresponding to the electron layers 1s and 2sp of Mg and O with the basis sets STO-6G(the columns A), the optimized basis sets for Mg2+ and O2- ions, in ionic systems (the columns B) and optimized for MgO (the columnsC). The elements marked with * are eliminated from the layers 2sp in some cases and with them the layers 3sp are built up,corresponding to the transformation of the basis set 86-1G/8-51G in to 85-11G/8-411G.Shell Mg

GTO Exponents [bohr-2] GTO CoefficientsA B C A B C

1s1 − 68371.87500 68370.0 − 0.0002226 0.00022262 − 9699.34009 9661.0 − 0.0018982 0.0019013 3103.386324 2041.176786 2041.0 0.009164 0.0110451 0.0110424 569.002485 529.862906 529.6 0.049361 0.0500627 0.050055 159.186391 159.186000 159.17 0.168538 0.169123 0.16906 54.684822 54.6848 54.71 0.370563 0.367031 0.366957 21.235716 21.2357 21.236 0.416492 0.400410 0.40088 8.746041 8.74604 8.791 0.130334 0.149870 0.14872sp1 (2s ) (2p)

156.795231 156.795 143.7 −0.0132530.003760

−0.006240.00772

−0.006710.00807

2 (2s ) (2p)

31.033868 31.0339 31.27 −0.0469920.037679

−0.078820.06427

−0.079270.06401

3 (2s ) (2p)

9.645303 9.6453 9.661 −0.0337850.173897

−0.079920.21040

−0.080880.2092

4 (2s ) (2p)

3.710896 3.7109 3.726 0.2502420.418036

0.290630.34314

0.29470.3460

5 (2s ) (2p)

* 1.611645 1.61164 1.598 * 0.595117* 0.425860

0.571640.37350

0.57140.3731

6 (2s ) (2p)

* 0.738735 * 0.64294 − * 0.240706* 0.101708

* 0.30664* 0.23286

−

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Table 1. (continued).Shell O

GTO Exponents [bohr-2] GTO CoefficientsA B C A B C

1s1 − 4000.0000 8020.0 − 0.00144 0.001082 1355.584234 1355.5800 1338.0 0.009164 0.00764 0.008043 248.544886 248.5450 255.0 0.049361 0.05370 0.053244 69.533902 69.5339 69.22 0.168538 0.16818 0.16815 23.8868 23.8868 23.90 0.370563 0.36039 0.35816 9.275933 9.27593 9.264 0.416492 0.38612 0.38557 3.820341 3.82034 3.851 0.130334 0.14712 0.14688 − 1.23514 1.212 − 0.07105 0.07282sp1 (2s ) (2p)

52.187762 52.1878 49.43 −0.1325300.003760

−0.008730.00922

−0.008830.00958

2 (2s ) (2p)

10.329320 10.3293 10.47 −0.0469920.037679

−0.089790.07068

−0.09150.0696

3 (2s ) (2p)

3.210345 3.21034 3.235 −0.0337850.173879

−0.040790.20433

−0.04020.2065

4 (2s ) (2p)

1.235135 1.23514 1.217 0.2502420.418036

0.376660.34958

0.3790.347

5 (2s ) (2p)

* 0.536420 * 0.53642 − * 0.595117* 0.425860

* 0.42248* 0.27774

−

6 (2s ) (2p)

* 0.245881 − − * 0.240706* 0.101708

− −

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Table 2. The exponents of the electron shells 3sp, 4sp and 3d of Mg and O used in this work (S1-S10) or in other articles onionic systems.

Basis Set Mg O

Type ofBasis Set

CoreShells

3sp 4sp 3d Type ofBasis Set

CoreShells

3sp 4sp 3d

Present work6-6G/6-6G 66G A 6-6G A8-66G/8-6G 86-6G 1s-B, 2sp-A 8-6G 1s-B, 2sp-AS0 85-1G B* 0.680 − − 8-41G B* 0.479 − −S1S1'S1''

86-1G86-1G86-1G

BBB

0.4000.3000.094

− − 8-51G8-51G8-51G

BBB

0.2100.2100.224

− −

S2 86-1G B 0.400 − 0.7708 B 0.210 − 0.198S3 85-11G B* 0.680 0.280 − 8-411G B* 0.479 0.230 -S4S4'S4''

85-11G*85-11G*85-11G*

B*B*B*

0.6800.6800.680

0.2800.2800.280

0.8000.1750.456

8-411G8-411G8-411G

B*B*B*

0.4790.4790.479

0.2300.2300.230

0.8000.8000.649

S5 85-11G C 0.688 0.280 − 8-411G * C 0.500 0.191 −S6S6'

85-11G*85-11G*

CC

0.6880.680

0.2800.280

0.6500.647

8-411G8-411G

CC

0.5000.500

0.1910.191

0.6500.489

S7 DB-1G DB0 0.680 − − DB-41G DB0 0.479 − −

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Table 2. (continued).

S8S8'

DB-1GDB-1G

DB1DB1

0.4001.518

− − DB-41GDB-41G

DB1DB1

0.2100.207

− −

S9 DB-1G DB2 3s: 1.5293p: 3.042

− DB-51G DB2 0.207 −

S9' DB-1G DB0 3s: 1.5463p: 3.067

− − DB-51G DB0 0.217

Other worksLi2O [4] − − − − − 8-51G B* 0.450 0.150 −MgO [5] 8-61G B, B* 0.400 − − 8-51G B, B* 0.210 − −MgF2 [6] 8-511G B* 0.680 0.280 − − − − − −Mg(OH)2 [12] B 0.680 0.280 − B* 0.500 0.200 0.800MgH2 [13] B* 0.680 0.180

0.295 a0.450 − − − −

MgO, α-Al2O3,MgAl2O4 [15]

B* 0.680 0.280 0.650 b

0.540 cB

MgCl2 [11] 8-511G* C 0.650 0.110 0.400 − −MgO [7]MgO [14]Mg/NiO [16]

C 0.680 0.280 0.650 C 0.479 0.195 0.450

MgCO3 [8]CaCO3 [10]

C 0.680 0.280 0.800 C 0.4790.479

0.2300.185

0.8000.800

a- the authors had used two values for the α3dMg , b - in MgO, c - in MgAl2O4

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Fig. 1. The relations ships between the basis sets employed in this work.

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RESULTS AND DISCUSIONS

As a criterion of the basis set accuracy we used the reproducibility of theexperimentally determined lattice constant a0

exp = 4.210 Å. The parameter of theelementary cell is being modified in steps of 0.005 Å within the 4.180–4.260 Årange. The results of the parabolic fitting E(a) = c0 + c1a + c2a2 of the MgOprimitive cell total energy and the relative error (RE = ( a0 − a0

exp)/ a0exp), are

represented in Table 3. The R-square, defined as R-sq = ( S − Σ)/S, where S is thesum of the squares of the residuals around the arithmetic mean value and Σ is thesum of the squares of the difference between the data points and the fitted curvepoints, describes the correlation between the parabolic fitting curve and thecalculated energy E(a) values of the magnesium oxide primitive cell, at differentvalues of the lattice constant (see Table 3).

Using the STO-6G atomic basis set for both elements, the equilibrium latticeconstant is underestimated (RE = −12.104%). For the reference value a = 4.210 Å,the virial coefficient value is 0.998108 and the system ionic character is badlyreproduced as the electronic charges on magnesium and oxygen based on theMulliken's analysis were 11.196 and 8.804 |e-|, respectively. The programCRYSTAL allows the modification of the initial population of the electron shells,bringing the electronic distribution in the first self-consistent field (SCF) iterationin agreement with the ionic character of the system. By making use of this facilily,we impose a two electron transfer from the magnesium to the oxygen, thusreproducing the perfect ionic system. This procedure decreased the number of theSCF iterations from 10 to 8 and the calculation time in the SCF step, as theanalysed values, within the range of the studied lattice constants, were exactlyreproduced. This procedure of initial population of the electron shells ofmagnesium and oxygen is used in all the calculations of the present study.

The charge transfer is underestimated, because in the Mulliken analysis theelectronic population of the 2PµνSµν mixed term is equally distributed between thetwo atomic orbitals µ and ν, belonging to different chemical elements Mg and O,neglecting their different electron affinity. This underestimation of the electroniccharge transfer has also been observed in the calculation that involves optimizedbasis sets for atoms or molecules. When a cluster, cut out of MgO, is embedded ina lattice of point charges ±2 |e-| with the purpose to reproduce locally the crystallineCoulombian field, it improves the electronic charge distribution. However, such animprovement of the electronic charges is not exhibited by the Mulliken analysis.The charge analysis in the Natural Bond Orbital formalism reproduces theelectronic charges correctly, even in the absence of the crystalline Coulombianfield [25].

Table 3. The results in the parabolic fitting E(a) = c0 + c1a + c2a2 of the MgO primitive celltotal energy, function of the lattice constant a, R-sq - the correlation between the calculated

374

values of the total energy E for different values of a, amin - the predicted equilibrium latticeconstant,

REa=0exp

the relative error in the reproduction of the experimental value a0exp =

4.210 Å, E(amin) - the total energy corresponding to the equilibrium lattice constant and thevirial coefficient determined for a = 4.210 Å. The negative values of the relative errorsshow an underestimation of the equilibrium lattice constant, while the positive valuesindicate an overestimation.

Basis Sets c0 c1 c2 R-sq amin

[Å]

RE

[%]

E(amin)

[Hartree]

VirialCoeff.ata =4.21

STO-6G −270.41171 −1.74843 0.23625 0.99999 3.7004 −12.104 −273.6466785 0.998116-6G/6-6G −273.77944 0.13974 0.02359 0.99999 2.9622 −17.036 −273.9864128 1.0124986-6G/8-6G −270.54611 −1.72041 0.23352 0.99999 3.6836 −12.504 −273.7147500 0.99999S0 −274.17168 −0.40277 0.09648 1.00000 2.0873 −50.421 −274.5920205 1.00326S1 −271.11780 −1.69237 0.20190 0.99998 4.1911 −0.449 −274.6642140 0.99998S1' −271.20673 −1.64915 0.19663 0.99997 4.1935 −0.391 −274.6646118 1.00002S1'' −272.04408 −1.72695 0.20586 0.99790 4.1944 −0.369 −274.6658601 1.00005S2 −271.71149 −1.41089 0.16823 0.99465 4.1933 −0.396 −274.6696583 1.00006S3 −271.22709 −1.64439 0.19667 0.99998 4.1806 −0.698 −274.6643648 0.99974S4 −271.41269 −1.56335 0.18756 0.99997 4.1675 −1.008 −271.4284315 0.99961S4' −271.75575 −1.38985 0.16573 0.99897 4.1930 −0.403 −274.6695925 0.99972S4'' −271.64621 −1.44671 0.17302 0.99952 4.1810 −0.689 −274.6707177 0.99959S5 −271.01930 −1.73965 0.20689 0.98683 4.2043 −0.135 −274.6763359 0.99965S6 −271.40567 −1.56268 0.00216 0.99974 4.1926 −0.412 −274.6815232 0.99960S6' −271.12517 −1.69517 0.20207 0.99861 4.1952 −0.351 −274.6815498 0.99959S7 −15.89645 −0.45099 0.09730 1.00000 2.3176 −44.951 −16.4190538 0.93538S8 −12.62385 −1.91126 0.22471 0.99967 4.2527 +1.015 −16.6872763 0.87369S8' −11.97858 −2.18707 0.25757 1.00000 4.2456 +0.845 −16.6212744 0.87421S9 −12.08957 −2.14329 0.25216 0.99999 4.2498 +0.947 −16.6439125 0.87348S9' −11.92426 −2.21284 0.26060 1.00000 4.2453 +0.839 −16.6213869 0.87789

If the valence shell 3sp is removed from the Mg STO-6G basis (basis set 6-6G/6-6G), thus forcing the migration of two electrons from Mg to O, the Mullikenanalysis indicates exactly ten electrons on each element even under the conditionsof a higher contraction of the lattice (ER = −17.036%). By removing the 3sp AOs,the mixed terms disappear and the insufficiency of the Mulliken analysis is over-come as the virial coefficient reaches a value of 1.01249. However, it is necessaryto include the 3sp layer, described by only one GTO function in order to describethe covalent component, whose importance grows up with the decrease of themagnesium coordination,. The results of Ref. 5 prove that this procedure satisfiesall requirements when the exponent of this GTO is about 0.400 bohr−2.

Using Dovesi’s solution [4] for the improvement of the virial coefficient (ameasure of a good reproduction of the wave function), we added to the 1s orbital ofMg and O two GTO functions for each element (basis 8-66G and 8-6G, respect-ively). The virial coefficient then improves drastically to 0.99999, but the equilib-

375

rium lattice constant (RE = −12.504%) and the net charges (11.183 |e-| for Mg and8.817 |e-| for O) were poorly reproduced. Following these procedures, we built upthe reference basis set S0 (85-1G for Mg and 8-41G for O). For this set the value ofthe RE is: -50.421% and such underestimation is predictable, as these basis sets areincomplete and unoptimized. By adding one GTO to the 2sp shell of each one ofthe elements (the set S1 = 86-1G/8-51G), RE is reduced to –0.449% preserving the3sp exponents values of 0.400 bohr−2 and 0.210 bohr−2 for Mg and O, respect-ively.When the 3sp exponent of Mg is modified to 0.300 bohr−2 (the set S1') the REbecomes –0.391%. By simultaneously reoptimizing the exponents of the 3sp shellof both elements a very diffused orbital for Mg is obtained (0.094 bohr−2), whilethe exponent 3sp of oxygen affects to a small extent the energetics of MgO and theenergy minimum is calculated at 0.224 bohr−2 (this basis set is denoted S1''). Thecoefficient R-sq is reduced to 0.99790, in comparison to the 0.99998 and 0.999997values determined for S1 and S1'. For the lattice constant a = 4.210 Å, thecalculated primitive unit cell energy is reduced to –274.6658601 Hartree, while thecalculated cell energies for S1 and S1' were –274.6642140 and –274.6646118Hartree, respectively.

The 3d polarization shell is important for systems with covalent bonds, there-fore, we added 3d polarisation AO to the AO basis sets in S1 (basis set S2). Whenthe 3d exponents of both chemical elements are optimized (0.7708 and 0.1984bohr−2 for Mg and O, respectively), the value of RE is comparable to the valueobtained for the S1' set, with R-sq = 0.99465. The number of the SCF iterationsincreases drastically, indicating an inadequate behaviour of the valence electrons.

The basis set S0 may be modified by adding to each element a 4sp shell,described by one GTO function (set denoted 85-11G/8-411G). In Ref. 6 4sp shellsare used with the exponents 0.280 and 0.230 bohr−2 for Mg and O, respectively.This basis set (the set S3) results in an improved energetic behaviour of thecalculations for the MgO: RE is –0.698%. The use of a more diffuse Gaussianfunction in the description of the 4sp AO of the cation (exponent 0.300 bohr−2)reduces the RE to –0.319% (the set S3') and introducing the 3d polarization shell inthe S3 basis set both for Mg and O with the same exponent of 0.800 bohr−2, as inthe molecular systems, [2] (as indicated in Ref. 8), we obtain the basis set S4. Thecalculations with this set underestimate the equilibrium lattice constant by 0.008%.For the standard values of the 3d GTO exponents, 0.175 Hartree-2 for Mg and0.800 bohr−2 for O (the set S4'), the relative error is reduced to –0.403%. Thesimultaneous optimization of the exponents of the 3d layers of Mg and O (the basisset S4'': the exponents 0.456 and 0.649 bohr−2 for Mg and O, respectively)increases RE to –0.689%, as the correlation between the fitting curve and thecalculated cell energy values is decreased to 0.99952. As the 3d AO of the of thecation is very diffuse, after the third SCF iteration the system has metallic charac-ter. This effect may be avoided by imposing the level shift of the occupied orbitals(the mechanism LEVSHIFT, ILOCK = 1) [26].

376

Based on the above-presented results, it appears that the basis sets must betotally optimized. Dovesi et al. [7] optimized the basis set 85-11G/8-411G formagnesium oxide. From the fitting of the basis set S5 (the parameters of 1s and 2spfor both elements are presented in the columns C of Table 1) an exceptional RE of–0.135% is obtained, but with unsatisfactory fitting of the calculated values of thetotal energy of the primitive cell (R-sq = 0.98683). This correlation is slightlyimproved to a R-sq = 0.99974 by the addition of a 3d AO for Mg and O with equalexponents 0.650 bohr−2 (the basis set S6) as in Ref. 14. The optimization of theexponents of the 3d shell for both elements (the basis set S6': the exponents of0.646 and 0.648 bohr−2 for Mg and O, respectively) led to an improvement of theRE = −0.351%, but at the expense of a weaker correlation of the fitting curve andthe calculated energy values (R-sq = 0.99861).

The splitting of the layer 2sp (N11 scheme, N = 4 for anion and 5 for cation)may be interpreted as a mathematical operation only in a variational sense thatincreases the number of freedom degrees of the Bloch orbitals. Thus, the splittingN11 of the layers 2sp into 2sp/3sp/4sp should be considered only as an index,without physical importance, since the quantum number n of the electron shellindex is not directly involved in the AO spatial extension, but only indirectlythrough the values of the coefficients in the LCAO expansion and the exponents ofthe GTO functions. Since with the best N gaussian-type orbitals the coefficientsand the last two GTO the exponents are reoptimized, we can see this splitting as asplitting of the electron shell 2sp into subshells. From this point of view, regardingthe electron charge distribution in shells, we find that the AE basis sets with dif-ferent degrees of freedom have similar behaviour. The ECP determined for mole-cular systems should be cautiously used in ionic crystalline systems, where thechemical environment of the atoms strongly influences the valence electron distri-bution. In the magnesium oxide the magnesium tends to release the two valenceelectrons thus leaving no electrons in its valence shell. The ECP developed formagnesium atom treats its core as containing ten electrons corresponding to the 1sand 2sp shells. Therefore, difficulties are expected in the case of magnesium oxide.We test the ECP of DURAND-BARTHELAT (DB) [27] type, included in the programCRYSTAL, representing 1s2sp for Mg2+ and 1s for O2- cores and preserv-ing theelectron shell structure and parameters of the basis set S0 (the basis set S7). Therelative error is –50.421% for S0 and –44.951% for S7. We modified the valencestructure of the oxygen, according to the basis set of the DB-41G type optimisedfor nickel oxide [28] (the basis set S8). The relative error is drastically reduced to+1.015% and for the first time the equilibrium lattice constant of MgO wasoverestimated. By optimizing the exponents of the 3sp layer for both elements (theset S8': exponents 1.5179 and 0.2017 bohr−2 for Mg and O, respectively), areduction of RE to +0.845% is reached as the calculated energy points fit perfectlythe curve (R-sq = 1.00000). The system ionic character for the basis sets S7-S8 thatinclude ECP is higher than that obtained with the AE sets S1-S6, and the net

377

charges on ions are ±1.927 with S7 and ±2.005 with the optimized S8 sets. It hasbeen shown in a previous work that the exceeding electron transfer of 0.005 |e-|from the completely occupied shell 2sp of Mg2+ to the completely occupied shell2sp of O2- is an artificial one and results from the fitting scheme of the electrondensity of states in the energy domain of the 2p levels of the anion [29]. Since Mgvalence electrons do not have enough ‘degrees of freedom’, we split the 3sp mag-nesium layer into two subshells 3s and 3p and optimize the exponents of the 3s and3p shells of Mg and 3sp of O (basis set S9). This improves the results for ER =+0.839% and R-sq = 1.0000, while the total optimization of the valence exponentsfor both elements (the basis set S9') does not improve the results in the calculations(ER = +0.947% and R-sq = 0.99999). Additionally, the virial coefficient, whoseideal value is 1.00 was calculated to be 0.94 with S7 and about 0.88 with S8-S9. Apossible explanation may be the big size of the Mg core included in the ECP andthe lack of ECP optimisation to the highly ionic character of the MgO. The trans-ferability of the Mg ECP to ionic crystalline systems requires a higher computa-tional effort regarding the ECP structure and parameterization, therefore it will notbe analysed here.

Another important factor in the choice of the basis sets and parameters of thecalculation is the computational effort (CPU time, required space on hard disk -HD). The basis set S1 does not involve a long CPU time and large disk space andoffers quality results for the energy. S1' offers an optimum compromise betweenthe quality of the results (as in the case of S1'') and the computational effort(similar to S1). The addition of a diffuse polarization 3d AO to S1 increases thecomputational effort (four times the CPU time and three times the HD space), witha slightly better precision than S1 and comparable precision for S1'. The splittingscheme N11 leads to an increase of the computational effort – a direct increase ofthe used HD space and, implicitly, due to the slow access to the reading/writing ofthe data on the HD, leads to an increase of the computational time. Thus, in S3 thecomputational effort grows slightly in comparison to S1 with a higher contractionof the lattice. The addition of the polarization 3d AO (the basis set S4) doubles thecomputational effort. The use of more diffuse 3d AO for Mg (the basis set S4')grows further the computational effort, as the storage space for the diffuse AO islarger and the SCF convergence is slow. The computational effort is reduced whenoptimized 3d AO’s are used. As it is expected in advavce the more diffuse AOincrease the calculation time. However the computational effort is reasonable withthe basis set S2, therefore, it is attractive for the initial tests on systems such as theLDH, where the orbitals 3d play an important role in the description of the inter-layer Van der Waals interaction. By replacing the ionic cores with ECP, the com-putational effort is much reduced, but as the number of core atomic orbitalsinvolved in the exact calculation is small, the ratio of the CPU times necessary forAE and VE type of calculations in the program CRYSTAL is smaller than thatrequired for the calculations of molecules in quantum chemistry.

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From the analysis of the basis set influence upon different components of thetotal energy it follows that the most sensible component is the bielectronic energy.A reduced relative error is obtained with those basis sets that can reproduce theelectron charge polarization phenomena as completely as possible. The virial coef-ficients are close to unity with the AE basis sets. This shows, that the total wavefunctions in these basis sets have a consistent description. In the case of VE basissets, the virial coefficient has lower values, which indicates an inadequate descrip-tion of the total wave function. However, the total energy dependence on the latticeconstant in the case of the basis set S8 is satisfactory.

A consequence of the truncation scheme in the direct space [30] (the para-meters TOLINTEG) is the dependence of the calculation accuracy on the elemen-tary cell geometry. This approximation is the source for the deviation of the para-bolic fitting curve and the calculated values of the MgO total energy, as well asthose of the curve shift observed with some basis sets for a0 = 4.235 Å. The evo-lution of the primitive cell total energy, calculated using the basis set S5, isrepresented in the Fig. 2. The choice of adequate tolerance parameters in the directspace may reduce the above-described errors. Since a higher accuracy requires ahigher calculation cost, a compromise is necessary for the accuracy of the calcula-tion and the computational effort.

Fig. 2. The total energy function of the lattice constant, calculated using the basis set S1with the calculation parameters I and III. The continuum curve represents the parabolic

fitting for the case I.

A study has been performed in order to identify the influence of the toleranceparameters on the calculation accuracy and on the computational effort for threeoptimized basis sets AE (S1 and S5) and VE (S8), respectively. The sets oftolerance thresholds for the direct space integration, as well as the reciprocal spaceintegration parameters, are listed in Table 4.

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Table 4. The tolerance parameters of the direct space integrals TOLINTEG (10-ITOL1 –coulombian coverage tolerance,10-ITOL2 – coulombian penetration tolerance, 10-ITOL3 –exchange coverage tolerance, 10-ITOL4 – exchange pseudocoverage F(G), 10-ITOL5 –exchange pseudocoverage P(G)) and the tolerance parameters in the reciprocal space (IS –Monkhorst factors, ISHF –Gilat factor, ISP – the number of terms in the eigenvaluesdevelopment). The maximum numbers of terms in the multipolar development has beenthat default in the program POLEORDR = 4.

Case Tolerance parameters of integrals(ITOLINTEG)

Parameters of reciprocal spaceintegration

ITOL1 ITOL2 ITOL3 ITOL4 ITOL5 IS ISHF ISP

I 6 8 6 7 14 8 6 8II 6 6 6 6 12 8 6 8III 5 5 5 6 12 8 6 8IV 5 5 5 6 12 16 16 16

The different degrees of tolerance in the total energy calculation as a functionof the lattice constant alters the fitting parameters of the curve E(a) (see Table 5).The best correlation coefficient of the regression with the basis set S1 is 0.999981,obtained for parameters set I, followed closely by 0.999929 – the coefficient withthe parameter set II, obtained by a much lower computational effort. For the lowaccuracy parameters (the sets III and IV), the accuracy of the fitting diminishes(0.938232), with a tenfold decrease of the relative deviation from the experimentalvalue. A more refined integration in the reciprocal space does not improve thecalculation results, but increases the computational effort in the SCF step.

The smaller values of tolerance threshold for the basis set S5, also implieshigh computational efforts, but the lattice constant is estimated with an increasedaccuracy. It is very good with the set II (0.999702), good also with the set I(0.986833) and poor with the set III (0.141809) and the set IV (0.141829) as thecalculated values of the primitive cell total energy are shifted within the4.235−4.240 Å range of the lattice constant values (see Fig. 2). This shift is due tothe different approximations used for the two subdomains 4.180–4.235 Å and4.240−4.260 Å. A separate analysis in regard to the lower and upper subdomainsshows that the correlation between the total energy calculated values and the valuesof the fitting curve is very good in each subdomain. Using the tolerance parameters10−5 10−5 10−5 10−5 10−5 with the same basis set, Harrison avoided the appearance ofthis energetic shift [14]. We find the exceptional value –0.0806% with set II andthe very good value –0.135% with set I, for the relative deviation of the predictedequilibrium lattice constant and the experimentally determined value of the latticeconstant. The different integration conditions in each of these subranges give avery similar behaviour with the set III and IV in the lower range (with a bettercorrelation to the more accurate conditions with the set IV 0.999021, in comparisonto the set III 0.996807) and strictly identical behaviours in the upper part of therange, where a high correlation of the calculated values and the fitting curve(

380

0.999737) is achieved. In the case of the VE basis set S8, the correlation to thefitting curve is better with the parameter sets III and IV (0.999962) and lessaccurate with the other two truncation sets: 0.999670 - set I and 0.999276 - set II,respectively. The computational effort is comparable, the difference being less than10%. This is also valid for the reproduction of the MgO lattice constantexperimental value. The explanation is that the use of the effective core potentialsto replace the core electrons decreases the number of mono- and bi-electron core-valence integrals, which decrease the strong dependence of the computationaleffort on the integral truncation conditions.

Table 5. The results of the parabolic fitting parameters E(a) = c0 + c1a + c2a2, R-sq – thecorrelation between the calculated values of the total energy and the fitting curve, amin - thepredicted equilibrium lattice constant,

REa=0exp

- the relative error in the reproduction of

the experimental value a0exp = 4.210 Å, E(amin) - the total energy corresponding to the

equilibrium lattice constant, with different basis sets and tolerance parameters (a varyingbetween 4.180 and 4.260 or in the subdomains a 4.180-4.235 Å, b 4.240-4.260Å). Thenegative values of the relative errors show an underestimation of the equilibrium latticeconstant, while the positive values indicate an overestimation.

BasisSets

c0 c1 c2 R-sq amin[Å]

RE.[%]

E(amin)[Hartree]

S1 I −271.117796 −1.692374 0.201903 0.999981 4.1911 −0.449 −274.664214II −271.224192 −1.642056 0.195953 0.999929 4.1899 −0.476 −274.664236III −269.340925 −2.529972 0.300605 0.938232 4.2081 −0.044 −274.664155IV −269.340916 −2.529976 0.300605 0.938232 4.2081 −0.044 −274.664155S5 I −271.019303 −1.739655 0.206889 0.986833 4.2043 −0.135 −274.6763359II −271.195712 −1.654719 0.196681 0.999702 4.2066 −0.0806 −274.6760875III −273.590577 −0.514074 0.060867 0.141809 4.2229 0.3071 −274.6760259IIIa −270.928103 −1.782005 0.21182 0.996807 4.2064 −0.085 −274.6760279IIIb −271.283660 −1.614538 0.192077 0.999737 4.2028 −0.170 −274.6764829IV −273.590497 −0.514112 0.060871 0.141829 4.2229 0.3079 −274.6760350IVa −271.135730 −1.683585 0.200156 0.999021 4.2057 −0.102 −274.6760416IVb −271.284084 −1.614338 0.192054 0.999737 4.2028 −0.170 −274.6764725S8 I −12.623849 −1.911258 0.224711 0.99967 4.2527 +1.015 −16.6872763II −12.399847 −2.016425 0.237048 0.999276 4.2531 +1.026 −16.6879762III −12.607733 −1.918741 0.225574 0.999962 4.2530 +1.0218 −16.6879641IV −12.607733 −1.918741 0.225574 0.999962 4.2530 +1.0218 −16.6879641

The reciprocal space integration conditions affect first of all the bielectronicintegrals and very slightly the kinetic energy and the total Coulombian electron-nuclei interaction. Upon using a higher order of the multipolar expansion thedeviation of the calculated energy values from the fitting curve is reduced. Forexample, the calculated R-sq value is 0.99959 with POLEORDR = 6 compared tothe R-sq value of 0.98683 obtained with POLEORDR = 4, using the basis sets S5and the calculation conditions I.

381

The Mulliken population analysis shows that the cation and anion electroncharge does not depend on the bielectron integral approximation (TOLINTEG) oron the reciprocal space integration. Small changes are found only in the populationof the last valence layers. In conclusion, the reciprocal space integration conditionshave insignificant effect on the results of MgO crystal total energy interpolationand have no effect on the charge distribution. The computational effort is slightlyaffected by the tolerance thresholds for the direct space integration (~10%) and isslightly increased by the integration scheme.

CONCLUSIONS

In the present paper we use all-electron and valence-electron LCAO-GTObasis sets in HFT to investigate MgO. The basis set of the type 8-61G/85-1Greproduce very well the equilibrium lattice constant. The splitting of the valenceshell of Mg and O atoms in the 8-511G/8-411G scheme does not significantlyinfluence the calculated equilibrium lattice constant. However, the full optimi-zation of the basis sets 8-511G/8-411G offers better results at the expense of theincreased computational effort. The replacement of the core electrons by ECPdrastically reduces the computational effort, but with a lower precision. A generaltendency of the HFT with the AE basis sets to underestimate the lattice constantand overestimate it with the VE basis sets is observed. The tolerance parametersand the calculation schemes also influence the results.

Acknowledgements: The authors (V.C, W.-K., S.-H.S.) thank Brain Korea 21project, for the financial support. We thank also to the Computer Centre of theKeimyung University, for the computational support.

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ВЛИЯНИЕ НА БАЗИСНИТЕ ФУНКЦИИ И ТОЧНОСТТАНА ПАРАМЕТРИТЕ НА ПРЕСМЯТАНЕ ВЪРХУ ЕНЕРГЕТИЧНИТЕ

СВОЙСТВА НА MgO. ИЗЧИСЛЕНИЯ ПО МЕТОДАНА ХАРТРИ-ФОК

Виорел Чихая1*, Валентин Д. Алексиев2, Николай М. Нешев2,Габриел Мунтяну1, Костинел И.Лепадату1, Уонг-Ки Мин3, Суонг-Хю Су3

1Институт по физикохимия “И. Г. Мургулеску”, Румьнска академия, пл. “Независимост” 202, 77208 Букурещ, Румьния

2Институт по катализ, Бьлгарска академия на науките,ул. "Акад. Г. Бончев", бл. 11, 1113 София

3Факултет по инженерна химия, Университет на Кеймюнг,Таегу, 704-701, Ю. Корея

Посветена на паметта на проф. Димирър Шопов по случай 80-тата му годишнина

Постъпила на 11 септември 2002 г.

(Резюме)

В рамките на теорията на Хартри–Фок е изследвана кристалната структура наМgО. Намерени са оптималните набори от базисни функции отчитащи, както самовалентните така и всички електрони за тази система. Оптимизирани са численитепараметри възпроизвеждащи експериментално измерената константа на решетката исвойствата на МgО. Показано е, че при набор от базисни функции за всички елек-трони пресметнатата стойност за константата на решетката е подценена, докатотенденцията за оптимизираната константа на решетката, пресметната с набор отбазисни функции само за валентните електрони е противоположна.