ORIGINAL PAPER

M. Prencipe Æ F. Pascale Æ C. M. Zicovich-Wilson

V. R. Saunders Æ R. Orlando Æ R. Dovesi

The vibrational spectrum of calcite (CaCO3): an ab initioquantum-mechanical calculation

Received: 26 April 2004 /Accepted: 9 July 2004

Abstract The vibrational spectrum of calcite (CaCO3) isevaluated at an ab initio periodic quantum-mechanicallevel by using the CRYSTAL package. A localized basisset of Gaussian-type functions and the B3LYP hybridHamiltonian are adopted. The dynamical matrix isobtained by differentiating numerically the analyticalfirst derivatives of the energy. The accuracy with respectto all computational parameters is documented. Thecalculated frequencies are compared with available IRand RAMAN data (16 and 5 peaks, respectively), themean absolute error being less than 12 cm�1 (frequenciesrange from 100 to 1600 cm�1). Overall, the agreementwith experiment is very satisfactory, and shows thatsimulation can produce at a relatively low cost the fullspectra of crystalline compounds of mineralogicalinterest.

Keywords Vibrational spectroscopy �Quantum-mechanical calculations � Calcite

Introduction

The calculation of vibrational spectra of molecular sys-tems is a well-known procedure, implemented in severalrelevant computer codes (Schmidt et al. 1993; Frischet al. 2001). The method is based on the calculation ofthe Hessian matrix, either numerically or analytically,although the latter option is a very demanding compu-tational task (Stanton 1991). The situation is differentfor infinite periodic systems, where the development ofreliable and accurate computer codes is at an earlierstage than in molecular quantum chemistry and only fewab initio codes permit the calculation of vibrationalspectra of crystalline compounds.

The implementation in the periodic ab initio CRYS-TAL code (Saunders et al. 2003), that uses a basis oflocalized functions, is similar to the computationalscheme of molecular codes. It is based on the calculationof analytical energy gradients with respect to the nuclearpositions (Doll et al. 2001; Doll 2001). The Hessianmatrix is then obtained by numerical differentiation.

In a previous paper (Pascale et al. 2004) devoted tothe presentation of the method, the effect of the com-putational parameters controlling the accuracy of thecalculated vibrational frequencies is discussed at lengthwith reference to a-quartz.

In the present paper such a scheme is used to calculatethe full vibrational spectrum of calcite CaCO3 at the Cpoint; this permits a careful comparisonwith the availableIR and RAMAN spectra (Hellwege et al. 1970).

Calcite is an important polymorph of CaCO3. Itsstructure [rombohedral, space group R3c, with two for-mula units (Z) per cell, but often described in a hexag-onal cell with Z ¼ 6] consists of a stacking of planarCO2�

3 groups and Ca2þ ions along the c axis of thehexagonal cell. Its structural and elastic properties have

Phys Chem Minerals (2004) 31: 559–564 � Springer-Verlag 2004DOI 10.1007/s00269-004-0418-7

M. Prencipe (&)Dipartimento di Scienze Mineralogiche e Petrologiche,Universita di Torino, Via Valperga Caluso 35,10125 Torino, ItalyE-mail: mauro.prencipe@unito.itTel.: +390116705131Fax: +390116705128

F. PascaleLaboratoire de Cristallographie et Modelisation des MateriauxMineraux et Biologiques (LCM3B), UMR 7036,Groupe Biocristallographie, Universite Henri Poincare, Nancy I,Faculte des Sciences, BP 239, 54506 Vandoeuvre les NancyCEDEX, France

C. M. Zicovich-WilsonDepartamento de Fısica, Universidad Autonoma del Estado deMorelos, Av. Universidad 1001, Col. Chamilpa,62210 Cuernavaca (Morelos), Mexico

V. R. Saunders Æ R. DovesiDipartimento di Chimica IFM, Universita di Torino,Via P. Giuria 7, 10125 Torino, Italy

R. OrlandoDipartimento di Scienze e Tecnologie Avanzate,Universita del Piemonte Orientale, C.so Borsalino 54,15100 Alessandria, Italy

been investigated in the past by using either quantum-mechanical ab initio calculations (Catti and Pavese1997) or model potentials (Dove et al. 1992; Pavese et al.1992; Catti et al. 1993a; Catti and Pavese 1997; Pilati etal. 1998). In particular, by using a previous version ofCRYSTAL, Catti and Pavese (1997) were able to obtainthe frequency of the A1g C–O stretching mode within theHartree–Fock approximation. To date, however, noquantum-mechanical calculations of the whole vibra-tional spectrum of calcite have been performed, to theauthors’ knowledge.

The structure of the paper is as follows. We firstsummarize briefly the method employed for the calcu-lation of vibrational spectra; the stability of the calcu-lated frequencies with respect to some of thecomputational parameters will be checked, in order toverify if the experience (very limited, so far) gained witha semicovalent system like a-quartz (Pascale et al. 2004)can be transferred as such to the present system, that hasa different structure and symmetry, and contains acovalent part and a fully ionic part. The Results sectionpresents our calculated data and analyzes the vibrationalmodes.

Computational details

Geometry optimization and spectra calculations were performed bymeans of a development version of the ab initio CRYSTAL code(Saunders et al. 2003), which implements the Hartree–Fock andKohn–Sham, self consistent field (SCF) method for the study ofperiodic systems (Pisani et al. 1988).

Basis set

The multielectronic wave function is constructed as an anti-symmetrized product (Slater determinant) of monoelectroniccrystalline orbitals (CO) that are linear combinations of localfunctions (to be indicated as AOs) centred on each atom of thecrystal. In turn, AOs are linear combinations of Gaussian-typefunctions (GTF, the product of a Gaussian times a real solidspherical harmonic to give s�, p� and d-type AOs). In thepresent case, calcium is described with a 86-511d3G basis (seethe CRYSTAL Web page1). It consists of eight and six con-tracted GTFs for the description of the 1s and 2sp shells,respectively; contractions of 5, 1 and 1 GTFs are respectivelyused for the valence sp shells; the symbol d3 refers to thepresence of a contraction of three d GTFs. The exponents of thetwo outer sp GTFs have been reoptimized in the present case(0.453 and 0.295 bohr�2, respectively). C and O are described by6-21G* and 8-411G* basis sets, respectively (CRYSTAL page1;see also Catti et al. 1993b, 1994; the star indicates the presenceof a d shell). The reoptimized exponents of the outer sp GTFs(one for C and 2 for O) are 0.230, 0.486 and 0.1925 bohr�2,respectively. The exponents of the d GTFs for C and O are 0.8and 0.5 bohr�2, respectively.

Hamiltonian and computational parameters

The B3LYP Hamiltonian has been used, which contains a hybridHartree–Fock/density-functional exchange-correlation term; sucha Hamiltonian is widely and successfully used in molecular

quantum chemistry (Koch and Holthausen 2000) as well as forsolid-state calculations. The exchange-correlation contribution isevaluated numerically by integrating, over the cell volume (Vcell), afunction f ðrÞ of the electron density and its gradient. The choice ofthe grid of points for the integration is based on an atomic partitionmethod, originally developed by Becke (1988) for molecularsystems and then extended to periodic systems. In this scheme, theintegration domain Vcell of the function f ðrÞ is partitioned intoatomic volumes, each centred at nucleus A and associated withweight xAðrÞ. Spherical coordinates are introduced to separateradial and angular integrationZ

Vcell

f ðrÞdr ¼X

A

Z 10

drAr2A

Z 4p

0

dX xAðrA;XAÞf ðrA;XAÞ

�X

A

Xi;j

r2AixðrAi;XAjÞf ðrAi;XAjÞ;

where the A sum extends to all the nuclei in the zero cell,rA ¼ jr� RAj is the distance of a point in the cell from nucleus Aand X is the solid angle. In CRYSTAL, radial and angular pointsof the grid are generated through Gauss–Legendre and Lebedevquadrature schemes. The grid size has a strong impact on both theaccuracy and cost of calculation, so that the tuning of the grid isimportant to ensure numerical accuracy at an affordable cost. Agrid pruning is usually adopted, which is based on the observationthat the electron density at short radii has essentially sphericalsymmetry and that, far from nuclei, the weights xðrAi;XAjÞ arerelatively small. Accordingly, the angular quadrature in prunedgrids is not radially uniform, that is, index j in the above expressionvaries with i. In particular, the number of angular points is amaximum in the valence region and decreases gradually away fromit in both directions. If properly designed, a pruned grid is expectedto be about as accurate as the corresponding unpruned grid andmuch less expensive. In the following, the notation ðnr; nXÞ willindicate an unpruned grid with nr radial points and nX angularpoints; the notation ðnr; nXÞp is used to indicate a pruned gridhaving nX points on the Lebedev surface in the most accurateintegration region. The effect of the grid size on the calculation ofthe vibrational frequencies has been extensively tested by using the(55, 434)p, (75, 434)p, (75, 974)p, (75, 974) and (99, 1454) grids.

Results are shown in Table 1. As indicated by the maximum(Dmax), minimum (Dmin) and average of the absolute (jDj) differ-ences of the frequencies obtained with each grid with respect tothose of the most accurate (99, 1454) one (to be considered as fullyconverged with respect to the grid size), the (75, 974)p grid per-forms quite well at a reasonable cost, the latter depending upon thenumber of points (Np) at which the function of the electron densityis evaluated. The accuracy of the calculation with different gridscan be estimated from the difference (De) between the number ofelectrons in the cell (100, in the present case) and the value of theintegral of the electron density over the cell; as shown in Table 1,the (75, 974)p, (75, 974) and (99, 1454) grids are essentially equiv-alent. The effect of the grid size on the equilibrium geometry isnegligible: the calculated C–O and Ca–O distances change at mostby 0.0003 A when going from (55, 434)p to (99, 1454) and do notchange from (75, 974)p on.

The thresholds (ITOL1, ITOL2, ITOL3, ITOL4 and ITOL5)controlling the accuracy of the calculation of Coulomb andexchange integrals (Saunders et al. 2003) were set to 10�6 (ITOL1to ITOL4) and 10�12 (ITOL5).

The diagonalization of the Fock matrix was performed at 32points in the reciprocal space (Monkhorst net; Monkhorst andPack 1976) by setting the shrinking factor IS (Saunders et al. 2003)to 6.

Central zone phonon frequencies

The Born–Oppenheimer potential energy surface V ðxÞ of a systemis a function of the position x’s of the N nuclei. For a finite system,at the equilibrium nuclear configuration and within the harmonicapproximation, V ðxÞ takes the form1The Crystal Web page: http://www.crystal.unito.it

560

V ðxÞ ¼ 1

2

Xij

uiHijuj;

where the i; j summations run over all the 3N coordinates; ui rep-resents a displacement of the i-th Cartesian coordinate from itsequilibrium value; Hij is the ði; jÞ element of the matrix of thesecond derivatives of the potential, evaluated at equilibrium, withrespect to the displacement coordinates:

Hij ¼1

2

@2V ðxÞ@ui@uj

� �0

:

Phonon frequencies are then evaluated as the eigenvalues of theweighted Hessian matrix W , whose ði; jÞ element is defined asWij ¼ Hij=

ffiffiffiffiffiffiffiffiffiffiffiMiMj

p, where Mi and Mj are the masses of the atoms

associated with the i and j coordinates, respectively.In periodic systems, the W matrix can be given a block-diagonal

form, each block being associated with a different q point in thefirst Brillouin zone [W ðqÞ]. The q ¼ 0 point (C point) is of specialimportance, because IR and RAMAN spectra refer to this point.Central zone (q ¼ 0) phonon frequencies are the eigenvalues of theW ð0Þ matrix:

Wijð0Þ ¼XG

H0Gijffiffiffiffiffiffiffiffiffiffiffi

MiMjp :

where H 0Gij is the second derivative of V ðxÞ, at equilibrium, with

respect to atom i in cell 0 (translation invariance is exploited) andatom j in cell G. By the way, once the Hessian matrix H is calcu-lated, frequency shifts due to isotopic substitutions can readily becalculated, at no cost, by accordingly changing the masses Mi, inthe above formula. In the present case, the isotopic effects havebeen estimated in the cases of substitution of 42Ca for 40Ca, 13C for12C and 18O for 16O.

In CRYSTAL, energy first derivatives with respect to theatomic positions, vj ¼ @V =@uj, are calculated analytically (Dollet al. 2001) for all uj coordinates, whereas second derivatives atu ¼ 0 are calculated numerically using a ‘‘two-point’’ formula(N ¼ 2):

vj

ui

� �0

� vjð0; . . . ; ui; . . .Þui

;

or a ‘‘three-point’’ formula (N ¼ 3) to partially compensate foranharmonicity:

vj

ui

� �0

� vjð0; . . . ; ui; . . .Þ � vjð0; . . . ;�ui; . . .Þ2ui

:

In the present case, the influence of either the number of pointsused in the evaluation of the second derivatives (N ¼ 2; 3), or themagnitude of the displacement ui of each atomic coordinate(u ¼ 0:001; 0:002 A) on the calculated frequencies has beenchecked. Taking as a reference the N ¼ 2 and u ¼ 0:001 case,jDj ¼ 0:6, Dmax ¼ 1:7, Dmin ¼ �1:7 cm�1 when N ¼ 3 is used; asregards the step jDj ¼ 0:6; Dmax ¼ 1:0; Dmin ¼ �1:6 cm�1 whenu ¼ 0:002 A.

Since the energy variations for the displacements here consid-ered can be as small as 10�6 � 10�8 Hartree, the tolerance on theconvergence of the SCF cycles has been set to 10�10 Hartree.

In the case of ionic compounds, a correction must be added tothe Hessian in order to obtain the TO–LO splitting. Such a cor-rection takes into account long-range Coulomb effects due tocoherent displacement of the crystal nuclei (see Born and Huang1954, Sects. 5, 10, 34, 35; see also Umari et al. 2001) and dependsessentially on the electronic (clamped nuclei) dielectric tensor �1

and on the Born effective charge tensor associated with each atom.In the present case, the latter is evaluated through well-localizedWannier functions (Baranek et al. 2001; Zicovich-Wilson et al.2001, 2002; Noel et al. 2002) , whereas the former is from experi-mental data (�xx ¼ 2:716, �zz ¼ 2:208; Deer et al. 1992).

Geometry

The lattice constants have been determined by minimizing the totalenergy over the V ; c=a surface; for each ðV ; c=aÞ pair, the internalgeometry (x fractional co-ordinate of O) was optimized by ana-lytical gradient methods, as implemented in CRYSTAL (Civalleriet al. 2001). At this stage, the grid used was (55 434)p. Since in thecalculation of the vibrational frequencies the localization of thetrue energy minimum with respect to fractional coordinates is ofparamount importance, the internal geometry was reoptimizedwith each different grid, by keeping the lattice constants at thevalues refined at the first stage. Geometry optimization is consid-ered converged when the gradient (TOLDEG parameter inCRYSTAL) is smaller than 0.00001 hartree bohr�1 and the dis-placement (TOLDEX) with respect to the previous step is smallerthan 0.00004 bohr.

Results and discussion

The optimized geometry is reported in Table 2. The xfractional coordinate of oxygen and the C–O and Ca–Odistances displayed were calculated with the (75, 974)pgrid. The error with respect to the experimental datum issmall (about 1%) for the a parameter, dominated by thecovalent C–O interaction; it is larger (1.6%) for the clattice parameter, determined by the interaction betweenthe two ionic subunits Ca2þ and CO2�

3 . The errors in thecalculated lattice parameters mirror the correspondingones for the C-O and Ca–O distances, that are about 1%(C–O) and 1.4% (Ca–O) larger than the experimentaldistances.

Frequency of the normal modes

The rhombohedral cell of calcite contains two CaCO3

formula units, for a total of N ¼ 10 atoms; the numberof vibrational normal modes (3N � 3) is therefore 27.The frequencies of such modes are listed in Table 3,classified according to the symmetry they have in the 3mpoint group (see below). The A1g and Eg modes are

Table 1 Effect of the integration grid on the vibrational fre-quencies. jDj, Dmax and Dmin are the average of the absolute dif-ference, the maximum and minimum difference with respect to thereference case, the (99, 1454) unpruned grid, that can be consideredas fully converged. Np is the number of points in the grid and De (inl-electron) is the error in the integration of the charge density withrespect to the number of electrons per cell (100 in the present case).DE is the energy difference with respect to the reference case, in l-hartree; dC��O and dCa��O are distances, in A, as resulting from theoptimization of the x fractional coordinate of O with the variousgrids

(55, 434)p (75, 434)p (75, 974)p (75, 974) (99, 1454)

jDj 13.3 5.9 0.3 0.3Dmax 47.5 19.1 0.4 0.2Dmin )4.1 )1.3 )0.9 )0.9Np 60 954 120 198 246 264 515 912 1 010 024De 990 300 20 20 40DE )1436 )104 )24 )26dC�O 1.2948 1.2950 1.2947 1.2947 1.2947dCa�O 2.3928 2.3927 2.3929 2.3929 2.3929

561

Raman-active, the A2u and Eu are IR active, whereas theA1u and A2g are spectroscopically inactive (silent modes).The calculated frequencies are in general very close tothe experimental ones, the maximum error being38 cm�1 and the absolute mean error being only11:6 cm�1. Errors are larger at the two extremes of thevibrational spectrum, respectively in the range between100 and 200 cm�1 and above 1400 cm�1. In both casesthe calculated frequencies are higher than the experi-mental ones. The overestimation might be due, amongother reasons, to the following two factors: limitedflexibility of the basis set, that might influence in par-ticular the low-frequency modes, and the adoptedHamiltonian (B3LYP), which is known to overestimate

by some percent the calculated frequencies in molecules(Scott and Radom 1996), particularly at high frequencyvalues. It should be noted that, in the case of the verylow frequencies, temperature effects are known to havesome influence; as the calculated spectrum refers to 0 K,and the experimental one refers to room temperature,the differences might change if low-T experimental datawere available.

Catti and Pavese (1997) were able to calculate thevibrational frequency of the A1g stretching C–O mode, atthe Hartree–Fock level, by using a basis set similar tothe present one. The frequency value they obtained is1210 cm�1. The difference (about 120 cm�1) with respectto both the present B3LYP calculation and the experi-mental value, is likely to be due to the effect of theHamiltonian (Hartree–Fock is known to overestimatefrequencies by about 10%).

Lattice dynamical methods, based on parametricmodel potentials, perform less well than the present abinitio method: jDj¼17:0, Dmax ¼ 43 andDmin ¼ �109 cm�1 (Pilati et al. 1998), and jDj ¼ 19:2,Dmax ¼ 31 and Dmin ¼ �110 cm�1 (Pavese et al. 1992).In both cases the largest error is for the longitudinal Eumode at 1549 cm�1.

Analysis of the normal modes

According to Bhagavantam and Venkatarayudu (1969),in calcite, where four almost independent groups can beidentified (two Ca2þ ions and two CO2�

3 units), vibra-tional modes can be grouped into three categories,namely: translatory type (T) and rotatory type (R)external oscillations and internal (I) vibrations. Externalvibrations involve relative translations of the four unitsand librations of the carbonate ions, as a result of thefreezing within the crystal of real translations androtations of each single free unit. Internal vibrations arethose modes which involve deformation of the carbonategroups. A symmetry analysis in the 3m point group(Bhagavantam and Venkatarayudu 1969) shows that the27 normal modes, classified as A1g þ 2A1u þ 3A2gþ3A2u þ 4Eg þ 5Eu, group themselves into 9 T modes ofsymmetry A2g þ A1u þ A2u þ Eg þ 2Eu, 6 R modesof symmetry A2g þ A2u þ Eg þ Eu and 12 I modes ofsymmetry A1g þ A2g þ A1u þ A2u þ 2Eg þ 2Eu. Due to thevery different strength of the intra CO2�

3 and interionicforces, it might be expected that T and R modes havelower frequencies than those associated with I modes.

Inspection of the eigenvectors of the Hessian matrixand their graphical representations with the aid of the

Table 2 Calculated and experimental (Maslen et al. 1993a; Markgraf and Reeder 1985b) equilibrium geometry of calcite. The B3LYPHamiltonian and the (75, 974)p grid have been used

a (A) c (A) V (A3) c/a x dC�O(A) dCa�O (A)

Calc 5.0492 17.3430 382.9 3.435 0.25642 1.2947 2.3929Expa 4.991(2) 17.062(2) 368.1(3) 3.419(2) 0.2573(2) 1.284(1) 2.3590(8)Expb 4.988(1) 17.061(1) 367.6(1) 3.420(1) 0.2567(2) 1.280(1) 2.3595(5)

Table 3 Calculated (mcal) and experimental (mexp; Hellwege et al.1970) Raman and infrared active vibrational frequencies (cm�1). Dis the difference mcal � mexp. jDj, Dmax and Dmin are the average of theabsolute D, and the maximum and minimum D, respectively. Silentmodes are spectroscopically inactive. The B3LYP Hamiltonian andthe (75 974)p grid have been used

Mode Symmetry mcal mexp D

Raman1 Eg 159.5 156 3.52 Eg 281.4 284 )2.63 Eg 707.2 712 )4.84 A1g 1087.2 1086 1.25 Eg 1458.4 1434 24.4

Infrared6 Eu (TO) 129.3 102 27.3

Eu (LO) 145.6 123 22.67 A2u (TO) 130.1 92 38.1

A2u (LO) 159.1 136 23.18 Eu (TO) 221.5 223 )1.5

Eu (LO) 230.6 239 )8.49 Eu (TO) 286.7 297 )10.3

Eu (LO) 376.4 381 )4.610 A2u (TO) 293.7 303 )9.3

A2u (LO) 395.8 387 8.811 Eu (TO) 707.7 712 )4.3

Eu (LO) 708.4 715 )6.612 A2u (TO) 867.6 872 )4.4

A2u (LO) 890.6 890 0.613 Eu (TO) 1426.5 1407 19.5

Eu (LO) 1566.1 1549 17.1

Silent14 A2g 197.615 A1u 287.316 A2g 310.417 A2g 885.018 A1u 1087.4jDj 11.6Dmax 38.1Dmin )10.3

562

graphical MOLDRAW program (Ugliengo et al. 1993)shows that such a separation between external andinternal modes can indeed be made. Vibrations withfrequency in the range 700–900 cm�1 mainly corre-spond to bending I modes of the carbonate ions: thetwo Eg and Eu modes (707.2 and 707.7 cm�1, respec-tively) are in-plane deformation of the planar CO3

units, whereas the A2g and A2u modes (867.6 and 885.0cm�1) are out-of-plane deformations. The remaininghigher frequencies above 1000 cm�1 correspond to thesymmetrical (A1g and A1u) and asymmetrical (Eg andEu) stretching I modes of the carbonate. The vibra-tional frequencies of the isolated CO2�

3 have also beencalculated at similar computational conditions. Suchfrequencies are close to the corresponding ones in thecrystal, which is a confirmation of the above analysis;in particular, the in-plane and out-of-plane deforma-tions have frequencies of 658.1 and 877:9 cm�1,respectively; the symmetrical and asymmetrical stret-ching modes have frequencies of 1004.1 and1374:5 cm�1. The frequency shifts due to isotopicsubstitutions (isotopic effect), reported in Table 4, area further confirmation of the analysis of the I normalmodes: Ca is never involved in the modes discussed sofar, since no shift is observed when 42Ca is substitutedfor 40Ca; A1g and A1u stretching modes involve onlyoxygen motion, whereas all the other modes (bendingdeformations) involve both carbon and oxygen atoms.The small values of the difference in the frequencies ofthe g and u modes corresponding to the same defor-mation within each CO2�

3 unit are to be noted; this isa measure of the very weak coupling of the two car-bonate ions to produce in-phase (Eu, A2u and A1g) andout-of-phase (Eg, A2g and A1u) vibrations.

Concerning the external modes, apart from one case,a clear-cut distinction between T and Rmodes cannot bemade: in general, vibrations with similar frequency andsame symmetry can couple to each other to give morecomplex modes. The only pure T mode is the A1u at287.3 cm�1: indeed, since there are no A1u R modes, noT þ R coupling exists and this vibration corresponds to apure translation-type mode.

The low-frequency A1u þ 2A2g þ 2A2u modes corre-spond to relative translations of the carbonate and cal-cium ions along [111] in the rhombohedral cell (z axishenceforth) which are, except for the A1u mode, accom-panied by librations of the carbonate planar unit aroundthe same axis. In particular, the A2g modes at 197.6 and310.4 cm�1 are essentially out-of-phase translations of thecarbonate ions, with Ca ions at rest, whereas the two A2umodes at 130.1 and 293.7 cm�1 involve translations alongz of both calcium and carbonate ions. Within eachsymmetry, the frequency difference between the modes isrelated to the different phase of the libration coupled withthe translation. The A1u mode at 287.3 cm�1 is the out-of-phase translation of the two Ca ions along the z axis. Thisanalysis is corroborated by that based on the isotopiceffect, reported in Table 4: by substituting 42Ca for 40Ca,no frequency shifts are observed in the A2g modes, thusconfirming that calcium is at rest. On the other hand, thesame substitution has some effect on the A1u and A2umodes, as it should be expected on the groundof the aboveanalysis. The substitution of 18O for 16O has no effect onlyin the case of A1u mode that, indeed, is the only one whichdoes not involve librations of the carbonate ions.

Of the two low-frequency Eg modes, those at159:5 cm�1 correspond to out-of-phase translations ofthe carbonate ions parallel to the (111) plane of therhombohedral cell (xy plane henceforth); translations areaccompanied by in-phase librations of the two carbon-ates units around one of the twofold symmetry axisnormal to z. The other Eg modes at 281.4 cm�1 appear tobe in-phase librations of the carbonate ions around thetwofold symmetry axes. The Eu modes at 129.3 and221.5 cm�1 are translations of the Ca ions in the xy planeaccompanied by out-of-phase librations of the carbonateunits around the twofold symmetry axes. Finally, the Eumodes at 286:7 cm�1 are relative translations of both Caand CO3 units with out-of-phase librations of the latteraround the twofold symmetry axes. This analysis is inagreement with the calculated isotopic effect (Table 4),the latter showing that calcium ions are not involved inEg modes, whereas they are in the Eu ones. Moreover,the 13C-isotopic effect is stronger in all of the modesinvolving CO3 units translation, whereas it is weaker orabsent in the case of librations of the same units.

Conclusion

The feasibility of calculations of reliable vibrationalfrequencies of crystal, at an ab initio level, has beenshown. At least in the case of calcite, the average error

Table 4 Frequency shifts (Dm, in cm�1) due to isotopic substitu-tions of 42 Ca for 40Ca, 13C for 12C and 18O for 16O. The referencefrequencies mref (in cm�1) are those calculated for the most abun-dant isotopes 40Ca, 12C and 16O

Mode Symmetry mref Dm

42Ca 13C 18O

4 A1g 1087.2 0.0 0.0 62.415 A1u 287.3 7.0 0.1 0.018 1087.4 0.0 0.0 62.214 A2g 197.6 0.0 0.7 10.516 310.4 0.0 2.3 14.917 885.0 0.0 25.8 12.57 A2u 130.1 0.7 0.1 5.810 293.7 2.8 0.8 9.112 867.6 0.0 26.8 10.11 Eg 159.5 0.0 1.0 7.92 281.4 0.0 0.3 15.73 707.2 0.0 2.2 37.95 1458.4 0.1 42.1 20.86 Eu 129.3 0.8 0.0 5.58 221.5 4.9 0.0 1.19 286.7 3.0 0.6 9.411 707.7 0.1 2.4 37.713 1426.5 0.0 41.1 20.5

563

on the calculated frequencies, with respect to theexperimental data, is of the order of 10 cm�1, which iseven slightly smaller than the average error obtainedwith parametric model potential calculations. As a sideproduct of the calculation of the frequencies, theeigenvectors of the Hessian matrix are obtained, whoseanalysis proved to be a valuable tool in the interpreta-tion of Raman and IR absorption spectra. The isotopiceffect can also be easily simulated; its analysis is an aid inthe interpretation of vibrational modes.

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