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Indian Journal of ChemistryVol. 17A. January 1979. pp. 1-4

Crystal Field Stabilization Energies & Lattice Energies of Diatomic}Triatomic & Complex Crystals

LAMBODAR THAKUR*, ARUN KUMAR SINHA & B. B. SANDWARPostgraduate Department of Chemistry, Bhagalpur University, Bhagalpur 812007

Received 12 May 1978; accepted 1 July 1978

A new lattice energy equation is proposed to compute the lattice energies of 55 first-rowtransition-metal oxides and dihalides and of 21 complex crystals. The crystal field stabiliza-tion energies of the transition metal oxides and halides compare satisfactorily with thosein the literature. Present values yield an average error of 2-5% with the Born-Habercyclic data.

THE evaluation of the lattice energy of ioniccrystals has been a subject of extensive studyand various empirical as well as semi-empirical

methods have been developed with varying degree ofaccuracy>", The calculation of the lattice energyaccording to theoretical methods requires a largenumber of input data such as crystal structure,compressibility, interionic distance, etc. which arenot always accurately known for all the ioniccrystals. In the absence of any of these parametersone cannot compute the lattice energy values bythese methods. For further refinements a knowledgeof the zero-point energy, van der Waals and otherinteraction terms are required. To overcome thesedifficulties Kapustinskii" on the basis of Born-Landeand Born-Mayer potentials proposed a generalmethod (Eqs 1 and 2) for the calculation of thelattice energy of an ionic crystal utilizing only thesum of the six coordinated ionic radii (rc+ra).

UL= 120'16~nZIZ2(1 _.!) ...(1)

rc+ra m

= 120'16~nZIZ2 (1 __ P-) ...(2)rc+ra rc+ra

with P = 0·0345 nm.In Eqs (1) and (2), ~n is the total number of

ions in the molecule, ZI and Z2 are the ioniccharges, and m is the repulsion exponent of theBorn-Lande potential, which varies little aroundthe mean value of 9; Kapustinskii adopted avalue, m = 9.

Kapustinskii's equations proved useful in severalstudies. If we closely compare the Kapustinskii'scomputed values with the recent cyclic data wefind the limitations of his equations. Kapustinskii'sequations are based upon inverse power and ex-ponential repulsive terms, which suffer from draw-backs pointed out by different workers on differentfootings7-lI .. Ree and Ho1t12 have shown that thesemi-empirical effective pair potentials used by Tosiand Furni-s are too soft and require deeper minimawhen the elastic constants of the crystal are con-

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sidered. The recent perturbation calculations byBrumer and Karplust! have shown that the use ofonly one exponential form for the overlap repulsionis an inadequate representation. Dobbs and jones"also remarked that "the exponential form for therepulsive potential makes calculations of the latticeproperties rather complicated and in any case, is.perhaps, not valid in the region near the minimumof the total potential which is of course the essentialpart in considering the properties of the lattices."One argument that has been frequently advancedin support of the exponential function is thatsuch a term is predicted by quantum mechanicalcalculations. Theoretical treatments of the repul-sive force between closed-shell anions and pointcations15•16 and between inert gas atoms17-20 havetended to support the assignment of an exponentialterm. However, many of the theoretical results are-not in agreement 'with the experimental data20•Woodcock-! has used a generalized three-term poten-tial corresponding to a simple polarizable-ion model ;the repulsive core potential of Woodcock is ( harder'than the inverse power and exponential functions.

Jenkins too, in his recent papers21'22, produced a.repulsive-energy equation for complex crystals whichhas the advantage that the normal repulsion con-stant, P, occurs as a multiplicative constant ratherthan in its normal exponential Born-Mayer form.He ha~ used his equation in the term-by-termevaluation of the lattice energy of salt in the samemanner as the Huggins-Mayer equation. All this.suggests that the Born-Mayer form is to be modifiedsomehow to improve the results.

Logarithmic form of the overlap repulsive potential'has been found more suitable for accurately repro-ducing the observed properties of ionic crystals23-25._The general form is represented by

UR =P log (a+P/rn)where P and P are potential parameters whosevalues are determined by applying the crystalstability and compressibility conditions and a and nare constants whose values are selected arbitrarily

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INDIAN J. CHEM., VOL. 17A, JANUARY 1979',

o COMPLEX CRYSTALS

'iJ ALKALI HYORIOES

Cl ALKALI HALIO£S

l;. ALKALINE EARTH CHALICIOES0.20

0.15

v

_...l;L\- __ ~~ £Q.1

0.10

w;:)<,at

;:)

0.05

tJ.A

0 00

00.00

0.2 Q3 OA 0.5 Op

~+ rc (nm)Fig. 1 - Plot of rc + ra versus UR/UE for ionic crystals

by requiring them to fit the observed data. Therepulsive core potential of the logarithmic functionhas been found to be harder than the Woodcockpotential'v.

In. the light of the above facts we propose amodified Kapustinskii's equation of the logarithmicform

UL = 120'16~nZ1Z2 [1-log10{a + p 2}]. ~+~ ~+~

+10·5~nZ1Z2 •••(3)

where a and p are constants, whose values aredetermined empirically.

In the present paper we use two sets of datafor the parameters a and p: (i) for simple crystals.at rc+ra ~ 0'4 nm we obtain a = 1 and p = 0·02nms by requiring Eq. (3) to fit the cohesive energy-data for KF; and (ii) for complex crystals we-obtain a = 1·40 and p = 0·02, nrn- by requiringEq. (3) to fit the cohesive energy data for K2PtC16•

The ratio of the repulsive component, UR, to theelectrostatic component, UE, against the sum ofthe ionic radii (rc+ra) according to Kapustinskii's.Eqs (1) and (2) and the present Eq. (3) have been

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plotted in Fig. 1. Experimental values'[of UR havebeen obtained from Eq, (4)UR = UE-UCyclic ... (4)It has been found that the experimental UR/UEpoints for a large number of crystals are closer tothe curve obtained according to Eq. (3) but theyare far off the curves due to Eqs (1) and (2).For complex crystals the curve suggested accordingto Eq. (3) happens to be very close to the experi-mental UR/UE points. Such a plot also suggeststhat any equation with only one set of parameterscan never yield good results for simple crystalsas well as complex crystals. Lattice energy valuescalculated according to Eq. (3) for 40 diatomiccrystals have been reported earlier-", We reportin this work the lattice energies of 55 transitionmetal oxides and halides and of 21 complex crystals.

Results and DiscussionTransition-metal compounds - When the transition

metal ions are surrounded by an octahedron ofnegative ions or dipoles, the d-electrons are splitby the electrostatic field of the ligand into a triplydegenerate lower energy level which is stabilized

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.THAKUR et al.: A NEW EQUATION FOR CRYSTAL FIELD STABILIZATION ENERGY

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.and a doubly degenerate higher energy level whichis destabilized. This splitting gives extra stabilityto the crystal lattice known as the crystal field'stabilization energy (CFSE). Several workers27-29have pointed out that when the thermochemicallattice energies of transition metal compounds are'plotted against the increasing order of the atomicnumber of the cations, a double humped curve isobtained in each series showing a minima at M'l2+.The dO, d5 and d1o-ions have no CFSE and a smooth-curve is drawn through Ca2+, Mn2+ and Zn2+. As a.first approximation, deviation of the thermochemicallattice en ergy from this smooth curve has been'taken to be CFSE. Waddington-? has shown thatwhen the allowance for CFSE is made in the<computed value for the lattice energy of a transition-metal compound the addendum is very close to the-cyclic value.

Tables 1-3 present the thermochemical lattice-energies, CFSE, lattice energies according to Eq. (3).and the effective lattice energies of the first-row

TABLE 1 - CRYSTALFIELD STABILIZATIONENERGIES ANDLATTICE ENERGIES OF TRANSITION METAL OXIDES

(Values in k ] mol=')

Cyclic data CFSE UL(This work) (This work) (Eq. 3)

Crystals EffectiveUL

CaOScOTiOVOCrOMnOFeOCoONiOCuOZnOMeanerror %

3435(3611)38413895(3996)377838953954404240833991

o(105)184251(243)

o5875

121117

o

35813569368237033749379538583870389538913870

35813674386639543992379539163945401640083870

1'3

Values in parentheses are estimated from the graph.

TABLE 2 - CRYSTALFIELD STABILIZATIONENERGIES AND LATTICE ENERGIES OF TRANSITIONMETAL FLUORIDESAND CHLORIDES

(Values in kJ mol-t)

Fluorides Chlorides

Cyclic data CFSE UL Effective UL Cyclic data CFSE UL Effective UL(This work) {This work) (Eq.3) (This work) (This work) (Eq.3)

,Ca 2636 0 2720 2720 2268 0 2385 2385Sc (2694) (33) 2715 2748 (2364) (38) 2381 2419Ti 2770 71 2795 2866 2510 117 2443 2560V (2837) (109) 2812 2921 2586 142 2460 2602'Cr 2920 155 2845 :1000 2594 96 2485 2581Mn 2799 0 2883 2883 2544 0 2514 2514Fe (2912) (71) 2920 2991 2740 54 2544 2598Co 3025 142 2937 3079 2699 84 2561 2645Ni 3071 146 2958 3104 2782 117 2577 2694-Cu 3088 113 3075 3188 2820 117 2669 2786Zn 3025 0 2937 2937 2732 0 2561 2561Mean error % 2·6 2'3

Values in parentheses are estimated from the graphs.

TABLE 3- CRY3TAL FIELD STABILIZATIONENERGIES AND LATTICE ENERGIES OF TRANSITION METAL BROMIDESAND IODIDES

(Values in kJ mol-I)

Bromides Iodides

Cyc.ic data CFSE UL Effective UL Cyclic data CFSE UL Effective UL(This work) (This work) (Eq. 3) (This work) (This work) (Eq.3)

(;a 2184 0 2293 2293 2079 0 2167 2167Sc (2305) (54) 2293 2347 (2205) (50) 2163 2213Ti 2427 117 2351 2468 2335 109 2222 2331V 2531 159 2364 2523 2464 167 2234 2401c- 2531 105 2389 2494 2431 84 2255 2339Mn 2481 0 2418 2418 2389 0 2514 2514Fe 2577 46 2443 2489 2485 46 2305 2351-Co 2640 67 2460 2527 2552 75 2318 2393Ni 2720 109 2556 2665 2632 105 2330 2435-Cu 2774 125 2561 2686 2699 138 2410 2548Zn 2673 0 2460 2460 2598 0 2318 2318Mean error % 3,0 4'7

Values in parentheses are estimated from the graphs.

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INDIAN J. CHEM., VOL. 17A, JANUARY 1979

AcknowledgementThe authors express their thanks to Dr H. D. B._

Jenkins of the University of Warwick, Coventry(UK), for helpful suggestions.

References1. BORN, M. & LANDE, A., Verhandl, Dent. Physik Ges•.

20 (1918), 210. '2. BORN, M. & ·MAYER, J. E., Z. Physik., 1 (1932), 75.3. RITTNER, E., J. chem, Phys., 19 (1951), 1030.4. VARSHNI, Y. P. & SHUKLA, R. C., J. chem, Phys, 35 ..

(1961), 582. '5. REDINGTON, R. L., J. phys. cs-«, 74 (1970), 181.6. KAPUSTINSKII, A. F., Quart. Rev., 10 (1956), 283.7. DOBBS, E. R. & JONES, G. 0., Rep. Prog. Phys., 20<

(1957), 516.8. GOODISMAN, J., Diatomic interaction potential theory,.

Vols I & II (Academic Press, New York), 1973.9. THAKUR, K. P., Indian]. Chem., 12 (1974), 376.

10. SINHA, A. K, Ph.D. thesis, Bhagalpur University, 1977.-11. WOODCOCK,L. V., J. chem, Soc., Faraday II, 70 (1974)

1405. '12. REE, F. H. & HOLT, A. C., Phys. Rev., B8 (1973), 826.13. TOSI, M. P. & FUMI, F. G., J. phys. Chem, Solids, as-

(1964), 31; 45.14. BRUMER, P. & KARPLUS, M., J. chem, Phys., 58 (1973) •.

3903.15. UNSOLD, A., Z. Phys., 43 (1927), 563.16. BRUCK, H., Z. Phys., 51 (1928), 707.17. SLATER, J. C., Phys. Reu., 32 (1928), 349.18. BLEICKE, W. E. & MAYER, J. E., J. chem, Phys., 2'. (1934), 252.19. KUNIMUNE, M., Prog. theor, Phys., 5 (1950), 412.20. WADDINGTON, T. C., Adv. inorg. chem, Radiochem., r

(1959), 190.21. JENKINS, H. D. B., J. chem, Soc., Faraday II, 72 (1976)

1569. '22. JENKINS, H. D. B. & PRATT, K. F., J. chem. Soc ••_

Faraday II, 73 (1977), 812.23. PRAKASH, S. & BEHARI, J., Indian J. pure appl. Phys •.

7 (1969), 709. '24. PANDEY, J. D. & THAKUR, K. P., J. chim. Phys., 71

(1974), 850.25. THAKUR, K. P. & PANDEY, J. D., J. inorg. nucl, cie«:

37 (1975), 645.26. THAKUR, L. & SINHA, A. K., Indian J. pure appl. Phys.,

17 (1977), 794.27. HUSH, N. S. & PRYCE, M. N. L., J. chem, Phys., 28'

(1958), 244.28. ORGEL, L. E., An introduction to transition metal che--

mistry (Methuen, London), 1960.29. LADD, M. F. C. & LEE, W. H., J. chem, Soc., (1962), 2837.30. FRANKLIN, J. L. & HARLAND, F. W., Ann. Rev. Pbys ..

Chem., 24 (1975),485.31. CANTOR, S., J. chem, Phys., 59 (1973), 5189.32. ';'IAGMANN, D. D., EVANS, W. H., HALOW, 1., PARKER,

V. B., BAILEY, S. M. & SCHUMM,R. H., Selected valuesof chemical thermodynamic properties, NBS TechnicalNotes 270-1, 1965; 270-4, 1969; 270-5, 1971; 270-6,.1971.

33. LANGE, N. A., Handbook of chemistry (McGraw-Hill,New York), 1974.

34. GREENWOOP, N. N., Ionic crystals, lattice defects &>nonstoichiometry (Butterworths, London), 1968, 40.

35. BARBER, M., LINNETT, J. W. & TAYLOR,N. H., J. chem •.Soc., (1961), 3323.

36. COTTON, F. A., Acta Chern, Scand., 10 (1956), 1520.37. YATSIMIRSKII, K. B., J. gen. cu-«, 17 (1947), 2019.38. JENKINS, H. D. B., J. chem, Soc., Faraday I, 72 (1976) •.

353.

TABLE 4 - LATTICE ENERGIES OF COMPLEXCRYSTALSINkJ mol"!

Crystals rc + ra UL Kapustin- Cyclic(nm) (Eq.3) skii (ref. 36-38)

(Eq.2)

Mg(H.0)6C1• 0·416 1490 1590Mg(H.0)6(NOa). 0'440 1418 1510Ca(H.0)6(NOa). 0·416 1490 1590Sr(H.O).(NOa). 0·415 1493 1593Ni(H.O)6Cl• 0·410 1509 1610Ni(H.O)6(NOa). 0·434 1436 1529Co(H.O)6CI• 0·415 1493 1593CO(H.O)6Br• 0'430 1447 1542Cd(NHa)6Br• 0'462 1358 1444Cd(NHa)6I• 0'486 1299 1378Co(H2O)6(NOa)2 0·439 1421 1513Mn(H.O)6(NOa) 2 0'439 1421 1513Zn(H.O)6(NOa). 0'440 1418 1510Zn(NHa)6Br• 0,459 1366 1453 1350Zn(NHa)61• 0·484 1304 1384 1298Ni(NHa)6Br• 0·454 1380 1467 1365Co(NHa)6Br• 0·456 1374 1462 1358Fe(NHa)6Br• 0·459 1366 1453 1348K.PtCI6 0,421 1475 1572 1475Rb.SnC16 0,505 1255 1330 1267Rb.TeC16 0·512 1240 1314 1237Mean error % 1·2 7·3for the last 8crystals only

transition-metal oxides, fluorides; chlorides, bromidesand iodides. The present thermochemical latticeenergies for all the crystals have been recalculatedin view of the recent changes in the enthalpy terms.The electron affinities of halogens and oxygenand the heats of formation data have been compiledfrom recent sourcesso-ss. The sum of ionic radii,(Yc+yal. have been compiled from Greenwoodw andBarber et al.ss. It can be seen from Tables 1-3that the effective lattice energy values for all thecrystals are in close agreement with the experimentaldata giving very good agreement in the case ofoxides, fluorides and chlorides. For bromides andiodides discrepancies are larger which Waddingtonhas interpreted due to decrease of ionic characterin these crystals. The lattice energies of polyatomiccomplex crystals are presented in Table 4. Theionic radii of the complex ions and the cyclic valueshave been compiled from different sourcesS6-S8• Thepresent equation yields an overall mean error of2·5%. Such a simple equation can hardly beperfect. This equation does not have to give veryaccurate results. It can be used as a good firstapproximation; particularly for such crystals whosecrystal geometry are unknown. The present equationwhile predicting the lattice energy of such a largenumber of crystals, has rightly earned the claimthat it is a marked advance in lattice energycalculations for ionic crystals.

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