3 calculations for periodic systems

Volume 108. number

MIND0/3

CHEhIICAL

6

CALCULATIONS

PHYSICS

LE-lTERS

17 July

1984

FOR PERIODIC SYSTEMS

J.M. RlCART Dpr_ Qtlintica FtGca. Facttltat de Quitnim. lJtuk?rsitat de Barcelona. A~da. Dtigonat 617, Barcelona, Spain F. ILLAS Opt. Quimim Fisim. Faarltat de Quimim de Tarra.~otm. Universitat de Barcelona, Pqa he&l Tat-ram S/II. Tarragona. Spain and R. DOVESI. C. PISANI and C. ROE-f-I-1 di Clzimim Teorica. LJnir-ersirridi Torho . Via P. Giurtb 5. I-I 0125 Turin. Iral)

Isriruto

Rcccivcd

10 hIarc

1984:

in final form 15 May 1981

A 3IlND0/3 crystalline orbital (CO) LCAO XI- cotnputational scbcn~c hss been applied to tbr study of t\vo- and riuccdirncnsional periodic systctns and to the study of regular cbcrnisorption of hydrogen on gapbite. Tbc results arc in good agrccnxnt with previous ab initio Hartrcc-Fock ulculations, as well as with u\pcrinxntaI data. .L\ comparison \virb hllNDO/Z cluster calculations of’ bydro&!cn cbcntisorbed on graphite i.. also rcportcd.

for symbols and an extensive derivation. The Fock matrix in reciprocal space. F(k). is obtained by Fourier transfortning the corresponding tnatrix in direct space. Fg. whose elements are

I _ Introduction Tire hllNDO/3 method [I] _initially proposed to deal wit11 reactivity of organic conlpounds, has rccently been applied to cbemisorption studies on graphite, using finite cluster nrodels [3,3]. Many of the drawbacks shown by the CNDO/I, and EHT methods when applied to the same kind of problems [4.5] disappear w11s11 the MINDOj3 paran~eterization is used. For this reason we decided to implenrent a MIND0/3 version of the senn-entpirical crystalline orbital (CO) LCAO SCF computer program of Dovcsi ct al. 161) capable of treating

periodic

systcn~s

in two and tlnce

wberc 1, 2. 3. 3’ label the general 210: X3, extends to tire A0 of tbc atom to which 3’ belongs; o, I. g are translational vectors: 11 is the pseudo-bond-order IIX+ trk (see eq. (4) in ref. [ 61 j: H is the one-electron contribution to F. DC and Dx arc sum of symmetrized Coulonlb (C) and escbange (s) bielectronic integra!s (see eqs. (3). (5) and (6) in ref. [6])_ Tbc only difference between the CNDO/Z scbemc and the MINDO/3 one (apart obviously from the difference in the numerical values of tlic parameters) is represented by the last term in eq. (I), origiuated by the monoccntric bi-electronic integrals. whic!i are col-

dimen-

sions. The availability of a periodic program allows an alternative and complementary approach to the cluster studies in the chcmisorption problems: in particular it is possible to check the influence of the finite size of the cluster, and to contpare isolated with regular cbemisorption. The basic equations for the COhlIND0/3 are very similar to the CNDO/2 ones; here WC summarize them, and rcfcr to Dovesi et al. [6.7] 0 009-2614/84/S

03.00

0 Elsevier

Science

Publishers

B.V.

593

108, number

Volume

lected

6

in a unique

symmetrized symmetrized Ilaw

CHEMICAL

111tcrm,Al

Coulomb exchange

PHYSICS

actly the same as in ref. [6] _ This preliminary work is intended

being the corresponding

integral minus one half of the one. For the Coulomb part we

+W$, whislr t!Ic

=F,,263.&Y314,K~,l

11lSD0/3

•~~3sw]s-f~]

leil to the

muncric:ll

tics.

nanicly

N,,

= O.SIlG,,

follorvirtg

indices

-G,,,)

ia)

,)I;four

c‘xxzs when

possible

in 3 and 4 take al! the possibili-

s. p or p’:

.

:‘J,,, = ‘lf[,, = /r(G,,, - 0_5H,,,! . ;‘I,,,’ = It(Gl,l,. - 0.5M,,,,) + s ,,! ,,,,, (O.j(; ,,,, -c; ,I1

I’I’

2. Results

,,,, , - 0-51/,,i,)

.

\vhcre

-c;

G and I1 indicate

,,,,* - 0-m Coulomb

,,,‘I -

(5)

and cschgc

17 J _ I\ detailed CII

operator

s&wtr.y

derivation

in a subsequent

relation

equations

will hc giv-

Due to 111~!-act tllat MiXDO/ and CNDO/’ use 11112 wnic kind o!‘a!~!~rosinialioi~s for the inultict’ntci biclcct~onic integrals. tlx Lruncalion criteria here ado!~~cd liar Lllr Coulomb and cscl~angc series arc es-

wlucs

ct>r binding

is ini!I!icit!y

taken

into account

in the

kIlNDO/ stable tllan lations. in difference energ!’ per

pa!xx.

~‘~~mpz~riwn of calc~11~1cd and uspcrinwrltxl IllcnIal da1a t’rlllll WI’. 191.

energy

!)aramcterization. Note that graphite is more diamond according fo tlx MINDO/3 calcuagrecnicnt with experiment. although the in stability is overestimated. The binding carbon atom ofgraphite isabout 70% high tllan t!mt obtained with the same hl!NDO/_? metllod for a hydrogen-saturated C’, 6 model of‘ the graphite

in the basis of the A0

of the

and discussion

SW. Tlw binding crwgics for graphite and diamond xc in reasonable agrccrncnt with cspxinmt. better tlmn resulting from the IIF calculation. since the cor-

bi-

ciectroiiic mimoccntric RIINDO/3 parainctcrs [ 1j. I2 is llw oi-Jcr 01’ t!icJwiiit you!> of the crystal. and X, 23A = ,-A _,*=, Tf;Ti;,T".bemg the matrix representation 01 rile ls

the feasi-

Table 1 reports the most important energy data obtained for tllc four isoelectronic crystals. T!le STO-3G ab initio results and the available cx!min~cnta! data. botlt taken from ref. [9]_ are also given for compsri-

* = h(G,,*, p - O.W,,,,)

+ .\ ,,‘[,,, .,,m(;,,,,

to show

19s4

bility of the computations and to assess the quality of the corresponding results. First, the method is applied to the study of four isoelectronic systems for which ab initio results [S-IO] and a wealth of experimental data are availabie; they are the graphite monolayer. diamond, the hexagonal boron nitride monolayer (HBN) and cubic boron nitride (CBN). Such a test is stringent enough since these systems cover a range of structural and electronic situations: two- and threedimensional systems. covalent and partially ionic corn-!>ounds. insulators and semiconductors. We have nest considered the absorption of hydrogen on graphite in four diifcrcnt regular phases. for which previous ab initio [ 1 I] and CND0/2 [ 121 calculations existed. Also, comparison with MINDO/3 cluster results [ 31 was possible in this case.

(3)

IQ311

27 July

LETTERS

cncrgy.

lattice

parameter

and force

wnstzmt.

Ab initio and cxlwri-

~--

Nindiw

.systcill

.**rlipllitc diam<)lld Ilc\:l!Klrlal

cubiC !%K

119

~rvx”v z_ (cV)

Lattice

pxnmctcr

(A)

Force

\llND0/3

311 initio

c\pt.

XlINDOJ3

31) iniliu

cxpt_

l-1.1 12.7 152

10.6s 11.3s 9.s

15.2 15.1 13.0

2.53 3.72 2.54

2.5 1 3.59 1.60

2.46 3.57 2.5 1

17.7

10.6s

13.0

3.65

3.59

3.61

-

constant

XlIND0/3 s.07 6.4 1 9.71

10.1

(mdyn/A\)

31) initio

cspt.

7.90 6.30 7.99

6.40 4.79

7.31

-

CHEhllCAL PHYSICS LETTERS

Volume 108. number 5

monolayer [3] _This fact is not surprising: Bagus et al. [ 131. using ab initio calculations and large basis sets, have shown that a large number of atoms must be included in a cluster model in order to obtain a proper estimate of the bulk binding energy, although properties concerning cheniisorption are less sensitive to the cluster size. The binding energy of the two BN crystals is somewhat too bigb, possibly due to defects in the parameterization, which exa,,Doerates the covalent cliaracter of those compounds. In all cases the equilibrium lattice parameters are reproduced to within 4% The calculated force constants are alsoreasonable. although they seem to be too bigb for cubic and hexagonal BN. Tire band structure for tlie four systems are topologically very similar to the corresponding ab initio ones; as an example, the MIND0/3 and HF band structure for graphite are shown in fig. 1_In the four cases the MINDO/3 valence bandwidths (40.2, 45.0 26.0 and 34.7 eV for graphite, diamond, HBN and CBN) are higher tlian tire ab initio ones (31.5, 30.3.23.3 and 29.4); on the other hand, the conduction bands are narrower and are located lower in energy. As a result. the gap between valence and conduction bands in the present semi-empirical approach is consistently smaller with respect to ab initio calculations; since the latter esaggerate the distance between levels in the prosilnity of the Fermi energy, tire present results are nearer to

the experimental data. For diamond, in particular, we found a gap of 5.9 eV, to be compared with the experimental and ab initio values of 5.4 and 13.9 eV, respectively. The cbemisorption calculations bave concerned the four regular phases of hydrogen on graphite already studied by Dovesi et al_ using CNDO [ 121 and ab initio [ 1 l] techniques: one-to-one on-atom adsorption (phase A’), hydrogen above alternani atoms (A), at the midpoint of every third C-C bond(B), and at the center of each surface liesagon (C); the latter three

4 2 c c

I

3

2

1

P

Fig. 1. Comparison ab initio

STO-3G

structures

between MIND0/3

1

Q

I (continuous

P tic)

and

(dashed line, taken flom ref. 181) SCF band

for graphite.

F&T 2. Binding energy Eb per hydrogen atom for different structures of chemisorbed hydrogen as a function of the distance of the adsorbed layer from the graphite surface. The binding energies have been obtained from the difference between the total ener&y of the system, and the sum of the totsl enegy of the graphite monolayer per unit cell and of the isolated hydrogen atom (in the hfINDO/3 parameterization).

For the labels of the different curves, see the test. 595

CHEMICAL

Volume 108, number 6

7 hydrogen-to-carbon stoichiphases correspond to 1:_ omerry. The chemisorption energies per hydrogen atom as a function of the distance of hydrogen from the graphite layer are reported in fig. 3. The curves are very similar to those obtained with the ab initio approach: the order of energies is the same and the location of the minima is practically coincident. However, for all phases, tlie present results correspond to an increase in chcmisorption energy by 2.5-3 eV; this brings the important qualitative consequence that the phase A’ appears to be stable, while it is not in the HF calculation. The chemisorption energy was espetted to be underestimated in the previous STO-3C calculation. both because of the lack of correlation corrections and because of the poverty of the basissct. especially

for hydrogen. In addition, Iong-range multipole terms were neglected at the time of that calculation. which could have a stabilizing effect [ 14]_ The

present results therefore appear to be reasonable. From an energetic point of view. they arc intermediate between the ab initio and the CNDO/Z ones, the latter nprosimarion systematically leading to excessive high chcmisorption encrgics. It is finally of interest to make a comparison with tile MIND0/3 results of dissociative chemisorption of a hydrogen molecule on graphite, using a cluster modei of ttlC substrate (C, 61-l ,o)_ The geometry was optimized using the same techniques described in ref. [3]; in erder to obtain results comparable with those of the present study, a simple IIHF calculation was performed_ In the horizontal approach a relative minimum was found at a distance of 1.76 ,fi 1.rom the model surface, corresponding to a I I-H bond icngth of 1.76 a, and with tllc molecular axis along the central C-C bond. Such a configuration rcsemblcs closely that of the A’ phase, although the l-1-H distance is somewhat shorter in the periodical calculation. The chcmisorption cnergics are also surprisingly (and perhaps fortuitously) similar: they are 0.69 and 0.75 eV per hydrogen atom, with reference to the sum oi the energies of the adsorbing system and of the isolated hydrogen atoms, calculated in the same hllND0/3 approximation. In

conclusion,

The study

596

the present

of relatively

results are encouraging. complicated chemisorbed

27 July 1984

PHYSICS LET-TERS

phases on single layers or thin films appears to be feasible at limited cost and with reasonable accuracy_ Such investigations are in a sense complementary to

those performed

with cluster models and using the

semi-empirical approach. The latter can in fact provide useful information concerning isolated chemisorption and the mechanism of adsorption, which cannot be obtained when considering systems which are translarionally periodic in two dimensions. same

Acknowledgement One of us (J.M.R.) is indebted to the CIRIT of the “Generaiitat de Catalunya” for agrant in aid. Calcuia-

tions were supported by the Computer Universitat de Barcelona.

Center of the

References

[ 11 R.C. Bin;man, M.J.S. Dewar and D.H. Lo, J. Am. Chem. sot. 97 (1975) 1285. 171 F. Ill;rs, 1:. Sanz nnd J. Virgili, J. Xl01 Strucr. (Thsochem) 94 (1983) 79. 131 J. Ca.siiias, I‘. Illas, I‘. Sanz and J. Virgili. Surface Sci. 133 (1983) 29. 141 A.J. Bennett, B. McCarroU and R.P. blcssmer, Surface sci 74 (1971) 141. 151 h1.R. Ilsyns,Theoret. Chim. Acta 39 (1975) 61: R. Dovesi, C. Pisani I-. Ricca and C. Roetti, Surface Sci 72 (1978) 140. 161 R. Dovcsi. C. Pisani and C. Roetti, Surface Sci 103 (1981) 4S2. 171 C. Pislni and R. Dovcsi. Intern. J. Quantum Chem. 17 (1980) 501. 181 R. Dovcsi, C. Pisani and C. Roetti. Intern. J. Quantum Chcm. 17 (1980) 517. 191 R. Dovcsi, C. Pkni, C. Roctti ;iod P. Dellarole, l’hyr Rev. B24 (1981) 1170. IlO1 R. Dovesi, C. Pisltni, I‘. Ricca ;tnd C. Roetti, Phys Rev. B2.5 (1982) 3731.

1111 R. Dovcsi. C. Pisani and C. Roctti, Chem. Phys. Letters 81 (1981)

498.

1111 R. Dovesi, C. Pisani, F. Riccll and C. Roetti, J. Chem. Phys. 65 (1976) 3075.4116. [ 131 P.S. Bagus, H.F. Schaefer III and C.W. Bnuschlicher Jr.. J. Chcm. Phys. 78 (1983) 1390. [ 141 R. Dovesi, C. Pisani, C. Roetti and V.R. Saunders, Phys. Rev. B28 (1983) 5761.