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Minimal-basis-adapted pseudopotentials for transition metal atoms
Volume 91, number 2
CHEMICAL
MNMALBASIS-ADAPTED
PHYSICS
LETKRS
3 September
1982
PSEUDOPOTENTIALS FOR TRANSITION METAL ATOhlS
G.H. JEUNC, J.C. BARTHELAT and M. PELISSIER Laboratorre de Physrque Quannque (ERA SZl), UnicersitC Paul Sabaner, 118, route de Narbome, 31062 Toulouse Cedex. France
Rccervcd1 Aped 1982, In Tutal form 28 June 1982
A pscudopotcnhal tcchquc
IS prcscntcd IO bc used w~lli P smglc lunctron constructed from three pruu~l~c GTOs for
the rcprcxntatlon of d orbrtals of tiansltron-metal atoms. Apphcd to Cu, Cuz. and CUT, It has pcrmlttcd 3 ruducllon In the number of 3d-type CT0 prunwcs without reducmg the quabty of the results.
1. Introduction Ab initlo molecular orbltal studies of compounds contaming transition-metal atoms such as clusters [l-3] are computationally expenswe because of the large number of electrons mvolved. The pseudopotential approach that restricts its working space to valence electrons through a frozen-core approxtmabon allows a reduction of the computational effort. For mstanco, valence calculations of transttlonmetal drmers [4-7] have proved the usefulness and accuracy of pseudopotential methods in ths field. However, all these stu&es have shown that a great number of gaussran pnmi~~ves are necessary to descnbe the d orbrtals. At least five or six gaussIan functlons are required to avold basis set superpoatlcr, errors. Consequently the time taken to evaluate the two-photon integrals over the atomic d basis functions becomes rapidly prolubltlve when the number of transitron-metal atoms is mcreased. Thrs calculatlonal handicap has discouraged the mterestmg study of metal clusters for a long ttme. For this reason we believe it to be of considerable Importance to investigate procedures leadtng to the use of a mmimal* representation for the d orbttals, e.g. no more than three gaussian pnmttives. In this work we take benefit from the arbitrariness l
In thus paper the term “minimal”
signitics that all d GTOs
are fully contracted,
0 009.2614/82/0000-0000/8
02.75 0 1982 North-Holland
mherent to the philosophy of all pscudopotentlal methods for derivmg a modlficd pscudopotcntral spcually prepared to be used mth nummal d basrs sets consistmg of three gaussian pnnutivcs. Of course It IS not the purpose of tlus pseudopotentlal to be used with a larger bavs set, m this cast the classical pseudopotentlal can bc used. In addltlon we hope that the use of the modlficd d pseudopotentrsl m molecular calculations could ovcrcomc the dlfficultlcs usually encountered with the use of mmimal d basis SCIS, I.e. a basis set extension cffcct. This paper IS devoted to the prescntatlon of the method. The modlficd pscudopotenbal and corrcspondlng t-mnlmal basis set are tcstcd by USCIn SCF
and Cl calculations on the Cu, and CuF molcculcs. In a subsequent paper small copper atom clusters urlll be dealt with m more dctal by ths method.
2. Method Pseudopotentials are dctermmed accordmg to the general scheme proposed by Barthclat and Durdnd [S]. WCbriclly recall the two steps of the above method: (I) The gencratron of valence nodcless pscudoorbltals 9, which reproduce best the outer parts of the true Fock valence orbltals pv. (II) The determination of the atormc pscudopotentlal Wpsby requirmg that the solution of the pseudo81
Voh~mc 91.
hadtonm
nulnbcr ?.
CHCIIICAL PHYSICS LEITJZRS
should bc E,, ~5” where E, IS the true
clgenvalue of the Fock opc~ator and $J, the abovcdefined pseudo-orbW In [ha work the fitst step is modified so as to reprcscnt each pseudo-orbM by only three gaussian functions. Ths choice seemed to be rather satlsfactory cvcn for d orbitills. This process forces that the rcsultlng
clgcnvector
gaussian
functions.
using as rafcrcncc
be representable
by three
vduc thu true e~gwfdue
P’
whcrc P, IS the projectIon operator over the Ith subspace of the spherical harmomcs, z IS the number of
d
1.79194
1; 2
1.79194 1.79194
0.113683 9.015S8 9.12989
-2 -I 0
0.59650 0.59640 0.59640
0.129380 7.86936 -2.91459
7
0 0 0 0 0 0 0 0 0
3) Tfahun from ref. [ 161. 82
Exponent
CoeKicicnt
5
0.628775 0.122365 0.041263
-0.175163 0.501686 1.0
P”)
0.1914
0.3799
0.0776 0.0316
0.5085 0.22 I 9
3.181954 0.952081 0.280435
0.637474 0.389569 0.215197
E, of the
pOd = _
P J,
Orbital
The second step is carried out by
ncor-Hartrce-Focksolutionsand the new nunimal tcprcscntatlon q$ of the pseudo-orbit&. The modified pseudopotential is extracted usmg a nearly complete space m order 10 avoId basis set superposition errors. An ad&tronal conslramt IS mamtamed to ensure 3 rapidly decrcasmg value of the pseudopotential for values of r greater than a core &us R,. This con&tion Is necessary to remove any spurious corcco~c mteraction m molecular calculations. The moMed pseudopotent& are wrItten m he usual semi-local form
5
3 Scptcmber 1982
28.0 20.0 15.0 10.0 7.0 5.0 3.5 2.0 1.0
3404.73 -12196.1 16384.3 -1393a.1 9953.17 -4 166.81 734.787 -17.7554 0.00409577
d
3) Taken from ref. 161.
valence electrons and WI(r) IS a radial function dependmg on I angular symmetry. W,(r) = _%,Pi I
exp(-ap*) .
The s and d components of the morhfied pseudopotenhal for the copper atom were obtamed from the *S(4~t3d~~) state. The values of the parameters are &splayed m table 1, together with the p component of the usual pseudopotential 16). The correspondmg optimized CT0 valence basis set used in tha work ISlisted in table 2. It is of mmimal quality, except for s where the three gaussian prinutives are contracted at a double-zeta level. Au SCF calculations were carried out by using the PSHONDOalgorithm [9] whch introduces pseudopotentials into the HONDOprogram IlO]. Valenceshell Cl calculations were performed accordmg to an Improved version [I I j of the CIPSI algorithm [Iz].
3. Resdf s and discussion As a prelimmary test, atomic SCF calculations usmg the modified pseudopotentral and the corresponding mmimal d basis set are compared with srrular crdculatrons carried out v&h the classuA pseudopotential. In the latter cases, two minimal d bass sets were used for companson, the fust one consHs of three GTOs and the second of five GTOs. We report in table 3 the d valence energy level and the results concerning the energy difference between
Volume
91, number
2
CHEhllCAL
PHYSICS
1982
I-able 4
the 2S and *D states, the lomzatlon potential and the electron affinity for copper. Usually minimal basis sets are expected to Bve energy separations In poor agreement with values obtamed from extended basis or numerical Hartree-Fock calculatmns (for example, see refs. [ 1,3]i. However we can see from table 3 that for the modified pseudopotentd, the
Wcncc orblkd encrg~cs (au) for mal d bars SCIS
Cuz PI r = 5.6 au ualngIIIIW Modified
ClklSSlCal pscudopotcntral
to an appreclablc modilicatlon of the predlcted values. This IS not the case wzth the classical pseudopotential which yields a noticeable change In the prc&cted values when the number of prlrmtlves IS rcduced from five to three. Slmdar behaviour of the modllied pscudopotential used with the corresponding d basis set IS obtamed in calculations on the t$ ground state of Cu,. SCF and Cl results are summarrzed rn tables 4 and 5. It is easily seen that the classlcal pseudopotentlal used with three d CTOs leads to too small an equihbrrum distance at the SCF level and an exaggerated value of the dlssocratlon energy (compared to the cze of five d GTOs). The good agreement with the experimental values IS fortuitous and must be consldered as an artifact due to a spurious basis set superposition effect. Another proof of that can be found in the values of the valence MO energes of d character obtamed in this case; they are in error (compared to the case of five d GTOs) by ~30%. Our purpose m thus work IS to avoid such an undesirable effect generally obtamed by using mmlnlmald representatlons. Indeed, the modified pseudopotentral used w-h three d CTOs exhibits SCF results in fairly good agreement with usual results Involving five d GTOs. Furthermore, when the number of d GTOs is Increased to five, the modified pseudopotcntial
pscudoporcnlwl
5 CT0
3 GYO
5 CT0
-0.110
-0.232
-0.22’
-0.1%
-0298
-0.439
-0.440
-0.4-15
-0.313
-0.348
-0.4J6
-0.450
-0.315
-0.455
-0.453
-0.456
-0.316
-0.459
-0.457
-0.460
-0.323
-0.472
-0.468
-0.47’
-0.322
-0.479
-0.476
-0.480
3 CT0
reduction m the number of prinutlvesdoes not lead
Table
3 sqllcmber
LETTERS
gives results which are close to those of tbrec d GTOs It proves hat this pscudopotentlalcan be used wttli three d GTOs without any basis set cxtcnslon cffcct. This agreement is also good after rntroductlon of Cl correlation energies. However It IS known that Cl effects on the spectroscopic constants are strongly dfferent dependmg on the quahty of the d basis set. The use ofa rmmmal reprcscntatlon for the d orbltals does not mclude the mtra-atomic d correlation cnerw which has been previously mentloncd 161 to play an important role m bond formation. Therefore the cquihbnum distance IS only shghtly altcrcd by mcluslon of energy correlation and remams too long (compared to the cxpenmental value) by 9%, but 60% of the bondmg energy IS recovcrcd. Bendes, calculations of Cul as a twoAectron system have been carned out considermg 3d orbltals as. frozen in the atomic copper core. For that purpose, a new pseudopotential was constructed m the usual
3
Atomicencrglcsfor copper
wltb various d baas Sets and pscudopotcnt& Bass set type ford orbM.s
a&electron 3)
chsucal pseudopotcntul modlied pseudopotential
a) See ref.
[ 131.
b) VJcnce
extended mmimal(3 mimmal(5 mimmzd (3 minunal(5
CTO) CTO) CTO) GTO)
cakulatcd
‘d b)
c) c) c) c,
d orbital energy for Cu(‘S).
-0.4907 -0.2819 -0.4605 -0.4758 -0.4825
C) Bans optimized
at tbc SW
E(‘D}
lcvcl (m au)
- E12S)
EKu+) - E(Cu)
E(Cu -) -E(Cu)
0.0132 -0.0072 0.1469 0.1577 0.1616
0.2352 0.2123 0.2340 0.2352 0.2379
-0.0005 0.0346 0.0284 0.0282 0.0265
for Cu(?S).
83
CtIEMlCAL
Volume 91 number 2
3 Scptcmbcr
PHYSICSLETTERS
1982
fnblc 5
Comparatw
spcctroscoptc
COIIS~II~S TOI
the ’ sg ground state of Cuz (cupcrimcntal l
Pseudopotcntul
values a) arc gwc.n m
rc (A)
3d orbkd
reprcscntnuon tucnty-t\~oclcctron
system
ChSSlC~l
motilicd
two-ckctron
n)
system
classlwl
DC WI
we
(cm-‘)
2.40
0.55
195
2.25 (2.22)
1.93 (2.05)
265 (266)
scr
2.30
1.61
263
munmal (5 GTO)
SCF Cl
2.46
0.45
183
2.41
1.28
178
muumal (3 GTO)
SCT Cl
2.46 2.42
0.40 1.22
190 186
mmunnl (5 CTO)
scr
Cl
2.41 2.43
0.37 1.14
171 200
no 3d
scr
2.48
Cl
2.58
-0.11 0.54
152 127
trrplc-zctLl
SCF b)
(6 GTO)
CI b)
nunlmat (3 GTO)
Ref. 1141. b, Ref. 161.
manner l-or tltc oneelectron atom, and a double-zeta s basts set was opumtzed for the ‘S(4st) state. For the two-electron model, the core-core interaction cannot bc approximated by the coulombtc repulsion of two potnt charges. So the 3d-3d electrostattc mtcracuon energy has been added IO the valence SCF energy for each value of r. In the SCF approxrmatron Cu2 ISnot bonded wtth respect to two separated Cu atoms. After tnclusion of the s correlatton energy, the dtssocration energy rematns only 26% of the expcrmlental value, and the equthbrium dntancc IS too long. This result emphastzes the promtnent role played by the d electrons m Cu2 bondmg. Thcrcforc tha fact dtscourages USCof the oneelectron copper atonuc model wtthout tahng account of the d contrrbuttons. These contrtbutions can bc analyzed as two major parts, at first the statrc deformatton effect of the d electron cloud due to the valence s electrons, and secondly the s-d correlation effect. As far as the d electrons can be viewed as rather locahzcd, a promrsmg way may be to maintain the 3d closed shell mto the atormc core and to model the core-valence polanzation and corrclatron effects. A srmplc pcrturbattve procedure has been 84
p~enum~s)
developed and successfully applted to alkali dtatoms [15] in order to treat these effects. The apphcation to small copper clusters is in progress along these lines [16]. Finally, we apphed our mo&fied copper pseudopotenttal to the ground state of CuF, which corresponds to a more iomc and stronger bonding wtth respect to Cu,. A classical pseudopotential and a standard double-zeta basis set were used for fluorine. For the sake of comparison we have again performed Calculations treating copper as a oneelectron system. All the results are gathered in table 6. Same trends in equrhbrium distance and dissociatton energy can be noticed for the SCF and CI results mvolving the modified pseudopotential for CuF and Cu2. Compared to experimental values, the CuF bond distance is 6% longer, and the bonding energy is ~75% of the experimental value at the CI level. Although the degree of polarizatton of the copper 3d orbitals is much greater than m Cuz, the use of the modfied pseudopotentral with three d GTOs leads to results which are of the same order of precision as the classical pseudopotential used w& five d GTOs.
Volume 91. number 2
CHEMICALPHYSICS
LCI-fCRS
3 Scprrmbcr
1982
Table 6 Comparatrvc
spcctroscop~c conslants
for the Cur ground Z.WX (cxpcnmcntnl
v-alucs 3) xc gvcn
Copper
Elcven-cicctron
m p~cnthcrs) D, (cv)
rc (A)
model
pscudopotcntrP
3d orbttal reprcsentctron
ClaSSICal
double-zeta
scr
(5GTO)
wc (cm
)
628
1.a2
2.15
(1.74)
t-1.46)
(623)
scr
1.77
3.30
654
mmunal
scr
(5 GTO)
Ci
I.83 1.83
2.10 3.81
539
muumaJ
scr
!.85
1.96
574
(3 CTO)
Cl
1.85
3.33
563
no 3d
SCI‘ Cl
2.02 2.02
0.98 2.46
480 475
mmrmaJ
-I
(3 CTO)
modified
one-clecuon
model
classical
553
a) Ref. [14].
4. Conclusion
[5l J.O. NocU,M.D.Newton, P.J. Hay, F.W. Bobrowcz,
We have seen that the proposed techruque yrelds pseudopotentials wluch are ad hoc for USCwith a mtntmal basis. Desptte the hmrted flexrbdtty in 3 nummal valence basis set, the use of three contracted d GTOs combtned wrth our modified pseudopotenttal leads to results of the same accuracy as those obtained from a classical pseudopotential working wrth a great number of d GTOs contracted to one function. These minimal representations for transitron-metal d orbrtds wdl make calculations possible on small metaLk clusters without a drasttc Increase of computattonal effort.
161 hf. Pchsstcr, J. Chcm. Phys. 75 (1981) 775. [7[ G. Das, Chcm. Phys. Lcttcrs 86 (1982) 482. [S] Ph. Durand and J.C. Uarthclat, ‘fhcorct. Chnn. Acta 38 (1975) 283; J.C. Barth&t, Ph. Durand and A. Scmtinr, Mol. Phye 33 (1977) 159, J.C. Barthclat and Ph. Durand, Cau. Chml. ItaL IO8 (1978) 225: hl. Rbssier and Ph. Durand, Thcorct Clmn. Acta 55 (1980) 43. 191 J.P. Daudcy, prtvatc commumcatton. [lo] hf. Dupurs, J. Rys and H.t-. Rmg. J. Chcm. Phys. 65
[ 111
[I]
H. Basch. M.D. Newton
(1976) 111. QCPC program 338. hL Pelisner, Thesis, UmvcrsttC Paul Sabstrcr, Toulouse ( 1980).
1121 B. Huron, J.P. hlalricu and P. Rancurcl, 58 (1973) 5745.
[ 131
E. Clcmcnti 14 (1974)
References and J.W. hloskowitz,
and C. Roctti,
1. Chcm. Phys.
At. Data NucL DataTablcs
177.
[ 141
KP. Hubcr ,tnd C. Hcnbcrg, Molecular spectra and molecular structure, Vol. 4 (Van Nostmnd, Prtnccton, 1979).
[IS]
G.H. Jeung, J.P. Malticu Phys, to be pubhsbcd.
[ 161
G.H. Jcung and J.C Barthclat, tton
J. Chcm.
Phys. 73 (1980) 4492. [2] J. Demuynck, MM. Rohmer, A. Stnch and A. Veflard, J. Chem. Phys 75 (1981) 3443. [31 H.Ba.wh,J. Am.Chcm.Soc. 103(1981)4657. [4]RN. Drxon and LL Robertson, MoL Phys. 36 (1978)
R.L. Martm and J. Chcm. Phys. 73 (1980) 2360.
and J.P. Daudey, J. Chcm. submitted
for pubhu-
1099.
85
CHEMICAL
MNMALBASIS-ADAPTED
PHYSICS
LETKRS
3 September
1982
PSEUDOPOTENTIALS FOR TRANSITION METAL ATOhlS
G.H. JEUNC, J.C. BARTHELAT and M. PELISSIER Laboratorre de Physrque Quannque (ERA SZl), UnicersitC Paul Sabaner, 118, route de Narbome, 31062 Toulouse Cedex. France
Rccervcd1 Aped 1982, In Tutal form 28 June 1982
A pscudopotcnhal tcchquc
IS prcscntcd IO bc used w~lli P smglc lunctron constructed from three pruu~l~c GTOs for
the rcprcxntatlon of d orbrtals of tiansltron-metal atoms. Apphcd to Cu, Cuz. and CUT, It has pcrmlttcd 3 ruducllon In the number of 3d-type CT0 prunwcs without reducmg the quabty of the results.
1. Introduction Ab initlo molecular orbltal studies of compounds contaming transition-metal atoms such as clusters [l-3] are computationally expenswe because of the large number of electrons mvolved. The pseudopotential approach that restricts its working space to valence electrons through a frozen-core approxtmabon allows a reduction of the computational effort. For mstanco, valence calculations of transttlonmetal drmers [4-7] have proved the usefulness and accuracy of pseudopotential methods in ths field. However, all these stu&es have shown that a great number of gaussran pnmi~~ves are necessary to descnbe the d orbrtals. At least five or six gaussIan functlons are required to avold basis set superpoatlcr, errors. Consequently the time taken to evaluate the two-photon integrals over the atomic d basis functions becomes rapidly prolubltlve when the number of transitron-metal atoms is mcreased. Thrs calculatlonal handicap has discouraged the mterestmg study of metal clusters for a long ttme. For this reason we believe it to be of considerable Importance to investigate procedures leadtng to the use of a mmimal* representation for the d orbttals, e.g. no more than three gaussian pnmttives. In this work we take benefit from the arbitrariness l
In thus paper the term “minimal”
signitics that all d GTOs
are fully contracted,
0 009.2614/82/0000-0000/8
02.75 0 1982 North-Holland
mherent to the philosophy of all pscudopotentlal methods for derivmg a modlficd pscudopotcntral spcually prepared to be used mth nummal d basrs sets consistmg of three gaussian pnnutivcs. Of course It IS not the purpose of tlus pseudopotentlal to be used with a larger bavs set, m this cast the classical pseudopotentlal can bc used. In addltlon we hope that the use of the modlficd d pseudopotentrsl m molecular calculations could ovcrcomc the dlfficultlcs usually encountered with the use of mmimal d basis SCIS, I.e. a basis set extension cffcct. This paper IS devoted to the prescntatlon of the method. The modlficd pscudopotenbal and corrcspondlng t-mnlmal basis set are tcstcd by USCIn SCF
and Cl calculations on the Cu, and CuF molcculcs. In a subsequent paper small copper atom clusters urlll be dealt with m more dctal by ths method.
2. Method Pseudopotentials are dctermmed accordmg to the general scheme proposed by Barthclat and Durdnd [S]. WCbriclly recall the two steps of the above method: (I) The gencratron of valence nodcless pscudoorbltals 9, which reproduce best the outer parts of the true Fock valence orbltals pv. (II) The determination of the atormc pscudopotentlal Wpsby requirmg that the solution of the pseudo81
Voh~mc 91.
hadtonm
nulnbcr ?.
CHCIIICAL PHYSICS LEITJZRS
should bc E,, ~5” where E, IS the true
clgenvalue of the Fock opc~ator and $J, the abovcdefined pseudo-orbW In [ha work the fitst step is modified so as to reprcscnt each pseudo-orbM by only three gaussian functions. Ths choice seemed to be rather satlsfactory cvcn for d orbitills. This process forces that the rcsultlng
clgcnvector
gaussian
functions.
using as rafcrcncc
be representable
by three
vduc thu true e~gwfdue
P’
whcrc P, IS the projectIon operator over the Ith subspace of the spherical harmomcs, z IS the number of
d
1.79194
1; 2
1.79194 1.79194
0.113683 9.015S8 9.12989
-2 -I 0
0.59650 0.59640 0.59640
0.129380 7.86936 -2.91459
7
0 0 0 0 0 0 0 0 0
3) Tfahun from ref. [ 161. 82
Exponent
CoeKicicnt
5
0.628775 0.122365 0.041263
-0.175163 0.501686 1.0
P”)
0.1914
0.3799
0.0776 0.0316
0.5085 0.22 I 9
3.181954 0.952081 0.280435
0.637474 0.389569 0.215197
E, of the
pOd = _
P J,
Orbital
The second step is carried out by
ncor-Hartrce-Focksolutionsand the new nunimal tcprcscntatlon q$ of the pseudo-orbit&. The modified pseudopotential is extracted usmg a nearly complete space m order 10 avoId basis set superposition errors. An ad&tronal conslramt IS mamtamed to ensure 3 rapidly decrcasmg value of the pseudopotential for values of r greater than a core &us R,. This con&tion Is necessary to remove any spurious corcco~c mteraction m molecular calculations. The moMed pseudopotent& are wrItten m he usual semi-local form
5
3 Scptcmber 1982
28.0 20.0 15.0 10.0 7.0 5.0 3.5 2.0 1.0
3404.73 -12196.1 16384.3 -1393a.1 9953.17 -4 166.81 734.787 -17.7554 0.00409577
d
3) Taken from ref. 161.
valence electrons and WI(r) IS a radial function dependmg on I angular symmetry. W,(r) = _%,Pi I
exp(-ap*) .
The s and d components of the morhfied pseudopotenhal for the copper atom were obtamed from the *S(4~t3d~~) state. The values of the parameters are &splayed m table 1, together with the p component of the usual pseudopotential 16). The correspondmg optimized CT0 valence basis set used in tha work ISlisted in table 2. It is of mmimal quality, except for s where the three gaussian prinutives are contracted at a double-zeta level. Au SCF calculations were carried out by using the PSHONDOalgorithm [9] whch introduces pseudopotentials into the HONDOprogram IlO]. Valenceshell Cl calculations were performed accordmg to an Improved version [I I j of the CIPSI algorithm [Iz].
3. Resdf s and discussion As a prelimmary test, atomic SCF calculations usmg the modified pseudopotentral and the corresponding mmimal d basis set are compared with srrular crdculatrons carried out v&h the classuA pseudopotential. In the latter cases, two minimal d bass sets were used for companson, the fust one consHs of three GTOs and the second of five GTOs. We report in table 3 the d valence energy level and the results concerning the energy difference between
Volume
91, number
2
CHEhllCAL
PHYSICS
1982
I-able 4
the 2S and *D states, the lomzatlon potential and the electron affinity for copper. Usually minimal basis sets are expected to Bve energy separations In poor agreement with values obtamed from extended basis or numerical Hartree-Fock calculatmns (for example, see refs. [ 1,3]i. However we can see from table 3 that for the modified pseudopotentd, the
Wcncc orblkd encrg~cs (au) for mal d bars SCIS
Cuz PI r = 5.6 au ualngIIIIW Modified
ClklSSlCal pscudopotcntral
to an appreclablc modilicatlon of the predlcted values. This IS not the case wzth the classical pseudopotential which yields a noticeable change In the prc&cted values when the number of prlrmtlves IS rcduced from five to three. Slmdar behaviour of the modllied pscudopotential used with the corresponding d basis set IS obtamed in calculations on the t$ ground state of Cu,. SCF and Cl results are summarrzed rn tables 4 and 5. It is easily seen that the classlcal pseudopotentlal used with three d CTOs leads to too small an equihbrrum distance at the SCF level and an exaggerated value of the dlssocratlon energy (compared to the cze of five d GTOs). The good agreement with the experimental values IS fortuitous and must be consldered as an artifact due to a spurious basis set superposition effect. Another proof of that can be found in the values of the valence MO energes of d character obtamed in this case; they are in error (compared to the case of five d GTOs) by ~30%. Our purpose m thus work IS to avoid such an undesirable effect generally obtamed by using mmlnlmald representatlons. Indeed, the modified pseudopotentral used w-h three d CTOs exhibits SCF results in fairly good agreement with usual results Involving five d GTOs. Furthermore, when the number of d GTOs is Increased to five, the modified pseudopotcntial
pscudoporcnlwl
5 CT0
3 GYO
5 CT0
-0.110
-0.232
-0.22’
-0.1%
-0298
-0.439
-0.440
-0.4-15
-0.313
-0.348
-0.4J6
-0.450
-0.315
-0.455
-0.453
-0.456
-0.316
-0.459
-0.457
-0.460
-0.323
-0.472
-0.468
-0.47’
-0.322
-0.479
-0.476
-0.480
3 CT0
reduction m the number of prinutlvesdoes not lead
Table
3 sqllcmber
LETTERS
gives results which are close to those of tbrec d GTOs It proves hat this pscudopotentlalcan be used wttli three d GTOs without any basis set cxtcnslon cffcct. This agreement is also good after rntroductlon of Cl correlation energies. However It IS known that Cl effects on the spectroscopic constants are strongly dfferent dependmg on the quahty of the d basis set. The use ofa rmmmal reprcscntatlon for the d orbltals does not mclude the mtra-atomic d correlation cnerw which has been previously mentloncd 161 to play an important role m bond formation. Therefore the cquihbnum distance IS only shghtly altcrcd by mcluslon of energy correlation and remams too long (compared to the cxpenmental value) by 9%, but 60% of the bondmg energy IS recovcrcd. Bendes, calculations of Cul as a twoAectron system have been carned out considermg 3d orbltals as. frozen in the atomic copper core. For that purpose, a new pseudopotential was constructed m the usual
3
Atomicencrglcsfor copper
wltb various d baas Sets and pscudopotcnt& Bass set type ford orbM.s
a&electron 3)
chsucal pseudopotcntul modlied pseudopotential
a) See ref.
[ 131.
b) VJcnce
extended mmimal(3 mimmal(5 mimmzd (3 minunal(5
CTO) CTO) CTO) GTO)
cakulatcd
‘d b)
c) c) c) c,
d orbital energy for Cu(‘S).
-0.4907 -0.2819 -0.4605 -0.4758 -0.4825
C) Bans optimized
at tbc SW
E(‘D}
lcvcl (m au)
- E12S)
EKu+) - E(Cu)
E(Cu -) -E(Cu)
0.0132 -0.0072 0.1469 0.1577 0.1616
0.2352 0.2123 0.2340 0.2352 0.2379
-0.0005 0.0346 0.0284 0.0282 0.0265
for Cu(?S).
83
CtIEMlCAL
Volume 91 number 2
3 Scptcmbcr
PHYSICSLETTERS
1982
fnblc 5
Comparatw
spcctroscoptc
COIIS~II~S TOI
the ’ sg ground state of Cuz (cupcrimcntal l
Pseudopotcntul
values a) arc gwc.n m
rc (A)
3d orbkd
reprcscntnuon tucnty-t\~oclcctron
system
ChSSlC~l
motilicd
two-ckctron
n)
system
classlwl
DC WI
we
(cm-‘)
2.40
0.55
195
2.25 (2.22)
1.93 (2.05)
265 (266)
scr
2.30
1.61
263
munmal (5 GTO)
SCF Cl
2.46
0.45
183
2.41
1.28
178
muumal (3 GTO)
SCT Cl
2.46 2.42
0.40 1.22
190 186
mmunnl (5 CTO)
scr
Cl
2.41 2.43
0.37 1.14
171 200
no 3d
scr
2.48
Cl
2.58
-0.11 0.54
152 127
trrplc-zctLl
SCF b)
(6 GTO)
CI b)
nunlmat (3 GTO)
Ref. 1141. b, Ref. 161.
manner l-or tltc oneelectron atom, and a double-zeta s basts set was opumtzed for the ‘S(4st) state. For the two-electron model, the core-core interaction cannot bc approximated by the coulombtc repulsion of two potnt charges. So the 3d-3d electrostattc mtcracuon energy has been added IO the valence SCF energy for each value of r. In the SCF approxrmatron Cu2 ISnot bonded wtth respect to two separated Cu atoms. After tnclusion of the s correlatton energy, the dtssocration energy rematns only 26% of the expcrmlental value, and the equthbrium dntancc IS too long. This result emphastzes the promtnent role played by the d electrons m Cu2 bondmg. Thcrcforc tha fact dtscourages USCof the oneelectron copper atonuc model wtthout tahng account of the d contrrbuttons. These contrtbutions can bc analyzed as two major parts, at first the statrc deformatton effect of the d electron cloud due to the valence s electrons, and secondly the s-d correlation effect. As far as the d electrons can be viewed as rather locahzcd, a promrsmg way may be to maintain the 3d closed shell mto the atormc core and to model the core-valence polanzation and corrclatron effects. A srmplc pcrturbattve procedure has been 84
p~enum~s)
developed and successfully applted to alkali dtatoms [15] in order to treat these effects. The apphcation to small copper clusters is in progress along these lines [16]. Finally, we apphed our mo&fied copper pseudopotenttal to the ground state of CuF, which corresponds to a more iomc and stronger bonding wtth respect to Cu,. A classical pseudopotential and a standard double-zeta basis set were used for fluorine. For the sake of comparison we have again performed Calculations treating copper as a oneelectron system. All the results are gathered in table 6. Same trends in equrhbrium distance and dissociatton energy can be noticed for the SCF and CI results mvolving the modified pseudopotential for CuF and Cu2. Compared to experimental values, the CuF bond distance is 6% longer, and the bonding energy is ~75% of the experimental value at the CI level. Although the degree of polarizatton of the copper 3d orbitals is much greater than m Cuz, the use of the modfied pseudopotentral with three d GTOs leads to results which are of the same order of precision as the classical pseudopotential used w& five d GTOs.
Volume 91. number 2
CHEMICALPHYSICS
LCI-fCRS
3 Scprrmbcr
1982
Table 6 Comparatrvc
spcctroscop~c conslants
for the Cur ground Z.WX (cxpcnmcntnl
v-alucs 3) xc gvcn
Copper
Elcven-cicctron
m p~cnthcrs) D, (cv)
rc (A)
model
pscudopotcntrP
3d orbttal reprcsentctron
ClaSSICal
double-zeta
scr
(5GTO)
wc (cm
)
628
1.a2
2.15
(1.74)
t-1.46)
(623)
scr
1.77
3.30
654
mmunal
scr
(5 GTO)
Ci
I.83 1.83
2.10 3.81
539
muumaJ
scr
!.85
1.96
574
(3 CTO)
Cl
1.85
3.33
563
no 3d
SCI‘ Cl
2.02 2.02
0.98 2.46
480 475
mmrmaJ
-I
(3 CTO)
modified
one-clecuon
model
classical
553
a) Ref. [14].
4. Conclusion
[5l J.O. NocU,M.D.Newton, P.J. Hay, F.W. Bobrowcz,
We have seen that the proposed techruque yrelds pseudopotentials wluch are ad hoc for USCwith a mtntmal basis. Desptte the hmrted flexrbdtty in 3 nummal valence basis set, the use of three contracted d GTOs combtned wrth our modified pseudopotenttal leads to results of the same accuracy as those obtained from a classical pseudopotential working wrth a great number of d GTOs contracted to one function. These minimal representations for transitron-metal d orbrtds wdl make calculations possible on small metaLk clusters without a drasttc Increase of computattonal effort.
161 hf. Pchsstcr, J. Chcm. Phys. 75 (1981) 775. [7[ G. Das, Chcm. Phys. Lcttcrs 86 (1982) 482. [S] Ph. Durand and J.C. Uarthclat, ‘fhcorct. Chnn. Acta 38 (1975) 283; J.C. Barth&t, Ph. Durand and A. Scmtinr, Mol. Phye 33 (1977) 159, J.C. Barthclat and Ph. Durand, Cau. Chml. ItaL IO8 (1978) 225: hl. Rbssier and Ph. Durand, Thcorct Clmn. Acta 55 (1980) 43. 191 J.P. Daudcy, prtvatc commumcatton. [lo] hf. Dupurs, J. Rys and H.t-. Rmg. J. Chcm. Phys. 65
[ 111
[I]
H. Basch. M.D. Newton
(1976) 111. QCPC program 338. hL Pelisner, Thesis, UmvcrsttC Paul Sabstrcr, Toulouse ( 1980).
1121 B. Huron, J.P. hlalricu and P. Rancurcl, 58 (1973) 5745.
[ 131
E. Clcmcnti 14 (1974)
References and J.W. hloskowitz,
and C. Roctti,
1. Chcm. Phys.
At. Data NucL DataTablcs
177.
[ 141
KP. Hubcr ,tnd C. Hcnbcrg, Molecular spectra and molecular structure, Vol. 4 (Van Nostmnd, Prtnccton, 1979).
[IS]
G.H. Jeung, J.P. Malticu Phys, to be pubhsbcd.
[ 161
G.H. Jcung and J.C Barthclat, tton
J. Chcm.
Phys. 73 (1980) 4492. [2] J. Demuynck, MM. Rohmer, A. Stnch and A. Veflard, J. Chem. Phys 75 (1981) 3443. [31 H.Ba.wh,J. Am.Chcm.Soc. 103(1981)4657. [4]RN. Drxon and LL Robertson, MoL Phys. 36 (1978)
R.L. Martm and J. Chcm. Phys. 73 (1980) 2360.
and J.P. Daudey, J. Chcm. submitted
for pubhu-
1099.
85