Electron correlation in the nickel atom

Volume 75. number 2

ELECTRON Rrchard

CHFMICAL PHYSICS LETTERS

CORRELATION

IN THE NICKEL

ATOM

L. MARTIN

lhoretrcal Dw~sron, MS.569, Los Alamos Los Alamos, New Mexico 87.545. USA Recewed

15 October 1980

Sclentlfzc

Laborator),

17 June 1980

The large ddferenbal correlatlon eneraes assocmted with 4s - 3d trangtlons m NI are mvestlgated wthm the framework of confiiuratton mteractton Partnxlar attentton IS patd to the importance of f orbltals and the role of hlghersrder ewitatlons on the theoretical transltlon energies

1. Introduction The chemistry of the nickel atom is made both interesting and camphcated by the near degeneracy of the 3d and 4s orbitals. The 3d84s2(3F) and 3dv4s(3D) states are separated

by only 0.03

eV;

the 3dt0(lS)

state rs~ust 1.7 eV lugher [I] _ Dependmg upon the specifics of the molecular envuonment, all three configurntrons might be expected to play some role in the bondmg. For example, the 3dl” configuration IS domrnant m Ni(CO), [2], but plays only a minor role m NIH, where the bond mvolves pnmarlly the 3dg4s configuration [3]. UnFortunately, an unbiased theoretlcal descnptlon of the interplay among these configurations in a molecule IS made chfficult by the large correlation energies associated with the 3d shell. Hartree-Fock theory [4] favors the 3d84s2(3F) state by 1.28 eV relatrve to 3dg4s(3D). The error 1s even more pronounced in the 3d1°(tS) state, where numerrcal Hartree-Fock calculations [4] predict it to lie 5.47 eV above 3d84s2(3F). The inclusion of these large correlation effects vra traditlonal single and double excltatlon configuration mteraction techniques would seem to be lmpractlcal For all but the smallest systems In fact, even if such calculations were possible 111a “complete” one-electron basis, it is not at all clear that thrs level of approximation would suffice. The development of altematrve approafhes. such as the incorporation of atomic correlation by an effectwe harmltoman [5-71, requires infor290

matlon about the accuracy which can be expected at vanous levels of ngor. The calculatrons reported m 011s letter were motivated by a desire to know the srgnificance off orbrtals and the role of higher-order excrtatlons on the low-lying transitions in Ni.

2. Results The First assumption made m this work, whrch was later relaxed, was that the 3s and 3p shells should be considered part of a transferable core as far as 4s + 3d excrtatrons are concerned. The original basis set was therefore designed to correlate only the 3d and 4s electrons. Wachters’ [S] (14s, 9p, Sd) set of cartesran gausslan Functions, optimrzed For the dgs2(3F) state, was augmented with the diffuse d Function suggested by Hay [4] (ol = O-1316), and two additional “eventempered” p functions (c~ = 0 10,O 04). These exponents are similar to those found appropriate for spanning the 4p orbrtal m the 3d84s4p(5G) excited state (ol= 0.15,0.045) [S] . This (14s, 1 lp, 6d) pnmitrve set was contracted (Ss/4p/3d) by associatmg the “mner” members wth the atomic orbltals of the 3d84s2(3F) state, and Ieavmg the most diffuse s and the two most &Ffuse p and d functions completely Free. It IS doublezeta m the 4s and 4p regions, and triple-zeta m the 3d. The SCF results presented in table 1A are in reasonably good agreement wrth riumerical Uartree-Fmk results.

A slight bias favoring the d8s2 configuration ISapparent

Volume 75. number 2 Table 1 TheoretIcal evatatlon

CHEMICAL PHYSICS LETTERS

and loruzation energies m NI a)

State

Eupt. b)

NHF [4]

scr

(A) basis set 1 dss2(3F) d9s(3D) dt’(‘S) d9(*D)

0.0 03 171 7.59

-1506.8713 1.28 5.47 -

-1506

-0

-1506

-1506

(B) basis d8s2(3F) d9s(3D) dt’(‘S) d9(* D) d8(3F)

15 October 1980

+fd)

C1(3d, 4~) c) 8224 1 32 5.62 7.68

-1506

+3s. 3p e)

+E7A

f)

-1507

- 1507.248 0.2 2.5 7.3 24.6

9482 0 32 2 71 7.08

set 2 00 -0.03 171 7 59 25 79

8713 1 28 5 47

-.

8228 1 28 551 7 63 22.91

-1506.9687 0.46 2.87 7 33 24.33

-1507

-

037 0 30 2.72 7.22 24.43

222 02 28 7.2 24.3

a) The excltatlon energies are given III eV relative to the energy of dSs2(3F). When approprrate, the entry for dasz(3F) is the total energy tn au b) An average over spm-orbtt components from ref. [ 11. C)All smgle and double evcltatlons mvolving the 3d and 4s electrons d) An f orbital (0~= 2.3) was added to the basis set and the calculattons of the previous column were repeated. e, Smgle and double evcttattons involvmg the 3s and 3p electrons were appended to the confiiuration lists of the previous column. f, An estimate of the conkibutton of unhnked clusters [ 131.

III the 0.15 eV OverestImate of the excltatlon energy to the dl”(l S) state. Electron correlatron was mcorporated by contiguratron interactlon calculatrons includmg all smgle and double orbital replacements (SDCI) from the 3d and 4s shells of the appropriate Hartree-Fock configuratlon. The SCF orbitals for the state in questron define the one-electron basrs The improvements III the excrtation energes are dramatic (table 1A). The error UI AE(d8s2 --f dt”) has been reduced from 1.3 eV to 0.3 eV, and that for AE(d8s2 + dgs) from 4 9 eV to 1 .O eV. Even so, the remaimng errors, particularly for the d10 configuratron, are chemically quote significant. Before mvestigatmg the most obvious deficiency m the theoretical treatment thus far, the lack off functions in the basrs set, the s, p and d symmetries were tested for saturation. In the first set of calculations, the d space was grven more flexlbtity in the mner reBon by recontractmg the primitives (3,1,1, 1) to form a (%/4p/4d) basis, wMe III the second set the basis was made more flexible in the outer re@on by adding a more drffuse (even-tempered, a! = 0.036) d functton followed by a (4, 1, 1, 1) contractlon to (S&~/&I). These Lnprovements Jfected both the SCF and the SDCI excttatron energies by less than 0.1

eV *, and so the original

triple-zeta

d basrs was used

in

all

subsequent calculahons. The s and p symmetries were exammed next. An additional s function (a = 0.36) was placed in the “gap” between the exponents important for the 3s and 4s orbrtals. Similarly, the two most diffuse p functions were made sigmficantly tighter (ol = 0.28,O. 12). The frost of these now spans the region between the 3p and 4p orbltals. Thus (1 Ss, 1 1p, 6d) primitive set was contracted (8s/Sp/3d) by leaving the 4 most diffuse s functions and the 3 most diffuse p functions completely free. The most rmportant qualitative aspect of ths new basis is the double-zeta vanatronal freedom given to the 3s and 3p orbitals. They need no longer be descrrbed by a single contracted functron defined by the 3d84s2(3F) state. This added flexlbrhty removes nearly all the error at the Hartree-Fock level (table I B), but the net effect is to slightly degrade agreement with experiment. r” The (5s/4p/4d)

basis formed by contracting the d functions (3, 1.1, 1) ytelds ESCF(~F) = -1506.82256. AESCF(~D) = 1.31 eV, AESCF(‘S) = 5 60 eV, ESDCI(~F) = -1506 98559. AEs~clt~D) = 0.31 eV. AEso,&S) = 2.74 eV. The results for the (4. 1.1, 1) contraction -1506

82241,1.32.5

60.

--1.506_94835.0_32.

and

are 266

eV, respectively. 291

Volume 75, number -1506

3

= w

4506

90

CHEMICAL

2 I

1

I

I

95 L

-150700 F

The magnitude of the first term on the nght can be estimated from the secondarder par energy calculations of Jankowski et al. [9]. They report 1.2 eV fo_r the contribution off waves to e(3d2,3F) in the Isoelectronic Zn2+ Ion. This is a dynamical correlation commg almost entirely from d2 + f2 excitations, and it should be directly transferabie to Ni. ~(4~2, 1~) should be nearly unaffected by f orbitals. It is more lfficult to estimate the last two terms. If the semiemplrical pair energes as a function of Z denved by Pittel and Schwarz

t

_l5o,.i7 IO

I5

20

25

15 October 1980

PHYSICS LETTERS

[lo]

from optical

30

=f

Fig 1. The SDCI energy as a function of f-orbital exponent The arrows mark the exponent which satlsfles the condition tr2)f = (r’)sd.

shell correlation energy could easdy be as large as 1 .O eV, much of which should come from 4s3d + pf contnbutions *_ The present results are consistent with the upper range of this estrmate, that IS, toward large f-orbital contributions to the mtershell correlatron **. One of the surprismg aspects of this work was the large vanatlon observed m the SCF “core energy”, 30

The SDCI excitation energies are uniformly larger (~0.15 eV), probably as a result of an improved descnption of the 4s2 pair correlation energy by the trghter p functions. In the final motiicatlon of the oneelectron basis, an optlmrzed f-orbital exponent was determmed for each state by SDCI calculations. As expected, the optlmum exponent for dlo (ol = 2.0) IS somewhat more diffuse than that for d8s2 (cr = 2.3), but the energy IS not a very strong function of the exponent (fig. I). The results reported in the column labeled (+f) m table 1 B were obtamed with a common exponent of (Y = 2.3. Note that the addition of a single gaussian f orbitai has lzttle effect on the differentml correlahon enby ergies. The dgs and d 10 states are both stabtized =0.1.5 eV relative to d%*. This result, which is not expected to be substantially modified by a further expansion of the f space, can be rationalized m terms of a competltlon between mtra- and inter-shell correlation energy. Consider the difference in correlation energy between d8s2 and dl” m terms of symmetry adapted pair correlation energies: Ec(3d10,

lS) - Ec(3d84s2,

- 6(4S2, l S) - e(4s3d, 292

3F) * e(3d2, 3D) - e(4s3d,

3F) ‘D).

data for Cr, Mn, and Fe are

extrapolated to NI, the sum of the 4s3d par energres is 0.45 eV. This IS only a rough estimate, and the inter-

E core =

30

izshl +, zls(Jf, - +KJ, 9

as a function of the number of d electrons. As electrons are transferred from the 4s to the 3d shell, the 3s and 3p orbltals become shghtly more Iffuse, and the core energy mcreases. The effect is dramatic, the core energres are -1467.866, -1467.637, and -1467.492 au for d8s2, dgs, and dlO, respectively. llus 1s an average increase of 5 eV/d electron *! The

* The largest amphtudes

for mtershell

evcltahon

III the pres-

ent calculations come from 4s3d + pf terms Thuswas also found to be the case m the extenswe ST0 basis set studies on V<3s2 3d’) by Munch and Davidson [ 111 It should be noted that these pau correlation concepts can be used to sgmficantly reduce the extent of the calcula-

**

l

tion by neglectmg the “transferable” potion of the d2dynanucal correlatron. If only those double evcltations which represent the 3, the sd, and the changmg potion of the d2 correlation are considered, one finds E(3F) = -1506 9559 au, AE(3D) = 0.45 eV, and AE(‘S) = 2.88 eV. These results are wrthif~ 0.15 eV of the complete SDCI calculations In view of these results, the successful reproduction of Hartree-Fock 3d- 4s excltatlon energies by effective core potentials which include the 3s and 3p electrons imphes an interesting cancellatron of errors and deserves some study. For a discussion of several Ni effective potentials, see ref. [ 121.

Volume 75. number 2

series of cakulatrons therefore examined the transferability of the correlatron energy associated with the 3s and 3p electrons. The configuration hsts which result from relaxing the restrictions of a “frozen” 3s and 3p core are very large, mvolvmg nearly 20 000 spm funchons for the d8s2 and dgs states, and were reduced m length by drscardmg all contiguratrons which contrrbute less than 4 X 1 O-6 au to the correlatron enfinal

ergy determined by secondarder perturbation theory. The total contribution of ali &carded configurations at this selection threshold was never more than 2 mhartree.

As table 1 B shows, the transferabrlity assumpapproximatron to wrthin the estimated uncertainty of the selectron procedure (50.1 eV). tion

15 October L980

CHEMICAL PHYSICS LETTERS

excitations or theu mcorporation into the orbital space through a multlconfiguration SCF approach [IS] _

Note added in proof Consideratron of relatrvrstic corrections make the error m the SDCI approximation even larger. Relativis-

tic numerical Hartree-Fock calculations, using the of Cowan and Gnffin [i 61, give excitation energies from d8s2(3F) of 1.63 eV(d9s, 3D) and 6.04 eV(d*O, 1 S). method

is a good

References 3. Conclusions Although the one-electron basis set is certainly far from being complete m the sense of recovermg the total correlatron energy available from single and double excrtatlons, It is doubtful that further refinements wrlJ affect the drfferentral correlatron energy, AE,SD, by more than a few tenths of an electron volt. While an rmprovement of this magnitude might bring the d8s2-d9s sphtting and the fust iomzatron potential (table 1 B) mto agreement with experiment, it would sttll leave an error of approxtmately 0.8 eV for the d8s2-die difference. It seems more probable that the remammg discrepancy involves a differential contnbution from hrgher-order excitations, AEzQ . In fact, the unhnked cluster terms m the SDCI energy, when estimated by Davrdson’s formula [ 131, differ by 0.3 eV between d8s2 and d18 (table 1B). If all the remainmg dlscrepancles were to be associated with AETQ , it would have to be roughly 30% of AE,SD, which is

certainly not unreasonable. Sasaki and Yoshimine [14]

[ I] C E. Moore, Atomic Energy Levels. Nat1 Bur. Std. (U S ) Cucuiar 467, Vols 1 and 2 (1949). [2] 1-H Hdber and V.R. Saunders, Mol. Phys. 22 (1971)

1025.

[ 31 C F. Metius, B.D. Ok&on and W.A. Goddard111.Chem. Phys Letters 28 (1974) 457. [4] P.J. Hay, J. Chem. Phys. 66 (1977) 4377. IS] B.H. Brandow, Intern. J. Quantum Chem. 15 (1979) 207. [6] K.F. Freed, in Semiempuical methods of electroruc structure calculation. part A: Techniques, ed. G.A. SegaI (Plenum Press, New York, 1977). I71 T.H. Upton and W.A. Goddard HI, J. Am. Chem. Sot. 100 (1978) 5659; M SoUenberger, MS. Thesis, Cahfomia Institute of Technology (1976). [81 A.J.H. Wachters, J. Chem. Phys. 52 (1970) 1033. 191 K. Jankowski, P. hlahnowski and M. PoIasik. J. Phys. B12 (1979) 345. [lOI B Pittel and W.H E. Schwartz, J. Phys. Bl I (1978) 769, ill1 D. hlunch and E-R. Davrdson. J. Chem. Phys. 63 (1975) 980. t121 J 0. Noell, M.D. Newton, PJ Hay, R.L. Martin and F.W. Bobrowra. J. Chem. Phys (1980). to be published. t131 E R. Davrdson and D.W. Sdver, Chem. Phys. Letters52

(1977) 403.

F. Sa& and M. Yoshhnine. Phys. Rev. A9 (1974) 26. T.H. Dunning Jr., B.H. Botch and J-F. Harrison.J. Chem.

fAEtD = 0.2 m their studies of the electron afftitles of first-row atoms. It therefore seems

[I41 IIS1

safe to conclude that significantly greater accuracy will require either the exphclt consideration of tugherarder

1161 R-D. Cowan and J-0. 1010

found

AEFQ

Phys 72 (1980) 3419. Crrffii.

J. Opt

8oc. Am. 60 (1976)

293