On correlation treatments of the nickel atom

Volume

115.

number

CHEMICAL

1

ON CORFUiXATION Celeste

McMlchael

TREATMENTS ROI-ILFING

Theorerrcal D~ursron. MS J569. Los Alamos, NM 87545, USA Recewed

26 December

PHYSICS

OF THE

and hchard

22 March

LE-I-I-ERS

NICKEL

1985

ATOM

L. MARTIN

Los Alamos Narronal i_nbora:ov,

1981

Uoller-PIesset periurbatron theory to finite order In ail terms (MPil(SDTQ)) and to rnfinlreorder in selecred terms (CCD) LS used IO exanune the Importance of higher-order excIIaUons In correlarlon treatments of the mclel atom Allowing Tar relatwwrc effects, good agreement wth expenmem 1s achieved for the ‘F-‘D spllttmg but not for Ihe ‘IF-‘S s.eparaIlon The MP perlurbation expansion etib1t.s slow wnvergence for the ruckel a:om, and suggests that the coupled cluster (CC) techmque wll be necessary IO obian a balanced descnptlon of zhe alormc stales

1. Introduction The Hartree-Fock approximation can be rather poor for transition metal atormc excitation energies, and so the effect of electron correlation on such sphttings has been examined by several authors [I -33 _ The ruckel atom is a dramatic kxample, with three states (3F fd8s2], 3D: [d%l] , and f S: [dl*]) Iymg experimentally within a 1.7 eV range [4]. Typical SCF calculations give a 5.5 eV spread wNe configuration interaction with smgle and double excitations (CISD) reduces this to 2.7 eV. More detailed results from one study [3] are @ven tn table 1. Use of Davtdson’s correction to approximate quadruples offers little unprovement, and the inclusion of reIativlstic corrections [5] further increases the dispanty between theory and expenment. One hkely explanation for the farlure of such standard CISD techniques on the nickel atom is the omission of tigher-order excitauons [1,3] _Two methods are employed here to investigate this possibility. M+%ler-PIesset perturbation theory through fourth order including single, double, triple, and quadrupIe excitations (m4(SDTQ)) and coupled cluster theory limited to double excitations (CCD). The CCD method is a perturbation cah~lation to infinite order in the operator Tz, while the “quadruples” in MP4 arise 104

from (r2)* only. Hence CCD at fourth order is MP4(DQ), and rapid convergence of the MP expansion is implied when MPIyDQ) approximates the CCD result [6] _

2. Computational methods The Wachters GTO basis set for r&kel(14s, 9p, Sd) was used [7], with the last three s exponents replaced by five (a = 1 .O, 0.333,0.133,0.0533, and 0.0213). Thus basis was augmented by two diffuse p functions (a = 025 and O-063), one diffuse d (ff = 0.1316) [S] and one f function (ru = 2.2) [l] . The resulting (16s, 1 lp, 6d, If) basis was contracted to [lOs, 6p, 4d, 1 f] with the outermost functions free. At the SCF level, this basis correctly reproduces the non-relativlshc numerical Martree-Fock excitation energies, and the use of even larger basis sets is expected to produce little change in these splittigs. UHF wavefunctions were used for the triplet states of the nickel atom, so spin contamination was checked and found to be negligible_ Five components of each d function and seven components of the f function were used. All calculations were performed with the GAUSSIAN 82 series of programs [9]. In ad&lion, ROHF and CISD calculations [IO] were done in order to compare with &e results from earlier studies. 0 009-26 14/85/S 03.30 0 Eisevier Sezence Publishers B.V. (North-Holland Physics Publishmg Bvision)

Volume 115. number 1

CHEMICAL

PHYSICS

3. Results

orbit& are not truly ‘*core” orbitals, and their effect on the magrutude of the CISD excitation energies is to change them by about 0.05--0.10 eV. A more impor-

In order to confidently compare theoretical and expenmental atomic term separations for nickel, one or the other set of data must be “corrected” for reIativistic effects. The results m table 1 from numerical Hartree-Fock calculations [S] with and without the inclusion of relativrty give an estimate of the size of this correction. Applymg this to the observed splittings yields “non-relativistic” experimental values that can be directly compared to non-relativistic calculations. This method can be justified only by discountmg any sizable coupling between relativity and correlation. In fact, a calculation [i 11 of the excited states of the nickel atom considered here found that the differential relat@&ic correcuon computed at the SCF level is wthin 0.01 e V of that ccmputed \nth a CX!?L) wavefunction. Hence, generous error bars for the “non-relativistic” experimental values in table 1 are +0.05 eV. Table 1 also lists +&e ROHF and CISD excitation energies of this work and of Bausch&her, Walch and Parrridge 133. The notations (3d4s) and (3s3p3d4s) indicate which orbitals are correlated. The 3s and 3p

Table 1 Excitation

energies for the Ni atom [III eV r&tzve

tent

effect IS the expansion of the basis set to include f functions. The CISD 3F-3D splitting for the largest

ST0 basis, which contains three f functions and one g function, is in error by OX eV and for 3F-1S, by 1.28 eV. Use of Davidson’s correction to account for unlinked clusters reduces these to 0.29 and 0.93 eV, respectively. These are still very large errors considermg the extensiveness of the basis set and the method of correlation employed. There are a few more points to be made by table I Comparing the CISD results of the present work with the best ST0 calculations. it is evidenr that an additional O-2-0.4 eV of differential correlation energy is available upon expansion of the GTO basis set. Most of this can be traced to the need for additional f functions. However, the errors associated with the incompleteness of the basis used here are smaller than those apparently inherent in the CISD procedure. Hence this ~ves~gation was undertaken in order to address the ecfects of higher-order excitations on the mckel atom. Table 2 summarizes the atortuc splittings obtained

to the 3F state) a)

Baas

Calculation

3D

CT0 [6s. 6p. 3d] b)

ROW CISD(3d4s)

1.28 0.47(0.48) 0.28(0.30) 0.24(0.23) 0.4 l(O.44) 0.09(0.10~ 0.18(0.t6) 0.05(0.01~ -o&N--0.10) 1.26 0.24

[6s. 6p, 3d. 1f’J b) (6% 6p, 4d. I6s. 6p, 4d, ST0 (8% 7p, Sd. (Ss, 7p, Sd, GTO

if] b) 2fl b) If) b) zf) W (8% 7p, 5d,3f, I& b) IlOs. 6p. 4d. lfl cl

CISD(3d4s) CISD(3s3p3d4s) CISD(3d4s) CISD(3d4s) CISD(3s3p3d4s) CISD(3s3p3d4s) ClSD(3s3p3d4s) ROW ClSD(3d4s)

non-rehtrvistic numeriti HF d) relativistic numerkaI HF a)

1.27 1.63

experiment e) expermxent, -‘corrected” for relativistic effects

a)_cCISD

22 March 1985

LETTERS

results with Davidson’s correctitionare in parentheses.

h) Ref. 131.

‘S

2.72(2_44) 2.42c2.07) 5151 2.68 5.47 6.04

-0.03

1.71

-0.39

1.14

C) This work.

d) Ref. [SJ.

e) Ref. 141.

105

Volume

CHEMICAL

115, number 1

PHYSICS

Table 2 Excitation energies derived from perturbation theory for the Ni atom (in eV relatwe to the ‘F state), usmg the [lOs, 6p. 4d, If] basis 3D UHIpostSCl- (3d4s) lblP2 MP3 MP4(DQ)

1.35

5.66

-0.59 0.66 0.16

-0.18 4.28 1.09 0.25 -1.24

hlP4(SDQ) MP4(SDTQ

-0.01 -0.30

CCD CCD + ST(4)

0.35 -0.11

post-SCF (3s3p3d4s) MP2 MP3 hlP4(DQ) MP4(.SDQ) MP4(SDTQ) CCD CCD + ST(4)

‘s

2.57 (0.24)

-1.16 1.10 -0.48 -0.75 -1.15

-1.43

0.22 -0.46

2.35 (-0.70)

5.68 -1.17 -2.38 -4.23

w-101 various

orders of perturbation theory. Frost examme the 3F-3D excrtatron energres with correlation of the 3d and 4s orbitals only. The hW4(SDTQ) result of -0.30 eV IS within 0.1 eV of the “non-relat.rvistic” experimental value. Triple excitations account for 0.29 eV and are probably overestimated at this level. The MP perturbation expansron IS not fully converged, however, as demonstrated by the sizable oscillations even through full fourth order. Inclusron of the 3s and 3p orbrtals in the correlating space increases the magnitude of the oscillatron, and the MP4(SDTQ) excitation energy is much too large. The failure of MP perturbation theory to reach convergence IS even more pronounced for the SF-1s splitting. Changes of several eV m the excrtation energy occur at successive orders, and allowing the 3s and 3p orbit& to be correlated also increases the size of the oscillations. In fact, MP4(SDTQ) gives the 1s state as the ground state of the nickel atom! Also in table 2 are the CCD results for both the 3F -3D and 3 F-l S excitation energies. They are similar

to the CISD calculations performed with this basis, although the CCD values mdrcate the contribution from 106

22 March 1985

LETTERS

unlinked clusters may be on the order of 0.25 eV for the 3 F-l S splitting. This agrees well with the average size of Davidson’s correctron found for this splitting in the CISD ST0 study [3] . The “3~3~” eff,zt lowers the CCD excitation energies by 0.1-03 eV. However, the 0veraIi agreement with “non-relativrshc” expenment is still poor so additional correlation energy must be obtained from consideration of other excitation operators in infinite order. At present we are unable to do this, but an estimate can be made of this contnbu-

tion as follows. The correlation corrections at each level of perturbation theory are grven in table 3. As expected, *he

second-order

contribution,

Eg),

is large and negative

while ,563) is always positive_ All Et41 contributions are smalI and negative, although in an absolute sense the sum of these contributions is still quite large. Triples amount to roughly 1.7 times the singles. For the 3 F and 3D terms, the total correlation energy through MP4(DQ) is nearly equal to the CCD correctron, implymg that the magnitude of double excitations m higher even orders is small. Thus it is reasonable to estimate the total srngles and triples contnbutron from Ei4) + E.$?) and add this to the CCD energy [6]. The 3F-3D splittings derived from this CCD + ST(4) approximation are listed in table 2. Presumably the fourth-order contribution of singles and tri-

ples from MP4 is overestimated,

so the CCD + ST(4)

values are actually too low. The best estimate of the 3F-3D excitation energy with this basis set that includes correlation of the 3s and 3p orbitals is -0.46 eV, within 0.07 eV of the “non-relativistic” experimental value. Note that the CCD + ST(4) approach is not justified for the 3F-1S excitation energy because CCD and MP4(DQ) yield quite different correlation energies for the lS state.

4. snmmary

In conclusron, atomic states of the nickel atom are not treated adequately by conventional CiSD techniques Both the unlinked cluster error in the CISD method and the contribution of triple excrtations have been found to be quite important. MblIer-PIesset perturbation theory is in fact worse than CISD because of convergence difficulties. A multireference approach suffers from the fact that there is no small subset of

Volume 115, number 1 Table 3 Correhtion

CHEMiCAL

to the total energy of the NI atom (m au relatxve to the SfIF energy). usmg the [ lOs, Sp, 4d, Iq

corrections

JD

3F

E$)

‘S

:3s3p3d4s)

(3d4s)

(3s3p3d4s)

(3d4s)

(3s3p3d4s)

-0X8

-0572

-0.340

-0.665

-0.483

-0.833

0.056

0.063

-0.006

/3(4)

bans a)

(3d4s)

3.010

Eg

22 March 1985

PHYSICS LETTERS

0.146

0.174

0.325

-0.035

-0.025

-0.093

-0.123

-0.287

ES” 2

-0.004

-0.008

-0.010

-0.019

-0.034

-0.053

.f$’

-0.007

-0.015

-0.018

-0.029

-0.062

-0.083

,+CD)

-0.264

-0.534

-0.302

-0.576

-0.378

-0.656

a) The UHF eaer8ies of the ‘F and 3D states mh gy of the ‘S state is -1506.521663 a”.

this basis are -1506.729777

domm~t con~gurat~ons on which to base such a scheme. The preferred method appears to be the coupled-clusters procedure, which combines the advantages of a linked fiite

order

expansion

approach.

Such

with

the necessity

calculations

of an m-

mclutig

and triples

sin-

(CCSDT) may help determine the source of the rem~~g error in the 3H-‘S energy

gles, doubles,

difference in the nickel atom.

References [ 11 R.L. Martm, Cbem. Phys. Letters 75 (1980) 290, [2] B_H. Botch, T.H. DUM& Jr. and J.F. Harrison, J. Chem. Pbys. 75 (1981) 3466.

and -1506.680012

au, respectwely,

The RHF ener-

131 C-W. Bausc~cher Jr., SP. W&h and H. Partridge, J, #em. Phys. 76 (1982) 1033. [4] C.E. Moore, Atomic Energy Levels NBS circular 467, Vol. 2 (NatL Bur. Stand. Washington, 1952). [5] R.L. hkt-tin and PJ. Hay, J. Chem. Phys. 75 (1981) 4539. [6] RJ. Bartlett, Ann. Rev. Phys. C&em. 32 (1981) 359. [7] A.J.H. Wachters, J. C&em. Phys. 52 (1970) 1033. [S] PJ. Hay, J. Chem. Phys. 66 (1977) 4377. 19) J.S. Binkley, MJ. Friwh, D.J. DeFrees. K. Raghavachari, R.A. Whiteade, H.B. Schlegel, E.M. Fluder and J.A. Pople, GAUSSIAN 82, Carnegie-Mellon University, P&tsbur&. IlO] H.-J. Werner and E.-A. Reinsch, J. Chem. Pnys. 76 (1982) 3144; H--J. Werner, unpubbshed results. [ 111 H.-J_ Werner, private communication.

107