Ab initio MRD CI calculation of the zero-field splitting of the 2Π ground state of the CBr molecule

Ab initio MRD CI calculation of the zero-field splitting of the 2Π ground state of the CBr molecule

Volume 119, CHEMICAL PHYSICS LETTERS number 5 AB INITIO MRD Cl CALCULATION OF THE ZERO-FIELD OF THE *II GROUND STATE OF THE CBr MOJ,ECULE Bemd A_ ...

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Volume 119,

CHEMICAL PHYSICS LETTERS

number 5

AB INITIO MRD Cl CALCULATION OF THE ZERO-FIELD OF THE *II GROUND STATE OF THE CBr MOJ,ECULE Bemd

A_ HESS.

Praphull

Lehrsruhl fur Thcorerrsrhe Rrcs~vcd

CHANDRA’

Cheme

Bererrche

and Robert

13 September 1985

SPLImING

J. BUENKER

Unruerrir~r-Ge~ut,lrhocl~scl~ule

Wupperkd.

Wupperful.

Werr Germmn

3 July 1984; in linal form I8 Junr 1985

The zersfirld spin-orbIt splrning of the ‘n ground sm~e of CBr is compuwd by means of an ab Imlio MRD Cl ~rcannent employing the Breir-Pauli formnhsm. The choice of the one-elecLron basis IS found KI b e Importanl. bum lhe resulw are seen LO be relauvcly msensirive 10 the number of core slcc~ons employed in the CI compuIation\

1. Introduction The ab init calculation of spin-orblt and other spmdependent phenomena has attained considerable interest because of the availability of computer programs capable of treating both one- and two-electron interactions of this type at both the SCF and largescale CI levels [l-3] _ Such computations for molecules have nonetheless been confined almost exclusively to systems containing atoms of the fast two rows of the periodic table, since it is generally acknowledged that other relativistic effects of various kinds begin to have a significant influence on the electromc charge distributions of still heavier elements. Hence if the conventional one- and two-electron spin-orbit Hamilto_nian IS to be treated with the usual two-component spinor functions commonly employed m molecular quantum chemistry studies for lighter systems without the benefit of core potentials or the bke, a way needs to be found to include relativistic effects in these calculations which can account for the necessary changes in the electronic charge distribution relative to standard non-relativistic (purely electrostatic) treatments. In pxticular, attention must be given to additional quantum-mechanical operators, generally deduced from the Foldy-Wouthuysen transformation [4] but also derivable from more classical considerations [5-71,



Permanent address: Depanmem Hmdu University. Vamnasi, India

0 009-2614/85/S (North-Holland

of

Chemistry.

Banaras

03 30 0 Elsevier Science Pubhshers B-V. Physics Publishing Division)

which serve as relativistic corrections to the kinetic and potential energy of the electronsTo estimate the effect of such relativistic corrections for heavier systems, it is useful to carry out calculations employing highquality non-relativistic CI wavefunctions to evaluate spin-orbit splittings and compare these results with experimental data. In prewous work [S] , for example, the zero-field splitting in the ground and first excited Rydberg states of the bromine atom was computed. In order to obtain some experience with molecules containing third-row atoms, the present study deals with the CBr system in its 211 ground state.

2. Details of the calculations Since the CBr ground state is well separated from all other electronic states [9], the use of first-order perturbation theory based on a non-relativistic wavefunction, as prescribed by the Breit-Pauli approximation, should be appropriate. The calculations are thus divided into two parts: fust, the CBr ‘-II electronic wavefunction is obtained in the non-relativistic BornOppenheimer approximation employing the MRD CI program package including the table Cl algorithm [ 10, 1 I] ; subsequently, all molecular integrals over the microscopic spin-orbit operator are computed m a given A0 basis and then transformed to an orthogonal (usually SCF MO) basis, whereupon the spin-orbit 403

Volume 119. number

5

CHEMICAL PHYSICS LEl-l-ERS

matrix element between the above CI wavefunctions is obtained [ 12]_ A (165,12p,5d) + [ 1 ls,SpJd] A0 basis set derived from Dunning 1131 was used for the bromine atom in this work; details of the contraction are given in ref_ [S]. For carbon a (1 Os,6p) basis set given by Huzinaga [ 141 was contracted to [4s,?p] according to a contraction scheme suggested by Langhoff [2]. Since the spin-orbit splitting is calculated in first-order PT with a CI wavefunction based on a selected set of configurations, it is clear that the results will depend on a number of technical factors other than the composition of the A0 basis, however. In particular the choice of orthogonal one-electron transformation (MO basis) can have an important mfluence, particularly if the CI expansion is far from converged_ Moreover experience with non-relativistic computations indicates that optimum results are not always obtained by simply employing the self-consistent orbitals of the parent state rn the CI treatment_ For this reason a number of different one-electron basis sets have been tested, optimized in the fields of both CBr+ and CBr- as well as for the neutral ground state. In general the contiguration space employed is generated by taking ail single and double cxcrtations relative to a series of reference configurations [lo]. A subset of the most important configurations is then selected on the basis of second-order perturbative energy correctrons; a threshold T 1s designated for this purpose and in general the results of more than one root are considered in making the selection [IO] - The SCF calculations in this study are carried out in the D= Abelian point group and thus do not constrain the components of the n and 8 pairs to be equivalentTo ensure the desired degeneracy characteristic under these circumstances it is imperative to treat electronic states in which the components of ‘ITand 6 orbit& are equally occupied. This requirement IS fulfilled for the CBr-(n4n*2,3~-) and the CBr+ closed-shell (7~~) ground states, but not for the 211 CBr species itseIf (rr4x* contiguratron). A set of symmetric natural orbitals (NOs) was generated instead for the neutral species, based on a CI treatment of the CBr 2~ state employing the above CBr- SCF MOs. Finally the SCF orbitals of the 5x* (rr2rr*2) state of CBr+ have been considered as welI in order to study further the dependence of the zero-field splitting on the choice of one-electron basis. 404

3

Results

13 September 1985 of the calculations

The first set of results to be considered is that obrained with the SCF MOs of :he n4rr*2 CBr- ground state_ Not surprisingly the T? variational orbrtal IS rather diffuse m this instance, with the result that a singles and doubles Cl for the CBr ground state (rr4n*) leads to a rather low zero-filed splitting (282.0 cm-l, see table 1), i.e. because the key MO is effectively constrained to have a relative large atomic radius; for this calculation a core of 22 electrons is assumed (seven D and two rr MOs of lowest orbital energy)_ The situation is greatly improved, however, by including five reference configurations to generate the MRD CI space and also by selecting on the basis of more than just the lowest 211 root With the five leading terms and four-root selection (5M4R), the ZFS based on the CBr- MOs increases to 381.6 cm -1 , for example, as compared to the experimental value of 466 cm-’ [9]. The next set of Cl calculations to be considered employs natural orbitals for the X 211 CBr state itself, as obtained by diagonahzing the firstorder density matrix for a CI wavefunction for this state obtained using CBr+ SCF MOs No secondary configuratron appears to be important in this CI treatment, indicating that a sunably state-specific set of CBr orbitals had indeed been achieved in this procedure. The resulting ZFS value is almost identical to that obtained in calculation no. 2 with CBr- MOs (table l), four less electrons were included in the core in this case (two less u MOs than before). Especially since tlus finding is still some 80 cm-l less than the experimental value, however, it was decided to expand the study further to include consideration of SCF MOs of the positive ion of CBr, be@nning with those of its closed-shell (jr 4, ground state. A further increase of 11 cm-l in the X 211 ZFS is noted as a result (392.2 cm-l, calculation no_ 4), in which the same number of active electrons (23) is allowed in the CI. Increasing the size of the reference set and the number of roots in which the configuration selectron is based has only a small effect (1.0 cm-l) on the magnitude of the zero-field splitting. Moreover a further reduction in the number of core electrons (from 18 to I2), achieved by correlating all rr (and 6) electrons (i-e the lowest-lymg six u MOs are 111core), also caused only a slight increase in the ZFS, to a value of 395.6 cm-l, still some 70 cm-l smaller than the ex-

Volume 119, number 5

CHEMICAL

Table 1 Results of MRD Cl cakuIations No.

MO basis

for the CBr molecuk

No. of correlated

PHYSICS

at the gronndrtatc

CBr- =)

19

NOs d) CBr+ e,

23 23

cIlr+g)

290 29

equilibrium geomem

MRD CI energy (extrapolated to T = 0)

Calculated ZFS (cm-‘)

lMl& SM4R, IMIR, 1MlS 5M4R 1MlR. 6hIlR

-2609.960448 -2609 966153 -2610.041424 -2610 028397 -2610.043618 -2610.0894SO -2610.092509

282 0 381.6 381.2 392.2 393.2 395.6 434.3

37357,124l 295275,3208 57860,1395 57 860.1733 815710.4817 87412,2147 498781,2728

1985

a)

CI treatment b)

&33XOD5

1 2 3 4 5 6 7

13 September

LETl-ERS

a) R = 1.823 A = 3.44497 au. b) ‘Ihe notatron XM YR denotes reference configurations relative to which ~II single and double excitations have been generated and that configuration selection is carried out with respect to the Y lowest roots The size of the generated space and of the selected space is also given The selection threshold was 20 phartrec throughout. C) The SCF energy for CBr- was -2609.789382 au. d) NOs have been generated from the CI wavefuncuon of calculation no. 4. e) The SCF energy for CBr*is -2609.404547 au. D All orbital.5 of m-type have been correlated. 9) MOs of the lowest *~+(n%g) state have been employed; the SCF energy for this state was -2609 lS7187 au.

perimental value. Throughout threshold was ftied at a value previous experience with ZFS indrcated that such results are

this work the selection of 20 fiartree, since calculations

[3,12]

has

quite insensitive to this

quantity_

Finally a fourth one-electron basis has been employed which results in an SCF treatment of the 5E+(51277*2) state of CBr+, with the rationale that such orbitals may be preferable to at least the previous CBr+ set (lx’) because in this case both the x and rr* are deternuned variationally- This CI treatment has also been carried out with 29 active electrons and leads to the largest ZFS value for the CBr 211 ground state obtained in this study, namely 4343 cm-l- The corresponding extrapolated MRD CI total energy [lo] is also the lowest value obtained in the present study, which fact tends to support the conclusion that the ZFS value is also the most accurate obtained, as turns out to be the case judging from the available experiment data [9]_

4. Conclusion The present study of the zero-field splitting of the CBr X 211 ground state indicates that the choice of one-electron basis for the CI expanson IS the most

crrtical factor in obtaimng high accuracy in a BreitPauh formulation. For typical MRD CI treatments ZFS values ranging from 380 to 435 cm-l have been obtained with drfferent MO sets, compared wrth the experimental value of 466 cm-l_ The size of the frozen core employed in the CI treatment is found to be a much less critical factor, as should be expected in view of the fact that the property in question is a property of a valence-shell orbital. Relativistic effects are known generally to decrease the size of p-type orbit&, which in the present case would lead to an increase in the ZFS value for the CBr X 211 state. Such a result would be consistent with an earlier study of the bromine atom [S] employing similar methods. At the same tune It should be recalled that the present A0 basis lacks polarization functions and this deficiency may also contribute to the observed discrepancy in the computed spin-orbit splitting. Both these considerations also raise the possibility that the better agreement obtained through the use of CBr+ ?L? MOs may result to a good extent because of a fortuitous cancellation of errors. In particular it is not at all unlikely that at the A0 basis limit for the CI treatment (full CI) the ZFS value most closely corresponds to the results obtained with the CBr ground state NOs In any event it appears that the type of Breit-Pauli treatment with large-scale CI wavefunc40.5

Volume

119. number 5

CHEMICAL

PHYSICS

tions employed in this work is capable ofpredicting ZFS values with 85-90% accuracy for systems containing atoms in the Z = 30-50 range. In future work particular attention should be paid to the question of whether the underestimation of ZFS values for CBr and the bromine atom itselfis general wrth such nonrelativistic

wavefunctions.

Acknowledgement The authors would like to thank Dr. C. Marian for various valuable discussions_ The financial support of the Deutschc Forschungsgemeinschaft given to this work in the framework of the Sonderforschungsbereich 42 is gratefully acknowledged, the computations have been carried out on the minicomputer of the Sonderforschungsbererch 42 in Wuppertal.

ReFerences W-G. Richards, H.P. Trivedl and D-L. Cooper, Spinorbit coupling in molecules (CIarendon Press, Oxford, 1981)_ [2] S-R Langhoff, J. Chem. Phys 61 (1974) 1708; S-R Langhoff and E-R. Davidson. Intern_J. Quantum Chem. 7 (1973) 759_ [I]

406

LEl-JERS

13 September 1985

[31 P- Chandra and RJ. Buenker, J. Chem. Phyr 79 (1983)

358; 79 (1983) 366; B-A Hess, RJ. Buenker, C.M. Marian and S-D. Peyerimhoff, Chem. Phys Letters 89 (1982) 459; C-M. Maian. R Marian, S-D. Peyerimhoff, B.k Hess, R-l- Buenker and G. Se&r. MoL Phys. 46 (1982) 779. [41 L-L. Foldy and S-A. Wouthuysen. Phys Rw. 78 (1950) 29. c51 J-C_ Slater. Quantum theory of atomic strocfure, VoL 2 (McGraw-Hill, New York, 1960). [61 J-E_ Hanim an, Theoretical foundations of eleclxonic spin resonance (Academic Press, New York, 1978)_ [73 R-J. Buenker. P. Chandra and B.A. Hess. Chem. Phys.

84 (1984) 1. [‘31 B A. Hess. P. Chanda

and R-J. Buenker. MoL Phys_ 52 (1984) 1177. 191 K-P_ Huber and G. HerrbeE. hfolecular spectra and molecular structure_IV. Constants of dlatomrcmolecules

wan Nostrand, Princeton, 1979).

Cl01 R.J. Buenkerand SD. Peyerirrihoff, Theoret. Chim.

Acta 35 (1974) 33;39 (1975) 217; R.J. Buenker. S D. Peyerimhoff and W. Butaher, MoL Phyr 35 (1978) 771Clli R-J. Buenker, in: Quantum Chemistry into the SO’s, Proceedings of the Workshop in WoUongong. Australia, Februzu-y 1980, ed. P. Burton; RJ. Buenker, in: Current aspects of quantum chemistry, 1981, ed. R. Carbd (-Elsetier. Amsterdam, 1982); FU. Buenker and R A. Phillips, J. Mol. Struct. THEOCHEM, to be published. 1121 B.A. Hess, Doctoral Thesis, Bonn (1981); C M Marian, Doctoral The&, Bonn <1981)_ 1131 T-H. Dunning Jr., J. Chem. Phys 66 (1977) 1382_ 1141 S Huzinaga. J. Chem. Phys 42 (1965) 1293_