Using JCP format - Washington State University

JOURNAL OF CHEMICAL PHYSICS

VOLUME 112, NUMBER 13

1 APRIL 2000

Convergence of Breit–Pauli spin–orbit matrix elements with basis set size and configuration interaction space: The halogen atoms F, Cl, and Br Andreas Nicklass and Kirk A. Petersona) Department of Chemistry, Washington State University and the Environmental Molecular Sciences Laboratory, Pacific Northwest National Laboratory, Richland, Washington 99352

Andreas Berning and Hans-Joachim Wernerb) Institut fu¨r Theoretische Chemie, Universita¨t Stuttgart, 70550 Stuttgart, Germany

Peter J. Knowlesc) School of Chemistry, University of Birmingham, Edgbaston, Birmingham B15 2TT, United Kingdom

共Received 20 September 1999; accepted 7 January 2000兲 Systematic sequences of basis sets are used to calculate the spin–orbit splittings of the halogen atoms F, Cl, and Br in the framework of first-order perturbation theory with the Breit–Pauli operator and internally contracted configuration interaction wave functions. The effects of both higher angular momentum functions and the presence of tight functions are studied. By systematically converging the one-particle basis set, an unambiguous evaluation of the effects of correlating different numbers of electrons in the Cl treatment is carried out. Correlation of the 2p-electrons in chlorine increases the spin–orbit splitting by ⬃80 cm⫺1, while in bromine we observe incremental increases of 130, 145, and 93 cm⫺1, when adding the 3d, 3p, and 2p electrons to the set of explicitly correlated electrons, respectively. For fluorine and chlorine the final basis set limit, all-electrons correlated results match the experimentally observed spin–orbit splittings to within ⬃5 cm⫺1, while for bromine the Breit–Pauli operator underestimates the splitting by about 100 cm⫺1. More extensive treatment of electron correlation results in only a slight lowering of the spin–orbit matrix elements. Thus, the discrepancy for bromine is proposed to arise from the nonrelativistic character of the underlying wave function. © 2000 American Institute of Physics. 关S0021-9606共00兲31413-1兴

Douglas–Kroll 共DK兲 transformation,7,8 are also currently popular and benefit from the same advantage, namely the usage of mature program packages, assuming that the additional molecular integrals required by the DK method are available. In many systems, especially with moderately heavy elements and closed electronic shells, the scalar effects represent the major contribution of relativity. However for very heavy elements, spin–orbit effects cannot be neglected since their impact is then of the same order as electron correlation or binding energies. On the other hand, for light elements spin–orbit coupling is also important—often even more so than scalar relativistic effects—especially when one is interested in excited state dynamics or high accuracy spectra, which naturally depict fine structure. The most rigorous way to account for spin–orbit coupling is to use a four-component method based on the Dirac equation. Some program packages are now available to perform Dirac–Fock configuration interaction 共CI兲 calculations for molecules, e.g., MOLFDIR,9,10 but actual calculations are still a formidable task and these results are mostly used as benchmark studies 共see, e.g., Refs. 11–13兲. More approximate methods do not use the four-component scheme of the Dirac equation, but introduce a two-component spin–orbit operator, which possesses the same symmetry properties as the Dirac operator. Thus the eigenstates will transform according to the corresponding double-group symmetry; how-

I. INTRODUCTION

With the availability of increasingly powerful computers, theoretical chemists not only turn their interest to larger systems, but many are pushing the limits of achievable accuracy. One consequence of this, however, is that effects that could be neglected in less sophisticated calculations can turn into relatively large sources of error. The most prominent among these are core–valence correlation and relativistic effects. The latter can be divided into scalar terms 共mass velocity and Darwin operators兲 and a nonscalar part 共mostly the spin–orbit interaction兲. The nature of scalar relativistic effects allows one to introduce them in a formally nonrelativistic scheme using the Russell–Saunders coupling model. The most popular and at the same time almost effortless way to consider scalar relativistic effects are through effective core potentials or pseudopotentials,1–6 because the elaborate methods and program packages that have been designed for nonrelativistic quantum chemistry can also be used with these effective potentials. Thus, pseudopotentials have been the means of choice in many investigations which accounted for relativistic effects during the last two decades. So-called no-pair all-electron methods, which are based on the a兲

Author to whom correspondence should be addressed. Electronic mail: [email protected] b兲 Electronic mail: [email protected] c兲 Electronic mail: [email protected] 0021-9606/2000/112(13)/5624/9/$17.00

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© 2000 American Institute of Physics

Downloaded 22 Feb 2001 to 192.101.100.146. Redistribution subject to AIP copyright, see http://ojps.aip.org/jcpo/jcpcpyrts.html

J. Chem. Phys., Vol. 112, No. 13, 1 April 2000

Convergence of spin-orbit matrix elements

ever an original one-component ab initio program has to undergo severe changes in order to deal with spin–orbit coupling in this way. Consequently the introduction of spin– orbit effects is often postponed to the very last step of a calculation and then treated as a simple perturbation. For heavier elements it is assumed to be advantageous to introduce spin–orbit coupling at an earlier stage of the calculation. One possibility used by different programs is to include spin–orbit effects when setting up the CI matrix.14–17 The most desirable path, of course, is to already account for spin–orbit effects when optimizing the orbitals/spinors.18 With these last two possibilities one has to be aware, however, that not all known spin–orbit operators are bound from below and may therefore cause problems in these variational procedures. The simplest spin–orbit operator one can choose is an effective one-electron operator19–21 1 H SO⫽ ␣ 2 2

兺 i␭

冋

Z 共eff␭ 兲 3 r i␭

册

li␭ •si ,

共1兲

where ␣ denotes the fine-structure constant, Ii␭ and si are angular momentum and spin operators, respectively. The ef(␭) fective nuclear charge Z eff is adjusted to a reference spin– orbit splitting for each element ␭ and usually depends on the basis set. This kind of operator can also be used in connection with effective core potentials as long as the parameter (␭) is chosen appropriately. Other forms of effective oneZ eff electron spin-orbit operators, namely Gaussian expansions, are in use together with effective core potentials22 or pseudopotentials but will not be discussed here. The most commonly used operator in all-electron calculations is the Breit–Pauli spin–orbit operator, which results from a truncated Foldy–Wouthuysen transformation23 of the Dirac–Breit equation to order ␣ 2 and also includes twoelectron terms, the so-called ‘‘spin–same-orbit’’ and ‘‘spin– other-orbit’’ interactions 1 H SO⫽ ␣ 2 2

兺 i␭

冋

Z␭ 3 r i␭

冋

共 ri␭ ⫻pi 兲 •si

册

册

1 1 ⫺ ␣2 共 r ⫻pi 兲 • 共 si ⫹2s j 兲 . 2 i⫽ j r 3i j i j

兺

共2兲

Notice that Z ␭ denotes the actual charge of the nucleus ␭ and that neglecting the two-electron terms can lead to a substantial overestimation of the spin–orbit effect. A very similar operator can be derived in the framework of the no-pair theory.24 It has some additional kinematic prefactors, but its most noteworthy feature is that it is bound from below and thus variationally stable. Naturally it is used together with the spin-free Douglas–Kroll–Hess 共no-pair兲 Hamiltonian. The functional forms of the discussed spin–orbit operators suggest that they might be rather sensitive to variations of the core electron density. In particular, core–valence and core–core correlation change the electron density close to the nuclei. Therefore it is very surprising that only a few previous publications25–32 addressed the interplay of core electron correlation 共‘‘core polarization’’兲 and spin–orbit coupling. One prerequisite of such an investigation, how-

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ever, is to be able to differentiate between correlation effects and basis set deficiencies. In this work we look at the ground-state spin–orbit splittings ( 2 P 3/2 – 2 P 1/2) of the halogen atoms fluorine, chlorine, and bromine. The splittings are calculated using the Breit–Pauli operator in first-order perturbation theory with nonrelativistic one-component CI wave functions and correlation consistent basis sets. The accuracy of the resulting spin–orbit matrix elements is then systematically investigated in regards to both the basis set and electron correlation treatment. A subsequent article33 describes the very same procedure at the Dirac–Fock CI level and compares the findings from these two investigations. In the following section the computational details are given, in particular the basis sets used and the ways they were modified in order to be compatible with the spin–orbit integral code. In the main section the dependence of the calculated splittings on the basis set and both valence and core correlation are discussed. We also briefly address the influence of the choice of orbitals and compare our results to some previous work. A summary of our results is presented in Sec. IV.

II. COMPUTATIONAL DETAILS A. Spin–orbit splittings

All calculations of spin–orbit splittings in this article were derived by applying the Breit–Pauli spin–orbit operator, Eq. 共2兲, on nonrelativistic internally contracted CI wave functions34,35 in first-order perturbation theory. If not otherwise stated, the underlying atomic orbitals were taken from restricted Hartree–Fock calculations for the atomic ground states. In the determination of these orbitals the spherical symmetry of the atoms was enforced by averaging the three spatial components. Using these orbitals a series of singlereference CISD 共CI with single and double excitations兲 calculations were carried out with varying numbers of active electrons; starting with calculations where only excitations from the valence p-orbitals were allowed, then correlating all valence electrons, subsequently including one subshell of core electrons after the other, and finally correlating all the electrons. Matrix elements of the Breit–Pauli spin–orbit operator were calculated between different spatial components of the spin-free CI wave functions.36 In the current implementation36 the full Breit–Pauli operator is used only for self-consistent field 共SCF兲 and multiconfiguration SCF 共MCSCF兲 wave functions, while for CI wave functions a mean-field one-electron Fock operator is employed for the contributions of the external configurations. The errors from this approximation are expected to be only on the order of 1 cm⫺1. The reader is directed to Ref. 36 for more details. The present calculations do not include effects due to the socalled anomalous magnetic moment of the electron, which has been shown37 to increase the splittings of F and Cl by ⬃1 and 2 cm⫺1, respectively. For the halogen atoms, the spin– orbit splittings that can be compared to experiment are simply proportional to the one unique matrix element, i.e., ⌬E SO⫽3 具 ⌽ x 兩 H zSO兩 ⌽ y 典 .


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For all calculations the ab initio program package MOLPRO 共Ref. 38兲 was used. To investigate the effects of more extensive electron correlation on the computed SO splittings, a few multireference CI 共MRCI兲 calculations were also carried out. The orbitals for these calculations were obtained from complete active space self-consistent-field 共CASSCF兲 calculations39,40 with various choices of active spaces 共see below兲. All orbitals were optimized, and the core orbitals were constrained to be doubly occupied. Test calculations were also carried out with the state-averaged SCF orbitals used above for the CISD splittings, and the different choice of orbitals in this case led to only very minor changes in the correlated results.

B. Basis sets

The development of systematic series of basis sets during the last decade—namely atomic natural orbital basis sets41 and correlation consistent basis sets42–45—has greatly advanced the achievable accuracy and reliability of quantum chemical calculations. In particular, the systematic convergence behavior observed in calculations with the correlation consistent basis sets often allows the extrapolation of various properties to their complete basis set limits 共see, e.g., Ref. 46 and citations therein兲. This not only improves the accuracy of the calculations, but can also enable a more unambiguous evaluation of different correlation methods, since the coupling between the one- and N-particle basis sets is effectively removed.47 One goal of the present work is to investigate the convergence behavior with respect to basis set for spin–orbit splittings using a series of correlation consistent basis sets. The influence of higher angular momentum functions, as well as tight 共large exponent兲 functions, are also of interest. On the other hand, the addition of diffuse functions via the aug-cc-pV nZ basis sets were found to have a completely negligible effect on the SO splittings and are not discussed further. Because of the r ⫺3 dependence of the spin–orbit operator, it was not obvious that the standard valence correlation consistent basis sets could provide a reasonable description of spin–orbit effects. Thus, most of the calculations employed core–valence correlation consistent basis sets, cc-pCV nZ, where additional tight functions have been added that were optimized to recover core–core and core– valence correlation energy.48,49 For fluorine, these sets consisted of the cc-pVDZ⫹1s1p, cc-pVTZ⫹2s2 p1d, cc-pVQZ⫹3s3p2d1 f , and cc-pV5Z⫹4s4p3d2 f 1g. The large exponent functions added to the standard cc-pV nZ chlorine sets were 1s1 p1d共DZ兲, 2s2p2d1 f 共TZ兲, 3s3p3d2 f 1g共QZ兲, and 4s4p4d3 f 2g1h共5Z兲. Preliminary development sets50 were used for bromine, and in analogy to the chlorine case, were optimized to minimize the total core– core and core–valence correlation energy of the n⫽3 electrons only (3s3p3d). Following the usual prescription, the augmenting functions were 1s1p1d1 f , 2s2 p2d2 f 1g, 3s3p3d3 f 2g1h, and 4s4 p4d4 f 3g2h1i for DZ, TZ, QZ, and 5Z, respectively. To test the effects of these additional tight functions, calculations were also carried out with successive removal of these functions.

In the current implementation of the spin–orbit matrix elements36 in the MOLPRO program, the SO integrals are due to Palmieri51 by modification of the CADPAC 共Ref. 52兲 second derivative integrals, which restricts the basis sets to segmented contractions with a maximum contraction length of ten primitive functions. In addition, no basis functions with angular momentum higher than f can be used. Hence in the present work all functions of g, h, and i symmetry have been removed from the basis sets outlined above. The impact of these neglected higher angular momentum functions on the calculated spin–orbit splittings is estimated below. Due to the other limitation of the available spin–orbit integral code, the correlation consistent basis sets had to be transformed from the original scheme of general contractions to segmented ones. Following Davidson’s suggestion,53 the resulting segmented contraction lengths were minimized, e.g., the correlation consistent polarized valence double zeta 共ccpVDZ兲 basis set for fluorine was converted from the original (9s4 p1d)/ 关 3s(1.9,1.9,9.9)2p(1.4,4.4)1d 兴 contraction to a (9s4p1d)/ 关 3s(1.7,2.8,9.9)2p(1.3,4.4)1d 兴 contraction scheme without affecting the total energy. The nomenclature indicates that in the original case the first two s-type basis functions were each constructed by forming a linear combination of all nine primitive functions, while in the modified scheme the first function was formed only from primitives 1 through 7, and the second function from primitives 2 through 8. In the case of chlorine and bromine the decrease of contraction lengths by applying this procedure was much more pronounced due to the larger number of primitive functions involved. Nevertheless, some of the resulting segmented contracted functions comprised more than ten primitive functions, which were still not compatible with the SO integral code. In these cases 共VTZ, VQZ, V5Z兲 one of the contractions was split into two functions. The extra flexibility introduced by this step proved to lower the energy in valence CISD calculations by less than 0.2 m E h in the worst case. The new segmented contraction schemes for the correlation consistent basis sets applied in this investigation are summarized in Table I. The respective contraction coefficients can be obtained from one of the authors 共K.A.P.兲 on request.

III. RESULTS AND DISCUSSION

Figure 1 shows the calculated SCF, valence CISD 共valCISD兲, and all-electron CISD 共ae-CISD兲 spin–orbit splittings for F, Cl, and Br as a function of the core–valence correlation consistent basis sets. These results are also shown explicitly in Table II. Some general trends with respect to electron correlation are easily observed, namely that valence correlation lowers the splittings compared to the SCF results, and core correlation increases the splittings. In regards to the convergence with respect to increasing the basis set, with the exception of F both the SCF and val-CISD splittings are rapidly convergent. In the case of fluorine the convergence is dominated by the SCF portion which is relatively slow. In all three cases, however, correlation of the core electrons brings the computed splittings closer to experiment, and for Cl and Br these results show the largest dependence on basis set.




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TABLE I. Modified contraction schemes for the correlation consistent basis sets from double zeta through sextuple zeta for fluorine, chlorine, and bromine. The numbers and ranges denote the primitives included in the different contracted functions. Functions that were uncontracted in the regular 共standard兲 scheme are also uncontracted in the modified scheme and are not explicitly shown.

Fluorine

Basis set

Prim.

cc-pVDZ

(9s) (4p) (10s) (5p) (12s) (6p) (14s) (8p) (16s) (10p)

1–9/1–9/ 1–4/ 1–10/1–10/ 1–5/ 1–12/1–12/ 1–6/ 1–14/1–14/ 1–8/ 1–16/1–16/ 1–10/

1–7/2–8/ 1–3/ 1–7/2–7,9/ 1–3/ 1–8/2–9/ 1–3/ 1–9/2–10/ 1–4/ 1–10/2–11/ 1–5/

(12s) (8p) (15s) (9p) (16s) (11p) (20s) (12p) (21s) (14p)

1–12/1–12/1–12/ 1–8/1–8/ 1–15/1–15/1–15/ 1–9/1–9/ 1–16/1–16/1–16/ 1–11/1–11/ 1–20/1–20/1–20/ 1–12/1–12/ 1–21/1–21/1–21/ 1–14/1–14/

1–9/2–10/3–11/ 1–6/2–7/ 1–3/4–11/5–12/6–12,14/ 1–6/2–6,8/ 1–3/4–11/5–12/6–13/ 1–7/2–8/ 1–4/5–14/6–15/7–16/ 1–7/2–8/ 1–4/5–14/6–15/7–16/ 1–8/2–9/

(14s) (11p) (20s) (13p) (21s) (16p) (26s) (17p)

1–14/1–14/1–14/1–14/ 1–11/1–11/1–11/ 1–20/1–20/1–20/1–20/ 1–13/1–13/1–13/ 1–21/1–21/1–21/1–21/ 1–16/1–16/1–16/ 1–26/1–26/1–26/1–26/ 1–17/1–17/1–17/

1–10/2–11/3–12/5–13/ 1–8/2–9/3–10/ 1–5/6–15/7–16/8–17/9–17,19/ 1–9/2–10/3–10,12/ 1–5/6–15/7–16/8–17/9–18/ 1–10/2–11/3–12/13/ 1–9/10–19/11–20/12–21/13–22/ 1–10/2–11/3–12/13/

cc-pVTZ cc-pVQZ cc-pV5Z cc-pV6Z Chlorine

cc-pVDZ cc-pVTZ cc-pVQZ cc-pV5Z cc-pV6Z

Bromine

cc-pVDZ cc-pVTZ cc-pVQZ cc-pV5Z

Regular scheme

With the cc-pCV5Z basis set the ae-CISD splittings differ from experiment by ⫺5, ⫹4, and ⫺102 cm⫺1 for F, Cl, and Br, respectively. Our results for the chlorine atom are very similar to the previous uncontracted basis set results of Berning et al.36 The SCF/cc-pCV5Z splitting calculated for the bromine

Modified scheme

atom 共3323 cm⫺1兲 shown in Table II is in excellent agreement with the numerical Hartree–Fock value of 3324 cm⫺1 共Ref. 54兲. Hess and co-workers24,31 have recently computed the spin–orbit splitting of the bromine atom using Cl methods with the Breit–Pauli operator. Their value of 3918 cm⫺1 is considerably larger than our results and is due to their use

FIG. 1. Convergence behavior of the calculated spin–orbit splittings 共cm⫺1兲 of fluorine 共a兲, chlorine 共b兲, and bromine 共c兲 for the cc-pCV nZ basis sets at the SCF 共䊏兲, valence CISD 共䊉兲, and all-electron CISD 共䉱兲 levels of theory.


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TABLE II. Dependence of halogen atom spin–orbit splittings 共in cm⫺1兲 on basis set size using core–valence correlation consistent basis sets.a Basis set Fluorine

SCF

val-CISD

ae-CISD

cc-pCVDZ cc-pCVTZ cc-pCVQZ cc-pCV5Z

376.53 392.55 398.80 401.33

366.95 388.64 394.77 397.14

367.42 390.22 396.75 399.22

cc-pCVDZ⫹p cc-pCVTZ⫹p cc-pCVQZ⫹p cc-pCV5Z⫹p

396.66 401.11 401.61 402.21

386.49 397.22 397.60 398.02

386.99 398.83 399.60 400.11

Expt.b Chlorine

cc-pCVDZ cc-pCVTZ cc-pCVQZ cc-pCV5Z

404 816.74 820.50 822.73 822.99

Expt.b Bromine

cc-pCVDZ cc-pCVTZ cc-pCVQZ cc-pCV5Z Expt.b

795.81 797.33 798.98 799.32

865.32 878.08 884.00 885.47

881 3315.58 3320.74 3322.42 3322.74

3214.75 3201.95 3202.67 3202.39 3685

3489.57 3554.14 3581.06 3583.39

a

Functions with angular momenta higher than f-type have been omitted. See the text. b Reference 55.

of 4 P g orbitals instead of the symmetrized 2 P u orbitals of the present work. In fact within the present treatment, calculations using a cc-pCVQZ basis set 共plus a diffuse s function with an exponent of 0.02兲 employing Br( 2 P u ), Br( 4 P g ), and Br⫺ ( 1 S) orbitals result in all electron CISD splittings of 3581, 3817, and 3316 cm⫺1, respectively. Obviously from these results, and also as previously pointed out by Hess et al.,31 the choice of molecular orbitals can have a substantial effect on the resulting spin–orbit splittings. As shown in Fig. 1共a兲 the SO splittings for F show seemingly anomalous slow convergence with respect to increases in the basis set. The origin of this behavior was found to be due to the lack of a sufficiently tight p function in the basis sets which was not accounted for until the cc-pCV5Z set. In fact inclusion of just one tight p function to the cc-pCV nZ sets 共denoted cc-pCV nZ⫹ p兲 yields a much improved convergence rate as shown in Fig. 2. In these calculations the single, extra p functions were energy optimized at the SCF level 共exponents of 128.2, 250.8, 465.3, and 1415.0 for DZ, TZ, QZ, and 5Z, respectively兲, and it was found to be sufficient to include them in the first contracted p functions so that the total size of the contracted basis did not increase. As shown in Fig. 2, with the augmenting p functions included, the F splittings are nearly converged at the cc-pCVTZ⫹ p level; after correlating all the electrons, the cc-pCVTZ⫹ p splitting is just 5 cm⫺1 below experiment, just 1 cm⫺1 below the cc-pCV5Z⫹p result. To address the effects of the limitation of the spin–orbit integral code to basis functions of s-, p-, d-, or f-types, spin– orbit splittings were calculated with the cc-pCV nZ basis sets (cc-pCV nZ⫹p for F兲 that included s through f functions, s through d, as well as s and p functions only. The results of

FIG. 2. The basis set convergence of the fluorine spin–orbit splittings 共cm⫺1兲 is shown for the standard cc-pCV nZ 共dotted lines兲 and these sets with an additional tight p function 共solid lines兲.

these calculations are shown in Fig. 3, where the contributions to the SO splittings from all d functions and all f functions are broken down into their effect on the valence-only and all-electron CISD results. Of course d 共F and Cl兲 and f 共F, Cl, and Br兲 functions have no effect on the SCF splittings. In the case of d polarization functions, their effect on the valence-only CISD results is to decrease the splittings by about 2–3 cm⫺1 for F and nearly 25 cm⫺1 for Cl. Upon correlating all the electrons, the total effect of d-type functions is about 1.5 cm⫺1 for F and ⬃10 cm⫺1 for Cl. For both F and Cl, the effect of f-type functions in the basis set is nearly negligible and the loss of accuracy from not including functions of higher angular momentum 共g, h, etc.兲 should not be a concern. In the case of Br, the inclusion of f functions to the basis set decreases the val-CISD SO splittings over the spd-only results by only about 10 cm⫺1. On the other hand, when the core electrons are correlated, and in particular the 3d electrons, the effect of f-type functions is substantial, ⬃50 cm⫺1. In estimating the effect of g and higher angular momentum functions for Br, it might be expected that they will be smaller than that of the f functions by about an order of magnitude, much like the difference between the d and f function contributions in Cl. Thus, the neglect of g and higher angular momentum functions should lead to only relatively small errors 共5–10 cm⫺1兲 for Br as well. In the above discussion of the effects of d and f functions on the SO splittings, both the standard cc-pV nZ polarization functions and the tight augmenting functions of the cc-pCV nZ sets were included. Figure 4 displays the effect of just the tight sp, d, and f functions in the cc-pCV nZ basis sets on the SCF, val-CISD, and ae-CISD SO splittings. In




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FIG. 3. The effects of all d 共䊏, 䊐兲 and f 共䊉, 䊊兲 polarization functions on the calculated val-CISD 共filled symbols兲 and ae-CISD 共open symbols兲 spin– orbit splittings 共cm⫺1兲 for 共a兲 fluorine, 共b兲 chlorine, and 共c兲 bromine. The cc-pCV nZ⫹p basis sets are used for fluorine.

general, the addition of tight functions increase the calculated splittings. In the case of fluorine, additional tight functions 共beyond the extra tight p function兲 have very little effect on the SO splittings for TZ and larger basis sets and are generally on the order of less than 1 cm⫺1 even for the aeCISD results. In fact for all three atoms additional tight functions are not required for accurate SCF splittings if at least the cc-pVTZ basis set is used. For the chlorine and bromine splittings, only additional tight s and p functions yield a noticeable effect when just the valence electrons are correlated. If all the electrons are correlated in chlorine, additional s and p functions are very important, while tight d functions have only a small effect of 5–6 cm⫺1. For both F and Cl the contributions of tight f functions are negligible. In the case of Br with all electrons correlated, tight s and p functions have

the strongest effect, over 200 cm⫺1 at the TZ level. Tight f functions increase the Br splittings by about 50 cm⫺1, while tight d functions have an effect of just 20–30 cm⫺1 for basis sets between TZ and QZ. It should also be noted that the effects of adding tight functions to the standard cc-pV nZ sets is fairly insensitive to the exponents of the added functions. Calculations with the weighted core–valence basis sets, cc-pwCV nZ, where the exponents are not nearly as tight as those of the cc-pCV nZ sets since they are optimized with a stronger weighting toward intershell correlation 共as opposed to core–core correlation兲, yield SO splittings for all three atoms within just a couple of wavenumbers of the cc-pCV nZ results. As shown in Fig. 1 and discussed briefly above, correlation of the valence electrons tends to decrease the SO split-

FIG. 4. The effect of adding tight sp 共䊏兲, d 共䊉兲 and f 共⽧兲 functions to the cc-pV nZ (cc-pV nZ⫹p for fluorine兲 basis sets on the calculated SCF 共top panels兲, val-CISD 共middle panels兲, and ae-CISD 共bottom panels兲 spin– orbit splittings 共cm⫺1兲 for 共a兲 fluorine, 共b兲 chlorine, and 共c兲 bromine.


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FIG. 5. The dependence of the calculated spin–orbit splittings 共cm⫺1兲 on the CI active space and basis set; DZ 共䊉兲, TZ 共䉱兲, QZ 共䊏兲, and 5Z 共⽧兲; for 共a兲 fluorine (⫹p basis sets兲, 共b兲 chlorine, and 共c兲 bromine.

tings compared to the SCF values, while core correlation increases them. Hence compared to experiment, valence electron correlation results in worse agreement compared to just SCF, while core correlation works to bring the splittings

into better agreement. Figure 5 decomposes the correlation contributions into which electrons are being correlated. For the effects of valence correlation, correlation of the s and p electrons both decrease the splitting over the SCF values, a

TABLE III. The dependence of halogen atom spin–orbit splittings 共in cm⫺1兲 on the CI active space using the cc-pCVQZ (⫹p for fluorine兲 basis set.a

Method Fluorine

Electrons correlated

CASSCF

MRCI

2s2p

1s2s2p

Chlorine

CASSCF

MRCI

3s3p

2p3s3p

Bromine

CASSCF

MRCI

4s4p

3p3d4s4p

2p3s3p3d4s4p

Active space

Splitting

2s2p 2s2p⫹3p 2s2p⫹3s3p3d 2s2p 2s2p⫹3p 2s2p⫹3s3p3d b 2s2p 2s2p⫹3p 2s2p⫹3s3p3d b 3s3p 3s3p⫹4p 3s3p⫹3d4s4p 3s3p 3s3p⫹4p 3s3p⫹3d4s4p b 3s3p 3s3p⫹4p 3s3p⫹3d4s4p b 4s4p 4s4p⫹5p 4s4p⫹4d 4s4p 4s4p⫹5p 4s4p⫹4d 4s4p 4s4p⫹5p 4s4p⫹4d 4s4p 4s4p⫹5p

401.6 397.2 398.9 397.6 398.2 397.8 399.6 399.5 399.5 822.7 820.4 791.2 799.0 804.0 774.7 879.0 875.1 870.7 3322.4 3291.7 3262.6 3202.7 3212.8 3165.6 3476.7 3451.5 3447.4 3579.3 3549.0

⌬ from SCF or CISD ⫺4.4 ⫺2.7 ⫹0.6 ⫹0.2 ⫺0.1 ⫺0.1 ⫺2.3 ⫺31.5 ⫹5.0 ⫺24.3 ⫺3.9 ⫺8.3 ⫺30.7 ⫺59.8 ⫹10.1 ⫺37.0 ⫺25.2 ⫺29.3 ⫺30.3

a

Values shown for the 2s2p, 3s3p, and 4s4p active spaces correspond to SCF and CISD results. Stateaveraged CASSCF orbitals were used throughout. b In these reference functions a maximum of two electrons were allowed into the specified nonvalence orbitals.



FIG. 6. The errors 共cm⫺1兲 from neglecting double excitations from the core orbitals for F, Cl, and Br. For bromine, results where singles-only are allowed from the 3d orbitals, designated by 关 Ar兴 3d, as well as where singles and doubles are taken from the 3d orbitals, designated by just 关Ar兴, are both shown.

total of ⫺4, ⫺24, and ⫺120 cm⫺1 for F, Cl, and Br, respectively, with the cc-pCV5Z basis sets. For F the 2p electrons represent most of the total valence correlation effect, while for Cl and Br the difference between the SCF and val-CISD SO splittings is nearly equally shared between correlation of the s and p valence electrons. As shown in Fig. 5, when the core-like electrons are included, the contributions of the p-type electrons clearly dominate the total core correlation effects on the SO splittings. In the case of chlorine, correlation of the 2p electrons increases the splitting by about 80 cm⫺1 while correlation of the remaining 2s and 1s electrons increase the splitting by a total of just ⬃5 cm⫺1 共cc-pCV5Z basis set兲. Likewise, correlation of the 1s-like electrons of fluorine increases the SO splitting by just 2 cm⫺1 over the val-CISD result. We note that this effect for F is entirely consistent with the spin-extended Hartree–Fock calculations of Veseth.28 The incremental core correlation effects for the SO splitting of Br with the cc-pCV5Z set amount to ⫹130(3d), ⫹145(3p), ⫹10(3s), ⫹93(2p), ⫹2(2s), and ⫹0.4(1s) cm⫺1. Thus, for an accurate spin–orbit splitting for the bromine atom, correlation down to the 2p electrons is required, with a total core correlation effect on the SO splitting of 381 cm⫺1 at the CISD level. However, even with all electrons correlated with the cc-pCV5Z basis set, the resulting splitting for Br is smaller than the experimental value by just over 100 cm⫺1. Up to this point only single reference CI wave functions have been utilized in calculating spin–orbit splittings. Table III shows results from various multireference CI calculations with the cc-pCVQZ (⫹p for fluorine兲 basis set. In the case


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of the fluorine atom, very little changes are observed between the CISD splittings and the MRCI ones obtained with reference functions including the valence orbitals (2s2p) and extra 3p or even a full set of 3s3 p3d orbitals. For chlorine, enlarging the active space results in CASSCF splittings lower by as much as 30 cm⫺1 when compared to SCF. When only the valence electrons are correlated, the MRCI splittings are either only slightly larger 共⫹5.0 cm⫺1 with the 3s3 p⫹4 p active space兲 or much lower 共⫺24.3 cm⫺1 with the 3s3 p⫹3d4s4 p active space兲 compared to val-CISD results. When the 2p core electrons are correlated, the MRCI calculations are only slightly lower than the ae-CISD results by about 8 cm⫺1. The results for bromine are calculated to be very similar to those of chlorine. The largest MRCI calculations for Br show a lowering of the SO splittings compared to the CISD values by about 30 cm⫺1. Hence the ⫺100 cm⫺1 error compared to experiment for the ae-CISD/cc-pCV5Z splittings mentioned above are most probably due to the nonrelativistic nature of the underlying wave function and not missing electron correlation effects. Given the strong dependence of the calculated spin– orbit splittings on core–valence correlation it was of interest to determine if this was a true correlation effect or an increase in the splittings due to just orbital relaxation 共‘‘core polarization’’兲. Calculations were then carried out in which only single excitations were allowed from the core orbitals and both single and double excitations from the valence orbitals. Figure 6 shows the difference between these results and those where double excitations were also allowed from the core using the cc-pCV5Z (⫹p for fluorine兲 basis sets. In the case of F, the effect of neglecting doubles from the 1s core is observed to be completely negligible 共⫹0.2 cm⫺1兲. The results for Cl show a slightly larger effect, but the differences from the CISD values are still only ⬃8 cm⫺1. Two series of results are shown in Fig. 6 for the bromine atom, one in which only singles were allowed from all the core electrons, 3d through 1s, and one in which doubles were also allowed from the 3d orbitals. In both cases the errors resulting from neglecting double excitations are relatively small, a total of ⬃20 cm⫺1 for the 关Ar兴 core and about twice that for when only singles were allowed from both the 3d and 关Ar兴 core orbitals. Hence, neglecting double excitations from the core orbitals result in errors only on the order of about 10%, indicating that the strong core–valence correlation effect on the SO couplings is essentially due to orbital relaxation of the core. A fortunate side effect of only needing single excitations from the core orbitals is a reduction in much of the added computational expense, which can be very substantial particularly for the bromine atom. IV. SUMMARY

Spin–orbit splittings of the halogen atoms F, Cl, and Br have been calculated within the framework of first-order perturbation theory with the Breit–Pauli operator and nonrelativistic CI wave functions. Using systematic sequences of correlation consistent basis sets, the role of both regular and tight polarization functions were investigated, as well as the effects of electron correlation. The most important results presented here can be summarized as:


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共i兲共ii兲

共iii兲

共iv兲共v兲共vi兲共vii兲

SCF splittings underestimate experiment and valence correlation lowers them even further. Core correlation, i.e., ‘‘core polarization,’’ especially of the p and d electrons, significantly increases the calculated splittings and is a requirement for accurate results. Including only single excitations from the core orbitals recovers nearly 90% of the CISD results. More extensive electron correlation obtained by increasing the size of the CI active space 共beyond the usual full valence CAS to include higher-lying orbitals兲 generally results in only a slight lowering of the calculated spin–orbit splittings compared to CISD. Accurate spin–orbit splittings for F require the addition of a very tight p function to the cc-pV nZ contracted 2p function. The presence of f-type functions is only important for Br and then only when correlating the 3d electrons where tight f functions are important. Correlation of the core electrons requires the addition of tight s and p functions to the standard cc-pV nZ basis sets 共plus tight d and f for Br兲. The all-electrons correlated basis set limit for the bromine atom CISD spin–orbit splitting is about 100 cm⫺1 lower than the experimental value due to the nonrelativistic nature of the underlying wave function.

ACKNOWLEDGMENTS

This work was supported by the National Science Foundation 共CAREER award CHE-9501262 to K.A.P.兲 and the Chemical Sciences Division in the Office of Basic Energy Sciences of the U.S. Department of Energy. This research was performed in the W. R. Wiley Environmental Molecular Sciences Laboratory, a national scientific user facility sponsored by the Department of Energy’s Office of Biological and Environmental Research and located at Pacific Northwest National Laboratory. Pacific Northwest National Laboratory is operated for the Department of Energy by Battelle. The development of the spin–orbit code by P.J.K. and H.J.W. was supported by the European research network ‘‘Theonet,’’ TMR-FMRX-CT96-088. 1

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