Relativistic and correlation effects on molecular properties. I. The dihalogens F 2,Cl 2,Br 2,I 2, and At 2

Relativistic and correlation effects on molecular properties. I. The dihalogens F 2,Cl 2,Br 2,I 2, and At 2 L. Visscher Laboratory of Chemical Physics and Material Science Center, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands K. G. Dyall Thermosciences Institute, NASA Ames Research Center, Moffett Field, California 94035-1000 Received 17 November 1995; accepted 6 March 1996 A benchmark study of a number of relativistic correlation methods is presented. Bond lengths, harmonic frequencies, and dissociation energies of the molecules F 2,Cl 2,Br 2,I 2, and At 2 are calculated at various levels of theory, using both the Schrödinger and the Dirac Coulomb Gaunt Hamiltonian. 1996 American Institute of Physics. S0021-9606 96 02122-8 I. INTRODUCTION The additivity of relativistic and electron correlation effects has long been questioned. Until recently few methods were available to quantitatively assess the interplay of both deviations of the nonrelativistic mean-field theory in molecular calculations. The so-called scalar relativistic methods 1,2 usually assume that corrections of higher order than c 2 can be neglected for chemical accuracy. Furthermore it is assumed that the effect of the spin orbit coupling on the form of the orbitals may be neglected, allowing a partition of the Hamiltonian into a spin-independent and a spin-dependent part. The latter part is then only used in the final stage of the calculation to couple the correlated many-electron functions. Since all these methods can be derived from the manyelectron Dirac Coulomb or Dirac Coulomb Breit equation, a straightforward way to provide a benchmark for the different approximations involved is to solve this equation directly. We here use relativistic analogues of a number of standard ab initio methods, which allows us to also study the influence of the choice of method to the calculated relativistic effects. The starting point is the Dirac Hartree Fock method, from which we get a set of 4-component molecular spinors. These spinors provide a natural basis for the socalled no-pair treatment of electron correlation. This means we do not consider states that have one or more positron electron pairs. In our treatment we do not include additional terms derived from the underlying theory, quantum electrodynamics QED. From experience with atomic calculations these corrections are known to have only very small differential effects on the valence shell properties. We, however, do consider the effect of the two-electron Gaunt interaction that describes the magnetic interaction between the electrons. Estimates of the importance of this term on the properties calculated will be given on basis of perturbation theory. The methods used are Hartree Fock HF, second-order Mo ller Plesset perturbation theory MP2, configuration interaction with single and double substitutions CISD, coupled cluster with single and double substitutions CCSD and the latter method, perturbatively corrected for the effect of triple excitations CCSD T. This gives a range of correlation treatments, from no electron correlation in the HF method to a fairly high level of correlation in the CCSD T method. II. COMPUTATIONAL DETAILS Nonrelativistic calculations were performed using GAUSSIAN92/DFT, 3 MOLPRO94 Ref. 4, and the nonrelativistic option of MOLFDIR. 5 All 4-component relativistic calculations were also performed with the latter code, using a newly-developed relativistic CCSD T code. 6 Two different models of the nuclei were used. The nonrelativistic calculations were done using the conventional point-nucleus model. Since this unphysical model gives rise to a weak singularity at the origin for the relativistic wave function, we used a Gaussian charge distribution to represent the spatial extent of the nucleus in the relativistic calculations. The exponents used in this work were 0.535 493 007 4E 09, 0.384 827 492 2E 09, 0.243 199 191 6E 09, 0.184 523 891 6E 09, and 0.136 902 770 0E 09 for F, Cl, Br, I, and At, respectively. The speed of light in atomic units was taken to be 137.035 989 5. All molecular calculations were performed using D 4h symmetry, except for MOLPRO94, where D 2h symmetry was used. The atomic calculations were carried out in O h.to prevent spurious discrepancies between the nonrelativistic and relativistic dissociation energies, we calculated both the nonrelativistic and the relativistic atomic asymptotes in a basis of spinors optimized for the spherical average of the four 2 P 3/2 states. In the MP2, CI, and CC calculations the halogens were treated as 7 valence electron atoms, i.e., correlating only the valence s and p-electrons. The virtual space used was truncated slightly by removing the virtual spinors that consisted of corelike functions. Spectroscopic constants were in general obtained by fitting the potential energy curve to a 4th order polynomial in the internuclear distance. For the nonrelativistic calculations on F 2,Cl 2, and Br 2 we used the frequency option of GAUSSIAN92. 9040 J. Chem. Phys. 104 (22), 8 June 1996 0021-9606/96/104(22)/9040/7/$10.00 1996 American Institute of Physics

L. Visscher and K. G. Dyall: Relativistic effects on dihalogens. I 9041 TABLE I. Summary of the basis set sizes. L, large component basis; S, small component basis. pvdz pvtz Primitive Contracted Primitive Contracted F L: 9s,5p,1d L: 3s,2p,1d L: 10s,6p,2d,1f L: 4s,3p,2d,1f S: 5s,10p,5d,1f S: 2s,4p,2d,1f S: 6s,12p,7d,2f,1g S: 3s,6p,4d,2f,1g Cl L: 12s,8p,1d L: 4s,4p,1d L: 15s,9p,2d,1f L: 5s,5p,2d,1f S: 8s,13p,8d,1f S: 3s,5p,4d,1f S: 9s,17p,10d,2f,1g S: 4s,7p,6d,2f,1g Br L: 15s,12p,6d L: 5s,5p,2d L: 17s,14p,9d,1f L: 6s,7p,3d,1f S: 12s,21p,12d,6f S: 4s,7p,6d,3f S: 14s,26p,15d,9f,1g S: 5s,9p,8d,4f,1g I L: 16s,12p,7d L: 6s,7p,3d L: 21s,17p,12d,1f L: 7s,8p,4d,1f S: 12s,23p,12d,7f S: 5s,9p,8d,4f S: 17s,33p,18d,12f,1g S: 6s,11p,10d,5f,1g At L: 23s,20p,15d,10f L: 7s,9p,5d,1f L: 25s,22p,17d,11f L: 7s,10p,6d,2f S: 20s,23p,20d,15f,10g S: 6s,9p,11d,6f,2g S: 22s,25p,22d,17f,11g S: 7s,11p,11d,7f,3g III. BASIS SETS The basis sets used for F and Cl were the correlation consistent polarized valence cc-pvdz and cc-pvtz basis sets of Dunning and co-workers. 7,8 The fluorine basis sets were augmented by a tight p function, with exponents 128.185 40 and 250.834 91 for the cc-pvdz and cc-pvtz basis sets, respectively. This addition reduced the error made in the 2p spin orbital splitting as compared to the numerical Dirac Hartree Fock value by a factor of 10 to less than 1%. For the Cl basis no additional functions were required to obtain this accuracy in the 3p spin orbital splitting. Both basis sets for both atoms were recontracted on basis of numerical Dirac Hartree Fock calculations using a modification 9 of GRASP. 10 For Cl one extra contracted large component p-function relative to the nonrelativistic contraction scheme was used to describe the 2 p spin orbit splitting. The small component primitive basis set was generated using unrestricted kinetic balance and contracted according to the atomic balance scheme. For Br we used the basis sets of Schäfer et al. 11 and F,gri 12 to supply a double and triple zeta description of the valence basis. The former basis set was extended with a tight s and a tight p function, exponents 6.234 303 1E 06 and 2.502 973E 04, while the latter, in addition to a tight s and a tight p function with exponents 1.693 86E 07 and 5.728E 05 was also extended with a correlating d and f function, exponents 0.24 and 0.54. These exponents were obtained by optimizing the average of the nonrelativistic CISD energies of Br and Br. The outermost primitive function in the original d set was also used for correlation to produce a (2d,1f ) correlating set. The basis sets were contracted using a similar scheme as for the F and Cl basis sets. The iodine double zeta basis was that of Stromberg et al. 13 extended with a tight s and p function and a correlating d function, with exponents 4.35E 06, 1.85E 04, and 0.28. For the iodine triple zeta description we used F,gri s basis set, 14 again extended in a similar manner to the Br basis with a tight s and p, exponents 1.098 643 0E 08 and 2.509 375 9E 05, and a correlating d and f, exponents 0.22 and 0.39, and recontracted as was done for Br. Since the use of extended nonrelativistic primitive basis sets for very heavy atoms is questionable and the number of small component primitive functions generated by the kinetic balance procedure becomes a burden, we followed a somewhat different procedure to obtain the At basis sets. We optimized two sets of exponents at the Dirac Hartree Fock level using a version of GRASP developed by Dyall and F,gri 15 under the constraints that the d exponents from subset of the s exponents and that the f exponents from a subset of the p exponents. This ensures a reasonably compact small component basis set without relaxing the kinetic balance concept. The resulting basis sets were contracted according to the cc-pvxz scheme described above. After completion of the calculations a flaw was discovered in the open-shell exponent optimization routine. This means that the exponents were not completely optimized, though the description is still better than would be obtained with a pure nonrelativistic basis set. A fully optimized set of exponents is to be published shortly, the basis sets used in this work are available upon request. For most of the nonrelativistic calculations we used the primitive basis sets as described above, but without the additional tight s and p-functions. For F 2 we also used the extended cc-pvdz basis, because comparing the nonrelativistic results in the original basis with the relativistic results in the extended basis gave rise to a spurious difference of 3 cm 1 in the frequency. The sizes of the primitive and contracted basis sets are summarized in Table I. IV. RESULTS AND DISCUSSION Calculated spectroscopic constants for the molecules F 2, Cl 2,Br 2,I 2, and At 2 are given in Tables II VI. Table VII gives the relativistic effect on these properties, calculated as the difference x rel x nonrel. The relativistic decrease of the dissociation energy of the molecules is clearly seen in all molecules. This is driven by the spin orbit coupling in the valence p-shell. For the lighter elements the atomic spin orbit coupling is almost completely quenched in the molecule. This gives a decrease of the atomic asymptotes relative to the molecule of two times 1/3 of the valence p-splitting. For all halogen molecules we find a relativistic destabilization of the bond. This destabilization, which is also found

9042 L. Visscher and K. G. Dyall: Relativistic effects on dihalogens. I TABLE II. Properties of F 2, calculated at various levels of theory. TABLE IV. Properties of Br 2, calculated at various levels of theory. NR-HF 1.347 1.329 1179 1267 33.7 25.3 DC-HF 1.347 1.329 1179 1266 34.6 26.3 DC G-HF 1.347 1.329 1178 1266 34.6 26.2 NR-MP2 1.424 1.398 930 1017 30.3 42.3 DC-MP2 1.424 1.398 931 1016 29.5 41.5 NR-CISD 1.406 1.371 982 1110 9.4 11.8 DC-CISD 1.406 1.371 982 1110 8.5 10.9 NR-CCSD 1.432 1.395 883 1013 22.1 28.3 DC-CCSD 1.432 1.395 884 1012 21.3 27.5 NR-CCSD T 1.458 1.416 780 919 27.0 34.8 DC-CCSD T 1.458 1.416 782 920 26.2 34.0 Experiment a 1.4119 916.6 38.2 NR-HF 2.311 2.279 337 354 16.4 23.4 DC-HF 2.309 2.277 334 351 9.0 15.7 DC G-HF 2.311 2.278 334 350 9.1 15.8 NR-MP2 2.334 2.291 317 336 39.2 49.5 DC-MP2 2.332 2.291 314 333 32.6 42.5 NR-CISD 2.338 2.292 315 338 27.4 29.1 DC-CISD 2.337 2.292 311 335 20.9 22.3 NR-CCSD 2.352 2.307 302 324 36.2 42.9 DC-CCSD 2.351 2.306 299 321 29.6 36.0 NR-CCSD T 2.361 2.315 293 316 38.4 46.1 DC-CCSD T 2.361 2.315 289 312 31.8 39.2 Experiment a 2.281 325.3 45.9 in other p-block elements like thallium and bismuth, 16 can be explained by considering the hybridization of the orbitals involved. The six possible linear combinations of the atomic p-orbitals may be labeled 1/2g, 1/2u, 3/2u, 1/2u *, 1/2g *, and 3/2g *. The orbitals are degenerate in the nonrelativistic limit, but since in the relativistic case the 1/2 and 1/2 are allowed to mix, six nondegenerate orbitals will be found. The bonding and antibonding orbitals are partly and partly with the amount of mixing determined by the spin orbit splitting. If the energy difference between the p 1/2 and p 3/2 is small, the 5 occupied linear combinations will be a pure -bond 1/2g, two pure -bonds 1/2u and 3/2u and 2 pure -antibonds 1/2g * and 3/2g *. The LUMO is the pure -antibonding 1/2u * orbital and the total bond order is one -bond. This is the case for F 2. On the other hand, if the spin orbit splitting is large the hybridization energy will make the formation of pure - or -bonds less favorable. In the hyper-relativistic limit no hybridization will occur and we combine the atomic p 1/2 s with each other and the atomic p 3/2 s also separately. The highest (1/2u) combination is the TABLE III. Properties of Cl 2, calculated at various levels of theory. Molecular NR-MP2, NR-CCSD, and NR-CCSD T values were taken from Ref. 18. NR-HF 2.007 1.984 592 613 17.1 26.5 DC-HF 2.007 1.985 592 612 15.3 24.5 DC G-HF 2.008 1.985 592 612 15.4 24.5 NR-MP2 2.030 1.997 548 578 43.6 56.9 DC-MP2 2.030 1.998 548 576 41.9 55.0 NR-CISD 2.033 1.997 548 585 29.1 32.1 DC-CISD 2.033 1.997 548 584 27.4 30.3 NR-CCSD 2.045 2.009 526 562 39.5 48.6 DC-CCSD 2.045 2.010 525 561 37.8 46.8 NR-CCSD T 2.054 2.018 507 546 41.9 52.3 DC-CCSD T 2.055 2.019 507 544 40.2 50.4 Experiment a 1.987 559.7 58.0 p 3/2,1/2 p 3/2,1/2 which is one-third -bonding and two-thirds -antibonding. This makes the 3/2g * a more favorable LUMO since it is pure -antibonding. The bond order is now one -bond, which means that the effect of relativity is to change the bond from pure into pure. In astatine the hyper-relativistic limit is not yet reached; the lowest determinant that can be formed is still the one with the (1/2u) LUMO. The amount of mixing is, however, appreciable as can be seen by looking at the amount of *-character in the occupied (1/2u) orbital. This percentage grows from 0.00% in F, 0.01% in Cl, 0.19% in Br, 1.10% in I to 10.20% in At. A. Fluorine For the lighter elements comparison may be made with a wealth of nonrelativistic calculations. We initially compared our relativistic results for F 2 to the nonrelativistic calculations done by Peterson et al., 17 since they employed the same basis set. This gave rise to a surprisingly large relativistic effect of minus 1 to 2 cm 1 in the frequency for the ccpvdz calculations. Since errors in the fitting procedure are of the order of 0.05 cm 1, the difference between fits of TABLE V. Properties of I 2, calculated at various levels of theory. NR-HF 2.714 2.681 227 237 16.7 23.8 DC-HF 2.714 2.682 219 228 2.9 9.2 DC G-HF 2.716 2.683 219 228 3.1 9.3 NR-MP2 2.736 2.686 215 230 34.9 45.6 DC-MP2 2.739 2.688 206 221 22.6 32.2 NR-CISD 2.742 2.691 213 229 24.6 26.8 DC-CISD 2.746 2.693 203 220 12.8 14.3 NR-CCSD 2.756 2.704 205 222 32.4 39.5 DC-CCSD 2.761 2.708 195 212 20.4 26.7 NR-CCSD T 2.764 2.712 200 217 34.3 42.3 DC-CCSD T 2.771 2.717 188 206 22.4 29.6 Experiment a 2.666 214.5 35.9

L. Visscher and K. G. Dyall: Relativistic effects on dihalogens. I 9043 TABLE VI. Properties of At 2, calculated at various levels of theory. the data to 4th and 6th order polynomials, the difference should originate from another source. We found this deviation to depend to a large extent on the extra tight p function that was added in the relativistic calculation. Redoing the nonrelativistic calculations with the extended basis gave frequencies that were 3 cm 1 smaller than reported by Peterson et al. The relativistic effect changed sign and was found to be maximally 1.7 cm 1 on the CCSD T level. This relativistic frequency shift is significantly larger than at the CCSD T /cc-pvtz level, where it is 0.3 cm 1. Adding the tight p-function to the triple zeta basis and redoing the calculation only alters the result by 0.1 cm 1. The relativistic effect on the frequency is thus exaggerated due to the basis set incompleteness and no conclusion can be drawn on basis of the double zeta results. On the triple zeta level the relativistic frequency shift is more reliable and found to be smaller than 1 cm 1. The relativistic effect on the dissociation energy is 0.8 to 0.9 kcal mol 1 which is in good agreement with the value of 0.77 kcal mol 1 that is obtained by correcting the nonrelativistic result only for the spin orbit effect on the asymptote this approximation will hereafter be referred to as ASO-only. For these light elements the two-electron relativistic effects are of the same order as the one-electron effects. Adding the Gaunt interaction as a perturbative correction to the two-electron integrals at the HF level shifts the frequency by 0.3 cm 1 and increases the bond length by 0.0003 Å, resulting in total relativistic shift of 0.8 cm 1 and 0.0003 Å. Its effect on the D e is 0.05 kcal mol 1. B. Chlorine NR-HF 2.982 2.904 163 169 13.3 22.2 DC-HF 3.033 2.973 127 131 11.5 8.2 DC G-HF 3.034 2.974 128 132 11.4 8.1 NR-MP2 3.012 2.914 154 164 29.9 42.1 DC-MP2 3.081 2.984 117 126 9.2 15.0 NR-CISD 3.024 2.919 150 163 24.0 24.7 DC-CISD 3.093 2.993 114 125 2.8 0.9 NR-CCSD 3.038 2.934 145 158 29.4 36.4 DC-CCSD 3.123 3.022 105 116 9.1 11.4 NR-CCSD T 3.046 2.942 141 154 30.8 39.0 DC-CCSD T 3.153 3.046 95 108 11.2 14.6 The nonrelativistic Cl 2 results were taken from Woon and Dunning, 18 except for the dissociation energies. For this number we recalculated the atomic Cl energy to avoid spurious effects from the use of double-group symmetry in the relativistic calculation. This affects the D e on the CCSD level by 0.2 kcal mol 1 ; we find 39.5 kcal mol 1, while Woon and Dunning report 39.3 kcal mol 1. Again the relativistic effect is slightly dependent on the basis set used. The cc-pvdz basis shows no significant effect on the bond length and frequency, while the cc-pvtz basis gives a small TABLE VII. Relativistic effects on the properties of the halogens at various levels of theory. Molecule but consistent shift for both of these spectroscopic constants. The shift in D e is slightly larger than the ASO-only value of 1.68 kcal mol 1, and quite consistent for all methods. Twoelectron relativistic effects are of the same order of magnitude as in F 2, we find a frequency shift of 0.2 cm 1, a bond expansion of 0.0004 Å and an increase of the D e by 0.07 kcal mol 1 upon adding the perturbative Gaunt correction. C. Bromine r e Å e cm 1 D e kcal mol 1 F 2 HF 0.000 0.000 1 0 0.9 0.9 MP2 0.000 0.000 1 0 0.8 0.8 CISD 0.000 0.000 1 0 0.8 0.9 CCSD 0.000 0.000 0 0 0.8 0.9 CCSD T 0.000 0.000 2 0 0.8 0.8 Cl 2 HF 0.000 0.001 0 1 1.8 2.1 MP2 0.000 0.001 0 1 1.7 1.9 CISD 0.000 0.000 0 0 1.7 1.9 CCSD 0.000 0.001 0 1 1.7 1.9 CCSD T 0.000 0.001 0 1 1.7 1.8 Br 2 HF 0.002 0.001 3 3 7.4 7.7 MP2 0.001 0.001 3 3 6.7 7.0 CISD 0.001 0.001 3 3 6.6 6.8 CCSD 0.001 0.001 3 3 6.6 6.9 CCSD T 0.000 0.000 4 4 6.5 6.8 I 2 HF 0.000 0.001 9 9 13.7 14.7 MP2 0.003 0.003 9 9 12.3 13.4 CISD 0.003 0.002 9 9 11.9 12.5 CCSD 0.005 0.004 11 10 12.0 12.8 CCSD T 0.008 0.006 12 11 11.9 12.7 At 2 HF 0.051 0.069 36 38 24.7 30.4 MP2 0.069 0.070 37 38 20.7 27.1 CISD 0.068 0.074 36 38 21.2 23.8 CCSD 0.085 0.088 40 42 20.3 25.0 CCSD T 0.106 0.104 46 46 19.7 24.4 In bromine a slight bond contraction is found. This is not an indication of a relativistically enhanced bond as the frequency decreases by 3 cm 1. The lowering of the dissociation energy, with values ranging from 6.6 to 7.7 kcal mol 1, is still close to the ASO-only value of 7.0 kcal mol 1. Adding the Gaunt interaction again gives only minor changes; a bond length increase of 0.0010 Å, a frequency shift of 0.1 cm 1, and an increase of the D e by 0.12 kcal mol 1. The results may be compared with the results of Lee and Lee, 19 who did RECP-MP2 calculations with and without spin orbit coupling. Using Christiansen and Ermler s RECP Ref. 20 they find a bond length of 2.31 Å which is increased upon inclusion of the spin orbit effect by 0.01 Å to 2.32 Å. The effect of the spin orbit operator on the frequency is 6 cm 1, twice as large as the total relativistic effect calculated by us. Schwerdtfeger et al. 21 used different pseudopotentials and did HF and CI calculations. At the nonrelativistic HF level they find an r e of 2.302 Å that decreases to 2.298 Å upon addition of scalar relativistic effects and to 2.300 Å upon addition of both scalar and spin orbit effects. The total

9044 L. Visscher and K. G. Dyall: Relativistic effects on dihalogens. I contraction of 0.002 Å agrees well with our HF/DF results. At the CI level the only semiempirical pseudopotential that they used includes core-polarization effects. This gives an r e of 2.296 Å, which is in between our double and triplet zeta CISD results. D. Iodine As the relativistic effects increase on going to iodine the dependence on the quality of the basis set becomes relatively less important. For iodine both the DZP and the TZP basis give the same shifts in frequency and a similar shift in the bond length. The method used to calculate the relativistic effect is more important the relativistic effect on the r e value ranges from 0.001 Å at the HF level to 0.006 Å at the CCSD T level of theory. Relativity leads to significant weakening of the bond, causing longer bond lengths and smaller dissociation energies than those calculated at the nonrelativistic level of theory. The calculated relativistic effect on the D e ranges from 14.7 to 11.9 kcal mol 1, whereas 14.5 kcal mol 1 is the atomic effect of 2/3 of the SO splitting. This indicates that the quenching of the spinorbit coupling in the molecule is still fairly complete. Twoelectron relativistic effects may be neglected since the absolute size of the two-electron Gaunt correction to the properties remains almost the same as for the previous molecules. We find a bond length increase of 0.0014 Å, a frequency decrease of 0.04 cm 1, and a D e increase by 0.12 kcal mol 1. We can again compare with several RECP-MP2 results. Lee and Lee 19 use Christiansens and Ermlers RECP Ref. 22 and find a spin orbit induced bond length change of 0.02 Å. The RECP-MP2 value is 2.73 Å. The RECP-MP2 spin orbit induced change in frequency is 11 cm 1, slightly larger than the all electron DC-MP2 total relativistic value of 9 cm 1. Schwerdtfeger s results are also available for this molecule. They find a relativistic expansion of 0.002 Å at the HF level, slightly larger than our calculated value. Their CISD r e value is 2.687 Å, which is slightly shorter than our CISD values. This can be due to the increasing importance of corevalence correlation that is neglected in our approach, since we do not correlate the d-electrons. When the d-electrons are included in the valence space a contraction of 0.02 Å both at the MP2/pVTZ level and CCSD/pVTZ of theory is found. 23 An extensive treatment of the ground and excited states of I 2 is given by Teichteil and Pelissier. 24 They give results for the ground state using a relativistic averaged pseudopotential and an effective one-electron spin orbit operator with a valence basis of triple zeta quality plus 2s, 2p, and 2d polarization functions. At the MCSCF/MRSDCI Q level of theory they find a bond distance of 2.75 Å without the SOoperator and 2.77 with the SO-operator. The e values are 199 and 185 cm 1, respectively. Again these changes are larger than the total relativistic effect found by us which may either point to an overestimation of the spin orbit effect by the effective one-electron spin orbit operator or to a cancellation of scalar relativistic and spin orbit effects. Teichteil and Pelissier also studied the importance of core-valence polarization effects by comparing 7-electron pseudopotential results with 17-electron pseudopotential results and found a small effect 0.003 Å at the MCSCF/ MRSDCI Q level of theory, but a 0.02 Å contraction when using second-order perturbation theory after the MCSCF step. The latter result is consistent with the finding of Schwerdtfeger et al. who report a contraction of 0.02 Å due to core-valence polarization and correlation effects. Recent work by Dolg, 25 who compared Douglas Kroll MP2 results with and without correlating the d-electrons also gives a contraction of 0.02 Å. This makes it likely that the core-valence correlation is at least partly responsible for the discrepancy between our CCSD T results and experiment. Teichteil and Pelissier estimated the effect of adding f and g functions to the valence basis and found the f function to decrease the bond length by 0.06 Å and the g function by an additional 0.02 Å. The effect of adding an f function agrees well with the difference of 0.05 Å between our double and triple zeta basis set results, which also differ by an f function, in addition to the extra s, p, and d function. E. Astatine Finally we consider the heaviest halogen treated here, astatine. The bond length expansion is 0.1 Å and the molecule is unbound at the Dirac Fock level of theory. The different methods again give somewhat different relativistic shifts for the r e values but show a rather consistent relativistic shift for the e and D e. As we saw for iodine, the relativistic effect is largest at the CCSD T level of theory. We see that the spin orbit effect is no longer completely quenched as the expected relativistic effect on the D e,45 kcal mol 1, is significantly larger than the calculated values. Again two-electron relativistic effects are very small, giving a bond length increase of 0.0015 Å, a frequency decrease of 0.15 cm 1, and a D e increase by 0.14 kcal mol 1. No literature values for the spectroscopic constants could be found. Though the diatomic form is found to be the ground state of neutral At, 21 all isotopes are radioactive and experimental information on the spectroscopic constants seems hard to obtain. Theoretical data is also very scarce and limited to semi-empirical extrapolation of values for other elements. Dirac Fock 26 and pseudopotential calculations 27 are performed for HAt but not for At 2. On the basis of our CCSD and CCSD T results we may predict the properties of At 2. The calculated bond length is consistently too long in our CCSD T /pvtz approach. Using the correction for I 2 and assuming a conservative error margin of 0.1 Å, we predict a bond length of 3.0 0.1 Å. The calculated frequency is consistently too low at this level of theory for Cl 2,Br 2, and I 2, so we again correct with the error found in I 2 and obtain an e value of 115 10 cm 1. For the D e the error is 7.5, 6.7, and 6.3 kcal mol 1 for Cl 2,Br 2, and I 2, respectively. Correction with the I 2 error of 6.3 kcal mol 1 and assuming an error bar of twice the range of the errors for the D e in these molecules gives an estimated value of 20 3 kcal mol 1.

L. Visscher and K. G. Dyall: Relativistic effects on dihalogens. I 9045 Summarizing the trends, we see that the D e is decreased by relativity. Up to bromine, this effect can be modelled by considering the spin orbit coupling as completely quenched in the molecule and correcting only the atomic asymptote. For iodine and especially astatine this simple model fails and scalar relativistic and molecular spin orbit effects should be taken into account. The influence of spin orbit coupling and other relativistic effects on the bond length and harmonic frequency is somewhat less predictable. In chlorine, iodine, and astatine we find an expansion of the bond length relative to the nonrelativistic result, while for bromine we find a small contraction. The effect on the harmonic frequency is more consistent. In all cases we find a decrease, that is more pronounced when we allow for a higher level of electron correlation. The relativistic correction to the bond length and frequency is fairly independent of the basis set employed. The remaining errors, when comparing our highest level CCSD T /pvtz results with experiment, are probably mostly due to the one-particle basis set incompleteness and the neglect of d-correlation for the heavier elements. Higher order relativistic effects are likely to be small, as can be seen from the magnitude of the Gaunt interaction corrections. The comparison with other theoretical work is not always straightforward due to method and basis set differences and the use of a semiempirical core-valence polarization potential in some schemes. 21 The question whether the relativistic pseudopotential schemes do reproduce the effects of relativity in a reliable way is therefor not always unambiguously answered. Comparison with the 2-component RECP- MP2 scheme of Lee and Lee 19 with our DC-MP2/pVDZ results is well possible, since both the basis set and the method are similar. The results for Br 2 and I 2 are in good agreement for the r e and the e and in reasonable agreement for the D e. The larger discrepancy in the D e is understandable since Lee and Lee used a different method to treat the open-shell atoms than we did. Schwerdtfeger et al. 21 also treated both molecules. Their HF results using a 2-component pseudopotential should be comparable to our pvdz or pvtz results since they employed a (9s,6p,2d) valence basis. For both Br 2 and I 2 the PP-HF results are in between our DC-HF/ pvdz and DC-HF/pVTZ results, with the D e s close to the pvtz result and the r e close to the pvdz result. Teichteil and Pelissier 24 employed a contracted [5s,5p,2d] basis and MCSCF followed by a selected CI method. Because this method is quite different from all the methods that we used a straightforward comparison is not possible. Their r e and e for I 2 are 2.77 Å and 185 cm 1, which is closest to our DC-CCSD T /pvdz result. V. CONCLUSIONS The effect of relativity on spectroscopic properties of the dihalogens has been considered by comparing nonrelativistic with relativistic all-electron methods. The halogen halogen bond is weakened by the increasing relativistic hybridization energy of the p-orbitals which leads to an increase in bond length and a decrease in the frequency and the dissociation energy. The additivity of relativistic and correlation effects that is often implicitly used in calculations on heavy systems has been confirmed for the lighter elements, but for the heaviest halogens the nonadditivity is apparent. The similarity of the relativistic corrections, obtained with different methods to treat electron correlation, implies that the relativistic-correlation cross term is well described at low order in the correlation treatment. Consequently one may estimate the relativistic effect at a certain level of theory by a correction calculated on a lower, more economical, level of theory. For the dihalogens treated here it has been found that such a scheme would lead to a small underestimation of the relativistic effects, as the higher level methods tend to show somewhat larger relativistic corrections. The effect of two-electron relativistic corrections, i.e, the Gaunt interaction, has been found to be unimportant at the level of accuracy reached in the calculated spectroscopic constants. It may be of interest for highly accurate calculations of the ligther elements since the effect on the frequency of F 2 and Cl 2 is of the order of a few tenths of a wave number. One has to be careful when comparing results from different methods, especially when the basis set limit is not reached as is usually the case in compounds like the ones described here. When comparison was possible we found the relativistic pseudopotential calculations in good agreement with our all-electron calculations. ACKNOWLEDGMENTS L.V. was supported by a National Research Council postdoctoral fellowship and a Cray Research Grant. K.G.D. was supported by a prime contract NAS2-14031 from NASA to Eloret and by contract BPNL 291140-A-A3 from Pacific Northwest Laboratory to Eloret. Pacific Northwest Laboratory is a multiprogram laboratory operated by the Batelle Memorial Institute for the U.S. Department of Energy under Contract No. DE-AC06-76RLO-1830; the work was performed under the auspices of the Office of Scientific Computing and under the Office of Health and Environmental Research, which funds the Environmental and Molecular Sciences Laboratory Project, D-384. L.V. wishes to thank Michael Dolg and Bert de Jong for providing him with their preliminary I 2 results. 1 L. L. Foldy and S. A. Wouthuysen, Phys. Rev. 78, 29 1950. 2 M. Douglas and N. M. Kroll, Ann. Phys. NY 82, 89 1974. 3 GAUSSIAN 92/DFT, Revision G.2, M. J. 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