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The mean absolute error in MNDO/d heats of formation amounts to 5.4 kcal/mol for the complete validation set of 575 molecules and is identical for the...
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J. Phys. Chem. 1996, 100, 616-626

Extension of MNDO to d Orbitals: Parameters and Results for the Second-Row Elements and for the Zinc Group Walter Thiel* and Alexander A. Voityuk Organisch-chemisches Institut, UniVersita¨ t Zu¨ rich, Winterthurerstrasse 190, CH-8057 Zu¨ rich, Switzerland ReceiVed: July 27, 1995X

The extension of the MNDO formalism to d orbitals is outlined. MNDO/d parameters are reported for Na, Mg, Al, Si, P, S, Cl, Br, I, Zn, Cd, and Hg. According to extensive test calculations covering more than 600 molecules and several properties, MNDO/d provides significant improvements over established semiempirical methods, especially for hypervalent compounds. The mean absolute error in MNDO/d heats of formation amounts to 5.4 kcal/mol for the complete validation set of 575 molecules and is identical for the subsets of 508 normal valent and 67 hypervalent compounds. In addition to the statistical evaluations, several specific applications are briefly discussed to illustrate the performance of MNDO/d in selected areas and to comment on problematic cases.

1. Introduction MNDO,1

AM1,2

PM33

and are widely used in theoretical studies of molecular structure and reactivity.4-6 These semiempirical methods employ an sp basis without d orbitals in their present implementation. Therefore, they cannot be applied to most transition metal compounds, and difficulties are expected for hypervalent compounds of main-group elements where the importance of d orbitals for quantitative accuracy is well documented at the ab initio level. The present paper outlines the extension of the MNDO formalism to d orbitals, which has been discussed in more detail elsewhere.7 Previous publications have described the parametrization of the resulting MNDO/d approach for the halogens (Cl, Br, I)8 and for silicon.9 Here we report MNDO/d parameters for all second-row elements except argon and for the elements of the zinc group. The results of extensive test calculations are compared with those from standard MNDO,1,10 AM1,2,11 and PM33,12 in order to assess the performance of MNDO/d. Some statistical comparisons with SAM1 and SAM1d13-15 are also included. 2. Method The established MNDO formalism and parameters remain unchanged for hydrogen, helium, and the first-row elements. The inclusion of d orbitals for the heavier elements requires a generalized semiempirical treatment of the two-electron interactions. The two-center two-electron integrals are calculated by an extension7 of the original point-charge model16 that is currently used in MNDO, AM1, and PM3. These integrals are expanded in terms of semiempirical multipole-multipole interactions. For an spd basis, there are 45 distinct one-center charge distributions that are associated with multipoles up to hexadecapoles. All monopoles, dipoles, and quadrupoles of these charge distributions are included whereas all higher multipoles are neglected.7 Using suitable point-charge representations the remaining multipole-multipole interactions are calculated by applying the Klopman-Ohno formula,17 which does not involve any new adjustable parameters. The computational implementation of the extended point-charge model has been specified in full detail.7,18 X

Abstract published in AdVance ACS Abstracts, December 15, 1995.

0022-3654/96/20100-0616$12.00/0

All nonzero one-center two-electron integrals are retained to ensure rotational invariance.19 Because of the large number of such unique integrals in an spd basis,7 it is generally not feasible to determine all of them from experimental data or to optimize all of them in the semiempirical parametrization. In our approach the one-center Coulomb integrals gss, gpp, and gdd are optimized in the parametrization and used to derive corresponding orbital exponents (ζ˜ s, ζ˜ p, ζ˜ d) which analytically reproduce these semiempirical integrals. Using these orbital exponents, all other one-center two-electron integrals are calculated analytically, except for gsp and hsp, which are optimized independently to allow for a finer tuning of the results. All other basic definitions in MNDO remain unchanged when including d orbitals. In particular, the resonance integrals are taken to be proportional to the corresponding overlap integrals, and the two-center core-electron attractions and core-core repulsions are expressed in terms of two-center two-electron integrals over s orbitals.1 For Coulomb interactions involving the core, the original MNDO formalism may be generalized by introducing a separate adjustable additive term Fcore into the extended point-charge model.7 This may become important in cases where the simulation of the atomic core by the valenceshell charge distribution ss is no longer sufficient to achieve a realistic balance of the electrostatic interactions. We adopt a separate Fcore value only for Na and Mg, where the results are improved significantly by this option. For the remaining elements we use Fcore ) F°ss, as in the standard MNDO formalism.1 The inclusion of d orbitals affects the results for different elements to a different extent. For example, d orbitals are clearly more important for P or S than for Na. Careful analogous parametrizations show that the results with an sp and an spd basis are essentially of the same quality in the case of Na, Mg, Zn, Cd, and Hg, which is not too surprising in a semiempirical framework. Therefore we adopt an sp basis for these five elements in MNDO/d. Since compounds of these elements often contain atoms described by an spd basis in MNDO/d (e.g. halogens), it would be inconsistent to employ any available standard MNDO parameters, e.g. for Zn10h or Hg,10i which have been derived using an sp basis for all elements. Hence, a new parametrization is required even if an sp basis is adequate in MNDO/d for a given heavy element (Z > 10). © 1996 American Chemical Society

Extension of MNDO to d Orbitals

J. Phys. Chem., Vol. 100, No. 2, 1996 617

TABLE 1: MNDO/d Parameters for Elements with an spd Basisa Uss (eV) Upp (eV) Udd (eV) ζs (au) ζp (au) ζd (au) βs (eV) βp (eV) βd (eV) R (Å-1) ζ˜ s (au) ζ˜ p (au) ζ˜ d (au) gss (eV) gpp (eV) gdd (eV) gsp (eV) gp2 (eV) hsp (eV) a

Al

Si

P

S

Cl

Br

I

-28.961 830 -22.814 474 -6.690 879 1.625 160 1.319 936 0.952 667 -5.402 719 -3.406 734 -1.778 777 1.439 879b 1.348 458 0.929 479 0.916 383 9.471 485 7.080 002 6.894 865 8.791 266 6.252 903 0.778 059

-36.051 529 -27.535 692 -14.677 439 1.915 655 1.681 611 0.966 772 -8.210 734 -4.884 620 -2.608 011 1.660 069 1.529 292 0.976 281 0.938 164 10.741 647 7.436 497 7.058 750 7.560 667 6.567 751 0.877 539

-47.055 531 -38.067 059 -23.691 597 2.266 463 1.940 015 1.100 109 -8.902 104 -9.386 110 -2.091 701 1.852 551 1.634 376 1.082 912 1.006 515 11.479 753 8.248 723 7.573 017 8.557 570 7.285 092 2.107 804

-56.889 130 -47.274 746 -25.095 118 2.225 851 2.099 706 1.231 472 -10.999 545 -12.215 437 -1.880 669 2.023 060 1.736 391 1.121 182 1.050 847 12.196 301 8.540 233 7.906 571 8.853 901 7.542 547 2.646 352

-69.622 971 -59.100 731 -36.674 572 2.561 611 2.389 338 1.251 398 -6.037 292 -19.183 386 -1.877 782 2.180 300 1.880 875 1.181 042 1.140 616 13.211 148 8.996 201 8.581 992 9.419 496 7.945 248 3.081 499

-65.402 779 -54.553 753 -13.728 099 2.590 541 2.330 857 1.357 361 -8.314 976 -10.507 041 -0.962 599 2.091 050 2.235 816 1.432 927 1.242 578 12.222 356 8.535 464 7.310 953 8.263 721 7.482 167 2.749 522

-62.765 354 -50.292 114 -12.248 305 2.756 543 2.253 080 1.502 335 -10.699 487 -4.941 178 -2.350 461 1.906 174 2.672 411 1.572 299 1.258 848 11.980 782 7.709 372 6.097 299 7.855 902 6.718 557 2.071 475

In standard notation. See text. b 1.387 880 for Al-H, Al-C, and Al-Al.

A final modification of the formalism is motivated by a comparison of the results obtained after parametrizing the MNDO/d approach (as described up to this point) for all elements studied presently. As it turned out, the results for Na, Mg, and Al were consistently less satisfactory than those for the other elements, with systematic deviations for certain types of bonds. In this situation we consider it legitimate in a semiempirical context to introduce a small number of bondspecific parameters in order to ameliorate such problems. In the spirit of AM12 and PM33 it seems best to include this refinement in the core-core repulsion function and to avoid a bond-specific modification of the electronic wave function. We thus extend the definition of the MNDO/d formalism by allowing a small number of independent bond-specific R parameters in the usual core-core repulsion function1 (presently one separate R value for Na and Al, and two separate R values for Mg, see section 3). Such bond-specific R parameters will also be needed for transition metals in MNDO/d. 3. Parametrization For an element with an spd basis, there are typically 15 parameters to be optimized: i.e. Uss, Upp, and Udd (one-center one-electron energies); gss, gpp, gdd, gsp, and hsp (one-center twoelectron integrals); βs, βp, βd, βs, ζs, ζp, ζd (to calculate the resonance integrals), and R (to evaluate the core-core repulsions). In the case of an sp basis, there are no parameters for d orbitals (Udd, gdd, βd, ζd), so that typically 11 parameters remain to be optimized. The special conventions concerning Fcore and R (see section 2) lead to slightly higher numbers for Na (sp, 13), Mg (sp, 14), and Al (spd, 16). It should be noted for comparison that PM3 employs 18 adjustable parameters per element and that AM1 employs between 7 and 21 (around 15 on average), even though these methods use only an sp basis. Hence, MNDO/d is less highly parametrized than AM1 or PM3 (in the sp part) and resembles the original MNDO method in this respect. This has been a deliberate choice which should enable us to assess the inherent performance of the MNDO/d model. The parametrizations were carried out in the usual manner.1,8 The reference data consisted of heats of formation, molecular geometries, ionization potentials, and dipole moments for representative sets of ground-state compounds. In most cases it was feasible to use only experimental reference data. For important classes of Na, Mg, Al, and Si compounds, however,

reliable experimental data were not available, so that we admitted theoretical reference data (heats of formation and geometries) from high-level ab initio calculations to cover the full range of bonding situations (see section 4 for details). Generally speaking the selection of reference data was guided by the attempt to include as many different types of molecules as possible, and a balanced overall performance was considered more important than an optimum performance for certain classes of compounds. Concerning properties, most of the emphasis was on heats of formation, while ionization potentials and dipole moments were given relatively little weight. With respect to geometries it was considered essential to reproduce the correct molecular shapes, e.g. for hypervalent compounds, whereas some systematic errors in bond lengths and angles were regarded as less severe and were therefore tolerated to a certain extent. For several elements, one or two of the (formally optimized) one-center parameters were essentially fixed by insisting on correct heats of formation for selected atomic ions (Na+, Mg+, Mg2+, Al+, Cl-, Br-, I-, Zn+, Zn2+, Cd+, Cd2+, Hg+, Hg2+). Each of the elements was parametrized separately, first the nonmetals in the order Cl, Br, I, S, P, Si and then the metals (more or less simultaneously), so that each optimization could include reference molecules with previously parametrized atoms. For each element, many nonlinear least-squares parametrization runs were carried out starting from fairly different initial parameter values. The resulting optimized parameter sets with a low error function were tested in extensive survey calculations in order to choose the set with the most balanced results. Tables 1 and 2 list the final optimized parameters for the elements with an spd and an sp basis, respectively. Previously published parameters8,9 are included in Table 1 for comparison. In addition to the optimized parameters, Tables 1 and 2 contain several derived parameters for convenience (ζ˜ s, ζ˜ p, ζ˜ d, gp2). Generally, the parameters show a regular and fairly smooth variation with atomic number in the sequences Na-Cl, Cl-I, and Zn-Hg. The orbital exponents ζs and ζp are reasonably close to the optimum ab initio exponents20 obtained from an energy minimization of the atomic ground state whereas the semiempirical d exponents ζd are smaller than expected.8 The optimized values of the other parameters generally appear to be of a realistic magnitude. A previously distributed version of our semiempirical computer program21 contained MNDO/d parameters for the elements treated here. These parameters which had been marked as

618 J. Phys. Chem., Vol. 100, No. 2, 1996

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TABLE 2: MNDO/d Parameters for Elements with an sp Basisa Uss (eV) Upp (eV) ζs (au) ζp (au) βs (eV) βp (eV) R (Å-1) ζ˜ s (au) ζ˜ p (au) gss (eV) gpp (eV) gsp (eV) gp2 (eV) hsp (eV) Fcore (au)d

Na

Mg

Zn

Cd

Hg

-5.201 000 -2.712 573 0.987 508 0.893 350 -1.087 382 -0.486 239 1.170 102b 0.654 113 0.564 409 4.594 445 4.299 198 4.147 574 3.796 957 0.534 409 1.530 553

-15.097 000 -10.650 000 1.448 904 0.952 930 -1.895 884 -2.141 089 1.621 470c 1.050 000 0.925 272 7.375 133 7.047 954 6.888 907 6.224 599 0.726 734 1.350 776

-18.023 001 -12.242 166 1.731 503 1.393 583 -5.017 261 -0.712 060 1.517 637 1.566 000 0.862 840 8.560 728 5.139 648 7.490 036 4.505 403 0.532 946 1.589 234

-16.969 700 -12.400 965 1.748 806 1.563 215 -2.771 544 -1.805 650 1.424 613 1.763 148 1.525 519 7.904 434 7.480 000 7.515 707 6.518 664 0.636 744 1.721 186

-18.815 649 -13.397 114 2.333 107 1.708 311 -2.218 722 -2.909 786 1.382 242 2.186 000 1.705 005 8.315 649 7.115 259 8.212 173 6.171 250 0.835 941 1.636 072

a In standard notation. See text. b 1.052 252 for Na-H and Na-C. c 1.350 530 for Mg-H, 1.481 721 for Mg-C and Mg-S. d Optimized only for Na and Mg. See text.

TABLE 3: Heats of Formation (kcal/mol) of Sodium Compounds molecule +

Na Na2 Na2+ NaH NaCH3 NaCN Na(NH3)+ cation NaO Na2O NaOH Na2(OH)2 NaF Na2F2 NaCl Na2Cl2 NaBr Na2Br2 NaI Na2I2 Na(H2O)+ cation Na(H2O)2+ cation Na(H2O)3+ cation Na(H2O)4+ cation mean error mean absolute error no. of comparisons

exp

MNDO/d

ref

145.6 34.0 146.7 29.7 26.3 22.5 105.5 20.8 -8.6 -47.3 -145.2 -69.4 -202.3 -43.4 -135.3 -34.4 -116.2 -19.0 -84.7 63.8 -13.6 -87.2 -158.8

145.6 30.5 163.4 27.1 23.4 27.2 119.2 38.6 -10.8 -41.5 -139.8 -68.5 -221.3 -36.1 -108.4 -29.8 -99.3 -24.9 -92.3 65.7 -13.1 -90.0 -163.5

a a b a c a d e e a a a a a a a a f f d d d d

3.14 7.57 23

a Reference 23. b Reference 24. c Bu ¨ hl, M.; Thiel, W. Unpublished G2 results (1995). d Reference 43. e Steinberg, M.; Schofield, K. J. Chem. Phys. 1991, 94, 3901. f Reference 25.

preliminary are identical to those in Tables 1 and 2, with the exception of Na, Mg, and Al, where recent refinements have been made (see section 2) that are included in the present version of our program. 4. Statistical Evaluations Survey calculations have been carried out for more than 600 compounds in order to assess the performance of MNDO/d in comparison with standard MNDO,1,10 AM1,2,11 and PM3.3,12 For each method, all test molecules have been calculated using the same computer program, the same geometry input, and the same reference data in the statistical evaluation to avoid any technical errors. Tables 3-10 list the results for the heats of formation of compounds containing Na, Mg, Al, P, S, Zn, Cd, and Hg, respectively. The corresponding tables for molecular geometries, ionization potentials, and dipole moments are given as supporting information. Analogous results for Si, Cl, Br, and

I have been published8,9 and are therefore not reproduced again. Tables 11-15 report statistical evaluations for heats of formation, bond lengths, bond angles, ionization potentials, and dipole moments, respectively. These evaluations include all the data given in Tables 3-10, in the supporting information, in Tables II-VII of ref 8, and in Tables 2-5 of ref 9. The statistical results certainly depend on the choice of the test molecules, but we believe that the number and diversity of the test molecules are large enough to provide a fair qualitative assessment. The selection of reliable experimental reference data is often a difficult practical problem that may involve a judgement on the accuracy of published experimental results. To be as unbiased as possible, we have preferentially taken the experimental data from recognized standard compilations for heats of formation,22-26 molecular geometries,27-31 ionization potentials,32 and dipole moments.33-36 In the case of multiple entries, we have usually adopted the following priorities: refs 22 > 23 > 24 > 25 > 26, refs 27 > 28 > 29 > 30 > 31, and refs 33 > 34 > 35 > 36. In addition, experimental reference data have been taken from specialized reviews for heats of formation37-43 and occasionally also from previous semiempirical evaluations.3,10-12 Generally we have attempted to exclude experimental reference data of low or dubious accuracy. In particular, we have normally accepted experimental heats of formation only if the quoted experimental error does not exceed 5 kcal/ mol. High-level ab initio calculations can nowadays provide fairly accurate heats of formation for small molecules. For example, the G2 method44 shows a mean absolute error of only 1.2 kcal/ mol for 125 energies of first-row and second-row molecules, with a maximum error of 5.1 kcal/mol for SO2. Therefore it is clearly legitimate to use theoretical G2 heats of formation as reference data when reliable experimental data are not available. This was done in 35 cases, with G2 values derived from literature data44-47 and from our own calculations. In an additional 22 cases, we have adopted theoretical heats of formation from other high-level ab initio calculations which are believed to be accurate to at least 5 kcal/mol (see Tables 4 and 5 and original references for more detail). Finally, for several important Na, Mg, and Al compounds with unknown experimental structures, we have used reliable ab initio geometries as reference data (normally at least MP2/DZP, see supporting information). Having specified the choice of the reference data, we now discuss the statistical results. Table 11 contains the statistical evaluations for heats of formation. The mean absolute error of MNDO/d amounts to 5.4 kcal/mol for the complete validation

Extension of MNDO to d Orbitals

J. Phys. Chem., Vol. 100, No. 2, 1996 619

TABLE 4: Heats of Formation (kcal/mol) of Magnesium Compounds molecule +

Mg Mg2+ MgH+ MgH MgH2 CH3MgH CH3Mg radical Mg(CH3)2 MgH(CHdCH2) MgH(CCH) Mg(C5H5)2 MgH(CN) Mg(CN)2 MgCN MgNH2 MgH(NH2) Mg(NH2)2 Mg(NH3)+ cation MgO MgH(OH) Mg(OH)2 Mg(OH) radical Mg(OH)+ cation Mg(H2O)+ cation Mg(Acac)2 MgCN(OH) MgF MgHF CH3MgF MgF2 Mg2F4 MgCl MgHCl MgClF MgCl2 Mg2Cl4 MgBr MgBr2 Mg2Br4 MgI MgI2 MgS MgSH radical MgH(SH) Mg(SH)2 Mg(OH)(SH) mean error mean absolute error no. of comparisons

exp 213.1 561.3 220.1 57.1 37.5 28.4 44.7 15.3 57.6 80.3 31.2 51.4 60.5 66.2 31.3 14.8 -6.2 164.3 36.0 -43.5 -136.8 -28.8 146.0 123.1 -273.2 -30.8 -56.6 -70.3 -74.6 -173.0 -410.7 -11.6 -29.9 -138.1 -93.8 -228.1 -8.4 -72.4 -183.5 5.9 -38.3 49.3 22.5 5.0 -26.5 -76.2

MNDO/d 213.1 561.2 239.9 63.8 55.9 34.9 41.8 14.2 50.5 74.3 94.1 43.7 32.8 49.5 31.3 24.8 -4.7 166.2 43.7 -31.5 -112.8 -26.8 152.2 114.2 -264.9 -41.7 -51.4 -54.4 -74.3 -160.3 -424.9 -12.3 -15.2 -122.1 -83.0 -208.7 -4.6 -64.2 -178.1 5.0 -39.9 41.1 18.5 15.7 -22.9 -67.5 4.45 9.61 46

PM3 217.9 555.1 230.8 41.8 49.3 23.8 14.2 2.9 49.6 86.9 31.6 62.8 80.8 54.6 -1.5 10.8 -24.7 144.3 5.5 -42.5 -121.9 -55.0 131.5 112.8 -280.1 -22.5 -71.3 -58.9 -79.1 -160.7 -408.6 -32.9 -21.8 -120.4 -82.4 -232.1 -12.5 -51.4 -162.0 -6.3 -44.6 42.4 14.8 24.4 -2.1 -64.8

TABLE 5: Heats of Formation (kcal/mol) of Aluminum Compounds ref a a b c c b b b b b d b b b b b b e f b g c f e h b g b b g g c b b g g g g g g g c b b b b

-1.81 12.71 46

a Reference 25. b Reference 45. c G2 values derived from the data in ref 44. d Reference 37. e Bauschlicher, C. W.; Sodupe, M.; Partridge, H. J. Chem. Phys. 1992, 96, 4453. f Operti, L.; Tews, E. C.; MacMahon, T. J.; Freiser, B. S. J. Am. Chem. Soc. 1989, 111, 9153. g Reference 23. h Ribeiro, M. In Proceedings of the 11th Summer School on Coordination Chemistry and Catalysis; Ziolkowski, J., Ed.; World Scientific: Singapore, 1988.

set of 575 molecules and is the same for the subsets of 508 normal valent and 67 hypervalent molecules. This balanced treatment of normal valent and hypervalent molecules achieves one of the design objectives of MNDO/d. The original evaluation of the standard MNDO method48 shows a mean absolute error of 6.3 kcal/mol for typical organic molecules containing H, C, N, and O. It would certainly be unrealistic to expect that the MNDO/d error can be pushed much below this value because most of the test molecules contain H, C, N, or O, which are treated in MNDO/d exactly as in the standard MNDO method (see section 2). In this sense we feel that we are at the limit of what can be reached for heats of formation at the MNDO/d level.

molecule Al+ Al2 3Πu AlH AlH2 radical AlH3 Al2H6 Al(CH3)3 Al(C2H5)3 Al(C3H7)3 AlO radical Al2O AlOH AlF AlF3 Al2F6 AlFO AlCl AlCl2 radical AlCl3 Al2Cl6 ClAlO AlF2Cl AlFCl2 AlBr AlBr3 Al2Br6 AlI AlI3 Al2I6 mean error mean absolute error no. of comparisons

exp 218.1 116.4 57.7 63.0 29.1 24.6 -19.4 -39.1 -57.1 16.0 -34.7 -43.0 -63.5 -289.0 -629.5 -139.0 -12.3 -67.0 -139.7 -309.7 -83.2 -238.8 -189.0 3.8 -98.1 -223.9 16.2 -46.2 -117.0

MNDO/d MNDO 218.1 119.9 63.0 51.8 26.9 27.6 -20.5 -40.6 -52.4 3.9 -34.0 -34.5 -65.7 -291.9 -628.4 -126.7 -12.7 -71.4 -149.2 -296.3 -78.6 -243.0 -195.4 4.5 -104.3 -223.1 25.3 -47.3 -126.8 -0.25 4.93 29

193.9 131.3 46.0 50.4 24.2 34.8 -40.1 -55.9 -69.6 -1.8 -71.7 -61.1 -83.6 -291.3 -631.5 -113.6 -27.8 -74.6 -140.3 -295.3 -68.7 -241.6 -191.4 0.7 -60.3 -132.6 31.3 12.0 8.3 6.02 22.07 29

AM1

PM3

ref

213.3 118.7 52.9 45.4 13.6 -15.5 -27.5 -46.1 -65.9 8.4 -39.2 -40.1 -77.9 -285.8 -645.9 -110.1 -20.8 -74.0 -140.3 -318.9 -66.4 -236.2 -186.7 -0.3 -96.4 -248.9 29.0 -29.7 -107.9

279.8 215.3 70.1 10.7 84.5 6.7 -5.7 -34.5 -47.0 9.8 -28.6 -33.4 -50.1 -291.5 -631.4 -124.7 -5.5 -68.2 -122.1 -311.2 -72.4 -234.6 -178.0 12.5 -85.8 -224.9 30.3 -39.9 -117.2

a a b b b c d d d a a a a a a a a a a a a a a a a a a a a

-3.63 10.45 29

10.59 16.44 29

a Reference 23. b G2 values derived from the data in ref 44. c Derived from the heat of formation of AlH3 at 0 K (30.46 kcal/mol, G2) and the dimerization energy of AlH3 (32 kcal/mol, CCSD(T)/ANO) from: Rendell, A. P.; Lee, T. J.; Komornicki, A. Chem. Phys. Lett. 1991, 178, 462. The resulting heat of formation of Al2H6 at 0 K (28.9 kcal/ mol) has been converted to 298 K, which yields 24.6 kcal/mol. d Reference 37.

MNDO/d represents a significant improvement over standard MNDO, AM1, and PM3 for calculating heats of formation (see Table 11). The fairest comparison involves the common subset of 488 molecules consisting of elements which have been parametrized in all these methods (i.e. excluding Na, Mg, and Cd compounds). For this subset the mean absolute error is considerably lower in MNDO/d (4.9 kcal/mol) than in standard MNDO, AM1, and PM3 (29.2, 15.3, and 10.0 kcal/mol, respectively). It is obvious from Table 11 that the methods with an sp basis perform reasonably well for normal valent compounds (although the errors are somewhat larger than in MNDO/d). However, they often give unrealistic heats of formation for hypervalent compounds (much too high). This is true even for PM3, where a determined effort has been made in the parametrization3 to overcome the deficiencies for hypervalent compounds. We conclude that the inclusion of d orbitals is essential for a balanced treatment of normal valent and hypervalent compounds in MNDO-type semiempirical methods. Table 12 lists the statistical results for bond lengths. The mean absolute error of MNDO/d is 0.056 Å (441 comparisons), which is clearly higher than desirable but still somewhat lower than the errors of MNDO, AM1, and PM3 (see data for common subset in Table 12). The deviations of MNDO/d bond lengths from experiment are fairly systematic, however. Na-H, NaC, Mg-H, and Mg-C bonds are calculated much too short, typically by 0.15-0.20 Å. Other M-C bonds are normally reproduced to within 0.03 Å (M ) Al, Si, P, S, Cl, Br, Hg).

620 J. Phys. Chem., Vol. 100, No. 2, 1996

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TABLE 6: Heats of Formation (kcal/mol) of Phosphorus Compounds molecule P2 P4 PH triplet PH2 radical PH3 PH4+ P2H4 HCP H2CdPH P(CH3)H2 P(CH3)3 P(CH3)3H+ P(C6H5)3 PN PNH HPN PNH2 singlet HPNH trans N3P3Cl6 HPO PO radical P4O6 P(OCH3)3 P(OC2H5)3 PF triplet Σ PF2 radical PF3 PCl3 PBr3 PI3 P(C2H5)Cl2 P4O10 (CH3)3PO (C2H5O)3PO (OCN)3PO (C2H5)Cl2PO PF5 F3PO PCl5 Cl3PO Br3PO PS P(SiH3)H2 mean error mean absolute error no. of comparisons

exp 34.3 14.1 56.3 33.1 1.3 178.0 5.0 52.7 28.4 -4.1 -24.2 114.0 76.5 42.8 56.6 75.3 48.1 33.7 -175.9 -20.9 -5.6 -375.5 -168.6 -194.4 -12.5 -114.4 -228.8 -68.4 -33.2 1.1 -66.0 -694.1 -103.8 -282.4 -158.6 -139.7 -381.1 -299.8 -89.6 -133.8 -94.0 42.2 1.8

MNDO/d MNDO 35.4 18.4 57.6 34.2 5.0 181.4 2.4 62.3 31.1 -6.9 -28.1 124.5 67.5 51.4 52.6 71.2 56.0 34.1 -196.7 -34.0 -14.0 -375.9 -178.8 -193.6 -32.7 -140.3 -247.1 -81.4 -25.2 20.4 -69.5 -681.6 -105.5 -273.5 -159.0 -136.3 -376.4 -316.7 -90.7 -141.1 -82.6 32.6 8.9 -1.55 7.62 43

41.1 36.2 55.3 31.4 3.9 210.0 -2.9 42.2 17.2 -14.7 -48.2 136.4 51.6 33.9 31.6 67.1 23.3 21.0 -62.4 -26.7 -21.0 -521.1 -205.2 -221.3 -37.8 -138.0 -229.3 -96.4 -38.1 25.3 -86.0 -431.7 -42.9 -209.5 -78.0 -82.0 -248.8 -199.5 -41.9 -79.6 -28.6 39.7 1.5 15.14 38.72 43

AM1

PM3

ref

24.6 50.4 55.9 34.2 10.2 197.1 6.6 47.4 27.5 -1.2 -22.0 102.5 99.2 32.5 29.3 68.2 11.4 24.3 -200.4 -21.3 -16.5 -321.5 -187.2 -204.4 -36.1 -133.2 -228.9 -89.0 -23.3 24.4 -69.8 -664.5 -101.5 -277.4 -168.4 -115.9 -379.3 -292.7 -71.8 -117.9 -29.0 40.3 15.1

32.0 42.9 54.2 29.3 0.2 117.2 -3.7 46.5 28.1 -9.5 -29.8 108.7 90.1 32.9 31.4 55.2 20.5 22.6 -192.7 -30.0 -19.4 -511.0 -191.8 -205.4 -36.9 -144.4 -252.2 -88.5 -27.3 38.8 -73.6 -712.6 -82.7 -252.9 -158.2 -129.4 -386.9 -297.7 -111.6 -140.2 -79.8 39.2 6.6

a a b b a c d e e e d c d f f f f f g e a d d d a h d d d d d a i j g i a a k a g e d

2.36 14.54 43

-9.26 17.08 43

a Reference 23. b Berkowitz, J.; Curtiss, L. A.; Gibson, S. T.; Greene, J. P.; Hillhouse, G. L.; Pople, J. A. J. Chem. Phys. 1986, 84, 375. c Reference 24. d Reference 38. e G2 values derived from the data in refs 44. f G2 values derived from the data in ref 47. g Hartley, S. B.; Holmes, W. S.; Jacques, J. K.; Mole, M. F.; McCoubrey, J. C. Q. ReV. 1963, 17, 204. h Berkowitz, J.; Greene, J. P.; Foropoulos, J., Jr.; Neskovic, O. M. J. Chem. Phys. 1984, 81, 6116. i Reference 37. j Reference 40. k Reference 25.

Some of the M-H bonds are generally too short (Al-H by 0.07 Å, Si-H by 0.06 Å, P-H by 0.05 Å) whereas some of the M-F bonds are usually too long (Al-F by 0.08 Å, Si-F by 0.09 Å, P-F by 0.08 Å, S-F by 0.07 Å). The deviations for most M-Cl bonds are again below 0.03 Å (M ) Mg, Al, Si, P, S). The errors for the MNDO/d bond lengths in hypervalent compounds are of similar magnitude as in normal valent compounds (see Table 12) and also systematic: Bonds involving hypervalent atoms (P, S, Cl, Br, I) tend to be too long in MNDO/d by 0.04-0.06 Å. When judging these results, it should be kept in mind that one of the bonding partners is usually hydrogen or a first-row element, i.e. an atom which is described in MNDO/d exactly as in MNDO. The resulting bond length will therefore largely be determined by the given and rather inflexible MNDO parametrization for this atom. Hence, the MNDO/d parametrization for the heavier elements cannot

be expected to yield consistently accurate bond lengths, and some systematic errors in MNDO/d bond lengths have to be tolerated (see supporting information for more details). Table 13 summarizes the statistical evaluations for bond angles. The mean absolute error of MNDO/d is 2.5° (243 comparisons), which is again an overall improvement over MNDO, AM1, and PM3. For hypervalent compounds, MNDO/d generally predicts structures that are qualitatively correct (with regard to the molecular shape): e.g. C2V for SF4, ClF3, and BrF3; C4V for ClF5, BrF5, and IF5; or Cs for ClF3O. By contrast, the other three methods often produce qualitatively wrong structures:8 e.g. D3h for ClF3 and ClF5. Consequently, the bond angles in hypervalent compounds show much smaller errors in MNDO/d (2.1°) than in MNDO, AM1, and PM3 (5.6°, 5.1°, and 6.5°, respectively). On the other hand, the bond angles in normal valent compounds are calculated with similar accuracy by MNDO/d, MNDO, and AM1 (see Table 13) whereas the PM3 statistics are much worse due to some qualitative failures for Mg, Al, and Br compounds (see section 5). All four methods show systematic errors for the bond angles involving P(III) and S(II) which tend to be overestimated by 3-6°. This also applies to angles at Si(II) in MNDO/d and AM1. Table 14 reports the statistical data for ionization potentials. The mean absolute error of MNDO/d is 0.45 eV (200 comparisons), slightly lower than in AM1 or PM3 and considerably lower than in MNDO. In the normal valent molecules studied, the first ionization often involves lone-pair electrons. In the case of P, S, and Br, MNDO/d usually underestimates the corresponding ionization potentials (on average by about 0.3 eV) while there is no such clear trend for the lone pairs of Cl and I. For hypervalent P(V) and S(VI) compounds, MNDO/d tends to overestimate the first ionization potential. Concerning the higher ionizations, MNDO/d normally predicts ionization potentials for σ(X-Y) orbitals with X ) P, S, Cl, Br, I which are too low (often by 1 eV or more). In such cases, great care is required when applying MNDO/d orbital energies to the interpretation of photoelectron spectra. Table 15 contains the statistical results for dipole moments. The mean absolute error of MNDO/d is 0.35 D (133 comparisons), which again represents an improvement over MNDO, AM1, and PM3. For normal valent molecules, the MNDO/d error is still lower (0.27 D), which would seem to indicate that the calculated charge distributions are reasonable. In the case of hypervalent molecules, the deviations between observed and computed dipole moments are much higher, for each of the four methods (see Table 15). MNDO/d usually overestimates the dipole moments of hypervalent molecules, which might suggest that it exaggerates their ionicity. The corresponding net atomic charges are indeed fairly large, e.g. for sulfur 1.62 in SF4 and 2.52 in SF6, or for chlorine around 0.8-1.1 in Cl(III) compounds, 1.8-2.2 in Cl(V) compounds, and 2.7-2.8 in Cl(VII) compounds. On the other hand, natural population analysis (NPA) of ab initio wave functions yields charges of comparable magnitude in hypervalent compounds,49,50 e.g. in the case of SF6, where the calculated NPA charge is about 2.9. In fact, ab initio studies have generally stressed the importance of ionic bonding in hypervalent compounds, so that the quoted MNDO/d charges would not appear to be too high. A related question concerns the importance of d orbital participation in hypervalent compounds.49-51 In all molecules studied presently, the total d population of any given atom is below 1.00 in MNDO/d. For typical hypervalent compounds, the total d populations from MNDO/d are 0.54 in SF4, 0.79 in

Extension of MNDO to d Orbitals

J. Phys. Chem., Vol. 100, No. 2, 1996 621

TABLE 7: Heats of Formation (kcal/mol) of Sulfur Compounds molecule

exp

S2 triplet Σ S3 S5 S6 S7 S8 HS- anion HS radical H2S H2S2 HS2 radical H-S-S-S-H HS3 radical H-S-S-S-S-H H-S-S-S-S-S-H CH2S CH3S radical CH3SH C2H5SH trans C2H3S radical C2H3SH 1-propanethiol 2-propanethiol 1-butanethiol cyclohexanethiol HS-C2H4-SH 1,2-propanedithiol HS-C4H8-SH S(CH3)2 CH3-S-S radical CH3-S-S-CH3 CH3-S-S-S-CH3 C2H5-S-C2H5 C2H5-S-S-C2H5 thiirane thiethan 2,5-dihydrothiophene tetrahydrothiophene thiophene 1,3-dithiolan-2-one 1,3-dithiol-2-one 1,3-dithiolane-2-thione 1,3-dithiole-2-thione C6H5S radical C6H5SH (C6H5)2S C6H5-S-S-C6H5 SO triplet Σ SOH radical S(OH)2 COS CS2

30.7 33.9 26.1 24.4 27.2 24.5 -15.0 33.3 -4.9 3.7 22.1 7.3 25.3 10.6 13.8 28.0 34.2 -5.4 -11.1 53.0 21.0 -16.2 -18.2 -21.1 -23.0 -2.3 -7.1 -12.0 -8.9 17.3 -5.8 -3.0 -20.0 -17.9 19.6 14.5 20.8 -8.2 27.5 -30.1 -3.6 22.4 60.5 56.8 26.9 55.3 58.2 1.2 5.0 -67.0 -33.9 27.9

a

MNDO/d MNDO AM1 29.4 43.1 28.3 22.5 22.9 16.9 -6.1 35.9 0.0 1.6 21.7 2.7 26.2 4.3 6.2 34.3 28.7 -4.5 -10.2 49.7 17.5 -13.8 -12.5 -19.4 -21.5 0.5 -2.6 -8.9 -8.5 17.6 -6.5 -5.4 -19.0 -17.3 29.1 4.4 12.9 -14.3 31.2 -25.4 -2.9 29.8 50.9 60.4 26.1 55.6 58.6 3.0 -3.8 -69.9 -11.2 48.3

34.8 73.7 32.1 27.6 28.1 23.3 6.9 37.3 3.8 6.5 29.0 8.4 33.1 10.6 12.9 27.8 22.6 -7.3 -13.4 43.8 15.2 -17.7 -16.2 -23.0 -25.8 -6.4 -10.7 -16.0 -17.1 18.8 -14.8 -13.2 -28.5 -26.8 18.9 -5.1 4.3 -24.1 26.5 -44.5 -19.3 11.4 36.4 54.9 23.4 46.8 48.0 4.2 -13.4 -77.9 -22.9 36.9

24.4 32.2 32.8 35.2 20.6 15.3 -11.9 40.1 1.2 8.6 27.2 7.8 19.1 3.1 8.2 29.9 29.3 -4.3 -10.6 49.1 17.3 -17.4 -15.6 -24.8 -30.4 -3.9 -9.4 -18.2 -9.3 19.0 -4.1 -4.6 -22.0 -16.6 30.7 7.2 12.7 -16.5 27.4 -27.9 -8.9 20.1 37.4 60.1 25.7 53.3 58.8 -11.5 -11.9 -92.1 -29.0 17.5

PM3 ref 28.7 44.0 46.8 48.7 25.9 18.2 -15.9 38.2 -0.9 8.7 27.2 5.7 21.1 -0.3 2.6 37.6 28.7 -5.5 -8.7 54.0 18.4 -14.7 -14.4 -20.7 -21.7 0.1 -4.9 -11.9 -11.0 19.9 -4.7 -6.7 -18.0 -10.9 28.8 7.5 14.8 -10.3 30.7 -21.1 -3.0 40.5 58.6 65.2 27.7 58.9 65.1 -13.6 -14.7 -94.4 -23.7 36.9

a a a a a a b a a c d c d c c e d f f d d f f f f f f f f d f b f f f f f f f f f f f d f f f a d d f f

molecule

exp

CH3-CO-SH CH3-CO-S-C2H5 HNCS CH3NCS CH3SCN NCS-SCN SF2 S2F2 CH3SCl CH3-S-SCl C6H5SCl C6H5-S-SCl SCl2 S2Cl2 SBr2 S2Br2 SO2 SO3 H2SO3 HOSO2 radical H2SO4 (CH3)2SO (C2H5)2SO (C6H5)2SO (CH3O)2SO (CH3O)(C2H5O)SO (C2H5O)2SO (CH3)2SO2 (CH3)(C2H5)SO2 (C2H5)2SO2 (C6H5)2SO2 (CH3O)2SO2 (C2H5O)2SO2 (CH3)2SdS SO2F radical HOSO2F SO2F2 SdSF2 SF4 SOF4 SF5 radical SF5OH SF6 SF5Cl HOSO2Cl SOCl2 SO2Cl2

-41.8 -54.5 30.0 31.3 38.3 83.6 -70.9 -68.4 -6.8 -5.1 25.3 27.0 -4.7 -4.0 5.0 9.0 -70.9 -94.6 -127.0 -98.0 -177.0 -36.2 -49.1 25.5 -115.5 -125.2 -132.0 -89.2 -97.7 -102.6 -28.4 -164.2 -180.8 4.0 -102.3 -180.0 -181.3 -71.0 -182.4 -228.0 -218.1 -289.9 -291.7 -248.3 -133.0 -50.8 -86.2

mean error mean absolute error no. of comparisons

MNDO/d MNDO -36.6 -46.0 53.4 47.6 32.8 80.8 -84.6 -71.1 -9.3 -7.6 22.2 24.6 -9.0 -7.2 11.1 10.7 -46.8 -105.3 -120.8 -102.4 -181.9 -33.6 -44.7 24.3 -112.3 -117.1 -122.7 -85.7 -90.5 -94.5 -32.7 -174.6 -184.9 12.8 -108.7 -186.6 -189.5 -70.4 -179.6 -226.9 -226.0 -281.5 -284.4 -228.2 -129.8 -42.6 -78.9 1.34 5.57 99

-41.3 -56.8 43.4 36.9 23.1 70.5 -52.9 -41.3 -21.2 -18.8 9.1 12.2 -23.8 -20.8 -0.2 1.5 4.5 58.5 -71.2 -0.2 -13.1 4.0 -8.5 66.2 -65.1 -70.5 -74.2 53.7 47.7 42.4 119.4 -5.5 -15.1 32.8 14.0 4.0 22.0 8.0 -48.1 33.6 -6.5 13.0 29.3 54.9 15.6 -22.1 44.6 40.22 48.43 99

AM1

PM3

ref

-39.2 -50.2 27.4 27.6 22.1 64.1 -102.1 -86.8 -17.9 -14.3 13.9 16.8 -26.2 -24.6 -2.1 -4.4 -47.0 -97.3 -150.6 -104.5 -186.3 -39.4 -51.7 22.4 -139.1 -145.2 -148.6 -70.3 -75.7 -80.7 -8.8 -174.7 -186.9 15.2 -110.8 -191.7 -195.8 -80.1 -225.5 -253.9 -278.7 -331.8 -330.7 -249.9 -126.1 -64.3 -69.3

-38.8 -46.5 39.5 36.1 28.3 78.4 -91.9 -73.8 -11.8 -6.6 23.2 28.1 -10.9 -7.7 24.9 21.8 -50.8 -104.7 -140.5 -104.3 -188.1 -38.8 -45.4 31.9 -129.2 -136.2 -136.5 -76.3 -79.0 -81.2 -5.8 -172.1 -181.9 18.7 -102.5 -186.4 -184.3 -56.1 -192.3 -236.3 -230.8 -305.1 -304.5 -243.4 -135.5 -47.6 -80.9

f f d d d f g g d d d d a a d d a a d d d f f f f f f f f f f f f d g d g g g g g g g a d c a

-5.79 10.34 99

0.55 7.50 99

Reference 23. b Reference 24. c Reference 25. d Reference 41. e G2 values derived from the data in ref 44. f Reference 22. g Reference 42.

SF6, 0.42 in ClF3, 0.65 in ClF5, and 0.67 in HClO4. Taking SF6 as an example,49 the Mulliken d population from a minimalbasis-set ab initio calculation (STO-3G*) is 1.63 and thus higher than the MNDO/d value of 0.79. However, moving from a minimal to a double-ζ basis in the sp part reduces the d contributions to the ab initio wave function significantly and leads to a total NPA occupancy of the sulfur d orbitals of about 0.25.49 This indicates that the role of d orbitals is exaggerated in combination with a minimal sp basis set (e.g. in MNDO/d or STO-3G*) because the d orbitals will partly compensate for deficiencies in the sp basis. In a semiempirical context, the parametrization determines the extent of d orbital participation such that the reference data (i.e. mainly experimental heats of formation and geometries) are reproduced in an optimum and balanced manner. We believe that the resulting d populations in MNDO/d (see above) are not excessive but still too high compared with the ab initio data.

Vibrational frequencies may also be used for validation purposes. We have selected more than 100 small molecules from our test set and checked about 400 vibrational modes involving significant motion of the elements currently parametrized (see Tables 1 and 2). Without presenting the numerical results in detail, we find that MNDO/d performs well in these tests. The vibrational frequencies are normally overestimated, on average by about 10%, similar to the case for ab initio SCF calculations. The scale factors which minimize the least-squares deviations between the observed and the calculated MNDO/d frequencies are around 0.9, both for normal valent and hypervalent compounds and also for different types of vibration, with one exception: The low-frequency XMY bending modes (M, element from Table 1 or 2; X, Y, other non-hydrogen elements) require a scale factor of about 1.1. Compared with MNDO/d the older semiempirical methods show somewhat larger mean absolute errors and a higher scatter in the results, with correlation

622 J. Phys. Chem., Vol. 100, No. 2, 1996

Thiel and Voityuk

TABLE 8: Heats of Formation (kcal/mol) of Zinc Compounds molecule Zn+ Zn2+ Zn2 ZnH ZnH+ cation Zn(CH3) radical Zn(CH3)+ cation Zn(CH3)2 Zn(CH3)2+ cation Zn(C2H5)2 Zn(C3H7)2 Zn(C4H9)2 Zn(Acac)2 Zn(H2O)+ cation ZnCl ZnCl2 ZnBr2 ZnI2

exp

MNDO/d MNDO

249.4 249.4 665.1 665.0 57.9 55.7 62.7 65.6 246.3 247.8 46.9 48.2 213.9 227.4 12.7 8.6 221.7 228.1 12.1 0.2 -3.3 -6.9 -12.3 -17.6 -198.2 -197.2 152.6 147.5 6.3 10.7 -63.5 -63.3 -44.4 -35.7 -15.6 0.2

mean error mean absolute error no. of comparisons

1.30 4.89 18

239.6 720.2 62.3 43.2 258.8 28.2 227.7 19.9 246.8 13.0 6.1 -4.7 -169.0 127.3 -13.8 -48.7 2.7 42.7 10.68 21.04 18

TABLE 10: Heats of Formation (kcal/mol) of Mercury Compounds

AM1 244.3 729.5 56.5 51.0 255.1 39.9 224.4 19.8 230.8 14.2 2.6 -11.1 -94.0 159.6 -4.5 -54.6 -63.1 4.7

PM3

ref

235.4 662.7 53.0 56.9 266.2 38.9 228.9 8.2 234.2 5.6 -6.4 -11.5 -117.1 167.3 0.0 -52.8 -21.2 15.7

10.82 16.91 18

a a b c c d d a a a e e f g b a b b

8.53 14.70 18

a Reference 25. b Reference 26. c Simoes, J. A. M.; Beauchamp, J. L. Chem. ReV. 1990, 90, 629. d Georgiadis, R.; Armentrout, P. B. J. Am. Chem. Soc. 1986, 108, 2119. e Reference 37. f Ribeiro, M. In Proceedings of the 11th Summer School on Coordination Chemistry and Catalysis; Ziolkowski, J., Ed.; World Scientific: Singapore, 1988. g Magnera, T. F.; David, D. E.; Michl, J. J. Am. Chem. Soc. 1989, 111, 4100.

TABLE 9: Heats of Formation (kcal/mol) of Cadmium Compounds molecule +

Cd Cd2+ Cd2 CdH doublet CdH+ cation Cd(CH3)+ cation Cd(CH3)2 Cd(CH3)2+ cation Cd(C2H5)2 Cd(Acac)2 CdF doublet CdF2 CdCl doublet CdCl2 CdBr doublet CdBr2 CdI doublet CdI2 mean error mean absolute error no. of comparisons

exp

MNDO/d

PM3

ref

235.7 627.0 51.4 63.2 239.8 213.9 25.3 223.2 25.2 -184.0 -27.3 -94.4 7.2 -46.5 16.4 -33.5 20.2 -14.4

235.8 627.1 51.4 62.2 249.3 223.5 14.7 225.3 6.2 -182.5 -19.8 -88.1 6.3 -51.3 17.3 -30.4 31.2 -1.8

179.4 544.4 53.4 36.8 139.7 188.7 30.6 223.3 25.9 -101.4 24.2 -17.3 5.0 -48.6 4.1 -37.6 -3.5 -23.8

a a b b b a a a a c b b b b b b b b

1.56 5.59 18

-6.94 31.32 18

a Reference 25. b Reference 26. c Ribeiro, M. In Proceedings of the 11th Summer School on Coordination Chemistry and Catalysis; Ziolkowski, J., Ed.; World Scientific: Singapore, 1988.

coefficients of 0.991 (MNDO/d), 0.986 (MNDO), 0.968 (AM1), and 0.930 (PM3) between the observed and calculated vibrational frequencies. Recently the SAM1 semiempirical method has been introduced.13,14 To our knowledge, the SAM1 formalism and parameters have not yet been published. We have used the distributed AMPAC (5.0) software15 to carry out SAM1 and SAM1d calculations for all test molecules where the corresponding parameters are available in AMPAC (5.0) (i.e. Si, P, S, Cl, Br, I). SAM1 employs an sp basis for the halogens and an spd basis for Si, P, and S. The only difference between SAM1 and SAM1d concerns Cl and Br, which are described

molecule Hg+ Hg2+ Hg2 Hg2+ cation HgH radical Hg(CH3)+ cation Hg(CH3) radical Hg(CH3)2 Hg(C2H5)2 Hg(i-C3H7)2 Hg(C3H7)2 Hg(C6H5)2 Hg(CN)2 Hg(Acac)2 HgCl radical HgCl2 Hg(CH3)Cl Hg(C2H5)Cl C3H7HgCl i-C3H7HgCl Hg(C6H5)Cl HgBr2 Hg(CH3)Br Hg(C2H5)Br C3H7HgBr i-C3H7HgBr Hg(C6H5)Br HgClBr HgI radical HgI2 Hg(CH3)I Hg(C2H5)I C3H7HgI i-C3H7HgI Hg(C6H5)I HgClI HgBrI mean error mean absolute error no. of comparisons

exp

MNDO/d MNDO

256.8 256.8 690.8 690.7 26.0 28.4 244.3 249.7 57.0 64.1 222.1 225.5 39.9 36.8 22.3 23.0 18.0 13.7 9.6 8.9 8.5 3.7 93.5 95.4 91.1 88.8 -141.2 -141.2 18.7 14.8 -35.0 -35.4 -12.5 -10.0 -15.0 -14.3 -20.6 -20.0 -20.1 -16.4 24.6 28.0 -20.4 -22.7 -4.4 -3.0 -7.2 -7.4 -12.7 -13.2 -12.6 -9.6 32.8 34.9 -28.0 -29.1 31.9 26.9 -3.9 -2.6 5.2 8.1 3.3 3.7 -1.2 -2.1 -1.1 1.4 42.6 46.1 -19.6 -18.9 -12.1 -12.6 0.53 2.15 37

222.5 679.3 30.2 216.7 37.8 218.8 28.9 10.2 11.8 22.5 6.9 101.6 69.1 -104.2 -6.3 -36.9 -18.0 -16.7 -20.2 -10.9 29.8 3.1 2.8 4.0 0.5 9.7 50.4 -17.0 24.6 21.5 13.7 14.6 13.1 20.2 60.5 -8.8 11.8 3.40 13.69 37

AM1

PM3

ref

225.5 685.4 28.0 220.7 55.9 220.8 38.2 27.4 19.7 15.8 7.8 101.2 84.3 -124.8 -2.7 -44.8 -12.5 -16.4 -23.1 -18.2 26.4 -43.9 -11.0 -14.9 -21.2 -16.8 27.3 -44.8 25.7 19.1 21.2 17.1 10.8 15.0 59.1 -13.9 -12.6

271.5 681.1 25.9 237.4 48.1 251.5 38.0 28.4 15.8 2.7 3.0 98.4 93.1 -176.8 4.9 -32.7 -3.4 -9.8 -18.9 -16.3 32.3 -26.9 -1.1 -7.7 -16.9 -14.5 34.5 -30.5 -32.2 -4.2 9.5 2.9 -5.6 -3.9 45.1 -21.6 -16.0

a a a a b a a a a c c c a d b b a a e e e b a a e e e f b b a a e e e f f

-1.10 8.99 37

-2.33 7.67 37

a Reference 25. b Reference 23. c Reference 37. d Ribeiro, M. In Proceedings of the 11th Summer School on Coordination Chemistry and Catalysis; Ziolkowski, J., Ed.; World Scientific: Singapore, 1988. e Reference 40. f Reference 26.

by an spd basis in SAM1d. Table 16 reports a brief statistical comparison of the results from MNDO/d, SAM1, and SAM1d. In an overall view, MNDO/d and SAM1d appear to be of similar accuracy for molecular geometries and ionization potentials, whereas MNDO/d is superior for heats of formation and dipole moments. The mean absolute errors for heats of formation in MNDO/d, SAM1d, and SAM1 are 5.1, 8.2, and 9.3 kcal/mol, respectively (404 comparisons, see Table 16). The inclusion of d orbitals for Cl and Br in SAM1d leads to considerable improvements over SAM1 (for Cl and Br compounds), which is completely analogous to our previous experience8 with MNDO and MNDO/d. Summarizing the results in this section, MNDO/d performs quite well in the extended test calculations that have been carried out for compounds containing Na, Mg, Al, Si, S, P, Cl, Br, I, Zn, Cd, and Hg. The deviations between theory and experiment are usually of similar magnitude as in MNDO calculations for organic molecules containing H, C, N, and O,48 e.g. for heats of formation, bond angles, ionization potentials, dipole moments, and vibrational frequencies (but not for bond lengths, where the MNDO/d errors are larger). The statistical evaluations show consistent advantages of MNDO/d over other semiempirical methods in calculations on second-row, halogen, and zinc group compounds (see Tables 11-16).

Extension of MNDO to d Orbitals

J. Phys. Chem., Vol. 100, No. 2, 1996 623

TABLE 11: Mean Absolute Errors for Heats of Formation (kcal/mol)a compound

Nb

MNDO/d

Na Mg Al Si P S Cl Br I Zn Cd Hg allc commond normal valente P S Cl Br I hypervalent P S Cl Br I

23 46 29 84 43 99 85 51 42 18 18 37 575 488 508 33 68 70 46 39 67 10 31 15 5 3

7.6 9.6 4.9 6.3 7.6 5.6 3.9 3.4 4.0 4.9 5.6 2.2 5.4 4.9 5.4 7.9 5.0 3.9 3.6 4.1 5.4 6.8 6.9 3.5 1.1 2.1

MNDO

AM1

22.1 12.0 38.7 48.4 39.4 16.2 25.4 21.0

10.4 8.5 14.5 10.3 29.1 15.2 21.7 16.9

13.7 29.2 29.2 11.0 22.1 8.9 4.0 5.4 7.6 143.2 93.4 135.2 204.4 115.9 256.8

9.0 15.3 15.3 8.0 13.6 7.1 4.4 6.3 5.6 61.3 17.8 17.4 144.1 97.7 231.8

PM3 12.7 16.4 6.0 17.1 7.5 10.4 8.1 13.4 14.7 31.3 7.7 10.9 10.0 9.6 18.3 6.5 5.7 5.2 8.5 19.9 13.0 9.7 32.1 34.7 76.2

a

On the basis of the results in Tables 3-10 and in refs 8 and 9. Number of comparisons. c N ) 488 for MNDO and AM1, N ) 552 for PM3. d Common subset without Na, Mg, and Cd compounds. e N ) 421 for MNDO and AM1, N ) 485 for PM3. b

TABLE 12: Mean Absolute Errors for Bond Lengths (Å)a compound

Nb

MNDO/d

Na Mg Al Si P S Cl Br I Zn Cd Hg allc commond hypervalent

16 55 20 68 58 77 63 32 22 10 7 17 441 367 90

0.120 0.120 0.067 0.049 0.048 0.040 0.032 0.032 0.062 0.061 0.073 0.044 0.056 0.047 0.055

MNDO

AM1

0.103 0.070 0.064 0.068 0.062 0.065 0.133 0.047

0.118 0.054 0.073 0.054 0.065 0.051 0.101 0.049

0.071 0.072 0.072 0.082

0.059 0.065 0.065 0.072

PM3 0.118 0.096 0.071 0.057 0.032 0.036 0.042 0.139 0.029 0.092 0.065 0.065 0.056 0.052

a On the basis of the results in the supporting information and in refs 8 and 9. b Number of comparisons. c N ) 367 for MNDO and AM1, N ) 429 for PM3. d Common subset without Na, Mg, and Cd compounds.

5. Specific Results In this section we discuss selected results for certain molecules or classes of molecules from our validation set (section 4) to illustrate strengths and weaknesses of the methods studied and in particular, to comment on problematic cases in MNDO/d. We also present some additional results for further assessment, mostly through comparisons with recent ab initio investigations. The relevant elements are treated consecutively in the following. Sodium. The successive hydration energies of the sodium cation are reproduced well (exp43 -24.0, -19.8, -15.8, -13.8 kcal/mol vs MNDO/d -19.0, -17.9, -15.9, -12.6 kcal/mol, for n ) 1-4) whereas the association energy between Na+ and NH3 is underestimated (exp43 -29.1 kcal/mol vs MNDO/d -20.0 kcal/mol). The dimerization energies of NaOH, NaF, and NaCl are calculated with rather large relative errors (+12%,

TABLE 13: Mean Absolute Errors for Bond Angles (deg)a compound

Nb

MNDO/d

MNDO

AM1

PM3

Na Mg Al Si P S Cl Br I Zn allc subsetd normal valente hypervalent

4 18 7 41 37 55 49 21 9 2 243 214 179 64

5.9 0.3 4.5 1.4 3.7 3.6 2.3 1.6 1.2 3.6 2.5 2.6 2.7 2.1

2.3 2.1 3.5 4.3 4.4 4.2 3.1 5.9 3.7 3.7 2.9 5.6

3.9 1.9 3.3 4.0 4.8 2.2 2.8 2.4 3.4 3.4 2.7 5.1

29.4 35.0 2.0 4.9 4.3 4.2 9.0 8.3 5.9 7.4 4.6 7.7 6.5

a

On the basis of the results in the supporting information and in refs 8 and 9. b Number of comparisons. c N ) 221 for MNDO and AM1, N ) 239 for PM3. d Subset without Na, Mg, and Al compounds. e N ) 157 for MNDO and AM1, N ) 175 for PM3.

TABLE 14: Mean Absolute Errors for Ionization Potentials (eV)a,b compound

Nc

MNDO/d

Na Mg Al Si P S Cl Br I Zn Cd Hg alld commone hypervalent

2 4 8 15 28 39 48 18 17 4 3 14 200 191 29

0.31 0.35 0.54 0.52 0.68 0.57 0.35 0.38 0.24 0.49 0.36 0.25 0.45 0.45 0.91

MNDO

AM1

1.07 0.83 0.93 0.63 0.92 0.99 1.30 0.67

0.81 0.61 0.60 0.50 0.40 0.35 1.05 0.42

0.88 0.89 0.89 0.74

0.29 0.53 0.53 0.84

PM3 0.61 0.75 1.23 0.82 0.46 0.69 0.42 0.29 1.03 0.43 0.59 0.64 0.64 0.85

a On the basis of the results in the supporting information and in refs 8 and 9. b First vertical ionization potential calculated via Koopmans’ theorem. c Number of comparisons. d N ) 191 for MNDO and AM1, N ) 198 for PM3. e Common subset without Na, Mg, and Cd compounds.

TABLE 15: Mean Absolute Errors for Dipole Moments (D)a compound

Nb

MNDO/d

MNDO

AM1

PM3

Na Mg Al Si P S Cl Br I Hg allc commond normal valente hypervalent

5 1 1 14 17 25 33 19 10 8 133 127 117 16

0.31 0.12 0.94 0.43 0.33 0.42 0.33 0.26 0.26 0.42 0.35 0.35 0.27 0.93

1.22 1.01 0.92 0.56 0.37 0.37 0.38 0.28 0.55 0.55 0.48 1.04

0.53 0.31 1.09 0.60 0.36 0.35 0.40 0.30 0.50 0.50 0.38 1.31

1.17 1.76 0.66 0.81 0.66 0.60 0.38 0.44 0.34 0.60 0.60 0.50 1.29

a On the basis of the results in the supporting information and in refs 8 and 9. b Number of comparisons. c N ) 127 for MNDO and AM1, N ) 128 for PM3. d Common subset without Na and Mg compounds. e N ) 111 for MNDO and AM1, N ) 112 for PM3.

+33%, and -28%, respectively), but the shapes of the dimers are qualitatively correct. When compared with those of an ab initio study on small sodium chloride clusters52 (NaCl, Na2Cl, Na2Cl2, Na3Cl3 D3h, Na3Cl3 C2V, Na4Cl4), the MNDO/d bond lengths are too short by typically 0.1 Å while, on average, the bond angles are reproduced to within 3°, the ionization potentials

624 J. Phys. Chem., Vol. 100, No. 2, 1996

Thiel and Voityuk

TABLE 16: Mean Absolute Errors for SAM1 and SAM1da propertyb

compound

Nc

MNDO/d

SAM1

SAM1d

∆Hf

Si P S Cl Br I all normal valent hypervalent Si P S Cl Br I all Si P S Cl Br I all all all

84 43 99 85 51 42 404 337 67 68 58 77 63 32 22 320 41 37 55 49 21 9 212 165 118

6.3 7.6 5.6 3.9 3.4 4.0 5.1 5.1 5.4 0.049 0.048 0.040 0.032 0.032 0.062 0.042 1.4 3.7 3.6 2.3 1.6 1.2 2.6 0.46 0.34

8.0 14.4 8.3 11.1 8.7 6.6 9.3 7.0 21.0 0.051 0.046 0.042 0.056 0.085 0.074 0.054 1.7 3.4 2.2 3.9 4.6 8.3 3.2 0.51 0.47

11.2 15.0 7.9 4.7 5.2 6.6 8.2 7.4 12.0 0.039 0.044 0.042 0.032 0.038 0.073 0.041 1.9 3.3 2.2 2.4 2.0 8.3 2.6 0.44 0.51

R

θ

IP µ

a For compounds of Si, P, S, Cl, Br, and I. MNDO/d results are included for comparison. b Heats of formation ∆Hf (kcal/mol), bond lengths R (Å), bond angles θ (deg), ionization potentials IP (eV), and dipole moments µ (D). c Number of comparisons.

to within 0.2 eV, and the (not very accurate) dipole moments with regard to their qualitative variations. The cohesive energies of these clusters are underestimated by typically 30%. Magnesium. According to the available experimental29 and ab initio45 reference data, divalent magnesium compounds prefer an XMgY angle of 180°. Unconstrained geometry optimizations yield this linear configuration for all relevant test molecules in MNDO/d but not in PM3, where angles between 108° and 157° are found in about half of the test cases (see supporting information). For example, MgF2 and MgCl2 are known to be linear whereas PM3 produces bent C2V structures with angles of 110° and 157°, respectively. This erratic behavior causes the large PM3 errors for Mg in Table 13. The dissociation energy of the sandwich complex magnesocene, Mg(C5H5)2 f Mg + 2C5H5, is given very poorly at the MNDO/d level, since the experimental value37 of 119.8 kcal/ mol is underestimated by 55.3 kcal/mol (by far the worst error in our whole validation set). Ab initio studies indicate53 that the bonding in magnesocene is predominantly ionic, with a quoted Mg charge of 1.12 (Mulliken population analysis), which is considerably higher than the MNDO/d value of 0.88. For further analysis, we have calculated NPA charges for a whole series of Mg compounds at the R(O)HF/6-311+G* level and find that these ab initio NPA charges for Mg are much higher than the MNDO/d charges, typically by a factor of 1.8 (cf. the NPA charge of 1.82 for magnesocene). From this perspective, MNDO/d underestimates the dissociation energy in magnesocene strongly because there are 10 ionic Mg‚‚‚C interactions (Coulomb attractions), which are all too weak due to the underestimated charge separation. This problem is much less severe in divalent compounds such as Mg(CH3)2, since there are only two Mg-C interactions. In the parametrization, such effects can be balanced only in an average sense, and it is therefore not surprising that we could fit the energetics either for magnesocene or for divalent organometallic Mg compounds, the latter choice being made in the parameters adopted. In this context, the question arises why the MNDO/d formalism does

not allow the buildup of large positive charge close to +2 at Mg (cf. magnesocene again, ab initio NPA 1.82 vs MNDO/d 0.88). We believe that this cannot be due to the chosen atomic parameters, which exactly reproduce the experimental heats of formation of Mg, Mg+, and Mg2+. In a molecule, a partial charge of +2 at Mg would imply that the 2s and 2p orbitals remain empty in spite of the inherent tendency of the SCF procedure to make use of available basis functions. In ab initio calculations there is a mechanism to keep these orbitals approximately empty, i.e. interactions with the core electrons or, equivalently, the use of an effective core potential. Such mechanisms are missing in the current MNDO/d formalism, which may cause problems with the partial charges of very electropositive elements (e.g. also Na). We believe that the problems may best be alleviated by introducing explicit effective core potentials in semiempirical methods. Compared with the other elements in Tables 1 and 2, the Mg parametrization relies most heavily on ab initio reference data.45 The scarce experimental data can also be fitted with other previous parameter sets,21 which turned out to be less adequate when applied to bonding situations considered in the ab initio work45 and not covered by the available experimental data. To a lesser extent, this is also true for Na and Al, which explains why the previous parameters21 for Na-Al have now been replaced by the values in Tables 1 and 2. Aluminum. MNDO/d predicts the two lowest states of Al2 (3Πu and 3Σg-) to be almost degenerate (within 1 kcal/mol), in agreement with recent experimental54 and ab initio55 results. The MNDO/d value for the dimerization energy of AlH3 (-26.3 kcal/ mol) is reasonably close to the preferred ab initio value (-32.0 kcal/mol56). In general, however, MNDO/d does not describe the relative energies of bridged nonclassical aluminum hydrides correctly (much too unstable when compared with reliable ab initio data e.g. for Al2H257). Analogous problems with bridged nonclassical species occur e.g. for silicon hydrides (MNDO/d)9 and are also encountered with other semiempirical methods (MNDO, AM1, PM3). MNDO/d reproduces the ab initio Al+ligand binding energies58 quite well whenever they are electrostatic in origin, particularly for water, ammonia, and carbonyl ligands. Similar remarks apply to the Mg+-ligand binding energies.59 Concerning the other semiempirical methods, the performance of AM1 for Al is fairly balanced whereas the behavior of PM3 is more erratic. For example, the energy difference between the 3Πu and 3Σg- states of Al2 has an error of more than 100 kcal/mol in PM3, and the geometry optimizations for AlH2 (C2V) and Al2H6 (D2h) yield structures with H2 moieties. This leads to the large PM3 errors for Al in Table 13. The PM3 problems for Al are probably caused by the choice of the one-center parameters, which favor the occupation of the 2p orbitals (rather than 2s). Silicon. The MNDO/d results for silicon have been discussed in detail.9 In addition to the statistical evaluations, comparisons with high-level ab initio calculations have been carried out (for silicon hydrides, for SiHnO and SiHnO2 compounds, and for proton affinities). These comparisons show that MNDO/d preserves its usual accuracy (see Tables 11-15) when treating a large variety of molecules with unusual bonding situations, whereas the other MNDO-type methods without d orbitals suffer from much larger deviations in such cases.9 Phosphorus. Concerning heats of formation the standard set of test molecules (see Table 6) is relatively small because there is only a limited number of P compounds where reliable experimental data or theoretical G2 data are available. For additional validation we have used another set of 41 normal

Extension of MNDO to d Orbitals valent P compounds60 which covers many types of bonds not included in the standard set (Table 6). Theoretical heats of formation for these compounds have been derived60 from correlated ab initio calculations with large basis sets using MP4 energies of hydrogenation reactions or MP2 energies of isodesmic reactions. These ab initio reference data are reproduced by MNDO/d, MNDO, AM1, and PM3 with mean absolute deviations of 5.3, 12.5, 6.1, and 7.9 kcal/mol, respectively, which is satisfactory and better than expected from the statistical data for normal valent molecules in Table 11. Inversion in tricoordinate P(III) compounds and pseudorotation in pentacoordinate P(V) compounds have received considerable attention. PH3 inverts through a planar trigonal D3h transition state, with a barrier of 37.7 kcal/mol in MNDO/d, which is close to the preferred ab initio value of 34.1 kcal/ mol.61 On the other hand, according to ab initio studies,61,62 the inversion in PF3 proceeds through a planar T-shaped C2V transition state rather than through a D3h structure (barriers61 of 52.9 and 85.3 kcal/mol, respectively). MNDO/d concurs and yields barriers of 56.3 and 65.2 kcal/mol, respectively. Ab initio SCF calculations62 lead to analogous qualitative conclusions for PCl3 and PBr3 while MNDO/d prefers the C2V process for PCl3 and the D3h process for PBr3 and PI3. Regarding the Berry pseudorotation in PX5 (X ) H, F, Cl, Br, I), the available experimental and theoretical ab initio data63 indicate low barriers of 2-4 kcal/mol, which are reproduced very well by MNDO/d (2.5-5.0 kcal/mol). MNDO, AM1, and PM3 predict reasonable values for PF5 and PCl5 but very high unrealistic values for PH5 (AM1 also for PBr5 and PI5). Sulfur. The structures and stabilities of sulfur clusters Sn have been studied by many groups. We have checked the MNDO/d results against ab initio data64 for S3-S8. In the difficult case of the S4 isomers, where specific correlation effects are very important at the ab initio level, MNDO/d gives incorrect stabilities. For the other clusters, MNDO/d predicts the stabilities qualitatively in the right order, with energy differences that are too low compared with the ab initio results. For example, both MNDO/d and the ab initio calculations predict the open C2V isomer of S3 to be the ground state and to lie below the cyclic D3h form, by 4.0 kcal/mol (MNDO/d) and 8.5 kcal/ mol (QCISD(T)/6-31G*), respectively. Likewise, for S6, both approaches predict the same stability order (chair < twist ≈ boat < half-chair), but the energy differences in MNDO/d are only about 60% of those at the MP4/6-31G* level.64 The MNDO/d structures of the Sn clusters are generally correct in a qualitative sense. However, as expected from the statistical evaluations (see section 4), the S-S-S angles tend to be too large by a few degrees, which causes the structures to be slightly too flat. As a second example65 we comment on the gas-phase reaction of H2O and SO3, which proceeds via an intermediate complex H2O‚SO3 and a transition state (TS) to give H2SO4. The energies of H2O‚SO3 and H2SO4 relative to the educts are calculated to be -3.3 and -15.6 kcal/mol by MNDO/d, compared with -7.9 and -20.5 kcal/mol at the highest ab initio level,65 respectively. MNDO/d also predicts a realistic TS structure, since the lengths of the forming and breaking bonds differ by less than 0.05 Å from the corresponding MP2/ 6-31+G* values.65 In the TS structure the migrating hydrogen is placed almost symmetrically between two oxygen atoms (O‚‚‚H‚‚‚O). It is well-known66,67 that the standard MNDO method strongly exaggerates the barriers for such hydrogen transfers involving O‚‚‚H‚‚‚O, and it is therefore not surprising that this is also found in the present case (calculated barriers relative to the educts: 62.8 kcal/mol in MNDO/d and 19.1 kcal/

J. Phys. Chem., Vol. 100, No. 2, 1996 625 mol at the highest ab initio level65). We mention this extreme example to emphasize again that MNDO/d treats hydrogen and the first-row elements exactly as in MNDO and that the MNDO/d parametrization for the heavier elements cannot remove shortcomings from the MNDO treatment of the light elements. Cl, Br, I. The MNDO/d results for the halogens have been discussed in detail,8 particularly with regard to the improvements in the description of hypervalent compounds and their structures, so that further remarks are not necessary. Zn, Cd, Hg. Reliable experimental reference data for these elements are quite limited (especially for Cd, but also for Zn), and accurate theoretical reference data are also normally not yet available. We have tried to remain on the safe side by insisting on correct MNDO/d heats of formation for all atomic cations and dications. The known heats of formation for zinc group compounds are reproduced surprisingly well by MNDO/d (see Tables 8-10). Inorganic and organometallic species are treated with similar accuracy, since the mean absolute errors in MNDO/d heats of formation are slightly below 4 kcal/mol for both classes of compounds. Additional comparisons have been carried out with ab initio M-C dissociation energies for M(CH3) and M(CH3)2 obtained at the CCSD(T)//MP2 level68 and lead to similar agreement (typical MNDO/d deviations around 5 kcal/ mol). 6. Conclusions Using the now available MNDO/d parameters (Tables 1 and 2), extensive test calculations have been carried out for more than 600 molecules containing second-row elements, halogens, and zinc group elements. Comparisons with reliable experimental and ab initio reference data show that MNDO/d represents a significant improvement over other semiempirical MNDO-type methods. For most properties, the MNDO/d predictions for these compounds with heavy elements (Z > 10) are of similar accuracy as the corresponding MNDO predictions for organic compounds (H, C, N, O). There are obvious directions for further work: The MNDO/d parametrization has been completed for some transition metals69 and is in progress for others. MNDO/d parameters are also needed for other important heavier main-group elements. Further gains in performance may be expected from an analogous AM1/d parametrization which is expected to benefit from the more accurate AM1 description of the lighter elements (Z < 10). Acknowledgment. This work was supported by the Schweizerischer Nationalfonds. We thank Dr. Ju¨rgen Breidung and Dr. Michael Bu¨hl for carrying out ab initio calculations to generate reference data. Supporting Information Available: Tables of molecular geometries, ionization potentials, and dipole moments for Na, Mg, Al, P, S, Zn, Cd, and Hg compounds (21 pages). Ordering information is given on any current masthead page. References and Notes (1) Dewar, M. J. S.; Thiel, W. J. Am. Chem. Soc. 1977, 99, 4899. (2) Dewar, M. J. S.; Zoebisch, E. G.; Healy, E.; Stewart, J. J. P. J. Am. Chem. Soc. 1985, 107, 3902. (3) Stewart, J. J. P. J. Comput. Chem. 1989, 10, 209, 221. (4) Dewar, M. J. S. J. Phys. Chem. 1985, 89, 2145. (5) Thiel, W. Tetrahedron 1988, 44, 7393. (6) (a) Stewart, J. J. P. In ReViews in Computational Chemistry; Lipkowitz, K. B., Boyd, D. B., Eds.; VCH: New York, 1990; pp 45-81. (b) Boyd, D. B. In ReViews in Computational Chemistry; Lipkowitz, K. B., Boyd, D. B., Eds.; VCH: New York, 1990; pp 321-354.

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