摘要:

316 J. Chem. SOC. (A), 1967 Molecular-orbital Theory of Organometallic Compounds. Part VIII.l A Comparison of the 4d and 5d Radial Wave Functions of the Second and Third Transition Series By David A. Brown and Noel J. Fitzpatrick, Department of Chemistry, University College, Belfield, Dublin 4, Ireland A comparison is made of the various radial wave functions for the 4d and 5d atomic orbitals of the second and third transition series. Selected outer and inner properties such as Slater-Condon integrals and cusp values are compared. It is concluded from comparisons of the overlap integrals S(2p,ndU) and S(2pand,) between the metal ndorbital and carbon 2p orbital, that it is most advisable in calculations on organometallic complexes of these elements to employ the new analytic functions of Basch and Gray.Previous qualitative discussions in terms of Slater functions are likely to be incorrect even in the prediction of trends. IN Part VII of this Series, a comparative study of the various radial functions available for the 3d wave func- tions of the first transition series was made and it was concluded that even for this series it is essential, in calculations of the electronic structures of organometallic systems, to use the multi-parameter type function of either Richardson or Watson when the central metal atom is varied. The problem of the choice of central metal wave functions is more acute for the second and third transition series and, consequently, very few cal- culations of electronic structures have been made for organometallic complexes of these elements.To date, only Slater orbitals* have been used, but it was never claimed that they are even approximate functions for the heavy elements, so such calculations are, at best, very qualitative. However, the Burns single-parameter func- tions 5 which are a marked improvement over the Slater functions for the first series may also be applied to the lower series but to the best of our knowledge have not been used in any such calculations. Recently, a new set of approximate analytical functions have been given by Basch and Gray for second- and third-row transition metals. These were obtained as orthonormal linear combinations of Slater-type orbitals by fitting to the published Herman-Skillman Hartree-Fock-Slater numerical wave functions.' Accordingly, we decided to extend the comparative study of Part VII to the above functions, that is, Slater, Burns, Basch and Gray, for the 4d and 5d radial functions of the second and third transi- tion series.Unlike the first series we have no really accurate set of functions available for the lower series but there is available for Mo(I) an extremely accurate eight-parameter analytic function due to Roothaan and Synek8 and we have used this as a standard for the second series. As in Part VII, we compare the functions with respect to (a) radial behaviour, (b) K ( n d ) , the sum of the kinetic energy and nuclear potential energy, (c) the Slater-Condon repulsion integrals F2(nd,nd), and (d) the cusp value. These latter pro- perties were chosen for comparison since they were shown in Part VII to contrast the outer and inner be- Part VII, D.A. Brown and N. J. Fitzpatrick, J . Chem. SOC. J. W. Richardson, W. C. Nieuwpoort, R. R. Powell, and J. C . Slater, Phys. Rev., 1930, 36, 57. (A), 1966, 941. W. F. Edgell, J . Chem. Phys., 1962, 36, 1057. * R. E. Watson, Phys. Rev., 1960, 119, 1934. haviour of the wave function. Finally, since our main interest in these functions is in their application to the calculation of the electronic structures of organometallic systems, a comparison of the following overlap integrals, S(2$,4d,), S(2$,4d$, S(2$,5d,), and S(2$,5d,) was made. FIGURE 1 Plot of R(r) against Y (in atomic units) for Mof (4d5). G = Basch and Gray function, R = Roothaan and Synek function, B = Burns function, and S = Slater function Figure 1 shows a plot of R(Y) versus Y for the above types of radial function for the molybdenum(1) ion (4d5).It can be seen immediately that of the nodeless single parameter functions, the Slater function is much too extended but that the Burns function gives good agree- ment with the accurate Roothaan function at distances greater than about 1.4 a.u. which is as expected since the Burns functions are obtained by a comparison of expect- ation values which gives better agreement in the outer regions. Neither of the nodeless functions attempts an accurate description of the inner part of the wave func- tion. In general, the Basch and Gray function agrees well with the accurate Roothaan function especially near the origin and at distances greater than 2 a.u., but in the region 0.4-1-4 a.u.there is some difference which we shall note reflected in other properties below. The value of K(nd), the sum of the kinetic and nuclear potential energies was shown in Part VII to give some indication of inner region behaviour and so values of K(4d) and K(5d) were calculated for the above functions 5 G. Burns, J . Chem. Phys., 1964, 39, 1521. H. Basch and H. B. Gray, Theor. Chim. Acta, 1966, 4, 367. F. Herman and S. Skillman, " Atomic Structure Calcul- C. C. J . Roothaan and M. Synek, Phys. Rev., 1964, 133 ations," Prentice-Hall, New Jersey, 1963. A1263.Inorg. Phys. Theor. 317 and are given in Figure 2. series : the Burns functions. As for the first transition screening constant to the 4f electrons in this series for K(nd) = T + Vn (1) The Slater-Condon repulsion integrals, P(n1,nl) : where where the general expression for the analytic wave functions of Roothaan and Synek9 and Basch and Gray take the form, where cti are the orbital exponents of a set of Slater type orbitals of integral mi values.In the case of the second and third series, the expression for T is complicated due to terms in mi - 3 not cancelling as discussed in the Appendix. The kinetic energy was also calculated by an independent method (see Appendix) as a check. These expressions were also used to calculate K(3d) for the 3d Slater function of iron (3d64s2) and gave agree- ment with the results of Part VII. As observed pre- viously,l there is a linear relationship between K(nd) and 2, the atomic number of the element.There is reasonable agreement between the Burns and the Basch and Gray functions for the second transition series with the latter function giving slightly better agreement with the value of K(4d) obtained from the accurate Roothaan and Synek function for Mo(I). This conclusion is where r< and r, are the lesser and greater of rl and r2 and R, and R, are the radial functions, were shown in Part VII to be more outer properties of the radial func- tion than K(nd); we have calculated F2(4d,4d) and d" FIGURE 3 Fa(nd, a d ) . G(2) etc., as in Figure 2 supporied by the behaviour di the radical functions F2(5d,5d) for the above functions and the results are given in Figure 3. These integrals were evaluated by two different methods as described in the Appendix for the single-parameter Burns function.However, use of the formula recently derived by Ros and Schuit (in which a factorial sign is omitted) proved very lengthy for the multi-parameter functions so only one method was employed to find F2(4d,4d) for the Roothaan function for MO(I). This value agreed very closely with an unpublished value due to Watson lo which gives us added confidence in our calculated values. The values of F2(4d,4d) calculated from both the Burns and Basch and Gray functions are in good agreement; the Burns func- tions reproduces more closely the accurate value for Mo(I\ due to both the Roothaan and Svnek and the :::: - 4 0 '-------".I:: n - 3 4 5 6 7 8 9 10 d" - ~ O ~ ( ~ ~ " region. However, for the third transition series the Burns functions are again inaccurate, presumably due as discussed above for K(5d)* sllown in ~i~~~~ 1.H ~ ~ ~ ~ ~ ~ , there is a marked dis- Burns and the Basch and Gray functions for the third transition series in monovalent states with the latter This difference also occurs for the F2 integrals and un- doubtedly arises because of the assignment of a unit crepancy between the values of K ( 5 4 obtained from the to the excessive screening assigned to the 4f As an indication of the inner properties of the radial giving a value of K(54 almost twice that of the former. functions, the cusp values, defined by the left-hand side p. Ros and G. c. A. Schuit, Theor. Ckim. Acts, 1966, 4, 10 R. E. Watson, unpublished results.318 J. Chem. SOC. (A), 1967 of the expression below, were compared for the above where pt = 5, 6 = 1 which is equivalent to n = 4, functions.However, it was also found necessary to evaluate For an exact Hartree-Fock function : 6 = 0. where In the case of the Roothaan function for Mo(I) this condition is satisfied exactly and, as shown in the Appen- dix, the value for the Basch and Gray function is simply -3q where m, = 3 and all other values of mi are greater than 3; however, for the Burns function since all mi are greater than 3 the limit does not exist. The results for the Basch and Gray 4d and 5d functions are given in Table 1. It follows that these functions are not very TABLE 1 Cusp condition for the Basch and Gray functions Zr ........................ 40 -26.193 0-655 Xb ..................... 41 - 37.204 0.664 Mo .....................42 -28.212 0.672 Tc ........................ 43 -29.212 0.679 Ru ..................... 41 - 30.216 0.687 Rh ..................... 45 -31.212 0.694 - 32.205 0-700 Pa 46 Hf ..................... 72 - 57.837 0.803 Ta ..................... 73 -58.812 0.806 w ........................ 74 - 59.787 0.808 Re ..................... 75 - 60.765 0.810 0 s ........................ 76 -61.740 0.812 Ir ........................ 77 - 62.715 0.814 Pt ........................ 78 - 63.690 0.817 Au ..................... 79 - 64.668 0.819 Ti * ..................... 22 - 20.343 0-925 Fe * ..................... 26 - 23.090 0.888 cu ........................ 29 -25.183 0.868 * Watson functions and 3dn-2 4s2 configurations. Element z cusp - Cusp/% ..................... Ag .....................47 - 33.195 0.706 suitable for calculations of inner properties such as spin- orbit coupling constants although the cusp test is very stringent. It is interesting to note that in contrast to the Watson functions for the first transition series, the Basch and Gray functions give improved cusp behaviour as 2, the atomic number, increases. In view of these differences between the single- and multi-parameter analytic functions for the second and third transition elements it is important to compare their respective overlap integrals with a carbon 2p orbital over an internuclear range embracing that found in known organometallic compounds of these elements. The overlap integral between a carbon 2p Slater orbital and the Burns or Basch and Gray 4d functions for the second transition series can be expressed in terms of overlap integrals between simple Slater orbitals of the form S(2p55d,) and S(2@,5d,) since these employ orbitals of the type: &(Y) = Nnrn - 1 - %-ar constituent overlap integrals t where n = 5, 6 = 0 and the formulz for these are given in the Appendix.Values of S(2pumiu> and S(2$,ndT) for both the Burns and the Basch and Gray functions for elements of the second 1.7 2.1 0 2.5 2.9 3.3 A- FIGURE 4 Plot of S(2p,4d5) against internuclear distance (in Angstroms) using various functions for Mo+(4d6). B using Burns function, R using Roothaan and Synek function, G using Basch and Gray function, and S using Slater function 1.7 2.1 ?.5 2.9 3.3 A- FIGURE 5 Plot of S(2p,ndu) against internuclear distance. W(B) using Burns function for W, W(G) using Basch and Gray function for W, W(S) using Slater function for W.and Cr(R) using Richardson function for Cr (n = 4) and third ( r , = 5) transition series were calculated by interpolation using the above calculated constituent tables for internuclear distances of 2.2 and 2.4 A. These integrals were also computed directly by special programmes and good agreement noted. The range of values of distance was then extended from 1.7 to 3-3 A and all the integrals computed directly. Overlap integrals based on the Roothaan and Synek function for Mo(I) 4d5 were calculated as a basis for comparison with the above functions. Figures 4, 5, 6, t Tables of these and other constituent overlap integrals will l1 C.C. J. Roothaan, L. M. Sachs, and A. W. Weiss, Rev. be published separately. ( J . Chem. Phys., in the press.) Mod. Phys., 1960, 52, 186.Inorg . Phys . Theor . . . I 2 0 and 7 show values of S(2$,ndu) and S(29.n.d.). n = 4 and 5 . for all these functions and. for comparison. the values of S(2$.3d. ) and S(2$.3+. ) based on the Richardson function for Cr(1). 3d5. are included for the same range . . ~~~~ W(8, CAR1 w(G) I I 1 J A- FIGURE 6 Plot of S(2pAdr) against internuclear distance . B etc . as in Figure 4 of internuclear distance . The trends observed in Figures 4 . 5. 6. and 7 are reproduced by all elements of 1.7 Zr ........................... 0.1339 Nb ........................... 0-1486 MO ........................... 0.1715 TC ...........................0.1786 Ru ........................... 0.1824 Rh ........................... 0-1754 Pa ........................... 0.1702 Ag ........................... 0.1612 Hf ........................... 0.1260 Ta ........................... 0.1517 W ........................... 0.1630 KC ........................... 0.1786 0 s ........................... 0.1825 Ir ........................... 0.1832 Pt ........................... 0.1811 Au ........................... 0.1756 Zr ........................... Nb ........................... Mo ........................... Tc ........................... Ku ........................... Rh ........................... Pa ........................... Ag ........................... Hf ........................... Ta ...........................w ........................... Re ........................... 0 s ........................... Ir ........................... Pt ........................... .4 u ........................... 0-2884 0.2577 0-231 1 0.2039 0.1811 0.1626 0.1408 0.1293 0.2877 0.2710 0.2486 0.2280 0.2077 0.1884 0.1706 0.1558 the second and third transition series but only the values for Cr(1). Mo(I). and W(I) are given in the figures as an illustration of the general trend in these series . How- ever. for reference. the values of S(2j5u4du). S(29.4d.). S(2$.5dU). and S(2$.5d. ) using the Basch and Gray functions. for all monovalent members of the second TABLE 2 Overlap integrals using Basch and Gray functions 1.9 0.1608 0.1681 0.1817 0.1799 0.1756 0.1645 0-1530 0.1430 0.1529 0.1758 0.1802 0.1880 0.1850 0.1793 0.1717 0.1626 0.2309 0.2007 0.1743 0.1494 0.1291 0.1142 0.0963 0.0878 0.2317 0.2123 0.1908 0.1702 0.1520 0.1352 0.1202 0.1083 S (2po4du) 2.1 2.3 0.1706 0.1676 0.1705 0.1610 0.1738 0.1548 0.1645 0.1406 0.1542 0.1266 0.1412 0.1138 0.1264 0.0983 0.1 168 0*0900 S(2po6dU) 0.1647 0.1645 0.1809 0.1721 0.1786 0.1642 0.1782 0.1568 0.1697 0.1449 0.1695 0.1324 0.1485 0.1201 0.1375 0.1095 S (2pAdv) 0.1814 0.1400 0.1525 0.1194 0.1281 0.0921 0.1066 0.0743 0.0894 0.0605 0.0780 0.0520 0.0640 0.0415 0.057!) 0.0373 S(2n5dn) 0.1833 0.1424 0.1624 0.1215 0.1428 0.1045 0.1236 0.0875 0.1081 0.0749 0.0942 0-0640 0.0822 0.0548 0.0731 0.0481 2.5 0.1557 0.1441 0.1308 0.1140 0.0989 0.0872 0.0729 0.0661 0.1554 0.1544 0.1425 0.1304 0.1171 0.1042 0.0923 0.0828 0-1063 0.0867 0.0649 0.0507 0.0400 0.0339 0.0264 0.0236 0.1087 0.0891 0.0748 0.0606 0.0507 0.0425 0.0357 0.0310 2.7 0.1385 0.1246 0.1058 0.0887 0.0741 0.0643 0.0520 0.0468 0.1402 0.1320 0.1182 0.1036 0.0905 0.0786 0-0680 0.0601 0.0795 0.063 1 0.0448 0.0340 0.0260 0.02 17 0.0165 0.0146 0.081 7 0.0641 0.0525 0.04 1 1 0.0337 0.0276 0.0228 0.0195 2.9 0.1189 0.1028 0.0827 0.0667 0.0537 0.0459 0.0360 0.9322 0.1217 0.1086 0.0943 0.0793 0.0675 0-0572 0.0485 0.0422 0-0586 0.0451 0.0306 0.0224 0.0166 0.0137 0-0101 0.0089 0.0604 0.0453 0.0362 0.0274 0.0219 0.0177 0.0143 0.0121 3.1 0.099 1 0.0815 0-0627 0.0487 0-0379 0.03 18 0.0243 0.02 16 0.1022 0.0864 0.0728 0.0587 0.0488 0.0404 0-0335 0.0288 0.0426 0.0317 0.0205 0.0145 0.0104 0.0085 0-0061 0.0054 0.0440 0.0315 0.0246 0.0179 0.0141 0*0111 0.0088 0-0074 3.3 0.0804 0-0648 0.0464 0.0347 0.0260 0.02 16 0.0 160 0.0141 0.0834 0.0668 0.0547 0.0423 0.0343 0.0278 0.0226 0.0192 0.0306 0.0222 0.0136 0.0093 0-0065 0-0052 0.0037 0.0032 0.0316 0.0216 0.0164 0.01 15 0.0089 0.0069 0.0054 0.0045J. Chem. SOC. (L4), 1967 and third transition series are given in Table 2 for the internuclear range of 1.7-3.3 A in intervals of 0.2 k. In the case of S(2pu4du), there is reasonable agreement between the Burns and the Basch and Gray functions although in the chemically important regions in which the internuclear distance is less than 2.3 k the Burns values are too large. The Basch and Gray values lie close to those of the accurate Roothaan function over the whole range and this suggests that at least for the second transition series they are quite accurate.In the case of S(2pO5d,) the Burns values are much larger than those based on the Basch and Gray functions and, as noted above, these functions are not suitable for the third transition series. Figures 6 and 7 show that in the case of the x type overlap integrals there is little to choose between the Burns and the Basch and Gray functions as both agree well with the value based on the accurate Roothaan function, except that again the former value of S(2p,5dn) is much too large. In all cases, the Slater functions grossly overestimate the S(2$,nd,) integrals. The values of S(2pumi,) based on Slater functions are again quite different from those based on the Basch and Gray functions with the maxima of the former lying at much greater internuclear dis- tances (245-2-9 A compared with 1.9 A).I t follows that Slater functions are quite unsuitable for calcul- ations on complexes of the second and third row transi- tion series. It is also interesting that the maximum value of S(2pUndu) moves to shorter internuclear distances as 2 increases and so, other things being equal one might expect shorter bonds for the later members of the period. Chemical ApfiZications.-Although few detailed cal- culations of the electronic structures of compounds of the second and third transition series have appeared, the overlap integrals (based on the accurate Basch and Gray functions) given. in this Paper permit a general comment on work to date. Taking a typical internuclear distance of 2.1 A for the metal-carbon distance in organo- metallic compounds as in Part 1’11, it is instructive to compare the various overlap integrals based on Slater, Richardson, and Basch and Gray radial d functions as shown in Table 3.From these values, it is obvious TABLE 3 Comparison of overlap integrals Element Cr (1) Mo (1) W(I) Overlap integral S(2p,ndU) 0-149 (R.) * 0.174 (B. and G.) * 0.179 (B. and G.) 0.228 (S.) * 0.144 ( S . ) 0.097 (S.) S(2pnndn) 0.148 (R.) 0.128 (B. and G.) 0.143 (B. and G.) 0.194 (S.) 0-307 ( S . ) 0.328 ( S . ) * R. = Richardson, B. and G. = Basch and Gray, S. = Slater. that the Slater functions do not give even the correct trend in overlap in the sequence Cr(I)-Mo(I)-W(I) and so even qualitative conclusions based on these functions when applied to members of different transition series S.F. A. Kettle and R. Mason, J . Organometallic Chem., 1966, 5, 673. are very doubtful indeed. This important conclusion casts doubt on some recent qualitative discussions l2 of bonding in organometallic complexes of the heavier elements. We conclude then that even in comparative studies of compounds of the second and third transition series it is essential to use multi-parameter analytic functions for the d orbitals. APPENEIX (a) K(nd). Using the expression for T given previously 1 one finds For the first transition series this expression is simplified as (mi - 3) and (mj - 3) equal zero. Alternatively from l3 mi(mi - l)(mi + mj - 2) ! 6(mi + mj - 2 ) ! - (ai + aj)mi+mj-l + (ai + q.mi+?ni-l Using these two expressions for T one gets K(nd) in two ways. where a = Cai For the first transition series this simplifies to the equation (given previously).2 Alternatively, one can use Ros and Schuit’s equation^.^ (c) cusp.Inorg. Phys. Theor. .. * If mi = 3 and all other mi’s > 3 [(l/fA)(dfAld~)J,=O = -El .*. ((1 + l)(l/f~)(df~ldr)l,=0 = --a, for the Basch and Gray functions. 32 1 S(2PAdn) = (3/4)(5)’~Vz, 5 x [(B4 - B 6 ) ( A 0 + + (B3 - B1)(A5 + f(B5 + B 7 ) ( A 1 - - 2(A1B3 - x (BO + B 2 ) ( A 6 - f 2 ( A 4 B 6 - B,A6)i where 3’2>5 = ( 1440-’(42)-1p8( 1 + T ) ~ / ~ ( 1 - T ) ~ ~ / ~ Wre thank Professor H. B. Gray for a copy of the manu- script of reference 6 prior to publication and Professor R. E. Watson for unpublished results of some of the F2 integrals. [6/998 Received, August 8% 1966l l3 J . C . Slater, “ Quantum Theory of Atomic Structure,” McGraw-Hill, New York, 1960, vol. I .

ISSN:0022-4944    

DOI:10.1039/J19670000316

出版商:RSC    

年代:1967

数据来源: RSC