Maráz, Vilmos

English

Maráz Vilmos

University of Szeged

APPROXIMATE FUNCTIONS FOR 4s, 4/7 SLATER-ORBITALS FOR TRANSITION METAL IONS HAVING d" ELECTRON CONFIGURATIONS By V. MARÁZ Institute of Theoretical Physics, Attila József University, Szeged (Received September 1, 1976) In this paper we give approximate functions for 4s, 4p Slater-orbitals for transition metal ions having d" electron configurations. In molecular orbital calculations, especially on transition metal compounds, it is very often necessary to calculate overlap and Coulomb integrals with 4s and 4p Slater-orbitals having the radial part ip{tx) = N{a)r™ e~xr, where N(a) i A2*)8-4 K 7.4! 3.7a0 and Z—a is the effective nuclear charge which can be determined from the well known'Slater-rules. The overlap integrals (nl\n'l') with n—4; n' = 1 ,2,3 can be expressed with the aid of integrals my±*) = G(n\y+x) = ^ -cy±xf+1 ' K " (y+x)"' where r(n) is the complete, (n\y±x)\ is the incomplete r-function [3] and x=ctR, y=a'R (R is the bond length between two atoms) but these expressions make the calculation of these integrals possible only if y>x. If y<x, and that is what takes place in transition metals in most cases, these integrals as well as the integrals {nl\n'l') with n=4; n'=4 ( j = x) cannot be expressed in closed forms. This is the reason why we give approximate functions for 45, 4p Slater-orbitals for transition metal ions making it possible to calculate these integrals with the aid of the integrals A„(a) and Bn(b) [4]. 2 2 6 V. M A R Á Z Constructions of approximate functions The simplest approximation for 4J, 4p Slater-orbitals can be obtained if the radial part of these functions are approximated with the function <p(P) = mP)-iae-», where N(fS) is the normalization factor determined by j qfii*dr=\y and P is an appropriate parameter which can be determined in various ways. If we require that the value of cp(P) at the point of the maximum value of "Ka) ^ m a x — — j be-equal to the maximum value of ij/(a), i.e, < P ( r m a x ) = ^ m a x , we get the following equation for /3: - — e P9e ' = 0.005 804 5019a9, from which ft can be determined for various a's. For transition metal ions having d" electron configurations we get the values for from this equation which are given in Table I. (In the table the numbers in the first column refer to the ionization states of the ions).-Following that we approximated i/*(a) with the function • <p(ot) = JV(a){ar2+fcr,}e^";, and we determined a and b from two requirements. a) First, we prescribed that (p (a) be normalized and the equation < P ( r m a x ) = "Amax be fulfilled. From this condition we get the following expressions for a and b: a = 0.759 826 1 - a - 0 7 , b = 0.460 9024- a+0-3. b) Then we required the normalization condition and the condition CO J (\j/ — (p)2r2dr = min. o VC loi 1 2 3 .4 5 1 2 3 4 5 6 1 2 3 4 5 6 7 í 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 SLATER-ORBITALS FOR TRANSITION METAL IONS 2 2 7 TaWe / 1.430 077 1 1.864 207 6 2.298 338 2 2.732 468 7 3.166 599 3 1.506 688 4 1.940 818 9 2.374 949 5 2.809 080 Ó 3.243 210 6 3.667 341 1 1.539 740 7 2.007 161 6 2.474 582 9 2:942 004 0 3.409 425 1 1.622 226 5 2.089 647 7 2.557 069 1 ' 3.024 490 3 3.491 911 6 3.959 332 2 1.583 299 6 2.017 430 2 2.451 560 7 2.885 691 3 3.319 821 8 3.753 952 4 4.188 082 9 1.704 712 7 2.172 133 8 2.639 555 1 3.106 976 2 3.574 397 4 4.041 818 8 4.509 240 1 1.659 2.094 2.528 2.962 3.396 3.830 4.264 4.698 9109 041 5 172 0 302 6 ' 433 1 563 6 694 2 824 7 1.736 2.170 2.604 3.038 3.473 3.907 4.341 4.775 5.209 522 2 652 7 783 3 913 8 0444 174 9-305 5 436 0 566 6 1.787 2.254 2.722 3.189 3.656 4.124 4.591 5.059 198 8 6201 041 1 462 6' 883 4 304 7 726 3 146 8 1.869 2.337 2.804 3.271 3.739 4.206 4.674 5.141 5.609 684 9 106 8 527 3 948 4 369 6 791 0 212 3 633 3 055 1 1.379 1.813 2.247 2.681 3.115 3.549 3.983 4.417 4.852 5.286 5.720 002 9 133 5 264 0 394 6 525 1 655 6 786 2 9167 047 3 177 8 308 4 1.484 1.952 2.419 2.887 3.354 3.821 4.289 4.756 5.224 5.691 6.158 749 8 171 0 592 2 0134 434 7 855 5 276 8 698 4 1198 5401 962 4 2 2 8 V. M A R Á Z Table I Cu, Ag, Au Zn, Cd, Hg I o n 1 2 3 4 5 6 7 8 9 10 11 2 3 4 5 6 7 8 9 10 11 1 2 1.889 2.323 2.758 3.192 3.626 4.060 4.494 4.928 5.362 5.796 6.231 744 7 875 3 005 8 1364 266 9 397 5 528 0 658 6 789 1 9197 050 2 2.400 2.834 3.268 3.702 4.137 4.571 5.005 5.439 5.873 6.307 6.741 486 6 617 1 747 6 878 2 088 7 1393 269 8 400 4 530 9 661 5 792 0 P 2.034 657 1 2.502 078 2 2.969 499 4 3.436 920 5 3.904 341 7 4.371 763 4 4.839 184 4 5.306 605 7 5.774 026 4 6.241 447 9 6.708 869 4 2.584 3.051 3.519 3.986 4.454 4.921 5.389 5.856 6.323 6.791 7.258 564 4 985 6 406 9 827 8 248 9 670 8 091 4 5129 933 5 354 6 776 5 These conditions give the following results for a and b: a — 0.710 569 7 - a - 0 ' , fo = 0.473 713 1 . a0-3 . In the course of testing the approximations, it has turned out (see Table IV) that the first requirements give better approximate functions than the second ones, so we determined a and b for transition metals from the first expressions. These values are given in Table II. Better approximate functions can be written in the form <2>(a) = N(a){c1r + c2r2 + c3r3 + ciri}e~xr. a) If we prescribe the normalization condition and the equations • Z K ' m a x ) = ^ m a x = ^ m a x i (1 •5rmax) = (1.5rmax), •we get the following expressions for the ck's: Cl = -0.450 026 5-a"1-7 , c2 = 0.870 987 7-a-°-7 , c3 = 0.505 035 3-a0-3, c4 = - 0 . 0 0 8 730 304 2-a 1 3 . OC on 1 2 3 4 5 1 2 3 4 5 6 1 2 3 4 5 6 7 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 10 SLATER-ORBITALS FOR TRANSITION METAL IONS 2 2 9 Table- Il 0.591 5107 0.491 324 3 0.424 350 3 0.375 946 1 0.339 077 6 0.570 292 7 0.477 666 4 0.414 721 1 0.368 739 2 0.333 450 7 0.305 379 9 0.550 0.464 0.405 0.361 0.328 0.301 0.278 833 1 895 4 606 0 859 0 045 4 003 9 806 9 0.532 911 3 0.452 923 2 0.396 962 6 0.355 282 4 0.322 848 0 0.296 777 1 0.275 291 4 0.257 230 1 0.516 0.441 0.388 0.348 0.317' 0.292 0.271 0.254 0.239 342 6 673 2 753 3 988 7 846 2 691 6 881 7 334 4 305 7 0.513 1175 0.555 593.7 0.591 606 8 0.623 125 5 0.651 308 8 0.521 214 0 0.562 347 2 0.597 455 2 0.628 316 2 0.655 996 5 0.681 191 4 0.529 0.568 0.603 0.633 0.660 0:685 0.708 027 1 9165 172 9 408 6 607 4 4182 2940 0.536 0.575 0.608 0.638 0.665 0.689 0.712 0.733 579 9 3135 766 8 407 3 144 4 585 0 156 4 171 8 0.543 0.581 0.614 0.643 0.669 0.693 0.715 0.736 0.756 892 6 548 7 243 4 3162 6103 693 9 970 5 737 7 222 3 0.606 0.500 0.431 0.380 0.342 0.313 0 .288 0.268 0.251 0.236 0.224 762 3 971 9 078 5 944 5' 959 1 028 5 7400 572 8 5166 872 7 140 6 0.507 Ö.550 0.587 0.619 0.648 0.674 0.697 0.719 0.740 0.759 0.777 549 7 982 7 631 7 608 3 139 3 007 7 746 7 737 7 263 8 541 5 740 7 2 3 0 V. MARÁZ Table II Cu, Ag, Au Zn. Cd, Hg 1 2 3 4 5 6 7 8 9 10 11 2 3 4 5 6 7 8 9 10 11 12 0.486 0.421 0.373 0.337 0.308 0.284 0.265 0.248 0.234 0.222 0.211 667 2 080 o 506 0 1764 3844 9156 359 9 773 5 498 9 063 0 1167 0.411 0.366 0.331 0.303 0.281 0.262 0.246 0.232 0.220 0.209 0.199 627 8 4107 625 0 904 2 211 9 238 9 101 9 182 0 031 4 3179 789 0 0.557 0.593 0.624 0.652 0.678 0.701 0.723 0.743 0.762 0.780 0.797 866 1 571 2 866 9 880 1 339 2 745 4 459 5 751 1 827 3 851 0 953 1 0.599 0.630 0.657 0.682 0.705 0.727 0.747 0.766 0.783 0 . 8 0 0 0.817 375 2 024 3 541 9 607 1 691 6 137 1 200 7 080 3 932 7 883 8 036 8 (Here the distance r=1.5rmax is the coordinate of the point where the value of \jj(a) is about the half of the maximum value). b) The conditions J &r*dr=\\ f {4/-<PYr2dr = min. 0 0 give the following results: Ci = -0.241 228 4-a-1-7 Cg - 0.703 008 5-a-°-7 C3 = 0.546 784 4-a0-3, . II -0 .012 017 239-a1" Table IV shows that with the latter parameters we get somewhat (if only a little) better approximate functions, so we give the values of ck-s in Table III accord-ing to these expressions. FUNCTIONS FOR 4j, 4p SLATER-ORBITALS FOR TRANSITION METAL IONS 231 Table III Ion Cj • 102 c2 ft C4 • lo2 V, Nb, Ta 1 -13.13159 0.547 279 1 0.608 728 8 -1.9132508 2 - 8.367 353 7 0.454 584 4 0.659 119 8 -2.700 519 3 - 5.861714 8 0.392 618 5 0.701 843 4 -3.545 216 5 4 - 4.368 019 5 0.347 833 9 0.739 235 2 -4.439 4221 5 - 3.399 539 5 0.313 7223 0.772 669 9 -5 .377 441 7 Cr, Mo, W 1 —12.016 79 0.527 647 7 0 618 333 9 -2.047 552 8 2 - 7.813 648 2 0.441 947 8 0.667 131 7 -2.845 674 3 3, - 5.543 907 2 Ö.383 709 4 0.708 781 5 -3.699 605 4 - 4.167 440 6 0.341 165 9 0.745 392 9 -4.601 908 3 5 . - 3.264154 0.308 516 2 0.778 231 3 -5.547 1824 6 2.636457 2. 0.282 544 4 0.808 120 6 -6.531 285 9 Mn, Tc, Re 1 -11.045 14 0.509 643 5 0.627 603 2 -2.183 9207 2 - 7.315 953 1 , 0.430131 8 0.674 925 2 -2 .992 559 3 3 - 5.252 618 9 0.375 275 9 0.715 654 6 -3 .855 494 6 . 4 - 3.981 104 8 0.334 800 1 0.751434 3 -4.765 7306 5 - 3.137 1355 . 0.303 515 0.783 701 1 -5.718 1275 6 - 2.545 643 0.278 495 6 0.813 135 -6.708 725 5 7 - 2.113 500 9 0.257 958 5 0.840 273 5 -7 .734 362 8 Fe, Ru, Os 1 . -10.192 58 0.493 661 4 0.636 562 9 -2.322 280 7 2 - ' 6.866 784 6 0.419 054 8 0.682 5141 -3.141 1261 3- - 4.984 908 0.367 278 8 0.722 200 9 -4.012 853 4 - 3.807 662 0.328 715 3 0.757 364 4 -4.930 862 7 5 - 3.017 791 2 0.298 7Ó6 3 0.789 083 6 -5.890 263 6 - 2.459 698 3 0.274 584 9 0.818 078 3 -6.887 254 9 7 - 2.049 363 4 0.254 705 9 0.844 855 5 • -7.918 792 8 - 1.737 987 6 0.237 9951 0.869 786 8 -8.982 3642 Co, Rh, Ir 1 -10.524 01 0.532 590 9 0.719 332 4 - 2.745 355 3 2 - 7.201 687 5 0.455 571 7 0.769 1351 - 3.669 286 1 3 • - 5.282 337 2 0.400986 6 0.812 375 8 - 4.650 687 7 4 - 4.064 588 5 0.359 9707 0.850 826 6 - 5.682 612 5 5 — '3.239 1441 0.327 848 1 0.885 601 9 - 6.759 856 5 6 ' - 2.651 374 0 0.301 9020 0.917 454 2 - 7.878 360 5 7 ' - 2.216 579 2 - 0.280437 3 0.946 916 5 - 9.034 843 5 8 - 1.885 018 8 0.262 337 8 0.974 382 4 -10.226 60 9 - 0.016 258 3 0.246 836 2 1.000152 0 -11.45134 Ni, Pd, Pt 0 -.15.573 17. 0.625 855 9 0.671 266 7 - 2.034 458 7 1 - 9.779 287 3 0.516 736 5 0.728 709 6 - 2.903 841 3 2 - 6.789 313 6 0.444 643 6 0.777 180 1 - 3.838 525 0 3 - 5.028 339 6 0.392 932 i 0.819 471 4 - 4.829 288 3 4 - 3.896141 4 0.353 751 3 0.857 205 3 - 5.869 548 6 5 - 3.121 197 6 0.322 878 9 0.891418 1 - 6.954 345 6 6 - 2.565 278 9 0.297 826 1 0.922 814 3 - 8.079 769 0 7 ' - 2.151 632 3 0.277 024 3 0.951 898 8 - 9.242 658 5 8 - 1.834 701 0 0.259 431 4 0.979 046 0 -10.440 39 9 - 1.585 976 8 0.244 3264 1.004 541 9 -11.670 74 10 - 1.386 835 0 0.231 193 9 1.028 611 5 -12.93181 2 3 2 V. MARÁZ Table III Ion c, • 10s c2 c4 • 10* Cu, Ag, Au 1 9.1149125 0.501 9816 0.737 813 3 - 3.064 349 0 2 — 6.4132185 0.434 331 2 0.785 035 7 - 4.009 505 7 3 — 4.793 205 3 0.385 259 5 0.826 426 2 - 5.009 424 7 4 — 3.7385172 0.347 786 7 0.863 475 4 - 6.057 8709 5 — 3.009 928 9 0.318 088 6 0.897 146 6 - 7.150094 6 6 — 2.483 540 8 0.293 881 3 0.928 102 7 - 8.282 342 6 7 — 2.089 656 4 0.273 710 3 0.956 821 2 - 9.451 559 7 8 — 1.785 483 4 0.256 601 9 0.983 658 0 -10.655 20 9 — 1.547 653 1 0.241 878 1 1.008 887 7 . -11.89110 10 — 1.355 821 3 0.229 050 8 1.032 724 9 -13.15741 11 — 1.1991782 0.217 759 5 1.055 343 7 -14.452 53 Zn, Cd, Hg 2 _ 6.069166 4 0.424 580 9 0.792 711 7 - 4.182184 2 3 — 4.575 065 8 0.377 9409 0.833 247 2 - 5.191069 8 4 — 3.590 785 6 0.342 0606 0.869 641 0 - 6.247 553 7 5 — 2.904 830 6 0.313 4674 0.902 791 3 - 7.347 0906 6 — 2.405 8630 0.290 061 1 0.933 321 9 - 8.486 067 7 7 — 2.030468 5 0.270 491 0 0.961 685 0 - 9.661 529 9 8 — 1.560 995 5 0.227 699 1 0.886429 8 - 9.751 253 4 9 - 1.355 163 3 0.214 8200 0.908 827 1 -10.864 78 10 — 1.189 3225 0.203 578 1. 0.930006 2 -12.005 32 11 — 1.053 542 7 0.193 665 7 0.950116 0 -13.171 45 12 — 0.940 828 1 0.184 849 3 0.969 278 8 -14.36193 Table IV Overlaps «1 (45, 2s) 0 . 6 9 1 5 7 2 0 . 5 6 9 5 3 2 • 0 . 4 4 8 9 4 6 0 . 3 3 9 3 7 5 0 . 2 4 6 8 4 2 {'P(ß), 2s) 0 . 6 8 5 6 0 8 0 . 5 6 3 2 2 5 0 . 4 4 2 3 7 1 0 . 3 3 2 7 8 3 . 0 . 2 4 0 5 9 9 (*.(«). 2S) 0 . 6 9 4 8 1 7 0 . 5 7 2 6 3 0 0 . 4 5 3 0 0 4 0 . 3 4 2 3 8 9 0 . 2 4 9 6 9 1 Ob(«), 2s) 0 . 6 9 5 7 7 9 0 . 5 7 3 9 2 8 0 . 4 5 3 4 5 2 0 . 3 4 3 8Ö4 . 0 . 2 5 0 9 4 5 № , , 2 s) 0 . 6 9 1 3 8 0 0 . 5 6 9 521 0 . 4 4 9 0 2 9 0 . 3 3 9 4 9 0 0 . 2 4 6 9 4 7 « P „ , 2 s) Ó.691 4 7 0 0 . 5 6 9 398 0 . 4 4 8 7 6 7 0 . 3 3 9 1 5 5 0 . 2 4 6 5 8 9 (4s , 2p z) 0 . 5 0 9 3 0 3 0 . 5 1 9 7 0 0 0 4 7 5 8 8 3 0 . 4 0 0 2 9 0 . 0 . 3 1 4 7 0 0 (Hß), 2Pl) 0 . 5 1 1 5 9 0 0 . 5 2 2 873 0 4 7 7 8 6 3 0 . 4 0 0 0 8 2 0 . 3 0 2 2 2 4 0Pa(<*), 2 p z ) 0 . 5 0 9 9 5 5 0 . 5 1 9 396 0 . 4 7 5 123 0 . 4 0 0 3 6 4 0 . 3 1 5 6 9 2 O b ( a ) , 2 p z ) 0 . 5 0 7 5 1 4 0 . 5 1 8 183 0 . 4 7 4 9 8 9 . 0 . 4 0 0 9 5 6 0 . 3 1 6 6 3 0 « P . , 2 P = ) 0 . 5 0 8 2 3 0 0 . 5 1 9 6 5 1 0 . 4 7 5 6 9 1 . 0 . 4 0 0 3 4 0 0 . 3 1 4 8 5 6 ( < P „ , 2 p . ) 0 . 5 0 9 3 0 6 0 . 5 2 0 401 0 . 4 7 6 109 0 . 4 0 0 5 1 9 0 . 3 1 4 8 4 8 ( 4 i , 3d2 %) 0 . 2 4 3 2 4 3 0 . 3 3 4 104 0 . 3 8 9 5 4 7 0 . 4 0 1 6 3 8 . 0 . 3 7 6 4 8 4 (<p(ßl K 2 ) 0 . 2 4 2 7 8 8 0 . 3 3 4 573 0 . 3 9 3 8 7 7 0 . 4 0 6 0 6 1 0 . 3 7 9 5 4 6 («».(«), 3dzz) 0 . 2 4 6 5 1 8 0 . 3 3 4 9 7 0 0 . 3 8 8 6 9 2 0 . 4 0 0 1 3 0 0 . 3 7 5 2 6 8 (<pb(a), 3d„z) 0 . 2 4 3 4 1 2 0 . 3 3 2 0 7 0 0 . 3 8 6 5 4 4 0 . 3 9 8 9 2 2 0 . 3 7 4 9 0 0 ( < P . , 3 rf2*) 0 . 2 4 2 0 0 5 0 . 3 3 2 9 1 6 0 . 3 8 8 7 1 6 0 . 4 0 1 155 0 . 3 7 6 2 9 0 ( * b , H O 0 . 2 4 3 2 9 2 0 . 3 3 4 0 3 9 ' 0 . 3 8 9 6 2 6 0 . 4 0 1 8 5 0 0 . 3 7 6 7 7 0 FUNCTIONS FOR 4s, Ap SLATER-ORBITALS FOR TRANSITION METAL IONS 2 3 3 Test of approximations In order to test the various approximations we have calculated the overlap integrals [1, 2] (4s, 2s); (4s,2pz); (4s,3dz*) exactly and with the approximate functions with SLATER-exponents a=2.5; a'= 1.5 and with the bond lengths: Rj, - 1.734 358 6 Ä, R2 = 2.167 948 3Ä, i?3 = 2.601 538 0Ä, R4 = 3.035 127 6 A, R5 = 3.468 717 3Ä. These bond lengths have been so chosen that in the integrals [3] (/>=3.7): / ( » i . p ) - - 1 ^ ! ) -Hply-x), ! • n^h, p) = Hp'y+x), the parameter should have the following values: . ux = 0.8; 1; 1.2; 1.4; 1.6 and u2 should be: u2=4u±. With these parameters we get the values for the over-laps summarized in Table IV. This table shows that ' the function cp(ß) in itself is already a good enough appoximation for 4s and 4p Slater-orbitals. Considering that these orbitals are also approximate-functions for the atomic orbitals we can use q>(ß) instead of Slater-orbitals in molecular orbital calculations. References [1] Gilde, F. J.: Acta Phys. et Cbem, Szeged 7, 3 (1961). [2] Maräz, V.: To be published in Acta Phys. et Chem. Szeged 22. [3] Pearson, K.: Tables of the Incomplete/^-function, University College, London, 1922. [4] Preuss, H.: Integraltafeln zur Quantenchemie, Springer, Berlin, I. Bd. 1956; II. Bd. 1957. ПРИБЛИЖЕНИЕ ФУНКЦИЙ 4Í, 4р ОРБИТАЛЕЙ СЛЕТЕРА ДЛЯ ИОНОВ ПЕРЕХОДНЫХ МЕТАЛЛОВ ИМЕЮЩИХ ЭЛЕКТРОННУЮ КОНФИГУРАЦИЮ dn В. Мараз В работе мы задаем приближение функции 4з,Лр орбиталей Слетера для ионов переходных металлов имеющих электронную конфигурацию dn. 

Approximate functions for 4s, 4p slater-orbitals for transition metal ions having dn electron configurations

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Approximate functions for 4s, 4p slater-orbitals for transition metal ions having dn electron configurations

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