Third-order diamagnetic susceptibilities of hydrogenlike atoms

Regular approach for generating van der WaalsCscoefficients to arbitrary orders

“…As s and k get larger it is noted that the convergence rate slows. What happens for a (6) 0,22 and a (8) 0,26 is that calculations with two or three basis functions lead to significant underestimations, with the underestimation being more prominent the smaller the value of N. The convergence pattern here is that the dispersion parameters increase while the first basis functions are being added, with about five pseudo-states being required before 1% accuracy is attained. Thereafter, convergence is more rapid with the accuracy then improving by a factor of 10 as each additional basis function is added.…”

“…1.243 990 835 836 2 10 3.285 828 414 967 3 12 1.214 860 208 969 5 1.241 587 803 846 3 14 6.060 772 689 192 6 3.009 633 559 003 5 16 3.937 506 393 999 8 4.745 455 287 408 7 18 3.234 218 715 848 10 7.009 061 179 013 9 1.228 997 100 211 6 20 3.278 573 440 415 12 1.083 922 187 504 12 1.267 151 498 881 9 22 4.021 082 847 680 14 1.832 218 340 959 14 7.003 975 407 187 11 24 5.868 996 334 529 16 3.444 924 740 16 2.944 024 287 593 14 3.481 462 326 9 26 1.005 294 993 316 19 7.249 736 601 18 1.114 254 753 17 1.053 141 411 12 28 1.996 944 940 747 21 1.709 242 841 21 4.120 005 317 19 1.536 284 745 16 30 4.553 288 865 668 23 4.507 033 23 1.553 198 22 1.498 025 19 2.3039 13 Odd orders s k = 3 k = 5 k = 7 k = 9 11 3.474 898 037 882 3 13 3.269 869 240 441 5 15 2.839 558 063 300 7 17 2.726 099 889 088 9 1.318 176 405 179 7 19 3.020 900 832 283 11 4.992 376 160 968 9 21 3.900 228 016 834 13 1...…”

“…The strategy for doing this is best exhibited by reference to a specific example. The sum rule for any of the eighth-order terms, a (8) s , with 14 sums over intermediate states can be written as P l 1 l 1 ,l 2 l 2 ,...,l 7 l 7 ,l 8 l 8 0;…”

“…The a (8) p;24+2λ matrix element is a (8) p;24+2λ = l 1 ...l 8 l 1 ...l 8 J 2 ...J 6 J 2 ...J 6 k 2 ...k 6 Q J 2 J 3 J 4 ,J 2 J 3 J 4 ,k 2 k 3 k 4 l 1 l 2 l 3 l 4 ,l 1 l 2 l 3 l 4 × Q J 6 J 5 J 4 ,J 6 J 5 J 4 ,k 6 k 5 k 4 l 8 l 7 l 6 l 5 ,l 8 l 7 l 6 l 5 P l 1 l 1 ,l 2 l 2 ,l 3 l 3 ,l 4 l 4 ,l 5 l 5 ,l 6 l 6 ,l 7 l 7 ,l 8 l 8 p;J 1 J 1 ,J The radial matrix elements are P l 1 l 1 ,l 2 l 2 ,l 3 l 3 ,l 4 l 4 ,l 5 l 5 ,l 6 l 6 ,l 7 l 7 ,l 8 l 8 0;J 1 J 1 ,J 2 J 2 ,J 3 J 3 ,J 4 J 4 ,J 5 J 5 ,J 6 J 6 J 7 J 7 = ρ l 1 l 1 ,l 2 l 2 ,l 3 l 3 ,l 4 l 4 ,l 5 l 5 ,l 6 l 6 ,l 7 l 7 ,l 8 l 8 J 1 J 1 ,J 2 J 2 ,J 3 J 3 ,J 4 J 4 ,J 5 J 5 ,J 6 J 6 J 7 J 7 (A.9) and P l 1 l 1 ,l 2 l 2 ,l 3 l 3 ,l 4 l 4 ,l 5 l 5 ,l 6 l 6 ,l 7 l 7 ,l 8 l 8 1;J 1 J 1 ,J 2 J 2 ,J 3 J 3 ,J 4 J 4 ,J 5 J 5 ,J 6 J 6 J 7 J 7 = 2ρ l 1 l 1 ,l 2 l 2 ,l 3 l 3 ,l 4 l 4 ,l 5 l 5 ,l 6 l 6 ,l 7 l 7 ,l 8 l 8 (J 1 J 1 ) 2 ,J 2 J 2 ,J 3 J 3 ,J 4 J 4 ,J 5 J 5 ,J 6 J 6 J 7 J 7 + 2ρ l 1 l 1 ,l 2 l 2 ,l 3 l 3 ,l 4 l 4 ,l 5 l 5 ,l 6 l 6 ,l 7 l 7 ,l 8 l 8 J 1 J 1 ,(J 2 J 2 ) 2 ,J 3 J 3 ,J 4 J 4 ,J 5 J 5 ,J 6 J 6 J 7 J 7 + 2ρ l 1 l 1 ,l 2 l 2 ,l 3 l 3 ,l 4 l 4 ,l 5 l 5 ,l 6 l 6 ,l 7 l 7 ,l 8 l 8 J 1 J 1 ,J 2 J 2 ,(J 3 J 3 ) 2 ,J 4 J 4 ,J 5 J 5 ,J 6 J 6 J 7 J 7 + ρ l 1 l 1 ,l 2 l 2 ,l 3 l 3 ,l 4 l 4 ,l 5 l 5 ,l 6 l 6 ,l 7 l 7 ,l 8 l 8…”

“…The culmination of this process was the development of an ansatz for deriving corrections to arbitrarily high orders for radiation transition probabilities between arbitrary parabolic states [6]. A similar program has been accomplished for the hydrogen-like diamagnetic states in a magnetic field on the orbital momentum basis of states, both for the energy spectrum [8] and for the radiation transition probabilities [9].…”