The quadrupole polarizabilities and nuclear shielding factors of the helium sequence By A. Dalgarno, W. D. Davison* and A. L. Stewart Department of Applied Mathematics and Department of Physics, The Queen's University of Belfast (Communicated by D. R. Bates, F.R.S—Received 27 February 1960) The quadrupole polarizabilities and nuclear shielding factors of the helium sequence are calculated to high accuracy by a combination of variational and perturbation methods. The results suggest that the simpler methods usually employed are satisfactory for computing polarizabilities but less so for computing shielding factors.
116 A. Dalgarno, W. D. Davison and A. L. Stewart gradient at the nucleus of the distortion of the electron distribution (Stemheimer 1950) and is given by the expression Too= (r) ^0) h (r) ^0) (9^i> 0o)}> (6) where ^(r) = 4 S (7) i—1ri or alternatively (Das & Bersohn 1956) by the expression Too = i{(0i> M**) 9^o) (^o> ^ 9^o) (0i> 9»o)}> (8) where (#-^o)0U r) + {^'(r)M9*o>^o)}9Mr) = <>• (9) 3. The variational method It is in general impossible to solve (5) exactly and variational methods must be used. A suitable functional is J = (fa, [. H- E0] fa) + {h - (fa, fa (10) for it yields a value of ccQ which is stationary with respect to first-order variations of fa(r).
The helium sequence 117 we must for consistency take as the effective Hamiltonian /_„ „0 2Z 2Z\ ir — (v!+v |+ - + - ) (15) and as the eigenvalue E°0=-2& . (16) The corresponding solution of (5) is O (^ ■+ f |) r,(oos,0,) ■+ + ||) P2(cos 02)} Si8(r) (17) M II1 so that «; = 30/z8 (18) and r t = 2/3Z. (19) To correct (18) and (19) we observe that the difference between the actual Hamiltonian H and the effective Hamiltonian H' is v(r) = 2/ru (20) and we expand all wave functions (}> and eigenvalues according to ^ = (2i) s—0 s=0 the superscript indicating the order in v(r). Then it can be shown (Dalgamo & Stewart 1958) that the first-order correction to is aj - - #1) + #2) - #0) (A (22) where v = v—(<$, (23) and <1>%{t) is the solution of (H' — Eq) ) + (&(r) — ($>, ^0)} ^2(r) + 4a® $)(r) = 0. (24) Expression (22) avoids the determination of the first-order corrections to $}(r) and 0}(r) to ^2(r).
118 A. Dalgarno, W. D. Davison and A. L. Stewart If we write i}r\ = r\r|P2(cos 0X) P2(cos g(rl2f 9r2), ( ) equation (28) reduces to dg (4Z iZ\ /£ Sr, 0, (30) 3or{+ Scr%+ (\rr5t - * */ )o|r?x +\r(2- - “ dr, \»i + r J g+\Z^ + 3Z + 3Z which has the solution . , 1 2(r,+ r2) 4r,r2 9(ri,r2) - 7^+ ^ + 9^2" (31) From (22), elementary integration yields oc\ = 81-1015625/Z7 (32) 30 81-1015625 and hence + 0(Z~8). (33) aa ^6^" Since the formulae (6) and (8) for involve two external perturbations, some extension of the methods of Dalgarno & Stewart (1958) is necessary in order to obtain an expression for y« which does not involve the first-order corrections and ^i(r). In the following analysis the perturbations r) and r) are not restricted to the forms (4) and (7) and may be left unspecified.
The helium sequence 119 Returning now to the evaluation of the nuclear shielding factor, we note that the function ^5(r) is given by equation (17). Solving (37), we find two possible solutions W(r) 2 + ^ P 2(cos01) + ( ^ + ^ P 2(cos02)}$(r) (43) 3r, and $'(r) “ {(^S + p^coa °i) + (^ i + 3^3) P2 («» ft,)} $(r), (44) of which only the former is admissible in view of the derivation of (42) from (35). The solution of (41) is achieved in a similar manner to that of (24). We write 0u(r) = $>(r) {A(ri) P4(°os 6X) +/1(r2) P4(cos 6»2) +/2(r1) P2(cos dx) +/2(r2) P2(cos d2) +fa(rl)+f&(r*) + tfril(V)} (45) and note that only/3 and \jflx contribute to (42). Then d/3 8Zr2 8r 32 4 + (46) dr2 1 \r dr ^ 45 ^3 15Z^5Z2r ’ 4f3 £r2 2r with solution (47) 3 = 135 + ¥Z ~~ 5Z2 and 8 8 1 (& + 1 (&_! + *!l + r21_ (B '-B 8)(fu$ ) - r3 \4r| 3Z/ r| \Z2 3Z/ 1,372 + T / 8 8 + r\ Is—+ P2(cos dx) P2( cos 02) $(r) = 0. (48) ,3^ 9 If we write fxi = r73r23 P 2(cos 6^) P2(cos 02) r2), (49) (48) reduces to If+sf_ (r,+2Z) (5+2Z) ^ +(f+f ) ^+^ w+^ o o + ^(^i + r|)+|r|rl(rf + rl) + — rfrl(r| + rl) = 0. (50) The solution is Jr) (12ZVf 5 J5(12ZV2 J(r„r,) = ^ j l + + 8Z3rf) + + + 8^i)}. (SI) From (42) by elementary integration , 1 7o {480 In 2-329} (52) 18Z2 and hence y « = ^ {4801n2-329} + 0(Z“3). (53) The series (53) may be compared to the series 2 1 ( 121211 ^ = 3Z + l8Z5{4801n2— 40-} <64> derived by Schwartz (1959).
120 A. Dalgarno, W. D. Davison and A. L. Stewart 5. Discussion If the unperturbed wave function 0o(r) of helium is represented by a product of orbitals u(rx) u(r2) and it is assumed that (1) is satisfied, the perturbed equation (5) can be solved exactly. Taking for u(r) an analytic representation due to Lowdin (1953), Sternheimer (1957,1959) obtained ag = 0-0993A5 and yM = 0-424 and taking for u(r) an analytic representation due to Green, Mulder, Lewis & Woll (1954), Das & Bersohn (1956) obtained ccq = 0-0949 A5 and y^ = 0-416. Of the two re presentations, that of Green et al. more closely reproduces the orbital derived according to the Hartree approximation.