Polarizability and Hyperpolarizability of the Helium Atom BY A. D. BUCKINGHAM AND P. G. HIBBARD School of Chemistry The University of Bristol Received 18th October 1968 The zeroth fist and second order perturbed Schrodinger equations for the helium atom in an external electric field have been solved to high accuracy through the variation principle. A ground- state energy differing by only about 1 part in lo9 from Pekeris's extrapolated figure was obtained with a wave function do containing 181 adjustable parameters (compared to Pekeris's 1078). The first-order wave function d1 and hence the second-order energy and polarizability of the atom was obtained with up to 84 adjustable parameters in dl. The polarizability a is 1.38319 a.u. = 0.204956 x cm3 = 0.228044~ C2m2 J -'. The dipole shielding factor differs from its exact value of unity by a few parts in lo5.The second-order wave function d2 and hence the fourth-order energy and hyperpolarizabilityy were obtained for various wave functions do and dl and with up to 106 adjustable parameters in d2. Smooth convergence was obtained yielding y = 43.10 a.u. = 2-171 x C4m4J-3. However an extension to d1 may be needed before an accurate value ofy can be computed. Accurate values have also been obtained for the %values of the unperturbed atom and for the quadrupole and octopole polarizabilities. e.s.u. = 2.688 x Polarizabilities and hyperpolarizabilities describe the changes in the charge distributions of atoms and molecules due to external electric fields. Polarizabilities have been of interest for a long time and their calculation has been reviewed by Dalgarno hyperpolarizabilities describing nonlinear distortions by strong fields are of special importance because of their relationship to " non-linear optics ".2 If an atom is in a uniform static electric field F its energy E and dipole moment YM may be written as power series in Fz E = Eo+E2Fz+E4F:+ ...m = - (aE/aF,) = aF + &F + . . . where the polarizability a = - 2E and the " second hyperpolarizability " y = - 24&. The differential polarizability n = am/dF is x = a + 3 ' y F 2 + ... (3) The Kerr effect is due to anisotropy in E viz. Zzz-EXx = ~(yzZ,,,-y,,,,,)F,2 ; for static fields this reduces to 3yF2 but dispersion may be significant. Measurements of the Kerr constant of helium have been r e p ~ r t e d ~ . ~ and yield y = 2.7+0.2 x e.s.u.= 3.3 f 0 - 2 x 10-63C4 J-3 m for an optical wave-length of 632.8 nm. Various attempts have been made to calculate y for helium; 4 9 6-8 the present work was undertaken in the hope of providing a definitive non-relativistic quantum-mechanical description of a helium atom in a strong static electric field. The total electric field at a stationary nucleus is zero ; thus the field due to the other charges in a molecule in a field precisely compensates F,. If this total field at the nucleus FiN) = F,( 1 - ol) then o1 is the " dipole shielding factor " and is unity in a neutral atom. 41 42 HYPERPOLARIZABILITY OF THE HELIUM ATOM Similarly if the atom is in a potential V which is symmetric about the z-axis the multipole moment of order I (i.e. the expectation value of xeirfPl (cos 0,)) is in the linear approximation where a is the appropriate polarizability* and Fzzz .. . = - (8 V/azz). The corres- ponding shielding factor ol is defined by and is the expectation value of zZ!eir,('+ l)PI(cos Bi)/Fzzz . . . Shielding factors are also equal to minus the ratio of the Zth order moment that would be induced in the molecule by a nuclear moment of order Z to the nuclear moment itself. i (4) r ( 0 s z z z . . . = UlFzzz. . . 3 Fir;. . . = Fzzz . . .(I - 0,) ( 5 ) I THE PERTURBATION EQUATIONS Schrodinger's equation for a helium atom in an electric field Fz is ( 2 0 +XlF,)Y = EY where is the Hamiltonian for the free atom and the perturbation Since F is a variable it is convenient to write the ground-state wavefunction Y as a power series in Fz %IFz = [r,Pl(cos 01) + ~ ~ P ~ ( c o s '82)]Fz.(7) Y = Y o + Y l F z + Y 2 F ~ + Y 3 F ~ + ... The identity (6) and eqn. (1) and (8) lead to the equations Eo = (Yo I *o I ' u o > / ( ~ o I 'yo> E2 = ( Y o I X l I Yl>/(YO I y o > E.4 = [(Wl I 2 1 I ' u 2 > -E2((Yo I ' u 2 > +(Yl I ~ l > ) I / ( ' Y O I y o > * (9) (10) (1 1) No analytic solution to the zeroth-order equation #oYo = EoYo exists and it is usual to obtain an approximation 4o to Yo through the variational condition A suitable 4o may be constructed from a power series in the three interparticle distances rl r2 and r12 with a factor exp [ - +k(rl + r2)] the best set of coefficients for any particular expansion being chosen by minimizing c0. The introduction of a scaling parameter k for all interparticle distances and minimization of c0 with respect to k ensures that the virial theorem is ~atisfied.~ Schwartz lo has demonstrated that functions of r l r2 and r12 are particularly appropriate.Expansions involving only positive powers of distances (Hylleraas-type expansions) give rather slow convergence to Eo when high accuracy is sought ; the rate of convergence is governed then by the ability of 4o to approximate Yo in the regions of the singularities in the potential (rl = 0 r2 = 0 rI2 = 0). Kinoshita l 1 has shown that the inclusion of negative powers of the distances (subject to restrictions l 2 placed on all satisfactory $o) improves 4o near these singularities. E o e o = <40 I 2 0 I40>/(+0 1 4 0 ) . (12) * The polarizability ccz is I ! times that defined in ref.(l). A . D. BUCKINGHAM A N D P .G . HIBBARD 43 The perturbed wave function is obtained in a similar way expansions in powers of r l r2 and r12 having the appropriate angular symmetry being used. The computa- tion of an approximate ith order wave function 4i is simplified if it is written as 40& and the variation theorem used to find the coefficients in 4;. This product form for 4i is valid provided 4o is nodeless. While Yo is nodeless there could be " accidental " nodes in q50 but as the expansion length of 4o increases these should occur in ener- getically less important regions and convergence to EZi should be satisfactory. Whereas E~ obtained from eqn. (12) is an upper bound to Eo the value of c l i would only necessarily be if all 4 j ( j < i ) were exact. The variational condition is that E< E for all F,.In practice it was found that E and E~ converge from above as 40 $1 and 4 are made more accurate. THE UNPERTURBED WAVEFUNCTION A N D ENERGY OF HELIUM Apart from the wavefunctions of Pekeris,13 no sets are available for the purpose of obtaining smooth convergence of properties. However Pekeris's coordinate system has not been found convenient for the atom in a field. Preliminary studies of published wavefunctions showed that considerable variations in computed properties exist between wavefunctions of similar length and energy; some of these variations are due to insufficient minimization. In this work new sets of wavefunctions for the helium atom in a field have been calculated (following Pekeris,13 using Kinoshita- type Since a and y depend on the long-range behaviour of Y no attempt was made to obtain the best possible c0 for any particular expansion length but rather to choose the functions so that E and E~ were gradually improved.expansions) and the convergence of atomic properties studied. The functions used may be written 40(n) = exp (-is)[ f cpQrsPtqu' + Kinoshita-type terms where s = k(r + r 2 ) t = k( -rl +rz) u = kr, and all n>p,q,r>O (because the spatial part of the singlet ground state wavefunction must p + q + r = O ( 4 even) -1 with q even be symmetric with respect to electron exchange) are included in the sum ; the Kinoshita-type func- tions are 20 of the most important terms with negative powers of s introduced by Kinoshita 1 1 in his 80 parameter 40. All 40(n) for JZ = 2-10 (27-181 adjustable parameters in all) were prepared using the fast direct minimization procedure of Fletcher and Powell,14 with double length arithmetic throughout.Full convergence for all +o was not achieved accumulation of round-off errors causing premature termination of the iterations for y1 = 5 and 8. The other $o(yl) probably have energies less than a.u. from the true minimum for the particular expansion. The energies and k values for these wavefunctions are in table 1 with the best four results of Pekeris l 3 (who used no negative exponents). The energies for our comparatively simple wavefunctions compare well with the accurate results though there is evidence from the rate of convergence that the number of Kinoshita-type terms is becoming disproportionately small. Values for are in table 2. Our &values are more accurate and converge more strongly than Pekeris's because of the inclusion of Kinoshita-type terms.44 HYPERPOLARIZABILITY OF THE HELIUM ATOM The original and re-minimized energies and functions for the six parameter 4o of Hylleraas l5 (40 = exp (- +s)[l+ clu + c2t2 + c3s+ c4s2 + c,u2]) are in table 3. This re-minimization gave considerable changes in calculated properties. TABLE EN ENERGIES co (IN A.u.) AND SCALING PARAMETERS k FOR TRIAL UNPERTURBED WAVE- n 2 3 4 5 6 7 8 9 10 12 15 18 21 original reminimized THE number of parameters 27 33 42 54 70 90 115 145 181 252 444 71 5 1078 FUNCTIONS ($0 (n) EO - 2.903714647 - 2.90371 9257 23329 23988 4236 4329 4354 4367 4372 4290 4356 4370 4375 k 3.6612 3.6352 3.8362 3.8461 3.8639 3,9432 3.9287 3.9500 3.9493 (3.4081) (Pekeris 13) (3.4081) (Pekeris 13) (3,4081) (Pekeris 13) (3.4081) (Pekeris 13) TABLE 2.-&VALUES FOR VARIOUS UNPERTURBED WAVEFUNCTIONS n 7 8 9 10 15 18 21 0 1) 1 -8 10447 1.810403 1 *8 1 0430 1 08 10433 1*810389* 1 *810410* 1.810419" * from Pekeris.13 d(r 12) 0,106369 0,106359 0.106355 0.106352 0.1 06377" 0*106362* 0.1 063 55 * TABLE 3 .-THE SIX-PARAMETER HYLLERAAS WAVEFUNCTION k c1 c 2 c3 c 4 C5 EO 3.63586 0,0972 0,0097 -0.0277 0.0025 -0.0024 - 2.90324 3.51 352 0.0961 0.0092 -0.0409 0.0019 - 0.0030 - 2,90333 FIRST ORDER WAVEFUNCTION AND POLARIZABILITY The first order wavefunction Y satisfies the equation and E2 is given by eqn.(10). Because of the presence of rl in Z0 there may be terms in Y1 with angular symmetries of the form of Yll,m(Ol,41) Yz,m(82,42). Those contributingto c2 can be generated from Yl,o(81 4,) Y ,0(02,42) through multiplica- tion by powers of u in the trial function.A suitable function l6 for obtaining g2 is (Xo-Eo)Y,+X1Yo = 0 (14) 41 = 404Lwhere 4;(n> = f c;qrsPtqur(rlP1(cos el) + r,~,(cos e,)> + p+q+r=O (4 even) 5 ckqrsPtqur(rlP1(cos 0,) - r,~,(cos 0,)). (15) p+q+r= 1 (4 odd) A . D. BUCKINGHAM AND P . G . HIBBARD 45 Results for E~ and crl for IZ = 0-6 are in table 4. Since 1 a.u. of polarizability is (b2/lme2)>" = 0.148176 x C2 m2 J- l the polarizability of the helium atomis a = 1.383193 = 0.204956 x C2 m2 J-l. The dipole shielding factor o1 provides a valuable check on 41 since ol is 1 ; owing to computational problems only about five figure accuracy was achieved for crl. Our result for a is to be compared with the value 1.384 obtained by Chung and Hurst." cm3 = 0.164863 x cm3 = 0.228044 x TABLE 4.-THE SECOND ORDER ENERGY Ez FOR THE 181 PARAMETER 40(10) AND THE DIPOLE SHIELDING FACTOR FOR THE 90 PARAMETER $0(7) FOR VARIOUS 4 l ( n ) with no terms in r12 n parameters ep (a.u.) 01 0 1 -0,5662528 1-53649 1 3 -0'6911853 0.98333 2 6 -0.6912874 0.99 195 3 10 -0.6914775 0-99735 4 15 -0.6915125 0.99948 5 21 - 0.691 5279 1 -00009 6 28 -0.6915349 1 -0002 1 with terms in r12 parameters 62 (a.u.) 0 1 1 -0,5662528 1 *5 3649 4 -0.6911886 0.98291 0.99490 10 -0.6914819 20 -0.6915716 0.99899 35 - 0.691 5897 0.99993 56 - 0.691 5944 1.00005 84 - 0.691 5959 1 -00003 TABLE 5.-THE FOURTH-ORDER ENERGY E4 FOR THE 181 PARAMETER +0(10) AND VARIOUS 4dn1) AND 42(n2) \nl 0 1 2 3 4 5 6 m\ O* 1* 2* 3* 4* 5* 0 1 2 3 4 5 - 10727567 - .07508832 - -091 5776 1 - -09212278 - '0921 3712 - e09214761 el0727567 -*37157956 - .43 5 8 8434 - -43743006 -*43751062 -125991 54 - 056828633 - *744 1 1092 - '74988604 - -749921 16 - '74994987 - 12599154 - 1'2587623 - 1.7514343 - 1'7723120 - 1.7724772 -126091 87 -12655203 *I2660158 -a57746122 - e57528826 - '57528369 7'76045337 -*75556923 - e75576099 - a76701894 - a76154030 - '761 84665 - -76706149 - '76160061 - .76190417 -76709204 -.76163687 -.76194189 *12609 1 87 -1 2655203 -1 2660 1 58 -1.2731227 - 1.2705417 - 1.2705538 - 1.7820801 - 1.7736862 - 1.7740656 - 1.8053507 - 1.7950649 - 1.7957793 - 1.8055247 - 1.7953062 - 1.79601 17 * with diqr = e;qr = 0.* 1266 1534 - -57527346 -*75577051 - -761 85908 - '7619 1710 - -76 195674 *12661534 - 1.2705490 - 1.7740688 - 1.7957708 - 1.7960018 * 12661 944 - -57526823 - -75576609 - -761 85153 - -76190986 - -76 194821 * 1266 1944 - 1.2705448 - 1'7740633 - 1.7957630 - 1'7959926 THE SECOND ORDER WAVE FUNCTION AND THE HYPERPOLARIZABILITY The second-order wave function Y satisfies the equation ( 2 0 - WY2 + 2 1 % = E2Y0 (16) and E4 is given by eqn.(11). zeroth- and first-order equations + 2 = $04i may be found where Using the functions q50 and q51 determined from the n +i(n) = 1 [c;qrsPtqur + d ~ q r ~ P t 4 ~ r ( r ~ P 2 ( ~ ~ ~ 0,) + r2P2(cos 0,)> + p + q + r = 1 e~qrsPt4urrlr2P1(cos 0,)P1(cos O,)] + n 1 &~qrsPtqur(r~P2(~os 0,) - riP2(cos 62)>. (17) o + a + r = 1 - (<odd) The calculation of z4 and 4; is similar to that of c2 and 41 and no difficulties were found beyond the need for care with accumulation of round-off errors.Values of c4 for &(n) for n = 0-5 are in table 5 for the 181 parameter 40(10) and various $,(n) both, 46 HYPERPOLARIZABILITY OF THE HELIUM ATOM for the full expansion (17) and also for restricted expansions involving the spherical terms only (i.e. (17) with d& = edbr = 0). The final result for c4 is - 1.796 a.u. equivalent to y = 43.10 a.u. = 2.171 x e.s.u. = 2.688 x The scaling parameter k has so far been considered to retain its field-free value ; greater flexibility is permitted by lettering k = ko + k F + k4F$ + . . . . This has no effect on #1 or E, but c4 is reduced by minimization with respect to k2.4 The im- provement is valuable for simple functions #$(n) but is insignificant with the larger expansions which have sufficient flexibility without this additional nonlinear variation- al parameter.Our hyperpolarizability y is in good agreement with Grasso Chung and Hurst’s value of 42.8 1 a.u. (these authors obtained E~ = - 2.903721 a.u. with a 25 parameter 40 and a = 1.3830 a.u.) and with Sitz and Yaris’s 42.6 a.u. However the experi- mental value from the Kerr effect is 53.6 +4 a.u. for a wave length of 632.8 nm ; a “ constant excitation energy ” model indicates that this experimental optical y is equivalent to a static value of 52.8 a.u. There is therefore a significant discrepancy between theory and experiment. This may be due to insufficient flexibility in q51 and 4,; it is possible that there could be additional terms in 41 contributing to E ~ though not to E or to the dipole shielding constant. We are investigating this possibility.We have found that it is particularly important to use a very accurate unperturbed wavefunction 40. C4 m4 J-3. THE QUADRUPOLE A N D OCTOPOLE POLARIZABILITIES In an axial field gradient Fzz = -2F, = -2F,,, the perturbation is where a, = - [r?P,(cos 61) + Y P (cos 6,)] is the quadrupole moment operator. The energy is given by the equivalent to eqn. (10) and the quadrupole polarizability C defined in ref. (4) is C = a2 = - 4 ~ ~ . Values for both a and the octopole polar- izability a3 = - 12c2 are shown in table 6 for the 181 parameter unperturbed wave- function 40(10) and for various perturbed functions 41(n). Using Kinoshita’s 80 parameter 40 Davison l 7 obtained a = 1-2202 a5 for helium. TABLE 6.-THE QUADRUPOLE AND OCTOPOLE POLARIZABILITIES FOR 40(10) AND VARIOUS (bl(n) with terms in r-12 n with no terms in r12 az(aS) a3(a7) a2(a5) a3(a7> 0 1 *05 1206 1.531 1 *05 1206 1.531 1 1.221 389 1.561 1.222038 1-561 2 1 -221 841 1 *567 1 -222388 1 -568 3 1 ~ 2 2 191 7 1 -570 1 -222499 1-571 4 1.221995 1 *222533 A.Dalgarno Adv. Physics 1962 11 281. N. Bloembergen Nonlinear Optics (Benjamin New York 1965) ; see also A. D. Buckingham and B. J. Orr Quart. Rev. 1967 21 195. A. D. Buckingham and J. A. Pople Proc. Physic. SOC. A 1955 68,905 ; A. D. Buckingham Proc. Roy. SOC. A 1962 267 271. L. L. Boyle A. D. Buckingham R. L. Disch and D. A. Dunmur J. Chem. Physics 1966 45 1318. A. D. Buckingham and D. A. Dunmur Trans. Faraday Soc. 1968 64 1776. G. W. F. Drake and M. Cohen J. Chem. Physics 1968 48 1168 (and references therein). ’ M. N. Grasso K. T. Chung and R. P. Hurst Physic. Reu. 1968,167,l (and references therein). A . D. BUCKINGHAM AND P . G . HIBBARD P. Sitz and R. Yaris J. Chem. Physics 1968 49 3546. E. A. Hylleraas Adv. Quantum Chem. 1964 1 1. lo C. Schwartz Physic. Rev. 1962 126 1015. l 1 T. Kinoshita Physic. Rev. 1957,105,1490; 1959,115 366. l 2 T. Kato Trans. Amer. Math. Soc. 1951 70 195 and 212. l3 C. L. Pekeris Physic. Rev. 1958 112 1649; 1959 115 1216. l4 R. Fletcher and M. J. D. Powell Cump. J . 1963 6 163. l5 E. A. Hylleraas Z. Physik 1929 54 347. l6 K. T. Chung and R. P. Hurst Physic. Rev. 1966 152 35. l7 W. D. Davison Proc. Physic. SOC. 1966 87 133. 47