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The polarised frozen-core approximation: Oscillator strengths for the boron isoelectronic sequence
.I Quant Spectmsc Radlat Trans.fer Vol 27 No 2, pp 111-117 1982
0022~1073182/020111-0750300/0 Pergamon Press Ltd
Prmled m Greal Britain
THE POLARISED FROZEN-CORE APPROXIMATION: OSCILLATOR STRENGTHS FOR THE BORON ISOELECTRONIC SEQUENCE R. P MCEACHRAN~-and M. COHEN Department of Physical Chemistry, The Hebrew University, Jerusalem, Israel (Recetved 1 June 1981)
Abstract--Orbital wave functions of a large number of ns-, rip- and nd-levels of the first five members of the boron isoelectromc sequence have been calculated using a frozen-core Hartree-Fock procedure augmented by an/-dependent core polansanon potential The calculated lonisatlon energies are generally in very good agreement with observations Electric dipole oscillator strengths have been derived from the calculated orbltals and energies and have been used to yield simple mterpolahon formulae for ]-values of higher members of the sequence They are expected to be of high accuracy m many cases 1 INTRODUCTION
Iomsanon energies of excited 2L hi-levels of the first few members of the boron isoelectronlc sequence are, with very few exceptions, rather well determined using the simple frozen-core (FC) variant of the Hartree-Fock (HF) approximation. 1 This is presumably a reflection of the adequacy of the physical model of a single valence electron moving in the HF field of the ls22s2JS core However, the single configuration FC model is considerably less successful for the ground 2s22p 2pO states, presumably due to the rather large interaction between the core 2s electrons and the valence 2p electron. A limited multiconfiguration frozen core (MCFC) treatment, which includes the "near degeneracy" effect of the 2p 3 2pO configuration, yields a small improvement in this case, 2 and only a very elaborate superpositlon of configurations (SOC) calculation 3 provides a satisfactory description of these ground states. An important physical effect, which is completely neglected in almost all HF treatments, involves the polarisatlon of the spherically symmetric core by the orbiting valence electron. The resulting slight distortion of the HF field experienced by the valence electron can be treated easily through the introduction of a polarisation potential Vpo~whose asymptotic form is given rigorously as 4 Vpol- - a / 2 r 4,
(1)
where a ts the dipole polarisability of the ion core. This form is clearly unsatisfactory at small radial distances, and various empirical forms have been suggested 5'6 to avoid the singularity at small r We have adopted the form 7 Vpoj = - (a12r4)[1 - exp (-x6)],
x = rip,
(2)
where O is an empirical cut-off parameter; we also chose a different p so as to reproduce the observed ionisation energy of the lowest member of each series of valence states for each ion considered. Our polarised frozen-core (PFC) procedure thus requires reliable energy level data on the lowest state of each series, which is often readily available For the boron sequence, this has been taken from a variety of recent sources, s'12 Accurate core dipole polarisabilities have been obtained from model potential calculations on B(I), C(II), N(III), O(IV), and Ne(VI); 1~we interpolated for F(V) (see below) The case of neutral boron has been treated previously by the PFC procedure, 14 but with a polarisability value (12.55 a.u.) from another source, 15which is almost 30% larger than the value adopted here (9.72 a u ). However, this relatively large difference has very little effect on the excited ns- and np-level ionisation energies once appropriate cut-off parameters p are introduced Since the ordinary FC ionisation energy of the 3d-level is already larger than the tPermanent address Department of Physics, York Umvers~ty, Toronto, Canada M3J 1P3
112
R P MCEAcHRANand M COHEN
experimental value, the PFC procedure is not appropriate for the nd-series of B(I); thus, our results for this series only are based on FC wave-functions and energies The use of a different empirical parameter p for each series introduces some flexibility mto the model and may compensate both for the limitations of the resulting model potential and for any maccuracy in the adopted values of the core dipole polarisabilities. Our procedure is
evidently successful for calculating ionlsation energies, but the accuracy of our ]'-values is much more difficult to assess
2. ENERGY LEVELS
Table 1 contains our PFC ionlsation energies (FC for the nd sertes of B(I) only) together with the most reliable observations. "'m For each series, we list the value of p adopted, and it will be noted that these differ appreciably from series to series for a given ion As we proceed along the isoelectronic sequence, p is found to vary systematically for each series like a mean radius, (r). The asymptotic behaviour is 16 p
=
(3)
A I ( Z - or),
where Z is the nuclear charge, while A and 0" are constants. In practme, Eq. (3) may be used to obtain good initial estimates of p for the higher members of the sequence from results for ions of lower Z. The same kind of analysis '7 leads to the following form for the dipole polarisability for a four electron ls22s 2 core: a = A/(Z
-
0")3,
(4)
Eq. (4) was used in this work to obtain an interpolated value of a for F(V). It will be seen from Table 1 that our PFC ionisation energies are generally very accurate, except for a few levels which may be perturbed by neighbours of the same overall symmetry
Table 1 Calculated and observed lomsatlon energies (m atomic umts) of nl2L levels of the boron isoelectromc sequence, perturbed levels are identified by superscript t BI
3s 4s 5s 6s 7s 8s
CII
Present results
Ref. 8
Present results
Ref
0.12252 0 05391 0.03043 0 01955 0 01362 0 01003
0 12252 0.05430 0.03090 0.02026 t 0.01263 t 0.00973
0.36510 0 17702 0 10481 0.06932 0 04924 0_03678
0 36510 0.1796? 0 10624 0 07017 0.04981 -
2.30926
2p 3p 4p 5p
6p 7p 8p
0.30492 0 08163 0.04088 0.02464 0 01648 0.01180 0.00886
3.52616
0.30492 0.08343 0.04165
-
2 80141
3d 4d 5d 6d 7d 8d
0 05617 0.03160 0.02020 0.01401 0.01028 0.OO787
9
0.89592 0 29162 0115126 0_09280 0.06275 0 04527 0 03419
0 23291 0.12951 0,08232 0,05691 0.04167 0,03183 3.30814
0.73543 0.36899 0 22235 0.14866 0.10640 0 07991
NIII Ref
I0
Present results
0.73543 0 37191 0.22323 0.14981 0 10733 0.08069
1 21552 0 62379 0 38024 0.25606 0 18416 0 13881
0 89592 0 29587 0.15557 0 09738 t 0.06229 0 04520
1.74327 0.61838 0 32657 0.20221 0 13754 0 09961 0.07547
1 74327 0.62431 t 0.32350 t 0.20297
-
-
0.52610 0 29206 0 18553 0.12820 0.09386 0.07167 2.43706
1.21552 0 63135 t 0.38737 t
2 84382 1.05864 0 56540 0.35216 0.24042 0 1/457 0.13250
0 52610 0 29589 t 0.18570 0.12837 o 09404 0.07183
0-93334 0 51822 0.32928 0 22759 0.16664 0 12727 1.98482
FV Present results
Ref.
12
1.8o755 0.94170 0.57852 0139148 o 28250 0_21345
1-80755 0.95009 t
1 28422
2.84382 Ii06696 t 0.56664 0.35344
-
1 55801
1 79385
0.23291 0 13005 0.08279 0.05737
OIV Ref. 11
1 55044
1.98355
2_15651
0.05542 0.03160 0.02024 0.01405 0.01031 0.00789
Present results
4.19629 1.61099 0 86719 0_54236 0.37123 0 27002 0 20523
4.19629 1.62194 t
1 39264
0 93334 0.51857 0.32967 0.22789
1.45329 0.8O750 0 51335 0.35493 0.25995 0.19857 1 69103
1.45329 0 80843 0.51387 0 35519
The polarlsed frozen-coreapprox,matlon
113
but different orbital structure. In this category, we include, in particular, the following (the probable pertmbing level is indicated in each case): B(I) 6s, 7s, 8s (2s2p2;2S); C(II) 5p (2s2p(3p°)3s; 2pO), N(III): 3p (2p3;ZP°), 4p (2s2p(3p°)3s;ZP°), 4d (2s2p(3p°)3p,2D); O(IV): 4s, 5s(2s, 2p2,2S), 3p(2p3,2P°), F(V). 4s(2s2p2; 2S)" 3p(2p3; 2pO). All of these levels have been marked by a dagger m Table 1. With these exceptions, agreement is sufficiently good that we feel that the PFC model may be used with confidence to confirm assignments of the term energies of more highly excited levels. 3 OSCILLATOR STRENGTHS Table 2 contains results of calculations of electric dipole oscillator strengths (f-values) for B(I) through F(V). The PFC orbitals and energies were used (except for the 2D series of B(I)) to compute the standard "length" and "velocity" forms of the f-value, according to A = ~ (21r _
1) AE~(P'IrlPr)2
(5)
and
v=3(2lt-l)
(6)
Here, P,/r and Pr[r are the initial and final valence electron radial orbitals, Ae~ = (el - e,) is the excitation energy of the transition, and li(= 1, + 1) is the angular momentum quantum number of the final state valence orbital. The effect of modifying the dipole length operator r so as to take account of core polarisation TM is known to be slight and has been neglected here. There is a slight formal difficulty, arising from the use of different cut-off parameters p in the initial and final states, so that the modified operator cannot even be defined unambiguously; however, the numerical [-values which result from different possible choices of P in the transition element are insignificant For the vast majority of transitions considered here, ft and fo agree to within a few percent, and we therefore present only the geometric mean fg of each pair of results. This mean has the merit of being independent of the excitation energy Ae~, but in view of the generally high quality of our PFC ionisation energies, any inaccuracy in the [-values must be due primarily to errors in the transition matrix elements. As with the earlier FC calculations of f-values, J differences between fl and fo are much larger for transitions which involve a valence 2p electron, although agreement improves steadily with increasing Z A comparison of our length and velocity •-values for N(III) and O(IV) together with results of a parametrised independent particle model (IPM), 19 which employs observed energy level data to fix the parameters, is presented in Table 3. In spite of the good agreement between the PFC velocity values and the IPM (length) values, and the fact that use of a modified length operator should reduce (albeit slightly) the present ft values, bringing them into better agreement with the fu-values, these results are among our least reliable [-values The 2p-3s transitions of the boron sequence are notoriously sensitive to the effects of other competing configurations, with a near cancellation of the oscillator strength at C(II); 3 our PFC model cannot reproduce such an effect. The discrepancies between our PFC •-values and refined SOC results for the 2p-3d, 3s-3p and 3p-3d transitions are generally much smaller, and we present a comparison for several members of the sequence in Table 4 In order to obtain our estimates for the higher ions, we have fitted the PFC •-values at Z = 8, 9 to the asymptotic Z-dependent formula having the form of a [1/1] Pad6 approximant: 2°
f = A + B/(Z- ~).
(7)
2
-0.210 -0.029 -0.010 -0.005 -0.003 -0.002
-0.160 -0.028 -0.010 -0.005 -0.003 -0.002
-0.139 -0 025 -0.009 -0.0~5 -0.003 -0.002
n/m
2 3 4 5 6 7 8
2 3 4 5 6 7 8
2 3 4 5 6 7 8
0.803 -0.356 -0.058 -0.002 -0.011 -0.006
0.951 -0.431 -0.064 -0.023 -0.011 -0.007
1.133 -0.615 -0.058 -0.019 -0.009 -0.005
3
0,056 1.110 -0 582 -0.094 -0.035 -0.018
0.018 1.311 -0.707 -0.102 -0.037 -0.018
0.012 1.585 -1 025 -0 089 -0.029 -0.014
4
0.024 0.053 1.411 -0.809 -0.129 -0.048
0.010 0.014 1.666 -0.983 -0.140 -0.051
0.002 0.028 2.022 -1.430 -0.118 -0.038
5
ns2S - mp2p°
0.012 0 024 0.052 1.710 -1.037 -0.165
0.006 0.009 0.011 2.017 -1.258 -0.177
~ 0 005 0 044 2 451 -1.831 -0.147
6
0.007 0.013 0.024 0.052 2.006 -1.265
0.003 0.005 0.008 0.009 2.367 -1.534
t 0.002 0.008 0.060 2.875 -2 228
7
8
0.002 0.003 0.005 0.008 0.007 2.715
t 0.001 0.003 0.012 0.076 3.297
0.004 0.008 0.013 0.025 0.052 2.302
NIII
CII
BI
0.395 0.411 -0.099 -0.014 -0.005 -0.002 -0.001
0.290 0.572 -0.146 -0.018 -0.006 -0.003 -0.002
0 114 0 857 -0 258 -0 019 -0.006 -0.003 -0.002
3
0.112 0.201 0 659 - 0 219 -0.033 -0.012 -0.006
0.095 0.100 0.888 -0.313 -0.041 -0.014 -0.007
0.049 0.001 1.240 -0.528 -0.041 -0.013 -0 006
4
0 048 0 074 0 151 0.875 -0.347 -0.054 -0 020
0 043 0 046 0 058 1.164 -0.487 -0.065 -0 023
0.025 0.004 0 004 1 571 -0.797 -1.063 -0.020
5
0 026 0 036 0.061 0.128 1.079 -0.479 -0.076
0,024 0.024 0.032 0.040 1.423 -0 665 -0 089
0.014 0 003 % 0.014 1.882 -1.066 -0.084
6
np2p ° - md2D
0.015 0.020 0 031 0 055 0.117 1.275 -0.614
0 014 0 014 0 018 0 024 0.030 1.673 -0 884
0.009 0.002 0.001 0.001 0.026 2.181 -1.333
7
Table 2 Electric dipole oscillator strengths for the boron isoelectronic sequence; weak transitions with I/I < 10 3 are identified by ["
0.010 0.013 0.018 0.029 0.051 0.110 1.467
0 009 0 009 0 011 0 015 0.020 0 024 1 918
0.006 0.002 0.001 t 0.003 0.038 2 473
('] 0
e~
Z
-0,121 -0,022 -0.008 -0.004 -0.002 -0.002
-0 108 -0,020 -0.008 -0 004 -0 002 -0.001
2 3 4 5 6 7 8
2 3 4 5 6 7 8
0.593 -0.267 -0.049 -0.0]9 -0.009 -0.006
0 683 -0.303 -0.053 -0.020 -0,010 -0 006
0.136 0.819 -0 436 -0 079 -0 030 -0 015
0.098 0.944 -0 494 -0 086 -0 032 -0.016
0 047 0.141 ].041 -0.607 -0,110 -0 042
0 037 0 099 1.200 -0.688 -0,119 -0.045
0 022 0 051 0 149 1 262 -0,780 -0 141
0.018 0.039 0.103 1.454 -0 883 -0,152
0.013 0.025 0.054 0.158 1 481 -0 953
0 010 0.019 0.040 0.107 1.707 -1 078
FV
0.006 0.011 0.021 0.042 0,112 1,958
0 OO8 0 014 0.027 0.057 0.168 1.699
OIV
0 499 0 260 -0.063 -0.010 -0.004 -0,002 -0 001
0.458 0 318 -0.076 -0 012 -0.004 -0.002 -0,001
0 122 0.326 0 428 -0 143 -0.025 -0.009 -0.005
0.119 0.274 0 520 -0.171 -0.028 -0.010 -0 005
0 050 0,101 0 279 0 573 -0.231 -0.041 -0.015
0.050 0.09] 0.224 0 695 -0.274 -0.045 - 0 017
0.026 0.046 0.093 0.261 0.710 -0.324 -0.058
0.026 0.042 0.080 0.203 0.859 -0,382 -0.065
0.015 0.025 0.044 0 090 0 256 0.843 -0 420
0.015 0.024 0.039 0.075 0.194 1.018 -0,493
0,010 0.016 0.025 0.044 0.089 0 256 0 973
0.073 0.191 1.174
0.038
0.010 0.015 0.023
=o
R P McEAcHRANand M COHEN
116
Table 3. PFC and IPM/-values In N(III) and O(IV), data from Re/ [19] are ldentllied by superscript t NIII
OIV
Transition
PFC(£)
PFC(v)
IPMt(,)
PFC(V)
PFC(v)
IPMt(Z)
2p-3s 4s 5s 6s 7s
0-0547 0.0098 0.0036 0,0018 0.0010
0 0390 0 0068 0 0025 0 0012 0.0007
0.0342 0.0065 0 0024 0 0012 0 00069
0.0465 0.0085 0 0032 0 0016 0 0009
0 0349 0.0063 0.0024 0 0012 0,0007
0 0303 0,0059 0.0022 0 0011 0,00064
2p-3d 4d 5d 6d 7d
0 4216 0 0520 0 0520 0.0274 0.0164
0 3693 0.0449 0.0449 0.0237 0.0141
0.3740 0.0470 0.0470 0.0248 0 0148
0.4823 0.0524 0.0524 0.0272 0.0161
0 4339 0 0468 0 0468 0 0243 0.0144
0 4385 0.0486 0 0486 0 0253 0.0150
Table 4 A comparison of/-values for the boron lsoelectronlc sequence Transition 2pi3s
2p-3d
3s-3p
BI
CIl
(1) (2) (3)
0 070 0.067 4) 0.055 ~
(I) (2) (3)
0.114
(I) (2)
1 133 1 199
o.197(q~ 0.175"-"
NIII
OIV
0 053 0 008(6, 0.018 )
0 045 0 028t_,
0 040
0.026(Y
0.022 (9)
290 330 _319 (6)
0-395 0,422 ¢ 0 375 (8)
0 458 0.486 t , 0.526 ( %
I
0.028t_,
FV
NeVI
0.036 0 027 t
0.033 t 0.026
0.499
, 0.528 t
0.533 t~ i
0_951 01702
! I
0.567(10) 0.313"
NaVII
0 031 t 0 025 t
0.549 t 0 593 ¢
0 803 0 618 t
0.683 0 536 t
0 593 0 468 t/
o.523 t 0.416
0.469 ¢ 0 365 t
0 411
0 318 0 319 t
0 260 0 251 t
0 219 t 0 203
0 190 t 0 169 t
I
3P-3d
(I) (2)
0 857 0 786
0 572 0 568
I
I 1
0 419 t
I
(I) (2) (3)
PFC values, interpolated values denoted by superscript SOC values from Bef. 3, interpolated values denoted by superscript ¢ Experimental beam-loll values," sources are as follows (4) Ref 22, (3) Ref. 23, (6) Ref 24, (7) Bef_ 25, (8) Ref. 26, (9) Ref 27, (10) Bef 2b
The hmiting hydrogenic value of / is zero for An = 0 transitmns, and numerical values of the hydrogenic transition element have been tabulated 2' for all transitmns of interest, so that exact values of A appear in Table 5. Thus, if our results for O(IV) and F(V) were exact, Eq. (7) would provide a very accurate interpolation formula for higher Z. Many of the SOC comparison data of Table 4 were derived by similar procedures from direct calculations on B(I), C(II), and Ne(VI)
Table 5 Parameters for the asymptotic formula [ Transition
2p-3s 3s-3p 3s-4p 3p-4s 4s-4p 4s-Sp 4p-5s 5s-5p 5ai6p 5p-6s 6s-6p 6s-7p 6p-7s 7s-7p 7s-Sp 7p-Ss
8siBp
=
A + B I ( Z - or)
Transition
0.0136 0.0 o.4847 0 0323 0.0 0.5442 o_o529 0 0 0.6078 0 0745 0 0 0 6736 0.0967 0 0 0 7406 0 1192 0 0
0.1389 4 472 -3.677 0 3267 6 174 -4 288 0_5291 7 868 -4 998 0-7360 9 534 -5.711 0.9393 11.193 -6 435 I 144 12 858
2.795 1.456 -1_360 3 237 1.460 -1.641 3.267 1.445 -1.890 3 245 I_444 -2 074 3 246 I 441 -2 232 3 236 1 433
2p-3d 3P-3d 3p-4d 3d-4p 4p-4d 4p-5d 4d-5p 5p-5d 5p-6d 5d-6p 6p-6d 6p-Td 6di7p 7p-7d 7p-8d 7d-8p 8p-8d
0.6958 0.0 0-6183 0_0110 0.0 O.6O93 0.0278 0_0 0.6247 0 0480 O0 0 6514 0,0702 O0 0.6843 0.0939 0.0
-I.140 I _406 -I 920 0 1143 2 411 -2 301 0 2582 3 280 -2 627 0 4107 4 I]9 -2 934 0.5685 4-895 -3.246 0 7278 5 683
3 216 3 585 2 432 4_688 3_362 2 036 4.548
3 278
1 768 4.479 3,204 1 585 4 427 3 191 1 426 4 393 3 158
The polarlsed frozen-coreapproximation
117
Table 4 also contains a few selected results of beam foil experiments of transitions involving the ground states. 22-28 While the absolute accuracy of any mdividual measurement is difficult to assess, we see that the PFC model is not completely satisfactory for these transitions. But, for all excited states not subject to significant perturbation effects, the PFC one-electron model clearly provides a good description of the levels, and it may therefore be expected to yield /-values similar to those of the Coulomb approximation 29 The present method enjoys an important advantage over the Coulomb approximation, since the latter requires a priori knowledge of the term energies for its application. In summary, our present results, taken together with earlier calculations on the beryllium lsoelectronic sequence, 3° suggest that the PFC procedure is capable of yielding reliable/-values for most transitions between excited states. Acknowledgements--We are grateful to Messrs G Hermanand J. Stlberstein for computationalassistance The work was supported, in part, by the Natural Sciences and EngineeringResearch Councdof Canada under Grant A-3629and, in part, by the Central Research Fund of the Hebrew Unlverszty
REFERENCES I R P McEachranand M Cohen, JQSRT II, 1819(1971) 2 M Cohen and J Nahon, J Phys B' Atom_ Molec Phys 13, 4325 (1980) 3 A W Weiss, Phys Rev. 188, 119 (1%9) 4 J H Van Vleck and N G. Whltelaw, Phys Rev 44, 551 (1933) 5 D R Bates, Proc R Soc (London)AI~, 350 (1947) 6 J Mlgdalekand W E Bayhs,JQSRT 22, 113 (1979) 7 L Blermannand E. Trefftz, Z. Astrophys 30, 275 (1953) 8 G A Odlntzovaand A R Stnganov,J Phys Chem Re/. Data 8, 63 (1979) 9 C_ E. Moore, Selected Tables o/Atomic Spectra, NSRDS-NBS3,Section 3 (1970) 10 C. E Moore, Selected Tables o/Atomic Spectra, NSRDS-NBS3,Section 5 (1975) 11. J Bromander,Ark. Fys 40. 257 (1%9) 12 C E Moore,Atomic Energy Levels, bIBS Circular 467, Vol I (1949) 13 C. Laughhn, E_ R Constantlnldes and G A. Victor, J Phys B Atom Molec Phys 11, 2243 (1978) 14 M Cohen and R. P. McEachran, Adv Atom Molec Phys 16, 1 (1980) 15 S. A Adelmanand A Szabo,./_ Chem Phys 58, 687 (1973) 16. M Cohen and A. Dalgarno, Proc R_ Soc (London) A261,565 (1%1) 17. G W F Drake and M Cohen,J Chem. Phys 48, 1168(1%8) 18 T C Caves and A Dalgarno,JQSRT 12, 1539(1972). 19 P S Ganas, Phys Lett 73A, 161 (1979), J Chem Phys 71, 1981(1979) 20 M Cohen and R P McEachran,lnt J Quant Chem Syrup. 12, 59 (1978)_ 21 R Herdan and T. P Hughes,Astrophys J 133, 294 (1%1) 22 I Bergstrom, J Bromander,R Buchta, L Lundln.and l Martmson, Phys. Lett. 28A,721 (1%9) 23 G M Lawrenceand B D Savage,Phys Rev 141,67 (1%6) 24 M C Poullzacand J. P Buchet, Phystca Scnpta 4, 191 (1971) 25 L Heroux, Phys Rev 180, 1(1%9) 26 J A_ Kernahan, E H Pmnington,and K E Donnelly,Phys. Lett 57A,323 (1976). 27 J P Buchet,M C Buchet-Poulizac,and M Druetta, J Opt Soc Am 66, 842 (1976) 28 J P Buchet and M Druetta, J Opt Soc Am 65, 991 (1975) 29 D. R Bates and A Damgaard,Phil Trans R Soc (London)242, 101 (1949) 30 E MarklewJcz,R P McEachran,and M Cohen, Phystca Scripta 23, 828 (1981)
0022~1073182/020111-0750300/0 Pergamon Press Ltd
Prmled m Greal Britain
THE POLARISED FROZEN-CORE APPROXIMATION: OSCILLATOR STRENGTHS FOR THE BORON ISOELECTRONIC SEQUENCE R. P MCEACHRAN~-and M. COHEN Department of Physical Chemistry, The Hebrew University, Jerusalem, Israel (Recetved 1 June 1981)
Abstract--Orbital wave functions of a large number of ns-, rip- and nd-levels of the first five members of the boron isoelectromc sequence have been calculated using a frozen-core Hartree-Fock procedure augmented by an/-dependent core polansanon potential The calculated lonisatlon energies are generally in very good agreement with observations Electric dipole oscillator strengths have been derived from the calculated orbltals and energies and have been used to yield simple mterpolahon formulae for ]-values of higher members of the sequence They are expected to be of high accuracy m many cases 1 INTRODUCTION
Iomsanon energies of excited 2L hi-levels of the first few members of the boron isoelectronlc sequence are, with very few exceptions, rather well determined using the simple frozen-core (FC) variant of the Hartree-Fock (HF) approximation. 1 This is presumably a reflection of the adequacy of the physical model of a single valence electron moving in the HF field of the ls22s2JS core However, the single configuration FC model is considerably less successful for the ground 2s22p 2pO states, presumably due to the rather large interaction between the core 2s electrons and the valence 2p electron. A limited multiconfiguration frozen core (MCFC) treatment, which includes the "near degeneracy" effect of the 2p 3 2pO configuration, yields a small improvement in this case, 2 and only a very elaborate superpositlon of configurations (SOC) calculation 3 provides a satisfactory description of these ground states. An important physical effect, which is completely neglected in almost all HF treatments, involves the polarisatlon of the spherically symmetric core by the orbiting valence electron. The resulting slight distortion of the HF field experienced by the valence electron can be treated easily through the introduction of a polarisation potential Vpo~whose asymptotic form is given rigorously as 4 Vpol- - a / 2 r 4,
(1)
where a ts the dipole polarisability of the ion core. This form is clearly unsatisfactory at small radial distances, and various empirical forms have been suggested 5'6 to avoid the singularity at small r We have adopted the form 7 Vpoj = - (a12r4)[1 - exp (-x6)],
x = rip,
(2)
where O is an empirical cut-off parameter; we also chose a different p so as to reproduce the observed ionisation energy of the lowest member of each series of valence states for each ion considered. Our polarised frozen-core (PFC) procedure thus requires reliable energy level data on the lowest state of each series, which is often readily available For the boron sequence, this has been taken from a variety of recent sources, s'12 Accurate core dipole polarisabilities have been obtained from model potential calculations on B(I), C(II), N(III), O(IV), and Ne(VI); 1~we interpolated for F(V) (see below) The case of neutral boron has been treated previously by the PFC procedure, 14 but with a polarisability value (12.55 a.u.) from another source, 15which is almost 30% larger than the value adopted here (9.72 a u ). However, this relatively large difference has very little effect on the excited ns- and np-level ionisation energies once appropriate cut-off parameters p are introduced Since the ordinary FC ionisation energy of the 3d-level is already larger than the tPermanent address Department of Physics, York Umvers~ty, Toronto, Canada M3J 1P3
112
R P MCEAcHRANand M COHEN
experimental value, the PFC procedure is not appropriate for the nd-series of B(I); thus, our results for this series only are based on FC wave-functions and energies The use of a different empirical parameter p for each series introduces some flexibility mto the model and may compensate both for the limitations of the resulting model potential and for any maccuracy in the adopted values of the core dipole polarisabilities. Our procedure is
evidently successful for calculating ionlsation energies, but the accuracy of our ]'-values is much more difficult to assess
2. ENERGY LEVELS
Table 1 contains our PFC ionlsation energies (FC for the nd sertes of B(I) only) together with the most reliable observations. "'m For each series, we list the value of p adopted, and it will be noted that these differ appreciably from series to series for a given ion As we proceed along the isoelectronic sequence, p is found to vary systematically for each series like a mean radius, (r). The asymptotic behaviour is 16 p
=
(3)
A I ( Z - or),
where Z is the nuclear charge, while A and 0" are constants. In practme, Eq. (3) may be used to obtain good initial estimates of p for the higher members of the sequence from results for ions of lower Z. The same kind of analysis '7 leads to the following form for the dipole polarisability for a four electron ls22s 2 core: a = A/(Z
-
0")3,
(4)
Eq. (4) was used in this work to obtain an interpolated value of a for F(V). It will be seen from Table 1 that our PFC ionisation energies are generally very accurate, except for a few levels which may be perturbed by neighbours of the same overall symmetry
Table 1 Calculated and observed lomsatlon energies (m atomic umts) of nl2L levels of the boron isoelectromc sequence, perturbed levels are identified by superscript t BI
3s 4s 5s 6s 7s 8s
CII
Present results
Ref. 8
Present results
Ref
0.12252 0 05391 0.03043 0 01955 0 01362 0 01003
0 12252 0.05430 0.03090 0.02026 t 0.01263 t 0.00973
0.36510 0 17702 0 10481 0.06932 0 04924 0_03678
0 36510 0.1796? 0 10624 0 07017 0.04981 -
2.30926
2p 3p 4p 5p
6p 7p 8p
0.30492 0 08163 0.04088 0.02464 0 01648 0.01180 0.00886
3.52616
0.30492 0.08343 0.04165
-
2 80141
3d 4d 5d 6d 7d 8d
0 05617 0.03160 0.02020 0.01401 0.01028 0.OO787
9
0.89592 0 29162 0115126 0_09280 0.06275 0 04527 0 03419
0 23291 0.12951 0,08232 0,05691 0.04167 0,03183 3.30814
0.73543 0.36899 0 22235 0.14866 0.10640 0 07991
NIII Ref
I0
Present results
0.73543 0 37191 0.22323 0.14981 0 10733 0.08069
1 21552 0 62379 0 38024 0.25606 0 18416 0 13881
0 89592 0 29587 0.15557 0 09738 t 0.06229 0 04520
1.74327 0.61838 0 32657 0.20221 0 13754 0 09961 0.07547
1 74327 0.62431 t 0.32350 t 0.20297
-
-
0.52610 0 29206 0 18553 0.12820 0.09386 0.07167 2.43706
1.21552 0 63135 t 0.38737 t
2 84382 1.05864 0 56540 0.35216 0.24042 0 1/457 0.13250
0 52610 0 29589 t 0.18570 0.12837 o 09404 0.07183
0-93334 0 51822 0.32928 0 22759 0.16664 0 12727 1.98482
FV Present results
Ref.
12
1.8o755 0.94170 0.57852 0139148 o 28250 0_21345
1-80755 0.95009 t
1 28422
2.84382 Ii06696 t 0.56664 0.35344
-
1 55801
1 79385
0.23291 0 13005 0.08279 0.05737
OIV Ref. 11
1 55044
1.98355
2_15651
0.05542 0.03160 0.02024 0.01405 0.01031 0.00789
Present results
4.19629 1.61099 0 86719 0_54236 0.37123 0 27002 0 20523
4.19629 1.62194 t
1 39264
0 93334 0.51857 0.32967 0.22789
1.45329 0.8O750 0 51335 0.35493 0.25995 0.19857 1 69103
1.45329 0 80843 0.51387 0 35519
The polarlsed frozen-coreapprox,matlon
113
but different orbital structure. In this category, we include, in particular, the following (the probable pertmbing level is indicated in each case): B(I) 6s, 7s, 8s (2s2p2;2S); C(II) 5p (2s2p(3p°)3s; 2pO), N(III): 3p (2p3;ZP°), 4p (2s2p(3p°)3s;ZP°), 4d (2s2p(3p°)3p,2D); O(IV): 4s, 5s(2s, 2p2,2S), 3p(2p3,2P°), F(V). 4s(2s2p2; 2S)" 3p(2p3; 2pO). All of these levels have been marked by a dagger m Table 1. With these exceptions, agreement is sufficiently good that we feel that the PFC model may be used with confidence to confirm assignments of the term energies of more highly excited levels. 3 OSCILLATOR STRENGTHS Table 2 contains results of calculations of electric dipole oscillator strengths (f-values) for B(I) through F(V). The PFC orbitals and energies were used (except for the 2D series of B(I)) to compute the standard "length" and "velocity" forms of the f-value, according to A = ~ (21r _
1) AE~(P'IrlPr)2
(5)
and
v=3(2lt-l)
(6)
Here, P,/r and Pr[r are the initial and final valence electron radial orbitals, Ae~ = (el - e,) is the excitation energy of the transition, and li(= 1, + 1) is the angular momentum quantum number of the final state valence orbital. The effect of modifying the dipole length operator r so as to take account of core polarisation TM is known to be slight and has been neglected here. There is a slight formal difficulty, arising from the use of different cut-off parameters p in the initial and final states, so that the modified operator cannot even be defined unambiguously; however, the numerical [-values which result from different possible choices of P in the transition element are insignificant For the vast majority of transitions considered here, ft and fo agree to within a few percent, and we therefore present only the geometric mean fg of each pair of results. This mean has the merit of being independent of the excitation energy Ae~, but in view of the generally high quality of our PFC ionisation energies, any inaccuracy in the [-values must be due primarily to errors in the transition matrix elements. As with the earlier FC calculations of f-values, J differences between fl and fo are much larger for transitions which involve a valence 2p electron, although agreement improves steadily with increasing Z A comparison of our length and velocity •-values for N(III) and O(IV) together with results of a parametrised independent particle model (IPM), 19 which employs observed energy level data to fix the parameters, is presented in Table 3. In spite of the good agreement between the PFC velocity values and the IPM (length) values, and the fact that use of a modified length operator should reduce (albeit slightly) the present ft values, bringing them into better agreement with the fu-values, these results are among our least reliable [-values The 2p-3s transitions of the boron sequence are notoriously sensitive to the effects of other competing configurations, with a near cancellation of the oscillator strength at C(II); 3 our PFC model cannot reproduce such an effect. The discrepancies between our PFC •-values and refined SOC results for the 2p-3d, 3s-3p and 3p-3d transitions are generally much smaller, and we present a comparison for several members of the sequence in Table 4 In order to obtain our estimates for the higher ions, we have fitted the PFC •-values at Z = 8, 9 to the asymptotic Z-dependent formula having the form of a [1/1] Pad6 approximant: 2°
f = A + B/(Z- ~).
(7)
2
-0.210 -0.029 -0.010 -0.005 -0.003 -0.002
-0.160 -0.028 -0.010 -0.005 -0.003 -0.002
-0.139 -0 025 -0.009 -0.0~5 -0.003 -0.002
n/m
2 3 4 5 6 7 8
2 3 4 5 6 7 8
2 3 4 5 6 7 8
0.803 -0.356 -0.058 -0.002 -0.011 -0.006
0.951 -0.431 -0.064 -0.023 -0.011 -0.007
1.133 -0.615 -0.058 -0.019 -0.009 -0.005
3
0,056 1.110 -0 582 -0.094 -0.035 -0.018
0.018 1.311 -0.707 -0.102 -0.037 -0.018
0.012 1.585 -1 025 -0 089 -0.029 -0.014
4
0.024 0.053 1.411 -0.809 -0.129 -0.048
0.010 0.014 1.666 -0.983 -0.140 -0.051
0.002 0.028 2.022 -1.430 -0.118 -0.038
5
ns2S - mp2p°
0.012 0 024 0.052 1.710 -1.037 -0.165
0.006 0.009 0.011 2.017 -1.258 -0.177
~ 0 005 0 044 2 451 -1.831 -0.147
6
0.007 0.013 0.024 0.052 2.006 -1.265
0.003 0.005 0.008 0.009 2.367 -1.534
t 0.002 0.008 0.060 2.875 -2 228
7
8
0.002 0.003 0.005 0.008 0.007 2.715
t 0.001 0.003 0.012 0.076 3.297
0.004 0.008 0.013 0.025 0.052 2.302
NIII
CII
BI
0.395 0.411 -0.099 -0.014 -0.005 -0.002 -0.001
0.290 0.572 -0.146 -0.018 -0.006 -0.003 -0.002
0 114 0 857 -0 258 -0 019 -0.006 -0.003 -0.002
3
0.112 0.201 0 659 - 0 219 -0.033 -0.012 -0.006
0.095 0.100 0.888 -0.313 -0.041 -0.014 -0.007
0.049 0.001 1.240 -0.528 -0.041 -0.013 -0 006
4
0 048 0 074 0 151 0.875 -0.347 -0.054 -0 020
0 043 0 046 0 058 1.164 -0.487 -0.065 -0 023
0.025 0.004 0 004 1 571 -0.797 -1.063 -0.020
5
0 026 0 036 0.061 0.128 1.079 -0.479 -0.076
0,024 0.024 0.032 0.040 1.423 -0 665 -0 089
0.014 0 003 % 0.014 1.882 -1.066 -0.084
6
np2p ° - md2D
0.015 0.020 0 031 0 055 0.117 1.275 -0.614
0 014 0 014 0 018 0 024 0.030 1.673 -0 884
0.009 0.002 0.001 0.001 0.026 2.181 -1.333
7
Table 2 Electric dipole oscillator strengths for the boron isoelectronic sequence; weak transitions with I/I < 10 3 are identified by ["
0.010 0.013 0.018 0.029 0.051 0.110 1.467
0 009 0 009 0 011 0 015 0.020 0 024 1 918
0.006 0.002 0.001 t 0.003 0.038 2 473
('] 0
e~
Z
-0,121 -0,022 -0.008 -0.004 -0.002 -0.002
-0 108 -0,020 -0.008 -0 004 -0 002 -0.001
2 3 4 5 6 7 8
2 3 4 5 6 7 8
0.593 -0.267 -0.049 -0.0]9 -0.009 -0.006
0 683 -0.303 -0.053 -0.020 -0,010 -0 006
0.136 0.819 -0 436 -0 079 -0 030 -0 015
0.098 0.944 -0 494 -0 086 -0 032 -0.016
0 047 0.141 ].041 -0.607 -0,110 -0 042
0 037 0 099 1.200 -0.688 -0,119 -0.045
0 022 0 051 0 149 1 262 -0,780 -0 141
0.018 0.039 0.103 1.454 -0 883 -0,152
0.013 0.025 0.054 0.158 1 481 -0 953
0 010 0.019 0.040 0.107 1.707 -1 078
FV
0.006 0.011 0.021 0.042 0,112 1,958
0 OO8 0 014 0.027 0.057 0.168 1.699
OIV
0 499 0 260 -0.063 -0.010 -0.004 -0,002 -0 001
0.458 0 318 -0.076 -0 012 -0.004 -0.002 -0,001
0 122 0.326 0 428 -0 143 -0.025 -0.009 -0.005
0.119 0.274 0 520 -0.171 -0.028 -0.010 -0 005
0 050 0,101 0 279 0 573 -0.231 -0.041 -0.015
0.050 0.09] 0.224 0 695 -0.274 -0.045 - 0 017
0.026 0.046 0.093 0.261 0.710 -0.324 -0.058
0.026 0.042 0.080 0.203 0.859 -0,382 -0.065
0.015 0.025 0.044 0 090 0 256 0.843 -0 420
0.015 0.024 0.039 0.075 0.194 1.018 -0,493
0,010 0.016 0.025 0.044 0.089 0 256 0 973
0.073 0.191 1.174
0.038
0.010 0.015 0.023
=o
R P McEAcHRANand M COHEN
116
Table 3. PFC and IPM/-values In N(III) and O(IV), data from Re/ [19] are ldentllied by superscript t NIII
OIV
Transition
PFC(£)
PFC(v)
IPMt(,)
PFC(V)
PFC(v)
IPMt(Z)
2p-3s 4s 5s 6s 7s
0-0547 0.0098 0.0036 0,0018 0.0010
0 0390 0 0068 0 0025 0 0012 0.0007
0.0342 0.0065 0 0024 0 0012 0 00069
0.0465 0.0085 0 0032 0 0016 0 0009
0 0349 0.0063 0.0024 0 0012 0,0007
0 0303 0,0059 0.0022 0 0011 0,00064
2p-3d 4d 5d 6d 7d
0 4216 0 0520 0 0520 0.0274 0.0164
0 3693 0.0449 0.0449 0.0237 0.0141
0.3740 0.0470 0.0470 0.0248 0 0148
0.4823 0.0524 0.0524 0.0272 0.0161
0 4339 0 0468 0 0468 0 0243 0.0144
0 4385 0.0486 0 0486 0 0253 0.0150
Table 4 A comparison of/-values for the boron lsoelectronlc sequence Transition 2pi3s
2p-3d
3s-3p
BI
CIl
(1) (2) (3)
0 070 0.067 4) 0.055 ~
(I) (2) (3)
0.114
(I) (2)
1 133 1 199
o.197(q~ 0.175"-"
NIII
OIV
0 053 0 008(6, 0.018 )
0 045 0 028t_,
0 040
0.026(Y
0.022 (9)
290 330 _319 (6)
0-395 0,422 ¢ 0 375 (8)
0 458 0.486 t , 0.526 ( %
I
0.028t_,
FV
NeVI
0.036 0 027 t
0.033 t 0.026
0.499
, 0.528 t
0.533 t~ i
0_951 01702
! I
0.567(10) 0.313"
NaVII
0 031 t 0 025 t
0.549 t 0 593 ¢
0 803 0 618 t
0.683 0 536 t
0 593 0 468 t/
o.523 t 0.416
0.469 ¢ 0 365 t
0 411
0 318 0 319 t
0 260 0 251 t
0 219 t 0 203
0 190 t 0 169 t
I
3P-3d
(I) (2)
0 857 0 786
0 572 0 568
I
I 1
0 419 t
I
(I) (2) (3)
PFC values, interpolated values denoted by superscript SOC values from Bef. 3, interpolated values denoted by superscript ¢ Experimental beam-loll values," sources are as follows (4) Ref 22, (3) Ref. 23, (6) Ref 24, (7) Bef_ 25, (8) Ref. 26, (9) Ref 27, (10) Bef 2b
The hmiting hydrogenic value of / is zero for An = 0 transitmns, and numerical values of the hydrogenic transition element have been tabulated 2' for all transitmns of interest, so that exact values of A appear in Table 5. Thus, if our results for O(IV) and F(V) were exact, Eq. (7) would provide a very accurate interpolation formula for higher Z. Many of the SOC comparison data of Table 4 were derived by similar procedures from direct calculations on B(I), C(II), and Ne(VI)
Table 5 Parameters for the asymptotic formula [ Transition
2p-3s 3s-3p 3s-4p 3p-4s 4s-4p 4s-Sp 4p-5s 5s-5p 5ai6p 5p-6s 6s-6p 6s-7p 6p-7s 7s-7p 7s-Sp 7p-Ss
8siBp
=
A + B I ( Z - or)
Transition
0.0136 0.0 o.4847 0 0323 0.0 0.5442 o_o529 0 0 0.6078 0 0745 0 0 0 6736 0.0967 0 0 0 7406 0 1192 0 0
0.1389 4 472 -3.677 0 3267 6 174 -4 288 0_5291 7 868 -4 998 0-7360 9 534 -5.711 0.9393 11.193 -6 435 I 144 12 858
2.795 1.456 -1_360 3 237 1.460 -1.641 3.267 1.445 -1.890 3 245 I_444 -2 074 3 246 I 441 -2 232 3 236 1 433
2p-3d 3P-3d 3p-4d 3d-4p 4p-4d 4p-5d 4d-5p 5p-5d 5p-6d 5d-6p 6p-6d 6p-Td 6di7p 7p-7d 7p-8d 7d-8p 8p-8d
0.6958 0.0 0-6183 0_0110 0.0 O.6O93 0.0278 0_0 0.6247 0 0480 O0 0 6514 0,0702 O0 0.6843 0.0939 0.0
-I.140 I _406 -I 920 0 1143 2 411 -2 301 0 2582 3 280 -2 627 0 4107 4 I]9 -2 934 0.5685 4-895 -3.246 0 7278 5 683
3 216 3 585 2 432 4_688 3_362 2 036 4.548
3 278
1 768 4.479 3,204 1 585 4 427 3 191 1 426 4 393 3 158
The polarlsed frozen-coreapproximation
117
Table 4 also contains a few selected results of beam foil experiments of transitions involving the ground states. 22-28 While the absolute accuracy of any mdividual measurement is difficult to assess, we see that the PFC model is not completely satisfactory for these transitions. But, for all excited states not subject to significant perturbation effects, the PFC one-electron model clearly provides a good description of the levels, and it may therefore be expected to yield /-values similar to those of the Coulomb approximation 29 The present method enjoys an important advantage over the Coulomb approximation, since the latter requires a priori knowledge of the term energies for its application. In summary, our present results, taken together with earlier calculations on the beryllium lsoelectronic sequence, 3° suggest that the PFC procedure is capable of yielding reliable/-values for most transitions between excited states. Acknowledgements--We are grateful to Messrs G Hermanand J. Stlberstein for computationalassistance The work was supported, in part, by the Natural Sciences and EngineeringResearch Councdof Canada under Grant A-3629and, in part, by the Central Research Fund of the Hebrew Unlverszty
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