Hartree-Fock Multiplet Strengths for K I, Ca II, and Sc III

Abstract

Analytical approximations to Hartree-Fock wavefunctions have been computed for the 4s, 4p, and 3d states of K I, Ca II, and Sc III. Multiplet strengths for the 4s–4p and 4p–3d transitions in these ions are also computed. Examination of the results indicate that relaxation of the core in the field of the series electron accounts for a large portion of the core polarization effect, at least for the 3d level. Serious cancellation is found to invalidate the dipole velocity results of the 4p–3d transition in Ca II.

1. Introduction

In view of the astrophysical importance of singly ionized calcium (Ca II), it is not surprising that a number of calculations should have been made for the most important transition integrals, notably for the 4s–4p and 4p-3d transitions. The current status of these calculations is summarized in the paper by Douglas and Garstang [1962]. Attention is drawn there to the still rather large uncertainty in the 4p–3d transition, as indicated by the scatter in the computed values, as well as the discrepancy between the length and velocity forms, where they have been calculated.

In view of the basic simplicity of the spectrum, it seems reasonable to expect the self-consistent field model to provide a fairly good representation of the ion and, perhaps, shed some light on this problem. This paper, therefore, reports on a study of the Hartree-Fock approximation as applied to Ca II. Hartree-Fock wavefunctions, and the accompanying transitions, are computed for the 4s, 4p, and 3d levels. To better assess the results, the same calculations were done on the neighboring isoelectronic systems, neutral potassium (K I) and doubly ionized scandium (Sc III).

All of these calculations were done by the expansion method, using the University of Chicago program for the IBM 7094 computer [Roothaan and Bagus, 1963].

2. Review of Theory

The configuration for members of the potassium isoelectronic sequence consists of a closed shell core plus a single series electron, i.e.,

$${\left(1s\right)}^2{\left(2s\right)}^2{\left(2p\right)}^6{\left(3s\right)}^2{\left(3p\right)}^{6}nl,$$

and, here, nl = 4s, 4p, and 3d. In an independent particle model, the total wave function is a single determinant of one-electron functions (orbitals) with each electron assigned to a single function in accordance with the aufbau principle. Application of the variational principle to this type of trial function then leads to the Hartree-Fock self-consistent field (SCF) equations for the orbitals,

$$F_i\left(\phi \right){\phi }_i={ϵ}_i{\phi }_i,{\phi }_i=1s,2s,\dots ,3p,nl.$$ (1)

F_i, the Fock operator, has the form of a one-electron Hamiltonian whose potential represents the average field of all the other electrons of the atom as seen by an electron in the ith orbital. There is a different equation for all the occupied orbitals, and the usual procedure is to solve this set of equations iteratively until the entire set of orbitals is self-consistent.

This brief description of the SCF procedure is not entirely out of place, since this is the point where the present calculations differ from all those previously done on Ca II. In Hartree’s original work, both with and without exchange (Hartree and Hartree, 1938), the core functions were carried over directly from the SCF calculations on the core ion, Ca⁺², and only the last of the eqs (1) were solved for the series electron. The same is true of the “core polarization” calculations, where a term representing polarization of the core by the outer electron is added onto the potential in the Fock operator. Neglecting the relaxation of the core is usually a reasonably good approximation for alkali type atoms and ions, since the series electron is normally well separated from the core; e.g., the K-shell of Li and Li⁺ are essentially the same. However, as will be seen, this spatial separation is not always achieved for potassium-like systems, with some interesting consequences.

The multiplet strength for an electric dipole transition nl → n′l′ is given by

$$S=\mathcal{S}{\sigma }^2$$ (2)

where $\mathcal{S}$ is an angular factor readily available in literature tabulations [Allen, 1963], and σ is the transition integral. If the core functions don’t change significantly for the two states, σ is simply given by

$$\sigma ={\left(4l_{>}^2-1\right)}^{-1/2}{\int }_0^{\infty }dr{r}^3{R}_{nl}{R}_{n\text{'}l\text{'}},$$ (3)

where l_> is the greater of the two l-values and R_nl is the radial function for the nl-electron. If the core changes enough to make the nonorthonormality of the orbitals from the two states significant, then σ is altered by the inclusion of nonorthogonality and exchange terms. For example, for the 4p–3d transition σ would become

$$\sigma =\sigma \left(4p,3d\right)-\langle 4p|3p\rangle \sigma \left(3p,3d\right)$$ (4)

where 〈4p|3p〉 is an overlap integral. While this change of the core orbitals from excited state to excited state is normally small for alkali-like systems, it does make some difference in a few cases, which will be discussed as they arise.

An alternative form of the transition integral (3) is the dipole velocity form [Chandrasekhar, 1945; Bethe and Salpeter, 1957], which is given by

$$\sigma =\frac{1}{\text{Δ}E}{\left(4l_{>}^2-1\right)}^{-1/2}{\int }_0^{\infty }drr\left[R_{nl}\frac{d}{dr}\left(r{R}_{n\text{'}l\text{'}}\right)\pm l_{>}{R}_{nl}{R}_{n\text{'}l\text{'}}\right],$$ (5)

where ΔE is the energy difference of the transition, and the upper sign is taken if l′ is l_> and the lower if l′ is l_<. While these two forms must give the same value for exact wavefunctions, they need not, and usually do not agree exactly for approximate wavefunctions. Although their agreement is only a necessary and not sufficient condition, the relative disparity does provide another handle on assessing the reliability of a calculation. For alkali-like systems, where an independent particle model should be particularly good, one might attach a little more significance to the length-velocity agreement. In fact, one of the problems here is the fact that a number of independent particle type of calculations give a wide discrepancy between the length and velocity forms for the 4p–3d transition in Ca II.

3. Results and Discussion

Analytical approximations to the Hartree-Fock wavefunctions were computed for the 4s, 4p, and 3d states of K I, Ca II, and Sc III [Roothaan and Bagus, 1963]. Based on comparisons with other calculations, where available [Clementi, 1965; Synek and Roothaan, 1966], and calculations with smaller basis sets, they are estimated to be accurate to about 3 decimal places; i.e., there may be a deviation from the true Hartree-Fock function of about 0.001. The computed total energies are given, in atomic units,¹ in table 1, where the spacing with respect to the ground state is also computed and compared with the spectroscopically observed level positions [Moore 1949]. The wavefunctions themselves are given in the appendix.

Table 1.

Computed and observed energies (in a.u.) for the potassium isoelectronic sequence

Atom or ion		Total energy	Energy relative to ground state
		Hartree-Fock	Hartree-Fock	Observeda
K I	3p64s 2S	−599.1646	0.0	0.0
	4p 2P	−599.1130	.0516	.0593
	3d 2D	−599.0755	.0891	.0981
Ca II	3p64s 2S	−676.5698	0.0	0.0
	3d 2D	−676.4981	.0717	.0624
	4p 2P	−676.4635	.1063	.1155
Sc III	3p63d 2D	−759.0921	0.0	0.0
	4s 2S	−758.9805	.1116	.1158
	4p 2P	−758.8222	.2699	.2839

^aMoore, 1949.

The positions of the energy levels are computed with fair accuracy – the errors range from 0.1 to 0.3 eV. This is about the level of accuracy one can reasonably expect from the Hartree-Fock approximation for such systems. Differences in the correlation energy between the outer electron and the core can easily amount to the order of 0.1 eV. This correlation error can be taken into account by configuration mixing, of course. Alternatively, since it is due to the deviations from the Hartree-Fock averaging over electronic interactions, one can also include the effect by introducing a polarization potential into the series electron equation, which represents the “instantaneous distortion” of the core from spherical symmetry due to the outer electron. As expected, polarization potential calculations do indeed give an improved term scheme [Douglas, 1956, Trefftz and Biermann, 1952]. As mentioned earlier, however, all the calculations done on Ca II used the frozen core approximation, with orbitals taken from Ca III, and it should be somewhat instructive to look more closely at the effects of relaxation.

Table 2 shows the location of the principal maxima of the series electron orbitals compared with that of the 3p core function. The 3p value given here is that of the argon-like ion, the change in the principal maximum for potassium-like systems being small. What one sees, of course, is the well-known phenomenon of the 3d electron being buried in the body of the atom as you move along the isoelectronic sequence in the direction of increasing positive ionicity. Only for neutral potassium can the 3d electron be thought of as being well separated from the core, and for doubly ionized scandium, it occupies essentially the same region of space as the 3p core electrons.

Table 2.

Location of principle maxima (in a.u.) for potassium-like wavefunctions

	3p	4s	4p	3d
K I	1.16	4.34	5.92	8.30
Ca II	1.05	3.25	3.96	1.46
Sc III	0.97	2.72	3.15	1.10

While the relaxation of the 3p function is small, it introduces enough of a change in the field seen by the series electron to make a significant change over the frozen core approximation. This is shown in figure 1, where the Hartree-Fock 3d for Ca II is compared with the frozen core SCF with exchange. Also shown is the 3d function of Douglas and Garstang [1962] computed with a polarization term in the potential, which agrees remarkably well with the Hartree-Fock function, suggesting that, to a large extent, the polarization is serving to mock up the core relaxation effect. The difference between the Hartree-Fock and polarization 3d is now comparable to the changes found by Douglas for the 4s and 4p orbitals.

Figure 1.

Radial 3d-functions for Ca II.

P_3d = rR_3d.

The Hartree-Fock multiplet strength data for the 4s–4p and 4p–3d transitions in these ions are given in table 3. The area ratio column refers to the ratio of the positive and negative parts of the principal transition integral for either eq (3) or (5), which, of course, is a measure of the degree to which cancellation is important. As can be seen, the agreement of the length and velocity values ranges from fairly good to excellent – the mean of the two ±20 percent easily encompasses both values. The sole exception to this observation is the 4p–3d transition in Ca II, which stands out like the proverbial sore thumb. It seems quite clear that cancellation in the velocity integral is giving a wrong result, and, since this makes the principal transition integral small, the core nonorthogonality effects are accentuated in this case. The inclusion in σ of an exchange term as shown in (4) changes S from 5.0 to 9.7. In the case of Sc III the velocity form of S is changed from 3.89 to 4.86. This cancellation has also been found in a recent calculation by Garstang and Hill [Garstang and Hill, 1966] of the velocity integral for Hartree’s SCF functions with frozen core.

Table 3.

Hartree-Fock results for the potassium isoelectronic sequence transitions

Transition	Wave-length	Type	Multiplet strength	Area ratio
4s-4p	Å
K I	7684	Length	62.1	2.8 ×104
		Velocity	44.9	17.3
Ca II	3945	Length	30.7	1.9 ×104
		Velocity	24.6	9.16
Sc III	2711	Length	20.0	2.7 ×104
		Velocity	16.8	6.72
4p-3d
K I	11743	Length	224	933
		Velocity	218	9.86
Ca II	8581	Length	21.6	29.2
		Velocity	5.0	1.2
Sc III	1605	Length	5.51	11.4
		Velocity	3.89	1.94

Finally, table 4 collects together for comparison purposes the transition integral calculations on Ca II by a number of other authors. It can be seen that there is substantial agreement among all these calculations, including the present Hartree-Fock values. Furthermore, it appears that, for the 4p–3d transition, the effect of relaxing the core in the field of the series electron (Hartree-Fock) is essentially the same as including a polarization potential with a frozen core (as illustrated by both the Douglas and Garstang and Trefftz and Biermann results). The Hartree-Hartree value is for a frozen core without polarization.

Table 4.

Comparison of calculations of the transition integral (σ²) for calcium II

	4s-4p		4p-3d
	Length	Velocity	Length	Velocity
Hartree-Fock (this paper)	5.13	4.09	0.36	0.084
Douglas & Garstang 1962	4.74	4.45	.35	.095
Trefftz & Biermann 1952	4.76	4.47	.31	.11
Hartree & Hartree 1938	5.15	4.1	.46	.005
Bates & Damgaard 1949	4.38		.25

4. Conclusions

The first conclusions to be drawn concern the wavefunctions. It seems clear, a priori, that neglecting core relaxation when there is a significant amount of penetration should have a marked effect on the wavefunctions, and the present calculations bear this out for the rather simple case of an alkali-like spectrum. It is interesting to note that the ad hoc introduction of a polarization potential in such a case serves to reproduce the bulk of the relaxation effect.

As for the multiplet strengths themselves, the values computed here are probably fairly reliable. There are three considerations on which this is based: (1) the overall agreement with a number of other authors on Ca II, (2) the generally good agreement of the length and velocity forms of the multiplet strengths, and (3) the fundamental soundness of an independent particle model for such a system. It is suggested, therefore, that one should take the mean of the length and velocity results of table 3, with an uncertainty of about 25 percent, which encompasses all the directly computed numbers. The sole exception to this, of course, is the 4p–3d transition of Ca II, where the large cancellations occur for the velocity form. In this case, the length value is probably good to about 25 percent.

5. Appendix

The Hartree-Fock orbitals are represented, in the expansion method, as a linear combination of analytical basis functions,

$${\phi }_p=\sum _i{C}_i^p{\chi }_i$$ (1A)

where the $C_i^p$ are the expansion coefficients and χi the basis functions. The basis functions used here are the normalized Slater type orbitals, whose radial parts are of the form,

$${\chi }_n={(2\zeta )}^{n+1/2}{\left[\left(2n\right)!\right]}^{-1/2}{r}^{n-1}{e}^{-\zeta r}$$

when n is an integer, and ζ is simply a variational parameter.

The detailed results of the present calculations are given in the accompanying tables. Table A1 gives the orbital energies, and tables A2 to A4 give the basis function parameters and expansion coefficients.

Table A1.

Orbital energies

State	Orbital	K I	Ca II	Sc III
3p64s 2S	1s	133.5332	149.6095	166.7134
	2s	14.4901	17.0668	19.9253
	3s	1.7488	2.4852	3.3591
	4s	0.1475	0.4163	0.7670
	2p	11.5194	13.8742	16.5093
	3p	0.9544	1.5817	2.3462
3p64p 2p	1s	133.5873	149.6572	166.7556
	2s	14.5435	17.1134	19.9658
	3s	1.8010	2.5290	3.3959
	2p	11.5736	13.9226	16.5527
	3p	1.0081	1.6291	2.3885
	4p	0.0956	0.3095	0.6078
3p63d 2D	1s	133.6251	149.4658	166.4673
	2s	14.5803	16.8979	19.6392
	3s	1.8357	2.3264	3.1168
	2p	11.6104	13.7095	16.2301
	3p	1.0428	1.4403	2.1294
	3d	0.0581	0.3570	0.9045

Table A2.

Basis function parameters and expansion coefficients for the 3p⁶4s ²S state

K I						Ca II					Sc III
n	ζ	1s	2s	3s	4s	ζ	1s	2s	3s	4s	ζ	1s	2s	3s	4s
1	16.51	0.94593	−0.37901	0.12755	−0.02456	17.5	0.95615	−0.38207	0.13404	−0.03780	18.27	0.96562	−0.38648	0.14080	−0.04766
1	28.3	.11551	.01461	−.00603	.00115	30.31	.10429	.01556	−.00644	.00183	32.33	.10188	.01414	−.00614	.00210
2	7.85	−.00193	.39322	.04237	−.02316	8.413	−.00302	.22229	.16638	−.05761	8.6	−.00644	.14781	.43430	−.17477
2	6.6	.00204	.71064	−.52685	.12159	7.26	.00290	.87831	−.67436	.20656	7.85	.00567	.95158	−.97252	.36531
3	20.54	−.07064	−.02936	.00979	−.00128	21.85	−.06935	−.03115	.01020	−.00300	22.93	−.07782	−.03014	.01013	−.00359
3	3.55	−.00024	.01267	.62787	−.15762	3.87	−.00030	.01908	.60049	−.20050	4.3	−.00020	.02492	.51871	−.20523
3	2.35	.00008	−.00263	.52163	−.07810	2.69	.00012	−.00615	.54961	−.17587	3.08	.00001	−.00883	.64088	−.26931
4	1.16	−.00002	.00058	.00163	.46183	1.59	−.00003	.00160	.00236	.37627	1.84	.00003	.00254	.00429	.37737
4	.71	.00001	−.00027	.00016	.61784	1.1	.00001	−.00072	.00062	.70947	1.414	−.00002	−.00134	.00041	.72176
		2p	3p				2p	3p				2p	3p
2	6.286	0.71952	−.25478			6.784	.73446	−.28229			7.342	.76631	−.31149
2	9.92	.28940	−.07577			10.66	.27464	−.07414			11.575	.24083	−.06541
2	17.2	.01941	−.00647			18.45	.01684	−.00626			19.2	.01547	−.00645
3	3.28	.00964	.64835			3.59	.01012	.68092			3.95	−.00034	.51232
3	1.565	.00094	.14319			1.8	.00093	.10055			2.61	.04655	.69577
4	2.486	−.00275	.34640			2.78	−.00291	.35341			3.02	−.03876	−.10589

Table A3.

Basis function parameters and expansion coefficients for the 3p⁶4p ²P state

K I					Ca II				Sc III
n	ζ	1s	2s	3s	ζ	1s	2s	3s	ζ	1s	2s	3s
1	16.51	0.94593	−0.37898	0.12748	17.5	0.95614	−0.38200	0.13392	18.27	0.96560	−0.38636	0.14065
1	28.3	.11551	.01460	−.00603	30.31	.10429	.01555	−.00644	32.33	.10189	.01413	−.00613
2	7.85	−.00188	.39201	.04019	8.413	−.00289	.21844	.16287	8.6	−.00648	. 13813	.42781
2	6.6	.00199	.71191	−.52395	7.26	.00277	.88234	−.66979	7.85	.00571	.96161	−.96477
3	20.54	−.07064	−.02926	.00993	21.85	−.06935	−.03088	.01037	22.93	−.07780	−.02975	.01031
3	3.55	−.00022	.01207	.62380	3.87	−.00025	.01706	.59471	4.3	−.00023	.02159	.51465
3	2.352	.00006	−.00199	.52552	2.69	.00007	−.00393	.55575	3.072	.00004	−.00535	.64643
		2p	3p	4p		2p	3p	4p		2p	3p	4p
2	6.286	0.71945	−0.25546	0.03567	6.784	0.73446	−0.28226	0.06933	7.348	0.76567	−0.31147	0.09933
2	9.92	.28947	−.075811	.01013	10.66	.27466	−.07476	.01803	11.575	.24061	−.06584	.01987
2	17.2	.01940	−.00650	.00092	18.45	.01684	−.00620	.00146	19.2	.01545	−.00635	.00199
3	3.29	.00958	.63426	−.09549	3.59	.01002	.67754	−.18545	3.95	.01238	.53467	−.18053
3	1.713	.00144	.19711	−.01853	1.9	.00108	.11528	.00469	2.65	.00794	.57297	−.23873
4	2.51	−.00317	.30062	−.04244	2.79	−.00302	.33920	−.10153	3.25	−.01014	−.00267	.03481
4	.88	−.00005	.00100	.37912	1.11	.00007	−.00124	.68637	1.39	.00012	.00253	.83349
4	.541	.00001	−.00010	.68670	0.818	−.00007	.00097	.35988	1.073	−.00011	−.00021	.23930

Table A4.

Basis function parameters and expansion coefficients for the 3p⁶3d ²D state

K I					Ca II				Sc III
n	ζ	1s	2s	3s	ζ	1s	2s	3s	ζ	1s	2s	3s
1	16.51	0.94593	−0.37899	0.12748	17.5	0.95618	−0.38196	0.13225	18.27	0.96566	−0.38647	0.13823
1	28.3	.11551	.01461	−.00603	30.31	.10429	.01553	−.00631	32.33	.10188	.01417	−.00603
2	7.85	−.00188	.39207	.04193	8.4	−.00315	.18815	.19668	8.6	−.00682	.13839	.42668
2	6.6	.00199	.71187	−.52605	7.3	.00300	.91131	−.69812	7.85	.00602	.96141	−.95457
3	20.54	−.07064	−.02926	.00981	21.85	−.06938	−.03027	.00940	22.93	−.07784	−.02962	.00980
3	3.55	−.00022	.01201	.62716	3.87	−.00024	.01783	.61535	4.28	−.00023	.02090	.54995
3	2.347	.00005	−.00196	.52287	2.596	.00006	−.00393	.53941	2.94	.00004	−.00460	.61398
		2p	3p			2p	3p			2p	3p
2	6.286	0.71947	−0.25626		6.784	0.73405	−0.27727		7.338	0.76397	−0.3042
2	9.92	.28947	−.07533		10.66	.27507	−.07295		11.575	.24297	−.06405
2	17.2	.01941	−.00659		18.45	.01681	−.00615		19.2	.01516	−.00627
3	3.29	.00952	.64124		3.58	.01013	.67860		3.93	.01324	.64050
3	1.67	.00120	.16721		1.62	.00088	.07911		2.26	.00289	.22938
4	2.49	−.00292	.32621		2.7	−.00264	.38395		3.12	−.00565	.25999
		3d				3d				3d
3	0.359	0.97724			2.37	0.41387			2.85	0.42944
3	1.4	.07898			1.06	.63671			1.59	.52111
3	7.88	.00162			8.72	.01173			9.26	.01865
3	3.4	.02887			4.53	.12310			4.85	.16484