A universal Gaussian basis set for positive and negative ions from H through Xe

Chemical Physics 253 (2000) 21±26

www.elsevier.nl/locate/chemphys

A universal Gaussian basis set for positive and negative ions from H through Xe F.E. Jorge *, M.L. Franco Departamento de Fõsica ± CCE, Universidade Federal do Espõrito Santo, 29060-900 Vit oria, ES, Brazil Received 6 July 1999

Abstract The generator coordinate Hartree±Fock (GCHF) method is used to generate a universal Gaussian basis set (UGBS) for positive He‡ ±Xe‡ and negative Hÿ ±Iÿ ions. For all ions studied, our ground state Hartree±Fock (HF) total energies are better than those obtained with a smaller UGBS generated previously by da Silva and Trsic [Can. J. Chem. 74 (1996) 1526]. For the cations from He‡ through Xe‡ and the anions from Hÿ through Iÿ , the largest di€erence between the corresponding energies calculated with the GCHF and the numerical HF method is always lower than 7.850 mhartree. For some cations, the ionization potential is evaluated and compared with the corresponding experimental values. Ó 2000 Elsevier Science B.V. All rights reserved.

1. Introduction Historically, it has been necessary to restrict the size of basis sets employed in molecular calculations to a reasonably small number of functions in order to keep the computation cost tractable. However, to achieve high accuracy, moderately large basis sets are ultimately required, especially if a signi®cant fraction of the molecular correlation energy is to be recovered. As the ¯exibility of a basis set is generally better as the number of functions is increased, the need to optimize parameters becomes less important. This led to the suggestion of Silver et al. [1±3] that a single moderately large basis set, with a reasonable choice of parameters, might be suitable for various atoms.

* Corresponding author. Tel.: +5527-335-2821; fax: +5527335-2460. E-mail address: [email protected] (F.E. Jorge).

Such a basis set was termed a universal basis set. This concept has been explored at the matrix Hatree±Fock (HF) level for a series of atom sets [1±4]. In 1986, Mohallen et al. [5] introduced the generator coordinate HF (GCHF) method. Using this method, da Silva and Trsic [6] presented a universal Gaussian basis set (UGBS) for positive and negative ions of the atoms H through Xe. Recently, Jorge et al. have used the GCHF method with success to generate UBSs [7±10] and adapted Gaussian basis sets [11,12] for light and heavy atoms. In the following, we present an accurate UGBS for mono-positive and mono-negative ions of the atoms H through Xe. The HF total energies of the ground states are calculated and compared with the best HF results available in the literature. Besides this, we compare the ionization potential (IP) obtained in this work for some cations with experimental values.

0301-0104/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 ( 9 9 ) 0 0 3 7 8 - X

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F.E. Jorge, M.L. Franco / Chemical Physics 253 (2000) 21±26

2. Theory

3. Results and discussion

The GCHF method [5] is based on choosing the one-electron functions as the continuous superpositions Z …1† Wi …1† ˆ /i …1; a†fi …a† da; i ˆ 1; . . . ; n;

The HF self-consistent ®eld calculations are performed, for the cations He‡ ±Xe‡ and the anions Hÿ ±Iÿ , by employing the GTF exponents generated with the GCHF method presented in Section 2. Throughout the calculations, we used the scaling factor A (see Eq. (3)) equal to 6.0. For each ion, we searched the best starting point (Xmin ) for each s, p, and d symmetry. For all symmetries of all ions, the best increment value DX found in this work is equal to 0.127 (full details about these wave functions are available upon request to the e-mail address: [email protected]). This optimum DX value was obtained after several tests with all ions studied here, and considering an error in our total energies always smaller than 10 mhartree. Table 1 gives the exponents of our UGBS. The

where n is the number of one-electron functions of the system, /i are the generator functions (Gaussian-type functions ± GTSs ± in our case), fi are the weight functions and a is the generator coordinate. Using Eq. (1) to build a Slater determinant for the multielectronic wave functions, we minimize the total energy E with respect to the fi (a) and arrive at the Grin±Hill±Wheeler±HF (GHWHF) equations Z ‰ F …a; b† ÿ ei S…a; b†Šfi …b† db ˆ 0; i ˆ 1; . . . ; n; …2† where F and S are Fock and overlap kernels, respectively (for more details about these kernels see Ref. [5]). The GHWHF equations are integrated using a procedure known as integral discretization, ID [13]. The ID technique is implemented through a relabeling of the generator coordinate space, i.e., X ˆ ln

a ; A

A > 1;

…3†

where A is a scaling factor that is numerically determined. In the new generator coordinate space X, an equally spaced N-point mesh {Xi } is selected, and the integration range is characterized by a starting point Xmin , an increment DX, and N (number of discretization points). The highest value (Xmax ) for the generator coordinate is related to the former values through Xmax ˆ Xmin ‡ …N ÿ 1† DX:

…4†

The choice of discretization points determines the exponents of the GTFs. An extensive discussion about the similarity between the even-tempered formula [14] and the ID technique used in the GCHF method is given in Ref. [9].

Table 1 The exponents of the UGBF 0.00738725 0.01582760 0.03391154 0.07265740 0.15567263 0.33353749 0.71462311 1.53112078 3.28051361 7.02868758 15.05936414 32.26554685 69.13077495 148.11662941 317.34832892 679.93690017 1456.80360061 3121.28482837 6687.53082139 14328.41632413 30699.44944418 65775.32190966 140927.37982765 301944.95151791 646934.28529397 1386093.61535888 2969784.65079438 6362933.04749865 13632947.07450281 29209366.89869088

F.E. Jorge, M.L. Franco / Chemical Physics 253 (2000) 21±26

23

our UGBS, with the UGBS from Ref. [6], and the energy values (in hartree) obtained with the numerical HF (NHF) method [15]. For all positive ions studied, Table 2 shows that our UGBS is equal to or larger in size than the UGBS from Ref. [6]. From this table, we can see that our total energy results are always lower than those obtained by da Silva and Trsic [6]. Besides this, we can also see from Table 2 that the DE values obtained by us for Xe‡ ±Sr‡ , Y‡ ±Tc‡ and Ru‡ ±Xe‡ are approximately 10, 40, and 100 times,

exponents of all symmetries of all ions studied here are subsets of the 30 values given in this table. From our calculations, we veri®ed that the Xmin values for all symmetries of the anions are in general lower than the corresponding ones of the cations. This result was expected because the electronic wave functions of the anions are more di€used than those of the cations. Tables 2 and 3 give, respectively, the deviations DE (in mhartree) of the energies for the cations He‡ ±Xe‡ and the anions Hÿ ±Iÿ computed with

Table 2 Deviations DE (in mhartree) of the UGBS ground state total energies from the numerical HF values (in hartree) for positive ions Z 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38

Ion ‡ 2

He ( S) Li‡ (1 S) Be‡ (2 S) B‡ (1 S) C‡ (2 P) N‡ (3 P) O‡ (4 S) F‡ (3 P) Ne‡ (2 P) Na‡ (1 S) Mg‡ (2 S) Al‡ (1 S) Si‡ (2 P) P‡ (3 P) S‡ (4 S) Cl‡ (3 P) Ar‡ (2 P) K‡ (1 S) Ca‡ (2 S) Sc‡ (3 D) Ti‡ (4 F) V‡ (5 D) Cr‡ (6 S) Mn‡ (7 S) Fe‡ (6 D) Co‡ (3 F) Ni‡ (2 D) Cu‡ (1 S) Zn‡ (2 S) Ga‡ (1 S) Ge‡ (2 P) As‡ (3 P) Se‡ (4 S) Br‡ (3 P) Kr‡ (2 P) Rb‡ (1 S) Sr‡ (2 S)

Con®guration 1

1s 1s2 [He]2s1 [He]2s2 [He]2s2 2p1 [He]2s2 2p2 [He]2s2 2p3 [He]2s2 2p4 [He]2s2 2p5 [He]2s2 2p6 [Ne]3s1 [Ne]3s2 [Ne]3s2 3p1 [Ne]3s2 3p2 [Ne]3s2 3p3 [Ne]3s2 3p4 [Ne]3s2 3p5 [Ne]3s2 3p6 [Ar]4s1 [Ar]4s1 3d1 [Ar]4s1 3d2 [Ar]4s0 3d4 [Ar]4s0 3d5 [Ar]4s1 3d5 [Ar]4s1 3d6 [Ar]4s0 3d8 [Ar]4s0 3d9 [Ar]4s0 3d10 [Ar]4s1 3d10 [Ar]4s2 3d10 [Ar]4s2 3d10 [Ar]4s2 3d10 [Ar]4s2 3d10 [Ar]4s2 3d10 [Ar]4s2 3d10 [Ar]4s2 3d10 [Kr]5s1

4p1 4p2 4p3 4p4 4p5 4p6

No. of GTFs

DE (UGBS)

DE (UGBS)a (18s 12p 11d)

18s 18s 18s 18s 18s 18s 18s 18s 18s 18s 19s 19s 19s 18s 18s 18s 18s 20s 21s 20s 20s 20s 20s 21s 21s 20s 20s 20s 22s 22s 22s 22s 21s 22s 22s 21s 23s

0.000145 0.000565 0.00835 0.02173 0.04749 0.03960 0.07615 0.1361 0.0868 0.1285 0.6739 0.8802 1.3085 2.2095 2.6424 3.5684 4.5672 1.1868 2.6523 5.2012 5.2164 1.4708 1.729 3.499 4.319 2.763 3.133 3.931 4.934 3.840 4.103 5.857 5.398 2.990 2.869 6.774 7.615

0.007 0.02 0.07 0.11 0.15 0.27 0.34 0.41 0.58 0.79 1.1 1.4 1.9 2.6 3.3 4.0 5.2 6.7 8.7 10 12 13 15 17 20 24 28 32 37 41 45 50 55 61 69 78 90

Present work

13p 13p 13p 13p 13p 13p 13p 13p 14p 14p 14p 14p 15p 13p 14p 13p 13p 13p 13p 13p 13p 13p 13p 13p 13p 13p 16p 15p 16p 16p 16p 16p 16p

10d 10d 10d 10d 10d 10d 10d 10d 10d 10d 10d 10d 11d 11d 11d 11d 11d 11d

c

NHFb )2 )7.236415201 )14.27739481 )24.23757518 )37.29222377 )53.88800501 )74.37260568 )98.83172020 )127.8178141 )161.6769626 )199.3718097 )241.6746705 )288.5731311 )340.3497759 )397.1731828 )459.0485907 )526.2745343 )599.0175794 )676.5700126 )759.5391440 )848.2034008 )942.6707837 )1043.139393 )1149.649383 )1262.213012 )1381.128750 )1506.591099 )1638.728242 )1777.567545 )1923.059722 )2075.086491 )2233.888335 )2399.558574 )2572.045211 )2751.567394 )2938.219931 )3131.373777

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F.E. Jorge, M.L. Franco / Chemical Physics 253 (2000) 21±26

Table 2 (continued) Z 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54

Ion ‡ 1

Y ( S) Zr‡ (4 F) Nb‡ (5 D) Mo‡ (6 S) Tc‡ (7 S) Ru‡ (4 F) Rh‡ (3 F) Pd‡ (2 D) Ag‡ (1 S) Cd‡ (2 S) In‡ (1 S) Sn‡ (2 P) Sb‡ (3 P) Te‡ (4 S) I‡ (3 P) Xe‡ (2 P)

Con®guration 2

[Kr]5s [Kr]5s1 [Kr]5s0 [Kr]5s0 [Kr]5s1 [Kr]5s0 [Kr]5s0 [Kr]5s0 [Kr]5s0 [Kr]5s1 [Kr]5s2 [Kr]5s2 [Kr]5s2 [Kr]5s2 [Kr]5s2 [Kr]5s2

4d2 4d4 4d5 4d5 4d7 4d8 4d9 4d10 4d10 4d10 4d10 4d10 4d10 4d10 4d10

5p1 5p2 5p3 5p4 5p5

No. of GTFs

DE (UGBS)c

DE (UGBS)a (18s 12p 11d)

23s 23s 22s 23s 26s 25s 25s 24s 24s 25s 25s 25s 25s 24s 25s 26s

6.602 7.182 3.351 3.111 2.241 2.646 2.110 2.595 2.824 3.775 4.247 5.076 5.455 7.197 6.021 5.818

110 120 140 160 180 200 230 250 280 310 330 370 410 460 510 580

Present work 16p 16p 16p 16p 16p 16p 17p 17p 17p 16p 16p 18p 18p 18p 18p 18p

11d 12d 13d 13d 13d 13d 14d 13d 13d 14d 14d 13d 13d 13d 13d 13d

NHFb )3331.472882 )3538.809305 )3753.389513 )3975.333703 )4204.594360 )4441.321956 )4685.664172 )4937.675930 )5197.481334 )5464.878609 )5739.978392 )6022.678323 )6313.165941 )6611.503394 )6917.627273 )7231.708947

a

Results obtained from Ref. [6]. Numerical HF total energies obtained from Ref. [15]. c Results obtained by using our UGBS. b

respectively, better than the corresponding ones obtained with the UGBS from Ref. [6]. Table 2 shows that the errors in the energy values computed with our UGBS are always smaller than 7.620 mhartree. For the cations Li‡ ±Zr‡ and Te‡ ± Xe‡ , our energy values are larger than the results obtained by Koga et al. [16] using fully optimized Slater-type functions (STFs) basis sets, whereas for Nb‡ ±Sb‡ , the opposite occurs. For the case of the negative ions (Table 3), we see that our UGBS is equal to or larger in size and that our energy values are always better than the UGBS from Ref. [6]. From Table 3, we can also see that for Snÿ ±Iÿ ions, the DE values obtained by da Silva and Trsic [6] are approximately 400 times larger than our corresponding results computed and that, for all anions studied, the di€erences between the total energies calculated with our UGBS and those obtained with the NHF method [15] are always smaller than 7.850 mhartree. For the anions from Hÿ through Yÿ and Iÿ , the energy results obtained by Koga et al. [16] with their fully optimized STFs basis sets are better than those computed with our UGBS, whereas for Zrÿ ±Teÿ , the opposite occurs. We recall that, for all cations and anions studied here, our HF total energy values (see Tables 2

and 3) are better than the corresponding ones calculated with a previous UGBS [6], and that our basis set sizes are always equal to or larger than the UGBS size of Ref. [6]. We consider that the main di€erence between the two UGBSs is not due to the basis set sizes, but to the fact that in Ref. [6], the calculations used ®xed values for the starting points of the s, p, and d symmetries of all cations and anions (see Eq. (4): Xmin (s) ˆ )0.55, Xmin (p) ˆ )0.40, and Xmin (d) ˆ )0.55 and Xmin (s) ˆ )0.75, Xmin (p) ˆ )0.55, and Xmin (d) ˆ )0.75), while we have searched the best Xmin value for each s, p, and d symmetry of each ion. From our results, we observe that the Xmin values for a given symmetry can be very di€erent for each ion. For instance, Xmin (s) ˆ )0.818 for Hÿ and Xmin (s) ˆ )0.437 for Iÿ , and these results show that the electronic wave functions of s symmetry for Hÿ are more di€used than the corresponding ones for Iÿ . On the contrary, in the generation of the UGBS of Ref. [6], the Xmin value for the s electronic wave functions of these two anions is the same ()0.75), and thus, this drastic approximation is re¯ected in the energy values. For Hÿ and Iÿ , the deviations of the UGBS of Ref. [6] from the NHF energies are 0.00068 mhartree (our value is 0.000117) and 2400 mhartree (our value is 6.291), respectively.

F.E. Jorge, M.L. Franco / Chemical Physics 253 (2000) 21±26

25

Table 3 Deviations DE (in mhartree) of the UGBS ground state total energies from the numerical HF values (in hartree) for negative ions Z 1 3 5 6 7 8 9 11 13 14 15 16 17 19 21 22 23 24 25 26 27 28 29 31 32 33 34 35 37 39 40 41 42 43 44 45 46 47 49 50 51 52 53

Ion Hÿ (1 S) Liÿ (1 S) Bÿ (3 P) Cÿ (4 S) Nÿ (3 P) Oÿ (2 P) Fÿ (1 S) Naÿ (1 S) Alÿ (3 P) Siÿ (4 S) Pÿ (3 P) Sÿ (1 P) Clÿ (1 S) Kÿ (1 S) Scÿ (3 F) Tiÿ (4 F) Vÿ (5 D) Crÿ (6 S) Mnÿ (5 D) Feÿ (4 F) Coÿ (3 F) Niÿ (2 D) Cuÿ (1 S) Gaÿ (3 P) Geÿ (4 S) Asÿ (3 P) Seÿ (1 P) Brÿ (1 S) Rbÿ (1 S) Yÿ (3 F) Zrÿ (4 F) Nbÿ (5 D) Moÿ (6 S) Tcÿ (5 D) Ruÿ (4 F) Rhÿ (3 F) Pdÿ (2 D) Agÿ (1 S) Inÿ (3 P) Snÿ (4 S) Sbÿ (3 P) Teÿ (2 P) Iÿ (1 S)

Con®guration 1s2 [He]2s2 [He]2s2 2p2 [He]2s2 2p3 [He]2s2 2p4 [He]2s2 2p5 [He]2s2 2p6 [Ne]3s2 [Ne]3s2 3p2 [Ne]3s2 3p3 [Ne]3s2 3p4 [Ne]3s2 3p5 [Ne]3s2 3p6 [Ar]4s2 [Ar]4s2 3d2 [Ar]4s2 3d3 [Ar]4s2 3d4 [Ar]4s2 3d5 [Ar]4s2 3d6 [Ar]4s2 3d7 [Ar]4s2 3d8 [Ar]4s2 3d9 [Ar]4s2 3d10 [Ar]4s2 3d10 [Ar]4s2 3d10 [Ar]4s2 3d10 [Ar]4s2 3d10 [Kr]4s2 3d10 [Kr]5s2 [Kr]5s2 4d2 [Kr]5s2 4d3 [Kr]5s2 4d4 [Kr]5s2 4d5 [Kr]5s2 4d6 [Kr]5s2 4d7 [Kr]5s2 4d8 [Kr]5s2 4p9 [Kr]5s2 4p10 [Kr]5s2 4d10 [Kr]5s2 4d10 [Kr]5s2 4d10 [Kr]5s2 4d10 [Kr]5s2 4d10

4p2 4p3 4p4 4p5 4p6

5p2 5p3 5p4 5p5 5p6

No. of GTFs

DE (UGBS)

DE (UGBS)a (18s 12p 11d)

18s 18s 19s 19s 19s 19s 19s 21s 19s 19s 19s 19s 19s 23s 23s 23s 24s 24s 24s 24s 24s 24s 24s 24s 24s 24s 24s 24s 25s 25s 25s 25s 26s 26s 26s 26s 26s 27s 27s 27s 27s 25s 25s

0.000117 0.086412 0.02420 0.04572 0.04304 0.07765 0.06237 1.2109 2.9031 2.0935 2.3979 3.2983 2.3047 2.3038 2.6288 2.8934 2.1912 2.487 1.830 2.101 1.939 3.374 2.886 7.844 2.354 2.292 2.636 2.188 6.777 7.423 5.455 5.969 5.567 4.397 3.903 4.236 4.575 4.007 5.843 4.585 4.857 5.927 6.291

0.00068 0.35 1.1 0.19 0.22 0.28 0.43 1.3 5.1 2.7 3.3 4.1 5.2 10 14 17 20 24 29 35 43 51 61 87 97 110 130 160 220 310 370 440 510 600 690 800 920 1100 1400 1600 1900 2100 2400

Present work

15p 14p 14p 14p 14p 14p 15p 14p 14p 14p 14p 14p 14p 14p 14p 14p 14p 14p 15p 15p 15p 16p 17p 17p 17p 17p 16p 16p 16p 16p 15p 16p 17p 17p 17p 17p 19p 19p 19p 19p 19p

10d 10d 10d 10d 12d 11d 12d 10d 11d 10d 10d 10d 10d 12d 12d 13d 14d 14d 14d 14d 14d 14d 14d 15d 15d 15d 15d 15d 15d

c

NHFb )0.4879297344 )7.428232061 )24.51922137 )37.70884362 )54.32195889 )74.78974593 )99.45945391 )161.8551260 )241.8782653 )288.8896602 )340.6988736 )397.5384302 )459.5769253 )599.1619170 )759.6887738 )848.3725498 )942.8631322 )1043.337097 )1149.729110 )1262.367074 )1381.351810 )1506.821133 )1638.964145 )1923.260381 )2075.394742 )2234.222940 )2399.904726 )2572.536273 )2938.354900 )3331.659116 )3538.994500 )3753.578216 )3975.526268 )4204.764631 )4441.528477 )4685.875582 )4937.891544 )5197.700050 )5740.175141 )6022.972657 )6313.481518 )6611.827949 )6918.075883

a

Results obtained from Ref. [6]. Numerical HF total energies obtained from Ref. [15]. c Results obtained by using our UGBS. b

It is worth mentioning that the GCHF method is one of the most ¯exible formalisms in the design of UBS, and that the ID technique of the GCHF method allows easy generation of UBSs that are

able to describe HF total energies for a large number of atoms and ions with good accuracy [7±10]. HF orbital energies can be used, according to KoopmanÕs theorem, to estimate the IP of an

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F.E. Jorge, M.L. Franco / Chemical Physics 253 (2000) 21±26

Table 4 Ionization potential (M‡ ® M2‡ in kJ/mol) for some positive ions Z

Ion

Present work

Experimental [17]

2 3 10 11 18 19 36 37 54

He‡ (2 S) Li‡ (1 S) Ne‡ (2 P) Na‡ (1 S) Ar‡ (2 P) K‡ (1 S) Kr‡ (2 P) Rb‡ (1 S) Xe‡ (2 P)

5251.0 7331.3 4218.0 4718.5 2744.2 3073.2 2401.9 2647.2 2063.9

5250.4 7298.1 3952.3 4562.4 2665.8 3051.4 2350.3 2632.0 2046.0

atomic system. Table 4 shows the IPs (M‡ ® M2‡ in kJ/mol) calculated in this work for some cations and the corresponding experimental values [17]. From this table, we can see that all our IP results are in good agreement with the experimental values [17]. 4. Conclusions The HF total energies reported in this work for the cations He‡ )Xe‡ and the anions Hÿ ±Iÿ show that with a careful numerical integration of the GHWHF equations in the GCHF method, we are able to generate an accurate UGBS to be used in HF calculations. The accuracy of the HF energy values calculated in this work has not been obtained, so far, with a UBS using GTFs. For all ions studied, the largest di€erence between the corresponding energies computed with our UGBS and the NHF method is lower than 7.850 mhartree. The IPs

obtained by us for some cations are in good agreement with the experimental values [17]. Acknowledgements We would like to acknowledge the ®nancial support of CNPq and CAPES (Brazilian Agencies). References [1] D.M. Silver, W.C. Nieuwpoort, Chem. Phys. Lett. 57 (1978) 421. [2] D.M. Silver, S. Wilson, J. Chem. Phys. 69 (1978) 3787. [3] D.M. Silver, S. Wilson, W.C. Nieuwpoort, Int. J. Quant. Chem. 14 (1978) 635. [4] E. Clementi, G. Corongiu, Chem. Phys. Lett. 90 (1982) 359. [5] J.R. Mohallem, R.M. Dreizler, M. Trsic, Int. J. Quant. Chem. Symp. 20 (1986) 45. [6] A.B.F. da Silva, M. Trsic, Can. J. Chem. 74 (1996) 1526. [7] F.E. Jorge, E.V.R. de Castro, A.B.F. da Silva, J. Comp. Chem. 18 (1997) 1565. [8] F.E. Jorge, E.V.R. de Castro, A.B.F. da Silva, Chem. Phys. 216 (1997) 317. [9] F.E. Jorge, R.F. Martins, Chem. Phys. 233 (1998) 1. [10] E.V.R. de Castro, F.E. Jorge, J. Chem. Phys. 108 (1998) 5225. [11] F.E. Jorge, P.R. Librelon, A.C. Neto, J. Comp. Chem. 19 (1998) 858. [12] F.E. Jorge, E.P. Muniz, Int. J. Quant. Chem. 71 (1999) 307. [13] J.R. Mohallem, Z. Phys. D 3 (1986) 339. [14] C.M. Reeves, J. Chem. Phys. 39 (1963) 11. [15] T. Koga, S. Watanabe, K. Kanayama, R. Yasuda, A.J. Thakkar, J. Chem. Phys. 103 (1995) 3000. [16] T. Koga, Y. Seki, A.J. Thakkar, H. Tatewaki, J. Phys. B 26 (1993) 2529. [17] C.E. Moore, Ionization potentials and ionization limits derived from the analyses of optical spectra, NSRDS-NBS 34, National Bureau of Standards, Washington, 1970.