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Compact Hylleraas-type wavefunctions for the lithium isoelectronic sequence
Chemical Physics Letters 366 (2002) 95–99 www.elsevier.com/locate/cplett
Compact Hylleraas-type wavefunctions for the lithium isoelectronic sequence Ajit J. Thakkar b
a,*
, Toshikatsu Koga b, Tomomi Tanabe b, Hirohide Teruya
c,1
a Department of Chemistry, University of New Brunswick, Fredericton, NB, Canada E3B 6E2 Department of Applied Chemistry, Muroran Institute of Technology, Muroran, Hokkaido 050-8585, Japan c Kumamoto University College of Medical Science, Kuhonji, Kumamoto 862-0976, Japan
Received 22 August 2002
Abstract Compact Hylleraas-type wavefunctions of three types, denoted A, B and C, are constructed for the 2 S ground states of the lithium atom and its isoelectronic ions Beþ , B2þ ; . . . ; Si11þ . All the exponents and powers of the basis functions are optimized. The 20-term type B and 50-term type C wavefunctions are as accurate as several significantly larger expansions reported in the literature. The 50-term type C wavefunctions predict energies accurate to better than 3 parts in 107 . Ó 2002 Elsevier Science B.V. All rights reserved.
1. Introduction The solution of the Schr€ odinger equation for the lithium atom and its isoelectronic ions has attracted a considerable amount of effort. These ions are the simplest systems with both a closed and an open shell, and serve as prototypes for alkali-metal atoms and alkaline-earth cations. They constitute a good test of quantum mechanical methods. Pioneering calculations on the ground state of the lithium atom were made by Eckart [1], Guillemin and Zener [2], Wilson [3], and James and
*
Corresponding author. Fax: +1-506-453-4981. E-mail addresses: [email protected] (A.J. Thakkar), koga@mmm. muroran-it.ac.jp (T. Koga). 1 Deceased.
Coolidge [4,5]. Since then the ground state energy of Li has been refined by Weiss [6], Larsson [7], Muszy nska et al. [8], King et al. [9–11], McKenzie and Drake [12], L€ uchow and Kleindienst [13], Yan and Drake [14], and Yan et al. [15]. All these studies, except that of Weiss [6], use Hylleraas-type expansions ranging in length from 100 terms [7] to 3502 terms [15]. The most recent results [15] have converged to an accuracy of a few parts in 1012 . Important work on the lithium atom using more mainstream methods includes the configuration interaction studies of Jitrik and Bunge [16], and Chung [17], and the multi-configuration Hartree– Fock study reported by Tong et al. [18]. A much more complete list of references to the extensive literature on the ground state of the lithium atom can be found in an excellent review article [19].
0009-2614/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 2 ) 0 1 5 4 4 - 0
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A.J. Thakkar et al. / Chemical Physics Letters 366 (2002) 95–99
The purpose of this brief Letter is to report Hylleraas-type wavefunctions that contain only 50 terms but provide energies with an accuracy that is better than 3 parts in 107 for the ground state of the lithium atom and its isoelectronic ions. No attempt is made to produce the most accurate energies and wavefunctions to date. Hartree atomic units are used throughout.
2. Wavefunctions The wavefunctions are taken to be of the form w¼A
N X
ð1Þ
Cl /l vl
l¼1
in which N is the number of terms, A is the threeparticle antisymmetrizer, the Cl are variationally determined expansion coefficients, the /l are spatial basis functions, and the vl are spin functions. The spatial basis functions are of the conventional Hylleraas form i
j
k
l
m
n
/l ðr1 ; r2 ; r3 Þ ¼ r1l r2l r3l r23l r13l r12l eal r1 bl r2 cl r3
ð2Þ
in which rp ¼ jrp j are the electron–nucleus distances, rpq ¼ jrp rqj are the interelectronic distances, the powers il , jl , kl , ll , ml , and nl are non-negative integers, and the exponents al , bl , and cl are positive variational parameters. There are two linearly independent, doublet, spin functions with MS ¼ 1=2 for a three-electron system. They can be chosen [20] as H1 ¼ 21=2 ðaba baaÞ;
ð3Þ
and H2 ¼ 61=2 ð2aab baa abaÞ:
ð4Þ
We use only one spin function and simply set all vl ¼ H 1 . Three different types of wavefunctions of the form of Eq. (1) were constructed. In the first type, denoted A, all the terms share a common set of exponents: al ¼ a, bl ¼ b and cl ¼ c. Then each term must have a unique set of powers, {il , jl , kl , ll , ml , nl }, to ensure that the terms are linearly independent. In the second type, denoted B, the exponents al , bl and cl differ from term to term,
and each term has a unique set of powers, {il , jl , kl , ll , ml , nl } even though this is no longer required for linear independence. In the third type, denoted C, the exponents al , bl and cl differ from term to term, and two or more terms are allowed to share the same set of powers {il , jl , kl , ll , ml , nl }. The variational optimization of these wavefunctions involved an iterative procedure nested to a maximum of four levels. The outermost (level 1) iteration simply looped over the level 2 iteration until the energy had converged to our satisfaction. The level 2 iteration looped over the N terms of the wavefunction. In the level 3 iteration, the powers of a single term were varied, by increments of 1, to minimize the energy. The level 3 iterations were omitted if the powers were not being optimized. In the innermost (level 4) iteration, the exponents {al , bl , cl } were varied to minimize the energy, for the powers chosen in the current level 3 iteration, by the method of conjugate directions [21]. This procedure is computationally demanding but leads to compact wavefunctions of relatively high accuracy because both the powers and exponents are optimized. Unlike previous work, the powers are neither ad hoc nor chosen by conditions like il þ jl þ kl þ ll þ ml þ nl 6 x, where x is some integer.
3. Lithium ground state We began by constructing several wavefunctions of type A, with expansion lengths N ¼ 10, 20, 30, 40 and 50, for the ground state of the lithium atom. Then type B counterparts were constructed by starting with a type A wavefunction, keeping all the powers {il , jl , kl , ll , ml , nl } fixed, and minimizing the energy with respect to the exponential parameters al , bl and cl . Finally wavefunctions of type C were constructed from scratch without reference to the wavefunctions of types A and B. The energies of our Li wavefunctions are shown in Table 1. The convergence of the energy with respect to the best available value, Eref ¼ 7:4780603236 from Yan et al. [15], is shown in Fig. 1 which plots the logarithmic energy error, ¼ log10 ð1 E=Eref Þ, as a function of the expan-
A.J. Thakkar et al. / Chemical Physics Letters 366 (2002) 95–99
97
Table 1 Energies for the 2 S ground state of the lithium atom from wavefunctions of types A, B and C with varying numbers of terms N N
A
B
C
10 20 30 40 50
)7.477525394 )7.477888149 )7.477981459 )7.478015913 )7.478026970
)7.477880794 )7.478031140 )7.478044830 )7.478056917 )7.478057528
)7.478001477 )7.478047580 )7.478056937 )7.478058730 )7.478059609
sion length N . The type A functions have energy errors decreasing from 1 part in 104 for 10 terms to less than 1 part in 105 for 50 terms. The type B functions have energy errors decreasing from less than 1 part in 104 for 10 terms to less than 1 part in 106 for 50 terms. The type C functions have energy errors decreasing from less than 1 part in 105 for 10 terms to less than 1 part in 107 for 50 terms. To demonstrate the compactness of our wavefunctions, we make comparisons with previously reported, Hylleraas-type, wavefunctions that yield comparable energies. Our 10A (10-term, type A) wavefunction has an energy that is 1:4 103 hartrees lower than the 17-term Hylleraas-type wavefunction of James and Coolidge [5]. The 20B wavefunction has an energy 6 106 hartrees lower than the 100-term Hylleraas-type wavefunction of Larsson [7]. The 20C wavefunction has
an energy that is 1:6 105 hartrees lower than the 92-term Hylleraas-type wavefunction of Ho [22]. Our 50C wavefunction has an energy lower by 1:6 106 hartrees than the 352-term Hylleraastype wavefunction of King and Shoup [9], lower by 6 107 than the 602-term Hylleraas-type function of King [10], lower by 1 107 than the 296term Hylleraas-type function of King [11], and lower by 2 107 than the 256-term Hylleraastype function of Yan and Drake [14] among others. The 50C energy is only 0:7 106 hartrees higher than the best 3502-term Hylleraas-type results of Yan et al. [15]. Clearly, we have achieved our goal of producing compact and accurate wavefunctions for the ground state of Li. Thorough optimization of both the powers and exponents was crucial to the achievement of this goal.
4. Isoelectronic ions
Fig. 1. Energy errors, ¼ log10 ð1 E=Eref Þ, of lithium wavefunctions (M: type A, }: type B, : type C) as a function of the number of basis functions N .
Encouraged by the results for Li, we then constructed 20B and 50C wavefunctions for the ground states of the ions from Beþ to Si11þ using the same powers as in the corresponding Li wavefunctions but with the exponents optimized separately for each ion. The resulting energies are shown in Table 2. Our 20B energies are lower than the 60-term Hylleraas-type results of Ho [22], shown in Table 2, by amounts increasing from 1:3 105 hartrees for Beþ to 4:5 105 hartrees for O5þ . The 50C energy for Beþ is, respectively, 12 106 lower and 1 106 hartrees higher than the 340-term and 401-term results of King [23]. Our 50C energies for B2þ to Ne7þ are, respectively, 1 106 hartrees lower and higher than the 400-term and 500-term (503 for B2þ ) results of King [10]. The
98
A.J. Thakkar et al. / Chemical Physics Letters 366 (2002) 95–99
Table 2 Energies for the 2 S ground states of the lithium-like ions
Li Beþ B2þ C3þ N4þ O5þ F6þ Ne7þ Na8þ Mg9þ Al10þ Si11þ
Hoa
20Bb
50Cc
Bestd
– )14.324696 )23.424523 )34.775418 )48.376798 )64.228435 – – – – – –
)7.478031 )14.324709 )23.424537 )34.775432 )48.376841 )64.228480 )82.330272 )102.682163 )125.284120 )150.136123 )177.238161 )206.590225
)7.478059609 )14.324759257 )23.424603104 )34.775508354 )48.376895004 )64.228539182 )82.330335075 )102.682228315 )125.284187497 )150.136193236 )177.238233098 )206.590298687
)7.478060323 )14.324763176 )23.424605720 )34.775511275 )48.376898319 )64.228542082 )82.330338097 )102.682231482 )125.284190753 )150.136196604 )177.238236559 )206.590302212
a
60-term Hylleraas-type wavefunctions. Ho [22]. 20-term, type B wavefunctions. This work. c 50-term, type C wavefunctions. This work. d 3502-term, Hylleraas-type wavefunctions. Yan et al. [15]. b
50C energies for Beþ to Si11þ are between 2:6 106 and 3:9 106 hartrees higher than the best 3502-term Hylleraas-type results of Yan et al. [15]. Thus the 50C energy errors range from 3 parts in 107 to 2 parts in 108 . Clearly, our 50-term, type C wavefunctions for the ground states of the cations Beþ to Si11þ are both compact and accurate.
type C wavefunctions. Table 4 shows that there are two sets of three terms that have the same powers, and three sets of two terms with the same powers. Table 4 also reveals that the 50C wavefunctions contain seven terms in which two of {ll , ml , nl } are odd, and only one term in which all three of these powers are non-zero (two even and one odd). This leads us to hope that compact and accurate results can also be obtained from wavefunctions in which the restriction is imposed that no term has more than one of the powers of the interelectronic distances, {ll , ml , nl }, equal to an odd number. It is hoped that the loss of accuracy resulting from this restriction can be averted by performing more thorough optimizations and by using expansions with more than 50 terms. This, in turn, will be made computationally feasible because the restriction will speed up the computations significantly. The reduction in computational effort arises because integrals in which two, and espe-
5. Discussion It is of some interest to examine the powers of the basis functions included in our wavefunctions. The powers in the 20B wavefunctions are shown in Table 3. Observe that they contain only two terms in which two of the powers of the interelectronic distances, {ll , ml , nl }, are odd, and none in which all three of these powers are non-zero. More than one term can have the same set of powers in the Table 3 Powers of the terms in the 20B wavefunctions i
j
k
l
m
n
i
j
k
l
m
n
i
j
k
l
m
n
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 1 2
0 1 1 2 2 0 0
0 0 4 0 1 3 2
0 0 0 0 0 0 0
0 0 0 0 0 0 1
1 1 1 1 1 2 0
0 0 0 0 0 0 0
0 0 0 0 2 0 0
0 1 2 3 0 1 1
0 1 1 1 2 2
1 0 0 1 0 1
0 0 1 1 1 0
1 0 0 0 1 0
0 1 0 0 0 0
0 1 1 0 0 1
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Table 4 Powers of the terms in the 50C wavefunctions i
j
k
l
m
n
i
j
k
l
m
n
i
j
k
l
m
n
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 1 1 2 2 2 3 0 0 0 0 0 0 0 0 0
0 1 0 1 0 1 2 0 0 0 0 1 1 2 2 2 3
0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1
0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0
1 1 1 1 1 1 2 2 2 3 1 2 0 0 0 0 1
0 0 0 0 0 1 0 0 1 0 0 2 0 0 0 0 0
0 0 0 1 2 1 0 1 3 0 0 2 0 1 1 1 0
3 4 6 0 0 0 1 2 0 0 2 1 3 0 1 2 0
1 1 1 1 1 1 1 1 2 2 2 2 2 3 3 4
0 0 0 1 1 1 1 2 0 0 0 0 1 0 2 0
1 1 1 0 0 1 2 3 0 1 1 1 0 1 1 1
0 0 0 0 1 0 1 1 0 0 0 0 2 0 0 0
0 0 0 0 0 0 0 0 1 0 0 2 1 0 0 0
1 1 3 0 1 0 5 1 0 0 1 0 0 2 1 0
cially three, of these powers are odd require more computational effort than integrals in which no more than one of these powers is odd. Wavefunctions with this restriction have the significant advantage that it is possible to extract the oneelectron density as a relatively simple, analytical expression. We are currently exploring this possibility in our laboratories. Acknowledgements We thank Philip Regier for his assistance at a very early stage of this work undertaken at the University of Waterloo. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada, and in part by a Grant-in-Aid for Scientific Research from the Ministry of Education of Japan. References [1] C. Eckart, Phys. Rev. 36 (1930) 878. [2] V. Guillemin Jr., C. Zener, Z. Phys. 61 (1930) 199.
[3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
E.B. Wilson Jr., J. Chem. Phys. 1 (1933) 210. H.M. James, A.S. Coolidge, Phys. Rev. 47 (1935) 700. H.M. James, A.S. Coolidge, Phys. Rev. 49 (1936) 688. A.W. Weiss, Phys. Rev. 122 (1961) 1826. S. Larsson, Phys. Rev. 169 (1968) 49. J. Muszy nska, D. Papierowska, W. Woznicki, Chem. Phys. Lett. 76 (1980) 136. F.W. King, V. Shoup, Phys. Rev. A 33 (1986) 2940. F.W. King, Phys. Rev. A 40 (1989) 1735. F.W. King, M.P. Bergsbaken, J. Chem. Phys. 93 (1990) 2570. D.K. McKenzie, G.W.F. Drake, Phys. Rev. A 44 (1991) R6973. A. L€ uchow, H. Kleindienst, Int. J. Quantum Chem. 51 (1994) 211. Z.-C. Yan, G.W.F. Drake, Phys. Rev. A 52 (1995) 3711. Z.-C. Yan, M. Tambasco, G.W.F. Drake, Phys. Rev. A 57 (1998) 1652. O. Jitrik, C.F. Bunge, Phys. Rev. A 43 (1991) 5804. K.T. Chung, Phys. Rev. A 44 (1991) 5421. M. Tong, P. J€ onsson, C.F. Fischer, Phys. Scr. 48 (1993) 446. F.W. King, J. Mol. Struct. (Theochem.) 400 (1997) 7. R. Pauncz, Spin Eigenfunctions: Construction and Use, Plenum Press, New York, 1979. M.J.D. Powell, Comput. J. 7 (1964) 155. Y.K. Ho, Int. J. Quantum Chem. 20 (1981) 1077. F.W. King, Phys. Rev. A 38 (1988) 6017.
Compact Hylleraas-type wavefunctions for the lithium isoelectronic sequence Ajit J. Thakkar b
a,*
, Toshikatsu Koga b, Tomomi Tanabe b, Hirohide Teruya
c,1
a Department of Chemistry, University of New Brunswick, Fredericton, NB, Canada E3B 6E2 Department of Applied Chemistry, Muroran Institute of Technology, Muroran, Hokkaido 050-8585, Japan c Kumamoto University College of Medical Science, Kuhonji, Kumamoto 862-0976, Japan
Received 22 August 2002
Abstract Compact Hylleraas-type wavefunctions of three types, denoted A, B and C, are constructed for the 2 S ground states of the lithium atom and its isoelectronic ions Beþ , B2þ ; . . . ; Si11þ . All the exponents and powers of the basis functions are optimized. The 20-term type B and 50-term type C wavefunctions are as accurate as several significantly larger expansions reported in the literature. The 50-term type C wavefunctions predict energies accurate to better than 3 parts in 107 . Ó 2002 Elsevier Science B.V. All rights reserved.
1. Introduction The solution of the Schr€ odinger equation for the lithium atom and its isoelectronic ions has attracted a considerable amount of effort. These ions are the simplest systems with both a closed and an open shell, and serve as prototypes for alkali-metal atoms and alkaline-earth cations. They constitute a good test of quantum mechanical methods. Pioneering calculations on the ground state of the lithium atom were made by Eckart [1], Guillemin and Zener [2], Wilson [3], and James and
*
Corresponding author. Fax: +1-506-453-4981. E-mail addresses: [email protected] (A.J. Thakkar), koga@mmm. muroran-it.ac.jp (T. Koga). 1 Deceased.
Coolidge [4,5]. Since then the ground state energy of Li has been refined by Weiss [6], Larsson [7], Muszy nska et al. [8], King et al. [9–11], McKenzie and Drake [12], L€ uchow and Kleindienst [13], Yan and Drake [14], and Yan et al. [15]. All these studies, except that of Weiss [6], use Hylleraas-type expansions ranging in length from 100 terms [7] to 3502 terms [15]. The most recent results [15] have converged to an accuracy of a few parts in 1012 . Important work on the lithium atom using more mainstream methods includes the configuration interaction studies of Jitrik and Bunge [16], and Chung [17], and the multi-configuration Hartree– Fock study reported by Tong et al. [18]. A much more complete list of references to the extensive literature on the ground state of the lithium atom can be found in an excellent review article [19].
0009-2614/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 2 ) 0 1 5 4 4 - 0
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A.J. Thakkar et al. / Chemical Physics Letters 366 (2002) 95–99
The purpose of this brief Letter is to report Hylleraas-type wavefunctions that contain only 50 terms but provide energies with an accuracy that is better than 3 parts in 107 for the ground state of the lithium atom and its isoelectronic ions. No attempt is made to produce the most accurate energies and wavefunctions to date. Hartree atomic units are used throughout.
2. Wavefunctions The wavefunctions are taken to be of the form w¼A
N X
ð1Þ
Cl /l vl
l¼1
in which N is the number of terms, A is the threeparticle antisymmetrizer, the Cl are variationally determined expansion coefficients, the /l are spatial basis functions, and the vl are spin functions. The spatial basis functions are of the conventional Hylleraas form i
j
k
l
m
n
/l ðr1 ; r2 ; r3 Þ ¼ r1l r2l r3l r23l r13l r12l eal r1 bl r2 cl r3
ð2Þ
in which rp ¼ jrp j are the electron–nucleus distances, rpq ¼ jrp rqj are the interelectronic distances, the powers il , jl , kl , ll , ml , and nl are non-negative integers, and the exponents al , bl , and cl are positive variational parameters. There are two linearly independent, doublet, spin functions with MS ¼ 1=2 for a three-electron system. They can be chosen [20] as H1 ¼ 21=2 ðaba baaÞ;
ð3Þ
and H2 ¼ 61=2 ð2aab baa abaÞ:
ð4Þ
We use only one spin function and simply set all vl ¼ H 1 . Three different types of wavefunctions of the form of Eq. (1) were constructed. In the first type, denoted A, all the terms share a common set of exponents: al ¼ a, bl ¼ b and cl ¼ c. Then each term must have a unique set of powers, {il , jl , kl , ll , ml , nl }, to ensure that the terms are linearly independent. In the second type, denoted B, the exponents al , bl and cl differ from term to term,
and each term has a unique set of powers, {il , jl , kl , ll , ml , nl } even though this is no longer required for linear independence. In the third type, denoted C, the exponents al , bl and cl differ from term to term, and two or more terms are allowed to share the same set of powers {il , jl , kl , ll , ml , nl }. The variational optimization of these wavefunctions involved an iterative procedure nested to a maximum of four levels. The outermost (level 1) iteration simply looped over the level 2 iteration until the energy had converged to our satisfaction. The level 2 iteration looped over the N terms of the wavefunction. In the level 3 iteration, the powers of a single term were varied, by increments of 1, to minimize the energy. The level 3 iterations were omitted if the powers were not being optimized. In the innermost (level 4) iteration, the exponents {al , bl , cl } were varied to minimize the energy, for the powers chosen in the current level 3 iteration, by the method of conjugate directions [21]. This procedure is computationally demanding but leads to compact wavefunctions of relatively high accuracy because both the powers and exponents are optimized. Unlike previous work, the powers are neither ad hoc nor chosen by conditions like il þ jl þ kl þ ll þ ml þ nl 6 x, where x is some integer.
3. Lithium ground state We began by constructing several wavefunctions of type A, with expansion lengths N ¼ 10, 20, 30, 40 and 50, for the ground state of the lithium atom. Then type B counterparts were constructed by starting with a type A wavefunction, keeping all the powers {il , jl , kl , ll , ml , nl } fixed, and minimizing the energy with respect to the exponential parameters al , bl and cl . Finally wavefunctions of type C were constructed from scratch without reference to the wavefunctions of types A and B. The energies of our Li wavefunctions are shown in Table 1. The convergence of the energy with respect to the best available value, Eref ¼ 7:4780603236 from Yan et al. [15], is shown in Fig. 1 which plots the logarithmic energy error, ¼ log10 ð1 E=Eref Þ, as a function of the expan-
A.J. Thakkar et al. / Chemical Physics Letters 366 (2002) 95–99
97
Table 1 Energies for the 2 S ground state of the lithium atom from wavefunctions of types A, B and C with varying numbers of terms N N
A
B
C
10 20 30 40 50
)7.477525394 )7.477888149 )7.477981459 )7.478015913 )7.478026970
)7.477880794 )7.478031140 )7.478044830 )7.478056917 )7.478057528
)7.478001477 )7.478047580 )7.478056937 )7.478058730 )7.478059609
sion length N . The type A functions have energy errors decreasing from 1 part in 104 for 10 terms to less than 1 part in 105 for 50 terms. The type B functions have energy errors decreasing from less than 1 part in 104 for 10 terms to less than 1 part in 106 for 50 terms. The type C functions have energy errors decreasing from less than 1 part in 105 for 10 terms to less than 1 part in 107 for 50 terms. To demonstrate the compactness of our wavefunctions, we make comparisons with previously reported, Hylleraas-type, wavefunctions that yield comparable energies. Our 10A (10-term, type A) wavefunction has an energy that is 1:4 103 hartrees lower than the 17-term Hylleraas-type wavefunction of James and Coolidge [5]. The 20B wavefunction has an energy 6 106 hartrees lower than the 100-term Hylleraas-type wavefunction of Larsson [7]. The 20C wavefunction has
an energy that is 1:6 105 hartrees lower than the 92-term Hylleraas-type wavefunction of Ho [22]. Our 50C wavefunction has an energy lower by 1:6 106 hartrees than the 352-term Hylleraastype wavefunction of King and Shoup [9], lower by 6 107 than the 602-term Hylleraas-type function of King [10], lower by 1 107 than the 296term Hylleraas-type function of King [11], and lower by 2 107 than the 256-term Hylleraastype function of Yan and Drake [14] among others. The 50C energy is only 0:7 106 hartrees higher than the best 3502-term Hylleraas-type results of Yan et al. [15]. Clearly, we have achieved our goal of producing compact and accurate wavefunctions for the ground state of Li. Thorough optimization of both the powers and exponents was crucial to the achievement of this goal.
4. Isoelectronic ions
Fig. 1. Energy errors, ¼ log10 ð1 E=Eref Þ, of lithium wavefunctions (M: type A, }: type B, : type C) as a function of the number of basis functions N .
Encouraged by the results for Li, we then constructed 20B and 50C wavefunctions for the ground states of the ions from Beþ to Si11þ using the same powers as in the corresponding Li wavefunctions but with the exponents optimized separately for each ion. The resulting energies are shown in Table 2. Our 20B energies are lower than the 60-term Hylleraas-type results of Ho [22], shown in Table 2, by amounts increasing from 1:3 105 hartrees for Beþ to 4:5 105 hartrees for O5þ . The 50C energy for Beþ is, respectively, 12 106 lower and 1 106 hartrees higher than the 340-term and 401-term results of King [23]. Our 50C energies for B2þ to Ne7þ are, respectively, 1 106 hartrees lower and higher than the 400-term and 500-term (503 for B2þ ) results of King [10]. The
98
A.J. Thakkar et al. / Chemical Physics Letters 366 (2002) 95–99
Table 2 Energies for the 2 S ground states of the lithium-like ions
Li Beþ B2þ C3þ N4þ O5þ F6þ Ne7þ Na8þ Mg9þ Al10þ Si11þ
Hoa
20Bb
50Cc
Bestd
– )14.324696 )23.424523 )34.775418 )48.376798 )64.228435 – – – – – –
)7.478031 )14.324709 )23.424537 )34.775432 )48.376841 )64.228480 )82.330272 )102.682163 )125.284120 )150.136123 )177.238161 )206.590225
)7.478059609 )14.324759257 )23.424603104 )34.775508354 )48.376895004 )64.228539182 )82.330335075 )102.682228315 )125.284187497 )150.136193236 )177.238233098 )206.590298687
)7.478060323 )14.324763176 )23.424605720 )34.775511275 )48.376898319 )64.228542082 )82.330338097 )102.682231482 )125.284190753 )150.136196604 )177.238236559 )206.590302212
a
60-term Hylleraas-type wavefunctions. Ho [22]. 20-term, type B wavefunctions. This work. c 50-term, type C wavefunctions. This work. d 3502-term, Hylleraas-type wavefunctions. Yan et al. [15]. b
50C energies for Beþ to Si11þ are between 2:6 106 and 3:9 106 hartrees higher than the best 3502-term Hylleraas-type results of Yan et al. [15]. Thus the 50C energy errors range from 3 parts in 107 to 2 parts in 108 . Clearly, our 50-term, type C wavefunctions for the ground states of the cations Beþ to Si11þ are both compact and accurate.
type C wavefunctions. Table 4 shows that there are two sets of three terms that have the same powers, and three sets of two terms with the same powers. Table 4 also reveals that the 50C wavefunctions contain seven terms in which two of {ll , ml , nl } are odd, and only one term in which all three of these powers are non-zero (two even and one odd). This leads us to hope that compact and accurate results can also be obtained from wavefunctions in which the restriction is imposed that no term has more than one of the powers of the interelectronic distances, {ll , ml , nl }, equal to an odd number. It is hoped that the loss of accuracy resulting from this restriction can be averted by performing more thorough optimizations and by using expansions with more than 50 terms. This, in turn, will be made computationally feasible because the restriction will speed up the computations significantly. The reduction in computational effort arises because integrals in which two, and espe-
5. Discussion It is of some interest to examine the powers of the basis functions included in our wavefunctions. The powers in the 20B wavefunctions are shown in Table 3. Observe that they contain only two terms in which two of the powers of the interelectronic distances, {ll , ml , nl }, are odd, and none in which all three of these powers are non-zero. More than one term can have the same set of powers in the Table 3 Powers of the terms in the 20B wavefunctions i
j
k
l
m
n
i
j
k
l
m
n
i
j
k
l
m
n
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 1 2
0 1 1 2 2 0 0
0 0 4 0 1 3 2
0 0 0 0 0 0 0
0 0 0 0 0 0 1
1 1 1 1 1 2 0
0 0 0 0 0 0 0
0 0 0 0 2 0 0
0 1 2 3 0 1 1
0 1 1 1 2 2
1 0 0 1 0 1
0 0 1 1 1 0
1 0 0 0 1 0
0 1 0 0 0 0
0 1 1 0 0 1
A.J. Thakkar et al. / Chemical Physics Letters 366 (2002) 95–99
99
Table 4 Powers of the terms in the 50C wavefunctions i
j
k
l
m
n
i
j
k
l
m
n
i
j
k
l
m
n
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 1 1 2 2 2 3 0 0 0 0 0 0 0 0 0
0 1 0 1 0 1 2 0 0 0 0 1 1 2 2 2 3
0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1
0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0
1 1 1 1 1 1 2 2 2 3 1 2 0 0 0 0 1
0 0 0 0 0 1 0 0 1 0 0 2 0 0 0 0 0
0 0 0 1 2 1 0 1 3 0 0 2 0 1 1 1 0
3 4 6 0 0 0 1 2 0 0 2 1 3 0 1 2 0
1 1 1 1 1 1 1 1 2 2 2 2 2 3 3 4
0 0 0 1 1 1 1 2 0 0 0 0 1 0 2 0
1 1 1 0 0 1 2 3 0 1 1 1 0 1 1 1
0 0 0 0 1 0 1 1 0 0 0 0 2 0 0 0
0 0 0 0 0 0 0 0 1 0 0 2 1 0 0 0
1 1 3 0 1 0 5 1 0 0 1 0 0 2 1 0
cially three, of these powers are odd require more computational effort than integrals in which no more than one of these powers is odd. Wavefunctions with this restriction have the significant advantage that it is possible to extract the oneelectron density as a relatively simple, analytical expression. We are currently exploring this possibility in our laboratories. Acknowledgements We thank Philip Regier for his assistance at a very early stage of this work undertaken at the University of Waterloo. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada, and in part by a Grant-in-Aid for Scientific Research from the Ministry of Education of Japan. References [1] C. Eckart, Phys. Rev. 36 (1930) 878. [2] V. Guillemin Jr., C. Zener, Z. Phys. 61 (1930) 199.
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