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Basis-set expansion method for relativistic atoms in the atomic-nucleus model of a finite sphere of constant electric field
Volume 172, number 2
CHEMICAL PHYSICS LETTERS
3 I August I990
Basis-set expansion method for relativistic atoms in the atomic-nucleus model of a finite sphere of constant electric field Osamu Matsuoka Deportmentof Physics.The Universityof Electra-Communications, Chojk Tokyo182, Japan Received 30 May 1990; in final form 18 June 1990
Use of a new atomic-nucleus model is proposed for relativistic atomic and molecular calculations; the atomic nucleus has a sphere of finite size and the electric field in the sphere is assumed constant. It is shown that for the basis-set expansion method basedon this model, the Slater-type functions of integral principal quantum numbers can be appropriately adopted and the Slatertype functions of “n=l” quantum numbers (e.g., lp, 2d, 3f, etc.) appearing in the point-nucleus model are not needed even if we use the scheme of kinetic balance. Test calculations on one-electron hydrogenic atoms imply the applicability of the proposed atomic-nucleus model to many-electron systems.
1. Introduction
It is becoming apparent that the variational collapse [ 1 ] in the Dirac-Fock-Roothaan calculations (or the relativistic Hartree-Fock calculations using the basis-set expansion method) can be circumvented in practice by the use of the following two schemes: ( 1) the kinetic balance (KB) between the large and the small component basis spinors [ 2 1, and (2) the fulfillment of the boundary condition (BC) on the wavefunctions at the atomic nuclei [ 3,4]. The former eliminates the appearance of the spurious states and the latter prevents a small upper-bound failure of the calculated energy levels. One can also add the condition that (2’ ) when the above BC is approximately satisfied, we should not use the basis functions of very large exponent values which will induce an insufficient cancellation of the kinetic and the nuclear-attraction energies. Thus, using the Gaussian-type functions (GTF) and the KB scheme, the present author and coworkers [ 5-8 ] could successfully generate the relativistic basis sets for the atoms helium through radon. In the calculations we adopted the uniformly charged sphere model of the atomic nucleus [ 91 so that the BCs on the wavefunctions are fulfilled in the lowest order of
the expansions in terms of the radial variables. Mohanty and Clementi [ lo,1 1 ] adopted similar schemes and could compute the relativistic GTF sets for helium through radon. These basis sets are expected to be used in molecular calculations. For the Slater-type functions (STF), systematic methods of generating the relativistic basis sets have not been presented, which pay particular attention to both the KB and the BC. Grant, Quiney, and Wilson, however [ 4,121 used the “Sspinor” functions as basis functions and obtained excellent wavefunctions for hydrogen atoms, helium, and argon. However, since the point-charge model was adopted for the atomic nuclei, the basis functions of the nonintegral principal quantum numbers had to be used to satisfy the BC. Computation of molecular integrals over these kinds of basis functions would not be so easy that their wavefunctions could be easily employed in molecular calculations. In the present Letter, we propose to adopt for the atomic nucleus a new model which has a sphere of finite size where it is assumed that the electric field in the inside of the sphere is constant, and that outside the potential is Coulombic as in the other atomicnucleus models. Thus, we can appropriately employ the STFs of integral principal quantum numbers,
0009-2614/90/$ 03.50 0 1990 - Elsevier Science Publishers B.V. (North-Holland)
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Volume 172, number 2
CHEMICAL
3 1 August I990
PHYSICS LETTERS
which fulfill the BC. For the small-component basis spinors which satisfy the KB and the BC, we can use the basis functions appearing in nonrelativistic calculations and need not adopt such unwieldy “n=l” functions (e.g., Ip, 2d, 3f, etc.) [ 13,141. Calculations using this new atomic-nucleus model have been performed on one-electron hydrogenic atoms, which imply the applicability of the proposed nucleus model to relativistic many-electron systems.
2. Atomic-nucleus model and basis-set expansion method We assume that the atomic nucleus of atomic number Z has a finite sphere of radius R, and that in the inside of the sphere the electric field is constant and that outside it is Coulombic. Hence, in terms of the radial variable r, the nuclear potential V(r) is written in atomic units as
Table 1 Convergence of energy levels E (in atomic units) for H, Age+, and Hg’9+ with respect to the number of exponentparameters iV.Kinetically balanced basis sets (5a), (5b), (7a), and (7~) in the text are used. Energy levels of the variational collapse are marked with asterisks
-E(%z) H
5 6 7
a
A&6+
0.12499851 0.12499961 0.12499962 0.12499962 0.12499963 0.12499963 0.12499963 0.12499963 0.12499963 0.12500208
12 13 14 15 PNa’
0.50000578 0.50000580 0.50000582 0.50000582 0.50000666
a 9
I 138.9397
8 9 10 11 12 13 14 15 16 17 18 PN ‘I
176
0.49998626 0.49999967 0.50000380 0.50000527 OSOOCKI560
16 17 18 PN ‘)
‘) Point-nucleus
0.12499850 0.12499851
9 10 11
10 11 12 13 14 IS
Hg”+
0.49981459 0.49993417
model.
1138.8412 1138.9769 1138.9946 1139.0019 1139.0040 1139.0049 1139.0048 1139.0048 I 139.0048 1139.0048 1139.0442 3527.817 3529.588 3530.070 3530.388 3530.455 3530.460 3530.460 3530.460 3530.460 3530.460 3530.460 3532.186
-
-E(%d
0. I2499592 0. I2499697 0.12499727 0.12499731 0.12499731 0.12499732 0. I2499732 0. I2499732 0.12499732 0. I2499732 0.12499732 0.12500042
0.05555064 0.05555070
0.05554859 0.05554885
0.05555074 0.05555076
0.05554885 0.05554886
0.05555077 0.05555077 0.05555078 0.05555078 0.05555078 0.05555078 0.05555078 0.05555580
0.05554886 0.05554886 0.05554886 0.05554886 0.05554886 0.05554886 0.05554886 0.05555563
286.95543. 286.95444. 286.95538* 286.95491, 286.95525; 286.95506* 286.955 17. 286.955 12+ 286.95513* 286.95513* 286.955 13” 286.95344
278.18510 278.18510
123.94449 123.94449 123.94449 123.94450 123.94450 123.94450 123.94450 123.94450 123.94450 123.94450 123.94450 123.94458
123.12545 123.12547 123.12547 123. I2548
904.87869*
8 17.80645
904.a5104* 904.88460” 904.86850* 904.87934. 904.87377, 904.87539* 904.87559. 904.87507’ 904.a7507* 904.87517* 904.84588
8 17.80649 817.80660 8 17.80663 817.80663 817.80663 817.80664 817.80664 8 17.80664 8 17.80664 8 17.80664 8 17.80722
123.14284* 366.14282= 366.14282’
358.98566 358.98570 358.98573 358.98574 358.98575 358.98575 358.98576 358.98576 358.98576 358.98576 358.98576 358.98680
278.18513 278.18514 278.18514 278.18515 278.18515 278.18515 278.18515 278.18515 278.18515 278.18546
366.14282* 366.14282* 366.14282* 366.14282’ 366.14282* 366.14282* 366.14282* 366.14282* 366.14255
123.12548 123.12548 123.12549 123.12549 123.12549 123.12549 123.12549 123.12586
V(r)=V,+V,r
(rcR),
V(r)= -Z/r
(OR).
(1)
The constants V, and V, could be so determined that at r=R the potential V and its first derivative are continuous: V, = -2Z/R
and
3 1 August 1990
CHEMICAL PHYSICS LETTERS
Volume 172, number 2
V, =Z/R2.
(2)
For the one-electron relativistic atoms subjected to the potential ( 1), the radial wavefunctions of the large components P(r) and of the small components Q(r) near the nucleus are shown to take the forms for the states of negative quantum number K, P,(r)=r”+‘(p,+p,r2+p,,r3+...),
(3a)
Qc(r)=F2
(3b)
(k0+4dr+hr*+...),
Furthermore, we should note that the small-component basis functions (5b) are the ordinary functions appearing in the nonrelativistic calculations (e.g. 2p, 3d, 4f, etc.). This is in contrast to the kinetically-balanced small-component functions in the point-nucleus model where we have to use the “0 = P’ functions (e.g. lp, 2d, 3f, etc.) [ 13,141. For K>O the basis functions satisfying the BC (4) to the order of rl”+’ should take the forms qL(r)=NLr’*+l
exp( -&)
,
$(r)=N+(lt~r)exp(-Tr).
(7a) (7b)
However, the functions (7a) and (7b) do not satisfy the KB (6) so that instead of the functions (7b) we should take the kinetically balanced functions
and for the states of positive quantum number K, P,(r)=++’
(Plli,+fk~r+hd2+...)y
QK(r)=rb(ql,+q,,rZ+qK)r3+...)
,
W
Table 2
(4b)
Hg’9+ with respect to the number of exponent parameters N. “Extended” kinetically balanced basis sets (7a), (7d), (Ye) are used. Energy levels of the variational collapse are marked with asterisks
where 1, is the angular quantum number of large component (i.e. l-, =O, 1+,= 1, 1L2= 1, etc.) and {P,,, QKi}are constants. A peculiar feature of the wavefunctions is that, for ICC0, pKl=O and, for w 0, qKI= 0. (The corresponding Schrijdinger wavefunctions take the form of P,( r) described by (3a).) The leading terms of the expansions (3 ) and (4 ) are of integral powers of r in contrast to those in the pointnucleus model where they are of the form of non-integral powers of r. Since the potential (1) has a cusp on the nucleus, we could simulate the behaviors ( 3) and (4) near the origin in terms of the STFs of integral principal quantum number n( = 1,2, ...). P-I exp( -[r), with {being the exponent parameter. In more detail, the radial basis functions of the large components $-(r) and those of the small components $(r) could be chosen for K
exp(-lr),
(5a)
,
(5b)
-
where N L and NS are normalization constants. The basis functions (5 ) fulfill the BC (3 ) to the order of dr+‘. Besides, they are the kinetically balanced ones: $(r)a(d/dr+rc/r)qL(r).
(6)
Convergenceof energy levels E (in atomic units) for Ag”+, and
Atom
N
-WP,,d
A&?+
a 9 10 11 12 13 14 I5 16 17 I8 PN”
286.94124 286.946 10 286.94953 286.95132 286.95236 286.95296 286.95330 286.95356, 286.95380* 286.95401. 286.95413. 286.95334
Hg’9+
8 9 10 11 12 13 14 15 16 17 18 PN”
90452572 904.67565 904.75694 904.81121 904.85420= 904.9cOoo* 904.95661’ 905.01506+ 905.02067* 905.01952* 905.02264* 904.84588
-U3&,2)
366.14187 366.14187 366.14190 366.14191 366.14191 366.14191 366.14192 366.14192 366.14192 366.14192 366.14192 366.14255
‘) Point-nucleus model.
177
Volume 172, number 2
CHEMICAL PHYSICS LETTERS
rjS(r)=NSrL[(lx+l+rc)-0-1
exp(-lr),
(7c)
which satisfy the BC (4b) only to the lowest order Yebut not in the next higher order #-+I. To remedy this situation, we could employ the “extended” KB scheme [ 3,15 ] so that we split the basis functions (7~) into two functions $(‘)(r)
=N sc’)rjKexp(
qs(z)(r) =P(*)r~+’ Thus,
-(i)
,
exp( -[r)
we can use existing
(7d)
Ue)
.
molecular-integral pack-
31 August 1990
ages in the nonrelativistic calculations with minor modifications of the nuclear-attraction integrals.
3. Calculations and results Calculations have been performed on the one-electron hydrogenic atoms. The exponent parameters < were calculated by the even-tempered scheme [ 16 1, [i=a$-’ (i= 1, 2, ...) N). We have optimized only the parameter (Y,fixing the parameter /3= 1.5, since
Table 3 Optimized parameters (Yused for the calculations of the energy levels shown in tables 1 and 2. The parameter /?was fixed at /?= 1.5.KB and EKE are the kinetically balanced and the “extended” kinetically balanced basis sets Atom KEl
H
Age+
5 6 7 8 9 10 11 12 13 14 15
1a22970 1.12943 1.14460 1.11782 1.11533 1.11533 1.11533 1.10657 1.10657 1.10657 1.10657
0.49824 0.49824 0.49824 0.49727 0.49727 0.49727 0.49727 0.49727 0.49727 0.49727 0.49727
8
53.72426 54.87834 52.47097 33.46322 52.43507 52.62788 52.62788 52.62788 52.62788 52.62788 52.62788
23.74103 24.02116 24.02116 24.02116 24.02116 24.02116 24.02116 24.02116 24.02116 24.02116 24.02116
99.02359 102.67208 93.49335 96.25950 89.49931 91.82281 91.82281 91.82281 91.82281 91.82281 91.82281
40.99096 44.31105 42.08355 43.57523 42.8 1463 43.13998 43.13998 43.13998 43.13998 43.13998 43.13998
9 10 II 12 13 14 15 16 17 18 Hg’9+
8 9 10 11 12 13 14 15 16 17
18
178
EKB
KB
EKB
0.56022 0.56022 0.55591 0.55591 0.55591 0.55591 0.55591 0.55591 0.55591 0.55591 0.55591
0.32917 0.329 I7 0.32917 0.32917 0.329 17 0.32917 0.32917 0.32917 0.32917 0.32917 0.32917
0.36609 0.37886 0.36886 0.36886 0.36886 0.36886 0.36886 0.36886 0.36886 0.36886 0.36886
16.68952 15.96400 15.28528 15.98986 15.98986 15.98986 15.98986 15.98986 15.98986 15.98986 15.98986
26.06650 26.06650 26.06650 26.06650 26.06650 26.06650 26.06650 26.06650 26.06650 26.06650 26.06650
15.70551 15.70551 15.70511 15.705 11 15.70511 15.70511 15.70511 15.70511 15.70511 15.70511 15.70511
17.28697 17.28697 17.28697 17.28697 17.28697 17.28697 17.28697 17.28697 17.28697 17.28697 17.28697
42.98506 43.77439 42.94266 43.13237 43.13237 43.13237 43.13237 43.13237 43.13237 43.13231 43.13237
44.79039 44.79039 44.79039 44.79039 44.79039 44.79039 44.79039 44.79039 44.79039 44.79039 44.79039
27.13676 27.13676 27.13676 27.13676 27.13676 27.13676 27.13676 27.13676 27.13676 27.13676 27.13676
27.03396 27.03396 27.03396 27.03396 27.03396 27.03396 27.03396 27.03396 27.03396 27.03396 27.03396
29.55652 29.55652 29.55652 29.55652 29.55652 29.55652 29.55652 29.55652 29.55652 29.55652 29.55652
Volume 172,number 2
CHEMICALPHYSICSLETTERS
in the course of the calculations we found that in the
case of STFs, the linear dependency among the basis functions is a severe problem and we could not see the convergence property of the calculated energy values if we varied the parameters (Yand /3 simultaneously. The nuclear radius R in eqs. ( 1) and (2) has been estimated [9] as 2.2677~ 10-5A1’3 (in atomic units) where A is the atomic mass number. The speed of light has been taken as 137.037 atomic units. Table 1 shows the convergence behaviors of the calculated energy levels for the one-electron atoms H, Ag46+and Hg79+where the atomic ma3s numbers A are assumed to be 1, 107, and 202, respectively. In the calculations the kinetically balanced basis sets (5a), (5b), (7a), and (7b) are used. For the neutral H atom, the energies seem to converge uniformly with increasing number of the exponent parameters. However, referring to the energy values in the pointnucleus model, the variational collapse [ 1 ] seems to occur in the K= 1 state for Ag4”+ and the K= 1 and the K= 2 states for Hg79+. Calculations with the “extended” KI3 [3,15] sets (7a), (7d), and (7e) are shown in table 2, which indicate that the variational collapse is remedied in the K= 2 state of Hg79+while in the K= 1 states of both Ag46+and Hg79+, it is still seen when the large number of basis functions are used. It could be avoided if we use exponent parameters of smaller values. However, we need not worry about this failure too much since we rarely deal with such highly ionized states. Resides, in the Dirac-Fock calculations on many-electron atoms which are our real interest, the field due to the electrons adjusts the bare nuclear field so that the simple KB scheme works very well as many Dirac-Fock calculations [ 5-8, 10, 111 show. Tables 1 and 2 show furthermore that the energy
31 August 1990
values of lower atomic numbers and of higher states are more similar to those in the point-nucleus model, as we can naturally understand. Table 3 collects the optimized Q!values employed for the calculations shown in tables 1 and 2.
Extensions to many-electron atoms are being planned. In nonrelativistic calculations, the STFs are now employed for linear molecules. Thus, the relzitivistic STF basis sets generated by the present scheme are also expected to be adopted for linear molecules. References [I ] W. Kutzelnigg,Intern. .I. Quantum Chem. 25 (1984) 107. [2] R.E. Stanton and S. Havriliak, J. Chem. Phys. 81 ( 1984) 1910. [ 31Y. Ishikawa and H.M. Quiney, Intern. J. Quantum Chem Symp. 21 (1987) 523. [4] I.P. Grant and H.M. Quiney, Advan. Atom. Mol. Phys. 23 (1988) 37. [ 510. Matsuoka and S. Huzinaga, Chem. Phys. Letteis 140 (1987) 567. [ 610. Matsuoka and S. Okada, Chem. Phys. Letters 155( 1989) 547. [ 71 S. Okada and 0. Matsuoka, J. Chem. Phys. 9 1 ( 1989) 4193. [S] S, Okada, M. Shinada and 0. Matsuoka, 5. Chem. Phys., submitted for publication. [9] J.P. Desclaux, At. DataNucl. DataTables 12 (1973) 311. IO] AK. Mohanty and E. Clementi, Chem. Phys. Letters 157 (1989) 348. I1 ] A.K. Mohanty and E. Clementi, Modem techniques in computational chemistry (ESCOM,Leidcn, 1989) p. 169. 121H.M. Quiney, LP. Grant andS. Wilson, J. Phys B 22 ( 1989) L15. 1310. Matsuoka, N. Suzuki, T. Aoyama and G. Malli, J. Chem.
Phys. 73 (1980) 1320. [ 14lY.S.LeeandA.D. McLean, J. Chem. Phys. 76 (1982) 735. [ 151 P.J.C. Aerts and W.C. Nieuwpoort, Intern. J. Quantum Chem. Symp. 19 (1985) 267. [ 161 R.C. Raffenetti and K. Ruedenberg, J. Chem. Phys. 59 (1973) 5978.
179
CHEMICAL PHYSICS LETTERS
3 I August I990
Basis-set expansion method for relativistic atoms in the atomic-nucleus model of a finite sphere of constant electric field Osamu Matsuoka Deportmentof Physics.The Universityof Electra-Communications, Chojk Tokyo182, Japan Received 30 May 1990; in final form 18 June 1990
Use of a new atomic-nucleus model is proposed for relativistic atomic and molecular calculations; the atomic nucleus has a sphere of finite size and the electric field in the sphere is assumed constant. It is shown that for the basis-set expansion method basedon this model, the Slater-type functions of integral principal quantum numbers can be appropriately adopted and the Slatertype functions of “n=l” quantum numbers (e.g., lp, 2d, 3f, etc.) appearing in the point-nucleus model are not needed even if we use the scheme of kinetic balance. Test calculations on one-electron hydrogenic atoms imply the applicability of the proposed atomic-nucleus model to many-electron systems.
1. Introduction
It is becoming apparent that the variational collapse [ 1 ] in the Dirac-Fock-Roothaan calculations (or the relativistic Hartree-Fock calculations using the basis-set expansion method) can be circumvented in practice by the use of the following two schemes: ( 1) the kinetic balance (KB) between the large and the small component basis spinors [ 2 1, and (2) the fulfillment of the boundary condition (BC) on the wavefunctions at the atomic nuclei [ 3,4]. The former eliminates the appearance of the spurious states and the latter prevents a small upper-bound failure of the calculated energy levels. One can also add the condition that (2’ ) when the above BC is approximately satisfied, we should not use the basis functions of very large exponent values which will induce an insufficient cancellation of the kinetic and the nuclear-attraction energies. Thus, using the Gaussian-type functions (GTF) and the KB scheme, the present author and coworkers [ 5-8 ] could successfully generate the relativistic basis sets for the atoms helium through radon. In the calculations we adopted the uniformly charged sphere model of the atomic nucleus [ 91 so that the BCs on the wavefunctions are fulfilled in the lowest order of
the expansions in terms of the radial variables. Mohanty and Clementi [ lo,1 1 ] adopted similar schemes and could compute the relativistic GTF sets for helium through radon. These basis sets are expected to be used in molecular calculations. For the Slater-type functions (STF), systematic methods of generating the relativistic basis sets have not been presented, which pay particular attention to both the KB and the BC. Grant, Quiney, and Wilson, however [ 4,121 used the “Sspinor” functions as basis functions and obtained excellent wavefunctions for hydrogen atoms, helium, and argon. However, since the point-charge model was adopted for the atomic nuclei, the basis functions of the nonintegral principal quantum numbers had to be used to satisfy the BC. Computation of molecular integrals over these kinds of basis functions would not be so easy that their wavefunctions could be easily employed in molecular calculations. In the present Letter, we propose to adopt for the atomic nucleus a new model which has a sphere of finite size where it is assumed that the electric field in the inside of the sphere is constant, and that outside the potential is Coulombic as in the other atomicnucleus models. Thus, we can appropriately employ the STFs of integral principal quantum numbers,
0009-2614/90/$ 03.50 0 1990 - Elsevier Science Publishers B.V. (North-Holland)
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Volume 172, number 2
CHEMICAL
3 1 August I990
PHYSICS LETTERS
which fulfill the BC. For the small-component basis spinors which satisfy the KB and the BC, we can use the basis functions appearing in nonrelativistic calculations and need not adopt such unwieldy “n=l” functions (e.g., Ip, 2d, 3f, etc.) [ 13,141. Calculations using this new atomic-nucleus model have been performed on one-electron hydrogenic atoms, which imply the applicability of the proposed nucleus model to relativistic many-electron systems.
2. Atomic-nucleus model and basis-set expansion method We assume that the atomic nucleus of atomic number Z has a finite sphere of radius R, and that in the inside of the sphere the electric field is constant and that outside it is Coulombic. Hence, in terms of the radial variable r, the nuclear potential V(r) is written in atomic units as
Table 1 Convergence of energy levels E (in atomic units) for H, Age+, and Hg’9+ with respect to the number of exponentparameters iV.Kinetically balanced basis sets (5a), (5b), (7a), and (7~) in the text are used. Energy levels of the variational collapse are marked with asterisks
-E(%z) H
5 6 7
a
A&6+
0.12499851 0.12499961 0.12499962 0.12499962 0.12499963 0.12499963 0.12499963 0.12499963 0.12499963 0.12500208
12 13 14 15 PNa’
0.50000578 0.50000580 0.50000582 0.50000582 0.50000666
a 9
I 138.9397
8 9 10 11 12 13 14 15 16 17 18 PN ‘I
176
0.49998626 0.49999967 0.50000380 0.50000527 OSOOCKI560
16 17 18 PN ‘)
‘) Point-nucleus
0.12499850 0.12499851
9 10 11
10 11 12 13 14 IS
Hg”+
0.49981459 0.49993417
model.
1138.8412 1138.9769 1138.9946 1139.0019 1139.0040 1139.0049 1139.0048 1139.0048 I 139.0048 1139.0048 1139.0442 3527.817 3529.588 3530.070 3530.388 3530.455 3530.460 3530.460 3530.460 3530.460 3530.460 3530.460 3532.186
-
-E(%d
0. I2499592 0. I2499697 0.12499727 0.12499731 0.12499731 0.12499732 0. I2499732 0. I2499732 0.12499732 0. I2499732 0.12499732 0.12500042
0.05555064 0.05555070
0.05554859 0.05554885
0.05555074 0.05555076
0.05554885 0.05554886
0.05555077 0.05555077 0.05555078 0.05555078 0.05555078 0.05555078 0.05555078 0.05555580
0.05554886 0.05554886 0.05554886 0.05554886 0.05554886 0.05554886 0.05554886 0.05555563
286.95543. 286.95444. 286.95538* 286.95491, 286.95525; 286.95506* 286.955 17. 286.955 12+ 286.95513* 286.95513* 286.955 13” 286.95344
278.18510 278.18510
123.94449 123.94449 123.94449 123.94450 123.94450 123.94450 123.94450 123.94450 123.94450 123.94450 123.94450 123.94458
123.12545 123.12547 123.12547 123. I2548
904.87869*
8 17.80645
904.a5104* 904.88460” 904.86850* 904.87934. 904.87377, 904.87539* 904.87559. 904.87507’ 904.a7507* 904.87517* 904.84588
8 17.80649 817.80660 8 17.80663 817.80663 817.80663 817.80664 817.80664 8 17.80664 8 17.80664 8 17.80664 8 17.80722
123.14284* 366.14282= 366.14282’
358.98566 358.98570 358.98573 358.98574 358.98575 358.98575 358.98576 358.98576 358.98576 358.98576 358.98576 358.98680
278.18513 278.18514 278.18514 278.18515 278.18515 278.18515 278.18515 278.18515 278.18515 278.18546
366.14282* 366.14282* 366.14282* 366.14282’ 366.14282* 366.14282* 366.14282* 366.14282* 366.14255
123.12548 123.12548 123.12549 123.12549 123.12549 123.12549 123.12549 123.12586
V(r)=V,+V,r
(rcR),
V(r)= -Z/r
(OR).
(1)
The constants V, and V, could be so determined that at r=R the potential V and its first derivative are continuous: V, = -2Z/R
and
3 1 August 1990
CHEMICAL PHYSICS LETTERS
Volume 172, number 2
V, =Z/R2.
(2)
For the one-electron relativistic atoms subjected to the potential ( 1), the radial wavefunctions of the large components P(r) and of the small components Q(r) near the nucleus are shown to take the forms for the states of negative quantum number K, P,(r)=r”+‘(p,+p,r2+p,,r3+...),
(3a)
Qc(r)=F2
(3b)
(k0+4dr+hr*+...),
Furthermore, we should note that the small-component basis functions (5b) are the ordinary functions appearing in the nonrelativistic calculations (e.g. 2p, 3d, 4f, etc.). This is in contrast to the kinetically-balanced small-component functions in the point-nucleus model where we have to use the “0 = P’ functions (e.g. lp, 2d, 3f, etc.) [ 13,141. For K>O the basis functions satisfying the BC (4) to the order of rl”+’ should take the forms qL(r)=NLr’*+l
exp( -&)
,
$(r)=N+(lt~r)exp(-Tr).
(7a) (7b)
However, the functions (7a) and (7b) do not satisfy the KB (6) so that instead of the functions (7b) we should take the kinetically balanced functions
and for the states of positive quantum number K, P,(r)=++’
(Plli,+fk~r+hd2+...)y
QK(r)=rb(ql,+q,,rZ+qK)r3+...)
,
W
Table 2
(4b)
Hg’9+ with respect to the number of exponent parameters N. “Extended” kinetically balanced basis sets (7a), (7d), (Ye) are used. Energy levels of the variational collapse are marked with asterisks
where 1, is the angular quantum number of large component (i.e. l-, =O, 1+,= 1, 1L2= 1, etc.) and {P,,, QKi}are constants. A peculiar feature of the wavefunctions is that, for ICC0, pKl=O and, for w 0, qKI= 0. (The corresponding Schrijdinger wavefunctions take the form of P,( r) described by (3a).) The leading terms of the expansions (3 ) and (4 ) are of integral powers of r in contrast to those in the pointnucleus model where they are of the form of non-integral powers of r. Since the potential (1) has a cusp on the nucleus, we could simulate the behaviors ( 3) and (4) near the origin in terms of the STFs of integral principal quantum number n( = 1,2, ...). P-I exp( -[r), with {being the exponent parameter. In more detail, the radial basis functions of the large components $-(r) and those of the small components $(r) could be chosen for K
exp(-lr),
(5a)
,
(5b)
-
where N L and NS are normalization constants. The basis functions (5 ) fulfill the BC (3 ) to the order of dr+‘. Besides, they are the kinetically balanced ones: $(r)a(d/dr+rc/r)qL(r).
(6)
Convergenceof energy levels E (in atomic units) for Ag”+, and
Atom
N
-WP,,d
A&?+
a 9 10 11 12 13 14 I5 16 17 I8 PN”
286.94124 286.946 10 286.94953 286.95132 286.95236 286.95296 286.95330 286.95356, 286.95380* 286.95401. 286.95413. 286.95334
Hg’9+
8 9 10 11 12 13 14 15 16 17 18 PN”
90452572 904.67565 904.75694 904.81121 904.85420= 904.9cOoo* 904.95661’ 905.01506+ 905.02067* 905.01952* 905.02264* 904.84588
-U3&,2)
366.14187 366.14187 366.14190 366.14191 366.14191 366.14191 366.14192 366.14192 366.14192 366.14192 366.14192 366.14255
‘) Point-nucleus model.
177
Volume 172, number 2
CHEMICAL PHYSICS LETTERS
rjS(r)=NSrL[(lx+l+rc)-0-1
exp(-lr),
(7c)
which satisfy the BC (4b) only to the lowest order Yebut not in the next higher order #-+I. To remedy this situation, we could employ the “extended” KB scheme [ 3,15 ] so that we split the basis functions (7~) into two functions $(‘)(r)
=N sc’)rjKexp(
qs(z)(r) =P(*)r~+’ Thus,
-(i)
,
exp( -[r)
we can use existing
(7d)
Ue)
.
molecular-integral pack-
31 August 1990
ages in the nonrelativistic calculations with minor modifications of the nuclear-attraction integrals.
3. Calculations and results Calculations have been performed on the one-electron hydrogenic atoms. The exponent parameters < were calculated by the even-tempered scheme [ 16 1, [i=a$-’ (i= 1, 2, ...) N). We have optimized only the parameter (Y,fixing the parameter /3= 1.5, since
Table 3 Optimized parameters (Yused for the calculations of the energy levels shown in tables 1 and 2. The parameter /?was fixed at /?= 1.5.KB and EKE are the kinetically balanced and the “extended” kinetically balanced basis sets Atom KEl
H
Age+
5 6 7 8 9 10 11 12 13 14 15
1a22970 1.12943 1.14460 1.11782 1.11533 1.11533 1.11533 1.10657 1.10657 1.10657 1.10657
0.49824 0.49824 0.49824 0.49727 0.49727 0.49727 0.49727 0.49727 0.49727 0.49727 0.49727
8
53.72426 54.87834 52.47097 33.46322 52.43507 52.62788 52.62788 52.62788 52.62788 52.62788 52.62788
23.74103 24.02116 24.02116 24.02116 24.02116 24.02116 24.02116 24.02116 24.02116 24.02116 24.02116
99.02359 102.67208 93.49335 96.25950 89.49931 91.82281 91.82281 91.82281 91.82281 91.82281 91.82281
40.99096 44.31105 42.08355 43.57523 42.8 1463 43.13998 43.13998 43.13998 43.13998 43.13998 43.13998
9 10 II 12 13 14 15 16 17 18 Hg’9+
8 9 10 11 12 13 14 15 16 17
18
178
EKB
KB
EKB
0.56022 0.56022 0.55591 0.55591 0.55591 0.55591 0.55591 0.55591 0.55591 0.55591 0.55591
0.32917 0.329 I7 0.32917 0.32917 0.329 17 0.32917 0.32917 0.32917 0.32917 0.32917 0.32917
0.36609 0.37886 0.36886 0.36886 0.36886 0.36886 0.36886 0.36886 0.36886 0.36886 0.36886
16.68952 15.96400 15.28528 15.98986 15.98986 15.98986 15.98986 15.98986 15.98986 15.98986 15.98986
26.06650 26.06650 26.06650 26.06650 26.06650 26.06650 26.06650 26.06650 26.06650 26.06650 26.06650
15.70551 15.70551 15.70511 15.705 11 15.70511 15.70511 15.70511 15.70511 15.70511 15.70511 15.70511
17.28697 17.28697 17.28697 17.28697 17.28697 17.28697 17.28697 17.28697 17.28697 17.28697 17.28697
42.98506 43.77439 42.94266 43.13237 43.13237 43.13237 43.13237 43.13237 43.13237 43.13231 43.13237
44.79039 44.79039 44.79039 44.79039 44.79039 44.79039 44.79039 44.79039 44.79039 44.79039 44.79039
27.13676 27.13676 27.13676 27.13676 27.13676 27.13676 27.13676 27.13676 27.13676 27.13676 27.13676
27.03396 27.03396 27.03396 27.03396 27.03396 27.03396 27.03396 27.03396 27.03396 27.03396 27.03396
29.55652 29.55652 29.55652 29.55652 29.55652 29.55652 29.55652 29.55652 29.55652 29.55652 29.55652
Volume 172,number 2
CHEMICALPHYSICSLETTERS
in the course of the calculations we found that in the
case of STFs, the linear dependency among the basis functions is a severe problem and we could not see the convergence property of the calculated energy values if we varied the parameters (Yand /3 simultaneously. The nuclear radius R in eqs. ( 1) and (2) has been estimated [9] as 2.2677~ 10-5A1’3 (in atomic units) where A is the atomic mass number. The speed of light has been taken as 137.037 atomic units. Table 1 shows the convergence behaviors of the calculated energy levels for the one-electron atoms H, Ag46+and Hg79+where the atomic ma3s numbers A are assumed to be 1, 107, and 202, respectively. In the calculations the kinetically balanced basis sets (5a), (5b), (7a), and (7b) are used. For the neutral H atom, the energies seem to converge uniformly with increasing number of the exponent parameters. However, referring to the energy values in the pointnucleus model, the variational collapse [ 1 ] seems to occur in the K= 1 state for Ag4”+ and the K= 1 and the K= 2 states for Hg79+. Calculations with the “extended” KI3 [3,15] sets (7a), (7d), and (7e) are shown in table 2, which indicate that the variational collapse is remedied in the K= 2 state of Hg79+while in the K= 1 states of both Ag46+and Hg79+, it is still seen when the large number of basis functions are used. It could be avoided if we use exponent parameters of smaller values. However, we need not worry about this failure too much since we rarely deal with such highly ionized states. Resides, in the Dirac-Fock calculations on many-electron atoms which are our real interest, the field due to the electrons adjusts the bare nuclear field so that the simple KB scheme works very well as many Dirac-Fock calculations [ 5-8, 10, 111 show. Tables 1 and 2 show furthermore that the energy
31 August 1990
values of lower atomic numbers and of higher states are more similar to those in the point-nucleus model, as we can naturally understand. Table 3 collects the optimized Q!values employed for the calculations shown in tables 1 and 2.
Extensions to many-electron atoms are being planned. In nonrelativistic calculations, the STFs are now employed for linear molecules. Thus, the relzitivistic STF basis sets generated by the present scheme are also expected to be adopted for linear molecules. References [I ] W. Kutzelnigg,Intern. .I. Quantum Chem. 25 (1984) 107. [2] R.E. Stanton and S. Havriliak, J. Chem. Phys. 81 ( 1984) 1910. [ 31Y. Ishikawa and H.M. Quiney, Intern. J. Quantum Chem Symp. 21 (1987) 523. [4] I.P. Grant and H.M. Quiney, Advan. Atom. Mol. Phys. 23 (1988) 37. [ 510. Matsuoka and S. Huzinaga, Chem. Phys. Letteis 140 (1987) 567. [ 610. Matsuoka and S. Okada, Chem. Phys. Letters 155( 1989) 547. [ 71 S. Okada and 0. Matsuoka, J. Chem. Phys. 9 1 ( 1989) 4193. [S] S, Okada, M. Shinada and 0. Matsuoka, 5. Chem. Phys., submitted for publication. [9] J.P. Desclaux, At. DataNucl. DataTables 12 (1973) 311. IO] AK. Mohanty and E. Clementi, Chem. Phys. Letters 157 (1989) 348. I1 ] A.K. Mohanty and E. Clementi, Modem techniques in computational chemistry (ESCOM,Leidcn, 1989) p. 169. 121H.M. Quiney, LP. Grant andS. Wilson, J. Phys B 22 ( 1989) L15. 1310. Matsuoka, N. Suzuki, T. Aoyama and G. Malli, J. Chem.
Phys. 73 (1980) 1320. [ 14lY.S.LeeandA.D. McLean, J. Chem. Phys. 76 (1982) 735. [ 151 P.J.C. Aerts and W.C. Nieuwpoort, Intern. J. Quantum Chem. Symp. 19 (1985) 267. [ 161 R.C. Raffenetti and K. Ruedenberg, J. Chem. Phys. 59 (1973) 5978.
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