numerical study of the iterative solution of the one-electron dirac equation based on ‘direct...

EL!SEVIER

17 January 1997

Chemical Physics Letters 264 (1997) 495-501

CHEMICAL PHYSICS LETTERS

Numerical study of the iterative solution of the one-electron Dirac equation based on ‘direct perturbation theory’

Robert Franke Lehrstuhl fir Theorerische Chemie, Ruhr-Universitiit Bochum, D-44780 Bochum, Germany

Received 2 September 1996; in final form 28 October I996

Abstract

The one-electron Dirac equation is solved in an iterative manner starting with the solution of the Schriidinger equation. The method is applied in a basis of atom-centred Gaussian-type functions to the ground state of selected hydrogen-like ions up to Eka PtlW+ and the heavy quasi-molecules T?I\~~+ , NiPblog+ and Tl15~~+ (in D,_,, and D,, symmetry). An overall g-figure accuracy in the absolute relativistic energies is achieved. The iterative procedure converges better than linearly for light systems and linearly for systems containing nuclear charges greater than Z = 40.

1. Introduction

The perturbation theory of relativistic corrections in powers of c- ’ , if based on a Foldy-Wouthuysen- type transformation, leads to serious singularities ’ in the presence of a Coulomb potential [l] which are not present in the Dirac operator [2]. If one uses ‘direct perturbation theory’ (DPT) these spurious singularities are avoided. DPT is a method in terms of four-component spinors with the natural perturbation parameter c-*. It was first proposed by Sewell who investigated the ground state of the hydrogen atom using first-order DPT [3]. Rutkowski proposed a similar method and carried out calculations for

’ To answer a comment of a referee, the singularities in the Foldy-Wouthuysen transformation can be avoided if one does not perform this transformation at an ‘operator level’ but on the matrix representation of the Dirac operator in a kinetically balanced basis.

one- and two-electron systems using Slater- and Gaussian-type basis sets [4-71, Kutzelnigg gave an alternative formulation of DPT starting from a Dirac equation with c-dependent metric [8]. As the straightforward nonrelativistic limit one then gets the L&y-Leblond equation. Furthermore, a resolvent formulation of DPT is presented in Ref. [8] which is in close relation to the method of Gesztesy, Grosse and Thaller [9-121. Recently DPT was combined with the ZORA/CPD scheme [ 13-151 and with density functional theory [ 161.

It has been demonstrated that DPT is a tool for achieving highly accurate results for the relativistic corrections for the energies of the ground state and several exited states of the light one-electron molecules Hl and HeH*+ [ 171. Concerning ground state energies the same holds for the DPT expansion of the Dirac-Fock Coulomb equation [ 181 for atoms 1191 and molecules [20] containing nuclei up to Z= 50.

0009-2614/97/$17.00 Copyright 0 1997 Published by Elsevier Science B.V. All rights reserved. PII SOOOS-2614(96)01361-9

496 R. Franke/Chemical Physics Letters 264 (1997) 495-501

In the last two years there has been growing interest in calculations of the relativistic energy of heavy one-electron quasi-molecules like NiPb”‘+ [21] or Thi7’+ [22,23]. A DPT calculation for the wavefunction up to @(c-~), which enables one to get the relativistic correction to the energy up to @‘(c-~), leads to absolute relativistic energies with an error of = 10 E,, Chartree) for these species. To demonstrate the reliability of DPT even for systems containing nuclei with large nuclear charge we de- cided to revisit one-electron systems and report calculations in the framework of DPT for the selected hydrogenic ions, Thi79+, NiPbJog+ and Tht6’+ which are of @‘Cc-“) in the model of the one-electron Dirac equation with n + x for a given basis set and machine accuracy.

2. Method

Using DPT, there are two possible ways of calcu- lating relativistic corrections to the energy higher than @CC-~> which are outlined in Ref. [8].

(a) Generalisation of the formulae of DPT up to the order of interest. This is possible to arbitrary order without the occurrence of divergent integrals.

(b) ‘Non-perturbative iterative solution’ (NIS) of the Dirac equation starting with the nonrelativistic wavefunction.

We focus on (b) because no tedious algebraic manipulations need to be done to get the working equations for the calculation of relativistic corrections up to arbitrary order. Furthermore, it has not been tested numerically. For a theoretical foundation we refer to Ref. [8], where it is derived for a resolvent formulation of DPT.

We start with the solution of the Schrijdinger equation

(HO-EO)@=O. (1)

We choose 8’ = E” and calculate the first relativistic correction using

8’=8~+(l/c~)(~~~v-8~l x0>, (2)

with

x0 = ;apqP. (3)

The following steps are carried out iteratively, begin- ning with i = 1 up to self-consistency:

(4)

cpi=‘pO+4i, (5) xir fapqYo’+ (1/2c2)(v-@)xi-‘, (6)

8 i+‘=~“+(l/c2)(x01v-~i]xi), (7)

if I @+ ’ -8’1 >E * i=i+l (8)

To solve Eqs. (l)-(7) we expand cp” in a basis of atom-centred Gaussians {-yk), the spinor 4’ in a basis of atom-centred Gaussian spinors (nk)

c##= zb:qk k

Pa)

and the spinor xi in the kinetically balanced basis

1 xi = z zd;apqk.

k (9b)

The matrix representation of Eq. (6) is then a system of linear equations. Solving these equations yields the coefficients d:.

One could be tempted to expand 4’ in a regular basis and use the identity (6) directly (without expanding xi in a basis of functions regular at r = 0) for the construction of xi, which means that one would satisfy the condition (6) pointwise, in particu- lar, at r = 0. If carried out that way the NIS procedure would diverge due to terms that arise in Eq. (7) which include V!, with n > 2. A similar behaviour was first observed by Rutkowski in ordinary DPT applied to the H atom [4], who found a slow convergence of E4 and a divergence of E6 by expanding rp2 in a basis of Slater- or Gaussian-type functions and using the identity

x2 = +upcp2 + +(V- E”)xo (10)

directly to evaluate matrix elements containing x2. While the exact (p2 in the case of the H atom is of the form

+ - (Z2/2) In r. cp” + btp’, (11)

with b = constant one ignores the logarithmic singularity by expanding ‘p2 in common ST0 or GTO basis sets with the consequence of a l/r singularity

R. Franke/ Chemical Physics Letters 264 (1997) 495-501 491

in(10).Hencethematrixelement(x2 IV-E01 x2> needed for the calculation of E6 diverges. One avoids this unbalanced treatment of q2 (representation without In r singularity) and x2 (representation with l/r singularity) by expanding x2 in an appropriate basis. Our DPT calculations are always based on a representation of (p2 and x2 according to (9) which was not stressed in Ref. [ 171. If one used basis functions for the representation of cp2 with the correct behaviour at r + 0, direct application of identity (6) would lead to well-behaved E4 and E6 with an upper bound property for E4 14,241. However, this implies dealing with basis functions which include terms in In r. For a foundation of the stationarity approach to DPT and a detailed formal analysis of the regularisation procedure including a discussion of upper bound properties we refer to Ref. [24].

Using the NIS scheme cpi, xi and 8’ in Eqs. (4)-(9) are, in contrast to DPT, not the coefficients of a perturbation expansion in terms of c2. While 8’ is the DPT energy up to order b(ce2 )

8’ = E” + ( 1/c2)E2, ( 12)

this correspondence does not hold for 8’ and the DPT energies up to order @(c-~‘) with i > 1, except in the limit i + m supposing that the perturbation expansion converges. As in DPT the correct nonrelativistic limit c + 03 is guaranteed and thus no prob- lems of variational collapse arise.

We generate the integrals over Cartesian Gauss- ians with a slightly modified version of the HERMIT program [25]. The DPT and NIS schemes for the calculation of relativistic corrections and the eigen- value problem solver in the basis of spherical Gauss- ians are part of a code developed in our laboratory [26]. For the velocity of light a value of c = 137.0359895 au [27] was used. We assume that the nuclei are point particles. All energy values for the quasi-molecules are given without the addition of nuclear repulsion energies. The exponents generated with even-tempered prescriptions [28] are used without any cut-off. Energies calculated with NIS are given converged to E = 5 X lo-l2 E,. All calculations are performed in 64bit arithmetic on the SGI PowerChallenge-XL-8 of the computer center of the Ruhr-Universitat Bochum.

3. Results and discussion

Calculations are documented for the hydrogen atom, the hydrogen-like ions Neg+, Sn4’+, Thsg+, Eka Pt”‘+ and the heavy quasi-molecules Thy’+, NiPb’Og+ and Th;pg+ to show the applicability and generality of the NIS approach.

3.1. The hydrogen atom and hydrogen-like ions

In Table 1 results of DPT and NIS calculations in a Gaussian basis for the ground state of the hydrogen atom and some selected hydrogen-like ions are dis- played. The nonrelativistic, relativistic and DPT energies are exactly known. To show the numerical behaviour of the iterative procedure we report the NIS energies of the calculations without the application of a convergence accelerator.

Inspection of Table 1 shows that one achieves sub-FE, ( < 10e6E,) accuracy for the relativistic energy of the hydrogen atom and for NegC with DPT calculations up to &‘(ce6) in the energy while the relativistic energy differs by approximately 24 mE, (10-3E,) from the exact value for Sn4’+. The NIS energy for the latter (- 1294.62615530 Eh) lies approximately 0.6 p,E, above the exact value for the relativistic energy. To achieve this accuracy within a DPT calculation the energy beyond b(c-“1 has to be taken into account. For Th*‘+ and Eka Pt”‘+ a DPT calculation up to @(cm6> in the energy gives errors of approximately 11 and 115 E, and up to a(c- lo> 1 and 30 E, respe ctively. NIS calculations, performed with the same basis, give a 9-figure accuracy in the absolute relativistic energy for Thsg+ and an g-figure accuracy for Eka Pt”‘+.

In all cases the NIS calculations converge to the exact values from above. The larger the nuclear charge of the system the more iterations are needed to achieve a given numerical accuracy for the relativistic energy. The analysis of various calculations for one-electron systems, not all of them documented here, indicates that the NIS scheme shows the at- tributes of a linearly convergent iteration procedure for systems with a nuclear charge greater Z = 40, while it converges better than linearly for light systems. The convergence rate

q= I(,‘+’ -E)/(g”+)I, (13)

498 R. Franke / Chemical Physics Letters 264 (1997) 495-501

Table 1 Ground-state energies of the hydrogen atom and hydrogen-like ions calculated in a 50s even-tempered basis (exponents according to crpc,..., aj349) with ‘direct perturbation theory’ (DPT) up to @(?‘) in the energy and ‘non-perturbative iterative solution’ (NIS) of the Dirac equation up to self-consistency. The values are given in hartree (Et,). E’ denotes the coefficients of the expansion in powers of cei, 8’ the NIS energy of iteration step i and E the exact relativistic energy

Ha Ne9+ b c&)49+ c ?ns9+ E Eka PtiW+ d

EO

E’+c-*E*

E” + c- 2E2 + c- 4E4

E’+~-*E*+c-~E~+c-~E~

;:

83

;:

16

;:

ET0

824 z4’

exact values E” + c-*E* E” + c-*E* + c-~E~ E”+c-2E2+c-4E4+c-6E6

E’+c-*E*+...+c-*Es E”+c-2E2+...+c-‘0E’0

E

- 0.5- - 0.5000066564202 - 0.5000066565974 - 0.5000066565974 - 0.5000066564202 - 0.5000066565974

- 0.5000066564202 - 0.5000066565975 - 0.5000066565975 - 0.5000066565975 - 0.5000066565975 - 0.5000066565975

- 5o.ooooooooo - 50.066564201 - 50.066741433 - 50.066742023 - 50.06656420 1 - 50.066741011 - 50.066742020 - 50.066742025

- 125o.OOOOO - 1291.60263 - 1294.37187 - 1294.60229 - 1291.60263 - 1294.21147 - 1294.5763 1 - 1294.61963 - 1294.62530 - 1294.62604 - 1294.62614 - 1294.626 15 - 1294.62616

- 50.066564202 - 1291.60263 - 50.066741434 - 1294.37187 - 50.066742024 - 1294.60229 - 50.066742026 - 1294.62376 - 50.066742026 - 1294.62590 - 50.066742026 - 1294.62616

- 4050.OOOOO - 4486.72773 - 4580.91593 - 4606.30766 - 4486.72773 - 4564.23999 - 4597.93 116 - 4609.72437 - 4614.49746 -4616.40785 -4617.19460 -4617.520% - 46 17.65764 -4617.71521 - 4617.75765

- 4486.72773 - 4580.91594 - 4606.30766 - 4613.97434 -4616.45451 - 4617.75765

- 6049.99993 - 7024.56648 - 7338.54327 - 7464.98595 - 7024.56648 - 7258.60582 - 7407.32688 - 7478.78685 - 7520.55835 - 7544.22244 - 7558.24849 - 7566.60325 -7571.65416 - 7574.72908 - 7579.68352

7579.69100

- 7024.56649 - 7338.54329 - 7464.98597 - 7522.01666 - 7549.57702 - 7579.69109

’ Even-tempered parameters of Ref. 1281 Table III. b Even-tempered parameters of Ref. [28] Table III. ’ a= 8.1, p= 1.9. d (I = 15.0, fl= 2.05.

with E the exact relativistic energy, strongly de- pends on the nuclear charge. For Sn49f the convergence rate is approximately 0.13 implying that one gets one correct decimal place per at least two iteration steps. For ‘Ih 89+ the convergence rate is approximately 0.4 and for Eka Ptto9+ 0.6. Thus one needs between four and five iteration steps to get one decimal place of the relativistic energy of the super- heavy Eka Pt”‘+.

3.2. The quasi-molecule Thi79 ’

While an analytic solution of the Schriidinger equation for the two-centre one-electron systems is known 1291 this does not hold for the corresponding Dirac equation. One is therefore limited to ‘purely numerical’ calculations, e.g. the finite-difference

method (FDM) or the finite-element method @EM), or solving the Dirac equation using a finite basis-set expansion.

Pyykkii recommended the one-electron quasi- molecule ‘III\‘~+ as a benchmark 2. Recently, Parpia and Mohanty [23] investigated the variational solution of the Dirac equation for the ground state of the system Thi79+ for an internuclear distance of R = 2/90 a0 using atom-centred Gaussian-type functions and the prescription of ‘strict kinetic balance’

x-w (14)

* Remarked during the discussion following the talk given by Mohanty at the European Science Foundation Conference on Relativistic Effects in Heavy Element Chemistry and Physics held in 11 Ciocco, Italy, 30 March-4 April 1995.

R. Franke/Chemical Physics Letters 264 (1997) 495-501 499

Table 2 Electronic energies (in Et,) for the ground state of the Th- ‘79+ molecular ion with an internuclear distance of R = 2/90 as calculated with ‘direct perturbation theory’ (DPT) up to c~(c-~) and ‘non-perturbative iterative solution’ (NISI of the Dirac equation. E,, denotes the nonrelativistic energy, E, denotes the relativistic energy, E” + c- *E2 + c- 4E4 + c - 6Z? the relativistic energy up to @CC-~) using DPT and Em the relativistic enerev calculated with the NIS method

EM E “r E”+c-2E2+c-4E4+c-6E6 E NIS 6 this work this work this work

50s a -8836.6987099 -8836.698714 b [23] -9359.0106167 - 9368.5223984 - 9368.522398 [23] 5os41 a p -8929.6193799 -8929.619385 b [23] -9491.0274327 - 9499.8949939 - 9499.894994 [23] 50~41 p32d a - 893 1.3056729 - 893 I .305678 b [23] - 9495.75 15625 - 9504.5711558 - 9504.571156 [23] 50s41p32d23f ’ - 8931.3363150 - 8931.336320 b [23] - 9495.9317396 - 9504.7484532 - 9504.748453 [23] 50s41p32d23f14g ’ - 893 1.3370449 - 893 1.337050 b [23] - 9495.9395710 - 9504.7561552 - 9504.756155 [23] 50s41p32d23f14g5h a - 893 1.3370959 - 893 1.337100 b [23] - 9495.9400627 -9504.7566412 -9504.756641 [23] 6Os5Op4Od3Of2Og 10h ’ - 893 1.3370959 - 9495.9400636 - 9504.7566429 6Os5Op5Od5Ot2Og 10h d - 893 1.3370995 -9495.9401340 - 9504.7567 155 variational solution ’ [32] - 8929.99 - 94%.04 variational solution ’ [33] - 9476.7 variational solution [23] -8931.337103 b - 9504.756696 FDM g (221 -8931.337137 - 9461.9833 FEM ’ [34] -8931.337137413 - 9504.756759 FBM i (341 -8931.33713741(O) - 9504.75675(4) exact i [35] -8931.3371374087

’ Even-tempered parameters: a = I .O, p = 2.1. The short notation “1: i-j” to specify the basis sets implies that the set of functions with I-symmetry includes the exponents a/3’- ‘, . . , apj- ‘. s: l-50, p: 3-43; d: 5-36; f: 7-29; g: 9-22; h: 11-15. b Calculations carried out with c = 1370359.895 au as assumed in Ref. 1231 to describe the nonrelativistic limit. DPT or NIS calculations performed with this value for the velocity of light reproduce the given values. ’ Even-tempered parameters: a = I .O, p = 2.1. S: I-60, p: 3-52; d: 5-44; f: 7-36; g: 9-28; h: 1 l-20. d Even-tempered parameters: a = 1.0, p = 2.1. S: l-60, p: 3-52; d: 3-52; f 3-52; g: 9-28; h: 1 l-20. ’ R = 2/9O au; c = 137.0362 au. ’ R = 0.02222 au; c unknown. g R = 0.022222222 au; c = 137.03599 au. h R = 2/W au; c = 137.0359895 au; calculated with 6561 grid points. i Extrapolated (see Ref. [21]).1 Energy for Hz at R = 2.0 au of Ref. [35] times 8100.

of Stanton and Havriliak [30]. In Table 2 we give the results of DPT calculations up to @‘(cm61 in the energy and NIS calculations mainly using the same

basis sets as in Ref. [23] and compare them with various values from the literature. The NIS calculations reproduce the relativistic energies of Ref. [23]

Table 3 Electronic energies (in E,) for the ground state of the NiPb”” molecular ion with an internuclear distance of R = 0.002 au calculated with ‘direct perturbation theory’ (DPT) up to 8(C6) and ‘non-perturbative iterative solution’ (NIS) of the Dirac equation. E’ denotes the coefficients of the expansion in powers of c-j, E, the relativistic energy which has been determined in this work with the NIS method

Basis EO E” + c-.~E’ E’+c-~E’+c-~E~ E”+c-2EZ+c-4E4+~-6E6 E r 50s a - 5990.60547 -6866.12444 - 7085.34111 -7143.01336 -7161.53741 5os41p p - 5990.64295 - 6866.68 105 - 7087.20905 -7146.11170 -7165.74592 5Os41p22d a - 5990.64298 - 6866.68234 - 7087.21723 -7146.12920 - 7165.77663 50~41 p22dl2f a - 5990.64298 - 6866.68236 - 7087.21738 -7146.12953 -7165.77728 5Os41p22d12fSg ’ - 5990.64298 - 6866.68236 - 7087.21739 -7146.12954 - 7165.77730 FEM [21] -7165.7773011

’ Same basis set for Ni and Pb with even-tempered parameters: cr = 1.0, @ = 2.1. For the specification of the basis sets see foonote a to Table 2. s: l-50. p: 3-43; d: 13-34; f: 12-23; g: 14-18.

500 R. Franke/Chemical Physics Letiers 264 (1997) 495-501

Table 4 Electronic energies (in I?,,) for the ground state of the Th:6J+ molecular ion in ah and D,, symmetry with an internuclear distance of R = 2/90 au calculated using the ‘non-perturbative iterative solution’ (NIS) of the Dirac equation. E, denotes the nonrelativistic energy, E, the relativistic energy

I D,h

E”, -5

50s a - 12314.28170 - 12887.16599 - 13687.03453 - I41 12.63765 5Os4lp a - 12342.84477 - 12934.94924 - 13908.81654 - 14433.27797 50~4 I p32d a - 12345.68585 - 12942.73060 - 13914.06372 - 14447.12169 50s4lp32d23f a - 12345.69516 - 12942.80144 - 13914.15815 - 14447.66986 50s4lp32d23f14g ’ - 12345.69657 - I 2942.8 I 62 I - 13914.16010 - 14447.69146 50s4lp32d23fl4g5h a - 12345.69659 - I 2942.8 I 650 - 13914.16024 - 14447.69282 floating Gaussians b [361 - 12345.69661 - 13914.16029

a Even-tempered parameters: a = I .O, p = 2.1. For the specification of the basis sets see footnote a to Table 2. s: l-50, p: 3-43; d: S-36; f: 7-29; g: 9-22; h: I l-15. b Variational calculation using basis functions of the form exp( - ai 1 r - Ai 1’) with individual optimized exponents ai and positions of the functions Ai. ?he calculations are performed in l28-bit arithmetic. All given decimal places are correct concluded from the convergence behaviour of the calculations reported by W. Cencek (private communication 1996).

obtained by variational calculations in the same basis set to all given decimal places, while there is a significant disagreement for the nonrelativistic energies. The nonrelativistic values of Ref. [23] lie between 4 and 5 k E, below the nonrelativistic energies reported in this work. We were able to reproduce them by DPT and NIS calculations with a value for the velocity of light chosen as c = 1370359.895 au as was assumed in Ref. [23] to model the nonrelativistic limit. A DPT calculation performed with the 5Os41 p32d23f 14g5h basis (description see Table 2) and c = 1370359.895 au gives a relativistic correction of 4.833 pE, to order ce2 and < lo-l4 E,, to order c- 4. To yield nonrelativistic energies up to 10 figures for the system Thi79+ at R = 2/90 u0 using a relativistic method, a value for c taken as the actual velocity of light times lo4 is not large enough to model the limit c --) CQ.

Our best NIS calculation is within 38 FE,, of the exact nonrelativistic energy and 39 pEh of the extrapolated relativistic energy calculated with the FEM method. It is the most accurate result available so far using atom-centred Gaussians.

Comparison with our NIS calculations and the most reliable values in the literature shows that the error for the relativistic energy of Thy9+ calculated with DPT up to &(F6) is of the same order of magnitude as found for the hydrogen-like ion Th89+.

3.3. The quasi-molecule NiPb””

Recently Dlisterhiift et al. [21] reported calculations of the relativistic energy of the quasi-molecule NiPb’09+ at the small critical internuclear distance R = 0.002 au3 using the FEM method. The expected accuracy for the energy of the ground state is better than eight decimal places. Our relativistic energy calculated with the NIS method and a sufficiently large basis - presented in Table 3 - is in excellent agreement with this value. The absolute error of the relativistic correction of the DPT calculation up to d(ce6 ) in the energy is approximately 20 E,.

3.4. The quasi-molecule Thg69+

In Table 4 NIS calculations for the ground state of linear CD,,,) and equilateral triangular (D3,,) Th:69+

3 The nonrelativistic electronic energy of a one-electron het- eronuclear diatomic molecule in the vicinity of the united atom can be estimated by a perturbation expansion in powers of the internuclear distance R. Byers Brown and Steiner derived an expression for the nonrelativistic energy of the ground state up to order @(R5) (Eq. (63) of Ref. (311). which gives for NiPb”‘+ at R = 0.002 au the value - 5990.53030 E,. The deviation from our nonrelativistic calculation (see Table 3) is 0.11 E,. The author is not aware of an analogous expression for the relativistic energy.

R. Frunke / Chemical Physics Letters 264 (I 997) 495-501 501

with an internuclear distance of R = 2/!90 au are documented. We give them without comment.

4. Conclusions and outlook

In this work pilot calculations of relativistic energies are given for one-electron systems based on a ‘non-perturbative iterative solution’ (MS) of the Dirac equation starting from the solution of the Schrodinger equation. Using atom-centred Gauss- ians, we achieve an overall S-figure accuracy in the relativistic energies for the investigated systems. The computer time needed for a NIS calculation approximately equals the time for solving the nonrelativistic problem. Work on solving the Dirac-Fock equation by applying the NIS scheme is in progress.

Acknowledgement

The author thanks W. Cencek, U. Fleischer and W. Kutzelnigg for helpful discussions. W. Cencek, C. Dusterhiift and D. Kolb contributed by providing the author with unpublished results. Financial sup- port of the Deutsche Forschungsgemeinschaft is gratefully acknowledged.

References

[l] J.D. Morrison and R.E. Moss, Mol. Phys. 41 (1980) 491. [2] W. Kutzehrigg, Z. Phys. D 15 (1990) 27. [3] G.L. Sewell, Proc. Cambridge Phil. Sot. 45 (1949) 631. [4] A. Rutkowski, J. Phys. B 19 (1986) 149, 3431, 3443. [5] A. Rutkowski and D. Rutkowska, Phys. Ser. 36 (1987) 397. [6] K. Jankowski and A. Rutkowski, Phys. Ser. 36 (1987) 464. [7] A. Rutkowski, J. Phys. B 21 (1988) 6147. [8] W. Kutzemigg, Z. Phys. D I1 (1989) 15. [9] S. Gesztesy, H. Grosse and B. Thaller, Phys. I&t. B 116

(1982) 155.

[lOI

[ill

iI21

1131

1141

I151

(161 [171

iI81

[I91

DO1

ml

La [x31

D41 [251 [W [271

I281

1291 I301

[311

[321

[331

I341 [351

S. Gesztesy, H. Grosse and B. Thaller, Phys. Rev. Lett. 50 ( 1983) 62. S. Gesztesy, H. Grosse and B. Thaller, Ann. Inst. Henri Poincare 40 ( 1984) 159. S. Gesztesy, H. Grosse and B. Thaller, Adv. Appl. Math. 6 (1985) 159. A.J. Sadlej and J.G. Snijders, Chem. Phys. Lett. 229 (1994) 435. A.J. Sadlej, J.G. Snijdets, E. van Lenthe and E.J. Baerends J. Chem. Phys. 102 (1995) 1758. J.G. Snijders and A.J. Sadlej, Chem. Phys. Lett. 252 (1996) 51. C. van Wbllen, J. Chem. Phys. 103 (1995) 3589. R. Franke and W. Kutzelnigg, Chem. Phys. Lett. 199 (1992) 561. W. Kutzelnigg, E. Ottschofski and R. Franke, J. Chem. Phys. 102 (1995) 1740. E. Ottschofski and W. Kutzelnigg, J. Chem. Phys. 102 (1995) 1740. W. Kutzelnigg, R. Franke, E. Ottschofski and W. Klopper. in: New challenges in computational quantum chemistry, eds. R. Broer, P.J.C. Aerts and P.T. van Duijnen (Rijksuni- versiteit Groningen, Groningen, 1994) p. I 12. C. Diisterhiift, L. Yang, D. Heinemann and D. Kolb, Chem. Phys. Len. 229 (1994) 667. D. Sundholm, Chem. Phys. Len. 223 (1994) 469. F.A. Parpia and A.K. Mohanty, Chem. Phys. Lett. 238 ( 1995) 209. W. Kutzelnigg, Phys. Rev. A 54 (1996) 1183. T.U. Helgaker, Hermit program (1986), unpublished. R. Franke, DPTIII program (1995). unpublished. E.R. Cohen and B.N. Taylor, Rev. Mod. Phys. 59 (1987) 1121. M.W. Schmidt and K. Ruedenberg, J. Chem. Phys. 71 (1979) 3951. E. Teller, Z. Phys. 61 (1930) 458. R.E. Stanton and S. Havriliak, J. Chem. Phys. 81 (1984) 1910. W. Byers Brown and E. Steiner, J. Chem. Phys. 44 (1966) 3934. F. Mark, H. Lischka and F. Rosicky. Chem. Phys. Lett. 71 (1980) 507. L. LaJohn and J.D. Talman, Chem. Phys. Lett. 189 (1992) 383. C. Diisterhoft and D. Kolb, private communication (19%). M.M. Madsen and J.M. Peek, At. Data 2 (1971) 171.

numerical study of the iterative solution of the one-electron dirac equation based on ‘direct...

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