Charge renormalization at the large-d limit for N-electron atoms and weakly bound systems

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1 Charge renormalization at the large-d limit for N-electron atoms and weakly bound systems S. Kais and R. Bleil Department of Chemistry, Purdue University, West Lafayette, Indiana Received 25 January 1995; accepted 7 February 1995 We develop a systematic way to determine an effective nuclear charge Z R D such that the Hartree Fock results will be significantly closer to the exact energies by utilizing the analytically known large-d limit energies. This method yields an expansion for the effective nuclear charge in powers of (1/D), which we have evaluated to the first order. This first order approximation to the desired effective nuclear charge has been applied to two-electron atoms with Z 2 20, and weakly bound systems such as H. The errors for the two-electron atoms when compared with exact results were reduced from 0.2% for Z 2 to 0.002% for large Z. Although usual Hartree Fock calculations for H show this to be unstable, our results reduce the percent error of the Hartree Fock energy from 7.6% to 1.86% and predicts the anion to be stable. For N-electron atoms N 3 18, Z 3 28, using only the zeroth order approximation for the effective charge significantly reduces the error of Hartree Fock calculations and recovers more than 80% of the correlation energy American Institute of Physics. I. INTRODUCTION The Hartree Fock approximation, which is based on the idea that we can approximately describe an interacting fermion system in terms of an effective single-particle model, remains the major approach for quantitative calculations. This self-consistent field approximation usually yields good zeroth order approximation results and accounts for more than 99% of the total energy. Recovering the remaining 1% error in the total energy, which is the correlation energy, is the main driving force for introducing new methods for calculating the electronic structure. A wide variety of techniques are available for predicting the correlation energy, including configuration interaction, many-body perturbation theory, multiconfiguration Hartree Fock, and coupled cluster methods. Most of these methods, however, are quite computationally expensive relative to Hartree Fock calculations. 1 Dimensional scaling offers an effective means to treat such a nonseparable many-body problem. 2 Among recent applications of dimensional scaling are large order dimensional perturbation expansions for few-body systems, 3 including complex dimensional scaling for resonances and unstable states, 4 correlated electronic structure models for atoms and solids based on the sub-hamiltonian approximation, 5 and dimensional renormalization for atoms 6 and simple molecular systems. 7 Our goal is to utilize dimensional renormalization to obtain an effective charge in a systematic procedure to enhance the accuracy of Hartree Fock calculations. Herschbach 2 suggested the use of charge renormalization for the two-electron atom as another means to exploit the simplicity of the D limit. He looked for an effective charge Z which makes the large-d limit energy the same as the D 3 energy. However, this requires that the solution is known at D 3. But, he did find an approximate effective charge, Z HF, which makes the Hartree Fock solution at D the same as the Hartree Fock solution at D 3. Calculating the exact energy at the large-d limit with Z HF gave a good approximation to exact energy at D 3. Eventually, this procedure was generalized to yield about 2/3 or more of the correlation energy for all neutral atoms and cations. 6 In the present approach, we are looking for a systematic way to determine an effective nuclear charge which will force the Hartree Fock calculations to give exact results. A means to find this effective charge has been developed utilizing the large-d limit where the energies are known analytically. We have applied this effective nuclear charge to N-electron atoms N 2 18 electrons with nuclear charge Z 2 28 and weakly bound systems such as H. The results show that the use of the effective nuclear charge obtained from our procedure in Hartree Fock calculations recovers more than 80% of the correlation energy. The general outline of this paper is as follows: in Sec. II we describe the systematic procedure which will lead to the two equations that will give the effective nuclear charge to first order used in our calculations. Section III gives a brief overview of the details of our Hartree Fock calculations in three dimensions. This is followed by a description of our calculations for two-electron atoms with Z varying from 2 to 20 in Sec. IV. Section V describes the procedure to calculate the effective charge for unstable systems such as H. Many electron atoms are covered in Sec. VI for N from3to18and Z from 3 to 28. Finally, the discussion follows in Sec. VII, where we justify our choice in scaling Z. II. CHARGE RENORMALIZATION Our goal is to find a nuclear charge renormalization procedure in which the Hartree Fock energy will yield the exact energy, and yet is independent of arbitrarily adjustable parameters or experimental input. In order to accomplish this goal in a systematic way, we take advantage of the fact that both the Hartree Fock and the exact energy of N-electron atoms have known analytical expressions in the large-d limit. By assuming that the effective charge is a function of the dimensionality D, and requiring that 7472 J. Chem. Phys. 102 (19), 15 May /95/102(19)/7472/7/$ American Institute of Physics

2 S. Kais and R. Bleil: Charge renormalization 7473 E HF D Z R D E exact D Z, 1 we can find an expression for the renormalized charge, Z R D, that will correct for correlation energy in the Hartree Fock theory in three dimensions. Expanding both sides of Eq. 1 in powers of (1/D) allows us to obtain the renormalized charge Z R D in a systematic way. We begin by first expanding the renormalized charge, Z R D, which gives Z R D Z 1 Z Z 2 ; D Substituting this expansion for Z R D in Eq. 1 and expanding both sides to first order in 1/, leads to the following conditions for Z and Z 1 : Z is readily found from the condition E Z E HF Z 0 3 and Z 1 is determined by Z 1 E 1 Z E HF 1 Z E HF 1. 4 Z Z Z For two electron atoms, both the large-d limit energy E (Z) and E HF (Z) and the first order correction E 1 (Z) and E 1 HF (Z) are known analytically. 2,12 For N-electron atoms, Loeser 8 derived simple analytical solutions for both E (Z) and E HF (Z). These analytical expressions, along with Eq. 3 above, form a systematic and self-consistent way to find the renormalized charge Z. III. HARTREE FOCK COMPUTATIONS The program used for the computations in this paper were performed utilizing an algorithm originally designed by Roothaan and Bagus. 9 This algorithm was altered slightly to allow for determination of an effective charge which will give a known exact energy for atoms, as well as determination of the Hartree Fock energy for a given effective nuclear charge. We used the well-tempered Gaussian-type functions, modified to account for the new renormalized nuclear charge the basis set exponents were changed to [(Z R D ) 2 /Z 2 ], as our basis set based upon the Roothaan Hartree Fock atomic orbital expansion for atoms from helium (Z 2) to argon (Z 18). 10 For those atoms with known near degeneracy mixing as in the beryllium isoelectronic series, for example, our Hartree Fock calculations were modified to approximate the complete active valence space multiconfigurational Hartree Fock energy, E CAS (Z). Following Davidson et al., 11 this modified energy can be written to a good approximation as E CAS Z E HF Z B 1 N Z B N, 5 where B 1 (N) has been tabulated in Ref. 11 and B(N) was fixed to give the exact CAS energies. 11 IV. TWO-ELECTRON ATOMS We would like to find the effective charge which makes the Hartree Fock solution the same as the exact solution in all dimensions. In the D limit, the exact energy of a two-electron atom is found from the minimum of the effective potential to be 12 2 E Z Z Z Z 4 1 Z /2 Z. The corresponding large-d limit for the Hartree Fock approximation 12 is obtained by imposing the constraint that the angle between the electron-nucleus radii, r 1 and r 2,is 90 ; this gives E HF Z Z /2 Z The first order correction to the large-d limit, E 1 (Z) and E HF 1 (Z), can be found analytically 2,12 by summing the corresponding normal mode frequencies around the minimum of the effective potential at the large-d limit. Using this expansion to first order, we fix the renormalized charge Z R D to be where and Z D R Z 1 Z 1, Z 1 2 3/2 Z Z Z 4 1 Z /2 Z 1/2 Z 1 E 1 Z E HF 1 Z 2 1/2 2Z. 10 In Table I we show that the renormalized charge to first order is within only 0.08% error compared with the exact renormalized charge for Z 2. This error drops to only 0.001% as Z increases. These results are shown graphically in Fig. 1. Here, we have shown how the error decreases dramatically from the large D limit of the renormalized charge Z by including the first order correction (Z Z 1 ). In Table II, we show the calculated Hartree Fock energies using the renormalized charges from Table I, and compare these energies with known exact energies. The percent error shows that the Hartree Fock energies utilizing the renormalized charges differ from the exact energies by less than 0.2% for small Z, corresponding to the recovery of approximately 85% of the correlation energy. In Fig. 2, it can also be seen that the percent error between the Hartree Fock energies with the renormalized charge and the exact energies decrease even further for increasing Z. V. WEAKLY BOUND SYSTEMS The Hartree Fock theory fails as a zero-order model for highly correlated systems such as open-shell systems, multiply excited states, or negative ions. The ground state energy 6 8 9

3 7474 S. Kais and R. Bleil: Charge renormalization TABLE I. Renormalized charges for two electron atoms. Z Z Z Z 1 Z 3 R % error of the hydride ion, H, is a prototype example of this inadequacy. Large-order dimensional perturbation theory has been used to obtain highly accurate energies for two-electron atoms. 2 However, systems that are only weakly bound at D 3 become unstable in the D limit. 13 The large-d limit solution of the two-electron atom with nuclear charge Z is stable only if Z Z c Doren and Herschbach 13 have shown that interpolation between the large-d limit solution and the D 1 solution gives an energy which is correct to within a few tenths of a percent. Recently, Watson and FIG. 1. Percentage error of the renormalized charge as function of the nuclear charge Z for two-electron atoms. The dashed line is the renormalized charge at infinite dimension (Z ), and the solid line include the first order correction (Z 1 ) to this renormalized charge. Goodson 14 have shown that accurate energies can be obtained from large-order perturbation theory in (1/D) if the repulsive part of the potential is attenuated with a D-dependent factor that prevents the large-d dissociation. Our results show that by using charge renormalization, one can drastically improve the Hartree Fock energy. For H, Eq. 3 gives Z Using this effective charge instead of the actual nuclear charge in the Hartree Fock equations reduces the error from 7.6% to 3%, and gives E HF (Z ) To carry out the first order correction to the renormalized charge, Z 1, special treatment is needed. Although the HF large-d limit solution remains stable for Z 1 (Z c HF ), it becomes unstable for the exact large-d solution. Since Z 1 is below the critical charge Z c, the antisymmetric stretching vibrational mode becomes negative and the first order correction to the energy E 1 (Z) becomes complex. Thus, the symmetric configuration corresponds to a saddle point. Rost 15 showed that by using complex dimensional scaling, one can handle this situation and obtain real energies to the first order in (1/D). Using this complex dimensional scaling procedure allows us to obtain the first order correction to the renormalized charge, Z 1,as shown in Table I. This gives a stable corrected Hartree Fock energy of E 3 HF (Z Z 1 ) , reducing the percentage error for H to just 1.9% as shown in Table II. VI. MANY-ELECTRON ATOMS For S electronic states with totally symmetric configurations where all electrons are equivalent, Loeser 8 obtained an analytical solution for the ground state energy of N-electron atoms in the large-d limit. In such a system, the electrons lie at the corners of a regular N-point simplex, while the nucleus lies along an axis which passes perpendicularly through the center of that figure. The total groundstate energy with hydrogenic shell structure is given by 8 n max E N,Z n 1 N n,z N n 1,Z n 2, 11 where N n is the number of electrons with principal quantum numbers less than or equal to n. To justify the hydrogenic shell structure for the total energy, Germann 16 has shown with a quantum defect analysis that the energy remains almost unchanged when one introduces a quantum defect to the quantum number n. The Hartree Fock total energy, E HF (N,Z), is given by a corresponding formula. For the exact energy, E (N,Z), each large-d solution for the energy of a given shell,, has the form 1 2 Z2 N N N 1 3 N 3, 12 where is the smallest positive root of the quartic equation 8NZ N The corresponding Hartree Fock formula can be obtained by fixing all interelectronic angles to 90, 6 which gives 2 2 Z 3/2 HF NZ2 2 1 N 1 14

4 S. Kais and R. Bleil: Charge renormalization 7475 TABLE II. Renormalized energies for two electron atoms. Z E 3 HF (Z) E 3 HF (Z ) E 3 HF (Z Z 1 ) E 3 (Z) a % error a Exact energies were taken from Ref. 11. to be the large-d solution for the Hartree Fock energy of a given shell. We find that renormalization of the nuclear effective charge by means of Z yields much improved Hartree Fock energies for N-electron atoms in D 3. The renormalized charges used in our Hartree Fock calculation are given in Table III along with the modified contracted basis sets used for the Hartree Fock calculations. 10 These nuclear charges have been obtained through the use of Eq. 3 as the condition to fix Z and Eq. 10 for the solution of E (N,Z) tobe used by this condition. We have also included in Table III the effective charges, Z 3 R, which force the Hartree Fock energy to be equal to the exact energies. Comparing Z with Z 3 R typically yields a percent error of less than 0.03% for small atoms, and 0.002% for large N. Figure 3 shows Z 3 and Z where Z D Z D R Z. 15 In this figure, it can be seen that Z has the same general structure as Z 3 as N varies from 2 to 18. This indicates that in the large-d limit, the electronic shell structure is maintained. Using only this large-d limit solution without the first order correction, and evaluating the Hartree Fock ener- TABLE III. Renormalized charges for N-electron atoms Li through Ar. Atom State Basis set a Z Z 3 R % error FIG. 2. Percentage error of the total Hartree Fock energy compared with the known exact energy with and without renormalized charges as a function of the nuclear charge Z for two-electron atoms. Li 2 S (14s/14 ) Be 1 S (14s/14 ) B 2 P (14s,10p/15 ) C 3 P (14s,10p/15 ) N 4 S (14s,10p/15 ) O 3 P (14s,10p/15 ) F 2 P (14s,10p/15 ) Ne 1 S (14s,10p/15 ) Na 2 S (17s,10p/17 ) Mg 1 S (17s,10p/17 ) Al 2 P (17s,13p/18 ) Si 3 P (17s,13p/18 ) P 4 S (17s,13p/18 ) S 3 P (17s,13p/18 ) Cl 2 P (17s,13p/18 ) Ar 1 S (17s,13p/18 ) a The basis sets were those taken from Ref. 10 and modified as described in the text.

5 7476 S. Kais and R. Bleil: Charge renormalization FIG. 3. Z as defined in the text, Eq. 15 as a function of the number of electrons N for both D and D 3. FIG. 4. Percentage error of the total Hartree Fock CAS energy compared with the known exact energy with and without renormalized charges as a function of the number of electrons N. gies with the effective charge, Z at D 3, we obtain a typical accuracy of 0.05% for small atoms and 0.003% for large atoms, as shown in Table IV. These results are shown graphically in Fig. 4. This figure also shows the percentage error of a typical Hartree Fock energy calculation using the actual nuclear charge. In Fig. 5, we show that the relationship between Z 3 and Z, which will allow us to find the correct effective nuclear charge at D 3, is a simple linear relation for a fixed number of electrons N. Figures 6 8 show that the use of our renormalization procedure to find an effective charge for N 4, 10, and 18, respectively, decreases the error of the calculated Hartree Fock energy to 0.06% as Z increases compared with exact results. VII. DISCUSSION Hartree Fock theory continues to be the major approach for quantitative calculations. We offer a systematic method in correcting the Hartree Fock energy, accounting for most of the missing correlation energy by taking an effective nuclear charge with no significant changes to the computational time. By taking advantage of high dimensional limit results, known analytically for all N-electron atoms, we have fixed this effective charge systematically. To test our procedure, TABLE IV. Renormalized energies for atoms Li through Ar. N E 3 CAS (Z) E 3 CAS (Z ) E 3 (Z) % error FIG. 5. Relationship between Z 3 and Z for He, N, and Ar.

6 S. Kais and R. Bleil: Charge renormalization 7477 FIG. 6. Percentage error of the total Hartree Fock and CAS energy compared with the known exact energy with and without renormalized charges as a function of the number of the nuclear charge Z for N 4. FIG. 8. Same as Fig. 6 but with N 18. we have applied the method to N-electron atoms N 2 18, with Z 2 28 and weakly bound systems such as H. We have found that our procedure significantly reduces the error of the Hartree Fock energy and recovers approximately 80% of the correlation energy. The worst error for the two-electron atoms when compared with exact results is for Z 2, and in this case the error was still reduced to less than 0.2%. As Z increases, this error decreases rapidly to 0.002% for large Z. We have also applied our method to FIG. 7. Same as Fig. 6 but with N 10. the study of H, which is predicted to be unstable by using standard Hartree Fock calculations. Our procedure reduces the percent error of Hartree Fock calculation from 7.6% to 1.86% and predicts H to be stable. Finally, for N-electron atoms N 3 18, Z 3 28, we restricted ourselves to using only the zeroth order approximation for the effective charge. We found that this significantly reduces the error of Hartree Fock calculations and recovers more than 80% of the correlation energy. This method can be extended to cover the entire Periodic Table without significant additional computational time as compared with standard Hartree Fock calculations. The basic idea in this paper was to vary the basic parameters of the Hartree Fock theory such that this theory will give the exact energy. There are four parameters which we could have chosen to vary; the dimensionality of the system D, the nuclear charge Z, the mass of the electrons m e, and the number of electrons N. The mass of the electron can be scaled out of the Hamiltonian, so varying this parameter would have no effect other than changing the overall units of the energy. We choose to vary the dimensionality because both the exact and Hartree Fock energies for atoms are known analytically in the large-dimensional limit. Therefore, we are left with the number of electrons N and the nuclear charge Z for use as adjustable parameters to equate the Hartree Fock energy to the exact energy. We choose to vary the nuclear charge Z to compensate for the change between attractive and repulsive contributions by averaging over the electron electron interactions in the Hartree Fock theory. Although we have focused upon the ground state energy of N-electron atoms and weakly bound systems, research is underway to examine the impact of effective nuclear charges on other atomic properties and molecular systems.

7 7478 S. Kais and R. Bleil: Charge renormalization ACKNOWLEDGMENTS We are very grateful to Dudley Herschbach for initiating the idea of renormalizing the charge to improve the large-d limit energies. We would like to thank John Loeser and Tim Germann for helpful discussions. This work has been supported by the Chemistry Department of Purdue University. 1 A. Szabo and N. S. Ostlund, Modern Quantum Chemistry Macmillan, New York, 1982, and references therein. 2 D. R. Herschbach, J. Avery, and O. Goscinski, Dimensional Scaling in Chemical Physics Kluwer, Dordrecht, M. Dunn, T. C. Germann, D. Z. Goodson, C. A. Traynor, J. D. Morgan III, D. K. Watson, and D. R. Herschbach, J. Chem. Phys. 101, S. Kais and D. R. Herschbach, J. Chem. Phys. 98, ; T.C. Germann and S. Kais, ibid. 99, J. G. Loeser, J. H. Summerfield, A. L. Tan, and Z. Zheng, J. Chem. Phys. 100, ; J. G. Loeser, Chap. 9 of Ref S. Kais, S. M. Sung, and D. R. Herschbach, J. Chem. Phys. 99, ; S. Kais and D. R. Herschbach, ibid. 100, ; S. Kais, S. M. Sung, and D. R. Herschbach, Int. J. Quantum Chem. 49, S. Kais, T. C. Germann, and D. R. Herschbach, J. Phys. Chem. 98, J. G. Loeser, J. Chem. Phys. 86, C. C. J. Roothaan and P. S. Bagus, Methods in Computational Physics, edited by B. Alder Academic, New York, S. Huzinaga and M. Klobukowski, J. Mol. Struct. 167, E. R. Davidson, S. A. Hagstrom, and S. J. Chakravorty, Phys. Rev. A 44, ; S. J. Chakravorty, S. R. Gwaltney, and E. R. Davidson, ibid. 47, D. Z. Goodson and D. R. Herschbach, J. Chem. Phys. 86, ; J. G. Loeser and D. R. Herschbach, ibid. 86, D. J. Doren and D. R. Herschbach, J. Chem. Phys. 87, D. K. Watson and D. Z. Goodson, Phys. Rev. A 51, R J. M. Rost, J. Phys. Chem. 97, T. C. Germann, S. Kais, and D. R. Herschbach unpublished.

Charge renormalization at the large-d limit for diatomic molecules

Charge renormalization at the large-d limit for diatomic molecules R. Bleil, A. Faliks, M. Miletic, and S. Kais Department of Chemistry, Purdue University, West Lafayette, Indiana 47907 Received 2 March