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

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 47907 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. 1995 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 1995 0021-9606/95/102(19)/7472/7/$6.00 1995 American Institute of Physics

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 1 1 2 Z 2 ; D 1. 2 2 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 1 5 32Z 2 1 2048 1 Z 4 1 Z 128 1 2 3/2 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 1 2 1/2 Z 1 8. 7 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 1 5 32Z 2 1 2048 1 Z 4 1 Z 128 1 2 3/2 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

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 1 1.017 14 1.021 636 1.028 551 9 0.672 39 2 2.008 173 9 2.014 096 50 2.012 427 5 0.084 48 3 3.005 366 5 3.008 482 46 3.008 103 6 0.012 59 4 4.003 994 5 4.006 145 00 4.005 995 1 0.003 74 5 5.003 181 2 5.004 830 84 5.004 812 3 0.000 37 6 6.002 643 1 6.003 983 50 6.004 011 9 0.000 48 7 7.002 260 7 7.003 390 31 7.003 446 7 0.000 80 8 8.001 974 9 8.002 951 62 8.003 027 3 0.000 94 9 9.001 753 3 9.002 614 02 9.002 704 4 0.001 01 10 10.001 576 10.002 345 0 10.002 448 8 0.001 04 11 11.001 432 11.002 126 6 11.002 241 8 0.001 05 12 12.001 312 12.001 946 4 12.002 071 1 0.001 04 13 13.001 210 13.001 792 9 13.001 928 5 0.001 04 14 14.001 123 14.001 663 2 14.001 807 7 0.001 04 15 15.001 048 15.001 550 6 15.001 704 4 0.001 02 16 16.000 982 16.001 451 4 16.001 615 3 0.001 03 17 17.000 924 17.001 365 6 17.001 537 9 0.001 01 18 18.000 872 18.001 287 4 18.001 470 1 0.001 02 19 19.000 826 19.001 218 8 19.001 410 5 0.001 01 20 20.000 785 20.001 157 7 20.001 357 9 0.001 01 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 1.2279. 12 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 1.017 14. 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 ) 0.511 560 1. 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 0.8839), 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 ) 0.517 901 8, 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 2 2 2 2 N 3 0. 13 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

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 1 0.487 729 70 0.511 560 1 0.517 901 8 0.527 751 1.866 2 2 2.861 633 73 2.889 283 45 2.909 401 25 2.903 724 0.196 30 3 7.236 291 85 7.265 164 19 7.281 954 82 7.279 913 0.028 05 4 13.609 723 9 13.639 190 9 13.655 068 2 13.655 566 0.003 64 5 21.985 834 1 22.015 667 3 22.031 145 5 22.030 972 0.000 79 6 32.360 596 4 32.390 668 0 32.405 923 6 32.406 247 0.001 00 7 44.735 333 6 44.765 575 0 44.780 689 6 44.781 445 0.001 69 8 59.110 040 8 59.140 408 2 59.155 429 8 59.156 595 0.001 97 9 75.484 715 4 75.515 181 5 75.530 140 0 75.531 712 0.002 08 10 93.859 355 8 93.889 892 7 93.904 796 6 93.906 807 0.002 14 11 114.233 961 114.264 571 114.279 422 114.281 884 0.002 16 12 136.608 530 136.639 199 136.654 031 136.656 948 0.002 13 13 160.983 063 161.013 768 161.028 561 161.032 003 0.002 13 14 187.357 603 187.388 303 187.403 092 187.407 050 0.002 11 15 215.732 020 215.762 805 215.777 572 215.782 091 0.002 09 16 246.106 443 246.137 253 246.151 984 246.157 126 0.002 09 17 278.480 828 278.511 667 278.526 408 278.532 158 0.002 07 18 312.855 177 312.886 024 312.898 293 312.907 186 0.002 85 19 349.229 488 349.260 360 349.275 042 349.282 211 0.002 04 20 387.603 762 387.634 674 0 387.649 350 387.657 234 0.002 03 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 ) 3.007 207 7 3.007 920 1 0.023 27 Be 1 S (14s/14 ) 4.007 224 6 4.005 995 1 0.030 69 B 2 P (14s,10p/15 ) 5.007 552 1 5.007 921 5 0.007 38 C 3 P (14s,10p/15 ) 6.007 984 8 6.009 435 0 0.024 14 N 4 S (14s,10p/15 ) 7.008 445 1 7.010 267 7 0.026 00 O 3 P (14s,10p/15 ) 8.008 900 7 8.011 567 1 0.033 28 F 2 P (14s,10p/15 ) 9.009 337 6 9.012 237 2 0.032 18 Ne 1 S (14s,10p/15 ) 10.009 750 10.012 547 0.027 93 Na 2 S (17s,10p/17 ) 11.009 430 11.011 167 0.015 77 Mg 1 S (17s,10p/17 ) 12.009 197 12.010 172 0.008 12 Al 2 P (17s,13p/18 ) 13.009 032 13.009 455 0.003 25 Si 3 P (17s,13p/18 ) 14.008 920 14.009 026 0.000 76 P 4 S (17s,13p/18 ) 15.008 849 15.008 667 0.001 21 S 3 P (17s,13p/18 ) 16.008 810 16.008 715 0.000 60 Cl 2 P (17s,13p/18 ) 17.008 797 17.008 675 0.000 72 Ar 1 S (17s,13p/18 ) 18.008 804 18.008 579 0.001 25 a The basis sets were those taken from Ref. 10 and modified as described in the text.

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 3 7.432 727 7.473 975 7.478 06 0.054 82 4 14.616 845 14.677 733 14.667 36 0.070 72 5 24.563 760 24.649 716 24.653 91 0.017 01 6 37.706 287 37.824 554 37.845 0 0.054 03 7 54.400 934 54.555 825 54.589 2 0.061 14 8 74.809 550 75.007 552 75.067 3 0.079 59 9 99.409 371 99.656 988 99.733 9 0.077 12 10 128.547 098 128.850 448 128.937 6 0.067 60 11 161.858 912 162.192 998 162.254 6 0.037 96 12 199.646 899 200.014 018 200.053 0.019 48 13 241.925 259 242.327 114 242.346 0.007 79 14 288.914 589 289.353 758 289.359 0.001 81 15 340.789 760 341.268 809 341.259 0.002 87 16 397.594 359 398.115 606 398.110 0.001 41 17 459.589 676 460.155 803 460.148 0.001 69 18 526.942 053 527.555 691 527.540 0.002 97 FIG. 5. Relationship between Z 3 and Z for He, N, and Ar.

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.

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, 1993. 3 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, 5987 1994. 4 S. Kais and D. R. Herschbach, J. Chem. Phys. 98, 3990 1993 ; T.C. Germann and S. Kais, ibid. 99, 7739 1993. 5 J. G. Loeser, J. H. Summerfield, A. L. Tan, and Z. Zheng, J. Chem. Phys. 100, 5036 1994 ; J. G. Loeser, Chap. 9 of Ref. 2. 6 S. Kais, S. M. Sung, and D. R. Herschbach, J. Chem. Phys. 99, 5184 1993 ; S. Kais and D. R. Herschbach, ibid. 100, 4367 1994 ; S. Kais, S. M. Sung, and D. R. Herschbach, Int. J. Quantum Chem. 49, 657 1994. 7 S. Kais, T. C. Germann, and D. R. Herschbach, J. Phys. Chem. 98, 11 015 1994. 8 J. G. Loeser, J. Chem. Phys. 86, 5635 1987. 9 C. C. J. Roothaan and P. S. Bagus, Methods in Computational Physics, edited by B. Alder Academic, New York, 1963. 10 S. Huzinaga and M. Klobukowski, J. Mol. Struct. 167, 1 1988. 11 E. R. Davidson, S. A. Hagstrom, and S. J. Chakravorty, Phys. Rev. A 44, 7071 1991 ; S. J. Chakravorty, S. R. Gwaltney, and E. R. Davidson, ibid. 47, 3649 1993. 12 D. Z. Goodson and D. R. Herschbach, J. Chem. Phys. 86, 4997 1987 ; J. G. Loeser and D. R. Herschbach, ibid. 86, 2114 1987. 13 D. J. Doren and D. R. Herschbach, J. Chem. Phys. 87, 433 1987. 14 D. K. Watson and D. Z. Goodson, Phys. Rev. A 51, R5 1995. 15 J. M. Rost, J. Phys. Chem. 97, 2461 1993. 16 T. C. Germann, S. Kais, and D. R. Herschbach unpublished.