Anomalous Electron Correlation Due To Near Degeneracy Effects: Low-lying Ionic States of Ne and Ar

Transcription

1 Anomalous Electron Correlation Due To Near Degeneracy Effects: Low-lying Ionic States of Ne and Ar Paul S. Bagus 1), Ria Broer 2), and Fulvio Parmigiani 3) 1 Department of Chemistry, University of North Texas, Denton, TX , USA 2 Department of Chemical Physics and Materials Science Centre, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands 3) Department of Physics, Università degli Studi di Trieste, Via Valerio, 2 I-34127, Trieste, Italy and Sincrotrone Trieste, I Basovizza, Trieste, Italy ABSTRACT: This letter addresses a long-standing problem related to non-dynamical electron correlation effects. The origin of the large differential electronic correlation energy among the neutral 1 S ground state, the lowest, 2 P, ionic state and the first excited, 2 S, ionic state of the Ne and Ar atoms is explained in terms of the near degeneracy of low-lying excited configurations. There is an anomalous correlation for the 2 S state that is shown to be due to non-dynamical correlation involving a low-lying excited configuration. The conceptual framework used here is also appropriate to be used for other atomic and molecular systems.

2 In the Hartree-Fock, HF, ΔSCF method for calculating ionization potentials, IP s, the IP is determined by subtracting the HF self-consistent field, SCF, energies calculated for the initial, ground state, of the system and for an appropriate state of the ion.[1] Specifically, IP(ΔSCF) = E SCF (Ion) E SCF (Ground State). (1) Especially for the IP to low-lying states of the ion, it is expected that IP(ΔSCF) will be smaller than the observed IP; this was first pointed out by Mulliken. [2] The explanation for this bounding of the IP arises from the decomposition of the electron correlation energy, E corr, into contributions from pairs of electrons as being the dominant terms for E corr. [3] Since the ion has N 1 electrons when the initial, neutral, system has N electrons; the ion has N-1 fewer pairs and hence a smaller E corr than the initial state. However, Bagus [4] pointed out that this was not the case for the ionization of Ne and Ar to the lowest 2 S state of the ions. This state is the first excited state of the ionic species. [5] For these states, the IP(ΔSCF) is actually larger than the experimental value; for the 3s ion of Ar, the IP(ΔSCF) is nearly 4 ev larger than experiment. This means that there must be unusual correlation effects for the 2 S states of Ne + and Ar + and related systems that lead to a larger E corr for the system with N 1 electrons than for the system with N electrons. The situation is summarized in Table I where experimental and theoretical values for the lowest 2 P(np 1 ) and 2 S(ns 1 ) ions of Ne and Ar are presented. The IP(ΔSCF) for np ionization are smaller than experiment as expected by the general rules for the magnitude of the correlation energy. [2] On the other hand, the IP(ΔSCF) for ns ionization are larger than the experimental values. The principle object of this work is to investigate and identify the origin of the unexpectedly large electron correlation effects for the 2 S(ns 1 ) ions. It is shown that by taking account of a near degeneracy electron correlation effect one returns, with a suitably extended definition of IP(ΔSCF), to the expected situation that the calculated IP s are smaller than those 2

3 measured. The near degeneracy effect that is included can arise only for the 2 S state of the ion but not for the 2 P state of the ion or for the 1 S ground state of the neutral atom. For representative studies of atomic correlation effects within the theoretical framework of the mixing of nearly degenerate configurations see Refs A classification scheme normally used to distinguish near degeneracy from other types of electron correlation effects is to include the near degeneracy contributions in the category of non-dynamical, or static, correlation as opposed to dynamical correlation effects. [11-14] While, as is usually the case with non-dynamical correlation, the definition of near degeneracy is qualitative, [12, 13, 15] this paper uses a simple relationship from perturbation theory to estimate when such near degeneracy may be important. Thus, this work may have general value, beyond the specific results for the low-lying ions of Ne and Ar. The conceptual framework used for these systems is also appropriate to use to estimate, for other atomic and molecular systems, when there may be anomalous electron correlation effects. The analysis is based on the properties of wavefunctions, WF s, for the initial and ionic states of Ne and Ar. The WF s are determined by numerical integration of the HF and multiconfiguration HF, MCHF, equations for the isolated atoms. The calculations were performed with programs developed by C. F. Fischer; [16] the WF s are non-relativistic and the usual spin and space symmetry restrictions are used. For the neutral atoms and for the ions, HF WF s are determined. In addition for the 2 S ions a two configuration MCHF WF was determined. The first configuration for the MCSCF WF, Φ 1 =(ns) 1 (np) 6, is just the HF configuration; the second configuration, Φ 2 =(ns) 2 (np) 4 (nd) 1, involves the excitation np 2 ns3d. Only the coupling of the open (np) 4 to the 1 D multiplet can lead to the required total coupling of 2 S for Φ 2. This configuration is a low-lying excitation since one np electron is lowered in energy by being 3

4 placed into the lower ns sub-shell while the other np electron is raised in energy by being placed into the higher 3d sub-shell. The net energetic cost involves a cancellation of these two terms; specific numerical values will be discussed below. Such a low-lying excitation is not possible for the 2 P (ns) 2 (np) 5 configuration. While the excitation of np 3d may not involve a large excitation energy, especially for Ar, the configuration (ns) 2 (np) 4 (3d) 1 has a different parity from (ns) 2 (np) 5 and these two configurations cannot mix. The configurations for the double excitations (np) 2 (3d) 2 will mix with the HF 2 P configuration for the ion, but it is expected that this configuration will involve a large excitation energy and will not be nearly degenerate. Similar arguments apply to the 1 S (ns) 2 (np) 6 ground state of the neutral atoms and show that there will not be near degeneracy in this case either. The type of excitation involved in the second configuration has been used previously in the treatment of correlation effects for the core-level, 3s, ionization of 3d series transition metal ions. [8,17] The inclusion of such near degeneracy configurations in the WF is necessary in order to correctly describe the complex structure observed with X-ray photemission spectroscopy, XPS, for these ions. [18] However, this is the first time that such an excitation has been used to describe low lying excited states. As was noted above, this configuration is expected to be nearly degenerate since there is a gain of energy when an np electron is moved to fill the partly occupied ns shell while there is an energetic cost when the second np electron is moved into the empty 3d shell. Furthermore, the off-diagonal Hamiltonian matrix element between the first and second configurations, H 12, is proportional to a Slater R integral, R 1 (ns,3d;np,np). [19] This exchange-like R integral could easily have a large value and, when H 12 is sufficiently large, Φ 2 4

5 will represent important static correlation effects. These qualitative considerations about near degeneracy will be placed in a quantitative context in the discussion that follows. A clear indication of the importance of the MCHF WF that includes Φ 2 is seen from the IP(ΔMCHF) values given in Table I. The Ne 2s IP is reduced by >2.5 ev from the ΔSCF value. The IP(ΔMCHF) is smaller than the experimental value for the IP as is to be expected when the differences of dynamical electron correlation effects for the neutal and the ionic states dominate the error of the theoretical IP. In fact, the error for Ne of the 2s IP(ΔMCHF) is about the same as the error of the 2p IP(ΔSCF). This suggests that the remaining electron correlation effects for these IP s may be due to dynamical correlation for the neutral and ionic states. The Ar 3s IP(ΔMCHF) is reduced from IP(ΔSCF) by much more than was the case for Ne; the MCHF IP is ~6.25 ev smaller indicating a greater importance for the configuration mixing of Φ 2 than in the case of Ne. The reason for the greater importance in the case of Ar will be discussed below. For the case of Ar, as well as for Ne, the IP(ΔMCHF) is smaller than experiment suggesting that dynamic correlation may be dominant for the remaining errors. However, for Ar, the error of the 3s IP(ΔMCHF) is over 1 ev larger than the error of the 3p IP(ΔSCF). This suggests that there may be additional static, or near degeneracy, correlation not yet accounted for; see discussion below. Tables II and III give information about the MCHF H matrix, describe the energetics of the H matrix eigenvalues, and characterize the configuration interaction, CI, WF for the lower 2 S state. The information in these tables makes it possible to examine and quantify the concepts of near degeneracy for the np 2 ns3d excited configuration, Φ 2. In particular, the ratio, R, of the 5

6 off-diagonal matrix element, H 12, to the excitation energy, ΔH=H 22 H 11, provides a direct measure of whether the second configuration is nearly degenerate and whether perturbation theory can be used. When the ratio is small, the configuration Φ 2 makes a small contribution to the WF and perturbation theory can be used; the relevant formulas for the contributions of Φ 2 to the energy of the first root of the two configuration CI, ΔE, and to the CI WF, C 2, are: [20] ΔE= H 2 12 /(H 22 H 11 ) and C 2 = H 12 /(H 22 H 11 ). (2) As the ratio, R=H 12 /ΔH, increases, the relations of Eq.(2) become less accurate and, for sufficiently large ratios, they cannot be used at all. The results for Ne are considered first, then those for Ar. From Table II, the ratio for the Ne MCHF, is R= Although the diagonal excitation energy, ΔH=38.0 ev, is fairly large, the off-diagonal matrix element, H 12 = 10.4 ev, is also large giving a value for R that is not small. The reasonably large value of R indicates that Φ 2 will have modest importance in the WF although the perturbation formulas of Eq.(2) may still be approximately valid. The perturbation theory value for the correlation energy lowering is 2.8 ev or about 7% larger than the value directly calculated. Similarly, perturbation theory predicts a slightly larger magnitude for the CI mixing coefficient, C 2, than obtained by direct diagonalization. This modest mixing leads to changes in the natural orbital, NO, occupation numbers [21] that are slightly different from the HF occupations. The 2s occupation increases from 1 to 1.06 and the 2p occupation decreases from 6 to Thus, the static correlation due to the 2p 2 2s3d excitation is significant and the remaining ~2 ev errors in the 2s and 2p IP s of Ne are probably due to differential dynamical correlation effects between the 10 electron WF for Ne 6

7 and the 9 electron WF s for Ne +. Overall, there is a modest contribution to the WF from Φ 2 and perturbation theory is approaching the limits of its accuracy. The situation is quite different for Ar; see Table III. The the diagonal excitation energy, ΔH=4.3 ev, is reasonably small, especially compared to the off-diagonal matrix element, H 12 = 8.2 ev. This is hardly surprising since the 3d shell will begin to fill only a few elements in the periodic table past Ar and one should expect that Φ 2 will be nearly degenerate with the HF configuration, Φ 1. It is interesting that, although the absolute value of the excitation energy has decreased for Ar from that in Ne by almost an order of magnitude, from 38 to 4 ev, the offdiagonal matrix elements for Ar and Ne are rather similar. The consequence of the near degeneracy is that the ratio of the off-diagonal matrix element to the diagonal excitation energy is quite large in magnitude, R= 1.9; for this magnitude, perturbation theory fails and direct diagonalization of the Hamiltoinian matrix is necessary. This diagonalization shows that a large correlation energy of E corr =6.3 ev is recovered and that the second configuration has a weight of almost 40% in the MCHF WF; clearly, it is appropriate to consider this as a non-dynamical correlation effect. Given the very low energy of the 3p 2 3s3d excitation for the 2 S state of Ar +, it is to not unreasonable to expect that the 3p 2 3d 2 excitation may also be important for the neutral Ar and for both the 3p 1 and 3s 1 ionic Ar + states. Thus the 2C MCHF WF may not include all of the static correlation effects for the low-lying ionizations of Ar + and 3p 2 3d 2 excitations may be required to fully treat the static correlation, especially the differential static correlation among these states. However, it is also clear that a very large part of the static correlation has been included for the 3s 1 2 S state of Ar + and that this inclusion is sufficient to correct the anomalous differential correlation energy found for the HF WF s for Ar and Ar + ( 2 S) 7

8 It is worth noting that near degeneracy, or non-dynamical, correlation effects arise frequently. They are a common occurrence in molecules where they normally must be treated to permit correct dissociation to ground state open shell atoms; see, for example Refs. 14 and 22. They are also important for many atomic systems. They need to be taken into account to correctly describe the energy splittings of different multiplets of the ground state electronic configuration of open shell atoms; see, for example, Refs. 6 and 7. For the excitations from atomic ground to excited states, near degeneracy effects lead to major changes in the transition probabilities. [9] For the ionization of core levels in transition metal systems, [8, 10, 17, 23, 24] dramatic changes, due to atomic near degeneracy effects, are found in relative energies and in the number and intensities of the states observed in XPS. In the present letter, a new consequence of near degeneracy has been shown to be the large differential correlation energy for the lowest 2 P and 2 S states of Ne + and Ar +. Given the pervasive importance of such effects, it is useful and important to understand the role that they have for various properties of atomic and molecular systems. An important advance in this understanding has been presented here. In summary, this letter has discussed the anomalous differential correlation energy of the lowest 2 S states of Ne + and Ar +. The anomaly is that the N 1 electron ionic state has a larger correlation energy than the N electron ground state of the neutral atom. This anomaly leads to a ΔSCF IP that is larger than experiment; this sign of the error of the IP is contrary to general experience and expectation. The origin of this anomalous correlation energy has been shown to be a non-dynamical effect that involves a nearly degenerate configuration, Φ 2 =(ns) 2 (np) 4 (3d) 1, with an excitation of np 2 ns3d with respect to the HF configuration, Φ 1 =(ns) 1 (np) 6. The 8

9 analysis for the importance of Φ 2 uses first order perturbation theory and is based on the magnitude of the ratio R=H 12 /(H 22 H 11 ); in effect, this ratio is used as a quantitative measure of the near degeneracy of Φ 2. The type of analysis used here is not restricted to the particular atoms treated but may be applied in general. The considerations presented here can be used qualitatively to identify potential anomalous correlation effects for ions of other atoms and molecules. Thus, one may be able to identify a priori limitations and irregular behavior in the IP s predicted by one electron models. Indeed, the knowledge that such qualitative estimates can be made may be one of the most important consequences of the results presented in this letter. This research was supported, in part, by the Geosciences Research Program, Office of Basic Energy Sciences, U. S. Department of Energy (DOE). One of us (P.S.B.) is pleased to acknowledge partial computer support from the National Center for Supercomputing Applications, Urbana-Champaign, Illinois. References: 1. See for example, P. S. Bagus et al. J. Elec. Spec. and Related Phen. 100, 215 (1999). 2. R. S. Mulliken, J. Chim. Phys. 46, 497 (1949). 3. P.-O. Löwdin, Adv. Chem. Phys. 2, 207 (1959). 4. P. S. Bagus, Phys. Rev. 139, A619 (1965). 5. C. E. Moore, Atomic Energy Levels, Vol. 1 Natl. Bur. Stand. No., 467, U. S. GPO, Washington, D. C., 1952 and 6. D. R. Hartree, W. Hartree, and B. Swirles, Phil. Trans. Roy. Soc. London A238, 233 (1939). 7. P. S. Bagus and C. M. Moser, Phys. Rev. 167, 13 (1968); P. S. Bagus, N. Bessis, and C. M. Moser, ibid. 179, 39 (1969); P. S. Bagus, A. Hibbert, and C. Moser, J. Phys. B 4, 1611 (1971). 8. P. S. Bagus, A. J. Freeman, and F. Sasaki, Phys. Rev. Lett. 30, 850 (1973). 9. Yong-Ki Kim and P. S. Bagus, Phys. Rev. A 8, 1739 (1973). 9

10 10. P. S. Bagus et al., Phys. Rev. Lett. 84, 2259 (2000). 11. O. Sinano_lu, Adv. Chem. Phys. 6, 315 (1969). 12. P. E. M. Siegbahn in Methods in Computational Physics ed by G. H. F. Diercksen and S. Wilson (D. Reidel, Dordrecht, 1983) p J. Cioslowski, Phys. Rev. A 43, 1223 (1991). 14. P. Knowles, M. Schütz, and H.-J. Werner in Modern Methods and Algorithms of Quantum Chemistry Vol. 3 ed. by J. Grotendorst (j. von Neumann Inst. Jülich, 2000) p. 97; also E. Valderrama, E. V. Lude_a, and J. Hinze. J. Chem. Phys. 110, 2343 (1999). 16. C. F. Fischer, Comput. Phys. Commun. 64, 431 (1991) and Refs. therein. 17. E.K. Viinikka and Y. Ohrn, Phys. Rev.B 11 (1975) B. Hermsmeier et al., Phys. Rev. Lett. 61, 2592 (1988). 19. J. C. Slater, Quantum Theory of Atomic Structure, Vol. II, (McGraw-Hill, New York, 1960). 20. H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms, (Academic Press, New York, 1957). 21. P.-O. Löwdin, Phys. Rev. 97, 1474 (1955) and B. O. Roos, P. R. Taylor, and P.E. M. Siegbahn, Chem. Phys. 48, 157 (1980). 22. H. F. Schaefer III, The Electronic Structure of Atoms and Molecules (Addison-Wesley, Reading, 1972) 23. A. J. Freeman, P. S. Bagus, and J. V. Mallow, Int. J. Magnetism, 4, 35 (1973). 24. R. P. Gupta and S. K. Sen, Phys. Rev. B, 10, 71 (1974); ibid. 12, 15 (1975). 10

11 Table I. Theoretical and experimental IP s for ionization to the lowest 2 P and 2 S states of Ne + and Ar +. In the one configuration description, these states have the configurations ns 2 np 5 ( 2 P) and ns 1 np 6 ( 2 S) where n=2 for Ne and n=3 for Ar. They are, respectively, the lowest and the first excited states of the cation. Theoretical IP s are obtained with SCF HF WF s, for the neutral and ionized states and with two configuration MCHF WF s for the 2 S ionic state. The theoretical IP s are labeled IP(ΔSCF) when the SCF WF is used for the ionic state and IP(ΔMCHF) when the MCHF WF is used for the ionic state. The difference of the theoretical and experimental IP s, ΔIP, is defined so that ΔIP<0 means that the theoretical IP is less than experiment. All IP s are in ev. Experiment a IP(ΔSCF)/ΔIP IP(ΔMCHF)/ΔIP Ne Ions 2 P / S / / 1.80 Ar Ions 2 P / S / / 2.31 a See Ref

12 Table II. Energies and other properties of the two configuration MCHF WF for the lowest 2 S state of Ne +. The diagonal and off-diagonal Hamiltonian matrix elements, H ij, and the CI eigenvalues, E 1 and E 2, are given. The difference ΔH=H 22 H 11 and the ratio ΔH/H 12 are given to show the near degeneracy of the two configurations. The correlation energy recovered with the 2C MCHF WF, E corr, and the difference of the MCHF energy eigenvalues, ΔE=E 2 E 1, are given as additional measures of the importance of the near degeneracy CI. Finally the CI coefficients, C 1 and C 2, and the natural orbital, NO, occupation numbers, ω, for the lower state are given. Total energies and diagonal energies are given in hartrees; all other energies are given in ev. Matrix Elements Eigenvalues Lowest ω Eigenvector H 11 = E 1 = C 1 = s H 22 = E 2 = C 2 = s ΔH= ΔE= p H 12 = E corr =2.64 3d H 12 /ΔH=

13 Table III. Energies and other properties of the two configuration MCHF WF for Ar + ( 2 S). See the caption to Table II. Matrix Elements Eigenvalues Lowest ω Eigenvector H 11 = E 1 = C 1 = s& 2s H 22 = E 2 = C 2 = p ΔH=+4.28 ΔE= s H 12 = 8.17 E corr =6.26 3p H 12 /ΔH= d