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Aspects of separability in the coupled cluster based direct methods for energy differences

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Summary

In this paper we have discussed in detail the aspects of separability of the energy differences obtained from coupled cluster based “direct” methods such as the open-shell Coupled Cluster (CC) theory and the Coupled Cluster based Linear Response Theory (CC-LRT). It has been emphasized that, unlike the state energiesper se, the energy differences have a semi-local character in that, in the asymptotic limit of non-interacting subsystemsA, B, C, etc., they are separable as ΔE A , ΔE B , ΔE A + ΔE B , etc. depending on the subsystems excited. We classify the direct many-body methods into two categories: core-extensive and core-valence extensive. In the former, we only implicitly subtract the ground state energy computed in a size-extensive manner; the energy differences are not chosen to be valence-extensive (separable) in the semi-local sense. The core-valence extensive theories, on the other hand, are fully extensive — i.e., with respect to both core and valence interactions, and hence display the semi-local separability. Generic structures of the wave-operators for core-extensive and core-valence extensive theories are discussed. CC-LRT is shown to be core-extensive after a transcription to an equivalent wave-operator based form. The emergence of valence disconnected diagrams for two and higher valence problems are indicated. The open-shell CC theory is shown to be core-valence extensive and hence fully connected. For one valence problems, the CC theory and the CC-LRT are shown to be equivalent. The equations for the cluster amplitudes in the Bloch equation are quadratic, admitting of multiple solutions. It is shown that the cluster amplitudes for the main peaks, in principle obtainable as a series inV from the zeroth order roots of the model space, are connected, and hence the energy differences are fully extensive. It is remarkable that the satellite energies obtained from the alternative solutions of the CC equations are not valence-extensive, indicating the necessity of a formal power series structure inV of the cluster amplitudes for the valence-extensivity. The alternative solutions are not obtainable as a power series inV. The CC-LRT is shown to have an effective hamiltonian structure respecting “downward reducibility”. A unitary version of CC-LRT (UCC-LRT) is proposed, which satisfy both upward and downward reducibility. UCC-LRT is shown to lead to the recent propagator theory known as the Algebraic Diagrammatic Construction. It is shown that both the main and the satellite peaks from UCC-LRT for the one valence problems are core-valence extensive owing to the hermitized nature of the underlying operator to be diagonalized.

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References

  1. Coester F (1958) Nucl Phys 7:421; Coester F, Kümmel H (1960) Nucl Phys 17:477; Kümmel H (1961) Nucl Phys 22:177

    Google Scholar 

  2. Kümmel H, Lührmann KH, Zabolitzky JG (1978) Phys Rep 36C:1 for an extensive survey of the closed-shell CC approach to nuclear structure

    Google Scholar 

  3. Cizek J (1966) J Chem Phys 45:4256; (1969) Adv Chem Phys 14:35

    Google Scholar 

  4. Bartlett RJ (1981) Ann Rev Phys Chem 32:359 for an exhaustive survey of the closed shell CC theory for electronic structure. Open shell CC developments upto the time are also covered

    Google Scholar 

  5. Mukherjee D, Moitra RK, Mukhopadhyay A (1975) Mol Phys 30:1861; (1977) Mol Phys 33:955

    Google Scholar 

  6. Mukherjee D (1979) Pramana 12:203; Haque A, Mukherjee D (1984) J Chem Phys 80:5058

    Google Scholar 

  7. Offerman R, Ey W, Kümmel H (1976) Nucl Phys A273:349; Ey W (1978) Nucl Phys A296:189

    Google Scholar 

  8. Lindgren I (1978) Int J Quantum Chem S12:33; Solomonson S, Lindgren I, Martensson AM (1980) Phys Scripta 21:351; Solomonson S, Martensson Pendrill AM (1984) Phys Rev A30:712; Lindgren I (1985) Phys Rev A31:1273

    Google Scholar 

  9. Kutzelnigg W (1982) J Chem Phys 77:3081; Kutzelnigg W, Koch S (1983) J Chem Phys 79:4315

    Google Scholar 

  10. Haque A, Kaldor U (1985) Chem Phys Lett 117:347; (1985) Chem Phys Lett 120:261; (1986) Chem Phys Lett 128:45; Kaldor U (1987) J Comp Chem 8:448; (1987) J Chem Phys 87:467; (1986) Int J Quantum Chem S20:445

    Google Scholar 

  11. Mukherjee D (1986) Chem Phys Lett 125:207; (1986) Proc Ind Acad Sci 96:145; (1986) Int J Quantum Chem S20:409; Mukherjee D (1988) in: Arponen J, Bishop R, Mannien M (eds) Condensed matter theories, Vol 3. Plenum Press, NY

    Google Scholar 

  12. Lindgren I, Mukherjee D (1987) Phys Rep 151:93

    Google Scholar 

  13. Kutzelnigg W, Mukherjee D, Koch S (1987) J Chem Phys 87:5902; Mukherjee D, Kutzelnigg W, Koch S (1987) J Chem Phys 87:5911; Koch S, Mukherjee D (1988) Chem Phys Lett 145:321

    Google Scholar 

  14. Sinha D, Mukhopadhyay S, Mukherjee D (1986) Chem Phys Lett 129:369; Sinha D, Mukhopadhyay S, Chaudhuri R, Mukherjee D (1989) Chem Phys Lett 154:544; Chaudhuri R, Mukhopadhyay D, Mukherjee D (1989) Chem Phys Lett 162:393

    Google Scholar 

  15. Pal S, Rittby M, Bartlett RJ, Sinha D, Mukherjee D (1987) Chem Phys Lett 137:273; (1988) J Chem Phys 88:4357

    Google Scholar 

  16. Kroto HW, Matti GY, Suffolk RJ, Watts JD, Rittby M, Bartlett RJ (1990) J Am Chem Soc 112:3779

    Google Scholar 

  17. Ben-Shlomo S, Kaldor U (1988) J Chem Phys 89:956; Kaldor U (1990) Chem Phys 140:1; (1990) J Chem Phys 92:3680

    Google Scholar 

  18. Stolarczyk LZ, Monkhorst HJ (1985) Phys Rev A32:725, 743; (1988) Phys Rev A37:1980, 1926

    Google Scholar 

  19. Nakatsuji H (1985) J Chem Phys 83:5743; (1987) Theor Chim Acta 71:201

    Google Scholar 

  20. Jeziorski B, Paldus J (1988) J Chem Phys 88:5673

    Google Scholar 

  21. Monkhorst HJ (1977) Int J Quantum Chem S11:421; Dalgaard E and Monkhorst HJ (1983) Phys Rev A28:1217

    Google Scholar 

  22. Paldus J, Cizek J, Saute M, Laforgue A (1978) Phys Rev A17:805; Saute M, Paldus J, Cizek J (1979) Int J Quantum Chem 15:463

    Google Scholar 

  23. Nakatsuji H (1978) Chem Phys Lett 59:362; (1979) Chem Phys Lett 67:329; Nakatsuji H, Hirao K (1978) J Chem Phys 68:2053; (1978) J Chem Phys 68:4279; Nakatsuji H (1983) Int J Quantum Chem S17:241

    Google Scholar 

  24. Mukherjee D, Mukherjee PK (1979) Chem Phys 39:325; Ghosh S, Mukherjee D, Bhattacharyya SN (1981) Mol Phys 43:173; Ghosh S, Mukherjee D, Bhattacharyya SN (1982) Chem Phys 72:161; Ghosh S, Mukherjee D (1984) Proc Ind Acad Sci 93:947

    Google Scholar 

  25. Emrich K (1981) Nucl Phys A351:379

    Google Scholar 

  26. Sekino H, Bartlett RJ (1984) Int J Quantum Chem S18:255

    Google Scholar 

  27. Takahashi M, Paldus J (1986) J Chem Phys 85:1486

    Google Scholar 

  28. Geertsen J, Rittby M, Bartlett RJ (1989) Chem Phys Lett 164:57

    Google Scholar 

  29. Koch H, Jensen HJ Aa, Jorgensen P, Helgaker T (1990) J Chem Phys 93:3345; Koch H and Jorgensen P (1990) J Chem Phys 93:3333

    Google Scholar 

  30. Banerjee A, Simons J (1981) Int J Quantum Chem 19:207; (1983) Chem Phys 81:297; (1984) Chem Phys 87:215

    Google Scholar 

  31. Laidig WD, Saxe P, Bartlett RJ (1987) J Chem Phys 86:887; Bartlett RJ, Dykstra CE, Paldus J (1984) in: Dykstra CE (ed) Advanced theories and computational approaches to the electronic structure of molecules. Reidel, Dordrecht

    Google Scholar 

  32. Chaudhuri R, Mukherjee D, Prasad MD (1987) in: Mukherjee D (ed) Lecture Notes in Cheimstry, Vol 50. Springer, Berlin

    Google Scholar 

  33. Mukherjee D, Pal S (1989) Adv Quantum Chem 20:291

    Google Scholar 

  34. Bloch C (1958) Nucl Phys 6:329

    Google Scholar 

  35. Goldstone J (1957) Proc Roy Soc A239:267

    Google Scholar 

  36. Hubbard J (1957) Proc Roy Soc A240:539

    Google Scholar 

  37. Hugenholtz NM (1957) Physica 23:481

    Google Scholar 

  38. Brandow B (1967) Rev Mod Phys 39:771; (1977) Adv Quantum Chem 10:187

    Google Scholar 

  39. Lindgren I (1971) J Phys B7:2441; Lindgren I, Morrison J (1981) Atomic many body theory. Springer, Heidelberg

    Google Scholar 

  40. Mukherjee D (1988) in: Arponen J, Bishop RF, Mannien M (eds) Condensed matter theories, Vol 3. Plenum, N.Y.

    Google Scholar 

  41. Hose G, Kaldor U (1979) J Phys B12:3827; (1980) Phys Scripta 21:357; (1982) J Phys Chem 86:2133

    Google Scholar 

  42. Schucan TH, Weidenmuller Y (1972) Ann Phys 73:108; (1973) Ann Phys 76:483

    Google Scholar 

  43. Jeziorski B, Monkhorst HJ (1981) A24:1668

  44. Chaudhuri R, Sinha D, Mukherjee D (1989) Chem Phys Lett 163:165

    Google Scholar 

  45. Mukhopadhyay D, Mukherjee D (1989) Chem Phys Lett 163:171

    Google Scholar 

  46. Mukhopadhyay D, Mukherjee D (1991) Chem Phys Lett 177:441

    Google Scholar 

  47. Meissner L, Kucharski SA, Bartlett RJ (1989) J Chem Phys 91:6187; (b) Meissner L, Bartlett RJ (1990) J Chem Phys 92:561

    Google Scholar 

  48. Meissner L, Bartlett RJ (1989) J Chem Phys 91:4800

    Google Scholar 

  49. Mukhopadhyay S, Chaudhuri R, Mukhopadhyay D, Mukherjee D (1990) Chem Phys Lett 173:181; (b) Mukhopadhyay S (1989) PhD Thesis, Jadavpur University (India)

    Google Scholar 

  50. Bloch C, Horowitz J (1958) Nucl Phys 8:91

    Google Scholar 

  51. Coope JAR, Sabo DW (1977) J Comput Phys 23:404

    Google Scholar 

  52. Lindgren I (1985) Phys Scripta 32:291, 611

    Google Scholar 

  53. Mukhopadhyay D, Mukherjee D, to be published

  54. Frantz LM, Mills RL (1960) Nucl Phys 15:16

    Google Scholar 

  55. Meissner L, Bartlett RJ (1990) preprint and the proceedings of the Harvard Symposium on Coupled Cluster Theory (1990); J Chem Phys 94:6670

  56. Schirmer J (1982) Phys Rev A26:2395. Schirmer J, Cederbaum LS, Walter O (1983) Phys Rev A28:1237; Tarantelli A, Cederbaum LS (1989) in: Kaldor U (ed) Lecture Notes in Chemistry, Vol 52. Springer, Heidelberg

    Google Scholar 

  57. Mukherjee D, Kutzeinigg W (1989) in: Kaldor U (ed) Lecture Notes in Chemistry, Vol 52. Springer, Heidelberg

    Google Scholar 

  58. Prasad MD, Pal S, Mukherjee D (1985) Phys Rev A31:1287

    Google Scholar 

  59. Bartlett RJ, Kucharski SA, Noga J (1989) Chem Phys Lett 155:133; Watts JD, Trucks GW and Bartlett RJ (1989) Chem Phys Lett (1989) 157:359

    Google Scholar 

Download references

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Mukhopadhyay, D., Mukhopadhyay, S., Chaudhuri, R. et al. Aspects of separability in the coupled cluster based direct methods for energy differences. Theoret. Chim. Acta 80, 441–467 (1991). https://doi.org/10.1007/BF01119665

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  • DOI: https://doi.org/10.1007/BF01119665

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