TECHNICAL PAPERS
Jan 1, 2007

Quantum Notions of Stress

Publication: Journal of Aerospace Engineering
Volume 20, Issue 1

Abstract

Existing notions of stress in a quantum mechanical framework are reviewed and discussed in a semitutorial manner suitable for a mechanics audience. Notwithstanding early fundamental work in this area since the 1930s, the increasing availability of computational tools to perform ab initio quantum mechanical calculations with high accuracy and efficacy has renewed interest in this field especially in the context of computational mechanics and materials science. Although some unresolved issues remain, the subject has evolved considerably in the past two decades with various authors offering their own unique viewpoint. In the present paper, we summarize the debate over the “definition” of stress in a quantum mechanical setting and discuss some controversial issues such as the uniqueness of the stress.

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References

Alerhand, O. L., Vanderbilt, D., Meade, R. D., and Joannopoulos, J. D. (1988). “Spontaneous formation of stress domains on crystal surfaces.” Phys. Rev. Lett., 61(17), 1973–1976.
Averill, F. W., and Painter, G. S. (1981). “Virial theorem in the density-functional formalism: Forces in H2 .” Phys. Rev. B, 24(12), 6795–6800.
Bach, C. E., Giesen, M., and Ibach, H. (1997). “Stress relief in reconstruction.” Phys. Rev. Lett., 78(22), 4225–4228.
Bamzai, A. S., and Deb, B. M. (1981). “The role of single-particle density in chemistry.” Rev. Mod. Phys., 53(1), 95–126.
Born, M., Heisenberg, W., and Jordan, P. (1926). “Zur quantenmechanik (The quantum mechanics).” Z. Phys., 35(8-9), 557–615.
Car, R., and Parinello, M. (1985). “Unified approach for molecular dynamics and density-functional theory.” Phys. Rev. Lett., 55(22), 2471–2474.
Chetty, N., and Martin, R. M. (1992). “First-principles energy density and its applications to selected polar surfaces.” Phys. Rev. B, 45(11), 6074–6088.
Collinson, C. D. (1969). “Proof of the uniqueness of the electromagnetic energy momentum tensor.” Proc. Cambridge Philos. Soc., 66(2), 437–438.
Dal Corso, A., and Resta, R. (1994). “Density-functional theory of macroscopic stress: Gradient-corrected calculations for crystalline Se.” Phys. Rev. B, 50(7), 4327–4331.
Dreizler, R. M., and Gross, E. K. U. (1990). Density functional theory, Springer, Berlin.
Ehrenfest, P. (1927). “Bermerkung über die angenäherte gültigkeit der klassischen mechanik innerhalb der quantenmechanik.” Z. Phys., 45(7-8), 455–457.
Feibelman, P. J. (1994). “Calculation of surface stress in a linear combination of atomic orbitals representation.” Phys. Rev. B, 50(3), 1908–1911.
Feynman, R. P. (1939a). “Forces in molecules.” Phys. Rev., 56(40), 340–343.
Feynman, R. P. (1939b). “Forces and stresses in molecules.” Bachelor’s dissertation, Massachusetts Institute of Technology, Cambridge, Mass.
Filippetti, A., and Fiorentini, V. (1999). “Faceting and stress of missing-row reconstructed transition-metal (110) surfaces.” Phys. Rev. B, 60(20), 14366–14371.
Filippetti, A., and Fiorentini, V. (2000). “Theory and applications of the stress density.” Phys. Rev. B, 61(12), 8433–8442.
Finkelstein, B. (1928). “Uber den virialsatz in der wellenmechanik.” Z. Phys., 50(3-4), 293–294.
Fiorentini, V., Methfessel, M., and Scheffler, M. (1993). “Reconstruction mechanism of FCC transition metal (001) surfaces.” Phys. Rev. Lett., 71(7), 1051–1054.
Fock, V. (1930). “Bemerkung zum virialsatz.” Z. Phys., 63(11-12), 855–858.
Folland, N. O. (1981). “Dynamical derivation of microscopic stress tensors.” Int. J. Quantum Chem., Quantum Chem. Symp., 15(1), 369–379.
Folland, N. O. (1986). “Microscopic stress tensors in atoms.” Phys. Rev. B, 34(12), 8296–8304.
Folland, N. O. (1988). “Reply to ‘Comment on microscopic stress tensors in quantum systems.’ ” Phys. Rev. B, 37(18), 10908–10911.
Fuchs, K. (1935). “A quantum mechanical investigation of the cohesive forces of metallic copper.” Proc. R. Soc. London, Ser. A, 151(874), 585–602.
Godby, R. W., Schutler, M., and Sham, L. J. (1986). “Accurate exchange-correlation potential for silicon and its discontinuity on addition of an electron.” Phys. Rev. Lett., 56(22), 2415–2418.
Godby, R. W., Schutler, M., and Sham, L. J. (1987). “Quasiparticle energies in GaAs and AlAs.” Phys. Rev. B, 35(8), 4170–4171.
Godfrey, M. J. (1988). “Stress field in quantum systems.” Phys. Rev. B, 37(17), 10176–10183.
Greiner, W. (1994). Quantum mechanics: An introduction, 3rd Ed., Springer, Berlin.
Griffiths, D. J. (1994). Introduction to quantum mechanics, 1st Ed., Prentice-Hall, Englewood Cliffs, N.J.
Hellmann, H. (1937). Einfuhrung in die quantenchemie, Deuticke, Leipzig, Germany (in German).
Hohenberg, P., and Kohn, W. (1964). “Inhomogeneous electron gas.” Phys. Rev., 136(3B), B864–B871.
Horsch, P., von der Linden, W., and Lukas, W. D. (1987). “On the self-energy of electrons in metals.” Solid State Commun., 62(5), 359–363.
Hybertsen, M. S., and Louie, S. G. (1986). “Electron correlation in semiconductors and insulators: Band gaps and quasiparticle energies.” Phys. Rev. B, 34(8), 5390–5413.
Hybertsen, M. S., and Louie, S. G. (1987). “Ab initio static dielectric matrices from the density-functional approach. I: Formulation and application to semiconductors and insulators.” Phys. Rev. B, 35(11), 5585–5601.
Hylleraas, E. (1929). “Neue berechnung der energie des heliums im grundzustande, sowie des tiefsten terms von ortho-helium.” Z. Phys., 54(5-6), 347–366.
Ibach, H. (1997). “Role of surface stress in reconstruction, epitaxial growth, and stabilization of mesoscopic structures.” Surf. Sci. Rep., 29(5-6), 193–263.
Jackson, J. D. (1999). Classical electrodynamics, 3rd Ed., Wiley, New York.
Jones, R. O., and Gunnarsson, O. (1989). “The density functional formalism, its applications, and prospects.” Rev. Mod. Phys., 61(3), 689–746.
Kerrighan, D. B. (1982). “On the uniqueness of the energy-momentum tensor for electromagnetism.” J. Math. Phys., 23(10), 1979–1980.
Kohn, W., and Sham, L. J. (1965). “Self-consistent equations including exchange and correlation effects.” Phys. Rev., 140(4), A1133–A1138.
Kosevich, A. M., Lifshitz, E. M., Landau, L. D., and Pitaevskii, L. P. (1986). Theory of elasticity, Vol. 7, 3rd Ed., Butterworth-Heinemann, Oxford.
Kugler, A. (1967). “Exact expressions for the energy current and stress tensor operators in many-particle systems.” Z. Phys., 198(3), 236–241.
Kulikov, N. I., Alouani, M., Khan, M. A., and Magnitskaya, M. V. (1987). “Self-energy corrections to the ab initio band structure: Chromium.” Phys. Rev. B, 36(2), 929–938.
Liberman, D. A. (1974). “Simplification of total-energy and pressure calculations in solids.” Phys. Rev. B, 9(10), 3985–3988.
Lovelock, D. (1974). “The electromagnetic energy momentum tensor and its uniqueness.” Int. J. Theor. Phys., 10(1), 59–65.
Martin, P. C., and Schwinger, J. (1959). “Theory of many-particle systems. I.” Phys. Rev., 115(6), 1342–1373.
Mavrikakis, M., Hammer, B., and Norskov, J. K. (1998). “Effect of strain on the reactivity of metal surfaces.” Phys. Rev. Lett., 81(13), 2819–2822.
Mazin, I. I., Maksimov, E. G., Savrasov, S. Y., and Uspenskii, Yu. A. (1984). “Problem of the band gap in the density functional theory.” Sov. Phys. Solid State, 29(9), 1516–1520.
McLellan, A. G. (1974). “Virial theorem generalized.” Am. J. Phys., 42(3), 239–243.
McLellan, A. G. (1984). “An exact quantum statistical formulation of the elastic constants of a solid.” J. Phys. C, 17(1), 1–12.
Meade, R. D., and Vanderbilt, D. (1989). “Origins of stress on elemental and chemisorbed semiconductor surfaces.” Phys. Rev. Lett., 63(13), 1404–1407.
Meunier, I., Tre’glia, G., Gay, J. M., and Aufray, B. (1999). “ AgCu(111) structure revisited through an extended mechanism for stress relaxation.” Phys. Rev. B, 59(16), 10910–10917.
Mitsura, L. (1985). “The pressure tensor in nonuniform fluids.” J. Chem. Phys., 83(7), 3633–3637.
Mitsura, L. (1987). “The definition of the pressure tensor in the statistical mechanics of nonuniform classical fluids.” Int. J. Thermophys., 8(3), 397–403.
Moruzzi, V. L., Janak, J. F., and Williams, A. R. (1978). Calculated electronic properties of metals, Pergamon, New York.
Mottet, C., Tre’glia, G., and Legrand, B. (1992). “Structures of a Ag monolayer deposited on Cu(111), Cu(100), and Cu(110) substrates: An extended tight-binding quenched-molecular-dynamics study.” Phys. Rev. B, 46(24), 16018–16030.
Narasimham, S., and Vanderbilt, D. (1992). “Elastic stress domains and the herringbone reconstruction on Au(111).” Phys. Rev. Lett., 69(10), 1564–1567.
Needs, R. J., and Godfrey, M. J. (1987). “The origin and possible implications of surface stress on metals.” Phys. Scr., T, T19B, 391–398.
Needs, R. J., Godfrey, M. J., and Mansfield, M. (1991). “Theory of surface stress and surface reconstruction.” Surf. Sci., 242(1–3), 215–221.
Nielsen, O. H., and Martin, R. M. (1983). “First-principles calculation of stress.” Phys. Rev. Lett., 50(9), 697–700.
Nielsen, O. H., and Martin, R. M. (1985a). “Quantum-mechanical theory of stress and force.” Phys. Rev. B, 32(6), 3780–3791.
Nielsen, O. H., and Martin, R. M. (1985b). “Stresses in semiconductors: Ab initio calculations on Si, Ge, and GaAs.” Phys. Rev. B, 32(6), 3792–3805.
Nielsen, O. H., and Martin, R. M. (1988). “Comment on ‘Microscopic stress tensors in quantum systems.’ ” Phys. Rev. B, 37(18), 10905–10907.
Nye, J. F. (2001). Physical properties of crystals, Oxford University Press, New York.
Parr, R. G., and Yang, W. (1989). Density-functional theory of atoms and molecules, Oxford University Press, New York.
Passerone, D., Tosatti, E., Chiarotti, G. L., and Ercolessi, F. (1999). “Variable-curvature-slab molecular dynamics as a method to determine surface stress.” Phys. Rev. B, 59(11), 7687–7696.
Pauli, W. (1958). Handbuch der physik, Band XXIV, Teil 1, Vol. V, Part 1, Springer, Berlin, 83–272.
Payne, M. C., Teter, M. P., Allan, D. C., Arias, T. A., and Joannopoulos, J. D. (1992). “Iterative minimization techniques for ab initio total-energy calculations: Molecular dynamics and conjugate gradients.” Rev. Mod. Phys., 64(4), 1045–1097.
Pendas, A. M. (2002). “Stress, virial, and pressure in the theory of atoms in molecules.” J. Chem. Phys., 117(3), 965–979.
Perdew, J. P., Burke, K., and Ernzerhof, M. (1996). “Generalized gradient approximation made simple.” Phys. Rev. Lett., 77(18), 3865–3868.
Perdew, J. P., and Wang, Y. (1986). “Accurate and simple density functional for the electronic exchange energy: Generalized gradient approximation.” Phys. Rev. B, 33(12), 8800–8802.
Pettifor, D. G. (1976a). “Individual orbital contributions to the SCF virial in homonuclear diatomic molecules.” J. Chem. Phys., 69(6), 2930–2931.
Pettifor, D. G. (1976b). “Pressure-cell boundary relation and application to transition-metal equation of state.” Commun. Phys. (London), 1(5), 141–146.
Ramer, N. J., Mele, E. J., and Rappe, A. M. (1998). “Theoretical examination of stress fields in Pb(Zr0.5Ti0.5)O3 .” Ferroelectrics, 206–207(1–4), 31–46.
Rogers, C. L., and Rappe, A. M. (2002). “Geometric formulation of quantum stress fields.” Phys. Rev. B, 65(22), 224117–224124.
Ross, M. (1969). “Pressure calculations and the virial theorem for modified Hartree-Fock solids and atoms.” Phys. Rev., 179(3), 612–615.
Schrödinger, E. (1927). “Der energieimpulssatz der materiewellen.” Ann. Phys., 387(2), 265–272.
Sham, L. J., and Kohn, W. (1966). “One-particle properties of an inhomogeneous interacting electron gas.” Phys. Rev., 145(2), B561–567.
Slater, J. C. (1933). “The virial and molecular structure.” J. Chem. Phys., 1(10), 687–691.
Slater, J. C. (1963). Quantum theory of molecules and solids, Vol. 1, McGraw-Hill, New York.
Slater, J. C. (1967). Quantum theory of molecules and solids, Vol. 3, McGraw-Hill, New York.
Sommerfeld, A. (1950). Mechanics of deformable bodies, Lectures on Theoretical Physics Vol. II, Academic, New York.
Wilson, A. A. (1957). Thermodynamics and statistical mechanics, Cambridge University Press, Cambridge.
Yin, M. T., and Cohen, M. L. (1982). “Theory of static structural properties, crystal stability, and phase transformations: Application to Si and Ge.” Phys. Rev. B, 26(10), 5668–5687.
Ziesche, P., Grafenstein, J., and Nielsen, O. H. (1988). “Stress, virial, and pressure in the theory of atoms in molecules.” Phys. Rev. B, 37(14), 8167–8178.
Ziesche, P., and Lehmann, D. (1987). “Local virial-relations and pressure.” Phys. Status Solidi B, 139(2), 467–483.

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Go to Journal of Aerospace Engineering
Journal of Aerospace Engineering
Volume 20Issue 1January 2007
Pages: 22 - 37

History

Received: Aug 26, 2005
Accepted: Jan 21, 2006
Published online: Jan 1, 2007
Published in print: Jan 2007

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R. Maranganti
Dept. of Mechanical Engineering, Univ. of Houston, Houston, TX 77204.
Dept. of Mechanical Engineering and Dept. of Physics, Univ. of Houston, Houston, TX 77204 (corresponding author). E-mail: [email protected]
L. Wheeler
Dept. of Mechanical Engineering, Univ. of Houston, Houston, TX 77204.

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