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The Vibration-rotation Energies of Molecules and their Spectra in the Infra-red

  • H. H. Nielsen
Part of the Encyclopedia of Physics / Handbuch der Physik book series (HDBPHYS, volume 7 / 37 / 1)

Abstract

The spectra of substances may, in a general way, be assigned to one of three categories, namely; the continuous spectra, the bright line spectra and the band spectra. The first of these occur only in emission and are produced by bodies heated to incandescence and are incapable of resolution into lines regardless of the resolving power of the available instruments. The second type are the spectra of atoms. They may be produced as bright line emission spectra by suitable excitation of the atoms, as for example by placing them in the crater of an arc or by passing an electric discharge through their vapors. When radiation from an incandescent source is allowed to pass through an atomic vapor and is examined with a spectroscope the spectra occur in absorption as dark line spectra against a continuous bright background.

Keywords

Polyatomic Molecule Fermi Resonance Angular Momentum Vector Contact Transformation Electric Moment 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Symbols and notation used

X, Y, Z

space-fixed coordinate system

X0, Y0, Z0

coordinates of center of mass in space-fixed coordinate system

Xi, Yi, Zi

coordinates of nuclei relative to center of mass in space-fixed coordinate system

x, y, z

body-fixed coordinate system

ϑ, φ, ψ

Eulerian angles relating x, y, z and X, Y, Z axes

α (β, γ)

symbol to denote x, y, or z

α’i, α’j

coordinates of nuclei and electrons, respectively, relative to center of mass in body-fixed system

αi, αj

\(M_i^{\frac{1} {2}} \alpha '_i = \alpha _i ,m^{\frac{1} {2}} \alpha '_j = \alpha _j ,\), M i , and m being, respectively, the masses of the i-th nucleus and of an electron

α0i

equilibrium value of α i .

Iαα

moment of inertia about α-axis

Iαβ

product of inertia

I’αα

effective moment of inertia about α axis. See Eq. (12.19)

I’αβ

effective product of inertia. See Eq. (12.19)

ωα

angular velocity about α-axis

Q, q

normal coordinates. \( {Q_{s\sigma }} = {\left( {{{{\hbar ^2}} \mathord{\left/ {\vphantom {{{\hbar ^2}} {{\lambda _s}}}} \right. \kern-\nulldelimiterspace} {{\lambda _s}}}} \right)^{\frac{1}{4}}}{q_{s\sigma }};{\lambda _s} = {\left( {2\pi c{\omega _s}} \right)^2} \)

ωs

normal vibration frequency in cm-1

lisσ

transformation coefficients relating δα i to normal coordinatesQ s σ

p

linear momentum conjugate to q s σ

rt, χt, ϑt

polar coordinates used to describe degenerate vibrations. (Note difference between ϑand ϑt.)

Prt, Pχt, Pϑt

momenta conjugate to r t , x t , and ϑt

pΞ, PH, Pz

linear momenta conjugate to coordinates of center of mass \( \Xi = {\left( {\sum\limits_i {{M_i} + Nm} } \right)^{\frac{1}{2}}}{X_0} \), etc

pα

component of internal angular momentum of nuclei directed along α-axis

πα

component of angular momentum of electrons directed along α-axis

Πα

Π α = Π α + p α

sX, sY, sZ

components of spin angular momentum directed along space-fixed X, Y, Z axes

sx, sy, sz

components of spin angular momentum directed along body-fixed x, y, z axes

\( {S_\alpha } = \sum\limits_j {{S_{j\alpha }}} \)

a component of total spin angular momentum directed along the α-axis

Mα

Mα = Π α + S α total internal angular momentum directed along the α-axis

Pα

component of total angular momentum directed along the α-axis

smn

distance between two atomic nuclei m and n

smn0

equilibrium value of s mn (s mn - s 0 mn = δs mn )

Asσsσ(αβ) ,a(αβ)

etc., see definitions (18.24)

ζsσs’σ’(α)

Coriolis coupling factor. See definition (18.24)

vs

total vibration quantum number for a harmonic oscillator

ls

quantum number of total vibrational angular momentum associated with a two- or three-fold degenerate vibration

ms

component of vibrational angular momentum associated with a three-fold degenerate oscillator directed along an axis fixed in the molecule

J

quantum number of angular momentum of a molecule exclusive of spin

F

quantum number of angular momentum of a molecule inclusive of spin

K

quantum number associated with the component of J directed along the z-axis

M

the magnetic quantum number of rotation of a molecule

Λ

quantum number of electronic angular momentum directed along z axis of a linear molecule

Σ

quantum number of spin angular momentum directed along z axis of a molecule

L

angular momentum of the molecular framework directed along the z axis (not quantized)

gs

weight factor assuming the values 1, 2, or 3, respectively, as ω s is one-, two-, or threefold degenerate

xss’, xlsls,

etc. anharmonic constants (corresponding to ω e x e in diatomic molecules)

DJ, DJK, DJL

etc. centrifugal stretching coefficients. See definitions (34.15) and (34.16)

Be(αα)

reciprocal of inertia (h/8 π 2 I αα (e) c)

Bv(αβ)

effective reciprocal of inertia or reciprocal product of inertia

λαB

direction cosines relating body-fixed axes to space-fixed axes

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Bibliography

Chapter I

  1. Bjerrum, Niels: Nernst-Festschrift, Halle 1912.Google Scholar
  2. Deslandres, H.: C. R. Acad. Sci. Paris 103, 375 (1886).Google Scholar
  3. Drude, P. K. L.: Ann. Physik 4, 677 (1904).ADSGoogle Scholar
  4. Heurlinger, T.: Ark. Mat. Astron. Fysik 12, 1 (1917).Google Scholar
  5. Heurlinger, T.: Phys. Z. 20, 188 (1919).Google Scholar
  6. Heurlinger, T.: Z. Physik 1, 82 (1920).ADSGoogle Scholar
  7. Hund, F.: Z. Physik 36, 657 (1926).ADSGoogle Scholar
  8. Kemble, E. C.: J. Amer. Opt. Soc. 12, 1 (1926).ADSGoogle Scholar
  9. Kramers, H. A., and W. Pauli: Z. Physik 13, 351 (1923).ADSGoogle Scholar
  10. Kratzer, A.: Z. Physik 3, 289 (1920).ADSGoogle Scholar
  11. Lord Rayleigh: Phil. Mag. 34, 410 (1892).Google Scholar

Chapter II

  1. Casimir, H. B. G.: Rotation of a Rigid Body in Quantum Mechanics. Groningen, Den Haag, Batavia: J.B.Walters Uitgevers-Maatschappij 1931.Google Scholar
  2. Condon, E. U., and G. H. Shortley: The Theory of Atomic Spectra, Cambridge: University Press 1935.Google Scholar
  3. Kemble, E. C.: Fundamental Principles of Quantum Mechanics. New York, N.Y.: McGraw-Hill Book Co. Inc. 1937.Google Scholar
  4. Vleck, J. H. Van: Rev. Mod. Phys. 23, 213 (1951).ADSzbMATHGoogle Scholar

Chapter III

  1. Born, M., and R. Oppenheimer: Z. Physik 84, 457 (1927).zbMATHGoogle Scholar
  2. Born, M., and P. Jordan: Elementare Quantenmechanik. Berlin: Springer 1930.zbMATHGoogle Scholar
  3. Burrau, O.: Kgl. danske Vid. Selsk. Mat.-fys. Medd. 7, 14 (1927).Google Scholar
  4. Darling, B. T., and D. M. Dennison: Phys. Rev. 57, 128 (1940).ADSGoogle Scholar
  5. Eyring, H., J. Walter and G. E. Kimball: Quantum Chemistry. New York: John Wiley Sons, Inc. 1944.Google Scholar
  6. Heitler, W., and F. London: Z. Physik 44, 455 (1927).ADSGoogle Scholar
  7. Hund, F.: Z. Physik 36, 657 (1926);ADSGoogle Scholar
  8. Hund, F.: Z. Physik 42, 93 (1927).ADSGoogle Scholar
  9. Kronig, R. de L.: Band Spectra and Molecular Structure. London: Cambridge University Press 1930.Google Scholar
  10. Meister, A. G., and F. F. Cleveland: Amer. J. Phys. 14, 13 (1946).ADSGoogle Scholar
  11. Nielsen, H. H.: Rev. Mod. Phys. 23, 90 (1951).ADSzbMATHGoogle Scholar
  12. Pauli jr., W.: Z. Physik 43, 601 (1927).ADSGoogle Scholar
  13. Podolsky, B.: Phys. Rev. 32, 812 (1928).ADSGoogle Scholar
  14. Vleck, J. H. Van: Phys. Rev. 33, 467 (1929).ADSGoogle Scholar
  15. Wang, S.: Phys. Rev. 31, 579 (1928).ADSGoogle Scholar
  16. Whittaker, E. T.: A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 3rd edition, p. 77. London: Cambridge University Press 1927.zbMATHGoogle Scholar
  17. Wilson jr., E. B., and J. B. Howard: J. Chem. Phys. 4, 262 (1936).Google Scholar

Chapter IV

  1. Born, M., and P. Jordan: Elementare Quantenmechanik. Berlin: Springer 1930.zbMATHGoogle Scholar
  2. Bourgin, D. G.: Phys. Rev. 29, 794 (1927).ADSGoogle Scholar
  3. Casimir, H. B. G.: Rotation of a Rigid Body in Quantum Mechanics. Groningen, Den Haag, Batavia: J. B. Walters Uitgevers-Maatschappij 1931.Google Scholar
  4. Childs, W. H. J., and H. A. Jahn: Proc. Roy. Soc. Lond., Ser. A 169, 451 (1939).ADSzbMATHGoogle Scholar
  5. Condon, E. U., and G. H. Shortley: The Theory of Atomic Spectra. London: Cambridge University Press 1935.Google Scholar
  6. Cross, P. C., R. M. Hainer and G. W. King: J. Chem. Phys. 12, 210 (1944).ADSGoogle Scholar
  7. Dennison, D. M.: Phys. Rev. 28, 318 (1926).ADSGoogle Scholar
  8. Dennison, D. M.: Phys. Rev. 31, 503 (1928).ADSGoogle Scholar
  9. Dennison, D. M.: Rev. Mod. Phys. 3, 280 (1931).ADSzbMATHGoogle Scholar
  10. Dennison, D. M.: Phys. Rev. 44, 304 (1932).ADSGoogle Scholar
  11. Dunham, J. L.: Phys. Rev. 34, 438 (1929).ADSGoogle Scholar
  12. Erlandson, G.: Ark. Fysik 10, 65 (1955).Google Scholar
  13. Golden jr., S.: J. Chem. Phys. 16, 78 (1948).MathSciNetADSGoogle Scholar
  14. Hainer, R. M., P. C. Cross and G. W. King: J. Chem. Phys. 17, 826 (1949).ADSGoogle Scholar
  15. Johnston, M., and D. M. Dennison: Phys. Rev. 48, 868 (1935).ADSzbMATHGoogle Scholar
  16. Kemble, E. C, and D. G. Bourgin: Nature, Lond. 117, 789 (1926).ADSGoogle Scholar
  17. Klein, O.: Z. Physik 58, 730 (1929).ADSGoogle Scholar
  18. King, G. W., R. M. Hainer and P. C. Cross: J. Chem. Phys. 11, 27 (1943).ADSGoogle Scholar
  19. Kramers, H. A., and G. P. Ittmann: Z. Physik 53, 553 (1929);ADSGoogle Scholar
  20. Kramers, H. A., and G. P. Ittmann: Z. Physik 58, 217 (1929);ADSGoogle Scholar
  21. Kramers, H. A., and G. P. Ittmann: Z. Physik 60, 663 (1930).ADSGoogle Scholar
  22. Kronig, R. de L., and J. J. Rabi: Phys. Rev. 29, 262 (1927).ADSGoogle Scholar
  23. Ray, B. S.: Z. Physik 78, 74 (1932).ADSzbMATHGoogle Scholar
  24. Reiche, F., and H. Rademacher: Z. Physik 39, 444 (1926).ADSGoogle Scholar
  25. Schrödinger, E.: Ann. Physik 79, 361 (1926).zbMATHGoogle Scholar
  26. Shaffer, W. H., H. H. Nielsen and L. H. Thomas: Phys. Rev. 56, 895 (1939).ADSzbMATHGoogle Scholar
  27. Shaffer, W. H., H. H. Nielsen and L. H. Thomas: Phys. Rev. 56, 1051 (1939).ADSzbMATHGoogle Scholar
  28. Shaffer, W. H.: Rev. Mod. Phys. 16, 245 (1944).MathSciNetADSGoogle Scholar
  29. Teller, E.: Hand- und Jahrbuch der Chemischen Physik, vol. 9, p. 125. 1934.Google Scholar
  30. Tolman, R. C.: Statistical Mechanics with Applications to Physics and Chemistry. New York 1927.Google Scholar
  31. Vleck, J. H. Van: Phys. Rev. 33, 467 (1929).ADSGoogle Scholar
  32. Vleck, J. H. Van: Rev. Mod. Phys. 23, 213 (1951).ADSzbMATHGoogle Scholar
  33. Wang, S. C.: Phys. Rev. 34, 243 (1929).ADSGoogle Scholar
  34. Winter, C. van: Physica, Haag 20, 274 (1954).ADSzbMATHGoogle Scholar
  35. Witmer, E. E.: Proc. Nat. Acad. Sci. U.S.A. 13, 60 (1927).ADSzbMATHGoogle Scholar

Chapter V

  1. Hansen, G. E., and D. M. Dennison: J. Chem. Phys. 20, 313 (1952).ADSGoogle Scholar
  2. Herman, R. C., and W. H. Shaffer: J. Chem. Phys. 16, 453 (1948).ADSGoogle Scholar
  3. Keller, Fred: Dissertation for Ph. D. degree, University of Tennessee, Knoxville, Tenn., 1955.Google Scholar
  4. Kemble, E. C.: Fundamental Principles of Quantum Mechanics. New York, N. Y.: McGraw-Hill Book Co., Inc. 1937.Google Scholar
  5. Kronig, R. de L: Band Spectra and Molecular Structure. London: Cambridge University Press 1930.Google Scholar
  6. Nielsen, H. H.: Phys. Rev. 60, 794 (1941).ADSzbMATHGoogle Scholar
  7. Nielsen, H. H.: Rev. Mod. Phys. 23, 90 (1951).ADSzbMATHGoogle Scholar
  8. Shaffer, W. H., H. H. Nielsen and L. H. Thomas: Phys. Rev. 56, 895 (1939).ADSzbMATHGoogle Scholar
  9. Vleck, J. H. Van: Phys. Rev. 33, 467 (1929).ADSGoogle Scholar

Chapter VI

  1. Dennison, D. M.: Astrophys. J. 62, 84 (1925).ADSGoogle Scholar
  2. Dennison, D. M.: Rev. Mod. Phys. 3, 280 (1931).ADSzbMATHGoogle Scholar
  3. Kaylor, H. M. and A. H. Nielsen J. Chem. Phys. 23, 2139 (1955).ADSGoogle Scholar
  4. Kivelson, D., and E. B. Wilson jr.: J. Chem. Phys. 20, 1575 (1952).ADSGoogle Scholar
  5. Lord, R. C., and R. E. Merrifield: J. Chem. Phys. 20, 1348 (1952).ADSGoogle Scholar
  6. Nielsen, A. H., and H. H. Nielsen: Phys. Rev. 48, 864 (1935).ADSGoogle Scholar
  7. Nielsen, A. H., H. H. Nielsen: Phys. Rev. 59, 565 (1941).ADSGoogle Scholar
  8. Sayvetz, A.: J. Chem. Phys. 7, 282 (1939).Google Scholar
  9. Straley, J. W., C. H. Tindal and H. H. Nielsen: Phys. Rev. 62, 161 (1942).ADSGoogle Scholar
  10. Teller, E.: Hand- und Jahrbuch der Chemischen Physik, vol. 9, p. 125. 1934.Google Scholar
  11. Tindal, C. H., J. W. Straley and H. H. Nielsen: Phys. Rev. 62, 151 (1942).ADSGoogle Scholar
  12. Winter, C. van: Physica, Haag 20, 274 (1954).ADSzbMATHGoogle Scholar

Chapter VII

  1. Amat, G., and M. Goldsmith: J. Chem. Phys. 23, 6, 1171–1172 (1955).ADSGoogle Scholar
  2. Amat, G., and M. Goldsmith and H.H. Nielsen: J. Chem. Phys. 27, 838 (1957).ADSGoogle Scholar
  3. Amat, G., and H. H. Nielsen: J. Chem. Phys. 27, 845 (1957);ADSGoogle Scholar
  4. Amat, G., and H. H. Nielsen: J. Chem. Phys. 29, 665 (1958).ADSGoogle Scholar
  5. Cole, Good and Hughes: Phys. Rev. 79, 224A (1950).Google Scholar
  6. Courtoy, C. P.: Canad. J. Phys. 35, 608 (1957).ADSGoogle Scholar
  7. Darling, B. T., and D. M. Dennison: Phys. Rev. 57, 128 (1950),ADSGoogle Scholar
  8. Dennison, D. M.: Phys. Rev. 41, 304 (1932).ADSzbMATHGoogle Scholar
  9. Ebers, E. S., and H. H. Nielsen: J. Chem. Phys. 5, 822 (1937).ADSGoogle Scholar
  10. Fermi, E.: Z. Physik 71, 251 (1931).ADSGoogle Scholar
  11. Fung, L. W., and E. F. Barker: Phys. Rev. 45, 238 (1934).ADSGoogle Scholar
  12. Garing, J., K. Narahari Rao and H. H. Nielsen: J. Mol. Spect. 1958 (in Press).Google Scholar
  13. Goldsmith, M., G. Amat and H. H. Nielsen: J. Chem. Phys. 24, 1178 (1956).ADSGoogle Scholar
  14. Hansen, G. E., and D.M. Dennison: J. Chem. Phys. 20, 313 (1952).ADSGoogle Scholar
  15. Hanson, H., and H. H. Nielsen: J. Chem. Phys. 25, 591 (1956).ADSGoogle Scholar
  16. Heer, J. de, and H. H. Nielsen: J. Chem. Phys. 20, 101 (1952).ADSGoogle Scholar
  17. Herman, R. C., and R. J. Rubin: Astrophys. J. 121, 533 (1955).ADSGoogle Scholar
  18. Herman, R. C., and R. F. Wallis: J. Chem. Phys. 23, 631 (1955).Google Scholar
  19. Herzberg, G.: Rev. Mod. Phys. 14, 219 (1942).ADSGoogle Scholar
  20. Jahn, H.: Phys. Rev. 56, 680 (1939).ADSzbMATHGoogle Scholar
  21. Keller, Fred: Dissertation for Ph. D. degree, University of Tennessee, Knoxville, Tenn. 1955.Google Scholar
  22. Kemble, J. C.: Phys. Rev. 25, 1 (1925).ADSGoogle Scholar
  23. Kessler, Ring, Trambarulo and Gordy: Phys. Rev. 79, 54 (1950).ADSGoogle Scholar
  24. Kronig, R. de L.: Z. Physik 50, 347 (1928).ADSGoogle Scholar
  25. Maes, S.: Thèse de doctorat (to be published).Google Scholar
  26. McConaghie, V. M., and H. H. Nielsen: J. Chem. Phys. 21, 1836 (1953).ADSGoogle Scholar
  27. Miller, C. H., and H. W. Thompson: Proc. Roy. Soc. Lond., Ser. A 200, 1 (1949).ADSGoogle Scholar
  28. Nielsen, H. H.: J. Chem. Phys. 5, 8l8 (1937).Google Scholar
  29. Nielsen, H. H.: Phys. Rev. 68, 181 (1945).ADSGoogle Scholar
  30. Nielsen, H. H.: Phys. Rev. 75, 1961 (1949).Google Scholar
  31. Nielsen, H. H.: Phys. Rev. 77, 130 (1950).ADSGoogle Scholar
  32. Nielsen, H. H.: J. Chem. Phys. 21, 142 (1953)ADSGoogle Scholar
  33. Nielsen, H. H.: J. Chem. Phys. 22, 1383 (1954).ADSGoogle Scholar
  34. Nielsen, H. H. and W. H. Shaffer: J. Chem. Phys. 11, 140 (1943).ADSGoogle Scholar
  35. Oppenheimer, J. R.: Proc. Cambridge Phil. Soc. 23, 327 (1926).ADSzbMATHGoogle Scholar
  36. Rao, K. Narahari, and A. H. Nielsen: In press.Google Scholar
  37. Renner, R.: Z. Physik 95, 172 (1934).ADSGoogle Scholar
  38. Silver, S.: J. Chem. Phys. 9, 565 (1941).Google Scholar

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© Springer-Verlag OHG. Berlin · Göttingen · Heidelberg 1959

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  • H. H. Nielsen

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