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The Nature of the Chemical Bond of Hydrogen-Type Molecules and Molecular Ions

411

corresponding to simple harmonic oscillation of the electron and nuclei, Eosc , is given by the sum of the corresponding energies,

ED and EKvib . Using Eq. (11.187) and Evib from Eq. (11.163) gives 1 k (11.188) Eosc ED EKvib ED p 2 2 1 (11.189) Eosc p 3 0.118755 eV p 2 0.29282 eV 2 To the extent that the MO dimensions are the same, the electron reentrant orbital energies EK are the same independent of the isotope of hydrogen, but the vibrational energies are related by Eq. (11.148). Thus, the differences in bond energies are essentially given by 1/2 the differences in vibrational energies. Using Eq. (11.187) with the deuterium reduced electron mass for ET and ED , and Evib for D2 1/ p given by Eq. (11.173), that corresponds to the deuterium reduced nuclear mass (Eq.

(11.170)), the corresponding Eosc is 1 Eosc p 3 0.118811 eV p 2 0.20714 eV 2

(11.190)

TOTAL, IONIZATION, AND BOND ENERGIES OF HYDROGEN AND DEUTERIUM MOLECULAR IONS

The total energy of the hydrogen molecular ion which is equivalent to the negative of the ionization energy is given by the sum of ET (Eqs. (11.121) and (11.125)) and Eosc given by Eqs. (11.185-11.188). Thus, the total energy of the hydrogen molecular ion having a central field of pe at each focus of the prolate spheroid molecular orbital including the Doppler term is (11.191) ET Ve V p T Eosc 2e 2 3 4 0 2aH ET p 2 2 2 me e 1 p (4 ln 3 1 2 ln 3) 8 a me c 2 0 H

1 k 2

(11.192)

1 2 k p 2 From Eqs. (11.189) and (11.191-11.192), the total energy for hydrogen-type molecular ions is ET p 216.28033 eV Eosc p 216.2803 eV p 3 0.118755 eV

p 216.28033 eV p 3 0.118755 eV

1 2 p 0.29282 eV 2

(11.193)

p 216.13392 eV p 3 0.118755 eV The total energy of the deuterium molecular ion is given by the sum of ET (Eq. (11.125)) corrected for the reduced electron mass of D and Eosc given by Eq. (11.190): ET p 216.284 eV Eosc p 216.284 eV p 3 0.118811 eV

1 2 p 0.20714 eV 2

(11.194)

p 216.180 eV p 3 0.118811 eV The bond dissociation energy, ED , is the difference between the total energy of the corresponding hydrogen atom or H 1/ p atom [19-20], called hydrino atom having a principal quantum number 1/ p where p is an integer, and ET .

ED E ( H 1/ p ) ET where [19] E ( H 1/ p ) p 213.59844 eV and [20] E ( D 1/ p ) p 213.603 eV

(11.195) (11.196) (11.197)

The hydrogen molecular ion bond energy, ED , is given by Eq. (11.193) with the reduced electron mass and Eqs. (11.19511.196):