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  1. E. C. Kemble, Phys. Rev. 19, p. 394; 1922. (Abstract.)
    [CrossRef]
  2. Foote and Mohler and give the value 13.7 volts. (Origin of Spectra, p. 186.)Their value has since been confirmed by Knipping (Zs. f. Phys. 7, p. 328; 1921)and by Mackay (Phys. Rev. 24, p. 319, 1924.)
    [CrossRef]
  3. E. F. Barker and O. S Duffendack, Phys. Rev. 26, p. 339; 1925.
    [CrossRef]
  4. F. L. Mohler, Phys. Rev. 25, p. 583, 1925 (Abstract).
  5. Phil. Mag. 47, p. 549; 1924.
    [CrossRef]
  6. j is the quantum number which measures the total angular momentum of the molecule and j−m is equivalent to Kratzer’s ρ. (Cf. Eq. 3a below.) Bell uses the symbols m′, m, m0′ for the quantities here designated as m, j, ρ respectively.
  7. ρ is usually assumed to be ½ for the HCl bands. The analysis given below makes 0.4925 the most probable value of ρ.The large discrepancy between Mr. Bell’s value of ρ and the others is due to the fact that he uses negative values of j for the negative branch of the band and fails to change the sign of ρ with that of j. Thus the absolute values of m which he uses in connection with the negative branch are not the same as those for the positive branch.
  8. W. F. Colby, Astrophys. J. 58, p. 303; 1923.
    [CrossRef]
  9. A. Kratzer, Annalen d. Physik 71, p. 72; 1923.
    [CrossRef]
  10. Kratzer gives larger values of δ for the upper electronic level associated with cyanogen, mercury, and zinc bands analysed by him. In the case of cadmium the reverse is true, but δ is here fairly small for both initial and final states.
  11. A. Kratzer: Zs. f. Phys. 3, p. 289; 1920.
    [CrossRef]
  12. M. Born and E. Hückel: Physikal. Zs. 24, p. 1; 1923.
  13. E. C. Kemble: Proc. Nat. Acad. Sci. 7, p. 283; 1921.
    [CrossRef]
  14. The author is informed by Professor Birge that the analysis of the vibrational energy values associated with electronic bands indicates that the coefficient δ of the cubic term in the expression for An is generally negligible, even though the terms of higher degree must be taken into account.
  15. F. W. Loomis: Astrophys. J. 52, p. 248; 1920.
    [CrossRef]
  16. Colby, Meyer, and Bronk, Astrophys. J. 57, p. 7; 1923.
    [CrossRef]
  17. Primes refer to the upper energy level and double primes to the lower one.
  18. R. S. Mulliken: Phys. Rev. 25, p. 259; 1925.
    [CrossRef]
  19. M. Born and W. Heisenberg: Zs. f. Phys. 23, p. 388; 1924.
    [CrossRef]
  20. K. Fajans and G. Joos: Zs. f. Phys. 23, p. 1; 1924.
    [CrossRef]
  21. Cf. M. Born: Die Naturwissenschaften 12, p. 1203;1924.E. V. Angerer: Zs. f. Phys. 11, p. 167; 1922.
    [CrossRef]
  22. O. Oldenberg: Zs. f. Phys. 25, p. 2; 1924.
    [CrossRef]

1925 (3)

E. F. Barker and O. S Duffendack, Phys. Rev. 26, p. 339; 1925.
[CrossRef]

F. L. Mohler, Phys. Rev. 25, p. 583, 1925 (Abstract).

R. S. Mulliken: Phys. Rev. 25, p. 259; 1925.
[CrossRef]

1924 (5)

M. Born and W. Heisenberg: Zs. f. Phys. 23, p. 388; 1924.
[CrossRef]

K. Fajans and G. Joos: Zs. f. Phys. 23, p. 1; 1924.
[CrossRef]

Cf. M. Born: Die Naturwissenschaften 12, p. 1203;1924.E. V. Angerer: Zs. f. Phys. 11, p. 167; 1922.
[CrossRef]

O. Oldenberg: Zs. f. Phys. 25, p. 2; 1924.
[CrossRef]

Phil. Mag. 47, p. 549; 1924.
[CrossRef]

1923 (4)

W. F. Colby, Astrophys. J. 58, p. 303; 1923.
[CrossRef]

A. Kratzer, Annalen d. Physik 71, p. 72; 1923.
[CrossRef]

M. Born and E. Hückel: Physikal. Zs. 24, p. 1; 1923.

Colby, Meyer, and Bronk, Astrophys. J. 57, p. 7; 1923.
[CrossRef]

1922 (1)

E. C. Kemble, Phys. Rev. 19, p. 394; 1922. (Abstract.)
[CrossRef]

1921 (1)

E. C. Kemble: Proc. Nat. Acad. Sci. 7, p. 283; 1921.
[CrossRef]

1920 (2)

F. W. Loomis: Astrophys. J. 52, p. 248; 1920.
[CrossRef]

A. Kratzer: Zs. f. Phys. 3, p. 289; 1920.
[CrossRef]

Barker, E. F.

E. F. Barker and O. S Duffendack, Phys. Rev. 26, p. 339; 1925.
[CrossRef]

Born, Cf. M.

Cf. M. Born: Die Naturwissenschaften 12, p. 1203;1924.E. V. Angerer: Zs. f. Phys. 11, p. 167; 1922.
[CrossRef]

Born, M.

M. Born and W. Heisenberg: Zs. f. Phys. 23, p. 388; 1924.
[CrossRef]

M. Born and E. Hückel: Physikal. Zs. 24, p. 1; 1923.

Bronk,

Colby, Meyer, and Bronk, Astrophys. J. 57, p. 7; 1923.
[CrossRef]

Colby,

Colby, Meyer, and Bronk, Astrophys. J. 57, p. 7; 1923.
[CrossRef]

Colby, W. F.

W. F. Colby, Astrophys. J. 58, p. 303; 1923.
[CrossRef]

Duffendack, O. S

E. F. Barker and O. S Duffendack, Phys. Rev. 26, p. 339; 1925.
[CrossRef]

Fajans, K.

K. Fajans and G. Joos: Zs. f. Phys. 23, p. 1; 1924.
[CrossRef]

Foote,

Foote and Mohler and give the value 13.7 volts. (Origin of Spectra, p. 186.)Their value has since been confirmed by Knipping (Zs. f. Phys. 7, p. 328; 1921)and by Mackay (Phys. Rev. 24, p. 319, 1924.)
[CrossRef]

Heisenberg, W.

M. Born and W. Heisenberg: Zs. f. Phys. 23, p. 388; 1924.
[CrossRef]

Hückel, E.

M. Born and E. Hückel: Physikal. Zs. 24, p. 1; 1923.

Joos, G.

K. Fajans and G. Joos: Zs. f. Phys. 23, p. 1; 1924.
[CrossRef]

Kemble, E. C.

E. C. Kemble, Phys. Rev. 19, p. 394; 1922. (Abstract.)
[CrossRef]

E. C. Kemble: Proc. Nat. Acad. Sci. 7, p. 283; 1921.
[CrossRef]

Kratzer, A.

A. Kratzer, Annalen d. Physik 71, p. 72; 1923.
[CrossRef]

A. Kratzer: Zs. f. Phys. 3, p. 289; 1920.
[CrossRef]

Loomis, F. W.

F. W. Loomis: Astrophys. J. 52, p. 248; 1920.
[CrossRef]

Meyer,

Colby, Meyer, and Bronk, Astrophys. J. 57, p. 7; 1923.
[CrossRef]

Mohler,

Foote and Mohler and give the value 13.7 volts. (Origin of Spectra, p. 186.)Their value has since been confirmed by Knipping (Zs. f. Phys. 7, p. 328; 1921)and by Mackay (Phys. Rev. 24, p. 319, 1924.)
[CrossRef]

Mohler, F. L.

F. L. Mohler, Phys. Rev. 25, p. 583, 1925 (Abstract).

Mulliken, R. S.

R. S. Mulliken: Phys. Rev. 25, p. 259; 1925.
[CrossRef]

Oldenberg, O.

O. Oldenberg: Zs. f. Phys. 25, p. 2; 1924.
[CrossRef]

Annalen d. Physik (1)

A. Kratzer, Annalen d. Physik 71, p. 72; 1923.
[CrossRef]

Astrophys. J. (3)

W. F. Colby, Astrophys. J. 58, p. 303; 1923.
[CrossRef]

F. W. Loomis: Astrophys. J. 52, p. 248; 1920.
[CrossRef]

Colby, Meyer, and Bronk, Astrophys. J. 57, p. 7; 1923.
[CrossRef]

Die Naturwissenschaften (1)

Cf. M. Born: Die Naturwissenschaften 12, p. 1203;1924.E. V. Angerer: Zs. f. Phys. 11, p. 167; 1922.
[CrossRef]

Phil. Mag. (1)

Phil. Mag. 47, p. 549; 1924.
[CrossRef]

Phys. Rev. (4)

E. C. Kemble, Phys. Rev. 19, p. 394; 1922. (Abstract.)
[CrossRef]

E. F. Barker and O. S Duffendack, Phys. Rev. 26, p. 339; 1925.
[CrossRef]

F. L. Mohler, Phys. Rev. 25, p. 583, 1925 (Abstract).

R. S. Mulliken: Phys. Rev. 25, p. 259; 1925.
[CrossRef]

Physikal. Zs. (1)

M. Born and E. Hückel: Physikal. Zs. 24, p. 1; 1923.

Proc. Nat. Acad. Sci. (1)

E. C. Kemble: Proc. Nat. Acad. Sci. 7, p. 283; 1921.
[CrossRef]

Zs. f. Phys. (4)

A. Kratzer: Zs. f. Phys. 3, p. 289; 1920.
[CrossRef]

M. Born and W. Heisenberg: Zs. f. Phys. 23, p. 388; 1924.
[CrossRef]

K. Fajans and G. Joos: Zs. f. Phys. 23, p. 1; 1924.
[CrossRef]

O. Oldenberg: Zs. f. Phys. 25, p. 2; 1924.
[CrossRef]

Other (6)

Primes refer to the upper energy level and double primes to the lower one.

The author is informed by Professor Birge that the analysis of the vibrational energy values associated with electronic bands indicates that the coefficient δ of the cubic term in the expression for An is generally negligible, even though the terms of higher degree must be taken into account.

Foote and Mohler and give the value 13.7 volts. (Origin of Spectra, p. 186.)Their value has since been confirmed by Knipping (Zs. f. Phys. 7, p. 328; 1921)and by Mackay (Phys. Rev. 24, p. 319, 1924.)
[CrossRef]

j is the quantum number which measures the total angular momentum of the molecule and j−m is equivalent to Kratzer’s ρ. (Cf. Eq. 3a below.) Bell uses the symbols m′, m, m0′ for the quantities here designated as m, j, ρ respectively.

ρ is usually assumed to be ½ for the HCl bands. The analysis given below makes 0.4925 the most probable value of ρ.The large discrepancy between Mr. Bell’s value of ρ and the others is due to the fact that he uses negative values of j for the negative branch of the band and fails to change the sign of ρ with that of j. Thus the absolute values of m which he uses in connection with the negative branch are not the same as those for the positive branch.

Kratzer gives larger values of δ for the upper electronic level associated with cyanogen, mercury, and zinc bands analysed by him. In the case of cadmium the reverse is true, but δ is here fairly small for both initial and final states.

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Figures (1)

F. 1
F. 1

Potential Energy of HCl Molecule as a Function of Nuclear Separation

Equations (43)

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J HCl = Lim r = [ U ( r ) U ( r 0 ) ] .
m = j 0.25 ,
F n ( j ) = A n + B n ( j 1 2 ) 2 + D n ( j 1 2 ) 4 ,
F n ( j ) = A n + 2 δ ( j ) + B n ( j ) 2 + D n ( j ) 4 .
F n ( j ) = A n + B n ( j ρ ) 2 + D n ( j ) 4 +
ρ = δ B n
A n = A n + 2 δ δ 2 B n .
u = k ξ 2 ( 1 + a ξ + b ξ 2 + c ξ 3 + d ξ 4 + e ξ 5 + ) .
ω 0 = 1 2 π q 2 k μ r 0 2 = 1 2 π q 2 k J 0 .
ω 1 = h 4 π 2 J 0 q ; u = ω 1 ω 0 = h π 8 k J 0
A n = β n + γ n 2 + δ n 3 B n = a + β n + γ n 2 + δ n 3 D n = a + β n + γ n 2 +
β = ω 0 ; γ = 3 4 ω 0 u [ b 5 4 a 2 ] δ = ω 0 u 2 8 [ 10 d 35 a c 17 2 b 2 + 225 4 a 2 b 705 32 a 4 ] }
a = ω 1 2 ; β = 3 2 ω 1 u ( 1 + a ) γ = 3 4 ω 1 u 2 [ 5 + 10 a 3 b + 5 c 13 a b + 15 2 ( a 2 + a 3 ) ] δ = 5 4 ω 1 u 3 [ 7 + 21 a 17 2 b + 14 c 9 2 d + 7 e + 225 8 a 2 45 a b + 105 4 a c 51 2 a d + 51 8 b 2 45 2 b c + 141 4 a 3 945 16 a 2 b + 435 8 a 2 c + 411 8 a b 2 1509 16 a 3 b + 3807 128 ( a 4 + a 5 ) }
a = ω 1 u 2 2 ; β = 3 4 ω 1 u 3 [ 19 2 + 9 a 4 b + 9 a 2 / 2 ] γ = 3 4 ω 1 u 4 [ 65 125 a + 61 b 30 c + 15 d 495 4 a 2 + 117 a b 26 b 2 95 2 a c + 207 2 a 2 b 90 ( a 3 + a 4 2 ) ] }
β = 2938.73 ; γ = 52.58 ; α = 10.4499 ; β = 0.30795 ; γ = 0.00695
ω 1 = 2 α = 20.8998 u = ω 1 β = 0.007112 a = 2 β 3 ω 1 u 1 = 2.38 ; b = 5 4 a 2 + 4 γ 3 ω 1 = 3.722 ; α = 0.000529 ; β = + 0.0000075 ; D 0 = 0.000529 ; D 1 = 0.000522 ; D 2 = 0.000514 .
P n n ( M ) = F n ( M ) F n ( M + 1 ) ;
( M ) = F n ( M + 1 ) F n ( M ) .
H ( M ) = 1 2 [ P 0 1 ( M ) + R 0 1 ( M ) ] + ( D 0 D 1 ) M 4 .
H ( M ) = A 1 A 0 + B 1 B 0 4 + D 1 D 0 16 + [ B 1 B 0 + 2 3 ( D 1 D 0 ) ] M 2
A 1 = 2887.19 ± 0.053 , B 0 B 1 = 0.3042 ± 0.00034.
R 0 1 ( M ) P 0 1 ( M + 1 ) = F 0 ( M + 2 ) F 0 ( M ) ;
R 0 1 ( M + 1 ) P 0 1 ( M ) = F 1 ( M + 2 ) F 1 ( M ) .
w 0 ( M ) = R 0 1 ( M ) P 0 1 ( M + 1 ) 8 D 0 ( M + 1 2 ) 3 .
w 0 ( M ) = u + υ M ,
υ = 4 B 0 8 D 0 ; u = υ ( 1 ρ ) .
w 0 ( M ) = R 0 1 ( M + 1 ) P 0 1 ( M ) + [ 4 ( B 0 B 1 ) + 8 ( D 0 D 1 ) ] ( M + 1 2 ) 8 D 1 ( M + 1 2 ) 3 .
B 0 = 1 2 ω 1 = 10.4469 ± 0.0050 ρ = 0.4925 ± 0.0012.
B 1 = 10.4469 0.3042 = 10.1427 , δ = B 0 ( 1 2 ρ ) = 0.078 ± 0.012.
w 2 ( M ) = P 0 2 ( M ) + 2 δ + B 0 ( M + 1 2 ) 2 + D 0 ( M + 1 2 ) 4 D 2 ( M 1 2 ) 4 ;
= R 0 2 ( M ) 2 δ + B 0 ( M 3 2 ) 2 + D 0 ( M 3 2 ) 4 D 2 ( M 1 2 ) 4 ;
= P 1 2 ( M ) + 2 δ + A 1 + B 1 ( M + 1 2 ) 2 + D 1 ( M + 1 2 ) 4 D 2 ( M 1 2 ) 4
R 0 1 ( M ) P 0 1 ( M + 1 ) = R 0 2 ( M ) P 0 2 ( M + 1 )
A 2 = 5667.23 ± 0.17 B 2 = 9.8624 ± 0.0023
β = ω 0 = 2940.77 u = 0.007105 γ = 53.58 β = 0.3042 γ = 0.012
a = 2.421 b = 3.905 c = 2.86
2 α e 2 r 5
σ = r 0 r .
U 1 ( σ ) = e 2 r 0 [ σ + α 2 r 0 3 σ 4 ] + U ( o )
J 0 = μ r 0 2 = 2.65 × 10 40 gm cm 2 r 0 = 1.28 × 10 8 cm .
U 2 ( σ ) = k ( 1 σ σ ) 2 [ 1 + a ( 1 σ σ ) + b ( 1 σ σ ) 2 + c ( 1 σ σ ) 3 ] .
U = U ( o ) e 2 r 0 [ σ + .75 σ 4 + 8.126 σ 5 28.877 σ 6 + 35.293 σ 7 19.41 σ 8 + 4.095 σ 9 ]
U ( o ) = e 2 r 0 [ 1 + 0.75 + 8.126 28.877 + 35.293 19.41 + 4.095 ] = 0.9782 × e 2 r 0