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Use of isotope frequency shift due to small mass change in determining intramolecular force field
JOURNAL OF MOLECULAR SPECTROSCOPY 19, 4-6
(1966)
Use of Isotope Frequency Shift Due to Small Mass in Determining intramolecular Force Field ~,IASAMICHI
of Pharmaceutical
Faculty
A
formula
Jacobian change
has
matrix of a force
beeu
Sciences,
obtained
TSUBOI
University
by
which
showing
how
ronst,aut
in a molecule.
Change
an isotope
of Tokyo, Bunkyo-ku,
one
can
frequency
calculate shift
Tokyo, Japan
an element is sensitive
of to
a
As is now well known, the force constants in a molecule can be calculated from a set of normal frequencies of the molecule obtained from observations of infrared absorptiolls and/or Raman effect. If the normal frequencies of the isotopes of the molecule in question are ~IIOWI~, and if they are used in the calculation, the reliability of the calculated force constants may be greatly increased. When the number of t’he data (normal frequencies) is greater than the number of the force constants to be determined, a device such as the least square method is often adopted. In this procedure a Jacobian matrix whose elements are dkk/8Kh is required (I), where Xk = 4a2c2vk2 is the frequency parameter of the kth normal vibration (c: light velocity, vk : normal frequency in cm-‘) and Kh is the hth force constant. If the normal frequencies vk’ are known of an isotopic molecule, elements h'/dKhcould be added to the Jacobian matrix. Except in the case of hydrogen, substitution of isotopic atoms ( 15N or 180, for example) is accompanied by a fairly small percentSage change quency shift Av~ or
in the normal
AX&k
frequency.
= (Xk’ -
X,)/X,
In such a case, isotopic
fre-
(1)
is considered to be more significant than the frequency itself as an independent new datum (2,s). To apply the method to such a case, therefore+? ( AXk/hk)/dKh is required rabher than d&'/aK,, .It is the purpose of this note to show how to calculate this a (A&/k,) /dKhvalue. According to a perturbation treatment for the isotope shift’ due to small mass change (4)(14N --$ 15pIJ, for example) the value of AX&k for the klh normal vibrae tion may be taken to be equal to the diagonal element Akk of the matrix A, wherA = L-‘AGE-’
(2)
ISOTOPE FRE:QIlXCL*
5
SIIIFT
in which L is the L-matrix for the ordinury ( “Y- ) n~olc~de t’hat, c~o~lnccd * :I wt ot i~~t~crnalwortlinates Ri and the set of mrn~al cwortli~l:~tw(+jk, ant1
AG = ‘“G _ 14G,
iAl
14G mcl “G being the inverse kinetic energy nlatriws (‘4.1 for the 14K :mtl ‘“S species. Therefore, in order to obtain dAkkJdkTh , \vc Iieecl t,o exmiine how L-’ changes clue to t,he force constant change 6k’h, or due to the change (6F) in I he polent#i:rl cwcrgy matrix F. Let 11s:tssunle that, Lee1bwonw ( E + 6P)Lp’ :~wortling ns F bc~~omcs F + SF. Here, E is :I unit mltris. 6P w:w wldnt~cd by SC kag:Lwa and Shimnnouchi (5 ) :I* 61’,, = (e6FL)/,J(Xp
-
XI)
( .-JI
nntl 6Pi.i. = 0.
( (i I
Since F ant1 K,, c’:m he relat,etl by the A,, matrices :w F = c
ii
A/&/,,
6F can be expressed as 6F = AJh,,
T
iii
iS)
.
Hence, ( 5 I ~II be rewritten :ts
6Pl,l = [(EALL)&X~
-
hl)]S&.
(9)
IA us now :wswne that A becomes A + 61 according as K/tbecomes k’,, + 6k’,, . A + 6A = (E + 6P)L = (E + SP)A(E
‘AGE--‘(m) + %,
(10)
= A + &PA + A%; .*. 6A = iiPA + A%;
(11)
(121
6
TSUBOI
Therefore,
as the required
formula, (13) ACKNOWLEDGEMENTS
The writer wishes to express his sincere thanks to Professor Takehiko his suggestions. This work was supported by a grant from the Ministry Japan and a grant from the U. S. Public Health Service GM 10024-2. RECEIVED:
Hhimanouchi for of Education of
July 22, 1965 REFERENCES
T. SHIM~NOUCHI AND I. SUZUKI, J. Chem. Phys. 42, 29G (1965). M. TSUBOI, Spectrochim. Acta 16, 505 (1960); M. TSUBOI, T. T~KENISHI, AND A. NAK.4MUR.4,Spectrochim. Acta 17, 634 (1961); M. TSUBOI, T. TbKENISHI, BND A. NAK~~MURA,Spectrochim. Acta 19, 271 (1963). T. MIYAZAWA, J. Mol. Spectry. 13,321 (1964). E. B. WILSON, J. C. DECITJS,AND P. C. CROSS, “Molecular Vibrations,” p. 188. McGrawHill, New York, 1955. I. NAKAGAWA AND T. SHIMANOUCHI, Nippon
Kagaku
Zasshi
80, 128 (1959).
(1966)
Use of Isotope Frequency Shift Due to Small Mass in Determining intramolecular Force Field ~,IASAMICHI
of Pharmaceutical
Faculty
A
formula
Jacobian change
has
matrix of a force
beeu
Sciences,
obtained
TSUBOI
University
by
which
showing
how
ronst,aut
in a molecule.
Change
an isotope
of Tokyo, Bunkyo-ku,
one
can
frequency
calculate shift
Tokyo, Japan
an element is sensitive
of to
a
As is now well known, the force constants in a molecule can be calculated from a set of normal frequencies of the molecule obtained from observations of infrared absorptiolls and/or Raman effect. If the normal frequencies of the isotopes of the molecule in question are ~IIOWI~, and if they are used in the calculation, the reliability of the calculated force constants may be greatly increased. When the number of t’he data (normal frequencies) is greater than the number of the force constants to be determined, a device such as the least square method is often adopted. In this procedure a Jacobian matrix whose elements are dkk/8Kh is required (I), where Xk = 4a2c2vk2 is the frequency parameter of the kth normal vibration (c: light velocity, vk : normal frequency in cm-‘) and Kh is the hth force constant. If the normal frequencies vk’ are known of an isotopic molecule, elements h'/dKhcould be added to the Jacobian matrix. Except in the case of hydrogen, substitution of isotopic atoms ( 15N or 180, for example) is accompanied by a fairly small percentSage change quency shift Av~ or
in the normal
AX&k
frequency.
= (Xk’ -
X,)/X,
In such a case, isotopic
fre-
(1)
is considered to be more significant than the frequency itself as an independent new datum (2,s). To apply the method to such a case, therefore+? ( AXk/hk)/dKh is required rabher than d&'/aK,, .It is the purpose of this note to show how to calculate this a (A&/k,) /dKhvalue. According to a perturbation treatment for the isotope shift’ due to small mass change (4)(14N --$ 15pIJ, for example) the value of AX&k for the klh normal vibrae tion may be taken to be equal to the diagonal element Akk of the matrix A, wherA = L-‘AGE-’
(2)
ISOTOPE FRE:QIlXCL*
5
SIIIFT
in which L is the L-matrix for the ordinury ( “Y- ) n~olc~de t’hat, c~o~lnccd * :I wt ot i~~t~crnalwortlinates Ri and the set of mrn~al cwortli~l:~tw(+jk, ant1
AG = ‘“G _ 14G,
iAl
14G mcl “G being the inverse kinetic energy nlatriws (‘4.1 for the 14K :mtl ‘“S species. Therefore, in order to obtain dAkkJdkTh , \vc Iieecl t,o exmiine how L-’ changes clue to t,he force constant change 6k’h, or due to the change (6F) in I he polent#i:rl cwcrgy matrix F. Let 11s:tssunle that, Lee1bwonw ( E + 6P)Lp’ :~wortling ns F bc~~omcs F + SF. Here, E is :I unit mltris. 6P w:w wldnt~cd by SC kag:Lwa and Shimnnouchi (5 ) :I* 61’,, = (e6FL)/,J(Xp
-
XI)
( .-JI
nntl 6Pi.i. = 0.
( (i I
Since F ant1 K,, c’:m he relat,etl by the A,, matrices :w F = c
ii
A/&/,,
6F can be expressed as 6F = AJh,,
T
iii
iS)
.
Hence, ( 5 I ~II be rewritten :ts
6Pl,l = [(EALL)&X~
-
hl)]S&.
(9)
IA us now :wswne that A becomes A + 61 according as K/tbecomes k’,, + 6k’,, . A + 6A = (E + 6P)L = (E + SP)A(E
‘AGE--‘(m) + %,
(10)
= A + &PA + A%; .*. 6A = iiPA + A%;
(11)
(121
6
TSUBOI
Therefore,
as the required
formula, (13) ACKNOWLEDGEMENTS
The writer wishes to express his sincere thanks to Professor Takehiko his suggestions. This work was supported by a grant from the Ministry Japan and a grant from the U. S. Public Health Service GM 10024-2. RECEIVED:
Hhimanouchi for of Education of
July 22, 1965 REFERENCES
T. SHIM~NOUCHI AND I. SUZUKI, J. Chem. Phys. 42, 29G (1965). M. TSUBOI, Spectrochim. Acta 16, 505 (1960); M. TSUBOI, T. T~KENISHI, AND A. NAK.4MUR.4,Spectrochim. Acta 17, 634 (1961); M. TSUBOI, T. TbKENISHI, BND A. NAK~~MURA,Spectrochim. Acta 19, 271 (1963). T. MIYAZAWA, J. Mol. Spectry. 13,321 (1964). E. B. WILSON, J. C. DECITJS,AND P. C. CROSS, “Molecular Vibrations,” p. 188. McGrawHill, New York, 1955. I. NAKAGAWA AND T. SHIMANOUCHI, Nippon
Kagaku
Zasshi
80, 128 (1959).