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A method for obtaining the roots of the ζναζναΔ matrices: The number of degrees of freedom
JOURNAL
OF MOLECULAR
A Method
SPECTROSCOPY
48, 214-217 (1971)
For Obtaining the Roots of the &OLGI Matrices: The Number of Degrees of Freedom ANKA ULNAE WALSH BAKKE~ Newton
College
of the Sacred
Heart,
hewlon,
Muss.
An equation is presented which will allow calculation of the number of degrees, freedom, nia, contributed by atomic displacements along the OL direction to the 6th irreducible representation. The quantity thus obtained is used in the determination of the roots of the cye{Ft matrices. The result of using this method is reported for several molecular structures. In their work on vibration-rotation interaction in polyatomic molecules Meal and Polo (1) have discussed several properties of the lV” mat,rices, the element,s of which are the Coriolis coupling constant’s, and of the product matrices designated by the symbol 3_YU{Zt.In particular, it has been shown t’hat the roots of the ~_,,~
214
TABLE
I
DEGREES OF FREEDOM OF NORMAL MODES OF VIBRATION FOR SEVERAL MOLECULAR CONFIGURATIONS Configuration XY2
62s Al An RI Re
XYS
XY,
c3u Al A? E
1
0
2
0
1
0
2 0
0 2
1 0 2
1
0
I
2
6
Td Al
1 0 2 3 9
A? E Fl F2
XSY4
cw Al -42
E
5 0 2
1 1 12
2
1
0
1
XYZ2
A,” A*” E’ E” (XYd42
BAKKE
“16
degrees of freedom in a given direction t.hat, is comributed to a given irreducible representation. What is needed is a relation which will predict t,he number of degrees of freedom in the a-direction, (CY= z, y, z), that a given mode of vibration contributes to its respective irreducible representation. The motions of atoms are described by displacement vect,ors. The character of bhe represent’ation for the complete vibrational speckurn (overall translations and rotations included) is x(R)
= VR(Ikl
+ 2 coscp),
(1)
where U, is t’he number of atoms unaffected by the symmetry operation R, (o is t.he angle of rotation about a p-fold symmetry axis of the molecule and the + and - signs refer to proper and improper rotations. The representation for a displacement vector, in particular that of a kanslation vector, is well known and its character is
x’“(R)
= fl
+ 2 cos ‘p,
(2 1
so that x(R)
= &x’“(R).
The number of degrees of freedom, n;“, contributed by x*~ (R) displacsment,s the a-direction (CY= 2, y, x) to the i-th representation, will be given by (2)
(5) in
(4) where di = degeneracy of the i-th represent’ation and h = number of elements in the group. For the doubly degenerate vibrations one always has two orthogonal displacement vectors corresponding to the two orthogonal normal coordinates. Since T, and T, will then belong to the same doubly degenerate irreducible representation, (the z direction is always along the principal axis of symmetry and the displacement for the degenerate modes always lie in the zy plane) one obtains nix + nY
and for the triply
degenerate
= ; 7 vibrat.ions
11;’+ n;’ + niz=
Equation (4), (4a and 4b molecules and the results for a for example, of X2Y4 the form berg (3) and inspection of his
U,XT”‘T”(R)~;(R), one has
$ CR
~RxTz~T~~Tz(R)x~(R).
when appropriate), have been applied t,o several few configurations are given in Table I. In the case, of the normal vibrations has been given by Herzfigures will yield the same information.
THE NUMBER OF DEGREES OF FREEDOM
217
ACKNOWLEDGMENTS I would like to acknowledge with deep appreciation the assistance and encouragement of Dr. Alfons Weber. I would also like to express my gratitude to the Research Corporation and to the National Science Foundation for their support of parts of this research RECEIVED: September
3,197.O REFERENCES
1. J. H. MEAL AND S. R. POLO,J. Chem. Phys. 84, 1119, 1126 (1956). 2. This formula is a modified form of an expression which was first presented in the author’s thesis. A. M. WALSH, Ph.D. thesis, Fordham University, Bronx, New York, 1965. 3. G. HERZBERG,“Molecular Spectra and Molecular Structure II: Infrared and Raman Spectra of Polyatomic Molecules, ” D. Van Nostrand and Co., Princeton, N.J., 1959.
OF MOLECULAR
A Method
SPECTROSCOPY
48, 214-217 (1971)
For Obtaining the Roots of the &OLGI Matrices: The Number of Degrees of Freedom ANKA ULNAE WALSH BAKKE~ Newton
College
of the Sacred
Heart,
hewlon,
Muss.
An equation is presented which will allow calculation of the number of degrees, freedom, nia, contributed by atomic displacements along the OL direction to the 6th irreducible representation. The quantity thus obtained is used in the determination of the roots of the cye{Ft matrices. The result of using this method is reported for several molecular structures. In their work on vibration-rotation interaction in polyatomic molecules Meal and Polo (1) have discussed several properties of the lV” mat,rices, the element,s of which are the Coriolis coupling constant’s, and of the product matrices designated by the symbol 3_YU{Zt.In particular, it has been shown t’hat the roots of the ~_,,~
214
TABLE
I
DEGREES OF FREEDOM OF NORMAL MODES OF VIBRATION FOR SEVERAL MOLECULAR CONFIGURATIONS Configuration XY2
62s Al An RI Re
XYS
XY,
c3u Al A? E
1
0
2
0
1
0
2 0
0 2
1 0 2
1
0
I
2
6
Td Al
1 0 2 3 9
A? E Fl F2
XSY4
cw Al -42
E
5 0 2
1 1 12
2
1
0
1
XYZ2
A,” A*” E’ E” (XYd42
BAKKE
“16
degrees of freedom in a given direction t.hat, is comributed to a given irreducible representation. What is needed is a relation which will predict t,he number of degrees of freedom in the a-direction, (CY= z, y, z), that a given mode of vibration contributes to its respective irreducible representation. The motions of atoms are described by displacement vect,ors. The character of bhe represent’ation for the complete vibrational speckurn (overall translations and rotations included) is x(R)
= VR(Ikl
+ 2 coscp),
(1)
where U, is t’he number of atoms unaffected by the symmetry operation R, (o is t.he angle of rotation about a p-fold symmetry axis of the molecule and the + and - signs refer to proper and improper rotations. The representation for a displacement vector, in particular that of a kanslation vector, is well known and its character is
x’“(R)
= fl
+ 2 cos ‘p,
(2 1
so that x(R)
= &x’“(R).
The number of degrees of freedom, n;“, contributed by x*~ (R) displacsment,s the a-direction (CY= 2, y, x) to the i-th representation, will be given by (2)
(5) in
(4) where di = degeneracy of the i-th represent’ation and h = number of elements in the group. For the doubly degenerate vibrations one always has two orthogonal displacement vectors corresponding to the two orthogonal normal coordinates. Since T, and T, will then belong to the same doubly degenerate irreducible representation, (the z direction is always along the principal axis of symmetry and the displacement for the degenerate modes always lie in the zy plane) one obtains nix + nY
and for the triply
degenerate
= ; 7 vibrat.ions
11;’+ n;’ + niz=
Equation (4), (4a and 4b molecules and the results for a for example, of X2Y4 the form berg (3) and inspection of his
U,XT”‘T”(R)~;(R), one has
$ CR
~RxTz~T~~Tz(R)x~(R).
when appropriate), have been applied t,o several few configurations are given in Table I. In the case, of the normal vibrations has been given by Herzfigures will yield the same information.
THE NUMBER OF DEGREES OF FREEDOM
217
ACKNOWLEDGMENTS I would like to acknowledge with deep appreciation the assistance and encouragement of Dr. Alfons Weber. I would also like to express my gratitude to the Research Corporation and to the National Science Foundation for their support of parts of this research RECEIVED: September
3,197.O REFERENCES
1. J. H. MEAL AND S. R. POLO,J. Chem. Phys. 84, 1119, 1126 (1956). 2. This formula is a modified form of an expression which was first presented in the author’s thesis. A. M. WALSH, Ph.D. thesis, Fordham University, Bronx, New York, 1965. 3. G. HERZBERG,“Molecular Spectra and Molecular Structure II: Infrared and Raman Spectra of Polyatomic Molecules, ” D. Van Nostrand and Co., Princeton, N.J., 1959.