A new derivation for the Teller—Redlich isotopic product rule

VIBRATIONAL

SPEcrROSCOPY ELSEVIER

Vibrational Spectroscopy 7 (1994) 191-196

A new derivation for the Teller-Redlich isotopic product rule V.1. Pupyshev a

a,

Yu.N. Panchenko

b.*,

N.F. Stepanov

1

a

Laboratory of Molecular Structure and Quantum Chemistry, Department of Chemistry, M. V. Lomonosov Moscow State University, Moscow 119899, Russian Federation b Laboratory of Molecular Spectroscopy, Department of Chemistry, M. V. Lomonosov Moscow State University, Moscow 119899, Russian Federation (Received 3rd January 1994)

Abstract A straightfOlward proof of the Teller-Redlich isotopic product rule which is based on the explicit form of the determinant of the kinetic energy matrix expressed in terms of nuclear masses and corresponding moments of inertia is given. Key words: Isotopes; Teller-Redlich rule

1. Introduction In 1935 Redlich published work [1] in which he expressed the product of frequencies of the molecular normal vibrations in terms of the product of the nuclear masses, molecular mass, and product of moments of inertia. However, Angus et a1. [2] noted in 1935 and later in 1936 [3] that Teller had communicated them already in 1934 in the fulfillment of the theorem relating the ratio of the vibrational frequencies belonging to a given symmetry species of isotopomers to the ratio of the above-mentioned quantities. In these papers [1-3] the equation which is now referred to as the

• Corresponding author. 1 Dedicated to the 60th anniversary of the discovery of the Teller-Redlich isotopic product rule.

Teller-Redlich rule was defined. It is interesting that the equation given in Refs. 2 and 3 has a much more contemporary form than the one given in Ref. 1. According to Refs. 2 and 3 this statement has been used to analyse the real molecular spectra starting from 1934. In manuals and textbooks the Teller-Redlich rule is usually mentioned either without any proof (see, for example, Ref. 4) or in a simple form of a relation connecting the vibrational frequency product ratio with the ratio of determinants of the corresponding G matrices [5]. This is accompanied by the commonly accepted comments taken from Ref. 1 and the description of the limit transition from a molecule in a matrix (external field) to a free molecule [6]. Unfortunately, this proof neglects some subtle aspects of the problem, for example, the choice of the coordinate origin. In addition, the detailed proof, as sug-

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V.I. Pupyshev et al.j Vibrational Spectroscopy 7 (/994) 191-196

192

gested in Ref. 1, is rather complicated and cumbersome. For this reason it seems useful to offer a reasonably simple proof of the rule; this follows from the explicit equation for the calculation of the determinant of the G matrix. This equation is also involved in the analysis of isotopic effects since the exact "product rules" for isotopomers are expressed using the ratio of determinants of G matrices [7].

dimension 3N, respectively. The i-th components of these vectors (i = 1, ... , N) are (T+z);=zand(R+z);=[zxr;]

(4)

Let g' be a matrix of dimension 6 x 3N constructed by combining the lines of the matrices T and R, i.e.

g'=(~)

(5)

Let us consider the 6 x 6 matrix defined as 2. Discusssion

0' =

2.1. Integral expression for det G

Lm;[r; x 8;] = 0

(2)

Here T and R are matrices of dimension 3 x 3N as defined below and M is the square 3N x 3N matrix of nuclear masses. Elements of the T matrix are defined by the following condition: if 4 is a vector containing the N three-dimensional components 8; (i = 1, ... , N) then 4 is transformed to the following three-dimensional vectors, viz. T4= L8; andR4= L[r;x8;]

det

0'

= JA,3n m

where JL = L.;m; is the total molecular mass and TIm is the product of the principal moments of inertia, is fulfilled. Proof. The 0' matrix has a block structure corresponding to the (Tij blocks of the dimension 3 x 3. Really,

(1)

where m; are nuclear masses and [r; x 8;] means the vector product. Provided that 4 is a 3N-dimensional vector composed from N three-dimensional components 8;, the conditions for Eq. 1 reduce to the form TM4=0 and RM4=0

(6)

Assertion 1. The equation

In the following derivation, partitioning the vibrational motions and the translation-rotational ones is carried out using the well-known Eckart conditions (see, for example, Refs. 8 and 9), represented usually as the equations for the vibrational nuclear displacements 8; with respect to the equilibrium molecular configuration described by the nuclear radius-vectors r; (i = 1,2, ... N)

Lm;l); = 0 and

g'Mg'+

(3)

i

It is easy to show that by using the matrices T+ and R + one may convert a three-dimensional vector z into the vectors T +Z and R +z of the

where 0'\1 = TMT+, 0'12 = TMR+, 0'22 = RMR\ = O'i2' From the well-known expression in the theory of determinants [10] one can obtain

0'21

det

0'

= det

0'\1

det( 0'22

- 0'210'1"/0'12)'

Let us calculate these determinants. For an arbitrary vector z from the three-dimensional space, it is possible to obtain from Eqs. 3 and 4 the following equation TMT+z =

(Lm;)z = JLZ I

Le. 0'\1 = JLI where I is the unit matrix. Therefore, det 0'\1 = JL3. Further, for the three-dimensional vectors y one can obtain TMR+y = Lm;[r; x y] = [s X Y]JL

(7)

where s = L.;m;r;/JL is the radius-vector of centre of the nuclear masses.

V.I. Pupyshev et al. / Vibrational Spectroscopy 7 (1994) 191-196

Consider now the scalar product of two threedimensional vectors x and Ay, where A = C122 or C1 21 C1 1/C1 12' The following relations may be written, viz.

ments of nuclei. This matrix possesses the property B~+=O

(~

which expresses the invariance of internal coordinates under translations and rotations of a molecule as a whole. The matrix B contains 3N columns and 3N - 6 rows. According to Eq. 8 the application of Assertion 2 to the matrices

(x, C122Y) = (R+x, MR+y) =

193

Emj([rj , x], [rj , yl), i

and accounting for Eq. 7: (x, C121C1111C112Y) = ([s, x], [s, Y])JL. It is also easy to see that

(x, [C1 22 - C1 21 C1 1/C1 12 ]Y) =

Em;([r; - s, x], [r; - s, y])

yields =

(x, Jy)

i

where J is the matrix (tensor) of moments of inertia in the coordinate system with the center of masses as the origin [8]. That is why

det HI det H 2 = det BB+ det g'Mg-+ = det BB+ det C1

(9)

Let us consider the matrix H 3 = HIM-I. It is apparent that

det C1 = JL3 det J = JL 3n m

(10)

q.e.d. The crucial point of the construction proposed is the following.

Moreover,

Assertion 2. Let the square matrices Q I and Q 2

Using goB += 0 (see Eq. 8) and applying Assertion 2 to H 3 and H 2 one can find with the help of the relation G = BM -I B + that

be divided into rectangular blocks in a similar fashion,

det H 3 det H 2 = det G det g-g-+. If CID;= 0 (or DIC;= 0), then det Q I det Q2 = det CIC; det DID;'

As a corollary, the following relations are derived from Eqs. 9, 10, and 11: det G = det H 3 det H 2/det g'g-+ = det HI det H 2/(det M det g'g-+)

Proof. Since

= det C1 det BB+/(det M det g'g-+) CIC; QIQ;= ( D c+ I

(11)

2

CID;) D D+ I

2

it follows from CID;= 0 that det Q I det Q 2 = det Q I det Q; = det QIQ; = det clct det DID; We tum now to the consideration of the matrix B which transforms the cartesian displacement to internal coordinates, i.e. the rows of which define the expansion of the internal coordinates over the initial set of cartesian displace-

(12)

Keeping in mind Assertion 1 we find that JL3 det

det G =

J det BB +

det M det

g'g'+

(13)

Note that det M is the cubed product of all of the nuclear masses and det g' g-+ is equal to N 3 n I where N is the number of atoms, n I is the product of the principal moments of inertia of a hypothetical molecule with unit masses. It is clear that B B + is an analog of the G matrix provided that all the masses are unitary.

V.I. Pupyshev et al. / Vibrational Spectroscopy 7 (J994) 191-196

194

2.2. Symmetry factorisation

Relations like Eq. 12 or Eq. 13 can be derived easily accounting for the symmetry of the molecular nuclear configuration. It is sufficient to prove the fulfillment of these relations for one choice of the internal coordinates. Really, if B# and G# correspond to another choice of coordinates then B# = PB for some non-degenerate matrix P and, therefore, det Gil'/det G = det B#(B#) +/det BB+

= (det p)2 Therefore, Eqs. 12 and 13 are true for any internal coordinates. The same is true for any type of symmetry a. Let us choose the symmetry coordinates in such a manner that they can mix the cartesian displacements of atoms of the same type a with masses mao Let D be an orthogonal matrix performing this factorisation according to the molecular symmetry. Obviously, from the chosen system of coordinates it follows that the matrices D and M are commutative. Then the mass m a corresponds to the "effective mass" of any symmetrized displacement a which is realized by atoms of the a type. It is important that the translations and rotations are transformed according to certain irreducible representations of the symmetry group. Let the matrix Bs be the matrix of internal coordinates in the basis set of the symmetrized displacements and Ba be its block corresponding to the type a. It is possible without any changes to repeat the derivation of Eq. 12 for the diagonal block G a of the matrix G. In the following discussion the index a refers to the motions of type a or the corresponding block of the a matrix. Then det G a = det

aa

det BaB;/det M a det

g'ag'a+

(14) If the symmetry group contains a rotation-inversion axis of an order higher than 2, the centre of inversion, two crossed axes of symmetry or symmetry plane and axis crossed in one point, then the centre of masses and the origin of a coordinate frame coincide. In this case, in the

course of proof of Assertion 1, the i vector is zero and the Assertion 1 is obvious. Only the groups Cn and Cnv require special analysis since the i vector is situated on the symmetry axis. This means that the choice of the origin of the coordinate frame in the point s"* 0 only affects the values of the moments of inertia with respect to the (equivalent) axes orthogonal to the axis Cn' But in this instance it is possible to repeat the proof of the Assertion 1 for the corresponding two coordinates. The same is true for the case of the symmetry group C s • Therefore, det aa is defined as a product of the moments of inertia for the r a axes, the rotations around which belong to the a species, and the ta-fold product of sum of masses corresponding to the translation coordinates of the a species, i.e. tr M a . Inasmuch as all of the nuclei contribute to translations, tr M a = JL if t a "* O. In addition, obviously, (15) where m a is the mass of an atom of the a subset. The mass m a occurs in this product as many times (n aa ) as the a subset creates the displacement of the a type. Thus, the Teller type relation is obtained in complete analogy to Eq. 13, viz.

where for the a species displacements: t a is the number of translational coordinates of the a species; is the product of quantities corresponding to the a species (when rotating around the corresponding axis); Ba is a block of the B matrix which corresponds to the coordinates of the a species; J; are the principal moments of inertia for the axis i; Jl are the principal moments of inertia for the hypothetical system with the unit masses; n a is the product of quantities for all species of the equivalent atoms; m a is the mass of one of the equivalent atoms of the a subset; n aa is the number of coordinates of the a species induced by the subset of the equivalent atoms of the type a.

nr

V./. Pupyshev et al. / Vibrational Spectroscopy 7 (/994) /9/-/96

Note that N,aOaJII = det g-ag-+ a ;

where Wa is the block of W analogous to Ba • As was noted at the beginning of this Section, Eq. 16 is true not only for the special case of the coordinates chosen, but for any choice of internal coordinates. 2.3. Teller-Redlich product rule

From Eq. 16 the simple rule for isotopomers referred to as the Teller-Redlich rule follows explicitly. Because the vibrations are described by the relation GFL=LA it is obvious that det A= det G det F For isotopically substituted molecules (notations marked by symbol *) in the adiabatic approximation it is possible to consider the matrix F as invariant, Le. for the isotopomers the ratio of products of the squared frequencies is

(n Wi )2 I

det G*

=

2.4. Simplified discussion

To complete the consideration we note the elementary interpretation of Eq. 13 (and obviously Eq. 17). Indeed, the matrix of kinetic energy in cartesian coordinates is M. When going over to the _generalised coordinates qj it transforms into T = WMW+ where W is the square matrix of derivatives of qj with respect to cartesian coordinates. That is why det T = det M det WW+. As is well known, in the harmonic approximation the use of the Eckart conditions permits the partitioning of the vibrational and translation-rotational motions [8,9]. In this approximation l' does not change on displacing some coordinates by a constant vector. If G - I is the kinetic energy matrix of the vibrational motions then obviously ;

)3

(j.L* 0 (Ji IJ;) 0 (mi Im;)-3 j.L; ; (17)

(Redlich version [1]) or for the symmetry species a of both isotopomers as considered for the (lowest) group of symmetry common to both molecules,

(Teller version according to Refs. 2 and 3).

It is worth noting (1) that the moments of inertia of each isotopomer are calculated in its own coordinate system (the origin of the coordinate frame is situated in the centre of mass in each case) and (2) that Eq. 16 can be used for the other isotopic rules (see Ref. 7) as well.

det l' = (det G) -1j.L3 0J;

det G

W;

195

where j.L3 and 0 J; represent the contributions ;

of translations and rotations to the kinetic energy, respectively. Therefore, det G = j.L3 0 J;/(det M det WW+) ;

All of the considerations given above are really the calculation of det WW+ and the proof that this magnitude does not depend on nuclear masses. However, the latter is less obvious since three of the coordinates are the coordinates of the centre of mass and the other three are connected with the tensor of inertia, so that W does depend on the masses although partitioning of the translational, rotational and vibrational coordinates is carried out. This means that the nontrivial step in the consideration given above is actually the Assertion 1.

V.I. Pupyshev et al. / Vibrational Spectroscopy 7 (J994) 191 -196

196

3. Conclusion

References

Accounting for det Wg'+= N 3 nl' Eq. 13, in general, has the form

[1] O. Redlich, Z. Physik. Chern. B, 28 (1935) 371. [2] W.R. Angus, CR. Bailey, J.L. Cleave, AH. Leckie, CG. Raisin, C.L. Wilson and C.K. Ingold, Nature, 135 (1935) 1033;136 (1935) 680. [3] W.R. Angus, CR. Bailey, J.B. Hale, CK. Ingold, AH. Leckie, CG. Raisin, J.W. Thompson and CL. Wilson, J. Chern. Soc., (1936) 971. [4] G. Herzberg, Infrared and Raman Spectra of Polyatomic Molecules, Van Nostrand, Princeton, New York, 1945. [5] S.J. Cyvin, Molecular Vibrations and Mean Square Amplitudes, Elsevier, Amsterdam, 1968. [6] E. Wilson, J. Decius and P. Cross, Molecular Vibrations, McGraw Hill, New York, 1955. [7] A Miiller, in AJ. Barnes and W.J. Orvill-Thomas (Eds.), Vibrational Spectroscopy, Modern Trends, Elsevier, Amsterdam, 1977, Chap. 12. [8] P.A Braun, AA Kiseliov, Vvedenie v teoriyu molekularnykh spektrov, (Introduction in Theory of Molecular Spectra), Leningrad University Publisher, Leningrad, 1983 (in Russian). [9] L.C. Biedenharn and J.D. Louck, Angular Momentum in Quantum Physics, Theory and Application (Encyclopedia of Mathematics and its Applications, Vol. 8). AddisonWesley, Reading, MA, 1981. [10] F.R. Gantmacher, Theory of Matrices, Chelsea, NY, 1959, Chap. X, Sec. 7, Subsec. 6.

detG detG) =

(1L)3 N

)

I/(J;/J;)I!(m j )

-3

where G) and J) are the magnitudes corresponding to the case of the unit masses for all of the nuclei. This means that the Eqs. 13 or 16 are not stronger statements than the Teller-Redlich rule. The goal of this work was only a new derivation of this rule using the simpler, but quite correct consideration.

Acknowledgements

The authors wish to thank Dr. George R. De Mare (Universite Libre de Bruxelles) for his helpful discussion of the problem. Financial support by the Russian Foundation for Fundamental Investigations (Grant 93-03-18386) is gratefully acknowledged.