Isotopic relations for tetrahedral and octahedral molecules

Journal of Molecular Structure 1206 (2020) 127729

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Journal of Molecular Structure journal homepage: http://www.elsevier.com/locate/molstruc

Isotopic relations for tetrahedral and octahedral molecules €te, C. Richard, V. Boudon* M. Loe Laboratoire Interdisciplinaire Carnot de Bourgogne, UMR 6303 CNRS - Universit e Bourgogne Franche-Comt e, 9 Av. A. Savary, BP 47870, F-21078 Dijon Cedex, France

a r t i c l e i n f o

a b s t r a c t

Article history: Received 31 October 2019 Received in revised form 7 January 2020 Accepted 13 January 2020 Available online 17 January 2020

The study and analysis of heavy spherical-top molecules is often not straightforward. The presence of hot bands and of many isotopologues can lead to a high line congestion very difficult for assignment. In this work, using a low-order model we have derived very simple isotopic relations in order to determine initial parameters of the analysis. We also show that an identical approach can be used for XY4 and XY6 molecules and all these results are illustrated by the comparison of numerical computations and experiments. © 2020 Elsevier B.V. All rights reserved.

This paper is dedicated to the memory of Dr. Jon T. Hougen. Keywords: Spherical-top molecules Infrared absorption Isotopic relations

1. Introduction Spherical-top molecules constitute an important class of molecular species whose study requires extensive use of symmetry [1] and group theory methods [2e4]. Besides the iconic example of methane [5], there exist many heaver species of this type, either tetrahedral like CF4 [6], SiH4 [7,8], GeH4 [9,10], OsO4 [11,12], RuO4 [13], P4 [14], C10H16 [15], C6N4H12 [16], …or octahedral like SF6 [17], SeF6 [18], WF6 [19], UF6 [20], C8H8 [21], …. Most of them possess, in natural abundance, several isotopologues, some being sometimes quite rare [22] or even radioactive [13]. It is thus desirable to derive at least approximate relations to calculate isotopic shift relations for the lower order molecular parameters in order to get a first estimate and simulation of the absorption spectrum of lowabundance or radioactive isotopologues [9,10,13,23]. Isotopic relations, however, have never been published in detail for such molecules, although there exist some partial studies [24e26] (some studies are part of old French theses which are difficult to find). In the present paper, we reinvestigate this question using a simple low-order model which is sufficient for the above-described purpose, that is to estimate initial parameters to allow initial simulations of infrared-active bands of minor isotopologues, based on the main isotopologue results. This method is

applicable to all isotopologues with the same symmetry. The problem is more complicated when the isototopic substitution lowers the symmetry (for instance, the passage from CH4 to CH2D2). In the latter case, there exist other techniques, see for instance the work of Ulenikov et al. [27]. Quadratic isotopic relations, for all molecules, are usually established using the GF matrix method [24], since force constants are invariant through isotopic substitution (within the BornOppenheimer approximation). Here, we apply this method to infrared absorption active triply degenerate modes of tetrahedral and octahedral species using the approach of Hoy et al. [28]. This one leads, a priori, to two systems of linear equations that allow to calculate harmonic frequencies (or wavenumbers) and Coriolis constants for the different isotopologues. We also propose an alternative approach based on force constants in the Cartesian coordinate system [29]. Section 2 describes the internal coordinate approach, while Section 3 presents the Cartesian coordinate point of view that leads to the isotopic relations derived in Section 4. Finally, we illustrate this through simple examples for tetrahedral and octahedral molecules in Sections 5 and 6, respectively.

2. XY4 force constants in symmetrized internal coordinates * Corresponding author. E-mail address: [email protected] (V. Boudon). https://doi.org/10.1016/j.molstruc.2020.127729 0022-2860/© 2020 Elsevier B.V. All rights reserved.

Let’s consider a molecule made up of one nucleus X of mass M

M. Lo€ete et al. / Journal of Molecular Structure 1206 (2020) 127729

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and four nuclei Y of mass m (see Fig. 1). The symmetry group is the Td group and the molecule possesses four vibration modes; s ¼ 1 of symmetry A1, s ¼ 2 of symmetry E and s ¼ 3,4 of symmetry F2. The quadratic potential, restricted to the modes 3 and 4, is expressed as:

  2 2 1 X F33 i S3a þ F44 i S4a þ 2F34 i S3a i S4a ; V34 ¼ 2 a¼x;y;z

(1)

2

where the oriented symmetrized coordinates i Ssa are expressed as a function of internal coordinates (see equation (2)).

1 S3x ¼ ðDr2  Dr3  Dr4 þ Dr5 Þ; 2 1 i S3y ¼ ð  Dr2 þ Dr3  Dr4 þ Dr5 Þ; 2 1 i S3z ¼ ð  Dr2  Dr3 þ Dr4 þ Dr5 Þ; 2 1 i S4x ¼ pffiffiffire ðDQ34  DQ25 Þ; 2 i

i

i

(2)

1 S4y ¼ pffiffiffire ðDQ24  DQ35 Þ; 2 1 S4z ¼ pffiffiffire ðDQ23  DQ45 Þ: 2

To determine force constants F33, F34 and F44, we use the GF method of Wilson [24] which allows to link these constants to the molecular masses and harmonic frequencies us following the secular equation:

  det ðGFÞij  ldij ¼ 0;

(3)

with

(4)

For F2 type vibrations we obtain:

   F F  G G  l  lðF33 G33 þ 2F34 G34 þ F44 G44 Þ þ  33 34  33 34  ¼ 0; F43 F44 G43 G44 2

(5)

6F34

6F44

(6)

  G33 G44  G234 F33 F44  F 234 ¼ l3 l4 :

(7)

A third equation is needed to determine the three force constants. We use here this equation from Ref. [28]:

C33 F33 þ C44 F44 þ 2C34 F34 ¼ l3 z3 þ l4 z4 ;

(8)

where C is a matrix only depending of masses and geometry of the molecule and where Coriolis constants z3 and z4 are not independent and follow sum rules. Nevertheless, the above equation (7) is not linear. So, by combining equations (6), (7) and (8), Hoy et al. [28] have obtained a third linear equation:

ðG33 C34  G33 C33 ÞF33 þ ðG34 C44  G44 C34 ÞF44 þ ðG33 C44 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  G44 C33 ÞF34 ¼ εðl3  l4 Þ z3 z4 jGj  jCj;

(9)

where |G| and |C| stand for the G and C matrix determinants and ε ¼ ±1, leading to two possible solutions. For XY4 molecules, the G matrix has been given by Simanouti [30] and Aboumajd [26], while the C matrix was given by Mills [31]. Considering our choice of symmetrized coordinates, these matrices can be expressed as follows:

0

4 1 B 3M þ m G¼B @ 8 3M

0 1 8 4 B 3M C 3M C ;C ¼B @ 8 16 2A 1 þ þ 3M m 3M m

1 8 1 þ C 3M m C : 16 1A þ 3M m

(10)

Therefore (see Appendix 1), equation (9) leads to

ls ¼ 4p2 c2 u2s :

6F33

G33 F33 þ G44 F44 þ 2G34 F34 ¼ l3 þ l4 ;

F33  2F44 þ F34 ¼ mmD sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   1 þ z4 : ¼ εðl3  l4 Þ 2mmð1  z4 Þ 2

Then, we obtain force constants F33, F44 and F34 by solving the system of 3 linear equations (6), (8) and (11). Their expressions are given in equation (12) and we have two solutions depending of the value of ε:

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " #  1 1 ð2m  mÞð4z4  1Þ þ 8ε 2mmð1  z4 Þ þ z4 ðl3  l4 Þ; ¼ ðm þ 2mÞðl3 þ l4 Þ þ 3 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " #  1 1 þ z4 ¼ ðm  mÞðl3 þ l4 Þ þ ðm þ mÞð1  4z4 Þ þ 2ε 2mmð1  z4 Þ ðl3  l4 Þ; 3 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " #   m 1 m 1 ðl3 þ l4 Þ þ  m ð4z4  1Þ  4ε 2mmð1  z4 Þ þ z4 ðl3  l4 Þ: ¼ mþ 2 3 2 2

where (F) and (G) are symmetric matrices and (G) depends of the choice of the symmetrized coordinates. The solutions are l3 and l4 with the following relations:

(11)

(12)

Knowing numerical values of these constants from l3, l4 and z4 of one isotopologue, we can compute l30 , l40 and z40 for other isotopologues by using equations (5) and (8) and by adapting the G and C matrices.

M. Lo€ete et al. / Journal of Molecular Structure 1206 (2020) 127729

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3 2 3 z4 ¼ 1  sin2 g; 2

z3 ¼ 1  cos2 g; (18)

hence,

1 2

z3 þ z4 ¼ :

(19)

For XY4 molecules [25,29], we have:

2

31=2

1 6 k2  k1 7 cos g ¼ pffiffiffi 41 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 2 2 2 ðk2  k1 Þ þ 4k3 2

In a symmetrized Cartesian coordinate system, the quadratic potential, restricted to modes s ¼ 3 and s ¼ 4, is expressed as [29]:

1 X  c 2 2 k33 S3a þ k44 c S4a þ 2k34 c S3a c S4a ; 2 a¼x;y;z

(13)

2

where oriented symmetrized coordinates c Ssa are expressed as a function of Cartesian coordinates:

c

c

c

c

5 1  X S3a ¼ pffiffiffiffiffiffiffi m ai  Ma1 ; 4 mm i¼2

1 S4x ¼ pffiffiffi ð  y2  y3 þ y4 þ y5  z2 þ z3  z4 þ z5 Þ; 2 2 S4y

(14)

k1 þ k2 ; m

k1 k  2; m 2m 1  l3 l4 ¼ 2 k1 k2  k23 : m

l3 z3 þ l4 z4 ¼

(21)

The system of equations (6), (8) and (11) becomes:

k1 þ k2 ; m k k C33 F33 þ C44 F44 þ 2C34 F34 ¼ 1  2 ; m 2m rffiffiffiffiffiffi 3 2m F33  2F44 þ F34 ¼ εjk j 2 m 3

(22)

By solving the system, we obtain:

1 S4z ¼ pffiffiffi ðy2  y3  y4 þ y5  x2 þ x3  x4 þ x5 Þ: 2 2 The l0 s are solutions of the secular equation:

(15)

At that point, let k33 ¼ k1, k44 ¼ k2 and k34 ¼ k3 in order to simplify the notations. We thus write:

ðk1  mlÞðk2  mlÞ  k23 ¼ 0

l3 þ l4 ¼

G33 F33 þ G44 F44 þ 2G34 F34 ¼

1 ¼ pffiffiffi ð  x2  x3 þ x4 þ x5 þ z2  z3  z4 þ z5 Þ; 2 2

  det kij  mldij ¼ 0:

;

noting that for the XY6 molecules, Moret Bailly [29] switched cosg and sing. Some additional relations are given in Appendix 2. We can also deduce from formulas (17)e(20):

3. Force constants of XY4 in the symmetrized cartesian coordinate system

V34 ¼

(20)

31=2

1 6 k2  k1 7 sin g ¼ pffiffiffi 41  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 2 ðk2  k1 Þ2 þ 4k23

Fig. 1. An XY4 molecule in its reference configuration.

;

(16)

2 2 F33 ¼ ak1 þ k2 þ bεjk3 j; 3 3 1 1 F44 ¼ ak1 þ k2  bεjk3 j; 6 3 1 1 F34 ¼ ak1  k2 þ bεjk3 j; 3 6

(23)

with

rffiffiffiffiffiffi 2m : a¼ ; b ¼ 3m m

m

(24)

and

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðk1  k2 Þ2 þ 4k23 ; l3 ¼ 2m qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k1 þ k2 þ ðk1  k2 Þ2 þ 4k23 : l4 ¼ 2m k1 þ k2 

We also obtain:

4. Isotopic relations in cartesian coordinates

(17) As for the force constants Fij, the k2 constant is invariant relative to isotopic substitutions as far as c S4a coordinates do not depend of masses. Therefore, we can write:

M. Lo€ete et al. / Journal of Molecular Structure 1206 (2020) 127729

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2 2 3 3 1 1 ak1  bεjk3 j ¼ a0 k1 0  b0 ε0 jk3 0 j 3 3

ak1 þ bεjk3 j ¼ a0 k1 0 þ b0 ε0 jk3 0 j; (25)

and so:

bεjk3 j ¼ b0 ε0 jk3 0 j:

(26)

However, in the relations from (17) to (20), allowing to calculate frequencies and Coriolis constants, only the parameter k23 is needed. Consequently,









b2 ε2 k3 2 ¼ b02 ε02 k3 0 2

(27)

and thus,

b2 k23 ¼ b02 k3 02 ;

(28)

2

Then k3’ can be expressed as

k3 02 ¼

m m’ 2 k : m’ m 3

(29)

Also, from equation (25) we show that:

k1 0 ¼

m m’ k ; m’ m 1

(30)

and

k2 0 ¼ k2 :

(31)

From formula (32) below, deduced from relations (21), we can calculate k1, k2 and k23 constants for one isotopologue that we know l3 and l4 frequencies and z3 or z4 Coriolis constants, and apply isotopic relations (29) to (31) to compute frequencies and Coriolis parameters of other isotopologues using formula (17) to (20):





3 k1 1 þ l4 ¼ l3 z3 þ 2 m 2



 1 z4 þ ; 2

k2 k ¼ l3 þ l4  1 ; m m

(32)

k23 ¼ k1 k2  m2 l3 l4 : Moreover, we have shown that isotopic relations (29) to (31) are independent of ε. Then, using GF method, both set of parameters Fij (for ε ¼ ±1) obtained by (12) lead to the same frequency values and Coriolis constants (ls and zs) for one isotopologue. 5. Application to XY4 molecules We present here applications of the previous formulas to three tetrahedral spherical-top molecules: methane (CH4), germane (GeH4) and ruthenium tetroxide (RuO4). Methane is of course a very important hydrocarbon molecule for many applications in atmospheric and astrophysical sciences, whose spectroscopy is exensively studied (see Refs. [5,36] and references therein). There already exist many studies concerning its different isotopologues. We use it here as a simple illustration example. Recent studies [5,36] use sophisticated effective Hamiltonians that take into account the complex polyad structure of this molecule, with many intra-polyad interactions (Fermi resonances, Coriolis interactions between different bands, etc). Their parameters are not easily useable for the present simple application since

one would need to extract decorrelated band centers and Coriolis constants. We rather use here simpler old studies that considered fits of the n3 and n4 bands as isolated. If the accuracy is not as high as what can be done nowadays, this is largely sufficient for simple low order calculations like the present ones. We thus start from 12CH4 values taken from Ref. [32] and then deduce, using the method explained in Sections 3 and 4, the corresponding constants for 13 CH4, 12CD4 and 12CT4. The results are compared to the literature [32e35]. This is presented in Table 1. Some important points should be noticed:  The method described in Section 2 using internal coordinates gives exactly the same results, within numerical rounding errors. The two choices for ε (±1) give different force constants F, but the same band center and Coriolis constant values. The cartesian coordinate method (Section 4), as it does not depend on ε, does not have this drawback.  In principle, calculations should use the harmonic wavenumbers u. As an approximation, we can also use the band center n, knowing that the fitted effective hamiltonian parameter n corresponds to the true band center (J ¼ 0 level) only when bands are studied as isolated. As we can see in Table 1, both calculations work well, the one using harmonic wavenumbers u being significantly better, considering the relative isotopic shift error. For instance, the error on the 12CH4/12CD4 shift is 3.3% for n3 and better for u3, as expected for our simple model. The result is similar for n4.  For high-resolution, more elaborated but heavier models should be used, like for instance that of Rey et al. [37]. We can notice, however, from Table 9 of this reference, that we get very similar 12 CH4/12CD4 isotopic shifts for n3 and n4 band centers, taking into account the expected accuracy of our simple model.  According to the correspondance between the Dijon tensorial formalism [2e4] (used for the fits) and “classical” notations [2,38], we have

pffiffiffi 1ð1;0F Þ t figfig 1 ¼ 3 2ðBzÞi ;

(33)

 (with i ¼ 3 or 4) for the tensorial Coriolis parameter. We use this relation to estimate the z value, with the approximation

ðBzÞi xB0 zi ;

(34)

 B0 being the ground state rotational constant.  The Coriolis constants z should follow the sum rule (19). However, when using such constants resulting from independent fits of n3 and n4 spectroscopic data, this sum rule is generally only approximately respected (within a few %). The present calculations proved to give better results if we make the following choice: (i) we take the experimental z3 value obtained through (33) and (34) and (ii) we use

1 2

z4 ¼  z3 :

(35)

 This only slightly differ from the experimental z4 value and is more consistent with the present method that is based on the sum rule. Our second exemple is GeH4. The fundamental bands of this molecule, which is of importance in planetology (see Ref. [9] and references therein) have been recently studied in our group [9,10].

M. Lo€ete et al. / Journal of Molecular Structure 1206 (2020) 127729

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Table 1 Band center and Coriolis z constant for the n3 and n4 fundamental bands of different tetrahedral methane isotopologues. We use experimental values for12CH4 from Ref. [32] as the starting point to calculate values for the other isotopologues and compare with literature values. Two calculations are made in each case: one using experimentally fitted band centers (n) and z constants and one using calculated harmonic frequencies (u, estimated by authors from the experimental values of n and of the anharmonic constants X) and z constants. For these two calculations, the z constants are denoted zn and zu, respectively.

exp

calc

exp

calc

exp

calc

exp

calc

exp

calc

exp

calc

exp

calc

exp

calc

12

CH4[32] n n ðz3 þ z4 Þexp ¼ 0:508

3019.2

3019.20

3156.8

3156.80

0.054

0.056

0.046

0.046

1310.8

1310.80

1367.4

1367.40

0.446

0.444

0.454

0.454

13

CH4[32e34] ðzn3 þ zn4 Þexp ¼ 0:509

3009.5

3008.45

3146.6

3145.76

0.045

0.046

0.036

0.036

1302.8

1302.77

1358.9

1358.94

0.455

0.454

0.464

0.464

12 CD4[32] ðzn3 þ zn4 Þexp ¼ 0:500

2260.1

2234.47

2336.2

2334.86

0.163

0.173

0.165

0.165

997.8

990.10

1034.1

1033.49

0.337

0.327

0.335

0.335

12

1937.0

1906.52

1993.9

1991.10

0.252

0.275

0.265

0.266

858.0

847.45

880.1

885.07

0.248

0.225

0.236

0.234

a

zn3

n4/cm1

u4/cm1

zu4

n3/cm1

CT4[35] n n ðz3 þ z4 Þexp ¼ 0:500

u3/cm1

zu3

Isotopologue

zn4 a

We take here zv4 ¼ 1=2  zv3 as the “experimental” value; the “true” experimental z sum is given at the beginning of the line.

In this case, we do not have anharmonic constants and thus u values cannot be obtained. But band centers are well determined and n are convenient. Moreover, for this molecule, no strong Fermi resonance is observed between bending and stretching modes. The n3 and n4 constants are obtained through fits of two dyads, n2/n4 [10] and n1/n3 [9], which only present limited rovibrational interactions of Coriolis type. Thus, the fitted values of n3 and n4 are almost identical to band centers as it can be verified by calculating the J ¼ 0 levels. Germanium in natural abundance presents 5 isotopes. We can thus calculate germane force constants using experimental n3, n4, z3 and z4 values (with the same remarks as for methane above) for

70

GeH4 and then deduce these constants for 72GeH4, 73GeH4, GeH4 and 76GeH4 and compare them to the experiment. The results are presented on Fig. 2. Taking into account the many approximations made, the results are very satisfying. The relative error on isotopic shifts is small. We also notice on the lower right panel the limited error on the z sum rule. The experimental z4 value is shifted from the 1/2z3 value, but follows the same tendency. The third example that we show here is ruthenium tetroxide. This molecule is of importance for nuclear power plant security and other industrial applications (see Refs. [13,23,39] and references therein). In natural abundance, it possesses 5 isotopologues, due to the different ruthenium isotope (we do not consider here oxygen 74

pffiffiffi Fig. 2. The band center and Coriolis constant (3 2ðBzÞ), see text) as a function of the germanium isotope for the GeH4 molecule. Blue curves show the values fitted on experimental spectra; the red curves show the calculated values determined from the 70GeH4 experimental value. In the case of z4, we used the 1/2z3 value (green curve), the experimental value being farther away from this sum rule. We also indicate the resulting relative error for the 76GeH4 itotopologue.

6

M. Lo€ete et al. / Journal of Molecular Structure 1206 (2020) 127729

pffiffiffi Fig. 3. The band center and Coriolis constant (3 2ðBz3 Þ), see text) as a function of the ruthenium isotope for the n3 band of the RuO4 molecule. We use symbols to distinguish between observed and extrapolated data for the different isotopologues.

isotopes): 99RuO4, 100RuO4, 101RuO4, 102RuO4 (the main one) and 104 RuO4. There are also two isotopologues with very low abundance, namely 97RuO4 and 98RuO4 and two radioactive, short-lived, isotopologues, 103RuO4 and 106RuO4. In Ref. [13], we analyzed in detail the n3 band for all observable isotopologues and extrapolated the band center and Coriolis parameter for the others. In Ref. [23], we analyzed the n2/n4 bending dyad for 102RuO4 only. We can thus apply the same method by starting with 102RuO4 values. As for germane, we use band centers, the u values being unknown and we also consider z4 as 1/2z3 (the z sum with the experimental z4 value [23] being 0.484). The calculated results are compared with experimental and extrapolated values from Ref. [13] on Fig. 3. Again, the results are very satisfying, with small errors on isotopic shifts; values are very close to those of Ref. [13]. It is then possible to deduce values for the n4 band for the different isotopologues that, apart from 102RuO4, have not yet been analyzed up to now. As an illustration, Table 2 gives the n4 predicted band centers and Coriolis constants. For these ones, we can calculate the difference between the experimental parameter for 102RuO4 [23] and the 1/2z3 value used in the present calculation; then we predict z4 value, say zp4 thanks to the calculated value zc4 obtained in the present paper through:

zp4 ¼ zc4 þ Dz4 :

(36)

As we saw for GeH4 (lower right panel of Fig. 2), the variation of the experimental and calculated (starting from 1/2z3 for the reference isotopologue) value of z4 are parallel; it is thus justified to apply the same correction Dz4 to all isotopologues. The tensorial pffiffiffi p constant is also evaluated by multiplying z4 by 3 2B0.

Table 2 Predicted n4 band centers and Coriolis constants (see text for notations) for the different isotopologues of RuO4. The reference isotopologue is102RuO4, so the values given here for this one are identical to the experimental ones [13]. 1ð1;0F Þ

Isotopologue

n4/cm1

zp4

t f4gf4g1 = cm1

97

337.047 336.683 336.325 335.972 335.623 335.281 334.942 334.609 333.957

0.2808 0.2828 0.2848 0.2867 0.2886 0.2904 0.2922 0.2940 0.2976

0.1618 0.1629 0.1641 0.1652 0.1663 0.1673 0.1684 0.1694 0.1714

RuO4 RuO4 RuO4 100 RuO4 101 RuO4 102 RuO4 103 RuO4 104 RuO4 106 RuO4 98 99

6. Application to XY6 molecules Formula established for XY4 molecules can be applied to XY6 molecules by simply replacing G, C and m by Refs. [26,40e42]:

0

2 1 BM þ m B G¼@ 4 M

4 M

1

0

2 B C M C; C ¼ B @ 1 8 2A 4 þ   M m m M

1 1 4   C m MC 1 8 A þ m M

(37)

and

m¼

mM : M þ 6m

(38)

In this case, the n3 and n4 normal mode coordinates possess the F1u symmetry in the Oh point group.

M. Lo€ete et al. / Journal of Molecular Structure 1206 (2020) 127729

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pffiffiffi Fig. 4. The band center and Coriolis constant (3 2ðBzÞ), see text) as a function of the sulfur isotope for the SF6 molecule. The legend is the same as for Fig. 2. In this case, no data 36 33 exist for the n4 band of SF6. As for SF6 the n4 band center is an extrapolated value from Ref. [43], while the Coriolis constant is no known.

We can thus apply the present theory to the SF6 molecule. This one is a very strong greenhouse gas, see Ref. [17] and references therein. It possesses four isotopologues in natural abundance, namely 32SF6, 33SF6, 34SF6 and 36SF6, the latter one having a very small abundance of only 0.02% [22]. We can thus use the very accurate values for 32SF6 (using band centers as for germane above) to deduce the constants for the other isotopologues and compare them to experimental values, when available. The results are presented on Fig. 4 and, again, are very satisfying. We see here that this method allows to estimate constants that are not known for some minor isotopologues and serve as a starting point for their future analysis.

provided that one main isotopologue is well known.

Appendix 1. Formulas for the GF method Following the G and C matrix definition from Equation (10), Equation (9) becomes

F33  2F44 þ F34 ¼ mmD; with

m¼ 7. Conclusion We have shown that isotopic relations, when established in the Cartesian coordinate system for tetrahedral and octahedral molecules, lead to simple calculations for harmonic and Coriolis constants. As a matter of fact, these calculations depend only on masses (m and m) and not on the whole G and C matrices as when using the internal coordinate system. We have also shown that the ambiguity on the determination of force constants is not involved in these isotopic relations. Moreover, it should be noticed that the ambiguity in Cartesian coordinates only resides in the sign of the interaction parameter k34 ¼ k3, while for internal coordinates, the three force constants lead to three different values when changing the sign of ε. Finally, we have illustrated the use of the simple isotopic relations developed here for a few tetrahedral molecules and for octahedral SF6. Such relations can be of help to make initial predictions for various rare isotopologues of spherical top molecules,

(39)

mM ; M þ 4m

D ¼ εðl3  l4 Þ

(40) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z3 z4 jGj  jCj;

jGj ¼  2jCj ¼

2

;

(41)

(42)

mm

whence

 1=2 D ¼ εðl3  l4 ÞG

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 z3 z4 þ : 2

(43)

According the sum rules for spherical tops [28,40,44,45].

z3 þ z4 ¼ and

1 2

(44)

M. Lo€ete et al. / Journal of Molecular Structure 1206 (2020) 127729

8

  1 z3 z4 þ 1 ¼ ð1  z4 Þ þ z4 : 2

(45)

[9]

In addition, we have:

1 4

[8]

1 4

l3 z3 þ l4 z4 ¼ ðl3 þ l4 Þ  ð4z4  1Þðl3  l4 Þ:

(46)

[10]

[11]

Appendix 2. Relations in terms of Cartesian force constants [12]

From z3 and z4 (cf. Eq. (18)) and noticing that sin 2g is positive by definition, we deduce:

1 ð4z4  1Þ; 3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi  4 1 ð1  z4 Þ þ z4 : sin 2g ¼ 3 2

cos 2g ¼

[13]

[14]

(47) [15] [16]

Then, from cosg and sing from Eq. (20) we obtain the following relations:

k2  k1 ; k 2jk3 j ; sin 2g ¼ k

[17]

cos 2g ¼

(48)

[19]

with

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k ¼ ðk2  k1 Þ2 þ 4k23 :

[20]

(49) [21]

In addition,

mðl3  l4 Þ ¼ k:

[18]

(50)

[22]

So,

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   1 mmD ¼ εðl3  l4 Þ 2mmð1  z4 Þ þ z4 2 rffiffiffiffiffiffi 3 2m εjk j: ¼ 2 m 3

[23]

(51)

[24] [25]

[26]

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9

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