The parameters of the vibration–rotational hamiltonian of XY3 type molecules at the local mode limit

Spectrochimica Acta Part A 53 (1997) 845 – 853

The parameters of the vibration–rotational hamiltonian of XY3 type molecules at the local mode limit O.N. Ulenikov a,*, R.N. Tolchenov a, Qing-shi Zhu b b

a Laboratory of Molecular Spectroscopy, Physics Department, Tomsk State Uni6ersity, Tomsk 634050, Russia Department of Chemical Physics, Uni6ersity of Science and Technology of China, Hefei 230026, People’s Republic of China

Received 17 July 1996; accepted 12 November 1996

Abstract Present study derived the complete ‘a-relations’ by taking the bending mode into considerations as well as simple relation between the quartic centrifugal distortion constants for XY3 type molecule at the local mode limit. They agree well with the experimental results. © 1997 Elsevier Science S.A. Keywords: Local mode limit; Vibration–rotation interaction parameters; XY3 type molecule

1. Introduction It is very well known that the symmetry conditions and the details of the intramolecular potential function totally determine the spectroscopic properties of a molecule. In fact, if one writes the vibration–rotational Hamiltonian by using normal mode coordinates [1 – 3]: (hc) − 1H 1 1 = % vl (p 2l +q 2l ) + % mab (Ja −Ga )(Jb −Gb ) 2 l 2 ab '2 − % maa +Vanh(kl…m ; ql …qm ) 8 a



(1)

(all notations in (1) are usual and corresponded to Refs. [2,3], namely: vl, kl…m are the harmonic * Corresponding author. E-mail: [email protected]. su

frequencies and anharmonic normal potential constants; Ja and Ga are the operators of components of the total and vibrational angular momenta, respectively; mab are the elements of the matrix of inverse inertia moments), then any parameter of the kinetic part of this Hamiltonian can be expressed as a simple function of the equilibrium structural parameters and the transformation coefficients lNal, which can be determined from the Eckart conditions, orthogonality relations and conditions on the potential function ( 2V (ql(qm



= 0,

at l"m

(2)

q=0

(relations (2) correspond to the absence of cross terms in quadratic part of intramolecular potential function). It is interesting to note that, in some special cases, the symmetry of the molecule and the details of potential function lead to so large simplifi-

1386-1425/97/$17.00 © 1997 Elsevier Science S.A. All rights reserved. PII S 1 3 8 6 - 1 4 2 5 ( 9 6 ) 0 1 8 5 2 - 5

O.N. Uleniko6 et al. / Spectrochimica Acta Part A 53 (1997) 845–853

846

cation of the lNal coefficients and the relations between parameters, that it becomes possible to obtain simple relations between vl, kl…m, different spectroscopic parameters, and, as a consequence, to simplify the analysis of real spectra. As one of the important examples in Ref. [4] we have derived the simple relationship between the parameters of the vibration-rotational Hamiltonian of XY2 type molecules. The purpose of the present work is to derive a similar relationship for XY3 type molecules. The conditions used here for XY3 type molecules are: (i) My /Mx “0, (ii) ÚYXY“ p/2, and (iii) the first constants between bond stretch and bond angle deformation, fra and frb, are zero. The conditions (i) and (ii) were used by Halonen and Robiette [6], and they are so ‘strict’, that the XY3 type molecules under them are actually accidental spherical tops (Ref. [6], of course, is not the only one which was devoted to the study of local mode effects. In recent paper [7] Lukka and Halonen presented an exhaustive review on that subject. Therefore, we do not propose to carry out the same review in this paper). It should be mentioned, that in all previous works on local mode effects the bending vibrational modes were completely omitted, therefore they have only found the special features of stretching vibrational states. The present work and earlier one [4] try to expand their results to include the bending vibrational modes. As the first step, we also use the strict local mode conditions for the sake of simplicity. This paper is organized as follows. In Section 2 some formulas of vibration-rotational theory are given for the reader’s convenience, and relations between the transformation coefficient lNal for general XY3 molecule are presented, which are important for the further analyses. In Section 3, lNal coefficients are analyzed, as applied to the case of near local mode limit. Sections 4 and 5 derive relations between quartic centrifugal coefficients and new relations between aparameters of vibration – rotation Hamiltonian operator % a lm abqlqm Ja Jb. lmab

2. Transformation coefficients and parameters of the XY3 (C3V symmetry) molecule Hamiltonian In this section we discuss briefly some formulas for the parameters of the Hamiltonian (1), they will be needed in the subsequent derivation. In Eq. (1) operators Vanh, mab, and Ga can be expressed in general form as [1–3]: Vanh(Ql …Qm ) = % klmnqlqmqn + % klmnjqlqmqnqj + …, lmn

 

   

lmnj

dag 2pc +% B ea l 'vl

1/2

dgd ddb 2pc +% B eg B eb m 'vm

1/2

mab = 2 % gd

a lagql

 

(3)

−1

a mdbqm

−1

m eab + % m labql + % m lm abqlqm + …, abl

and

  

Ga = % z alm qlpm lm

(4)

ablm

vm vl

1/2

− qmpl

 n vl vm

1/2

,

(5)

where B ea =

' ; 4pcI eaa

(6)

and ql are dimensionless normal coordinates. In turn, the values a lab, z alm, and I eab of Eqs. (4)–(6) can be expressed as functions of the transformation coefficients lNal and of equilibrium nuclear positions r eNa in the following form [1–3]: 1/2 e a ab l = 2 % eagkebdk % m N r NglNdl, gdk

(7)

N

z alm = % eabglNbllNgm,

(8)

e2 I eab = 12 dab % % mN eagd (r e2 Ng + r Nd ).

(9)

bg

N gd

Here eabg and eabg denote totally antisymmetric tensor and its absolute value, respectively; transformation coefficients lNal and equilibrium nuclear positions r eNa can be obtained from the Eckart conditions

O.N. Uleniko6 et al. / Spectrochimica Acta Part A 53 (1997) 845–853

%m 1/2 N lNal =0,

(10)

l3yl = − l2yl = − 3l2xl = − 3l3xl = 3l1xl /2

N

% m

e

= l (1) l /2,

1/2 e N abg Nb Ngl

r

l

(11)

=0,

Nbg

from the orthogonality relations % lNallNam =dlm,

(12)

(13)

N

% mNr eNbr eNg =0,

at b "g,

(14)

N

from the conditions in Eq. (2), and from

  (V (ql



q=0

(18)

where l, m =1, 2. Eqs. (17) and (18) determine the parameters lNal (l=1, 2) as functions of one ambiguous parameter which should be obtained from the additional conditions of Eq. (2). (c) For the lNals (l=3, or 4, s= 1, 2): l2xl2 = − l3xl2 = 1/3(l1xl1 − l2xl1), l3xl1 = l2xl1,

Eqs. (13) and (14) define the orientation of the molecule-fixed coordinate system and Eq. (15) states the presence of stable nuclear configuration. XY3 pyramidal molecules have four vibrational modes, two of them (q1 and q2) are nondegenerate (A1 symmetry), and two (q3s and q4s, s= 1, 2) are doubly degenerate (E symmetry). Two modes (q1 and q3) can be considered as stretching motions, and two others (q2 and q4) are deformations. In case of an XY3 molecules, the solution of the general Eq. (10) – Eq. (14) gives the following results: (a) e 4x



M+ 3m (2) (2) l l l m = dlm M

l4xl2 = l1xl2 = 0, (15)

=0.

l1zl = l2zl = l3zl =l (2) l . (17)

(2) Four parameters l (1) l and l l in Eq. (17) should satisfy three conditions (1) l (1) l lm +3

% mNr eNa =0,

l4xl = l4yl = l1yl = 0;

l4zl = − 3 m/Ml (2) l ,

Na

from the equations

847

e 1x

e 2x

e 3x

r = 0,

r = − 2r = − 2r =re/ 3;

r e4y = 0,

r e1y =0,

r e3y = − r e2y =re/2; r e1z = r e2z =r e3z = −h

r e4z =h









where re =2re sin ae and h= re{1 − 43 sin2 ae}1/2 (b) For the lNal (l =1 or 2):

3l1yl2 = 4l2xl1 − l1xl1,

3l2yl1 = − 3l3yl1 = l1xl1 − l2xl1, l4yl1 = l1yl1 = 0,

3l2yl2 = 3l3yl2 = 2l1xl1 +l2xl1,

l4zl1 = l4zl2 = l1zl2 = 0, 2 3l2zl1 = 2 3l3zl1 = 2l2zl2 = − 2l3zl2 = − 3l1zl1 = 2h/re(l1xl1 + 2l2xl1).

(19)

Like in Eq. (17), four parameters l1xl1 and l2xl1 (l=3, 4) are not determined by Eq. (19), however, they should satisfy three orthogonality conditions l 21xl1 + 2l 22xl1 + l 24xl1 + 23 (l1xl1 − l2xl1)2 + 6l 22zl1 =1 (20)

3m , M + 3m

3m , M + 3m

l4xl1 = l4yl2 = − m/M(l1xl1 + 2l2xl1),

(16)

and % lNa31lNa41 = 0,

(21)

Na

therefore only one of these four parameters is not determined. It should be mentioned that transformational coefficients in the above form correspond to the following symmetry properties of the vibrational coordinates qls (l=3, 4; s=1, 2): ql1 = r cos f= (ql )E2 and ql2 = r sin f=(ql )E1 ,

O.N. Uleniko6 et al. / Spectrochimica Acta Part A 53 (1997) 845–853

848

which are transforming according to second and first rows, respectively, of irreducible representation E of C3v symmetry group. Substituting into Eqs. (7) – (9), one obtains the following nonzero values of the equilibrium inertia moments I eaa, Coriolis constants z alm, and of the parameters a ab l : (a) I exx = I eyy =





n

mMr 2e 3m 3 +2 sin2 ae −1 M +3m M

Now we estimate the values of two ambiguous lNal parameters of XY3 molecule by considering the quadratic part V(2) of the potential function, which has six force constants frr, frr%, fra, frb, fab, and faa in the following manner: V(2) = 12 frr (Dr 21 + Dr 22 + Dr 23)

,

I ezz = 4mr 2e sin2 ae;

3. Ambiguous parameters sin g and sin d of the XY3 molecule

+frr%(Dr1Dr2 + Dr1Dr3 + Dr2Dr3)

(22)

+frare[Dr1(Da12 + Da13)+ Dr2(Da12 +Da23)

(b) For l = 1, 2:

+ Dr3(Da13 + Da23)]

yy (1) (2) a xx l =a l = m(rel l −6hl l ), zz l

(1) e l

a = 2 mr l ;

+frbre[Dr1Da23 + Dr2Da13 + Dr3Da12] +fabr 2e[Da12Da13 + Da12Da23 + Da13Da23]

(23)

+ 12 faar 2e[Da 212 + Da 213 + Da 223].

and for l= 3, 4: yy xy yx xx a xx l1 = −a l1 = − a l2 = − a l2 a l

= −2 mrel2yl1, yz zy xz a = a zx l1 =a l2 =a l2 a l = −2 3mrel2zl1, xz l1

(24) (c) For l=1, 2 and m =3, 4: z ylm1 = −z xlm2 = − z ym1l =z xm2l zlm (2) =l (1) l 3l2zm1 +l l {l1ym2 +2l2ym2 1/2

−3(m/M) l4xm1};

Here Dri (i= 1, 2, 3) are the changes of the bond lengths 4-i (atom X is denoted as 4), and Daij are the changes of valence angles between lengths 4-i and 4-j. If one uses L-tensor formulas [5], which provide transformation from internal coordinates Dri and Da to normal coordinates ql, it is possible to obtain two more equations for determining the lNal transformation coefficients by means of Eq. (2): W12 = [( 2V/(q1(q2]q = 0 = 0

(25)

(28)

(29)

and

for l, m = 3, 4:

W3s4s = [( 2V/(q3s(q4s ]q = 0 = 0

z zl1m2 = − z zm2l1 = − z zl2m1 =z zm1l2 z (z) lm

(in this case Eq. (30) will be totally equivalent for s= 1 and s= 2). Generally left sides of these two equations are complicated functions of parameters lNal. However, if the molecule under consideration satisfies the conditions: (1) the mass of atom X is much larger then the mass of atom Y; (2) bond angle ÚXYZ= p/2 (in the strict local mode approximation it is assumed also that the only frr, frrr, frrrr, etc. parameters are presented in the potential function, Eq. (28); however, as our analysis shows, this last limitation is very strong, and is not necessary for the purpose of this paper), then, (2) (1) (2) one can show that the parameters l (1) 1 , l1 , l2 , l2 ,

=l4xl1l4xm1 −l1xl1l1xm1 + 2{l1xl1l2xm1 +l2xl1l1xm1}

(26)

z x3142 = −z x4231 = − z x4132 =z x3241 =z y3141 = − z y4131 = z y4232 = − z y3242 z x34 =

M 2 3h (l l −l l ). M +3m re 1x31 2x41 2x31 1x41

(27)

The Hamiltonian of axially symmetric XY3 molecule will be obtained if one substitutes Eqs. (22)–(27) into Eqs. (4) and (5).

(30)

O.N. Uleniko6 et al. / Spectrochimica Acta Part A 53 (1997) 845–853

and l1x31, l2x31, l1x41, l2x41, which satisfy Eqs. (18), (20) and (21), will have the following forms: (1)

3l (2) 2 = 9 l 1 =sin g, (2) l (1) 2 =  3l 1 =cos g,

2l1x31 =  2l2x41 =sin d, 2l2x31 = 9 2l1x41 =cos d.

(31)

Here sin g and sin d are two ambiguous parameters mentioned in Section 2. Any combination of signs in Eq. (31) satisfies Eqs. (18), (20) and (21). We in our further analysis used the upper signs. In this case, it can be shown, that general equations are simplified considerably, and can be transformed to the forms:

849

Firstly the expressions for lNal coefficients, Eqs. (17)–(21), can be simplified significantly. Such simplified expressions for these coefficients are presented in Table 1. In its turn, substituting data from Table 1 into Eqs. (23)–(27) leads to the following simple relations for the parameters a ab l and Coriolis constants z alm: 2 yy zz a xx 1 = a 1 = a 1 = 3 6Ie, 1 yz xx yy xy a xz 31 = a 32 = − 2a 31 = 2a 31 = 2a 32 = 3 6Ie, 1 zz 1 yy a xx 2 = a 2 = − 2 a 2 = − 3 3Ie, yy xy xz yz a xx 41 = − a 41 = − a 42 = 2a 41 = 2a 42

= − 13 6Ie;

(36)

and

( frr + 2frr% −2faa −4fab )( 2 cos g −sin g) ( 2 sin g+ cos g) + (2fra + frb )(4 sin 2g − 2 cos 2g) =0

(32)

( frr − frr% −2faa −2fab )( 2 sin d −4 cos d) ( 2 cos d+ 4 sin d) + (2fra − 2frb )(7 2 cos 2d − 8 sin 2d)= 0.

(33)

From these equations, parameters sin g and sin d can be determined.

4. The relationship between the parameters of the vibration – rotational Hamiltonian of XY3 (C3v) molecules at the strict and ‘expanded’ local mode approach If, in addition to the approximation used in the above section, one further assumes that only constant frr of the potential function (28) is not equal to zero (in this case v1 =v3 v), then Eqs. (32) and (33) have the following solutions

2 cos g= sin g

or 2 sin g = −cos g

(34)

and

2 sin d =4 cos d

or 2 cos d = − 4 sin d. (35)

These four pairs of values sin g and sin d, derived at the strict local mode limit, can be used to simplify the general relations of Section 2.

Table 1 Values of the lNal parameters in the strict local modea N

a

l

1 2 3 1 2 3 1 2 3

x x x y y y z z z

1 1 1 1 1 1 1 1 1

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

x x x y y y z z z x x x y y y z z z

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

a

s

1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2

lNal

N

a

l

2/3 − 2/6 − 2/6 0 −1/ 6 1/ 6 −1/3 −1/3 −1/3

1 2 3 1 2 3 1 2 3

x x x y y y z z z

2 2 2 2 2 2 2 2 2

2/3 1/6 1/6 0 1/2 3 −1/2 3 − 2/3 1/3 2 1/3 2 0 1/2 3 −1/2 3 0 1/2 1/2 0 1/ 6 −1/ 6

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

x x x y y y z z z x x x y y y z z z

4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

All parameters l4al are equal to zero.

s

lNal 1/3 −1/6 −1/6 0 −1/2 3 1/2 3

2/3

2/3

2/3

1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2

1/3 2 − 2/3 − 2/3 0 1/ 6 −1/ 6 1/3 −1/6 −1/6 0 1/ 6 −1/ 6 −1/ 2 0 0 0 −1/2 3 1/2 3

O.N. Uleniko6 et al. / Spectrochimica Acta Part A 53 (1997) 845–853

850

z y1,31 =z x1,32 =z y1,41 =z x1,42 =0,

values of u2 and u4 can be used to estimate the frequencies v2 and v4 in case of lack of information on bending. For example Eq. (41) compares the estimation and observation for the AsH3 molecule:

2z y2,31 = − 2z x2,32 =1, 2z y2,41 = −2z x2,42 = −1, z z31,32 =0,

2z z41,42 = −1,

2z z31,42 = 2z z41,32 = − 1,

2z x31,42 =2z x32,41 =2z y31,41 = −2z y32,42 = −1,

(37)

where Ie =I exx =I eyy =I ezz =2mr 2e.

(38)

It is interesting to note, that Eqs. (36)–(38) directly lead to the results of the strict local mode limit, Ref. [6]. However, when frr%, faa, or fab are not equal to zero, the solutions of the equations Eqs. (32) and (33) also give rise to same values of sin g and sin d (Eqs. (34) and (35)) and therefore Eqs. (36)–(38) are important in studying XY3 molecule at the ‘expanded’ local mode limit with bending modes included, i.e. frr% "0, faa "0 or/and fab " 0. In particular, substituting of Eq. (36) into the Eq. (4), and using the expression for centrifugal distortion coefficients: 1 1 tabgd = − % m labm lgdv − l , 2 l

(39)

one can derive some nontrivial and interesting relations for DJ, DJK and DK centrifugal parameters and parameter e, which describes Dk= 3 splitting of doubly generated vibrational states: DJ =

B 3e 2 −2 {9 + u − 2 +2u 4 }, 3v 2

DJK = DK =

n exp. = 906.75 cm − 1, 2

v predict. = 941.7 cm − 1, 4

n exp. = 999.22 cm − 1, 4 (41)

here the parameters u2 and u4 are obtained from experimental values of DJ, DJK, DK, and e splitting constants of the ground vibrational state [8]. One can see satisfactory accuracy of predictions in comparison with the observations of bands centers n2 and n4 from Ref. [9]. If one will assume u2 = u4 (this last condition is satisfied with good accuarcy for the most from XY3 (C3v) molecules), then following simple relations can be derived: 3 1 B 3e 3 B 3e DK = − DJK = DJ − 4 2 = e− 2 2. 2 7 v v

2

(42)

To illustrate the validity of these relations we used the experimental value of DK 104 =1.1166 cm − 1 constant of the AsH3 molecule from Ref. [8] for estimation of other three parameters: DJK, DJ, and e. Below the results of such extimation are presented. For comparison, experimental values of the same parameters from Ref. [8] are also presented: D predict. 104 = − 0.957 cm − 1, JK

2B 3e 2 {1 − u − 2 }, v2

4 −1 D exp. , JK 10 = − 1.240 cm

104 = 0.949 cm − 1, D predict. J

3 e 2

B 2 −2 { − 7 +9u − 2 −2u 4 }, 3v

2B 3e 2 e= {1 + u − 4 }. 3v 2

= 931.9 cm − 1, v predict. 2

D exp. 104 = 0.975 cm − 1, J e predict.105 = 3.36 cm − 1, (40)

where parameters u2 =v2/v and u4 = v4/v can be considered as semiempirical ones, i.e. on the one hand, they can be determined from the values of v2 and v4 harmonic frequencies; on the other hand, u2 and u4 can be determined also from experimental values of centrifugal distortion and Dk= 3 splitting parameters. In this last case the

e exp.105 = 2.10 cm − 1, One can see that, in spite of simplicity of the used model, predicted values of parameters are not far from corresponding experimental ones. Introduction of semiempirical parameters u2 and u4 gives possibility not only to derive centrifugal distortion constants and e parameters, but also to (a) improve the relations between vibra-

O.N. Uleniko6 et al. / Spectrochimica Acta Part A 53 (1997) 845–853

tion–rotation interaction parameters a ba , a xx 13 and a xz , q and r and (b) obtain interesting results for 13 3 3 zx spectroscopic parameters a b2 , a b4 , a xx , a , q and 24 24 4 r4 connected with the deformational vibrational states. In fact, substituting Eqs. (36) –(38) into the formulas (see, e.g. Refs. [6,10 – 12]) for vibration– xz rotation interaction parameters a bl , a xx 13 , a 13, q3 and r3, one obtains: (1) xz a z1 = a x1 = 2a xx 13 = −a 13

=−

 

4B 2e 2B 2e − v v

3/2

6K111

(43)

which consistent with the results of Ref. [6]. (2) a z3 = a z1 −

4B u , v 1 − u 24

a x3 = a z1 −

2B 2e u 24 u 22 + , v 1 −u 24 1 −u 22

2 e

2 4



q3 = − 12 a z1 +



4B 2e u 22 , v 1 − u 22

(44) (45) (46)

and r3 = −

1

a z1 +

2B 2e u 24 , v 1 − u 24

(47) 4 2 which differ from corresponding relations of Ref. [6] by the presence of corrections (second terms) in the right hand sides. It is important, since in some cases the corrections are not small in comparison with the first terms. In particularly, for the AsH3 molecules:

a z3 = 0.0326 ( − 0.0064) cm − 1); a x3 = 0.0326 ( − 0.0063) cm − 1); q3 = − 0.0163 (0.0062) cm − 1); r3 = − 0.0058 (0.0024) cm − 1), where the values in parenthesis are corrections. In the above estimations of main parts we used the value of a z1 =0.0326 cm − 1, which can be easily obtained from rotational parameters C of the (0000) [8] and the (1000) [13] vibrational states. Table 2 presents, for illustration, results of estimations of some spectroscopic constants by means of relations Eqs. (43) – (47) both with (column 3), and without (column 2) taking into

851

Table 2 Specroscopic parameters of stretch vibrational states of the AsH3 molecule (in cm−1) Parameter

Calculated

Calculated

Experimental

a z1 a x1 a z3 a x3 q3 r3 a xx 13 a zx 13

0.0326 0.0326 0.0326 −0.0163 −0.0058 0.0256 −0.0326

0.0326 0.0262 0.0263 −0.0101 −0.0034 0.0182 −0.0326

0.0326 0.0366 0.0163 0.0374 0.0066 0.0050 −0.0179 −0.0282

account semiemperical parameters u2 =0.4396 and u4 = 0.4442. The lasts were estimated according to Eq. (40) from experimental values of DJ, DJK, DK centrifugal parameters of the ground vibrational state [8]. Column 4 presents experimental values of corresponding parameters estimated from the data of Ref. [13]. It is seen that: (a) the predicted value of the parameter a x1 is close to the corresponding experimental one; (b) differences between calculated (column 3) and experimental values of parameters a z3 and a x3 are + 0.0098 cm − 1 and − 0.0111 cm − 1, respectively. At the same time, as can be seen from exact formulas for a bl parameters (see, e.g. Ref. [10]), terms omitted in expressions (44)–(45) are just proportional to + K233 for a z3 and − K233 for a x3 parameters, respectively. It is important, that this situation is totally equivalent to the corresponding situation with H2Se molecule [4]. (c) Off-diagonal parameters q3, r3 and a xx 13 in Table 2 have signs opposite to that of corresponding experimental values. However, the signs of off-diagonal zx parameters q3, r3, a xx 13 , and a 13 can be changed by corresponding choice of signs in Eq. (31). (3) a z2 = −

2B 2e − 1 u2 , v

a x2 = −

2B 2e − 1 1 1 1+ 3u 22 u2 − , v 4 2 1− u 22

(49)

a z4 = −

2B 2e − 1 1 1 1+ 3u 24 u4 − , 4 2 1− u 24 v

(50)

! !

(48)

" "

!

852

" "

O.N. Uleniko6 et al. / Spectrochimica Acta Part A 53 (1997) 845–853

2B 2e − 1 5 1 1 +3u 24 u4 − , v 8 4 1 − u 24 B 2e 1 +3u 24 1 u− 1+ , r4 = − 4 1 −u 24 2 2v B 2e − 1 q4 = − u4 , 2v 1 −1 B 2e u − 1 +2u2u4 + u 22 2 u4

2a xx 2+ 24 = v 1 − u 22 a x4 = −

!

!

"

1 + 2u2u4 +u 24 , 1 − u 24 1 −1 B 2e u − 1 +2u2u4 +u 22 2 u4 a zx 1+ 24 = − 2v 1 − u 22 +

+

! "

1 + 2u2u4 +u 24 . 1 −u 24

(51) (52) (53)

(z z41,42B ez )(iJz )(q4 − p4 + − q4 + p4 − )

(56)

where (ql 9 = ql1 9 ql2 and pl 9 = pl1 9pl2 for l= 3 or 4, and ql1}, ql2, pl1, pl2 are taken from Ref. [11], and Coriolis interaction between (61626364) and (6162 9 16364  1) states is determined by the operator (z y2,41B ex )(i − )(q2p4 − − q4 − p1)

(54)

− (iJ + )(q2p4 + − q4 + p1),

(57)

where J 9 = Jx  iJy. For the conditions of present Section, as is seen from Eqs. (37) and (38), (z z41,42B ez)= (z y2,41B ex)= − 1/2Be. (55)

It worth noting that Eqs. (48) – (55) were obtained under the assumption that (u2 − u4)2  1. In Table 3, column 2 presents the parameters calculated with Eqs. (48) – (55), and, for comparison, column 3 presents experimental values of corresponding parameters from Ref. [9]. One can see satisfactory agreement between calculated and experimental values (we should like to remind that the parameters, Eqs. (48) – (55) are obtaned without any initial information about deformational states). As in Table 2, differences in signs of calculated and experimental values of parameter a xx 24 can be understanded if one takes into account ambiguities in Eq. (31). (4) It also interesting to mention that the main part of (k −l) splitting in exited bending vibrational states generally is determined by the operator Table 3 Specroscopic parameters of deformational vibrational states of the AsH3 molecule (in cm−1) Parameter

Calculated

Experimental

a z2 a x2 a z4 a x4 q4 r4 a xx 24 a zx 24

−0.0302 0.0220 0.0221 −0.0039 −0.0075 0.0105 0.0628 −0.0369

−0.0276 0.0129 0.0181 −0.0145 −0.0061 0.0146 −0.0568 −0.0235

5. Conclusion As is shown in the above consideration, expanded local mode approach gives as simple as the strict local mode one, but considerably more adequate description of physcal effects and phenomenons in near local mode molecules. Numerical results for the AsH3 molecule illustrate validity of derived formulas. And all of this gives possibility to expect that the expanded local mode approach’s results will be useful in analysis of spectra of poorly studied near local mode molecules and free radicals.

Acknowledgements This work was supported by the Russian Foundation on Fundamental Research and the National Natural Science Foundation of China.

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