Linear Triatomic Molecules - BClH+ (HBCl+) - COSe (OCSe)

532 Pages·1995·27.397 MB·English

by  G. Guelachvili, K. Narahari Rao (auth.), G. Guelachvili (eds.)

Contents
Subvolume Bl: Linear Triatomic Molecules
BCIH+ (HBCl+) .. . COSe (OCSe)
Introduction . . . . . _ . . . . . . . . . . . . . . . . . . _ . . . . . . . . . . . _ . . . . . . . IX
I Energy level designations ................................... IX
II Effective Hamiltonians .................................... X
II.1 Energy matrix ........................................ X
II.l.l Diagonal elements ...................................... X
11.1.2 Off-diagonal elements ................................... XII
II.2 Energy expressions referred to the ground state. ...................... XII
11.2.1 Vibrational states ...................................... XIII
11.2.2 Rotational states ...................................... XIII
II.3 Conversion table for energy-related units and selected fundamental constants ....... XIV
III Formulas for determining rotational constants ....................... XV
III. 1 Effective parameters ..................................... XV
III.2 Band center and band origin ................................ XVI
III.3 Comments on BHO (HBO) (see Chap. 6). ......................... XVII
III.4 Some specifics related to carbonyl sulfide, COS (OCS) (see Chap. 38) .......... XVIII
111.4.1 Diagonalizing the energy matrix ............................. XVIII
111.4.2 Effective molecular parameters .............................. XVIII
111.4.3 Unperturbed vibrational states ............................... XIX
111.4.4 Effects of perturbations ................................... XIX
III.5 Quadrupole coupling .................................... XXI
IV Potential energy function (PEF). .............................. XXI
IV.1 PEF expanded as a Taylor series .............................. XXI
IV.2 Curvilinear valence coordinates and Morse functions .................... XXII
IV.3 Dimensionless normal coordinates ............................. XXII
IV.4 Specific forms of the PEF. ................................. XXII
V Dipole moment ...................................... XXIV
v.l General equations. ..................................... XXIV
v.2 Specifics related to COS (OCS) (see Chap. 38) ...................... XXV
VI Intensities. ........................................ XXVII
VI. 1 Intensities of spectral lines ................................ XXVII
VI.2 Integrated absorption intensities ............................. XXVII
VI.3 Total internal partition sum ................................ XXIX
VI.4 F-factors (Herman-Wallis factors) ............................ XXIX
VI.5 Intensity expressions ................................... XXX
VI.6 Intensity units and conversion table ............................ XXX1
VI.7 Line profiles ..................................... XXXVIII
VI.7.1 Lorentz profile .................................... XXXVIII
VI.7.2 Doppler profile .................................... XXXVIII
v1.7.3 Voigt profile ..................................... XXXVIII
VI.7.4 Galatry profile ..................................... XXXIX
Contents
VI.8 Miscellaneous topics . . . . _ . . . . . . . .................
VI.8.1 Some definitions related to collisions . . . . .................
VI.8.2 Foreign gas broadening . . . . . . . . . . . .................
VI.8.3 Line coupling. . . . _ . . _ . . . . . . . . .................
VI.8.4 Temperature dependence of broadening . . .................
VI.9 Einstein coefficient of spontaneous emission
VI.10 Rotational state transfer. . _ . . . . . . . .
VII Renner-Teller effect (some aspects). . . . . . . . . . . . . . . . . . . . . . . . . . . XL11
VIII List of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XLIV
Data ........................... . . . . . . . . . . . . . . . . . . . . . . 1
1 BClH+ (HBCl+) ....... 1 21 CFN (FCN). . . . . . . . 59
2 BCIH+ (BClH+) ....... 9 22 CFN(FNC). . . . . . . . 76
3 BCIO (OBCl) ........ 10 23 CFP (FCP) . . . . . . . . 78
4 BFH+ (HBF+) ........ 13 24 CHN(HCN) . . . . . . . 79
5 BFO (FBO). ........ 16 25 CHN(HNC) . . . . . . . 183
6 BHO (HBO) ........ 17 26 CHO+ (HCO+) . . . . . . 198
7 BHS (HBS) ......... 26 27 CHO+ (HOC+) . . . . . . 202
8 BOz (OBO) ......... 34 28 CHP (HCP). . . . . . . . 206
9 BeFz (FBeF) ........ 38 29 CHS+ (HCS+) . . . . . . 212
10 CBaN (BaCN) ....... 39 30 CIN (ICN) . . . . . . . . 216
11 CBaN (BaNC) ....... 40 31 CLiN (LiCN) _ . . . . . . 222
12 CBeN (BeCN) ....... 40 32 CLiN (LiNC) . . . . . . . 224
13 CBeN (BeNC) ....... 40 33 CMgN (M&N) . . . . . 226
14 CBrN (BrCN) ....... 40 34 CMgN (MgNC) . . . . . 227
15 CBrN+ (BrCN+) ...... 43 35 CNO- (NCO-) . . . . . . 230
16 CCaN (CaCN) ....... 46 36 CNS (NCS). . . . . . . . 231
17 CCaN (CaNC) ....... 46 37 CNS- (NCS-) . . . . . . 233
18 CClN (ClCN) ........ 46 38 cos (OCS). . . . . . . . 238
19 CClN+ (ClCN+) ...... 51 39 COSe (OCSe) . . . . . . 416
20 CCIP (CICP) ........ 52
References ....................... . . . . . . . . . . . . . . . . . . 447
Ref. p. 4471 Introduction IX
Introduction
Molecular parameters of linear triatomic molecules are being presented in three separate volumes.
These parameters have been determined by studies made mostly in the infrared region of the
electromagnetic spectrum. In this region, the data are mainly concerned with the vibration rotation
energy changes of molecules occurring in their electronic ground states. Molecular speciesh ave been
listed by adopting the Hill system [OOHil].
I Energy level designations
A linear molecule with N atoms has two rotational and (3N - 5) vibrational degrees of freedom,
whereas a nonlinear (bent) molecule has three rotational and (3N - 6) vibrational degreeso f freedom.
The number of vibrational degreeso f freedom gives the number of normal modes of vibration. In the
caseo f linear triatomic molecules (i.e. for N = 3) there are four vibrational degreeso f freedom. Two of
these involve only the stretching of the bonds and the remaining two belong to a degenerate pair
associatedw ith the bending of the molecule. The stretching fundamentals correspond to what are called
the vr and vg states and their associated vibrational quantum numbers are o1 and vg. The doubly
degenerate bending fundamental corresponds to the v2 state and its associated vibrational quantum
number is v2.
In Herzberg’s book on the Infrared and Raman spectra of polyatomic molecules [45Her] the
numbering for thesev ibrational modes appears as indicated below in Table 1 by choosing a few specific
molecules as illustrations.
Table 1. Locations of some vibration rotation fundamentals (in cm-r).
Molecular species
In the Hill system Commonly used 1o ”o (2I) 01’0 (rI) oo” 1 (E)
chemical formulas Vl
v2 v3
CHN HCN 2089 712 3312
cos ocs 859 527 2079
NNO N,O 1285 589 2224
As can be seen,t he doubly degenerate v2 state has the lowest wavenumber. Among the remaining
two, the higher wavenumber mode is named the v3 state and the one at the lower wavenumber is named
the v1 state. In the course of the past many years, investigators have been using this scheme of
vibrational numbering. However, in recent years, papers have appeared making use of a notation that
interchanges v1 and v3 (also equivalent to interchanging u1 and v3) as compared to what is indicated in
the above table. This other schemeo f vibrational numbering corresponds to a recommendation made
by the IAU-IUPAP joint commission on spectroscopy [55Mul]. In the tables of parameters of linear
triatomic molecules presented in the three volumes of this series, specific mention has been made
whenever v1 and v3 are interchanged as compared to the vibrational numbering schemeo f Herzberg’s
book [45Her].
The states of the doubly degenerate mode v, are specified not only by the vibrational quantum
number v (v2 in this case)b ut also by 1,t he quantum number for vibrational angular momentum. For
a certain u2, the quantum number 1 (1, in this case) assumest he values u2, r2 - 2, r2 - 4, , . . 1 or
0 depending on whether v is odd or even. Each vibrational state of a linear molecule is represented by
Landolt-BGmstein
New Series 11/20bl
X Introduction [Ref. p. 447
the designation (uluZ1~J. In this scheme the vi, v2 and v3 states correspond to the vibrational levels
denoted by (10’ 0), (0 1’ 0) and (0 0’ l), respectively as indicated in Table 1. Levels with 1= 0, 1,2,3,. . .
are referred as C, II, A, 0, . . . states.
A vibrational state is composed of one or more I substates,e ach of which contains rotational levels,
labelled by the quantum number of the overall angular momentum J > 1.T hese 1s ubstatesa re coupled
with each other by vibration-rotation interactions, called I-type resonance. For I= 1, this resonance
causes the removal of the degeneracy of the levels. This splitting of the 1= 1 levels is called Z-type
doubling. The I-doublet components are labelled e and f adopting the recommendations of Brown and
coworkers [75Bro]; see the comments following Eq. II.9 for further information pertaining to this
notation.
Sometimes,e specially when using computers, the capital letters E and F are being used instead of
e and f for labelling the l-doublet components. Also, it may be recalled that prior to the recommenda-
tions in [75Bro], the letters c and d were used instead of e and f.
II Effective Hamiltonians
Vibrational and rotational spectroscopic parameters are defined by effective Hamiltonians, the matrix
elementso f which provide the energy expressions suitable for the analysis of experimental spectra. For
describing the degeneracy of the v2 state and l-type doubling effectso n the spectra of linear triatomic
molecules, the basic theory was developed by Amat and Nielsen [58Amal, 58Ama2,71Ama] (seea lso
[42Her]). Subsequently, this theory has been applied and extended by numerous investigators, Maki
and Lide [67Makl], Pliva [72Pli], Winnewisser and Winnewisser [72Win], Hietanen [83Hie],
Yamada, Birss and Aliev [85Yam], and Herman et al. [91Her]. Aspects of the notations used by all
these theorists have slight variations which are mainly a consequence of individual habits and
preferences.A ttempts have been made to clarify the various symbolisms used so far. Hopefully, this will
enable one to make meaningful comparisons of the molecular parameters generated by the experimen-
talists, who in turn showed preference to using one or the other of the Hamiltonian models.
II.1 Energy matrix
In the Hamiltonian of a linear triatomic molecule, there are diagonal as well as non-diagonal matrix
elements. The unperturbed vibration rotation energy E, to a good approximation, is given by:
E = Evib+ E,,t> (11.1)
the subscripts vib and rot referring to vibration and rotation, respectively. The diagonal matrix
elements lead to the following energy expressions (Eqs. 11.2-11.6).T hey are followed by several
explanatory notes related to the symbols appearing in them. The matrix elements off diagonal in 1a re
summarized in Eqs. 11.7-11.9a ccompanied by a few clarifying comments.
11.1.1 Diagonal elements
(11.2)
Land&-Bhstein
New Series 11/20bl
Ref. p. 4473 Introduction XI
E,,, = B,[J(J + 1) - I;] - D,[J(J + 1) - 1”,]’ + H,[J(J + 1) - l;]“, (11.3)
where
(11.4)
(11.5)
(11.6)
Explanatory notes for the above energy expressions
Some general comments:
(1) The energy expressions are given up to sixth order. The word ‘order’ refers to the highest power of
J occurring in E,,,. In this case,t he highest power for v in the expression for Evib is four, two less
than for J. Aspects of the notation of Amat and Nielsen [58Amal, 58Ama2], are retained here. For
instance, the vibrational indices s, s’, s” and s”’ are used for both non-degenerate and degenerate
modes. These vibrational indices assume the values 1,2 or 3.
(2) For the symbol d, representing the degeneracy, d, = 1, d, = 2 and d, = 1.
In Evib (Eq. 11.2):
(3) o, refers to the harmonic oscillator frequency for the sth normal mode.
(4) xss,y, ,,,., and z,,, 5s1il1la re anharmonicity constants.
(5) gZ2, y,““, zz: and z2222 are anharmonic parameters which describe the contribution of the
vibrational angular momentum I, to the vibrational energy. In this notation, the superscript 2 is
the same as the vibrational index 2 of the degenerate bending mode v2. The number of times this
superscript occurs is the same as the power of 1, which it multiplies. For instance, in z22221,4t,h e
power of I, viz. 4 is the number of times 2 is repeated in the superscript of z. In this type of notation,
strictly, according to Pliva [72Pli], the commonly used symbol g22 should be written as x22. For
this parameter, investigators have also used xII [58Amal, 58Ama2] and more recently xL
[85Yam]. In other words, g22,g 22, x22, xI1, and xL all refer to the same parameter.
(6) Yamada, Birss and Aliev [85Yam] write G, for the I-independent part of Evib.
In E,,, (Eqs. 11.3-11.6):
(7) B,, D, and H, are the rotational parameters, v indicating the particular vibrational state.B ,, D, and
H, give the values of the same parameters for the equilibrium structure of the molecule.
(8) a,, Y,,,a nd Y,,~,i~n ~t he expression for B,; fi, and j?,,,i n the expression for D,; and E, in the expression
for H, are rovibration interaction parameters.
(9) y22 and y”. m the expression for B, and b”” in the expression for D, describe the contributions of
the vibrational angular momentum to the rotational energy.
(10) In Eq. II.5 some investigators have been using a minus sign after D;
i.e., write D, = D, - C&(vs + d,/2) + . . . .
(11) In Eq. II.6 H, has be& used instead of .a,;i .e., write H, = H, + CH,(v, + d,/2).
s
Landolt-BBmstein
New Series 11/20bl
XII Introduction [Ref. p. 447
11.1.2 Off-diagonal elements
(lJ,,1,,J(A;(u,,z, 22,J) =$q2{(u* f l,)(v, IL 1, + 2)IIJ(J+ l).l,V, 2 111
’ [J(J + 1) - (1, f lN2 IL 211p2> (11.7)
where
+ q&J(J + 1) + q\(l, ) 1)2+ qy(J + 1)2; (11.8)
. [J(J + 1).1,(1,* 111C J(J + 1) - (12t - l)U, IL 91
. [J(J + 1) - (1, L- 2)(12I f: 3)l CJ(J + 1) - (1, -t 3)(1, f 4)1)1’2. (11.9)
Comments on off-diagonal matrix elements
(1) Matrix element II.7 connects components of vibrational states with a given v2 and different values
of the vibrational angular momentum quantum number I, and are responsible for rotational I-type
resonance. For 1, = 1 (occurring for odd v,), the element II.7 connects levels (02, 1, = + 1, J) and
(v,, 1, = - 1, J) which are degenerate, and thus causes splitting of these levels. For example, for
u2 = 1, the split levels are (~1~= 1, 1, = 1, J) f (1/2)q,J(J + 1). This removal of the degeneracy is
referred to as (rotational) l-type doubling. The upper sign here referst o the so-called e-levels and the
lower sign to the f-levels conforming to the recommendations mentioned earlier [75Bro]. According
to this recommendation, the e-levels have parity + (- 1)’ and the f-levels have parity - (- l)J. For
a linear triatomic molecule in the ground electronic state, this results in the labelling of the split
u2 = 1 levels given here. Obviously, for v2 = 1, the separation between the split levels is given by:
Av = q2J(J + 1). (11.10)
(2) The term qyJ*(J + 1)2 in Eq. II.8 is, in principle, of higher order than the terms included in
Eqs. 11.2-11.5b ut it has been included in the analysis of data by some investigators. Incidentally,
q;J is of the same order as E, appearing in Eq. 11.6.A lso, the J(J + 1) term has been used with
a negative sign. For instance, the dependence of q, on J has been written as follows [91Makl]:
q,;J = 4: - q”JJcJ + l) + qc.JJ52(5 + l)*,
or (II.1 1)
qDJ= qf - qiJ(J + 1) + qYJ2(J + 1)2. I
(3) The matrix element II.9 is also of higher order than the other terms. It can cause splitting of 1, = 2
levels (even u2).
(4) The following three expressions used for Av, the separation between the l-type doubling splittings
(Eqs. 11.12-11.14)a re being presented to draw attention to the slight variations in the symbolism
adopted for the same parameters.
In [85Jon]: Av = q,J(J + 1) - q1J2(J + 1)2. (11.12)
In [91Fru]: Av = qJ(J + 1) +qD[J(J + l)]” + .... (11.13)
In [93Mey]: Av = q,J(J + 1) - qo,J2(J $ 1)2+ qH,J3(J + 1)3. (11.14)
II.2 Energy expressions referred to the ground state
A majority of the spectra generated in the infrared are absorption spectra originating from the ground
vibrational state. The transition wavenumbers of interest would therefore be calculated as differences
Landolt-B&stein
New Series 11/20bl
Ref. p. 4471 Introduction XIII
between the energy levels of an excited state and those of the ground state. Therefore, it is useful to work
out energy expressions Evib and E,,, relative to the ground state. That is what has been done in
Eqs. 11.15-11.19.
11.2.1 Vibrational states
Evib- Etib = G(v,,v,“v,) - G(O,O’,O)
(11.15)
11.2.2 Rotational states
E,,, - EFo,= (B, - B, + 20,1,2 - 3H,ld;) [J(J + 1) - $1 - (D, - Do + 3H,$) [J(J + 1) - l;]”
+ (H” - H,)[J(J + 1) - lf]” - B,1; +0,1; - H,1,6, (11.16)
with
B, - B, = - c c~,ov+, c y,oss,v,v+, . ~~~1; + 1 &qvs~vs~~ + c y~~;~v,l;, (11.17)
s SSS’ s2 S5’ sfl s
D, - Do = c /$I, + c ~,oss.v,v+, . f1221;, (11.18)
s SSS’
H, - Ho = c (11.19)
E,V,.
s
Comments on energy expressions 11.15-11.19
(1) The superscript 0 is used to indicate that we are concerned with expressions relative to the ground
state. These formulas have to be used when only a subset of vibrational modes is investigated as,f or
instance, when a fundamental and its overtones have been studied. In this case,t he summations
over the vibrational indices s extend over the subset of modes studied.
(2) The parameters wz, x$ etc., are not the same as those appearing in Eq. II.2 (viz. wS,x ,,., etc.). They
can be converted into the latter when constants for all three vibrational modes have been
determined. This conversion can be achived by using the relations given below (Eqs. 11.20-11.25)
which are correct to the 4th order. For more extensive data requiring higher order terms, the energy
expressions II.2 referred to the equilibrium state are normally used. Note that in the symbol y$, of
Eq. 11.17,t he subscript (0) has the same meaning as the superscript 0 appearing elsewhere.
co, = co,”- (x,“, - 2 y,,,,.,d,)d, - 3 C (x,“,, - y,,,.d,)d,. + + c y,s<,..d,tds,,, (11.20)
S’fS S’<S”#S
x,, = x,“, - $ y,,,d, - : t: y,,,4~ (11.21)
s’+ s
X ss=’ x,Osa - ( y,,,,d, + y,,.,.d,J- 4 c y,,y d,-> (s’ Z 4 (11.22)
S”#S,S’
922 = s;z - i C y:24, (11.23)
(11.24)
(11.25)
Landolt-BBmstein
New Series 11/20bl
II.3 Conversion table for energy-related units and selected fundamental constants
In the Data part of this volume the units for energy-related quantities are mostly given in cm -I or MHz. Conversion from MHz to cm-’ is obtained by
dividing by 29 979.2458 or from cm-l to MHz by multiplying by 29 979.2458.
J m-l Hz eV hartree
W4 l/W Me> WKmW
lJ&l
5.034 112 5(30). 1oz4 1.509 188 97(90).1033 6.241 5064(19).10” 2.293 7104(14) 1or7
WJL)
Im-l Jhc) 1 {c> { WeI
- 1.9864475(12)~10-25 299 792 458 1.239 84244(37).W6 4.556 335 267 2(54). lo-’
PI4 WLc~
1Hz-~:260755(40)10-34 :‘!;;64095210-9 ’ 4.135 669 2(12). lo- l5 1.5198298508(18).10-‘6
WQ ~@JLhcl
1
’ eV G $02 177 33(49)lO-I9 k6+;)54 lO(24) 2.417 988 1Ol4 0.036 749 309( 11)
:%463.067(26) 6tw.5m7c9) 683 3869(97 92()7. 8). 1015 2(K7,.h2c1l1e3~9 6 l(81) 1
’ hartree G &$$2(26).10- l8
1 hartree = 1 a.u. (atomic unit) = 2 Ry (Rydberg)
Ref. p. 4471 Introduction xv
Selectedf undamental constants *)
Quantity Symbol Value Units
SI CD
-
Speed of light C 2.9979 24 58 (exactly) lO*ms-’ 10” cm s-l
Fine structure constant CI 7.2973 53 08 (33) 1o-3 10-3
Lx-l 137.0359895 (61)
Electron charge e 1.6021 77 33 (49) lo-‘9 c 10m20em u
4.803 206 8 (15) lo-‘Oesu
Planck’s constant h 6.6260 75 5 (40) 10-34J.s 1O-27e rg.s
k = h/2x 1.0545 72 66 (63) 10-34J.s 1O-27e rg.s
Avogadro’s number N 6.022 136 7 (36) 1O23m ol-r 1O23m ol-’
Boltzmann constant k, 1.380658 (12) 1O-23J K-r lo-l6 erg K-l
Universal gas constant R 8.314 510 (70) J mol-’ K-l 10’ erg mol-’ K-l
Molar volume L 22.414 10 (19) 10m3m 3 mol-i lo3 cm3 mol-’
at T=273.15K and
p = 101325 Pa
Standard atmosphere 1.0132 5 10’ Pa lo6 dyn cmm2
Atomic mass unit ‘) m,=lu 1.6605402 (10) 1O-27k g 1o-24g
Electron rest mass me 9.1093 89 7 (54) 1O-31k g 1o-28g
Proton rest mass 1.672623 l(l0) 1O-27k g 10-24g
mP
Neutron rest mass m, 1.674928 6 (10) 1O-27k g 1o-24g
Rydberg constant Rm 1.097373 1534 (13) lo7 m-l 105cm-l
Bohr radius 5.291772 49 (24) lo-“m 10eg cm
a0
Electron magnetic moment pe/pB 1.001159 652 193 (10)
in Bohr magnetons
Bohr magneton 9.2740154 (31) 1O-24J T-’ 1O-21e rg Gauss-’
PB
Nuclear magneton 5.0507866 (17) 1O-27J T-’ 1O-24e rg Gauss-’
PN
Electron magnetic moment ~1, 9.284 770 1 (31) 1O-24J T-l 1O-21e rg Gauss-’
Proton magnetic moment pLp 1.410607 61 (47) lO-=j JT-1 10Wze3r g Gauss-’
*) After E.R. Cohen and B.N. Taylor: CODATA Bulletin No. 63, 1986.
‘) The atomic mass unit is also called 1 amu (= (1/12)m(12C)= 1.6605 40 2 (10).10-27kg).
III Formulas for determining rotatiocal constants
III.1 Effective parameters
From the energy expressions given above, several polynomial relations have been derived and they are
presented in Eqs. 111.3-111.6E. xperimental data for the rovibrational lines are fitted to these poly-
nomials to determine values for the band origin (vo) and rotational constants, B, D, H and 4. In these
polynomials, the single prime (‘) refers to the upper energy state and the double prime (“) to the lower
energy state involved in a transition. For developing these formulas, the selection rules applicable for
the rovibrational spectra are [45Her]:
for vibrational transitions:
Al=O, k 1; C++C-; g+g; u+t+u, (111.1)
Landolt-Bhstein
New Series 11120bl
XVI Introduction [Ref. p. 447
for rotational transitions:
AJ=J’-J”=O,*l (J=O+J=O); +cf-; sea; AJ#O for &=O+-+l,=O. (111.2)
P, Q and R lines correspond to AJ = - 1, 0, and + 1, respectively. In degenerate vibrational states
where 1# 0, the levels J = 0, 1,2, . . . , I - 1 do not occur.
The formula used for C - C bands is given in Eq. 111.3:
v, = v. + (II’ + B”)m + [(B’ - H’) - (II’ - D”)] m2 - [2(D’ + D”) - (H’ + W)] m3
- [(D’ -II”) - 3(H’ - W)]m4 + 3(H’ + H”)m5 + (H’ - H”)m6. (111.3)
Here v0 = G’(v) - G”(v); vmi s the wavenumber of the P and R branch lines with m = - J for P-branch
lines and m = J + 1 for R-branch lines.
Eqs. 111.4I,I I.5 and III.6 give respectively the formulas for II -II, II - C and C - II bands, which
seemt o be adequate for most analyses.
II - H bands:
VW= V. + [(B’ + B”) * +(q’ + 4”) + 2(0’ + II”)] m + [(B’ - I?“) ) +(q’ - 4”) + (D’ - II”)] (m2- 1)
- 2(D’ + D”)m3 -(D’ - D”)]m4. (111.4)
In II -II bands, Q branches are allowed but they are weak.
In the case of II -C and X -II bands, P and R branch lines and Q branch lines are usually
analyzed separately. This is becauseP and R lines terminate on one component of the l-doublet levels
and the Q lines terminate on the other component. The polynomial expression used to fit the P and
R branch lines is:
II - X bands, P, R branch lines:
v, = v. - (B’ + D’) + [B’ + B” + +q’ + 20’1 m + [(II’ - F) + $4’ + (II’ + II”)] m2
- 2(0’ +D”)m3 - (D’ - D”)m4. (111.5)
Q branch lines are fitted to the following polynomial expression:
II - C bands, Q branch lines:
v= v. -(B’+D’)+ [(B’-IS”)-$q’+2D’]J(J+ l)-(D’-D”)J2(J+ 1)2. (111.6)
III.2 Band center and band origin
It may be noted that the terms band center, band origin and vibrational band origin have all been used
while identifying the symbol vo. Recalling Eq. II.3 which states that
E,,, = B[J(J + 1) - Z”] - D[J(J + 1) - Z2]’ + H[J(J + 1) - 1213+ ... (111.7)
the wavenumber v of a vibration rotation transition is given by:
v = v. + E;,, - E”to t’ (111.8)
Making use of these relations, the rotational analysis of a band determines the band center vBc,
which is given by:
\IBC= v. - gp + B,rp (plus centrifugal terms). (111.9)
Here v. would be the band origin. Let us consider an example to show the need to be careful in
comparing the v. values quoted by different investigators. In the case of HCN, the grating measure-
Land&B&stein
New Series II/ZObl