Linear Triatomic Molecules - CCH

328 Pages·2001·4.903 MB·English

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

Introduction IX
Introduction
Molecular parameters of linear triatomic molecules are being presented in several separate books.
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 species have been
listed by adopting the Hill system [ 1900Hill.
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 degrees of freedom.
The number of vibrational degrees of freedom gives the number of normal modes of vibration. In the
case of linear triatomic molecules (i.e. for N = 3) there are four vibrational degrees of freedom. Two of
these involve only the stretching of the bonds and the remaining two belong to a degenerate pair
associated with the bending of the molecule. The stretching fundamentals correspond to what are called
the 1~~a nd yj states and their associated vibrational quantum numbers are ~‘i and c3. The doubly
degenerate bending fundamental corresponds to the vZ state and its associated vibrational quantum
number is u2.
In Herzberg’s book on the Infrared and Raman spectra of polyatomic molecules [45Her] the
numbering for these vibrational 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- ‘).
Molecular species
In the Hill system Commonly used 1 o”o (I) 01’0 (II) oo”l (X)
chemical formulas Vl V2
V3
CHN HCN 2089 712 3312
cos ocs 859 527 2079
NNO N20 1285 589 2224
As can be seen, the doubly degenerate Ye 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 v, 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 vi and v3 (also equivalent to interchanging ui and u3) as compared to what is indicated in
the above table. This other scheme of 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 sub-volumes of this series, specific mention has been made
whenever ui and u3 are interchanged as compared to the vibrational numbering scheme of Herzberg’s
book [45Her].
The states of the doubly degenerate mode v2 are specified not only by the vibrational quantum
number u (u2 in this case) but also by I, the quantum number for vibrational angular momentum. For
a certam uZ, the quantum number 1 (1, in this case) assumes the values ZJ~,u 2 - 2, u2 - 4, . . . 1 or
0 depending on whether u is odd or even. Each vibrational state of a linear molecule is represented by
X Introduction
the designation (uluzl~J. In this scheme the vi, v2 and v3 states correspond to the vibrational levels
denoted by (1 O’O), (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 1s ubstates, each of which contains rotational levels,
labelled by the quantum number of the overall angular momentum J 2 1. These 1 substates are coupled
with each other by vibration-rotation interactions, called I-type resonance. For 1= 1, this resonance
causes the removal of the degeneracy of the levels. This splitting of the I= 1 levels is called I-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, especially when using computers, the capital letters E and F are being used instead of
e andffor labelling the I-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 andf.
II Effective Hamiltonians
Vibrational and rotational spectroscopic parameters are defined by effective Hamiltonians, the matrix
elements of which provide the energy expressions suitable for the analysis of experimental spectra. For
describing the degeneracy of the v2 state and I-type doubling effects on the spectra of linear triatomic
molecules, the basic theory was developed by Amat and Nielsen [58Amal, 58Ama2,71Ama] (see also
[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. Attempts 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 off-diagonal matrix
elements. The unperturbed vibration rotation energy E, to a good approximation, is given by:
E = .&, + E,,,, (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). They are followed by several
explanatory notes related to the symbols appearing in them. The matrix elements off diagonal in 1 are
summarized in Eqs. 11.7-11.9 accompanied by a few clarifying comments.
11.1.1 Diagonal elements
(11.2)
X Introduction
the designation (uluzl~J. In this scheme the vi, v2 and v3 states correspond to the vibrational levels
denoted by (1 O’O), (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 1s ubstates, each of which contains rotational levels,
labelled by the quantum number of the overall angular momentum J 2 1. These 1 substates are coupled
with each other by vibration-rotation interactions, called I-type resonance. For 1= 1, this resonance
causes the removal of the degeneracy of the levels. This splitting of the I= 1 levels is called I-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, especially when using computers, the capital letters E and F are being used instead of
e andffor labelling the I-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 andf.
II Effective Hamiltonians
Vibrational and rotational spectroscopic parameters are defined by effective Hamiltonians, the matrix
elements of which provide the energy expressions suitable for the analysis of experimental spectra. For
describing the degeneracy of the v2 state and I-type doubling effects on the spectra of linear triatomic
molecules, the basic theory was developed by Amat and Nielsen [58Amal, 58Ama2,71Ama] (see also
[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. Attempts 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 off-diagonal matrix
elements. The unperturbed vibration rotation energy E, to a good approximation, is given by:
E = .&, + E,,,, (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). They are followed by several
explanatory notes related to the symbols appearing in them. The matrix elements off diagonal in 1 are
summarized in Eqs. 11.7-11.9 accompanied by a few clarifying comments.
11.1.1 Diagonal elements
(11.2)
Introduction XI
Lt = B”CJV + 1) - /,“I - D”[J(J + 1) - li]” + H”[J(J + 1) - Ii]“, (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, the 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 ,,y and z,,.,..,,..a re anharmonicity constants.
(5) gz2, y,“‘, z2? and z2”’ are anharmonic parameters which describe the contribution of the
vibrationarangular momentum 1, 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. Fdr instance, in z22221t, the
power of 1, 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 gz2 should be written as x22. For
this parameter, investigators have also used xl1 [58Amal, 58Ama2] and more recently xL
[85Yam]. In other words, gz2, g 22 , x 22 , xll, 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) cr,, yss8a nd Y,,,- in the expression for B,; j?, and b,,, in the expression for D,; and E, in the expression
for H, are rovibration interaction parameters.
(9) y22 and yf2 in the expression for B, and 8” 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, - Cps(us + dJ2) + ‘...
(11) In Eq. II.6 H, has be& used instead of cs; i.e., write H, = H, + ~H,(u, + dJ2).
s
XII Introduction
11.1.2 Off-diagonal elements
(u,,l,,J~R~Iu,,I, 312, J) =$q2{(u2 T I,)@, r!zI , + 2) [J(J+ 1) -&(b * l>l
*[J(J + 1) - (1, fi l)(& I!I 3-J) 1’2, (11.7)
where
q2=q;+Cqzs +q;J(J+1)+q;(12+1)2+qJ,JJ2(J+1)2; (11.8)
(t;2,l2,J,~f,~2,~2~4,J)=~Plj(~2rL)(~2~12+2)(~2fI2-2)(~2+12+4)
. [J(J + 1).1,(1, _+ l)][J(J + 1) -(I, AI 1)(4 It 31
. [J(J + 1) - (I, + 2)(1, + 3)] [J(J + 1) - (12+ 3)U2 k 4)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 Iz and are responsible for rotational I-type
resonance. For 1, = 1 (occurring for odd v,), the element II.7 connects levels (v,, 1, = + 1, J) and
(u2, I, = - 1, J) which are degenerate, and thus causes splitting of these levels. For example, for
u2 = 1, the split levels are (~7~= 1, 1, = 1, J) + (1/2)q, J(J + 1). This removal of the degeneracy is
referred to as (rotational) I-type doubling. The upper sign here refers to 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 + ( - l)J and the f-levels have parity - ( - 1)“. 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 c2 = 1, the separation between the split levels is given by:
Av = q2 J(J + 1). (11.10)
(2) The term q:-‘J’(J + 1)2 in Eq. II.8 is, in principle, of higher order than the terms included in
Eqs. 11.2-11.5 but 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. Also, 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 [91 Makl]:
q,J = 4; - 4,yJJ(J + 1) + qrJJJ2(J + 112,
or (11.11)
= q,o- qfJ(J + 1) + qf-‘J2(J + 1)2.
1
q,J
(3) The matrix element II.9 is also of higher order than the other terms. It can cause splitting of I, = 2
levels (even u2).
(4) The following three expressions used for Av, the separation between the I-type doubling splittings
(Eqs. 11.12-11.14) are 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 = q J(J + 1) +qD[J(J + 1)12 + ..a. (II. 13)
In [93Mey]: Av =q, J(J + 1) - qD,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
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
Evih - L-fib = G(c,, L’~~‘L:~) - G(O,O’,O)
= 1 o,“v,+ c .Yps.V+s &VIsf . + c y:‘&v,+ + 1
Y,:;,vJ:
s 5 s’ s s S’I s”
S
(II. 15)
; 1 z,,.,..,...vsv,,vs~.v,... + 1 z~s~vsvs.i~ +z22221;.
s 5 s’ < s” 5 s”’ SSl ’
11.2.2 Rotational states
‘k - ‘% = (B,. - Bo + 2D,lf - 3H,l;)[J(J + 1) - I;] - (D, - Do + 3Hof;)[J(~ + 1) - l;]’
(11.16)
f (H, - H,)[J(J + 1) - l;]” - B,l; + D,I; - &1,6,
with
B,. - B, = - c u,ov, + 1 “~~.v,c,. + ;,22l; + c ~~~s~~~s~s~t.s~~ + 1 j’;;,t’sl& (11.17)
s 5 9’ s5 s’$ s”
S S
D,. - Do = c ,@v, + c ~~q~~. + pz21;, (11.18)
sI s’
S
(11.19)
H,. - Ho = 1 E,~c,.
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, for
instance, when a fundamental and its overtones have been studied. In this case, the summations
over the vibrational indices s extend over the subset of modes studied.
(2) The parameters of, x$, etc., are not the same as those appearing in Eq. II.2 (viz. LO,x, sS.e, tc.). They
can be converted into the latter when constants for all three vibrational modes have been
determined. This conversion can be achieved by using the relations given below (Eqs. 11.20-11.25)
which are correct to the 4th order. For more extensive data requiring higher order terms, the energy
expressions II.2 referred to the equilibrium state are normally used. Note that in the symbol y$, of
Eq. 11.17,t he subscript (0) has the same meaning as the superscript 0 appearing elsewhere.
y,,.,.. (11.20)
Q, = o$ - (x;% - $ d,)d, - ; 1 (.$ - ysss,ds)ds. + $ 2 ~ss,s..ds,ds..y
S’#S s’ $ s” # s
(11.21)
x,, = .u; - $ y,,,d, - + 1 .c;,,.d,.,
S’#S
(11.22)
xss. = -u,;~. - ( y,,,.d, + yss.,ds~, - + c yss.s,,ds., (s’ # S)
s” # s.s’
(11.23)
cx=, x,0+ ‘,‘,,d+, + c y,,.d,., (11.24)
S’fS
fl,= P ,-” P ss- di cs
(11.25)
&A-
S’#S
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-’ or MHz. Conversion from MHz to cm-’ is obtained by
dividing by 29 979.2458 or from cm-’ to MHz by multiplying by 29 979.2458.
J m-* Hz eV hartree
l/b4 l/V4 WI W%hc)
lJ&l
5.034 1125(30). 1oz4 1.509 18897(90).1033 6.241 5064(19).1018 2.2937104(14).10"
{Cl { WeJ 1/{2Rm~
lm-l pJ 1
1.9864475(12)~10-25 299792458 1.23984244(37).10-'j 4.5563352672(54).1O-8
WR,c~
1.5198298508(18)~10-‘6
W4
(e/2R,hc}
leVe 1
~:t!O217733(49)W'" iii%4 lO(24) 2.417988 36(72). 1014 0.036749309(11)
Wdl {W&c/e)
1
6.5796838999(78).10" 27.2113961(81)
1 hartree = 1 a.u. (atomic unit) = 2 Ry (Rydberg)
The symbol L has been used to mean “corresponds to” adopting the recommendations of the IUPAP. For example 1 Hz is not equal to
3.335640952. 10-9m-1 but only corresponds to it.
The above table should be read starting from the left column. In each line the conversion factors (which are multiplicative factors) are given in terms of
the fundamental constant(s) (upper entries) and also by their numerical values (lower entries), e.g.
l/(c) m-i
1HZA
3.335640952. 10-9m-i
01 1
[due to the physical relation v ; = - ,
a
where v is the frequency, il is the wavelength and c is the speed of light].
Another example is:
leV4 (e/2R,hc} hartree
0.036749309(11> hartree
Introduction xv
Selected fundamental constants *)
Quantity Symbol Value Units
SI cgs
Speed of light 2.997 924 58 (exactly) lo8 ms-’ 10”cms-’
C
Fine structure constant 7.297 353 08 (33) 10-3 10-3
CL
137.035989 5 (61)
a-l
Electron charge e 1.602 177 33 (49) lo-‘9 c 10e20 emu
4.8032068 (15) lo-” esu
Planck’s constant h 6.626 075 5 (40) 10-34J.s 1O-27 erg.s
A = h/2n 1.054 572 66 (63) 10-34Js 1O-27 erg.s
Avogadro’s number N 6.022 136 7 (36) 1O23 mol-’ 1O23 mol-’
Boltzmann constant kB 1.380658 (12) 1O-23 J K-i lo-l6 erg K-’
Universal gas constant R 8.314 510 (70) J mol-’ K-’ lo7 erg mol-’ K-’
Molar volume vrrl 22.414 10 (19) 10e3 m3 mol-’ lo3 cm3 mol-’
at T = 273.15 K and
p = 101325 Pa
Standard atmosphere atm 1.013 25 lOsPa lo6 dyn cmd2
Atomic mass unit ‘) m,=lu 1.6605402 (10) 1O-27 kg 1o-24 g
Electron rest mass me 9.109 389 7 (54) 1O-31 kg 1o-2a g
Proton rest mass mlJ 1.672 623 1 (10) 1O-27 kg lo-24g
Neutron rest mass 1.674928 6 (10) 1O-27 kg 1o-24 g
Rydberg constant 2m 1.097373 1534(13) lo7 m-’ 10’ cm-’
Bohr radius 5.291772 49 (24) lO-l’m lop9 cm
a0
Electron magnetic moment ~~/,u,,, 1.001159 652 193 (10)
in Bohr magnetons
Bohr magneton 9.2740154 (31) 1O-24 JT-’ 10e2’ erg Gauss-’
PB
Nuclear magneton 5.0507866 (17) 1O-27 JT-’ 1O-24 erg Gauss-’
PN
Electron magnetic moment pu, 9.284 770 1 (3 1) 1O-24 JT-’ 10m2’ erg Gauss-’
Proton magnetic moment pLp 1.410607 61 (47) 1O-26 JT-’ 1O-23 erg 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)= 1.6605402 (10).10-27kg).
III Formulas for determining rotational constants
III.1 Effective parameters
From the energy expressions given above, several polynomial relations have been derived and they are
presented in Eqs. 111.3-111.6. Experimental 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 q. In these
polynomials, the single prime (‘) refers to the upper energy state and the double prime (‘I) to the lower
energy state involved in a transition. For developing these formula’s: the selection rules applicable for
the rovibrational spectra are [45Her]:
for vibrational transitions:
Al =O, ) 1; X++C-; g+g; u+u, (111.1)
Introduction xv
Selected fundamental constants *)
Quantity Symbol Value Units
SI cgs
Speed of light 2.997 924 58 (exactly) lo8 ms-’ 10”cms-’
C
Fine structure constant 7.297 353 08 (33) 10-3 10-3
CL
137.035989 5 (61)
a-l
Electron charge e 1.602 177 33 (49) lo-‘9 c 10e20 emu
4.8032068 (15) lo-” esu
Planck’s constant h 6.626 075 5 (40) 10-34J.s 1O-27 erg.s
A = h/2n 1.054 572 66 (63) 10-34Js 1O-27 erg.s
Avogadro’s number N 6.022 136 7 (36) 1O23 mol-’ 1O23 mol-’
Boltzmann constant kB 1.380658 (12) 1O-23 J K-i lo-l6 erg K-’
Universal gas constant R 8.314 510 (70) J mol-’ K-’ lo7 erg mol-’ K-’
Molar volume vrrl 22.414 10 (19) 10e3 m3 mol-’ lo3 cm3 mol-’
at T = 273.15 K and
p = 101325 Pa
Standard atmosphere atm 1.013 25 lOsPa lo6 dyn cmd2
Atomic mass unit ‘) m,=lu 1.6605402 (10) 1O-27 kg 1o-24 g
Electron rest mass me 9.109 389 7 (54) 1O-31 kg 1o-2a g
Proton rest mass mlJ 1.672 623 1 (10) 1O-27 kg lo-24g
Neutron rest mass 1.674928 6 (10) 1O-27 kg 1o-24 g
Rydberg constant 2m 1.097373 1534(13) lo7 m-’ 10’ cm-’
Bohr radius 5.291772 49 (24) lO-l’m lop9 cm
a0
Electron magnetic moment ~~/,u,,, 1.001159 652 193 (10)
in Bohr magnetons
Bohr magneton 9.2740154 (31) 1O-24 JT-’ 10e2’ erg Gauss-’
PB
Nuclear magneton 5.0507866 (17) 1O-27 JT-’ 1O-24 erg Gauss-’
PN
Electron magnetic moment pu, 9.284 770 1 (3 1) 1O-24 JT-’ 10m2’ erg Gauss-’
Proton magnetic moment pLp 1.410607 61 (47) 1O-26 JT-’ 1O-23 erg 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)= 1.6605402 (10).10-27kg).
III Formulas for determining rotational constants
III.1 Effective parameters
From the energy expressions given above, several polynomial relations have been derived and they are
presented in Eqs. 111.3-111.6. Experimental 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 q. In these
polynomials, the single prime (‘) refers to the upper energy state and the double prime (‘I) to the lower
energy state involved in a transition. For developing these formula’s: the selection rules applicable for
the rovibrational spectra are [45Her]:
for vibrational transitions:
Al =O, ) 1; X++C-; g+g; u+u, (111.1)
XVI Introduction
for rotational transitions:
AJ=J’-J”=O,+l (J=O+t+J=O); -t++-; sttta; AJfO for I,=Ottl,=O. (111.2)
P, Q and R lines correspond to AJ = - 1, 0, and + 1, respectively. In degenerate vibrational states
where I # 0, the levels J = 0, 1,2, . . . , 1- 1 do not occur.
The formula used for C - C bands is given in Eq. 111.3:
v, = v. + (B’ + B”)m + [(B’ - B”) - (D’ - D”)] m2 - [2(D’ + D”) - (H’ + H”)]m3
- [(D’ - D”) - 3(H’ - H”)]m4 + 3(H’ + H”)m5 + (H’ - If”)m6. (111.3)
Here ve = G’(v) - G”(o); v, is 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. 111.4,I II.5 and III.6 give respectively the formulas for II -II, II - X and C - II bands, which
seem to be adequate for most analyses.
II - II bands:
V, = Vg + [(E + F) + +(q’ + 4”) + 2(0’ + D”)]m + [(B’ - B”) ) i(q’ - 4”) +(D’ - LY’)](m” - 1)
- 2(0’ + 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 C - II bands, P and R branch lines and Q branch lines are usually
analyzed separately. This is because P 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 - C bands, P, R branch lines:
Vm= - (B’ + D’) $ [B’ + B” + +q’ + 2D’]m + [(B’ - B”) + :q’ + (D’ + D”)]m’
V.
- 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’ - ,,,) - +q’ + 2D’]J(J + 1) -(D’ - D”)J’(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 ve. Recalling Eq. II.3 which states that
E,,, = B[J(J + 1) - I”] - D[J(J + 1) - 1212 + H[J(J + 1) - 1213 + ... (111.7)
the wavenumber v of a vibration rotation transition is given by:
v = v. + E;,, -E” rot* (111.8)
Making use of these relations, the rotational analysis of a band determines the band center vBc,
which is given by:
Bc = v. - By’2 + )yl”2 (plus centrifugal terms). (111.9)
V
Here ve would be the band origin. Let us consider an example to show the need to be careful in
comparing the ve values quoted by different investigators. In the case of HCN, the grating measure-