Linear Triatomic Molecules - OCO. Part b

404 Pages·1997·3.358 MB·English

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

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2
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Preface
With the advent of modern instruments and theories, a considerable amount of spectroscopic
information has been collected on molecules during this last decade. The infrared, in particular, has seen
extraordinary activity. Using Fourier transform interferometers and infrared lasers, accurate data have
been measured often with extreme sensitivity. These data have also been analyzed and accurate molecular
parameters determined.
Volume II/20 "Molecular Constants mostly from Infrared Spectroscopy" is a recent Landolt-Börnstein
publication series. It is made up of several subvolumes (A, B, C, D) with comprehensive compilation of
critically evaluated molecular constants of diatomic (A), linear triatomic (B), other triatomic (C) and other
polyatomic (D) molecules. The first subvolume, II/20B1, published in 1995, deals with 39 linear triatomic
molecules and their isotopic species, from BCIH+ (HBCI+) to COSe (OCSe), given in the alphabetical
order of their Hill's formulas.
Subvolume II/20B2 is devoted to the carbon dioxide molecule CO , which has been the subject of
2
extensive studies both from theoretical and experimental points of view. Due to the tremendous amount of
information generated in these studies, subvolume B2 has been split into two parts, a and b . The
subvolume II/20B2a has dealt exclusively with the normal isotopic species of carbon dioxide 12C16O16O
(16O12C16O). The present subvolume, II/20B2b deals with thirteen isotopic varieties of this linear
triatomic molecule.
The introduction essentially reports molecular theories and equations, based on which most of the
evaluated data are established. Specific comments related to some of the molecules considered in the
Chaps. 1...39 of subvolume II/20B1 are retained in the present introduction due to their possible general
applicability. Additional information of practical interest (list of symbols with their definitions, units, table
of conversion factors, notations for the bands and energy levels, table of energy-related units and selected
fundamental constants, ...) are also given. The tables are preceded by an additional index to help the
search for specific information. In order to keep their consistency and their optimum ability to reproduce
data, molecular constants are reported when possible from the same calculation of a given set of
measurements. The subvolume ends with a reference section.
We gratefully acknowledge Dr. N. Lacome for her incisive comments on the "Line mixing theories".
The editorial staff of Landolt-Börnstein, particularly Mrs. H. Hämmer and Dr. H. Seemüller, played
an essential role in the realization of this volume. Finally, all of us appreciate the untiring efforts of
Dr. D.S. Parmar in aspects of this undertaking.
Orsay, April 1997 The Editor
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2
16O12C16O (cid:20)(cid:3)(cid:21)(cid:4)(cid:8)a
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CS (SCS) ... N (NNN) (cid:14)(cid:12)(cid:15)(cid:16)(cid:5)(cid:10)(cid:12)(cid:6)(cid:11)(cid:8)(cid:18)(cid:22)
2 3
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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 [OOHil].
I Energy level designations
A linear molecule with N atomd 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 v, and v3 states and their associated vibrational quantum numbers are u, and v3. The doubly
degenerate bending fundamental corresponds to the v2 state and its associated vibrational quantum
number is oz.
In Herzberg’s book on the Infrared and Raman spectra of polyatomic molecules [4.5J-Ier] t&
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 (C) 01’0 (I-I) 0001 (C)
chemical formulas VI
v2 v3
CHN HCN 2089 712 3312
cos ocs 859 527 2079
NNO N20 1285 589 2224
t;
As can be seen, the 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 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 three volumes of this series, specific mention has been made
whenever u1 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 1, the quantum number for vibrational angular momentum. For
a certain u2, the quantum number 1 (I2 in this case) assumes the values u2, u2 - 2, u2 - 4, . . . 1 or
0 depending on whether u is odd or even. Each vibrational state of a linear molecule is represented by
Landolt-B&m&in
New Series I1120B2a
X Introduction
the designation (uluzl~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 1 substates, 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 = .&, + Et,,, (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
Land&-Bhufein
New Series IIl20B2a
Introduction XI
-%t = u-J(J -I 1) - lfl- D”[J(J + 1) - l:]” + 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):
o, refers to the harmonic oscillator frequency for the sth normal mode.
(3)
xss’, y,,,. and z,,.,..,.. are anharmonicity constants.
(4)
(5) 9227 Y2s2, z,‘,t and z2222 are anharmonic parameters which describe the contribution of the
vibrational angular 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 I, which it multiplies. For instance, in z22221& 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 g22 should be written as x2’. For
this parameter, investigators have also used xii [58Amal, 58Ama2] and more recently xL
[85Yam]. In other words, gz2, g22 , x2 2, xil, 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,, yssr and Y,,~,~ in the expression for B,; & and fl,,, 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(vs + dJ2) + ....
(11) In Eq. II.6 H, has be& used instead of Es; i.e., write H, = H, + C H,(v, + d,/2).
s
Landolt-Bbmstein
New Series III2OB2a
XII Introduction
11.1.2 Off-diagonal elements
(u,,I,,JIA:Iu,,I,~2,J)=~q,{(u,fI,)(u,+I,+~~CJ(J+ l).bu,+ 111
.[J(J + 1)- (1,I !z1 )O;I tI .Ny, (11.7)
where
+ qJ,J(J + 1) + q\(& * II2 + q;JJ2(J + 112; (11.8)
. [J(J + 1).l,(Z, & l)][J(J + 1) -(I, -t 1)U2 t- 211
. [J(J + 1) - (I, * 2)(1, f 311 CJ(J + 1) - (1, + 3N2 + 4)lI”“. (11.9)
Comments on off-diagonal matrix elements
(1) Matrix element II.7 connects components of vibrational states with a given u2 and different values
of the vibrational angular momentum quantum number i2 and are responsible for rotational l-type
resonance. For I, = 1 (occurring for odd u,), the element 11.7 connects levels (u,, I, = + 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 (u2 = 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
v2 = 1 levels given here. Obviously, for v2 = 1, the separation between the split levels is given by:
Av = q25(J + 1). (11.10)
(2) The term q<‘J2(J + 1)’ 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 qv on J has been written as follows [91Makl]:
%.I = 4: - 4,JV + 1) + 4,JJ.w + 112>
or (II.1 1)
qo5 = q; - q$J(J + 1) + qfv(J + 1)2. i
(3) The matrix element II.9 is also of higher order than the other terms. It can cause splitting of 1, = 2
levels (even v2).
(4) The following three expressions used for Av, the separation between the I-type doubling splittings
(Eqs. 11.12-11.14) are being presented to draw attention to the slight variations in the symbolism
adopted for the same parameters.
In [85Jon]: Av = q,J(J + 1) - q1J2(J + 1)2. (11.12)
In [91Fru]: Av = qJ(J + 1) +qD[J(J + l)]’ + .... (11.13)
In [93Mey]: Av =q,J(J + 1) - qD,J2(J + 1)2 +qH,J3(J + 1)3. (11.14)
II.2 Energy expressions referred to the ground state
A majority of the spectra generated in the infrared are absorption spectra originating from the ground
vibrational state. The transition wavenumbers of interest would therefore be calculated as differences
Landolt-Bdrnstein
New Series 11/20B2a
Introduction XIII
between the energy levels of an excited state and thpse of the ground state: Therefore, it is useful to work
out energy expressions Evib and E,,, relative to the ground state. That is what has been done in
Eqs. 11.1.5-11.19.
11.2.1 Vibrational states
Evi, - E$, = G(v,, u2”v3) - G(O,O’,O)
11.2.2 Rotational states
L - e,, = (4: - &I + 2D& - 3H,lxw + 1) - 1:-j -(D, - Do + 3H,lf)[J(J + 1) - l;]”
+ (H, - H,)[J(J + 1) - I;]” - B,l; + D,1,4 - H,1,6, (11.16)
with
B, - B, = - 1 a,Ov, + c y,o,.v,c, + ~~~1; + c y~s.s~.v,v,~v,+~ ~1 y~~&v& (11.17)
s SSS’ ss 5’5 s” s
D,. - Do = c /!?,“v,+ c f$v,v,~ + /3221;, (11.18)
s SSS’
H, - Ho = c E,v,. (11.19)
s
Comments on energy expressions 11.15-11.19
(1) The superscript 0 is used to indicate that we are concerned with expressions relative to the ground
state. These formulas have to be used when only a subset of vibrational modes is investigated as, for
instance, when a fundamental and its overtones have been studied. In this case, the summations
over the vibrational indices s extend over the subset of modes studied.
(2) The parameters wf, x$, etc., are not the same as those appearing in Eq. II.2 (viz. o,, xSs’,e tc.). 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
exprt$$ons II.2 referred to the equilibrium state are normally used. Note that in the symbol y$, of
Eq. 11.17, the subscript (0) has the same meaning as the superscript 0 appearing elsewhere.
w, = co,0- (x,“, - ; y,,.sd,)ds - + C (xfs, - y,,,.d,)d,. + d 1 y,,.,.d,.d,., (11.20)
s’+s S’SS”#S
x,, = x,“, - P y,,,ds - : c y,,,4~ (11.21)
s‘f.9
x,,. = x,9,.- ( y,,,4 + y,,y4 - t c y,.,4,, 6’ + 4 (11.22)
S”#Ss ’
g22 = s;~-f~y?24, ’ (11.23)
s
(Y,= ~1: + y,$s + : 1 y,,d,., (11.24)
s’z s
B, = P: - B,,d, - t 1 Bss4. (11.25)
S’#S
Landolt-BBmstein
New Series 11120B2a
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-l is obtained by
dividing by 29 979.2458 or from cm-’ to MHz by multiplying by 29 979.2458.
J m-l Hz eV hartree
W4 WI l/G9 WWcl
lJ&l
5.034 112 5(30).10Z4 1.509 188 97(90). 1o33 6.241 5064(19).10’8 2.2937104(14).10”
{cl {We) W%)
lm-l ,CW 1
- 1.9864475(12).10-*’ 299 792 458 1.239 84244(37)10-‘j 4.556 335 267 2(54). 1O -8
C&l WRmc)
1Hz-~~260755(40)-10-34 ~k~64095210-9 ’ 4.1356692(12).10-15 1.5198298508(18).1 o-16
Celh) {e/2R,hc}
1
1ev-i!~0217733(49)10-‘9 :&I,,,,,, 2.417988 36(72).1014 0.036 749 309( 11)
{2RWCl iW&le)
’ hartree-!~;?32(26)10-18 :;&63.067(26) 1
6.579 683 899 9(78). 10” 27.211 396 l(81)
1 hartree = 1 a.u. (atomic unit) = 2 Ry (Rydberg)
The symbol 4 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-l
1HZ+
3.335640952. 1O-9 m-i
[due to the physical relation v f = i ,
0
where v is the frequency, a is the wavelength and c is the speed of light].
Another example is:
1 evs {e/2R,hc} hartree
0.036749309(11) hartree