Linear Triatomic Molecules - SCS, SCS+, SCS++, SeCSe, CCN, CNC, CCN+, CNC+

309 Pages·2000·2.557 MB·English

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

X Introduction
the designation (uluzl~J. In this scheme the vi, v2 and v3 states correspond to the vibrational levels
denoted by (10’ U), (0 1’ U) 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
cl 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
Erot = B,CJV + 1) - /,“I - D”[J(J + 1) - l;]” + H”[J(J + 1) - Q”, (11.3)
where
+
(11.4)
CYS”
s
(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) x,~, Y~~,~., and z,,.,..,.. are anharmonicity constants.
(5) g22, yi2, 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 .z****l~, 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 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) as, yssr and Y,,,,~ in the expression for B,; p, 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, - Cfis(us + dJ2) + .s..
(11) In Eq. II.6 H, has be& used instead of Es; i.e., write H, = H, + C H,(u, + dJ2).
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-l Hz eV hartree
l/V4 l/v4 l/G9 W%hc)
lJ&l
5.034 112 5(30)- 10Z4 1.509 188 97(90)lo33 6.241 5064(19)10’8 2.2937104(14)lO”
{cl W/e) 1/{2Rm)
lm-l JW 1
- 1.9864475(12).10-*’ 299 792 458 1.239 84244(37)lO-‘j 4.5563352672(54)+10+
(h/e) b’WL4
1Hz-~;260755(40)10-34 %64O952.lO-9 ’ 4.1356692(12).10-15 1.5198298508(18)~10-‘6
w4 1 {e/2R,hc}
1eV-~!~0217733(49)W19 k:;:410,,4) 2.417988 36(72).10i4 0.036 749 309( 11) ,
CWZJc~ lW&le)
’ hartree-::;:c4,2(26)10-18 k!463.067(26) 1
6.579 683 899 9(78). 1015 27.211396 l(81)
1 hartree = 1 a.u. (atomic unit) = 2 Ry (Rydberg)
The symbol B 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
1HZS
3.335640952. 1O-9 m-l
[due to the physical relation v i = i ,
0
where v is the frequency, il is the wavelength and c is the speed of light].
Another example is:
1 Eve {e/2R,hc} hartree
0.036749309(11) hartree
Introduction xv
Selected fundamental constants *)
Quantity Symbol Value Units
SI w
Speed of light 2.997 924 58 (exactly) lo8 ms-’ 10”cms-’
C
Fine structure constant a 7.297 353 08 (33) 1O-3 1o-3
a-l 137.035989 5 (61)
Electron charge e 1.602 177 33 (49) lo-i9 c 10m20 emu
4.8032068 (15) lo-‘Oesu
Planck’s constant h 6.626 075 5 (40) 10-34J.s 1O-27 erg.s
A = hj2n 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 b 1.380658 (12) 1O-23 J K-’ lo-l6 erg K-’
Universal gas constant R 8.314 510 (70) J mol-’ K-’ 10’ erg mol-’ K-’
Molar volume vrrl 22.414 10 (19) 10e3 m3 mol-’ lo3 cm3 mol-’
at T=273.15Kand
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 3897 (54) 1O-31 kg 1o-28 g
Proton rest mass mlJ 1.672623 l(10) 1O-27 kg 10^.‘24g
Neutron rest mass % 1.674928 6 (10) 1O-27 kg 1o-24 g
Rydberg constant Rm 1.097373 1534(13) lo7 m-’ 10’ cm-’
Bohr radius 5.291772 49 (24) 1O-l’m 10e9 cm
a0
Electron magnetic moment p,/pa 1.001159 652 193 (10)
in Bohr magnetons
Bohr magneton h 9.2740154 (31) 1O-24 JT-” 10e2’ erg Gauss-’
Nuclear magneton 5.0507866 (17) 1O-27 JT-’ 1O-24 erg Gauss-’
PN
Electron magnetic moment pe 9.284 7701 (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 Formhas 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:
XVI Introduction
for rotational transitions:
AJ=J’-J”=O,+l (J=O+t+J=O); -t+-+--; sea; AJfO f o r I,=Ottl,=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, . . . , 1- 1 do not occur.
The formula used for C - C bands is given in Eq. 111.3:
v, = vg + (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’ - H”)m6. (III.3)
Here v0 = G’(v) - G”(u); 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, III.5 and III.6 give respectively the formulas for II -II, II - I; and X - II bands, which
seem to be adequate for most analyses.
II - II bands:
v, = v. + [(B’ + F) + +(q’ + 4”) + 2(D’ + D”)]m + [(I?’ - B”) ) i(q’ - 4”) +(D’ - D”)](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:
v, = v. - (B’ + D’) + [B’ + B” + +q’ + 20’1 m + [(B’ - B”) + :q’ + (0’ + D”)] m2
- 2(D’ + D”)m3 -(D’ - Dtr)m4. (111.5)
Q branch lines are fitted to the following polynomial expression:
II - C bands, Q branch lines:
v = v. -(B’ + D’) + [(B’ -I?“) - +q’ + 2D’]J(J + I) -(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 v,,. Recalling Eq. II.3 which states that
E,,, = B[J(J + 1) -l”] - D[J(J + 1) - 1212 + H[J(J -+- 1) - 1213 f ... (111.7)
the wavenumber v of a vibration rotation transition is given by:
v = v. + E;,, -E”m, (111.8)
Making use of these relations, the rotational analysis of a band determines the band center vBc,
which is given by:
vBc = v. - B’112 + B”1”2 (plus centrifugal terms). (111.9)
Here v0 would be the band origin. Let us consider an example to show the need to be careful in
comparing the v0 values quoted by different investigators. In the case of HCN, the grating measure-
Introduction XVII
ments of Yin et al. [72Yin] quote the following results (in units of cm-‘):
Band Band origin Rotational constants
I
ol’o-oo”o v. =713.459 B(Ol’0) = 1.481756
02°0-01’o v. =697.958 B(O2’0) = 1.485 80
02-20-01’0 v. =719.014 B(0220) = 1.484 9.5
Starting with these data and making use of Eq. III.9 we can evaluate vBc for the different transitions.
This is donein the following:
forO1’O-OOOO:
sc=713.459-1.481756(12)+B”(02)=711.977cm-’, (111.10)
V
for 02°0-01’O:
Bc = 697.958 - B’(02) + 1.481 756(12) = 699.440cm-‘, (III.1 1)
V
andfor0220-01’0:
Bc = 719.014 - 1.48495(22) + 1.481 756(12) = 714.556cm-‘. (111.12)
V
Now, Duxbury et al. [89Dux] gave the following results for HCN from data obtained by the high
resolution studies using Fourier spectroscopy (in cm-‘):
State 01’0 02Oo 0220
711.97985 1411.413 76 1426.53045
VO
These data of [89Dux] are the vBc values and they are rearranged below by taking appropriate
differences for the transitions indicated:
Ol’O-00’0 v,,=711.97985cm-’ (111.13)
02°0-Ol’0 v,,=699,43391cm-’ (111.14)
0220-Oc0 v,,=714.55060cm-‘. (111.15)
It is clear that i?ere is excellent agreement between the band centers from grating spectroscopy given in
Eqs. 111.10, III.11 and III.12 and the ones in Eqs. 111.13, III.14 and III.15 obtained by more sophis-
ticated Fourier spectroscopy studies made 17 years later. The Fourier experimental results quote more
significant digits reflecting the advances made in technology over the period.
III.3 Comments on BHO (HBO)
(see Chap. 6)
In this item, some corrections and clarifications are given for the parameters pertaining to the
molecular species “BH160 (H”B160), 11BD160 (D”B160), “BH”jO (H”B“jO), and l”BD160
(D’OB160), Chap. 6. They became available during some private communications with E. Hirota.
First, the errors indicated in (i) and (ii) below need to be corrected.
(i) In Table 7, the value of q2 should be changed to - 181.995 MHz.
(ii\ Thp Tahlp ln the-- va11w nf II- chnnd he rhanod tn ldd 120 MFTLJ
XVIII Introduction
Second, in Table 4, in Tables 9,10 and 11, in Tables 14,15 and 16, and in Tables 19 and 20, the values of
the I-type doubling constant q2 in MHz appear as 181.995,144.139,192.3879 and 147.879, respectively.
All ofthem have been deduced from pure rotational spectra measured in the microwave region. It is not
possible to determine the sign of these parameters from such measurements alone. The same parameter
has been quoted in Tables 7, 12, 17 and 21 as all negative values, viz. - 181.995, - 144.139, - 192.388
and - 147.879, respectively. This negative sign has been determined from theoretical considerations.
III.4 Some specifics related to carbonyl sulfide, COS (OCS)
(see Chap. 38)
111.4.1 Diagonalizing the energy matrix
The parameters of Table 14, Chap. 38, have been determined by diagonalizing the energy matrix
obtained from an effective Hamiltonian expressed by [87Yam]:
2 = .f$ + A, + 42 + 44, (111.16)
where R, is the diagonal part,
R, = G,> + xJ; + yJ; + {B, + d,,J;? + h&} (J’ - Jf) - {D, + hJLJf} (J’ - J;)”
+ H,(P - JZ,” + L”(P -J,‘)“, (111.17)
R, is the Al = 0 interaction term (vibrational I doubling) which vanishes for a linear triatomic molecule
like OCS,
A, is the Al = + 2 interaction term (rotational 1 doubling and 1 resonance),
A* = {L+ + (*,J- (4 + 4JJ2 + 4J4V- + L - -J+ (4 + 4JJ2 + 4J4V+ l/T (111.18)
and A, is the Al = + 4 interaction term,
J4_ + L z -(&,. (111.19)
~4=G:+(,,
The rotational step operators are defined as
J, = J, * iJy, (111.20)
and the vibrational step operators are
L*+ =(q2* -ip2+k2+ +h,Y49 (111.21)
with
q2* = q2x + iq2yy (111.22)
and
(111.23)
P2& = P2x i- iPzy,
where q2x and q2y are the normal coordinates of the bending vibration, and pzx and pay are their
conjugate momenta.
11.4.2 Effective molecular parameters
While treating large amounts of data which became available for the carbonyl sulfide molecule, it has
been found useful [86FayJ to determine effective molecular parameters by employing vibration
rotation energy expressions, E,, expanded in powers of J(J + 1). In the case of COS (OCS) the
symbolism appearing in Tables 25, 70 and 83, Chap. 38, has been defined in the following formula