Proc. Indian Acad. Sci. (Chem. Sci.), Vol. 109, No. 4, August 1997, pp. 257-265.
© Printed in India.
Effect of kinetic energy terms on the vibrational frequencies (VI, V3)
in the ~B2 (~A') excited state of SO2
C I N D U a a n d Ch V S R A M A C H A N D R A R A O *b
aFountain House, 505 Ventura Lane, Whitewater, WI 53190, USA
bDepartment of Mathematics and Computer Science, University of Wisconsin, Whitewater, WI 53190, USA
e-mail: RAOC@UWWVAX.UWW.EDU MS received 2 April 1997
Abstract. The effect of various kinetic energy terms on the solution of the two- dimensional Schr&linger equation; involving the two large-amplitude stretching modes v 1 and v 3 of S02 in its 1B 2 excited state, is discussed. Employing two large amplitude Hamiltonian H ° (P l, P3 ) obtained earlier, three sets of force constants were obtained. In obtaining set 1, all the 6 codticients of the kinetic energy A, H, B, G, F, and C were taken into account and varied with (pt, P3)- For set 2, only the three coefficients A, H, and B, evaluated at the absolute minima (p] + 6, ___ p~) were considered. In obtaining the set 3 constants, only the two coefficients A and B evaluated at the saddle point (p], 0) were retainecL The nine force constants of the potential V0(pt, P3) which includes a double minimum function in P3, were obtained in each case by a least squares fit to the 12 vibrational frequencies corresponding to the levels (vl, %) = (0,2), (1,0), (1,2), (2,0), (0,6) and (3,0) of $1602 and StSO2. It is found that set 2 is superior to set 3, and sets 1 and 2 fit the frequencies essentially to the same degree of accuracy.
Keywords. Two-dimensional Schrrdinger equation; excited states of SO2;
vibrational frequencies.
1. Introduction
It is reasonably well established (Brand a n d R a o 1976; H o y and Brand 1978; Mezey and R a o 1980) that the 1B 2 state of the SO 2 molecule has an unsymmetrical equilibrium structure corresponding to a double m i n i m u m potential in the antisymmetrical strength coordinate P3 (see figure 1). A two-dimensional model for the H a m i l t o n i a n with a potential
V(q 1, q3)
was developed ( H o y and Brand 1978) and used successfully to explain a group of observed quantities including vibrational term values of $1602 and S 1 so2" Subsequently, the vibrational frequencies of S 1602 and S ~ 8 0 2 were calculated (Mezey and R a o 1980) b y solving a m o r e rigorous two-large amplitude equation obtained earlier (Brand and Rao 1976). This two-large amplitude H a m i l t o n i a n H ° ( P 1, P 3 ) contains a potential V o (p 1, P 3 ) which no t only has fewer parameters than the observed vibrational term values but is m o r e nearly isotopic invariant than any other model potential considered before.In the two large-amplitude formalism (Brand and Rao 1976), the kinetic energy expression has six terms with coefficients A, H, B, G, F, and C which are functions of Pl and P3- T h e potential energy
V°(pl, P3)
is taken to contain nine constants.* For correspondence
257
(/ = 103.75 r;= 1.6376
A
r"F i g a r e 1. U n s y m m e t r i c a l e q u i l i b r i u m structure o f SO z i n its IB2(ZA') excited state. 2pz = r z -{- r2, 2p3 = r 1 - r 2 a n d =o is fixed at 103.75 °.
The corresponding Schr6dinger equation is an elliptic eigenvalue partial differential equation in two variables px and P3 satisfying the constraint
H 2 - AB < O.
In the present state of the art, all the three vibrations of a nonlinear triatomic molecule are dealt with rigorously and with high accuracy using a variety of techniques (Choi and Light 1992; Jensen and Kozin 1993; Polyansky et al 1993). In particular, Handy and coworkers (Handy 1987; Bramley et al 1991; Whitehead and Handy 1995) have treated the three vibrations of a triatomic molecule rigorously with an exact expression for the kinetic energy operator. Such an exact treatment becomes relevant if the complete potential for the 3 modes is deduced from experiment or is constructed from ab initio computation of electronic energy.
In the present communication, we consider only (v 1, v3) frequencies of the excited state as data are available only for these frequencies. We will study the effect of various terms of the kinetic energy on the vibrational frequencies (v x , v 3) of SO 2 . We obtained three sets of force fields (set 1, set 2 and set 3) each fitted to 12 frequencies ofS~60 2 and S ~ s o 2. In obtaining set 1, all the six terms of the kinetic energy were retained and varied with (Pt, P3) and the nine potential constants were refined to fit the 12 frequencies. For set 2, only terms with coefficients A, H, and B of the kinetic energy, evaluated at the absolute minima, were retained and the force field refined. Set 3 is the more traditional one (Carreira et a11972; Hoy and Brand 1978; Wilson et a11980) and its kinetic energy has terms with coefficients A and B evaluated at the saddle point.
2. The Hamiltonian
The two large-amplitude zeroth order Hamiltonian describing the two stretching motions Pl and P3 of any bent triatomic molecule with masses ml, m2 and m 3 is given by (Brand and Rao 1976; Mezey and Rao 1980),
-- -- h2 / 0 02 910 02
~ I 0 0/92H°(Pl, P 3 ) - 2AO----~ -133 63p2
"=13Op 10p 3
110p32
+ 2G ~--T + 2F ~-~3 + C ) + Vo(p,, p3 ). (1) Here pt and P3 describe the two stretching motions given by
2 p l = r l + r 2, 2 P 3 = r 1 - r 2 (2)
where r~ and r 2 are the instantaneous values of the bond lengths of the reference
configuration (Brand and Rao 1976, figure 1). In what follows, we shall omit, for
Two-dimensional vibrational problem at SO 2 259 brevity, the superscript 0 for the quantities in (1) denoting zeroth order. The coefficients A = 111, H = - 113, B = 133, G, F and C in (1) are functions of pl and P3. They are given by (35) (Brand and Rao 1976). The value of A13 is given by
A13 = I11/33 --
(113) 2
= 4m lm 2 maiM. (3)
The units of H, arc ergs when Ploand Ps are expressed in/~, 111,113 and Is3 in ainu, A13 in (ainu) 2, G and F in (ergs. A), C and V 0 in ergs. H J h c is in cm -1.
The potential function Vo(pl, P3) is given by
1 1 1
Vo(pl, r.(pl + rxlx(p, + - rl.l(p,
1 2 - -k,pl 1 4 1
1 e 2 2
-I-
.~Kl,=33(Pl -Pl) P3. (4)
Here V o is expanded at the saddle point S(p I - - P l , P3 = ), and the absolute minima
0M 1 M 2 are assumed to occur at (Pl = P~ + 6, P3 = --- P[) where 6 is a small positive number. The constants p~ and p~ are equilibrium values. The method of finding 6 as well as the extremal points (critical points) for the potential Vo(pl, P3) is explained in appendix A of Mezey and Rao (1980). The quadratic and Gaussian terms in P3 of(4) give a double minimum potential in P3. Even though the complete vibrational problem of a triatomic molecule requires the inclusion of bending angle also, it is not taken into account here as there are no observed values corresponding to levels (v I v2v3). The treatment of the vibrational problem given here is inadequate to that extent.
For computational convenience, the Hamiltonian (1) is transformed into dimension- less coordinates ql and q3 given by
ql = (~llt 1)1/2(Pl - P~),
q3 = 0~3133)1/2p3' (5)
where
Yi = 47t2CtOi/h (i = 1, 3). (6)
The values 71 (erg s-1 s - 2 ) are the scaling factors, and 111 and Iaa are the "reduced masses", both being functions of (P:, Pa). Harmonic frequencies o~ 1 and co a of (6) are obtained from the harmonic force constants K l l and K33 using the formula
tO 2 = Kii/4~z2c21u (i = 1, 3). (7)
When the coefficients of the kinetic energy terms are evaluated at a fixed point (saddle point or absolute minimum), coefficients G, F and C, being essentially partial deriva- tives of constants I,B(at, fl = x, y, z, 1, 3) (Brand and Rao 1976), vanish.
Thus the Hamiltonian, in terms of ql and qa and the conjugate momenta
0 0
~Oq3 ~ ' -
Pl = - iT"Y-_ and P3 = -
t/q1 becomes
1 2
Hs(qx,q3) = -~(aPx + 2hPtP3 + bp 2) + Vo(qx,q3), (8)
where
and
a = 11
t I336D1/A13,
h = ( -
113/A13)(I11
I3atOX("03) 1/2
(9)b = 111113c°a/Ala •
The values a, h and b of (8) are evaluated at absolute minima (p~ + 6, +_ p~). When they are evaluated at the saddle point (p~, 0), 113 = 0 and (8) reduces to the traditional Hamiltonian (Carreira et al 1972; H o y and Brand 1978; Wilson et al 1980) for two vibrations,
1 2 2
H~(ql,q3 ) = ~(colp 1 + r.oap3) + Vo(ql,q3). (10)
The potential Vo(q~, q3) (in c m - t) in these two cases is given by (Hoy and Brand 1978;
Coon et al 1966)
I 2 1 1 4 1 2 2
Vo(ql,q3 ) = ~o91q I + -~qblllq31 + ~-~ ~b1111q1 + } o93q3 + f l e-/•q"
1 4 1 1 2 2 ,
+ 2"4 (~1111ql + ~b133qlq 2 + ~$113aqlq3
where f l = Bo93ea/D, f2 = D / 2 B and D = e p - - p - - 1.
(11)
The nine force constants of V o resulting from (8) and (10) are termed set 2 and set 3. Bto 3 is the barrier height and p is the shape parameter of the double minimum potential V(q3) in q3. These parameters were originally introduced by Coon et al (1966) and subsequently used by Hoy and Brand (1978). The maximum of the one-dimensional potential,
1 . r - f ~ d
V(q3) = ~to3q 2 -f-j1 e , (12)
occurs at q3 = 0 and the minima occur at
qa = +- [ln(2 f l f2/oga)/f2] I/2 = +- q~" (13) giving the barrier height (in c m - 1 ),
V ÷ - V - = V(q 3 = O) - V(q 3 = + q3 u ) = Bto 3. (14) But the barrier height of the two-dimensional potential Vo(ql, q3) is given by the height of the saddle point above the absolute minima (Hoy and Brand 1978; Mezey and Rao 1980),
Barrier height = Vo(P~,O) - Vo(p~ + 6, + p~). (15)
3. C o m p u t a t i o n a l p r o c e d u r e
A variational technique (Brand and Rao 1976; Mezey and g a o 1980; Polyansky et al 1993) has been employed to obtain the energy levels G(v I , v[*'n), where v I and v a a r e
Two-dimensional vibrational problem at SO 2 261 quantum numbers corresponding to the two stretching modes v 1 and v 3. We have employed 200 basis functions, 100 functions for Ivl,v~ yen> and 100 functions for
V 1, 3 >. For the complete Hamiltonian (1), all the 200 functions have to be employed V ° d d
in obtaining the energy levels, since there are contributions to (v' 1V'~wnl H~Iv 1 v~ v~" >
coming from terms with coefficients H and F 2. For all other terms, there is no mixing of the functions vlv3 I .. . . "~" / a n d l v l -°dd" v 3 / to obtain (v~, v~ ~ ) levels. For the Hamiltonian (8) or (10), it is enough to use the basis set Ivy, v~ ve~ > with even v 3 as we are only interested in obtaining levels with even v 3 in this work.
Three sets of force constants were obtained. In obtaining set 1, all the 6 terms of the kinetic energy with coefficients A, H, B, G, F and C were taken into account.
These are functions of (Pl,P3) and their matrix elements were evaluated using the Gauss-Hermite quadrature formula (Carnahan et al 1969; Press etal 1989).
We have used 20 zeros of the Hermite polynomial H2o(q ) = 0 and the corresponding weight functions to evaluate the integrals of these functions. The complete (100 x 100) matrix of the Hamiltonian (v'lv'~lH~lv~v'a w~> with 15 terms (6 kinetic energy and 9 potential energy terms) was then diagonalized to obtain the energy levels Gtvl,v~'en).
In set 2, the six coefficients of the kinetic energy were fixed at the absolute minima (p~ + 6, + p~). This eliminates terms with coefficients G, F and C and we are left with the Hamiltonian (8). In set
3,
the coefficients were evaluated at the saddle point (p~ 0) and the resulting Hamiltonian (10) contains only two kinetic energy terms. This is the conventional Hamiltonian used for a 2-dimensional vihrational problem (Hoy and Brand 1978; H a n d y 1987; Bramley et al 1991).4. Adjustment of force constants
p~ and ct ° (the bond angle) were fixed at 1.5525/~ and 103.75 ° respectively (Hoy and Brand 1978; Mezey and Rao 1980). The initial set of nine force constants K 1 l, K11 l, K1111, K33, kl, k2, K3333, K133 and K 11 a 3 were obtained (Mezey and Rao 1980) where they were refined to fit the 12 vibrational frequencies corresponding to the levels (v 1 , v3) = (0, 2), (1, 0), (1, 2), (2, 0), (0, 6) and (3, 0) of Si602 and S1802 . Unlike in Mezey and Rao (1980), the potentialis here expanded at (pl - P~' 0) and hence the geometry is slightly different.
All computations were carried out using a VAX 4000/200 computer. It required approximately 10 minutes to compute the 12 frequencies of $1602 and $1sO2 for set 1, and 2 minutes each for sets 2 and 3. It was not always easy to do a least squares calculation using normal equations for the adjustment of force constants. Therefore, we have used method D (Shimanouchi and Suzuki 1965; Ueda and Shimanouchi 1967) which circumvents the inversion of an ill-conditioned normal matrix even though it does not remove the inherent problems associated with it. In their method D, each force constant is adjusted in succession by a certain factor 2 so that the sum of the squared deviations between the observed and calculated frequencies becomes smaller with each successive iteration. When this no longer happens, a new 2 is obtained by multiplying it with #(0 < # < 1) and the computations are continued. The adjustment procedure is terminated when 2 by successive multiplications with ;t(2,/~i,/~24...) becomes so small that the change of the force constants by 2 no longer affects the sum of squared
12
deviations [SES - squared error sum: ~.(Vob s -- Vital )2].
i
Table 1. Kinetic energy coefficients (111, I~3,133) and force constants of SO 2 in large amplitude coordinates
(Pl, P3) a.
See (4) forVo(Pl
P3)-Force constants
Set 2 Set 3
Set 1 81602 81802 81602 S1802
I l l (amu) 113(amu) Iaa(amu)
K l l ( m d y n / ~ ) 13.41405 K11 l(mdyn/~2) - 124-49120 K 1111 (mdyn/~a ) 329-01305
K33 (mdyn/~) 4-48724
Ka333(mdyn//~ a) 306"14337
kl(mdyn./~ ) 0-03766
k2 (/~k -2 ) 88-43281 K133(mdyn//]~2) - 58-88147 Kl133(mdyn//]t 3) 384"56279
25.894810 28.734259 25.899934 28.740817
0.109044 0.139562 0.0 0"0
19.781843 21.245190 19.777470 21.239664
13.405051 13.474051
-123.751198 -123-751198
318"013051 321"563051
4.479237 4.495237
305-093372 304.643372
0.037555 0.037655
87"582814 88.412814
-58.941471 -59.036471
384.112786 377.562790
aSet 2 constants were obtained at absolute minima (p~ + 6, __+ p~), and set 3 constants at saddle point (p], 0)
Table 2.
Kineticenergycoefficients(a,h,b)andforceconstants(incm-1)of
SO2 in dimensionless coordinates(ql,
q3) a" See (11) forVo(ql q3).
Set 2 Set 3
Parameters $1602 $1802
81602
81802a 937"40 889"89 939"70 892.04
h - 3 " 6 7 2 2 -4-1206 0-0 0"0
b 619"95 598-22 621" 13 599-35
o91 937"36 889-86 939"70 892"04
~b111 - 322-50 - 298-30 - 321"16 -297"10
~b 1111 30-8874 27"8352 31"0661 27-9954
o93 619"94 598"20 621"13 599"35
tk3333 116-0822 108-0862 1 1 5 " 5 2 3 9 107"5706 f l 1890"70 1890"72 1895"73 1895"73
f2 0"2408 0-2323 0"2426 0"2341
q5133 - 304-0255 - 285-8350 - 303.6017 - 285-4400
~b 11 a 3 73" 8403 67"6398 72"2669 66-1988 aln set 1, all the six kinetic energy terms were varied with respect to (Pl, P3) and hence are not included here; set 2 constants were evaluated at absolute minima (p] + 6, ___ p~) and set 3 constants at saddle point (p$, 0);
bf~ is dimensionless.
5. Results and discussion
T h e k i n e t i c e n e r g y coefficients for sets 2 a n d 3 a n d force c o n s t a n t s for all t h e t h r e e sets a r e given in t a b l e 1. C o r r e s p o n d i n g v a l u e s in c m - 1 for sets 2 a n d 3 a r e given in t a b l e 2.
T h e two t a b l e s d o n o t s h o w kinetic e n e r g y coefficients for set 1 as t h e y were v a r i e d w i t h
Two-dimensional vibrational problem a t S O 2 263 (Pl, P3)- As m e n t i o n e d before, all the kinetic energy coefficients were c o m p u t e d at 20 points c o r r e s p o n d i n g to 20 zeros of the H e r m i t e p o l y n o m i a l s in the G a u s s - H e r m i t e q u a d r a t u r e p r o c e d u r e ( C o o n et al 1966; Carreira et al 1972). T h e values o f table 2 c o r r e s p o n d to (8) a n d (10). Observed a n d difference frequencies for b o t h $ 1 6 0 2 and
$ 1 s o 2 for the three sets of constants are given in table 3. Also included in the table are the SES values for the three sets giving the s u m of the squared deviations between observed and calculated frequencies. Set 2, which takes into a c c o u n t three kinetic energy terms 111, I13 a n d
I33,
is superior to set 3 which employs the c o n v e n t i o n a l expression for the kinetic energy. Barring a n y unforseen numerical errors in the lengthly c o m p u t a t i o n , it is s o m e w h a t d i s a p p o i n t i n g that set 1, which is the result of a rigorous t r e a t m e n t of kinetic energy, is only as g o o d as set 2. It is h o p e d that for some other molecules, set 1 will turn o u t to be superior to sets 2 a n d 3.The deviations between observed a n d calculated values for S 1602 are in the opposite direction to $ 1 8 0 2 indicating the correctness of the calculation ( S h i m a n o u c h i and Suzuki 1965)*. There are n o significant differences a m o n g the three sets of force cons- tants given in table 1 o r table 2 as the starting set is very close to the final set. T h e s u m of the squared deviations for the 12 frequencies is 382.09 c m - 2 for set 2 a n d 408"86 c m - 2 for set 3. This difference amounts to an improvement of [(408-86 - 382~9)/12] 1/2 ,~ 1.5 c m - 1, o n the average, for each one of the 12 frequencies. I n set 2 of table 2, the small differences between a a n d ta 1 on the o n e hand, a n d b a n d 093 o n the other are due to the fact that I13 ~ 0 for this set (see (9)).
Table 4 contains the structural p a r a m e t e r s a n d various vibrational constants o b t a i n e d with set 2 constants. T h e saddle p o i n t a n d potential m i n i m a o c c u r at (p~, 0)
+ e
a n d (p~ + 6, _ P3), where p~ = 1.5525/~ a n d ~ = 0.0116/~. T h e value of ~ was obtained from the nine force c o n s t a n t s o f set 2 using the equations given in the appendix of M e z e y a n d R a o (1980). The equilibrium b o n d lengths are calculated to be
Table
3. Observed and difference frequencies (obs-calc) (in c m - ~) for the 3 sets of constants a.81602 81802
Obs-Calc. Obs-Calc.
(v l, v3) Obs b Set 1 Set 2 Set 3 Obs b Set 1 Set 2 Set 3 (0, 2) 561-30 - 0"83 - 0"83 - 1 - 1 4 533.30 0.60 0.52 0.19 (1,0) 960-30 -5.21 -2"95 - 6 . 8 2 920.90 2.39 4-85 1.14
(1,2) 1655.70 0-16 1.39 0.80 1582-30 6-01 7.10 7-09
(2,0) 1918.10 - 4 . 2 8 - 4 . 6 8 -6-31 1840-00 10.55 9.76 9.17 (0, 6) 1965"90 - 6.04 - 5.94 - 7"21 1880.20 6.53 6.41 4.65 (3, 0) 2921.20 - 9.34 - 9.58 - 6.96 2798-40 4.03 3"36 7'93 Squared error sum for 12 frequencies (cm-2): 382"13 for set 1; 382.09 for set 2; 408'86 for set 3.
a See table 1, bSee Hoy and'Brand (1978).
÷ For HC1 and DCI molecules with reduced masses/ano = ff979593 amu and PDO = 1"904322 amu and harmonic force constants KHC ~ = 5"1574 mdyn/A and Kt,c~ = 4.9043 mdyn/A, the harmonic frequencies 2989 cm -~ and 2090cm -1 are fitted exactly. But with K = (Kna + KDO)/2, the devia- tions between observed and calculated values are + 37 c m - ~ for HC1 and - 27 c m - ~ for DCI.
Table 4. Various vibrational constants obtained with set 2 force constants.
p] = 1-5525/~, p; = + 0.0735/~, ~° = 103.75 ° 6 = 0.0116/~
Saddle point (S) = (p], 0) Absolute minima (Mx,
M2)
V o a t S V o at M1,M2 Barrier height
= (1.5525, 0)
=(p~ + 6, + p;) = (1.5641, + 0-0735)/~
= 1, 890"70cm -1
= 1 , 7 6 0 - 1 6 c m -
= (1890-70- 1760.16) = 130.54cm -1 Zero point energy G(0, 0) = 618.47 c m - 1 for S
1602
= 590"54cm -x for $1sO2 Isotopic shift = - 20-81 c m - 1
1 2
Squared error sum ~ (rob, - Y/eal) 2 = 382"09 c m - 2
i
~o i ...... POTENT~- Vo = p, =~ j
- - POTEN'r~ %= p,
=~,~
1870
";~ 1920
" 1870
> o \ /
. /
I-- • s
C) 1770.
1 7 2 0 '
~ I I I I I I I I l l l l ~ I l l I I I I l ~ I I l ' l l l l l l l l t l I l l I l l I l I I I I I
-0.12 -0.07 -0.tw 0.03 0.08 0.13
p~ (A)
F i g u r e 2 . Cross-sections of the potential function Vo(p~,p3 ). The broken curve
represents V o at Px = P~ = 1-5525/~, P3 = 0. The solid curve represents Vo at the absolute minima Pl = P~ + 6 = 1"5641/~, P3 = + 0.0745/~,.
p ~ + 6 + Pae - 1"6376/~ a n d - P1"+6 - P 3 " = 1.4906/~ (see figure 1). T h e b a r r i e r h e i g h t is t h e h e i g h t of t h e s a d d l e p o i n t (S) a b o v e t h e a b s o l u t e m i n i m a ( M 1 , M2):
b a r r i e r h e i g h t = Vo(p~,O ) - Vo(p" 1 + 6,p'3)
= 1 8 9 0 . 7 0 - 1760.16 = 130.54 c m - 1 .
F i g u r e 2 s h o w s t h e cross sections o f Vo(pl,p3) for P l = P ~ ( d a s h e d curve) a n d
_ e + 6 (solid curve). T h e s e c u r v e s a l s o s h o w the s a d d l e p o i n t (S) a n d t h e a b s o l u t e P l - P l
Two-dimensional vibrational problem at SO 2 265 minima (Mx,M2). The zero point energy G(0,0) is 618-47cm -1 for $1602 and 590.54cm- 1 for $1sO2 .
Acknowledgement
We would like to thank M a r y Jo deMeza for her help in plotting the figures.
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