Bobby G. Sumpter and Gregory S. Ezra- Semiclassical Rotation-Vibration Energies for a Triatomic...

8/3/2019 Bobby G. Sumpter and Gregory S. Ezra- Semiclassical Rotation-Vibration Energies for a Triatomic Molecule: H2O

1/5

Volume 142, number 3,4 CHEMICAL PHYSICS LETTERS 11 December 1987

SE MICLASSICAL ROTATION-VIBRATION EN ERGIE S FOR A TRIATOMIC MOLECULE :Ih O

Bobby G. SUM PTER and Gregory S. EZRA Department of Chemistry, aker Laboratory, Cornell Un iversity, thaca, NY 1 48 53, US AReceived 2 1 uly 1987

We report primitive semiclassical rotation-vibrational energy levels for a triatomic molecule, H 20, with inclusion of all threevibrational degrees of freedom. Both the Lai-Hagstrom and Hoy -Bunker potential surfaces are used. Vibrational and rotationalactions are calculated usinn the fast Fourier transform Sorbie-Handy method. Overall, encouraging agreement w ith the quantumvariationa l results of Chen, Maessen, and W olfsberg is found.

1. Introduction

Much effort has been devoted in recent years tothe problem of semiclassical quantization of energyeigenvalues for non-separable molecular potential-energy surfaces [ 11, with the aim of providing prac-tical alternatives to quantum -mechan ical variationalcalculations [ 21. Most studies in this area have fo-cused on vibrational states in the absence of rota-tions. Nevertheless, incorporation of rotationaldegrees of freedom is essen tial for elucidation of thespectroscopic a nd dynamical consequefices of the in-teraction b etween rotation and vibration (for a re-cent review, see ref. [ 31 ) .

Wo rk on the sem iclassical quantization of the rigidasymm etric top [4-81 provides a foundation forsubsequent efforts to extend semiclassical methodsto the full rotation-vibra tion problem (for wave-packet-based approaches, see refs. [ 9,10 ] ) . A key in-gredient is the canonical transformation of Aug ustinand M iller [ 111 (see also ref. [ 12]), which explicitlyreduces the rigid-rotor H amiltonian to a system withone degree of freedom. Exploiting the Augustin-Miller transformation, Frederick and McC lellandhave calculated both primitive [ 131 and uniform[ 13,14 semiclassical energy levels for a rigid-bender[ 151 model of HzO , in which stretching m otions werefrozen out. Good ag reement was obtained with the Alfred P. Sloan Fellow

14 2

quantum results of Bunker [ 131 in a regime wherethe usual perturbation theory of rotation-vibrationinteraction [ 161 is inapplicable. The rigid-bendermodel for Hz0 is effectively a two-dim ensional sys-tem, whose phase-space dynamics can be exploredusing the Poincarb surface-of-section method[ 17,181.

In this paper w e report primitive EBK [ 11 semi-classical rotation-vibration energies for a triatomicmolecule, H20, with inclusion of all three vibra-tional degrees of freedom. The full rotation-vibration problem for a triatomic is effectively fourdimension al, and requires a method suitable forquantizing systems with several degrees of freedom.We have applied the FFT SH method [ 191 (see alsoref. [ 201) to calculate good vibrational and rota-tional action variables. The semiclassical results re-ported here are obtained using the potential-energysurfaces of Lai-Hagstrom [2 1,221 and Hoy-Bunker[ 231, and are compared with the quantum varia-tional results of Chen, Maessen and Wolfsberg [ 221.

2. MethodClassical equations of motion for the rotating-

vibrating triatomic are integrated in lab-fixed Carte-sian coordinates, with the potential surface ex-pressed in terms of geometrically defined bondlengths and interbond angles [ 241. Both the Lai-


2/5

Volume 14 2, number 3,4 CHEMICAL PHYSICS LETTERS 11 December 1987

Hagstrom [2 1 and Hoy-Bunker [ 23 ] potentials areTaylor expansions in bend and stretch displacementcoordinates containing terms up to quartic, withslightly different equilibrium geom etries (see table 1of ref. [ 221). A transformation to molecule-fixed(Eckart) coordinates [ 251 is made to follow the timeevolution of vibrational (normal-mode coordinatesand mom enta) and rotational (molecule-fixed com -ponents of the total angular momentum vector J)variables.

The orientation of the rotating Eckart frame is cal-culated using a symm etric o rthonormalization pro-cedure [25,26]. Let the 3N compo nents {up}, QY =13 **a,N, i=x, y, z, define the static molecular model(equilibrium structure) for an N-atom system[ 25,261. For an arbitrary distorted configuration ofnuclei {P}, define the three Eckart vectors

Table IRotationa l energies for J= 10 (cm-) for the Lai-Hag strom po-tential-energy surface of water. The energies are given with re-spect to the rotationless vibrational state (0 , 0, 0). T he dashedline divides A- from C-type states

k, k, AE QMb FFT SH c)10 0 3.76 2672.964 2669.2010 1 3.16 2672.964 2669.209 I - 0.480 2451.850 2452.339 2 -0.480 2451.850 2452.338 2 0.178 2243.318 2243.148 3 0.177 2243.317 2243.147 3 -2.51 2051.021 2053.517 4 -2.53 2050.981 2053.516 4 -7.61 1879.253 1886.496 5 -7.98 1878.511 1886.495 5 0.216 1736.246 1736.035 6 -7.95 1728.075 1736.03_..-_..._..-._.._.-..~.~..~.-~~.....-.-..~.~~....-~4 6 -2.38 1635.853 1638.234 7 22.3 1593.975 1571.713 I -13.4 1558.337 1571.713 8 3.49 1458.701 1455.212 8 -2.49 1452.721 1455.212 9 -0.108 1304.352 1304.461 9 -0.524 1303.936 1304.461 10 2.07 1123.481 1121.410 10 2.06 1123.470 1121.41

AE=QM-FFI SH.b, Quantum-mechanical energies from ref. [22].Cl Present work .d Eigenv alue obtained by interpolation from the higher-energyA-type states.

k;= C myayry,a (1)and the associated Gram matrix I,rij=~i..Fj. (2)The three unit vectors {f;} defining the Eckart frameare then given byf;= C F,(I? -2)ji )i (3)where the positive definite square root of r is used.This method involves only numerical matrix alge-bra, and avoids use of the Euler angle parameteri-zation of the direction cosine matrix. (For a planarmolecule such as HzO, one of the Eckart vectors isidentically zero. The corresponding 2 x 2 Gram ma-trix is used in this case.)

Normal-mo de coordinates are obtained by lineartransformation of molecule-fixed Cartesian displace-ments [ 241. Instantaneous values of the canonicalpair of variables (k,,~.) associated with A-type ro-tation about the axis of least inertia are calculatedfrom the body-fixed compon ents of the total angularmomentum vector J, with the replacementJ(J+l)+(J+J)Z [5,7]:J x=-[(J+$)2-ki]12 sin&), (da)J,=-[(J+f)-k:]* cos(x.), (4b)J,=k,. (4c)A corresponding pair of variables (k,, x,) can be de-fined for C-type motion.

The FFT SH method [ 191 is used to calculate goodaction variables from the normal-mode and rota-tional variable time series. The FFT SH approach isan approximate procedure, in which it is necessaryto identify a set of coordinates in which the motionis approximately separable. For the states reportedhere, the good actions are assumed to be topologi-tally equivalent to the uncoupled vibrational and ro-tational actions, i.e. it is assumed that there are nostrong vibration-vibration or rotation-vibrationresonances. A lack of low-order rotation-vibrationresonances is consistent with the quantum calcula-tions of Chen, Maessen and Wolfsberg [ 221, in whichonly a limited amount of centrifugal mixing of J= 0vibrational eigenstates is found for (0, 0,O) and(0, 1,O) states with J= 10. Also, for the vibrational

143


3/5

Volume 142 , number 3,4 CHEMICAL PHYSICS LETTERS 1 December 1987quantum num bers considered, the rotationless stateshave normal-mode character, and no transition tolocal-mode dyn amics is evident. O ur assum ption ofquasiseparability is further supp orted by examiningplots of zeroth-order actions (for the vibration s, har-monic normal-mode energies) v ersus time for quan-tizing trajectories; although the zeroth-order actionsexhibit fluctuations due to anharm onic and vibra-tion-rotation couplings, there are no large-scale os-cillations characteristic of low-order resonances.Nevertheless, some trajectories with k-values in thevicinity of the A- to C-type separatrix (see below)appear to be weakly chaotic on a 1 6 ps timescale (theFIT SH algorithm will still yield values for ac-tions in such cases). A detailed classical trajectorystudy of rotation-vibra tion interactions in three-mode H 20 is in progress. For quasiperiodic trajec-

tories, use of the FFT EBK [ 19,201 method ratherthan FFT SH would allow more accurate calculationof good actions or treatment of strong low-order res-onant motions.

Initial conditions are iterated to find trajectoriessatisfying the primitive EBK quantization conditions:Ni=(ni+i)fi 3 i=1,2,3 3 vibrations, (5a)Ni =n;A , i=A or C , rotation . (5b)As discussed in detail by Duchovic and Schatz forthe case 6f a rigid asymm etric top [ 71, the appro-priate quantization condition (5b) for rotationalmotion depends on which side of the separatrix di-vidingd- from C-type motion the trajectory lies. Theprimitive quan tization procedure used here is un-able to obtain the k-splittings of the rotational levels;

Table 2 Table 3Rotational energies for J= 1 0 (cm-) for the Hoy-Bunker po- Rotational energies for J= 10 (cm- ) for the Lai-Hagstrom po-tential-energy surface of water. The energies are given with re- tential-energy surface of water. The energies are given with re-spect to the rotationless vibrational state (0, 0, 0). Th e dashed spect to the rotationless vibrational state (0, 1, 0). The dashedline divides A- from C-type states line divides A- from C-type states

k, k, Ai? e QM b FFT SH )10 0 -0.088 2710.710 2710.7910 1 - 0.088 2710.710 2710.799 1 1.31 2478.836 2477.539 2 1.31 2478.836 2477.538 2 - 0.026 2260.733 2260.768 3 -0.025 2260.732 2260.767 3 -2.81 2059.779 2056.997 4 -2.78 2059.775 2056.996 4 - 5.68 1879.999 1885.686 5 -6.17 1879.512 1885.685 5 2.51 1728.485 1725.985 6 - 3.46 1722.509 1725.98---.-_.---__II__--____111___1__1_______1----_-_.-----4 6 13.0 1619.653 1606.64 d4 7 5.72 1584.526 1578.813 7 -37.8 1541.021 1578.813 8 -4.03 1448.839 1452.872 8 - 12.3 1440.598 1452.872 9 -0.66 1295.952 1296.611 9 -1.28 1295.328 1296.611 10 -2.17 1116.621 1118.790 10 -2.19 1116.603 1118.79

a AE=QM-FFTSH.) Quantum-m echanical energies from ref. [221.) Present work.d, Eigenv alue obtained by interpolation from the higher-energy

A-type states.

k, k, AE a QMb FFTSHC10 0 3.06 2829.225 2826.1710 1 3.06 2829.225 2826.179 1 0.92 2590.400 2589.489 2 0.92 2590.400 2589.488 2 -0.25 2362.718 2362.978 3 -0.25 2362.718 2362.977 3 1.34 2150.503 2149.167 4 1.31 2150.471 2149.166 4 1.30 1958.681 1957.386 5 0.67 1958.054 1957.385 5 - 1.42 1795.943 1797.365 6 -8.68 1788.678 1797.361~1.~~~~1~~~~~~-_~-~-------~-__-_-~~~__-_--~~----4 6 2.64 1678.624 1675.95 d,4 7 29.09 i637.899 1608.813 7 - 16.96 1591.850 1608.813 8 6.84 1487.755 1480.912 8 - 1.81 1479.098 1480.912 9 -1.61 1317.366 1318.981 9 -2.28 1316.698 1318.981 10 4.32 1116.360 1112.040 10 4.10 1116.339 1112.04

a) AE=QM-FFT SH.) Quantum-m echanical energies from ref. [221.c, Present work.d, Eigenv alue obtained by interpolation from the higher-energyA-type states.

14 4


4/5

Volume 1 42, number 3,4 CHEMICAL PHYSICS LETTER!? 11December 1987

Table 4Rotational energies for J= 10 (cm-) for the Hoy-Bunker PO-tential-energy surface. The energies are given with respect to therotationless vib rational state (0, 1 , 0). The dashed line dividesA- from C-type states

k, k< AE a QM b mSH)10 0 0.148 2915.048 2914.9010 1 0.148 2915.048 2914.909 1 -0.160 2656.970 2657.139 2 -0.160 2656.970 2657.138 2 1.67 2411.744 2410.078 3 1.67 2411.744 2410.077 3 -2.19 21 83.339 2185.537 4 - 2.20 2183.325 2185.536 4 2.38 1976.248 1973.876 5 2.07 1975.939 1973.875 5 -2.83 1797.621 1800.455 6 - 7.07 1793.383 1800.45_________________-_-_________l__l_____ll-..--_ll-l4 6 3.91 1663.377 1 659.47 *4 7 21.5 1633.731 1612.193 7 40.8 1571.396 1612.193 8 12.1 1481 .055 1468.962 8 -2.20 1466.760 1468.962 9 1.38 1312.081 1310.701 9 0.097 1310.801 1310.701 10 1.50 1112.713 1111.210 IO 1.46 1112.670 1111.21

a AE=QM -FFT SH.LJJ uantum-mechanical energies from ref. [221.c) Present work.d, Eigenv alue obtained by interpolation from the higher-energy

A-type states.

for k, XJ or k,x J, each primitive energy level cor-responds to a pair of near-degenerate quantum lev-els. For qu antum states in the vicinity of theseparatrix, there can either be two primitive quan-tizing trajectories corresponding to the same state, ornone. Duchovic and Schatz discuss several inter-polation schemes for quantizing states near the se-paratrix [ 71. For the resu lts reported below, we usea simple extrapolation scheme to determine primi-tive semiclassical energies for such states. More ac-curate treatment requires a uniform quantizationprocedure [ 5,7,13,14].

Total trajectory integration time was 16 ps. Goodactions were converged to 1O-2, while quantizingenergies, as corrected by linear extrapolation [ 271,were converged to 10e4.

3. ResultsRotation-vibration energies for total angular mo-

mentum J= 10 are given in tables 1 and 2 for theground (0, 0, 0) vibrational state of Hz0 using theLai-Hagstrom [ 211 and Hoy-Bunker [ 231 poten-tials, respectively. The energ ies shown are relative tothe corresponding semiclassical eigenvalue for theJ= 0 vibrational ground state. Results for the excitedbending (0, 1, 0) state are given in tables 3 and 4.The qua ntum variational levels of Chen , Ma essen andWolfsberg [22] are shown for comparison. Theoverall level of agreement is very encouraging, as canbe seen from figs. 1 and 2, where differences betweenthe quantum and semiclassical eigenvalues are plot-ted against the quantum variational energies. Thereis however a noticeable d eterioration in the qualityof the primitive semiclassical eigenvalues in the vi-cinity of the separatrix. This disagreem ent is a con-sequence of the use of primitive quantizationconditions, and also possibly the presence of weakchaos for states with rotational motion near the A/Cseparatrix.

tj YI I I I I I I

1000 0 1390 0 1760.0 2170.0 2560.0 2950.0Energy

I:---;.;

+I , , , ,1000 0 1390 0 1760 0 2170.0 25600 2:Energy 0. 0

Fig. 1. Difference AE between prim itive semiclassical rotation-vibration energies and quan tum va riational eigenv alues ver-sus quantum energies for& IO, ( 0 , 0 , O ) statesofH,O. (a) H oy-Bunk er potential. (b) Lai-Hag strom potential. All energies incm-.

145


5/5

Volume 142, number 3,4 CHEMICAL PHYSICS LETTERS 11 December 1987

ii1000.0 1390.0 1760.0 2170.0 2560.0 2950.0Energy

2I I I I I

1000.0 1390.0 1760.0 2170.0 2560.0 2950.0

EnergyFig. 2. Difference AE between primitive semiclassical rotation-vibration energies and quan tum variationa l eigenvalues ver-sus quantum energies for J= 10, ( 0 , 1,O) states of H20. (a) Hoy-Bunk er potential. (b) Lai-H agstrom potential. All energies incm-.

4. C onclusionThe work reported here establishes the feasibility

of semiclassical quan tization of rotation-vibrationlevels for triatomic molecules w ith inclusion of allvibrational degrees of freedom. Improvements cur-rently under investigation are accurate calculation ofactions using the FFT EBK method [ 201, and uni-form quantization of rotation-vibration levels using,for example, the method suggested by Frederick [ 141,

AcknowledgementThis work was supported by NSF Grant CHE-

8410 685. Computations reported here were per-formed in part on the Cornell National Supercom-puter Facility, which is supported in part by the NSFan d IBM Corporation. GSE acknowledges partialsupport of the Alfred P. Sloan Foundation.

References[ 11 G.S. Ezra, C.C. Martens and L.E. Fried, J. Phys. Cbem. 9 I

(1987 ) 3721 , and references therein.[2] G.D . Camey, L .L. Sprandel and C.W. Kern, Advan. Chem.

Phys. 37 (I 978) 305:J. Tennyson, Comp uter Phys. Rept. 4 (198 6) 1.[3] G.M. McClelland, GM. Nathanson, J.H. Frederick andF.W . Farley, in: Excited states, Vol. 7, eds. EC. Lim andK.K. Innes (Academic Press, New York, 1987 ).

[4] G.W . King, J. Chem. Phys. 15 (1947 ) 8 20.[ 51SM. Colwell, N.C. Handy and W .H. Miller, J. Chem. Phys.

68 (1978) 745.[61 W.G. Harter and C.W. Patterson, J. Chem. Phys. 80 (1984 )

4241.[71 R.J. Duchovic and G .C. Schatz, J. Chem. Phys. 8 4 (1986)

2239.[81 C.W. Patterson, R.S. Smith and R .B. Shirts, J. Chem. Phys.

85 (1986) 7241.[9] M. Blanc0 and E.J. Heller, J. Chem. Phys. 78 (1983) 2504;83 (1985) 1149.

[ lo] D. Huber, E.J. Heller and W.G. Harter, J. Chem. Phys. 87(1987) 1116.

[ 1 I ] S.D. Augustin and W.H. M iller, J. Chem. Phys. 61 (197 4)3155.

[ 121 A. Deprit, Am. J. Phys. 35 (1967 ) 424.[ 131 J.H. Frederick and G .M. McClelland, J. Chem. Phys. 84

(1986) 876.[ 141 J.H. Frederick, C hem. Phys. Letters 13 1 (1986) 60.[ 151 P.R. Bunker, Ann. Rev. Phys. Chem. 3 4 (1983) 59.[ 161 D. Papousek and M.R. Aliev, Molecular vibrational-

rotational spectra (Elsevier, New York, 1 982).[ 171 J.H. Frederick and G.M. M cClelland, J. Chem. Phys. 84

(1986) 4347.[ 181 G.S. Ezra, Chem. Phys. Letters 127 (1986) 492.[ 191 C.W. Eaker, G.C. Schatz, N. DeLeon and E.J. Heller, J.

Chem. Phys. 81 (1984) 5913;C.W.EakerandG.C.Schatz, J.Chem.Phys. 81 (1984 ) 2394 ;J. Phys. Chem. 89 (1985) 2 612.

[201 C.C. Martens and G.S. Ezra, J. Chem. Phys. 83 (198s) 2990 ;86 (1987) 279.

[2 11 E.K. Lai, M.S. Thesis, Bloomington, Indiana (1975 ), quotedin ref. [221.

[221 C.-L. Chen, B. Maessen and M. Wolfsberg, J. Chem. Phys.83 (1985) 1795.

[231 A.R. Hoy and P.R. Bu nker, J. Mol. Spectry. 52 (1974)43%74 (1979) 1.

[24] G.C. Schatz, in: Molecular collision dynamics, ed. J. M.Bowman (Springer, Berlin, 1983).

[251 G.S. Ezra, Lecture Notes in Chemistry, Vol. 28 (Springer,Berlin, 1982) p. I.

[261 J.D. Lou ck and H.W . Galbraith, Rev. Mod. Phys. 48 (1976)69.

[27] N. DeLeon and E.J. Heller, J. Chem. Phys. 78 (1983) 4005.

146

Date post:	06-Apr-2018
Category:	Documents
Upload:	omasazz
View:	215 times
Download:	0 times

Bobby G. Sumpter and Gregory S. Ezra- Semiclassical Rotation-Vibration Energies for a Triatomic...

Documents