Volume 96, number 5 CHEMICAL PHYSICS LETTERS -22 April 1983

UNIFORM SEMICLASSICAL SELF-CONSISTENT-FIELD CALCULATIONS OF VIBRATIONAL RESONANCE ENERGIES AND WIDTHS

David FARRELLY, Richard M. HEDGES Jr. and Wiiam P_ REINHARDT Department of Chemistry. University of Colorado. Boulder. Colorado 80309. USA and Joint Institute $ocor. Laboratory Asrroplysics, Uni~*ersit_v of Colorado and National Bureau of Standards. Boulder. Colorado 80309. USA

Received 20 December 1982; in final form 4 Februvy 1983

Using the Breit-Wigner definition of a resonance, we calculate both resonance positions and widths for a model non- separable hamiltonian. using a uniform semiclassical self-consistent-field (SCF) method. Excellent agreement with recent comples-coordinate quantum SCF calculations is obtained

I. Introduction

There has recently been a flurry of activity into the practical development of semiclassical methods for the calculation of resonance energies (positions) and widths for a variety of one-dimensional systems_ Specifically, the work of Connor and Smith [l-3], Borsch and co-workers [4-61 and Strand and Rein- hardt [7] has established the semiclassical approach to be a highly accurate and efficient way of extract- ing resonance information either via Wigner-Eisenbud [ 1,2], Breit-Wigner [ 11. or complex energy eigen- value [3-71 calculations. Central to the success of the method is the use of appropriate semiclassical connection formulas in deriving uniformly valid quantization conditions [83. Concurrent with. but largely independent of this work has been interest in the accurate calculation of vibrational resonance ener- gies and widths in coupled oscillator systems [9-151.

The study of both shape and Feschbach resonances in coupled oscillator systems provides insight into processes where quantum-mechanical tunneling is im- portant, for example, vibrational predissociation of van der Waals molecules [ 161 and chemically reactive scattering 1171. A variety of methods have been used to calculate resonance positions and widths ranging from “golden-rule” formulas [ 1 S] to large complex- coordinate [ 19,201 configuration-interaction (Cl)

calculations [g-14]. In this letter however. we shall restrict our discussion to shape resonances (i.e. tunnel- ing through a physical rather than a dynamical bar- rier), since the theory cannot be directly applied to Feschbach resonances [3_1]_

A particularly simple yet accurate method for the calculation of bound-state energies or resonance posi- tions in arbitrary non-separable coupled oscillator systems is the quantum-mechanical self-consistent- field (SCF) approach developed for these systems largely by Carney et al. [22] and Bowman and co- workers [23-X]_ By means of complex-coordinate rotation, Christoffel and Bowman [12] have extend- ed the method to allow for the calculation of reso- nance widths. The quantity extracted from the cal- culation is the complex energy eigenvalue, E = E, - W/2, Er being identified with the resonance posi- tion and F with the width. This identification is ex- pected to be good for the case of fairly narrow reso nances [ 1 ] _ In a calculation on a model hamiltonian supporting only memstable (resonance) states, Christoffel and Bowman obtained excellent agree- ment with accurate quantum results. A clear compu- tational drawback to the method, however, is rhe need to numerically evaluate highly oscillatory inte- grals over the wavefunctions. which requires the accu- rate evaluation of wavefunctions at a large number of points. Also, since a finite-basis method is employed,

0 009-2614/83/0000-0000/S 03.00 0 1953 North-Holland 599

Volume 96. number 5 CHEMICAL PHYSICS LETTERS 22 April 1963

IIW cigenvalues are not independent of the rotation mgiic and consequently the behaviour of the cornples

eigcavaloe as A function of rotation angle must be

cxirni~wd f0 e_xiracf 9 conver@ result.

The probimts corrected with convergence of flritle-h.isis calculations. and with the highly oscillatory n~tuit of the \\awfunction have been successfully .~\cklrd bl Gerber and Ratner [76] and by Ratner et ~1 12’1 v.910 developed a printitlve semiclassical SCF mxh~kl for the c~lculntion of approximate resonance ~hmIIl~llS_ i$ xviu& requires neither the explicit m,Amiorr of rhe x\ a\efunction nor comples-coordi- ~LIIC’ I~W~KW~ and yet maintains a good level of accu- UC>. The method is prhnirivc in two senses: a prirni- ti\c 3pp~miniatioti for the w.ivefunction is used to ~cpl.icc the quamum wtivefunctions in the quantum SCF ~netirod and also to derive the sen~iclassicsl qu.mtl/~tml condltivrls. The sirnplicit_v of the method ICIIC~ tw the use of a primiti\‘e ltpproxintation for the ~~,~\ef~~tttms. The c:dcular~on of resonance wkirhs I\. huv3ctcr. prechtdcd hy the use of a primitive quart- ttldttm III rhzs iime wc show frow_ while retaining 3. p~umti\e .tpproxm~tlou for the ~axefunction but ~ICII~~ a unlforrnly \~liJ quantiwtion condition. reso- IIJ!IL+C 11 itflhs IIIJ~ be calculated with little sdditiotlal cfltict mcr the completely primitive SCF method.

in section 2 v.e describe the primitive semiclassical SC1 method \\ith specific reference to the hamiiton- ~.m [‘I] used by Christoffel and Bo~nan [ 1 -?I_ itt .&htion. we slio~ ho\\ J umforni quantization. of the 13mt- \\‘I~IICI tj pe. IIU~ be introduced into the in~thod. Cskui~tions itre in section 3 where we corn p.uc’ prmiitivc and uniform seniicl~ssical SCF calcula- tlonr with lhe caku~Jtions of Christoffel and Bowman .md B nh xcurate quantum resuhs. Good agreement between rhc umforxn and quantum SCF methods is &>erxed. in partxculrir. we note that the error intro- drxed by use of’ a semicfassica1 approximation is ~b~t.~rn~.~lly Irss than the difference between the qu.mIuni SCF and .iccur;lte quantum results. Conclu- afons dre 111 scctilw 4.

2. The semicla~sird SCF method

The model harniltoni3n used in refs. \3,12 J is

ff = il, + ‘I> + rr&.y,. (1)

GOG

In the absence of the coupling potential. Y&z._JJI). they mode potential is simply a harmonic oscillator and the-u mode potential is a double-barrier problem as illustrated in fig_ 1_ It is clear from the form of the coupling potential and from fig. 1 that the hatnil- tonian can support no bound states: only metastable resonance states. We now briefly outline the primitive semiclassical SCF nlethod of refs. [26.X’] with specif- ic reference to the hamiltonian of eqs. (1) and (2).

III the SCF method a liartree product =a\efunction of the form

\I1 ,,x,,, @.y,1’) = 4&) &_*.(1?) (3)

is as&ed in the Schrijdinger equation

~l\k,,, (s. ,i’I = E, ,r %zx’x-“,.ck Y)- xt (4)

The condition that the energy be stationary subject to the orthonormaiity conditions

V,(x)

rig. I. The uncoupled x mode potent&I defined by eq. (?a>. Three of the ckissical turning points, xa, xb and xc are shown at an arbitrary enegy.

Volume 96. number 5 CHEMICAL PHYSICS LETTERS 37 April 1983

W

(-9

(6)

(7)

The corrections to the effective potentials in the Schrildinger-like equations (6) and (7) B$?(.x) and W$?(IJ) are given by

(8)

(9)

&G “Jx ‘I>

= c$’ + e(i) “Ir. - rut’, x “Ix I*_,, - (10)

together with

I&’ ,*_.&, = M;;,I_,. I V&. ?W&,)- (11)

To evaluate eqs. (S), (9) and (11). the wavefunc- tions @,1X and $J,, , are required_ In the primitive semi- classical SCF ap{roach. Gerber and Ratner make the approsiniation

IQ(c1~12 = m&d, q,G(IGqb;

= 0. otherwise, (19

where (1 is a coordinate and cl, and qb are the classical turning points bounding the potential well. Substitu- tion of eq. (12) into eqs. (S), (9) and (11) reduces them to simple integrals of the form

(13

which are normalized using eq. (13) [26] _ The single-mode eigenvdues ~$2 and e$ may now

be obtained from the primitive quantizat&n condi- tions,

1 Xb

3iT s p$>(_lr) dx = (IZx i- +,7r, (14)

-5

(1%

Y.J_x) = +JJZ&C2 exp(--x7),

7 3

(18)

V,.(v) = +JJZGJ~-, (19)

and the limits of integration are the classical turning points, illustrated for the x mode potential in fig_ l_ The total energy is given by eqs. (10) and (1 l)_

The SCF procedure is to solve eqs. (14), (15), (10) and (11) until self-consistency is achieved- It is appar- ent that the primitive quantization condition (14) es- eludes the calculation of resonance widths, a uniform approximation being required_

[ii) UJZifOrJJZ SCF In this subsection we describe the replacement of

the primitive quantization formula (14) with a uni- fomrly valid espression. The appropriate Breit- Wigner uniform quantization formula for thex mode effective potential has been given by Connor [X] and is

1 -3

ti s p,(X) dU = (JZx + $) X + Q(E).

“a

where

(20)

Q(E) = l + arg F(f + ie) - E In ]e].

and

(21)

wherex,, x,, and xc are the classical turning points as illustrated in fig. 1, and the branch ofp, is chosen such that E < 0 for energies below the barrier top_

In writing these formu!as, advantage has been taken of the symmetry of the effective potential about x = 0 (see fig. 1). Eq. (30) is a uniformly valid

601

Volume 96. nunlber 5 CHEMICAL PHYSICS LETTERS 22 April 1963

quantifition condition for thex !node resonance posltio!!. The v. idth may be obtained from the for- !lnIla [‘S]

q)

u 111cli is exp~c!ed 10 he sufficiently accura!e for nar- row resu!!rl!!ces 11 i!!g helo\v the top of the barrier. For st.!tes above the barrier top .! !nore complete t~e.it~twnt IIIJ> hc ~leedect [3] together with a uni- for!!! .qm!~!\!!n.itio!! for the wavefunction. rather tltdii the pri!n!t!ve approhimatio!! of eq. (l-1). Since. III llii?: letter. u e wish to retain the simplicily of using 3 pri!ii!ti\r dpproxi!nat!o!i for tlie wavefu!!ctio!!. we h11l JWI consider this problem here. Krtther. we rc- strict ourselves to reso!!a!!ces Iyi!!g below the barrier lop. 1;!! \c’I!icl! published results of co!i!ples-coordi- ndlc qu~!itu!i~ SCF c.!kul.!tio!is are available [ 12]_

3. Kes1111s

III t.~hic 1 we w!npxe the pri!nitive a!!d uniform XT 11wt1ld~ u ith the ~omplc.\-coordinate quantu!!! S<‘F C.~CAJIIOI~ oi Christoffel md Bowtmt [ 131 .ml tlw xcur.ltc ~UJIII tmt results ofM’~Jite and Miller ]‘I] gnen 111rcl-. [I'].

I i!r pimti~~ results for the positions are in fair

agree!nent with the quantum SCF but the primitive method cannot provide any estimates for the widths. A substantial improvement in the resonance posi- tion is obtained by going to the uniform approxima- tion and the width is also seen to be in excellent agreement with the quantum SCF results. even for states close to the top of the potential barrier.

It is important to note that the error introduced by use of the SCF method itself is much greater than that induced by using a se!niclassical approsima- rion within the SCF formalism.

4. Conclusions

In this note we have extended the primitive semi- classical SCF method by use of a suitable unifor!!! se!niclnssical quantization to calculate resonance positions and widths for a model hamiltonian. Good agreement with quantum SCF and accurate quan- t um calculations of complex energy eigenvalues was obtained. eve!! though we used a pri!nitive wavefunc- tio!! to approkiinate the quantum wavefunction in the SCF !nethod. It is likely that the !nethod ca!! be extended to resonances above the top of the poten- tial barrier by use of a suitable uniform approxima- t io!! for the wavefunct ion combined with a semi- classical complex energy eigenvalue calculation [3] _

JJ , . “I_, 1. r J ’ 1‘ ‘It L; b, 1- b) Er c) _Er d) I‘ d)

-- ---. - __._.~_ _-

;. 0 X.167 4.07 x 10-7 25.105 3.63 x lo-’ 25.269 25.234 3.49 x 10-Y 3. I 3 1 1 6 3 2.36 x 10-s 31.176 1x3 x 10-e 31.341 31.309 1.82 x 1O-6 -I. 11 30.236 1.70 x 10-J 30.192 1.51 x 10-a 30.37 1 30.326 1.53 x 10-a

1. 1 35 836 7.10 x 10-J 36.013 7.0s x 10-3 36.098 36.049 7.06 x IO-! 5. 0 x 630 2.16 x 10-I 34.680 2.05 x 10’ 34.809 34.7 15 2.08 x 10-I 5. 1 3Y.766 3.26 x 10-l 39.954 7.40 x 10-Z e) 39.999 7.49 x 10-Z

J) :\LcurJ!c quJn!wn rcsuirs of \\‘Ji!c and Miller 19 J. “)QuJn!un! SW rrsulrz oiClirisioffc1 Jnd Box\ !!lJn [ 12 1. cj l’r!!llir!rc sa!liclJ~\icJl SCI-‘. d) L!!!ifor!!l w!n!classiul SCI-. c) CJlculJ!ion did IlO! con\r?rgc.

602

Volume 96. number 5 CHEMICAL PHYSICS LETTERS 22 April 1983

Acknowledgement

RMH Jr_ acknowledges a useful conversation with Dr. MA. Ratner on the semiclassical SCF method. We thank Z.I referee for suggestions which have result- ed in an improved manuscript. This work was sup- ported in part by grants CHE80-I 1412 and PHY82- 00805 from the National Science Foundation_ The calculations were carried out on the CYBER 720 computer of the University of Colorado Computer Center.

References

[l] J-N-L. Connor and A.D. Smith, Mol. Phys. 43 (1981) 397.

[ZJ J.N.L. Connor and AD. Smith;, Mot Phys. 45 (1982)

149. [3] J.N.L. Connor and A.D. Smith. Chem. Phys Letters

88 (1982) 559. [4] NJ. Borsch, Ii. Laurent and R_ Mblenkamp, MoL

Phys. 43 (1981) 1441. IS] H.J. Korsch, H, Laurent and R. ~l~ie~arnp, J. Pbys.

Bl5 (1982) 1. [6] H.J. Korsch, H. Laurent and R. Miilenkamp, Phys. Rev.

A26 (1982) 1802. 17) &l.P. Strand and W.P_ Reinhardt, J. Chem. Phys., to be

published. [S] M.S. Child, MolecuLr collision theory (Academic Press,

New York, 1974) appendix C. (91 B.A.Waite and W.H. hfiller, J, Chem. Phys. 73 (1980)

3713.

B-AA. Waite and W.H. Miller, J. Chem. Phys. 74 (1980) 3910. BAA. Waite and W-H. Miller, J. Chem Phys. 76 (1982) 2412. KM. Chriioffel and J-M. Bowman, J. Chem. Phys. 76 (1982) 5370. R.&I. Hedges Jr- and W.P_ Reinhardt. Chem. Phys. Letters91 (1982) 241. R-M. Hedges Jr- and W.P. Reinhardt, J. Chem. Phys. (1983), to be pubIished. B.C. Garrett and D-G. Truhlar, C&em. Phys. Letters 92 (1982) 6C J_ Jortner and R.D. Levine, Advan Chem. Phys_ 47 (1981) 1. B.C. Garrett, D.G. Truhlar, R.S. Grev, G-C. Schatz and R-B. WaIker, J- Chem. Phyr 85 (1981) 3806. A.P. Hickman, A.D. Isacson and W-H. MiIler, Chem. Phys. Letters 37 (1976) 63. W-P. Reinhardt, Ann. Rev. Phys Chem. 33 (1982) 223. B-R. Junker, Advan. At. hlol Phys 18 (19S2) 207. C-W_ hicCurdy, TX Rescigno, E.R. Davidson and J-G_ Lauderdale, J. Chcm. Phys. 73 (1980) 3268. G-D. Carney, L-1. Spmndel and C-W_ Kern. Advsn. Chem. Phys. 37 (1978) 305. J-h%_ Bowman, KM. Christoffel ami F_ Tobin, J. Phyr Chem. 83 (1979) 905. K.M. Christoffel and J.hl. Bowman, J. Chem. Phys. 74 (1981) SO57- J.hl. Bowman, 1. Chem. Phys. 68 (1978) 608. R-B. Gerber and hf.A. Rstner. Chem. Phys. Letters 68 (1979) 195 hIA Ratner. V_ Buch and RB. Gerber. Chem. Phyr 53 (1980) 345. J-N-L. Connor, hlol. Phys_ IS (1968) 37.

603