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Rotational predissociation of Ar—HD van der Waals complex: space-fixed complex-coordinate coupled-channel calculations
Volunw
95. number
CHEMICAL
1
ROTATIONAL
P~D~SSOCIATION
SPACE-FIXED
COMPLEX-COORDINATE
PHYSICS
OF Ar-HD
LETTERS
1983
VAN DER WAALS COMPLEX:
COUPLED-CHANNEL
Krishna K. DATTA and Shih-I CHU* of Clictttisrry. iiltinmit~ 0-f Kansas. lmsrettce. Kattsas 66&j.
lk~p~rllIlc~lil
18 February
CALCULATIONS
USA
The sp;ts&ixrd c~,mplcs~~orditl~~~ couptrd-chxmet (SFCCCC) merhod is npplicd tn the zccumte determination of the Ic~cl cncqir‘s zmd widrhs flit~~tim~s) of prcdissocisting st;lfes of Ar-llD(r~ = 1.j = 2) van der IVanIs molecule, usins the (‘arlcy--Lr Rt+- BC3(,6.S) potrntisi for AI-H_. , The calcuktted widths are in good i~armony with the experimental data of McKcllar 3nd in cscrllcnt z~precmcnt with the close-coupling scattering results of Hutson and Le Roy.
I. Introduction tnuch interest in the study of spectroscopy, structure and predissociation dynamics of weakvan dcr Waals (vdW) molecules [ I_?]_ Reliable anisotropic potential surfaces for several vdW complexes, particularly the rare-gas-H2 systems, are now becomin, 0 available. In this paper. we are concerned with accurate crttctrlations of rite resonance energies and rotational predissociatiotl (RP) linewidths (iifetimes) of the Ar-HD vti\Vmoleculr_ This study is prompted by the recent successful experimental determination of the RP linewidths 11~ XicKcllar [ 31 from the ~i~tlproved) IR high-resolution spectra of Ar-HD and &-Hz systems. The direct compak~n bctwc‘cn theoretical and esperimental results allows a critical assessment on the quality of the theOretiCtd Tltcrc
is currctttly
ly bound
snctiwtis as wctl 3s rhe reliability of the potential surfaces used in the calculations. Presiws theoretical approaches for the treatment of predissociation by internal
vibration/rotation
have been
summarized by Bcswick and Jortner [Z]. Recently, we have developed two alternative practical and accurate appt~~~ltcs for the study of rotational predissociation of vdW molecules. They are the comples-coorditzare coupledthutttcV (CCCC) ~nethods, one formulated in the s,uacc-fived (SF). the other in the body’-fhd (BF) frames of COordinates. The SFCCCC theory 14.51 is more appropriate for the treatment of weak-coupling vdth’ compleses (such as Ar--112). whereas the BFCCCC theory [6] more natural for strong-coupling complexes (such as Ar-HCl). The mitiry and adranrage of the CCCC methods may be summarized as foflows: fl) It is an ab initio method (given 3 defined -‘exact” hamiltonian). (2) Only bound-state structure calculations are required and no asymptotic conrtirions need to be enforced. (3) The resonance energies are obtained directl& from eigenvalue analysis ofappropri;ttr non-hcrmite;ln matrices. the imaginary parts of the complex eigenvnlues being related directly to the tifctimcs of the vdW complexes. (4) It is applicable to many (open and closed) channel problems involving muitiply coupled continua. These methods have been applied successfully to 8 number of vdW molecules [4-71. In section 2. rhe SFCCCC theory appropriate for the Ar-HD study is briefly outiined, In section 3. we discuss ths potential-energy surface chosen in the current study_ The predicted RP linewidths are compared with the expcrimcntal data 3s well 3s other theoretical results in section 4.
0 009-2614/83/0000-0000/S
03-00 0 1983 Nor&Holland
CHEhWAL PHYSICS LETTERS
Volume 95, number 1 2. Complexcoordinate
coupledchannel
approach to predissociation
18 February 1983
by internal vibration and rotation
The hamiltonian ofa triatomic vdW molecule A.__BC, after separating out the motion of the centre of mass, may be expressed in the SF frame as H(R,r)
= (1/~~)[-r2’R-l(a2/aR2)R
+Z2(k)/R2]
+H&)+
V(R. r, 0) -
(1)
Here p =mA(mB + mC)/(T?zA +mg + nrC) is the reduced mass, R is the axis of the complex, r that of the diatom BC, Z is the angular momentum of BC and A about each other, Hd is the vibration-rotation hamiltonian of the diatom, V(R. r. tl) is the interaction potential of A and BC. and 6 = COS-~(~^-~)_Th e interaction potential for weak-
coupling complexes may be conveniently expressed as an expansion in terms of a Legendre polynomial as well as a power series expansion in the diatom stretching coordinate t(r) [S] :
where g(r) = (r - r&/r0 and r. is an isotope-independent constant. The eigenfunctions of Hd(r) may be written aa& I&&i), where u andj are respectively the vibrational and rotational quantum numbers of the diatom BC. qt,lf is the spherical harmonics, and Q(r) satisfies the radial Schr6dinger equation
{-(tl'/~Bc)['-1(a2/a32)3 - j(j + 1)/G]
+ V&-)
-E,(u,
j)}Q&)=
0 _
(3
are the vibration-rotation eigenvalues of the free diatom, flgc is the reduced mass, and vg&) In eq. (3, &(U) is the diatom potential_ In the spaced-fixed total angular momentum (J,M) representation,/ =I +j, a convenient angular basis for wavefunction expansion is the total angular momentum eigenfunction defined by
where ( I ) is the Clebsch-Gordan
coefficient and Yx_,,*~ is the spherical harmonics_ It is expedient to define a scheme for labeling of the eigenstates of the complex uniquely_ In the zero-coupling limit, each isotropic state is an eigenfunction ofHd, Z2, 52 and Jz, and may be labelled by ~ujUiI~_The latter may be decomposed into radial and angular functions IvjuAfl= $&l(R)
@Jr)YnJ:,-(F,k)
-
(3
When the coupling is turned on, the state will be only an eigenfunction ofJ2 and Jz and only3, 111and the parity p = (_Qi+‘+J will be good quantum numbers in the perturbed state. However, for weak-coupling complexes such
as Ar--HZ and Ar-HD, v, j and I remain nearly good quantum numbers. Thus the isotropic function IL$VM)provides a good zeroth-order picture for level energies and widths. We now discuss the complex-coordinate coupled-channel (CCCC) formulation in the SF frame. According to the theory of dilatation or cqmplex-coordinate transformation [9-l l] the energy (ER) and the width (I’) associated with a metastable state of vdW molecule may be determined by the solution of the complex eigenvaluc of a non-hermitean hamiltonianI-I,(Rei”, r), obtained by applying the complex-coordinate transformation [9] to the vdW bond (dissociating) coordinate, R + R e’“. In the CCCC formalism, the total wavefuncrion of the coordinate-roiated hamiltonian Ha(ReiC’, r), for a given J, M and p, is expanded in terms of the complete set of the
isotropic state functions IujZJM)allowed by the symmetry. We further expand the radial function $&(R) square-integrable (L’) basis function Ix,,(r)],
in terms
of an orthonormalized
39
Volume 95, number 1
IS February 1983
CHEMICAL PHYSICS LETTERS
where y specifies the channel quantum numbers, y = (u$V.M),iV, is the size of the truncated (x,~ Ix,,, > = 6,,,,,_ For convenience, let us defme the channel basis function to be ^ I -PI ) = x,, CR1 Q&l Y ?$j Cr. RI
radial basis, and
(7)
and arnnge the order of the matrix elements ofH,(ReiU, r) in such a way that II is allowed to vary from 1 toN_,. witl1in each channel block 7 = (u$VM). The mat&v element in the jvz) representation is then (7’,l’JHc,(Rein-,r)17rl) = eeziu (fi’/$)
+ ‘;;1@‘j?~UU’
6jj’611’6,,,,‘6JJf6,~jnl
- R-‘(a’/aR’)R
1*q
F
+ l(Z + l)/R21X~,)6,,~6jj’6,~~,.~~~~~,
(Q,*j’ I Pi Q,j)(Xnll v,x_(Re’*)lX,,)f,(li”,
wherefk is the Percival-Seaton coefficient [I 21. The resulting matrix ofH, complex eigenv3lues may be determined via the secular determinant DetI(U&,,~_~,,
is
3
lit-0 ~_TJJ,,J,* T
symmetric
complex
one whose
- Et I = 0 -
The dtsired metast3ble pies cigenvalues.
3. Potential-energy
@I
(9)
st3tes arc then identified
by the stationary
points
[4-7,101
of the (Ytrajectories
of com-
surface
111the present study. the Ar-HD potential surface was derived from the Buckingham-Corner-type BC3(6,8) A-l l2 poteI:ti3l of Carley and Le Roy [S]. The reasons for this choice are as follows: (i) In one recent SFCCCC study. WC have carried out detailed RP linewidths calculations for the Ar-H2 system, using six realistic anisotropic potentials obtained recently from experiments [5]. It was found that the RP linewidths are sensitive to the surfaces used. 111particular. the widths predicted for the three (less accurate) Lennard-Jones (LJ) potentials adopted vary as lsrge 3s 3 fxtor of four. However, the agreement among the more recent potentials (namely, the BC potential of Zandee and Reuss [ 131. the BC, potential of Carley and Le Roy [S], and the semi-empirical potential of Tang and Toennies [ 141). are much closer - to within 30%. (ii) The BC3(6,S) potential of Carley and Le Roy is the only surfxc which contains the information about its dependence on the length of the diatom bond. (iii) Recently Hutson and Le Roy [ 151 have independently performed the RP calculations for the Ar-HD system using the BC3(6,8) potentitl and the traditional close-coupling scattering method, allowing for 3 detailed comparison between different theoretic31 3pproaclies. The BV3(6,S) Ar-H, potential [S] is expanded in the form
where ,t = (r - ro)/rn
with I-~ = 0.7666348
VAX-(R)= AXkexp(-PAR)
A_ The radial strength function
- D(R)(C~‘/R6
t C;“/R”)
*
where D(R) = exp[3(RolR =I, 40
- 1)3] .
forR
=Rz’
forR >Rg
_
.
VA,(R) 113sthe form
Volume 95, number 1
CHEMICAL PHYSICS LETTERS
:
Ar-HD
Fig_ l_ The diagonal vibrationally preavemged anisotropy strengh radial funcrions ?~(u,ilR) for Ar-HDtu = 1.j = 2).
The parameters (A”“, B k_._ete.) are listed in ref. [6]. To obtain the Ar-ND potential, we first performed the vibrational averaging over the 5 coordinate of eq. (IO), using exact ~,i(~) eigenfunctions of Hd(r). The diatom vibrational wavefunctions GUj(r) were obtained by the numerical solutions of the accurate HD potential data of Bishop and Shih [16]_ Nest we carried out the well-known asymmetric isotope frame transformation [17] to a coordinate system based on the centre of mass of HD, which is located =0_166556r from the bond midpoint. This transformation introduces Legendre terms of odd order into the potential expansion, so that the diagonal vibrationally preaveraged Ar-HD potential, for esample, has the form ?‘ttKIX ir(u,ilR,
8) = h z ~ _ ~ACu,ilRV’,fcos = , .---
The vibrationally j = 3) are plotted
averaged anisotropy in fig. 1.
8).
strength
00 radial functions
&(u,jJR)
for the case of current
interest
(u = 1.
and discussion
4.. Calculations The Ar-HD
(u = 1 ,j = 2) predksociating
states have been observed experimentally
[3] in the SL(0) band of
the vdW complex, which corresponds to transitions between states of Ar-HD(u” = OJ” = 0) and Ar-HD(u’ = 1, 3*’= 2)_ The allowed transitions consist of four branches: N branch P branch
, ,
J” 1 I” + I’ = I” -
3 , J’ = J” -
1,
J”
1 , J’
1 , J”,
=: 1” +
I’
= 1” -
=J”
-
J”
-f. 1 ,
41
\‘olumc
18 Februarv 1953
CHE~llCAL PHYSICS LETTERS
95, number I
I&(E). CM- I
I:ig. 3. A typicsl rc.son~~uxQ (rotational an$c) trajectory
(dg) for the complex eifenwiue associated with the mettlstable (I = ~5.1~ 3) state of the Ar-HI)@ = 1, j = 2) system.
R briIItctl .
J”
= I”
T branch
J”
zt 1” --L 1’ =Z”+j
.
+ I’
=
I”
-I-
1
_J’=J”
- 1 .f”
_J’=J“+
1 _
.J” + 1 .
Only the N- and T-branch spectral lines are well resolved and their widths can be deterlnined exp~r~lent~ly_ The SIzCCCC catcula~ions were carried out for all N- and T-branch predissociating states observed experimentally. The basis set of Ml includes all drannels corresponding to u = 1, i = 0, 1, 2, and 3_ Cllannels corresponding to llD(v = 0-i) swtes can be safely ignored, as the vibrational predissociation (VP) rates from u = 1 to u = 0 states \verc‘ ford to he aI least three orders of magnitude smaller than the rotational predissociation (RP) rates in the II= 1 stafts 11S]. Fig. 2 depicls an example of the isotropic-channel potential curves (specified by] and Z) COTresponliin~ to 111ccase cd-J = 8. The experinlentally observed AI---HD(v = 1 ,I = 2) SI(0) RP widths correspond ICI rhosc of rhc mcrasrable levels associated with thej = :! potential curve manifold. Both open v = 0 andi= 1) a& &FXJ (i = 3) channels arc illustrated here. Simx the Ar- llD surface used in this work was obtained by first vibrational preaveraging [with @V9v=I,~=2(~)] over tlte t coordinate. tltc coupling-matrix detnents [the last term in eq_ (S)] beconle ~m3.t
A=?1___ (x,:,,+(R)I iFACu = 1.i , .
= 2iReia)lX,,(R))lx.(l~‘:li:J)
11 was found 11~1 A,,,., = 3 is sufficient
6,J4_,,Ie
-
to obtain converged results to within 0.001 cm_1 _ The matrix
of interest
CHEhlICAL
Volume 95, number 1
PHYSICS
18 February 1983
LElTERS
in the I-& representation is ofIV,, byN, block form, whereN, is the number of isotropic basischannels included in the SFCCCC calculations. Within each diagonal block specified by the channel quantum number 7 (=~$Uil4), we use the orthonormal harmonic-oscillator (HO) L2 basis 14-71
XAR) = @/~1’22n”!)“2jfJ-p) exp(_~p’x’)
,
where AY_,, is a Hermite polynomial and x =R - Ro ( with R,-, and fi being adjustable parameters), to expand the bound as well as to discktize the continuum radial wavefunctions I,!&(R) defined in eq.(5). The kinetic matrix elements in the HO basis can be worked out analytically. Since the radial potentials v,(R) were generated in IZUnzerical rather than in analytic form, the radial coupling-matrix elements (x,~(R)I ~A(Rei”)lx,(R)) were computed most conveniently and accurately using the new computational scheme designed for the extension of complex-coordinate transformation to numericai potentials [7] _ Using the abovementioned procedure and the potential surface shown in fig. 1~ we have performed the SFCCCC calculations for all N- and T-branch predissociating states of Ar-HD(u = 1, i = 2) observed experimentally_ Shown in fig_ 3 is a typical example of the resonance LYtrajectory of the complex eigenvalue correlated with the metastable state [.Z= 5, u = 1, i = 2, Z= 3] _The desired resonance position (E R, --r/2) is clearly identified by the stationary point in the diagram [E8 = Re(E), P/2 = -ImQ] _Th e results of the present SFCCCC calculations are summarized in table 1. Shown in table 1 are also the theoretical
results obtained recently by Hutson and Le Roy [ 151 using the close-
coupling scattering method and a similar procedure to generate the Ar-HD surface from the BC3(6,S) Ar-HI potential_ The overall agreement of the predicted RP widths (I’) is seen to be excellent (to within l--2%), providing a mutual check on the reliability of both methods. The predicted resonance energies (Ed) are also in close
agreement, except they are uniformly differed by ~0.15 cm-r. This displacement is mainly attributed to the fact that slightly different diatom level energies were used in both computations_ (In our SFCCCC calculations, the HD(u = I,i=
O-3)
vibration-rotation
energies (in cm-l)
used were -2X407,
-170_073,0_00,253_629,
respec-
Table 1
Ccmparbcn of SFCCCC a) with close-couplinS scatter@ (CCS) b) level enrrgies (ER) and widths (P) (in Cm-‘) for the Ar-HD (U = 1 ,,i = 7) resonances. The zero of energy is defmed asj = 2 rotational energy threshold 1 0 1 2 3 4 5 6 7 8 9
2 3 4 1 5 2 6 3 7 4 8 5 9 6 7 -.-..___
SFCCCC ER
-ccs bR
&R
-26.477 -25.775 -24.254 -22.1 a2 -21.950 -19.067 -18.689 -15.243 -15.093 - 10.728
-26.322 -25.619 -24.097 -22.027 -21.794 -18.912 -18.733 -15.085 -14.939 -10.570
-5.559 -5.447 +0.195 +6.416 ______.___
-5.407 -5.299
-10.598
a) Molecular constants used are p(Ar-HD)
b)
-10.446
.._ ._
= 2.8094762
_.. _____. ___._.
c)
r (SFCCCC)
l.(CCS)
0.155 0.156 0.157 0.155 0.156 0.155 0.156 0.158 0.154 0.158
0.667 0.728 0.739 0.955 0.731 0.787 0.705 0.682 0.666 0.591
0.690 0.720 0.730 0.948 0.721 0.784 0.696 0.676 0.656 0582
0.152 0.148
0.501 0.543 0.413 0.336 .___~___. -.. ..-..-
0.489 0.535
0.157
0.610
_ ..-
0.602
amu. p(HD) = 0.671711245 amu.
Ref. [15]_
C) GR
ditom
_ Eil-c_ This appro.ximnte constant shift in energies is mainly due to the fact that sliihtly different level energies were used in both calculations. See test for details.’
= Egcs
43
CHEMICAL PHYSICS LETTERS
Volume 95. number 1
;i -- -..__. i .__i..___.~__~.._I_i__
: I
6
!
AI: 10
i
18 Februm
1983
Fig. 4. Comparison of cAculated (SFCCCC) and esperimen131 widths for predissociating states of Ar-HD(u = 1.j = 3)_
tively. relative IO the (.u = 1.j = 2) level. Whereas the corresponding ones used by Hutson and Le Roy were -255.54, - 170.16. 0.00, and 253.77.) Since these diatom level energies appear oniy in the diagonal matrix elements, they have negligible effect cm the width ca!culations. The SFCCCC calculated N- and T-branch widths are compared with the experimental ones of McKellar [3] in iig. 4. The two sets of data appear in good harmony, considering the sensitivity of the width calculations with respect to potenti surfaces used [S] _The discrepancy between theoretical and experimental widths ’ as well as the recent hyperfine spectroscopy of Ar-Hz [ 191 suggest that further refinement of the BC3(6,S) anisotropic potential may be possible. In conclusion. the present work lends further srrong supports to previous conclusions [4-j] that the CCCC formalisms provide reliable and highly efficient methods for direct prediction of predissociating resonance energies grid widths of v&V ~i~oleculrs. Acknowledgement The authors are grrltrful to Dr. McKellar for providing us his experimental data and to Dr. Hutson for his close-coupliri~ scuttcrin g results. This research was supported in part by the United States Department of Energy (Division of Chemical Sciences) under contract No_ DE-ACOZSOER1074S, by the Research Corporation and by the Alfred 1’. Sloan Fotmdation. Acknowledgement is also made to the Donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of this work. We are grateful to the United Telcom Computing Croup (Kansas City) for generous support of the CRAY computer time. ’ Affcr thr rumpletion of this lvork. the authors received 3 preprint from flutson and Le Roy [ZO], who reperformed the RP calculations using different procedures to obtGn the XI-IID
potential. Their new widths differ =9_7% from rhcir previous ones [ 15 1.
References 111 D.11.Levy. _-idun. Cire~n. I’hys. 17 (1961) 323. 121J.A. lkswick zmd J. Jortncr. Advan. Chem. Phys. 47 (1981) 363. [ 3 ] A.R.W. McKcllar. lkaday l)isrussions Chem. Sot. 73 (1982). to be published. 44
Volume 95, number 1.
CHEMICAL PHYSICS LETTERS
18 February
1983
]4] S-1 Chu, J_ Chem. Phys. 72 (1980) 4772.
[5] S.1Chuand K-K_ Datta, J. Chem. Phyr 76 (1982j 5307_ [6] S-1 Chu, Chem. Phys. Letters 88 (1982) 213. [7] K-K. Datta and S-1 Chu, Chem. Phys. Letters 87 (1982) 357.
[S] R.J. Le Roy andJ_C_ Carley. Advan. Chem. Phys. 42 (1980) 353. [9] E. Baislev and J.hf. Combes, Commun. Math. Phys. 22 (1971) 280; J. Aguilar and J-hi. Combes, hlath. Phys. 22 (1971) 265; B. Simon, Commun. Math. Phys. 27 (1972) 1; Ann. Math. 97 (1973) 247. [lo] Proceedings of the 1977 Sanibel Workshop on Complex Sealin,.0 Intern. J. Quantum Chem. 14 (1978) [ 111 W.P. Reinhardt, Ann. Rev_ Phys. Chem. 33 (1982) 223. [ 121 1-C. Percival and M.J. Seaton, Proc. Cambridge PhiL Sot. 53 (1957) 654. [ 131 L. Zandee and J_ Reuss, Chem. Phys. 26 (1977) 345_ [ 141 K-T_ Tang and J-P. Toennies. J. Chem. Phys. 74 (1981) 1148_ [ 151 J.M. Hutson and R-J_ Le Roy, Faraday Discussions Chem. Sot. 73 (1982). to be publisbcd. [ 161 D-hi. Bishop and SK. Shih, J. Chem. Phys. 64 (1976) 162_ [ 171 S-1 Chu, J. Chem. Phys. 62 (1975) 4089. [ 181 K-K. Datta and S-1 Chu, unpublished results. [ 191 M. Waaijer, Ph.D. thesis. Katholicke Universiteit, Nijmqen,The Nederlands (1981)_ [IO] J_hl. Hutson and R-L Le Roy. J. Chem. Phys., submitted for publication.
No. 4.
45
95. number
CHEMICAL
1
ROTATIONAL
P~D~SSOCIATION
SPACE-FIXED
COMPLEX-COORDINATE
PHYSICS
OF Ar-HD
LETTERS
1983
VAN DER WAALS COMPLEX:
COUPLED-CHANNEL
Krishna K. DATTA and Shih-I CHU* of Clictttisrry. iiltinmit~ 0-f Kansas. lmsrettce. Kattsas 66&j.
lk~p~rllIlc~lil
18 February
CALCULATIONS
USA
The sp;ts&ixrd c~,mplcs~~orditl~~~ couptrd-chxmet (SFCCCC) merhod is npplicd tn the zccumte determination of the Ic~cl cncqir‘s zmd widrhs flit~~tim~s) of prcdissocisting st;lfes of Ar-llD(r~ = 1.j = 2) van der IVanIs molecule, usins the (‘arlcy--Lr Rt+- BC3(,6.S) potrntisi for AI-H_. , The calcuktted widths are in good i~armony with the experimental data of McKcllar 3nd in cscrllcnt z~precmcnt with the close-coupling scattering results of Hutson and Le Roy.
I. Introduction tnuch interest in the study of spectroscopy, structure and predissociation dynamics of weakvan dcr Waals (vdW) molecules [ I_?]_ Reliable anisotropic potential surfaces for several vdW complexes, particularly the rare-gas-H2 systems, are now becomin, 0 available. In this paper. we are concerned with accurate crttctrlations of rite resonance energies and rotational predissociatiotl (RP) linewidths (iifetimes) of the Ar-HD vti\Vmoleculr_ This study is prompted by the recent successful experimental determination of the RP linewidths 11~ XicKcllar [ 31 from the ~i~tlproved) IR high-resolution spectra of Ar-HD and &-Hz systems. The direct compak~n bctwc‘cn theoretical and esperimental results allows a critical assessment on the quality of the theOretiCtd Tltcrc
is currctttly
ly bound
snctiwtis as wctl 3s rhe reliability of the potential surfaces used in the calculations. Presiws theoretical approaches for the treatment of predissociation by internal
vibration/rotation
have been
summarized by Bcswick and Jortner [Z]. Recently, we have developed two alternative practical and accurate appt~~~ltcs for the study of rotational predissociation of vdW molecules. They are the comples-coorditzare coupledthutttcV (CCCC) ~nethods, one formulated in the s,uacc-fived (SF). the other in the body’-fhd (BF) frames of COordinates. The SFCCCC theory 14.51 is more appropriate for the treatment of weak-coupling vdth’ compleses (such as Ar--112). whereas the BFCCCC theory [6] more natural for strong-coupling complexes (such as Ar-HCl). The mitiry and adranrage of the CCCC methods may be summarized as foflows: fl) It is an ab initio method (given 3 defined -‘exact” hamiltonian). (2) Only bound-state structure calculations are required and no asymptotic conrtirions need to be enforced. (3) The resonance energies are obtained directl& from eigenvalue analysis ofappropri;ttr non-hcrmite;ln matrices. the imaginary parts of the complex eigenvnlues being related directly to the tifctimcs of the vdW complexes. (4) It is applicable to many (open and closed) channel problems involving muitiply coupled continua. These methods have been applied successfully to 8 number of vdW molecules [4-71. In section 2. rhe SFCCCC theory appropriate for the Ar-HD study is briefly outiined, In section 3. we discuss ths potential-energy surface chosen in the current study_ The predicted RP linewidths are compared with the expcrimcntal data 3s well 3s other theoretical results in section 4.
0 009-2614/83/0000-0000/S
03-00 0 1983 Nor&Holland
CHEhWAL PHYSICS LETTERS
Volume 95, number 1 2. Complexcoordinate
coupledchannel
approach to predissociation
18 February 1983
by internal vibration and rotation
The hamiltonian ofa triatomic vdW molecule A.__BC, after separating out the motion of the centre of mass, may be expressed in the SF frame as H(R,r)
= (1/~~)[-r2’R-l(a2/aR2)R
+Z2(k)/R2]
+H&)+
V(R. r, 0) -
(1)
Here p =mA(mB + mC)/(T?zA +mg + nrC) is the reduced mass, R is the axis of the complex, r that of the diatom BC, Z is the angular momentum of BC and A about each other, Hd is the vibration-rotation hamiltonian of the diatom, V(R. r. tl) is the interaction potential of A and BC. and 6 = COS-~(~^-~)_Th e interaction potential for weak-
coupling complexes may be conveniently expressed as an expansion in terms of a Legendre polynomial as well as a power series expansion in the diatom stretching coordinate t(r) [S] :
where g(r) = (r - r&/r0 and r. is an isotope-independent constant. The eigenfunctions of Hd(r) may be written aa& I&&i), where u andj are respectively the vibrational and rotational quantum numbers of the diatom BC. qt,lf is the spherical harmonics, and Q(r) satisfies the radial Schr6dinger equation
{-(tl'/~Bc)['-1(a2/a32)3 - j(j + 1)/G]
+ V&-)
-E,(u,
j)}Q&)=
0 _
(3
are the vibration-rotation eigenvalues of the free diatom, flgc is the reduced mass, and vg&) In eq. (3, &(U) is the diatom potential_ In the spaced-fixed total angular momentum (J,M) representation,/ =I +j, a convenient angular basis for wavefunction expansion is the total angular momentum eigenfunction defined by
where ( I ) is the Clebsch-Gordan
coefficient and Yx_,,*~ is the spherical harmonics_ It is expedient to define a scheme for labeling of the eigenstates of the complex uniquely_ In the zero-coupling limit, each isotropic state is an eigenfunction ofHd, Z2, 52 and Jz, and may be labelled by ~ujUiI~_The latter may be decomposed into radial and angular functions IvjuAfl= $&l(R)
@Jr)YnJ:,-(F,k)
-
(3
When the coupling is turned on, the state will be only an eigenfunction ofJ2 and Jz and only3, 111and the parity p = (_Qi+‘+J will be good quantum numbers in the perturbed state. However, for weak-coupling complexes such
as Ar--HZ and Ar-HD, v, j and I remain nearly good quantum numbers. Thus the isotropic function IL$VM)provides a good zeroth-order picture for level energies and widths. We now discuss the complex-coordinate coupled-channel (CCCC) formulation in the SF frame. According to the theory of dilatation or cqmplex-coordinate transformation [9-l l] the energy (ER) and the width (I’) associated with a metastable state of vdW molecule may be determined by the solution of the complex eigenvaluc of a non-hermitean hamiltonianI-I,(Rei”, r), obtained by applying the complex-coordinate transformation [9] to the vdW bond (dissociating) coordinate, R + R e’“. In the CCCC formalism, the total wavefuncrion of the coordinate-roiated hamiltonian Ha(ReiC’, r), for a given J, M and p, is expanded in terms of the complete set of the
isotropic state functions IujZJM)allowed by the symmetry. We further expand the radial function $&(R) square-integrable (L’) basis function Ix,,(r)],
in terms
of an orthonormalized
39
Volume 95, number 1
IS February 1983
CHEMICAL PHYSICS LETTERS
where y specifies the channel quantum numbers, y = (u$V.M),iV, is the size of the truncated (x,~ Ix,,, > = 6,,,,,_ For convenience, let us defme the channel basis function to be ^ I -PI ) = x,, CR1 Q&l Y ?$j Cr. RI
radial basis, and
(7)
and arnnge the order of the matrix elements ofH,(ReiU, r) in such a way that II is allowed to vary from 1 toN_,. witl1in each channel block 7 = (u$VM). The mat&v element in the jvz) representation is then (7’,l’JHc,(Rein-,r)17rl) = eeziu (fi’/$)
+ ‘;;1@‘j?~UU’
6jj’611’6,,,,‘6JJf6,~jnl
- R-‘(a’/aR’)R
1*q
F
+ l(Z + l)/R21X~,)6,,~6jj’6,~~,.~~~~~,
(Q,*j’ I Pi Q,j)(Xnll v,x_(Re’*)lX,,)f,(li”,
wherefk is the Percival-Seaton coefficient [I 21. The resulting matrix ofH, complex eigenv3lues may be determined via the secular determinant DetI(U&,,~_~,,
is
3
lit-0 ~_TJJ,,J,* T
symmetric
complex
one whose
- Et I = 0 -
The dtsired metast3ble pies cigenvalues.
3. Potential-energy
@I
(9)
st3tes arc then identified
by the stationary
points
[4-7,101
of the (Ytrajectories
of com-
surface
111the present study. the Ar-HD potential surface was derived from the Buckingham-Corner-type BC3(6,8) A-l l2 poteI:ti3l of Carley and Le Roy [S]. The reasons for this choice are as follows: (i) In one recent SFCCCC study. WC have carried out detailed RP linewidths calculations for the Ar-H2 system, using six realistic anisotropic potentials obtained recently from experiments [5]. It was found that the RP linewidths are sensitive to the surfaces used. 111particular. the widths predicted for the three (less accurate) Lennard-Jones (LJ) potentials adopted vary as lsrge 3s 3 fxtor of four. However, the agreement among the more recent potentials (namely, the BC potential of Zandee and Reuss [ 131. the BC, potential of Carley and Le Roy [S], and the semi-empirical potential of Tang and Toennies [ 141). are much closer - to within 30%. (ii) The BC3(6,S) potential of Carley and Le Roy is the only surfxc which contains the information about its dependence on the length of the diatom bond. (iii) Recently Hutson and Le Roy [ 151 have independently performed the RP calculations for the Ar-HD system using the BC3(6,8) potentitl and the traditional close-coupling scattering method, allowing for 3 detailed comparison between different theoretic31 3pproaclies. The BV3(6,S) Ar-H, potential [S] is expanded in the form
where ,t = (r - ro)/rn
with I-~ = 0.7666348
VAX-(R)= AXkexp(-PAR)
A_ The radial strength function
- D(R)(C~‘/R6
t C;“/R”)
*
where D(R) = exp[3(RolR =I, 40
- 1)3] .
forR
=Rz’
forR >Rg
_
.
VA,(R) 113sthe form
Volume 95, number 1
CHEMICAL PHYSICS LETTERS
:
Ar-HD
Fig_ l_ The diagonal vibrationally preavemged anisotropy strengh radial funcrions ?~(u,ilR) for Ar-HDtu = 1.j = 2).
The parameters (A”“, B k_._ete.) are listed in ref. [6]. To obtain the Ar-ND potential, we first performed the vibrational averaging over the 5 coordinate of eq. (IO), using exact ~,i(~) eigenfunctions of Hd(r). The diatom vibrational wavefunctions GUj(r) were obtained by the numerical solutions of the accurate HD potential data of Bishop and Shih [16]_ Nest we carried out the well-known asymmetric isotope frame transformation [17] to a coordinate system based on the centre of mass of HD, which is located =0_166556r from the bond midpoint. This transformation introduces Legendre terms of odd order into the potential expansion, so that the diagonal vibrationally preaveraged Ar-HD potential, for esample, has the form ?‘ttKIX ir(u,ilR,
8) = h z ~ _ ~ACu,ilRV’,fcos = , .---
The vibrationally j = 3) are plotted
averaged anisotropy in fig. 1.
8).
strength
00 radial functions
&(u,jJR)
for the case of current
interest
(u = 1.
and discussion
4.. Calculations The Ar-HD
(u = 1 ,j = 2) predksociating
states have been observed experimentally
[3] in the SL(0) band of
the vdW complex, which corresponds to transitions between states of Ar-HD(u” = OJ” = 0) and Ar-HD(u’ = 1, 3*’= 2)_ The allowed transitions consist of four branches: N branch P branch
, ,
J” 1 I” + I’ = I” -
3 , J’ = J” -
1,
J”
1 , J’
1 , J”,
=: 1” +
I’
= 1” -
=J”
-
J”
-f. 1 ,
41
\‘olumc
18 Februarv 1953
CHE~llCAL PHYSICS LETTERS
95, number I
I&(E). CM- I
I:ig. 3. A typicsl rc.son~~uxQ (rotational an$c) trajectory
(dg) for the complex eifenwiue associated with the mettlstable (I = ~5.1~ 3) state of the Ar-HI)@ = 1, j = 2) system.
R briIItctl .
J”
= I”
T branch
J”
zt 1” --L 1’ =Z”+j
.
+ I’
=
I”
-I-
1
_J’=J”
- 1 .f”
_J’=J“+
1 _
.J” + 1 .
Only the N- and T-branch spectral lines are well resolved and their widths can be deterlnined exp~r~lent~ly_ The SIzCCCC catcula~ions were carried out for all N- and T-branch predissociating states observed experimentally. The basis set of Ml includes all drannels corresponding to u = 1, i = 0, 1, 2, and 3_ Cllannels corresponding to llD(v = 0-i) swtes can be safely ignored, as the vibrational predissociation (VP) rates from u = 1 to u = 0 states \verc‘ ford to he aI least three orders of magnitude smaller than the rotational predissociation (RP) rates in the II= 1 stafts 11S]. Fig. 2 depicls an example of the isotropic-channel potential curves (specified by] and Z) COTresponliin~ to 111ccase cd-J = 8. The experinlentally observed AI---HD(v = 1 ,I = 2) SI(0) RP widths correspond ICI rhosc of rhc mcrasrable levels associated with thej = :! potential curve manifold. Both open v = 0 andi= 1) a& &FXJ (i = 3) channels arc illustrated here. Simx the Ar- llD surface used in this work was obtained by first vibrational preaveraging [with @V9v=I,~=2(~)] over tlte t coordinate. tltc coupling-matrix detnents [the last term in eq_ (S)] beconle ~m3.t
A=?1___ (x,:,,+(R)I iFACu = 1.i , .
= 2iReia)lX,,(R))lx.(l~‘:li:J)
11 was found 11~1 A,,,., = 3 is sufficient
6,J4_,,Ie
-
to obtain converged results to within 0.001 cm_1 _ The matrix
of interest
CHEhlICAL
Volume 95, number 1
PHYSICS
18 February 1983
LElTERS
in the I-& representation is ofIV,, byN, block form, whereN, is the number of isotropic basischannels included in the SFCCCC calculations. Within each diagonal block specified by the channel quantum number 7 (=~$Uil4), we use the orthonormal harmonic-oscillator (HO) L2 basis 14-71
XAR) = @/~1’22n”!)“2jfJ-p) exp(_~p’x’)
,
where AY_,, is a Hermite polynomial and x =R - Ro ( with R,-, and fi being adjustable parameters), to expand the bound as well as to discktize the continuum radial wavefunctions I,!&(R) defined in eq.(5). The kinetic matrix elements in the HO basis can be worked out analytically. Since the radial potentials v,(R) were generated in IZUnzerical rather than in analytic form, the radial coupling-matrix elements (x,~(R)I ~A(Rei”)lx,(R)) were computed most conveniently and accurately using the new computational scheme designed for the extension of complex-coordinate transformation to numericai potentials [7] _ Using the abovementioned procedure and the potential surface shown in fig. 1~ we have performed the SFCCCC calculations for all N- and T-branch predissociating states of Ar-HD(u = 1, i = 2) observed experimentally_ Shown in fig_ 3 is a typical example of the resonance LYtrajectory of the complex eigenvalue correlated with the metastable state [.Z= 5, u = 1, i = 2, Z= 3] _The desired resonance position (E R, --r/2) is clearly identified by the stationary point in the diagram [E8 = Re(E), P/2 = -ImQ] _Th e results of the present SFCCCC calculations are summarized in table 1. Shown in table 1 are also the theoretical
results obtained recently by Hutson and Le Roy [ 151 using the close-
coupling scattering method and a similar procedure to generate the Ar-HD surface from the BC3(6,S) Ar-HI potential_ The overall agreement of the predicted RP widths (I’) is seen to be excellent (to within l--2%), providing a mutual check on the reliability of both methods. The predicted resonance energies (Ed) are also in close
agreement, except they are uniformly differed by ~0.15 cm-r. This displacement is mainly attributed to the fact that slightly different diatom level energies were used in both computations_ (In our SFCCCC calculations, the HD(u = I,i=
O-3)
vibration-rotation
energies (in cm-l)
used were -2X407,
-170_073,0_00,253_629,
respec-
Table 1
Ccmparbcn of SFCCCC a) with close-couplinS scatter@ (CCS) b) level enrrgies (ER) and widths (P) (in Cm-‘) for the Ar-HD (U = 1 ,,i = 7) resonances. The zero of energy is defmed asj = 2 rotational energy threshold 1 0 1 2 3 4 5 6 7 8 9
2 3 4 1 5 2 6 3 7 4 8 5 9 6 7 -.-..___
SFCCCC ER
-ccs bR
&R
-26.477 -25.775 -24.254 -22.1 a2 -21.950 -19.067 -18.689 -15.243 -15.093 - 10.728
-26.322 -25.619 -24.097 -22.027 -21.794 -18.912 -18.733 -15.085 -14.939 -10.570
-5.559 -5.447 +0.195 +6.416 ______.___
-5.407 -5.299
-10.598
a) Molecular constants used are p(Ar-HD)
b)
-10.446
.._ ._
= 2.8094762
_.. _____. ___._.
c)
r (SFCCCC)
l.(CCS)
0.155 0.156 0.157 0.155 0.156 0.155 0.156 0.158 0.154 0.158
0.667 0.728 0.739 0.955 0.731 0.787 0.705 0.682 0.666 0.591
0.690 0.720 0.730 0.948 0.721 0.784 0.696 0.676 0.656 0582
0.152 0.148
0.501 0.543 0.413 0.336 .___~___. -.. ..-..-
0.489 0.535
0.157
0.610
_ ..-
0.602
amu. p(HD) = 0.671711245 amu.
Ref. [15]_
C) GR
ditom
_ Eil-c_ This appro.ximnte constant shift in energies is mainly due to the fact that sliihtly different level energies were used in both calculations. See test for details.’
= Egcs
43
CHEMICAL PHYSICS LETTERS
Volume 95. number 1
;i -- -..__. i .__i..___.~__~.._I_i__
: I
6
!
AI: 10
i
18 Februm
1983
Fig. 4. Comparison of cAculated (SFCCCC) and esperimen131 widths for predissociating states of Ar-HD(u = 1.j = 3)_
tively. relative IO the (.u = 1.j = 2) level. Whereas the corresponding ones used by Hutson and Le Roy were -255.54, - 170.16. 0.00, and 253.77.) Since these diatom level energies appear oniy in the diagonal matrix elements, they have negligible effect cm the width ca!culations. The SFCCCC calculated N- and T-branch widths are compared with the experimental ones of McKellar [3] in iig. 4. The two sets of data appear in good harmony, considering the sensitivity of the width calculations with respect to potenti surfaces used [S] _The discrepancy between theoretical and experimental widths ’ as well as the recent hyperfine spectroscopy of Ar-Hz [ 191 suggest that further refinement of the BC3(6,S) anisotropic potential may be possible. In conclusion. the present work lends further srrong supports to previous conclusions [4-j] that the CCCC formalisms provide reliable and highly efficient methods for direct prediction of predissociating resonance energies grid widths of v&V ~i~oleculrs. Acknowledgement The authors are grrltrful to Dr. McKellar for providing us his experimental data and to Dr. Hutson for his close-coupliri~ scuttcrin g results. This research was supported in part by the United States Department of Energy (Division of Chemical Sciences) under contract No_ DE-ACOZSOER1074S, by the Research Corporation and by the Alfred 1’. Sloan Fotmdation. Acknowledgement is also made to the Donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of this work. We are grateful to the United Telcom Computing Croup (Kansas City) for generous support of the CRAY computer time. ’ Affcr thr rumpletion of this lvork. the authors received 3 preprint from flutson and Le Roy [ZO], who reperformed the RP calculations using different procedures to obtGn the XI-IID
potential. Their new widths differ =9_7% from rhcir previous ones [ 15 1.
References 111 D.11.Levy. _-idun. Cire~n. I’hys. 17 (1961) 323. 121J.A. lkswick zmd J. Jortncr. Advan. Chem. Phys. 47 (1981) 363. [ 3 ] A.R.W. McKcllar. lkaday l)isrussions Chem. Sot. 73 (1982). to be published. 44
Volume 95, number 1.
CHEMICAL PHYSICS LETTERS
18 February
1983
]4] S-1 Chu, J_ Chem. Phys. 72 (1980) 4772.
[5] S.1Chuand K-K_ Datta, J. Chem. Phyr 76 (1982j 5307_ [6] S-1 Chu, Chem. Phys. Letters 88 (1982) 213. [7] K-K. Datta and S-1 Chu, Chem. Phys. Letters 87 (1982) 357.
[S] R.J. Le Roy andJ_C_ Carley. Advan. Chem. Phys. 42 (1980) 353. [9] E. Baislev and J.hf. Combes, Commun. Math. Phys. 22 (1971) 280; J. Aguilar and J-hi. Combes, hlath. Phys. 22 (1971) 265; B. Simon, Commun. Math. Phys. 27 (1972) 1; Ann. Math. 97 (1973) 247. [lo] Proceedings of the 1977 Sanibel Workshop on Complex Sealin,.0 Intern. J. Quantum Chem. 14 (1978) [ 111 W.P. Reinhardt, Ann. Rev_ Phys. Chem. 33 (1982) 223. [ 121 1-C. Percival and M.J. Seaton, Proc. Cambridge PhiL Sot. 53 (1957) 654. [ 131 L. Zandee and J_ Reuss, Chem. Phys. 26 (1977) 345_ [ 141 K-T_ Tang and J-P. Toennies. J. Chem. Phys. 74 (1981) 1148_ [ 151 J.M. Hutson and R-J_ Le Roy, Faraday Discussions Chem. Sot. 73 (1982). to be publisbcd. [ 161 D-hi. Bishop and SK. Shih, J. Chem. Phys. 64 (1976) 162_ [ 171 S-1 Chu, J. Chem. Phys. 62 (1975) 4089. [ 181 K-K. Datta and S-1 Chu, unpublished results. [ 191 M. Waaijer, Ph.D. thesis. Katholicke Universiteit, Nijmqen,The Nederlands (1981)_ [IO] J_hl. Hutson and R-L Le Roy. J. Chem. Phys., submitted for publication.
No. 4.
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