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Ab initio study of the isomerization HNC → HCN. I. Ab initio calculation of the HNC ⇌ HCN potential surface and the corresponding energy levels
Chehxl PhysicsRZ(19X3) 317-336 Norch-Holland
Puhlishin_e Company
AB INlTllO STUDY OF THE lSOMERlZATlON HNC -+ HCN. 1. AB INlTIO CALCULATION OF THE HNC = HCN POTENTIAL AND THE CORRESPONDING ENERGY LEVELS
SURFACE
and Robert J. BUENKER
Received
3 May 1983: in final form 4 Augusr 19S3
The results of an ab initio SCF+ Cl calculation of the porcntls; rwfxr for rhe reaction HSC = HCS arc prcssnrcd. Ths soordinacs~ for J approximative minimum-energy parh is obtained minimizing the cnsrgy nirh rcspccr lo both swkhing numbrr of values of the bending coordinate 0. This potential CUT\‘C is corrsctcd by adding Ihc zero-point snsrgics corresponding to the stretching vibrations. assuming rhsm to be B dcpsndcnt. The corresponding vihr3Iionsl-ror~rionrtl energ>levels and wavefunctions are calculmcd using an appropriate vihrxion-rorarion hxniltonian whsrehy 111e porsnrhland kinetic-energy terms. as well as the wnvsfuncrions are represented by Fourier series in 8.
1. introduction The isomerization reactions RNC * RCN. representing classical esamples of unimolecular reactions_ have been the subject of extensive experimental and theoretical studies for some time. Holvever. due to the fact that the higher members in the RNC series have too many degrees of freedom thus preventing an accurate ab initio calculation of the potential surface for isomcrization. and that the simplest member of this family, HNC, till recently has been observed only under non-equilibrium conditions or at very low temperatures, there has been no opportunity to compare the correspondin g experimental and theoretical results concerning the kinetics of the rearrangements. This situation has considerably changed due to the results of latest experimental and theoretical works. giving new impulses for investigations in this direction. A spectrum of the HNC was first observed twenty years ago by Milligan and Jacos [l] who observed the IR absorption in an argon matrix at 4 K. and in both argon and nitrogen matrices at 14 K [2]. Snyder and Buhl [3.4] observed a new interstellar emission line at 90665 MHz which was attributed to the J = 1 f- 0 transition of HNC. Arrington and Ogryzlo [5] obtained the JR emission spectrum of HNC and gave the values for both the band center and the AB term for the N-H stretching vibmtions. An extensive ab initio calculation of the HNC G= HCN potential surface has been performed by Pearson et al. [6]. A double-zeta plus polarization basis has been used. The equilibrium geometries of HNC. HCN 0301-0104/83/0000-OOOO/SO3.00
8 1983 North-Holland
and the critical conformation have been determined. The SCF calculations predicted HNC to lie 9.5 kcal/mol above HCN. while the corresponding Cl result was 14.6 kcal/mol. For the barrier height. seen by the HNC molecule. the values of 40.2 and 34.9 kcal/mole have been obtained in SCF and CI calculations. resprctively. An attempt to check the validity of the predicted value for the zero-point difference between HNC and HCN has been made by Blackman et al. [7]. Since the microwave spectrum of HNC did not appear under the experimental conditions applied. it was concluded that the lower limit for the isomerizaby Botschwina tion energy must lie at = 11 kcal/mol. The force constants for HNC have been determined et al. [g]_ Taylor et al. [9]. Creswell and Robiette [lo] and Murrell et al. [ll]. Gray et al. [12] applied the reaction-path hamiltonian in order to study the tunneling in the reaction HCN + HCN. In the work by Bunker and Howe 1131 the potential surface calculated by Pearson et al. [6] has been used for calculation of the bending-energy levels in the HNC * HCN system. It was found that the tunneling can play a significant role in the isomerization process only in the vicinity of the potential barrier. The results recently reported by Maki and Sams [14] opened the discussion about the value of the isomerization energy. These authors performed spectral measurements on HNC at temperatures ranging From 900 to 1250 K. On the basis of the temperature dependence of the absorption intensity the authors estimate an isomerization energy value of 3600 f 400 cm-’ (10.3 f 1 kcal/mole). significantly smaller than the ab initio results obtained after the effects of correlation are included_ In order to find the reasons for such discrepancies. Redmon et al. [15] have undertaken a very extensive study of the effect of various factors (size of the basis. method of calculation. etc.) on the calculated isomerization energy and have come to the conclusion that the best value obtained (15.1 kcal/mole) agrees with the results of previous theoretical works. but differs significantly from the value [14] deduced from experiment_ New experimental results of Pau and Hehre 1161 (14.S & 0.2 kcal/mole) support the theoretical predictions. Lehmann et al. 1171 have recently observed several highly excited stretching states using an intracavity photoacoustic technique In the present paper we report the results of a new ab initio calculation of the potential surface for the reaction HNC --‘ HCN. These results are used in the following paper to calculate the unimolecular rate constants according to a slightly modified RRKM model. The proposed approach requires the knowledge of the vibrational-rotational energy levels in the entire range of the reaction coordinate (being predominantly bending lvibration). in which case the couplin g between this degree of freedom with the others cannot be neglected. For an efficient and accurate calculation one needs the form of the hamiltonian which minimizes this coupling and the reaction surface must be known in more detail, allowing for a consideration of the variation of the bond lengths. as well as the correspondin g force constants with the bending coordinate. An appropriate form of the vibration-rotation hamiltonian is presented in section 2 of this paper. In section 3 it is shown how the problem of obtaining the energy levels can ultimately be reduced to a number of one-dimensional treatments not neglecting the important coupling of various degrees of freedom. The results of these calculations are presented in section 5.
2. Vibrational-rotational
hamiltonian
In a more rigorous theoretical treatment of the isomerization reactions in triatomic molecules it is necessary to calculate the vibration-rotation energy levels * going beyond the simple harmonicoscillator-rigid-rotor approximation. Therefore one needs to solve the nuclear Schrijdinger equation with the rotation-vibration hamiltonian allowing for large-amplitude displacements in all the modes of motion. Since a solution of this problem is connected with considerable computational expenditure (it should not be forgotten that a calculation of a three-dimensional potential surface must precede) it is necessary for * 11 is assumed
that the effect of excited electronic slates can be neglected.
practical applications to find a wav to reduce the dimension (six) of the nuclear Schriidinger equation by treating. of course if physically justified. some of the degrees of freedom as independent from one another. The key role in the isomerization process in a triatomic molecule is usually played by the bending vibrations which. when excited enough. can lead to the rearrangement_ Therefore it is convenient to use such a vibration-rotation hamiltonian which minimizes the coupling between the bending vibrations and other motion modes. Obviously_ the hamiltonian in terms of the generally ustzd internal coordinates (three internuclear distances [lS] or two internuclear distances and the angle between them [ 19.20]) does not fulfil this requirement because the internuclear distances can he drastically changed during the large-amplitude bending vibrations leading to the molecular rearrangement (e.g. the C-H distance is changed from roughly 2 to 1 A in the HNCe HCN reaction)_ On the other hand. in the course of a highly excited bending vibration the molecule passes twice through the linear nuclear configuration lvhereby the gradual transformation of a rotational degree of freedom of the non-linear molecule into the second component of the degenerate bending vibrational mode of the linear molecule takes place. In order to ensure a unified it is necessary to consider the treatment over the entire range of variations of the bendin, 0 coordinate rotations with the smallest moment of inertia (becomin g zero at linear molecular geometry) together \vith the bending vibrations. In order to keep this problem in one dimension (this is possible due to the usial symmetry of the potential if polar coordinates are used) the moving system should be attached to the molecule in such a way that at alllinear geometries one of its axes coincides \vith the molecular axis. This condition is not satisfied if the Eckart conditions [21] are used. and also not in the method developed by Bunker and co-workers [22-241. Therefore in the present paper a set of internal coordinates is employed which minimizes the coupling of this one (bending) mode mostly contributing to the reaction: coordinate with the other two. and the axes of the molecule-fixed system art chosen IO be coincidenr with the instantaneous principal axes of inertia (IPAI system)_ The form of the corresponding vibration-rotation hamiltonians satisfies both of the above mentioned conditions. The classical expression for the nuclear-kin&c-energy operator of a triatomic molcculs in rhe Space-fixed system has the form 2T13,, = 2
,I,;( ji’
+ ?,I + i,z)_
(1)
r-1
Introduction
of the center-of-mass 1
x0
=
-
3 c
??I,X;.
&=-
Iv;=,
leads to the separation
coordinates 1 M
X0_ Yh_ Z0 and relative
3 c I=1
of the translation
z,,=-
AI
s;_ J;_ :;.
~5. .I_<_;:
3
1 ,?l,Y#.
coordinates
c
,?l,Z,.
,=I
part of the kinetic
energy:
with PI = tnl(ml
defined
/.Lz= I?12( tn 2 + t111)/I?1 3.
+ tnz)/tnl.
choice of the Euler transformation of a vector
The
j.lL:=
tlI1t?l,/t)lj_
angles is made according to the recommendation (I’ defined in the space-fixed system into the vector
given in ref. 1251: rhe u in the moving frame is
by
a = R,(y)
R,(P) R,(cy)a’+
a;.
(4)
Fig. 1. Coordinates
used in the present
work:
r. R. B - internal
coordinates: y - third Euler angle. s. _r (s”. _I-“) is the molecular plane. x, _r coordinate axes are parallel to the X. Y axes of the space-fixed
system:
10
0 sin p
-sin
cos p
0
0
cos p
/I
x”. J”
axes of the moving
1 [ _ R,(y)=
frame.
cos y
-sin-y 0
1 (9
sin y
0
cosy
0 _
0
1
In the present paper the internal coordinates shown in fig. 1 are used: the distance from particle 3 to the center of mass of 1 and 2 is denoted by r. the distance between 1 and 2 is denoted by R. and the angle between these two lines is 0. The z axis of the moving system is perpendicular to the molecular plane and the third Euler angle y is defined such that the s axis of the moving system represents the molecular axis ” whenever the molecu!e reaches a linear nuclear configuration. The same set of internal coordinates has
already heen used by other authors [26-321. but accompanied used in the present paper. On the other hand. several authors
with the moving system different from that have employed such a body-fixed system. but
in connection with other internal coordinates [18.19.33.34]. From fig. 1 it follows that the internal coordinates and the condition inertia I, I‘ are given by the relations: r = [ ( us,
+
bxJ + ( u_l’,
)?I“‘.
+ fy,
for vanishing
of the product
[(_~2-_~,)1+(.,.~-).,)2]“2,
R=
of
(6)
with u =
r?l,M/ltl,(t?l,
+ m,).
b = ttt2hf/tnj(
tn, + tn2)_
Q is the angle between the 15 line and the .X axis. related to the Euler angle y by y = + - +’ with Cp’being * The = axis of the moving comparison
system
of the hamiltonians
is chosen derived
to be perpendicular
to the molecular
in ref. [IS] and in the present
paper.
plane as in ref. [IS] in order
the
to make SimPlier
a
I!.
angle
between
Peril- cl 01. / Srrr~r of rhc i.wntcri-_urionif>VC -
72 and the s axis in the IPAI system.
IICX
It is easy to show that
tg 2+’ = c~I-’ sin Xl/( c, R’ + c3t-’ cos 20). where c, =
111, I)1
,/( m, + m
z
)_
CJ =
tnl(nl,
+ r11,)/Al.
(7)
described in the appendis. the folloxvinp Applying the usual procedure [24.35] or usin g the approach expression for the quantum-mechanical vibration-rotation kinetic-energy operator is derived:
with
where
Z& and I_:!, are the moments ZJ.1,. = 4 [ ( c, R’ + c,r’) I::=f[(c,R’
of inertia
+ (c;R”
in the IPAI
+ c;r4 + Zc,c,r’R= cfr’
+c,r’)-(cfRJ+
+ 2c,c,r2R2
system: cos 20)““]
_
cos 28 )‘.‘2].
1:: = 0.
(IU)
The vibrational part of the kinetic-energy operator (9). ?, . possesses no cross terms rhus coniirmin~ rhc convenience of the chosen internal coordinates. The cross terms can. of course. appear in the hamiltonian through the potential. but the very simple form of TV represents a great advantage by an approsimative separation of the vibrational modes from one another. The rotational part of the kinetic ensmy operator is identical with that of Diehl et al. [18] as a consequence of the same definition of the Euler angles. The expression (9) corresponds to the volume element dr = r’R’ sin 8 sin flfId(~, (jacobian = 1) by integration of wavefunctions. The same rotational part. ?<_ has also the hamiltonian corresponding to the volume element dT = sin B sin /3TIdq,. which is actually used in the present paper. The vibrational and vibrational-rotational parts take in this case the form:
4 sin e
i;., =
--c,rR’
cos e-
( Z& - z::J
3. Calculation
of the potential
a ar
+ c,r’R
cos ea
aR
i ( c,R’-c-
(11)
surface
The calculation of the potential surface for the reaction HNC = HCN has been performed by means of the ab initio SCF and CI method under the assumption of the validity of the Born-Oppenheimer
approximation. The basis set consists of 49 contracted Cartesian gaussian functions: five s and three p 7 1. 1. l/3. 1. 1) given by Dunning [36] in addition to a set of Cartesian d groups in the contraction (4. _. functions on each the C and N atoms, with the exponents ad = 0.75 for C and ad = 1 .O for N; for the
hydrogen atom two s groups in the contraction
(4. 1) with the scaling factor 9’ = 2.0 [37] augmented by one finally one s and one p function (a.\ = 1.45. a,, = 0.5) are located in the set of p functions ( aP = 0.735); middle of the C-N bond. The configuration-interaction treatment is of the standard MRD Cl type [38.39]. with configuration selection and energy extrapolation. A core of two MOs corresponding to the Is shells of carbon and nitrogen respectively is kept doubly occupied in all configurations, the corresponding two complementary MOs with the highest orbital energies are entirely excluded from consideration; all other MOs are in principal allowed variable occupation. Only one reference configuration. corresponding to the ground state of the molecule in the SCF approximation. has been employed in order to generate the double-excitation configuration space. since all other species appear with small coefficients (less than 0.5%) in the final Cl
functions while the This procedure led to configuration spaces of = 10000 symmetry-adapted sizes of the secular equaticns actually diagonalized after configuration selection (a selection threshold of T= 10 FH was employed) were in the order of 4000. The contribution of the remaining generated configurations is accounted for by the usual extrapolation procedure_ The effect of still higher excitations is also estimated according to the formula AE = (1 - ci)( EscF - E,,,D =,) [40] which upon addition to the corresponding Cl result is referred to as the estimated full-Cl limit in the ensuing discussion.
expansion.
1. Vibrational
treatment
The vibrational-rotational energy levels are calculated diagonalizing the hamiltonian derived in section 2 of this paper. The most appropriate form of this hamiltonian is obtained transforming the expression corresponding to the volume element d7 = sin 0 sin fi lldq,. given by eqs. (11) (for TV and F,,) and (9) (for Tr) in such a way that the molecular axis at linear nuclear arrangements coincides with the z axis of the moving frame. i.e. by interchanging the A-and 2 axes. The vibrational part of the hamiltonian:
(12) is invariant i;, =
with respect 4 sin 8 ( I,,. - I,,
to this change. -c,rR’
cos 0%
The vibrational-rotational a
+
c,r’R
a
cos 6---
aR
1’
iJr, while the rotational
part can be written
pari becomes
03)
in the form 04)
with I?, + i,, + fr = - 2? and all the quantities (c?. Ii,.) referring to the new frame. Expression (14) is identical with the H6 term of the hamiitonian derived by Freed and Lombardi [19]. For an approximate calculation of the eigenvalues of the hamiltonian (12)-(14) we neglect the vibrational-rotational part (13), having only the terms that couple the vibrations with the rotations about the x axis. For a molecule possessing a hydrogen atom (m,), the relations I,, = IA -=KI,, = I_,._,. = I, hold so that it can be treated as a nearly symmetric top; the part of the hamiltonian describing x and _Y rotations
has in this case the form:
(15) It is easy to show [41] that the eigenvaiues of a symmetric top can be ohtainrd IO :I gc~xl approsimation as sums of eigenvalues of a one-dimensional pius a t\~,o-dimensional rotator with the ~mmlt’nts of inertia I, and I,, respectively. The degeneracy of a particular cncrgy level is then obtained hv multiplying the degeneracies of these two hypothetical rotators (1 and 2 for K = 0 md K f 0 rcipcctivcly for the one-dimensional rotator and 25 + 1 for the two-dimensional one). In such :I trcatmcnt the values of ths quantum numbers J and I\’ (K = I) are independent of one another. In the prcxnt paper XVL’ use this idea and calculate the energy levels corresponding to (15) as (1/7fH).I( J t 1) \\-hilt the part of the rotational hamiltonian including j_’ term is treated together with the vibrations. For a number of villues of the bending coordinate the eigenvalucs of the stretching part of the hamiltonian
i a2 -2A&+-CI c?R’ 3
i
ii
k,, (0 ).S,.S, -i higher terms of I,‘.
(S,.
$=I-_
K).
(16)
using a variational method involving expansion of the vibrational \\avefunctions in products of eigenfunctions of two harmonic oscil!ators and employing a po~ynnmial fit to the potential. as described in more detail elsewhere [42]. As indicated in eq. (16) the coefficients of these pol~nomivl expansions are assumed to be dependent on 8. The calculation of stretchin, 0 vibrational levels at home 8 v~tluss for xvhich. for the sake of computer time saving. the complete ttvo-dimensional potsntial surface \V;IS not constructed. is performed only in the harmonic approximation. The eigenvalues of the operator (16). when added to the pure bending potential obtained by optimizing the r and R values at each value of the bending coordinate 8. represent the effective bending potrntials L&(8). Thus. the last part of the hamiltonian including the bending vibrations. their interaction with stretching vibrations and the rotations about the : axis (becoming at linear geometries the molecular xxis)
are calculated
has the form: - 2fii,.,
=
I,,I, ,.
--
1 ’ (7:
--2-l;ii(c3). I:., ! a&.
(17)
with I,.,.=+
i
(qR’+
(I;= + I~,,T)/I,,!,,.
qr’)
+ (c;R’
= (l/sin’
+ +’
0)(1/c,
+ 2~ ,c3r2R2 R’ + l/c-,r’).
cos ze)l. ‘1. (IS)
and the angle Q describing now the rotations about the I axis. If the sigrnfunctions +(0. 0) of H,_, are represented as usual with R( 8) esp(i/+), the second term on the right-hand side of (17) can be transformed into
so that the problem
is reduced to one dimension only. In the present paper special attention is paid to the calculation of energy Isvsls and \vavefunctions corresponding to the hamiltonian given by eq. (17). Because of some peculiarities of the problem. the methods usually applied in a treatment of large-amplitude bending vibrations [22.23,43] are not quite appropriate. Some difficulties arise from the form of the potential having (in rhe HNC G= HCN case) t\vo
minima
separated
by a maximum between them. but not going to infinity for any B value. in contrast to the in more familiar problems. This prevents the use of the polynomial ;epresentation of the potential [43]. Another problem is a consequence of the form of the kinetic-energy operator. having singularities at 0 = 0 and 8 = n. In ordrr to avoid these difficulties a variational method is developed which makes use of the expansion of all terms in the kinetic- and potential-energy operators into Fourier series in 8 - the same idea was already applied in some simpler cases (42.441. Let us write the hamiltonian (17) in the form: usual assumption
~I,.,=T,(e)a’/ae2+T,(e)a/ae-127;(e)+V(e). with
(
-$ L+c-,R2
T,(B)=
1 CT
1
1 . T2(f?)=
(19)
1
’ -+-1
-+
( C,R2
c3r2
1
cos sin-
e
T,(B)=
-7
1
‘.,yl.+ L, -- 1 J.Lyy i I,,.I=, i -
(20) The terms T, and T? go to infinity when 0 + 0 or T. Since the volume element dT = sin 0 ITdq, is used. in a hasis consisting of sin( n13) and/or COS(N~?) functions the matrix elements corresponding to T2 would have iinite values but those of T3 would be infinite. For I= 0 the term with T3 disappears so that the basis with cos( IJo) functions is appropriate in this case “. Thus. for I = 0 the following matrix elements must be calculated: ,
7
cos( ~0)
cos( I,JO) T, 5
/0
Cod
a aB
JJJ 6 ) T2
7 1
sin f? de =
/0
Sill 8
JJo)
de =
COS(lJJe)~‘COS(fZ@)
Sin
@df?=/=
0
COS(
--II
cos(
/
0
t?Jo)[
I/
Sin
d)[T, o]
e] c05-(ne) de.
sin
CoS(d)[T,
1 COS(
I
6
-IJ’
COS(
sin
e]
IIf?)
sin( 110) de.
de.
(21)
This can be achieved easily if the functions T, sin 8 and I/sin 0 are represented by series in sin(k6). while an appropriate representation of T, sin B = T, cos 0 is a series in cos(X.0). In order to do that, the optimized values of r and R calculated for a few values of 8 are fitted first with second-order polynomials in 8. These in turn are used for calculation of the interpolated values of r and R and thereafter for generating (T, sin 8) and (T2 sin 8) values for a number of 0 values (e.g. for 0 = 0. 1, 2,. __, 1800). The obtained points are fitted with sin(k8) or cos(k0) according to the least-squares scheme. In order to obtain a brttrr representation of the potential the cubic-spline technique can be used for computing a number of supplementary points; the potential points are multiplied by sin 0 and then fitted in a series in sin(k8). For the case in which If 0 there are two terms in the kinetic-energy operator, namely T2 and 7;. which have singularities at B = 0 and T. The second of these being proportional to sin-’ 8 is particularly difficult IO handle when B + 0( 5). In the basis used for I = 0 this term would produce infinite matrix elements. To avoid this difficulty_ bases consisting of sin”’ B sin( no) and sin ‘y2 B cos(110) are used in the calculations of vibrational levels and wavefunctions for I # 0. Furthermore, it was found that a basis consisting of both Sin(jJ8)
sin( JJo
ud 1 or
)J8)
functions
cost
Y
/0
terms
COS(JJ6) cos(
sin”’
8
sin(
)>I e ) tile)
has computationally
no serious
alone, so that the following
sin”’ 6
advantage
matrix
elements
over
a basis
consisting
of either
appear:
1
sin 8 de
y A sin(n8)basis is not suirable because rhcse functions arc zero at 0 = 0 and T and thus cannot non-zero values al these angles.
represent the R,_, functions having
=
JzCoS(“ze)ITt(l
-’
0 sin(zzz8)
+ II z /
sin
ecos(J)*e)T
sin,,2
77 cos(nze)
=’ J0
sin(m9)
z J0
COS( sin”’
z J 0
sin”’
0
8
2
B]~~~~~~~dtl_
1 cos(tze)
cos(J*e) _ sin”’
_
[T2sinBc0sf?]
sm( lze)
I?ze) I~T,sin',~ecoso*e)
sin( ftze) cam sin( zzz8)
LJ sin z9do
' [ ae sizr(,zz9)
sin( zzzB)
0
B cos
V sin”’
d8in Juz:~~~~~~~)) [< sin’81
sinede=,'
sin( dl)
8COS(~~~) sm_ sin( zz8)
6 de =
-coso,Ie) r Jo sin(,lIB) [T sin’ e] $::,“I = COS(
/ 0
de.
~~~~~Og))
Jil e )
sin( zzzf?)
[ V sin’
COS( e]
Ize
)
sin( tz8 )
do.
de.
(2’)
The functions in brackets on the right-hand side of eqs. (22) are fitted in terms of sin{ X-8 ) ( T Tz size B CM 0 and i T2 sin’ 6 = T T, sin 6 cos 8). and cos( X-8) (all others) series. Both. the sin’..’ f9 cos( zzf?) md sin’/’ 0 sin(&) bases have been employed in expansions of vibrational \vavefunctions becatzse it \v;zs speculated that the first one would be more appropriate for even 1 values (in analogy with the \velI-kno\vzl properties of the eigenfunctions of a two-dimensional harmonic oscillator) and the srcozzd one for odd I values. However. the numerical results to be presented in section 5 show that the choice of a cos(zzQ) or sin(zz8) basis is quite irrelevant_ The small disadvantage caused by the rather exotic forzn of the basis functions. namely their zznn-arthogonality, is fully compensated by the capability of sin(zzQ) and COS(JZ6) series to fit very 1~41 all the functions given by (21) and (22) with only a few expansion terms and the very siznple computation of all integrals
required.
5. Results 5. I. Calculuriotz of llze porerzlial-ejzergr
szrrfuce
Three parameters required to define the molecular geometry are chosen to he the C-N bond length R. the distance r between the hydrogen atom and the C-N center of mass (M). and the angle between these two lines 8. The main parameter varied was the angle 0. playing a crucial role in the HNC = HCN rearrangement. For B values of O” (HCN). 12.6”. 30”. 65”. 70°. SO” S5”. 1ZO”. 150” and 1SO” (HNC) the other two coordinates were varied over a relatively large range. starting \vith the initial ~ahzes chosen on the basis of experimentally obtained data or taken from previous calculations [6]. This strategy w;\s chosen to enable not only the calculation of the optimal z- andR values for each 8. but also of some of the lowest-lying stretching energy levels. Special attention was paid to the regions in the vicinity of the HCN and HNC equilibrium geometries as well as to the expected location of the transition state: these parts of the three-dimensional potential surface were calculated completely. while for some other 6 values only its one-dimensional sections, corresponding to the variations of r or R (where the other two pammeters were kept fixed) were obtained. The calculated optimized values for r and R and the corresponding diagonal force constants are presented in table 1 and figs. 2 and 3. The results corresponding to three different levels of calculation arc
‘ILhlc 1 Op~irnkd XI-II and C-N lengths. corresponding diagonal force consmnls and energies as functions of the hrnding coordinate 0. All v.rlue\ in shwlic units mglcs in dqqxs
SC’I-
0
3.195
2.120
0.440
1.50
- 92.90 15
13 30 65
3.153 3.010 2.385
2.146
1.32
- 92.s950 - 92.86%
2.190
0.445 0.330 0.270
70
2.287
2.200
0.270
1.07 1.08
- 92.8212 - 92.S’OO
75 80
2.200 7.115
2.200 2.110
0.240 0.140
- 92.8205 - 92.R21X
x5
2.0x4
-.-_ 7 730
0.220
1.10 1.10 1.14
120 150 167 18(l
2.170 2.720 2.862 2.890
2.192 2.190
0.290 0.400 0.556 0.540
1.21 1.20
- 92.RS34 - 92.8860
E”,(TS-llCN) hll
L-,,(TS-HNC)
= 0.0660
EJHNC-HCN)
= 0.0155
0
3.200
2.170
0.420
1.35
- 83.1950
13 30 65
3.164 3.020 2.400
2.188
1.31
2.220
0.445 0.330 0.257
1.11
- a.1 90X -93.1703 -93.1177
70
2.297
_.__ -J ‘-JO
0.255
1.11
-93.1149
75
2.204
-.-_ ‘) ‘30
0.223
1.10
- 93.1141
X0
2.125
2.240
0.240
1.10
-93.1150
x5
2.073
2.262
0.255
1.12
-93.1183
120 150
2.180 2.750
167
2.897
2.223
0.553
1.19
- 93.1706
180
2.900
2.220
0.550
1.20
- 93.1724
E”,(-lsHCN) c’( lull
= 0.0x15
- 82.8263 - 92.852s - 92.8740
= O.OYO9
0.305 0.400
&,(-l-S-HNC)
0 13 30 65
3.200 3.169 3.020 2.410
2.190 2.204
70 75
3.308 2.218
80 85
2.140 2.085
= 0.0583
- 93.1435 - 93.1624
E,(HNC-HCN)
= 0.0226
0.420 0.440 0.330 0.250
1.35 1.29 1.10
- 93.2220 -93.21Sl - 93.2000 - 93.1475
2.240 2.250
0.250 0.223
1.10 1 .on
- 93.1422
2.255 2.276
0.237 0.252
1.07 1.11
- 93.1425 - 93.1452
2.240
93.1440
120
2.200
0.330
150
2.750
0.400
167
2.886
2.236
0.548
1.18
-93.1962
180
2.910
2.230
0.540
1.20
- 93.1980
E,,(TS-HCN)
= 0.01100
E,(TS-HNC)
= 0.0560
- 93.1698 - 93.1885
E,(HNC-HCN)
= 0.0240
SCF. MRD CI (CI extrapolated to zero selection threshold) and the estimated full CI energy obtained by applying the Davidson correction [40]. The general behavior of the r and R variations is similar to that already found by Pearson et al. [6]: the MH distance has the largest values at linear nuclear conformations being appreciably smaller for bent geometries. The variations of the C-N length with 6 are
given:
much smaller and the C-N bond length becomes slightly larger with the increase of the bent character of the molecule. Similarly, the value of the diagonal force constant for the M-H vibrations alters significantly (roughly by a factor of 2) in the course of the variation of the angle 8 from 0’ (or 18OO) to B = SO0 as
dJ---‘MM
I
‘
'CN
SCF
Fig.
2.
Dependence
of
the
M-H
and
C-N
lengths
on
the
bending coordinate 8 for the lowest HCN-HNC surface at the three levels of theoretical treatment SCF. MRD Cl (CI SKcrapolated
IO selection
threshold
zero) and full CI rstimare.
apparent from fig. 3. while the value for the other (kc_ S ) remains nearly const:~n: hrtwcen I SO” and 90” (a decrease of = 10% in its value in the interval O”-90” is found). These facts are reflected in the \*nriation.s of the stretching frequences with 0 (fig. 4): for M-H vibrations the values of 3400 and X00 cm-’ for linear species and of roughly 2600 cm-t for strongly bent geometries are obtained. On the other hand. the values of the C-N frequency remain between 2100 and 2300 cm -’ in the entire range of 8 \.;12ues. The values of the electronic energy as a function of 0 are presented in table 1 and fig. 5. Each point corresponds to the optimized r and R values for this particular 8. Takin, 0 into account that the reaction coordinate for the isomerization is predominantly determined by the bendin, 0 angle 8. rhe CUIA-es sho\vn in path” for this reaction. The results &rained in the present fig. 5 represent roughly the “minimum-energy conformation j. barrier height (correspondin, 0 10 the geometrical study for the isomerization energy (I&, (HNC) and the ‘-product” (HCN) are collected in table 2 of the transition state) relative to the “reactant” and compared with the results of other authors. Very good agreement with the results of other theoretical energ? obtained in Cl studies can be noted: for example the value of = 0.023 au for the isomerization calculation is in excellent agreement with the correspondin, 0 result given in refs. [6.9.49]. This result is also only from the value obtained bv in agreement with the experimental findings [7.16]. differin, 0 significantly Maki and Sams [14]. Taking into account that this discrepancy much exceeds the usual calculation rrro;. the reliability of the result of ref. [14] seems to be questioned. The dependence of the energy on B obtained by means of the estimated full Cl energies and the cubic-spline procedure is presented in fig. 6. For comparison. the results published by Pearson [6] are shown too. The upper curve given in fig. 6 represent
a a
.r;thk 2 Comparison
of the properlies
of 1he HNC-HCN
polenrial
surface
obtained
HCN C-H S(‘l‘
r, I61
(A)
1 .O86 1 AI624
C-N
(A,
1.122 1.1366
N-H
1.057 I .0655
1.126 1.1532
- 92.9015 - 92.88972
0.9946 0.9859
1.159 1.1586
0.985
1.15x
0.9837 0.9940
1.148 1.16X9
b,
1.075
WI
1.069 I .066 1.064
1.14x 1.159 I.150 1.148
;47j
I .064
1.148
- 93.1950 - 93.2220 -93.19241 -93.21707
1.151
- 92.8860 - 92.87454
- 92.88179
-92.S913S 0.9926 0.9954 0.9955
1.175 1.180 1.1696
0.987 0.9954
1.204 1.168
”
- 93.1724 - 93.1980 -93.16916
- 93.13707
- 93.19290
- 93.16893 - 93.159
- 92.97383
1121 1151
1501
E(au)
- 93.21707
1491 1171
C&P.
(A,
- 93.16376 1.065
studies
- 92.85533
- 92.89980
1451 14x1 191
N-C
- 92.90834 ”
”
WI
(A,
- 92.87520
1151 currcla:ion
with the resul1s of earlier
E(au)
1x1
SCI-‘ f
paper
HNC
I451 191 IlO]
in the present
- 93.23751 1.066
- 93.21363
=’
=’
1.153 0.987
IStl
1.171
I71 1141
1161 *’
Prcsen1 worh - SCF.
” Presen1 work - MRD
Cl.
c’ Presen1 work - full Cl rscimatr.
” X-basis
[15].
more realistic *’minimum-energy path” obtained by correcting the originally calculated curve with the zero-point energies corresponding to the C-N and M-H stretching vibrations, taken as a function of 0. The bending curve becomes slightly !latter due to this correction, having as a consequence somewhat sniallrr separations between bending-energy levels. 5.2. Tesr of the method for wearing the bending
uibrutions
The next series of calculations is performed to test the efficiency of the method for calculating vibrational energy levels and wavefunctions, as described in section 4 of this paper. In order to have a possibility to check the accuracy of the results obtained with the proposed method, the potential curve for the ground state of CJ, published by Jungen and Merer [52] is employed as a test example; for better comparison the large-amplitude bending hamiltonian (H”‘) by Barrow et al. (531 is used, differing from that given by eqs. (19) and (20) in the form of the 7”(e) coefficients. In these test calculations the potentialand kinetic-energy terms are fitted with seven sin(rr8) or COS(II~) terms each, which seemed entirely sufficient for a good representation of these energy functions. The bending hamiltonian is diagonal&d in bases consisting of twenty sin(n0) or cos(n0) terms for each value of 1. For even I z 0 values cos(n8) terms, for odd I values sin(&) series are normally used, but some test
TS
HNC F_,Ul”l s:)
M-H
(A,
C-N
1.201 1.202
(A,
Eh.lr
Hc-N el
E hr, s’
B(d%)
E(u)
71 70.2
- 92.8200 - 92.51054
9.7 9.5
‘II.4 30.2
51.1 19.7
- 92.79195
12.5
39.7
51.2
1.164 1.174
11.3
1.166
1.180
1.157 1.178
1.192 1.1x1
75 77 73.7
1
- 92.83179
10.6
‘7.1
JS.
-93.1141
14.1
36.6
50.X
- 93.1420 -93.1335
15.1 14.6
35.1 33.9
50.’ 19.5
- 93.05213
16.8
31.1
51.’
15.0 i 4.7 16
71
73.75
1.1945
1.1709
36.3 15.2 r’
-93.16034C
34.1 f’
.iU.? r’
> 1o.s 10.3 13.5 =’ SDQ
MBPT
Table 3 Comparison taken
[4] results.
of vibrational
” Final
results 11.51.
energy values (in cm-’
B’ In kcal/moir.
) obtained
employing
the prescm
and other wchniquss.
Potential
and geometry
from ref. [49] for ‘Ep’ state of C,
I value:
I=0
Merhod:
cos
hamiltonian:
BDD[53]
po’i431 BDD[53]
2.45
2.57
0.07
134.11 287.76 458.46
134.53 288.41 459.36
643.49
643.29
I value:
I=2
Method: Hamilronian:
COS
I=1 num[52]
sin
COL
pd[-l3]
pol[43]
m.ln~[52]
HBJ[22]
BDD[53]
BDD[53]
BDDi53]
HBJIX]
HBJ[Z]
0.00
65.57
65.54
65.34
63.26
h3.10
132.7S 286.99 458.15
132.45 286.42 457.35
209.97 373.64 553.01
209.95 373.77 553.51
209.17 372.16 550.5’
207.64 370% 549.10
207.22 370.21 5-18.62
642.23
642.54
745.92
747.24
polI431 HBJ[22]
I=3 pou431 BDD[53]
poll431 HBJ[22]
num[52]
sin
HBJ[22]
BDD[ZZ]
polI43] BDD[ZZ]
po1143) HBJ[ZZ]
numI57]
BDD[53] 134.21
134.45
132.85
132.65
209.41
209.67
208.37
2os.17
288.62 459.86 645.42
288.96 460.30 646.03
287.64 459.15 645.01
287.18 458.51 643.98
373.28 552.23 744.28
373.73 552.67 744.90
372.60 551X5
372.11 551.20
HBJ[X]
data
Ebu)l -9280~
----
----
I
-------I
Fig. 5. Electronic cnrr~y of the ground state of the HCN-HNC s_vstsm as ;t func:ion of the bending coordinate 0. Each calculated point corresponds to the optimal M-H and C-N distances for this 0 vnluc.
calculations with the reverse choice are also performed to investigate the sensitivity of the results with respect to the form of vibrational basis functions. The results of calculations are presented in table 3 for I = 0, 1.2 and 3. In the first columns are given the results obtained using the method described in the present paper. In the I = 1 case the energy values calculawd employing both sin( 110) and cos(fr0) bases are shown. It can be concluded that the agreement between 111~‘corresponding vibrational energy values is excellent, especially for lowest levels. The same is also true for the form of the wavefunctions being indistinguishable in fig. 7 from one another for the first I= 1 level. and differing negligibly for the fifth level. In the second columns of table 3 (in the third for I = I) the results are given which are obtained employing the method described in ref. [43]. In this case the eigenfunctions of a two-dimensional harmonic oscillator are used as the basis in which the bending hamiltonian is diagonalized. while all kinetic- and potential-energy terms are expanded in power series in bending coordinate. The accuracy of this method is discussed in more detail in ref. 1431 and is also illustrated by the last two columns in each part of table 3. Both series of results are obtained using the same form of the hamiltonian (derived by Hougen et al. [22]), but the last one by a numerical solution of the Schrodinger equation [52]. Differences between the results given in the first two columns of table 3 and these in the last two ones are a consequence of a slight difference in the BDD and HBJ hamiltonians. Having this in mind and noting that the overall agreement between the results presented in the two first columns of table 3 is better than 1 cm-‘. one can conclude that the method proposed in the present paper
--___-f _ ;_ ..;: .-Y ‘A_--i “\
--xl
‘1. --I
\
is quite computer processor
capable
to yield very accurate
time needed for the described system for a particular I value.
vibrational calculations
energies and \vavctfuncrions. was roughly
3 CPU
AI rhc same rime rhs s on an IBX1 i70-16s singls-
_i__T.Cnlcuk~tiot~of benditlg-energy lecels it1 ihe HNC + HCN qsrrttl The final calculations are performed to obtain the bending-energy levels and \vavrfuncrions corresponding to the potential curve presented in fig. 6. In this case the hamiltonian (19) + (10) is used. Due IO a larger range of the bending-coordinate values (O”-1 SO”) and a more complicared form of Ihe potential (comparrtd with the C, case). nine sin(pz8) or cos(rzB) terms are used to represent rhe kin&csnd potential-energ! terms and bases consisting of fifty functions are employed in calculation of the vibrational energies and wavefunctions. The 8 dependence of the M-H distance. occurrin, 0 in the ?; functions. is taken inro account correcting rhs net-bending curve as described in section 4 and the stretch-bend couplin g is incorporated with the o-dependent zero-point energy corresponding IO the hl-H vibrations. It is assumed rhar rhs C-N bond length. as well as the zero-point energy of the correspondin, 0 vibrtlrions. do not depend on 0. The calculated bending-energy levels for I = 0. 1 and 2 are shown in fig. 6. The values of the bending frequencies
w. obtained
by taking
the difference
betwzsn
the second
and rhe firsr I = 0 term value divided
by two are 760 and 485 cm-’ for HCN and HNC respectivel_v. These results are in significantly better agreement with the experimental data than those obtained by Bunker and Howe [13] who used rhe potential surface calculated by Pearson et al. 161. This can be arrribured IO a more appropriate form of our hamiltonian which takes into account the coupling between the stretching and bending vibrations in a more complete way. The number of I = 0 levels below the barrier is 13 for HNC and 12 for HCN. in accord with the result obtained by Bunker and Howe. The energy differences between the successive levels with the Same I value decrease when one goes from the minima corresponding to the linear geometries to the barrier and incrtXiSe again above the barrier.
hl. Perk EI cd. / S~IIC~~01 rhc isonwizorion
33’
~.__ _ .-.__-
Hh’C -
HCh’
-J I
c3.
!
01. I
I
0.'. 03
I OJ
1 Lo
0
.--..L---Lo_
._._...
30
50
--iz
150
e(o)
160
Fig. 7. Vibrational wavefunction for the first (lower part) and fifth (upper part) I= 1 bending level obtained for the X ‘2; SI~ISof C, hy applying the BDD [53] hamihonian and basis functions of solely cos 110 (dashed line) or solely sin nfI (solid line) type. The curves are indistinguishable for the first level.
30
60
90
120
I Iso
lso-er~
Fig. 8. Shape of the vibrational wavefunctions for the bending levels of the HNC = HCN system near the top of the barrier. The potential barrier V is also indicated.
The form of some wavefunctions is shown in fig. 8. It can be seen that the levels far below the barrier can always be attributed to one of the potential valleys, so that the tunneling from HNC to HCN and vice
versa is negligible. as already stated by Bunker and Howe 1131. The magnitude of of the wavefunctions for the levels lying very close to the barrier is of the same order of magnitude on each side of the barrier indicating an appreciable penetration of it. Above the barrier, where the bending vibrations become actually free rotations of an atom (hydrogen) about the diatomic core (C-N), there exists a high density of states resembling the rotational structure.
Adnowledgement
The financial support of the DFG in the framework of the SFB 42 is gratefully acknowledged. The services and computer time made available by the University of Bonn Co-mputation Center have been essential to the present study.
Appcndis
The change of the coordinates in the hamiltonian is usually achieved in one of the two ~vays [24]: (i) the classical kinetic-energy operator in cartesian coordinates is transformed into the quantum-mechanical operator and only thereafter the replacement of the coordinatrs is prrformed. (ii) the classical kinetic-energy operator in new coordinates is constructed first and the corresponding quantum-mechanical operator is then obtained using the Podolsky transformation. In the present paper \ve propo.se an approach being between the two mentioned above: according to it the classical kinetic-energy operator is constructed in terms of the moments conjugate to the&r Cartesian coordinates defining the positions of the nuclei in the molecular plane and to the IWOEuler angles (Y. fi leadin g the 9Y plane of the space-fixed coordinate system into coincidence with the molecular plane. This is motivated by the fact that this classical espression can be used for derivation of various quantum-mechanical operators_ differing from one another in the internal coordinates employed and due to the definition of the third Euler angle y_ Starting with that classical hamiltonian. which when multiplied with - 1 (h = 1) represents at the same time the c;lradratic part of the quantum-mechanical operator. for any particular choice of internal coordinates arid can he obtained by a 4 x 4 coordinate transformation instead one of dimension 6 X 6. Applying the transformation R, (8) R,(a) [eq. (5)] on the Cartesian coordinates in (2) and inverting the obtained matrix T, in order expression for T is obtained:
to replace
the velocities
by correspondin,
* conjugate
impulses.
the follo\ving
with
where Z.,.,. I.I:,. and I,,. represent the moments and the product of inertia in the moving systems xvhose _YJ’ planes coincide with the molecular plane:
and the jacobian for the transformation from the space-fised system J = (J,s~ - s,_rz) sin /3 i\13,/t~~~J. The above expression representing the classical kinetic energy in the hamiltonian form. and if all pc, p‘, pairs are ‘ I replaced by - d’/i3qjilq, also the quadratic part of the corresponding quantum-mechanical operator_ can be employed as the starting point for construction of various forms of the full quantum-mechanical vibration-rotation hamiltonian for triatomic molecules. As an illustration of this approach let us derive the vibration-rotation hamiltonian in the IPXl system.
We want to perform
a transformation
of the expression (A-4)
where 5,. <, represent
the Cartesian coordinates.
4 2i- = C 7;,,,1,aVaqA x-J= 1
into
a+
(A-5)
with qA, q, = (r, R. 8, y ). The CC zfficients q,(, and q,,,,, are related to one another
From
T_,=
2 l.j-1
(A.l).
(A-6)
- z$”
by
T,,,,(aq~;/ag,)(aq,/as,)-
(A-6)
and (6) it is easy to obtain:
= Cj, +- 4
-k 4.
- ( I:r,. - I :y, ) sin 2 y cot fl&
+
(f:r,. - I.::,.) sin 2 y
a’
sin p
aaap
(A-7)
l
1
with the jacobian of the transformation J = tn~rR/M’. Therefore. the jacobian the space-fixed system to the final system (with I-, R, 8. y. a. /3) is
for the transformation
J = r’) R' sin 0 sin /3_
from
(A-8)
The linear terms of the full vibration-rotation hamiltonian can simply be obtained by applying the Podolsky transformation [35]. We consider first the case when the volume element in the integration of the
wavefunction be represented
is d7 = ndq, by
(q, = r, R, 6, y, a, j3). In this case the full nuclear kinetic-energy
operator
can
=
a’
4 wq, +f ar,,G( T-
1~
‘I
I
a4 <
aT, a aq aq
+ [
I
a4,
aq
ac ,
)I
(
-----
.,
aG ppaG __
ac a 84, aq,
ac a h, aci,
+g,G-’
a4
,
\
a'G
TfjG-‘pa4. a4,
+
(A.91
. j 1
sin’ ~9sin’ p is the determinant of the T”’ matrix. The where T, = T, y and G = (M/l,,,))l,,~l_~)~/rJRJ second term in’ 6e square bracket vanishes after summing over i and j SO that the coefficients of the terms linear in moments can be obtained from the corresponding quadratic coefficients hy means of relations
@'=Ca7j,/aq,,. - ,
(‘4.10)
_i Applying
one obtains
(A-10)
(X.1 1)
The constant
term
T’“’
can also be obtained
from (A.9).
or much
simpler
by the procedure
bslo\v. The
result is
The kinetic-energy operator corresponding to the volume = 1) can be constructed applying the transformation r-‘R-’
sin-‘/‘e
on the expression
sin-‘/’
-_ T(?”
p jrR
+ f”‘)
sin’” given
dT = r’R’
sin B sin p ndq,
e sin’/?fl_
w
(jacobian
(X.13)
by eqs. (.4.7)
+ ( I_:!, COG y + I.:;,_ sin’ y )cot 0 2 C\ I;.;.
element
and (X.11).
The
term linear
+ ( I_;:,. - I_::, ) sin 2y (1 + cos’ 2 I::, I:.:. sin’ /3
in momenta
fi )
a 5’
becomes
(X.14)
terms not involving any derivatives appear also_ Since the constant term ?(“’ must be equal to zero in this case (jacobian = 1). this multiplied with - 1 represents exactly the constant term of the kinetic-energy operator corresponding to the volume element d7 = !Jdq,_ given by eq. (A-12). Introducing the components of the rovibronic angular momentum in terms of the Euler angles [lS.23]. ssprcssion (9) is obtained.
Some
References [l] D.E. hlillig;rn and hl.E. J~~cox. J. Chem. Phys. 39 (1963) 712. [Z] D.1:. Rlillig;tn itnd h1.E. Jacoxq J. Chrm. Phys. 47 (1967) 278. 131 L.E. Snyder atid D. Buhl. Bull. Am. Astron. Sot. 3 (1971) 38s. 141 L.E. Snyder and D. Buhl. Ann. NY Acad. Sci. 194 (1972) 17. 1.51 C.A. Arrington and EA. Ogryz!Jo. J. Chem. Phys. 63 (1975) 3670. 161 P.R. Pearson. H.F. Schaefer 111 and U. Wahlgren. J. Chem. Phys. 62 (1975) 350. 171 G.L. Blackman. R.D. Brown. P.D. Godfrey and H.I. Gunn. Chem. Phys. Letters 34 (1975) 241. [Xl 1’. Botschwina. E. Nachhauer and B.hI. Rode. Chem. Phys. Letters 41 (1976) 4R6. 19) P.R. Taylor. G.B. Bacskay and N.S. Hush. J. Chem. Phys. 69 (1978) 1971. \lO] R.A. Crrswll and A.G. Robiettr. Mol. Phys. 36 (1978) g69. [I 11 J.N. hlurrell. S. Carter and L.O. Halonen. J. Mol. Spcctry. 93 (1982) 307. 1121 S.K. Gray. W.H. hlillrr. Y. Samaguchi and H.F. Schaefer 111. J. Chem. Phys. 73 (1980) 2733. 1131 P.R. Bunkcr and D.J. Howe. J. Mol. Spectry. S3 (IYRO) 2%. 1141 A.G. htnki and R.L. Sums. J. Chem. Phgs. 75 (1981) 417s. 1151 L.T. Redmon. G.D. Purvis 111 nnd RJ. Bartlett. J. Chem. Phys. 72 (1980) 9X6. [ I61 C.-F. Pau and W.J. Hehre. J. Phys. Chem. X6 (1982) 321. 1171 K.K. Lehmann. G.J. Schrrrer and W. Klemperrr. J. Chem. Phys. 77 (19X2) 2853. [IS] t1. Dichl. S. Fliiggc. A. Viilkel and A. Weiguny. Z. Physik 162 (1961) 1. 1191 K.F. Freed and J.R. Lombardi. J. Chem. Phys. 45 (1966) 591. [ZOl R. Walace nnd Ch.V.S. Ramachandra Rae. Z. Naturforsch. 3% (1977) 1450. 1211 C. Eckart. Phys. Rev. 47 (1935) 552. 122) J.T. Hougrn. P.R. Bunker and J.W.C. Johns. J. hlpl. Spectry. 34 (1970) 136. 1231 P.R. Bunker and C. Landsberg. J. Mol. Spectry. 67 (1977) 374. 124) P.R. Bunker. hlolecular symmetry and spectroscopy (Academic Press. New York. 1979). 1251 tl. hlargenau and G.hl. Murphy. The mathematics of physics and chemistry (Van Nostrand. Princeton. 1261 C.F. Curtiss. J.O. Hirschfelder and F.T. Adler. J. Chem. Phys. 18 (1950) 163% 1271 R.J. le Roy and J. van Kranendonk. J. Chem. Phys. 61 (1974) 4750. 1281 S.L. Holmgren. hl. Waldman and W. Klrmprrer, J. Chem. Phys. 67 (1977) 4414. 129) V.A. Istomin. N.F. Istomin and B.1. Zhilinskii. J. Mol. Spectty. 67 (1977) 265; 31 (1978) 413. 1301 J. Tennyson and B.T. Sutcliffe. J. Chem. Phys. 77 (1982) 4061. 1311 D.J. Veuetri and S.I. Rubinow. Ann. Phys. 35 (1965) 373. 1321 R.T Pack J. Chem. Phys. 60 (1974) 633. 1331 hi. de Cellas and B.T. Darling. J. Mol. Spectry. 29 (1969) 66. 1341 AS. Skalozub and A.R. Caunc Optika i Spektroskopija 50 (1981) 3. l3Sj B. Podolsky. Phys. Rev. 32 (1928) 812. 1361 T.H. Dunning Jr.. J. Chem. Phys. 53 (1970) 2823. 1371 J.L. Whitten. J. Chem. Phys. 44 (1966) 359. [3S] R.J. Bucnkcr and S.D. Peyerimhoff. Theoret. Chim. Acta 39 (1975) 217: 35 (1974) 33. 1391 R.J. Bucnker. S.D. Peyerimhoff and W. Butscher. Mol. Phys. 35 (1978) 771. 1401 S.R. Langhoff and E.R. Davidson. Intern. J. Quantum Chem. 8 (1974) 61. 1411 R.J. Robinson and K.A. Holbrook, Unimolecular reactions (Wiley-lnterscience, New York. 1972). 1421 hf. P&k. R. Runau. J. R~mrlt. S.D. Peyerimhoff and R.J. Buenker, J. Mol. Spectry. 78 (1979) 309. 1431 hl. Perit. R.J. Buenker and S.D. Peyerimhoff. Mol. Phys, 49 (1983) 379. 1441 S.D. Peyrrimhoff and R.J. Buenker, Theoret. Chim. Acta 27 (1972) 243. [4S] J.A. Pople. R. Krishnan. H.B. Schlegel and J.S. Binkley. Intern. J. Quantum Chem. 14 (1978) 545. 146) U. Wahlgren. J. Pacansky and P.S. Bagus, J. Chem. Phys. 63 (1975) 2874. 1471 W.P. Kraemer and G.H.F. Diercksen. Asrrophys. Letters 205 (1976) 97. 14x1 G.M. Schwenzer. H.F. Schaefer HI and CF. Bender. J. Chem. Phys. 63 (1975) 569. 1491 L.T. Redmon. G.D. Purvis and R.J. Bartlett. J. Chem. Phys. 69 (1978) 5386. [SO] G. Strey and LM. Mills. Mol. Phys. 26 (1973) 129. 1511 R.A. Creswell. E.F. Pearson. M. Winnewisser and G. Winnewisser. Z. Naturforsch. 31a (1976) 221. I521 Ch. Jungen and A.J. Merer. Mol. Phys. 40 (1980) 95. 1531 T. B3rrow. R.N. Dixon and G. Duxbuty. Mol. Phys. 27 (1974) 1217.
1956).
Puhlishin_e Company
AB INlTllO STUDY OF THE lSOMERlZATlON HNC -+ HCN. 1. AB INlTIO CALCULATION OF THE HNC = HCN POTENTIAL AND THE CORRESPONDING ENERGY LEVELS
SURFACE
and Robert J. BUENKER
Received
3 May 1983: in final form 4 Augusr 19S3
The results of an ab initio SCF+ Cl calculation of the porcntls; rwfxr for rhe reaction HSC = HCS arc prcssnrcd. Ths soordinacs~ for J approximative minimum-energy parh is obtained minimizing the cnsrgy nirh rcspccr lo both swkhing numbrr of values of the bending coordinate 0. This potential CUT\‘C is corrsctcd by adding Ihc zero-point snsrgics corresponding to the stretching vibrations. assuming rhsm to be B dcpsndcnt. The corresponding vihr3Iionsl-ror~rionrtl energ>levels and wavefunctions are calculmcd using an appropriate vihrxion-rorarion hxniltonian whsrehy 111e porsnrhland kinetic-energy terms. as well as the wnvsfuncrions are represented by Fourier series in 8.
1. introduction The isomerization reactions RNC * RCN. representing classical esamples of unimolecular reactions_ have been the subject of extensive experimental and theoretical studies for some time. Holvever. due to the fact that the higher members in the RNC series have too many degrees of freedom thus preventing an accurate ab initio calculation of the potential surface for isomcrization. and that the simplest member of this family, HNC, till recently has been observed only under non-equilibrium conditions or at very low temperatures, there has been no opportunity to compare the correspondin g experimental and theoretical results concerning the kinetics of the rearrangements. This situation has considerably changed due to the results of latest experimental and theoretical works. giving new impulses for investigations in this direction. A spectrum of the HNC was first observed twenty years ago by Milligan and Jacos [l] who observed the IR absorption in an argon matrix at 4 K. and in both argon and nitrogen matrices at 14 K [2]. Snyder and Buhl [3.4] observed a new interstellar emission line at 90665 MHz which was attributed to the J = 1 f- 0 transition of HNC. Arrington and Ogryzlo [5] obtained the JR emission spectrum of HNC and gave the values for both the band center and the AB term for the N-H stretching vibmtions. An extensive ab initio calculation of the HNC G= HCN potential surface has been performed by Pearson et al. [6]. A double-zeta plus polarization basis has been used. The equilibrium geometries of HNC. HCN 0301-0104/83/0000-OOOO/SO3.00
8 1983 North-Holland
and the critical conformation have been determined. The SCF calculations predicted HNC to lie 9.5 kcal/mol above HCN. while the corresponding Cl result was 14.6 kcal/mol. For the barrier height. seen by the HNC molecule. the values of 40.2 and 34.9 kcal/mole have been obtained in SCF and CI calculations. resprctively. An attempt to check the validity of the predicted value for the zero-point difference between HNC and HCN has been made by Blackman et al. [7]. Since the microwave spectrum of HNC did not appear under the experimental conditions applied. it was concluded that the lower limit for the isomerizaby Botschwina tion energy must lie at = 11 kcal/mol. The force constants for HNC have been determined et al. [g]_ Taylor et al. [9]. Creswell and Robiette [lo] and Murrell et al. [ll]. Gray et al. [12] applied the reaction-path hamiltonian in order to study the tunneling in the reaction HCN + HCN. In the work by Bunker and Howe 1131 the potential surface calculated by Pearson et al. [6] has been used for calculation of the bending-energy levels in the HNC * HCN system. It was found that the tunneling can play a significant role in the isomerization process only in the vicinity of the potential barrier. The results recently reported by Maki and Sams [14] opened the discussion about the value of the isomerization energy. These authors performed spectral measurements on HNC at temperatures ranging From 900 to 1250 K. On the basis of the temperature dependence of the absorption intensity the authors estimate an isomerization energy value of 3600 f 400 cm-’ (10.3 f 1 kcal/mole). significantly smaller than the ab initio results obtained after the effects of correlation are included_ In order to find the reasons for such discrepancies. Redmon et al. [15] have undertaken a very extensive study of the effect of various factors (size of the basis. method of calculation. etc.) on the calculated isomerization energy and have come to the conclusion that the best value obtained (15.1 kcal/mole) agrees with the results of previous theoretical works. but differs significantly from the value [14] deduced from experiment_ New experimental results of Pau and Hehre 1161 (14.S & 0.2 kcal/mole) support the theoretical predictions. Lehmann et al. 1171 have recently observed several highly excited stretching states using an intracavity photoacoustic technique In the present paper we report the results of a new ab initio calculation of the potential surface for the reaction HNC --‘ HCN. These results are used in the following paper to calculate the unimolecular rate constants according to a slightly modified RRKM model. The proposed approach requires the knowledge of the vibrational-rotational energy levels in the entire range of the reaction coordinate (being predominantly bending lvibration). in which case the couplin g between this degree of freedom with the others cannot be neglected. For an efficient and accurate calculation one needs the form of the hamiltonian which minimizes this coupling and the reaction surface must be known in more detail, allowing for a consideration of the variation of the bond lengths. as well as the correspondin g force constants with the bending coordinate. An appropriate form of the vibration-rotation hamiltonian is presented in section 2 of this paper. In section 3 it is shown how the problem of obtaining the energy levels can ultimately be reduced to a number of one-dimensional treatments not neglecting the important coupling of various degrees of freedom. The results of these calculations are presented in section 5.
2. Vibrational-rotational
hamiltonian
In a more rigorous theoretical treatment of the isomerization reactions in triatomic molecules it is necessary to calculate the vibration-rotation energy levels * going beyond the simple harmonicoscillator-rigid-rotor approximation. Therefore one needs to solve the nuclear Schrijdinger equation with the rotation-vibration hamiltonian allowing for large-amplitude displacements in all the modes of motion. Since a solution of this problem is connected with considerable computational expenditure (it should not be forgotten that a calculation of a three-dimensional potential surface must precede) it is necessary for * 11 is assumed
that the effect of excited electronic slates can be neglected.
practical applications to find a wav to reduce the dimension (six) of the nuclear Schriidinger equation by treating. of course if physically justified. some of the degrees of freedom as independent from one another. The key role in the isomerization process in a triatomic molecule is usually played by the bending vibrations which. when excited enough. can lead to the rearrangement_ Therefore it is convenient to use such a vibration-rotation hamiltonian which minimizes the coupling between the bending vibrations and other motion modes. Obviously_ the hamiltonian in terms of the generally ustzd internal coordinates (three internuclear distances [lS] or two internuclear distances and the angle between them [ 19.20]) does not fulfil this requirement because the internuclear distances can he drastically changed during the large-amplitude bending vibrations leading to the molecular rearrangement (e.g. the C-H distance is changed from roughly 2 to 1 A in the HNCe HCN reaction)_ On the other hand. in the course of a highly excited bending vibration the molecule passes twice through the linear nuclear configuration lvhereby the gradual transformation of a rotational degree of freedom of the non-linear molecule into the second component of the degenerate bending vibrational mode of the linear molecule takes place. In order to ensure a unified it is necessary to consider the treatment over the entire range of variations of the bendin, 0 coordinate rotations with the smallest moment of inertia (becomin g zero at linear molecular geometry) together \vith the bending vibrations. In order to keep this problem in one dimension (this is possible due to the usial symmetry of the potential if polar coordinates are used) the moving system should be attached to the molecule in such a way that at alllinear geometries one of its axes coincides \vith the molecular axis. This condition is not satisfied if the Eckart conditions [21] are used. and also not in the method developed by Bunker and co-workers [22-241. Therefore in the present paper a set of internal coordinates is employed which minimizes the coupling of this one (bending) mode mostly contributing to the reaction: coordinate with the other two. and the axes of the molecule-fixed system art chosen IO be coincidenr with the instantaneous principal axes of inertia (IPAI system)_ The form of the corresponding vibration-rotation hamiltonians satisfies both of the above mentioned conditions. The classical expression for the nuclear-kin&c-energy operator of a triatomic molcculs in rhe Space-fixed system has the form 2T13,, = 2
,I,;( ji’
+ ?,I + i,z)_
(1)
r-1
Introduction
of the center-of-mass 1
x0
=
-
3 c
??I,X;.
&=-
Iv;=,
leads to the separation
coordinates 1 M
X0_ Yh_ Z0 and relative
3 c I=1
of the translation
z,,=-
AI
s;_ J;_ :;.
~5. .I_<_;:
3
1 ,?l,Y#.
coordinates
c
,?l,Z,.
,=I
part of the kinetic
energy:
with PI = tnl(ml
defined
/.Lz= I?12( tn 2 + t111)/I?1 3.
+ tnz)/tnl.
choice of the Euler transformation of a vector
The
j.lL:=
tlI1t?l,/t)lj_
angles is made according to the recommendation (I’ defined in the space-fixed system into the vector
given in ref. 1251: rhe u in the moving frame is
by
a = R,(y)
R,(P) R,(cy)a’+
a;.
(4)
Fig. 1. Coordinates
used in the present
work:
r. R. B - internal
coordinates: y - third Euler angle. s. _r (s”. _I-“) is the molecular plane. x, _r coordinate axes are parallel to the X. Y axes of the space-fixed
system:
10
0 sin p
-sin
cos p
0
0
cos p
/I
x”. J”
axes of the moving
1 [ _ R,(y)=
frame.
cos y
-sin-y 0
1 (9
sin y
0
cosy
0 _
0
1
In the present paper the internal coordinates shown in fig. 1 are used: the distance from particle 3 to the center of mass of 1 and 2 is denoted by r. the distance between 1 and 2 is denoted by R. and the angle between these two lines is 0. The z axis of the moving system is perpendicular to the molecular plane and the third Euler angle y is defined such that the s axis of the moving system represents the molecular axis ” whenever the molecu!e reaches a linear nuclear configuration. The same set of internal coordinates has
already heen used by other authors [26-321. but accompanied used in the present paper. On the other hand. several authors
with the moving system different from that have employed such a body-fixed system. but
in connection with other internal coordinates [18.19.33.34]. From fig. 1 it follows that the internal coordinates and the condition inertia I, I‘ are given by the relations: r = [ ( us,
+
bxJ + ( u_l’,
)?I“‘.
+ fy,
for vanishing
of the product
[(_~2-_~,)1+(.,.~-).,)2]“2,
R=
of
(6)
with u =
r?l,M/ltl,(t?l,
+ m,).
b = ttt2hf/tnj(
tn, + tn2)_
Q is the angle between the 15 line and the .X axis. related to the Euler angle y by y = + - +’ with Cp’being * The = axis of the moving comparison
system
of the hamiltonians
is chosen derived
to be perpendicular
to the molecular
in ref. [IS] and in the present
paper.
plane as in ref. [IS] in order
the
to make SimPlier
a
I!.
angle
between
Peril- cl 01. / Srrr~r of rhc i.wntcri-_urionif>VC -
72 and the s axis in the IPAI system.
IICX
It is easy to show that
tg 2+’ = c~I-’ sin Xl/( c, R’ + c3t-’ cos 20). where c, =
111, I)1
,/( m, + m
z
)_
CJ =
tnl(nl,
+ r11,)/Al.
(7)
described in the appendis. the folloxvinp Applying the usual procedure [24.35] or usin g the approach expression for the quantum-mechanical vibration-rotation kinetic-energy operator is derived:
with
where
Z& and I_:!, are the moments ZJ.1,. = 4 [ ( c, R’ + c,r’) I::=f[(c,R’
of inertia
+ (c;R”
in the IPAI
+ c;r4 + Zc,c,r’R= cfr’
+c,r’)-(cfRJ+
+ 2c,c,r2R2
system: cos 20)““]
_
cos 28 )‘.‘2].
1:: = 0.
(IU)
The vibrational part of the kinetic-energy operator (9). ?, . possesses no cross terms rhus coniirmin~ rhc convenience of the chosen internal coordinates. The cross terms can. of course. appear in the hamiltonian through the potential. but the very simple form of TV represents a great advantage by an approsimative separation of the vibrational modes from one another. The rotational part of the kinetic ensmy operator is identical with that of Diehl et al. [18] as a consequence of the same definition of the Euler angles. The expression (9) corresponds to the volume element dr = r’R’ sin 8 sin flfId(~, (jacobian = 1) by integration of wavefunctions. The same rotational part. ?<_ has also the hamiltonian corresponding to the volume element dT = sin B sin /3TIdq,. which is actually used in the present paper. The vibrational and vibrational-rotational parts take in this case the form:
4 sin e
i;., =
--c,rR’
cos e-
( Z& - z::J
3. Calculation
of the potential
a ar
+ c,r’R
cos ea
aR
i ( c,R’-c-
(11)
surface
The calculation of the potential surface for the reaction HNC = HCN has been performed by means of the ab initio SCF and CI method under the assumption of the validity of the Born-Oppenheimer
approximation. The basis set consists of 49 contracted Cartesian gaussian functions: five s and three p 7 1. 1. l/3. 1. 1) given by Dunning [36] in addition to a set of Cartesian d groups in the contraction (4. _. functions on each the C and N atoms, with the exponents ad = 0.75 for C and ad = 1 .O for N; for the
hydrogen atom two s groups in the contraction
(4. 1) with the scaling factor 9’ = 2.0 [37] augmented by one finally one s and one p function (a.\ = 1.45. a,, = 0.5) are located in the set of p functions ( aP = 0.735); middle of the C-N bond. The configuration-interaction treatment is of the standard MRD Cl type [38.39]. with configuration selection and energy extrapolation. A core of two MOs corresponding to the Is shells of carbon and nitrogen respectively is kept doubly occupied in all configurations, the corresponding two complementary MOs with the highest orbital energies are entirely excluded from consideration; all other MOs are in principal allowed variable occupation. Only one reference configuration. corresponding to the ground state of the molecule in the SCF approximation. has been employed in order to generate the double-excitation configuration space. since all other species appear with small coefficients (less than 0.5%) in the final Cl
functions while the This procedure led to configuration spaces of = 10000 symmetry-adapted sizes of the secular equaticns actually diagonalized after configuration selection (a selection threshold of T= 10 FH was employed) were in the order of 4000. The contribution of the remaining generated configurations is accounted for by the usual extrapolation procedure_ The effect of still higher excitations is also estimated according to the formula AE = (1 - ci)( EscF - E,,,D =,) [40] which upon addition to the corresponding Cl result is referred to as the estimated full-Cl limit in the ensuing discussion.
expansion.
1. Vibrational
treatment
The vibrational-rotational energy levels are calculated diagonalizing the hamiltonian derived in section 2 of this paper. The most appropriate form of this hamiltonian is obtained transforming the expression corresponding to the volume element d7 = sin 0 sin fi lldq,. given by eqs. (11) (for TV and F,,) and (9) (for Tr) in such a way that the molecular axis at linear nuclear arrangements coincides with the z axis of the moving frame. i.e. by interchanging the A-and 2 axes. The vibrational part of the hamiltonian:
(12) is invariant i;, =
with respect 4 sin 8 ( I,,. - I,,
to this change. -c,rR’
cos 0%
The vibrational-rotational a
+
c,r’R
a
cos 6---
aR
1’
iJr, while the rotational
part can be written
pari becomes
03)
in the form 04)
with I?, + i,, + fr = - 2? and all the quantities (c?. Ii,.) referring to the new frame. Expression (14) is identical with the H6 term of the hamiitonian derived by Freed and Lombardi [19]. For an approximate calculation of the eigenvalues of the hamiltonian (12)-(14) we neglect the vibrational-rotational part (13), having only the terms that couple the vibrations with the rotations about the x axis. For a molecule possessing a hydrogen atom (m,), the relations I,, = IA -=KI,, = I_,._,. = I, hold so that it can be treated as a nearly symmetric top; the part of the hamiltonian describing x and _Y rotations
has in this case the form:
(15) It is easy to show [41] that the eigenvaiues of a symmetric top can be ohtainrd IO :I gc~xl approsimation as sums of eigenvalues of a one-dimensional pius a t\~,o-dimensional rotator with the ~mmlt’nts of inertia I, and I,, respectively. The degeneracy of a particular cncrgy level is then obtained hv multiplying the degeneracies of these two hypothetical rotators (1 and 2 for K = 0 md K f 0 rcipcctivcly for the one-dimensional rotator and 25 + 1 for the two-dimensional one). In such :I trcatmcnt the values of ths quantum numbers J and I\’ (K = I) are independent of one another. In the prcxnt paper XVL’ use this idea and calculate the energy levels corresponding to (15) as (1/7fH).I( J t 1) \\-hilt the part of the rotational hamiltonian including j_’ term is treated together with the vibrations. For a number of villues of the bending coordinate the eigenvalucs of the stretching part of the hamiltonian
i a2 -2A&+-CI c?R’ 3
i
ii
k,, (0 ).S,.S, -i higher terms of I,‘.
(S,.
$=I-_
K).
(16)
using a variational method involving expansion of the vibrational \\avefunctions in products of eigenfunctions of two harmonic oscil!ators and employing a po~ynnmial fit to the potential. as described in more detail elsewhere [42]. As indicated in eq. (16) the coefficients of these pol~nomivl expansions are assumed to be dependent on 8. The calculation of stretchin, 0 vibrational levels at home 8 v~tluss for xvhich. for the sake of computer time saving. the complete ttvo-dimensional potsntial surface \V;IS not constructed. is performed only in the harmonic approximation. The eigenvalues of the operator (16). when added to the pure bending potential obtained by optimizing the r and R values at each value of the bending coordinate 8. represent the effective bending potrntials L&(8). Thus. the last part of the hamiltonian including the bending vibrations. their interaction with stretching vibrations and the rotations about the : axis (becoming at linear geometries the molecular xxis)
are calculated
has the form: - 2fii,.,
=
I,,I, ,.
--
1 ’ (7:
--2-l;ii(c3). I:., ! a&.
(17)
with I,.,.=+
i
(qR’+
(I;= + I~,,T)/I,,!,,.
qr’)
+ (c;R’
= (l/sin’
+ +’
0)(1/c,
+ 2~ ,c3r2R2 R’ + l/c-,r’).
cos ze)l. ‘1. (IS)
and the angle Q describing now the rotations about the I axis. If the sigrnfunctions +(0. 0) of H,_, are represented as usual with R( 8) esp(i/+), the second term on the right-hand side of (17) can be transformed into
so that the problem
is reduced to one dimension only. In the present paper special attention is paid to the calculation of energy Isvsls and \vavefunctions corresponding to the hamiltonian given by eq. (17). Because of some peculiarities of the problem. the methods usually applied in a treatment of large-amplitude bending vibrations [22.23,43] are not quite appropriate. Some difficulties arise from the form of the potential having (in rhe HNC G= HCN case) t\vo
minima
separated
by a maximum between them. but not going to infinity for any B value. in contrast to the in more familiar problems. This prevents the use of the polynomial ;epresentation of the potential [43]. Another problem is a consequence of the form of the kinetic-energy operator. having singularities at 0 = 0 and 8 = n. In ordrr to avoid these difficulties a variational method is developed which makes use of the expansion of all terms in the kinetic- and potential-energy operators into Fourier series in 8 - the same idea was already applied in some simpler cases (42.441. Let us write the hamiltonian (17) in the form: usual assumption
~I,.,=T,(e)a’/ae2+T,(e)a/ae-127;(e)+V(e). with
(
-$ L+c-,R2
T,(B)=
1 CT
1
1 . T2(f?)=
(19)
1
’ -+-1
-+
( C,R2
c3r2
1
cos sin-
e
T,(B)=
-7
1
‘.,yl.+ L, -- 1 J.Lyy i I,,.I=, i -
(20) The terms T, and T? go to infinity when 0 + 0 or T. Since the volume element dT = sin 0 ITdq, is used. in a hasis consisting of sin( n13) and/or COS(N~?) functions the matrix elements corresponding to T2 would have iinite values but those of T3 would be infinite. For I= 0 the term with T3 disappears so that the basis with cos( IJo) functions is appropriate in this case “. Thus. for I = 0 the following matrix elements must be calculated: ,
7
cos( ~0)
cos( I,JO) T, 5
/0
Cod
a aB
JJJ 6 ) T2
7 1
sin f? de =
/0
Sill 8
JJo)
de =
COS(lJJe)~‘COS(fZ@)
Sin
@df?=/=
0
COS(
--II
cos(
/
0
t?Jo)[
I/
Sin
d)[T, o]
e] c05-(ne) de.
sin
CoS(d)[T,
1 COS(
I
6
-IJ’
COS(
sin
e]
IIf?)
sin( 110) de.
de.
(21)
This can be achieved easily if the functions T, sin 8 and I/sin 0 are represented by series in sin(k6). while an appropriate representation of T, sin B = T, cos 0 is a series in cos(X.0). In order to do that, the optimized values of r and R calculated for a few values of 8 are fitted first with second-order polynomials in 8. These in turn are used for calculation of the interpolated values of r and R and thereafter for generating (T, sin 8) and (T2 sin 8) values for a number of 0 values (e.g. for 0 = 0. 1, 2,. __, 1800). The obtained points are fitted with sin(k8) or cos(k0) according to the least-squares scheme. In order to obtain a brttrr representation of the potential the cubic-spline technique can be used for computing a number of supplementary points; the potential points are multiplied by sin 0 and then fitted in a series in sin(k8). For the case in which If 0 there are two terms in the kinetic-energy operator, namely T2 and 7;. which have singularities at B = 0 and T. The second of these being proportional to sin-’ 8 is particularly difficult IO handle when B + 0( 5). In the basis used for I = 0 this term would produce infinite matrix elements. To avoid this difficulty_ bases consisting of sin”’ B sin( no) and sin ‘y2 B cos(110) are used in the calculations of vibrational levels and wavefunctions for I # 0. Furthermore, it was found that a basis consisting of both Sin(jJ8)
sin( JJo
ud 1 or
)J8)
functions
cost
Y
/0
terms
COS(JJ6) cos(
sin”’
8
sin(
)>I e ) tile)
has computationally
no serious
alone, so that the following
sin”’ 6
advantage
matrix
elements
over
a basis
consisting
of either
appear:
1
sin 8 de
y A sin(n8)basis is not suirable because rhcse functions arc zero at 0 = 0 and T and thus cannot non-zero values al these angles.
represent the R,_, functions having
=
JzCoS(“ze)ITt(l
-’
0 sin(zzz8)
+ II z /
sin
ecos(J)*e)T
sin,,2
77 cos(nze)
=’ J0
sin(m9)
z J0
COS( sin”’
z J 0
sin”’
0
8
2
B]~~~~~~~dtl_
1 cos(tze)
cos(J*e) _ sin”’
_
[T2sinBc0sf?]
sm( lze)
I?ze) I~T,sin',~ecoso*e)
sin( ftze) cam sin( zzz8)
LJ sin z9do
' [ ae sizr(,zz9)
sin( zzzB)
0
B cos
V sin”’
d8in Juz:~~~~~~~)) [< sin’81
sinede=,'
sin( dl)
8COS(~~~) sm_ sin( zz8)
6 de =
-coso,Ie) r Jo sin(,lIB) [T sin’ e] $::,“I = COS(
/ 0
de.
~~~~~Og))
Jil e )
sin( zzzf?)
[ V sin’
COS( e]
Ize
)
sin( tz8 )
do.
de.
(2’)
The functions in brackets on the right-hand side of eqs. (22) are fitted in terms of sin{ X-8 ) ( T Tz size B CM 0 and i T2 sin’ 6 = T T, sin 6 cos 8). and cos( X-8) (all others) series. Both. the sin’..’ f9 cos( zzf?) md sin’/’ 0 sin(&) bases have been employed in expansions of vibrational \vavefunctions becatzse it \v;zs speculated that the first one would be more appropriate for even 1 values (in analogy with the \velI-kno\vzl properties of the eigenfunctions of a two-dimensional harmonic oscillator) and the srcozzd one for odd I values. However. the numerical results to be presented in section 5 show that the choice of a cos(zzQ) or sin(zz8) basis is quite irrelevant_ The small disadvantage caused by the rather exotic forzn of the basis functions. namely their zznn-arthogonality, is fully compensated by the capability of sin(zzQ) and COS(JZ6) series to fit very 1~41 all the functions given by (21) and (22) with only a few expansion terms and the very siznple computation of all integrals
required.
5. Results 5. I. Calculuriotz of llze porerzlial-ejzergr
szrrfuce
Three parameters required to define the molecular geometry are chosen to he the C-N bond length R. the distance r between the hydrogen atom and the C-N center of mass (M). and the angle between these two lines 8. The main parameter varied was the angle 0. playing a crucial role in the HNC = HCN rearrangement. For B values of O” (HCN). 12.6”. 30”. 65”. 70°. SO” S5”. 1ZO”. 150” and 1SO” (HNC) the other two coordinates were varied over a relatively large range. starting \vith the initial ~ahzes chosen on the basis of experimentally obtained data or taken from previous calculations [6]. This strategy w;\s chosen to enable not only the calculation of the optimal z- andR values for each 8. but also of some of the lowest-lying stretching energy levels. Special attention was paid to the regions in the vicinity of the HCN and HNC equilibrium geometries as well as to the expected location of the transition state: these parts of the three-dimensional potential surface were calculated completely. while for some other 6 values only its one-dimensional sections, corresponding to the variations of r or R (where the other two pammeters were kept fixed) were obtained. The calculated optimized values for r and R and the corresponding diagonal force constants are presented in table 1 and figs. 2 and 3. The results corresponding to three different levels of calculation arc
‘ILhlc 1 Op~irnkd XI-II and C-N lengths. corresponding diagonal force consmnls and energies as functions of the hrnding coordinate 0. All v.rlue\ in shwlic units mglcs in dqqxs
SC’I-
0
3.195
2.120
0.440
1.50
- 92.90 15
13 30 65
3.153 3.010 2.385
2.146
1.32
- 92.s950 - 92.86%
2.190
0.445 0.330 0.270
70
2.287
2.200
0.270
1.07 1.08
- 92.8212 - 92.S’OO
75 80
2.200 7.115
2.200 2.110
0.240 0.140
- 92.8205 - 92.R21X
x5
2.0x4
-.-_ 7 730
0.220
1.10 1.10 1.14
120 150 167 18(l
2.170 2.720 2.862 2.890
2.192 2.190
0.290 0.400 0.556 0.540
1.21 1.20
- 92.RS34 - 92.8860
E”,(TS-llCN) hll
L-,,(TS-HNC)
= 0.0660
EJHNC-HCN)
= 0.0155
0
3.200
2.170
0.420
1.35
- 83.1950
13 30 65
3.164 3.020 2.400
2.188
1.31
2.220
0.445 0.330 0.257
1.11
- a.1 90X -93.1703 -93.1177
70
2.297
_.__ -J ‘-JO
0.255
1.11
-93.1149
75
2.204
-.-_ ‘) ‘30
0.223
1.10
- 93.1141
X0
2.125
2.240
0.240
1.10
-93.1150
x5
2.073
2.262
0.255
1.12
-93.1183
120 150
2.180 2.750
167
2.897
2.223
0.553
1.19
- 93.1706
180
2.900
2.220
0.550
1.20
- 93.1724
E”,(-lsHCN) c’( lull
= 0.0x15
- 82.8263 - 92.852s - 92.8740
= O.OYO9
0.305 0.400
&,(-l-S-HNC)
0 13 30 65
3.200 3.169 3.020 2.410
2.190 2.204
70 75
3.308 2.218
80 85
2.140 2.085
= 0.0583
- 93.1435 - 93.1624
E,(HNC-HCN)
= 0.0226
0.420 0.440 0.330 0.250
1.35 1.29 1.10
- 93.2220 -93.21Sl - 93.2000 - 93.1475
2.240 2.250
0.250 0.223
1.10 1 .on
- 93.1422
2.255 2.276
0.237 0.252
1.07 1.11
- 93.1425 - 93.1452
2.240
93.1440
120
2.200
0.330
150
2.750
0.400
167
2.886
2.236
0.548
1.18
-93.1962
180
2.910
2.230
0.540
1.20
- 93.1980
E,,(TS-HCN)
= 0.01100
E,(TS-HNC)
= 0.0560
- 93.1698 - 93.1885
E,(HNC-HCN)
= 0.0240
SCF. MRD CI (CI extrapolated to zero selection threshold) and the estimated full CI energy obtained by applying the Davidson correction [40]. The general behavior of the r and R variations is similar to that already found by Pearson et al. [6]: the MH distance has the largest values at linear nuclear conformations being appreciably smaller for bent geometries. The variations of the C-N length with 6 are
given:
much smaller and the C-N bond length becomes slightly larger with the increase of the bent character of the molecule. Similarly, the value of the diagonal force constant for the M-H vibrations alters significantly (roughly by a factor of 2) in the course of the variation of the angle 8 from 0’ (or 18OO) to B = SO0 as
dJ---‘MM
I
‘
'CN
SCF
Fig.
2.
Dependence
of
the
M-H
and
C-N
lengths
on
the
bending coordinate 8 for the lowest HCN-HNC surface at the three levels of theoretical treatment SCF. MRD Cl (CI SKcrapolated
IO selection
threshold
zero) and full CI rstimare.
apparent from fig. 3. while the value for the other (kc_ S ) remains nearly const:~n: hrtwcen I SO” and 90” (a decrease of = 10% in its value in the interval O”-90” is found). These facts are reflected in the \*nriation.s of the stretching frequences with 0 (fig. 4): for M-H vibrations the values of 3400 and X00 cm-’ for linear species and of roughly 2600 cm-t for strongly bent geometries are obtained. On the other hand. the values of the C-N frequency remain between 2100 and 2300 cm -’ in the entire range of 8 \.;12ues. The values of the electronic energy as a function of 0 are presented in table 1 and fig. 5. Each point corresponds to the optimized r and R values for this particular 8. Takin, 0 into account that the reaction coordinate for the isomerization is predominantly determined by the bendin, 0 angle 8. rhe CUIA-es sho\vn in path” for this reaction. The results &rained in the present fig. 5 represent roughly the “minimum-energy conformation j. barrier height (correspondin, 0 10 the geometrical study for the isomerization energy (I&, (HNC) and the ‘-product” (HCN) are collected in table 2 of the transition state) relative to the “reactant” and compared with the results of other authors. Very good agreement with the results of other theoretical energ? obtained in Cl studies can be noted: for example the value of = 0.023 au for the isomerization calculation is in excellent agreement with the correspondin, 0 result given in refs. [6.9.49]. This result is also only from the value obtained bv in agreement with the experimental findings [7.16]. differin, 0 significantly Maki and Sams [14]. Taking into account that this discrepancy much exceeds the usual calculation rrro;. the reliability of the result of ref. [14] seems to be questioned. The dependence of the energy on B obtained by means of the estimated full Cl energies and the cubic-spline procedure is presented in fig. 6. For comparison. the results published by Pearson [6] are shown too. The upper curve given in fig. 6 represent
a a
.r;thk 2 Comparison
of the properlies
of 1he HNC-HCN
polenrial
surface
obtained
HCN C-H S(‘l‘
r, I61
(A)
1 .O86 1 AI624
C-N
(A,
1.122 1.1366
N-H
1.057 I .0655
1.126 1.1532
- 92.9015 - 92.88972
0.9946 0.9859
1.159 1.1586
0.985
1.15x
0.9837 0.9940
1.148 1.16X9
b,
1.075
WI
1.069 I .066 1.064
1.14x 1.159 I.150 1.148
;47j
I .064
1.148
- 93.1950 - 93.2220 -93.19241 -93.21707
1.151
- 92.8860 - 92.87454
- 92.88179
-92.S913S 0.9926 0.9954 0.9955
1.175 1.180 1.1696
0.987 0.9954
1.204 1.168
”
- 93.1724 - 93.1980 -93.16916
- 93.13707
- 93.19290
- 93.16893 - 93.159
- 92.97383
1121 1151
1501
E(au)
- 93.21707
1491 1171
C&P.
(A,
- 93.16376 1.065
studies
- 92.85533
- 92.89980
1451 14x1 191
N-C
- 92.90834 ”
”
WI
(A,
- 92.87520
1151 currcla:ion
with the resul1s of earlier
E(au)
1x1
SCI-‘ f
paper
HNC
I451 191 IlO]
in the present
- 93.23751 1.066
- 93.21363
=’
=’
1.153 0.987
IStl
1.171
I71 1141
1161 *’
Prcsen1 worh - SCF.
” Presen1 work - MRD
Cl.
c’ Presen1 work - full Cl rscimatr.
” X-basis
[15].
more realistic *’minimum-energy path” obtained by correcting the originally calculated curve with the zero-point energies corresponding to the C-N and M-H stretching vibrations, taken as a function of 0. The bending curve becomes slightly !latter due to this correction, having as a consequence somewhat sniallrr separations between bending-energy levels. 5.2. Tesr of the method for wearing the bending
uibrutions
The next series of calculations is performed to test the efficiency of the method for calculating vibrational energy levels and wavefunctions, as described in section 4 of this paper. In order to have a possibility to check the accuracy of the results obtained with the proposed method, the potential curve for the ground state of CJ, published by Jungen and Merer [52] is employed as a test example; for better comparison the large-amplitude bending hamiltonian (H”‘) by Barrow et al. (531 is used, differing from that given by eqs. (19) and (20) in the form of the 7”(e) coefficients. In these test calculations the potentialand kinetic-energy terms are fitted with seven sin(rr8) or COS(II~) terms each, which seemed entirely sufficient for a good representation of these energy functions. The bending hamiltonian is diagonal&d in bases consisting of twenty sin(n0) or cos(n0) terms for each value of 1. For even I z 0 values cos(n8) terms, for odd I values sin(&) series are normally used, but some test
TS
HNC F_,Ul”l s:)
M-H
(A,
C-N
1.201 1.202
(A,
Eh.lr
Hc-N el
E hr, s’
B(d%)
E(u)
71 70.2
- 92.8200 - 92.51054
9.7 9.5
‘II.4 30.2
51.1 19.7
- 92.79195
12.5
39.7
51.2
1.164 1.174
11.3
1.166
1.180
1.157 1.178
1.192 1.1x1
75 77 73.7
1
- 92.83179
10.6
‘7.1
JS.
-93.1141
14.1
36.6
50.X
- 93.1420 -93.1335
15.1 14.6
35.1 33.9
50.’ 19.5
- 93.05213
16.8
31.1
51.’
15.0 i 4.7 16
71
73.75
1.1945
1.1709
36.3 15.2 r’
-93.16034C
34.1 f’
.iU.? r’
> 1o.s 10.3 13.5 =’ SDQ
MBPT
Table 3 Comparison taken
[4] results.
of vibrational
” Final
results 11.51.
energy values (in cm-’
B’ In kcal/moir.
) obtained
employing
the prescm
and other wchniquss.
Potential
and geometry
from ref. [49] for ‘Ep’ state of C,
I value:
I=0
Merhod:
cos
hamiltonian:
BDD[53]
po’i431 BDD[53]
2.45
2.57
0.07
134.11 287.76 458.46
134.53 288.41 459.36
643.49
643.29
I value:
I=2
Method: Hamilronian:
COS
I=1 num[52]
sin
COL
pd[-l3]
pol[43]
m.ln~[52]
HBJ[22]
BDD[53]
BDD[53]
BDDi53]
HBJIX]
HBJ[Z]
0.00
65.57
65.54
65.34
63.26
h3.10
132.7S 286.99 458.15
132.45 286.42 457.35
209.97 373.64 553.01
209.95 373.77 553.51
209.17 372.16 550.5’
207.64 370% 549.10
207.22 370.21 5-18.62
642.23
642.54
745.92
747.24
polI431 HBJ[22]
I=3 pou431 BDD[53]
poll431 HBJ[22]
num[52]
sin
HBJ[22]
BDD[ZZ]
polI43] BDD[ZZ]
po1143) HBJ[ZZ]
numI57]
BDD[53] 134.21
134.45
132.85
132.65
209.41
209.67
208.37
2os.17
288.62 459.86 645.42
288.96 460.30 646.03
287.64 459.15 645.01
287.18 458.51 643.98
373.28 552.23 744.28
373.73 552.67 744.90
372.60 551X5
372.11 551.20
HBJ[X]
data
Ebu)l -9280~
----
----
I
-------I
Fig. 5. Electronic cnrr~y of the ground state of the HCN-HNC s_vstsm as ;t func:ion of the bending coordinate 0. Each calculated point corresponds to the optimal M-H and C-N distances for this 0 vnluc.
calculations with the reverse choice are also performed to investigate the sensitivity of the results with respect to the form of vibrational basis functions. The results of calculations are presented in table 3 for I = 0, 1.2 and 3. In the first columns are given the results obtained using the method described in the present paper. In the I = 1 case the energy values calculawd employing both sin( 110) and cos(fr0) bases are shown. It can be concluded that the agreement between 111~‘corresponding vibrational energy values is excellent, especially for lowest levels. The same is also true for the form of the wavefunctions being indistinguishable in fig. 7 from one another for the first I= 1 level. and differing negligibly for the fifth level. In the second columns of table 3 (in the third for I = I) the results are given which are obtained employing the method described in ref. [43]. In this case the eigenfunctions of a two-dimensional harmonic oscillator are used as the basis in which the bending hamiltonian is diagonalized. while all kinetic- and potential-energy terms are expanded in power series in bending coordinate. The accuracy of this method is discussed in more detail in ref. 1431 and is also illustrated by the last two columns in each part of table 3. Both series of results are obtained using the same form of the hamiltonian (derived by Hougen et al. [22]), but the last one by a numerical solution of the Schrodinger equation [52]. Differences between the results given in the first two columns of table 3 and these in the last two ones are a consequence of a slight difference in the BDD and HBJ hamiltonians. Having this in mind and noting that the overall agreement between the results presented in the two first columns of table 3 is better than 1 cm-‘. one can conclude that the method proposed in the present paper
--___-f _ ;_ ..;: .-Y ‘A_--i “\
--xl
‘1. --I
\
is quite computer processor
capable
to yield very accurate
time needed for the described system for a particular I value.
vibrational calculations
energies and \vavctfuncrions. was roughly
3 CPU
AI rhc same rime rhs s on an IBX1 i70-16s singls-
_i__T.Cnlcuk~tiot~of benditlg-energy lecels it1 ihe HNC + HCN qsrrttl The final calculations are performed to obtain the bending-energy levels and \vavrfuncrions corresponding to the potential curve presented in fig. 6. In this case the hamiltonian (19) + (10) is used. Due IO a larger range of the bending-coordinate values (O”-1 SO”) and a more complicared form of Ihe potential (comparrtd with the C, case). nine sin(pz8) or cos(rzB) terms are used to represent rhe kin&csnd potential-energ! terms and bases consisting of fifty functions are employed in calculation of the vibrational energies and wavefunctions. The 8 dependence of the M-H distance. occurrin, 0 in the ?; functions. is taken inro account correcting rhs net-bending curve as described in section 4 and the stretch-bend couplin g is incorporated with the o-dependent zero-point energy corresponding IO the hl-H vibrations. It is assumed rhar rhs C-N bond length. as well as the zero-point energy of the correspondin, 0 vibrtlrions. do not depend on 0. The calculated bending-energy levels for I = 0. 1 and 2 are shown in fig. 6. The values of the bending frequencies
w. obtained
by taking
the difference
betwzsn
the second
and rhe firsr I = 0 term value divided
by two are 760 and 485 cm-’ for HCN and HNC respectivel_v. These results are in significantly better agreement with the experimental data than those obtained by Bunker and Howe [13] who used rhe potential surface calculated by Pearson et al. 161. This can be arrribured IO a more appropriate form of our hamiltonian which takes into account the coupling between the stretching and bending vibrations in a more complete way. The number of I = 0 levels below the barrier is 13 for HNC and 12 for HCN. in accord with the result obtained by Bunker and Howe. The energy differences between the successive levels with the Same I value decrease when one goes from the minima corresponding to the linear geometries to the barrier and incrtXiSe again above the barrier.
hl. Perk EI cd. / S~IIC~~01 rhc isonwizorion
33’
~.__ _ .-.__-
Hh’C -
HCh’
-J I
c3.
!
01. I
I
0.'. 03
I OJ
1 Lo
0
.--..L---Lo_
._._...
30
50
--iz
150
e(o)
160
Fig. 7. Vibrational wavefunction for the first (lower part) and fifth (upper part) I= 1 bending level obtained for the X ‘2; SI~ISof C, hy applying the BDD [53] hamihonian and basis functions of solely cos 110 (dashed line) or solely sin nfI (solid line) type. The curves are indistinguishable for the first level.
30
60
90
120
I Iso
lso-er~
Fig. 8. Shape of the vibrational wavefunctions for the bending levels of the HNC = HCN system near the top of the barrier. The potential barrier V is also indicated.
The form of some wavefunctions is shown in fig. 8. It can be seen that the levels far below the barrier can always be attributed to one of the potential valleys, so that the tunneling from HNC to HCN and vice
versa is negligible. as already stated by Bunker and Howe 1131. The magnitude of of the wavefunctions for the levels lying very close to the barrier is of the same order of magnitude on each side of the barrier indicating an appreciable penetration of it. Above the barrier, where the bending vibrations become actually free rotations of an atom (hydrogen) about the diatomic core (C-N), there exists a high density of states resembling the rotational structure.
Adnowledgement
The financial support of the DFG in the framework of the SFB 42 is gratefully acknowledged. The services and computer time made available by the University of Bonn Co-mputation Center have been essential to the present study.
Appcndis
The change of the coordinates in the hamiltonian is usually achieved in one of the two ~vays [24]: (i) the classical kinetic-energy operator in cartesian coordinates is transformed into the quantum-mechanical operator and only thereafter the replacement of the coordinatrs is prrformed. (ii) the classical kinetic-energy operator in new coordinates is constructed first and the corresponding quantum-mechanical operator is then obtained using the Podolsky transformation. In the present paper \ve propo.se an approach being between the two mentioned above: according to it the classical kinetic-energy operator is constructed in terms of the moments conjugate to the&r Cartesian coordinates defining the positions of the nuclei in the molecular plane and to the IWOEuler angles (Y. fi leadin g the 9Y plane of the space-fixed coordinate system into coincidence with the molecular plane. This is motivated by the fact that this classical espression can be used for derivation of various quantum-mechanical operators_ differing from one another in the internal coordinates employed and due to the definition of the third Euler angle y_ Starting with that classical hamiltonian. which when multiplied with - 1 (h = 1) represents at the same time the c;lradratic part of the quantum-mechanical operator. for any particular choice of internal coordinates arid can he obtained by a 4 x 4 coordinate transformation instead one of dimension 6 X 6. Applying the transformation R, (8) R,(a) [eq. (5)] on the Cartesian coordinates in (2) and inverting the obtained matrix T, in order expression for T is obtained:
to replace
the velocities
by correspondin,
* conjugate
impulses.
the follo\ving
with
where Z.,.,. I.I:,. and I,,. represent the moments and the product of inertia in the moving systems xvhose _YJ’ planes coincide with the molecular plane:
and the jacobian for the transformation from the space-fised system J = (J,s~ - s,_rz) sin /3 i\13,/t~~~J. The above expression representing the classical kinetic energy in the hamiltonian form. and if all pc, p‘, pairs are ‘ I replaced by - d’/i3qjilq, also the quadratic part of the corresponding quantum-mechanical operator_ can be employed as the starting point for construction of various forms of the full quantum-mechanical vibration-rotation hamiltonian for triatomic molecules. As an illustration of this approach let us derive the vibration-rotation hamiltonian in the IPXl system.
We want to perform
a transformation
of the expression (A-4)
where 5,. <, represent
the Cartesian coordinates.
4 2i- = C 7;,,,1,aVaqA x-J= 1
into
a+
(A-5)
with qA, q, = (r, R. 8, y ). The CC zfficients q,(, and q,,,,, are related to one another
From
T_,=
2 l.j-1
(A.l).
(A-6)
- z$”
by
T,,,,(aq~;/ag,)(aq,/as,)-
(A-6)
and (6) it is easy to obtain:
= Cj, +- 4
-k 4.
- ( I:r,. - I :y, ) sin 2 y cot fl&
+
(f:r,. - I.::,.) sin 2 y
a’
sin p
aaap
(A-7)
l
1
with the jacobian of the transformation J = tn~rR/M’. Therefore. the jacobian the space-fixed system to the final system (with I-, R, 8. y. a. /3) is
for the transformation
J = r’) R' sin 0 sin /3_
from
(A-8)
The linear terms of the full vibration-rotation hamiltonian can simply be obtained by applying the Podolsky transformation [35]. We consider first the case when the volume element in the integration of the
wavefunction be represented
is d7 = ndq, by
(q, = r, R, 6, y, a, j3). In this case the full nuclear kinetic-energy
operator
can
=
a’
4 wq, +f ar,,G( T-
1~
‘I
I
a4 <
aT, a aq aq
+ [
I
a4,
aq
ac ,
)I
(
-----
.,
aG ppaG __
ac a 84, aq,
ac a h, aci,
+g,G-’
a4
,
\
a'G
TfjG-‘pa4. a4,
+
(A.91
. j 1
sin’ ~9sin’ p is the determinant of the T”’ matrix. The where T, = T, y and G = (M/l,,,))l,,~l_~)~/rJRJ second term in’ 6e square bracket vanishes after summing over i and j SO that the coefficients of the terms linear in moments can be obtained from the corresponding quadratic coefficients hy means of relations
@'=Ca7j,/aq,,. - ,
(‘4.10)
_i Applying
one obtains
(A-10)
(X.1 1)
The constant
term
T’“’
can also be obtained
from (A.9).
or much
simpler
by the procedure
bslo\v. The
result is
The kinetic-energy operator corresponding to the volume = 1) can be constructed applying the transformation r-‘R-’
sin-‘/‘e
on the expression
sin-‘/’
-_ T(?”
p jrR
+ f”‘)
sin’” given
dT = r’R’
sin B sin p ndq,
e sin’/?fl_
w
(jacobian
(X.13)
by eqs. (.4.7)
+ ( I_:!, COG y + I.:;,_ sin’ y )cot 0 2 C\ I;.;.
element
and (X.11).
The
term linear
+ ( I_;:,. - I_::, ) sin 2y (1 + cos’ 2 I::, I:.:. sin’ /3
in momenta
fi )
a 5’
becomes
(X.14)
terms not involving any derivatives appear also_ Since the constant term ?(“’ must be equal to zero in this case (jacobian = 1). this multiplied with - 1 represents exactly the constant term of the kinetic-energy operator corresponding to the volume element d7 = !Jdq,_ given by eq. (A-12). Introducing the components of the rovibronic angular momentum in terms of the Euler angles [lS.23]. ssprcssion (9) is obtained.
Some
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1956).