- Email: [email protected]
Vibration spectrum of H3+: A model hamiltonian
Chet@cal Physics 83 (1984) North-Holland. Amsterdam
: i _
.-:
-.
83-88
--
._
-_ :__. _. _.: -;
;
~_.
___:,._I. _j
-. ‘, _-
.- __ :
.
P-G_ BURTtiN,
E; VOti
_.
: ..,8!;
.1
1.-
-. .-
‘--.-
-.
VIBRATION.SP~~UMOF~H::-~-MODEL
_,:
:
.
.--: ._:- ,’ <‘ .__-
:
--1..-
z
_-
HAMLT+IAN
--
NAGY-FELSOBUKI
Chemisny Department, -University of Wollong&g, N.S. FK. 2500, Atkalia
and G. DOHERTY Mathe&ics
and M. HAMILTON
Department. Uniuersity of Wollo&ong, N.S. IV.. 2500, Australia
Received 11 April 1983
A model hamiltonian is proposed which is similar to that used by Camey term is expiicitiy expanded in a Taylor series. Using the mode1 hamiltonian
and Porter but which differs in that the Watson (in which the Watson term is truncated to third
order) a 320-configuration variational CI vibration calculation yields Hz eigenenergiesand eigenfunctionsin excellent agreement
with experiment
and with the results of Camey
1. Introduction
To aid experimental efforts in the spectroscopic detection of Hz in laboratory studies [1,2] and in searches of interstellar space 131, accurate ab initio calculations are required of its infrared spectrum. To this end, the work of Camey and Porter [4-71 has proved invaluable in modelling the observed vibration-rotation spectra of H,f [2] and DT [8] in their lowest vibrational states. For Hz and Dz Camey and Porter [5-71 have adapted Watson’s [9] normal coordinate hamiltonian to ma&e use of the high symmetry (Djh --, C,,) associated with these. molecules. Historically, a normal coordinate analysis of the Eckart framework was desirable since by assuming a harmonic function,for the potential energy, the multidimensional vibration problem could be separated into a more tractable set of &e-dimensional problems. While this approach was and still is the key in our understanding of molecular vibritions; it has become clear [lo-141. that the normal-mode ,-representation is far more appropriate. for low-lying vibrational states than for states of high excitation energies_ In this sense, the.+ney and Porter (CP)
and Porter.
hamiltonian assumes a highly diagonal representation of vibrations of small amplitudes in terms of the normal modes, at the expense of a strongly mixed representation of more highly excited modes of greater amplitude. A problem that arises from the normal-mode representation, however, is the well-known singularities [9,15-201 associated with the ndrmal coordinate description of the Eckart hamiltonian. Basicaily the problem is that the inertia tensor (p) becomes singular whenever a large-amplitude vibration distorts the bent molecule into a linear configuration (i.e. for 8 = 0 or T). Recently seve%al attempt+ have been made to circumvent this pioblem. For example, Camey and co-workers [5,21,22] incorporated the Watson term into their potential-energy expansion by computing a cqrreqtion to each grid point of the electronic potential, thereby avoiding singular regions. On- dther hand, Bartholomae et al. [17] mtiltiplied their bqsis:functions with- an auxiliary function which -vanishes strongly .& IRI vanishes. Of the two forms they investigated, [l - exp(--alR12)] and IRI, the latter was fqtid to be less defeitive. In a similar vein, Reimers and Watts [18] employed ‘an. auxiliary
0301-0104/84/$03.00 @ Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
54
P-G. Burto~~et aL /
function of the form. exp( - 1061pt,1), which they found to be satisfactory for HzO_ In a different direction Carter and Handy [19] avoided the singularity at linearity by employing Watson’s [23] linear hamiltonian for quasi-linear configurations. More recently, Carter and Handy [20] have rejected the normal coordinate hamiltonian altogether. utilizing instead a hamiltonian based on bond lengths and the included angle. Whilst such a hamiltonian avoids the singularity at 8 = 0 or IT. it is not appropriate for states with J > 0 and moreover. for states where J = 0. the optimal form of the angular basis functions still remains unclear (241.
Vibration spectrum
I
ofH3+
.-
where R, is the equilibrium bjqnd length, nr the proton mass and S, (a = 1, 2, 3) the vibration symmetry coordinates. From eq.(2) it is clear that there is an ellipse of singularities [(R, + S,)2 = S,’ + $1 in the Watson term of the hamiltdnian. By restricting our model to small vibrational amplitudes, eq. (2) can be expanded in a Taylor series yielding (see appendix A): 6’;“=
+i'(l/mR~),
(34
tip= -~(h2/m~~)(~e-2s,), h”
$a=-
L
- 5
2m i 4RZ
this paper we present results obtained from our model hamiltonian which is based on CP’s hamiltonian, but in which the inertia tensor has been explicitly expanded as a Taylor series in the normal-mode coordinates (as suggested by Watson [9]). Thus, our model hamiltonian is applicable to vibrational amplitudes which are small compared to bond distances and so lie in coordinate space in which the normal-mode representation is quasi-diagonal. In
(
-1
*s,
3s;
c!
Rf
+R--
iI
(3b)
(34
(34 2. Model
hamiltonian
For a molecule Eckart framework.
and solution
methodology
with D3,, symmetry within the CP’s vibration hamiltonian is
[5].
Thus our proposed
model hamiltonian
is,
3
Iti=
c
T_+ f,+ ti(s,, s,, s,)+
tiy,
(4)
3
Fj=
c i,+ i;+
il,+ 3,
(1)
a=1
where i;, is the kinetic energy, ?! the vibration angular momentum and i’, the Watson operator (the latter term acts as a mass-dependent contribution to the potential energy)_ The mass and geometry dependent Watson term is given by [5,9].
i’, = -
-h2 8
4[(R,+S$+.S;+S;] [
wheren=1,2and3 --- etc. It is self evident that eqs. (3a)-(3d) cannot contain singularities. We have adopted a solution procedure analogous to that of CP. That is, our three-dimensional functions are expanded in terms of products of eigenfunctions of our finite element solution to each of the one-dimensional normal mode vibrational equations [25]. On applying the variational principle to the secular equation for the three-dimensional vibrational wavefunctions IH - ES1 ICI = 0,
“‘[(R,+S1)2-.s+s.$
(2)
(5)
we obtaiti the vibrational eigenvalues Ei and the corresponding configurational expansion coefficients Cii_ While a more detailed description of our solu-
P-G. Burton et al: J Vibration spectrum of H3+
,85
tion procedure will be given elsewhere 1251 it is necessary to point out- the following differences from CPs approach: (i) Our basis- is a -finite element_ rather than an analytical basis, converged to the level specified in (ii). (ii) The basis employed can be described as [5 x +(S,), 8 x +(S,), 8 x +(S,)] which yields 100 more configurations than CP [5]. This basis set
truncations (without-the- Y, integrals) on the vibrational eigenenergks and -eigenfunctions_- It is evident that the vibrational assignment of successive eigenvalues is invariant to the degree of truncation and is consistent with the assignment given by the zero-order hamiltonian. Furthermore, the per-. centage weight changes insignificantly (by less than 0.2%) across the entries of table 1 which indicates the negligible effect of the mass-dependent coirec-
was chosen on an energy criterion, retaining all one-dimensional eigenfunctions lying within = 12000 cm-’ of the zero-point energy. (iii) This configuration space involves no selection within the 5 X 8 x 8 full CI space. (iv) The potential energy grid is -based on 78 symmetry-independent PNO CI calculations using an electronic orbital basis of 81 functions and the analytical potential fitted to the resultant 112 points is expanded in terms of a sixth degree Simons-Parr-Finlan (SPF) [26] force field. (v) In our case, all integrals are numerically evaluated and moreover, no approximations (inner projections) have been made to reduce the powers of the operators. (vi) For integrals involving the ?,,, ?, and 02”) operators, we have used a six-point GaussLegendre quadrature scheme within the domain of each finite-element basis-function interval. (vii) We have employed a 2O(S,) X 20(&) X 2O(S,) Harris, Engerholm and Gwinn (HEG) [27] potential-energy integrator. This is 7000 more quadrature points than the CP scheme [5].
tion to the character of the low-lying vibrational wavefunctions. For HT (with an equilibrium bond length of 1.65247 au ? the zero-order Watson term (/o(O)[see eq. (3a)] lowers the vibrational energy levels by 27.37 cm-‘. Table 1 indicates that UL’), UJ*) and Ut3’ corrections lower the vibrational eigeneneriies on an average by 20.87, 27.47 and 27.01 cm-’ respectively_ Moreover, the average difference of ]U,Cr’- Uj’)] and ]CrJ’;- Ud”] is 4.31 and 0.46 cm-’ respectively, indicating that the UL3) correction is closely approaching convergence. In table 2 we show the effect of the Watson-term truncations (without the T, integrals) on the degenerate components of the assigned E modes. It is evident that the U, perturbation does not add significantly to the non-degeneracy (= 0.09 cm-‘). Furthermore, table 2 indicates that the average energy- of the (quasi) degenerate pairs has converged at the third-degree level. Table 3 gives the effect of the Watson-term truncations (with T, integrals) on the vibrational eigenenergies and eigenfunctions. Table 3 shows that the order of the assignment is independent of the truncations and is completely consistent with table 1. Furthermore a comparison of the % weight of fi(‘) (table 1) and k(O) + ?‘, (table 3) indicates that the character of the vibrational wavefunctions has not significantly changed (within 0.5%) with the inclusion of the T, integrals. As has been previously reported by CP [5] the inclusion of the T, integrals raises the vibrational energy levels and in our particular case the increase is on an average of 27.46 cm-‘. Table 3 also indicates that the character of the vibrational wavefunctions changes only within
3. Results The Watson-term truncations affect both the three-dimensional eigenvalues and eigenfunctions. In order to assess the effect on the character of the vibrational eigenfunction, we shall define (in the usual manner) the weight of the basis function to be,
where Cii is thejth basis-function expansion coefficient in the ith vibrational eigenfunction. In table 1 we show the effect of the Watson-term
* For H;, PNO CI calculations yield a minimum energy of - 134188 hartree for an equilateral &ometry.
36
P.G. Burton et al_ /
Vibration specnwn of H3+
Table 1 Effect of the Watson-term truncations (without the T, integrals) on the vibrational eigenenergies and eigenfunctions=) Dominant configuration
fi’0’
Ato, + @?I 0
fit01 + 0”) Y
% weight
eigen-
eigen-
R weight
$0, + eigen-
% weight
0
56 weight
eigenenergy (cm- ’ )
energy
energy (cm- ’ )
energy (cm-‘)
C(3)
(cm-‘)
ooo 001
96.9 s5_7
4397.39 6887.88
96.9 85.8
4372.16 6863.84
96.9 85.8
4370.37 6860.50
96.9 85.8
4370.46 6860.67
010
88.7
6890.11
88.8
6866.06’
86.4 32.2 + 31.7 46.6 - 32.9 63.0
7606.56 9163.60 9316.71 9324.46
86.5 32.2+31.7 46.6 - 32.9 63.1
7583.15 9140.52 9293.84 9301.58
88.8 86.5 32.2t31.6 41.6 - 32.9 63.0
6862.72 7579.70 9135.30 9288.75 9296.59
88.8 86.4 32.2 + 31.6 46.6 - 32.9 63.0
6862.89 7580.17 9135.55 9289.01 9296.85
100 (020)+(002) (020)~(002) 011 101 110 200
63.5 66.2 76.9
9955.81 9981.19 10709.95
9933.60 9958.91 10688.40
64.0 66.4 77.1
9928.26 9953.69 10683.05
63.9 66.3 77.0
9928.92 9954.33 10684.70
63.9 66.3 77.0
il’ % weight is given by sq. (6). 8”) = X’,_ ,?,- + r’( S,. Sz_ S,) where 9 is a sixth-degree SPF force field generated from a 112 point PNO CI grid i25]. For a definition of oLn’ see eq. (3).
Table 2 Effect of the Watson-term truncations (w.thout the T, integrals) (cm-‘) with respect to the degenerate components of the E modes ‘) E modes (oOl)--(010)
010 -(Oil) ( oo’, 1
(lOl)-(110)
fi’O,
fi’0’ l (/!I> fi’“‘* $21 L 0
2.23
2.22
2.22
7.75
7.74
7.84
7.84
25.38
15.31
25.43
25.41
corrections
lower
to the third-order correcthat I/A’), UA’) and Q3)
the vibrational
eigenenergies
=’ See footnote to table 1. The splittings in these components of E modes is due to lack of convergence in the 3D vibration
which
is in
results
of table
Table
4
truncations
rxpnnsion at the 5 x S x S level [ZSl and the choice of Sz. S,.
accordance
with
the
corresponding
1.
gives (with
the effect of the Watson-term the T, integrals) on the degener-
Table 3 Effect of the Watson-term
Dominant configuration
truncation (with the Tl integrals) on the vibrational eigenenergies and eigenfunctions A)
#“I + i‘ 1 % weight
$0’ + fr + CC
fi’“’ + i, + @I) % weight
AC01+ f* +
w
0
eigenenergy (cm-‘)
eigenenergy
% weight
(cm-‘)
eigenenergy 4370.76 6883.35
OOG 001
96.9 85.9
4397.78 6910.71
96.9 85.9
010 100
58.7 86.4
6913.02 7607.24
88.8 86.5
688898 7583.82
88.7 86.5
6885.65 7580.38
32.7 + 32.1 4&l- 32.9 62.7
9189.33 9391.44 9401.68
32.7 -I-32.1 46.2 - 32.9 62.8
9166.27 9368.62 9378.83
32.7 + 32.1 46.2 - 32.9 62.7
9161.11 9363.51 9373.84
101 110 200
64.2 66.2 77.1
a) See footnote to table 1.
9979.66 10006.29 10711.12
64.3 66.4 77.2
9957.47 9984.01 10689.56
% weight
(cm-‘) 96.9 85.9
4372.55 6886.69
(020)+(002) (020)-(002) 011
on
an average by 23.14, 27.45 and 27.03 cm-’ respectively is in accordance with the results of table 1. Once again, the average difference of IUi”U,‘“[ and I(/:” - clA3’I is 4.30 and 0.42 cm- * respectively
fi’“’ + O’(3) 0
2.22
0.1% from the first-order tions. It is also evident
64.2 66.3 77.1
9952.14 9978.82 10684.22
96.9 85.9 88.7 86.5 32.7 + 32.1 46.2 - 32.9 62.7 64.3 66.3 77.1
(J’3b
w
eigenenergy (cm-‘) 4370.86 6883.52 6885.85 7580.84 9161.35 9363.77 9374.10 9952.80 9979.46 10685.40
--.87
RG. Burtonetal / Vibatioispectrum ofH3+ Table 4 Effect of the Watson-term truncation (with the T, integrals) ‘“._“,“ith respect to the degenerate components of l he E
(ooL)-(010)
( ) E
i(O11)
(i01)~(110)
231
2.29
2.30
2.33
10.24
10.21
10.33
10.33
26.63
26.54
26.68
26.66
xl See footnote to tables 1 and 3.
ate components of the E modes. As anticipated the effect is almost identical to that shown by table 2, indicating that the inclusion of the T,-term.manifests itself as almost translating the average energy of the degenerate components in a linear fashion. That is, the increase of the non-degeneracy of the (001) and (OlO), (&“,) and (011) and (101) and (110) components by the T term is = 0.08, 2.49 and 1.25 cm-’ respectively across the entries of table 4. Table 5 compares experimental and-ab initio calculated vibrational band origins. The CP calculation [5] yields basically the same order of the dominant configuration, the minor exceptions being that the order of the degenerate components (001) and (010) and (101) and (110) are reversed. it will be shown elsewhere [25] that the difference can be accounted for by the form of the basis set used (analytical as opposed to numerical) as well as the extent of the CI truncation (220 as opposed to 320 configurations). Whilst our calculation is more accurate in terms of the basic potential
surfa&, and the numerical techniques used in sol@ing the vibrational Sdhrijd&ger equation that? CP [25], there is nevertheless excellent agreement between the_kalculated vibratiq+l band origins. As antikipaied the majo;. discrepaxicies .a& for. the -high& excitation energies.. For example, the, difference between the calculated band origins &-,l.k,, ~l.O,O~ vo.z.o*%.1.* 1 and 5.o.o is only 2.25, 24.66,. 8.70, 6.05, 27.46 and 46.77 cm-’ respectively_ Comparison. with experiment shows that the more-refined calculation yields a FE fundamental vibrational band origin within = 14 cm-‘, slightly larger than the CP result of = 11 cm-‘.
4. Conclusion In this report we have reformulated Carney and Porter’s normal coordinate hamiltonian for HT, by explicitly expanding the Watson term in a Taylor series. In doing so, we have avoided singularities associated with large-amplitude vibrations distorting a bent molecule into a linear configuration. However, the trade-off is that the hamiltonian is now restricted to vibrational amplitudes which are small compared to the bond length. This appears to present little difficulties with .the description of low-lying vibrational states df H,+ despite the extensive tunnelling due to the low nuclear
masses
involved_
We have also shown that within our solution algorithm the vibrational eigenvahtes and eigenfunctions are converged at the third-order Watson-term truncation. Moreover, we have dem-
Table 5 ’ Co.mparison of vibrational band origins (cm-‘) Designation *’
Exp. b,
Sixth SPF PNO CI ” FEM
Fifth SPF SD CI ‘) HHH
fio.1.* 1
2522 [3196] 147971
2513.83 3209.98 4790.49
2516.08 3185.32
[4986] [5572] 162701
4998.08 5595.27 6314.54
%.o,o Fo.20
Fo.z.*z ~l.l.il p20.0
a) b, c, d,
4799.19 5004.13 5567.81 6267.77
For an explanation of the notation see ref. [28] From refs. [2] and [29]; only the yE fundamental has been definitively assigned. Finite element basis with I? = x:,T, + ?‘,+ cl” + p. A total of 320 configurations was used. From Camey and Porter[S]_
onstrated the effectiveness of our model hamiltonian. by showing that the results from our most accurate calculation are in agreement with experiment and with the calculation of Camey and Porter. Attainment of fully converged 3D vibrational CI wavefunctions for these states is reported elsewhere [25].
References [l] [2] [3] [4] [5] (61
Acknowledgement
[7]
We wish to acknowledge the University of Wollongong Computer Centre for its excellent service. We also acknowledge the support of the Commonwealth Postgraduate Research award held by M. Hamilton.
[S] [9]
A. Canifigton, J. Buttenshawand R. Kennedy. Mol. Phys. 45 (1982) 753. T. Oka, Phys. Rev. Letters 45 (1980) 531. T. Oka. Phil. Trans. Roy Sot. London A303 (1981) 543. G.D. Camey and R.N. Porter, J. Chem. Phys. 60 (1974) 4251. G.D. Camey and R.N. Porter. J. Chem. Phys. 65 (1976) 3547. G.D. Camey and R.N. Porter, Chem. Phys. Letters 50 (1977) 327. G.D. Camey and R.N. Poner. Phys. Rev. Letters 45 (1980) 537. J.T. Shy. J.W. Farley, W.E. Lamb and W.H. Wing, Phys. Rev. Letters 45 (19800 535. J.K. Watson, Mol. Phys. 18 (1968) 479.
[lOI W. Siebrand and D.F_ Williams. J. Chem. Phys. 49 (1968) 1860.
VII B.R. Henry. Accounts
Chem.
Res.
10 (1976)
207. and
references therein.
Appendis A According to Watson [9] the mass-dependent correction to the potential energy is given by i/:=
-$c/l,,,. 0
(A-1)
where the p tensor is in terms of the normal coordinates. The expansion of the effective reciprocal inertia tensor in the Taylor series is given by. !.l= (IO)_’
+ f(l”)-‘a(P)-‘a(l”)-’ - +(I”)-‘a(l”)-‘a(l”)-‘a(l’)-
f
- - - ,(A.2)
where a is the interaction coefficient matrix. terms of the vibration symmetry. the form (IO) -I and a is given by 1 0
0 1
o0
mR2, o
o
+
[
s,+sz a= mR e - SJ 0
D. Martin [I71 R. Bartholomae. Specuy. 57 (1981) 367.
1181 J. Reimers and R. 1191 S. Carter and NC VI S. Carter and NC. VI G.D. Games. L.L.
and
B.T. Sutcliffe,
J. Mol.
Watts, private communication (1953). Handy, J. Mol. Spectry. 87 (1981) 367.
Handy, Mol. Phys. 47 (1982) 1445. Snrandel and C.W. Kern, Advan. Chem. Phys. 37 (1678) 305:
[E] G-C. Camey. Mol. Phys. 39 (1980) 923.
-(IO)-‘a(l”)-’
(1())-Lz
H.S. Moller and D. Sonnichmortenxn, Chem. Phys. Letters 66 (1979) 539. (131 R.T. Lawton and M.S. Child. Mol. Phys. 44 (1981) 709. 1141 R. Wallace, Chem. Phys. 71 (1982) 173. 1151 A. Sayvvctz. J. Chsm. Phys. 7 (1939) 383. [161 B.T. Sutcliffe. in: Quantum dynamics of molecules. ed. R.G. Woolley (Plenum Press. View York. 1979).
molecules (Van Nostrand. Princeton. 1948). [291 U. Steinmetzger. Ph.D. Thesis. Berlin (1982).
,
_I
-s; 0 s,--s, 0 0
In of
[231 J.K. Watson. Mol. Phys. 19 (1970) 465. [241 B.T. Sutcliffe, Mol. Phys. 48 (1983) 561. I=1 P.G. Burton. E. von Nagy-Felsobuki, G. Doherty and M. Hamilton, to be published. 1261 G. Simons, R. Parr and J.M. Finlan, J. Chem. Phys. 59 (1973) 3229. P71 D. Harris, G. Engerholm and L.D. Gwinn. J. Chem. Phys. 43 (1965) 1515. 1281 G. Herzberg, Infrared and Raman spectra of polyatomic
2s,
.1
(A-3)
: i _
.-:
-.
83-88
--
._
-_ :__. _. _.: -;
;
~_.
___:,._I. _j
-. ‘, _-
.- __ :
.
P-G_ BURTtiN,
E; VOti
_.
: ..,8!;
.1
1.-
-. .-
‘--.-
-.
VIBRATION.SP~~UMOF~H::-~-MODEL
_,:
:
.
.--: ._:- ,’ <‘ .__-
:
--1..-
z
_-
HAMLT+IAN
--
NAGY-FELSOBUKI
Chemisny Department, -University of Wollong&g, N.S. FK. 2500, Atkalia
and G. DOHERTY Mathe&ics
and M. HAMILTON
Department. Uniuersity of Wollo&ong, N.S. IV.. 2500, Australia
Received 11 April 1983
A model hamiltonian is proposed which is similar to that used by Camey term is expiicitiy expanded in a Taylor series. Using the mode1 hamiltonian
and Porter but which differs in that the Watson (in which the Watson term is truncated to third
order) a 320-configuration variational CI vibration calculation yields Hz eigenenergiesand eigenfunctionsin excellent agreement
with experiment
and with the results of Camey
1. Introduction
To aid experimental efforts in the spectroscopic detection of Hz in laboratory studies [1,2] and in searches of interstellar space 131, accurate ab initio calculations are required of its infrared spectrum. To this end, the work of Camey and Porter [4-71 has proved invaluable in modelling the observed vibration-rotation spectra of H,f [2] and DT [8] in their lowest vibrational states. For Hz and Dz Camey and Porter [5-71 have adapted Watson’s [9] normal coordinate hamiltonian to ma&e use of the high symmetry (Djh --, C,,) associated with these. molecules. Historically, a normal coordinate analysis of the Eckart framework was desirable since by assuming a harmonic function,for the potential energy, the multidimensional vibration problem could be separated into a more tractable set of &e-dimensional problems. While this approach was and still is the key in our understanding of molecular vibritions; it has become clear [lo-141. that the normal-mode ,-representation is far more appropriate. for low-lying vibrational states than for states of high excitation energies_ In this sense, the.+ney and Porter (CP)
and Porter.
hamiltonian assumes a highly diagonal representation of vibrations of small amplitudes in terms of the normal modes, at the expense of a strongly mixed representation of more highly excited modes of greater amplitude. A problem that arises from the normal-mode representation, however, is the well-known singularities [9,15-201 associated with the ndrmal coordinate description of the Eckart hamiltonian. Basicaily the problem is that the inertia tensor (p) becomes singular whenever a large-amplitude vibration distorts the bent molecule into a linear configuration (i.e. for 8 = 0 or T). Recently seve%al attempt+ have been made to circumvent this pioblem. For example, Camey and co-workers [5,21,22] incorporated the Watson term into their potential-energy expansion by computing a cqrreqtion to each grid point of the electronic potential, thereby avoiding singular regions. On- dther hand, Bartholomae et al. [17] mtiltiplied their bqsis:functions with- an auxiliary function which -vanishes strongly .& IRI vanishes. Of the two forms they investigated, [l - exp(--alR12)] and IRI, the latter was fqtid to be less defeitive. In a similar vein, Reimers and Watts [18] employed ‘an. auxiliary
0301-0104/84/$03.00 @ Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
54
P-G. Burto~~et aL /
function of the form. exp( - 1061pt,1), which they found to be satisfactory for HzO_ In a different direction Carter and Handy [19] avoided the singularity at linearity by employing Watson’s [23] linear hamiltonian for quasi-linear configurations. More recently, Carter and Handy [20] have rejected the normal coordinate hamiltonian altogether. utilizing instead a hamiltonian based on bond lengths and the included angle. Whilst such a hamiltonian avoids the singularity at 8 = 0 or IT. it is not appropriate for states with J > 0 and moreover. for states where J = 0. the optimal form of the angular basis functions still remains unclear (241.
Vibration spectrum
I
ofH3+
.-
where R, is the equilibrium bjqnd length, nr the proton mass and S, (a = 1, 2, 3) the vibration symmetry coordinates. From eq.(2) it is clear that there is an ellipse of singularities [(R, + S,)2 = S,’ + $1 in the Watson term of the hamiltdnian. By restricting our model to small vibrational amplitudes, eq. (2) can be expanded in a Taylor series yielding (see appendix A): 6’;“=
+i'(l/mR~),
(34
tip= -~(h2/m~~)(~e-2s,), h”
$a=-
L
- 5
2m i 4RZ
this paper we present results obtained from our model hamiltonian which is based on CP’s hamiltonian, but in which the inertia tensor has been explicitly expanded as a Taylor series in the normal-mode coordinates (as suggested by Watson [9]). Thus, our model hamiltonian is applicable to vibrational amplitudes which are small compared to bond distances and so lie in coordinate space in which the normal-mode representation is quasi-diagonal. In
(
-1
*s,
3s;
c!
Rf
+R--
iI
(3b)
(34
(34 2. Model
hamiltonian
For a molecule Eckart framework.
and solution
methodology
with D3,, symmetry within the CP’s vibration hamiltonian is
[5].
Thus our proposed
model hamiltonian
is,
3
Iti=
c
T_+ f,+ ti(s,, s,, s,)+
tiy,
(4)
3
Fj=
c i,+ i;+
il,+ 3,
(1)
a=1
where i;, is the kinetic energy, ?! the vibration angular momentum and i’, the Watson operator (the latter term acts as a mass-dependent contribution to the potential energy)_ The mass and geometry dependent Watson term is given by [5,9].
i’, = -
-h2 8
4[(R,+S$+.S;+S;] [
wheren=1,2and3 --- etc. It is self evident that eqs. (3a)-(3d) cannot contain singularities. We have adopted a solution procedure analogous to that of CP. That is, our three-dimensional functions are expanded in terms of products of eigenfunctions of our finite element solution to each of the one-dimensional normal mode vibrational equations [25]. On applying the variational principle to the secular equation for the three-dimensional vibrational wavefunctions IH - ES1 ICI = 0,
“‘[(R,+S1)2-.s+s.$
(2)
(5)
we obtaiti the vibrational eigenvalues Ei and the corresponding configurational expansion coefficients Cii_ While a more detailed description of our solu-
P-G. Burton et al: J Vibration spectrum of H3+
,85
tion procedure will be given elsewhere 1251 it is necessary to point out- the following differences from CPs approach: (i) Our basis- is a -finite element_ rather than an analytical basis, converged to the level specified in (ii). (ii) The basis employed can be described as [5 x +(S,), 8 x +(S,), 8 x +(S,)] which yields 100 more configurations than CP [5]. This basis set
truncations (without-the- Y, integrals) on the vibrational eigenenergks and -eigenfunctions_- It is evident that the vibrational assignment of successive eigenvalues is invariant to the degree of truncation and is consistent with the assignment given by the zero-order hamiltonian. Furthermore, the per-. centage weight changes insignificantly (by less than 0.2%) across the entries of table 1 which indicates the negligible effect of the mass-dependent coirec-
was chosen on an energy criterion, retaining all one-dimensional eigenfunctions lying within = 12000 cm-’ of the zero-point energy. (iii) This configuration space involves no selection within the 5 X 8 x 8 full CI space. (iv) The potential energy grid is -based on 78 symmetry-independent PNO CI calculations using an electronic orbital basis of 81 functions and the analytical potential fitted to the resultant 112 points is expanded in terms of a sixth degree Simons-Parr-Finlan (SPF) [26] force field. (v) In our case, all integrals are numerically evaluated and moreover, no approximations (inner projections) have been made to reduce the powers of the operators. (vi) For integrals involving the ?,,, ?, and 02”) operators, we have used a six-point GaussLegendre quadrature scheme within the domain of each finite-element basis-function interval. (vii) We have employed a 2O(S,) X 20(&) X 2O(S,) Harris, Engerholm and Gwinn (HEG) [27] potential-energy integrator. This is 7000 more quadrature points than the CP scheme [5].
tion to the character of the low-lying vibrational wavefunctions. For HT (with an equilibrium bond length of 1.65247 au ? the zero-order Watson term (/o(O)[see eq. (3a)] lowers the vibrational energy levels by 27.37 cm-‘. Table 1 indicates that UL’), UJ*) and Ut3’ corrections lower the vibrational eigeneneriies on an average by 20.87, 27.47 and 27.01 cm-’ respectively_ Moreover, the average difference of ]U,Cr’- Uj’)] and ]CrJ’;- Ud”] is 4.31 and 0.46 cm-’ respectively, indicating that the UL3) correction is closely approaching convergence. In table 2 we show the effect of the Watson-term truncations (without the T, integrals) on the degenerate components of the assigned E modes. It is evident that the U, perturbation does not add significantly to the non-degeneracy (= 0.09 cm-‘). Furthermore, table 2 indicates that the average energy- of the (quasi) degenerate pairs has converged at the third-degree level. Table 3 gives the effect of the Watson-term truncations (with T, integrals) on the vibrational eigenenergies and eigenfunctions. Table 3 shows that the order of the assignment is independent of the truncations and is completely consistent with table 1. Furthermore a comparison of the % weight of fi(‘) (table 1) and k(O) + ?‘, (table 3) indicates that the character of the vibrational wavefunctions has not significantly changed (within 0.5%) with the inclusion of the T, integrals. As has been previously reported by CP [5] the inclusion of the T, integrals raises the vibrational energy levels and in our particular case the increase is on an average of 27.46 cm-‘. Table 3 also indicates that the character of the vibrational wavefunctions changes only within
3. Results The Watson-term truncations affect both the three-dimensional eigenvalues and eigenfunctions. In order to assess the effect on the character of the vibrational eigenfunction, we shall define (in the usual manner) the weight of the basis function to be,
where Cii is thejth basis-function expansion coefficient in the ith vibrational eigenfunction. In table 1 we show the effect of the Watson-term
* For H;, PNO CI calculations yield a minimum energy of - 134188 hartree for an equilateral &ometry.
36
P.G. Burton et al_ /
Vibration specnwn of H3+
Table 1 Effect of the Watson-term truncations (without the T, integrals) on the vibrational eigenenergies and eigenfunctions=) Dominant configuration
fi’0’
Ato, + @?I 0
fit01 + 0”) Y
% weight
eigen-
eigen-
R weight
$0, + eigen-
% weight
0
56 weight
eigenenergy (cm- ’ )
energy
energy (cm- ’ )
energy (cm-‘)
C(3)
(cm-‘)
ooo 001
96.9 s5_7
4397.39 6887.88
96.9 85.8
4372.16 6863.84
96.9 85.8
4370.37 6860.50
96.9 85.8
4370.46 6860.67
010
88.7
6890.11
88.8
6866.06’
86.4 32.2 + 31.7 46.6 - 32.9 63.0
7606.56 9163.60 9316.71 9324.46
86.5 32.2+31.7 46.6 - 32.9 63.1
7583.15 9140.52 9293.84 9301.58
88.8 86.5 32.2t31.6 41.6 - 32.9 63.0
6862.72 7579.70 9135.30 9288.75 9296.59
88.8 86.4 32.2 + 31.6 46.6 - 32.9 63.0
6862.89 7580.17 9135.55 9289.01 9296.85
100 (020)+(002) (020)~(002) 011 101 110 200
63.5 66.2 76.9
9955.81 9981.19 10709.95
9933.60 9958.91 10688.40
64.0 66.4 77.1
9928.26 9953.69 10683.05
63.9 66.3 77.0
9928.92 9954.33 10684.70
63.9 66.3 77.0
il’ % weight is given by sq. (6). 8”) = X’,_ ,?,- + r’( S,. Sz_ S,) where 9 is a sixth-degree SPF force field generated from a 112 point PNO CI grid i25]. For a definition of oLn’ see eq. (3).
Table 2 Effect of the Watson-term truncations (w.thout the T, integrals) (cm-‘) with respect to the degenerate components of the E modes ‘) E modes (oOl)--(010)
010 -(Oil) ( oo’, 1
(lOl)-(110)
fi’O,
fi’0’ l (/!I> fi’“‘* $21 L 0
2.23
2.22
2.22
7.75
7.74
7.84
7.84
25.38
15.31
25.43
25.41
corrections
lower
to the third-order correcthat I/A’), UA’) and Q3)
the vibrational
eigenenergies
=’ See footnote to table 1. The splittings in these components of E modes is due to lack of convergence in the 3D vibration
which
is in
results
of table
Table
4
truncations
rxpnnsion at the 5 x S x S level [ZSl and the choice of Sz. S,.
accordance
with
the
corresponding
1.
gives (with
the effect of the Watson-term the T, integrals) on the degener-
Table 3 Effect of the Watson-term
Dominant configuration
truncation (with the Tl integrals) on the vibrational eigenenergies and eigenfunctions A)
#“I + i‘ 1 % weight
$0’ + fr + CC
fi’“’ + i, + @I) % weight
AC01+ f* +
w
0
eigenenergy (cm-‘)
eigenenergy
% weight
(cm-‘)
eigenenergy 4370.76 6883.35
OOG 001
96.9 85.9
4397.78 6910.71
96.9 85.9
010 100
58.7 86.4
6913.02 7607.24
88.8 86.5
688898 7583.82
88.7 86.5
6885.65 7580.38
32.7 + 32.1 4&l- 32.9 62.7
9189.33 9391.44 9401.68
32.7 -I-32.1 46.2 - 32.9 62.8
9166.27 9368.62 9378.83
32.7 + 32.1 46.2 - 32.9 62.7
9161.11 9363.51 9373.84
101 110 200
64.2 66.2 77.1
a) See footnote to table 1.
9979.66 10006.29 10711.12
64.3 66.4 77.2
9957.47 9984.01 10689.56
% weight
(cm-‘) 96.9 85.9
4372.55 6886.69
(020)+(002) (020)-(002) 011
on
an average by 23.14, 27.45 and 27.03 cm-’ respectively is in accordance with the results of table 1. Once again, the average difference of IUi”U,‘“[ and I(/:” - clA3’I is 4.30 and 0.42 cm- * respectively
fi’“’ + O’(3) 0
2.22
0.1% from the first-order tions. It is also evident
64.2 66.3 77.1
9952.14 9978.82 10684.22
96.9 85.9 88.7 86.5 32.7 + 32.1 46.2 - 32.9 62.7 64.3 66.3 77.1
(J’3b
w
eigenenergy (cm-‘) 4370.86 6883.52 6885.85 7580.84 9161.35 9363.77 9374.10 9952.80 9979.46 10685.40
--.87
RG. Burtonetal / Vibatioispectrum ofH3+ Table 4 Effect of the Watson-term truncation (with the T, integrals) ‘“._“,“ith respect to the degenerate components of l he E
(ooL)-(010)
( ) E
i(O11)
(i01)~(110)
231
2.29
2.30
2.33
10.24
10.21
10.33
10.33
26.63
26.54
26.68
26.66
xl See footnote to tables 1 and 3.
ate components of the E modes. As anticipated the effect is almost identical to that shown by table 2, indicating that the inclusion of the T,-term.manifests itself as almost translating the average energy of the degenerate components in a linear fashion. That is, the increase of the non-degeneracy of the (001) and (OlO), (&“,) and (011) and (101) and (110) components by the T term is = 0.08, 2.49 and 1.25 cm-’ respectively across the entries of table 4. Table 5 compares experimental and-ab initio calculated vibrational band origins. The CP calculation [5] yields basically the same order of the dominant configuration, the minor exceptions being that the order of the degenerate components (001) and (010) and (101) and (110) are reversed. it will be shown elsewhere [25] that the difference can be accounted for by the form of the basis set used (analytical as opposed to numerical) as well as the extent of the CI truncation (220 as opposed to 320 configurations). Whilst our calculation is more accurate in terms of the basic potential
surfa&, and the numerical techniques used in sol@ing the vibrational Sdhrijd&ger equation that? CP [25], there is nevertheless excellent agreement between the_kalculated vibratiq+l band origins. As antikipaied the majo;. discrepaxicies .a& for. the -high& excitation energies.. For example, the, difference between the calculated band origins &-,l.k,, ~l.O,O~ vo.z.o*%.1.* 1 and 5.o.o is only 2.25, 24.66,. 8.70, 6.05, 27.46 and 46.77 cm-’ respectively_ Comparison. with experiment shows that the more-refined calculation yields a FE fundamental vibrational band origin within = 14 cm-‘, slightly larger than the CP result of = 11 cm-‘.
4. Conclusion In this report we have reformulated Carney and Porter’s normal coordinate hamiltonian for HT, by explicitly expanding the Watson term in a Taylor series. In doing so, we have avoided singularities associated with large-amplitude vibrations distorting a bent molecule into a linear configuration. However, the trade-off is that the hamiltonian is now restricted to vibrational amplitudes which are small compared to the bond length. This appears to present little difficulties with .the description of low-lying vibrational states df H,+ despite the extensive tunnelling due to the low nuclear
masses
involved_
We have also shown that within our solution algorithm the vibrational eigenvahtes and eigenfunctions are converged at the third-order Watson-term truncation. Moreover, we have dem-
Table 5 ’ Co.mparison of vibrational band origins (cm-‘) Designation *’
Exp. b,
Sixth SPF PNO CI ” FEM
Fifth SPF SD CI ‘) HHH
fio.1.* 1
2522 [3196] 147971
2513.83 3209.98 4790.49
2516.08 3185.32
[4986] [5572] 162701
4998.08 5595.27 6314.54
%.o,o Fo.20
Fo.z.*z ~l.l.il p20.0
a) b, c, d,
4799.19 5004.13 5567.81 6267.77
For an explanation of the notation see ref. [28] From refs. [2] and [29]; only the yE fundamental has been definitively assigned. Finite element basis with I? = x:,T, + ?‘,+ cl” + p. A total of 320 configurations was used. From Camey and Porter[S]_
onstrated the effectiveness of our model hamiltonian. by showing that the results from our most accurate calculation are in agreement with experiment and with the calculation of Camey and Porter. Attainment of fully converged 3D vibrational CI wavefunctions for these states is reported elsewhere [25].
References [l] [2] [3] [4] [5] (61
Acknowledgement
[7]
We wish to acknowledge the University of Wollongong Computer Centre for its excellent service. We also acknowledge the support of the Commonwealth Postgraduate Research award held by M. Hamilton.
[S] [9]
A. Canifigton, J. Buttenshawand R. Kennedy. Mol. Phys. 45 (1982) 753. T. Oka, Phys. Rev. Letters 45 (1980) 531. T. Oka. Phil. Trans. Roy Sot. London A303 (1981) 543. G.D. Camey and R.N. Porter, J. Chem. Phys. 60 (1974) 4251. G.D. Camey and R.N. Porter. J. Chem. Phys. 65 (1976) 3547. G.D. Camey and R.N. Porter, Chem. Phys. Letters 50 (1977) 327. G.D. Camey and R.N. Poner. Phys. Rev. Letters 45 (1980) 537. J.T. Shy. J.W. Farley, W.E. Lamb and W.H. Wing, Phys. Rev. Letters 45 (19800 535. J.K. Watson, Mol. Phys. 18 (1968) 479.
[lOI W. Siebrand and D.F_ Williams. J. Chem. Phys. 49 (1968) 1860.
VII B.R. Henry. Accounts
Chem.
Res.
10 (1976)
207. and
references therein.
Appendis A According to Watson [9] the mass-dependent correction to the potential energy is given by i/:=
-$c/l,,,. 0
(A-1)
where the p tensor is in terms of the normal coordinates. The expansion of the effective reciprocal inertia tensor in the Taylor series is given by. !.l= (IO)_’
+ f(l”)-‘a(P)-‘a(l”)-’ - +(I”)-‘a(l”)-‘a(l”)-‘a(l’)-
f
- - - ,(A.2)
where a is the interaction coefficient matrix. terms of the vibration symmetry. the form (IO) -I and a is given by 1 0
0 1
o0
mR2, o
o
+
[
s,+sz a= mR e - SJ 0
D. Martin [I71 R. Bartholomae. Specuy. 57 (1981) 367.
1181 J. Reimers and R. 1191 S. Carter and NC VI S. Carter and NC. VI G.D. Games. L.L.
and
B.T. Sutcliffe,
J. Mol.
Watts, private communication (1953). Handy, J. Mol. Spectry. 87 (1981) 367.
Handy, Mol. Phys. 47 (1982) 1445. Snrandel and C.W. Kern, Advan. Chem. Phys. 37 (1678) 305:
[E] G-C. Camey. Mol. Phys. 39 (1980) 923.
-(IO)-‘a(l”)-’
(1())-Lz
H.S. Moller and D. Sonnichmortenxn, Chem. Phys. Letters 66 (1979) 539. (131 R.T. Lawton and M.S. Child. Mol. Phys. 44 (1981) 709. 1141 R. Wallace, Chem. Phys. 71 (1982) 173. 1151 A. Sayvvctz. J. Chsm. Phys. 7 (1939) 383. [161 B.T. Sutcliffe. in: Quantum dynamics of molecules. ed. R.G. Woolley (Plenum Press. View York. 1979).
molecules (Van Nostrand. Princeton. 1948). [291 U. Steinmetzger. Ph.D. Thesis. Berlin (1982).
,
_I
-s; 0 s,--s, 0 0
In of
[231 J.K. Watson. Mol. Phys. 19 (1970) 465. [241 B.T. Sutcliffe, Mol. Phys. 48 (1983) 561. I=1 P.G. Burton. E. von Nagy-Felsobuki, G. Doherty and M. Hamilton, to be published. 1261 G. Simons, R. Parr and J.M. Finlan, J. Chem. Phys. 59 (1973) 3229. P71 D. Harris, G. Engerholm and L.D. Gwinn. J. Chem. Phys. 43 (1965) 1515. 1281 G. Herzberg, Infrared and Raman spectra of polyatomic
2s,
.1
(A-3)