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The vibrations and rotations of the pseudogaussian oscillator
In this paper we discuss
the energy
levels and wavefunctions
of a rotatmg
diatomic
molecule
using a three-parameter
model
potential called the pseudogaussian potential which is reasonably behaved for both small and large mtemuclear sepamttons. The solutions to this model problem provide a reasonable description of the rotating diatomic_molecule and also ~111 be useful in dtscussing large amplitude vtbrations in triatomic and larger molecules. A comparison is made with the Morse oscdlator.
1. Introduction The Morse-potential is one of the most widely used model potentials for discussing large amplitude stretching vibrations [1,2]. In this paper we introduce a new model potential called the pseudogaussian or PG potential for these vibrations which, like the Morse, depends on three parameters. In contrast wrth the Morse oscillator, energy levels and \Navefunctions of the PG osctllator can be found for the three-dimensional rotating system using the same methods as for the one-dimensional oscillator. The PG potential is
where j3 = - 2 + (4 + krt/D)‘/* with k, r,, and D the force constant at equilibrium, the equilibrium bond length and the dissociation energy, respectively *_ Fig. 1 illustrates the PG and Morse potentials with values of k, ro, and D appropriate to the electronic ground state of the OH molecule. Included in the diagram are the- RKB [6,7] values of the potential determined from experiment and an effective Morse potential which uses the correct values of equilibrium distance and force constant but picks the “dissociation energy” which yieIds the experimental anharmonicity correction for the observed vibrational levels. For large r and near theequilibrit.+ position the PG and Morse potentials are the same. Everywhere else the PG potential lies above the Morse. Spacings between succeeding energy levels decrease more rapidly and there are more bound states for the Morse potential than for the PG potential. While the-changes introduced by-the PG potential may not be-quantitatively accurate they are in the direction needed to give agreement with experiment: Modifications of the PG potential which are capable of accurately representing the true internuclear potential are described in ._ __I section 4.
In analytic
qxlculations
using the Morse _poiential
we consider
oneTdimensional
motion
tihich’is
equivalent to solving for r times‘the radial function for a thre~&limensiona.l oscillator [8]. The- Morse wavefunction must clearly vanish for r = 0, However, we usually assume that it vanishes at r = L 00 instead
which makes little difference since the wavefunction
is small for negative the large value _ z-values of:r due to__.-
* Another k&D
three-parameter = 2 for his potential
potential
with
which dkgrees
the same
behavior
ti_th observation.
as r -, 0 and as r-+ CD was introduced For-example
0301-0104~84&3.00 @_Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) _ - ._c_ _ __ _-
in the ground
-
_
by Vat-shim
smte of OH k&D
-
__-
_
[3,4]_ However,
= 9.9 [5]. - _
_
_-
--
.
I
M L Sage /
Vzbrar~onsand rorarionsof rhe pseudogaussian oscdlaror
EFFECTIVE linOR SE
0
t
2
3 R
r.i\,
4
hg I. Pseudogaussian, Morse. potentictls for the OH nolecules. ated by x.
effecttre Morse, and RKR The RKR potcntrai is mdt-
of the IMorse potenti& in this non-physical regton. However. this approximation may lead to difficulties if we consxder the bond to be part of a three-dimensional molecule. For the diatomic molecule no problems are encountered provided we restrict our attention to s states. For other values of the angular momentum we must find the radial wavefunction for the Morse oscillator with a centrifugat potential. This can be done numencally or by a perturbatron technique introduced by Pekeris (91 in which the centrifugal part of the potential is expanded about the equilibrium position. The s-type basis functions are not useful for describing the radial wavefunction of states with rotational angular momentum since many matrix elements of the kinetic energy will diverge when evaluated in this basis.’ In many respects the PG potential is easier to use than the Morse potential. Calculations are no more difficuh for a rotating molecule than for the non-rotating molecule. The PG potential eigenfunctions will be obtined by expansion of the Schrijdinger equation in terms of a compIete set of three-dimensional pseudoharmonic (PH) oscillator functions_ These functions, which will be introduced in section 2, are well known and have reasonable behavior at the origin. near the equilibrium bond length, and at co [lO.ll]. Furthermore they can be used to drscuss states near and even above the dissociation Limit. For small amplitude motion they correspond to the rotating and harmonically vibrating diatomic molecuIe. All necessary matrix elements can be calculated analytically using the PH basis set. 2. The pseudoharmonic oscillator basis The spherically symmetric potential
(2)
. _
. _
_
-
-_
_
-
M-L
.._
-
-_
_=-
_-
Sage /
-
_
_<_
__
.
-
_ - _-I
--
.
Vibrations hod rotankzsof the pseudogau&.m ox&to? _. -_ ^_ _ .-_ -_ ;- -- ‘r-
-
-_
:.
_--
-__
_
_
._ .
--
i b__.
_
_
I-
$33
-
-
corresponds-to a vibrating diatomic_ moIectiIe with equihbriumbond~ Iength’-r, and for&constant &:-tie-energy levels and wavefunctions for this. oscihator~can be found-exactly for any_@&iJar~monientum using the~polynomial method [lO,lZJ_-The sohrtions for I gO.a$5 given in the Russian probIem_book by_Gopdmari et al. [ll]. For conve@ience we shall-measure mass in terms of the reduced-mass ~1,force constants in units _ of k, and anguhu momentum in units of &The unit of length is (A2/&k)1’4 and-of energy is ti(k/~)*‘*. Calling the wavefunction with vibrational quantum number u and angular momentum quantum numbers 1 and m
where YI, is a normalized spherical harmonic, we find the radial Schrddinger equation R:[;(r)
+(2/r)&(r)
- [I(/+
l)/rz]
R,,(r)
-*r&/r,
- r,/r)2R”,(r)
+ 2E,,R,,(r)
= 0.
(4)
The normalized eigenfunction is R,,(r)
= rnf-‘1/2 [ “!/(
a, + u)!2”‘] “2L2($r2)
eup( -tr’),
(5)
where otl= $[(21+ 1)’ + r:J*/* and E
rl
=U+~+$([(2f+l)2+r,4]“2-r,z}.
(6)
For large r,,
l,,a~+Qfl(l+
1)/2r:+l/Sr,Z,
(7)
which are the energy levels of the corresponding harmonic oscillator and rigrd rotator except for the small constant l/8$. For small values of r, ~,,av+$I+$,
(8)
which are the energy levels of the isotropic three-dimensional harmonic oscillator with principal quantum number 2u + I and force constant $. This value of the force constant is the value obtained by setting r, = 0 in the PH potential. The functions L:(x) are generalized Laguerre polynomials [13] and satisfy the following orthogondity relation: (9) wavefunctions are similar in form to Morse oscihator functions [8]_ For the PH oscihator we have polynomials times an exponential in the variable r’ while for the Morse oscillator the variable is exp[ -a(r - rO)]. Matrix elements may be evaluated using the same methods previously used for the Morse oscihator 181. In the mathematical appendix we outline the method and give a table of useful integrals. The kinetic energy operator for a polyatomic molecule may include terms involving r- ’ and r-z. We can evaluate the necessary matrix elements using the PH basis set.
These
3. Energy level of the pseudogaussian
oscillator
We shall use the pseudoharmonic basis set to find the eigenvalues and eigenfunctions of the pseudogaussian oscihator. The hamiltonian for the PG oscihator can be written _ HPG=HPH+
V,-
VPH=HpH+Ati,
434
M-L
Sage /
Vibahms
and rotarions of the pseudogaussian oscruOror
se
reducea= = L- w ” m-A-‘- J~m~ntc are &aeonaI in angular momentum quantum numbers. The obvious choice of PH basis set corresponds’to functions with the same equilibrium force constant, k, and bond length, r,, as the PG oscillator. Using this basis set and the units described in sectioh 2 we find
01) and (UlAViU +J)
= ( D i-$/4)6,0 --D(l
+ [r,(Pl(af)~~~,.,i,(a,)]
+j3/2)rs/‘.J$,(aI:
P/r:)
-in{
o&-o’ eBCJ~.;L~(af;
-1,!!i+,(a,)/4-
B/r:)/4
r~1~;~,(aJ)/16),
02)
where the I and J integrats are given in the appendix_ We have suppressed the angular momentum quantum numbers in the wavefunctionsThe parameters j3 and al are defined following eqs. (1) and (5) respectively_ Table 1 presents the energy levels for the non-rotating OH molecule for a number of model potentials including the Morse, effectrve Morse. and PG which are illustrated in fig. 1. Since the PG potential lies above the Morse curve except at r = r0 and 00 we find each bound state for the PG oscillator at higher energy than the corresponding Morse state. The non-rotating Morse oscillator has 20 bound states while the PG oscillator has only 16. Near the dissociation limit the differences are most pronounced with the spacing between the Morse levels becoming much smaller than the spacing of the PG levels. The effective Morse oscillator potential crosses the PG potentia! near the dissociation limit and thus there is a crossover of the energy levels_ Results for the PG potential reported in table 1 were obtamed from a linear variational calculation using a PH basis set with up to fifty basis functions_ For the PH basis described above, convergence was slow for the upper bound levels. With 25 functions the lowest 8 energy levels were determined to 0.1 cm-’ while all states with u z= 10 had errors greater than 100 cm- ‘. In fact only 13 levels appear to be bound. For 32 linear basis functions large errors occurred for u > 12. Even for 50 functions accurate energies are only found for u < Il. The remaining bound states in order are in error by 1.1, 29.6. 26.8, and 1016 cm-‘. The
Table 1 Vibrauonal energy lsvds
of the non-rotating
OH molecule m cm-’
RKR
Morse
Effectwe Morse
PH
PG
0 1 2 3
1845 5418 8822 12062
1845 5395 8757 11931
1848 5420 8526 12067
1876 5611 9349 13087
1852 5448 8898 12199
4 5 6 7 8 9
1.5139 lSO55 20806 23392 ‘~SQS 28051
14916 17714 20324 22746 24980 27025
15142 17551 20794 23371 25783 28029
16825 20562 24300 28038 31776
10
28883
30109
39251
15346 18332 21153 23800 26266 28541 30612
11 12 13
30553 32035 33329
32023 33772 35355
14 15
34434 35351
36772 38023
42989 46727 50465 54202
Quantum number
35513
57940
32466 34081 35431 36474 37152
__ _-_
-.-
I__.-__-
-
.
_-
_’ i : ‘-
-
ML-
Si+~+$~ -.
_
-_
ancrrnr.~~o~~~~~~e~-~-~i~JLL,,:-=.e
-,
_: :
t
_
-_
_
_
-
_
_
-_.
last does not appear to be 6&d, The klo&&k&rg&sc~-fir hi& leCe1 r 1 ee r. ~Acco+i@ly -njany PH- functi$s Bie* nee UPp-~~~c~~f~c~~~_~__~ . --+7&,lCX‘-f.$ -&e-&t&a&& overcome these difficult&s Iby u&zig PH I&&-.__constant or larger valua_of eq&brimu bond length, On; ;&& -&@ although they do not have t&e exact f&w-gikeh in && (11) z&d fact that there are n6.w two “natural” systemi.of units; one_fi, oscillator. Numerical calculations have been performed using constants ranging from k to 0.2kand equilibrium bond lengths rangink from rc to lArd_.: ^ Substantial improvements in calculational efficiency are made by these-shifts. For- example with force constant = 0.6kand i- = 1.2& twenty-five PH functions gave timparable‘resulfs *tOthe original -dalctil~tions with forty PH functions. For u >, 11 these new results were more accurate. Values that yielded the best energy for the highest bound states led to slight errors for other states. For example using forty PH functions we found the most accurate value of the energy of the u = 15 state with force constant = 0.2k and I;S = 1.3r,, while-states with 9 < u f 14 were in error by up to 0.7 cm-’ with these parameters. On the other hand with force>constant = 0.3kand rE4= 1.3t, all states with u < 13 were accurately determined to 0.1 cm-’ while the two uppermost bound states (U = 14, 15) were in error by 0.5 and 35.3 cm-’ respectively. A tied PH basis could also be used, with force constant decreasing and reqincreasing with increasing PH quantum number. These functions would not be orthogonal and might not even form a complete set. An example of where a scale change with quantum number converts a compLete set of Laguerre functions into an incomplete set occurs in the hydrogen atom where the scale depends on the principal quantum number. The bound hydrogenic functions are incomplete. We now shall consider the rotating PG oscillator. Fig. 2 shows the value of the equilibrium bond length and the position of the predissociation barrier maximum as a function of angular momentum quantum number I for the parameters appropriate to OH for all rotational states which have a local minimum in the PG potential including centrifugal terms. Fig. 3 plots the value of the effective potential at equilibrium and
RMIN
I-
01 0
40 50 60 L FigsI. Position of the potential minimum and centrifugal bamer maximum for the PG oscillator as a function of angular momentuni quantum number I for the OH molecule. 10
20
30
Fig. 3. Values of the potential mini&mm and Cen+fugal barrier inaximum for the PG Gciiator as a ftiction of angular quantum number 1 for the OH molecule.
_
436
M. L Sage /
Table 2 Rotational enere I
0 1
2 3 4 5
Vtbrarions and roranons of rhe pseudogaussian oscdlcuor
contributions to the PG oscillator in cm-’
0
0
1
2
3
4
5
0.0 37.3 111.7 223 3 371.9 557 3
0.0 36.1 108.4 216.6 360.7
0.0 350 104.9 209 6 349.1 523.0
00 33 8 101.3 202.4 337.1 505 0
00 32 5 97.5 194.9 324.6
0.0 31.2 936 187.1 311.6 466-S
5404
486.3
at the top of the barrier versus 1. We find an effective potential with a minimum below the dissociation limtt only for I < 50. For values of I between 50 and 62. the potential has no true bound states but does allow metastable rotationally predissociating states. The depth of the well, including the centrifugal barrier, decreases monotonically from the full dissociation energy of 37271 cm-’ for I = 0 to 149 cm-’ for I = 62. The centrifugal barrier height reaches a maximum of 7306 cm-’ at i= 49. Table 2 presents the first six rotational energy levels for each of the first six vibrational states of the PG oscillator. Effects of centrifugal distortion show up clearly in these results. Calculations have also been carried out usmg 30 PH basis functions for high values of 1. We find that the number of true bound states decreases from two at I = 45 and 46 to one at i = 47 and 45. The number of metastable states decreases from four at I = 45 to one for each I between 58 and 62. For I = 62 the metastable state lies 118 cm-’ above the minimum of the effective potential and 31 cm ml below the top of the barrier. Once again we shouId emphasize that calculations of energy levels for bound and metastable states are no more difficult for any values of the anguiar momentum than for the non-rotating PG oscillator. For the metastable states we can readily extend our calculations to evaluate lifetimes.
4. Conclusions
and extensions
We have introduced the pseudogaussian potential to represent the vrbrations of a diatomic molecule. This potential has calculational advantages over the Morse potential if we consider motion in three dimensions. Furthermore we have demonstrated that the pseudoharmonic basis set is useful in studying molecular vrbrations. If we compare experimental or RKR potential for a diatomic molecule with the PG and Morse potentials, as is done in fig. 1 for OH, we find that the Morse potential approaches the dissociation energy too slowly while the PG approaches this energy too rapidly. To use a Morse potential to more accurately describe the lower energy levels we choose constants that fit experiment_ This procedure leads to an effective Morse potential which fits observed levels well but does not dissociate at the proper energy. We can easily modify the PG potential without appreciably complicating calculations by noting that any PH basis set can be used for calculations with any PG potential. We used this fact in section 3 to select a basis which more effectively describes the highest bound vxbrational levels. Accordingly we may carry out calculations using a modified PG potential which is the sum of a number of PG potentials with differing parameters. The modified PG potential can have the proper values of equilibrium bond length, force constant, and dissociation energy. Fig. 4 plots several modified PG potentials each of which is the sum of the two PG potentials. Each of the PG potentials has a minimum at the equilibrium bond length r,,. For fixed equilibrium bond length the shape of a PG potential curve depends only on the ratio of equilibrium force constant to dissociation energy while the magnitude is proportional to the dissociation energy.
-. ML
Sage /
_
.
--_. __
--
Vibratiom.and rofattonsof ihe pseudogaussianoscdiator-_ _ _
_ __ _
_ _: _ . -_ _ _ _437-
4-
3
2 R 6,
4
Fig. 4. Pseudogaussian, RKR, and two modified pseudogausSian potenuals for the OH molecule The RKR potential is indicated by X _ The parameters for MODPGl and MODPGZ are described m the text.
Therefore by combining one broader and one narrower PG curve we can arrive at curves which are close to the RKR or the Morse potential. The potential MODPGl is the sum of one PG potential with force constant and dissociation energy 0.6k and 0.40 and one with 0.4k and 0.60. MODPG2 is the sum of one PG potential with 0.68k and 0.32D and one with 0.32kand 0.680. In this paper our aim was to present a new model potential rather than to accurately describe molecule rotations and vibrations. However as can be seen from fig. 4 greater accbracy can readily be achieved_ In subsequent papers we will use the PG oscillator to consider the coupling of large amplitude stretching vibrations with bending vibrations.
Mathematical appendix In this appendix we shall consider several types of integrals namely
(A-1) and
(A-2) which arise in using the pseudoharmonic (PH) basis set to fmd the eigenfunc@ons and eigenvalues of the pseudogaussian (PG) oscillator_ These integrals can be obtained using the many relationships- for the associated Laguerre polynomials 1131. We previously used this procedure in our discussion of the Morse
hf. L. Sage /
43s
Vibrations and rotations of the pseudogaursian oscillator
oscihator 181. However, we now find it convenient to use a generating function [13] procedure to evaluate the integrals. 5 LF(y)s” U--O
= (1 - ~)-~-r
exp[y.s/(s
-
l)] ,
IsI c 1.
(A-3)
Accordingly we can evaluate
P’(s,
t; a) =
2 fJI,‘~)(a)s”t? n-o
(A-4
m=O
The result is
Fk’(s.
t; a) = (k + a)!(1 -
sy(1 -
t)“/(1
- St)=Ck+l.
(A-5)
From this expression we can readily determine the integrals needed. In table 3 we give all integrals for k = 0, 1, and -1. We only include I,, ,+,(a) forj>, 0.
Similarly we can evaluate Gtk’(s,
f; a;
K)
=
f
n=O
g
J,‘z(a;
K)S~~~
(A-6)
m=O
to find G’“)(s,r;cu;~)=(k+a)!(l
-s)X(l-~)L/[l-s~+~(l-s)(l-t)]atli~l_
(A-7)
An equivalent procedure to find the J,, integrals would be to return to the definition and make a variable change (1 + k)y = z. The integra: becomes J,(_‘A(a; K) =
(1 + K)-~-‘-’
/mLz[r/(l 0
+~)]L~[z/(l+~)]t~~~e-~dz,
Table 3 Integrals mvolving Laguerre polynomials I~~~+,(a)=(a+n)‘/n!
=o z~~~+,(a)=(~+2n+lXa+n)!/n!
J-=0
otherwise J=o
= -(a+n+l)!/rz!
J=l
=o
otherwise
I,$~‘,‘,(a)=(a+n)!/m!
(A-g)
_
_-
Substitution into eq. (A18).-leads to’ -
_
- _ 1
(A-10)
which is equivalent to (A-7) but easier to use. Other integrals which may be needed for use-with polyatomic molecules include generalizations of the two integrals given above in eqs. (A-1) and (A-2) to allow for different parameters (Y for-each generalized Laguerre polynomial and also integrals involving derivatives of the Laguerre polynomials_ The former may be obtained using the methods previously described while the latter may be found using the relationship
[I31 (A-11)
(A-12)
References [I] [2] [S] [4] [5] [6] [7] [S] (91 [lo] [ll]
M.L. Sage and J. .lortner. in: Photoselective chemistry. Part 1, ed. J. Jortner (Whey, New York. 1981) pp_ 293-322. B R. Henry, in: Vtbrattonal spectra and structure, Vol. 10. ed J.R. Durig (Eisevrer, Amsterdam. 1981) pp 269-319. Y P. Varshini. Rev- Mod. Phys. 29 (1957) 664 D. Steele, RR. Lippincott and J-T. Vanderslice, Rev. Mod. Phys. 34 (1962) 239. K.P. Huber and G. Her&erg. Constants of diatomic molecules (Van Nostrand-Reinhold, New York, 1979)
A.L.G. Rees. Proc. Phys. Sot. London 59 (1947) 998. RJ FaIlon. I. Tobias and J-T. Vanderslice, J. Chem. Phys. 34 (1961) 167. M L. Sage, Chem. Phys 35 (1978) 375. CL. Pekeris, Phys. Rev. 45 (1934) 98. M.L. Sage and J. Gcxxiisman. Am. J. Phys , to be published. I I. Gol’dman, V.D Krivchenkov, V.I. Kogan and V.H. Gahtskii, Problems in quantum mechanics, ed. D. ter Haar (Infosearch, London, 1960) p_ 8. [12] G.M. Murphy, Ordinary differential equations and their solutions (Van Nostrand, Princeton, 1960) section 8-2. 119 V.W. Hochstrasser, in: Handbook of mathematical functions, eds. M. Abramowitz and I A. Stegun (Natl. Bur. Std., Washington. 1964) ch 22. [14] A Erdelyi, ed , Higher transeendentaI functions, Vol. II (McGraw-Hill. New York, 1953) ch 10.
the energy
levels and wavefunctions
of a rotatmg
diatomic
molecule
using a three-parameter
model
potential called the pseudogaussian potential which is reasonably behaved for both small and large mtemuclear sepamttons. The solutions to this model problem provide a reasonable description of the rotating diatomic_molecule and also ~111 be useful in dtscussing large amplitude vtbrations in triatomic and larger molecules. A comparison is made with the Morse oscdlator.
1. Introduction The Morse-potential is one of the most widely used model potentials for discussing large amplitude stretching vibrations [1,2]. In this paper we introduce a new model potential called the pseudogaussian or PG potential for these vibrations which, like the Morse, depends on three parameters. In contrast wrth the Morse oscillator, energy levels and \Navefunctions of the PG osctllator can be found for the three-dimensional rotating system using the same methods as for the one-dimensional oscillator. The PG potential is
where j3 = - 2 + (4 + krt/D)‘/* with k, r,, and D the force constant at equilibrium, the equilibrium bond length and the dissociation energy, respectively *_ Fig. 1 illustrates the PG and Morse potentials with values of k, ro, and D appropriate to the electronic ground state of the OH molecule. Included in the diagram are the- RKB [6,7] values of the potential determined from experiment and an effective Morse potential which uses the correct values of equilibrium distance and force constant but picks the “dissociation energy” which yieIds the experimental anharmonicity correction for the observed vibrational levels. For large r and near theequilibrit.+ position the PG and Morse potentials are the same. Everywhere else the PG potential lies above the Morse. Spacings between succeeding energy levels decrease more rapidly and there are more bound states for the Morse potential than for the PG potential. While the-changes introduced by-the PG potential may not be-quantitatively accurate they are in the direction needed to give agreement with experiment: Modifications of the PG potential which are capable of accurately representing the true internuclear potential are described in ._ __I section 4.
In analytic
qxlculations
using the Morse _poiential
we consider
oneTdimensional
motion
tihich’is
equivalent to solving for r times‘the radial function for a thre~&limensiona.l oscillator [8]. The- Morse wavefunction must clearly vanish for r = 0, However, we usually assume that it vanishes at r = L 00 instead
which makes little difference since the wavefunction
is small for negative the large value _ z-values of:r due to__.-
* Another k&D
three-parameter = 2 for his potential
potential
with
which dkgrees
the same
behavior
ti_th observation.
as r -, 0 and as r-+ CD was introduced For-example
0301-0104~84&3.00 @_Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) _ - ._c_ _ __ _-
in the ground
-
_
by Vat-shim
smte of OH k&D
-
__-
_
[3,4]_ However,
= 9.9 [5]. - _
_
_-
--
.
I
M L Sage /
Vzbrar~onsand rorarionsof rhe pseudogaussian oscdlaror
EFFECTIVE linOR SE
0
t
2
3 R
r.i\,
4
hg I. Pseudogaussian, Morse. potentictls for the OH nolecules. ated by x.
effecttre Morse, and RKR The RKR potcntrai is mdt-
of the IMorse potenti& in this non-physical regton. However. this approximation may lead to difficulties if we consxder the bond to be part of a three-dimensional molecule. For the diatomic molecule no problems are encountered provided we restrict our attention to s states. For other values of the angular momentum we must find the radial wavefunction for the Morse oscillator with a centrifugat potential. This can be done numencally or by a perturbatron technique introduced by Pekeris (91 in which the centrifugal part of the potential is expanded about the equilibrium position. The s-type basis functions are not useful for describing the radial wavefunction of states with rotational angular momentum since many matrix elements of the kinetic energy will diverge when evaluated in this basis.’ In many respects the PG potential is easier to use than the Morse potential. Calculations are no more difficuh for a rotating molecule than for the non-rotating molecule. The PG potential eigenfunctions will be obtined by expansion of the Schrijdinger equation in terms of a compIete set of three-dimensional pseudoharmonic (PH) oscillator functions_ These functions, which will be introduced in section 2, are well known and have reasonable behavior at the origin. near the equilibrium bond length, and at co [lO.ll]. Furthermore they can be used to drscuss states near and even above the dissociation Limit. For small amplitude motion they correspond to the rotating and harmonically vibrating diatomic molecuIe. All necessary matrix elements can be calculated analytically using the PH basis set. 2. The pseudoharmonic oscillator basis The spherically symmetric potential
(2)
. _
. _
_
-
-_
_
-
M-L
.._
-
-_
_=-
_-
Sage /
-
_
_<_
__
.
-
_ - _-I
--
.
Vibrations hod rotankzsof the pseudogau&.m ox&to? _. -_ ^_ _ .-_ -_ ;- -- ‘r-
-
-_
:.
_--
-__
_
_
._ .
--
i b__.
_
_
I-
$33
-
-
corresponds-to a vibrating diatomic_ moIectiIe with equihbriumbond~ Iength’-r, and for&constant &:-tie-energy levels and wavefunctions for this. oscihator~can be found-exactly for any_@&iJar~monientum using the~polynomial method [lO,lZJ_-The sohrtions for I gO.a$5 given in the Russian probIem_book by_Gopdmari et al. [ll]. For conve@ience we shall-measure mass in terms of the reduced-mass ~1,force constants in units _ of k, and anguhu momentum in units of &The unit of length is (A2/&k)1’4 and-of energy is ti(k/~)*‘*. Calling the wavefunction with vibrational quantum number u and angular momentum quantum numbers 1 and m
where YI, is a normalized spherical harmonic, we find the radial Schrddinger equation R:[;(r)
+(2/r)&(r)
- [I(/+
l)/rz]
R,,(r)
-*r&/r,
- r,/r)2R”,(r)
+ 2E,,R,,(r)
= 0.
(4)
The normalized eigenfunction is R,,(r)
= rnf-‘1/2 [ “!/(
a, + u)!2”‘] “2L2($r2)
eup( -tr’),
(5)
where otl= $[(21+ 1)’ + r:J*/* and E
rl
=U+~+$([(2f+l)2+r,4]“2-r,z}.
(6)
For large r,,
l,,a~+Qfl(l+
1)/2r:+l/Sr,Z,
(7)
which are the energy levels of the corresponding harmonic oscillator and rigrd rotator except for the small constant l/8$. For small values of r, ~,,av+$I+$,
(8)
which are the energy levels of the isotropic three-dimensional harmonic oscillator with principal quantum number 2u + I and force constant $. This value of the force constant is the value obtained by setting r, = 0 in the PH potential. The functions L:(x) are generalized Laguerre polynomials [13] and satisfy the following orthogondity relation: (9) wavefunctions are similar in form to Morse oscihator functions [8]_ For the PH oscihator we have polynomials times an exponential in the variable r’ while for the Morse oscillator the variable is exp[ -a(r - rO)]. Matrix elements may be evaluated using the same methods previously used for the Morse oscihator 181. In the mathematical appendix we outline the method and give a table of useful integrals. The kinetic energy operator for a polyatomic molecule may include terms involving r- ’ and r-z. We can evaluate the necessary matrix elements using the PH basis set.
These
3. Energy level of the pseudogaussian
oscillator
We shall use the pseudoharmonic basis set to find the eigenvalues and eigenfunctions of the pseudogaussian oscihator. The hamiltonian for the PG oscihator can be written _ HPG=HPH+
V,-
VPH=HpH+Ati,
434
M-L
Sage /
Vibahms
and rotarions of the pseudogaussian oscruOror
se
reducea= = L- w ” m-A-‘- J~m~ntc are &aeonaI in angular momentum quantum numbers. The obvious choice of PH basis set corresponds’to functions with the same equilibrium force constant, k, and bond length, r,, as the PG oscillator. Using this basis set and the units described in sectioh 2 we find
01) and (UlAViU +J)
= ( D i-$/4)6,0 --D(l
+ [r,(Pl(af)~~~,.,i,(a,)]
+j3/2)rs/‘.J$,(aI:
P/r:)
-in{
o&-o’ eBCJ~.;L~(af;
-1,!!i+,(a,)/4-
B/r:)/4
r~1~;~,(aJ)/16),
02)
where the I and J integrats are given in the appendix_ We have suppressed the angular momentum quantum numbers in the wavefunctionsThe parameters j3 and al are defined following eqs. (1) and (5) respectively_ Table 1 presents the energy levels for the non-rotating OH molecule for a number of model potentials including the Morse, effectrve Morse. and PG which are illustrated in fig. 1. Since the PG potential lies above the Morse curve except at r = r0 and 00 we find each bound state for the PG oscillator at higher energy than the corresponding Morse state. The non-rotating Morse oscillator has 20 bound states while the PG oscillator has only 16. Near the dissociation limit the differences are most pronounced with the spacing between the Morse levels becoming much smaller than the spacing of the PG levels. The effective Morse oscillator potential crosses the PG potentia! near the dissociation limit and thus there is a crossover of the energy levels_ Results for the PG potential reported in table 1 were obtamed from a linear variational calculation using a PH basis set with up to fifty basis functions_ For the PH basis described above, convergence was slow for the upper bound levels. With 25 functions the lowest 8 energy levels were determined to 0.1 cm-’ while all states with u z= 10 had errors greater than 100 cm- ‘. In fact only 13 levels appear to be bound. For 32 linear basis functions large errors occurred for u > 12. Even for 50 functions accurate energies are only found for u < Il. The remaining bound states in order are in error by 1.1, 29.6. 26.8, and 1016 cm-‘. The
Table 1 Vibrauonal energy lsvds
of the non-rotating
OH molecule m cm-’
RKR
Morse
Effectwe Morse
PH
PG
0 1 2 3
1845 5418 8822 12062
1845 5395 8757 11931
1848 5420 8526 12067
1876 5611 9349 13087
1852 5448 8898 12199
4 5 6 7 8 9
1.5139 lSO55 20806 23392 ‘~SQS 28051
14916 17714 20324 22746 24980 27025
15142 17551 20794 23371 25783 28029
16825 20562 24300 28038 31776
10
28883
30109
39251
15346 18332 21153 23800 26266 28541 30612
11 12 13
30553 32035 33329
32023 33772 35355
14 15
34434 35351
36772 38023
42989 46727 50465 54202
Quantum number
35513
57940
32466 34081 35431 36474 37152
__ _-_
-.-
I__.-__-
-
.
_-
_’ i : ‘-
-
ML-
Si+~+$~ -.
_
-_
ancrrnr.~~o~~~~~~e~-~-~i~JLL,,:-=.e
-,
_: :
t
_
-_
_
_
-
_
_
-_.
last does not appear to be 6&d, The klo&&k&rg&sc~-fir hi& leCe1 r 1 ee r. ~Acco+i@ly -njany PH- functi$s Bie* nee UPp-~~~c~~f~c~~~_~__~ . --+7&,lCX‘-f.$ -&e-&t&a&& overcome these difficult&s Iby u&zig PH I&&-.__constant or larger valua_of eq&brimu bond length, On; ;&& -&@ although they do not have t&e exact f&w-gikeh in && (11) z&d fact that there are n6.w two “natural” systemi.of units; one_fi, oscillator. Numerical calculations have been performed using constants ranging from k to 0.2kand equilibrium bond lengths rangink from rc to lArd_.: ^ Substantial improvements in calculational efficiency are made by these-shifts. For- example with force constant = 0.6kand i- = 1.2& twenty-five PH functions gave timparable‘resulfs *tOthe original -dalctil~tions with forty PH functions. For u >, 11 these new results were more accurate. Values that yielded the best energy for the highest bound states led to slight errors for other states. For example using forty PH functions we found the most accurate value of the energy of the u = 15 state with force constant = 0.2k and I;S = 1.3r,, while-states with 9 < u f 14 were in error by up to 0.7 cm-’ with these parameters. On the other hand with force>constant = 0.3kand rE4= 1.3t, all states with u < 13 were accurately determined to 0.1 cm-’ while the two uppermost bound states (U = 14, 15) were in error by 0.5 and 35.3 cm-’ respectively. A tied PH basis could also be used, with force constant decreasing and reqincreasing with increasing PH quantum number. These functions would not be orthogonal and might not even form a complete set. An example of where a scale change with quantum number converts a compLete set of Laguerre functions into an incomplete set occurs in the hydrogen atom where the scale depends on the principal quantum number. The bound hydrogenic functions are incomplete. We now shall consider the rotating PG oscillator. Fig. 2 shows the value of the equilibrium bond length and the position of the predissociation barrier maximum as a function of angular momentum quantum number I for the parameters appropriate to OH for all rotational states which have a local minimum in the PG potential including centrifugal terms. Fig. 3 plots the value of the effective potential at equilibrium and
RMIN
I-
01 0
40 50 60 L FigsI. Position of the potential minimum and centrifugal bamer maximum for the PG oscillator as a function of angular momentuni quantum number I for the OH molecule. 10
20
30
Fig. 3. Values of the potential mini&mm and Cen+fugal barrier inaximum for the PG Gciiator as a ftiction of angular quantum number 1 for the OH molecule.
_
436
M. L Sage /
Table 2 Rotational enere I
0 1
2 3 4 5
Vtbrarions and roranons of rhe pseudogaussian oscdlcuor
contributions to the PG oscillator in cm-’
0
0
1
2
3
4
5
0.0 37.3 111.7 223 3 371.9 557 3
0.0 36.1 108.4 216.6 360.7
0.0 350 104.9 209 6 349.1 523.0
00 33 8 101.3 202.4 337.1 505 0
00 32 5 97.5 194.9 324.6
0.0 31.2 936 187.1 311.6 466-S
5404
486.3
at the top of the barrier versus 1. We find an effective potential with a minimum below the dissociation limtt only for I < 50. For values of I between 50 and 62. the potential has no true bound states but does allow metastable rotationally predissociating states. The depth of the well, including the centrifugal barrier, decreases monotonically from the full dissociation energy of 37271 cm-’ for I = 0 to 149 cm-’ for I = 62. The centrifugal barrier height reaches a maximum of 7306 cm-’ at i= 49. Table 2 presents the first six rotational energy levels for each of the first six vibrational states of the PG oscillator. Effects of centrifugal distortion show up clearly in these results. Calculations have also been carried out usmg 30 PH basis functions for high values of 1. We find that the number of true bound states decreases from two at I = 45 and 46 to one at i = 47 and 45. The number of metastable states decreases from four at I = 45 to one for each I between 58 and 62. For I = 62 the metastable state lies 118 cm-’ above the minimum of the effective potential and 31 cm ml below the top of the barrier. Once again we shouId emphasize that calculations of energy levels for bound and metastable states are no more difficult for any values of the anguiar momentum than for the non-rotating PG oscillator. For the metastable states we can readily extend our calculations to evaluate lifetimes.
4. Conclusions
and extensions
We have introduced the pseudogaussian potential to represent the vrbrations of a diatomic molecule. This potential has calculational advantages over the Morse potential if we consider motion in three dimensions. Furthermore we have demonstrated that the pseudoharmonic basis set is useful in studying molecular vrbrations. If we compare experimental or RKR potential for a diatomic molecule with the PG and Morse potentials, as is done in fig. 1 for OH, we find that the Morse potential approaches the dissociation energy too slowly while the PG approaches this energy too rapidly. To use a Morse potential to more accurately describe the lower energy levels we choose constants that fit experiment_ This procedure leads to an effective Morse potential which fits observed levels well but does not dissociate at the proper energy. We can easily modify the PG potential without appreciably complicating calculations by noting that any PH basis set can be used for calculations with any PG potential. We used this fact in section 3 to select a basis which more effectively describes the highest bound vxbrational levels. Accordingly we may carry out calculations using a modified PG potential which is the sum of a number of PG potentials with differing parameters. The modified PG potential can have the proper values of equilibrium bond length, force constant, and dissociation energy. Fig. 4 plots several modified PG potentials each of which is the sum of the two PG potentials. Each of the PG potentials has a minimum at the equilibrium bond length r,,. For fixed equilibrium bond length the shape of a PG potential curve depends only on the ratio of equilibrium force constant to dissociation energy while the magnitude is proportional to the dissociation energy.
-. ML
Sage /
_
.
--_. __
--
Vibratiom.and rofattonsof ihe pseudogaussianoscdiator-_ _ _
_ __ _
_ _: _ . -_ _ _ _437-
4-
3
2 R 6,
4
Fig. 4. Pseudogaussian, RKR, and two modified pseudogausSian potenuals for the OH molecule The RKR potential is indicated by X _ The parameters for MODPGl and MODPGZ are described m the text.
Therefore by combining one broader and one narrower PG curve we can arrive at curves which are close to the RKR or the Morse potential. The potential MODPGl is the sum of one PG potential with force constant and dissociation energy 0.6k and 0.40 and one with 0.4k and 0.60. MODPG2 is the sum of one PG potential with 0.68k and 0.32D and one with 0.32kand 0.680. In this paper our aim was to present a new model potential rather than to accurately describe molecule rotations and vibrations. However as can be seen from fig. 4 greater accbracy can readily be achieved_ In subsequent papers we will use the PG oscillator to consider the coupling of large amplitude stretching vibrations with bending vibrations.
Mathematical appendix In this appendix we shall consider several types of integrals namely
(A-1) and
(A-2) which arise in using the pseudoharmonic (PH) basis set to fmd the eigenfunc@ons and eigenvalues of the pseudogaussian (PG) oscillator_ These integrals can be obtained using the many relationships- for the associated Laguerre polynomials 1131. We previously used this procedure in our discussion of the Morse
hf. L. Sage /
43s
Vibrations and rotations of the pseudogaursian oscillator
oscihator 181. However, we now find it convenient to use a generating function [13] procedure to evaluate the integrals. 5 LF(y)s” U--O
= (1 - ~)-~-r
exp[y.s/(s
-
l)] ,
IsI c 1.
(A-3)
Accordingly we can evaluate
P’(s,
t; a) =
2 fJI,‘~)(a)s”t? n-o
(A-4
m=O
The result is
Fk’(s.
t; a) = (k + a)!(1 -
sy(1 -
t)“/(1
- St)=Ck+l.
(A-5)
From this expression we can readily determine the integrals needed. In table 3 we give all integrals for k = 0, 1, and -1. We only include I,, ,+,(a) forj>, 0.
Similarly we can evaluate Gtk’(s,
f; a;
K)
=
f
n=O
g
J,‘z(a;
K)S~~~
(A-6)
m=O
to find G’“)(s,r;cu;~)=(k+a)!(l
-s)X(l-~)L/[l-s~+~(l-s)(l-t)]atli~l_
(A-7)
An equivalent procedure to find the J,, integrals would be to return to the definition and make a variable change (1 + k)y = z. The integra: becomes J,(_‘A(a; K) =
(1 + K)-~-‘-’
/mLz[r/(l 0
+~)]L~[z/(l+~)]t~~~e-~dz,
Table 3 Integrals mvolving Laguerre polynomials I~~~+,(a)=(a+n)‘/n!
=o z~~~+,(a)=(~+2n+lXa+n)!/n!
J-=0
otherwise J=o
= -(a+n+l)!/rz!
J=l
=o
otherwise
I,$~‘,‘,(a)=(a+n)!/m!
(A-g)
_
_-
Substitution into eq. (A18).-leads to’ -
_
- _ 1
(A-10)
which is equivalent to (A-7) but easier to use. Other integrals which may be needed for use-with polyatomic molecules include generalizations of the two integrals given above in eqs. (A-1) and (A-2) to allow for different parameters (Y for-each generalized Laguerre polynomial and also integrals involving derivatives of the Laguerre polynomials_ The former may be obtained using the methods previously described while the latter may be found using the relationship
[I31 (A-11)
(A-12)
References [I] [2] [S] [4] [5] [6] [7] [S] (91 [lo] [ll]
M.L. Sage and J. .lortner. in: Photoselective chemistry. Part 1, ed. J. Jortner (Whey, New York. 1981) pp_ 293-322. B R. Henry, in: Vtbrattonal spectra and structure, Vol. 10. ed J.R. Durig (Eisevrer, Amsterdam. 1981) pp 269-319. Y P. Varshini. Rev- Mod. Phys. 29 (1957) 664 D. Steele, RR. Lippincott and J-T. Vanderslice, Rev. Mod. Phys. 34 (1962) 239. K.P. Huber and G. Her&erg. Constants of diatomic molecules (Van Nostrand-Reinhold, New York, 1979)
A.L.G. Rees. Proc. Phys. Sot. London 59 (1947) 998. RJ FaIlon. I. Tobias and J-T. Vanderslice, J. Chem. Phys. 34 (1961) 167. M L. Sage, Chem. Phys 35 (1978) 375. CL. Pekeris, Phys. Rev. 45 (1934) 98. M.L. Sage and J. Gcxxiisman. Am. J. Phys , to be published. I I. Gol’dman, V.D Krivchenkov, V.I. Kogan and V.H. Gahtskii, Problems in quantum mechanics, ed. D. ter Haar (Infosearch, London, 1960) p_ 8. [12] G.M. Murphy, Ordinary differential equations and their solutions (Van Nostrand, Princeton, 1960) section 8-2. 119 V.W. Hochstrasser, in: Handbook of mathematical functions, eds. M. Abramowitz and I A. Stegun (Natl. Bur. Std., Washington. 1964) ch 22. [14] A Erdelyi, ed , Higher transeendentaI functions, Vol. II (McGraw-Hill. New York, 1953) ch 10.