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Harmonic oscillator representation for anharmonic vibrational wavefunctions
Volume 16, number I
CHEMICAL PHYSICS LETTERS
HARMONIC FOR
OSCILLATOR
ANHARMONIC
1.5 September
1972
REPRESENTATION
VIBRATIQNAL
WAVEFUNCTIONS”
G .W . F . DRAKE Deparftnetlt
ofPllsvsics.
iJrlil,ersiry of‘ IVirrdsor, It’indsar I I, Onrario, Received
12 June
Canada
1972
It is shown that a harmonic oscillator basis set provides a compact analytic representation for the complete set of bound and continuum vibrational states of nn unhnrmonic oscillator in the inner region. Calculations oivibration01 excitation cross sections based on n harmonic oscik~tor model can thus bc improved by choosing linear combinations of harmonic eircnfunctions such that the vibrational hnmiltonian is diagonal.
1. Introduction Secrest and Johnson [ 11 have suggested a method for the quantum mechanical calculation of the probabilities for the vibrational excitation of a diatomic mokcule in a head-on collision with an atom. In their model, the diatomic molecule is approximated by a harmonic oscillator. The model is therefore limited to the excitation of the first few vibrational quantum states and cannot describe at all the collisional dissociation of the molecule or the effects of anharmonicity. In this letter, we suggest a simple modification of Secrest and Johnson’s calculation which allows for anharmonicity and permits dissociation at sufficientiy high impact energies.
2. Theory
used to expand an arbitrary vibrational eigenfunction of a real diatomic molecule. The Schr’ddinger equation for the radial motion of the nuclei in the Born-Oppenheimer approximation is
with H= dzJdR2 + J(J + 1)/R2 - (2~/@)
V(R),
p is the reduced mass and J the rotational quantum number. For the bound states, a variationa! approximation to the kth eigenfunciion Ok(R) is obtained by expanding in terms of a truncated set of harmonic osciilator eigenfunctions G,,(R) as follows: ?7 QX-(R) = C
Q,,+,,(R),
(3)
r1=!
The hamlonic oscillator eigenfunctions form a complete discrete set of functions and can therefore be * Research Canada.
supported
by the National
Research Council of
where N is the number of temls in the basis set and the ~k,~ are linear variational coefficients de terrnined by diagonalizing the Itamiltonian matrk H,,,,, = ($,,lI HI&J in the finite basis set. The GrI(R) are given
~Tolumc 16,
nomber 1
15 September 1972
CHEMICAL PHYSICS LETTERS
Rx- = (C-’ RC)kg,
by (4)
(5)
where C is an orthogonal transfom~ation, and the diagonal matrix elements P’&,.(R~> are evaluated by interpolation. The potential energy matrix V can then be
5=
c#(R - Ro), cr = (~}1/2~~and H, is a Hermite polynorniai. The constants R. and k are the equilibrium separation and the force constant of the oscillator,
transformed back to the($,,) transformation
but they can be treated as nor.-linear variational parameters. Zetik and Matsen [?I have shown that the above technique yields bound state eigenvalues as good as or better than those obtained by direct nunterioal integm tion of eq. (1) with basis sets of manageable size. In additions the eigenfunct~ons ~utonlatica~ly form an
V’=CVC-1.
orthonormatl set and are in a convenient analytic form for further calculations. It is interesting to investigate also the variationally dc ternlined eigenfunctions lying above the dissociation limit. Although there is no longer a variationat minimum principle for continuum eigenvalues, we show below that the low-lying continuum eigenfunctions arc in close agreement with the exact continuum eigenfunctions evaluated at the same energy, provided that R is not too large. Thus transition integrals from
bound to continuum states are accurately determined since only small values of R contribute significantly to tfle integral. Since the number of bound states is finite (and often small), increasing the size of the basis set increases the density and accuracy of the continu-
um eigenfunctions. Hazi and Taylcr f3] have obtained results similar to the above for the resonant and nonresonant scattering states in a :;impIe model potential with a barrier and containing no bound states. We show that their conclusions can be extended to include the iow-Iying continuum eigcnfunctions in realistic molecular potentials containing several bound states and no barrier. The bound pIus continuum variational cigenfunctions foml an 0rthonomnLl set of functions which tend to the exact eigenfunctions as the size of the basis set is increased. It cften happens that the potential energy function V(R) in (3) is known only in numerical form. It is then convenient to work in a representation in which R is diagonal in order to avoid a large number of numerical integrations [4]. In this procedure, the matrix &I, = i$r_,rlR l&1,>is diagonahzed to yield eigenvalues
36
basis set by the inverse
(6)
The diagonalization
of R need not be repeated for different potentials, provided that the same basis set is used in each case. The eigenvalues R, provide a direct indication of the range ofR-values adequately represent. ed by the basis set. The above procedure was used in the trial calculation described in the following section. 3. Results
and discussion
The vibrational eigenfunctions of the I-$ molecule were calculated using the potential energy function of Kofos and Wofniewicz ES] . The molecule possesses 15 bound vibrational states. The calculation yielded 14 bound states with a 40-term harmonic oscillator basis set, and the full 15 states with a 5@-term basis set. In the latter case, all the eigenvslues but the highest were in good agreement with the values obtained by numeric-
Comparison
of dissociation
Table 1 energies
(in cm-“)
from
the har-
monic oscilfator basis set with the n~rneric~ integration results of Kotos and Wolnicwicz [S]
1’
0 1 2 3 4 5 6 7 8 9 10 11 12
13 14
Harmonic oscillator
Numericat integration
36111.89 31948.09 28020.22
36118.09 31956.03 28039.39
24523.00 20851.89 17609.90 14595.90 11813.90 9270.74 6978.35 4952.62
24333.25 20864.57 17622.33 14607.84 11825.02 9281.13 6987.48 4960.67 3223.54
32 16.97 1803.03
755.98 113.86
18OS.00 759.02 138.86
Volume
I-
16, number
1
CHEMICAL PHYSICS LETTERS
15 September
1972
n
Fig. 1. Continuum wavel‘unction at E = 0.18 eV above the dissociation limit. The dashed line is obtained with a 50-term hsrmonic oscillator basis set and the full line is obtained by numerical integration. Also shown is the wavcfunction for I’= 0. al integration [S] , except for a nearly constant shift of about S cm-l. The non-linear parameters R, = 3.3789 a,-, and ,112 = 3.125 a; 1 were chosen to give a good fit over the entire range of bound states with particular emphasis on the highly excited ones. The eigenvalues are listed in table 1. Of particular interest is the accuracy of the continuum eigenfunctions. The functions obtained from the finite basis sets are compared in figs. 1 and 2 with the exact eigenfunctions obtained by direct numerical integration of (1) or the same energy. Fig. 1 shows the lowest 50-term continuum eigenfunction at E = 0.18 eV above the dissociation limit. Agreement with the numerical solution is quite satisfactory out to!7 = 4 Qo, aftei which the anaiytic solution decays exponentially as expected. The agreement in the inner region becomes progressively worse with increasing energ/ above the dissociation limit, but remains acceptable up to about 5.5 eV as shown in fig. 2b. The phase of the analytic solution is progressively shifted relative to the numerical solution and the analytic solution looses amplitude in the range of R-vaiues where the bound states are most prominent (R 2 1.8 ao). However the agreement at higher energies tiproves as the size of the basis set is increased, as shown by a comparison of figs. 2a and 2b. A number of formal identities can be used to check the completeness of the basis set in the inner region. For example, the exact eigenfunctions satisfy
(7) where the sum over IZ includes continuum. As shown in table very well satisfied by the finite higher vibrational siates where
an integration over the 2, the above identity is basis set, even for the the potential is strong-
II
1
1
,
6.0 RF&. 2. Conrinuum wavefunction 5.5 CV above the dissociation limit. The full line is obtained by numerical integration. The 1.0
210
310
4.0
5.0
dashed line in (a) is obtained with B SO-term harmonic oscillator basis set and the dashed line in (b) is obtained with a 60term basis set.
ly anharmonic. In summary, the harmonic oscillator basis set Frovides a compact analytic representation for the complete set of bound and continuum states of an anharmanic oscillator in the inner region. Calculations of the cross sections for vibrational excitation are in progress.
Espcctntion -------
---_-._._
Tab!e 2 values of?
v
(1;lR2 iv)
0
2.1257155 2.473 1153
1 8 9 10 11 12 13 14
(in ui) 5
If *= % 2.1257155 2.3731153
6.4525954
6.4515954
7.4845094 8.7986412 10.576 1035 13.1974590 17.70407OQ 25.6255580
7.4845091 8.7986396 10.5760995 13.1974520 17.7040663 25.6253917
References (11 D. Secrest and B.R. Johnson, [21 I31 [41 ISI
J. Chem. Phys. 45 (1966) 4556. D.F. Zetik and F.A. XIntsen, J. Mol. Spectry. 24 (1967) 122. A.U. Hazi and H.S. Taylor, Phys. Rev. Al (1970) 1109. D-0. Harris, G.G. Engerhohn and W.D. Gwinn, J. Chem. Phys. 43 (1965) 1515. W. Kotos and L. Wolniewicz, J. Chem. Phys. 43 (1965) 2429; 49 (1968) 404.
37
CHEMICAL PHYSICS LETTERS
HARMONIC FOR
OSCILLATOR
ANHARMONIC
1.5 September
1972
REPRESENTATION
VIBRATIQNAL
WAVEFUNCTIONS”
G .W . F . DRAKE Deparftnetlt
ofPllsvsics.
iJrlil,ersiry of‘ IVirrdsor, It’indsar I I, Onrario, Received
12 June
Canada
1972
It is shown that a harmonic oscillator basis set provides a compact analytic representation for the complete set of bound and continuum vibrational states of nn unhnrmonic oscillator in the inner region. Calculations oivibration01 excitation cross sections based on n harmonic oscik~tor model can thus bc improved by choosing linear combinations of harmonic eircnfunctions such that the vibrational hnmiltonian is diagonal.
1. Introduction Secrest and Johnson [ 11 have suggested a method for the quantum mechanical calculation of the probabilities for the vibrational excitation of a diatomic mokcule in a head-on collision with an atom. In their model, the diatomic molecule is approximated by a harmonic oscillator. The model is therefore limited to the excitation of the first few vibrational quantum states and cannot describe at all the collisional dissociation of the molecule or the effects of anharmonicity. In this letter, we suggest a simple modification of Secrest and Johnson’s calculation which allows for anharmonicity and permits dissociation at sufficientiy high impact energies.
2. Theory
used to expand an arbitrary vibrational eigenfunction of a real diatomic molecule. The Schr’ddinger equation for the radial motion of the nuclei in the Born-Oppenheimer approximation is
with H= dzJdR2 + J(J + 1)/R2 - (2~/@)
V(R),
p is the reduced mass and J the rotational quantum number. For the bound states, a variationa! approximation to the kth eigenfunciion Ok(R) is obtained by expanding in terms of a truncated set of harmonic osciilator eigenfunctions G,,(R) as follows: ?7 QX-(R) = C
Q,,+,,(R),
(3)
r1=!
The hamlonic oscillator eigenfunctions form a complete discrete set of functions and can therefore be * Research Canada.
supported
by the National
Research Council of
where N is the number of temls in the basis set and the ~k,~ are linear variational coefficients de terrnined by diagonalizing the Itamiltonian matrk H,,,,, = ($,,lI HI&J in the finite basis set. The GrI(R) are given
~Tolumc 16,
nomber 1
15 September 1972
CHEMICAL PHYSICS LETTERS
Rx- = (C-’ RC)kg,
by (4)
(5)
where C is an orthogonal transfom~ation, and the diagonal matrix elements P’&,.(R~> are evaluated by interpolation. The potential energy matrix V can then be
5=
c#(R - Ro), cr = (~}1/2~~and H, is a Hermite polynorniai. The constants R. and k are the equilibrium separation and the force constant of the oscillator,
transformed back to the($,,) transformation
but they can be treated as nor.-linear variational parameters. Zetik and Matsen [?I have shown that the above technique yields bound state eigenvalues as good as or better than those obtained by direct nunterioal integm tion of eq. (1) with basis sets of manageable size. In additions the eigenfunct~ons ~utonlatica~ly form an
V’=CVC-1.
orthonormatl set and are in a convenient analytic form for further calculations. It is interesting to investigate also the variationally dc ternlined eigenfunctions lying above the dissociation limit. Although there is no longer a variationat minimum principle for continuum eigenvalues, we show below that the low-lying continuum eigenfunctions arc in close agreement with the exact continuum eigenfunctions evaluated at the same energy, provided that R is not too large. Thus transition integrals from
bound to continuum states are accurately determined since only small values of R contribute significantly to tfle integral. Since the number of bound states is finite (and often small), increasing the size of the basis set increases the density and accuracy of the continu-
um eigenfunctions. Hazi and Taylcr f3] have obtained results similar to the above for the resonant and nonresonant scattering states in a :;impIe model potential with a barrier and containing no bound states. We show that their conclusions can be extended to include the iow-Iying continuum eigcnfunctions in realistic molecular potentials containing several bound states and no barrier. The bound pIus continuum variational cigenfunctions foml an 0rthonomnLl set of functions which tend to the exact eigenfunctions as the size of the basis set is increased. It cften happens that the potential energy function V(R) in (3) is known only in numerical form. It is then convenient to work in a representation in which R is diagonal in order to avoid a large number of numerical integrations [4]. In this procedure, the matrix &I, = i$r_,rlR l&1,>is diagonahzed to yield eigenvalues
36
basis set by the inverse
(6)
The diagonalization
of R need not be repeated for different potentials, provided that the same basis set is used in each case. The eigenvalues R, provide a direct indication of the range ofR-values adequately represent. ed by the basis set. The above procedure was used in the trial calculation described in the following section. 3. Results
and discussion
The vibrational eigenfunctions of the I-$ molecule were calculated using the potential energy function of Kofos and Wofniewicz ES] . The molecule possesses 15 bound vibrational states. The calculation yielded 14 bound states with a 40-term harmonic oscillator basis set, and the full 15 states with a 5@-term basis set. In the latter case, all the eigenvslues but the highest were in good agreement with the values obtained by numeric-
Comparison
of dissociation
Table 1 energies
(in cm-“)
from
the har-
monic oscilfator basis set with the n~rneric~ integration results of Kotos and Wolnicwicz [S]
1’
0 1 2 3 4 5 6 7 8 9 10 11 12
13 14
Harmonic oscillator
Numericat integration
36111.89 31948.09 28020.22
36118.09 31956.03 28039.39
24523.00 20851.89 17609.90 14595.90 11813.90 9270.74 6978.35 4952.62
24333.25 20864.57 17622.33 14607.84 11825.02 9281.13 6987.48 4960.67 3223.54
32 16.97 1803.03
755.98 113.86
18OS.00 759.02 138.86
Volume
I-
16, number
1
CHEMICAL PHYSICS LETTERS
15 September
1972
n
Fig. 1. Continuum wavel‘unction at E = 0.18 eV above the dissociation limit. The dashed line is obtained with a 50-term hsrmonic oscillator basis set and the full line is obtained by numerical integration. Also shown is the wavcfunction for I’= 0. al integration [S] , except for a nearly constant shift of about S cm-l. The non-linear parameters R, = 3.3789 a,-, and ,112 = 3.125 a; 1 were chosen to give a good fit over the entire range of bound states with particular emphasis on the highly excited ones. The eigenvalues are listed in table 1. Of particular interest is the accuracy of the continuum eigenfunctions. The functions obtained from the finite basis sets are compared in figs. 1 and 2 with the exact eigenfunctions obtained by direct numerical integration of (1) or the same energy. Fig. 1 shows the lowest 50-term continuum eigenfunction at E = 0.18 eV above the dissociation limit. Agreement with the numerical solution is quite satisfactory out to!7 = 4 Qo, aftei which the anaiytic solution decays exponentially as expected. The agreement in the inner region becomes progressively worse with increasing energ/ above the dissociation limit, but remains acceptable up to about 5.5 eV as shown in fig. 2b. The phase of the analytic solution is progressively shifted relative to the numerical solution and the analytic solution looses amplitude in the range of R-vaiues where the bound states are most prominent (R 2 1.8 ao). However the agreement at higher energies tiproves as the size of the basis set is increased, as shown by a comparison of figs. 2a and 2b. A number of formal identities can be used to check the completeness of the basis set in the inner region. For example, the exact eigenfunctions satisfy
(7) where the sum over IZ includes continuum. As shown in table very well satisfied by the finite higher vibrational siates where
an integration over the 2, the above identity is basis set, even for the the potential is strong-
II
1
1
,
6.0 RF&. 2. Conrinuum wavefunction 5.5 CV above the dissociation limit. The full line is obtained by numerical integration. The 1.0
210
310
4.0
5.0
dashed line in (a) is obtained with B SO-term harmonic oscillator basis set and the dashed line in (b) is obtained with a 60term basis set.
ly anharmonic. In summary, the harmonic oscillator basis set Frovides a compact analytic representation for the complete set of bound and continuum states of an anharmanic oscillator in the inner region. Calculations of the cross sections for vibrational excitation are in progress.
Espcctntion -------
---_-._._
Tab!e 2 values of?
v
(1;lR2 iv)
0
2.1257155 2.473 1153
1 8 9 10 11 12 13 14
(in ui) 5
I
6.4525954
6.4515954
7.4845094 8.7986412 10.576 1035 13.1974590 17.70407OQ 25.6255580
7.4845091 8.7986396 10.5760995 13.1974520 17.7040663 25.6253917
References (11 D. Secrest and B.R. Johnson, [21 I31 [41 ISI
J. Chem. Phys. 45 (1966) 4556. D.F. Zetik and F.A. XIntsen, J. Mol. Spectry. 24 (1967) 122. A.U. Hazi and H.S. Taylor, Phys. Rev. Al (1970) 1109. D-0. Harris, G.G. Engerhohn and W.D. Gwinn, J. Chem. Phys. 43 (1965) 1515. W. Kotos and L. Wolniewicz, J. Chem. Phys. 43 (1965) 2429; 49 (1968) 404.
37