J. Phys. Chem. 1991, 95, 1058-1062
1058
A Computational Study of Kratzer Oscillator Basis Sets Don Secrest School of Chemical Sciences, University of Illinois, Urbana, Illinois 61801 (Received: June 21, 1990)
A formula is given for computing integrals of exponentials and powers of r between eigenfunctions of two different Kratzer oscillators. Such integrals would be of use in using mixed basis sets of Kratzer oscillator functions. A practical, numerically stable recursion relation is given for computing these integrals for both mixed and pure basis sets. The use of Kratzer functions is illustrated in the computation of the eigenstates of a rotating-vibrating Morse oscillator. The eigenstates are also computed by the use of a harmonic oscillator basis set. The Kratzer oscillator functions have the advantage over the harmonic oscillator basis in that all integrals can be performed analytically. It is found that the Kratzer basis set size is only 2/3 as large as that of the harmonic basis set required to give the same accuracy.
The Kratzer oscillator1 functions are well-suited to SimonsParr-Finlan (SPF)2*3type potentials and converge rapidly for potential of this sorts4 Since the S P F functions are similar in form to the Kratzer oscillator potential, this is perhaps not surprising. It is of some interest to explore their use for a potential with somewhat different properties than the Kratzer oscillator potential. The Morse potentialS makes an ideal test case for the use of these functions as a basis set. The Morse potential is much shorter range than the Kratzer potential and always rises more rapidly. We shall solve the equations for a diatomic rotating Morse oscillator by the variation method using the Kratzer oscillator functions as a basis set. For comparison, we shall also use a harmonic oscillator basis set. The advantage of the Kratzer oscillator functions is that all integrals may be computed analytically, while the harmonic oscillator integrals must be computed numerically. It is conceivable that a method could be found to compute the harmonic oscillator integrals in closed form for the J = 0 states, but they must clearly be approximated for finite J . The Schriidinger equation for a rotating diatomic Morse oscillator may be written in the reduced form as
where (2)
r=re+x
For the nonrotating molecule, J = 0, the eigenvalue is given approximately by
E,,,, = ( n
+
i) - -+(n + i)
For a reasonably manageable number of bound states we will choose 0 = 5, giving 25 bound states. The H2 molecule has 13 bound states, and oxygen has around 45 in the ground electronic state. If we choose a value of 25 for re, and use the vibrational frequency of N,, we obtain a value for the rotation constant, Be, of 1.89 cm-'. The Be for N2 is 2.01 cm-I. We might then consider this to be a model for N2,though the dissociation energy is a bit low for N2. The Kratzer oscillator basis functions are solutions to the equation
and the harmonic oscillator basis set is solutions to the equation (7) Here B is the location of the minimum with respect to the minimum of the Morse potential in eq 1. The K determines the depth of the Kratzer oscillator well, and the form of the coefficient was chosen for convenient comparison with 0. The k in the harmonic oscillator determines the width of the harmonic well. A value of k = 1 causes the harmonic and Morse potentials to match at the minimum. That is to say, the second derivative of both potentials is I . We may write the Krazter oscillator Hamiltonian in reduced form as7
2
(3) The eigenfunction is then
+, = N,,xXe-*alF , (-n, 2X;2P,,x)
for the bound states. For this potential, there are int (P2) bound states. If the boundary condition +(x) = 0, x--m
(4)
where h(X - I )
is used, eq 3 is exact, but for the more proper boundary condition at r = 0 or x = -re $(x) = 0 (5) eq 3 is only approximately correct though the error is small for realistic parameted and is completely negligible in the present case. ~~
~
Pn
=
~ + n
and
~~~
(1) Kratzer. A . Z.Phys. 1920. 3, 289. (2) Rquena, A.; Zbfiiga, J.; Fuentes, L. M.; Hidalgo, A. J . Chem. Phys. 1986, 85, 3939. (3) Simons, G.; Parr, R. G.;Finlan, J . M. J . Chem. Phys. 1973.59. 3229. (4) Chang, B. H. Ph.D. Thesis, University of Illinois, Urbana, Illinois, 1990. Chang, B. H.; Secrest, D. J . Chem. Phys., in press. ( 5 ) Morse, P. Phys. Reu. 1929, 34, 57. (6) Secrcst, D.; Cashion, K.; Hirschfelder. J . 0.J . Chem. Phys. 1962, 37, 830.
0022-3654/9l/2095-l058$02.50/0
The , F , is the confluent hypergeometric function and (a), is a Pochhammer's symbol defined as ( a ) , = o ( a + l ) ( a 2) ... (a + n - 1) = r ( a + n)/I'(a). The eigenvalue corresponding to this wave function is
+
E, = -Pn2
(12)
I t may at times prove useful to use a mixed basis set. By a mixed
0 1991 American Chemical Society
Kratzer Oscillator Basis Sets
The Journal of Physical Chemistry, Vol. 95, No. 3, 1991
basis we mean some elements of the basis set correspond to one value of K and E and other elements to a different value of these parameters. In the reduced Hamiltonian, eq 8, this would give rise to different values of A. The integrals are given analytically by ~m+*n(X’,x)xPe-”+m(X,~) dx = P’nPm (
2 x 9n
m!n!(2A),(m + X)(n + A’)
(2~m)’(2~’n)”’
]’Iz (a
+ p’, +
X
P,)h‘+A+p+I
1059
Kratzer 1 8
-t \ W
I
Pi
-12
12
/
11 0
10
*
30
where
where
- 1)
X’(X’
h(X - 1)
and
+
@’n=
Pm
=
I Basis
Size
50
Figure 1. Base IO logarithm of the error in the eigenvalues of a diatomic Morse oscillator plotted against basis set size. The Morse oscillator well depth is 12.5 reduced units, and the minimum of the well is located 25 units from the origin. The eigenvalues are E l o = 8.295, E,, = 8.855, and EI2= 9.375. The ordinate is log (E,,calc The basis set consists of a Kratzer oscillator with K = 18 and B = 0 (eq 6). For this figure J = 0.
h+m
For X’ = A, eq 13 reduces to the known result.’ For practical values of A, the generalized hypergeometric function, eq 14, has two problems. First, since h is typically large, the F2 tends to underflow. Second, the alternating sign in the sum leads to a rapid loss of significant figures. Even for rather modest values of m and n, F2 computed with 64 bits of precision can be left with no significant figures. These problems may be overcome by computing (2X),,Fz where m > n. Since the integral in eq 13 is symmetric in n and m, we may without loss of generality take m > n. The problem of loss of significance may be overcome by evaluating the sum over j as the inner sum by a stable recursion relation to give
(
I
(2X),F2 X+X’+p+ I ;-n,-m;2hr,2X; 2P’n a+
i
Pm
2Pm
+ P’n’a + P m + P’n
+ +
(-n)i(h + X’ p (2X’)j!
)
2pn ( a
+ Dm +
40
= i
p n ) Tm(i,m) (15)
where
+
Tk+,(i,m) = [2k 2X - (A k(x - 1)(2X - 1
+ A’ + 1 + i + k)x]Tk(i,m) + + k)Tk-,(i,m), To(i,m) = 1 (16)
and x =
I
11
60 Figure 2. Logarithm of the error in the eigenvalues of the Morse osBasis
Size
cillator against the number of harmonic oscillator basis functions. The harmonic functions are for a force constant k = 0.48 and a minimum located I unit beyond the minimum of the Morse oscillator. The minimum of the Morse oscillator is at r = 25, and the harmonic oscillator minimum is at r = 26 units. J = 0. See the caption of Figure I .
which may be used for problems for which the rounding becomes important. By the use of eq 13 with eq 15 and the recursion, eq 16, we may quickly and easily compute all of the potential integrals for quite general potentials. Kinetic energy integrals may also be computed by these techniques. Using eq 8, we may show that
2Pm a
+ P m + P’n
Notice here that T,(i,m) is the internal sum, but the earlier terms of that rccursion do not correspond to internal sums. It is necessary to repeat the cntirc recursion for each value of i and m . This is no more effort, however, than the original sum, and the recursion relation is completely stable. When a is small, we find that x is very nearly I , so some loss of significant figures may occur in computing 1 - x and also the coefficient of Tkin eq 16. This can be overcome, however, by computing these coefficients analytically, and no loss of significance is experienced in computing T,. There is some loss of significance in performing the outer sum. This has been rather minor in the practical calculations we have done. A two-dimensional recursion has been developed by R. Tuzuna (7) Secrest, D. J . Chem. fhys. 1988, 89, 1017.
(8) Tuzun, R. E.;Secrest, D. Manuscript in preparation.
When the kinetic energy operator involves first derivatives, we may follow Requena et aL9 and we use hypervirialsI0 to obtain
(9) FrancEs, J. M.; Zdiiiga, J.; Alacid. M.; Requena, A. J . Chem. fhys. 1989, 90. 5536.
1060 The Journal of Physical Chemistry, Vol. 95, No. 3, I991 -4 r
Kratzer
Secrest Harmon i c
16
0,3+20
! -4 c
q
LJ
,"I
:,
24
,"I
22
w + -.e - 1 L.3
-
12
/
* + /
70 B a s i s Size 150 Figure 5. Error in the five highest eigenstates of the Morse oscillator as a function of the number of harmonic oscillator functions with k = 0.3 and E = 20. See the caption of Figure 4 and eq 7.
t
1
70
I I 1 1 B a s i s Size '50
Figure 4. Error in the five highest states of the Morse oscillator as a function of the number of Kratzer oscillator functions with K = 10 and E = 4. The eigenstates are E,, = 12.095, E,, = 12.255, = 12.375, EZ3= 12.455. and = 12.495. The well depth is 12.5 reduced units. J = 0.
Results and Discussion The lower vibrational states of the molecule may be easily fit by either harmonic oscillator functions or Kratzer oscillator functions. We first investigate computing the lower half of the bound eigenvalues for J = 0. The parameters k of eq 7 for the harmonic oscillator and K of eq 6 for the Kratzer oscillator are chosen experimentally to give the most rapid convergence for the eigenfunction expansion. A few computations with a small basis set suffice to determine the proper values to use for k. K , and B of eqs 6 and 7 . In Figures 1 and 2 we plot the logarithm of the error as a function of basis set size. The Kratzer oscillator K in Figure 1 is 18, which is found to give good results for the n = 12 state of the Morse oscillator. The Kratzer well minimum is taken to coincide with the Morse minimum. A more extensive search was necessary to find optimal parameters for the harmonic oscillator basis set in Figure 2. The harmonic oscillator force constant k = 0.48 was found to be optimum with the minimum of the harmonic oscillator at a bond length I unit farther from the origin than that of the Morse oscillator. As can be seen, the Kratzer oscillator gives results correct to 8 figures after the decimal point with 35 basis functions while the harmonic basis set requires 50 to achieve the same accuracy. The harmonic oscillator basis set is seen to be more uniform in convergence for the different quantum numbers. That is, the curves in Figure 2 have more nearly the same slopes. The Kratzer functions, however, commonly exhibit the behavior illustrated in Figure 1. When the K parameter is optimized for one eigenvalue, the lower ones tend to be deop(IO) Hirschfelder, J . 0. J . Chem. Phys. 1960, 33, 1462.
Figure 6. Plot of the Morse potential (thick line) with a well depth of 12.5 and a minimum re = 25, a harmonic oscillator with k = I , and a Kratzer oscillator with k = 5. Thus, both the Morse oscillator and the harmonic oscillator have the same curvature at the bottom of the well. The broad curve is a Kratzer oscillator well with re = 25 and a well depth of 12.5 units.
timized. In Figure 1 K = 18, which is optimum for E I 2 . The K = 16 basis set plottediin Figure 3 is not optimum for any of the eigenvalues shown, and the results have more of the qualitative behavior of the harmonic oscillator functions. Even in this nonoptimum case, the Kratzer functions are converged to 8 figures after the decimal point with 40 basic functions, while the optimum harmonic oscillator basis set requires 50 basis functions for the same accuracy. The highest bound eigenvalue for the Morse oscillator is difficult to compute with either basis set. In Figure 4 we plot the optimum basis set for the Kratzer functions. These functions are broader with a K = 10 and the minimum 4 units farther from the origin than the Morse oscillator. This offset is detrimental to the accuracy of the lower eigenvalues, but they are still better converged than E24. This eigenvalue is harder to determine with harmonic functions. The optimal basis set for E24has a force constant k = 0.3 and a minimum 20 units farther from the origin than the Morse oscillator (Figure 5). The Morse oscillators origin is at 25 units, and the harmonic oscillator functions minimum is at 45 units. Even with this optimum basis set, the eigenvalue is only determined to 2 figures after the decimal point with 150 basis functions. With such a strongly displaced minimum, the lower eigenvalues are even more poorly determined than when a 90-function basis set is used. This suggests that a mixed basis set might be more suitable for representing the Morse oscillator in terms of harmonic or Kratzer functions. We tried using a force parameter and displacement suitable for the lower eigenvalues along with the higher functions of sets suitable for the higher eigenvalues. This had been a successful technique in a different application." For both ( 1 I ) Scherzinger, A.
L.;Secrest, D. J . Chem. Phys. 1980, 73,
1706.
Kratzer Oscillator Basis Sets
The Journal of Physical Chemistry, Vol. 95, No. 3, 1991
-2r +
Kratzer
1061
15+1
Figure 7. Plot of the Morse potential as in Figure 6, with the Kratzer potential of Figure 1 and the harmonic potential of Figure 2. For the Kratzer potential K = 18 and B = 0. For the harmonic potential k = 0.48 and B = I . This corresponds to an re of 26 for the harmonic oscillator.
J
20
Basis S i z e
50
Figure 9. Logarithm of the error in eigenvalues of the diatomic Morse oscillator as a function of basis set size. The basis set is Kratzer oscillator functions with K = 15 and B = 1. J = 88. The eigenvalues are E7 = 1 1.453 247, E8 = 1 1.970 734, and E9 = 12.440 597. E l o is a metastable long-lived state with an energy of 12.861 342.
-2
r
Harmon i
t
0,4+2
Figure 8. Plot of the Morse potential as in Figure 6 and 7 with the Kratzer potential of Figure 4 and the harmonic potential of Figure 5. For the Kratzer potential K = IO and B = 4. For the harmonic potential k = 0.3 and B = 20. harmonic and Kratzer functions, more basis functions were required with these mixed nonorthogonal basis sets than with a set based on a single parameter pair. We were unable to find a mixed basis set of any sort that was superior to a single orthogonal set. It is certainly possible that a mixed set could be useful in other situations, such as a two-well problem. Neither the harmonic potential nor the Kratzer potential mimics the Morse potential, as can be seen from Figure 6. Here we have plotted a harmonic potential that has the same curvature as the Morse potential at the minimum, k = 1 .O, and a Kratzer oscillator of the same depth as the Morse potential. Of course, neither of these parameter sets is optimal for use as basis sets for eigenvalues of the Morse potential. The harmonic oscillator functions need to be a good bit broader and the Kratzer oscillator functions more localized, which implies a deeper well. When the parameters of the basis functions are optimized, the corresponding potentials of the harmonic oscillator and the Kratzer oscillator still do not mimic the Morse oscillator. It is not necessary that the potential which defines the basis functions be similar to the potential for which the basis functions are used. Those basis functions that best fit the classical region of the target potential will converge fastest. In Figure 7, we have plotted the optimum parameter harmonic and Kratzer oscillator wells for representing the lowest 12 Morse oscillator eigenstates. The symmetry of the harmonic oscillator around the minimum requires it to be displaced to larger r to give a satisfactory basis set. To compute the highest bound eigenvalue, it is necessary to displace both the Kratzer and harmonic oscillators to higher r to obtain a suitable basis set (Figure 8). The Kratzer oscillator is displaced 4 units beyond the equilibrium position while the harmonic oscillator is displaced 20 units. It would certainly appear that a mixed set of harmonic oscillator functions consisting of functions that match the lower part of the well along with higher functions of this displaced well would be superior. We always found, however, that with mixed sets of this sort more functions were required for the same accuracy for all states. The examples we have discussed so far have been for J = 0. For J # 0, the bound states are easier to compute by either the harmonic or Kratzer basis sets. The centrifugal barrier tends to
30
Basis s i z e
60
Figure IO. Logarithm of the error in eigenvalues of the diatomic Morse oscillator as a function of basis set size. The basis set is harmonic oscillator functions with k = 0.4 and B = 2. J = 88. See Figure 9. localize the bound states and gives rise to some metastable states. In Figures 9 and IO, we plot the error as a function of basis set size for J = 88. For this high angular momentum, there are only 10 bound states for n = 0, ..., 9. The n = 10 state is a metastable state mainly localized on the origin side of the centrifugal barrier. The basis functions used here are too localized to show any of the continuum states between 9 and the metastable state labeled 10. For these states, the behavior is much the same as for the lower bound states of J = 0. The Kratzer functions give 7-9 figures after the decimal point with 35 basis functions while the order of 50 harmonic oscillator functions are required for the same accuracy. As before, the behavior of the harmonic basis is somewhat more uniform than that of the Kratzer functions. Since the effective minimum of the potential is pushed to larger r by the centrifugal term in the Hamiltonian, both the optimal Kratzer and harmonic functions have their origins moved out compared to the J = 0 minimum of the Morse oscillator. For the J # 0 case, the Morse oscillator cannot be solved in closed form, and so even if we used Morse oscillator functions of the nonrotating Morse oscillator as a basis, it would be necessary to compute the centrifugal integral approximately numerically. The centrifugal integrals with J = 0 Morse functions are singular due to the singularity at the origin, but they may be truncated at small r to give acceptable results. Both the Kratzer oscillator and harmonic oscillator functions may be used efficiently for solving single-well problems. The symmetry of the harmonic oscillator functions seems to be a disadvantage in representing wave functions of nonsymmetric wells,
J. Phys. Chem. 1991, 95, 1062-1066
1062
and the Kratzer functions are better suited to this purpose. The Kratzer functions have the additional advantage that all integrals may be computed analytically.
Acknowledgment. The author acknowledges many useful
discussions with Bernard H. Chang and Robert E. Tuzun. This work was supported by a grant from the National Science Foundation, the donors of the Petroleum Research Fund, administered by the American Chemical Society, and the NCSA at the University of Illinois.
Dioxygen Complexes of 3d Transition-Metal Atoms: Formation Reactions in the Gas Phaset Carl E. Brown,: S.A. Mitchell,* and Peter A. Hackett Laser Chemistry Group, Division of Chemistry, National Research Council of Canada, 100 Sussex Drive, Ottawa, Ontario, Canada, K I A OR6 (Received: June 25, 1990)
Reactions of ground-state 3d transition-metal atoms including Ti, Mn, Co, Ni, and Cu with molecular oxygen in Ar buffer gas have been investigated in the pressure range 5-700 Torr at 296 K. Attention has been given to termolecular association reactions in which mono(dioxygen) complexes are formed. A pulsed laser photolysis-laser fluorescence technique is used where metal atoms are produced by visible multiphoton dissociation of a volatile organometallic precursor in a static pressure reaction cell, and reactions of metal atoms are monitored by resonance fluorescenceexcitation at variable time delay following the photolysis pulse. The present study completes a survey of the reactions of 3d transition-metal atoms with O2under room temperature conditions, from which it emerges that reactivity with respect to complex formation is correlated with a d"s' valence electron configuration of the metal atom. Simplified RRKM calculations have been used to interpret termolecular rate constants for the association reactions in terms of a trend in the binding energies of the dioxygen complexes.
Introduction It is known from matrix-isolation infrared' and ESR2,3spectroscopic studies that certain transition-metal (TM) atoms react with O2to form mono- and bis(dioxygen) complexes, T M ( 0 2 ) and TM(02)2. Very little is known about the formation of such complexes in the gas phase. Of the first-row (3d) T M atoms, only Sc, Ti, and V have exothermic 0-atom-transfer reactions with 02.4 Therefore, under moderate temperature conditions ground-state Cr, Mn, Fe, Co, Ni, and Cu atoms may only form an association complex with 02.Ritter and Weisshaar5 have reported rate constants for 0-atom-transfer reactions of Sc, Ti, and V atoms with O2 near room temperature in a flow reactor. The reaction Cu O2 Ar Cu(02) Ar at room temperature and IO Torr total pressure has been investigated by Vinckier et a1.,6 who also used a flow reactor. We have previously described a pulsed laser photolysis-laser fluorescence technique for kinetic studies of Cr7 and Fes atoms under room temperature conditions and at total pressures to -760 Torr. This technique is based upon visible multiphoton dissociation (MPD) of a volatile organometallic compound for production of metal atoms. Whereas Cr atoms react rapidly with 02,7 Fe atoms were found to be unreactive with respect to O2 at room temperature.s We report here a survey of the reactivity of 3d T M atoms with respect to complex formation with O2in the gas phase under room temperature conditions. Together with previous work on S C ,Ti,5 ~ V,5 Cr,7 Fe,8 and Cu6 atoms, the present investigation of Mn, Co, and Ni atoms completes the survey of the 3d T M series. The present study also includes new measurements for the reactions of Ti and Cu atoms with 02.Our objective was to identify trends in reactivity across the 3d T M series and to seek correlations between reaction rate constants and molecular properties of the complexes. It emerges clearly that TM atoms with singly occupied valence s-orbitals are reactive with respect to complex formation with O2 at room temperature, and those with doubly occupied valence s-orbitals are not. A rationalization of this trend is given in terms of simple molecular orbital concepts. An attempt is made to correlate the magnitude of the termolecular rate constant for
-
+ +
-
+
the association reaction with the binding energy of the complex, using a simplified form of RRKM theory together with molecular parameters obtained &om experimental or theoretical investigations. This approach appears satisfactory for Cu O2but meets with difficulties for Ni + 02.It is suggested that the results for Ni O2 point to the occurrence of reaction on more than one potential energy surface.
+
+
Experimental Section The experimental setup has k e n described in detail previo~siy.~*~ Briefly, T M atoms are produced by visible MPD of volatile organometallic precursors in a static pressure reaction cell using the focused output of pulsed dye laser (photolysis laser). T M atoms are detected by saturated resonance fluorescence excitation using a second, independently triggered pulsed dye laser (probe laser). The photolysis and probe laser beams are aligned collinear and counterpropagating through the center of the reaction cell, with the photolysis beam focused in the fluorescence viewing zone at the center of the cell. Laser-induced fluorescence (LIF) is viewed through a 10 cm focal length monochromator by a gated photomultiplier tube. To obtain kinetic data, the interpulse delay is scanned in preset increments and the LIF signal from 50 to 200 shots averaged at each delay setting. The entire delay range was swept 4-6 times to allow monitoring of the sample stability during the data acquisition period. The repetition rate of the laser pulses was 20 Hz. Titanium atoms were produced by MPD of TiCI., at 445 nm, using a pulse energy of 2.5 mJ focused with a 25 cm focal length ( I ) (a) Moskovits, M.; Ozin, G. A. In Cryochemistry; Moskovits, M., Ozin, G. A., Eds.; Wiley: New York, 1976; Chapter 8, and references cited therein. (b) Huber, H.; Klotzbucher, W.; Ozin, G.A,; Vander Voet, A. Can.J. Chem. 1973, 51, 2722. (2) Kasai, P. H.; Jones, P. M. J . Phys. Chem. 1986, 90, 4239. (3) Howard, J. A.; Sutcliffe, R.; Mile, B. J . Phys. Chem. 1984,88, 4351. (4) For a compilation of bond energies for 3d transition metal oxide diatomic molecules, see: Merer, A. J. Annu. Reo. Phys. Chem. 1989, 40, 407. (5) Ritter, D.; Weisshaar, J. C. J . Phys. Chem. 1989, 93, 1576; 1990. 94, 49117.
(6) Vinckier, C.; Corthouts, J.; De Jaegere, S.J . Chem. Soc., Faraday Trans. 2 1988,84, 1951.
'Issued as NRCC No. 32289. t NRCC Research Associate 1989-present.
(7) Parnis, J. M.;Mitchell, S.A.; Hackett, P. A. J . Phys. Chem. 1990, 94, 8 152. (8) Mitchell, S . A.; Hackett, P. A. J. Chem. Phys. 1990, 93,7813,7822.
0022-3654/91/2095-1062%02.50/0 0 1991 American Chemical Society
