Dissociative Adsorption of O2 on Cu(110) and Cu(100): Three

Kinetic Barriers of the Phase Transition in the Oxygen Chemisorbed Cu(110)-(2 × 1)-O as a Function of Oxygen Coverage. Liang Li , Qianqian Liu , Jona...
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J. Phys. Chem. 1996, 100, 11432-11437

Dissociative Adsorption of O2 on Cu(110) and Cu(100): Three-Dimensional Quantum Dynamics Studies Jiu-Yuan Ge,† Jiqiong Dai, and John Z. H. Zhang*,‡ Department of Chemistry, New York UniVersity, New York, New York 10003 ReceiVed: March 13, 1996X

The dynamics of dissociative chemisorption of oxygen on Cu(110) and Cu(100) has been studied using a time-dependent quantum wave-packet approach. Dissociation probabilities for O2 on both (110) and (100) surfaces of copper are calculated for ground state as well as rovibrationally excited oxygen molecules. The present calculation simulates O2 dissociation on nascent copper surfaces with no consideration for surface reconstruction. The dynamics calculation is based on a flat-surface model in which three molecular degrees of freedom are explicitly included while the lateral coordinates of the molecule are neglected. The interaction potential energy surface (PES) for the dynamics calculation is constructed using the LEPS (London-EryingPolyni-Sato) PES form with potential parameters fitted to some available experimental and theoretical data. The barrier of the LEPS PES for oxygen dissociation on copper is 0.11 eV for Cu(110) and 0.08 eV for Cu(100). Relatively speaking, the saddle point of the O2/Cu(110) PES is located near the entrance channel, while that of the O2/Cu(100) is near the product channel. This feature is primarily responsible for the difference in calculated dissociation probabilities of oxygen on two surfaces. Specifically, the dissociation probability of O2 on Cu(110) is large and less sensitive to the vibrational excitation of the molecule, while that of O2 on Cu(100) is much smaller and more sensitive to vibrational excitation of the molecule.

I. Introduction The understanding of the mechanism of adsorption of oxygen on transition metals is important to our understanding of its relationship to heterogeneous catalysis and corrosion.1 Much experimental work has been devoted to the study of O2 chemisorption on copper surfaces using a variety of experimental techniques,2-6 mostly on kinetics studies. Theoretical calculations using both ab initio and empirical methods have also been reported that are aimed to characterize the structure and energetics for oxygen adsorption on Cu.7-17 However, very little dynamics information is known for chemisorption of oxygen on copper surfaces. Although progress has been made in quantum dynamics calculations for dissociative adsorption of hydrogen on metal surfaces such as copper,18 the dynamics of the chemisorption of oxygen molecule on metals including copper surfaces has not been theoretically studied in detail. The adsorption of oxygen on copper is highly exothermic and irreversible. On Cu(110), molecular oxygen adsorbs dissociatively above 100 K,4 and the sticking coefficient is independent of surface temperature for molecular beam energies above 0.2 eV.2 These experimental results indicate that the dissociation of O2 on Cu(110) is a direct process at high energies, while the trapping mechanism plays a role only at low energies. The dissociation probability of O2 on Cu(110) is experimentally found to be about an order of magnitude larger than that on Cu(100).5 Theoretical calculations using cluster models have determined that the lowest adsorption site of the oxygen atom is the long bridge site on Cu(110) and the 4-fold center site on Cu(100). In this paper, we study the dynamics of dissociative chemisorption of oxygen on Cu(110) and Cu(100) using the direct dynamics model. Our theoretical study uses a time-dependent quantum wave-packet approach to calculate the dissociation †

Department of Physics, New York University. Alfred P. Sloan Fellow and Camille Dreyfus Teacher-Scholar. X Abstract published in AdVance ACS Abstracts, June 15, 1996. ‡

S0022-3654(96)00776-9 CCC: $12.00

probability as a function of translation energy and internal state of O2. The current quantum calculation is based on a 3D flatsurface model in which three molecular degrees of freedom are explicitly included in the quantum treatment. The flat-surface model has been shown in previous studies to be quite successful in providing qualitative and even some quantitative dynamics information for hydrogen dissociation on metals such as the role of hydrogen vibration and rotation in dissociative adsorption on Cu(111).18-20 The present dynamics calculation is carried out on a potential energy surface that models the dissociation process of O2 on Cu. We model the PES using an empirical LEPS potential surface. The parameters of the LEPS surface are chosen to be compatible with some available theoretical calculations and with experimental adsorption data. The saddle point geometry and dissociation barrier height are adjusted by varying the Sato parameter such that the calculated dissociation probabilities are in reasonable agreement with experimental data. This paper is organized as follows: Section II presents the time-dependent theory for dissociation of a diatomic molecule on a rigid crystal surface with the lateral coordinates of the center-of-mass of the diatom frozen at a given impact site. The LEPS potential energy surface for O2 on Cu is used, and its parameters are fitted to the available experimental and theoretical data for oxygen dissociation on both (110) and (100) surfaces. In section III, the dissociation probabilities of O2 on both surfaces are calculated, and different behaviors of the probabilities on two surfaces are discussed. Finally, section IV provides a brief discussion of the results of this study and concludes. II. Theory A. Time-Dependent Approach to Adsorption Dynamics. In the flat-surface model (no corrugation), there are three molecular degress of freedom denoted by (Z,r,θ) (cf. Figure 1). The molecular Hamiltonian in the flat-surface model is defined as © 1996 American Chemical Society

Dissociative Adsorption of O2 on Cu(110) and Cu(100)

J. Phys. Chem., Vol. 100, No. 27, 1996 11433 to the coordinate r. The full TI scattering wave function is + normalized as 〈ψ+ iE|ψiE′〉 ) 2πpδ(E - E′). Equation 4 can be simplified as

[〈 |

| 〉]

p ∂ + + PiR(E) ) Im ψiE δ(r - r0) ψiE µ ∂r

(6)

The TI scattering wave function is obtained by the Fourier transform Figure 1. Jacobi coordinates for the diatom on a flat surface. Z is the perpendicular distance from the surface to the diatomic center, r is the diatomic distance, and θ is the polar angle.

B J p2 ∂2 p 2 ∂2 H)+ + V(Z,r,θ) 2M∂Z2 2µ∂r2 2µr2 2

)∑ t,n,j

m Ftnj,V (t) 0j 0

utn(Z)

Pjm(θ) χn(r)

the where (V0j0) denotes the initial rovibrational state, translational basis function, and χn(r) the vibrational basis function. We use Pjm(θ) to denote normalized associate Legendre polynomials. In the flat-surface model, the rotation projection quantum number m of the diatom along the axis perpendicular to the surface (Z-axis) is conserved. We refer the reader to refs 18 and 20 for the definition of translational and vibrational functions. The time-dependent wave function is propagated using the split-operator method,21

Ψ(Z,r,θ,t+∆) ) e-iH0∆/2e-iU∆e-iH0∆/2Ψ(Z,r,θ,t)

(3)

where U is the interaction potential plus centrifugal potential as defined in ref 20 and H0 contains only the kinetic energy operators in eq 1. The split-operator propagator is explicitly unitary so that the normalization of the wave packet is preserved during the propagation. To avoid artificial boundary reflection of the wave function due to a finite numerical grid, absorbing potentials are used to absorb the wave function at the edges of the grid as is done in previous studies.20 The initial state-selected total dissociation probability of the diatom is obtained by projecting out the energy-dependent reactive flux. If ψ+ iE denotes the time-independent (TI) full scattering wave function, where i and E are the labels for initial state and energy, respectively, the total dissociation probability from an initial state i can be obtained by the flux formula + + |Fˆ |ψiE 〉 PiR ) 〈ψiE

(4)

In the above equation, Fˆ is the flux operator defined as

1 Fˆ ) [δ(rˆ - r0)Vˆ r + Vˆ rδ(rˆ - r0)] 2

(5)

where r is the O2 bond distance, r0 defines the dividing surface for flux evaluation, and Vˆ r is the velocity operator corresponding

(7)

+ ai(E) ) 〈ψiE |ψi(0)〉

(8)

) lim 〈φiE|e(i/p)H0te-(i/p)Ht|ψi(0)〉

(9)

) 〈φiE|ψi(0)〉

(10)

tf-∞

where the free wave function is given by

φiE ) -

exp(-ikiZi)

xVi )

(2)

unt (Z)

∞ (i/p)(E-H)t 1 e |ψi(0)〉 dt ∫ -∞ ai(E)

The coefficient ai(E) in the above equation is evaluated from the free asymptotic function as follows:

(1)

where B J is the angular momentum operator, M the total mass of O2, µ the reduced mass of O2, and V the potential. Our dynamics calculation consists of propagating an initial wave packet in a chosen quantum state that is located in the asymptotic region where O2 is far away from the copper surface. The detailed procedure has been described previously,20 and here only some main formulas are summarized below. The time-dependent wave function satisfying the Schro¨dinger equation ip(∂/∂t)Ψ(t) ) Hψ(t) is expanded in terms of complete rovibrational basis functions as

ΨVm0j0(Z,r,θ,t)

+ |ψiE 〉)

2i

+

exp(ikiZi)

xVi

sin(kiZi)

xVi

(11)

(12)

and is normalized as 〈φiE|φiE′〉 ) 2πpδ(E - E′). B. Potential Energy Surfaces for O2/Cu(110) and O2/Cu(100). Accurate ab initio calculation of the PES for molecular interaction with metals is a tremendously challenging task. At present, no sufficient ab initio calculations are available for us to fit an accurate PES for studying the dynamics of O2 dissociation on Cu surfaces. Thus, our dynamics study uses an empirical LEPS potential energy surface to model the chemisorption of oxygen on Cu. The LEPS potential energy surface is a reasonable representation for dissociation of a molecule on a rigid surface that involves a simple barrier. Therefore, the direct dissociation of O2 on Cu should be reasonably well modeled by the LEPS empirical potential surface, while the possible complication of the dissociation of O2 at very low energies due to molecular adsorbed states is not considered in the present study. The LEPS PES has the following general form:

V ) U1 + U2 + U3 - [Q12 + (Q2 - Q3)2 - Q1(Q2 + Q3)]1/2 (13) where Ui and Qi are defined, respectively, as

Ui )

1 D {(3 + ∆i) exp[-2Ri(qi - qi0)] 4(1 + ∆i) i (2 + 6∆i) exp[-Ri(qi - qi0)]} (14)

Qi )

1 D {(1 + 3∆i) exp[-2Ri(qi - qi0)] 4(1 + ∆i) i (6 + 2∆i) exp[-Ri(qi0)]} (15)

where U1 and Q1 describe interactions between two oxygen atoms, while U2,3 and Q2,3 describe interactions between an oxygen atom and the copper surface. All the Morse and anti-Morse potential parameters are obtained from several references, as shown in Table 1. In particular, diatomic potential parameters of O2 are given in refs

11434 J. Phys. Chem., Vol. 100, No. 27, 1996

Ge et al.

TABLE 1: LEPS Potential Surface Parameters Used in This Work

a

O2/Cu(110)

D1 ) 5.2136 eVa D2(3) ) 4.0431 eVb

R1 ) 2.653 87 Å-1a R2(3) ) 1.0094 Å-1b

D1 ) 5.2136 eVa D2(3) ) 5.5836 eVc

O2/Cu(100) R1 ) 2.653 87 Å-a R2(3) ) 0.899 78 Å-1c

r0 ) 1.208 Åa z10(20) ) 0.314 Åb

∆1 ) -0.470 ∆2(3) ) 0.273

r0 ) 1.208 Åa z10(20) ) 0.910 Åc

∆1 ) -0.770 ∆2(3) ) 0.471

Reference 23. b Reference 17a. c Reference 17b.

Figure 2. Contour plot of the potential energy surface for O2 approaching Cu(110) with the molecular axis parallel to the surface. The saddle point is located at Z ) 1.3 au and r ) 2.6 au.

22 and 23, and the O-Cu potential parameters are based on ref 17. The only adjustable parameters in the current LEPS PES are two Sato parameters ∆1 and ∆2, which determine the location and the height of the reaction barrier. For O2 on Cu(110), we utilize the experimental result of ref 2 to help determine the Sato parameters. The parameters given in Table 1 produce the saddle point located at Z ) 1.3 au and r ) 2.6 au with a barrier height of 0.11 eV. This is consistent with the experimental finding of ref 2, which indicates a saddle point near the entrance channel for O2 dissociation on Cu(110). Of course, the choice of Sato parameters is by no means unique without detailed ab initio calculations. The Sato parameters are similarly determined for O2 dissociation on Cu(100). Here, we also based our choice of parameters on the experimental results of refs 3 and 5 that indicate a barrier near the product channel and an activation energy around 0.1-0.2 eV. Our choice of Sato parameters for O2/Cu(100) in Table 1 results in the saddle point at Z ) 2.4 au and r ) 2.8 au with the barrier height of 0.08 eV. Since the dissociation probability for O2 on Cu(100) is more than an order of magnitude smaller than on Cu(110) at low energies, one could also account for this difference by choosing a larger barrier for the O2/Cu(100) PES. However, without reliable ab initio data, we stick with experimental hindsight and try to account for the different behavior of O2 on two Cu surfaces by the difference in location of the saddle point on the PES. Figure 2 shows a PES contour plot for dissociation of O2 on Cu(110) with the polar angle θ fixed at 90° (parallel approach). A similar contour plot is also shown for O2 on the Cu(100) surface in Figure 3. The lowest adsorption site for the oxygen atom is known to be on the long bridge site on Cu(110)17 and the 4-fold site on Cu(100).9 Our model PES gives a dissociation barrier of 0.11 eV, and the saddle point is located at Z ) 1.3 au and r ) 2.6 au on the (110) surface. On Cu(100), the saddle point is at Z ) 2.4 au and r ) 2.8 au and the barrier height is 0.08 eV. We notice the difference of saddle point geometry between (110) and (100) surfaces: the saddle point on Cu(110) is located near the entrance channel, while that on Cu(100) is

Figure 3. Same as Figure 2 but for the Cu(100) surface. The saddle point is located at Z ) 2.4 au and r ) 2.8 au.

located near the product channel. We will see later that this difference in saddle point geometry gives large differences in calculated dissociation probabilities of oxygen on these two surfaces. III. Results of Calculation A. Dissociation of O2 on Cu(110). Numerical details of the present calculation are given in this section. The wavepacket dynamics calculation uses 400 grids in the Z coordinate, 200 vibrational function for the r coordinate, and 40 rotation functions in the angular coordinate (jmax ) 79). The time increment in wave-packet propagation is 50 atomic units (au), with which the calculated dissociation probabilities are satisfactory. Most dissociation probabilities are converged after a propagation of about 35 000 au except at very low kinetic energies that require longer time of propagation. Extensive numerical tests are performed to ensure that the calculated dissociation probabilities are stable with respect to changes of various numerical parameters. Since the ground electronic state of the oxygen molecule is O2(3∑g ), we only calculate the dissociation probability from odd rotational states of O2.22 The probabilities for O2 dissociation on Cu(110) in ground vibrational and j ) 1 rotational states are shown in Figure 4, where they are plotted as a function of O2 kinetic energy. There are two orientational states for j ) 1 with m ) 0 and 1 (the m ) -1 state gives the identical result as that of the m ) 1 state). Figure 4 shows that the j ) m ) 1 state produces larger dissociation probability than the j ) 1, m ) 0 state. This is due to the orientational dependence of the dissociation probability that favors parallel approach of the molecule.18,20 Also plotted in Figure 4 is the orientationaveraged dissociation probability defined as

P h (j) )

1

m)j

∑

2j + 1m)-j

P(jm)

(16)

where P(jm) is the rotation state-selected dissociation probability. The orientation-averaged dissociation probability P h (j) is more relevant for comparison with experiment. In Figure 5, we show

Dissociative Adsorption of O2 on Cu(110) and Cu(100)

Figure 4. Dissociation probabilities of O2 on Cu(110) as a function of translation energy for O2 at ground vibration and j ) 1 rotation states. The three numbers in parentheses denote three quantum numbers (Vjm), and the diamonds are the results averaged over orientational states m.

J. Phys. Chem., Vol. 100, No. 27, 1996 11435

Figure 6. Similar to Figure 4 except for j ) 5 and m ) 0, 1, 2, 3, 4, and 5.

Figure 7. Comparison of orientationally averaged dissociation probabilities for O2 on Cu(110) for ground O2(V)0,j)1) and vibrationally excited O2(V)1,j)1). Figure 5. Comparison of calculated orientation-averaged probability Pj for O2 dissociation on Cu(110) with the experimental measurement of ref 2. Circles are the theoretical results, and squares are the experimental results.

a comparison of P h (j) (j ) 1) with the measured sticking probability from ref 2. We should keep in mind that the experimental results are not energy selected but rather averaged over the Boltzmann distribution at the corresponding beam temperature.2 In addition, the experimental results are also Boltzmann averaged over rotational states and possibly also have contributions from vibrationally excited states. In view of all the uncertainties, the calculated dissociation probabilities are in quite reasonable agreement with experimental measurements. We also show in Figure 6 the orientation-dependent dissociation probability P(jm) for initial orientational states of j ) 5 and m ) 0, ..., 5 of O2. The dissociation probability for the j ) m ) 5 state (helicopter mode) gives the largest dissociation probability, while the j ) 5, m ) 0 state (cartwheel mode) generally gives small dissociation probabilities. This is very similar to the case of hydrogen dissociation on Cu, as discussed quite fully in refs 18 and 20. Another important feature of the dynamics in O2 dissociation over Cu surfaces is the role of molecular vibration. As is generally understood, the effect of reagent vibration on reaction is closely related to the position of the saddle point of the PES.

Generally speaking, the reagent vibration is not very effective in enhancing the reactivity if the PES has an early barrier (near the entrance channel). Contrarily, if the PES has a later barrier (near the product channel), the reagent vibration could significantly enhance the dissociation probability. As shown in Figure 2, the saddle point of the PES for O2/Cu(110) has a relatively earlier barrier; one thus would not expect O2 vibration to have a significant effect on dissociation. This is indeed the case, as shown in Figure 7, where dissociation probabilities from both ground and excited vibrational states are plotted, except at very low kinetic energies. B. Dissociation of O2 on Cu(100). The dissociation probability of O2 on the Cu(100) surface is also calculated and shown in Figures 8-10. In Figure 8, the dissociation probabilities from ground vibrational O2 at j ) 1 are plotted as a function of kinetic energy. We first notice that the dissociation probability on the Cu(100) surface is smaller by orders of magnitude than the corresponding one on Cu(110) of Figure 3 at low energies, although the dissociation barrier height of O2 on Cu(100) is very similar to that on Cu(110): 0.08 eV on Cu(100) versus 0.11 eV on Cu(110). Therefore, the much smaller dissociation probability of O2 on Cu(100) is not directly related to the height of the potential barrier. Rather it is the position of the barrier that is most likely responsible for the smaller dissociation probability of O2 on Cu(100). This assertion is

11436 J. Phys. Chem., Vol. 100, No. 27, 1996

Figure 8. Similar to Figure 4 but for O2 dissociation on Cu(100).

Ge et al.

Figure 11. Comparison of dissociation probability of O2 on Cu(110) and Cu(100) for ground vibrational O2 at j ) 1 and averaged over orientational states.

comparison with the present calculation of dissociation probability for O2 on Cu(100). However, the experimental data suggest that the dissociation probability of O2 on Cu(100) is at least an order of magnitude smaller than on the corresponding Cu(110) surface at room temperature.5 Our theoretical calculation is consistent with the experimental evidence. The orientational dependence of the dissociation probability for O2/Cu(100) is also shown in Figure 10, and it clearly shows a preference for O2 dissociation when it approaches the metal surface parallelly (m ) j). This result is essentially similar on both (110) and (100) surfaces. IV. Conclusions

Figure 9. Similar to Figure 7 but for O2 dissociation on Cu(100).

Figure 10. Similar to Figure 6 but for O2 dissociation on Cu(100).

consistent with the calculation of dissociation probability for vibrationally excited O2, as shown in Figure 9. As seen in Figure 9, the dissociation probability of vibrationally excited O2 is orders of magnitude larger than ground vibrational O2 at low energies. Thus, the vibrational excitation of O2 has a significant effect on the dissociation of oxygen on Cu(100) at low energies. There are currently no explicit experimental adsorption data (to our knowledge) that are available for direct

We presented the first three-dimensional quantum dynamics study to investigate the chemisorption of oxygen on (110) and Cu(100) surfaces of copper using the flat-surface model. The dynamics calculations are carried out on empirical LEPS potential energy surfaces. The dissociation probability of oxygen on Cu(110) is much larger than that on Cu(100), as compared in Figure 11. The main difference between the PES on two copper surfaces is the location of the saddle point or transition state. On the other hand, the dissociation barrier on both surfaces is small and comparable in size. Therefore, the difference in the calculated dissociation probability of O2 on Cu(110) and Cu(100) is directly related to the location of the potential barrier on two PESs. Future study calls for more accurate ab initio calculations of the PES, especially in the transition state region, to accurately determine the location of the saddle point. The dynamics calculation will also need to investigate the role of molecularly adsorbed states that may be effective at low energies. In addition, the effect of Cu surface reconstruction should also be investigated. Acknowledgment. This work is supported by the National Science Foundation under the Presidential Faculty Fellows program and also in part by the Petroleum Research Fund, administered by the American Chemical Society. J.D. is a recipient of the Dean’s dissertation fellowship of New York University. References and Notes (1) See: Brundle, C. R.; Broughton, J. Q. In Chemical Physics of Solid Surfaces and Heterogeneous Catalysis; King, D. A., Woodruff, D. P., Eds.; Elsevier: New York, 1991; Vol. 3. (2) Hodgson, A.; Levin, A. K.; Nesbitt, A. Surf. Sci. 1993, 293, 211.

Dissociative Adsorption of O2 on Cu(110) and Cu(100) (3) Pudney, P.; Bowker, M. Chem. Phys. Lett. 1990, 171, 373. (4) (a) Mundenar, J. M.; Baddorf, A. P.; Plummer, E. W.; Sneddon, L. G.; Didio, R. A.; Zehner, D. M. Surf. Sci. 1987, 188, 15. (b) Mundenar, J. M.; Plummer, E. W.; Sneddon, L. G.; Baddorf, A. P.; Zehner, D. M.; Gruzalski, G. R. Surf. Sci. 1988, 198, L309. (5) Balkenende, A. R.; den Daas, H.; Huisman, M.; Gijzeman, O. L. J.; Geus, J. W. Appl. Surf. Sci. 1991, 47, 341. (6) Harbaken, F. H. P. M.; Mesters, C. M. A. M.; Bootsma, G. A. Surf. Sci. 1980, 97, 262. (7) Bagus, P. S.; Batra, J. P.; Bauschlicher, C. W., Jr.; Broer, R. J. Electron Spectrosc. Relat. Phenom. 1983, 29, 225. (8) Bauschlicher, C. W., Jr. J. Chem. Phys. 1986, 84, 250. (9) Madhaven, V.; Newton, M. D. J. Chem. Phys. 1987, 86, 4030. (10) Upton, T.; Steven, P.; Madix, R. J. J. Chem. Phys. 1988, 88, 3988. (11) Panas, I.; Siegbahn, P.; Wahlgren, U. J. Chem. Phys. 1989, 90, 6791. (12) Fischer, C.; Whitten, J. Phys. ReV. B 1989, 40, 5745. (13) Bagus, P. S.; Illas, F. Phys. ReV. B 1990, 42, 10852. (14) Hellsing, B.; Gao, S. Chem. Phys. Lett. 1991, 187, 137.

J. Phys. Chem., Vol. 100, No. 27, 1996 11437 (15) Chan, A. W. E.; Hoffman, R.; Ho, W. Langmuir 8 1992, 1111. (16) (a) Hellsing, B. Surf. Sci. 1993, 282, 216. (b) Lou, L.; Nordlander, P.; Hellsing, B. Surf. Sci. 1994, 320, 320. (17) (a) Ricart, J. M.; Torras, J.; Clotet, A.; Sueiras, J. E. Surf. Sci. 1994, 301, 89. (b) Ricart, J. M.; Torras, J.; Illas, E.; Rubio, J. Surf. Sci. 1994, 307, 107. (18) See: Dai, J.; Zhang, J. Z. H. J. Chem. Phys. 1995, 102, 6280 and references therein. (19) Sheng, J.; Zhang, J. Z. H. J. Chem. Phys. 1993, 99, 1373. (20) (a) Dai, J.; Sheng, J.; Zhang, J. Z. H. J. Chem. Phys. 1994, 101, 1555. (b) Dai, J.; Zhang, J. Z. H. Surf. Sci. 1994, 319, 193. (21) Fleck, J. A., Jr.; Morris, J. R.; Feit, M. D. Appl. Phys. 1976, 10, 129. (22) Hertzberg, G. Molecular Spectra and Molecular Structure: I. Spectra of Diatomic Molecules; Van Nostrand: New York, 1950. (23) Lin, J.-H.; Garrison, B. J. J. Chem. Phys. 1984, 80, 2904.

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