J. Phys. Chem. 1996, 100, 19257-19267
19257
Radiationless Transitions between the First Excited Triplet State and the Singlet Ground State in Ethylene: A Theoretical Study Bernhard Gemein and Sigrid D. Peyerimhoff* Institut fu¨ r Physikalische und Theoretische Chemie, UniVersita¨ t Bonn, Wegelerstrasse 12, D-53115 Bonn, Germany ReceiVed: NoVember 6, 1995; In Final Form: September 17, 1996X
A theoretical analysis is reported for the first triplet (T) to ground state (N) radiationless transition of the ethylene molecule. This work is based on ab initio calculations of the potential energy surfaces for the T and N states taking into account CC and CH bond stretching as well as HCH bending and torsion of the CH2 groups. The radiationless transition rates are calculated for different molecular vibrational levels using the Fermi-Wentzel golden rule formalism. The interaction matrix element 〈T|Vint|N〉 is approximated by the electronic spin-orbit coupling matrix element between the N and T states, integrated over the torsional coordinate; the effect of the three totally symmetric normal modes on the calculated rates is accounted for by their Franck-Condon factors. The transition rates calculated in this work are on the order of 106-108 s-1 if one or two levels of the totally symmetric modes are excited, which is in good agreement with the global values predicted earlier by Salem and Rowland.
I. Introduction
II. Global Aspects of the Calculations
Radiationless transitions play a major role in many photochemical processes and determine frequently the lifetime of molecules in their electronically excited states. Many experimental and theoretical studies of such transitions have been carried out over the past 30 years,1-4 but only in very recent times has it become possible by use of modern laser techniques to perform accurate measurements of fast decay rates for experimentally prepared unstable initial states of molecules in condensed or low-pressure phase.5-7 Likewise quantum chemical ab initio calculations have progressed in recent years so that they are now able to describe reliably probabilities not only for radiative but also for radiationless transitions,8 as for example in OH+ 9 and ArH.10 The advantage of the theoretical treatment is that it will yield not only the lifetime but also details about the underlying decay mechanism, in particular information on the states involved in the process. If the states involved possess the same multiplicity, the radiationless transition is generally referred to as internal conversion; if they have different spin, one refers to it as intersystem crossing. From a theoretical point of view only the operator describing the interaction (nuclear kinetic energy or spin-orbit operator) causes the difference, and sometimes competing processes are likely if several states are involved. In the present work we will study the intersystem crossing between vibrational levels of the first excited 3ππ* or T state with those of the 1π2 or N state in the ethylene molecule. The spin-orbit interaction makes the transition from one state to the other possible. The ethylene molecule is the first member in the series of unsaturated hydrocarbons CnHn+2 whose photochemical behavior is of importance for technical and biological aspects. The presentation will start with the potential surfaces for the N and T states calculated as a function of important internal coordinates. The computation of the spin-orbit interaction matrix elements between these states will follow. Vibrational wave functions will be generated on the basis of the potentials, and finally the transition rates will be evaluated.
Electronic States. The potential energy surfaces and the electronic wave functions are generated by employing the multireference configuration interaction method (MRD-CI).11-13 Single and double excitations are generated with respect to a set of reference configurations. This MRD-CI space is partitioned: all symmetry-adapted configuration functions (SAF) are tested one-by-one, and those that lower the energy of the reference space by more than a given threshold are included directly in the zero-order Hamiltonian matrix which is diagonalized, while the effect of the unselected configurations is accounted for in a perturbation-like treatment. The energy corresponding to the full-CI treatment is estimated in analogy with the formula by Langhoff and Davidson14 as
X
Abstract published in AdVance ACS Abstracts, November 1, 1996.
S0022-3654(95)03263-1 CCC: $12.00
ref
est Efull-CI ) EMRD-CI + (1 - ∑cp2)(EMRD-CI - Eref-CI) (1) p
where EMRD-CI is the energy corresponding to the generated MRD-CI space and Eref-CI is the energy of reference configurations. Sections of the potential surface for the T and N state were calculated along the internal coordinates RCC (carbon-carbon distance), rCH (carbon-hydrogen distance), RHCH (bending angle of the CH2 group), and the torsion angle φ (torsion of the CH2 groups around the CdC bond). Thus apart from this torsional motion only totally symmetric displacements of the carbon and hydrogen atoms are considered following the arguments given earlier by Siebrand et al.15 It was found that the vibronic structure of the N-V and N-T transitions can be modeled by invoking the ν2 (CC stretch) and ν3 (scissoring motion of the CH2 groups) in addition to the obvious ν4 torsional mode.4 The carbon and hydrogen atoms are described by the Gaussian basis set given by Huzinaga16 in the contraction suggested by Dunning.17 The carbon (9s,5p) set is contracted to [4s,2p] and augmented by one (six-component) d polarization function of exponent 0.75;18 one Rydberg s (exponent 0.023), one Rydberg p (exponent 0.028), and one Rydberg d function (exponent 0.015) are furthermore added to this basis.19 The hydrogen (4s) basis set is contracted to [2s] and one p polarization function with exponent 1.0 is added. The inclusion of carbon Rydberg © 1996 American Chemical Society
19258 J. Phys. Chem., Vol. 100, No. 50, 1996
Gemein and Peyerimhoff
functions allows for the strong Rydberg-valence interaction between the first excited singlet states 1ππ* and 1π3py of the ethylene molecule. Although these states are not of primary interest, the basis set is thus adequate for the description of the lower electronically excited states. The coordinate system adopted for the ethylene molecule is that recommended by Mulliken.20 The molecule is chosen to lie in the yz plane with the CC and z axis collinear. Unless otherwise indicated, the CH bond length (HCH angle) is fixed to 1.08 Å (120°). At planar geometry the nuclear frame possesses D2h symmetry, in the perpendicular arrangement of CH2 groups the pertinent symmetry group is D2d, and for arbitrary torsional angles the symmetry is D2. To keep the same technical treatment for all points of the energy surface during torsion, these calculations are undertaken in D2 symmetry, i.e. from φ ) 0.2° instead of φ ) 0° in D2h to φ ) 89.8° instead of φ ) 90° (D2d). The MRD-CI calculations use the SCF orbitals of the lowest 3B state, which correlates with 3B1u and 3B states in the higher symmetry. In contrast to the ground 2 state SCF orbitals, the antibonding π* orbital, which has Rydberg-valence character for small twisting angles,21-23 is not a virtual orbital but is optimized in this state at the SCF level. In this manner the excited state T (3ππ*) should be described in a more compact CI expansion than by the use of N state MOs. The two lowest orbitals corresponding to the two carbon 1s shells (or rather 1sc ( 1sc in 1a1g and 1b1u symmetry) are kept doubly occupied in all MRD-CI calculations, while the two orbitals of highest energy corresponding to the AOs with the largest exponents (simulating the 1s cusp) are excluded entirely from the MRD-CI space. The AO basis comprises 72 functions, while in the MRD-CI treatment the 12 valence electrons are distributed among 68 molecular orbitals. Vibrational Functions. The total wave function is written b,R) and the nuclear part as a product of the electronic part φel(r χN(R). In treating the the vibrational part, it is convenient to use normal coordinates. If the modes are considered to be uncoupled, the vibrational wave function can be written in the form
χV(Q1,Q2,...,Q3N-6) ) χV1(Q1) χV2(Q2) ... χV3N-6(Q3N-6)
(2)
In this manner each normal coordinate Qi refers to a single onedimensional oscillator χVi(Qi). In the harmonic approximation the energy of the vibrational motion is given as
( )
3N-6
EV )
∑ i)1
Vi +
1
2
2πpcνi
(3)
where Vi denotes the number of vibrational quanta and νi the fundamental frequency of the ith mode. In the present context the most important mode is the torsion mode ν4 of Au symmetry in addition to the three totally symmetric Ag modes labeled ν1 to ν3 involving the internal coordinates rCH, RCC, and angle RHCH, as mentioned earlier. The modes ν5 to ν12 are not considered explicitly; their form can be found in the literature.23 Spin-Orbit Interaction. The interaction mechanism that makes the singlet-triplet intersystem crossing possible is the spin-orbit interaction. In the present work the spin-orbit Hamiltonian is employed in the Breit-Pauli form:
e2p ZR b s i‚(b r iR × b p i) HSO ) ∑∑ 2 2 3 R i 2m c r iR
∑ i*j
e2p 1
(b r ij × b p i)‚(b s i + 2b s j) (4) 2m2c2rij3
The summation over nuclei (electrons) is indicated by greek (roman) letters. All one- and two-electron integrals are evaluated explicitly. The interaction matrix element between the N and T state is taken in the form
VNT ) 〈N|HSO|T〉
(5)
or in the Born-Oppenheimer approximation
VNT ) 〈φN(b,Q) r χN(Q)|HSO|χT(Q) φT(b,Q)〉 r
(6)
The electronic wave functions are MRD-CI wave functions with the mS ) S component. According to Fermi’s golden rule, the transition rate for the radiationless transition between the T and N state can be obtained as
KT-N )
2π |V |2F(EN) p NT
(7)
where F(EN) represents the density of states. The corresponding lifetime τ is 1/K. III. Potential Energy Surfaces In its planar ground state (D2h symmetry) the ethylene molecule possesses the 1a21g1b21u2a21g2b21u1b22u3a21g1b23g1b23u electronic configuration. According to Mulliken, this state is referred to as the N state. The highest occupied orbital (b3u type) is a π orbital, which is represented by the in-phase linear combination of carbon px orbitals oriented perpendicular to the molecular plane. In addition there is a small contribution from the hydrogen p functions to this orbital. In D2 symmetry the corresponding electronic configuration is 1a21b122a22b121b223a21b322b32. Totally Symmetric Motions. The energies of the ground state N and the excited triplet state T are calculated at six different CC values between 1.37 and 1.8 Å, four CH bond lengths between 1.0 and 1.16 Å, and three values for the bending angle RHCH between 94° and 140°. Hence a grid of 72 energy points is obtained for each state. The torsion angle was fixed at φ ) 89.8° (pertaining to formally D2 symmetry), which approximates realistically the optimal angle of φ ) 90° for the T state. Both N and T states are represented by a set of 16 reference configurations. To get tractable sizes for the secular problems, the energy selection threshold is chosen to be ET ) 15 µhartree (1 hartree is the atomic unit of energy, 27.21 eV). The dimensions of the Hamiltonian matrices corresponding to the MRD-CI spaces were on the order of 430 000 (1A) and 720 000 (3B1), while matrices that have to be diagonalized do not exceed 10 000 (17 000) for the 1A (3B1) irreducible representations. The contribution of the reference configurations amounted always to about 92% of the total wave function. The lowest four roots of the Hamiltonian matrix in 1A and 3B symmetry were calculated, but only the N, T, and V states 1 are of interest at the perpendicular geometry. The T state is described to a very large extent solely by the 3ππ* configuration, while the N and V state wave functions show essentially equal mixing of π2 and π*2 configurations close to D2d symmetry due to the degeneracy of the corresponding molecular orbitals. The calculated potential energy surfaces are presented as a function of two internal coordinates (the third one is kept constant, the explicit value is indicated in the corresponding figure) in Figures 1-6. In general it is seen that the surfaces for the N and T states are fairly parallel for the perpendicular arrangement of CH2 groups (Figures 1-3) with respect to slopes and their minima. The most anharmonic behavior is seen for
Radiationless Transitions in Ethylene
Figure 1. Calculated two-dimensional rCH-RHCH potential energy surfaces for the T and N states (φ ) 89.8°).
J. Phys. Chem., Vol. 100, No. 50, 1996 19259
Figure 3. Calculated two-dimensional rCH-RCC potential energy surfaces for the T and N states (φ ) 89.8°).
Figure 2. Calculated two-dimensional RHCH-RCC potential energy surfaces for the T and N states (φ ) 89.8°).
variations in RCC followed by changes in rCH. Changes in the bending angle RHCH can be represented safely by the harmonic approximation. These results suggest that there is no substantial difference between the form of the vibrational normal modes in the two electronic states. The more or less parallel course of the potential surfaces in the area close to the D2d nuclear conformation should yield similar vibrational frequencies for the totally symmetric normal modes of the N and T states. The situation is quite different for the planar ethylene molecule, whose corresponding potential surfaces are displayed in Figures 4-6. For this geometry both states are energetically well separated and their energy surfaces exhibit greater differences in their shape. Torsional Motion. Calculations are performed for various CC distances between 1.1 and 1.7 Å and torsional angles φ from 0.2° to 89.8°. The remaining internal coordinates rCH and RHCH are held fixed in all calculations at 1.08 Å and 120°. The sets of reference configurations and energy selection thresholds are identical to those employed in the previous calculations. The numbers of selected configurations vary for the singlet ground state (triplet state) between 13 000 (16 000) and 9000 (16 000) with respect to twisting angles of 0.2° and 89.8°. In what follows only calculations with RCC ) 1.321 and 1.48 Å are taken into closer inspection because these distances cor-
Figure 4. Calculated two-dimensional rCH-RHCH potential energy surfaces for the T and N states (φ ) 0.2°).
respond to the equilibrium distance of the T state (1.48 Å) with perpendicular CH2 groups and of the N state (1.321 Å) with planar CH2 groups. All other calculations serve primarily to further characterize the potential surfaces. The SCF-MO energies calculated at 1.48 Å are represented in Figure 7 in terms of the irreducible representations of D2 as a function of the torsion angle. As one would expect, orbitals containing valence px and py contributions exhibit a distinct change in energy upon torsion. No significant change in energy is observed for σ-type valence orbitals (oriented along the CC axis) or for Rydberg orbitals. Figures 8 and 9 represent the CI results for 1 1A, 2 1A, and 1 3B1 at RCC ) 1.321 Å and RCC ) 1.48 Å. Here the potential energy curves are given as a function of the torsion angle together with the squared coefficients of the most important configurations. Both calculations yield similar results, but some differences are apparent upon closer inspection. First, the competition of the π2 and π2 f π*2 configuration during torsion is somewhat different. At larger bond separation the influence of the π2 f π*2 configuration in the lowest state increases faster with torsion than at the smaller CC separation. In the limit of perpendicular CH2 groups both
19260 J. Phys. Chem., Vol. 100, No. 50, 1996
Figure 5. Calculated two-dimensional RHCH-RCC potential energy surfaces for the T and N states (φ ) 0.2°).
Gemein and Peyerimhoff
Figure 7. Calculated SCF-MO energies at RCC ) 1.48 Å for the MOs in the four different irreducible representations as a function of the torsional angle φ. The dominant character of the MOs (x+x for 2px + 2px, and 3px for the first Rydberg px function, etc.) is indicated.
Figure 6. Calculated two-dimensional rCH-RCC potential energy surfaces for the T and N states (φ ) 0.2°).
singlet wave functions contain an equal, bond length independent mixing of the two configurations. Secondly, for small twisting angles φ, the first excited 1A1 state is of Rydberg character (π f 3px); the torsional angle that characterizes the change from Rydberg to valence-type π2 f π*2 character is about 10° less (at 45°) for the larger CC bond separation. As a third point, one observes that the torsional potential curves for the T and N states are more shallow at the larger CC separation and their energy separation at 0° is smaller. Finally, at a torsional angle of 89.8° the T state is found slightly lower in energy than the N state in both calculations. A comparison of calculated energies is given in Table 1. The computed vertical singlet-triplet transition energy of 37 000 cm-1 is in line with results of earlier calculations23 and with the measured energy splitting. Likewise the rotational barrier of the ground state of 26 800 cm-1 (1.321 Å) and 20 000 cm-1 (1.48 Å) or 22 400 cm-1 if taken at the optimal CC
Figure 8. Calculated energy (est. full-CI) for the 11A, 21A, and 13B1 states (RCC ) 1.321 Å) as a function of the torsional angle φ and the weight of various configurations (measured by the square of their coefficient) in the corresponding wave functions.
distance at the planar and perpendicular geometry compares very well with the generally accepted experimental value of 20 900 cm-1 for this barrier.4,25-27 The triplet is found slightly below the singlet at the perpendicular geometry of CH2 groups which has also been observed in earlier calculations.1,28,29 For completeness the calculated potential energy surfaces of the lowest 1A and 3B1 states are represented in Figure 10 as a function of CC bond length and torsional angle φ.
Radiationless Transitions in Ethylene
J. Phys. Chem., Vol. 100, No. 50, 1996 19261
Figure 10. Calculated two-dimensional potential energy surfaces for the lowest two 1A and 3B1 states as a function of CC stretch and torsion (rCH ) 1.08 Å, RHCH ) 120°).
TABLE 2: Calculated Spin-Orbit Matrix Elements (in cm-1) between the Electronic Wave Functions of the 1A and 3B States as a Function of the Torsional Angle φ (CC Kept 1 Fixed at 1.48 Å)
Figure 9. Calculated energy (est. full-CI) for the 11A, 21A, and 13B1 states (RCC ) 1.48 Å) as a function of the torsional angle φ and the weight of various configurations (measured by the square of their coefficient) in the corresponding wave functions.
TABLE 1: Estimated Full-CI Energies Calculated for Different RCC Values in Planar and Perpendicular Geometrya Emin Emin ∆E(89.8°-0.2°) ∆E(0.2°-89.8°) ∆E(0.2°) ∆E(89.8°) ∆E(min)
state
1.321 Å
1.48 Å
1 1A (0.2°) 1 3B1 (89.8°) 1 1A 1 3B1 1 3B1-1 1A 1 3B1-1 1A 1 3B1-1 1A
-78.347 128 -78.226 464 26 877 10 586 37 175 -288 26 588
-78.336 110 -78.245 527 20 013 6041 25 922 -133 19 880
aE min values (in au) refer to minima of potential energy near planar (1 1A) or perpendicular (1 3B1) geometry. For spectroscopic considerations the differences in energy ∆E are given in cm-1. The difference between the maximum and minimum of each state is listed in rows 3 and 4. The vertical differences in energy between the T and N state for planar and perpendicular geometry appear in rows 5 and 6. The negative sign indicates the triplet with perpendicular CH2 groups to be lower in energy than the singlet.
IV. Spin-Orbit Coupling between 1A and 3B1 States One of the first theoretical works29 that treated spin-orbit coupling in biradicals (e.g. electronic excited ethylene) was published by Salem and Rowland in 1972. Considering spinorbit coupling between the ground state and lowest lying triplet state, they pointed out three important aspects. First, the magnitude of spin-orbit coupling (in the case of homosymmetric biradicals) should be proportional to the ionic character of the singlet wave function since the lowest triplet state is purely covalent. Secondly, due to the angular momentum part of the spin-orbit Hamiltonian, the relative orientation in space of the singly occupied atomic orbitals is important. And finally, an increase in the interatomic distance or a decrease of orbital
φ (deg)
〈11A|HSO|13B1〉
〈21A|HSO|13B1〉
2 20 40 50 60 70 80 88
0.092 0.865 1.730 1.890 1.904 1.696 1.147 0.295
0.150 1.383 3.308 4.754 5.290 5.517 5.829 5.983
overlap should lead to a strong decrease in the one-electron twocenter spin-orbit integrals. Keeping these aspects in mind, spin-orbit matrix elements were calculated as a function of the CH2 torsional angle (CC bond fixed at 1.48 Å) since this motion changes the orientation of the orbitals in space as well as the character of the singlet wave function. The third aspect, pointed out by Salem and Rowland, is studied by employing different combinations of internal coordinates (RCC, rCH, RHCH). In these calculations the torsional angle φ is fixed at 50° since the maximum of the 1A3B spin-orbit coupling is found between 50° and 60°. 1 The calculated data for the pure torsional motion are given in Table 2. The contribution of the various configurations to this matrix element is displayed in Figure 11. It is seen that the total coupling matrix element (ME) between the N and T states exhibits a sinusoidal shape as a function of the torsional angle φ. It vanishes at φ ) 0° (planar D2h) and φ ) 90° (perpendicular D2d) geometry because of symmetry. The maximum is around 55° with a magnitude of 1.9 cm-1. The data are similar to those obtained from CAS-MCSCF calculations by Caldwell et al.;30 they obtained a maximum at the torsion angle φ ) 50°, with a value of 0.6 cm-1, which is only about one-third of the matrix element computed here. This fact is not necessarily alarming, since the earlier calculation employs only a 2-in-2 CAS and, more importantly, only a 3-21G basis set. It is well-known that such small basis sets do not give reliable values for the absolute size of the splitting.31 In addition it is likely that the spin-orbit operator (not given explicitly) in the previous work is different from the form employed in the present work, in particular since a footnote30 mentions three
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Figure 11. Decomposition of the spin-orbit matrix elements for the twisting motion at RCC ) 1.48 Å into contributions from the dominant configurations of the 1A CI expansion. The 3B1 is characterized by ππ* throughout.
components while in the present treatment only one component (T01, of the same symmetry as the B1 state) contributes to the singlet-triplet interaction. The qualitative behavior of the matrix element during torsion can be explained by looking at the composition of the 1A1 wave function. The T state is described solely by the 3ππ* configuration at all angles φ, so that all differences in the spin-orbit matrix elements during torsion must be traced to the singlet wave function, whose main components are the π2 and π*2 configurations plus a (y+y) f π* contribution corresponding to a single excitation. The 1A-3B1 spin-orbit couplings between π2 and π*2 configurations dominant in the singlet wave function with the leading configuration ππ* of the triplet state increase in magnitude when the CH2 groups are twisted away from the planar geometry. For the π2 configuration this holds up to a torsional angle of φ ≈ 70° as seen from Figure 11, but for larger torsional angles the expansion coefficient of this configuration in the CI wave function decreases markedly (Figures 8, 9). The π*2 configuration enters into the ground state wave function expansion with a negative sign, and therefore the spin-orbit contribution to the total 1A-3B1 matrix element is also negative, as seen in Figure 11. In the limit of φ ) 90° the contribution of the π2 and the π*2 configuration to the ME cancel so that the resulting ME is zero. In the range between φ ) 40° and φ ) 70° a spin-orbit contribution of a third configuration (y+y) f π* is not negligible (0.4 cm-1), even though the mixing of this configuration into the ground state wave function (measured by its expansion coefficient) is very small. The spin-orbit matrix element between the V (2 1A) and T state is about 3 times larger than the N-T matrix element in the region for large torsional angles (φ > 70°). An analysis similar to that given above shows a fairly complex situation. First, the number of important configurations characterizing the V state increases to five. Secondly, each matrix element contributes between 0 and 2 cm-1, with a positive sign. As a consequence, in the limit of perpendicular CH2 groups, the total matrix element is nonzero. It is also seen that the contribution of the π f 3py configuration becomes small at large torsional angles and that the contribution of the closed-shell π2 configuration is proportional to the expansion coefficient in the CI wave function. As stated in the beginning of this section spin-orbit coupling in principle must be regarded as a function of all internal coordinates. During vibrations these coordinates will change, and consequently the coupling matrix element will also be affected. Since besides the torsional motion the totally symmetric normal modes are of primary interest, elongations of the CC and CH bonds are considered as well as a change of the
Figure 12. Calculated 〈11A|HSO|13B1〉 matrix elements as a function of RCC and φCH2 (rCH ) 1.08 Å).
Figure 13. Calculated 〈11A|HSO|13B1〉 matrix elements as a function of RCC and RHCH (rCH ) 1.08 Å).
bending angle RHCH. The torsional angle φ is fixed at 50° in these investigations because the maximum of spin-orbit coupling is located nearby. The results are displayed in Figures 12 and 13. Bond stretching yields smooth variations for the matrix element. As expected, the matrix element decreases with CC and CH (not shown) elongation mainly due to decreasing overlap. A different situation occurs by varying the bending angle RHCH. The shape of the matrix element curve now resembles to a first approximation a slightly distorted parabola. The minimum is centered near 120°, which corresponds to the equilibrium value of RHCH. As can be seen from Figure 14 the shape of the total matrix element ME is mostly determined by 〈π2|HSO|ππ*〉. Furthermore, the decreasing contribution due to 〈π*2|HSO|ππ*〉 is partially cancelled by 〈(y+y)π*|HSO|ππ*〉. If these results are taken into account in a vibrational averaging, it seems justified to consider only the dependence of 〈HSO〉 with respect to torsional motion. The remaining internal coordinates are fixed with respect to the equilibrium values of the T state (RCC ) 1.48 Å, rCH ) 1.08 Å, RHCH ) 120°). In this connection it is interesting that the external heavy atom effect, i.e. X+C2H4 (X ) F-, Cl-, etc.) on the enhanced radiative singlet-triplet
Radiationless Transitions in Ethylene
J. Phys. Chem., Vol. 100, No. 50, 1996 19263 TABLE 3: Calculated Energy Surface Data for the N and T States Obtained from the Three-Dimensional Fitting Procedure at φ ) 89.8° RCC (Å) rCH (Å) RHCH (deg) kRR (mdyn/Å) krr (mdyn/Å) kRR (dyn.Å/rad2) kRR (mdyn/Å) kRR (mdyn/rad) krR (mdyn/rad) ν˜ 1 (cm-1) ν˜ 2 (cm-1) ν˜ 3 (cm-1)
Figure 14. Decomposition of the spin-orbit matrix elements for CC and CH stretch and HCH bending into contributions from the dominant configurations of the 1A CI expansion. The 3B1 is characterized by ππ* throughout.
transitions can be interpreted as an increased (S-T) spin-orbit coupling due to back-charge-transfer from the heavy atom.32 V. Radiationless Transitions From the results in the previous section it appears sufficient to consider the variation in the spin-orbit coupling between the N and T states only as a function of the torsional ν4 mode with a fixed CC and CH framework because the major change in the coupling appears in this mode. The coupling matrix element can thus be written in this approximation as
VNT ) 〈χN(Q1)|χT(Q1)〉〈χN(Q2)|χT(Q2)〉〈χN(Q3)|χT(Q3)〉 × r 4)|HSO|φTel(b,Q r 4)〉el|χT(Q4)〉 (8) 〈χN(Q4)|〈φNel(b,Q so that the totally symmetric modes enter in the final evaluation of VNT simply as Franck-Condon factors. Since the sections of the potential surfaces are computed along the internal coordinates RCC ≡ R, RCH ≡ r, and Rˆ ≡ 1/2∠(HCH), the normal coordinates Q1, Q2, and Q3 have to be expressed in these terms. The three-dimensional potential surface computed at the angle kept fixed at φ ) 89.8°, because the interaction of states is most important for twisted CH2 groups, is fitted for both N and T states whereby the standard deviation is 0.000 19 au for the N and 0.000 21 for the T state. The form of the normal vibrational modes and the vibrational frequencies are obtained in the framework of the Wilsons FG formalism.33 The first vibrational mode represents mainly the CH stretching, coupled with some CC stretch. In the second, the CC stretching is dominant, but appreciable coupling with HCH bending is seen, while the third mode possesses primarily bending character and mixes the least with the other two displacements. The situation is similar in both electronic states. The fitting procedure (in this restricted optimization) leads to very nearly identical structural parameters (geometries and force constants) for both the N and T states at perpendicular CH2 geometry; both minima are separated by only 70 cm-1, the T state being the lower of the two. This is in contrast to planar ethylene, for which the optimal T and N state geometries are quite different. Corresponding data are also computed from the potential surfaces at φ ) 0.2° employing an equivalent grid of energy points. In this fitting procedure based on a parabola the calculated frequencies depend somewhat on the section of the surface employed in the fitting procedure. If only CC stretching potential points between 1.37 and 1.58 Å are used,
N state
T state
1.484 1.104 117.8 134.2 24.2 6.59 0.996 -1.79 0.577 3258 1857 1414
1.484 1.103 118.0 13.8 24.7 5.88 0.545 -2.08 0.750 3298 1878 1451
the parabola becomes steeper (enhancing the inner part of the potential); if the values for the larger CC separation are also included, a mode ν2 frequency of 1857 cm-1 results for such calculations (Table 3). The CH stretching and HCH bending potentials possess quasiharmonic behavior and therefore show little sensitivity to the special choice of coordinates. This could also be concluded from Figures 1-3. A summary of experimental data34,35 and data from other theoretical works36,15 for the N and T states is given in Table 4. The symmetry of states is given in parentheses. The frequencies computed in the present work do not differ significantly from previous theoretical data; in comparison with experiment the ground state frequencies ν˜ 1 to ν˜ 3 are overestimated by about 100-400 cm-1. In this comparison one should keep in mind that only nearly planar or perpendicular geometries were used and that possible uncertainties in the potential surface calculation employing the theoretical treatment discussed can easily amount to 250 cm-1. The torsional frequency, also included in Table 4 for comparison, is seen to depend on the CC bond lengths. Shortening of the CC bond in the N state from 1.484 Å (D2d structure) to 1.328 Å leads to an increase of 110 cm-1 in ν4 resulting in 942 cm-1. Such behavior can directly be seen from the torsional potentials in Figures 8 and 9. The torsional mode is the most important mode in the present context; it is the only mode of au symmetry and therefore represents a normal mode by itself. In contrast to the modes described above, the torsional motion is determined by a nonharmonic periodic potential with a finite barrier height. The minima of the potential energy are located at 0° (N) and 90° (T state). Since rotation of one CH2 group by 180° leaves the molecule unchanged, generally the torsional wave functions are periodic in 2π or 4π, respectively; therefore, these torsional wave functions exist with symmetric or antisymmetric behavior. wave functions are classified in accordance with the double group G16(2) of the permutation inversion group of Longuet-Higgins.37,38 The Hamiltonian for the torsional motion is
H)-
1 ∂2 + Eel(φ) 2Iφ ∂φ2
(9)
where Eel(φ) is the estimated full-CI energy of the N or T state. The reduced moment of inertia Iφ is given by
Iφ )
I1‚I2 I1 + I2
(10)
while Ii is the inertial moment with respect to one CH2 group. As pointed out by Koehler and Dennison,39 the torsional wave functions eiµφ are characterized by the quantum number
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TABLE 4: Comparison of Vibrational Frequencies and Potential Surface Data Obtained from Various Methods mode ν1 (cm-1) ν2 (cm-1) ν3 (cm-1) ν4 (cm-1) RCC (Å) rCH (Å) RHCH (deg) E (au)
exp34,35 N (D2h) 3026 1630d 1344d 1027d
this work N (D2h) 3004 1790 1406 942a
1.334 1.081 117.37
T (D2h) 2898 2037 1448 851a
N (D2d) 3258 1857 1414 832b
ref 36 T (D2d)
N (D2h)
3298 1878 1451 672b
3413 1742 1433 903
Geometries Employed 1.484c 1.484c 1.104c 1.102c 117c 118c
1.328c 1.10c 117c
1.56c 1.09c 121c
-78.3452c
Calculated Energies for Torsional Minimum -78.2196c -78.2451c -78.2454c
ref 15
T (D2h) 3407 1511 1324 593
1.33 1.09 123
N (D2h) 3320 1830 1480 1135
T (D2d) 3288 1991 1167 638
1.321 1.076 121.7
1.470 1.076 121.1
-78.0432
-77.9751
Calculated with RCC ) 1.321 Å, ν4 computed as difference between V ) 1 and V ) 0. b Calculated with RCC ) 1.48 Å, ν4 computed as difference between V ) 1 and V ) 0. c Obtained by three-dimensional fit in harmonic approximation. d Reference 4. a
TABLE 5: Correlation between the Symmetry Groups, Symmetry Classification of States, the Lz Component Occurring in HSO and of Basis Functions Employed G16(2)
D2h
D2
D2d
basis
+
Ag B1u Ag Au B1g Au
A B1 A A B1 A
A1 A2 B1 A1 A2 B1
cos 2mφ
A1g A2u+ B1g+ A1uA2gB1u-
Figure 15. Vibrational energies and symmetry-adapted vibrational wave functions for torsional motion of the T and N states. Solid lines correspond to vibrational wave functions with cos 2m or sin 2m, dashed lines to ungerade arguments.
I1 µ)m-k I1 + I2
(11)
where m is an integer and k is the (integral) overall rotational quantum number. In the case of C2H4, µ is an integer or halfinteger and the periodicity of the torsional functions is 2π or 4π, respectively. The molecule has been considered in its overall rotational ground state (k ) 0). So µ equals m and the functions are periodic in 2π. The vibrational wave functions are constructed as symmetry-adapted functions of gerade (2m) or ungerade (2m+1) parity in the basis of cos 2mφ or sin 2mφ and cos(2m+1)φ or sin(2m+1)φ functions. Symmetries of states, wave functions, and the Lz component of the angular momentum operator are given in Table 5. The estimated full-CI energies as a function of the torsional angle for fixed values of RCC ) 1.48 Å, rCH ) 1.08 Å, and RHCH ) 120° serve as the potential for the torsional motion. These curves are fitted in the cos 2nφ series (2n ) 0 to 2n ) 10) to match the torsional function basis. To guarantee convergence for the high-lying vibrational levels, terms up to n ) 75 are included in the wave function expansion. Figure 15 gives a graphical representation of the N and T state potentials and the corresponding vibrational functions and eigenvalues. Wave functions of “g” parity are drawn by solid lines, while those of “u” character are indicated by dashed lines. Levels V′′ >24 are formally obtained from the secular equations but are not shown since they correspond to a free rotor above the ground state potential. For the ground state each pair of torsional gerade and ungerade vibrational levels are accidentally degenerate up to
cos(2m+1)φ sin(2m+1)φ
state T N Lz
sin 2mφ
an energy of about 16 000 cm-1. All functions corresponding to even quantum numbers V are characterized by a cosine series of gerade or ungerade parity (formally a1g+ and b1g+ symmetry), while the corresponding sine series (formally a1u- and b1u- symmetry) correspond to odd quantum numbers V. The degeneracy disappears around V ) 20. Low-lying T state vibrational levels are also nearly degenerate (up to V ) 8). In this case the degenerate components are series of sine and cosine functions (b1u-, b1g+ and a1u-, a1g+), and the parity of these functions alternates from sine to cosine and from even to odd quantum numbers. This behavior is explained by the different location of the potential energy minima. At φ ) 0° a ground state vibrational wave function must possess either zero or extreme values. As a consequence, only a sine expansion in the first case or a cosine expansion in the second is possible. For the excited state with its minimum at φ ) 90° the vibrational wave functions can be expanded both in a series of cosine and a series of sine functions (with arguments of opposite parity). Vibrational wave functions that contribute to the spin-orbit matrix elements
∫∫χV′φe′HSOφe′′χV′′ dτedτV
(12)
are fixed by symmetry considerations; combinations of cos 2mφ and sin 2mφ or cos(2m+1) and sin(2m+1) wave functions (Table 5) are only able to give nonzero elements. For the actual calculation of the total matrix element the electronic part between the N and T state is (Figure 11) expanded in a sin 2mφ series up to order 2m ) 12. The coupling matrix elements involving the vibrational quantum numbers between V′ ) 0-3 of the triplet state and V′′ ) 0-34 of the ground state were calculated. Because of symmetry considerations and the degeneracy the combination of V′ ) 0 and V′′ ) 0 is based on sinu - cosu type wave functions in the triplet levels. The matrix elements that connect V′ ) 1 and V′′ ) 1 are given by contributions due to cosg - sing wave functions since for V′ ) 0 sinu and cosg type functions are furthermore degenerate. The calculated squares of the total coupling matrix elements are displayed in Figure 16. A large interaction zone is seen as a result of the displaced potential
Radiationless Transitions in Ethylene
J. Phys. Chem., Vol. 100, No. 50, 1996 19265
Figure 16. Calculated squared coupling matrix elements between the lowest four torsional levels of the T state (V′) and all N state torsional levels (V′′).
TABLE 6: Franck-Condon Factors for the Coupling of the T and N State in the Three Symmetric Modes (W′ Refers to T, W′′ to N State) mode 1 mode 2 mode 3
V′ ) 0 V′ ) 1 V′ ) 2 V′ ) 0 V′ ) 1 V′ ) 2 V′ ) 0 V′ ) 1 V′ ) 2
V′′ ) 0
V′′ ) 1
V′′ ) 2
0.999 739 0.000 251 0.000 010 0.999 907 0.000 014 0.000 080 0.999 667 0.000 299 0.000 034
0.000 253 0.999 218 0.000 500 0.000 014 0.999 720 0.000 027 0.000 304 0.999 000 0.000 592
0.0 0.000 508 0.998 679 0.000 079 0.000 028 0.999 374 0.000 030 0.000 612 0.998 271
energy minima. The maximum of the interaction that occurs for V′ ) 0 with V′′ ) 24 is shifted to lower lying V′′ levels for increasing V′. In case of V′ ) 3 the largest matrix element arises from a combination of V′ ) 3 with V′′ ) 16. In addition to the calculated vibronic coupling due to torsion the effect of the totally symmetric normal modes has to be considered also. In agreement with the normal mode formalism, the potentials were expanded up to second order in each normal coordinate. Since there is a small but distinct displacement of the origin of the oscillators in the N and T states, one should take this shift into account. This has been done by a linear transformation. The Franck-Condon factors (FCF) are close to unity for ∆(V′,V′′) ) 0 and very small otherwise (Table 6). The transition rate for the intersystem crossing is governed by three major quantities: first, the vibronic coupling matrix element in the torsional mode between the T and N state including spin-orbit interaction as a driving factor; secondly, the influence of the normal modes that were neglected in first order (here the totally symmetric ones), which is introduced via FCF; and finally, the density of final singlet vibrational states, which are isoenergetic to the initial triplet states. Different methods of calculating the density of states such as Laplacian transformation of partition functions,40 saddle point
method,41 or direct counting algorithms42 have been proposed. In this work a modified direct counting algorithm is used. The total vibrational energy contains first the levels of the totally symmetric modes 3
( )
E ) ∑ Vi + i)1
1
2
pωi
(13)
then the energy levels resulting from torsional motion, and finally the zero-point energies of the remaining modes ν5 to ν12. The program generates the sum of vibrational energies for different sets of quantum numbers Vi < 10 (i ) 1-3), keeping the quantum numbers V ) 0 for modes ν5 to ν12. The FCF between levels higher than 10 of the N state and corresponding low-lying triplet levels are very small so that levels of the totally symmetric modes higher than Vi ) 10 can be neglected. The density of final singlet state levels is, in principle, given by
F(EN) ) n/∆E
(14)
i.e. the number of generated levels within an energy range ∆E. Between 30 000 and 32 000 cm-1 the mean density of singlet state levels is calculated to be F(EN) ) 0.2781/cm-1. This value is used in the following calculations. If only three levels of each totally symmetric mode are considered, except all torsional levels and the zero-point levels of the remaining modes, the density of states reduces to 0.033/cm-1 for vibrational energies between 30 000 and 32 000 cm-1. Although the potential surfaces for the N and T states intersect in the calculations, there is no real degeneracy between the torsional levels of the N and T states. Therefore during a radiationless transition, energy is expected to be transferred to the totally symmetric modes, which act as “acceptor modes”. Since there is no significant geometrical displacement for these normal mode oscillators in the N and T states, the transition rate will decrease drastically for transitions involving different
19266 J. Phys. Chem., Vol. 100, No. 50, 1996
Gemein and Peyerimhoff
TABLE 7: Radiationless Transition Rates KTfN for Pure Torsional Transitions (Energies in cm-1) E(V′)
V1′-V3′
V4′
E(V′′)
V1′′-V3′′
V4′′
K (s-1)
30 806
0
0
29 786 30 450 31 126 28 430 31 211 31 859 26 259 31 859 23 977 32 578 21 635
0 0 0 0 0 0 0 0 0 0 0
24 25 26 22 27 28 19 28 16 29 13
7.7 × 1010 6.5 × 1010 4.5 × 1010 1.8 × 1011 6.9 × 109 1.6 × 1010 2.1 × 1011 1.7 × 1010 2.2 × 1011 1.3 × 1010 2.3 × 1011
31 478
0
1
32 087
0
2
32 647
0
3
33 170
0
4
totally symmetric normal mode quanta compared to hypothetical pure torsional transitions. Calculated transitions rates due to pure torsional transitions are given in Table 7 together with the total vibrational energy E(V) (in cm-1). In this case the first row for each different V4′ value lists the transitions from the V4′ triplet to V′′ singlet levels with the largest coupling matrix elements (Figure 16). From Figure 15 it is seen that these transitions can be considered as “vertical” with the largest Franck-Condon overlap. The other rows show the rates to the ground state torsional levels which are closest in energy to the respective T state level; these are levels above the N state barrier (Figure 15). The transition rate to these is only slightly reduced compared to the maximum. All these hypothetical pure torsional transitions would possess a rate of approximately 1011 s-1. The condition of (near) isoenergetic levels for which a realistic transition is most likely requires additional acceptor modes involving excitation of the totally symmetric modes. Since the FCF will become unfavorable, the rates are decreased by orders of magnitude. The results obtained are presented in Table 8 up to the vibrational energy of |V1′V2′V3′V4′〉 ) |1000〉. In these calculations singlet levels lying within (600 cm-1 (taking also into account possible errors in the calculation of the vibrational level position) relative to triplet levels are considered. Furthermore the combinations of initial and final levels are restricted by the condition ∆Vi e (2 since larger differences in vibrational quanta yield very small FranckCondon factors and therefore unlikely transitions. The same holds for transitions with two active acceptor modes that differ in one or more vibrational quanta. The list of data shows that the transition probabilities are all on the order of 106-108 s-1. The assumption of Salem and Rowland,29 who estimated the rate of intersystem crossing T f N for biradicals on the order of 108-106 s-1, is thus confirmed by the present ab initio work. The lifetime of a level decaying by a radiationless transition due to intersystem crossing decreases considerably with the actual energetic mismatch between the T and N energy levels. Since the calculations do not yield accuracies of a few wavenumbers, no attempt was made for a quantitative calculation of the lifetime of individual T state levels. The transition probabilities of Table 8 to levels within (600 cm-1 of an upper state triplet level suggest lifetimes on the order of 10-6-10-8 s. In comparison to the calculated43 phosphorescence lifetime of (2-5) × 10-3 s the nonradiative transition is an effective relaxation channel as long as only a few quanta of the acceptor modes are active. VI. Summary and Conclusions The purpose of the present work was to calculate the intersystem crossing rate between the lowest excited triplet state and the ground state of the ethylene molecule, considering the spin-orbit interaction as a driving factor in this process.
TABLE 8: Calculated Rates KTfN for Radiationless Transitions from Distinct Upper Triplet Levels E(W′) into Lower Singlet Levels Spaced about (600 cm-1 around the Upper Level E(V′)
V1′ V2′ V3′ V4′
30 806
0
0
0
0
31 478
0
0
0
1
32 087
0
0
0
2
32 257
0
0
1
0
32 647
0
0
0
3
32 684
0
1
0
0
32 930
0
0
1
1
33 170 33 356
0 0
0 1
0 0
4 1
33 539
0
0
1
2
33 667 33 709
0 0
0 0
0 2
5 0
33 966
0
1
0
2
34 099
0
0
1
3
34 104
1
0
0
0
E(V′′) 30 257 30 288 30 531 30 714 31 258 30 257 30 975 30 981 31 200 31 258 31 439 31 644 31 688 32375 31 688 31 945 32 145 32 375 31 945 32 396 32 375 32 388 32 375 32 388 32 832 33 045 33 297 32 388 32 832 32 839 32 615 32 672 33 058 33 102 33 045 33 058 33 297 33 546 33 102 33 789 34 459 33 045 33 359 33 803 33 810 34 029 33 058 33 502 34 233 33 789 34 459 33 789 34 233 33 240 34 459
V1′′ V2′′ V3′′ V4′′ 1 0 0 0 0 1 0 1 0 0 0 0 1 1 1 0 0 1 0 1 1 0 1 0 0 1 0 0 0 1 0 0 0 1 1 0 0 1 1 1 1 1 0 0 1 0 0 0 1 1 1 1 1 2 1
0 1 0 2 0 0 1 0 0 0 2 1 0 0 0 0 2 0 0 0 0 1 0 1 2 0 3 1 2 1 0 0 1 0 0 1 3 1 0 0 0 0 0 1 0 0 1 2 1 0 0 0 1 0 0
0 0 1 0 2 0 0 0 1 2 0 0 0 0 0 2 0 0 2 1 0 1 0 1 0 0 0 1 0 0 2 3 1 1 0 1 0 0 1 1 1 0 3 2 2 3 1 0 0 1 1 1 0 0 1
20 22 23 20 22 20 23 21 24 22 21 24 22 23 22 23 22 23 23 21 23 23 23 23 23 24 21 23 23 21 24 22 24 22 24 24 21 22 22 23 24 24 23 23 21 24 24 24 23 23 24 23 23 21 24
K (s-1) 8.5 × 106 9.2 × 105 2.3 × 107 2.8 × 106 9.8 × 106 3.8 × 107 2.2 × 106 4.4 × 107 3.3 × 107 5.4 × 106 1.3 × 107 1.5 × 106 4.5 × 107 4.0 × 107 2.3 × 107 1.2 × 107 1.3 × 106 1.0 × 107 4.6 × 107 1.2 × 107 2.2 × 107 1.0 × 106 7.4 × 106 1.3 × 102 2.4 × 106 1.6 × 107 1.2 × 107 2.3 × 107 2.1 × 106 1.2 × 107 6.6 × 107 1.6 × 107 1.5 × 106 4.5 × 107 1.4 × 107 3.3 × 107 4.1 × 107 4.5 × 107 2.3 × 107 1.0 × 107 1.9 × 106 3.0 × 106 7.0 × 107 1.0 × 106 1.2 × 107 7.1 × 107 2.3 × 106 2.1 × 105 1.0 × 107 7.9 × 106 1.6 × 107 1.5 × 107 1.0 × 106 2.5 × 107 2.3 × 107
The potential hypersurfaces for both electronic states were calculated in the Born-Oppenheimer approximation employing multireference CI calculations. For the description of the totally symmetric motions the CC and CH bond lengths and the HCH bending angle are varied simultaneously. The potential energy surfaces with respect to torsional motion are calculated for different CC bond lengths (1.2-1.8 Å). The number of selected configuration state functions is on the order of 13 000-18 000, while the sum of the squared coefficients of the reference configurations is 0.91 and higher for both singlet and triplet state. Spin-orbit coupling matrix elements, which are calculated in the same MO basis, exhibit large variations under torsion as a consequence of changed px,y orbital orientation. The maximum is found around a torsional angle of 50°. Considerably smaller
Radiationless Transitions in Ethylene changes in the spin-orbit coupling are found for CC and CH bond stretching and HCH bending (here the torsional angle is fixed to 50°). The vibrational wave functions, which are necessary for the calculation of the rates for radiationless transitions, are expressed in normal coordinates. For this reason the corresponding potential energy hypersurfaces for the symmetric vibrational modes are approximated in a polynomial series up to second order in each internal coordinate (harmonic approximation); the internal coordinates are transformed to the appropriate normal coordinates in a subsequent step. Since the unitary transformations for the sets of internal coordinates are only slightly different for the N and T state, only one transformation is chosen to calculate the displacement between the normal mode oscillators of the N and T state. The total interaction matrix element that couples the N and T states in first order and induces a radiationless transition involves three Franck-Condon factors for the modes ν1 to ν3 and one vibrationally averaged (ν4) electronic spin-orbit coupling matrix element. Here the totally symmetric normal modes serve as energy acceptor modes for the radiationless transition between isoenergetic levels. The geometrical displacement of the totally symmetric normal mode oscillators is very small and therefore the rate constants will decrease in a predictable manner if the isoelectronic levels correspond to excitations in the acceptor modes. In contrast to hypothetical pure torsional transitions, in which the totally symmetric modes are restricted to the vibrational ground states, the calculated rate constants decrease from 2 × 1011 to a range between 106 and 5 × 107 s-1 if the vibrational quantum of one acceptor mode is changed by one or two. In this case the corresponding FranckCondon factors are rate determining. The intersystem crossing between the T and N states is, in principle, a process competitive with phosphorescence, but since the radiative lifetime of T state levels was calculated43 on the order of (2-5) × 10-3 s, the radiationless transitions are more effective for the intramolecular energy transfer, at least as long as only few levels of the totally symmetric modes are excited. Acknowledgment. The authors wish to thank Dr. Miljenko Peric´ for a great number of very stimulating discussions and his help in the course of this work. B.G. wants to thank C. M. Marian and B. A. Hess for the introduction to the spin-orbit programs of this laboratory. The services and computer time made available by the University of Bonn Computer Center have been essential for the present study and are gratefully acknowledged. This work was financially supported by the Deutsche Forschungsgemeinschaft (Sonderforschungsbereich 334). References and Notes (1) See for example: Michl, J.; Bonacic-Koutecky, V. Electronic Aspects of Organic Photochemistry; John Wiley & Sons Inc.: New York, 1990, which includes many references. (2) Turro, N. J. Modern Molecular Spectroscopy; Benjamin: Menlo Park, NJ, 1978. (3) Avouris, Ph.; Gelbhart, W. M.; El-Sayed, M. A. Chem. ReV. 1977, 77, 793. (4) Orlandi, G.; Zerbetto, F.; Zgierski, M. Z. Chem. ReV. 1991, 91, 867.
J. Phys. Chem., Vol. 100, No. 50, 1996 19267 (5) Quack, M., Kutzelnigg, W., Eds.; Ber. Bunsen-Ges. 1995, 99, special issue Molecular Spectroscopy and Molecular Dynamics: Theory and Experiment. (6) Dai, H.-L., Field, R. W., Ed.; Molecular Dynamics and Spectroscopy by stimulated Emission Pumping; Advanced Series in Physical Chemistry 4; World Scientific: Singapore, 1995. (7) Smith, P. G.; McDonald, J. D. J. Chem. Phys. 1990, 92, 1004. (8) A survey of modern methods is found in: Yarkony, D. R., Ed.; Modern Electronic Structure Theory; Advanced Series in Physical Chemistry 2; World Scientific: Singapore, 1995. (9) de Vivie, R.; Marian, C. M.; Peyerimhoff, S. D. Chem. Phys. 1987, 112, 349. (10) van Hemert, M. C.; Dohmann, H.; Peyerimhoff, S. D. Chem. Phys. 1986, 110, 55. (11) Buenker, R. J.; Peyerimhoff, S. D. Theor. Chim. Acta 1974, 35, 33. (12) Buenker, R. J.; Peyerimhoff, S. D. Theor. Chim. Acta 1975, 39, 217. (13) Buenker, R. J.; Peyerimhoff, S. D.; Butscher, W. Mol. Phys. 1978, 35, 771. (14) Langhoff, S. R.; Davidson, E. R. Int. J. Quantum Chem. 1974, 8, 61. (15) Siebrand, W.; Zerbetto, F.; Zgierski, M. Z. J. Chem. Phys. 1989, 91, 5926. (16) Huzinaga, S. J. Chem. Phys. 1965, 42, 1293. (17) Dunning, T. H., Jr. J. Chem. Phys. 1970, 53, 2823. (18) Poirier, R.; Kary, R.; Czismadia, I. G. Handbook of Gaussian Basis Sets; Elsevier: Amsterdam, 1985. (19) Cave, R. J. J. Chem. Phys. 1990, 92, 2450. (20) Mulliken, R. S. J. Chem. Phys. 1953, 23, 1997. (21) Huzinaga, S. J. Chem. Phys. 1962, 36, 453. (22) Buenker, R. J.; Peyerimhoff, S. D.; Kammer, W. E. J. Chem. Phys. 1971, 55, 814. (23) Buenker, R. J.; Peyerimhoff, S. D.; Hsu, H. L. Chem. Phys. Lett. 1971, 11, 65. (24) Herzberg, G. Molecular Spectroscopy and Molecular Structure; Infrared and Raman Spectroscopy, Vol. II); van Nostrand Reinhold: New York, 1945; p 326. (25) Douglas, J. E.; Rabinovitch, B. S.; Looney, F. S. J. Chem. Phys. 1955, 23, 315. (26) Rabinovitch, B. S.; Michel, K. W. J. Am. Chem. Soc. 1959, 81, 5065. (27) Benson, S. W.; O’Neal, H. E. Kinetic Data on Gas Phase Unimolecular Reactions; NSRDS-NBS 21; 1970. (28) Buenker, R. J.; Hsu, H. L.; Peyerimhoff, S. D. Chem. Phys. Lett. 1971, 11, 65. (29) Salem, L.; Rowland, C. Angew. Chem., Int. Ed. Engl. 1972, 11, 92. (30) Caldwell, R. A.; Carlacci, L.; Doubleday, C. E., Jr.; Furlani, T. R.; King, H. F.; McIver, J. W., Jr. J. Am. Chem. Soc. 1988, 110, 6901. (31) Hess, B. A.; Marian, C. M.; Peyerimhoff, S. D. In Modern Electronic Structure Theory; Yarkony, D. R., Ed.; World Scientific: Singapore, 1995; p 152. (32) Minaev, B. F.; Knuts, S.; A° gren, H. Chem. Phys. 1994, 15, 181. (33) Wilson, E. B.; Decius, J. C.; Cross, P. C. Molecular Vibrations; McGraw-Hill: New York, 1955. (34) Hehre, W. J.; Randon, L; von Rague Schleyer, P.; Pople, J. A. Ab initio Molecular Orbital Theory; Wiley: New York, 1986. (35) Herzberg, G. Molecular Spectroscopy and Molecular Structure; Electronic Spectra and Electronic Structure of Polyatomic Molecules, Vol. III; van Nostrand Reinhold: New York, 1966; p 626. (36) Klessinger, M.; Po¨tter, T.; van Wu¨llen, C. Theor. Chim. Acta 1991, 80, 1. (37) Merer, A. J.; Watson, J. K. G. J. Mol. Spectrosc. 1973, 47, 499. (38) Petrongolo, C.; Buenker, R. J.; Peyerimhoff, S. D. J. Chem. Phys. 1982, 76, 3655. (39) Koehler, J. S.; Dennison, D. N. Phys. ReV. 1940, 57, 1006. (40) Schlag, E. W.; Sandsmark, R. S.; Valance, W. G. J. Phys. Chem. 1965, 69, 1431. (41) Moore, M. R.; Ruijrok, T. W. J. Chem. Phys. 1979, 52, 113. (42) Stein, S. E.; Rabinovitch, B. S. J. Chem. Phys. 1972, 58, 2438. (43) McClure, D. S. J. Chem. Phys. 1949, 17, 905.
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