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J. Phys. Chem. C 2010, 114, 5312–5320
The Photoelectron Spectrum of Pyrrolide: Nonadiabatic Effects due to Conical Intersections† Xiaolei Zhu and David R. Yarkony* Department of Chemistry, Johns Hopkins UniVersity, Baltimore, Maryland 21218 ReceiVed: May 11, 2009; ReVised Manuscript ReceiVed: July 10, 2009
The negative ion photoelectron spectrum of the pyrrolide-h4 anion and the completely deuterated analogue, pyrrolide-d4, are computed employing the multimode vibronic coupling approach, based on a two-state, fully quadratic, quasi-diabatic Hamiltonian, Hd, which accurately represents the vicinity of the ab initio determined equilibrium geometry of the ground 2A2 state as well as the minimum energy crossing point (MECP) on the symmetry-allowed 2A2-2B1 accidental seam of conical intersection. The ab initio data are obtained from multireference configuration interaction wave functions based on triple-ζ quality atomic orbital bases. The determined photoelectron spectra, which are converged with respect to the vibronic basis, compare favorably with previous spectroscopic results, for pyrrolide-h4. The principal impact of the seam of conical intersection is a significant perturbation of the portion of the spectrum attributable to the 2B1 state, in agreement with previous analyses. However, despite the fact that the MECP is approximately 0.5 eV above the 2A2 minimum, its manifestations are shown to be evident in the photoelectron spectrum near threshold. By comparing the simulated and measured photoelectron spectrum, it is deduced that the electronic transition moment for the production of the 2A2 state of pyrrolyl from pyrrolide is approximately twice that for the production of the 2B1 state. I. Introduction Recently, there has been considerable interest1-7 in the azolyls, five-member carbon-nitrogen heterocycles of the form (CH)5-mNm, m ) 1-5. These planar five-member rings can be viewed as being derived from the cyclopentadienyl radical,8-10 (CH)5, in which one or more of the CH moieties is replaced by a nitrogen atom. The ground electronic state of the cyclopentadienyl radical is a 2E1′′ state,8 arising from five electrons in five π-orbitals. However, this high symmetry nuclear configuration is rendered unstable by the static Jahn-Teller interaction.11 The presence of nitrogen atoms in the heterocycles further complicates the electronic structure, as low energy excitations from the nitrogen lone pairs to the half filled or vacant π-orbitals are also possible. As a result, conical intersections of two2,4,5 and even three6 low-lying electronic states profoundly influence the electronic structure of these heterocycle radicals. In this work, we consider the photoelectron spectrum of pyrrolide (yielding the vibronic spectrum of pyrrolyl), which has been studied previously.2,5,12 Two electronic states must be considered, the ground 2A2 state and an excited 2B1 state. These states can be viewed as being derived from the 2E1′′ state of the cyclopentadienyl radical, as a single CH to N substitution lowers the point group symmetry from C5V to C2V. Previous electronic structure calculations on pyrrolyl have (approximately) located points on a low-lying seam of conical intersection of the 2A2 and 2B1 states.2,5 The existence of this seam of conical intersection reflects the symmetry-required seam in the cyclopentadienyl radical. The electronic structure of the azolyls has been studied both computationally4,5,7 and by photoelectron spectroscopy1-4,13 of the corresponding azolide, (CH)5-mNm-. Photoelectron detachment spectra have been reported and analyzed for m ) 1, †
Part of the “Barbara J. Garrison Festschrift”.
pyrrolide,2,5 m ) 2, imidazolide,4,14 and pyrazolide,4,7 and m ) 3, triazolide.3 The existence of low-lying conical intersections complicates the interpretation of these spectra using standard adiabatic techniques. The most successful analyses of the experimental photodetachment spectra have been reported for pyrrolide,5 imidazolide,4 and pyrazolide4,7 using the time independent, multimode vibronic coupling model, developed by Cederbaum, Domcke, and Ko¨ppel.15 Substituent substitution alters the topography near a conical intersection. Consider the 2A′ and 2A′′ states of the alkoxys,16-22 ethoxy and isopropoxy, derived from the 2E state of methoxy (CH3O) by replacing one or two hydrogens by methyls. Here, the Jahn-Teller stabilization energy is small and the relevant extrema can be found in close proximity to the conical intersection which is expected to impact both the 2A′ and 2A′′ states in an approximately equivalent manner. On the other hand, in pyrrolyl, the 2A2 minimum is much lower in energy than the 2 B1 extremum. The 2B1 extremum, a saddle point, is much closer to the minimum energy point of conical intersection than is the 2 A2 extremum, which is found in a different region of nuclear coordinate space.23 Thus, in pyrrolyl, the effects of the conical intersection on the 2A2 state and on the 2B1 state are expected to be quite different. Previously, we have shown how these geometric and energetic differences between the 2A2 and 2B1 extrema in pyrrolyl complicate the construction of the quasi-diabatic Hamiltonian Hd which describes the electronic structure aspects of the system and is used to determine the photoelectron spectrum. We then showed how a recently introduced pseudonormal equations approach can overcome those challenges.23 In this work, we focus on the implications for the photoelectron spectrum of these geometric and energetic differences. A recent experimental/theoretical treatment by Lineberger’s group,2 denoted GIHKBL below, illustrates some of the practical consequences of these differences. They observe that features
10.1021/jp904379q 2010 American Chemical Society Published on Web 10/20/2009
Photoelectron Spectrum of Pyrrolide attributable to both the 2A2 and 2B1 states are predicted by an adiabatic states analysis. However, only features attributable to the 2A2 state are observed. In the region where features attributable to the 2B1 state are expected, only a weak diffuse continuum is observed. It is the proximity of the 2B1 state extremum to the 2A2-2B1 conical intersection seam in pyrrolyl that leads to this diminution of the intensity of the 2B1 state in the photoelectron spectrum of pyrrolide. This was shown in a perturbative analysis by GIHKBL and in an analysis based on the multimode vibronic coupling approach in ref 5, denoted MLWD below. As noted by MLWD, a second factor may be relevant to the limited contribution from the 2B1 state to the spectrum, the strength of the bound-free electronic transition moments. This same issue has been raised in analyses of photodetachment spectra of CH3CC- (refs 24 and 25) and NO3-.26 In addition, the question of the impact of the 2A2-2B1 conical intersection seam on the 2A2 portion of the photoelectron spectrum has been largely ignored. To address these two issues in a quantitative manner, an accurate treatment of the nonadiabatic effects induced by the seam of conical intersection is required. To this end, the present work extends the previous work MLWD. Significant methodological advances in the implementation of the multimode vibronic coupling model, since the work of MLWD, in refs 27-29 enable us to provide the most in-depth treatment of the nonadiabatic effects in the photoelectron spectrum of pyrrolide reported to date, explicitly considering all 21 internal modes, and reporting coupled state vibronic wave functions converged with respect to all relevant active internal coordinates. Section II briefly reviews the methodology used to determine the photoelectron spectrum. Section III describes the electronic structure aspects of the treatment. Our determination and analysis of the minimum energy conical intersection point in pyrrolyl reported in this section addresses questions raised in a more approximate treatment of this issue.2 Section IV presents our determination and analysis of the pyrrolide-h4 spectrum. Also in that section, we report the photoelectron spectrum of the fully deuterated isotopologue pyrrolide-d4, which has yet to be measured experimentally. Section V summarizes and concludes. II. Theoretical Approach In this section, we summarize our implementation of the time independent multimode vibronic coupling method, so that we can discuss our level of treatment and explain how it extends the previous determination of the pyrrolide electron detachment spectrum, by MLWD, which used the same formal approach.15 The photoelectron spectrum is characterized in two steps. First, a two coupled state, quasi-diabatic Hamiltonian, Hd, is constructed by least-squares fitting of energy gradient and derivative coupling data at representative nuclear configurations using a pseudonormal equations approach.29 Then, the eigenvalues and eigenvectors of the vibronic Hamiltonian Hvib ) TNI + Hd (where TN is the nuclear kinetic energy operator) are determined and used to construct the spectral intensity distribution function. Hvib is constructed in a basis chosen to limit the size of the vibronic expansion required to converge the spectral intensity envelope. The issue of the choice of the vibronic basis was also considered by MLWD. A. Vibronic Wave Function. The wave function for the neutral species ΨKT is expanded as
J. Phys. Chem. C, Vol. 114, No. 12, 2010 5313 Nstate
ΨKT(q, w)
(w) ∑ ΨdI (q; w)ζ(n),K I
)
(1a)
I)1
where ΨId is the quasi-diabatic electronic wave function. The vibrational functions are, in turn, expanded as
ζ(n),K (w) ) I
(n) ∑ dI,K m ξm (w)
(1b)
m
and the ξ(n) m are expressed as multimode products Nint
ξ(n) m (w)
)
∏ χm(n),j(wj) j)1
(1c)
j
with
0 e mj < Mj
(1d)
vib The dI,K m in eq 1b are the eigenvectors of H , which in turn is Hvib in the vibronic basis ξm(w), m ) 0, ... , M. In these equations, q denotes the electronic coordinates and w ) T(Q - Q0), T invertible, denotes the set of internal coordinates chosen to describe the neutral molecule. Q is an arbitrary set of Nint internal nuclear coordinates.30 The choice of origin Q0 is discussed below. χm(n),j is the mth harmonic oscillator function associated with the jth mode. The w and vibrational functions χm(n),j will be referred to below as the neutral biased basis, hence the superscript (n) in eqs 1b and 1c. From eqs 1a-1d, the size Nint Mj. of this vibronic basis is NT) NstateNvib, where Nvib ) ∏j)1 In this work, all necessary internal coordinates will be explicitly included in ΨTJ . The ΨdJ, quasi-diabatic electronic states, are specified by their matrix elements of the electronic (coulomb) Hamiltonian, He(q; w), which are
d HK,L (w) ≡ 〈ΨKd(q; w)|He(q; w)|ΨLd(q; w)〉q ) Nint
EK(Q )δK,L + 0
∑
k)1
VK,L k wk
1 + 2
Nint
∑
K,L Ak,l wkwl
(2)
k,l)1
Here, EK(Q0) is the energy of the Kth adiabatic state at Q0. Since Hd includes all terms through second order, it gives rise to the fully quadratic vibronic coupling model. The determination of the coefficients VK,L and AK,L from ab initio electronic structure data has been discussed in detail previously.23 In the previous study by MLWD, the linear vibronic coupling (LVC) model, which includes only linear interstate coupling terms, was used. In this work, the appropriateness of the LVC model will be established. B. Anion Wave Function. The anion is assumed to be in its ground state
ΨT,(an) (q, w ¯ ) ) Ψ(an),a (q; w ¯ )ζ(an),0 (w ¯) 0 0 0 where Ψ0(an),a is an adiabatic electronic state and
(3a)
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Zhu and Yarkony
Nint
ζ(an),0 (w ¯) 0
)
(w j j) ∏ χ(an),j 0
(3b)
j)1
j (Q - Q j 0), where Q j 0 is the ground state equilibrium Here, w j )T geometry of the anion, the w j are its normal coordinates, and χm(an),j is the mth harmonic oscillator function associated with the jth normal coordinate of the anion. C. Spectral Intensity Distribution. The quantity of interest in this study is I(E), the spectral intensity distribution:15
I(E) )
∑ |AK|2δ(E - EK)
(4)
K
where Nstate
AJ )
¯ )|µ0,I |ζ(n),J (w)〉w ) ∑ 〈ζan,0 0 (w I I)1
(n) (w ¯ )|µ0,I |dI,J ∑ 〈ζ(an),0 0 m ξm (w)〉w
(5)
I,m
µ0,I is the transition moment connecting the ground electronic state of the anion with the Ith diabatic state of the neutral. µ0,I depends on both nuclear coordinates and the wave vector for the detached electron. Neglecting, as is routinely the case,15 the coordinate and energy dependencies of µ0,I, AJ becomes
AJ )
∑ µ0,Is0,mdI,Jm
(6a)
I,m
where using eqs 1c and 3b Nint
s0,m ) 〈
∏
Nint
χ(an),j (w j j)| 0
j)1
∏ χm(n),j(wj)〉 j)1
j
(6b)
As discussed in ref 27, the large number (Nvib) of multidimensional Franck-Condon overlaps in eq 6b are evaluated using standard generating function techniques.31-34 It is the exact evaluation of the s0,m that allows us to use the neutral biased basis which reduces the size of the vibronic expansion used to compute the AJ without approximation. The s0,m satisfy the normalization constraint:
D(M1, ..., MNint) )
∑ |s0,m|2 e 1
(7)
m
where the equality indicates that the negative ion ground state is fully described by the neutral biased basis. The condition D(M1, ..., MNint) ) 1, which as we show in section IV is readily satisfied to a good approximation, is a necessary, but not sufficient, condition for the adequacy of the neutral biased basis. III. Electronic Structure Aspects In this section, we describe the electronic structure of the A2 and 2B1 states of pyrrolyl and how accurately the two state quasi-diabatic state Hamiltonian Hd reproduces the ab initio data. The construction of the optimal Hd has been discussed elsewhere.23 2
A. Level of Treatment. The electronic structure data used to construct Hd was obtained from multireference configuration interaction (MRCI) wave functions using orbitals obtained from state-averaged multiconfiguration self-consistent field (SAMCSCF) wave functions. The SA-MCSCF treatment averaged two states with equal weights and used wave functions obtained from a five-electron in five-orbital complete active space expansion. This active orbital space is comprised of the five π-orbitals from the CH and N moieties. The molecular orbitals are expanded in a basis made up of Dunning’s cc-pVTZ bases on nitrogen and carbon and a Dunning’s polarized double-ζ (DZP) basis on the hydrogens. Dynamic correlation was included at the second order configuration interaction level, with the interacting space restrictions included. The resulting MRCI expansion consisted of 108.5 million configuration state functions (CSFs). Since, as is routinely the case,15 the µ0,I are not computed in this work, the anion calculations are independent of those for the neutral. The anion wave functions were computed using Dunning’s aug-cc-pVTZ bases on the carbons and nitrogen and a Dunning’s DZP basis on the hydrogens. The molecular orbitals were determined from a single configuration SCF procedure. Dynamic correlation was included at the single and double excitation CI levels. The CI expansion consists of 27.5 million CSFs. All electronic structure calculations reported in this work employed the COLUMBUS suite of electronic structure codes.35,36 B. Surface Extrema. In this work, the following extrema j 0 ) Qmin(1A1), the local energy minimum were determined: Q of the anion surface; Qmin1(2A2), the local energy minimum of the neutral ground state potential energy surface; Qts1(2B1), a saddle point with 2B1 symmetry on the ground state potential energy surface; and Qmex(2A2-2B1) ) Q0, the minimum energy point on the 2A2-2B1 seam of conical intersection. Note that as indicated by the term symbols these structures have C2V symmetry. Internal coordinates for these extrema are given in Table 1. The atom numbering refers to that used in Figure 1a. Also summarized in Table 1 are the Qmin and Qmin1 values reported by GIHKBL and MLWD and Qmex values estimated by GIHKBL. The Qmin and Qmin1 values reported in Table 1 are in reasonable accord with the results of GIHKBL, which were based on a B3LYP/6-311++G(d,p) treatment. Agreement is less satisfactory with the results of MLWD, who used MP2/aug-ccpVDZ level calculations for pyrrolide and B3LYP/6-311G(d,p) for pyrrolyl. Most significantly, MLWD predicts little difference, ∼0.008 Å, between the C3-C4 bond distance, d(C3-C4), for Qmin1 and that for Qmin, that is, d(C3-C4)(Qmin1) d(C3-C4)(Qmin) ) -0.008 Å, whereas the present work [that of GIHKBL] predicts a much larger difference, d(C3-C4)(Qmin1) - d(C3-C4)(Qmin) ) -0.054[-0.058] Å. Simliarly, for d(C2-C3) our work [that of GIHKBL] predicts d(C2-C3)(Qmin1) - d(C2-C3)(Qmin) ) 0.063 [0.056] Å, whereas for MLWD, d(C2-C3)(Qmin1) - d(C2-C3)(Qmin) ) -0.009 Å. These differences can be expected to be reflected in the vibronic spectra reported in section IV and are germane to our analysis of the local topography of Qmex discussed below. Note that as expected Qmex lies between Qmin1 and Qts1 which is in accord with the results of GIHKBL. Finally, our prediction of Qmex agrees well with the estimate of GIHKBL. Table 2 reports the energetics at the extrema in Table 1, relative to the energy at Qmex. From this table, it is seen that Qmex lies 0.56 eV above the ground state minimum but is energetically only 130 cm-1 above Qts1 and geometrically (see Table 1) quite close to Qts1. This observation is in good accord
Photoelectron Spectrum of Pyrrolide
J. Phys. Chem. C, Vol. 114, No. 12, 2010 5315
TABLE 1: Extrema: Bond Distances in Å, Bond Angles in Degreesa Qmin1 d(C2-N1)
d(C2-C3)
d(C3-C4)
d(C2-H6)
d(C4-H8)
C2-N1-C5
N1-C2-C3
C2-C3-C4
N1-C2-H6
C2-C3-H7
1.3359 1.3358 1.3441 1.369 1.4527 1.4527 1.4598 1.410 1.3551 1.3548 1.3610 1.422 1.0765 1.0765 1.0834 1.089 1.0736 1.0736 1.0798 1.087 104.84 104.84 104.74 104.72 112.36 112.36 112.38 112.27 105.21 105.21 105.25 105.37 121.06 121.07 120.99 120.33 126.20 126.20 126.26 127.16
Qts1
Qmex
1.3783 1.3783 1.3856
1.3712 1.3712 1.3804
1.3594 1.3539 1.3639
1.3718 1.3718 1.3759
1.4848 1.4771 1.4947
1.4646 1.4645 1.4779
1.0735 1.0732 1.0794
1.0739 1.0739 1.0798
1.0740 1.0738 1.0806
1.0739 1.0739 1.0804
106.59 106.38 106.84
106.31 106.31 106.59
111.23 111.17 111.04
111.41 111.41 111.21
105.47 105.64 105.54
105.43 105.43 105.50
120.05 119.87 120.05
120.24 120.24 120.17
128.12 128.17 128.29
127.85 127.85 128.03
TABLE 2: Energies in cm-1 at Qmin1, Qts1, and Qmin Relative to E1(Qmex) ) E2(Qmex)a
Qmin min1
1.3483
Q
1.3652 1.379 1.3890
Qts1
1.4035 1.419 1.4096
Qmin
E1
E2
-4530.34 -4483.06 -130.70 -133.27 -2810.29 -2738.56
6043.76 6132.71 1651.23 1719.42 1441.73 1501.74
a Ab initio MRCI energies above Hd energies. In each case, the ab initio MRCI (Hd) energies are calculated at the ab initio MRCI (Hd) determined geometries.
1.4195 1.430 1.0783 1.0865 1.096 1.0763 1.0847 1.095 105.12 105.13 104.75 112.17 112.06 112.20 105.27 105.38 105.42 120.61 120.54 120.37 127.07 127.20 127.15
a
For each geometric parameter: first row, ab initio MRCI results; second row, Hd determined results; third row, GIHKBL results; fourth row, MLWD results. All structures have C2V symmetry.
Figure 1. g and h vectors in Cartesian coordinates. Atom numbering used in text indicated.
with the estimate of GIHKBL who predicted Qmex to be 80 cm-1 above Qts1 with bond distances quite close to those for Qts1. The predicted vertical 2B1-2A2 energy separation, that is, the energy separation at Qmin, of 4252 (4235) cm-1 is in good accord with the MRCI result of MLWD given parenthetically. C. Intersection Adapted Coordinate Analysis. Additional insights into the topography in this region can be obtained by the use of intersection adapted coordinates.37 Intersection adapted coordinates, w, are an orthogonal transformation of the internal coordinates, Q, with a conical intersection, here Qmex, as the origin. At a point of C2V symmetry such as Qmex, there are eight
Figure 2. 1,22A potential energy surfaces in the branching plane. Qmex, the minimum energy crossing point; Qmin1, the 12A2 minimum; Qts1, the 12B1 transition state; and Qmin, the pyrrolide minimum, are shown.
a1, three a2, three b1, and seven b2 internal coordinates. Since Qmex is an accidental conical intersection of a 2A2 state and a 2 B1 state, h, the interstate coupling vector, transforms as b2. g, the energy difference gradient vector, transforms as a1. In intersection adapted coordinates, the x-coordinate, the first a1 coordinate, points along the direction of g, while the y-coordinate, the first b2 coordinate, points along h. These two coordinates, which are pictured in Cartesian coordinates in Figure 1a and b, respectively, define the branching37 or g-h38 plane. It is the explicit identification of the branching plane coordinates that gives intersection adapted coordinates their conceptual value. It is in this plane that the degeneracy is lifted linearly. Figure 2 illustrates this conical topography reporting the energies of the 1,22A states as a function of (x,y). The remaining coordinates, which are chosen to be mutually orthogonal and orthogonal to the branching plane, but are otherwise arbitrary, define the seam space. Expressing Qmin1(2A2), Qts1(2B1), and Qmin(1A1) in intersection adapted coordinates, we find Qmin1(2A2) ) [0.3073(0.1063)], Qts1(2B1) ) [-0.0538,(0.0299)], and Qmin(1A1) ) [0.1200,(0.0692)]. Here, the first number in the square brackets is x and the number in the parentheses is the norm of the seam space contribution. Since as noted above each of these extrema has C2V symmetry, only the a1 contribution is nonvanishing. It is seen from this decomposition that each of these points has a significant projection in the branching plane and necessarily lies along the g direction in that plane. The arrangement of these points is depicted in Figure 2. It is interesting to compare this analysis with a related analysis by MLWD. MLWD represent nuclear motions in terms of the normal modes of pyrrolide. MLWD (see their Figure 3) define
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a path from Qmin to a point of conical intersection. This path is along their pyrrolide mode labeled Q4 which is pictured in their Figure 2. According to the analysis of the preceding paragraph, see Figure 2, in this work, our path from Qmin to Qmex lies principally along g, which is pictured in Figure 1a. Q4 of MLWD is somewhat different from our g vector. At least two explanations, separately or combined, are possible for this difference: (i) the intersection reported by MLWD may not be near the minimum energy crossing point or (ii) it may reflect the previously noted differences between the present work and that of MLWD for the bond distances defining structures Qmin and Qmin1. D. The Quasi-Diabatic Hamiltonian. (i) Choice of Origin. In this work, Qmex(2A2-2B1) is chosen as the origin for Hd. The reasons for this choice are as follows. It guarantees that the properties of this conical intersection are well reproduced by Hd. Further, as was noted above, Qmex is quite close to Qts1. Dynamics near Qts1 are likely to be relevant to the spectral region associated with the threshold of the 2B1 state, which is of particular interest in this work. Finally, the more frequent choice5,15 of origin, Qmin, is not particularly close to Qts1 or Qmin1. See Figure 2. This makes Qmin a comparatively poor choice for expanding Hd. (ii) Linear Ws Quadratic Coupling. It is significant, as pointed out by MLWD, to assess the importance of quadratic coupling terms in the vibronic coupling model. Intersection adapted coordinates are particularly useful in this regard. At Qmex, using intersection adapted coordinates, all of the nonvanishing VK,L are in the branching plane and the nonvanishing linear interstate 2,2 1,2 couplings are V1,1 x - Vx ) 2|g| and Vy ) |h|. Here, we find |g| ) 0.077754 and |h| ) 0.088717. The corresponding second 2,2 order interstate coupling terms are (A1,1 x,x - Ax,x )/2 ) -0.002571, 1,1 2,2 1,2 (Ay,y - Ay,y )/2 ) 0.0038465, and Ax,y ) 0.001292. Thus, the second order coupling terms in the g-h plane are small compared to the linear terms, consistent with the LVC model used by MLWD. We comment further on the size of the second order terms in section IV. (iii) Performance of the Quasi-Diabatic Hamiltonian. Since the construction of Hd has been described in a previous work,23 here we summarize how well Hd reproduces the ab initio data. Table 1 reports the Hd determined geometric parameters for the extrema discussed above. Since, as discussed above, Qmex is taken as the origin for Hd, its geometric parameters are reproduced almost exactly. The Hd determined parameters for the remaining extrema are seen to be in excellent agreement with the ab initio values, with the largest errors being 0.007 Å and 0.2°. Table 2 reports the Hd determined energetics. Again, the agreement between the ab initio and Hd determined energies is seen to be excellent, with the largest error being less than 4%. Finally, we consider in Tables 3 and 4 the ab initio and Hd determined harmonic frequencies at Qmin1(2A2) and Qts1(2B1), respectively. Also reported is the overlap of the ab initio and Hd determined normal mode eigenvectors. Table 3 also reports the pyrrolide harmonic frequencies. Given the importance of the harmonic frequencies at Qmin1(2A2) in determining the low energy portion of the photoelectron spectrum, Hd was constructed to favor reproduction of the ab initio harmonic frequencies at Qmin1 at the expense of those at Qts1. Consequently, it is seen from Table 3 that the ab initio and Hd results are in quite good accord at Qmin1(2A2). The root-mean-square error is
