J. Phys. Chem. 1994,98, 3467-3471
3467
Ab Initio Vertical Ionization Potentials of trans-Butadiene and Cyclobutadiene Using the Effective Valence Shell Hamiltonian Method Charles H. Martin, Richard L. Graham,? and Karl F. Freed’ The James Franck Institute and Department of Chemistry, The University of Chicago, Chicago, Illinois 60637 Received: September 28, 1993; In Final Form: December 9, 1993’ Moderate-size basis set a b initio calculations for the r-electron vertical ionization potentials of trans- and cyclobutadiene (1 16 and 128 functions, respectively) are performed using the effective valence shell Hamiltonian (H’) method. These calculations demonstrate the flexibility and unique features of the Hu method because the ionization potentials emerge from previous computations of the neutral valence-like and Rydberg state excitation energies using the same H”.The computed ionization potentials agree well with experiment and with previous calculations, including highly correlated ones for trans-butadiene. New predictions are provided for ionizations to quartet and shake-up states. Furthermore, in contrast to current beliefs about quasidegenerate perturbation theory, the third-order H’calculations do not encounter intruder state problems.
I. Introduction Various modern ab initio electronic structure packages exist for describing the electronic excited states of small molecules. The methods range from crude semiempirical approachesto highly accurate correlated ab initio techniques. Despite the plethora of easily used methods, these commonly available packages cannot treat many molecular systems because the packages lack important, sophisticated features. For examples, many electronic structure problems demand a multireference configuration treatment, require large amounts of dynamical correlation, and suffer from size-extensivityproblems. Furthermore, traditional ab initio methods involve a separate calculation for each excited state (or class of states), and as a result, different types of calculations may be required to describe a wide class of states with comparable accuracy. Additionally, most of the highly accurate methods significantlycomplicatesimpleintuitive models which non-specialisttsuse to think about electronic structure. To alleviate these problems, we have developed the effectivevalence shell Hamiltonian (Hu)method-an ab initio, multireference configuration, size-extensive electronic structure method. We have recently reported ab initio Hwcalculations for the *-electron vertical excitation energies of trans- and cyclobutadiene.12 These calculations employ moderate-size basis sets (126 and 116 functions for trans- and cyclobutadiene, respectively), treat both valence-like and Rydberg states on an equal footing, and require only a single large-scale H+’ calculation to generate all the *-electron vertical excitation energies for each molecule. The present work demonstrates that these same previously computed H’’s provide, in addition to the already presented accurate ab initio neutral state excitationenergies, highly accurate vertical *-electron ionization potentials. We begin with a brief review of the H’calculations for transand cyclobutadiene, demonstrating how the Ifumethod readily contains the necessary multireference configuration character, treats a variety of qualitatively different states equalIy well, and provides unique insights into electronic structure problems. The first large basis set Wcalculations have been performed for transbutadiene. As evidence to the accuracy of the HVmethod, the computed trans-butadiene r-electron vertical excitation energies deviate from experiment on average by only 0.10 eV. These calculations continue to be the most accurate ab initio work available for the vertical excitation energies of this compound.’ The difficulty in treating trans-butadiene is evident from the ~
~
Present address: Cray Research Inc., 655 E. Lone Oak Drive, Eagan, M N 55121. * Abstract published in Aduance ACS Absrracrs, March 1 , 1994.
0022-3654/94/2098-3461~04.50/0
experimental evidence suggesting that the low-lying r-electron spectrum of trans-butadiene contains both valence-like and Rydberg states. Since these states contain substantially different degrees of electron correlation, an accurate ab initio description of this spectrum therefore requires a method that can treat both valence-like and Rydberg states with comparable accuracy. Consequently, in order to obtain the differential correlation between nearby valence-like and Rydberg states, the computations must employ larger basis sets which include both standard valence basis functions and additional Rydberg functions. Furthermore, theab initioreferencevalenceorbitals must contain bothvalencelike and Rydberg orbitals, and many states demand a multiconfigurational treatment. The Hp method is ideally suited for such systems since the Hu method employs large reference spaces that can include all the relevant valence orbitals. Additionally, becauseof the(Fock-space) many-bodynatureofHp, both valence like and Rydberg states emerge simultaneously from a single H” calculation. In addition, the Hp calculations have helped in properly identifying some experimentally observed low-lying transitions. Unlike the case for trans-butadiene, very little experimental information is available for cyclobutadiene. Thus we have used the H’method togenerate aprioripredictionsabout the excitation spectrum of this elusive compound and hence to challenge both experimentalists and other ab initio researchers to either verify or rectify our predictions. Given the extremely good Hu results for trans-butadiene, the H’method also should treat cyclobutadiene extremely well. Indeed, even basic organic chemistry teaches that both trans- andcyclobutadiene are simpleconjugated hydrocarbons which can be understood well with simple *-electron theories such as Hiickel theory or other semiempirical methods. Of course, semiempirical methods cannot completely treat the *-electron states of trans-butadiene due to the Rydberg-valence mixing? and hence similar problems are expected in cyclobutadiene. Additionally, cyclobutadiene is believed to interconvert between two ground-stateconformersvia a square intermediate.5-7 Consequently, an accurate description of the high symmetry of the transition state should thus employ a zero-order description involving at least two reference configurations. (Note however that, although single reference (CCSTD-1 b) coupled cluster calculations perform well, they do require triple excitations to compensate for the true multiconfigurational character of the reference state.8 See below.) The Hp method is well suited for multireference calculations with many more than two reference configurations, and furthermore, no new theory or programming is needed. 0 1994 American Chemical Society
3468 The Journal of Physical Chemistry, Vol. 98, No. 13, 1994
We have therefore computed the the low-lying *-electron vertical excitation energies of cyclobutadiene at its rectangular ground-state geometry and at the predicted square transitionstate geometry. The barrier height for the ground-state automerization is computed to be 8.3 kcal/mol, which agrees well (to within about 1 kcal/mol) with previous single reference configuration coupled cluster calculations (CCSDT-1 b, which includes triples) and two-configuration generalized valence bond multireferencesingles and doubles configuration interaction (CISD-GVB) calculations. As expected, the Hvcomputations for cyclobutadiene reveal a number of previously unpredicted lowlying valence-like and Rydberg states. The numerous states arise naturally from the Hvmethod becauseall states are simultaneously produced with a single calculation. The Rydberg-valence mixing is found to be weak for most low-lying states because when Rydberg orbitals are not explicitly included in the valence space, excitation energies to valence-like states change by at most 0.13 eV. Hence, we conclude that no intruder state problems are encountered in treating these low-lying states (see discussion in section IV). Finally, wealso offer new predictionsfor this system. In particular, the lowest lying optically allowed transition is calculated as occurring for the 11B3, state at 5.99 eV. Experimental *-electron vertical ionization potentials of both trans- and cyclobutadiene are available for comparison with our Hycomputations. In addition, there are a number of previous calculations and experiments for the ionization spectrum of both trans-butadienes16 and cy~lobutadiene.~J~J* Our computed vertical ionization potentials generally agree well with the best previous theoretical determinations and with experiment. The present series of computations serves to underscore the wealth of information available from a single computation of Hv for a given system. Section I1 briefly sketches necessary background concerning Hvtheory and computational methods. The computed results are presented and discussed in sections I11 and IV, respectively.
11. The Effective Valence Shell Hamiltonian ab Initio Metbod The theory underlying the effective valence shell Hamiltonian method has been described previou~ly.1+~~ Details of prior Hv computations for both trans- and cyclobutadiene are given elsewhere.1J Also, as described previously, the basis set used is a [9s5p2d/4s2pJ contraction of Gaussian functions, augmented with two diffuse carbon p functions.*J Here we review briefly those aspects of the basic theory and computational methodology relevant to the ionization potential calculations. Consider the exact, nonrelativistic, time-independent electronic SchrMinger equation
Martin et al. in a unique fashion. The effective valence shell Hamiltonian Hv is a many-body operator, which may be written in the form
where Ec is the fully correlated energy of theoccupied coreorbitals and K , q,, and q j k are the one-, two-, and three-body effective valence shell Hamiltonian operators, respectively. Matrix elementsof the H'n-body operatorsare termed Hvn-bodyparameters (wherenrangesfrom zero to twice thenumberofvalenceorbitals). Each H' n-body parameter contains some portion of correlation contributions from states lying outside the valence space (dynamical correlation), and the valence orbitals are the same for every state in thevalence space. Diagonalizing Hvthen accounts for near degeneracy or nondynamical correlation. The dynamically correlated Hvparameters couple the valence configurations and producea balanced, fully correlated, sizaextensivedescription of the valence states. The dynamical contributions in the q and qj.operators represent a rigorous ab initio counterpart to semiempirical parameters, and this allows H' to be used to interpret and help correct semiempirical theories. The Hvmethod adds dynamical correlation to the bare one- and two-electron interactions via diagramatic perturbation theory.22 Thus, matrix elements of the correlated q and q,operators are actually ab initio "parameters". In contrast, semiempirical methods modify the one- and two-electron parameters by employing empirical data and simplified valence shell CI equations. H*ab initio parameters can be directly compared to semiempiricalparameters, and future papers will provide large-scaleab initiocalculationsof the PariserParr-Pople (PPP) a-electron Hamiltonianz5 as a test of the fundamental assumptions of a-electron semiempirical methods (alsoseeref 4). Theab initiocalculationspresentedin the present paper serve, in part, as reference calculations for future analysis of semiempirical methods. The Hvcannot be evaluated exactly for nontrivial basis sets. Hence, the Hv n-body parameters are computed using third-order quasidegenerate (multireferenceconfiguration) RayleighSchrB dinger perturbation theory.zz The many-body, multireference perturbation expansion can, in principle, describe all the valence states simultaneously. However, to accomplish this enormous feat, the Hamiltonian must describe all valence states equally well, and hence the calculations require a unique set of valence orbitals and energies. To this end, the full Hamiltonian is decomposed as
H=Ho+V in which the Hamiltonian H acts on the complete space of manyelectron wave functions I$). Classifythe orbitals as core, valence, and excited. Define a valence space (or primary space) with projector P. The valence space is a complete active space (CAS) of configurations in which the core orbitals are fully occupied and any occupancy of the valence orbitals is permitted. The valence space, hence, contains all neutral and ion valence states of the system. Given a valence space, the full Hamiltonian H i s transformed into a Hermitian effectivevalenceshell Hamiltonian H*which generates the exact eigenvalues E of H but acts only within the valence space:
The effective valence shell Hamiltonian is constructed with diagrammatic perturbation theory and thus takes a many-body form.zz As a result, the Hy equations treat electron correlation
(4)
where HOis a zero-order Hamiltonian, and the perturbation is V - H- Ho. The present calculationstake Hoas a sum of modified one-electron Fock matrices. Therefore, HOis defined only by the molecular orbitals and orbital energies. The orbitals used in the Hycalculations are not restricted to be canonical HartreeFock orbitals or MCSCF-type orbitals, but generally these do perform very well in Hv calculations. Again, the best H' orbitals should simultaneously described all valence states of the system equally well, so that the valence space Hystates correspond as closely as possible to the true H eigenvectors. The trans-butadiene and cyclobutadienecalculationsemploy a fairly general scheme which works nearly automatically and sufficesfor highly accurate thirdorder H v calculations.l.z The scheme is as follows: The core and valence molecular orbitals that are occupied in the SCF or MCSCF approximateground state aredetermined from a groundstate SCF or MCSCF calculation. The valence orbitals which are not occupied in the (MC)SCF ground state are determined using a series of improved virtual orbital (IVO) calculations, as described The excited orbitals are generated by
Ab Initio Vertical Ionization Potentials
The Journal of Physical Chemistry, Vol. 98, No. 13, 1994 3469
TABLE 1: Vertical r Ionization Potentials* of hrPo4Butadiene As Computed with the Effective Valence Shell Hamiltonian H Method. Commrison of 4V and 6V Valence Spaces 4V space 6V space 1st order 2nd order 3rd order 1st order 2nd order 3rd order 12Bg 8.97 (0.00) 9.05 (0.00) 8.78 (0.00) 8.95 (0.00) 9.14 (0.00) 8.72 (0.00) 11.09 (2.31) 11.46 (2.51) 1ZAu 11.50 (2.53) 11.67 (2.62) 11.76 (2.62) 10.98 (2.26) 13.85 (4.90) 14.93 (5.79) 13.54 (4.76) 1 4 ~ ~ 13.90 (4.93) 14.80 (5.75) 13.36 (4.64) 22Au 14.65 (5.68) 14.21 (5.16) 14.63 (5.68) 14.27 (5.13) 13.71 (4.99) 13.78 (5.00) 16.79 (7.94) 16.43 (7.29) 15.61 (6.83) 22B, 16.84 (7.87) 16.37 (7.32) 15.47 (6.75) 16.89 (8.11) 18.12 (9.17) 1 4 ~ ~ 18.17 (9.20) 17.97 (8.92) 18.04 (8.90) 16.71 (7.99) a Energies in electronvolts. Values in parenthesis are excitation energies relative to the 12B, state. diagonalizing the remaining portion of the ground-state Fock operator. The orbital energies are taken as diagonal matrix elements of the (MC)SCF Fock operator appropriate for the (MC)SCF calculations that define the orbitals. The valence orbital energies are averaged, however, in order to help avoid numerical instabilities from intruder state problems (see below). Because corrections to averaging the valence orbital energies do not appear until third order, accurate ab initio calculations require a third-order HV. In order to understand the HVmore fully, we compare the HV method to other, more common a b initio CI techniques. Firstorder H' computations are just complete active space (CAS) configuration interaction calculations within the valence space. However, keep in mind that the valence orbitals need not be CASSCF orbitals, and in fact, CASSCF orbitals are not generally used for HVcalculations.' The second- and third-order contributions incorporate dynamical correlation into the CAS CI in a size-extensive manner. Essentially, third-order Hp compuations are a perturbative, size-consistent/size-extensive equivalent of multireference single and double CI (MRSDCI) with a CAS corresponding to the Hu valence space. Indeed, many MRSDCI calculations require large reference and thus an important feature of the HVcomputations lies in the capacity to use rather large reference spaces. Both the trans- and cyclobutadiene Hu calculations employ two valence spaces, one consisting only of valence-like orbitals and the other containing additional Rydberg orbitals. A comparison of calculations with both types of spaces provides inferences on the extent of Rydberg-valence mixing in these systems. The valence-like-only spaces consist of the four lowest u molecuiar orbitals for each system. Both molecules have the lowest two u molecular orbitals occupied in the SCF approximation to the ground state, and the next two lowest are generated by IVO calculations for low-lying excited triplet configurations. The Rydberg-extended valence spaces each contain a single additional (IVO) Rydberg orbital, each of ?r symmetry, thus placing six orbitals in the trans-butadiene (6V) valence space and eight in the cyclobutadiene (8V) valence space. Both 4V spaces should be adequate for treating the lowest ?r-electron ionization potentials since neither the neutral ground states nor the low-lying ion states should contain significant Rydberg function contributions. The difference between the 4V and 8V space ionization potentials for these states serves as an excellent test of this expectation. The ion state HVcomputations involve only miniscule labor (compared to an SCF calculation or actual generation of Hv), since the HVmatrix is diagonalized by the states with three u-electrons. 111. Results and Discussion A. trans-Butadiene Vertical Ionization Potentials. Table 1 presents the computed vertical ionization potentials for transbutadiene. A comparison of the first- and third-order HV computations exhibits the influence of dynamical correlation, Rydberg-valence mixing, and valence space dependence. The computed lowest ionization potential differs very little from order to order for each valence space, indicating that dynamical
TABLE 2 Experimental Determinations of the r Ionization Potentials' of hrPo4Butadiene state VIP6 UPSc PDSd UV/vis(Ar)# PES' 12B, 9.07 1zAu 11.5 (2.46) (2.3) (2.3) (2.45) 22Au (4.0) (4.2) Energiesin electronvolts. Values in parenthesis are excitation energies relativeto the l2B, state. Verticalionizationpotentials.9J0 Ultraviolet photoelectron spectra." The UPS method does not exactlyyield vertical ionization potentials because the ion states may relax their geometry (see ref 15). Gas-phase photodissociation spectra.12 e Ultraviolet/visible absorptionspectra of trans-butadiene radical cation in an argon matrix.I3 f Gas-phase He I photoelectron spectra.14 TABLE 3: Previous Calculations of the Vertical r Ionization Potentials' of hrPo4Butadiene state GFb CIC CASSCFd MRSCDP MRSCDI+Qf MRACPP 12B, 8.55 (0.00) 8.69 (0.00) l*Au (2.7) (2.8) 11.02 (2.47) 10.91 (2.22) Z2A, (5.8) (5.7) 13.72 (5.17) 13.48 (4.79) 22B, 15.45 (6.90) 15.17 (6.48)
8.79 (0.00) 8.79 (0.00) 11.10 (2.31) 11.08 (2.29) 13.69 (4.90) 13.67 (4.88) 15.19 (6.40)
a Energies in electronvolts. Values in parenthesis are excitation energie relative to the 12Bgstate. Green's function method,32 as estimated by Cave er aZ.15 CCI calculations.16 Full r CASSCF employing a 6-31G basis set.15 e Multireference singlesand doublesconfigurationinteraction with DZP+basis and large referencespace.15f MRSCDI using Davidson size-consistency correctionwith DZP+basis and large referencespace.15 Multireference averaged pair functional theory with .DZP+basis and large reference space.l5
correlation is not terribly important for obtaining this vertical ionization potential. Even our CAS CI-first-order descrption suffices for this state. Furthermore, the4V and 6V computations are contrasted to assess the extent of Rydberg-valence mixing. The ionization potentials are virtually identical for both valence spaces because the lowest ion state is spatially contracted and should not involve Rydberg contributions. The H' trans-butadiene computations also agree well with available experimental data, which are summarized in Table 2 (also see Cave and Perrott).15 The lowest computed r-electron ionization to the 12B, state differs from the experimental value of 9.07 eV by only about 0.1 eV at first and second orders and about 0.3 eV at third order for both valence spaces. The second ion u state, the 12A, state, is found experimentally by various methods to lie at an excitation energy between 2.3 and 2.45 eV and is computed here to lie in the range between 2.26 and 2.62 eV. As with the lowest 12B, ionization, the deviations between first, second, and third order are small. The 22A, state is computed to lie around 5.0 eV above the ground ion state, which is 1 eV higher than that found from experiment. This is consistent with prior theoretical determinations, as discussed below. Previous a b initio calculations of the vertical ionization potentials of trans-butadiene are listed in Table 3. (These calculations are summarized by Cave and Perrott, who also review the semiempirical and experimental determination^.'^) The HV predictions are generally comparable to those of the other very powerful correlated a b initio MRSDCI, MRSDCI+Q, and MRACPF methods, and differences are explained by the choice
Martin et al.
3410 The Journal of Physical Chemistry, Vol. 98, No. 13, 19'94 TABLE 4 Vertical r Ionization Potentials' of Cyclobutadiene As Computed with the Effective Valence Shell Hamiltonian H'Method. Comparison of 4V and 8V Valence Spaces for the Lowest Lying Doublets and Quartets of Each Spatial Symmetry 4V space 8V space 1st order 2nd order 3rd order 1st order 2nd order 3rd order 8.18 8.25 7.67 7.60 8.08 12B2, 8.28 9.76 10.20 10.12 10.54 9.68 12B3, 10.58 12A, 14.41 13.63 13.53 14.27 13.64 13.51 12.61 12.63 12.58 13.45 13.50 12.55 12B1, 20.09 19.47 21.17 20.07 19.59 14B2, 21.38 17.73 17.19 17.26 18.40 17.71 14B3, 18.61 12.64 13.01 13.09 13.02 12.63 14A, 13.06 22.13 23.36 24.36 24.43 25.16 14B1, 18.92 a Energies in electronvolts. of Hu orbitals. All methods underestimate the first ionization potential, although the Hu results are slightly better than the others. Furthermore, all methods greatly overestimate the 22A, state excitation energy, and the H' method is 0.2 eV worse for this state than the MRSDCI, MRSDCI+Q, and MRACPF calculations. The poorer H' treatment of this state probably arises because the first-order Hu does not describe this state as well as the CASSCF calculations. Indeed, while the CASSCF method optimizes orbitals for each individual state, our valence orbitals must describe all states simultaneously, and the perturbation expansion must alleviate any valence orbital deficiencies for particular states (as well as include other forms of correlation). The third-order expansion performs well even though the firstorder Hu inadequately treats certain high-lying ion states. Similarly, the first-order 22B, ionization deviates strongly from the CASSCF. The first-order Hu 22B, ion excitation energy is 7.87 eV with the4V space and7.94eV with the 8V space, whereas the CASSCF result is 6.90 eV. The third-order excitation energy, however, differs only by 0.4 eV from the MRSDCI+Q and MRACPF calculations, and thus the poorer first-order description is still corrected quite well by the HUperturbation expansion. Cave and Perrott express concern that their calculations do not recover enough of the uu' and UT correlation enery for states with large amounts of multireference character,15and as a result, they suggest that the inadequate treatment of correlation may cause the discrepancies between experiment and theory. The H' method, however, includes both uu' and UT correlation contributionsin second and third order, and the H'and the MRSDCI+Q and MRACPF calculations agree well. This suggests, on the contrary, that the discrepancies with experiment are not due to absent u d and UT correlation contributions. The above comparisons also indicate a slight disadvantage in using a single set of orbitals in H'calculations, but a countervailing advantage of these calculations is that all states of the system emerge from a single calculation. To demonstrate this further, Table 1 also reports the lowest lying quartet states of each symmetry, relative to the neutral ground state. No other experimental or theoretical assignments are available for these multiconfigurational, high spin states. B. Cyclobutadiene Vertical Ionization Potentials. The calculated vertical ionization potentials for cyclobutadiene are presented in Table 4 for both the lowest lying doublets and quartets of each spatial symmetry. The relationship between the 4V and 8V Hu computations is similar to that for trans-butadiene. Although the first-order calculations reproduce the first ?r ionization potential to within 0.4 and 0.1 eV of experiment for the 4V and 8V spaces, respectively, the second ?r ionization potential in first order is off from experiment by over 1 eV in both valence spaces. The third-order treatment of the second r ionization potential, however, is much closer to experiment, deviating by only about 0.3 eV for both valence spaces. Again, the Humethod is capable of providing a good description of all
TABLE 5 Vertical r Ionization Potentials. of Cvclobutadiene. hevious Calculations and Experiment state EOMlb EOMZC PERTCIld PERTCI2' MP2f exptr 12B2, 7.77 7.90 8.67 7.58 7.53 8.24 (8.16) 12B3g 9.78 9.41 l2Bl, 12.57 12.71 11.88 12.68 12.2 a Energies in electronvolts. Equations of motion/propagator method withan [9~5pld+4slp]/(3~2pld+ 2slp) +poiarization(ac=0.75and ap = 1.0) basis.'* Same as b, but with different polarization functions (ac= 0.3 and a~ = 0.75). Semiempirical MNDO PERTCI." e Semiempirical LNDO/S PERTCI.I7f6-31+G*/MP2.7 8 Gas-phase photoelectron spectra by Schwieg et d . I 7 and (in parenthesis) by Kohn and Chen.'
the states of the system by using a single set of valence orbitals and allowing the perturbation expansion to correct for many valence orbital deficiencies. Table 5 summarizes previous theoretical calculations and experimental determinations of the vertical ionization potentials of cyclobutadiene. The third-order Hv computations agree well with experiment, reproducing the lowest 7r ionization to within 0.16 and 0.06 eV for the 4V and 8V spaces, respectively, and the second lowest observed u ionization to within about 0.4 eV for both spaces. The H' calculations compare well to ab initio equations of motion (EOM) calculations, being somewhat better (although the Hvcalculations use a slightly larger basis set). Semiempirical methods actually fare rather well compared to the previous a b initio calculations, and in fact, the semiempirical results seem reasonable for aiding in identifying the experimental ionizations potentials. The 6-3 1+G*/MP2 calculations7perform more poorly, missing the experimental value of the lowest ionization potential by over 0.5 eV. The 6-3 1+G*/MP2 ionization potential is, however, nearly identical to thesecond-order Hvresult. Thiscorrespondence is expected since H U is simply a multireference generalization of the very popular single-reference configuration MP2/3 method: Given that MP2 and second-order Hvcompare so well, that neither lowest ion state or the neutral ground state are multiconfigurational, and that the third-order H' performs extremely well, an MP3 calculation should likewise reproduce the lowest cyclobutadiene ionization extremely well. Other experimental cyclobutadiene ionizations have been assigned as u symmetry according to semiempirical17J8 and equations of motion (EOM) calculations,ls but some assignments overlap with the Hv*-electron spectrum. In particular, a peak a t 13.4 eV has been assigned to *B3,, shake-up ionization. The present Hvcalculations predict that a shake-up l2AUionization occurs at 13.5 eV, and a 4A, state also appears in this region. Because we have not attempted to calculate spectral intensities or u ionizations, however, we do not suggest any reassignment a t this time. Additionally, without information on u excitations, nothing may be said concerning how these numerous states contribute to the photoelectron spectra. In principle, we could compute a cyclobutadiene H' which contains u valence orbitals, but this would involve generating a new series of calculations which are not required for the *-electron valence states. The single Hu *-electron calculation does, however, readily generate new information for quartet states and shake-up doublets. Since the Hu method provides both excitation energies for the neutral molecule and the ion with a single, large-scale calculation, we contend that the Huapproach is a unique and remarkably flexible a b initio method.
IV. Discussion The *-electron vertical ionization potentials of rruns- and cyclobutadiene are computed using the effective valence shell Hamiltonian (Hu) method. Agreement with previous highquality theoretical calculations and with experiments is excellent, and a number of unique and desirable features of the calculation are
Ab Initio Vertical Ionization Potentials highlighted. For example,all the %-electronionizationpotentials, including those with shake-up character and those to quartet states, are determined with the same Hv calculation that is also used to describeexcitationenergies in the neutral molecules. These calculations employ large multireference configuration valence spaces which include either only valence-like orbitals or, additionally, Rydberg orbitals. In addition, our effective Hamiltonian is useful for understanding semiempirical theories of valence, as will be discussed in some forthcoming papers. The present calculations further demonstrate the power of the ab initio H v electronic structure method. The literature contains a widespread belief that the H’method should contain “so-called” intruder state problems,26-27 but in fact, even though these problems may arise in other forms of quasidegenerate perturbation theory, the H v method has been designed from the onset to avoid these difficulties. The original H”formulation considers potential sources of these problems and offers some remedies as well, such as valence orbital energy averaging and diagram resummation techniques.22 The valence orbital energy averaging appears to work well in third-order Hu calculationsto alleviate intruder state problems, and additionally, Hp calculations always employ large reference spaces in order to explicitly treat all possible states that might intrude into similar, but much smaller, valence space perturbation theory calculations. These H v reference spaces are much larger than the miniscule ones previously used in model studies.26.27 As a result of these features of the Hv method, intruder state problems do not appear in the present H’calculations. Related concerns about intruder state problems have likewise been raised for the MRACPF calculations,yet the cited MRACPF trans-butadiene calculations do not demonstrate any such problems,lS and other forms of multireference variational perturbation theory have also been developed to explicitly avoid intruder state problems.28 It is unfortunate that these methods are considered less “robust” than multireference configuration interaction methods because of a misunderstanding about the Occurrence of intruder state problems in large-scalecomputations. The present computations generate accurate eigenenergy differences for a large class of the valence states, both neutral and ion, with a single Hu calculation. The method simultaneously treats states of varying spatial character, such as Rydberg and valence-like states, and recent computational advances allow us to employ very large basis sets and valence spaces. H’computations have also been performed for the excited Rydberg and valence-like potential energy surfaces for the photodissociation of methyl mercaptan (CH&Y), and no intruder state problems Occur here either.29 Rather, the large valence spaces and the size extensivity of the Hv calculation produce remarkably reliable and useful potential energy surfaces. The results of these calculations suggest that the intruder state problem should be reinvestigated for quasidegenerateperturbation theory calculations employing large valence space and averaged valence orbital energies (and with, perhaps, some resummed diagrams), so that means may be developed which can systematically avoid and/or compensate for these difficulties.
The Journal of Physical Chemistry, Vol. 98, No. 13, I994 3471
Acknowledgment. This research is supported, in part, by the National Science Foundation Grant No. CHE 93-07489, the Petroleum Research Fund Grant No. PRF 23610-AC6, and a computing allocation from the National Center for Supercomputing Applications (NCSA), Grant No. TRA92075N.We also thankDanny Yeagerfor theuseofhis MCSCFcomputerprogram and the theoretical chemistry group at Argonne National Laboratory for the use of the COLUMBUS package and SIFS library routine^.^^.^^ References and Notes (1) Graham, R. L.; Freed, K. F. J . Chem. Phys. 1992, 96, 1304. (2) Martin, C. H.; Graham, R. L.; Freed, K. F. J . Chem. Phys. 1993, 99, 7833. (3) Serrano-And&, L.; Marchin, M.;Nebot-Gil, I.; Lindh, R.; Roos, B. 0. J. Chem. Phys. 1993, 98, 3151. (4) Lee, Y. S.;Freed, K. F.; Sun, H.; Yeager, D. L. J . Chem. Phys. 1983, 79, 3862. (5) Hess, B. A. H., Jr.; Carsky, P.; Shaad, L. J. J . Am. Chem. SOC.1983, 105, 695. (6) Whittman, D. W.; Carpenter, B. K. J . Am. Chem. Soc. 1982, 104, 6473. (7) Kohn, D. W.; Chen, P. J. Am. Chem. Soc. 1993, 115, 2884. (8) Carsky, P.; Bartlett, R. J.; Fitzgerald, G.; Noga, J.; Spirko, V. J . Chem. Phys. 1988,89, 3008. (9) Reddish. T.: Comer. B. W. J. Chem. Phvs. 1986.. 208.. 159. (10) McDiarmid, R. J . Chem. Phys. 1976, ai, 514. (11) Eland, J. H. Int. J. Mass. Spectrom. Ion Phys. 1969, 2, 471. (12) Dunbar, R. C. Chem. Phys. Lett. 1975,32, 508. (13) Bally, T.; Nitsche, S.; Roth, K.; Haselbach, E. J . Am. Chem. SOC. 1984,106, 3927. (14) Koenig, T.; Klopfenstein, C. E.; Southworth, S.; Hoobler, J. A,; Wielesek, R. A,; Balle, T.; Snell, W.; Imre, D. J. Am. Chem. SOC.1983,105, 2256. (15) Cave, R. J.; Perrott, M. G. J. Chem. Phys. 1992, 96, 3745. (16) Kimura, K.; Katsumata, S.;Achiba, Y.; Yamazaki, T.; Iwata, S. Handbook of HeI Photoeleciron Spectra of Fundamental OrganicMolecules; Halstead: New York, 1981. (17) Kreile, J.; Munzel, N.; Schwieg, A.; Specht, H. Chem. Phys. Lett. 1986,124, 140. (18) Haddon, R. C.; Williams, G. R. J. J . Am. Chem. SOC.1975,97,6582. (19) Iwata, S.;Freed, K. F. J. Chem. Phys. 1976, 65, 1071. (20) Iwata, S.;Freed, K. F. J . Chem. Phys. 1977, 66, 1765. (21) Sun,H.; Freed, K. F.; Herman, M.; Yeager, D. J . Chem. Phys. 1980, 72,4158. (22) Sheppard, M.G.; Freed, K. F. J. Chem. Phys. 1981, 75, 4507. (23) Sheppard, M. G.; Freed, K. F. J. Chem. Phys. 1981, 75,4525. (24) Freed, K. F.; Sheppard, M. G. J . Phys. Chem. 1982,86, 2130. (25) Parr, R. The Quantum Theory of Molecular Electronic Structure; Benjamin: New York, 1963. (26) Bauschlicher,C. W., Jr.; Langoff,S. R.; Korminicki,A. Theor.Chim. Acta 1990, 77, 263. (27) Evangelisiti, S.; Daudly, J. P.; Malrieu, J.-P. Phys. Rev. A 1987,35, 4930. (28) Cave, R. J.; Davidson, E. R. J . Chem. Phys. 1988, 89, 6798. (29) Stevens, J.; Graham, R. L.; Freed, K. F.; Arent, M. Manuscript in preparation. (30) Shepard, R.; Shavitt, I.; Pitzer, R. M.; Comeau, D. C.; Pepper, M.; Lischka, H.; Szalay, P. G.; Ahlrichs, R.; Brown, F. B.; Zhao, J.-G. Int. J . Quantum Chem. 1988, S22, 149. (31) Shepard, R. Int. J. Quantum. Chem. 1991, 40, 865. (32) Cederbaum, L. S.;Domcke, W.; Schirner, J.; von Nissen, W.; Diercksen, G. H. F.; Kramer, W. P. J . Chem. Phys. 1978,69, 1591.
