14310
J. Phys. Chem. 1996, 100, 14310-14315
Highly Accurate ab Initio π-Electron Hamiltonians for Small Protonated Schiff Bases Charles H. Martin† Beckman Institute, UniVersity of Illinois at Urbana-Champaign, Urbana, Illinois 61801 ReceiVed: February 29, 1996X
The effective valence shell Hamiltonian (Hν) method is used to compute the ab initio π-electron Hamiltonians for the two small protonated schiff bases propeniminium (C3H4NH2+) and pentaniminium (C5H6NH2+), which are small analogs of the photoreceptor chromophore retinal. Because these small polyenes lack low lying, ionic excited states, the Hν calculations perform remarkably well even though they employ constrained, semiempirical-like valence spaces. In contrast to studies on other polyenes, even the second-order Hν performs extremely well, while the third-order Hν calculations yield an effective π-electron Hamiltonian that resembles the familiar semiempirical Pariser-Parr-Pople (PPP) Hamiltonian but which also retains ab initio accuracy for the vertical excitation energies. The ab initio π-electron Hamiltonian need only retain the PPP-type Ri, βi,j, and γi,j effective integrals to produce excitation energies that compare well with previous multireference single and double configuration interaction (MRSDCI) calculations. Since the ab initio π-electron effective integrals arise from first principles, they do not require any empirical corrections. However, they do lack the simple transferability generally assumed in semiempirical models for these short-chain polyenes.
I. Introduction The excited states of polyenes play a key role in the function of many proteins involved in photobiology, such as the pigment in bacteriorhodopsin,1 the light-harvesting antennae in the photosynthetic reaction center,2 and even our own, mammalian photoreceptors.3 Understanding the excited states of such polyenes has motivated a number of developments in electronic structure theory. Historically, the community has pursued two different routes. The first employs semiempirical π-electron theories, such as the familiar Pariser-Parr-Pople (PPP) method,4-9 whereas the second utilizes ab initio electronic theories. Until very recently, the PPP model had provided a quite reliable description of the excited states compared to many ab initio treatments, and at a fraction of the cost. Now it appears the ab initio techniques have finally caught up. Readily available and efficient correlated ab initio methods, such as CASSCF plus second-order perturbation theory (CASPT2,10,11 CASSCF+MP2,12,13 etc.), can treat polyene excited states with chemical accuracy. But despite the advances in computational speed and accuracy, most ab initio theories offer little insight into the accuracy of the simple but effective semiempirical models. So while researchers continue to develop and use semiempirical methods, especially for the study of biological chromophores in their native environments,3,14-18 most ab initio theories provide little assistance in furthering the development of the still useful semiempirical methods. The bridge between semiempirical and ab initio theories lies in the effective valence shell Hamiltonian (Hν) theory of Freed and co-workers.19-21 The Hν theory offers a unique solution to developing a new and highly accurate PPP-like π-electron theory. Very accurate Hν calculations for small polyenes yield an ab initio PPP Hamiltonian, termed the Hν - PPP Hamiltonian, which acts only as a minimal basis set of carbon atom pπ orbitals, treats all valence-like π-electron states simultaneously, and provides semiempirical-like one- and two-electron integrals which do not, in principle, require empirical modifications,19,22-28 because the correlation contributions arise from first principles. † Present address: WM Keck Center for Molecular Electronics and the Department of Chemistry, Syracuse University, Syracuse, NY 13244. X Abstract published in AdVance ACS Abstracts, July 15, 1996.
S0022-3654(96)00617-X CCC: $12.00
The Hν calculations perform as well as, if not better than, the best correlated ab initio methods. Recent Hν calculations for trans-butadiene,22,24-26, cis-butadiene,29 cyclobutadiene,23,24 benzene,27 and hexatriene28 provide vertical excitation energies (EE) and ionization potentials (IP) that compare well with multireference single and double configuration interaction (MRSDCI30,31), CASPT2,10,32 and other correlated ab initio calculations. Hence, the Hν ab initio calculations are accurate enough to begin rebuilding semiempirical π-electron theory from first principles. Second, Hν calculations do not use different orbitals for each state, but, like their semiempirical counterparts, treat all the valence-like states simultaneously with a small set of valence orbitals and with only a few valence shell integrals. Orbital (or polarization) corrections, as well as all other electron-electron correlation contributions, arise directly in the computed Hν as effective integrals. When using an appropriate valence space, the effective Hν integrals may be directly related to and compared with the traditional semiempirical PPP oneand two-electron Ri, βi,j, and γi,j integrals. Like their semiempirical counterparts, the ab initio βi,j and γi,j transfer exceptionally well between different polyenes and display simple bond length dependencies. Being, in principle, exact, however, the true effective π-electron Hamiltonians contain numerous other non-negligible effective integrals, such as all the two-electron integrals that are traditionally neglected in the zero-differentialoverlap (ZDO) approximation, as well as integrals of an effective three-electron operator, a component which arises solely from the Hν ab initio correlation contributions. Many of these additional integrals may be neglected or averaged over without greatly affecting the polyene spectra, while others might be included in new and improved PPP model. Although the Hν theory was developed nearly 20 years ago,33-35 only recently have extensive Hν studies focused on the ab initio basis of semiempirical π-electron theory in large polyenes.22-29,36,37 While the pure ab initio calculations perform extremely well, studies employing semiempirical PPP-like valence spaces lack accuracy for the lowest lying but most interesting singlet states. The errors arise because the low lying singlet states mix with low lying Rydberg states. To resolve this mixing, accurate ab initio calculations require large π-electron multiconfigurational active spaces containing both © 1996 American Chemical Society
π-Electron Hamiltonians for Protonated Schiff Bases valence-like and diffuse orbitals. Such diffuse (or Rydberg) states fall outside of the domain of semiempirical treatments because semiempirical valence spaces do not contain Rydberg orbitals22,25,27,28,38 (see, however, the RINDO method.39). The Hν calculations designed to mimic the PPP model must therefore employ less optimal, “constrained” valence spaces that consist of a minimal basis set of atomic pπ orbitals. Unfortunately, these more approximate polyene calculations represent all of the low lying states well except the optically allowed singlet states. Thus, the current Hν ab initio calculations will not serve as an optimal ab initio model for an improved PPP theory. The errors associated with using semiempirical-like, constrained valence spaces in Hν theory do not reflect a conceptual error with the use of small valence spaces in the Hν theory. Rather, the problems characterize a general difficulty with ab initio techniques (and with perturbative methods in particular) in finding a good zeroth-order description that correctly identifies and orders the low lying, singlet states of polyenes, states which vary in spatial character and thus in their amounts of dynamical electron correlation. Crudely speaking, the closer spatially the electrons are in a state, the more electron correlation they require. The difficulties with computing polyene singlet states occur because less dynamical electron correlation is present in diffuse, covalent states than valence-like, ionic states. It is well-known that models deficient in static electron-electron correlation may misorder the ionic and covalent excited singlet states.9 But even calculations that include static electron correlation, such as CASSCF treatments, may still misorder the low lying excited states due to the absence of dynamical electron correlation, placing the less correlated diffuse states below the true, lowest lying, ionic excited singlets.10,11,31 To correct these misorderings, it is necessary to correctly locate the desired state and then append large amounts of electron-electron correlation using a well-chosen valence space (which resolves nearly degenerate valence states) and a practically convergent method for treating the dynamical interactions. If the proper state can be identified (sometimes a nontrivial task31), then methods such as MRSDCI and CASPT2 perform extremely well at appending the dynamical electron correlation. As an added complication, perturbative methods, such as CASPT2, CASSCF+MP2, and Hν, may suffer from convergence problems (the so-called intruder state problems). In essence, the perturbation expansion cannot find the proper state. The Hν calculations may suffer from these difficulties more than the CASSCF+MP2 variants because the Hν theory treats all states simultaneously and thus requires the same orbitals and zeroth-order Hamiltonian H0 (i.e. orbital energies) for all valence states (for a more detailed discussion see ref 40). Formally, the Hν theory can employ identical valence spaces as the semiempirical models and still treat the low lying valence-like states exactly, but higher accuracy constrained calculations may require a better H0 and higher orders in the perturbation expansion for many systems. Despite the efforts to study semiempirical π-electron theory, Hν calculations designed to mimic the PPP model remain inaccurate for the lowest excited singlet states of small polyenes because of the inadequacy of the zeroth-order Hamiltonian and the congested spectra of these compounds.26-28 One solution is to try reformulating the multireference perturbation theory, as has been done to produce the CASPT2 and CASSCF+ MP2 methods. Another approach is to identify interesting polyenes that do not contain low lying ionic and diffuse states and study the π-Hamiltonian for these systems. Fortunately, in turning to the conjugated, protonated Schiff bases, which are analogs of the biologically interesting retinal
J. Phys. Chem., Vol. 100, No. 34, 1996 14311 chromophore, the problems associated with low lying ionic states disappear. The excited states of these and other charged polyenes consist of mostly covalent configurations.9 Also, as noted by Davidson,30,31 small protonated Schiff bases like propeniminium (CH2dCH-CHdNH2+) have large IPs (compared to trans-butadiene) and thus lack low lying diffuse (Rydberg) states. Indeed, in previous MRSDCI studies, Du and Davidson found that it is quite easy to locate the lowest lying excited state of a “truncated” retinal analog (C11H12NH2+) using a minimal basis set π-CI, whereas a similar calculation cannot even locate the actual 11Bu state of dodecahexane within the first seven excited singlets.31 In our continuing studies of the ab initio basis of semiempirical π-electron theory,19,25-28,37 Hν ab initio calculations are performed for the protonated Schiff bases propeniminium and pentaniminium (CH2dCH-CH2dCH-CHdNH2+) in order to derive and to study properties of the first principles PPP-like Hν effective Hamiltonian for these biologically relevant polyenes. To preview the results, it is found that the Hν calculations on these polyenes perform remarkably better than Hν calculations for other small, neutral polyenes. These Hν calculations describe the lowest π-electron excited states quite well even though the calculations use only constrained, PPP-like valence spaces. These results starkly contrast with previous criticisms of semiempirical models, which specifically claim that “atomic orbitals don’t work”.41 And in stark contrast to previous constrained Hν polyene calculations, even the second-order theory performs quantitatively well, while the third-order ab initio calculations provide a highly accurate PPP Hamiltonian for each molecule. The third-order ab initio π-electron Hamiltonian need only retain the one- and two-electron Ri, βi,j, and γi,j type ab initio parameters to describe the low lying π-electron excitation energies with the accuracy of the most accurate, correlated, ab initio electronic structure methods. II. Theory and Methods Being described in lucid detail before,20,21,28 let us just review the relevant components of the Hν theory necessary for treating the protonated Schiff bases. The PPP π-electron Hamiltonian has the form4,5 N
1
i
2! i*j
HPPP ) Eσ + ∑UiPPP +
N
∑Vi,jPPP
(1)
where Eσ is the correlated energy of the σ framework, UPPP i and VPPP i,j are one- and two-electron operators, respectively, and the sums are over the π-electrons. The standard theory depicts HPPP as an approximate, minimal basis set π-CI, but from the conception of the PPP model it was recognized that even the complete π-CI could not accurately describe electronic spectra.4,6 To correct the π-CI, the integrals of the Ui and Vi,j operators enter the model as empirical parameters, * (1)UPPP RiPPP ) ∫pπ,i 1 pπ,i(1) dr(1)
(2)
PPP * βi,j ) ∫pπ,i (1)UPPP 1 pπ,j(1) dr(1)
(3)
PPP * * PPP γi,j ) ∫pπ,i (1) pπ,j (2)V1,2 pπ,i(1) pπ,j(2) dr(1) dr(2) (4) PPP PPP ) 〈i|UPPP or, in Dirac notation, RPPP 1 |i〉, βi,j ) 〈i|U1 |j〉, and i PPP PPP γi,j ) 〈ij|V1,2 |ij〉. The Hν method uses a generalized version of multireference configuration many-body Rayleigh-Schro¨dinger perturbation theory20,42,43 and standard ab initio techniques to generate an
14312 J. Phys. Chem., Vol. 100, No. 34, 1996
Martin
effective operator that acts only on the minimal basis π-CI space. Hence, the ab initio π-electron Hν resembles the standard PPP Hamiltonian HPPP in structure and yet includes all dynamical correlation contributions in its many-electron matrix elements. The effective operator Hν acts only on the valence (π-electron) states Ψπ and generates the exact (full CI) state energies (Eπ);
Hν|Ψπ〉 ) Ev|Ψπ〉
(5)
To set up the perturbation expansion for Hν, write the full Hamiltonian as H ) H0 + V, with
H0 ) ∑|i〉i〈i|
(6)
V ) H - H0
(7)
i
The sum ranges over all orbitals i, |i〉 are the molecular orbitals, and i are the orbital energies. Define the valence space by partitioning the molecular orbitals into three sets: core {c}, valence {v}, and excited {e}. For producing PPP-like effect integrals, the Hν valence orbitals must consist of a minimal basis set of nv atomic pπ orbitals (LCAOs). With the orbitals specified, now define the projection operators
P ) ∑|k〉〈k|
(8)
Q)1-P
(9)
k
where k ranges over all valence space Slater determinants. Standard Rayleigh-Schro¨dinger many-body perturbation theory now yields Hν to third order as44
{
{
1
{
}
Q P(k)V VP(k′) + hc + Ek - H0 1 Q Q P(k)V V VP(k′) + hc ∑ 2 k,k′ Ek - H0 Ek - H0 1 Q P(k)V VP(k′)VP(k′′) + hc + ∑ 2k,k′,k′′ (Ek′ - H0)(Ek′′ - H0)
Hν ) PHP +
∑ 2 k,k′
}
}
4
O(V ) (10) where P(k) ) |k〉〈k|, Ek is the zeroth-order energy of the P-space configuration k, and hc designates the Hermitian conjugate of the proceeding term in braces. Notice that the first contribution to Hν, PHP, is just the bare π-space CI Hamiltonian and is therefore equivalent to the traditional “theoretical” model for HPPP. The remaining terms in eq 10 rigorously contain the “dynamical” correlation contributions to the individual integrals which enter semiempirical Hamiltonian integrals empirically. To obtain the corresponding correlated Hν valence shell integrals, eq 10 must be converted to a form that resembles the HPPP of eq 1. Since the P-space is a complete active space and given the simple choice of H0 in eq 6, the sums over k in eq 10 may be converted into sums over valence electrons (with the aid of some many-body theory43). The resulting Hν takes the form20,21 N
Hν ) Ec + ∑ Uiν + i
1
N
1
N
ν + ∑ Vi,jν + 3! i*j*k ∑ Wi,j,k 2! i*j
1
N
ν + ... ∑ Xi,j,k,l 4! i*j*k*l
(11)
Figure 1. Geometry of protonated Schiff bases propeniminium and pentaniminium used in Hν calculations. ν where EC is the correlated core energy and Uνi , Vi,j , Wνi,j,k, and ν Xi,j,k,l are the one-, two-, three-, and four-electron effective operators (the third-order perturbation expansion only includes up to four-body operators). We may identify the ab initio Ri, βi,j, and γi,j with the matrix elements of Uνi and Vνi,j: Ri ) 〈i|Uν1|j〉, βi,j ) 〈i|Uν1|j〉, and γi,j ) 〈ij|Vν1,2|ij〉. The three- and four-electron operators Wνi,j,k and Xνi,j,k,l have no analog in traditional semiempirical theory. As a first approximation, the Wνi,j,k and Xνi,j,k,l may be neglected. For greater accuracy they usually must be either explicitly included or averaged into the related one- and two-electron integrals.19,28 Although Hν spans a minimal basis set of atomic pπ orbitals, the Hν computations proceed according to normal ab initio procedures, requiring an extended basis set (including a split valence shell, polarization functions, etc.), HF calculations to define the core and excited orbitals, and intelligent choices of the somewhat arbitrary orbital energies, all to ensure good convergence.25-28,40 The molecular orbitals are represented in a standard Gaussian basis set. Given a choice of semiempiricallike valence orbitals {v}, the core {c} and excited {e} orbitals are optimized as much as possible in order to provide a good starting point for the perturbation expansion. The core orbitals and orbital energies arise from a ground state HF calculation in which all (c) f (v) and (v) f (c) excitations are prohibited. This allows the core to relax in the presence of the less-thanoptimal valence space and the valence orbitals to mix among themselves. The excited orbitals and orbital energies are then obtained from diagonalizing the excited orbital block of an N - 1 electron ground state Fock operator. Thus, the excited orbitals span the remainder of the basis set not present in the core and the valence orbitals, including the contributions from the split-valence shell, polarization functions, σ* orbitals, etc. Most importantly, the valence orbital energies, which are taken as matrix elements of an IVO-type Fock operator,22,25,27,28 must be averaged in order to ensure at least asymptotic convergence of the perturbation series.40,42,45 The Hν calculations are performed for the protonated Schiff bases propeniminium and pentaniminium. The molecular geometries are depicted in Figure 1. The propeniminium structure is that used in the MRSDCI studies.30 The Hν calculations utilize the correlation consistent PVDZ basis set.46
π-Electron Hamiltonians for Protonated Schiff Bases TABLE 1: Excitation Energy and Ionization Potential Calculations of the π-Electron States of the Propeniminium Cation:a Comparison of Hν Results with MRSDCIb Calculations state
Hν1stc
Hν2ndd
Hν3rde
Hν3rd-PPPf
MRSDCI
21A′ 31A′ 13A′ 23A′ 12A′′
6.58 8.81 4.09 7.68 15.98
5.98 8.46 3.87 6.89 15.62
6.01 7.85 3.60 6.17 15.59
5.91 7.76 3.54 5.99 15.03
5.96(5.67) 7.63(7.76) 3.75 6.35 15.78
a Energies in eV. b Multireference single and double configuration interaction calculations.31 Values in parentheses correspond to limited excitation CI plus Davidson correction. c First-order constrained Hν, or PHP, which is equivalent to π-space CI calculation. d Second-order constrained Hν. e Third-order constrained Hν. f Third-order constrained Hν, but retaining only the PPP Ri, βi,j, and γi,j effective integrals.
TABLE 2: Comparison of π-Electron Excitation Energy and Ionization Potential Calculations of the π-Electron States of the Pentaniminium Cation:a Comparison of Hν Results with CASSCF and CASSCF+MP2b Calculations state
Hν2ndc
Hν3rdd
Hν3rd-PPPe
CASSCFb
CASSCF+MP2b
21A′ 31A′ 13A′ 23A′ 12A′′
4.51 6.08 2.87 5.16 13.50
4.50 5.80 2.76 4.68 13.25
4.66 6.03 2.85 4.78 12.56
2.00
4.77
a Energies in eV. b As implemented in Gaussian94.13 Both calculations employ a 6-31G* basis, 4 electrons, and 4 π electrons. c Secondorder constrained Hν. d Third-order constrained Hν. e Third-order constrained Hν, but retaining only the ab initio PPP-like Ri, βi,j, and γi,j effective integrals.
All calculations employ valence spaces constructed from linear combinations of atomic pπ orbitals, where the basis set coefficients for the pπ orbital are taken from isolated carbon or nitrogen atom HF calculations. The Hν calculations provide all valence states, including the ion states, with a single calculation. Comparisons are made against the MRSDCI studies and against CASSCF+MP2 Gaussian94 calculations.13 III. Results Generally third-order Hν (denoted as Hν3rd) calculations on small polyenes perform as well as, if not better than, the best correlated ab initio methods.11,22-24,27,28 On the other hand, the second-order (Hν2nd) calculations perform less satisfactory for these molecules. For propeniminium and pentaniminium, however, the Hν2nd calculations readily describe the low lying vertical excitation energies (EEs) and ionization potentials (IPs), comparing well with both the Hν3rd and other, highly correlated ab initio calculations. Table 1 summarizes the EEs and IPs of propeniminium computed with Hν and MRSDCI calculations. The Hν3rd and MRSDCI results differ by a mere 0.18 eV at most, and that is for the higher lying and less interesting 23A′ state. Given that the MRSDCI studies involve a number of approximations and somewhat arbitrary choices, the Hν3rd calculations predict virtually identical spectra. This comes as no surprise. In addition, except for the 23A′ state, the Hν2nd values are likewise of nearly comparable accuracy to the Hν3rd and MRSDCI calculations. Although no MRSDCI studies of pentaniminium are available, Table 2 demonstrates that the Hν2nd performs equally well for this molecule, as compared to Hν3rd, CASSCF, and CASSCF+MP2 calculations. The CASSCF calculation, employing only four valence electrons, underestimates the lowest 21A′ by about 2.5 eV or greater as compared to the CASSCF+MP2 or the Hν3rd calculations. In contrast, the Hν2nd calculations yield this lowest EE as 4.51 eV, which is extremely
J. Phys. Chem., Vol. 100, No. 34, 1996 14313 TABLE 3: Comparison of Theoretical (Hν1st-PPP) and Hν3rd-PPP Parameters for the Propeniminium (C3H4NH2+) and Pentaniminium (C5H6NH2+) Cationsa parameter C3H4NH2+ 1st C5H6NH2+ 1st C3H4NH2+ 3rd C5H6NH2+ 3rd R1 R2 R3 R4 R5 R6 β2,1 β3,2 β4,3 β5,4 β6,5 γ1,1 γ2,2 γ3,3 γ4,4 γ5,5 γ6,6 γ2,1 γ3,1 γ3,2 γ4,1 γ4,2 γ4,3 γ5,1 γ5,2 γ5,3 γ5,4 γ6,1 γ6,2 γ6,3 γ6,4 γ6,5 a
-44.8856 -39.8806 -37.4894 -31.0230 -3.7034 -3.0428 -3.6622
19.8640 16.7331 16.8996 16.3438 9.3445 5.6459 8.5409 3.8452 5.5679 8.8026
-50.4378 -46.6699 -47.1333 -44.7955 -41.7294 -34.9557 -3.71182 -3.02734 -3.63172 -3.09041 -3.70944 19.8745 16.7194 16.8944 16.8928 16.8920 16.3550 9.3433 5.7452 8.5384 3.9010 5.6211 8.9133 2.9691 3.7537 5.6333 8.5607 2.3740 2.9231 3.8175 5.5716 8.8088
-40.8675 -37.5950 -35.0309 -30.4671 -3.5390 -2.8958 -3.4118
15.2553 13.9359 13.4712 12.9528 8.1730 5.8642 7.7159 4.3925 5.7099 7.6239
-47.1835 -44.8118 -44.2670 -42.6878 -39.7809 -34.8368 -3.6184 -2.9126 -3.4648 -2.8604 -3.3768 15.0405 13.3508 12.8183 12.8049 12.6269 12.3566 8.3304 5.9344 7.6767 4.3728 5.6198 7.5860 3.4575 4.2043 5.5782 7.4810 2.8168 3.3695 4.2050 5.5808 7.5855
Effective integrals in eV.
close to the Hν3rd result of 4.50 eV. And again and except for the 23A′ state, the Hν2nd and Hν3rd EEs and IP differ by at most 0.28 eV. An even better approximation than the Hν2nd is found from ν H 3rd calculations that only retain the PPP-like Ri, βi,j, and γi,j effective integrals. This approximation is denoted as the Hν3rdPPP model. When compared to the highly accurate Hν3rd calculations, the Hν3rd-PPP EEs differ by at most 0.18 eV for propeniminium and 0.23 eV for pentaniminium. Such excellent results could not be obtained for the neutral polyenes without retaining many more Hν3rd effective integrals, even including some three-electron terms.27,28 The vertical ionization potentials show greater deviations in the Hν3rd-PPP model for both molecules, although the first-, second-, and third-order calculations exhibit considerably smaller differences. The IP is more sensitive to the neglect of the threeelectron integrals. A more accurate, approximate π-electron Hamiltonian must retain more parameters, as is done in the simple first-order calculations, in order to reproduce both excitation energies and ionization potentials. It is of interest to compare the very accurate, correlated Hν3rd ab initio Ri, βi,j, and γi,j parameters with the first-order Hν (Hν1stPPP) integrals, the so-called “theoretical” integrals in the traditional PPP theory, in order to evaluate the various assumptions in standard PPP methods. Table 3 presents both sets of integrals for propeniminium and pentaniminium in the Lo¨wdin orthogonalized basis of pπ atomic orbitals. One of the most important assumptions of any semiempirical Hamiltonian is that the parameters transfer well between different molecules, just as the bare interactions do. (Of course, the individual bond lengths must be exactly the same for a direct evaluation of the transferability of the integrals across a given
14314 J. Phys. Chem., Vol. 100, No. 34, 1996 bond.) While the bare interactions do not transfer exactly because of the Lo¨wdin orthogonalization, the correlated Hν3rdPPP parameters deviate between different molecules by at least an order of magnitude more than the “theoretical” integrals. The first-order γ1,1 (d〈11|(1/R1,2)|11〉) integral is 19.8640 in propeniminium and 19.8745 in pentaniminium, a difference of a mere 0.01 eV, whereas the correlated Hν3rd-PPP γ1,1 differ by an order of magnitude more. The γ1,1 for propeniminium and pentaniminium are 15.0405 and 15.2553, respectively. Likewise, while the bare, carbon γi,i integrals differ by about 0.01 eV between the two Schiff bases for each i, the Hν3rd-PPP γi,i values differ by over 0.5 eV for γ2,2 and γ3,3, while γ4,4 differs by 0.15 eV. As above, the Hν3rd-PPP parameters transfer less well within the same molecule when compared to the first-order integrals. The first-order integrals indicate that a good HPPP needs only three different carbon γi,i values for similar protonated Schiff bases: γ2,2, γend,end, and γmid,mid, where end denotes the end carbon, and mid denotes all other (middle) carbons, except 2. At first order, γ2,2 lies in the range 16.72-16.73 eV, γend,end spans 16.34-16.35 eV, and γmid,mid is in the range 16.90-16.89 eV. The differences between the first-order γi,i reflect only the Lo¨wdin orthogonalization. The Hν3rd-PPP γi,i differ by greater amounts because of the ab initio correlation corrections; however, similar trends do arise in the larger protonated schiff base. For pentaniminium, γ2,2 is 13.35 eV, γmid,mid ranges from 12.82 to 12.63 eV, and γend,end is 12.36 eV. In contrast to the first-order integrals, however, these values do not transfer between the two polyenes. It is unclear whether the pentaniminium values will transfer to larger protonated schiff bases. Only more extensive ab initio Hν calculations will elucidate the subtle transferabilities of the ab initio PPP-like effective integrals. The two-center Hν3rd-PPP βi,j and γi,j transfer better than the effective γi,i integrals, no doubt because they contain substantially less correlation contributions and, consequently, depend less on their molecular environments. Both the first- and thirdorder βi,j values transfer to within about 0.5 eV, and the Hν1stPPP and Hν3rd-PPP γi,j both transfer to within 0.03 eV, except for the Hν3rd-PPP γ3,2 and γ5,4 integrals, which differ by 0.2 eV. In fact, the first-order and third-order βi,j integrals differ by about a constant 0.2 eV. This is consistent with the Hν βi,j integrals of other polyenes and most likely represents a general trend for any improved ab initio PPP model. Also, as with other polyenes, the βi,j are significantly (about 0.5 eV) less than typical semiempirical PPP values (although keep in mind that this difference may arise because the semiempirical PPP βi,j may effectively include averages of the Hν 〈ii|Vν1,2|ij〉 and 〈iii|Wν1,2,3|iij〉 parameters, as described briefly in reference 28). IV. Discussion The ab initio effective valence Hamiltonian (Hν) theory provides both the proper theoretical basis for an improved semiempirical π-electron theory and a prescription for computing all of the π-electron effective integrals from first principles.21,28 When applied to small polyenes, Hν theory explains the major components of the Pariser-Parr-Pople (PPP) semiempirical π-electron theory, but the resulting constrained ab initio Hν calculations fail to reproduce the lowest lying singlet states with high accuracy.26-28 The fault lies not in the Hν theory per se, but appears because of difficulties in finding a good ab initio zeroth-order model for the low lying singlet states of many polyenes. Consequently, while Hν calculations perform exceptionally well if designed solely to reproduce spectral data,22-24,27,28 calculations designed to mimic semiempirical models employ
Martin more approximate orbitals and valence spaces. The calculations perform less adequately because of the presence of and the mixing between low lying ionic and Rydberg states. The pathological problems in simple polyenes do not appear for protonated Schiff bases, such as retinal31 and its smaller analogs,30 because their low lying excited states contain only covalent configurations and lack low lying Rydberg states. Thus, the ground and excited states require similar amounts of electron-electron correlation, and even the simplest ab initio models, such as open shell HF, π-CI, and semiempirical PPP9 calculations, provide a qualitatively sufficient picture of the lowest excited states of these systems. For this reason and because of their biological relevance, small analogs of retinal serve as the perfect candidates for further trials of the Hν theory, both as pure ab initio method and as a device for extracting corrected semiempirical models from ab initio calculations. New Hν calculations are performed for propeniminium (C3H4NH2+) and pentaniminium (C5H6NH2+), two small protonated Schiff bases, and the computed vertical excitation energies and ionization potentials are compared to MRSDCI,30,31 CASSCF, and CASSCF+MP2 (OVB MP2) calculations.12,13 The Hν method performs extremely well, even at second order, whereas previous Hν calculations for polyenes require third-order calculations for a reasonable description of the π-electron spectra. For propeniminium, second-order Hν (Hν2nd) calculations with constrained valence spaces treat the two lowest excited states as well as the ionization potential quite well, differing from the MRSDCI and Hν3rd calculation by no more than 0.2 eV. Pentaniminium Hν2nd calculations provide equally adequate values for the ionization potential (12A′′) and the lowest four π-electron excited state excitation energies (21A′, 31A′, 11A′′, 21A′). In addition and more remarkably, an even better approximate description of the excitation energies arises from Hν3rd constrained calculations which only retain the PPP-like one- and two-electron effective integrals Ri, βi,j, and γi,j. For the first time, the Hν ab initio theory provides an ab initio PPP Hamiltonian which accurately describes all low lying states, including the lowest singlet states, of a moderate size molecule. It is now possible to examine all of the assumptions of the simple PPP model in full detail within the framework of a suitably accurate ab initio model. For instance, the parameters with the largest correlation contributions, the γi,i and the shortrange γi,j, do not transfer well between different molecules nor between intramolecular carbon atoms, whereas the βi,j thirdorder integrals transfer well and only differ by about a constant 0.2 eV from the bare first-order integrals. The pentaniminium γi,i carbon parameters do exhibit a trend which also arises with the bare, Lo¨wdin orthogonalized integrals: γ3,3, γ4,4, and γ5,5 all lie within 0.02 eV, whereas γ2,2 and γ6,6 take on markedly different values. Most recent calculations indicate that many of these trends hold for even larger protonated schiff bases.47 The current success demonstrates the feasibility of the Hν approach and will hopefully stimulate support for and new interest in continuing studies to develop a new and improved, ab initio based π-electron theory suitable for treating biological chromophores. In closing, it is most fortunate that these large biological chromophores, the protonated Schiff bases, lend themselves so readily to ab initio studies. The ability to perform accurate Hν calculations for the protonated Schiff bases lies in the fact that these polyenes lack low lying ionic valence-like states and therefore require less electron-electron correlation and do not suffer the Rydberg-valence mixing problems common in other polyenes. It is hypothesized that Hν2nd and Hν3rd-PPP calcula-
π-Electron Hamiltonians for Protonated Schiff Bases tions will perform equally well for other polyenes placed in condensed phases or proteins because the electrostatic interactions with the environment should drive any low lying diffuse states far up in energy as well as stabilize the low lying singlet states. Such behavior has been described for biological chromophores.48 An avenue of possibly very fruitful research lies in studying these biological polyenes, both with Hν theory and other ab initio methods. Acknowledgment. The authors thanks the National Science Foundation Postdoctoral Fellowship in Chemistry to pursue this research. All calculations have been performed utilizing the resources in the Freed group at the University of Chicago and in the Theoretical Biophysics Group at the Beckman Institute, University of Illinois at Urbana-Champaign. The author also thanks Karl Freed and Isiah Shavitt for useful discussions and encouragement. References and Notes (1) Stuart, J. A.; Vought, B. W.; Zhang, C.-F.; Birge, R. R. Biospectroscopy 1995, 1, 8. (2) McDermott, G.; et al. 1995, 374, 517. (3) Barlow, R. R.; Birge, R. R.; Kaplan, E.; Talent, R. J. Nature 1993, 366, 64. (4) Pople, J. A. Int. J. Quantum Chem. 1990, 37, 349. (5) Jug, K. Theor. Chim. Acta 1969, 74, 91. (6) Parr, R. G. The Quantum Theory of Molecular Electronic Structure; Benjamin: New York, 1963. (7) Warshel, A.; Karplus, M. J. Am. Chem. Soc. 1974, 96, 5677. (8) Tavan, P.; Schulten, K. J. Chem. Phys. 1974, 70, 5407. (9) Schulten, K.; Dinur, U.; Honig, B. J. Chem. Phys. 1980, 73, 3978. (10) Serrano-Andre´s, L.; Marcha´n, M.; Nebot-Gil, I.; Lindh, R.; Roos, B. O. J. Chem. Phys. 1993, 98, 3151. (11) Roos, B. O.; Fu¨lscher, M. P.; Malmqvist, P.-A° .; Mercha´n, M.; Serrano-Andre´s, L. Theoretical studies of electronic spectra of organic molecules. In Quantum Mechanical Electronic Structure Calculations with Chemical Accuracy; Langhoff, S. R., Ed.; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1995. (12) McDouall, J. J.; Peasly, K.; Roob, M. A. Chem. Phys. Lett. 1988, 148, 183. (13) As implemented in the following: Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Gill, P. M. W.; Johnson, B. G.; Robb, M. A.; Cheeseman, J. R.; Keith, T.; Petersson, G. A.; Montgomery, J. A.; Raghavachari, K.; Al-Laham, M. A.; Zakrzewski, V. G.; Ortiz, J. V.; Foresman, J. B.; Peng, C. Y.; Ayala, P. Y.; Chen, W.; Wong, M. W.; Andres, J. L.; Replogle, E.
J. Phys. Chem., Vol. 100, No. 34, 1996 14315 S.; Gomperts, R.; Martin, R. L.; Fox, D. J.; Binkley, J. S.; Defrees, D. J.; Baker, J.; Stewart, J. P.; Head-Gordon, M.; Gonzalez, C.; Pople, J. A. Gaussian 94, Revision B.3; Gaussian, Inc.: Pittsburgh, PA; 1994. (14) Zerner, M. C. In ReViews in Computational Chemistry; Boyd, D., Ed.; VCH: New York, 1991; Vol. II. (15) Warshel, A.; Chu, Z. T.; Hwang, J.-K. 1991, 158, 303. (16) Ho, L. L.; MacKerell, A. D.; Bash, P. A. J. Am. Chem. Soc., in press. (17) Thompson, M. J. Phys. Chem. 1995, 99, 6374. (18) Xu, D.; Martin, C.; Schulten, K. Biophys. J. 1996, 70, 453. (19) Sun, H.; Freed, K. F.; Herman, M. F.; Yeager, D. L. J. Chem. Phys. 1980, 72, 4158. (20) Freed, K. F. Lect. Notes Chem. 1989, 52, 1. (21) Freed, K. F. In Conceptual Trends in Quantum Chemistry; Kayachko, E., Calais, J.-L., Eds.; Kluwer: Amsterdan, 1995; Vol. II. (22) Graham, R. L.; Freed, K. F. J. Chem. Phys. 1992, 96, 1304. (23) Martin, C. H.; Graham, R. L.; Freed, K. F. J. Chem. Phys. 1993, 99, 7833. (24) Martin, C. H.; Graham, R. L.; Freed, K. F. J. Phys. Chem. 1994, 98, 3467. (25) Martin, C. H.; Freed, K. F. J. Chem. Phys. 1994, 100, 7454. (26) Martin, C. H.; Freed, K. F. J. Chem. Phys. 1994, 101, 4011. (27) Martin, C. H.; Freed, K. F. J. Chem. Phys. 1994, 101, 5929. (28) Martin, C. H.; Freed, K. F. J. Phys. Chem. 1995, 99, 2701. (29) Lee, S. Y.; Freed, K. F. J. Chem. Phys. 1996, 104, 3260. (30) Du, P.; Racine, S. C.; Davidson, E. R. J. Phys. Chem. 1990, 94, 3944. (31) Du, P.; Davidson, E. R. J. Phys. Chem. 1990, 94, 7013. (32) Roos, B. O.; Andersson, K.; Fulscher, M. P. Chem. Phys. Lett. 1992, 192, 5. (33) Freed, K. F. Chem. Phys. Lett. 1972, 13, 181. (34) Freed, K. F. Acc. Chem. Res. 1983, 16, 137. (35) Iwata, S.; Freed, K. F. J. Chem. Phys. 1976, 65, 1071. (36) Graham, R. L.; Freed, K. F. unpublished work. (37) Martin, C. H.; Freed, K. F. J. Chem. Phys., in press. (38) Hay, P. J.; Shavitt, I. J. Chem. Phys. 1974, 60, 2865. (39) Haque, W. J. Chem. Phys. 1977, 67, 3631. (40) Finley, J. P.; Chaudhuri, R. K.; Freed, K. F. J. Chem. Phys. 1995, 103, 4990. (41) Davidson, E. R. In ReViews In Computational Chemistry; Boyd, D., Ed.; VCH: New York, 1990; Vol. I. (42) Freed, K. F.; Sheppard, M. G. J. Phys. Chem. 1982, 86, 2130. (43) Brandow, B. H. Int. J. Quantum Chem. 1979, 15, 207. (44) Sheppard, M. G.; Freed, K. F. J. Chem. Phys. 1981, 75, 4507. (45) Kaldor, U. In Many Body Methods in Quantum Chemistry; Kaldor, U., Ed.; Springer-Verlag: Berlin, 1988. (46) Dunning, T. H. J. Chem. Phys. 1989, 90, 1007. (47) Martin, C. H. Chem. Phys. Lett., accepted. (48) Birge, R. R. Acc. Chem. Res. 1986, 19, 138.
JP960617W
