Atomic integral driven second order polarization propagator calculations of the excitation spectra of naphthalene and anthracene

Transcription

1 JOURNAL OF CHEMICAL PHYSICS VOLUME 112, NUMBER 9 1 MARCH 2000 Atomic integral driven second order polarization propagator calculations of the excitation spectra of naphthalene and anthracene Keld L. Bak UNI-C, Olof Palmes Allé 38, DK-8200 Aarhus N, Denmark Henrik Koch and Jens Oddershede Department of Chemistry, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark Ove Christiansen Theoretical Chemistry, Chemical Centre, University of Lund, P.O. Box 124, S Lund, Sweden Stephan P. A. Sauer a) Chemistry Laboratory IV, Department of Chemistry, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen O, Denmark Received 16 August 1999; accepted 1 December 1999 An atomic integral direct implementation of the second order polarization propagator approximation SOPPA for the calculation of electronic excitation energies and oscillator strengths is presented. The SOPPA equations are solved iteratively using an integral direct approach and, contrary to previous implementations, the new algorithm does not require two-electron integrals in the molecular orbital basis. The linear transformation of trial vectors are calculated directly from integrals in the atomic orbital basis. In addition, the eigenvalue solver is designed to work efficiently with only three trial vectors per eigenvalue. Both of these modifications dramatically reduce the amount of disk space required, thus, increasing the range of applicability of the SOPPA method. Calculations of the lowest singlet excitation energies and corresponding dipole oscillator strengths for naphthalene and anthracene employing basis sets of 238 and 329 atomic orbitals, respectively, are presented. The overall agreement of our results with experimental spectra is good. The differences between the vertical excitation energies calculated by SOPPA and the position of the maximum intensity peaks in the experimental spectra are within the range of 0.35 ev with two exceptions, the 4 1 A g state of naphthalene and anthracene where a 0.85 ev and 0.41 ev deviation is found, respectively. The relatively large discrepancy for this transition is due to large contributions from two-electron excitations which cannot accurately be described in SOPPA. For naphthalene we find additional excitations to Rydberg states of 1 A u and 1 B 2u symmetry as compared with previous calculations American Institute of Physics. S I. INTRODUCTION Since the early 1980 s the second order polarization propagator approximation SOPPA 1 has been successfully applied to the calculations of electronic excitation energies, oscillator strengths and several linear response properties of small molecules. With a computational scaling comparable to second order Mo ller Plesset perturbation theory, SOPPA is also well suited for studies of electronic excitation spectra and response properties of larger molecules. The difficulty in applying SOPPA to larger molecules has previously been due to limitations caused by the program implementations. For example the implementations of SOPPA in the MUNICH 2,3 and RPAC 4,5 program packages are based on a partitioned form of the matrices that define the generalized eigenvalue equation for excitations and the linear equations for response properties, and they require the construction and storing of all matrix elements. In the most recent implementation of SOPPA, 6 in the DALTON program package, 7 the equations are solved iteratively, and in contrast a Corresponding author, electronic mail: sps@ithaka.ki.ku.dk to both the RPAC and the MUNICH programs, the unpartitioned form of the SOPPA matrices is used. The paired structure of the matrices is used to stabilize and enhance the convergence. 8,9 Linear transformations of the trial vectors are calculated directly from the molecular orbital integrals without an explicit construction of the matrices, thereby reducing the operation count from N 6 to N 5 where N is the number of orbitals. The DALTON implementation has lead to a significant increase in the application range of SOPPA as demonstrated in recent publications. 6,10,11 However, the DALTON implementation requires a full integral transformation and the number of trial vectors which must be stored for the iterative solution of the eigenvalue problem 8,9 increases with every iteration. Thus disk space requirements severely restrict the size of the molecules which can be handled by this implementation. We present a second implementation of SOPPA in the DALTON program, where the disk space requirements are reduced by: performing the linear transformation of the trial vectors directly from the atomic orbital two-electron integrals; /2000/112(9)/4173/13/$ American Institute of Physics

2 4174 J. Chem. Phys., Vol. 112, No. 9, 1 March 2000 Bak et al. using only a fixed number of trial vectors in each iteration, typically three per excitation. The linear transformation of a trial vector is calculated directly from the atomic orbital integrals using a scheme similar to the one previously used in the integral-direct coupled cluster singles and doubles CCSD implementation by Koch et al. 12 The integrals are processed as distributions with one fixed and three free orbital indices. Only one distribution at a time is needed in memory and can be read from a file containing all integrals or calculated when needed. Only a fixed number of optimal trial vectors of the current and previous iterations are kept. Together with the new trial vectors these define the basis for the next iteration. 13 This significantly reduces the number of trial vectors to be stored and it hardly increases the number of iterations. The new implementation is applied to the calculation of the lowest vertical electronic excitation energies and oscillator strengths of naphthalene and anthracene. Naphthalene has previously been investigated with SOPPA, 6 however, using a basis set that was inferior to the basis set used for benzene in the same study. This motivated us to reinvestigate the excitation spectrum of naphthalene with a better basis set and also to investigate anthracene using the same quality basis set. For anthracene this is the first SOPPA calculation of the lowest electronic excitation energies and oscillator strengths. II. THEORY A. The SOPPA equations The electronic excitation energies l are in SOPPA 1,6 calculated by solving the generalized eigenvalue equation E [2] l S [2] X l 0, 1 and the corresponding transition moments are obtained from the eigenvectors X l as P [1] X l. The Hessian matrix E [2], the metric S [2], and the property vector P [1] are defined as and E [2] A(0,1,2) C (1) B (1,2)* 0 C (1) D (0) 0 0 B (1,2) 0 A (0,1,2)* C (1)* (0)*, C (1)* D S S(0,2) [2] 0 0 S (0,2)* , P [1] 0 Pˆ,q 0 (0,2) 0 Pˆ,R 0 (1) 0 Pˆ,q 0 (0,2) 0 Pˆ,R 0 (1), 4 respectively. The orders, in terms of the fluctuation potential, to which the matrix elements are evaluated are indicated by the superscripts n. The orders are based on a partitioning of the nonrelativistic Hamiltonian Ĥ into the inactive Fock operator Fˆ and the fluctuation potential Vˆ Ĥ Fˆ Vˆ, such that the reference state 0 is expanded in a Mo ller Plesset perturbation series 14,15 0 HF (0) 0 (1) 0 (2). The Hartree Fock ground state HF (0) is zeroth order in the fluctuation potential while the corrections are first and second order. The SOPPA method is defined to include the orders indicated, such that the calculated excitation energies and transition moments as well as the propagator, i.e., the response, itself are correct through second order in the fluctuation potential for single excitations, but not for higher excitations. 1 The row vectors q, R, q and R consist of single and double excitation operators and single and double deexcitation operators with respect to the Hartree Fock ground state, respectively. The operator Pˆ denotes the one-electron operator for which the transition moment is calculated; in the present work this means components of the dipole operator. Explicit expressions for the matrix elements on the right hand side of Eqs. 2 4 are not given here but may be found elsewhere. 6,16 Notice that the C (1) 1 matrix in Eq. C16 of Ref. 16 has the incorrect sign. In accordance with Eqs. 2 4 the eigenvector is often written as a combination of four components Xl X l ph X l 2p2h hp X l X 2h2p, l where the components refer to single and double excitation amplitudes, X l ph and X l 2p2h, and to single and double deexcitation amplitudes, X l hp and X l 2h2p. The eigenvectors are usually 16 normalized with the S [2] matrix as metric, i.e., 1 X l S [2] X l X l ph S (0,2) X l ph X l hp S (0,2) X l hp X l 2p2h X l 2p2h X l 2h2p X l 2h2p. As described in Sec. C we solve the generalized eigenvalue problem by an iterative algorithm using trial vectors bl b l ph b l 2p2h hp b l b 2h2p, l to span the eigenvectors X l. In our implementation, as in the work of Packer et al., 6 the SOPPA matrices E [2] and S [2] are not constructed explicitly, but instead the linear transformed trial vectors, E [2] b l and S [2] b l, are evaluated directly from basic entities such as integrals, orbital energies and amplitudes. The expressions for the linear transformed trial vectors have been given previously. 6 However, in this work we use a biorthonormal representation of the doubly excited configurations 17,18 and therefore we give expressions for the elements of the linear

3 J. Chem. Phys., Vol. 112, No. 9, 1 March 2000 Atomic orbital driven SOPPA 4175 transformed trial vectors that are influenced by this choice. For the remaining elements we refer to Packer et al. 6 The biorthonormal representation of the doubly excited states are generated by the double excitation and deexcitation operators C (1)* and D (0)* are analogous to Eqs The double excitation part of the property vector is also influenced by the use of the biorthonormal basis and reads 0 Pˆ,R 0 (1) aibj 0 Pˆ,S aibj R 0 (1) S aibj R 1 2 E ai E bj, k P ik ab kj P jk ab ik 1 S iajb L 3 1 ab ij 2E iae jb E ja E ib, 11 where ai bj and E ai, a a a i is the standard orbital rotation operator with a p, a p being the secondquantization creation and annihilation operators, respectively. The indices i, j,k,l and a,b,c,d denote spatial molecular orbitals which are occupied and unoccupied, respectively, in the Hartree Fock ground state. The parts of the linear transformed trial vector that results from multiplications with C (1), C (1), D (0) and the complex conjugated of those, are the ones that are influenced by the biorthonormal basis. For these we obtain C (1) b ph aibj ck 0 S jbia L, Ĥ,E ck 0 (1) ph b ck 2 1 ij ab c k C (1) b 2p2h ai 1 and 2 bjck ki bj b ak ac bj b ph ci ai bc b ph cj ph ai kj b ph bk, 12 0 E ia, Ĥ,S bjck R 0 (1) 2p2h b bjck 1 2p2h 2 kc ab kb ac b bick 2 bck 2p2h 2 jb ac jc ab b bjci bcj 2p2h 2 kc ji ki jc b ajck cjk 2p2h, bjk 2 ki jb kb ji b bjak 13 D (0) b 2p2h aibj 0 S jbia L, Ĥ,S ckdl R 0 (0) 2p2h b ckdl ckdl a b i j b 2p2h 1 ab ij aibj b 2p2h bjai, 14 where (pq rs) is a molecular orbital two-electron integral and p is the canonical Hartree Fock molecular orbital energy. The terms involving the complex conjugated C (1)*, 1 P 2 ca cb ij P cb ac ij, 15 c where P pq p Pˆ q is a molecular integral over the one electron property operator Pˆ. The ab ij vector is defined from the first order Mo ller Plesset double excitation correlation coefficients 14 ab ij ai bj, i j a b 16 as ab ij 4 ab ij 2 ab ji. 17 The double de-excitation part of the property vector is analogous to the expression in Eq. 15. The elements in Eqs and the ones in the work of Packer et al. 6 are evaluated from two-electron integrals in the molecular orbital basis. In our implementation the calculation is driven over atomic orbital integrals and the transformation of the atomic orbital integrals is performed on the fly either directly or by back transforming amplitudes or trial vectors with the molecular orbital coefficients, C i and C a, for the occupied and unoccupied molecular orbitals, respectively. Greek letters,, and are used to label atomic orbitals. B. The atomic orbital integral direct algorithm The atomic orbital two-electron integrals are handled the same way as in the recent integral-direct implementation of the coupled cluster single and doubles CCSD model. 12 The linear transformation of the trial vector is implemented with the outer loop over the atomic shells in the molecule. All atomic orbital integrals ( ) for belonging to the atomic shell are calculated and written to disk. The next loop is over belonging to the shell. For a given all atomic orbital two-electron integrals are read into memory and processed. As an example of how this is implemented, we show the loop structure and the individual steps involved in calculating the linear transformed (C (1) b ph ) aibj elements of Eq. 12 which we first rewrite as C (1) b ph aibj 2 ai bj bj ai 1 ij ab with ai bj c ac bj b ph ci k ki bj b ph ak In our implementation this term is calculated by the following structure

4 4176 J. Chem. Phys., Vol. 112, No. 9, 1 March 2000 Bak et al. If available disk capacity is sufficient, the two-electron integrals will be calculated only once and stored on disk. The loop over atomic shells can then be removed and instead the loop over is extended to cover all atomic orbitals. C. The equation solver The algorithm used for solving the generalized eigenvalue problem is similar to the generalized Davidson Liu algorithm described by Olsen et al. 8,9 One difference is that we only use a maximum of usually three trial vectors per excitation. As advised by Olsen 13 these are 1 a new trial vector generated from the residual vector, 2 the optimal trial vector from the current iteration and 3 the optimal trial vector from the previous iteration. Experience has shown that with this number of trial vectors the number of iterations needed to converge the excitations is identical to, or only slightly higher than, the number of iterations needed if all generated trial vectors are used to expand the solution vector. Additional optimal trial vectors from previous iterations can be kept, if more than three trial vectors per excitation is requested. As in the algorithm by Olsen et al. 8,9 the paired structure of the matrices and eigenvectors is exploited. This implies that for each trial vector there is also a so-called partner trial vector, which is generated by permuting the excitation and de-excitation parts of the original trial vector. The importance of the partners for the stability and convergence of the iterative scheme has been discussed by Olsen et al. 8,9 After the first two iterations the total number of trial vectors is thus 3 2 N, where N is the number of excitations to be calculated. Assuming that only three trial vectors per excitation have been requested the implemented algorithm involves the following steps: 1 Generate a trial vector b l for each excitation l. Three possibilities for choosing these initial trial vectors are currently implemented. i Solution vectors for the l lowest eigenvalues of Eq. 1 in a diagonal approximation. ii Solution vectors from the corresponding RPA eigenvalue problem augmented with zeros in double excitation and de-excitation parts of the trial vectors. iii Solution vectors from the corresponding RPA D vectors defined in Ref Orthonormalize the new set of N trial vectors and their partners against the optimal trial vectors b l opt including partners from the previous iterations if any and against each other. The S [2] matrix is hereby used as metric. 3 Generate the linear transformed trial vectors u l E [2] b l and m l S [2] b l. Linear transformations of the partners to the b l s are not needed as they are known implicitly from the u l s and m l s. 4 Calculate as scalar products of the b l s and u l s or the b l s and m l s and their partners, the elements of E [2] and S [2] in the space spanned by the trial vectors. This space is called the reduced space with a maximum dimension of 3 2 N. 5 Solve the eigenvalue problem in the reduced space and keep the N lowest eigenvalues and corresponding reduced eigenvectors.

5 J. Chem. Phys., Vol. 112, No. 9, 1 March 2000 Atomic orbital driven SOPPA 4177 TABLE I. Iterations needed to converge the singlet excitations in benzene using three trial vectors per excitation and, in parentheses, using all generated trial vectors. a Symmetry b A g B 3u B 2u B 1g B 1u B 2g B 3g A u 1. excitation a Excitation vectors are converged to 10 4 a.u., i.e., the excitation energies to The RPA D solution vectors are used as initial trial vectors. b In D 2h point group symmetry. 6 Orthonormalize the N reduced eigenvectors and their partners of the previous iteration if any against the N new reduced eigenvectors and their partners and against each other. 7 Calculate new optimized trial vectors b opt opt l,i,b l,i 1 of the current, i, and previous if any, i 1, iteration as a linear opt opt combination of the trial vectors b l,b l,i 1,b l,i 2. The coefficients for the linear combinations are given by the reduced eigenvectors obtained in step 5 and 6. New optimized linear transformed trial vectors, u opt opt l,i,u l,i 1 s and m opt opt l,i,m l,i 1 are obtained similarly. 8 Calculate from the linear transformed trial vectors the residual vectors r l u opt l,i l m opt l,i. The l are the eigenvalues obtained from the reduced eigenvalue problem in 5. Calculate also the norm of each of the residual vectors. 9 If all the norms of the residual vectors r l are less than the defined threshold stop the iteration, otherwise go to Calculate a new trial vector b l for all excitations with norms larger than threshold. These are obtained by dividing the corresponding residual vectors with the zeroth order diagonal parts of E [2] l S [2]. 11 Go to 2. To illustrate the convergence behavior of the implemented algorithm we have performed a calculation on benzene. The geometry and basis set were the same as in the previous study by Packer et al. 6 and we consider the same 15 singlet excitations as determined in that study. Seven of the excitations are degenerate and show up twice in our calculations since we use D 2h point group symmetry. Therefore, in total 22 excitations are calculated and the norm of the residual vector for these excitations are converged to 10 4 a.u., i.e., the excitation energies are converged to 10 8 a.u. The convergence criterium is more strict than really needed for the excitation energies, but is chosen to better illustrate the efficiency of the algorithm. The RPA D vectors 22 are used as initial trial vectors. The number of iterations for each of the excitations are listed in Table I. They vary between 7 and 13 iterations. For 17 out of the 22 excitations, the number of iterations is the same when using 3 trial vectors per excitation as when using all calculated trial vectors. For the remaining five excitations one more iteration is needed when using only three instead of all trial vectors. Thus, it appears that the number of iterations needed to converge with the current algorithm is identical to or only slightly higher than the number of iterations needed if all trial vectors are kept and used in every iteration. III. COMPUTATIONAL DETAILS The atomic orbital driven SOPPA program as implemented in a local version of Dalton has been used to calculate the lowest singlet singlet vertical excitation energies and corresponding oscillator strength for naphthalene and anthracene. For comparison we have also calculated the excitations energies of anthracene using RPA. The excitations are characterized by considering the dominant amplitudes in the excitation vectors and the x 2, y 2, and z 2 expectation values of the corresponding molecular orbitals. It should be kept in mind that such assignments always are qualitative and to some extent debatable. To reduce the computational effort, excitations out of the core orbitals are neglected in the calculations and the core orbitals corresponding to carbon 1s) are also kept frozen when computing the Mo ller Plesset correlation coefficients. To determine the implication of this frozen core approximation we have applied it in a calculation on benzene. The geometry and basis set were the same as in the previous study by Packer et al. 6 For the 15 singlet excitations previously reported the deviations between the frozen core and the all orbital excitation energies are less than ev in RPA and less than 0.01 ev in SOPPA. The oscillator strengths deviate less than in RPA and less than 0.01 in SOPPA. The effect of the frozen core approximation on the naphthalene and anthracene results is therefore expected to be negligible. For naphthalene we used the ground state equilibrium geometry calculated by Martin et al. 23 using density functional theory at the level of B3LYP/cc-pVTZ. The molecular geometry is planar and has D 2h symmetry. The molecule has been placed in the xy plane with the long molecular axis aligned along the x axis. The basis set consists of the 4s3p1d / 2s1p ANO basis set 24 augmented with diffuse functions to enable the description of Rydberg states. The diffuse functions are located at the center of the molecule and consist of two s exponents , , two p exponents , , and two d exponents , functions taken from the previous CASPT2 calculations on naphthalene. 25 In total the basis set consists of 238 CGTO s and is denoted 4s3p1d / 2s1p /2s2p2d. Excitations to the five lowest

6 4178 J. Chem. Phys., Vol. 112, No. 9, 1 March 2000 Bak et al. states of each symmetry have been determined. An exception is made for 1 B 2u symmetry where eight states are determined to include important valence states. For anthracene we used the ground state equilibrium geometry calculated by Martin et al. 23 using density functional theory at the level of B3LYP/cc-pVDZ. The geometry is planar and has D 2h symmetry. The molecule has again been placed in the xy plane with the long molecular axis aligned along the x axis. The 4s3p1d / 2s1p ANO basis set 24 augmented with diffuse functions is used. The diffuse functions are located at the center of the molecule and consist of three s exponents , , , three p exponents , , , and three d exponents , , functions. In total the basis set consists of 329 CGTO s. Excitations to the five lowest states of each symmetry have been determined. For all calculations in this work the available disk capacity was sufficient such that the total list of atomic twoelectron integrals could be stored on disk and was therefore only calculated once. IV. RESULTS Before comparing our calculated results to experiments, it is relevant to discuss what is actually being compared and what kind of agreement to expect. The vertical excitation energy is a theoretical quantity which measures the energy difference between the electronic ground state and the electronic excited state at the same geometry. These excitation energies cannot be measured experimentally, but may be viewed as approximations to the positions of the maxima in the experimental UV absorption peaks. This has proven to be a useful approximation for larger molecules such as naphthalene and anthracene. Presently we are not able to go beyond this approximation without a tremendous increase in computational requirements. Thus we use this approximation in the present work. The errors introduced by this way of comparing theoretical and experimental energy differences varies from state to state and molecule to molecule. For benzene and furan it has been shown that the uncertainties in the comparison are of the orders ev, but they could be significantly larger for other molecules. We may estimate the errors for the lowest singlet transitions of naphthalene and anthracene. Jas and Kuczera 29 have investigated structures and vibrations for the S 1 states of both benzene, naphthalene and anthracene using configuration interaction singles CIS calculations and found that the 0-0 transition energies are 0.25 ev, 0.47 ev and 0.47 ev below the vertical excitation energies for the three molecules, respectively. Differences in zero-point vibrational energies of the two states make up 0.12, 0.13 and 0.11 ev of the total energy differences. For benzene the CIS result is in reasonable agreement with the 0.29 ev correction to the vertical excitation energy obtained in more accurate coupled cluster singles and doubles CCSD calculations. 28 Thus, despite the rather poor vertical excitation energies commonly obtained with CIS, the CIS calculations may be used to give an estimate of the effect of the vertical approximation, also for naphthalene and anthracene. The experimental absorption peak is energetically positioned at the 0-0 transition or at higher energy. For the S 1 states of naphthalene and anthracene the offsets of the maximum intensity peaks relative to the 0-0 transitions is only 0.05 ev and zero, respectively, judging from accurate gas phase experiments. 30,31 By comparing this information with the CIS corrections mentioned above the exact vertical excitation energies are both estimated to be of order ev higher in energy than the maximum intensity peaks for the naphthalene and anthracene S 1 states. Besides the error due to the approximation made in the comparison of theory and experiment there are additional errors resulting from the approximative calculations of the vertical excitation energies. These arise from i the use of a limited basis set and ii the approximate treatment of electron correlation. The ANO basis sets used are essentially of double zeta with polarization and diffuse function quality. Prior work indicate that basis set errors are of the order ev for such basis sets. 26,27 Comparisons with full configuration interaction calculations have for a number of small systems 22 shown that SOPPA for excitations dominated by single-electron excitations tend to give vertical excitation energies that are too low by 0.1 to 1.1 ev with an average correlation error of about 0.5 ev. In Refs. 6 and 22, it was also shown that the accuracy of SOPPA, as expected, decreases with the importance of contributions from twoelectron excitations. Based on the considerations above the discrepancies between the calculated numbers and the experimental values might be as large as 1 ev or more, but since some of the errors tend to have opposite sign, cancellations of errors are also likely to happen. A very good agreement between calculation and experiment might therefore be fortuitous. Nevertheless, keeping the aforementioned approximations in mind, we still think that the comparison of vertical excitation energies with experiments is useful. Indeed, we shall see that the calculations for naphthalene and anthracene correctly predict the major trends in the spectra for both molecules. A. Naphthalene The computed vertical excitation energies and oscillator strengths for naphthalene are presented in Table II together with selected experimental data. 30,32 35 The excitations are listed in ascending order according to our SOPPA values and the excitations are characterized as valence or Rydberg excitations. Since we have only calculated five excitations of each symmetry the SOPPA spectrum is only complete up till about 6.92 ev. In the previous SOPPA work on benzene and naphthalene Packer et al. 6 write that the naphthalene basis set is clearly inferior to that of benzene, which will be reflected in the results. In the present study the naphthalene basis set 4s3p1d / 2s1p /2s2p2d is of the same quality as the basis set Packer et al. used for benzene. It would therefore be interesting to compare our results to the previous SOPPA values. However, the geometry used by Packer et al., a small variation (R C4 C Å instead of Å of the experimental geometry of Ketkar and Fink, 36 differs from the one used in our study as well as from the experimental geometry of Innes et al., 37 which has been used in the previous

7 J. Chem. Phys., Vol. 112, No. 9, 1 March 2000 Atomic orbital driven SOPPA 4179 TABLE II. Singlet vertical excitation energies in ev and oscillator strengths for naphthalene. Oscillator strength Excitation energy SOPPA a State Characterization SOPPA a Expt. L b V b Expt. 1 1 B 3u Valence c c 1 1 B 2u Valence d d 1 1 A u 1a u 3s d 1 1 B 1g Valence e 2 1 A g Valence e 2 1 B 3u Valence d d 1 1 B 2g 1a u 3p y B 3g 1a u 3p x B 1u 2b 1u 3s B 1g 1a u 3p z f 2 1 B 2u Valence d d 2 1 A u 1a u 3d xx 3d yy B 1g Valence B 1u 1a u 3d xy A u 1a u 3d zz B 2g 2b 1u 3p x B 3g 2b 1u 3p y B 2u 1a u 3d xz B 3u 1a u 3d yz A u 1a u 4s A g 2b 1u 3p z B 2g 1a u 4p y B 3g 1a u 4p x B 2g Valence B 1g 1a u 4p z A g Valence e 3 1 B 1u 2b 1u 3d xx 3d yy B 3g Valence A u 1a u 4d zz B 1u 1a u 4d xy B 2u 1a u 4d xz B 3u 1a u 4d yz B 1u 2b 1u 4d zz B 3u 2b 1u 3d xz B 2u 2b 1u 3d yz A g Valence B 3g 1b 3g 3s B 2g 2b 1u 4p x A g 2b 1u 4p z B 2u 7 1 B 2u 5 1 B 1g 8 1 B 2u 1b 3g Valence 3p z 2b 1u 4d yz Valence 1b 3g Valence 3p z d g a 4s3p1d / 2s1p ANO basis set Ref. 24 augmented with 2s2p2d diffuse functions Ref. 25 located at the center of the molecule ( s , ; p , ; d , B3LYP/cc-pVTZ optimized geometry Ref. 23 Cartesian coordinates in Å of the symmetry unique atoms: C1: 0.0, , 0.0; C2: , , 0.0; C3: , , 0.0; H1: , , 0.0; H2: , , 0.0. b Calculated in length L and velocity V representations. c Data from gas-phase absorption, Ref. 30. d Data from gas-phase absorption, Ref. 32. e Data from two-photon absorption in ethanol solution, Ref. 33. f Data from excited state absorption in cyclohexane solution, Ref. 34. g Data from absorption in n-heptane solution, Ref. 35. CASPT2 study of Rubio et al. 25 We performed therefore additional calculations using the geometry of Innes et al., 37 with the 4s3p1d / 2s1p /2s2p2d basis set as well as with the smaller basis set employed by Packer et al. These results are presented in Table III together with the CASPT2 results of Rubio et al. 25 Comparing the two SOPPA calculations we observe the same energy ordering but that the larger basis set gives excitation energies lower by ev. The only exception is the 4 1 A u state which is found 0.03 ev higher in the large basis set calculation and which is therefore also

8 4180 J. Chem. Phys., Vol. 112, No. 9, 1 March 2000 Bak et al. positioned slightly higher in the corresponding SOPPA spectrum. This small irregular behavior is possibly due to the fact that diffuse d functions are not included in the small basis set and as a result, the 2 1 A u and 3 1 A u states from the large basis set calculation are not observed using the small basis set. A second difference concerns the characterization of the transitions to the 2 1 B 1g and 3 1 B 1g states as 1a u 3p z and valence, respectively. This is reversed in the small basis set calculation. We note that a strong mixing of valence and Rydberg character is observed for these two states. The CASPT2 method also offers a second order treatment of electron correlation and we find accordingly an overall reasonable agreement between the SOPPA and CASPT2 results. Nonetheless some differences persist and the energy ordering is similar but not exactly the same in all cases. In particular most of the Rydberg states resulting from 1a u n 3 transitions are found to be lower in the SOPPA spectrum than in the CASPT2 spectrum. The differences of SOPPA and CASPT2 excitation energies SOPPA CASPT2, using the same basis set and geometry, are in the range from 0.15 to 0.30 ev rms 0.2 ev with two exceptions. The first exception is the 6 1 B 2u state with a difference of 0.56 ev and the other exception is the valence 4 1 A g state, which is 1.03 ev lower in the CASPT2 spectrum. From the work of Rubio et al. 25 it is seen that the 4 1 A g state has large contributions from double excitations and this is confirmed in our SOPPA calculations. Since SOPPA only provides a zeroth order description of double excitations the discrepancy in this case between SOPPA and CASPT2 is probably due to a poor SOPPA result. In Table IV the SOPPA eigenvectors, X l, for the valence excitations are analyzed. From the third column we can see that the total contribution of single excitations and de-excitations to the norm of the eigenvector, Eq. 8, for the 4 1 A g state is rather low with 85%, only surpassed by the 5 1 B 1g state with 84%. Furthermore we can see that three double excitations contribute to the 1 1 A g 4 1 A g excitation with larger coefficients, X 2p2h aibj, than for most other states. However, the largest doubles coefficients, X 2p2h aibj, are found for the two double excitations which contribute to the 1 1 A g 5 1 B 1g transition. We therefore expect that also the 5 1 B 1g state is not particularly well represented by SOPPA. To a large extent our SOPPA calculations in Table II confirm the assignments of the experimental spectra made by Rubio et al. 25 and the present study does not give rise to any adjustments. With this assignment the deviations of the maximum of the experimental peaks and SOPPA vertical excitation energies exp. SOPPA are in the range from 0.34 to 0.34 ev, except for the 4 1 A g state which is off by 0.85 ev for the reason discussed above. Eight n 4 Rydberg states resulting from transitions 1a u n 4 are identified in our calculations and have excitation energies ranging from 6.56 to 6.98 ev. This is in excellent agreement with the experimental assignment by Huebner et al. 38 which suggests that the n 4 states are in the region 6.5 to 7.0 ev. Huebner et al. 38 expects the Rydberg states resulting from 1a u n 3 transitions to lie in the TABLE III. Singlet excitation energies ev in naphthalene. A comparison of basis set and correlation effects. a 4s3p1d / 2s1p / 2s2p2d b 3s2p1d / 2s / 2s2p(2d) c SOPPA SOPPA CASPT2 This This Ref. State Characterization Work Work A g Valence A g 2b 1u 3p z A g Valence A g Valence A g 2b 1u 4p z B 1g Valence B 1g 1a u 3p z B 1g Valence B 1g 1a u 4p z B 1g Valence B 2g 1a u 3p y B 2g 2b 1u 3p x B 2g 1a u 4p y B 2g Valence B 2g 2b 1u 4p x B 3g 1a u 3p x B 3g 2b 1u 3p y B 3g 1a u 4p x B 3g Valence B 3g 1b 3g 3s A u 1a u 3s A u 1a u 3d xx 3d yy A u 1a u 3d zz A u 1a u 4s A u 2b 1u 3d xy B 1u 2b 1u 3s B 1u 1a u 3d xy * 6.50* 3 1 B 1u 2b 1u 3d xx 3d yy * 4 1 B 1u 1a u 4d xy * 5 1 B 1u 2b 1u 4d xx 4d yy * 1 1 B 2u Valence B 2u Valence B 2u 1a u 3d xz * 6.34* 4 1 B 2u 1a u 4d xz * 5 1 B 2u 2b 1u 3d yz * 6 1 B 2u 1b 3g Valence 3p z * 7.16* 7 1 B 2u 2b 1u 4d yz * 8 1 B 2u Valence B 3u Valence B 3u Valence B 3u 1a u 3d yz * 6.58* 4 1 B 3u 1a u 4d yz * 5 1 B 3u 2b 1u 3d xz * a Experimental geometry of Innes et al. Ref. 37 Cartesian coordinates in Å of the symmetry unique atoms: C1: 0.0, 0.705, 0.0; C2: , , 0.0; C3: , , 0.0; H1: , , 0.0; H2: , , 0.0. b 4s3p1d / 2s1p ANO basis set Ref. 24 augmented with 2s2p2d diffuse functions Ref. 25 located at the center of the molecule ( s , ; p , ; d , c Basis set BS5 of Rubio et al. Ref. 25 : 3s2p1d / 2s ANO basis set Ref. 24 augmented with 2s2p diffuse functions located at the center of the molecule ( s , ; p , The excitation energies mark with* are calculated with basis set BS6 of Rubio et al. Ref. 25 : 3s2p1d / 2s ANO basis set Ref. 24 augmented with 2s2p2d diffuse functions located at the center of the molecule ( s , ; p , ; d ,

9 J. Chem. Phys., Vol. 112, No. 9, 1 March 2000 Atomic orbital driven SOPPA 4181 TABLE IV. Analysis of SOPPA a singlet vertical valence excitation energies in ev for naphthalene. 5 to 6 ev range. We calculate these nine Rydberg states to lie between 5.26 and 6.50 ev. Of these the 1a u 3s,3p states are within the energy range suggested by Huebner et al. 38 but all 1a u 3d states are above 6.0 ev. However, the transition energies calculated for the 1a u 3d transitions agree with the value of 6.52 ev estimated from an autoionization spectrum by Syage and Wessel. 39 The 2b 1u n 3 Rydberg states are calculated to lie in the range from 5.95 to 7.18 ev. This is approximately 0.7 ev higher than the corresponding 1a u n 3 states, reflecting the difference of 0.7 ev between the 2b 1u and 1a u MO energies. A very accurate A g 1 1 B 3u gas-phase excitation energy of 3.97 ev has been recorded 30 which is only one vibrational quantum below the vibrational transition with largest intensity at 4.02 ev. According to the CIS calculations of Jas and Kuczera 29 quoted above a 0-0 transition energy is obtained from our numbers by subtracting 0.47 ev from the vertical excitation energy. This gives a calculated 0-0 transition energy of 3.35 ev which is 0.55 ev below the experiment. This suggests that the good agreement between SOPPA and experiment for the 1 1 A g 1 1 B 3u transition seen in Table II, might be somewhat fortuitous even though the CIS corrections of Ref. 29 may be somewhat inaccurate. B. Anthracene The calculated SOPPA and RPA singlet vertical excitation energies and SOPPA oscillator strengths for anthracene are presented in Table III. Since we have only calculated five excitations of each symmetry the SOPPA spectrum is only complete below 5.82 ev. To facilitate a comparison, selected experimental excitation energies 35,40,41 are also included in the table. The excitations are listed in ascending order according to the SOPPA values and they are characterized as valence or Rydberg. Since s, d zz and d xx d yy orbitals transform as the same irreducible representation in D 2 h symmetry, the transitions characterized as 2b 2g 3s, 2b 2g 3d zz and 2b 2g 3d xx 3d yy are mixed in our analysis. Therefore, the specific characterization for these transitions is unclear and the same apply to all other transitions to s, d zz and d xx d yy orbitals. The experimental near ultraviolet absorption spectrum of anthracene consists of four bands 35,40,42 from which Inoue et al. 40 have assigned five excitations as listed in Table V. The results of our SOPPA calculations are in good agreement with all of these assignments. The deviations exp. SOPPA are 0.20 ev or less. The most intense peak in the experimental spectrum is assigned to the 2 1 B 3u excitation at 4.82 ev. 40 The SOPPA calculation also places the 2 1 B 3u excitation at 4.82 ev and in very good agreement with experiment 35 the oscillator strength for this excitation is more than ten times larger than the second largest oscillator strength found for the 4 1 B 2u excitation. SOPPA also reproduces the experimental finding that the weak 1 1 B 3u state is above the 1 1 B 2u state. The two-photon excitation spectrum have been recorded from to cm 1 by Dick and Hohlneicher. 41 They assign two excitations of 1 A g symmetry and two exci-

10 4182 J. Chem. Phys., Vol. 112, No. 9, 1 March 2000 Bak et al. TABLE V. Singlet excitation energies in ev and oscillator strength in anthracene. Excitation energy ev Oscillator strength Expt. SOPPA a Expt. State Characterization SOPPA a RPA a Ref. 35 b Ref. 40 c Ref. 41 d L e V e Ref. 35 b 1 1 B 2u Valence B 3u Valence B 1g Valence B 3g 2b 3g 3s B 1g Valence B 3u Valence B 2u Valence B 1u 2b 3g 3p y A g Valence A u 2b 3g 3p x B 2u 2b 3g 3p z B 3g 2b 3g 3d xx 3d yy B 2g 2b 3g 3d xy B 3g 2b 3g 3d zz B 2u Valence A g 2b 3g 3d yz B 1g 2b 3g 3d xz A g Valence B 3u Valence B 2u Valence B 2g 2b 2g 3s B 3g 2b 3g 4s B 1u 2b 3g 4p y A u 2b 3g 4p x B 1u Valence B 3g 2b 3g 4d zz B 2g 2b 3g 4d xy B 1u 2b 2g 3p x A u 2b 2g 3p y A g 2b 3g 4d yz B 1g 2b 3g 4d xz A u Valence B 1u 2b 3g 5p y A u 2b 3g 5p x A g Valence B 2g 2b 3g 5d xy B 1g 2b 3g 5d xz B 3u 2b 2g 3p z B 2g 2b 2g 3d xx 3d yy B 3u Valence a 4s3p1d / 2s1p ANO basis set Ref. 24 augmented with 3s3p3d diffuse functions located at the center of the molecule ( s , , ; p , , ; d , , B3LYP/cc-pVDZ optimized geometry Ref. 23 Cartesian coordinates in Å of the symmetry unique atoms: C1: 0.0, , 0.0; C2: , , 0.0; C3: , , 0.0; C4: , , 0.0; H1: , , 0.0; H2: , , 0.0; H3: , , 0.0. b Data from absorption in n-heptane solution. c Data from absorption in a PVA sheet. d Data from two-photon excitation in ethanol solution. e Calculated in length L and velocity V representations. tations of 1 B 1g symmetry as listed in Table III. The deviations to the SOPPA results are slightly larger than for the one-photon absorption excitations. For the 2 1 A g and the 4 1 A g transitions the deviations exp. SOPPA are 0.28 and 0.41 ev and for the 1 1 B 1g and 2 1 B 1g excitations 0.20 and 0.13 ev. Thus, the SOPPA results support the assignments of the experimental spectra made in Refs. 40 and 41. The lowest nine Rydberg excitations result from 2b 3g n 3 transitions and have excitation energies ranging from 4.65 to 5.67 ev. Eight 2b 3g n 4 Rydberg transitions are observed in the more narrow energy range ev. Six Rydberg excitations, resulting from the second highest occupied molecular orbital and characterized as 2b 2g n 3, are found in the energy range 5.83 to 6.70 ev. In the case of the 1 1 A g 1 1 B 2u transition for anthracene the 0-0 transition is also the maximum intensity peak and has been recorded at 3.43 ev in a comprehensive study of Lambert et al. 31 According to the CIS calculations of Jas and Kuczera 29 quoted above a 0-0 transition energy is obtained from our numbers by subtracting from the vertical excitation energy 0.47 ev. This gives a calculated 0-0 tran-

11 J. Chem. Phys., Vol. 112, No. 9, 1 March 2000 Atomic orbital driven SOPPA 4183 TABLE VI. Valence singlet excitation energies in ev and oscillator strengths in anthracene. A comparison of correlation effects. Excitation energy ev Oscillator strength L a RPA b SOPPA b CASSCF c MRMP c SOPPA b MRMP c State This work This work Ref. 43 Ref. 43 This work Ref B 2u B 3u B 1g B 1g B 3u B 2u A g B 2u A g B 3u a Calculated in length representation. b 4s3p1d / 2s1p ANO basis set Ref. 24 augmented with 3s3p3d diffuse functions located at the center of the molecule ( s , , ; p , , ; d , , B3LYP/cc-pVDZ optimized geometry Ref. 23 Cartesian coordinates in Å of the symmetry unique atoms: C1: 0.0, , 0.0; C2: , , 0.0; C3: , , 0.0; C4: , , 0.0; H1: , , 0.0; H2: , , 0.0; H3: , , 0.0. c cc-pvdz basis set Ref. 44 without a polarization function on hydrogen: 3s2p1d / 2s. Geometry optimized at the CASSCF level. sition energy of 2.61 ev which is 0.82 ev below the experiment. This suggests that the good agreement between SOPPA and experiment for the 1 1 A g 1 1 B 2u transition seen in Table V, might be somewhat fortuitous, as was the case for the S 1 state of naphthalene. The calculated RPA excitation spectrum deviates significantly from the SOPPA spectrum. The largest discrepancies are found for the valence transitions where the deviations RPA SOPPA range from 0.60 to 2.10 ev, the same trend as found for benzene and naphthalene. 6 In addition, many of the computed SOPPA valence transitions are not found among the calculated RPA transitions, but must be found at higher energies in the RPA spectrum. All Rydberg excitations determined in SOPPA are also observed in RPA. For these, the deviations range from 0.08 to 0.58 ev. Only the 2b 3g 3p z transition has lower excitation energy in RPA than in SOPPA. The largest deviations between RPA and SOPPA Rydberg excitation energies are found for the transitions from the 2b 2g orbital. In contrast to SOPPA the RPA spectrum is in poor agreement with experiment. Recently a multireference Mo ller Plesset perturbation theory MRMP study of the valence excitation energies of anthracene has been published by Hirao and co-workers. 43 They used a smaller basis set, the cc-pvdz basis set 44 without the polarization function on the hydrogen atoms, and optimized the geometry at the CASSCF level. In Table VI their CASSCF and MRMP results are compared with our RPA and SOPPA results. The first observation is that Hirao and co-workers 43 predict the 1 1 B 3u state to be lower in energy than the 1 1 B 2u state, which is in disagreement with the experimental findings and our SOPPA results. Second the differences between MRMP and SOPPA are within a range of 0.07 ev for the states (2 1 B 1g,2 1 B 3u,2 1 A g,4 1 B 2u ), whereas for the (1 1 B 1g,4 1 A g,3 1 B 3u,2 1 B 2u ) the deviation is larger 0.33 ev 0.56 ev and SOPPA predicts higher excitation energies than MRMP. From the analysis of the SOPPA eigenvectors X l in Table VII we can see that the total weight of the single excitations in the 1 1 A g 1 1 B 1g and 1 1 A g 4 1 A g transitions is rather low and that a double excitation with large coefficients X 2p2h aibj contribute to both excitations. As the agreement between SOPPA and the experimental two-photon values 41 is also less favorable for those two states, we conclude that SOPPA has problems describing these state due to their double excitation character. For the 2 1 B 2u and 3 1 B 3u states, however, the SOPPA eigenvectors in Table VII show no sign of double excitation character and the SOPPA value for the 2 1 B 2u state is actually in excellent agreement with the one-photon absorption value. 40 V. SUMMARY A new implementation of the second order polarization propagator approximation SOPPA is presented, where the disk space requirement is dramatically reduced. This is achieved by performing the direct linear transformations of the trial vectors from the two-electron integrals in the atomic orbital basis, thereby eliminating a full two-electron integral transformation and opening for integral-direct calculations. In addition the iterative algorithm for the solution of the eigenvalue problem is implemented with a limited number of trial vectors, typically three, per excitation. In each iteration an optimal trial vector is constructed for each eigenvector which together with the optimal trial vector of the last iteration and a new trial vector calculated from the residual error vector builds after orthonormalization the new set of trial vectors for the next iteration. Test calculations on benzene showed that the number of iterations required for conver-

12 4184 J. Chem. Phys., Vol. 112, No. 9, 1 March 2000 Bak et al. TABLE VII. Analysis of SOPPA a singlet vertical valence excitation energies in ev for anthracene. Dominant single excitation Dominant double excitation Excitation Total weight of State energy single de- exc. b orbitals i a X ai c orbitals ij ab X aibj 1 1 B 2u % 2b 3g 6b 1u b 1u 1a u 6b 3g 10a u B 3u % 2b 3g 2a u a g 1b 1u 47b 1g 23b 3g B 1g % 1a u 6b 1u b 3g 3b 3u 30b 1g 4a u B 1g % 2b 3g 6b 2g b 3g 4b 3u 20b 1g 4a u B 3u % 2b 3g 2a u b 1g 2b 2g 7b 2g 54b 2u B 2u % 2b 3g 7b 1u b 1g 2b 3g 4b 3g 32b 3u A g % 1a u 2a u b 2g 2b 2g (2a u ) B 2u % 2b 2g 2a u b 1u 4b 2u 20b 2g 35b 3u A g % 2b 1u 6b 1u b 2g 2b 3g 6b 1u 2a u B 3u % 2b 2g 7b 1u b 1g 2b 2g 7b 2g 44b 2u B 2u % 1b 3g 6b 1u b 3g 6b 2u 67a g 4b 3g B 1u % 8a g 6b 1u b 3u 5b 3u 17b 1g 12a u A u % 5b 1g 6b 1u b 2g 2b 2g 66a g 40b 1g A g % 1a u 2a u 0.61 (2b 3g ) 2 (2a u ) B 3u % 1b 3g 2a u b 1g 2b 2g 7b 2g 44b 2u a 4s3p1d / 2s1p ANO basis set Ref. 24 augmented with 3s3p3d diffuse functions located at the center of the molecule ( s , , ; p , , ; d , , B3LYP/cc-pVDZ optimized geometry Ref. 23 Cartesian coordinates in Å of the symmetry unique atoms: C1: 0.0, , 0.0; C2: , , 0.0; C3: , , 0.0; C4: , , 0.0; H1: , , 0.0; H2: , , 0.0; H3: , , 0.0. b The total contribution of single excitation and de-excitations X ph l S (0,2) X ph l X hp l S (0,2) X hp l to the norm of the eigenvectors in Eq. 8. c Absolute value of the appropriate element of the SOPPA eigenvector X ph l or X hp l in Eq. 7. d Absolute value of the appropriate element of the SOPPA eigenvector X 2p2h l or X 2h2p l in Eq. 7. 2p2h d gence is at most increased by one with the new algorithm compared to an algorithm where the number of trial vectors increases with each iteration. With the new implementation SOPPA can be applied to significantly larger molecules and using larger basis sets than previously was the case. In this study we have presented calculations on several electronic states of naphthalene and anthracene using up to 329 basis functions. Calculations on larger molecules with larger basis sets are also possible with the new implementation of SOPPA. 45 However, as the computational time is largely proportional to the number of iterations in the iterative calculation of excitation energies it might be necessary to limit the number of the calculated excitation energies for molecules considerably larger than anthracene in order to keep the calculations computationally feasible on the available computers. Comparing previous SOPPA results for naphthalene with the results of our calculations using an extended basis set shows that extending the basis set leads I to slightly smaller excitation energies and II that we find four more excitations to Rydberg states, two of A u (1a u n 3) and two of B 2u (1a u n 4 and 2b 1u n 3) symmetry. Despite the various approximations involved in the direct comparison of the vertical excitation energies with the experimental spectra and the intrinsic approximation in finite basis set calculations of SOPPA vertical excitation energies the final comparison is encouraging. For naphthalene the differences between the calculated vertical excitation energies and the position of the peaks of maximum absorption in the experimental spectra are of order 0.3 ev apart from the 4 1 A g state, which contains large contribution from double excitations. A similar comparison for anthracene show differences of 0.2 ev or less apart from the 4 1 A g state and the 2 1 A g state, where the differences are 0.41 ev and 0.28 ev, respectively. We are not aware of experimental data on the anthracene Rydberg states and therefore predict three Rydberg series in the ranges ev (2b 3g n 3), ev (2b 3g n 4) and ev (2b 2g n 3). ACKNOWLEDGMENTS We thank Jeppe Olsen for helpful advice concerning the design of the generalized eigenvalue solver. This research was made possible by grants from the Danish Natural Science Research Council Grant Nos to SPAS and to JO and SPAS, which sponsored the implementation of the new algorithm. Danish Natural Science Research Council Grant No SPAS provided for the necessary computer time on the supercomputer facilities of UNI-C. HK acknowledges the Carlsberg foundation. JO acknowledges the Danish Natural Science Research Council Grant No E. S. Nielsen, P. Jo rgensen, and J. Oddershede, J. Chem. Phys. 73, G. H. F. Diercksen and W. P. Kraemer, MUNICH, Molecular Program System, Max-Planck-Institut für Physik und Astrophysik, Munich. 3 G. H. F. Diercksen, N. E. Grüner, and J. Oddershede, Comput. Phys. Commun. 30, T. D. Bouman and Aa. E. Hansen, RPAC, Molecular Properties Package, Version 9.0, Copenhagen University, Copenhagen, T. D. Bouman and Aa. E. Hansen, Chem. Phys. Lett. 175, M. J. Packer, E. K. Dalskov, T. Enevoldsen, H. J. Aa. Jensen, and J. Oddershede, J. Chem. Phys. 105, T. Helgaker, H. J. Aa. Jensen, P. Jo rgensen, J. Olsen, K. Ruud, H. Ågren, T. Andersen, K. L. Bak, V. Bakken, O. Christiansen, P. Dahle, E. K. Dalskov, T. Enevoldsen, B. Fernandez, H. Heiberg, H. Hettema, D.