J. Phys. Chem. 1996, 100, 6131-6137
6131
On the Nature of Electronic Transitions in Radicals: An Extended Single Excitation Configuration Interaction Method David Maurice and Martin Head-Gordon* Department of Chemistry, UniVersity of California, and Chemical Sciences DiVision, Lawrence Berkeley Laboratory, Berkeley, California, 94720 ReceiVed: September 19, 1995; In Final Form: December 27, 1995X
We propose that the generalization of pure spin single reference excited state theories from molecules with closed shell ground states to radicals be based on a redefinition of the meaning of excitation levels. Open shell single substitutions are expanded to include spin-adapted double excitations which traverse the singly occupied orbitals. These excitations are deleted from the expanded open shell double substitutions, which include the corresponding spin-adapted triple substitutions (e.g., a pair of the excitations traverses the singly occupied orbitals), etc. Applied to single excitation configuration interaction (CIS), this approach defines a spin-adapted extended CIS (XCIS) theory for the excited states of radicals. XCIS is size consistent and variational and has computational complexity that scales in the same way as CIS with molecular size. More low-lying excited states of radicals are correctly described by XCIS than by the restricted open shell CIS method. This is demonstrated by calculations on excited states of seven diatomic radicals and the methyl, nitromethyl, benzyl, phenoxyl, and anilino radicals.
1. Introduction Significant effort is being directed toward the development of methods for calculating electronic excited states of molecules. Due to the simplicity of application and ease of comparison of results between different molecules, single reference methods are an ideal choice for a variety of problems of chemical interest. This paper focuses on the question of how to formulate single reference excited state theories for species with open shell ground states, which leads in simplest form to an extended single excitation configuration interaction theory for radicals. In the context of honoring the contributions of Dr. Samuel Francis Boys and Professor Isaiah Shavitt to molecular quantum mechanics, we note that this new theory, and indeed virtually all excited state methods currently in wide use or under development, employs the tools and language of the theory of configuration interaction, which they have each done so much to develop. We are fortunate to be following in their footsteps. First we briefly review the progress on single reference excited state methods for molecules with closed shell ground states. A hierarchy of single reference excited state theories is now available which offers a variety of distinct tradeoffs between accuracy and computational cost. The simplest and least accurate theory is single excitation configuration interaction (CIS).1,2 CIS is an excited state treatment roughly comparable to the ground state Hartree-Fock (HF) method because, by the Brillouin theorem, single substitutions are Hamiltonian orthogonal to the HF reference. Therefore the CIS states are approximate zero order descriptions of electronic excited states. Higher level theories incorporate the mixing of double (and sometimes higher) subsitutions either self-consistently by linear response3-5 or equation of motion6,7 coupled cluster theory at the single-double substitution level (CCSD) or by low-order perturbative expansions.8 * David and Lucile Packard Fellow, 1995-2000; Alfred P. Sloan Fellow, 1995-7; NSF Young Investigator 1993-8; and author to whom correspondence should be addressed. X Abstract published in AdVance ACS Abstracts, March 15, 1996.
0022-3654/96/20100-6131$12.00/0
The relative success of these single reference methods is all based on the qualitative validity of CIS as the starting point for the description of excited states. For example, the CCSD excited state approach effectively treats states that are primarily one-electron promotions from the ground state, by accurately describing the differential effect of electronic correlation on the two states. By contrast, excited states that are two-electron promotions are completely absent in CIS (by definition!) and they are correspondingly treated in a very unbalanced way via CCSD. The electron pair that is simultaneously promoted from the ground state is correlated in the ground state but not properly so in the excited state, leading to an excitation energy that can be significantly too high.9 The question of an effective single reference treatment of these two-electron promotions can in principle be solved by inclusion of connected triple excitations, either self-consistently10 or perturbatively,11 but such CCSDT excited state methods are computationally very expensive and hence applicable only to very small molecules. The problem of developing a hierarchy of single reference theories of this type for excited states of molecules whose ground state is an open shell reduces to an initial question of how best to generalize CIS for such molecules. Two obvious possibilities are to employ single excitation configuration interaction from either of the widely used single determinant treatments of open shell ground states. Thus, beginning from unrestricted Hartree-Fock (UHF), we are led to unrestricted CIS (UCIS), while beginning from a restricted open shell HF ground state (ROHF), we are then led to a restricted open shell CIS (ROCIS) theory. We have recently reported a preliminary study of the effectiveness of UCIS and ROCIS for describing the excited states of radicals12 and have formulated corresponding second order perturbation theories,13 which we term UCIS(D) and RCIS(D), respectively, where the notation is analogous to that employed for ground state Moller-Plesset perturbation theory for radicals. The main conclusion of that work was that the effectiveness of ROCIS was superior to that of UCIS but overall inferior to the performance of CIS for excited states of closed shell molecules. In cases where ROCIS and UCIS © 1996 American Chemical Society
6132 J. Phys. Chem., Vol. 100, No. 15, 1996
Maurice and Head-Gordon
provided an adequate starting point, the perturbative corrections were effective. These definitions of single excitations are also the ones that are generally employed in semiempirical molecular orbital methods for excited states of radicals. In particular, semiempirical methods that are essentially carefully parametrized versions of ROCIS with a minimal basis are the CNDO/S14,15 and INDO/S methods,16 although not all single excitations are included. Other semiempirical methods, such as the LNDO/ PERT-CI approach,17 include double substitutions which are selected according to a perturbative estimate of their importance. This diminishes the significance of how the singles are defined. Finally, we note that for ab initio multireference methods, based either on a complete active space approach18 or configuration selection,19 the definition of the singles is also irrelevant since it is not employed in the selection process. While there are some systems which are inherently multiconfigurational, and attempting to apply single reference methods yields rather poor results, the price to be paid for the flexibility of multireference methods is that assessing the accuracy of a calculation becomes harder and comparing results on different molecules calculated using different protocols can be problematic. Let us return to the question of why ROCIS and UCIS are less effective for radicals than CIS is for closed shell molecules. We have recently introduced a method for the analysis of electronic transitions, which is independent of the methods employed to calculate wave functions for the ground and excited states. Using this attachment-detachment density analysis,20 we were able to clearly identify the origin of the most dramatic failures of ROCIS (and UCIS) that were observed in our previous paper.12 For the nitromethyl radical, ROCIS excitation energies to the three lowest excited states were in error by over 2 eV in two cases and 4 eV in the third case. The origin of this failure is the neglect of a particular class of double substitution which involves the simultaneous promotion of an R spin electron from the singly occupied orbital and the promotion of a β spin electron into the singly occupied orbital. Thus spin-adapted configurations of the form
|Φ ˜ ia(1)〉 )
1 2 asj (Φajji - Φia) + Φsıj x6 x6
(1)
are of crucial importance in the three lowest excited states of this unsaturated radical. It is quite likely that similar excitations are also very significant in other radicals of interest. We note from the definition of the spin-adapted configuration, eq 1, that the effective number of electrons promoted from the Hartree-Fock reference determinant is actually 5/3, based on the weighted average of the three terms. Therefore in a strict sense, this spin-adapted configuration is neither single nor double but a mixture of the two which is of the correct multiplicity to describe doublet excited states. We cannot form this CSF in ROCIS, due to explicit exclusion of the last term of eq 1. Given its possible importance, it may instead be logical to group all the determinants entering the configuration in a single class: an extended definition of single excitations for radicals. This will be the central thesis of this paper. We propose that a more satisfactory generalization of CIS to excited states of open shell molecules than the strict singlesonly excitation based methods explored previously12 is to simultaneously include the restricted class of double substitutions similar to those given in eq 1. A corollary of the appropriateness of this definition is that without the inclusion of the special class of double substitutions given by the last term of eq 1 it is
not possible to form a spin-adapted state of higher multiplicity. For example, to generate a quartet, we must take linear combinations of determinants of the form
1 1 asj |Φ ˜˜ ia〉 ) (Φia - Φajjı ) + Φsıj x3 x3
(2)
Since this approach is a spin-pure extension of CIS for open shell systems, we suggest that it be termed XCIS, for an extended restricted open shell configuration interaction with single substitutions. The ground state remains the ROHF determinant, while the XCIS excited state wave function is a linear combination of the single excitations included in ROCIS plus the double substitutions that traverse the singly occupied orbital(s). These substitutions are the promotion of a β electron from a doubly (d) occupied orbital to a singly (s) occupied orbital (d f s β) coupled with the promotion of an R electron from a singly occupied orbital to a virtual (V) orbital (s f V R). Equation 1 illustrates the spin-adapted form of one of these excitations for a doublet state. For doublet states, the size of the new class of substitutions is equal to the single excitations obtained by promotions from doubly occupied to virtual orbitals. Therefore the increase in the overall number of CSFs is by less than a factor of 2 relative to ROCIS. This in turn suggests that the overall scaling of the method will be similar to that of ROCIS, apart from a constant multiplicative factor. The XCIS method is most suited for use with spin-pure wave functions. There would be several drawbacks in applying XCIS in an unrestricted sense: namely an inability to uniquely define the singly occupied orbitals in a UHF wave function and the fact that the β unoccupied orbitals through which the spin-flip double excitations proceed may not match the half-occupied R orbitals in either character or even symmetry. The central idea is more general than the specific method we pursue in the remainder of this paper. If XCIS is established as a qualitative improvement over ROCIS, as will become clear from the applications, then the implication is that the operating definition of “single” and “double” substitutions for single reference excited state theories of open shell molecules should be modified. The “extended singles” should include the double substitutions which go through singly occupied orbitals, for example, as defined by eq 1 in the doublet case. This is reasonable as they could alternatively be viewed as single excitations where the excitation is accompanied by a paired spin flip (a spin exchange). The “extended doubles” should then comprise the regular doubles, excluding those which were classified as “extended singles”, and also spin-adapted triple excitations in which a pair of the excitations proceeds through the singly occupied orbitals. The value of this reclassification for higher level single reference excited state theories of radicals, such as coupled cluster response methods, and associated loworder perturbation schemes will be directly related to the improvement XCIS exhibits over ROCIS. The outline of the remainder of this paper is as follows. In section 2 we shall present the general theory of XCIS and discuss its formal properties which include size consistency and variationality. In section 3, a brief discussion of our implementation of the XCIS method is given, as a result of which it will be possible to assess the increase in computational cost relative to the ROCIS method. Section 4 contains comparative applications of the XCIS method to two separate classes of problems. First, we consider the performance of the XCIS method for the series of small radicals for which we have previously reported UCIS, ROCIS, UCIS(D), and RCIS(D)
Electronic Transitions in Radicals
J. Phys. Chem., Vol. 100, No. 15, 1996 6133
excitation energies. Second, we apply the ROCIS and XCIS methods to several prototypical larger radicals: the benzyl radical (PhCH2), the phenoxyl radical (PhO), and the anilino (PhNH) radicals. In the final section, we discuss future prospects and summarize our conclusions. 2. Formulation and Properties of Extended Single Excitation Configuration Interaction We assume that a restricted open shell Hartree-Fock calculation has been performed for the ground state of the species of interest. The result of the ROHF calculation is three sets of uniquely defined orbital spaces: doubly occupied (d), singly occupied (s), and unoccupied or virtual (V). The ROHF energy is invariant to unitary transformations within each of these three subspaces of the full orbital space. Beginning from this starting point, three types of single excitations of the same multiplicity as the ground state can be distinguished: d f s, s f V, and d f V. Thus the spin-adapted ROCIS wave function is
|ΦROCIS〉 )
dV
1
sV
ds
aia(Φia + Φajjı ) + ∑aapΦap + ∑apjjı Φpjjı ∑ ia pa ip
x2
(3)
where Φai is a determinant formed by replacing occupied orbital i with unoccupied orbital a, with an amplitude of aai . Here p is used to signify the singly occupied orbitals, either occupied (R) or unoccupied (β). The extension of CIS theory to incorporate higher excitations, which when spin-adapted give configuration state functions (CSFs) of the correct multiplicity, is relatively straightforward. The ground state as defined by the reference ROHF determinant is unaltered. However, the excited state wave function is augmented by permitting these CSFs to mix with the ROCIS function defined by eq 3. In the case of a general multiplet, the wave function includes terms corresponding to the addition of the CSFs similar to eq 1, as well as ones where the double excitation is through different orbitals in the R and β space:
|ΦXCIS〉 )
1
dV
sV
ds
pa
ip
∑aia(Φia + Φajjı ) + ∑aapΦap + ∑apjjı Φpjjı +
x2 ia
dVs
dV,ss
∑a˜ ia(p)Φ˜ ia(p) + iap
j aqj aaq ∑ pıj Φpıj ia,p*q
(4)
For the doublet case, the fourth term in eq 4 is given by eq 1, while the last term vanishes. For higher multiplets, the form of the spin-adapted CSFs given by the fourth term is somewhat arbitrary (since we are spanning a degenerate subspace of the S2 operator). One CSF can be chosen as analogous to the doublet CSF:
|Φ ˜ ia(1)〉 )
Ns
x2N
2
s
(Φajjı - Φia) +
+ 4Ns
2
x2N
2
s
s
∑p Φappıjj
+ 4Ns
(5) where Ns is the number of singly occupied orbitals. The remaining Ns CSFs can be readily constructed through orthonormality relations. The equations for the transition energy from the ROHF ground state to the excited states defined by eq 4 are obtained
as the solution to an eigenvalue problem, which is simply the diagonalization of the Hamiltonian matrix in the space of the classes of CSFs included in the XCIS wave function, eq 4. General prescriptions for the resulting matrix elements are available in standard references, while efficient evaluation of the elements for the case of doublet excited states will be discussed further in the implementation section. While the rest of this paper will focus on the doublet case it is useful to briefly consider higher multiplets. We suppose that there are s singly occupied orbitals and seek the configuration state function formed through linear combinations of all possible spin-exchange double excitations and the original d f V singles. There are in general (s2 + 2)dV such determinants in total. The majority of these determinants are already spin-adapted: that is, the (s2 - s)dV cases where s′ * s. There remain (s + 2)dV determinants to spin-adapt, which can be easily proven to yield (s + 1)dV configuration state functions of the same multiplicity as the ground state, and a set of dV CSFs which have a total spin that is one quantum higher. The general form of the eigenfunctions is also not difficult to obtain by the standard techniques of configuration interaction theory. Therefore the XCIS prescription is general to radicals with arbitrary numbers of unpaired electrons, with the number of additional CSFs relative to ROCIS being s2dV. We next briefly discuss the main properties of the XCIS excited state wave functions. In general we find that all desirable attributes of CIS excited state wave functions for molecules with closed shell ground states are retained: (1) Variationality. Since XCIS is a linear configuration interaction method, it is automatically variational. This of course applies to the total energy of the electronic excited states, rather than the excitation energies, which do not obey any variational principle. (2) Size consistency. If XCIS is applied consistently to an ensemble of non-interacting molecules, it is guaranteed that the XCIS excitation energies of the individual molecules will be included among the XCIS excitation energies of the ensemble. The proof requires only the fact that ground state energy is size consistent (being ROHF), and therefore the orbitals of the individual molecules are recovered from the ensemble calculation. The ensemble XCIS Hamiltonian matrix then includes that of individual molecules as separate blocks. (3) Invariance to unitary transformations. The XCIS excitation energies are invariant to unitary transformations of the ROHF orbitals within any of the three subspaces: the doubly occupied space, the singly occupied space, and the virtual space. Proof follows from the fact that complete sets of excitation energies to or from each subspace are retained in the construction of the theory. (4) Computational complexity. The XCIS method will be shown in the next section to scale in the same way with molecular size as the ROCIS method, with our specific implementation being approximately 4 times slower for doublet ground states. This scaling is approximately quartic for small molecules, falling to between quadratic and cubic for large molecules, similar to self-consistent field theory for the ground state. By contrast, more sophisticated approaches for correcting the deficiencies of CIS inevitably involve much greater increases in computational complexity, including steeper scaling with molecular size. One important remaining question is how well balanced is the treatment of the ground state by the ROHF method, and excited states by the XCIS procedure? This will be answered more definitively by the numerical results in section 4, but it is important to consider the question in terms of the defining theory
6134 J. Phys. Chem., Vol. 100, No. 15, 1996 first. To begin with, we note that the treatment of the ground and excited states is not as uniform as for CIS in the closed shell ground state case. For molecules with closed shell ground states, both HF ground and CIS excited states emerge from diagonalizing the Hamiltonian in the space of the HF reference and singly excited CSFs. By contrast, in the XCIS case, the restricted class of double excitations can mix with the ground state and therefore lower its energy. We choose to exclude this mixing because if permitted it will violate the size consistency of the ground state energy. Since these CSFs are permitted in the excited states but not in the ground state, we should expect that some fraction of the excited state correlation energy may be recovered, which could cause excitation energies to be too low. However, the magnitude of this effect should be small. This is because the CSFs introduced cannot account for the pair correlation energy of any pair of electrons, and the fraction of double substitutions involved is roughly the square root of the total number of double substitutions which is very small indeed. Therefore in very qualitative terms, the XCIS approach to excitation energies of radicals appears tolerably balanced, and detailed numerical studies are warranted. This task is commenced in section 4.
Maurice and Head-Gordon (d+s)V R P ˜ λσ )
(6a)
d(s+V) β ) P ˜ λσ
aj c* ∑ λibjı cσa ıa
(6b)
dV
d ) ∑c* ˜ iacσa P ˜ λσ λib
(6c)
ia
sV
ds
pa
ip
x a pj ) ∑c* P ˜ λσ λibjı cσp λpbpcσa - ∑c*
(6d)
These density matrices are used to make the J and K matrices: R β ˜ λσ +P ˜ λσ ) JµV ) ∑(µV|λσ)(P
(7)
λσ
R R ) ∑(µσ|λV)P ˜ λσ KµV
(8a)
β β ) ∑(µσ|λV)P ˜ λσ KµV
(8b)
d d ) ∑(µσ|λV)P ˜ λσ KµV
(8c)
x x ) ∑(µσ|λV)P ˜ λσ KµV
(8d)
λσ
3. Implementation of the XCIS Method for Radicals The XCIS method for excited states of radicals has been implemented in the Q-Chem molecular orbital program package,21 as a generalization of our efficient ROCIS algorithm.12 The implementation we describe below is restricted to doublet states, but extension to excited states of molecules whose ground state is of higher multiplicity is quite readily possible, as sketched above in section 2. XCIS excitation energies are obtained in a double-direct fashion, via iterative Davidson diagonalization22 of the Hamiltonian matrix, which is achieved by repeated contractions of the approximate amplitudes with the two-electron integrals.23 Iterative diagonalization of the Hamiltonian is an efficient way to obtain the lower excited states, which are the states that CIS-like theories treat most adequately. Direct XCIS nominally scales with the fourth power of system size (in practice further reduced by two-electron integral prescreening, as mentioned previously), enabling application of the method to systems far larger than would be possible via diagonalization of the full XCIS matrix, which scales as the sixth power of the size of the system. The fourth order scaling of this direct implementation of XCIS is possible because of the very restricted number of double substitutions; methods that include all doubles scale as the sixth power of molecular size. We divide the types of integrals that need to be evaluated into two classes. The integrals over two or more singly occupied orbital indices are calculated once and stored in memory (they will be “in-core integrals”). This corresponds to a main memory (or disk space) requirement that only grows quadratically with the size of the radical. The formation of these integrals is at worst a fourth order process, since it is accomplished by assembling Fock-like matrices. However, this transformation need only be done once, and hence its relative cost is small compared to the iterative fourth order nature of the full problem. The contributions from integrals with one or zero singly occupied orbitals are evaluated directly on each iteration of the Davidson diagonalization procedure. Evaluation of the direct integrals requires the construction of four different pseudodensity matrices, P ˜ , by transformation of the current subspace amplitudes, b, into the atomic orbital basis:
a c* ∑ λibi cσa ia
λσ
λσ
λσ
which are summed and transformed into the molecular orbital basis: (d+s)V R ) F ˜ ai
∑ ia
R d c*µa[JµV - KµV + (N1 + N2δis)KµV ]cVi
(9)
β d c* ∑ µa[JµV - KµV - (N1 + N2δsa)KµV]cVi ıa
(10)
d(s+V)
F ˜ βajjı )
dV
d R β d x ) ∑c* F ˜ ai µa[N1KµV - N1KµV - KµV + N2KµV]cVi
(11)
ia
where N1 and N2 are the normalization constants for the singly and doubly excited determinants in eq 5. The remaining integral pieces are the in-core pieces and can be added to these Focklike matrices at the same time as the one-electron terms (from the usual spin-orbital Fock matrix) to form the subspace representation of the XCIS Hamiltonian contracted with the current transition amplitudes. The XCIS method we have implemented for doublets is roughly 4 times slower than the corresponding fast ROCIS method we outlined previously. The origin of the 4-fold increase in computational cost relative to ROCIS is the formation of the exchange matrices from four distinct density matrices, rather than the one matrix required in ROCIS. For higher multiplets, the implementation will be slower still, as the number of density and exchange matrices will grow roughly with the square of the number of singly occupied orbitals. However, for fixed multiplicity, the total cost of a calculation will still scale as the fourth power of molecular size (and lower with integral cutoffs). By contrast, there is a sixth power dependence for methods that include all double excitations, such as CCSD- and CISD-like methods.
Electronic Transitions in Radicals
J. Phys. Chem., Vol. 100, No. 15, 1996 6135
TABLE 1: Vertical Excitation Energies for a Series of Small Radicals, Calculated via the ROCIS and XCIS Methods with the 6-311(2+,2+)G** Basis, at Geometries Given in Ref 12 excitation energies radical BeH
state 2Π 2Π
BeF
2Π
2 +
Σ
2Σ+
CH CH3
2∆ 2A
CN CO+ NO OH
1′
2′′ Π 2Σ+ 2Π 2 + Σ 2Σ+ 2Σ+ 2 + Σ 2Σ+ 2A 2
transition V: R: V: R: R: V: R: R: V: V: V: V: R: R: V: V:
a
sσ f vπ sσ f v3p sσ f vπ sσ f v3s sσ f v3p dσ f sπ s2p f v3s s2p f v3p dπ f sσ dσ f sσ dπ f sσ dσ f sσ sπ f v3s sπ f v3p dσ f sπ dσ f vσ
ROCIS
XCIS
exptb
2.715 6.544 4.260 6.349 6.600 3.466 6.300 7.713 2.185 6.378 5.656 10.404 7.094 8.249 4.454 9.549
2.661 6.468 4.249 6.345 6.587 3.249 6.243 7.651 1.150 4.802 4.267 8.192 6.842 8.028 4.151 9.134
2.484 6.318 4.138 6.158 6.271 2.880 5.729 7.436 1.315 3.219 3.264 5.819 5.921 7.027 4.087 8.651
oscillator strength
XCIS character,c %
ROCIS
XCIS
1.0 1.6 0.1 0.0 0.2 0.7 0.2 0.6 4.5 14.2 5.8 37.5 1.9 1.8 0.7 26.0
0.056 0.006 0.199 0.072 0.097 0.020 0.043 0.000 0.008 0.015 0.010 0.005 0.004 0.014 0.004 0.040
0.054 0.006 0.198 0.073 0.095 0.018 0.044 0.000 0.004 0.012 0.007 0.001 0.003 0.015 0.003 0.028
a The type of electronic transition. First whether the transition is principally valence (V) or Rydberg (R) in character, followed by the dominant electron promotion in terms of doubly occupied (d), singly occupied (s), and unoccupied (v) molecular orbitals. b Vertical excitation energies were calculated from spectroscopic data in ref 12. c Percentage of the XCIS wave function from the CSFs, eq 1, which are excluded in ROCIS.
4. Applications to Vertical Excitation Energies of Radicals and Discussion The first tests of the value of the XCIS procedure for excited states of radicals should be applications to vertical excitation energies. Subsequently, as has recently been emphasized,24 it is of course essential to assess the performance of a candidate theory of excited states on an entire range of properties. In the present work, we take the first step, which is to assess the comparative performance of XCIS relative to the parent method, ROCIS, which excludes the special class of double substitutions. This begins with repeating a series of calculations which we recently employed to assess the performance of the ROCIS method12 and a second order perturbative correction,13 RCIS(D). The results of performing XCIS calculations on this selection of low-lying excited states of seven diatomic molecules and methyl are summarized in Table 1. Two different types of cases may be broadly distinguished in Table 1. First are molecules where ROCIS itself was moderately successful, which include BeH, BeF, CH3, and OH. In these cases, it is evident from table 1 that XCIS does not substantially change the excitation energies obtained at the ROCIS level. This is consistent with the expectation that the ROCIS wave functions are qualitatively correct in these molecules, and therefore little change should occur at the XCIS level. It is also a corollary of the assertion in section 3 that we expect relatively little extra correlation energy of these excited states to be recovered via XCIS, as is necessary for a wellbalanced treatment of ground and excited states. The second, and more interesting, set of cases in table 1 are molecules where ROCIS performed relatively poorly. Examples in this class include CN and CO+. Here much larger changes in the excitation energies are seen between ROCIS and XCIS, with the XCIS results generally moving in the direction of experiment. As a second, more demanding, test of XCIS, we consider calculations of the three lowest excited states of the nitromethyl radical in Table 2. For this molecule, we have previously inferred from high-level correlated calculations, and our attachment-detachment density analysis, that the double excitations excluded from the ROCIS wave function are qualitatively very important in all three excited states.20 As is evident from Table 2, the XCIS results show very large differences relative to the ROCIS calculations, with a correction of over 4 eV for the 2A2
TABLE 2: Calculations of Vertical Excitation Energies (eV) to the Lowest Excited States of Different Symmetry to the 2B Ground State in the Nitromethyl Radical by the ROCIS 1 and XCIS Methods with the 6-31+G* Basis, versus Benchmark Calculations Employing G2 Theory, from Ref 20. The UMP2/6-31G* Optimized Geometry from Ref 12 Was Used for the ROCIS and XCIS Calculations excitation energies state 2
B2 A1 2 A2 2
ROCIS
XCIS
G2
XCIS character,a %
4.557 4.688 6.183
2.607 2.928 1.512
1.990 2.473 2.476
42.61 36.36 52.70
oscillator strength ROCIS
XCIS
0.000 0.000 0.148
0.000 0.000 0.006
a Percentage of the XCIS wave function from the CSFs, eq 1, which are excluded in ROCIS.
state being the largest. Comparing with the highest level calculations we reported, which are of the G2 type,25 appropriately modified for excitation energies, the XCIS results are clearly greatly improved over ROCIS. For the three lowest excited states of this unsaturated radical, we conclude that the ROCIS excited states are not of qualitatively correct character but the XCIS excited states almost certainly are qualitatively correct. This point is reinforced by the magnitude of the contribution of CSFs of the form of eq 1, which are included in XCIS but omitted in ROCIS. They range between 36% and 53% for these three states. As a further exploration of the importance of employing XCIS in preference to ROCIS to obtain a zero order description of the excited states of larger radicals, we consider the isoelectronic series composed of the benzyl, phenoxyl, and anilino radicals. This series of prototypical unsaturated radicals has been extensively studied using both experimental and theoretical methods, although it is beyond our present scope to fully review this literature. In Table 3 we summarize calculations using the ROCIS and XCIS methods with the 6-31+G* basis, which is of valence double ζ quality, enhanced by including a diffuse sp shell and a d shell of polarization functions on non-hydrogen atoms. Since the experiments with which we shall compare are not in the vapor phase26-28 we have excluded Rydberg states from Table 3. Valence excited states involving π f π* excitations (also mixed with π f π excitations into the singly occupied orbital) are to some extent transferable between the three species, and Table 3 is organized to emphasis this commonality.
6136 J. Phys. Chem., Vol. 100, No. 15, 1996
Maurice and Head-Gordon
TABLE 3: Valence Electronic Transitions (eV) in the Benzyl, Anilino, and Phenoxyl Radicals statea 2B
2
(2A′)
2A (2A′′) 2
2B
1
(2A′′)
2A 1
2A (2A′′) 2
2B
2B
1
1
(2A′′)
(2A′′)
methodb ROCIS (strength) XCIS (strength) XCIS character, % experiment ROCIS (strength) XCIS (strength) XCIS character, % experiment ROCIS (strength) XCIS (strength) XCIS character, % experiment ROCIS (strength) XCIS (strength) XCIS character, % experiment ROCIS (strength) XCIS (strength) XCIS character, % experiment ROCIS (strength) XCIS (strength) XCIS character, % experiment ROCIS (strength) XCIS (strength) XCIS character, % experiment
benzylc
anilinod
phenoxyle
2.29 (0.003) 0.38 (0.000) 2.07 (0.002) 0.20 (0.000) 0.9 0.5 5.13 (0.002) 2.99 (0.000) 32.8 2.7 5.41 (0.180) 2.36 (0.001) 42.9 2.7
5.11 (0.041) 2.65 (0.004) 31.7 2.9 5.20 (0.161) 2.22 (0.007) 40.8 2.9
5.51 (0.037) 2.76 (0.007) 38.00 3.0 5.60 (0.156) 2.38 (0.013) 49.2 3.0 4.69 (0.003) 4.36 (0.003) 2.0 4.0 6.44 (0.342) 6.75 (0.427) 7.43 (0.461) 4.22 (0.054) 4.26 (0.059) 4.82 (0.041) 37.9 43.0 57.0 3.9 3.9 4.0 4.76 (0.003) 82.8 4.8 5.55 (0.054) 5.08 (0.229) 10.2 4.8
4.71 (0.008) 4.70 (0.012) 80.2 85.4 6.29 (0.000) 7.00 (0.261) 5.36 (0.220) 5.65 (0.115) 8.3 17.6
a The symmetry of the state in the C2V point group for benzyl and phenoxyl and in the Cs point group for anilino (given in parentheses). B1 symmetry is defined as antisymmetric in the plane of the molecule only, while A2 symmetry is antisymmetric in the mirror plane perpendicular to the molecular plane also. Some previous calculations have reverse definitions for B1 and B2. States which are common between all three radicals are primarily of π f π* character mixed with π f π excitations into the singly occupied orbital. b Calculations are vertical excitation energies by the ROCIS and XCIS methods using the 6-31+G* basis set at UHF/6-31G* optimized geometries. Oscillator strengths are given in parentheses. The XCIS character is the percentage contribution of CSFs of the form of eq 1 (that are omitted in ROCIS) to the XCIS wave function. Total ROHF/6-31+G*//UHF/ 6-31G* ground state energies (in au) are -269.120168, -285.114293, and -304.949051 for benzyl, anilino, and phenoxyl, respectively. c Experimental data for the benzyl radical are from refs 26, 29, and references therein. Note that where a given absorption is close to two calculated XCIS excited states, we have chosen to include it twice in the Table. d Experimental data for the anilino radical are from refs 28 and 34. See also footnote c. e Experimental data for the phenoxyl radical are from ref 27. See also footnote c.
The main observation from Table 3 is that the ROCIS method performs very poorly for a majority of these excited states. So poorly, in fact, that the character of the π f π* states in particular cannot be regarded as being correctly given by ROCIS, and there is no real mapping between the XCIS and ROCIS excited states. The origin of the difference is no more and no less than the additional CSFs, eq 1, which enter the XCIS wave function. They contribute significantly to the zero order wave function in almost all of these states (that is, roughly 30% or more of the amplitude, as documented in Table 3). On the basis of these dramatic changes, ROCIS is clearly inadequate as a zero order description of this entire manifold of π f π* states! The next question to consider is whether XCIS is, or is not, itself adequate as a zero order description of these states. Beginning with π f π* excitations in benzyl (2B1 ground state), it has been known for nearly 20 years that the lowest 2A2 state and the first 2B1 excited state are nearly degenerate,29 with the 2A being lower by roughly 0.1 eV or less. The center of the 2 experimental absorption is roughly between the XCIS values
for these two states, with both in error by 0.4 eV or less (by contrast the ROCIS states are both in error by over 2.5 eV, a factor of 6 greater). We note that XCIS predicts the ordering of states incorrectly and gives a splitting that is substantially too large. Nevertheless, we consider the level of agreement between XCIS and experiment to be acceptable, given that electron correlation effects are neglected in the XCIS method. CIS for closed shell molecules yields errors of this magnitude or larger but is nevertheless generally accepted as giving qualitatively correct descriptions of one-electron excitations. Multireference CI calculations give a better balanced description of these two states,30,31 although of course such calculations involve greatly increased computational complexity and choices of active spaces or configurations which are specifically customized for the benzyl molecule. By contrast XCIS can be generically applied in an unambigous fashion to any radical. Moving to the higher π f π* excited states of the benzyl radical, we note that there is excellent agreement between the XCIS value for the second 2A2 state and the corresponding experimental assignment, while ROCIS exhibits a serious discrepancy of over 2 eV. The next valence excitations are two 2B states calculated in the vicinity of 5 eV by the XCIS method, 1 which is very close to the shortest wavelength absorption seen experimentally, which is known from polarization data to be of 2B1 symmetry. One of these XCIS states is entirely absent in the ROCIS method, as it is dominated by the double substitutions. Overall, we conclude that the XCIS method yields a greatly improved description of the absorption spectrum of the benzyl radical relative to the ROCIS method, although not of course of the same quality achievable by customized multireference CI calculations. For the phenoxyl radical, the two lowest excited states are also believed to give overlapping absorptions,32 with the maxima separated by not much more than 0.1 eV. If this interpretation is correct then XCIS performs similarly for this molecule, giving a splitting between the states that is too large but far better agreement with experiment than the ROCIS method. The second absorption in phenoxyl27 at 3.9 eV has been assigned to the 22A2 r X2B1 transition on the basis of semiempirical calculations33 and multireference CI calculations.32 Our XCIS results appear moderately consistent with this assignment, although the XCIS calculations (and also for that matter the ROCIS calculation) shows that there is an additional valence excitation to the 12A1 state (primarily an excitation into the singly occupied orbital) which might also be a candidate. For the anilino radical, XCIS also gives similar positions for the two lowest π f π* excitations, and given the similarity of the experimental absorption maxima,34 the band is quite likely to be of identical origin to the lowest absorption in benzyl and phenoxyl. The second, shorter wavelength absorbance in anilino is consistent with the A′′ state correlating with the 22A2 states of benzyl and phenoxyl. Overall, from the above data, it appears that XCIS is an acceptable zero order theory for describing low-lying electronic transitions of open shell molecules. The marked improvements over ROCIS in cases where the latter was grossly in error underscore the importance of the special class of double excitations included in the XCIS theory. The fact that XCIS retains variationality and size consistency, while keeping roughly balanced treatments of the ground and excited states (in terms of correlation), suggests that it is likely to emerge as the preferred open shell equivalent to closed shell CIS. It is interesting to briefly compare the XCIS method against the performance of simple semiempirical methods such as
Electronic Transitions in Radicals CNDO/S for radicals. While CNSO/S is explicitly restricted to single excitations (it is a parametrized version of ROCIS with a minimum basis), it still performs remarkably well for the benzyl, anilino, and phenoxyl radicals studied here. Published CNDO/S calculations15,33 are of at least comparable accuracy to our extended basis XCIS calculations for excitation energies. While these results are most impressive, the use of CNDO/S for radicals must nevertheless be viewed with caution, because the fundamental character of the π f π* electronic excitations has been established as having a substantial component which is not single excitation in nature (the CSFs of eq 1). 5. Conclusions and Outlook In this paper we have advanced the idea that a generalized definition of excitation levels for single reference excited state theories of radicals is beneficial, in the sense of leading to methods that are better approximations to the exact solution for truncation at a given excitation level. We suggest that the definition of single excitations should include configuration state functions that are based on double excitations that traverse the singly occupied orbitals, with obvious extensions to the definitions of higher substitutions. Truncating at a given excitation level, the number of these generalized excitations is only a constant multiple of the number of original excitations for excited states of a given multiplicity. The simplest realization of this approach is an extended single excitation configuration interaction method for excited states of radicals. We have discussed the formulation and implementation of this XCIS method and shown that, like CIS for excited states of closed shell molecules, the XCIS energy is size consistent, variational, and invariant to unitary transformations to which the Hartree-Fock ground state energy is also invariant. Its computational cost is roughly 4 times higher than CIS for closed shell molecules or restricted open shell CIS for radicals (doublets as implemented). We have shown through numerical studies of vertical excitation energies for radicals that XCIS is an improvement over ROCIS in terms of leading to results for the majority of low-lying excited states that are in qualitative agreement with experiment. By contrast, ROCIS fails, in the sense of being qualitatively incorrect, for a fairly high proportion of excited states of the unsaturated radicals tested here. There are many cases where the new CSFs contribute from 30% to above 50% of the probability amplitudes in XCIS states, and they must hence be regarded as crucial to a correct description of radical excited states. Therefore XCIS seems to be preferred over ROCIS as the counterpart of closed shell CIS for studying excited states of large radicals. Like CIS itself for excited states of molecules with closed shell ground states, XCIS does not provide quantitative accuracy for excited states of radicals. It is best regarded as a zero order description. The fact that it is qualitatively successful suggests that there are encouraging prospects for developing higher level single reference excited state theories of radicals that do approach quantitative accuracy, on the basis of the modified definition of excitation levels proposed here. These may take the form of linear response coupled cluster theory or equation of motion methods carried to the generalized single and double
J. Phys. Chem., Vol. 100, No. 15, 1996 6137 substitution level or a low-order perturbation expansion of those methods. We hope to report on developments of this type in the future. Acknowledgment. This work was supported by the Director, Office of Energy Research, Office of Basic Energy Sciences, Chemical Sciences Division, of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098. We acknowledge the National Energy Research Supercomputer Center for a grant of computer time. References and Notes (1) Del Bene, J.; Ditchfield, R.; Pople, J. A. J. Chem. Phys. 1971, 55, 2236. (2) Foresman, J. B.; Head-Gordon, M.; Pople, J. A.; Frisch, M. J. J. Phys. Chem. 1992, 96, 135. (3) Monkhorst, H. J. Int. J. Quantum Chem., Symp. 1977, 11, 421. (4) Koch, H.; Jensen, H. J. Aa.; Jorgensen, P.; Helgaker, T. J. Chem. Phys. 1990, 93, 3345. (5) Rico, R. J.; Head-Gordon, M. Chem. Phys. Lett. 1993, 213, 224. (6) Sekino, H.; Bartlett, R. J. Int. J. Quantum Chem., Symp. 1984, 18, 255. (7) Stanton, J. F.; Bartlett, R. J. J. Chem. Phys. 1993, 98, 7029. (8) Head-Gordon, M.; Rico, R. J.; Oumi, M.; Lee, T. J. Chem. Phys. Lett. 1994, 219, 21. (9) Rico, R. J.; Lee, T. J.; Head-Gordon, M. Chem. Phys. Lett. 1994, 218, 139. (10) Watts, J. D.; Bartlett, R. J. J. Chem. Phys. 1994, 101, 3073. (11) Watts, J. D.; Bartlett, R. J. Chem. Phys. Lett. 1995, 233, 81. (12) Maurice, D.; Head-Gordon, M. Int. J. Quantum Chem., Symp. 1995, 29, 361. (13) Head-Gordon, M.; Maurice, D.; Oumi, M. Chem. Phys. Lett. 1995, 246, 114. (14) Del Bene, J.; Jaffe´, H. H. J. Chem. Phys. 1968, 48, 1807. (15) Chang, H. M.; Jaffe´, H. H.; Masmanidis, C. A. J. Phys. Chem. 1975, 79, 1118. (16) Ridley, J.; Zerner, M. Theor. Chim. Acta 1973, 32, 111; 1972, 42, 223. (17) Hanekamp, G.; Heidenreich, A.; Krause, D.; Meyer, H.; Mu¨ller, T.; Schweig, A. J. Mol. Struct. 1994, 327, 193. (18) Roos, B. O.; Fulscher, M. P.; Malmqvist, P. A.; Merchan, M.; Serrano-Andres, L. In Quantum Mechanical Electronic Structure Calculations with Chemical Accuracy; Langhoff, S. R., Ed.; Kluwer: Dordrecht, 1994. (19) Bruna, P. J.; Peyerhimoff, S. D. AdV. Chem. Phys. 1987, 67, 1. (20) Head-Gordon, M.; Gran˜a, A. M.; Maurice, D.; White, C. A. J. Phys. Chem. 1995, 99, 14261. (21) Johnson, B. G.; Gill, P. M. W.; Head-Gordon, M.; White, C. A.; Maurice, D.; Adamson, R. A.; Adams, T. R.; Oumi, M. Q-Chem, Q-Chem, Inc.: Pittsburgh, 1995 (unpublished). (22) Davidson, E. R. J. Comput. Phys. 1975, 22, 571. (23) Almlof, J.; Faegri, K., Jr.; Korsell, K. J. Comput. Chem. 1982, 3, 385. (24) Stanton, J. F.; Gauss, J.; Ishikawa, N.; Head-Gordon, M. J. Chem. Phys. 1995, 103, 4160. (25) Curtiss, L. A.; Raghavachari, K.; Trucks, G. W.; Pople, J. A. J. Chem. Phys. 1991, 94, 7221. (26) Hiratsuka, H.; Okamura, T.; Tanaka, I.; Tanizaki, Y. J. Phys. Chem. 1980, 84, 285. (27) Land, E. J.; Porter, G.; Strachan, E. Trans. Faraday Soc. 1960, 56, 1885. (28) Land, E. J.; Porter, G. Trans. Faraday Soc. 1963, 59, 2027. (29) Cossart-Magos, C.; Leach, S. J. Chem. Phys. 1976, 64, 4006. (30) Rice, J. E.; Handy, N. C.; Knowles, P. J. J. Chem. Soc., Faraday Trans. 2 1987, 83, 1643. (31) Krauss, M. J. Mol. Struct. (THEOCHEM) 1994, 307, 47. (32) Takahashi, J.; Momose, T.; Shida, T. Bull. Chem. Soc. Jpn. 1994, 67, 964. (33) Chang, H. M.; Jaffe´, H. H. Chem. Phys. Lett. 1973, 23, 146. (34) Leyva, E.; Platz, M. S.; Biu, B.; Wirz, J. J. Phys. Chem. 1987, 91, 2293.
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