State specific multireference Møller–Plesset perturbation theory: A few applications to ground, excited and ionized states

Chemical Physics 401 (2012) 15–26

Contents lists available at ScienceDirect

Chemical Physics journal homepage: www.elsevier.com/locate/chemphys

State specific multireference Møller–Plesset perturbation theory: A few applications to ground, excited and ionized states Sudip Chattopadhyay a,⇑, Uttam Sinha Mahapatra b, Rajat K. Chaudhuri c a

Department of Chemistry, Bengal Engineering and Science University, Shibpur, Howrah 711103, India Department of Physics, Maulana Azad College, 8 Rafi Ahmed Kidwai Road, Kolkata 700013, India c Indian Institute of Astrophysics, Bangalore 560034, India b

a r t i c l e

i n f o

Article history: Available online 28 April 2011 Keywords: State specific multireference Møller–Plesset perturbation theory Ground, or excited/ionized states Spectroscopic constants Excitation or ionization energy

a b s t r a c t We provide further tests and illustrations of the complete active space based state specific multireference Møller–Plesset perturbation theory (SS-MRMPPT) which opens the way for the treatment of dynamic correlations in situations containing significant static correlations also in an accurate, size- extensive and intruder free manner enjoying at the same time a very favorable computational cost. We have investigated various interesting and computationally challenging systems [H2O, H2O+, BeC, MgC, CO+, Be3, benzene, trimethylenemethane and 1,2,3-tridehydrobenzene] in either their ground, or excited/ ionized states. It is found that SS-MRMPPT calculations provide very encouraging results which can be meaningfully compared with other state-of-the-art theoretical estimates. Present results convincingly indicate that the SS-MRMPPT method is not only successful in portraying situations that warrant MR description but also performs acceptably good in cases where a naive single-reference method is enough which reinforces the claim that the SS-MRMPPT is a very useful and flexible ab initio method. Ó 2011 Elsevier B.V. All rights reserved.

1. Introduction Despite various methodological developments and subsequent computational implementations at the state-of-the-art level, contemporary electronic structure theory is still lacking a rigorously size-extensive method that describes quasidegenerate situations or phenomena (characterized by large non-dynamic correlation effects) in a proper manner through accurate and balanced computation of dynamical and non-dynamical correlation effects. The most typical examples where nondynamical correlation effects are of utmost importance include radicaloids, excited and ionized states, transition states and reaction intermediates, energy surfaces along bond breaking coordinates and so on. Amongst the various approaches to describing quasidegenerate situations, the multireference perturbation theory (MRPT)[see recent reviews [1–3] and references therein] is usually considered as a potentially useful tool. The major problem of the conventional MRPT treatment arises from the difficulties in combining numerical stability (due to the notorious intruder state problem [4,5]) and satisfactory low-order results with the strict size consistency. Recently, Kowalski and Piecuch [6] argued that the intruder-states and intruder-solutions problems are an innate feature of the ⇑ Corresponding author. E-mail addresses: [email protected] (S. Chattopadhyay), [email protected] (U.S. Mahapatra), [email protected] (R.K. Chaudhuri). 0301-0104/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2011.04.011

non-linear nature of the generalized Bloch equation based MR methods and this leads to the problem of numerical attainability of a desired solution. In order to obtain smooth energies over a wide range of reaction coordinates and other cases of electronic quasidegeneracies, it has been necessary to employ intruder state free techniques. The search for viable intruder-free size-extensive theories thus continues to be a major thrust of MR correlation methods. In order to circumvent or at least to attenuate the intruder effect, one has to either incorporate level shift technique (adjusted in an ad hoc manner) [7,8] or rely on the incomplete model space (IMS) [9,10] approaches. An alternative approach to avoid intruder states, in part, is by employing an intermediate Hamiltonian formalism that only considers a subset of the reference space states [11]. It has been found that the choice of the shift parameter can destroy the size consistency and influence significantly the quality of results indicating serious doubts on the validity of the shift techniques in multireference perturbation theory [8]. The effective valence-shell Hamiltonian method of Sheppard and Freed [12] appears to be relatively free of such problems imposing a well defined energy gap between the reference energies of the states corresponding to the model space and the states corresponding to the secondary space. In this context, we should also mention the development of generalized Van Vleck perturbation theory (GVVPT2) of Hoffmann [13]. Needless to say, it is highly desirable to have a genuine MR approach that is capable of handling quasidegenerate situations

16

S. Chattopadhyay et al. / Chemical Physics 401 (2012) 15–26

without facing intruder effects in a natural manner. For the very reason, at present, a consensus is emerging in the literature that it appears best to use a ‘state-selective’ or ‘state-specific’ MRPT approach, in which one state is obtained at a time. Although there has been substantial success to numerous chemical and spectroscopical problems with the MRMPPT [of Hirao and co-workers [14]] and CASPT2 [due to Roos and co-workers [2,15]] methods, these MRPT methods are not fully commensurate with a number of useful theoretical concepts [16], the principal problems encountered include the size-extensivity issue [17,18]. The extensivity of the method is crucial when we deal with extended systems. We also mention that the MCQDPT (a multi-root version of MRMPPT [14]) of Nakano [19] and multi-state version of the CASPT2 method formulated by Finley et al. [20] are not free from such an objection. Moreover, the CASPT2 and MRMPPT (as well as its different variants) are not always immune from intruder states, especially when applied to large complete active space (CAS) or to the entire range of nuclear coordinates [10,21]. The multiconfiguration perturbation theory (MCPT) due to Surján, Szabados, and co-workers [22] is inherently size-inconsistent. BWPT of Hubac´, Wilson and coworkers [23] which avoids intruders using BrillouinWigner (BW) scheme, however, does so at the cost of a complete loss of sizeextensivity which makes a posteriori or iterative corrections necessary [24]. The size-extensivity is also not manifest in GVVPT2 [25]. As discussed by Dyall [26], the definition of a partially bi-electronic zeroth-order Hamiltonian can be effective in removing the intruder states. The n-electron valence space perturbation theory (NEVPT) [27] which makes use of such a Hamiltonian is shown to be free from the problem of intruder state. CIPSI (configuration interaction by perturbation with multiconfigurational zeroth-order wave functions selected by iterative process) approach [28] has established itself as a method to avoid the intruder problem. New areas for using the MRPT method have opened due to the development of state specific MRPT (SS-MRPT) method by Mahapatra, Datta, and Mukherjee [29] using their state-specific multireference coupled-cluster theory (SS-MRCC) [30,31]. The SS-MRPT is explicitly size-extensive, spin free [31] and yields size-consistent energies only when the model space adopted is able to correctly describe the fragments and the orbitals are localized on either of the fragments. Thus, the SS-MRPT method encompasses almost all the desirable properties that a MRPT method should have and thus it seems worthwhile to investigate its performance for various systems as is evident from the published works [29,31–34]. Recently, Evangelista et al. [34] demonstrated that the SS-MRPT theory is particularly useful in MR focal point extrapolations to determine ab initio limits. SS-MRPT is generalizable to the use of incomplete active space [31] (although currently all the applications are based on CAS). In recent years, our group has worked on several applications of SS-MRPT using various partitioning and expansion schemes [32,33]. Among the various versions, in the present paper, we employ the second-order SS-MRPT within the framework of multipartitioning Møller–Plesset scheme using Rayleigh-Schrödinger (RS) perturbative expansion and use the abbreviation SS-MRMPPT. The SS-MRMPPT method can be considered as a consistent generalization of multipartitioning MP originally published by Zaitevskii and Malrieu [35]. In passing, we also mention the geminal ansatz based developments [36,37]. Recently, Ten-no [38] has proposed and tested a simple F12 geminal correction in multireference perturbation theory. At the end, we also mention the works relating to numerical comparison of various MR-based PT methods on representative examples [25,39]. In the present paper, we have applied the SS-MRMPPT method to investigate molecular systems of various sizes and characters [namely H2O, H2O+, BeC, MgC, CO+, Be3, benzene (C6H6), trimethylenemethane (TMM) and 1,2,3-tridehydrobenzene (TDB)]. In addition to optimizing geometrical parameters, we also focus on the

excited and ionized states. It should be stated that the problem of calculating electronic excited or ionized states is a challenging task in contemporary quantum chemistry. The SS-MRMPPT method can be applied to calculate both ground and excited states as long as the target energy is rather well removed from the energies of the virtual functions. The test of accuracy of a genuine MR-based method in the single-reference situation is also an important probing ground to establish the generality of the method. It is thus important to investigate how well the SS-MRMPPT gradient method performs in such circumstances and how reliable are the results it yields when compared with contemporary ab initio molecular electronic structure studies. As is well known, the ground states of C6H6 and H2O (equilibrium region) possess single reference character. It is not our aim to provide the best possible results for the systems considered here as it would require a detailed monitoring and analysis of the basis set and reference space effects. Instead, we want to show the extent of accuracy that can be achieved via SS-MRMPPT method with relatively small basis set and conceptually optimal reference space to study electronic states of molecules displaying pronounced multireference as well as single reference characters. Indeed, at this level of theory, it would be meaningless to aim for a high accuracy. 2. SS-MRPT theory in brief: structure of the working equations The origin and formalism of the SS-MRPT have been discussed in detail in Refs. [29,31] from which one can say that the SS-MRPT method involves the following three steps: (i) choose a proper zero-order wave function in order to incorporate static correlation and construct the corresponding reference coefficients and energy (ii) carry out the iterative solution to obtain cluster amplitudes using the reference coefficients and energy of step (i), and (iii) use the converged cluster amplitudes of step (ii) to construct the effective Hamiltonian and compute the (pseudo) second order energy through diagonalization. In the SS-MRPT method (which employ the the Jeziorski– Monkhorst ansatz for the wave operator [40]), the Schrödinger equation is converted into an effective eigenvalue problem as

X

e ð2Þ cm ¼ Eð2Þ cl ; H lm

ð1Þ

m

e ð2Þ where H reference specific dressed Hamiltonian and lm denotes P lð1Þ e ð2Þ ¼ H H lm þ lm l Hll t m ðmÞ. The reference coefficients cl of wave function and the target energy E are obtained by diagonalizing an effective Hamiltonian, Eq. (1). In Eq. (1), only one root corresponding to the target state is acceptable due to the state specific nature of weighting coefficients, cl and cluster amplitudes (tl) while the remaining eigenvalues have no physical meaning. Consequently, the SS-MRPT equations must be solved anew for each target state. Cluster amplitudes appear in Eq. (1) are determined by solving the following equations [29]:

H ll tl ðlÞ ¼ þ ½E0  Hll  lð1Þ

Pm–l m

  0 0 tlð1Þ l ðmÞHlm cm =cl ½E0  Hll 

;

ð2Þ

l

where fvl g (for all l) are the set of virtual functions spanning the D E l space complementary to the model space. Here, vl jT lð1Þ l j/l ¼ D E D E  l l lð1Þ l l lð1Þ tlð1Þ l ðlÞ; vl jT m j/l ¼ t l ðmÞ vl jHj/l ¼ Hll , and vl jH0 jvl ¼ Hll . The cluster operators, Tl are excitation operators for each reference configuration (/l) that promote electrons from the occupied to the virtual orbital space of /l. The unperturbed state energy E0 is deP fined by m Hlm c0m ¼ E0 c0l . In Eq. (2), we have used MP partitioning scheme within the framework of RS-perturbative expansion. In SS-MRPT method all the variables [cl, Tl] are determined

S. Chattopadhyay et al. / Chemical Physics 401 (2012) 15–26

self-consistently. We note that the state-specificity is explicit in the working equation, Eq. (2), via the dependence on the model coefficients of the reference state. For a CMS or CAS, all excitation operators are external by definition. Containing only connected terms in Eqs. (1) and (2) with a complete model space (CMS) guarantees a rigorous size extensivity of the theory. In our present work, the reference coefficients have been kept fixed in the course of iteration of the cluster finding equation while the coefficients get updated i.e. relaxed during the diagonalization step via Eq. (1). In view of this, the computed energy is not a rigorously second-order, rather a pseudo - second order in nature and the formalism provides a relaxed description of the nondynamical correlation in presence of dynamical correlation. The form of the denominator in Eq. (2) ensures that SSMRMPPT is not prone to intruder-state problems for low-lying electronic states whenever the zeroth-order energies (E0) of the target state and the external space are well separated. We employ the following Fock operator [32,33]:

fl ¼

X ij

" ij fcore þ

# X  ju 1 uj  l n j o V iu  V iu Duu Ei : 2 u

ð3Þ

Here, u represents both a doubly occupied and a singly occupied active orbital in /l, and Dls are the densities labelled by the active orbitals. Since, our H0 is always diagonal for MP scheme, the zerothP l order Hamiltonian operator is: H0 ¼ i flii fEii g. Thus, H0 is build on one-particle operators i.e. it is monoelectronic in nature. 3. Molecular applications and discussion Although the current and previous applications of SS-MRMPPTalong with other versions of it have shown considerable promise [29,31–34], much work remains to be done in order to outline the range of applicability of the approach. In our present applications, we examine (a) selected spectroscopic constants of the ground H2O and ground as well as excited H2O+, (b) vertical ionization energies of H2O to different electronic states of H2O+, (c) state energies of ground as well as low-lying excited states and (vertical) excitation energies of triangular structure for the Be3 cluster and finally, (d) equilibrium geometries for 11A1g benzene, 1 1A1 trimethylenemethane diradical and 1,2,3-tridehydrobenzene triradical. One of the ultimate goals of structural chemistry is the determination of equilibrium molecular geometries which are defined by local or global minima on the adiabatic energy surface. Additionally, accurate equilibrium structures are also helpful to properly calibrate the accuracy of various quantum chemical approaches. There has been a great deal of interest shown to the systems considered here, resulting in a large number of both experimental and theoretical studies, some combining theory with experiment. Consequently, it is possible to make a thorough comparison with the existing theoretical studies and thus assess the performance of our SS-MRMPPT approach. We highlight the fact that the results collected here (from literature) are not in an attempt to make any ‘direct comparison’ with our present results. The only intention to illustrate this lies in our attempt to bring onto fore how the present method fares vis-á-vis the others in predicting aspects related to electron correlation. In that sense, we mentioned the results of various established methods just as reference values. To calibrate our findings, we examine molecular constants such as equilibrium bond length (Re), harmonic vibrational frequency (xe), and the dissociation energy (De). The spectroscopic constants have been obtained by numerically oriented gradient based SS-MRMPPT approach [41]. The calculated dissociation energies have been computed by subtracting the energy at a large interatomic distance from that at Re. In our calculations, the CAS is described by the rep-

17

resentation CAS (m, n). The notation reflects the fact that the reference CAS functions include m active electrons and n active orbitals. It is important to note that the essence of the MR-based theory is to employ as small an active space as possible. The choice of the model space in the MRPT calculations is an important issue that can affect the accuracy and the convergence of the calculations. In the applications of the SS-MRMPPT method, molecular orbitals, two-electron integrals and the core-Hamiltonian matrices have all been computed via freely available GAMESS (US) program [on a MacBook Pro Laptop]. The basis sets have been obtained from the EMSL database [42]. 3.1. Ground state H2O and the doublet states of H2O+ We first consider the prototypical H2O molecule, which has been extensively studied in the past [43–53]. There are two bonding electron pairs involving two O–H bonds and two lone pairs near the valence shell of the oxygen atom and hence a proper reference wavefunction calls for a large reference space. In our previous publication [32], we had considered C2v symmetry adapted energy surface calculations using relatively small basis set, 6–31G and CAS (6, 6) [with six active electrons in six active orbitals (1b1, 3a1, 1b2, 4a1, 2b2, 2b1)]. The same CAS scheme has been adopted in our present calculation of geometry optimization with aug-cc-pVDZ basis set [42]. 1s core orbital of oxygen has been excluded from the correlation treatment. In recent past, various groups have used different CAS to investigate the H2O system [47,48,50,51]. The results of SS-MRMPPT calculations, together with a number of conventional methods have been presented in Table 1. The accuracy of the geometrical parameters calculated with the SSMRMPPT is roughly comparable to that of the other established theoretical predictions that clearly highlights that the predictive capability of the SS-MRMPPT method is not limited to computation of the energetics but is also capable of providing a highly accurate description of molecular constants. We also extended our SS-MRMPPT gradient calculations on H2O+ to its three lowest doublet states (the ground electronic state of H2O+, 2B1 and its two first excited states, 2A1 and 2B2) [54–58]. Needless to say, there has been a great deal of interest in the excited/ionized states of water resulting in a large number of both experimental and theoretical studies, some combining both theory and experiment (see Ref. [45] and references therein). In Table 2, we have provided the geometrical parameters for 12B1, 12A1 and 12B2 states of H2O+. The states considered here have been extensively studied using semi-empirical methods and modern ab initio methods by various authors [45,59]. These calculations can be used to emphasize the predictive ability of the SS-MRMPPT approach to molecular constants. The present calculations are carried out using C2v symmetry. The minimum of the 12B1 state, according to our calculations, has been found at ROH = 1.000 Å and \H–O–H = 107.8° (close to that of the neutral ground state). The geometry of the 12B2 state minimum is found by our calculations to be for ROH = 1.14 Å and \H–O–H = 56.2°. The ionized state 12A1 calculated by us shows a minimum at ROH = 0.99 Å and \H–O–H = 172.3°, indicating that the first excited state of H2O+ is approximately linear (D1h). High level theoretical calculations on 12A1 state have determined it to be nearly linear [45,60]. As per Reutt et al., the ionic ground state of H2O (12B1) possesses an equilibrium geometry with \H–O–H = 110.46° and ROH = 0.9988 Å. Reutt et al. stated that the first electronically excited state, 12A1 is near linear, with bond length ROH = 0.98 Å. For 1 2 B2 state, \H–O–H = 54.98°, and ROH = 1.140 Å. Hence, the equilibrium geometry of the first three electronic states of H2O+ provided by SS-MRMPPT are vastly different as is also evident from other theoretical [45] and experimental results [60]. From the ta-

18

S. Chattopadhyay et al. / Chemical Physics 401 (2012) 15–26

Table 1 Equilibrium molecular constants of the ground state water molecule. Bond lengths and bond angles are given in Å and degree (°), respectively. Values enclosed in parentheses due to the TZ2P basis. References

Methods

Basis

Re

\H–O–H

Present work Ref. [44]

SS-MRMPPT/CAS (6, 6) CCSD (T) CCSD (T) WuppertalBonn MRDCI CISDTQ CCSD OO-CCD CCSD (T) CCSD 2R-RMR-CCSD 3R-RMR CCSD FCI W4 Experiment

aug-cc-pVDZ cc-pVDZ cc-pVQZ

0.968 0.9663 0.9579 0.952 0.9688 0.9610 (0.9563) 0.9609 (0.9563) 0.9618 (0.9586) 0.965 0.965 0.965 0.975 0.9576 0.95782

103.2 101.91 104.12 104.5 103.6 104.63 (104.74) 104.64 (104.74) 104.49 (104.52) 102.0 102.0 102.0 110.6 104.50 104.485

Ref. [48] Ref. [46] Ref. [50]

Ref. [47]

Ref. [43] Ref. [52] Ref. [53]

DZP DZP DZP DZP cc-pVDZ cc-pVDZ cc-pVDZ cc-pVDZ aug’-cc-pV (n + d) Z

MRDCI: Multireference double configuration interaction. The generalized Davidson correction has been included. A large Rydberg basis set has been used. RMRCCSD: Reduced multireference (RMR) coupled-cluster method with singles and doubles (CCSD). OO-CCD: Optimized-orbitals coupled cluster doubles.

Table 2 Selective structural constants for H2O+ using CAS (6, 6). Bond lengths and bond angles are given in Å and degree, respectively. Ref.

State

Methods

Re

\H-O-H

Present work Ref. [45] Ref. [61] Present work Present work Ref. [45]

12B1

SS-MRMPPT MRCI Experiment SS-MRMPPT SS-MRMPPT MRCI

1.000 1.000 1.000 0.99 1.14 1.16

107.8 107.4 109.3 172.3 56.2 54.0

12A1 12B2

Table 3 Vertical ionization energies (eV) of H2O to different electronic states of H2O+. Ref.

State

Methods (eV)

IP

Present work Ref. [45] Ref. [54] Ref. [55] Ref. [61] Present work Ref. [45] Ref. [54] Ref. [55] Ref. [61] Present work Ref. [45] Ref. [54] Ref. [55] Ref. [61]

12B1

SS-MRMPPT MRCI CEPA MO-SCF Experiment SS-MRMPPT MRCI CEPA MO-SCF Experiment SS-MRMPPT MRCI CEPA MO-SCF Experiment

12.66 12.30 12.48 11.07 12.61 14.78 14.60 14.68 13.01 14.73 18.92 18.70 18.86 17.94 18.55

12A1

12B2

ble, it is clear that our computed geometrical parameters for various states of oxoniumyl ion are in well agreement with the MRCI (multireference configuration interaction) [45] and experimental results (whenever available). To judge the consistency and robustness of the SS-MRMPPT calculations for energies of the three lowest doublet states of H2O+ in comparison with those from the previously mentioned H2O results, we computed the theoretical ionization potentials or energies (IPs) of H2O with the same basis set (aug-cc-pVTZ ) at the computed ground state minimum geometry. Our calculated IPs for the 12B1, 12A1 and 12B2 states are summarized in Table 3 with the experimental values [61,60]. It can be seen from the entries of Table 3 that the ionized energies provided by SS-MRMPPT method are noticeably close with the experimental values [61].

3.2. Diatomics alkaline-earth metal carbides (3R BeC and 3R MgC) The BeC and MgC molecules have gained considerable attention in the realm of electronic structure theory over the past few years due to the existence of strong quasidegeneracy of their ns and np orbitals (of alkaline-earth metal) resulting in a multireference wavefunction with significant biexcited ns2 M 3np2 character, even at the equilibrium geometry [62–66]. To the best of our knowledge, these two molecules are unknown experimentally, however, several theoretical studies have been devoted to them [62–66]. Previous calculations [63–66] for BeC and MgC using different methodologies demonstrated a triplet (3R) ground state with a quintet (5R) state lying very close above. These features recommend it for present study. In the current paper, we focus on the investigation of the ground states (3R) of these two molecules. In planning the calculations, we used CAS (4, 4) and aug-cc-pVDZ basis set. 1 s orbitals of Mg and C have been excluded from the correlation treatment in the MgC system; whereas, in the BeC system, only 1 s orbital of C was frozen. In Table 4, we have summarized spectroscopic constants calculated for 3R BeC via SS-MRMPPT method. The results are compared against the traditional SR-based CCSD approach, as well as with other single and multireference methods employing the cc-pVQZMg/aug-cc-pVQZC basis [65]. The equilibrium bond length and vibrational frequency of BeC using GVB-CI + Q are 1.700 Å and 648.65 cm1 respectively [63]. Borin and Ornellas [64] obtained a dissociation energy of 55.1 kcal/mol and an equilibrium bond length of 1.667 Å via the MRCI method. Teberekidis et al. [65] observed that the best extrapolated fv MR-CI + core + Q Re and De values for the ground state are 1.680 Å and 49.7 kcal/mol respectively. Clearly, our Re (1.6704 Å) and De (51.65 kcal/mol) values are in acceptably good agreement with the best fvMR-CI + core + Q CBS values due to Teberekidis et al. [65]. Teberekidis et al. [65] found that the ROHF (restricted openshell HartreeFock), CISD (configuration interaction singles-doubles), and RCCSD (restricted coupled cluster singles and doubles) methods fail in predicting the correct ground state (leads to 5R). As that for the MR-BWCCSD method, our SS-MRMPPT calculation yields the correct ground state,3R and leads to a fair agreement with the current generation calculations such as fvMRCI + Q, MRACPF, MR-BWCCSD, and RCCSD (T). The dissociation energy and the vibrational frequency reported by the present SS-MRMPPT method appears to be a bit overestimated with respect to the other methods reported in the table. On the whole, our results are very close to the findings of previous results obtained by established methods.

19

S. Chattopadhyay et al. / Chemical Physics 401 (2012) 15–26 Table 4 Spectroscopic constants [equilibrium bond length: Re(Å), vibrational frequency: xe (cm1) and dissociation constant, De (kcal/mol)] for the ground state BeC system. Ref.

Basis

Methods

Re

xe

De

Present work

aug-cc-pVDZ aug-cc-pVDZ cc-pVQZBe aug-cc-pVQZC

SS-MRMPPT (4, 4) CASSCF (4, 4)

1.6704 1.666

974.27 947.4

51.65 45.0

MRCI MRCI + Q fvCAS-SCF fvMR-CI fvMR-CI + Q MR-BWCCSD (4, 4) MR-BWCCSD (6,6) RCCSD RCCSD (T) GVB-CI + Q

1.680 1.633 1.692 1.684 1.684 1.677 1.669 1.700 1.677 1.700

906.7 894.7 900.4 909.4 906.2 905.2 877.3 901.0 648.65

47.7 46.5 42.5 48.5 49.2 45.3 46.5 37.8 46.9 n.a

MRACPF MRCI

1.684 1.683

908 925

49.3

Ref. [65]

Ref. [66]

(6s4p2d)Be (6s5p2d)C cc-pVQZ+[3s3p2d1f] 6–311 + G (3d1f )

MRACPF: Multireference averaged coupled pair functional. GVB-CI: Generalized valence-bond configuration interaction. fvMR-CI: Internally contracted fvCAS-SCF + single + double replacements. fvMR-CI + Q: fvMR-CI + multireference Davidson correction.

Table 5 Spectroscopic constants [equilibrium bond length: Re(Å), vibrational frequency: xe (cm1) and dissociation constant, De (kcal/mol)] for the ground state MgC system. Ref.

Basis

Methods

Re

xe

De

Present work Present work Ref. [65]

aug-cc-pVDZ aug-cc-pVDZ cc-pVQZMg aug-cc-pVQZC

SS-MRMPPT (4, 4) CASSCF (4, 4)

2.0707 2.069

666.23 578.5

40.56 29.7

MRCI MRCI + Q MR-BWCCSD/CAS (4, 4) MR-BWCCSD/CAS (6,6) RCCSD RCCSD (T) fvCAS-SCF fvMR-CI fvMR-CI + Q GVB-CI + Q

2.083 2.090 2.086 2.077 2.097 2.085 1.692 1.684 2.090 2.103

556.0 546.5 546.2 526.5 554.7 900.4 909.4 547.3 492.51

35.3 34.7 32.7 33.0 26.4 33.8 42.5 48.5 36.4 89.2

MRACPF MRCI

2.089 2.089

558 555

35.5

Ref. [66]

(6s4p2d)Mg (6s5p2d)C cc-pVQZ+[3s3p2d1f] 6–311 + G (3d1f )

RACPF: Multireference averaged coupled pair functional. MRCI: Configuration-selected MR-CI. fvMR-CI: Internally contracted fvCAS-SCF + single + double replacements. fvMR-CI + Q: fvMR-CI + multireference Davidson correction.

Table 5 summarizes the different spectroscopic constants of 3

R MgC computed numerically at various levels of theory. The

MRCI and MR-BWCC results at different levels of sophistication, employing the cc-pVQZMg/aug-cc-pVQZC basis of Teberekidis et al. [65], are also listed for comparison. To monitor our results, values obtained by fvMR-CI calculations are also reported in the same table. Teberekidis et al. [65] found that the ROHF calculation produces repulsive 3R state and CISD computation offers erroneously 5R as the ground state. In contrast, our SS-MRPPT approach predicts correctly the ground-state symmetry as that predicted by the computationally highly demanding RCCSD (T) (restricted coupled cluster singles-doubles-with perturbative triples corrections) and MR-BWCC methods. Compared with previous theoretical results collected in Table 5, the agreement is quite encouraging. The overall agreement between the SS-MRMPPT and MR-BWCC results is quite promising. The vibrational frequency calculated by SS-MRMPPT is slightly higher than that of previous RCCSD,

RCCSD (T), MRACPF, MRCI + Q and MR-BWCC studies for MgC. It is important to note that the CASSCF (4, 4) frequency is closer to the MR-BWCCSD/CAS (4, 4) value than the SS-MRMPPT/CAS (4, 4) one. At present, we do not understand why the deviation of the vibrational frequency with respect to other higher order methods increases when we account for dynamical correlation effect (via SS-MRMPPT) of the state considered. The SS-MRMPPT equilibrium bond length is slightly shorter than the values obtained from other MRCI and MR-BWCC methods. Bauschlicher it et al. [62] obtained a bond length of 2.0881 Å, which further reduced to 2.0702 Å after inclusion of core-valence correlation and polarization effects. As that of our results, the equilibrium distance calculated in their work is slightly shorter than those of MR-BWCC and RCCSD (T) calculations due to Teberekidis et al. [65]. For the ground state, our results indicate that the equilibrium bond distance increases in the order of BeC followed by MgC. Keeping in mind the perturbative method, the overall performance of

20

S. Chattopadhyay et al. / Chemical Physics 401 (2012) 15–26

the SS-MRMPPT method with respect to the best estimated fvMRCI + core + Q and MR-BWCCSD to investigate 3R state of the experimentally unknown alkaline earth metal carbides, BeC and MgC, is reasonably promising. 3.3. Carbon monoxide positive ion in ground state (X 2R+ CO+) Next, we test SS-MRMPPT on the challenging problem of computing the spectroscopic constants of the ground state carbon monoxide positive ion (CO+) displaying a significant multireference character. The ground electronic state of CO+ is doublet in nature (X 2R+). A number of investigations have been carried out to study the electronic structure of CO+ [67–70]. Major difficulties encountered when investigating X2R+ CO+ are: (a) the multireference character of the system even near equilibrium region, (b) the presence of significant dynamic electron correlation effects and (c) open-shell nature. Molecular orbitals have been obtained from a CASSCF (7, 7) computation using cc-pVTZ basis set. In Table 6, we report the SS-MRMPPT dissociation energies, equilibrium distances, and harmonic vibrational frequencies of X2R+ CO+. The results have been compared with the corresponding experimental results [67]. Comparisons show that our computed values are in close proximity with the previous CASSCF + MRCI and full valence CASSCF calculations due to Martin and Fehér [70]. It must be emphasized here that the truncated MRCI is not size-extensive while the SS-MRMPPT method is fully size-extensive. It is notable that our results of the spectroscopic constants are in promising agreement with the experimental results quoted in Ref. [67]. The SS-MRMPPT estimated value is in agreement with the value of 177.56 kcal/mol (7.70 ± 0.19 eV) given by Reddy and Viswanath [71]. The molecular constants due to Misra et al. [72] are also listed in Table 6 which are very close to our SS-MRMPPT values. The estimated accurate values of spectroscopic constants of X2R+ CO+ are in close agreement with the values estimated here using SSMRMPPT/cc-pVTZ scheme, indicating the effectiveness of the SSMRMPPT method in handling of the ionized non-singlet state. 3.4. Ground and excited states of equilateral triangular beryllium trimer (Be3) Despite the small size, a quantitatively correct description of the energetics of the ground state of Be3 is a demanding test case for the standard electronic structure methods [73–75,77,76] because of the near-degeneracy of Be 2s and 2p atomic orbitals which is significantly enhanced upon the formation of the Be trimer (gives rise to significant MR character in the cluster) and the flatness of the energy surface (causes the structural parameters to be exceedingly sensitive to the adequacy of the chosen computational method to describe the binding)(see Ref. [75]). The full CI (FCI) [75] and and EOMCC (equation-of-motion coupled-cluster)

Table 6 Spectroscopic constants [equilibrium bond length: Re(Å), vibrational frequency: xe (cm1) and dissociation constant, De (kcal/mol)] for the ground state CO+ system. Methods

Re

xe

De

SS-MRMPPT/cc-pVTZ FVS-CASSCF/[8s6p3d]a MCSCF-CI/[7s5p2d]a Misra et al. Experiment

1.1251 1.1201 1.1240 1.1150 1.1150

2271.37 2203 2246 2214.27 2214.2

189.04 n.a n.a 180.77 192.18

n.a: Not available. FVS-CASSCF: full valence-space CASSCF. Experiment:Ref. [67] a Ref. [70]

[77] calculations on Be3 demonstrate that the cluster is characterized by significant high order correlation effects in the ground electronic state. Be3 cluster is not bound at all at the Hartree–Fock level. In view of this, the bonds involved in this system are difficult to handle computationally. FCI results due to Junquera-Hernández et al. [75] for ground as well as excited states of Be3 employing the atomic natural orbital (ANO) [3s2p1d] basis set [optimized by Widmark et al. [78]] have been used as a reference in our present computations. In the present paper, we have reported the SS-MRMPPT energies for the ground and few low-lying excited states along with vertical electronic transition or excitation energies using the same basis and scheme as that of Junquera-Hernández et al. [75] for equilateral triangular Be3 cluster. The MR nature of the states considered here is clear from Table-I of Ref. [75]. In our calculations, RHF orbitals and CAS (6,6) have been used. The Be atom has only two valence electrons, and thus, in Be3, there will be six electrons for three bonds or two electrons per bond available. Tables 7 and 8 summarize respectively, state energies for various states and excitation energies among themselves obtained from the SS-MRPT method along with FCI calculations. The excitation energies have also been compared with the EOMCC results available in the bibliography [77]. All EOMCC calculations reported in Table 7 have been performed by Piecuch et al. [76,77] with RHF orbitals using the same geometry and ANO basis set as used in Ref. [75]. The CAS (4,6)-SDCI vertical excitation energies are also reported in Table 8 that can provide an additional test on the quality of our computed values. The results shown in Table 7 demonstrate clearly that the values of state energies due to the SS-MRMPPT calculations are acceptably close to the FCI results of Junquera-Hernández et al. [75]. The SS-MRMPPT approach provides a correct ordering of excited states. Table 8 displays the performance of the SS-MRPT to be satisfactorily close to the FCI values. The closeness of the SS-MRMPPT results to those of more extended treatments of electron correlation via EOMCC and CAS (4,6)-SDCI methods demonstrate the ability of our approach to treat the correlation energy (dispersion effects) in highly delicate weakly bound Be3. It is worth noting that the EOMCCSD and CR-EOMCC (completely renormalized EOMCC) methods completely fail for the excited state having significant doubly excited contributions (mentioned in Table 8 as 11 A001 ). This should be contrasted with the performance of our SS-MRMPPT calculation, which describes the 11 A001 state with acceptable accuracy (0.06 eV error with respect to the FCI value). From the foregoing discussion it is found that the SS-MRMPPT approach as that of the computationally demanding EOMCCSDt (I) theory [76,79] provides a useful solution to investigate ground as well as excited states of triangular Be3 cluster having significant single and doubly excited contributions, which may help us to understand the electronic structure of clusters of alkali earth elements. The equilibrium geometry corresponding to a Be–Be distance (RBeBe) of Be3 obtained by our SS-MRMPPT (6, 6)/[3s2p1d] gradient scheme is 2.2760 Å which is close to the actual FCI value (2.2741 Å). For the sake of comparison, we also mention the RBeBe values corresponding to the optimized equilibrium geometry

Table 7 Ground and excited state energies (a.u.) of the equilateral triangular Be3 system, as described by the ANO basis set of Ref. [75] obtained with the SS-MRMPPT approach, with those resulting from the FCI calculations [75]. States 1 1 1 1

1

A1 1 00 E 1 00 A1 1 0 E

SS-MRMPPT

FCI

43.854401 43.791364 43.791068

43.882330 43.820925 43.816867

43.774887

43.807524

21

S. Chattopadhyay et al. / Chemical Physics 401 (2012) 15–26 Table 8 Comparison of vertical excitation energies (eV) of the equilateral triangular Be3 system, as described by the ANO basis set of Ref. [75] obtained with the SS-MRMPPT approach, with those resulting from the CAS (4,6)-SDCI, EOMCCSDt, EOMCCSD, CR-EOMCCSD (T), and full CI calculations. States 00

1 1E (S) 1 1 A001 (D) 0 1 1E (S)

SS-MRMPPT

CAS (4,6)-SDCI

FCI

EOMCCSD

CREOMCCSD (T)

EOMCCSDt (I)

1.715 1.723

1.73 1.80

1.67 1.78

1.718

1.647

1.658 1.877

2.164

2.04

2.05

2.122

1.988

2.031

FCI: Ref. [75]. EOMCCSD and CR-EOMCCSD (T): Ref. [77]. EOMCCSDt (I): Ref. [76]. Letter S (D) indicates the single (double) excitation character of excited state.

Table 9 Results of equilibrium geometry for the ground state C6H6 system. Ref.

Basis

Methods

RCC(Å)

RCH(Å)

\C-C-H (°)

Present work Ref. [83] Ref. [86] Ref. [82] Ref. [84]

cc-pVDZ MRMP ANO-RCC

SS-MRMPPT cc-pVDZ + CM[888/111] CASPT2/(6,6) CASPT2 CCSD (T) CCSD (T) CCSD (T) CCSD (T) Experiment

1.4054 1.397 1.394 1.395 1.4107 1.392 1.3911 1.3914 1.397

1.0932 1.084 1.081 1.085 1.0978 1.081 1.0800 1.0802 1.084

120.0

Ref. [85] Ref. [80]

cc-pVDZ Best estimated cc-pVQZ Best estimated

120.0 120.0

120.0

GVB: Gradient method for general valence bond wavefunction. CASPT2: The basis set used was Dunning’s cc-pVTZ basis set for carbon atoms and cc-pVDZ for hydrogen atoms, augmented with Rydberg functions (8s8p8d/1s1p1d) placed at the center of molecule. Ref. [86]: With shifted zeroth-order Hamiltonian.

of Be3 cluster due to Rendell–Lee–Taylor [73] with [5s3p2d1f] basis: RBeBe[CCSD] = 2.243 Å, RBeBe[CCSD (T)] = 2.232 Å and RBeBe[MRCI] = 2.223 Å. As we have observed in many other situations, the SS-MRMPPT method offers a promising description of complicated electronic quasidegeneracies present in the Be3 cluster. 3.5. 11A1g Benzene (C6H6) Benzene is an alternant conjugated hydrocarbon and is one of the most investigated molecules both experimentally [80,81] and theoretically [82–86]. The basis set used for carbon and hydrogen of benzene is Dunnings cc-pVDZ [42]. CAS (6, 6) has been used: the six p electrons (treated as active electrons) are distributed among the six valence p orbitals (three bonding p and three antibonding p⁄ orbitals). The optimized geometrical parameters calculated using the SSMRMPPT method with cc-pVDZ basis are listed in Table 9. In Table 9, we have also provided data computed via the CCSD (T) [84] method to assess our computational results as the method is referred to as a ‘gold standard’ in computational chemistry. Here, CCSD (T) results provide convenient benchmarks for calibrating the accuracy of our results. As shown in the table, the RCC and RCH bond lengths via full CCSD (T)/cc-pVDZ [84] are as much as 0.005 Å longer than our values. The power of the SS-MRMPPT method is once more attested by a fairly small difference between the SS-MRMPPT and the best estimated CCSD (T) values of Martin and co-workers [84]. The data presented in Table 9 show that the SS-MRMPPT method with cc-pVDZ basis set overestimates the bond lengths RC C and RCH by 0.013 and 0.012 Å respectively compared to the best estimated values at the CCSD (T) level reported by Martin and co-workers [84]. The C–C and C–H bond lengths obtained by CASPT2 method [82] are 1.395 Å and 1.085 Å respectively, which are close to the present SS-MRMPPT values. Our geometrical constants are very close to the MRMP values of Hirao and co-workers [83]. From the entries of Table 9, we find that

the RC-H distance that results from CCSD (T)/cc-pVQZ procedure of Gauss and Stanton appears to be too long. In this context, we also mention the optimized geometries [RCC = 1.3988 Å and RCH = 1.1005 Å] obtained from the work of Gauss and Stanton [85] using CCSD (T)/cc-pVQZ approach with vibrational corrections calculated at the SDQ-MBPT (4)/cc-pVTZ level. The RCC distance due to Gauss and Stanton is in excellent agreement with that of the present study, but there is a rather large difference in the RCH distance. RC-C value provided in our present study is in agreement with the empirically corrected recommendation of 1.3914 Å due to Gauss and Stanton [85]. In passing, it is important to note that the RCH distance provided by Gauss and Stanton in Ref. [85] is shorter than the experimental data as well as our computed value. Such a large difference seems rather unlikely. The deviation of our computed bond lengths with respect to the corresponding experimental findings is 0.009 Å which is quantitatively small, indicating that the recommended structural parameters are very accurate. It should be noted that at a fraction of the computational cost of CCSD (T), SS-MRMPPT correctly produces the equilibrium geometrical parameters of 11A1g benzene. As a whole, the performance of the SS-MRMPPT method for the estimations of molecular constants of H2O and C6H6 molecules is very similar to that of the best estimated ‘gold standard’ CCSD (T) values. This is an encouraging finding, considering the singlereference character. This is also very promising from the point of view of applying the SS-MRMPPT method in calculations aimed at treatment of dynamical correlation with a relatively low computational cost. We next present and discuss the results (obtained by numerically oriented SS-MRMPPT gradient scheme) for radicaloid species of various sizes and character, namely, the 11A1 trimethylenemethane (TMM) diradical and 2B1 1,2,3-tridehydrobenzene (TDB) triradical. TDB features more extensive electronic degeneracies than TMM. The interaction of unpaired electrons in radicals results in unusual bonding patterns. An increased multiconfigurational character challenges both theory and experiment. Di- and tri-radicals

22

S. Chattopadhyay et al. / Chemical Physics 401 (2012) 15–26

Table 10 Selected structural parameters of the 11A1 state of trimethelynemethane (TMM) calculated using the SS-MRMPPT method with cc-pVDZ and cc-pVTZ basis sets. Bond lengths and bond angles are given in Å and degrees, respectively. Parameters

SS-MRMPPT cc-pVDZ

R (C-C) (r1) R (C-C) (r2) R (C-H) (r3) R (C-H) (r4) R (C-H) (r5) \C-C-C(a1) \C-C-H(a2)

SS-MRMPPT cc-pVTZ

1.3600 1.4703 1.0839 1.0817 1.0811 120.07 121.09

MCSCF (10,10) cc-pVDZ

1.3477 1.4569 1.0803 1.0774 1.0771 119.88 121.22

1.370 1.496 1.082 1.080 1.080 121.1 121.2

BP86 cc-pVDZ 1.381 1.450 1.104 1.097 1.098 114.1 121.4

RMR-CCSD (T) 6–31G⁄ 1.349(1.345) 1.486(1.492) 1.082 1.080 1.080 121.1 121.2

SF-DFT 6–31G⁄ 1.3384 1.4526 1.0773 1.0749 1.0744 120.54 121.90

SS-MRCCSD cc-pVDZ 1.3615 1.4710 1.0961 1.0941 1.0934 120.07 120.17

MCSCF (10,10) and BP86: Ref. [87]. R-MRCCSD (T): Results obtained with the cc-pVTZ basis set are given in parentheses, Ref. [90]. and SF-DFT: Ref. [88].

thus represent excellent probing grounds for testing the efficiency of the SS-MRMPPT method in its ability to account for quasidegeneracy.

3.6. 11A1 trimethylenemethane (TMM) diradical The lowest excited singlet state (1A1) of TMM (non-Kekulé alternant hydrocarbon) represents a challenging subject for various ab initio calculations [87–90] due to the presence of significant degeneracy (for a detailed review of previous TMM studies, see Refs. [88]). The complexity of the electronic structure of TMM is due to the fact that all four of its p orbitals are close in energy. The point group symmetry of the 1A1 state is C2v in the equilibrium geometry. 1A1 TMM is a planar system like the 3 A02 (ground state) TMM, but one of the C–C bonds is longer than the others [87,88,90]. The HOMO orbital is doubly occupied yielding a closed-shell configuration. 2R-RMRCCSD (T) [tailored particularly for quasidegenerate systems] optimized geometry of the TMM in 1 A1 state has been published recently by Li and Paldus [90]. Li and Paldus [90] observed that the differences between the results of 6–31G⁄ and cc-pVTZ optimized geometrical parameters are not significant. Brabec and Pittner [89] studied energies of the singlet and triplet states of TMM by the multireference BWCC method [3] which is computationally tractable and does not suffer from intruder states. Krylov and co-workers also computed equilibrium geometry of 1 A1 TMM using SF-DFT method in the 6–31G⁄ basis set. Thus, we can use these previously published results to assess ours provided by SS-MRMPPT method. Two basis sets have been used viz. cc-pVDZ and cc-pvTZ with CAS (2, 2) to calculate equilibrium geometrical parameters. This allows us to investigate the dependence of the basis set size on the equilibrium molecular constants. The results obtained with SS-MRMPPT method are summarized in Table 10. Fig. 1 depicts the geometry and labelling of atoms of

TMM system. From the table, it is found that the structure of 11A1 TMM is characterized by one short and two-long CC bonds. This is because in the 11A1 state, two p electrons participate in the short bond and the remaining two p electrons constitute the two longer bonds. This observation agrees well with the structure indicated in the database[87,88,90]. We also found that the SSMRMPPT value of the shorter C–C bond is close in proximity with the C–C bond length in ethylene, while the value of longer C–C bonds are slightly lower than that of the corresponding bond in twisted ethylene. For quantitative and qualitative assessment of the performance of the SS-MRMPPT approach, we also tabulate the results obtained by other methods. We observed that the SFDFT (spin-flip density functional theory) [88,91] geometries agree reasonably well with the SS-MRMPPT results for the 1A1 state. 1 A1 TMM optimized structure provided by SS-MRMPPT calculations is characterized by one short and two-long CC bonds. As shown in Table 10, the SS-MRMPPT geometries are very close to the state-of-the-art SS-MRCCSD and 2R-RMRCCSD (T) results. This feature is very interesting since SS-MRMPPT is computationally far less expensive than the full-blown SS-MRCCSD and 2RRMRCCSD method. Table 10 shows that the predictions at the BP86/cc-pVDZ level is not very promising with respect to the other methods that accounts both nondynamic and dynamic correlation effects. From the table, we observe that the \C  C  C valence angles obtained by SS-MRMPPT, SS-MRCCSD, R-MRCCSD (T), and SF-DFT method are close to 120°. Thus, we accept that in the 11A1 state of TMM, the carbons are sp2 hybridized. In any case, the agreement (of our findings with contemporary predictions) observed is yet another example of the general utility of the SS-MRMPPT numerical gradient scheme in predicting equilibrium geometry of singlet diradicals (that benefit from a MR description). Finally, increasing the basis set size from cc-pVDZ to cc-pVTZ has a non-negligible effect on the SS-MRMPPT geometrical constants for the 11A1 TMM. 3.7. Two lowest electronic states of 1,2,3 tridehydrobenzene (TDB)

H

H

r3

2b 1

a2

-λ 1a 2

a1

H

r1

H

a3

r2

C 2v planar

a4 r5 H

r4 1A

1

H

Fig. 1. Depiction of the geometry and labelling of atoms of TMM system and the p system of TMM (considered here).

Lastly, the SS-MRMPPT gradient method is employed to 1,2,3 tridehydrobenzene (1,2,3-TDB) considered as a prototypical transient triradical [92] (open-shell system) and posing a significant challenge (exceed even those of diradicals) for any ab initio method due to orbital degeneracies [three unpaired electrons distributed over three nearly degenerate orbitals: 10a17b211a1(C2v labels are used)] and the electrons reside on three different atomic centers. The resultant complex electronic structure contains several close lying low spin (doublet) and high spin (quartet) states and that consequently requires a MR treatment. The doublet states are lower in energy than the quartet counterpart [93]. In recent past, the molecular and electronic structure of 1,2,3-TDB has been investigated in detail via different computational methods [94–96].

23

S. Chattopadhyay et al. / Chemical Physics 401 (2012) 15–26

Krylov’s [95,96] comprehensive investigations illustrate that (a) the 2A1 state is energetically less stable than the 2B2 state (the global minimum) of C2v symmetry and (b) the 2A1 state is monocyclic whereas, the 2B2 state is bi-cyclic. the 1,2,3-TDB system featuring two nearly degenerate (adiabatically) doublet states: 2 A1 and 2B2. The 2A1 state has the leading electronic configuration (10a1)2(11a1)1 while (10a1)2(7b2)1 is the leading electronic configuration for the 2B2 state [see Fig. 2]. In this paper, we have reported the optimized geometrical parameters of these two doublet states

11a1

7b2

11a1

7b2

10a1

10a1

2

2

A1

B2

RC(1)-C(3) Fig. 2. Molecular orbitals of adiabatically nearly degenerate doublet states (2A1 and 2 B2) of 1,2,3-TDB as a function of separation between the C (1) and C (3) radical centers.

82.6 82.5 74.8 75.4 78.2 77.3

C(2)

1.358 1.374 1.352 1.352 1.342 1.354

C(6)

1.401 1.415 1.395 1.385 1.394 1.402

C(3)

C(1) 150.2 150.2 157.3 157.0 154.6 155.2

H(9)

113.1 113.2 113.0 112.7 113.1 113.1

111.0 110.9 108.7 109.0 109.8 109.6

C(4)

H(7)

1.373 1.387 1.376 1.368 1.371 1.380

H(8)

C(5) 1.084 1.097 1.089 1.078 1.084 1.085

1.084 1.092 1.083 1.072 1.078 1.079

1.403 1.415 1.404 1.395 1.404 1.410

C(1)-C(3) SS-MRMPPT/cc-pVDZ SS-MRMPPT/cc-pVTZ SF-CCSD/6-31G(d) SF-DFT/6-31G(d) B3LYP/cc-pVTZ RCCSD(T)/cc-pVTZ

characterized by different interactions of the unpaired electrons using cc-pVDZ and cc-pVTZ basis sets and CAS (3, 3). Figs. 3 and 4 list selected structural parameters obtained at various levels of theory for 2A1 and 2B2 states respectively. They reveal large structural differences. Our calculations are compared with those by Krylov and co-workers [95,96] in the same figures. In agreement with the previous theoretical estimates due to Krylov and co-workers [95,96], we also found different equilibrium structures for the 2A1 and 2B2 doublet states as is evident from the computed C1–C3 and \C1–C2–C3. The present SS-MRMPPT optimization predicts 2A1 state to be bicyclic with a relatively short C(1)–C(3) bond of 1.812 Å whereas, the 2B2 state has a mono-cyclic structure with a C(1)–C(3) distance of 2.410 Å. Recent comprehensive calculations [93,96] show that the 2A1 state exhibits a C(1)– C(3) bonding interaction at a small distance (1.68–1.69 Å) whereas, the 2B2 state has a larger separation (2.36–2.37 Å). In view of the present and earlier investigations, it is evident that the doublet ground state exhibits partial bond formation between the radical centers. Experimental studies by Venkataramani et al. [93] also suggested the bi-cyclic and mono-cyclic structure of 2A1 and 2B2 respectively. Here, it is important to mention that a comparison with the experimentally observed IR transitions supports the assignment of the 2A1 ground state [93,96]. Entries in Figs. 3 and 4 also illustrate that the SS-MMPPT predicts geometries display the same trends as those appearing in the studies of Krylov and co-workers [96], namely, the C(5)–C(6) bond length in the 2A1 state is shorter than that of the 2B2 state, and the C(4)–C(6) bond length in the 2A1 state is slightly longer than the C(4)–C(6) bond length in the 2B2 state. As seen from Figs. 3 and 4 our results are in

1.793 1.812 1.642 1.629 1.692 1.692

E(SS-MRMPPT/cc-pVDZ) = -229.536418 E(SS-MRMPPT/cc-pVTZ) = -229.744462 Fig. 3. Optimized structures of the bicyclic 2A1 state of 1,2,3-tridehydrobenzene triradical obtained at various levels of theories [SS-MRMPPT, SF-CCSD, SF-DFT, B3LYP and RCCSD (T) respectively]. Calculated SS-MRMPPT energies are also listed for each basis sets. Bond lengths, angles and energies are in Angstroms, degrees and a.u. respectively.

1.080 1.094 1.075 1.086 1.081 1.082

1.295 1.313 1.295 1.281 1.287 1.300 C(1)

C(6)

C(2)

115.7 115.1 117.8 117.4 117.2 116.8

122.9 123.3 123.0 122.7 122.9 123.1

H(8)

132.8 132.6 129.1 130.3 130.9 131.1 116.2 C(3) 116.5 116.1 115.9 115.9 116.0 C(4) H(9)

C(5)

1.402 1.420 1.408 1.398 1.406 1.412

H(7)

SS-MRMPPT/cc-pvDZ SS-MRMPPT/cc-pVTZ SF-CCSD/6-31G(d) SF-DFT/6-31G(d) B3LYP/cc-pVTZ RCCSD(T)/cc-pVTZ

1.083 1.097 1.089 1.077 1.083 1.084

C(1)-C(3) 2.405 2.374 2.360 2.326 2.342 2.367

E(SS-MRMPPT/cc-pVDZ) = -229.526627 E(SS-MRMPPT/cc-pVTZ) = -229.735285 Fig. 4. Optimized structures of the monocyclic 2B2 state of 1,2,3-tridehydrobenzene triradical obtained at various levels of theories [SS-MRMPPT, SF-CCSD, SF-DFT, B3LYP and RCCSD (T) respectively]. Calculated SS-MRMPPT energies are also listed for each basis sets. Bond lengths, angles and energies are in Angstroms, degrees and a.u. respectively.

24

S. Chattopadhyay et al. / Chemical Physics 401 (2012) 15–26

acceptable agreement with the spin-adapted CCSD (T) [termed RCCSD (T)] one with the cc-pVTZ basis set. Despite a modest basis sets used here, the SS-MRMPPT structures are in close agreement with the findings of the recent study due to Venkataramani et al. [93]. The adiabatic energy gap (DEadb) between the 2A1 and 2B2 states can be used as a testing ground to establish the applicability of any correlation recovery methods because D Eadb is very small due to geometrical relaxation, the near degeneracy of the two states, and the strongly differing radical character of the two doublets. The interesting point of the D Eadb calculation is that the value is so small that any approximation can noticeably affect it (like basis set, method, etc.). The adiabatic state ordering is very sensitive to the extent of incorporation of dynamical correlation. The adiabatic energy gap is also an important parameter for analyzing studies of the reactivity of the TDB. In addition to R-CCSD and R-CCSD (T) data, Table 11 compares the results of various previously published multireference calculations due to Krylov and co-workers [96]. As shown in the table, the MR-CISD + Q (Davidson correction) and AQCC methods lead to a nearly vanishing DEadb, however, the ACPF method places the 2A1 state above the 2B2 (a moderate preference of 0.03 eV for the 2A1 state). Krylov and co-workers [96] demonstrated that all equation-of-motion SF coupled-cluster with single and double substitutions [EOM-SFCCSD or EOM-SF (2, 2)] calculations favor the 2B2 state while, partial or full inclusion of triples [EOM-SF-CCSD (fT)] place the 2A1 below the 2B2 state. Our SSMRMPPT calculation places 2A1 state below the 2B2 state by 0.266 and 0.249 eV with cc-pVDZ and cc-pTZ basis sets respectively which are in excellent agreement with the value of 0.26 eV obtained by UHF-CCSD (T)/cc-pVTZ calculation. The table depicts that the R-CCSD and R-CCSD (T) estimations are in accordance with a 2 A1 ground state whereas, the 2B2 excited state is 0.09–0.22 eV higher in energy. Even though the state energies provided by SSMRMPPT/cc-pVTZ corresponding to the optimized geometry for both 2A1 and 2B2 states are lower than those obtained with SSMRMPPT/cc-pVDZ one (illustrate the effect of basis sets on the SS-MRMPPT state energies is not negligible), the DEadb value with

Table 11 Adiabatic energy separation [DEadb(eV)  E(2A1)  E(2B2] of 1,2,3-tridehydrobenzene multireference calculations employ of methods with cc-pVDZ and cc-pVTZ basis sets. Reference

Methods

cc-pVDZ

cc-pVTZ

Present work Ref. [96]

SS-MRMPPT CASSCFa MR-CISDa MR-CISD + Qa AQCCa RCCSDa RCCSDb RCCSD (T)a RCCSD (T)b CAS-RS2a CAS-RS2b CAS-RS3a CAS-RS3b ACPFa EOM-SF-CCSD EOM-SF-CCSD (fT) UB3PW91 UBPW91 UB3LYP UBLYP

0.266 0.66 0.18 0.00 0.00 0.22 0.22 0.10 0.09 0.02 0.01 0.03 0.02 0.03 0.24 0.24 0.18 0.12

0.249 068 0.20 0.01 0.00 0.23 0.23 0.10 0.10 0.03 0.03 0.04 0.04 0.03 0.07 0.03 -

Ref. [93]

In Ref. [96] all multireference calculations employ a CASSCF (9,9) reference space. RCC: spin-adapted coupled cluster. fT: Fock triples. RS2 and RS3: Dynamical electron correlation included by perturbation theory to second (RS2) and third (RS3) order respectively. a B3LYP/cc-pVTZ equilibrium geometry. b R-CCSD (T)/cc-pVTZ equilibrium geometry.

cc-pVDZ is in close agreement with the cc-pVTZ one. From the foregoing analysis, it is found that the SS-MRMPPT method provides the correct energy ordering of the adiabatically nearly degenerate doublet states of 1,2,3-TDB. The geometries of both doublet states and adiabatic energy gap are obtained (via SS-MRMPPT approach) with an accuracy close to that achieved from established and current generation state-of-the-art calculations. Another important thrust of the present analysis is to ascertain the applicability of the SS-MRMPPT method from well-behaved systems (e.g., closed-shell molecules around the equilibrium geometries) to situations plagued by electronic degeneracy, e.g. radicaloids. Usually enough providence is needed while performing computations for open-shell species, even for relatively wellbehaved states as the quality of the results can be affected by spin-contamination. However, the SS-MRPT method is flexible, allowing the use of different reference spaces of arbitrary spin multiplicity [31–33]. The combination of accuracy and low computational expense of the SS-MRMPPT calculations provide a means for accurate computations on much larger (closed and open-shell) radical chemical systems. From our present and previous investigations we observed that the SS-MRMPPT method also provides very satisfactory results in the non-degenerate situations. It is well appreciated by the practitioners of electronic structure theory that formulating a MR method which reproduces the success of the single-reference approach in the case of degenerate or near-degenerate electronic states is still an open problem. The success and advantages of SS-MRPT method are counterweighed by the lack of energy invariance with respect to certain orbital rotations (see Ref. [33]) as that of its parent SS-MRCC formalism [30]. It is worth noting that the lack of invariance is inherent in the Jeziorski–Monkhorst ansatz [40], and there does not seem to exist an easy solution without modifying the ansatz [97,98]. Another undesirable feature which has also been discussed recently [25,99,100] is the convergence issue of the cluster finding equations (different from the inherent intruder problem in the conventional effective hamiltonian based MR methods) when some of the reference coefficients tend to take on very small values. Considering Eq. (2), we see that division by reference expansion coefficient c0l in the numerator may give rise to numerical instability when c0l is much smaller compared to c0m . The variety of systems and CAS that we have explored till now [32,33] (including the present work) to implement the SS-MRMPPT method have all equivocally revealed that the method is free from any numerical instability of the amplitude determining equations even when all the coefficients are being included. The computation of energy for the excited or ionized states at the SS-MRMPPT level is done by converging on the second (or excited) root of the effective Hamiltonian. It is worth mentioning, the convergence of the cluster finding equations for these states may be slower, thus requiring a larger number of iterations when solving the SSMRMPPT equations than is the case for the ground state. We also point out that ‘root-homing’ scheme used here might seem to be useful for the kind of systems that have been dealt with in this work, however, the generality of the approach is lost in cases where the coefficients rapidly change sign viz. ‘mixed electronic states’.

4. Conclusions Rigorously size-extensive and size-consistent (with localized orbitals) state specific multireference perturbation theory (SSMRPT) has the desired flexibility needed for the accurate treatment of correlation effects in molecular systems enjoying at the same time a very encouraging (performance/cost) ratio with respect to the dimensions of the system. This method takes into

S. Chattopadhyay et al. / Chemical Physics 401 (2012) 15–26

account both the static and dynamic electron correlation effects in an effective and balanced way without encountering any difficulties due to the intruders in studies of chemical reaction pathways and relative energetics of systems characterized by varying degrees of quasidegenerate character [32–34]. Intruder states cause problems with converging the desired solution of the MRPT equations and are blamed for the significant deterioration of the quality of the results when the target state becomes nondegenerate (i.e. single reference situation). The SS-MRPT considered here appears to be a practical and effective solution to overcome the failure of the standard MRPT approach [29,31–34]. In this paper, we have further explored the applicability of SSMRMPPT method by reporting calculations for systems where static correlation is strong on top of the dynamical one. It is also the purpose of this paper to examine through benchmark calculations the performance of SS-MRMPPT method in a few examples where static correlation effects are very weak (characterized by singlereference description). Present numerical tests clearly demonstrate the accuracy and numerical stability of the SS-MRMPPT method in the ground, ionized and excited-state calculations in rather complicated situations. We should also mention that the present works reveal that the performance of the SS-MRPT method is promising when dealing with exactly or almost exactly degenerate states (corresponding to the multireference situations), it too is effective when the state of interest acquires a single-reference character if an appropriate active space is chosen. On the whole, the presented results depict the usefulness of the SS-MRMPPT method in handling of the ground and exited states of singlet and non-singlet nature in a target specific (i.e. intruder free) manner, even for medium sized molecular systems. In general, we find an acceptably good agreement of our SS-MRMPPT results with the recent generation calculations (several widely used packageable and state-of-the-art many-body methods). No evidence of divergence or cases yielding physically unacceptable values in the numerical solution of the equations has been found in the study of the systems mentioned, although, the cluster finding equation of the SS-MRMPPT explicitly contains the reference coefficient weight in the denominator. It is worth mentioning that the main aim of the present work is not to solve the still open questions of interpretation of electronic structure of the molecules studied but rather to verify the ability of SS-MRMPPT to provide a viable means of calculation of ground, excited and ionized states. As far as calculations for excited states are concerned, because of the unknown behavior of the denominator factors, the possibility of the convergence problem may not be excluded in the case of SS-MRPT method. Moreover, for excited states, the problems of choosing the appropriate eigenvalue of the effective Hamiltonian from several eigenvalues in the case of our SS-MRMPPT method is not be an ambiguous matter as it is in the case of the ground state. Although the scope of the examples considered here is very limited and more investigations are needed to fully judge the accuracy and applicability of the SS-MRMPPT approach in excited or ionized state calculations, the present results are encouraging and illustrate the potential of SS-MRMPPT as a tool for the study of realistic chemical problems or systems requiring multireference zeroth-order wave functions. It would be certainly very important to test its capabilities for more and lower-lying excited/ionized states of chemically interesting and challenging real systems.

Acknowledgements This research has been financed by the Department of Science and Technology of India [Grant No. SR/S1/PC-61/2009]. We also

25

wish to acknowledge the help of Debi Banerjee in preparing the manuscript. References [1] J.-P. Malrieu, J.-L. Heully, A. Zaitsevskii, Theor. Chim. Acta. 90 (1995) 167; M.R. Hoffmann, in: D.R. Yarkony (Ed.), A dvanced Series in Physical Chemistry, Modern Electronic Structure Theory, Part I, Vol. 2, World Scientific, River Edge, NJ, 1995, p. 1166; M.W. Schmidt, M.S. Gordon, Annu. Rev. Phys. Chem. 49 (1998) 233. [2] B.O. Roos, K. Andersson, M.P. Fülscher, P.-Å. Malmqvist, L. Serrano-Andrés, K. Pierloot, M. Merchán, Adv. Chem. Phys. 93 (1996) 219; L. Gagliardi, B.O. Roos, Chem. Soc. Rev. 36 (2007) 893. [3] I. Hubac´, S. Wilson, Progress in Theoretical Chemistry and Physics, BrillouinWigner Methods for Many-Body Systems, Vol. 21, Springer, New York, 2010. [4] T.H. Schucan, H.A. Weidenmüller, Ann. Phys. (NY) 76 (1973) 483; G. Hose, U. Kaldor, J. Phys. B: Atom. Molec. Phys. 12 (1979) 3827. [5] J.P. Finley, R.K. Chaudhuri, K.F. Freed, J. Chem. Phys. 103 (1995) 4990; J.P. Finley, R.K. Chaudhuri, K.F. Freed, Phys. Rev. A 54 (1996) 343. [6] K. Kowalski, P. Piecuch, Phys. Rev. A 61 (2000). 052506; K. Kowalski, P. Piecuch, Int. J. Quantum Chem. 80 (2000) 757. [7] J.P. Finley, K.F. Freed, J. Chem. Phys. 102 (1995) 1306. and references therein; B.O. Roos, K. Andersson, Chem. Phys. Lett. 245 (1995) 215; N. Forsberg, P.Å. Malmqvist, Chem. Phys. Lett. 274 (1997) 196. [8] C. Camacho, R. Cimiraglia, H.A. Witek, Phys. Chem. Chem. Phys. 12 (2010) 5058. [9] A. Hose, U. Kaldor, J. Phys. B 12 (1979) 3827; G. Hose, Theor. Chim. Acta. 72 (1987) 303. [10] H. Nakano, J. Nakatani, K. Hirao, J. Chem. Phys. 114 (2001) 1133; Y.-K. Choe, H.A. Witek, J.P. Finley, K. Hirao, J. Chem. Phys. 114 (2001) 3913. [11] J.P. Malrieu, Ph. Durand, J.P. Daudey, J. Phys. A (Math. Gen.) 18 (1985) 809; M.R. Hoffmann, Chem. Phys. Lett. 210 (1993) 193; Y.G. Khait, M.R. Hoffmann, J. Chem. Phys. 108 (1998) 8317. [12] M.G. Sheppard, K.F. Freed, J. Chem. Phys. 75 (1981) 4507; J.E. Stevens, R.K. Chaudhuri, K.F. Freed, J. Chem. Phys. 105 (1996) 8754. [13] M.R. Hoffmann, Chem. Phys. Lett. 195 (1992) 127; W. Jiang, Y.G. Khait, M.R. Hoffmann, J. Phys. Chem. A. 113 (2009) 4374. [14] K. Hirao, Chem. Phys. Lett. 190 (1992) 374; K. Hirao, Chem. Phys. Lett. 196 (1992) 397; K. Hirao, Int. J. Quantum Chem. S26 (1992) 517; H. Nakano, T. Nakajima, T. Tsuneda, K. Hirao, J. Mol. Struct. (Theochem) 573 (2001) 91. [15] K. Andersson, P. Å Malmqvist, B.O. Roos, A.J. Sadlej, K. Wolinski, J. Chem. Phys.94 (1990) 5483; K. Andersson, P.Å. Malmqvist, B.O. Roos, J. Chem. Phys. 96 (1992) 1218. [16] J.-P. Malrieu, J.-L. Heully, A. Zaitzevskii, Theor. Chim. Acta 90 (1995) 167. [17] J.M. Rintelman, I. Adamovic, S. Varganov, M.S. Gordon, J. Chem. Phys.122 (2005). 044105. [18] Z. Azizi, B.O. Roos, V. Veryazova, Phys. Chem. Chem. Phys. 8 (2006) 2727. [19] H. Nakano, J. Chem. Phys. 99 (1993) 7983. [20] J. Finley, P.-A. Malmqvist, B.O. Roos, L. Serrano-Andres, Chem. Phys. Lett. 288 (1998) 299. [21] B.O. Roos, in: M.K. Shukla, J. Leszczynski, (Eds.), Radiation Induced Molecular Phenomena in Nucleic Acids, Springer Science Business Media, 2008. [22] Z. Rolik, Á. Szabados, P.R. Surján, J. Chem. Phys. 119 (2003) 1922; Á. Szabados, Z. Rolik, G. Tóth, P.R. Surján, J. Chem. Phys. 122 (2005). 114104. [23] I. Hubacˇ, P. Mach, S. Wilson, Mol. Phys. 100 (2002) 859; P. Papp, P. Mach, J. Pittner, I. Hubacˇ, S. Wilson, Mol. Phys. 104 (2006) 2367; P. Papp, P. Mach, I. Hubacˇ, S. Wilson, Int. J. Quantum Chem. 107 (2007) 2622. [24] Size-extensive version of BWPT (by converting the BW form to the RS form) incipiently sense intruders as in the traditional effective Hamiltonian theory. [25] M.R. Hoffmann, D. Datta, S. Das, D. Mukherjee, Á. Szabados, Z. Rolik, P.R. Surján, J. Chem. Phys. 131 (2009). 204104. [26] K.G. Dyall, J. Chem. Phys. 102 (1995) 4909. [27] C. Angeli, R. Cimiraglia, S. Evangelisti, T. Leininger, J.-P. Malrieu, J. Chem. Phys. 114 (2001). 10252; C. Angeli, M. Pastore, R. Cimiraglia, Theor. Chem. Acc. 117 (2007) 743. [28] B. Huron, J.-P. Malrieu, P. Rancurel, J. Chem. Phys. 58 (1973) 5745; R. Cimiraglia, M. Persico, J. Comp. Chem. 8 (1987) 39. [29] U.S. Mahapatra, B. Datta, D. Mukherjee, J. Phys. Chem. 103 (1999) 1822. This article has been concerned with an analysis of the main features and developments of the approach. [30] U.S. Mahapatra, B. Datta, D. Mukherjee, Mol. Phys.94 (1998) 157; U.S. Mahapatra, B. Datta, D. Mukherjee, J. Chem. Phys. 110 (1999) 6171. [31] D. Pahari, S. Chattopadhyay, S. Das, D. Mukherjee, U.S. Mahapatra, in: C.E. Dytkstra, G. Frenking, K.S. Kim, G.E. Scuseria (Eds.), Elsevier, Amsterdam, 2005, p. 581. [32] U.S. Mahapatra, S. Chattopadhyay, R.K. Chaudhuri, J. Chem. Phys. 129 (2008). 024108; U.S. Mahapatra, S. Chattopadhyay, R.K. Chaudhuri, J. Chem. Phys. 30 (2009). 014101; S. Chattopadhyay, U.S. Mahapatra, R.K. Chaudhuri, J. Phys. Chem. A 113 (2009) 5972;

26

[33] [34] [35] [36] [37] [38] [39]

[40] [41]

[42]

[43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53]

[54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66]

[67] [68]

S. Chattopadhyay et al. / Chemical Physics 401 (2012) 15–26 U.S. Mahapatra, S. Chattopadhyay, R.K. Chaudhuri, J. Comput. Chem. 32 (2011) 325. U.S. Mahapatra, S. Chattopadhyay, R.K. Chaudhuri, J. Chem. Theory Comput. 6 (2010) 662. F.A. Evangelista, A.C. Simmonett, H.F. Schaefer III, D. Mukherjee, W.D. Allen, Phys. Chem. Chem. Phys. 11 (2009) 4728. A. Zaitevskii, J.P. Malrieu, Chem. Phys. Lett. 233 (1995) 597; Z. Rolik, Á. Szabados, Int. J. Quantum Chem. 109 (2009) 2554. E. Rosta, P.R. Surján, J. Chem. Phys 116 (2002) 878. A. Köhn, D. P. Tew, vol. 132, 2010, p. 024101. S. Ten-no, Chem. Phys. Lett. 447 (2007) 175; S. Ten-no, J. Chem. Phys. 126 (2007). 014108. R.K. Chaudhuri, K.F. Freed, G. Hose, P. Piecuch, K. Kowalski, M. Wloch, S. Chattopadhyay, D. Mukherjee, Z. Rolik, A. Szabados, G. Toth, P.R. Surjan, J. Chem. Phys. 122 (2005). 134105. B. Jeziorski, H.J. Monkhorst, Phys. Rev. A 24 (1981) 1668. S. Chattopadhyay, U.S. Mahapatra, R.K. Chaudhuri, Chem. Phys. Lett. 488 (2010) 229; U.S. Mahapatra, S. Chattopadhyay, R.K. Chaudhuri, J. Phys. Chem. A 114 (2010) 3668. D. Feller, J. Comput. Chem. 17 (1996) 1571; K.L. Schuchardt, B.T. Didier, T. Elsethagen, L. Sun, V. Gurumoorthi, J. Chase, J. Li, T.L. Windus, J. Chem. Inf. Model. 47 (2007) 1045. see . C.W. Bauschlicher Jr., P.R. Taylor, J. Chem. Phys. 85 (1986) 2779. J.M.L. Martin, Chem. Phys. Lett. 242 (1995) 343. F. Schneider, F. Di Giacomo, F.A. Gianturco, J. Chem. Phys. 105 (1996) 7560. R.A. King, C.D. Sherrill, H.F. Schaefer III, Spectrochimica Acta Part A 53 (1997) 1163. X. Li, J. Paldus, J. Chem. Phys. 110 (1999) 2844. R. van Harrevelt, M.C. van Hemert, J. Chem. Phys. 112 (2000) 5777. K. Yagi, T. Taketsugu, K. Hirao, M.S. Gordon, J. Chem. Phys. 113 (2000) 1005; K. Yagi, T. Taketsugu, K. Hirao, J. Chem. Phys. 116 (2002) 3963. A.I. Krylov, C.D. Sherrill, J. Chem. Phys. 116 (2002) 3194. T. Yanaia, G. K-L. Chan, J. Chem. Phys. 124 (2006). 194106; J.A. Parkhilla, M. Head-Gordon, J. Chem. Phys. 133 (2010). 124102. A. Karton, J.M.L. Martin, J. Phys. Chem. 133 (2010). 144102. R.A. Mc Clatchey et al., AFCRL Atmospheric absorption line parameters compilation, AFCRL-TR-73-0096 environmental research paper No. 434, Air force Cambridge research laboratories, edited by L.G. Hanscom Field, Bedford, Massachusetts, 1973.; O.L. Polyansky, Császár, S.V. Shirin, N.F. Zobov, P. Bartletta, J. Tennyson, D.W. Schwenke, P.J. Knowles, Science 299 (2003) 539; A.G. Császár, G. Czakó, T. Furtenbacher, J. Tennyson, V. Szalay, S.V. Shirin, N.F. Zobov, O.L. Polyansky, J. Chem. Phys. 122 (2005). 214205. W. Meyer, Int. J. Quantum Chem. 5 (1971) 341. J.A. Smith, P. Jørgensen, Y. Öhrn, J. Chem. Phys. 62 (1975) 1285. B. Das, J.W. Farley, J. Chem. Phys. 95 (1991) 8809; M. Staikova, B. Engels, M. Peric, S.D. Peyerimhoff, Mol. Phys. 80 (1993) 1485. J. Olsen, P. Jørgensen, H. Koch, A. Balkova, R.J. Bartlett, J. Chem. Phys. 104 (1996) 8007. S. Hirata, P.-D. Fan, Å. Auer, M. Nooijen, P. Piecuch, J. Chem. Phys. 121 (2004). 12197. F. Illas, P.S. Bagus, J. Rubio, M. Gonzalez, J. Chem. Phys. 94 (1991) 3774; W.N. Whitton, S. Mahling, P.J. Kuntz, Chem. Phys. 162 (1992) 379. J.E. Reutt, L.S. Wang, Y.T. Lee, D.A. Shirley, J. Chem. Phys. 85 (1986) 6928. C.R. Brundle, D.W. Turner, Proc. R. Soc. London, Ser. A 307 (1968) 27. C.W. Bauschlicher Jr., S.R. Langhoff, H. Partridge, Chem. Phys. Lett. 216 (1993) 341. C.O. da Silva, F.E.C. Teixeira, J.A.T. Azevedo, E.C. Da Silva, M.A.C. Nascimento, Int. J. Quantum Chem. 60 (1996) 433. A.C. Borin, F.R. Ornellas, J. Chem. Phys. 98 (1993) 8761; J.S. Wright, M. Kolbuszewski, J. Chem. Phys. 98 (1993) 9725. ˇ ársky, A. Mavridis, Int. J. V.I. Teberekidis, I.S.K. Kerkines, C.A. Tsipis, P. C Quantum Chem., 102 (2005). 762 and references therein. J. Kalcher, A.F. Sax, J. Mol. Struct. (Theochem) 498 (2000) 77; M. Pelegrini, O. Roberto-Neto, F.R. Ornellas, F.B.C. Machado, Chem. Phys. Lett. 383 (2004) 143. K.P. Huber, G. Herzberg, Constants of Diatomic Molecules, Van Nostrand Reinhold, New York, 1979. M. Calsson - Göthe, B. Wannberg, F. Falk, L. Karlsson, S.SvenssonP. Baltzer, Phys. Rev. A 44 (1991) R17; A.W. Potts, T.A. Williams, J. Electron Spectrosc. 3 (1974) 3;

[69]

[70] [71] [72] [73] [74] [75]

[76] [77] [78] [79]

[80] [81]

[82] [83] [84] [85] [86] [87]

[88] [89] [90] [91]

[92]

[93] [94]

[95] [96]

[97] [98] [99] [100]

S. Svensson, M. Carisson-Göthe, L. Karlsson, A. Nilsson, N. Mårtensson, U. Gelius, Phys. Scr. 44 (1991) 184. P.W. Langhoff, S.R. Langhoff, T.N. Rescigno, J. Schirmer, L.S. Cederbaum, W. Domcke, W. Von Niessen, Chem. Phys. 58 (1981) 71; P. Rosmus, H-J. Werner, Mol. Phys. 47 (1982) 661; H. Lavendy, J.M. Robbe, J.P. Filament, Chem. Phys. Lett. 205 (1993) 456. P.A. Martin, M. Féhdr, Chem. Phys. Lett. 232 (1995) 491. R.R. Reddy, R. Viswanath, J. Astrophys. Astr. 11 (1990) 67. P. Misra, D.W. Ferguson, K. Narahvi Rao Jr., W. Willians, C.W. Mathews, J. molec. Spectrosc. 125 (1987) 54. T.J. Lee, A.P. Rendell, P.R. Taylor, J. Chem. Phys. 92 (1990) 489; A.P. Rendell, T.J. Lee, P.R. Taylor, J. Chem. Phys. 92 (1990) 7050. J.D. Watts, I. Cernusak, J. Noga, R.J. Bartlett, C.W. Bauschlicheri Jr., T.J. Lee, A.P. Rendell, P.R. Taylor, J. Chem. Phys. 93 (1990) 8875. J.M. Junquera-Hernández, J. Sánchez-Marín, G.L. Bendazzoli, S. Evangelisti, J. Chem. Phys. 121 (2004) 7103; J.M. Junquera-Hernández, J. Sánchez-Marín, G.L. Bendazzoli, S. Evangelisti, J. Chem. Phys. 120 (2004) 8405. K. Kowalski, S. Hirata, M. Włoch, P. Piecuch, T.L. Windus, J. Chem. Phys. 123 (2005). 074319. P. Piecuch, S. Hirata, K. Kowalski, P.-D. Fan, T.L. Windus, Int. J. Quantum Chem. 106 (2006) 79. P.-O. Widmark, P.-A. Malmqvist, B.O. Roos, Theor. Chim. Acta. 77 (1990) 291. The active space used in the EOMCCSDt calculations consisted of the three highest-energy occupied orbitals and nine lowest-energy unoccupied orbitals that correlate with the 2s and 2p shells of the Be atoms. G. Herzberg, Molecular Spectra and Molecular Structure: Electronic Spectra and Electronic Structure of Polyatomic Molecules, Van Nostrand, NY, 1966. A. Langseth, B.P. Stoicheff, Can. J. Phys. 34 (1956) 350; K.K. Innes, J.E. Parkin, D.K. Ervin, J. Mol. Spectrosc. 16 (1965) 406; S.G. Grubb, C.E. Otis, R.L. Whetten, E.R. Grant, A.C. Albrecht, J. Chem. Phys. 82 (1985) 1135; A. Hiraya, K. Shobatake, J. Chem. Phys. 94 (1991) 7700. B.O. Roos, K. Andersson, M.P. Fülscher, Chem. Phys. Lett. 192 (1992) 5. K. Hirao, H. Nakano, T. Hashimoto, Chem. Phys. Lett. 235 (1995) 430; K. Hirao, H. Nakano, K. Nakayama, M. Dupuis, J. Chem. Phys. 105 (1996) 9227. J.M.L. Martin, P.R. Taylor, T.J. Lee, Chem. Phys. Lett. 275 (1997) 414. J. Gauss, J.F. Stanton, J. Phys. Chem. A 104 (2000) 2866. G. Ghigo, B.O. Roos, P-Å. Malmqvist, Chem. Phys. Lett. 396 (2004) 142. P.G. Wenthold, J. Hu, R.R. Squires, W.C. Lineberger, J. Am. Chem. Soc. 118 (1996) 475; C.J. Cramer, B.A. Smith, J. Phys. Chem. 100 (1996) 9664. L.V. Slipchenko, A.I. Krylov, J. Chem. Phys. 117 (2002) 4694; L.V. Slipchenko, A.I. Krylov, J. Chem. Phys. 118 (2003) 6874. J. Brabec, J. Pittner, J. Phys. Chem. A 110 (2006). 11765. X. Li, J. Paldus, J. Chem. Phys. 129 (2008). 174101. The spin-flip (SF) approach implemented within the timedependent (TD) density functional theory (DFT) extends DFT to multireference situations with no cost increase relative to the non-SF TD-DFT. The SF-DFT model is formally exact and therefore will yield exact answers with the exact density functional. Krylov and his group [88] demonstrated that SF-DFT yields accurate equilibrium properties in radicaloids. Interest in triradicals and multi-spin species stems from their potential role as building blocks of organic ferromagnets. It is expected that in these materials, electron spin could be used as a carrier of information rather than electron charge as it is in more conventional electronics. S. Venkataramani, M. Winkler, W. Sander, Angew. Chem. Int. Ed. 44 (2005) 6306. H.F. Bettinger, P.V.R. Schleyer, H.F. Schaefer III, J. Am. Chem. Soc. 121 (1999) 2829; H.A. Lardin, J.J. Nash, P.G. Wenthold, J. Am. Chem. Soc. 124 (2002). 12612. A.I. Krylov, J. Phys. Chem. A 109 (2005). 10638. L.V. Slipchenko, A.I. Krylov, J. Chem. Phys. 118 (2003) 9614; L. Koziol, M. Winkler, K.N. Houk, S. Venkataramani, W. Sander, A.I. Krylov, J. Phys. Chem. A 111 (2007) 5071; P.U. Manohar, A.I. Krylov, J. Chem. Phys. 129 (2008). 194105; A.M.C. Cristian, Y. Shao, A.I. Krylov, J. Phys. Chem. A 108 (2004) 6581. L. Kong, Int. J. Quantum Chem. 110 (2010) 2603. F.A. Evangelista, J. Gauss, J. Chem. Phys. 133 (2010). 044101. M. Hanrath, J. Chem. Phys. 123 (2005). 84102; A. Engels-Putzka, M. Hanrath, Mol. Phys.107 (2009) 143. S. Das, D. Mukherjee, M. Kállay, J. Chem. Phys. 132 (2010) 074103.