Multireference perturbation theory can predict a false ground state

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Multireference perturbation theory can predict a false ground statew

Cristopher Camacho,

a

Renzo Cimiraglia

b

and Henryk A. Witek*

a

Received 5th January 2010, Accepted 25th February 2010

First published as an Advance Article on the web 26th March 2010 DOI: 10.1039/c000196a

Prediction of a false ground state with popular variants of multireference perturbation theory (CASPT2 and MRMP) is reported for a remarkably simple chemical system: the Sc2

molecule.

In a recent publication, Matxain et al.1_{used quantum diﬀusion}

Monte Carlo (DMC) and second-order multireference Møller–Plesset perturbation theory2,3(MRMP) to show that the ground state of Sc2 is 3Su instead of the previously

predicted 5Su. Their ﬁnding was rather surprising, because

the consensus concerning the symmetry of the ground state of the scandium dimer dates back to 30 years ago. This result motivated us to reinvestigate the same problem using popular variants of multireference perturbation theory. It has been found that the multireference perturbation theory (MRPT) ground state calculations (either for3Suor

5

Su) suﬀer severely

from the intruder state problem. In order to obtain smooth potential energy surfaces (PESs) it has been necessary to employ intruder state removal techniques. The resultant curves—shown in Fig. 1 for various magnitudes of the shift parameter—are continuous and closely spaced. Surprisingly enough, the energy ordering of the3_S

uand5Sustates depends

on the magnitude of the shift parameter, s. The energy separation between the two curves is not large, but clearly smaller values of the shift parameter, s, promote the3Sustate

as the ground state, while larger values of s produce similar order like at the CASSCF level. A discussion of this behavior is presented below. At present, MRPT is routinely used for finding the energetics and properties of a wide spectrum of chemical systems in their ground and excited states. Often these calculations are plagued by intruder states and the intruder state removal technique is a standard tool to deal with such problems. It was found previously that the choice of the shift parameter can influence significantly the values of spectroscopic properties of the ground state of Mn2.4,5 The

present study raises further doubts as to the appropriateness of using the shift techniques in MRPT. A choice of too small a value of the shift parameter can produce false assignment of the ground state and consequently distort the energy spectrum of a given molecule in a considerable manner. We stress again that the MRPT calculations employing the intruder state removal techniques should be performed at least twice

(with diﬀerent values of s) to verify the independence of the results on the shift magnitude.

The present calculations have adopted the same, moderate-size basis set as used previously by Matxain et al.1A detailed speciﬁcation of this basis is given in the ESI.w The (reduced valence) active space consists of the molecular orbitals derived from the 4s and 3d AOs of scandium. The orbitals are optimized state-speciﬁcally. The MRPT calculations correlate the inner 3s and 3p orbitals. The CASPT26,7_{calculations have}

been performed using the MOLCAS8 and MOLPRO9 quantum chemistry programs whereas the MRMP2,3calculations used the GAMESS10 program. The CASPT2/MOLCAS employed the IPEA shift11 of 0.25 a.u. in conjunction with the imaginary shift technique,12 _{the CASPT2/MOLPRO} _{used the real shift}

technique,13 and MRMP used the intruder state avoidance (ISA) technique.14All calculations were performed with the standard, unmodiﬁed zeroth-order Hamiltonian. Dissociation energies, De, have been computed using the MRPT energy of

the 5_S

u at 40.00 a.u., which corresponds to the 2Dg + 4Fg

atomic asymptote.

At the CASSCF level, the 5Su state lies 0.335 eV lower

than the 3Su state. The experimentally determined

spectro-scopic parameters of the ground state are oe= 238.9 cm 1

Fig. 1 Potential energy curves for the5_S

u and3Su states obtained

from the MRMP/GAMESS, CASPT2/MOLPRO, and CASPT2/MOLCAS

calculations using diﬀerent values of shift parameter, s.

a_{Institute of Molecular Science and Department of Applied}

Chemistry, National Chiao Tung University, 1001 Ta-Hsueh Road, Hsinchu 30010, Taiwan. E-mail: [email protected]

b_{Dipartimento di Chimica, Universita` di Ferrara, Via Borsari 46,}

I-44100 Ferrara, Italy

w Electronic supplementary information (ESI) available: Potential energy curves for the5Su and

3

Su states obtained with the NEVPT2

method; basis set used. See DOI: 10.1039/c000196a

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Published on 26 March 2010. Downloaded by National Chiao Tung University on 25/04/2014 07:35:08.

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and D0= 1.05 0.2 eV; no accurate estimate of reis available.

Accounting for dynamical correlation using MRPT leads to highly perturbed potential energy curves caused by numerous intruder states. Using intruder state removal techniques improves the situation considerably; the resultant curves— shown in Fig. 1—are smooth. For small values of the shift parameter,3Suis lower in energy, while for larger values of the

shift parameter, the situation is opposite. A possible explana-tion of this behavior can be given by taking into account the following facts. In the limit of an inﬁnitely large shift, the second-order perturbation energy contribution is equal to zero, and the reproduced energy order is exactly the same as the CASSCF level (i.e.5Su is the ground state). On the other

hand, for small values of the shift parameter, the intruder states’ inﬂuence can be expected to be larger for the triplet

state, since the number of intermediate states appearing in the perturbation expansion is much larger for the triplet than for the quintet space (7 295 672 and 3 506 188, respectively). Consequently, the number of intermediate states with quasi-degenerated zeroth-order energy is statistically larger for the triplet state, causing much stronger defects and artiﬁcial energy lowering than for the quintet state. This behavior is most visible in the CASPT2/MOLCAScurve with s = 0.01 and s = 0.015. Analogous information can be inferred from the CASSCF reference weight plotted for the CASPT2/MOLCAS wavefunctions in Fig. 2 for various values of s. The CASSCF reference weight in the3_S

u wavefunction is almost constantly

lower than for5

Su over the whole studied range of distances.

This analysis allows us to conclude that the energy order found by Matxain et al.1in their MRPT calculations—i.e. with

3

Su being the ground state and 5

Su the ﬁrst excited state—is

merely an artifact produced by the intruder state removal technique. In the previously studied case of the manganese dimer,5we showed that the spectroscopic parameters of this molecule depend strongly on the magnitude of the shift parameter. A similar situation occurs for both of the investigated states of Sc2. The spectroscopic parameters

obtained from the CASPT2/MOLCAS, CASPT2/MOLPRO, and MRMP/GAMESS calculations with various values of the shift parameter are tabulated in Table 1. As can be seen, the change is largest for small values of the shift parameter, while for larger values of s, the spectroscopic parameters tend toward those of CASSCF.

Intruder states appearing in MRPT originate from a poor choice of the zeroth-order Hamiltonian. In all the studied cases, the zeroth-order Hamiltonian is chosen as a multi-reference generalization of one-electron Møller–Plesset

Fig. 2 Reference weights for the5_S

uand3Sustates in CASPT2/MOLCAS

calculations for diﬀerent values of imaginary shift, s.

Table 1 Spectroscopic parameters of the5_S

u and3Su states of Sc2computed using CASPT2 and MRMP with diﬀerent values of the shift

parameter, s. For details, see text. Equilibrium distances (re) are given in A˚, harmonic frequencies (oe) in cm 1

and dissociation energies (De) in eV

Method s 5_S u 3Su DE(5Su -3 Su) re oe De re oe De MRMP/GAMESS 0.002 2.610 237.1 1.51 2.641 240.0 1.54 0.035 0.005 2.605 237.7 1.51 2.636 241.1 1.52 0.017 0.007 2.603 238.0 1.50 2.633 241.6 1.51 0.006 0.009 2.600 238.4 1.50 2.631 242.0 1.50 0.005 0.020 2.593 239.8 1.48 2.622 245.6 1.43 0.049 0.050 2.586 242.4 1.43 2.605 248.8 1.32 0.116 N 2.665 256.2 1.20 2.644 268.5 0.87 0.335 CASPT2/MOLPRO 0.180 2.637 221.4 —a 2.691 208.0 —a 0.059 0.200 2.632 226.2 —a 2.678 218.0 —a 0.043 0.250 2.623 232.3 —a _2.654 _229.8 _—a _0.013 0.300 2.616 235.7 1.65 2.641 236.5 1.64 0.011 0.400 2.607 239.7 1.63 2.629 242.9 1.58 0.048 0.500 2.601 242.3 1.61 2.619 247.7 1.53 0.077 N 2.665 256.2 1.20 2.644 268.5 0.87 0.335 CASPT2/MOLCAS 0.010 2.555 1528.0 2.45 2.609 1767.2 2.55 0.102 0.015 2.535 439.6 1.87 2.596 436.6 1.89 0.016 0.020 2.550 301.6 1.78 2.604 318.6 1.80 0.013 0.080 2.609 238.8 1.69 2.631 249.5 1.69 0.005 0.150 2.607 240.0 1.69 2.626 249.2 1.67 0.018 0.300 2.599 243.6 1.67 2.611 252.5 1.60 0.078 N 2.665 256.2 1.20 2.644 268.5 0.87 0.335

a_{At 40.00 a.u. the calculation is aﬀected by strong intruder states.}

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Hamiltonian. This choice often produces a large number of quasi-degeneracies in the zeroth-order energy spectrum as there is no guarantee that the energy order in the zeroth-order Hamiltonian spectrum would be the same as for the original two-electron operator. Consequently, the zeroth-order energy of the reference state can be embedded in the zeroth-order spectrum of the intermediate states, which brings about the appearance of the intruder states in the perturbation calcula-tions. We want to stress here that these problems are purely artifactual and have very little connection to the physical reality of a given system. Nor does the inclusion of additional orbitals in the active space provide a viable way to the solution of the intruder state problem. The intruder state removal techniques discussed here, due to the arbitrariness in choosing the magnitude of the energy shift, result in a corresponding arbitrariness in the final MRPT results. As discussed by Dyall,15the definition of a partially bielectronic zeroth-order Hamiltonian can be effective in removing the intruder states. A recent variant of MRPT, NEVPT,16 which makes use of such a Hamiltonian, is shown to be free from the problem of quasi-degeneracies in the zeroth-order spectrum. Preliminary results obtained with NEVPT2 for the two states of Sc2

studied here and given in the ESIw show that the ground state of this system is5Su. An extensive study of low-lying states

of the scandium dimer using the NEVPT method will be published shortly.

In this communication, we have shown that the calculations recently published by Matxain et al.1are aﬀected by a serious intruder state problem. The energy order of the two low-lying states of the scandium dimer,5Su and3Su, depends strongly

on the magnitude of the shift parameter used to remove the intruder states from the perturbation expansion. For small shift values, multireference perturbation theory predicts 3Su

to be the ground state, while for larger values,5_S

uis lower in

energy. The strong shift-dependence of the computed results cast serious doubts on the validity of the shift techniques in multireference perturbation theory.

This work was supported by National Science Council (grant NSC 96-2113-M-009-022-MY3), Ministry of Education MOE-ATU project, and by the National Center for High-Performance Computing NCHC, Hsinchu, Taiwan.

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5060 | Phys. Chem. Chem. Phys., 2010, 12, 5058–5060 This journal is c _{the Owner Societies 2010}