Highly accurate potential energy surfaces by COLUMBUS: dissociation of ozone

Accurate potential energy surfaces are required for many problems in chemistry, in particular for spectroscopy. High accuracy also means high computational expense, therefore such calculations, even with efficient implementations as in COLUMBUS, can be utilized only for small

hor’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI:10.1063/1.5144267

molecules.⁹¹ Here we show, through an application on ozone dissociation, how some unique features of COLUMBUS allow accurate treatment of complicated situations.

Accurate quantum chemical results require the consideration of not only dynamic but often also static correlation.¹³ Despite significant previous efforts,⁹² MR versions of coupled cluster (CC) methods are not yet routinely available. Therefore, MRCI methods seem to be one of the best choices, in particular at the MR-CISD and MR-AQCC levels.¹³

One important shortcoming of MR-CISD is the linearly increasing size of the parameter space with the number of reference functions. If a CAS^93-95 reference is used, the cost increases exponentially with the number of active orbitals involved. To avoid this explosion in computational cost, often internally contracted (ic) functions are used^96-99 (see Ref. ¹³ for a detailed review). This procedure is very efficient and has been applied successfully in many cases.

However, ic methods have the disadvantage that the treatment of static and dynamic correlation appears in separated steps of the calculation. Therefore, if strong interaction with another state is present and there are (avoided) crossings between different potential energy surfaces, the separation of the static and dynamic correlation is problematic because the crossing happens at different geometries for the reference and the subsequent MR-CISD calculations.¹⁰⁰

Such a problem is largely solved if, instead of internal contraction, an uc MRCI wavefunction is used with all reference functions individually included in the ansatz. The advantage of the uc vs. ic calculations to avoid unphysical reef-like structures on dissociation curves has been shown in Refs. ^{101, 102} and will be demonstrated with an example below. We note that flexible selection of reference functions in COLUMBUS allows a reduction of the cost compared to the full CAS reference calculations; it is possible to use one wavefunction in the

MCSCF step to obtain orbitals and then use a different set of reference configurations in the correlated MR-CISD calculation³ for both single points and energy gradients.⁶

Contrary to CC methods, MRCI methods are not size-extensive,¹⁰³ i.e., the energy does not scale properly with the system size,^{77, 104} an error which certainly needs to be corrected in high accuracy calculations. There are several possibilities, both a priori and a posteriori, for this correction, which are summarized in Refs. ^{13, 103}. The most often used a posteriori correction is due to Davidson (MR-CISD-QD),⁷⁷ but we usually suggest its Pople version (MR-CISD-QP).¹⁰⁵ A priori corrections are more advantageous since not only the energy, but also the wavefunction, is corrected, and also analytic gradients are available. The two most popular versions are MR-ACPF¹⁰⁶ and MR-AQCC;^{14, 15} the latter seems to be more stable.¹⁰³

We show the importance of both the uncontracted ansatz and the size-extensivity correction on the accurate potential energy surface of ozone. Here, the potential along the minimum energy path (MEP) leading to dissociation of one O atom is important for accurate prediction of highly excited vibration levels as well as to describe the scattering of an O atom by an O2 molecule.¹⁰⁷ The latter process shows an unusual isotope effect,^{108, 109} which is difficult to explain theoretically.

Theoretical methods often predict a “reef-like” structure with a small barrier and a van-der-Waals minimum along the MEP, see e.g. Refs. ^{107, 110} for reviews. In the case of ozone, Holka et al.¹⁰⁷ showed the significant effect of size-extensivity correction on the size of the barrier, while Dawes et al.¹¹⁰ demonstrated that including several internally contracted reference functions in a multistate ic-MRCI calculation causes the reef to disappear. Fig. 6 shows how the effects mentioned above, i.e., uncontraction of the reference space (left panel) and the inclusion of size-extensivity corrections in form of the Davidson (QD) and Pople (QP) corrections, as well as by MR-AQCC (right panel), lower the barrier and lead essentially at the uc-MR-AQCC level to its

disappearance. As discussed by Tyuterev et al.^{111, 112} and Dawes et al.,^{102, 113} only the barrierless potential is capable of reproducing the experimental findings both for the vibrational levels and the scattering.

Fig. 6 Potential energy curves calculated by MR methods along the minimum energy path (MEP) to dissociation for ozone.¹¹⁴ The one-dimensional cut along one O-O distance is shown, while the other two coordinates are fixed at R1=2.275 a.u. and =117. The left panel demonstrates the effect of internal contraction showing ic- and uc-MR-CISD and MR-AQCC results, while the right panel shows how the barrier disappears when including size-extensivity corrections. The calculations have been performed using a full valence CAS reference space in all MR calculations with the frozen core approximation. The orbitals have been obtained using a full valence CAS, with the 1s orbitals frozen. These latter orbitals have been obtained from a preceding MCSCF calculation with only the 2p orbitals included in the CAS. The cc-pVQZ basis was used. The uc-MR calculations included over 1 billion configurations.

In document The Generality of the GUGA MRCI Approach in COLUMBUS for Treating Complex Quantum Chemistry (Pldal 22-26)