Internally Contracted Multireference Coupled Cluster Calculations with

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Internally Contracted Multireference Coupled Cluster Calculations with a Spin-Free Dirac–Coulomb Hamiltonian: Application to the Monoxides of Titanium, Zirconium, and Hafnium Filippo Lipparini, Till Kirsch, Andreas Köhn, and Jürgen Gauss J. Chem. Theory Comput., Just Accepted Manuscript • Publication Date (Web): 13 Jun 2017 Downloaded from http://pubs.acs.org on June 13, 2017

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Journal of Chemical Theory and Computation

Internally Contracted Multireference Coupled Cluster Calculations with a Spin-Free Dirac–Coulomb Hamiltonian: Application to the Monoxides of Titanium, Zirconium, and Hafnium Filippo Lipparini,∗,† Till Kirsch,† Andreas Köhn,‡ and Jürgen Gauss† †Institut f¨ ur Physikalische Chemie, Universität Mainz, Duesbergweg 10-14, D-55128 Mainz, Germany ‡Institut f¨ ur Theoretische Chemie, Universit¨ at Stuttgart, Pfaffenwaldring 55, D-70569 Stuttgart, Germany E-mail: [email protected]

Abstract In this contribution, we combine internally-contracted multireference coupled cluster theory with a four-component treatment of scalar-relativistic effects based on the spin-free Dirac-Coulomb Hamiltonian. This strategy allows for a rigorous treatment of static and dynamic correlation as well as scalar-relativistic effects, which makes it viable to describe molecules containing heavy transition elements. The use of a spin-free formalism limits the impact of the four-component treatment on the computational cost to the non rate-determining steps of the calculations. We apply the newly developed method to the lowest singlet and triplet states of the monoxides of titanium, zirconium, and hafnium and show how the interplay between electronic correlation and relativistic effects explains the electronic structure of such molecules.

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1

Introduction

The accurate description of molecules containing transition metals is often a particularly challenging task for computational quantum chemistry. Such systems may contain heavy elements and can exhibit a complex electronic structure, with degenerate or quasi-degenerate ground states, low-lying excited states, and open-shell configurations, requiring thus that one not only performs a high-level treatment of both dynamic and static electronic correlation, but also takes into account relativistic effects 1–9 that can play a fundamental role for secondand third-row transition elements. Among all of the methods of modern ab initio quantum chemistry, single-reference coupled cluster (CC) theory is one of the most accurate, 10–14 to the point that CC with single and doubles excitations 15 augmented with a perturbative treatment of triple excitations, 16 CCSD(T) is usually referred to as the gold standard of quantum chemistry. Truncated CC theory requires, however, in order to be accurate, that the system is well represented by a single reference, usually the Hartree-Fock (HF) determinant, from which the CC wavefunction is generated. When this is not the case, such as for many transition metal compounds, a more general approach is required. While some effort has been made to extend the applicability of single-reference CC methods to those kind of problems, 17–24 all of these extensions suffer from limitations, 25,26 in particular the bias towards the chosen reference determinant. For that reason, a genuine multireference (MR) treatment of CC theory is needed and our preferred choice. In this work, we focus on combining a state-specific MRCC treatment of the electronic problem with a relativistic approach and aim at formulating and implementing a computational strategy viable for describing systems containing heavy transition metals in an accurate, yet affordable way. In particular, we combine internally-contracted MRCC theory 27–33 with the four-component spin-free Dirac-Coulomb (SFDC) Hamiltonian 34 to treat scalarrelativistic effects. We apply such a strategy to the monoxides of titanium, zirconium, and hafnium, of which we study the lowest singlet and triplet states. These systems have been 2


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extensively studied in the literature, both experimentally 35–57 and computationally. 57–68 In particular, TiO and ZrO have been investigated due to their astrophysical interest. The electronic structure of these molecule presents an interesting trend, as while the ground state of TiO is a triplet state, ZrO and HfO ground states are both singlet states. Furthermore, while the lowest triplet state has Λ = 2 for all three molecules (resulting in a 3 ∆ state), the lowest singlet state of TiO and those of ZrO and HfO differ with respect to Λ: For TiO, a 1

∆ state is found, while for ZrO and HfO the lowest singlet state is 1 Σ + . Many correlated multireference strategies have been developed during the last decades,

including perturbative approaches such as the complete active space second-order perturbation theory 69–71 (CASPT2) and n-electron valence state perturbation theory 72,73 (NEVPT), multireference configuration interaction 74,75 (MRCI) and multireference CC theory. 26,76 We choose MRCC as it offers an accuracy comparable to its well-established single-reference counterpart, even when small active spaces are used. 26 Many different formulations exist for MRCC, which can be divided into two main families. The first is based on the Jeziorski–Monkhorst (JM) ansatz, 77 which defines a specific cluster operator for each determinant in the reference. Examples of JM MRCC methods are Mukherjee’s state-specific MRCC, 78 state-universal MRCC, 79 Brillouin-Wigner CC 80,81 and the MRExpT method. 82 The main issues of JM MRCC methods are their lack of orbital invariance 33,83,84 and the large number of amplitudes that need to be determined, as for each determinant in the reference an independent cluster operator is defined, which limits their applicability to small active spaces. 26 The second family, internally-contracted MRCC theory 27–33 (icMRCC), uses a single cluster operator that acts on a multideterminantal reference, such as the one generated by a previous complete active space self-consistent field (CASSCF) calculation. The icMRCC ansatz has the advantage of providing an energy which is fully orbital-invariant, size extensive 32 and spin-adapted. 85 Furthermore, the number of amplitudes to be determined is independent of the number of reference determinants and thus similar to that of a single-reference CC treatment. For these reasons, we prefer the icMRCC strategy, despite the

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more complex implementation work that it requires. 28,29 It is here worth mentioning another strategy to extend the CC method to multireference systems, namely, the Fock-space CC method 86–88 (FSCC). Such a method, however, suffers from various limitations that make it not completely satisfactory 25,26 and in particular from the use of non-optimized orbitals for the configurations in the reference. 26,86 An extensive comparison between the various state-specific MRCC schemes can be found in refs. 26,76. In order to include relativistic effects in the description, numerous strategies have been developed, 2–9 among which the most rigorous are those based on Dirac’s four-component formalism, which defines the appropriate setting to treat electrons in a relativistic regime. The most popular choice is the Dirac-Coulomb (DC) Hamiltonian, possibly in combination with correction terms for the electron-electron interaction such as the Breit contributions. 89 Several implementations of CC methods in conjunction with the DC Hamiltonian exist in the literature, 90–92 including a few that allow for MR ansätze. 93–96 However, the latter are either based on FSCC or on GASCC which, as discussed previously, possess a few shortcomings that render them less satisfactory than icMRCC. The use of a four-component relativistic framework comes, however, at a price, especially when a correlated method such as CC theory is employed. As an example, the rate-determining step of a DC CC calculation with single and double excitations is roughly 30 times more expensive than its non-relativistic counterpart. 97 This difference arises mainly from the presence, in the DC Hamiltonian, of the spin-orbit interaction, which breaks spin and point-group symmetry and forces one to use complex algebra. One possible way of reducing the impact of the relativistic treatment on the computational cost thus involves neglecting this interaction, which is much smaller in magnitude than scalar-relativistic effects, 1 by writing the DC Hamiltonian as the sum of a purely spin free (SF) and a spin orbit part and then dropping the latter. 34,97–101 Recently, we presented an implementation of the CASSCF method based on the SFDC Hamiltonian. 102 CASSCF is the natural starting point for icMRCC theory. Combining a SF relativistic approach with a high-level, correlated treatment is particularly attractive, as the

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cost-dominating correlated step can be performed at the same cost and using the same code as for the non-relativistic case, by virtue of spin separation. A SFDC-icMRCC calculation consists of three main steps. First, the reference is determined by solving the SFDC-CASSCF equations. 102 Then, using the converged CASSCF orbitals, one- and two-electron integrals are transformed to the molecular basis. Finally, the icMRCC equations are solved and the energy is computed. Again, thanks to spin separation, only the first two steps are more expensive than their non-relativistic counterparts. 102,103 The SFDC-icMRCC method can therefore be used for any system for which a corresponding non-relativistic calculation is possible. This work is organized as follows. In section 2, the SFDC Hamiltonian is introduced and the SFDC-CASSCF and SFDC-icMRCC equations are recapitulated. The implementation is summarized in section 3 and the numerical results on the monoxides of titanium, zirconium, and hafnium are presented in section 4. Section 5 finally contains the conclusions and gives some perspectives on future development of the methods introduced in this work.

2

Theory

In this section, the SFDC Hamiltonian is described and the optimization technique used to compute a relativistic CASSCF wavefunction is briefly discussed. The no-pair approximation is then invoked in order to define a setting for the treatment of electronic correlation and the main equations of the icMRCC method are introduced.

2.1

The Spin-Free Dirac-Coulomb Hamiltonian

Using the formalism of second quantization, the DC Hamiltonian reads 3

H DC =

X PQ

ˆQ + hD ˆ†P a P Qa

1 X ˆS a ˆR , ˆ†Q a GP QRS a ˆ†P a 2 P QRS

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(1)


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where P, Q, . . . refer to one-electron Dirac four-component spinors, which can in turn be expressed in terms of two two-component spinors which are called large component ψPL and small component ψPS



 ψP = 

with



 ψPL = 

ψPL,α ψPL,β

ψPL ψPS



(2)

 , 



 ψPS = 

 ,

ψPS,α ψPS,β



 .

(3)

hD is the one-particle Dirac operator, whose matrix elements are

hD PQ =

*

  

ψPL ψPS

     + L c~σ · p~   ψQ    V      c~σ · p~ V − 2c2 ψ S Q

L S L S = hψPL |V |ψQ i + c(hψPL |~σ · p~|ψQ i + hψPS |~σ · p~|ψQ i) + hψPS |V − 2c2 |ψQ i,

(4)

where ~σ = (σx , σy , σz )T is a vector collecting the three Pauli matrices, p~ the momentum operator, V the nuclear electrostatic potential and c the speed of light. In addition, we have shifted the energy by −c2 in order to match the non-relativistic energy scale. G is the (instantaneous) Coulomb interaction operator, whose matrix elements are

GP QRS =

Z

dr1

Z

dr2

ψP (r1 )† ψR (r1 )ψQ (r2 )† ψS (r2 ) = (P R|QS) |r1 − r2 |

L L S S L L S S = (ψPL ψRL |ψQ ψS ) + (ψPL ψRL |ψQ ψS ) + (ψPS ψRS |ψQ ψS ) + (ψPS ψRS |ψQ ψS ),

(5)

where we used the Mulliken notation for the electron repulsion integrals (ERIs). Spin separation 34 is obtained as follows. First, we introduce the pseudo-large component φLP by applying the following metric change to the small component wavefunction: ψPS =

~σ · p~ L φ . 2c P

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(6)

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Then, we write the DC Hamiltonian matrix elements in terms of the large and pseudo-large component and use Dirac’s identity

(~σ · ~u)(~σ · ~v ) = ~u · ~v + i~σ · (~u × ~v ).

(7)

to decompose the DC Hamiltonian into the sum of a spin-free and a spin-orbit part D,SF D,SO hD P Q = hP Q + hP Q ;

SO GP QRS = GSF P QRS + GP QRS .

(8)

The spin-free Dirac-Coulomb Hamiltonian is obtained by neglecting the SO terms

H SFDC =

X

ˆQ + ˆ†P a hPD,SF Q a

PQ

1 X SF ˆS a ˆR , ˆ† a G a ˆ† a 2 P QRS P QRS P Q

(9)

where L L L L L L L hPD,SF Q = hψP |V |ψQ i + hψP |T |φQ i + hφP |T |ψQ i + hφP |

p~ · V p~ − T |φLQ i, 2 4c

1 L L ([~pφLP ] · [~pφLR ]|ψQ ψS ) 4c2 1 1 ([~pφLP ] · [~pφLR ]|[~pφLQ ] · [~pφLS ]). + 2 (ψPL ψRL |[~pφLQ ] · [~pφLS ]) + 4c 16c4

(10)

L L L L GSF P QRS = (ψP ψR |ψQ ψS ) +

(11)

The properties of the SFDC Hamiltonian allow one to use the same basis functions to expand both the large and pseudo-large component (this is equivalent to using a kinetic-balanced basis set 104 ) and to use real basis functions. Furthermore, as the SFDC Hamiltonian does not depend on the spin operator, a spin-restricted formalism can be adopted. Omitting the SF labels, the SFDC Hamiltonian can therefore be re-written as follows:

H SFDC =

X

ˆ hD P Q EP Q +

PQ

1 X (P Q|RS)ˆ eP QRS , 2 P QRS

(12)

where the indices P, Q . . . now run over a set of 2Nb orthonormal, real spatial functions, Nb 7



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of which are associated with positive energy solutions (PES) and Nb of which are associated with negative energy solutions (NES), and where EˆP Q and eˆP QRS are the spin-free one- and two-body excitation operators. 105

2.2

Optimization of the SFDC-CASSCF wavefunction

In this section, the SFDC-CASSCF state-specific and state-averaged optimization problems are briefly recapitulated. For state-specific optimizations, we follow the norm-extended optimization (NEO) scheme 106–108 by Jensen et al., adapted to the SFDC Hamiltonian as described in ref. 102. For state-averaged optimization, we implemented a generalization to the SFDC Hamiltonian of the algorithm proposed by Werner and Meyer 109 and Werner and Knowles. 110 In the NEO scheme, the CASSCF wavefunction is parametrized as follows:

|Ψ i = e−κ

In eq. 13, |0i =

PN det I=1

|0i + Pˆ |ci . k|0i + Pˆ |cik

(13)

c0I |ΦI i is the current expansion point (CEP), Pˆ = 1 − |0ih0| is the

orthogonal projector onto the orthogonal complement to |0i and

|ci =

N det X

cI |ΦI i

I=1

is a CI correction vector. Note that, due to the normalization condition, this parametrization is redundant. The variations of the orbital coefficients are parametrized in eq. 13 by a unitary transformation Uˆ = e−ˆκ . Such a transformation mixes orbitals belonging to different spaces, i.e., internal, active and external. For a SFDC-CASSCF optimization, NES have to be taken into account as well, by including in the definition of κ ˆ the NES-internal and NES-active

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orbital rotations. The CASSCF energy

E(κ, c) =

hΨ (κ, c)|H SFDC |Ψ (κ, c)i hΨ (κ, c)|Ψ (κ, c)i

(14)

is then optimized with respect to all the variational parameters. Note that, as a difference to the non-relativistic case, the optimization procedure requires one to minimize the energy with respect to the CI correction and the PES/PES orbital rotation parameters, and to maximize it with respect to the PES/NES orbital rotation parameters. This is consistent with the quantum electrodynamics (QED) interpretation of the NES as empty positronic states. 3 A detailed description of the NEO procedure can be found elsewhere, 108 including its generalization to saddle point optimization problems. 102,111 Here, it suffices to say that the NEO is quadratically convergent and that convergence is guaranteed for the ground state. 112,113 For the saddle point problem associated with a relativistic calculation, convergence is also guaranteed in practice due to the large separation between the eigenvalues of the Hessian associated with NES/PES and PES/PES rotations, respectively. 102 In order to describe linear molecules, it is necessary to be able to correctly account for doubly-degenerate electronic states such as Π or ∆ states. If only Abelian point-group symmetry can be enforced, a state-specific approach gives rise to symmetry-broken solutions. The proper symmetry of the solution can be restored by computing a state-averaged 109 (SA) CASSCF wavefunction where both degenerate states are optimized at the same time. In SA CASSCF, the full CI problem within the active space is solved for several states |Ψµ i and averaged one- and two-body density matrices are computed as follows

ave γxy =

X

ωµ hΨµ |Eˆxy |Ψµ i,

ave Γxyuv =

µ

X

ωµ hΨµ |ˆ exyuv |Ψµ i,

(15)

µ

where the indices x, y, . . . run over the active orbitals, the index µ runs over the states that are simultaneously optimized, and the ωµ coefficients weight the various states and are chosen by the user. The averaged density matrices are then used to compute the orbital gradient 9



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and Hessian, which are used for the orbital optimization. As the NEO scheme cannot be easily adapted to the state-averaged case, we follow a different strategy. The Meyer-WernerKnowles (MWK) algorithm decouples the orbital and CI optimization problems, which makes a SA implementation straightforward. Given a set of orbitals, the full CI problem is solved for all the desired states and the averaged density matrices are built as in eq. 15. These matrices are then used to carry out a second-order optimization step for the orbitals. Furthermore, the MWK strategy introduces a model for the energy which is higher order in the orbital rotations and which has the advantage that the gradient and Hessian of the model energy can be easily computed without the need for an expensive four-index integral transformation. This high-order parametrization ensures robust and rapid convergence. All the details can be found in refs 109,110,114. The MWK scheme is easily extended to a four-component relativistic treatment. In particular, the orbital optimization step, which is performed by using an augmented Hessian strategy, needs to be modified in order to make sure that the energy is maximized with respect to the PES/NES rotation parameters and minimized with respect to the PES/PES ones. This is achieved by choosing the appropriate eigenvector of the augmented Hessian as a step, as described in detail in ref. 102.

2.3

Internally contracted Multireference Coupled Cluster with the SFDC Hamiltonian

The most common approach for treating electron correlation is to expand the wavefunction into a linear combination of Slater determinants, which are defined by replacing one or more occupied orbitals in the reference wavefunction with the same number of virtual orbitals. In the following, we invoke the no-pair approximation 115,116 as it is usual in relativistic quantum chemistry, and discard all the determinants that contain occupied NES, rewriting thus the

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SFDC Hamiltonian in the basis of the optimized CASSCF PES

SFDC HNP =

X

ˆ hD pq Epq +

pq

1X (pq|rs)ˆ epqrs , 2 pqrs

(16)

where the sums run over the PES. The icMRCC method 26–29 is defined by the following Ansatz ˆ

|Ψ i = eT |0i,

(17)

where |0i is the SFDC-CASSCF reference described in section 2.2, Tˆ =

X

tµ τˆµ

(18)

µ

is the cluster operator, τˆµ an excitation operator and the index µ runs over the excitation manifold. One of the main difficulties of icMRCC is related to the definition of such a manifold, as the active orbitals in the reference function are neither always empty nor always doubly occupied. In other words, excitations from the active space and into the active space need both to be considered. As the associated excitation operators do not commute, the icMRCC equations become rather complex and their implementation can be only achieved by means of automatic derivation techniques 29 or by exploiting a full-CI code. 28 The excitation manifold is usually truncated at a given level, for instance, including single and double excitations with respect to the reference, which defines the icMRCCSD method. Triple excitations can be accounted for in a perturbative way, giving rise to the icMRCCSD(T) method. 30 We remark here that, consistent with the use of the no-pair approximation, excitations are restricted to PES only. The excitation manifold also contains, in principle, excitations among active orbitals. These are not considered, as their effect is accounted for by relaxing the CI coefficients of the reference wavefunction during the solution of the icMRCC equations. 29 The icMRCC energy and amplitudes equations are obtained by inserting the Ansatz eq.

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ˆ

17 into the Schrödinger equation and multiplying it on the left by e−T

H |0i = E|0i,

(19)

where we introduced the similarity transformed SFDC no-pair Hamiltonian ˆ

ˆ

SFDC T H = e−T HNP e .

(20)

Projection on the reference state gives the icMRCC equation for the energy

E = h0|H |0i,

(21)

while projection on the excited determinants gives the equations for the amplitudes

h0|˜ τµ† H |0i = 0.

In addition, the expansion coefficients of the reference state |0i = solving the equations X

hΦI |H |ΦJ icJ = cI E.

(22)

P

I

cI |ΦI i are updated by

(23)

J

Relaxing the reference state is important for reaching the inherent accuracy of the icMRCC approach, in particular in the vicinity of state crossings. 33 Note that in eq. 22, a modified deexcitation operator τ˜µ† appears. This is due to another main difficulty encountered in icMRCC theory, as the functions obtained by applying the excitation operators τˆµ to the reference are non-orthogonal and, in general, not linearly independent. 28,29 Redundant parameters need to be removed, which can be done by computing the singular value decomposition of the metric matrix

Sµν = h0|ˆ τµ† τˆν |ri

12


(24)

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and by dropping all the singular vectors associated with singular values smaller than a threshold. The remaining singular vectors X are used to define the excitation operators used in eq. 22 as τ˜ = τˆX.

(25)

The way this operation is performed is critical, as it can affect the size-consistency of the method. 32 A complete discussion of the various procedures is beyond the aims of this work, the reader can find all the details in refs. 29,32. We note that the excitation operators can be easily restricted to pure singlet operators, which guarantees a final correlated state with defined spin-symmetry, as determined by the reference function.

3

Implementation

In this section, the implementation of the interface between the SFDC-CASSCF code and the icMRCC code is described. We use the SFDC-CASSCF module in the CFOUR 117 suite of programs, described in detail in ref. 102, together with its state-averaged extension. The icMRCC calculations are performed using the General Contraction Code (GeCCo). 29,30,32,118 As mentioned in the introduction, a SFDC-icMRCC calculation is performed in three main steps. The first is the optimization of a SFDC-CASSCF wavefunction. This is done by the CFOUR SFDC-CASSCF program that returns optimized MO coefficients for both PES and NES and the optimized CI coefficients. The second step prepares the information needed to run the icMRCC calculation and assembles the one- and two-electron integrals required to represent the Hamiltonian in the molecular basis. The number of basis functions per irreducible representation, the symmetry of the desired solution, and the definition of the various spaces (core, internal, active and external orbitals) are written on a file. The one- and two-electron integrals are transformed to the semi-canonical CASSCF MO basis. Note that for the two-electron integrals we need to accumulate the transformed integrals as the result of four subsequent transformations, that 13



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involve (LL|LL), (LL|SS), (SS|LL), (SS|SS) integrals, respectively, where L stands for large component and S for pseudo-large, with the pseudo-large component representing here the small component. Because of the no-pair approximation, only the MO integrals with PES indices are computed, i.e., hpq and (pq|rs). The transformed integrals are then dumped on a file which can be read by the GeCCo code. This second step is also performed by CFOUR. The third and final step is the solution of the icMRCC equations performed by the GeCCo program. Note that the latter step is unaffected by the choice of the Hamiltonian, i.e., the only steps of the calculations that are affected by using a four-component SFDC Hamiltonian are the first two, consistent with the use of a SF formalism and with the no-pair approximation.

4

Numerical Results

In this section, we apply the SFDC-icMRCC method to the study of the lowest singlet and triplet states of the monoxides of titanium, zirconium, and hafnium. The first part of this section is devoted to a qualitative investigation of the interplay between relativistic effects and electronic correlation. Using HfO as an example, we discuss how both relativity and dynamic electronic correlation play a role in defining what is the ground state of HfO, and comment on the additivity of relativistic effects and dynamic correlation. The second part of this section presents more quantitative results obtained with our SFDC-icMRCC method and compares the performance of the SFDC approach with those of the somewhat simpler spin-free exact two-component method in its one-electron variant (X2C1e). 103,119–121

4.1

Computational details

The lowest singlet state is 1 ∆ for TiO and 1 Σ + for ZrO and HfO, while the lowest triplet state is 3 ∆ for all three molecules. In order to minimize the computational effort required by the icMRCC calculation, we chose for each system the smallest active space able to capture

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the main features of the wavefunction. Such a choice is justified by the effectiveness of icMRCC in capturing dynamic correlation, which allows one to limit the size of the active space to what is required to correctly describe static correlation. 28,29,32,118 In order to select the most appropriate active space, we performed a full-valence CASSCF calculation and dropped from the active space all the orbitals with occupation numbers close to either 2 or 0. Furthermore, as we are interested in describing Σ or ∆ states, we also dropped all the π orbitals from the active space. In order to obtain a wavefunction with the correct symmetry, we use a state-averaged CASSCF description for the ∆ states, where we optimize both degenerate electronic states at the same time. This results in a CAS(4,5) description for TiO, with the active space consisting of the bonding-antibonding σ pair, one further σ orbital and two degenerate δ orbitals. In C2v symmetry, our choice corresponds to 4 a1 and 1 a2 active orbitals. For both the singlet and the triplet, one σ and one δ orbital have occupation numbers very close to one, meaning that the TiO 1 ∆ state is an open-shell singlet. For ZrO and HfO, a larger active space is needed to describe the 1 Σ + states, namely, a CAS(4,6) treatment is required. The active space contains two bonding-antibonding σ pairs and two degenerate δ orbitals, which, in C2v symmetry, correspond to 5 a1 orbitals and one a2 orbital. In order to be able to compare the energy of the singlet and of the triplet, we use the same active space for the state-averaged description of the 3 ∆ states. We employ for all elements large, uncontracted basis sets suitable for a correlated treatment of the core electrons. For TiO, we use Dunning’s cc-pwCVQZ basis set for both oxygen 122 and titanium 123 and correlate all electrons. In order to reduce the computational cost of the calculations, we only correlate the external core electrons for ZrO (4s and 4p electrons for Zr) and HfO (5s, 4f, 5p electrons for Hf). We use Dyall’s quadruple-zeta basis set for zirconium 124 and hafnium, 125 augmented with the functions needed for correlating the external core electrons, 124,125 and the cc-pwCVQZ basis set for oxygen. For each molecule and for each state, the equilibrium geometry is determined by computing a potential energy curve (PEC), using 11 uniformly spaced grid points, and by fitting

15



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it with a fourth-order polynomial. Convergence of the fit was tested by using a different number of points or a sixth-order polynomial. The use of uncontracted basis sets, which is required for kinetic-balanced basis sets, 104 sharply increases the number of basis functions. Due to the presence of primitives with very large exponents, very high-lying virtual orbitals are produced in the calculations. In order to reduce the computational cost of the icMRCC treatment, all orbitals with orbital energy above 1000 EH were dropped. This has been shown to have a very limited effect on the final results. The computational setup of the calculations is summarized in table 1. All the icMRCC calculations were performed at the icMRCCSD(T) level of theory. 30 The Table 1: Definition of the various orbital spaces for the calculations on the lowest triplet and singlet states of TiO, ZrO, and HfO. For each molecule, the number of orbitals per irreducible representation (a1 , b1 , b2 , a2 ) that define the core (i.e., non-correlated electrons), internal, active, and external spaces is reported. The number of external orbitals is the number of functions actually used in the correlated treatment, i.e., after dropping all orbitals with orbital energy higher than 1000 atomic units.

Core Internal Active External

TiO

ZrO

HfO

0,0,0,0 7,3,3,0 4,0,0,1 111,65,65,34

8,3,3,1 3,2,2,0 5,0,0,1 133,73,73,37

12,5,5,2 5,4,4,1 5,0,0,1 150,90,90,43

computations use the additional approximation of neglecting all triple-excitation operators τˆ3 with more than three active indices to avoid the diagonalization of higher-order reduced density matrices in the orthogonalization procedure. 30 For the present type of systems, with only four active electrons, this should be an uncritical approximation. 30 We use a convergence threshold of 10−10 for the SFDC-CASSCF equations (RMS norm of the gradient) and of 10−7 for the icMRCCSD amplitude equations (RMS norm of the residual). A threshold of 10−4 was used in order to remove redundant parameters during the solution of the icMRCC equations, as described in section 2.3. Scalar-relativistic effects are treated by using both the four-component SFDC Hamiltonian and the two-component SFX2C1e Hamiltonian.

16


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Page 17 of 43

4.2

Electronic Correlation and Scalar-Relativistic Effects: a Subtle Interplay

The PECs for the lowest singlet (1 Σ + ) and triplet (3 ∆) of HfO, computed at the CASSCF level of theory, are reported in figure 1. In figure 2, the same curves are reported for the icMRCCSD(T) level of theory.

The non-relativistic CASSCF treatment predicts the

Figure 1: Non-relativistic and SFDC potential energy curves for the 1 Σ + and 3 ∆ states of HfO computed at the CASSCF level of theory with Dunning’s uncontracted cc-pwCVQZ basis set O and Dyall’s uncontracted QZ basis set for Hf, augmented with external-core correlation functions. HfO − non−relativistic CASSCF −14396.18

HfO − SFDC CASSCF −15155.44

Singlet Triplet

−14396.19

Singlet Triplet

−15155.46

−14396.21

Energy (EH)

Energy (EH)

−14396.20

−14396.22 −14396.23

−15155.48 −15155.50 −15155.52

−14396.24 −15155.54

−14396.25 −14396.26 1.5

1.6

1.7 1.8 Hf − O distance (Å)

1.9

−15155.56 1.5

2.0

1.6


1.9

2.0

Figure 2: Non-relativistic and SFDC potential energy curves for the 1 Σ + and 3 ∆ states of HfO computed at the icMRCCSD(T) level of theory with Dunning’s uncontracted ccpwCVQZ basis set for O and Dyall’s uncontracted QZ basis set for Hf, augmented with external-core correlation functions. HfO − non−relativistic icMRCCSD(T) −14397.31

HfO − SFDC icMRCCSD(T) −15156.60

Singlet Triplet

−14397.32

−15156.64 Energy (EH)

−14397.34 −14397.35 −14397.36

−15156.66 −15156.68 −15156.70

−14397.37

−15156.72

−14397.38 −14397.39 1.5

1.6

1.7

1.8

1.9

−15156.74 1.5

2.0

Hf − O distance (Å)

3

Singlet Triplet

−15156.62

−14397.33 Energy (EH)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 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


1.6

1.7

1.8

1.9

2.0


∆ state to be the ground state, which is in contradiction to experiment. Scalar-relativistic

effects play a very important role and reverse the order of the states, predicting the 1 Σ + state 17



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 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

to be lower than the triplet, giving a qualitatively correct picture. This can be explained in terms of the stabilization of the σ orbitals and destabilization of the δ ones due to relativistic effects. 1 The effect is here particularly large, with the predicted adiabatic energy difference between the singlet and triplet going from -2816 cm−1 in the non-relativistic picture to 8659 cm−1 in the SFDC one. Relativistic effects are, however, not the only driving force in determining the relative stability of the singlet and triplet states. Electronic correlation plays an important role, too, and it is the subtle interplay between the two effects that gives the final result. A non-relativistic, correlated treatment finds in fact the singlet and triplet states to be almost degenerate, with the triplet being more stable by only 358 cm−1 , as can be seen from the left panel of fig. 2. In other words, dynamic correlation stabilizes the Σ state more than it stabilizes the ∆ one, which is also to be expected. It is here interesting to notice that the effects of electronic correlation and relativity are almost, but not exactly, additive. In order to better clarify this aspect, we report in figure 3 the non-additivity error on the energy, defined as the difference between the SFDCicMRCCSD(T) energy and its additive approximation, i.e., the energy obtained as

E add = E nrel (CAS) + [E SFDC (CAS) − E nrel (CAS)] + [E nrel (icMRCC) − E nrel (CAS)] (26)

The non-additivity error is sizable and not constant as a function of the Hf-O distance and can be interpreted as the relativistic contribution to the dynamic correlation energy. Nevertheless, such a contribution is small when compared with the total energy, which makes its effect on a computed property small, or even negligible. As a consequence, a relativistic treatment where such a contribution is neglected or approximated is expected to be very accurate. This is the case for the SFX2C1e approximation. 103,119–121 In the SFX2C1e method, the SF one-electron Dirac Hamiltonian is block-diagonalized in order to decouple the large- and small-component wavefunctions, and the latter is then dropped. Therefore,

18


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Page 19 of 43

Figure 3: Non-additivity energy error for HfO computed as the difference between the SFDCicMRCCSD(T) energy and its additive approximation defined in eq. 26. All the computations were performed using Dunning’s uncontracted cc-pwCVQZ basis set for O and Dyall’s uncontracted QZ basis set for Hf, augmented with external-core correlation functions. HfO non−additivity energy error −0.036 −0.038 Energy difference (EH)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 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


Singlet Triplet

−0.040 −0.042 −0.044 −0.046 −0.048 −0.050 1.5

1.6


1.9

2.0

the effects of relativity on the two-electron integrals are included only indirectly, affecting the description of the relativistic effects on electronic correlation. As these effects are small, the SFX2C1e approximation is expected to be sufficiently accurate in most cases. A direct comparison of the SFDC and SFX2C1e treatment is offered in section 4.3.

4.3

Structure and Properties of the Lowest Singlet and Triplet States of the Ti Group Monoxides

In this section, we report the computed equilibrium distances and harmonic stretching frequencies for the lowest singlet and triplet states of TiO, ZrO, and HfO, as well as the adiabatic singlet-triplet splittings. All studied systems exhibit some multireference character, which is particularly marked in the TiO 1 ∆ state, which is an open-shell singlet, but also in the 1 Σ + states of ZrO and HfO, where the occupation numbers of the δ orbitals are non-negligible (0.22 and 0.12, respectively, at the equilibrium distance). The multireference character is less pronounced for the 3 ∆ states, where the HF wavefunction weights are always about 90% in the CAS-CI expansion. 19



We report the icMRCCSD(T) energy curves for TiO, ZrO, and HfO in figures 4, 5, and 6, respectively. Figure 4: SFDC and SFX2C1e potential energy curves for the 1 ∆ and 3 ∆ states of TiO computed at the icMRCCSD(T) level of theory with Dunning’s uncontracted cc-pwCVQZ basis set. TiO − SFDC −928.96

−928.98

−928.88

−928.99

−928.89

−929.00 −929.01

−928.90 −928.91

−929.02

−928.92

−929.03

−928.93

−929.04 1.3

1.4

1.5 1.6 1.7 Ti − O distance (Å)

1.8

Singlet Triplet

−928.87

Energy (EH)

Energy (EH)

TiO − SFX2C1e −928.86

Singlet Triplet

−928.97

−928.94 1.3

1.9

1.4

1.5 1.6 1.7 Ti − O distance (Å)

1.8

1.9

Figure 5: SFDC and SFX2C1e potential energy curves for the 1 Σ + and 3 ∆ states of ZrO computed at the icMRCCSD(T) level of theory with Dunning’s uncontracted cc-pwCVQZ basis set for O and Dyall’s uncontracted QZ basis set for Zr augmented with external-core correlation functions. ZrO − SFDC −3672.60

ZrO − SFX2C1e −3671.91

Singlet Triplet

−3672.61 −3672.62

−3671.93

−3672.63

−3671.94

−3672.64 −3672.65

−3671.95 −3671.96

−3672.66

−3671.97

−3672.67

−3671.98

−3672.68 1.5

1.6

1.7 1.8 Zr − O distance (Å)

1.9

Singlet Triplet

−3671.92

Energy (EH)

Energy (EH)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 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

Page 20 of 43

−3671.99 1.5

2.0

1.6

1.7 1.8 Zr − O distance (Å)

1.9

2.0

The computed and experimental adiabatic singlet-triplet splittings are reported in table 2. The predicted singlet-triplet splittings are in good agreement with the experimental ones. Note that spin-orbit effects play an important role here, especially for the heavier Zr and Hf oxides, so a quantitative agreement cannot be expected, as we are only including scalar-relativistic effects in our treatment. Nevertheless, the order of the states is correctly reproduced and a semi-quantitative agreement is observed. 20


Page 21 of 43

Figure 6: SFDC and SFX2C1e potential energy curves for the 1 Σ + and 3 ∆ states of HfO computed at the icMRCCSD(T) level of theory with Dunning’s uncontracted cc-pwCVQZ basis set for O and Dyall’s uncontracted QZ basis set for Hf augmented with external-core correlation functions. HfO − SFDC −15156.60

HfO − SFX2C1e −15151.60

Singlet Triplet

−15156.62

Energy (EH)

−15151.64

−15156.66 −15156.68

−15151.66 −15151.68

−15156.70

−15151.70

−15156.72

−15151.72

−15156.74 1.5

Singlet Triplet

−15151.62

−15156.64 Energy (EH)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 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


1.6

1.7

1.8

1.9

−15151.74 1.5

2.0

1.6


1.7

1.8

1.9

2.0


Table 2: SFDC and SFX2C1e adiabatic singlet-triplet splittings (in cm−1 ) for TiO, ZrO, and HfO, computed at the icMRCCSD(T) level of theory, using Dunning’s uncontracted cc-pwCVQZ basis set for Ti and O and Dyall’s uncontracted QZ basis set for Zr and HF augmented with external-core correlation functions. Experimental data for TiO are taken from ref. 42, for ZrO from ref. 126, and for HfO from ref. 55. For ZrO, the values reported correspond to the Ω = 1 and Ω = 3 terms, while for HfO to the Ω = 1 and Ω = 2 terms.

TiO ZrO HfO

SFX2C1e

SFDC

Exp

3065 1752 10392

3174 1885 10458

3430 1099-1724 9230-10152

21



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Page 22 of 43

The fitted equilibrium geometries and frequencies are reported in table 3 and 4, respectively, together with the experimental values. Table 3: SFDC and SFX2C1e (denoted “X2C”) equilibrium distances (in ˚ A) for the lowest singlet and triplet states of TiO, ZrO, and HfO computed at the icMRCCSD(T) level of theory, using Dunning’s uncontracted cc-pwCVQZ basis set for Ti and O and Dyall’s uncontracted QZ basis set for Zr and Hf, augmented with external-core correlation functions. Experimental data for TiO are taken from refs. 35, 36, for ZrO from refs. 53, 54 and for HfO from refs 53, 56. TiO Singlet Triplet

ZrO

HfO

X2C

SFDC

Exp

X2C

SFDC

Exp

X2C

SFDC

1.6258 1.6277

1.6258 1.6277

1.619 1.620

1.7186 1.7343

1.7185 1.7347

1.712 1.728

1.7220 1.7412

Exp

1.7220 1.723 1.7411 1.742

Table 4: SFDC and SFX2C1e (denoted “X2C”) harmonic stretching frequencies (in cm−1 ) for the ground states of TiO, ZrO and HfO computed at the icMRCCSD(T) level of theory, using Dunning’s uncontracted cc-pwCVQZ basis set for Ti and O and Dyall’s uncontracted QZ basis set for Zr and Hf, augmented with external-core correlation functions. Experimental data for TiO are taken from refs. 47, 39, for ZrO from ref. 36, for HfO from ref. 127, 55. Note that a frequency of 1000 cm−1 is reported for the 1 ∆ state of TiO in ref. 42. TiO Singlet Triplet

ZrO

HfO

X2C

SFDC

Exp

X2c

SFDC

Exp

X2c

SFDC

Exp

963 969

963 969

1016 1009

958 923

959 923

969 936

985 914

982 912

974 918

The agreement between the computed and experimental equilibrium distances for the singlet and triplet states of TiO, ZrO, and HfO is good, with errors always smaller than 0.01˚ A and of about 0.001˚ A for HfO. Furthermore, the agreement between the SFDC and SFX2C1e treatment of scalar-relativistic effects is remarkable, with virtually no difference between the two sets of results. In order to understand the good performances of the SFX2C1e method, we report in figure 7 the differences between the SFDC and SFX2C1e CASSCF and correlation energies for the 1 Σ + state of HfO. The SFX2C1e error on the CASSCF energy is sizable and much larger than the error on the correlation energy, as expected from the results shown in figure 3. Nevertheless, if one decomposes the error as the sum of a constant part 22


Page 23 of 43

Figure 7: Differences between the SFDC and SFX2C1e CASSCF (left) and correlation (right) energies for the 1 Σ + state of HfO, computed using Dunning’s uncontracted cc-pwCVQZ basis set for O and Dyall’s uncontracted QZ basis set for Hf augmented with external-core correlation functions. SFDC vs SFX2C1e − CASSCF energy

SFDC vs SFX2C1e − correlation energy

−5.00764

−0.00003

−5.00765

−0.00004

−5.00766

−0.00005

Energy difference (EH)

Energy difference (EH)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 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


−5.00767 −5.00768 −5.00769 −5.00770 −5.00771 −5.00772 −5.00773 1.5

−0.00006 −0.00007 −0.00008 −0.00009 −0.00010 −0.00011

1.6


1.9

−0.00012 1.5

2.0

1.6


1.9

2.0

and a distance-dependent part, one immediately notes that the distance-dependent part is very small. As the equilibrium distances and frequencies depend on the shape of the PECs and not on the absolute energy, the SFX2C1e results are in very good agreement with the SFDC ones. The agreement between the computed frequencies is good for ZrO and HfO, but less satisfactory for TiO, for which we observe a larger difference between the predicted and experimental values. As a general trend, we observe that the agreement between the computed and experimental distances and frequencies improves going from TiO to HfO, being very good for the latter and poorer for the former. The observed discrepancies can stem from three different sources, namely, the lack of spin-orbit interaction in our treatment, the incompleteness of the basis set, and the truncation of the cluster operator in the correlated treatment. As errors are larger for the lighter elements, it is safe to assume that the spinorbit contribution to the bond distances and frequencies is much smaller than the observed error. In order to understand the effect of the basis set size on the computed properties, we compute a PEC for both the singlet and triplet states of TiO at the complete basis set (CBS) limit. We perform a CBS extrapolation on TiO at the non-relativistic level, as this allows us

23



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Page 24 of 43

to use contracted basis sets, which mitigates the cost of the calculations. Relativistic effects, which are expected to be small for TiO, can be added a posteriori as a correction, computed as the difference between the non-relativistic and SFDC energies using the uncontracted cc-pwCVQZ basis set. We use Feller’s three point, exponential extrapolation scheme for the CASSCF energy 128 E(X) = E∞ + Ae−BX ,

(27)

where X = 3, 4, 5 corresponds to the results obtained with the cc-pwCVXZ basis sets, and a two point, cubic extrapolation 129 for the correlation energy

E(X) = E∞ +

C , X3

(28)

using the cc-pwCVXZ results with X = 4, 5. The CBS results are reported in table 5, with and without relativistic corrections. As expected, relativistic corrections are small, although Table 5: Non-relativistic and SFDC-corrected equilibrium distances (in ˚ A) and harmonic stretching frequencies (in cm−1 ) for the 1 ∆ and 3 ∆ states of TiO computed at the icMRCCSD(T)/CBS level. Experimental data from refs. 35, 47, 39. Note that a frequency of 1000 cm−1 is reported for the 1 ∆ state of TiO in ref. 42. Bond Distance Singlet Triplet

Frequency

NR

SFDC

Exp

NR

SFDC

Exp

1.6228 1.6252

1.6223 1.6244

1.619 1.620

964 969

965 970

1016 1009

not negligible, which justifies the use of a composite scheme. The CBS extrapolated results are only in slightly better agreement with experiment than the uncontracted cc-pwCVQZ ones reported in tables 3 and 4. We can therefore assume that such results are close to be converged with respect to the basis-set size, and that the observed discrepancies are due to approximations in the treatment of electronic correlation. The quality of such a treatment depends on two related factors, i.e., the choice of the active space and the truncation of the icMRCC cluster operator. To investigate the impact of the former choice, we computed 24


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the equilibrium distances and frequencies for TiO also using a CAS(8,9) reference, i.e., a full-valence CAS from which we removed the oxygen 2s orbital. In C2v symmetry, this corresponds to 4a1 , 2b1 , 2b2 , and 1a2 orbitals. With respect to our previous choice, we include in the active space two additional pairs of π orbitals. In order to limit the computational requirements of the calculations, they were performed at the non-relativistic level, using the contracted cc-pVQZ 122 basis set and the frozen-core approximation. The results are compared in table 6 with the results obtained with the smaller CAS(4,5) reference using the same setup, i.e., the cc-pVQZ basis set, the frozen-core approximation and a non-relativistic treatment. The choice of the active space has apparently a large effect on the computed ˚) and harmonic stretching frequencies Table 6: Non-relativistic equilibrium distances (in A −1 (in cm ) for the lowest singlet and triplet states of TiO computed at the icMRCCSD(T) level of theory with a CAS(4,5) or a CAS(8,9) reference, using Dunning’s cc-pVQZ basis set and the frozen-core approximation. The difference between the results is also reported (denoted “δ”). Bond Distance Singlet Triplet

Frequency

CAS(4,5)

CAS(8,9)

δ

CAS(4,5)

CAS(8,9)

δ

1.6465 1.6478

1.6326 1.6352

0.0139 0.0126

927 931

1004 1000

77 69

bond distances and frequencies, contrary to what is usually observed for icMRCC. 28,29,32,118 In contrast to that, the singlet-triplet adiabatic splitting is not much affected, varying from 2953 cm−1 with the CAS(4,5) reference to 3017 cm−1 with the CAS(8,9) one, which corresponds to a change in energy of less than 0.8 kJ/mol. Note that, while the frequencies computed with the larger active space seem in very good agreement with the experimental results, the computation is performed with the frozen-core approximation, which is expected to have a large impact on both the distances and the frequencies. Hence, we expect the use of a larger active space not to be sufficient to achieve a quantitative agreement with experiment. A similar, large impact on the results for TiO can be observed by changing the level at which dynamic correlation is described. Due to hardware limitations, icMRCCSSDT calculations, with full, iterative triples, are currently not possible if not for the smallest 25



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Page 26 of 43

systems. Nevertheless, the effect of the perturbative treatment of triples excitation on the computed properties can be used to estimate whether the correlation treatment is adequate. We report in table 7 the computed bond distances and in table 8 the computed frequencies for TiO, ZrO and HfO at the SFDC-icMRCCSD level of theory and we compare such results with the corresponding SFDC-icMRCCSD(T) ones. For both the distances and the frequencies, Table 7: SFDC equilibrium distances (in ˚ A) for the lowest singlet and triplet states of TiO, ZrO, and HfO computed at the icMRCCSD (denoted “SD”) and icMRCCSD(T) (denoted “SD(T)”) level of theory, using Dunning’s uncontracted cc-pwCVQZ basis set for Ti and O and Dyall’s uncontracted QZ basis set for Zr and Hf, augmented with external-core correlation functions. The difference between the results is also reported (denoted “δ”). TiO Singlet Triplet

ZrO

HfO

SD

SD(T)

δ

SD

SD(T)

δ

SD

SD(T)

δ

1.6015 1.6045

1.6258 1.6277

0.0243 0.0232

1.7064 1.7212

1.7185 1.7347

0.0121 0.0131

1.7134 1.7291

1.7220 1.7412

0.0086 0.0121

Table 8: SFDC harmonic stretching frequencies (in cm−1 ) for the lowest singlet and triplet states of TiO, ZrO and HfO computed at the icMRCCSD (denoted “SD”) and icMRCCSD(T) (denoted “SD(T)”) level of theory, using Dunning’s uncontracted cc-pwCVQZ basis set for Ti and O and Dyall’s uncontracted QZ basis set for Zr and Hf, augmented with external-core correlation functions. The difference between the risults is also reported (denoted “δ”). TiO Singlet Triplet

ZrO

HfO

SD

SD(T)

δ

SD

SD(T)

δ

SD

SD(T)

δ

1078 1068

963 969

115 93

1002 977

959 923

43 53

1007 947

982 912

25 35

the perturbative triples correction becomes smaller going from TiO to HfO. The effects of the (T) corrections are particularly marked for the TiO harmonic stretching frequencies, which change by as much as 100 cm−1 due to the (T) correction. These results, together with the ones obtained by varying the dimension of the active space, suggest that, in order to get a quantitative agreement, a more accurate treatment of both the static and dynamic electronic correlation for TiO is required, i.e., one needs to include full iterative triples or even higher excitations in the cluster expansion, possibly in conjunction with the use of a 26


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larger active space for the reference.

5

Conclusions and perspectives

In this contribution, we use for the first time the icMRCC method in conjunction with the four-component spin-free Dirac Coulomb Hamiltonian to describe scalar-relativistic effects. The implementation is based on the SFDC-CASSCF implementation in the CFOUR suite of programs, which has been extended in order to allow for state-averaged SFDC-CASSCF computations, and on the General Contraction Code (GeCCo), which has been interfaced with CFOUR, for icMRCC. The use of a SF formalism concentrates the computational effort necessary to treat scalar-relativistic effects with a rigorous four-component formalism in the non rate-determining step of computing the reference CASSCF wavefunction and transforming the integrals to the MO basis. Once this has been done, the expensive MRCC correlation treatment is performed with the same computational cost as for non-relativistic computations. As MRCC is clearly the limiting part of the computation, our implementation allows one to use a SFDC relativistic treatment whenever a non-relativistic calculation is feasible with the same basis set. The high-level treatment of electronic correlation guaranteed by MRCC theory can therefore be extended to compounds of heavy transition metals, which often exhibit not only strong relativistic effects, but also a marked multireference character, without significant computational overhead. We have then applied the SFDC-icMRCC methodology to study the lowest singlet and triplet states of the monoxides of titanium, zirconium and hafnium. A first, qualitative analysis on HfO shows how a correct description of both correlation and relativistic effects is needed in order to predict the relative stability of the electronic states of such a molecule. The interplay between correlation and relativistic effects has been investigated, and it has been shown that scalar relativistic effects have a small, but non negligible influence on the correlation energy. In order to compute the geometry and the harmonic stretching

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frequencies of the oxides investigated in this work, we have applied the SFDC-icMRCC methodology in conjunction with large, uncontracted basis sets, correlating both valence and the external core electrons. This approach has proven to be very accurate for the heaviest HfO and accurate for ZrO, but not entirely satisfactory for TiO. After investigating the convergence of our results for TiO with respect to the basis set size, we concluded that the discrepancies between the computed and experimental results mainly stem from the correlation treatment, namenly, icMRCCSD(T). The dramatic effect of the perturbative correction for triple excitations on the bond distances and frequencies lead us to conclude that the inclusion of higher excitation is required in order to obtain a more quantitative agreement. Nevertheless, the computed SFDC-icMRCCSD(T) distances are in good agreement with the experimental data, and for ZrO and HfO the predicted harmonic frequencies agree with the experimental ones within about 10 cm−1 . Finally, we compared our SFDC results to the ones obtained by treating scalar relativistic effects with an approximate, two-component Hamiltonian, namely, the spin-free X2C1e Hamiltonian. The excellent agreement between such results confirms the viability of two-component spin-free schemes, to which the SFDC method provides a benchmark. The good agreement is justified in terms of the small relative magnitude of the relativistic effects on the correlation energy. We remark here that, in the present case, the computational advantages of using the SFX2C1e with respect to the SFDC one are limited to the non rate-determining step and, as a consequence, nearly negligible. The SFDC-icMRCC method we presented suffers from two main shortcomings. The use of uncontracted basis sets, which is required when using restricted kinetic balanced basis sets in order to avoid variational collapse problems, 104 significantly increases the number of basis functions that are needed with respect to a non-relativistic calculation. This problem is only partially mitigated by the fact that one can drop the highest lying virtual orbitals in the correlated step without significantly affecting the energy and affects significantly the CASSCF step. Overcoming this difficulty will further extend the applicability of the proposed SFDC-icMRCC scheme.

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The accurate treatment of scalar-relativistic effects and of electronic correlation presented here lacks a last, important ingredient in order to offer a fully quantitative tool for describing heavy elements, i.e., the inclusion of spin-orbit effects. A perturbative approach based on the SFDC Hamiltonian is currently under investigation in our group.

Acknowledgments The authors would like to thank Dr. Y. A. Aoto for initial help with performing the icMRCC computations and for helpful discussions about titanium monoxide. F.L. is grateful to the Alexander von Humboldt foundation for financial support. The work was supported by the Deutsche Forschungsgemeinschaft (DFG Grant GA 370/6-1 and 6-2, as well as KO 2337/4-1).

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