Four-Component Relativistic State-Specific Multireference

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Four Component Relativistic State-Specific Multireference Perturbation Theory with a Simplified Treatment of Static Correlation Anirban Ghosh, Suvonil Sinha Ray, Rajat Kumar Chaudhuri, and Sudip Kumar Chattopadhyay J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.6b11348 • Publication Date (Web): 23 Jan 2017 Downloaded from http://pubs.acs.org on January 25, 2017

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Revised Version: I Manuscript ID: jp-2016-11348z-R1

Four Component Relativistic State-specific Multireference Perturbation Theory with a Simplified Treatment of Static Correlation Anirban Ghosh,1 Suvonil Sinha Ray,1 Rajat K Chaudhuri,2 and Sudip Chattopadhyay1, ∗ 1

Department of Chemistry, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103, India

2

Theoretical Physics, Indian Institute of Astrophysics, Bangalore 560034, India (Dated: January 20, 2017)

1



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Abstract The relativistic multireference (MR) perturbative approach is one of the most successful tools for the description of computationally demanding molecular systems of heavy elements. We present here the ground state dissociation energy surfaces, equilibrium bond lengths, harmonic frequencies, and dissociation energies of Ag2 , Cu2 , Au2 , and I2 computed using the four-component (4c) relativistic spinors based state-specific MR perturbation theory (SSMRPT) with improved virtual orbital complete active space configuration interaction (IVO-CASCI) functions. The IVO-CASCI method is a simple, robust, useful and lower cost alternative to the complete active space selfconsistent field approach for treating quasidegenerate situations. The redeeming features of the resulting method, termed as 4c-IVO-SSMRPT, lies in (i) manifestly size-extensivity, (ii) exemption from intruder problems, (iii) the freedom of convenient multipartitionings of the Hamiltonian, (iv) flexibility of the relaxed and unrelaxed descriptions of the reference coefficients, and (v) manageable cost/accuracy ratio. The present method delivers accurate descriptions of dissociation processes of heavy element systems. Close agreement with reference values has been found for the calculated molecular constants indicating that our 4c-IVOSSMRPT provides a robust and economic protocol for determining the structural properties for the ground state of heavy element molecules with eloquent MR character as it treats correlation and relativity on equal footing.

∗

Electronic address: Correspondingauthor:[email protected]

2


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I.

INTRODUCTION

The concomitant interplay of three physical effects such as relativistic, nondynamical (long-range) and dynamical (short-range) correlations has made the study of bond-breaking of molecular systems containing heavy atoms a challenging proposition [1–9]. Molecules having heavy atoms have energetically proximate (as compared to the lighter atoms) s, p, d, and f shells, and hence the nondynamic/static electron correlation has a pivotal role to play. This effect enhances the chances of having quasi-degenerate configurations, which gets further augmented by the increased density of states via the spin-orbit coupling. In the relativistic domain, with the s and p orbitals undergoing contraction and the d and f orbitals exhibiting expansion, the level of complexity increases, see, the discussion in Refs. 10, 11. Such a modification of the spatial distribution of the relativistic orbitals, in comparison to the corresponding non-relativistic counterparts, modifies the nature of electron correlation, in addition to affecting the orbital energies. Such changes ultimately render non-additivity to the relativistic and electron correlation corrections. One has to account for these effects in a judicious manner. For a balanced and accurate treatment of both dynamical and nondynamical electron corrections, it is useful to employ a multireference (MR) correlation method that is potentially unfluctuating and faithful over the entire potential energy surfaces (PES) of chemical reactions to as much extent as is permissible. For investigating systems involving heavy elements and energy surfaces of their chemical reactions, relativistic MR perturbation theory (MRPT) has now become a promising method of choice[12–25] combining twin advantages - ease of implementation and simultaneous treatment of relativistic and correlation effects at the expense of a low computational cost. Various MRPT methods based on the four component (4c) Dirac-Hartree-Fock (DHF) Hamiltonian have already been developed by various workers which include the work of Abe et al. [18] who proposed a relativistic version of second-order complete active space perturbation theory (CASPT2) of Roos and co-workers [26, 27]. Abe et al. argued that the IVO (improved virtual orbital) scheme for virtual spinors are effective for 4c-CASPT2 scheme. The relativistic two-component CASPT2 has also been suggested and implemented [22] using Kramers restricted complete active space self-consistent field (CASSCF) reference function. Nakano and coworkers [15–17, 19] have proposed a relativistic quasidegenerate (QD) perturbation theory using 4c general multiconfiguration (GMC) reference functions(GMC-

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QDPT)[28, 29]. Tatewaki et al. [20] have performed calculations for the electron affinity using the 4c-MCQDPT. Shiozaki and Mizukami [23] were the first to report the theory and algorithms for relativistic 4c-ic-MRCI, and 4c-NEVPT2. Vilkas et al. [12, 13] have come up with a relativistic generalization of the MRMP2 method of Hirao [30, 31] based on DHF-MRCI reference functions to study atomic systems. More recently, the GVVPT2 (MR generalized van Vleck second-order perturbation theory) [32, 33] level of theory, with relativistic effects included via the spin-free exact two-component (sf-X2C) Hamiltonian, has been suggested by Tamukong et al. [21]. Various MRPT schemes have also been suggested to describe quasi-degenerate electronic states with strong spin-orbit effects [34–36]. In addition, relativistic MR-configuration interaction (CI) [37, 38], and coupled cluster(CC) [39–44] approaches that have the ability to yield highly accurate energies have also been suggested. A variety of two-component electron correlation methods using effective core potentials [45] and relativistic density functional theory, [46–49] have appeared as other efficient alternatives. In an attempt to describe the near-degenerate and dissociative regions of the electronic states of molecules involving atoms of heavy elements, we have very recently proposed a relativistic SSMRPT method (targeting one specific state of interest) involving the improved virtual orbital-complete active space configuration interaction (IVO-CASCI) reference wavefunction for the four-component (4c) relativistic Hamiltonian (spinors) under the name of 4c-IVOSSMRPT [25]. It is a safe bet to say that this is the first development of the relativistic SSMRPT method with the 4c spinors. The 4c-IVOSSMRPT can be reckoned as a relativistic generalization of the IVO-SSMRPT method[50, 51]. 4c-based SSMRPT can be viewed as a quasilinearized version of our recently suggested 4c-SSMRCC (state specific multireference coupled cluster theory ) theory[52] as that of its non-relativistic counter part. We have recently developed a flexible and useful form of the SSMRPT method[53–56] by invoking IVO-CASCI as a reference function instead of a plain CASSCF one (termed as IVO-SSMRPT)[50, 51]. Based on our previous works [25, 50, 51, 57–59] we may argue that IVO-MRPT schemes are reliable and robust to provide acceptable accuracy with a low computational expense for describing the electronic structure of molecular systems perturbed by intruders and/or quasidegeneracies. The 4c-IVOSSMRPT method is endowed with various desirable formal properties such as (i) rigorous size-extensivity and size-consistency (with localized orbitals) (ii) applicable to both ground and excited states, (iii) intruder free in a 4


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natural way, irrespective of whether the target state is ground or excited, as long as the focussed state is estranged from the virtual functions, (iv) freedom of convenient partitionings of the Hamiltonian, (v) flexibility, allowing the use of relaxed and unrelaxed descriptions of the reference space coefficients, (v) manageable cost/accuracy ratio, and (vi) multipartitioning of Hamiltonian allowing different partitioning for different states. These features can make the 4c-IVOSSMRPT method a reliable contender for efficient manipulation of simultaneous consideration of relativistic and correlation effects towards providing a reliable computation of the PES. Although 4c-based MCQDPT, MRMP and CASPT2 methods are very useful, they are not strictly size-extensive [60, 61] and are not immune to intruder state problems. The existence of intruder states[62, 63] in these methods can be tackled by using shift techniques[64–66] which can significantly affect the computed PESs, spectroscopic constants and the order of electronic states [67, 68]. Another defect common to the CASPT2 and MRMP2 methods is that they do not allow the coefficients of the reference configurations to relax in the correlation treatment. Unrelaxed descriptions of the model space functions can be conspicuous to describe the mixed electronic states, the avoided crossings and overall PESs where the relative contributions of the different reference configurations tend to vary strongly with variations in the molecular geometry. Note that the stability of the MRPT equations and their successful description of electronic states is sensitive to the choice of orbitals, reference space, zero-order Hamiltonian, and of so called perturbers (zero-order states allowed to interact with reference state wavefunction in PT treatment) used in the calculations[69, 70]. Although a reorganization of CASSCF within the 4c relativistic framework is possible, it needs complicated calculations [71–73]. Moreover, a nonlinear nature of the CASSCF working equations can yield multiple solutions, convergence difficulties in conjunction with the appearance of unwanted kinks in the computed energy surfaces, and symmetry breaking of the wavefunction. The convergence problem that may arise in the orbital optimization of the CASSCF wavefunction can be erased effectively in the IVO-CASCI method by avoiding the orbital optimization step altogether and thereby reducing the computational cost considerably[74, 75]. The IVO-CASCI method is simpler, efficient and more stable than the CASSCF, inspite of inheriting all the redeeming characters of the CASSCF wavefunction. Thus, IVO-SSMRPT represents a good approximation to the parent CASSCF-SSMRPT method[50, 51]. Here, the ground state of coinage metal dimers (Ag2 , Cu2 , and Au2 ) and I2 5



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molecule are selected as test molecules as an appropriate account of the entire dissociation process requires accurate and pragmatic description of the combined interplay of various physical effects: (i) electron correlation, both static and dynamic, and (ii) relativistic contributions, both scalar and spin-orbit. We have compared the spectrohelioscopic constants extracted from the 4c-IVO-SSMRPT PESs with the available database results in an attempt to decide the quality of our estimates. We enunciate the results obtained from a plethora of theoretical approaches, implementing a host of different basis sets, for the purpose of a complete reference, which indicate that a comparison between the experimental and the estimated results depends critically on the level of correlation and relativistic effects embedded in the approach used.

II.

THEORY

The formulation of the 4c-IVOSSMRPT method is almost the same as the nonrelativistic IVO-SSMRPT [50, 51]. However, for the completeness of our discussion, we commence with a brief review of the salient features of the theories used here. A more detailed description may be found in our recent review [70]. To treat relativistic effects theoretically within the framework of Born-Oppenheimer approximation, the 4c-Dirac-Coulomb (DC) Hamiltonian (DCH) used in our present work can be written as: H=

X

hD (i) +

i

1 X Î4 · Î4 2 i6=j rij

(1)

where Î4 describes a 4×4 unit matrix. The one-electron Dirac Hamiltonian customarily can be written as hD = βˆ0 mc2 + c(ˆ α · p) + Vˆ

(2)

where, m, c, p, and Vˆ represent the rest mass of the electron, the speed of light, momentum, and the external potential (characterizes the interaction with the field of nuclei), respectively. The electron–electron repulsion is considered as the Coulomb interaction and the interactions between electron and positron can be neglected by invoking “no pair approximation”. α ˆ and βˆ (βˆ0 = βˆ − Î4 ) appearing in the above equation are Pauli matrices in the usual relativistic theory. As the Fermi distribution is not representable in a closed form, one assumes the 6


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nuclear charge distribution for a finite nucleus to be Gaussian in nature. While purely Coulombic terms shape the inter-electronic interactions, the no-pair approximation is in force that gives one the liberty to neglect the electron-positron interaction terms. In an attempt to circumvent the issue of variational collapse [76] during the solution of the Dirac equation in a kinetically balanced finite basis, one imposes intuitively motivated constraints [77] on the small component of the basis functions. We now put forth a concise outline of the scheme for generating the IVOs [74, 75]. The procedure [having a close kinship to the approach proposed long ago by Huzinaga[78]] for constructing relativistic IVOs is the same as that of its nonrelativistic counterparts except for the fact that the dimension of the matrix to be diagonalized in the relativistic case is double, since we have to deal with relativistic 4c-spinors. To reduce the numerical instability of the working equations of the IVOCASCI equation, occupied and improved virtual orbitals, which are proximate to the Fermi level, are usually used to construct the CAS. To generate orbitals via IVOCASCI scheme, we start with (i) a single ground state HF computation, followed by (ii) the diagonalization of a single modified Fock operator, and finally (iii) a CASCI is performed. The IVO-CASCI method needs no iterations in either steps (i) and (ii), and several various CASs (including ones for the ions) may be checked without repeating the first two steps. The latter is crucial as the selection of an appropriate CAS is, perhaps, the most intricate part of a MR-based strategy. The IVO-CASCI offers significant savings in the computer effort without hampering the accuracy when compared with the analogous CASSCF calculations. Although the primary aspect of making the IVOs is the same for both restricted and unrestricted HF orbitals, here, we address the restricted HF case only, which is used herein. In IVO-CASCI method, the state energies and the corresponding wave functions can be obtained by diagonalizing the Hamiltonian matrix in the CI space to generate the desired roots: H|ΨI i = E|ΨI i Here, |ΨI i =

P

i,u

Ciu |Φui i+

P

i≥j,u≥v

(3)

Cijuv |Φuv ij i+· · · The symbols i, j, k, · · · } and u, v, w, · · · }

describe the occupied and unoccupied HF molecular orbitals (MOs), respectively. When the ground state of the system is a closed shell, one can employ the HF molecular orbitals for the corresponding wave function: Φ0 = A[φ1 φ¯1 φ2 φ¯2 · · · φn φ¯n ] where A is an appropriate antisymmetrizer. In the IVO-CASCI method, all the MOs are generated via 7



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diagonalisation of a modified Fock operator constructed from the field of (N - 1) electrons (termed as V(N −1) potential) rather than the normal HF potential (in order to optimize the CASCI predictions for low lying electronic states and thereby to minimize the higher order perturbative corrections): 1

Flm = hφl |H1 +

occ X

(2Jk − Kk )|φm i = δlm l ,

(4)

k=1

Here, l and m refer to occupied or unoccupied HF orbitals and l is the HF orbital energy. Note that H1 describes the one-electron portion of the Hamiltonian, and Jk and Kk are Coulomb and exchange operators, respectively, for the occupied HF orbitals (φk ). We obtain a new set {χ} of MOs, as a result of an excited state HF computation, that produce the lowest possible energies for the low lying singly excited Ψα→µ state, Ψ(α → µ) = A[χ1 χ¯1 χ2 χ¯2 · · · (χα χ¯µ ± χµ χ¯α ) · · · χn χ¯n ],

(5)

involving the excitation of an electron from the orbital χα to χµ , where the + and - signs correspond to triplet and singlet states, respectively. A linear combination of the ground state MOs {φi , φu } can be used to express the new MOs {χα } and {χµ }. Another possibility is to restrict the orbitals further such that the {χα } are linear combinations of only the occupied ground state MOs {φα } and the {χµ } are expanded only in terms of the unoccupied {φu }, |χα i =

occ X

aαi |φi i;

|χµ i =

unocc X

cµu |φu i

(6)

u=1

i=1

which leads to a new set of orbitals {χα , χµ } that not only leaves the ground state wave function unchanged but also ensures the orthogonality and applicability of Brillouin’s theorem between the HF ground state and the Ψα→µ excited states. Such a choice is, in addition, benefitted by having a common set of MOs for the ground and excited states, a choice that simplifies the evaluation of oscillator strengths, etc. A computationally expensive reoptimization of the occupied orbitals is avoided by setting {χα } ≡ {φα }, i.e., by choosing aαj = δαj , thereby enormously simplifying the procedure for generating the IVOs. This effectively reduces the coupled equations determining the coefficients aαj and cµν to a single eigenvalue equation of the form F 0 C = CΓ, where the one-electron operator F 0 is given by, 0 Fvw =1 Fvw + Aαvw ,

8


(7)

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1

F is the ground state Fock operator, with the additional term Aαvw taking into account the

excitation of an electron out of an orbital φα , Aαvw = hχv | − Jα + Kα ± Kα |χw i.

(8)

The minus sign in Eq. (12) applies when 3 Ψα→µ is a triplet state, while the plus sign is for the singlet 1 Ψα→µ state [78]. The corresponding transition energy is 1,3

∆E(α → µ) = E0 + γµ −1 Fαα ,

(9)

where E0 is the HF ground state energy and γµ is the eigenvalue of F 0 C = CΓ for the µth orbital. It is important to note that when the highest occupied MOs exhibit doubly degeneracy, the construction of F 0 is then modified by employing Huzinaga’s scheme[78] in order to retain molecular symmetry of {χl }. If φα and φβ are the highest occupied degenerate HF MOs, then the matrix element Aαvw in Eq. (12) is replaced for these degenerate systems byAαβ vw , where 1 1 Aαβ vw = hχv | − Jα + Kα ± Kα |χw i + hχv | − Jβ + Kβ ± Kβ |χw i 2 2

(10)

The generation of the virtual orbitals is an eigenvalue problem with dimension equal N − No where N and No are the total number of basis functions and the number of occupied orbitals in the reference state SCF approximation, respectively. The effort involved in this step scales with the number of two-electron integrals No Nu2 required for the formation of the Fock operator, where Nu stands for the number of unoccupied orbitals. It is to be noted that the IVO energies yield a leading Koopmans-like approximation to the excitation energies of singly excited states which is computationally more effective than the CIS scheme, whose computational cost increases with the size of the No Nu -dimension CI matrix. The computational time generally scales approximately as the number of two-electron integrals [(No +Nv )4 −No4 ] , 8

where Nv is the number of IVOs in the CAS. This process involves the same

computational cost as in a single CASSCF integral transformation. However, one envisages the benefits of the IVO-CASCI strategy when one realises that in the IVO-CASCI, all the states get evaluated in a single shot from just one IVO-CASCI computation, whereas the traditional CASSCF needs several partial transformations (or direct integral generation) in

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its iteration scheme and a separate CASSCF computation is often performed for each individual electronic state (or symmetry). Thus, the IVO-CASCI is numerically more tractable and easy to perform as compared to the traditional CASSCF. We now briefly review SSMRPT method. Although many widely used MRPT methods [69, 70] have been proposed from physical considerations, however, the SSMRPT [53–56] method has emerged from the quasilinearized version of the SSMRCC of Mukherjee and coworkers [79, 80] preserving all the essential tenets of the parent. The only judicious innovation required is selection of a physically motivated zero order description of the Hamiltonian (H0 )[81] so that the perturbative approximation remains faithful (vide post). At this end, we must reiterate that the 4c-IVOSSMRPT method can be inspected as a relativistic generalization of a Hilbert space MRPT theory for 4c relativistic Hamiltonian within the framework of the state specific parametrization of multiexponential Jeziorski- Monkhorst (JM) ansatz [82] . In the SSMRPT formalism, the exact wavefunction (ψ) is written as: X exp(Tµ )|φµ icµ (11) |ψi = µ

where each model space function, φµ yields the various virtual functions χl via the action of a reference-specific cluster operator Tµ assuming the existence of a CAS wavefunction. In fact, Tµ s describe the dynamic correlation effects specific for a state described by a given reference (model) space. In SSMRPT method, the energy (E) can be evaluated as an eigenvalue of the (2) e µν effective Hamiltonian matrix H and the expansion coefficients (cµ ) defined in Eq.(11)can be computed from the corresponding eigenvector via the following equation: X (2) e µν H cν = E (2) cµ

(12)

ν

with the principal constraint that only one root of the eigensystem in Eq.(12) has physical meaning. Here H represents the electronic Hamiltonian operator. It is important to note that the main hurdle faced by state specific parametrization of JM-based MR effective Hamiltonian formalism is the mismatching between the number of unknowns, cluster amplitudes (wavefunction parameters) and the number of conditions that can be derived from projections of the JM ansatz-based Schrödinger equation. By invoking physically motivated sufficiency conditions to remove redundant wave function parameters[83], one can obtain the following unique solution for the cluster finding operators (hold for each µ and l): P l(1) Hlµ + ν6ν =µ tµ (ν)Hµν (c0ν /c0µ ) l(1) (13) tµ (µ) = (ECAS − Hµµ ) + (E0, µµ − E0, ll ) 10


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l(1)

l(1)

l(1)

l(1)

with hχµl |Tµ |φµ i = tµ (µ), hχµl |Tν |φµ i = tµ (ν), hφµ |H|φν i = Hµν , and hχµl |H|φµ i = Hlµ . In the above equation, ECAS describes the energy of the CAS state corresponding to the coefficients c0ν . Here, E0 is the expectation value of H0 with respect to a specific configuration 0 or hχl |H0 |χl i = Hll0 ]. We take H0 to be diagonal. However, the [i.e. hφµ |H0 |φµ i = Hµµ

actual expression of H0 depends on the type of partitioning. The denominator of Eq. (13) is manifestly immune to intruders as long as the energy, ECAS , is well-separated from the energies of the virtual functions. In our present work (based on the Rayleigh–Schrödinger (RS) perturbation scheme), the coefficients in Eq. (13) are kept frozen at the IVO-CASCI P values c0µ obtained from diagonalization of ν Hµν c0ν = ECAS c0µ . Thus, the RS-based energy allows the relaxation of the reference coefficients via Eq. (12) where (2) e µν H = Hµν +

X

Hµl tl(1) ν (ν).

(14)

l

Thus, substituting the first order amplitudes [obtained via Eq. (13)] into Eq. (14), the energy of the target state is obtained as an eigenvalue. Eqs. (13) and (12) are the principal working equations for the present work. The second-order energy of the IVO-SSMRPT contains higher order contributions owing to the relaxation of the reference space coefficients. Therefore, the final coefficients perceive the impact of both nondynamic and dynamic correlations. The GVVPT2 method of Hoffmann et al. [32, 33] also has this feature [which adopts the “diagonalize-perturb-diagonalize” approach]. At this stage, one may argue that the diagonalization of the effective Hamiltonian in a very small model space will not provide sufficient revisions of the coefficients of the reference state for the RS-based SSMRPT method. In this context, the recent works of iterative CI method [84] bescribe to which allow revision of the primary, external, and secondary states iteratively. It is important to note that the presence of (c0ν /c0µ ) in the coupling term of Eq.(13) may invite numerical instability during the computation of the cluster amplitudes. As the contribution from a configuration state function of small coefficients is usually negligible, setting all corresponding cluster amplitudes (controled by a threshold value) to zero is the simplest scheme to avoid such difficulty. However such strategy may yield nonphysical kinks on the PES provided by the method, as different sets of tµ s may be neglected at different points on the energy surface. Tikhonov regularization[85] is an effective way to get rid of this problem. It is indeed a happy feeling for us that the systems we have dealt with, so far, by the 4c-IVOSSMRPT method have performed very well, and have been able to avoid any numerical instability of the amplitude 11



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determining equations, even when we have opted to incorporate all the coefficients. Note that special care of small values of c0µ are not needed for the unrelaxed description of the RS-based SSMRPT (belongs to the “diagonalize-perturb” philosophy), akin to CASPT2 and MRMP methods. At this stage, it is instructive to state that the unrelaxed IVOSSMRPT has close resemblance with the CASCI-MRMP method of Hirao and coworkers [86] as both methods are based on MP partitioning and have configuration coefficients fixed at CASCI values. The extra computational labor entailed in the IVOSSMRPT method [due to a small iterative step implied in Eq (13)] as compared to Hirao’s MRMP one is not noticeably large which leads to rigorous size-extensivity of the method. It should be mentioned here that various target specific MRPT methods differ from each other in the choice of unperturbed Hamiltonian, nature of the orbitals and scheme of perturbers which shape also numerical demands of the methods. For the calculations with the CAS of large dimension, the unrelaxed version is computationally cost effective with respect to the relaxed one as diagonalization of large matrix is not required in the former case. It would be interesting to explore this scheme in the near future. In the present paper, we have considered only the relaxed description of the IVOSSMRPT method [can be described as diagonalize-perturb-diagonalize scheme]. It is relevant to note that the relaxed state specific technique is usually a valuable tool to compute the surfaces often plagued by (weakly) avoided crossings. As that of nonrelativistic works [50, 51, 53–56], here, we employ a reference-related (vacuum-dependent) Fock-like operator within the framework of multipartitioning scheme of Zaitsevskii-Malrieu [81]: " fµ =

X ij

# X ju 1 uj j µ ij Ei fcore + Viu − Viu Duu 2 u

(15)

where u stands both for a doubly and a singly occupied active orbital in Φµ , and Dµ s are the one-particle (first-order) density matrix elements in the CSF space labeled by the active orbitals. The operators in curly brackets, {E} denote the normal ordering with respect to the to φ0µ taken as the vacuum, φ0µ being the largest closed-shell component of φµ . This leads to a spin-adapted version of IVO-SSMRPT, where the cluster operators are described in terms of the spin-free unitary generators. Here, i; j; · · · refer to spatial orbitals for the ii nonrelativistic case and fcore is the diagonal element of the core Fock operator. Viuiu (andViuui )

are two-electron molecular integrals. The one- and two-electron integrals appearing in the above equation are computed using the IVOs. As our H0 is always diagonal for MP scheme, 12


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the zeroth-order Hamiltonian operator (monoelectronic in nature) is: H0µ =

P

i

fµii {Eii }.

Eq.(15) is the widely used expression. See Refs.53, 54, for a detailed discussion about the significance of the symbols appeared in the above equations. As that of other perturbation methods such as MRMP, CASPT2, and MCQDPT, the non-relativistic SSMRPT or 4c-IVOSSMRPT method is not variational and thus, the felicitous selection of the zerothorder Hamiltonian is very crucial to the accurate and successful description of electronic states considered. The present study also reasserts the cogency of the zeroth-order multipartitioning MP Hamiltonian for the 4c relativistic Hamiltonian. Note that both MRMP and CASPT2 methods use one-particle Fock-like zeroth-order Hamiltonian. Note that in the relativistic SSMRPT method, the required matrix elements are evaluated in the spinor basis which makes its computational cost larger than in the nonrelativistic case. In contrast to state-universal MR-based PT, only those sets of cluster operator amplitudes sharing a common set of orbitals are coupled together in the working equations of the SSMRPT formulation within the framework of the choice of the unperturbed Hamiltonian that we have opted to work with. This is an appealing feature of the method from the computational point of view as it leads to the flexibility that each type of excitation involving a specific set of orbitals can be taken into account, and the cluster amplitudes for different µs with the same set of orbitals can be computed employing Eq. (13). Once this is done, the (2) e µν H is updated to incorporate the contributions of all these amplitudes. This is followed by starting the same with a different set of excitation. This eludes the need to store any of the T amplitudes, and all of them get evaluated on the fly. It is one of the major computational benefits compared to the parent SSMRCC method[79, 80] in handling the large systems as SSMRCC stores all amplitudes to the disk that can be circumvented in the SSMRPT scheme considered here. It should be mentioned that our newly developed 4c-IVOSSMRPT method has the ability to be a budding MRPT method for describing the real chemical situations/processes, in which (i) the HF determinant is not a good zero-order wave function, and (ii) both the dynamical correlation as well as relativistic effects are very important. Documentation of the numerical results in the next section endorse this demand. In our method, a simplified treatment of the dynamic and nondynamic correlations, in cooccurrence with size-extensive and intruder free nature combined together to form a good computational tool that is particularly useful for the bond breaking processes and able to provide good quality energy surfaces for the whole range of the internuclear separations. 13



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The limitations of the usefulness of the method is the same as in the nonrelativistic case (IVO-SSMRPT) [50, 51]. Like other MRPT methods, our 4c-IVOSSMRPT is not orbital invariant with respect to certain orbital rotations which leave the energy of the reference wave function unchanged. Note that the lack of orbital invariance is the Achilles heel for JM ansatz based methods. It is important to note that whether or not a method is orbital invariant depends not only on the ansatz used but also the resulting amplitude finding conditions. It should be noted that computational cost of the IVO-SSMRPT, like other CAS-based MRPT methods including MRMP CASPT2 and their multi-root versions grows exponentially with the dimension of the model/active space which inhibits the wide practical implementation of the approaches to chemically useful molecular systems. This is still a challenging problem in quantum chemistry [87–90].

III.

NUMERICAL RESULTS

Here, we have confined ourselves to CAS(2,2) described by the configurations:Φ1 = [· · · ]nσg2 and Φ2 = [· · · ]nσu2 in which the two active orbitals σg and σu belong to two different symmetries. This smallest active space not only allows for a qualitatively correct treatment of the single-bond breaking but also provides reference function characterizing the ground state which is size consistent in nature. Using the expressions of 4c-IVOSSMRPT given in the previous section, the present computations have been performed using our in-house code which we have interfaced with the DIRAC package[91]. In the present work, the orbitals with single particle energy ≥ 100 a.u. are not incorporated in the correlated computations as the deep lying core- as well as high lying virtual-spinors give insignificant contributions when electronic structure properties of spectroscopic interest are calculated in this basis. We have already illustrated that optimization of the virtual orbitals using 4c-IVOCASCI method is very effective to get a qualitatively correct description of nondegenerate and nearly degenerate region including the dissociation one[25]. In contrast, the 4c-CASCI method yields an undesirable hump in the far equilibrium region for Li2 , Na2 and so on. We have performed the SSMRPT calculations using (i) Dyall.v2z and Dyall.v3z basis sets [92] for Au2 and Ag2 , (ii) Dyall.pVDZ.unc and Dyall.pVTZ.unc basis sets [92] for I2 , as well as (iii) cc-pVDZ and cc-pVTZ basis sets of Dunning and co-workers [93] for Cu2 . The selective spectroscopic constants such as equilibrium bond length [Re , in ˚ A], and the vibrational 14


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frequency [ω2 in cm−1 ] has been extracted from a fourth to sixth order Dunham fit [94] polynomial of the computed PESs, which gave consistent results. The dissociation energy [De , in kcal/mol] have been estimated as the difference of the energy at an asymptotically large bond length and the fitted energy minimum. Here, it should be noted that the aim of the present work is to illustrate the traits and feasibility of the method vis-à-vis other standard and established methods in routine use rather than provide definitive potential functions or spectroscopic constants, and basis sets of moderate size have therefore been employed. In order to get results with very high accuracy, the basis should be extended by the functions of higher angular momentum.

A.

Coinage Metal Dimers: Cu2 , Ag2 , and Au2

The first application of the 4c-IVOSSMRPT method considered here is the dissociation of Au2 and its lighter homologs Cu2 and Ag2 . These systems have also been investigated using different methods with various level of sophistication [95–113]. They are electronically the simplest transition-metal dimers, with no open d sub-shells, and play a very useful role for investigations of transition-metal clusters which are very pertinent in catalysis and thin-film growth. When moving from Cu to Au, the relativistic stabilization of s and p orbitals and destabilization of d and f orbitals can be characterized by the estimation of the properties of spectroscopic interest using relativistic and nonrelativistic methods. The numerical analysis reported in the literature [104, 105, 107] indicate that the effects due to electron correlation dominate over the relativistic effects for the spectroscopic constants of Cu2 , whereas for Au2 , the effects due to relativity are more prominent than the correlation ones. For Ag2 , effects of electron correlation and relativity are almost competitive. From the previous works [104, 105, 107, 111] for Cu2 , it is found that the spectroscopic constants obtained by involving the calculations with the relativistic and non-relativistic effects provide almost similar results. Note that the recent theoretical investigations using different correlated-relativistic schemes yield a relativistic bond contraction between 0.18 and 0.30 ˚ A for Au2 [102]. The relativistic and non-relativistic cp-CCSD(T)calculations [107] provide a variation in equilibrium bond lengths (˚ A) which are 0.03 for Cu2 , 0.11 for Ag2 , and 0.29 for Au2 . Concomitant to the simultaneous interplay of three physical effects such as (i) the relativistic contraction of 6s orbitals, (ii) expansion of the 5d orbitals, and (iii) the contribution of the (n − 1)d orbitals 15



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to the bonding, give a bond length of Au2 between those of Cu2 and Ag2 [114, 115]. The convulsion of the optimized bond length due to relativistic effects is contemplated to be greater in the gold dimer, than in the silver one. Literature results [111] based on the Mulliken population analysis on the basis of atomic orbitals illustrate that due to the small energy gap between d and s atomic orbitals in the case of Cu2 and Au2 compared to Ag2 , the d-electrons have more significant contribution to HOMO for Cu2 and Au2 than that of Ag2 . As a result of this, for the ground state of Ag2 , the σu orbitals (LUMO) and σg orbital (HOMO) are exclusively possessed of pure 4d and 5s atomic orbitals, respectively whereas for Au2 and Cu2 , these MOs emerge from the mixing of the corresponding d and s orbitals. The “aurophilic effect” mainly emerges from the counter-intuitive electron correlation effects of the closed-shell components which has close similarity with van der Waals interactions with unusual strong bond-energy due to the strong relativistic effects [1, 2]. In Figures 1, 2, and 3, we present the PESs for Cu2 , Ag2 , and Au2 , respectively. In the present paper the state energies relative to their energy at the equilibrium geometry, that is, E(R)- E(Re ), are plotted against the bond length. The figures attest that the transition of the performance of the IVO-SSMRPT method from the nondegenerate zone to degenerate ones over the whole PES including the dissociation limit is quite smooth and uniform. Therefore, our 4c-IVOSSMRPT can handle mutual modulation of both nondynamical and dynamical correlations under the influence of the relativistic effects quite well. The differences of the topology of 4c-IVOCASCI and 4c-IVOSSMRPT plots clearly demonstrate the impact of the dynamical correlation effect especially for Cu2 system. The 4c-IVOCASCI calculation as the zeroth-order perturbation approximation yields at least topologically correct description of the surfaces, although the 4c-IVOCASCI scheme provides a quantitatively good description for the whole region. This fact is reflected in the deviations of 4c-IVOCASCI and 4c-IVOSSMRPT spectroscopic constants from the reference data. Prediction of spectroscopic constants for these dimers is a difficult task owing to the shallowness of their potential surfaces. In Tables (I), (II), and (III), we have assembled the spectroscopic constants which have extracted from the computed PESs for Au2 , Ag2 , and Cu2 , respectively using IVOCASCI and 4c-IVOSSMRPT methods. For comparison, the results of other highly correlated relativistic calculations such as the Fock-space MR coupled cluster (FS-MRCC), CASPT2 and the GMCQDPT are listed. In addition, a comparison with the experimental data is provided. Our 16


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4c-IVOSSMRPT results showed a better accuracy than the DFT results[100, 106, 108, 109]. From the tables, 4c-IVOSSMRPT methods are seen to be suitable in describing Re , De , and ωe . Note that our 4c-IVOSSMRPT calculation provided better estimates than the relativistic CCSD method as the former performs better for larger bond distances than the latter one which can be attributed to the fact that IVOSSMRPT method can treat the varying degrees of neardegenracies (i.e MR character) picked up by the system upon dissociation. Although in some cases, Re and ωe can be estimated with reasonable accuracy at the static correlation level with our 4c-IVOCASCI calculations, to get these molecular constants with satisfactory accuracy, proper description of dynamical correlation is needed as is evident from the values of the 4c-IVOSSMRPT calculations. The present study demonstrates that the nonrelativistic computations yield inferior values and exhibit large errors, especially for the Ag and Au dimers as also is evident from the previous works[103, 107]. Note that the correlation associated with the underlying 5d electrons mainly provides the major correction for the too long SCF/CASCI bond distance [107, 110]. Incorporation of the semi-core 5p electron also provides some improvement. The spectroscopic constants provided by our 4c-IVOSSMRPT showed fairly good agreement with the experimental estimates for Au2 system. The deviations from the experiment values at the levels of (i) 4c-IVOSSMRPT/Dyall.v2z, (ii) 4c-IVOSSMRPT/Dyall.v3z, (iii) AE-DKH3-CASPT2 [113], (iv) DKH-cp-CCSD(T) [103], and (v) DK3-CCSD(T) [DK3-cpCCSD(T)] [107] calculations are: (i) ∆ Re = -0.039 ˚ A, ∆ωe = -12 cm−1 , and ∆De = 3.12 kcal/mol, (ii) ∆ Re = -0.030 ˚ A, ∆ωe = -8 cm−1 , and ∆De = 4.62 kcal/mol, (iii) ∆ Re = 0.001 ˚ A, ∆ωe = -3 cm−1 , and ∆De = -1.39 kcal/mol, (iv) ∆ Re = -0.016 ˚ A, ∆ωe = 4 cm−1 , and ∆De = 3.92 kcal/mol, and (v) ∆ Re = -0.012 [-0.022] ˚ A, ∆ωe = -1.2 [2.4] cm−1 , and ∆De = 2.54 [2.77] kcal/mol, respectively. For Au2 , the computed values via 4c-IVOSSMRPT method displayed proximity with the AE-DKH3-CASPT2 [113] estimates. The close correspondence of our estimates with the results of Peterson[104, 105] is reasonable. 4c-IVOSSMRPT values displayed nice accordance with DKH2-CCSD(T) [110] and DKH-ACPF [101] estimates. Overall, computations at the level of 4c-IVOSSMRPT demonstrate a tendency to overestimate the vibrational frequencies and underestimate the optimized bond-length and dissociation energies compared with the experimental values, whereas AE-DKH3-CASPT2 [113] calculations showed overestimation for spectroscopic constants. Note that as that of the work of Hirao and coworkers [107], we have also observed that the impact of both the 17



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correlation and relativistic effects in our IVO-SSMRPT based calculations lead to a decrease of the bond length of Au2 and the concomitant increase of the vibrational frequency and the dissociation energy. In addition, we illustrate that the nonrelativistic IVO-CASCI(2,2) Re (˚ A), De (kcal/mol), and ωe (cm−1 ) are 2.785, 25.1 and 120, respectively. The bond length is decreased by 0.130 ˚ Adue to the relativistic effects and 0.115 ˚ Adue to the effect of electron correlation. On the other hand, relativistic effect increases the IVO-CASCI dissociation energy (and harmonic frequency) by 4.2 kcal/mol (and 24 cm−1 ), whereas electron correlation effect reduces it to 3.6 kcal/mol (and 18 cm−1 ). For Au2 , the electron correlation effects are relatively less pronounced than the relativistic effects. The spectroscopic constants collected in Table (II) indicate that effects of electron correlation and relativity are competitive in the case of Ag2 . The nonrelativistic IVO-CASCI(2,2) gives Re =2.782 ˚ A, De = 22.9kcal/mol, and ωe = 99 cm−1 . Incorporation of relativistic effect reduces the mean-field bond length by 0.081 ˚ Awhile it is decreased by 0.129 ˚ Adue to the electron correlation effects. The relativity and electron correlation corrections lead to a amplification of the dissociation energy (and harmonic frequency) relative to NR-IVOCASCI by 3.5 (32 cm−1 ) and 2.2 (30 cm−1 ) kcal/mol, respectively. As that of the gold dimer, we also find a relativistic stabilization of Ag-Ag bond. The error of our 4c-IVOSSMRPT estimates with respect to the experimental values[116, 117] decreases with increase in the size of the basis sets as that of gold dimer. The deviations of Re , ωe , and De provided by 4c-IVOSSMRPT/Dyall.v3z scheme from the experimental values are -0.040 ˚ A, -5 cm−1 , and -4.5 kcal/mol, respectively. The error of 4c-IVOSSMRPT/Dyall.v2z estimates in Re is -0.056 ˚ A, for ωe the error is -8 cm−1 , and for De it is -5.7 kcal/mol illustrating the quality of the 4cIVOSSMRPT approach. The corresponding deviations for the benchmark DK3-CCSD(T) [DK3-cp-CCSD(T)] calculations are -0.02˚ A[-0.035], 0.4 [4.1] cm−1 , 0.84 [2.23] kcal/mol, respectively. As a whole, calculated 4c-IVOSSMRPT spectroscopic constants for the ground state silver dimer are in harmony with the corresponding experimental values and highly accurate relativistic methods as that of Au2 . As shown in Table (III), increasing the size of the basis set from cc-pVDZ to cc-pVTZ correct the three properties, Re , ωe , and De toward the experimental values for the ground state Cu2 . From the comparison, it might be inferred that for Cu2 correction of both relativistic and correlation effects brings the IVO-CASCI values closer to the experimentally estimated values by reducing bond length along with the increase in the dissociation energy 18


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indicating that both the effects strengthen the Cu-Cu bond (relativistic stabilization of the bond) akin to the other coinage metal dimers. At the nonrelativistic IVO-CASCI(2,2)/ccpVTZ level we observe an Re of 2.410 ˚ Athat decreases to 2.314 ˚ A(contraction of 0.096 ˚ A) upon addition of 4c-relativistic effects and to 2.239 ˚ A(contraction of 0.171 ˚ A) upon addition of electron correlation effects. This observation is consistent with the finding of Hirao and co-workers [107] who report a contraction of 0.116 and 0.158 ˚ Aemerge form the correction of relativistic and electron correlation effects, respectively. Note that the electron correlation effect has more impact on the computed properties than the relativistic one in the case of Cu2 . The extent of the impact dependent on the basis set used. Our 4cIVOSSMRPT/cc-pVTZ spectroscopic constants tally well with the experimental results, but the remaining errors -0.018 ˚ Ain Re , 5.37 cm−1 in ωe , and -1.11 kcal/mol in De that could mainly be due to absence of higher-order correlation effects that are not recovered by the second-order MRPT method. The errors due to the F12∗ -CCSD(T)/aug-cc-pVQZ-PP calculations [118] from experimental ones are 0.005 ˚ Afor equilibrium distance and 0.4 cm−1 for vibrational frequency. The DK3-CCSD(T) level of computations [107] provide errors as ∆ Re =-0.002˚ A, ∆ωe = -5.5 cm−1 , and ∆De = 2.08 kcal/mol. The deviation from experiment at the CCSD(T)/CBS(aQ5)/cc-pVXZ-PP [104, 105] level is 0.00 ˚ A, 2.62 kcal/mol and 0.4 cm−1 , respectively. The corresponding errors for DK3-CC CCSD(T)/aug-cc-pV[Q5]Z-PP [118] are 0.004 ˚ A, 1.16 kcal/mol and -2.1 cm−1 , respectively. Therefore, we come to the conclusion that the 4c-IVOSSMRPT method shows promise in its numerical performance. The above analysis undoubtedly suggest that the highly relativistic system considered here Au2 is as well characterized as the weakly relativistic system Cu2 by our 4cIVOSSMRPT method. For the coinage metal dimers addressed here, upon adding the relativistic correction via 4c-spinors at the IVO-SSMRPT level of calculations, we find a decrement of the bond length (relativistic bond stabilization) which is consistent with a increase of frequency and the dissociation energy[103, 107]. The error of our estimates with respect to the database values can be attributed to the sources, such as the basis set truncation errors, the lack of extensive treatment of the dynamical correlation effect, and so on.

19



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B.

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Iodine molecule, I2

The ground state I2 is well characterized experimentally and has already been chosen in many various studies [15, 119–131] that can be used to gauge our estimates. Figure 4 shows the ground state energy [relative to the energy at the equilibrium geometry] for I2 . The figure in the inset describes the PES computed via 4c-IVOCASCI level of calculation with Dyall.pVTZ basis set. As for the coinage metal dimers, here also we have obtained consistently correct topological descriptions for I2 along the entire range of I-I bond coordinates for 4c-IVO-based calculations. The IVO-based calculations do not exhibit any unphysical hump in the entire range of the computed dissociation PES. From the figure, it is evident that the topology of the 4c-IVOSSMRPT PESs with different basis are very similar in shape at both large and small bond lengths as they closely obey the surfaces provided by recent benchmark calculation via relativistic GMC-QDPT method in conjunction with GCS(10,24) [Figure 2 of Ref. 15]. This is quite encouraging since our 4c-IVOSSMRPT(2,2) requires only a fraction of the computational requirements of GMC-QDPT(10,24). We observed that the influence of the size of the basis sets and the dynamical correlation effect on the topology of the calculated surface is substantial which is evident from the derived principal spectroscopic constants. The results of the computed spectroscopic constants for the ground state I2 are given in Table (IV). It is found that the 4c-IVOSSMRPT values move towards the experimental results with the enlargement of the basis set. It can be stressed that switching over from the Dyall.pVDZ to Dyall.pVTZ, leads to a shortening of bond length by 0.079 ˚ A, and an enhancement in the frequency as well as the dissociation energy by 10 cm−1 and 9.4 kcal/mol, respectively at the level 4c-IVOSSMRPT. Teichteil and Pelissier[121] observed that the addition of f and g functions leads to a decrease in the equilibrium bond distance by 0.06 ˚ Aand 0.08 ˚ A, respectively. From our numbers, it is found that the incorporation of the relativistic effects leads to weakening of the bond as is evident from the enhancement of the bond length and lowering of the dissociation energies in comparison to the values obtained from the calculations at the corresponding nonrelativistic level. This fact is also gets corroborated with the previous [120, 121, 123, 124]. To illustrate, we point out that at the level of 4c-IVOSSMRPT/Dyall.pVTZ calculation, the incorporation of the effect of relativity culminates in a significant weakening of the I-I bond, causing a length-

20


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ening of the bond (by 0.018 ˚ A) and a consequent lowering of the bond-dissociation energy (by 3.3 kcal/mol) as well as harmonic frequency (by 13 cm−1 ) than those computed at the NR-IVOSSMRPT/Dyall.pVTZ level of theory. Note that the estimated relativistic effect on the equilibrium bond length (in ˚ A) varies from 0.001 at the level of RHF calculation to 0.006 at the level of CCSD(T) computation[120]. For the dissociation energies, the values change from -14.7 to -11.9 kcal/mol [120]. Table (IV) also shows the values provided by 4c-based GMC-CI and GCS-CI calculations significantly differ from reference values. Incorporation of the dynamic correlation effects by QDPT corrects the bond lengths, dissociation energies and harmonic frequencies of GMC/GCS towards the experimental values. This fact clearly illustrates the impact of the dynamical correlation effects. The 4c-IVOSSMRPT/Dyall.pVTZ dissociation energy is lower than the experimental data with a deviation of 2.6 kcal/mol. In contrast, the 4c-IVOSSMRPT equilibrium bond length and harmonic frequency with Dyall.pVTZ basis set are smaller than the corresponding experimental values by 0.029 ˚ Aand 5.5 cm−1 , respectively. The spectroscopic constants obtained with 4c-IVOSSMRPT(2,2)/Dyall.pVTZ and 4c-GMC-QDPT agree well with each other.

At the GMC-QDPT(10,12) level, the deviation from the

experimental estimates for De (kcal/mol), Re (˚ A), and ωe (cm−1 ) are 4.08, -0.03, and 9.5, respectively. Note that these deviations were reduced to 3.92 kcal/mol, 0.03 ˚ A, 9.5 cm−1 , respectively due to the GMC-QDPT(10,24) calculations indicating size of the active space is not significant at the level of GMC-PT calculation. The difference between the experimental and DC-FSMRCCSD (and DCG/FSMRCCSD+BSSE) [123] values of equilibrium bond length (˚ A), vibration frequency (cm−1 ) and dissociation energy (kcal/mol), are -0.024(-0.045) ˚ A, 0.50(5.50) cm−1 , and 2.00(5.46) kcal/mol, respectively. It should be noted that the FSMRCC method, an all-order MR correlation approach, is rigorously sizeextensive in nature for treating targets states with varying numbers of valence electrons through the use of a single wave operator and thus very effective to study the properties of spectroscopic interest. Note that the errors of 4c-IVOSSMRPT(2,2)/Dyall.pVTZ, 4c-GMC-QDPT(10,24), and DCG/FSCCSD(DCG/FSMRCCSD+BSSE) are very similar and rather close.

Closeness of our estimates with the values of relativistic-DMC[128],

MRSDCI+Q/ECP28MDF[130], CCSD(T)/pVTZ[131] REP-KRCCSD(T) [125, 126], DCCCSD(T)[120], all-electron (AE) DC-CCSD(T) [122], AE-CCSD(T)/cc-pV5Z-DK[124] and state-of-the-art AREP-FS-MRCCSD [125, 126] level of calculations is also noticeable. Note 21



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that dissociation energy value is not well reproduced by the two-component SOREP spinor based FSMRCC with single and double excitations (SOREP-FS-MRCCSD) scheme. However, the dissociation energy provided by DC-CCSD(T) calculations is slightly worse[120]. DC-CCSD(T) results illustrate that the contribution from triple excitations is very important to estimate experimental data [122, 125, 126]. The results of (i) composite CCSD(T)based relativistic calculations [considering the corrections of a core/valence effect, a Douglas–Kroll–Hess scalar relativistic effect, a first-order and second-order spin-orbit corrections] [127] and (ii) MCSCF/MRCISD+Q [using relativistic averaged pseudopotential and an effective one-electron spin-orbit operator with a valence basis of triple zeta quality plus 2s, 2p, and 2d polarization functions][121] show nice accordance with the 4c-IVOSSMRPT results. In contrast to the coinage metal dimers, for I2 molecule we find a relativistic destabilization of the bond which also corroborates with the previous observations [120, 123–131]. The bond weakening due to the relativistic effects, which also features not only in other halogen molecules but also in other p-block atoms such as thallium and bismuth, can be unraveled by considering the hybridization of orbitals involved in bond formation [120]. Therefore,the results for I2 provided by 4c-IVOSSMRPT are conclusive in nature as that of the spectroscopic constants of coinage metal dimers considered here. To get a better feeling of the performance of the method used here, in Tables (I), (II), (III), and (IV), we compare the spectroscopic constants obtained by the 4c-IVOSSMRPT method with those obtained from our recently published full-blown 4c-SSMRCCSD level of calculations[52] using the same basis and the same model space.

4c-IVOSSMRPT

spectroscopic constants are found to be comparable in accuracy to the corresponding 4cSSMRCCSD results despite the second-order perturbative nature of the former method. It should be stressed here that the SSMRPT approach is structurally and computationally much simpler than the corresponding SSMRCC method. This fact indicates the high quality and good balance of the 4c-SSMRPT calculations in describing the ground state total energies in the whole range of bond dissociation. For the sake of completeness, values provided by various DFT calculations [100, 106, 108, 109] and SR-based MP2/CC methods have also been collected in the supplementary material for a comparative analysis. In the supplementary material, we have summarized the performance of the 4c-IVOSSMRPT method vis-á-vis these methods. Present numerical analysis exhibit that the method used to treat both the electron cor22


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relations and relativistic effects is crucial to get values closer to the experimental estimates. One should emphasize at this point that apart from the computational labor of FSMRCC method, its conventional form is very vulnerable to the so-called intruder states. In contrast, our 4c-IVOSSMRPT method is exclusively built on a SS Hilbert space strategy and uses reference functions spanning a Hilbert space of a fixed number of valence electrons which makes it possible to avoid the annoying intruder state problem. At this point, we want to state that as the basis sets used here are not large enough, BSSEs (basis set superposition errors) are non-negligible and require to be accounted for. Varying the definition of the Fock-operator of the zeroth order Hamiltonian and the modification of the expression of the renormalization term appearing in the cluster finding equation through Tikhonov regularization (damping) approach[85] may be expected to affect the accuracy of the computed estimates. Although the present results are quite encouraging when viewed as an initial application, nevertheless, far-reaching conclusions regarding its general applicability cannot be drawn from these results. We would comment that our current code for the 4c-IVOSSMRPT method cannot (at present) handle MR-situations with CAS of arbitrary dimension. To fully illustrate the accuracy and utility of this method, multiple bond-breaking processes of heavy-element molecular systems have to be investigated because these situations require larger CAS (involving closed as well as open-shell reference functions), than the minimal CAS(2,2) one employed here, for a qualitatively correct description, and the results obtained with larger active spaces may be much more sensitive to the basic flaw of the method attributed to the small coefficients in the CAS wavefunction which appear in the denominator of the cluster finding equation. Moreover, the most rigorous test for the utility of the method would be in cases where the orbitals vary rapidly as a function of minor geometrical changes. Alas, we are not able to apply the method immediately for such situations. Thus, further development of our present code is quite desirable in the near future.

IV.

CONCLUSION

A relativistic realization of the widely used state-specific multireference perturbation theory (SSMRPT) method has been achieved through our recently proposed four-component (4c) improved virtual orbital based (IVO) SSMRPT approach (named as 4c-IVOSSMRPT). 23



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This method handles quasidegeneracy by employing several reference functions provided by the IVO complete active space configuration interaction (IVO-CASCI) calculations. The dynamical correlation is then incorporated by using the SSMRPT scheme. The method is used for the computation of PESs and spectroscopic constants for the ground states of Cu2 , Ag2 , Au2 and I2 as they are critically influenced by both relativity and electron correlation. Although 4c-IVOCASCI yields qualitatively correct unperturbed wavefunction, the lack of dynamical electron correlation effects generally averts quantitative accordance with reference values. Over the whole range of internuclear distances (both equilibrium and dissociation) considered here, the obtained 4c-IVOSSMRPT energy surface is both smooth and consistent indicating robustness of the method. For all the dimers, the 4c-IVOSSMRPT spectroscopic constants are in good agreement with the reference values from the literature. For the lighter diatomics Cu2 and Ag2 , it should be mentioned that 4c-IVOSSMRPT computation overestimates the dissociation energy, in comparison with the experimental results and those from established theoretical methods. Keeping in mind the differences in the level of the treatments of relativistic as well as correlation together with the quality of the basis sets, the consensus between computationally expensive relativistic CC and 4c-IVOSSMRPT methods for Cu2 , Ag2 , and Au2 is very encouraging. Due to the counter intuitive impact of relativistic contraction of 6s orbitals along with the expansion of the 5d orbitals, the 4c-IVOSSMRPT method provides maximum bond length for Ag2 [Re (Cu2 ) < Re (Au2 ) < Re (Ag2 )] as that of other previously reported highlevel ab initio relativistic methods. For the ground state of I2 , an elongation of the bond is accompanied by a reduction of the harmonic frequency and dissociation energy. For Cu2 , Ag2 , and Au2 , however, the relativistic effects decrease the bond length and increase the harmonic frequency and the dissociation energy. This observation is interesting as the correction due to the relativistic effect can affect the bond formation of the systems treated here with other chemical moiety. The present estimates depict the predictive ability of our relativistic 4c-IVOSSMRPT method in the high-Z regime. Finally, one should mention that the blend of accuracy and low computational cost offered by the present 4c-IVOSSMRPT method asseverates that the fusion of the three schemes, four component relativistic Hamiltonian, IVO-CASCI and SSMRPT will seemingly be the method of choice for simultaneous treatment of interplay of relativistic, dynamical and nondynamical correlation effects in an accurate manner and in our opinion, our 4c-IVOSSMRPT 24


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may become a computationally attractive tool for larger heavy-element systems.

V.

ACKNOWLEDGMENT

We gratefully acknowledge financial support from the DST (EMR/2015/000124), India. A.G. acknowledges the DST, India, for an Inspire Fellowship. S.S.R. thanks the UGC, India, for his research fellowship.

25



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Figure Captions Figure 1. The ground state potential energy profiles [in terms of E(R) - E(Re ) ] along the dissociation coordinate of Au2 as obtained using 4c-IVOSSMRPT(2,2) and 4cIVOCASCI(2,2) level of calculations indicating the effect of dynamical correlation on the topology of energy surfaces. All calculations have been performed using the Dyall.v2z basis set. Figure 2. The ground state potential energy profiles [in terms of E(R) - E(Re ) ] along the dissociation coordinate of Ag2 as obtained using 4c-IVOSSMRPT(2,2) and 4cIVOCASCI(2,2) level of calculations indicating the effect of dynamical correlation on the topology of energy surfaces. All calculations have been performed using the Dyall.v2z basis set. Figure 3. The ground state potential energy profiles [in terms of E(R) - E(Re ) ] along the dissociation coordinate of Cu2 as obtained using 4c-IVOSSMRPT(2,2) and 4cIVOCASCI(2,2) level of calculations indicating the effect of dynamical correlation on the topology of energy surfaces. All calculations have been performed using the cc-pVTZ basis set. Figure 4. The ground state potential energy profiles [in terms of E(R) - (-14230)] along the dissociation coordinate of I2 as obtained using 4c-IVOSSMRPT(2,2) and 4cIVOCASCI(2,2) level of calculations with Dyall.pVDZ and Dyall.pVTZ basis sets illustrating the effect of basis set size on the topology of energy surfaces. The figure in the inset represents the 4c-IVOCASCI(2,2) ground state of I2 to illustrate the effect of dynamical correlation on the topology of energy surface.

26


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Table I: Spectroscopic constants [equilibrium internuclear distances (Re in ˚ A), harmonic vibrational frequencies (ωe in cm−1 ), dissociation energies (De in kcal/mol)] of ground state Au2 molecule at various calculation levels along with experiment values. Reference

Re (˚ A) De (kcal/mol) ωe (cm−1 )

Method

Present Work NR-IVOCASCI/Dyall.v2z

Ref.52

2.785

25.1

120

NR-IVOSSMRPT/Dyall.v2z 2.670

28.7

138

4c-IVOCASCI/Dyall.v2z

2.655

29.3

144


2.620

32.4

151

4c-IVOSSMRPT/Dyall.v2z

2.511

51.3

203


2.502

49.8

199

4c-SSMRCCSD/Dyall.v3z

2.481

57.1

196


2.490

58.4

194

Ref.113

AE-DKH3-CASPT2

2.473

55.81

194

Ref.103

DKH-cp-CCSD(T)

2.488

50.50

187

DKH-cp-ACPF

2.496

45.89

DKH-MP2

2.418

DK3-CCSD(T)

2.484

51.89

192.2

DK3-cp-CCSD(T)

2.494

50.04

188.6

Ref.110

DKH2-CCSD(T)

2.477

52.58

192.1

DKH2-MRCISD

2.509

44.51

180.0

Ref.101

DKH-ACPF

2.530

44.28

Ref.114, 115

Experiment

2.472

54.42

Ref.107

185.4 212.4

191

4c: four-component spinors, Dirac-Coulomb Hamiltonian cp: denotes counterpoise-corrected calculations DKH method: spin-free Douglas-Kroll-transformed nopair Hamiltonian. ACPF: Averaged coupled pair functional.

27



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Table II: Spectroscopic constants [equilibrium internuclear distances (Re in ˚ A), harmonic vibrational frequencies (ωe in cm−1 ), dissociation energies (De in kcal/mol)] of ground state Ag2 molecule at various calculation levels along with experiment values. Reference

re (˚ A) De (kcal/mol) ωe (cm−1 )

Method

Present Work NR-IVOCASCI/Dyall.v2z

2.782

22.9

99

NR-IVOSSMRPT/Dyall.v2z 2.653

25.1

129


2.701

26.4

131


2.662

28.8

144


2.585

43.9

200


2.569

42.7

197

Ref.52


2.540

41.9

194


2.542

40.8

197

Ref.107

DK3-CCSD(T)

2.549

37.36

191.6

DK3-cp-CCSD(T)

2.564

35.97

187.9

2.529

38.2

192

Ref. 116, 117 Experiment

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Table III: Spectroscopic constants [equilibrium internuclear distances (Re in ˚ A), harmonic vibrational frequencies (ωe in cm−1 ), dissociation energies (De in kcal/mol)] of ground state Cu2 molecule at various calculation levels along with experiment values. The spectroscopic constants calculated with cc-pVDZ basis are listed in the parentheses to illustrate the impact of basis set size. Reference

De (kcal/mol)

ωe (cm−1 )

2.410

29.99

180.51

NR-IVOSSMRPT/cc-pVTZ

2.240

44.91

246.81

4c-IVOCASCI/cc-pVTZ

2.314 (2.325) 36.76 (33.58) 201.10 (188.82)

4c-IVOSSMRPT/cc-pVTZ

2.237 (2.241) 49.08 (44.11) 261.13 (257.61)

4c-SSMRCCSD/cc-pVDZ

2.239

45.2

261

4c-SSMRCCSD/cc-pVTZ

2.235

47.06

265

NR-cp-CCSD(T)

2.267

41.51

251.1

DK3-CCSD(T)

2.221

45.89

272.0

Re (˚ A)

Method

Present Work NR-IVOCASCI/cc-pVTZ

Ref.52

Ref.107

Ref.104, 105

Ref.118

DK3-cp-CCSD(T)

2.234

44.05

265.8

NR-CCSD(T)/cc-pVTZ

2.2642

41.47

246.2

DK-CCSD(T)/cc-pVTZ

2.2610

41.22

244.9

CCSD(T)/CBS(aQ5)/cc-pVXZ-PP

2.2186

45.35

266.1

CCSD(T)/CBS(aQ5)/aug-cc-pVXZ-PP 2.2146

46.62

268.6

CCSD(T)/aug-cc-pV[Q5]Z-PP

2.2146

46.81

268.6

F12b-CCSD(T)/aug-cc-pVQZ-PP

2.2138

F12∗ -CCSD(T)/aug-cc-pVQZ-PP

2.2143

-

266.1

2.219

47.97

266.5

Ref. 116, 117 Experiment

29


266.4


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Table IV: Spectroscopic constants [equilibrium internuclear distances (Re in ˚ A), harmonic vibrational frequencies (ωe in cm−1 ), dissociation energies (De in kcal/mol)] of ground state I2 molecule at various calculation levels along with experiment values. Reference

Re (˚ A) De (kcal/mol) ωe (cm−1 )

Method

Present Work NR-IVOSSMRPT/Dyall.pVTZ 4c-IVOSSMRPT/Dyall.pVDZ

Ref.52

Ref.15

Ref.123

Ref.128

2.619

41.8

222

2.716

29.1

199

4c-IVOSSMRPT/Dyall.pVTZ

2.637

38.5

209

4c-SSMRCCSD/Dyall.pVDZ

2.725

31.1

192

4c-SSMRCCSD/Dyall.pVTZ

2.675

34.5

216

4c-GCS-CI(10,12)

2.75

19.14

168

4c-GMC-QDPT(10,12)

2.70

32.05

205

4c-GCS-CI(10,24)

2.78

23.29

163

4c-GMC-QDPT(10,24)

2.70

31.82

205

DC/FS-MRCCSD

2.691

33.90

214

DCG/FS-MRCCSD+BSSE

2.711

30.44

209

Rel-DMC/1-det

2.663

46.30

246

Rel-DMC/multideterminant

2.674

45.76

205

Ref.121

PP-MCSCF-MRCISD

2.769

17.526

185

Ref.120

NR-CCSD(T)/ pVTZ

2.712

42.3

217

DC-CCSD(T)/ pVTZ

2.717

29.6

206

DHF-CCSD(T)[Ext. Basis]

2.685

35.283

217

Ref.122

DC-CCSD(T)

2.685

35.283

217

Ref.125, 126

REP-KRCCSD

2.692

27.90

210

REP-KRCCSD(T)

2.686

43.12

204

AREP-FS-MRCCSD

2.678

39.89

SOREP-FS-MRCCSD

2.692

29.52

Ref.124

AE-CCSD(T)/cc-pV5Z-DK

2.6810

45.99

220.9

Ref.130

MRCI+Q

2.687

44.9

219.5

Ref.131

2c-CCSD(T)/pVTZ

2.703

Sc. Rel. CCSD(T)/pVTZ

2.686

214

Ref.127

CCSD(T)/aug-cc-pRV5Z

2.6728

220.8

Ref.114

Experiment

204

CCSD(T)/CBS(FC)+CV+ est. FCI 2.6473 2.6663

225.2 35.90

214.5

NR: Nonrelativistic. DCG: Dirac-Coulomb-Gaunt. Rel-DMC: Relativistic Diffusion Monte Carlo (used energy-consistent relativistic Hartree–Fock pseudopotentials)[128]. KRCCSD and KR CCSD(T): Kramers restricted HF based CCSD and CCSD(T). REP: Relativistic effective potential. AREP: Average REP. SOREP-FS-MRCCSD: Spin-orbit REP based FS-MRCCSD. Sc. Rel.: Scalar relativistic.

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0.18 0.16 0.14

Energy (a.u.)

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

0.12 0.10 0.08 0.06


0.04 0.02 0.00 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0

RAu-Au(Å) Figure 1


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Page 47 of 50

0.14 0.12

Energy (a.u.)

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



0.10 0.08 0.06 0.04 0.02 0.00

2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0

RAg-Ag(Å) Figure 2 ACS Paragon Plus Environment

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

Energy (a.u.)


0.16 0.15 0.14 0.13 0.12 0.11 0.10 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00 -0.01 1.5

Figure 3

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2.0

2.5

3.0

3.5

4.0

4.5

5.0

RCu-Cu (Å) ACS Paragon Plus Environment

5.5

6.0

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-0.80

4c-IVOSSMRPT

-0.82 -0.84 -0.86

Dyall pVDZ Dyall pVTZ

-0.88 -0.90 -1.56 -1.540

-1.58

-1.542 -1.544

Energy (a.u.)

Energy (a.u.)

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


-1.60 -1.62

-1.548 -1.550 -1.552

4c-IVOCASCI

-1.554 -1.556 -1.558 -1.560

-1.64

2.50

2.75

3.00

3.25

3.50

3.75

4.00

RI-I(Å)

2.5 Figure 4

-1.546

3.0

3.5

4.0

4.5

RI-I(Å)

5.0

5.5


6.0

6.5

7.0

The Journal of Physical Chemistry Table of Contents Graphic 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


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Four-Component Relativistic State-Specific Multireference

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