Valence-universal Multireference Research Articles

Effect of three-body transformed Hamiltonian ( $$\tilde{H} _3 $$ H ~ 3 ) using full connected triple excitation coupled cluster operators on valence ionisation potentials of $$\hbox {Cl}_{2}$$ Cl 2 and $$\hbox {F}_{2}$$ F 2 computed via EIP-VUMRCCSD $$\uptau $$ τ scheme

Valence universal multireference coupled cluster (VUMRCC) method via eigenvalue independent partitioning has been applied to estimate the effect of three-body transformed Hamiltonian (\(\widetilde{{H}}_3\)) on ionisation potentials through full connected triple excitations \(S_{3}^{(1,0) }\). \(\widetilde{H}_3 \) is constructed using CCSDT1-A model of Bartlett et al for the ground-state calculation. Contribution of transformed Hamiltonian through full connected triples \(\overline{\widetilde{H}_3 S_3^{\left( {1,0} \right) }}\) involves huge amount of computational operations that is time-consuming. Investigation on \(\hbox {Cl}_{2}\) and \(\hbox {F}_{2}\) molecules using cc-pVDZ and cc-pVTZ basis sets shows that the above effect varies from 0.001 eV to around 0.5 eV, suggesting that inclusion of \(\overline{\widetilde{H} _3 S_3^{\left( {1,0} \right) } }\) is essential for highly accurate calculations.

Spectroscopic constants relating to ionization from the strongest bonding and inner valence molecular orbital 2σg of N2: An EIP-VUMRCC search

Valence universal multireference coupled cluster (VUMRCC) method suffers from numerical divergence owing to intruder states in inner valence region of molecules. Eigenvalue independent partitioning (EIP) technique in Fock space converts VUMRCC equations to a non-Hermitian eigenvalue problem [29]. EIP makes access to the roots invaded by intruders much easier. The EIP-VUMRCC determines with significant precision the potential energy curve of the strongest bonding and inner valence 2σg MO of N2+ ion. Inclusion of full connected triple excitation CC operators is found very crucial. In ground state, perturbative Tˆ3 is considered by CCSDT1-a approximation.

Search of truncation of (N−1) electron basis containing full connected triple excitations in computing main and satellite ionization potentials via Fock‐space coupled cluster approach

A valence-universal multireference coupled cluster (VUMRCC) theory, realized via the eigenvalue independent partitioning (EIP) route, has been implemented with full inclusion of triples excitations for computing and analyzing the entire main and several satellite peaks in the ionization potential spectra of several molecules. The EIP-VUMRCC method, unlike the traditional VUMRCC theory, allows divergence-free homing-in to satellite roots which would otherwise have been plagued by intruders, and is thus numerically more robust to obtain more efficient and dependable computational schemes allowing more extensive use of the approach. The computed ionization potentials (IPs) as a result of truncation of the (N-1) electron basis manifold involving virtual functions such as 2h-p and 3h-2p by different energy thresholds varying from 5 to 15 a.u. with 1 a.u. intervals as well as thresholds such as 20, 25, and 30 a.u. have been carefully looked into. Cutoff at around 25 a.u. turns out to be an optimal threshold. Molecules such as C2H4 and C2H2 (X = D,T), and N2 and CO (X = D,T,Q) with Dunning's cc-pVXZ bases have been investigated to determine all main and 2h-p shake-up and 3h-2p double shake-up satellite IPs. We believe that the present work will pave the way to a wider application of the method by providing main and satellite IPs for some problematic N-electron closed shell systems.

Valence universal multireference coupled cluster calculations of the properties of indium in its ground and excited states

In view of its importance in high precision spectroscopy, the valence universal multireference coupled cluster (VU-MRCC) method with four-component relativistic spinors has been applied to compute ionization potential (IP) and excitation energies (EEs) of the indium atom (In I). The effect of electron correlations on the ground and excited state properties is investigated using different levels of CC approximations and basis sets. This study reveals that for a given basis, the linearized VU-MRCC method tends to underestimate the IP, EEs and other one-electron properties such as magnetic hyperfine constant (A) compared to the full blown VU-MRCC method. Our computed results have been compared with available theoretical and experimental data. The IP, EEs, A and oscillator strengths (f) determined at the VU-MRCC level are in excellent agreement with the experimental results. The properties reported here further demonstrate that a basis set with at least h-type of orbitals is ubiquitous to achieve converged results.

Effect of three body transformed Hamiltonian ( [formula omitted]) through full connected triples on main and satellite ionization potentials computed via valence universal EIP-MRCC method

Valence universal multi-reference coupled cluster (VUMRCC) method via eigenvalue independent partitioning has been applied to estimate the effect of H ∼ 3 on ionization potentials through full connected triple excitations S 3 ( 1 , 0 ) . H ∼ 3 is constructed using CCSDT1-A model for the ground state calculation. H 3 ∼ S 3 ( 1 , 0 ) ¯ involves n vir 4 n occ 4 operations that may lead to large time consumption. Our investigation on HF, HCl, N 2 and CO molecules using cc-pVDZ and cc-pVTZ basis sets indicates that the above effect varies from 0.001 eV to around 0.5 eV (∼11.5 kcal/mol), thus suggesting that inclusion of H 3 ∼ S 3 ( 1.0 ) ¯ is essential in high accuracy calculation.

Use of a convenient size-extensive normalization in multi-reference coupled cluster (MRCC) theory with incomplete model space: A novel valence universal MRCC formulation

We present in this paper a size-extensive formulation of a valence universal multi-reference coupled cluster (VU-MRCC) theory which uses a general incomplete model space (IMS). The earlier formulations by Mukherjee [D. Mukherjee, Chem. Phys. Lett. 125 (1986) 207] led to size-extensive Heff which was both connected and ‘closed’, thereby leading to size-extensive energies. However, this necessitated abandoning the intermediate normalization (IN) for the valence universal wave-operator Ω when represented as a normal ordered exponential cluster Ansatz Ω≡{exp(S)} with S as the cluster operator. The lack of IN stemmed from the excitation operator Sq-op which leads to excitations into the complementary model space by their action on at least one model function. The powers of Sq-op can in general bring a model function ϕi back to another model function ϕj, and this is the reason why Ω does not respect IN. Sq-op are all labelled by active orbitals only. To achieve connectivity of Heff, it must be a ‘closed’ operator. A closed operator is one which always produces a model function by its action on another model function. Since the decoupling conditions Lq-op=0, and Lop=0 for the transformed operator L=Ω-1HΩ would be in conflict with Ωq-op=1q-op, the model space projection of Ω, PΩP=P cannot be maintained for the normal ordered Ansatz. This leads to a somewhat awkward expression for Heff. Bera et al. [N. Bera, S. Ghosh, D. Mukherjee, S. Chattopadhyay, J. Phys. Chem. A 109 (2005) 11462] recently tried to simplify the expression for Heff, and accomplished this by introducing suitable counter-terms Xcl in Ω to enforce Ωcl=1cl. We show in this paper that Heff in this formulation leads to a disconnected Heff, though it is equivalent by a similarity transformation to a connected effective hamiltonian H∼eff. Guided by the insight gleaned from this demonstration, we have proposed in this paper a new form of the wave-operator which never generates any powers of Sq-op, which is closed. This ‘externally projected’ wave-operator does not need counter-terms Xcl and automatically ensures Ωcl=1cl, thereby yielding directly a closed connected H∼eff. The desirable features of the traditional normal ordered Ansatz, such as the valence universality, subsystem embedding conditions hierarchical decoupling of the VU-MRCC equations for decreasing valence ranks are all satisfied by this new Ansatz for the wave-operator.

Explicitly intruder-free valence-universal multireference coupled cluster theory as applied to ionization spectroscopy

Although it is quite promising to compute the spectroscopic energies [say, ionization potential (IP)] via the traditional valence-universal multireference coupled cluster (VUMRCC) method based on the description of the complete model space being seriously plagued by the perennial intruder state problem, the eigenvalue independent partitioning (EIP) based VUMRCC (coined as EIP-MRCC) method is quite effective to predict the spectroscopic energies in an intruder-free manner. Hence, the EIP-MRCC method is suitable for generating both the principal IPs and the satellite IPs of the inner-valence region. An EIP strategy converts the nonlinear VUMRCC equations for M(m,n) dimensional model space of m hole and n particle to a non-Hermitian eigenproblem of larger dimension whose M(m,n) roots are only physically meaningful. To increase the quality of the computed energy differences in the sense of chemical accuracy and to locate the correct position of it in the spectrum, the inclusion of higher-body cluster operators on top of all the standard singles-doubles is not the only pivotal issue, the effect of the size of the basis set is also equally important. This paper illustrates these issues by calculating the principal and satellite IPs of HF and HCl molecules using various basis sets (viz., Dunning's cc-pVDZ, cc-pVTZ, and cc-pVQZ) via EIP-MRCC method with full inclusion of triples (abbreviated as EIP-MRCCSDT). The results seem quite encouraging in comparison with the experimental values. The controversial 2Pi satellite at 28.67 eV of HCl of Svensson et al. [J. Chem. Phys. 89, 7193 (1988)] is also reported.

A novel VU-MRCC formalism for the simultaneous treatment of strong relaxation and correlation effects with applications to electron affinity of neutral radicals

We present and implement in this paper a novel spin-free valence-universal multi-reference coupled cluster (VU-MRCC) formalism for energy differences which can capture orbital relaxation and correlation relaxation to all orders. Unlike in the traditional normal ordered cluster Ansatz for computing energy differences, this cluster expansion formalism allows contractions between various valence excitation operators with valence spectator lines. These contractions simulate the orbital relaxation and correlation relaxation effects for the ionized/excited states via Thouless-like exponential type of operators. Generally such operators are non-commuting. To ensure that each distinct excitation generated by contracted composites formed by these operators appear only once in the wave-operators, the factors accompanying these composites have to be judiciously chosen. Hence, the combinatoric factors accompanying such contracted composites are not taken to be 1/ n! for nth-power, but rather the inverse of the automorphic factor (the number of ways the n operators can be connected in various permutations generating the same composite). It is shown that this Ansatz leads to a set of VU-MRCC equations for the valence cluster amplitudes, in which all the cluster operators are attached to the hamiltonian by at least one non-spectator line (a strongly connected series). The series is thus terminating at the quartic power. Illustrative applications are presented by computing electron affinity of neutral doublet radicals (viz., NH 2, OH, F, BO and CN), where the orbital relaxation effect attendant on the anion formation is considerable. Several basis-sets capable of describing the anions have been studied. It has been found that aug-cc-pVTZ basis gives the best overall results, while aug-cc-pVQZ overestimates the electron affinity, presumably because of an imbalance in describing the neutral radicals. The method performs consistently much better then the one with the traditional normal ordered Ansatz.

Applications of core-valence extensive multi-reference coupled cluster theory and core-extensive coupled cluster-based linear response theory

The single reference coupled cluster based linear response theory (CCLRT) and valence-universal multi-reference coupled cluster (VU-MRCC) theory for direct difference energy calculations have been presented. The similarities and differences in the structure of these two formalisms have also been addressed. The core-extensive CCLRT is applied to compute the ionization potentials (IPs) of HCl and excitation energies of CuH using Hartree–Fock (HF) orbitals. We further report the relativistic applications of the VU-MRCC theory through the computation of the ground and excited/ionized state energies and related properties of Ag and Hg atoms using Dirac–Hartree–Fock (DHF) orbitals. The IPs, excitation energies, and the spectroscopic constants reported here are in favorable agreement with experiment and with other correlated calculations.

Reappraisal of the Role of Size-Extensive Normalization for Multireference Coupled Cluster (MRCC) Theory Using General Model Space: A Valence Universal MRCC Approach

We present a brief description of a valence-universal multireference coupled cluster (VU-MRCC) theory that can handle completely general incomplete model spaces, remaining close to the intermediate normalization (IN) condition for omega as much as possible without violating extensivity and without the use of a post facto correction. In this formalism, the connectedness of the cluster operators as well as effective Hamiltonian and hence the extensivity of the corresponding roots is achieved by invoking appropriate decoupling conditions on the special type of wave operator omega = {exp(S + X(cl))} satisfying the Bloch equations in the Fock-space S in an excitation operator and X is a closed operator (denoted by cl). This special type of wave-operator leads to a unique partition of the excitations from the model space into those generated by the cluster operators (open and quasi-open) and those generated by the effective Hamiltonian (closed). In this formulation, for every X(cl), there is a counterterm from {exp(S)}(cl) canceling each other. This leads to a connected expressions for cluster amplitudes, using the constraint omega(cl) = 1. The new form of the effective Hamiltonian preserves the extensivity of the target energies. Our analysis implies that IN for omega is a valid size-extensive normalization for certain special IMS such as the quasi-complete model space and the isolated incomplete model space.

Full effect of triples in a valence universal multi-reference coupled cluster calculation

Full three-electron coupled cluster (CC) operators (triples) along with all singles and doubles are included in a valence universal multi-reference CC calculation to estimate the change in the correlation effect for ionization potential (IP) calculations. We study the vertical IPs of two typical systems, N 2 and CO, which go in the opposite direction, i.e., the results were worsened on inclusion of triples when working with relatively smaller basis. However, upon reaching the basis set saturation the expected trend in the results is observed and they are close within 0.1 eV to experimental values almost in every IP.

Applicability of valence-universal multireference coupled-cluster theories to quasidegenerate electronic states. I. Models involving at most two-body amplitudes

The recently developed algebraic formulation of valence-universal coupled-cluster (VU-CC) theories [Jeziorski and Paldus, J. Chem. Phys. 90, 2714 (1989)] for open-shell systems has been employed in a systematic derivation of explicit equations defining cluster amplitudes assuming Lindgren’s normal ordered exponential ansatz for the wave operator. The latter is approximated by its one- and two-electron components. Various aspects of the applicability of this version of the VU-CC theory to quasidegenerate electronic states are studied for a model system consisting of two slightly stretched, interacting hydrogen molecules. A single parameter that determines the geometry of this system makes it possible to vary the extent of quasidegeneracy of the two lowest-energy states over a wide range. Along with the complete theory, the linear version (VU-LCC) is also examined. The results are compared with the full configuration interaction results as well as with those obtained using other approaches. It was found that, at least in the strongly quasidegenerate region, the VU-CC energies are less accurate than those obtained with other multireference CC theories. It is shown that the VU-CC equations for cluster amplitudes possess multiple solutions representing various pairs of states. The individual solutions that arise for a system possessing n-valence electrons may be characterized in terms of a genealogical scheme that involves states of systems with 0,1,...,n valence electrons.