Development and applications of a unitary group adapted state specific multi-reference coupled cluster theory with internally contracted treatment of inactive double excitations

Any multi-reference coupled cluster (MRCC) development based on the Jeziorski-Monkhorst (JM) multi-exponential ansatz for the wave-operator Ω suffers from spin-contamination problem for non-singlet states. We have very recently proposed a spin-free unitary group adapted (UGA) analogue of the JM ansatz, where the cluster operators are defined in terms of spin-free unitary generators and a normal ordered, rather than ordinary, exponential parametrization of Ω is used. A consequence of the latter choice is the emergence of the "direct?term" of the MRCC equations that terminates at exactly the quartic power of the cluster amplitudes. Our UGA-MRCC ansatz has been utilized to generate both the spin-free state specific (SS) and the state universal MRCC formalisms. It is well-known that the SSMRCC theory requires suitable sufficiency conditions to resolve the redundancy of the cluster amplitudes. In this paper, we propose an alternative variant of the UGA-SSMRCC theory, where the sufficiency conditions are used for all cluster operators containing active orbitals and the single excitations with inactive orbitals, while the inactive double excitations are assumed to be independent of the model functions they act upon. The working equations for the inactive double excitations are thus derived in an internally contracted (IC) manner in the sense that the matrix elements entering the MRCC equations involve excitations from an entire combination of the model functions. We call this theory as UGA-ICID-MRCC, where ICID is the acronym for "Internally Contracted treatment of Inactive Double excitations." Since the number of such excitations are the most numerous, choosing them to be independent of the model functions will lead to very significant reduction in the number of cluster amplitudes for large active spaces, and is worth exploring. Moreover, unlike for the excitations involving active orbitals, where there is inadequate coupling between the model and the virtual functions in the SSMRCC equations generated from sufficiency conditions, our internally contracted treatment of inactive double excitations involves much more complete couplings. Numerical implementation of our formalism amply demonstrates the efficacy of the formalism.     

Development of a relaxation-inducing cluster expansion formalism for treating strong relaxation and correlation effects

We present in this paper a comprehensive account of an explicitly spin-free coupled cluster theory for treating energy differences of open-shell states relative to a closed-shell ground state, where the open-shell states of interest are dominated by a few simple configuration state functions. We develop a valence-universal coupled cluster formalism to achieve this via a novel cluster expansion ansatz for the valence part of the wave operator, where the orbital relaxation and the correlation relaxation accompanying ionization/excitation from the ground state are taken care of to all orders in compact, efficient, and explicitly spin-free manner. The essential difference of our proposed ansatz from the ordinary and the normal-ordered cluster ansatz in vogue is that (a) we allow the valence cluster operators to be connected among themselves with spectator valence lines only and (b) we use suitable combinatoric factors accompanying powers of cluster operators thus connected, which are equal to the number of ways the operators can be joined, leading to the same excitation (the automorphic factor). We emphasize that such an ansatz does not generate terms (diagrams) with chains of cluster operators joined among themselves via spectator lines only. Barring only a few, almost all the terms in the working equations determining the cluster amplitudes involve contraction of the Hamiltonian with the cluster operators via at least one nonspectator line, leading to what we call a "strongly connected" series. The structure of the working equation is remarkably similar to the single-reference closed-shell equation, with a few additional terms. The presence of contractions among cluster operators via spectator lines introduces the additional physical effects of orbital and correlation relaxation using low-body cluster operators. As an illustrative application of the new multireference coupled cluster (CC) theory, we consider in this paper computation of ionization potentials (IPs) of one-valence problem with only one active orbital. The numerical applications are made for both the core- and the inner- and outer-valence IPs for several molecular systems. The numerical values demonstrate the superiority of the relaxation-inducing CC theory, as compared to the normal-ordered ansatz.     

Inclusion of selected higher excitations involving active orbitals in the state-specific multireference coupled-cluster theory

The parent state-specific multireference coupled-cluster (SS-MRCC) theory proposed by Mukherjee et al. [J. Chem. Phys. 110, 6171 (1999)], though rigorously size-extensive and also size-consistent with localized orbitals, has some deficiencies in the minimal truncation scheme, viz. at the singles and doubles (SD) level (SS-MRCCSD). SS-MRCCSD does not involve the direct coupling of all the model functions with a given virtual function belonging to the uncontracted multiconfiguration CISD space. It also does not involve, even in the linear power of a cluster operator T(μ), the direct coupling of the virtual functions χ(l(μ)), which are up to doubly excited with respect to a model function φ(μ) to the other virtual functions of the MRCISD space which can be generated by triple and quadruple excitations from φ(μ). We argue that inclusion of a selection of triples and quadruples involving at most two inactive orbital excitations from every φ(μ) would ameliorate the shortcoming of the incomplete coupling of the triply and quadruply excited virtual functions which can couple with the singly and doubly excited ones. This extended ansatz for our SS-MRCC theory, to be called SS-MRCCSDtq by us, would still miss the direct coupling of the manifold of the model functions {φ(λ),λ ≠ μ} to singly and doubly excited virtual functions. However, this effect is expected to be less significant than the lack of the more complete virtual space couplings, these functions being many more numerous, suggesting the new methods to be significantly improved schemes. Excellent results on the potential energy surfaces of small molecules involving single, double, and triple bond dissociation bear out our expectations fully.     

The linked-cluster theorem in the open-shell coupled-cluster theory for incomplete model spaces

The linked-cluster theorem (LCT) holds good in open-shell coupled-cluster theory for incomplete model spaces provided the intermediate normalization condition (IN) for the eigenfunctions is abandoned. The crucial requirement for proving the LCT is the valence universality of the wave operator Ω. Thus Ω contains not only m-valence operators S̃^(m) for an m-valence problem but also all the lower-valence S̃S^(k) operators to correlate k <m valence problems. LCT is proved using a particular scheme for choosing the S̃S^(k) operators for which IN does not hold good.     

Cubic assembly of a geometrically frustrated {Fe12} spin cluster

We describe an S(4)-symmetric {Fe(12)} spin cluster [Fe(12)O(4)(OH)(2)(L)(4)(OAc)(8)][Cl](2) {H(4)L = (HOCH(2)CH(2))(2)NCH(2)CH(2)N(CH(2)CH(2)OH)(2)} where the iron(III) centres describe a squashed hexagonal antiprism. The clusters pack into a large cubic cell with circular cavities, lined by weak C-H···O interactions, and a unit cell volume of over 60,000 ?(3) containing large solvent accessible voids. The core of the cluster is stable in solution, as confirmed by electrospray mass spectrometry. The cluster possesses a non-trivial, frustrated S = 0 ground state, due to the presence of multiple competing antiferromagnetic interactions. The finite temperature Lanczos method has been employed to calculate the temperature dependent magnetic properties of an analogous dodecanuclear S(i) = 3/2 model spin system, in order to reduce the very large Hilbert space. Three archetypal models with two independent exchange coupling parameters have been employed that render a low temperature feature possible, as seen in the χ vs. T plot for the {Fe(12)} spin cluster.     

Controlling transformations in the assembly of polyoxometalate clusters: {Mo11V7}, {Mo17V8} and {Mo72V30}

The reaction of molybdate with vanadium(V) in the presence of sulfite anions is explored showing how, via cation control, stepwise assembly through the {Mo(11)V(7)} cluster yields a {M(25)} cluster-based compound, [Mo(VI)(11)V(V)(5)V(IV)(2)O(52)(μ(9)-SO(3))(Mo(VI)(6)V(V)O(22))](10-) (1a), which was first discovered using cryospray mass spectrometry, whereas switching the cation away from ammonium allows the direct formation of the spherical 'Keplerate' {Mo(72)V(30)} cluster.     

Discontinuous approximate molecular electronic wave-functions

Following Kohn (reference 4), Schlosser and Marcus (reference 3), and Weare and Parr (reference 2), an energy functional is defined for a molecular problem which is stationary in the neighborhood of the exact solution and permits the use of trial functions that are discontinuous. The functional differs from the functional of the standard Rayleigh–Ritz method in the replacement of the usual kinetic energy operators T?(μ) with operators T?′(μ) = T?(μ) + Î(μ), where Î(μ) generates contributions from surfaces of nonsmooth behavior. If one uses the ?Ψ · ?Ψ way of writing the usual kinetic energy contributions, one must add surface integrals of the product of the average of ?Ψ and the change of ?Ψ across surfaces of discontinuity.     

Fluoride bridges as structure-directing motifs in 3d-4f cluster chemistry

The use of kinetically robust chromium(III) fluorido complexes as synthons for mixed 3d-4f clusters is reported. The tendency toward linear {Cr(III)-F-Ln(III)} units dictates the cluster topology. Specifically, we show that reaction of cis-[Cr(III)F(2)(NN)(2)]NO(3) (NN = 1,10-phenanthroline ("phen") or 2,2'-bipyridine ("bpy")) with Ln(NO(3))(3)·xH(2)O produces isostructural series of molecular {Ln(2)Cr(2)} squares (1-9) with linear fluoride bridges. In a parallel fashion, fac-[Cr(III)F(3)L], where L = N,N',N″-trimethyl-1,4,7-triazacyclononane ("Me(3)tacn"), reacts with Nd(NO(3))(3)·6H(2)O to form a fluoride-centered penta-nuclear complex and fac-[Cr(III)F(3)L'], with L' = 1,1,1-tris-((methylamino)methylethane) ("Me(3)tame"), reacts with [Ln(hfac)(3)(H(2)O)(2)] (hfacH = 1,1,1,5,5,5-hexafluoroacetylacetone) to yield an isostructural series of {Ln(3)Cr(2)} (10-14) trigonal bipyramids with no central ligand. The formation of the latter is accompanied by a partial solvolysis of the Cr(III) precursor but without formation of insoluble LnF(3). The magnetic properties of the gadolinium containing clusters allow quantification of fluoride-mediated, antiferromagnetic Gd-Cr exchange interactions of magnitude between 0.14 cm(-1) and 0.71 cm(-1) (? = J(12)?(1)·?(2) formalism) and vanishingly small J(Gd-Gd) of 0.06(0) cm(-1). The large spin and small anisotropy together with weak exchange interactions in the {Gd(3)Cr(2)} (11) cluster give rise to a very large magneto-caloric effect of -ΔS(m) = 28.7 J kg(-1) K(-1) (μ(0)H = 90 to 0 kOe).     

ENTROPIC CONTRIBUTION TO THE INTERACTION PARAMETER xIN THE POLYMER／OLIGOMER SYSTEM： A LATTICE CLUSTER THEORY CONSIDERATION

Entropic contribution to the interaction parameter xeff in the model incompressible polymer/oligomer system is calculated by the lattice cluster theory (LCT). It is found that in the oligomer solvent, there exists a wide concentration range that the non-combinatorial “entropic interaction” term xeff φ1φ2 perceptibly counteracts the mean field combinary entropy ΔSMF. With the increase of the solvent size, both xeff and the ratio xefc φ1φ2/ΔSMF first reach their maximum and finally become trivially to zero. It is worth noting that no any demixing was found in the current calculation. This makes the controversial idea “entropically driven demixing” even elusive. However, we propose that further work on compressible polymer solution with structured monomer will witness the demixing owning to an increased configurational correlation.     

From platonic templates to Archimedean solids: successive construction of nanoscopic {V16As8}, {V16As10}, {V20As8}, and {V24As8} polyoxovanadate cages

Supramolecular coordination cages provide unique restricted inner cavities that can be exploited for molecular recognition purposes and catalysis. Their syntheses often involve complex self-organization processes and rely on the identification of preorganized, kinetically stable building units that provide ligand-accessible coordination sites. Here we report a highly effective protocol for the successive buildup of symmetrical nanoscopic polyoxometalate (POM) cages. Our methodology takes advantage of a supramolecular templating effect and utilizes the structure-directing influence of octahedral {X(x)(H(2)O)(6-x)} (X = Br(-), Cl(-); x = 2, 4, 6) assemblies that reside inside the hollow cluster shells and determine the arrangement of di- and tetranuclear vanadate units. The approach allows the preparation of a series of high-nuclearity POM cages that are characterized by {V(16)As(8)}, {V(16)As(10)}, {V(20)As(8)}, and {V(24)As(8)} core structures. In the latter cluster cage, the vanadium centers adopt a truncated octahedral topology. The formation of this Archimedean body is the direct result of the assembly of six square {V(4)O(8)} units that cap the vertices of the encapsulated Platonic {Cl(6)} octahedron. To the best of our knowledge, this {V(24)As(8)} cage is the largest hybrid vanadate cluster reported to date.     

Tuning the light absorption of a molecular vanadium oxide system for enhanced photooxidation performance

A new mechanism for enhancing the visible light absorption of a homogeneous polyoxovanadate system is described. The photoactive pentanuclear {V(5)} isopolyoxovanadate cluster is formed in situ by a thermally-induced condensation reaction starting from a tetranuclear {V(4)} precursor. Upon irradiation with visible light, {V(5)} undergoes a light induced reduction reaction resulting in the formation of a 2-electron reduced {V(10)} cluster. Simultaneously, the oxidant methanol is selectively oxidized to formaldehyde. The {V(10)} cluster can subsequently be re-oxidized using H(2)O(2) or O(2).     

Anion induced diversification from heptanuclear to tetranuclear clusters: syntheses, structures and magnetic properties

Three new polynuclear complexes, [Co(7)(bm)(12)]·(ClO(4))(2)·13H(2)O (1), [Co(4)(bm)(4)Cl(4)(C(3)H(7)OH)(4)] (2), and [Co(4)(bm)(4)(μ-HCO(2))(2)(μ(2)-HCO(2))(2)(C(3)H(7)OH)(2)] (3) (Hbm = (1H-Benzimidazol)-methanol), have been synthesized and characterized by elemental analysis, IR, powder X-ray diffraction and X-ray single-crystal diffraction. Compound 1 features a centrosymmetric wheel-like heptanuclear Co(II) cluster. Compound 2 having a I4(1)/a space group exhibits a tetranuclear Co(II) cluster with a cubane topology in which the central Co(II) ion and oxygen atoms from bm occupy the alternate vertices of the cube. However, compound 3 has a tetranuclear Co(II) cluster with a C2/c space group different from that of compound 2. These results show that the geometries and sizes of the corresponding anions as well as their coordinating and hydrogen-bonding properties are essential in determining the final structures of the assemblies. Magnetic properties of 1-3 in the 2-300 K have also been discussed. The {Co(7)} (1) and {Co(4)} (2) cores display dominant ferromagnetic interactions while the {Co(4)} (3) core displays dominant anti-ferromagnetic interactions.     

Unusual oxidation state distributions observed for two mixed-valence heptanuclear manganese disc-like clusters

The synthesis, structures and magnetic properties of two new mixed-valence heptanuclear manganese clusters are described. Both complexes utilize triethanolamine (teaH(3)) as a bridging ligand, displaying near planar, disc-like metal topologies and are of formulae [Mn(II)(4)Mn(IV)(3)(tea)(teaH(2))(3)(peolH)(4)](BF(4))(2)·solv (1) and [Mn(II)(4)Mn(III)(3)F(3)(tea)(teaH)(teaH(2))(2)(piv)(4)(Hpiv)(chp)(3)]·0.5MeCN (2). Compound 1 is a rare mixed-valence compound containing Mn(II) and Mn(IV) ions only and is the first example of a heptanuclear disc with a {Mn(II)(4)Mn(IV)(3)} oxidation state distribution. Compound 2 is a {Mn(II)(4)Mn(III)(3)} complex and displays a unique arrangement of oxidation states within the disc, when compared to other known {Mn(II)(4)Mn(III)(3)} examples. Variable temperature DC and AC magnetic susceptibility studies were carried out for 1 and 2 in the 2-300 K temperature range. Compound 1 displayed an increase in the χ(M)T susceptibility values as the temperature is decreased indicating dominant ferromagnetic interactions are present within the cluster. Fits of the χ(M)T vs. T data reveals an S = 23/2 ground state, with several close lying excited states within 1 cm(-1). Compound 2 displays an overall decrease in the χ(M)T value as the temperature is decreased down to 2 K indicating dominant antiferromagnetic interactions present with a probable S = 4 ground state as determined from the DC and AC susceptibility data.     

Structures and magnetism of {Ni2Na2}, {Ni4} and {Ni6(II)Ni(III)} 2-hydroxy-3-alkoxy-benzaldehyde clusters

Three polynuclear complexes, [NiNa(μ(1,1,1)-N(3))(μ-hmb)(2)(DMF)](2), (1), [Ni(4)(μ(3)-OMe)(4)(heb)(4)(MeOH)(1.05)(H(2)O)(2.95)], (2) and [Ni(III)(OH)(6)(hmb)(6)Ni(II)(6)]·(ClO(4))(3) (3) (Hhmb = 2-hydroxy-3-methoxy-benzaldehyde; Hheb = 2-hydroxy-3-ethoxy-benzaldehyde), were prepared by reaction of the appropriate ligand with nickel(II) perchloride hexahydrate under solvothermal conditions. All compounds were characterized by elemental analysis, IR spectroscopy and X-ray single-crystal diffraction. Compound 1 exhibits a centrosymmetric heterotetranuclear cluster which represents the first nickel complex to possess two connected face-sharing cubes structure {Ni(2)Na(2)N(2)O(4)}. Compound 2 has a tetranuclear Ni cluster with a cubane topology in which the Ni(II) and the oxygen atoms from the methanol ligands occupying alternate vertices of the cube. Compound 3 consisits of a mixed-valence [Ni(III)(OH)(6)(hmb)(6)Ni(II)(6)](3+) subunits and it represents the first nickel {Ni(II)(6)Ni(III)} complex to possess a planar hexagonal disc-like structure. The results show that the minor ligand modifications or solvent change have a key role in the structural control of the self-assembly process. Magnetic properties of 1-3 in the 300-2 K have been discussed. The {Ni(2)Na(2)} (1) and {Ni(4)} (2) core display dominant ferromagnetic interactions from the nature of the binding modes through μ(3)-N(3)(-) or μ(3)-OCH(3)(-), while {Ni(II)(6)Ni(III)} core (3) displays dominant anti-ferromagnetic interactions from the nature of the binding modes through μ(3)-OH(-).     

Towards the Development and Applications of Manifestly Spin-free Multi-reference Coupled Electron-pair Approximation-like Methods: A State Specific Approach

As a practical tool of being applicable to bigger molecules, a full-blown state-specific multi-reference coupled cluster formalism developed by us (Mahapatra et al. in J Chem Phys 110:6171, 1999) would be rather demanding computationally, and it is worthwhile to look for physically motivated approximation schemes which capture a substantial portion of the correlation of the full-blown theory. In this spirit, we have recently proposed coupled electron-pair approximation (CEPA)-like various approximants to the parent spin-adapted state-specific multi-reference coupled cluster (SS-MRCC) theory which depend on the inclusion of EPV terms to various degree. Here, the space of excitations is confined to the first order interactive virtual space generated by the cluster operator, but the EPV terms are included exactly. We call them spin-free state specific multi-reference CERA (SS-MRCEPA) theories. They work within the complete active space (CAS) and have been found to be very effective in bypassing the intruders, similar in performance to that of the parent SS-MRCC theory. The spin-adaptation of the working equations of both the SS-MRCC and the CEPA-like approximants is a non-trivial exercise. In this paper, we delineate briefly the essentials of a spin-free formulation of the SS-MRCC and SS-MRCEPA theories. This allows us to include open-shell configuration state functions (CSF) in the CAS. We consider three variants of SS-MRCEPA method. Two are explicitly orbital invariant: (1) SS-MRCEPA(0), a purely lineralized version of the SS-MRCC theory, (2) SS-MRCEPA(I), which includes all the EPV terms explicitly and exactly in an orbital invariant manner and (3) the SS-MRCEPA(D), which emerges when we keep only the diagonal terms of a set of dressed operators in the working equations. Unlike the first two, the third version is not invariant under the orbital transformation within the set of doubly occupied core, valence and virtual orbitals. The SS-MRCEPA methods produce very encouraging results as was evidenced in the applications on the computation of potential energy surfaces for the ground states of LiH and HF molecules.     

Van der Waals complexes of Cu, Ag, and Au with hydrogen sulfide. The bonding character

The electronic and structural features of the Cu...SH2, Ag...SH2, and Au...SH2 complexes are investigated by using the spin-adapted restricted open-shell HF coupled cluster CCSD(T) method combined with the second-order spin-free Douglas-Kroll-Hess (DKH) relativistic approach. M...SH2 complexes are nonplanar with bonding energies -5.99, -1.99, and -9.08 mHartree, respectively. Comparison with analogous M...OH2 and M...NH3 complexes allows us to establish general features of the bonding between coinage metal atoms and ligand molecules with the participation of their lone electron pairs. Consistent interpretation of the interaction effects can be obtained by using the molecular orbital picture of the M...L region. The bonding character is explained by stressing the importance of the charge transfer from the lone pair of the ligand to the metal atom. Relativistic changes of the metal element electron affinity and polarizability facilitate the understanding of major trends in the pattern of interactions between the coinage metal atoms and different lone pair donating ligands.     

Solution identification and solid state characterisation of a heterometallic polyoxometalate {Mo(11)V(7)}: [Mo(VI)(11)V(V)(5)V(IV)(2)O(52)(mu(9)-SO(3))](7-)

A polyoxomolybdenum/vanadium-sulfite {M(18)} cluster-based compound, [Mo(VI)(11)V(V)(5)V(IV)(2)O(52)(mu(9)-SO(3))](7-), is reported that exhibits a unique structural motif, arising from the incorporation of five V(V) and two V(IV) ions into a {M(18)} cluster framework templated by SO(3)(2-); this cluster compostion was first identified using cryospray mass spectrometry.