Debasis Mukhopadhyay

Aspects of separability in the coupled cluster based direct methods for energy differences

SummaryIn this paper we have discussed in detail the aspects of separability of the energy differences obtained from coupled cluster based “direct” methods such as the open-shell Coupled Cluster (CC) theory and the Coupled Cluster based Linear Response Theory (CC-LRT). It has been emphasized that, unlike the state energiesper se, the energy differences have a semi-local character in that, in the asymptotic limit of non-interacting subsystemsA, B, C, etc., they are separable as ΔEA, ΔEB, ΔEA + ΔEB, etc. depending on the subsystems excited. We classify the direct many-body methods into two categories: core-extensive and core-valence extensive. In the former, we only implicitly subtract the ground state energy computed in a size-extensive manner; the energy differences are not chosen to be valence-extensive (separable) in the semi-local sense. The core-valence extensive theories, on the other hand, are fully extensive — i.e., with respect to both core and valence interactions, and hence display the semi-local separability. Generic structures of the wave-operators for core-extensive and core-valence extensive theories are discussed. CC-LRT is shown to be core-extensive after a transcription to an equivalent wave-operator based form. The emergence of valence disconnected diagrams for two and higher valence problems are indicated. The open-shell CC theory is shown to be core-valence extensive and hence fully connected. For one valence problems, the CC theory and the CC-LRT are shown to be equivalent. The equations for the cluster amplitudes in the Bloch equation are quadratic, admitting of multiple solutions. It is shown that the cluster amplitudes for the main peaks, in principle obtainable as a series inV from the zeroth order roots of the model space, are connected, and hence the energy differences are fully extensive. It is remarkable that the satellite energies obtained from the alternative solutions of the CC equations are not valence-extensive, indicating the necessity of a formal power series structure inV of the cluster amplitudes for the valence-extensivity. The alternative solutions are not obtainable as a power series inV. The CC-LRT is shown to have an effective hamiltonian structure respecting “downward reducibility”. A unitary version of CC-LRT (UCC-LRT) is proposed, which satisfy both upward and downward reducibility. UCC-LRT is shown to lead to the recent propagator theory known as the Algebraic Diagrammatic Construction. It is shown that both the main and the satellite peaks from UCC-LRT for the one valence problems are core-valence extensive owing to the hermitized nature of the underlying operator to be diagonalized.

The construction of a size-extensive intermediate Hamiltonian in a coupled-cluster framework

Abstract A size-extensive formulation for an intermediate Hamiltonian H int , furnishing size-extensive energies for the main roots, is presented. The working model space, comprised of the main and intermediate space, is taken as complete. The starting point is a shifted Bloch equation derived by us, involving shift operators generated by the projector spanned by the intermediate eigenvectors of H int . It is shown that a manifestly size-extensive H int can be constructed in the coupled-cluster framework provided we choose the shift operator explicitly as additively separable and the associated intermediate wave operator R as multiplicatively separable.

Applications of open-shell coupled cluster theory using an eigenvalue-independent partitioning technique: Approximate inclusion of triples in IP calculations

Abstract Using our eigenvalue-independent partitioning (EIP) approach for the calculation of open-shell coupled cluster (CC) energy differences, we have computed the ionization potentials of HF and H 2 O using basis sets with and without polarization functions. Our results include the three-body cluster operator for the ionized states at the lowest order of approximation. It is found that a CCSD calculation for the ground state, followed by a CCSD calculation for the ionized states - with additional inclusion of triples using the converged CCSD amplitudes - produces results that are accurate up to third order and recovers the relaxation and differential correlation energies consequent on ionization in a balanced and compact manner.

Size-extensive effective Hamiltonian formalisms using quasi-Hilbert and quasi-Fock space strategies with incomplete model spaces

Abstract An open-shell coupled cluster (CC) theory is developed using an incomplete model space (IMS) that works entirely within one particular n -valence Hilbert space sector. It is proved that the theory is size-extensive since both H eff and the energies obtained therefrom are connected. The method may be viewed as the Fock space CC theory for an IMS projected onto a given Hilbert space, and is denoted here as a “quasi-Hilbert” space theory. A “quasi-Fock“ formalism is also outlined where additional information on the lower valence model spaces are deliberately retained.

Molecular applications of size-extensive quasi-Hilbert- and quasi-Fock-space coupled-cluster formalisms using incomplete model spaces

Abstract In this paper, we report the first numerical applications of the size-extensive quasi-Hilbert- and quasi-Fock-space coupled-cluster theories, developed by us recently, to the ground and the first excited 1,3 Σ states of the molecule LiH. Unlike the corresponding Fock-space theories, there is no need to solve for the cluster amplitudes of lower-valence model spaces characterizing LiH + and LiH − species in these formulations. The results are compared with the CISD values, those from the Fock-space theories and the coupled-cluster-based linear-response theory, and are found to be quite satisfactory. This indicates the computational viability of the two formulations.

Renner-Teller intersections along the collinear axes of polyatomic molecules: H2CN as a case study.

The tetra-atomic C(2)H(2)(+) cation is known to form Renner-Teller-type intersections along its collinear axis. Not too long ago, we studied the nonadiabatic coupling terms (NACTs) of this molecule [G. J. Halász et al., J. Chem. Phys. 126, 154309 (2007)] and revealed two kinds of intersections. (i) By employing one of the hydrogens as a test particle, we revealed the fact that indeed the corresponding (angular) NACTs produce topological (Berry) phases that are equal to 2pi, which is a result anticipated in the case of Renner-Teller intersections. (ii) However, to our big surprise, repeating this study when one of the atoms (in this case a hydrogen) is shifted from the collinear arrangement yields for the corresponding topological phase a value that equals pi (and not 2pi). In other words, shifting (even slightly) one of the atoms from the collinear arrangement causes the intersection to change its character and become a Jahn-Teller intersection. This somewhat unexpected novel result was later further analyzed and confirmed by other groups [e.g., T. Vertesi and R. Englman, J. Phys. B 41, 025102 (2008)]. The present article is devoted to another tetra-atomic molecule, namely, the H(2)CN molecule, which just like the C(2)H(2)(+) ion, is characterized by Renner-Teller intersections along its collinear axis. Indeed, we revealed the existence of Renner-Teller intersections along the collinear axis, but in contrast to the C(2)H(2)(+) case a shift of one atom from the collinear arrangement did not form Jahn-Teller intersections. What we found instead is that the noncollinear molecule was not affected by the shift and kept its Renner-Teller character. Another issue treated in this article is the extension of (the two-state) Berry (topological) phase to situations with numerous strongly interacting states. So far the relevance of the Berry phase was tested for systems characterized by two isolated interacting states, although it is defined for any number of interacting states [M. V. Berry, Proc. R. Soc. London, Ser. A 392, 45 (1984)]. We intend to show how to overcome this limitation and get a valid, fully justified definition of a Berry phase for an isolated system of any number of interacting states (as is expected).

Nonlinear optical and electroabsorption spectra of polydiacetylene crystals and films

Vibronic structure of nonlinear optical (NLO) coefficients is developed within the Condon approximation, displaced harmonic oscillators, and crude adiabatic states. The displacements of backbone modes of conjugated polymers are taken from vibrational data on the ground and 1B excited state. NLO resonances are modeled by three excitations and transition moments taken from Pariser–Parr–Pople (PPP) theory and optimized to polydiacetylene (PDA) spectra in crystals and films, with blue‐shifted 1B exciton. The joint analysis of third‐harmonic‐generation, two‐photon absorption, and nondegenerate four‐wave‐mixing spectra of PDA crystals and films shows weak two‐photon absorption to 2A below 1B, leading to overlapping resonances in the THG spectrum, strong two‐photon absorption to an nA state some 35% above 1B, and weak Raman resonances in nondegenerate FWM spectra. The full π‐π* spectrum contributes to Stark shifts and field‐induced transitions, as shown by PPP results for PDA oligomers. The Stark shift dominates...

A tri-atomic Renner-Teller system entangled with Jahn-Teller conical intersections

The present study concentrates on a situation where a Renner-Teller (RT) system is entangled with Jahn-Teller (JT) conical intersections. Studies of this type were performed in the past for contours that surround the RT seam located along the collinear axis [see, for instance, G. J. Halász, Á. Vibók, R. Baer, and M. Baer, J. Chem. Phys. 125, 094102 (2006)]. The present study is characterized by planar contours that intersect the collinear axis, thus, forming a unique type of RT-non-adiabatic coupling terms (NACT) expressed in terms of Dirac-δ functions. Consequently, to calculate the required adiabatic-to-diabatic (mixing) angles, a new approach is developed. During this study we revealed the existence of a novel molecular parameter, η, which yields the coupling between the RT and the JT NACTs. This parameter was found to be a pure number η = 22/π (and therefore independent of any particular molecular system) and is designated as Renner-Jahn coupling parameter. The present study also reveals an unexpected result of the following kind: It is well known that each (complete) group of states, responsible for either the JT-effect or the RT-effect, forms a Hilbert space of its own. However, the entanglement between these two effects forms a third effect, namely, the RT/JT effect and the states that take part in it form a different Hilbert space.

A comparative study of core-extensive and core—valence-extensive coupled-cluster theories for energy differences: Excitation energies

Abstract We have investigated the relative computational efficacies of the open-shell coupled-cluster (OSCC) theory and the CC-based linear response theory (CC-LRT) for computing excitation energies directly. It is emphasized that the core—valence-extensive nature of OSCC and the core-extensive nature of CC-LRT stem essentially from the differences in the cluster structures of the wave operators. Results obtained from CH + , H 2 O and N 2 indicate that both methods perform well for low-lying excited states. The model space used in our OSCC method consists of h—p determinants, and consequently, the excited states dominated by the 2h—2p determinants are poorly described. In contrast, these states are well described in CC-LRT, since the working space in this case has both h—p and 2h—2p determinants. We comment on the relative ease of implementation of the two methods.

Vibronic analysis of overlapping resonances and the third‐harmonic‐generation spectrum of β‐carotene

Vibronic contributions to third‐harmonic‐generation (THG) are obtained in the Condon approximation for displaced harmonic oscillators in the region of overlapping three‐ and two‐photon resonances, where enhanced THG intensity and explicit dependence on the relative signs of the displacements are found. The THG intensity and phase of β‐carotene are then modeled in terms of four electronic states, including overlapping 2 1Ag and 1 1Bu resonances whose displacements are taken from two‐photon and linear spectra, respectively, and a high‐energy Ag state based on a Pariser–Parr–Pople (PPP) sum rule for transition dipoles. Relations between THG and other spectra show the limitations of three‐state models and provide useful constraints on the excited‐state structure of related conjugated systems such as polyenes and β‐carotene.