State-specific Theory Research Articles

Excited State-Specific CASSCF Theory for the Torsion of Ethylene.

State-specific complete active space self-consistent field (SS-CASSCF) theory has emerged as a promising route to accurately predict electronically excited energy surfaces away from molecular equilibria. However, its accuracy and practicality for chemical systems of photochemical interest have yet to be fully determined. We investigate the performance of the SS-CASSCF theory for the low-lying ground and excited states in the double bond rotation of ethylene. We show that state-specific approximations with a minimal (2e,2o) active space provide comparable accuracy to state-averaged calculations with much larger active spaces, while optimizing the orbitals for each excited state significantly improves the spatial diffusivity of the wave function. However, the incorrect ordering of state-specific solutions causes excited state solutions to coalesce and disappear, creating unphysical discontinuities in the potential energy surface. Our findings highlight the theoretical challenges that must be overcome to realize practical applications of state-specific electronic structure theory for computational photochemistry.

Excited States, Symmetry Breaking, and Unphysical Solutions in State-Specific CASSCF Theory.

State-specific electronic structure theory provides a route toward balanced excited-state wave functions by exploiting higher-energy stationary points of the electronic energy. Multiconfigurational wave function approximations can describe both closed- and open-shell excited states and avoid the issues associated with state-averaged approaches. We investigate the existence of higher-energy solutions in complete active space self-consistent field (CASSCF) theory and characterize their topological properties. We demonstrate that state-specific approximations can provide accurate higher-energy excited states in H2 (6-31G) with more compact active spaces than would be required in a state-averaged formalism. We then elucidate the unphysical stationary points, demonstrating that they arise from redundant orbitals when the active space is too large or symmetry breaking when the active space is too small. Furthermore, we investigate the singlet-triplet crossing in CH2 (6-31G) and the avoided crossing in LiF (6-31G), revealing the severity of root flipping and demonstrating that state-specific solutions can behave quasi-diabatically or adiabatically. These results elucidate the complexity of the CASSCF energy landscape, highlighting the advantages and challenges of practical state-specific calculations.

Comparing Theories of Resource Distribution: The Case of Iran

This study addresses inequality through resource distribution in Iranian provinces with the use of new data collected and compiled from various sources using multilevel modeling. The models compare predictions of the various resource distribution theories using Iran’s 31 provincial budgets over 10 years. This resource distribution study provides a rare look at inequality in a country that, to a large degree, prohibits such examinations. The authors find that although ethnicity, religiosity, and allegiance to the regime have no influence on provincial budgets, geographic proximity to the central government is the strongest and only consistent predictor of resource allocation. The authors find that the greater the distance from the capital, the larger the respective per capita budget, net of other factors. This study contradicts both the centrality theory of state economic distribution and specific Iranian economic distribution theory and provides a new body of knowledge on state resource distribution.

Aspects of Size-Consistency of Orbitally Noninvariant Size-Extensive Multireference Perturbation Theories: A Case Study Using UGA-SSMRPT2 as a Prototype.

Profiling a potential energy surface (PES), all the way to dissociate a molecular state into particular fragments and to display real or avoided crossings, requires a multireference description and the maintenance of size-consistency. The many body methods, which suit this purpose, should thus be size-extensive. Size-extensive theories, which are invariant with respect to transformation among active orbitals are, in principle, size-consistent. Relatively cheaper size-extensive theories, which do not possess this invariance, can still be size-consistent if the active orbitals are localized on the asymptotic fragments. Such methods, if perturbative in nature, require the use of an unperturbed Hamiltonian, which has orbital invariance with respect to the transformation within active, core, and virtual orbitals. The principal focus of this paper is to numerically realize size-consistency with localized active orbitals using our recently developed orbitally noninvariant Unitary Group Adapted State Specific Multireference second order Perturbation Theory (UGA-SSMRPT2) as a prototype method. Our findings expose certain generic potential pitfalls of size-extensive but orbitally noninvariant MRPT theories, which are mostly related to the inability of reaching proper localized active orbitals in the fragments due to the artifacts of the orbital generation procedure. Despite the invariance of the zeroth order CAS function, lack of invariance of the MRPT itself then leads to size-inconsistency. In particular, reaching symmetry broken fragment active orbitals is an issue of concern where suitable state-averaging might ameliorate the problem, but then one has to abandon full orbital optimization. Additionally, there can be situations where the orbitals of the fragment reached as an asymptote of the supermolecule are not the same as those obtained from the optimization of the fragments individually and will require additional transformation. Moreover, for a certain PES, one may either abandon the use of optimized orbitals for that state to preserve proper symmetry and degeneracy in the fragment orbitals or be satisfied with the use of optimized orbitals, which generate broken symmetric orbitals in the fragmentation limit. All these pathologies are illustrated using the PES of various electronic states of multiply bonded systems like N2, C2H2, HCN, C2, and O2. Subject to such proviso, the UGA-SSMRPT2 turns out to be an excellent theory for studying the PES leading to fragmentation of strongly correlated systems satisfying the requirements of size-consistency with localized active orbitals. An unexpected spin-off of our studies is the realization that the size-inextensive MRMP2, which bears a close structural similarity with our theory, might under certain situations display size-consistency. We analyze this feature concretely in our paper. Our studies may serve as a benchmark for monitoring numerically the size-consistency of any state specific multireference theory which is size-extensive but not invariant with respect to transformation of active orbitals.

Discontinuities-free complete-active-space state-specific multi-reference coupled cluster theory for describing bond stretching and dissociation.

The earlier proposed multi-reference state-specific coupled-cluster theory with the complete active space reference [CASCC; Lyakh et al., J. Chem. Phys. 122, 024108 (2005)] suffered from a problem of energy discontinuities when the formal reference state was changing in the calculation of the potential energy curve (PEC). A simple remedy to the discontinuity problem is found and is presented in this work. It involves using natural complete active space self-consistent field active orbitals in the complete active space coupled-cluster calculations. The approach gives smooth PECs for different types of dissociation problems, as illustrated in the calculations of the dissociation of the single bond in the hydrogen fluorine molecule and of the symmetric double-bond dissociation in the water molecule.

Ab initio wavefunctions in bioinorganic chemistry: More than a succès d’estime?

Ab initio wavefunctions in bioinorganic chemistry: More than a succès d’estime?

Potential energy surface studies via a single root multireference coupled cluster theory

We have employed complete active space based single root multireference coupled cluster method (the resulting method is referred to by the acronym sr-MRCC) to compute the potential energy surfaces (PESs) of some well studied "protypical model" systems for which a highly accurate and reliable database is available for comparison. As that of state-specific theory, the sr-MRCC approach focuses and correlates one state while using a multiconfigurational reference and thus it naturally avoids intruder states. The present method is structurally different from the well known state specific multireference coupled cluster (SS-MRCC) method introduced by Mahapatra et al. [Mol. Phys. 94, 157 (1998)]. As that of the SS-MRCC theory, the present method is also based on the Jeziorski-Monkhorst ansatz where a different exponential cluster operator exp(T(mu)) acts on its corresponding model function phi(mu). The final cluster finding equations contain coupling between the cluster operators for all the mu, which are mainly responsible to prove the extensivity of both the cluster amplitudes and the energy. The present sr-MRCC theory is size-extensive and size-consistent when localized orbitals are used. The systems considered here exhibit varying degrees of degeneracy at different regions of PES. The treatment of these systems via traditional effective Hamiltonian based methods suffers from divergence problems in the iterative solution of the CC equations (the issue termed as "intruder state"). The sr-MRCC results lie closer to the ones obtained by the SS-MRCC method for these systems. To judge the efficacy of the present method, we have compared our results with other previously published theoretical estimations, which clearly indicate that the present method is reliable in studying the dissociation PES of states plagued by electronic degeneracy as well as notorious intruder effects. The highly satisfactory performance of the sr-MRCC method, vis-a-vis the other sophisticated methods, in describing the lowest and the first excited singlet states of BeH(2) at points of high degeneracy is noticeable.

Multireference state-specific coupled-cluster methods. State-of-the-art and perspectives

This work reviews the state-specific multireference coupled-cluster (CC) approaches which have been developed as approximate methods for performing high-level quantum mechanical calculations on quasidegenerate ground and excited states of atomic and molecular systems. The term "quasidegenerate" refers to a state that cannot be described even in the first approximation by a single-determinant wavefunction (a Slater determinant), but requires two or more determinants for this purpose. The main challenge with applying the coupled-cluster theory to such states is in describing the electron correlation effects in the wavefunctions representing these states in a manner that is size-extensive, yet accurate and simple enough so the method can be routinely applied to small and medium-size molecular systems. We are describing how this can be accomplished within a theory that focuses on only one state of the system in a single CC calculation (the state-specific theory).

Applications of linear response theories to compute the low-lying potential energy surfaces: state-specific MRCEPA-based approach

Linear response theories based on the state specific multi-reference coupled electron-pair approximation-like methods (SS-MRCEPA) (Chattopadhyay and Mahapatra 2004 J. Phys. Chem. A 108 11664) (MRCEPA-LRT) have been amply utilized for computing excited-state potential energy surfaces (PES). Our MRCEPA-LRT methods provide energies in a size-intensive manner. As a pilot application, the effectiveness of the method is demonstrated by computing the PES of some low-lying excited states including the ground state of the BeH2 molecule. This system is very effective and widely used to judge the potentiality of any state-specific theory since its ground state possesses degeneracy/quasi-degeneracy and also faces intruders at different regions of the PES.

Excited states in the multireference state-specific coupled-cluster theory with the complete active space reference

The recently proposed multireference state-specific coupled-cluster theory with the complete active space reference has been used to study electronically excited states with different spatial and spin symmetries. The algorithm for the method has been obtained using the computerized approach for automatic generation of coupled-cluster diagrams with an arbitrary level of the electronic excitation from a formal reference determinant. The formal reference is also used to generate the genuine reference state in the form of a linear combination of determinants contracted to a configuration with the spin and spatial symmetries of the target state. The natural-orbital expansions of the one-electron configuration inferaction density matrix allowed us to obtain the most compact orbital space for the expansion of the reference function. We applied our approach in the calculations of singlet and triplet states of different spatial symmetries of the water molecule. The comparisons of the results with values obtained using other many-particle methods and with the full configuration interaction results demonstrate good ability of the approach to deal with electronic excited states.

An externally-corrected size-extensive single-root MRCC formalism: Its kinship with the rigorously size-extensive state-specific MRCC theory

The rigorously size-extensive intruder-free state-specific multi-reference coupled cluster theory (SSMRCC) developed by Mukherjee and coworkers [J. Chem. Phys. 110 (1999) 6171] makes use of the Jeziorski–Monkhorst Ansatz [ Ω = ∑ μ exp ( T μ ) | ϕ μ 〉 〈 ϕ μ | ] involving a different cluster operator exp( T μ ) acting on its corresponding model function ϕ μ . The resulting equations involve a coupling between the cluster operators for all the μ as demanded by the rigorous requirement of size-extensivity. If one wishes a size-extensive formalism where such couplings are minimal, one will have to replace the exact coupling term by hand where only T μ for a given ϕ μ appears. In this paper, we propose such an externally corrected size-extensive single-root multi-reference CC (ecsr-MRCC) formalism, which is intruder-free and simpler in structure as compared to the parent SSMRCC theory. Our intention in this paper is not to replace or underplay the efficacy of the rigorous SSMRCC theory, but to indicate how a simple external correction leads to results of similar quality where the coupling of T μ s for various μ are avoided. Kinship of the ecsr-MRCC formalism with the parent SSMRCC, and certain other allied MRCC theories will also be discussed. The performance of the method will be assessed by applying it to H4, BeH 2, F 2 and CH 2, and comparing them against the SSMRCC results, benchmark full CI results (when available), and those from the allied MRCC formalisms. The encouraging results indicate that this simple modification leads to equations with the minimal coupling without unduly sacrificing the accuracy.

Calculation ofn=3intrashell resonance states ofHe−and of isoelectronic atoms

We discuss the theory and computation of the lowest three, $n=3$ intrashell triply excited resonance states of ${\mathrm{He}}^{\ensuremath{-}}$ of Li, and of positive ions of the sequence, up to ${\mathrm{N}}^{4+}.$ These are the ${3s}^{2}3p{}^{2}{P}^{o},$ ${3s3p}^{2}{}^{4}P,$ and ${3s3p}^{2}{}^{2}D$ states, for which wave function characteristics, energies, and widths are reported. Contrary to recently published results for ${\mathrm{He}}^{\ensuremath{-}}$ and to earlier ones for ${\mathrm{N}}^{4+},$ we found that electron correlation and orthogonality to lower states are such that they make the ${}^{2}{P}^{o}$ state the lowest $n=3$ triply excited state (TES), as is the case with the $n=2$ shell. Our predictions for these states of ${\mathrm{He}}^{\ensuremath{-}}$ are in harmony with the measurements of Roy [Phys. Rev. Lett. 38, 1062 (1977)], which were interpreted only recently [C. A. Nicolaides and N. A. Piangos, J. Phys. B 34, 99 (2001)]. In addition, our value for the position of the Li ${3s}^{2}3p{}^{2}{P}^{o}$ TES, 175.15 eV, agrees with the measurement $(175.165\ifmmode\pm\else\textpm\fi{}0.050\mathrm{eV})$ of Diehl et al. [Phys. Rev. A 56, R1071 (1997)]. Apart from specifics, the paper discusses or points to certain basic aspects of computational quantum mechanics of such multiply excited states. For example, it refers to the utility of open-channel-like configurations toward proper convergence to a local energy minimum in the continuous spectrum, where quasibound and unbound states of the same symmetry lie below, and for which the normal eigenvalue properties of the discrete spectrum do not apply. Also, we discuss the possibility that is given by the state-specific theory for carrying out economic and physically transparent calculations and for deducing semiquantitative conclusions about the interplay between electronic structure, interference, and autoionization widths.

He−2Dweakly bound triply excited resonances: Interpretation of previously unexplained structures in the experimental spectrum

A He plus electron scattering experiment by Gosselin and Marmet [Phys. Rev. A 41, 1335 (1990)] produced the previously known ${\mathrm{He}}^{\ensuremath{-}}$ ${2s2p}^{2}{}^{2}D$ resonance at $58.283\ifmmode\pm\else\textpm\fi{}0.003\mathrm{eV}$ with a width $\ensuremath{\Gamma}=59\ifmmode\pm\else\textpm\fi{}4\mathrm{meV},$ and two new structures at $58.415\ifmmode\pm\else\textpm\fi{}0.005$ and $58.48\ifmmode\pm\else\textpm\fi{}0.02\mathrm{eV}$ with $\ensuremath{\Gamma}l2\mathrm{meV}$ and $\ensuremath{\Gamma}=70\ifmmode\pm\else\textpm\fi{}20\mathrm{meV},$ respectively. The nature of these last two structures has since remained unexplained. Heuristic analysis of the spectrum of the triply excited states (TES) of ${\mathrm{He}}^{\ensuremath{-}}$ and of conditions of electron correlation and possible localization, and application of the state-specific theory (SST) for the calculation of electronic structures and properties led to the determination of three resonance states of ${}^{2}D$ symmetry that explain quantitatively the experimental data. The implementation of the SST involved the combination of suitable choices of optimized orbitals, of nonorthonormal configuration interaction, and of mixing of bound with scattering configurations. The first two ${}^{2}D$ TES contain the ${2s2p}^{2}$ configuration as a major component, but strong radial correlation results in the spatial separation of the two p spin orbitals. The third TES, computed systematically with up to 996 optimized configurations, is an open-channel like localized wave packet.

Existence and characterization of n = 3 triply excitedresonances of He-

We predict, via large-scale calculations carried out in theframework of the state-specific theory of resonance states andaccounting for all the localized electron correlations, thepositions of 11 new triply excited resonances of He- inthe region between 68.7 and 70.3 eV. These states can, inprinciple, be reached in electron-He 1s2 1S and1s2s 3S scattering, and have the symmetries2S, 4Po, 2Po,2D, 4Fo, 2Fo and2G. A few of these are essentially overlapping. Theirzero-order representations consist of strongly mixing n = 3configurations. Of special significance to achieving goodconvergence of the calculation to local energy minima in thecontinuous spectrum, while accounting for a large portion of thebound open-channel interactions, is the inclusion ofopen-channel-like configurations, includedself-consistently via the multi-configurational Hartree-Fockmethod. Our results allow the long overdue confirmation andinterpretation of Roy's measurements (Roy D 1977 Phys. Rev. Lett. 38 1062).

State-specific theory and computation of a polyelectronic atomic state in a magnetic field. Application to doubly excited state of H-

The problem of computing the intrinsic properties of the system 'many-electron atomic state plus magnetic field' is treated by formulating and solving state-specific, non-Hermitian matrices of manageable sizes. These are created by appropriate choices and optimization of different function spaces representing the localized and the asymptotic part of the wavefunction of the field-dressed state which is made square-integrable through the use of r to re-i theta for the coordinate of the outgoing electron. The localized parts of the wavefunction and of the Hamiltonian are expressed in terms of the real coordinates. Application was made to the closely lying H- doubly excited states 2p2 3P, '2s2+2p2' 1S and 2p2 1D for field strengths gamma =0.0-0.03 au and results were obtained for the variation of the binding energies with respect to the H n=2 threshold, the autoionization widths and the geometrical shapes of the electron density.

Computation of resonances by two methods involving the use of complex coordinates.

We have studied two different systems producing resonances, a highly excited multielectron Coulombic negative ion (the ${\mathrm{He}}^{\mathrm{\ensuremath{-}}}$ 2s2${\mathit{p}}^{2}$ $^{4}$P state) and a hydrogen atom in a magnetic field, via the complex-coordinate rotation (CCR) and the state-specific complex-eigenvalue Schr\"odinger equation (CESE) approaches. For the ${\mathrm{He}}^{\mathrm{\ensuremath{-}}}$ 2s2${\mathit{p}}^{2}$ $^{4}$P resonance, a series of large CCR calculations, up to 353 basis functions with explicit ${\mathit{r}}_{\mathit{i}\mathit{j}}$ dependence, were carried out to serve as benchmarks. For the magnetic-field problem, the CCR results were taken from the literature. Comparison shows that the state-specific CESE theory allows the physics of the problem to be incorporated systematically while keeping the overall size of the computation tractable regardless of the number of electrons.

Bound excited states of atomic negative ions

This brief review focuses on the physics and on the electronic structure theory and methods for the calculation of excited discrete states of atomic negative ions. These states are difficult to observe experimentally and, therefore, it is theory which in the past few years has taken the lead in predicting their existence and their properties (electron affinities, fine and hyperfine structure, relativistic autoionization, radiative autoionization, etc.). Our work toward the reliable calculation of such states and their properties has been carried out along the lines of a state-specific theory (SST) for the discrete as well as the continuous spectrum. According to SST, the aim is to obtain and apply, separately optimized, symmetry-adapted zeroth-order and one-, two-, three-etc. electron, virtual function spaces which are specific to the quantum-mechanical state or states involved in the problem. The SST for atoms is already developed at the Coulomb and Pauli-Breit levels, while a flexible, bound state, fully relativistic (Dirac-Breit) version is currently being tested. As examples of our approach to the calculation of wavefunctions and electron affinities, we present new results for excited bound states in Al −, Fe −, Mn − and Cu −.

Partial autoionization widths of inner hole states of O V from the complex-eigenvalue Schrödinger equation.

We have calculated the partial autoionization widths of the O v 1s2${s}^{2}$2p ${}^{3}$${P}^{0}$ and $^{1}\mathrm{P}^{0}$ states which were measured in recent experiments by Bruch et al. In order to account for electron correlation and interchannel coupling we have used the state-specific theory of autoionizing states with complex coordinates. Our results correspond to well-defined regions of stability of the widths and yield ratios for the partial widths which, contrary to existing theory, are in agreement with experiment.

Solution of the complex eigenvalue Schr�dinger equation for the rydberg series of inner hole excited autoionizing states. The Be 1s 2s 2 n p (n=2?5)3 P 0 and1 P 0 series

In the state-specific theory of autoionizing states, the square-integrable wavefunction of complex coordinates corresponding to the complex eigenvalue is separated into terms contributing to the stability of the state and terms contributing to its decay. The choice and optimization of the function spaces describing these terms are different. In this work, taking the inner hole excited Be 1s 2s2n p Rydberg series as an example, we compute the asymptotic correlation and the concomittant energy shifts and widths by expanding the open channels in terms ofback-rotated generalized Laguerre polynomials and by diagonalizing a complex Hamiltonian matrix where the coordinates of the operator and of the occupied and virtual orbitals are real. Interchannel coupling is determined by mixing the independently optimized asymptotic pair correlation functions via the diagonalization of the total matrix. We found that, as expected, the Auger widths of both3P0 and1P0 series converge as a function ofn to the value of the previously determined width of the Be+ 1s 2s22S state. We also computed the oscillator strengths of the transitions Be 1s2 2s21S→1s 2s2n p1P0, which had previously been calculated within the time-dependent Hartree-Fock approximation. Forn>2, the discrepancy is large.

Mechanisms of configuration interaction in relativistic autoionisation: threshold phenomena and the Li4PJlevels

The authors have calculated the relativistic autoionisation of the low-lying 4P/sub /J levels of Li using the state-specific theory with the Breit-Pauli Hamiltonian. The results provide a definitive explanation of the shorter lifetimes of metastable Li quartets above the Li+ 1s2s 3S threshold compared with the theoretical predictions from the radiative transition probabilities. This discrepancy between previous theory and experiment is explained in terms of the indirect mechanism of relativistic autoionisation, which is suddenly enhanced-as a result of symmetry, electron correlation and non-orthonormality effects-as the 1s2s 3S channel opens up. The explanation of the threshold effect is thus based on a configuration interaction mechanism and not on a low-energy electron emission effect recently proposed by Mannervik and Cederquist (1986).