Complete Model Space Research Articles

An integrated surrogate model constructing method: Annealing combinable Gaussian process

Surrogate models are widely used to mimic complex systems to reduce the high experimental cost. As the system becomes high-dimensional and complex, there is an increasing demand for building relatively simplified surrogate models that capture key variables and represent complex interactions. This study proposes the annealing combinable Gaussian process, an integrated solution for identifying key variables and constructing the high-precision surrogate model. Firstly, to identify redundant variables, this study optimises variables selection with a modified simulated annealing algorithm over the complete model space. This process is called the outer loop. Secondly, to improve the model accuracy and structure, this study constructs a non-parametric Gaussian process model with additive or multiplicative kernel, effectively extracting high-order interactions. Simultaneously, a Markov chain is proposed to sample the model space. The conditional entropy is used as the scoring rule for the simulated annealing algorithm of the outer loop. It is named the inner loop. This study also discusses the rationality of conditional entropy as a criterion. The annealing combinable Gaussian process performs well in various application scenarios, including regression and classification problems. Finally, our method is implemented with the particle-in-cell simulation to find out the key physics parameters.

Bayesian adaptive basket trial design using model averaging

In this article, we develop a Bayesian adaptive design methodology for oncology basket trials with binary endpoints using a Bayesian model averaging framework. Most existing methods seek to borrow information based on the degree of homogeneity of estimated response rates across all baskets. In reality, an investigational product may only demonstrate activity for a subset of baskets, and the degree of activity may vary across the subset. A key benefit of our Bayesian model averaging approach is that it explicitly accounts for the possibility that any subset of baskets may have similar activity and that some may not. Our proposed approach performs inference on the basket-specific response rates by averaging over the complete model space for the response rates, which can include thousands of models. We present results that demonstrate that this computationally feasible Bayesian approach performs favorably compared to existing state-of-the-art approaches, even when held to stringent requirements regarding false positive rates.

The determinants of CDS spreads: evidence from the model space

We apply Bayesian model averaging and a frequentistic model space analysis to assess the pricing determinants of credit default swaps (CDSs). Our study focuses on the complete model space of plausible models and thus supports ultimate robustness. Using a large dataset of CDS contracts we find that CDS price dynamics can be mainly explained by factors describing firms’ sensitivity to extreme market movements. More precisely, our results suggest that dynamic copula based measures of tail dependence incorporate most essential pricing information, making other potential determinants such as Merton-type factors or linear variables measuring the systematic market evolution negligible.

The Determinants of CDS Spreads: Evidence from the Model Space

We apply Bayesian Model Averaging and a frequentistic model space analysis to assess the pricing determinants of credit default swaps (CDS). Our study focuses on the complete model space of plausible models and thus supports ultimate transparency and robustness. Using a large dataset of CDS contracts we find that CDS price dynamics can be mainly explained by factors describing firms' sensitivity to extreme market movements. More precisely, our results suggest that dynamic copula based measures of tail dependence incorporate most essential pricing information, making other potential determinants such as Merton-type factors or linear variables measuring the systematic market evolution negligible. This is in line with intermediary asset pricing theory.

Efficacy of the SU(3) scheme for ab initio large-scale calculations beyond the lightest nuclei

We report on the computational characteristics of ab initio nuclear structure calculations in a symmetry-adapted no-core shell model (SA-NCSM) framework. We examine the computational complexity of the current implementation of the SA-NCSM approach, dubbed LSU3shell, by analyzing ab initio results for 6Li and 12C in large harmonic oscillator model spaces and SU3-selected subspaces. We demonstrate LSU3shell’s strong-scaling properties achieved with highly-parallel methods for computing the many-body matrix elements. Results compare favorably with complete model space calculations and significant memory savings are achieved in physically important applications. In particular, a well-chosen symmetry-adapted basis affords memory savings in calculations of states with a fixed total angular momentum in large model spaces while exactly preserving translational invariance.

The Determinants of CDS Spreads: Evidence from the Model Space

We apply Bayesian Model Averaging and a frequentistic model space analysis to assess the pricing-determinants of credit default swaps (CDS). Our study focuses on the complete model space of plausible models covering most of the variables and specifications used elsewhere in the literature, including different copula models. The approach followed supports ultimate transparency and robustness for the empirical study at hand. Using a large data-set of CDS contracts we find that CDS price dynamics can be mainly explained by factors describing firms' sensitivity to extreme market movements. More precisely, our results suggest that dynamic copula based measures of tail dependence incorporate almost all essential pricing information making other potential determinants such as Merton-type factors or variables measuring the systematic market evolution - based on simple means or principal component analysis - negligible.

Emergence of Simple Patterns in Complex Atomic Nuclei from First Principles

We study the structure of low-lying states in 6Li, 6He, 8Be, 8B, 12C, and 16O, using ab initio symmetry-adapted no-core shell model. The results of our study demonstrate that collective modes in light nuclei emerge from first principles. We investigate the impact of the symmetry-adapted model space on spectroscopic properties and, in the case of the ground state of 6Li, on elastic electron scattering charge form factor. The results confirm that only a small symmetry-adapted subspace of the complete model space is needed to accurately reproduce complete-space observables and the form factor momentum dependence.

Dominant Modes in Light Nuclei - Ab Initio View of Emergent Symmetries

An innovative symmetry-guided concept is discussed with a focus on emergent symmetry patterns in complex nuclei. In particular, the ab initio symmetry-adapted no-core shell model (SA-NCSM), which capitalizes on exact as well as partial symmetries that underpin the structure of nuclei, provides remarkable insight into how simple symmetry patterns emerge in the many-body nuclear dynamics from first principles. This ab initio view is complemented by a fully microscopic no-core symplectic shell-model framework (NCSpM), which, in turn, informs key features of the primary physics responsible for the emergent phenomena of large deformation and alpha-cluster substructures in studies of the challenging Hoyle state in Carbon-12 and enhanced collectivity in intermediate-mass nuclei. Furthermore, by recognizing that deformed configurations often dominate the low-energy regime, the SA-NCSM provides a strategy for determining the nature of bound states of nuclei in terms of a relatively small subspace of the symmetry-reorganized complete model space, which opens new domains of nuclei for ab initio investigations, namely, the intermediate-mass region, including isotopes of Ne, Mg, and Si.

Proposal of multi-root multi-reference coupled cluster formalisms

This work first returns on the intrinsic difficulties of multi-reference coupled cluster (MR-CC) formalisms. They may be formulated either in an effective Hamiltonian frame or in an intermediate effective Hamiltonian (IEH) one. In the former case complete model space approach being intractable, the incomplete model space approach is re-examined, and is formulated in terms of an IEH, despite the fact that the model space dimension is equal to the number of desired roots. Some of its drawbacks are illustrated on the magnetic systems problem. Then one proposes a multi-root complete active space (CAS)-based CC-SD, which only handles single and double excitation operators, generalising a previously proposed State Specific MR-CC formalism. The method proceeds through an iterative dressing of the matrix elements between the singles and doubles and the CAS determinants.

Ab initio No-core Shell Model Calculations in a SU(3)-based Coupling Scheme

We use powerful algorithms of computational group theory to perform ab initio configuration-interaction calculations in a SU(3)-based symmetry-adapted many-particle basis. We demonstrate that eigenfunctions for the low-lying states of 6Li, 8Be, 12C, and 16O exhibit a strong dominance of low proton, neutron, and total intrinsic spins that carry the same spatial deformation as the leading symplectic Sp(3,ℝ) irreducible representations. Our findings imply that only a small fraction of the complete model space is needed to model nuclear collective dynamics, deformation, and α-particle clustering even if one uses modern realistic interactions that do not preserve SU(3) symmetry. This in turn points to the importance of using a symmetry-adapted framework, one based on a LS coupling scheme with the associated spatial configurations organized according to deformation.

Crustal constraint through complete model space screening for diverse geophysical datasets facilitated by emulation

Deep crustal constraint is often carried out using deterministic inverse methods, sometimes using seismic refraction, gravity and electromagnetic datasets in a complementary or “joint” scheme. With increasingly powerful parallel computer systems it is now possible to apply joint inversion schemes to derive an optimum model from diverse input data. These methods are highly effective where the uncertainty in the system is small. However, given the complex nature of these schemes it is often difficult to discern the uniqueness of the output model given the noise in the data, and the application of necessary regularization and weighting in the inversion process means that the extent of user prejudice pertaining to the final result may be unclear. We can rigorously address the subject of uncertainty using standard statistical tools but these methods also become less feasible if the prior model space is large or the forward simulations are computationally expensive. We present a simple Monte Carlo scheme to screen model space in a fully joint fashion, in which we replace the forward simulation with a fast and uncertainty-calibrated mathematical function, or emulator. This emulator is used as a proxy to run the very large number of models necessary to fully explore the plausible model space. We develop the method using a simple synthetic dataset then demonstrate its use on a joint data set comprising first-arrival seismic refraction, MT and scalar gravity data over a diapiric salt body. This study demonstrates both the value of a forward Monte Carlo approach (as distinct from a search-based or conventional inverse approach) in incorporating all kinds of uncertainty in the modelling process, exploring the entire model space, and shows the potential value of applying emulator technology throughout geophysics. Though the target here is relatively shallow, the methodology can be readily extended to address the whole crust.

Nonadiabatic coupling terms for the GVVPT2 variant of multireference perturbation theory

A Lagrangian based approach was used to obtain analytic formulas for nonadiabatic coupling terms within a multireference perturbation theory description of molecular electronic structure. Specifically, formulas were developed for the second-order generalized Van Vleck perturbation theory (GVVPT2) description of electron correlation. The formalism can use either complete or incomplete model (or reference) spaces, and is limited, in this regard, only by the capabilities of the MCSCF program. Of particular interest, the suggested formalism can straightforwardly use state-averaged MCSCF descriptions of the reference space in which the states have arbitrary weights. Since GVVPT2 wave functions are not strictly orthogonal, the definition of nonadiabatic coupling terms is untrivial and a perturbation-order consistent definition is suggested herein from which working equations are developed.

Symmetry-adapted Ab Initio Theory for Many-body Correlations in Nuclei

We demonstrate that no-core shell-model results for low-lying states of light and medium mass nuclei, whether they are dilute or dense systems, reveal a strong dominance of low-spin and high-deformation configurations. This result is independent of whether the system Hamiltonian is phenomenological in nature or derived from a realistic interaction. It implies that only a small fraction of the complete model space is required for a description of such states, and this in turn points to the importance of using a symmetry-adapted, no-core shell-model framework for describing such nuclei, one based on an LS coupling scheme with the associated spatial configurations organized according to deformation.These results confirm that the pioneering work of early developers of the field, J. P. Elliott with his SU(3) model and M. Moshinsky with his U(3) many-body oscillator work, extends to more open, multi-shell environments. Specifically, algebraic methods are both relevant in such an environment and they can be used to quell the combinatorial growth in dimensionality that comes with the addition of oscillator shells to a model space. Indeed, our findings demonstrate the utility of a symmetry-adapted, no-core shell-model approach, one that takes advantage of group theoretical as well as advanced computational methods. And importantly, what at first glance appear to be a daunting task – casting complex algebraic expressions of a symmetry-adapted scheme into a user-friendly and efficient shell-mode code, turns out to be not only doable, but a logical framework that embraces constructs that can be made to execute efficiently on massively parallel, multi-processor (and core) systems. Early results for some light p-shell nuclei are presented. In addition, we will show that the method can be extended to heavier nuclei of the sd-shell and beyond, including some cases of special astrophysical interest in the upper fp- and lower gds-shells, like isotopes of Ge, Se, and even Kr.

GVVPT2 energy gradient using a Lagrangian formulation

A Lagrangian based approach was used to obtain analytic formulas for GVVPT2 energy nuclear gradients. The formalism can use either complete or incomplete model (or reference) spaces, and is limited, in this regard, only by the capabilities of the MCSCF program. An efficient means of evaluating the gradient equations is described. Demonstrative calculations were performed and compared with finite difference calculations on several molecules and show that the GVVPT2 gradients are accurate. Of particular interest, the suggested formalism can straightforwardly use state-averaged MCSCF descriptions of the reference space in which the states have arbitrary weights. This capability is demonstrated by some calculations on the ground and first excited singlet states of LiH, including calculations near an avoided crossing. The accuracy and usefulness of the GVVPT2 method and its gradient are highlighted by comparing the geometry of the near-C(2v) minimum on the conical intersection seam between the 1 (1)A(1) and 2 (1)A(1) surfaces of O(3) with values that were calculated at the multireference configuration interaction, including single and double excitations (MRCISD), level of theory.

Evaluation of the performance of single root multireference coupled cluster method for ground and excited states, and its application to geometry optimization

The complete model space (CAS) based "genuine" single root multireference (MR) coupled cluster (sr-MRCC) method [Mahapatra and Chattopadhyay, J. Chem. Phys. 133, 074102 (2010)] has been extended to enable geometry optimizations by adopting the numerical gradient scheme. The sr-MRCC theory is designed to treat quasidegeneracies of varying degrees through the computation of essential static and dynamic correlation effects in a balanced way while bypassing the intruder states problem in a size-extensive manner. The efficacy of our sr-MRCC gradient approach has been illustrated by the optimization of the geometries of N(2)H(2),CH(2),C(2)H(4),C(4)H(4),O(3) as well as trimethylenemethane (TMM) molecular systems, since such cases, by virtue of their complexity, warrant truly multireference description. We have explored the capability of the sr-MRCC approach to yield rotational energy surfaces for the ground and first singlet excited states of N(2)H(2). We also intend to explore the ground and the excited state energetics of some model systems (such as P4, H4, and H(8)) for the computation of excitation energies by relying on the sr-MRCC method. An analysis of the results and a comparison with previous pertinent theoretical works including state specific MRCC (SS-MRCC) theory of Mukherjee and co-workers have also been presented. Although in most of the cases, we observe a close behavior between the sr-MRCC and SS-MRCC method, the error in the sr-MRCC is lower than the overall error of the SS-MRCC calculations in the vicinity of the transition region (manifesting a significant quasidegenerate character). The present results show that the sr-MRCC method and its numerical gradient variant are generally applicable to very demanding model and realistic chemical problems at acceptable accuracy and affordable computational expense which together attests the efficacy and viability of the sr-MRCC formalism for handling of static and dynamic correlations simultaneously thereby ensuring a balanced description for bond-breaking and other quasidegenerate situations with a various degree of MR character. Our preliminary results illustrate that our sr-MRCC method is a potential competitor for other state specific MRCC theories.

Development of algebraic techniques for the atomic open-shell MBPT(3)

The atomic third-order open-shell many-body perturbation theory is developed. Special attention is paid to the generation and algebraic analysis of terms of the wave operator and the effective Hamiltonian as well. Making use of occupation-number representation and intermediate normalization, the third-order derivations are worked out by employing a computational software program that embodies the generalized Bloch equation. We prove that in the most general case, the terms of effective interaction operator on the proposed complete model space are generated by a maximum of eight types of the n–body (n ⩾ 2) parts of the wave operator. To compose the effective Hamiltonian matrix elements in an efficient way, the operators are written in irreducible tensor form. We present the reduction scheme which is applied throughout other effective operator approaches to many-body perturbation theories, such as the coupled-cluster approach.

Multireference general-model-space state-universal and state-specific coupled-cluster approaches to excited states

The concept of C-conditions, originally introduced in the framework of the multireference (MR), general-model-space (GMS), state-universal (SU), coupled-cluster (CC) approach with singles and doubles (GMS-SU-CCSD) to account for the internal amplitudes that vanish in the case of a complete model space, is applied to a state-selective or state-specific Mukherjee MR-CC method (MkCCSD). In contrast to the existing applications, the emphasis is on the description of excited states, particularly those belonging to the same symmetry species. The applicability of the C-conditions in all MR-SU-CC approaches is emphasized. Convergence problems encountered in the MkCCSD method when handling higher-lying states are pointed out. The performance of the GMS-SU-CCSD and MkCCSD methods is illustrated by considering low-lying vertical excitation energies of the ethylene molecule and para-benzyne diradical. A comparison with the equation-of-motion CCSD results, as well as with the available experimental data and recent multireference configuration interaction theoretical results, is also provided.

Lagrangian approach for geometrical derivatives and nonadiabatic coupling terms in MRCISD

A new formulation for multireference configuration interaction method, with single and double excitations (MRCISD), energy gradients and nonadiabatic coupling terms based on a Lagrangian approach is suggested and the gradients demonstrated to be accurate with calculations on several molecules. Of particular interest, the suggested formalism can straightforwardly use state-averaged-multiconfiguration self-consistent field (SA-MCSCF) descriptions of the reference space in which the states have arbitrary weights. This capability is demonstrated by some calculations on the ground and first excited 1Σ+ states of LiH, including calculations near an avoided crossing. The formalism can use either complete or incomplete model (i.e. reference) spaces, and is limited, in this regard, only by the capabilities of the MCSCF program. The formalism for nonadiabatic coupling shares many structural features with the energy gradient formalism, with the use of a function that corresponds to the energy for the coupling.

Model space incompleteness in multireference state-universal and state-selective coupled-cluster theories

The role of the so-called C-conditions, originally introduced in the general model space (GMS), state-universal (SU), coupled-cluster (CC) theory with singles and doubles (GMS SU CCSD) to account for the internal cluster amplitudes that vanish in the case of a complete model space, is explored in the context of the state-selective Mukherjee MR-CC method (MkCCSD). Using three examples that involve a considerable quasidegeneracy we show that the C-conditions can be usefully employed also in the MkCCSD method once an incomplete (or general) model space is used.

Generalized Møller-Plesset Partitioning in Multiconfiguration Perturbation Theory.

Two perturbation (PT) theories are developed starting from a multiconfiguration (MC) zero-order function. To span the configuration space, the theories employ biorthogonal vector sets introduced in the MCPT framework. At odds with previous formulations, the present construction operates with the full Fockian corresponding to a principal determinant, giving rise to a nondiagonal matrix of the zero-order resolvent. The theories provide a simple, generalized Møller-Plesset (MP) second-order correction to improve any reference function, corresponding either to a complete or incomplete model space. Computational demand of the procedure is determined by the iterative inversion of the Fockian, similarly to the single reference MP theory calculated in a localized basis. Relation of the theory to existing multireference (MR) PT formalisms is discussed. The performance of the present theories is assessed by adopting the antisymmetric product of strongly orthogonal geminal (APSG) wave functions as the reference function.