Multireference perturbation theory with optimized partitioning. I. Theoretical and computational aspects

Analysis and Assessment of Knowles' Partitioning in Many-Body Perturbation Theory

A detailed analysis of a new partitioning in many-body perturbation theory recently proposed by Knowles (J. Chem. Phys. 156, 011101, 2022), termed "perturbation adapted partitioning" (PAPT), is presented. Level shift and orbital rotation effects are identified as gears of the zero-order Hamiltonian. These two components are examined separately, revealing that, in themselves, neither of the two is competitive with the combined effect. The success of PAPT can be attributed to determining a set of molecular orbitals and corresponding orbital energies that can systematically outperform the canonical orbitals and Koopmans' energy-based Møller-Plesset partitioning. The self-consistent version of the method is also tested in terms of energy and convergence. Previous numerical studies are further complemented with an application to an inherent multireference example and an investigation of van der Waals interaction energies. In addition, a rigorous mathematical analysis of the consequence of the linear dependence of projection functions on the solution of the Knowles' equations is provided.

Quantum Chemical Approaches to the Calculation of NMR Parameters: From Fundamentals to Recent Advances

Quantum chemical methods for the calculation of indirect NMR spin–spin coupling constants and chemical shifts are always in progress. They never stay the same due to permanently developing computational facilities, which open new perspectives and create new challenges every now and then. This review starts from the fundamentals of the nonrelativistic and relativistic theory of nuclear magnetic resonance parameters, and gradually moves towards the discussion of the most popular common and newly developed methodologies for quantum chemical modeling of NMR spectra.

Downfolding of many-body Hamiltonians using active-space models: Extension of the sub-system embedding sub-algebras approach to unitary coupled cluster formalisms

In this paper, we discuss the extension of the recently introduced subsystem embedding subalgebra coupled cluster (SES-CC) formalism to unitary CC formalisms. In analogy to the standard single-reference SES-CC formalism, its unitary CC extension allows one to include the dynamical (outside the active space) correlation effects in an SES induced complete active space (CAS) effective Hamiltonian. In contrast to the standard single-reference SES-CC theory, the unitary CC approach results in a Hermitian form of the effective Hamiltonian. Additionally, for the double unitary CC (DUCC) formalism, the corresponding CAS eigenvalue problem provides a rigorous separation of external cluster amplitudes that describe dynamical correlation effects-used to define the effective Hamiltonian-from those corresponding to the internal (inside the active space) excitations that define the components of eigenvectors associated with the energy of the entire system. The proposed formalism can be viewed as an efficient way of downfolding many-electron Hamiltonian to the low-energy model represented by a particular choice of CAS. In principle, this technique can be extended to any type of CAS representing an arbitrary energy window of a quantum system. The Hermitian character of low-dimensional effective Hamiltonians makes them an ideal target for several types of full configuration interaction type eigensolvers. As an example, we also discuss the algebraic form of the perturbative expansions of the effective DUCC Hamiltonians corresponding to composite unitary CC theories and discuss possible algorithms for hybrid classical and quantum computing. Given growing interest in quantum computing, we provide energies for H₂ and Be systems obtained with the quantum phase estimator algorithm available in the Quantum Development Kit for the approximate DUCC Hamiltonians.

Application of multireference linearized coupled-cluster theory to atomic and molecular systems

Applications of the multireference linearized coupled-cluster single-doubles (MRLCCSD) to atomic and molecular systems have been carried out. MRLCCSD is exploited to calculate the ground-state energies of HF, H2O, NH3, CH4, N2, BF, and C2with basis sets, cc-pVDZ, cc-pVTZ and cc-pVQZ. The equilibrium bond lengths and vibration frequencies of HF, HCl, Li2, LiH, LiF, LiBr, BH, and AlF are computed with MRLCCSD and compared with the experimental data. The electron affinities of F and CH as well as the proton affinities of H2O and NH3are also calculated with MRLCCSD. These results are compared with the results produced with second-order perturbation theory, linearized coupled-cluster doubles (LCCD), coupled-cluster doubles (CCD), coupled-cluster singles and doubles (CCSD), CCSD with perturbative triples correction (CCSD(T)). It is shown that all results obtained with MRLCCSD are reliable and accurate.

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.

The restricted active space followed by second-order perturbation theory method: theory and application to the study of CuO2 and Cu2O2 systems

A multireference second-order perturbation theory using a restricted active space self-consistent field wave function as reference (RASPT2/RASSCF) is described. This model is particularly effective for cases where a chemical system requires a balanced orbital active space that is too large to be addressed by the complete active space self-consistent field model with or without second-order perturbation theory (CASPT2 or CASSCF, respectively). Rather than permitting all possible electronic configurations of the electrons in the active space to appear in the reference wave function, certain orbitals are sequestered into two subspaces that permit a maximum number of occupations or holes, respectively, in any given configuration, thereby reducing the total number of possible configurations. Subsequent second-order perturbation theory captures additional dynamical correlation effects. Applications of the theory to the electronic structure of complexes involved in the activation of molecular oxygen by mono- and binuclear copper complexes are presented. In the mononuclear case, RASPT2 and CASPT2 provide very similar results. In the binuclear cases, however, only RASPT2 proves quantitatively useful, owing to the very large size of the necessary active space.

Full configuration interaction calculation of the low lying valence and Rydberg states of BeH

The all-electron full configuration interaction (FCI) vertical excitation energies for some low lying valence and Rydberg excited states of BeH are presented in this article. A basis set of valence atomic natural orbitals has been augmented with a series of Rydberg orbitals that have been generated as centered onto the Be atom. The resulting basis set can be described as 4s2p1d/2s1p (Be/H) + 4s4p3d. It allows to calculate Rydberg states up to n= {3,4,5} of the s, p, and d series of Rydberg states. The FCI vertical ionization potential for the same basis set and geometry amounts to 8.298 eV. Other properties such as FCI electric dipole and quadrupole moments and FCI transition dipole and quadrupole moments have also been calculated. The results provide a set of benchmark values for energies, wave functions, properties, and transition properties for the five electron BeH molecule. Most of the states have large multiconfigurational character in spite of their essentially single excited nature and a number of them present an important Rydberg-valence mixing that is achieved through the mixed nature of the particle MO of the single excitations.

Comparison of low-order multireference many-body perturbation theories

Tests have been made to benchmark and assess the relative accuracies of low-order multireference perturbation theories as compared to coupled cluster (CC) and full configuration interaction (FCI) methods. Test calculations include the ground and some excited states of the Be, H(2), BeH(2), CH(2), and SiH(2) systems. Comparisons with FCI and CC calculations show that in most cases the effective valence shell Hamiltonian (H(v)) method is more accurate than other low-order multireference perturbation theories, although none of the perturbative methods is as accurate as the CC approximations. We also briefly discuss some of the basic differences among the multireference perturbation theories considered in this work.

Multiconfiguration perturbation theory: Size consistency at second order

A modified version of a previously elaborated multiconfiguration perturbation theory (MCPT) [Rolik et al. J. Chem. Phys. 119, 1922 (2003)] is presented. In the modified formulation size consistency is ensured at second order in energy, by omitting projectors from the zero order Hamiltonian operator. This MCPT formulation is abbreviated as SC2-MCPT (size consistent at second order). To ensure proper separability, we also require that energy denominators are constructed as differences of some one-particle energies. A similar choice for energy denominators also renders the well-known multireference Moller-Plesset (MRMP) energy size consistent at second order. The same thing applies to the related multireference perturbation theory by Witek, Nakano, and Hirao.

Multireference second-order perturbation theory: How size consistent is “almost size consistent”

A systematic study of the deviation from size consistency of the multireference second-order Moller-Plesset perturbation theory (MRMP2) method is presented. The size-consistency error is shown to depend on the number of monomers in a supermolecule calculation, size of basis set, number of correlated valence electrons, and size of active space. HF, F(2), and N(2) are used as test cases, with stretched bonds, to include simple, well-defined multireference character. This is essential in ensuring that MRMP2 is being tested as a multireference method. It is concluded that the MRMP2 and other multireference perturbation theory methods can exhibit significant size-consistency errors, and that the size of the error depends on the manner in which the perturbation theory is implemented.

Full configuration interaction calculation of singlet excited states of Be3

The full configuration interaction (FCI) study of the singlets vertical spectrum of the neutral beryllium trimer has been performed using atomic natural orbitals [3s2p1d] basis set. The FCI triangular equilibrium structure of the ground state has been used to calculate the FCI vertical excitation energies up to 4.8 eV. The FCI vertical ionization potential for the same geometry and basis set amounts to 7.6292 eV. The FCI dipole and quadrupole transition moments from the ground state are reported as well. The FCI electric quadrupole moment of the X (3)A(1) (') ground state has been also calculated with the same basis set (Theta(zz)=-2.6461 a.u., Theta(xx)=Theta(yy)=-1/2Theta(zz)). Twelve of the 19 calculated excited singlets are doubly excited states. Most of the states have large multiconfigurational character. These results provide benchmark values for electronic correlation multireference methods. (4ex6MO)CAS-SDCI values for the same energies and properties are also reported.

Convergence Enhancement in Perturbation Theory

The Frobenius norm of operator QW is minimized with respect to level shift parameters applied to the zero-order spectrum, where W is the perturbation while Q is the reduced resolvent of the zero-order Hamiltonian. The stationary condition leads to a simple formula for the level shifts which eliminates degeneracy-induced singularities. Such level shifts may increase the radius of convergence of the perturbation series, and may improve low-order perturbative estimations - as it is found in the cases of a simple matrix eigenvalue problem and the one-dimensional quartic anharmonic oscillator.

Multireference perturbation theory with optimized partitioning. II. Applications to molecular systems

The second-order multireference perturbation theory using an optimized partitioning, denoted as MROPT(2), is applied to calculations of various molecular properties-excitation energies, spectroscopic parameters, and potential energy curves-for five molecules: ethylene, butadiene, benzene, N(2), and O(2). The calculated results are compared with those obtained with second- and third-order multireference perturbation theory using the traditional partitioning techniques. We also give results from computations using the multireference configuration interaction (MRCI) method. The presented results show very close resemblance between the new method and MRCI with renormalized Davidson correction. The accuracy of the new method is good and is comparable to that of second-order multireference perturbation theory using Møller-Plesset partitioning.