Intruder State Problem Research Articles

Complete set of solutions of multireference coupled-cluster equations: The state-universal formalism

The homotopy method is used to obtain a complete set of solutions of multireference coupled-cluster equations within the framework of the state-universal coupled-cluster (SUCC) formalism. The solutions are obtained for an ab initio four-electron model system consisting of four hydrogen atoms in rectangular arrangement. Essential information about the structure of solutions of nonlinear SUCC equations is provided and analyzed, both numerically and analytically, in the form of two theorems. The existence of multiple solutions of SUCC equations provides new insights into the nature of convergence problems often encountered in SUCC calculations. The evidence for the possible association of some solutions of the SUCC equations with the intruder-state problem is presented and briefly discussed.

On a mixed Møller–Plesset Epstein–Nesbet partition of the Hamiltonian to be used in multireference perturbation configuration interaction

Within the frame of second-order multi-reference perturbation theory (MR-PT), drawing on previous work of Dyall, a new partition of the Hamiltonian, which can loosely be considered as intermediate between Møller–Plesset (MP) and Epstein–Nesbet (EN), is presented. The method, which is well suited for CAS-SCF zero-order wavefunctions, is expected to correct partially the defects of MP and EN while retaining their good properties. The new approach has been successfully applied to some test cases involving the homonuclear diatomics F 2, Cl 2 and N 2 where the faults of the EN treatment appear to be in large measure eliminated. Application to the Cr 2 diatomic shows that the dramatic intruder state problem suffered by the MP treatment is completely solved, affording a qualitatively good description of the potential energy curve for the ground state.

Ab initio correlation corrections to the Hartree-Fock quasi band-structure of periodic systems employing Wannier-type orbitals

A size-consistent ab initio formalism to calculate correlation corrections to ionization potentials as well as electron affinities of periodic systems is presented. Our approach is based on a Hartree-Fock scheme which directly yields local orbitals without any a posteriori localization step. The use of local orbitals implies non-zero off-diagonal matrix elements of the Fock operator, which are treated as an additional perturbation and give rise to localization diagrams. Based on the obtained local orbitals, an effective Bloch Hamiltonian is constructed to second order of perturbation theory with all third-order localization diagrams included. In addition, the summation of certain classes of diagrams up to infinite order in the off-diagonal Fock elements as well as the Epstein-Nesbet partitioning of the full Hamiltonian are discussed. The problem of intruder states, frequently encountered in many-body perturbation theory, is dealt with by employing the theory of intermediate Hamiltonians. As model systems we have chosen cyclic periodic structures up to an oligoethylene ring in double-zeta basis; however, the theory presented here straightforwardly carries over to infinite periodic systems.

Reduced multireference CCSD method: An effective approach to quasidegenerate states

Standard multireference (MR) coupled cluster (CC) approaches are based on the effective Hamiltonian formalism and generalized Bloch equation. Their implementation, relying on the valence universal or state universal cluster Ansatz, is very demanding and their practical exploitation is often plagued with intruder state and multiple solution problems. These problems are avoided in the so-called state selective or state specific (SS) MR approaches that concentrate on one state at a time. To preserve as much as possible the flexibility and generality offered by the general MR CC approaches, yet obtaining a reliable and manageable algorithm, we propose a novel SS strategy providing a size-extensive CC formalism, while exploiting the MR model space and the corresponding excited state manifold. This strategy involves three steps: (i) The construction of a variational configuration interaction (CI) wave function within the singly (S) and doubly (D) excited state manifold, (ii) the cluster analysis of this CI wave function providing the information about the higher than pair cluster amplitudes, and (iii) the exploitation of these amplitudes in the so-called externally corrected CCSD procedure. This approach is referred to as the reduced MR (RMR) SS CCSD method and is implemented at the ab initio level and applied to several model systems for which the exact full CI results are available. These include two four electron H4 systems (usually referred to as the H4 and S4 models), an eight electron H8 model and the singlet-triplet separation problem in CH2. It is shown that the RMR CCSD approach produces highly accurate results, is free from intruder state problems, is very general and effective and applicable to both closed and open shell systems.

Method of intermediate hamiltonians via eigenvalue-independent partitioning: application to theoretical spectroscopy

We develop and apply in this paper a coupled cluster (CC)-based intermediate hamiltonian method that is suitable for describing both the lower- and higher-lying excited/ionized states relative to a closed-shell ground state. Generation of the main roots corresponding to the lower-lying states is attempted via an open-shell CC expansion. This expansion dresses the hamiltonian appropriately to incorporate the effect of the virtual space Q in a size-extensive manner. We have shown that by recasting the CC equations in a pseudo-eigenvalue equation form, we may also generate the higher-lying states approximately. The space in which the dressed matrix works is the union of the starting model space (which now becomes the “main” model space, P m ) and the space reached by the action of the first power of the cluster operator on the main model space functions (they span the intermediate space P i ). Using the wave operator in the Fock space, the same kind of dressing is maintained for both the main and the intermediate functions via the use of the same transformed hamiltonian for both these types. This dressing thus incorporates the same decoupling of the Q space from both the P m and P i spaces. The P i , roots, however, miss the “extra” dressing which should arise because of extra hole-particle vacancies in the P i -space functions; to this extent, they are distorted. The pseudo-eigenvalue form of our working equations bypasses the difficult problem of intruder states in a straightforward and obvious way. Applications to compute the main and satellite Auger spectra of H 2O produce encouraging results, indicating the viability of the method.

Perturbation theory for electronic excited states: The low lying Rydberg states of water

The recently developed Λ-transformed perturbation theory offers a rigorous solution of the intruder state problem and can thus be employed for systematic construction of convergent perturbation series. The method is applied to the vertical excitation energies of the low lying Rydberg states of the water molecule and compared to results obtained by standard techniques for the calculation of excited electronic states. The values calculated by Λ-transformed perturbation theory are found to be of about the same quality as the best theoretical results available. It is shown that for practical applications the computational effort for a Λ-transformed perturbation series is only slightly greater than the cost for the original series.

Multi-partitioning quasidegenerate perturbation theory. A new approach to multireference Møller-Plesset perturbation theory

We propose a new approach to the construction of perturbative expansions for effective Hamiltonians, implying the use of multiple partitioning of the total Hamiltonian. This approach offers the possibility to formulate the state-universal multireference many-body perturbation theory retaining the most important features of conventional Møller-Plesset expansions. Restricted to the second order, the novel theory gives rise to a simple method of electronic structure calculations which combines a decontracted treatment of model-space components of wavefunctions with the strict size-consistency for complete model spaces and the ability to avoid, or at least attenuate, the intruder-state problem.

On Some Attempts to Generalize the Effective Hamiltonian Approach

A clear and simple method to derive various intermediate Hamiltonians is pro- posed. Several intermediate Hamiltonians usable for various purposes are presented. The new intermediate Hamiltonians have the same properties as the traditional effective Hamiltonians but they should present better convergence properties, and should avoid any intruder state problem P'

Explicit construction of a convergent MBPT series for the 1Δ state of C 2 and the H 2 ground state at large bond distance

We explicitly construct a convergent perturbation series for intrinsically divergent Møller—Plesset cases. The means to accomplish this goal is an explicit reduction of the size of the perturbation. This is achieved by means of a Λ transformation and an unlinked-graph shift. Generic multi-reference cases are treated by single-reference methods after a simple and feasible transformation of the Hamiltonian matrix to a basis comprising a suitable reference zero-order vector. This effectively solves the intruder-state problem. We present the 1Δ state of C 2 and the H 2 ground state at large bond distance as typical examples.

Convergence properties of the effective interaction

The convergence properties of two perturbative schemes to sum the so-called folded diagrams are critically reviewed in this work, with an emphasis on the intruder state problem. The methods we study are the approaches of Kuo and co-workers and Lee and Suzuki. The suitability of the two schemes for shell-model calculations is discussed

State-selective multireference coupled-cluster theory employing the single-reference formalism: Implementation and application to the H8 model system

The new state-selective (SS) multireference (MR) coupled-cluster (CC) method exploiting the single-reference (SR) particle-hole formalism, which we have introduced in our recent paper [P. Piecuch, N. Oliphant, and L. Adamowicz, J. Chem. Phys. 99, 1875 (1993)], has been implemented and the results of the pilot calculations for the minimum basis-set (MBS) model composed of eight hydrogen atoms in various geometrical arrangements are presented. This model enables a continuous transition between degenerate and nondegenerate regimes. Comparison is made with the results of SR CC calculations involving double (CCD), single and double (CCSD), single, double, and triple (CCSDT), and single, double, triple, and quadruple (CCSDTQ) excitations. Our SS CC energies are also compared with the results of the Hilbert space, state-universal (SU) MR CC(S)D calculations, as well as with the MR configuration interaction (CI) results (with and without Davidson-type corrections) and the exact correlation energies obtained using the full CI (FCI) method. Along with the ground-state energies, we also analyze the resulting wave functions by examining some selected cluster components. This analysis enables us to assess the quality of the resulting wave functions. Our SS CC theory truncated at double excitations, which emerges through selection of the most essential clusters appearing in the full SR CCSDTQ formalism [SS CCSD (TQ) method] provides equally good results in nondegenerate and quasidegenerate regions. The difference between the ground-state energy obtained with the SS CCSD(TQ) approach and the FCI energy does not exceed 1.1 mhartree over all the geometries considered. This value compares favorably with the maximum difference of 2.8 mhartree between the SU CCSD energies and the FCI energies obtained for the same range of geometries. The SS CCSD(T) method, emerging from the SR CCSDT theory through selection of the most essential clusters, is less stable, since it neglects very important semi-internal quadruple excitations. Unlike the genuine multideterminantal SU CC formalism, our SS CC approach is not affected by the intruder state problem and its convergence remains satisfactory in nondegenerate and quasidegenerate regimes.

A valence-universal coupled-cluster single- and double-excitations method for atoms. III. Solvability problems in the presence of intruder states

For pt. II see ibid., vol. 27, p. 829-42 (1994). To better understand the problems met when solving the equations of VU-CC approaches in the presence of intruder states, the authors are concerned with the following aspects of the solvability problem for sets of non-linear equations: the existence and properties of multiple solutions and the attainability of these solutions by means of various numerical methods. The study is concentrated on the equations obtained for Be within the framework of the recently formulated atomically oriented form of the valence-universal coupled-cluster theory accounting for one- and two-electron excitations (VU-CCSD/R) and based on the complete model space (2s2, 2p2). Six pairs of multiple solutions representing four 1S states are found and discussed. Three of these solutions provide amplitudes describing the 2p2 1S state for which the intruder state problem has been considered as extremely serious. Several known numerical methods have been applied to solve the same set of non-linear equations for the two-valence cluster amplitudes. It is shown that these methods perform quite differently in the presence of intruder states, which seems to indicate that the intruder state problem for VU-CC methods is partly caused by the commonly used methods of solving the non-linear equations.

A valence-universal coupled-duster single- and double-excitations method for atoms. II. Application to Be

For pt. I see Malinowski and Jankowski, ibid., vol. 26, p. 3035 (1993). The recently formulated atomically oriented form of the valence-universal coupled-cluster theory accounting for one- and two-electron excitations (VU-CCSD/R) is applied to several states or Be. The complete model space (CMS) is employed. Various approximate forms of this method including the lowest-order perturbation theory are also considered. The standard single-reference (SR) formulation of the CCSD, method, treated as a special case of the VU-CCSD/R approach, is applied to the ground state. The calculations for the 9s9p7d7f5g STO basis set are the most accurate ones performed so far for any SR- and VU-CCSD method formulated in terms of analytical orbitals. Proceeding from the SR- to the VU-formulation increased the magnitude of the ground-state correlation energy by 1.1%. Special attention is paid to the well known intruder state problem for the pair of 2s2 1S and 2p2 1S states. It is demonstrated that, despite earlier findings, the CMS based VU-CCSD, methods can be safely used in the presence of intruder states. The impact of the electronic structure of the series of systems considered in VU-CCSD calculations on the accuracy of results is also addressed.

Ab Initio Vertical Ionization Potentials of trans-Butadiene and Cyclobutadiene Using the Effective Valence Shell Hamiltonian Method

Moderate-size basis set ab initio calculations for the [pi]-electron vertical ionization potentials of trans- and cyclobutadiene (116 and 128 functions, respectively) are performed using the effective valence shell Hamiltonian (H[sup F]) method. These calculations demonstrate the flexibility and unique features of the H[sup F] method because the ionization potentials emerge from previous computations of the neutral valence-like and Rydberg state excitation energies using the same H[sup F]. The computed ionization potentials agree well with experiment and with previous calculations, including highly correlated ones for trans-butadiene. New predictions are provided for ionizations to quartet and shake-up states. Furthermore, in contrast to current beliefs about quasidegenerate perturbation theory, the third-order H[sup F] calculations do not encounter intruder state problems. 32 refs., 5 tabs.

Real-space quantum renormalization group and Anderson localization.

We discuss the application of a quantum real-space renormalization-group approach to the Anderson-localization problem. States obtained by applying combinations of fixed and free boundary conditions to the system are used as a truncated basis at each iteration. We discuss the problem of intruder states, which occurs when the renormalization-group procedure is used to find states at the center of the band; we also discuss some possible solutions. We present numerical results for the conductance, the localization length, and the scaling function for the Anderson-localization problem in one dimension.

Hilbert space multireference coupled-cluster methods. II. A model study on H8

The performance of various coupled-cluster (CC) approaches using both single and multideterminantal references is investigated for the (quasi-)degenerate states of molecular systems, where inclusion of higher excitations (or equivalently nondynamic correlation) proves to be needed. The prototype system H8 represents an adequate model for our study, where we can vary the degree of degeneracy from a completely degenerate situation to a nondegenerate one in a continuous way. To obtain a reliable benchmark for our CC results, the full configuration interaction (FCI) and large-scale complete active space configuration interaction (CAS CI) calculations, respectively, are performed for a variety of geometries and states. The convergence of the approximate single reference CC approaches is found to be extremely sensitive to the level of degeneracies involved. In the nondegenerate case the standard CC method with single and double excitations is found to be quite satisfactory; in the (quasi-)degenerate situations, however, the inclusion of triple excitations and noniterative quadruple excitations is needed to furnish semiquantitative values of correlation energies. The alternative treatment of nondynamic correlation using a multideterminantal Hilbert space coupled-cluster (MRCC) method demonstrates the power of this approach, which provides a balanced description of both dynamic and nondynamic correlation in the degenerate region for all the investigated states of H8. Its convergence for nondegenerate situations, however, is less satisfactory, being affected by an intruder state problem.

Negative-energy states in the Dirac—Hartree—Fock problem: the effect of omission of two-electron integrals involving the small component

The effect of omission of two-electron integrals involving basis functions for the small component of the wavefunction on the eigenvalue spectrum in the Dirac—Hartree—Fock problem is studied. From an analysis of the Fock matrix it is shown that omission of these integrals moves the negative-energy states down, not up. Their complete omission does not give rise to intruder states. The appearance of intruder states occurs when only some of the core integrals are omitted, due to the nature of particular contraction schemes used for the core basis functions. Use of radially localized functions rather than atomic functions alleviates the intruder state problem.

Rayleigh-Schrodinger QDPT for Hermitian intermediate Hamiltonians by the shift technique

A general approach to the construction of Rayleigh-Schrodinger-like perturbative expansions for Hermitian intermediate Hamiltonians using the shift technique is proposed. The shift operator depends on the projector onto the subspace spanned by the 'main' eigenvectors of the intermediate Hamiltonian which should be determined iteratively for a given PT order. The intruder state problem can be completely eliminated by an appropriate choice of an intermediate space and shift parameters without introducing any additional perturbation. Numerical examples are reported in order to demonstrate the convergency properties and approximate size consistency of the novel method.

A b i n i t i o study of the t r a n s-butadiene π-valence states using the effective valence shell Hamiltonian method

Low-lying π-electron vertical excitation energies of trans-butadiene are calculated using the effective valence shell Hamiltonian method. The results are compared with previous experimental and theoretical analyses of this molecule’s congested electronic spectra. The computations employ a large basis set (126 functions) which includes both diffuse functions on the carbon atoms and polarization functions on all atoms. Good agreement is obtained with the experimentally well known vertical excitation energies to the 1 3Bu, 1 3Ag, and 1 1Bu states where deviations from experiment are only 0.01, 0.01, and 0.22 eV, respectively. We confirm the experimental assignment of a valence like 1Ag state around 7.4 eV (calculated at 7.49 eV). Likewise, a member of a symmetry allowed 3p Rydberg series (of Au or Bu symmetry) in the electron impact spectrum with origin at 7.07 eV is assigned as the 2 1Bu state (with calculated vertical excitation energy of 7.00 eV). Most experiments place the 2 1Ag state above the 1 1Bu state; however, a resonance Raman assignment places it below. Our calculated excitation to the 2 1Ag state is 0.05 eV above the 2 1Bu state, about 0.5 eV lower than previous ab initio determinations. The computed vertical excitation energies are in good agreement with the interpretation of experimental electronic spectra, are in much better agreement with experiment than previously published ab initio calculations, provide the first definitive assignment of the 2 1Bu state at 7.08 eV, and conclusively assign the 3 1Ag state at 7.4 eV. The accuracy of the large basis effective valence shell Hamiltonian is, in part, due to retention of both valence and Rydberg orbitals in the valence space, a feature which has a bearing on intruder state problems and on current semiempirical π-electron theories.

The multireference coupled-cluster method in Hilbert space: An incomplete model space application to the LiH molecule

The first results from a Hilbert space, multireference coupled-cluster (CC) method in an incomplete model (active) space are reported for the five lowest states of LiH. The active space is spanned by several configurations at the level of single and double excitations, where the configuration(s) causing intruder state problems are excluded from the complete Hilbert reference space. Full inclusion of single- and double-excitation operators is considered in the expansion for the cluster operator, with all quadratic terms in the renormalization part. The multireference CC results for the ground (X 1Σ+) and four low-lying excited states (a 3Σ+, A 1Σ+, a 3Π, A 1Π ) of LiH are compared with the corresponding full configuration-interaction (FCI) energies. The agreement between FCI and CC values within a few hundredths of mH for the Π states proves the feasibility of the present method to describe, quantitatively, the quasicomplete reference space problem. Deviations of the incomplete multireference results from the exact FCI energies for for excited Σ states vary with respect to the quasidegeneracies involved from a few tenths of mH for the nondegenerate (near-equilibrium) geometries to several mH in the dissociation region.