Intruder State Problem Research Articles

GVVPT2 Multireference Perturbation Theory Description of Diatomic Scandium, Chromium, and Manganese

With relatively simple model spaces derived from valence bond models, a straightforward zero-order Hamiltonian, and the use of moderate-sized Dunning-type correlation consistent basis sets (cc-pVTZ, aug-cc-pVTZ, and cc-pVQZ), the second order generalized Van Vleck perturbation theory (GVVPT2) method is shown to produce potential energy curves (PECs) and spectroscopic constants close to experimental results for both ground and low-lying excited electronic states of Sc(2), Cr(2) and Mn(2). In spite of multiple quasidegeneracies (particularly for the cases of Sc(2) and Mn(2)), the GVVPT2 PECs are smooth with no discontinuities. Since these molecules have been identified as ones that widely used perturbative methods are inadequate for describing well, due to intruder state problems, unless shift parameters are introduced that can obfuscate the physics, this study suggests that the conclusion about the inadequacy of multireference perturbation theory be re-evaluated. The ground state of Sc(2) is predicted to be X(5)∑(u)(-), and its spectroscopic constants are close to the ones at the MRCISD level. Near equilibrium geometries, the 1(3)∑(u)(-) electronic state of Sc(2) is found to be less stable than the quintet ground state by 0.23 eV. The Cr(2) PEC has several features of the Rydberg-Klein-Rees (RKR) experimental curve (e.g., the pronounced shelf at elongated bond lengths), although the predicted bond length is slightly long (R(e) = 1.80 Å with cc-pVQZ compared to the experimental value of 1.68 Å). The X(1)∑(g)(+) ground state of Mn(2) is predicted to be a van der Waals molecule with a long bond length, R(e), of 3.83 Å using a cc-pVQZ basis set (experimental value = 3.40 Å) and a binding energy, D(e), of only 0.05 eV (experimental value = 0.1 eV). We obtained R(e) = 3.40 Å and D(e) = 0.09 eV at the complete basis set (CBS) limit for ground state Mn(2). Low lying excited state curves have also been characterized for all three cases (Cr(2), Mn(2), and Sc(2)) and show similar mathematical robustness as the ground states. These results suggest that the GVVPT2 multireference perturbation theory method is more broadly applicable than previously documented.

Constrained variational approach for energy derivatives in Intermediate Hamiltonian Fock-space coupled-cluster theory

Fock-space multi-reference coupled cluster theory is an efficient method for electronic structure of nearly degenerate cases. However, it suffers from the intruder state problem for large model space. The intermediate Hamiltonian formulation eliminates intruder state problem and helps in the convergence of equations. In this paper we have implemented intermediate Hamiltonian approach in Fock-space coupled cluster method for the response properties. We test our method for the dipole moments of doublet radicals. We report dipole moments of CN, SF and NS radicals.

Multi-reference Fock space coupled-cluster method in the intermediate Hamiltonian formulation for potential energy surfaces

The effective and intermediate Hamiltonian multi-reference coupled-cluster (CC) method with singles and doubles for the doubly ionized (0,2) sector of Fock space (FS) is formulated and implemented. The intermediate Hamiltonian realization of the (0,2) FS problem provides a robust computational scheme for solving the FS-CC equations free from the intruder state problem. By introducing an efficient factorization strategy, we obtain a very efficient tool that can be used for computing double ionization potentials but more significantly to describe multi-reference problems in CC theory, illustrated by twisted ethylene and the potential energy curve for F(2). The latter separates smoothly to two F atoms, while the former avoids the cusp behavior at the 90° dihedral. We also explore the double ionization potentials for several small molecules, H(2)O, CO, C(2)H(2), and C(2)H(4).

The spin-free analogue of Mukherjee's state-specific multireference coupled cluster theory

In this paper, we develop a rigorously spin-adapted version of Mukherjee's state-specific multireference coupled cluster theory (SS-MRCC, also known as Mk-MRCC) [U. S. Mahapatra, B. Datta, and D. Mukherjee, J. Chem. Phys. 110, 6171 (1999)] for reference spaces comprising open-shell configurations. The principal features of our approach are as follows: (1) The wave operator Ω is written as Ω = ∑(μ)Ω(μ)|φ(μ)>c(μ), where {φ(μ)} is the set of configuration state functions spanning a complete active space. (2) In contrast to the Jeziorski-Monkhorst Ansatz in spin-orbital basis, we write Ω(μ) as a power series expansion of cluster operators R(μ) defined in terms of spin-free unitary generators. (3) The operators R(μ) are either closed-shell-like n hole-n particle excitations (denoted as T(μ)) or they involve valence (active) destruction operators (denoted as S(μ)); these latter type of operators can have active-active scatterings, which can also carry the same active orbital labels (such S(μ)'s are called to have spectator excitations). (4) To simulate multiple excitations involving powers of cluster operators, we allow the S(μ)'s carrying the same active orbital labels to contract among themselves. (5) We exclude S(μ)'s with direct spectator scatterings. (6) Most crucially, the factors associated with contracted composites are chosen as the inverse of the number of ways the S(μ)'s can be joined among one another leading to the same excitation. The factors introduced in (6) have been called the automorphic factors by us. One principal thrust of this paper is to show that the use of the automorphic factors imparts a remarkable simplicity to the final amplitude equations: the equations consist of terms that are at most quartic in cluster amplitudes, barring only a few. In close analogy to the Mk-MRCC theory, the inherent linear dependence of the cluster amplitudes leading to redundancy is resolved by invoking sufficiency conditions, which are exact spin-free analogues of the spin-orbital based Mk-MRCC theory. This leads to manifest size-extensivity and an intruder-free formulation. Our formalism provides a relaxed description of the nondynamical correlation in presence of dynamical correlation. Pilot numerical applications to doublet systems, e.g., potential energy surfaces for the first two excited (2)A' states of asymmetric H(2)S(+) ion and the ground (2)Σ(+)state of BeH radical are presented to assess the viability of our formalism over an wide range of nuclear geometries and the manifest avoidance of intruder state problem.

The electronic structure of the triiodide ion from relativistic correlated calculations: A comparison of different methodologies

The triiodide ion I(3)(-) exhibits a complex photodissociation behavior, the dynamics of which are not yet fully understood. As a first step toward determining the full potential energy surfaces of this species for subsequent simulations of its dissociation processes, we investigate the performance of different electronic structure methods [time-dependent density functional theory, complete active space perturbation theory to second order (CASPT2), Fock-space coupled cluster and multireference configuration interaction] in describing the ground and excited states of the triiodide ion along the symmetrical dissociation path. All methods apart from CASPT2 include scalar relativity and spin-orbit coupling in the orbital optimization, providing useful benchmark data for the more common two-step approaches in which spin-orbit coupling is introduced in the configuration interaction. Time-dependent density functional theory with the statistical averaging of model orbital potential functional is off the mark for this system. Another choice of functional may improve performance with respect to vertical excitation energies and spectroscopic constants, but all functionals are likely to face instability problems away from the equilibrium region. The Fock-space coupled cluster method was shown to perform clearly best in regions not too far from equilibrium but is plagued by convergence problems toward the dissociation limit due to intruder states. CASPT2 shows good performance at significantly lower computational cost, but is quite sensitive to symmetry breaking. We furthermore observe spikes in the CASPT2 potential curves away from equilibrium, signaling intruder state problems that we were unable to curb through the use of level shifts. Multireference configuration interaction is, in principle, a viable option, but its computational cost in the present case prohibits use other than for benchmarking purposes.

Study of the Ground State Dissociation of Diatomic Molecular Systems Using State-Specific Multireference Perturbation Theory: A Brillouin-Wigner Scheme.

The size-extensive second-order state-specific (or single root) multireference (MR) perturbation theory (SS-MRPT) in the Brillouin-Wigner (BW) form using Mϕller-Plesset perturbative evaluations of orders up to 2 [termed as SS-MRMPPT(BW)] presents a viable, as well as promising, approach to include both nondynamic and dynamic correlations in the study of the bond-stretching (in multireference/quasidegenerate situations) of molecular species with a manageable cost/accuracy ratio. It combines numerical stability in the presence of an intruder state problem with strict size consistency (when localized orbitals are used). In this paper, the SS-MRMPPT(BW) method has been shown to properly break the bonds (in the ground state) of several diatomic molecules (such as F2, Cl2 and Br2, and BH) that have posed a severe challenge to any many-body theoretical approach due to the presence of quasidegeneracy of varying degrees in the ground state. A comparison of the resulting potentials with the various theoretical results reveals that the method represents a valuable tool that is capable of properly accounting even for very strong quasidegeneracies, while also performing well in nondegenerate situations. In this work, we have also calculated spectroscopic constants (such as equilibrium bond lengths, vibrational frequencies, and dissociation energies) of the ground state of these molecular systems. The SS-MRMPPT spectroscopic constants are compared with the most accurate available ab initio calculations and other theoretical estimates of previous works to calibrate the efficacy of the method. For the sake of completeness, we also compare the computed spectroscopic constants with the experimental observations. The accuracy of computed spectroscopic parameters appears to be rather consistent over a multitude of systems for various basis sets. The SS-MRMPPT enables quantitatively accurate and computationally affordable analysis of potential energy surfaces and spectroscopic constants of various multireference systems in the ground state. It is particularly visible for spectroscopic parameters and nonparallelism error (NPE) calculations. The calculations further reveal that the SS-MRMPPT(BW) method compared to its Rayleigh-Schrödinger counterpart [SS-MRMPPT(RS)] provides a more accurate and consistent solution for the whole dissociation path and spectroscopic constants.

Strongly contracted canonical transformation theory

Canonical transformation (CT) theory describes dynamic correlation in multireference systems with large active spaces. Here we discuss CT theory's intruder state problem and why our previous approach of overlap matrix truncation becomes infeasible for sufficiently large active spaces. We propose the use of strongly and weakly contracted excitation operators as alternatives for dealing with intruder states in CT theory. The performance of these operators is evaluated for the H(2)O, N(2), and NiO molecules, with comparisons made to complete active space second order perturbation theory and Davidson-corrected multireference configuration interaction theory. Finally, using a combination of strongly contracted CT theory and orbital-optimized density matrix renormalization group theory, we evaluate the singlet-triplet gap of free base porphin using an active space containing all 24 out-of-plane 2p orbitals. Modeling dynamic correlation with an active space of this size is currently only possible using CT theory.

A study of cumulant approximations to n-electron valence multireference perturbation theory

We investigate the possibility of reducing the complexity of multireference perturbation theory through cumulant based approximations to the high-order density matrices that appear in such theories. Our test cases show that while the cumulant approximated forms are degraded in accuracy relative to the parent theory and exhibit intruder state problems that must be carefully handled, they may provide a route to a simple estimation of dynamic correlation when the parent perturbation theory is infeasible. Nonetheless, further work is clearly needed on better approximations to the denominators in the perturbation theory.

Intruder states in multireference perturbation theory: The ground state of manganese dimer

A detailed analysis of a severe intruder state problem in the multistate multireference perturbation theory (MS-MRPT) calculations on the ground state of manganese dimer is presented. An enormous number of detected intruder states (> 5000) do not permit finding even an approximate shape of the X(1)Sigma(g) (+) potential energy curve. The intruder states are explicitly demonstrated to originate from quasidegeneracies in the zeroth-order Hamiltonian spectrum. The electronic configurations responsible for appearance of the quasidegeneracies are identified as single and double excitations from the active orbitals to the external orbitals. It is shown that the quasidegeneracy problem can be completely eliminated using shift techniques despite of its severity. The resultant curves are smooth and continuous. Unfortunately, strong dependence of the spectroscopic parameters of the X(1)Sigma(g) (+) state on the shift parameter is observed. This finding rises serious controversies regarding validity of employing shift techniques for solving the intruder state problem in MS-MRPT. Various alternative approaches of removing intruder states (e.g., modification of the basis set or changing the active space) are tested. None of these conventional techniques is able to fully avoid the quasidegeneracies. We believe that the MS-MRPT calculations on the three lowest A(g) states of manganese dimer constitute a perfect benchmark case for studying the behavior of MRPT in extreme situations.

Explicitly intruder-free valence-universal multireference coupled cluster theory as applied to ionization spectroscopy

Although it is quite promising to compute the spectroscopic energies [say, ionization potential (IP)] via the traditional valence-universal multireference coupled cluster (VUMRCC) method based on the description of the complete model space being seriously plagued by the perennial intruder state problem, the eigenvalue independent partitioning (EIP) based VUMRCC (coined as EIP-MRCC) method is quite effective to predict the spectroscopic energies in an intruder-free manner. Hence, the EIP-MRCC method is suitable for generating both the principal IPs and the satellite IPs of the inner-valence region. An EIP strategy converts the nonlinear VUMRCC equations for M(m,n) dimensional model space of m hole and n particle to a non-Hermitian eigenproblem of larger dimension whose M(m,n) roots are only physically meaningful. To increase the quality of the computed energy differences in the sense of chemical accuracy and to locate the correct position of it in the spectrum, the inclusion of higher-body cluster operators on top of all the standard singles-doubles is not the only pivotal issue, the effect of the size of the basis set is also equally important. This paper illustrates these issues by calculating the principal and satellite IPs of HF and HCl molecules using various basis sets (viz., Dunning's cc-pVDZ, cc-pVTZ, and cc-pVQZ) via EIP-MRCC method with full inclusion of triples (abbreviated as EIP-MRCCSDT). The results seem quite encouraging in comparison with the experimental values. The controversial 2Pi satellite at 28.67 eV of HCl of Svensson et al. [J. Chem. Phys. 89, 7193 (1988)] is also reported.

Correlated one-particle method: Numerical results

In a previous paper a correlated one-particle method was formulated, where the effective Hamiltonian was composed of the Fock operator and a correlation potential. The objective was to define a correlated one-particle theory that would give all properties that can be obtained from a one-particle theory. The Fock-space coupled-cluster method was used to construct the infinite-order correlation potential, which yields correct ionization potentials (IP's) and electron affinities (EA's) as the negative of the eigenvalues. The model, however, was largely independent of orbital choice. To exploit the degree of freedom of improving the orbitals, the Brillouin-Brueckner condition is imposed, which leads to an effective Brueckner Hamiltonian. To assess its numerical properties, the effective Brueckner Hamiltonian is approximated through second order in perturbation. Its eigenvalues are the negative of IP's and EA's correct through second order, and its eigenfunctions are second-order Brueckner orbitals. We also give expressions for its energy and density matrix. Different partitioning schemes of the Hamiltonian are used and the intruder state problem is discussed. The results for ionization potentials, electron affinities, dipole moments, energies, and potential curves are given for some sample molecules.

The coupled-cluster corrections to multi-reference configuration interaction method: the HF and F2 bond breaking description*

Potential energy curves may be difficult to describe by multi-reference (MR) coupled-cluster or perturbation methods because of the intruder state problem. To avoid intruder states one may have to change the reference space while going from one internuclear distance to another which can make the description inconsistent and cause discontinuities on the potential energy curve. Conversely to this, the MR configuration interaction method with singles and doubles (MR-CISD) does not require the reference space that is specially tailored for particular geometries to avoid intruders. A disadvantage of the MR-CISD scheme is the relatively poor description of the dynamic correlation which does not benefit from the efficiency of the cluster expansion. However, if the coupled-cluster corrections to MR-CISD are a posteriori used to improve that description then a very accurate potential energy curve can be obtained. In this preliminary study the corrections are applied in model calculations to investigate the HF and F2 bond breaking cases. Also the problem of size-inextensivity of the MR-CISD method is addressed. A numerical test shows that the corrections are capable of significant reduction of the MR-CISD inextensivity error.

Quadratic Padé approximants and the intruder state problem of multireference perturbation methods

Simple and quadratic Pade resummation methods are applied to high-order series from multireference many-body perturbation theory (MR-MBPT) calculations using various partitioning schemes (Moller–Plesset, Epstein–Nesbet, and forced degeneracy) to determine their efficacy in resumming slowly convergent or divergent series. The calculations are performed for the ground and low-lying excited states of (i) CH2, (ii) BeH2 at three geometries, and (iii) Be, for which full configuration interaction (CI) calculations are available for comparison. The 49 perturbation series that are analyzed include those with oscillatory and monotonic divergence and convergence, including divergences that arise from either frontdoor or backdoor intruder states. Both the simple and quadratic Pade approximations are found to speed the convergence of slowly convergent or divergent series. However, the quadratic Pade method generally outperforms the simple Pade resummation. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

A state-specific approach to multireference coupled electron-pair approximation like methods: development and applications.

The traditional multireference (MR) coupled-cluster (CC) methods based on the effective Hamiltonian are often beset by the problem of intruder states, and are not suitable for studying potential energy surface (PES) involving real or avoided curve crossing. State-specific MR-based approaches obviate this limitation. The state-specific MRCC (SS-MRCC) method developed some years ago can handle quasidegeneracy of varying degrees over a wide range of PES, including regions of real or avoided curve-crossing. Motivated by its success, we have suggested and explored in this paper a suite of physically motivated coupled electron-pair approximations (SS-MRCEPA) like methods, which are designed to capture the essential strength of the parent SS-MRCC method without significant sacrificing its accuracy. These SS-MRCEPA theories, like their CC counterparts, are based on complete active space, treat all the reference functions on the same footing and provide a description of potentially uniform precision of PES of states with varying MR character. The combining coefficients of the reference functions are self-consistently determined along with the cluster amplitudes themselves. The newly developed SS-MRCEPA methods are size-extensive, and are also size-consistent with localized orbitals. Among the various versions, there are two which are invariant with respect to the restricted rotations among doubly occupied and active orbitals separately. Similarity of performance of this latter and the noninvariant versions at the crossing points of the degenerate orbitals imply that the all the methods presented are rather robust with respect to the rotations among degenerate orbitals. Illustrative numerical applications are presented for PES of the ground state of a number of difficult test cases such as the model H4, H(8) problems, the insertion of Be into H(2), and Li(2), where intruders exist and for a state of a molecule such as CH(2), with pronounced MR character. Results obtained with SS-MRCEPA methods are found to be comparable in accuracy to the parent SS-MRCC and FCI/large scale CI results throughout the PES, which indicates the efficacy of our SS-MRCEPA methods over a wide range of geometries, despite their neglect of a host of complicated nonlinear terms, even when the traditional MR-based methods based on effective Hamiltonians fail due to intruders.

Alternatives to multireference methods for the molecular electronic structure problem

AbstractThe development of robust many‐body methods for the molecular electronic structure problem with respect to multireference functions has attracted much attention over the past two decades. In recent years, multireference methods based on the Brillouin–Wigner expansion have been shown to overcome the intruder state problems which have plagued similar approaches based on the Rayleigh–Schrödinger series. However, there are some problems in molecular electronic structure theory, which may be handled by means of multireference methods, but can be treated by methods that avoid the use of multireference functions. We consider two examples of alternatives to multireference methods for the molecular electronic structure problem. The study of excited states having the same symmetry as the ground state or some lower lying excited state often involves the use of a multireference function. We describe an alternative procedure based on the generalized Rayleigh–Ritz variation principle. Gidopoulos, Glushkov, and Wilson have termed this the optimized trace method. The generalized Rayleigh–Ritz principle for the relativistic formulation of the molecular electronic structure problem is briefly also considered. Molecular dissociative processes are frequently described by quantum chemical methods based on multireference functions. Single‐reference Hartree–Fock methods usually provide a qualitatively incorrect description of such processes even in the simplest of molecules, H2. However, approximate single configuration wave functions constructed from nonorthogonal orbitals can often afford a useful approximation to bond‐breaking processes. Each occupied nonorthogonal orbital is then an eigenfunction of a different Fock‐like operator. Each of these Fock‐like operators supports a spectrum of M single particle states, where M is the size of the basis set, and only the lowest of these is occupied. We briefly consider the relativistic formulation of the molecular electronic structure problem based on approximations involving products of nonorthogonal functions within the Furry bound state interaction picture of quantum electrodynamics. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

Can We Avoid the Intruder-State Problems in the State-Universal Coupled-Cluster Approaches While Preserving Size Extensivity?

Following the analysis of principal bottlenecks in the extension of the single-reference (SR) coupled-cluster (CC) methodology to the multireference (MR) case, we review and discuss some recent developments that facilitate the use of general model spaces (GMSs) within the state universal (SU) or Hilbert space MR CC formalism. The use of a GMS improves our ability to avoid the intruder state problems. This feature is further enhanced by generalizing the idea of the externally corrected (ec) SR CC formalism to the MR situations. In this latter approach we employ the cluster analysis to extract the most important higher-than-pair cluster amplitudes from a suitable set of known wave functions. Similarly to the SR case, the most convenient external source is represented by wave functions that are obtained via a modest size MR configuration interaction (CI), which employs an N-dimensional reference space. The resulting higher-than-pair cluster amplitudes are subsequently used in the SU CCSD method that is based on an M-dimensional GMS avoiding intruders. We discuss general aspects of these developments from various viewpoints and provide selective illustrations of the key concepts and ideas.

INTERMEDIATE HAMILTONIAN FOCK-SPACE COUPLED CLUSTER METHOD AND APPLICATIONS

An intermediate Hamiltonian Fock-space coupled cluster method is presented. This method allows the use of large and flexible P spaces without encountering intruder-state problems. The larger P spaces enhance the accuracy of calculated transition energies by factors of 3–6, and make accessible many states which could not be handled by other CC methods. Examples given include the pilot calculation on Sc + levels, where accuracy increased by a factor of 4; element 114, where ionisation potentials show considerable increase relative to lighter group-14 atoms, due to relativistic stabilisation of the 7s and 7p1/2 orbitals; and a benchmark calculation of the electron affinities of the alkali atoms Na–E119, where results agree within 5 meV or better with known experimental values, providing a new reference value of 491±5 meV for the experimentally unknown EA of Fr.

N -reference, M-state coupled-cluster method: Merging the state-universal and reduced multireference coupled-cluster theories

We propose a generalization of the reduced multireference coupled-cluster method with singles and doubles (CCSD) to the genuine MR, state-universal (SU) CC approaches. Two key ingredients of this generalization are (i) the algorithm for the cluster analysis of general MR configuration interaction (CI) wave functions that is based on the SU cluster ansatz of Jeziorski and Monkhorst, and (ii) the formulation of the SU CC method employing a general (incomplete) model space. These recent developments enable us to employ modest size MR CISD wave functions that are based on an N-dimensional reference space M1 as a source of higher-than-pair-cluster amplitudes in the externally corrected SU CCSD method that is based on an M-dimensional model space M0, forming a subspace of M1. An appropriate choice of M0 and M1 makes it then possible to avoid the most severe intruder-state problems. The method is illustrated on the often-investigated H4 and H8 model systems.

Molecular gradients for the second-order generalized Van Vleck variant of multireference perturbation theory

Recently, a revised second-order generalized Van Vleck perturbation theory (GVVPT2) for the description of molecular electronic structure has been reported [J. Chem. Phys. 117, 4133 (2002)] that is both state selective and of the “perturb-then-diagonalize” type of multireference perturbation theory (MRPT). Herein, formulas for analytic derivatives of the GVVPT2 energy with respect to nuclear perturbations are presented, as are illustrative calculations on model problems. Specifically, it is shown that the modification of the energy denominator, which addresses the so-called intruder-state problem of MRPT, is analytically differentiable with respect to nuclear perturbation and only requires use of matrices available, or directly obtainable, from the underlying multiconfigurational self-consistent field calculation. The developed formalism takes full advantage of the theoretical and computational characteristics of the GVVPT2 energy. In particular, the calculations are performed directly in a spin-adapted basis and utilize the recently introduced concept of macroconfigurations. Moreover, the full flexibility of the energy calculations with respect to arbitrariness of reference—i.e., no restriction to complete active space self-consistent field—is retained. Test calculations on N2 and O3 comparing the analytic derivatives with the results of finite-difference calculations corroborate the formulas and implementation.

Brillouin–Wigner coupled cluster theory. Fock-space approach

The generalization of coupled cluster (CC) theory is far from being a standard method to account for correlation effects of ubiquitous open-shell systems. This dilemma is largely due to three problems. The first concerns the incorporation of multiple reference “guess” wave functions into nonvariational theories. Next is the size-extensivity issue. Finally, and perhaps most importantly is the notorious intruder state problem. Brillouin–Wigner (BW) and generalized Brillouin–Wigner (gBW) perturbation theories are used to aid in the development of new Fock-space coupled cluster theories in an attempt to alleviate some of these problems. Bloch equations are derived which can be used to formulate BWCC and gBWCC theories of arbitrary dimension in all sectors of Fock space. Since this is our first study of Brillouin–Wigner coupled cluster theories in Fock space we have chosen to keep our approach very simple. Explicit effective Hamiltonian and amplitude equations for up to two-body S-amplitudes in the (0,1) and (1,0) sectors are given for the single reference case. Aspects concerning the connectivity of the amplitude equations are addressed.