Ground‐State and Excitation Spectra of a Strongly Correlated Lattice by the Coupled Cluster Method

The coupled cluster method is applied to a strongly correlated lattice Hamiltonian, and the coupled cluster linear response method is extended to the calculation of electronic spectra by finding an approximation to a resolvent operator which describes the spectral response of the coupled cluster solution to excitation operators. In this spectral coupled cluster method, the ground and excited states appear as resonances in the spectra, and the resolvent can be iteratively improved in selected spectral regions. The method is applied to a MnO₂ plane model which corresponds to previous experimental works.     

Approximate variational coupled cluster theory

We show that it is possible to construct an accurate approximation to the variational coupled cluster method, limited to double substitutions, from the minimization of a functional that is rigorously extensive, exact for isolated two-electron subsystems and invariant to transformations of the underlying orbital basis. This approximate variational coupled cluster theory is a modification and enhancement of our earlier linked pair functional theory. It is first motivated by the constraint that the inverse square root of the matrix that transforms the cluster amplitudes must exist. Low-order corrections are then included to enhance the accuracy of the approximation of variational coupled cluster, while ensuring that the computational complexity of the method never exceeds that of the standard traditional coupled cluster method. The effects of single excitations are included by energy minimization with respect to the orbitals defining the reference wavefunction. The resulting quantum chemical method is demonstrated to be a robust approach to the calculation of molecular electronic structure and performs well when static correlation effects are strong.     

Quasi-variational coupled cluster theory

We extend our previous work on the construction of new approximations of the variational coupled cluster method. By combining several linked pair functional transformations in such a way as to give appropriately balanced infinite-order contributions, in order to approximate (L) well at all orders, we formulate a new quantum chemical method, which we name quasi-variational coupled cluster. We demonstrate this method to be particularly robust in the regime of strong static electron correlation, improving significantly on our earlier approximate variational coupled cluster approach.     

Alternative single-reference coupled cluster approaches for multireference problems: the simpler, the better

We report a general implementation of alternative formulations of single-reference coupled cluster theory (extended, unitary, and variational) with arbitrary-order truncation of the cluster operator. These methods are applied to compute the energy of Ne and the equilibrium properties of HF and C(2). Potential energy curves for the dissociation of HF and the BeH(2) model computed with the extended, variational, and unitary coupled cluster approaches are compared to those obtained from the multireference coupled cluster approach of Mukherjee et al. [J. Chem. Phys. 110, 6171 (1999)] and the internally contracted multireference coupled cluster approach [F. A. Evangelista and J. Gauss, J. Chem. Phys. 134, 114102 (2011)]. In the case of Ne, HF, and C(2), the alternative coupled cluster approaches yield almost identical bond length, harmonic vibrational frequency, and anharmonic constant, which are more accurate than those from traditional coupled cluster theory. For potential energy curves, the alternative coupled cluster methods are found to be more accurate than traditional coupled cluster theory, but are three to ten times less accurate than multireference coupled cluster approaches. The most challenging benchmark, the BeH(2) model, highlights the strong dependence of the alternative coupled cluster theories on the choice of the Fermi vacuum. When evaluated by the accuracy to cost ratio, the alternative coupled cluster methods are not competitive with respect to traditional CC theory, in other words, the simplest theory is found to be the most effective one.     

Two-particle density matrix cumulant of coupled cluster theory

A two-particle density matrix obtained from the expectation value expression for the coupled cluster wave function is separated into the antisymmetrized product of the one-particle density matrices and the remaining cumulant part. It is demonstrated that the proposed formula for the coupled cluster cumulant is a valid cumulant expression, since it is a connected, and therefore size-extensive quantity. It is also shown that the density matrices from coupled cluster gradient theory, when used to define a cumulant, result in the expression containing disconnected terms. The proposed formulation of the coupled cluster cumulant makes it easy to develop size-extensive truncation schemes. As an example, explicit equations for the cumulant at the coupled cluster single, double and triple excitation level are presented.     

New way of describing static and dynamic deformations of the Jahn-Teller type in ring molecules

A new method is presented to describe deformations of an N-membered planar ring (N-ring) molecule in terms of deformation vectors that can be expressed by a set of 2N-3 deformation amplitudes and phase angles. The deformation coordinates are directly derived from the normal vibrational modes of the N-ring and referenced to a regular polygon (N-gon) of unit length. They extend the conceptual approach of the Cremer-Pople puckering coordinates (J. Am. Chem. Soc. 1975, 97, 1354) to the planar ring and make it possible to calculate, e.g., a planar ring of special deformation on a Jahn-Teller surface. It is demonstrated that the 2N-3 deformation parameters are perfectly suited to describe the pseudorotation of a bond through the ring as it is found in cyclic Jahn-Teller systems. In general, an N-membered planar ring can undergo N-2 different bond pseudorotations provided the energetics of such a process is feasible. The Jahn-Teller distortions observed in ring compounds correspond either directly to the basic pseudorotation modes or to linear combinations of them. Any deformed ring molecule can be characterized in terms of the new ring deformation coordinates, which help to identify specific electronic effects. The usefulness of the ring deformation coordinates is demonstrated by calculating the Jahn-Teller surfaces for bond pseudorotation in the case of the cyclopropyl radical cation and cyclobutadiene as well as the ring deformation surfaces of disulfur dinitride and its dianion employing multireference averaged quadratic coupled cluster (MR-AQCC) theory, equation-of-motion coupled cluster theory in form of EOMIP-CCSD, and single determinant coupled cluster theory in form of CCSD(T).     

Benchmark studies of variational, unitary and extended coupled cluster methods

Comparative benchmark calculations are presented for coupled cluster theory in its standard formulation, as well as variational, extended, and unitary coupled cluster methods. The systems studied include HF, N(2), and CN, and with cluster operators that for the first time include up to quadruple excitations. In cases where static correlation effects are weak, the differences between the predictions of molecular properties from each theory are negligible. When, however, static correlation is strong, it is demonstrated that variational coupled cluster theory can be significantly more robust than the traditional ansatz and offers a starting point on which to base single-determinant reference methods that can be used beyond the normal domain of applicability. These conclusions hold at all levels of truncation of the cluster operator, with the variational approach showing significantly smaller errors.     

MBPT and coupled cluster calculation on the neon atom with numerical orbitals

Our diatomic coupled cluster method with numerical orbitals has been applied to the Ne atom. Second, third, and full fourth order MBPT correlation corrections as well as coupled cluster correlation energies are reported.     

Testing the parametric two-electron reduced-density-matrix method with improved functionals: application to the conversion of hydrogen peroxide to oxywater

Parametrization of the two-electron reduced density matrix (2-RDM) has recently enabled the direct calculation of electronic energies and 2-RDMs at the computational cost of configuration interaction with single and double excitations. While the original Kollmar energy functional yields energies slightly better than those from coupled cluster with single-double excitations, a general family of energy functionals has recently been developed whose energies approach those from coupled cluster with triple excitations [D. A. Mazziotti, Phys. Rev. Lett. 101, 253002 (2008)]. In this paper we test the parametric 2-RDM method with one of these improved functionals through its application to the conversion of hydrogen peroxide to oxywater. Previous work has predicted the barrier from oxywater to hydrogen peroxide with zero-point energy correction to be 3.3-to-3.9 kcal/mol from coupled cluster with perturbative triple excitations [CCSD(T)] and -2.3 kcal/mol from complete active-space second-order perturbation theory (CASPT2) in augmented polarized triple-zeta basis sets. Using a larger basis set than previously employed for this reaction-an augmented polarized quadruple-zeta basis set (aug-cc-pVQZ)-with extrapolation to the complete basis-set limit, we examined the barrier with two parametric 2-RDM methods and three coupled cluster methods. In the basis-set limit the M parametric 2-RDM method predicts an activation energy of 2.1 kcal/mol while the CCSD(T) barrier becomes 4.2 kcal/mol. The dissociation energy of hydrogen peroxide to hydroxyl radicals is also compared to the activation energy for oxywater formation. We report energies, optimal geometries, dipole moments, and natural occupation numbers. Computed 2-RDMs nearly satisfy necessary N-representability conditions.     

Analytic gradient for the multireference Brillouin-Wigner coupled cluster method and for the state-universal multireference coupled cluster method

We present the analytic gradient theory and its pilot implementation for the multireference Brillouin-Wigner coupled cluster (BWCC) method and for the state-universal multireference coupled cluster method. The analytic gradient has been derived for three cases: (i) BWCC method without a size-extensivity correction, (ii) BWCC method with the iterative size-extensivity correction, and (iii) for the rigorously size-extensive state-universal method. The pilot implementation is based on full-configuration interaction expansions and is presently limited to single and double excitation levels; however, the resulting equations are general. For BWCC methods, they also do not contain terms explicitly mixing amplitudes of different reference configurations and can thus be implemented in an efficient way. The analytic gradients have been verified with respect to numerically computed ones on the example of CH2 molecule, and geometry optimizations of CH2 and SiH2 have been carried out.     

Block-correlated coupled cluster theory: the general formulation and its application to the antiferromagnetic Heisenberg model

The general formalism of the block-correlated coupled cluster (BCCC) method, an alternative multireference coupled cluster method for calculating the ground-state electronic structures of molecular systems, has been presented. The BCCC theory is constructed in terms of a complete set of many-electron states of individual blocks, assumed that the whole system could be partitioned into a set of blocks. The reference state in the BCCC is selected as a tensor product of the most important many-electron state of each system block. By truncating the cluster operator to a certain n-block correlation level, an approximate but size-extensive BCCC method, denoted as BCCCn, is defined. For reducing the computational effort but without much loss of accuracy, the reduced density matrix is introduced to generate an optimal subset of many-electron states for each block. I have implemented the BCCCn (n=2,3) methods within the S=1/2 Heisenberg Hamiltonian, and applied them to calculate the ground-state energies of one-dimensional spin chains and quasi-one-dimensional two-leg spin ladders. The calculated results show that with the appropriate partition of the studied systems the BCCC3 method can yield quite satisfactory ground-state energies for these spin systems.     

On the structure and stability of Ar n and Ar n + clusters at finite temperature

For Ar_2–29 and Ar _2–29 ⁺ clusters at 20 K in the polarization model presented here the electrodynamical dipole-dipole many-body problem is solved selfconsistently with the Monte-Carlo method (MC) at 20 K, i.e. the instantaneous dipole-dipole interaction is solved to infinite perturbation order and in cluster expansion to the order of the cluster size. The long range many-body dipole-dipole interaction is coupled to exchange interaction by a modified effective dipole polarizability. This model will be compared to the dimer model and classical MC simulation of Ar _n . The resulting different magic numbers in the binding energies are discussed in this connection with different experimental techniques of cluster ionization. By the mean square cluster diameter a shape parameter is introduced and it is found that with this parameter structural form transition in cluster growth can be resolved, and surprisingly do not correlate with the magic numbers.     

A relativistic 4-component general-order multi-reference coupled cluster method: initial implementation and application to HBr

We present the initial implementation of a determinant-based general-order coupled cluster method which fully accounts for relativistic effects within the four-component framework. The method opens the way for the treatment of multi-reference problems through a state-selective expansion of the model space. The evaluation of the coupled cluster vector function is carried out via relativistic configuration interaction expansions. The implementation is based on a large-scale configuration interaction technique, which may efficiently treat long determinant expansions of more than 10⁸ terms. We demonstrate the capabilities of the new method in calculations of complete potential energy curves of the HBr molecule. The inclusion of spin–orbit interaction and higher excitations than coupled cluster double excitations, either by multi-reference model spaces or the inclusion of full iterative triple excitations, lead to highly accurate results for spectral constants of HBr. An erratum to this article can be found at     

The relationship between double excitation amplitudes and Z vector components in some post-Hartree-Fock correlation methods

The relationship between Z vector components and excitation amplitudes is analyzed for several post-Hartree-Fock correlation methods limited to double excitation amplitudes. An analytical formula approximating the Z vector for the coupled cluster doubles method is presented and shown to be quite accurate. This approximation is also used to determine the prefactor of the norm of doubly excited states in averaged coupled pair functional-type energy functionals self-consistently leading to better agreement with coupled cluster results.     

Performance of SCAN density functional for a set of ionic liquid ion pairs

Computational chemistry is a powerful tool for the discovery of novel materials. In particular, it is used to simulate ionic liquids in search of electrolytes for electrochemical applications. Herein, the choice of the computational method is not trivial, as it has to be both efficient and accurate. Density functional theory methods with appropriate corrections for the systematic weaknesses can give precision close to that of the post‐Hartree–Fock coupled cluster methods with a fraction of their cost. Thence, we have evaluated the performance of a recently developed nonempirical strongly constrained and appropriately normed (SCAN) density functional on electronic structure calculations of ionic liquid ion pairs. The performance of SCAN and other popular functionals (PBE, M06‐L, B2PLYP) among with Grimme's dispersion correction and Boys–Bernardi basis set superposition error correction was compared to DLPNO‐CCSD(T)/CBS. We show that SCAN reproduces coupled‐cluster results for describing the employed dataset of 48 ion pairs.     

Recursive intermediate factorization and complete computational linearization of the coupled-cluster single,double, triple,and quadruple excitation equations

Summary The nonlinear CCSDTQ equations are written in a fully linearized form, via the introduction of computationally convenient intermediates. An efficient formulation of the coupled cluster method is proposed. Due to a recursive method for the calculation of intermediates, all computational steps involve the multiplication of an intermediate with aT vertex. This property makes it possible to express the CC equations exclusively in terms of matrix products which can be directly transformed into a highly vectorized program.This work has been supported by the U.S. Air Force Office of Scientific Research, Grant No. 90-0079     

A simple correction to final state energies of doublet radicals described by equation-of-motion coupled cluster theory in the singles and doubles approximation

A computationally inexpensive energy correction is suggested for radicals described by the equation-of-motion coupled cluster method for ionized states in the singles and doubles approximation (EOMIP-CCSD). The approach is primarily intended for doublet states that are qualitatively described by Koopmans' approximation. Following a strategy similar to those used in multireference coupled cluster theory, the proposed correction accounts for all correlation effects through third order in perturbation theory and also includes selected contributions to higher-order energies. As an initial test of the numerical performance of the method, total energies and energy splittings are calculated for some small prototype radicals.     

Ionization potentials and electron affinities of Cu,Ag, and Au: Electron correlation and relativistic effects

The electron correlation and relativistic effects on ionization potentials and electron affinities of Cu, Ag, and Au are investigated in the framework of the coupled cluster method and different 1-component approximations to the relativistic Dirac-Coulomb Hamiltonian. The first-order perturbation approach based on the massvelocity and Darwin terms is found to be sufficiently accurate for Cu and Ag while it fails for Au. The spin-averaged Douglas-Kroll no-pair method gives excellent results for the studied atomic properties. The ionization potentials obtained within this method and the coupled cluster scheme for the electron correlation effects are 7.733(7.735) eV for Cu, 7.461(7.575) eV for Ag, and 9.123(9.225) eV for Au (experimental values given in parentheses). The calculated (experimental) electron affinity results for Cu, Ag, and Au are 1.236(1.226), 1.254(1.303), and 2.229(2.309) eV, respectively. There is a marked relativistic effect on both the ionization potential and electron affinity of Ag which sharply increases for Au while Cu exhibits only a little relativistic character. A similar pattern of relativistic effects is also observed for electric dipole polarizabilities of the coinage metal atoms and their ions. The coupled cluster dipole polarizabilities of the coinage metal atoms calculated in this article in the Douglas-Kroll no-pair formalism (Cu: 46.50 au; Ag: 52.46 au; Au: 36.06 au) are compared with our earlier data for their singly positive and singly negative ions. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 63: 557–565, 1997     

Generation of potential energy curves for the X1Sigma(+)g, B1Delta(+)g, and B'1Sigma(+)g states of C2 using the effective valence shell Hamiltonian method

Calculations of the ground and excited state potential energy curves of C2 using the third-order effective valence Hamiltonian (Hv3rd) method are benchmarked against full configuration interaction and other correlated single-reference perturbative and nonperturbative theories. The large nonparallelity errors (NPEs) exhibited even by state-of-art coupled cluster calculations through perturbative triples indicate a serious deficiency of these single-reference theories. The Hv method, on the other hand, produces a much reduced NPE, rendering it a viable approximate many-body method for accurately determining global ground and excited state potential energy curvessurfaces.