Local (multiplicative) effective exchange potentials obtained from the linear-combination- of-atomic-orbital (LCAO) optimized effective potential (OEP) method are frequently unrealistic in that they tend to exhibit wrong asymptotic behavior (although formally they should have the correct asymptotic behavior) and also assume unphysical rapid oscillations around the nuclei. We give an algebraic proof that, with an infinity of orbitals, the kernel of the OEP integral equation has one and only one singularity associated with a constant and hence the OEP method determines a local exchange potential uniquely, provided that we impose some appropriate boundary condition upon the exchange potential. When the number of orbitals is finite, however, the OEP integral equation is ill-posed in that it has an infinite number of solutions. We circumvent this problem by projecting the equation and the exchange potential upon the function space accessible by the kernel and thereby making the exchange potential unique. The observed numerical problems are, therefore, primarily due to the slow convergence of the projected exchange potential with respect to the size of the expansion basis set for orbitals. Nonetheless, by making a judicious choice of the basis sets, we obtain accurate exchange potentials for atoms and molecules from an LCAO OEP procedure, which are significant improvements over local or gradient-corrected exchange functionals or the Slater potential. The Krieger–Li–Iafrate scheme offers better approximations to the OEP method.

We develop a combined coupled-cluster (CC) or equation-of-motion coupled-cluster (EOM-CC) theory and Rayleigh–Schrödinger perturbation theory on the basis of a perturbation expansion of the similarity-transformed Hamiltonian $\mathrm{H̄=}\exp \mathrm{(-T)H }\exp \mathrm{(T).}$ This theory generates a series of perturbative corrections to any of the complete CC or EOM-CC models and hence a hierarchy of the methods designated by $\mathrm{CC}\mathrm{(m)}\mathrm{PT}\mathrm{(n)}$ or $\mathrm{EOM-CC}\mathrm{(m)}\mathrm{PT}\mathrm{(n).}$ These methods systematically approach full configuration interaction (FCI) as the perturbation order $\mathrm{(n)}$ increases and/or as the cluster and linear excitation operators become closer to complete $\mathrm{(m}$ increases), while maintaining the orbital-invariance property and size extensivity of CC at any perturbation order, but not the size intensivity of EOM-CC. We implement the entire hierarchy of $\mathrm{CC}\mathrm{(m)}\mathrm{PT}\mathrm{(n)}$ and $\mathrm{EOM-CC}\mathrm{(m)}\mathrm{PT}\mathrm{(n)}$ into a determinantal program capable of computing their energies and wave functions for any given pair of m and n. With this program, we perform $\mathrm{CC}\mathrm{(m)}\mathrm{PT}\mathrm{(n)}$ and $\mathrm{EOM-CC}\mathrm{(m)}\mathrm{PT}\mathrm{(n)}$ calculations of the ground-state energies and vertical excitation energies of selected small molecules for all possible values of m and $\text{0⩽n⩽5.}$ When the Hartree–Fock determinant is dominant in the FCI wave function, the second-order correction to CCSD [CC(2)PT(2)] reduces the differences in the ground-state energy between CCSD and FCI by more than a factor of 10, and thereby significantly outperforms CCSD(T) or even CCSDT. The third-order correction to CCSD [CC(2)PT(3)] further diminishes the energy difference between CC(2)PT(2) and FCI and its performance parallels that of some CCSD(TQ) models. $\mathrm{CC}\mathrm{(m)}\mathrm{PT}\mathrm{(n)}$ for the ground state with some multideterminantal character and $\mathrm{EOM-CC}\mathrm{(m)}\mathrm{PT}\mathrm{(n)}$ for the excitation energies, however, appear to be rather slowly convergent with respect to n.

From coupled-cluster theory and many-body perturbation theory we derive the local exchange-correlation potential of density functional theory in an orbital dependent form. We show the relationship between the coupled-cluster approach and density functional theory, and connections and comparisons with our previous second-order correlation potential [OEP-MBPT(2) (OEP—optimized effective potential)] [I. Grabowski, S. Hirata, S. Ivanov, and R. J. Bartlett, J. Chem. Phys. 116, 4415 (2002)]. Starting from a general theoretical framework based on the density condition in Kohn–Sham theory, we define a rigorous exchange-correlation functional, potential and orbitals. Specifying initially to second-order terms, we show that our ab initio correlation potential provides the correct shape compared to those from reference quantum Monte Carlo calculations, and we demonstrate the superiority of using Fock matrix elements or more general infinite-order semicanonical transformations. This enables us to introduce a method that is guaranteed to converge to the right answer in the correlation and basis set limit, just as does ab initio wave function theory. We also demonstrate that the energies obtained from this generalized second-order method [OEP-MBPT(2)-f] and [OEP-MBPT(2)-sc] are often of coupled-cluster accuracy and substantially better than ordinary Hartree–Fock based second-order $\mathrm{MBPT}=\mathrm{MP}\text{2.}$

Using the optimized effective potential (OEP) method in conjunction with an orbital-dependent correlation functional developed on the basis of second-order many-body perturbation theory [MBPT(2)], we perform ab initio correlated density functional calculations. Unlike other density functional methods, this offers the first correlation approximation in a converging series of approximations analogous to that in wave function theory. The OEP-MBPT(2) correlation potentials of the helium isoelectronic series and the neon atom are in excellent agreement with prior quantum Monte Carlo (QMC) results, while the widely used conventional approximate density functional theory correlation potentials have qualitatively incorrect behavior. Furthermore, the correlation energies, total energies, and the highest occupied orbital energies calculated by the OEP-MBPT(2) method are also remarkably accurate. Unlike QMC, our procedure can be applied to general molecules.

Exchange-only ab initio (parameter-free) time-dependent density functional calculations for the vertical excitation energies of atoms and polyatomic molecules are performed by employing optimized effective potentials (OEP’s) and their corresponding adiabatic exchange kernels for the first time. Accurate OEP’s are obtained by a novel linear-combination-of-atomic-orbital (LCAO) algorithm [R. Colle and R. K. Nesbet, J. Phys. B 34, 2475 (2001)] in which a potential is represented as a sum of a seed potential having the correct $\text{−1/r}$ asymptotic behavior and a small and rapidly decaying correction, the latter being approximated accurately by a linear combination of Gaussian functions. The time-dependent OEP (TDOEP) methods with and without the Tamm–Dancoff approximation are implemented by using a trial-vector algorithm, which allows us to avoid the storage or manipulation of transformed two-electron integrals or the diagonalization of large matrices. No approximation is made to TDOEP, besides the adiabatic approximation to the exchange kernel, the LCAO expansion of the orbitals and potentials, and occasionally the Tamm–Dancoff approximation. The vertical excitation energies of the beryllium atom and the nitrogen and water molecules calculated by TDOEP are compared with those obtained from time-dependent density functional theory (TDDFT) employing conventional local or gradient-corrected functionals, configuration interaction singles (CIS), time-dependent Hartree–Fock (TDHF) theory, similarity-transformed equation-of-motion coupled-cluster with single and double substitutions, and experiments. TDOEP, which neglects electron correlation while treating the exchange contribution rigorously within the Kohn–Sham DFT framework, performs equally well as, or even appreciably better than, CIS or TDHF. The slightly better performance of TDOEP might be attributed to the local nature of the exchange potentials that allows the bare orbital energy differences to approximate excitation energies well. Nevertheless, TDDFT employing local or gradient-corrected functionals outperforms TDOEP for low-lying valence excited states, implying that the former somehow accounts for electron correlation effectively, whereas for high-lying and Rydberg excited states, the latter performs better than the former. By combining the desirable features of OEP and local or gradient-corrected exchange-correlation potentials, we arrive at a simple asymptotic correction scheme to the latter. TDDFT with the asymptotic correction yields uniformly accurate excitation energies for both valence and Rydberg excited states.

The recently developed algebraic formulation of valence‐universal coupled‐cluster (VU‐CC) theories [Jeziorski and Paldus, J. Chem. Phys. 90, 2714 (1989)] for open‐shell systems has been employed in a systematic derivation of explicit equations defining cluster amplitudes assuming Lindgren’s normal ordered exponential ansatz for the wave operator. The latter is approximated by its one‐ and two‐electron components. Various aspects of the applicability of this version of the VU‐CC theory to quasidegenerate electronic states are studied for a model system consisting of two slightly stretched, interacting hydrogen molecules. A single parameter that determines the geometry of this system makes it possible to vary the extent of quasidegeneracy of the two lowest‐energy states over a wide range. Along with the complete theory, the linear version (VU‐LCC) is also examined. The results are compared with the full configuration interaction results as well as with those obtained using other approaches. It was found that, at least in the strongly quasidegenerate region, the VU‐CC energies are less accurate than those obtained with other multireference CC theories. It is shown that the VU‐CC equations for cluster amplitudes possess multiple solutions representing various pairs of states. The individual solutions that arise for a system possessing n‐valence electrons may be characterized in terms of a genealogical scheme that involves states of systems with 0,1,...,n valence electrons.