Poul Jo

Linear and nonlinear response functions for an exact state and for an MCSCF state

We have examined the response of an exact and an MCSCF reference state to a general time‐dependent field. The time development of both the exact and the MCSCF reference state have been parametrized in terms of explicit exponential time‐dependent transformations. The time development has been determined by requiring the Ehrenfest theorem to be satisfied through each order in the interaction between the molecular system and the field. The response of the exact and the MCSCF reference state has been used to evaluate linear, quadratic, and cubic response functions. It has been shown how a large variety of molecular properties may be expressed in terms of these response functions. It has also been demonstrated that molecular properties containing the electric dipole operator may be expressed in equivalent forms involving the momentum operator both for the exact and the MCSCF state. The MCSCF response functions have been transformed to computationally attractive expressions which do not contain summation indices over intermediate states and which allow direct techniques to be straightforwardly applied.

Coupled cluster response functions

The linear and quadratic response functions have been determined for a coupled cluster reference state. From the response functions, computationally tractable expressions have been derived for excitation energies, first‐ and second‐order matrix transition elements, transition matrix elements between excited states, and second‐ and third‐order frequency‐dependent molecular properties.

Determinant based configuration interaction algorithms for complete and restricted configuration interaction spaces

A restricted active space (RAS) wave function is introduced, which encompasses many commonly used restricted CI expansions. A highly vectorized algorithm is developed for full CI and other RAS calculations. The algorithm is based on Slater determinants expressed as products of alphastrings and betastrings and lends itself to a matrix indexing C(Iα, Iβ ) of the CI vector. The major features are: (1) The intermediate summation over determinants is replaced by two intermediate summations over strings, the number of which is only the square root of the number of determinants. (2) Intermediate summations over strings outside the RAS CI space is avoided and RAS calculations are therefore almost as efficient as full CI calculations with the same number of determinants. (3) An additional simplification is devised for MS =0 states, halving the number of operations. For a case with all single and double replacements out from 415 206 Slater determinants yielding 1 136 838 Slater determinants each CI iteration takes ...

Excitation energies from the coupled cluster singles and doubles linear response function (CCSDLR). Applications to Be, CH+, CO, and H2O

The linear response function for a coupled cluster singles and doubles wave function is used to calculate vertical electronic energies for the closed shell system Be, CH+, CO, and H2O. It is shown that excitations of single electron replacement character can be described accurately in such an approach. Improved convergence is obtained using a preconditioned form of the coupled cluster linear response matrix.

Response functions in the CC3 iterative triple excitation model

The derivation of response functions for coupled cluster models is discussed in a context where approximations can be introduced in the coupled cluster equations. The linear response function is derived for the approximate coupled cluster singles, doubles, and triples model CC3. The linear response functions for the approximate triples models, CCSDT‐1a and CCSDT‐1b, are obtained as simplifications to the CC3 linear response function. The consequences of these simplifications are discussed for the evaluation of molecular properties, in particular, for excitation energies. Excitation energies obtained from the linear response eigenvalue equation are analyzed in orders of the fluctuation potential. Double replacement dominated excitations are correct through second order in all the triples models mentioned, whereas they are only correct to first order in the coupled cluster singles and doubles model (CCSD). Single replacement dominated excitation energies are correct through third order in CC3, while in CCSD...

The CC3 model: An iterative coupled cluster approach including connected triples

An alternative derivation of many-body perturbation theory (MBPT) has been given, where a coupled cluster parametrization is used for the wave function and the method of undetermined Lagrange multipliers is applied to set up a variational coupled cluster energy expression. In this variational formulation, the nth-order amplitudes determine the energy to order 2n+1 and the nth-order multipliers determine the energy to order 2n+2. We have developed an iterative approximate coupled cluster singles, doubles, and triples model CC3, where the triples amplitudes are correct through second order and the singles amplitudes are treated without approximations due to the unique role of singles as approximate orbital relaxation parameters. The compact energy expressions obtained from the variational formulation exhibit in a simple way the relationship between CC3, CCSDT-1a [Lee et al., J. Chem. Phys. 81, 5906 (1984)] CCSDT-1b models [Urban et al., J. Chem. Phys. 83, 4041 (1985)], and the CCSD(T) model [Raghavachari et...

Basis set convergence of the interaction energy of hydrogen-bonded complexes

The Hartree-Fock and correlation contributions to the interaction energy of the hydrogen-bonded complexes (HF)2, (HCl)2, H2OHF, HCNHF, and (H2O)2 are computed in conventional calculations employing the aug-cc-pVXZ series of basis sets at the levels of Hartree-Fock theory, second-order perturbation theory, and coupled-cluster theory with single and double excitations augmented by a perturbative triples correction. The basis set convergence of the interaction energy is examined by comparison with results obtained with an explicitly correlated wave function model. The counterpoise-corrected and uncorrected Hartree-Fock interaction energies both converge very unsystematically. The convergence of the uncorrected correlation contribution is also very unsystematic because the basis set superposition error and the error from the incomplete description of the electronic Coulomb cusp both are present. Once the former has been effectively removed by the counterpoise correction, the cusp dominates and the convergence...

An electronic Hamiltonian for origin independent calculations of magnetic properties

A gauge origin independent formalism for the calculation of molecular magnetic properties is presented. Origin independence is obtained by using London’s gauge invariant atomic orbitals, expanding the second quantization Hamiltonian in the external magnetic field and nuclear magnetic moments, and using the resulting expansion terms as perturbation operators in response function calculations. To ensure orthonormality of the molecular orbitals, a field‐dependent symmetrical orthonormalization is employed. In this way the gauge dependence of the London orbitals is transferred to the Hamiltonian. The resulting perturbation operators may be used to calculate magnetic properties from any approximate ab initio wave function.

Optimization of orbitals for multiconfigurational reference states

The problem of finding orbitals, which make the total energy stationary, when the ground state or reference state is of the multiconfigurational form, is analyzed in this paper. Particular attention is given to the quadratically convergent methods. Orbital variations are generated by unitary transformations and an expansion of the total energy through second order in the parameters used to define these transformations provides a characterization of the stationary point, which is reached at convergence, as well as estimates of second order properties. Effective one‐particle potentials are analyzed and it is emphasized that only certain blocks of the Fock matrix are determined by the generalized Brillouin theorem. In an analysis of the convergence properties of the various methods it is shown that the Hartree–Fock procedure can be brought to converge for stable states of a given symmetry if the blocks of the Fock matrix, which are not determined by the Brillouin theorem, are chosen in an appropriate fashion. The procedures, which we discuss, are finally compared to currently employed techniques including direct energy optimization procedures, energy weighted steepest descent methods and generalizations of Roothaans equations.The problem of finding orbitals, which make the total energy stationary, when the ground state or reference state is of the multiconfigurational form, is analyzed in this paper. Particular attention is given to the quadratically convergent methods. Orbital variations are generated by unitary transformations and an expansion of the total energy through second order in the parameters used to define these transformations provides a characterization of the stationary point, which is reached at convergence, as well as estimates of second order properties. Effective one‐particle potentials are analyzed and it is emphasized that only certain blocks of the Fock matrix are determined by the generalized Brillouin theorem. In an analysis of the convergence properties of the various methods it is shown that the Hartree–Fock procedure can be brought to converge for stable states of a given symmetry if the blocks of the Fock matrix, which are not determined by the Brillouin theorem, are chosen in an appropriate fashion...

The prediction of molecular equilibrium structures by the standard electronic wave functions

A systematic investigation has been carried out of the accuracy of molecular equilibrium structures of 19 small closed-shell molecules containing first-row atoms as predicted by the following standard electronic ab initio models: Hartree–Fock (HF) theory, Mo/ller–Plesset theory to second, third, and fourth orders (MP2, MP3, and MP4), coupled-cluster singles and doubles (CCSD) theory; CCSD theory with perturbational triples corrections [CCSD(T)], and the configuration-interaction singles and doubles (CISD) model. For all models, calculations were carried out using the correlation-consistent polarized valence double-zeta (cc-pVDZ) basis, the correlation-consistent polarized valence triple-zeta (cc-pVTZ) basis, and the correlation-consistent polarized valence quadruple-zeta (cc-pVQZ) basis. Improvements in the basis sets shorten the bond distances at all levels. Going from cc-pVDZ to cc-pVTZ, bond distances are on the average reduced by 0.8 pm at the Hartree–Fock level and by 1.6 pm at the correlated levels....