Variational energy derivatives and perturbation theory

In this note we discuss the variational forms of the energy derivatives and the method of obtaining them. We show that perturbation theory can be formulated in terms of stationary energy derivatives and that this formulation extends the idea of the Hylleraas functional to infinite order and excited states.     

Variational localized-site cluster expansions. VIII. Projection of spin symmetries

We investigate the total spin structure of an approximate localized-site wave function for a collection of paramagnetic sites interacting so as to favor a singlet ground state. As the number of sites becomes infinite we obtain the distribution of weights of the different symmetry components of the localized site wave function; further, although only a very small fraction of such nonsymmetric wave functions is actually singlet, we find that it generally yields the same bulk property expectation values as its singlet-projected component.     

Variational expectation values and perturbation theory corrections

The relationship between the Delves variational procedure for calculating expectation values and Dalgarno's perturbation method based on an interchange theorem are investigated in general. Calculations on model one-electron systems suggest that direct calculation of non-variational expectation values may yield more accurate results with less computational effort than either the variational or the perturbation method.     

Many-Electron problems. II. Energy expansions and internal fields

Energy relations are derived for neutral atoms and isoelectronic sequences. It is shown that the field of the electrons at the position of the nucleus follows a very simple law and the diamagnetic shielding constant may be given in simple analytic form as a function of the atomic number.     

Accurate eigenvalues of three- through ten-electron atomic isoelectronic sequences from low-order Z-dependent perturbation theory combined with variational constraints and screening

A method is developed, based on Rayleigh-Schrödinger perturbation theory combined with variational constraints and screening, for obtaining accurate atomic eigenvalues from third-order 1/Z expansions. Application of the procedure to the ground states of the 3NV10 electron atomic sequences yields energies of 99.95–100.05% or greater accuracy, a marked improvement over those obtained from other third-order summations including Padé approximants. In the important test cases of the Be and Ne atoms, our results are found to exceed in accuracy all but the most elaborate ab initio calculations.     

Comments on exciton—phonon coupling. II. Variational solutions

Two variational calculations of the energy and correlation functions for a simple excition—phonon coupled system are presented and contrasted to the adiabatic solution and the exact solution. The simpler variational solution leads to two minima and abrupt changes in the properties of the system; an asymmetric variational wavefunction, motivated by the form of perturbation theory for this problem, leads to smooth behavior in agreement with the exact result.     

Connections between perturbation theory and the Van Vleck transformation: Illustrative calculations on the perturbed harmonic oscillator

We continue to examine the connection between perturbation theory and the Van Vleck unitary transformation. Here we illustrate the formalism derived earlier by applying it to compute the stationary states of the perturbed harmonic oscillator. We find that each solution of the traditional Brillouin-Wigner perturbation theory equations gives rise to a different unitary transformation which, when operating on the unperturbed ground state, produces one or the other of the perturbed eigenstates. With any of the perturbed states able to be reached by a unitary transformation on the unperturbed ground state, we advise caution in using approximate solutions of the perturbation equations in general cases, lest an unexpected stationary state be obtained.     

On the use of symmetry in first-order perturbed HF theory. II

A method is described whereby molecular symmetry is employed to reduce the number of two-electron integrals in perturbed Hartree–Fock calculations of second-order properties. The method is a generalization of the Dacre–Elder procedure. First- and second-rank perturbing tensor operators are examined in the coupled HF approach to electric dipole polarizabilities, magnetic susceptibilities, quadrupole polarizabilities, and spin-dopolar contributions to spin–spin coupling constants. The procedure sketched here permits a large saving of computer efforts, which is shown by some illustrative examples.     

Multireference perturbation theory with optimized partitioning. II. Applications to molecular systems

The second‐order multireference perturbation theory using an optimized partitioning, denoted as MROPT(2), is applied to calculations of various molecular properties—excitation energies, spectroscopic parameters, and potential energy curves—for five molecules: ethylene, butadiene, benzene, N₂, and O₂. The calculated results are compared with those obtained with second‐ and third‐order multireference perturbation theory using the traditional partitioning techniques. We also give results from computations using the multireference configuration interaction (MRCI) method. The presented results show very close resemblance between the new method and MRCI with renormalized Davidson correction. The accuracy of the new method is good and is comparable to that of second‐order multireference perturbation theory using Møller‐Plesset partitioning. © 2003 Wiley Periodicals, Inc. J Comput Chem 24: 1390–1400, 2003     

Longitudinal acoustic mode in polymers. II. Solution of the general composite perturbed elastic rod

The theory of the longitudinal vibrations of a composite rod with perturbing forces at the ends has been generalized by the inclusion of inertial effects at the ends of the rod. The frequencies have been calculated for both symmetric and asymmetric forces and masses. The problem of coupled rods has also been treated, and its relation to the isolated rod is shown. These results should be applicable to the study of longitudinal acoustic modes in long chain molecular systems.     

Multiconfigurational expansions of density operators: equations of motion and their properties

 Multiconfigurational expansions of density operators which may be used in numerical treatments of the dynamics of closed and open quantum systems are introduced. The expansions of the density operators may be viewed as analogues of those used in the multiconfiguration time-dependent Hartree (MCTDH) method, which is a well-established and highly efficient method for propagating wavepackets in several dimensions. There is no unique multiconfigurational representation of a density operator and two sensible types of MCTDH-like expansions are studied. Equations of motion for these multiconfigurational expansions are presented by adopting the Dirac–Frenkel/McLachlan variational principle (or variants of thereof). Various properties of these sets of equations of motion are derived for closed and open system dynamics. The numerical and technical aspects of this approach have been recently discussed by us [(1999) J Chem Phys 111: 8759]. Here we discuss the formal aspects of the approach in a more general context. Received: 26 January 2000 / Accepted: 8 February 2000 / Published online: 12 May 2000     

Calculation of periodic molecular systems with perturbed periodicity within the PCILO framework. II. Model systems

The energetic behavior of one-dimensional (HF)_n and (H₂O)_n chains with added or inserted H₂O or HF was studied by means of the PCILOPSI method presented in Part I. The results were compared with the ones by the original PCILO method. The stabilization energies are in good qualitative agreement; energy differences due to impurity effects even quantitatively agree. The results show that the PCILOPSI method is well suited for the study of perturbed periodic molecular systems.