A multireference direct CI program based on the symmetric group graphical approach

Specific features of an SGGA-based multireference direct CI program working in large internal spaces are discussed. In particular, advantages resulting from the explicit separation of the orbital and the spin spaces are explored. Concepts allowing for the efficient creation of a flexible and symmetry-adapted CI basis, for the high-speed generation of the coupling coefficients and for structuring a simple permutation driven algorithm to handle the orbital space are briefly discussed.Alexander von Humboldt Fellow 1985/87     

A comparison between multireference CI and effective medium theories for diatomic FeN

Summary Internally contracted multireferece CI calculations have been performed for the diatomic molecules FeN, CrN and CrO. For the latter two molecules the calculated dissociation energies are 3.18 eV (3.9±0.2 eV) and 4.09 eV (4.5±0.1 eV) respectively, where the corresponding experimental results are given in parentheses. It is argued that the correct value for CrN must lie in the lower end of the range given by experiment or perhaps slightly below. The best calculated result for FeN, for which no experimental result exists, is 1.69 eV. This value can be compared to an older CI value of 0.9 eV and a recent result of 5.25 eV obtained by an effective medium approach. Based on the results for all the three molecules treated here the correct dissociation energy of FeN can be estimated to be about 2.1 eV. The relevance of the present results for modelling chemical reactions in the regions around on-top positions of transition metal surfaces is discussed.     

An efficient improvement of the string-based direct selected CI algorithm

Summary A modified version of the Direct Selected Configuration Interaction Using Strings (DISCIUS) code is presented. From a series of test calculations it is shown that the string-based direct-CI algorithm is also very efficient in larger examples than the ones we showed in our former version. The performance of this new code with respect to the computer time is reasonable as well as its storage requirements. Hence, we present a new tool able to treat more than 10⁶ determinants of a variational space that does not have a defined structure, i.e. a selected or any way truncated CI.     

Multireference perturbation CI III. Fast evaluation of the one-particle density matrix

Within the frame of multireference perturbation configuration interaction we have developed a fast algorithm, based on diagrammatic techniques, for the calculation of the first-order correction to the one-particle density matrix. As an example of an application we have chosen the evaluation of the dipole moment of the CO molecule, where utilization of the first-order density is shown to corroborate the variational calculation. Received: 4 August 1998 / Accepted: 21 September 1998 / Published online: 16 November 1998     

Accuracy of energy extrapolation in multireference configuration interaction calculations

The accuracy of extrapolation procedures in conjunction with energy-based configuration selection in CI calculations is examined. The normally high accuracy of such extrapolation can deteriorate in multireference CI calculations when configuration functions of low weight are included in the root (reference) set. This is due to the inadequacy of second-order energy contribution estimates for the very large number of discarded low-contribution functions generated as single and double excitations from the minor members of the root set. The problem may be overcome by increasing the number of configurations included in the zero-order function used for the energy contribution estimation process. Illustrative results are presented for excited states of the H₂O molecule and the H₂O⁺ ion.     

Analytical first derivatives of the energy for small CI expansions

Using the Z-vector formalism the analytical gradient of the energy in small CI expansions is derived and implemented for semiempirical MNDO-type methods. The computation time is shown to scale as O(N ³) with the size of the system, with the memory requirements growing as O(N ²). The evaluation of the analytical gradient is significantly faster than the underlying SCF and CI calculations, so that routine full geometry optimizations at the semiempirical CI level become possible for large systems. Received: 24 April 1997 / Accepted: 10 June 1997     

A new method for approximate estimates of the dispersion interaction between two molecules

A new method to evaluate the dispersion interaction between two weakly interacting closed shell molecules is presented. The method is based on a second-order sum-over-states perturbation method, where the nominator is approximated by products of one electron integrals.     

Geometry optimizations with explicit inclusion of electron correlation

Automatic techniques for geometry optimization are applied in conjunction with configuration interaction and perturbation treatments of electron correlation. The computational effort and numerical accuracy of the optimizations are discussed, as well as problems with approximate correlation methods concerning the continuity of the potential surface. The optimized geometries of fourteen molecules obtained with different correlation treatments (MNDO SCF MOs) are compared. The configuration interaction results are reproduced satisfactorily by simple perturbation approaches. The largest change of the optimized SCF geometry is found for hydrogen peroxide.     

Convergence property of multireference many-body perturbation theory analyzed by the use of a norm of effective Hamiltonian

Summary It is proposed to use a norm of anth order effective Hamiltonian, for analyzing the convergence property of the multireference many-body perturbation theory (MR-MBPT). The utilization of the norm allows us to employ only (1) asingle number for all the states that we are interested in, and (2) values which decreases from thepositive side to zero as the ordern of the perturbation increases. This characteristic features are in contrast to those in the usually used scheme whereseveral numbers, namely, the eigenvalues of the target states, should be used and they mayoscillate around exact eigenvalues. The present method has been applied to MR-MBPT calculations of the (H₂)₂, CH₂, and LiH molecules based on the multireference versions of Rayleigh-Schrödinger PT, Kirtman-Certain-Hirschfelder PT, and the canonical Van Vleck PT; and following features are found: (1) the above three versions of the perturbation theories have essentially the same convergence property judged from the lowering of the norm; (2) the lower order truncation of the perturbation series gives reasonable solutions; (3) the norm decreases irrespective of the perturbation expansion being convergent or divergent for the first several orders (up to about the sixth order).     

Theoretical study on the size consistency of the second and third order energies of the multireference perturbation theory

The size consistency of the second and third order energies of the multireference perturbation theory(Chen F, Davidson E, Iwata S. Int J Quant Chem, 2002, 86: 256) is investigated theoretically with a su-per-molecular model composed of N-hydrogen molecules separated by a large distance. It is found that the two perturbation series corresponding to two Hamiltonian partitions are not size consistent at the second and third order. However, two size consistent forms are suggested for two Hamiltonian parti-tions at the second order, if some approximations to the denominators of the original second order energies are assumed.     

On the Fe-CO molecule

Using a perturbational configuration interaction approach, it is found that the Fe-CO molecule has a low-spin ground state (i.e. ^3–), at variance with similar compounds formed by the first transition elements of this series (e.g. Sc and Ti). Binding energies, interatomic distances and vibration frequencies have been calculated for the ^3– state as well as for the ^5– high-spin state.Dedicated to Professor J. Koutecký on the occasion of his 65th birthday     

On the phase equilibrium Stockmayer fluids

The fluid phase equilibrium of the Stockmayer fluid is investigated using a thermodynamic perturbation theory approach. The reference and the perturbation potential are the Lennard–Jones potential and the dipolar–dipolar interactions, respectively. They are assumed to be represented by the modified Benedict–Webb–Rubin equation of state [J.K. Johnson, J.A. Zollweg, K.E. Gubbins, Mol. Phys. 78 (1993) 591–618] and the Padé approximant [G. Stell, J.C. Rasaiah, H. Narang, Mol. Phys. 27 (1974) 1393–1414], respectively. The asymmetry found in an analogous study [M.E. van Leeuwen, B. Smit, E.M. Hendriks, Mol. Phys. 78 (1993) 271–283] based on the BWR equation of state [J.J. Nicolas, K.E. Gubbins, W.B. Streett, D.J. Tildesley, Mol. Phys. 37 (1979) 1429–1454] is now not observed on the vapour–liquid equilibrium coexistence curves of Stockmayer fluids with dipolar strength of μ^*2 = 1, 2, 3, and 4. Results agree with computer simulations for dipolar strength of μ^*2 = 1; however as strength dipole increases, liquid densities are over-estimated.     

On the many-body contributions to the interaction polarisability and hyperpolarisability of Hen

The many-body contributions to the interaction polarisability and hyperpolarisability of He _{n (n}=3,4, 5), for various configurations and distances have been determined and analysed. Several cases have been found where the three-body terms contribute more than 20% to or of He _n . The remarkable dependence of the above interaction properties on the internuclear distances and the shape of the cluster has been demonstrated. The interaction hyperpolarisabilities are shown to be uniquely sensitive probes of the electronic structure changes induced by variation of the cluster configuration. The results were computed by employing a computational procedure which relies on anab initio wave function, McWeeny's et al. coupled Hartree-Fock perturbation theory and an efficient algorithm for the determination of hyperpolarisabilities starting from a non-orthogonal basis set. The function counterpoise method has been used to reduce the basis set superposition error.     

Multiconfigurational second-order perturbative methods: Overview and comparison of basic properties

Summary The multiconfigurational second-order perturbative treatments of molecular electronic calculations can be classified into four groups: i) quasi-degenerate perturbation theory (QDPT) in the basis of determinants, ii) non-degenerate perturbation theory applied to eigenvectors resulting from a truncated CI, ii) QDPT in a model space of non-interacting multiconfigurational functions, iv) intermediate Hamiltonians theory, and examined according to three criteria: i) risk of numerical instability due to intruder states, ii) ability to treat the effect of the outer-space on the model space component of the wavefunction, especially important for the treatment of weakly avoided crossings, iii) separability for (A* ... B) problems. None of the existing methods satisfies these three criteria, as shown both by model analysis and real ab initio calculations on LiF and CuF.     

On the number of spin functions in the first order interaction space

A proof is given that in a configuration interaction method the first-order interaction space contains at most only twice as many spin functions as the zeroth-order space. This allows for a dramatic reduction of the size of CI expansion. For most of the high-spin systems only two spin functions for each configuration are needed.     

On the invariance of the configuration interaction energy with respect to orbital rotations

Summary The invariance of the configuration interaction (CI) energy with respect to orbital rotation is considered. The inclusion of all spin couplings versus only those from the first-order interacting space is considered. A definition for the analog of a second-order CI calculation when inactive electrons are present is proposed.     

Perturbation calculation of the interaction energy using orthogonalized orbitals

The diagrammatic Rayleigh-Schr?dinger perturbation theory for the interaction of two closed-shell systems is developed up to the third order of pertur-bation using orthogonalized orbitals. The interaction energy is expressed by the Rayleigh-Schr?dinger perturbation expansion. A simple approach for the estimation of basis set superposition error is introduced. The preliminary calculations of the intermolecular interactions for the He dimer within the augmented cc-pVTZ basis set are compared with the supermolecular approach, perturbation calculation in biorthogonal basis sets and symmetry adapted perturbation theory results. Received: 17 December 1996 / Accepted: 5 November 1997     

The electronic states of 1,2,5-oxadiazole studied by VUV absorption spectroscopy and CI, CCSD(T) and DFT methods

The 1,2,5-oxadiazole VUV absorption spectrum in the range 5–11.5 eV, shows broad bands centred near 6.2, 7.1, 8.3, 8.8, 10.6 and 11.3 eV. Rydberg states associated with three ionisation energies (IE) were identified in the complex fine structure above 8.7 eV. Electronic vertical excitation energies for singlet and triplet valence, and Rydberg states were computed using ab initio multi-reference multi-root CI methods. There is generally a good correlation between the envelope of the theoretical intensities and the experimental spectrum. The nature of the more intense calculated Rydberg states, and positions of the main valence and Rydberg bands are discussed. The lowest triplet, singlet and Rydberg 3s excited states have equilibrium structures that are non-planar with C_S symmetry, in a chair-like orientation where the O and H atoms lie out of the NCCN plane. This finding is consistent with the doubling of the low energy UV spectral lines [B.J. Forrest, A.W. Richardson, Can. J. Chem., 50 (1972) 2088].The nearly degenerate IE of the UV-photoelectron spectrum (UV–PES, Palmer et al. 1977) makes analysis of the VUV spectrum difficult, leading to the necessity for reinvestigation. Vertical studies (IE_V) using CI, Tamm–Dancoff (TDA) and Green’s Function (GF) methods all gave similar results, with near degeneracy of the first 3IE_V confirming the earlier study. Studies of the adiabatic IE (IE_A) using CCSD(T) and B3LYP methods, showed the energy sequence ²A₂ < ²B₁ < ²B₂, but these states are all saddle points, in contrast to the 4th state (²A₁) which is a minimum. In contrast, MP2 study of the ²B₂ state showed a minimum, with only two saddle points.Complete minima were found after minor twisting of the structures. The lowest energy cationic state is ²A^″ (C_S), which closely resembles the ²B₂ state. The O–N–C–C skeleton is twisted by 8°. The corresponding ²A^′ state (C_S) is effectively identical to the ²B₁ state. Attempts to find minima for other symmetry states were unsuccessful.     

Exact size consistency of multireference Møller–Plesset perturbation theory

Single‐reference closed‐shell Møller–Plesset perturbation theory is well known for its size consistency, a quality that is essential for consistent comparisons of calculations on molecules of different size. However, it is far from obvious whether this quality can be retained in the multireference case. In this work it is shown that an exactly size consistently generalization to multireference perturbation theory can be constructed. The central result is that the zeroth‐order Hamiltonian should be constructed using separate projection operators for each excitation level, i.e., it should contain no couplings between different excitation levels. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 72: 549–558, 1999