On the theory of electron correlation in atoms and molecules. II. General cluster expansion theory and the general correlated wave functions method

An exact cluster expansion of many electron wave functions is derived, beginning with a finite linear combination of Slater determinants rather than the more usual single determinant. This general cluster expansion is found to apply both in the case where all possible Slater determinants from a finite set of spin orbitals are included in the linear combination, and in the case where the number of determinants is restricted. The special properties of that finite linear combination of determinants closest to the exact wave function in the least squares sense are studied. These properties lead to the derivation of a general correlated wave functions method, illustrating again the close relationship between methods of this type and cluster expansion theory. Additional approximations, necessary for practical calculations, are set out.     

Two-electron integrations in the quantum theory of atoms in molecules with correlated wave functions

A recent method proposed to compute two-electron integrals over arbitrary regions of space [Martin Pendas, A. et al., J Chem Phys 2004, 120, 4581] is extended to deal with correlated wave functions. To that end, we use a monadic factorization of the second-order reduced density matrix originally proposed by E. R. Davidson [Chem Phys Lett 1995, 246, 209] that achieves a full separation of the interelectronic components into one-electron terms. The final computational effort is equivalent to that found in the integration of a one determinant wave function with as many orbitals as occupied functions in the correlated expansion. Similar strategies to extract the exchange and self-interaction contributions from the two-electron repulsion are also discussed, and several numerical results obtained in a few test systems are summarized.     

Calculation of molecular quadrupole moments and a demonstration of the importance of overlap densities in the theory of polyatomic molecules

A computational method for calculating quadrupole moments from molecular wave functions in a Slater orbital basis set is described. Using both IEHT and CNDO wave functions quadrupole moments for a series of polyatomic molecules are calculated. They are compared with experimental results and the IEHT wave functions are found to give agreement with experiment while CNDO wave functions do not. The importance of bicentric densities (overlap densities) in the calculation of multipole moments is shown. This is followed by a discussion of the usefulness of these wave functions for a quantitative characterization of the electronic structure of large molecules.     

On bond orders and valences in the Ab initio quantum chemical theory

The concept of bond order and valence indices calculated at the ab initio level is discussed in connection with their close relationship to the nonclassical exchange effects in bonding. An improved definition of bond order and free valence indices is given for the open-shell SCF (UHF ) case, and the generalization of the bond order and valence indices to correlated wave functions is also introduced.     

Transition states in modern valence-bond theory: application to the Cope rearrangement

Modern valence-bond theory, in its spin-coupled form, is used to study the electronic structure of the transition structures in the Cope rearrangement. It is found that the transition structure described by a “chair” geometry with a “6-in-6” CASSCF/6-31G^* wave function is clearly aromatic while the CASSCF/6-31G^*“boat” transition structure corresponds more closely to two weakly interacting allyl radicals. Moreover, there is a striking resemblance between the CASSCF chair transition structure and the benzene molecule, arising from the modern valence-bond analysis in terms of Rumer spin functions. In agreement with previous works, dynamical correlated wave functions show shorter interallylic distances in the optimized transitions structures. The use of spin-coupled wave functions on the latter geometries results in diradical and aromatic character for the chair and boat transition structures, respectively. Received: 13 October 1998 / Accepted: 30 December 1998 / Published online: 7 June 1999     

Methods of composite molecular wave functions. I. Variational principle and multiconfiguration SCF theory

The Silverstone–Stuebing variational principle for the discontinuous wave functions of one-electron systems is generalized for many-electron systems. The variational functional of energy takes real or complex value. The condition that it is real is given. Using the generalized variational principle, a multiconfiguration SCF theory for the composite molecular wave function is formulated. According to the theory, we may divide the whole space into space-filling cells, solve the SCF equations in each cell and build up the wave functions of the system by gathering the wave functions obtained in the cells. For use in the basis-set expansion method, the SCF equations are rewritten as matrix forms in which only one- and two-center integrals appear if an expansion center is located in each cell.     

The local orbital energy and density functional theory

From the viewpoint of density functional theory, an expression is derived which improves the average energy of a trial density. Applications to atoms and molecules are made using wave function methods and are based on properties of the variance, which is defined as $ (\overline {\varepsilon ^2 } - (\overline \varepsilon)^2)^{1/2} $, where ? is the local orbital energy. Calculated results for both Hartree-Fock and correlated wave functions are quite encouraging.     

Utilization of deformations in molecular quantum chemistry and application to density functional theory

The aim of this article is to present in a way accessible to most quantum chemists a general mathematical method which consists in deforming wave functions and density functions (in the spirit of the local scaling transformation). This deformation method allows us to obtain several new results, including a characterization of the set of wave functions that have the same given density function (which gives a new insight on a result of G. Zumbach and K. Maschke, Phys. Rev. A 28 , 544 (1983)) and an N-representability result where symmetry is taken into account. We also propose new theoretical ways to generate approximations of the exact density functional and give a numerical example. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 68: 221–231, 1998     

Analytic correlated wave functions in momentum space and related properties of helium like ions

We obtain analytic correlated wave functions in momentum space as the Fourier transform of correlated wave functions which are able to incorporate almost all of the correlation energy for the ground-state of two-electron atoms. Then we study the atomic momentum-density, the Compton profile and the elastic and inelastic scattering factors for this kind of wave functions. The scattering factors are also compared with the ones provided by a more accurate correlated wave function. All the calculations can be analytically performed, provided the correlated wave function in position space has been determined.     

Expectation values and density matrices in the Coupled-Cluster theory

A scheme for calculating expectation values in coupled-cluster wave functions is given. The unlinked terms originating from the norm of the wave function are eliminated. Hermiticity and trace conditions for the density matrices are exhibited in such a way as to guarantee their conservation in any reasonable truncation scheme.     

Modified one-electron hamiltonian method in the MC SCF theory

For orbital optimization within the MC SCF theory a modification of the OEH method is proposed with the direction of descent determined according to the Fletcher–Reeves gradient method. The combined method developed on this basis ensures the convergence of the iterative process when the Hessian singularities occur. The convergence properties of the method proposed are studied by performing the ab initio water molecule calculations using two types of multiconfigurational wave functions.     

An open‐source framework for analyzing N‐electron dynamics. II. Hybrid density functional theory/configuration interaction methodology

In this contribution, we extend our framework for analyzing and visualizing correlated many‐electron dynamics to non‐variational, highly scalable electronic structure method. Specifically, an explicitly time‐dependent electronic wave packet is written as a linear combination of N‐electron wave functions at the configuration interaction singles (CIS) level, which are obtained from a reference time‐dependent density functional theory (TDDFT) calculation. The procedure is implemented in the open‐source Python program det CI@ORBKIT, which extends the capabilities of our recently published post‐processing toolbox (Hermann et al., J. Comput. Chem. 2016, 37, 1511). From the output of standard quantum chemistry packages using atom‐centered Gaussian‐type basis functions, the framework exploits the multideterminental structure of the hybrid TDDFT/CIS wave packet to compute fundamental one‐electron quantities such as difference electronic densities, transient electronic flux densities, and transition dipole moments. The hybrid scheme is benchmarked against wave function data for the laser‐driven state selective excitation in LiH. It is shown that all features of the electron dynamics are in good quantitative agreement with the higher‐level method provided a judicious choice of functional is made. Broadband excitation of a medium‐sized organic chromophore further demonstrates the scalability of the method. In addition, the time‐dependent flux densities unravel the mechanistic details of the simulated charge migration process at a glance. © 2017 Wiley Periodicals, Inc.     

On the accuracy of density functional theory for iron-sulfur clusters

A simple, yet powerful wave function manipulation method was introduced utilizing a generalized ionic fragment approach that allows for systematic mapping of the wave function space for multispin systems with antiferromagnetic coupling. The use of this method was demonstrated for developing ground state electronic wave function for [2Fe-2S] and [Mo-3Fe-4S] clusters. Using well-defined ionic wave functions for ferrous and ferric irons, sulfide, and thiolate fragments, the accuracy of various density functionals and basis sets including effective core potentials were evaluated on a [4Fe-4S] cluster by comparing the calculated geometric and electronic structures with crystallographic data and experimental atomic spin densities from X-ray absorption spectroscopy, respectively. We found that the most reasonable agreement for both geometry and atomic spin densities is obtained by a hybrid functional with 5% HF exchange and 95% density functional exchange supplemented with Perdew's 1986 correlation functional. The basis set seems to saturate only at the triple-zeta level with polarization and diffuse functions. Reasonably preoptimized structures can be obtained by employing computationally less expensive effective core potentials, such as the Stuttgart-Dresden potential with a triple-zeta valence basis set. The extension of the described calibration methodology to other biologically important and more complex iron-sulfur clusters, such as hydrogenase H-cluster and nitrogenase FeMo-co will follow.     

The use of explicitly correlated,partially antisymmetric wave functions in atomic and molecular calculations

Trial wave functions, written as the sum of a configuration interaction expansion and an explicitly correlated term which is not antisymmetric, are proposed for use in calculating the electronic properties of atoms and molecules. A variational principle, modified to allow the use for such partially antisymmetric wave functions, is developed. It is shown that the consequences of partial antisymmetry on calculated expectation values can be estimated. The method avoids difficult three-electron integrals which arise in other theories.     

Product of group-function expansions for correlated wave functions

A theorem is proved which demonstrates the relationship between a product of group functions describing the correlated motion of a particular group of electrons in an N-electron system and a wave function obtained from the exact wave function which describes the correlation of the same group of electrons. By considering such products of group functions as elements in a variational wave function, an expansion for correlated wave functions is suggested, which emphasizes the correlated motion of groups of electrons in the whole system.     

Variational Monte Carlo calculations for some cations and anions of the first‐row atoms using explicitly correlated wave functions

The variational Monte Carlo method is applied to calculate ground‐state energies of some cations and anions of the first‐row atoms. Accurate values providing between 80 and 90% of the correlation energy are obtained. Explicitly correlated wave functions including up to 42 variational parameters are used. The nondynamic correlation due to the 2s ? 2p near degeneracy effect is included by using a multideterminant wave function. The variational free parameters have been fixed by minimizing the energy that has shown to be a more convenient functional than the variance of the local energy, which is the most commonly employed method in variational Monte Carlo calculations. The energies obtained improve previous works using similar wave functions. © 2002 Wiley Periodicals, Inc.; DOI 10.1002/qua.10125     

Communication: Second-order multireference perturbation theory with explicit correlation: CASPT2-F12

An explicitly correlated complete active space second-order perturbation (CASPT2-F12) method is presented which strongly accelerates the convergence of CASPT2 energies and properties with respect to the basis set size. A Slater-type geminal function is employed as a correlation factor to represent the electron-electron cusp of the wave function. The explicitly correlated terms in the wave function are internally contracted. The required density matrix elements and coupling coefficients are the same as in conventional CASPT2, and the additional computational effort for the F12 correction is small. The CASPT2-F12 method is applied to the singlet-triplet splitting of methylene, the dissociation energy of ozone, and low-lying excited states of pyrrole.     

Correlated wave functions for the ground and some excited states of the iron atom

We study the states arising from the [Ar]4s(2)3d6 and [Ar]4s(1)3d7 configurations of iron atom with explicitly correlated wave functions. The variational wave function is the product of the Jastrow correlation factor times a model function obtained within the parametrized optimized effective potential framework. A systematic analysis of the dependence of both the effective potential and the correlation factor on the configuration and on the term is carried out. The ground state of both, the cation, Fe+, and anion, Fe-, are calculated with correlated wave functions and the ionization potential and the electron affinity are obtained.