Direct MRCI method for the calculation of relativistic many-electron wavefunctions. I. General formalism

The formalism of a quasi- or full-relativistic multireference CI method has been developed and implemented. The scheme is appropriate for the calculation of molecular systems in which the relativistic effects are of the same order of magnitude as the correlation contributions. In this contribution some important symmetry aspects of a relativistic many-electron wave function are discussed and the consequences for the CI matrix structure are shown. An efficient CI strategy in the form of a direct CI is presented, which avoids the construction of the whole CI matrix. Based on a determinantal expansion of molecular spinor products, the individual one- and two-electron molecular integrals are processed, and the molecular symmetry is easily accounted for by a proper linear combination of Slater determinants in the CI starting vector. For an efficient CI organization some powerful techniques of the graphical unitary group approach have been transferred to the relativistic case.     

Wigner molecules: the strong-correlation limit of the three-electron harmonium

At the strong-correlation limit, electronic states of the three-electron harmonium atom are described by asymptotically exact wave functions given by products of distinct Slater determinants and a common Gaussian factor that involves interelectron distances and the center-of-mass position. The Slater determinants specify the angular dependence and the permutational symmetry of the wave functions. As the confinement strength becomes infinitesimally small, the states of different spin multiplicities become degenerate, their limiting energy reflecting harmonic vibrations of the electrons about their equilibrium positions. The corresponding electron densities are given by products of angular factors and a Gaussian function centered at the radius proportional to the interelectron distance at equilibrium. Thanks to the availability of both the energy and the electron density, the strong-correlation limit of the three-electron harmonium is well suited for testing of density functionals.     

Computing the energy of a water molecule using multideterminants: a simple, efficient algorithm

Quantum Monte Carlo (QMC) methods such as variational Monte Carlo and fixed node diffusion Monte Carlo depend heavily on the quality of the trial wave function. Although Slater-Jastrow wave functions are the most commonly used variational ansatz in electronic structure, more sophisticated wave functions are critical to ascertaining new physics. One such wave function is the multi-Slater-Jastrow wave function which consists of a Jastrow function multiplied by the sum of Slater determinants. In this paper we describe a method for working with these wave functions in QMC codes that is easy to implement, efficient both in computational speed as well as memory, and easily parallelized. The computational cost scales quadratically with particle number making this scaling no worse than the single determinant case and linear with the total number of excitations. Additionally, we implement this method and use it to compute the ground state energy of a water molecule.     

Half-Projected and Projected Hartree-Fock Calculations for Singlet Ground States. i. four-Electron Atomic Systems

The half-projected Hartree-Fock function (HPHF ) for singlet states is defined as a linear combination of two Slater determinants, which contains only spin eigenfunctions with even quantum number. Using a self-consistent procedure based on the generalized Brillouin's theorem, the RHF , HPHF and PHF functions are deduced for the ground states of the Li^?, Be, B⁺, and C²⁺ systems, in a limited basis set. It is found that the HPHF function yields better energy values than the RHF function, very close to that of the PHF one. The HPHF scheme seems thus to be useful as a substitute for the PHF model, specially in the case of large electronic systems in which the latter method becomes unmanageable.     

Evaluation of expansion coefficients for translation of Slater‐type orbitals using complete orthonormal sets of exponential‐type functions

By the use of exponential‐type functions (ETFs) the simpler formulas for the expansion of Slater‐type orbitals (STOs) in terms of STOs at a displaced center are derived. The expansion coefficients for translation of STOs are presented by a linear combination of overlap integrals. The final results are of a simple structure and are, therefore, especially useful for machine computations of arbitrary multielectron multicenter molecular integrals over STOs that arise in the Hartree–Fock–Roothaan approximation and also in the Hylleraas correlated wave function method for the determination of arbitrary multielectron properties of atoms and molecules. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 81: 126–129, 2001     

Non-empirical LCAO-MO-SCF-CI calculations on organic molecules with Gaussian type functions

This paper (Part I) describes the theoretical and computational bases of some non-empirical calculations on small organic molecules to be reported in later papers (Parts II et seq). Approximate solutions for the usual fixed nucleus electronic Hamiltonian, in the term of wave functions composed of Slater determinants, are discussed, with particular emphasis on their computational utility. Possible choices of basis functions, from which to form the determinants are examined, and the advantages of Gaussian type functions (GTF) centered on the component atoms are pointed out. Some of the properties of molecules which can be calculated using such approximate wave functions are outlined. Finally an attempt is made to discuss the current limitations of non-empirical calculations of the type described here, and some guesses are made about their future. Brief outlines as a set of appendices are given of the mathematical formalism and computational details of the calculations.     

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.     

Analysis of molecular orbital wave functions in terms of valence bond functions for molecular fragments. I. Theory

A method of expansion of molecular orbital wave functions into valence bond (VB ) functions is extended to molecular fragments. The wave function is projected onto a basis of mixed determinants, involving molecular orbitals as well as fragment atomic orbitals, and is further expressed as a linear combination of VB functions, characteristic of structural formulas of the fragment but whose remaining bonds are frozen. Structural weights for the fragment are deduced from this expression. Delocalized molecular orbitals are used as a startpoint, as they are after an ordinary SCF calculation. Wave functions of medium-sized molecules may be analyzed with reasonable storage requirements in a computer.     

Calculation of transition matrix elements by nonsingular orbital transformations

A general strategy is described for the evaluation of transition matrix elements between pairs of full class CI wave functions built up from mutually nonorthogonal molecular orbitals. A new method is proposed for the counter‐transformation of the linear expansion coefficients of a full CI wave function under a nonsingular transformation of the molecular‐orbital basis. The method, which consists in a straightforward application of the Cauchy–Binet formula to the definition of a Slater determinant, is shown to be simple and suitable for efficient implementation on current high‐performance computers. The new method appears mainly beneficial to the calculation of miscellaneous transition matrix elements among individually optimized CASSCF states and to the re‐evaluation of the CASCI expansion coefficients in Slater‐determinant bases formed from arbitrarily rotated (e.g., localized or, conversely, delocalized) active molecular orbitals. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

An open‐source framework for analyzing N‐electron dynamics. I. Multideterminantal wave functions

The aim of the present contribution is to provide a framework for analyzing and visualizing the correlated many‐electron dynamics of molecular systems, where an explicitly time‐dependent electronic wave packet is represented as a linear combination of N‐electron wave functions. The central quantity of interest is the electronic flux density, which contains all information about the transient electronic density, the associated phase, and their temporal evolution. It is computed from the associated one‐electron operator by reducing the multideterminantal, many‐electron wave packet using the Slater‐Condon rules. Here, we introduce a general tool for post‐processing multideterminant configuration‐interaction wave functions obtained at various levels of theory. It is tailored to extract directly the data from the output of standard quantum chemistry packages using atom‐centered Gaussian‐type basis functions. The procedure is implemented in the open‐source Python program det CI@ORBKIT, which shares and builds on the modular design of our recently published post‐processing toolbox (Hermann et al., J. Comput. Chem. 2016, 37, 1511). The new procedure is applied to ultrafast charge migration processes in different molecular systems, demonstrating its broad applicability. Convergence of the N‐electron dynamics with respect to the electronic structure theory level and basis set size is investigated. This provides an assessment of the robustness of qualitative and quantitative statements that can be made concerning dynamical features observed in charge migration simulations. © 2017 Wiley Periodicals, Inc.     

Correlated wave-function theory for many-center many-electron problems

The correlated electronic wave-function theory developed by S. Obara and K. Hirao [Bull. Chem. Soc. Jpn. 66 , 3300 (1993)], as applied to two-electron molecular systems, is generalized to many-center many-electron problems. The exact formulas for effective Hamiltonian operators are given. The rules for the calculation of matrix elements with three-electron operators over Slater determinants are formulated. From the energy-minimum principle, the system of master equations is derived for variational coefficients of a trial wave function for the molecules with closed electronic shells. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 69: 639–648, 1998     

Improved partition–expansion of two‐center distributions involving slater functions

The calculation of the electronic structure of large systems is facilitated by the substitution of the two‐center distributions by their projections on auxiliary basis sets of one‐center functions. An alternative is the partition–expansion method in which one first decides what part of the distribution is assigned to each center, and next expands each part in spherical harmonics times radial factors. The method is exact, requires neither auxiliary basis sets nor projections, and can be applied to Gaussian and Slater basis sets. Two improvements in the partition–expansion method for Slater functions are reported: general expressions valid for arbitrary quantum numbers are derived and the efficiency of the procedure is increased giving analytical solutions to integrals previously computed by numerical quadrature. The efficiency of the new version is assessed in several molecules and the advantages over the projection methods are pointed out. © 2013 Wiley Periodicals, Inc.     

On the theory of electron correlation in atoms and molecules: Relation between cluster expansion theory and the correlated wave functions method

A relation between the cluster expansion theory of many electron wave functions and the correlated wave functions method is established. In this way, the theoretical basis of the method is elucidated and the approximations involved in its application become apparent. General forms of the correlated wave function, differing in certain important respects from that form usually assumed, are derived.     

Half-projected and projected Hartree-Fock calculations for singlet ground states. II. Lithium hydride

The half-projected Hartree–Fock function (HPHF ) for singlet states is defined as a linear combination of two Slater determinants which contains only spin eigenstates with even spin quantum numbers. The possible uses of such an approach for determining molecular properties are investigated computing the potential energy curve, binding energy, force constant, and dipole moment variation corresponding to the lithium hydride ground state. Full projected and restricted Hartree–Fock calculations (PHF and RHF ) are performed simultaneously for comparison purposes. It is found that the HPHF model yields very satisfactory results, very close to those of the PHF scheme. Both models predict properly the molecular behavior as a function of nuclear separation, whereas the RHF one fails. A discussion is given in terms of configuration equivalents. It is concluded that the HPHF scheme seems to be useful for determining molecular properties specially in the case of large systems in which the more sophisticated methods are unmanageable.     

An explicitly correlated Mukherjee's state specific coupled cluster method: development and pilot applications

This paper reports development of the explicitly correlated variant of Mukherjee's state specific multireference coupled cluster method (MkCC-F12). The current implementation is restricted to conventional single and double excitations and to pseudo-double excitations related to the Slater Type Geminal (STG) correlation factor using the SP ansatz. The performance of the MkCCSD-F12 was tested on calculations of singlet methylene, dissociation curve of the fluorine molecule, and the BeH(2) insertion pathway. As expected, the results of the newly developed method reconfirm the significantly faster convergence with respect to the basis set limit compared to the traditional expansion in Slater determinants. Results prove that treating the correlation factor separately for each reference is appropriate.     

Analysis of wave functions for open-shell molecules

During the past decade we have looked at several ways to track the distribution of unpaired electrons during chemical reactions and in different spin states. These methods were inspired by our previous work on singlet di-radicals where the spin density is zero yet there are clearly singly occupied orbitals. More recently we have been concerned with analysis of wave functions for single molecule magnets. This review discusses the mathematical framework by which open-shell systems can be described, in addition to methods that extract the effectively unpaired electron density, the spin state of atoms in a molecule, and other useful properties from a molecular wave function. Some of the difficulties associated with using broken spin Slater determinants to evaluate the exchange coupling parameters in the Heisenberg Hamiltonian are also mentioned.     

The BERKELEY system. III. General configuration-interaction methods for open-shell molecular electronic states

A system of general “open-ended” configuration-interaction (CI ) programs, specifically designed for the Harris Corporation Slash Four minicomputer, is described. These methods are general in the sense that an arbitrary list of configurations (linear combinations of Slater determinants) may be used, and open ended in that peripheral (i.e., disk) storage capacity determines the maximum size problem that can be solved. The largest variational calculations carried out to date using BERKELEY involve 7064 open-shell singlet configurations (31,898 Slater determinants). Detailed timing breakdowns are presented for four test cases, two of which involve the lowest ππ* singlet state of ethylene. The other two examples are the orthogonal or bisected singlet state of trimethylenemethane and the ⁸B₁ state of the MnCH₂ complex. In the latter case, it is found that the predicted Mn? CH₂ dissociation energy is only slightly increased by electron correlation effects.     

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     

Third-order many-body perturbation theory for the ground state of the carbon monoxide molecule

Many-body perturbation calculations have been performed for the ground state of the carbon monoxide molecule at its equilibrium internuclear separation. The calculations are complete through third order within the algebraic approximation; i.e., the state functions are parameterized by expansion in a finite basis set. All two-, three-, and four-body terms are rigorously determined, and many-body effects are found to be very important. A detailed comparison is made with a previously reported configuration interaction study. Padé approximants to the energy expansion are constructed. The many-body perturbative wave function is used in the Rayleigh quotient to produce upper bounds to the electronic energy.