All-pair wave function and reduced variational equation for electronic systems

A new type of wave function is proposed for atomic and molecular systems. This all-pair function is constructed of N(N – 1)/2 identical geminals for N electrons. For systems with the highest multiplicity this is the full space part of the wave function. For closed shell systems it has to be multiplied by a Slater determinant according to the antisymmetry condition. In the case of maximal multiplicity a reduced variational equation is derived for the geminal. This equation is independent of the dimensionality of the system and contains the particle number as a multiplicative factor only. The method is extended to the closed shell case where a restriction has to be fulfilled. The reduction of the variational equation can be done only approximately. The use of identical geminals can be treated as a first approximation. An extension of the method, called the pair interdependent configuration interaction (PICI), is proposed. The special features of the method are discussed briefly.     

A study of the relationships between unpaired electron density, spin-density and cumulant matrices

This work describes the derivation of simple relationships between the density matrix of effectively unpaired electrons and the spin-density matrix in N-electron systems. The link between both devices turns out to be the one-electron matrix arising from the diagonal contraction of the cumulant matrix corresponding to the second-order reduced density matrix. We study some features of this contracted matrix, showing its usefulness to describe the electronic correlation. Numerical determinations performed in selected systems with different spin symmetries confirm the theoretical predictions.     

Generalized density-functional theory: Conquering the N -representability problem with exact functionals for the electron pair density and the second-order reduced density matrix

Using the constrained search and Legendre-transform formalisms, one can derive “generalized” density-functional theories, in which the fundamental variable is either the electron pair density or the second-order reduced density matrix. In both approaches, theN-representability problem is solved by the functional, and the variational principle is with respect to all pair densities (density matrices) that are nonnegative and appropriately normalized. The Legendre-transform formulation provides a lower bound on the constrained-search functional. Noting that experience in density-functional and density-matrix theories suggests that it is easier to approximate functionals than it is to approximate the set ofN-representable densities sheds some light on the significance of this work.     

The non-N-representability of the Colle–Salvetti second-order reduced density matrix

The Colle–Salvetti second-order reduced density matrix (2-matrix) is an approximation to the 2-matrix obtained from a wave function that is a product of a reference wave function containing little or no correlation times a product of correlation factors that are functions of the coordinates of pairs of electrons. A formal proof is given for the non-N-representability for the Colle–Salvetti 2-matrix using the nonnegativity condition of the 2-matrix. The nonnegativity condition of the particle-hole overlap matrix (G matrix) is also not satisfied. The proof is valid for Colle–Salvetti 2-matrices obtained from both the Hartree–Fock and small multiconfigurational-self-consistent-field wave functions. Even though the Colle–Salvetti 2-matrix is not N-representable, it does satisfy the Pauli principle component of the G-matrix condition because it reduces to an N-representable first-order reduced density matrix. © 1993 John Wiley & Sons, Inc.     

The structure of the second-order reduced density matrix in density matrix functional theory and its construction from formal criteria

Some formal requirements for the second-order reduced density matrix are discussed in the context of density matrix functional theory. They serve as a basis for the ad hoc construction of the second-order reduced density matrix in terms of the first-order reduced density matrix and lead to implicit functionals where the occupation numbers of the natural orbitals are obtained as diagonal elements of an idempotent matrix the elements of which represent the variational parameters to be optimized. The numerical results obtained from a first realization of such an implicit density matrix functional give excellent agreement with the results of full configuration interaction calculations for four-electron systems like LiH and Be. Results for H2O taken as an example for a somewhat larger molecule are numerically less satisfactory but still give reasonable occupation numbers of the natural orbitals and indicate the capability of density matrix functional theory to cope with static electron correlation.     

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.     

Correlation energy,correlated electron density,and exchange‐correlation potential in some spherically confined atoms

We report correlation energies, electron densities, and exchange‐correlation potentials obtained from configuration interaction and density functional calculations on spherically confined He, Be, Be²⁺, and Ne atoms. The variation of the correlation energy with the confinement radius R_c is relatively small for the He, Be²⁺, and Ne systems. Curiously, the Lee–Yang–Parr (LYP) functional works well for weak confinements but fails completely for small R_c. However, in the neutral beryllium atom the CI correlation energy increases markedly with decreasing R_c. This effect is less pronounced at the density‐functional theory level. The LYP functional performs very well for the unconfined Be atom, but fails badly for small R_c. The standard exchange‐correlation potentials exhibit significant deviation from the “exact” potential obtained by inversion of Kohn–Sham equation. The LYP correlation potential behaves erratically at strong confinements. © 2016 Wiley Periodicals, Inc.     

Necessary conditions for the N‐representability of pair distribution functions

A necessary condition for the N‐representability of the electron pair density proposed by one of the authors (E. R. D.) is generalized. This shows a link between this necessary condition and other, more widely known, N‐representability conditions for the second‐order density matrix. The extension to spin‐resolved electron pair densities is considered, as is the extension to higher‐order distribution functions. Although quantum mechanical systems are our primary focus, the results are also applicable to classical systems, where they reduce to an inequality originally derived by Garrod and Percus. As a simple application, bounds to the average angle between an electron pair are derived. It is shown that computational methods based on variational minimization of the energy with respect to the electron pair density can give extremely poor results unless robust N‐representability constraints are considered. For reference, constraints for the N‐representability of the pair density are summarized. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

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.     

Partitioning schemes for the 2-matrix and the pair density

Several schemes are discussed for partitioning the second-order reduced density matrix Γ into two parts, Γ⁰ and Γ′. The Γ⁰s are based on the independent particle model and the Γ′s are corrections due to electron correlation. The difficulties of choosing a Γ⁰ that will serve as a suitable reference point for studying electron correlation are discussed. In order to compare alternative partitioning schemes, an atomic wave function for the ¹S ground state of the Be atom in the configuration-interaction approximation was selected. A fifty-two configuration wave function was computed and contour graphs were made of the total pair density Γ(1 2) and of the “correlation pair density” Γ′(1 2) for several choices of the reference Γ⁰.     

Calculation of X‐ray scattering intensities by means of the coupled cluster singles and doubles model

Total X‐ray scattering intensity σ_ee(q) is very sensitive to electron correlation effects. In this study σ_ee(q) of N₂, CO, and N₂O have been computed by the coupled cluster singles and doubles (CCSD) method and compared with configuration interaction singles and doubles (CISD) calculations as well as experimental observations. σ_ee(q) curves by CCSD calculations are rather close to those by CISD, but although small, there still exist some discrepancies between calculated and observed values. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 1315–1320, 2001     

Calculation of the exchange coupling constants of copper binuclear systems based on spin-flip constricted variational density functional theory

We have recently developed a methodology for the calculation of exchange coupling constants J in weakly interacting polynuclear metal clusters. The method is based on unrestricted and restricted second order spin-flip constricted variational density functional theory (SF-CV(2)-DFT) and is here applied to eight binuclear copper systems. Comparison of the SF-CV(2)-DFT results with experiment and with results obtained from other DFT and wave function based methods has been made. Restricted SF-CV(2)-DFT with the BH&HLYP functional yields consistently J values in excellent agreement with experiment. The results acquired from this scheme are comparable in quality to those obtained by accurate multi-reference wave function methodologies such as difference dedicated configuration interaction and the complete active space with second-order perturbation theory.     

Application of the unitary group approach to evaluate spin density for configuration interaction calculations in a basis of S2 eigenfunctions

We present an implementation of the spin‐dependent unitary group approach to calculate spin densities for configuration interaction calculations in a basis of spin symmetry‐adapted functions. Using S² eigenfunctions helps to reduce the size of configuration space and is beneficial in studies of the systems where selection of states of specific spin symmetry is crucial. To achieve this, we combine the method to calculate U(n) generator matrix elements developed by Downward and Robb (Theor. Chim. Acta 1977, 46, 129) with the approach of Battle and Gould to calculate U(2n) generator matrix elements (Chem. Phys. Lett. 1993, 201, 284). We also compare and contrast the spin density formulated in terms of the spin‐independent unitary generators arising from the group theory formalism and equivalent formulation of the spin density representation in terms of the one‐ and two‐electron charge densities.     

Closed-form expressions for correlated density matrices: application to dispersive interactions and example of He2

Empirically correlated density matrices of N-electron systems are investigated. Closed-form expressions are derived for the one- and two-electron reduced density matrices from a pairwise correlated wave function. Approximate expressions are then proposed which reflect dispersive interactions between closed-shell centrosymmetric subsystems. Said expressions clearly illustrate the consequences of second-order correlation effects on the reduced density matrices. Application is made to a simple example: the He(2) system. Reduced density matrices are explicitly calculated, correct to second order in correlation, and compared with approximations of independent electrons and independent electron pairs. The models proposed allow for variational calculations of interaction energies and equilibrium distance as well as a clear interpretation of dispersive effects on electron distributions. Both exchange and second order correlation effects are shown to play a critical role on the quality of the results.     

Some basic properties of the correlation matrices

On the basis of the properties of correlation matrices, it is shown here that the set of all the first‐order transition reduced density matrices of a system provide complete information about that system. Also, the interrelation between the properties of the correlation matrix and the 2‐RDM N‐representability conditions is studied. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

Dynamic correlation for biorthogonal valence bond reference states

Summary The use of biorthogonal valence bond reference functions in evaluating the correlation energy is investigated. Since the method is not variationally bound some care must be taken in defining the reference state to ensure that the variational bound is not violated, some discussion is given to this matter. The procedure adopted here is a matrix element driven configuration interaction scheme. To reduce the computational labour involved, a configuration selection criterion is introduced. The method is tested through its application to the symmetric stretching of HF, H₂O, (² B ₁) NH₂ and the singlet-triplet gap in CH₂. Comparison is made with other methods, including full CI. The results show that the current method is quite promising.     

Two-body Reduced Density Matrix Reconstruction for Van der Waals Systems

A method for the calculation of the electronic energy of a correlated system is presented. This approach is based on the reconstruction of the total two-body reduced density matrix by doing separate configurations interaction calculations on fragments. The method has been tested on Van der Waals systems and has been implemented by considering restrictive N-representability conditions. It is shown that the computational strategy presented in this work can describe with good accuracy weak dispersion interactions, and considerably lowers the size-consistency error of a classical configuration interaction calculation.     

Single determinant N‐representability and the kernel energy method applied to water clusters

The Kernel energy method (KEM) is a quantum chemical calculation method that has been shown to provide accurate energies for large molecules. KEM performs calculations on subsets of a molecule (called kernels) and so the computational difficulty of KEM calculations scales more softly than full molecule methods. Although KEM provides accurate energies those energies are not required to satisfy the variational theorem. In this article, KEM is extended to provide a full molecule single‐determinant N‐representable one‐body density matrix. A kernel expansion for the one‐body density matrix analogous to the kernel expansion for energy is defined. This matrix is converted to a normalized projector by an algorithm due to Clinton. The resulting single‐determinant N‐representable density matrix maps to a quantum mechanically valid wavefunction which satisfies the variational theorem. The process is demonstrated on clusters of three to twenty water molecules. The resulting energies are more accurate than the straightforward KEM energy results and all violations of the variational theorem are resolved. The N‐representability studied in this article is applicable to the study of quantum crystallography. © 2017 Wiley Periodicals, Inc.     

Density functional theory for open-shell systems using a local-scaling transformation scheme. II. Euler-Lagrange equation for f(r) versus that for ρ(r)

Following the previous article (Part I), we express the total nonrelativistic energy for spin manifolds of open-shell multielectronic systems, within an orbit θ^N induced by a model wave function (MWF) _Ψ using a single local-scaling transformation (LST) as an exact functional of the single-particle density ρ( r ) or, alternatively, of the LST scalar function f( r ). We derive the corresponding Euler–Lagrange variational equations: one implicit in ρ( r ), which can be solved iteratively through steps involving f( r ), and one explicit in f( r ), derived from the total energy as a functional of f( r ). Both equations fulfill the space and spin symmetries characterizing the system. The problems arising from the specificities of these two highly nonlinear integrodifferential equations are discussed. The optimal charge density ρ( r ) derived from these equations is N- and v-representable and determines the optimal spin density σ( r ) as well. Accurate optimal values of all observables can be derived from this scheme using standard procedures. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65 : 257–268, 1997     

A density matrix variational calculation for atomic Be

The ground-state energy of the beryllium atom is calculated using a variational procedure in which the elements of the two-body reduced density matrix (particle–particle matrix) are the variational parameters. It is shown that, for this problem and with the limited number of spin-orbitals used, the trace condition and the simultaneous nonnegativity conditions on the particle–particle, the particle–hole, and the hole–hole matrices form a complete solution to the N-representability problem. The energy obtained is – 14.61425 a.u., practically identical to the value given by a configuration interaction calculation which uses the same states. The effects of weakening the nonnegativity conditions on each of the matrices in turn were also explored.