The importance of Colle–Salvetti for computational density functional theory

The purpose of this presentation is to show the importance of the Colle–Salvetti (Theor Chim Acta 37:329, 1975) paper in the development of modern computational density functional theory. To do this we cover the following topics (1) the Bright Wilson understanding (2) the Kohn–Sham equations (3) local density exchange (4) the exchange-hole (5) generalised gradient approximation for exchange (Becke and Cohen) (6) left–right correlation and dynamic correlation (7) the development of the Lee–Yang–Parr dynamic correlation functional from the Colle–Salvetti paper (8) the early success of GGA DFT. Finally we observe that the the BLYP and OLYP exchange-correlation functionals are not semi-empirical; this may explain their great success.     

The self-interaction error and the description of non-dynamic electron correlation in density functional theory

The self-interaction error (SIE) plays a central role in density functional theory (DFT) when carried out with approximate exchange-correlation functionals. Its origin, properties, and consequences for the development of standard DFT to a method that can correctly describe multi-reference electron systems by treating dynamic and non-dynamic electron correlation on an equal footing, is discussed. In this connection, the seminal work of Colle and Salvetti on wave function-based correlation functionals that do no longer suffer from a SIE is essential. It is described how the Colle–Salvetti correlation functional is an anchor point for the derivation of a functional multi-reference DFT method.     

Application of the Colle–Salvetti model to the uniform electron gas

An application of the Colle–Salvetti wave function (Colle and Salvetti in Theor Chim Acta 37: 329, 1975) to the uniform electron gas model is made. Some different levels of approximation are used. Contrarily to the previous conclusions of other work (Tao et al. in Phys Rev A 63: 032513, 2001), the present result shows that the Colle–Salvetti wave function is able to reproduce, to a semiquantitative level, the properties of the uniform electron gas. The most important request for this result is an apropiate choice for the value of the only parameter entering in the Colle–Salvetti wave function. The present results are a good complement to those obtained by Moscardó et al. (Theor Chem Accounts 115: 334, 2006) for atoms. On the basis of the results obtained in this paper, a very simple functional for the correlation energy is put forward. Its application to the uniform electron gas, lead to a very reasonable set of results. It can be concluded that the Colle–Salvetti wave function remains being a good option to built, in an approximate way, the correlation component of a N-electron system. Contribution to the Serafin Fraga Memorial Issue.     

Approximately N-representable density functional density matrices

For the first time, we obtain practical density matrices approximately N-representable by correlated-determinant wave functions, which are functionals of the electron density and entirely defined by information obtainable from the X-ray coherent diffraction experiment. © 1994 John Wiley & Sons, Inc.     

The Colle–Salvetti Wavefunction Revisited: a Comparison Between Three Approaches for Obtaining the Correlation Energy

The correlation factor of Colle and Salvetti is studied by comparing the behavior of three different correlation functionals. The normalization, sum rule, Coulomb hole, correlation energy integrand, and the Wigner exclusion hole have all been analyzed by applying the three approaches. The results indicate that the correlation factor proposed by Colle–Salvetti is a very good choice for modeling electron correlation in atoms. The flaws appearing in the development of the Colle–Salvetti equations seem mainly due to an inadequate use of the first mean value theorem of integral calculus. The Gaussian summation used for the two-body density matrix seems to be a good approximation to obtain the correlation factor equations.     

Density matrix expansions and their application in the theory of the many-electron systems

A method of calculation of the correlation energy is proposed, which includes the superposition of configurations and the two particle approach. This method is based on the density matrix formalism. The approximate, but N-representable expressions for the reduced density matrices are used. The correlation energy of the beryllium atom is calculated as an example.     

Fundamental importance of the Coulomb hole sum rule to the understanding of the Colle-Salvetti wave function functional

In this paper we consider the general form of the correlated-determinantal wave function functional of Colle and Salvetti (CS) for the He atom. The specific form employed by CS is the basis for the widely used CS correlation energy formula and the Lee-Yang-Parr correlation energy density functional of Kohn-Sham density functional theory. We show the following: (i) The key assumption of CS for the determination of this wave function functional, viz., that the resulting single-particle density matrix and the Hartree-Fock theory Dirac density matrix are the same, is equivalent to the satisfaction of the Coulomb hole sum rule for each electron position. The specific wave function functional derived by CS does not satisfy this sum rule for any electron position. (ii) Application of the theorem on the one-to-one correspondence between the Coulomb hole sum rule for each electron position and the constraint of normalization for approximate wave functions then proves that the wave function derived by CS violates charge conservation. (iii) Finally, employing the general form of the CS wave function functional, the exact satisfaction of the Coulomb hole sum rule at each electron position then leads to a wave function that is normalized. The structure of the resulting approximate Coulomb holes is reasonably accurate, reproducing both the short- and the long-range behavior of the hole for this atom. Thus, the satisfaction of the Coulomb hole sum rule by an approximate wave function is a necessary condition for constructing wave functions in which electron-electron repulsion is represented reasonably accurately.     

A variational density matrix approach with nonlocal effective potential

We show that using the Colle–Salvetti correlation-energy functional (Colle and Salvetti in Theoret Chim Acta 37:329, 1975) in the Hartree–Fock-type procedure suggested by Kohn and Sham (Phys Rev 140:A1133PR, 1965), one can calculate quite accurately electronic properties of systems in which the “dynamical” correlation energy is dominant. We compare our results with those obtained by Grabo and Gross (Chem Phys Lett 240:141, 1995) using the optimized effective potential method, and we discuss characteristics and advantages of our procedure.     

Equation for the direct determination of the density matrix: Time-dependent density equation and perturbation theory

The density equation proposed previously for the direct determination of the density matrix, i.e. for the wave mechanics without wave, is extended to a time-dependent case. The time-dependent density equation has been shown to be equivalent to the time-dependent Schr?dinger equation so long as the density matrix, included as a self-contained variable, is N-representable. Formally, it is obtainable from the previous time-independent equation by replacing the energy E with . The perturbation theory formulas for the density equation have also been given for both the time-dependent and time-independent cases. Received: 16 June 1998 / Accepted: 2 September 1998 / Published online: 8 February 1999     

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.     

On the accuracy of correlation energy calculated by the correlation factor method: first- and second-row atoms

Summary The correlation energy of two- and four-isoelectronic series, a representative example for which the existing spin-density functionals fails, is calculated using the Colle and Salvetti method, considering mono- and multideterminantal wave functions. The results are in agreement with experimental data, and show the potentiality of this method when it is applied to wave functions including the most relevant configurational features. Also, results for the ionization energies and electron affinities of first- and second-row atoms are reported.     

Approximately N -representable density functional density matrices: The case of large N

Density matrices approximatelyN-representable by a correlated determinant wavefunction (CDWF) have been previously derived. For the case of helium it had been shown how to obtain these density matrices as density functionals. Here it is shown that the same procedures, used for helium, generalize to the case for the number of electronsN-large. The result is a procedure for obtaining approximately CDWFN-representable density matrices which are functionals of the density for largeN.     

A density functional for the correlation energy,deduced in the framework of the correlation factor approach

An approximate density functional is deduced from a wave function within the correlation factor method. The new functional does not include terms depending on the gradient of the density, but shows the simplicity of local density functionals without spin polarization. However, it includes correctly the inhomogeneity effects and, also, the nonlocal nature of an electronic system. The approach adopted here stresses the goodness of the expression taken by Colle and Salvetti for building a correlation factor and, at the same time, allows us to gain light on the nature of the deficiencies of those functionals obtained, up to now, from the perspective of the Hohenberg and Kohn theorem.     

Method for evaluation of density functional integrals in molecular calculations

Numerical methods for computing variationally optimized molecular orbitals within the Hartree–Fock approximation are augmented to include correlation functionals of the density in the energy and the numerical methods for carrying this out are described. The approach is applied explicitly to the Colle–Salvetti correlation energy functional. It is found that the gradient terms in the Colle–Salvetti functional present numerical problems associated with the low-density behavior, but also that they make a relatively small contribution to the correlation energy. In the three cases considered, HF, H₂O and N₂, it is found that the Colle–Salvetti correction considerably underestimates the correlation energies obtained in coupled-cluster theory.     

N-representability of diagonal elements of second-order reduced density matrices

The problem of pure-state N-representability of the two-particle spin-dependent density function ρ(x₁, x₂) is considered for an N-electron system, and a procedure for finding an N-representable ρ(x₁, x₂) is advanced. The problem is formulated in the framework of a family of N × N matrices formed from integrals of auxiliary two-particle functions θ_n(x₁, x₂) converging at n → ∞ to ρ(x₁, x₂)/[N(N−1)]. The simple requirement of positive definiteness of these matrices is shown to play a decisive role in finding an N-representable ρ(x₁, x₂). The results obtained may open new possibilities for using ρ(x₁, x₂) in the density-functional theory. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65 : 127–142, 1997     

Thomas–Fermi term as the simplest correction to the Weizsacker term

The nature of the correlation function G(1,2) appearing in the definition of the reduced first order density operator γ′(1,2) = ρ(1)^1/2ρ(2)^1/2G(1,2) is analyzed. It is shown that when G(1,2) is expanded in terms of plane waves in the context of a single-determinant approximation to the wave function, the correction to the Weizsacker term in the kinetic energy density expression is the Thomas–Fermi term.     

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     

Correlation energy of many-electron systems: a modified Colle-Salvetti approach

The Colle and Salvetti approach [Theo. Chim. Acta 37, 329 (1975)] to the calculation of the correlation energy of a system is modified in order to explicitly include into the theory the kinetic contribution to the correlation energy. This is achieved by deducing from a many electrons wave function, including the correlation effects via a Jastrow factor, an approximate expression of the one-electron reduced density matrix. Applying the latter to the homogeneous electron gas, an analytic expression of the correlation kinetic energy is derived. The total correlation energy of such a system is then deduced from its kinetic contribution inverting a standard procedure. At variance of the original Colle-Salvetti theory, the parameters entering in both the kinetic correlation and the total correlation energies are determined analytically, leading to a satisfactory agreement with the results of Perdew and Wang [Phys. Rev. B 45, 13244 (1992)]. The resulting (parameter-free) expressions give rise to a modified-local-density approximation that can be used in self-consistent density-functional calculations. We have performed such calculations for a large set of atoms and ions and we have found results for the correlation energies and for the ionization potentials which improve those of the standard local-density approximation.     

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.