Katarzyna Pernal

An improved density matrix functional by physically motivated repulsive corrections

An improved density matrix functional [correction to Buijse and Baerends functional (BBC)] is proposed, in which a hierarchy of physically motivated repulsive corrections is employed to the strongly overbinding functional of Buijse and Baerends (BB). The first correction C1 restores the repulsive exchange-correlation (xc) interaction between electrons in weakly occupied natural orbitals (NOs) as it appears in the exact electron pair density rho(2) for the limiting two-electron case. The second correction C2 reduces the xc interaction of the BB functional between electrons in strongly occupied NOs to an exchange-type interaction. The third correction C3 employs a similar reduction for the interaction of the antibonding orbital of a dissociating molecular bond. In addition, C3 applies a selective cancellation of diagonal terms in the Coulomb and xc energies (not for the frontier orbitals). With these corrections, BBC still retains a correct description of strong nondynamical correlation for the dissociating electron pair bond. BBC greatly improves the quality of the BB potential energy curves for the prototype few-electron molecules and in several cases BBC reproduces very well the benchmark ab initio potential curves. The average error of the self-consistent correlation energies obtained with BBC3 for prototype atomic systems and molecular systems at the equilibrium geometry is only ca. 6%.

The ground state of harmonium

A detailed analysis that benefits from a slate of new approximate numerical and exact asymptotic results produces highly accurate properties of the ground state of the harmonium atom as functions of the confinement strength ω and quantifies the domains of the weakly and strongly correlated regimes in this system. The former regime, which encompasses the values of ω greater than ωcrit≈4.011 624×10−2, is characterized by the one-electron density ρ(ω;r1) with a global maximum at r1=0. In contrast, the harmonium atom within the latter regime, which corresponds to ω<ωcrit, differs fundamentally from both its weakly correlated counterpart and Coulombic systems. Resembling a Wigner crystal of a homogeneous electron gas, it possesses a radially localized pair of angularly correlated electrons that gives rise to ρ(ω;r1) with a “fat attractor” composed of a cage critical point and a (1, −1) critical sphere. Allowing for a continuous variation in ω, the new compact representation of the ground-state wave function an...

Constraints upon natural spin orbital functionals imposed by properties of a homogeneous electron gas

The expression Vee[Γ1]=(1/2)∑p≠q[npnqJpq−Ω(np,nq)Kpq], where {np} are the occupation numbers of natural spin orbitals, and {Jpq} and {Kpq} are the corresponding Coulomb and exchange integrals, respectively, generalizes both the Hartree–Fock approximation for the electron–electron repulsion energy Vee and the recently introduced Goedecker–Umrigar (GU) functional. Stringent constraints upon the form of the scaling function Ω(x,y) are imposed by the properties of a homogeneous electron gas. The stability and N-representability of the 1-matrix demand that 2/3<β<4/3 for any homogeneous Ω(x,y) of degree β [i.e., Ω(λx,λy)≡λβΩ(x,y)]. In addition, the Lieb–Oxford bound for Vee asserts that β⩾βcrit, where βcrit≈1.1130, for Ω(x,y)≡(xy)β/2. The GU functional, which corresponds to β=1, does not give rise to admissible solutions of the Euler equation describing a spin-unpolarized homogeneous electron gas of any density. Inequalities valid for more general forms of Ω(x,y) are also derived.

Excitation energies with time-dependent density matrix functional theory: Singlet two-electron systems

Time-dependent density functional theory in its current adiabatic implementations exhibits three striking failures: (a) Totally wrong behavior of the excited state surface along a bond-breaking coordinate, (b) lack of doubly excited configurations, affecting again excited state surfaces, and (c) much too low charge transfer excitation energies. We address these problems with time-dependent density matrix functional theory (TDDMFT). For two-electron systems the exact exchange-correlation functional is known in DMFT, hence exact response equations can be formulated. This affords a study of the performance of TDDMFT in the TDDFT failure cases mentioned (which are all strikingly exhibited by prototype two-electron systems such as dissociating H(2) and HeH(+)). At the same time, adiabatic approximations, which will eventually be necessary, can be tested without being obscured by approximations in the functional. We find the following: (a) In the fully nonadiabatic (omega-dependent, exact) formulation of linear response TDDMFT, it can be shown that linear response (LR)-TDDMFT is able to provide exact excitation energies, in particular, the first order (linear response) formulation does not prohibit the correct representation of doubly excited states; (b) within previously formulated simple adiabatic approximations the bonding-to-antibonding excited state surface as well as charge transfer excitations are described without problems, but not the double excitations; (c) an adiabatic approximation is formulated in which also the double excitations are fully accounted for.

Systematic construction of approximate one-matrix functionals for the electron-electron repulsion energy

The Legendre transform of an (approximate) expression for the ground-state energy E0(η,g) of an N-electron system yields the one-matrix functional Vee[Γ(x′,x)] for the electron-electron repulsion energy that is given by the function Vee(n;g) of the occupation numbers n pertaining to Γ(x′,x) and the two-electron repulsion integrals g computed in the basis of the corresponding natural spinorbitals. Extremization of the electronic energy functional, which is a sum of Vee[Γ(x′,x)] and the contraction of Γ(x′,x) with the core Hamiltonian, produces the (approximate) ground-state energy even if E0(η,g) itself is not variational. Thanks to this property, any electron correlation formalism can be reformulated in the language of the density matrix functional theory. Ten conditions that have to be satisfied by Vee(n;g) uncover several characteristics of Vee[Γ(x′,x)]. In particular, when applied in conjunction with the homogeneity property, the condition of volume extensivity imposes stringent constraints upon the po...

Density matrix functional theory of weak intermolecular interactions

The known asymptotic behavior of the total energy of two weakly interacting systems imposes stringent conditions on the exchange-correlation energy as a functional of the one-electron reduced density matrix. Although the first-order conditions that involve Coulomb-type two-electron integrals are relatively trivial to satisfy, the exact functional should also conform to two second-order expressions, and consequently to certain sum rules. The primitive natural spin-orbital functionals satisfy the first-order conditions but, lacking terms quadratic in two-electron integrals, are found to be incapable of recovering the dispersion component of the interaction energy. Violating the sum rules, the recently proposed Yasuda functional yields nonvanishing dispersion energy with spurious asymptotic terms that scale like inverse fourth and fifth powers of the intersystem distance.

Response properties and stability conditions in density matrix functional theory

Expressions for the second-order energy variations in the density matrix functional theory (DMFT) are derived, resulting in a formalism for time-independent response properties (including absolute electronegativity and hardness) and stability conditions. A quadratically convergent scheme for a direct determination of natural spinorbitals and their occupancy numbers is developed and tested with the Goedecker–Umrigar and the exact two-electron functionals. The derivatives of the electronic energy with respect to the number of electrons are found to be very sensitive to the DMFT description of the exchange-correlation energy, providing a sensitive measure of accuracy that can be readily employed in testing and development of approximate functionals.

Excitation energies from extended random phase approximation employed with approximate one- and two-electron reduced density matrices

Starting from Rowes equation of motion we derive extended random phase approximation (ERPA) equations for excitation energies. The ERPA matrix elements are expressed in terms of the correlated ground state one- and two-electron reduced density matrices, 1- and 2-RDM, respectively. Three ways of obtaining approximate 2-RDM are considered: linearization of the ERPA equations, obtaining 2-RDM from density matrix functionals, and employing 2-RDM corresponding to an antisymmetrized product of strongly orthogonal geminals (APSG) ansatz. Applying the ERPA equations with the exact 2-RDM to a hydrogen molecule reveals that the resulting (1)Σ(g)(+) excitation energies are not exact. A correction to the ERPA excitation operator involving some double excitations is proposed leading to the ERPA2 approach, which employs the APSG one- and two-electron reduced density matrices. For two-electron systems ERPA2 satisfies a consistency condition and yields exact singlet excitations. It is shown that 2-RDM corresponding to the APSG theory employed in the ERPA2 equations yields excellent singlet excitation energies for Be and LiH systems, and for the N(2) molecule the quality of the potential energy curves is at the coupled cluster singles and doubles level. ERPA2 nearly satisfies the consistency condition for small molecules that partially explains its good performance.

Approximate one-matrix functionals for the electron–electron repulsion energy from geminal theories

A simple extension of the antisymmetrized product of strongly orthogonal geminals theory produces a “JK-only” one-matrix functional for the electron–electron repulsion energy of a closed-shell system that is exact for two-electron singlet ground states, size-extensive, and incorporates some intergeminal correlation and thus dispersion effects. The functional is defined only for one-matrices with occupation numbers that can be arranged into sets with elements that sum up to two. Its possible generalizations are discussed.

Intergeminal Correction to the Antisymmetrized Product of Strongly Orthogonal Geminals Derived from the Extended Random Phase Approximation.

We present a correction to the antisymmetrized product of strongly orthogonalized geminals (APSG) approach accounting for intergeminal correlation energy that APSG lacks. The correction is based on the fluctuation dissipation theorem formulated for geminals with transition density matrices obtained from the recently formulated extended random phase approximation. We show that the proposed intergeminal correlation correction greatly improves upon APSG energies by accounting for short- and long-range dynamical correlation. For covalently bonded molecules the potential energy curves are in good agreement with the exact results in the entire range of bond breaking. Also the description of weakly interacting systems is superior to that of APSG. In particular, we show that the proposed intergeminal correlation energy reduces to the correct form of the dispersion energy and asymptotically yields exact interaction energy of the helium dimer.