Reduced-density-matrix theory and algebraic structures

A survey of our recent work on algebraic structures and reduced-density-matrix theory is presented. Our approach leads to a method of classifying reduced density matrices and generalizes the notion of open and closed shells in many-body theory.     

Isomorphism in electron-pair densities of atoms

The spherically averaged electron-pair intracule (relative motion) h(u) and extracule (center-of-mass motion) d(R) densities are a couple of densities which characterize the motion of electron pairs in atomic systems. We study a generalized electron-pair density (q; a, b) that represents the probability density function for the magnitude of two-electron vector a r _j +b r _k of any pair of electrons j and k to be q, where a and b are nonzero real numbers. In particular, h(u)=g(u;1, −1) and d(R) = . It is shown that the scaling property of the Dirac delta function and the inversion symmetry of orbitals in atoms due to the central force field generate several isomorphic relations in the electron-pair density (q; a, b) with respect to the two parameters a and b. The approximate isomorphism d(R)≅8h(2R) known in the literature between the intracule and extracule densities is a special case of the present results. Received: 24 May 2000 / Accepted: 18 July 2000 / Published online: 27 September 2000     

Quantum transition state theory: featuring ensemble pre-equilibrium approximation

In this article, we use an ensemble pre-equilibrium approximation to vindicate the use of Heisenberg Equation of Motion and some other approximation for the rate of the change of quasi-equilibrium state in order to derive, based on the most precise and accurate existing theory of reaction rates to date, a rate law for any given elementary step of a chemical reaction that applies even to the condensed phase. The necessity of such a theory arises because first of all, quantum mechanics must be taken into account, and secondly, other molecules and especially the solvent may play an important role in the transition state of any elementary reaction, and we have devised a theory for doing so. Also, the correct theory for reaction rates critically depends on a complete set of wavefunctions of the transition state representing initially the products and finally the reactants, and we show a simple way for deriving it. We also show that several prevailing formulas in the literature for the cumulative reaction rate within the range of applicability of non-relativistic mechanics to the electrons and nuclei of the reacting systems within the reactor are exactly equivalent to each other, even in the condensed phase, and therefore the present theory is exact within the laws of quantum non-relativistic mechanics.     

Nanofiltration theory: good co-ion exclusion approximation for single salts

We applied an approximate analytic method, the good co-ion exclusion (GCE) approximation, to the hindered electrotransport theory describing salt and solution transport across charged nanofiltration membranes. This approximation, which should be valid at sufficiently low feed electrolyte concentration, leads to a considerable simplification of the exact parametrized equations obtained previously for single salt nanofiltration parameters (salt rejection, electric filtration potential, and volume flux density) and therefore provides further insight into ion transfer in nanoporous membranes. We also established the domain of validity of the GCE approximation as a function of the salt type for 1:1, 2:1, 1:2, and 2:2 salts. Our results for the volume flux density, obtained within an extended GCE approximation, confirm that the global osmotic reflection coefficient in the solution flux equation is not equal to the limiting salt rejection.     

Orbitally invariant internally contracted multireference unitary coupled cluster theory and its perturbative approximation: theory and test calculations of second order approximation

A unitary wave operator, exp (G), G(+) = -G, is considered to transform a multiconfigurational reference wave function Φ to the potentially exact, within basis set limit, wave function Ψ = exp (G)Φ. To obtain a useful approximation, the Hausdorff expansion of the similarity transformed effective Hamiltonian, exp (-G)Hexp (G), is truncated at second order and the excitation manifold is limited; an additional separate perturbation approximation can also be made. In the perturbation approximation, which we refer to as multireference unitary second-order perturbation theory (MRUPT2), the Hamiltonian operator in the highest order commutator is approximated by a Mo?ller-Plesset-type one-body zero-order Hamiltonian. If a complete active space self-consistent field wave function is used as reference, then the energy is invariant under orbital rotations within the inactive, active, and virtual orbital subspaces for both the second-order unitary coupled cluster method and its perturbative approximation. Furthermore, the redundancies of the excitation operators are addressed in a novel way, which is potentially more efficient compared to the usual full diagonalization of the metric of the excited configurations. Despite the loss of rigorous size-extensivity possibly due to the use of a variational approach rather than a projective one in the solution of the amplitudes, test calculations show that the size-extensivity errors are very small. Compared to other internally contracted multireference perturbation theories, MRUPT2 only needs reduced density matrices up to three-body even with a non-complete active space reference wave function when two-body excitations within the active orbital subspace are involved in the wave operator, exp (G). Both the coupled cluster and perturbation theory variants are amenable to large, incomplete model spaces. Applications to some widely studied model systems that can be problematic because of geometry dependent quasidegeneracy, H4, P4, and BeH(2), are performed in order to test the new methods on problems where full configuration interaction results are available.     

The electron-pair density of atomic systems. Rigorous bounds and application to helium

Firstly, the monotonicity properties of the electron-pair densityI(u) of atomic systems are investigated. Leth(u) denote the spherically-averaged electron-pair density of an arbitraryN-electron system, which essentially coincides withI(u) in the case of atoms. It is found that the interelectronic functiong _α(u)=h(u)/u ^α, α≧0, is not only monotonically decreasing from the origin for α≧α₁=max{uh′(u)/h(u)} but it also has the property of convexity for α≧α₂, where the value of α₂ is given in the text. Secondly, the Stieltjes technique is used to obtain rigorous, simple and compact inequalities which involve three interelectronic radial expectation values 〈u ^k 〉. These inequalities are universal in the sense that they are valid for both ground and excited states in the whole periodic table. Thirdly, for those systems with a unimodalh(u), i.e. having a single maximum atu=u _max , are found (i) upper bounds tou _max in terms of any number of moments 〈u ^k 〉 via the above-mentioned technique, and (ii) lower bounds to the maximal valueh _max ≡h(u _max ) by means of two arbitrary moments 〈u ^k 〉 in a variational way. A particular case of the latter bound leads to a rigorous upper bound to the total electro-electron repulsion energyE _ee of the system, namely \(E_{ee} \leqq \left[ {\frac{{9\pi }}{8}N^2 (N - 1)^2 h_{max} } \right]\frac{1}{3}\) . Finally, the electron-pair density of Helium is analysed in detail and the quality of the above mentioned inequalities is studied by means of theM-term Hylleraas-type wavefunctions, withM=1,2,3,6,10 and 20. We observe that in the 20-term case, which is shown to be very close to the exact one, α₁ and α₂ take the values 0.0414 and 0.2067, respectively. Moreover, in such a case we found that some of the above mentioned inequalities are very accurate.     

Graph theory of molecular orbitals beyond tight-binding approximation

The graph theory of molecular orbitals including second neighbor interactions (η) is considered here. Graphical methods of getting the characteristic polynomial for the π system of a conjugated molecule are given. It is shown that the characteristic polynomial can be factorized if there are symmetries in the molecule.     

A theory of polymer solutions without the mean-field approximation in Flory-Huggins theory

The concept of the volume fraction at chain end is proposed, which is the conditional probability for a site having been occupied, knowing that an adjacent site is occupied by polymer end. The overall entropy of polymer/solvent system is separated into two fundamentally different parts, i.e., the translational entropy and the conformational entropy. Based on these a theory of polymer solutions is established. When a mean-field approximation is introduced, Flory-Huggins (FH) theory is recovered. The FH interaction parameters and spinodal curves of the polystyrene/cyclohexane system are calculated and compared with the experimental data. The good prediction of them two is achieved.     

The "JK-only" approximation in density matrix functional and wave function theory

Various energy functionals applying the "JK-only" approximation which leads to two-index two-electron integrals instead of four-index two-electron integrals in the electron-electron interaction term of the electronic energy are presented. Numerical results of multiconfiguration self-consistent field calculations for the best possible "JK-only" wave function are compared to those obtained from the pair excitation multiconfiguration self-consistent (PEMCSCF) method and two versions of density matrix functional theory. One of these is derived making explicit use of some necessary conditions for N representability of the second-order density matrix. It is shown that this method models the energy functional based on the best possible "JK-only" wave function with good accuracy. The calculations also indicate that only a minor fraction of the total correlation energy is incorporated by "JK-only" approaches for larger molecules.     

Charge-shift bonding--a class of electron-pair bonds that emerges from valence bond theory and is supported by the electron localization function approach

This paper deals with a central paradigm of chemistry, the electron-pair bond. Valence bond (VB) theory and electron-localization function (ELF) calculations of 21 single bonds demonstrate that along the two classical bond families of covalent and ionic bonds, there exists a class of charge-shift bonds (CS bonds) in which the fluctuation of the electron pair density plays a dominant role. In VB theory, CS bonding manifests by way of a large covalent-ionic resonance energy, RE(CS), and in ELF by a depleted basin population with large variances (fluctuations). CS bonding is shown to be a fundamental mechanism that is necessary to satisfy the equilibrium condition, namely the virial ratio of the kinetic and potential energy contributions to the bond energy. The paper defines the atomic propensity and territory for CS bonding: Atoms (fragments) that are prone to CS bonding are compact electronegative and/or lone-pair-rich species. As such, the territory of CS bonding transcends considerations of static charge distribution, and involves: a) homopolar bonds of heteroatoms with zero static ionicity, b) heteropolar sigma and pi bonds of the electronegative and/or electron-pair-rich elements among themselves and to other atoms (e.g., the higher metalloids, Si, Ge, Sn, etc), c) all hypercoordinate molecules. Several experimental manifestations of charge-shift bonding are discussed, such as depleted bonding density, the rarity of ionic chemistry of silicon in condensed phases, and the high barriers of halogen-transfer reactions as compared to hydrogen-transfers.     

Correlated electron-pair properties of the Li atom in momentum space

On the basis of multiconfiguration Hartree–Fock calculations, correlated electron-pair intracule (relative motion) and extracule (center-of-mass motion) properties are reported for the Li atom in momentum space. The present results are more accurate and consistent than those in the literature. Received: 10 September 2001 / Accepted: 11 December 2001 / Published online: 22 March 2002     

Bounds to electron-pair relative and center-of-mass radii in many-electron atoms

When the electron-electron interaction is explicitly considered in many-electron atoms, the average electron radius (r) splits into the inner (r<) and outer (r>) radii. It is shown that the sum and difference of these radii constitute upper and lower bounds, respectively, to the electron-pair relative distance r(12)=(/r1-r2/). An analogous result is also derived for the electron-pair center-of-mass radius (R)=(/r1+r2//2). For the 102 atoms He through Lr in their ground states, the tightness of these bounds is numerically examined at the Hartree-Fock limit level. Good linear correlations are observed between (r12) and (r>), and between (R) and (r>)/2.     

Adiabatic approximation of time-dependent density matrix functional response theory

Time-dependent density matrix functional theory can be formulated in terms of coupled-perturbed response equations, in which a coupling matrix K(omega) features, analogous to the well-known time-dependent density functional theory (TDDFT) case. An adiabatic approximation is needed to solve these equations, but the adiabatic approximation is much more critical since there is not a good "zero order" as in TDDFT, in which the virtual-occupied Kohn-Sham orbital energy differences serve this purpose. We discuss a simple approximation proposed earlier which uses only results from static calculations, called the static approximation (SA), and show that it is deficient, since it leads to zero response of the natural orbital occupation numbers. This leads to wrong behavior in the omega-->0 limit. An improved adiabatic approximation (AA) is formulated. The two-electron system affords a derivation of exact coupled-perturbed equations for the density matrix response, permitting analytical comparison of the adiabatic approximation with the exact equations. For the two-electron system also, the exact density matrix functional (2-matrix in terms of 1-matrix) is known, enabling testing of the static and adiabatic approximations unobscured by approximations in the functional. The two-electron HeH(+) molecule shows that at the equilibrium distance, SA consistently underestimates the frequency-dependent polarizability alpha(omega), the adiabatic TDDFT overestimates alpha(omega), while AA improves upon SA and, indeed, AA produces the correct alpha(0). For stretched HeH(+), adiabatic density matrix functional theory corrects the too low first excitation energy and overpolarization of adiabatic TDDFT methods and exhibits excellent agreement with high-quality CCSD ("exact") results over a large omega range.     

Conductance theory of concentrated electrolytes in an MSA-type approximation

A closed system of equations for the nonequilibrium pair-distribution function is given which corresponds to the mean spherical approximation (MSA) in equilibrium. The electrophoretic contribution to the conductance is calculated exactly in the framework of the MSA. For the calculation of the remaining contributions the equilibrium distribution function due to Henderson and Smith was used. The final expressions for the conductance are given in analytic terms using the restricted primitive model. The calculated curves are in good agreement with experimental data for 1-1 electrolytes up to about 1 M and qualitatively show the right behaviour up to rather high concentrations.     

The adiabatic approximation in time-dependent density matrix functional theory: response properties from dynamics of phase-including natural orbitals

The adiabatic approximation is problematic in time-dependent density matrix functional theory. With pure density matrix functionals (invariant under phase change of the natural orbitals) it leads to lack of response in the occupation numbers, hence wrong frequency dependent responses, in particular α(ω→0)≠α(0) (the static polarizability). We propose to relinquish the requirement that the functional must be a pure one-body reduced density matrix (1RDM) functional, and to introduce additional variables which can be interpreted as phases of the one-particle states of the independent particle reference system formed with the natural orbitals, thus obtaining so-called phase-including natural orbital (PINO) functionals. We also stress the importance of the correct choice of the complex conjugation in the two-electron integrals in the commonly used functionals (they should not be of exchange type). We demonstrate with the Lo?wdin-Shull energy expression for two-electron systems, which is an example of a PINO functional, that for two-electron systems exact responses (polarizabilities, excitation energies) are obtained, while writing this energy expression in the usual way as a 1RDM functional yields erroneous responses.     

Local quantum chemistry: The local space approximation for Møller–Plesset perturbation theory

The local space approximation is an accurate technique for describing a relatively small cluster embedded within an extended system. It has previously been developed for the Hartree-Fock, local density functional, configuration interaction, and coupled cluster electronic structure methods. Here it is extended to Møller-Plesset perturbation theory. © 1995 John Wiley & Sons, Inc.