Order Density Matrix Research Articles

Efficient localized Hartree–Fock methods as effective exact-exchange Kohn–Sham methods for molecules

The form of the Kohn–Sham (KS) exchange potential, which arises from the approximation that the Hartree–Fock (HF) and the exchange-only KS determinant are equal, is derived. Two related procedures to determine the KS exchange potential follow from this approximation: a self-consistent localized HF procedure and a transformation localized HF procedure yielding the local KS exchange potential from HF orbitals. Both procedures can be considered as almost exact exchange KS methods which require only occupied orbitals and are invariant with respect to unitary transformations of the orbitals, i.e., depend only on the first order density matrix. The resulting local KS exchange potentials are free of Coulomb self-interactions and exhibit the correct long-range 1/r-behavior. The Krieger, Li, and Iafrate (KLI) procedure to determine the KS exchange potential can be considered as an approximation to the introduced localized HF procedures. Highly efficient methods to carry out the presented localized HF as well as KLI procedures are introduced. An efficient basis set approach to calculate the Slater potential is presented. The methods can easily be implemented in present standard quantum chemistry codes. Applications to small and medium size molecules and clusters are presented. The Hartree–Fock and the exchange-only KS determinant are found to be surprisingly close. Qualitatively correct, Coulomb self-interaction free KS orbitals and eigenvalue spectra are obtained.

Near-Diagonal Behaviour of First-order Density Matrix for N Closed Shells in a Bare Coulomb Field

When the first order density matrix is expanded to lowest order, it is shown that the lowest order term is characterized by a functionf(r) whose derivative can be expressed in terms of the diagonal electron density and the nuclear charge, for the case of N closed shells in a bare Coulomb field.

Dilute Bose gas revised.

The well-known results concerning a dilute Bose gas with the short-range repulsive interaction should be reconsidered due to a thermodynamic inconsistency of the method being basic to much of the present understanding of this subject. The aim of our paper is to propose another way of treating the dilute Bose gas with an arbitrary strong interaction. Using the reduced density matrix of the second order and a variational procedure, this way allows us to escape the inconsistency mentioned and operate with singular potentials of the Lennard-Jones type. The derived expansion of the condensate depletion in powers of the boson density n=N/V reproduces the familiar result, while the expansion for the mean energy per particle is of the form epsilon=2 pi planck(2)an/m[1+128/(15 square root of [pi]) square root of [na(3)](1-5b/8a)+...], where a is the scattering length and b> or =0 stands for one more characteristic length depending on the shape of the interaction potential (in particular, for the hard spheres a=b). All the consideration concerns the zero temperature.

Towards strong-coupling generalization of the Bogoliubov model

The well-known results concerning a dilute Bose gas with the short-range repulsive interaction should be reconsidered due to a thermodynamic inconsistency of the method being basic to much of the present understanding of this subject and nonrelevant behaviour of the pair distribution function at small boson separations. The aim of our paper is to propose a new way of treating the dilute Bose gas with an arbitrary strong interaction. Using the reduced density matrix of the second order and a variational procedure, this way allows us to escape the inconsistency mentioned and operate with singular potentials like the Lennard-Jones one. All the consideration concerns the zero temperature.

Non-local energy density functional for atoms and metal clusters

Starting from the Hartree-Fock (HF) exchange energy for a spin saturated system and expanding the first order density matrix we have built a non local energy density functional and its corresponding variational one body potential without problems of divergence at large distances. Fitting the “dummy” parameter k which appears due to the truncation of the expansion, our non-local Energy Density Functional (EDF) can be applied to atoms and metal clusters. In the atomic case the present approach improves significantly the total binding energies as well as the eigenenergies of the single particle levels. The first ionization potential of the alkaline metals approaches the experimental values better than the Local Density Approximation (LDA) ones. The application of this non-local approach to Na clusters gives also a significant improvement of the first ionization potentials compared with the LDA prediction.

The reduced multiplication scheme of the Rys-Gauss quadrature for 1st order integral derivatives

An implementation of the reduced multiplication scheme of the Rys-Gauss quadrature to compute the gradients of electron repulsion integrals is discussed. The study demonstrates that the Rys-Gauss quadrature is very suitable for efficient utilization of simplifications as offered by the direct computation of symmetry adapted gradients and the use of the translational invariance of the integrals. The introduction of the so-called intermediate products is also demonstrated to further reduce the floating point operation count. Two prescreening techniques based on the 2nd order density matrix in the basis of the uncontracted Gaussian functions is proposed and investigated in the paper. This investigation gives on hand that it is not necessary to employ the Cauchy-Schwarz inequality to achieve efficient prescreening. All the features mentioned above were demonstrated by their implementation into the gradient programalaska. The paper offers a theoretical and practical assessment of the modified Rys-Gauss quadrature in comparison with other methods and implementations and a detailed analysis of the behavior of the method as suggested above as a function of changes with respect to symmetry, basis set quality, molecular size, and prescreening threshold.

Uniform density theorem for the Hubbard model

A general class of hopping models on a finite bipartite lattice is considered, including the Hubbard model and the Falicov–Kimball model. For the half-filled band, the single-particle density matrix ρ(x,y) in the ground state and in the canonical and grand canonical ensembles is shown to be constant on the diagonal x=y, and to vanish if x≠y and if x and y are on the same sublattice. For free electron hopping models, it is shown in addition that there are no correlations between sites of the same sublattice in any higher order density matrix. Physical implications are discussed.

The LCAO representation of the first order density matrix in non-orthogonal basis sets: a note

It is argued that the matrix PS is to be considered as the proper linear combination of atomic orbitals (LCAO) representation of the first-order density matrix, P and S being the usual molecular “density matrix” and the overlap matrix, respectively. This conclusion is in line with the fact that Hermitian operators are represented by non-Hermitian matrices if an overlapping basis is used. (The matrix representation of an operator is to be distinguished from the matrix of its integrals.) The diagonal elements of the matrix PS are Mulliken's gross orbital populations and the elements of this matrix are also necessary to define the bond order between a pair of atoms and the actual total and free valences of an atom in a molecule.

Quantum beats between the light and heavy hole excitons in a quantum well

We have detected quantum beats in the coherent polarization of simultaneously excited heavy and light hole excitons in a GaAs/Al 0.3Ga 0.7As quantum well. The beat frequency and the light and heavy hole energy splitting agree well. The 3rd order density matrix theory of Yajima and Taira has been extended to the case of quantum beats and successfully applied to model our results.

Analysis of the omega function for a molecular system. Application to lithium hydride

The omega function for the lithium hydride molecule ground state is analyzed at the equilibrium and at an dissociative distance. For this purpose, the first order density matrix, the natural orbitals, the natural configurational equivalents, and the correlation factors are deduced and discussed. It is found that the omega function taken into account appreciable correlation effects especially in the bond region.

Coherence in Disordered Condensed Matter. I: Dissipative Structures in Liquids and Dynamics of Far‐Infrared Absorption

AbstractThe Complex Scaling Method (CSM), in connection with Prigogine's Subdynamics predicts the spontaneous emergence of large dissipative (or: coherent) structures in amorphous matter, like liquids. The connection between these structures and the appearance of a large Jordan block (see appendix) with Segré characteristic equal to the dimension or degrees of freedom of the emerged structure, in the CSM‐transformed reduced second order density matrix, is presented and analysed in some detail. The interdependence of the present theoretical considerations with Yang's concept of off‐diagonal long‐range order (ODLRO) is stressed. – The appearance of dissipative structures is accompanied by a quantization condition for the lifetimes of the states contributing to the creation of those structures. The minimum size nmin of a dissipative structure obeys the equation nmin ≅ τrot · (2πkBT/h), where τrot represents the molecular relaxation time that is to be determined by a suitable spectroscopic experiment. – Application to far‐infrared (FIR) absorption spectroscopy is made explicitly. The above formula allows for a surprising interpretation of the well‐known anomalous temperature dependence of the FIR bands from first principles. The “size” of the dissipative structures that contribute to the FIR absorption in liquids is estimated.

The variation of the energy functional with respect to the density matrix of first order

The Hohenberg-Kohn theorem may be extended to prove the existence of an universal energy functional based upon the reduced density matrix of first order. We use a set of parameters representing these density matrices uniquely: the eigenvalues (occupation numbers), the projection operators on the corresponding eigenspaces and their dimensions. Without solving the corresponding variational equations explicitly, it is possible to show that, in the stationary case, the eigenfunctions of an "effective Hamiltonian" may serve as natural spin orbitals. Thus the connection to the familiar orbital representation is established. Because interacting many particle systems may lead to partial occupation numbers, the number of involved natural orbitals may exceed the number of particles in this scheme. The unique parametrization of the density matrix, the consideration of all necessary constraints and the possibility for partial occupation numbers consequently lead to more general and complicated variational equations. From these we derive an "average energy theorem" generalizing results of other authors.

Use of a nonlocal density approximation for transformation from electron density to electron momentum density

A new method to extract atomic electron momentum density from the corresponding electron density has been proposed. The method is based upon the nonlocal density approximation (NLDA) due to Gunnarsson, Jonson, and Lundqvist; and to Alonso and Girifalco. The reduced first order density matrix Γ(r/r′) is estimated from the atomic electron density ρ (r) by use of an averaged density distribution, ρ̃ (r). The Γ(r′/r′+r) thereby obtained is integrated out to obtain the autocorrelation function B(r) enabling evaluation of electron momentum density γ(p) and 〈pn〉 values. The method has been developed for spherically symmetric densities and has been tested out for beryllium, nitrogen, neon, and argon atoms. The 〈pn〉 values and momentum densities thus obtained are in a very good agreement with the corresponding Hartree–Fock counterparts.

Compton profiles of atoms from electron densities via reciprocal form factors

A new method to extract the Compton profile from the electron density has been proposed. The method is based on the nonlocal density approximation (nlda) due to Gunnarssonet al and Alonso and Girifalco. Initially the reduced first order density matrix has been estimated from the knowledge of the electron density alone through the use of an averaged density distribution, $$\tilde \rho (r)$$ . The reduced first order density matrix thus obtained has then been employed to extract the reciprocal form factor,B(r). The Compton profile,J(q) may thereby be obtained by a cosine transformation ofB(r). These results for theJ(q) are in good agreement with their near Hartree-Fock counterparts.

Bond orders and valences in the SCF theory: a comment

The calculation of bond order and valence indices for ab initio and semi-empirical wave functions is discussed by emphasizing their relations to the exchange part of the second order density matrix. Comments are also given on a recent work of Gopinathan and Jug, as well as propositions made to avoid ambiguity in the nomenclature.

Characterization of bond and current correlation structures in a molecular many‐electron wave function

AbstractA method has been developed to analyzed the bond and current correlation structures of a molecular many‐electron wave function. It is shown that the second order density matrix contains information about the bond and current correlations in its off‐diagonal components with respect to the indices of orbital basis functions. We break down the off‐diagonal correlation functions into five kinds: charge, spin scalar, spin quadrupole, charge spin, and spin polar correlation functions. For a real wave function, the four correlation functions, except for the spin polar one, have only symmetric–symmetric and antisymmetric–antisymmetric components. The former components give site–bond and bond–bond correlations of charges and spins, while the latter components give current–current correlations of charges and spins. The spin polar correlation function has only symmetric–antisymmetric components that give site–current and bond–current correlations of spins. The five off‐diagonal correlation functions are expressed in terms of the off‐diagonal components of the second order density matrix. The linked off‐diagonal correlation functions are defined in that they give dynamical bond and current correlations. The method is applied to the analyses of the bond and current correlations in the low lying exact eigenstates of the PPP Hamiltonian of benzene.

Structure and spin‐purity conditions for reduced density matrices of arbitrary order

AbstractFor arbitrary k, the separation of spin variables is performed in the reduced density matrix of the kth order (RDM‐k) on the basis of the Fock coordinate function method. The independent spatial components of RDM‐k are analyzed. For RDM‐k of the total spin eigenstate, their number is proved never to exceed its spin multiplicity 2s + 1. Integral and other nontrivial interrelations between spatial components are established which turn out to be the necessary and sufficient conditions of spin purity of a wavefunction corresponding to a given RDM‐k. It is shown that the r‐rank k‐particle spin distribution matrix F, defined as a spatial coefficient at the spin‐tensorial operator of rank r in the RDM‐k expansion, can be obtained by reduction of the (k + r)‐particle charge density matrix F. In particular, all spatial components of RDM‐2 are explicitly expressed in terms of the four‐electron charge density matrix only. This allows us to purpose some approximative formulas for the McWeeny‐Mizuno spin–orbit and spin–spin coupling functions in the case of the weak spin contamination.

Spin dependent properties of perturbed wave functions: An analytic comparison of the exact, UHF, and spin-projected UHF states

A generally applicable diagrammatic representation for spin-annihilated wave functions is developed; the diagrammatic approach avoids the complex algebra usually associated with the application of spin-annihilation operators. The diagrammatic formulation is first applied to the perturbation expansion for the exact wave function to elucidate the diagrammatic origin of the (correct) spin eigenfunction properties of the exact perturbed state. Employing a previously derived perturbation expansion for the unrestricted Hartree–Fock (UHF) wave function, the diagrammatic spin-annihilation formalism is then used to analyze the effect of projecting unwanted spin states from the UHF wave function. Results obtained for the projected UHF state are compared to those appropriate to both the exact perturbed wave function and the unprojected UHF wave function. It is shown that results obtained by annihilating only the lowest unwanted spin multiplicity are expected, in a perturbation theory sense, to be very similar to those obtained by annihilating all unwanted spin states. However, it is further shown that additional terms are introduced into the UHF wave function by the spin projection procedure and that these terms are in general unrelated to the exact perturbative corrections to the UHF state. In particular, the wave function, energy, one and two-electron density matrices, and spin densities obtained for the projected state are all shown to be in error in the lowest order of perturbation theory; in contrast, the UHF state leads to the correct one-electron density matrix and spin densities in lowest order.

Hypevirial theorem for open dynamical assemblies in the density matrix representation

AbstractThe concepts of “open assembly” is defined, and the hypervirial theorem is derived in three steps: (i) An equation of motion of the reduced density matrix of the first order is derived. (ii) This equation is utilized to derive a general equation of continuity for single‐particle properties. (iii) The hypervirial equation for open assemblies is obtained as a long‐time average of the continuity equation. The hypervirial theorem is shown to be equivalent to a simple postulate: no physical property can keep on accumulating for ever in a limited volume. Finally, the hypervirial theorem is applied to the problems of force, torque, and virial balance in open dynamical assemblies, and the results are compared with the corresponding classical results.

A Comparison of the Super-CI and the Newton-Raphson Scheme in the Complete Active Space SCF Method

A density matrix formulation is presented of the super-C I and Newton-Raphson methods in complete active space SCF (CASSCF) calculations. The CASSCF method is a special form of the MC-SCF method, where the C I wave function is assumed to be complete in a subset of the orbital space (the active space), leaving the remaining orbitals doubly occupied in all configurations.Explicit formulas are given for all matrix elements in the super-C I method and the first and second derivatives in the Newton-Raphson formulation. The similarities between the two methods are pointed out and the differences in the detailed formulations are discussed. Especially interesting is the fact, that while the second derivatives can be expressed in terms of first and second order density matrices, the matrix elements between the super-C I states involve also the third order density matrix in some cases.