Optimization of the temple lower bound

The Temple formula is perhaps the most common method used in the uncommon endeavor of calculating a lower bound to the ground-state energy of an atomic or molecular system. We generalize the Temple formula by introducing a parameter that can be varied to optimize the lower bound. This generalization does not require any information that is not already used for the traditional Temple lower bound. Examples with the helium cation and neutral atom show that improvement is greatest when the approximate wave function poorly approximates the true ground-state wave function. The examples also show that in some cases the traditional Temple lower bound may already be optimal so that our generalization gives no improvement.     

Molecular properties from variational reduced-density-matrix theory with three-particle N-representability conditions

Molecular ground-state energies and two-electron reduced density matrices (2-RDMs) have recently been computed without the many-electron wave function by constraining the 2-RDM to satisfy a complete set of three-positivity conditions for N representability [D. A. Mazziotti, Phys. Rev. A 74, 032501 (2006)]. Energies at both equilibrium and nonequilibrium geometries are obtained within 0.3% of the correlation energy. In this paper the authors extend this work to examine the accuracy of molecular properties, including multipole moments and components of the ground-state energy, relative to full configuration interaction (FCI). Comparisons are also made with 2-RDM methods with two-positivity conditions and two-positivity plus the generalized T1T2 conditions as well as several approximate wave function methods. Using the 2-RDM method with three-positivity conditions, the authors obtain dipole, quadrupole, and octupole moments for BeH2, BH, H2O, CO, and NH3 at equilibrium geometries that are within 0.04% of their FCI values. In addition, for the potential energy surface of N2, the 2-RDM method with three-positivity yields not only accurate total ground-state energies but also accurate expectation values of the kinetic energy operator, the electron-nuclei potential, and electron-electron repulsion.     

Computation of correlation functions and wave function projections in the context of quantum trajectory dynamics

The de Broglie-Bohm formulation of the Schrodinger equation implies conservation of the wave function probability density associated with each quantum trajectory in closed systems. This conservation property greatly simplifies numerical implementations of the quantum trajectory dynamics and increases its accuracy. The reconstruction of a wave function, however, becomes expensive or inaccurate as it requires fitting or interpolation procedures. In this paper we present a method of computing wave packet correlation functions and wave function projections, which typically contain all the desired information about dynamics, without the full knowledge of the wave function by making quadratic expansions of the wave function phase and amplitude near each trajectory similar to expansions used in semiclassical methods. Computation of the quantities of interest in this procedure is linear with respect to the number of trajectories. The introduced approximations are consistent with approximate quantum potential dynamics method. The projection technique is applied to model chemical systems and to the H+H(2) exchange reaction in three dimensions.     

Zero-variance zero-bias quantum Monte Carlo estimators of the spherically and system-averaged pair density

We construct improved quantum Monte Carlo estimators for the spherically and system-averaged electron pair density (i.e., the probability density of finding two electrons separated by a relative distance u), also known as the spherically averaged electron position intracule density I(u), using the general zero-variance zero-bias principle for observables, introduced by Assaraf and Caffarel. The calculation of I(u) is made vastly more efficient by replacing the average of the local delta-function operator by the average of a smooth nonlocal operator that has several orders of magnitude smaller variance. These new estimators also reduce the systematic error (or bias) of the intracule density due to the approximate trial wave function. Used in combination with the optimization of an increasing number of parameters in trial Jastrow-Slater wave functions, they allow one to obtain well converged correlated intracule densities for atoms and molecules. These ideas can be applied to calculating any pair-correlation function in classical or quantum Monte Carlo calculations.     

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.     

Theoretical investigation on RuO2 nanoclusters adsorbed on TiO2 rutile (110) and anatase (101) surfaces

Minimizing the energy of an $N$ -electron system as a functional of a two-electron reduced density matrix (2-RDM), constrained by necessary $N$ -representability conditions (conditions for the 2-RDM to represent an ensemble $N$ -electron quantum system), yields a rigorous lower bound to the ground-state energy in contrast to variational wave function methods. We characterize the performance of two sets of approximate constraints, (2,2)-positivity (DQG) and approximate (2,3)-positivity (DQGT) conditions, at capturing correlation in one-dimensional and quasi-one-dimensional (ladder) Hubbard models. We find that, while both the DQG and DQGT conditions capture both the weak and strong correlation limits, the more stringent DQGT conditions improve the ground-state energies, the natural occupation numbers, the pair correlation function, the effective hopping, and the connected (cumulant) part of the 2-RDM. We observe that the DQGT conditions are effective at capturing strong electron correlation effects in both one- and quasi-one-dimensional lattices for both half filling and less-than-half filling.     

Making More Extensive Use of the Coupled-cluster Wave Function: from the Standard Energy Expression to the Energy Expectation Value

The standard coupled-cluster (CC) scheme with single and double excitations in the cluster operator (CCSD) includes only up to quadruple excitations in the equations. The CCSD exponential expansion generates, however, all possible excitations out of the reference function through products of the cluster operators. Clearly, in all standard approximate CC approaches only a part of the CC wave function is used in the equations. If the standard CCSD wave function is inserted into the energy expectation value expression then the complete CCSD wave function contributes to the energy. Such an energy expectation value expression can be presented as a sum of the standard CCSD energy formula plus correction terms. The correction terms provide an information about the quality of the total CC function. Contributions associated with the presence of higher than double excitations in the bra CCSD wave function supplement the CCSD energy obtained within the standard scheme. These contributions can be generated in a sequential way by considering intermediate excitation levels for the bra CCSD wave function in the expectation value expression before reaching the highest excitation level. In this way the importance of particular components differing in the standard and expectation value CCSD energies can be investigated. Some of the contributions can be recognized as close to or identical with the so-called renormalized noniterative corrections to the CC methods. We try to see to what an extent the nonstandard energy expressions, like the energy expectation value or the asymmetric energy formula, can be used to extend the applicability of the CCSD method illustrating our considerations with some numerical examples.Dedicated to Professor Jean-Paul Malrieu to honor his contribution to quantum chemistry and physics     

Local effective potential theory: nonuniqueness of potential and wave function

In local effective potential energy theories such as the Hohenberg-Kohn-Sham density functional theory (HKS-DFT) and quantal density functional theory (Q-DFT), electronic systems in their ground or excited states are mapped to model systems of noninteracting fermions with equivalent density. From these models, the equivalent total energy and ionization potential are also obtained. This paper concerns (i) the nonuniqueness of the local effective potential energy function of the model system in the mapping from a nondegenerate ground state, (ii) the nonuniqueness of the local effective potential energy function in the mapping from a nondegenerate excited state, and (iii) in the mapping to a model system in an excited state, the nonuniqueness of the model system wave function. According to nondegenerate ground state HKS-DFT, there exists only one local effective potential energy function, obtained as the functional derivative of the unique ground state energy functional, that can generate the ground state density. Since the theorems of ground state HKS-DFT cannot be generalized to nondegenerate excited states, there could exist different local potential energy functions that generate the excited state density. The constrained-search version of HKS-DFT selects one of these functions as the functional derivative of a bidensity energy functional. In this paper, the authors show via Q-DFT that there exist an infinite number of local potential energy functions that can generate both the nondegenerate ground and excited state densities of an interacting system. This is accomplished by constructing model systems in configurations different from those of the interacting system. Further, they prove that the difference between the various potential energy functions lies solely in their correlation-kinetic contributions. The component of these functions due to the Pauli exclusion principle and Coulomb repulsion remains the same. The existence of the different potential energy functions as viewed from the perspective of Q-DFT reaffirms that there can be no equivalent to the ground state HKS-DFT theorems for excited states. Additionally, the lack of such theorems for excited states is attributable to correlation-kinetic effects. Finally, they show that in the mapping to a model system in an excited state, there is a nonuniqueness of the model system wave function. Different wave functions lead to the same density, each thereby satisfying the sole requirement of reproducing the interacting system density. Examples of the nonuniqueness of the potential energy functions for the mapping from both ground and excited states and the nonuniqueness of the wave function are provided for the exactly solvable Hooke's atom. The work of others is also discussed.     

Power series expansion of the random phase approximation correlation energy: The role of the third- and higher-order contributions

We derive a power expansion of the correlation energy of weakly bound systems within the random phase approximation (RPA), in terms of the Coulomb interaction operator, and we show that the asymptotic limit of the second- and third-order terms yields the van der Waals (vdW) dispersion energy terms derived by Zaremba-Kohn and Axilrod-Teller within perturbation theory. We then show that the use of the second-order expansion of the RPA correlation energy results in rather inaccurate binding energy curves for weakly bonded systems, and discuss the implications of our findings for the development of approximate vdW density functionals. We also assess the accuracy of different exchange energy functionals used in the derivation of vdW density functionals.     

Normalization and Fermi–Coulomb and Coulomb hole sum rules for approximate wave functions

For approximate wave functions, we prove the theorem that there is a one‐to‐one correspondence between the constraints of normalization and of the Fermi–Coulomb and Coulomb hole charge sum rules at each electron position. This correspondence is surprising in light of the fact that normalization depends on the probability of finding an electron at some position. In contrast, the Fermi–Coulomb hole sum rule depends on the probability of two electrons staying apart because of correlations due to the Pauli exclusion principle and Coulomb repulsion, while the Coulomb hole sum rule depends on Coulomb repulsion. We demonstrate the theorem for the ground state of the He atom by the use of two different approximate wave functions that are functionals rather than functions. The first of these wave function functionals is constructed to satisfy the constraint of normalization, and the second that of the Coulomb hole sum rule for each electron position. Each is then shown to satisfy the other corresponding sum rule. The significance of the theorem for the construction of approximate “exchange‐correlation” and “correlation” energy functionals of density functional theory is also discussed. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Ab initio study on structural and electronic properties of BanOm clusters

Density-functional calculation within local density approximation, shows that the electronic property of a barium oxide cluster is strongly correlated with its equilibrium structure. The ground-state structures of BanOm (4 < or = n < or = 9,m < or = n) clusters can be classified into four categories: (a) compact, (b) dangling state, (c) F-center, and (d) stoichiometric. The compact cluster is metallic, almost no energy gap exists between the highest occupied and the lowest unoccupied molecular orbitals. The energy gap for the dangling state cluster is larger than that for the F-center cluster, while the stoichiometric cluster has the largest energy gap.     

Quantum Monte Carlo study of the Ne atom and the Ne+ ion

We report all-electron and pseudopotential calculations of the ground-state energies of the neutral Ne atom and the Ne(+) ion using the variational and diffusion quantum Monte Carlo (DMC) methods. We investigate different levels of Slater-Jastrow trial wave function: (i) using Hartree-Fock orbitals, (ii) using orbitals optimized within a Monte Carlo procedure in the presence of a Jastrow factor, and (iii) including backflow correlations in the wave function. Small reductions in the total energy are obtained by optimizing the orbitals, while more significant reductions are obtained by incorporating backflow correlations. We study the finite-time-step and fixed-node biases in the DMC energy and show that there is a strong tendency for these errors to cancel when the first ionization potential (IP) is calculated. DMC gives highly accurate values for the IP of Ne at all the levels of trial wave function that we have considered.     

Density functional resonance theory: complex density functions, convergence, orbital energies, and functionals

Aspects of density functional resonance theory (DFRT) [D. L. Whitenack and A. Wasserman, Phys. Rev. Lett. 107, 163002 (2011)], a recently developed complex-scaled version of ground-state density functional theory (DFT), are studied in detail. The asymptotic behavior of the complex density function is related to the complex resonance energy and system's threshold energy, and the function's local oscillatory behavior is connected with preferential directions of electron decay. Practical considerations for implementation of the theory are addressed including sensitivity to the complex-scaling parameter, θ. In Kohn-Sham DFRT, it is shown that almost all θ-dependence in the calculated energies and lifetimes can be extinguished via use of a proper basis set or fine grid. The highest occupied Kohn-Sham orbital energy and lifetime are related to physical affinity and width, and the threshold energy of the Kohn-Sham system is shown to be equal to the threshold energy of the interacting system shifted by a well-defined functional. Finally, various complex-scaling conditions are derived which relate the functionals of ground-state DFT to those of DFRT via proper scaling factors and a non-Hermitian coupling-constant system.     

Obtaining the two-body density matrix in the density matrix renormalization group method

We present an approach that allows to produce the two-body density matrix during the density matrix renormalization group (DMRG) run without an additional increase in the current disk and memory requirements. The computational cost of producing the two-body density matrix is proportional to O(M3k2+M2k4). The method is based on the assumption that different elements of the two-body density matrix can be calculated during different steps of a sweep. Hence, it is desirable that the wave function at the convergence does not change during a sweep. We discuss the theoretical structure of the wave function ansatz used in DMRG, concluding that during the one-site DMRG procedure, the energy and the wave function are converging monotonically at every step of the sweep. Thus, the one-site algorithm provides an opportunity to obtain the two-body density matrix free from the N-representability problem. We explain the problem of local minima that may be encountered in the DMRG calculations. We discuss theoretically why and when the one- and two-site DMRG procedures may get stuck in a metastable solution, and we list practical solutions helping the minimization to avoid the local minima.     

Accurate Heitler-London interaction energy for He2

A 75-term Gaussian geminal wave function for the helium atom that has a variational energy within 0.42 μhartree of the exact one is constructed. It predicts an electron density that agrees to better than 0.4% with the predictions of energetically superior Hylleraas wave functions to electron-nucleus distances as large as 6a₀. The first-order Heitler-London interaction energy E⁽¹⁾ between a pair of helium atoms was computed using an antisymmetrized product of this Gaussian geminal wave function for each of the atoms. This interaction energy is an essential component in the exchange-Coulomb model for the He₂ potential. Our E⁽¹⁾ is probably converged to within 0.03 μhartree for interatomic distances between 3.0a₀ and 7.5a₀. The Coulombic part of the interaction energy was checked by computations using even more accurate Hylleraas wave functions for the monomers. Comparison with an E⁽¹⁾ value computed from self-consistent-field atomic wave functions shows that intra-atomic correlation effects range between 4% and 9%.     

Molecular Structures from Density Functional Calculations with Simulated Annealing

The geometrical structure of any aggregate of atoms is one of its basic properties and, in principle, straightforward to predict. One chooses a structure, determines the total energy E of the system of electrons and ions, and repeats the calculation for all possible geometries. The ground state structure is that with the lowest energy. A quantum mechanical calculation of the exact wave function Ψ would lead to the total energy, but this is practicable only in very small molecules. Furthermore, the number of local minima in the energy surface increases dramatically with increasing molecular size. While traditional ab initio methods have had many impressive successes, these difficulties have meant that they have focused on systems with relatively few local minima, or have used experiment or experience to limit the range of geometries studied. On the other hand, calculations for much larger molecules and extended systems are often forced to use simplifying assumptions about the interatomic forces that limit their predictive capability. The approach described here avoids both of these extremes: Total energies of predictive value are calculated without using semi-empirical force laws, and the problem of multiple minima in the energy surface is addressed. The density functional formalism, with a local density approximation for the exchange-correlation energy, allows one to calculate the total energy for a given geometry in an efficient, if approximate, manner. Calculations for heavier elements are not significantly more difficult than for those in the first row and provide an ideal way to study bonding trends. When coupled with finite-temperature molecular dynamics, this formalism can avoid many of the energetically unfavorable minima in the energy surface. We show here that the method leads to surprising and exciting results.     

Real-time, local basis-set implementation of time-dependent density functional theory for excited state dynamics simulations

We present a method suitable for large-scale accurate simulations of excited state dynamics within the framework of time-dependent density functional theory (DFT). This is achieved by employing a local atomic basis-set representation and real-time propagation of excited state wave functions. We implement the method within SIESTA, a standard ground-state DFT package with local atomic basis, and demonstrate its potential for realistic and accurate excited state dynamics simulations using small and medium-sized molecules as examples (H(2), CO, O(3), and indolequinone). The method can be readily applied to problems involving nanostructures and large biomolecules.     

Square Barrier Model of a One-dimensional Inhomogeneous Electron Liquid in Relation to Relativistic Density Functional Theory

Abstract

A long-term objective of density functional theory (DFT) has been to obtain the electronic kinetic energy density directly from the ground-state density, without recourse to wave functions. This is the more important in relativistic DFT since Dirac wave functions have four components. The above aim is here achieved for the admittedly specialized square barrier model of a one-dimensional inhomogeneous electron liquid.     

Energy conserving approximations to the quantum potential: dynamics with linearized quantum force

Solution of the Schrodinger equation within the de Broglie-Bohm formulation is based on propagation of trajectories in the presence of a nonlocal quantum potential. We present a new strategy for defining approximate quantum potentials within a restricted trial function by performing the optimal fit to the log-derivatives of the wave function density. This procedure results in the energy-conserving dynamics for a closed system. For one particular form of the trial function leading to the linear quantum force, the optimization problem is solved analytically in terms of the first and second moments of the weighted trajectory distribution. This approach gives exact time-evolution of a correlated Gaussian wave function in a locally quadratic potential. The method is computationally cheap in many dimensions, conserves total energy and satisfies the criterion on the average quantum force. Expectation values are readily found by summing over trajectory weights. Efficient extraction of the phase-dependent quantities is discussed. We illustrate the efficiency and accuracy of the linear quantum force approximation by examining a one-dimensional scattering problem and by computing the wavepacket reaction probability for the hydrogen exchange reaction and the photodissociation spectrum of ICN in two dimensions.