Comment on “density and physical current density functional theory” by Xiao‐Yin Pan and Viraht Sahni

A recent paper by Xiao‐Yin Pan and Viraht Sahni [Int. J. Quant. Chem. 110, 2833 (2010)] claims that current density functional theory should be based on the physical current density rather than the paramagnetic current density, as in the standard Vignale‐Rasolt formulation. In this comment we show that the claims in the paper by Pan and Sahni are erroneous. © 2012 Wiley Periodicals, Inc.     

Inverting the external-potential-to-electron density map in systems of electrons by minimization of a least-squares fit functional for analytic potentials

In a system of electrons, there is a map connecting any external potential v with its electron density ρ _v. In this work, we describe a procedure for inverting this potential-to-density map, so that potentials (if any) corresponding to a target density ρ_t can be obtained. We give the trial external potential v _α , an analytic expression depending on a number of parameters α = (α₁, …) and then minimize the least-squares integral ∫(ρ _α − ρ_t)² d r by the conjugate gradient method, where ρ _α is the density corresponding to v _α . The implementation takes advantage of the analytic nature of v _α . The procedure can be applied to any system and quantum chemistry model, and works both for ground and excited states, as well as for ensembles of states. The method is tested with some excited states of the particle-in-a-box model, confirming the lack of a Hohenberg–Kohn theorem for excited states. It is also applied to the first singlet excited state of the helium atom, where, apart from the nucleus–electron attraction potential, some generalized external potentials are found.     

On the original proof by reductio ad absurdum of the Hohenberg–Kohn theorem for many‐electron Coulomb systems

It is shown that, for isolated many‐electron Coulomb systems with Coulombic external potentials, the usual reductio ad absurdum proof of the Hohenberg–Kohn theorem is unsatisfactory since the to‐be‐refuted assumption made about the one‐electron densities and the assumption about the external potentials are not compatible with the Kato cusp condition. The theorem is, however, provable by more sophisticated means, and it is shown here that the Kato cusp condition actually leads to a satisfactory proof. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Hohenberg–Kohn theorems in the presence of magnetic field

In this article, we examine Hohenberg–Kohn theorems for Current Density Functional Theory, that is, generalizations of the classical Hohenberg–Kohn theorem that includes both electric and magnetic fields. In the Vignale and Rasolt formulation (Vignale and Rasolt, Phys. Rev. Lett. 1987, 59, 2360), which uses the paramagnetic current density, we address the issue of degenerate ground states and prove that the ensemble‐representable particle and paramagnetic current density determine the degenerate ground states. For the formulation that uses the total current density, we note that the proof suggested by Diener (Diener, J. Phys.: Condens. Matter. 1991, 3, 9417) is unfortunately not correct. Furthermore, we give a proof that the magnetic field and the ensemble‐representable particle density determine the scalar and vector potentials up to a gauge transformation. This generalizes the result of Grayce and Harris (Grayce and Harris, Phys. Rev. A 1994, 50, 3089) to the case of degenerate ground states. We moreover prove the existence of a positive wavefunction that is the ground state of infinitely many different Hamiltonians. © 2014 Wiley Periodicals, Inc.     

On the proof by reductio ad absurdum of the Hohenberg–Kohn theorem for ensembles of fractionally occupied states of Coulomb systems

It is demonstrated that the original reductio ad absurdum proof of the generalization of the Hohenberg–Kohn theorem for ensembles of fractionally occupied states for isolated many‐electron Coulomb systems with Coulomb‐type external potentials by Gross and colleagues is self‐contradictory, since the to‐be‐refuted assumption (negation) regarding the ensemble one‐electron densities and the assumption regarding the external potentials are logically incompatible to each other due to the Kato electron‐nuclear cusp theorem. It is proved, however, that the Kato theorem itself provides a satisfactory proof of this theorem. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

The Liu‐Parr power series expansion of the Pauli kinetic energy functional with the incorporation of shell‐inducing traits: Atoms

An approximate expression for the Pauli kinetic energy functional T_p is advanced in terms of the Liu‐Parr expansion [S. Liu, R.G. Parr, Phys. Rev. A 1997 , 55, 1792] which involves a power series of the one‐electron density. We use this explicit functional for T_p to compute the value of the noninteracting kinetic energy functional T_s of 34 atoms, from Li to Kr (and their positive and negative monoions). In particular, we examine the effect that a shell‐by‐shell mean‐square optimization of the expansion coefficients has on the kinetic energy values and explore the effect that the size of the expansion, given by the parameter n, has on the accuracy of the approximation. The results yield a mean absolute percent error for 34 neutral atoms of 0.15, 0.08, 0.04, 0.03, and 0.01 for expansions with n = 3, 4, 5, 6, and 7, respectively (where ). We show that these results, which are the most accurate ones obtained to date for the representation of the noninteracting kinetic energy functional, stem from the imposition of shell‐inducing traits. We also compare these Liu‐Parr functionals with the exact but nonexplicit functional generated in the local‐scaling transformation version of DFT.     

Correlation of electron density and bond length to band gap for binary oxides and halides

The best quantity correlated to the electronic energy band gap is found for alkali and alkaline-earth metal oxides and halides with face centered cubic (fcc) structure based on density functional theory and Bader's atom-in-molecule theory. Previous studies show the correlation of the band gap to the ground state electron density at the bond critical point (BCP). Whereas, in quantum mechanics, the gap between the energy levels of one dimensional square well potential is inversely proportional to the square of the width of the well which is the metal–nonmetal chemical bond length in our case. These motivate the proposition of a new quantity Q, the ratio of the density at the BCP to the square of the bond length. Our study reveals that, for the aforementioned materials, the band gap has a strong correlation to Q when they are multiplied by the density at the BCP.     

Comment on “On the original proof by reductio ad absurdum of the Hohenberg–Kohn theorem for many‐electron Coulomb systems”

We argue with Kryachko's criticism [Int J Quantum Chem 2005, 103, 818] of the original proof of the second Hohenberg‐Kohn theorem. The Kato cusp condition can be used to refute a “to‐be‐refuted” statement as an alternative to the original proof by Hohenberg and Kohn applicable for Coulombic systems. Since alternative ways to prove falseness of the “to‐be‐refuted” statement in a reduction ad absurdum proof do not exclude each other, Kryachko's criticism is not justified. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Kohn–Sham potentials by an inverse Kohn–Sham equation and accuracy assessment by virial theorem

In the recent study, the authors have proposed an integral equation for solving the inverse Kohn–Sham problem. In the present paper, the integral equation is numerically solved for one-dimensional model of a He atom and an H₂ molecule in the electronic ground states. For this purpose, we propose an iterative solution algorithm avoiding the inversion of the kernel of the integral equation. To quantify the numerical accuracy of the calculated exchange-correlation potentials, we evaluate the exchange and correlation energies based on the virial theorem as well as the reproduction of the exact ground-state electronic energy. The results demonstrate that the numerical solutions of our integral equation for the inverse Kohn–Sham problem are accurate enough in reproducing the Kohn–Sham potential and in satisfying the virial theorem.     

Alternative Representations of the Correlation Energy in Density‐Functional Theory: A Kinetic‐Energy Based Adiabatic Connection

The adiabatic‐connection framework has been widely used to explore the properties of the correlation energy in density‐functional theory. The integrand in this formula may be expressed in terms of the electron–electron interactions directly, involving intrinsically two‐particle expectation values. Alternatively, it may be expressed in terms of the kinetic energy, involving only one‐particle quantities. In this work, we explore this alternative representation for the correlation energy and highlight some of its potential for the construction of new density functional approximations. The kinetic‐energy based integrand is effective in concentrating static correlation effects to the low interaction strength regime and approaches zero asymptotically, offering interesting new possibilities for modeling the correlation energy in density‐functional theory     

Reply to “Comment on the Comparative Use of the Electron Density and Its Laplacian”

In our reply to the preceding comment by Richard Bader we show that the statements of the author are not justified and that he contradicts his own previous work.     

Accelerating Kohn‐Sham response theory using density fitting and the auxiliary‐density‐matrix method

An extension of the formulation of the atomic‐orbital‐based response theory of Larsen et al., JCP 113, 8909 (2000) is presented. This new framework has been implemented in LSDalton and allows for the use of Kohn‐Sham density‐functional theory with approximate treatment of the Coulomb and Exchange contributions to the response equations via the popular resolution‐of‐the‐identity approximation as well as the auxiliary‐density matrix method (ADMM). We present benchmark calculations of ground‐state energies as well as the linear and quadratic response properties: vertical excitation energies, polarizabilities, and hyperpolarizabilities. The quality of these approximations in a range of basis sets is assessed against reference calculations in a large aug‐pcseg‐4 basis. Our results confirm that density fitting of the Coulomb contribution can be used without hesitation for all the studied properties. The ADMM treatment of exchange is shown to yield high accuracy for ground‐state and excitation energies, whereas for polarizabilities and hyperpolarizabilities the performance gain comes at a cost of accuracy. Excitation energies of a tetrameric model consisting of units of the P700 special pigment of photosystem I have been studied to demonstrate the applicability of the code for a large system.     

Density functionals in the presence of magnetic field

In this article, density functionals for Coulomb systems subjected to electric and magnetic fields are developed. The density functionals depend on the particle density ρ and paramagnetic current density j^p. This approach is motivated by an adapted version of the Vignale and Rasolt formulation of current density functional theory, which establishes a one‐to‐one correspondence between the nondegenerate ground‐state and the particle and paramagnetic current density. Definition of N‐representable density pairs (ρ,j^p) is given and it is proven that the set of v‐representable densities constitutes a proper subset of the set of N‐representable densities. For a Levy–Lieb‐type functional Q(ρ,j^p), it is demonstrated that (i) it is a proper extension of the universal Hohenberg–Kohn functional to N‐representable densities, (ii) there exists a wavefunction ψ₀ such that , where H₀ is the Hamiltonian without external potential terms, and (iii) it is not convex. Furthermore, a convex and universal functional F(ρ,j^p) is studied and proven to be equal the convex envelope of Q(ρ,j^p). For both Q and F, we give upper and lower bounds. © 2014 Wiley Periodicals, Inc.     

Toward the complete range separation of non‐hybrid exchange–correlation functional

In this study, we use a very simple scheme to achieve range separation of a total exchange–correlation functional. We have utilized this methodology to combine a short‐range pure density functional theory (DFT) functional with a corresponding long‐range pure DFT, leading to a “Range‐separated eXchange–Correlation” (RXC) scheme. By examining the performance of a range of standard exchange–correlation functionals for prototypical short‐ and long‐range properties, we have chosen B‐LYP as the short‐range functional and PBE‐B95 as the long‐range counterpart. The results of our testing using a more diverse range of data sets show that, for properties that we deem to be short‐range in nature, the performance of this prescribed RXC‐DFT protocol does resemble that of B‐LYP in most cases, and vice versa. Thus, this RXC‐DFT protocol already provides meaningful numerical results. Furthermore, we envisage that the general RXC scheme can be easily implemented in computational chemistry software packages. This study paves a way for further refinement of such a range‐separation technique for the development of better performing DFT procedures. © 2015 Wiley Periodicals, Inc.     

Reply to ‘comment on “extending Hirshfeld‐I to bulk and periodic materials”’

The issues raised in the comment by Manz are addressed through the presentation of calculated atomic charges for NaF, NaCl, MgO, SrTiO $$_3$$ , and La $$_2$$ Ce $$_2$$ O $$_7$$ , using our previously presented method for calculating Hirshfeld‐I charges in solids (Vanpoucke et al., J. Comput. Chem. doi: 10.1002/jcc.23088). It is shown that the use of pseudovalence charges is sufficient to retrieve the full all‐electron Hirshfeld‐I charges to good accuracy. Furthermore, we present timing results of different systems, containing up to over 200 atoms, underlining the relatively low cost for large systems. A number of theoretical issues are formulated, pointing out mainly that care must be taken when deriving new atoms in molecules methods based on “expectations” for atomic charges. © 2012 Wiley Periodicals, Inc.     

He 2++ molecular ion in a strong time-dependent magnetic field: a current-density functional study

The He molecular ion exposed to a strong ultrashort time‐dependent (TD) magnetic field of the order of 10⁹ G is investigated through a quantum fluid dynamics (QFD) and current‐density functional theory (CDFT) based approach using vector exchange‐correlation (XC) potential and energy density functional that depend not only on the electronic charge‐density but also on the current density. The TD‐QFD‐CDFT computations are performed in a parallel internuclear‐axis and magnetic field‐axis configuration at the field‐free equilibrium internuclear separation R = 1.3 au with the field‐strength varying between 0 and 10¹¹ G. The TD behavior of the exchange‐ and correlation energy of the He is analyzed and compared with that obtained using a [B‐TD‐QFD‐density functional theory (DFT)] approach based on the conventional TD‐DFT under similar computational constraints but using only scalar XC potential and energy density functional dependent on the electronic charge‐density alone. The CDFT based approach yields TD exchange‐ and correlation energy and TD electronic charge‐density significantly different from that obtained using the conventional TD‐DFT based approach, particularly, at typical magnetic field strengths and during a typical time period of the TD field. This peculiar behavior of the CDFT‐based approach is traced to the TD current‐density dependent vector XC potential, which can induce nonadiabatic effects causing retardation of the oscillating electronic charge density. Such dissipative electron dynamics of the He molecular ion is elucidated by treating electronic charge density as an electron‐“fluid” in the terminology of QFD. © 2011 Wiley Periodicals, Inc. J Comput Chem, 2011     

Investigation of dephasing in an open quantum system under chaotic influence via a fractional Kohn–Sham scheme

In this work, the dynamics of dephasing (without relaxation) in the presence of a chaotic oscillator is theoretically investigated. The time‐dependent density functional theory framework was used in tandem with the Lindblad master equation approach for modeling the dissipative dynamics. Using the Kohn–Sham (K–S) scheme under certain approximations, the exact model for the potentials was acquired. In addition, a space‐fractional K–S scheme was developed (using the modified Riemann–Liouville operator) for modeling the dephasing phenomenon. Extensive analyses and comparative studies were then done on the results obtained using the space‐fractional K–S system and the conventional K–S system. © 2014 Wiley Periodicals, Inc.     

Neural network and ReaxFF comparison for Au properties

We have studied how ReaxFF and Behler–Parrinello neural network (BPNN) atomistic potentials should be trained to be accurate and tractable across multiple structural regimes of Au as a representative example of a single‐component material. We trained these potentials using subsets of 9,972 Kohn‐Sham density functional theory calculations and then validated their predictions against the untrained data. Our best ReaxFF potential was trained from 848 data points and could reliably predict surface and bulk data; however, it was substantially less accurate for molecular clusters of 126 atoms or fewer. Training the ReaxFF potential to more data also resulted in overfitting and lower accuracy. In contrast, BPNN could be fit to 9,734 calculations, and this potential performed comparably or better than ReaxFF across all regimes. However, the BPNN potential in this implementation brings significantly higher computational cost. © 2016 Wiley Periodicals, Inc.