Relation between the Slater–Kohn–Sham orbitals generated by the one-body potential V(r) of DFT and the Löwdin natural orbitals for inhomogeneous electron liquids

The Löwdin natural orbitals (NO) are defined as those orbitals which bring the first-order density matrix of a correlated electron assembly into diagonal form. Another one-particle density matrix with the same diagonal elements is the single-particle idempotent Dirac density matrix generated by the one-body potential of density functional theory. Here, we compare the off diagonal form of γ expanded in terms of the Slater–Kohn–Sham (SKS) orbitals generated by V(r) with the NO expansion of Löwdin for general inhomogeneous electron liquids. In particular, the equation of motion of the correlated γ is corrected from that of γ _s , both now containing the one-body potential V(r). To illustrate the theory presented here, we first construct an approximate, albeit accurate, correlated 1DM for the ground state of the He atom and display connections between the resulting NOs and the SKS orbitals. The second example we discuss, but now quite briefly, is that of the inhomogeneous electron liquid in crystalline Si, where the NO expansion is available from the literature.     

The exchange-correlation potential of DFT obtained from a semiempirically fine-tuned Hartree–Fock density for inhomogeneous electron liquids

The present authors have given an exact theory of the exchange-correlation potential V _xc (r) in terms of (i) the exact ground-state electron density n(r) and (ii) the idempotent Dirac density matrix γ(r,?r′) generated by the DFT one-body potential V(r), having n(r) as its diagonal element. Here, we display two approximate consequences: (a) a form of V _xc (r) generated by the semiempirically fine-tuned HF density of Cordero et al. (N.A. Cordero, N.H. March, and J.A. Alonso, Phys. Rev. A 75, 052502 (2007)) and (b) the exchange-only potential V _x (r) determined solely by the HF ground state density for the Be atom.     

Correlated and idempotent Dirac first-order density matrices with identical diagonal Fermion density: a route to extract a one-body potential energy in TDDFT

After a brief introduction to the use of the idempotent Dirac first-order density matrix (DM), its time-dependent generalization is considered. Special attention is focused on the equation of motion for the time-dependent DM, which is characterized by the one-body potential V(r, t) of time-dependent density functional theory. It is then shown how the force –∇ V(r, t) can be extracted explicitly from this equation of motion. Following a linear-response treatment in which a weak potential V(r, t) is switched on to an initially uniform electron gas, the non-linear example of the two-electron spin-compensated Moshinsky atom is a further focal point. We demonstrate explicitly how the correlated DM for this model can be constructed from the idempotent Dirac DM, in this time-dependent example.     

Exact electronic properties in the classically forbidden region of a metal surface

We derive exact properties of the inhomogeneous electron gas in the asymptotic classically forbidden region at a metal–vacuum interface within the framework of local effective potential energy theory. We derive a new expression for the asymptotic structure of the Kohn–Sham density functional theory (KS‐DFT) exchange‐correlation potential energy v_xc(r) in terms of the irreducible electron self‐energy. We also derive the exact asymptotic structure of the orbitals, density, the Dirac density matrix, the kinetic energy density, and KS exchange energy density. We further obtain the exact expression for the Fermi hole and demonstrate its structure in this asymptotic limit. The exchange‐correlation potential energy is derived to be v_xc(z → ∞) = ?α_KS,xc/z, and its exchange and correlation components to be v_x(z → ∞) = ?α_KS,x/z and v_c(z → ∞) = ?α_KS,c/z, respectively. The analytical expressions for the coefficients α_KS,xc and α_KS,x show them to be dependent on the bulk‐metal Wigner–Seitz radius and the barrier height at the surface. The coefficient α_KS,c = 1/4 is determined in the plasmon‐pole approximation and is independent of these metal parameters. Thus, the asymptotic structure of v_xc(z) in the vacuum region is image‐potential‐like but not the commonly accepted one of ?1/4z. Furthermore, this structure depends on the properties of the metal. Additionally, an analysis of these results via quantal density functional theory (Q‐DFT) shows that both the Pauli W_x(z → ∞) and lowest‐order correlation‐kinetic W(z → ∞) components of the exchange potential energy v_x(z → ∞), and the Coulomb W_c(z → ∞) and higher‐order correlation‐kinetic components of the correlation potential energy v_c(z → ∞), all contribute terms of O(1/z) to the structure. Hence correlations attributable to the Pauli exclusion principle, Coulomb repulsion, and correlation‐kinetic effects all contribute to the asymptotic structure of the effective potential energy at a metal surface. The relevance of the results derived to the theory of image states and to KS‐DFT is also discussed. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Slater sum for an inhomogeneous liquid of Fermions generated by a sech2 x potential in one dimension

In one dimension, the Slater sum S(x, β), which is the diagonal element of the canonical density matrix, satisfies a known partial differential equation characterised by a one-body potential V(x). Here, for the case of a sech² x potential in one dimension, it is stressed that S(x, β) is explicitly related to the limit S ₀(β) as V(x) → 0 and to V(x) itself. This is the same input information as in the Thomas–Fermi result. The relevance to density functional theory is emphasised.     

Towards an exact orbital-free single-particle kinetic energy density for the inhomogeneous electron liquid in the Be atom

Holas and March [Phys. Rev. A51, 2040 (1995)] wrote the gradient of the one-body potential V(r) in terms of low-order derivatives of the idempotent Dirac density matrix built from a single Slater determinant of Kohn–Sham orbitals. Here, this is first combined with the study of Dawson and March [J. Chem. Phys. 81, 5850 (1984)] to express the single-particle kinetic energy density of the Be atom ground-state in terms of both the electron density n(r) and potential V(r). While this is the more compact formulation, we then, by removing V(r), demonstrate that the ratio t(r)/n(r) depends, though non-locally, only on the single variable n′(r)/n(r), no high-order gradients entering for the spherical Be atom.     

Explicit density–potential relation for 4-electron atoms in 2-orbital S states

An explicit relation is derived between the one-body potential energy and the electron density for the ground state of the Be atom in a nonrelativistic framework. This same relation applies to any four-electron atomic ion (or to Be itself) in a state where the electrons occupy two doubly filled orbitals. The relation is interpreted as an exact Hartree-like model of the Hohenberg–Kohn theorem within the general context of N electrons and a potential that is not necessarily spherically symmetrical.     

Theoretical study on the reaction of CN radicals with ClO radicals by density functional theory

The reaction mechanism of CN radicals with ClO radicals has been studied theoretically using ab initio and density functional theory (DFT). The result shows that the main reaction path is the O atom in radical ClO attacks the C atom in radical CN to compose the intermediate 1 ClOCN. Three thermodynamically accessible prodncts, P₁ (CO+ClN), P₃ (NO+CCl), and P₄ (ClNCO), were obtained from intermediate 1 through isomerization and decomposition reactions. P₄ is the primary product, and P₁ and P₃ are the secondary product. Compared with the singlet potential energy surface, the contribution of the triplet potential energy surface can be ignored.     

Density functional complete study of hydrogen bonding between the water molecule and the hydroxyl radical (H2O · HO)

The hydrogen bonding complexes formed between the H₂O and OH radical have been completely investigated for the first time in this study using density functional theory (DFT). A larger basis set 6‐311++G(2d,2p) has been employed in conjunction with a hybrid density functional method, namely, UB3LYP/6‐311++G(2d,2p). The two degenerate components of the OH radical ²Π ground electronic state give rise to independent states upon interaction with the water molecule, with hydrogen bonding occurring between the oxygen atom of H₂O and the hydrogen atom of the OH radical. Another hydrogen bond occurs between one of the H atoms of H₂O and the O atom of the OH radical. The extensive calculation reveals that there is still more hydrogen bonding form found first in this investigation, in which two or three hydrogen bonds occur at the same time. The optimized geometry parameter and interaction energy for various isomers at the present level of theory was estimated. The infrared (IR) spectrum frequencies, IR intensities, and vibrational frequency shifts are reported. The estimates of the H₂O · OH complex's vibrational modes and predicted IR spectra for these structures are also made. It should be noted that a total of 10 stationary points have been confirmed to be genuine minima and transition states on the potential energy hypersurface of the H₂O · HO system. Among them, four genuine minima were located. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

Correlation energy,correlated electron density,and exchange‐correlation potential in some spherically confined atoms

We report correlation energies, electron densities, and exchange‐correlation potentials obtained from configuration interaction and density functional calculations on spherically confined He, Be, Be²⁺, and Ne atoms. The variation of the correlation energy with the confinement radius R_c is relatively small for the He, Be²⁺, and Ne systems. Curiously, the Lee–Yang–Parr (LYP) functional works well for weak confinements but fails completely for small R_c. However, in the neutral beryllium atom the CI correlation energy increases markedly with decreasing R_c. This effect is less pronounced at the density‐functional theory level. The LYP functional performs very well for the unconfined Be atom, but fails badly for small R_c. The standard exchange‐correlation potentials exhibit significant deviation from the “exact” potential obtained by inversion of Kohn–Sham equation. The LYP correlation potential behaves erratically at strong confinements. © 2016 Wiley Periodicals, Inc.     

Chemical bond inside a fullerene cage: is it possible?

The geometries, electronic structures, and hyperfine coupling constants of azafullerene C₅₉N (a π-electron radical) and its derivatives, C₅₉NH and endofullerene H@C₅₉N, were calculated at the B3LYP level of the density functional theory. Analysis of calculated potential energy profiles along trajectories of the motion of encapsulated hydrogen atom from the center of the fullerene sphere toward different atoms of C₅₉N revealed formation of a chemical bond between the H atom and a carbon atom that is involved in the 6,6-bond with the N atom and bears the most part of the π-electron spin density. The C—H endo-bond length is 1.12 Å, the bond dissociation energy being equal to 26.4 kcal mol⁻¹. The C—H exo-bond involving the same carbon atom is 0.02 Å shorter than the endo-bond, the bond dissociation energy being much higher (78.4 kcal mol⁻¹).__________Published in Russian in Izvestiya Akademii Nauk. Seriya Khimicheskaya, No. 1, pp. 51–54, January, 2005.     

TixNy团簇结构的密度泛函研究

采用杂化密度泛函方法(B3LYP)和有效核势基组预测了Ti_xN_y (x≤3, y≤2)团簇的结构及稳定性, 并分析了可能存在构型的电子结构. 结果表明Ti₂N中体系的自旋多重度由Ti原子决定. Ti₃N中随着N的配位数增加, N的负电荷增加, 平均每个Ti向N提供约0.3个电子. 从Ti₂N₂可能稳定构型分析, 成键数目越多, 能量上越有利, 且Ti—N键的数目的增加, 将削弱N—N间的成键.     

Study of the magnetic state of transition element atoms in compounds and alloys usingKα 1 spectra

A review is given for investigations of the magnetic state of transition element atoms in compounds and alloys using K₁ spectra. The shift and width of the K₁ line are used for determining changes in the spin and charge density on the emitting atom depending on the type and concentration of the components. This method permits us to determine the local magnetic moment at an atom with precision of 0.1–0.2 _B . The electron state density in the conductance band affects the spin density formed on the resonance 3d level of an atom in the field of a 2p vacancy. Localization of 3d states was observed upon the transition of an atom to an impurity state. The contribution to the spin state on the emitting atom in Pauli paramagnetics is proportional to the state density on the Fermi level. The width of the K₁ line of transition element atoms was found to depend on the magnetic moment on the atoms of the immediate environment and on the magnitude and sign of the exchange interaction between them, which permits us to study magnetic phase transitions.Institute of the Mineralogy, Geochemistry, and Crystal Chemistry of Rare Elements. Translated from Zhurnal Strukturnoi Khimii, Vol. 30, No. 6, pp. 122–137, November–December, 1989.     

Self-consistent Thomas–Fermi–Dirac theory,extended by Gell-Mann and Brueckner correlation,in terms of density n and its two reduced gradients ∇2n/n and ∇n/n

We prove the following results, relevant for the density functional theory: the Thomas–Fermi–Dirac theory, generalized to include the contribution due to the high electron density result of Gell-Mann and Brueckner for the correlation energy, is shown to lead to a differential equation for the self-consistent ground-state density n( r ) in atoms and molecules in the form F(n, { ∇ n/n}², ∇²n/n)=1, where the function F is given explicitly. A straightforward extension yields a similar result for the equation determining the Pauli plus exchange–correlation potential and for the divergence of the many-electron force. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 69: 145–149, 1998     

How the Chalcogen Atom Size Dictates the Hydrogen-Bond Donor Capability of Carboxamides,Thioamides, and Selenoamides

The amino groups of thio- and selenoamides can act as stronger hydrogen-bond donors than of carboxamides, despite the lower electronegativity of S and Se. This phenomenon has been experimentally explored, particularly in organocatalysis, but a sound electronic explanation is lacking. Our quantum chemical investigations show that the NH₂ groups in thio- and selenoamides are more positively charged than in carboxamides. This originates from the larger electronic density flow from the nitrogen lone pair of the NH₂ group towards the lower-lying π*_C=S and π*_C=Se orbitals than to the high-lying π*_C=O orbital. The relative energies of the π* orbitals result from the overlap between the chalcogen np and carbon 2p atomic orbitals, which is set by the carbon-chalcogen equilibrium distance, a consequence of the Pauli repulsion between the two bonded atoms. Thus, neither the electronegativity nor the often-suggested polarizability but the steric size of the chalcogen atom determines the amide's hydrogen-bond donor capability.     

Local density functional calculations of the electronic structures of the intermetallic systems U2Fe2Sn and UFe2Ge2

The electronic structures of U₂Fe₂Sn and UFe₂Ge₂ are self-consistently calculated within the local density functional theory using the augmented spherical wave (ASW) method. Calculations are scalar relativistic. The experimentally observed Pauli paramagnetic behavior of the two systems is accounted for and the influence of hybridization between the different l-states on the chemical bonding is discussed from the site-projected densities of sates (DOS) as well as from the modulation of the DOS by the sign and magnitude of the overlap integral, i.e., with the so-called COOP. From this, we propose a mechanism for the evolution of bonding within the series to which the two compounds belong. © 1997 John Wiley & Sons, Inc.     

Isoelectronic changes in energy of quark atoms and molecules via the Levy equation

Summary Z-transition state calculations based on the Levy equation suggest that the isoelectronic changes in energy of quark atoms,Q, (ordinary atoms with extra nuclear charge in units of ±1/3 and/or ±2/3) can be expressed quantitatively in terms of the electrostatic potential at the nucleus of an isoelectronic ordinary atom. Numerical tests within the local density functional theory are presented for the quark atoms of Li-F. Theab initio MO (molecular orbital) calculations using STO-5G basis on the C₂ molecule and its quark derivatives lead to similar conclusions.     

Brief review related to the foundations of time-dependent density functional theory

The electron density n(r,t), which is the central tool of time-dependent density functional theory, is presently considered to be derivable from a one-body time-dependent potential V(r,t), via one-electron wave functions satisfying a time-dependent Schrödinger equation. This is here related via a generalized equation of motion to a Dirac density matrix now involving t. Linear response theory is then surveyed, with a special emphasis on the question of causality with respect to the density dependence of the potential. Extraction of V(r,t) for solvable models is also proposed.     

NMR shielding constants of CuX,AgX, and AuX (X = F,Cl, Br,and I) investigated by density functional theory based on the Douglas–Kroll–Hess Hamiltonian

Two‐component relativistic density functional theory (DFT) with the second‐order Douglas–Kroll–Hess (DKH2) one‐electron Hamiltonian was applied to the calculation of nuclear magnetic resonance (NMR) shielding constant. Large basis set dependence was observed in the shielding constant of Xe atom. The DKH2‐DFT‐calculated shielding constants of I and Xe in HI, I₂, CuI, AgI, and XeF₂ agree well with those obtained by the four‐component relativistic theory and experiments. The Au NMR shielding constant in AuF is extremely more positive than in AuCl, AuBr, and AuI, as reported recently. This extremely positive shielding constant arises from the much larger Fermi contact (FC) term of AuF than in others. Interestingly, the absolute values of the paramagnetic and the FC terms are considerably larger in CuF and AuF than in others. The large paramagnetic term of AuF arises from the large d‐components in the Au d_π –F p_π and Au sd_σ–F p_σ molecular orbitals (MOs). The large FC term in AuF arises from the small energy difference between the Au sd_σ + F p_σ and Au sd_σ–F p_σ MOs. The second‐order magnetically relativistic effect, which is the effect of DKH2 magnetic operator, is important even in CuF. This effect considerably improves the overestimation of the spin‐orbit effect calculated by the Breit–Pauli magnetic operator. © 2013 Wiley Periodicals, Inc.