密度泛函理论及其数值方法新进展

综述了密度泛函理论及其数值方法的最新进展.密度泛函理论的发展以寻找合适 的交换相关近似为主线,从最初的局域密度近似、广义梯度近似到现在的非局域泛函、自相 互作用修正,多种泛函形式的相继出现使得密度泛函理论可以提供越来越精确的计算结果. 除了交换相关近似的发展,近年来密度泛函理论向含时理论、相对论等方面的扩展也很活跃 .另外,在密度泛函理论体系发展的同时,相应的数值计算方法的发展也非常迅速.从古老 的有限差分、有限元到新兴的小波分析都被用来实现密度泛函理论的数值计算.与此同时, 线性标度的密度泛函理论算法日趋成熟,使得通过密度泛函理论研究诸如生物大分子之类的 体系成为可能.随着密度泛函理论本身及其数值方法的发展,它的应用也越来越广泛,一些新的应用领域和研究方向不断涌现.     

Numerical examination of performance of some exchange-correlation functionals for molecules containing heavy elements

The performance of 17 exchange-correlation functionals for molecules containing heavy elements are numerically examined through four-component relativistic density DFT calculations. The examined functionals show the similar accuracy as they do for the molecules containing light elements only except for bond lengths. LDA and OP86 produce good results for bond lengths and frequencies but bad bond energies. Different functionals do not show much different performance for bond energies except LDA. BP86 and GP86 produce results with average accuracy while LYP does not perform well. Although encouraging results are obtained with functional B97GGA-1, other heavily parameterized and meta-GGA functionals do not produce impressive results.     

The importance of electronic exchange-correlation in non-self-consistent frozen density approximation

The effects of different treatments for the exchange-correlation energy on the accuracy of non-self-consistent frozen density approximation (FDA) are discussed. Local spin density approximation (LSDA) and non-local spin density approximation (NLSDA) are employed, respectively. Corresponding results obtained by using full-self-consistent density functional theory (DFT) are also given for the purpose of comparison. Explicit calculations for hydrogen bonds, covalent bonds and ionic bonds indicate that, comparing with LSDA, NLSDA can improve the accuracy of FDA as well as that of DFT. This improvement attributed to the refinements in the treatment for the electronic exchange-correlation energy may help to extend the application of FDA.     

Orbital-dependent correlation energy in density-functional theory based on a second-order perturbation approach: success and failure

We have developed a second-order perturbation theory (PT) energy functional within density-functional theory (DFT). Based on PT with the Kohn-Sham (KS) determinant as a reference, this new ab initio exchange-correlation functional includes an exact exchange (EXX) energy in the first order and a correlation energy including all single and double excitations from the KS reference in the second order. The explicit dependence of the exchange and correlation energy on the KS orbitals in the functional fits well into our direct minimization approach for the optimized effective potential, which is a very efficient method to perform fully self-consistent calculations for any orbital-dependent functionals. To investigate the quality of the correlation functional, we have applied the method to selected atoms and molecules. For two-electron atoms and small molecules described with small basis sets, this new method provides excellent results, improving both second-order Moller-Plesset expression and any conventional DFT results significantly. For larger systems, however, it performs poorly, converging to very low unphysical total energies. The failure of PT based energy functionals is analyzed, and its origin is traced back to near degeneracy problems due to the orbital- and eigenvalue-dependent algebraic structure of the correlation functional. The failure emerges in the self-consistent approach but not in perturbative post-EXX calculations, emphasizing the crucial importance of self-consistency in testing new orbital-dependent energy functionals.     

The self-interaction error and the description of non-dynamic electron correlation in density functional theory

The self-interaction error (SIE) plays a central role in density functional theory (DFT) when carried out with approximate exchange-correlation functionals. Its origin, properties, and consequences for the development of standard DFT to a method that can correctly describe multi-reference electron systems by treating dynamic and non-dynamic electron correlation on an equal footing, is discussed. In this connection, the seminal work of Colle and Salvetti on wave function-based correlation functionals that do no longer suffer from a SIE is essential. It is described how the Colle–Salvetti correlation functional is an anchor point for the derivation of a functional multi-reference DFT method.     

Orbital energies and negative electron affinities from density functional theory: Insight from the integer discontinuity

Orbital energies in Kohn-Sham density functional theory (DFT) are investigated, paying attention to the role of the integer discontinuity in the exact exchange-correlation potential. A series of closed-shell molecules are considered, comprising some that vertically bind an excess electron and others that do not. High-level ab initio electron densities are used to calculate accurate orbital energy differences, Deltavarepsilon, between the lowest unoccupied molecular orbital (LUMO) and the highest occupied molecular orbital (HOMO), using the same potential for both. They are combined with accurate vertical ionization potentials, I(0), and electron affinities, A(0), to determine accurate "average" orbital energies. These are the orbital energies associated with an exchange-correlation potential that averages over a constant jump in the accurate potential, of magnitude Delta(XC)=(I(0)-A(0))-Deltavarepsilon, as given by the discontinuity analysis. Local functional HOMO energies are shown to be almost an order of magnitude closer to these average values than to -I(0), with typical discrepancies of just 0.02 a.u. For systems that do not bind an excess electron, this level of agreement is only achieved when A(0) is set equal to the negative experimental affinity from electron transmission spectroscopy (ETS); it degrades notably when the zero ground state affinity is instead used. Analogous observations are made for the local functional LUMO energies, although the need to use the ETS affinities is less pronounced for systems where the ETS values are very negative. The application of an asymptotic correction recovers the preference, leading to positive LUMO energies (but bound orbitals) for these systems, consistent with the behavior of the average energies. The asymptotically corrected LUMO energies typically agree with the average values to within 0.02 a.u., comparable to that observed with the HOMOs. The study provides numerical support for the view that local functionals exhibit a near-average behavior based on a constant jump of magnitude Delta(XC). It illustrates why a recently proposed DFT expression involving local functional frontier orbital energies and ionization potential yields reasonable estimates of negative ETS affinities and is consistent with earlier work on the failure of DFT for charge-transfer excited states. The near-average behavior of the exchange-correlation potential is explicitly illustrated for selected systems. The nature of hybrid functional orbital energies is also mentioned, and the results of the study are discussed in terms of the variation in electronic energy as a function of electron number. The nature of DFT orbital energies is of great importance in chemistry; this study contributes to the understanding of these quantities.     

Calculation of ionization potentials of small molecules: a comparative study of different methods

With the help of various theoretical methods, ionization potentials (IPs) have been computed for a panel of small molecules containing atoms of group 14, 15, or 16 and representing different singly, doubly, or triply bonded systems with or without an interacting heteroatom lone pair. Comparison of experimental IP values to theoretical results indicates that (i) the standard outer valence green function (OVGF), density functional theory (DFT), and DeltaSCF methods lead to rather accurate values, (ii) the CASPT2 method systematically underestimates IPs, (iii) the method of deducing IPs from a shift of some standard DFT eigenvalue spectrum is a straightforward approach leading to rather accurate IPs, (iv) the eigenvalue spectrum obtained with the so-called statistical average of different orbital model potential (SAOP) exchange-correlation model potential is an efficient approach leading directly to quite accurate IPs, and (v) a good prediction of the IP spectrum can be obtained from the shifted excitation spectra of the system calculated by the time-dependent DFT (TD-DFT) method. It is also shown that the TD-DFT calculations of the ionized species bring a significant improvement over the calculations of the neutral molecules, indicating that a great part of the electronic relaxation is already taken into account (in a similar way for all ionizations). Finally, in the case of TD-DFT calculations of neutral molecules, the statistical average of different orbital model potential (SAOP) functional does not lead to significantly better results than the B3LYP functional.     

The chemical Hamiltonian approach in density functional theory

The chemical Hamiltonian approach (CHA) for handling the basis set superposition error problem in intermolecular interactions has been implemented within density functional theory (DFT) using Gaussian atomic basis sets. As test examples, the potential curves of the water dimer were calculated using the Vosko-Wilk-Nusair, Becke-Perdew and Perdew exchange-correlation functionals. Comparisons with the counterpoise correction method show that CHA within DFT performs as well as previously for Hartree-Fock.     

Density functional energy decomposition into one- and two-atom contributions

The present work provides a generalization of Mayer's energy decomposition for the density-functional theory (DFT) case. It is shown that one- and two-atom Hartree-Fock energy components in Mayer's approach can be represented as an action of a one-atom potential V(A) on a one-atom density rho(A) or rho(B). To treat the exchange-correlation term in the DFT energy expression in a similar way, the exchange-correlation energy density per electron is expanded into a linear combination of basis functions. Calculations carried out for a number of density functionals demonstrate that the DFT and Hartree-Fock two-atom energies agree to a reasonable extent with each other. The two-atom energies for strong covalent bonds are within the range of typical bond dissociation energies and are therefore a convenient computational tool for assessment of individual bond strength in polyatomic molecules. For nonspecific nonbonding interactions, the two-atom energies are low. They can be either repulsive or slightly attractive, but the DFT results more frequently yield small attractive values compared to the Hartree-Fock case. The hydrogen bond in the water dimer is calculated to be between the strong covalent and nonbonding interactions on the energy scale.     

Derivative discontinuity, bandgap and lowest unoccupied molecular orbital in density functional theory

The conventional analysis of Perdew and Levy, and Sham and Schlu?ter shows that the functional derivative discontinuity of the exchange-correlation density functional plays a critical role in the correct prediction of bandgaps, or the chemical hardness. In a recent work by the present authors, explicit expressions for bandgap prediction with all common types of exchange-correlation functionals have been derived without invoking the concept of exchange-correlation energy functional derivative discontinuity at all. We here analyze the two approaches and establish their connection and difference. The present analysis further leads to several important results: (1) The lowest unoccupied molecular orbital (LUMO) in DFT has as much meaning in describing electron addition as the highest occupied molecular orbital (HOMO) in describing electron removal. (2) Every term in the total energy functional contributes to the energy gap because of the discontinuity of the derivative of the density (or density matrix) with respect to the number of electrons, ((?ρ(s)(r('),r))/?N)(v(s) ), at integers. (3) Consistent with the Perdew-Levy-Sham-Schlu?ter conclusion that the exact Kohn-Sham energy gap differs from the fundamental bandgap by a finite correction due to the functional derivative discontinuity of the exchange-correlation energy, we show that the exchange-correlation functional cannot be an explicit and differentiable functional of the electron density, either local or nonlocal. The last result is further strengthened when we consider Mott insulators. There, the exact exchange-correlation functional needs to have an explicitly discontinuous (nondifferentiable) dependence on the density or the density matrix. (4) We obtain exact conditions on the derivatives of total energy with respect to the spin-up and spin-down number of electrons.     

95Mo nuclear magnetic resonance parameters of molybdenum hexacarbonyl from density functional theory: appraisal of computational and geometrical parameters

Solid-state (95)Mo nuclear magnetic resonance (NMR) properties of molybdenum hexacarbonyl have been computed using density functional theory (DFT) based methods. Both quadrupolar coupling and chemical shift parameters were evaluated and compared with parameters of high precision determined using single-crystal (95)Mo NMR experiments. Within a molecular approach, the effects of major computational parameters, i.e. basis set, exchange-correlation functional, treatment of relativity, have been evaluated. Except for the isotropic parameter of both chemical shift and chemical shielding, computed NMR parameters are more sensitive to geometrical variations than computational details. Relativistic effects do not play a crucial part in the calculations of such parameters for the 4d transition metal, in particular isotropic chemical shift. Periodic DFT calculations were tackled to measure the influence of neighbouring molecules on the crystal structure. These effects have to be taken into account to compute accurate solid-state (95)Mo NMR parameters even for such an inorganic molecular compound.     

Hemibonding of hydroxyl radical and halide anion in aqueous solution

Molecular geometries and properties of the possible reaction products between the hydroxyl radical and the halide anions in aqueous solution were investigated. The formation of two-center three-electron bonding (hemibonding) between the hydroxyl radical and halide anions (Cl, Br, I) was examined by density functional theory (DFT) calculation with a range-separated hybrid (RSH) exchange-correlation functional. The long-range corrected hybrid functional (LC-ωPBE), which have given quantitatively satisfactory results for odd electron systems and excited states, was examined by test calculations for dihalogen radical anions (X(2)(-); X = Cl, Br, I) and hydroxyl radical-water clusters. Equilibrium geometries with hemibonding between the hydroxyl radical and halide anions were located by including four hydrogen-bonded water molecules. Excitation energies and oscillator strengths of σ-σ* transitions calculated by the time-dependent DFT method showed good agreement with observed values. Calculated values of the free energy of reaction on the formation of hydroxyl halide radical anion from the hydroxyl radical and halide anion were endothermic for chloride but exothermic for bromide and iodide, which is consistent with experimental values of equilibrium constants.     

Density functional theory for charge transfer: the nature of the N-bands of porphyrins and chlorophylls revealed through CAM-B3LYP, CASPT2, and SAC-CI calculations

While density functional theory (DFT) has been proven to be extremely useful for the prediction of thermodynamic and spectroscopic properties of molecules, to date most functionals used in common implementations of DFT display a systematic failure to predict the properties of charge-transfer processes. While this is explicitly manifest in Rydberg transitions of atoms and molecules and in molecular charge-transfer spectroscopy, it also becomes critical for systems containing extended conjugation such as polyenes and other conducting polymers, porphyrins, chlorophylls, etc. A new density functional, a Coulomb-attenuated hybrid exchange-correlation functional (CAM-B3LYP), has recently been developed specifically to overcome these limitations, and it has been shown to properly predict molecular charge-transfer spectra. Here, we demonstrate that it predicts qualitatively reasonable spectra for porphyrin, some oligoporphyrins, and chlorophyll. However, alternate density functionals developed to overcome the same limitations such as current-density functional theory are shown, in their present implementation, to remain inadequate. The CAM-B3LYP results are shown to be in excellent agreement with complete-active-space plus second-order M?ller-Plesset perturbation theory and symmetry-adapted cluster configuration interaction calculations: These depict the N and higher bands of porphyrins and chlorophylls as being charge-transfer bands associated with localization of molecular orbitals on individual pyrrole rings. The validity of the basic Gouterman model for the spectra of porphyrins and chlorophylls is confirmed, rejecting modern suggestions that non-Gouterman transitions lie close in energy to the Q-bands of chlorophylls. As porphyrins and chlorophylls provide useful paradigms for problems involving extended conjugation, the results obtained suggest that many significant areas of nanotechnology and biotechnology may now be realistically treated by cost-effective density-functional-based computational methods.     

甲烷晶体的晶格能和弹性性质: 不同方法及泛函的评估

通过对甲烷晶体进行结构、晶格能和弹性特性的研究, 评估了不包含和包含色散能量修正的密度泛函理论的性能. 我们分别利用不包含色散能量修正的密度泛函理论(DFT) (包含不同的标准泛函和杂化泛函)和包含色散能量修正的密度泛函理论(DFT-D)计算了甲烷晶体特性, 并与实验作对比. 尽管DFT-D 与传统密度泛函理论及杂化密度泛函理论相比, 修正了甲烷晶体中的范德华(vdW)相互作用, 但是一些修正方案过分修正了这种相互作用. 因此, 人们在使用DFT-D方法时务必谨慎.     

Atomistic Model of Uranium

The electronic state and potential data of U₂ molecules are performed by first principle calculations with B3LYP hybrid exchange-correlation functional, the valence electrons of U atom are treated with the (5s4p3d4f)/[3s3p2d2f] contraction basis sets, and the cores are approximated with the relativistic effective core potential. The results show that the ground electronic state is X⁹∑⁺_g . The pair potential data are fitted with a Murrell-Sorbie analytical potential function. The U-U embedded atom method (EAM) interatomic potential is deter-mined based on the generalized gradient approximation calculation within the framework of the density functional theory using Perdew-Burke-Ernzerhof exchange-correlation functional at the spin-polarized level. The physical properties, such as the cohesive energy, the lattice constant, the bulk modulus, the shear modulus, the sc/fcc relative energy, the hcp/fcc rela-tive energy, the shear modulus and the monovacancy formation energy are used to evaluate the EAM potential parameters. The U-U pair potential determined by the first principle calculations is in agreement with that defined by the EAM potential parameters. The EAM calculated formation energy of the monovacancy in the fcc structure is also found to be in close agreement with DFT calculation.     

NO双分子在Cu2O(111)面吸附与解离的理论研究

采用广义梯度密度泛函理论结合周期平板模型方法, 在DNP基组下, 研究了NO双分子在三重态和单重态两种电子组态下在Cu2O(111)完整表面的吸附情况. 考虑了Cu+(NO)(NO)、Cu+(NO)(ON)及Cu+(ON)(ON)这三种构型, 计算了它们的吸附能和Mulliken电荷, 分析并预测了吸附后可能产生的物种. 结果表明, 当两个NO分子都以O端吸附在Cu2O(111)表面时即Cu+(ON)(ON)构型, N—N键长很短, 只有124.4 pm, 吸附的两个NO分子形成了二聚体形式, 这种吸附构型有利于进一步离解产生N2或N2O并形成Cu-O表面物种.     

Performance of multi-configurational calculations for a 1,4-bis(phenylethynyl)benzene derivative conjugated molecule.

The theoretical challenge of finding a single method that quantitatively reproduces both the experimental low-lying excitation energies and the torsional barrier of a prototypical conjugated molecule, which could act as a molecular wire, has been addressed here. The results indicate that this goal can be reasonably achieved when multi-reference perturbation theory up to second order (MRMP2) based on a complete active space self-consistent field (CASSCF) wave function using large active spaces is used. The results obtained were also used to compare with less expensive Kohn-Sham (KS) density functional theory (DFT) calculations when applied to these properties. The results obtained with BLYP and B3LYP exchange-correlation functionals indicate that quantitative agreement with all the experimental data cannot be obtained with this methodology, with a clear dependence on the exchange-correlation form selected. We thus encourage a careful testing of pure and hybrid density functionals whenever KS DFT is used for the rational design of conjugated materials for charge conduits.     

A dual-level approach to density-functional theory

An efficient approximate scheme for density-functional theory (DFT) calculations, which eliminates the time-consuming self-consistent-field (SCF) procedure, is proposed using a dual-level DFT approach. In this approach, dual levels of basis sets and exchange-correlation functionals are adopted. The dual-level DFT approach is based on the idea that the total electron density in the ground state can be represented in terms of the density evaluated using the low-quality basis set and the low-cost exchange-correlation functional. Since the SCF procedure is avoided in the total energy evaluation, the dual-level DFT approach drastically reduces the computational cost. The applications of several dual-level DFT calculations to molecular systems show that our approach is more efficient than the self-consistent DFT approach with a moderate accuracy.     

Spin states at a tipping point: what determines the dz2(1) ground state of nickel(III) tetra(tbutyl)porphyrin dicyanide?

Density functional theory (DFT) calculations, regardless of the exchange-correlation functional, have long failed to reproduce the observed dz2(1) ground state of the [NiIII(TtBuP)(CN)2]- anion (where TtBuP is the strongly ruffled tetra(tbutyl)porphyrin ligand), predicting instead a dx2-y2(1) ground state. Normally, such failures are associated with DFT calculations on spin states of different multiplicity, which is not the case here. The calculations reported here strongly suggest that the problem does not lie with DFT. Instead environmental factors need to be taken into account, such as counterions and solvents. Counterions such as K+ placed against the cyanide nitrogens and polar solvents both result in a dz2(1) ground state, thus finally reconciling theory and experiment.