Spin contamination for Hartree-Fock, optimized effective potential, and density functional approximations

In an earlier paper [S. Thanos and A. K. Theophilou J. Chem. Phys. 124, 204109 (2006)], we found an explicit formula for the expansion of a Slater determinant |Phi(M)> in terms of eigenstates of S(2). In this paper, we use the same formula to determine the spin contamination S(con) of the unrestricted single determinant approximations, i.e., Hartree-Fock, optimized effective potential, and density functional theory. We derived an expression which gives S(con) in terms of the overlap of the spatial parts of the spin up and spin down "corresponding" orbitals. It was found that S(con) does not depend on M, the eigenvalue of S(z), at least for the lower order approximations, i.e., when || is large. In this case, the predominant coefficient of the expansion assumes its maximum value when S=M. However, for the class of solutions that || is small, the spin L of the largest coefficient increases with the number of unpaired electrons. We also derived the explicit form of the expansion states.     

Density-functional theory with effective potential expressed as a mapping of the external potential: applications to open-shell molecules

In this paper we apply the direct-mapping density-functional theory (DFT) to open-shell systems, in order to get many-electron wave functions having the same transformation properties as the eigenstates of the exact Hamiltonians. Such a case is that of spin, where in order to get the magnetic properties, the many-particle states must be eigenstates not only of S(z) but also of S2. In this theory the Kohn and Sham [Phys. Rev. A 140, 1133 (1965)] potential is expressed directly as a mapping of the external potential. The total energies of the molecules calculated were satisfactory as their relative deviations (deltaEE) from the exact Hartree-Fock ones were of the order of 10(-4). This accuracy is much higher than that of the standard DFT in its local exchange potential approximation. This method does not need an approximate density as input, as the effective potential is derived directly from the external potential.     

Spin relaxation in isolated molecules and clusters: the interpretation of Stern-Gerlach experiments

Intramolecular spin relaxation may occur in isolated molecules or clusters provided that the density of rovibrational eigenstates is sufficiently high to serve as an energy bath and angular momentum is conserved. In the coupled, zero-field limit, total angular momentum (J) is the sum of spin (S) and rotational (N) momenta such that J and M(J) are good angular momentum quantum numbers. In the coupled limit, transitions between Zeeman levels (Delta M(J)++0) cannot occur in the absence of an external torque. However, in the high-field limit, J and M(J) are no longer good quantum numbers, as N and S are decoupled and only their projections on the z axis defined by the external field are invariant. In this case M(N) and M(S) remain as good quantum numbers so that angular momentum conserving transitions can occur subject to the selection rule Delta M(N)=-Delta M(S). Determination of the magnetic moments of isolated molecules and clusters via a thermodynamics-based analysis requires that their magnetizations are measured at sufficiently large fields that spin-rotation effects become negligible and the Zeeman level structure approaches the free-spin case.     

Spin component scaling in multiconfiguration perturbation theory

We investigate a term-by-term scaling of the second-order energy correction obtained by perturbation theory (PT) starting from a multiconfiguration wave function. The total second-order correction is decomposed into several terms, based on the level and the spin pattern of the excitations. To define individual terms, we extend the same spin/different spin categorization of spin component scaling in various ways. When needed, identification of the excitation level is facilitated by the pivot determinant underlying the multiconfiguration PT framework. Scaling factors are determined from the stationary condition of the total energy calculated up to order 3. The decomposition schemes are tested numerically on the example of bond dissociation profiles and energy differences. We conclude that Grimme's parameters determined for single-reference M?ller-Plesset theory may give a modest error reduction along the entire potential surface, if adopting a multireference based PT formulation. Scaling factors obtained from the stationary condition show relatively large variation with molecular geometry, at the same time they are more efficient in reducing the error when following a bond dissociation process.     

Unrestricted-Hartree–Fock wave functions approximating low-lying covalent states of ring π systems

We show that spin projected unrestricted-Hartree–Fock (PUHF ) wave functions are able to well approximate some low-lying covalent states of ring π systems. The UHF wave functions belong to either the axial or torsional spin density wave class. Their spin structures are found to be approximations for the spin correlation structures of the corresponding exact wave functions. The PUHF wave functions become close to exact eigenstates with homopolar valence bond characters in the strong correlation limit.     

Calculation of molecular electronic spectra by projected-unrestricted Hartree–Fock theory

This paper is concerned with a new application of projected-unrestricted Hartree–Fock theory, namely, the calculation of electronic spectra for symmetric molecules. The excited electronic state is represented by a single determinant whose unrestricted nature allows for orbital rearrangement relative to the self-consistent ground state. The self-consistent calculation must be followed by spin projection to obtain appropriate spin eigenstates. It was necessary to develop modified procedures for portions of the spin projection calculation because our method of constructing the wave functions produces degeneracies among the natural orbitals. Illustrative calculations using the all-valence-electron INDO approximations produced results which compared favorably with configuration-interaction treatments. The method described here should be most useful, however, in conjunction with ab initio calculations using flexible basis sets.     

Spin-state splittings, highest-occupied-molecular-orbital and lowest-unoccupied-molecular-orbital energies, and chemical hardness

It is known that the exact density functional must give ground-state energies that are piecewise linear as a function of electron number. In this work we prove that this is also true for the lowest-energy excited states of different spin or spatial symmetry. This has three important consequences for chemical applications: the ground state of a molecule must correspond to the state with the maximum highest-occupied-molecular-orbital energy, minimum lowest-unoccupied-molecular-orbital energy, and maximum chemical hardness. The beryllium, carbon, and vanadium atoms, as well as the CH(2) and C(3)H(3) molecules are considered as illustrative examples. Our result also directly and rigorously connects the ionization potential and electron affinity to the stability of spin states.     

Accurate Kohn–Sham ionization potentials from scaled‐opposite‐spin second‐order optimized effective potential methods

One important property of Kohn–Sham (KS) density functional theory is the exact equality of the energy of the highest occupied KS orbital (HOMO) with the negative ionization potential of the system. This exact feature is out of reach for standard density‐dependent semilocal functionals. Conversely, accurate results can be obtained using orbital‐dependent functionals in the optimized effective potential (OEP) approach. In this article, we investigate the performance, in this context, of some advanced OEP methods, with special emphasis on the recently proposed scaled‐opposite‐spin OEP functional. Moreover, we analyze the impact of the so‐called HOMO condition on the final quality of the HOMO energy. Results are compared to reference data obtained at the CCSD(T) level of theory. © 2016 Wiley Periodicals, Inc.     

Tests of Monte Carlo perturbation theory for the free energy of liquid copper

Monte Carlo perturbation theory, in which terms in the thermodynamic perturbation series are evaluated by Monte Carlo averaging, has potentially large advantages in efficiency for calculating free energies of liquids from ab initio potential surfaces. In order to test the accuracy of perturbation theory for liquid metals, a series of calculations has been done on liquid copper, modeled by an embedded atom potential. A simple 1/r(12) pair potential is used as the reference system. The free energy is calculated to third order in perturbation theory, and the results are compared to an exact formula. It is found that for optimal reference potential parameters, second order perturbation theory is essentially exact. Second and third order theories give accurate results for significantly nonoptimal reference parameters. The relation between perturbation theory and reweighting is discussed, and an approximate formula is derived that shows an exponential dependence of the efficiency of reweighting on the second order free energy correction. Finally, techniques for application to ab initio potentials are discussed. It is shown that with samples of 100 configurations, it is possible to obtain accuracy and precision at the level of approximately 1 meV/atom.     

Construction of stationary quantum states with targeted energies

Given an exact stationary state and its energy-eigenvalue, we devise a simple way of generating new families of exact eigenstates with the same eigenenergy, but for different potentials. While this recipe of designing isoergic states is quite general in the context of arriving at new exact solutions from a known premise, relevance of this route with the scheme of construction of isospectral potentials is noted under specific situations. The idea is extended to finding new exact eigenstates with eigenvalues as the sum of energies of two or more separate but known stationary states in dissimilar potentials. Implication of the latter endeavor in the contexts of manifold energy minimization is discussed.     

Correlation potentials for molecular bond dissociation within the self-consistent random phase approximation

Self-consistent correlation potentials for H(2) and LiH for various inter-atomic separations are obtained within the random phase approximation (RPA) of density functional theory. The RPA correlation potential shows a peak at the bond midpoint, which is an exact feature of the true correlation potential, but lacks another exact feature: the step important to preserve integer charge on the atomic fragments in the dissociation limit. An analysis of the RPA energy functional in terms of fractional charge is given which confirms these observations. We find that the RPA misses the derivative discontinuity at odd integer particle numbers but explicitly eliminates the fractional spin error in the exact-exchange functional. The latter finding explains the improved total energy in the dissociation limit.     

DFT and CASPT2 study on the mechanism of ethylene dimerization over Cr(II)OH+ cation

In this work, the ethylene coordination and dimerization mechanism over Cr(II)OH(+) cation were systematically investigated using density functional theory (DFT) and complete active space second-order perturbation theory (CASPT2). It was found that Cr(II)OH(+) cation can coordinate with up to four ethylene molecules which gives seven possible stable Cr(II)OH(+)·(C(2)H(4))(n) (n = 1-4) π-complexes. We investigated whether ethylene dimerization over Cr(II)OH(+) cation proceeds through either a carbene mechanism or a metallacycle mechanism. The potential energy surfaces were characterized using four different functionals (M06L, BLYP, B3LYP, and M06). It was found that the potential energy profiles calculated at the M06 level agreed well with the CASPT2 energy profiles. Since the intermediates involved in the proposed catalytic cycles showed different ground spin states, a reaction pathway involving a spin crossing between two potential energy surfaces was observed. The minimum-energy crossing points (MECPs) that connect the two potential energy surfaces were successfully located. The two-state metallacycle reaction pathway with the formation of chromacyclopentane as the rate-determining step was found to be energetically more favorable than the carbene reaction pathway. 1-Butene was formed from the chromacyclopentane by a two-step reductive elimination pathway through a chromium(IV) hydride intermediate.     

Stochastic search of the quantum conformational space of small lithium and bimetallic lithium-sodium clusters

In this paper we report the results obtained by an implementation and application of the simulated annealing optimization procedure to the exploration of the conformational space of small neutral and charged lithium clusters (Li(n)(q), n = 5, 6, 7; q = 0, +/-1) and of the bimetallic lithium/sodium clusters (Li5Na) in their lowest spin states. Our methodology eliminates the structure guessing procedure in the process of generating cluster configurations. We evaluate the quantum energy, typically with the Hartree-Fock Hamiltonian, of randomly generated points in the conformational space and use a modified Metropolis test in the annealing algorithm to generate candidate structures for atomic clusters. The structures are further optimized by analytical methods (gradient following) at the M?ller-Plesset second order perturbation theory level (MP2), in conjunction with basis sets including polarization functions with and without diffuse functions. High accuracy ab initio energies at the coupled clusters level, with single, double, and triple substitutions from the Hartree-Fock determinant (CCSD(T)), on the MP2 geometries were calculated and used to establish the relative stability of the isomers within each potential energy surface. Various cluster properties were computed and compared to existing values in order to validate our methods. Our results show excellent agreement with previous experimental and theoretical reports. Even at these small sizes, evidence for 10 new structures never reported before for the lithium clusters and four new structures for the bimetallic clusters is presented.     

Density functional theory for excited states and special symmetries

For Hamiltonians which are invariant under a group of transformations, one can restrict the search for the energy eigenstates in spaces whose functions transform according to the irreducible representations of the group. However, the construction of a Slater determinant to represent the equivalent noninteracting system of DFT, with the proper transformation properties, is not trivial. Further such a determinant does not always exist. The use of the subspace theory [J. Phys. C 12, 5419 (1979)] developed initially to deal with the density functional theory for excited states overcomes this difficulty and an equivalent system of one-particle Kohn and Sham equations is derived with nonintegral occupation numbers in the expression of the density. In this article, we derive the explicit form of the subspace density for systems with spherical symmetry. The density does not depend on the Clebsch-Gordan coefficients, but only on the radial part of the orbitals entering the determinant of the noninteracting state with largest 1. © 1997 John Wiley & Sons, Inc.     

Calculation of alkaline metal dimers in terms of model perturbation theory

The problem of calculating diatomic alkaline metal (homo- and heteronuclear) molecules KM (M=Li, Na, K, Rb, Cs) is treated in terms of a pseudopotential approach in the framework of a formally exact model perturbation theory of Rayleigh-Schrödinger type with a test zero-order potential. A Gell-Mann type local model potential is used as a zero-order potential. The results of calculations of energy parameters, in particular, dissociation energies, are given; some of them were obtained for the first time. The calculation demonstrated that two major effects of second-order perturbation theory: polarizing interaction of valence particles across the core and mutual screening of particles must be taken into account to achieve an acceptable accuracy of calculations.     

FeMo cofactor of nitrogenase: a density functional study of states M(N), M(OX), M(R), and M(I).

The M(N) S = (3)/(2) resting state of the FeMo cofactor of nitrogenase has been proposed to have metal-ion valencies of either Mo(4+)6Fe(2+)Fe(3+) (derived from metal hyperfine interactions) or Mo(4+)4Fe(2+)3Fe(3+) (from M?ssbauer isomer shifts). Spin-polarized broken-symmetry (BS) density functional theory (DFT) calculations have been undertaken to determine which oxidation level best represents the M(N) state and to provide a framework for understanding its energetics and spectroscopy. For the Mo(4+)6Fe(2+)Fe(3+) oxidation state, the spin coupling pattern for several spin state alignments compatible with S = (3)/(2) were generated and assessed by energy and geometric criteria. The most likely BS spin state is composed of a Mo3Fe cluster with spin S(a) = 2 antiferromagnetically coupled to a 4Fe' cluster with spin S(b) = (7)/(2). This state has a low DFT energy for the isolated FeMoco cluster and the lowest energy when the interaction with the protein and solvent environment is included. This spin state also displays calculated metal hyperfine and M?ssbauer isomer shifts compatible with experiment, and optimized geometries that are in excellent agreement with the protein X-ray data. Our best model for the actual spin-coupled state within FeMoco alters this BS state by a slight canting of spins and is analogous in several respects to that found in the 8Fe P-cluster in the same protein. The spin-up and spin-down components of the LUMO contain atomic contributions from Mo(4+) and the homocitrate and from the central prismane Fe sites and muS(2) atoms, respectively. This qualitative picture of the accepting orbitals for M(N) is consistent with observations from M?ssbauer spectra of the one-electron reduced states. Similar calculations for the Mo(4+)4Fe(2+)3Fe(3+) oxidation state yield results that are in poorer agreement with experiment. Using the Mo(4+)6Fe(2+)Fe(3+) oxidation level as the most plausible resting state, the geometric, electronic and energetic properties of the one-electron redox transition to the oxidized state, M(OX), catalytically observed M(R) and radiolytically reduced M(I) states have also been explored.     

On the accuracy of DFT-SAPT, MP2, SCS-MP2, MP2C, and DFT+Disp methods for the interaction energies of endohedral complexes of the C(60) fullerene with a rare gas atom

Selected points on the potential energy surface for the complexes Rg@C(60) (Rg = He, Ne, Ar, Kr) are calculated with various theoretical methods, like symmetry-adapted perturbation theory with monomers described by density functional theory (DFT-SAPT), supermolecular M?ller-Plesset theory truncated on the second order (MP2), spin-component-scaled MP2 (SCS-MP2), supermolecular density functional theory with empirical dispersion correction (DFT+Disp), and the recently developed MP2C method that improves the MP2 method for long-range electron correlation effects. A stabilization of the endohedral complex is predicted by all methods, but the depth of the potential energy well is overestimated by the DFT+Disp and MP2 approaches. On the other hand, the MP2C model agrees well with DFT-SAPT, which serves as the reference. The performance of SCS-MP2 is mixed: it produces too low interaction energies for the two heavier guests, while its accuracy for He@C(60) and Ne@C(60) is similar to that of MP2C. Fitting formulas for the main interaction energy components, i.e. the dispersion and first-order repulsion energies are proposed, which are applicable for both endo- and exohedral cases. For all examined methods density fitting is used to evaluate two-electron repulsion integrals, which is indispensable to allow studies of noncovalent complexes of this size. It has been found that density-fitting auxiliary basis sets cannot be used in a black-box fashion for the calculation of the first-order SAPT electrostatic energy, and that the quality of these basis sets should be always carefully examined in order to avoid an unphysical long-range behavior.     

A new approach to the determination of good action-angle variables for coupled oscillator systems

A theory of action-angle variables for coupled oscillator systems is developed which involves solving the Schrödinger equation using a basis of WKB eigenfunctions, then using the logarithm of the resulting wavefunction to define the generator for the canonical transformation which determines the action-angle variables. This theory is based on the marriage between Miller's method for solving the Hamilton-Jacobi equation using the logarithm of the generating function, and the Ratner-Buch-Gerber method for solving the Schrödinger equation using WKB basis functions. A perturbation-theory analysis of this theory indicates that the semiclassical eigenvalues and canonical transformations obtained from it should become identical to their exact classical counterparts in the limit of large actions for each vibrational mode. Two methods for systematically improving the theory for the lower eigenstates are also proposed. Numerical applications of the theory are presented for two systems, the Morse oscillator and the Henon-Heiles two-mode hamiltonian. The resulting semiclassical eigenvalues are in excellent agreement with their exact quantum counterparts, with the magnitude of the error roughly independent of the energy of the eigenstate. Analogous good agreement is found in comparing the approximate and exact classical canonical transformations. In particular, for the Morse oscillator, good results are obtained for certain higher energy states where second-order classical perturbation theory makes serious errors. Other information examined includes surfaces of section for the Henon-Heiles system (comparing the analytical functions obtained from the present theory with results based on exact trajectory calculations) and vibrational distributions chosen to simulate trajectory calculations (using the present theory to determine bin boundaries for a histogram calculation). Again, the comparison in each case with accurate results is excellent, with maximum errors in action calculations of 0.02 h, and in angle calculations of 0.01 rad.     

Au12M (M=Na,Mg, Al,Si, P,S, Cl)团簇的结构、稳定性和电子性质

采用基于密度泛函理论的第一性原理方法系统地研究了Au12M(M=Na,Mg,Al,Si,P,S,Cl)团簇的结构、稳定性和电子性质.对团簇的平均结合能、镶嵌能、垂直离化势、最高占据分子轨道(HOMO)和最低未占据分子轨道(LUMO)的能级差、电荷布居分析、自然键轨道(NBO)进行了计算和讨论.对于Au12M(M=Na,Mg,Al)团簇,它们形成了内含M原子的最稳定的笼状结构.然而对于Au12M(M=Si,P,S,Cl)团簇,它们却形成了以M元素为顶点的稳定锥形结构.在这些团簇中发现Au12S团簇相对是最稳定的,这是由于Au12S团簇形成了稳定的满壳层的电子结构.自然电荷布居分析表明:对于所有的Au12M(M=Na,Mg,Al,Si,P,S,Cl)团簇电荷总是从Au原子转向M原子.自然键轨道和HOMO分析表明Au12M团簇中发生了Au原子的s-d轨道和M原子的p轨道间的杂化现象.