Density functional perturbational orbital theory of spin polarization in electronic systems. I. Formalism

A perturbational approach is presented for the general analysis of spin-polarization effect on electronic structures and energies within spin-density functional formalism. Explicit expressions for the changes in Kohn-Sham [Phys. Rev. 140, 1133 (1965)] orbital energies and coefficients as well as for the change in total electronic energy are derived upon using the local spin density and self-interaction-corrected exchange-correlation functionals. The application of the method for atoms provides analytical expressions for the exchange splitting energy and spin-polarization energy. The atomic exchange parameters are obtained from the expressions for the elements with Z=1-92 and they match well with Stoner exchange parameters for 3d metal elements.     

Nonspherical model density matrices for Rung 3.5 density functionals

"Rung 3.5" exchange-correlation functionals for Kohn-Sham density functional theory depend linearly on the nonlocal one-particle density matrix of the noninteracting Kohn-Sham reference system. Rung 3.5 functionals also require a semilocal model for the one-particle density matrix. This work presents new model density matrices for Rung 3.5 functionals. The resulting functionals give reasonable predictions for total energies, molecular thermochemistry and kinetics, odd-electron bonds, and conjugated polymer bandgaps. Global-hybrid-like combinations of semilocal and Rung 3.5 exchange, and empirical density matrix models, also show promise.     

Analytical evaluation of Fukui functions and real-space linear response function

Many useful concepts developed within density functional theory provide much insight for the understanding and prediction of chemical reactivity, one of the main aims in the field of conceptual density functional theory. While approximate evaluations of such concepts exist, the analytical and efficient evaluation is, however, challenging, because such concepts are usually expressed in terms of functional derivatives with respect to the electron density, or partial derivatives with respect to the number of electrons, complicating the connection to the computational variables of the Kohn-Sham one-electron orbitals. Only recently, the analytical expressions for the chemical potential, one of the key concepts, have been derived by Cohen, Mori-Sánchez, and Yang, based on the potential functional theory formalism. In the present work, we obtain the analytical expressions for the real-space linear response function using the coupled perturbed Kohn-Sham and generalized Kohn-Sham equations, and the Fukui functions using the previous analytical expressions for chemical potentials of Cohen, Mori-Sánchez, and Yang. The analytical expressions are exact within the given exchange-correlation functional. They are applicable to all commonly used approximate functionals, such as local density approximation (LDA), generalized gradient approximation (GGA), and hybrid functionals. The analytical expressions obtained here for Fukui function and linear response functions, along with that for the chemical potential by Cohen, Mori-Sánchez, and Yang, provide the rigorous and efficient evaluation of the key quantities in conceptual density functional theory within the computational framework of the Kohn-Sham and generalized Kohn-Sham approaches. Furthermore, the obtained analytical expressions for Fukui functions, in conjunction with the linearity condition of the ground state energy as a function of the fractional charges, also lead to new local conditions on the exact functionals, expressed in terms of the second-order functional derivatives. We implemented the expressions and demonstrate the efficacy with some atomic and molecular calculations, highlighting the importance of relaxation effects.     

Magnetic exchange couplings from constrained density functional theory: an efficient approach utilizing analytic derivatives

We introduce a method for evaluating magnetic exchange couplings based on the constrained density functional theory (C-DFT) approach of Rudra, Wu, and Van Voorhis [J. Chem. Phys. 124, 024103 (2006)]. Our method shares the same physical principles as C-DFT but makes use of the fact that the electronic energy changes quadratically and bilinearly with respect to the constraints in the range of interest. This allows us to use coupled perturbed Kohn-Sham spin density functional theory to determine approximately the corrections to the energy of the different spin configurations and construct a priori the relevant energy-landscapes obtained by constrained spin density functional theory. We assess this methodology in a set of binuclear transition-metal complexes and show that it reproduces very closely the results of C-DFT. This demonstrates a proof-of-concept for this method as a potential tool for studying a number of other molecular phenomena. Additionally, routes to improving upon the limitations of this method are discussed.     

Spin densities from subsystem density-functional theory: assessment and application to a photosynthetic reaction center complex model

Subsystem density-functional theory (DFT) is a powerful and efficient alternative to Kohn-Sham DFT for large systems composed of several weakly interacting subunits. Here, we provide a systematic investigation of the spin-density distributions obtained in subsystem DFT calculations for radicals in explicit environments. This includes a small radical in a solvent shell, a π-stacked guanine-thymine radical cation, and a benchmark application to a model for the special pair radical cation, which is a dimer of bacteriochlorophyll pigments, from the photosynthetic reaction center of purple bacteria. We investigate the differences in the spin densities resulting from subsystem DFT and Kohn-Sham DFT calculations. In these comparisons, we focus on the problem of overdelocalization of spin densities due to the self-interaction error in DFT. It is demonstrated that subsystem DFT can reduce this problem, while it still allows to describe spin-polarization effects crossing the boundaries of the subsystems. In practical calculations of spin densities for radicals in a given environment, it may thus be a pragmatic alternative to Kohn-Sham DFT calculations. In our calculation on the special pair radical cation, we show that the coordinating histidine residues reduce the spin-density asymmetry between the two halves of this system, while inclusion of a larger binding pocket model increases this asymmetry. The unidirectional energy transfer in photosynthetic reaction centers is related to the asymmetry introduced by the protein environment.     

Plane wave/pseudopotential implementation of excited state gradients in density functional linear response theory: a new route via implicit differentiation

This work presents the formalism and implementation of excited state nuclear forces within density functional linear response theory using a plane wave basis set. An implicit differentiation technique is developed for computing nonadiabatic coupling between Kohn-Sham molecular orbital wave functions as well as gradients of orbital energies which are then used to calculate excited state nuclear forces. The algorithm has been implemented in a plane wave/pseudopotential code taking into account only a reduced active subspace of molecular orbitals. It is demonstrated for the H(2) and N(2) molecules that the analytical gradients rapidly converge to the exact forces when the active subspace of molecular orbitals approaches completeness.     

Spin densities in two-component relativistic density functional calculations: noncollinear versus collinear approach

With present day exchange-correlation functionals, accurate results in nonrelativistic open shell density functional calculations can only be obtained if one uses functionals that do not only depend on the electron density but also on the spin density. We consider the common case where such functionals are applied in relativistic density functional calculations. In scalar-relativistic calculations, the spin density can be defined conventionally, but if spin-orbit coupling is taken into account, spin is no longer a good quantum number and it is not clear what the "spin density" is. In many applications, a fixed quantization axis is used to define the spin density ("collinear approach"), but one can also use the length of the local spin magnetization vector without any reference to an external axis ("noncollinear approach"). These two possibilities are compared in this work both by formal analysis and numerical experiments. It is shown that the (nonrelativistic) exchange-correlation functional should be invariant with respect to rotations in spin space, and this only holds for the noncollinear approach. Total energies of open shell species are higher in the collinear approach because less exchange energy is assigned to a given Kohn-Sham reference function. More importantly, the collinear approach breaks rotational symmetry, that is, in molecular calculations one may find different energies for different orientations of the molecule. Data for the first ionization potentials of Tl, Pb, element 113, and element 114, and for the orientation dependence of the total energy of I+2 and PbF indicate that the error introduced by the collinear approximation is approximately 0.1 eV for valence ionization potentials, but can be much larger if highly ionized open shell states are considered. Rotational invariance is broken by the same amount. This clearly indicates that the collinear approach should not be used, as the full treatment is easily implemented and does not introduce much more computational effort.     

Multireference space without first solving the configuration interaction problem

We further develop an idea to generate a compact multireference space without first solving the configuration interaction problem previously proposed for the ground state (GS) (Glushkov, Chem. Phys. Lett. 1995, 244, 1). In the present contribution, our attention is focused on low‐lying excited states (ESs) with the same symmetry as the GS which can be adequately described in terms of an high‐spin open‐shell formalism. Two references Møller–Plesset (MP) like perturbation theory for ESs is developed. It is based on: (1) a main reference configuration constructed from the parent molecular orbitals adjusted to a given ES and (2) secondary double excitation configuration built on the GS like orbitals determined by the Hartree–Fock equations subject to some orthogonality constraints. It is shown how to modify the MP zeroth‐order Hamiltonian so that the reference configurations and corresponding excitations are eigenfunctions of it and are compatible with orthogonality conditions for the GS and ES. Intruder states appearance is also discussed. The proposed scheme is applied to the GS, ES, and excitation energies of small molecules to illustrate and calibrate our calculations. © 2013 Wiley Periodicals, Inc.     

Electronic structure of the ground state of the benzyl radical

Results from the -electron approximation are given for the total energy, spin-density distribution, and electron density, which have been derived by configuration interaction via self-consistent orbitals for closed and open shells and with allowance for all singly excited configurations and some doubly excited ones. Some singly excited configurations do not mix with the ground-state configuration in the first order of perturbation theory but make a contribution to the latter greater than do some of the configurations that do mix. Incorporation of all singly excited configurations results in a lower ground-state energy if orbitals for a closed shell are used instead of those for an open one. Calculations via closed-shell orbitals lead to an inhomogeneous electron-density distribution, which is gradually smoothed out as the set of configurations is expanded. The spin-density distribution is very much dependent on the number of configurations used and on the orbitals employed in them.     

Covalency in the 4f shell of tris-cyclopentadienyl ytterbium (YbCp3)--a spectroscopic evaluation

Evidence is presented of significant covalency in the ytterbium 4f shell of tris-cyclopentadienyl ytterbium (YbCp(3)) in its electronic ground state, that can be represented by the superposition of an ionic configuration Yb(III):4f(13)(Cp(3)) and a charge-transfer configuration Yb(II):4f(14)(Cp(3))(-1). The relative weights of these configurations were determined from (i) the difference in their 4f photoionization cross sections, (ii) the accumulation of spin-density centered on the (13)C atoms of the Cp ring, as measured by a pulsed EPR (HYSCORE) experiment, (iii) the reduction in the spin-density in the 4f shell, manifest in the (171)Yb hyperfine interaction, and (iv) the principal values of the g-tensor, obtained from the EPR spectrum of a frozen glass solution at 5.4 K. Each of these methods finds that the spin density attributable to the charge transfer configuration is in the range 12-17%. The presence of configuration interaction (CI) also accounts for the highly anomalous energies, intensities, and vibronic structure in the "f-f" region of the optical spectrum, as well as the strict adherence of the magnetic susceptibility to the Curie law in the range 30-300 K.     

The ab initio calculation of molecular electric, magnetic and geometric properties

We give an account of some recent advances in the development of ab initio methods for the calculation of molecular response properties, involving electric, magnetic, and geometric perturbations. Particular attention is given to properties in which the basis functions depend explicitly both on time and on the applied perturbations such as perturbations involving nuclear displacements or external magnetic fields when London atomic orbitals are used. We summarize a general framework based on the quasienergy for the calculation of arbitrary-order molecular properties using the elements of the density matrix in the atomic-orbital basis as the basic variables. We demonstrate that the necessary perturbed density matrices of arbitrary order can be determined from a set of linear equations that have the same formal structure as the set of linear equations encountered when determining the linear response equations (or time-dependent self-consistent-field equations). Additional components needed to calculate properties involving perturbation-dependent basis sets are flexible one- and two-electron integral techniques for geometric or magnetic-field differentiated integrals; in Kohn-Sham density-functional theory (KS-DFT), we also need to calculate derivatives of the exchange-correlation functional. We describe a recent proposal for evaluating these contributions based on automatic differentiation. Within this framework, it is now possible to calculate any molecular property for an arbitrary self-consistent-field reference state, including two- and four-component relativistic self-consistent-field wave functions. Examples of calculations that can be performed with this formulation are presented.     

The spin-unrestricted molecular Kohn-Sham solution and the analogue of Koopmans's theorem for open-shell molecules

Spin-unrestricted Kohn-Sham (KS) solutions are constructed from accurate ab initio spin densities for the prototype doublet molecules NO(2), ClO(2), and NF(2) with the iterative local updating procedure of van Leeuwen and Baerends (LB). A qualitative justification of the LB procedure is given with a "strong" form of the Hohenberg-Kohn theorem. The calculated energies epsilon(isigma) of the occupied KS spin orbitals provide numerical support to the analogue of Koopmans' theorem in spin-density functional theory. In particular, the energies -epsilon(ibeta) of the minor spin (beta) valence orbitals of the considered doublet molecules correspond fairly well to the experimental vertical ionization potentials (VIPs) I(i) (1) to the triplet cationic states. The energy -epsilon(Halpha) of the highest occupied (spin-unpaired) alpha orbital is equal to the first VIP I(H) (0) to the singlet cationic state. In turn, the energies -epsilon(ialpha) of the major spin (alpha) valence orbitals of the closed subshells correspond to a fifty-fifty average of the experimental VIPs I(i) (1) and I(i) (0) to the triplet and singlet states. For the Li atom we find that the exact spin densities are represented by a spin-polarized Kohn-Sham system which is not in its ground state, i.e., the orbital energy of the lowest unoccupied beta spin orbital is lower than that of the highest occupied alpha spin orbital ("a hole below the Fermi level"). The addition of a magnetic field in the -z direction will shift the beta levels up so as to restore the Aufbau principle. This is an example of the nonuniqueness of the mapping of the spin density on the KS spin-dependent potentials discussed recently in the literature. The KS potentials may no longer go to zero at infinity, and it is in general the differences nu(ssigma)( infinity )-epsilon(isigma) that can be interpreted as (averages of) ionization energies. In total, the present results suggest the spin-unrestricted KS theory as a natural one-electron independent-particle model for interpretation and assignment of the experimental photoelectron spectra of open-shell molecules.     

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.     

The chemisorption of spin polarised NO on Ag{111}

Spin-density functional theory calculations are presented for NO adsorbed on Ag{111}. The ground state for the monomeric species is chemisorbed in an upright configuration, but retains 90% of the spin-density of the free molecule, in the molecular 2π^* orbital. In constrast, two NO molecules in upright configuration chemisorbed at neighbouring fcc and hcp sites have zero spin-density, and charge density difference plots demonstrate π bonding as in the free dimer, (NO)₂.     

Spin-spin contributions to the zero-field splitting tensor in organic triplets, carbenes and biradicals-a density functional and ab initio study

An evaluation study for the direct dipolar electron spin-spin (SS) contribution to the zero-field splitting (ZFS) tensor in electron paramagnetic resonance (EPR) spectroscopy is presented. Calculations were performed on a wide variety of organic systems where the SS contribution to the ZFS dominates over the second-order spin-orbit coupling (SOC) contribution. Calculations were performed using (hybrid) density functional theory (DFT), as well as complete active space self-consistent field (CASSCF) wave functions. In the former case, our implementation is an approximation, because we use the two-particle reduced spin-density matrix of the noninteracting reference system. In the latter case, the SS contribution is approximated by a mean-field method which, nevertheless, gives accurate results, compared to the approximation free computation of the SS part in a CASSCF framework. For the case of the triplet dioxygen molecule, it was shown that restricted open-shell density functional theory (RODFT), as well as CASSCF, can provide accurate spin-spin couplings while spin-unrestricted DFT leads to much larger errors. Furthermore, 15 organic radicals, including several 1,3 and 1,5 diradicals, dinitroxide biradicals, and even a chlorophyll a model system, were examined as test cases to demonstrate the accuracy and efficiency of our approach within a DFT framework. Accurate D values with root-mean-square deviations of 0.0035 cm(-1) were obtained. Furthermore, all trends, including those due to substituent effects, were correctly reproduced. In a different set of calculations, the polyacenes benzene, naphthalene, anthracene, and tetracene were studied. Applying DFT, the absolute D values were noticeably underestimated, but it was possible to correctly reproduce the trend to smaller D values with larger size of the systems. Finally, it was demonstrated that our approach is also well-suited for the study of carbenes. The smaller organic radicals of this work were also studied, through the use of CASSCF wave functions. This was a special advantage in the case of the triplet polyacenes, where the CASSCF approach gave better results than the DFT method. In comparing spin-restricted and spin-unrestricted results, it was shown through a natural orbital analysis and comparison to high-level ab initio calculations that even small amounts of spin polarization introduced by the unrestricted calculations lead to large deviations between the unrestricted Kohn-Sham (UKS) and restricted open-shell Kohn-Sham (ROKS) approaches. It is challenging to understand why the ROKS results show much better correlation with the experimental data.     

Local-scaling transformation version of density functional theory

The local-scaling transformation version of density functional theory (LS-DFT ) is reviewed. It is shown that in the context of LS-DFT it is possible to construct N-representable energy density functionals and that the theory provides systematic ways for calculating strict upper bounds to the exact energies. The importance of the concept of “orbit” in LS-DFT is indicated and several approaches leading to intraorbit and interorbit optimization are discussed. Results of the application of these optimization procedures to the determination of upper bounds for the ground-state energy of the beryllium atom are given. Also, numerical results are reported on the use of local scaling transformations for the direct solution of the Kohn-Sham equations via the density-constrained minimization of the kinetic energy of a noninteracting system. © 1995 John Wiley & Sons, Inc.     

Density functional restricted-unrestricted approach for nonlinear properties: application to electron paramagnetic resonance parameters of square planar copper complexes

The density functional restricted-unrestricted approach for treatments of spin polarization effects in molecular properties using spin restricted Kohn-Sham theory has been extended from linear to nonlinear properties. It is shown that the spin polarization contribution to a nonlinear property has the form of a quadratic response function that includes the zero-order Kohn-Sham operator, in analogy to the lower order case where the spin polarization correction to an expectation value has the form of a linear response function. The developed approach is used to formulate new schemes for computation of electronic g-tensors and hyperfine coupling constants, which include spin polarization effects within the framework of spin restricted Kohn-Sham theory. The proposed computational schemes are in the present work employed to study the spin polarization effects on electron paramagnetic resonance spin Hamiltonian parameters of square planar copper complexes. The obtained results indicate that spin polarization gives rise to sizable contributions to the hyperfine coupling tensor of copper in all investigated complexes, while the electronic g-tensors of these complexes are only marginally affected by spin polarization and other factors, such as choice of exchange-correlation functional or molecular structures, will have more pronounced impact on the accuracy of the results.