Variational Density Functional Theory Research Articles

Comprehensive Study of Equilibrium Structure of Trans-Azobenzene: Gas Electron Diffraction and Quantum Chemical Calculations

The geometrical re parameters of trans-azobenzene (E-AB) free molecule were refined by gas electron diffraction (GED) method using available experimental data obtained previously by S. Konaka and coworkers. Structural analysis was carried out by various techniques. First of all, these included the widely used molecular orbital constrained gas electron diffraction method and regularization method. The results of the refinements using different models were also compared—a semirigid model, three variants of one-dimensional dynamic models, and a two-dimensional pseudoconformer model. Several descriptions have been used due to the fact that E-AB has a shallow potential energy surface along the rotation coordinates of phenyl groups. Despite this, it turned out that the semirigid model is suitable for use for E-AB and allows good agreement with experimental data to be achieved. According to the results of GED structural analysis, coupled with the results of DLPNO-CCSD(T0) calculations, E-AB has a planar structure. Based only on GED data, it is impossible to unambiguously determine the rotational angle of the phenyl group due to the facts that (i) with rotation over a wide range of angles, the bonded distances in the molecule change insignificantly and (ii) potential function in a structural analysis within a dynamic model is not determined with the necessary accuracy. This work also examines the sensitivity of the GED method to structural changes caused by trans-cis isomerization. The paper also analyzes the applicability of different variants of density functional theory (DFT) calculations in GED structural analysis using E-AB as an example. There are not enough similar methodological works in the literature. This experimental and methodological information is especially important and relevant for planning and implementing GED experiments and corresponding processing of the results for azobenzene derivatives, in which the conformer and isomeric diversity are even more complicated due to the presence of different substituents.

Energy barriers of Be and B in passing through the C60 fullerene cage

We have studied the potential barriers for the penetration of atomic beryllium or boron inside the C60 fullerene by performing density functional theory (DFT) calculations with three variants for the exchange and correlation: B3LYP (hybrid functional), PW91, and PBE. Four principal trajectories to the inner part of C60 for the penetrating atom have been considered: through the center of six-member-carbon ring (hexagon), five-member-carbon ring (pentagon), and also through the center of the double C–C bond (D-bond) and the center of the single C–C bond (S-bond). Averaging over the three DFT variants yields the following barriers for beryllium penetrating inside a deformable fullerene: 3.2 eV (hexagon), 4.8 eV (S-bond), 5.3 eV (D-bond), 5.9 eV (pentagon). These barriers correspond to the slow and adiabatic penetration of Be, in contrast to the fast (non-adiabatic) penetration through the rigid cage of C60 resulting in 5.6 eV (hexagon), 16.3 eV (pentagon), 81.8 eV (S-bond), and 93.4 eV (D-bond). The potential barriers for the boron penetrating inside deformable/rigid C60 are 3.7/105.4 eV (D-bond), 4.0/86.8 eV (S-bond), 4.7/7.8 eV (hexagon), 6.8/14.0 eV (pentagon).

Variational density functional perturbation theory for metals

Variational density functional perturbation theory for metals

Spherical densities and potentials in exactly solvable model molecules.

A recently initiated variant of density functional theory utilizes a set of spherically symmetric densities instead of the density. The exact functionals are unknown in the new theory akin to the standard density functional theory. In order to test approximate functionals exactly solvable models are introduced. A harmonic molecular ion, the analogue to the hydrogen molecule ion and a harmonic two-electron molecule showing analogy to the hydrogen molecule are proposed. It has been found that the wave function and the density can be given analytically. The exact spherical densities and the effective potentials of the Euler equations also have analytical form. It has been shown that the models can be easily extended to several "nuclei."

Spectroscopic investigation and density functional theory prediction of first and second order hyperpolarizabilities of 1-(4-bromophenyl)-3-(2,4-dichlorophenyl)‑prop-2-en-1-one

This work presents the spectroscopic analysis and nonlinear optical (NLO) behaviour of 1-(4-Bromophenyl)-3-(2,4-dichlorophenyl)‑prop-2-en-1-one (DCBC). The motivation behind choosing chalcone derivative DCBC for NLO studies is its crystallization in non-centrosymmetric structure, excellent second harmonic generation efficiency (SHG) and flexibility in structure of chalcones due to donor-acceptor groups. The compound was characterized by FT-IR, FT-Raman, UV–vis and NMR techniques. The structural and spectroscopic features of the molecule were calculated using density functional theory (DFT) employing B3LYP/6–311++G(d,p) basis set and compared with the experimental values. The complete vibrational assignments were made on the basis of potential energy distribution (PED). The Natural Bond Orbital (NBO) analysis of DCBC has been done in order to provide the detailed insight into the nature of electronic conjugation between the bonds in this molecule. The second-order perturbation theory analysis of Fock matrix in the NBO basis of the molecule shows the strong intramolecular hyperconjugative interactions between σ (Cl3-C28) to σ*(C26-C28), n(1)(Cl3) to σ*(C26-C28) and π(C6-C8) to π*(C5-C13) resulting in large stabilization energies leading to large delocalization. A time-dependant variant of DFT (TD-DFT) is applied to calculate the electronic properties such as oscillator strengths, absorption maxima and HOMO and LUMO energies. The HOMO-LUMO energy gap is found to be 3.882 eV and is used to calculate global reactivity parameters. Isotropic chemical shifts were calculated using gauge invariant atomic orbital (GIAO) method and these matched well with the experimental results, indicating the accuracy of calculation method. Molecular Electrostatic Potential (MEP) surface has been depicted to know the chemically active regions of the molecule. The nonlinear optical properties of title molecule were addressed theoretically by the determination of two contributions, vibrational and electronic, to polarizability (α) and the first and the second order hyperpolarizabilities (β and γ). The electronic first and second order hyperpolarizabilities of DCBC determined by Finite Field approach method are found to be 17.805 × 10−30 and 6.0 × 10−41 e.s.u. respectively. These values are comparable to other chalcones and organic NLO materials which indicate that the studied molecule is responsive to nonlinear optical (NLO) effect and has enhanced applicability in the development of nonlinear optical devices.

The metal–insulator phase change in vanadium dioxide and its applications

Vanadium dioxide is an unusual material that undergoes a first-order Metal–Insulator Transition (MIT) at 340 K, attracting considerable interest for its intrinsic properties and its potential applications. However, the nature of MIT has not been fully determined. Variants of density functional theory (DFT) have been widely used to study the MIT in pure and doped VO2. A full description of MIT is complicated by several related factors such as V–V dimerization, magnetic properties, and spin correlations. Each of these requires careful attention. In this Perspective, we explain why DFT fails, introduce a spin-pairing model of MIT, and propose a new way to estimate the transition temperature. We then use the method to study the doping and alloying process. Finally, we give an overview of some applications of MIT. This work aims to provide insight into and stimulate more research studies in this promising field.

A Theoretical Study of the Occupied and Unoccupied Electronic Structure of High- and Intermediate-Spin Transition Metal Phthalocyaninato (Pc) Complexes: VPc, CrPc, MnPc, and FePc.

The structural, electronic, and spectroscopic properties of high- and intermediate-spin transition metal phthalocyaninato complexes (MPc; M = V, Cr, Mn and Fe) have been theoretically investigated to look into the origin, symmetry and strength of the M–Pc bonding. DFT calculations coupled to the Ziegler’s extended transition state method and to an advanced charge density and bond order analysis allowed us to assess that the M–Pc bonding is dominated by σ interactions, with FePc having the strongest and most covalent M–Pc bond. According to experimental evidence, the lightest MPcs (VPc and CrPc) have a high-spin ground state (GS), while the MnPc and FePc GS spin is intermediate. Insights into the MPc unoccupied electronic structure have been gained by modelling M L2,3-edges X-ray absorption spectroscopy data from the literature through the exploitation of the current Density Functional Theory variant of the Restricted Open-Shell Configuration Interaction Singles (DFT/ROCIS) method. Besides the overall agreement between theory and experiment, the DFT/ROCIS results indicate that spectral features lying at the lowest excitation energies (EEs) are systematically generated by electronic states having the same GS spin multiplicity and involving M-based single electronic excitations; just as systematically, the L3-edge higher EE region of all the MPcs herein considered includes electronic states generated by metal-to-ligand-charge-transfer transitions involving the lowest-lying π* orbital (7eg) of the phthalocyaninato ligand.

Variational Density Functional Calculations of Excited States via Direct Optimization

The development of variational density functional theory approaches to excited electronic states is impeded by limitations of the commonly used self-consistent field (SCF) procedure. A method based on a direct optimization approach as well as the maximum overlap method is presented, and the performance is compared with previously proposed SCF strategies. Excited-state solutions correspond to saddle points of the energy as a function of the electronic degrees of freedom. The approach presented here makes use of a preconditioner determined with the help of the maximum overlap method to guide the convergence on a target nth-order saddle point. This method is found to be more robust and converge faster than previously proposed SCF approaches for a set of 89 excited states of molecules. A limited-memory formulation of the symmetric rank-one method for updating the inverse Hessian is found to give the best performance. A conical intersection for the carbon monoxide molecule is calculated without resorting to fractional occupation numbers. Calculations on the excited states of the hydrogen atom and a doubly excited state of the dihydrogen molecule using a self-interaction corrected functional are presented. For these systems, the self-interaction correction is found to improve the accuracy of density functional calculations of excited states.

Variational Multistate Density Functional Theory for a Balanced Treatment of Static and Dynamic Correlations.

Here, an approach to variational multistate density functional theory (vMSDFT) is explored. In this approach, the Kohn-Sham orbitals as well as configuration coefficients were simultaneously optimized, thus yielding a full variational minimum. Furthermore, this work also proposes two important improvements on the MSDFT framework. First, a "point-to-point correction" is used to correct the static correlation present in the DFT framework. Therefore, double counting of static correlation in vMSDFT is mitigated. Second, a general form to construct the transition density functional in the vMSDFT framework is proposed, which allows for the properties of vMSDFT wave functions to be standardized to the complete active space self-consistent field properties. The utility of vMSDFT is illustrated on molecular systems of interest including bond breaking, diradicals, excited states, and conical intersections. The numerical results suggest that the accuracy of vMSDFT is in close agreement with the high-level multireference methods.

Phase transitions of titanite CaTiSiO5 from density functional perturbation theory

Phonon dispersion of titanite ${\mathrm{CaTiSiO}}_{5}$ has been calculated using the variational density functional perturbation theory. The experimentally known out-of-center distortion of the Ti atom is confirmed. The distortion is associated with a ${B}_{u}$ mode that is unstable for wave vectors normal to the octahedral chain direction of the $C2/c$ aristotype structure. The layer of wave vectors with imaginary mode frequencies also comprises the Brillouin zone boundary point $Y$ $(0,1,0)$, which is critical for the transition to the $P{2}_{1}/c$ ground-state structure. The phonon branch equivalent to the imaginary branch of the titanite aristotype is found to be stable in malayaite ${\mathrm{CaSnSiO}}_{5}$. The unstable phonon mode in titanite leads to the formation of transoriented short and long Ti-O1 bonds. The Ti as well as the connecting O1 atom exhibit strongly anomalous Born effective charges along the octahedral chain direction [001], indicative of the strong covalency in this direction. Accordingly and in contrast to malayaite, LO-TO splitting is very large in titanite. In the $C2/c$ phase of titanite, the Ti-O1-Ti distortion chain is disordered with respect to neighboring distortion chains, as all chain configurations are equally unstable along the phonon branch. This result is in agreement with diffuse x-ray scattering in layers normal to the chain direction that is observed at temperatures close to the $P{2}_{1}/c$ to $C2/c$ transition temperature and above. The resulting dynamic chains of correlated Ti displacements are expected to order in two dimensions to yield the $P{2}_{1}/c$ ground-state structure of titanite.

Tunable electron and phonon properties of folded single-layer molybdenum disulfide

A unique feature of transition metal dichalcogenides is their single-layer form, which enables folding. Although folding has been found to significantly affect the photoluminescence spectrum and some in-plane properties, only limited insight has been gained on how to modulate those properties. In this report, we examine the structure of folds of a single sheet of MoS2 and the dependence of the ground-state electronic and phonon transport properties on the wrapping length. As the folded structure is effectively a bilayer that terminates in a loop, the wrapping length modulates the relative size of the bilayer region to the closed loop along the edge. A combination of computational methods, including approaches based on variational mechanics, classical potentials, and density functional theory, are employed. Highly accurate calculations of the reference folded structure are first carried out to show that the folded structure is largely insensitive to the wrapping length. The folded structures are subsequently used to estimate the electronic band gap, which is found to vary significantly as a function of the wrapping length, and converges from below to the limit value corresponding to an infinite bilayer. The gap values range from 0.43 to 1.09 eV, with a crossover to an indirect gap, which suggests that the transitions must be lattice-assisted, similar to the transitions in the bilayer and bulk forms. However, the phonons, while affected by the formation of the folded structure, are insensitive to the wrapping length. In fact, the overall thermal transport behavior along the folding axis is unchanged. The possibility of modulating the gap value while keeping the thermal properties unchanged opens up new exciting avenues for further applications of this emerging material.

Constricted variational density functional theory for spatially clearly separated charge-transfer excitations.

Constricted Variational Density Functional Theory (CV-DFT) is known to be one of the successful methods in predicting charge-transfer excitation energies. In this paper, we apply the CV-DFT method to the well-known model systems ethylene-tetrafluoroethylene (C2H4 × C2F4) and the zincbacteriochlorin-bacteriochlorin complex (ZnBC-BC). The analysis of the CV-DFT energies enables us to understand the -1/R charge-transfer behaviour in CV-DFT for large separation distances R. With this we discuss the importance of orbital relaxations using the relaxed version of CV(∞)-DFT, the R-CV(∞)-DFT method. Possible effects of the optimization of the transition matrix for the relaxed self-consistent field version of CV(∞)-DFT, RSCF-CV(∞)-DFT in the case of large fragment separations are shown and we introduce two possible gradient restrictions to avoid the unwanted admixing of other transitions.

Self-Consistent Constricted Variational Theory RSCF-CV(∞)-DFT and Its Restrictions To Obtain a Numerically Stable ΔSCF-DFT-like Method: Theory and Calculations for Triplet States.

In this paper, the relaxed self-consistent field infinite order constricted variational density functional theory (RSCF-CV(∞)-DFT) for triplet calculations is presented. Here, we focus on two main features of our implementation. First, as an extension of our previous work by Krykunov and Ziegler ( J. Chem. Theory Comput. 2013 , 9 , 2761 ), the optimization of the transition matrix representing the orbital transition is implemented and applied for vertical triplet excitations. Second, restricting the transition matrix, we introduce RSCF-CV(∞)-DFT-based numerically stable ΔSCF-DFT-like methods, the most general of them being SVD-RSCF-CV(∞)-DFT. The reliability of the different methods, RSCF-CV(∞)-DFT and its restricted versions, is examined using the benchmark test set of Silva-Junior et al. ( J. Chem. Phys. 2008 , 129 , 104103 ). The obtained excitation energies validate our approach and implementation for RSCF-CV(∞)-DFT and also show that SVD-RSCF-CV(∞)-DFT mimics very well ΔSCF-DFT, as the root-mean-square deviations between these methods are less than 0.1 eV for all functionals examined.

Excited State Studies of Polyacenes Using the All-Order Constricted Variational Density Functional Theory with Orbital Relaxation.

For the polyacenes series from naphthalene to hexacene, we present the vertical singlet excitation energies 1 (1)La and 1 (1)Lb, as well as the first triplet excitation energies obtained by the all-order constricted variational density functional theory with orbital relaxation (R-CV(∞)-DFT). R-CV(∞)-DFT is a further development of variational density functional theory (CV(∞)-DFT), which has already been successfully applied for the calculation of the vertical singlet excitation energies (1)La and (1)Lb for polyacenes,15 and we show that one obtains consistent excitation energies using the local density approximation as a functional for singlet as well as for triplet excitations when going beyond the linear response theory. Furthermore, we apply self-consistent field density functional theory (ΔSCF-DFT) and compare the obtained excitation energies for the first triplet excitations T1, where, due to the character of the transition, ΔSCF-DFT and R-CV(∞)-DFT become numerically equivalent, and for the singlet excitations 1 (1)La and 1 (1)Lb, where the two methods differ.

On the relation between adiabatic time dependent density functional theory (TDDFT) and the ΔSCF-DFT method. Introducing a numerically stable ΔSCF-DFT scheme for local functionals based on constricted variational DFT

In ΔSCF density functional theory studies of a i → a transition one performs separate fully self-consistent field calculations on the ground state configuration (i)n (n = 1,2) and the excited state configuration (i)n − 1a. The excitation energy for the transition i → a is subsequently determined as the Kohn-Sham energy difference ΔEi → a = E[in − 1a] − E[in] between the ground state (i)n and the excited state configuration (i)n − 1a. The ΔSCF scheme has been applied extensively and works well for lower energy excitations provided that they can be represented by a single orbital replacement or transition i → a. However, for excitations of higher energy ΔSCF tends to become numerically unstable with a variational collapse to transitions of lower energy. We demonstrate here a numerically stable ΔSCF scheme for local functionals that is guaranteed not to collapse on excited configurations of lower energy as well as the ground state. The new scheme is based on constricted variational density functional theory in which the canonical ground state orbitals are allowed to relax (R-CV(∞)-DFT). Since it is restricted to a single orbital replacement i → a it is termed SOR-R-CV(∞)-DFT.

Modeling of the Amorphous Phase of Poly-CO with He and N

ABSTRACTDensity functional theory (DFT) and variational density functional perturbation theory (DFPT) simulations of amorphous poly-CO structures were performed to understand the stability of the polymerized structure at low pressures and to study the mechanism of destruction of the extended network at low pressures. Charge population analyses accompanied the search of the “weakest link” in the covalently bonded network. IR and Raman spectra of amorphous p-CO, calculated at 15 and 5.02 GPa, show significant contributions of CO molecules, carbonyl groups fragments decorating chains, and lactones of amorphous p-CO structures. DFT simulations of formation of amorphous polymeric structures were also done with the addition (as a result of replacement of CO molecules) of N or He atoms to the crystalline delta phase of CO. For the CO-N mixtures, the concentration of N was varied in the range from 6.25 % to 50% with different distribution patterns of N atoms in the unit cell. For all studied CO-N concentrations, isotropic compression led to CO polymerization beginning at a pressure of 11 GPa; the N was incorporated in the random network in low concentration. In CO-He mixtures He atoms appear to facilitate complete formation of the random structure which is almost completely polymerized at a pressure of 18 GPa. He atoms also help stabilize the structure at low pressures.

Use of density functional theory orbitals in the GVVPT2 variant of second-order multistate multireference perturbation theory.

A new variation of the second-order generalized van Vleck perturbation theory (GVVPT2) for molecular electronic structure is suggested. In contrast to the established procedure, in which CASSCF or MCSCF orbitals are first obtained and subsequently used to define a many-electron model (or reference) space, the use of an orbital space obtained from the local density approximation (LDA) variant of density functional theory is considered. Through a final, noniterative diagonalization of an average Fock matrix within orbital subspaces, quasicanonical orbitals that are otherwise indistinguishable from quasicanonical orbitals obtained from a CASSCF or MCSCF calculation are obtained. Consequently, all advantages of the GVVPT2 method are retained, including use of macroconfigurations to define incomplete active spaces and rigorous avoidance of intruder states. The suggested variant is vetted on three well-known model problems: the symmetric stretching of the O-H bonds in water, the dissociation of N2, and the stretching of ground and excited states C2 to more than twice the equilibrium bond length of the ground state. It is observed that the LDA-based GVVPT2 calculations yield good results, of comparable quality to conventional CASSCF-based calculations. This is true even for the C2 model problem, in which the orbital space for each state was defined by the LDA orbitals. These results suggest that GVVPT2 can be applied to much larger problems than previously accessible.

Restricted open-shell configuration interaction cluster calculations of the L-edge X-ray absorption study of TiO(2) and CaF(2) solids.

X-ray metal L-edge spectroscopy has proven to be a powerful technique for investigating the electronic structure of transition-metal centers in coordination compounds and extended solid systems. We have recently proposed the Restricted Open-Shell Configuration Interaction Singles (ROCIS) method and its density functional theory variant (DFT/ROCIS) as methods of general applicability for interpreting such spectra. In this work, we apply the ROCIS and DFT/ROCIS methods for the investigation of cluster systems in order to interpret the Ca and Ti L-edge spectra of CaF2 and TiO2 (rutile and anatase), respectively. Cluster models with up to 23 metallic centers are considered together with the hydrogen saturation and embedding techniques to represent the extended ionic and covalent bulk environments of CaF2 and TiO2. The experimentally probed metal coordination environment is discussed in detail. The influence of local as well as nonlocal effects on the intensity mechanism is investigated. In addition, the physical origin of the observed spectral features is qualitatively and quantitatively discussed through decomposition of the dominant relativistic states in terms of leading individual 2p-3d excitations. This contribution serves as an important reference for future applications of ROCIS and DFT/ROCIS methods in the field of metal L-edge spectroscopy in solid-state chemistry.

Describing long-range charge-separation processes with subsystem density-functional theory.

Long-range charge-transfer processes in extended systems are difficult to describe with quantum chemical methods. In particular, cost-effective (non-hybrid) approximations within time-dependent density functional theory (DFT) are not applicable unless special precautions are taken. Here, we show that the efficient subsystem DFT can be employed as a constrained DFT variant to describe the energetics of long-range charge-separation processes. A formal analysis of the energy components in subsystem DFT for such excitation energies is presented, which demonstrates that both the distance dependence and the long-range limit are correctly described. In addition, electronic couplings for these processes as needed for rate constants in Marcus theory can be obtained from this method. It is shown that the electronic structure of charge-separated states constructed by a positively charged subsystem interacting with a negatively charged one is difficult to converge - charge leaking from the negative subsystem to the positive one can occur. This problem is related to the delocalization error in DFT and can be overcome with asymptotically correct exchange-correlation (XC) potentials or XC potentials including a sufficiently large amount of exact exchange. We also outline an approximate way to obtain charge-transfer couplings between locally excited and charge-separated states.

Introducing constricted variational density functional theory in its relaxed self-consistent formulation (RSCF-CV-DFT) as an alternative to adiabatic time dependent density functional theory for studies of charge transfer transitions.

We have applied the relaxed and self-consistent extension of constricted variational density functional theory (RSCF-CV-DFT) for the calculation of the lowest charge transfer transitions in the molecular complex X-TCNE between X = benzene and TCNE = tetracyanoethylene. Use was made of functionals with a fixed fraction (α) of Hartree-Fock exchange ranging from α = 0 to α = 0.5 as well as functionals with a long range correction (LC) that introduces Hartree-Fock exchange for longer inter-electronic distances. A detailed comparison and analysis is given for each functional between the performance of RSCF-CV-DFT and adiabatic time-dependent density functional theory (TDDFT) within the Tamm-Dancoff approximation. It is shown that in this particular case, all functionals afford the same reasonable agreement with experiment for RSCF-CV-DFT whereas only the LC-functionals afford a fair agreement with experiment using TDDFT. We have in addition calculated the CT transition energy for X-TCNE with X = toluene, o-xylene, and naphthalene employing the same functionals as for X = benzene. It is shown that the calculated charge transfer excitation energies are in as good agreement with experiment as those obtained from highly optimized LC-functionals using adiabatic TDDFT. We finally discuss the relation between the optimization of length separation parameters and orbital relaxation in the RSCF-CV-DFT scheme.