Lagrangian approach to molecular vibrational Raman intensities using time-dependent hybrid density functional theory

The authors propose a new route to vibrational Raman intensities based on analytical derivatives of a fully variational polarizability Lagrangian. The Lagrangian is constructed to recover the negative frequency-dependent polarizability of time-dependent Hartree-Fock or adiabatic (hybrid) density functional theory at its stationary point. By virtue of the variational principle, first-order polarizability derivatives can be computed without using derivative molecular orbital coefficients. As a result, the intensities of all Raman-active modes within the double harmonic approximation are obtained at approximately the same cost as the frequency-dependent polarizability itself. This corresponds to a reduction of the scaling of computational expense by one power of the system size compared to a force constant calculation and to previous implementations. Since the Raman intensity calculation is independent of the harmonic force constant calculation more, computationally demanding density functionals or basis sets may be used to compute the polarizability gradient without much affecting the total time required to compute a Raman spectrum. As illustrated for fullerene C60, the present approach considerably extends the domain of molecular vibrational Raman calculations at the (hybrid) density functional level. The accuracy of absolute and relative Raman intensities of benzene obtained using the PBE0 hybrid functional is assessed by comparison with experiment.     

Multigrid methods in density functional theory

A multigrid method for real-space solution of the Kohr-Sham equations is presented. By using this multiscale approach, the problem of critical slowing down typical of iterative real-space solvers is overcome. The method scales linearly in computer time with the number of electrons if the orbitals are localized. Here, we describe details of our multigrid method, present preliminary many-electron numerical results illustrating the efficiency of the solver, and discuss its strengths and limitations. © 1997 John Wiley & Sons, Inc.     

Accurate calculation of transport properties for organic molecular semiconductors with spin-component scaled MP2 and modern density functional theory methods

At ambient temperatures, intermolecular hopping of charge carriers dominates the field effect mobility and thus the performance of organic molecular semiconductors for organic-based electronic devices. We have used a wide variety of modern and accurate computational methods to calculate the main parameters associated with charge transport, taking oligoacenes, and its derivatives as the exemplary organic materials. We tackle the problem from a combined inter- and intramolecular approach, in which the parameters are calculated for an isolated single molecule concomitantly with the stability of the dimers found in experimentally determined crystalline structures. Considering that most of the future applications within the field would need a full understanding of the transport mechanism, we assess the reliability of the methods to be employed according to the nature of the problem. Finally, we perform a computationally guided molecular engineering of a new set of materials derived from tetracene (rubrene and highly twisted oligoacenes) which allows to robustly anticipate the reasons for their expected performance in organic-based electronic devices.     

Lattice density functional theory of molecular diffusion

A density functional theory of diffusion is developed for lattice fluids with molecular flux as a functional of the density distribution. The formalism coincides exactly with the generalized Ono-Kondo density functional theory when there is no gradient of chemical potential, i.e., at equilibrium. Away from equilibrium, it gives Fick's first law in the absence of a potential energy gradient, and it departs from Fickian behavior consistently with the Maxwell-Stefan formulation. The theory is applied to model a nanopore, predicting nonequilibrium phase transitions and the role of surface diffusion in the transport of capillary condensate.     

Performance of density functional theory methods to describe intramolecular hydrogen shifts

The performance of three exchange and correlation density functionals, LDA, BLYP and B3LYP, with four basis sets is tested in three intramolecular hydrogen shift reactions. The best reaction and activation energies come from the hybrid functional B3LYP with triple-ζ basis sets, when they are compared with high-level post-Hartree-Fock results from the literature. For a fixed molecular geometry, the electrophilic Fukui function is computed from a finite difference approximation. Fukui function shows a small dependence with both the exchange and correlation functional and the basis set. Evolution of the Fukui function along the reaction path describes important changes in the basic sites of the corresponding molecules. These results are in agreement with the chemical behavior of those species.     

Some applications of local density functional theory to the calculation of reaction energetics

Summary We report the results of a local density functional investigation of the energetics of some isomerization reactions, involving the conversions of several unsaturated systems to highly strained molecules related to triprismane and tetrahedrane. The program DMol was used at the DNP level to compute the activation barriers and total energy changes associated with these processes. We also show, for more than 70 first- and second-row atoms and molecules, that the errors (non-local corrections) in their energies correlate very well with the number of electrons, within isonuclear series. This should provide a useful empirical means for improving dissociation energies obtained within the local approximation.     

Suitability of density functional methods for calculation of electrostatic properties

A systematic analysis was performed on the suitability of the molecular electrostatic potential (MEP) and MEP-derived properties determined by means of density functional (DFT) methods. Attention was paid to the electrostatic potential (ESP) derived charges, the ESP and exact quantum mechanical dipole moments, the depth of MEP minima, and the MEP distribution in layers around the molecule for a large series of molecules. The electrostatic properties were determined at either local or nonlocal DFT levels using different functionals. The results were compared with the values estimated from quantum mechanical calculations performed at Hartree–Fock, Møller–Plesset up to fourth order, and CIPSI levels. The suitability of the MEP-derived properties estimated from DFT methods is discussed for application in different areas of chemical interest. © 1997 John Wiley & Sons, Inc. J Comput Chem 18 : 980–991, 1997     

Polarizability and second hyperpolarizability evaluation of long molecules by the density functional theory with long-range correction

Polarizabilities and second hyperpolarizabilities of polyacetylene and a hydrogen chain are evaluated by density functional theory (DFT) using a hybrid generalized gradient approximation functional with correct long-range electron-electron interactions. The well known catastrophic overestimate of the hyperpolarizabilities for molecular systems of enhanced length is corrected by the two-electron repulsion operator decomposition technique, integrating the distance-dependent nonlocal exchange effects for long-range interaction, while neither the asymptotically corrected exchange functional for long-range interaction nor ordinary hybrid methods seem to be capable of overcoming the serious drawback of the DFT in polarizability/hyperpolarizability evaluation.     

The choice of appropriate density functional for the calculation of static first hyperpolarizability of azochromophores and stacking dimers

The effect of the stacked azo‐chromophore dimer formation on the values of static first hyperpolarizability is studied in the framework of the DFT‐based approach; calculations were also performed at the MP2 level. A number of dispersion‐corrected density functionals—В97D, ωВ97X‐D, and M06‐2X—is tested to calculate the structure of the dimer, the value of binding energy, and molecular nonlinear optical characteristics. According to the QTAIM analysis, the presence of bond critical points is revealed in the intermolecular region, the signs and values of topological characteristics giving evidence for the noncovalent van der Waals interaction between the chromophores. The formation of stacks results in moderate increase of dimer static first hyperpolarizability as compared to that of a single chromophore, the effect depending on the relative shift of the chromophores in dimer. In a special case of greatly shifted chromophores, this enhancement of the first hyperpolarizability becomes appreciable and achieves 72%. © 2015 Wiley Periodicals, Inc.     

Application of the Sakurai-Sugiura projection method to core-excited-state calculation by time-dependent density functional theory

The Sakurai-Sugiura projection (SS) method was implemented and numerically assessed for diagonalization of the Hamiltonian in time-dependent density functional theory (TDDFT). Since the SS method can be used to specify the range in which the eigenvalues are computed, it may be an efficient tool for use with eigenvalues in a particular range. In this article, the SS method is applied to core excited calculations for which the eigenvalues are located within a particular range, since the eigenvalues are unique to atomic species in molecules. The numerical assessment of formaldehyde molecule by TDDFT with core-valence Becke's three-parameter exchange (B3) plus Lee-Yang-Parr (LYP) correlation (CV-B3LYP) functional demonstrates that the SS method can be used to selectively obtain highly accurate eigenvalues and eigenvectors. Thus, the SS method is a new and powerful alternative for calculating core-excitation energies without high computation costs.     

Direct calculation of electron transfer parameters through constrained density functional theory

It is shown that constrained density functional theory (DFT) can be used to access diabatic potential energy surfaces in the Marcus theory of electron transfer, thus providing a means to directly calculate the driving force and the inner-sphere reorganization energy. We present in this report an analytic expression for the forces in constrained DFT and their implementation in geometry optimization, a prerequisite for the calculation of electron transfer parameters. The method is then applied to study the symmetric mixed-valence complex tetrathiafulvalene-diquinone radical anion, which is observed experimentally to be a Robin-Day class II compound but found by DFT to be in class III. Constrained DFT avoids this pitfall of over-delocalization and provides a way to find the charge-localized structure. In another application, driving forces and inner-sphere reorganization energies are calculated for the charge recombination (CR) reactions in formanilide-anthraquinone (FA-AQ) and ferrocene-formanilide-anthraquinone (Fc-FA-AQ). While the two compounds have similar reorganization energies, the driving force in FA-AQ is 1 eV larger than in Fc-FA-AQ, in agreement with experimental observations and supporting the experimental conclusion that the anomalously long-lived FA-AQ charge-separated state arises because the electron transfer is in the Marcus inverted region.     

The use of the hybrid density functional theory (DFT) approach for calculation of vibrational spectra: nitromethane

The molecular geometry of nitromethane was optimized and its force field and vibrational spectrum were calculated by the BECKE3LYP method. The accuracy of optimization of the geometry of MeNO₂ obtained by this method using the 6–311G(d,p) and 6–311++G(d,p) basis sets is not poorer than that obtained at the second-order Møller-Plesset level of perturbation theory (MP2). The vibrational frequencies of nitromethane and its d₁, d₂, and d₃ isotopomers obtained by the BECKE3LYP method are in much better agreement with the experimental data than those calculated at the MP2 level using the same basis set. The average absolute error of calculations performed without the use of any scaling factors is ～2% for frequencies; the maximum deviation is ～4%.     

Utilization of deformations in molecular quantum chemistry and application to density functional theory

The aim of this article is to present in a way accessible to most quantum chemists a general mathematical method which consists in deforming wave functions and density functions (in the spirit of the local scaling transformation). This deformation method allows us to obtain several new results, including a characterization of the set of wave functions that have the same given density function (which gives a new insight on a result of G. Zumbach and K. Maschke, Phys. Rev. A 28 , 544 (1983)) and an N-representability result where symmetry is taken into account. We also propose new theoretical ways to generate approximations of the exact density functional and give a numerical example. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 68: 221–231, 1998     

Sensitivity analysis and uncertainty calculation for dispersion corrected density functional theory

The precision of binding energies and distances computed with dispersion-corrected density functional theory (DFT-D) is investigated by propagation of uncertainties, yielding relative uncertainties of several percent. Sensitivity analysis is used to calculate the geometry-dependent relative importance of each input parameter for the dispersion correction. While DFT-Ds are exact at asymptotically large distances, their damping functions are shown to play a significant role in binding geometries. This is demonstrated in detail for the interlayer binding of graphite. The techniques presented allow practitioners to quickly compute error bars and to get an a posteriori estimate about the transferability of their results. They can also aid the development of future dispersion corrections.     

Applicability of density functional theory in reproducing accurate vibrational spectra of surface bound species

The structural equilibrium parameters, the adsorption energies, and the vibrational frequencies of the nitrogen molecule and the hydrogen atom adsorbed on the (111) surface of rhodium have been investigated using different generalized‐gradient approximation (GGA), nonlocal correlation, meta‐GGA, and hybrid functionals, namely, Perdew, Burke, and Ernzerhof (PBE), Revised‐RPBE, vdW‐DF, Tao, Perdew, Staroverov, and Scuseria functional (TPSS), and Heyd, Scuseria, and Ernzerhof (HSE06) functional in the plane wave formalism. Among the five tested functionals, nonlocal vdW‐DF and meta‐GGA TPSS functionals are most successful in describing energetics of dinitrogen physisorption to the Rh(111) surface, while the PBE functional provides the correct chemisorption energy for the hydrogen atom. It was also found that TPSS functional produces the best vibrational spectra of the nitrogen molecule and the hydrogen atom on rhodium within the harmonic formalism with the error of ?2.62 and ?1.1% for the N? N stretching and Rh? H stretching frequency. Thus, TPSS functional was proposed as a method of choice for obtaining vibrational spectra of low weight adsorbates on metallic surfaces within the harmonic approximation. At the anharmonic level, by decoupling the Rh? H and N? N stretching modes from the bulk phonons and by solving one‐ and two‐dimensional Schrödinger equation associated with the Rh? H, Rh? N, and N? N potential energy we calculated the anharmonic correction for N? N and Rh? H stretching modes as ?31 cm^?1 and ?77 cm^?1 at PBE level. Anharmonic vibrational frequencies calculated with the use of the hybrid HSE06 function are in best agreement with available experiments. © 2014 Wiley Periodicals, Inc.     

Ab initio calculation of the electronic absorption of functionalized octahedral silsesquioxanes via time-dependent density functional theory with range-separated hybrid functionals

Recent advances in the ability to functionalize octahedral silsesquioxanes with different photoactive ligands, and thereby tune their optical properties, suggest that these molecules may serve as potential building blocks of light-harvesting, photovoltaic, and photonic devices. In this paper we report extensive ab initio calculations of the excitation energies underlying the absorption spectra of these systems. The calculations are based on density functional theory for the ground electronic state and time-dependent density functional theory for the excited electronic states. The ability of the commonly used B3LYP functional to reproduce the experimentally observed absorption excitation energies is compared to that of recently developed range-separated hybrid functionals. The importance of pairing the range-separated hybrid functionals with basis sets that include diffuse and polarization basis functions is demonstrated in the case of vinyl-functionalized silsesquioxanes. Absorptive excitation energies are then calculated and compared with experiment for octahedral silsesquioxanes functionalized with larger ligands. The tunability of optical properties is demonstrated by considering the effect on the excitation energies of functionalizing the ligands with electron-donating or -withdrawing groups.     

Application of second-order density functional methods to the calculation of the BeFH potential energy surface

Second-order density functional methods (where the correlation energy depends on the second-order density matrix and on a density functional) are used to introduce the electron correlation in two-configuration direct minimization (TCDM) ab initio electronic energy calculations of three-dimensional potential energy surfaces (PES). We analyzed the behavior of these methods in PES calculations by applying them to the Be + FH reaction. This system was studied also by the usual techniques, allowing a full comparison for the lowest ¹A′ adiabatic state. In particular, we compared the results obtained using the TCDM and multiple reference single and double excitations configuration interaction (MRDCI) methods with the corresponding results obtained using the Colle-Salvetti (CS) and Moscardó-San Fabián (MSF) procedures, within the correlation factor method, using as starting point the TCDM results. We found that the CS and MSF results are in a good overall agreement with the more accurate ab initio results, including the heights of the saddle points and the transition state. © 1997 John Wiley & Sons, Inc.     

Koopmans' theorem for large molecular systems within density functional theory

It is shown that in density functional theory (DFT), Koopmans' theorem for a large molecular system can be stated as follows: The ionization energy of the system equals the negative of the highest occupied molecular orbital (HOMO) energy plus the Coulomb electrostatic energy of removing an electron from the system, or equivalently, the ionization energy of an N-electron system is the negative of the arithmetic average of the HOMO energy of this system and the lowest unoccupied molecular orbital (LUMO) energy of the (N - 1)-electron system. Relations between this DFT Koopmans' theorem and its existing counterparts in the literature are discussed. Some of the previous results are generalized and some are simplified. DFT calculation results of a fullerene molecule, a finite single-walled carbon nanotube and a finite boron nitride nanotube are presented, indicating that this Koopmans' theorem approximately holds, even if the orbital relaxation is taken into consideration.     

Application of second-order density functional methods to the calculation of the LiFH potential energy surface

Second-order density functional methods are used to introduce the electron correlation in Hartree-Fock (HF ) ab initio electronic energy calculations of three-dimensional potential energy surfaces (PES ). We analyze the behavior of these methods in PES calculations by applying them to the Li + FH reaction, which has been considered a prototype of the elementary atom–diatom reactions. This system has been studied also by the usual techniques, allowing a point-by-point (for a total of 317 grid points) comparison for the lowest ²A' adiabatic state. In particular, we compare the results obtained using the HF , Møller–Plesset (MP 3 level), and configuration interaction (CISD and MRDCI levels) methods with the corresponding results obtained using the Colle–Salvetti (CS ) and Moscardó-San Fabián (MSF ) procedures using the HF results as the starting point. We found that the CS and MSF procedures support the prediction of a shallow well in the entrance channel that deepens slightly away from collinearity and disappears for a bond angle Θ < 74°. We also found that the constrained saddle-point positions remain essentially constant from Θ = 180°–90° and are clearly in the exit channel as for the MRDCI approach (corresponding to the best results). In conclusion, there is a good overall agreement, but there is a question in which this agreement is less pronounced: the heights of the saddle points including the transition state. In particular, the transition-state height is about 3 kcal/mol higher than the more accurate value obtained with the MRDCI approach. However, the second-order density functional methods have been capable of reducing the HF barrier from 18 to 9 kcal/mol (all of these values obtained by spline interpolation), the latter value being very similar to the CISD result. © 1994 John Wiley & Sons, Inc.