Time-resolved THz Stark spectroscopy of molecules in solution.

For decades, it was considered all but impossible to perform Stark spectroscopy on molecules in a liquid solution, because their concomitant orientation to the applied electric field results in overwhelming background signals. A way out was to immobilize the solute molecules by freezing the solvent. While mitigating solute orientation, freezing removes the possibility to study molecules in liquid environments at ambient conditions. Here we demonstrate time-resolved THz Stark spectroscopy, utilizing intense single-cycle terahertz pulses as electric field source. At THz frequencies, solute molecules have no time to orient their dipole moments. Hence, dynamic Stark spectroscopy on the time scales of molecular vibrations or rotations in both non-polar and polar solvents at arbitrary temperatures is now possible. We verify THz Stark spectroscopy for two judiciously selected molecular systems and compare the results to conventional Stark spectroscopy and first principle calculations.

Bonding in the helium dimer in strong magnetic fields: the role of spin and angular momentum.

We investigate the helium dimer in strong magnetic fields, focusing on the spectrum of low-lying electronic states and their dissociation curves, at the full configuration-interaction level of theory. To address the loss of cylindrical symmetry and angular momentum as a good quantum number for nontrivial angles between the bond axis and magnetic field, we introduce the almost quantized angular momentum (AQAM) and show that it provides useful information about states in arbitrary orientations. In general, strong magnetic fields dramatically rearrange the spectrum, with the orbital Zeeman effect bringing down states of higher angular momentum below the states with pure σ character as the field strength increases. In addition, the spin Zeeman effect pushes triplet states below the lowest singlet; in particular, a field of one atomic unit is strong enough to push a quintet state below the triplets. In general, the angle between the bond axis and the magnetic field also continuously modulates the degree of σ, π, and δ character of bonds and the previously identified perpendicular paramagnetic bonding mechanism is found to be common among excited states. Electronic states with preferred skew field orientations are identified and rationalized in terms of permanent and induced electronic currents.

Magnetic-Field Density-Functional Theory (BDFT): Lessons from the Adiabatic Connection.

We study the effects of magnetic fields in the context of magnetic field density-functional theory (BDFT), where the energy is a functional of the electron density ρ and the magnetic field B. We show that this approach is a worthwhile alternative to current-density functional theory (CDFT) and may provide a viable route to the study of many magnetic phenomena using density-functional theory (DFT). The relationship between BDFT and CDFT is developed and clarified within the framework of the four-way correspondence of saddle functions and their convex and concave parents in convex analysis. By decomposing the energy into its Kohn-Sham components, we demonstrate that the magnetizability is mainly determined by those energy components that are related to the density. For existing density functional approximations, this implies that, for the magnetizability, improvements of the density will be more beneficial than introducing a magnetic-field dependence in the correlation functional. However, once a good charge density is achieved, we show that high accuracy is likely only obtainable by including magnetic-field dependence. We demonstrate that adiabatic-connection (AC) curves at different field strengths resemble one another closely provided each curve is calculated at the equilibrium geometry of that field strength. In contrast, if all AC curves are calculated at the equilibrium geometry of the field-free system, then the curves change strongly with increasing field strength due to the increasing importance of static correlation. This holds also for density functional approximations, for which we demonstrate that the main error encountered in the presence of a field is already present at zero field strength, indicating that density-functional approximations may be applied to systems in strong fields, without the need to treat additional static correlation.

The importance of current contributions to shielding constants in density-functional theory.

The sources of error in the calculation of nuclear-magnetic-resonance shielding constants determined by density-functional theory are examined. Highly accurate Kohn-Sham wave functions are obtained from coupled-cluster electron density functions and used to define accurate-but current independent-density-functional shielding constants. These new reference values, in tandem with high-accuracy coupled-cluster shielding constants, provide a benchmark for the assessment of errors in common density-functional approximations. In particular the role of errors arising in the diamagnetic and paramagnetic terms is investigated, with particular emphasis on the role of current-dependence in the latter. For carbon and nitrogen the current correction is found to be, in some cases, larger than 10 ppm. This indicates that the absence of this correction in general purpose exchange-correlation functionals is one of the main sources of error in shielding calculations using density functional theory. It is shown that the current correction improves the shielding performance of many popular approximate DFT functionals.

Charge-constrained auxiliary-density-matrix methods for the Hartree-Fock exchange contribution.

Three new variants of the auxiliary-density-matrix method (ADMM) of Guidon, Hutter, and VandeVondele [J. Chem. Theory Comput. 6, 2348 (2010)] are presented with the common feature that they have a simplified constraint compared with the full orthonormality requirement of the earlier ADMM1 method. All ADMM variants are tested for accuracy and performance in all-electron B3LYP calculations with several commonly used basis sets. The effect of the choice of the exchange functional for the ADMM exchange-correction term is also investigated.

Revisiting the density scaling of the non-interacting kinetic energy.

Scaling relations play an important role in the understanding and development of approximate functionals in density functional theory. Recently, a number of these relationships have been redefined in terms of the Kohn-Sham orbitals [Calderín, Phys. Rev. A: At., Mol., Opt. Phys., 2013, 86, 032510]. For density scaling the author proposed a procedure involving a multiplicative scaling of the Kohn-Sham orbitals whilst keeping their occupation numbers fixed. In the present work, the differences between this scaling with fixed occupation numbers and that of previous studies, where the particle number change implied by the scaling was accommodated through the use of the grand canonical ensemble, are examined. We introduce the terms orbital and ensemble density scaling for these approaches, respectively. The natural ambiguity of the density scaling of the non-interacting kinetic energy functional is examined and the ancillary definitions implicit in each approach are highlighted and compared. As a consequence of these differences, Calderín recovered a homogeneity of degree 1 for the non-interacting kinetic energy functional under orbital scaling, contrasting recent work by the present authors [J. Chem. Phys., 2012, 136, 034101] where the functional was found to be inhomogeneous under ensemble density scaling. Furthermore, we show that the orbital scaling result follows directly from the linearity and the single-particle nature of the kinetic energy operator. The inhomogeneity of the non-interacting kinetic energy functional under ensemble density scaling can be quantified by defining an effective homogeneity. This quantity is shown to recover the homogeneity values for important approximate forms that are exact for limiting cases such as the uniform electron gas and one-electron systems. We argue that the ensemble density scaling provides more insight into the development of new functional forms.

Molecular Binding in Post-Kohn-Sham Orbital-Free DFT.

Molecular binding in post-Kohn-Sham orbital-free DFT is investigated, using noninteracting kinetic energy functionals that satisfy the uniform electron gas condition and which are inhomogeneous under density scaling. A parameter is introduced that quantifies binding, and a series of functionals are determined from fits to near-exact effective homogeneities and/or Kohn-Sham noninteracting kinetic energies. These are then used to investigate the relationship between binding and the accuracy of the effective homogeneity and noninteracting kinetic energy at the equilibrium geometry. For a series of 11 molecules, the binding broadly improves as the effective homogeneity improves, although the extent to which it improves is dependent on the accuracy of the noninteracting kinetic energy; optimal binding appears to require both to be accurate simultaneously. The use of a Thomas-Fermi-von Weizsäcker form, augmented with a second gradient correction, goes some way toward achieving this, exhibiting molecular binding on average. The findings are discussed in terms of the noninteracting kinetic potential and the Hellmann-Feynman theorem. The extent to which the functionals can reproduce the system-dependence of the near-exact effective homogeneity is quantified, and potential energy curves are presented for selected molecules. The study provides impetus for including density scaling homogeneity considerations in the design of noninteracting kinetic energy functionals.

Density Scaling of Noninteracting Kinetic Energy Functionals.

The influence of imposing an approximate density scaling condition on a noninteracting kinetic energy functional is investigated. A simple generalized gradient approximation (GGA) is presented, which satisfies both the density scaling condition and the usual coordinate scaling condition; the remaining multiplicative constant is determined from an energy criterion. In post-Kohn-Sham calculations, noninteracting kinetic energies of the closed-shell molecules of the G1 set determined using the GGA are a modest improvement over those determined using the corresponding local functional, which does not satisfy the density scaling condition. Potential energy curves of CO, F2, and P2 exhibit binding with the GGA, compared to purely repulsive curves with the local functional. Adjusting the exponent in the GGA form in order to optimize energy accuracy violates the density scaling condition, and two of the diatomics no longer exhibit binding. Results are compared with those from other local/GGA functionals in the literature.

Negative electron affinities from DFT: influence of asymptotic exchange-correlation potential and effective homogeneity under density scaling.

The influence of the asymptotic exchange-correlation potential and density-scaling homogeneity on negative electron affinities determined using the approach of Tozer and De Proft [J. Phys. Chem. A2005, 109, 8923] is investigated. Application of an asymptotic correction to the potential improves the accuracy for several of the systems with the most negative affinities, reflecting their diffuse lowest unoccupied orbitals. For systems with modest affinities, it reduces the accuracy marginally. Enforcing a near-exact effective homogeneity through a simple shift in the potential leads to improved correlation with experimental values but significantly overestimated affinities. Optimal effective homogeneities are therefore determined, and a simple scheme is proposed for enforcing an average optimal value. Application of the scheme to a series of organic molecules maintains the excellent correlation with the experimental values while significantly reducing the absolute errors.

Effective homogeneity of the exchange-correlation and non-interacting kinetic energy functionals under density scaling.

Correlated electron densities, experimental ionisation potentials, and experimental electron affinities are used to investigate the homogeneity of the exchange-correlation and non-interacting kinetic energy functionals of Kohn-Sham density functional theory under density scaling. Results are presented for atoms and small molecules, paying attention to the influence of the integer discontinuity and the choice of the electron affinity. For the exchange-correlation functional, effective homogeneities are highly system-dependent on either side of the integer discontinuity. By contrast, the average homogeneity-associated with the potential that averages over the discontinuity-is generally close to 4/3 when the discontinuity is computed using positive affinities for systems that do bind an excess electron and negative affinities for those that do not. The proximity to 4/3 becomes increasingly pronounced with increasing atomic number. Evaluating the discontinuity using a zero affinity in systems that do not bind an excess electron instead leads to effective homogeneities on the electron abundant side that are close to 4/3. For the non-interacting kinetic energy functional, the effective homogeneities are less system-dependent and the effect of the integer discontinuity is less pronounced. Average values are uniformly below 5/3. The study provides information that may aid the development of improved exchange-correlation and non-interacting kinetic energy functionals.

Information carriers and (reading them through) information theory in quantum chemistry.

This Perspective discusses the reduction of the electronic wave function via the second-order reduced density matrix to the electron density ρ(r), which is the key ingredient in density functional theory (DFT) as a basic carrier of information. Simplifying further, the 1-normalized density function turns out to contain essentially the same information as ρ(r) and is even of preferred use as an information carrier when discussing the periodic properties along Mendeleev's table where essentially the valence electrons are at stake. The Kullback-Leibler information deficiency turns out to be the most interesting choice to obtain information on the differences in ρ(r) or σ(r) between two systems. To put it otherwise: when looking for the construction of a functional F(AB) = F[ζ(A)(r),ζ(B)(r)] for extracting differences in information from an information carrier ζ(r) (i.e. ρ(r), σ(r)) for two systems A and B the Kullback-Leibler information measure ΔS is a particularly adequate choice. Examples are given, varying from atoms, to molecules and molecular interactions. Quantum similarity of atoms indicates that the shape function based KL information deficiency is the most appropriate tool to retrieve periodicity in the Periodic Table. The dissimilarity of enantiomers for which different information measures are presented at global and local (i.e. molecular and atomic) level leads to an extension of Mezey's holographic density theorem and shows numerical evidence that in a chiral molecule the whole molecule is pervaded by chirality. Finally Kullback-Leibler information profiles are discussed for intra- and intermolecular proton transfer reactions and a simple S(N)2 reaction indicating that the theoretical information profile can be used as a companion to the energy based Hammond postulate to discuss the early or late transition state character of a reaction. All in all this Perspective's answer is positive to the question of whether an even simpler carrier of information than the electron density function ρ(r) can be envisaged: the shape function, integrating to 1 by construction fulfils this role. On the other hand obtaining the information (or information difference) contained in one (or two) systems from ρ(r) or σ(r) can be most efficiently done by using information theory, the Kulback-Leibler information deficiency being at the moment (one of) the most advisable functionals.

Alternative Kullback-Leibler information entropy for enantiomers.

In our series of studies on quantifying chirality, a new chirality measure is proposed in this work based on the Kullback-Leibler information entropy. The index computes the extra information that the shape function of one enantiomer carries over a normalized shape function of the racemate, while in our previous studies the shape functions of the R and S enantiomers were used considering one as reference for the other. Besides being mathematically more elegant (symmetric, positive definite, zero in the case of a nonchiral system), this new index bears a more direct relation with chirality oriented experimental measurements such as circular dichroism (CD) and optical rotation measurements, where the racemate is frequently used as a reference. The five chiral halomethanes holding one asymmetric carbon atom and H, F, Cl, Br, and I as substituents have been analyzed. A comparison with our calculated optical rotation and with Avnir's Continuous Chirality Measure (CCM) is computed. The results show that with this index the emphasis lies on the differences between the noncoinciding substituents.

Confinement effects on excitation energies and regioselectivity as probed by the Fukui function and the molecular electrostatic potential.

When a molecule is placed as a guest inside a zeolite pore, its electronic structure will be altered, among others by the effect of the so-called "confinement". It has been established that the compression of the molecular orbitals influences a system's reactivity. In this work we use a simple potential barrier method to quantify the importance of confinement effects on chemical reactivity. In the first part, excitation energies and molecular orbital energy gaps are evaluated for molecules placed in cavities of different sizes, resembling a zeolite pore. Our results for ethylene and formaldehyde reveal an increase in excitation energy and the gap between the occupied and the unoccupied levels. In the case of the larger molecules naphthalene and anthracene, the HOMO-LUMO gap shows very little sensitivity to the confinement. To investigate the role of confinement effects on local aspects of chemical reactivity and on regioselectivity, we evaluated its effect on the Fukui function and the molecular electrostatic potential, reactivity indices that are central in the description of orbital and charge controlled reactions. The results indicate that confinement can influence the regioselectivity and that the reactivity of anions is expected to change, due to the artificial binding of the excess electron.

Analyzing Kullback-Leibler information profiles: an indication of their chemical relevance.

The Kullback-Leibler information deficiency is evaluated along molecular internal rotational or vibrational coordinates and along the intrinsic reaction coordinate for several reactions (intra- and intermolecular proton transfer and SN2 reaction). For the first time an in depth analysis of the information deficiency along the reaction path is reported. The results are consistent with the Hammond postulate, indicating that the information profiles contain relevant chemical information. A local version of the information deficiency is defined by considering Hirshfeld's partitioning of atoms in molecules. The analysis of the local information profiles permits the identification of the atoms taking part in the electron reorganization processes.

Information theoretical study of chirality: enantiomers with one and two asymmetric centra.

In this work, the Kullback-Leibler information deficiency is probed as a chirality measure. It is argued that the information deficiency, calculated using the shape functions of the R and S enantiomers, considering one as reference for the other, gives an information theory based expression useful for quantifying chirality. The measure is evaluated for five chiral halomethanes possessing one asymmetric carbon atom with hydrogen, fluorine, chlorine, bromine, and iodine as substituents. To demonstrate the general applicability, a study of two halogen-substituted ethanes possessing two asymmetric carbon atoms has been included as well. The basic expression of the sum of the local information deficiency over all atoms can be decomposed into separate summations over coinciding and noncoinciding atoms, or into a global and a mixing entropy term, or into a local entropy contribution for each atom individually based on the Hirshfeld partitioning. Avnir's continuous chirality measure (CCM) has been computed and confronted with the information deficiency. Finally, the relationship between chirality and optical rotation is used to study the proposed measure. The results illustrate Mezey's holographic electron density theorem with an intuitively appealing division of the strength of propagation of the atomic chirality from an asymmetric carbon atom throughout the molecule. The local information deficiency of the carbon atom is proposed as a measure of chirality; more precisely, the difference in information between the R and the S enantiomer turns out to be a quantitative measure of the chirality of the system. It may be evaluated as the arithmetic mean of the different alignments, or considering only the alignment resulting in the highest similarity value, or using the QSSA alignment.

Computing Fukui functions without differentiating with respect to electron number. I. Fundamentals.

By using perturbations in the molecular external potential, the authors deduce the Fukui function from the change in Kohn-Sham orbital energies, avoiding the troublesome differentiation of the density with respect to electron number. Though this paper focuses on the Fukui function, the same general technique can be used to compute the functional derivative of any observable with respect to the external potential. In this paper, the method is used to compute the Fukui function for the beryllium atom and the formaldehyde molecule. The follow-up paper (part II) addresses the problem of computing condensed reactivity indicators.